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**Chapter 5 **

**Columns**

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 = σ_{sc} A _{sc} + σ_{cc} . A_{ c}

A= A_{sc} + A_{cc}

Where, A_{c} = Area of concrete σ_{sc} = Stress in compression steel

σ_{cc} = Stress in concrete

A = Total area A_{sc} = Area of compression steel

4. Load carrying capacity for long column

P = C_{r }(σ_{sc} A _{sc} + σ_{cc} . A_{ c} )

where,C_{r} = Reduction factor

or

where,l_{eff} = Effective length of column

B = Least lateral dimension

i_{min} = 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 (σ_{sc} A _{sc} + σ_{cc} . A_{ c}) — For short column

P = 1.05C_{r} (σ_{sc} A _{sc} + σ_{cc} . 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, d_{c} = core diameter = dg – 2 × clear cover to helical reinforcement

A_{g} = gross area =

d_{g} = gross diameter

V_{h} = Volume of helical reinforcement in unit length of column

φ_{h} = diameter of steel bar forming the helix

V_{c }= A_{c} × 1

d_{h} = centre of centre dia of helix = d_{g} – 2 clear cover – φ_{h}

**6. Some other Indian Standards Recommendation **

**Longitudinal reinforcement**

(a) Minimum area of steel = 0.8% of the gross area of column

(b) Maximum area of steel

(i) when bars are not lapped A_{max}= 6% of the gross area of column

(ii) when bars are lapped A_{max}= 4% of the gross area of column **Minimum number of bars for reinforcement**For rectangular column — 4 For circular column — 6 - Minimum diameter of bar = 12 mm
- Maximum distance between longitudinal bar = 300 mm
**Pedestal :**It is a short length whose effective length is not more than 3 times of least lateral dimension. In case of Pedestal minimum percentage of steel = 0.15%. **Transverse reinforcement (Ties)**

φ = maximum

where, φ_{main} = dia of mainbar

φ = dia of bar for transverse reinforcement

**Pitch (p)**

φ = minimum

where, φ_{min} = minimum dia of bar

- Slenderness limit

(i) Unsupported length between end restrains > 60 times least lateral dimension.

(ii) If in any given plane one end of column is unrestrained than its unsupported length >

All column should be designed for a minimum eccentricity of e_{min} = maximum

20mm

**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

e_{min} = 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 P_{u} = 0.4 f_{ck} A_{c} + 0.67 f_{y} A_{sc}

where, P_{u} = axial load on the column

**(4) Short axially loaded column with helical reinforcement**

Strength of the column is increased by 5% P_{u} = 1.05 (0.4 f_{ck} A_{c} + 0.67 f_{y} A_{sc})

**(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.

length >

**(6) Concentrically Loaded Columns**

where e = 0, i.e., the column is truly axially loaded.

P_{uz} = 0.45 f_{ck} A_{c} + 0.75 f_{y} A_{ac }

Where P_{uz} = 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

1.P_{u }

2. (M_{ux} < M_{u min}) where Mu(min) = P_{u}e_{(min)}]

3. (M_{ux} < M_{u min})

Then, As per IS 456 : 2000

To consider the slenderness effect additional moments are added with given design moments.

I_{ex }= effective length in respect of major axis.

I_{ey} = 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**

1. P_{u}

2. (M_{ux} + M_{ax})

3. (M_{uy} + M_{ay})

The values M_{ax} and M_{ay} may be multiplied by a factor

where, P_{v} = axial load or compression member

P_{uz} = ultimate load carrying capacity of column

P_{b} = 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.

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