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2E4: SOLIDS & STRUCTURES Lecture 15 Dr. Bidisha Ghosh Notes: lids & Structures

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Buckling of Columns What is buckling? Buckling is a large deformation produced under compressive load in a direction or plane normal to the direction of application of the load. Buckling is a form of instability, it occurs suddenly with large changes in deformation but little change in loading. For this reason it is a dangerous phenomenon that must be avoided in structural design. Buckling is not failure through yielding. Due to the shape of a structural element it can buckle under a load below the ultimate strength. Whether a column will buckle or not depends on the length of the column.

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Long Columns Long columns usually fail by elastic buckling. The failure load is below ultimate strength of the material. The Euler formula is used to calculate failure strength in long columns.

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Short Columns Short columns generally don’t fail by elastic buckling. The failure stress is close to yield stress of the material. The true short-columns do not have much practical application.

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How do we know which is a short/long column?

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Classification of Columns CD is Euler curve showing behaviour of long columns Euler formula should not be used for slenderness ratio <120 For slenderness ratio less than 30, columns are called short columns. They fail by yielding or crushing, generally not buckling.

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End Conditions What was the pinned-pinned condition mentioned in connection with Slenderness ratio? Euler derived all formulae related to column buckling for pinned-pinned condition and later for other end support conditions, those formula were altered by using a constant, C Instead of the actual length of the columns a new length termed as the ‘effective length’ was used. Effective length,

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Different End-Conditions Check the load required to buckle!

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Failure Stress

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The radius of gyration provides a measure of the resistance provided by a cross-section to lateral buckling. The radius of gyration is not a physical entity in itself. It is a relationship derived to make prediction of column behaviour easy. The radius of gyration is related to the size and shape of the cross-section. Columns will buckle in the direction of least cross- sectional stiffness (minimum value of I ). A rectangular column will buckle in the direction of the smaller dimension in cross-section. A square column cross-section will be equally prone to buckling in both x and y directions. This is because the cross section will offer equal resistance to buckling in the direction x and y. Cross-sectional Shape & Buckling

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Example Determine the thickness of a round steel tubular strut, 375mm external diameter and 2.5m long, pin-jointed at the ends, to withstand an axial load of 39000kN. E=200GPa.

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Moment curvature equation A quick touch on bending before learning buckling! 1. Moment at any section 2. Moment-curvature equation: 3. Buckling is an effect of combined compression & bending

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Compression on A Slender Column From knowledge of bending, Solve this equation… So,

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Compression on A Slender Column For any buckling to happen the second condition has to be true and that means,

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