Tham khảo tài liệu 'finite element analysis - thermomechanics of solids part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 9 Element Fields in Linear Problems This chapter presents interpolation models in physical coordinates for the most part for the sake of simplicity and brevity. However in finite-element codes the physical coordinates are replaced by natural coordinates using relations similar to interpolation models. Natural coordinates allow use of Gaussian quadrature for integration and to some extent reduce the sensitivity of the elements to geometric details in the physical mesh. Several examples of the use of natural coordinates are given. INTERPOLATION MODELS One-Dimensional Members Rods The governing equation for the displacements in rods also bars tendons and shafts is EA 2u pA dx2 dt2 in which u x t denotes the radial displacement E A and P are constants x is the spatial coordinate and t denotes time. Since the displacement is governed by a second-order differential equation in the spatial domain it requires two timedependent constants of integration. Applied to an element the two constants can be supplied implicitly using two nodal displacements as functions of time. We now approximate u x t using its values at xe and xe 1 as shown in Figure . The lowest-order interpolation model consistent with two integration constants is linear in the form u xt q 1 ti Ym Y m1 t ue t A I ue 1 t PL x 1 x . We seek to identify m1 in terms of the nodal values of u. Letting ue u xe and ue 1 u xe 1 furnishes ue t 1 xe AJO ue 1 t 1 xe 1 K1Y m1 t . 121 2003 by CRC CRC Press LLC 122 ue Finite Element Analysis Thermomechanics of Solids Ue 1 xe e FIGURE Rod element. Wef We 1 we L we 1 x xe Xe 1 FIGURE Beam element. However from the meaning of Ym1 t . we conclude that -1 Í1 xe 1 1 Xe 1 - xe 1 1 xe 1_ l _-1 1 _ . le x - xe. Beams The equation for a beam following Euler-Bernoulli theory is EI dw PA d2u 0. dx dt in which w x. t denotes the transverse displacement of the beam s neutral axis. and I is a constant. In the spatial domain. there are
