Virtual Work and Variational Formulations

We can similarly obtain the other integral forms. Let be the virtual (transverse) displacement at cross-section x . Further, let belong to the same class of admissible functions from which w ( x ) is chosen. Then setting w equal to in eq. (18.8), we get

(18.9)

Now, we treat the symbol in the expression as the operator called as the variational operator . The quantity is called as the variation of . Using the properties of - operator from section 5 of Lecture 2, we can convert eq. (18.9) to the following form :

(18.10)

where

(18.11)

The functional I is called as the Variational Functional of the boundary value problem (18.1(a), 18.1(b), 18.1(c)). This is the integral form to be used for the Variational Formulation . This integral needs to be extremized to obtain the solution of the boundary value problem.

The extremizing function needs to satisfy the following three conditions, which are similar to the conditions on the weight function w ( x ) :

1. The function must satisfy the kinematic boundary conditions 18.1(b). Thus and at   x = 0. Further, at this point, the variation must satisfy the conditions and at x = 0.
2. The function , its variation and the first derivatives of and must be unconstrained where the force boundary conditions (eq. 18.1(c)) are specified. Thus, and must be unconstrained at x = L .
3. The function must be smooth enough to make the functional I finite. Thus, be must be such that is finite at every point of the internal (0, L ).

The equation 18.1 (a) is called as the Euler Equation of the functional I (eq. 18.11). Further, the boundary conditions 18.1(b) are called as the Essential Boundary Conditions , and the conditions 18.1(c) are called as the Natural Boundary Conditions.

Physically, the quantity I (eq. 18.11) is the total potential energy of the beam. Therefore, the variational formulation states the following principle. The solution of the boundary value problem (18.1(a), 18.1(b), 18.1(c)) extremizes the total potential energy of the beam. This is called as the Principle of the Stationary Value of the Total Potential Energy . In eq. (18.9), the left hand side represents the virtual work of the internal forces while the right hand side represents the virtual work of the external forces q ( x ) and V and the moment M . Thus, equation (18.9) states the following. For to be a solution of the boundary value problem (18.1(a), 18.1(b), 18.1(c)), the virtual work of the external forces must be equal to the virtual work of the internal forces.