Derivatives of Shape Functions

Evaluation of the element stiffness matrix [k]^k involves x -derivative of the shape functions (equation 9.8). Now, we have expressed the shape functions as functions of the natural coordinate . Therefore, x -derivative of these shape functions can be obtained by the chain rule. Thus,

(11.41)

To evaluate , we differentiate expression (11.7) for x with respect to .

(11.42)

Using the following expression for the element length

(11.43)

we get

(11.44)

Then, the expression (11.41) for the x-derivation of the shape function becomes

(11.45)