Resolução hibbeler 5ª ed cap 5

Resolução capitulo 5

1. Sl. A shaft is made of a steel alloy baving an allowable
shea tes of rye 124s I the Sameer ofthe shat
15 in, determine the maximum torque T that can be
‘tassated. What would be the maxima torque Tift
‘in-diameter hole is bored though the salt Sketch the
shearstres distribution slong a rad ine in each exe

2 Aloweble Sher Stas: pig eno forma

Tas em

al ows

Toros. Am

Dy Allowable Shear Ste; ep hin formal

Taco pi Ei Aus

re. ses
7" ROMA

5-2. The solid shaft of radius rs subjected to a torque T.
Determine the radius y o the inner core of the shaft that
resists one-quatter of the applied torque (74). Solve the
problem two ways: (a) by using the torsion formula, (9) by
finding the resultant of the shear stress distribution,

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prnl ch Len arme

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2. $- The shaft has an outer diameter of 125 in. and an
inner diameter of 1 in its subjected othe applied torques
as shown, determine the absolute maximum shear stress

‘developed in the shaft The smooth bearings at A and B do
not resist torque.

TEA The shaft has an outer diameter of 125 in and am
inner diameter of in. Vitis subjected othe applied torques
35 shown, plot the shearstress distribution acting along a
Fadia line lying within region FA of the shaft. The smooth
bearings at À and B do not resis torque.

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5-5. The sold Somm-diametar shaft 5 wed to transmit
tbe torques applied to the gear Determine the shear stress
developed inte shat a points C and D. Indicate the sbeat
stesso volume element located ae

eT ites

3. 5-6. The assembly consists of two sections of galvanized
steel pipe connected together using a reducing coupling at
B. The smaller pipe has an outer diameter of 075 in and an
inner diameter of 068 in, whereas the larger pipe has an
‘outer diameter of Lin. and an inner diameter of 086 in.
the pipe is tightly secured to the wall at C, determine the
‘maximum shear stress developed in each section ofthe pipe

when the couple shown is applied to the handles of the
ren.

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5-7. The sold aleman shat bar a
fod an allowable shea ste of tue
{he ages torque, that an be applied tothe sat iit
subjected fo te othe trio! loudigs Ts require
{at T, ct in the direcion shown. Alo, determine the
"maximo shear seu within regions CD and DE,

Mama Shar Smet hoping be ron Fert

[TE

Tes

sens
Fou

sium rs wi jon BC inn
pa

aq e mal een D DE
TAN me GUN mar,

Tos AKO yoy

m Am

4. 5-8 The sold alumigum shaft bas a diameter of
50 mm, Determine the absolute maximum shea sess in
he Saft and sketch the shearstress datrbulen alo a
radial ine of the hat where the shear tes maxi

S87, Nm

neral Torque: Pom deuda, be
Mesina Shar Sue à Aa es tor

¿Bono “ica A

CS

59 The sold shaft has a diameter of 075 in. it w

subjected to the torques shown, determine the maximum

shear stress developed in regions BC and DE ofthe shat

‘The bearings a A and F allow tree rotation of the shor
sat He

Th

an
ee

Sp = 50 Ama

Son pin ak Am

5-10. The solid shaft has a diameter of 075 in. I it is
subjected to the torques shown, determine the maximum
shear stress developed in regions CD and EF of the shat.
‘The bearings at À and Fallow free rotation ofthe shaft

on = Ewe Am

Roe, ETS) tase
MEG TN
AS

5. 5-11. À steel tube having an outer diameter of 25 in. is
‘wed to transmit 35 hp when turning at 2700 revimin,
Determine the inner diameter d ofthe tbe to the nearest |
in. le allowable shear stes 8 on = 10 Ki

on FON „ana
a
35650/=TORIO

rason

dern

Una Am

6. 5-18. Tee copper pipe az an outer diameter of 250 i,

And a inner diameter of 20 in Ci tighlly secured to

wall aC anda uniformly dourbuted torque i applied

Ko se town, determine the shea stress developed points

{A and B Toes pointe on he pipe ote surface, Sketch
res ou volume elements located at A and.

Marian Sher Serr: AE Bon ome

he
mode espada door hal
EDS sean O

mi de

Sc, Toe copper pipe hs as ose meter of 23010
and an inner dameter of 20 in Ii tightly secre to
aná is subjected tothe if dub

is entire length, determino the absolute

‘maximum shear ses inthe pipe. Discuss the valid of
(hire .

Ina Tora mine ra on

E

Shear Sr; Aa Senn rs

$-15. The 6Ommdiameter sold shalt subjected tothe
distributed and concentrated tomomal loas shown
Determine the sar ares at point À und Band sketch the
shear ses on volume clement located at thee pol.
Inımal Terque: A os FDO.

Mazina Shas Srs à Aro de nf

BE £000 an
eee Lu

ne. 400,

Feen”

7. 5-16. The 6mm diameter soli shaft is subjected to tbe
dstibuted and conomntated torional Tendigs shown,
Determine te absolute maximum and minimum bear stress

in the shat an spec their locations, measured from the a
fed end nf S
neral Torque: ron de ore gi De or

ce o ad rons de

= Ben urn S

Sheer un: Ang bein fm

SEIT. The solid safe subjected fo the distributed and
Scareayated tonal loop shown. Determine the
fequred diameter do the sat he alowable sear sess

forthe mates Ton = 175 MPa

tal Targus Fr e eg dag. te man
Pan a MD warden $e,

tnt Sher Sorts: Ai e aio fre de

vet) =
PO
ARTO .

cg o San nat pú penton te ion
‘Sema pis fin re om eg
‘Som cc ong Terr estore wa

5-18. The steel shafts subjected 10 the torsional loading
shown. Determine the shear stress developed at points À and
Band sketch the shear stress on volume elements
located at these points The shaft where A and B are located
as an outer rads of 60 mm,

PE casera ame
an 7

8. 5219. ‘The ste shat is subjected 1 the torsional loading
‘shown. Determine the absolute maximum shear stes in the
shaft and sketch the shearstres distribution along a radial
line where iis maximum.

