# OS-V: 0534 Laminated Shell Strength Analysis Thermal Load 1

This problem analyzes the strength of laminated composite shells when subjected to a uniform constant temperature.

When plies in a laminate have different orientation, the expansion will not be identical. The net expansion of the laminates will be determined from the compatibility of all the plies. Therefore, this benchmark addresses the failure indices under uniform temperature loading. The model and boundary conditions are described by Hopkins (2005). The resulting ply failure indices and reserve factors are compared against analytical solutions from classical lamination theory (CLT). The results show a good correlation between OptiStruct and CLT.

## Benchmark Model

Table 2 lists out all the computed ply failure indices from OptiStruct (OS) based on Tsai-Wu, Hill and Hoffman failure criteria. 800 mesh elements of CQUAD4 element type were used in this study. The model is pinned at point A using a SPC card and a uniform thermal load of 150°C is applied throughout the domain using a TEMP card.

The material properties are:
Property
Value
Longitudinal Young’s Modulus, El (GPa)
207.0
Transverse Young’s Modulus, Et (GPa)
7.6
Longitudinal Shear Modulus, Glt (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
Longitudinal co-efficient of thermal expansion, αl (per 0°C)
0.0
Transverse co-efficient of thermal expansion, αt (per 0°C)
30.0 x 10-6
Table 1. Laminate Properties of the Composite Model
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
The geometry of the composite laminate:
Dimension
Value
Length (m)
0.2
0.1

## Results

Table 2 compares the average midplane strains computed from OptiStruct with CLT. The average midplane strains from CLT presented in Table 2 are of the homogenized composite; therefore, STRAIN I/O should be used, which, gives the midplane strains of individual plies. The identical results show that OptiStruct calculates the midplane strains accurately.
Table 2. Comparison of Midplane Strains between OptiStruct (OS) and Classical Lamination Theory (CLT)
Midplane Strains Theory OptiStruct Result
$\text{ε}$ x -0.698 x 10-3 -0.698 x 10-4
$\text{ε}$ y -0.698 x 10-3 -0.698 x 10-4
$\text{ε}$ xy -0.1661 x 10-10 -2.611 x 10-13**
** Represents maximum
The FI shows a good correlation between the finite element results and analytical solution with a maximum difference of 0.01% in ply 2 and 3, 0.02% in ply 2 and ply 3, and -0.01% in ply 1 when Tsai-Wu, Hill and Hoffman failure criteria are used, respectively.
Table 3. Comparison of Failure Index in OptiStruct and CLT
Failure Criteria Ply 1 Ply 2 Ply 3 Ply 4
Theory OptiStruct Result Theory OptiStruct Result Theory OptiStruct Result Theory OptiStruct Result
Tsai-Wu 5.875 5.875 6.7875 6.788 6.7875 6.788 5.875 5.875
Hill 22.073 22.07 26.104 26.1 26.104 26.1 22.073 22.073
Hoffman 5.9177 5.918 6.5938 6.594 6.5938 6.594 5.9177 5.9177
Table 4. Comparison of Reserve Factor for each Ply between OptiStruct and CLT
Reserve Factor Ply 1 Ply 2 Ply 3 Ply 4
Theory OptiStruct Result Theory OptiStruct Result Theory OptiStruct Result Theory OptiStruct Result
Tsai-Wu 0.21342 0.2134 0.19239 0.1924 0.19239 0.1924 0.21342 0.2134
Hill 0.21285 0.2128 0.19573 0.1957 0.19573 0.1957 0.21285 0.2128
Hoffman 0.21304 0.2130 0.19369 0.1937 0.19369 0.1937 0.21304 0.2130

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

## Model Files

Refer to Access the Model Files to download the required model file(s).

The model files used in this problem include:
• lssat1_tsai.fem
• lssat1_hill.fem
• lssat1_hoff.fem

## 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