Spring at One End

Consider a bar, fixed at the left end and supported by a spring at the right end (Fig. 7.5).

Figure 7.5 Bar supported by a spring at one end ( = pre-compression of the spring)

The above problem can be modeled by replacing the spring with a force as shown in Fig. 7.6 .

Figure 7.6 Equivalent problem

The expression for the spring force is

(7.41)

where is the spring force and is the initial compression in the spring. Since the force is compressive, the boundary condition (1.1c) becomes

, at .

(7.42)

Then the weighted residual integral becomes

(7.43)

or

(7.44)

If we develop the finite element equations for this problem using the 6-element mesh of Fig. 7.1, then the global stiffness matrix becomes

(7.45)

Further, the global force vector becomes

(7.46)

Then, the finite element equations become

(7.47)

or

(7.48)

Now the essential boundary condition

(7.49)

is applied by modifying the matrix and vector as described by equations (7.14) and (7.15). Note that here .