A sagging floor is jacked up and a steel girder 10 ft long whose cross sectional area is 6in^2 is put underneath. When the jack is removed, a strain gauge shows that the girder has been compressed by 0.0080in. Find the weight the girder is supporting in lbs and tons.
delta L/L = w /(E A)
.008/120 = w/(E*6)
w = 6 E * .008/120
To find the weight the girder is supporting, we need to calculate the stress and then use it to determine the weight.
Step 1: Calculate the stress:
The stress (σ) can be calculated using the formula:
σ = F / A
where F is the force acting on the girder and A is the cross-sectional area of the girder.
Given:
Length of the girder (L) = 10 ft
Cross-sectional area (A) = 6 in^2
Compression (Δx) = 0.0080 in
Step 2: Convert the units:
Since the given length is in feet and the cross-sectional area is in square inches, we need to convert them to a consistent unit. Let's convert the length to inches and the cross-sectional area to square feet.
Length (L) = 10 ft = 120 inches (1 foot = 12 inches)
Cross-sectional area (A) = 6 in^2 = 6 / 144 ft^2 (1 square foot = 144 square inches)
Step 3: Calculate the force:
The force (F) is equal to the difference in length (strain) multiplied by the cross-sectional area and the modulus of elasticity (E) of the girder.
The modulus of elasticity for steel is typically around 30,000,000 psi. Let's assume that value.
F = E * A * Δx
Given:
Modulus of elasticity (E) = 30,000,000 psi
Δx = 0.0080 in
A = (6/144) ft^2
Substituting the values:
F = 30,000,000 * (6/144) * 0.0080
Step 4: Calculate the weight:
The weight (W) is equal to the force (F) divided by the acceleration due to gravity (g) which is approximately 32.2 ft/s^2.
W = F / g
Given:
Acceleration due to gravity (g) = 32.2 ft/s^2
Substituting the values:
W = F / 32.2
Now, we can calculate the weight supported by the girder.
Note: 1 pound (lb) is equal to 0.0005 tons.
Let's calculate the weight in pounds first:
Step 5: Calculate weight in pounds:
W_lb = F / 32.2
Step 6: Convert weight to tons:
W_ton = W_lb * 0.0005
You can now substitute the values and calculate the weight in pounds and tons.
To find the weight the girder is supporting, we need to use the concept of stress and strain.
Let's start by calculating the compression strain:
ε = ΔL / L
Where:
ε is the strain
ΔL is the change in length
L is the original length
Given that the girder is compressed by 0.0080 inches and the original length is 10 feet (or 120 inches), we can substitute these values into the equation:
0.0080 = ΔL / 120
Now, let's solve for the change in length:
ΔL = 0.0080 * 120
ΔL = 0.96 inches
Next, we can calculate the stress using the formula:
σ = F / A
Where:
σ is the stress
F is the force
A is the cross-sectional area
We need to find the force F that the girder is supporting. Rearranging the formula, we have:
F = σ * A
To find the stress, we can use the Hooke's Law:
σ = E * ε
Where:
σ is the stress
E is the Young's modulus (elastic modulus)
ε is the strain
The Young's modulus for a steel girder is typically around 29,000,000 psi (pounds per square inch).
Substituting the values:
σ = 29,000,000 * 0.0080
σ = 232,000 psi
Now we can calculate the force F:
F = 232,000 * 6
F = 1,392,000 lbs
The girder is supporting a weight of 1,392,000 lbs. To convert this to tons, divide by 2,000:
Weight in tons = 1,392,000 / 2,000
Weight in tons = 696 tons
Therefore, the girder is supporting a weight of 1,392,000 lbs or 696 tons.