Question

Q2 (50 pts): Design, verify and detail a reinforced concrete beam with fixed dimensions of b = 16 in and h= 24 in with f'c = 5,000 psi and fy = 60,000 psi subjected to the bending moments: (a) Mu= 460 ft.k, and (b) Mu- 590 ft.k

Answer 1

(a)

Given:

Mu=460 ft-kips

Fc=5000 Psi

Fy=60 Ksi

Step 1:

Mu=460 ft-kips

b=16 in, h=24 in

d=21 in (considering 3 in as cover)

Rn=Mu/(0.9bd²)

Rn=460x12x1000/(0.9x16x21x21)

Rn=869.2 Psi

Step 2:

$\rho$=0.85fc/fy[1-(1-2Rn/0.85fc)^0.5]

$\rho$=0.85x5000/60000[1-(1-2x869.2/0.85x5000)^0.5]

$\rho$=0.016

Ast=0.016xbxd

Ast=0.0164x16x21=5.5 in² (provide #9 6 number of bars)

Step 3:

Now procceed for verification

$\beta$=0.85-(fc-4000/1000)x0.05=0.8

a=AsFy/(0.85fcb)

a=6x60/(0.85x5.5x16)

a=4.81 in

c=a/0.8

c=4.81/0.8=6.01 in

$\epsilon$_t=d-c(0.003)/c

$\epsilon$_t=21-6.01(0.003)/6.01=0.0074>0.005, so

Step 4:

Resisting moment is :

$\phi$Mn=0.9AsFy(d-a/2)

=0.9x6x60(21-4.81/2)

=6024.78 in-kips=6024/12=502 ft-kips

As $\phi$ Mn>Mu (502 ft-kips>460 ft-kips), so it is safe

(b)

Given:

Mu=590 ft-kips

Fc=5000 Psi

Fy=60 Ksi

Step 1:

Mu=590 ft-kips

b=16 in, h=24 in

d=21 in (considering 3 in as cover)

Rn=Mu/(0.9bd²)

Rn=590x12x1000/(0.9x16x21x21)

Rn=1114.8 Psi

Step 2:

$\rho$=0.85fc/fy[1-(1-2Rn/0.85fc)^0.5]

$\rho$=0.85x5000/60000[1-(1-2x1114.8/0.85x5000)^0.5]

$\rho$=0.022

Ast=0.022xbxd

Ast=0.022x16x21=7.39 in² (provide #11 5 number of bars, Ast=7.81 in²)

Step 3:

Now procceed for verification

$\beta$=0.85-(fc-4000/1000)x0.05=0.8

a=AsFy/(0.85fcb)

a=7.81x60/(0.85x5.5x16)

a=6.26 in

c=a/0.8

c=6.26/0.8=7.83 in

$\epsilon$_t=d-c(0.003)/c

$\epsilon$_t=21-7.83(0.003)/7.83=0.00504>0.005, so

Step 4:

Resisting moment is :

$\phi$Mn=0.9AsFy(d-a/2)

=0.9x7.81x60(21-6.26/2)

=7536 in-kips=7536/12=628 ft-kips

As $\phi$ Mn>Mu (628 ft-kips>590 ft-kips), so it is safe

Note : Everything in (b) except we provide reinforcement in 1st case and negative reinforcement in 2nd case