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 | \(\circ \) | \(\circ \) | \(\bullet \) | \(\frac{1}{2} \alpha ^4 - \alpha ^3 + \frac{1}{2} \alpha ^2\) | \(\frac{1}{2} \alpha ^5 - \alpha ^4 + \frac{1}{2} \alpha ^3\) | \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\) | |