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

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

COLUMNS AND STRUTS

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Slender Column ?

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Typical failure of columns

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

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

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.

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

SLENDERNESS RATIO (λ)

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

CLASSIFICATION OF COLUMNS

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Short Compression

Member

Short Column

Essentially fails by bulging or crushing

and not by buckling

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Short Column

Essentially fails by bulging or crushing and not by buckling

P

P

Ductile Material

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Short Column

Essentially fails by bulging or crushing and not by buckling

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Long Column

Essentially fails by buckling and not by crushing

Stress at failure < yield stress

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Long Column

Essentially fails by buckling and not by crushing

Stress at failure < yield stress

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Intermediate Column :

Fails by a combination of crushing and buckling

if

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

CRITICAL LOAD AND BUCKLING

Long column : P – Axial load F – a small test load – lateral direction

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

CRITICAL LOAD AND BUCKLING

Long column : P – Axial load F – a small test load – lateral direction

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

CRITICAL LOAD AND BUCKLING

Long column : P – Axial load F – a small test load – lateral direction

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

CRITICAL LOAD AND BUCKLING

Long column : P – Axial load F – a small test load – lateral direction

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Effective Lengths for some standard cases

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Buckling behaviour for different end conditions

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

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

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

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)

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

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Boundary condition (i)

y = 0 at x =0

From eq. (3)

--------(3)

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

--------(3)

y = 0 at x = L

From eq. (3)

Here either or

Boundary condition (ii)

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….....

COLUMNS AND STRUTS

Strength of MaterialsProgram No. 23

Euler’s Critical Load for Long Columns

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

Best of Luck

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.