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Column Design: The J.B. Johnson Formula for Short Columns (Part 5)

The Danger of the Wrong Formula

In Column Design (Part 4), we introduced the Euler formula. However, Euler's equation assumes the column fails purely by elastic instability (buckling).

If you try to apply Euler's formula to a Short Column (where the slenderness ratio KL/r is less than the transition value Cc), the results are dangerous. The formula will predict a critical load much higher than the column can actually support. In reality, the material will yield (crush) long before it buckles theoretically.

The J.B. Johnson Formula

To accurately predict failure in short or intermediate columns, we use the J.B. Johnson parabolic formula. This equation accounts for the fact that failure is a combination of buckling geometry and material yielding.

J.B. Johnson Column Formula

Physical Interpretation

Look closely at the structure of the formula above. It tells us an important story about how short columns fail:

  1. The Baseline: The formula starts with A × Sy. This represents simple crushing failure (Area times Yield Strength). If the length (L) was zero, the column would simply squash.
  2. The Deduction: The second term subtracts from that strength based on the slenderness ratio. As the column gets longer (higher KL/r), the capacity drops in a parabolic curve until it meets the Euler curve at point Cc.

Euler vs. Johnson: The Material Difference

This brings us to a critical design insight:

  • For Long Columns (Euler): The Yield Strength (Sy) does not appear in the formula. Only Stiffness (E) matters. High-strength steel offers no advantage over mild steel.
  • For Short Columns (Johnson): The Yield Strength (Sy) is a primary variable. Using a heat-treated alloy steel or a higher grade material will significantly increase the load capacity.

Next Step: Automating Calculation

We have covered the theory, the decision rules, and the formulas. Now, let's put it all together into a practical tool. In the next post, we will build an Excel calculator that automatically checks the Cc value and applies the correct formula.

Continue to Part 6:
Column Design (Part 6): Excel Calculator for Critical Loads

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