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 [(02/90)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).

a. Predict the critical buckling load [latex]N_{xx}^c[/latex] and mode for this laminate, if 0 ≤ [latex]N_{yy}[/latex] ≤ 400 kN/m.

b. Predict the critical buckling load [latex]N_{yy}^c[/latex] and mode for this laminate, if 0 ≤ [latex]N_{xx}[/latex] ≤ 400 kN/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 [(02/90)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.

a. It is noted that n = 1, since [latex]N_{yy} ≥ 0[/latex]. Equation 11.39

[latex]N_{xx}=\frac{-\pi ^2}{(ma)^2}\left[D_{11}m^4+2(D_{12}+2D_{66})(mnR)^2+D_{22}(nR)^4+N_{yy}\left\lgroup\frac{naR}{\pi } \right\rgroup ^2\right][/latex] (11.39)

becomes in this case:

[latex]N_{xx}=\frac{-\pi ^2}{0.09m^2}\left[40.1m^4+65.34m^2+172.8+N_{yy}\left\lgroup\frac{0.6}{\pi } \right\rgroup^2 \right][/latex]

A plot of the critical buckling load for [latex]0 ≤ N_{yy} ≤ 400 kN/m[/latex] is presented in Figure 11.9a. As expected, [latex]N_{xx}^c[/latex] is increased as [latex]N_{yy}[/latex] is increased. A change in buckling mode also occurs as [latex]N_{yy}[/latex] is increased. The plate buckles in mode [2,1] over the range [latex]0 ≤ N_{yy} < 35 kN/m[/latex], in mode [3,1] over the range [latex]35 kN/m ≤ N_{yy} < 150 kN/m[/latex], and in mode [4,1] over the range [latex]150 kN/m ≤ Nyy < 400 kN/m.[/latex] These buckling modes are illustrated in Figures 11.9b,c,d, respectively.

b. It is noted that m = 1, since [latex]N_{xx }≥ 0[/latex]. Equation 11.41 becomes in this case:

[latex]N_{yy}=\frac{-\pi ^2}{0.36n^2}\left[40.1m^4+65.34n^2+172.8(n)^2+N_{xx}\frac{0.09}{\pi^2 } \right][/latex]

A plot of the critical buckling load for [latex]0 ≤ N_{xx} ≤ 400 kN/m[/latex] is presented in Figure 11.10a. As expected, [latex]N_{yy}^c[/latex] is increased as [latex]N_{xx}[/latex] is increased. A change in buckling mode also occurs as [latex]N_{xx}[/latex] is increased. The plate buckles in mode [1,1] over the range [latex]0 ≤ N_{xx} < 72 kN/m[/latex], and in mode [1,2] over the range [latex]72 kN/m ≤ N_{xx} < 400 kN/m[/latex]. These buckling modes are illustrated in Figures 11.10b,c, respectively.

Note that the magnitudes of [latex]N_{yy}^c[/latex] calculated in part (b) are far lower than those calculated for [latex]N_{xx}^c[/latex] in part (a). This pronounced difference is largely due to the stacking sequence involved. For the [latex][(02/90)2]_s[/latex] laminate under consideration, 8 of 12 plies are 0° plies (i.e., plies with fibers parallel to the x-axis). Hence, the resistance to buckling due to a compressive load [latex]N_{xx}[/latex] is far higher than resistance to buckling due to a compressive [latex]N_{yy}[/latex].

Question 11.5: A [(02/90)2]s graphite–epoxy laminate is cured at 175°C and ...

Related Answered Questions