Samer Adeeb

Approximate Methods: Virtual Work Method

The statement of the equilibrium equations applied to a set D\subset\mathbb{R}^3 is as follows. Assuming that at equilibrium \sigma:D\rightarrow\mathbb{M}^3 is the symmetric Cauchy stress distribution on D and that u:D\rightarrow\mathbb{R}^3 is the displacement vector distribution and knowing the relationship \sigma=\sigma(u), then the equilibrium equation seeks to find u\in U such that the associated \sigma satisfies the equation:

    \[ \mathrm{div}\sigma+\rho b= 0 \]

where b:D\rightarrow\mathbb{R}^3 is the body forces vector distribution on D, \rho is the mass density, and U is the space of all possible displacement functions applied to D, i.e., U=\{a:D\rightarrow\mathbb{R}^3| a\mbox{ is kinematically admissible}\}. The term “Kinematically admissible” in U indicates that the space of possible solutions must satisfy the boundary conditions imposed on \partial D_u (as stated below) and any differentiability constraints.

The boundary conditions for the equations of equilibrium are usually given on two parts of the boundary of D denoted \partial D. On the first part, \partial D_n, the external traction vectors t_n are known so we have the boundary conditions for \sigma since \sigma^Tn=t_n (n is the normal vector to the boundary). On the second part, \partial D_u, the displacement is given.

Alternatively, the statement of the principle of virtual work states that at equilibrium, \forall v\in U\cup V where V is the space of at least once differentiable vector functions on D, i.e., V=\{a:D\rightarrow\mathbb{R}^3| a\mbox{ is once differentiable}\}:

    \[ \int_{\partial D} \! t_n\cdot v \,\mathrm{d}s+\int_D \! \rho b\cdot v \,\mathrm{d}x=\int_D \!\sum_{i,j=1}^3\sigma_{ij}(u)\varepsilon_{ij}(v)\,\mathrm{d}x \]

where

    \[ \varepsilon_{ij}(v)=\frac{1}{2}\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right) \]

As shown in the principle of virtual work section, one of its major applications is to find approximate solutions for the equations of equilibrium. This can be obtained by assuming that the solution u has a particular form with a finite number of unknowns, i.e., by looking for u in a subset U_{fd}\subset U that is finite dimensional but still able to approximate functions in U. The example in the principle of virtual work section utilizes polynomial approximations to find an approximate solution for the displacement field. The finite element method in the next sections utilizes piecewise affine functions to approximate the displacements. In this case the set V is expanded to allow for possible displacement functions that are not differentiable across element boundaries: V=\{a:D\rightarrow\mathbb{R}^3| a\mbox{ is once differentiable within the elements' domain and continuous across elements}\}.

Problems

Solve the problems in the Rayleigh Ritz method section using the Virtual Work Method.

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