Approximate Methods: Final Note

The the Rayleigh Ritz method, the virtual work method, the weighted residuals method, and the Galerkin methods are designed to find approximate solutions using assumed trial functions for the displacement field sought. The terminology “weak formulation” is often used to describe the equations developed in those methods in comparison to the “strong formulation”, which is used to describe the original differential equation. The displacement function that satisfies the differential equation at every point is called the “strong form solution”, while the approximate displacement function that satisfies the weak formulation is termed the “weak form solution”. The term “strong” refers to the strict requirement that the solution is exact, while the term “weak” refers to the fact that the obtained solution satisfies different (sometimes fewer) requirements than the “strong form solution”.

Another major difference between the “strong form” and the “weak form” of the problem is that the strong form often involves a differential equation, which is harder to deal with at locations of discontinuities. When dealing with a strong form, the equations often involve dividing the domain into different locations at points of discontinuities. When dealing with a weak form, however, the equations involve an integral, which can sustain discontinuities, and thus, while the exact solution (strong form) is discontinuous, the weak form obtained can still be continuous.

General Notes on the Weak Form of Structural Mechanics Problems:

• The trial functions assumed should be as close as possible to the exact solution.
• Since in the weak form the system is constrained to a specified displacement shape, the displacements are under-predicted and the total energy stored in the system is also under-predicted. On the other hand, if the boundary conditions involve prescribed displacements, then the loads producing such displacements are over-predicted using the weak formulation.
• As with any approximate solution, the accuracy of the principal unknown is higher than the accuracy of its derivatives. For the problems described in this text, the displacement is the primary unknown function. Thus, the accuracy of the approximate displacement function is better than the accuracy of its derivatives (the strain components), and thus, is better than the accuracy of the stress components (which are functions of the strain components).
• The term “essential boundary conditions” in solid mechanics problems is used to refer to the boundary conditions of the displacements in solids while it refers to displacements and rotations in Euler Bernoulli beams and in shells. The term “non-essential boundary conditions” is used to refer to the boundary conditions of the stress in solids and of bending moments and shearing forces in Euler Bernoulli beams and shells.