A 6-ft-long cylindrical pressure vessel (Figure 5.2) with an inner diameter of 35 in. is subjected to an internal gauge pressure of 150 psi. The vessel operates at room temperature and curing residual stresses are neglected. The cost of a graphite/epoxy lamina is 250 units/lbm and cost of a glass

Question

A 6-ft-long cylindrical pressure vessel (Figure 5.2) with an inner diameter of 35 in. is subjected to an internal gauge pressure of 150 psi. The vessel operates at room temperature and curing residual stresses are neglected. The cost of a graphite/epoxy lamina is 250 units/lbm and cost of a glass/epoxy lamina is 50 units/lbm. The following are other specifications of the design:

1. Only 0°, +45°, –45°, +60°, –60°, and 90° plies can be used.
2. Only symmetric laminates can be used.
3. Only graphite/epoxy and glass/epoxy laminae, as given in Table 2.2, are available, but hybrid laminates made of these two laminae are allowed. The thickness of each lamina is 0.005 in.
4. Calculate specific gravities of the laminae using Table 3.3 and Table 3.4 and fiber volume fractions given in Table 2.2.
5. Neglect the end effects and the mass and cost of ends of the pressure vessel in your design.
6. Use Tsai–Wu failure criterion for calculating strength ratios.
7. Use a factor of safety of 1.95.

Design for ply orientation, stacking sequence, number of plies, and ply material and give separate designs (laminate code, including materials) based on each of the following design criteria:

1. Minimum mass
2. Minimum cost
3. Both minimum mass and minimum cost

You may be unable to minimize mass and cost simultaneously — that is, the design of the pressure vessel for the minimum mass may not be same as for the minimum cost. In that case, give equal weight to cost and mass, and use this as your optimization function:

[latex] F=\frac{A}{B}+\frac{C}{D} [/latex] (5.24)

where
A = mass of composite laminate
B = mass of composite laminate if design was based only on minimum mass
C = cost of composite laminate
D = cost of composite laminate if design was based only on minimum cost

TABLE 2.2
Typical Mechanical Properties of a Unidirectional Lamina (USCS System of Units)

Property	Symbol	Units	Glass/epoxy	Boron/epoxy	Graphite/epoxy
Fiber volume fraction	[latex]V_f[/latex]	—	0.45	0.50	0.70
Longitudinal elastic modulus	[latex]E_1[/latex]	Msi	5.60	29.59	26.25
Transverse elastic modulus	[latex]E_2[/latex]	Msi	1.20	2.683	1.49
Major Poisson’s ratio	[latex] u_{12}[/latex]		0.26	0.23	0.28
Shear modulus	[latex]G_{12}[/latex]	Msi	0.60	0.811	1.040
Ultimate longitudinal tensile strength	[latex](\sigma_1^T)_{ult}[/latex]	ksi	154.03	182.75	217.56
Ultimate longitudinal compressive strength	[latex](\sigma_1^C)_{ult}[/latex]	ksi	88.47	362.6	217.56
Ultimate transverse tensile strength	[latex](\sigma_2^T)_{ult}[/latex]	ksi	4.496	8.847	5.802
Ultimate transverse compressive strength	[latex](\sigma_2^C)_{ult}[/latex]	ksi	17.12	29.30	35.68
Ultimate in-plane shear strength	[latex]( au_{12})_{ult}[/latex]	ksi	10.44	9.718	9.863
Longitudinal coefficient of thermal expansion	[latex]\alpha_1[/latex]	μin./in./°F	4.778	3.389	0.0111
Transverse coefficient of thermal expansion	[latex]\alpha_2[/latex]	μin./in./°F	12.278	16.83	12.5
Longitudinal coefficient of moisture expansion	[latex]\beta_1[/latex]	in./in./lb/lb	0.00	0.00	0.00
Transverse coefficient of moisture expansion	[latex]\beta_2[/latex]	in./in./lb/lb	0.60	0.60	0.60

