01 Buckling, Strain Energy, Virtual Work - Dr Oyelade.pdf

This is a document explaining the concepts behind Energy Methods, virtual work, Strain energy and buckling.

1. MECHANICS
OF
MATERIALS
OYELADE, AkintoyeOlumide
Department of Civil and
Environmental Engineering

2. References
StructuralAnalysis1:StaticallyDeterminateStructuresbySALAH
KHALFALLAH
STABILITY OF STRUCTURES : Principles and Applications by CHAI
H. YOO and SUNG C. LEE
STABlLlTYOF STRUCTURES: Elastic, lnelastic, Fracture and
Damage Theories by ZDENEK BAZANT and LUIGI CEDOLIN
3

3. Content
Introduction Euler Theory
Buckling Instability of
Struts/ Columns
Strain Energy Methods
Thin Plates and Shells:
Application
Summary
3

4. Introduction
Failuresofmanyengineeringstructuresfallintooneoftwosimple
categories:
•materialfailure:equilibriumconditionsorequationsofmotionthat
arewrittenfortheinitial,undeformedconfigurationofthestructure
•structuralinstability:equationsofequilibriumormotiontobe
formulatedonthebasisofthedeformedconfigurationofthe
structure
3

5. Introduction
Dynamicstabilityanalysisisessentialforstructuressubjectedto
nonconservativeloads,suchaswindorpulsatingforces.
Structuresloadedinthismannermayfalselyappeartobestable
accordingtostaticanalysiswhileinrealitytheyfailthrough
vibrationsofeverincreasingamplitudeorsomeother
acceleratedmotion.
4

6. Introduction
Aphysicalphenomenonofareasonablystraight,slendermember
(orbody)bendinglaterally(usuallyabruptly)fromitslongitudinal
positionduetocompressionisreferredtoasbuckling.
There are two kinds of buckling:
•bifurcation-type buckling; and
•deflection-amplification-type buckling.
In fact, most, if not all, buckling phenomena in the real-life
situation are the deflection amplification type
5

7. Introduction
Structuralmembersresistingtension,shear,torsion,orevenshort
stockycolumnsfailwhenthestressinthememberreachesacertain
limitingstrengthofthematerial.Therefore,oncethelimitingstrength
ofmaterialisknown,itisarelativelysimplemattertodeterminethe
loadcarryingcapacityofthemember.Buckling,boththebifurcation
andthedeflection-amplificationtype,doesnottakeplaceasaresult
oftheresistingstressreachingalimitingstrengthofthematerial.
6

8. Introduction
Thestressatwhichbucklingoccursdependsonavarietyoffactors
rangingfromthedimensionsofthemembertotheboundary
conditionstothepropertiesofthematerialofthemember.
Determiningthebucklingstressisafairlycomplexundertaking.
7

9. Introduction
Ifbucklingdoesnottakeplacebecausecertainstrengthofthe
materialisexceeded,then,doesacompression
memberbuckle?
8

10. Introduction
Aslendercolumnshortenswhencompressedbyaweightappliedtoitstop,and,in
sodoing,lowerstheweight’sposition.Thetendencyofallweightstolowertheir
positionisabasiclawofnature.Itisanotherbasiclawofnaturethat,whenever
thereisachoicebetweendifferentpaths,aphysicalphenomenonwillfollowthe
easiestpath.Confrontedwiththechoiceofbendingoutorshortening,thecolumn
findsiteasiertoshortenforrelativelysmallloadsandtobendoutforrelatively
largeloads.Inotherwords,whentheloadreachesitsbucklingvaluethecolumn
findsiteasiertolowertheloadbybendingthanbyshortening.Structurein
Architecture
9

11. Introduction: Neutral Equilibrium
Theconceptofthestabilityofvariousformsofequilibriumofacompressed
barisfrequentlyexplainedbyconsideringtheequilibriumofaball(rigid
body)invariouspositions
stable equilibrium
unstableequilibriumneutralequilibrium 10

12. Introduction: Neutral Equilibrium
Thestraightconfigurationofthecolumnisstableatsmallloads,butitis
unstableatlargeloads.Itisassumedthatastateofneutralequilibriumexists
atthetransitionfromstabletounstableequilibriuminthecolumn.
Thentheloadatwhichthestraightconfigurationofthecolumnceasestobe
stableistheloadatwhichneutralequilibriumispossible.Thisloadisusually
referredtoasthecriticalload.
11

