Stresses

1. Chapter 2 MacromechanicalAnalysis of a Lamina
Hygrothermal Stresses and Strains
Dr.Autar Kaw
Department of Mechanical Engineering
University of South Florida,Tampa, FL 33620
Courtesy of the Textbook
Mechanics of Composite Materials by Kaw

2.  For a unidirectional lamina
 Thermally induced strains:
 Moisture induced strains:




























































0
+
0
+
S
0
0
0
S
S
0
S
S
= C
2
C
1
T
2
T
1
12
2
1
66
22
12
12
11
12
2
1



































0
T
=
0
2
1
T
2
T
1





























0
C
=
0
2
1
C
2
C
1




(2.174)
(2.175)
(2.176)

3.  For a unidirectional lamina




























































0
0
0
0
0
0
2
1
2
1
12
2
1
66
22
12
12
11
12
2
1
ε
ε
+
ε
ε
+
τ
σ
σ
S
S
S
S
S
=
γ
ε
ε
C
C
T
T




































γ
ε
-
ε
-
ε
ε
-
ε
-
ε
Q
Q
Q
Q
Q
=
τ
σ
σ
C
T
C
T
12
2
2
2
1
1
1
66
22
12
12
11
12
2
1
0
0
0
0
(2.174)
(2.177)

4.  For an angular lamina
 Thermally induced strains:
 Moisture induced strains:
































































γ
ε
ε
+
γ
ε
ε
+
τ
σ
σ
S
S
S
S
S
S
S
S
S
=
γ
ε
ε
C
xy
C
y
C
x
T
xy
T
y
T
x
xy
y
x
xy
y
x
66
26
16
26
22
12
16
12
11

































xy
y
x
T
xy
T
y
T
x
T
=

































xy
y
x
C
xy
C
y
C
x
C
=
(2.178)
(2.179)
(2.180)

5.  For an angular lamina
























0
2
2
1
1
α
α
]
= [T
/
α
α
α
-
xy
y
x
















s
c
sc
sc
sc
c
s
sc
s
c
=
T
2
2
2
2
2
2
1
2
2
]
[












s
-
c
sc
sc
-
2sc
-
c
s
2sc
s
c
=
[T]
2
2
2
2
2
2
)
(
=
c 
Cos
)
(
=
s 
Sin
(2.181)
(2.95) (2.96)
(2.97a,b)

6.  For an angular lamina
























0
]
[T
=
/2
2
1
1
-
xy
y
x





















s
c
sc
sc
2sc
c
s
2sc
s
c
=
]
[T
2
2
2
2
2
2
1












s
-
c
sc
sc
-
2sc
-
c
s
2sc
s
c
=
[T]
2
2
2
2
2
2
)
(
=
c 
Cos
)
(
=
s 
Sin
(2.95) (2.96)
(2.97a,b)
(2.182)

7. Find the following for a 600
angle lamina of Glass/Epoxy
a) coefficients of thermal expansion,
b) coefficients of moisture expansion,
c) strains under a temperature change of -1000
C and a moisture absorption of 0.02 kg/kg.
Use properties of unidirectional Glass/Epoxy lamina from Table 2.1.

8. a) From Table 2.1,
C,
/
m/m
10
8.6
= 0
-6
1 

C.
/
m/m
10
22.1
= 0
-6
2 

Using Equation (2.181), gives
,
0
10
22.1
10
8.6
0.5000
-
0.4330
-
0.4330
0.8660
0.2500
0.7500
0.8660
-
0.7500
0.2500
=
/2
6
-
-6
xy
y
x







































C.
/
m/m
10
11.69
-
10
11.98
10
18.73
= 0
6
-
6
-
-6
xy
y
x






























.
0
]
[T
=
/2
2
1
1
-
xy
y
x





























(2.181)

9. b) From Table 2.1
m/m/kg/kg,
0
=
1

m/m/kg/kg.
0.6
=
2

Using Equation (2.182) gives
































0
0.6
0.0
0.5000
-
0.4330
-
0.4330
0.8660
0.2500
0.7500
0.8660
-
0.7500
0.2500
=
/2
xy
y
x



m/m/kg/kg
0.5196
-
0.1500
0.4500
=
xy
y
x

















































0
]
[T
=
/2
2
1
1
-
xy
y
x





(2.182)

10. c) Now using Equations (2.179) and (2.180) to calculate the strains as.
(0.02)
0.5196
-
0.1500
0.4500
+
(-100)
10
11.69
-
10
11.98
10
18.73
=
6
-
6
-
-6
xy
y
x








































m/m
10
0.9223
-
10
0.1802
10
0.7127
=
2
2
2



























































































C
xy
C
y
C
x
T
xy
T
y
T
x
xy
y
x
66
26
16
26
22
12
16
12
11
xy
y
x
+
+
S
S
S
S
S
S
S
S
S
= (2.178)