ˉ ˉ ? ? ? ? ? ? 16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #10 - Principle of minimum potential energy and Castigliano’s First Theorem Principle of minimum potential energy The principle of virtual displacements applies regardless of the constitutive law. Restrict attention to elastic materials (possibly nonlinear). Start from the PVD: σ ij ? ij dV = t i u i dS + f i u i dV, ?ˉ u = 0 on S u (1)ˉ ˉ u/ˉ V S V Replacing the expression for the stresses for elastic materials: ?U 0 σ ij = ?? ij and assuming that the virtual displacement ?eld is a variation of the equili- brated displacement ?eld ˉu = δu, ? ij = δ? ij . ?U 0 δ? ij dV = t i δu i dS + f i δu i dV V ?? ij V ? ?? ? S 1 ? ? ? ? The expression over the brace is the variation of the strain energy density δU 0 : ?U 0 δU 0 = δ? ij ?? ij Using the properties of calculus of variations δ () = δ(): ? ? ? ? ? ? δU 0 dV = δ U 0 dV = δU = δ t i u i dS + f i u i dV = δ(?V ) S V where V is the potential of the external loads. Therefore: δΠ = δ(U + V ) = 0 which is known as the Principle of minimum potential energy (PMPE). In fact this expression only says that Π is stationary with respect to variations in the displacement ?eld when the body is in equilibrium. We can prove that it is indeed a minimum in the case of a linear elastic 1 material: U 0 = 2 C ijkl ? kl . We want to show: Π(v) ≥ Π(u), ?v Π(v) = Π(u) ? v = u Consider ˉu = u + δu: ? ? 1 ? Π(u + δu) = C ijkl (? ij + δ? ij )(? kl + δ? kl ) dV 2 V ? ? ? t i (u i + δu i )dS ? t i (u i + δu i )dV S ? V ? 1 1 =Π(u)+ ? 2 V ? 2 C ? ijkl ? ij δ? kl dV + V 2 C ijkl δ? ij δ? kl dV ? t i δu i dS ? f i δu i dV S V The second, fourth and ?fth term disappear after invoking the PVD and we are left with: 1 Π(u + δu) = Π(u) + C ijkl δ? ij δ? kl dV V 2 2 ? The integral is always ≥ 0, since C ijkl is positive de?nite. Therefore: Π(u + δu) = Π(u) + a, a ≥ 0, a = 0 ? δu = 0 and Π(v) ≥ Π(u), ?v Π(v) = Π(u) ? v = u as sought. Castigliano’s First theorem Given a body in equilibrium under the action of N concentrated forces F I . The potential energy of the external forces is given by: V = ? N I=1 F I u I where the u I are the values of the displacement ?eld at the point of applica- tion of the forces F I . Imagine that somehow we can express the strain energy as a function of the u I , i.e.: U = U (u 1 ,u 2 ,...,u N ) = U (u I ) 3 ? ? ???? ? ? ? Then: Π = Π(u I ) = U (u I ) + V = U (u I ) ? N I=1 F I u I Invoking the PMPE: δΠ = 0 = N I =1 ?U δu I ? ?u I ?u I F I ?u J δu J N I=1 ?U δu I ? ?u I F I δ IJ δu J = N I=1 ?U δu I ? ?u I F I δu I = ? ?U ? F I ?u I δu I = ?U ?u I ? δu I ? F I = Theorem: If the strain energy can be expressed in terms of N displacements corresponding to N applied forces, the ?rst derivative of the strain energy with respect to displacement u I is the applied force. Example: 4 ? ? (L + u) 2 + v 2 u ? I = ? 1 ～ L 2 L (L + u) 2 + (L ? v) 2 1 u ? v ? I I = ? 1 ～ 2L 2 2 L 1 ? ? u ? 2 √ ? 1 ? u ? v ?? 2 ? U = AEL + AE 2L 2 L 2 L Note that we have written U = U (u,v). According to the theorem: ?U 0 = ?u ?U F = ?v See solution in accompanying mathematica ?le. 5