Consider a rectangular reinforced concrete beam with an effe

Consider a rectangular reinforced concrete beam with an effective depth, d of 20 in. f\'_c= 5000 psi, and f_y = 75,000 psi and A_s = 2.00 in². Determine the moment capacity Mn of this beam for the following cases:

a) section width, b = 15 in.

b) b = 17.5 in.

c) b = 20 in (square section)

Tabulate your results for different M_n values (column) and b values (along rows). In the bottom row of the table, show the percentage increase in M_n.

Solution

Solution:-

(1) when b =15 inch

x_{u lim} = 0..48 * d

x_{u lim} = 0.48 *20 = 9.6 inch

x_u = 0.87 f_y A_st/(0.36 f_ck b)

         = 0.87 * 75000 * 2 /(0.36 * 5000*15)

       = 4.83inch less than x_{u lim}

Section is under reinforced

M_n = 0.87 f_y A_st (d – 0.42 x_u)

      =0.87 * 75000 * 2 *(20 – 0.42 *4.83)

      = 2.3452 * 10⁶ psi- inch       Answer

(2) when b = 17.5 inch

     Then x_u = 4.14 inch

M_n = 0.87 * 75000 * 2 (20 – 0.42*4.14)

      = 2.383 * 10⁶ psi – inch

(3) when b= 20 inch

Then   x_u = 0.87 * 75000 *2 /(0.36 * 5000*20)

                 = 3.6 inch

M_n = 0.87 * 75000 * 2 * (20 – 0.42 * 3.6) = 2.41268 * 10⁶ psi –inch

b(inch)                    M_n(psi –inch)                                                     % increase in (M_n)

15                                 2.3452 * 10⁶

17.5                              2.383 *10⁶                                                         1.6

20                                 2.41268 *10⁶                                                     2.87