1. Unit-5. Torsion in Shafts and Buckling of Axially Loaded Columns
Lecture Number-4
Mr. M.A.Mohite
Mechanical Engineering
S.I.T., Lonavala
Strength of Materials

2. Buckling of Axially Loaded Columns …Contd
CHAPTER OBJECTIVES
Discuss the behavior of columns.
Discuss the buckling of columns.
Determine the axial load needed to buckle an ideal column.
Analyze the buckling with bending of a column.
CHAPTER OUTLINE
Critical Load Calculations for
1. Ideal Column with Pin Supports
2. Columns Having Various Types of Supports.
Strength of Materials

3. What is Buckling? Members under compression are susceptible to
Buckling means Loss of stability.
Axial loads cause lateral deformations
(bending-like deformations)
Axial force that causes Buckling is called Critical Load.
It is associated to the column strength
Critical Load, Pcr depends on
Length of member
Material Properties
Section Properties
Strength of Materials

4. Ideal Column With Pin Supports
To determine critical load and buckled shape of column,
we apply Equation,
Recall that this eqn assume the slope of the elastic curve is small
deflections occur only in bending.
The material behaves in a linear-elastic manner
The column bends in a
single plane.
Strength of Materials

5. Euler’s Formula for Long Column
Y= Lateral Deflection for Column with both Ends
Pinned, The Bending Moment due to crippling load
Column with both Ends
Pinned
Strength of Materials

6. Euler’s Formula for Long Column..contd
Strength of Materials

7. After deriving, we get Critical load
Ideal Column With Pin Supports
After deriving, we get Critical load
Smallest value of P is obtained for n = 1, so critical load for column is
Strength of Materials

8. Ideal Column With Pin Support
A column buckle about principal axis x-section
having the least moment of inertia (weakest axis).
For example, the meter stick shown will buckle about the a-a axis and not the b-b axis.
Thus, circular tubes made excellent columns
Square tube or those shapes
having Ix ≈ Iy are selected for columns.
Strength of Materials

9. Slenderness Ratio Expressing I = Ak2 , A is sectional area of column
k is radius of gyration of sectional area.
cr = critical stress, an average stress just before
column buckles. This stress is an elastic stress
and therefore cr  Y
L = unsupported length of pinned-end columns.
k = smallest radius of gyration of column, determined from k = √(I/A
Strength of Materials

10. Slenderness ratio…. contd
The ratio L/k is known as the slenderness ratio.
Buckling occur where this ratio greatest value.
It is a measure of the column’s flexibility.
Buckling occur about axis where ratio greatest.
Columns are long slender members.
Columns subjected to axial loads.
Strength of Materials

11. Columns having various types of supports
Strength of Materials

12. Effective length of Column,
Many design codes provide column formulae
Use a dimensionless coefficient K,
known as the effective-length factor.
Effective length is the unsupported length of the column exposed to buckling
Thus, Euler’s formula
Here (KL/r) is the column’s effective-slenderness ratio.
Strength of Materials