On Integrating by parts the second integral and keeping the other integral same, we have

                 

                                                                                     (12.20)  

The highest order of derivative with respect to x in equation (12.20) is third, so the completeness of   and   is required. The prime represents derivatives with respect to spatial coordinate x . Hence, a polynomial of the third degree will be able to satisfy above conditions.

Highest order of derivative with respect to x in integral terms of equation (12.20) is second, so compatibility up to w and is required at each node. For most simple line element as shown in Figure 12.3, each node will have two dof so total degrees of freedom(dof) of the element will be four. Hence ,the interpolation should have four constants to uniquely define the interpolation function. Hence, the cubic polynomial will be sufficient for the beam element, which satisfies both the completeness and compatibility conditions.