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Chapter 9 D eflections using Energy MethodsPowerPoint Presentation

Chapter 9 D eflections using Energy Methods

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### External Work

### External Work – Force/Moment

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

Conservation of Energy Principle

Ue = Ui

- Linear Elastic Response

Strain Energy – Axial Force

- Hooke’s Law
- σ = EЄ

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

- σ = N/A
- Final Strain
- Є=Δ/L
- L=Length

- Є=Δ/L
- Δ = NL/AE
- Ui = N2L/2AE

Strain Energy – Bending

- Strain Energy
- dUi =M2dx/2EI

- Strain Energy over Beam Length

Principle of Work and Energy

- 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

Method of Virtual Work Procedure

- 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 Notes

- You must retain the algebraic sign for each of the n and N values when placing them into the virtual work equation.
- 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.
- For Temperature Equation:
- ΔT = Positive with temperature increase, Negative with temperature decrease.

- For Fabrication Equation:
- ΔL = Positive with increase in length, Negative with decrease in length.

- MAKE SURE YOU PAY ATTENTION TO YOUR UNITS.

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