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

From: Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

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