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**CIVL3310 STRUCTURAL ANALYSIS**

Chapter 9: Deflections Using Energy Methods CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang

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**Work and Energy Principles**

Work done by external forces F F F F F

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**Work and Energy Principles**

Where does the work go? Saved in the beam in terms of “Strain energy” Member force i-th component Member deformation F F Work - Energy Total strain energy

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**Virtual Work Principle**

Under equilibrium, perturb the structure Work - Energy F F F Perturb DF by d DF Perturb F by dF

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**Virtual Work Principle**

F F (Complementary) virtual work principle Perturb F by dF Virtual work principle Perturb DF by d DF

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**Virtual Work Principle**

(Complementary) virtual work principle Perturb F by dF fi di F F The complementary virtual work done by an external virtual force system under the actual deformation of a structure is equal to the complementary strain energy done by the virtual stresses under the actual strains The complementary virtual work done by an external virtual force system under the actual deformation of a structure is equal to the complementary strain energy done by the virtual stresses under the actual strains

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**Virtual Work Principle**

fi di F F Superposition Virtual system Actual system

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**Virtual Work Principle**

Looking for DF 1 Virtual system Actual system fiLi EiAi fi di F F 1

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**Virtual Work Principle**

dfi F fi F

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**Virtual Work Principle**

Temperature effect dfi DT F

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**Virtual Work Principle**

Misfit effect dfi DL F

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**Virtual Work Principle**

For Beams M=0, q≠0, k=0 dF dM(x) M(x) M ≠ 0, q≠0, k ≠ 0 DF F x dq Perturbation M(x) Integrate for the whole beam

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**Virtual Work Principle**

DF dMA qA M(x) dF dM(x) Moment due to actual loads Deformation due to actual loads Virtual force corresponds to actual deformation Virtual moment induced by The virtual force

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**Virtual Work Principle**

For Frames w DF dF P Negligible

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**Castigliano’s Principle**

Work-Energy Fi Fi Fi 1 k

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**Castigliano’s Principle**

For Trusses For Beams For Frames

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**Betti’s Law of Reciprocal Deflections**

Location 1 Location 2 Virtual work principle P Q P Q Betti’s Law

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**Betti’s Law of Reciprocal Deflections**

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**Betti’s Law of Reciprocal Deflections**

M

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**Betti’s Law and Flexibility Coefficients**

Location j Location i 1 Flexibility coefficient Deflection at j due to a unit load at i 1 Deflection at i due to a unit load at j

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**9. Deflections Using Energy Methods**

Work-energy principle Virtual work principle Castigliano’s principle Betti’s law

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