2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Displacement: analytical solution obtained by serial decomposition of the form:

\(w=\underset{i}{\Sigma }\underset{j}{\Sigma }{w}_{\mathrm{ij}}\mathrm{sin}(\frac{i\pi x}{a})\mathrm{sin}(\frac{j\pi y}{b})\)

Constraints: numerical solution [bib1], [bib2]

The baseline results are as follows:

\(w(\mathrm{0,}\mathrm{0,}0)\)	\(0.01507m\)	Move \(w\) to the center of the plate (point \(A\)),
\(\mathrm{SIXX}(\mathrm{0,}\mathrm{0,}h/2)\)	\(2.4216{10}^{7}\mathrm{Pa}\)	Stress \({\sigma }_{\mathit{xx}}\) on the upper skin of the upper skin from layer \(3\) (\(z=h/2\)) to the center of the plate (point A),
\(\mathrm{SIYY}(\mathrm{0,}\mathrm{0,}h/6)\) layer to \(90°\)	\(5.7810{10}^{6}\mathrm{Pa}\)	Stress \({\sigma }_{\mathit{yy}}\) on the upper skin of the layer \(2\) (\(z=h/6\)) at the center of the plate (point \(A\)),
\(\mathrm{SIXY}(a/\mathrm{2,}a/\mathrm{2,}h/2)\)	\(1.2825{10}^{6}\mathrm{Pa}\)	Stress \({\sigma }_{\mathrm{xy}}\) at point \(C\) on the upper skin of the layer \(3\),
\(\mathrm{SIXZ}(a/\mathrm{2,}\mathrm{0,}0)\)	\(2.3526{10}^{5}\mathrm{Pa}\)	Stress \({\sigma }_{\mathrm{xz}}\) at point \(D\) on the middle skin of the layer \(2\) (\(z=0\)),
\(\mathrm{SIYZ}(\mathrm{0,}a/\mathrm{2,}0)\)	\(8.8950{10}^{4}\mathrm{Pa}\)	Stress \({\sigma }_{\mathrm{yz}}\) at point \(B\) on the middle skin of the layer \(2\) (\(z=0\)),

The reference solution is given for a number of terms in the series equal to 25.
The transverse shear correction factor used is 5/6.
With significant slenderness (\(a/h=100\)), the level of transverse shear is low and therefore difficult to obtain accurately. There is then uncertainty about the stress values \({\sigma }_{\mathrm{ij}}\) calculated during the validation of the \(\mathit{VPCS}\) test, the differences obtained by the software on the shear components are of the order of \(\text{10\%}\).

VPCS: Software for calculating composite structures; Validation examples. Review of Composites and Advanced Materials, Volume 5 - special issue/ 1995. Hermes edition.
PUTCHA, N.S. and REDDY, J.N.: A mixed shear flexible finite element for the analysis of laminated plates, computer meth. in applied mech. Eng. 44 (1984).