Question 11.S.P.1: A [(±45/0)2]s graphite-epoxy laminate is cured at 175°C and ...
A [(±45/0)_{2}]_{s} graphite-epoxy laminate is cured at 175°C and then cooled to room temperature (20°C). After cooling, the flat laminate is trimmed to in-plane dimensions of 300×150 mm and is mounted in an assembly that provides type S4 simple supports along all four edges. The x axis is defined parallel to the 300-mm edge (i.e., a = 0.3 m, b = 0.15 m). The laminate is then subjected to a uniform transverse load q(x,y) = 30 kPa. No in-plane loads are applied (i.e., N_{xx} = N_{yy} = N_{xy} = 0). Determine the maximum out-of-plane displacement based on a Ritz analysis and plot the out-of-plane displacement field. Use the properties listed for graphite-epoxy in Table 3 of Chap. 3 and assume each ply has a thickness of 0.125 mm.
| Table 3 Nominal Material Properties for Common Unidirectional Composites | |||
| Property | Glass/epoxy | Kevlar/epoxy | Graphite/epoxy |
| E_{11} | 55 GPa (8.0 Msi) | 100 GPa (15 Msi) | 170 GPa (25 Msi) |
| E_{22} | 16 GPa (2.3 Msi) | 6 GPa (0.90 Msi) | 10 GPa (1.5 Msi) |
| ν_{12} | 0.28 | 0.33 | 0.30 |
| G_{12} | 7.6 GPa (1.1 Msi) | 2.1 GPa (0.30 Msi) | 13 GPa (1.9 Msi) |
| σ_{11}^{fT} | 1050 MPa (150 ksi) | 1380 MPa (200 ksi) | 1500 MPa (218 ksi) |
| σ_{11}^{fC} | 690 MPa (100 ksi) | 280 MPa (40 ksi) | 1200 MPa (175 ksi) |
| σ_{22}^{yT} | 45 MPa (5.8 ksi) | 35 MPa (2.9 ksi) | 50 MPa (7.25 ksi) |
| σ_{22}^{yC} | 120 MPa (16 ksi) | 105 MPa (15 ksi) | 100 MPa (14.5 ksi) |
| σ_{22}^{fT} | 55 MPa (7.0 ksi) | 45 MPa (4.3 ksi) | 70 MPa (10 ksi) |
| σ_{22}^{fC} | 140 MPa (20 ksi) | 140 Msi (20 ksi) | 130 MPa (18.8 ksi) |
| τ_{12}^{y} | 40 MPa (4.4 ksi) | 40 MPa (4.0 ksi) | 75 MPa (10.9 ksi) |
| τ_{12}^{f} | 70 MPa (10 ksi) | 60 MPa (9 ksi) | 130 MPa (22 ksi) |
| α_{11} | 6.7 μ/m °C
(3.7 μin./in. °F) |
-3.6 μm/m °C
(-2.0 μin./in. °F) |
-0.9 μm/m °C
(-0.5 μin./in. °F) |
| α_{22} | 25 μ/m °C
(14 μin./in. °F) |
58 μm/m °C
(32 μin./in. °F) |
27 μm/m °C
(15 μin./in. °F) |
| β_{11} | 100 μm/m %M
(100 μin./in. %M) |
175 μm/m %M
(175 μin./in. %M) |
50 μm/m %M
(50 μin./in. %M) |
| β_{22} | 1200 μm/m %M
(1200 μin./in. %M) |
1700 μm/m %M
(1700 μin./in. %M) |
1200 μm/m %M
(1200 μin./in. %M) |
Based on the properties listed in Table 3 of Chap. 3 for graphite-epoxy, the [ABD] matrix for a [(±45/0)_{2}]_{s} laminate is
[ABD] = \left[\begin{matrix}145.2 \times 10^{6} & 35.3 \times 10^{6} &0 &0 & 0&0\\ 35.3 \times 10^{6} & 64.8 \times 10^{6} &0 &0 & 0&0 \\0& 0 & 50.2 \times 10^{6} &0 & 0& 0 \\ 0 & 0& 0 & 22.3& 7.97 & 2.20 \\0 & 0& 0 &7.97 &14.3 & 2.20 \\ 0 & 0& 0 &2.20 &2.20 &10.8 \end{matrix} \right]where the units of A_{ij} are Pa m and the units of D_{ij} are Pa m³ . Notice that neither D_{16} nor D_{26} equals zero; hence, the laminate is generally ortho-tropic. The 12-ply laminate has a total thickness t = 1.5 mm and the aspect ratio R = a/b = 2.0.
The computer program SYMM (described in Sec. 6) can be used to perform the required Ritz analysis. Several analyses were performed using increasing values of M (and N ) to evaluate whether the solution has con-verged to a reasonably constant value. Solutions were obtained using values of M (and N ) ranging from 1 through 10 (i.e., analyses were performed in which the number of terms used to describe the displacement field ranged from 1 through 100). Maximum predicted displacement is plotted as a function of M and N in Fig. 3. As indicated, the maximum displacement con-verges to a value of 8.03 mm when M=N=10.
A contour plot of out-of-plane displacements predicted using M=N= 10 is shown in Fig. 4. As would be expected, the maximum displacement occurs at the center of the plate (i.e., at x = 150 mm, y = 75 mm). Careful examination of these contours will reveal that the contours are very slightly distorted. This distortion (which is barely discernible in Fig. 4) occurs be-cause the plate is generally orthotropic. That is, for a [(±45/0)_{2}]_{s} laminate D_{16}, D_{26} ≠ 0. However, for this problem, the magnitudes of D_{16} and D_{26} (relative to D_{11} and D_{22}) are very small. Specifically, for the laminate considered in this problem D_{16}/D_{11} = D_{26}D_{11} = 0.0986 and D_{16}/D_{22} = D_{26} D_{22} = 0.153. Consequently, distortion of out-of-plane displacements is very slight. The out-of-plane displacement induced by a uniform transverse load applied to a laminate with relatively higher values of D_{16} and D_{26} is considered in Sample Problem 3. As will be seen, the distortion of displacement contours is much more pronounced in that case due to the relatively higher values of D_{16} and D_{26}.