TABLE 3.3
Typical Properties of Fibers (USCS System of Units)

Property	Units	Graphite	Glass	Aramid
Axial modulus	Msi	33.35	12.33	17.98
Transverse modulus	Msi	3.19	12.33	1.16
Axial Poisson’s ratio	—	0.30	0.20	0.36
Transverse Poisson’s ratio	—	0.35	0.20	0.37
Axial shear modulus	Msi	3.19	5.136	0.435
Axial coefficient of thermal expansion	μin./in./°F	-0.7222	2.778	-2.778
Transverse coefficient of thermal expansion	μin./in./°F	3.889	2.778	2.278
Axial tensile strength	ksi	299.7	224.8	200.0
Axial compressive strength	ksi	289.8	224.8	40.02
Transverse tensile strength	ksi	11.16	224.8	1.015
Transverse compressive strength	ksi	6.09	224.8	1.015
Shear strength	ksi	5.22	5.08	3.045
Specific gravity	—	1.8	2.5	1.4

TABLE 3.4
Typical Properties of Matrices (USCS System of Units)

Property	Units	Epoxy	Aluminum	Polyamide
Axial modulus	Msi	0.493	10.30	0.5075
Transverse modulus	Msi	0.493	10.30	0.5075
Axial Poisson’s ratio	—	0.30	0.30	0.35
Transverse Poisson’s ratio	—	0.30	0.30	0.35
Axial shear modulus	Msi	0.1897	3.915	0.1885
coefficient of thermal expansion	μin./in./°F	35	12.78	50
Coefficient of moisture expansion	in./in./lb/lb	0.33	0.00	0.33
Axial tensile strength	ksi	10.44	40.02	7.83
Axial compressive strength	ksi	14.79	40.02	15.66
Transverse tensile strength	ksi	10.44	40.02	7.83
Transverse compressive strength	ksi	14.79	40.02	15.66
Shear strength	ksi	4.93	20.01	7.83
Specific gravity	—	1.2	2.7	1.2

Accepted Answer

LOADING. For thin-walled cylindrical pressure vessels, the circumferential stress or hoop stress [latex]σ_y[/latex] and the longitudinal or axial stress [latex]σ_x[/latex] is given by

[latex] \sigma_{x}=\frac{p r}{2 t} [/latex] (5.25a)

[latex] \sigma_{y}=\frac{p r}{t} [/latex], (5.25b)

where

p = internal gage pressure, psi
r = radius of cylinder, in.
t = thickness of cylinder, in.

For our case, we have

[latex]p = 150 psi \ r=\frac{35}{2}=17.5 in[/latex]

giving

[latex] \sigma_{x}=\frac{(150)(17.5)}{2 t} \ \space \ =\frac{1.3125 imes 10^{3}}{t} \ \space \ \sigma_{y}=\frac{(150)(17.5)}{t} \ \space \ =\frac{2.625 imes 10^{3}}{t}. [/latex]

For the forces per unit length,

[latex] N_{x}=\sigma_{x} t \ \space \ =\frac{1.3125 imes 10^{3}}{t} t [/latex] (5.26a)

[latex] =1.3125 imes 10^{3} \frac{l b}{i n} \ \space \ N_{y}=\sigma_{y} t \ \space \ =\frac{2.625 imes 10^{3}}{t} t[/latex] (5.26b)

[latex]=2.625 imes 10^{3} \frac{l b}{i n}[/latex].

MASS OF EACH PLY. The mass of a graphite epoxy ply is

[latex] m_{G r / E p}=V_{G r / E p} ho_{G r / E p}, [/latex]

where
[latex]V_{Gr/Ep}[/latex] = volume of a graphite epoxy ply, in.³
[latex]ρ_{Gr/Ep}[/latex] = density of a graphite/epoxy ply, lbm/in.³
[latex]V_{Gr/Ep} = πLdt_p[/latex]

where
L = length of the cylinder, in.
d = diameter of the cylinder, in.
[latex]t_p[/latex] = thickness of graphite/epoxy ply, in.