13. Euler
Theory
13

14. Euler Theory: Theory of Bending
IntheEuler–Bernoulliorthinbeamtheory,the
rotationofcrosssectionsofthebeamisneglected
comparedtothetranslation.Theangular
distortionduetoshearisconsiderednegligible
comparedtothebendingdeformation.Thethin
beamtheoryisapplicabletobeamsforwhichthe
lengthismuchlargerthanthedepth(atleast10
times)andthedeflectionsaresmallcomparedto
thedepth.
14

15. Euler Theory: Theory of Bending
where u, v, and wdenote the components of displacement parallel
to x, y, and z directions, respectively. The components of strain and
stress corresponding to this displacement field are given by

,
,
0,
,
w x t
uz
x
v
w w x t




 2
2
,
= = = = 0
xx
yy zz xy yz zx
uw
z
xx





 2
2xx
w
Ez
x




14

16. Euler Theory: Theory of Bendingz


 A
M zdA z
EE

   2
2
AA
z E EI
M E dA z dA
  
 EI
M


15

17. Euler Theory: Theory of Bendingz


 A
M zdA z
EE

   2
2
AA
z E EI
M E dA z dA
  
 EI
M


16

18. Euler Theory: Theory of Bending1
''
M
w
EI
 ''MEIw ''0EIyPy
Equation above is a second-order linear differential equation
with constant coefficients. Its boundary conditions are0 at 0, y x xl  
17

19. Euler Theory: Theory of Bending2
2
''0
'' 0
''0
EIyPy
P
EIyy
EI
yky
P
k
EI




The general solution for P>O
(compression) issincosyAkxBkx 0,00yxA 0, sin0yxlBkl
Now we observe that the last equation allows a
nonzero deflection if and only if ,2,3,.....kl
18

20. Euler Theory: Theory of Bending,2,3,.....kl n
n
k
l



 2 22
22
2
,1,2,3.......
Pl nEI
n P n
EI l

  22
2
, 1,2,3.......
nEI
Pn
l


If a pinned prismatic column of length L is going to buckle, it will buckle at
n= 1 unless external bracings are provided in between the two ends.sin
nx
yB
l


19

21. Euler Theory: Theory of Bending
2
2
cr
cr
P E
Alr


Critical stress
slenderness ratiol
r
radiusof gyration
of the cross sectionI
r
A
 22 22
22
eigenvalue:
eigen pair
eigenvector: sin
cr
e
nEInEI
P
ll
nx
yB
l











20

22. Buckling
Instability
of Struts/
Columns
22

23. Buckling Instability of Struts/ Columns 0
y
FVdVVqdx
Bifurcation-typebuckling is essentially flexural
behaviour. Therefore, the free-body diagram must
be based on the deformed configuration as the
examination of equilibrium is made in the
neighbouring equilibrium position. Summing the
forces in the horizontal direction indVVqdx
22

24. Buckling Instability of Struts/ Columns 
2
top
dx
MMdMMVdxPdyqdx
Summing the moment at the top of the free body gives
Neglectingthe second-order term leads to0dMVdxPdy dMdy
PV
dxdx
 MPyV


23

25. Buckling Instability of Struts/ Columns
Fundamental beam-column governing differential equation.MPyV

 Vqdx ''MEIy ''
iv
EIyPyqdx
24

26. Buckling Instability of Struts/ Columns
The homogeneous solution of governs the bifurcation
buckling of a column (characteristic behaviour). The
concept of geometric imperfection (initial crookedness),
material heterogeneity, and an eccentricity is equivalent
to having nonvanishingq(x) terms''
iv
EIyPyqdx 22
''0,
iv P
ykyk
EI

25

27. Assignment : Civil
Derive the beam-column governing differential equation of the
following free body diagram.
26

28. Assignment: System
Derive the beam-column governing differential equation of the
following free body diagram.
27

29. Assignment: Building/QS
Derive the beam-column governing differential equation of the
following free body diagram.
28

30. Buckling Instability of Struts/ Columns
Assuming the solution to be of a form22
''0,
iv P
ykyk
EI
 mx
ye  
4 22
0
mx
emmk   
4 22
0; 0; 0
mx
e mmk  24
;
mxiv mx
ymeyme
29

