Expressions of Element Stiffness Matrix and Force Vector for Piecewise Quadratic Basis Functions

We use the Galerkin method to develop the finite element equations. Therefore, the integral form to be used is given by equation (2.5b) or (3.68). This equation is rearranged as  

(8.19)

For the Galerkin method, we choose the basis functions as the weight functions. Thus,

for i = 1, 2., Nn

(8.20)

where N_n is the number of nodes of the domain. Note that these functions are linearly independent and satisfy all the constraints arising out of the three admissibility conditions on w(x). Substituting the expression (8.20) for w(x) in equation (8.19), we get

for i = 1, 2., Nn

(8.21)

Note that the domain (0, L ) is divided into N elements. For k^th element, the domain is

(8.22)

(See Fig. 8.3). Substituting the approximation (8.9) for u ( x ) into equation (8.21) and writing the integral from 0 to L as a sum of the N integrals over (2k -1, 2k +1), k = 1,. N , we get the following equations:

for i = 1, 2, 3.., Nn .

(8.23a)

In matrix form, this can be written as

(8.23b)

Here, the stiffness matrix is given by

for i = 1,2,3.., Nn , for j = 1,2,3.., Nn ,

(8.24)

where

for i = 1,2,3.., Nn , for j = 1,2,3.., Nn .

(8.25)

Similarly, the force vector is given by

for i = 1, 2, 3..., Nn ,

(8.26)

where

for i = 1, 2, 3..., Nn ,

(8.27)

The derivation of equation (8.23) is similar to the derivations of sections (5.2) and (6.1) for the case of piecewise linear basis functions.

Note that and represent the contributions of the k^th element to the stiffness matrix and the force vector . To simply the expressions (8.25) and (8.27) for and , we note that the only values of ' i ' and ' j ' for which  and are non-zero over the interval  are    i = 2k-1, 2k , 2k+1 and j = 2k-1, 2k , 2k +1. Thus, the matrix becomes:

   (8.28)

Further, the vector becomes:

(8.29)

Note that the elements of and are zero when i ≠ 2k-1, 2k , 2k+1 and j ≠ 2k-1, 2k , 2k+1. Therefore, the matrix is often condensed as 3x3 matrix with the notation . Similarly the vector is often condensed as 3x1 vector with the notation . Next, we use the local notation for the basis functions , and and the coordinates , and . In local notation, they are written as , and and , and . Then, the condensed form of becomes:

(8.30)

Similarly, the condensed form of becomes:

(8.31)

The matrix (8.30) is called as the element stiffness matrix and the vector (8.31) is called as the element force vector.