The minimum axial compressive load, P, required to initiate buckling for a pinned-pinned slender column with bending stiffness EI and length L is

The correct answer is:

$P=\frac{{\pi }^{2}EI}{L^2}$

(which corresponds to the formula shown in the second image, Image 1).

Step-by-Step Explanation:

1. Understanding Euler's Buckling Theory
Euler's buckling formula determines the maximum axial compressive load that a slender, ideal column can support before it undergoes sudden bending or buckling. The general formula for the critical buckling load ($P_{\text{cr}}$) is given by:

$P_{\text{cr}}=\frac{{\pi }^{2}EI}{{\left(KL\right)}^2}$

where:
• $E$ is the Young's modulus (modulus of elasticity) of the material.
• $I$ is the minimum area moment of inertia of the cross-section (bending stiffness is represented by $EI$).
• $L$ is the actual length of the column.
• $K$ is the effective length factor, which depends on the end support conditions.

2. Determining the Effective Length Factor (K)
The boundary conditions of the column determine the value of the effective length factor $K$:
• For a column with both ends pinned (pinned-pinned), the column is free to rotate at both ends but is constrained against lateral translation. The deflected shape represents a single half-sine wave.
• For this standard pinned-pinned configuration, the effective length factor is:

$K=1$

3. Calculating the Critical Load (P)
Substituting $K=1$ into the Euler buckling formula gives:

$P=\frac{{\pi }^{2}EI}{{(1\cdot L)}^2}=\frac{{\pi }^{2}EI}{L^2}$

Thus, the minimum axial compressive load required to initiate buckling for a pinned-pinned column is indeed $P=\frac{{\pi }^{2}EI}{L^2}$, as shown in Image 1.