9780203009512.ch9

9780203009512.ch9 - 0749_Frame_C09 Page 121 Wednesday 5:09...

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121 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 interpola- tion 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. 9.1 INTERPOLATION MODELS 9.1.1 O NE -D IMENSIONAL M EMBERS 9.1.1.1 Rods The governing equation for the displacements in rods (also bars, tendons, and shafts) is (9.1) in which u ( x , t ) denotes the radial displacement, E, Α and ρ 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 (time- dependent) 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 x e and x e + 1 , as shown in Figure 9.1. The lowest-order interpolation model consistent with two integration constants is linear, in the form (9.2) We seek to identify Φ m1 in terms of the nodal values of u . Letting u e = u ( x e ) and u e + 1 = u ( x e + 1 ), furnishes (9.3) 9 E A u x A u t = 2 2 2 2 , uxt x t t ut xx m T mm m e e m T (,) () , () ( ) . == = + ϕ Φ γ γ ϕ 11 1 1 1 1 1 , x t u t x t e e mm e e ) , . ++ 1 1 11 Φγ © 2003 by CRC CRC Press LLC
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122 Finite Element Analysis: Thermomechanics of Solids However, from the meaning of γ m 1 ( t ), we conclude that (9.4) 9.1.1.2 Beams The equation for a beam, following Euler-Bernoulli theory, is: (9.5) 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 four constants of integration. In an element, they can be supplied implicitly by the values of w and w = w /∂ x at each of the two element nodes. Referring to Figure 9.2, we introduce the interpolation model for w ( x , t ): (9.6) FIGURE 9.1 Rod element. FIGURE 9.2 Beam element. u e x +1 w t e+1 Φ m e e e ee e x x l xx lx x 1 1 1 1 1 1 1 1 11 = = =− + + + ,. E I w x A u t + = 4 4 2 2 0 ρ , wxt x t x x x x t w w w w bb b b m e e e e (,) () , , . == = + + ϕ Φ γ ϕ γ Τ Τ 1 1 23 1 1 1 1 © 2003 by CRC CRC Press LLC
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Element Fields in Linear Problems 123 Enforcing this model at x e and at x e + 1 furnishes (9.7) 9.1.1.3 Beam Columns Beam columns are of interest, among other reasons, in predicting buckling according to the Euler criterion. The z –displacement w of the neutral axis is assumed to depend only on x and the x –displacement. Also, u is modeled as (9.8) in which u 0 ( x ) represents the stretching of the neutral axis. It is necessary to know u 0 ( x ), w ( x ) and at x e and x e + 1 . The interpolation model is now (9.9) and 9.1.1.4
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This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

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9780203009512.ch9 - 0749_Frame_C09 Page 121 Wednesday 5:09...

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