ejercicios de estructuras isostáticas. analisis estructural

UNIVERSIDAD MAYOR DE SAN SIMON
FACULTAD DE CIENCIAS Y TECNOLOGIA
CARRERA DE INGENIERIA CIVIL

“APOYO DIDÁCTICO PARA LA ENSEÑANZA Y
APRENDIZAJE DE LA ASIGNATURA DE ESTRUCTURAS
ISOSTÁTICAS”
“TEXTO ESTUDIANTE ”

TRABAJO DIRIJIDO, POR ADSCRIPCIÓ N, PARA OPTAR AL
DIPLOMA ACAD ÉMICO DE LICENCIATURA EN
INGENIERÍA CIVIL

PRESENTADO POR:
 DENNIS BERNARDO TORRICO ARAUCO
 RAUL LIENDO UDAETA
TUTOR:
Ing. Oscar Antezana Mendoza

COCHABAMBA – BOLIVIA
MARZO DE 2006

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  1  -U
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CAPITULO 1

FUNDAMENTOS DE LA ESTATICA Y
ENFOQUE VECTORIAL
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1.1   OBJETIVO GENERAL.
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niUversda vUMyoSmósodmiUSsUsFdsUcmuadyivUsFUlysUsiUsFdySamodsUcvovtcmUivFUMyoSmósodvFUSsUimU
sFdCdacmUcvoUyoUsoMvlysU scdvYamiTUgUsFdsUSsFmYYviisUSsFdYstmFUumYmUsiUcCicyivUSsiUslyaiaeYavUsFdCdacvíU
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1.2   OBJETIVOS ESPECIFICOS.
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U d64mAo7oA (-+ 1(6At+ 6- 2IASo-34 )6 4( S3)(.3L )oA6mmof-L i 46-2o)3n
U  vóuYsoSsYU gUcmicyimYUsiUcvocsudvUSsUóvósodvUcvoUYs.+mof- + (- P(-23 i (- 656L +)6SB4 6.
sMscdvUlysUsFdsUaóuiacmíU
U ISlyaYaYUSsFdYstmFUumYmU!miimYUimUYsFyidmodsUSsUyoUcvoryodvUSsUMysYtmFUgUóvósodvFUsoUsiUsFumcavUgU
SsUóvósodvFíU
1.3 FUERZA.
c+ 1(6At+ 64 .+ +mmof- )6 (- m(6AP3 437A6 32A3L =(6 PA3)(m6 6- 6426 M.2oS3 (-+ S3)o1om+mof-
soUFyUsFdmSvUSsUYsuvFvUvUSsUóv aóasodvíUU
"mUyoaSmSUSaósoFavomiUsoUsiUmoCiaFaFUSsUimFUsFdYycdyYmFUsFUiaeYmTU$aiv%YmóvTUdvosimSmíUUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  2  -U
U
UU &oU scdvYUMysYtmUsFUyoUFs%ósodvUvYasodmSvUsoUsiUsθ→√([ω+θβ+,βδβ+√→∑→√γβ∑(•→∑+[∑+γβ⇒β∑<+
U OrigenUUmUcvoFaSsYmYUcymoSvUaodsYsFsUcvovcsYUsiUuyodvUSsU→⇔(√→√(⇒+δβ⇔+)β√γ[∑⋅+
U Dirección+[+⇔ ⇒β→+δβ+→√√(⇒+√[(⇒√(δβ⇒γβ+√[⇒+⇔→+δβ+⇔→+∑β√γ→+′ysUimUcvodasosUvUcymilyasYUvdYmU
YscdmUumYmisimíU
U SentidoUU asosUSsdsYóaomSvUuvYUimUuyodmUSsUMisc!mUivcmiatmSmUsoUsiUs(dYsóvUSsiU scdvYíU
U MóduloUsFUimUSaFdmocamUsodYsUsiUvYa%soUgUsiUs(dYsóvUSsiU scdvYíUUUUUUU
UUUUUUUUU
)odYvSycasoSvUivFU scdvYsFUyoadmYavFUaTUrTU$UUvYasodmSvFUsoUimFUSaYsccavosFUSsUivFUsrsFU(TUgTUtTU
YsFuscda mósodsTU*Ma%yYmU+í+,TUuvSsóvFUs(uYsFmYU-Us⇒+⇔→+−[∑→<+
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UUU kFjFiFF
ZYX
...
rrr
++= UUUUUExpresión vectorial de la fuerzaU
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Figura 1.1U
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[⇒δβ+δβ−(⇒(∑β[θ+√→δ→+γ∑(⇒[<+
UUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U  vóuvosodsFUsFcmimYsFUSsUF
r
+βθγ∴⇒+δβ−(⇒(δ→θ+[∑+⇔→θ+∑β⇔→√([⇒βθ<+
ZYXFFF TTUmUimY%vUSsUivFU
dYsFUsrsFUSsUcvvYSsomSmFíU
U β√γ[∑βθ+,⇒(γ→∑([θ+β⇒+δ(∑β√√(⇒++δβ+⇔[θ+√→∑γβθ(→⇒[θ<+
kji
rrr
00 U
U  vóuvosodsFU scdvYamisFUSsU
kFjFiFF
ZYX
...
rrr
++= íU
U β⇔→√(⇒+β⇒γ∑β+⇔→+→2⇒(γ,δ+δβ++∫+θ,θ+√[[⇒β⇒γβθ+βFcmimYsFUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
ZYXFFFF
222
++=⇔
r
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  3  -U
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U β√γ[∑+,⇒(γ→∑([+β⇒+⇔→+δ(∑β√√(⇒+δβ[ ]
ZYXFFFF
F
eF TT
+
< r
rr
r= U
U "vFUdYsFUCo%yivFU
α(TUβgTUγt+δβ−(⇒β⇒+⇔→+δ(∑β√√(⇒+δβ+⇔→+−,β∑•→+
F
r
ω+⇔[θ+√θβ⇒[θ+δβ+α(TUβgTUγtUUFsU
√[⇒[√β⇒+√[[+⇔[θ+√θβ⇒[θ+δ(∑β√γ[∑βθ+δβ+⇔→+−,β∑•→+F
r
<+
F
F
F
F
F
F
Z
Y
X
r
r
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=
=
=γ
β
αcvF
cvF
cvF
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→∑→+ ⇔→+ βα[∑+ √[⇒√β√(⇒+ δβ⇔+ →⇒γβ∑([∑+ √[⇒√βγ[ω+ ∑[[⇒β[θ+ →+ √[⇒γ(⇒,→√(⇒+ β⇔+ θ(2,(β⇒γβ+
srsYcacavíU
Ejercicio #1
→δ→+⇔→+−,β∑•→+++≤+[U2TU56TU7U]+β⇒+γ[⇒β⇔→δ→θω+δβγβ∑(⇒→∑+⇔→+→2⇒(γ,δ+∫+δ(∑β√√(⇒+δsUimUMysYtmU
Solución.-
5U8sUdasosUivFUcvóuvosodsFUsFcmimYsFU*2dTU56dTU7d,U
5U8sUdasosUivFUcvóuvosodsFU scdvYamisFU*2UaTU56UrTU7U$,U
U "mUóm%oadySUSsU
F
r
θβ∑∴<++
()()() tonF +9í9762
222
=+−+=
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U
=⇒γ[⇒√βθ<+
cvFα
2
9 +9,
:62,
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cvFβ
6−
9 +9,
:−;<,
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cvFγ
7
9 +9,
:=+,
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+ 3→+δ(∑β√√(⇒+δβ⇔+)β√γ[∑+βθ<+
[ ]70602
+9í9
+
−=
F
e
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U
[ ]=+í:0;<í:062í:−=⇒
F
e
r
r
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  4  -U
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1.4  VECTOR DE POSICION.
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8aU dsosóvFU yoU uyodvU cymilyasYmU 3
++ θβ+ δβ−(⇒β+ β⇔+ )β√γ[∑+ δβ+ [θ(√(⇒+ δβ⇔+ ,⇒γ[+ +U cvóvU siU
Fs%ósodvUvYasodmSvUlysUSsdsYóaomUsiUvYa%soUSsUcvvYSsomSmFU>UgUsiUuyodvU3
+íU
nFUsiUFs%ósodvUYscdai'osvUvYasodmSvUSaYa%aSvUSsiUvYa%soUmiUuyodvUcvóvU sóvFUsoUimUMa%+í2íU
U
Figura 1.2
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voFaSsYmóvFUsiUFaFdsómUSsUMysYtmFUlysUmcd#moUFveYsUyoUcysYuvUY'%aSvUsoUivFUuyodvFU?
+TU@+TU
A
+ω+δβ−(⇒(δ[θ+[∑+⇔[θ+)β√γ[∑βθ+[θ(√(⇒
+r
r
íU
3
+U4U*?+TU@+TUA+,U
U
UUU
[ ]
++++00ZYXr=
r
UsFUimUóm%oadySTUgUYsuYsFsodmUimUSaFdmocamUsodYsU>UgU3 +U
U
∞(+ θβ+ δβθβ→+ β⇒√[⇒γ∑→∑+ ⇔→+ →2⇒(γ,δ+ ∫+ δ(∑β√√(⇒+ δβ+ ⇔→U -ysYtmU
r
r
U uysSsU cmicyimYFsU SsU imFU
cvóuvosodsFUY
(TUYgTUYtU√[[+θβ+β⇔(√→+→+√[⇒γ(⇒,→√(⇒⋅++
-CcaiósodsU uvSsóvFU sYaMacmYU lysU ivFU scdvYsFU SsU uvθ(√(⇒+ []βδβ√β⇒+ ⇔→+ ⇔β∫+ δβ+ θ,→+ →∑→+
scdvYsFíU voFaSsYsóvFTUuvYUsrsóuivTUivFU scdvYsFUSβ+[θ(√(⇒+∑+∫+∑
B
UcvoUYsFuscdvUmUivFUuyodvFUSsU
YsMsYsocamU>UgU>
B
+∫+β⇔+)β√γ[∑+δβ+[θ(√(⇒+δβ+sUgU>UcvoUYsFuscdvUmU>
B
USsUimUMa%yYmU+í6U sYaMacmóvFUlysU
β⇔+ )β√γ[∑+ δβ+ [θ(√(⇒+ ∑
B
U 4U >
B
∆+ ,βδβ+ []γβ⇒β∑θβ+ δβ+ ⇔[θ+ )β√γ[∑βθ+ δβ+ [θ(√(⇒+ θ+ ≤+ Τ
B
>U gU YU 4U >IU
muiacmoSvUimUYs%imUSsiUdYamo%yivUumYmUimUFyómUSsU scdvYsFíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  5  -U
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O
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Figura 1.3
vóvUFasóuYsUsFUuvFaeisUdYmtmYUyoUFs%ósodvUYscdai'osvTUsodYsUsiUvYa%soUgUyoUuyodvUmYeadYmYavU
SmSvTUsodvocsFTUFasóuYsUsFUuvFaeisUmFvcamYUmUSac!vUtreXjf vr dYvtYcXExj xYcXjd sY tjvEcE3ez

1.5  VECTOR DE DEZSPLAZAMIENTO.

/móvFUmUsóustmYUuvYUimUcmodaSmSU scdvYamiUóCFUFaóuisTUsiU desplazamientoTUlysUovUsFUóCFU
orY YF cóniEj sY tjvEcE3e sY re treXj ó jXdj MgXYecE3ef YvXY treXj trYsY vYd re njsYFj orY
YsuYsFsodmU yomU umYd'cyimU vU yoU uslysCvU cysYuvU lysU FsU dYmFimSm,íU niU SsFuimtmóasodvU sFU yoU scdvYU
tjdorY ej vjFónYeXY ióvXó sYcEd ó or2 sEvXóecEó vY njxE3 vEej Ye or2 sEdYccE3ez Cj Yv Fj nEvnj
FmiaYU SsU imU uysYdmU SsU cmFmU gU óv sYFsU 2U cymSYmFU !mcamU imU SsYsc!mU lysU !mcamU imU atlyasYSmíU niU
SsFuimtmóasodvU
noUsFUsiUóaFóvíU
niU SsFuimtmóasodvU mU ósoySvU ivU YsuYsFsodmóvFU uvYU yomU FvimU isdYmU ómg#FcyimU lysU mly'U imU
óvFdYmYsóvFUsoUos%YadmU3TUusYvU!mgUóyc!mFUvdYmFUómosYmFíUU
noUimUMa%U+í;UóvFdYmóvFUlysUsiUSsFuimtmóasodvUumYmUaYUSsUIU!mFdmUEUsFUyomUi'osmUYscdmUlysU
yosUsFdvFUuyodvFTUsóuastmUsoUIUgUdsYóaomUsoUEUSaYa%aSmU!mcamUEíU ymoSvUsiUcysYuvUFsUóys sUSsU
ómosYmUlysU mgmUgU ysi mUmiUuyodvUaoacamiTUsiUSsFuimtmóasodvUsFUcsYvíUnFUaóuvYdmodsUSmYFsUcysodmU
lysUsiUSsFuimtmóasodvUovUsFdCUYsimcavomSvUcvoUimUSaFdmocamUYscvYYaSmíU
A
B
P
UU U U U
A
B
P
U
Figura 1.4 (a) Figura 1.4 (b)

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  6  -U
U
/móvFUmUYsuYsFsodmYUimUmagnitudUSsUyoU scdvYU*imUivo%adySUsoUsiUcmFvUSsiUSsFuimtmóasodv,U
uvYUimUóaFómUisdYmUSsiU scdvYUusYvUovUsoUos%YadmUvU](β⇒<+
F→2⇒(γ,δ+[+δ,⇔[+δβ+
P,U4
PP
rr
=U
[∑+δβ−(⇒(√(⇒+β⇔+δ,⇔[+δβ++βθ+,⇒+βθ√→⇔→∑+,⇒+⇒0sYv,UgUFasóuYsUsFUuvFada víU
8yuvosóvFU m!vYmU lysU yomU umYd'cyimU dasosU yoU SsFuimtmóasodvU
PTU Fs%yaSvU uvYU yoU
SsFuimtmóasodvU
QíU niU YsFyidmSvU sFU siU óaFóvU lysU FaU FsU !yeasYmU cvoFaSsYmSvU umYdasoSvU SsiU óaFóvU
uyodvUaoacamiUyoU#oacvUSsFuimtmóasodvU
RUcvóvUuvSsóvFU sYUsoUimUMa%yYmíU
U
U
Figura 1.5
U
"vU lysU soU F'óevivFU uvSsóvFU s(uYsFmYU
R = P + QTU mU sFdsU scdvYU FsU ivU iimómUsuma vU
resultante.+[⇒β∑+→γβ⇒√(⇒+′,β+→′, +βθγ→[θ+θ,→⇒δ[+)β√γ[∑βθ+∫UovUsFUimUFaóuisUFyómUmi%seYmacmUSsU
θ,θ+δ,⇔[θ+θ(⇒[+′,β+δβ]β[θ+γ[→∑+β⇒+√,β⇒γ→+θ,θ+δ(YsccavosFíUU
U
U
Figura 1.6U
U
(β⇒δ[+⇔→+−(2+⋅↑+θβ+δβδ,√β+′,β<+++
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
()
+2+2+2rrrr
rrrrr
−=−+=δ U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  7  -U
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UUUUUUUUUUUUUUUUU+2δ
r
UU"imómSvU scdvYUSsUSsFuimtmóasodvUsodYsU
2+PyPU
U
8asóuYsUFsUuysSsUmFvcamYUmUyoUuyodvUFyUYsFuscda vU β√γ[∑+δβ+[θ(√(⇒+
[ ] [ ]
2222++++00000ZYXrZYXr ==
rr
U
[ ][ ]
+++222+20000ZYXZYX −=δ
r
U
[ ]
+2+2+2+200ZZYYXX −−−=δ
r
U
( ) ( ) ( )
2
+2
2
+2
2
+2
ZZYYXX −+−+−⇔δ
r
U
δ
r
+=θ+⇔→+−∑,⇔→+δβ+δ(θγ→⇒√(→+β⇒γ∑β+δ[θ+,⇒γ[θ++UgU32U
Ejercicio #2
→δ[θ+⇔[θ+,⇒γ[θ++USsUcvvYSsomSmFU*2T56T:,UósdYvFTU32USsUcvvYSsomSmFU*:T+T6,UósdYvFTU36USsU
cvvYSsomSmFU*5+T:T6,UósdYvFíU
m,
G→⇔⇔→∑+β⇔+)β√γ[∑+δβ+[θ(√(⇒+β⇒γ∑β+2UgU36U
e,
GmiimYUsiU scdvYUSsUSsFuimtmóasodvUsodYsU32UgU36U
c,
GmiimYUimUSaFdmocamUsodYsU3+UgU32U
Solución.-                                                                                      A
m,U→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUU→UUUUUUUUUUUU→U
UUUUY+UU⇔U[2T56TU:]UU⇔UU2UaUUU56UrUUUHU:U$U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUvU3+U
UU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUU→UUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU δ26U
UUY
2UUU⇔U[:TU+TU6]U⇔U:UaUHU+UrUH6U$UUUUUUUUUUUUUUUUUUUUUU36U>UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU>U3 2U
U
UUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUU→UUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU @UU
UUUY
6UU⇔U[5+TU:TU6]U⇔U5+UaUH:UrUHU6U$U
U
e,UU
→UUUUUUUUUUU→UUUUUU→U
UUUUUδ26U⇔UY6UIUY2U⇔[5+TU:TU6]U5U[:TU+TU6]UUUUUUUUUUUUUUUUUUUUUUUU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU?U
δ
26U4U[5+T5+TU:]UUU
c,UUU
→UUUUUUUUUUUUU→UUUUU→U
UUUUUUδ +2U⇔UUY2UIUY+U⇔U[:TU+TU6]U5U[2T56TU:] UUU
U
UUUU→UUUUUUUUUUUU
U
UUδ
+2UUU⇔U[52TU;TU6]U
U
U
δ
+2
→
U4U( )
222
6;2++−
UUU
4U7TU6=U*ó,USsUSaFdmocamU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  8  -U
U
1.6   MOMENTO DE UNA FUERZA CON RESPECTO A UN PUNTO .
voFaSsYsóvFUyomUMysYtmU-UlysUmcd#mUFveYsUyoUcysYuvUY'%aSvUcvóvUFsUóysFdYmUsoUimUMa%yYmUU
+íJíU vóvUFmesóvFUimUMysYtmU-UFsUYsuYsFsodmUuvYUyoU)β√γ[∑+′,β+δβ−(⇒β+θ,+→2⇒(γ,δ+∫+δ(∑β√√(⇒⋅+∞(⇒+
sóemY%vTUsiUsMscdvUSsUimUMysYtmUFveYsUsiUcysYuvUY'%(δ[+δββ⇒δβ+γ→](⇒+δβ+θ,+,⇒γ[+δβ+→⇔(√→√(⇒+⋅+
3→+[θ(√(⇒++δβ++θβ+δβ−(⇒β+√[⇒)β⇒(β⇒γββ⇒γβ++[∑+βiU scdvYUYUlysUyosUsiUuyodvUSsUYsMsYsocamUMarvU>U
cvoUsiUuyodvU30UsFdsU scdvYUFsUcvovcsUcvóvU scdvYUSβ+[θ(√(⇒+δβ+⋅+=⇔+)β√γ[∑+δβ+[θ(√(⇒+
rUgUimU
MysYtmU
FUSsMaosoUsiUuimovUóvFdYmSvUsoUimUMa%yYmU+íJU
U
Figura 1.7
β−(⇒(∑β[θ+β⇔+[β⇒γ[+δβ++√[⇒+∑βθβ√γ[+→+Τ+√[[+βiUuYvSycdvU scdvYamiUSsU rUgUF.

M
o = r * FU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUKU
YU⇔+β√γ[∑+δβ+[θ(√(⇒+δβ++
                                                                                                          →→→→            →→→→     →→→→
M P1 = δδδδ * F1
                                                                   →→→→
δU⇔U/scdvYUSsUSsFuimtmóasodvUsodYsU3+UgU32U
U
niUóvósodvUMoUSsesUFsYUusYusoSacyimYUmiUuimovUlysUcvodasosUmU>UgUmU FíUniUFsodaSvUSsUMo UFsU
δβ−(⇒β+[∑+β⇔+θβ⇒γ(δ[+δβ+∑[γ→√(⇒+′,β+[δ∑ →+⇔⇔β)→∑UsiU scdvYU
rUmUFsYUcviaosmiUcvoUsiU scdvYUF0UusYvU
βθγ→+ ∑[γ→√(⇒+ βθ+ ⇔→+ ∑[γ→√(⇒+ ′,β+
FU dasoSsU mU aóuYaóaYU miU cysYuvíU IF'TU siU FsodaSvU SsiU óvósodvU MoU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  9  -U
U
√→∑→√γβ∑(•→+ β⇔+ θβ⇒γ(δ[+ δβ+ ⇔→+ ∑[γ→√(⇒+ ′,β F+ γ(β⇒δβ+ →+ (∑((∑+ →⇔+ √,β∑[+ ∑ 2(δ[+ βθγ→+ ∑[γ→√(⇒+FsYCU
veFsY mSmUcvóvUcvodYmYamUmUimUSsUimFUm%yrmFUSsiUYsivrUuvYUyoUveFsY mSvYUFadymSvUsoUsiUs(dYsóvUSsU
MoíU
Τγ∑→+→⇒β∑→+δβ+βθγ→]⇔β√β∑+⇔→+∑β⇔→√(⇒+β⇒γ∑β+β⇔+θβ⇒γaSvUSsU
Mo ∫+β⇔+θβ⇒γ(δ[+δβ+⇔→+∑[γ→√(⇒+δβ⇔+√,β∑[+
Y'%aSvUsFUsiUFyóaoaFdYmSvUuvYUimUYs%imUSsUimUómovUSβ∑β√ϕ→<+√(β∑∑β+θ,+→⇒[+δβ∑β√ϕ→+→⇒γβ⇒(β⇒δ[+β⇔+
δβδ[+,⇔2→∑+βγβ⇒δ(δ[ω+θ[θγ⇒2→⇔→+δβ+γ→⇔+[δ[+′,β+θ,θ+δβδ[θ+(⇒δ(′,β⇒+β⇔+θβ⇒γ(δ[+δβ+∑[γ→√(⇒+′,β+⇔→+
MysYtmU
FUdasoSsUmUaóumYdaYisUmiUcysYuvUY'%aSv0UFyUSsSvUuyi%mYUaoSacmYmUsiUFsodaSvUSsiUóvósodvU MoíU
-aomiósodsTUiimómoSvU
θ+→⇔+∆⇒2,⇔[+√[∑β⇒δ(δ[+β⇒γ∑β+⇔→θ+⇔ ⇒β→θ+δβ+→√√(⇒+δsiU scdvYUSsU
[θ(√(⇒+
rUgUimUMysYtmUFU!miimóvFUlysUsiUóvósodvUSsUF+√[⇒+∑βθβ√γ[+→+Τ+βθ<+
Mo = r F. sen θθθθ = F d

Ejercicio #3

→δ→+⇔→+−,β∑•→+[+TU2T52]UdvosimSmFTUlysUumFmUuvYUsiUuyodvU3++ωωφ+βγ∑[θ⋅+βγβ∑(⇒→∑+β⇔+
óvósodvUSsUimUMysYtmUYsFuscdvUsiUuyodvU3
2U*:TU2T5+,íU
Solución.-                                                                UUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUAU
UUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUδ2+U
UUUY
+⇔U[2T+T6]UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU3 2UvUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUvU3+UUU
UUUU→U U
UUUY6UU⇔U[:TU2T5+]UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU@U
→UUUUUUUUUUUU→UUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
δ2+U⇔UY+UIUY2U⇔U[2TU+T6]5[:TU2T5+]UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU-U
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
δ2+U⇔U[2T5+T;]U scdvYUSsFuimtmóasodvUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU-U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU?U
→UUUUUUUUUU→UUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUU
F
2U4Uδ2+ULU-U
U
U→UUUUUUUUU→UUUUUUUUU→U
UUUUUUU2UU5+UUU;UU
F
2U4Uδ2+ULU-U4UUUUU+UUU2UU52UUUUU4UU59UaUHU=UrUHU7U$U4U[59TU=T7U]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUrUUU$U
U
→
U
F
2U4UU√U*59,
2
UHU*=,
2
UHU*7,
2
U
→U
F2U4U++TU=9UdvoMóU
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  10  -U
U
1.7   MOMENTO DE UNA FUERZA RESPECTO DE UN EJE.
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
I!vYmUlysUFsU!moUmyósodmSvUoysFdYvFUcvovcaóasodvFUSsUmi%seYmU scdvYamiTUaodYvSycaYsóvFUyoU
oys vU cvocsudvTU siU cvocsudvU SsiU óvósodvU SsU yomU MysYtmU cvoU YsFuscdvU mU yoU srsíU voFaSsYmYsóvFU
oys mósodsU yomU MysYtmU -U lysU mcd#mU FveYsU yoU cysYuvU Y'%aSvU gU siU óvósodvU F
vU SsU sFmU MysYtmU cvoU
YsFuscdvUmUimU*Ma%U+í=,íUU
U
Figura 1.8
m,UnoUsiU%YmMacvUFsUuysSsUveFsY mYUlysUsiUsrsUsFUusYusoSacyimYUmiUuimovU))U
e,UniUuyodvU3UusYdsoscsUmiUuimovU))U
c,UniUuyodvUmUsFUimUdYmtmUSsiUsrsUsoUsiUuimovU
S,U
+F
r
UsFUimUcvóuvosodsUSsUF
r
UcvodsoaSmUsoUsiUuimovU))U
s,U
2F
r
UsFUimUcvóuvosodsUSsUF
r
UumYmisimUmiUsrsU
δ[⇒δβ<+
F
mU =UFvósodvUSsUMysYtmTUYsFuscdvUSsUaUvUcymilyasYUuyodvUSsUsFsUsrsíUUUUUUUUUUUUUUUUUU
UUUUUUUUUU
UUUUU
Ee
r
++≤+=θ+β⇔+)β√γ[∑+,⇒(γ→∑([+β⇒+⇔→+δ(∑β√√(⇒+δβ⇔+βαβ⋅+
U
[⇒δβ+βθγβ+[β⇒γ[+θβ+⇔[+β∑βθ→+√[[<+
EE
eMM
rr
UL
m
=
∞β+ √[⇒√⇔,∫β+ δβ+ ⇔→+ δβ−(⇒(√(⇒+ δβ+ [β⇒γ[+ δβ+ ,⇒→+ −,β∑tmU cvoU YsFuscdvU mU yoU srsTU lysU siU
óvósodvUSsU-UcvoUYsFuscdvUmUyoUsrsUcvvYSsomSvUsFUa%ymiUmimUcvóuvosodsUF
mUcvoUYsFuscdvUmUSac!vU
srsíU8yFdadygsoSvUFycsFa mósodsUcmSmUyovUSsUivFU scdvYsFUyoadmYavFUaTUrTU$TUuvYUs
nTUcvóuYvemóvFUlysU
imFU s(uYsFavosFU mFaU vedsoaSmFU SsU ivFU óvósodvFU SsU -UcvoU YsFuscdvU mU ivFU srsFU cvvYSsomSvFU FvoTU
YsFuscda mósodsTUa%ymisFUmUimFUs(uYsFavosFUSsUimFUcvóuvosodsFUSsiUóvósodvUF
mUSsU-UcvoUYsFuscdvUmU
3íUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  11  -U
U
Ejercicio #4
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU U UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU →UUUUUUUUUUUUUUUUUUUUU
UUUUUUUUU"vFU3yodvFU3
+U*:T52T+,UósdYvFUgU32U*6T52T+,USsMaosoUyoUsrsUsoUsiUsFumcavTUcasYdmUMysYtmUU-U [2TU6T+]U
N%íUumFmUuvYUsiUuyodvU3
:+πωπω⌠⋅+βγβ∑(⇒→∑+β⇔+[β⇒γ[+δβ+⇔→+−,β∑•→+∑βθscdvUSsiUsrsíU
U
Solución.-
U→U
3
23+U⇔U[56TU:T+]UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU3 +UvU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUnU
UUUU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUsUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
323+U4U
+: U4U6T2U⇔UUUUUU 4U[5:T<;0U:0U:T6+]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU3
+32U
→U
3
+32UU⇔U[6TU:TU5+]UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUvU3 2U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UU
3+32U4U
+: 4U6T2U⇔UUUUUUUUUsUUUUUUU4UUUUUUU[:T<;0U:0U5:T6+]UUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU3
+32U
→U
δ+:U4U[5+TU:TU52]U
UU U →UUUUU→UUUUU→U
F
3+U4Uδ+:ULU-U
U
UUUUUUU U U U4U
[5+TU:T52]ULU[2TU6TU+]U
U UUUUUUUUUUUUUUaUUUUrUUUUU$U
UUUUUU U UU4UUU5+UUU:UUU52UUUUUUU4U9aUIU6rU56$UUUU
UUUUUUUUU2UUU6UUUU+U
→UUUUUUUUUUUUUUU→U
F
nU4UFu+ULUsUu2u+ULUnU4U[9TU56TU56]ULU[5:í<;0U:0U5:í6+]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UU 4U
[57í9;UHU:í<6]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UU 4U9í7;UN%MóU
F
nU⇔U[9TU56TUU56]ULU[:í<;0U:0U5:í6+]U4U:í7;UN%MóUUUUUUUUUUUUU
UUUUUUU U UUUUU
→UUUUUUUUU→U
F
u2U⇔Uδ2U:ULU-U
UUUUUUUUUU U U U 4U
[5;TU:TU+]ULU[2TU6TU+]U
UUUUUUUUUU
UUUUUUUUUUUUUUU U U 5U;UUU:UUU+U
UUUUUU U U UUUUU4UUUUUU2UUU6UUU+UUUUUU4U6UaUHU2UrUIU+2U$U
UUUUUUUUUUUUUUaUUUUrUUUU$U
F
nU⇔U[6TU2T5+2]ULU[5:í<;0U:0U:í6+]U
UUUUUUUU U U U 4U52í=2UHU:UHU*56íJ2,U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  12  -U
U
UUUUUUUU U U U 4U59í7;UN%MóíU
F
nU⇔[6TU2TU5+2]ULU[:í<;0U:0U5:í6+]U
U
UUUUUUUU U U U 4U2í=2UHU6íJ2U
U
UUUUUUUU U U U 4U9í7;UN%MóíU
U
1.8   TEOREMA DE VARIGNON.

"mU uYvuasSmSU SaFdYaeyda mU SsU ivFU uYvSycdvFU scdvYamisFU uysSsU ydaiatmYFsU umYmU SsdsYóaomYU siU
óvósodvUSsUimUYsFyidmodsUSsU mYamFUMysYtmFUcvocyYYsodsFíU8aU mYamFUMysYtmFU-
+TU-2TOOOíFsUmuiacmoUmiU
(θ[+,⇒γ[+ω+∫+θ(+⇔⇔→→[θ+∑+→⇔+)β√γ[∑+δβ+[θ(√(⇒+δβ+ω+θβ+√[⇒√⇔,∫β+(⇒βδ(→γ→β⇒γβ+′,β<+
YULU*-
+UHU-2UHUOOí,U4UYULU-+UHUYULU-2UHUOOOU
U
Figura 1.9
noUumimeYmFTUsiUóvósodvUcvoUYsFuscdvUmUyoUuyodvUSmSvU>USsUimUóaFómUYsFyidmodsUSsU mYamFU
MysYtmFUcvocyYYsodsFUsFUa%ymiUmUimUFyómUSsUivFUóvósodvFUSsUimFUMysYtmFUcvoUYsFuscdvUmiUóaFóvUuyodvU
Τ⋅+ 3→+ ∑β⇔→√(⇒+ δβ+ ⇔→+ 2∑→−(√→+ ϕ→√β+ [θ(]⇔β+ ⇔→+ δβγβ∑(⇒→√(⇒+ δβ⇔+ [β⇒γ[+ δβ+ ,⇒→+ −,β∑•→+ + [∑+ β⇔+
cmicyivUSsUivFUóvósodvFUSsUSvFUvUómFUMysYtmFUcvóuvosodsFíUUU
"mUMysYtmU-UFsUSsFcvóuvosUsoUUU-U4U-
+UHU-2UH-6U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUU→UUUUUU→UUUUUUUU→UUUUUUUU→UUUUUUUUUUUU→U
⇒UFnU4UFULUsU4U[YULU*-+UHU-2UHU-6,]ULUsU
[⇒δβ<++++++++++++++++++++++++
                  →→→→      →→→→     →→→→       →→→→      →→→→    →→→→       →→→→      →→→→     →→→→ U
ME  =  (r * F1) .e + (r * F2).e + (r * F3). e

PsvYsómUóvósodvUSsUyomUMysYtmUYsFuscdvUSsUyoUuyodvU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  13  -U
U
1.9   PAR DE FUERZAS – MOMENTO DEL PAR.

8sUSacsUlysUSvFUMysYtmFUFUgU–FUMvYómoUyoUumYUFaUdasosoUimUóaFómUóm%oadySTUSaYscca3e tYdj
FsodaSvUcvodYmYavTU sYUMa%yYmU+í+:U
U
Figura 1.10

U nFUs aSsodsUlysUimUFyómUSsUivFUcvóuvosodsFUSsUimFUsjv prYdaóv Ye cróForEYd sEdYccE3e Yv
csYvíU8aoUsóemY%vTUimUFyómUSsUivFUóvósodvFUSsUimFUSvFUMysYtmFUcvoUYsFuscdvUmUyoUuyodvUSmSvUovUsFU
csYvíUIyolysUimFUMysYtmFUovUSsFuimtmoUmiUcysYuvUFveYsUsiUlysUmcd#moUsiimFUdasoSsoUmUaóuYaóaYisUyoU
njxEnEYeXj sY djXócE3e

Figura 1.11

U 1suYsFsodmoSvUuvYUrAUgUrB Fjv xYcXjdYv sY tjvEcE3e sY Fjv treXjv sY ótFEcócE3e sY FU@U– FUSsU
imUMa%yYmU+í++UsocvodYmóvFUlysUimUFyómUSsUivFUóvósoXjv sY Fóv sjv prYdaóv cje dYvtYcXj ó w Yv1
rA * F + rB * (-F) = (rA – rB) * F

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  14  -U
U
U 1sóuimtmoSvUrAUIUrB 4UrTUSvoSsUYUsFUsiU scdvYUlysUyosUivFUuyodvFUSsUmuiacm√(⇒+δβ+⇔→θ+δ[θ+
MysYtmFTUcvociyaóvFUlysUimUFyómUSsUivFUóvósodvFUSsU-UgUI-UcvoUYsFuscdvUmU>UFsUYsuYsFsodmoUuvYUsiU
)β√γ[∑<+
Mo = r * FUUUUUUUUUUUUUUUUUUUUUUUUU*+,U
U niU scdvYU
MoUFsUisUiimómUóvósodvUSsiUumY0UsFUyoU scdvYUusYusoSacyimYUmiUuimovUlysUcvodasosU
⇔→θ+δ[θ+−,β∑•→θ+∫+θ,+→2⇒(γ,δ+βθ<+
Mo = r.F.senθθθθ = FdUUUUUUUUUUUUUU*2,U
U
+ [⇒δβ+δ+βθ+⇔→+δ(θγ→⇒√(→+β∑β⇒δ(√,⇔→∑+β⇒γ∑β+⇔→θ+⇔ ⇒β→θ+δβ+→√√(⇒+δβ+
FUgU–FTUgUsiUFsodaSvUSsUMoU
FsUsocysodYmUósSamodsUimUYs%imUSsUimUómovUSsYsc!mUSsiU%YmMacvíU
U vóvU siU scdvYU
r
r
+ β⇒+ + βθ+ (⇒δββ⇒δ(β⇒γβ+ δβ+ ⇔→+ β⇔β√√(⇒+ δβ⇔+ [∑(2β⇒+>U SsU ivFU srsFU
cvvYSsomSvFTUFsUvedsoSY'mUsiUóaFóvUYsFyidmSvUFaUivFUóvósodvFUSsU
FU@U– FUFsU!yeasYmoUcmicyimSvUcvoU
YsFuscdvUmUyoUuyodvUSaMsYsodsU>
′
íU3vYUdmodvTUsiUóvósodvUFUSsUyoUumYUsFUyoU scdvYUiaeYsUlysUuysSsU
muiacmYFsUsoUcymilyasYUuyodvUcvóvUFsU sUsoUimU%YmMacmíU
U
Figura 1.12
+ β+⇔→+δβ−(⇒(√(⇒+δβ⇔+[β⇒γ[+δβ+,⇒+→∑ω+θβ+√[⇒√⇔,∫sUlysUSvFUumYsFTUyovUMvYómSvUuvYUimFU
MysYtmFU
F1UgU-F1TUgUsiUvdYvUuvYUimFUMysYtmFUF2UgU5F2  cvóvUFsU sUsoUimUMa%yYmU+í+6U*m,UgU*e,,UdsoSYCoU
óvósodvFUa%ymisFUFaUgUFvivUFa0UivFUSvFUumYsFUsFdCoUsoUuimovFUumYmisivFUvUsoUsiUóaFóvUuimovUgUdasosoU
β⇔+(θ[+θβ⇒γ(δ[<+++++++++++++
UUUUUUUUUUUUUUUU-
+S+U4U-2S2U
U U 5- +UU - 2U
U UUUUS +U U SU
U -
+U UU5- 2U

Figura 1.13 (a)                                 Fi gura 1.13 (b)
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→→→→          →→→→     →→→→
3→+β∑βθ(⇒<    MO = δδδδ * F  YsuYsFsodmUsiUóvósodv Ssi PAR DE FUERZAS
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  15  -U
U
Ejercicio #5U
+ →δ[θ+⇔[θ+→∑βθ+δβ+⇔→+−(2,∑→+⋅+θβ+(δβ+δβγβ∑(⇒mYUimUFyómUSsUivFUóvósodvFUumYsFíU
+ [⇒δβ<+{
+UgU 6UusYdsoscsoUmiUuimovU@ATU 2UusYdsoscsUmiUuimovUαUumYmisivU?AíUU
U
U
Figura 1.14
+ ∞β+γ(β⇒β<+
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UUUUUUUUUUUUUUU→U
m,UFvósodvUSsUivFUumYsFUUUUUUFc+U4U9UaU4U[9TU:TU:]UdíóUUUUUUUUUUUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
UU→UUUUUUUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUFc2U4U5U=UrU4U[:TU5=TU:]UdíóU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UUUUUUUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUFc6U4U2:UaU4U[2:TU:TU:]UdíóUUUU
e,U8yómUSsUivFUumYsFUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UUUU→UUUUUUUUUUUU→UUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU8U4UFc +UHUFc2UHUFc6U4U[29TU5=TU:]UdíóU
+++++++++=θ+δβ√(∑<+
para sumar pares se debe sumar los momentos de los pares U
1.10   TRASLACION DE UNA FUERZA A UNA POSICION PARA LELA.

U 8sUóysFdYmUyomUMysYtmUgUFsUSsFsmUdYmFimSmYimUmUyom+[θ(√(⇒+→∑→⇔β⇔→+′,β+→θ→+[∑+β⇔+,⇒γ[+⋅+
vóvUFsU sUsoUimUMa%yYmU+í+7U
U
Figura 1.15

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  16  -U
U
UU niUuYvcsSaóasodvUumYmUdYmFimSmYUyomUMysYtmUmUcasYγ→+[θ(√(⇒+→∑→⇔β⇔→+δ→δ→+√[⇒θ(θγβ+β⇒++++
+⇔[θ+θ(2,(β⇒γβθ+→θ[θ<+
+,
∞β+γ∑→θ⇔→δ→+⇔→+−,β∑•→+→+⇔→+⇒,β)→+[θ(√(⇒+→⇒γβ⇒(β⇒δ[+θ,+δ(∑β√√(⇒ω+→2⇒(γ,δω+∫+θβ⇒γ(δ[⋅+
2,
∞β+→C→δβ+⇔→+−,β∑•→+θ(γ∑(√→+β⇒+⇔→+δ(∑β√√(⇒+δβ+γ∑→FimSvU
6,
8sUaociygsUsiUóvósodvUSsiUumYUvcmFavomSvUuvYUimUMys∑•→+∫+θ,+θ(γ∑(√→⋅+
U U
U U
U
U U U
U U
U
U U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U U U →U UUUUUU→
U UUcU4UsFUsiUumYUMvYómSvUuvYU-UgU5-U
U U
U
U U
U
UUUUUUUUUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUU
U
U
U
                     figura 1.16UUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U voUsFdvFUumFvFUFsUvYa%aomUyoUoys vUFaFdsómTUcvoFdadyaSvUuvYU
F
r
+β⇒+⇔→+[θ(√(⇒+→∑→⇔β⇔→+∫+β⇔+
umYUcUMvYómSvUuvYUF
r
UgUIF
r
UcvóvUFsU sUsoUimUMa%U+í+9íU
+ {,→⇒δ[+ θβ+ γ∑→θ⇔→δ→+ ,⇒→+ −,β∑•→+ →+ ,⇒→+ [θ(√(⇒+ →∑→⇔simU SmSmTU SsesU mSsóCFU SsU imU MysYtmTU
dvómYFsUsoUcysodmUsiUumYíU
+ [θ+ θ(θγβ→θ+ δβ+ −,β∑•→θ+ θ[⇒+ β′,()→⇔β⇒γβθ+ θ(+ γ(β⇒β⇒U mU uYvSycaYU siU óaFóvU sMscdvU FveYsU yoU
cysYuvíU8sU!mU aFdvUlysUyosUMysYtmUFsUuysSsUSsFuimt→∑+θ[]∑β+θ,+⇔ ⇒β→+δβ+→√√(⇒+∫+β⇔+β−β√γ[+θ[]∑β+β⇔+
√,β∑[+⇒[+θβ+[δ(−(√→+γ→⇒γ[+→∑→+γ∑→θ⇔→√(⇒+√[[+→∑→+∑[γ→√(⇒⋅+∆θ +(θ[+θβ+ϕ→+δ(√ϕ[+′,β+,⇒+→∑ω+
uvYU FsYU yoU scdvYU iaeYsTU FsU uysSsU dYmFimSmYU mU cymil,(β∑+ [θ(√(⇒+ θ(⇒+ ′,β+ θβ+ √→](β+ β⇔+ β−β√γ[+ ′,β+
uYvSycsUFveYsUsiUcysYuvíU
Ejercicio #6UUUU
+ →δ→+⇔→+−,β∑•→+
F
r
U4U[2TU+TU:]UdvosimSmFUlysUumFmUuvYUsiUuyodvU3+U*6TU2TU5+,UósdYvFUgUsiUuyodvU
3
2U*;TU:TU2,UósdYvFTUFsUuaSsUdYmFimSmYUimUMysYtmUmUyo→+[θ(√(⇒+→∑→⇔β⇔→+′,β+→θ→+[∑+ 2<UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→+γ∑→θ⇔→√(⇒+δβ+
F
r
+→+⇔→+[θ(√(⇒+→∑→⇔β⇔→+′,β+→θ→+[∑+2U
UUUUUUUUUUUUUU U U
UUUUUUUUU U U U UUUUUUU F
r
UsoU3UUUF
r
U4[U2TU+T:]UdU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  17  -U
U
UUU→UUUUUUUUUUUU→ U
UUU3
23+UU4UδU4[5+TU2TU56]UdU
U
UUU
→UUUUUUUU→UUUUUUU→U UUU5+UUU2UUU56U
UUUF
32U4UUδUU4UU-U4UUUUUU2UUU+UUUU:UUUUU4U[6TU59TU57]UdíóU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUrUUUU$U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UU
→U
U 3vYUivUdmodvTUsiUoys vUFaFdsómUlysSmUcvoFdadyaSvUuvYU-U4U
[2TU+TU:]UtonUdYmFimSmSmUmUimU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUU
→U
+ [θ(√(⇒+→∑→⇔β⇔→+′,β+→θ→+[∑+
2UgUsiUóvósodvUSsiU3mYUF32U4U[6TU59TU57]íU

1.11   RESULTANTE DE FUERZAS Y MOMENTOS.

U
                            Figura 1.17 (a)                                              Figura 1.17 (b)
U
U noU imU Ma%yYmU +í+JU *m,U FsU veFsY mU yoU FaFdsómU %sosYmiU SsU MysYtmFTU cvoFdadyaSvU uvYU MysYtmFU gU
óvósodvFUSsUumYsFíUU
U FsSamodsU siU uYvcsSaóasodvU SsFcYadvU soU siU modsYavYUdadyivíU 8asóuYsU sFU uvFaeisU dYmFimSmYU imFU
−,β∑•→θ+→+,⇒→+[θ(√(⇒+→∑→⇔β⇔→+′,β+→θ→+[∑+β⇔+[∑(%soTUve amósodsUdsoasoSvUsoUcysodmUivFUóvósodvFU
cvYYsFuvoSasodsFUgUYscvYSmoSvUlysUivFUóvósodvFUSsUivFUumYsFUFvoU scdvYsFUiaeYsFíU
+ =⇒+⇔→+−(2,∑→++⋅J+]+θβ+γ(β⇒β<+
→UUUUUUUU→UUUUUUUU→UUUUUUUU→ U
-
1U4U-+UHU-2UHU-6U
→UUUUUUUUUU→UUUUUU→UUUUUUUU→UUUUUUU→UUUUUUU→UUUUUU→UUUUUUUUU→UUUUUUUUU→U
F1U4UY+ULU-+UHUY2ULU-2UHUY6ULU-6UHU +UHU 2U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  18  -U
U
U imYmósodsTU ivFU FaFdsómFU SsU imFU Ma%yYmFU +í+JU *m,U gU *e,U FvoU EQUIVALENTES PARA
CIERTA CAPACIDAD.
U 8aoUaóuvYdmYUimUcvóuisraSmSUSsUyoUFaFdsómUSsUMysYtmFUgUóvósodvFTUsFdsUFasóuYsUuysSsU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UFsYUYsSycaSvUmUyomUFvimUMysYtmU-1UgUmUyoUFvivUóvósodvUF1íU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
-
1U4UFsUSsovóaomU1sFyidmodsUSsUMysYtmFU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUUUF1U4UFsUSsovóaomU1sFyidmodsUSsUóvósodvFU
U
+ =⇒γ[⇒√βθ<+3→+∑βθ,⇔γ→⇒γβ+δβ+,⇒+θ(θγβ→+βθ+⇔→+=Q5∆"nD )IUóCFUFaóuisíU
Ejercicio #7
+ →δ[+β⇔+θ(θγβ→+δβ+⇔→+−(2,∑→+⋅ο+δβγβ∑(⇒→∑+⇔→+∑βFyidmodsUSsUóvósodvFíU
U
Figura 1.18
UUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
[⇒δβ<++U4U[6TU2T+]UdUgU-2U4U[5;TU+T6]Uó0U32U*;TU7T9,UóUgUsiU3mYU UFsUsocysodYmUsoUsiUuimovU(g0+θβ+(δβ<+
→+βθ,⇔γ→⇒γβ+δβ+−,β∑•→θ+β⇒+β⇔+[∑(2β⇒<+
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUU→UUUUUUUU→U
U U - 1U4U-+UHU-2U4U[5+TU6T;]UdU
]+βθ,⇔γ→⇒γβ+δβ+[β⇒γ[θ+β⇒+β⇔+[∑(2β⇒<+
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
Y+U4U[2T52T+]U0UY2U4U[;TU7T9]U
[∑+γ→⇒γ[<++
U UUUUUUUUUUUUUUUUUUUUU
→U
U UUUUUUUUUUUUUUUUUUUUUF
1U4U[2T52T+]ULU[6TU2T+]UHU[;TU7T9]ULU[5;TU+T6]UHU[:TU:T9]U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  19  -U
U
UU U →U
U U F
1U4U[7T567T;:]Uton_mU
U
√+βθ,⇔γ→⇒γβ+δβ⇔+θ(θγβ→<+
U
U
Figura 1.19U
U
UUUUUU >e amósods0UsiUFaFdsómUSsUimUMa%yYmU+í+<UsFUnQ&)/I"nDPnUmiUFaFdsómUSsUimUMa%yYmU+í+=U
umYmUcasYdmUcmumcaSmSíU
Ejercicio #8
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU U U U U U U UUUU2UUUUUUUUUUUUUUUUUUUUUU56U
UUUUU/sYaMacmYUimUa%ymiSmSUSsUivFUSsFuimtmóasodvFUsodYsUivFUuyodvFU3
+UUU:UU*ó,UUgUU32UUUUUU;UU*ó,U
U U U U U U UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU6UUUUUUUUUUUUUUUUUUUUUUU+U
Solución.-
UUUUUUUUUUUUUUUUUU→U
m,UUUUUUUUUδ+2U scdvYUSsUSsFuimtmóasodvUU
UUUUU
UUUUUUUUUUU→UUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUUUδ+2U⇒UY+U4U[2TU:T6]U0UY2U4U[56TU;T+]U
UUUUUUUUUUUU
→UUUUUUUUUUUU→UUUUUUUUUUUU→U
UUUUUUUUUUUUδ+2U⇔UY2UHU*5UY+,U⇔U[56TU;T+]UHU*5,[2TU:T6]U
UUUUUUUUUUUUUUUUUU→U
δ+2U⇔U[57TU;T52]U
e,UUUUUUU
U→UUUUUUUUUUUU→UUUUUUU→U
UUUUUUUUUUUδ2+U⇔UY+UHU*5Y2,UU⇔UU[7T5;T2]U
UUUU
UUUUUUUUUU→U
UUUUUUUUUUUUUUUU
δ2+U4U[7T5;T2]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUδ+U2UUUUURUUUUδU2U+U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  20  -U
U
Ejercicio #9
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUU U5+UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU →U
UU "mUMysYtmU-⇔U[:TU6TU52U]UUdíUumFmUuvYUsiUuyodvU32UUUUU2UU*ó,USsdsYóaomYUsiUóvósodvUSsU-U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU6U
U
Solución.-
U
m,
1sFuscdvUSsiUvYa%soU*:T:T:,UóU
U
UUUUUUU [ ]6T2T+
2−⇔⇒r
r
U
kji
FrM
O 26:
62+
L
2 −
−
==
rU
()()()kjiM
O
.L6.L2.L;< −+−+−+−= U
kjiM
O
.6.2.+6 −−−= U [ ] mtonM
O
M6T2T+6−−−= U
e,U1sFuscdvUSsU3
+U*5;TU2T:,UóU
U
FM
P L
+2+δ
r
= U
[ ] [ ]6T2T+0:T2T;
2+−=−−= rr
rr
U
[]6T:T6
+2⇔δ
r
U
6:6
26:
+ −=
kji
M
p
4S5<T9T<TUUton_mU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  21  -U
U
Ejercicio #10 UUU
++++++++++++++→δ→+⇔→+−,β∑•→+F
r
⇔[5+TU2TU6U]UdUlysUumFmUuvYUsiUuyodvU
6
2
+−
P
m+βγβ∑(⇒→∑+β⇔+[β⇒γ[+δβ+
F
r
U
Solución.-
m, 1sFuscdvUSsiUvYa%soUU*:TU:T:,UóíU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
U -U
>•U
U
UUUUUUUUUUUUUUUUUUUUUU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→ U
UUUUUUUUUUY
2UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUY 2UU⇔U[5+TU2TU6]U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU•UU3
2U
UU
U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUU→ UUUUUUU5+UUU2UUU6U
U F
>U4UYULU-U4UUUUU:UUU6UUU52U
U aUUUUrUUUU$U
U
4U5<UHU*5;,UaUHU*52,UrUHU*56,U$U
U
4U5+6UaUIU2UrU56U$U
UUUUUUU→U
⇒UF
>U4U[5+6TU52TU56]Uton_mU
U
e,U1sFuscdvUSsU3
+U*5;TU2TU:,UFU
U
UUUUUUUUUUUUUUU
:U3+U
UUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUY
+U -UUUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUU→U
:U
UδU Fu +U4Uδ+2ULU-U
UUUUUU
→UUUUUUUUUUUUUUUU
UUUUUUY2UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→ U
U UU
:UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUY+U4U[5;TU2TU:]U0UY 2U4U[5+TU2TU6]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU3
2UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUδ
+2UU⇔U[6TU:TU6]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUaUUUUrUUUUU$U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUFu
+U4UUUUU:UUU6UUU52UUUU4U[5<TU9TU<]Uton_mU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU6UUU:UUUU6UUUUUUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  22  -U
U
Ejercicio #11 UUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU2UUUUUUUUUUUUUUUUUUUU56U
asYdvUsrsUsFdmUSsMaoaSvUuvYUivFUuyodvFU3
+UUU6U*ó,UUgUUU32UUUU2U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU:UUUUUUUUUUUUUUUUUUUUU+U
UUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU;U
UUUUUUUUU-U⇔[;TU:TU+]UN%UUlysUumFmUuvYUsiUuyodvU3
vUUU52U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU5+U
++βγβ∑(⇒→∑+β⇔+[β⇒γ[+δβ+⇔→+−,β∑•→+∑βθβ√γ[+δβ⇔+βrsíU
Solución.-
UUUUUUUUUU→U
UUUUUUUUU-UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU:U32UUUUUUUUUUUUUUUUUUUUUUUU3+32U⇔U[57TU5+TU+]U
UUUU3
>U →UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU →U
UUUUUUUUUδUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU3
+32UUUUU4U7ULU+<U
UUUUUUUUUUUUU:U
UUUUUUUUU3+UUUUUUUU
UUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUU+
U
snU4UUUUUUUUUUUUUUUULU[57TU5+TU+]U⇔U[5:í<9TU5:í+<TU:í+<]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU7Uí+<U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUδ
2:U⇔U[JTU5;TU52]U
U
UUUUUUUUUUUUUUUUUU→UUUUUU→UUUUUUUUJU5;UUU52U
UUUUUUFu
2U⇔Uδ 2:ULU-UU⇔UUUU;UUU:UUUU+U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUrUUUUU$U
U
UUUUUUFu
2U4U[5;TU5+7TU+9]U
UUUUUUUUUUUUUUUUUUUUU UUUUUU UUUUUUUUUUUUUU→UUUUUU→U
UUUUUUUUUUUUUUUU UU∴UUF
nU4UFu2ULUsnU
⇒UF
nU4U[*5;,U*5:í<9,UHU*5+7,U*5:í+<,UHU*+9,U*:í+<,]U
UUUUUU U U ∴F
nU4U<íJ6UN%íóU
U
Ejercicio #12
8sUuaSsUSsdsYóaomYUimUYsFyidmodsUóCFUFaóuisUSsiUFaFdsómUm,USsMaoaSvUsoUMvYómUcvuimomYU
UUUUUUUUUUUUU UU -
+U⇔U[6TU52TU:]U0U3 +U*2TU;TU:,U
U U U -
2U⇔U[5+TU5+í7TU:]U0U3 2U*5+TU6TU:,U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  23  -U
U
U

Solución.-

→
UUUUUUUUUUUUUU U Y +UU⇔U[2TU;TU:]UU
UUUUUUUUUUUUU U
→U
UUUUUUUUUUUUUUUUU U Y 2U⇔U[5+TU6TU:]UUU
→→→→               U
UF
1UU⇔U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UUUUUU→ UUUUUUUUaUUUUUUrUUUUU$U
Y
+ULU-+U4UUUU2UUUU;UUUUU:UUUUU4U[:TU:TU5+9]Uton_mU
UUUUUUUUUUUUUUUUU6UUU52UUUUU:U
UU
→UUUUUUU→ UUUUUUUUaUUUUUUUUrUUUUUU$U
Y
2ULU-2U4UUU5+UUUUUU6UUUUUU:UUUUU4U[:TU:TU;í7]Uton_mU
UUUUUUUUUUUUUUUUU5+UUU5+í7UUUU:U
→U
3mYU4U[:TU:TU;í7]Uton_mU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUU→UUUUUUU→UUUUUUU→UUUUUU→UUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUF1U4UY+ULU-+UHUY2ULU-2UHU3mYU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUF1U4U[:TU:TU5+9]UHU[:TU:TU;í7]UHU[:TU:TU57]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUF1U4U[:TU:TU5+9í7]Uton_mU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
U FvósodvUSsU-
1U⇔UFvósodvUSsiUFaFdsómU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  24  -U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUgU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUU→UUUUUU→U
U Y
1ULU-1U⇔UF 1U
U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUF
1U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU(U
U
                                                  →→→→
U - 1U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
U 8smU3USsUcvvYSsomSmFU*(TUgTUv,UóUyoUuyodvUSsUimUi'⇒β→+δβ+→√√(⇒+δβ+
1U
U
3U*(TUgTUv,UóU⇒UUY
1UU⇔U[(TUgTUv]U
U
UUUUUUUUU→UUUUUU→UUUUUUUUUUUU→UUUUUUU
UUUUUUUUU(UUUUUgUUUUUvU
⇒UUY
1ULU-1U⇔UF 1U⇒UUUUU52UU56í7UUU:UUUU4U[:TU:TU5+9í7]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUUUrUUUUU$U
U
⇔U[:TU:TU56í7U(U52Ug]UUUUUUUUUUULU
+ 3→+β∑βθ(⇒+L+∑β∑βθβ⇒γ→+⇔→+β√,→√(⇒+δβ+,⇒→+∑β√γ→+′,β+δβ−(⇒β+⇔→+⇔ ⇒β→+δβ+→√√(⇒+δβ+⇔→+
YsFyidmodsíU
UU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUU6í7U(UHU2UgU4U+9í7U
UUUUUUUUUUUUUUUUUUUU=í27U
UUUUUUUUUUUUUUUUUU•UUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU8aUUU(U4U:UU⇒UgU4U=í27UU
UUUUUUUUUUUUUUUUUUUUUUUU-
1UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU 8aUUUgU4U:UU⇒U(U4U7íJU
U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU•U7íJU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
∞∞P=F∆+FU∞+=5{Τ+
U
Ejercicio #13UUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U5+
U "mUMysYtmU-U⇔U[2TU6TU56U]dUUumFmUuvYUsiUuyodvU3 2+++π+++βγβ∑(⇒→∑+β⇔+[β⇒γ[++
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU56U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU:U
SsUSac!mUMysYtmUYsFuscdvUSsiUuyodvU3
+UU52UUU*ó,íUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU2U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  25  -U
U
Solución.-

UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU>U3+UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUU→UUUUUU→U
U UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUδ +2U4UY2UIUY+U
UUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUU→U
UUUUUUUUUUY+UUUUUUUUUUUUUUUUUUUUUUUUUUUUδ +2UUUUUUUUUUUUUUUUUUUUUUUUUUUUUU-UUUUUUUUUUUY2U4U[? 2TU@2TUA2]U⇔U[5+TU52TU6]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUY +U4U[? +TU@+TUA+]U⇔U[:TU52TU2]U
UUUUUUUU>UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU> U
UUU>UUUUUUUUUUUUUUUUUUUUUU
→U 3 2U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUY
2U
U
U
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUU→UUUUUUUUU→U
δ+2U4U[5+TU:TU+]U0UUUFu +U4Uδ+2ULU-U
UU
UUUUUU→UUUUUUUUU5+UUUUU:UUUUU+U
⇒UUFu
+U4UUUU2UUUUU6UUUU56UUUU4U[56TU5+TU56]Uton_mU
UUUUUUUUUUUUUUUUUUUUUaUUUUUUrUUUUUU$U
→UUUUU
→UUUU→UUUUUUUUU→UUUUUU→UUUUUUUUU→UUUUUUUUU→U
F3U4UYULU-U⇔UYULU*- ?UHU-@UHU-A,U
→→→→
     →→→→     →→→→        →→→→     →→→→        →→→→     →→→→
MP = r * FX + r * FY + r * FZ      TEOREMA DE VARIGNON

U
Ejercicio #14
UUUUUUUUUUUUUUUUUUUUUUUUU:UUUUUUUUUUUUUUUUUUU56UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU →U
"vFUuyodvFU3+UUU2UU*ó,UgU32UUUU+UUU*ó,USsMaosoUyoUsrsUsoUsiUsFumcavTUyomUMysYtmU-UU
+UUUUUUUUUUUUUUUUUUUU:U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→U
soUN%íUumFmUuvYUsiUuyodvU3U*6TU2TU5;,íU8sUuaSsUSsdsYóaomYUsiUóvósodvUSsU-U
UUUUUUUUUUUUYsFuscdvUSsiUsrsíU
Solución.-

U UUUUUUUU:UUUUUUUUUUUUUUUUUUUUUU56UUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU6U
3
+UUUU2UUUUUUgUUUUUUU32UUUUU+UUU0UUUU-U⇔U[2TU6TU+]U$%UUU0U3UUUU2UU*ó,UUU
UUUUUUUU+UUUUUUUUUUUUUUUUUUUUUUU:UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU5;U
U
U
∞,[⇒β∑+,⇒→+[∑(β⇒γ→√(⇒<++

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  26  -U
U
U nU
U→
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
3+32U⇔U[*565:,TU*+52,TU*:5+,]U⇔U[56TU5+TU5+]UóUUUU UUU3 2UUUUUUUUUUUUUUUUUUUUUUUUUUUUUU-U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU:UUUUUUUUUUUUUUUUUUU
U→U
U3
+32UUUU⇔U√U6
2
UHU+
2
UHU+
2
U4U6í62UóUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUδUUUUUUUUUUUUvU3U
UU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU3 +vU
U&
nU4UUUUUU+V6í62UUUUUU4U[56TU5+TU+]U4U[5:í<:TU5:í6:TU5:í6:]U
→U
δ3+U3U⇔U[6TU:TU57]U
U
UUUUUUUU→UUUUUUUUUUUUUU→UUUUUUUUUUUUUU→UUUU UUUUUU6UUUUUU:UUUU57U
⇒UF
3+U⇔Uδ 3+U3UULU-U⇔UUUU2UUUUUU6UUUUUU+UUU⇔U[+7TU5+6TU<]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUUUUrUUUUUU$U
U
∴UF
nU⇔U[+7TU5+6TU<]ULU[5:í<:TU5:í6:TU5:í6:]U⇔U[+7L*5:í<:,UHU*5+6,L*5:í6:,UHU<L*5:í6:,]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU4U[5+6í7UHU6í<U52íJ]UU
UUU U U U U F
nUUU4U5+2í6Ukg_mU
U
EL TEOREMA DE VARIGNON DEMOSTRADO EN EL EJERCICIO A NTERIOR
ES TAMBIEN VALIDO EN EL CASO DE MOMENTO DE UNA FUER ZA RESPECTO DE
UN EJE (SE DEJA AL ESTUDIANTE SU DEMOSTRACION).
Ejercicio #15
U
→δ[+β⇔+θ(θγβ→+→+√[⇒θγ(γ,(δ[+[∑+δ[θ+−,β∑•→θ+∫+,oUumY0U8sUuaSsUSsdsYóaomYUyoUFaFdsómU
YsFyidmodsUcvoFdadyaSvUuvYUyomUFvimUMysYtmUsoUsiUvYa%soUgUyoUFvivUóvósodvíU
UU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  27  -U
U
Solución.-
U →U
UUUUUUUUUUU +U3sYdsoscsUmiU3imovU?@U
→UUUUUUUUU→UUUUUUUU→U
-1U4U-+UHU-2U⇔U[+TU;TU+]U
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
Y+U⇔U[56TU5+TU2]U0UY 2U⇔U[7TU6TU+]U
→UUUUUUUUUUUUU→UUUUUU→UUUUUUUU→UUUUUUU→UUUUUUU→UUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUU→U
F1U⇔UY+ULU-+UHUY2ULU-2UHU +UUUU0UUUUUU +UUU4UUUU-UUUULUSU4U2ULU6U4U9Uton_mU
UU U
UUUUUUUUUUUUUUUUUUUUUUUUUUAU
UUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUU-1UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUF1UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUU→UU UUUUUU56UUU5+UUUU52U
U UUUUU ⇒UUUUY
+ULU-+U4UUUU;UUUU2UUUUUU:UUU4U[;TU5=TU52]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUUUrUUUUUU$U
U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU@U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUUU→UUUUUUU→UUUUUUUUUUUUU 7UUUU6UUUUU+UUUU
U *e,UUUUUUUUUUUUUUUUUUUUUY
2ULU-2U4UUU56UUUU2UUUUU+UUUU4UU[+TU5=TU+<]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUUUrUUUUU$U
U
?UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU +U4U[:TU:TU59]U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUF
1U4U[7TU5+9TU++]Uton_mU
U
Ejercicio #16   U
→δ→θ+⇔→θ+φ+−,β∑•→θ+∫+⇔[θ++→∑βθ⋅+G→⇔⇔→∑+,⇒+θ(θγβmUYsFyidmodsíUUUUUUU U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  28  -U
U
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU56U
-
+U⇔U[2TU:TU;]Ud0UU3 +UUUU5+UUU*ó,U
UUUUUUUUUUUUUUUUUUUUUU52U
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU2U
-
2U⇔U[7TU5+TU+]Ud0U3 2UU5+UUU*ó,U
UUUUUUUUUUUUUUUUUUUUUUU2U
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU7U
-
6U⇔U[2TU;TU5+]Ud0U3 6UUU6UUU*ó,U
UUUUUUUUUUUUUUUUUUUUUUU+U
Solucion.-
→U
+U3sYdsoscsUmiUuimovU@AU
→U
2U3sYdsoscsUmiUuimovU?@U
→UUUUUUUUUUU→UUUUUUUU→UUUUUUUU→U
-1U⇔U-+UHU-2UHU-6U⇔U[<TU6TU;]Ud UU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
Y+U⇔U[56TU5+TU52]0UY 2U⇔U[2TU5+TU2]0UY 6U⇔U[7TU6TU+]UU
U
→UUUUUU→U UUUUUUU56UUU5+UUU52U
Y
+ULU-+U4UUUUU2UUUU:UUUUU;UUUU4U[5;TU5=TU2]U
UUUUUUUUUUUUUUUUUUUaUUUUUrUUUUU$U
U
→UUUUUU→U UUUUUUUU2UUU5+UUUU2U
Y
2ULU-2U4UUUUU7UUU5+UUUU+UUUUU4U[+TU=TU6]U
UUUUUUUUUUUUUUUUUUUaUUUUUrUUUUU$U
→UUUUUU→U UUUUUUUU7UUUU6UUUU+U
Y
6ULU-6U4UUUUU2UUUU;UUU5+UUUUU4UU[5JTUJTU+;]U
UUUUUUUUUUUUUUUUUUUaUUUUUrUUUUU$U
U
→UUUUUUUUUU→UUUUUU→UUUUUUUU→UUUUUU→UUUUUUUU→UUUUUU→UUUUUUUU→UUUUUUUU→U
F1U4UY+ULU-+UHUY2ULU-2UHUY6ULU-6UHU +UHU 2U
UUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
U⇒UU +UUU4U7ULU;U4U2:UUU0UUUUU +U4U[2:TU:TU:]UU
U
UUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
U⇒UU 2UUU4U;ULU2U4U=UUUU0UUUUU 2U4U[:TU:TU5=]UU
UUUUUUUUUUUUUUU→U
UUU∴UUUF1U4U[+:TUJTU++]Uton_mU
U
Ejercicio #17
→δ→+⇔→+−,β∑•→+
F
r
U4U[6TU52TU;]+β⇒+γ[⇒β⇔→δ→θω+θβ+(δβ+δβγβ∑(⇒→∑<+
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  29  -U
U
Solución.-
U m,U vóuvosodsFUsFcmimYsFU⇔U6UdTU52UdTU;UdU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UUUUUUUUUU→UUUUUUUU→U
U e,U vóuvosodsFU scdvYamisFU⇔U6UaUTU5U2UrUTU;U$U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→U
U c,UFm%oadySUSsU-U4U ( )
222
;26 +−+ U4U7í6=UdU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUU→UU→U
+ δ+β√γ[∑+,⇒(γ→∑([+β⇒+⇔→+δ(∑β√√(⇒+δβ++⇔U sU-U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→UU→UUUUUUUUUUUUUUUU
U U U
sU-U4UUUUUUUU
+
UUUUUUULU[6TU52TU;]U4U[:í77TU5:í6JTU:íJ;]UUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU7í6=U
U
+ β+{θβ⇒[θ+δ(∑β√γ[∑βθ<+
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU6UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU52UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU;U
U UUUUUUUUUUUcvFUαU4UUUUUUUUUUUU4U:í77UUUUcvFUβU4UUUUUUUUUUUU4U5U:í6JUUUcvFUγU4UUUUUUUUUUUUU4U:íJ;UUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU7í6=UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU7í6=UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU7í6=UUU
U
U
Ejercicio #18
→δ[θ+⇔[θ+,⇒γ[θ++U*2TU56TU5;,UgU32+⌠ω+ϖω+πω+β⇒+βγ∑[θω+θβ+(δβ<+
Solución.-
U
+ →+β√γ[∑βθ+δβ+[θ(√(⇒+δβ+
+UgU32UU
U U U U
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
U U U U Y +U4U[2TU56TU5;]Uó0UY 2U4U[:TU7TU5+]UUUUUUU
U
U e,U/scdvYUSsUSsFuimtmóasodvUsodYsU3
+UgU32U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUU→UUUUUU→U
U U U δU4UY 2UIUY+U4U[:UIU2í7UIU*56,TU5+UIU*5U;,]U4U[52TU=TU6]UóU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUU→U
+ √+(θγ→⇒√(→+β⇒γ∑β++UgU32U⇔UUUUUδU
U
UUUUUU→U
U U UUUUUUUUUUUUUUδUUU4U ( )
222
6=2 ++− U4U=íJJUóU
U
Ejercicio #19U
→δ[+β⇔+,⇒γ[+ω+φω+πϖ+ω+∫+⇔→+−,β∑•→+
+U4U[:TU5;TU6]UdTUlysUumFmUuvYUsiUuyodvU3 2U*9TU5;TU2,UóTU
θβ+(δβ<+
Solución.-

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  30  -U
U
U
U m,UFvósodvUSsU-
+UYsFuscdvUSsiUvYa%soU>U*:TU:TU:,UU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUU
→U
U U U >3 2U4U[9TU5;TU2]U⇔UYU⇔+)β√γ[∑+δβ+[θ(√(⇒+δβ+ 2U
UUUUUUUUUUUUUUUUUUUUUUUU→UUUUU
→UUU→UUUUUUUU9UUU5U;UUUU2U
U U F
>U4UYULU-+U4UUUU:UUU5U;UUUU6UU4U[5U;TU5U+=TU5U2;]Uton_mU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUUUrUUUUU$U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
U U Fm%oadySUSsUF
>U⇔UUUF>UUUUU4U6:í29Uton_mU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U→U
U e,UFvósodvUSsU- +UYsFuscdvUSsU3+U
UUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUU→U
U U 3 +32U4U[;TU5JTUJ]U⇔UδU⇔U scdvYUSsUSsFuimtmóasodvUSsU3 +UmU32U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU;UUUU5UJUUUUJU
U U F
3+U4UδULU-+U4UUU:UUUU5U;UUUU6UUUU4U[JTU5U+2TU5U+9]Uton_mU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUUUUrUUUUU$UUUUU
[⇒δβ<+++++++++++++→UUUUUUUU
→U
U U F ?U4UJUaU
U U →UUUUUUUUUU
→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
U U F
@U4U+2UrUUUUUUUUUUUUU vóuvosodsFU scdvYamisFUSsUFu+U
UUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUU
→U
U U F AU4U5U+9U$U
∆δβ∴θ<+F→2⇒(γ,δ+δβ+F
+U4UUUFu+UUUU4U2+í+<Uton_mU U
U
Ejercicio #20     U
++++++→δ[θ+⇔[θ+,⇒γ[θ+
+U*2TU:TU6,UóUgU32U*5;TU7TU:,UóUlysUSsMaosoUyoUsrsUgUimUMysYtmU
F
r
4U[7TU9TU52]UdU
lysUumFmUuvYU3
6+⌠ω+πω++ω+θβ+(δβ<+
U
Solución.-
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
→U
U m,UFvósodvUSsU-UYsFuscdvUSsiUsrsU⇔UF nU
U U UUUUUUUUUUUUU→UUUUUU
→U
U U U 3 +36U4UδU4U[52TU52TU52]UóUUUUUUUUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUU
→U
U U U 3 236U4Uδ+U4U[;TU5JTU+]+++++++++β⇒γ[⇒√βθ<+++
UUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUU
→UUU→UUUUU5U2UUUU5U2UUUUU5U2U
U U Fu
+U4UδULU-U4UUUUU7UUUUUUU9UUUUU52UUU4U[+9TU5U+;TU5U2]Uton_mU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUUUUrUUUUUUUU$UU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  31  -U
U
U
U UUUUUUUUUUUU→UUUUUUU
→UUU→UUUUUUUU;UUUU5UJUUUUUU+U
U U Fu
2U4UδULU-U4UUUUU7UUUUUU9UUUUU52UUUU4U[=TU+6TU7<]Uton_mU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUaUUUUUUUrUUUUUUU$UU
UUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
∆δβ∴θ<++32U4U[59TU7TU56]U⇒UU3 +32UUU4U=í6JUóTUuvYUdmodv0UsUnU4U[5:íJ2TU:í9TU5:í69]UU
β⇒γ[⇒√βθ<
UUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUU→ U
U U F
nU4UFu+ULUsnUUUUUUUvUeasoUUUUUFnU4UFu2ULUsnU
U
=θ+δβ√(∑<+
F
nU4U[+9TU5+;TU52]ULU[5:íJ2TU:í9TU5:í69]U4U5U+<í2Uton_mU
vUeasoUU
F
nU4U[=TU+6TU7<]ULU[5:íJ2TU:í9TU:í69]U4U5U+<í2Uton_mUU UUUU
U
Ejercicio #21
{[⇒+⇔[θ+δ→γ[θ+δβ⇔+→⇒γβ∑([∑+βαβ⇔[+θβ+(δβ<+
Solución.-
U m,UFvósodvUSsU-UYsFuscdvUmUdYsFUsrsFUvYdv%vomisFUcvocyYYsodsFUsoU3 +UgUmUivFUsrsFUcmYdsFamovFU
⇔UF
nWn8UUUUUUUUUUUUUUUUUUUUUU→UFu +U4U[+9TU5U+;TU52]U
U U U F
nWnU?U4U[+9TU:TU:]ULU[aTU:TU:]U4U+9UUton_mU
U U U F
nWnU@U4U[:TU5+;TU:]ULU[:TUrTU:]U4U5+;UUton_mU
U U U F
nWnUAU4U[:TU:TU52]ULU[:TU:TU$]U4U5U2UUton_mU
U 8sUovdmUmSsóCF0UlysUsiUóvósodvUSsUyomUMysYtmUYsFuscdvUSsUyoUuyodvTUsFUa%ymiUmUimUFyómUSsU
ivFUóvósodvFUSsUSac!mUMysYtmTUYsFuscdvUSsUdYsFUsrsF+[∑γ[2[⇒→⇔βθ+′,β+→θ→⇒+[∑+β⇔+,⇒γ[+βθ+δβ√(∑<+
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUU→UUUUUUUUUU→UUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U U U F
3+U4UFnWnU?UHUFnWnU@UHUFnWnUAU4U[+9TU5U+;TU5U2]UUUton_mU
U
U
U
1.12   PROBLEMAS PROPUESTOS.
U
Problema #1
U →UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
-U4U[6TU;TU52]+γ+∫++ω+⌠ω+πφ++,⇒+,⇒γ[+δβ+⇔→+⇔ ⇒β→+δβ+→√√(⇒+SsU-TUmSsóCFUyoUuyodvUmYeadYmYavU
3
++ϖω+πφω+π++∞β+(δβ<+
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
m,UFvósodvUSsU-UYsFuscdvUSsiUvYa%soíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  32  -U
U
UUUUUUUUUUUUUUUUUUUUUUUUUU→U
e,UFvósodvUSsU-UYsFuscdvUSsU3+U
Respuesta<+→+F
>U4U[+2TU5+TU+9]Uton_mTUe,UFu +U4U2UaUIU=UrUIU+6U$Uton_mU
U
Problema #2U
U
"vFUSsiUsrsYcacavU+í+UgUsiUuyodvUmYeadYmYavU3
2+⌠ω+ω+π++∞β+(δβ<+
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
m,UFvósodvUSsU-UYsFuscdvUSsiUsrsU3+U32íU
U
Respuesta:Um,UF nU4U5U92+Uton_mU

Problema #3

→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
-U4U[56TU7TU:]+γ+∫++⌠ω+ω+++,⇒+,⇒γ[+δβ+⇔→+⇔ ⇒β→+δβ+→√√(⇒+δsU-TUmSsóCFUyoUuyodvUmYeadYmYavU
3
++ω+φω+π++∞β+(δβ<+
U
m,
FvósodvUSsU-UYsFuscdvUmUdYsFUsrsFUvYdv%vomisFUcvocyYYsodsFUsoU3 +UgUumYmisivFUUmUivFUsrsFU
cmYdsFamovFíU
U
Respuesta<
a)UF(U4U527Uton_m0UF gU4U5+7Uton_m0UF tU4U5+6Uton_mU

Problema #4
UUU
β[θγ∑→∑+β⇔+γβ[∑β→+δβ+→∑(2⇒[⇒ω+→∑→+β⇔+√→θ[+δβ⇔+óvósodvUSsUyomUMysYtmUYsFuscdvUSsUyoU
uyodvíU
U
Problema #5
→∑→+⇔→+θ(2,(β⇒γβ+−(2,∑→+θβ+(δβ+δβγβ∑(⇒→∑<+
m,U"vFUóvósodvFUSsUivFUumYsFU
+TU 2UgU 6íU
e,U"mUFyómUSsUivFUóvósodvFUSsUivFUumYsFU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  33  -U
U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→UUUUUUUUUU→UUUUUUUUUU→U
Respuesta<+→+F +U4U9Ur0UF2U4U5Jí7Ua0UF6U4U;U$UsoUton_mTU b)UF+UHUF2UHUF6U4U[5Jí7TU9TU;]Uton_mU
U
Problema #6
{[⇒+⇔[θ+δ→γ[θ+δβ⇔+∑[]⇔β→+φ+θβ+(δβ<+
UUUUUUUUUUUUUUUUUUUU
→U
→+P∑→θ⇔→δ→∑++→+,⇒→+[θ(√(⇒+→∑→⇔β⇔→+′,β+→θβ+[∑U3 +íU
]+⇒δ(√→∑+⇔→+√[[θ(√(⇒+δβ⇔+⇒,β)[+θ(θγβ→⋅+
UUUUUUUUUUUUUUUUUU→UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
Respuesta<+→ )U-UdYmFimSmUmU3+UsoUMvYómUumYmisimTUb)U-UsoU3+UómFUFu+4U[527TU5+7TU5+6]Uton_mU

Problema #7

++++++++++++=⇔+θ(θγβ→+δβ+⇔→+−(2,∑→<+
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  34  -U
U
∞β+(δβ<+→+βθ,⇔γ→⇒γβ+δβ+−,β∑•→θ+β⇒+β⇔+[∑(2β⇒⋅+
UUUUUUUUUUUUUUUe,U1sFyidmodsUSsUóvósodvFíUU
Respuesta:Um,U-
1U4U[;TUJTU<]UdTU b)UF1U4U[59TU9TU+;]Uton_mU
Problema #8U
noUsiUFaFdsómUcvuimomYUSsUimUMa%yYmUU
U
∞β+(δβ<++
UUUUUUUUUUUUUUm,U1sFyidmodsUSsiUFaFdsómíU
+ ++]+3 ⇒β→+δβ+→√√(⇒+δβ+⇔→+∑βθ,⇔γ→⇒γβ⋅+
UUUUUUUUUUUUUUUUUUUUUUUU
→U
Respuesta: a)U-1U4U[6TUJTU:]UdTU b)UFs%#oUimUYscdmUJ(UIU6gU4U=U
U
U
Problema #9
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUniUFaFdsómUcvuimomYUSsUimUMa%yYmU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  35  -U
U
∞β+(δβ<++
m,
1sFyidmodsUSsiUFaFdsómíU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU→U
Respuesta< a)UsiUumYUF>U4U[:TU:TU+]Uton_mU
U
U
1.13  EVALUACION DIAGNOSTICO DEL CAPITULO.

noUsFdsUmumYdmSvUFsUuYvuvosUlysUsiUsFdySamodsUYsmia√β+,⇒→+β)→⇔,→√(⇒+δβ+γ[δ[+⇔[+→∑β⇒δ(δ[+β⇒+
sFdsUcmuadyivTUumYmUivUcymiUFsUoscsFadmUlysUYsFuvoSmUimFUFa%yasodsFUuYs%yodmFUFaoUYs aFmYUsiUcvodsoaSvU
SsiUuYsFsodsUcmuadyivíU
U  vóvUFsUsocysodYmUsiUSsFuimtmóasodvUSsUyoUuyodvUYsFuscdvUSsiUvYa%soXU
U QysUsFUimUóm%oadySUSsUyoU scdvYXU
U β−(⇒(β⇒δ[+⇔→+→2⇒(γ,δω+√[[+θβ+β⇒√,β⇒γ∑→+⇔→+δ(θγ→⇒camUsodYsUsiUvYa%soUgUyoUuyodvXU
U  vóvUFsUSsMaosUsiUóvósodvUSsUyomUMysYtmUcvoUYsFuscdvUmUyoUuyodvXU
U β[θγ∑→∑+β⇔+γβ[∑β→+δβ+VARIGNON YsFuscdvUSsUyoUsrsíU
U β−(⇒(∑+′,β+βθ+,⇒+→∑+δβ+−,β∑•→θ⋅+
3mYmUcvociyaYUsFdsUSam%ovFdacvUFsUuaSsU!miimYUimUYsFyidmodsUSsiUFa%yasodsUFaFdsómUSsUMysYtmFíU
U
U
U
íU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  36  -U
U

CAPÍTULO 2

INTRODUCCION AL ANÁLISIS DE LAS
ESTRUCTURAS
U

2.1   OBJETIVO GENERAL.
niUversda vUMyoSmósodmiUSsUsFdsUcmuadyivUsFU!mcsYUlysUsiUsFdySamodsUcvovtcmUsiUcvocsudvUSsU
642A(m2(A+ m3S3 2+S7oI- .34 )o16A6-264 2oP34 )6 m+A%mFU gU muvgvFU lysU mcd#moU soU yoU FaFdsómU
sFdYycdyYmiíU
2.2   OBJETIVOS ESPECIFICOS.
e. 26ASo-+A 6. m+Po2(.3 6. 642()o+-26 P3)ABa
U l.+4o1om+AB .+4 642A(m2(A+4 P3A 4( 13AS+ v63SI2Aom+TUuvYUFyUFaFdsómUSsUcmY%mFTUuvYUFyUFaFdsómUSsU
muvgvFUgUuvYUFyFUcvoSacavosFUSsUaFvFdmdacaSmSíUU
U d626ASo-+A+ .+4 A64(.2+-264 )6 m+Av+4 )o42Ao7(o)+4 i .+ P34omof- 6- .+ =(6 642+ +m2M+n
U d626ASo-+A+ .+4 A6+mmo3-64 )6 +P3i3 )6 )o42o-234 2oP34 )6 642A(m2(A+4 *yov+4L PfA2om34L +Am34L
sFdYycdyYmFUóa(dmF,íU
2.3UUINTRODUCCION.U
l3- (- A+t3-+So6-23 .fvom3 i 3A)6-+)3 46 .6 PA3P3AmavomUmiUiscdvYUivFUsisósodvFUSsUmoCiaFaFU
P+A+ =(6L + P+A2oA )6 .+ )61o-omof- )6. m3-m6P23 )6UsFdYycdyYm0UimFUcimFaMalys0USsUimUóaFómUMvYómUuvYU
4(4 m3-)omo3-64 )6 o4342+2omo)+)L o)6-2o1om+-)3 +a imFUsFdYycdyYmFU!auvFdCdacmFU*aosFdmeisF,TUcymoSvUsiU
-MS6A3 )6 .+4 o-mfv-o2+4 )6. 4o426S+ 64 S6-3A + .+4UcvoSacavosFUSsiUslyaiaeYavUsFdCdacv0UsFdYycdyYmFU
o4342B2om+4 *642+7.64C m(+-)3 6. -MS6A3 )6 .+4 o-mf%oadmFU SsiU FaFdsómU sFU a%ymiU miU o#ósYvU SsU imFU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  37  -U
U
scymcavosFUSsUslyaiaeYavUsFdCdacv0UsFdYycdyYmFU!ausA642B2om+4L m(+-)3 6. -MS6A3 )6 .+4 o-mfv-o2+4 64
ómgvYUmiUo#ósYvUSsUimFUscymcavosFUSsiUslyaiaeYavUsFdCdacvíUU
U
2.4   CONCEPTOS BÁSICOS Y DEFINICIONES.
d(A+-26 6. PA3m643 )6 13AS+mof- )6. o-v6-o6A3 moyo.TUsiUcvocsudvUSsUsFdYycdyYmUFsUsocysodYmU
YsMsYaSvUmUyoUcvoryodvUSsUsisósodvFUlysUimUcvoMvYóm-0 6SP6A3L I42+ +46y6A+mof- -3 )6S(642A+ (-+
)61o-omof- P+A+ 6-26-)6A.3 6- 23)+ .+ 6826-4of- )6 FyUFa%oaMacmSvíU3vYUsoSsTUmiUuimodsmYFsUimUuYs%yodmU
SsUlysUsFUyomUsFdYycdyYmTUimUYsFuysFdmUuysSsUSmYFsUyoUdmodvUFyersda míU3vYUivUlysTUumYmUsodsoSsYivUgU
mFaóaimYivUsFUoscsFmYavUcvoFaSsYmYUcvóvUuYsóaFmUsiUsodvYovUlysUimUYvSsmíUU
8o 2(yoIA+S34 m(Ao34o)+) )6 46A 3746Ay+)3A64 26-o6-SvU#oacmósodsUoysFdYvUósSavUmóeasodsU
cvóvUi'óadsTUvUsoUvdYmFUumimeYmFTUsiUi'óadsUSsUoysFdYmU aFdmUgUYmtvomóasodv0UovFUSmY'móvFUcysodmUlysU
uvSsóvFU aSsodaMacmYU SaMsYsodsFU sFdYycdyYmFTU SaMsYsomoB-)3.+4 P3A 4( (2o.ot+mof- P+A+ .3 =(6 1(6A3-
SaFsCmSmF0U mF'TU soU siU CóeadvU casod'MacvU SsU imU ao%soo6Ab+ moyo.L 6- 13AS+ -+2(A+. )o4mAoSo-+Ab+S34a
mFmFU !meadmcavosFTU sSaMacavFU SsU vMacaomFTU sFcysimFTU uysodsFTU dmolysFTU sSaMacavFU SsumYdmósodmisFTU
moyocavFUsFuscdmcyimYsFTUmsYvuysYdvFTUuYsFmFTUSaFdYadvUSsUYas%vTUcmYYsdsYmFTUmicmodmYaiimSvTUi'osmFUSsU
m3-)(mmof- i )o42Ao7(mof- )6 +v(+ P32+7.6L 6-2A6 32A+4L 23)+4 6..+4 m(SP.o6-)3 (- PA3Pf4o23
umYdacyimYUumYmUivUlysUMysYvoUSaFsCmSmFíU U
33A .3 2+-23a
U
U ESTRUCTURA, ES UN CONJUNTO DE ELEMENTOS CON FORMA G EOMÉTRICA
QUE, UNIDOS ENTRE SÍ POR MEDIO DE NODOS, SOPORTAN C ARGAS QUE SON
TRANSMITIDAS A SUS APOYOS A TRAVÉS DE LOS ELEMENTOS
ESTRUCTURALES QUE LA INTEGRAN.
U APOYO, ES UN DISPOSITIVO CONSTRUCTIVO QUE PERMITE E NLAZAR ENTRE
SI LOS ELEMENTOS ESTRUCTURALES Y/O SUJECIÓN ENTRE L A ESTRUCTURA
Y EL SISTEMA TIERRA, CON EL PROPÓSITO DE EVITAR DES PLAZAMIENTOS.
d6 46A +m6P2+)+ .+ )61o-omof-L 6. m3-5(-23 )6 6.6S6odvFUlysUaods%YmoUyomUsFdYycdyYmUSsesUSsU
m3-26-6A 4oS(.2B-6+S6-26a 13AS+ v63SI2Aom+L 4o426S+U SsU cmY%mFU gU FaFdsómU SsU muvgvFTU YsMsYsocamFU
lysUMmcaiadmoUdsosYUsisósodvFUSsUmoCiaFaFUumYmUuimo26+A (-+ m.+4o1om+mof- 6- 1(-mof- )6 4( v63S62Ab+L
SsUFyFUmuvgvFUgUSsUFyFUcmY%mFíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  38  -U
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2.5  TIPOS DE ESTRUCTURAS.
c+  m.+4o1om+mof- 6- 6. 6-23A-3 )6 .+4 642A(m2(A+4L 46 )+ 6- 1(-mof- )6 .+ (7om+mof- )6 4(4
sisósodvFTUlysUuysSsoUsFdmYUivcmiatmSvFUsoUsiUuimovUvUsoUsiUsFumcav0UsoUcvoFscysocamTUimFUsFdYycdyYmFU
46 )oyo)6- 6-a
2.5.1 Estructuras planas.
8voUmlysiimFUcymoSvUFyFUsisósodvFUFsUsocysodYmoUsoU(- P.+-30 (-+ A6PA646-2+mof- )6 I42+4 46
srsóuiaMacmUaSsmiatmoSvUyomU a%mUFaóuisósodsUmuvgmSmUlysUFsUsocysodYmUivcmiatmSmUFveYsUsiUuimovU
?@TUcvóvUFsUóysFdYmUsoUimUMa%yYmU2í+UcvoUimUemYYmUIEíU
U
Figura 2.1
2.5.2 Estructuras en el espacio.
8sUaSsodaMacmoUcvóvUdmisFUcymoSvUmiUósovFTUyovUSsUFyFUsisósodvFUFsUsocysodYmUsoUsiUsFumcavíU
8aUFsUveFsY mUimUMa%yYmU2í2TUsiUómYcvUIE5>nUFsUsocy6-2A+ 6- 6. P.+-3 ?@0 i P6AP6-)om(.+A + I426L
FveYsUsiUsrsUAUFsUyeacmUsiUómYcvUYG5N"UYsFyidmoSvUsoUcvoFscysocamUyomUsFdYycdyYmUsoUsiUsFumcavíU
PvómoSvUSsUYsMsYsocamUsiUmicmocsUlysUFsUuYsdsoSsUsoUsiUmoCiaFaFUSsUimFUsFdYycdyYmFUaFvFdCdacmFTU
FsUmevYSmoU#oacmósodsUimFUsFdYycdyYmFUuimomF0UaSsodo1om+-)3 6-2A6 6..+4 .+ 4ov(o6-26 m.+4o1om+mof-a

U
Figura 2.2

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  39  -U
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2.6 CLASIFICACION DE LAS ESTRUCTURAS

"mF cFóvEpEcódYnjv tjd vr pjdnó 5Yjn2XdEcóf tjd vr vEvXsómUSsU muvgvFTU uvYUFyUFaFdsómUSsU
cmY%mFíU
U
2.7 POR SU FORMA GEOMÉTRICA. U
)óv YvXdrcXrdóv tjd vr pjdnó 5Yjn2XdEcó Ye YF cjeXY(dvUSsUFyFUsisósodvFUlysUimFUaods%YmoUFsU
cimFaMacmoUsoUYscdmFTUcyY mFUgUSsUcvodmcdv0UumYmUFy EsYeXEpEcócE3e vY EesEcódle Ye óF5reóv Fóv cód5óv gU
ótjujv beEcónYeXY tódó vr EsYeXEpEcócE3ef uó orY tódó vr sYpEeEcE3e vY Y7tjeYe Ye Fóv vE5rEYeXYv
vYccEjeYv sY 2vXY cótyXrFjz
2.7.1 Viga Horizontal. "mU Ma%yYmU YsuYsFsodmU yomU a%mU FaóuisósodsU muvgmSmU cvoU cmY%mU
cvocsodYmSmUZ3ZUgUyomUcmY%mUyoaMvYósósodsUSaFdYaeyaSmUZ[Zí* sYUMa%í2í6,íU
U
Figura 2.3

2.7.2 Viga Inclinada. "mUMa%yYmU2í9UYsuYsFsodmUmUyomU a%mUFaóuisósodsUmuvgmSmUcvoUcmY%mU
cvocsodYmSmTUveFsY moSvUlysUimUemYYmUIEUdasosUyomUSEdYccE3e dYvtYcXj óF YZY (jdEajeXóF Ye
precE3e sYF le5rFj \
U
Figura 2.4

2.7.3 Columnas. nFUyoUsisósodvUsFdYycdyYmiU sYdacmiUvUaociaomSvíU voUimUMa%yYmU2í7UlysUFsU
óysFdYmUFsUsFlysómdatmUyomUcviyóomU sYdacmiUIEUcvoUcmY%mUcvocsodYmSmUm(amiíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  40  -U
U
U
Figura 2.5

2.7.4 Marcos o Pórticos. nFdCoU cvoFdadyaSvFU uvYU yoU cvoryodvU SsU sisósodvFU Yscdai'osvFTU
!vYatvodmisFTU sYdacmisFU gVvU aociaomSvFíU nisósodvFUlysU FvuvYdmoU cmY%mFU cvocsodYmSmFU vU
SaFdYaeyaSmFUcvóvUFsUóysFdYmUsoUimFUFa%yasodsFUMa%yYmFíU
U
Fig 2.6 (a)                       Fig 2.6 (b)
2.7.5 Armaduras. "mFU aods%YmoU yoU cvoryodvU SsU sisósodvFU Yscdai'osvFU Ysimda mósodsU
sFesidvFU lysU sFdCU yoaSvFU Y'%aSmósodsU soU FyFU s(dYsóvFU lysU FvuvYdmoU cmY%mFU cvocsodYmSmFU
SaYscdmósodsUFveYsUimFUyoavosFUvUovSvFUcvóvUFsUóysFdYmUsoUimUMa%yYm
.
U
Figura 2.7

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  41  -U
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2.7.6 Arcos. nFdCoU aods%YmSmFU uvYU yoU cvoryodvU SsU sisósodvFU cyY vFU SsU MvYómU caYcyimYTU
YFytXEcó j tódói3FEcóz gF dYvtYcXj vY nrYvXdóe Fóv vE5rEYeXYv pE5rdóv1

Fig 2.8 (a)                   Fig 2.8 (b)
2.7.7 Cables. :YcXEFyeYjv Ye crujv treXjv sY EepFY7E3e vjtjdXóe cmY%mFU cvocsodYmSmFíU
Sódói3FEcjv orY vjtjdXóe cód5óv dYtódXEsóvf 4óiFYv lysUFvuvYdmoUFyUusFvUuYvuavíUUU

Fig 2.9 (a)                   Fig 2.9 (b)
U
Fig 2.9 (c)
2.8 POR SU SISTEMA DE APOYOS
¿QUE ES UN APOYO?
vóvUgmU!me'móvFUósocavomSvUyoUmuvgvUsFUyoUSaFuvFada vUcvoFdYycda vUlysUusYóadsUsoimtmYU
YeXdY vE Fjv YFYnYeXjv YvXdrcXrdóFYv uVj vrZYcE3e YodYsU imU sFdYycdyYmU gU siU FaFdsómU dasYYmTU cvoU siU
tdjt3vEXj sY YxEXód sYvtFóaónEYeXjvz

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  42  -U
U
nFdvFUmuvgvFUdasosoUuvYUversdvUYsFdYao%aYUivFUSsFuimtmóasodvFUsoUmi%yovUSsUivFUsrsFUsoUivFUlysU
sFdsUyeacmSvTUgUFs%#oUmUivFUdauvFUSsUmuvgvUcvoUimFUYsFdYaccavosFUYsFuscda mFUimFUcimFaMacmYsóvFUsoUsiU
tFóej sY Fó vE5rEYeXY nóeYdó1
Tabla 2.1 Sistemas de apoyosU
Nombre del apoyo Vínculos Grados de libertad diagrama
,3xEF 6ej .jv
U
_EZj .jv 6ej
U
nóuvdYmóasodvU PYsFU sYvU
U
Yryó .jv 6ej
U
U U
Sódó Fó nYZjd cjecYtcE3e sYF vYeXEsj sY Fóv dYóccEjosFUSsUmuvgvUsoUsiUuimovUuYvuvosóvFUimU
vE5rEYeXY XóiFó1
Tabla 2.2   Grados de libertad en apoyosU
Esquema Nombre Grados de libertad Diagrama
U
,3xEF .jv
RY
U
U
-arvU &ovU
R
Y
U
U
nóuvdYmóasodvU sYvU
M
R
Y
Z
U
U
Yy'mU &ovU
M
R
X
Z
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  43  -U
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2.9 POR SU SISTEMA DE CARGAS

r-+ PAoS6A+ m.+4o1om+mof- )6 .+4 642A(m2(A+4 P3A 4( 4o426S+ )6 m+Av+4 46 o)6-2o1om+-a
U c+4 m+Av+4 6826A-+4a +m2M+- 437A6 .+ 4(P6A1omo6 )6.UsisósodvUsFdYycdyYmiíU
U c+4 m+Av+4 o-26A-+4a +m2M+- )6-2A3 )6. 6.6S6-23 642YycdyYmiU*cvóvUsFUsiUcmFvUSsUivFU
sisósodvFUóscCoacvFUlysUFsUmomiatmoUsoUsiUFa%yasodsUcmuadyiv,U
voU emFsU soU ivU modsYavYTU cymoSvU imU sFdYycdyYmU FvuvYdmU yoU FaFdsómU SsU cmY%mFU s(dsYovU FsU
o)6-2o1om+-a 33A 6. 2oP3 )6 m+Av+L P3A 4( P6AS+-6-mamUgUuvYUimUMvYómUsoUlysUmcd#míU
U
2.9.1 Por el tipo de carga.
2.9.1.1   Cargas concentradas.
83- +=(6..+4 =(6 2o6-6- (- 43.3 P(-23 )6 +P.om+mof-U
2.9.1.2   Cargas Axiales.  83- +=(6..+4 =(6 642B- +m2(+-)3 6- 6. m6-2A3 v63SI2YacvUSsU
(-+ 46mmof- 2A+-4y6A4+.n
U
Figura 2.10
2.9.1.3 UUCargas no axiales. 83- +=(6..+4 =(6 +m2M+- 1(6A+ )6. m6-2A3 v63SI2AomvUSsU
(-+ 46mmof- 2A+-4y6A4+.L v6-6A+-)3 (-+ 68m6-2Aomo)+SU*s,íU sYUMa%U2í++U
U
U
Figura 2.11

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  44  -U
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2.9.1.4   Cargas distribuidas. )ó xódEócE3e sY Fó cód5ó Yv cjevXóeXY vjidY YF cFóYvU
"L" sjesY YvXó ócXróesj cjnj vY jivYdxó Ye Fó pE5rdó1U
U
Figura 2.12

2.9.1.5   Carga no uniforme. )ó xódEócE3e sY Fó cód5ó ej Yv cjevXóeXY vjidY YF cimYvU
"L"USvoSsUsFdmUmcdymoSvíU3vYUsrsóuivTUsoUimUMa%yYmUlysUFsUóysFdYmUFsUSaFdao%ysUyomUcmY%mUSsU
xódEócE3e FEeYóF vjidY YF Xdónj
"a" u cje reó xódEócE3e crósdlXEcó vjidY YF Xdónj "b"íU
U
Figura 2.13
2.9.2 Por su permanencia. 8sU cimFaMacmoU soU cmY%mFU a mFTU cmY%mFU óysYdmFU gU cmY%mFU
mccaSsodmisFíU
2.9.2.1 Cargas vivas. Sódó reó vEntFY EsYeXEpEcócE3e u re XóeXj cjFjorEóiTUFsUSsMaosoU
cjnj óorYFFóv orY vY nrYxYeR tjd YZYntFj Ye re vóF3oU SsU cimFsFU imU cmY%mU a mU sFdmU
YsuYsFsodmSmU uvYU siU usFvU SsU ivFU miyóovFU lysU FsU s(uYsFmoU soU yoaSmSsFU SsU ivo%adySU miU
cymSYmSvíU
2.9.2.2 Cargas muertas. U nFdCoU YsuYsFsodmSmFU uvYU siU usFvU uYvuavU SsiU sisósodvU
sFdYycdyYmiU
2.9.2.3 Cargas accidentales.U 8voU mlysiimFU lysU sFdCoU YsuYsFsodmSmFU uvYU imU MysYtmU
F'FóacmTUuvYUimUMysYtmUSsiU asodvTUuvYUsiUusFvUSsUimUoas sTUsdcíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  45  -U
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2.9.3 Por la forma en que actúan.U
8voUimFUcmY%mFUmcda mFTUimFUcmY%mFUYsmcda mFUgUimFUcmY%mFUaodsYomFí* sYUMa%yYmU2í+;,

U
Figura 2.14
2.9.3.1 Cargas activas. nFdCoUYsuYsFsodmSmFUuvYUsiUFaFdsómUSsUcmY%mFUs(dsYovU*gU4Uó(0U
gU4Uó(],UlysUsFdCoUmcdymoSvUFveYsUsiUsisósodvUsFdYycdyYmií

2.9.3.2 Cargas reactivas.
nFdCoUYsuYsFsodmSmFUuvYUimFUcvóuvosodsFUSsUcmSmUyovUSsU
ivFU muvgvFU lysU FvuvYdmoU miU sisósodvU sFdYycdyYmiU *1I+ E+ (δβ⇒γ(−(√∴⇒δ[θβ+ γ→](⇒+ √[[+
'ocyivFUvUYsmccavosFíU
2.9.3.3 Cargas internas.U8voUimFUlysUmcd#moUSsodYvUSsiUsisósodvUsFdYycdyYmiTUóaFómFU
′,β+θβ+[[⇒β⇒+→+⇔→+→√√(⇒+δβ+⇔→θ+√→∑2→θ+βγβ∑⇒→θ⋅+β⇒γ∑[+δβ+θγ→+√⇔→θ(−(√→√(⇒+θβ+δ(θγ(⇒2,β⇒<+
"mUMysYtmUovYómiTUsFUyomUMysYtmUaodsYomUlysUsFdmUmcγ,→⇒δ[+β∑β⇒δ(√,⇔→∑β⇒γβ+→+⇔→+θβ√√(⇒+
dYmoF sYFmiU SsiU sisósodvU sFdYycdyYmiíU "mU MysYtmU cvYdmodsTU sFU yomU MysYtmU aodsYomU lysU sFdmU
→√γ,→⇒δ[+→∑→⇔β⇔→β⇒γβ+→+⇔→+θβ√√(⇒+γ∑→⇒θ)β∑θ→⇔+δβ⇔UsisósodvUsFdYycdyYmiíU3vYUivFUsMscdvFUlysU
2β⇒β∑→⇒+ θγ→θ+ −,β∑•→θ+ (⇒γβ∑⇒→θω+ θβ+ ⇔βθ+ →θ[√(→+ √[⇒+ ⇔vFU sisósodvFU óscCoacvFU SsiU sisósodvU
sFdYycdyYmi0UcvocsudvFUlysUFsUmomiatmoUsoUsiUFa%yasodsUcmuadyivíU
U
2.10 CONDICIONES DE EQUILIBRIO.

noUsiUcvods(dvUSsUimUdsvY'mUSsUimUnFdCdacmTUFsUSacsUlysUyoUFaFdsómUSsUMysYtmFUFsUsocysodYmUsoU
slyaiaeYavUcymoSvUFyUYsFyidmodsUsFUa%ymiUmUcsYv0UsFγ[+βθ<+
:: ====
∑∑ RRMMyFF U
U
Τ+](β⇒+δβθδβ+β⇔+,⇒γ[+δβ+)(θγ→+βθ√→⇔→∑<+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  46  -U
U
L+=⇒+β⇔+βθ→√([+β(θγβ⇒+↑+β√,→√([⇒βθ<+
U
∑∑∑ === :0:0:
ZYX
FFF U
∑∑∑ === :0:0:
ZYX
MMM U
U
L+=⇒+β⇔+⇔→⇒[+β(θγβ⇒+φ+β√,→√([⇒βθ<+
∑∑∑ === :0:0:
ZYX
FFF U
noUemFsUSsUivUmodsYavYTUimFUcvoSacavosFUSsUslyaiaeYavUumYmUyomUsFdYycdyYmUsoUsiUuimovUsFdCoU
YsuYsFsodmSmFUuvYUdYsFUscymcavosFUlysUovFUFsY aYCoUumYmUSsdsYóaomYUimFUYsmccavosFUSsUmuvgvíU
U
8aoUsóemY%vTUmUsFdmFUscymcavosFUSsUimUsFdCdacmTUyFymiósodsUuysSsoUmCmSaYFsUvdYmFUscymcavosFTU
iimómSmFU^sFuscamisF_UvYa%aomSmFUuvYUimUuYsFsocamUSsU^mYdacyimcavosF_U sYUMa%U2í+7U
U
articulación
U
Figura 2.15
3→+→∑γ(√,⇔→√(⇒+(⇒γ∑[δ,√β+,⇒→+β√,→√(⇒+βθβ√(→⇔ω+→+θ→]β∑<+
UUUUUUUUUUUUUUUUUUU
:=
∑M +→+⇔→+(•′,(β∑δ→+[+δβ∑β√ϕ→+δβ+⇔→+→∑γ(√,⇔→√(⇒+
2.11     GRADO DE ISOSTATICIDAD.
=⇔+ ^2∑→δ[+ δβ+ ∞Τ∞P∆P{∆_+ δβ+ ,⇒→+ βθγ∑,√γ,∑→+ θβ+ δβMaosU ósSamodsU imU Fa%yasodsU
β∑βθ(⇒<+
U
YU4U1UIUnUIsU
U
[⇒δβ<+
U R => Numero de reacciones (depende del numero y del tipo de apoyos)
E => Ecuaciones de la estática (en el plano 3 y en el espacio 6)
e => Ecuaciones especiales

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  47  -U
U
.Y Fj óeXYdEjdnYeXY Y7trYvXjf vY (ó tjsEsj sYnjvXdóYU ivFU dYsFU sFcsomYavFU lysU FsU uysSsoU
uYsFsodmYU soU siU moCiaFaFU SsU yomU sFdYycdyYmU umYmU SsdsYóaomYU imFU cvoSacavosFU SsU aFvFdmdacaSmSTU lysU mU
nóeYdó sY dYvrnYe vY Y7tjeY ó cjeXEerócE3e1
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Tabla 2.3   Grados de isostaticidad de una estructura
voSacavosFUSsU
EvjvXóXEcEsós1
.EpYdYecEó YeXdY Fóv Eec35eEXóv u
scymcavosFUSsUslyaiaeYavU
PauvUSsUsFdYycdyYmU
U
YU`U:U
nU5U)U4U%YmSvUSsUiaesYdmSUSsUimU
sFdYycdyYmU
G)3>8PUP) IU
vUsFdYycdyYmUaosFdmeisU
YU4U:U n1>U
)8>8PUP) IU
vUsFdCdacmósodsU
SsdsYóaomSmU
YUaU:U
)U5UnU4U%YmSvUSsUaFvFdmdacaSmSUSsU
imUsFdYycdyYmU
G)3n1n8PUP) IU
vUsFdCdacmósodsU
aoSsdsYóaomSmU

Donde:
I  => numero de incógnitas
E => numero de ecuaciones de la estática
2.12  CARACTERIZACION DE LAS CARGAS.
vóvUgmU!me'móvFUósocavomSvUsoUsiUmoCiaFaFUSsUsFdYycdyYmFUimFUcmY%mFU FsUuysSsoUmomiatmYU
cvóvUcmY%mFUuyodymisFUvUcvocsodYmSmFTUcmY%mFUm(amisFTUcmY%mFUovUm(amisFU*cvoUs(csodYacaSmS,TUcmY%mFU
SaFdYaeyaSmFUgUcmY%mFUovUyoaMvYósFíU
U
2.13   RESULTANTE DE CARGAS DISTRIBUIDAS LINEALMENTE. U
Sódó óF5rejv tdjt3vEXjv Yv cjexYeEYeXY vrvXEXrEd remUcmY%mUSaFdYaeyaSmUiaosmiósodsTUuvYUyomU
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tjvEcE3e sY sEc(ó cód5óz
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SaFdYaeyaSmUiaosmiósodsUivUs(uYsFmóvFUSsUimUFa%yasoXY nóeYdó1

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  48  -U
U
R
y
x
dx
dR
X (m)
L(m)
U
dxxdR ,*ω= U
∫
=
l
dxxR
:
,*ω        [t]
Además:

∫
∫
⋅
⋅
=
l
l
dxx
dxxx
x
:
:
,*
,*ω
ω
   [m]
Donde:

⇒x es la posición aproximada donde se encuentra la carga resultante R a lo largo de la
viga.
2.14  DETERMINACIÓN DE REACCIONES DE APOYO.
noUsFdsUFyedadyivUFsUóvFdYmYCUsiUuYvcsSaóasodvUlysUFsUFa%ysUumYmUSsdsYóaomYUimFUYsmccavosFUsoU
⇔[θ+→[∫[θ+δβ+⇔→θ+βθγ∑,√γ,∑→θω+=Τ+∆DP=∞+=+DPΤ& )1D>8UnDUn8PnUIFE)P>UsFdmóvFU
veia%mSvFU mU Ys%aYovFU mU yomU
convención de signosTU lysU ovFU usYóadmU FviycavomYU imFU scymcavosFU
sFdCdacmFUSsUyomUómosYmUómFUFsocaiimUsFuscamiósodsUFaUFs%yaóvFUyoUuYvcsSaóasodvUmoCiaFaFUsFcmimYíU

=θγ→+βθ+⇔→+√[⇒)β⇒√(⇒+δβ+θ(2⇒[θ+δβ+⇔→+βθγ∴γ(√→ω+ β∑vUcymoSvUFsUSsFcvovcsUsiUFsodaSvUSsUimU
∑β→√√(⇒+βθ+√[⇒)β⇒(β⇒γβ+→θ,(∑+→∑→+β⇔⇔→+β⇔+θβ⇒γ(δ[UuvFada vTUgUFaUmiUYsmiatmYUsiUcCicyivUvedsosóvFUyoU
YsFyidmSvUuvFada vUFa%oaMacmYmUlysUsiUFsodaSvUmFyóaSvUFsYCUsiUcvodYmYavíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  49  -U
U
2.15    EJERCICIOS RESUELTOS.
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PA646-2+A6S34 + m3-2o-(+mof- + (-+ 46Ao6 )6 656Amomo34 A64(6.234L P3A .34 SI23)34 i+ S6-mo3-+)34
modsYavYósodsTUumYmUmF'UYsmiatmYU sYaMacmYUsiUdauvU)6 642A(m2(A+ i 2+S7oI- u+..+A A6+mmo3-64 )6 +P3i34U
soUivFUSaFdaodvFUdauvFUSsUsFdYycdyYmFUlysUFsUóysFdY+- + m3-2o-(+mof-n
Ejercicio #1
d+)+4 .+4 4ov(o6-264 642A(m2(A+4 6- 6. P.+-3L )626ASo-+A 4( vA+)3 )6 Q8>8PePQlQdedn
m,
UR1 R3
R2
UU
vóvUsiU%YmSvUSsUaFvFdmdacaSmSUsFdmUSsMaoaSvUuvYUsiUo#ósYvUSsUYsmccavosFUósovFUsiUo#ósYvU
)6 6m(+mo3-64 )6 .+ 642B2om+ 26-6S34a
G = 3 – 3 = 0; Estructura ISOSTATICA
e,
R2
R1
R1
R2
M1
U
16+.ot+S34 6. So4S3 +-B.o4o4a Y 4 1 I 0 I 6
YU4U7UIU6U4U2
; Estructura HIPERESTATICAU
nFdmUsoUyomUsFdYycdyYmUGausYsFdCdacmUSsU%YmSvU2UvUsFdCdacmósodsUaoSsdsYóaomSmíU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  50  -U
U
c, U
R2
R1 R3
U

16m3A)6S34 =(6 m3- .+ PA646-mo+ )6 (-+ +A2om(.+mof- 46 +C+)6 (- 6m(+mof- 64P6mo+. *6C
G = 4 – 3 – 1 = 0 ; Estructura ISOSTATICA

S,
R3R1
R2
U

G = 3 - 3 – 1 = 0 ; Estructura HIPO-ESTÁTICA

s,
R2
R1 R3
R4
U
G = 4 - 3 – 1 = 0; Estructura ISOSTÁTICA
o arco ISOSTATICO

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  51  -U
U
,
M1
R5
R4
R3
R1
R2
U
YU4U=UIU6UIU+UU4UU;
; Estructura HIPERESTATICA DE GRADO 4
Ejercicio #2
U "mFUFa%yasodsFUsFdYycdyYmFUovUFvoUmSscymSmFUumYmUFvuvYdmYUcmY%mFTUgmUlysUFsUimFUcvoFaSsYmU
Yn>FnP1) IFnDPnU)Dn8PIE"n8íU
m,
U
h
U
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PA3Po6)+)64 v63SI2Aom+4 )6 642+7o.o)+)n
e,
U
h
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  52  -U
U
noUsFdsUdauvUSsUsFdYycdyYmUuvSsóvFU sYUlysUdasosUyoU%YmSvUaFvFdCdacvUSsUYU4U9UusYvUSsUa%ymiU
→⇒β∑→+∑βθβ⇒γ→+,⇒→+D=∞P∆E3∆+Y=ΤFbP{∆⋅+

Ejercicio #3
U 3mYmU imFU Fa%yasodsFU sFdYycdyYmFU lysU uYsFsodmoU cmY%mU SaFdYaeyaSmTU SsdsYóaomYU siU mivYU SsU imU
∑βθ,⇔γ→⇒γβ+∫+⇔→+,](√→√(⇒+β→√γ→+δβ+⇔→+(θ→<+
m,
U
L
dx
dRR
X
y
x
w(x)
U
U
3→+β√,→√(⇒+δβ+w(x)UcvYYsFuvoSsUcimYmósodsUmUimUSsUyomUYscdmUumYmisimUmiUsrsU(TUsodvocsFU
w(x)4UcvoFdmodsU
n′⋅+[∑+⇔[+γ→⇒γ[<+
lqRlqxqdxqRdxqdR
l
l
⋅=⇒⋅=⋅=⋅=⇒⋅=∫
::
U
noUsFdsUcmFvUsiU mivYUSsU1TUcvaocaSsUcvoUsiU mivYUSsUimUFyusYMacasU1n PIDY&"I1USsMaoaSmU
[∑+⇔→+√→∑2→+δ(θγ∑(],(δ→ω+→δβ∴θ<+
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U
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U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  53  -U
U
[∑+⇔[+γ→⇒γ[+⇔→+β√,→√(⇒+δβ+⇔→+⇔ ⇒β→+δβ+→√√(⇒+δβ++θβ∑∴<+ ∑β√γ→+ →∑→⇔β⇔→+ →⇔+
srsU^
y”.

