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Figure 1: The Critical Stress curve. Note how the J.B. Johnson parabola is tangent to the Euler curve at Cc, creating a perfectly smooth transition between failure modes.
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.
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The J.B. Johnson Formula
To accurately predict failure in short or intermediate columns, we use the J.B. Johnson parabolic formula.
Recall: The Column Constant (Cc)
Before calculating the load, recall that Cc defines the transition point based on material properties:
Cc = √
2 π2 E
Sy
The J.B. Johnson formula subtracts the "buckling risk" from the pure material strength:
Pcr = A Sy
[
1 -
(KL / r)2
2 Cc2
]
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Physical Interpretation
Look closely at the structure of the formula above. It tells us an important story about how short columns fail:
- 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.
- 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 regarding material selection:
Long Columns (Euler)
Yield Strength (Sy) is NOT in the formula. Only Stiffness (E) matters.
Result: High-strength steel offers no benefit over mild steel for very long columns.
Short Columns (Johnson)
Yield Strength (Sy) is a primary variable.
Result: Using a heat-treated alloy or higher grade material WILL significantly increase 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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