R0031(3): Three-layer sandwich shell under normal pressure loading

This problem provides evidence that Abaqus can reproduce the result from the benchmark defined by NAFEMS and cited as the reference solution.

The following topics are discussed:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/Explicit

Elements tested

S4R

S8R

SC6R

SC8R

Problem description

Mesh:

One-quarter model with a 4 × 4 mesh of composite S8R elements in Abaqus/Standard, an 8 × 8 mesh of composite S4R elements in Abaqus/Explicit, and an 8 × 8 mesh of continuum shell elements in Abaqus/Standard. Two types of continuum shell models are provided: (1) a single composite element and (2) three single-layer elements stacked in the thickness direction.

Material:

Face sheets: $E_{1}$ = 1.0 × 10⁷ psi, $E_{2}$ = 4.0 × 10⁶ psi, $ν_{12}$ = 0.3, $G_{12}$ = 1.875 × 10⁶ psi, $G_{13}$ = 1.875 × 10⁶ psi, $G_{23}$ = 1.875 × 10⁶ psi

Core: $E_{1}$ = 10.0 psi, $E_{2}$ = 10.0 psi, $ν_{12}$ = 0, $G_{12}$ = 10.0 psi, $G_{13}$ = 3.0E4 psi, $G_{23}$ = 1.2E4 psi. The thickness moduli in continuum shell models are chosen sufficiently high to avoid pinching effects, which are neglected in the analytical solution.

Boundary conditions:

Simply supported on all four edges. The continuum shell models use an equation constraint to provide an equivalent midsurface constraint.

Loading:

Uniform normal pressure of 100 psi.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test R0031/3 from NAFEMS publication R0031, “Composites Benchmarks,” February 1995.

Results and discussion

The results are given in Table 1 and Table 2. The values enclosed in parentheses are percentage differences with respect to the reference solution. The displacements reported for the stacked continuum shell model are the average displacements of the bottom and top skins.

Table 1. Abaqus/Standard analysis.
Model	$u_{z}$ at C	$σ_{11}$ at C	$σ_{22}$ at C	$σ_{12}$ at E
NAFEMS	−0.123	34449	13350	−5068
S8R	−0.122 (0.6%)	35307 (2.5%)	13802 (3.4%)	−5236 (3.3%)
Composite SC6R	–0.120 (–2.4%)	34687 (0.7%)	13675 (−1.8%)	–5132 (2.5%)
Stacked SC6R	–0.129 (4.8%)	35382 (2.7%)	13745 (−1.3%)	–5219 (3.0%)
Composite SC8R	–0.122 (–0.8%)	35312 (2.5%)	13805 (−0.9%)	–5237 (3.3%)
Stacked SC8R	–0.131 (6.5%)	36118 (5.7%)	13900 (4.1%)	–5312 (4.8%)

Table 2. Abaqus/Explicit analysis.
Model	$u_{z}$ at C	$σ_{11}$ at C	$σ_{22}$ at C	$σ_{12}$ at E
NAFEMS	−0.123	34449	13932	−5068
S4R	−0.135 (9.8%)	36272 (5.3%)	13287 (5.8%)	−5696 (12.3%)*

*Nodal stress value obtained by averaging integration point stress values of all adjoining elements.

Input files

Abaqus/Standard input files

nco3s8rx.inp: S8R elements.
r313_std_sc6r_composite.inp: Composite SC6R analysis.
r313_std_sc6r_stacked.inp: Stacked SC6R analysis.
r313_std_sc8r_composite.inp: Composite SC8R analysis.
r313_std_sc8r_stacked.inp: Stacked SC8R analysis.

Abaqus/Explicit input file

r313shl.inp: S4R elements.