Theory of Columns Civil Engineering (CE) Notes | EduRev

Civil Engineering SSC JE (Technical)

Civil Engineering (CE) : Theory of Columns Civil Engineering (CE) Notes | EduRev

The document Theory of Columns Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Civil Engineering SSC JE (Technical).
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Chapter 10

THEORY OF COLUMNS

Compression Member : A compression member is a structural member which is straight and subjected to two equal and opposite compressive forces applied at its ends.
Column: It is defined as a vertical member which is fixed at both the ends and subjected to an axial compressive load.Column, is a compression member that is so slender compared to its length that under gradually increasing loads, it fails by buckling at loads considerably less that those required to cause failure by crushing.  A compression member is generally considered to be a long column when its unsupported length is more than 10 times its least lateral dimension.  Column, Stanchion or post is a vertical compression member supporting floors or girders in a building.  Principal rafter is a top chord member in a roof truss.  Boom is the principal compression member in a crane.  Strut is commonly used for compression member in a roof  truss. It may either be in vertical position or in inclined position.

 

MODES  OF  FAILURE  OF  A COLUMN
Elastic instability A column subjected to axial load, may fail under the following modes: (i) Crushing (ii) Buckling (iii) Mixed mode of Buckling and Crushing.

 

Euler's Theory (Buckling failure)  Assumption of Euler's theory: (i) Axis of the column is perfectly straight  when unloaded. (ii) The line of thrust coincides exactly with the unstrained of the column. (iii) Flexural rigidity EI is uniform (iv) Material is isotropic and homogeneous. (v) The buckling value of load is assumed to obtain for all degrees of flexure. (vi) Length is assumed to be too large with respect to cross section.

 

Limitation of Euler's Formula: (i) The critical load that causes buckling depends not on strength of the material, but only on its dimensions and modulus of elasticity. This converts the strut problem from the problem of instability to a problem of stress, of which Euler's theory takes no account. (ii) In order for Euler's formula to be applicable, the critical stress p1 must not exceed the proportional limit.

 

IDEAL END CONSITION & EFFECTIVE LENGTH
Depending upon various combinations of restraints, there may be the following four cases of end condition: End condition Euler's load or critical load Equivalent length (Le)

Case 1 : Both  ends hinged

 Theory of Columns Civil Engineering (CE) Notes | EduRev

Case 2 : One end fixed and other end free

Theory of Columns Civil Engineering (CE) Notes | EduRev

Case 3 : Both ends fixed

 Theory of Columns Civil Engineering (CE) Notes | EduRev
Theory of Columns Civil Engineering (CE) Notes | EduRev

PRACTICAL END CONSITION  AND EFFECTIVE LENGTH FACTORS:
As per IS : 800 - 1984, following table can be used for finding effective length: Effective length of compression members Radius of Gyration and Slenderness Ratio: 

  • The radius of gyration of a section is given by

 Theory of Columns Civil Engineering (CE) Notes | EduRev

where, I = Moment of Inertia A = area of section 

  • Rankine's formula The Rankine's formula is applicable for all columns  ranging form very short to very long struts.

Ranking proposed,

 Theory of Columns Civil Engineering (CE) Notes | EduRev

Where, Pc = fc.A= ultimate load for a strut

 Theory of Columns Civil Engineering (CE) Notes | EduRev

= Euler  crippling load for the standard case After rearranging above relations,

 Theory of Columns Civil Engineering (CE) Notes | EduRev
Where,
Theory of Columns Civil Engineering (CE) Notes | EduRev

= Rankine's constant for a particular material

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