Manu qe EEN; hen o cons
‘fe at ou coria ns ren

510x099 yep,

Foo Team,
EA
on ETS

Matimam Shar Siri: hep e on om
BRETTEN

es am

Tom Ss “E

5-21. The soi shat i subjected to the distributed and
Sorcentated torsional loadings shown. Determine the
absolute maximum shar sess a the shalt and spec ts
location, measured fom the fied ead O Take d=
mm

eee as rae en
Shear Sis seg Se io aa

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oa

9. 5-22. The solid shaft is subjected to the distributed

‘concentrated torsional loadings shown, Deiemiae the
Fequred diameter dof he shaft the alowable sear tees
forthe material ram = 150 MPa.

Han,
wate She Sr: Martin tem
wa) ca
er e. ay

TER Determine o the nearest in, te diameter ofa shalt
that is require to transmit 150 hp a 400 revi. The
Tater a alwabe she es Of rye = HSL

(E) cr

AA

Allwable Su Sres + Apoyo Stan foma
TO

EUX

ECS

u.

ente

meet am

EI
and a ios
Pipe Sag an outer ameter af 45 la. and ica
DE ir the pipe uming ot 650 reina whe being
powered by a Sp mote, determine the maximum shear

eas nthe pipe
ern Tore

ns EM) eaten

semi

Jaron

Maximum She Sea + Ag ein ose

aan,

DEE am

10. 5-25. The gear motor can develop 210 bp when letras at
300 revinin Ifthe shalt has a diameter off ins determing
‘he marimum shar sires tat wilde developed the sat

+ he RM) son

Maximum Shear Sires: gpg on om

„samen

fou) wer Am

5-26. The gear motor can develop 110 bp when ur at
80 revimin tthe alowable shear es forthe 00 à
‘ae = Ss determise the smallest diameter of the shalt
15e nearest| in tat canbe sed

pare Allabie Shar See: sep ein om

eto

pow

“u

anus

5-27, “The coupling is used to connect the two shats
together. Assuming that the shear Sres in the bolt Y
‘uniform, determine the number of bolts necessary to make
the maximum shear stress inthe shaft equal 10 Ihe shece
stress in the bots Each bolt has a dlameter d

mis Ge umber oft a Fe he rein ch o

Toate rat

11. +5-28. The steel shafts are connected together using fillet
‘weld as shown, Determine the average shear stress inthe
weld along section a-aif the torque applied 0 the shafts is.
‘T= 60 Nm. Note: The critical section where the weld falls

is along section a-a
ome al

conos +0006),
O]

5-29. A motor deliver 50 bp to the see aha I is
Fibular and has so outer diameter of 2 a. 1 the shalt i
{luting e200 a, determine its largest inner diameter to
the nearest] i ifthe allowable her tres fo the material
rae = 25

al

reson

Altoabie Sha Sit: bein esi oat

PRET

50 The pump peras lg be tar Esas por
15 W if the impeller a is trang at 15 rem,
determine the maximum shear stress developed lo tbe
So-mmedlameter warsmision sha at A

Inna Torque

en (EIER essa

Sara

en

12. SB. A steel tbe having an outer diameter ef d=

25 in.is sed to transmit 35 bp sing at 2700 rev/min. DP
ep
De ee MOT
an ve inden e ms
2 mn wc
PR) os MES

ow ate ma dant me eed

32 A steel tube having an outer

ameter of
di 25 in and an ier ameter of d = 2 in ted fo
{ran 45 bp, Deere is maximum pte of real Y

the alowabie

5:38. A ship hase propele dive shat that ering at
1500 eva wide developing 1800 hp. It 8 Tong and
ameter of4 a. determine the maximum shen sess

Maxima Shar Sis: bein orm,

00 una)
1 ,

13. 5-34. The shaft is subjected to a distributed torque along
its length of t= (10x) N mim, where x sin meters Ifthe
‘maximum stress in he shafts o remain constant at 80 MPa,
determine the required variation of the radius cof the shaft
ford = 253m.

re fear fo

co Fes un ES
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amame Am

5-36. The shafthas a diameter of 80mm and due to fiction
at its surface within the hole, itis subjected 10 a variable
torque described bythe function = (2Sxe")N mim, where
in meer Determine the minimum tous a ned o
overcome friction cause it 1 twist Also,
determine the absolute maximum stes in the shaft.

ETC

Anna wing Spine ok weet
messe ON Am

(WR
7° Foon

= sun Am

7336 Determine the diameter d of a sold satt
qd to trank a power P be allowable shear sess
[Stans and the angular velocity ofthe shaft is

Altea Sha Sres + Mpg ion om

14. 5-37. The solid shaft has a linear
to ry at the other. Derive. an
‘maximum shear stress inthe shaft,
shafts axis

taper from ra at one end
equation that gives the
ata lation x slong the

og yatta nt ae

rta + mx 2

„wien —m
ES
A

a L- m
Rp aaa oP

5-38. The drive shat ofan automobile is made of «set
{ube having 20 alowable shear set of tre = 8 ba I
{We outer diameters Sin andthe engine deers 200 hp to
{he sft when it turning at 110 rev, determi toe
minimum required hike ofthe salt wal

Altea Sher Sen 3 epee eon mia

CETTE
Risen

moe

Reina

Amis Am

15. 5-39. The drive shaft of an automobile is to be
designed as a thin-walled abe, The engine delivers 150 hp
‘we the shat i turning at 1500 revi, Determine
Minimum thicknese ofthe tube wall ie outer diameter
325 ln The material bas on allowable sbear sles of
Tan = Ti

Saunas

ne

hen-netss-hiam-oiin An

75-30. The motor delivers SO hp while wring ata constant |

te of 1350 rpm at A. Using the bet and pulley sytem this
Loading is delivered 10 the steel blower shaft BC. Determine
to the neatet fin. he smallest diameter ofthis shaft if the
allowable shear stress for ih steel ru

a «rs mann Sy

Ge

Per

PACE)

ense

Teen

Tus) as pe

mes ago) = 68e
ET

eros

16. 5-41. A oyindial spring consiste of a aber
bonded o arg en

torque Ts ap
Scar cs he rubber

sus
shaft If the rings eld ied and
shat determi the manna

Guen nano ne

Sci ue hema Le = 4,

5-42. The propellers of a ship are connected to a solid

36 steel haft thats 60 m long and has an outer diameter
of 340 mm and inner diameter of 260 mm. If the pquer
Output is 45 MW when the shaft rotates at 20° adie

determine the maximum torsional stress in th shaft and its
angle of twist,

EE sure Amo
7 Honor]

E A

17. 5-4. A shaft is subjected to a torque T. Compare the
effectiveness of using the tube shown in the figure with that
‘of sold section of radins e. To do this compute the percent
increase in torsional stress and angle of twist per unit length
for the tube versus the solid section

