Strength of Singly Reinforced Concrete beam

Problem 9-1

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

singly reinforced beam

Figure 9-1

Solution:

The given section is singly reinforced. The computation of nominal flexural strength Mn 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  fc′  grerater than 4000 psi the value  of β1 is calculated as given below;

 β1  = 0.85 - 0.05 {( fc′ -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.

       As = 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;

  Cc = 0.85 fcba = 0.85 (5) (15) a = 63.75 a

  T =  As fy  =  (4) (50) = 200 kips

 Applying static equilibrium, we get  Cc = 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

          Cc = 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.

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

     εy = fy /Es = 50 / 29000 = 0.00172

   εs  is greater than εy , 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) 

  Mn = Cc  (d - a/2)

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

    Mn = 349.15 ft-kips 

 Mn  can also be calculated by ; Mn = 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.

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