Small displacement - Statics - Weak formulation and “energy” theorems

7. Small displacement - Statics - Weak formulation and “energy” theorems#

This section presents different weak formulations in the elastic problem, for both equilibrium conditions and congruence conditions, for different models/structural elements.

Models. In this section, weak formulations for different models of elastic structures are explicitly derived:

Approaches. The results shown here can be classified into two different approaches, depending on the independent variables of the methods:

displacement approach: 1) weak form of the equilibrium equations, 2) principle of virtual work, and 3) stationariety of the total potential energy
force approach: 1) weak form of the congruence conditionss, 2) principle of complementary virtual work, and 3) stationariety of the total complementary potential energy.

A force approach can be efficiently used to evaluate hyperstatics and point displacements/rotations, common questions in simple problems/exercises of the introductory courses in structural mechanics, as a result of low-dimensional linear systems that can be solved with a pen-and-paper approach. Displacement methods usually require some assumptions about the displacement field, and provides a common general methods for solving the whole structure. As an example, finite element method can be formulated as a discrete counterpart (usually a projection over a finite-dimensional basis) of the the weak form of equilibrium equations: FEM usually results in a “higher”-dimensional linear system1 (if compared with those of a force approach), that makes from little to no sense at all to solve by-hand, but can be easily built and solved with a computer.

Examples, problems and exercises. Some problems about beam structures can be conveniently approached with the contents of this section.

Reading guidance

There is no mandatory or recommended order for addressing these paragraphs. The treatment of the elastic solid requires some familiarity with tensors. Beam models can be treated as simplified models of the 3-dimensional solid, under assumptions on stress (or strain) fields. The formulation of the general elastic beam introduces a vector-based framework applicable to any elastic beam: it may appear obscure or cryptic at first, but it offers a level of generality that greatly simplifies the numerical implementation of arbitrary beam models. The structurally decoupled elastic beam is a very specific case of this general treatment, yet it enables the construction of simple models suitable for solving structural problems by hand, using only paper, pencil, and manageable amounts of algebra. Simple models become even simpler for slender beams with Bernoulli’s kinematic assumption - i.e. if displacement due to axial and shear actions are negligible if compared with bending. This approach helps internalize the core principles by allowing practice on simpler problems without unnecessary algebraic complexity — after all, computers exist for the heavy calculations.

1: Finite element methods for linear structural problems usually provide a linear problem, \(\mathbf{K} \mathbf{u} = \mathbf{f}\), with some useful properties that can be exploited while building and solving the problem: the stiffness matrix \(\mathbf{K}\) is usually sparse (this properties allows to save memory, using sparse matrix format to avoid storing in an inefficient way lots of information — lots of zero; efficients algorithms exist for matrix operations with sparse matrices) and symmetric (many algorithms work well with symmetric matrices).