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Chapter 9 D eflections using Energy Methods

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 D eflections using Energy Methods

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  1. Chapter 9 Deflections using Energy Methods Reading Assignment: Chapters 9.1 - 9.5

  2. External Work Used for more complicated loadings or for structures such as trusses and frames. Conservation of Energy Principle Ue = Ui

  3. External Work – Force/Moment - Linear Elastic Response

  4. 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

  5. Strain Energy – Bending • Strain Energy • dUi =M2dx/2EI • Strain Energy over Beam Length

  6. Example #1 Truss Example Real Work Method

  7. 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.

  8. 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.

  9. 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

  10. Method of Virtual Work: Trusses • External Loading • Temperature • Fabrication Error and Camber

  11. 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)

  12. 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.

  13. Example #2 Truss Example Virtual Work Method

  14. Example #3 Truss Example Virtual Work Method

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