Chapter 9 D eflections using Energy Methods

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# Chapter 9 D eflections using Energy Methods - PowerPoint PPT Presentation

Chapter 9 D eflections 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 U e = U i. External Work – Force/Moment. - Linear Elastic Response.

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### Chapter 9 Deflections using Energy Methods

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

- Linear Elastic Response

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
Example #1

Truss Example

Real Work Method

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
• 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)

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.
Example #2

Truss Example

Virtual Work Method

Example #3

Truss Example

Virtual Work Method