# OS-V: 0533 Laminated Shell Strength Analysis Mechanical Load 3

This problem demonstrates when the loads applied in OS-V: 0532 are factored by 1.1491, a Hill failure index (FI) and a reserve factor (RF) of unity for ply 1 are obtained.

Examination of the FI and RF for each criterion confirm that only for the Hill
criteria is the FI directly related to reserve factor, RF = 1/(FI)^{0.5}.
For all other plies, the RF are reduced by a factor of 1.149^{-1}, but the
FI is not. The model is subjected to a uniform longitudinal, transverse, shear,
bending and torsional load per unit length. The geometry of 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). ^{5} The results show a good correlation
between OptiStruct and CLT.

## 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 26.573 N/m, uniform
transverse force per unit length
(`N`_{y}) of 28.728 N/m and shear
force per unit length (`N`_{xy}) of
5.7455 N/m are applied using a FORCE card. Bending moments per
unit length (`M`_{x} = 0.45964 N and
`M`_{y} = -0.8618 N) and torsional
load per unit length (`T`_{xy} =
-0.20109 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} |
-1.9902 x 10^{-4} |
-1.990 x 10^{-4} |

$\text{\epsilon}$
_{y} |
-6.3794 x 10^{-4} |
-6.4 x 10^{-4} |

$\text{\epsilon}$
_{xy} |
-4.5136 x 10^{-4} |
-4.51 x 10^{-4} |

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

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

Tsai-Wu | -2.12900 | -2.127 | -2.76890 | -2.77 | -2.13250 | -2.133 | -1.13540 | -1.134 |

Hill | 1.00000 | 1.00000 | 0.29949 | 0.29950 | 0.08470 | 0.08470 | 0.64777 | 0.64780 |

Hoffman | -2.56460 | -2.56500 | -2.51860 | -2.51900 | -1.99770 | -1.99800 | -1.36110 | -1.36100 |

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

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

Tsai-Wu | 1.6123 | 1.612 | 3.5651 | 3.568 | 6.3911 | 6.402 | 2.2331 | 2.232 |

Hill | 0.99998 | 1.000 | 1.8273 | 1.827 | 3.436 | 3.436 | 1.2425 | 1.242 |

Hoffman | 1.7718 | 1.772 | 2.983 | 2.983 | 4.9355 | 4.933 | 2.6439 | 2.644 |

This document addresses the verification of numerical results for the criteria and
does not address the merits of a particular criteria.
For
details of particular failure criteria. ^{2}
^{3}
^{4}

## Model Files

The model files used in this problem include:

- /lssam3_tsai.fem
- /lssam3_hill.fem
- /lssam3_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