To determine the lightest square shape that would be safe for the load under both short and long column conditions, we need to use the provided formulas for each case.

Given Data:

• Length of column, $L = 15 \, \text{ft} = 180 \, \text{in}$ (converting to inches for consistency in units)
• Modulus of Elasticity, $E = 29,600 \, \text{ksi} = 29,600 \times 1000 \, \text{psi}$
• Critical load, $P_{cr}$ (this needs to be determined based on allowable stress or provided load, but it’s not given explicitly in the problem).

Let's proceed by calculating each column type with respect to its conditions.

Step-by-Step Solution Outline

For Short Columns

1. Formula for Short Column: $P_{cr} = F_y \times A$
   • Where $F_y$ is the yield stress, and $A = b^2$ for a square cross-section with side length $b$.
2. Rearranging, we can solve for $b$ once $P_{cr}$ and $F_y$ are defined.

For Long Columns

1. Formula for Long Column (Euler’s Buckling): $P_{cr} = \frac{\pi^2 \cdot E \cdot I}{(K \cdot L)^2}$
   • Where $I$ is the moment of inertia, given by $I = \frac{b^4}{12}$ for a square cross-section.
   • $K$ is the effective length factor (often 1.0 for pinned columns unless otherwise specified).
2. Substitute $I = \frac{b^4}{12}$ into the equation and solve for $b$ based on $P_{cr}$.

Could you please confirm if there is a given load or allowable stress value $F_y$ that we should use for this problem? This will help in proceeding with precise calculations for $b$.

Let me know if you have questions or need further clarification.

Related Questions:

1. What is the typical range for yield stress $F_y$ in similar materials?
2. How does the effective length factor $K$ influence the stability of a column?
3. What are the key differences in behavior between short and long columns under load?
4. Why is Euler’s formula preferred for calculating critical load in long columns?
5. How does changing the cross-sectional shape impact the load-bearing capacity of columns?

Tip:

When dealing with buckling calculations, always check the boundary conditions of the column (fixed, pinned, or free) as they directly affect the effective length factor $K$ and subsequently the critical load.