Minimum Area of Reinforcement Required for Reinforced Concrete Beams

In reinforced concrete beams, the minimum area of longitudinal reinforcement required is given by;

A_s,min = 0.26 (f_ctm / f_yk) b_t d   ≥ 0.0013 b_t d —- (1)

where:
f_ctm is the mean value of axial tensile strength of concrete at 28 days, see Table 3.1 of EN 1992-1-1:2004
f_yk is the characteristic yield stress of reinforcement steel
b_t is the mean width of the tension zone; for a T-beam with the flange in compression, only the width of the web is taken into account in calculating the value of b_t
d is the effective depth of concrete cross-section.

Any section containing less than A_s,min as given in equation (1) should be treated as an unreinforced section.

The minimum area of reinforcement for different classes of concrete strengths assuming f_yk = 500 MPa is given in the Table below;

fck (MPa)	fctm (MPa)	0.26fctm/fyk
25	2.6	0.13%
28	2.8	0.14%
30	2.9	0.15%
32	3.0	0.16%
35	3.2	0.17%
40	3.5	0.18%
45	3.8	0.20%
50	4.2	0.21%

For a T-beam with the flange in tension (such as upstand beams or ground beams), the minimum area of reinforcement is given by equation (2);

A_s,min = 0.26 (f_ctm / f_yk) A_c,t —– (2)

Where A_c,t is the area of the tension zone above the neutral axis of the reinforcement.

BS 8110-1:1997 made express provisions for flanged beams for different stress states of the web or the flange. The requirements are given in the table below;

Flanged beam in flexure (tension reinforcement)	Definition of Percentage	Minimum Percentage (fy = 500 MPa)
(a) Web in tension
bw/b < 0.4	100As/bwh	0.18%
bw/b ≥ 0.4	100As/bwh	0.13%
(b) Flange in tension
T-beam	100As/bwh	0.26%
L-beam	100As/bwh	0.20%

Maximum area of longitudinal reinforcement
The maximum area of tension and compression reinforcement in reinforced concrete beams is;

A_s,max = 0.04A_c —- (3)

Where;
A_c is the area of the cross-section

Shear Reinforcements (links and stirrups)
The minimum area of shear reinforcement in beams should be calculated from;

A_sw/sb_w ≥ ρ_w,min —– (4)

Where;
ρ_w,min = (0.08√f_ck)/f_yk
A_sw = Area of shear reinforcement
s = spacing of shear reinforcement
b_w = width of the web

Solved Example
For the concrete section shown below, find the minimum area of longitudinal reinforcement required for;
(a) When the web is in tension
(b) When the flange is in tension
f_ck = 25 MPa, f_yk = 500 MPa

(a) When the web is in tension
Using Eurocode;
A_s,min = 0.26 (f_ctm / f_yk) b_t d 
A_s,min = 0.26 x (2.6/500) x 250 x 695 = 235 mm² > 0.0013b_td (0.0013 x 250 x 695) = 226 mm²

Using Table 3.25, BS 8110-1:1997;
b_w/b = 250/1500 = 0.167 < 0.4
A_s,min = 0.18b_wh/100 = (0.18 x 250 x 750)/100 = 338 mm²

(b) When the flange is in tension
Using Eurocode;
A_s,min = 0.26 (f_ctm / f_yk) A_c,t
A_c,t = (550 x 250) + (1500 x 145) = 355000 mm² 
A_s,min = 0.26 x (2.6/500) x 355000 = 480 mm²

Using Table 3.25, BS 8110-1:1997;
A_s,min = 0.26b_wh/100 = (0.26 x 250 x 750)/100 = 488 mm²