BFC 20903 Mechanics of Materials Chapter 5 Compression

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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