91Ó°ÊÓ

When designing a column, one of the critical factors we consider is the slenderness ratio, which helps to understand its tendency to buckle. The slenderness ratio \( \lambda \) is calculated using the formula \( \lambda = \frac{KL}{r} \). In this formula:

• \( K \) represents the effective length factor. For a pin-ended column, \( K \) is usually taken as 1.
• \( L \) is the actual length of the column, and in this exercise, is given as 2400 mm.
• \( r \) is the radius of gyration, found from the cross-sectional properties of the column.

The radius of gyration \( r \) helps to understand the distribution of the cross-sectional area around an axis and is fundamental to identify the column's resistance to buckling. Understanding and calculating the slenderness ratio allows engineers to predict and enhance the stability of columns used in construction.

After determining the slenderness ratio, the next step in allowable stress design is to calculate the critical stress \( \sigma_c \). This is the stress level needed to cause the column to buckle, and it is computed using Euler's Formula: \[ \sigma_c = \frac{\pi^2 E}{\lambda^2} \]Where:

• \( E \) is Young's modulus, the measure of the stiffness of a material (200 GPa or 200,000 MPa in this exercise).
• \( \lambda \) is the slenderness ratio.

With the calculated \( \sigma_c \), engineers need to ensure it does not exceed the allowable yield stress \( \sigma_Y \), which is 250 MPa in this example. Balancing these stresses ensures the column remains safe under load, protecting against buckling before reaching yield stress limits.

The final step in assuring a column's stability is determining the required cross-sectional area \( A_{req} \). This ensures the column can handle the given load without exceeding the critical stress. The required area is calculated using the formula: \[ P = \sigma_c \times A_{req} \]With:

• \( P \) being the load the column must support (180 kN or 180,000 N in this problem).
• \( \sigma_c \) being the previously calculated critical stress.

To find an appropriate cross-sectional area, engineers compare \( A_{req} \) to available column sections. For this exercise, the thicknesses of the angles (6.4 mm, 9.5 mm, 12.7 mm) are considered, seeking the lightest configuration meeting or exceeding the required area, providing both safety and efficiency in design. Selecting the optimal cross-section ensures the structure can withstand the specified load without failure.