Читать книгу Modern Trends in Structural and Solid Mechanics 1 - Группа авторов - Страница 17

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. In E₁ and E₂, N₃ was set equal to either 3 or 4 and N₁ = N₂ was assigned values 4, 6 and 8; in E₃ and E₄, N₁ = N₂ = 4 and N₃ was given values 3, 4, 6 and 8; and in E₅, N₁ = N₂ = N₃ were assigned values 4, 6 and 8. Thus, a total of 13 combinations of N₁, N₂ and N₃ were evaluated for each value of a/h, and values of u, v, w, σxx, σyy, σxy and σxz at typical points were compared with their analytical values. For a/h = 10, N₁ = N₂ = 8, N₃ = 3 (total degrees of freedom (DOFs) = 9,477) provided an error of less than 0.1% in the value of each of these seven variables. Keeping N₁ = N₂ = 8 and increasing N₃ = 4, 5 marginally decreased the error, suggesting that there is no benefit derived from the increased computation cost. For a/h = 4, all 13 combinations of values of N₁, N₂ and N₃ resulted in an error of 7.25% in w and the error in σxz decreased from 37.7% for N₁=N₂=N₃ = 4 to 0.03% for N₁ = N₂ = N₃ = 8 (DOFs = 24,057).