# OS-V: 0532 Laminated Shell Strength Analysis Mechanical Load 2

This problem analyzes the strength of laminated composite shells when subjected to a more general combination of membrane and bending load.

The model and boundary conditions are described by Hopkins (2005). The resulting ply failure indices, reserve factor and midplane strains are compared against analytical solutions from classical lamination theory (CLT). The results indicate a good correlation between OptiStruct and CLT.

## Model Files

## Benchmark Model

800
mesh elements of CQUAD4 element type were used in this study. The
model is fixed at point A using a SPC card; a uniform
longitudinal force per unit length
(`N`_{x}) of 23.125 N/m, uniform
transverse force per unit length
(`N`_{y}) of 25.0 N/m and shear
force per unit length (`N`_{xy}) of 5
N/m are applied using a FORCE card. Bending moments per unit
length (`M`_{x} = 0.4 N and
`M`_{y} = -0.75 N) and torsional
load per unit length (`T`_{xy} = -0.175
N) are applied along the edges of the laminate using a MOMENT
card.

**Property****Value**- Longitudinal Young’s Modulus, E
_{l}(GPa) - 207.0
- Transverse Young’s Modulus, E
_{t}(GPa) - 7.6
- Longitudinal Shear Modulus, G
_{lt}(GPa) - 5.0
- Major Poisson’s ratio,
$\upsilon $
_{12} - 0.3
- Longitudinal Tensile Strength,
$\sigma $
_{lt}(MPa) - 500.0
- Longitudinal Compressive Strength,
$\sigma $
_{lc}(MPa) - 350.0
- Transverse Tensile Strength,
$\sigma $
_{tt}(MPa) - 5.0
- Transverse Compressive Strength,
$\sigma $
_{tc}(MPa) - 75.0
- In-plane shear strength,
$\tau $
_{lt}(MPa) - 35.0

Ply | Orientation (°) | Thickness ( $\mu $ m) |
---|---|---|

1 | 90.0 | 0.05 |

2 | -45.0 | 0.05 |

3 | 45.0 | 0.05 |

4 | 0.0 | 0.05 |

**Dimension****Value**- Length (m)
- 0.2
- Breadth (m)
- 0.1

## Results

Midplane Strains | Theory | OptiStruct Result |
---|---|---|

$\text{\epsilon}$
_{x} |
-1.732 x 10^{-3} |
-1.732 x 10^{-3} |

$\text{\epsilon}$
_{y} |
-5.552 x 10^{-4} |
-5.552 x 10^{-4} |

$\text{\epsilon}$
_{xy} |
-3.928 x 10^{-4} |
-3.928 x 10^{-4} |

^{0.5}. The minimum RF for this laminate and load condition occurs in ply 1 for the Hill criteria, RF = 1.491 and FI = 0.75736. Therefore, if the loads are increased by a factor of 1.1491, a RF of 1 and FI of 1 should be obtained. This illustrates the benefit of being able to compute a RF and that the FI only provides a pass or fail condition with no direct indication of how changes in the load will affect the strength of the laminate. This is illustrated by the benchmark problem in OS-V: 0533 Laminated Shell Strength Analysis Mechanical Load 3.

Failure Criteria | Ply 1 | Ply 2 | Ply 3 | Ply 4 | ||||
---|---|---|---|---|---|---|---|---|

Theory | OptiStruct Result | Theory | OptiStruct Result | Theory | OptiStruct Result | Theory | OptiStruct Result | |

Tsai-Wu | -2.35980 | -2.36000 | -2.54390 | -2.54400 | -1.90380 | -1.90400 | -1.13300 | -1.13300 |

Hill | 0.75736 | 0.75740 | 0.22681 | 0.22680 | 0.06410 | 0.06415 | 0.49058 | 0.49060 |

Hoffman | -2.68970 | -2.69000 | -2.35430 | -2.35400 | -1.80170 | -1.80200 | -1.30400 | -1.30400 |

Reserve Factor | Ply 1 | Ply 2 | Ply 3 | Ply 4 | ||||
---|---|---|---|---|---|---|---|---|

Theory | OptiStruct Result | Theory | OptiStruct Result | Theory | OptiStruct Result | Theory | OptiStruct Result | |

Tsai-Wu | 1.8527 | 1.853 | 4.0967 | 4.096 | 7.344 | 7.343 | 2.5661 | 2.566 |

Hill | 1.1491 | 1.149 | 2.0997 | 2.100 | 3.9483 | 3.948 | 1.4277 | 3.038 |

Hoffman | 2.0359 | 2.036 | 3.4277 | 3.428 | 5.6690 | 5.669 | 3.0381 | 1.428 |

This document addresses the verification of numerical results for the criteria and does not address the merits of a particular criteria. ESDU data sheet (1986), Soden et al. (1998) and ESA PSS-03-1101 (1986) address the details of particular failure criteria.

## Reference

NAFEMS R0092 - Benchmarks for membrane and bending analysis of laminated shells. Part 1, Stiffness matrix and thermal characteristics

NAFEMS R0093 - Benchmarks for membrane and bending analysis of laminated shells. Part 2, Strength analysis