Forde abe, \

Godan =
Ford so shalt

(oan 2 TE

%

coa EE ao

nr set =

(en
A aa EEC Go
a
Frente,
re
E

A2 ogg = LE OD gn)

7 Tem

ECC

18. 15-44. The hydrofoil boat hasan À tel propeller stat,
{hat is 100 ft log, Is connected to an inne diesel engine
that delivers a maximum power of 50) bp and cases the
sat rotate at 1700 pm If the outer diameter o the batt
is Bin andthe wal thickness] in determine the maximum.
shea stes developed inthe sha Also, wht the "wind
up” or angle of titi the sha a fll power?

een SESE) EF sooner

rm (ES)

1: on in Fm

COTES
jasa

zn, mom
TOS

5-48. The slinedends and gears attached to the A-36stecl
shaft are subjected 10 the torques shown. Determine the
angle of twist of end B with respect to end A. The shaft has

a diameter of 40 mm, =

untl. 38103) , 2005), 2009)

A
16 * TOA

19. sold ection BC supported on smooth bearings bat
‘Sow ito rote ely, Ifthe ends ae subjected 1 85 >
orgues, determine the ange of tt of end À relative o end
De ter have an outer diameter of 0 um end sn ier
‘ameter of 30mm The aid section basa diameter 0 mun.

PEUT

msm |,
tua | Massen |”
Te
Tao

rom Am

5-47. The À 36 see ane is made from tubes AB and CD
Andaschdsecion BC supported on smooth bearing tht
flow ito rotate frech If ends A and D are subjected lo
BEN. mtorques, determino the angle of twit of end ofthe
‘old section rene to end Ca The tes have an outer
“amet 30mm and an inner diameter 020 mm. The sold
section has diameter of 40 xm,

Amie of Tos
The, 15909
DE CA

+5044 The gears attached to the 304 sales tel sat
sre subjected tothe torques shown, Determine the ange of
A led awh respect toend The Bt bus deter

tra Torque he anse FD

Aer

DRE TU

pn Dig nono
nom

FO ONE
omiso Am

20. 5-49. The gean attached to th 304 sine tel bat are
Subject the torques shown, Determane the angl tut
spec 10 gear BL The sbat has à dameter

slag, ROI teens
Be gana)

5-50. ‘The rotating Mywheel-and shat, when brought to a
sudden stop at D, begins 10 oscillate clockwise:
‘counterclockwise such that a point A on the outer edge of
the Mywhcel is displaced through a 6 mumare Determine the
‘maximum shear stress developed inthe tubular A'36 steel
shat due to this oscillation. The shaft has an inner diameter of
24 mm an outer diameter of 32 mm. The bearings at B and
Clow the shaft to rotate freely, whereas the support at D
holds the shaft xed,

6n 56 6 000

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a KO)
mus
Dore]
reas

PETT

= om Am

21. 8-51. The motor delivers 40 hp othe 204 sis seta
While routes at 20 Hz The shat supported on smooth
earings at A and, hich allow free solution ofthe bat.
‘The gears C a D fied to be sat remove 25 bp and 15 bp,
respectively Determine the amer fe bat the nearest
Vin i the alowable shear stress & te = SAS and the

a pans
em gm

strat Apple Torque: Ags Te
CONTRE BEM mann
ES %

ES

ÉD na

Bases nj
ost Show Sr are es bass.

Ala of Tost: oa ae o iin

te len
sean tae toon Bas
EEN

ah) tn
er ru
dette ai (oman

15-52, The motor delivers 40 bp tothe 304 sine see
soli shaft whl it rotates at 20 Hz. The sat has diameter
Of 15 in. and is supported on smooth bearings at A 04 2,
‘ich allow ree rotation of he bat The gars Cand D fixed
tothe shaft remove 25 hp aad 15 bp, respectively Detemine
‘he absolute maximum tesa the hat ande ange ows
‘gear wih respect o pear D.

sein To ome

sa mann
no a
ss
ern
Imre Torque: so où DO. ST
Alone Sher Sn; Damme eg can in R
tegen ACL awe ashes MOT Be
E A
as | ee de Len
ange Totes: Besssuh
deviate

Fea
TOS

22. 5-53, The A-36 steel shafts 2 m long and has an outer
‘iamete of 40 mm When ti rotating a 80 rad it transmits
32 KW of power from the engine £ to the generator G.

mine the smallest thickness ofthe shaft if the allowable
SESS iS tapa #40 MPa and the shaft is
restricted not to twist more than 008 rad

ir

pete
3200 = ra)
Toon.

nmooinsm

EST MAS solid steel shaft à 3 m long and has a
¡diameter of 50 mm tis require o transmit 35 KW of power
from the engine E to the generator G. Determine the

smallest angular velocity the shaft can have iit is restricted
not to twist more than 1.

—=

vn, m
Te" FOO

TS

3500) 267240)
ee de

23. 5-55, The A36 see shaft rotates atu = 125 rad and
leansmits the power shown. Determise the abla
‘maximum hear mes inthe salt and the angle of wit
vih respect to F. The ier and outer diameters ofthe
‘eat are = 30 mm and d, = 40 mun The joural bearings
SUB and G are smoot.

nul Apple Torque: Ang TE shaw, sede

uv) 2200)
ES

UN ar

Irma Torque: A down on FD.
animan Shea Sie Te mano on win
‘noon Erte teh on Tigo

yy le a
a
nse of Po

toned

A

RANA

OS

pad
772

Fame Tarzan

Ge5tonm

24. 15-56. The A-36 steel shaft rotas at = 125 rad and
transmis the power shown, Determine the inner and ter
diameters of the shat Y d/d, = 06, the allowable she

‘iret rane = 1S0MPa, and the allowable relative ele

ot wists Ba. = 27m. The journal bearings at Band Gare
moot,

sternal Applied Targa: Aoi To E. tae

Tessonm

raro
77

4 000050

‘Ante of Tit Ase fa de movi cio,

Fein Tasio

1278

EEN TA)

Testonn
4200 m= 380mm (mn Ae

4 =064 ICAA Am

5-57. The 36 steel assembly consists ofa tube having an
‘outer radius of 1 in. and a wall thickness of 0.125 in Using
a rigid plate at B, ti connected to the Solid l-in diameter
Shaft AB, Determine the rotation of the tubes end C if a
torque of 200 1b in. is applied to the tube at this end. The
end A of the shaft i fixed supported

