Finite Element Method - The Basis (Volume 1)

24 A direct approach to problems in elasticityTo make the nodal forces statically equivalent to the actual boundary stresses anddistributed body forces, the simplest procedure is to impose an arbitrary (virtual)nodal displacement and to equate the external and internal work done by the variousforces and stresses during that displacement.Let such a virtual displacement be a e at the nodes. This results, by Eqs (2.1) and(2.2), in displacements and strains within the element equal tou ˆ N a e and e ˆ B a e …2:6†respectively.The work done by the nodal forces is equal to the sum of the products of the individualforce components and corresponding displacements, i.e., in matrix languagea eT q e…2:7†Similarly, the internal work per unit volume done by the stresses and distributedbody forces isorye T r ÿ u T b…2:8†a T …B T r ÿ N T b†…2:9†Equating the external work with the total internal work obtained by integratingover the volume of the element, V e , we have …a eT q e ˆ a…V eT B T r d…vol†ÿ N T b d…vol†…2:10†e V eAs this relation is valid for any value of the virtual displacement, the multipliersmust be equal. Thus……q e ˆ B T r d…vol†ÿ N T b d…vol†V e V e …2:11†This statement is valid quite generally for any stress±strain relation. With the linearlaw of Eq. (2.5) we can write Eq. (2.11) aswhereandq e ˆ K e a e ‡ f e…K e ˆ B T DB d…vol†V e………f e ˆÿ N T b d…vol†ÿV e B T De 0 d…vol†‡V e B T r 0 d…vol†V e…2:12†…2:13a†…2:13b†y Note that by the rules of matrix algebra for the transpose of products…AB† T ˆ B T A T