Problem $4\mathrm{.}3$

Derive the shape functions for a two$-$node one$-$dimensional element which is $C^1$ continuous. Note that the

shape functions derived in Chapter $4$ are $C^0$ continuous. To enforce $C^1$ continuity, it is necessary to enforce

continuity of displacements and their derivatives. Start by considering a complete cubic approximation

$u^e={}_0^e+{}_1^{e}x+{}_2^e{x}^2+{}_3^e{x}^3$ and derive four shape functions corresponding to the displacements and their

derivatives at each node. For clarity of notation, denote the derivatives at the nodes by ${}_i,i=1,2.$