MATH: Reinforced Concrete Design 01

Problem 2.7
Design a rectangular beam for a 10-m simple span to support a dead load of 18 kN/m (not including its own weight) and a live load of 24 kN/m. Use f1=21MPa and fy=276MPa. Assume weight of concret is 23.5kN/m3.
Solution
Assume weight of beam to be 21% of (DL +LL)
Assume weight of beam = 0.21(18 + 21) = 8.82 kN/m
ωDL=18=8.82
ωDL=26.82 kN/m
ωu=1.4ωDL=1.7ωLL
ωu=1.426.82+ 1.724
ωu=78.348 kN/m
Maximum moment: Mu=ωuL28
Mu=78.3481028
Mu=979.35 kN-m
Try p=0.5 pa
p=0.50.85fc1β1600fy600+fy
p=0.85210.85600276600+276
p=0.0188
pmin=1.4fy
pmin=1.4276
pmin=0.0051
m=pfyf((c
m=0.018827621
m=0.247
pm=fc m 1-0.59m
pm=320.2471-0.590.247
pm=4.433MPa
Mx=ϕRubd2
979.35 x 106=0.904.431bd2
bd2=245,580,381
Try d=1.75b
b1.75b2=245,580,381
d=755 say 760mm
approximate depth of beam, h=760+100=860mm
Note: The value 100 is the distance (estimated) from the e.g. of the bars to extreme concrete fiber.
Beam weight=bfc=23.50.430.86
Beam weight=8.89kNmOK
Minimum beam thickness from TABLE 2.1:
h=L160.4+fy700
h=10,000160.4+276700= 496mm OK
Tension steel area:
A1=phd
A1=0.0188430760
A1=6144mm2
Using # 11 bars (35 mm)
nl 353 N=6144
N+6.4 say 7
Actual A1=6,735


Locating the centroid of the bars:
Taking the area of the bar as A:
7Ay=5AU+ 2.460
y=17mm
Check for actual d:
d=720+60-17
d=763>760(OK)
Checking the spacing x:
x=340-3554
x=41.25>35mm(OK)
If we check the capacity of this beam with b=440, d=763, and A1=6734mm2(for 7 #11bars, the moment capacity is 1,078 kN-m, with p=0.02P=0.028.
INVESTIGATION ANALSISPROBLEMS WHERE STEEL YIELDS (fs=fy)