where a and b are unknowns to be determined from the force distribution conditions, e.g. the amplitude of forces at ends.

For such a linear force distribution over the element, at least two nodes are required on an element and corresponding force values f₁ (t) and f₂(t) must be specified, i.e.

which can be solved to give

Then, the assumed form of the element force becomes

where N_f(z) is the shape function for the force . Hence the consistent load vector would be

A quadratic variation of the force : In a quadratic polynomial one needs three constants to be determined. Hence, at least three nodes are required as shown in Figure 9.10 with node 3 as an internal node . At these nodes (i.e., at z = 0, 0.5 l and l ) corresponding force values f₁(t), f₂(t) and f₃(t) must be sp ecified. So that the element force can be written as

Figure 9.10 A beam element with a quadratic variation of the force

Hence, the nodal external force vector is defined as

To obtain the explicit expression of the same, it is left to the reader as an exercise.