Calculation of Element Stiffness Matrix and Element Force Vector

When p^th order approximation is used, there are ( p +1) nodes per element. Further, there is only 1 degree of freedom (dof) per node. Thus, there are ( p +1) dof per element. As a result, the size of the element shiftiness matrix is while that of the element force vector is For p = 1 (i.e., the 1st order approximation with piecewise linear shape functions), the sizes of and are and 2 respectively. The corresponding expressions are given by equations (6.25-6.26). For p = 2 (i.e., the 2 nd order approximation with piecewise quadratic shape functions), the sizes of and are and 3 respectively. The corresponding expressions are given by equations (8.30-8.31). Generalizing these equations for , and writing them in the index notation, we get the following expressions for the elements stiffness matrix and the element force vector :

(9.8)

and

(9.9)

The area of cross-section ( A ) of the bar, the Young's modulus ( E ) of the bar material and the distributed force ( f ) acting on the bar are given as functions of x . Further, the shape functions are given by the expression (9.6). Using all this information, the element stiffness matrix and the element stiffness vector are calculated using the expressions (9.8-9.9). This is done for all the elements.