Tat

Tak. 20000,

Pao

e

= 0001852 rat

ans)

fe Te” FoR OT

= = 00001119

06 2 + G00
= 0001882 + aoooun
ETES

25. ad hey ate supported by Bearings at À
Band €. wich slow tee rotation, tthe support at Di
fixed, determine be sale wit end wen the toques
te applied tothe sembly as shown.

e

A IDEE ana

error farma ot

Sac bon ge pere Fan

DEP

5-89. The two bats are made of A-36 steel Each bas à
diameter 0 ln and they ae supported by bearings a À
B. and €, which allow fre rotation, I the support at D
fied, determine the angle of wit fend Anden tbe torques
are applied tothe assembly as shown,

Ante ap Pee ”

x

FEA ann mono)
ETC de OT

orion foam «onen

Tae

ocre
Faro
= US ra Du a

Den

CTP RES ase

26. *S-60. The Gin-diameter L-2 steel shaft on the turbine is
supported on journal bearings at A and B. IC is held fixed
and the turbine blades create a torque on the shaft that
increases linearly from zero at C to 2000 Ib: at D,
determine the angle of twist of the shat of end D relative
Lo end C. Also, compute the absolute maximum shear stress
inthe shaft. Neglect the size ofthe blades

Ta = 41201200020) = 14410)
Te 14006 Jugo009 «1410-1000

vf

Sf anette

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a $0083 Am
Fara A

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Te LO)

E

Moi am

5-61. The assembly is made of A steel and consis of
a soi rod 15 mm is diameter connected tothe inside of a
{ube using a rigid sk at 8. Determine the ange of tit
A. The tube Bas an outer diameter of 30 mm and wall
tics of man

neo Tel
e
$0103) won

Om SOTO” FORTRON STOTT

27. 5-62. The device shown is used to mix sols in order to
provide init stabilization. I the mixer s connected to an
‘A:36 steel tubular shaft that has an inner diameter of 3 in
and an outer diameter of 45 in determine the angle of twist
of the shaft of À relative to € if each mining blade à
subjected 10 the torques shown,

00412415412) RIA
ESTIMAR FEINA

=00Snde 545 Ama

8-63. The device serves as a compact torsional spin, It
ls made of À 36 sect and consists of sold me shaft CB
ich is surrounded by and attached toa tube AB wing a
ii rng at B. The ag a Acan ao be asumod rigid tnd
is fied trom rotating Ia torque of T= 2k in. applied
Lo the shat, determine the ange of twist atthe end and
the maximum shea tes inthe tube and shaft.

Tice 20405)
For

Com

Mg, 2000
CE ED A
Pro
Angle Pett:

on
Fan on

000208

200129
FOND “Ptos
aban use
Brome Am

28. 8-64. The device serves as a compact torsion spring It is
mude of A steel and consists ofa soi inner sat CB
‘which i surrounde by and attached t a tube AB wing à
ti ing at B. The ring tA can alo be asumed gd and
1 Ted from rota, I the allowable bear ses for the
ratas gg = 12 stand he angle of stat United
10 due "determine the maximum torque Patan be
Apple abe eed C

mr
am x)

ro mer

a Pete: hae ae apt ren.

Tale Tun

E ET

= scour

CC
da

rozas a com A

5-48 The contour of the suce ofthe hats defined by
the equation y =e", where a a cost. Ifthe sat
Sujeto oroe Tatu end determine the angle of
‘tend À Wa rrpct to end, The shear modus 6.

tof Fos Darme 4009

29. 5-66, The tapered shafts made 020146 aleminumalioy

14 has a raus which can be described by tne function
= 002(1 +.) m where x i in meters Determine the
Ange of twist of is end À iit mbjeied 16 à torque of
SONS

At Pot: ep meme vers 1) «Sans
ETES

‘rang ener Spb, mere
#4 008631500479 ná

5-67. “The shaft of radius € is subjected to a distributed:
torque 1 measured as torquelength of shaft Determine the
angle of twist at end A. The shear modulus is G

nf

PEA

2

euer Eolo co bo us
DE

Lyd 2
Ella wor Lor
reos pn

30. 15-68. The A-36 steel shaft has a diameter of SO mm and
is subjected to the distributed and concentrated loadings
shown. Determine he absolute maximum shear stress the
shat and plot a graph of the angle of twist of the shaft in

neral Torque: A shown ca RD.
Eee Tree tps

Kant REN ones am
o

Anglo Pout

ne
fee

Nessim png
for OAmcr<dm Sie TU) =O, en

Pecos

Taye

Ar 05m Xb im

31. 5-68, The glas tube is confined within a rubber stopper,
So that when the tube is twisted at constant angular velocity
the stopper creates aconstandisiburion of frictional torque
along the contacting length AB of the tube Ifthe tube has
an inner diameter 012 mm and an outer diameter of 4 mm,
determine the shear stress developed at a point located
its inner and outer walls at a seston through level C. Show
the shearstress distribution acting along a radial line
segment at this section, Aso, determine the angle of twist at
A with respect 10 8.G,

A
EM 01S-1(00000 sean

Ana FED) a)
Sheer Ses: ping ern fr weave

ooo
“oa

unes Am

ose

eee fon

“common Am

5-70. The dm damte sid sha is made of A36
a eed (te ded aed ent
{oie loadings stow, Detemine he ange rer
fie ad À ofthe sat de to thes odas

neral Targus ba shown cD.

Andi pre:
nz

oo (0-20

ral, pa

000 sa 04024 ant

32. 5-71. The tapered shaft has a length L anda radi rat
né A and 2ratend 8 Ci Be tend Band is subjected
fora torque Y determine the angle of tis of ead A. Toe
shear modul 6.

waver tga thers

ID

Angle] To:

| ns

Ef a

el sem].

