This test case tests the laminate integration and stress computation of the MITC shell elements The gradients are calculated for each layer individually, independent of the number of integration points specified, with gradients computed at the lower surface, the mid-point, and at the upper surface of each layer.

The model is a rectangular plate defined in the xy-plane and modeled with 4 elements in the x-direction and 3 elements in the y-direction, with the platebeing clamped on the left edge and loaded in the x-direction on the right edge. Out-of plane displacements and in-plane rotations are locked. The following test input options can be set:

**Table 26. Input options**

`matid` |
Material id number: matid=1 refers to the laminate definition, matid=2 is the material data used by the laminate, and matid=13 refers to the equivalent istropic material definition, as the laminate is quasi-isotropic (90/0/+45/-45). |

`elemtype` |
elemtype=4 selects the Q4.S.MITC.E4 shell element. elemtype=8 selects the Q8.S.MITC shell element. elemtype=9 selects the Q9.S.MITC shell element. |

Displacements for the different through-thickness integration methods for the laminate as well as the isotropic material model must give identical results for the same element type. The stresses are independent of the through-thickness laminate integration method used. No stresses are checked for the isotropic material model. There are some slight differences between the 4-node and 9-node element results, as is to be expected. The tables below list the results checked for the 4-node and 9-node elements:

**Table 27. Results to be expected.**

Result | Position | Value |

Axial displacement (Ux) | Tip (x=8.7, y=0) | 7.40291E-02 |

Side compression (Uy) | Root | 3.61251E-03 |

Axial stress (Sxx) | Element 8, integration point 3, layer 1 | 92.1 |

Axial stress (Sxx) | Element 8, integration point 3, layer 2 | 1255. |

Axial stress (Sxx) | Element 8, integration point 3, layer 3 | 312. |

Axial stress (Sxx) | Element 8, integration point 3, layer 4 | 317. |

This test case tests the bending behavior of the MITC shell elements. The modeled problem is a rectangular plate under a sinusoidal load. The edges of the plates are simply supported. In order to exploit the symmetry of the considered problem only a quarter of the plate is modeled. The B2000++ solution is validated with the analytic solution according to Reddy [1]. Therefore, the transverse displacement in the z-direction is compared at the point (0, 0, 0), where the deflection is maximal. The analytic solution is only valid for orthothropic laminates. Two different laminates are tested.

**Table 28. Laminate 1: symmetric cross-ply**

Layer | Thickness | Angle |

1 | 0.02 | 0^{o} |

2 | 0.04 | 90^{o} |

3 | 0.02 | 0^{o} |

**Table 29. Laminate 2: asymmetric cross-ply**

Layer | Thickness | Angle |

1 | 0.02 | 0^{o} |

2 | 0.02 | 90^{o} |

3 | 0.02 | 0^{o} |

4 | 0.02 | 90^{o} |

[1] J. N. Reddy. *Mechanics of Laminated Composite Plates
and Shells.* Boca Raton: CRC Press, 2. edition, 2004