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please justify the crushing compared to the given allowable stress

To check if the column is acceptable for crushing, we need to compare the actual compressive stress generated by the load with the allowable compressive stress of the timber. 

Here are the steps to justify the compressive stress calculation:

### 1. Calculate the cross-sectional area of the column
The column has a circular cross-section with a diameter of 4 inches.

The area \( A \) of a circle is given by the formula:

\[
A = \frac{\pi d^2}{4}
\]

where:
- \( d = 4 \) inches is the diameter.

\[
A = \frac{\pi (4^2)}{4} = \frac{16\pi}{4} = 4\pi \approx 12.566 \, \text{in}^2
\]

### 2. Calculate the actual compressive stress
The load applied by the sculpture is 1.2 kips (1 kip = 1000 lbs).

The actual compressive stress \( \sigma_{\text{actual}} \) is:

\[
\sigma_{\text{actual}} = \frac{\text{Load}}{A} = \frac{1.2 \, \text{kips} \times 1000}{12.566} \approx \frac{1200}{12.566} \approx 95.47 \, \text{psi}
\]

### 3. Apply the safety factor
The problem specifies a safety factor of 1.5. Therefore, the actual stress must be multiplied by this factor to account for safety:

\[
\sigma_{\text{with safety}} = 95.47 \times 1.5 \approx 143.21 \, \text{psi}
\]

### 4. Compare with the allowable stress
The allowable compressive stress of the timber is given as 1200 psi.

Since the actual compressive stress with the safety factor (143.21 psi) is **much less than** the allowable stress of 1200 psi, the column is acceptable for crushing.

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Do you need more details or have any questions?

### Related Questions:
1. How do you calculate the buckling stress for this column?
2. What is the importance of the safety factor in structural design?
3. How does the modulus of elasticity affect the buckling behavior of a column?
4. What role does the diameter of the column play in determining compressive stress?
5. How would the result change if the load were increased?

**Tip:** Always check if the load applied to a column exceeds the material's allowable stress to ensure structural safety.

Is the column acceptable for crushing (compressive stress)? Is the column acceptable for buckling?

This page provides a step-by-step analysis of a timber column with a diameter of 4 inches and a height of 8 feet, subjected to a load of 1.2 kips. The analysis includes compressive stress, buckling behavior using Euler's buckling theorem, and checks against an allowable stress of 1,200 psi with a safety factor of 1.5.

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