3→+ ⇔ ⇒β→+ δβ+ →√√(⇒+ δβ+ + →θ→+ [∑+ β⇔+ {=DPΤ+ =+ Y∆=∆+ δβ+ ⇔→+ θ,β∑−(√(β+
1n PIDY&"I1USsMaoaSmUuvYUimUcmY%mUSaFdYaeyaSmíU U
U
e,
U
x
y
2/3L
L(m)
R
w(x)
q(t/m)
U
w(x)+∑β∑βθβ⇒γ→+→+,⇒→+∑β√γ→+√,∫→+β√,→√(⇒+γ(β⇒β+⇔→+−[∑mU x
l
q
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L
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l
l
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l
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2
lq
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R cvaocaSsUcvoUsiU mivYUSsUimUFyusYMacasUP1)IDY&"I1U
SsMaoaSmUuvYUimUcmY%mUSaFdYaeyaSmTUumYmU!miimYUimUu[θ(√(⇒+β→√γ→+δβ+⇔→+√→∑2→+δ(θγ∑(],(δ→+(⇒γβ2∑→[θ+
oys mósodsUgUYssóuimtmóvFUsoUimUMvYóyimUgmUósocavomδ→<+
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⇒=⋅⋅=∫
l
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TS
6
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TS
2
m
l
x=

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  54  -U
U
TS
6
2
m
l
x=
U
+ [∑+γ→⇒γ[+⇔→+β√,→√(⇒+δβ+⇔→+⇔ ⇒β→+δβ+→√√(⇒+δβ+
RUFsYCUU
U
+ ∞(+⇒[θ+δ→[θ+√,β⇒γ→+⇔→+[θ(√(⇒+δβ+⇔→+∑βθ,⇔γ→⇒γβ+
RUumFmUuvYUsiUcsodYvUSsU%Ym sSmSUSsUimU
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c,
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L(m)
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x
q(x)
p(x)
dx
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w(x)
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+ β⇒γ∑[+δβ+⇔→+ββ∑(β⇒√(→+β⇒+→θ(2⇒→γ,∑→θ+→⇒γβ∑([∑βθUFmesóvFUlysU ,*
x
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β√,→√(⇒+′,β+βθγ→+∑β∑βθβ⇒γ→δ→+[∑+⇔→+θ(2,(β⇒γβ+β√,→√(⇒<+
x
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SaFdYaeyaSmUaods%YmóvFUoys mósodsUgUYssóuimtmóvFUso+⇔→+−[∑,⇔→+∫→+β⇒√([⇒→δ→<+
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  55  -U
U
+ 3→+⇔ ⇒β→+δβ+→√√(⇒+δβ++→θ→+[∑+β⇔+√β⇒γ∑[+δβ+2∑→)sSmSUSsUimUFyusYMacasUdYmustvaSmiUSsMaoaSmU
uvYUimUcmY%mUSaFdYaeyaSmíU
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Ejercicio #4
U 3mYmUimUFa%yasodsUsFdYycdyYmTUSsdsYóaomYUsiU mivYUSsUimFUYsmccavosFUSsUmuvgvU*YsFvi sYUuvYUsiU
γ[δ[+)β√γ[∑(→⇔+∫+βθ√→⇔→∑⋅+
R1 R3
R2
2
6
3 t
A B
U
mUumYdaYUSsUsFdmUsFdYycdyYmUsFUcvo soasodsUcvoFdYya∑+,⇒+δ(→2∑→→+δβ+√,β∑[+⇔(]∑β+⋅{⋅3⋅ω+→∑→+→θ(+
uvSsYUyeacmYovFUSsUósrvYUómosYmUsoUsiUmoCiaFaFUSsUsFdmUsFdYycdyYmU
R1 R3
R2
3 t = P
A B
U
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:T:T
++
RR=
r
UUdUUUU0UUU:TT:
22
RR=
r
UUdUUU0UUUUU:TT:
66
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r
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r
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U vóvUFvimósodsUovFUcvYYsFuvoSsUmomiatmYUsoUsiUuimo[ω+,γ(⇔(•→[θ+⇔→+θ(2,(β⇒γβ+√[⇒δ(√(⇒+δβ+
β′,(⇔(]∑([<+
:
=
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r
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:
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=+++ PRRR
rrrr
UUUUUUUUUUUUUUUUscí*+,U
∆ϕ[∑→+∑β√,∑∑([θ+→+,⇒→+θβ2,⇒δ→+√[⇒δ(√(⇒+δβ+β′,(⇔(]∑([+→∑→+βθγ∑,√γ,∑→θ+√[⇔→⇒∴∑βθ<+ ⌠
=
A
M
r
U
[][ ][][ ]::TT::T:T9:T6T::T:T2
6
=+− RxxU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  56  -U
U
[⇒δβ+[]:T:T2Um βθ+,⇒+)β√γ[∑+δβ+[θ(√(⇒+δβ+,⇒+,⇒γ[+δβ++∫+[]:T:T9m+βθ+,⇒+)β√γ[∑+δβ+[θ(√(⇒+δβ+,⇒+
uyodvUSsU
6R
r
TUsodvocsFUYsFvi asoSvUsiUFaFdsómU scdvYamiUdsosóvF<+
[ ]:99T:T:
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rr
U
+ {[[+∫→+γβ⇒β[θ+√→⇔√,⇔→δ[+β⇔+)→⇔[∑+δβ+⇔→+∑β→√√(⇒+soUE*16,TUYssóuimcsóvFUsFdsU mivYUsoUimU
β√,→√(⇒+ω+→θ(+θβ+γ(β⇒β<+
:T:T::T6T::T+T::TT::T:T
2+ =−+++RR U
[ ][] tRytRRR 2MM::T:T::,T6+*T
2+2+==⇔=−+⇒U
+ [∑+γ→⇒γ[+γβ⇒β[θ+⇔[θ+θ(2,(β⇒γβθ+∑βθ,⇔γ→δ[θ<+
[]:T:T:
+
=R
r
UUt
[]:T2T:
2
=R
r
  t
[]:T+T:
6=R
r
  t
U Procedimiento Escalar:
+ →∑→+⇔→+θ[⇔,√(⇒+δβ+βθγ→+βθγ∑,√γ,∑→+[∑+β⇔+γ[δ[+sFcmimYUsFUoscsFmYavUlysUYscvYSsóvFUimU
√[⇒)β⇒√(⇒+δβ+θ(2⇒[θ+→θ,(δ[θ+→+∑(⇒√(([+δβ+βθγβ+√muadyivíU
R1 R3
R2
3 t = P
A B
U
U nóuscsóvFUmomiatmoSvUimFUscymcavosFUMyoSmósodmisFUδβ+β′,(⇔(]∑([<+
U U U
::: =⇒=⇔=
∑∑ AxHHF UUUUUUUUUUUUUUUUUUUUUUUscí*+,U
:6:: =+−⇒=⇔= ∑∑ BAyVVVF UUUUUUUUUUUUUUscí*2,U
U U U
TS+:,*9,2,*6*:: tVVMM
BBAZ=⇒=−⇒=⇔=
∑∑ UUUUUUUscí*6,U
U 8yFdadygsoSvUimFUscymcavosFU*6,UgU*+,UsoUimUscymca⇒++γβ⇒β[θ<+
AV4U2UdU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  57  -U
U
+ Dγβθβ+′,β+→⇔+√→⇔√,⇔→∑+β⇔+)→⇔[∑+δβ+⇔→+∑β→√√(⇒+
AVω+δββ⇒δ(+δβ⇔+√∴⇔√,⇔[+δβ+⇔→+∑β→√√(⇒+
BVTU
θβ→+′,β+θ(+θβ+ϕ,](β∑→+√[βγ(δ[+,⇒+β∑∑[∑+→⇔+√→⇔√,⇔→∑+⇔→+∑(β∑→++∑β→√√(⇒ω+θβ+γ∑→⇒θ(γ(∑ →+√[[+,⇒+
2∑→⇒+β∑∑[∑+→∑→+⇔→+θβ2,⇒δ→+∑β→√√(⇒ω+→∑→+β)(γ→∑+βθ→+(⇒√2⇒(γ→+θβ+⇔→⇒γβ→+β⇔+θ(2,(β⇒γβ+∑[√βδ((β⇒γ[+
δβ+→⇒∴⇔(θ(θ<+
::: =⇒=⇔=
∑∑ AxHHF U
TS+:,*9,2,*6*: tVVM
BBA=⇒=−⇒= ∑ U
TS2:,*9,;,*6*: tVVM
AAB=⇒=−⇒= ∑ U
>DP1>"UU
:+62: =+−⇒=
∑V U
+ =θγβ+∑[√βδ((β⇒γ[+δβ+)β∑(−(√→√(⇒+√[⇒θ(θγβ+β⇒+∑β→iatmYUyomUFyómdvYamUSsUdvSvFUivFUsFMysYtvFU
s(aFdsodsFUsoUsiUsrsU^g_UvUdvSmFUimFUcvóuvosodsFUSsUcmY%mFUsoUFsodaSvU sYdacmiTUSac!mUFyómdvYamUSsesU
SmYU uvYU YsFyidmSvU csYvU *:,TU uvYU lysU mF'U sFdmYCU cvoMaYómSvU lysU ivFU sFMysYtvFU SsU imFU cmY%mFU mcda mFU
sFdmYCoUcvóusoFmSmFUuvYUimFUcmY%mFUYsmcda mFU*YsmccavosF,TUóvFdYmoSvUlysUs(aFdsUyoUslyaiaeYavUdmodvU
2β[γ∑(√[+√[[+(⇒γβ∑⇒[⋅+β⇒γ∑[+δβ+→∑√[θ+→√βγ→]⇔βθ+θβ+,βδβ+→√βγ→∑+′,β+β⇒+⇔→+)β∑(−(√→√(⇒+δβ+
β′,(⇔(]∑([+,⇒+β′,βC[+→∑2β⇒+δβ+β∑∑[∑<+HU:í:2TUumFmSvUsFdsUsYYvYUimUsFdYycdyYmUFsYCUcvoFaSsYmSmUcvóvU
sFdCdacmósodsUaosFdmeisíU
Ejercicio #5
U 3mYmUimUFa%yasodsUsFdYycdyYmTUSsdsYóaomYUsiU mivYUSsUimFUYsmccavosFUSsUmuvgvíU
2
1.5 m
3 2
2 t
q=0.5 t/m
U
3YaósYmósodsUYsmiatmóvFUyoUmoCiaFaFUumYmU sYaMacmYUsiU%YmSvUSsUaFvFdmdacaSmSU
U
G = 3 – 3   = 0 ; Estructura ISOSTATICA

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  58  -U
U
U 3mYmUsMscdvFUSsUMmcaiaSmSUSsUcmicyivUuYsFsodmóvFUs⇒+β⇔+θ(2,(β⇒γβ+{3ω+⇒γβθβ+′,β+θβ+∑β→⇔(•→+β⇔+
moCiaFaFUFvivUcvoUimUYsFyidmodsUSsUimUcmY%mUSaFdYaeyaSmTUsFdmUsFUslya misodsUmUimUmodsYavYUsFdYycdyYmUusYvU
FvivUumYmUsMscdvFUSsUcmicyivUSsUYsmccavosFTUusYvUimFUsFdYycdyYmFUD>U8>DU)Y&I"n8íU
2 t
R=0.5x2=1.0 t
V
A
H
A
V
B
U
U nóuscsóvFUmomiatmoSvUimFUscymcavosFUMyoSmósodmisFUδβ+β′,(⇔(]∑([<+
U U U
:2:: =+⇒=⇔=
∑∑ AxHHF U
TS66í+:,7í+,*2*,*6,+,*+*: tVVM
BBA=⇒=+−⇒= ∑ U
U U U
TS66í::,2,*+*,*6,7í+,*2*: tVVM
AAB−=⇒=−−⇒=
∑ U
+ Dγβθβ+β⇔+θ(2⇒[+π+δβ+⇔→+∑β→√√(⇒+
ITUsFdvUovUlyasYsUSscaYUlysU!mgmóvFUYsmiatmSvUSsUómosYmU
β∑∑⇒β→+⇔[θ+√∴⇔√,⇔[θ+δβ+βθγ∴γ(√→ω+θ[⇔[+′,β+θβ+γ(β⇒βUlysUcmóeamYUsiUFsodaSvUlysU!me'móvFUmFyóaSvUumYmU
δ(√ϕ→+∑β→√√(⇒+β⇒+∑(β∑→+(⇒θγ→⇒√(→⋅+
2 t
1.0 t
0.33 t
2 t
1.33 t
U
>DP1>"UU
:66í:+66í+: =+−⇒=
∑V U
+ =θγβ+∑[√βδ((β⇒γ[+δβ+)β∑(−(√→√(⇒+√[⇒θ(θγβ+β⇒+∑β→iatmYUyomUFyómdvYamUSsUdvSvFUivFUsFMysYtvFU
s(aFdsodsFUsoUsiUsrsU^g_UvUdvSmFUimFUcvóuvosodsFUSsUcmY%mFUsoUFsodaSvU sYdacmiTUSac!mUFyómdvYamUSsesU
SmYU uvYU YsFyidmSvU csYvU *:,TU uvYU lysU mF'U sFdmYCU cvoMaYómSvU lysU ivFU sFMysYtvFU SsU imFU cmY%mFU mcda mFU
sFdmYCoUcvóusoFmSmFUuvYUimFUcmY%mFUYsmcda mFU*YsmccavosF,TUóvFdYmoSvUlysUs(aFdsUyoUslyaiaeYavUdmodvU
2β[γ∑(√[+√[[+(⇒γβ∑⇒[⋅+β⇒γ∑[+δβ+→∑√[θ+→√βγ→]⇔βθ+θβ+,βδβ+→√βγ→∑+′,β+β⇒+⇔→+)β∑(−(√→√(⇒+δβ+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  59  -U
U
I2TODOC5OS TU 1I2TICS a”5%IU “I I55S50 HU:í:+TUumFmSvUsFdsUsYYvYUimUsFdYycdyYmUFsYCUcvoFaSsYmSmUcvóvU
sFdCdacmósodsUaosFdmeisíU
Ejercicio #6
U 3mYmUimUFa%yasodsUsFdYycdyYmTUSsdsYóaomYUsiU mivYUSsUimFUYsmccavosFUSsUmuvgvíU
1.5 t
2t.m
q=1 t/m
A
B
C
4
3
223
1
U
O fidiOPZPlR'uOtPOZilYrYtitOtPOlirlXr'O(O'suPdMilY2aOdPirYyic'uOPrO1m4mhmOtPOriOOPuRdXlRXdiO
dvómoSvUsoUcysodmUFvivUimFUYsFyidmodsFUSsUimUcmY%mUSaFdYaeyaSmUusYvU sYaMacmoSvUFasóuYsUsiU%YmSvUSsU
aFvFdmdacaSmSíU
V
H
V
M
1.5 t
2t.m
R=0.5(5)(1)
A
B
C
1.5 t
2 t
2
2,67
A
A
B
B
U
G = 4 - 3 – 1   = 0 ; Estructura ISOSTATICA
O AaOPuRPOliu'O'suPdMic'uOlXiRd'OYal2qaYRiuOód'MPaYPodsFUuvYUcmSmUyomUSsUimFUYsmccavosFU
socvodYmSmFTUumYmUSsdsYóaomYUFyU mivYUcvodmóvFUcvoURdPuOPlXilY'aPuOtPOriOPuRxRYliO(OXaiOPlXilY2aO
PuóPlYirOnXPOPuRiOtitiOó'dOriOidRYlXrilY2aOPaOPrOóXodvU íU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  60  -U
U
U nóuscsóvFUmomiatmoSvUimFUscymcavosFUMyoSmósodmisFUδβ+β′,(⇔(]∑([<+
U U ∑=:HU⇒U :2=+
AH U ⇒UUU 2−=
AH Ut
U U ∑ =:
AM U⇒UU :2,7*7í+,2*7í+,9Jí2*2 =−++++
BBVM U
U U ∑ =:
BM UU⇒UU :2,6*7í+,9Jí+*27,+,*2* =++−++−−
BAMV U
U vóvUgmU!me'móvFUósocavomSvTUsFdmFUdYsFUscymcavosFUSsUimUsFdCdacmUovUFvoUFyMacasodsFUumYmU
YsmiatmYU ivFU cCicyivFU oscsFmYavFTU usYvU FaU SsesóvFU yθ→∑+ ⇔→+ β√,→√(⇒+ βθβ√(→⇔+ ′,β+ βθγ→+ δ→δ→+ [∑+ ⇔→+
→∑γ(√,⇔→√(⇒+β⇒+β⇔+,⇒γ[+{ω+γ[→⇒δ[+βθγ→+β√,→√(⇒+βFuscamiUuvSsóvFUmomiatmYU!mcamUsiUimSvUlysUóCFU
dsUcvo so%mUumYmUMmcaiaSmSUSsUcCicyivU*atlyasYSmUvUSsYsc!m,íU
∑ =:M
C
DERECHA
UU
⇒UU :,*+2,+*7í+ =−++
BBVM U
+ {[⇒+βθγ→+β√,→√(⇒+[δβ[θ+∑βθ[⇔)β∑+β⇔+θ(θγβ→ω+⇔,β2[+γβ⇒β[θ+⇔[θ+θ(2,(β⇒γβθ+∑βθ,⇔γ→δ[θ<+
2t.m
R=0.5(5)(1)
A
B
C
1.5 t
2 t
2 t
0.59 t
3.59 t
0.11 t_m
U
>DP1>"UU
:7<í67í+7í+7<í:: =+−−−⇒=
∑V U
Ejercicio #7
3mYmUimUFa%yasodsUsFdYycdyYmTUSsdsYóaomYUsiU mivYUSsUimFUYsmccavosFUSsUmuvgvíU
33
3
V
H
A
A H
B
VB
A B
1 t.m
0.6 t/m
C
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  61  -U
U
+ β→⇔(•→[θ+β⇔+⋅{⋅3⋅+δβ+⇔→++βθγ∑,√γ,∑→+γ[→⇒δ[+β⇒+cysodmUFvivUimFUYsFyidmodsFUSsUimUcmY%mU
SaFdYaeyaSmUgU sYaMacmoSvUFasóuYsUsiU%YmSvUSsUaFvFdmdacaSmSíU
U
V
H
A
A H
B
VB
A B
1 t.m
C
1.5
1
R=0.5(0.6)(3)
U
+ 3→+→∑γ(√,⇔→√(⇒+(⇒γ∑[δ,√β+→⇔+θ(θγβ→+,⇒→+β√,→√(⇒+SsUóvósodvTUlysUSsesUdvómYFsUmUimU
(•′,(β∑δ→+[+→+⇔→+δβ∑β√ϕ→+δβ+⇔→+→∑γ(√,⇔→√(⇒+β∑[+⇒[UmUmóevFUimSvFíUU
G = 4 - 3 – 1   = 0 ; Estructura ISOSTATICA

IomiatmoSvUimFUscymcavosFUMyoSmósodmisFUSsUslyaiaeY([+γβ⇒β[θ<+
U U
∑=:H UU0UU :<í:=−+
BAHH U
U U ∑ =:
AM UU0UU :9,+*<í:,7í+*2í++ =−−+
BV U
U U U ⇒UUUU62í:=
BV UUtU
U U ∑ =:
BM UU0UU :9,+*<í:,7í;*2í++ =+−−
BV U
U U U ⇒UUUUU==í:=
AV UUtU
+ =√,→√(⇒+βθβ√(→⇔+++++++++∑ =:M
C
DERECHA
U
U U
⇒UUU :6,62í:*6,2*<í: =−−
BH UU⇒UU 2=í:=
BH UtUU0UUUU92í:=
AH Ut

3mYmU sYaMacmYUYsmiatmóvFUsiUcvodYviUcvYYsFuvoSasodsíU
>DP1>"UU
:62í:2í+==í:: =+−⇒=
∑V U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  62  -U
U
Ejercicio #8
U 3mYmUimUFa%yasodsUsFdYycdyYmTUSsdsYóaomYUsiU mivYUδβ+⇔→+−,β∑•→+∑βθ,⇔γ→⇒γβ+∫+θ,+⇔ ⇒β→+δβ+→√√(⇒+
umYmUsiUFaFdsómU*m,UFmeasoSvUlysUM
*(,U4UumYCevimUSsU2SvU%YmSvíU
2 2 3
x
P1 P2
0.5 t 0.7 t
z3
1 2
6 t/m
f(x)
1 t.m
M1
U
*m,U
+ 3→++⇔ ⇒β→+δβ+→√√(⇒+βθγ→+∑β∑βθβ⇒γ→δ→+[∑+,⇒→+β√,→√(⇒ω+δβ+[δ[+′,β+βθ+∑β√(θ[+δβ√(δ(∑+,⇒+
FaFdsómUSsUcvvYSsomSmFTUsiUvYa%soUSsUSac!vUFaFdsómUuysSsUFadymYFsUsoUcymilyasYUuyodvUSsiUuimovUlysU
cvodasosUmUimFUMysYtmFTUusYvUsFUuYsMsYaeisUlysUsFdsUFsUFadymYmUsoUsiUiy%mYUómFUmSscymSvíU
x
R
dx
dR
x
3 m
6 t/m
f(x)
3
U
+ β→⇔(•→[θ+,⇒+√∴⇔√,⇔[+∑β)([<+
U U U U U S
RU4UM*(,Udx
y =
UM*(,U4Uk x
2

U U U U U 9U4U k LU6
2
U
⇒k =9V<UUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  63  -U
U
U U U U UUUUU ⇒UUk =UU:í9JUU
U U U U UUUUU ⇒UUM*(,U4U:í9JULUx
2
USt/mTU
+ ⇒γβ2∑→⇒δ[+γβ⇒β[θ+β⇔+)→⇔[∑+δβ+⇔→+∑βθ,⇔γ→⇒γβ<+
U U U U U
∫
=
6
:
2
L9Jí:dxxR U
U U U U UUUUUU
⇒U tRR9<L9Jí: =⇒= UUU0UU vaocaSsocamU
+ ∆ϕ[∑→+ϕ→⇔⇔→[θ+⇔→+[θ(√(⇒+β→√γ→+δβ+⇔→+∑βθ,⇔γ→⇒γβUSsUimUcmY%míU
U U U 29í2
9
7Jí+6
9
9Jí:L;V6
,9Jí:*
;
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:
2
====
∫
R
dxxx
x UUm
+ →∑→+(→2(⇒→∑⇒[θ+δβ+βα[∑+→⇒β∑→<+
xR(-P,0) Q(P,0)
P(x,y)
U
3QU4U31U
U I!vYmUuYvcsSsóvFUmU scdvYamiatmYUimFUMysYtmFíU
U U U
tP T:T7í:T:S
+ −=
r
U 0U T:T:T;S
+ −=r
r
U
U U
tPT:TJí:T:S
2
−=
r
U 0U T:T:T2S
2 −=r
r
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+
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r
UU 0U T:T:T29í2S
6
=r
r
U
+ ∞→]β[θ+′,β<++
T:T2íJT:S
+2+ −⇔++= RPPF
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rrrr
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++622++
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.
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9
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.
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:
.
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R MT+9í<T:T:ST+T:T:ST79í+6T:T:ST;íí+T:T:ST2T:T:S
−=+−++⇔
r
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  64  -U
U
U 8smUsiUuyodvUPUSsUcvvYSsomSmFUP 4USx ,0 ,0TUmω+,⇒+β⇔ββ⇒γ[+δβ+⇔→+⇔ ⇒β→+δβ+→√√(⇒+δβ+
RF
r
TU
sodvocsFU
RRPMFxr
rrr
=0UnFUSscaYUsiUóvósodvUSsUimUMysYtmUYsFyidmodsUF RUSsesUFsYUa%ymiUmUimUYsFyidmodsU
SsUóvósodvFíU
U U U∴U
kji
x
M
R
.
:
:
.
2íJ
:
.
:−=
r
U
⇒ T+9í<T:T:ST2íJT:T:S −=−x U
U U U U U ⇒ +9í<2íJ−=−x U
U U U U U ⇒ mx2Jí+= U
+ =θγ→+βθ+⇔→+β√,→√(⇒+δβ+,⇒→+∑β√γ→+→∑→⇔β⇔→+→⇔+βαβ+yíU
x
F=7.2 t
1.27
R
(b)
U
U 3vYU dmodvU siU FaFdsómU *m,U *cvóuisrv,U uysSsU FsYU FyFdadyaSvU uvYU siU FaFdsómU *e,U *Faóuis,TU FaoU
sóemY%vUmóevFUFaFdsómFUovUFvoUa%ymisFTUFvoUslya misodsFUumYmUcasYdvFUsMscdvFUSsUcCicyivíU
U
Ejercicio #9
U "mUMa%yYmUóysFdYmUyomU ómoa simU>nTUsFmUuastmU sFdmUFyrsdmUmiUuimovU?AU ósSamodsU2U cmeisFU
*dsoFvYsF,TUsiUs(dYsóvU>UsFUyomUumFmSvYUsoUsiUsFumcavTUFsUuaSsUSsdsYóaomYUimFUYsmccavosFUSsUmuvgvUsoU
siUumFmSvYUgUsoUivFUdsoFvYsFíU
+ 3→θ+√[[∑δβ⇒→δ→θ+δβ+⇔[θ+,⇒γ[θ+β⇒+β⇔+βθ→√([+θ[⇒<+
U
6
:
2
A
UUSmTU
6
:
2−
B
SmTU
:
<
:
C
SmTU
6
<
:
D
SmTU
6
+6
:
E
SmTU
:
;
:
G
SmTU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  65  -U
U
q2=0.2 t/m
q1=0.5 t/m
R=0.6t
R=0.8
0.8 t.m
T1
T2
O G C
D E
Z
X
Y
A
B

Vectorializar:
U U6=í7T6T;T2S =⇒−⇔
→→
GAGA
U
U U U
T79í:TJ;í:T6Jí:S−⇔
→
uGA
r
U
U U
{
T79í:TJ;í:T6Jí:S
+++
aaaTuTTGA
a
−⇔⇒⋅⇔⇒
→
rrrr
U
U U 6=í7T6T;T2S =⇒−−⇔
→→
GBGB
U
U U U
T79í:TJ;í:T6Jí:S −−⇔
→
uGB
r
U
U U
{
T79í:TJ;í:T6Jí:S
222
bbbTuTTGB
b
−−⇔⇒⋅⇔⇒
→
rrrr
U
U U U
T:T:TS
++RR=
r
U U 0U T:T:T=í:S
+ −=M
r
U
U U
T:TT:S
22RR=
r
U U 0U T9í:T:T:S−=
∆R
r
U
U U U T T:T:S
66
RR=
r
U U 0U T=í:T:T:S−=
rec
R
r
U
U U U
T9í:T:T:S
−=Q
r
UU U
L
{ { {
TT:T:T:ST=í:9í:9í:79í:79í:TJ;í:J;í:T6Jí:6Jí:S:
77í2
6
9
2
:
+
=−−−+++−−+−⇒=
−
RbaRbaRbaF
t
R
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  66  -U
U
LLU ⇒=:
RM
r
U
:T:T:T=í:S
.
=í:
:
.
:
++
.
:
:
.
9í:
:
.
:
66í+:
.
:
:
.
9í:
:
.
:
;
.
:
:
.
79í:
:
.
J;í:
;
.
6Jí:
:
.
79í:
:
.
J;í:
;
.
6Jí:
:
=−+
+−+−+−+−−+−
kjikjikjik
b
j
b
i
b
k
a
j
a
i
a
U
∴ :T:T:T=í:ST:T:T=í=ST:T:T+<=í9ST:T:T;í2ST;=í+T:T2;í2ST;=í+T:T2;í2S =−+−+−+−+++− bbaa U
U U TT:T:T:S,T*;=í+T:T+<=í+=,*2;í2S =−−+⇒ abba U
U U ba=⇒ U
SolíU
2+:9í; TbtTa
rr
=⇔==⇒U
Ejercicio #10
U 8sUuaSsUSsdsYóaomYUimFUYsmccavosFUSsUmuvgvUumYmUimUFa%yasodsUMa%yYmíU
A B
3 3 2,5
H
A
VA
H
B
VB
0.5 t/m
2 t/m
4
C
D
U
U /sYaMacmóvFUsiU%YmSvUSsUaFvFdmdacaSmSU
YU4U;UIU6UIU+UU4:UU
;  Estructura ISOSTATICAU
U 1smiatmóvFUsiUFa%yasodsUcCicyivUuYs avíU
U U U U
mLL 7;6
22
=⇒+= U
+ =⇔+∴⇒2,⇔[+δβ+(⇒√⇔(⇒→√(⇒+θβ∑∴<+
c6í76
7
;
7
;
==⇒= arcsensen
θθ U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  67  -U
U
U 1smiatmóvFUsiUcCicyivUSsUimUYsFyidmodsUSsUimUcmY%mUSaFdYaeyaSmUiaosmiósodsUmUivUimY%vUSsiUdYmóvU
{+⇔,β2[+∑β→⇔(•→[θ+β⇔+√∴⇔√,⇔[+δβ+⇔→θ+√[[⇒β⇒γβθ+)sYdacmisFUgU!vYatvodmisFíU
R
R
X
R
Y
U
U U tR
R
sen
Y
Y
27í2L,=:í:*
7í2
c6í76 ==⇒= U
U U tR
R
X
X
7í+7í2L,7<=í:*
7í2
c6í76cvF ==⇒= U
+ ∆⇔(√→⇒δ[+⇔→θ+β√,→√([⇒βθ+δβ+β′,(⇔(]∑([+γβ⇒β[θ<+
U U∑ =⇒=−⇒= tHHH
AA7í+:7í+:U
U U ∑ −=⇒−+−−⇒= tVVM
BBA 26í:7í=+2L7í+7í;L27í+L9:U
U U ∑ =⇒=++−+−⇒= tVVM
AAB26í;:+;L22L7í+7í=JL9:U
U 1smiatmóvFUsiUcvodYviUYsFuscda víU
>DP1>"UU
:26í:2926í;: =−+−⇒=
∑V U
2.16   EJERCICIOS PROPUESTOS.
Problema #1
βγβ∑(⇒→∑+⇔→θ+∑β→√√([⇒βθ+δβ+→[∫[+δβ+⇔→+θ(2,(β⇒γβ+Ma%yYmU
0.5 t/m
1 t/m
2 t
2,52,522222
A B E F C D
U
Respuesta:
HA=0 t   ;       VA=0.75 t
V
B=1.75 t   ;      VC=1.95 t      ;      VD=1.05 t

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  68  -U
U
Problema #2
A B
1 t/m
1 t/m
C
33
3
U
Respuesta:
HA=1.5 t   ;    HB=1.5 t
V
A=3 t     ;    VB=0 t

Problema #3
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj1
0.5 t/m
2 t
1.5 t/m
2.5 m
1.5 m
1 m
2 m
1 m
1.5
m 2.5 m 1.5 m
2 m
1 t/m
A
B
C
U
Respuesta:
HA=0.42 t     ;   HB=1.33 t
V
A=2.91 t     ;   VB=-4.246 t
V
C=1.957 t

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  69  -U
U

Problema #4
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj sY Fó vE5rEYeXY Ma%yYmU
2 m
4 m
1.5 m
1.5 m
2 t
2 m 2 m 2 m
A
C D E F
B
0.5 t/m
U

Respuesta:
HA=2 t
V
A=1.75 t
V
B=1.083 t
M
B=7.298 t

Problema #5
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj sY Fó vE5rEYeXY Ma%yYmU
0.5 t/m
A D
C
B
0.3 t/m
4225
13
0.3 t.m
2
0.2 t
E
U

Respuesta:
HC=0.2 t
M
C=0.7 t_m
V
B=1.31 t
V
A=0.14 t
V
C=0.25 t
Problema #6
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj sY Fó vE5rEYeXY Ma%yYmU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  70  -U
U
D E
0.3 t/m
4
1126
0.2 t/m
0.4 t/m
A C
B
0.5 t
U
Respuesta:
HA=1.1 t
V
A=0.8 t
V
B=0.17 t
V
C=0.27 t

Problema #7
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj sY Fó vE5rEYeXY Ma%yYmU
A B
0.6 t/m
0.2 t/m
C
33
4
U

Respuesta:
HA=2.30 t   ;    VA=3.06 t
H
B=0.7  t   ;    VC=0.13 t

Problema #8
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj sY Fó vE5rEYeXY Ma%yYmU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  71  -U
U
B
1 t/m
3 3
6
A
U
Respuesta:
HA=6 t
V
A=5.25 t
V
B=2.25 t

Problema #9
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj sY Fó vE5rEYeXY Ma%yYmU
B
1 t/m
2 t
2
3
2
4
A
U

Respuesta:
HA=2 t      ;    VA=2.08 t
H
B=0 t     ;     VB=0.75 t
M
A=4.6 t_m

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  72  -U
U
Problema #10
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj sY Fó vE5rEYeXY Ma%yYmU
B
0.5 t/m
A
C D
1 t.m
33
4
1
U
Respuesta:
HA=4 t      ;     VA=2.54 t
V
B=1.04 t

Problema #11
.YXYdnEeód Fóv dYóccEjeYv sY ótjuj sY Fó vE5rEYeXY Ma%yYmU
0.2 t/m
Y
Z
1 t
x
EMPOTRAMIENTO
C B
A
C
D
O
2 t
1 t.m
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  73  -U
U
Respuesta: Rx= 2.5 t     ;    Mx=8.25 t_m
R
y= -0.5 t    ;    My=15.5 t_m
R
z= 2 t      ;       Mz=-3.0  t_m

Problema #12
βγβ∑(⇒→∑+⇔→θ+∑β→√√([⇒βθ+δβ+→[∫[+δβ+⇔→+θ(2,(β⇒γβ+−(2,∑→+δ→δ[θ+⇔[θ+θ(2,(β⇒γβθ+δ→γ[θ<+
7
:
6
7
:
6
2
J
:
+
7
: −
DCBA
U
Z
1 t
X
Y
T2
T1
C
D
A
B
U
Respuesta:
HA=0.531    ;    VA=0.55 t t
H
B=0.031 t    ;    VB=2.55 t

2.17  EVALUACION DIAGNOSTICO DEL CAPITULO.

→∑→+ ∑β→⇔(•→∑+ ,⇒→+ β)→⇔,→√(⇒+ δ(→2⇒[θγ(√[+ ∑β−β∑β⇒γβ+miU uYsFsodsU cmuadyivTU FsU uYvuvosU miU
sFdySamodsUYsFuvoSsYUimFUFa%yasodsFUuYs%yodmFíU
U β−(⇒(∑+β⇔+√[⇒√βγ[+δβ+ESTRUCTURAíU
U QysUsFUyoUmuvgvXU
U QysUdauvUSsUsFdYycdyYmFUs(aFdsoXU
U {,→⇔+βθ+⇔→+√⇔→θ(−(√→√(⇒+δβ+⇔→θ+βθγ∑,√γ,∑→θX+
U QysUdauvFUSsUmuvgvFUs(aFdsoUgUcymisFUFvoUivFU%YmSvFUSsUiaesYdmSUSsUcmSmUyovXU
U QysUsFUsiU%YmSvUSsUaFvFdmdacaSmSXU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  74  -U
U
U 3mYmU cvociyaYU sFdsU Sam%ovFdacvU FsU uaSsU SsdsYóaomYUimFU YsmccavosFU SsU muvgvU SsU imU Fa%yasodsU
Ma%yYmíU
2 t
0.5 t
5
3
A B
1,5 10 1,5
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  75  -U
U

CAPÍTULO 3

ESFUERZOS INTERNOS EN LAS
ESTRUCTURAS

3.1   OBJETIVO GENERAL.
U
niUsFdySamodsUmuYsoSsYCUmUsimevYmYUivFUSam%YmómFUSs y+Ao+mof- )6 .34 6.6S6-234 S6mB-om34
lysUFsUuYsFsodmoUsoUcymilyasYUuyodvUSsUyomUsFdYycdyA+ o4342B2om+ + 2A+yI4 )6 .+ )6)(mmof- )6 4(4
1(-mo3-64 i )6 .+ A6.+mof- =(6 68o426 6-2A6 .+4 m+A%mFUs(dsYomFUcvoUimFUMysYtmFUaodsYomFUSsUimU
sFdYycdyYmíUU
U
3.2   OBJETIVOS ESPECIFICOS.

U
IuYsoSsYCUcymisFUFvoUivFUsFMysYtvFUaodsYovFUSsUyomU642A(m2(A+ i 4+7AB 6- =(6 )oA6mmof- +m2M+-
sFdmFU*ovYómiTUcvYdmodsUgUóvósodv,íU
U 0. 642()o+-26 +P.om+A+ (-+ m3-y6-mof- )6 4ov-34 +PAvuamSmUumYmUmuiacmYUmiUcCicyivUSsUivFU
sisósodvFUgmUósocavomSvFíU
U 0. 642()o+-26 +PA6-)6AB .34 )o42o-234 SI23)34 )6 +-B.o4o4 P+A+ .+ )626ASo-+mof- )6 .34 641(6At34
aodsYovFUSsUivFUSaMsYsodsFUdauvFUSsUsFdYycdyYmFU* av+4L PfA2om34L +Am34L 642A(m2(A+4 So82+4Cn
U 86vM- m+)+ SI23)3 +PA6-)o)3 A6+.ot+A+ )6 S+-6A+ m3S7o-+)+ .+ 6.+73A+mof- )6 .34 )o+vA+S+4 )6U
sFMysYtvFUaodsYovFíU
U 0. 642()o+-26 m3- .34 SI23)34 )6 +-B.o4o4 S6-mo3-+)vFTUSsóvFdYmYmUlysUimFUsFdYycdyYmFU
momiatmSmFUsFdCoUsoUslyaiaeYavUsFdCdacvíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  76  -U
U
3.3   INTRODUCCION.
) CNDCLPLSLDOICTEUICPECEQQ cDCMLCNDCY YOLSECMLCFNLJtmFUs(dsYovUFsUsocysodYmUsoUslyaiaeYavTU
QNEPRN LJCVIJQ cDCMLPCS YSIGCOEST rDCMLTLJBCMLCLDQIodYmYFsUsoUslyaiaeYavíUnFdsUsFUsiUuYaocauavUSsU
uyodvUSsUYsMsYsocamUumYmUlysUsiUiscdvYUsodasoSmUimUcmyFm5sMscdvUSsUyoUFaFdsómUSsUMysYtmFUs(dsYovUcvoU
yoU FaFdsómU SsU MysYtmFU aodsYov0U uYaocauavU lysU ovFU SsóysFdYmU imU MYvodsYmU sodYsU imU óscCoacmU SsU ivFU
cysYuvFUY'%aSvFUmFvcaCoSvimUmiUslyaiaeYavUs(dsYovUgUimUóscCoacmUSsUivFUcysYuvFUSsMvYómeisFTUcvoUsiU
slyaiaeYavUaodsYovíUU
noUsFdsUcmuadyivUFsUSsóysFdYmUcvóvUsocvodYmYUsiUsly P TJ IC DOLJDICECOJE1rYCMLCPECJLPEQ cDC
sodYsUimFUMysYtmFUs(dsYomFUgUimFUMysYtmFUaodsYomF0UJLYNPOEDMICMLCLYOECJLPEQ cDCPIYCLPLSLDOIYCSLQBD QIYU
RNLCYIDC5LDLJEMIYCVIJCPECEQQ cDCMLPCY YOLSECMLCFNLJtmFUs(dsYov0UgUlysTUuvYUFyFUsMscdvFUFsUcimFaMacmoU
IU0 -TI53” US5a”D ^D_ -SaS D” ETI53” OUPI5U” 2TI ”-O4ECVLJVLDM QNPEJCECPECYLQQ cDCOJEDY1LJYEPGCQIDC
NDC LFLQOIC MLC OLDY cDC IC MLC QISVJLY cD3C tNLJ.EC QIJOEDdsU ^/_TU cvóvU yomU MysYtmU aodsYomU lysU mcd#mU
VEJEPLPEC ECPECYLQQ cDCOJEDY1LJYEPGCQIDCNDCLFLQOIC MLCQIJOLCICMLFIJSEQ cDCMLCPECYLQQ cDCOJEDY1LJYEP3C
FvósodvUMis(avomodsU^F_TUcvóvUyoUóvósodvUaodsYovUlyLCEQO4ECVEJEPLPICECPECYLQQ cDGC5LDLJEDMICNDC
LFLQOICMLCFPLl cDGCVIJCPICOEDOIGCPEYCF TJEYCMLPCLPLSLDOICLYOJNQONJEPCLYOBDCYISLO MEYCECQISVJLY cDCIC
OLDY cD3CyISLDOICOIJY IDEDOLC^á_GCQISICNDCSISLDOIC DOLJDICRNLCEQO4ECVLJVLDM QNPEJCECPECYLQQ cDC
dYmoF sYFmiTUaSsodaMacCoSvFsUuvYUFyUsMscdvUSsU^dvYcsY_UmiUsrsUivo%adySaomiUFveYsUsiUcymiUsFdmUmcdymoSvíUU
iIDCPECMLF D Q cDGC MLDO F QEQ cDC2CQPEY F QEQ cDCMsUivFUsisósodvFUóscCoacvFUuvYUFyFUsMscdvFTU
FsU isU uYvuvYcavomU miU iscdvYU sisósodvFU SsU moCiaFaFU uEJEC LDQIDOJEJC PEC 1EJ EQ cDC MLC PIYC LPLSLDOIYC
SLQBD QIYC YITJLC LYOJNQONJEYC LDC FIJSEC MLC 1 5EYGC VcJOacvFU FaóuisFU gU mYcvFTU cvoFaSsYmoSvU cmY%mFU
YsumYdaSmFUsoUMvYómUYscdmo%yimYTUsoUMvYómUdYamo%yimJGCQEJ5EYCQIDQLDOJEMEYC2CSISLDOIYGCECOJE1rYCMLCPEC
MLMNQQ cDCMLCPEYCFNDQ IDLYCMLCPIYCLPLSLDOIYCSLQBD QvFUFsCmimoSvUivFUuyodvFUSsUSaFcvodaoyaSmSUSsU
PEYC QEJ5EYC LlOLJDEY=C 2C VIJC PEC JLPEQ cDGC LDOJLC PEC QEY%mU s(dsYomU cvoU imU MysYtmU cvYdmodsTU gU imU MysYtmU
cvYdmodsUcvoUsiUóvósodvUMis(avomodsíUU
3.4   DEFINICION Y ANÁLISIS DE LOS ELEMENTOS MECANICOS INTERNOS.
U
8aU aSsmiatmóvFU yoU sisósodvU sFdYycdyYmiU lysU FsU socysDOJEC LDC LRN P TJ IC TEUIC PEC EQQ cDC MLPC
FaFdsómUSsUMysYtmFUs(dsYovUYsuYsFsodmSvFUuvYUimFUcmY%mFU^3_0UcvóvUFsUóysFdYmUsoUimUMa%yYmU6í+*m,UgU
*e,TUdQysUFycsSsUFaUFveYsUsiUuimovUSsUcvYdsUaóm%aom5OS gI1”5”aSg ID -TI51SX0
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  77  -U
U

Figura 3.1 (a) Figura 3.1 (b)
dnFdmY'mUSsUmcysYSvUsoUmcsudmYUlysUsiUsMscdvUlysUFsU%sosYmUmiUFsumYmYUsiUcysYuvUFveYsUsiUuimovU
SsUcvYdsUaóm%aomYavUsFUsiUYvóuaóasodvUSsiUslyaiaeYavUs(dsYovXTUFaUsFUmF'TUsodvocsFUovUFsUcyóuisUsiU
tdEecEtEj sY orY ^vE re crYdtj ióZj Fó óccE3e sY reU FaFdsómU SsU MysYtmFU s(dsYovU FsU socysodYmU soU
YorEFEidEjf cróForEYd tjdcE3e sYF crYdtj XóniE2e sYesYCUSsUsocvodYmYFsUsoUslyaiaeYav_0UuvYUivUdmodvTU
vYdl eYcYvódEj YorEFEidód Fó tjdcE3e EaorEYdsó u sYdYc(ó ó Xdóx2v sY re vEvXYnó dYvrFXóeXY EeXYdej
FjcóFEaósj Ye YF cYeXdjEsY sY Fó vYccE3e XdóevxYdvóF orY vYó YorExóFYeXY ó Fó óccE3e sYF vEvXYnó sY
prYdaóv Y7XYdej sY cósó tjdcE3e z Sódó Fj óeXYdEjd jivYdxY Fó pE5rdó >zT1
U
Figura 3.2
Porción izquierda: ájidY YF cYeXdjEsY sY Fó vYccE3e XdóevxYdvóF vY FjcmiatmUyoUFaFdsómUYsFyidmodsU
aodsYovUaods%YmSvUuvYUimUMysYtmUYsFyidmodsU-1aUgUuvYUsiUóvósodvUYsFyidmodsUaodsYovUF1aTUslya misodsU
ó Fó óccE3e sYF vEvXYnó sY prYdaóv Y7XYdej orY ócXbó Ye Fó nEvnó tjdcE3e EaorEYdsóz
Porción derecha: ke pjdnó óelFj5ó u vjidY YF cYeXdjEsY sY Fó vYccE3oUdYmoF sYFmiTUFsUivcmiatmU
yoU FaFdsómU YsFyidmodsU aodsYovU slya misodsU mU imU mcca3e sY Fóv prYdaóv Y7XYdeóv orY ócXbóe vjidY Fó
tjdcE3e sYdYc(óR sEvXEe5rE2esjvY 2vXY vEvXYnó dYvrFdmodsUaodsYovTUuvYUdsosYUFsodaSvUcvodYmYavUmiUSsUimU
tjdcE3e EaorEYdsó tódó orY Y7EvXó YF YorEFEidEj EeXsYovUsodYsUmóemFUFsccavosFíUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  78  -U
U
noU emFsU SsU ivU modsYavYU gU SmSvU lysU sFdmóvFU mevYSmoSvU siU cvocsudvU SsU YsFyidmodsTU sFU SsU
vrtjeYdvY orY 2vXY XEYeY vrv cjntjeYeXYv Ye Fó sEdYccE3e sY Fjv YZYv qf ”f u H dYvtYcXExónYeXYf
dmodvUumYmUimUMysYtmUcvóvUumYmUsiUóvósodvUYsFyidmodYR Xjnóesj cjnj dYpYdYeXY Fó tjdcE3e EaorEYdsóf
Fóv cjntjeYeXYv vY jivYdxóe Ye Fó pE5rdó >z>1
U
Figura 3.3
noUcvoFscysocamTUimFUcvóuvosodsFUSsiUYsFyidmodsUaodsYovU*dYsFUMysYtmFUcvóuvosodsFUaodsYomFUgU
dYsFUóvósodvFUcvóuvosodsFUaodsYovF,0UFsUisFUSsMaosUcvóvUsisósodvFUóscCoacvF0UlysUuvYUFyFUsMscdvFU
vY EsYeXEpEcóe1
_rYdaó ejdnóF MC01 αóniE2e dYcjejcEsó cjnj _??UsFUimUMysYtmUcvóuvosodsUaodsYomUlysUmcd#mU
tYdtYesEcrFód ó Fó vYccE3e XdóevxYdvóFR vY EsYeXEpEcó tjd sjv YpYcXjv1 XYevE3e j cjntdYvE3ef
sYUMa%U6í;íUU

Figura 3.4

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  79  -U
U
_rYdaó cjdXóeXY MO01UnFdmUYsuYsFsodmSmUuvYU-?@U-?AU*sFdsUyidaóvUsoUsiUsFumcav,0UcvóuvosodsFU
EeXYdeóv orY ócXbóe tódóFYFónYeXY ó Fó vYccE3e XdóeF sYFmi0UFsUaSsodaMacmoUuvYUFyU sMscdvUSsU
cvYdsíU sYUMa%U6í7íU

Figura 3.5
,jnYeXj XjdvEjeóeXY Mα01
UF??UóvósodvUcvóuvosodsUaodsYovUlysUmcd#mUusYusoSacyimYUmUimU
vYccE3e XdóevxYdvóFR vY FY EsYeXEpEcó tjd vr YpYcXj sY XjdvE3e óF YZY Fje5EXrsEeóF sYF YFYnYeXj
sFdYycdyYmiíU
,jnYeXj pFY7EjeóeXY M,01U F?@U F?AU óvósodvFU cvóuvosodsFU aodsYovFU lysU mcd#moU
umYmisimósodsUmiUsrsUivo%adySaomiTUlysUFsUisFUmFvcaó re YpYcXj sY pFY7E3ez

Figura 3.6

3.5   CONVENCIÓN DE SIGNOS DE ESFUERZOS INTERNOS.
g cjeXEerócE3e tdYvYeXónjv Fó cjexYecE3e sY vE5ejv umYmU ivFU sisósodvFU óscCoacvFU gU FyFU
YpYcXjvf 2vXY vYdl sY ócrYdsj óF YpYcXj orY 5YeYdó FveYsU siU sisósodvU sFdYycdyYmiíU 3mYmU siivTU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  80  -U
U
cvoFaSsYsóvFU yomU a%mU FaóuisósodsU muvgmSmU cvoU cmY%mU cvocsodYmSmU miU csodYvU SsiU cimYvU cvóvU FsU
óysFdYmUsoUimUMa%yYmU6íJ*m,íUU
A C
30°
L/2L/2
P
B

Figura 3.7(a)
voU yoU Sam%YmómU SsU cysYuvU iaeYsTU ovFU usYóadaYCU aSsodaMacmYU siU FaFdsómU mcda vU gU siU FaFdsómU
Ysmcda vU sYUMa%U6íJU*e,U
R1=Psen30/2
R3=cos30
R2=Psen30/2
l/2 l/2
l/2 l/2
A CB
A C
30°
P
B

Figura 3.7 (b)
PvómoSvU SsU YsMsYsocamU siU Sam%YmómU SsU cysYuvU iaeYsTUFveYsU siU dYmóvU IEU FsU ivcmiatmU yomU
vYccE3e tYdtYesEcrFód óF YZY Fje5EXrsEeóF ó reó sEvXóecEó q sYF YZY gf Fó tjdcE3e EaorEYdsó sY Fó
sFdYycdyYmUFsUveFsY mUsoUimUfigura 3.8íUU

Figura 3.8

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  81  -U
U
ke ióvY sY Fj óeXYdEjdf Fó tjdcE3e EaorEYdsó ej vY socysodYmUsoUslyaiaeYav0UuvYUivUlysTUumYmU
slyaiaeYmYimUFsUYslyasYsUyoUFaFdsómUaodsYovUaods%Ymsj tjd sEpYdYeXYv YFYnYeXjv nYcleEcjvf YvXj Yv1 )ó
óccE3e sY :
+U%sosYmUyomUMysYtmUaodsYomU sYdacmiUlysUmcd#mUumYmiYFónYeXY ó Fó vYccE3e XdóevxYdvóF Fó
lysUFsUSsMaosUcvóvUMysYtmUcvYdmodsU/IBU0UmFaóaFóvTUimUMysYtmUs(dsYomU1
+TUgUimUaodsYomU/IBUuYvSycsoU
re 5Edj ó Fó tjdcE3e sY Fó YvXdrcXrdó 5YeYdóesj re umYUaodsYovUYsuYsFsodmSvUuvYUFIBTUlysUFsUSsMaosU
cjnj njnYeXj pFY7EjeóeXYR pEeóFnYeXYf Fó óccE3e sY imUMysYtmUs(dsYomU16U%sosYmUyomUMysYtmUaodsYomU
tYdtYesEcrFód ó Fó vYccE3e XdóevxYdvóF CgBf orY vY SsMaosU cvóvU imU MysYtmU ovYómi0U ivU modsYavYU FsU
YsuYsFsodmUsoUimUfig >z]1

Figura 3.9

.Y Fj óeXYdEjd vY sYsrcY orY Fjv YFYnYeXjv nYcleEcjFU%sosYmSvFUuvYUsiUFaFdsómUs(dsYovUlysU
ócXbó vjidY Fó tjdcE3e sYdYc(ó sY Fó YvXdrcXrdóf vY YorEFEidóe cje Fjv 5YeYdósjv Ye Fó tjdcE3e
EaorEYdsóR YvXj Yv1

Figura 3.10

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  82  -U
U
8sYóFEaóesj vjidY Fó vYccE3e gB re YFYnYeXj sY Fje5adySU S(TU ivFU sisósodvFU óscCoacvFU
5YeYdósjv tjd YF vEvXYnó Y7XYdej sY cósó tjdcE3e vYUóysFdYmUsoUimUfig >z991

Figura 3.11
g cjeXEerócE3e óijdsódYnjv cósó rej sY Fjv YFYnYeXjFU óscCoacvFU umYmU aFymiatmYU FyFU
sMscdvFUlysUovFUusYóadmoUmFvcamYisUsiUFa%ovUcvYYsFujesEYeXY1
3.5.1   Fuerza normal:
UIUimUMysYtmUovYómiUFsUisUmFvcamUsiUFa%ovUuvFada vUcymoSvUsiUsMscdvU
SsUimFUMysYtmFUs(dsYomFUlysUmcd#moUFveYsUsiimUFvoUSY XYevE3ez

Figura 3.12
3.5.2   Fuerza cortante:
UIUimUMysYtmUcvYdmodsUFsUisUmFvcamUsiUFa%ovUuvFada vUcymoSvUimUcmY%mU
Y7XYdeó orY ócXbó vjidY Fó tjdcE3e EaorEYdsó sY Fó YvXdrcXrdó XEYeY reó sEdYccE3e xYdXEcóF cje
FsodaSvUmFcsoSsodsíU

Figura 3.13

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  83  -U
U
3.5.3 Momento flexionante: UUIiUóvósodvUMis(avomodsUFsUisUmFa%omUsiUFa%ovUuvFada vU
cymoSvUsiUsMscdvUFveYsUimUMaeYmUFyusYavYUsFUSsUcvóuA6-4of-n

Figura 3.14
noUYsFyósoUdsosóvFUlysUla convención de signos en sentido positivo 46ABa
M M
N
Q
Q
N
(+)
IZQ DER
U
Figura 3.15

