8.6 Neural Development
475
where N g is the growth force, and A = bh is the cross-sectional area of the beam.
Thus, P = aEA.
In the buckled configuration, moment equilibrium for an arbitrary beam section
(Fig. 8.59c) gives the bending moment
M = P v,
and Eq. (8.59) takes the form
v
+ β
2 v = 0,
where
β
2
=
P
EI
.
The general solution of this equation is
v(x) = c 1 sin βx + c 2 cos βx,
and the boundary conditions v(0) = v(L) = 0 give c 2 = 0 and
c 1 sin βL = 0.
(8.66)
The conditions of zero moment at the pinned supports involve second derivatives of
v(x), so they cannot be used to determine the constants of integration for a secondorder equation. However, because v = 0 at both ends, these boundary conditions
already are satisfied by the moment equilibrium relation M = P v.
Equation (8.66) yields c 1 = 0 (v = 0) or sin βL = 0, which has roots
βL = nπ,
n = 1, 2, . . . .
(8.67)
With β defined above, the eigenvalues are
P n =
n 2 π 2 EI
L 2 ,
n= 1, 2, . . . ,
(8.68)
and the corresponding mode shapes are
v n (x) = c 1 sin βx = c 1 sin
nπ x
L
.
(8.69)
The minimum P n corresponds to n = 1, giving the critical load
P cr =
π 2 EI
L 2 = a cr (EA),
(8.70)