72. The solid steel shat conte of two tapered ends,
AB and CD, and à centra porton BC having cost
Samer I i supported by two smoot beatings. which
Mow it to rotate freely Ifthe geam fined o 1 ends are
fbjocted to couaterbalancing Torque. of 1700 Ib in.
‘determine the angle of est of one end ofthe hat relative
tov the oiber end. Min: Use the results of Prob
Sn Ge 100) ka

Anal Te before ca Pas 37,

toe Ferma) ran

US, omo |
FOO "mean

On 800 Ane

33. STB. Show that when the bar is subjected to the
torque 7, the length’ L shortens by an amount

La - Vi = RIA06))

The shat is made of material

ng asar modular À
us oro
à

Geomen

COPA ET
ENT

BC

eu h

E em

m

a

fel

EN

SH. A ovlndial spring consists of a ruber anautes
‘bonded to avigi ring and shaft If the ring is eld aed so
“torque Tis applied tothe rgd sat. determine the sop

beri 6.
the clement
y tis
expression along with + = TiQ2nrh), trom Prob Salto
in the era.
nr a.

um

Sher Sires and Sher Sr Relationship From

Fo Set er ty «then

a a

Angle] ein: Soi ETES

34. 5-18. The steel shat ha a diameter of 40 mm and is fixed
atts ends A and 8. If itis subjected othe couple, determine
‘he maximum shear stress in regions AC and CB of the shat.
Ga = 10800") bs

ube
nr, 30000.1 «0 o

Compay conte de
pS Tes

no „mim
a WG

nest a

Saving Es) and (ya

BE su Am
12000)

35. 76. The steel shaft is made from two segments: AC has
a diameter of 0. in, and CB has a diameter of 1 in Itt
is fixed at its ends A and B and subjected to a torque of
500 bi determine the maximum shear stress inthe shaft.
Ga = 108010) ks

+H 5020 w

Sapa ¡FEA

eo = dos Eee?
RO 2 „nun

For * E Toms

HO = 19275 o

Satin En 100) ió
non T= wane

202029

te = m Am

77 7035)
tos oF OD. zur
3° Fass

5-77. The brome CA6100 pipe has an outer diameter of
15in and a thickness 010125 I. The coupling en rat Cs
being tightened using a wrench the torque developed at
‘A is 125 lolo, determine the magnitude F ofthe couple
Forces. The pipe i xed supported at end.

Equi:

RD 2m 0 wm
Compara +
dor ton eo Y

809
HR F

Gaine
Deo
rn) -18625-15-0

rene aw

36. 5-78. The bronze C86100 pipe has an outer diameter of
15 im and a hikoess of 0.128 a. The coupling on tat Cis
being tightened wing a wrench. Ifthe applied force is
P= 20), determine the maximum shear stress athe pipe

A) u
Compay +

ea a
Soin Ea et

COTE CELLES

Maximum seer ares

e. 302

Pe

15-79. A rod is made from two segment AB À 36 tel
Sou has à diameter of 30 mm and BD is C8400 red brass
AS basa diameter of SO mom Tt fixed at its ends and
ibjeced to a torque of 7 = 500 N° m. Detemise the
Trienal rections at the walls A and D.

Erin
net-s0=0 u
compat
too tou gone
ron PT gary 7
PE" RRA
PE
wa 4
nom, a
Sage (i il
TE Am

37. 15-80. Determine the absolute maximum shear ses ia
the shat of Prob. 5-79, ss
En
ns m
compat:
toot ton
ran "man
PARA TENA
EX)
“E
roo a
sein. (ie
pate A

CRTC
5 ams

re. nooo

Sao

(ean

yt fan unes Ane

a

5-42. The A 6 tee shaft made fom wo segments AC
bas diameter of i. and CB bas diameter of. titi
lied at is ends A and B and subjected to a torque of
T2 500 Ibs, determine the absolute maximun she
first nthe sat

tia

uw

Beam,
Ses a

Rear meso
cda Le BATIDOS ya
CAE

O ou am
E

Pre

38. 5:42. Determie the tomiona reactions atthe ends A and
B of he shat in Prob, 5-8

m

Company:

bandos
OD, TADA, EUA
fase “Ke "FING

ems, a

Sas (mé

5-8. Toe A état made rom mo segun
has a diameter of 1 in. at de
eed ten À snd ern the magne 1
ed torque Tithe recon As OR.

Eyton
soon-7=0 a
Compas:
dea tou
O. non ‚wunasın
fe" Foss HNO

From 0
OS

En

39. 5-84. The composite shaft consists of a mid-section that

includes the 1-in-diameter solid shaft and a tube tht is
welded tothe rigid flanges at A and B. Neglect the thickness
of the flanges and determine the ange of twist of end C of
the shaft relativ to end D. The shat is subjected 10 a torque

01800 1 ft The material is A-36 steel

Egaña

ATT [O

Fasr-aama” 1090
mean

ham

ID 2255807940 gonna

TODA FOS LOKI)

S88. The two star made of AS sel Each us à
diameter of 25mm ad y re can ai he an

Pearce enue fed
a ae supported by urs
which allow oro of heat

50 Nm applied tthe
carat à own, dermis the eacion ear

gatita +
n+ron-uo-0

R009 «0 u
re lad)
Rem -s000 a
Compas
Ders

au fau

nec, 1

San E (ad te

isan a

40. 5-86. Determine the rotation ofthe gear at Ein Prob. 5-85.

esti
Teno 20000 w
%-R005 20 a
Font m
Remo o
14,0054,

Ka
eee]

nem sa
Stig. (ad io
SS
Ange of Tt

en
ROSES

somerset a

41. 5-87. The shaft is made from a solid steel section AB and
2 tubular portion made of steel and having a bras core. I it
is fixed to a righ support at A, and a torque of = $0 1b- ft
is applied toi at C, determine the angle of twist that occurs
at € and compute the maximum shear and maximum shear
Strain in the brass and steel, Take Gy = 11510") ksi,
Gay = 56010) ks

aba

mémo o

Bd be suce anda or ue same ai it Gon
TaN)
ES

y= omas, a

Saving Es. (Gp
mean Aminen

a, ID, 020
"os “TACA
gd « O16 Ane

Le. LID puisque spot) Ane
O Le

+ PRG, a mont Ane

(den,

42. *S-88, “The shaft of radius cis subjected 10 a distributed
torque measured as torque/length of shaft Determine the
reactions at the fixed supports A und B

mo fu ar Pare eus do ©
By soi
ee
AAA
Dur Ae Ew
Pere
Foe Ea D.
near Lo
Tel Mat
eens?
Det m

43. 8-49. The tapere shaft is confined by the fixed supports
MCA and Ie torque Ts applied at midpoint, determine
€ restons at the support,

gut

nener-0 uw Lay

neiges)

+
”
=
if

Hao Tae T

5-90. The aluminum rod hasa square eros section of 10 mm
by 10 mm. it is $ m long, determine the torque T that is
required to rotate one end rel