3.6   ANÁLISIS DE LAS ESTRUCTURAS.
niUmoCiaFaFUSsUimFUsFdYycdyYmFTUuysSsUsoU%sosYmiUFy75)oyo)oA46 6-a
U Análisis de vigas.  c+4 yov+4 43- 7+AA+4L 6- .+4 m(+.64 .+ )oA6mmof- )6iUsrsUcsodYvaSmiUsFU
umYmisimUmiU!vYatvodsU%sv%YCMacvíU
U Análisis de pórticos. c34 PfA2om34 43- 642A(m2(A+4 6- .+4 m(+.64L .+ )oA6mmof- )6 .34 6564
csodYvaSmisFUSsUimFUemYYmFUovUsFUumYmisimUmiU!vYatvodsU%sv%YCMacvUFaovUlysUsiUsrsU(UcvaocaSsU
cvoUsiUsrsUivcmiUSsUimUemYYmíU
U Análisis de arcos. c34 +Am34 43- 642A(m2(A+4 6- .+4 m(+.64 .+ 6m(+mofoUSsiUsrsUcsodYvaSmiU
cvYYsFuvoSsUmUyomUcyY mU*ovUiaosmi,íU
U Análisis de estructuras mixtas. 83- .+4 =(6 642B- m3SP(642+4 P3A yov+4L PfA2om34 iU
mYcvFíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  84  -U
U
3.7   ANÁLISIS DE VIGAS.
3.7.1 PROCEDIMIENTO DE ANÁLISIS
&omU stUsodsoSaSvUlysUivFUsisósodvFUóscCoacvFUFvoU%6-6A+)34 P3A .+ +mmof- )6 (- 4o426S+
SsUMysYtmFUs(dsYovUlysUmcd#mUsoUyomUsFdYycdyYmTUsoUI426 +P+A2+)3 P.+-26+A6S34 6. PA3m6)oSo6-23 )6
+-B.o4o4 P+A+ 2A+t+A .34 )o+vA+S+4 )6 y+Ao+mof- m3AYsFuvoSasodsTU cvodsóuimoSvU umYmU siivU SvFU
PA3m6)oSo6-2340 .+ )6)(mmof- )6 .+4 1(-mo3-64 )6 .34 6.6S6-234 S6mB-om34 *SI23)3 )6 .+4 46mmo3-64
3 +-B.o4o4 1(-mo3-+.C i .+ A6.+mof- )6 .+ m+Av+ 682sYomUcvoUimUMysYtmUcvYdmodsUgUimUMysYtmUcvYdmodsUcvoU
6. S3S6-23 1.68o3-+-26 *SI23)3 )6 4(S+ )6 BA6+4 3 +oCiaFaFUYsimcavomi,íU3mYmUsiivTUgUdvómoSvUSsU
7+46 .+ +P.om+mof- )6 (-+ m+Av+ m3-m6-2A+)+L )6 (-+UcmY%mUyoaMvYósósodsUSaFdYaeyaSmUgUSsUyomUcmY%mU
SaFdYaeyaSmUSsUMvYómUdYamo%yimYUFveYsUyomUsFdYycdyY+ *yov+ 3 7+AA+CL o)6-2o1om+A6S34 .+ y+Ao+mof- )6 .mU
MysYtmUcvYdmodsTUimUMysYtmUovYómiUgUsiUóvósodvUMis(avomodsUs(uYsFCoSvimUsoUFyUYsFuscda vUSam%YmómíUU
3.7.1.1    MÉTODO DE LAS SECCIONES.
U U PasosU cvóvU versda vU socvodYmYU imU
óm%oadySUSsUivFUsisósodvFUóscCoacvFUlysUFsUuYsFsodm- 6- (-+ 642A(m2(A+ o4342B2om+ + 2A+yI4
)6 .+ )6)(mmof- )6 4(4 1(-mo3-64 A64P6m2oy+4 6- 6. cvods(dvUSsUyoUumYCósdYvUSsMaoaSv0UsoU
m3-46m(6-mo+L 6. +-B.o4o4 P+A+ )626ASo-+A I42+4 1(-mo3-64 642B .oSo2+)3 P3A .+ y+Ao+mof- )6 .+4
cmY%mFUs(dsYomFUlysUmcd#moUsoUimUsFdYycdyYmíU3mYmUS6S342A+A .3 +-26Ao3A PA3P3-6S34a
PRIMERO.-UIUumYdaYUSsUimUsFdYycdyYmUuimodsmSmTUFsUcvoFdYygsUyoUSam%YmómUSsUcysYuvU
.o7A6 P+A+ o)6-2o1om+Aa
U El sistema de fuerzas.- d+)3 =(6 46 +-+.ot+- 642A(m2(A+4 P.+-+4L 6. 4o426S+U SsU
MysYtmFU soU FyU cvoryodvU uvSYCU FsYU yoU FaFdsómU SsU MysYtmFU umYmisivU vU yoU FaFdsómU SsU
MysYtmFU%sosYmisFUsoUsiUuimovíUU
U El número de ecuaciones de equilibrio estático.-UnoUcvoFscysocamUSsUivUmodsYavYTUsiU
o#ósYvU SsU imFU scymcavosFU SsU slyaiaeYavU sFdCdacvU umYmU siU FaFdsómU SsU MysYtmFU soU FyU
m3-5(-23L 46 o)6-2o1om+AB- d>8 P+A+ 6. 4o426S+ )6 1ysYtmFUumYmisivUvUP1n8UumYmUsiU
FaFdsómUSsUMysYtmFU%sosYmisFUsoUsiUuimovíUU
U El número de incógnitas.-U nFdCoU YsuYsFsodmSmFU uvYU imFU cvóuvosodsFU Ysmcda mFUvU
YsmccavosFUsoUcmSmUyovUSsUivFUmuvgvFUSsUimUsFdYycdyYmUgUcvoFaSsYmoSvUFyUFaFdsómUSsU
+P3i34 46 P3)AB o)6-2o1om+Aa rDe o-mfv-o2+ A6PA646-dmSmUuvYUimUcvóuvosodsUYsmcda mU
)6 (- +P3i3 Sfyo.0 d>8 o-mfv-o2+4 A6PA646-2+)+4 P3AUimFUcvóuvosodsFUYsmcda mFUsoU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  85  -U
U
(- +P3i3 1o530 i P108 o-mfv-o2+4 m(+-)3 46 2o6-6 (-U muvgvU aSsodaMacmSvU cvóvU
sóuvdYmóasodvíUU
U Las condiciones de isostaticidad.- Q)6-2o1om+)3 6. -MS6A3 )6 .+4 o-mfv-o2+4 i 6.
o#ósYvU SsU imFU scymcavosFU SsU slyaiaeYavU sFdCdacvU lysU cvodasosU siU FaFdsómU soU FyU
cvoryodvTU FsU dsoSYCoU sisósodvFU SsU moCiaFaFU umYmU mevYSmYU sFdYycdyYmFU cymoSvU imFU
m3-)omo3-64 )6 o4342+2omo)+) 46+- ov(+.640 6423 64aU)U4UníUU
U El tipo de estructura.-U voU siU YsFyidmSvU modsYavYTU sFdmYsóvFU soU cvoSacavosFU umYmU
mevYSmYUsiUmoCiaFaFUSsUyomUsFdYycdyYmUlysUFsUSsMaosUcvóvU)FvFdCdacmíUU
SEGUNDO.-UIuiacmoSvUimFUcvoSacavosFUSsUslyaiaeYavUFveYsUsiUSam%YmómUSsUcysYuvU
iaeYsTUFsUSsdsYóaomoUimFUYsmccavosFUumYmUsocvodYmYUsiUslyaiaeYavUs(dsYovíUU
TERCERO.-UIomiatmoSvUimUsFdYycdyYmUSsUatlyasYSmUmUSsYsc!mUvU acs sYFmUFs%#oUFsmU
siUcmFvUFsUaSsodaMacmoUivFUdYmóvFUlysUuYsFsodsoUcmóeavFUSsUMvYómUsoUimUcmY%mUs(dsYomTU
umYmUlysTUFveYsUcmSmUyovUSsUsiivFTUFsUcvoFdYygmUyoUSam%YmómUSsUcysYuvUiaeYsUaoSacmoSvUimU
+mmof- )6 .+4 m+Av+4 6826A-+4 *=(6 +m2M+- 437A6 .+ P3Amof- =(6 46 +-+.om6C .+4 =(6 46
!meYCoUSsUslyaiaeYmYUcvoUyoUYsFyidmodsUaodsYovTUaSs-2o1om+-)3 6- I426L .+4 m3SP3-6-264 =(6
SsMaosoUmUivFUsisósodvFUóscCoacvFUlysUmcd#moUsoUsiUdYmóvUYsFuscda víUU
CUARTO.-U IuiacmoSvU imFU cvoSacavosFU SsiU slyaiaeYavU sFdCdacvUFsU SsSycsoU imFU
MyocavosFUSsUivFUsisósodvFUóscCoacvFUlysUFsUuYsFsodmoUsoUcmSmUdYmóvTUgUmiUFyFdadyaYUsoU
I42+4 .34 P+ABS62A34 )6 y+Ao+mof- )6 .+ )o42+-mo+ ?U FsU vedasosoU imFU óm%oadySsFU
YsFuscda mFíUU
QUINTO.-U voUsiUYsFyidmSvUSsiUuyodvUmodsYavYUFsUuYvcsSsYCUmUdYmtmYUsiUSam%YmómUSsU
y+Ao+mof- P+A+ m+)+ 6.6S6-23 S6mB-om3 m3-4o)6A+-)3aUU
U Línea baseUSsUimUMysYtmUovYómiTUMysYtmUcvYdmodsUvUóvósodvUMis8o3-+-26a e P+A2oA )6
yoUómYcvUSsUYsMsYsocamTUFveYsUsiUsrsU!vYatvodmiUFsUdYmtmUimUi'osmUemFsUFveYsUimUlysUFsU
o-)om+AB- .34 P+ABS62A34 )6 y+Ao+mof- )6 .+ )o42+-mamU^?_0UlysUFsUYsimcavomYCoUFveYsU
siUsrsU sYdacmiUsoUsiUlysUFsUivcmiatmYCoUimFUóm%oadySsFUSsUivFUsisósodvFUóscCoacvFU
YsFuscda vFíUU
U Trazo de la variación de los elementos mecánicos.-  voUivFUuyodvFUsocvodYmSvFUSsiU
A64(.2+)3 +-26Ao3AL 46 (-6 P(-23 + P(-23 3726-oI-)346 .+ y+Ao+mof- =(6 642+AB )61o-o)+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  86  -U
U
uvYUimFUMyocavosFUSsUivFUsisósodvFUóscCoacvFTUaSsodo1omB-)346 6- I42+4L .+4 4ov(o6-264
m+A+m26Ab42om+4a
3mYmU cymoSvU imU
carga externa es nulaL .+ 1(-mof- )6. )o+vA+S+ )6 .+ 1(6At+
cvYdmodsUsFUyomUi'osmU!vYatvodmiUgUumYmisimUmiUsrsU.3-vo2()o-+. + 2A+yI4 )6 (-+ 6m(+mof- )6
imUMvYómU
“y =”UgUimUSsiUSam%YmómUSsiUóvósodvUMis(avomodsUFsUYsuYsFsodmUuvYUyomUi'osmU
o-m.o-+)+ =(6 376)6m6 + (-+ 6m(+mof- )6 .+ 13AS+
” y = mx + b”; íUU
3mYmU cymoSvU imU
carga externa es uniformemente distribuida .+ 1(-mof- )6.
Sam%YmómU SsU imU MysYtmU cvYdmodsU FsU YsuYsFsodmU uvYU yomU i'osmU aociaomSmU uvYU sFdmYU
m3-)omo3-+)+ + (-+ 6m(+mof- )6 .+ 13AS+
“y = mx + b”UgUimUSsiUSam%YmómUSsiUóvósodvU
Mis(avomodsUuvYUyomUi'osmUcyY mUSsUFs%yoSvU%YmSvUós)o+-26 (-+ 6m(+mof- )6 .+ 13AS+
“y
= ax
2
+ bx + c”UU
3mYmUcymoSvUimU
carga externa es repartida de forma triangularL .+ 1(-mof- )6.
Sam%YmómUSsUimUMysYtmUcvYdmodsUFsUYsuYsFsodmUuvYUyomUi'osmUcyY mUSsUFs%yoSvU%YmSvUuvYU
m3-26-6A (-+ 6m(+mof- )6 .+ 13AS+
“y = ax
2
+ bx + c”UgUimUSsiUSam%YmómUSsiUóvósodvU
Mis(avomodsUuvYUyomUi'osmUcyY mUSsUdsYcsYU%YmSvUcvo (-+ 6m(+mof-
“y = ax
3
+ bx
2
+ cx +
d”
íUU
U Valores críticos.- 0- m+)+ )o+vA+S+ )6 y+Ao+mof- )6 .34 6.6S6-234 S6mCoacvFUcvóvU
YsFyidmSvU SsiU uyodvU modsYavYTU FsU !meYCU SsU aSsodaMacmYU siU
valor máximoU *uvFada vU gU
os%mda v,TU
mínimoUgUcero 23)34L A616Ao)34 + .+ .b-6+ 7+46 A64P6m2oy+0 6- I4dsUFyuysFdvUFsU
2o6-6a l(+-)3 .+ 1(6At+ m3A2+-26 P+46 P3A m6A3L 46 mFvcamUsoUsiUSam%YmómUSsiUóvósodvU
Mis(avomodsUyoU mivYUóC(aóvUlysUuysSsUFsYUuvFada vUvUos%mda v0UgUcymoSvUsiUóvósodvU
Mis(avomodsUumFsUuvYUcsYvUsiUuyodvUFsUSsMaosUcvóvUu(-23 )6 o-1.68of-n
3.7.1.2 ANALISIS RELACIONAL.
U U U nFU yoU uYvcsSaóasodvU SsU moCiaFaFU umYmU
)626ASo-+A .+ y+Ao+mof- )6 .34 6.6S6-234 S6mB-om34 YsimcavomoSvU imU cmY%mU s(dsYomU cvoU imU
MysYtmUcvYdmods0UgUimUMysYtmUcvYdmodsUcvoUsiUóvósodvUMis(avomodsTUsodYsUSvFUuyodvFUSsMaoaSvFU
soUyomUsFdYycdyYmíUU
3mYmU SsóvFdYmYU ivU modsYavYTU dvómYsóvFU SsU YsMsYsocamU imU sFdYycdyYmU aSsmiatmSmU lysU mU
m3-2o-(+mof- 46 S(642A+a
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  87  -U
U
W t/m
R1
W t/m R2
12
P ton
dx
X

Figura 3.16
U
3mYdasoSvUSsiUFyuysFdvUlysUimUsFdYycdyYmUFsUsocysodYmUsoUslyaiaeYavTUsFdmUFsUSsMaosUcvóvUyomU
a%mUFaóuisósodsUmuvgmSmUcvoUcmY%mUYsumYdaSmU*SsUMvYómUdYamo%yimYUgUyoaMvYós,UgUcmY%mUcvocsodYmSmU
imFUlysUsoUFyUcvoryodvTUYsuYsFsodmoUmiUFaFdsómUYsmcda v0UgUsiUFaFdsómUYsmcda vUsFdmUYsuYsFsodmSvUuvYU
imFUYsmccavosFU1
+UgU12UíU
)oacamYsóvFU siU moCiaFaFU mU umYdaYU SsU yoU sisósodvU SaMsYsocamiU SsU imU cmY%mU yoaMvYósósodsU
SaFdYaeyaSmUivcmiatmSvUmUyomUSaFdmocamU^?_UgUiaóadmSvUuvYUSvFUFsccavosFU*+UgU2U,UusYusoSacyimYsFUmiU
srsUivo%adySaomiUgUFsumYmSmFUyomUSaFdmocamUS(íU"vFUsisósodvFUóscCoacvFUlysUFsUuYsFsodmoUsoUcmSmU
yomUSsUimFUFsccavosFUFsCmimSmFTUFsUóysFdYmoUsoUimUFE5rEYeXY pE5rdó1
U
Figura 3.17U
niU FaFdsómU óvFdYmSvU sFdmU cvóuysFdvU uvYU yoU
sistema externoTU YsuYsFsodmSvU uvYU imU MysYtmU
YsFyidmodsUslya misodsUmUimUcmY%mUyoaMvYósósodsUSaFdYaeyaSmU
FR = W dx gUuvYUyoUsistema internoU
YsuYsFsodmSvUuvYUivFUsisósodvFUóscCoacvFU
M gUV orY vY tdYvYeXóe Ye Fó vYccE3e 9f u tjd (V + dV)UgU
(M + dM) Ye Fó vYccE3e TR Ye cjevYcrYecEóf YF vEvXYnó Ye vrUcvoryodvUFsUSsMaosUcvóvUyoUFaFdsómUSsU
MysYtmFUumYmisivUsoUsiUuimovíUU
gtFEcóesj Fóv YcrócEjeYv sY YorEFEidEj Y7XYdej1 árnmUSsUMysYtmFU sYdacmisFíUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  88  -U
U
:,* =+−+ dVVWdxV UUUUUU0UUUUUUU : =−−+ dVVWdxV U
:=−dVWdx UUUUUUUUUUUUUU∴UUUUUUUUUUUUdVWdx=UUUUUUUscí*+,U
β+⇔→+β√,→√(⇒+++[]γβ⇒β[θ+
dx
dV
W
=ω+[∑+⇔[+γ→⇒γ[+θβ+,βδβ+→−(∑→∑+′,β<+
“La pendiente de la fuerza cortante esta representada por la carga externa”.
⇒γβ2∑→⇒δ[+⇔→+β√,→√(⇒++[δβ[θ+[]γβ⇒β∑<+
∫∫
=
2
+
2
+
x
x
V
V
WdxdVU
∞(+⇔→+∑(β∑+(⇒γβ2∑→⇔+∑β∑βθβ⇒γ→+⇔→+)→∑(→√(⇒+δβ+⇔→UMysYtmUcvYdmodsUsodYsUimFUFsccavosFU+U
gU 2U TU gU imU Fs%yoSmTU siU CYsmU emrvU imU MvYómU SsU imU cmY%mU s(dsYomU sodYsU imFU FsccavosFU +U gU 20U
β⇒γ[⇒√βθ<+
“La variación de la fuerza cortante entre las secciones 1 y 2 es igual al área bajo la forma de la
carga externa entre ambas secciones”.
{[⇒+]→θβ+β⇒+⇔[+→⇒γβ∑([∑+θβ+γ(β⇒β+′,β<++
+2VVV −=∆UU0UCYsmUSsUimUcmY%mUs(dsYomUsodYsU+52U
∆⇔(√→⇒δ[+→ϕ[∑→+⇔→+θβ2,⇒δ→+√[⇒δ(√(⇒+δβ⇔+β′,(⇔(]∑([+βθγ∴γ(√[ω+√[⇒+θ,→+δβ+[β⇒γ[θ<++
:,*
2
=+−++ dMM
dx
WdxVdxMUUUUUU0UUUUUUU :: =−−++ dMMVdxM U
∴UUUUUUdMVdx=UUUUUUUscí*2,U
β+⇔→+β√,→√(⇒+++[]γβ⇒β[θ+
dx
dM
V=
ω+[∑+⇔[+γ→⇒γ[+θβ+,βδβ+→−(∑→∑+′,β<+
“La pendiente del momento flexionante esta representado por la fuerza cortante”.
⇒γβ2∑→⇒δ[+⇔→+β√,→√(⇒+<++
∫∫
=
2
+
2
+
x
x
VdxdM
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  89  -U
U
3→+∑(β∑→+(⇒γβ2∑→⇔+∑β∑βθβ⇒γ→+⇔→+)→∑(→√(⇒+δβ⇔+[sodvUMis(avomodsUsodYsUimFUFsccavosFU+UgU
20UgUimUFs%yoSmUsiUCYsmUSsUimUMysYtmUcvYdmodsUsodYs+⇔→θ+(θ→θ+θβ√√([⇒βθ++∫++[∑+⇔[+γ→⇒γ[<++

“La variación del momento flexionante entre las secciones 1 y 2 es igual al área del diagrama de
la fuerza cortante entre ambas secciones respectivamente”.
=θγ[+βθ<++
+2MMM −=∆UU0UCYsmUSsiUSam%YmómUSsUMysYtmUcvYdmodsUsodYsUU+52U
{[⇒+]→θβ+δβ+⇔[+→⇒γβ∑([∑ω+⇔→+∑β⇔→√(⇒+β⇒γ∑β+⇔→+√→∑2→Us(dsYomUgUimUMysYtmUcvYdmods0UgUimU
MysYtmUcvYdmodsUcvoUsiUóvósodvUMis(avomodsTUumYmUSsγβ∑(⇒→∑+⇔→+)→∑(→√(⇒+δβ+⇔[θ+β⇔ββ⇒γ[θ+
β√∴⇒(√[θ+β⇒γ∑β+δ[θ+,⇒γ[θ+θβ+∑βθ,β<++
PRIMERO.-  "mUusoSasodsUSsUimUMysYtmUcvYdmodsTUsFdmUYsuYsFsodmSmUuvYUimUcmY%mUs(dsYomíUU
SEGUNDO.-++3→+)→∑(→√(⇒+δβ+⇔→+−,β∑•→+√[∑γ→⇒γβ+β⇒γ∑β+δ[θ+,⇒γvFTUsFUa%ymiUmiUCYsmUSsUimUcmY%mU
s(dsYomUsodYsUivFUóaFóvFUuyodvFUSsUYsMsYsocamíUU
TERCERO.- U "mU usoSasodsU SsiU óvósodvU Mis(avomodsTU sFdmU YsuYsFsodmSmU uvYU imU MysYtmU
cvYdmodsíUU
CUARTO.- + 3→+)→∑(→√(⇒+δβ⇔+[β⇒γ[+−⇔β([⇒→⇒γβ+β⇒γ∑β+δ[θ+,⇒dvFTUsFUa%ymiU miUCYsmUSsiU
δ(→2∑→→+δβ+)→∑(→√(⇒+δβ+⇔→+−,β∑•→+√[∑γ→⇒γβ+β⇒γ∑β+⇔vFUóaFóvFUuyodvFUSsUYsMsYsocamíUU
→∑→+γβ⇒β∑+,⇒→+√[⇒√β√(⇒+2∑→−(√→+δβ+⇔→θ+→⇔(√→√([⇒sFUSsU a%mFUsoUsFdYycdyYmFUcvóuisrmFU sYU
mos(vFUSsiUcmuadyivU6íU
U
EJERCICIOS RESUELTOS.
→∑→+ [δβ∑+ γβ⇒β∑+ ,⇒→+ βα[∑+ β∑√β√(⇒+ δβ+ ⇔→+ (⇒γ∑[δ,√√(⇒+ →⇔+ →⇒∴⇔(θ(θ+ δβ+ ⇔→θ+ βθγ∑,√γ,∑→θ+
∑βθβ⇒γ→∑β[θ+→+√[⇒γ(⇒,→√(⇒+→+,⇒→+θβ∑(β+δβ+βαβ∑√(√([θ+∑βθ,β⇔γ[θ+[∑+⇔[θ+γ[δ[θ+∫→+β⇒√([⇒→δ[θ+
→⇒γβ∑([∑β⇒γβω+ →∑→+ →θ + ∑β→⇔(•→∑+ ⇔→+ )β∑(−(√→√(⇒+ δβU cmSmU dauvU SsU sFdYycdyYmU gU momiatmYU siU
√[[∑γ→(β⇒γ[++βθγ∑,√γ,∑→⇔+δβ+√→δ→+)(2→ω+γ→](⇒+→uYsoSsYUmU%YmMacmYUivFUSam%YmómFUSsUDTUQUgUFíU
Ejercicio #1
3mYmU SsóvFdYmYU ivU modsYavYósodsU s(uysFdvTU FsU uYvuvoβ+ →+ √[⇒γ(⇒,→√(⇒+ →⇔(√→∑+ β⇔+ →⇒∴⇔(θ(θ+
−,⇒√([⇒→⇔+[+γ[δ[+δβ+θβ√√([⇒βθ+α,⇒γ→β⇒γβ+√[⇒+β⇔+γ[δ[+∑β⇔→√([⇒→⇔⋅+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  90  -U
U
a) Método de las secciones o análisis funcional.
l /2 l /2
BA C
U
3YaósYmósodsUYsmiatmóvFUyoUmoCiaFaFUumYmU sYaMacmYUsiU%YmSvUSsUaFvFdmdacaSmSU
G = 3 – 3   = 0 ; Estructura ISOSTATICA
+ β→⇔(•→[θ+β⇔+√∴⇔√,⇔[+δβ+⇔→θ+∑β→√√([⇒βθ+δβ+→[∫[<+
TS
2
:,*,*7í:: t
WL
VVWLM
AAC
=⇒=+−⇒=∑
TS
2
:7í:,*: t
WL
VVWLM
CCA
=⇒=−⇒=∑ U
>DP1>"UU
:2V2V: =−−⇒=
∑ WLWLWLV U
∆ϕ[∑→+θβ2,([θ+√[⇒+⇔→+[]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δsUsFMysYtvFUaodsYovFUemFCoSvovFUsoUimU
√[⇒)β⇒√(⇒+δβ+θ(2⇒[θ+′,β+∫→+ϕβ[θ+[θγ∑→δ[⋅+
M M
N
Q
Q
N
(+)
IZQ DER
U
I!vYmTUsFUcvo soasodsUsFdmeiscsYUyoUFaFdsómUSsUYsMsYsocamTU!mcasoSvUcvaocaSaYUsiUsrsU
xUcvoUimU
δ(∑β√√(⇒+δβ⇔+βαβ+√β⇒γ∑[(δ→⇔+δβ+⇔→+)(2→+
w(x)
l /2 l /2
BA C
X
WL/2 WL/2U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  91  -U
U
8sUsMscd#mUyoUcvYdsUsoUimU a%mTU^8_TUóaFómUlysUFsUmomiatmYmUyomUivo%adySU?UsoUyoUYmo%vUSsU
ivo%adySUSsU a%mU^"_íU
Τ]θβ∑)β[θ+β⇔+∑(β∑[+√[∑γβω+θβ√√(⇒+^∞_+
w(x)
B
S
WL/2
x
Q
N
M
R
U
DOMINIO ⇔ Lx<<:
vYdmodsUQUU





=⇒=
−=⇒=
⇒−⇒
2
:
2
2
WL
Qx
WL
QLx
ParaWX
WL
U
8sU veFsY mU lysU cymoSvU ?4:TU imU MysYtmU cvYdmodsU dasosU yoU sMscdvTU uvFada vTU gU cymoSvU ?4"U
os%mda víU3vYUivUmodsYavYUFsUSsSycsUlysUimUMysYtmUcvYdmodsUSsUIUmU TUFveYsUmi%#oUuyodvU!meYCUSsUumFmYU
uvYUcsYv0UsoUcvoFscysocamTUimUSaFdmocamUSvoSsUvcyYYβ+⇔[+→⇒γβ∑([∑ω+⇔→+−,⇒√(⇒+δβ+⇔→+−,β∑•→+√[∑γ→⇒γβ+θβ+
(2,→⇔→+→+√β∑[ω+→∑→+δβγβ∑(⇒→∑+βθ[+βθ<+
VWX
WL
=−
2
+++β∑[+√,→⇒δ[+≤⌠ω+γβ⇒β[θ+′,β<+
22
L
XWX
WL
=⇒=
U
3YvcsSsóvFU miU
trazo del diagrama de la fuerza cortante uYaósYvU FsU dYmtmU yoU ómYcvU SsU
YsMsYsocamU cvóuysFdvU uvYU yoU srsU !vYatvodmiU ^?_U gU yoU srsU sYdacmiU ^@_TU FveYsU siU srsU !vYatvodmiU FsU
ivcmiatmUivFUuYaósYvFUumYCósdYvFUSsUimU mYameisU^?_UgUFveYsUsiUsrsU sYdacmiUFsUivcmiatmUimUóm%oadySUSsU
imUMysYtmUcvYdmodsUumYmUcmSmUumYCósdYvUSsUimU mYameisU^?_íU
&oasoSvUivFUuyodvFUSsFSsUIUcvoU?4:UgU/4:U!mFdmTUYsF,⇔γ→+⇔→+)→∑(→√(⇒+δβ+⇔→+−,β∑•→+√[∑γ→⇒γβ+
′,β+[]βδβ√β+→+⇔→+−,⇒√(⇒+′,β+θβ+[]γ,)[⋅+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  92  -U
U
BA C
(+ )
(-)
V=0
WL/2
-WL/2
V=0
x
NR
X=0 X=L/2 X=L
U
FvósodvUFUU



=⇒=
=⇒=
⇒−⇒ ::
:
22
2
Mx
MLx
Para
WX
X
WL
U
3mYmUdmiUsMscdvUSsesóvFUdsosYUcvovcaóasodvUlysUcymoSvUFsUveFsY sUyoU mivYUó'oaóvUSsUimU
cvYdmodsU*:,TUsFUm!'USvoSsUFsUsocysodYmUsiU mivYUóC(aóvUSsiUSam%YmómUSsUóvósodvFíU
FvósodvUFUU



=⇒=⇒−⇒
=
2V
22
22
WL
MLxPara
WX
X
WL
U
3YvcsSsóvFU miU
trazo del diagrama del momento flexionanteTU cvóvU gmU ósocavomóvFU
dYmtmoSvUyoUsrsU!vYatvodmiU^?_UgUyoUsrsU sYdacmiU^@_TUFveYsUsiUsrsU!vYatvodmiUFsUivcmiatmUivFUuYaósYvFU
umYCósdYvFUSsUimU mYameisU^?_UgUFveYsUsiUsrsU sYdacmiUimUóm%oadySUSsiUóvósodvUMis(avomodsUumYmUcmSmU
umYCósdYvUSsUimU mYameisU^?_íU
BA C
M=0
WL /8
y
x
NR
X=0 X=L/2 X=L
(+)
M=0
2
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  93  -U
U
&omU stUYsmiatmSvUsiU%YmMacvUSsUivFUsFMysYtvFUaodsYovFUSsUimUsFdYycdyYmUuYvcsSsóvFUmUmovdmYU
ivFU mivYsFUcY'dacvFUSsUimUsFdYycdyYmíU
Fuerza cortante máxima0 IgP” 5I15IgIUP”“” IU DSg ”1S7Sg IU ?4: -SU gO%US uvFada vUgUsoU
?4"UcvoUFa%ovUos%mda vTUsoUmóevFUcmFvFUcvoUyoU mivYUSsU["V2U
Fuerza cortante cero0 gI 5I15IgIUP” ”D -IUP5S “ID -D”5S “I D” NO%”6 1TUdvUsoUlysUsiUSam%YmómU
SsiUóvósodvUMis(avomodsUsFUyoU mivYUóC(aóvíU
Momento flexionante cero0 gI 15IgIUP” IU DSg ”1S7Sg 5Ig1I-PONSg8
Momento flexionante máximo0 gI 15IgIUP” ”D -IUP5S “ID -D”5S -SU TU IEI-PS 1Sgada vU*UimU
MaeYmU FyusYavYU sFdmU FvósdaSmU mU yoU sMscdvU SsU cvóuYsouY2aHO óXaR'O nXP O PaO PrO tYiqdiciO tPO ZXPdyiO
cvYdmodsUsFdmUumFmUuvYUcsYvíU
b) Método relacional.
l /2 l /2
BA C
U
1smiacsóvFUyoUSam%YmómUSsUcysYuvUiaeYsUlysUovFUusYóadmUaSsodaMacmYUimFUMysYtmFUs(aFdsoUsoUimU
a%míU
w(x)
l/2 l/2
WL/2 WL/2
BA C
R
U
3vYUSmdvUFmesFUlysUimFUYsmccavosFUSsUmuvgvUsoU/
IUgU/ gSU0

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  94  -U
U
TS
2
:,*,*7í:: t
WL
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=⇒=−⇒=∑ U
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el
diagrama de variación de la fuerza cortante
<+
{,→⇒δ[+?≤⌠ω+⇔→+−,β∑•→+√[∑γ→⇒γβ+β⇒+∆+βθ<+
/IU4U/IUU4U["V2U
{,→⇒δ[+?≤3Vω+⇔→+−,β∑•→+√[∑γ→⇒γβ+β⇒+E+βθ<++
/EU4U/IUHU*CYsmUSsUimUcmY%mUs(dsYomUmcdymoSvUSsUIUm+E+βθγ[+βθ<+
/EU4U["V2UHU*5[ULU"V2,U4U["V2UIU["V2U4U:U
U
ymoSvU?4"TUcvoFaSsYmoSvUyoUuyodvUSaMsYsocamiUmodsF+δβ⇔+,⇒γ[+{+βθ<+
/ U4U/EUHU*CYsmUSsUimUcmY%mUs(dsYomUmcdymoSvUSsUEUm+{+βθγ[+βθ<+
/BcU4U:UHU*5[ULU"V2,U4UIU["V2UUUUmodsFUSsU U
U
FINALMENTE+⇔→+−,β∑•→+√[∑γ→⇒γβ+β⇒+{+βθ<+{+≤+B√+H+∑β→√√(⇒+βoU ,UU
/ U4U5["V2UHU["V2U4U:U
voUivFUYsFyidmSvFUmodsYavYsFTUsiUSam%YmómUSsU mYam√(⇒+δβ+βθ−,β∑•[θ+δβ+⇔→+−,β∑•→+√[∑γ→⇒γβ+θβ+
,βθγ∑→+β⇒+⇔→+θ(2,(β⇒γβ+−(2,∑→<+
BA C
(+)
(-)
V=0
WL/2
-WL/2
V=0
x
NR
X=0 X=L/2 X=L
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  95  -U
U
P'ciat'Ol'c'OdPZPdPalYiOPrOtYiqdiciOtPOMidYilY2aOtPUimUMysYtmUcvYdmodsUYsmiatmóvFUm!vYmU el
diagrama de variación del momento flexionante
U
T”U“S ?4:6 ID aSaIUPS  IU 9 Ig0 F9 4 :
T”U“S ?4"V26 ID aSaIUPS IU E Ig0
FEU4UFIUHU*CYsmUSsiUSam%YmómUSsUimUMysYtmUcvYdmodsU“I 9 ” E, IgPS Ig0
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T”U“S ?4"6 ID aSaIUPS IU E Ig0
F U4UFEUHU*CYsmUSsiUSam%YmómUSsUimUMysYtmUcvYdmodsU“I E ”  , IgPS Ig0
F U4U["2V=UHU*5["V2,*"V2,*+V2,U4U["
2
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V=U4U:U
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D” gO%TOIUPI EO%T5”0
BA C
M=0
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x
NR
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Ejercicio #2
8PO óYtPO iairYyidO riO uYqXYPaRPO PuRdXlRXdiO ó'dO PrO c3RvSvU SsU FsccavosFU vU moCiaFaFU MyocavomiU
BXaRicPaRPOl'aOPrOc3R't'OdPrilY'airmO
a) Método de las secciones o análisis funcional.

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  96  -U
U
R
W
VBVA
BA
L
2/3L
U
3YaósYmósodsUYsmiatmóvFUyoUmoCiaFaFUumYmU sYaMacmYUsiU%YmSvUSsUaFvFdmdacaSmSU
G = 3 – 3   = 0 ; Estructura ISOSTATICA
+ β→⇔(•→[θ+β⇔+√∴⇔√,⇔[+δβ+⇔→θ+∑β→√√([⇒βθ+δβ+→[∫[<+
1U4UeL"L[U4aUU14["V2U TS
6
:
26
2
:
t
WL
VLV
WL
LM
CCA
=⇒=−⇒=∑
U
TS
9
:
26
+
:
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WL
VLV
WL
LM
AAC
=⇒=+−⇒=∑ U
>DP1>"UU
:
692
: =++−⇒=∑
WLWLWL
V
U
8sUsMscd#mUyoUcvYdsUsoUimU a%mTU^8_TUóaFómUlysUFsUmomiatmYmUyomUivo%adySU?UsoUyoUYmo%vUSsU
ivo%adySUSsU a%mU^"_íU
R
W
VBVA
BA
S
X
2/3L
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  97  -U
U
X
2/3X
R
VA
A
S
Y
Τ]θβ∑)β[θ+β⇔+∑(β∑+√[∑γβω+θβ√√(⇒+^∞_+
U
∆⇔(√→⇒δ[+⇔→θ+β√,→√([⇒βθ+δβ+β′,(⇔(]∑([+P=D=FΤ∞<+
UUUUUUUUUUUU/BIU4U["V9UIU
Y?V2U
+++++++++[∑+γ∑(∴⇒2,⇔[θ+θββα→⇒γβθ+γβ⇒β[θ<++[V3+≤+@V?U
U
=⇒γ[⇒√βθ<+
X
L
W
Y=
ω+θ,θγ(γ,∫β⇒δ[+β⇒+⇔→+−,⇒√(⇒+δβ+√[∑γ→⇒γβ+∑β−β∑(δ→+→iUuyodvUIBU
29
f
29
f
2
X
L
WWL
V
X
X
L
WWL
V
CC
−=∴−=
U
DOMINIO
⇔

Lx<<:

vYdmodsUQUU





=⇒=
−=⇒=
⇒−⇒
9
:
6
29
2
WL
Qx
WL
QLx
Para
X
L
WWL
U
βγβ∑(⇒→[θ+⇔→+δ(θγ→⇒√(→+ϕ[∑(•[⇒γ→⇔+[∑+δ[⇒δβ+θβ+βMscd#mUsiUcmóeavUUSsUuvFada vUmUos%mda vU
soUimUcvYdmodsíU
V
L
WXWL
=−
29
2
+++β∑[+√,→⇒δ[+≤⌠ω+γβ⇒β[θ+′,β<+
LXoLX
W
LWL
X7JJí:
6
+2
9
2
==⇒= U
3YvcsSsóvFUmiUdYmtvUSsiUSam%YmómUSsUimUMysYtmUcvYdmodsíU
BA C
(+)
(-)
V=0
WL/6
-WL/3
V=0
X=0 X=0.577L X=L
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  98  -U
U
FvósodvUFUU



=⇒=
=⇒=
⇒−⇒ ::
:
99
6
Mx
MLx
Para
L
WX
X
WL
U
3mYmUdmiUsMscdvUSsesóvFUdsosYUcvovcaóasodvUlysUcymoSvUFsUveFsY sUyoU mivYUó'oaóvUSsUimU
cvYdmodsU*:,TUsFUm!'USvoSsUFsUsocysodYmUsiU mivYUóC(aóvUSsiUSam%YmómUSsUóvósodvFíU
FvósodvUFUU
{
2
2
:9;í:7JJí:
22
WLMLxPara
WX
X
WL
=⇒=⇒−⇒U
3YvcsSsóvFU miU
trazo del diagrama del momento flexionanteTU cvóvU gmU ósocavomóvFU
dYmtmoSvUyoUsrsU!vYatvodmiU^?_UgUyoUsrsU sYdacmiU^@_TUFveYsUsiUsrsU!vYatvodmiUFsUivcmiatmUivFUuYaósYvFU
umYCósdYvFUSsUimU mYameisU^?_UgUFveYsUsiUsrsU sYdacmiUimUóm%oadySUSsiUóvósodvUMis(avomodsUumYmUcmSmU
umYCósdYvUSsUimU mYameisU^?_íU
BA C
M=0
0.0064WL
(+)
M=0
2
X=0 X=0.577L X=L
U
&omU stUYsmiatmSvUsiU%YmMacvUSsUivFUsFMysYtvFUaodsYovFUSsUimUsFdYycdyYmUuYvcsSsóvFUmUmovdmYU
ivFU mivYsFUcY'dacvFUSsUimUsFdYycdyYmíU
Fuerza cortante máxima<+√,→⇒δ[++?≤3+√[⇒+→2⇒(γ,δ+[3Vφ⋅+
Fuerza cortante mínima<+√,→⇒δ[++?≤⌠+√[⇒+→2⇒(γ,δ+[3V↑++
Fuerza cortante cero<+ θβ+ ∑βθβ⇒γ→+ √,→⇒δ[+ ?+ ≤+ ⌠⋅ϖJJ+ 3+ δ(θγ→⇒√(→+ ′,β+ θβ+ →FvcamU soU siU
Sam%YmómUSsiUóvósodvUMis(avomodsUcvoUyoU mivYUóC(aóvUSsUsFdsíU
Momento flexionante cero<+θβ+∑βθβ⇒γ→+β⇒+⇔[θ+→[∫[θ+∑βθβ√γ()[θ⋅+
Momento flexionante máximo<+θβ+∑βθβ⇒γ→+√,→⇒δ[+?≤⌠⋅ϖJJ+3+δ(θγ→⇒√(→+′,β+β⇒+⇔→+lysUsiU
Sam%YmómUSsUMysYtmUcvYdmodsUumFmUuvYUcsYvíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  99  -U
U
b) Solución por el método relacional.
R
W
VBVA
BA
L
2/3L
U
noU aFdmUSsUlysUsiUcCicyivUSsUimFUYsmccavosFUSsUmuvgvUsFUsiUóaFóvTUFsUdsoSYCoUivFUFa%yasodsFU
)→⇔[∑βθ<+
1U4UeL"L[U4aUU14["V2U
TS
6
:
26
2
:
t
WL
VLV
WL
LM
BBA
=⇒=−⇒=∑
U
TS
9
:
26
+
:
t
WL
VLV
WL
LM
AAB
=⇒=+−⇒=∑ U
1smiatmóvFUsiUcCicyivUSsUivFUsisósodvFUSsUcvYdsíU
{,→⇒δ[+?≤⌠ω+⇔→+−,β∑•→+√[∑γ→⇒γβ+β⇒+∆+βθ<+
/IUU4U["V9U
ymoSvU?4"TUcvoFaSsYmoSvUyoUuyodvUSaMsYsocamiUmodsF+δβ⇔+,⇒γ[+E+βθ<+
/ U4U/IUHU*CYsmUSsUimUcmY%mUs(dsYomUmcdymoSvUSsUIUm+E+βθγ[+βθ<+
/
EBU4U["V9UUHU*5[ULU"V2,U4UIU["V6UUUUmodsFUSsUEU
=⇒+]→θβ+→+⇔[+→⇒γβ∑([∑+θβ+[]θβ∑)→+γ→](⇒+′,β+√,→⇒δ[U?4:TUimUMysYtmUcvYdmodsUdasosUyoUsMscdvTU
uvFada vTUgUcymoSvU?4"Uos%mda víU3vYUivUmodsYavYUFsUSsSycsUlysUimUMysYtmUcvYdmodsUSsUIUmUETUFveYsU
mi%#oU uyodvU !meYCU SsU umFmYU uvYU csYv0U soU cvoFscysocamTU imU SaFdmocamU SvoSsU vcyYYsU ivU modsYavYTU imU
−,⇒√(⇒+δβ+⇔→+−,β∑•→+√[∑γ→⇒γβ+θβ+(2,→⇔→+→+√β∑[ω+→∑→+δβγβ∑(⇒→∑+βθ[+βθ<+
U
3→+ −,⇒√(⇒+ δβ+ ⇔→+ −,β∑•→+ √[∑γ→⇒γβ+ (2,→⇔→δ→+ →+ √β∑[+ βθ<++++++++++++
:2V+
9
2
=−
L
WXWL
UUU4aUUUXU4U:í7JJUU"U
X
2/3X
R
VA
A
Y

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  100  -U
U
IomiatmoSvUSsUa%ymiUómosYmUumYmUsiUóvósodvUMis(avom⇒γβ<+





=∴=
=∴=
=∴=
⇒=−=
::
:9;í:7JJí:
í:
:
99
2
6
MX
WLMLX
MLX
L
WX
X
WL
M
U
Y∑→−(√→⇒δ[+γβ⇒β[θ<+
BA C
(+)
(-)
V=0
WL/6
-WL/3
V=0
X=0 X=0.577L X=L
BA C
M=0
0.0064WL
(+)
M=0
2
X=0 X=0.577L X=L
U
CONCLUSIÓN:+3→+)→∑(→√(⇒+δβ⇔+δ(→2∑→→+γ→⇒γ[+δβ+⇔→+−,β∑•→+√[∑γ→odsUcvóvUSsiUóvósodvUMis(avomodsU
FvoUFaóaimYsFTUaoSsusoSasodsósodsUSsiUmoCiaFaFUlysUFsUmevYSs0UsFUSscaYTUSsiUmoCiaFaFUMyocavomiUvUsiUmoCiaFaFU
∑β⇔→√([⇒→⇔⋅+Τγ∑[+→θβ√γ[+([∑γ→⇒γβ+→+θβC→⇔→∑<+√,→⇒SvUimUcmY%mUs(dsYomUsFUyoaMvYósósodsUYsumYdaSmTUimU
−,⇒√(⇒+δβ+⇔→+−,β∑•→+√[∑γ→⇒γβ+βθ+,⇒→+⇔ ⇒β→+∑β√γ→+β+(⇒√⇔(⇒→δ→+∫+⇔→+−,⇒√(⇒+δβ⇔+[β⇒γ[+−⇔β([⇒→⇒γβ+βθ+,omU
i'osmUcyY mUSsUFs%yoSvU%YmSv0UmFaóaFóvTUcymoSvUFveYβ+⇔→+βθγ∑,√γ,∑→+βθγ→+→√γ,→⇒δ[+,⇒→+√→∑2→+,⇒γ,→⇔ω+θdmU
%sosYmU yoU cmóeavU eYyFcvU SsU imU MysYtmU cvYdmodsU gU Mao→⇔β⇒γβω+ β⇔+ [β⇒γ[+ β⇒+ ∆ω+ γ→](⇒+ 2β⇒β∑→+ ,⇒+
cmóeavUeYyFcvUSsiUóvósodvUMis(avomodsíU
→∑→+[δβ∑+∑β→⇔(•→∑+,⇒+→⇒∴⇔(θ(θ+,⇒+[√[+→θ+βγδ(√vUuimodsmóvFUmiUiscdvYTUyoUYscsdmYavUumYmUimU
∑βθ[⇔,√(⇒+δβ+)(2→θ+(θ[θγ∴γ(√→θω+(θ[+′,β+θβ∑∴+β⇔(√→δ[+→+√[⇒γ(⇒,→√(⇒⋅+
Ejercicio #3
∞β+(δβ+→⇒→⇔(•→∑+⇔→+θ(2,(β⇒γβ+βθγ∑,√γ,∑→+[∑+β⇔+γvSvUSsUFsccavosFUvUmoCiaFaFUMyocavomiUryodmósodsU
√[⇒+β⇔+γ[δ[+∑β⇔→√([⇒→⇔⋅++

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  101  -U
U
A B
1 t/m
C
0.8 t.m
D
1.5 t
0.5 t1 t
1,5213
U
Paso 1)U1smiatmóvFUsiUcCicyivUSsUYsmccavosFUSsUmuvgvTUusYvUFvivUumYmUsMscdvFUSsUcmicyivUYsmiatmóvFU
siUcmóeavUSsUimUcmY%mUSaFdYaeyaSmUmUcmY%mUuyodymiíU
C
0.8 t.m
D
1.5 t
0.5 t1 t
VBVA
HA
R=3.0 t
U
U
1U4U6L+U4aUU14+U
tU
7í+:7í+: =⇒=−⇒=
∑ AAHHH
tVVM
BBA+=í2:,7íJ*7í:,*9=í:,;*+,7í+,*6*: =⇒=+−++⇒=
∑
tVVM
AAB62í2:,7í+*7í:=í:,2*+,7í;*6,*9: =⇒=++−−⇒= ∑ U
>DP1>"UU
:7í:+=í2+662í2: =−+−−⇒=
∑V U
Paso 2)+β→⇔(•→[θ+⇔→+[]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+βθ−,βYtvFUaodsYovFU
S S S S
C
V =2.32 t V =2.18 t
H =1.5 t
A
A
B
?f
?f
?
?
1 2 4 3
1 t/m 0.8 t.m
D
1.5 t
0.5 t1 t
U
=⇔+{3+′,β+θβ+,βθγ∑→+β⇒+β⇔+→θ[++∑β→⇔(•→[θ+β⇔+√→óeavUSsUimUcmY%mUSaFdYaeyaSmUmUcmY%mFU
uyodymisFTUusYvUFvivUumYmUsMscdvFUSsUMmcaiaSmSUSsUc∴⇔√,⇔[ω+→∑→+⇔→+[]γβ⇒√(⇒+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θ+βθ+
oscsFmYavUdYmemrmYUcvoUimUcmY%mUSaFdYaeyaSmíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  102  -U
U
noU sFdsU FsU !moU aodYvSycaSvU ;U FsccavosFU umYmU FyU moCiaFaFTU gU mU SaMsYsocamU SsU ivFU srsYcacavFU
modsYavYsFUsFdsTUFsUmomiatmYmUSsUmóevFUimSvFUSsUimU a%mUgUFsUYsmiatmYmUyoUcvodYviUsoUsiUoySvUaodsYósSavU
íUumYmUivUcymiUaodYvSycaóvFUyoUsrsUSsUcvvYSsomSmFU(UIUgUlysUaYmUSsUatlyasYSmUmUSsYsc!mTUgUvdYvU(BUIU
gBTUlysUaYmUSsUSsYsc!mUmUatlyasYSmíU
SECCIÓN S
1 +++ΤFDΤ+
⇔
U:U`Ux `U6U
Normal   UDUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para 7í+:
7í+6
7í+ U
CortanteUQUU



−=⇒=
=⇒=
⇒⋅−⇒
tQx
tQx
Parax 9=í:6
62í2:
+62í2 U
Momento FUU



⋅=⇒=
⋅=⇒=
⇒⋅⋅−⋅⇒
mtQx
mtQx
Para
x
xx ;9í26
::
2
+62í2 U
1 t/m
V =2.32 t
H =1.5 t
A
A
?
1
Q
N
M
U
SECCIÓN S2 +++ΤFDΤ+
⇔
U6U`Ux `U;U
Normal   UDUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para 7í+;
7í+6
7í+ U
CortanteUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para 9=í:;
9=í:6
662í2 U
Momento FUU



⋅=⇒=
⋅=⇒=
⇒−−⋅⇒
mtMx
mtMx
Paraxx J=í+;
;9í26
,7í+*662í2 U
V =2.32 t
H =1.5 t
A
A
?
2
R=3.0 t

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  103  -U
U
SECCIÓN S3 +++ΤFDΤ+
⇔
U:U`Ux’ `U+í7U
Normal   UDUU



−=⇒=
−=⇒=
⇒−⇒
tNx
tNx
Para 7í+7í+f
7í+:f
7í+ UU
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para 7í:7í+f
7í::f
7í: U
Momento FUU



⋅−=⇒=
⋅=⇒=
⇒−⇒
mtMx
mtMx
Parax J7í:7í+f
::f
f7í: U
D
1.5 t
0.5 t
S
? f
3
Q
N
M
U
SECCIÓN S4 +++ΤFDΤ+
⇔
U+í7U`Ux’ `U6í7U
U
0.8 t.m
D
1.5 t
0.5 t
S
V =2.18 t
B
?f
4
Q
N
M
U
Normal   UDUU



−=⇒=
−=⇒=
⇒−⇒
tNx
tNx
Para 7í+7í6f
7í+7í+f
7í+ UU
CortanteUUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para 9=í+7í6f
9=í+7í+f
+=í27í: U
Momento FUU



⋅=⇒=
⋅−=⇒=
⇒−+−−⇒
mtMx
mtMx
ParaXx J=:í+7í6f
77í+7í+f
,7í+f*+=í2=í:f7í: U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  104  -U
U
Paso 3)U1smiatmóvFUsiUDIAGRAMA DE ESFUERZOS INTERNOS U
1 t/m 0.8 t.m
D
1.5 t
0.5 t1 t
C
V =2.32 t V =2.18 t
H =1.5 t
A
A
B
N
Q
M
1.5 t
1.5 t
1.5 t
1.5 t
1.5 t
2.69 t.m
2.46 t.m
1.78 t.m
1.55 t.m
0.75 t.m
(-)
(-)
(+)
(+)
(+)
(-)
U

Paso 4)
+ β∑(−(√→√(⇒+ δβ+ ⇔[θ+ βθ−,β∑•[θ+ βγ∑β[θ+ θβ20⇒+ →+ ⇔[θ+ γ[δ[θ+ ∫→+ β⇒√([⇒→δ[θ+ β⇒+ β⇔+
uYvcsSaóasodvUSsUcCicyivíU
DUU
tMin
tMax7í+
:−=
=
⇒ UUU QUU
tMin
tMax9=í+
62í2−=
=
⇒ U UUUUUUFUU
tMin
tMax77í+
9<í2−=
=
⇒ U
Ejercicio #4
∞β+ (δβ+ →⇒→⇔(•→∑+ ⇔→+ θ(2,(β⇒γβ+ βθγ∑,√γ,∑→+ [∑+ β⇔+ γvSvU SsU FsccavosFU vU moCiaFaFU MyocavomiU
α,⇒γ→β⇒γβ+√[⇒+β⇔+γ[δ[+∑β⇔→√([⇒→⇔⋅++
2 t/m 1 t
A B
C D
E F
2 1 2 1 1 1
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  105  -U
U
Paso 1)++{U3{53Τ+=+=∆{{ΤD=∞+=+∆Τ@Τ⋅+
U
V
A
R=4 t
V
B V
E V
F
4 t
2 t 2 t
1 t
C D
V
A V
B V
E V
F
2 t 2 t
H
A
PARTE IZQUIERDA
PARTE DERECHA
1 t
C D
U
U
∆P=+AQ5=∆++
C
V
A V
B
2 t
H
A
U
U
U U U
tHH
A:: =⇒=
∑ U
tVVM
BBA6:,6,*2*,*2: =⇒=+⇒= ∑ U
U U U
tVVM
AAB+:,2,*+*,*2: −=⇒=+⇒=
∑ U
∆P=+=={G∆++
1 t
D
V
E V
F
2 t
U
U
U U U
tVVM
FFE 7í::,*2,+*+,+*2: −=⇒=−+−⇒=
∑ U
U U U
tVVM
EEF7í6:,+*+,6*2,*2: =⇒=−−⇒=
∑ U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  106  -U
U
U
1 t
C D
1 t
R=4 t
3 t 3.5 t 0.5
U
>DP1>"UU
:7í:+7í6;6+: =−−+−+−⇒=
∑V U
Paso 2)+ΤEP=D{ΤD+=+={5∆{ΤD=∞+=+=∞5=AΤ∞+DP=DΤ∞+
2 t/m 1 t
C D
1 t 3 t 3.5 t 0.5
1 2 3 6 5 4
x
x
x
x'
x'
x'
U
β+(2,→⇔+→⇒β∑→+θβ+(⇒γ∑[δ,√(∑∴+δ[θ+θ(θγβ→θ+δβ+√[[∑SsomSmFUumYmUlysUuysSmUFsYUmomiatmSvU
uvYUmóevFUimSvFTUyoUsrsUSsUcvvYSsomSmFU(UIUgUlysUaYmUSsUatlyasYSmUmUSsYsc!mTUgUvdYvU(BUIUgBTUlysUaYmU
SsUSsYsc!mUmUatlyasYSmíU
SECCIÓN S
1 +++ΤFDΤ+
⇔
U:U`Ux `U2U
U
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para :2
::
: 0U
N=0UUUumYmUdvSmFUimFUFsccavosFU
CortanteUUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para +2
+:
+ U
Momento FUU



⋅−=⇒=
⋅=⇒=
⇒−⇒
mtMx
mtMx
Parax 22
::
+ U
U
SECCIÓN S
2 +++ΤFDΤ+
⇔
U2U`Ux `U6U
CortanteUUQUU



=⇒=
=⇒=
⇒+−⇒
tQx
tQx
Para 26
22
6+ U
Momento FUU



⋅=⇒=
⋅−=⇒=
⇒−+−⇒
mtMx
mtMx
Paraxx :6
22
,2*6,*+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  107  -U
U
niU mivYUSsUM=0UsoU(46TUFsUSmUuvYUlysUFsUsocysodYmUsoUimUmYdacyimc(⇒ω+√[[+∑β√[∑δ→∑β[θ+
,⇒→+→∑γ(√,⇔→√(⇒+[√→θ([⇒→+,⇒+β√,→√(⇒+βθβ√(→⇔+δβ+⇔aesYdmSUSsUóvósodvUmUyoUSsUivFUimSvFíU
SECCIÓN S
3 +++ΤFDΤ+
⇔
U6U`Ux `U7U
CortanteUUQUU



−=⇒=
=⇒=
⇒−−+−⇒
tQx
tQx
Parax 27
26
,6*26+ U
Momento FUU



⋅=⇒=
⋅=⇒=
⇒
−
−−−+−⇒
mtMx
mtMx
Para
x
xxx :7
:6
2
,6*
,6*2,2*6+ U
U
SECCIÓN S
4 +++ΤFDΤ+
⇔
U:U`Ux’ `U+U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para 7í:+f
7í::f
7í: U
Momento FUU



⋅−=⇒=
⋅=⇒=
⇒−⇒
mtMx
mtMx
Parax 7í:+f
::f
f7í: U
U
SECCIÓN S
5 +++ΤFDΤ+
⇔
U+U`Ux’ `U2U
CortanteUUQUU



