[(02/90)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

Question

[latex]A [(0_{2}/90)_{2}]_{s}[/latex] 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 [latex]N_{xx}[/latex] 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
[latex]E_{11}[/latex]	55 GPa (8.0 Msi)	100 GPa (15 Msi)	170 GPa (25 Msi)
[latex]E_{22}[/latex]	16 GPa (2.3 Msi)	6 GPa (0.90 Msi)	10 GPa (1.5 Msi)
[latex]ν_{12}[/latex]	0.28	0.33	0.30
[latex]G_{12}[/latex]	7.6 GPa (1.1 Msi)	2.1 GPa (0.30 Msi)	13 GPa (1.9 Msi)
[latex]σ_{11}^{fT}[/latex]	1050 MPa (150 ksi)	1380 MPa (200 ksi)	1500 MPa (218 ksi)
[latex]σ_{11}^{fC}[/latex]	690 MPa (100 ksi)	280 MPa (40 ksi)	1200 MPa (175 ksi)
[latex]σ_{22}^{yT}[/latex]	45 MPa (5.8 ksi)	35 MPa (2.9 ksi)	50 MPa (7.25 ksi)
[latex]σ_{22}^{yC}[/latex]	120 MPa (16 ksi)	105 MPa (15 ksi)	100 MPa (14.5 ksi)
[latex]σ_{22}^{fT}[/latex]	55 MPa (7.0 ksi)	45 MPa (4.3 ksi)	70 MPa (10 ksi)
[latex]σ_{22}^{fC}[/latex]	140 MPa (20 ksi)	140 Msi (20 ksi)	130 MPa (18.8 ksi)
[latex]τ_{12}^{y}[/latex]	40 MPa (4.4 ksi)	40 MPa (4.0 ksi)	75 MPa (10.9 ksi)
[latex]τ_{12}^{f}[/latex]	70 MPa (10 ksi)	60 MPa (9 ksi)	130 MPa (22 ksi)
[latex]α_{11}[/latex]	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)
[latex]α_{22}[/latex]	25 μ/m °C (14 μin./in. °F)	58 μm/m °C (32 μin./in. °F)	27 μm/m °C (15 μin./in. °F)
[latex]β_{11}[/latex]	100 μm/m %M (100 μin./in. %M)	175 μm/m %M (175 μin./in. %M)	50 μm/m %M (50 μin./in. %M)
[latex]β_{22}[/latex]	1200 μm/m %M (1200 μin./in. %M)	1700 μm/m %M (1700 μin./in. %M)	1200 μm/m %M (1200 μin./in. %M)

Accepted Answer

A [latex][(0_{2}/90)_{2}]_{s}[/latex] 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:

[latex]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\} ]}[/latex] (25)

[latex]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) [/latex] (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, [latex]ΔT = ΔT^{c}[/latex] , under these conditions the maximum out-of-plane displacement is given by:

[latex]w|_{max} = \frac{324}{\pi ^{4}\left(278 + \frac{0.099}{\pi ^{2}}N_{xx} \right) } [/latex] (meters)

A plot of maximum out-of-plane displacement as a function of [latex]N_{xx}[/latex] is shown in Fig. 6(a). As would be intuitively expected, the plate is stiffened as [latex]N_{xx}[/latex] is increased. A maximum deflection of 12 mm occurs when [latex]N_{xx}[/latex] = 0, whereas the maximum deflection is reduced to 3.4 mm when [latex]N_{xx}[/latex] = 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 [latex]N_{xx}[/latex]= 50 kN/m is applied, the maximum out-of-plane displacement is given by:

[latex]w|_{max} = \frac{324}{\pi ^{4}[781 + \frac{82.5}{\pi ^{2}}(\Delta T^{c} - \Delta T)] }[/latex] (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 [latex]N_{yy}[/latex] that develops as temperature is decreased. At the lowest temperature considered (-50°C), a maximum deflection of 2.4 mm is predicted.

Question 10.S.P.3: [(02/90)2]s graphite–epoxy laminate is cured at 175°C and th...

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