Understanding the Hollow Steel Column Problem

This problem asks us to determine the necessary sectional area for a hollow steel column that must safely support a specific axial load. We are given the axial load the column needs to carry, the ultimate stress the steel can withstand, and a load factor that provides a safety margin.

In structural engineering, a load factor is a safety factor applied to the expected working load to determine the load for which the structure must be designed (the ultimate load). Similarly, an allowable stress is a reduced stress value obtained by dividing the ultimate stress by a load factor or safety factor. The design ensures that under the working load, the stress in the material does not exceed this allowable stress.

Given Data for the Steel Column

• Axial Load (P): 2,00,000 kg
• Ultimate Stress ($\sigma_{ultimate}$): 4800 kg/cm²
• Load Factor (LF): 4

Calculating the Allowable Stress

The load factor is used to calculate the allowable stress from the ultimate stress. The allowable stress is the maximum stress the material is permitted to experience under normal working load conditions. It is calculated as:

\$ \text{Allowable Stress} (\sigma_{allowable}) = \frac{\text{Ultimate Stress}}{\text{Load Factor}} \$

Substituting the given values:

\$ \sigma_{allowable} = \frac{4800 \text{ kg/cm}^2}{4} \$

\$ \sigma_{allowable} = 1200 \text{ kg/cm}^2 \$

So, the steel column must be designed such that the stress under the 2,00,000 kg axial load does not exceed 1200 kg/cm².

Determining the Required Sectional Area of the Column

The relationship between axial load, stress, and sectional area is given by:

\$ \text{Stress} = \frac{\text{Load}}{\text{Area}} \$

To find the required sectional area (A), we rearrange the formula:

\$ \text{Area} = \frac{\text{Load}}{\text{Stress}} \$

We use the working axial load (2,00,000 kg) and the calculated allowable stress (1200 kg/cm²) to find the required minimum sectional area for the column:

\$ \text{Required Sectional Area} (A) = \frac{\text{Axial Load}}{\text{Allowable Stress}} \$

\$ A = \frac{2,00,000 \text{ kg}}{1200 \text{ kg/cm}^2} \$

Let's perform the calculation:

\$ A = \frac{200000}{1200} \text{ cm}^2 \$

\$ A = \frac{2000}{12} \text{ cm}^2 \$

\$ A = \frac{500}{3} \text{ cm}^2 \$

\$ A \approx 166.666... \text{ cm}^2 \$

The required sectional area for the hollow steel column is approximately 166.66 cm².

Comparison with Options

Let's compare our calculated area with the given options:

Our calculated value of approximately 166.66 cm² matches Option 2.

Revision Table: Steel Column Design

Additional Information: Structural Safety Factors

Safety factors, like the load factor used here, are crucial in structural engineering design. They account for uncertainties in several factors:

• Variations in material properties (e.g., the actual strength of the steel might be slightly different from the specified ultimate stress).
• Inaccuracies in load estimation (actual loads might exceed predicted loads).
• Approximations in structural analysis methods.
• Variations in workmanship during construction.
• Degradation of materials over time.

Using a load factor ensures that the structure has a reserve capacity beyond the expected working loads, making it safer and more reliable.

For columns, especially long ones, buckling is also a critical failure mode besides yielding (reaching ultimate stress). The design of actual columns involves checking for both stress failure and buckling failure based on slenderness ratio, end conditions, and material properties, using relevant design codes (like IS, AISC, Eurocode, etc.). This problem simplifies the scenario by focusing purely on the stress criterion under axial load.