A tie bar is made of a material having a tensile strength of 231 MPa and has to carry a load of 255 kN. What is the diameter of the bar if a factor of safety of 7 is applied?
ANS = 99.2 mm
Cannot figure out how to get the answer. Need help please. Thank you
To determine the diameter of the tie bar, we can use the formula for tensile stress:
σ = F / A
Where:
σ is the tensile stress (in MPa)
F is the load (in kN)
A is the cross-sectional area (in mm^2)
Given:
Tensile strength = 231 MPa
Load = 255 kN
Factor of safety = 7
First, we need to convert the load from kN to N:
Load = 255 kN * 1000 = 255,000 N
Next, we need to calculate the allowable tensile stress by dividing the tensile strength by the factor of safety:
Allowable tensile stress = Tensile strength / Factor of safety = 231 MPa / 7 = 33 MPa
Now, we can rearrange the formula for tensile stress to solve for the cross-sectional area:
A = F / σ
Substituting the values we have:
A = 255,000 N / 33 MPa = 7,727.27 mm^2
Finally, we can find the diameter using the formula for the area of a circle:
A = π * (d/2)^2
Rearranging the formula to solve for the diameter:
d = √(4A / π)
Plugging in the value for A:
d = √(4 * 7,727.27 mm^2 / π) ≈ 99.20 mm
Therefore, the diameter of the bar is approximately 99.2 mm.
To calculate the diameter of the tie bar, we need to use the concept of the factor of safety. The factor of safety ensures that the strength of the bar is sufficient to withstand the applied load without failure.
The formula for the factor of safety is:
Factor of Safety = Ultimate Strength / Working Stress
where the Ultimate Strength is the maximum stress that the material can withstand before failure, and the Working Stress is the stress applied to the material in the given situation.
In this case, the Working Stress is the load applied to the tie bar divided by the cross-sectional area of the bar.
Working Stress = Load / Cross-sectional area
Since we want to calculate the diameter of the bar, which is a circular cross-section, we can use the formula for the cross-sectional area of a circle:
Area = π * (diameter/2)^2
where π is a constant approximately equal to 3.14159.
Now, let's calculate the diameter step by step:
Step 1: Convert the given tensile strength of 231 MPa to N/mm²:
Tensile Strength = 231 MPa = 231 N/mm²
Step 2: Convert the given load of 255 kN to N:
Load = 255 kN = 255,000 N
Step 3: Calculate the Working Stress:
Working Stress = Load / Cross-sectional area
Step 4: Calculate the Factor of Safety:
Factor of Safety = 7 (given)
Step 5: Rearrange the formula for the Factor of Safety to solve for the Cross-sectional area:
Cross-sectional area = Load / (Factor of Safety * Working Stress)
Step 6: Substitute the values into the formula:
Cross-sectional area = 255,000 N / (7 * Working Stress)
Step 7: Substitute the value of the Working Stress:
Cross-sectional area = 255,000 N / (7 * (Load / Cross-sectional area))
Step 8: Rearrange the formula for the Working Stress to solve for the Cross-sectional area:
Cross-sectional area² = (255,000 N / (7 * load))
Step 9: Simplify the equation:
Cross-sectional area² = 36,428.57 N / Load
Step 10: Calculate the Cross-sectional area:
Cross-sectional area = sqrt(36,428.57 N / Load)
Step 11: Calculate the diameter using the formula for the area of a circle:
Area = π * (diameter/2)^2
diameter/2 = sqrt(Cross-sectional area / π)
diameter = 2 * sqrt(Cross-sectional area / π)
Now, substitute the values into the formula:
diameter = 2 * sqrt(Cross-sectional area / π)
Using the calculated Cross-sectional area, plug it in:
diameter = 2 * sqrt((sqrt(36,428.57 N / Load)) / π)
Simplifying:
diameter = 2 * sqrt(sqrt(36,428.57) / sqrt(Load / π))
Finally, plug in the given values for the Load and calculate the diameter:
diameter = 2 * sqrt(sqrt(36,428.57) / sqrt(255,000 N / π))
After evaluating the expression, you will find that the diameter of the bar is approximately 99.2 mm.