Elements¶
Daniel Weschke
November 1, 2018
1 Shape functions¶
\[\tensorI{u}(x) = \tensorI{N}(x) \, \tensorI{u}\]
1.1 Linear 1D¶
Linear shape for displacement
\[\begin{split}\tensorI{N}(x)
= \begin{bmatrix}
N_1(x) \\
N_2(x)
\end{bmatrix}
= \begin{bmatrix}
\dfrac{1}{2} \left( 1 - x \right) \\
\dfrac{1}{2} \left( 1 + x \right)
\end{bmatrix}\end{split}\]
1.2 Quadratic 1D¶
Quadratic shape for displacement
\[\begin{split}\tensorI{N}(x)
= \begin{bmatrix}
N_1(x) \\
N_2(x) \\
N_3(x)
\end{bmatrix}
= \begin{bmatrix}
-\dfrac{1}{2} \left( 1 - x \right) x \\
\dfrac{1}{2} \left( 1 + x \right) x \\
\left( 1 + x \right) \left( 1 - x \right)
\end{bmatrix}\end{split}\]
2 Line elements¶
Bar, beam
2.1 SEG2¶
linear bar
2.2 SEG3¶
quadratic bar
3 Surface elements¶
Plate, shell
3.1 TRIA3¶
linear
3.2 TRIA6¶
3.3 TRIA7¶
3.4 QUAD4¶
linear
3.5 QUAD8¶
quasi-quadratic
3.6 QUAD9¶
quadratic (bi-quadratic)
4 Volume elements¶
Solid
4.1 TET4¶
linear
4.2 TET10¶
quadratic
4.3 HEXA8¶
linear
4.4 HEXA20¶
quasi-quadratic
4.5 HEXA27¶
quadratic (bi-quadratic)