=⇒=
=⇒=
⇒+⇒
tQx
tQx
Para 7í+2f
7í++f
+7í: U
Momento FUU



⋅−=⇒=
⋅−=⇒=
⇒−−−⇒
mtMx
mtMx
Paraxx 22f
7í:+f
,+f*+f7í: U
U
U
SECCIÓN S
6 +++ΤFDΤ+
⇔
U2U`Ux’ `U6U
CortanteUUQUU



−=⇒=
−=⇒=
⇒−+⇒
tQx
tQx
Para 26f
22f
7í6+7í: U
Momento FUU



⋅=⇒=
⋅−=⇒=
⇒−+−−−⇒
mtMx
mtMx
Paraxxx :6f
22f
,2f*7í6,+f*+f7í: U
U
U
Dys mósodsUsiU mivYUSsUF4:UcymoSvUiis%móvFUmUimUmYd(√,⇔→√(⇒⋅+

Paso 3)
UDIAGRAMA DE ESFUERZOS INTERNOS U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  108  -U
U
2 t/m 1 t
C D
1 t 3 t 3.5 t 0.5
N
Q
M
1 t
2 t
2 t
1.5 t
0.5 t
2 t.m 2 t.m
2 t.m
1 t.m
(+)
(+)
(+)
(-) (-)
(-) (-)
U
Paso 4)+β∑(−(√→√(⇒+δβ+⇔[θ+βθ−,β∑•[θ+βγ∑β[θ+θβ20⇒+→+⇔[θ+γ[δ[θ+∫→+β⇒√([⇒→δ[θ+β⇒+β⇔+
uYvcsSaóasodvUSsUcCicyivíU
DUU
tMin
tMax:
:=
=
⇒ UU QUU
tMin
tMax2
2−=
=
⇒ U UUUUUUFUU
tMin
tMax2
+−=
=
⇒ UU
Ejercicio #5
∞β+ (δβ+ →⇒→⇔(•→∑+ ⇔→+ θ(2,(β⇒γβ+ βθγ∑,√γ,∑→+ [∑+ β⇔+ γvSvU SsU FsccavosFU vU moCiaFaFU MyocavomiU
α,⇒γ→β⇒γβ+√[⇒+β⇔+γ[δ[+∑β⇔→√([⇒→⇔⋅++
B
0.4 t.m
A
0.2 t/m
0.5 t/m
2 2 6
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  109  -U
U
Paso 1)++{U3{53Τ+=+=∆{{ΤD=∞+=+∆Τ@Τ⋅+
U
0.4 t.m
0.2 t/m
0.3 t/m R=0.9 t
R=1.2 t
1
2
M
V
A V
B
H
B
B
4
3
C
U
U
3mYmUMmcaiadmYUsiUcCicyivUFsumYsóvFUimUcmY%mUdYmustvaSmiUsoU2UcmY%mFUSaFdYaeyaSmFTUyomU
dYamo%yimYUgUyomUYscdmo%yimYTUimFUlysUdsoSYCoUimFUFa%yasodsFUYsFyidmodsFTUcvóvUFsUóysFdYmUsoUimUMa%yYmU
→⇒γβ∑([∑<+
UtRR<í:,6í:*,9*
2
+
++
=⇒⋅⋅= U U tRR 2í+,2í:*,9*
+2=⇒⋅= U
U U
tHH
B:: =⇒=
∑ U
U U
tVVM
AA
IZQ
C
+í::,+:,*+í:*,*;:−=⇒=+⇒=
∑ U
U U
mtMRRMM
BBB ⋅=⇒=−−+−⇒=
∑ =íJ:,2V+,*9*,9,*6V2*;í:,+:*+í::
2+
U
U U
tVVM
BB
DER
C
2í2:9=íJ2í+,2V+*6V,<í:*:=⇒=−++⇒=
∑ U
U U U U
>DP1>"UU
:2í22í+<í:+í:: =+−−−⇒=
∑V U
Paso 2)+ΤEP=D{ΤD+=+={5∆{ΤD=∞+=+=∞5=AΤ∞+DP=DΤ∞+
0.4 t.m
0.2 t/m
0.3 t/m
M
V
A V
B
H
B
B
C
S
1 S
2
S
3
X
X
X'
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  110  -U
U

SECCIÓN S1 +++ΤFDΤ+
⇔
U:U`Ux `U2U
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para :2
::
: 0U
N=0UUUumYmUdvSmFUimFUFsccavosFU
CortanteUUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para +í:2
+í::
+í: U
Momento FUU



⋅−=⇒=
⋅=⇒=
⇒−⇒
mtMx
mtMx
Parax 2í:2
:í::
+í: U
SECCIÓN S
2 +++ΤFDΤ+
⇔
U2U`Ux `U;U
CortanteUUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para +í:;
+í:2
+í: U
Momento FUU



⋅=⇒=
⋅=⇒=
⇒+−⇒
mtMx
mtMx
Parax :;
2í:2
;í:,*+í:

SECCIÓN S
3 +++ΤFDΤ+
⇔
U:U`Ux’ `U9U
CortanteUUQUU



−=⇒=
−=⇒=
⇒⋅+⋅+−⇒
tQx
tQx
ParaXx +í:9f
2í2:f
f:27í:f2í:2í2
2
U
Momento F



⋅=⇒=
⋅−=⇒=
⇒−⋅⋅⋅−⋅−⇒
mtMx
mtMx
Paraxx
x
xx
:9f
=íJ:f
=íJ,f
6
+
*f:27í:
2
f
f2í:f2í2
2
U
Dys mósodsUsiU mivYUSsUF4:UcymoSvUiis%móvFUmUimUmYd(√,⇔→√(⇒⋅+

Paso 3)
+∆Y∆F∆+=+=∞5=AΤ∞+DP=DΤ∞+
+ Τ]θβ∑)β[θ+β⇔+δ(→2∑→→+δβ+)→∑(→√(⇒+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θω+β⇒+⇔→+−(2,∑→+θ(2,(β⇒γβ<+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  111  -U
U
0.4 t.m
0.2 t/m
0.3 t/m
M
V
A V
B
H
B
B
C
N
Q
M
0.1 t
2.2 t
0.2 t.m
0.2 t.m
7.8 t.m
(-)
(-)
U
Paso 4)+ β∑(−(√→√(⇒+ δβ+ ⇔[θ+ βθ−,β∑•[θ+ βγ∑β[θ+ θβ20⇒+ →+ ⇔[θ+ γ[δ[θ+ ∫→+ β⇒√([⇒→δ[θ+ β⇒+ β⇔+
uYvcsSaóasodvUSsUcCicyivíU
DUU
tMin
tMax:
:=
=
⇒ UU QUU
tMin
tMax+í:
2í2−=
−=
⇒ UUUUUUUFUU
tMin
tMax=íJ
2íí:−=
=
⇒ U
3.8   ANÁLISIS DE PÓRTICOS
Ejercicio #6
PvómoSvU cvóvU YsMsYsocamU siU ómYcvU lysU FsU óysFdYmU soUimU Fa%yasodsU Ma%yYm0U SsdsYóaomYU imU
)→∑(→√(⇒+δβ+⇔[θ+β⇔ββ⇒γ[θ+β√∴⇒(√[θ+′,β+θβ+∑βθβ⇒γmoUsoUimUsFdYycdyYmTUlysUFvoU%sosYmSvFUuvYUimU
→√√(⇒+δβ+⇔→+√→∑2→+,⇒γ,→⇔+δβ+↑+γ[⇒⋅+∆√γ,→⇒δ[+β⇒+β⇔UuyodvU íU
4
2 2
4
A
B
C
D
E
6 t
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  112  -U
U
3.8.1 PROCEDIMIENTO DE ANÁLISIS. U
"vFUcYadsYavFUSsUmoCiaFaFUumYmUSsdsYóaomYUsiUSam%YmS+ )6 y+Ao+mof- )6 .34 6.6S6-234
óscCoacvFUlysUFsUuYsFsodmoUsoUyomUsFdYycdyYmUsoUsiUm3-26823 v63SI2Aom3 )6 (- S+Am3L 43-
FaóaimYsFU mU ivFU lysU FsU muiacmYvoU soU ivFU mumYdmSvFU 9í+í+U gU 9í+í2U YsFuscda mósods0U sFU SscaYTU
ydaiatmoSvUsiUmoCiaFaFUMyocavomiUvUsiUmoCiaFaFUYsimcavomiTUsoUsFdYycdyYmFUsoUMvYómUSsU a%mTUusYvTU
sFUaóuvYdmodsUFsCmimYUSvFU mYamodsFUómFUlysUFsUydaiot+- P+A+ 6..3a
U Primera
a 86 )61o-6 (- S+Am3 )6 A616A6-mo+ v6-6A+. 6- 6. m3ods(dvU SsU imU sFdYycdyYmTU
o-26vA+)3 P3A )34 6564 A6m2+-v(.+A64a (-3 u3Aot3-2+.a ? i 6. 32A3 y6A2om+.a @L 26-o6-)3 m3S3
vYa%soUsiUovSvUIUdmiUcvóvUFsUóysFdYmUsoUimUMa%yYmíU
U Segunda:
UniUmoCiaFaFUSsUivFUsisósodvFUlysUaods%YmoUimUsFdYycdyYmU*dYmóvUIETUdYmóvUE TU
2A+S3 ldL i 2A+S3 d0CL 46 +73A)+- m3-4o)6A+-)3 (- 4sodaSvUSsUYscvYYaSvUuvYUimUumYdsUaodsYomU
)6 .+ 642A(m2(A+ i )6 ot=(o6A)+ + )6A6mu+ )61o-oI-)visU mU cmSmU yovU SsU siivFU yoU ómYcvU SsU
YsMsYsocamUumYdacyimY0UuvYUsrsóuivTUumYmUsiUdYmóvUE TUFyUvYa%soUsFdCUivcmiatmSvUsoUsiUovSvUEU
SsUdmiUómosYmUlysUsoUsiUFsodaSvUSsUEUmU TUFsUaSsodaMacmUsiUsrsU!vYatvodmiUuvFada vU?0UgUsoUsiU
FsodaSvU sYdacmiUmUumYdaYUSsUEUsiUsrsU@íUU
3.8.2 DIAGRAMA DE CUERPO LIBRE U
D34 P6ASo26 o)6-2o1om+Aa c+4 m+Av+4 +m2oy+4 i .+4 cmY%mFU Ysmcda mFíU niU FaFdsómU SsU
MysYtmFUlysUFsUmFvcamUcvoUimUsFdYycdyYmíUniUo#ósYvUSsUimFUscymcavosFUSsUslyaiaeYavUumYmUsiU
FaFdsómUSsUMysYtmFUSsdsYóaomSvíUniUo#ósYvUSsUimFUaomfv-o2+4 =(6 m3-2o6-6 6. 4o426S+n c+4
cvoSacavosFUSsUaFvFdmdacaSmSUumYmUSsdsYóaomYUsiUdauvUSsUsFdYycdyYmUlysUFsUsFdCUmomiatmoSvíU
U Cargas activas:
8sUaSsodaMacmU#oacmósodsUimUcmY%mUuyodymiUSsU9UdvoíUmcdymoSvUsoUsiUuyodvU
íU
U Cargas reactivas:
U nFdCoU YsuYsFsodmSmFU uvYU imFU YsmccavosFU soU cmSmU yovU SsU ivFU muvgvF0U
6423 64a 6- e 6. +P3i3 Sfyo. =(6 m3-2o6-6 (-+ m3SP3-6-26 A6+m2oy+ 6- .+ )oA6mmof- y6A2om+.
1IíU noU nTU siU muvgvU MarvTU lysU cvodasosU SvFU cvóuvosod64 A6+m2oy+40 (-+L 6- .+ )oA6mmof-
y6A2om+. i .+ 32A+L 6- .+ )oA6mmof- u3Aot3-2+.L P6A3L )+)3 =(6 -3 68o426 m+Av+ 6826A-+ 6- I42+
)oA6mmof-L .+ m3SP3-6-26 64 ov(+. + m6A3n
U 8o426S+ )6 1(6At+4 437A6 .+ 642A(m2(A+a 3+A+.6.3 6-UsiUuimovíUU
U Número de ecuaciones que contiene el sistema: d340 4(S+ )6 1(6At+4 y6A2om+.64 i
FyómUSsUóvósodvFíUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  113  -U
U
U Número de incógnitas en el sistema:+[θ+⇔→+∑β→√√(⇒+)β∑γ(√→⇔+β⇒+∆ω<+∆ω+∫+⇔→+∑β→√√(⇒U
)β∑γ(√→⇔+β⇒+=<+=⋅++
U Condiciones de isostaticidad:
2,→⇔βθ+βθγ[+βθ<+θ(+β⇔+⇒0β∑[+δβ+(⇒√2⇒(γ→θ++≤+ωU gUsiU
⇒0β∑[+δβ+β√,→√([⇒βθ+=+≤+ω+β⇒+√[⇒θβ√,β⇒√(→<++≤+=UuvYUivUdmodvUFsUdYmdmUSsUyomUsFdYycdyYmU
aFvFdCdacmíUU
PvómoSvUcvóvUYsMsYsocamUivUmodsYavYTUsiUSam%YmómUSs+√,β∑[+⇔(]∑β+δβ+⇔→+βθγ∑,√γ,∑→+βθ<++
B
C
D
V
A V
E
x
6 t
U
Paso 1)UCálculo de reacciones:U IuiacmoSvU imFU cvoSacavosFU SsU slyaiaeYavU FveYsU siU Sam%YmómU SsU
√,β∑[+⇔(]∑β<++
tVVM
EEA6:,*;,9*2: =⇒=−⇒=
∑ U
mtVVM
AAE ⋅=⇒=−−⇒= ∑ 6:;,9*2: U
Paso 2) Análisis de los elementos estructurales.
Análisis del tramo AB:
U PvómoSvU cvóvU YsMsYsocamU siU moCiaFaFU MyocavomiTU imFU MyocavosFU SsU ivFU
sisósodvFU óscCoacvFU FsU vedasosoU muiacmoSvU FveYsU siUSam%YmómU SsU cysYuvU iaeYsU SsiU dYmóvTU imFU
cvoSacavosFUSsUslyaiaeYavUSsiUFaFdsómUSsUMysYtmFUlysUFsUisUmFvcamUsoUFyUcvoryodvíU3mYmUSsóvFdYmYUivU
modsYavYUdvósóvFUcvóvUYsMsYsocamUimUMa%yYmUlysUmUcv⇒γ(⇒,→√(⇒+θβ+,βθγ∑→<++

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  114  -U
U
TRAMO A-B +++ΤFDΤ+
⇔
U:U`Ux `U;U
U

Normal
UDUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para 6;
6:
6 U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para :;
::
: U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⇒
mtMx
mtMx
Para :;
::
: U
U
U
Análisis del tramo BC:U3mYmUsiUsMscdvTUdvómYsóvFUcvóvUYsMsYsocamUimUsFdYy√γ,∑→+β′,(⇔(]∑→δ→<++
niUSam%YmómUSsUcysYuvUiaeYsUumYmUsiUdYmóvUE UFsUóysθγ∑→+→+√[⇒γ(⇒,→√(⇒<++
voFaSsYmoSvUsiUómYcvUSsUYsMsYsocamUumYdacyimYUumYm+β⇔+γ∑→[+E{+∫+√[⇒+[∑(2β⇒+β⇒+β⇔+,⇒γ[+E<+
siUFaFdsómUSsUMysYtmFUlysUFsUaods%YmUsoUsiUSam%Ymóm+δβ+√,β∑[+⇔(]∑β+βθ<+
paralelo en la dirección del
eje Y incluyendo pares
íU
PvómoSvU cvóvU YsMsYsocamU siU Sam%YmómU SsU cysYuvU iaeYsω+ ∫+ →⇔(√→⇒δ[+ θ[]∑β+ θγβ+ ⇔→θ+
cvoSacavosFUSsUslyaiaeYavUsFdCdacvUumYmUyoUFaFdsómUSsUMysYtmFUumYmisivTUFsUvedasosoUimFUMyocavosFUSsU
ivFUsisósodvFUóscCoacvFUlysUFsUuYsFsodmoUsoUsiUdYmó[+E{ω+′,β+→⇔+θ,θγ(γ,(∑+β⇒+θγ→θ+⇔[θ+→∑∴βγ∑[θ+δβ+
imUSaFdmocamU?TUYsFyidmUsoUcvoFscysocamUimUóm%oadySUSsUcmSmUyovUSsUivFUsisósodvFUóscCoacvFUlysUFsU
∑βθβ⇒γ→⇒+β⇒+β⇔+γ∑→[<++
TRAMO B-C
+++ΤFDΤ+
⇔
U:U`Ux `U2U

Normal
UDUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para :2
::
: U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para 62
6:
6 U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⋅⇒
mtMx
mtMx
Parax 92
::
6 U

B
3 t
x
y
X
4
2
Q
N
M
B
C
3 t
x
X

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  115  -U
U
Análisis del tramo CD:+=⇔+δ(→2∑→→+δβ+√,β∑[+⇔(]∑β+→∑→+β⇔+γ∑→[+{+βθ<++
Función de los elementos mecánicos: 8aUsiUSam%YmómUSsUcysYuvUiaeYsUaods%YmUyoUFaFdsómUSsUMysYtmFU
umYmisivUsoUsiUuimovUaociygsoSvUumYsFTUimFUMyocavosFUSsUimUMysYtmUcvYdmodsUgUSsiUóvósodvUMis(avomodsU
FsUSsdsYóaomoUmuiacmoSvUimFUcvoSacavosFUSsUslyaiaeYavUFyFdadygsoSvUsoUimFUMyocavosFUYsFuscda mFTUivFU
→∑∴βγ∑[θ+δβ+⇔→+δ(θγ→⇒√(→+?<++
TRAMO C-D
++++ΤFDΤ+
⇔
U2U`U(U`U;U
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para :;
:2
: U
CortanteUUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para 6;
62
6 U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒+⋅−⇒
mtMx
mtMx
Parax :;
92
+26 U
(x-2)
x
Q
N
M
B
C
3 t
X
6 t
U
Análisis del tramo DE:U Dys mósodsU dvómYsóvFU cvóvU YsMsYsocamU siU Sam%YmómU SsU imU sFdYycdyYmU
β′,(⇔(]∑→δ→ω+ →∑→+ ′,β+ →+ →∑γ(∑+ δβ+ θγ→ω+ θβ+ δβ−(⇒→+ ,⇒+ δ(→2∑→→+ δβ+ √,β∑[+ ⇔(]∑β+ δβ⇔+ γ∑→[+ βθγ[+ βθ<+
Funciones de los elementos mecánicos:U voFaSsYmoSvU lysU siU Sam%YmómU SsU cysYuvU iaeYsU lysU FsU
vedy vUsFdmUaods%YmSvUuvYUyoUFaFdsómUSsUMysYtmFUumYmisivUsoUsiUuimovUlysUaociygsUumYsFTUimFUMyocavosFU
SsUivFUsisósodvFUóscCoacvFTUFsUvedasosoUmiU muiacmYUimFU cvoSacavosFUSsUslyaiaeYavíU voUemFsUsoUivU
modsYavYUgUdvómoSvUcvóvUYsMsYsocamUivFUumYCósdYvFUSsUimUSaFdmocamU?TUimUóm%oadySUSsUivFUsisósodvFU
óscCoacvFUFsUSsdsYóaomoUmiUFyFdadyaYUYsFuscda mósodβ+β⇒+⇔→θ+−,⇒√([⇒βθ+⇔→+δ(θγ→⇒√(→+?<+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  116  -U
U
TRAMO D-E +++ΤFDΤ+
⇔
U:U`Ux `U;U
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para :;
::
: U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para 6;
6:
6 U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⇒
mtMx
mtMx
Para :;
::
: U
Q
N
M
X
4
2 2
B
C
D
3 t
X
6 t
U
Paso 3)UDIAGRAMA DE ESFUERZOS INTERNOS: 3mYmUsiUsMscdvTUFsUdvómUcvóvUYsMsYsocamUivFU
mivYsFUmovdmSvFUsoUivFUmoCiaFaFUSsUcmSmUdYmóvíU3mY→+γ→⇔+∑β→⇔(•→∑β[θ+βθγ[θ+δ(→2∑→→θ+β⇒+β⇔+∑γ(√[+
uvYUFsumYmSvíU
Diagrama de variación de la fuerza normal:
(-) (-)
E
D3 t 3 t
4
B
C
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  117  -U
U
Diagrama de variación de la fuerza cortante:

A E
D
3 t
3 t
B C
4
2 2
4
(+)
(-)
U
Diagrama de variación del momento flexionante:
A E
D
6 t.m
B C
4
2
4
0 t.m 0 t.m
(+)
2
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  118  -U
U
Ejercicio #7
PvómoSvU cvóvU YsMsYsocamU siU ómYcvU lysU FsU óysFdYmU soUimU Fa%yasodsU Ma%yYm0U SsdsYóaomYU imU
MidYilY2aOtPOr'uOPrPcPaR'uOcPlxaYl'uOnXPOuPOódPuPaR”U IU D” IgP5T-PT5”0
2 t
1 t/m
3
3
4
A
B
C D
E
1
U
Procedimiento de análisis:U IuiacmYsóvFU siU moCiaFaFU YsimcavomiU umYmU SsdsYóaomYU ivFU sisósodvFU
óscCoacvFíUU
Primero:U 8sU cvoFdYygsU yoU Sam%YmómU SsU cysYuvU iaeYsU SsU imU sFdYycdyYmTU umYmU aSsodaMacmYU imFU cmY%mFU
mcda mFUgUimFUcmY%mFUYsmcda mFTUSsUdmiUMvYómUlysUovFUusYóadmUaSsodaMacmYUsiUFaFdsómUSsUMysYtmFUsoUFyU
cvoryodv0U siU o#ósYvU SsU scymcavosFU SsU slyaiaeYavU lysO uPO rPO iu'lYP9O PrO awcPd'O tPO Yal2qaYRiuO nXPO
cvodso%mU siU FaFdsómU soU %sosYmi0U imFU cvoSacavosFU SsUaFvFdmdacaSmSU umYmU sYaMacmYU FaU imU sFdYycdyYmU FsU
1TI“I ”U”DO3”5 -SaS OgSgP4PO-”0 IgPS Ig0
2 t
1 t/m
3
A
B
C D
E
VV
H
A
B
B
1
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  119  -U
U
Segundo:  CicyivU SsU YsmccavosF0U FsU vedasosoU muiacmoSvU FveYsUsiU Sam%YmómU SsU cysYuvU iaeYsU imFU
√[⇒δ(√([⇒βθ+δβ+β′,(⇔(]∑([<++
U U U
tHHH
EE762: =⇒++⇒=
∑
U
tVVM
EEA +Jí;:,*6,;*2,6*7í+: =⇒=−+⇒= ∑
U
U U U
tVVM
AAE+Jí;,*6,;*2,6*7í+: =⇒−+⇒=
∑
U
Tercero:U voUsiU mivYUSsUimFUYsmccavosFTUFsUSsMaosUimUsFdYycdyYmUslyaiaeYmSmUdmiUcvóvUFsUóysFdYmUsoUimU
θ(2,(β⇒γβ+−(2,∑→<++
2 t
1 t/m
3
3
4
A
B
C D
E
4.17
4.17
5
1
U
Cuarto:UIuiacmoSvUsiUmoCiaFaFUYsimcavomiTUgUdvómoSvUcvóvUYsMsYsocamUimUsFdYycdyYmUslyaiaeYmSmTUFsU
δβγβ∑(⇒→∑∴⇒+ ⇔[θ+ δ(→2∑→→θ+ δβ+ )→∑(→√(⇒+ δβ+ ⇔→+ −,β∑•mU ovYómiTU MysYtmU cvYdmodsU gU óvósodvU
−⇔β([⇒→⇒γβ<++
TRAMO A-B
+++ΤFDΤ+
⇔
U:U`Ux `U6U

Normal
UDUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para +Jí;6
+Jí;:
+Jí; U
CortanteUUQUU



−=⇒=
=⇒=
⇒⋅−⇒
tQx
tQx
Parax 66
::
+ U
MomentoUFUU



⋅−=⇒=
⋅=⇒=
⇒⋅⇒
mtMx
mtMx
Parax
x 7í;6
::
2
U
U
U
1 t/m
3
A
B
4.17
x
x

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  120  -U
U
Análisis del tramo BC:U3mYmUsiUsMscdvTUdvómYsóvFUcvóvUYsMsYsocamUimUsFdYy√γ,∑→+β′,(⇔(]∑→δ→<++
niUSam%YmómUSsUcysYuvUiaeYsUumYmUsiUdYmóvUE UFsUóysθγ∑→+→+√[⇒γ(⇒,→√(⇒<++
TRAMO B-C
+++ΤFDΤ+
⇔
U6U`Ux `U;U
U

Normal
UDUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para +Jí;;
+Jí;6
+Jí; U
CortanteUUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para 6;
66
6 U
MomentoUFUU



⋅−=⇒=
⋅−=⇒=
⇒−⋅⇒
mtMx
mtMx
Parax 7íJ;
7í;6
,7í+*6 U

Análisis del tramo CD:
+=⇔+δ(→2∑→→+δβ+√,β∑[+⇔(]∑β+→∑→+β⇔+γ∑→[+{+βθ<++
TRAMO C-D
+++ΤFDΤ+
⇔
U:U`Ux `U6U
Normal   UDUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para 76
7:
7 U
CortanteUUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para +Jí;6
+Jí;:
+Jí; U
MomentoUFUU



⋅−=⇒=
⋅−=⇒=
⇒−⋅−⇒
mtMx
mtMx
Parax 2:6
7íJ:
,6*7í2+Jí; U

1 t/m
3
A
B
4.17
x
2 t
1
3
B
C
3
D
x
U
1 t/m
3
A
B
4.17
x
x
2 t
1
3
B
C

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  121  -U
U
Análisis del tramo DE:
TRAMO D-E +++ΤFDΤ+
⇔
U:U`Ux `U;U
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para :;
::
: U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para 6;
6:
6 U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⇒
mtMx
mtMx
Para :;
::
: U
x
2 t
1 t/m
3
3
A
B
C D
E
4.17
1
x
U
DIAGRAMA DE ESFUERZOS INTERNOS: 3mYmUsiUsMscdvTUFsUdvómUcvóvUYsMsYsocamUivFU mivYsFU
movdmSvFU soU ivFU moCiaFaFU SsU cmSmU dYmóvíU 3mYmU dmiU U Yβ→⇔(•→∑β[θ+ βθγ[θ+ δ(→2∑→→θ+ β⇒+ β⇔+ ∑γ(√[+ [∑+
FsumYmSvíU/sYUimFUFa%yasodsFUMa%yYmFíU
Diagrama de variación de la fuerza normal:
A
B
C D
E
5 t
4.17 t 4.17 t
0 t
0 t
0 t
(+) (+)
(-)
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  122  -U
U

Diagrama de variación de la fuerza cortante:

A
B
C D
E
4.17 t
3 t
5 t
0 t
0 t
0 t
(-)
(-)
(+)
U
Diagrama de variación del momento flexionante:
A
B
C
D
E
20 t.m
4.5 t.m
20 t.m
0 t.m
0 t.m
0 t.m
DC
(-)
(-) (-)7.5 t.m
7.5 t.m
U
8s%#oU siU uYvcsSaóasodvU gmU ósocavomSvTU sFU cvo soasodsU sFcv%sYU yoU uyodvU U aodsYósSavU SsU
l'alPaRdilY2aOtPrOPuRXtY'mOAaOPuRPOóXaR'OuPOdPirYyidiOPrOl'aRd'rOdPuóPlRYM' O2uPiOriOMPdYZYlilY2aOtPO
dvSvFUivFUsisósodvFUsFdYycdyYmisFU*DvYómiTU vYdmodsTUFvósodv,íU
U
Ejercicio #8
PvómoSvU cvóvU YsMsYsocamU siU ómYcvU lysU FsU óysFdYmU soUimU Fa%yasodsU Ma%yYm0U SsdsYóaomYU imU
MidYilY2aOtPOr'uOPrPcPaR'uOcPlxaYl'uOnXPOuPOódPuPaR”U IU D” IgP5T-PT5”0

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  123  -U
U
A
B
αααα
4
3
U
G = 3 – 3   = 0 ; Estructura ISOSTATICA
Paso 1)
U CicyivUSsUYsmccavosFUSsUmuvgvíU
U RR
V
R
H
A
B
αααα
α
4 t
0
4 t
U
1U4U2L;U4aUU14U=U
tU
:: =⇒=
∑ BHH U
tVVM
BBA;:,*;,=,*2*: =⇒=−⇒= ∑ U
tVVM
AAB=í::,2*=,*;: =⇒=−−⇒= ∑ U
UUU >DP1>"UU
:2í+;í:=í:: =−+⇒=
∑V U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  124  -U
U
3mYmUYsmiatmYUsiUmoCiaFaFUSsUimUemYYmTUSaFdYaeyaYsóvFUimUcmY%mUSaFdYaeyaSmUsoUSvFUSaYsccavosFTUsoU
⇔→+δ(∑β√√(⇒+δβ+⇔→+]→∑∑→+∫+β∑β⇒δ(√,⇔→∑+→+β⇔⇔→ω+→YmUmF'USsdsYóaomYUivFUsFMysYtvFUaodsYovFíU
EmYYmU"U mLL 7;6
22
=⇒+=⇒ U
c=9í69
;
6
dmo
=⇒=αα U
tR
V
;í9cvF==⋅=α UUUU0UUU tsenR
H =í;= =⋅=α U
mtq
R
qqR
H
H
HHH
V<9í:
7
=í;
7
L7 =⇒==⇒= U
mtq
R
qqR
V
V
VVV
V2=í+
7
;í9
7
L7 =⇒==⇒= U
Paso 2)+Τ]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θ⋅U
A
B
4 t
R
V
RH
0.96 t/m
1.28 t/m
36.86°
4 t
U
U
βθ√[[⇒β[θ+γ→](⇒+⇔→θ+∑β→√√([⇒βθ+δβ+→[∫[+β⇒+δ(∑β√√(⇒+δβ+⇔→+)(2→⋅++
UUUUUUUUUUUUUU
N = 4 * 0.6 = 2.4  t
Q = 4 * 0.8 = 3.2  t
U
U

FN
FQ
36.86°

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  125  -U
U
TRAMO A-B +++ΤFDΤ+
⇔
U:U`Ux `U7U
U
Normal   UDUU



+=⇒=
−=⇒=
⇒⋅+−⇒
tNx
tNx
Parax ;í27
;í2:
<9í:;í2 U
CortanteUUQUU



−=⇒=
=⇒=
⇒⋅−⇒
tQx
tQx
Parax 2í67
2í6:
2=í+2í6 U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⋅−⋅⇒
mtMx
mtMx
Paraxx :7
::
2
2=í+
2í6
2
U
nodvocsFUsiU
DIAGRAMA DE ESFUERZOS INTERNOS +→∑→+β⇔+γ∑→[+∆+→+E+θβ∑∴<++
A
B
4 t
R
V
RH
0.96 t/m
1.28 t/m
N
Q
M
2.4 t
2.4 t
3.2 t
3.2 t
4 t.m
(+)
(+)
(+)
(-)
(-)
U
U U
∑βθβ⇒γ→[θ+→+√[⇒γ(⇒,→√(⇒+,⇒+∑βθ,β⇒+δβ+⇔[θ+)→⇔[∑βFUs(dYsóvFUs(aFdsodsFUsoUimUemYYmíU
DUU
tMin
tMax;í2
;í2−=
=
⇒ UUU 0UUUQUU
tMin
tMax2í6
2í6−=
=
⇒ UUUUU0UUUUU UFUU
mtMin
mtMaxM:
M;=
=
⇒ U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  126  -U
U
Ejercicio # 9
8sUuaSsUmomiatmYUimUsFdYycdyYmíU
1 t
1.5 t
0,5
3
1
4
6
A B
D
C
E
F
3
0.5 t/m
1.5 t/m
U
G = 4 – 3 - 1  = 0 ; Estructura ISOSTATICA
Paso 1)
U CicyivUSsUYsmccavosFUSsUmuvgvíU
D
C
0.5 t/m
1.5 t/m
H
B
ββββ
R=1.5 t
R=9.12 t
R
V
RH
VA VB
αααα
1 t
1.5 t
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  127  -U
U



=
=
⇒= <<í:cvF
+9í:
9
+
dmoα
α
α
sen
U
βαβ=⇒=
6
7í:
dmo U
tHHH
BB :;í+:,+9í:*+2í<7í++: −=⇒=+−+⇒=
∑
tVVM
BBA6=í9:9,6,*<<í:*+2í<,7í;,*+9í:*+2í<,7í+*7í+,=*+,7*7í+: =⇒=⋅−+−++⇒= ∑
tVVM
AAB+7í;:9,7*7í+,=*+,7í;*7í+,6,*<<í:*+2í<,7í;,*+9í:*+2í<: =⇒=+++−−−⇒= ∑ U
>DP1>"UU
:+7í;<<í:L+2í<7í+6=í9: =+−−⇒=
∑V U
Paso 2)+Τ]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θ+
1.5 t/m
VA
H
B
VB
ββββ
R=1.5 t
R=9.12 t
R
V
RH
y
x
y
x
x
y
x
1 t
1.5 t
y
x
αααα
D
C
0.5 t/m
UUU
U
8sUYsmiatmUyoUcvodYviUMaomiUsoUsiUoySvU UumYmU sYaM(√→√(⇒⋅++
TRAMO A-C
UUDOMINIO
⇔
0 < x < 2.5
Normal   UDUU4U5;í+7Ut
vYdmodsUUQUU4U:UdU
FvósodvUFUU4U:UdMóU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  128  -U
U
C
1.5 t
4.15 t
4.15 t
N Q M
4 t
(-)
U
TRAMO B-D +++ΤFDΤ+
⇔
U:U`Ux `U;U
Normal   UDUU t6=í9−⇒U
CortanteUUQUU t:;í+⇒U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⋅⇒
mtMx
mtMx
Parax +9í;;
::
:;í+ U
6.38 t
1.04 t
6.38 t
6.38 t
1.04 t
1.04 t
N Q M
6.38 t
1.04 t 4.16 t.m
(-) (+)
(+)
4.16 t.m
4.16 t.m
U
TRAMO D-C +++ΤFDΤ+
⇔
U:U`Ux `U9í:=U
U
Normal   UDUU



−=⇒=
−=⇒=
⇒−⇒
tNx
tNx
Para :7í2:=í9
:7í2:
:7í2 U
CortanteUUQUU



=⇒=
−=⇒=
⇒⋅+−⇒
tQx
tQx
Parax <Jí2:=í9
+7í9:
7í++7í9 U
MomentoUFUU



⋅=⇒=
⋅−=⇒=
⇒⋅−⋅+−⇒
mtMx
mtMx
Para
x
x 27í7:=í9
+9í;:
2
7í++7í9+9í;
2
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  129  -U
U
1.04 t
4.16 t.m
C
1.5 t
6.38 t
1.04 t
4.16 t.m
2.05 t
4.16 t
4.16 t.m
N
Q
M
(-)
(-)
(-)
(+)
(+)
2.05 t
6.15 t
2.97 t
8.45 t.m
5.25 t.m
4.16 t.m
4,33
2.05 t
2.97 t
5.25 t.m
6.38 t
U
TRAMO F-E +++ΤFDΤ+
⇔
U:U`Ux `U6U
U
Normal   UDUU



−=⇒=
=⇒=
⇒⋅−⇒
tNx
tNx
Parasenx 2;í:6
::
,*7í:
β U
CortanteUUQUU



=⇒=
=⇒=
⇒⋅⇒
tQx
tQx
Parax ;=í+6
::
7í: U
MomentoUFUU



⋅−=⇒=
⋅=⇒=
⇒⋅−⇒
mtMx
mtMx
Para
x 27í26
::
2
7í:
2
U
0.5 t/m
1 t
E
F
N
Q
M
0.24 t
1.48 t
2.25 t.m
(-)
(-)
(+)
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  130  -U
U

TRAMO E-C +++ΤFDΤ+
⇔
U:U`Ux `U6U
U
Normal   UDUU t7í+−⇒U
CortanteUUQUU t+⇒U
MomentoUFUU



⋅−=⇒=
⋅−=⇒=
⇒⋅−−⇒
mtMx
mtMx
Parax 27í76
27í2:
+27í2 U
C
1 t
1.5 t
1.5 t
2.25 t.m
1 t
1.5 t
2.25 t.m
2.25 t.m
1.5 t
1 t
1 t
1.5 t
5.25 t.m
N Q M1 t
1.5 t 5.25 t.m
2.25 t.m
(-)
(-)
(+)
U
Paso 3)+β∑(−(√→√(⇒+β⇒+β⇔+⇒,δ[+δβ+√[⇒γ∑[⇔ω+⇒,δ[+{<+
C
1.5 t
4.15 t
5.25 t.m
1.5 t
1 t
2.5 t
2.65 t
5.25 t.m
U
U
∑=:HU0UU∑=:VU0UUU∑=:MU
LA ESTRUCTURA ESTA EN EQUILIBRIO.

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  131  -U
U
Ejercicio # 10
8sUuaSsUmomiatmYUimUsFdYycdyYmíU
A B
1.5 t/m
0.5 t/m
1 t
1 t
2 t.m
3 2 2 3
4
1
1
C
E
G
F
D
U
G = 4 - 3 – 1   = 0 ; Estructura ISOSTATICA
Paso 1)
U CicyivUSsUYsmccavosFUSsUmuvgvíU 1 t
1 t
2 t.m
C D
1.5 t
2 t 6 t
H
B
VB
H
A
VA
2.5 t
E
G
F
U
tHHHHH
ABBA7í<:++97í+: −−=⇒=−−−−+⇒= ∑
tVVM
BBA =í+:+:,2*92,6*+,7í*+: −=⇒=⋅−−+−−⇒= ∑ U
tVVM
AAB=í6:2,2*9,6*+,7*+,=*7í2,*+:: =⇒=+−−−−⇒= ∑ U
∑ =⇒=⋅−⋅⇒= tHHV
AAA
IZQ
C M
=7í2:;6:U
UU
tH
B97í9=⇒U
>DP1>"UU
:=í+2=í6: =−−⇒=
∑V U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  132  -U
U
Paso 2)+Τ]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θ+
A B
1.5 t/m
0.5 t/m
1 t
1 t
2 t.m
6.65 t
1.8 t
2.85 t
3.8 t
y
x
y
x
y
x
y
x
y
x
y
x
UUU
vóvUgmU!sóvFUósocavomSvUYsmiatmYsóvFUyoUmoCiaFaFUuvYUFsccavosFTUYsmiatmoSvUyoUcvodYviUMaomiUsoU
,⇒+⇒,δ[ω+⇒,δ[+{+→∑→+)β∑(−(√→√(⇒⋅++
TRAMO A-C
UU8sUYsmiatmYmoU2UFsccavosFíU
DOMINIO
⇔
0 < x < 5
Normal   UDUU



−=⇒=
−=⇒=
⇒⋅+−⇒
tNx
tNx
Parax 27í27
J7í;:
7í:J7í; U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para :7
::
: U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⇒
mtMx
mtMx
Para :7
::
: U
nodvocsFUsiU
DIAGRAMA DE ESFUERZOS INTERNOS +→∑→+β⇔+γ∑→[+∆+→+{+θβ∑∴<++
A
0.5 t/m
x
2.85 t
3.8 t
2.85 t
3.8 t
2.85 t
3.8 t
4.75 t
0 t
2.25 t
0 t
0 t.m
4.75 t
2.25 t
(-)
N
Q
M
1.35 t
1.8 t
C
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  133  -U
U

TRAMO B-D UU8sUYsmiatmYmoU2UFsccavosFíU
mtq
mtq
tP
tsenP
VJ2í:
7
9í6
V<9í:
7
=í;
9í6,9í:*9cvF9
=í;,=í:*99
+
+
==
==
===
===
α
α
U
2 t.m
1.8 t
4.24 t
5.43 t
1.83 t
N
Q
M
0.96 t/m
x
0.72 t/m
1.8 t
1.8 t
6 t
P
P
1
6.65 t
6.65 t
5.43 t
0.65 t
1.8 t
9.2 t.m
9.2 t.m
1.83 t
0.56 t
9.2 t.m
1.8 t
0.65 t
1.5 t/m
2.25 t
4.24 t
9.2 t.m
9.36 t.m
4.42
B
(-)
(+)
(+)
(+)
6.65 t
D
U
DOMINIO
⇔ 0 < x < 5
Normal   UDUU



=⇒=
=⇒=
⇒⋅−⇒
tNx
tNx
Parax
=6í+7
;6í7:
J2í:;6í7 U
CortanteUUQUU



=⇒=
−=⇒=
⇒⋅+−⇒
tQx
tQx
Parax
79í:7
2;í;:
<9í:2;í; U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒−⇒
mtMx
mtMx
Para
x
2í<7
::
2
<9í:2;í;
2
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  134  -U
U
TRAMO D-E UU8sUYsmiatmYmoU2UFsccavosFíU
2 t.m
9.2 t.m
1.8 t
0.65 t
D
7.2 t.m
0.65 t
1.8 t
7.2 t.m
0.65 t
1.8 t
3.6 t.m
0.65 t
1.8 t
E
N
Q
M
(+)
(+)
(+)
1.8 t
0.65 t
7.2 t.m
3.6 t.m
3.6 t.m
0.65 t
1.8 t

DOMINIO
⇔
0 < x < 2
Normal   UDUU t97í:−⇒U
CortanteUUQUU
t=í+⇒U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒+−⇒
mtMx
mtMx
Para
9í62
2íJ:
2íJ=í+ U
TRAMO F-E
UU8sUYsmiatmYmoU2UFsccavosFíU
F
1 t
N Q M1 t
1 t.m
x
(-)
(+)
1 t
1 t.m
E
U
DOMINIO
⇔
0 < x < 1
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tNx
tNx
Para
:+
::
: U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  135  -U
U
CortanteUUQUU



−=⇒=
−=⇒=
⇒−⇒
tQx
tQx
Para
++
+:
+ U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⇒
mtMx
mtMx
Parax
++
::
+ U
TRAMO G-E
UU8sUYsmiatmYmoU2UFsccavosFíU
1 t N Q M1 t
1 t.m
x
(-)
(+)
1 t
1 t.m
E
G

DOMINIO
⇔
0 < x < 1
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tNx
tNx
Para
:+
::
: U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para
++
+:
+ U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⇒
mtMx
mtMx
Parax
++
::
+ U
TRAMO E-C
UU8sUYsmiatmYmoU2UFsccavosFíU
U
E
3.6 t.m
0.65 t
1.8 t
1 t
1 t.m
1 t
1 t.m
C
3.6 t.m
1.8 t
1.35 t
3.6 t.m
1.35 t
1.8 t
N
Q
M
(+)
(+)
(-)
3.6 t.m
1.8 t
1.35 t
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  136  -U
U
DOMINIO
⇔
0 < x < 2
Normal   UDUU 67í+−⇒Ut
CortanteUUQUU
=í+⇒Ut
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒+⋅−⇒
mtMx
mtMx
Parax
:2
9í6:
9í6=í+ U
Paso 3)+β∑(−(√→√(⇒+β⇒+β⇔+⇒,δ[+δβ+√[⇒γ∑[⇔ω+⇒,δ[+{<+
U
C
1.35 t
1.8 t
1.35 t
1.8 t
0 t.m
0 t.m
U
U
∑=:HU0UU∑=:VU0UUU∑=:MU
UUUUUUUUUUUU
=⇔+D5Τ+∞=+=D{5=DP∆+=D+=Q53EΤ+
"ys%vU SsiU moCiaFaFU YsmiatmSvTU uvSsóvFU mMaYómYU lysU
la ESTRUCTURA ESTA EN
EQUILIBRIO
íU
→∑→+γβ⇒β∑+,⇒→+√[⇒√β√(⇒+2∑→−(√→+δβ+⇔→+→⇔(√→√(⇒+∑γ(√[θ+β⇒+⇔→+)(δ→+βθγ∑,√γ,∑→⇔+)β∑+→⇒β[θ+
SsiUcmuadyivU6U
U
3.9   ANÁLISIS DE ARCOS
3.9.1 PROCEDIMIENTO DE ANÁLISIS
=⇒+θγβ+→→∑γ→δ[+→⇔(√→∑β[θ+β⇔+∑[√βδ((β⇒γ[+δβ+→⇒CiaFaFUumYmUdYmtmYUivFUSam%YmómFU
δβ+ )→∑(→√(⇒+ √[∑∑βθ[⇒δ(β⇒γβω+ √[⇒γβ⇔→⇒δ[+ →∑→+ β⇔⇔vU ivFU uYvcsSaóasodvFU SsU moCiaFaFU gmUU
ósocavomSvFU soU ivFU modsYavYsFU mumYdmSvF0U imU SsSycca⇒+ δβ+ ⇔→θ+ −,⇒√([⇒βθ+ δβ+ ⇔[θ+ β⇔ββ⇒γ[θ+
óscCoacvFUFsUYsmiatmYmUemrvUsiUcyóuiaóasodvUSsUivFUθ(2,(β⇒γβθ+→θ[θ<++

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  137  -U
U
PRIMERO.-UIUumYdaYUSsUimUsFdYycdyYmUuimodsmSmTUFsUcvoFdYygsUyoUSam%YmómUSsUcysYuvU
⇔(]∑β+→∑→+(δβ⇒γ(−(√→∑<++
U El sistema de fuerzas.-+→δ[+′,β+θβ+→⇒→⇔(•→⇒+βθγ∑,√γ,∑→θ+⇔→⇒→θω+β⇔+θ(θγβmUSsUMysYtmFUsoU
FyUcvoryodvUuvSYCUFsYUyoUFaFdsómUSsUMysYtmFUumYmisivUvUyoUFaFdsómUSsUMysYtmFU%sosYmisFUsoU
siUuimovíUU
U Las condiciones de isostaticidad.-+δβ⇒γ(−(√→δ[+β⇔+⇒0β∑[+δβ+⇔→θ+(⇒√2⇒(γ→θ+∫+β⇔+⇒0βYvU
SsUimFUscymcavosFUSsUslyaiaeYavUsFdCdacvUlysUcvodasosUsiUFaFdsómUsoUFyUcvoryodvTUFsUdsoSYCoU
sisósodvFU SsU moCiaFaFU umYmU mevYSmYU sFdYycdyYmFU cymoSvU imFU cvoSacavosFU SsU aFvFdmdacaSmSU
θβ→⇒+(2,→⇔βθ+βθγ[+βθ<++≤+=⋅++
U El tipo de estructura.-U voUsiUYsFyidmSvUmodsYavYTUsFdmYsóvFUsoUcvoSacavosFUumYmUmevYSmYU
siUmoCiaFaFUSsUyomUsFdYycdyYmUlysUFsUSsMaosUcvóvU
Isostática.UU
SEGUNDO.-UIuiacmoSvUimFUcvoSacavosFUSsUslyaiaeYavUFveYsUsiUSam%YmómUSsUcysYuvU
iaeYsTUFsUSsdsYóaomoUimFUYsmccavosFUumYmUsocvodYmYUsiUslyaiaeYavUs(dsYovíUU
TERCERO.-UIomiatmoSvUimUsFdYycdyYmUSsUatlyasYSmUmUSsYsc!mUvU acs sYFmUFs%#oUFsmU
siUcmFvUFsUaSsodaMacmoUivFUdYmóvFUlysUuYsFsodsoUcmóeavFUSsUMvYómUsoUimUcmY%mUs(dsYomTUumYmU
lysTUFveYsUcmSmUyovUSsUsiivFTUFsUcvoFdYygmUyoUSam%Y→→+δβ+√,β∑[+⇔(]∑β+(⇒δ(√→⇒δ[+⇔→+→√√(⇒+
δβ+ ⇔→θ+ √→∑2→θ+ βγβ∑⇒→θ+ ′,β+ →√γ0→⇒+ θ[]∑β+ ⇔→+ [∑√(⇒U lysU FsU momiacs,U imFU lysU FsU !meYCoU SsU
slyaiaeYmYUcvoUyoUYsFyidmodsUaodsYovTUaSsodaMacmoSv+β⇒+θγβω+⇔→θ+√[[⇒β⇒γβθ+′,β+δβ−(⇒β⇒+→+⇔[θ+
sisósodvFUóscCoacvFUlysUmcd#moUsoUsiUdYmóvUYsFuscda víUU
CUARTO.-U IuiacmoSvU imFU cvoSacavosFU SsiU slyaiaeYavU sFdCdacvUFsU SsSycsoU imFU
MyocavosFUSsUivFUsisósodvFUóscCoacvFUlysUFsUuYsFsod→⇒+β⇒+√→δ→+γ∑→[ω+∫+→⇔+θ,θγ(γ,(∑+β⇒+θγ→θ+
⇔[θ+ →∑∴βγ∑[θ+ δβ+ )→∑(→√(⇒+ δβ+ ⇔→+ δ(θγ→⇒√(→+ →⇒2,⇔→∑U ^ϕ_ω+ ∑β)(→+ δβθ√[[θ(√(⇒+ δβ+ ⇔[θ+
sisósodvFUSsUimUcmY%mUs(dsYomUvedsoasoSvUimFUóm%oadySsFUYsFuscda mFíUU
QUINTO.-U voUsiUYsFyidmSvUSsiUuyodvUmodsYavYUFsUuYvcsSsYCUmUdYmtmYUsiUSam%YmómUSsU
)→∑(→√(⇒+→∑→+√→δ→+β⇔ββ⇒γ[+β√∴⇒(√[+√[⇒θ(δβ∑→⇒δ[<UU
U Línea baseUSsUimUMysYtmUovYómiTUMysYtmUcvYdmodsUvUóvósodvUMis([⇒→⇒γβ<+∆+→∑γ(∑+δβ+,⇒+
ómYcvUSsUYsMsYsocamTUóaFóvUlysUFsYCUSsUMvYómUSsiUmY√[+β⇒+→⇒∴⇔(θ(θ+√(∑√,⇔→∑+[+→∑→]⇔(√[+
uvYUsiUcymiUFsUdYmtmUimUi'osmUemFsTUFveYsUimUlysUFs+(⇒δ(√→∑∴⇒+⇔[θ+→∑∴βγ∑[θ+δβ+)→∑(→√(⇒+δβ+
imUSaFdmocamUmo%yimYU^ϕ_0UlysUFsUYsimcavomYCoUFveYsUimUi'osmUemFsTUUsoUsiUlysUFsUivcmiatmYCoU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  138  -U
U
imFU óm%oadySsFU SsU ivFU sisósodvFU óscCoacvFU YsFuscda vFU *ovYómiTU cvYdmodsU vU óvósodvU
Mis(avomods,íUU
U Trazo de la variación de los elementos mecánicos.-  voUivFUuyodvFUsocvodYmSvFUSsiU
A64(.2+)3 +-26Ao3AL 46 (-6 P(-23 + P(-23 3726-oI-)346 .+ y+Ao+mof- )6 641(6At34 =(6 642+AB
SsMaoaSmUuvYUimFUMyocavosFUSsUivFUsisósodvFUóscCoac34 u+..+)34 46vM- .+ y+Ao+mof- )6 .+
SaFdmocamUmo%yimYUcvYYsFuvoSasodsUmiUmYcvíUU

Ejercicio # 11
d626ASo-+A i vA+1om+A .34 6.6S6-234 o-26A-34 )6. 4o%yasodsUmYcvíU
C
A B
1 t/m
2 t
U
G = 4 - 3 – 1   = 0 ; Estructura ISOSTATICA
Paso 1)
U CicyivUSsUYsmccavosFUSsUmuvgvíU
U
C
R=3 t
2 t
V
H
A
A
V
H
B
B
1,5
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  139  -U
U
U
tHHHHH
ABBA2:2: +−=⇒=−+⇒= ∑
U tVVM
BBA 27í::9,6*2,7í+*6: −=⇒=⋅−−⇒= ∑ U
tVVM
AAB27í6:,6*2,7í;*6,*9: =⇒=−−⇒= ∑ U
∑ =⇒=⋅−−−⇒= tHH
BB
DER
C M
27í::66,27í:=:U
UU
tH
AJ7í+=⇒U
>DP1>"UU
:27í:627í6: =−−⇒=
∑V U
Paso 2)+Τ]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θ+
C
1 t/m
2 t
3.25 t
1.75 t
0.25 t
0.25 t
UUU
U
noUivFUmYcvFUcaYcyimYsFUsFUuYsMsYaeisUydaiatmYUCo%yivFUumYmUSsMaoaYUimFUFsccavosFíUU
TRAMO A-C
∞→]β[θ+′,β<+
,cvF66*+
6
cvF6
cvF66
2
+
ϕ
ϕ
ϕ ϕ
⋅−⋅=
⋅=
⋅=
⋅−=
R
send
d
d
U
+ ∞→]β[θ+γ→](⇒+′,β+δβ+⇔→+δβθ√[[θ(√(⇒+δβ+−,β∑•→θ+∑βθ,⇔γ→<+
ϕϕ
ϕϕ
ϕ
ϕ
ϕ
ϕ
senF
F
F
senF
F
senF
⋅−=
⋅−=
⋅=
⋅=
⋅=
⋅=
,cvF66*
cvF,cvF66*
cvFJ7í+
J7í+
cvF27í6
27í6
9
7
;
6
2
+
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  140  -U
U
1 t/m
3.25 t
1.75 t
R
SS +
ϕ
M
N
Q
3.25 t
1.75 t
3−3cvFϕ
-2
-+
-;
-6
-7
-9
U
DOMINIO
⇔
0 < ϕ  < 90º
Normal
⇔
      ::
762=−++⇒= ∑ FFFNF
N

ϕϕϕ
2
cvF6J7í+cvF27í: −−−=∴ senN
U
Cortante
⇔
      ::
9;+=++−⇒= ∑ FFFQF
Q

ϕϕϕϕcvF6cvFJ7í+27í: sensenQ +−=∴
U
Momento
⇔
      :
2
J7í+27í6:
2
=⋅+⋅+⋅−⇒=∑
d
RddMM
S

2
,cvF66*
,6*J7í+,cvF66*27í6
2
ϕ
ϕϕ−
−−−=∴ senM U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  141  -U
U
=⇒+⇔→+θ(2,(β⇒γβ+γ→]⇔→+[θγ∑→[θ+⇔→+)→∑(→√(⇒+δβ+)→⇔vYsFUlysUs(aFdsoUmUivUimY%vUSsiUdYmóvUI5 U
θβ+∑β√[(β⇒δ→+,⇒→+)→∑(→√(⇒+√→δ→+ϖc+
Normal “N”
[t]
Cortante “Q”
[t]
Momento “M”
[t_m]
:cU 56í27U 5+íJ7U :U
+7cU 56í;<U 5:í==U 5+í:6U
6:cU 56í6;U 5:í:<U 5+í;:U
;7cU 52í<+U :í;;U 5+í2;U
9:cU 52í6<U :í9;U 5:í=:U
J7cU 5+í<7U :í7;U 5:í62U
<:cU 5+íJ7U :í27U :U
TRAMO B-D
DOMINIO
⇔
0 <UϕU+ < 90º
ϕ
M
N
Q
0.25 d
-
+
-2
0.25 t
0.25 t
0.25 t
-;
-6
+
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  142  -U
U
∞→]β[θ+′,β<+
+;
+6
+2
++
cvF27í:
27í:
27í:
cvF27í:ϕ
ϕ
ϕ
ϕ
⋅=
⋅=
⋅=
⋅=
F
senF
senF
F
U
Normal
⇔
      ::
6+=−−⇒= ∑ FFNF
N

++27í:cvF27í:ϕϕsenN +=∴
Cortante
⇔
      ::
;2=+−⇒= ∑ FFQF
Q

++cvF27í:27í:ϕϕ−=∴ senQ
Momento
⇔
      :,6*27í:,cvF66*27í::
++=⋅−⋅−+⇒= ∑ ϕϕsenMM
S

,6*27í:,cvF66*27í:
++ϕϕsenM +⋅−−=∴ U
=⇒+⇔→+θ(2,(β⇒γβ+γ→]⇔→+[θγ∑→[θ+⇔→+)→∑(→√(⇒+δβ+)→⇔vYsFUlysUs(aFdsoUmUivUimY%vUSsiUdYmóvUE5 U
θβ+∑β√[(β⇒δ→+,⇒→+)→∑(→√(⇒+√→δ→+ϖc+
Normal “N”
[t]
Cortante “Q”
[t]
Momento “M”
[t_m]
:cU :í27U :í27U :U
+7cU :í6+U :í+=U :í+JU
6:cU :í6;U :í:<U :í2JU
;7cU :í67U :U :í6+U
9:cU :í67U 5:í:<U :í2JU
J7cU :í67U 5:í+=U :í+JU
<:cU :í27U 5:í27U :U

