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

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.

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.

The material properties are:

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, $υ$ ₁₂: 0.3
Longitudinal Tensile Strength, $σ$ _lt (MPa): 500.0
Longitudinal Compressive Strength, $σ$ _lc (MPa): 350.0
Transverse Tensile Strength, $σ$ _tt (MPa): 5.0
Transverse Compressive Strength, $σ$ _tc (MPa): 75.0
In-plane shear strength, $τ$ _lt (MPa): 35.0

Table 1. Comparison of Failure Index for each Ply between OptiStruct (OS) and Classical Lamination Theory (CLT)
Ply	Orientation (°)	Thickness ( $μ$ 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
Breadth (m): 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
$ε$ _x	-1.732 x 10^-3	-1.732 x 10^-3
$ε$ _y	-5.552 x 10^-4	-5.552 x 10^-4
$ε$ _xy	-3.928 x 10^-4	-3.928 x 10^-4

The results show a good correlation between the finite element results and analytical solution with a maximum difference of -0.011% in ply 3, -0.08% in ply 3 and -0.02% in ply 3 when Tsai-Wu, Hill and Hoffman failure criteria are used, respectively. Examination of the FI and RF for each failure criteria for ply 2 confirms that only for the Hill criteria is the FI directly related to reserve factor, RF = 1/(FI)^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.

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	-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

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	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 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:

lssam2_tsai.fem
lssam2_hill.fem
lssam2_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