COLUMNS and STRUTS
Prof. S. K. Prasad
Department of Civil Engineering
Sri Jayachamarajendra College of Engineering
JSS Science and Technology University
Mysuru – 570 006
Strength of Materials
Program No. 23
Learning Outcome
The students are introduced to
• the concepts of Elastic Stability of Columns
and struts
• Euler’s Theory for critical load in long
columns for different cases
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
COLUMNS AND STRUTS
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
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Strength of Materials
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Slender Column ?
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Typical failure of columns
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Structural members
Columns Compressive forces
and Lengths are large compared to lateral
Struts dimensions
Often subjected to axial forces
Although – loaded eccentrically
Columns are vertical compressive members
Struts are Inclined compressive members
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
Radius of Gyration
It is the distribution of the components of an object
around an axis. It is the perpendicular distance from
the axis of rotation to a point of mass that gives an
equivalent inertia to the original object.
It has the unit of length
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
Effective Length of Column (le)
It is the length of an imaginary column with both ends
hinged and whose critical load is same as that of
actual column with given end conditions.
Note - Material and geometric properties same in
above columns
Effective length depends on its end conditions
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Program No. 23
SLENDERNESS RATIO (λ)
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CLASSIFICATION OF COLUMNS
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Short Column
Essentially fails by bulging or crushing
and not by buckling
Short Compression
Member
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Short Column
Essentially fails by bulging or crushing and not by buckling
P
P
Ductile Material
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Short Column
Essentially fails by bulging or crushing and not by buckling
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Long Column
Essentially fails by buckling and not by crushing
Stress at failure < yield stress
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Long Column
Essentially fails by buckling and not by crushing
Stress at failure < yield stress
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Intermediate Column :
Fails by a combination of crushing and buckling
if
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CRITICAL LOAD AND BUCKLING
Long column : P – Axial load F – a small test load – lateral direction
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Program No. 23
CRITICAL LOAD AND BUCKLING
Long column : P – Axial load F – a small test load – lateral direction
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
CRITICAL LOAD AND BUCKLING
Long column : P – Axial load F – a small test load – lateral direction
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
CRITICAL LOAD AND BUCKLING
Long column : P – Axial load F – a small test load – lateral direction
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Effective Lengths for some standard cases
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Buckling behaviour for different end conditions
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EULER’S THEORY
Theoretical analysis to estimate critical load for
long columns
- Great Swiss mathematician Leonard Euler
(pronounced as Oiler),
- Developed in 1757
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Strength of Materials
Program No. 23
ASSUMPTIONS IN EULER’S THEORY
• The column is long and fails by buckling
• The column is axially loaded
• The column is perfectly straight and the cross
sections are uniform (prismatic)
• The column is initially free from stress
• The column is perfectly elastic, homogenous and
isotropic
COLUMNS AND STRUTS
Strength of Materials
Program No. 23
EULERS CRITICAL LOAD FOR LONG COLUMNS
Case (1) Both ends hinged
Long column with both ends hinged
subjected to critical load P
Bending moment in terms of load P and deflection y is
M=–Py ---------(1)
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Program No. 23
For beams / columns the bending moment is proportional to the
curvature of the beam, which, for small deflection can be expressed as
or --------------(2)
Where E – Young’s modulus, I – Moment of Inertia
Substituting eq.(1) in eq.(2)
or
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Second order differential equation The general solution is of form
--------(3)
Where C1 and C2 are constants
Constants can be evaluated by applying the boundary conditions
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--------(3)
Boundary condition (i)
y = 0 at x =0
From eq. (3)
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--------(3)
Boundary condition (ii) y = 0 at x = L
From eq. (3)
Here either or
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Hence or Here, n =0,1,2,3….....
Taking least significant value of n, i.e. n=1, we have
or
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Program No. 23
Euler’s Critical Load for Long Columns
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Summary
You were introduced to terminologies
• Columns & Struts
• Long, Intermediate & Short Columns
• Slenderness Ratio
• Effective Length of column
• Critical Load
We derived expression for critical load of column
with both ends hinged
COLUMNS AND STRUTS
Best of Luck
Strength of Materials
Program No. 23
Built-up Section
It is a structural member made from individual plates
or tubes or angles riveted / welded / bolted together to
improve its strength and stiffness in steel construction
industry.
COLUMNS AND STRUTS
