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Chapter 5 Elastic Strain, Deflection, and Stability 1
Chapter 5 Elastic Strain, Deflection, and Stability
Elastic StressStrain Relationship A stress in the xdirection causes a strain in the xdirection
by x also causes a strain in the ydirection & zdirection by

Chapter 5 Elastic Strain, Deflection, and Stability 2
E =
Resulting Strain Each Direction
Stress x y z
x
y
z

Chapter 5 Elastic Strain, Deflection, and Stability 3
Adding the columns to obtain the total strain in each direction Shear strain Note: shear strain on a given plane is _________________by the shear stresses on other planes.
Generalized Hookes Law Only ______________________elastic constants are needed for an__________ material.
x
y
z
xy = , yz = , zx =
G =

Chapter 5 Elastic Strain, Defl
Only two are independent elastic constant
Mohrs circle /2
from G =
Pure shear stress
ection, and Stability 4
1 =
2 =
3 =
=
/

Chapter 5 Elastic Strain, Deflection, and Stability 5
Example:
1. The stress that develops in the ydirection.
2. The strain in the zdirection.
3. The strain in the xdirection.
4. The stiffness E = z / z in the zdireciton.
Is this equal to E ?
y = & x =

Chapter 5 Elastic Strain, Deflection, and Stability 6
1. y = = 0
_______________________________________ y = 2. z =
_______________________________________ =
3. x = _______________________________________ =
4. E = z / z =

Chapter 5 Elastic Strain, Deflection, and Stability 7
Volumetric Strain & Hydrostatic Stress
Volume changes associated with __________________________. Shear strains cause only __________________________ Since V=LWH
=dV
=VdV
=
==VdV
v
== V 5.0 volumedecreasestresstensile> 5.0

Chapter 5 Elastic Strain, Deflection, and Stability 8
Hydrostatic stresses Invariant Volumetric strain hydrostatic stress
Constant ___________ modulus
B=
h = v =

Chapter 5 Elastic Strain, Deflection, and Stability 9
Castiglianos Method
Useful in computing elastic deflection and redundant reactions
Deflection Figure 5.15 General load deflection curve for elastic range
stored elastic energy is equal to _________________ times ___________________. Deflection, In general case,
== UU
== UddU
=
=

Chapter 5 Elastic Strain, Deflection, and Stability 10
Axial Loading Case
Sample problem 5.4
=
=
=
=
=
U
U
U

Chapter 5 Elastic Strain, Deflection, and Stability 11
Sample problem 5.4 cont. 1. 2. 3.
202LxxonlyvalidPxM
VM
===
==
Q
=
=
=
=U
==PU

Chapter 5 Elastic Strain, Deflection, and Stability 12
Problem 5.15 (page233) What are the angular and linear displacements of point A of Figure 5.15? Known: Figure P.15 is given. Find: Calculate the angular and linear displacements of point A.

Chapter 5 Elastic Strain, Deflection, and Stability 13
Problem 5.19 (page 234) Figure 5.19 shows a steel shaft supported by selfaligning bearings and subjected to a uniformly distributed load. Using Castiglianos method, determine the required diameter d to limit the deflection to 0.2mm.
Known: A steel shaft supported by selfaligning bearings is subjected to a uniformly distributed load. Find: Using Castiglianos Method, determine the required diameter, d, to limit the deflection to 0.2mm. Assumption: 1. The steel shaft remains in the elastic region. 2. The transverse shear deflection is negligible. Analysis:

Chapter 5 Elastic Strain, Deflection, and Stability 14
Problem 5.23(page 235) In order to reduce the deflection of the Ibeam cantilever shown, a support is to be added at S. (a). What vertical force at S is needed to reduce the deflection at this point to zero? (b). What force is needed to cause an upward deflection at S of 5mm? (c). What can you say about the effect of these forces at S on the bending stresses at the point of beam attachment? Assumptions: 1. The beam remains elastic. 2. Transverse shear deflection is negligible. Analysis:

Chapter 5 Elastic Strain, Deflection, and Stability 15
Redundant Reactions by Castiglianos Method
Reduntant reaction: _____________ force or moment that is ___________________
for equilibrium. As magnitude of a redundant reaction is varied, _____________________ changes,
But _________________ remains. Castiglianos theorem states that the _________________associated with any
reaction that can be varied without upsetting equilibrium. The deflection = ______________________.

Chapter 5 Elastic Strain, Deflection, and Stability 16
Sample Problem 5.9
Find: Determine the tension in the guy wire Assumption: 1. 2. 3. Analysis:
Figure 5.22
At point a

Chapter 5 Elastic Strain, Deflection, and Stability 17
M=
= Bending energy below point a The horizontal deflection at point a
=
== dyEIMu
3
0
2
2
===Fu0
= F

Chapter 5 Elastic Strain, Deflection, and Stability 18
Euler Column Buckling
Figure 5.24 B=0 x=L, y=0
Q
0sin = LA
=cr
==EIM
dxyd2
2
=
2AI =
=crS
=ESor cr

Chapter 5 Elastic Strain, Deflection, and Stability 19
Fig5.25 Loglog plot of Euler Eq. 5.11 (dimensionless, hence applies to all materials within their elastic range).
Fig5.26 Euler column buckling curves illustrated for two values of E and Sy.
ESpLe
cr 1.010/
==

Chapter 5 Elastic Strain, Deflection, and Stability 20
Figure 5.27 Equivalent column lengths for various end conditions
Figure 5.28 Euler and Johnson column curves illustrated for two valuses of E and Sy

Chapter 5 Elastic Strain, Deflection, and Stability 21
Secant formula for the ______________loading, taking the _______________________ into account. Where c denotes the distance from the neutral bending plane to the extreme fiber.
+==
AEPLec
SAPS
cre
ycrcr
4)(sec)(1 2

Chapter 5 Elastic Strain, Deflection, and Stability 22
Draft paper 1/2

Chapter 5 Elastic Strain, Deflection, and Stability 23
Draft paper 2/2
Chapter 5Elastic Strain, Deflection, and StabilityElastic StressStrain RelationshipA stress in the xdirection causes a strain in the xdirectionResulting Strain Each Direction
Adding the columns to obtain the total strain in each directionGeneralized Hookes Law
Example:Volumetric Strain & Hydrostatic StressCastiglianos MethodDeflection
Axial Loading Case
Sample problem 5.4Redundant Reactions by Castiglianos Method