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

Buckled configuration
If the compressive load P is less than the critical load P cr , then the column will remain
straight and will shorten in response to a uniform compressive axial stress σ = P/A. As long
as P < P cr , the column is in stable equilibrium. When the compressive load P is increased
to the critical load P cr , the column is at the transition point between stable and unstable
equilibrium—a situation called neutral equilibrium. At P = P cr , the deflected shape shown
in Figure 16.3b also satisfies equilibrium. The value of the critical load P cr and the shape
of the buckled column will be determined by an analysis of this deflected shape.
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buCkLINg OF PIN-ENdEd
COLuMNS
Equilibrium of the Buckled column
A free-body diagram of the entire buckled column is shown in Figure 16.3c. Summing
forces in the vertical direction gives A x = P, summing moments about A gives B y = 0, and
summing forces in the horizontal direction gives A y = 0.
Next, consider a free-body diagram cut through the column at a distance x from the
origin (Figure 16.3d). Since A y = 0, any shear force V acting in the horizontal direction on
the exposed surface of the column in this free-body diagram must also equal zero in order
to satisfy equilibrium. Consequently, both the horizontal reaction A y and a shear force V can
be omitted from the free-body diagram in Figure 16.3d.
Differential Equation for column Buckling
In the buckled column of Figure 16.3d, both the column deflection v and the internal bending
moment M are shown in their positive directions. As defined in Section 10.2, the bending
moment M creates positive curvature. From the free-body diagram in Figure 16.3d, the
sum of moments about A is
Σ MA = M + Pv = 0
(a)
From Equation (10.1), the moment–curvature relationship (assuming small deflections)
can be expressed as
M
= EI d 2
v
(b)
2
dx
Equation (b) can be substituted into Equation (a) to give
EI d 2
v + Pv = 0
2
dx
(16.1)
Equation (16.1) is the differential equation that dictates the deflected shape of an ideal
column. This equation is a homogeneous second-order ordinary differential equation with
constant coefficients that has boundary conditions v(0) = 0 and v(L) = 0.
Solution of the Differential Equation
Established methods are available for the solution of equations such as Equation (16.1). To
use these methods, Equation (16.1) is first simplified by dividing by EI to obtain
2
d v P
+ =
dx EI v 0
2