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Table 7 Shape functions employed in the numerical examples

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

Load case Node \(\varvec{I}\) Basis \(\varvec{r}\) Single shape function 1st shape function 2nd shape function
A 1 1 \(\alpha ^2\) \(\alpha ^4\) \(\alpha ^5 - \alpha ^4\)
   2 \(\alpha ^2\) \(\alpha ^4\) \(\alpha ^5 - \alpha ^4\)
  2 1 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
   2 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
B 1 1 \(\alpha ^2\) \(\alpha ^3\) \(\alpha ^4-\alpha ^3\)
   2 \(-3\alpha ^4 + 4\alpha ^3\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
  2 1 \(\alpha ^2\) \(\alpha ^4\) \(\alpha ^5 - \alpha ^4\)
   2\(^\mathrm{a}\) \(\alpha ^3-\alpha ^2\) \(\alpha ^5 - \alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
C 1 1 \(\alpha ^2\) \(\alpha ^3\) \(\alpha ^4 - \alpha ^3\)
   2 \(\alpha ^3\) \(\alpha ^4\) \(\alpha ^5 - \alpha ^4\)
  2 1 \(\alpha ^2\) \(\alpha ^3\) \(\alpha ^4-\alpha ^3\)
   2\(^\mathrm{a}\) \(\alpha ^3 - \alpha ^2\) \(\alpha ^4 - \alpha ^3\) \(\frac{1}{2} \alpha ^5 - \alpha ^4 + \frac{1}{2} \alpha ^3\)
D 1 1 \(\alpha ^2\) \(\alpha ^4\) \(\alpha ^5 - \alpha ^4\)
   2 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
  2 1 \(-2\alpha ^3 + 3\alpha ^2\) \(-3\alpha ^4 + 4\alpha ^3\) \(\frac{1}{2} \alpha ^5 - \alpha ^4 + \frac{1}{2} \alpha ^3\)
   2 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
E 1 1 \(-2\alpha ^3 + 3\alpha ^2\) \(-3\alpha ^4 - 4\alpha ^3\) \(\frac{1}{2} \alpha ^5 - \alpha ^4 + \frac{1}{2} \alpha ^3\)
   2 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
  2 1 \(\alpha ^2\) \(\alpha ^4\) \(\alpha ^5 - \alpha ^4\)
   2\(^\mathrm{a}\) \(\alpha ^3 - \alpha ^2\) \(\alpha ^5 - \alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
F 1 1 \(\alpha ^2\) \(\alpha ^3\) \(\alpha ^4 - \alpha ^3\)
   2\(^\mathrm{a}\) \(\alpha ^3 - \alpha ^2\) \(\alpha ^5 - \alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
  2 1 \(\alpha ^2\) \(\alpha ^3\) \(\alpha ^4 - \alpha ^3\)
   2\(^\mathrm{a}\) \(\alpha ^3 - \alpha ^2\) \(\alpha ^5 - \alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
G 1 1 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
   2 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
  2 1 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
   2 \(-2\alpha ^3 + 3\alpha ^2\) \(-4\alpha ^5 + 5\alpha ^4\) \(\frac{1}{2} \alpha ^6 - \alpha ^5 + \frac{1}{2} \alpha ^4\)
  1. The index I of the node refers the left node for the value 1, to the right node for the value 2. The bases r on the Lie algebra are dilatations for the value 1, rotations for the value 2. “Single shape function” refers to the formulation with only one shape function for each basis of the Lie algebra. The remaining columns contain the first and the second shape function related to the respective basis of the Lie algebra for the formulations based on two shape functions for each basis. For node 1, with \(\xi _1 = 0\), \(\alpha = 1 - \xi \), whereas for node 2, with \(\xi _2 = 1\), \(\alpha = \xi \)
  2. \(^\mathrm{a}\) An additional shape function, given by \(\alpha ^2\) in the case of one shape function and \(\alpha ^4\) in the case of two shape functions, is identically zero, as the beam is clamped in a horizontal orientation. The coefficient of this shape function exclusively depends on the boundary condition at the clamped node