Problem 1 A rectangular reinforced concrete beam has dimensi

Problem 1 A rectangular reinforced concrete beam has dimensions of b 12in, h 25 in, and effective depth d 23 in. It supports a service live load of 2,400 Ib/ft in addition to its self weight. The beam is 20 ft long and is simply supported. The reinforcement consists of 3 #9 Gr 60 bars in a single row and the concrete compressive strength (P) is 4,000 psi. Does the section crack in service? What are the stresses in the reinforcing steel and the extreme compression filber of the beam in service?

Solution

Solution:-

Given

fc =4000 psi

f_y = 60000 psi (Let)

b =12 inch

d = 23 inch

Area of steel = 3 * /4 *(1.128)² = 2.99 inch²

Service load = 2400 lb/feet

Length of beam =20 feet

Neutral axis (x_u) = 0.87*f_y * A_st/(0.36 * f_c *b)

                             = 0.87*60000*2.99/(0.36*4000*12)

                             = 9 inch

x_ulim/d = 0.48

x_ulim =0.48*23 = 11.04 inch

x_ulim greater than x_u

section is under reinforced

M_u = 0.87f_yA_st(d – f_yA_st/(f_ckb))

      = 0.87*60000*2.99(23 – 60000*2.99/(4000*12))

     = 3.006*10⁶ pound –inch

Moment from given service load = wl²/8

= 2400*20²/8

= 120000 lb –feet

= 1.44 *10⁶ lb – inch

Given service moment are less than ultimate moment

So there is no crack occur in section. Section is safe

Service load = 2400 *20 = 48000 lb

Stress in steel = 48000/area of steel

                        = 48000/2.99

                        = 16053.5 psi Answer

Stress in concrete in compression =48000/(bx_u)

= 48000/(12*9)

= 444.4 psi Answer