Because L = 6 ft, d = 35 in., and [latex]t_p[/latex] = 0.005 in,

[latex]V_{Gr/Ep}[/latex] = π(6 × 12)(35)(0.005)
= 39.584 in.³

The density of a graphite/epoxy lamina is

[latex] ho_{Gr/Ep}= ho_{Gr}V_f+ ho_{Ep}V_m[/latex].

From Table 2.2, the fiber volume fraction, [latex]V_f[/latex] , of the graphite epoxy is 0.7. Thus,

[latex]V_f=0.7[/latex]

The matrix volume fraction, [latex]V_m[/latex], then is

[latex]V_m=1-V_f \ = 1-0.7 \ =0.3[/latex]

The specific gravity of graphite and epoxy is given in Table 3.3 and Table 3.4, respectively, as [latex]s_{Gr} = 1.8 and s_{Ep} = 1.2[/latex]; given that the density of water is [latex]3.6127 × 10^{–2} lbm/in^3,[/latex]

[latex] ho_{G r / E p}=(1.8)\left(3.6127 imes 10^{-2} ight)(0.7)+(1.2)\left(3.6127 imes 10^{-2} ight)(0.3) \=5.8526 imes 10^{-2} \frac{ lbm }{ in ^{3}} [/latex]

Therefore, the mass of a graphite/epoxy lamina is

[latex] m_{G r / E p}=V_{G r / E p} ho_{G r / E p} \ =(39.584)(5.8526 imes10^{-2}) \ =2.3167 lbm[/latex].

The mass of a glass/epoxy ply is

[latex] m_{G l / E p}=V_{G l / E p} ho_{G l / E p} [/latex]

where
[latex]V_{Gl/Ep} [/latex] = volume of glass/epoxy, in.³
[latex]ρ_{Gl/Ep}[/latex] = density of glass/epoxy, lbm/in.³
[latex]V_{Gl/Ep} = πLdt_p[/latex] = 39.584 in.³

The density of a glass/epoxy lamina is

[latex] ho_{G l / E p}= ho_{G l} V_{f}+ ho_{E p} V_{m} [/latex]

From Table 2.2, the fiber volume fraction [latex]V_f[/latex] of the glass/epoxy is 0.45; thus,

[latex]V_f=0.45.[/latex]

The matrix volume fraction [latex]V_m[/latex] then is

[latex]V_m=1-V_f\ =1-0.45 \ =0.55. [/latex]

The specific gravity of glass and epoxy is given in Table 3.3 and Table 3.4, respectively, as

[latex]s_{Gl} =2.5,s_{Ep} = 1.2[/latex]

and, given that the density of water is [latex]3.6127 × 10^{–2} lbm/in^3[/latex],

[latex] ho_{G / E p}=(2.5)\left(3.6127 imes 10^{-2} ight)(0.45)+(1.2)\left(3.6127 imes 10^{-2} ight)(0.55) \ =6.4487 imes 10^{-2} \frac{ lbm }{ in ^{3}} . [/latex]

Therefore, the mass of glass/epoxy lamina is

[latex] m_{G l / E p}=V_{G l / E p} P_{G l / E p} \ =(39.584)\left(6.4487 imes 10^{-2} ight) \ =2.5526 lbm.[/latex]

COST OF EACH PLY. The cost of a graphite/epoxy ply is

[latex] C_{G r / E p}=m_{G r / E p} c_{G r / E p} [/latex]

where
[latex]m_{Gr/Ep}[/latex] = mass of graphite/epoxy ply
[latex]c_{Gr/Ep}[/latex] = unit cost of graphite/epoxy ply

