8.6 Neural Development
477
If n is an integer, this expression satisfies all boundary conditions, and substitution
into Eq. (8.58) yields
P =
π 2 EI
L 2 n
2
+
kL 2
π 2
1
n 2 .
(8.72)
To determine the value of n corresponding to the minimum value of P , we set
∂P
∂n
= 0,
which gives
n =
L
π
k
EI
1/4
.
(8.73)
Clearly, this relation does not produce an integer in general, so the actual value of n
is the closest integer to the computed value (Hetenyi 1971). Then, substitution into
Eq. (8.72) gives P cr = (EA)a cr .
For an infinite beam on an elastic foundation, n can take any value, even noninteger. In this case, substituting (8.73) into (8.72) gives
(P cr ) ∞ = 2
√
kEI .
(8.74)
Note that Eq. (8.71) can be written in the form v = c 1 sin 2πx//, where
=
2L
n
= 2π
EI
k
1/4
(8.75)
is the wavelength of the buckling pattern.
To help us understand the solution, we follow Hetenyi (1971) and define the
dimensionless quantities
α =
kL 4
EI
,
¯
P cr =
P cr
(P cr ) 0
,
where (P cr ) 0 = π 2 EI /L 2 is the critical load for a pinned-pinned beam without a
foundation [k = 0; see Eq. (8.70)]. Equation (8.72) then can be written in the form
α
2
= π
4 n
2 ( ¯
P cr − n
2 ),
which is plotted in Fig. 8.60b for the first four modes (n = 1, 2, 3, 4), as well as
for an infinite beam where α = π 2 ( ¯
P cr ) ∞ /2. Given the geometric and material
properties, we can compute α and use the graph to find which curve gives the