4.7 Bending of a Block
201
Eqs. (4.121) into (4.119) and integrating yield
p(r) = ¯
σ r (r) +
r 2
r
( ¯
σ θ − ¯
σ r )
dr
r
,
(4.126)
which, with σ r = ¯
σ r − p, satisfies the zero-stress boundary condition at r = r 2 . The
condition σ r = 0 at r = r 1 then gives
r 2
r 1
( ¯
σ θ − ¯
σ r )
dr
r
= 0.
(4.127)
With these relations, the solution procedure is the following:
1. Define a grid in the undeformed block for a 1 ≤ X ≤ a 2 .
2. With r 1 specified, solve Eq. (4.127) for r 2 using a root-finding routine that
specifies a trial value for r 2 at each iteration. For each step in the iteration, do
the following:
(a) Compute k, C, and r(X) using Eqs. (4.115) and (4.117).
(b) Compute the λ i (X) at each grid point from (4.116).
(c) Compute the ¯
σ i (X) from (4.109) and (4.122).
(d) Set dr = λ r dX in (4.127) and evaluate the integral from X = a 1 to a 2 .
Submit this value to the solver until convergence is achieved.
3. With dr = λ r dX and r(X) now known, integrate Eq. (4.126) from X to a 2 to
obtain p(X) at each grid point.
4. Compute the Cauchy stresses at each point using (4.121).
5. Compute the bending moment M from (4.125) 2 .
4.7.3 Illustrative Results
To show the effects of nonlinearity, results are compared with those given by linear
plate theory, which assumes that displacements, strains, and rotations are small. For
cylindrical bending of a thin rectangular plate with the geometry defined in Fig. 4.16,
the bending moment and bending stress, respectively, are given by (Szilard 1974) 8
M p = Dκ(2c)
σ p =
E
1 − ν 2 κ(r − ρ),
(4.128)
8 Beam and plate theories are discussed more fully in Chap. 8.
201
Eqs. (4.121) into (4.119) and integrating yield
p(r) = ¯
σ r (r) +
r 2
r
( ¯
σ θ − ¯
σ r )
dr
r
,
(4.126)
which, with σ r = ¯
σ r − p, satisfies the zero-stress boundary condition at r = r 2 . The
condition σ r = 0 at r = r 1 then gives
r 2
r 1
( ¯
σ θ − ¯
σ r )
dr
r
= 0.
(4.127)
With these relations, the solution procedure is the following:
1. Define a grid in the undeformed block for a 1 ≤ X ≤ a 2 .
2. With r 1 specified, solve Eq. (4.127) for r 2 using a root-finding routine that
specifies a trial value for r 2 at each iteration. For each step in the iteration, do
the following:
(a) Compute k, C, and r(X) using Eqs. (4.115) and (4.117).
(b) Compute the λ i (X) at each grid point from (4.116).
(c) Compute the ¯
σ i (X) from (4.109) and (4.122).
(d) Set dr = λ r dX in (4.127) and evaluate the integral from X = a 1 to a 2 .
Submit this value to the solver until convergence is achieved.
3. With dr = λ r dX and r(X) now known, integrate Eq. (4.126) from X to a 2 to
obtain p(X) at each grid point.
4. Compute the Cauchy stresses at each point using (4.121).
5. Compute the bending moment M from (4.125) 2 .
4.7.3 Illustrative Results
To show the effects of nonlinearity, results are compared with those given by linear
plate theory, which assumes that displacements, strains, and rotations are small. For
cylindrical bending of a thin rectangular plate with the geometry defined in Fig. 4.16,
the bending moment and bending stress, respectively, are given by (Szilard 1974) 8
M p = Dκ(2c)
σ p =
E
1 − ν 2 κ(r − ρ),
(4.128)
8 Beam and plate theories are discussed more fully in Chap. 8.