Chapter 9 d eflections using energy methods
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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

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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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