BFC 20903 (Mechanics of Materials)
Chapter 5: Compression Member
Shahrul Niza Mokhatar
shahruln@uthm.edu.my
Shahiron Shahidan
shahidan@uthm.edu.my
Chapter Learning Outcome
1. Determine the type of failure in compression
member
2. Determine the shape of buckling in
compression member
3. Analyse the compression member using
Euler’s theory and Secant formula
BFC 20903 (Mechanics of Materials)
Shahrul Niza Mokhatar (shahrul@uthm.edu.my
Introduction
• The selection of the column is often a very critical part of the design of
structure because the failure of the column usually has catastrophic
effects.
– If a column is long compared to its width, - fail in buckling (bending & deflection
laterally).
– The buckling may be either elastic or inelastic depends upon the slenderness of the
column.
• Column - vertical member carries compressive axial loads.
• The compressive axial load can be applied at the centroid and offset from
centroid.
Types of Failures Column
• Short column = when the cross-section large as compared to its height. It
fails due to ‘crushing’ of column material.
• long/slender column = when the cross-section small as compared to
its height. It fails due to ‘buckling’.
Critical Buckling Load – Euler’s Theory
• The maximum axial load that a column can support when it is on the
verge of buckling is called the critical buckling load, Pcr.
– To derive the Pcr, several assumption can be made:
• Column is “ideal column” - perfectly straight, made of homogeneous
material and the load is applied through the centroid of the cross
section.
• No lateral loads act along the height of column
• Material behaves within elastic region or ideal rigid-plastic or elasticplastic behavior
Effective length and support condition
Limitation of Euler’s Theory / Critical stress, σcr
• Euler’s formula can be used to determine the buckling load since the stress in
the column remains elastic.
• (L/r) - slenderness ratio, buckling will occur about the axis when the ratio gives
the greatest value. Measurement of the column’s flexibility.
• The graph below is used to identify the (L/r) for the column made of a structural
steel.
For a steel column if (L/r)s ≥ 89 , Euler’s formula
can be used to determine the buckling load since
the stress in the column remains elastic. But if
the (L/r)s < 89, the column’s stress will exceed
the yield point and the Euler formula is not valid
in this case.
Example 1
Example 2
Exercise
• Due to this condition of bracing, the column will buckle with
different axis namely are x and y axis.
Secant Formula
• The Euler formula was derived with the assumption;
i) The load,P is always applied through the centroid of the column’s cross
sectional area and;
ii) The column is perfectly straight.
• This is actually quite unrealistic since manufactured columns are never perfectly
straight. In actual condition, column never suddenly buckle, instead they begin to
bend slightly upon the application of the load.
• The actual criterion for load application will be limited either to a specified
deflection of the column or by not allowing the maximum stress exceed the
allowable stress in the column.
To investigate this effect load, P is
applied to the column at a short
eccentric distance, e from the
centroid of the cross section.
Secant Formula
Example 3
a) Determine the allowable eccentric load, P that can be applied.
Perry-Robertson Formula
• The formula used for structural steelwork is the Perry-Robertson formula that
represented as the average end stress to cause yield in a strut.
Rankine-Gordon Formula
Assignment
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