COLUMNS and STRUTS · The column is perfectly elastic, homogenous and isotropic. COLUMNS AND STRUTS...

Prof. S. K. PrasadDepartment of Civil Engineering

Sri Jayachamarajendra College of Engineering

JSS Science and Technology University

Mysuru – 570 006

COLUMNS and STRUTS

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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Slender Column ?

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Typical failure of columns

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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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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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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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SLENDERNESS RATIO (λ)

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CLASSIFICATION OF COLUMNS

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Short Compression

Member

Short Column

Essentially fails by bulging or crushing

and not by buckling

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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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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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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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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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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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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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--------(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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Boundary condition (i)

y = 0 at x =0

From eq. (3)

--------(3)

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--------(3)

y = 0 at x = L

From eq. (3)

Here either or

Boundary condition (ii)

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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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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

Best of Luck

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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 · The column is perfectly elastic, homogenous and isotropic. COLUMNS AND STRUTS...

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