A [(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 S4 simple supports along all four edges. The x-axis is defined

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 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 [latex]N_{xx} = N_{yy} = 50 kN/m[/latex] and uniform transverse loading q(x,y) = 100 kPa. Temperature remains constant and no change in moisture content occurs (ΔM = 0). Plot the out-of-plane displacements induced along the centerline defined by y = 0.075 m. Use the properties listed for graphite– epoxy in Table 3.1, and assume each ply has a thickness of 0.125 mm.

TABLE 3.1
Typical Properties of Common Unidirectional Composites

Property	Glass/ Epoxy	Kevlar/ Epoxy	Graphite/ Epoxy
[latex]E_{11}[/latex]	55 GPa	100 GPa	170 GPa
[latex]E_{11}[/latex]	(8.0 Msi)	(15 Msi)	(25 Msi)
[latex]E_{22}[/latex]	16 GPa	6 GPa	10 GPa
[latex]E_{22}[/latex]	(2.3 Msi)	(0.90 Msi)	(1.5 Msi)
[latex]ν_{12}[/latex]	0.28	0.33	0.3
[latex]G_{12}[/latex]	7.6 GPa	2.1 GPa	13 GPa
	(1.1 Msi)	(0.30 Msi)	(1.9 Msi)
[latex]\sigma _{11}^{fT}[/latex]	1050 MPa	1380 MPa	1500 MPa
[latex]\sigma _{11}^{fT}[/latex]	(150 ksi)	(200 ksi)	(218 ksi)
[latex]\sigma _{11}^{fC}[/latex]	690 MPa	280 MPa	1200 MPa
[latex]\sigma _{11}^{fC}[/latex]	(100 ksi)	(40 ksi)	(175 ksi)
[latex]\sigma _{22}^{ff}[/latex]	45 MPa	35 MPa	50 MPa
[latex]\sigma _{22}^{ff}[/latex]	(5.8 ksi)	(2.9 ksi)	(7.25 ksi)
[latex]\sigma _{22}^{fC}[/latex]	120 MPa	105 MPa	100 MPa
[latex]\sigma _{22}^{fC}[/latex]	(16 ksi)	(15 ksi)	(14.5 ksi)
[latex] au ^f_{22}[/latex]	40 MPa	40 MPa	90 MPa
[latex] au ^f_{22}[/latex]	(4.4 ksi)	(4.0 ksi)	(13.1 ksi)
[latex]\alpha _{11}[/latex]	6.7 μm/m−°C	−3.6 μm/m−°C	−0.9 μm/m−°C
[latex]\alpha _{11}[/latex]	[latex](3.7 μin./in.\boxtimes °F)[/latex]	(−2.0 μin./in.−°F)	(−0.5 μin./in.−°F)
[latex]\alpha _{22}[/latex]	25 μm/m−°C	58 μm/m−°C	27 μm/m−°C
[latex]\alpha _{22}[/latex]	(14 μin./in.−°F)	(32 μin./in.−°F)	(15 μin./in.−°F)
[latex]\beta _{11}[/latex]	100 μm/m−%M	175 μm/m−%M	50 μm/m−%M
	(100 μin./in.−%M)	(175 μin./in.−%M)	(50 μin./in.−%M)
[latex]\beta _{22}[/latex]	1200 μm/m−%M	1700 μm/m−%M	1200 μm/m−%M
	(1200 μin./in.−%M)	(1700 μin./in.−%M)	(1200 μin./in.−%M)
Ply	0.125 mm	0.125 mm	0.125 mm
Thickness	(0.005 in.)	(0.005 in.)	(0.005 in.)

Accepted Answer

[latex]A [(0_2/90)2]_s[/latex] graphite–epoxy laminate was also considered in Example Problem 11.1, 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 Section 10.4. The coefficients in the Fourier series expansion were found to be

[latex]q_{mn}=\frac{16q_0}{\pi ^2mn},m,n=[/latex] odd integers

Combining these coefficients with Equation 11.28c

[latex]w(x,y)=\frac{R^4b^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)}{\left[D_{11}m^4+2(D_{12}+2D_{66})(mnR)^2+D_{22}(nR)^4+(a^2/\pi ^2)\left\{N_{xx}m^2+N_{yy}(nR)^2\right\} \right] } }[/latex] (11.28c)

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 Figure 11.7. 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 converges rapidly. The maximum displacement predicted on the basis of a 1-, 4-, and 9-term expansion equals 5.27, 4.71, and 4.78 [latex] mm^3[/latex] , respectively. If 100 terms were used (m, n = 1,3, . . ., 19) the maximum predicted displacement is 4.77 mm.

Question 11.4: A [(02/90)2]s graphite–epoxy laminate is cured at 175°C and ...

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