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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

Chapter 9 Deflections using Energy Methods

Reading Assignment:

Chapters 9.1 - 9.5

external work

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

External Work – Force/Moment

- Linear Elastic Response

strain energy axial force
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 – Bending
  • Strain Energy
    • dUi =M2dx/2EI
  • Strain Energy over Beam Length
example 1
Example #1

Truss Example

Real Work Method

principle of work and energy
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
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 work1
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
Method of Virtual Work: Trusses
  • External Loading
  • Temperature
  • Fabrication Error and Camber
method of virtual work procedure
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
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
Example #2

Truss Example

Virtual Work Method

example 3
Example #3

Truss Example

Virtual Work Method

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