MATH: Reinforced Concrete Design 01

Problem 2.5
Design a rectangular beam reinforced for tension only to carry a dead load moment of 56 kN-m (including its own weight) and a live laod moment of 47 kN-m. Use fx=20.7 MPa and fy=276MPa.
Solution:
Ma=1.4MDL=1.7MLL
Ma=1.4(56=1.746
My=158.3 kN-m
(Note: this already includes the weight of the beam)
ρb=0.85fc1β1600fy600+fy
β1=0.85 since fr1.4fy
ρmin=1.4278
ρmin=0.00507
Note: This is just the author's assumption
Try ρ=0.6ρb
ρ=0.60.0071
ρ=0.02226
a=ρfyfx
a=0.0222627620.7
a=0.2968
Ra=f 1-0.59
Rb=20.70.29681-0.590.2968
Rb=5.068
Mc=ϕRabd2
158.3 x 106=0.505.068bd2
bd2=34,705.779
Try it=1.75b
b1.75b2=34,705,779
b=225mm say 230mm
230d2=34,705,779
d=388.4 say 390mm
Note: we can't check for the minimum beam thickness since there is no given beam length and type of support.
A1=9hd=0.02226230390
A1=1,997mm2
Summary:b=230 mm
d=390mm
A1=1,997 mm2