Modern Trends in Structural and Solid Mechanics 1. Группа авторов

is that they are insensitive to shear-locking effects, which means that reduced integration is not needed in the thickness direction.

      We note that the above formulation holds for a laminate, when the continuity of variables u₁, u₂, u₃, σ₄, σ₅, σ₆, e₁, e₂, e₃ is enforced by adding the appropriate residuals in equation [1.8] or using a layerwise theory. Here, we use a layerwise theory, where the contribution from each layer is included in the summation in equation [1.8] and the continuity of the variables in s at each layer interface is ensured.

1.3. Results and discussion

      1.3.1. Verification of the numerical algorithm

      To verify the algorithm and to establish the accuracy of computed results, we study the problem analytically analyzed by Pagano (1969). It involves a four-layered [0/90/90/0] simply supported square laminate of side length a, with the sinusoidal surface traction

[1.12]

      applied only on the top surface. The material of the layers has the following values of the moduli:

      [1.13]

      Here, E, G and ν denote Young’s modulus, shear modulus and Poisson’s ratio, respectively, and subscripts L and T indicate directions parallel and transverse to the fiber direction. Following Pagano, we express the results in terms of the non-dimensionalized quantities defined in equation [1.4] and employ (x, y, z) = (x₁, x₂, x₃) as the coordinate axes and (u, v, w) = (u₁, u₂, u₃)

      [1.14]

In Table 1.1, we compare our results of select quantities with those in Pagano (1969) for the plate aspect ratio a/h = 100, 10, 4 and 2. It is clear that the developed least-squares method algorithm yields highly accurate results for the simply supported laminate.

Table 1.1. Comparison of the results with the 3D exact solution of Pagano for the [0/90/90/0] laminate

a/h	-
2	Pagano	1.38841-0.91165	0.83508-0.79465	-0.086300.06732	0.15300	0.29458	5.0745
	Present	1.38020 -0.90607	0.83038-0.79049	-0.085990.06711	0.15311	0.29428	5.0643
4	Pagano	0.72026-0.68434	0.66255 -0.66551	-0.046660.04581	0.21933	0.29152	1.93672
	Present	0.72020-0.68427	0.66246-0.66541	-0.046650.04575	0.21939	0.29154	1.93660
10	Pagano	0.55861-0.55909	0.40095-0.40257	-0.027500.02764	0.30137	0.19595	0.73698
	Present	0.55862-0.55910	0.40096-0.40257	-0.027470.02761	0.30140	0.19597	0.73698
100	Pagano	0.53885-0.53887	0.27101-0.27103	-0.021350.02136	0.33880	0.13894	0.43460
	Present	0.53883-0.53885	0.27100-0.27102	0.021353-0.021355	0.33880	0.13894	0.43460

      For a/h = 100, the maximum error in the computed quantities equals 0.023% for the in-plane shear stress at point (0, 0, −h/2), and for a/h = 2, the maximum error is 0.612% for the in-plane axial stress at point (a/2, a/2, −h/2). The errors for a/h = 10 and 4 are between those for a/h = 100 and 2. The through-the-thickness plots of , , , and coincide well with those from Pagano’s solution and are omitted.

For a/h = 10 and 2, we conducted the following five numerical experiments, E₁, E₂, …, E₅, by varying N₁, N₂ and N₃.