A rectangular beam of concrete with fc′ of 3000 psi [20.7 MPa] and steel with fs of 20 ksi [138 MPa] has dimensions of b = 15 in. [380 mm] and h = 36 in. [910 mm]. Find the area required for the steel reinforcement for a moment of 200 kip-ft [271 kN-m].

Question

A rectangular beam of concrete with [latex]f_c^′[/latex] of 3000 psi [20.7 MPa] and steel with [latex]f_s[/latex] of 20 ksi [138 MPa] has dimensions of b = 15 in. [380 mm] and h = 36 in. [910 mm]. Find the area required for the steel reinforcement for a moment of 200 kip-ft [271 kN-m].

Accepted Answer

The first step in this case is to determine the balanced moment capacity of the beam with the given dimensions. If we assume the section to be as shown in Figure 15.9, we may assume an approximate value for d to be h minus 2.5 in. [64 mm], or 33.5 in. [851 mm]. Then, with the value for R from Table 15.2,

M=Rbd²=(0.226)(15)(33.5)²= 3804 kip-in.

or

[latex]M=\frac{3804}{12}=[/latex] 317 kip-ft [M=(1554)(0.380)(0.850)²= 427 kN-M]

Since this value is considerably larger than the required moment, it is thus established that the given section is larger than that required for a balanced stress condition. As a result, the concrete flexural stress will be lower than the limit of 0.45[latex]f_c^′[/latex], and the section is qualified as being underreinforced, which is to say that the reinforcement required will be less than that required to produce a balanced section (with moment capacity of 317 kip-ft). In order to find the required area of steel, we use equation (15.2.4), just as we did in the preceding example. However, the true value for j in the equation will be something greater than that for the balanced section (0.872 from Table 15.2).

As the amount of reinforcement in the section decreases below the full amount required for a balanced section, the value of k decreases and the value of j increases. However, the range for j is small: from 0.872 up to something less than 1.0. A reasonable procedure is to assume a value for j, find the corresponding required area, and then perform an investigation to verify the assumed value for j, as follows. Assume j = 0.90. Then

[latex]A_s=\frac{M}{f_sjd}=\frac{200×12}{(20)(0.90)(33.5)}=3.98[/latex] in.² [2567 mm²]

and

[latex]p=\frac{A_s}{bd}=\frac{3.98}{(15)(33.5)}=0.00792[/latex]

Using this value for p in Figure 15.8, find k = 0.313. Using equation (15.2.6), j is found to be

[latex]j=1-(\frac{k}{3})=1- ({\frac{0.313}{3}})=0.896[/latex]

which is reasonably close to the assumption, so the computed area is adequate for design.

For beams that are classified as under-reinforced (section dimensions larger than the limit for a balanced section), a check should be made for the minimum required reinforcement. For the rectangular section, the ACI Code (Ref. 4) specifies that a minimum area be

[latex]A_s=\frac{3\sqrt{f_c^′ } }{f_y}[/latex] (bd)

but not less than

[latex]A_s=(\frac{200}{F_y})[/latex](bd)

On the basis of these requirements, values for the minimum reinforcement for rectangular sections with tension reinforcement only are given in Table 15.3 for the two common grades of steel and a range of concrete strengths.

For the example, with [latex]f_c^′[/latex] of 3000 psi and [latex]f_y[/latex] of 40 ksi, the minimum area of steel is thus

[latex]A_s=0.005 (bd) = 0.005(15 imes 33.5 ) = 2.51[/latex] in.²

which is not critical in this case.

Question 15.2: A rectangular beam of concrete with fc′ of 3000 psi [20.7 MP...

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