Modeling of cylindrical composite shell structures based on the Reissner’s Mixed Variational Theorem with a variable separation method

Vidal, Philippe; Polit, Olivier; Gallimard, Laurent; D’Ottavio, Michele

doi:10.1186/s40323-019-0132-0

Advanced Modeling and Simulation in Engineering Sciences

Table 2 Influence of the order-approximation of \(1/\mu \) with respect to S—one layer \([0^{\circ } ]\)—\(\phi =\pi /3\)—mesh 26 \(\times \) 10—\(N_z = NC\)

From: Modeling of cylindrical composite shell structures based on the Reissner’s Mixed Variational Theorem with a variable separation method

S	Expansion	Model	\(\bar{{{\varvec{u}}}}{{\varvec{(0, e/2}}}{} \mathbf{)}\)	\(\bar{{{\varvec{w}}}}{{\varvec{(L,0)}}}\)	\(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}{{\varvec{(-e/2)}}}\)	\(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}{{\varvec{(0)}}}\)	\(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\) max
2	Zero-order	VS-LM4	3.9747	0.7982	\(-\) 1.4093	0.4688	0.8883
2	Zero-order	Error	16.67%	19.83%	42.59%	15.76%	11.17%
2	One-order	VS-LM4	4.9928	1.0179	\(-\) 2.2682	0.5908	0.9324
2	One-order	Error	4.68%	2.24%	7.59%	6.16%	6.76%
2	Two-order	VS-LM4	4.7940	0.9769	\(-\) 2.1818	0.5776	0.8967
		Error	0.51%	1.88%	11.12%	3.78%	10.33%
		Exact	4.7696	0.9956	\(-\) 2.4546	0.5565	1.0000
4	Zero-order	VS-LM4	2.3665	0.2792	\(-\) 1.0406	0.5287	0.9225
4	Zero-order	error	10.39%	10.50%	21.81%	7.87%	7.75%
4	One-order	VS-LM4	2.6745	0.3153	\(-\) 1.3452	0.5958	0.9163
4	One-order	error	1.28%	1.08%	1.08%	3.82%	8.37%
4	Two-order	VS-LM4	2.6386	0.3112	\(-\) 1.3261	0.5909	0.9062
		Error	0.09%	0.25%	0.36%	2.97%	9.38%
		Exact	2.6408	0.3120	\(-\) 1.3309	0.5739	1.0000
10	Zero-order	VS-LM4	2.5623	0.1094	\(-\) 0.8071	0.5558	\(-\) 1.3651
10	Zero-order	Error	4.54%	4.55%	9.30%	4.13%	9.35%
10	One-order	VS-LM4	2.6932	0.1150	\(-\) 0.8937	0.5838	\(-\) 1.4207
10	One-order	Error	0.33%	0.33%	0.44%	0.69%	5.66%
10	Two-order	VS-LM4	2.6846	0.1146	\(-\) 0.8908	0.5829	\(-\) 1.4165
		Error	0.01%	0.01%	0.11%	0.55%	5.94%
		Exact	2.6843	0.1146	\(-\) 0.8898	0.5798	\(-\) 1.5059
40	Zero-order	VS-LM4	9.2631	0.0770	\(-\) 0.7553	0.5564	\(-\) 7.0856
40	Zero-order	Error	1.21%	1.20%	2.25%	2.19%	0.23%
40	One-order	VS-LM4	9.3800	0.0780	\(-\) 0.7746	0.5629	\(-\) 7.1657
40	One-order	Error	0.04%	0.04%	0.25%	1.05%	0.89%
40	Two-order	VS-LM4	9.3775	0.0780	\(-\) 0.7744	0.5633	\(-\) 7.1654
		Error	0.01%	0.02%	0.22%	0.97%	0.89%
		Exact	9.3765	0.0780	\(-\) 0.7727	0.5688	\(-\) 7.1021

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