I=I(un)=U(u)+V=U(un)-> I=1 Invoking the pmpe dII=0 a8u11 ∑F Frou OU F)6 V SuI +Fr du Theorem: If the strain energy can be expressed in terms of n displacements corresponding to N applied forces, the first derivative of the strain energy with respect to displacement ur is the applied force Example: EA� � ���� � � � Then: Π = Π(uI ) = U(uI ) + V = U(uI ) − N I=1 FIuI Invoking the PMPE: δΠ = 0 = N I=1 ∂U δuI − ∂uI ∂uI FI ∂uJ δuJ N I=1 ∂U δuI − ∂uI = FI δIJ δuJ N I=1 ∂U δuI − ∂uI = FI δuI � ∂U − FI ∂uI = δuI ∂U ∂uI ∀ δuI ⇔ FI = Theorem: If the strain energy can be expressed in terms of N displacements corresponding to N applied forces, the first derivative of the strain energy with respect to displacement uI is the applied force. Example: 4