Numerical Integration Expressions for and

Now, we apply the Gauss quadrature rule (equation 12.1) to the expressions (11.48) and (11.51) for the element coefficient matrix and the element force vector . To select the number of Gauss point, we used to evaluate the degree of the integrands. For this purpose, we rewrite these expressions as:

,

(12.30)

and

,

(12.31)

where the integrands are given by

,

(12.32)

and

.

(12.33)

For the sake of concreteness, we make the following choices:

• is a constant function of x (constant Young's modulus)
• is a linear function of x (tapered bar)
• is a quadratic function of x (quadratic loading)
• The order of approximation is p . Thus, degree of is p and that of its derivative is

Now, the degree of the integrand becomes

,

(12.34)

and that of is given by

.

(12.35)

Let the maximum of and be . Thus,

.

(12.36)

Now, we use equation (12.29) to select the number of Gauss points. Using equations (12.34) and (12.36), we get the following values of n for

;	(12.37)
;
.

Similarly, we can obtain n for .

Application of the Gauss quadrature rule (equation 12.1) to the integrals (12.30) - (12.31) leads to the following expressions for and :

,

(12.38)

and

(12.39)

where and are given by equations (12.32)-(12.33) and n is given by equation (12.37). The Gauss point coordinates and the weights corresponding to n are to be taken from Table 12.1.