Q.No.5.22. Consider a spring with stiffness k as shown in Figure 5.12. Determine the
stiffness matrix of the spring using the direct method.
The force-displacement equations of a step constitute the required element equations. To derive these
equations for a typical element we isolate the element as shown in Figure.
In this figure, force (F) and displacement (q) are defined at each of the two nodes in the positive
direction of the x axis. The element equations can be expressed in matrix form as
Where [k] is called the stiffness or characteristic matrix, u is the vector of nodal displacements, and P
is the vector of nodal forces of the element. We shall derive the element stiffness matrix from the
basic definition of the stiffness coefficient, and for this no assumed interpolation polynomials are
needed. In structural mechanics, the stiffness influence coefficient kij is defined as the force needed at
node i (in the direction of x) to produce a unit displacement at node j (uj = 1) while all other nodes are
restrained. This definition can be used to generate the matrix [k]. For example, when we apply a unit
displacement to node 1 and restrain node 2 as shown in Figure below.
We can obtain the values of k11, k12, k21 and k22 as below.
Similarly when node 2 is restrained,
And the element equations can be expressed in matrix form as