a) Find the maximum allowable working moment.
b) Check whether the beam is under reinforced or over reinforced.
c) Calculate then maximum working load that can be carried safely by the beam. Consider the beam is cantilever having a span of 8 ft.
d) Check whether the micro-crack will be developed or not before reaching the service load.
Solution:
Given, b = 10 inch, h = 24 inch, L = 8 feet, Effective depth, d = 24 – 3.5 inch= 20.5 inch
f’c = 3000 psi=3 ksi, fy = 415W= 60000 psi=60 ksi
Working stress in steel, fs = 0.4 × fy= 0.4 × 60 =24 ksi
Working stress in Concrete, fc= 0.45 ×𝑓𝑐′= 0.45 × 3 = 1.35 ksi
Working Stress Ratio, r = 𝑓𝑠𝑓𝑐 = 241.35 = 17.77 ≈ 18
Modular Ratio, n = 𝐸𝑠𝐸𝑐 = 29×10657000×√3680 = 9.28 ≈ 9
As = (6 × 0.48) = 2.88 in2, Steel Ratio, 𝜌=𝐴𝑠𝑏𝑑=2.8810×20.5=0.014
Now, k=√(𝜌𝑛)2+2𝜌𝑛−𝜌𝑛= 99+18 = 0.39, j = 1−𝑘3= 1−0.303 = 0.87
Solution (a)
Moment Capacity due to Concrete Limit, Mc = 𝑓𝑐2𝑗𝑘𝑏𝑑2 = 1.352×0.87×0.39×10×20.52
= 962 kip-in
Moment Capacity due to Steel Limit, Ms = 𝐴𝑠𝑓𝑠𝑗𝑑 = 2.88 × 24 × 0.87 × 20.5 = 1232.7 kip-in
Allowable Working Moment = 962 kip-in(Ans)
Solution (b)
As Ms > Mc the beam is over reinforced.
Solution (c)
Maximum working load that can be carried safely, Mw = 𝑊𝐿22
W= 𝑀𝑤×2 𝐿2= 962×2(8×12)2 kip/in= 0.208 kip/in= 2.5 kip/ft(Ans.)
Solution (d)
Modulus of Rupture, fr = 7.5×√𝑓′𝑐 = 7.5×√3000 psi = 410 psi = 0.410 ksi.
Now, for maximum Moment Case, fr = 6𝑀𝑚𝑎𝑥𝑏𝑑2
𝑀𝑚𝑎𝑥 = 𝑏𝑑2×𝑓𝑟6= 10×20.52×0.4106= 287 kip-in
As Mmax < Mdeveloped, So micro-crack will be developed
NOTE
Ec= 57000√f′c for Normal weight Concrete (Stone Chips)
Ec = 45000√f′c for Light weight Concrete (Brick Chips)
The Working Stress Design (WSD) method, focusing on conditions at service load (that is, when the structure is being used), was the principal method used from the early 1900s until the early 1960s. Today, with few exceptions, the strength design method is used, focusing on conditions at loads greater than service loads when failure may be immanent. The strength design method is deemed conceptually more realistic to establish structural safety. Hence, only Strength Design method has been discussed elaborately in this course.
