CHAPTER: EIGHT
COLUMN THEORY
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BUCKLING
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 A long column subjected to direct load
deflects in lateral direction, is known as
buckling
 The maximum limiting load at which the
column tends to have lateral displacement
or tends to buckle is called buckling or
crippling load.
 Buckling takes place about the axis having
minimum radius of gyration or least
moment of inertia.
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 Radius of Gyration: Defined as the distance from a given
reference where the whole mass or area of the body is assumed
to be concentrated.
 Effective Length: Length between points of inflection in
deflection curve
SLENDERNESS RATIO
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 The longer a compression member becomes for the same
cross section, the greater becomes its tendency to buckle
and the smaller becomes the load it will support.
 The tendency of a member to buckle is usually measured
by its slenderness ratio.
 The slenderness ratio of a member is the ratio of the
effective length to the minimum radius of gyration.
CLASSIFICATION OF COLUMN
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CLASSIFICATION OF COLUMN
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SHORT
 usually fail by crushing or yielding.
 column shortens in the direction of applied force
 shortening of the column continues until the column
squashes
INTERMEDIATE
 fail by inelastic buckling (yielding and buckling)
 Failure occurs after the extreme fibers have reached
the yield point and the others remain elastic.
LONG
 fail by elastic buckling
 At the point of failure, the stress in a long column
will not exceed the proportional limit
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Analysis of Short Column (Eccentrically Loaded)
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Analysis of Long Column: Euler’s Theorem
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Assumptions:
Column is initially straight and of uniform cross section so EI
is constant.
Material of column is homogeneous, isotropic and obey’s
Hooke’s law.
Compressive load on column is axial.
Weight of column is neglected.
Direct stress due to compressive force is very small as
compared to bending stress due to buckling of column.
Column fails only by buckling.
Pin joints are frictionless and fixed ends are perfectly rigid.
Sign convention for Bending Moment
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1. Euler’s Formula When Both Ends Are Hinged or
Pinned
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2. One End is Fixed and Other End is Free
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3. One End Fixed and other End Hinged
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4. Both Ends Fixed
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Limitations of Euler’s Formula
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Note:
As λ → 0, critical stress tends to infinity. But this cannot happen. Before critical stress
tends to infinity, material reaches crushing stress and gets crushed. So, limiting value of
critical stress is crushing stress.
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