Given: A 20ft. Span, simple supported beam is uniformly loaded with 8kips/ft.
-Find shear and bending stresses for a rectangular 10in x 20in cross section.
-Design it as a balanced reinforced concrete cross section. Use concrete fc’= 4 ksi, steel fy = 40 ksi.
-Design it as a steel beam. Use plastic condition, steel Fy = 50 ksi.
-Design as an unbalanced RFC beam. Use concrete fc’= 4 ksi, steel fy = 40 ksi. Use 14 x 20in cross-section.
Solution:
Finding Shear and Bending Stresses:
Firstly, we need to calculate the maximum moment and maximum shear force acting on the beam.
The maximum moment will occur at the center of the span which is equal to:
M = (qL^2) / 8
where q = the uniformly distributed load, L = the span
M = (8 kips/ft) x (20 ft)^2 / 8
M = 800 kip-ft
The maximum shear force will occur at the supports and is equal to:
V = qL / 2
V = (8 kips/ft) x (20 ft) / 2
V = 80 kips
To determine the bending stress, we use the bending equation:
σ_b = M*c/I
where c = the distance from the neutral axis to the extreme fiber, I = moment of inertia
For a rectangular section,
c = d/2 = 10 in / 2 = 5 in
I = (bd^3) / 12
I = (10 in) x (20 in^3) / 12
I = 6667 in^4
σ_b = (800 kip-ft) x (5 in) / (6667 in^4)
σ_b = 0.6 ksi
To determine the shear stress, we use the shear equation:
τ = V/A
where A = the cross-sectional area
A = bd = (10 in) x (20 in) = 200 in^2
τ = (80 kips) / (200 in^2)
τ = 0.4 ksi
Therefore, the shear and bending stresses for a rectangular 10in x 20in cross section are:
τ = 0.4 ksi and σ_b = 0.6 ksi
Design as a Balanced Reinforced Concrete Beam:
To design a balanced reinforced concrete beam, we need to calculate the moment capacity of the section and the required steel reinforcement.
The moment capacity of the section can be calculated using the following equation:
Mn = 0.85*f'c*b*(d-0.4167*x)*x + As*fy*(d - 0.5*As/fy)
where f'c = the compressive strength of concrete, b = the width of the beam, d = the depth of the beam, x = the distance from the top of the section to the neutral axis, As = the area of steel reinforcement, fy = the yield strength of steel
Taking x = 8.33 in (approximately 2/5 of d),
Mn = 0.85*4 ksi*10 in*(20 in - 0.4167*8.33 in)*8.33 in + As*40 ksi*(20 in - 0.5*As/40 ksi)
Mn = 542 kip-ft
The required amount of steel reinforcement can be calculated using the following equation:
As = (M - 0.85*f'c*b*(d-0.4167*x)*x) / (fy*(d - 0.5*As/fy))
Taking the dead and live load combination factor as 1.2 and 1.6 respectively, the maximum moment on the beam would be 1.2(M_d) + 1.6(M_l) = 1600 kip-ft.
As = (1600 kip-ft - 0.85*4 ksi*10 in*(20 in - 0.4167*8.33 in)*8.33 in) / (40 ksi*(20 in - 0.5*As/40 ksi))
As = 2.92 in^2
Assuming 4-#8 bars, the area of steel reinforcement would be 3.14 in^2, which is greater than 2.92 in^2.
Therefore, a balanced reinforced concrete cross section for this beam would be a rectangular section of dimensions 10in x 20in with 4-#8 steel reinforcement bars.
Design as a Steel Beam:
To design a steel beam, we need to calculate the required section modulus and select a section. Assuming plastic condition, the required section modulus can be calculated using the following equation:
S = M / Fy
where Fy = the yield strength of steel
S = 800 kip-ft / 50 ksi
S = 16 in^3
Based on a standard steel section table, we can select a W12x40 section, which has a section modulus of 16.3 in^3.
Therefore, a W12x40 steel section would be suitable for this beam.
Design as an Unbalanced Reinforced Concrete Beam:
To design an unbalanced reinforced concrete beam, we need to calculate the moment capacity of the section and the required steel reinforcement. The maximum moment would be at the support where the load is applied.
The moment capacity of the section can be calculated using the same equation as above:
Mn = 0.85*f'c*bx*(d-0.4167*x) + As*fy*(d - 0.5*As/fy)
Taking x = 2 in (approximately 1/7 of d),
Mn = 0.85*4 ksi*14 in*20 in*(20 in - 0.4167*2 in) + As*40 ksi*(20 in - 0.5*As/40 ksi)
Mn = 2554 kip-ft
The required amount of steel reinforcement can be calculated using the same equation as above:
As = (M - 0.85*f'c*b*(d-0.4167*x)*x) / (fy*(d - 0.5*As/fy))
Assuming a dead and live load combination factor of 1.2 and 1.6 respectively, the maximum moment on the beam at the support would be 1.2(M_d) + 1.6(M_l) = 3200 kip-ft.
As = (3200 kip-ft - 0.85*4 ksi*14 in*20 in*(20 in - 0.4167*2 in)) / (40 ksi*(20 in - 0.5*As/40 ksi))
As = 4.35 in^2
Assuming 4-#11 bars, the area of steel reinforcement would be 4.91 in^2, which is greater than 4.35 in^2.
Therefore, an unbalanced reinforced concrete cross section for this beam would be a rectangular section of dimensions 14in x 20in with 4-#11 steel reinforcement bars.