Chapter 12 - Energy Theorems

Abstract

In this chapter the concepts and theorems are defined and illustrated for strength of materials problems. It is assumed that stress is linearly related to strain through the Young's modulus, which is a constant. Likewise, the load displacement relationship is linear. The energy theorems are based on strain energy, work done, and related concepts. The energy theorems discussed in this chapter are—principle of virtual work, principle of minimum potential energy, Castigliano's first theorem, unit displacement theorem, principle of complementary virtual work, principle of complementary energy, Castigliano's second theorem, unit load theorem, Betti's theorem, Maxwell's reciprocal theorem, and principle of superposition. Further, the ten basic concepts that are defined in this chapter are—strain energy, complementary strain energy, strain energy of total deformation, complementary strain energy of total deformation, work, complementary work, virtual displacement, virtual force, virtual work, and complementary virtual work. Strain energy is a key concept in the discipline of structural mechanics. It also plays a central role in failure theories and fracture mechanics. Equations of structural mechanics have been derived from a use of energy principles. The concept has been used to generate approximate solutions to difficult problems. Several analytical propositions referred to as “energy theorems” have been established utilizing strain energy and related concepts.