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FIGURE 2.84 Practical examples of plane strain problems.

2.10.5 Choice of Element Shapes and Sizes

A finite element generally has a simple one-, two-, or three-dimensional configuration. The boundaries of elements are often straight lines and the elements can be one-dimensional, two-dimensional, or three-dimensional, as shown in Figure 2.85. While subdividing the continuum, one has to decide the number, shape, size, and configuration of the elements in such a way that the original body is simulated as closely as possible. Nodes must be located in locations where abrupt changes in geometry, loading, and material properties occur. A node must be placed at the point of application of a concentrated load because all loads are converted into equivalent nodal-point loads.

It is easy to subdivide a continuum into a completely regular one having the same shape and size. But problems encountered in practice do not involve regular shape; they may have regions of steep gradients of stresses. A finer subdivision may be necessary in regions where stress concentrations are expected to obtain a useful approximate solution. Typical examples of mesh selection are shown in Figure 2.86.

2.10.6 Choice of Displacement Function

Selection of displacement function is an important step in finite element analysis, since it determines the performance of the element in the analysis. Attention must be paid to select a displacement function that

1. Has the number of unknown constants as the total number of degrees of freedom of the element

2. Does not have any preferred directions

3. Allows the element to undergo rigid-body movement without any internal strain

4. Is able to represent states of constant stress or strain

5. Satisfies the compatibility of displacements along the boundaries with adjacent elements

Elements that meet both requirements 3 and 4 are known as complete elements A polynomial is the most common form of displacement function. Mathematics of polynomials are easy to handle in formulating the desired equations for various elements and are convenient in digital computation. The degree of approximation is governed by the stage at which the function is truncated. Solutions closer to exact solutions can be obtained by including a greater number of terms. The polynomials are of the general form w(x) = a1 + a2x + a3x2 + ••• + an+1xn (2.154)

The coefficient a is known as a generalized displacement amplitude. The general polynomial form for a two-dimensional problem can be given as u(x, y) = a1 + a2x + a3 y + a4x2 + a5xy + a6y2 + ••• + amyn v(x, y) = am+1 + am+2x + am+3y + am+4x2 + am+5xy + am+6y2 H-----h a2myn

where n+1

These polynomials can be truncated at any desired degree to give constant, linear, quadratic, or higher-order functions. For example, a linear model in the case of a two-dimensional problem can be given as u = a1 + a2x + a3y

A quadratic function is given by

FIGURE 2.86 Typical examples of finite element mesh.

The Pascal triangle shown below can be used for the purpose of achieving isotropy, that is, to avoid displacement shapes that change with a change in the local coordinate system.

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