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Chapter 5 Elastic Strain, Deflection, and Stability 1
Chapter 5 Elastic Strain, Deflection, and Stability
Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction
by σx also causes a strain in the y-direction & z-direction by

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

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 Hooke’s Law Only ______________________elastic constants are needed for an__________ material.
εx
εy
εz
γxy = , γyz = , γzx =
G =

Chapter 5 Elastic Strain, Defl
Only two are independent elastic constant
Mohr’s circle σ → ε τ → γ/2
from G =
Pure shear stress
ection, and Stability 4
ε1 =
ε2 =
ε3 =
γ =
τ/γ ⇒

Chapter 5 Elastic Strain, Deflection, and Stability 5
Example:
1. The stress that develops in the y-direction.
2. The strain in the z-direction.
3. The strain in the x-direction.
4. The stiffness E′ = σz / εz in the z-direciton.
Is this equal to E ?
εy = & σx =

Chapter 5 Elastic Strain, Deflection, and Stability 6
1. εy = = 0
_______________________________________ ⇒ σy = 2. εz =
_______________________________________ =
3. εx = _______________________________________ =
4. E′ = σz / εz =

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
vε
=⇒= Vεν 5.0 volumedecreasestresstensile⇒> 5.0ν

Chapter 5 Elastic Strain, Deflection, and Stability 8
Hydrostatic stresses Invariant ∴∴∴∴ Volumetric strain ∝ hydrostatic stress
Constant → ___________ modulus
B=
σh = σv =

Chapter 5 Elastic Strain, Deflection, and Stability 9
Castigliano’s 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
δ

Chapter 5 Elastic Strain, Deflection, and Stability 11
Sample problem 5.4 con’t. 1. 2. 3.
202
LxxonlyvalidPxM
VM
=→==
==
Q
=
=
=
=U
=∂∂=PUδ

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 self-aligning bearings and subjected to a uniformly distributed load. Using Castigliano’s method, determine the required diameter d to limit the deflection to 0.2mm.
Known: A steel shaft supported by self-aligning bearings is subjected to a uniformly distributed load. Find: Using Castigliano’s 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 I-beam 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 Castigliano’s Method
• Reduntant reaction: _____________ force or moment that is ___________________
for equilibrium. • As magnitude of a redundant reaction is varied, _____________________ changes,
But _________________ remains. • Castigliano’s 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
=
== ∫ dyEI
Mu3
0
2
2
=∂∂==Fu0δ
=∴ F

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

Chapter 5 Elastic Strain, Deflection, and Stability 19
Fig5.25 Log-log 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 ρρ

Chapter 5 Elastic Strain, Deflection, and Stability 22
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Chapter 5 Elastic Strain, Deflection, and Stability 23
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