A. Working Stress Method 1. Slenderness ratio ( λ)
; If λ ≤ 12 then the column is short, if λ > 12 then the column is long.
3. Load carying capacity for short column
P = σsc A sc + σcc . A c
A= Asc + Acc
Where, Ac = Area of concrete σsc = Stress in compression steel
σcc = Stress in concrete
A = Total area Asc = Area of compression steel
4. Load carrying capacity for long column
P = Cr (σsc A sc + σcc . A c )
where,Cr = Reduction factor
where,leff = Effective length of column
B = Least lateral dimension
imin = Least radius of gryation and
where, I = Moment of inertia and A = Cross sectional area
5. Column with helical reinforcement
Strength of the column is increased by 5%
P = 1.05 (σsc A sc + σcc . A c) — For short column
P = 1.05Cr (σsc A sc + σcc . A c ) – For long column
Helical reinforcement is provided only for circular columns.
(i) Diameters of helical reinforcement is selected such that
(ii) Pitch of helical reinforcement: (p) (a) Fig. 39
where, dc = core diameter = dg – 2 × clear cover to helical reinforcement
Ag = gross area =
dg = gross diameter
Vh = Volume of helical reinforcement in unit length of column
φh = diameter of steel bar forming the helix
Vc = Ac × 1
dh = centre of centre dia of helix = dg – 2 clear cover – φh
6. Some other Indian Standards Recommendation
φ = maximum
where, φmain = dia of mainbar
φ = dia of bar for transverse reinforcement
φ = minimum
where, φmin = minimum dia of bar
All column should be designed for a minimum eccentricity of emin = maximum
B. Limit Stress Method
(1) Assumptions: All assumption for beams will be valid for column in addition to it there are two more assumptions.
(i) The maximum compressive strain in concrete in axial compression is take as 0.002
(ii) The maximum compresive stain at the highly compressed extreme fibre in concrete subjected to axial compression and bending and when there is no tension in the section shall be 0.0035 minus 0.75 times the strain at the least compressed extreme fibre.
(2) Minimum Eccentricity: All column should be designated for a minimum eccentricity of
emin = maximum
(3) Design of Short Columns : When the minimum eccentricity does not exceed 0.05 B or 0.05 D then load carrying capacity of column is given by Pu = 0.4 fck Ac + 0.67 fy Asc
where, Pu = axial load on the column
(4) Short axially loaded column with helical reinforcement
Strength of the column is increased by 5% Pu = 1.05 (0.4 fck Ac + 0.67 fy Asc)
(5) Some others I.S Recommendation
(a) Slenderness limit
(i) Unsupported length between and restrains > 60 times least lateral dimension.
(ii) If in any given plane one end of column is unrestrained than its unsupported.
(6) Concentrically Loaded Columns
where e = 0, i.e., the column is truly axially loaded.
Puz = 0.45 fck Ac + 0.75 fy Aac
Where Puz = Ultimate load carrying capacity of column This formula is also used for member subjected to combined axial load and bi-axial bending and also used when e > 0.05D.
Design of Long Columns
Long column is to be designed for moment + load if given values are
2. (Mux < Mu min) where Mu(min) = Pue(min)]
3. (Mux < Mu min)
Then, As per IS 456 : 2000
To consider the slenderness effect additional moments are added with given design moments.
Iex = effective length in respect of major axis.
Iey = effective length in respect of minor axis.
D = Depth of cross-section at right angle to the major axis.
B = Width of member
Final Design Values
2. (Mux + Max)
3. (Muy + May)
The values Max and May may be multiplied by a factor
where, Pv = axial load or compression member
Puz = ultimate load carrying capacity of column
Pb = axial load corresponding to the condition of maximum compressive strain of 0.0035 in concrete and tensile of 0.002 in outer most layer of tension steel.