Paso 3)
UDIAGRAMAS DE ESFUERZOS

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  143  -U
U
U ESFUERZO NORMAL
0°
15°
30°
45°
60°
75°
90°
15°
0°
15°
30°
45°
60°
75°
(-)
(+)
3.25 t
3.49 t
3.34 t
2.91 t
2.39 t
1.967 t
1.75 t
0.25 t
0.31 t
0.35 t
0.35 t
0.34 t
0.31 t
0.25 t
U
U ESFUERZO CORTANTE
0°
15°
30°
45°
60°
75°
90°
15°
0°
15°
30°
45°
60°
75°
1.75 t
0.88 t
0.09 t
0.44 t
0.64 t
0.54 t
0.25 t
0.25 t
0.18 t
0.09 t
0 t
0.09 t
0.18 t
0.25 t
(-)
(+)
(-)
U
U DIAGRAMA DE MOMENTOS
0°
15°
30°
45°
60°
75°
90°
15°
0°
15°
30°
45°
60°
75°
(-)
(-)
0 t.m
1.03 t.m
1.40 t.m
1.24 t.m
0.80 t.m
0.32 t.m
0 t.m
0.17 t.m
0.27 t.m
0.31 t.m
0.27 t.m
0.17 t.m
0 t.m
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  144  -U
U
A
B
S
dR
dR
dR
1
2
ϕ
ϕ
45°45°
2 t
Nota.-  =⇒+⇔→+)→∑(→√(⇒+δβ+⇔[θ+δ(→2∑→→θ+δβ+βθ−,β∑•[θ+(⇒γβ∑ovFU*DvYómiTU vYdmodsUgUFvósodv,UcmesU
ósocavomYUlysUovUFvoUYscdmFUimFUlysUFsU moUSaeyrmoS[+β⇒γ∑β+⇔[θ+,⇒γ[θ+δβ+)→∑(→√(⇒ω+θ[⇒+√,∑)→θ+′,β+
FsUimFUveFsY mY'mUósrvYUFaUsFUlysUsiUCo%yivUSsU mYa→√(⇒+→+⇔[+⇔→∑2[+δβ⇔+→∑√[+−,β∑→+→θ+∑βδ,√(δ[ω+[∑+
srsóuivUcmSmU+cí

Ejercicio # 12
βγβ∑(⇒→∑+∫+2∑→−(√→∑+⇔[θ+β⇔ββ⇒γ[θ+(⇒γβ∑⇒[θ+δβ⇔+θ(%yasodsUmYcvíU
U
lU4U+Ut/mU
YmSavU4U2Um
SU4U2U
2
2
4U2Um
U
U

Paso 1)
U CicyivUSsUYsmccavosFUSsUmuvgvíU
U 3Ys amósodsUFsUSsdsYóaomYmUsiU mivYUSsUimUYsFyidmodsUcvYYsFuvoSasodsUmUimUcmY%mUSaFdYaeyaSmUq
= 1 t/m
qrR
dsenRdRsendR
qrR
dRdRdR
rdqdsqdR
=
=⇒=
=
=⇒=
⋅=⋅=
∫
∫
2
<:
:
22
+
<:
:
++
cvFcvF
ϕϕϕ
ϕϕϕ
ϕ
U
+ [∑+γ→⇒γ[<+
2
2
2
+
RRR +=U
+ =⇒γ[⇒√βθ<+
tqrR =2=í22==U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  145  -U
U
+ + β→⇔(√β[θ+,⇒+{3+→∑→+γ∑→]→α→∑+√[⇒+→∫[∑+−→√(⇔(δmSU
2 t
H
A
VA
VB
2.828 t
U
U
tHHH
AA2:2: =⇒=−⇒=
∑
U tVVM
BBA
=2=í::,22*22,2*=2=í2: =⇒=+−−⇒= ∑ U
tVVM
AAB
::í2:2=2=í222,22*: =⇒=−−+⇒= ∑ U
>DP1>"UU
:=2=í:=2=í2::í2: =+−⇒=
∑V U
Paso 2)+Τ]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θ+
2 t
H
A
VA
VB
2.828 t
UUU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  146  -U
U
DOMINIO
⇔
0 < ϕ  < 45º
2 t
2 t
M
N
Q
S
S
+
ϕ
2 t
2 t
-2
-+
-;
-6
U
∞→]β[θ+′,β<+
UUUUUUUU
ϕ
ϕ
send
d
⋅=
⋅−=
2
cvF22
+
U
+ ∞→]β[θ+γ→](⇒+′,β+δβ+⇔→+δβθ√[[θ(√(⇒+δβ+−,β∑•→θ+∑βθ,⇔γ→<+
ϕ
ϕ
ϕ
ϕ
senF
F
senF
F
⋅=
⋅=
⋅=
⋅=
2
cvF2
2
cvF2
;
6
2
+
U
DOMINIO
⇔
0 < ϕ  < 45º
Normal
⇔
      ::
62=++⇒= ∑ FFNF
N

ϕϕcvF22−−=∴ senN
U
Cortante
⇔
      ::
;+=−+⇒= ∑ FFQF
Q

ϕϕsenQ2cvF2+−=∴

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  147  -U
U
Momento
⇔
      :22:
+ =−+⇒= ∑ ddMM
S

,cvF22*2;ϕϕ−+−=∴ senM U
=⇒+ ⇔→+ θ(2,(β⇒γβ+ γ→]⇔→+ [θγ∑→[θ+ ⇔→+ )→∑(→√(⇒+ δβ+ )→⇔vYsFU lysU s(aFdsoU mU ivU imY%vU SsiU dYmóvU
→⇒→⇔(•→δ[+δβ+⌠c+→+ϖc+θβ+∑β√[(β⇒δ→+,⇒→+)→∑(→√(⇒+cmSmU+7c,U
Normal “N”
[t]
Cortante “Q”
[t]
Momento “M”
[t_m]
:cU 52U 52U :U
+7cU 52í;7U 5+í;+U 5:í<U
6:cU 52íJ6U 5:íJ6U 5+í;9U
;7cU 52í=6U :U 5+í99U

DOMINIO
⇔
45 <UϕU+ < 135º
A
B
R
M
N
Q
ϕ
α
2 t
2 t
R
-+
-2
-6
-;
-7
-9
2 t
2 t
γ
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  148  -U
U
∞→]β[θ+′,β<+
c;7−=ϕα U
ISsóCF0U
ϕ
ϕsenF
F
⋅=
⋅=
2
cvF2
2
+
UUU U U UUUUU
ϕ
ϕcvF2
2
;
6
⋅=
⋅=
F
senF
U
+ →∑→+⇔→+δβθ√[[θ(√(⇒+δβ+⇔→+−,β∑•→+∑βθ,⇔γ→⇒γβ+θβ∑CíU
γ
γ
αγ
cvF
2V<:
9
7⋅=
⋅=
−=
RF
senRF
UU
+ β∑[++βθ<+
2
2
2
+
:
2
:
+
,cvF+*UUU
UcvF\
RRR
qrdsenqrR
qrsendαqrR
+=
−==
==∫
∫
ααα
α
α
α
U
Pβ⇒β[θ<+
Normal
⇔
      ::
9;2=+++⇒= ∑ FFFNF
N

,cvFcvF22*γϕϕRsenN ++−=∴ U

Cortante
⇔
      ::
76+=+−+⇒= ∑ FFFQF
Q

γϕϕRsensenQ −+−=∴2cvF2 U

Momento
⇔
      :,
2
*,cvF*22: =+−−+⇒=∑
α
ϕϕ rsenRrrrsenMM
S

=⇒+⇔→+θ(2,(β⇒γβ+γ→]⇔→+[θγ∑→[θ+⇔→+)→∑(→√(⇒+δβ+)→⇔vYsFUlysUs(aFdsoUmUivUimY%vUSsiUdYmóvU
→⇒→⇔(•→δ[+δβ+ϖc+→+φϖc+θβ+∑β√[(β⇒δ→+,⇒→+)→∑(→√(oUcmSmU+7c,U
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  149  -U
U
Normal “N”
[t]
Cortante “Q”
[t]
Momento “M”
[t_m]
;7cU 52í=6U :U 5+í99U
9:cU 52í=:U :í2+U 5+í9:U
J7cU 52íJ2U :í;+U 5+í;6U
<:cU 52í7<U :í7<U 5+í+JU
+:7cU 52í;+U :íJ2U 5:í=6U
+2:cU 52í2+U :í=:U 5:í;6U
+67cU 52U :í=6U :U

Paso 3)
UDIAGRAMAS DE ESFUERZOS
U ESFUERZO NORMAL
0°
15°
30°
45°
60°
75°
90°
135°
120°
105°
(-)
2 t
2.45 t
2.73 t
2.83 t
2.80 t
2.72 t
2.59 t
2.41 t
2.21 t
2 t
U
U ESFUERZO CORTANTE
75°
90°
135°
120°
105°
(-)
(+)
2 t
1.41 t
0.73 t
0 t
0.21 t
0.41 t
0.59 t 0.72 t
0.80 t
0.83 t
0°
15°
30°
45°
60°
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  150  -U
U
U DIAGRAMA DE MOMENTOS
0°
15°
30°
45°
60°
75°
90°
135°
120°
105°
(-)
0 t.m
0.90 t.m
1.46 t.m
1.66 t.m
1.60 t.m
1.43 t.m
1.17t.m
0.83 t.m
0.43 t.m
0 t.m
U
→∑→+γβ⇒β∑+,⇒→+√[⇒√β√(⇒+2∑→−(√→+δβ+⇔→+→⇔(√→√(⇒+mYcvFUsoUimU aSmUsFdYycdyYmiU sYUmos(vFUSsiU
cmuadyivU6U
3.10 TEORIA GENERAL DE ARCOS PARABOLICOS íU
→∑→+γβ⇒β∑+,⇒→+)(θ(⇒+→θ+√⇔→∑→+θ[]∑β+β⇔+→⇒∴⇔(θ(θ+δsUarcos parabólicosUgUvdYvFUdauvFUSsUmYcvFU
sFUuYsMsYaeisUSsFmYYviimYUyomUdsvY'mU%sosYmiTUumYmUuvSsYUMmcaiadmYUsiUdYmemrvUgUdYmemrmYUSsUómosYmUómFU
βγδ(√→⋅+
M
N
Q
xo
yo
M z
N x
Q y
ϕ
ϕ
X
Y
U
3→+ →⇒γβ∑([∑+ 2∑→−(√→+ ⇒[θ+ ,βθγ∑→+ ,⇒→+ [∑√(⇒+ δβ+ ,⇒→+casYdmU a%mU FsisccavomSmU soU imU umYdsU
SsYsc!mTUmSsóCFU
N, Q, y M  FvoUivFUsFMysYtvFUcvYYsFuvoSasodsFUmiUFaFdsómUivcmiUx o - yoUTUgUNx, Qy y MzU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  151  -U
U
FvoUivFUcvYYsFuvoSasodsFUmiUFaFdsómU%sosYmiUx – y. =⇒γ[⇒√βθ+∑β→⇔(•→⇒δ[+⇔→+∑[γ→√(⇒+√[∑∑βθ[⇒δ(β⇒γβ+
dsosóvFíU
MM
QNsenQ
QsenNN
z
y
x
=
+−=
+=
ϕϕ
ϕϕ
cvF
cvF
U
>UeasoUuysSsUFsYUs(uYsFmSmUsoUMvYómUómdYacamiíU



















 −
=










z
y
xM
Q
N
sen
sen
M
Q
N
+::
:cvF
:cvF
ϕϕ
ϕϕ U
U ISsóCFTUFaUYsmiatmóvFUsiUmoCiaFaFUSsUyoUuslysCvUFs2β⇒γ[+δ(−β∑β⇒√(→⇔ω+γβ⇒β[θ<+
d x
d y
d s
ϕ
dx
dy
ds
ϕ
U
=⇒γ[⇒√βθ<+
22
yxddds +=UUUUUUU0UUUUUU
ds
dy
sen=
ϕ UUUUUUU0UUUUUUUds
dx
=
ϕcvF U
U U U
ωϕ==
dx
dy
dmo U
3,β2[<+
UU
+
+
vFUUUUUUUU0UUUUUU
+
22
ω
ϕ
ω
ω
ϕ
+
=
+
= csen U
U "vFU mivYsFUvedsoaSvFUumYmUFso
ϕUgUcvFϕ+δβ]β⇒+θβ∑+θ,θγ(γ,(δ[θ+β⇒+⇔→+δ(θ[θ(√(⇒+→γ∑(√(→⇔⋅U
U U
→∑→+γβ⇒β∑+,⇒→+βα[∑+√[∑β⇒θ(⇒+δβ⇔+√→⇔√,⇔[+δβ+→∑√[θ+∫+⇔→+→⇔(√→√(⇒+δβ+⇔→+γβ[∑ →+2β⇒β∑→⇔+δβ+
⇔→θ+(θ→θω+⇔→⇒γβ→[θ+→+√[⇒γ(⇒,→√(⇒+,⇒→+θβ∑(β+δβ+srsóuivFUYsFysidvFíU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  152  -U
U
Ejercicio # 13
βγβ∑(⇒→∑+∫+2∑→−(√→∑+⇔[θ+β⇔ββ⇒γ[θ+(⇒γβ∑⇒[θ+δβ⇔+θ(2,(β⇒γβ+→∑√[+→∑→]⇔(√[⋅+
A
(0,0)
P
(4,7)
f(x)
0.5 t/m
1 t
U
U
Paso 1)U CicyivUSsUYsmccavosFUSsUmuvgvíU
1 t
R=3.5 t
3,5m
V
H
A
A
M
7m
4m
U
tHHH
AA 7í6:7í6:−=⇒=+⇒= ∑
U mtMMM
AAA ⋅−=⇒=++⇒= ∑ 27í+9:,;*+,7í6*7í6: U
tVVM
AAP+:,7í6*7í6,27í+9*J,7í6*;:=⇒=−−+−−⇒= ∑ U
UU >DP1>"UU
:++:=−⇒=
∑V U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  153  -U
U
Paso 2)+Τ]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θ+
P
(4,7)
f(x)
0.5 t/m
1 t
1 t
3.5 t
16.25 t.m
U
∑β)(→β⇒γβω+δβγβ∑(⇒→[θ+⇔→θ+√[⇒δ(√([⇒βθ+2β[γ∑(√mFUSsUf(x)TUsodvocsFU
::,U>*:TUU3mYmUUUUUUUU0UUUUUUU
2
=⇒++= ccbxaxy U
U 3vYUivUdmodvTU
:f;UcymoSvUUUU0UUUUU2f
2
=⇒=+=⇒+= yxbaxybxaxy U
3vYUivUdmodvU
U:=
=+ba U
U 3mYmUP (4,7), dsosóvFU
UJ;+9 =+ba U
U 3vYUivUdmodvTU
U
2
J
+9
J
2
xxy +−= U
U nodvocsFTU
U
2
J
=
J
f +−== xy
ω U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  154  -U
U
U 3vYUivUdmodvTU
22
2
J
=
J
+
2
J
=
J
+






+−+
+−
=
+
=
x
x
sen
ω
ω
ϕ
U
22
2
J
=
J
+
+
+
+
cvF






+−+
=
+
=
x
ω
ϕ
U
0.5 t/m
1 t
3.5 t
16.25 t.m
R=0.5y
M z
N x
Q y
x
g
U
β√[∑δβ[θ+′,β+⇔→+√[⇒)β⇒√(⇒+δβ+θ(2⇒[θ+β⇒+θβ⇒γ(δ[+[θ(γ()[+βθ<+
M M
N
Q
Q
N
(+)
IZQ DER
U
+ =⇒γ[⇒√βθ<+
27í+97í6
2
7í:
+
7í:7í6
2
−++−=
=
−=
xy
y
M
Q
yN
z
y
x
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  155  -U
U
+ β+⇔→+δ(θ[θ(√(⇒+→γ∑(√(→⇔+θβ+γ(β⇒β<+
z
yx
yxMM
QsenNQ
senQNN
=
+=
−=
ϕϕ
ϕϕ
cvF
cvFU
U 3mYmUsiUSvóaoavU ;:
≤≤xω+γβ⇒β[θ<+
Normal
⇔

2
2
2
J
=
J
+
=
J
62
J






+−+
−
=
x
xx
N

Cortante
⇔

2
2
2
J
=
J
+
+
2
J
=
J
;
J
62
J
7í6






+−+
+












+−





−+
=
x
xxx
Q

Momento
⇔
      27í+9
2
J
+9
J
7í6
2
J
+9
J
2
7í:
2
2
2
−+





+−+





+−−= xxxxxM

=⇒+⇔→+θ(2,(β⇒γβ+γ→]⇔→+[θγ∑→[θ+⇔→+)→∑(→√(⇒+δβ+)→⇔vYsFUlysUs(aFdsoUmUivUimY%vUSsiUdYmóvU
momiatmSvíU
Distancia
“m”

Normal “N”
[t]
Cortante “Q”
[t]
Momento “M”
[t_m]
:U :U 6í9;U 5+9í27U
:í9JU 5:í+9U 2í92U 5=í<7U
+í66U 5:í6+U +í=6U 57í+:U
2U 5:í6<U +í+6U 52íJJU
2í9JU 5:í7+U :í<7U 5+í;=U
6í66U 5:í;2U :í<+U 5:í9=U
;U :U +U :U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  156  -U
U
Paso 3)UDIAGRAMAS DE ESFUERZOS
U ESFUERZO NORMAL
0.67m 1.33m 2.0m 2.67m 3.33m 4m0m
0 t
0.16 t
0.31 t
0.39 t
0.51 t
0.42 t
0 t
(-)
U
U ESFUERZO CORTANTE
3.64 t
2.62 t
1.83 t
1.13 t
0.95 t
0.91 t
1 t
0.67m1.33m 2.0m 2.67m3.33m 4m0m
0 t
0 t
(+)
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  157  -U
U
U DIAGRAMA DE MOMENTOS
0.67m 1.33m 2.0m 2.67m 3.33m 4m0m
0 t.m
16.25 t.m
8.95 t.m
5.10 t.m
2.77 t.m
1.48 t.m
0.68 t.m
(-)
U
Nota.-  0- .+ y+Ao+mof- )6 .34 )o+vA+S+4 )6 641(6At34 o-26AovFU*DvYómiTU vYdmodsUgUFvósodv,UcmesU
ósocavomYUlysUovUFvoUYscdmFUimFUlysUFsU moUSaeyrmoS3 6-2A6 .34 P(-234 )6 y+Ao+mof-L 43- m(Ay+4 =(6
FsUimFUveFsY mY'mUósrvYUFaUsFUlysUimUSaFdmocamUSsU +Ao+mof- + .3 .+Av3 )6. +Am3 1(6A+ S+4 A6)(mo)3L P3YU
srsóuivUcmSmU:í+míU
Ejercicio # 14
d626ASo-+A i vA+1om+A .34 6.6S6-234 o-26A-34 )6. 4ovyasodsUmYcvUsiaudacvíU
A(0,0) B(6,0)
C(3,2)
X
Y
1 t/m
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  158  -U
U
U
Paso 1)U CicyivUSsUYsmccavosFUSsUmuvgvíU
C(3,2)
R=3 t
H
A
VA
H
B
VB
+í7Uó
U

::=+⇒=
∑ BAHHH U U
tVVM
BBAJ7í::9,7í+*6:=⇒=−⇒= ∑ U
tVVM
AAB27í2:,7í;*69:=⇒=−⇒= ∑ U
tHHM
BB
DER
C +27í+:2,J7í:*6:−=⇒=−−⇒= ∑ U
[∑+γ→⇒γ[<+
tH
A+27í+= U
>DP1>"UU
:J7í:627í2:=+−⇒=
∑V U
Paso 2)+Τ]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+βθ−,β∑•[θ+(⇒γβ∑⇒[θ+
X
Y
1 t/m
1.125 t
2.25 t
1.125 t
0.75 t
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  159  -U
U
+ βγβ∑(⇒→[θ+⇔→θ+√[⇒δ(√([⇒βθ+2β[γ∑(√→θ+δβ+⇔→+√,∑)mUsi'udacm.U
xxy
yx
9
6
2
+
;<
,6*
UUUUUUsiauFs
2
;2
+−=⇒=+
−
⇔ U
U 3vYUivUdmodvTU
xx
x
y
96
92
f
2
+−
+−
==ω U
X
1.125 t
2.25 t
M z
N x
Q y
U
U
+ β√[∑δβ[θ+′,β+⇔→+√[⇒)β⇒√(⇒+δβ+θ(2⇒[θ+β⇒+θβ⇒γ(δ[+[θ(γ()[+βθ<+
M M
N
Q
Q
N
(+)
IZQ DER
U
+ =⇒γ[⇒√βθ<+
x
x
yM
xQ
tN
z
y
x 27í2
2
+27í+
27í2
UU+27í+
2
+−−=
−=
−=
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  160  -U
U
+ D,β)→β⇒γβ+δβ+⇔→+δ(θ[θ(√(⇒+→γ∑(√(→⇔+θβ+γ(β⇒β<+
z
yx
yxMM
QsenNQ
senQNN
=
+=
−=
ϕϕ
ϕϕ
cvF
cvFU
U 3mYmUsiUSvóaoavU 6:
≤≤xω+γβ⇒β[θ<+
DvYómiUUU
⇔

( ) ( )
2
2
2
2
9296
,92,*27í2*96J7í6
+−++−
+−−−+−−
=
xxx
xxxx
N

Cortante
⇔

( )( )( )
( ) ( )
2
2
2
2
9296
9627í292+27í+
+−++−
+−−++−−
=
xxx
xxxx
Q

FvósodvUU⇔
      x
x
xxM 27í2
2
9
6
2
+27í+
2
2
+−





+−−= U
=⇒+ ⇔→+ θ(2,(β⇒γβ+ γ→]⇔→+ [θγ∑→[θ+ ⇔→+ )→∑(→√(⇒+ δβ+ )→⇔vYsFU lysU s(aFdsoU mU ivU imY%vU SsiU dYmóvU
momiatmSvíU
Distancia
“m”

Normal “N”
[t]
Cortante “Q”
[t]
Momento “M”
[t_m]
:U 52í27U 5+í+27U :U
+U 5+í9U :í7U :í:JU
2U 5+í+7U :U :í6=U
6U 5+í+27U 5:íJ7U :U
U
+ →∑→+β⇔+→⇒∴⇔(θ(θ+δβ⇔+θβ2,⇒δ[+γ∑→[ω+γβ⇒β[θ<+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  161  -U
U
X
1 t/m
1.125 t
2.25 t
M z
N x
Q y
U
=⇒γ[⇒√βθ<+
( ) xxyM
tQ
tN
z
y
x
27í27í+6+27í+
UUUJ7í:UU
UU+27í+
+−−−=
−=
−=
U
U 3mYmUsiUSvóaoavU 96≤≤xω+γβ⇒β[θ<+
Normal
⇔

( )
( ) ( )
2
2
2
2
9296
92J7í:96J7í6
+−++−
+−−+−−
=
xxx
xxx
N

Cortante
⇔

( )()( )
( ) ( )
2
2
2
2
9296
927í292+27í+
+−++−
+−++−−
=
xxx
xxx
Q

Momento
⇔
      ( ) xxxxM 27í27í+69
6
2
+27í+
2
+−−





+−−= U
=⇒+ ⇔→+ θ(2,(β⇒γβ+ γ→]⇔→+ [θγ∑→[θ+ ⇔→+ )→∑(→√(⇒+ δβ+ )→⇔vYsFU lysU s(aFdsoU mU ivU imY%vU SsiU dYmóvU
momiatmSvíU
Distancia
“m”

Normal “N”
[t]
Cortante “Q”
[t]
Momento “M”
[t_m]
6U 5+í+27U 5:íJ7U :U
;U 5+í2JU 5:í;=U :í92U
7U 5+í67U :U :í<6U
9U 5:íJ7U 5+í+27U :U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  162  -U
U
Paso 3)UDIAGRAMAS DE ESFUERZOS
U ESFUERZO NORMAL
6m5m4m3m2m1m
0 t
2.25 t
1.60 t
1.15 t
1.125 t
1.27 t
1.35 t
0.75 t 0 t
(-)
U
U
U ESFUERZO CORTANTE
1m 2m 3m 4m 5m 6m
0 t
(-)
(+)
(-)
(+)
1.125 t
0.5 t 0 t
0.75 t
0.48 t
0 t
1.125 t
U
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  163  -U
U
U DIAGRAMA DE MOMENTOS
1m 2m 3m 4m 5m 6m
0 t.m
0.07 t.m
(+)
(-)
0.38 t.m
0 t.m
0.62 t.m
0.93 t.m
U
U
U
Nota.-  0- .+ y+Ao+mof- )6 .34 )o+vA+S+4 )6 641(6At34 o-26AovFU*DvYómiTU vYdmodsUgUFvósodv,UcmesU
ósocavomYUlysUovUFvoUYscdmFUimFUlysUFsU moUSaeyrmoS3 6-2A6 .34 P(-234 )6 y+Ao+mof-L 43- m(Ay+4 =(6
FsUimFUveFsY mY'mUósrvYUFaUsFUlysUimUSaFdmocamUSsU +Ao+mof- + .3 .+Av3 )6. +Am3 1(6A+ S+4 A6)(mo)3L P3YU
srsóuivUcmSmU:í+míU
U
3.11  ANÁLISIS DE ESTRUCTURAS MIXTAS íU

86 A61o6A6 + .+ m3S7o-+mof- )6 23)+4 .+4 642A(m2(A+FUgmUmomiatmSmFTUSvoSsUsiUuYvcsSaóasodvUSsU
cmicyivUSsUcmSmUyovUSsUivFUdYmóvFTUgmUFsmoUsFdvFUYscdvFUvUmYcvFUSsesYCoUFsYUmomiatmSvFUSsUómosYmU
aoSsusoSasodsUcvóvUgmUFsU!mU aFdvU!mFdmUm!vYmíUU
3mYmUcvociyaYUsiUmoCiaFaFUSsUsFdYycdyYmFTUuimodsmóv4 + m3-2o-(+mof- .+ 43.(mof- )6 656SP.34
YsFysidvFUSsUsFdYycdyYmFUóa(dmFíU

Ejercicio # 15
d626ASo-+A i vA+1om+A .34 6.6S6-234 o-26A-34 )6 .+ 4a%yasodsUsFdYycdyYmUóa(dmíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  164  -U
U
E
C
B
A
D
q=1.5 t/m
q =0.5 t/m
1
2 t
1 t
1,53 3
1,5
2,5
1
U
Paso 1)U CicyivUSsUYsmccavosFUSsUmuvgvíU
3Ys amósodsUFsUSsdsYóaomYmUsiU mivYUSsUimUYsFyidmodsUcvYYsFuvoSasodsUmUimUcmY%mUSaFdYaeyaSmUSsU
q = 1.5 t/mTUlysUmcd#mUusYusoSacyimYósodsUmiUmYcvíU
dR
dR
dR
H
V
ϕ
ϕ
dϕ
ds
R
U
22
VHRRR += U
+ =⇒γ[⇒√βθ<+
ϕqrdqdsdR == U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  165  -U
U
+ [∑+⇔[+γ→⇒γ[<+
:UcvF
cvF
cvF
:
==
=
=
∫
π
ϕϕ
ϕϕ
ϕ
dqrR
dqrdR
dRdR
H
H
H
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
qrdqrsenR
dqrsendR
dRsendR
V
V
V
2U
:
==
=
=
∫
π
ϕϕ
ϕϕ
ϕ
U
=⇒γ[⇒√βθ<+
tR7í;,7í+,*7í+*2== U
U U
+ β](δ[+ →+ ⇔→+ θ(βγ∑ →+ δβ+ ⇔→+ √→∑2→+ δ(θγ∑(],(δ→+ →−(∑→óvFU lysU imU YsFyidmodsU SsU imU cmY%mU
SaFdYaeyaSmUusYusoSacyimYUmiUmYcvUumFmUuvYUsiUsrsU^g_íU
U
∆+√[⇒γ(⇒,→√(⇒+δβγβ∑(⇒→[θ+⇔→+∑βθ,⇔γ→⇒γβ+δβ+⇔→+√→∑2mUSaFdYaeyaSmUlysUmcd#mUmUivUimY%vUSsiUU
dYmóvUE – C.U
E C
q =0.5 t/m
1
3
R
1
U
() tLqRUU7í+67í:
++ === U
+ 3→+⇔ ⇒β→+δβ+→√√(⇒+δβ+
+UsFUcvaocaSsodsUcvoUimUYscdmUlysUumFmUuvYUsiUdmóvUE – C.U
+ F[θγ∑→[θ+β⇔+δβγ→⇔⇔β+δβ+⇔→θ+→∑γ(√,⇔→√([⇒βθ<+
E
D
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  166  -U
U
+ ∆ϕ[∑→+)β→[θ+⇔→+βθγ∑,√γ,∑→+β⇒+⋅{⋅3⋅+
E
C
B
A
D
2 t
1 t
R=4.5t
H
A
VA
H
B
VB
VC
R =1.5t
1
U
∆⇔(√→⇒δ[+⇔→θ+β√,→√([⇒βθ+δβ+β′,(⇔(]∑([+γβ⇒β[θ<+
:+7í+2:=−+++⇒=
∑ BAHHH U U
()() () :7íJ++67í2+27í+7í;:=−−−−⇒= ∑ CBBAVVHM U
()()()() :7í;7í6+7í+27í27í+7í+7í;7í26:=−−++−+⇒= ∑ CAABVHVM U
U
U IuiacmoSvUimFU2UscymcavosFUsFuscamisFUlysU%sosYmoUimFUmYdacyimcavosFTUYsFvi sóvFUsiUFaFdsómTU
usYvUuYs amósodsUU
() tVVM
CC
DER
E 66í::6++:−=⇒=−−⇒= ∑ U
() tHHM
BB
ABAJO
D =í::7í2+2:−=⇒=−−⇒= ∑ U
U
[∑+⇔[+γ→⇒γ[<++
tH
A Jí+−= UU tV
A:=6í2= UU tV
BJ7í2= U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  167  -U
U
E
C
B
A
D
2 t
1 t
R=4.5t
R =1.5t
1
1.7 t
2.083 t
2.75 t
0.8 t
0.33 t
U
>DP1>"UU
:66í:J7í27í;:=6í2:=−+−⇒=
∑V U
Paso 2)+Τ]γβ⇒√(⇒+δβ+⇔→θ+β√,→√([⇒βθ+δβ+ESFUERZOS INTERNOS
TRAMO A – D +++ΤFDΤ+
⇔
U:U`Ux `UgU
U
A
M
N
Q
R
-6
-;
-2
-+
-9
-7
1.7 t
2.083 t
γ
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  168  -U
U
∞→]β[θ+′,β<+
2V
cvF++=í6
cvF22ϕγ
ϕ
ϕ=
−=
−=
R
qrR
U
+ ∞→]β[θ+γ→](⇒+′,β+δβ+⇔→+δβθ√[[θ(√(⇒+δβ+−,β∑•→θ+∑βθ,⇔γ→<+
2VcvF
2V
cvFJí+
Jí+
:=6í2
cvF:=6í2
9
7
;
6
2
+
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
⋅=
⋅=
⋅=
⋅=
⋅=
⋅=
RF
senRF
F
senF
senF
F
U
"ys%vTUumYmUsiUSvóaoavUSsU:U`Ux `UgTUSsiUmYcvUSsiUdmóvUA – D θβ∑∴<+
Normal
⇔
      ::
76+=+−+⇒= ∑ FFFNF
N

2
cvF++=í6Jí+cvF:=6í2
ϕ
ϕϕϕsensenN ⋅−−+−=∴
CortanteUU
⇔
      ::
9;2=+−−⇒= ∑ FFFQF
Q

2
cvFcvF++=í6cvFJí+:=6í2
ϕ
ϕϕϕ⋅−−+=∴ senQ
Momento
⇔
      ( ) ( ) :Jí+cvF:=6í2
2
: =−−−+⇒=∑ ϕϕ
ϕ
rsenrrRrsenMM
S

( ) ϕϕ
ϕ
ϕ sensenM 77í2cvF++27í6
2
cvF+JJí; +−+⋅−−=∴ U
U
=⇒+⇔→+θ(2,(β⇒γβ+γ→]⇔→+[θγ∑→[θ+⇔→+)→∑(→√(⇒+δβ+βθ−ysYtvFUlysUs(aFdsoUmUivUimY%vUSsiUdYmóvUIU
I+⋅+
Distancia
“m”

Normal “N”
[t]
Cortante “Q”
[t]
Momento “M”
[t_m]
:cU 52í:=U +íJU :U
+7cU 5+í97U +í9U :í97U
6:cU 5+í29U +í6<U +í2;U
;7cU 5:í<6U +í:=U +íJ6U
9:cU 5:í9;U :íJ+U 2í:=U
J7cU 5:í79U :í2=U 2í2=U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  169  -U
U
<:cU 5:í77U 5:í+JU 2í6U
+:7cU 5:í97U 5:í9:U 2í+7U
+2:cU 5:í=9U 5:í<<U +í=;U
+67cU 5+í+9U 5+í62U +í6=U
+7:cU 5+í7;U 5+í79U :í=+U
+97cU 5+í<JU 5+í9=U :í+JU
+=:cU 52í;+U 5+íJ:+U 5:í;<U
U
U
DIAGRAMAS DE ESFUERZOS
U ESFUERZO NORMAL
0°
15°
30°
45°
60°
75°
90° 180°
165°
150°
135°
120°
105°
(-) (-)
1.65 t
1.26 t
0.93 t
0.64 t
0.56 t
0.55 t
0.65 t
0.86 t
1.16 t
1.54 t
1.97 t
2.41 t
U
U ESFUERZO CORTANTE
0°
15°
30°
45°
60°
75°
90°
180°
165°
150°
135°
120°
105°
(+)
(-)
1.7 t
1.6 t
1.39 t
1.08 t
0.71 t
0.28 t
0.17 t
0.60 t
0.99 t
1.32 t
1.56 t
1.68 t
1.701 t
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  170  -U
U
U DIAGRAMA DE MOMENTOS
0°
15°
30°
45°
60°
75°
90° 180°
165°
150°
135°
120°
105°
0 t.m
0.65 t.m
1.24 t.m
1.73 t.m
2.08 t.m
2.28 t.m
2.3 t.m
2.15 t.m
1.84 t.m
1.38 t.m
0.81 t.m
0.17 t.m
0.49 t.m
(+)
U
3mYmUsiUmoCiaFaFUSsUemYYmFUYscvYSsóvFUlysUimUcvo so√(⇒+δβ+θ(2⇒[θ+β⇒+θβ⇒γ(δ[+[θ(γ()[+βθ<+
M M
N
Q
Q
N
(+)
IZQ DER
U
TRAMO B-D +++ΤFDΤ+
⇔
U:U`Ux `U+í7U
U
Normal   UDUU
tJ7í2−⇒ U
CortanteUUQUU
t=í:⇒ U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⇒
mtMx
mtMx
Parax
2í+7í+
::
=í: U
ΤFDΤ+
⇔
U+í7U`Ux `U2í7U
U
Normal   UDUU
tJ7í2−⇒ U
CortanteUUQUU
t2í+2=í:−=−⇒ U
MomentoUFUU
( )



⋅=⇒=
⋅=⇒=
⇒−−⇒
mtMx
mtMx
Paraxx
:7í2
2í+7í+
7í+2=í: U
2 t
2.75 t
0.8 t
2.75 t
1.2 t
(-)
(+)
(+)
(-)
N Q M
2.75 t
0.8 t
1.2 t
1.2 t.m
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  171  -U
U
TRAMO D-E +++ΤFDΤ+
⇔
U:U`Ux `U+í7U
D
2.414 t
1.7 t
0.496 t.m
2.75 t
1.2 t
0.496 t.m
0.5 t
0.33 t
0.496 t.m
0.5 t
0.33 t
N
Q
M
(+)
(+)
(-)
0.5 t
0.33 t
0.496t.m
0.5 t
0.33 t
U
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tNx
tNx
Para
7í:7í+
7í::
7í: U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para
66í:7í+
66í::
66í: U
MomentoUFUU



⋅=⇒=
⋅−=⇒=
⇒+−⇒
mtMx
mtMx
Parax
:7í+
;<9í::
66í:;<9í: U
TRAMO F-C
+++ΤFDΤ+
⇔
U:U`Ux `U+U
1 t
N Q M1 t
1 t.m
x
(-)
(+)
1 t
1 t.m
C
F
U
Normal   UDUU



=⇒=
=⇒=
⇒⇒
tNx
tNx
Para
:+
::
: U
CortanteUUQUU



−=⇒=
=⇒=
⇒−⇒
tQx
tQx
Para
++
::
+ U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  172  -U
U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒⇒
mtMx
mtMx
Parax
++
::
+ U
TRAMO C-E +++ΤFDΤ+
⇔
U:U`Ux `U6U
0.5 t
0.5 t
0.33 t
E
1 t.m
1 t0.33 t
N
Q
M
(+)
(+)
1 t.m
0.33 t
1 t
q =0.5 t/m
1
(-)
(+)
U
Normal   UDUU



=⇒=
−=⇒=
⇒+−⇒
tNx
tNx
Parax
7í:6
+:
7í:+ U
CortanteUUQUU



=⇒=
=⇒=
⇒⇒
tQx
tQx
Para
66í:6
66í::
66í: U
MomentoUFUU



⋅=⇒=
⋅=⇒=
⇒−⇒
mtMx
mtMx
Parax
:6
+:
66í:+ U
Paso 3)+β∑(−(√→√(⇒+β⇒+β⇔+⇒,δ[+δβ+√[⇒γ∑[⇔ω+⇒,δ[+=<+
U
0.5 t
0.33 t
0.5 t
0.33 t
E
U
∑=:H U0UU∑=:VU0UUU∑=:M U
=⇔+D5Τ+∞=+=D{5=DP∆+=D+=Q53EΤ+
"ys%vUSsiUmoCiaFaFUYsmiatmSvTUuvSsóvFUmMaYómYUlysU
la ESTRUCTURA ESTA EN
EQUILIBRIO
íU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  173  -U
U
3.12   EJERCICIOS PROPUESTOS.
U 8sUuaSsUSsdsYóaomYUimFUscymcavosFUgUSam%YmómFUSsUsFMysYtvFUumYmUivFUFa%yasodsFUsrsYcacavFíU
Problema #1
q=1 t/m
3
M
V
H
U
Respuesta: UUíJ7í:0+ mtMtV == U
(→2∑→→++δβ+[β⇒γ[θ+
(-)
0.75 t.m
M
U
Problema #2

βθ[⇔)β∑+β⇔+θ(2,(β⇒γβ+βαβ∑√(√([+[∑+β⇔+γ[δ[+∑β⇔→√([⇒→⇔+∫+[∑+β⇔+γ[δ[+−,⇒√([⇒→⇔⋅+
222VA VB
0.5 t/m
2 t.m
A B
U
Respuesta:
tVtV
BAJ7í:UUUUU0UUUUU27í:== U
(→2∑→→+δβ+[β⇒γ[θ<+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  174  -U
U
2 t.m
2 t.m
0.5 t.m
0.06 t.m
1
(-)
U
Problema #3
213
3,5
3,5
2 t.m
H
A
VA
A B
2 t
VB
1.5 t/m
C D E
F
G
U
Respuesta:
tV
tVtH
B
AA
2<í6
2<í2UUUUU0UUUUUJ7í+
=
==
U
(→2∑→→++++γ∑→[+++++∆+I+{+
N
Q
M
2.29 t
1.75 t
8.17 t.m
(+)
(+)
(+)
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  175  -U
U

Problema #4
A B
1 t/m
0.5 t.m
1 t
2 t
D
F
E
E G
2,5
1,5
33
2
4
U
Respuesta:
tVtH
tVtH
BB
AA
JJí:UUUUU0UUUUU6í:
26í2UUUUU0UUUUU2í2
==
==
U
(→2∑→→++++γ∑→[+++++∆+I++
1 t/m
D
2.23
2.19
X
X
N
Q
M
-2.23 t
2.19 t
0.31 t
2.38 t.m
1.92 t.m
(-)
(+)
(-)
(+)
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  176  -U
U
Problema #5
A
B
H
V
A
VB
A
2 t
C D
132
3
1
0.8 t/m
U
Respuesta:
U72í+
72í6UUUUU0UUUUU2í6
tV
tVtH
B
AA
=
==
U
(→2∑→→++++γ∑→[+++++{+I++
BC D
(-)
(-)
(-)
N
Q
M
3.2 t
1.52 t
4.56 t.m
U
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  177  -U
U
Problema #6
R=2.5 m
C
A B
1 t/m
0.5 t
V
H
A
A
V
H
B
B
30°
R
U
Respuesta:
tVtH
tVtH
BB
AA
97=í:UUUUU0UUUUU97=í:
6;2í2UUUUU0UUUUU97=í:
==
==
U
(→2∑→→+D++γ∑→[+++++∆+I+{+
0°
15°
30°
45°
60°
75°
90°
(-)
2.34 t
2.35 t
2.07 t
1.25 t
0.87 t
0.63 t
0.66 t
1.63 t
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  178  -U
U
0.5m1.0m1.5m2.0m2.5m3m0m
0 t.m
1.25 t.m
2.0 t.m
2.25 t.m
2.0 t.m
1.25 t.m
(+)
0.5m1.0m1.5m2.0m0m
0 t.m
0.03 t.m
0.082 t.m
0.090 t.m
(+)
Problema #7UU
0.5 t/m
VA
VB
Arco Parabolico
A(0,0)
B(3,5)
U
Respuesta:
tVtV
BA6UUU0UUU6U==U
(→2∑→→++++F+++γ∑→[+++++∆+I+E+
Problema #8
A(0,0)
VA
B(4,0)
VB
C(2,3)
H
A
H
B
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  179  -U
U
Respuesta:
tVtH
tVtH
BB
AA
66í:UUUUU0UUUUU22í:
9Jí:UUUUU0UUUUU22í:
==
==
U
8sU%YmMacmUsiUSam%YmómUU^F_UUdYmóvUIUIU UU
Problema #9
R=2.5 m
C D
0.3 t/m 0.2 t
0.4 t.m
0.3 t
H
V
B
B
A
VA
B
2
34
1,5
1,5
U
Respuesta:
tVtH
tV
BB
A
+;í:UUUUU0UUUUU6í:
099í:U
==
=
U
(→2∑→→+D++γ∑→[+++++∆+I+{+
N
Q
M
0.14 t
0.3 t
0.45 t.m
(+)
(-)
(+)
(-)
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  180  -U
U
3.13   EVALUACION DIAGNOSTICO DEL CAPITULO.

3+A+ A6+.ot+A (-+ 6y+.(+mof- )o+v-342om3 A616A6-26 miU uYsFsodsU cmuadyivTU FsU uYvuvosU miU
sFdySamodsUYsFuvoSsYUimFUFa%yasodsFUuYs%yodmFíU
U  ymisFUFvoUivFUsisósodvFUSsUivFUsFMysYtvFUaodsYovFXU
U l(+. 64 .+ )61o-omof- )6 641(6At3 D3AS+.X
U  vóvUFsUSa aSsUsiUmoCiaFaFUSsUsFdYycdyYmFXU
U l(+.64 43- .34 SI23)34 P+A+ 6. +-B.o4o4 )6 yov+4X
U l(+. 64 .+ )o16A6-mo+ 6-2A6 6. +-B.o4o4 P3A 6. SI23)3 A6.+mo3-+. i 6. SI23)3 1(-mo3-+.X
U 0- 6. +-B.o4o4 )6 PfA2om34 =(6 SI23)3 )6 +-B.o4o4 6FUómFUmcvoFsrmeisUgUuvYUlysXU
U 0- =(6 m3-4o426 .+ y6Ao1om+mof- )6 641(6At34 o-26A-34 6- PfA2om34 i u+42+ m(+-)3 64 +m6P2+7.6
(-+ y+Ao+mof- 6- .34 A64(.2+)34X
U 08P.o=(6 .+ 263Ab+ v6-6A+. )6 +Am34 P+A+7f.om34X
U
U 3mYmU cvociyaYU sFdsU Sam%ovFdacvU FsU uaSsU SsdsYóaomYUimFU YsmccavosFU SsU muvgvU gU %YmMacmYU ivFU
Sam%YmómFUSsUsFMysYtvFUSsUimFUFa%yasodsFUMa%yYmFíU
Prob # 1
44
2
2
0.8 t.m
0.5 t.m
B
A
D
C
E
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  181  -U
U
Prob #2
B
0.5 t/m
A
0.8 t
0.6 t
26
3
4
2
C
D
U
Prob #3
A
B
0.5 t/m
0.8 t
C
1
26
3
U
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  182  -U
U

CAPITULO 4
U
U
U
U
U
U
ARMADURAS (CERCHAS)
U
U
U
U
4.1   OBJETIVO GENERAL.
0. 642()o+-26 P(6)6 m3-m6P2(+.ot+A .+ 1(-mof- )6 (-mUmYómSyYmTUivFUdauvFUSsUmYómSyYmFUgUFyU
+P.om+mof-n 08uo76 )642A6t+4 P+A+ A6+.ot+A 6. mB.m(ivUSsUsFMysYtvFUaodsYovFUgUs(dsYovFUsoUimUmYómSyYmíU
U
4.2 OBJETIVOS ESPECIFICOS.
e. 26ASo-+A 6426 m+Po2(.3 6. 642()o+-26 P3)ABa
U )SsodaMacmYUmYómSyYmFUaFvFdCdacmFUsU!ausYsFdmdacmFíU
U e)=(oAoA u+7o.o)+)64 P+A+ +P.om+A )oy6A434 SI23)34 PmYmUSsdsYóaomYUivFUsFMysYtvFUaodsYovFUgU
s(dsYovFUsoUivFUóasóeYvFíU
U 3ysSsUYscvYSmYUyoUdauvUSsUmYómSyYmUumYmUcmSmUcmFvíU
U
U
4.3   ARMADURAS.

&omUcsYc!mUuysSsUSsMaoaYFsUcvóvUyomUsFdYycdyYmUcvóuysFdmUSsUyoUo#ósYvUSsUsisósodvFUvU
emYYmFUyoaSvFUsoUFyFUs(dYsóvFUuvYUósSavUSsUumFmSvYs4 4o- 1Aommof- P+A+ 13AS+A (- +AS+tf- Abvo)3n
"mFUMysYtmFUs(dsYomFUgUYsmccavosFUFsUFyuvosUlysUsFdCoUsoUsiUóaFóvUuimovUSsUimUsFdYycdyYmUgUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  183  -U
U
mcd#moUFvimósodsUFveYsUivFUumFmSvYsFíUISsóCFTUFsUFyuvosUlysUcmSmUyovUcvaocaSsUcvoUimUi'osmUlysU
yosU ivFU csodYvFU SsU ivFU oySvFU SsU FyFU s(dYsóvFU gU lysUsiU usFvU SsU cmSmU emYYmU sFU SsFuYscameisU soU
m3SP+A+mof- m3- .+4 1(6At+4 6826A-+4 =(6 +m2M+- 437A6 .+ m6Amu+n d6 .+4 m3-)omo3-64 +-26Ao3A64 46
SacsU lysU cmSmU emYYmU SsU yomU csYc!mU sFU yoU sisósodvU FvósdaSvU mU SvFU MysYtmFU gU FyrsdvU FvimósodsU mU
1(6At+4 +8o+.64 )oA6m2+4 *2A+mmof- 3 m3SPA64of-Cn 0iUmoCiaFaFUcvóuisdvUSsUyomUcsYc!mUcvoFaFdsUsoUimU
)626ASo-+mof- )6 .+4 1(6At+4 +8o+.64 o-26A-+4 )6 23SmFUFyFUemYYmFíU
c+4 uoPf264o4 =(6 o)6+.ot+- 6. +-B.o4o4 )6 .+4 +AS+)(A+4 43-a
U
U 8voUsFdYycdyYmFUdYamo%yimYsFíU
U "mFUcmY%mFUmcd#moUsoUivFUoySvFí
U
"vFUoySvFUFvoUmYdacyimcavosFíU
U
"mFUemYYmFUsFdCoUFvósdaSmFU#oacmósodsUmUsFMysYtvFUo3AS+.64 )6 2A+mmof- 3 m3SPA64of-n
4.4  GRADO DE ISOSTATICIDAD
.
U noUimFUmYómSyYmFUFsUSaFdao%ysoUSvFU%YmSvFUSsUaFvFd+2omo)+)a
U
4.4.1U YYmSvU n(dsYovíU niU cvocsudvU SsU moCiaFaFU sFU FaóaimYU m. )6 .+ yov+L PfA2om34 i
UUUUUUUUUUU
IYcvFíU
4.4.2 YYmSvU)odsYovíU8sUSa aSsoUsoUSvFUumYdsFUIYómSyYmFU3imomFUgUIYómSyYmFUnFumcamisFU
lysUSsdsYóaomoUsiU%YmSvUaodsYovíU3mYmUcvóuYsoSsYUósrvYUFsU sUsoUimFU%YmMacmFU;í+U*m,U
gU*e,U
U
U
U
U U UUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U U
U
    Figura 4.1(a)        Figura 4.1(b)
            b barras  = 2 nnudos – 3                                     b = 3 nnudos - 6
                              En el plano                                             En el Espacio
d3-)6a
7 a  64 6. -MS6A3 )6 7+AA+4
- a  6. -(S6A3 )6 -()34n

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  184  -U
U
U 8sUóvFdYmYmUsiU%YmSvUSsUaFvFdmdacaSmSUcvoUmi%yovFUYZYntFjv1
Ejercicio #1    U
.ósó Fó ódnósrdó sYXYdnEeód YF 5dósj sY EvjvXóXEcEsmSUaodsYovUY
)U
UUUUUUUUUUUUUUUUUUU
Solución.-

áóiYnjv orY1   e Yv YF ernYdj sY ersjv  e G D
U UUUUUUUUUUUUUUUUUUUUUUUUUeUsFUsiUoyósYvUSsUemYYmFUUeU4U<U
kF 5dósj sY EvjvXóXEcEsós Yv1 Y
)U4UeUIU2oUHUYUU4U<UIU2*9,UHU6U4U:U
Y
)U4U:UUsodvocsFUsFU)8>8PIP) >U
Ejercicio #2

.ósó Fó ódnósrdó sYcEsEd vr 5dósj sY EvjvXóXEcEsós EodsYovUY )U

Solución.-
áóiYnjv orY1 e Yv YF ernYdj sY ersjv e G -
                              eUsFUsiUoyósYvUSsUemYYmFUeU4U++U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  185  -U
U
kF 5dósj sY EvjXóXEcEsós Yv1 Y)U4UeUIU2oUHUYU4U++UIU2*J,UHU6U4U:U
Y
)U4U:UsodvocsFUsFU)8>8PIP) >U

Ejercicio #3
.ósó Fó ódnósrdó sYcEsEd vr 5dósj sY EvjvXóXEcEsós EodsYovUY )U

Solución.-
áóiYnjv orY1 e Yv YF ernYdj sY ersjv e G [
                              eUsFUsiUoyósYvUSsUemYYmFUeU4U9U
kF 5dósj sY EvjXóXEcEsós Yv1
Y
)U4UeUIU2oUHUYU4U9UIU2*;,UHU6U4U+U
GI = 1 YeXjecYv Yv káαgα84g,kCαk 8C.kαk:,8Cg.w
Ejercicio #4
.ósó Fó ódnósrdóf sYcEsEd vr 5dósj sY EvjvXóXEcEsósUaodsYovUY )íU

Solución.-

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  186  -U
U
        áóiYnjv orY1 e Yv YF ernYdj sY ersjv e G ]
                               eUsFUsiUoyósYvUSsUemYYmFUeU4U+6
kF 5dósj sY EvjXóXEcEsós Yv1 Y)U4UeUIU2oUHUYU4U+6UIU2*<,UHU6U4U52U
Y
) G =T YeXjecYv Yv káα:64α6:g Cw g.k46g.g
3I1IU1n8)8P)1U I1YI8U
U
U ymoSvUsiU%YmSvUSsUaFvFdmdacaSmSUsFU4U:UsFdmóvFU!meimoSvUSsUyomUsFdYycdyYmUn8PIP) ITUsoU
cmFvU SsU lysU siU %YmSvU SsU aFvFdmdacaSmSU MysFsU hU +U sodvocsFU !meimóvFU SsU yomU sFdYycdyYmU
káαgα84g,kCαk 8C.kαk:,8Cg.gf u vE prYvY nYejd orY cYdj YedvocsFUFsYamUyomUsFdYycdyYmU
Cw g.k46g.g Sg:g :ká8áα8: 4g:Ygáz
U
αóF cjnj vY nrYvXdó Ye Fó vE5rEYeXY XóiFó1
Tabla 4.1 condiciones de isostaticidad
Condiciones de
isostaticidad:
Tipo de estructura
YU`U:U
nFdYycdyYmUaosFdmeisTUovUmSscymSmUumYmU
YsFaFdaYUcmY%mFU
YU4U:U nFdCdacmósodsUSsdsYóaomSmU
YUaU:U nFdCdacmósodsUaoSsdsYóaomSmU
U
4.5   DETERMINACION DE ESFUERZOS.