Gu = 28 GPa, (ny) = 240 MPa,

2 non

481079)
om

44. 5-92. The 6061-T6aluminumbar bas guar cross secon
9625 mmby 25 mm ii 2m leg, deter be minimum.

shear ses inthe bar andthe rotas of onc ond ei
to he otber end

was

Angles] rai

zum ras)
Fo” ROAD
+ Zion

Gone ee RO
RE © DE
CT

Cicas» ZE à 2009,

a
ETS a
Aout a Pott:
were
CD Es
(Mon +0)

+ REST ST MOL + MO

meseta Pre

PS om

‘5.98. Solve Prob. 5-92 for be maximum sbear sen

‘within regions AC and BC, andthe angle of ist $ ol end
Brel oC

Me , 2000,
Con a

ante of Pett:

MEA
(oof 002200119)
OO

ERPIEFFarr Ane

PS

Bronx

45. 5-94, The square shaft is used at the end ofa drive cable
in order to register the rotation ofthe cable on a gauge. If
it bas the dimensions shown and is subjected toa torque of
$ Nm determine the shear Stress inthe shaft at point 4.
‘Sketch the shear stress on a volume element located at this
point,

ar. sn

te a

8-98. The unio C36100 bronze rod has an elíptica
rss section, I i i subjected tothe Ave torque shows,
determine the dimension a of lis cros section zo that the
‘Shear rs the rod doe not exoted ane = 8 MPa Abo,
‘what the rotation af end A with esp to end BY

Allen bte Shar Siren: Tho dm os ot
‘moat ce in

Anne of Feat:

En
on ES

ATA
A THD

EN

nn ee Am

95-96 The uniform CB6100 bronze rod has an elliptical
‘ros section. It subjected tothe Ave torques shown,
‘determin the marimum shear stress in the rod ad tbe angle
Of twist ofend À wih respect o end B. Take a = 20 mm,

Marian Snr Sts: Tego ag swt ae
Cort)

anio of Tota

eon

an E
ha ee
AAN HOY

einsam wa

46. 5-97. The aluminum strut is fixed between the two walls
at A and Bt has a2 in by 2 in. square cros section, and
itis subjected to the torque of 80 Ib fat C, determine the
actions atthe fixed supports Also, what isthe angle of twist
at C? Gy = 3800") ks

By papes
Do

wth
où 2100 _ 7207545) ER
ae 6

+ de
GRA Am (== £
%
7432-00 ñ

Tee Am SL
ACTE =
PPP

amo

3-98. The bras wre has a wlangular arom seo,
2 mm on aid. If the yield stes for bras ry = 205 MP
determine the matimum torque T to wich can be
abeced 0 thatthe wie will not il. IC bis tongue

applied o segment 4m long, determine the rentes sole
Of ist of one ead ol the wire relative t theater end that

il ot aus permanent damage tothe wire G = 37 GPa

Antic of Pea:

ooo,
Bac

539, “The plastic tube is subjected toa torque oFISON cm.
Determine the mean dimension a ofits sides ifthe allowable
Shea Stes if uo = 60 MPa. Each side has a thickness of

1=3 mm. Neglect” stress concentrations. at the

47. "5-100, The plastic tube is subjected to a torque of 150
IN m. Determine the average shear stress in the tube if the
mean dimension a = 200 mm. Each side has a thickness of
= 3 mim, Neglect stress concentrations at the comers

EE

2m ne

5-101. A torque of2kip-in.is applied to the tube If the

wal cies 101 in, determine the average seat sens
in the tube. “

A
ur

5.102. The 304 sis steel tube has a thickness of
Yo mm. It ie allowable shear stress I Fa = 80 MPS,
{determin the maximum torque 7 that cas rast. As,
that the agle of twist fone end ofthe ube with respect
Wo the other ifthe tube ie 4 moog? Neglect the stress
Concentrations a the comes. The meso dimensions are
shows

A ao 00m105"

Ta = 00m +005 =0200m

DRE
romanes m
arre
a

son

48. 5-102. The 304 ssinles steel tube has a thickness of
10 mm fie applied torqu is 7 = SON m determine the
verge shear stress in tbe tube. Neglect theses:
concenraicas atthe corra The mean dimensions ate
sown

‘4.001000 «0corion"

[Reon

*5-104. The tube is made of plastic, is Smm thick, and has
the mean dimensions shown, Determine the average shear
Stress at points À and B if itis subjected 19 the torque of
T=5 Nem. Show the shear sress on volume elements
located at these points

4010009» 400900) 0010"

ÉTÉ OH Am

>
ton

CEE

$105. The sie abe has an lips cos secan ofthe
mean dimensions shown and 4 constant Mises of
#02 in. It the allowable shear ate is tae = 8 bs
(determine the necessary dimension D needed to est the
torque shown The mea aca Au fr to elie le (05)

Ar Tre en FAD mn ee

Ten
+ ge Js
PRET
D RE eos
, ele
soma y NS
se

be

49. 5-106, ‘The tube is made of plastic is $ mm thick, and has
‘the mean dimensions shown, Determine the average shear
stress at points À and Bil the tube is subjected o the torque
Of T= SON m, Show the shear stress on volume elements
located at these points Neglect stress concentrations atthe

A = 214000000) + 011000 = 00052

de 2 ne =

a
a ann “a
7 [Zu

5-107, For a given average shear sess, determine the
factor by whieh the torque-crrig capac is increased if
the balear section re reversed fom the daube ie
postas othe section show. The tbe 01 a thick

Seen Popes:

room [Ne mos

Average Shear Sete:

50. 15-108. Dueto afabicon error the inner ie ofthe
te cn ih pec oe ar ec, bt
percentage ls the torsional strength reduced when the
‘centri soneto of hs erence In he adi?”

TES

MAC
IC

5-108. The symmetn
feck having the mea
mm His sube
the average shear stress developed at points A and Be

Indicate the shear stress on volume elements located at ee
points

A= 4000400694008" 001120

ae a Aw

51. 5-110. The plastic hexagor
of 150 N «m. Determine th
ifthe allowable shear stress
‘thickness of ¢= 3 mm

ral tube is subjected toa torque
1 mean dimension a of its sides
iS aoe = 60 MPa. Each side has

año

To
N a

room

S-111. The steel shat is made from two segments AB and
‘BC, which are connected using a fillet weld having a radius
‘of 28mm, Determine the maximum shear stress developed

in the shat
oc „ lono0s A u
men = Zei „ 190)
ee Ee Tae
nes =
An
From Fig 5-36.K = 1308
7 Foor Toran 10

on)

52. SS-12. The shaft is used to transmit 9 hp while turing at
600 rpm. Determine the maximum shear ares in the aha

The segments are connected using a file weld having à
radius of 015 in

S-113. The members are connected with a fillet weld of
radius r = 4 mm, Determine the maximum shear stress in
the shaft #7 = 10N om.