31. Buckling Instability of Struts/ Columns 
2 2 2
0
0,
mmk
mmik

 00
1 2 3 4
kix kix
h
ycececxece

 412 3
sincos
kix kix
h
cxc
CxD
ycece
AkxBkx





30

32. Buckling Instability of Struts/ Columns 
2 2 2
0
0,
mmk
mmik

 00
1 2 3 4
kix kix
h
ycececxece

 412 3
sincos
kix kix
h
cxc
CxD
ycece
AkxBkx





31

33. Buckling Instability of Struts/ Columns0101
010
0
sincos 1
cossin10
k
D
klkll
kklkkl


For a nontrivial solution for A, B, C, and D (or the stability condition
equation), the determinant of coefficients must vanish. Hence, 2cos1sin0klklkl
32

34. Buckling Instability of Struts/ Columns 2cos1sinsincossin0
2222
klklklkl
klklkl

 

 sin0; tan
2 22
kl klkl
 ,2
2
kl
n 2
2
22
2
2
;
24
nP
kk
lEI
nP n
P EI
lEI l







33

35. Buckling Instability of Struts/ Columns0@0;
0@;
0@0;
0@
yx
yxl
yx
yxl




34sincos
h
DyAkxBkx Cx  22
sin cos
h
nn
DyAxBx
ll
Cx

  
22 2 2
222
4
eigenvalue:
2
eigen pair
2
eigenvector: cos1
cr
e
n EIEI
P EI
ll l
n
yB x
l



 








36. Assignment
35
Determine the
eigenvector and
eigenvalue for
the propped
column shown

37. Strain Energy
Methods
36

38. Introduction
37
Concepts of the work of external actions, internal energy
and the principle of virtual work.
The latter is widely used to calculate deflections in
trusses, beam and frames.

39. Introduction
38
The principle of virtual work is one or the most fundamental and
comprehensive principles of rational mechanics. For a rigid body
it may be stated:
If a rigid body under the action of any set of forces in
equilibrium be given a very small "virtual" displacement
(i.e., possible but not necessarily actual), the sum of the
work done by the force system will equal zero.

40. Introduction
39
If when a rigid body under the action of any set of force is
given a very small “virtual" displacement, the
total work of the forces vanishes, then the system of
forces is in equilibrium.

41. Introduction
40
The corresponding theorem for deformable bodies, such
as will be needed in analysis of statically indeterminate
stresses, may be stated:
If a structure in equilibrium under a set or forces be given
a very small, virtual deformation (i.e., one consistent with
continuity and elastic behaviorbut not necessarily actual),
the total summation of internal and external work will
vanish.

42. Introduction
41
The deflection of structures will be obtained from a
consideration of the work done by the forces acting on
the structures. The external work done by a force acting
through a deformation of the structure will be stored in
the structural material as potential energy of
deformation, or strain energy. This energy is recovered as
the structure returns to its original position, as the load or
loads are recovered.

43. Introduction
42
Law of the Conservation
of Energy

44. Work of external actions
43
The work of an external action is defined by the scalar
product of the forces vector (moments) and the
displacements vector (slopes), which are generated. The
expression of the elemental work of a force is given by

45. External mechanical work
44
??????=
1
2
�∆ ??????=
1
2
�??????
??????=
1
2
??????�
∆�??????�??????�??????������
??????rotation
s transverse slippings
??????angle
??????=
1
2
�
????????????

46. Internal or strain energy
45
The expression of a structure’s internal energy depends on the internal
action, as well as the geometric and mechanical characteristics of the
material constituting the structure.
�??????
????????????=�??????.�∆�
????????????=��??????
????????????=
�??????
??????
�??????=
∆�
�
�??????=
�??????∆�
�
�??????
????????????=
�??????∆�
�
.�∆�
??????
????????????=න
0
∆??????
�??????∆�
�
.�∆�=
�??????
2�
∆�
2

47. Internal or strain energy
46
??????
????????????=
��
2??????
∆�
2
=
????????????
2
∆�=
????????????
2
??????????????????
��
??????
????????????=
�??????
2
�
2�??????
Substitute �??????and ∆�
In the integral form
??????
????????????=න
0
??????
�
2
??????
2�??????
�??????

48. Bending Moment
47
�??????
????????????=�??????.�??????
�??????=�??????
??????
????????????=න
0
??????
�??????.�??????.=
�??????
2
??????
where k is the rigidity factor of the bar
�??????
????????????=�??????.�??????
??????
????????????=
1
2
න
0
??????
�
2
??????
�??????
�??????

