MATH: Reinforced Concrete Design 02

Problem 4.6
Design the spacing of a 10-mm U stirrups for the beam shown in Figure 4.3, for which DL = 60 kN/m and LL = 87 kN/m. Use f'c = 27.6 MPa and fy = 414 MPa.


Solution
wu= 1.4 (60) + 1.7(8.7)
wu= 231.9 kN/m

R= wuL2
R= 231.9(4.4)2

Vc= 16f'c bw d
Vc=16 27.6 (375)(570)
Vc=187,158 N
Vc=187.16 kN
ϕVc = 159.086 kN


Factored shear near the support:
Vu=R-wu d
Vu=510.18-231.9(0.57)
Vu=378 kN> ϕVc
Vu/ϕ= 378/0.85
Vu/ϕ= 444.7 kN

Vs= Vu / ϕ -Vc
Vs=444.7-187.16
Vs=257.54 kN

Verify if the section is adequate to carry the shear:
23 f'c bw d= 23 27.6 (375)(570)
23f'c bw d= 748,634 N> Vs

Thus, the section is adequate to carry the shear.
s= Av fy dVs
Av= π4(10)2 x 2
Av=157mm2
s= 157(414)(570)257,540
s= 144mm

Maximum spacing:
13bwd = 374,317N

Since Vs < 13 f'c bw d
Smax =d/2 = 570/2
Smax = 285 mm or 600 mm

Use s = 144 mm o.c. near the supports


x1510.18-159.1= 2.2510.18
x1=1.51 m
x1510.18-79.5= 2.2510.18
x2=1.86 m

@ 0.9 m from support:
Vu = 510.18 — 231.9(0.9)
Vu= 301.47kN > V

Vs= Vu / ϕ -Vc
Vs = 301.47/ 0.85 — 187.16
Vs= 167.5 kN

s= 157(414)(570) 167,500

s = 221 mm
@1.2 m from support:
Vu=510.18-231.9 1.2
Vu=231.9 kN> ϕVc
Vs=231.9/0.85 -187.16
Vs=85.66 kN
s= 157414(570)85,660
s= 433 mm > Smax (use Smax = 285mm)

Spacing
Length
Total distance
from support
1 @70 mm
70 mm
70 mm
7@l4Omm
980mm
1050mm
1@220mm
220mm
1270mm
The rest @ 285 mm