Assignment

SCIENCES

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.

Solution:

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

Or

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,

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Hence,

And the element equations can be expressed in matrix form as

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University:
Aligarh Muslim University

Subject:
Finite Element Method

Upload date:
08/07/2012