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Strength of Reinforced Concrete  - Singly Reinforced Section

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.

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.

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Analysis and design of reinforced concrete structures

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

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 Reinforced Concrete

You can also use our  Reinforced concrete calculator