Answer :
1. Stresses caused by a bending moment of M=25ft-kips is 33.7 ksi
2. Stresses caused by an increased bending moment of M=55ft-kips is 74.3 ksi
3. The calculations show that the beam is not strong enough to resist the increased bending moment of 55 ft-kips.
Determining the stress caused by bending moment
The section modulus of the beam can be calculated as:
[tex]S = (bd^2/6) + 4As((d-2a)/2)[/tex]
where
b is the width of the beam,
d is the overall depth of the beam,
a is the diameter of the reinforcing bars, and
As is the total area of the reinforcing bars.
Given
b = 10 ft, d = 18 in = 1.5 ft, a = 7/8 in = 0.0729 ft, As = 4 x [tex]\pi[/tex]/4 x (0.0729 [tex]ft^2[/tex] = 0.0073 [tex]ft^2[/tex]
Plug in the given values
[tex]S = (10 ft * (1.5 ft)^2/6) + 4 * 0.0073 ft^2 * (1.5 ft - 2 * 0.0729 ft) = 0.556 ft^3[/tex]
The maximum bending stress caused by a bending moment of M = 25 ft-kips can be calculated as
sigma = Mc/S
where c is the distance from the neutral axis to the extreme fiber of the beam.
For a rectangular beam, c = d/2.
Substitute the given values
c = 1.5 ft / 2 = 0.75 ft
sigma = (25 ft-kips * 0.75 ft) / 0.556[tex]ft^3[/tex] = 33.7 ksi
The allowable bending stress in concrete is given by:
sigma_c = 0.45f_c
where f_c is the compressive strength of concrete.
Substituting the given value
sigma_c = 0.45 x 4000 psi = 1800 psi = 1.8 ksi
The maximum bending stress of 33.7 ksi exceeds the allowable stress in the concrete, so the beam will fail in compression.
The yield stress of the reinforcing bars is given by
sigma_y = 60 ksi
The maximum bending stress of 33.7 ksi is less than the yield stress of the reinforcing bars, so the bars will remain elastic.
Repeating steps 2 and 3 for M = 55 ft-kips
sigma = (55 ft-kips x 0.75 ft) / 0.556 [tex]ft^3[/tex] = 74.3 ksi
The maximum bending stress of 74.3 ksi exceeds both the allowable stress in the concrete and the yield stress of the reinforcing bars, so the beam will fail in compression and the reinforcing bars will yield.
Comment on the findings
The calculations show that the beam is not strong enough to resist the increased bending moment of 55 ft-kips. The beam will fail in compression and the reinforcing bars will yield. To increase the strength of the beam, additional reinforcement or a larger beam size may be required.
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