Because

[latex] m_{G r / E p}=2.3167 lbm \ c_{G r / E p}=250 \frac{ ext { units }}{l b m},[/latex]

the cost of a graphite/epoxy ply is

[latex] C_{G r / E p}=(2.3167)(250) \ =579.17 unites.[/latex]

Similarly, the cost of a glass/epoxy ply is

[latex] C_{G l / E p}=m_{G l / E p} c_{G l / E p} [/latex]

Because

[latex] m_{G l / E p}=2.5526 lbm [/latex]

and

[latex] c_{G l / E p}=50 \frac{ ext { units }}{ ext { lbm }}, [/latex]

the cost of a glass/epoxy ply is

[latex] C_{G l / E p}=(2.5526)(50) \ =127.63 units.[/latex]

1. To find the design for minimum mass, consider a composite laminate made of graphite/epoxy with [latex][0/90_2]_s[/latex]. We simply choose this laminate as [latex]N_y = 2N_x[/latex] and thus choose two 90° plies for every 0° ply. For this laminate, from PROMAL we get the minimum strength ratio as

SR = 0.6649.

Because the required factor of safety is 1.95, we need

[latex]\frac{1.95}{0.6649} imes 6 \ \cong 18 ext { plies. } [/latex]

[latex][0/90_2]_{3s}[/latex] is a possible choice because it gives a strength ratio of 1.995. However, is this laminate with the minimum mass? Choosing some other choices such as laminates with ±60° laminae, a graphite/epoxy [latex][±60]_{4s}[/latex] laminate gives an SR = 1.192 and that is lower than the required SR of 1.95.

[latex]A [0/90_2]_{3s}[/latex] laminate made of glass/epoxy gives a strength ratio of SR = 0.5192 and that is also lower than the needed strength ratio of 1.95. Other combinations tried used more than 18 plies. A summary of possible combinations is shown in Table 5.9.

TABLE 5.9
Mass and Cost of Possible Stacking Sequences for Minimum Mass

Stacking sequence	No. plies	Minimum strength ratio	Mass (lbm)	Cost (units)
[latex][0/90_2]_{3s}[/latex] (Graphite/epoxy)	18	1.995	41.700	10,425
[latex][±60]_{4s}[/latex] (Graphite/epoxy)	16	1.192	—	—
[latex][0/90_2]_{3s}[/latex] (Glass/epoxy)	18	0.5192	—	—
[latex][±60]_{5s}[/latex] (Graphite/epoxy)	20	149.0	—	—
[latex][±45_2/±60_3]_s[/latex] (Graphite/epoxy)	20	2.332	46.334	11,583

TABLE 5.10
Mass and Cost of Possible Stacking Sequences for Minimum Cost

Stacking sequence	No. plies	Minimum strength ratio	Mass (lbm)	Cost (units)
[latex][0/90_2]_{3s}[/latex] (Graphite/epoxy)	18	1.995	41.700	10,425
[latex][±45_2/±60_3]_s[/latex] (Graphite/epoxy)	20	2.291	46.334	11,583
[latex][0/90_2]_{12s}[/latex] (Glass/epoxy)	72	2.077	183.79	9,189
[latex][90/±45]_{10s}[/latex] (Glass/epoxy)	60	1.992	153.16	7,658
[latex][±60]_{15s}[/latex] (Graphite/epoxy)	60	2.033	153.16	7,658

Thus, one can say that the laminate for minimum mass is the first stacking sequence in Table 5.9:
Number of plies: 18
Material of plies: graphite/epoxy
Stacking sequence: [latex][0/90_2]_{3s}[/latex]
Mass of laminate = (18 × 2.3167) = 41.700 lbm
Cost of laminate = (41.700 × 250) = 10425 units

