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

16.3 The Effect of End conditions
on column Buckling
The Euler buckling formula expressed by either Equation (16.5) or Equation (16.8) was
derived for an ideal column with pinned ends (i.e., ends with zero moment that are free to
rotate, but are restrained against translation). Columns are commonly supported in other
ways, as well and these different conditions at the ends of a column have a significant effect
on the load at which buckling occurs. In this section, the effect of different idealized end
conditions on the critical buckling load for a column will be investigated.
The critical buckling load for columns with various combinations of end conditions
can be determined by the approach taken in Section 16.2 to analyze a column with pinned
ends. In general, the column is assumed to be in a buckled condition and an expression for
the internal bending moment in the buckled column is derived. From this equilibrium equation,
a differential equation of the elastic curve can be expressed by means of the moment–
curvature relationship [Equation (10.1)]. The differential equation can then be solved with
the boundary conditions pertinent to the specific set of end conditions, and from the solution,
the critical buckling load and the buckled shape of the column can be determined.
To illustrate this approach, the fixed–pinned column shown in Figure 16.6a will be
analyzed to determine the critical buckling load and buckled shape of the column. Then,
the effective length concept will be introduced. This concept provides a convenient way to
determine the critical buckling load for columns with various end conditions.
Buckled configuration
The fixed support at A prohibits both translation and rotation of the column at its lower end.
The pinned support at B prohibits translation in the y direction, but allows the column to
rotate at its upper end. When the column buckles, a moment reaction M A must be developed,
because rotation at A is prevented. On the basis of these constraints, the buckled shape of the
column can be sketched as shown in Figure 16.6b. The value of the critical load P cr and the
shape of the buckled column will be determined from analysis of this deflected shape.
Equilibrium of the Buckled column
A free-body diagram of the entire buckled column is shown in Figure 16.6c. Summing
forces in the vertical direction gives A x = P. Summing moments about A reveals that a
horizontal reaction force B y must exist at the upper end of the column as a consequence of
the moment reaction M A at the fixed support. The presence of B y necessitates, in turn, a
horizontal reaction force A y at the base of the column, to satisfy equilibrium of forces in the
horizontal direction.
Next, consider a free-body diagram cut through the column at a distance x from the
origin (Figure 16.6d). We could consider either the lower or the upper portion of the column
for further analysis, but here we will consider the upper portion.
Differential Equation for column Buckling
In the buckled column of Figure 16.6d, both the column deflection v and the internal bending
moment M are shown in their positive directions. From the free-body diagram in
Figure 16.6d, the sum of moments about exposed surface O is
Σ MO = −M − Pv + By ( L − x) = 0
(a)
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