16.21 Techniques of Structural Analysis and esig 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 ai dv=ti S+/fii dv, Vi/i=0 on S. Replacing the expression for the stresses for elastic materials: aUo and assuming that the virtual displacement field is a variation of the equili brated displacement field i=du, Ei;= deir i Su ds+/fiuid¯ ¯ � � � � � � 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: σij �ijdV = tiu¯idS + fiu¯idV, ∀u/¯ u¯ = 0 on Su (1) V S V Replacing the expression for the stresses for elastic materials: ∂U0 σij = ∂�ij and assuming that the virtual displacement field is a variation of the equili￾brated displacement field u¯ = δu, �ij = δ�ij . ∂U0 δ�ij dV = tiδuidS + fiδuidV V ∂�ij V � �� � S 1