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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