# Forces and Moments Calculation

## Integration Points Throughout the Thickness

Coefficients | w_{1} |
w_{2} |
w_{3} |
---|---|---|---|

Radioss | 0.250 | 0.500 | 0.250 |

Simpson | 0.166 | 0.666 | 0.166 |

Coefficients | w_{1} |
w_{2} |
w_{3} |
---|---|---|---|

Radioss | -0.083 | 0. | 0.083 |

Simpson | -0.083 | 0. | 0.083 |

Number of Points | Position | Weight |
---|---|---|

1 | ±0.0000 | 2.0000 |

2 | ±0.5774 | 1.0000 |

3 | 0.0000 ±0.7746 |
0.8889 0.5556 |

4 | ±0.8611 ±0.3400 |
0.6521 0.3479 |

5 | ±0.9062 ±0.5385 0.0000 |
0.2369 0.4786 0.5689 |

6 | ±0.9325 ±0.6612 ±0.2386 |
0.1713 0.3608 0.4679 |

7 | ±0.9491 ±0.7415 ±0.4058 0.0000 |
0.1295 0.2797 0.3818 0.4180 |

8 | ±0.9603 ±0.7967 ±0.5255 ±0.1834 |
0.1012 0.2224 0.3137 0.3627 |

9 | ±0.9681 ±0.8360 ±0.6134 ±0.3243 0.0000 |
0.0813 0.1806 0.2606 0.3123 0.3302 |

10 | ±0.9739 ±0.8650 ±0.6794 ±0.4334 ±0.1489 |
0.0667 0.1495 0.2191 0.2693 0.2955 |

Number of Points | Position | Weight for Membrane w |
Weight for Bending w |
---|---|---|---|

1 | 0.0000 | 1.0000 | 0.0000 |

2 | ±0.5000 | 0.5000 | ±0.0833 |

3 | ±0.5000 0.0000 |
0.2500 0.5000 |
±0.0833 0.0000 |

4 | ±0.5000 ±0.1667 |
0.1667 0.3333 |
±0.0648 ±0.0556 |

5 | ±0.5000 ±0.2500 0.0000 |
0.1250 0.2500 0.2500 |
±0.0521 ±0.0625 0.0000 |

6 | ±0.5000 ±0.3000 ±0.1000 |
0.1000 0.2000 0.2000 |
±0.0433 ±0.0600 ±0.0200 |

7 | ±0.500 ±0.3333 ±0.1667 0.0000 |
0.0833 0.1667 0.1667 0.1667 |
±0.0370 ±0.0556 ±0.0278 0.0000 |

8 | ±0.5000 ±0.3750 ±0.2500 ±0.1250 |
0.0714 0.1429 0.1429 0.1429 |
±0.0323 ±0.0510 ±0.0306 ±0.0102 |

9 | ±0.5000 ±0.3750 ±0.2500 ±0.1250 0.0000 |
0.0625 0.1250 0.1250 0.1250 0.1250 |
±0.086 ±0.0469 ±0.0313 ±0.0156 0.0000 |

10 | ±0.5000 ±0.3889 ±0.2778 ±0.1667 ±0.0555 |
0.0556 0.1111 0.1111 0.1111 0.1111 |
±0.0257 ±0.0432 ±0.0309 ±0.0185 ±0.0062 |

For shell elements, integration points through the thickness are almost Lobatto points.

## Global Plasticity Algorithm

^{1}form:

${N}_{x}={\displaystyle \underset{-t/2}{\overset{t/2}{\int}}{\sigma}_{x}^{pa}dz}$ | ${M}_{x}={\displaystyle \underset{-t/2}{\overset{t/2}{\int}}{\sigma}_{x}^{pa}zdz}$ |

${N}_{y}={\displaystyle \underset{-t/2}{\overset{t/2}{\int}}{\sigma}_{y}^{pa}dz}$ | ${M}_{y}={\displaystyle \underset{-t/2}{\overset{t/2}{\int}}{\sigma}_{y}^{pa}zdz}$ |

${N}_{xy}={\displaystyle \underset{-t/2}{\overset{t/2}{\int}}{\sigma}_{xy}^{pa}dz}$ | ${M}_{xy}={\displaystyle \underset{-t/2}{\overset{t/2}{\int}}{\sigma}_{xy}^{pa}zdz}$ |

Where, $\beta $ and $\gamma $ are scalar material characteristic constants, function of plastic deformation. They can be identified by the material hardening law in pure traction and pure bending.

If no hardening law in pure bending is used, $\gamma $ is simply computed by $\gamma =\frac{{\sigma}_{y}/E+\frac{3}{2}{\epsilon}^{p}}{{\sigma}_{y}/E+{\epsilon}^{p}}$ varying between 1.0 and 1.5.

Where,
$H$
is the plastic module. The derivative of function
$f$
in Equation 7 is discontinuous
when
${\left\{N\right\}}^{t}$
$\left\{A\right\}\left\{M\right\}$
=0. This can be treated when small steps are used by putting
s=0 as explained in ^{2}.

^{1}Iliouchine A., “Plasticity”, Edition Eyrolles Paris, 1956.

^{2}Crisfield M.A., “Nonlinear finite element analysis of solids and structures”, J. Wiley, Vol. 2, 1997.