8.6 Neural Development
473
β
2
=
P
EI
.
(8.61)
The general solution (particular + complementary) of this equation is
v(x) = v 0 + c 1 sin βx + c 2 cos βx.
For a second-order differential equation, the boundary conditions should involve
derivatives of the dependent variable lower than 2. 16 The conditions M = V = 0
at the upper end do not qualify, as they involve second and third derivatives of v,
respectively. At the fixed end, the conditions v(0) = v (0) = 0 give c 1 = 0 and
c 2 = −v 0 , and the solution becomes
v(x) = v 0 (1 − cos βx),
(8.62)
which defines the shape of the beam in neutral equilibrium (see Fig. 8.58b).
To compute the critical load, we enforce the condition v(L) = v 0 to obtain the
relation
v 0 cos βL = 0.
The trivial solution v 0 = 0 yields the unbuckled state. For v 0 = 0, the roots of
cos βL = 0 are
βL =
nπ
2
,
n= 1, 3, . . . .
Substitution into (8.61) gives the eigenvalues
P n =
n 2 π 2 EI
4L 2 ,
n= 1, 3, . . . ,
(8.63)
and (8.62) yields the corresponding eigenfunctions (also called shape functions)
v n (x) = v 0
1 − cos
nπ x
2L
,
n= 1, 3, . . .
(8.64)
As in the rigid bar example, the amplitude v 0 is arbitrary (but small).
The eigenfunction v n defines an equilibrium configuration associated with the
load P n (n = 1, 3, . . .). The smallest P n corresponds to n = 1, giving the critical
load
P cr =
π 2 EI
4L 2 ,
16 In general, for a differential equation of order n, boundary conditions should involve derivatives
no higher than n − 1.
473
β
2
=
P
EI
.
(8.61)
The general solution (particular + complementary) of this equation is
v(x) = v 0 + c 1 sin βx + c 2 cos βx.
For a second-order differential equation, the boundary conditions should involve
derivatives of the dependent variable lower than 2. 16 The conditions M = V = 0
at the upper end do not qualify, as they involve second and third derivatives of v,
respectively. At the fixed end, the conditions v(0) = v (0) = 0 give c 1 = 0 and
c 2 = −v 0 , and the solution becomes
v(x) = v 0 (1 − cos βx),
(8.62)
which defines the shape of the beam in neutral equilibrium (see Fig. 8.58b).
To compute the critical load, we enforce the condition v(L) = v 0 to obtain the
relation
v 0 cos βL = 0.
The trivial solution v 0 = 0 yields the unbuckled state. For v 0 = 0, the roots of
cos βL = 0 are
βL =
nπ
2
,
n= 1, 3, . . . .
Substitution into (8.61) gives the eigenvalues
P n =
n 2 π 2 EI
4L 2 ,
n= 1, 3, . . . ,
(8.63)
and (8.62) yields the corresponding eigenfunctions (also called shape functions)
v n (x) = v 0
1 − cos
nπ x
2L
,
n= 1, 3, . . .
(8.64)
As in the rigid bar example, the amplitude v 0 is arbitrary (but small).
The eigenfunction v n defines an equilibrium configuration associated with the
load P n (n = 1, 3, . . .). The smallest P n corresponds to n = 1, giving the critical
load
P cr =
π 2 EI
4L 2 ,
16 In general, for a differential equation of order n, boundary conditions should involve derivatives
no higher than n − 1.