When building a CNC router or upgrading a 3D printer, the first question is usually: "Is NEMA 17 enough, or do I need NEMA 23?" Most beginners look at the Holding Torque and stop there. This is a mistake. A NEMA 23 motor isn't just "stronger"—it is physically different in ways that affect your speed, your driver choice, and your machine's ability to avoid missed steps. If you choose a NEMA 17 for a heavy gantry, it is far more likely to overheat or lose steps under cutting load. If you choose NEMA 23 for a fast 3D printer, it might actually run slower than the smaller motor. This guide explains the engineering limits of each frame size. Table of Contents 1. Physical Difference (The Frame Size) 2. Torque & Speed (The Inductance Trap) 3. Driver Compatibility 4. Selection Summary Advertisement 1. Physical Difference (The Frame Size) "NEMA" is just a standard for ...
Figure 1: The Column Constant (Cc) marks the boundary between Inelastic Buckling (where material yielding dominates) and Elastic Buckling (pure instability).
The Great Divide: Long vs. Short Columns
In Part 2, we learned how to calculate the Slenderness Ratio (Le/r). This ratio tells us the geometry of the column.
However, geometry isn't enough. We also need to account for the material properties. A steel column behaves differently than an aluminum one.
To decide whether to use the Euler Formula (for elastic instability) or the J.B. Johnson Formula (for inelastic buckling), we must calculate a transition value known as the Column Constant (Cc).
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Calculating the Column Constant (Cc)
The Column Constant represents the specific slenderness ratio where the critical stress equals half of the material's yield strength. It is the borderline between "Short" and "Long."
Cc = √
2 π2 E
Sy
Where:
- E = Modulus of Elasticity (Young's Modulus).
- Sy = Yield Strength of the material.
- π = Pi (3.14159...)
Note: Since Cc depends only on material properties (E and Sy), it is constant for any specific material regardless of the column's shape.
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The Step-by-Step Decision Algorithm
When designing a machine element under compression, follow this exact workflow to ensure safety:
- Analyze Geometry: Determine the actual Length (L) and End Fixity conditions.
- Determine K Factor: Select the correct Constant (K) based on the end supports (e.g., K=1 for pinned, K=2 for free end).
- Effective Length: Compute
Le = K × L. - Radius of Gyration: Calculate
r = √(I/A)for the weakest axis (minimum I). - Slenderness Ratio: Compute the actual ratio
SR = Le / r. - Column Constant: Calculate Cc using the formula above.
- The Final Check: Compare the actual Slenderness Ratio (SR) against the Column Constant (Cc).
The Decision Rule
1. Long Column (Slender)
IF:
Failure Mode: Elastic Buckling (Instability)
→ USE EULER FORMULA
IF:
(KL/r) > CcFailure Mode: Elastic Buckling (Instability)
→ USE EULER FORMULA
2. Short Column (Intermediate)
IF:
Failure Mode: Inelastic Buckling (Yielding)
→ USE J.B. JOHNSON FORMULA
IF:
(KL/r) < CcFailure Mode: Inelastic Buckling (Yielding)
→ USE J.B. JOHNSON FORMULA
Next Step: The Critical Load Formulas
Now that you know which formula to pick, we need to look at the formulas themselves. In the next post, we will present the equations for calculating the Critical Load (Pcr) for both cases.
Continue to Part 4:
Column Design: The Euler Formula for Long Columns (Part 4)
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