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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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