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

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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” F F i-th component Member force Member deformation Total strain energy Work - Energy

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Virtual Work Principle Under equilibrium, perturb the structure F F F Work - Energy Perturb F by F Perturb F by F

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Virtual Work Principle (Complementary) virtual work principle Perturb F by dF Virtual work principle Perturb F by F F F

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Virtual Work Principle (Complementary) virtual work principle Perturb F by F 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 f i d i

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Virtual Work Principle F F f i d i Superposition Actual system Virtual system

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Virtual Work Principle F F f i d i Actual system Virtual system Looking for F 1 1 fiLiEiAifiLiEiAi

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Virtual Work Principle F F fifi fifi

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F fifi TT Temperature effect

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Virtual Work Principle F fifi LL Misfit effect

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Virtual Work Principle For Beams F M=0, ≠0, =0 M ≠ 0, ≠0, ≠ 0 M(x) x dd Integrate for the whole beam Perturbation FF FF M(x)

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Virtual Work Principle P FF w Deformation due to actual loads Moment due to actual loads FF M(x) M(x) Virtual force corresponds to actual deformation Virtual moment induced by The virtual force AA A

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Virtual Work Principle For Frames Negligible w P FF FF

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Castigliano’s Principle Work-Energy FiFi FiFi FiFi 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 P Q Location 1Location 2 Virtual work principle Betti’s Law Q P

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Betti’s Law of Reciprocal Deflections P P

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P M

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Betti’s Law and Flexibility Coefficients 1 1 Location jLocation i Deflection at j due to a unit load at i Flexibility coefficient 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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