evaluation of the integral by using the Gauss–Lobatto quadrature rule of order Pn in the three directions, gives J as a function of Saijk. We deduce the needed linear algebraic equations by setting
We realize that expressions for the residuals have different units. When equation [1.11] is written as KA = F, it is likely that the use of non-dimensional variables throughout the chapter will improve the condition number of the matrix K and reduce error. However, we have not tried this. A feature of the equations derived from [1.11] using basis functions of type [1.10] 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
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 |
