Question 10.S.P.4: A [(02/90)2]s graphite–epoxy laminate is cured at 175°C and ...
A [(0_{2}/90)_{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 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 in-plane tensile loading N_{xx} = N_{yy} = 50 kN/m and uniform transverse loading q(x,y) = 100 kPa. The temperature remains constant and no change in moisture content occurs (ΔM = 0). Plot the out-of-plane dis-placements induced along the centerline defined by y = 0.075 m. 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) |
A [(0_{2}/90)_{2}]_{s} graphite–epoxy laminate was also considered in Sample Problem 1, and numerical values for the [ABD] matrix are listed there. As before, the 12-ply laminate has a total thickness t = 1.5 mm and aspect ratio R = a/b = 2.0.
The double-Fourier series expansion of a uniform transverse loading was discussed in Sec. 4 of Chap. 9. The coefficients in the Fourier series expansion were found to be:
q_{xx} = \frac{16 q_{o}}{π^{2} mn}, m,n = odd integers
Combining these coefficients with Eq. (28c) allows prediction of out-of-plane displacements. A plot of these displacements along the centerline of the plate defined by y = b/2 = 0.075 m is shown in Fig. 7. Curves are shown based on a 1-term expansion (i.e., m,n = 1), a 4-term expansion (m,n = 1,3), and a 9- term expansion (m,n = 1,3,5). As would be expected due to symmetry, out-of-plane displacement is maximum at the center of the plate. The solution rapidly converges. The maximum displacement predicted on the basis of a 1-, 4-, and 9-term expansion equals 5.27, 4.71, and 4.78 mm, respectively. If 100 terms were used (m,n = 1,3,…,19), the maximum predicted displacement is 4.77 mm.
w(x,y) = \frac{R^{4}b^{4}}{\pi ^{4}} \sum\limits_{m=1}^{\infty }{\sum\limits_{n=1}^{\infty }} \frac{q_{mn} sin(m\pi x/a) sin(n\pi y/b)}{[D_{11} m^{4} + 2(D_{12} +2D_{66}) (mnR)^{2}+ D_{22}(nR)^{4} + \frac{a^{2}}{\pi ^{2}} \left\{N_{xx} m^{2} +N_{yy} (nR)^{2}\right\} ]} (28c)
