# OS-V: 0531 Laminated Shell Strength Analysis Mechanical Load 1

This problem analyzes the strength of laminated composite shells when subjected to a uniform longitudinal load per unit length.

The model and boundary conditions are described by Hopkins (2005). The resulting ply
failure indices are compared against analytical solutions from classical lamination
theory (CLT). ^{5} The results indicate good agreement
between OptiStruct and CLT.

## Benchmark Model

800
mesh elements of CQUAD4 element type were used in this study. The
model is fixed at the point A using a SPC card and a uniform
longitudinal load per unit length
(`N`_{x}) of 1500 N/m is applied
along edges A-D and B-C using FORCE 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} |
3.176 x 10^{-4} |
3.176 x 10^{-4} |

$\text{\epsilon}$
_{y} |
-1.447 x 10^{-4} |
-1.447 x 10^{-4} |

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

^{2}

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

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

Tsai-Wu | 0.88402 | 0.8841 | 0.37308 | 0.3731 | 0.01990 | 0.01991 | -0.34309 | -0.343 |

Hill | 0.77952 | 0.77960 | 0.16323 | 0.16330 | 0.00435 | 0.00435 | 0.00136 | 0.00136 |

Hoffman | 0.88110 | 0.88140 | 0.37630 | 0.37610 | 0.02004 | 0.02000 | -0.34534 | -0.34510 |

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

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

Tsai-Wu | 1.1223 | 1.122 | 2.53671 | 2.537 | 14.304 | 14.3 | 31.879 | 31.89 |

Hill | 1.1325 | 1.133 | 2.4748 | 2.475 | 15.157 | 15.16 | 27.124 | 27.12 |

Hoffman | 1.1259 | 1.126 | 2.4944 | 2.494 | 14.101 | 14.1 | 37.869 | 37.88 |

## Model Files

The model files used in this problem include:

- /lssam1_tsai.fem
- /lssam1_hill.fem
- /lssam1_hoff.fem

^{1}Hopkins, P., 1986, Benchmarks for Membrane and Bending Analysis of Laminated Shells, Part 1, Stiffness Matrix and Thermal Characteristics, NAFEMS Publication

^{2}ESDU datasheet 83014, Failure Criteria for an Individual Layer of a Fibre Reinforced Composite Laminate under in-plane loading, 1986

^{3}ESA PSS-03-1101, 1986, Composite Design Handbook for Space Structure Application, Issue 1

^{4}A Comparison of the Predicted Capabilities of Current Failure Theories for Composite Laminate by P. D. Soden, M. J. Hinton and A. S. Kaddour, Composite Science and Technology, Volume 58, 1998, pages 1225-1254

^{5}Jones, R.M., 1975, Mechanics of Composites, McGraw Hill, New York