# Table 8 Polynomial functions for different boundary conditions and up to two degrees of freedom per Lie algebra element $${\mathbf {Z}}_{I,r}$$

Boundary cond. $$\varvec{r}$$ $$\varvec{q}$$ $$\varvec{{\partial ^\mu p_{I,r,q}} (1)}$$ $${\varvec{p_{I,r,q}} (\alpha )}$$
$$\varvec{\mu = 0}$$ $$\varvec{\mu = 1}$$ $$\varvec{\mu = 2}$$ $$\varvec{\bar{q}_J= 0}$$ $$\varvec{\bar{q}_J= 1}$$ $$\varvec{\bar{q}_J= 2}$$
clamped 1 0 $$\bullet$$ $$*$$ $$*$$ $$\alpha ^2$$ $$\alpha ^3$$ $$\alpha ^4$$
1 1 $$\circ$$ $$\bullet$$ $$*$$ $$\alpha ^3 - \alpha ^2$$ $$\alpha ^4 - \alpha ^3$$ $$\alpha ^5 - \alpha ^4$$
2 0 $$*$$ $$^{\mathrm{a}}$$ $$*$$ $$*$$ $$\alpha ^2$$ $$\alpha ^3$$ $$\alpha ^4$$
2 1 $$\circ$$ $$\bullet$$ $$*$$ $$\alpha ^3 - \alpha ^2$$ $$\alpha ^4 - \alpha ^3$$ $$\alpha ^5 - \alpha ^4$$
2 2 $$\circ$$ $$\circ$$ $$\bullet$$ $$\frac{1}{2} \alpha ^4 - \alpha ^3 + \frac{1}{2} \alpha ^2$$ $$\frac{1}{2} \alpha ^5 - \alpha ^4 + \frac{1}{2} \alpha ^3$$ $$\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4$$
pin (w/o load)$$^{\mathrm{b}}$$ 1 0 $$\bullet$$ $$\circ$$ $$*$$ $$-2\alpha ^3 + 3\alpha ^2$$ $$-3\alpha ^4 + 4\alpha ^3$$ $$-4\alpha ^5 + 5\alpha ^4$$
1 1 $$\circ$$ $$\circ$$ $$^{\mathrm{c}}$$ $$*$$ (not included in the element formulation)
1 2 $$\circ$$ $$\circ$$ $$\bullet$$ $$\frac{1}{2} \alpha ^4 - \alpha ^3 + \frac{1}{2} \alpha ^2$$ $$\frac{1}{2} \alpha ^5 - \alpha ^4 + \frac{1}{2} \alpha ^3$$ $$\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4$$
2 0 $$\bullet$$ $$\circ$$ $$*$$ $$-2\alpha ^3 + 3\alpha ^2$$ $$-3\alpha ^4 + 4\alpha ^3$$ $$-4\alpha ^5 + 5\alpha ^4$$
2 1 $$\circ$$ $$\circ$$ $$^{\mathrm{c}}$$ $$*$$ (not included in the element formulation)
2 2 $$\circ$$ $$\circ$$ $$\bullet$$ $$\frac{1}{2} \alpha ^4 - \alpha ^3 + \frac{1}{2} \alpha ^2$$ $$\frac{1}{2} \alpha ^5 - \alpha ^4 + \frac{1}{2} \alpha ^3$$ $$\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4$$
pin (moment) 1 0 $$\bullet$$ $$*$$ $$*$$ $$\alpha ^2$$ $$\alpha ^3$$ $$\alpha ^4$$
1 1 $$\circ$$ $$\bullet$$ $$*$$ $$\alpha ^3 - \alpha ^2$$ $$\alpha ^4 - \alpha ^3$$ $$\alpha ^5 - \alpha ^4$$
2 0 $$\bullet$$ $$*$$ $$*$$ $$\alpha ^2$$ $$\alpha ^3$$ $$\alpha ^4$$
2 1 $$\circ$$ $$\bullet$$ $$*$$ $$\alpha ^3 - \alpha ^2$$ $$\alpha ^4 - \alpha ^3$$ $$\alpha ^5 - \alpha ^4$$
pin (distr. load)$$^{\mathrm{d}}$$ 1 0 $$\bullet$$ $$*$$ $$*$$ $$\alpha ^2$$ $$\alpha ^3$$ $$\alpha ^4$$
1 1 $$\circ$$ $$\bullet$$ $$*$$ $$\alpha ^3 - \alpha ^2$$ $$\alpha ^4 - \alpha ^3$$ $$\alpha ^5 - \alpha ^4$$
2 0 $$\bullet$$ $$\circ$$ $$*$$ $$-2\alpha ^3 + 3\alpha ^2$$ $$-3\alpha ^4 + 4\alpha ^3$$ $$-4\alpha ^5 + 5\alpha ^4$$
2 1 $$\circ$$ $$\circ$$ $$^{\mathrm{c}}$$ $$*$$ (not included in the element formulation)
2 2 $$\circ$$ $$\circ$$ $$\bullet$$ $$\frac{1}{2} \alpha ^4 - \alpha ^3 + \frac{1}{2} \alpha ^2$$ $$\frac{1}{2} \alpha ^5 - \alpha ^4 + \frac{1}{2} \alpha ^3$$ $$\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4$$
1. Filled circles denote nonzero values, empty circles denote zeros, and asterisks denote arbitrary real numbers. For the meaning of $$\bar{q}_J$$, see Section “Full p-refinement and selective p-refinement”. For node I with $$\xi _I = 0$$, $$\alpha = 1 - \xi$$, whereas for node J, with $$\xi _J = 1$$, $$\alpha = \xi$$
2. $$^\mathrm{a}$$ Value of the coefficient associated with this shape function is externally given by the boundary condition
3. $$^\mathrm{b}$$ Simply supported node
4. $$^\mathrm{c}$$ Value is zero due to the local characteristics of the configuration
5. $$^\mathrm{d}$$ Distributed load in transversal direction in the neighborhood of the position of the node on the neutral axis