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Design of reinforced concrete columns

Type of columns

Failure of reinforced concrete columns

Short column or Long column?

ACI definition

Where k is slenderness factor, l_u is unsupported length,  and r is radius of gyration.  M1 and M2 are the smaller and larger end moments. The value, (M₁/M₂) is positive  if the member is bent in single curve, negative if the member is bent in double curve.

Determine the slenderness factor, k

The slender factor, k should be determined graphically from the Jackson and Moreland Alignment charts.

(Charts will be added later)

where y = å (E_cI_c/l_c) of column /å (E_bI_b/l_b) of beam, is the ratio of effective length factors.  

E_c and E_c are younger modulus of column and beams.

l_c and l_c are unbraced length of column and beams.

The cracked  moment of inertia, I_c is taken as 0.7 times gross moment of column and I_b is taken as 0.35 times gross moment of inertia of beam.

Alternatively, k can be calculated as follows:

1. For braced frame with no sway, 

k can be taken as the smaller value of the two equations below.

k = 0.7 + 0.05 (y_A+y_B) £ 1,

k = 0.8 + 0.05 (y_min) £ 1

y_A and y_B are the y at both ends, y_min is the smaller of the two y values.

2. For unbraced frame with restrains at both ends, 

For y_m  < 2

k =  [(20- y_m)/20] Ö(1+y_m)

For y_m  ³ 2

k = 0.9 Ö(1+y_min)

y_m is the average of the two y values.

2. For unbraced frame with restrain at one end, hinge at the other.

k = 2.0 + 0.3 y

y is the effective length factor at the restrained end.

Beam information:

Beam size: b = 18 in, h = 24 in

Beam unsupported length: l_b = 30 ft

Concrete strength: 4000 psi

Young's modulus, E_b = 57 Ö4000 = 3605 ksi

Moment of inertia of beam: I_b = 0.35bh³/12 = 7258 in⁴.

Column information:

Square Column: D = 18 in, unsupported length l_c =10 ft

Concrete strength: 5000 psi

Young's modulus: E_c = 57 Ö5000 = 4030 ksi

moment of inertia of column: I_c = 0.7D⁴/12 = 6124 in⁴.

Column top condition:

There are beams at both sides of column at top of column, no column stop above the beams

The effective length factor: y_A = (E_cI_c/l_c) /[2 (E_bI_b/l_b)] = 1.4

Column bottom condition:

There are beams at both sides of column at bottom of column and a column at bottom level

The effective length factor: y_A = [2 (E_cI_c/l_c)] / [2 (E_bI_b/l_b)] = 2.8

From chart:

If the column is braced: k » 0.84

If the column is unbraced: k » 1.61

From equation

If the column is braced: 

k = 0.7 + 0.05 (y_A+y_B) = 0.91 

k = 0.8 + 0.05 (y_min) = 0.92

If the column is unbraced: y_m = (y_A+y_B)/2 = 2.12

k = 0.9 Ö(1+y_min) = 1.6

Design of reinforced concrete columns

Column ties and spiral

ACI code requirements for column ties

1. No. 3 ties for longitudinal reinforcement no. 10 bars or less, no. 4 ties for no. 11 bars or larger and bundled bars.
2. Tie spacing shall not exceed 16 diameter of longitudinal bars, 48 diameters of tie bars, nor the least dimension of column.
3. Every corner bar and alternate bars shall have lateral tie provide the angle shall not exceed 135 degree.
4. No longitudinal bar shall be spacing more than 6 inches without a lateral tie.

ACI code requirements for spiral

1. Sprial shall be evenly space continuous bar or wire, no. 3 or larger.
2. Sprial spacing shall not exceeds 3 in, nor be less than 1 in.
3. Anchorage of spiral shall be provided by 1-1/2 extra turn.

Design of short columns

Design of long non-sway columns

Design of long column with sway