**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 β_{1} is calculated as given
below;

β_{1} = 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
in^{2}.

* A*_{s} = 4-#9
bars. = 4 (1.00) = 4
in^{2}.

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 */ β_{1}
= 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.

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