Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 131

2.20 The principle of virtual work expression for a flat member of thickness t with in-plane
loading is

δ xx σ xx + δ yy σ yy + δγ xy σ xy tdA

δ W = −

S p δ

u x p x dS −

S p δ

u y p y dS = 0

A

(a) Derive the equation of equilibrium in V .
(b) Derive the statical boundary condition on S p .
(c) Formulate the virtual work expression along with the derived relations in matrix
notation.
2.21 For a rod subject to extension (Chapter 1), the kinematic relation is 0 x = ∂ x u 0 , th e force-

displacement relation is N = EA 0 x , and the condition of equilibr iu m is ∂ x N + p x = 0.

(a) Derive A of the first-order governing equations ∂ x z = Az + P .

(b) Find D u of D u u .

(c) Derive k D of the principle of virtual work.
(d) Derive A from k D . Note that, since k D embodies the principle of virtual work,
it leads to the equilibrium condition only. This must be supplemented with the
kinematic and material relations in order to obtain A .

=

Generalized Principles

2.22 Use form AD of Section 2.3 to derive the governing differential equations for a bar
under extension.
2.23 Use the Hellinger-Reissner functional to derive the governing differential equations
for a bar under extension.
2.24 Obtain an expression in matrix form for the hybrid functional of Eq. (2.104) for a 2D
plane stress problem.