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

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

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

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

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

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

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

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

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

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

Columns having various types of supports Strength of Materials

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