Question 10.S.P.2: 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 (i.e., N_{xx} = N_{yy}) and transverse pressure given by q(x,y) = 40 {sin [(πx)/a]} {sin[(πy)/b]} (kPa). Temperature remains constant and no change in moisture content occurs (ΔM = 0).
(a) Plot the out-of-plane displacements induced along the centerline defined by y = 0.075 m, if in-plane loads N_{xx} = N_{yy} = 50 kN/m are applied.
(b) Plot the maximum out-of-plane displacement as a function of in-plane loads, over the range 0 < (N_{xx} = N_{yy}) <70 kN/m.
(c) Compare the maximum out-of-plane displacement caused by the specified transverse load at room temperature if the plate is sub-ject to
(i) type S1 simple supports (as discussed in Sec. 3), and
(ii) type S4 simple supports, with N_{xx} = N_{yy} = 0
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. Using these laminate stiffnesses, dimensions, and the specified transverse loading, Eq. (15) 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{a^{2}}{\pi ^{2}} (N_{xx} + N_{yy} R^{2}) ]} (15)
w(x,y) \left[\frac{324}{\pi ^{4} \left(278 +\frac{0.090}{\pi ^{2}}(N_{xx} + 4 N_{yy}) \right) } \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). Using N_{xx} = N_{yy} = 50 kN/m and y = 0.075 m, out-of-plane dis-placements are:
w(x,y) = [1.30 × 10^{-3}] sin\left(\frac{\pi x}{0.3} \right) (meters)
A plot of these displacements over the length of the plate (0<x<0.3m is shown inFig. 4(a). Displacements are zero at the edges defined by x= 0, 0.3 m, as dictated by the specified boundary conditions. As would be expected because of symmetry, out-of-plane displacement is maximum at the center of the plate, and equals 1.3 mm for the loading considered.
Part (b). 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 m, and at this point the maximum out-of-plane displacement is given by:
w|_{max} = \left[\frac{324}{\pi ^{4} \left(278 +\frac{0.090}{\pi ^{2}}(N_{xx} + 4 N_{yy}) \right) } \right](meters)
A plot of maximum out-of-plane displacement as a function of in-plane tensile loads is shown in Fig. 4b. As would be intuitively expected, the plate is stiffened by the application of in-plane loading. That is, the maximum out-of-plane displacement is decreased as in-plane tensile loads are increased. A maximum deflection of 12 mm occurs when N_{xx} = N_{yy} = 0, whereas the max-imum deflection is reduced to 0.96 mm when N_{xx} = N_{yy} = 70 kN/m.
Part (c). This same panel and transverse loading was considered in Sample Problem 1, except type S1 simple supports were assumed. Thus in Sample Problem 1 in-plane displacements were fixed and were not allowed to change when the transverse load was applied. In contrast, type S4 simple supports are assumed in this problem; in-plane stress resultants are specified rather than in-plane displacements.
Referring to the results presented in these two sample problems, we find that identical deflections are predicted, despite the differences in boundary conditions. That is, a maximum deflection of 12 mm is predicted at room temperature for type S1 condition, and an identical 12 mm deflection is predicted if N_{xx} = N_{yy} = 0 for type S4 conditions. This result may seem nonintuitive and (rigorously speaking) is incorrect. That is, for type S4 boundary conditions, a transverse loading will cause a change in in-plane displacements. Therefore one might anticipate that the out-of-plane displacement for type S4 conditions would be increased, relative to type S1 conditions. However, the relative increase is very small if displacement gradients are small. Thus the relative increase in out-of-plane displacements for type S4 conditions is not predicted because we have based our analysis on infinitesimal strains. The consequences of the infinitesimal strain as-sumption were alluded to in Sec. 2.2 of Chap. 9. It was noted there that this assumption ultimately leads to the conclusion that in-plane displacement fields u_{o}(x,y) and v_{o}(x,y) are independent of the transverse load, q(x,y). The comparison between the results of Sample Problems 1 and 2 presented here is an illustration of this independence.
Of course, results for the two different boundary conditions are iden-tical because we have considered the case in which N_{xx} = N_{yy} = 0. If N_{xx} and/or N_{yy} ≠ 0, then the transverse displacements for a plate supported by type S4 simple supports is quite different from that of a plate supported by type S1 supports.
