8.3 Linear Theory for Growing Beams and Plates
423
For all three cases, Eq. (8.15) provides the stress
σ =
E
1 − ν 2 [ 0 + yκ − g ]
=
E
1 − ν 2
0 + yκ −
0, −h/2 < y < 0
a, 0 < y < h/2
,
(8.35)
where 0 and κ = −v are given by the above results for each case.
Writing equations in dimensionless form often is a valuable exercise. For N > 0,
we define the following nondimensional quantities:
( ¯
x, ¯
y, ¯
v, ¯
L, ¯
c i ) = (x, y, v, L, c i )h
−1 ,
¯
β = βh
¯
M g = M g h/D,
¯
κ = κh,
¯
σ = σ/E.
In terms of these quantities, Eqs. (8.33) and (8.34) take the form
¯
v( ¯
x) = ¯
c 3 (sinh ¯
β ¯
x − ¯
β ¯
x) + ¯
c 4 (cosh ¯
β ¯
x − 1)
¯
c 3 = −
¯
M g
¯
β 2
cosh ¯
β ¯
L − 1
¯
β ¯
L cosh ¯
β ¯
L − sinh ¯
β ¯
L
¯
c 4 =
¯
M g
¯
β 2
sinh ¯
β ¯
L − ¯
β ¯
L
¯
β ¯
L cosh ¯
β ¯
L − sinh ¯
β ¯
L
.
The stress is given by
¯
σ =
1
1 − ν 2 [ 0 + ¯
y ¯
κ − g ],
where
¯
κ = −
d 2 ¯
v
d ¯
x 2 .
For case (c), Eqs. (8.19), (8.26), (8.27), and (8.29) yield
¯
M g = 3a/2,
¯
β =
√ −6a.
Results
Illustrative results for case (a) are shown for a = 0.1, ¯
L = L/ h = 10, and
ν = 0.4. As in the previous example, the plate bends upward into a circular shape
(Fig. 8.12a). The discontinuous growth between layers causes a discontinuity in
axial stress at the middle surface (Fig. 8.12b), and it is straightforward to show that
the computed distribution satisfies the equilibrium requirement N = M = 0. In
computing the moment, the compressive and tensile stresses in the small triangular
regions near the upper and lower surfaces produce a clockwise moment that opposes
423
For all three cases, Eq. (8.15) provides the stress
σ =
E
1 − ν 2 [ 0 + yκ − g ]
=
E
1 − ν 2
0 + yκ −
0, −h/2 < y < 0
a, 0 < y < h/2
,
(8.35)
where 0 and κ = −v are given by the above results for each case.
Writing equations in dimensionless form often is a valuable exercise. For N > 0,
we define the following nondimensional quantities:
( ¯
x, ¯
y, ¯
v, ¯
L, ¯
c i ) = (x, y, v, L, c i )h
−1 ,
¯
β = βh
¯
M g = M g h/D,
¯
κ = κh,
¯
σ = σ/E.
In terms of these quantities, Eqs. (8.33) and (8.34) take the form
¯
v( ¯
x) = ¯
c 3 (sinh ¯
β ¯
x − ¯
β ¯
x) + ¯
c 4 (cosh ¯
β ¯
x − 1)
¯
c 3 = −
¯
M g
¯
β 2
cosh ¯
β ¯
L − 1
¯
β ¯
L cosh ¯
β ¯
L − sinh ¯
β ¯
L
¯
c 4 =
¯
M g
¯
β 2
sinh ¯
β ¯
L − ¯
β ¯
L
¯
β ¯
L cosh ¯
β ¯
L − sinh ¯
β ¯
L
.
The stress is given by
¯
σ =
1
1 − ν 2 [ 0 + ¯
y ¯
κ − g ],
where
¯
κ = −
d 2 ¯
v
d ¯
x 2 .
For case (c), Eqs. (8.19), (8.26), (8.27), and (8.29) yield
¯
M g = 3a/2,
¯
β =
√ −6a.
Results
Illustrative results for case (a) are shown for a = 0.1, ¯
L = L/ h = 10, and
ν = 0.4. As in the previous example, the plate bends upward into a circular shape
(Fig. 8.12a). The discontinuous growth between layers causes a discontinuity in
axial stress at the middle surface (Fig. 8.12b), and it is straightforward to show that
the computed distribution satisfies the equilibrium requirement N = M = 0. In
computing the moment, the compressive and tensile stresses in the small triangular
regions near the upper and lower surfaces produce a clockwise moment that opposes