A Timber Beam Reinforced with a Steel Plate: Determining Maximum Load and Bending Moment
Calculating Maximum Bending Moment and Concentrated Load
A timber beam measuring 150 mm x 250 mm is reinforced at the bottom only by a steel plate. With a given value of n = 20, we are tasked to determine the following:
a. The Concentrated Load that can be applied at the center of the simply supported span, which is 6 meters long, without exceeding the stress of wood of 8 MPa. We need to calculate the beam's maximum bending moment using the given stress.
To determine the concentrated load that can be applied at the center of the simply supported span without exceeding the stress of wood, we need to calculate the maximum bending moment the beam can withstand and then use that to find the corresponding concentrated load.
The formula for calculating the maximum bending moment (M_max) in a simply supported beam subjected to a concentrated load at the center is:
M_max = (P * L) / 4
Where:
P is the concentrated load at the center
L is the span length
Given that the span length (L) is 6 m and the stress of wood (σ_wood) is 8 MPa, we can use the formula for bending stress in a rectangular beam:
σ = (M * c) / I
Where:
σ is the bending stress
M is the bending moment (M_max in this case)
c is the distance from the neutral axis to the outermost fiber (half of the beam's depth)
I is the moment of inertia of the beam cross-section
The moment of inertia (I) for a rectangular beam is:
I = (b * d^3) / 12
Given the dimensions of the beam (150 mm x 250 mm), we need to convert them to meters (1 m = 1000 mm) for consistency in units.
Let's calculate step by step:
1. Convert Beam Dimensions to Meters:
b = 150 mm = 0.15 m
d = 250 mm = 0.25 m
2. Calculate Moment of Inertia:
I = (0.15 * (0.25)^3) / 12
3. Calculate Maximum Bending Stress:
σ_wood = 8 MPa = 8 * 10^6 N/m^2
Now, rearrange the bending stress formula to solve for maximum bending moment:
M_max = (σ_wood * I) / c
Substitute the values and solve for M_max.
Finally, use the formula for maximum bending moment to find the concentrated load:
M_max = (P * L) / 4
Solve for P.
Keep in mind that this calculation assumes linear-elastic behavior and doesn't account for factors like safety margins, deflection limits, and possible beam imperfections.
A timber beam 150 mm x 250 mm is reinforced at the bottom only by a steel plate. If n = 20, determine the following: a. The concentrated load that can be applied at the center of the simply supported span 6 m long without exceeding the stress of wood of 8 MPa. Calculate the beam's maximum bending moment using given stress. Use M_max = (P * L) / 4 to find concentrated load. To determine the concentrated load that can be applied at the center of the simply supported span without exceeding the stress of wood, we need to calculate the maximum bending moment the beam can withstand and then use that to find the corresponding concentrated load. The formula for calculating the maximum bending moment (M_max) in a simply supported beam subjected to a concentrated load at the center is: M_max = (P * L) / 4 Where: P is the concentrated load at the center L is the span length Given that the span length (L) is 6 m and the stress of wood (σ_wood) is 8 MPa, we can use the formula for bending stress in a rectangular beam: σ = (M * c) / I Where: σ is the bending stress M is the bending moment (M_max in this case) c is the distance from the neutral axis to the outermost fiber (half of the beam's depth) I is the moment of inertia of the beam cross-section The moment of inertia (I) for a rectangular beam is: I = (b * d^3) / 12 Given the dimensions of the beam (150 mm x 250 mm), we need to convert them to meters (1 m = 1000 mm) for consistency in units. Let's calculate step by step: Convert beam dimensions to meters: b = 150 mm = 0.15 m d = 250 mm = 0.25 m Calculate moment of inertia: I = (0.15 * (0.25)^3) / 12 Calculate maximum bending stress: σ_wood = 8 MPa = 8 * 10^6 N/m^2 Rearrange the bending stress formula to solve for maximum bending moment: M_max = (σ_wood * I) / c Substitute the values and solve for M_max. Finally, use the formula for maximum bending moment to find the concentrated load: M_max = (P * L) / 4 Solve for P. Keep in mind that this calculation assumes linear-elastic behavior and doesn't account for factors like safety margins, deflection limits, and possible beam imperfections.