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ECIV 320 Structural Analysis I

Deflections

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Deflection Diagrams & Elastic Curve

ASSUMPTION

Linear Elastic Material ResponseA structure subjected to a load will return to its original

undeformed position after load is removed

Double Integration Method

Moment Area Theorems

Conjugate Beam

Energy Methods

OBJECTIVE

Calculate Slope and Deflection due to Bending at any point on a Beam

METHODS

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Last Time - Conjugate Beam Method

Same amount of Computations as Moment-Area Theorems

Use to find slopes and deflection due to

bending of beams

Method is based on

Principles of Statics Only

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Energy Methods

Double Integration Method

Moment Area TheoremsConjugate Beam

Energy Methods

Calculate Slope and Deflection

Beams

Simple Loadings

Complicated Loadings

Trusses

Frames

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External Work & Strain Energy

Ue=Ui

Work of External

Forces & Moments

Strain Energy

Internal Forces

Conservation of Energy(Elastic Material Behavior)

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External Work

Force

x

eFdxU

0

Gradually Applied

PUe

2

1

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External Work

x

e FdxU0

Due to Another Force

' PU

e

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External Work

Moment

0

MdUe

Gradually Applied

MUe

2

1

Due to Another Moment

'MU

e

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Strain Energy- Axial Force

Hookes Law:

Stress:

Strain:

Final Deflection:

AE

LNNU

i22

12

E

AN

L AE

NL

N

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Strain Energy - Bending

EI

M

dx

d

dxEI

MMddU

i

22

12

L

idx

EI

MU

0

2

2

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Example

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Principle of Virtual Work

Bernoulli 1717

External and Internal Loads are Related throughEquations of Equilibrium

External and Internal Displacements are Related through

Equations of Compatibility

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Principle of Virtual Work

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Principle of Virtual Work

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Virtual Work

udL1Virtual

Loads

Real

Displ.

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Virtual Work for Trusses - External Loads

AENLdL /

udL1Virtual

Loads

Real

Displ.

u=n

AE

nNL1

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Virtual Work for Trusses - Temperature

TLadL

udL1Virtual

Loads

Real

Displ.

u=n

LTna1

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Virtual Work for Trusses - Fabrication Errors

LdL

udL1Virtual

Loads

Real

Displ.

u=n

Ln1

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Virtual Work for Trusses - Combined Effects

LnLTnaAEnNL1

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Example

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Step 1

Remove ALL External Loads

Apply Virtual (Unit) Load in the Direction of

the Unknown Displacement

Calculate Internal Forcesn

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Step 2

Apply REAL Loads

Calculate Internal ForcesN

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Step 3

Virtual Work Equation

LnLTnaAE

nNL1

0

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Solution

0

00.75

-1.25

1

0

80

120

-100

80

0.009

6

86

10

8

58,000

58,00058,000

43,500

58,000

0.029

0.011

0.049

0

0

0

00

0

7.2e-4 7.2e-4

7.2e-4

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Virtual Work for Beams and Frames - BENDING

dx

EI

MddL

udL1Virtual

Loads

Real

Displ.

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Virtual Work for Beams and Frames - BENDING

udL1Virtual

Loads

Real

Displ.

u=m

L

dxxEI

xMxm

0 )(

)()(1

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Virtual Work for Beams and Frames - BENDING

dxEI

MddL udL1

Virtual

Loads

Real

Displ. u=m

L

dxxEIxMxm

0 )()()(1

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Virtual Work for Beams and Frames - BENDING

dxEI

MddL

dLu1Virtual

Loads

Real

Displ.

u=mq

L

dxxEI

xMxm

0 )(

)()(1

To Calculate Slope