Mechanics of Materials II
(ME323)
Columns
Chapter 11
11.3: Columns with pinned ends
- Introduction
• The behavior of an ideal column compressed by an axial
load P (Figs. a and b) may be summarized as follows:
• If P < Pcr: the column is in stable equilibrium in the
straight position.
• If P = Pcr: the column is in neutral equilibrium in either
the straight or a slightly bent position.
• If P > Pcr: the column is in unstable equilibrium in the
straight position and will buckle under the slightest
disturbance.
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11.3: Columns with pinned ends
- Introduction
• Column with pinned ends: (a) ideal column, (b) buckled
•shape, and (c) axial force P and bending moment M
acting at a cross section
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11.3: Columns with pinned ends
- Differential Equation for Column Buckling
• To determine the critical loads and corresponding
deflected shapes for an ideal pin-ended column, we use
one of the differential equations of the deflection curve of
a beam.
• These equations are applicable to a buckled column
because the column bends as though it were a beam
• The bending-moment equation
• M: Bending moment at any cross section
• ν: Lateral deflection
• EI: Flexural rigidity
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11.3: Columns with pinned ends
- Differential Equation for Column Buckling
• Therefore, from equilibrium of moments about point A,
we obtain
• The differential equation of the deflection curve
• By solving this equation, which is a homogeneous, linear,
differential equation of second order with constant
coefficients, we can determine the magnitude of the
critical load and the deflected shape of the buckled
column.
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11.3: Columns with pinned ends
- Solution of the Differential Equation
• For convenience in writing the solution of the differential
equation, we introduce the notation
• where, k is always taken as a positive quantity. Note that
k has units of the reciprocal of length, and therefore
quantities such as kx and kL are no dimensional.
• Using this notation, we can rewrite Eq. in the form
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11.3: Columns with pinned ends
- Solution of the Differential Equation
• From mathematics we know that the general solution of
this equation is
• where, C1 and C2 are constants of integration
• To evaluate the constants of integration appearing in the
solution, we use the boundary conditions at the ends of
the column; namely, the deflection is zero when x = 0 and
x=L
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11.3: Columns with pinned ends
- Solution of the Differential Equation
• The first condition gives C2 = 0, and therefore
• The second condition gives
• From this equation we conclude that either C1 = 0 or
sin kL = 0.
• We will consider both of these possibilities.
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11.3: Columns with pinned ends
- Solution of the Differential Equation
• Case I: If the constant C1 equals zero.
• The deflection v is also zero , and therefore the column
remains straight.
• In addition, we note that when C1 equals zero, First
Equation is satisfied for any value of the quantity kL.
Consequently, the axial load P may also have any value
[see Eq. 2].
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11.3: Columns with pinned ends
- Solution of the Differential Equation
• Case II:
• The second possibility is given by the following equation,
known as the buckling equation:
• This equation is satisfied when kL= 0, π, 2π,……..
• However, since kL = 0 means that P = 0, this solution is
not of interest.
• Therefore, the solutions we will consider are
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11.3: Columns with pinned ends
- Solution of the Differential Equation
• Case II:
• This formula gives the values of P that satisfy the
buckling equation.
• The values of P given by above Equation are the critical
loads for this column.
• The equation of the deflection curve
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11.3: Columns with pinned ends
- Critical Loads
• The lowest critical load for a column with pinned ends is
obtained when n = 1;
• The corresponding buckled shape (sometimes called a
mode shape) is
• The constant C1 represents the deflection at the
midpoint of the column and may have any small value,
either positive or negative.
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11.3: Columns with pinned ends
- Critical Loads
• Buckled shapes for an ideal column with pinned ends:
(a) initially straight column (b) buckled shape for n = 1,
and (c) buckled shape for n = 2
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11.3: Columns with pinned ends
- Critical Loads
• Buckling of a pinned-end column in the first mode is
called the fundamental case of column buckling.
•The type of buckling described in this section is called
Euler buckling, and the critical load for an ideal elastic
column is often called the Euler load.
• The famous mathematician Leonhard Euler (1707–1783),
was the first person to investigate the buckling of a slender
column and determine its critical load (Euler published
his results in 1744).
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11.3: Columns with pinned ends
- Critical Stress
• For the fundamental case of buckling , the critical stress
is critical load divided by the cross sectional area
• where, I is the moment of inertia for the principal axis
about which buckling occurs.
•This equation can be written in a more useful form by
introducing the notation
• where, r is the radius of gyration of the cross section in
the plane of bending.
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11.3: Columns with pinned ends
- Critical Stress
• The equation for the critical stress becomes
• in which L/r is a non-dimensional ratio called the
slenderness ratio:
• A column that is short and stubby will have a low
slenderness ratio and will buckle at a high stress.
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11.3: Columns with pinned ends
- Critical Stress
• The critical stress is the average compressive stress on the
cross section at the instant the load reaches its critical
value.
• The graph between Critical stress and slenderness ratio
is known as Euler Curve
• For structural steel
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11.3: Columns with pinned ends
- Example 11-2
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11.3: Columns with pinned ends
- Example 11-2
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11.3: Columns with pinned ends
- Example 11-2
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11.3: Columns with pinned ends
- Example 11-2
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11.3: Columns with pinned ends
- Example 11-2
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Columns
- Summary
 Mechanics of Materials by J. M. JERE & B. J. GOODNO, 8th Edition.
(Read Chapter No 11)
• Solve Problems: 11.3-1 , 11.3-4 , 11.3-5
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