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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 x 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
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 y-direction.
2. The strain in the z-direction.
3. The strain in the x-direction.
4. The stiffness E = z / z in the z-direciton.
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 self-aligning 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 self-aligning 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 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 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 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
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 Stress-Strain RelationshipA stress in the x-direction causes a strain in the x-directionResulting 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