2. To find the design for minimum cost, we found in part (1) that the [latex][0/90_2]_{3s}[/latex] graphite/epoxy laminate is safe, but the same stacking sequence for glass/epoxy gives a SR = 0.5192. Therefore, we may need four times more plies of glass/epoxy to keep it safe to obtain a factor of safety of 1.95. If so, would it be cheaper than the [latex][0/90_2]_{3s}[/latex] graphite/epoxy laminate? Yes, it would because a glass/epoxy costs 127.63 units per ply as opposed to 579.17 units per ply for graphite/epoxy. Choosing [latex][0/90_2]_{12s}[/latex] glass/epoxy laminate gives SR = 2.077. Are there other combinations that give an SR > 1.95 but use less than the 72 plies used in [latex][0/90_2]_{12s}[/latex]? Stacking sequences of 60 plies such as [latex][90/±45]_{10s} and [±60]_{15s}[/latex] were tried and were acceptable designs. The results from some of the stacking sequences are summarized in Table 5.10.
Therefore, we can say that the laminate for minimum cost is as follows
Number of plies = 60
Material of plies: glass/epoxy
Stacking sequence: [latex][±60]_{15s}[/latex]
Mass of laminate = 60 × 2.5526 = 153.16 lbm
Cost of laminate = 153.16 × 50 = 7658 units
3. Now, how do we find the laminate that minimizes cost and mass?
We know that the solutions to part (1) and (2) are different. Thus, we need to look at other combinations. However, before doing so, let us find the minimizing function for parts (1) and (2). The minimizing function is given as

[latex] F=\frac{A}{B}+\frac{C}{D} [/latex],

where
A = mass of composite laminate
B = mass of composite laminate if design was based only on minimum mass
C = cost of composite laminate
D = cost of composite laminate if design was based only on minimum cost

From part (1), B = 41.700 lbm and, from part (2), D = 7658 units; then, the minimizing function is

[latex] F=\frac{41.700}{41.700}+\frac{10425}{7658}=2.361 [/latex]

for the [latex][0/90_2]_{3s}[/latex] graphite/epoxy laminate obtained in part (1).

[latex] F=\frac{153.16}{41.700}+\frac{7658}{7658}=4.673 [/latex]

for the [latex][±60]_{15s}[/latex] glass/epoxy laminate obtained in part (2). Therefore, the question is whether a laminate that has an optimizing function value of less than 2.361 can be found. If not, the answer is the same as the laminate in part (1). Table 5.11 gives the summary of some of the laminates that were tried to find minimum value of F. The third stacking sequence in Table 5.11 is the one in which, in the [latex][0/90_2]_{3s}[/latex] graphite/epoxy laminate of part (1), six of the graphite/epoxy plies of [latex]0/90_2[/latex] sublaminate group are substituted with 24 glass/epoxy plies of the [latex]0/90_2[/latex] sublaminate group.

Thus, it seems that [latex][0/90_2]_{3s}[/latex] graphite/epoxy laminate is the answer to part (3). Although more combinations should have been attempted to come to a definite conclusion, it is left to the reader to try other hybrid combinations using the PROMAL program.

TABLE 5.11
Optimizing Function Values for Different Stacking Sequences

Stacking sequence	Mass (lbm)	Cost	Minimum strength ratio	F
[latex][0/90_2]_{3s}[/latex] graphite/epoxy (part a)	41.700	10,425	1.995	2.361
[latex][±60]_{18s}[/latex] glass/epoxy (part b)	153.16	7,658	2.0768	4.672
[latex][0_{Gr/Ep}/90_{2 Gr/Ep}/0_{Gr/Ep}/90_{2 Gr/Ep}/0_{Gl/Ep}/ 90_{2 Gl/Ep}/0_{Gl/Ep}/90_{2 Gl/Ep}/0_{Gl/Ep}/90_{2 Gl/Ep}/0_{Gl/Ep}/ 90_{2 Gl/Ep}]_s[/latex]	89.063	10,013	2.012	3.443

Question 5.7: A 6-ft-long cylindrical pressure vessel (Figure 5.2) with an...

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