)ó sYXYdnEeócE3e sY YvprYdajv vY dYsrcY  ó rXEFEaód YF vE5ej ML0 tódó Fó XdóccE3e  u M=0 tódó Fó
cjntdYvE3e óF orY YvXó vjnYXEsó reó ióddó sYXYdnEeóSmTUdmiUcvóvUFsUóysFdYmUsoUimUFa%yasodsUMa%yYmíU
U
Figura 4.2 (a)     Figura 4.2 (b)
U 3mYmU SsdsYóaomYU ivFU sFMysYtvFU soU imFU mYómSyYmFU s(avXYe xódEjv n2Xjsjv YeXdY Fjv cróFYv
nYecEjeódYnjv Fjv vE5rEYeXYv1
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  187  -U
U
4.5.1 METODO DE LOS NUDOS.-

8aUyomUmYómSyYmUsFdmUsoUslyaiaeYavTUsodvocsFUcmSmUy⇒[+δβ+θ,θ+⇒,δ[θ+δβ]β+γ→](⇒+βθγ→∑+β⇒+
β′,(⇔(]∑([⋅+ [∑+ γ→⇒γ[ω+ β⇔+ γ[δ[+ δβ+ ⇔[θ+ ⇒,δ[θ+ √[⇒θ(FdsU soU FmdaFMmcsYU imFU cvoSacavosFU SsU slyaiaeYavU
∑-
(4U:UgU∑- g4:UumYmUimFUMysYtmFUsrsYcaSmFUFveYsUsiUumFmSvYUsoUcmSmUoySvUSsUimFUmYómSyYmíU
++ =⇔+∑[√βδ((β⇒γ[+θ(2,(β⇒γβ+∑[[∑√([⇒→+,⇒+γ[δ[+umYmUmomiatmYUyomUmYómSyYmUósSamodsUsiU
γ[δ[+δβ+⇔[θ+⇒,δ[θ⋅+
+ βγβ∑(⇒β+∑(β∑[+⇔→θ+∑β→√√([⇒βθ+β⇒+⇔[θ+θ[[∑γβθ+√[oFaSsYmoSvUsiUslyaiaeYavUSsUimUmYómSyYmU
sodsYmíU"ys%vUSaeyrsUsiUSam%YmómUSsUcysYuvUiaeYsUSsUyoUoySvUlysUdso%mUuvYUivUósovFUyomUMysYtmU
√[⇒[√(δ→+ ∫+ √,→⇒δ[+ ∴θ+ δ[θ+ −,β∑•→θ+ (⇒√2⇒(γ→θ⋅+ ∞(+ βFdsU oySvU sFdmU soU yovU SsU ivFU FvuvYdsFTU FsYCU
oscsFmYavUcvovcsYUimFUYsmccavosFUs(dsYomFUsoUsiUFvuvYds,íUU
+ 5θβ+β⇔+γ[δ[+δβθ√∑(γ[+→⇒γβθ+→∑→+βθγ→]⇔β√β∑+β⇔+θβ⇒daSvUSsUyomUMysYtmUSsFcvovcaSmíU"vFUsrsFU
?U gU @U SsesoU vYasodmYFsU SsU ómosYmU lysU imFU MysYtmFU soU siU Sam%YmómU SsU cysYuvU iaeYsU uysSmoU
SsFcvóuvosYFsU MCcaiósodsU soU FyFU cvóuvosodsFU ?U gU @0U iys%vU muialysU imFU SvFU scymcavosFU SsU
slyaiaeYavUSsUMysYtmFU∑U-(U4U:Ug
U∑U-@+≤+⌠⋅+β]β+⇒[γ→∑θβ+γ→](⇒+′,β+θ(+,⇒[+δβ+⇔[θ+βαβθ+βFdmUvYasodmSvU
mU ivU imY%vU SsU yomU SsU imFU MysYtmFU SsFcvovcaSmFTU imU vdYmU MysYtmU SsFcvovcaSmU uysSsU SsdsYóaomYFsU
FyómoSvU MysYtmFU mU ivU imY%vU SsiU vdYvU srsTU SsFusrsU imFU SvFU MysYtmFU SsFcvovcaSmFU gU sYaMalysU FyFU
FsodaSvFUcvYYscdvFíU
+ {[⇒γ(⇒,+ →⇒→⇔(•→⇒δ[+ √→δ→+ ,⇒[+ δβ+ ⇔[θ+ [γ∑[θ+ ⇒,δ[θω+ βia%asoSvU oys mósodsU yovU lysU dso%mU
√,→⇒δ[+→θ+δ[θ+(⇒√2⇒(γ→θ+∫+[∑+⇔[+β⇒[θ+,⇒→+−,β∑•→+√[⇒[√(δ→⋅+βθβ+√,β⇒γ→+′,β+,⇒→+)β•+β⇒√[⇒γ∑→δ→+
yomUMysYtmUsoUyoUóasóeYvUuvYUsiUmoCiaFaFUSsiUoySvUsoUyovUSsUFyFUs(dYsóvFTUsiUYsFyidmSvUuysSsUyFmYFsU
umYmUmomiatmYUimFUMysYtmFUlysUmcd#moUsoUsiUoySvUsoUβ⇔+[γ∑[+βγ∑β[⋅+β]β+[]θβ∑)→∑θβ+δβθδβ+⇔,β2[+,⇒→+
βθγ∑(√γ→+ →δϕβ∑β⇒√(→+ →⇔+ ∑(⇒√(([+ δβ+ ⇔→+ →√√(⇒+ ∫+ δβ+⇔→+ ∑β→√√(⇒⋅+ β√,β∑δβ+ ′,β+ ,⇒+ (β]∑[+ β⇒+
√[∑βθ(⇒+β,α→+β⇔+⇒,δ[+∫+′,β+,⇒+(β]∑[+β⇒+γβ⇒θ(⇒+α→⇔→+β⇔+⇒,δ[⋅+
+ 3[θ+→θ[θ+′,β+θβ+β⇔β→⇒+θ[⇒<+
U βγβ∑(⇒→√(⇒+βθγ∴γ(√→++++++++++++++++++++++
U  micyivUSsUYsmccavosFUSsUmuvgvUUUUUUUU
U nFMysYtvFUsoUimFUemYYmFU
IuialysóvFUsFdsUuYvcsSaóasodvUumYmUivFUFa%yasodsFUsαβ∑√(√([θ<+
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  188  -U
U
V
M
(+)
Ejercicio #5
U IomiatmYUimUmYómSyYmUuimomUdauvUG>[níU

Solución.-
U
PASO 1) DETERMINACION ESTATICA íU
U YYmSvUs(dsYovUU Y
nU4U6UIU6UIU:U4U:U
U YYmSvUaodsYovUUUUUUUUUUUUUY
)U4UeUIU2oUHUYU4U+6UIU2*=,UHU6U4U:U
              Y)U4U:U
U U U U U
6í7
;
!
2
UU
U U U UU UUUUUU4aU!U4U+íJ7U

PASO 2) CÁLCULO DE REACCIONES DE APOYO
U
U U U U U U U U >D/nD )>DU
∑UFU4U:UU⇒UG
IUHU:í6U4U:UU
UUUUUUUUUUUUUUUUUUUUUG
IU4U5U:í6UdU
∑UF
IU4U:U⇒U:í7ULU*;,UHU:í6UL*+íJ7,UIU=/EU4U:UU
UUUUUUUUUUUUUUUU⇒U/
EU4U:í62UdU
∑UF
EU4U:U⇒U=/ IUIU:í7ULU*;,UHU:í6ULU*+íJ7,U4U:U
UUUUUUUUUUUUUUUU⇒U/
IU4U:í+=UdU
U
PASO 3) ESFUERZOS EN LAS BARRAS

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  189  -U
U
>D/nD )>DU
U
+ →∑→+→⇔(√→∑+βθγβ+γ[δ[+δβ]β+βθ√[2β∑θβ+⇒,δ[θ+β⇒+⇔vFUcymisFUmUivUFyóvUcvocyYYsoU2UemYYmFU
(⇒√2⇒(γ→θ⋅+
U
NUDO 1 (SUPUESTO)   NUDO 1 (REAL)
U
dmoθ
6í7
;
θ;++<,
U UU UUUUUUUUUUUUUUU8soUθU4U:í99UUU0UUUU vFUθU4U:íJ7U

U U
U
UUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U
U
U
U
U
U
∑UGU4U:U⇒U5:í6UHUe
+UHUe=U*:íJ7,U4U:U
∑U/U4U:U⇒U:í+=UHUe
=*:í99,U4U:U
⇒Ue
=U4U5U:í2JUdU
⇒Ue
+U4U:í7:UdU
U
NUDO 2 (SUPUESTO)   NUDO 2 (REAL)

U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U
U
U
UUUUUU

U
∑UGU4U:U⇒U5:57UHUe
2U4U:U
⇒Ue
2U4U:í7UdU
∑U/U4U:U⇒Ue
<U4U:

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  190  -U
U
NUDO 8 (SUPUESTO)           NUDO 8 (REAL)

U
U U U
U
U
U
U
U
U
U
U U U
U
U
∑UGU4U:U⇒U:í2JcvFφUHUe
JUcvFφUHUe +:cvFφU4U:U
⇒U:í2JL:íJ7UHUe
JL:íJ7UHUe+:L:íJ7U4U:U
∑U/U4U:U⇒U:í2JFsoφUHUe
JFsoφUHUe +:FsoφU4U:U
⇒U:í2JL:í99UHUe
JL:í99UHUe+:L:í99U4U:U
⇒Ue
JU4U5:í2JUdU
⇒Ue
+:U4U:UdU
NUDO 7 (SUPUESTO)            NUDO 7 (REAL)

U
U
U
U
U
U
U
U
U
U
U
∑UGU4U:U⇒U:í2JcvFφUHUe
9cvFφU4U:U
⇒U:í2JLcvFU:íJ7UHUe
9LUcvFU:íJ7U4U:U
⇒Ue
9U4U5:í2JUdU
∑U/U4U:U⇒U:í2JFsoφU5U:í7UIUe
++UIU*5e9,UFsoφU4U:U
⇒U:í2JLFsoU:í995:í75Ue
++U5:í2JLFsoU:í99U4U:U
⇒Ue
++U4U5:í+;UdU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  191  -U
U
NUDO 5 (SUPUESTO)   NUDO 5 (REAL)
U U UU
U UUUUUUUUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUU
U U
U
U

U
U
U
U
∑U/U4U:U⇒U:í62UHUe
7U*:í99,U
⇒Ue
7U4U5U:í;=UdU
∑UGU4U:U⇒U5e
;UIUe7U*:íJ7,U4U:U
⇒Ue
;U4U:í69UdU
NUDO 4 (SUPUESTO)                                                          NUDO 4 (REAL)
U
U e
+6U
[∑+θ(⇔β+[]θβ∑)→√(⇒+δβ√([θ+′,β+
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U
UUUe
6U U :í69UdUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU:í69UdUUUUUUUUUUUUUUUUUUUUUUUU:í69UdU
U
U
NUDO 6 (SUPUESTO)      NUDO 6 (REAL)
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U U U
U
U U U

∑UGU4U:U⇒U5e +2L:íJ7U5Ue9L:íJ7UHU:í6UIU:í;=L:íJ7U4U:U
∑U/U4U:U⇒U5e
+2*:í99,UHUe9*:í99,UHU:í;=*:í99,U4U:U
⇒Ue
+2U4U:í+<UdU
⇒Ue
9U4U5:í2=Ud

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  192  -U
U
RESUMEN DE ESFUERZOS
U
4.5.2  METODO DE LAS SECCIONES.
+ =⇔+ γ[δ[+ δβ+ ⇔→θ+ θβ√√([⇒βθ+ √[⇒θ(θγβ+ β⇒+ →θ→∑+ ,⇒→+ θβ√√(⇒+ + (→2(⇒→∑(→+ [∑+ ⇔→+ →∑→δ,∑→ω+
cvYdCoSvimUmF'UsoUSvFUumYdsFíU8aUimUmYómSyYmUsodsYmUsFdmUsoUslyaiaeYavTUsodvocsFUcmSmUyomUSsUimFUSvFU
→∑γβθ+ δβ]β+ γ→](⇒+ βθγ→∑+ β⇒+ β′,(⇔(]∑([ω+ β⇒+ √[⇒θβ√,socamU uysSsoU muiacmYFsU imFU dYsFU scymcavosFU SsU
slyaiaeYavUmUcymilyasYmUSsUsFmFUumYdsFUumYmUSsdsYóa⇒→∑+⇔→θ+−,β∑•→θ+β⇒+⇔[θ+(β]∑[θ+β⇒+⇔→+θβ√√(⇒+δβ⇔+
cvYdsTU
+ {,→⇒δ[+ θβ+ ,θ→+ β⇔+ γ[δ[+ δβ+ ⇔→θ+ θβ√√([⇒βθ+ →∑→+ δβγβYóaomYU imU MysYtmU soU yoU óasóeYvU
→∑γ(√,⇔→∑ω+ δβ]β+ γ[→∑θβ+ ,⇒→+ δβ√(θ(⇒+ θ[]∑β+ √[[+ √[YdmYU vU FsisccavomYU imU mYómSyYmíU vóvU FvivU
uysSsoUmuiacmYFsUdYsFUscymcavosFUaoSsusoSasodsFUSsUslyaiaeYavU*∑-
?U4U:TU∑- @U4U:TU∑F >U4U:,íUIUimU
umYdsUmaFimSmUSsUimUmYómSyYmTUdYmdsUSsUsFcv%sYUyomUθβ√√(⇒+′,β+β⇒+2β⇒β∑→⇔ω+→θβ+[∑+⇒[+∴θ+δβ+γ∑βθ+
óasóeYvFUcygmFUMysYtmFUFsmoUSsFcvovcaSmFíU
U nFdsU dauvU SsU mYómSyYmU FsU ydaiatmU cvoU MYscysocamU so+ ⇔→+ √[⇒θγ∑,√√(⇒+ δβ+ ,β⇒γβθω+ →∑→+
MsYYvcmYYaisFUgUumYmUcmóavosFíUIi%yomFU scsFUFsUivUyγ(⇔(•→+γ→](⇒+→∑→+√,]∑(∑+→](β⇒γβθ+δβ+∴]∑(√→θω+
yoa sYFaSmSsFUgUcymYdvFíU
+ 3[θ+→θ[θ+′,β+θβ+β⇔β→⇒+θ[⇒<+
U βγβ∑(⇒→√(⇒+βθγ∴γ(√→+
U  micyivUSsUYsmccavosFUSsUmuvgvUU
U nFMysYtvFUsoUimFUemYYmFU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  193  -U
U
Ejercicio #6

βγβ∑(⇒→∑+⇔[θ+βθ−,β∑•[θ+β⇒+⇔→θ+]→∑∑→θ+]+:UgUe2+U
U
Solución.-
U
PASO 1) DETERMINACION ESTATICA
U
U U U U U Y)U ⇒UeU4U2oUIU6U
⇔U27U4U2U*+;,UIU6U4U27U
U U U U U Y)U4U6UIU6UIU:U4U:U
PASO 2) CÁLCULO DE REACIONES DE APOYO
∑ GU4U:U⇒U:í;UHG EU4U:U⇒U HB = - 0.4 t
∑ FIU4U:U⇒U:í;U*6,UHU:í7U*7,UHU:í2U*+2í7,UIU+7U/EU4U:U
⇒U
VB = 0.41 t
∑UFEU4U:U⇒U+7U/ IUHU:í;U*6,UIU:í7U*+:,UIU:í2U*2í7,U4U:U
⇒U
VA = 0.29 t
vodYviU∑U/U4U:U⇒U:í2<UIU:í7UIU:í2UHU:í;+UdU4U:U
PASO 3) ESFUERZOS EN LAS BARRAS
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  194  -U
U
=⇔+ γ[δ[+ √[⇒θ(θγβ+ β⇒+ θβ⇔β√√([⇒→∑+ √[∑γ→∑+ ⇔→+ β∑√ϕ→⋅+ 3→+ θβ√√(⇒+ →+ ⇔[+ θ,[+ δβ]β+ √[∑γ→∑+ φ+
emYYmFUSsFcvovcaSmFíU 8sUFsumYmUimUusYc!mUcvYdmSmUgUFsUmomiatmUimUumYdsUlysUcvo so%mU*)tlyasYSmUgU
β∑β√ϕ→⋅+
U
U U
U U ∑UF
;U4U:U⇒U:í2<U*Jí7,UHU:í;U*6,UIU:í7U*2í7,UHUe+:U*6,U4U:U
U U U U e
+:+≤+⌠⋅J⌠+γ+√[∑βθ(⇒+
U
U U ∑UF
7U4U:U⇒U5:í;+U*7,UHU:í2U*2í7,U4Ue2+(U*6,UHU:íJU*6,U4U:U
U U U

e2+U(U4U:í+=UdU
dmo%θU4UUU6UUUU4U+í2U
UUU2í7U
FsoUθU4U:íJJU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  195  -U
U
cvFUθU4U:í9;U
⇒U vFUθU4UUe
2+U(UU⇒Ue 2+U4UU:í2=UUUU4U:í2=UdU
U
e
2+U4U:í2=UdUUUUTRACCIONU
U
4.5.3 ARMADURAS EN EL ESPACIO METODO GENERAL.
U
ymoSvU mYamFUemYYmFUFsUyosoUsodYsUFaUuvYUFyFUs(dYs[θ+→∑→+−[∑→∑+,⇒→+√[⇒−(2,∑→√(⇒+β⇒+γ∑βθ+
SaósoFavosFTUimUsFdYycdyYmUvedsoaSmUFsUiimómUmYómSyYmUsoUsiUsFumcamiíU
U "mU mYómSyYmU Y'%aSmU eaSaósoFavomiU ómFU sisósodmiU cvoFaFdsU SsU dYsFU emYYmFU yoaSmFU uvYU FyFU
s(dYsóvFTUlysUMvYómoUyoUdYamo%yivTUgUlysUm%Ys%moSvU→+⇔→+√[⇒−[∑→√(⇒+]∴θ(√→+δ[θ+]→∑∑→θ+→√[⇔→δ→θ+
mUyoUoys vUoySvTUsYmUuvFaeisUvedsosYUyomUsFdYycdyYmUY'%aSmUómgvYUlysU SsMaoaóvFUcvóvUmYómSyYmU
FaóuisíU IoCiv%mósodsTU imU mYómSyYmU sFumcamiU óCFU FsocaiimU cvoFaFdsU SsU FsaFU emYYmFU yoaSmFU uvYU FyFU
s(dYsóvFTUlysUMvYómoUimFUmYaFdmFUSsUyoUdsdYmsSYvUIE +δβ+⇔→+−(2,∑→+→⋅+∆2∑β2→⇒δ[+γ∑βθ+]→∑∑→θ+→+⇔→+
√[⇒−[∑→√(⇒+]∴θ(√→ω+γ→⇔βθ+√[[+∆=ω+E=+∫+{=ω++→√[⇔∴oSvimFUmUivFUoySvFUSaFdaodvFUgmUs(aFdsodsFUgU
,⇒(⇒δ[⇔→θ+β⇒+,⇒+⇒,β)[+⇒,δ[ω+[δβ[θ+[]γβ⇒β∑+,⇒→+βθdYycdyYmUY'%aSmUómgvYTUimUcymiUSsMaoaóvFUcvóvU
armadura simple. LU>eFsY moSvUlysUsiUdsdYmsSYvUeCFacvUdasosUFsaFUemYYmFUgUcymdYvUoySvFUgUlysTUcmSmU
stUlysUFsUm%Ys%ysoUdYsFUemYYmFTUFsUmyósodmUsoUyovUsiUo#ósYvUSsUoySvFTUcvociyaóvFUlysUsoUyomU
mYómSyYmUFaóuisUsFumcamiUsiUo#ósYvUdvdmiUSsUemYYmFUsFUb = 3n -6, FasoSvUnUsiUo#ósYvUdvdmiUSsUoySvFíUU
∞(+⇔→+→∑→δ,∑→+βθ→√(→⇔+δβ]β+∑βθβ⇒γ→∑+√[⇒θγ∑,√√(⇒UdvdmiUgUFaUimFUYsmccavosFUsoUFyFUmuvgvFU
FvoU sFdCdacmósodsU SsdsYóaomSmFTU ivFU muvgvFU SsesY'mo+ θβ∑+ ,⇒→+ √[](⇒→√(⇒+ δβ+ βθ−β∑→θω+ ∑[δ(⇔⇔[θ+ ∫+
∑γ,⇔→+ ′,β+ ∑[[∑√([⇒β⇒+ θβ(θ+ ∑β→√√([⇒βθ+ δβθ√[⇒[√(δ→FíU "mFU YsmccavosFU SsFcvovcaSmFU uysSsoU
socvodYmYFsUMCcaiósodsUYsFvi asoSvUimFUFsaFUscymcavosFUlysUs(uYsFmoUlysUimUmYómSyYmUdYaSaósoFavomiU
sFdCUsoUslyaiaeYavíU
IyolysU imFU emYYmFU SsU yomU mYómSyYmU sFumcamiU sFdCoU YsmiósodsU yoaSmFU sodYsU F'U ósSamodsU
cvos(avosFUFviSmSmFUvUYsómc!mSmFTUFsUcvoFaSsYmUlysU√→δ→+⇒,δ[+βθγ∴+√[⇒θγ(γ,(δ[+[∑+,⇒→+∑γ,⇔→⋅+β+
sFdmUómosYmTUovUFsUmuiacmYCUoao%#oUumYUmUimFUemYYmFUSsUimUmYómSyYmUgUcmSmUemYYmUuvSYCUdYmdmYFsUcvóvU
emYYmUFvósdaSmUmUSvFUMysYtmFíU"mFUcvoSacavosFUSsUslyaiaeYavUumYmUcmSmUoySvUFsUs(uYsFmYCoUuvYUimFUdYsFU
scymcavosFU-(U4U:TUU-g4U:TUgU-tU4U:íUnoUsiUcmFvUSsUyomUmYómSyYmUsFumcamiUFaóuisUlysUcvodso%moUn
oySvFTUmiUsFcYaeaYUimFUscymcavosFUSsUslyaiaeYavUumYmUcmSmUoySvUFsUdsoSYCoU6n scymcavosFíU
U 3ysFdvUlysUb = 3n – 6, sFdmFUscymcavosFUFvoUFyMacasodsFUumYmUSsdsYóaomYUdvSmFUimFUMysYtmFU
SsFcvovcaSmFU *MysYtmFU soUn emYYmFU gU FyFU YsmccavosFU soU ivFU muvgvF,íU 8aoU sóemY%vTU umYmU s admYU imU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  196  -U
U
∑βθ[⇔,√(⇒+δβ+,√ϕ→θ+β√,→√([⇒βθ+θ(,⇔γ∴⇒β→θω+⇔[θ+⇒,SvFUSsesYCoUFsisccavomYFsUcyaSmSvFmósodsUumYmU
SsFcmYdmYUmlysiivFUlysUcvodso%moUóCFUSsUdYsFUMysYtmFUSsFcvovcaSmFíU
U
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U UUUUUUUUUUU
Figura 4.3 (a)                                                        Figura 4.3 (b)
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U U U U U UUUUUUUU-U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU4U vs−(√(β⇒γβ+δβ+γβ⇒θ(⇒+
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU"UU
UUUUUUUUUUUUUU-?          "?U UUUUUUUUUU-U
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?U4UUUUUUUUUUUULU"?U4U ULU"?U
                           -UUUUUUUUUUUUUUUU"UUUUUUUUUUUUUUUUUUUUUUUUUUU"UU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  197  -U
U
UUUUUUUUUUUUUUUUUUUUUUUUUU-@          "@U UUUUUUUUUU-U
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                           -UUUUUUUUUUUUUUUU"UUUUUUUUUUUUUUUUUUUUUUUUUUU"U
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UUUUUUUUUUUUUUUUUUUUUUUUUU-
A          "AU UUUUUUUUUUU-U
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AU4UUUUUUUUUUUUUUUULU"AU4U ULU"AU
                           -UUUUUUUUUUUUUUUU"UUUUUUUUUUUUUUUUUUUUUUUUUUUU"U

=⇔+γ[δ[+′,β+δβθ√∑(](∑β[θ+→+√[⇒γ(⇒,→√(⇒+,βδβ+θβYUydaiatmSvUsoUsiUmoCiaFaFUSsUmYómSyYmFU
uimomFUgUsFumcamisFUcvóvUFsUóysFdYmUsoUsiUFa%yasodsUsrsóuiví

Ejercicio #7
U voFaSsYsóvFUyoUoySvUsoUsiUsFumcavUmiUlysUcvocyYYsoUdYsFUemYYmFUgUyomUUMysYtmU3U⇔[56TU2TU+]íU
βγβ∑(⇒→∑+θ,θ+βθ−,β∑•[θ+√[∑∑βθ[⇒δ(β⇒γβθ⋅+

Solución.-
U
UUUUUUUUUUU"
?U4U5+U52U4U56U
e
+U⇒UU"@U4U:U56U4U56UUUUUUUUUUUUUUU"e+U4UUUUUU56
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e
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2
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UUUUUUUUUUU"
AU4U2UIU+U4U+UUU
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e
6U⇒UU"@U4U52U56U4U57UUUUUUUUUUUUUU"e6U4UUUUU:UHU7
2
UHU*52,UUU⇒U"e 6U:U7í6<UóU
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AU4U5+U5+U4U52UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U U U U U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  198  -U
U
U U U U ∑U- +U4U:U
U U U -(
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+UHU*5;,U 2UHU*:,U 6UHU*56,U4U:U
UUUUUU U U56U
+UUHU*52,U 2UHU*57,U 6UHU*U2U,U4U:U
UUUUUUUU U +U
+UHU*+,U 2UHU*5U2,U 6UHU*+,U4U:U
+ 3,β⇒2[+γβ⇒β[θ<+{
+U4U+T67UUUUUUU 2U4UU5+TJ9UUUUUUU 6U4U:T2<U
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+ =⇒γ[⇒√βθ<+ 
e+U4U+í67ULU;í69U4U7í=<UdUUTRACCIONU
-
e2U4U5+íJ9ULU;í7=U4U=í:9UdUUCOMPRESIONU
-
e6U4U:í2<ULU7í6<UU+í79UUTRACCION
U
4.6   UTILIZACION DE ARMADURAS.
U
5⇒→+ →∑→δ,∑→+ βθ+ ,⇒→+ √[⇒θγ∑,√√(⇒+ ∑βγ(√,⇔→δ→+ √[⇒−[∑→SmU %sosYmiósodsU uvYU dYaCo%yivFU
MvYómSvFU uvYU sisósodvFU YscdvFU gU lysU FsU ydaiatmU umYmU FvuvYdmYU cmY%mFíU "mFU mYómSyYmFU uysSsoU FsYU
uimomFU vU sFumcamisFíUU nrsóuivFU d'uacvFU SsU mYómSyYmF+ θ[⇒<+ ,β⇒γβθω+ √β∑√ϕ→θω+ γ[∑∑βθ+ δβ+ γ∑→⇒θ(θ(⇒ω+
√0,⇔→θ+δβ+βθγ→δ([θω+βγ√⋅+∆+√[⇒γ(⇒,→√(⇒+θβ+∑βθβ⇒γ→oUmi%yovFUsrsóuivFUSsUmYómSyYmFUd'uacmF

4.6.1 Tipos de armaduras o Cerchas.
U n(aFdsoUivFUSaMsYsodsFUdauvFUSsUmYómSyYmFíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  199  -U
U

FaFómFUlysUómgvYósodsUFvoUydaiatmSmFUsoUimUcvoFdYycQ cDCMLCOLQoIYGCVNLDOLYGC2CQNT LJOEY3CC
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4.6.2 ARMADURAS PARA TECHOS .
C #EYCEJSEMNJEYCLl YOLDOLYCVEJECPECQIDYOJNQQ cDCMLCOsc!vFUómgvYósodsUFvoUmomiatmSmFUsoUsiU
VPEDIGCVLJICOEST rDCLl YOLDCEJSEMNJEYCJLEP .EMEYCLDCLPCLYVEQ IGCQISIC1LSIYCECQIDO DNEQ cDjC

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  200  -U
U
U >746Ay6S34 (-+ m6Amu+ )6 P6-)3.f- 6- 6. P.+-3L S+i3AS6-26 (4+)+ 6- .+ m3-42A(mmof-  )6
dsc!vUsoU a asoSmFíU
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Figura 4.4 Cercha tipo pendolón en el plano
U
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r4+-)3 6. 262A+6)A3L 6826-4of- 64P+mo+. )6. 2AoB-v(.vU gU cvóeaoCoSvivFU FsU vedasosoU imFU
SsovóaomSmFU
estructuras espacialesU vU sodYmómSvFU sFumcamisFU mYdacyimSvFTU yFmSvFU umYmU
FvuvYdmYU dsc!vFU SsU %YmoSsFU iycsFTU cvóvU ivFU lysU FsU uYsFsodmoU soU evSs%mFTU csodYmisFU SsU
dYmoFuvYdsTUmySadvYavFUgUsFdmcavosFUSsUFsY acavíU"vFUóasóeYvFUFsUcvoFdYygsoUcvoUsisósodvFU
2(7(.+A64L )6 46mmof- moAm(.+A 3 m(+)A+)+ )6 +m6A3 sFdYycdyYmiTU* sYUSsdmiisUSsUyoavosFUsoUimU
Ma%yYm,íU
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Figura 4.5 Cercha espacial en la terminal de transportes de Medellín

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  201  -U
U

Figura 4.6 Esquema de una estructura tipo tensigrity o espacial
4.6.3  ARMADURAS PARA PUENTES.
"mUcsYc!mUsFUyomUSsUimFUMvYómFUsFdYycdyYmisFUóCFUmótFEónYeXY rvósó Ye Fó cjevXdrccE3e sY
uysodsFUSsUiycsFUuslysCmFUgUósSamomFTUusYvUSaFsCmSvFUumYmUiycsFUaodsYósSamFU!mFdmUivFU7:UóUSsUiytU
FaóuisíUU
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Figura 4.7 Puente de cerchas vehicular, tablero inferior
I!vYmUlysU siUmcsYvU sFdYycdyYmiUsFdCUSsUóvSmTU misUimUusomUósocavomYUyomUSsUFyFUómgvYsFU
xYeXóZóv1 vr cjevXdrccE3e Ye YF XóFFYdf Ye tEYaóv tslysCmFUgUimUMmcaiaSmSUSsUdYmFimSvUmiUFadavUumYmUFyU
mYómSv0UsFdvUisUusYóadsUcvóusdaYUcvoUivFUuysodsFUSs cjecdYXj tdYYvpjdaósjf Ye vEXEjv Ee(3vtEXjv sY Fó
%sv%YmM'mUomcavomiTUvUcymoSvUsiUMmcdvYUdasóuvUSsUcvevXdrccE3e Yv reó xódEóiFY presónYeXóF tódó Fó
jidóz CjdnóFnYeXY Fó sEvtjvEcE3e sY Fjv trYeXYv sY csYc!mUsFUcvóvUFsUóysFdYmUsoUimUMa%yYmU;íJU8sU
cvivcmoUSvFUcsYc!mFUumYmisimFUlysUFsUmYYavFdYmoUsoddY vyR Fó XdóevnEvE3e sY Fóv cód5óv sY Fjv xY(ycrFjFU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  202  -U
U
46 u+m6 6- )34 2oP34a )6 2+7.6A3 o-16Ao3A *.+ 13AS+UóCFUcvó#o,UgUSsUdmeisYvUFyusYavYTUFs%#oUsiU%CiaevU
FveYsUsiUcmycsUivUusYóadmíUU
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Figura 4.8 estabilización lateral de las cerchas de tablero inferior
08o426- 2+S7oI- P(6-264 6.6y+)34 P6+23-+.64L )3-)6 46 A6+.ot+ .+ m3S7o-+mof- )6 S+26Ao+.64
m3S3 +m6A3L S+)6A+ i u3ASovf-L y6A 1ov(A+ ;n<n

Figura 4.9 Viga Vierendel de acero usada en puente peatonal
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yoavosFU FsU !mcsoU uvYU ósSavU SsU iCóaomFU iimómSmFU cmYdsimFTU lysU uysSsoU sFdmYU mdvYoaiimSmFTU
Ysómc!mSmFUvUFviSmSmFUcvoUivFUsisósodvFUSsUimUsFdYycdyYmíUU
U "mFUMysYtmFUlysU mUmUFvuvYdmYUFsUsrsYcsoUFveYsUimFUyoavosFíUU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  203  -U
U
U niUusFvUSsUivFUsisósodvFUsFUSsFuYscameisUsoUcvóumYmmof- m3- .+4 m+Av+4  +P.om+)+4n
U vóvU cvoFscysocamU SsU imFU cvoFaSsYmcavosFU modsYavYsFTU ivFU sisósodvFU SsU imU mYómSyYmU FvoU
cysYuvFUFvósdaSvFUmUSvFUMysYtmF0UsFdvUlyasYsUSscaYUlysUcmSmUsisósodvUFvivUuysSsUsFdmYUFvósdaSvUmU
26-4of- 3 + m3SPA64of-n
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4.7  PROBLEMAS RESUELTOS.

Ejercicio #8

IYómSyYmUdauvU31IPPíU8sUSsFsmUSsdsYóaomYUsiU sFMysYtvUm(amiUUsoUcmSmUyomUSsUimFUemYYmFíU
*F0P>d> d0 c>8 Drd>8C

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  204  -U
U
V
M
(+)

Solución.-
PASO 1) DETERMINACION ESTATICA
U U U Y )U4UeUIU2UoUHUYUUU
U U U Y
)U4U2+UIU2U*+2,UHU6U4U:UU)8>8PIP) >U
U
PASO 2) CÁLCULO DE REACCIONES DE APOYO
>D/nD )>DU
∑UGU4U:U⇒UUG
IUIU:í9U4U:U⇒U HA = 0.6 tU
∑UF
IU4U:U⇒U:í;U*;,UHU:í7U*9,UIU:í9U*6,U4U:U⇒UU VB = 0.23 tU
∑UF
EU4U:U⇒U+2U/ IUIU:í;*=,UIU:í7*9,UIU:í9*6,U4U⇒U VA = 0.67 t
U
U >DP1>"U ∑U/U4U:U⇒U:í9JUIU:í;UIU:í7UHU:í26U4U:U
U
PASO 3) ESFUERZOS EN LAS BARRAS
UUUUUUUUUUUUUUUUUUUUUU U U U UU >D/nD )>DU
U
U ymoSvUFsUSsFcvovcsUFsodaSvUSsUsFMysYtvFUmFyóaYUPY→√√(⇒⋅+
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  205  -U
U

NUDO 1 (SUPUESTO)
UU U U U UUUUUUUUUUUUUUUUU NUDO 1 (REAL)
U

UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
                                   U


U

U U
U
UU U U U U Pm%U θU4U6V2U
6

8soUθU4UUUUUUUUUU4U:í=6U
6í9+U
U
vFUθU4UUUUUU2UU
UUUU4U:í77U
6í9+U
∑UGU4U:UUUU:í9UHUe
+UHU:í77Ue+2U4U:U
∑U/U4U:UUU∴UU:í9JUHU:í=6Ue
+2U4U:U⇒Ue +2U4U5:í=+UdU
⇒Ue
+U4U5:í+7UdU
vóvUFsUuysSsUveFsY mYUsoUcmSmUoySvUFsUSaFuvosUFvimósodsUSsU2UscymcavosFU*U∑GU4U:UgUU
∑/4:,U∴ ymoSvUFsUsFcv%sUyoUoySvUSsesUcyaSmYFsUlysUmUsFsUoySvUcvocyYYmoUmUivUFyóvU2UemYYmFU
SsFcvovcaSmFíUUU
NUDO 2 (SUPUESTO)   NUDO 2 (REAL)

U


UUUUUU
U
⇒U:í+7UHUe
2U4U:U⇒Ue 2U4U5:í+7UdU
∑U/U4U:U⇒Ue
+6U4U:

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  206  -U
U
               NUDO 12 (SUPUESTO)        NUDO 12 (REAL)

UUUUUUUU
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U U U U U UUUUUUU2U
8soUϖU4UUUUUUUUUUUUU4U:í770U vFUϖU4U:í=6U
U U U U U UUUUU6í9+U
∑UGU4U:U⇒U:í=+U*:í=6,UHUe
++UHU:í77Ue+;U4U:U
∑U/U4U:U⇒U:í=+U*:í=6,UIUe
+;U*:í=6,U4U:U
⇒Ue
+;U4U:í=+UdU
⇒Ue
++U4U5:í=<+UdU
U
        NUDO 11 (SUPUESTO)         NUDO 11 (REAL)

UUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU U
U
∑UGU4U:U⇒U:í=<+UHUe
+:U4U:U
⇒Ue
+:U4U5:í=+UdU
∑U/U4U:U⇒Ue
+7UHU:í;U4U:U
⇒Ue
+7U4U5:í;UdU
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  207  -U
U

              NUDO 7 (SUPUESTO)   NUDO 7 (REAL)

UUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU U
U
8soUθU4U:í=60U vFUθU4U:í77U
∑UGU4U:U⇒Ue
9UHUeJU*:í77,U4U:U
∑U/U4U:U⇒U:í26UHUe
JU*:í=6,U4U:U
⇒Ue
JU4U5:í2=UdU
⇒Ue
9U4U:í+7;UdU
U

Ejercicio #9
U
+ ∆⇒→⇔(•→∑+⇔→+→∑→δ,∑→+γ([+∆PP⋅++Fγ[δ[+δβ+⇔[θ+D,δvF,U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  208  -U
U
Solución.-
PASO 1) DETERMINACION ESTATICA
U U U Y )U4UeUIU2oUHU6U4U:UU
UU U U Y
)U4U+JUIU2*+:,UHU6U4U:UUUU)8>8PIP) IU
PASO 2) CÁLCULO DE REACCIONES DE APOYO
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  209  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  210  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  211  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  212  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  213  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  214  -U
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U U ∑UGU4U:U⇒U5U+í=7UHU2í:<U*:í=<,UHUe ;U4U:U⇒Ue ;U4U:UdUU
U U ∑U/U4U:U⇒UIU:í<;UHUe
+JU*:í;7,U4U:U⇒Ue +JU4U2í:<UdU
NUDO 5 (SUPUESTO)                                     NUDO 5 (REAL) U

UUUUUUUUUUU e7UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe 7U4U:í<;UdUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
U U
UUUU
UUU
UUUUUUU:UdUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU:UdUUUUUUUUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUUUUUUU>UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU>U
UUUUUUU
U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUU:í<;UdUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU:í<;U
UU
U U ∑U/U4U:U⇒U:í<;UHUe
7U4U:U⇒Ue 7U4U5U:í<;UdU
U

Ejercicio #11
U
+ ∞β+(δβ+δβγβ∑(⇒→∑+β⇔+βθ−,β∑•[+β⇒+⇔→+]→∑∑→⌠+Fγ[SvUSsUimFUU8sccavosF,U
U
Solución.-
PASO 1) DETERMINACION ESTATICA

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  215  -U
U
U U U U YU4U6UIU6U4U:U
U U U Y
)U4U2+UIU2*+2,UHU6U4U:UUUUUUU)8>8PIP) >UU
U
PASO 2) CÁLCULO DE REACCIONES DE APOYO
U U ∑UGU4U:U⇒UG IUIU:í2U4U:U⇒U HA = 0.2 t
U U ∑UF IU4U:U⇒U:í;U*;,UHU:í6U*9,UIU*:í2,U+í66UIU+2U/EU4U:U
UUUUUUUUUUUUUUUUUUUUUUUUUUUU U UU∴
VB = 0.26 tU
U U ∑UF
EU4U:U⇒U+2U/ IUIU:í;U*=,UIU:í6U*9<UIU:í2U*+í66,U4U:U
UUUUUUUUUUUUUUUUUUUUUUUUUUUU U UU∴
VA = 0.44 t
{ΤDPΤ3<+++∑U/U4U:U⇒U:í;;UIU:í;UIU:í6UHU:í29U4U:U
PASO 3) ESFUERZOS EN LAS BARRAS
+ =⇔+γ[δ[+δβ+⇔→θ+θβ√√([⇒βθ+√[⇒θ(θγβ+β⇒+(⇒γ∑[δ,√(∑+mUimUmYómSyYmUcasYdvFUcvYdsFUUsoUiy%mYsFU
cvo soasodsFUsFdySasóvFUsiUcvYdsU8
+UlysUFsccavomUimFUemYYmFUe+:UgUe6íU

BARRA 10
<+∞+UmUimUSsYsc!m,U
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUU+í66U
U Pm%UθU4UUUUUUUUUUUUUUUUUU0UUUU8soUθU4U:í77UUUU0UUU vFUθU4U:í=6U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU2U
U
U ∑UF
U;U4U:U⇒U5*:í29,U*9,UIU:í2U*+í66,UIUe+:
(UU*;,U4U:U
U U UUUUUUUUUUU∴e
+:
(U4UU5U:í;7UdU
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
+:
(UUUUUUUUUUUUUUUUUUUUUU5U:í;7U
U vFUθU4UUUUUUUUUUUUUUUUU⇒Ue
+:U4UUUUUUUUUUUUUUUUUUUU⇒UUe +:U4U5U:í77UdU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
+:UUUUUUUUUUUUUUUUUUUUUUUUUU:í=6U
U U U ⇒Ue
+:U4U:í77UdUUUUUUCOMPRESION

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  216  -U
U
BARRA 3<+∞+UmUimUatlyasYSm,U
U
U U U U Pm%U θU4UUUUUUUUUS+UUUUUUMU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUU;U
U U UUUUU U U S
+U4U2í99UóU U UUUUUU
U U ∑UF
U;U4U:U⇒U:í;;U*;,UIU*:í2,U2í99UIUe6U*2í99,U4U:U
U UUUUUUUUUUUUUUUUU U U⇒Ue
6U4U:í;9UdUUUUUTRACCIONU
Ejercicio #12  UUU
+ ∞(+ θβ+ δβθβ→+ δβγβ∑(⇒→∑+ ,⇒→+ ]→∑∑→+ (⇒√⇔(⇒→δ→ω+ δβθ,FU SsiU cvYdsU FsU U SsFcvóuvosU gU FsU dvómU
[β⇒γ[θ+γ→⇔+′,β+θβ+β⇔((⇒β+,⇒→+√[[⇒β⇒γβ+Fγ[δ[+SsUimFUFsccavosF,U
U

Solución.-
PASO 1) DETERMINACION ESTATICA
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  217  -U
U
U Y )U4UeUIU2oUHU6U4U:UU
U Y
)U4U+7UIU2*<,UHU6U4U:UUU)8>8PIP) IU
PASO 2) CÁLCULO DE LAS REACCIONES U
U ∑UGU4U:U⇒UG
IUHUGEUIU=:U4U:U
U ∑UF
EU4U:U⇒U7UG IUHU9:U*6,UHU<U*7:,U4U:U
U UUUUUUUUUUUUUUUU⇒U
HA = -126 kU
U ∑UF
U4U:U⇒U57:U*6,UIU<U*9:,UHU+2U/IUHU7UGIU4U:U
U UUUUUUUUUUUUUUUU⇒U
VA = 110 k

⇒UHB = 206 kU
{ΤDPΤ3<+∑U/U4U:U⇒U++:UIU9:U57:U4U:U
PASO 3) ESFUERZOS EN LAS BARRAS U

NUDO 5 (SUPUESTO)                                          NUDO 5 (REAL)
U
U
U
UUUUUUUUUUUUUUUUUUUUUe
7UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUUUUUUUUUUUUUUUe7U4U:U$U
U
UUUUUUUUUUUUUUUUUUUUUUUUθUUUUUUUUUUUUUUUUUUUU
=:U$UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU =:U$U
UUUUUUUUUUUUUUUUe
;UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe;UUUUUUUUUU
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUU7UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU+2U
UU 8soUθU4UUUUUUUUUUUUUU4U:í6=UUU0UUU vFUθU4UUUUUUUUUUUU4U:í<2U
+6
+6U
UU∑UGU4U:U⇒Ue
;UHU:í<2Ue7UHU=:U4U:U
UU∑U/U4U:U⇒U:í6=Ue
7U4U:U
UUUUUUUUUUUUUUU UU⇒Ue
7U4U:U$U
UU U U ⇒Ue
;U4U5U=:U$UUU >F31n8)>DU
βγβ∑(⇒→∑+]
9<+
U U U 8soU ϖU4UUUUU+2UUUUU
U4U:í<2UUUU0UUUUU vFU4U:í6=U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUU UUUUUUU UUUUUU+6U U
UU U U U?UUUUU4UUUUUUU9UUUUUUU⇒UUU?U4U2í7UUU*!auvdsoyFm,UU U
UUUUUUUUU U UUUUUUUUUUUUU7UUUUUUUUUUUUUU+2U
U U U ⇒USU4U2í7ULU8soUϖU4U2í6+UóU U
U U ∑UF
U6U4U:U⇒U5+29U*7,UHU++:U*9,UIU9:*6,UHUe9U*2í6+,U4U:UU
UUUUUUUUUUUUUUUUUUUUUUU U UUUUUUU∴U⇒Ue
9U4U9;U<;U$UUUTRACCION

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  218  -U
U
Ejercicio #13
+ ∆∑→δ,∑→+ ⇔→⇒→+ γ([+ G[iβ+ δβ+ →⇔(→+ ,γ(⇔(•→√(⇒+ β⇒+ ⇔→+√[⇒θγ∑,√√(⇒+ δβ+ γβ√ϕ[θ+ ∞=+
FΤ∞P∆∆D+∞PDPΤ∞+{ΤP=∞⋅++
U

Solución.-

PASO 1) DETERMINACION ESTATICA
U
U U Y
)U4UeUIU2oUHU6U4U:UU
U U Y
)U4U+6UIU2*=,UHU6U4U:UUU)8>8PIP) IU
U
PASO 2) CÁLCULO DE LAS REACCIONES U
UUUUUUUUUUUUUUUUUUUU∑UGU4U:U⇒UG
IU5U:í7U4U:U
UUU∑UF
IU4U:U⇒U+U*2,UIU:í7U*2,UHU:í=U*9,UIU=U/EU4U:UU⇒U VB = 0.725 tU
∑UF
EU4U:U⇒U=U/ EU5+U*9,UIU:í7U*2,UIU:í=U*2,U4U:U⇒U VA = 1.075 tU
{ΤDPΤ3<++
∑U/U4U:U⇒U+í:J7UIU+UIU:í=UHU:íJ27U4U:U⇒U/U
U
PASO 3) ESFUERZOS EN LAS BARRAS U
a) cvYdsU8UIU8UlysUFsccavomUmUimFUemYYmFUe=UgUe+UsoUimUumYdsUSsYsc!mU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  219  -U
U
U
∆F∆5∆+=D+=Q53EΤ+
U
nFUuvFaeisUcvoFaSsYmYU#oacmósodsUimUumYdsUFadymSmUmUimUSsYsc!mUSsiUcvYdsU8U5U8U
U U ∑UF
=U4U:U⇒Ue +U*+,UIU:í7U*+,UHU:í=U*;,UIU:íJ27U*9,U4U:U
U U UUUUUUUUUUUUUUUU⇒Ue
+U4U+í97UdUUUUUP1I  )>DU
U ∑UF
2U4U:U⇒U5Ue
(
=
U*+,UIU:í7U*2,UHU:í=U*;,UIU:íJ27U*9,U4U:U⇒Ue
(
=
U4U5U2í+7UdU
⇒UUUUUUUUUUUUUUUUUUUUUUUUUUUUe
(
=
UUUUUUUUUUUUUUUUUUU5U2í+7U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUcvFU∞U4UUUUUUUUUUUUU⇒Ue
=U4UUUUUUUUUUUUUUU4UUe
(
=
U4UU5U2í;+U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
=UUUUUUUUUUUUUUUUUUU:í=<2U
[∑+⇔[+γ→⇒γ[<++++
          e=U4U2í;+UdUU >F31n8)>DU
b) cvYdsU8+U5U8+UlysUFsccavomUmUimFUemYYmFUe9Ue+2Ue6UUsoUimUumYdsUatlyasYSmU
U
∆F∆5∆+=D+=Q53EΤ++++++++++++++++++++

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  220  -U
U
U
∑UF
6U4U:U
[∑+⇔[+γ→⇒γ[<+
5Ue
(
9
U*2,UIU:í7U*2,UIU+U*2,UHU+í:J7U*;,U4U:U⇒Ue
(
9
U4U5U+í6UdU
U ⇒U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
(
9UUUUUUUUUUUUUU
UUUUUUUUUUUU5U+í6UUUU
UU U UcvFU ∞U4UUUUUUUUUUUUUU⇒Ue
9U4UUUUUUUUUUUUUUUUUU4U5U+í;7JUdUU U U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
9UUUUUUUUUUUUUUUUUUUUUUU:í=<2U
++ [∑+⇔[+γ→⇒γ[<+
e
9U4U+í;7JUdUUU >F31n8)>DU
U voFaSsYmoSvUimUumYdsUSsUimUmYómSyYmUFadymSmUmUimUSsYsc!mUSsiUcvYdsU8
+UIU8+íU
∑UF
9U4U:U⇒Ue 6U*+,UIU:íJ27U*2,U4U:U⇒Ue 6U4U+í;7UdUUUP1I  )>DU
U voFaSsYsóvFUSsUoys vUimUumYdsUFadymSmUmUimUatlyasYSmUSsiUcvYdsU8
+UIU8+UU
U U ⇒UU∑UF
JU4U:U⇒U5:í7U*2,UHU+í:J7U*2,UIUe6U*2,UIUe
(
+2
U*2,U4U:U
+ β∑[<+
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
(
+2UUUUUUUUUUUUUU
UUUUUUUUUUUU5U:í=UUUU
UU U UcvFU ∞U4UUUUUUUUUUUUUU⇒Ue
+2U4UUUUUUUUUUUUUUUUUU4U5U:í=<JUdUU U U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
+2UUUUUUUUUUUUUUUUUUUUUU:í=<2U
+ [∑+⇔[+γ→⇒γ[<+
U U U U e
+2U4U:í=<JUdUUU >F31n8)>DU
U
c)UcvYdsU82UIU82UlysUFsccavomUimFUemYYmFUe=TUe<TUe2TUsodvocsFUUU
U
∆F∆5∆+=D+=Q53EΤ+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  221  -U
U
∑UF+U4U:UU∴+*2,UIUe <U*2,U4U:U⇒Ue <U4U+UdUUUP1I  )>DU
U
d) cvYdsU86UIU86UlysUFsccavomUmUimFUemYYmFUeJTUe+:UgUe2U
U
∆F∆5∆+=D+=Q53EΤ+
U U
∑UF
=U4U:U⇒U5Ue 2U*+,UHU+í:J7U*2,UIU:í7U*+,U4U:U⇒Ue 2U4U+í97UdUUP1I  )>DU
∑UF
6U4U:U⇒U5Ue
(
J
U*2,UIU:í7U*2,UHU:í=U*2,UIU:íJ27U*;,U4U:U⇒Ue
(
J
U4U5U+í+7UdU
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
(
JUUUUUUUUUUUUUU
UUUUUUUUUUUU5U+í+7UUUU
UU U UcvFU ∞U4UUUUUUUUUUUUUU⇒Ue
JUU4UUUUUUUUUUUUUUUUUU4U5U+í2=<UdUU U U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
JUUUUUUUUUUUUUUUUUUUUUU:í=<2U
+ [∑+⇔[+γ→⇒γ[<+
e
JU4U+í2=<UU >F31n8)>DU
∑UF
2U4U:U⇒U5Ue
(
+:U
*+,UHU+í+7U*+,UHU+í:J7U*2,UIU:í7U*+,U4U:U⇒Ue
(
+:
U4U5U:í7UdU
U U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
(
+:UUUUUUUUUUUUUU
UUUUUUUUUUUU5U:í7UUUU
UU U UcvFU ∞U4UUUUUUUUUUUUUU⇒Ue
+:U4UUUUUUUUUUUUUUUUUU4U5U:í79UdUU U U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
+:UUUUUUUUUUUUUUUUUUUUUU:í=<2U
U
⇒e
+:U4U:í79UU >F31n8)>DU
U
e) cvYdsU8;UIU8;UlysUFsccavomUmUimFUemYYmFUe;UgUe7U
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  222  -U
U
U
U
U
U
∆F∆5∆+=D+=Q53EΤ+
U
U
U
∑UF
9U4U:U⇒U5Ue ;U*+,UHU:í7U*+,UHU+í:J7U*9,U4U:U⇒Ue ;U4U+í;7UdUUP1I  )>DU
∑UF
;U4U:U⇒U5Ue
(
7
U*+,UIU:í7U*2,UHU+í:J7U*9,UIU+U*;,U4U:U⇒Ue
(
7
U4U5U+í;7UdU
U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
(
7UUUUUUUUUUUUUUU
UUUUUUUUUUUU5U+í;7UUUU
UU U UcvFU ∞U4UUUUUUUUUUUUUU⇒UUe
7UU4UUUUUUUUUUUUUUUUUU4U5U+í927UdUU U U
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUe
7UUUUUUUUUUUUUUUUUUUUUUU:í=<2U
U
⇒UUUUUe
7U4U+í927UdUUU >F31n8)>DU

f)
cvYdsU87UIU87UlysUFsccavomUmUimFUemYYmFUeJTUe++TUe+6UgUe;U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  223  -U
U
U
∆F∆5∆+=D+=Q53EΤ+
U
∑UF
9U4U:U⇒U5Ue ++U*2,UHU:í7U*+,UHU+í+7U*+,UHU:í7J7U*2,UHU+í;7U*+,UIU:íJ27UU4U:UUUUUUUUUUUUUUUUUUU
⇒Ue
++U4U+í;7UdUUUP1I  )>DU
∑UF
JU4U:U⇒U:í=U*2,UHUe+6U*2,UIU:í;U*2,UIU:í=U*+,UHU+í;7U*2,UIU:íJ27U*;,U4U:UU
⇒Ue
+6U4U:UdU
+ ∆+√[⇒γ(⇒,→√(⇒+)β[θ+,⇒+∑βθ,β⇒+δβ+⇔[θ+βθ−,β∑•[θ+β⇒UimUemYYmFíU
U
1.075 t
0.5 t
e
U=U42í;+Ud
e
UJU4+í2=<Ud
e+U4+í97Ud e 2U4+í97Ud e 6U4+í;7Ud e ;U4+í;7Ud
0.725 t
eU<U4+í:Ud
e
U++U4+í=Ud
e
U+6U4:Ud
e
7U
4+í927Ud
e
U9U
4+í;7JUd
e
UU+2U4:í=<JUd
e
U+:U
4:í79Ud
1 t
0.5 t
0.8 t
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  224  -U
U
U niUsrsóuivUóysFdYmUimFUFa%yasodsFU sodmrmFUsoUsiUm⇒∴⇔(θ(θ+[∑+β⇔+γ[δ[+δβ+θβ√√([⇒βθ<+
U niU mivYU SsiU sFMysYtvU SsU yomU SsdsYóaomSmU emYYmU ovU SsusoSsU soU %sosYmiU SsU vdYvFU mivYsFU SsU
sFMysYtvFUuYs amósodsUcmicyimSvFTUsFUSscaYUyoUsYYvYUaoacamiUovUmMscdmUuvFdsYavYósodsíU
U nFUuvFaeisUSsdsYóaomYUsiUsFMysYtvUSsUyomUemYYmUFaoUoscsFaSmSUSsUYsFvi sYUdvSmUimUmYómSyYmíU
U 8aoUsóemY%vTUSsUsFdmFU sodmrmFUsFUuYsMsYaeisUydaia•→∑+θ(,⇔γ∴⇒β→β⇒γβ+β⇔+γ[δ[+δβ+⇔[θ+⇒,δ[θ+
gUSsUimFUFsccavosFíU

Ejercicio #14
U 8sUuaSsUSsdsYóaomYUivFUsFMysYtvFUsoUcmSmUyomUSsUimθ+]→∑∑→θ++Fγ[δ[+Yβ⇒β∑→⇔+

Solución.-

U "
(
e+
U4U*52UIU:,U4U520UU"
g
e+
U4U*56UIU2,U4U570UU"
t
e+
U4U*:UHU+,U4U+U
U
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U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  225  -U
U
U
U 5U2U
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Ejercicio #15
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U 8sUuaSsUSsdsYóaomYUivFUsFMysYtvFUsoUcmSmUyomUSsUimFUemYYmFíU
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UUUUUUUUUUUUUUUUUUUUUUU"
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e6
U4UJí9+UóU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  226  -U
U
U
+ =⇒+√→δ→+,⇒[+⇒,δ[+βθ+[θ(]⇔β+→⇔(√→∑+φ+β√,→√([⇒βθ<+
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                                                                                                       C 3 = 0.74

C
1 = - 1.31      ;   C2 = 0.205U
UUU
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U4U7í96UdUUUUTRACCIONU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  227  -U
U
Ejercicio #16 UU
+ βγβ∑(⇒→∑+⇔[θ+βθ−,β∑•[θ+β⇒+√→δ→+,⇒→+δβ+⇔→θ+]→∑∑→θ⋅U

Solucion.-
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U UUe
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                         "AU4U2UIU+U4U+UUUUUUUUUUUUUUUUUUUUUUU"AU4U2UIU+U4U+UUUUUUUUUUUUUUUUUUUUUUU"AU4U5+U5+U4U52U
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U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  228  -U
U
5U6U +UHU*5;,U 2UHU:U 6UHU*56,U4U:U
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4.8 PROBLEMAS  PROPUESTOS.
U
Problema #1UUU
U 8sUuaSsUSsdsYóaomYUsiU mivYUSsiUsFMysYtvUsoUimFUemYYmFíUU
U
U
Respuesta:
U U b
1 = 0.095 t  Comp.
b
2 = 3.120 t  Tracc.
b
3 = 0.279 t  Tracc.
b
4 = 2.553 t  Comp.
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  229  -U
U

Problema #2
UUU8sUuaSsUSsdsYóaomYUsiU mivYUSsiUsFMysYtvUsoUimFUemYYmFíU

Respuesta:    b1 = 6.684 t Tracc.
b
2 = 7.186 t CompíU
Problema #3UUU8sUuaSsUSsdsYóaomYUsiU mivYUSsiUsFMysYtvUsoUimFUemYYmFíU
U
U
UUUU
Respuesta:    b
1 = 6.111 t Tracc
b
2 = 6.630 t Comp.
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  230  -U
U
Problema #4UUU
U 8sUuaSsUSsdsYóaomYUsiU mivYUSsiUsFMysYtvUsoUimUemYYmíU
U
  Respuesta:
                                                     b
1 = 0.167 t CompíU
U
Problema #5UUUU
U 8sUuaSsUSsdsYóaomYUsiU mivYUSsiUsFMysYtvUsoUimFUemYYmFíU
U
Respuesta:
                                                                      b
1 = 0 t
b
2 = 2 t Comp.
b
3 = 2 t Comp.
b
4 = 2.066 t TraccíU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  231  -U
U
Problema #6UUUU
U 8sUuaSsUSsdsYóaomYUsiU mivYUSsiUsFMysYtvUsoUimFUemYYmFíU
U
Respuesta:
R
1 = 1 t                 T b1 = 1.665 t Comp.
R
2 = 1.333 t T b2 = T b3 = 0
R
3 = 0 t
R
4 = 2 t T b4 = 0
R
5 = 4 t T b5 = 4.47 t Comp.
R
6 = 6.333 t   T b6 = 6.333 t Tracc
.
Problema #7UUUU
U 8sUuaSsUSsdsYóaomYUsiU mivYUSsiUsFMysYtvUsoUimFUemYYmFíU
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  232  -U
U
  Respuesta:
                                                                          b
1 = 2.235 t Comp.
                                                                          b
2 = 2.50 t Comp.
                                                                          b
3 = 2.06 t Comp.

