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Ebook A first course in the finite element method (4th edition): Part 2

(BQ) Part 1 book "A first course in the infite element" has contents: Structural dynamics and time dependent heat transfer, thermal stress, heat transfer and mass transport, plate bending element, plate bending element, axisymmetric elements,. and other contents. | CHAPTER 8 Development of the Linear-Strain Triangle Equations Introduction In this chapter, we consider the development of the stiffness matrix and equations for a higher-order triangular element, called the linear-strain triangle (LST). This element is available in many commercial computer programs and has some advantages over the constant-strain triangle described in Chapter 6. The LST element has six nodes and twelve unknown displacement degrees of freedom. The displacement functions for the element are quadratic instead of linear (as in the CST). The procedures for development of the equations for the LST element follow the same steps as those used in Chapter 6 for the CST element. However, the number of equations now becomes twelve instead of six, making a longhand solution extremely cumbersome. Hence, we will use a computer to perform many of the mathematical operations. After deriving the element equations, we will compare results from problems solved using the LST element with those solved using the CST element. The introduction of the higher-order LST element will illustrate the possible advantages of higherorder elements and should enhance your general understanding of the concepts involved with finite element procedures. d 8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations d We will now derive the LST stiffness matrix and element equations. The steps used here are identical to those used for the CST element, and much of the notation is the same. 398 8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations d 399 Step 1 Select Element Type Consider the triangular element shown in Figure 8–1 with the usual end nodes and three additional nodes conveniently located at the midpoints of the sides. Thus, a computer program can automatically compute the midpoint coordinates once the coordinates of the corner nodes are given as input. Figure 8–1 Basic six-node triangular element showing .