49. Shear Force
48

50. Example
49AC Pb
Mx
L
 2
1
0
1
2
a
Pb
W xdx
EIL




 322
1 2
6
abP
W
EIL
 CB Pa
MX
L
 2
2
0
1
2
b
Pa
W XdX
EIL




 232
2 2
6
abP
W
EIL
 
322 232 222 222
12 2 2 2
666 6
in
abPabPabP abP
WWW ab
EILEILEIL EIL
 

51. Two Energy Methods
50
Virtual Work Principle
Castigliano’s Principle

52. Virtual Work Principle
51
JohnBernoulli introduced the principle of virtual work in
1717. It is also called the unit action method. This principle
is a very powerful tool for calculating deflections at
specified points on trusses, beams and plane frames.
The principle of virtual forces can be established as follows:
When a set of external actions, F
ex, is applied to a
deformable structure, internal actions, F
in, develop at any
point of the structure. Equilibrium equations ensure the
relationship between external and internal actions.

53. Virtual Work Principle
52
Externalactions cause externaldisplacementΔat
any point on the structure’s surface. In the same
context, internal actions generate internal
displacements d verifying the compatibility
relationships.
෍�
????????????∆=෍�
????????????�

54. Virtual Work Principle
53
�
????????????.�∆=෍�
????????????��
??????
��
????????????.∆=෍��
????????????.�
??????
Complementary
Virtual Work Principle
Virtual Work Principle
Perturb ∆by �∆
Perturb�
????????????∆by ��
????????????

55. Complementary Virtual Work Principle
54
��
????????????.∆=෍��
????????????.�
??????
Perturb ∆by �∆
The complementary virtual work done by an
external virtual forceunder actual
deformation of a structure is equal to the
complementary strain energy done by
virtual member forces under actual member
deformations.

56. Example
55;;
33
AC
FF
RR
Calculate the vertical displacement in joint B of the truss. EAis assumed to be
constant. 0.6();0.5();
0.6()
AB AC
BC
FFTRFT
RFC



57. Example
5611
;
22
AC
RR
Calculate the vertical displacement in joint B of the truss. EAis assumed to be
constant. 0.901();0.75();
0.901()
AB AC
BC
F TR C
RT



58. Example
57
��
????????????.∆=෍��
????????????.�
??????
�
??????=
�
??????�
??????
�??????
1.∆=෍��
????????????.
�
??????�
??????
�??????
∆=��
��.
���??????��
��
+��
��.
���??????��
��
+��
��.
���??????��
��
∆
=0.901×
0.6�×1.2�
�??????
+0.901
×
−0.6�×1.2�
�??????
−0.75×
0.5�×2�
�??????
∆=−
3��
4�??????

59. Thin Plates
and Shells:
Application
59

60. Thin Plates and Shells: Application
59
A plate is a structural element that is relatively thin in one
direction compared with the other two, and is flat.
plate has bending stiffness, whereas the membrane does
not. Typically, the flexural (bending) stiffness arises
because a plate is considerably thicker than a membrane
relative to its other dimensions.

61. Thin Plates and Shells: Application
60
Plates are important structural elements. They may exist
in many applications. In civil engineering, flat panels exist
in various steel or concrete structures (e.g., floor slabs).
They may be of various shapes (rectangular, circular,
rhombic, triangular, trapezoidal, and others).

62. Thin Plates and Shells: Application
61
Plates also occur in aerospace (e.g., aircraft, missile) and
naval (e.g.,ship, submarine) structures.

63. Thin Plates and Shells: Application
62
In mechanical engineering, plates can be seen as rotor
disks in brake systems, parts of various clutch and
other components. They can also exist as flat panels in
machine housings.

64. Thin Plates and Shells: Application
63
Geometrically viewed, a shell is like a plate, except that it
has curvature. Whereas a plate is flat, a shell is not.
Nevertheless, like a plate it has one dimension, which we
call its thickness (h), which is small compared to its other
dimensions. The thickness need not be constant, but in
many practical applications it is. And like a plate,
deformation of a shell is characterized entirely by what
happens at its mid surface and the normal to the mid-
surface.

65. Thin Plates and Shells: Application
64
Shell theory represents the deformations of a three
dimensional body by equations which are mathematically
two dimensional.
That is, only two independent space variables are
needed to unequivocally define what is occurring at
every point within the shell, instead of three.

66. Introduction
Euler Theory
Buckling Instability of
Struts/ Columns
Strain Energy Methods
Thin Plates and Shells:
Application
66
This Photoby Unknown
Author is licensed under CC

67. 67