Chapter 9 Deflections using Energy Methods

Reading Assignment:Chapters 9.1 - 9.5

External Work

•Used for more complicated loadings or for structures such as trusses and frames.

•Conservation of Energy Principle

Ue = Ui

External Work – Force/Moment

2

2

2

00

0

PU

xPxdxPFdxU

xPF

FdxU

e

XX

e

X

e

- Linear Elastic Response

MU

MdU

e

e

210

Strain Energy – Axial Force• Hooke’s Law

σ = EЄ• Normal Stress

σ = N/A A=Constant Cross-Sectional Area

Final Strain Є=Δ/L

L=Length Δ = NL/AE Ui = N2L/2AE

Strain Energy – Bending

• Strain Energy dUi =M2dx/2EI

• Strain Energy over Beam Length

L

e EIdxMU

0

2

2

Example #1

Truss Example

Real Work Method

Principle of Work and Energy

L

i EILP

EIdxPx

EIdxMU

0

3222

61

22

2

PUe

EIPL

EILPPUU ie 362

332

•Application of this method is limited•Only one load may be applied to the structure•Only the displacement under the force can be obtained.

Principle of Virtual Work• Also known as the “unit load method”

• Used to determine displacement and slope at a specific location on a structure.

• See Figure 9-6 in textbook.

Principle of Virtual Work• External Virtual Work

P’Δ • Internal Virtual Work

Σu*dL• ΣPΔ = Σuδ

P’ * Δ = Σu*dL• Moment

P’*θ = Σuθ *dL

Method of Virtual Work: Trusses• External Loading

• Temperature

• Fabrication Error and Camber

AEnNLP'

TLnP '

LnP'

Method of Virtual Work Procedure1) Place unit load on the truss at the joint where the desired

displacement is to be determined and remove all real loads.NOTE: Load should be in the same direction of

displacement.2) Calculate the internal forces in each truss member. (+ =

tension, - = compression)3) Remove unit load and apply real loads to truss.4) Calculate the internal forces in each truss member. (+ =

tension, - = compression)

Method of Virtual Work Important Notes1) You must retain the algebraic sign for each of the n and N values

when placing them into the virtual work equation.2) If the virtual work equation is positive, the displacement is in the

same direction as the unit load. If it is negative it is in the opposite direction.

3) For Temperature Equation:1) ΔT = Positive with temperature increase, Negative with temperature

decrease.4) For Fabrication Equation:

1) ΔL = Positive with increase in length, Negative with decrease in length.5) MAKE SURE YOU PAY ATTENTION TO YOUR UNITS.

Example #2

Truss Example

Virtual Work Method

Example #3

Truss Example

Virtual Work Method