4.9   EVALUACION DIAGNOSTICO DEL CAPITULO.

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U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  233  -U
U
U
U
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CAPITULO 5

U
LINEAS DE INFLUENCIA
U
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5.1   OBJETIVO GENERAL.
U
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5.2   OBJETIVOS ESPECIFICOS.
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U  voFdYyaYUimUso vi sodsUSsUQUgUFU
U 1scvovcsYUimFUFsccavosFUcY'dacmFUSsUQUgUFíU

5.3   PLANTEAMIENTO DEL PROBLEMA.
U
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Figura  5.1

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  234  -U
U

Figura 5.2
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cvYYsFuvoSsUivFU mivYsFU/
IU4U/EU4U:í7UdUsoUimUMa%yYmU7í2UimUcmY%mU3U4U+UdUovUFsUsocysodYmUsoUimUóadmSU
sYF cFódj u ó Yvó erYxó tjvEcE3e sY Fó cód5ó cjddYvuvoSsUvdYvFU mivYsFUSsU/
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cmYYsdsYvFUgUimFUdYvumFUóaiadmYsFUsoUómYc!mUcvóumcdmUlysUmcd#moUFveYsUivFUuysodsFTUFvoUsodYsUvdYvFTU
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5.4   LINEAS DE INFLUENCIA.
geXYv sY EeXdjsrcEd reó sYpEeEcE3e sY FyeYó sY EepFrsocamTU cvoFaSsYmYsóvFU siU Fa%yasodsU
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U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  235  -U
U
UUUUUUUUUUUUUUUUUUUUUUUUU3U4U+UdU
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UUUUUUUUUUUIuvgvíU

Ejercicio #1

=⇒+β⇔+βαβ∑√(√([+′,β+θβ+,βθγ∑→+→+√[⇒γ(⇒,→√(⇒+θβ+,βδβ+[]θβ∑)→∑+′,β+β⇔+)→⇔[∑+δβ+⇔→+∑β→√√(⇒+
)β∑γ(√→⇔+ β⇒+ ∆+ δββ⇒δβ+ δβ+ ⇔→+ [θ(√(⇒+ ′,β+ [√,→+ ⇔→+ √→Y%mU 3U FveYsU imU a%míU 8sU dYmdmU sodvocsFU SsU
δβγβ∑(⇒→∑+,⇒→+β√,→√(⇒+′,β+(⇒δ(′,β+⇔→+)→∑(→√(⇒+δβiU mivYUSsUimU sYdacmiUlysUcymoSvUimUcmY%mU3UlysU
dYmoFadsU*FsUUóys m,UFveYsUimU a%mUIEíU

Solución:

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  236  -U
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                       Figura 5.4

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  237  -U
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5.5   PROCEDIMIENTO DE ANALISIS DE LAS LINEAS DE IN FLUENCIA.
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  238  -U
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5.6   LINEA DE INFLUENCIA DE ESFUERZO CORTANTE.
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Figura 5.5U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  239  -U
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Ejercicio #2

βγβ∑(⇒→∑+⇔→+[θ(√(⇒+√[∑γ→⇒γβ+δβ⇔+γ∑β⇒⋅+
Solución:
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′,β+θβ+,βθγ∑→<+
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Ejercicio #3

=⇔+√→(⇒+′,β+θβ+[]θβ∑)→ω+γ∑→⇒θ(γ→+[∑+⇔→+)(2→+δβ+,β⇒γβ+∆Eω+θβ+(δβ<+++
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√[∑γ→⇒γβ+β⇒+⇔→+θβ√√(⇒+∞+θβ→+⌠+θ(+θβ+,βδβ+
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θβ√√(⇒+∞+θβ→+β⇔+→θ+β′,βC[+δβ+⇔[θ+[θ(γ()[θ⋅+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  240  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  241  -U
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5.7   LINEA DE INFLUENCIA DE MOMENTO FLECTOR.
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Figura 5.6

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  242  -U
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Ejercicio #4
O 4DcD3lóou3OiuO g.ó-ó6oOMDiOc3DoO
Solución:
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  244  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  246  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  247  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  248  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  249  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  250  -U
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Ejercicio #7

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Solución:
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Ejercicio #8

=⇔+γ∑β⇒+δβ+√→∑2→θ+√→(⇒+δβ+√→∑2→θ+γ∑→⇒θ(γ→+[∑+⇔→+)(2→+δβ+,β⇒γβ+∆Eω+θβ+(δβ<+++++++++++++++++++++++UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  251  -U
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Solución:
U 3 +U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  252  -U
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Ejercicio #9

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Solución:
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  253  -U
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Ejercicio #10
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Solución:
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  254  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  255  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  257  -U
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Solución
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  258  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  260  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  263  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  265  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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5.9   PROBLEMAS PROPUESTOS.

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  272  -U
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CAPÍTULO 6

CABLES

6.1  OBJETIVO GENERAL.

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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c)  4óiFYv dYcXEFyeYjv Ye crujv treXjv sY EepFY7E3e vjuvYdmoUcmY%mFUcvocsodYmSmFíU
U
Figura 6.3

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  274  -U
U
6.4  ANÁLISIS DE LOS TIPOS DE CABLES.
6.4.1 CABLES SOMETIDOS A CARGAS UNIFORMEMENTE DISTRIBUIDA S.
"vFUUcmeisFUuYvuvYcavomoUyoUósSavUóygUsMacmtUumYmUFvuvYdmYUsiUusFvUóysYdvUSsUimFUdYmesFUvU
cyeasYdmFUSsUuysodsFUSsUcimYvFUvUiycsFUóygU%YmoSsFTUyoUuysodsUcvi%modsUsFUyoUsrsóuivUd'uacvUsoUsiU
lysU imU cyeasYdmU sFdmU FyFusoSaSmU SsiU cmeisU yFmoSvU yomU FsYasU SsU cvi%modsFU ómFU csYcmovFU gU
yoaMvYósósodsUFsumYmSvFíU
3mYmUmomiatmYUsFdsUuYveisómUSsdsYóaomYsóvFUuYaósYvUimUMvYómUSsUyoUcmeisUFvósdaSvUmUyomU
cmY%mU sYdacmiU
ο
ωTUSaFdYaeyaSmU!vYatvodmiósodsUsoUMvYómUeaMvYósUcvóvUFsUóysFdYmUsoUimUMa%yYmU9í;U
*m,Umly'UivFUsrsFUx, yUdasosoUFyUvYa%soUsoUsiUuyodvUómFUemrvUSsiUcmeisTUSsUómosYmUlysUimUusoSasodsUsFU
csYvUsoUsFdsUuyodvíUniUSam%YmómUSsUcysYuvUiaeYsUSsUyoUuslysCvUFs%ósodvUSsUcmeisTUcvoUivo%adySUs
∆U
FsUóysFdYmUsoUimUMa%yYmU;í2U*e,íU vóvUimUMysYtmUSsUγβ⇒θ(⇒+β⇒+β⇔+√→]⇔β+√→](→+√[⇒γ(⇒,→β⇒γβ+γ→⇒γ[+β⇒+
→2⇒(γ,δ+√[[+β⇒+δ(∑β√√(⇒+→+⇔[+⇔→∑2[+δβ+⇔→+⇔[⇒2(γ,SUSsiUcmeisTUsiUcmóeavUFsUSsovdmUcvoUT∆UsoUsiU
Sam%YmómUSsUcysYuvUiaeYsTUimUcmY%mUSaFdYaeyaSmUFsUYsuYsFsodmUuvYUFyUMysYtmUYsFyidmodsU
R a%ymiUmU
ο
ωx
∆U
lysUmcd#mUmUyomUSaFdmocamUSsU
2Vx
∆USsiUuyodvU>íU
3mYmUMaosFUSsUMmcaiaSmSUSsUcCicyivUmFyóaóvFUimUFa%y(β⇒γβ+√[⇒)β⇒√(⇒+δβ+θ(2⇒[θ<+
VVVV
M  M  M  M
(+)(+)(+)(+)
HHHH
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U 1smiatmóvFUsiUmoCiaFaFUsoUsiUFa%yasodsUcmeisUgUFyUSam%YmómUSsUcysYuvUiaeYsíU
x
y
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Τ +∆Τ
θ+∆θ
θ
Τ
∆s
∆x
∆y
∆x/2
ο(∆x)
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U
U U UUUUUU
Figura 6.4 (a)UU U U U UUUUUUUUUUUU Figura 6.4 (b)U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  275  -U
U
∆⇔(√→⇒δ[+⇔→θ+β√,→√([⇒βθ+δβ+β′,(⇔(]∑([+θβ+[]γ(β⇒β<+
∑=0:
XF U : ,cvF*,*cvF =∆+∆++− θθθ TTT U
∑=0:
YF U :,*,*,* =∆+∆++∆+−θθωθsenTTxTsen
o
U
∑ =0:
OM U :cvF,2V,** =∆+∆−∆∆ xTsenyTxx
o
θθω U
()(δ(β⇒δ[+√→δ→+,⇒→+δβ+βθγ→θ+β√,→√([⇒βθ+β⇒γ∑β+
x
∆UgUUdvómoSvUiaóadsFUcymoSvU:→∆x TUuvYU
cvoFa%yasodsU
:
→∆y 0 :→∆θ UgU :→∆T TUvedsosóvFíU
:
,cvF*
=
dx
Td θ
U U U *+,U
O
dx
Tsend
ω
θ
=
,*U U U *2,U
UUUU UU
θdmo=
dx
dy
U U U U *6,U
⇒γβ2∑→⇒δ[+⇔→+β√,→√(⇒+ω+δ[⇒δβ+P+≤+
G+β⇒+≤⌠ω+γβ⇒β[θ<+
H
FT =θcvF UU4UUUU ddsU U *;,U
U 3vYUivUdmodvUimUcvóuvosodsU!vYatvodmiUSsUimUMysYtmUsoU
cualquier puntoUmUivUimY%vUSsiUcmeisU
usYómoscsU
constante. ⇒γβ2∑→⇒δ[+⇔→+β√,→√(⇒+ω+√[⇒θ(δβ∑→⇒δ[+′,β+P+θβ⇒θ+≤+⌠+β⇒+≤⌠ω+θβ+[]γ(β⇒β<+
xTsen
O
ωθ= UU U U *7,U
()(δ(β⇒δ[+⇔→+β√,→√(⇒+ϖ+β⇒γ∑β+⇔→+β√,→√(⇒++θβ+β⇔((⇒→+P⋅+,θ→⇒δ[+⇔,β2[+⇔→+β√,→√(⇒+φω+
uvSsóvFUvedsosYUimUusoSasodsUsoUcymilyasYUuyodvUmUi[+⇔→∑2[+δβ⇔+√→]⇔βω+θβ→⋅+
H
OF
x
dx
dy
ω
θ
==dmoU UU U *9,U
+ =−β√γ,→⇒δ[+,⇒→+θβ2,⇒δ→+(⇒γβ2∑→√(⇒+√[⇒+
y =0UsoUx =0,UYsFyidmU
2
2
x
F
y
H
O
ω
= U U U U U*J,U
=θγ→++β√,→√(⇒+βθ+√[∑∑βθ[⇒δ(β⇒γβ+→+,⇒→+
parábolaTUimUcvoFdmodsU-GUuysSsUsocvodYmYFsUyFmoSvU
⇔→+√[⇒δ(√(⇒+δβ+−∑[⇒γβ∑→+
y =h UsoUUx = LíUmF'UsodvocsFUdsoSYsóvFíU
h
L
F
O
H 2
2
ω
= U U U U *=,U
+ (⇒→⇔β⇒γβ+θ,θγ(γ,∫β⇒δ[+β⇒+⇔→+β√,→√(⇒+J+γβ⇒β[θ<+
2
2
xL
h
y=
U U U U *<,U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  276  -U
U
+ β+→√,β∑δ[+√[⇒+⇔→+β√,→√(⇒+ω+⇔→+γβ⇒θ(⇒+∴(→+βoUsiUcmeisUvcyYYsUcymoSvU θUsFUóC(aóm0U
sFdvUsFUsoU
x =LUíU3vYUdmodvUSsUimFUscymcavosFU*;,UgU*7,UdsosóvFíU 22
,*LFT
OHMAXω+= U U *+:,U
+ 5θ→⇒δ[+⇔→+β√,→√(⇒+οω+[δβ[θ+γ→](⇒+β∑βθ→∑+β⇔U
MAXT+β⇒+γ∑(⇒[θ+δβ+ωvUsFdvUsFíU
2
,2V*+ hLLT
OMAX
+=ω U U *++,U
+ Dγβθβ+′,β+ϕβ[θ+δβθ∑β√(→δ[+β⇔+βθ[+∑[([+δβ⇔+√→eisTUlysUsFU
uniformeUFveYsUimU longitudU
δβ⇔+√→]⇔β+∫+⇒[+θ[]∑β+θ,+∑[∫β√√(⇒+ϕ[∑(•[⇒γ→⇔⋅+=⇒+∑smiaSmSUyoUcmeisUFvósdaSvUmUFyUusFvUuYvuavUgUiaeYsU
SsUcymilyasYUvdYmUcmY%mTUdvómYmUyomUMvYómUSsUcyY mU
catenaria⋅+∞(⇒+β]→∑2[ω+θ(+⇔→+∑→•⇒+−⇔β√ϕ→+→+√⇔→∑[+
sFUuslysCmTUlysUsFUsiUcmFvUsoUimUómgvY'mUSsUimFUmuiacmcavosFUsFdYycdyYmisFTUsFdmUcyY mUFsUmuYv(aómU
]→θγ→⇒γβ+→+,⇒→+√,∑)→+→∑→]⇔(√→ω+√[[+θβ+δβγβ∑(⇒→+mly'íU
+ =⇒+β⇔+→⇒∴⇔(θ(θ+δβ+⇔[θ+√→]⇔βθ+βθ+([∑γ→⇒γβ+γ→](⇒UcvovcsYUimUivo%adySUdvdmiUSsiUóaFóvTUumYmU
γ→⇔+β−β√γ[+θβ+∑β√,∑∑β+→+⇔→+θ(2,(β⇒γβ+β∑βθ(⇒<+
∫






+=
l
S
dx
dy
L
:
2
+ U U U *+2,U
+ β+ ⇔[θ+ ∑βθ,⇔γ→δ[θ+ δβ+ βθγβ+ →⇒∴⇔(θ(θω+ θβ+ (⇒γβ∑−(β∑β+ ′ysU FaU yoU cmeisU
mantieneU yomU MvYómU
→∑→]⇔(√→ω+⇔→+√→∑2→+,β∑γ→+δβ+⇔→+√,](β∑γ→+δβ+,⇒+,sodsUcvi%modsUvUSsUyomUdYmesUsFdmYCUyoaMvYósósodsU
SaFdYaeyaSmUFveYsUimUivo%adySU!vYatvodmiUuYvgscdmSmUSsiUcmeisíU3vYUdmodvUFaUimUdYmesUlysUFsUóysFdYmUsoU
imUFa%yasodsUMa%yYmUsFdmUFvuvYdmSmUuvYUyomUFsYasUSsUcvi%modsFUcsYcmovFUgUyoaMvYósósodsUFsumYmSvFTUimU
cmY%mUsoUUcmSmUcvi%modsUSsesUFsYUimUóaFómUumYmU%mYm⇒γ(•→∑+′,β+β⇔+√→]⇔β+γβ⇒2→+,⇒→+−[∑→+→∑→]⇔(√→⋅+
ISsóCFUFaUFyuvosóvFUlysUimUdYmesUsFUY'%aSmUgUlysUFsU→⇒γ(β⇒β+⇔→+β⇒δ(β⇒γβ+→∑→]⇔(√→+δβ⇔+√→]⇔βω+
β⇒γ[⇒√βθ+√,→⇔′,(β∑+√→∑2→+)(⇔+
PUSsesUFsYUa%ymiósodsUcvóumYdaSmUuvYUcmSmUcvi%modsíU
U
+ 5θ→⇒δ[+βθγ→+ϕ(γβθ(θ+[δβ[θ+β−β√γ,→∑+β⇔+→⇒∴⇔(θ(θ+sFdYycdyYmiUSsUimUdYmesUvUSsUcymilyasYUvdYmU
sFdYycdyYmU lysU sFdsU iaeYsósodsU FyFusoSaSmU uvYU siU cmeisU mF'U cvóvU FsU óysFdYmU soU imU Ma%U 9í7*m,TU
sFuscamiósodsUFaUimUdYmesUsFdmUFaóuisósodsUFvuvYdmSm0UmF'UcvóvUFvuvYdmSmUuvYUsiUcmeisUsiUmoCiaFaFUFsYCU
sFdCdacmósodsUaoSsdsYóaomSvUSsUuYaósYU%YmSvTUFaoUsóeYm%vTUFaUimUdYmesUdasosUyoUumFmSvYUaodsYovUsoU
mi%#oUuyodvUaodsYósSavUmUivUimY%vUSsUFyUivo%adySTUsθγ[+δ→∑∴+,⇒→+√[⇒δ(√(⇒+δβ+[β⇒γ[+⇒,⇔[+∫+,βδβ+
sodvocsFUsMscdymYFsUyoUmoCiaFaFUsFdYycdyYmiUsFdCdacmósodsUSsdsYóaomSvUcvóvUFsUuysSsUveFsY mYUsoUimU
Ma%U9í7U*e,íU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  277  -U
U
UUUUUUUUUUU
P
trabe libremente
suspendida
colgantes
apoyos de cables
U
Figura 6.5(a)
UUUUUU
P
carga movil
a lo largo de la
trabe
U
UUUU
     Figura 6.5(b)U
U vóvUgmU!sóvFUósocavomSvUsiUyFvUSsUcmeisFUFvoUuYavdEsós Ye Fó cjevXdrccE3e sY trYeXYv
Y7XYevjv j sY FrcYv nru 5dóesYvf ó cjeXEerócE3e njvdYmYsóvFUmi%yovFUuysodsFUSvoSsUFsUveFsY moU
ivFUSaMsYsodsFUdauvFUSsUcmeisFíU
U
Puente de San francisco, muestra cables con colgantes fijos a la trabe

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  278  -U
U
40 m
20 m
100 - x´
A
C
B
x
x´

Puente de Normandia, muestra cables ajustados directamente a la trabe
3[θ+θ(2,(β⇒γβθ+βαβ⇔[θ+(⇔,θγ∑→∑→⇒+⇒,∑(√→β⇒γβ+βθdvFUcvocsudvFíU
Ejemplo #1
U noUimUFa%yasodsUMa%yYmUsiUcmeisUFvuvYdmUyomUdYmesU′,β+βθ→+οϖ⌠+.2V⋅+δβγβ∑(⇒β+⇔→+γβ⇒θ(⇒+β⇒+
siUcmeisUsoUivFUuyodvFU
A, B y C. IF'UóaFóvUd yCiUsFUimUMysYtmUsoUcmSmUyovUSsUivFUcvi%modsFXU
U
40 m
20 m
100 m
A
C
B
U
Solución.- siUvYa%soUSsUivFUsrsFUcvvYSsomSvFUFsUMarmUsoUsiUuyodvUETUlysUsFUsiUuyodvUómFUemrvU
SsiU cmeisU gU soU siU lysU FyU usoSasodsU sFU a%ymiU mU csYv0+ δβ+ ⇔→+ β√,→√(⇒+ J+ θ→]β[θ+ ′,β+ ⇔→+ β√,→√(⇒+
→∑→]⇔(√→+δβ⇔+√→]⇔β+βθ<+
222;27
2
V=7:
2
x
F
x
F
mkg
x
F
y
HHH
O
=== ω
U
U 8yuvoasoSvUlysUsiUuyodvU UsFdmUyeacmSvUmUyomUSaFdmocamU(kUSsU
Bω+γβ⇒β[θ<+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  279  -U
U
2
´
;27
2: x
F
H
= U
2
´27í2+xF
H
= U
+ ββ⇔→•→[θ+γ→](⇒+→∑→+β⇔+,⇒γ[+
AUgUYssóuimtmóvFUsiU mivYUSsUFH β⇒+⇔→+β√,→√(⇒⋅+
2
´,T+::*S
;27
;: x
F
H
−−=
2
2
´,T+::*S ´27í2+
;27
;: x
x
−−= U
:+::::´2::´
2
=−+xx
mx;2í;+´
=
+ =⇒γ[⇒√βθω+δβ+⇔→θ+β√,→√([⇒βθ+∫→+β⇒,⇒√(→δ→θ+γβ⇒β[θ<U
2í69;7<,;2í;+*27í2+
2
==
H
F UkgU
xx
dx
dy
:266+í:
2í69;7<
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==
noUsiUuyodvUA.
7=í7=,;2í;++::* −=−−=x Um
699í+,7=í7=*:266+í:dmo
7=í7=
−=−==
−=x
Adx
dy
θ U
o
A
J<í76−=θ
U
5θ→⇒δ[+⇔→+β√,→√(⇒++γβ⇒β[θ+
t
F
T
o
A
H
A
Jí9+
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2í69;7<
cvF
=
−
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θ
U
U
noUsiUuyodvUEUdsosóvFU
x=0.U
U U U
:dmo
:
==
=x
Adx
dy
θ ;  :=
Bθ

t
F
T
B
H
B
7í69
,:cvF*
2í69;7<
cvF
===
θ
U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  280  -U
U
noUsiUuyodvU UdsosóvF.
;2í;+=x Um
<97Jí:,;2í;+*:266+í:dmo
;2í;+
===
=x
Cdx
dy
θ U
o
C
:í;;=θ
t
F
T
o
C
H
C
Jí7:
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2í69;7<
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===
θ
U
U
vóvUimUdYmesUsFdmUFvuvYdmSmUuvYU++Ucvi%modsFTUimUMysYtmUsoUcmSmUyovUSsUsiivFUsFU
t
mmkg
FhJ9íJ
++
,+::*V=7:
== U
6.4.2 CABLES SOMETIDOS A SU PESO PROPIO
vóvUgmUFsU!mUósocavomSvUivFUcmeisFUuYvuvYcavomoUyoUósSavUóygUsMacmtUumYmUFvuvYdmYUsiU
usFvUóysYdvUSsUimFUdYmesFUvUcyeasYdmFUSsUuysodsFUSs+√⇔→∑[θ+[+⇔,√βθ+,∫+2∑→⇒δβθω+β∑[+γ→](⇒+√[[+
,⇒+2∑→⇒+δ(θ[θ(γ()[+δβ+,⇒(⇒+β⇒γ∑β+βθγ∑,√γ,∑→θ+→⇔βαmSmFUvUFsumYmSmFTUumYmUmomiatmYUmlysiivFUcmeisFUlysU
sFdCoUFvósdaSvFUmUFyUuYvuavUusFvUSsdsYóaomYsóvFUimUMvYómUSsiUcmeisUmUivUimY%vUSsUFyUivo%adySU
∆8íU
vóvUgmUFsU!mUósocavomSvTUyoUcmeisUlysUFvuvYdsUFvimósodsUFyUusFvUuYvuavUdvómYmUimUMvYómU
SsUyomUcyY mU
catenaria,+√,∫→+β√,→√(⇒+√[∑∑βθ[⇒δβ+→+,⇒→+→∑∴][⇔→+√[[+θβ+[θγ∑+→⇒γβ∑([∑β⇒γβ⋅+
∆+√[⇒γ(⇒,→√(⇒+√[⇒θ(δβ∑→[θ+β⇔+√→θ[+δβ+,⇒+√→]⇔β+′,β+FvuvYdmU#oacmósodsUFyUusFvUuYvuavTU
β⇒γ[⇒√βθ+⇔→+√→∑2→+,βδβ+δ→∑θβ+√[[+,⇒→+−,⇒√(⇒+δβ+→YcvU
SíUimFUscymcavosFUSsSycaSmFUmodsYavYósodsU
FvoU miaSmFUFaUFsUmomiatmoUcvóvU
∆(U4U∆θ+β⇒γ[⇒√βθ<+
Ejemplo #2
IomiatmYUyoUcmeisUlysUFvuvYdmU#oacmósodsUFyUusFvUuYvuavU w*F,U4U+:Uk/m
x
y
30 m
(-15;7.5) m (15;7.5) m
(0;0)
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  281  -U
U
+ =⇒+βθγβ+√→θ[<+
dy
dx
ds
U
HT =ϕcvF U
()dssTsenUUU ∫
=ωϕ U U U *+6,U
∫
= dss
Hdx
dy
UU,*
+ω U U U *+;,U
ISsóCFTU
( ) ( )
22
dxdsdy +=U
+
2
−





=
dx
ds
dx
dyU U U *+7,U
+ β+⇔→θ+β√,→√([⇒βθ++∫+ϖ+β⇒γ[⇒√βθ+
dss
Hdx
dsUUU,*
+
+
2
∫
=−




ω U U *+9,U
U
Paso 1) E√,→√(⇒+δβ+⇔→+√,∑)→+′,β+∑β∑βθβ⇒γ→+β⇔+√→]⇔β+
2+
+:+:
CS
H
CS
Hdx
dy
+=+=
U
ymoSvU
:UUT:==
dx
dy
S
+[∑+⇔[+γ→⇒γ[<+
:
2=CU
5γ(⇔(•→⇒δ[+⇔→+β√,→√(⇒+↑+γβ⇒β[θ+
S
Hdx
ds
+:
+
2
=−




 U
2
2
2+::
+
S
Hdx
ds
=−




U
+
+::
2
2
2
+=





S
Hdx
dsU
+
+::
2
2
+= S
Hdx
dsU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  282  -U
U
β+⇔→+β√,→√(⇒+δ(−β∑β⇒√(→⇔+→+)→∑(→]⇔βθ+θβ→∑→]⇔βθ+θβUdasosU
dx
S
H
ds
=
++
+::
2
2
U
6
+:
io
+:
+:
+:
2
2
2
2
C
H
SS
H
H
S
ds
H
x +






















++=






+
=∫
U
vUeasoUU
6
+:
+:
C
H
S
arcsenh
H
x +





= U
ymoSvUs = :UTUx 4U:UsodvocsFUC
3 4U:U
H
x
senh
H
S
H
S
arcsenh
H
x
+:+:+:
+:
=⇒





= U
3vYUivUdmodvU






=
H
x
senh
H
S
+:
+:
U
GmFdmUsFdmUumYdsUFsU!mUSsdsYóaomSvUS+β⇒+−,⇒√(⇒+δβ+x, sodvocsFíU
H
x
senh
H
x
senh
H
HH
S
dx
dy +:+:
+:
+:+:
=












== U
nodvocsFU
+:
:UUU0UU:
+:
cvF!
+:
;
;
H
Cyx
C
H
xH
y
−=⇒==
+





=
U
3vYUivUdmodvU
7íJUUU0UU+7
+:
+:
cvF!
+:
==
−





=
yx
H
H
xH
y
U
3vYUivUdmodvU






−





=⇒








−





= +
+7:
cvF!J7+
+7:
cvF!
+:
7íJ
H
H
H
H
U
β+δ[⇒δβ+H≤↑+→∑[(→δ→β⇒γβω+[∑+⇔[+γ→⇒γ[<+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  283  -U
U






−





= +
+9+
+:
cvF!
+:
+9+
xy U
gUMaomiósodsU
( )[ ]+:92í:cvF!+í+9 −= xyUCatenariaU
Paso 2)+ βγβ∑(⇒→√(⇒+δβ+⇔→+γβ⇒θ(⇒+∴(→
,:92í:*UdmoUUUU0UUUU
cvF
xsenh
dx
dyH
T ===ϕ
ϕ
3vYUivUdmodvU
( )[ ]xsenh:92í:mYcdmo=ϕ U
nodvocsFUumYmUx=15 mU
()[ ]<6í:mYcdmosenh=ϕ U
nodvocsFU
9=6í:cvF =ϕ U
β+[δ[+′,β+⇔→+γβ⇒θ(⇒+βθ<+
íUUUJ2í267kgrT= U
Paso 3) βγβ∑(⇒→√(⇒+δβ+⇔→+⇔[⇒2(γ,δ+δβ⇔+√→]⇔β+S,U












= xsenhS
+9+
+:
+:
+9+
2
mS UU76í6;= U
U
6.4.3 CABLES SOMETIDOS A CARGAS CONCENTRADAS
ymoSvUyoUcmeisUSsUusFvUSsFuYscameisUFvuvYdmU mYamFUcmY%mFUcvocsodYmSmFTUsiUcmeisUmSvudmUimU
MvYómU SsU mYavFU Fs%ósodvFU SsU i'osmU YscdmTU cmSmU yovUSsU ivFU cymisFU sFdmU FvósdaSvU mU yomU MysYtmU
√[⇒θγ→⇒γβ+δβ+γβ⇒θ(⇒⋅+{[⇒θ(δβ∑β+[∑+βαβ⇔[ω+β⇔+√→]⇔β+′,β+θβ+,βθγ∑→+β⇒+⇔→+θ(2,(β⇒γβ+−(2,∑→<+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  284  -U
U
D
3 KN
B
C
3 KN
L
l1
θ
l2 l3
yc yD
U
Ily'U
θTU YsuYsFsodmU siU Co%yivU SsU imU cysYSmUAB SsiU cmeisU gULU sFU siU cimYvU SsiU cmeisíU 8aU imFU
SaFdmocamFUL
1, L2, y L3 gU imFU cmY%mFUP 1 y P2 U FsU cvovcsoTUel problema es determinar las nueve
incógnitas que conforman la tensión  en cada uno de los tres segmentosTUimFUcymdYvUcvóuvosodsFUSsU
∑β→√√(⇒+β⇒+AUgUBUgUimFUMisc!mFUSsUy
UgUy+soUivFUSvFUoySvFUCUgUD⋅+→∑→+⇔→+θ[⇔,√(⇒+[δβ[θ+βθ√∑(](∑+
SvFUscymcavosFUSsUslyaiaeYavUSsUMysYtmFUsoUcmSmUyovUSsUivFUuyodvFUA,B,C y DíUsFdvUSmUyoUdvdmiUSsU
ocho+β√,→√([⇒βθ+→∑→+√[⇔βγ→∑+⇔→+θ[⇔,√(⇒+βθ+∑β√(θ[+FmesYUmi%vUFveYsUimU%svósdY'mUSsiUcmeisUumYmU
vedsosYU imUnovena ecuaciónU oscsFmYamíU 3vYU srsóuivU FaU FsU sFuscaMacmU imU ivo%adySULU dvdmiU SsiU cmeisU
uvSsóvFU yFmYU siU dsvYsómU SsU 3adC%vYmFU umYmU YsimcavomYU cmSmU yomU SsU imFU dYsFU ivo%adySsFU SsU ivFU
θβ2β⇒γ[θω+ βθ√∑(γ[θ+ β⇒+ γ∑(⇒[θ+ δβ+
θT y TUyTUL1, L2, y L3, cvoU imU ivo%adySU dvdmi LíU
SsFmMvYdyomSmósodsTUsFdsUdauvUSsUuYveisómUovUuysSsUYsFvi sYFsUMCcaiósodsUmUómovTUFaoUsóemY%vUvdYmU
uvFaeaiaSmSU sFU sFuscaMacmYU yomU SsU imFU Misc!mFTU Sa%móvFUy
U vUyTU soU stU SsU imU ivo%adySU SsiU cmeisíU
G→√(⇒δ[⇔[+ →θ ω+ ⇔→θ+ β√,→√([⇒βθ+ δβ+ β′,(⇔(]∑([+ θ[⇒+ β⇒dvocsFU FyMacasodsFU umYmU vedsosYU imFU MysYtmFU
SsFcvovcaSmFUgUimUMisc!mUYsFdmodsíU&omU stUvedsoaSmUimUMisc!mUsoUcmSmUuyodvUSsUcmY%mTUimUivo%adySUSsiU
cmeisUuysSsUSsdsYóaomYFsUuvYUdYa%vovósdY'míU
IiUsMscdymYUyoUmoCiaFaFUSsUslyaiaeYavUumYmUyoUuYveisómUSsUsFdsUdauvTUimFUMysYtmFUsoUsiUcmeisU
γ→](⇒+,βδβ⇒+[]γβ⇒β∑θβ+βθ√∑(](β⇒δ[+⇔→θ+β√,→√([⇒βθUSsUslyaiaeYavUumYmUsiUcmeisUsodsYvUvUcymilyasYU
[∑√(⇒+δβ+β⇔⋅+
3[θ+θ(2,(β⇒γβθ+βαβ⇔[θ+⇒[θ+(⇔,θγ∑→∑→⇒+⇒,∑(√→β⇒γsUsFdvFUcvocsudvFíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  285  -U
U
Ejemplo #3
βγβ∑(⇒β+⇔→+γβ⇒θ(⇒+β⇒+√→δ→+θβ2β⇒γ[+δβ⇔+√→]⇔β+[θγ∑→δ[+→+√[⇒γ(⇒,→√(⇒+dQ,+)→⇔[∑+γ(β⇒β+
⇔→+δ(β⇒θ(⇒+h?
1,522
3 KN
B
C
8
KN
2
2
h
D

Solución.- [∑+(⇒θβ√√(⇒+)β[θ+′,β+ϕ→∫+√,→γ∑[+∑β→√√([⇒βθ+βγβYomFUSsFcvovcaSmFTUSvFUsoU
cmSmUmuvgvUgUdYsFUdsoFavosFUSsFcvovcaSmFUsoUsiUcmeiβ⋅+=θ→θ+J+(⇒√2⇒(γ→θω+α,⇒γ[+√[⇒+⇔→+δ(θγ→⇒√(→+ϕω+
uysSsoU SsdsYóaomYFsU cvoU imFU =U scymcavosFU SsU slyaiaeYavU SaFuvoaeisFTU∑=0:
XF∑=:
YF TU
muiacmeisFUsoUivFUuyodvFUA al D.
5⇒+∑[√βδ((β⇒γ[+→θ+δ(∑β√γ[+δβ+θ[⇔,√(⇒+βθγ∑(]→+β⇒UcvoFaSsYmYUimUusoSasodsUSmSmUSsiUcmeisU
CD0U soU imU Fa%yasodsU Ma%yYmU FsU óysFdYmU yoU Sam%YmómU SsiU cysYuvU iaeYsU SsiU cmeisU sodsYvU gU SsU m!'U
[]γβ⇒β[θ+⇔→+γβ⇒θ(⇒+β⇒+β⇔+θβ2β⇒γ[+CD+√[[+θβ+)β+→+√[⇒γ(⇒,→√(⇒<+
:=
∑ AM ⇒ :,;*=,2*6,7í7,*7V;*,2,*7V6* =−−+ mtmtmTmT
CDCD
U
U U U U tT
CD
J<í9= U U
U 3vSsóvFUmomiatmYUm!vYmUsiUslyaiaeYavUSsUivFUuyodvFU UgUEUsoUFscysocamTUcvóvU sóvFUsoUsiU
θ(2,(β⇒γβ+δ(→2∑→→+δβ+√,β∑[+⇔(]∑β<+
A
D
2
2 2 1,5
3 KN
B
C
8 KN
V
H
A
A
3
45
Τ
C D
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  286  -U
U
U
,⇒γ[+{<+
C
KN
x
y
3
45
KN
θ
bC
Τ
bc
U
U
U U U
:=
∑XF ⇒ :cvF,7V6*J<í9 =−
BCBC
Ttθ U
:= ∑YF ⇒ :=,7V;*J<í9 =+−
BCBC
senTttθ U
⇒U tT
BC
=2í;= U
o
BC
6í62=θ U
,⇒γ[+E<+
3 KN
B
x
32.3°
ba
θ
ba
U
:=∑XF ⇒ :6í62cvF=2í;cvF =+−
o
BABA
tTθ U
:=
∑YF ⇒ :66í62=2í; =−− ttsensenT
o
BABA
θ U
⇒U
o
BA
=í76=θ U
tT
BA<:í9=U
nodvocsFUSsUimUMa%yYmUaoacamiUuvSsóvFU!miimYUsiU mivYUSsUhíU
mmh
o
J;í2=í76dmo,2* == U
U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  287  -U
U
6.5  EJERCICIOS PROPUESTOS.
Problema #1
βγβ∑(⇒→∑+⇔→θ+∑β→√√([⇒βθ+δβ+→[∫[+∫+⇔→+−β√ϕ→+√[∑∑βFuvoSasodsUmUP +UgUP2U
2233
2 t
1.5 t
Va
Ha
VB
HB
y1 y2
0.5 m
A B
1 t
U
Respuesta:
mymy
tVtH
tVtH
BB
AA
J22í:2UUUUU0UUUUU7í:+
J22í:UUUUU0UUUUU=í+:
=í+UUUUU0UUUUU=í+:
==
== ==
U
Problema #2
βγβ∑(⇒→∑+⇔→θ+∑β→√√([⇒βθ+β⇒+A+β√,→√(⇒+δβ⇔+√→]⇔β+∫+P max
A
Va
Ha
VB
HB
x
y
(b;4) m
(a;4) m
B
35 m
0.2 t/m
U
Respuesta:
tT
xy
tVtH
AA
UU;+6í=
:+6í:UUUUncymcavoUU
7í6UUUUU0UUUUU979íJ
óm(
2
=
=
==
U
Problema #3
βγβ∑(⇒→∑+⇔→θ+∑β→√√([⇒βθ+β⇒+B+⇔→+β√,→√(⇒+δβ⇔+√→]⇔βω+⇔[⇒2(γ,δ+δβ⇔+√→]⇔β+∫+P max, donde solo
actúa el peso propio del cable w(s)=2: k/m

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  288  -U
U
40 m
x
y
(0;0)
(-20;6) m (20;6) m
VB
HB
U
Respuesta:
( )[ ]
kLtT
xy
tVtH
S
BB
:9í=:;0UU;+6í=
+:2<í:cvF!2J7í6;UUUUncymcavoUU
27í;2:UUUUU0UUUUU7í9=7
óm(
==
−=
==
U
Problema #4
&oUcmeisUusFmU7Uk/mUgUsFdmUFyrsdvUsoUFyFUs(dYsóvFUmUSvFUmuvgvFUFadymSvFUmU+7UmUuvYUsocaómU
SsUimU!vYatvodmiUgUFsumYmSvFUsodYsUFaUmU+7: m⋅+βγβ∑(⇒→∑+⇔→+⇔[⇒2(γ,δ+δβ⇔+√→]⇔β+∫+⇔→+γβ⇒θ(⇒+∴aómí
U
150 m
x
y
(0;0)
U
Respuesta: k LtT
S
<2í+760UU+:27
óm(
== U
Problema #5
βγβ∑(⇒→∑+⇔→+β√,→√(⇒+δβ+⇔→+√,∑)→+δβ⇔+√→]⇔β+∫+Pmax
10 pie
5 pie
0.2 t/m
40 pie
U
Respuesta: tTxyEcuación UU+:270:+=2í:
óm(
2
== U

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  289  -U
U

6.6   EVALUCION  DIAGNOSTICO DEL CAPITULO.

3+A+ A6+.ot+A (-+ 6y+.(+mof- )o+v-342om3 A616A6-26 miU uYsFsodsU cmuadyivTU FsU uYvuvosU miU
sFdySamodsUYsFuvoSsYUimFUFa%yasodsFUuYs%yodmFU
U QysUdauvFUSsUcmeisFUs(aFdsoXU
U l(+. 64 .+ 6m(+mof- )6 (- m+7.6 =(6 43P3A2+ 4( P643UuYvuavXU
U QysUsFUyomUcmdsomYamXU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  290  -U
U

CAPITULO 7
U
U
U
U
U
U
TUTORIALES DE PROGRAMAS
U
U
U
U
7.1  OBJETIVO GENERAL.
niUsFdySamodsUmuYsoSsYCUmUydaiatmYUuYv%YmómFUsFdYycdyYmisFUlysUisUusYóadmoUYsFvi sYUSaMsYsodsFU
dauvFUSsUsFdYycdyYmFTUóaFómFUlysUsFdmYCoUmiUmicmocsUSsUFyFUómovFUFasóuYsUgUcymoSvUivFUvYasodsUsoUFyFU
P+434 )6 A643.(mof-n
U
7.2  OBJETIVOS ESPECIFICOS.
IiUdsYóaomYUsFdsUcmuadyivUsiUsFdySamodsUFsYCUcmumtU)6a
U 1sFvi sYU a%mFU aFvFdCdacmFU ósSamodsU siU uYv%YmómU SsUmoCiaFaFU sFdYycdyYmiU vigaGU umYmU
cmicyimSvYmUG3U;=YTUYHTUY?TU;<YíU
U 1643.y6A m6Amu+4 P3A 6. SI23)3 )6 -()34 S6)o+-26 6.U uYv%YmómU SsU moCiaFaFU sFdYycdyYmiU
CERCHASUumYmUcmicyimSvYmUG3U;=YTUYHTUY?TU;<YíUU
U
7.3  INTRODUCCION A PROGRAMA VIGA-G.

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So4S+4 =(6 -34 PA3P3Amo3-+- o-13AS+mof- 46v(A+ i m3oMameisUmcsYcmUSsUivUgmU aFdvUsoUsiUcmuadyivíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  291  -U
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Calculadora hp 49 G
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Paso1) Inicio del programa
a)
GmeYCUimUcmYusdmU a%mYUSsodYvUSsUFyUcmicyimSvYmUG3UgUiys%vUuYsFavosUimUdscimU-2UumYmUcYsmYU
yoUoys vUmoCiaFaFí

b)
"ys%vUsóustmóvFUmUaodYvSycaYUivFUSmdvFUSsUimU a%mUm +-+.ot+Aa
2 t/m 1 t
A B
C D
E F
2 1 2 1 1 1
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Paso2) Establecer la geometría
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cmFvU =U SmTL 2+S7oI- 26-6S34 =(6 o-2A3)(moA .+ Aovo)6t )6. 6.sósodvTU usYvU cvóvU sFdmóvFU
)+-)3 .34 PAoS6A34 P+434 6- .+ o-2A3)(mmof- )6. +-BiaFaFUSsUimFUsFdYycdyYmFTUovUdvómYsóvFUsoU
cysodmU mivYsFU SsU sisósodvFU lysU mMscdsoU oysFdYvU cmicyivTU usYvU aodYvSycaYsóvFU +TU cvóvU
6.6S6-23 o)6+. P+A+ .+ -3 y+Ao+mof- )6 .34 mB.m(.34n P+. m3S3 46 S(642A+ + m3-2o-(+mof-a

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  292  -U
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nóustmóvFU mU SsMaoaYU cymisFU FsYCoU imFU cvoSacavosFU SsU slyaiaeYavU gU ivFU muvgvFU SsU imU a%míU
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  vóvUFsUuysSsUveFsY mYUumYmUSsMaoaYUimFUcvoSacavosFUaoacamisFU gVvUMaomisFUFvivUFsUoscsFadmU
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&omU stUsFdmeiscaSmUimFU cvoSacavosFUaoacamisFU gU MaomisFUSsU a%mTU cvósotmóvFUmUSsMaoaYUivFU
muvgvFUgUimFUmYdacyimcavosFUs(aFdsodsFUsodYsUivFUmuvgvFUgmUSsMaoaSvFíU
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Paso3)  Cargado de datos
m, noU imU sodmomU lysU m!vYmU mumYscsU s(aFdsoU imFU vucavosFU SsU cvivcmYU muvgvFU aodsYósSavFU gU
mYdacyimcavosFTUuYsFavomoSvUimFUMyocavosFU-+UgU-2UuvSsóvFUsFdmeiscsYUivFUmuvgvFUlysUsFdmYCoU
+a 2 i 90 i .+4 +A2om(.+mo3-64 + 6 i 7 S )6 .+ m3-)omof- o-omo+. )6 .+ yov+

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  293  -U
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uYaocaumiíU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  294  -U
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Paso4)   Resultados del programa
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momiatmYU ivFU SmdvFTU yomU stU cvociyaSvFU dsU óvFdYmYmU,⇒→+ )β⇒γ→⇒→+ δ[⇒δβ+ γβ+ δ(2→<+ ^{∴⇔√,⇔[θ+
PsYóaomSvF_íU
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U
c,
β⇒γ∑[+δβ+⇔→θ+∑βθ,βθγ→θ+′,β+γβ+[−∑β√β+ βθγβ+∑[2∑→mUSsUmoCiaFaFUSsU a%mFUsF0UsoUimUuYaósU
usFdmCmU dvSvFU ivFU cvóuvosodsFU mFvcamSvFU mU imFU YsFuysFdmFU SsiU sFMysYtvFU SsU cvYdsU ^/_TU
óvósodvFU ^F_TU SsFuimtmóasodvU SsU ivFU oySvFU ^θ_ω+ δ(→2∑→→+ δβ+ δβ−[∑→√(⇒+ ^@_ω+ ⇔→θ+
YsmccavosFUSsUmuvgvíU
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  295  -U
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  vóumYmoSvUsiUovóeYsUSsUcmSmUmuvgvUgU sYaMacmoSvUcvoUsiUsrsYcacavUYsFysidvUSsiUcmuadyivU6TU
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  296  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  297  -U
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7.4  INTRODUCCION A PROGRAMA CERCHAS

niUuYv%YmómUsFdYycdyYmiUCERCHA sFUyoUuYv%YmómUSaFsCmSvUumYmUsiUcCicyivUSsUCERCHAS
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Calculadora hp 49 G                                               Calculadora hp 48 GX

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Pasos a seguir para la utilización del programa
a)
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b)
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sFdmYUsoyósYmSvF

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  298  -U
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d)
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e)
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f)
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g)
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niUuYv%YmómUusYóadsU mYamYUimFUcvoSacavosFUSsUimFUcmY%mFUumYmUimUóaFómUcsYc!mUFsisccavomoSvU
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Ejemplo
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MysYtmFUsoUimFUemYYmFU*srsYcacavUjU+6USsiUcmuadyivU;,íU

                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  299  -U
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Paso1) Establecer la geometría
a)
GmeYCUimUcmYusdmU sYc!mUSsodYvUSsUFyUcmicyimSvYmUG3C2CPNL5ICVJLY IDLCPECIVQ cDC^+ND_CVEJEC
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b)
IumYscsYCUimUFa%yasodsU sodmomUsoUFyUG3í

c)
*JLY IDEDMICLPCTIOcDCmGEVEJLQLJBCPECIVQ cDC^AHá+ZeZCeACeZáX)_3VJLY IDEJCAHáA+3

d)
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IUPSU-Ig0

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  300  -U
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Paso2)  Cargado de datos
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  301  -U
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b) &omU stUaodYvSycaSvFUsFdvFUSmdvFUuYsFavosóvFUnDPn1íU"ys%vUmly'UimUuYs%yodmUsFUcymodvFU
dauvFUSsUFsccavosFUdasosTUFaUFsUlyasYsUFsUuysSsUdsosYUSaFdaodvFUdauvUSsUFsccavosFUumYmUimFUemYYmFU
=sFdvUsFUómFUcvo soasodsUcymoSvUgmUFsU!atvUcvYYsYUsiUuYv%YmómUyomU stTUgUFsUSaFsCvUcvoUsFmFU
MysYtmFTUsodvocsFUumYmU sYaMacmYUsiUSaFsCvUFsUydaia.ECLYOECIVQ cDnGCVLJICVEJECLPCLULSVPICYIPIC
QIDY MLJESIYCNDECYLQQ cDGCPNL5ICVJLY IDESIYCAHáA+3

c)
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d)
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  303  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  304  -U
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

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                                 Universidad Mayor de San Simón
                             Facultad de Ciencias Y Tecnología
                             Carrera de Ingeniería Civil                                                                            U

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                               Pag  -  310  -U
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ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 1
CAPÍTULO 2

INTRODUCCIÓN AL ANÁLISIS
DE LAS ESTRUCTURAS

ESTRUCTURAS
Como ves, existe una gran variedad de estructuras. En ocasiones, nos encontramos con
estructuras que cumplen la misma función pero que están construidas utilizando distintos
materiales

Las estructuras también pueden diferenciarse notablemente por su forma.
Por ejemplo, un edificio tiene una estructura de armazón, mientras que un monitor tiene una
estructura de carcasa.
Según la forma y el material con que estén construidas pueden distinguirse tres  tipos de
estructuras: masivas, laminares o de carcasa y de armazón
  ESTRUCTURAS MASIVAS
Las estructuras masivas son estructuras muy pesadas y macizas formadas por superficies
anchas y resistentes. Para construirlas se emplea gran cantidad de material.
Los muros gruesos de los embalses son ejemplos de estructura s masivas.
Las bóvedas y techos de las antiguas iglesias de piedra o los enormes pilares y arcos de los
puentes o acueductos son otros ejemplos de estructuras masivas.

ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 2
  ESTRUCTURAS LAMINARES O DE CARCASA
Las estructuras laminares o de carcasa están constituidas por láminas resistentes que envuelven
al objeto, formando una caja o carcasa que protege y mantiene en su posición a las piezas que lo
componen.
Otros ejemplos de estructuras de carcasa son las carrocerías y fuselajes de coches y aviones, y la
mayoría de los envases como botellas de plástico o tetra briks.

  ESTRUCTURAS APORTICADAS DE ARMAZÓN O ARMADURAS
Estas estructuras están formadas por piezas alargadas, como barras, tubos, pilares, vigas,
travesaños o cables unidos entre sí para formar una especie de esqueleto o armazón. Según la
disposición de sus elementos, las clasificamos en: trianguladas, entramadas y colgadas.

o Las estructuras trianguladas se caracterizan por la disposición de barras formando
triángulos. Resultan muy resistentes y ligeras a la vez.

o Las estructuras entramadas están formadas por una malla de piezas verticales y
horizontales.

ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 3
o Las estructuras colgadas soportan el peso de la construcción mediante cables o barras
que van unidos a soportes muy resistentes.

APOYOS
Grados de libertad – Estabilidad
Hemos visto que las fuerzas en un plano pueden producir traslaciones y rotaciones. La traslación
puede expresarse por sus dos componentes, según ejes ortogonales, y la rotación alrededor de un
eje perpendicular al plano que contiene a las fuerzas. Diremos que una estructura plana posee tres
grados de libertad (dos de traslación y una de rotación). Es evidente que estos grados de libertad
deben ser restringidos para evitar toda tendencia al movimiento de la estructura y lograr su
equilibrio. Esta restricción está dada por los apoyos, los que deben impedir las diversas
posibilidades de movimiento; aparecen las reacciones en estos apoyos formando este conjunto
(de cargas y reacciones) un sistema de fuerzas en equilibrio.
La función de estos apoyos es restringir los grados de libertad de la estructura, apareciendo
reacciones en la dirección de los movimientos impedidos.
a) Apoyo de primer género (MÓVIL): impide desplazamiento en la dirección perpendicular al
plano de apoyo (figura A-1).

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 4

b) Apoyo de segundo género o articulación: impide traslaciones en cualquier dirección,
permitiendo sólo rotaciones

La dirección de la reacción puede ser cualquiera pero siempre podrá ser representada por sus dos
componentes V y H.  No es obligación descomponer la reacción de apoyo en ejes ortogonales, se
puede descomponer en dos direcciones cualquiera.
c) Apoyo de tercer género o empotramiento: este tipo de apoyo impide todo tipo de movimiento
de la estructura (dos traslaciones y una rotación) (figura 2).

ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 5
Observación
Existe también otro tipo de equilibrio inadmisible para las construcciones y es el equilibrio
indiferente. Es cuando, al actuar una pequeña fuerza, la estructura se traslada, y si deja de actuar
la fuerza, se restablece el equilibrio, pero en otro lugar (figura MI-19).

RECONOCIMIENTO DE ESTRUCTURAS Y APOYOS
  APOYO FIJO. Se pueden identificar por la distribución mas compleja en la base.


  EMPOTRADO.  Aquellas que carecen de grados de libertad

ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 6
  ARTICULACION. Generalmente se presentan en uniones empernadas o clavadas.




  TIRANTES. En estructuras que necesitan soporte para equilibrarlos, generalmente se
las observa en puentes atirantados y/o puentes colgantes.

  ENTRAMADOS. Se presenta en construcciones generalmente metálicas, en algunos
casos en construcciones de gran magnitud con en la que se presenta a continuación.

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 7

  ARCOS. Utilizados generalmente en la construcción de puentes, pero también se las
utiliza para del diseño de bóvedas y construcciones arqueadas como se muestra a
continuación.

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS” A - 8
CAPÍTULO 3

ESFUERZOS INTERNOS EN
LAS ESTRUCTURAS

Como ya tenemos conocimiento, los esfuerzos internos de una estructura bajo el efecto de
fuerzas externas es el siguiente

Vemos a continuación un ejemplo claro de los tipos de esfuerzos sobre una pequeña
estructura:

ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 9
VIGAS
Vistas en todo tipo de estructuras, son de forma geométrica y dependiendo de la estructura
el tipo de material.

En este grafico se muestra una viga de hormigón armado correspondiente a un puente de luz larga

En este grafico se muestra una viga de acero correspondiente a una construcción metálica

POTICOS
Según el tipo de pórtico se distinguen los siguientes:

En este grafico se muestra una estructura metálica aporticada

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS” A - 10

En este grafico se muestra una estructura aporticada de concreto armado

En este grafico se muestra una estructura aporticada de concreto armado

ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 11
ARCOS
Utilizados generalmente en la construcción de puentes, pero también se las utiliza para del
diseño de bóvedas y construcciones arqueadas como se muestra a continuación.

En este grafico se muestra un puente en forma de arco para uso viacular

En este grafico se muestra un puente viga con armazón metálico en forma de arco parabólico
A continuación damos una muestra de los tipos de diseño de estructuras arqueadas y
bóvedas

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS” A - 12
En este grafico se muestra una estructura aporticada de concreto armado

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS” A - 13

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS” A - 14
CAPÍTULO 4

ARMADURAS O CERCHAS

Como ya tenemos conocimiento, las armaduras o cerchas pueden ser identificadas de la
siguiente manera:

ANEXOS TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS” A - 15

Estos tipos de armaduras pueden ser combinadas y usadas en la construcción de paneles,
como también pueden ser usados independientemente para el uso de cerchas para techo,
puentes, casas, etc; veamos algunas estructuras de este tipo

En este grafico se muestra un panel hecho de madera, combinando distintos tipos de armadura.

En este grafico se muestra un puente de armadura metálica tipo Pratt plana

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS” A - 16
CAPÍTULO 5

LINEAS DE INFLUENCIA

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ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS” A - 32
CAPÍTULO 6

CABLES
Mas comúnmente visto en puentes de luces largas, viales y peatonales, como también en
sostenimiento de antenas metálicas.

En este grafico se muestra un puente atirantado de uso peatonal.

En este grafico se muestra un puente atirantado de vial.

ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 33
CAPÍTULO 7

TUTORIALES DE PROGRAMAS
PARA HEWLETT PACKARD

Para poder tener mayor practica con el uso de programas para la calculadora HEWLETT
PACKARD, aconsejamos los siguientes programas para bajar, junto a su descripción:

Nombre del
programa
Descripción
ARMADURAS
Permite la solución de armaduras o cerchas isostáticas e
hiperestaticas
ARMADURAS
ISOSTATICAS
Solución de armaduras por el método de nudos
FEM 49 Permite la solución de pórticos isostáticos e hiperestaticos
REACCIONES FR
Programa sencillo para el calculo de reacciones y esfuerzos
internos (esta en Francés)
ESF 3D
Para el calculo de esfuerzos internos de pórticos isostáticos o
hiperestaticos en 3D
QUERY Para el calculo de cortantes en vigas y pórticos
LCASE Para el cálculo de pórticos.
Todos estos programas y mas los puedes descargar de las siguientes paginas:
C www.hpcivil.tk
C www.HpCivil.org
C www.elprisma.com/apuntes
C www.foros.construaprende.com/
C www.pachanet.com/hpcivil/
C www.iupsmcivil.turincon.com

ANEXOS   TEXTO ESTUDIANTE

ADSCRIPCION DE ESTRUCTURAS “ISOSTATICAS”                                                            A - 34

DIRECCIONES DE INTERNET
RELACIONADOS CON INGENIERIA CIVIL
ESTRUCTURAS ISOSTATICAS

UNIVERSIDAD Dirección de Internet País
Universidad mayor de San Andrés www.umsanet.edu.bo BOLIVIA
Universidad de Valle www.univalle.edu BOLIVIA
Universidad Católica Boliviana www.ucbcba.edu.bo BOLIVIA
Universidad san Francisco Xavier www.usfx.edu.bo BOLIVIA
Universidad Técnica de Oruro www.uto.edu.bo BOLIVIA
Universidad Autónoma de México www.unam.mx   MEXICO
Harvard University www.harvard.edu USA
Princeton University www.princeton.edu/index.shtml USA
Yale University www.yale.edu USA
Universidad de Monterrey www.mty.itesm.mx/principal/html MEXICO
Universidad Nacional de Chile www.uchile.cl CHILE
George Town University www.georgetown.edu USA

ejercicios de estructuras isostáticas. analisis estructural

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