Basic FEA Theory – Part 1 – Principle of Virtual Work

The principle of virtual work is a global (integral) form of the equilibrium equation. The derivation starts by integrating the product of the equilibrium equation with a virtual displacement field over the body: \[ -\int_V (\sigma_{ji,j} + \bar{b}_i) \delta u_i dV + \int_{A_t} (t_i – \bar{t}_i) \delta u_i dA = 0. \] In this equation both integrands are identically zero, and quantities with a bar denotes a prescribed value. It many seem odd to formulate this equation since it is clear that is will always be zero, but as I will show below, the equation can be manipulated into a different form that is quite useful. Now consider the following integral in more detail \[ \begin{align} \int_{A_t} t_i \delta u_i dA &= \int_A t_i \delta u_i dA – \int_{A_u} t_i \delta u_i dA,\\ &= \int_V (\sigma_{ji}\delta u_i)_{,j}\, dV – \int_{A_u} t_i \delta_i dA,\\ &= \int_V \sigma_{ji,j} \delta u_i dV + \int_V \sigma_{ji}\delta u_{i,j}\, dV – \int_{A_u} t_i \delta u_i dA. \end{align} \] Inserting this equation into the first equation in this section yields $$ \int_V \sigma_{ji} \delta u_{i,j} dV – \int_V \bar{b}_i \delta u_i dV – \int_{A_t} \bar{t}_i \delta u_i dA – \int_{A_u} t_i \delta u_i dA = 0.$$ But \(\delta u_i=0\) on \(A_u\) and \(\sigma_{ij} \delta u_{i,j} = \sigma_{ij} \delta\varepsilon_{ij}\) giving \[ \int_V \sigma_{ij} \delta\varepsilon_{ij} dV – \int_V \bar{b}_i \delta u_i dV – \int_{A_t} \bar{t}_i \delta u_i dA = 0. \] Which also can be written \( \delta W_i + \delta W_e = 0 \), where \( \delta W_i \) is the internal virtual work, and \( \delta W_e \) is the external virtual work. The equation for the principle of virtual work can also be written extended to include external point forces (\(f_i\)): \[ \int_V \sigma_{ij} \delta \varepsilon_{ij} dV = \int_V b_i \delta u_i dV + \int_A t_i \delta u_i dA + \sum \delta u_i f_i, \] where I dropped the bar on the body force and surface tractions for simplicity.