columns and struts · the column is perfectly elastic, homogenous and isotropic. columns and struts...
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Prof. S. K. PrasadDepartment of Civil Engineering
Sri Jayachamarajendra College of Engineering
JSS Science and Technology University
Mysuru – 570 006
COLUMNS and STRUTS
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COLUMNS AND STRUTS
Strength of MaterialsProgram 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
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
COLUMNS AND STRUTS
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Slender Column ?
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Typical failure of columns
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Columns
and
Struts
Structural members
Compressive forces
Lengths are large compared to lateral
dimensions
Often subjected to axial forces
Although – loaded eccentrically
Columns are vertical compressive members
Struts are Inclined compressive members
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COLUMNS AND STRUTS
Strength of MaterialsProgram 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
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Note - Material and geometric properties same in
above columns
Effective length depends on its end conditions
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.
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
SLENDERNESS RATIO (λ)
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
CLASSIFICATION OF COLUMNS
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Short Compression
Member
Short Column
Essentially fails by bulging or crushing
and not by buckling
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Short Column
Essentially fails by bulging or crushing and not by buckling
P
P
Ductile Material
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Short Column
Essentially fails by bulging or crushing and not by buckling
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Long Column
Essentially fails by buckling and not by crushing
Stress at failure < yield stress
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Long Column
Essentially fails by buckling and not by crushing
Stress at failure < yield stress
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Intermediate Column :
Fails by a combination of crushing and buckling
if
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
CRITICAL LOAD AND BUCKLING
Long column : P – Axial load F – a small test load – lateral direction
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
CRITICAL LOAD AND BUCKLING
Long column : P – Axial load F – a small test load – lateral direction
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
CRITICAL LOAD AND BUCKLING
Long column : P – Axial load F – a small test load – lateral direction
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
CRITICAL LOAD AND BUCKLING
Long column : P – Axial load F – a small test load – lateral direction
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Effective Lengths for some standard cases
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Buckling behaviour for different end conditions
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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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COLUMNS AND STRUTS
Strength of MaterialsProgram 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
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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 = – P y ---------(1)
EULERS CRITICAL LOAD FOR LONG COLUMNS
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
For beams / columns the bending moment is proportional to the
curvature of the beam, which, for small deflection can be expressed as
Where E – Young’s modulus, I – Moment of Inertia
Substituting eq.(1) in eq.(2)
or
or --------------(2)
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
--------(3)
Constants can be evaluated by applying the boundary conditions
Where C1 and C2 are constants
Second order differential equation The general solution is of form
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Boundary condition (i)
y = 0 at x =0
From eq. (3)
--------(3)
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
--------(3)
y = 0 at x = L
From eq. (3)
Here either or
Boundary condition (ii)
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Hence or
Taking least significant value of n, i.e. n=1, we have
or
Here, n =0,1,2,3….....
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
Euler’s Critical Load for Long Columns
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COLUMNS AND STRUTS
Strength of MaterialsProgram No. 23
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
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Best of Luck
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COLUMNS AND STRUTS
Strength of MaterialsProgram 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.