Question 10.S.P.3: [(02/90)2]s graphite–epoxy laminate is cured at 175°C and th...
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 S2 simple supports along the two edges x = 0 and x = a, and type S1 simple supports along the two edges y = 0 and y = b. 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} along x = 0,a and a transverse pressure given by q(x,y) = 40 {sin[(πx)/a]}{sin[(πy)/b]} (kPa).
(a ) Plot the maximum out-of-plane displacement as a function of tensile load over the range 0 <N_{xx} < 70 kN/m, assuming temperature remains constant at room temperature.
(b) Plot the maximum out-of-plane displacement as a function of temperature, over the range -50°C < T < 20°C, assuming a con-stant in-plane tensile load N_{xx} = 50 kN/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 and effective thermal expansion coefficients 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. Using these laminate stiffnesses, dimensions, and the specified transverse loading, Eq. (25) becomes:
w(x,y) = \frac{q_{o} R^{4}b^{4} sin(\pi x/a) sin(\pi y/b)}{\pi ^{4} [D_{11} + 2R^{2}(D_{12} + 2D_{66}) + R^{4} D_{22} +\frac{R^{4} b^{2}}{\pi ^{2}}\left\{N_{xx} \left(\frac{1}{R^{2}}-\frac{A_{12}}{A_{11}} \right) + \frac{\overline{\alpha }_{yy}(A_{12}A_{22} – A_{12}^{2})}{A_{11}}(\Delta T^{c} – \Delta T) \right\} ]} (25)
w(x,y) [\frac{324}{\pi ^{4}\left(278 + \frac{0.099}{\pi ^{2}}N_{xx} + \frac{82.5}{\pi ^{2}}(\Delta T^{c} – \Delta T) \right) }]sin\left(\frac{\pi x}{0.3} \right) sin\left(\frac{\pi y}{0.15} \right) (meters)
This expression can be used to calculate the out-of-plane displacement induced at any point (x,y) over the surface of the plate.
Part (a). The maximum out-of-plane displacement occurs at the center of the plate, i.e., at x = a/2 = 0.15 m and y = b/2 = 0.075. Because the plate remains at room temperature, ΔT = ΔT^{c} , under these conditions the maximum out-of-plane displacement is given by:
w|_{max} = \frac{324}{\pi ^{4}\left(278 + \frac{0.099}{\pi ^{2}}N_{xx} \right) } (meters)
A plot of maximum out-of-plane displacement as a function of N_{xx} is shown in Fig. 6(a). As would be intuitively expected, the plate is stiffened as N_{xx} is increased. A maximum deflection of 12 mm occurs when N_{xx} = 0, whereas the maximum deflection is reduced to 3.4 mm when N_{xx} = 70 kN/m.
Part (b). As before, the maximum out-of-plane displacement occurs at x = a/2 = 0.15 m and y = b/2 = 0.075. Because a constant in-plane tensile load N_{xx}= 50 kN/m is applied, the maximum out-of-plane displacement is given by:
w|_{max} = \frac{324}{\pi ^{4}[781 + \frac{82.5}{\pi ^{2}}(\Delta T^{c} – \Delta T)] } (meters)
A plot of maximum out-of-plane displacement as a function of temperature is shown in Fig. 6(b). At room temperature (20°C), a maximum deflection of 4.3 mm is predicted. As would be expected, the plate becomes stiffer as the temperature is decreased. Out-of-plane displacements are decreased because of the in-plane tensile load N_{yy} that develops as temperature is decreased. At the lowest temperature considered (-50°C), a maximum deflection of 2.4 mm is predicted.
