8.10 Mesenchymal Morphogenesis
507
(e) Use the following baseline parameter values:
L = 10, h = 1, a = 4, E = 10, ν = 0.5, k = 1, p 0 = 2, α = 1, β = 10
• Plot p vs x. On a single graph, plot the exact pressure and the series
approximation for 5, 10, and 100 terms.
• Plot M g vs x. On a single graph, plot the exact growth moment and the
series approximation for 5, 10, and 100 terms.
• Plot −v vs x (100 terms).
• Plot σ vs x along the lower surface of the plate (100 terms).
(f) Including 100 terms in the series, plot −v vs x for the following cases:
• Effects of pressure: On a single graph, plot the curve for the baseline
parameter values, as well as curves for p 0 = 0 and p 0 = −2.
• Effects of growth force: On a single graph, plot the curve for the
baseline parameter values, as well as curves for α = 0 and α = 2.
• Effects of growth moment: On a single graph, plot the curve for the
baseline parameter values, as well as curves for β = 0 and β = −10.
8.5 Consider again the plate problem of Example 8.4. As before, growth is
specified only in the lower half of region 1 (0 ≤ x ≤ d) and the left end is
fixed, but now the right end (x = L) is supported by a roller, rather than a pin
(see Fig. 8.15a). Solve this problem using (a) Fourier series and (b) by finding
the “exact” solution. For L = 10h and g = a = 0.1, plot the deflection
(−v/ h) along the beam for both solutions on the same graph. Plot results for
d/L = 0.25, 0.5, 0.75, and 1.
Hint: For part (b), solve the 4th-order plate equation separately in regions
1 and 2 for the deflections v 1 (x) and v 2 (x), respectively, with the growth
moment being constant in each region. Determine the eight constants of
integration using four boundary conditions and the four continuity conditions
f 1 (d) = f 2 (d), where f = v, θ, M, V . Suggestion: Work in terms of the
dimensionless quantities defined in Eq. (8.36).
8.6 Using the series expansions
sinh x = x + x
3 /6 + . . .
cosh x = 1 + x
2 /2 + . . . ,
show that Eqs. (8.33) and (8.34) yield (8.32) as β → 0 (N → 0).
8.7 Consider a rigid, weightless vertical bar of length L that is pinned at its
lower end. The upper end is supported by a horizontal spring and loaded by
a downward force P (Fig. 8.81). As the bar rotates counter-clockwise, the
load remains vertical, and the tip displaces rightward by the distance . In
addition, the right end of the spring (stiffness k) follows the downward motion
of the tip, so that the spring remains horizontal. Whereas is assumed to be
small in the analysis of Sect. 8.6.4, consider here the “post-buckling” behavior
507
(e) Use the following baseline parameter values:
L = 10, h = 1, a = 4, E = 10, ν = 0.5, k = 1, p 0 = 2, α = 1, β = 10
• Plot p vs x. On a single graph, plot the exact pressure and the series
approximation for 5, 10, and 100 terms.
• Plot M g vs x. On a single graph, plot the exact growth moment and the
series approximation for 5, 10, and 100 terms.
• Plot −v vs x (100 terms).
• Plot σ vs x along the lower surface of the plate (100 terms).
(f) Including 100 terms in the series, plot −v vs x for the following cases:
• Effects of pressure: On a single graph, plot the curve for the baseline
parameter values, as well as curves for p 0 = 0 and p 0 = −2.
• Effects of growth force: On a single graph, plot the curve for the
baseline parameter values, as well as curves for α = 0 and α = 2.
• Effects of growth moment: On a single graph, plot the curve for the
baseline parameter values, as well as curves for β = 0 and β = −10.
8.5 Consider again the plate problem of Example 8.4. As before, growth is
specified only in the lower half of region 1 (0 ≤ x ≤ d) and the left end is
fixed, but now the right end (x = L) is supported by a roller, rather than a pin
(see Fig. 8.15a). Solve this problem using (a) Fourier series and (b) by finding
the “exact” solution. For L = 10h and g = a = 0.1, plot the deflection
(−v/ h) along the beam for both solutions on the same graph. Plot results for
d/L = 0.25, 0.5, 0.75, and 1.
Hint: For part (b), solve the 4th-order plate equation separately in regions
1 and 2 for the deflections v 1 (x) and v 2 (x), respectively, with the growth
moment being constant in each region. Determine the eight constants of
integration using four boundary conditions and the four continuity conditions
f 1 (d) = f 2 (d), where f = v, θ, M, V . Suggestion: Work in terms of the
dimensionless quantities defined in Eq. (8.36).
8.6 Using the series expansions
sinh x = x + x
3 /6 + . . .
cosh x = 1 + x
2 /2 + . . . ,
show that Eqs. (8.33) and (8.34) yield (8.32) as β → 0 (N → 0).
8.7 Consider a rigid, weightless vertical bar of length L that is pinned at its
lower end. The upper end is supported by a horizontal spring and loaded by
a downward force P (Fig. 8.81). As the bar rotates counter-clockwise, the
load remains vertical, and the tip displaces rightward by the distance . In
addition, the right end of the spring (stiffness k) follows the downward motion
of the tip, so that the spring remains horizontal. Whereas is assumed to be
small in the analysis of Sect. 8.6.4, consider here the “post-buckling” behavior