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

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). 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.

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. Laminate Properties of the Composite Model
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 midplane strains computed from OptiStruct (OS) with classical lamination theory (CLT). The midplane strains from CLT presented in Table 2 are of the homogenized composite; therefore, STRAIN I/O should be used. CSTRAIN I/O 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	3.176 x 10^-4	3.176 x 10^-4
$ε$ _y	-1.447 x 10^-4	-1.447 x 10^-4
$ε$ _xy	1.108 x 10^-4	1.108 x 10^-4

Table 3 and Table 4 illustrate the failure indices (FI) and reserve factor (RF) calculated from OptiStruct and CLT. The FI show a good agreement between the finite element results and analytical solution with a maximum difference of 0.2% in ply 3, -0.05% in ply 3 and 0.20% in ply 3 when Tsai-Wu, Hill and Hoffman failure criteria are used respectively. This verification of numerical results for the criteria is addressed and does not address the merits of a particular criteria.

Table 3. Comparison of Failure Index between 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	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

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	1.1223	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

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

The model files used in this problem include:

lssam1_tsai.fem
lssam1_hill.fem
lssam1_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