ABSTRACT
The governing equations of a continuum can be derived using the laws of physics. For solid mechanics problems, the laws of physics may be expressed in several alternative forms. For example, the principle of conservation of linear momentum, which leads to the equations of motion, can be derived using either Newton’s Second Law of motion or by using the principle of virtual displacements. The former is termed a vector approach and the latter an energy approach. The use of Newton’s laws to determine the governing equations of a structural
problem requires the isolation of a typical volume element of the structure with all its applied and reactive forces (i.e., the free-body diagram of the element). Then the vector sum of all static and dynamic forces and moments acting on the element is set to zero to obtain the equations of motion. For simple mechanical systems for which the free-body diagram can be set up, the vector approach provides an easy and direct way of deriving the governing equations. However, for complicated systems the procedure becomes more cumbersome and intractable. In addition, the type of boundary conditions to be used in conjunction with the derived equations is not always clear. In the energy approach, the total work done or energy stored in the body due
to actual forces in moving through virtual displacements that are consistent with the geometric constraints of a body is set to zero to obtain the equations of motion. The energy approach yields not only the equations of motion, but also the force boundary conditions as well as the form of the variables involved in the specification of geometric boundary conditions. In addition, the energy expressions are useful in obtaining approximate solutions by direct variational methods, such as the Ritz and finite element methods. In most of the present study, the principles of virtual work will be used to
derive the equations of motion of plates and shells. We begin with the concepts of virtual displacements and virtual forces and work and energy. Mostly bar and beam problems are used to illustrate the concepts.