53. $-114, The assembly is subjected to a torque of 71015 in
If the allowable shear stress for the material is tye =
12 ksi, determine the radius of the smallest size fillet that
can be used to transmit the torque

de

19.0379)

eo. LR
Keim

O

aa

5-115. The sel sed forthe abat has a allowable shear

Sesto sacs tte mambo co pad
toprol edt ain = 22 mm tine
‘Slain rue Fen be pled
era eS: mo (faz
jun Lg
° LECUDES

54. 5-116, sod shat subjected to he torque T, which
ses the materia ted Ifthe materiel prety
last show that Ihe oraue cane pre in Leim af
the angle of twst of the sh a Y 2 ID wf,
‘ete Ty and dy are the torque and ange of wi when the
material bis yl

Vienes 3-26100 bent
tt u

Pa) E)

esp ne. 6 nor. ento (H)

Ñ Vet se ue toscas qu
el m
on us PA) eu
o

SH. ‘The assembly is subjected to a torque of
1800 1 in. IC the allowable shear stress for the materials
Tos = 12 ki determine the radius ofthe smallest sie et
‘tha can be used to transmit the torque

(oxy

SB. A bar having a ciculr cross section of 3 in,
diameter i subjected o à torqu of 100 Kip iI the
metal is elsic perlecly plas, Wh hu = 16 kr
(determine be radius ofthe els core.

u al ee son
rue)

a]

55. S119, A sold shat having a diameter of 2 in is made of
«ic porlcty plas material ving a eld tress of
Sr = 1ekatand shear modula of = 1210 Ya Determine
‘he torque required to develop an case core in the shaft
sing diameter of in. Also wha ie plate torque?

Hla Plstl Targus: Aoi Eg, 5-26 tom des

= 229 (4¢0) 09]
se eme Am
Pa Forge ioe 3-7 eet
= Bao)
nie a

*S-120, The 2-m-ong tube is made from an elastic plastic
material as shown, Determine the applied torque T. which
Subject the material of the tube outer edge to a shearing
train of Yası = 0.008 rad. What would be the permanel
angle of twist of the tube when the torgue is removed?
Sketch the residual stress distribution of the tube

AS m)
un vay
Ni
x
5 ¿ua

¡dal &
i) 250 OM

56. S-121. Determine the torque needed 10 twist à short
S-om-diameterstel wire through several revolutions fits
made of steel assumed to be elastic perfectly plastic and
having a yield stress of ry = 80 MPa, Assume that the
‘material becomes fully plastic

‘asl Torque Va ml cont y pe, en

= Baw) (oan)

40mm and length
of 1m Tt made of an elastic perfect plastic male
having a yield stress of ry = 100 MPa Determine the
maximum last torque Ty an the corresponding angle of
{ist Whats the angle of (wit torque tente fo
T= 127) G = 8 GPA

Matimam Ele Forge a
he pase oa,
A “oem

MENCIONES Eu Pla ere: gt 3260050

Ko
DOTE

Am nein)
ass FÜ (av)

Anat of mia

eee ee aus

SR, Tue seras sa eed o

duces ying on the star of the large
Fanta Detrmae te tdi ofthe eae ce produced
{vt smaler diameter segment Neg

Sooo a

asian Ele Target et eg dc sg

ser

late Plane Torque E mat mets pen
ade (0009) = 1.6( 0) ar > 139109) ar.
Aopen Sale weh

Teilen)
135(10%) ny Eo) =p]
ram

Dom Am

57. “5-124 ‘The shaft is made from astrain-hardening material
having a r-ydiagram as shown, Determine the torque T that
must be applied to the shaft in order to create an elastic core
in the shaft having a radius of 9. = 03 in.

es
sen

ae 00h 7 + sa) a on

as

rou = 28 0005 = 0006
cane

7 Era» £0008 = 0019
Bra = Bono = 0

Satin mn (ad is
= 20e
5 10109 + 500)
Teufen
= anf mao" « ae 100000 + so!

AS

58. 5-128. The shear stress-strain diagram fora slid SO-mm

diameter shaft can be approximated as shown in the figure
Determine the torque required to cause a maximum shear sone
‘tess in the shaft of 125 MPa. Ifthe shaft 3 m long, what"

is the corresponding ange of twist?

» foe
ES but
ua 001
wie y= 0025
ot
META
7
su) A sonne
00062 or LOS
= no)
= said) 18500) auch eats

OS "0033 OO
= side + 2540

ral coe
u ro ap
+20 ia, tario + 25 dp

Ten = 327m Ane
Yay = lo
rn
= 120m = 688 Am

59. S-126 The 2mio0g tbe is made of an elastic perte
Pate material as shown. Determine the apple org F on
ht abject the material at the tube outer edge o a lar

‘stain OÙ Yuu = 0006 tad, What would be the permanent
Ange ol it of the tube when this torque removed? Shoe
the residual stress run inthe be

Plate Toque Te wei ay pase 7,3 7 «00 a

Te rason
aber pa
Belt
mee) = Ea
(cs au) we
ert am
Ant of rene
Feet «(CO ca 03016 a
CS)
ea en (Ryans
ETC vo,
ons) pompa

TOR ner“ on

emerge it,

»

a

60. 5-127. The solidshaft is made ofan elasti-perfecty plastic
material as shown. Determine the torque T needed to form
an elastic core in the shaft having a radius of py = 20 mm.
ithe shafts 3 m long, theough what angle does one end of
the shaft twist with respect to the other end? When the
torque is removed, determine the residual stress distribution
in he shaft and the permanent angle of twist

Este lane orga: AVE, 336 erat
O)

004) -008]

Aaa To
elec nan eme am
na a mi it
UN ora
O wags m8
OS
eel
Br
Comme Am
(Oe N rn

1020667467

Ts

a

28. The shat consists of two sections that are ri
connected. I the material is elastic perfectly plastic 35
shown, determine the largest torque 7 that can be applied
10 the shaft. Also, draw the shearstres distribution over a
radial tine for each section. Neglect the effect of stress

Past Torque: For mer ner een,

20 ep do
anos
on a

stain St Far mm

20100,

Fo

Ai 6 #1088 ORD

=.

gi

61. 5-129. The shaft is made of an elas pertecy plastic

material as shown. Plot the sheartress distribution tin
along a radial line if it ls subjected to a torque of
T ZEN m. Whats the residual res distribution in the mu

shaft when the torque is removed?

