Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

P< Pcr
B
x
P=Pcr
B
B y
P
B
B y
x
P
B
L −x
681
THE EFFECT OF ENd
CONdITIONS ON
COLuMN buCkLINg
L
v(x)
x
L
V
M
P
O
v
x
A
y, v
A
A y
A
y, v
(a) Fixed–pinned
column
(b) Buckled column in
neutral equilibrium
FIGURE 16.6 Buckling of a fixed–pinned column.
M A
A x
(c) Free-body diagram
of entire column
(d) Free-body diagram
of partial column
From Equation (10.1), the moment–curvature relationship (assuming small deflections)
can be expressed as
M
EI dv 2
= (b)
2
dx
which can be substituted into Equation (a) to give
EI d 2
v + Pv = By( L − x)
2
dx
(16.9)
By dividing both sides of Equation (16.9) by EI and again substituting the term k 2 = P/EI,
the differential equation for the fixed–pinned column can be expressed as
2
d v B
2 y
+ k v = −
dx EI ( L x)
(16.10)
2
Equation (16.10) is a nonhomogeneous second-order ordinary differential equation with
constant coefficients that has boundary conditions v(0) = 0, v′(0) = 0, and v(L) = 0.
Solution of the Differential Equation
The general solution of Equation (16.10) is
B
v = C sinkx + C cos kx + y
1 2 ( −
P L x )
(16.11)