DESIGN OF AXIALLY LOADED COLUMN

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DESIGN OF COMPRESSION MEMBERS A compression member subjected to pure axial load rarely occurs in practice. All columns are subjected to some moment which may be due to accidental eccentricity or due to end restraint imposed by monolithically placed beams or slabs. A column may be classified as short or long depending on its effective slenderness ratio. A short column has a maximum slenderness ratio of 12. Its design is based on the strength of columns and applied loads. A long column has a slenderness ratio greater than 12. However the maximum slenderness ratio of a column should not exceed 60. A long column is designed to resist the applied loads plus additional induced loads due to its tendency to buckle. Effective Length The effective length of a column is defined as the length between the points of contra-flexure of the buckled column. The code has given two charts to calculate the effective length of columns in a framed structure. For normal usage, the effective length in a given plane may be assumed from the table below assuming idealized conditions. Table: Effective Length of Column Members (IS 456:2000)

No.	Degree of end restraint of member	Effective Length
1	Effectively held in position and restrained against rotation at both ends	0.65L
2	Effectively held in position at both ends and restrained against rotation at one end.	0.80 L
3	Effectively held in position at both ends but not restrained against rotation.	L
4	Effectively held in position and restrained against rotation at one end and at the other end, restrained against rotation but not held in position.	1.20 L
5	Effectively held in position and restrained against rotation at one end and at the other end, partially restrained against rotation but not held in position.	1.50 L
6	Effectively held in position but not restrained against rotation at one end and at the other end, restrained against rotation but not held in position.	2.0 L
7	Effectively held in position and restrained against rotation at one end and at the other end, neither restrained against rotation nor held in position.	2.0 L

  Where L is the unsupported length of a compression member DESIGN ASPECT The relationship between stress and strain in concrete is assumed to be parabolic. Maximum compressive stress is equal to or 0.446. The tensile strength of concrete is ignored. The stress in reinforcement is derived from the representative stress-strain cure for the type of steel. Figure 1: Stress distribution across the cross section of a column The stress distribution lines for purely axial compression case and for limiting case when the strain varies from 0.0035 at one edge to zero at the opposite edge as shown in figure (1). These lines intersect each other at a depth of from the highly compressive edge. The point is assumed to act as a fulcrum for the strain distribution line when the neutral axis lies above the section as shown in figure (1). This leads to the assumption that the maximum compressive strain at the highly compressed extreme fibre in concrete is 0.002 minus 0.75 times the strain at the least compressed fibre. The maximum compressive strain at the highly compressed extreme fibre, when partly the section is in torsion is taken as 0.0035. SHORT COLUMN UNDER AXIAL COMPRESSION The factored axial load, is given by the equation , Where = area of concrete and, = area of longitudinal reinforcement of columns. This equation can be recast as Where P = percentage of reinforcement. Design charts are prepared based on this equation. REINFOCEMENT There are two kinds of reinforcement in a column, longitudinal and transverse reinforcement. The purpose of transverse reinforcement is to hold the vertical bars in position, providing lateral support so that individual bars can not buckle outward and split the concrete. 1. Longitudinal Reinforcement in columns (as per IS 456:2000) a) The cross-sectional area of longitudinal reinforcement shall be not less than 0.8 percent nor more than 6 percent of the gross cross-sectional area of the column. Note: the use of 6 percent reinforcement may involve practical difficulties in placing and compacting of concrete, hence lower percentage is recommended. Where bars from the columns below have to be lapped with those in the column under consideration, the percentage of reinforcement steel shall usually not exceed 4 percent. b) In any column that has a larger cross-sectional area than that required to support the load, the minimum percentage of steel shall be based upon the area of concrete required to resist the direct stress and not upon the actual area. c) The minimum number of longitudinal bars provided in a column shall be four in rectangular columns and six in circular columns. d) The bars shall not be less than 12mm in diameter. e) A reinforced concrete column having helical reinforcement shall have at least six bars of longitudinal reinforcement within the helical reinforcement. f) In a helically reinforced columns, the longitudinal bars shall be in contact with the helical reinforcement and equidistant around its inner circumference. g) Spacing of longitudinal bars measured along the periphery of the column shall not exceed 300mm. h) In case of pedestals in which the longitudinal reinforcement is not taken into account in strength calculations, nominal reinforcement not less than 0.15 percent of the cross-sectional area shall be provided. Note: Pedestal is a compression member, the effective length of the which does not exceed three times the least lateral dimension. (2) Transverse Reinforcement in columns: (a) A reinforced compression member shall have transverse reinforcement or helical reinforcement so disposed that every longitudinal bar nearest to the compression face has effective lateral support against buckling subject to provisions in (b) below. The effective lateral support is given by transverse reinforcement either in the form of circular rings capable of taking up circumferential tension or by polygonal links (lateral ties) with internal angles not exceeding . The ends of the transverse reinforcement shall be properly anchored. (b) Arrangement of transverse reinforcement:

1. If the longitudinal bars are not spaced more than 75mm on either side, transverse reinforcement need only to go round corner and alternate bars for the purpose of providing effective lateral supports. (figure 2)
2. If the longitudinal bars spaced at a distance of not exceeding 48 times the diameter of the tie are effectively tied in two directions, additional longitudinal bars in between these bars need to be tied in one direction, by open ties (figure 3)
3. Where the longitudinal reinforcing bars in a compression member are placed in more than one row, effective lateral support to the longitudinal bars at the inner rows may be assumed to have been provided, if
   1. Transverse reinforcement is provided for the outer row and
   2. No bar of the inner row is closer to the nearest compression face than three times the diameter of the largest bar in the inner row (figure 4)
4. Where the longitudinal bars in a compression member are grouped (not in contact) and each group adequately tied with transverse reinforcement, the transverse reinforcement for the compression member as a whole may be provided on the assumption that each group is a single longitudinal bar for purpose of determining the pitch and diameter of the transverse reinforcement. The diameter of such transverse reinforcement need not however exceed 20mm (figure 5).

(c) Pitch and diameter of lateral ties (as per IS 456:2000)

1. Pitch – The pitch of transverse reinforcement shall be not more than the least of the following distances:

a. The least lateral dimension of the compression member b. Sixteen time the smallest diameter of the longitudinal reinforcement bar to be tied c. Forty-eight times the diameter of the transverse reinforcement.

1. Diameter – The diameter of the polygonal links or lateral ties shall be not less than one-fourth of the diameter of the largest –longitudinal bar, and in no case less than 5mm.

(d) Helical Reinforcement (as per IS 456: 2000)

1. Pitch – Helical reinforcement shall be of regular formation with the turns of the helix spaced evenly and its ends shall be anchored properly by providing one and a half extra turns of the spiral bar. Where an increased load on the column on the strength of helical reinforcement is allowed for, the pitch of helical turns shall be not more than 77 mm nor more than one-sixth of the core diameter of the column, nor less than 25mm, nor less than 3 times the diameter of the steel bar forming the helix. In other cases, the requirements of transverse reinforcement shall be complied with.
2. Diameter – The diameter of the helical reinforcement shall be in accordance with para (c) above.

Figure 2 Figure 3 Figure 4 Figure 5 Short Columns with Helical Reinforcement: The permissible load for columns with helical reinforcement satisfying the following shall be 1.05 times the permissible load for similar member with lateral ties or rings. Ratio of the volume of helical reinforcement to the volume of the core should not be less than Where = gross area of the section = area of the core of the helically reinforced column measured to the outside diameter of the helix. = characteristic compressive strength of the concrete = characteristic strength of the helical reinforcement but not exceeding 415 .