Strength of Singly Reinforced Concrete beam

Problem 9-1

Compute the nominal flexural strength M_n of the reinforced concrete rectangular section given below in figure 9-1(a).  Take f_c′ = 5000 psi,      f_y = 50,000 psi, b = 15 in., d = 22.5 in., and A_s = 4-#9 bars.

Figure 9-1

Solution:

The given section is singly reinforced. The computation of nominal flexural strength M_n is based on the guidelines of ACI-318. The maximum value of usable strain at the extreme concrete  fiber is assumed  to be 0.003.

For  f_c′  grerater than 4000 psi the value  of β₁ is calculated as given below;

 β₁  = 0.85 - 0.05 {( f_c′ -4000)/1000} = 0.8

Assume that the steel has already yielded when the strength is reached (strain in concrete is 0.003).

Given that tension steel consists of 4 bars of #9 (dia 1.128 in.) .

       Area of one bar of #9 = 1 in².

       A_s = 4-#9 bars. = 4 (1.00) = 4 in².

The internal forces acting on the section  shown in figure 9-1(c) are calculated as given below;

  C_c = 0.85 f_c′ ba = 0.85 (5) (15) a = 63.75 a

  T =  A_s f_y  =  (4) (50) = 200 kips

 Applying static equilibrium, we get  C_c = T;

  therefore  63.75 a  = 200        

  Depth of  Whitney stress block,    a = 3.14 in.

  Depth of neutral axis   x = a / β₁  = 3.14/0.8 = 3.93 in

          C_c = 63.75 (3.14) = 200.18 kips

By straight line proportion (figure 9-1(b)) we can calculate the strain in tension steel when the extreme concrete fiber has a compressive strain of 0.003.

     ε_s = (d - x ) (0.003) / x = (22.5 - 3.93) (0.003) / 3.93 =  0.014

     ε_y = f_y /E_s = 50 / 29000 = 0.00172

   ε_s  is greater than ε_y , this means that steel has yielded before crushing of concrete, which is a safe condition as it gives warning before failure. The section is Tension-Controlled because the strain is more than 0.005, therefore the strength reduction factor will be taken as 0.9

The nominal flexural strength is computed below;

  Moment = (force) (lever-arm) 

  M_n = C_c  (d - a/2)

      =  200.18 (22.5 - 3.14/2) = 4189.77 in-kips

    M_n = 349.15 ft-kips 

 M_n  can also be calculated by ; M_n = T (d - a/2)

see another problem 9-2 and problem 9-3 on Strength of Reinforced Concrete beam.

You can also use our  Reinforced concrete calculator for quickly finding the strength of reinforced concrete sections.