Fac Ple Torque; To ms cir
Beine 2 41010") (019) = 188518 ant
epi is T= Fe = 3019) (003)
ASIN Se CF, pb E 326400
Se weh

yor
rizo)

200) ECS Kom) pf]

pr =0.01870m.

7. + reno

CREAS
Residual Shear Stress: Wen e eve rg = 2010
‘ph,

CORRECTE ONE

62. 5-130. The shat is made of an elsticperecy plastic
‘atrial a shown, Determine the torque thatthe sat <a
ans ifthe allowable angle of twat & 0375 rad, Aso,
determine the permanent angle of tet os tbe torque
removed. The shafts 2 mo

Anglo Pott

p= 001m

#)
= UP (002) ao
canoa a

Permanent Anti of Pest: When bere
TR Non els

one

mon

A
O
hno ans
RN

63. S-A3L The shear seso diagram for «tld Somm-
iameier shat can be approximated as shown in the figure
Determine the torque 7 required to cause a maximum bear
reste shaft of 125 MPa. Ifthe shaft 15m ong, what
is the corresponding ange of twist?

Sern Diagram

aio

FERNEN N
PAS “Tassos De
tan 4(10) pe25(10) a
The Ultimas org: %
Di

lo
cu. Be]

ON IN ane

Ange of Dota:

asma Am

*S-132. A torque is applied tothe shaft of radis .1 the
material has a shears where
‘isa constant, determine the maximum shear stress in the
shaft

64. S-A33. The 04 stainless see shaft 3 m ong and bad an

Outer dlameter (60 mm When it rotating a 60 rad it

transmits 30 LW of power rom the engine Eto he generntor

G. Determine the smallest tickesr ofthe satt if tbe

Allowable shear stress I rrp = 150 MPa and the shat o
008 ra,

Attal Sher Srs Asa es u he u

aa

Artie of Pte: Asan a e ge vt hd.

nos sn)

RCE ETC)

002093 m= 2.40 me

‘x00 (het trol = 8409

Mee Fa =a

‘0009903 = 29929 mn

OA Aa

as a dameter of $0 mm Ii require to tase 4 KW
‘ot power rom the engine Eto the generator 6. Determine
{the smallest angular velocity the aft can have i it à
restricted not to wast more than 15°

Ante of Pett:

Toa

5-135, If the soli shaft AB to which the valve handle is
attached is made of C&3400 red brass and has a diameter of
10mm. determine the maximum couple forces F that can be
applied to the handle just before the material start to fai,
“Take Tam = 40 MPa, What is the angle of twist of the
handle? The shaft is fixed at A.

aan = 2628
12 03F = 7885-0

m. 138019)

26 © Toro
000 = 186

65. 15-136, If the solid shaft AB to which the valve handle is
attached is made of C83400 re brass determine the smallest
diameter of the handle so thatthe angle of twist does hot
‘exceed 0" and the shear stress does not exceed 40 MPa
when F = 25 N

Paty tee

Te 2803) = 758m

ou. EE

aby mis

Be ox

=. 15019

e. er) 7 RNA

‘SCARY, The assembly is made of À see and conis of
A soli 20015 mm in diameter connect 1 the Lan
‚3, Determine the ange of twist D

‘Be ube has an outer diameter of 30 pun and eal ey

rat Torque As wn 80,

Ate of Tolo be ce 80,

lac

won
A

ema Am

5-138 The rotating Myuhecl and shaft is brought 10 a
Sudden stop at D when the bearing freezes This causes the
Aywheel to osilateclockwise-counterclockwise,t0 that à
Point A on the outer edge of the fiywheel is displaced
through a 10mm are Determine the maximum shear stress
developed inthe tubular 30 stainless tel shaft due to this
‘osilation. The shat has an inner diameter of 25 mm and
an outer diameter of 3 mm. The journal bearings a B and
low the Shaft 10 rotate freely

Angle] Toi

EN
FAA

resenne

ION agro ane

66. S138. The 6Osmm-diameter shaft rotates at 300 revimin
This motion is caused by the unequal belt tensions on the

pulley of 800 N and 450 N. Determine the power transmitted =
and the maximum shea stress developed in the shaft,

mar

T+ 002 moon = 0

DETTE
P= Ta = 35000 = 1100W LIO Aus

ERE pe so
MERE CU

of

05-100, The motor A develops a torque ur ge
500 Ib, whic is applied along the axis of the Zi
diameter A6 steel shaft CD. TAS torque 5 10 be
ranemited tothe pinion gears at Band Fes gea are
temporary fed, determine the maximum shear ses

‘Segment CB and BD ofthe bat, Also, what the angle of
A ofeach of these segments? The bearings at C and D

iy sue fee esos oh ht x
tratan simon She Srs à aa à
Benno r “a a
| CCE

ENE Angi pre
BR
Bi
Rom a e
Annan

Saba aaa ERRE

henna ranma Ono Am

5-141.| The coping consis of wo dis ied t separate
‘shat each 25 mum in diameter The aha are upperted og
jguraal bearings that allow fee rotation I onder 1 ligt 5
the torque That can be trasmite, “abc plo" Ph eed

Lo connect tbe dss together If tis pin cin seal an

average shear fore of $50 N before it fis determing te »
matiium constan torque Tit can be anne (am
one shaft o the other Alo, wht isthe maximum shoot
stress ln each shaft when the “shear pin" abou to ah

Agata

at
Burn reos Oe

Mas Shear Sent

Te, son

an nam Am
oa

67. 5-142, The aluminum tube bas a thickness of $ mm and
the outer cross-sectional dimensions shown. Determine the
maximum average shear stress in the tube. I the tube has a
length of § m, determine the angle of twist. Gy = 28 GPa

A = 14510095) = 0013795 wt

iM
ET GST SNOT

20m Ame

n
"or

4 1019:10099

er ge)

. D 2402)+ 14) 00049 mat 0.258" Ana

“AA