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: Buckling is a geometric instability failure, not just a material strength failure.
In a mechanical design situation, the expected load on a column and its length are usually known. The designer's job is to specify the structural parameters to prevent failure.
Advertisement
The 5 Key Design Parameters
- End Fixity: How is the column attached? (Pinned-Pinned, Fixed-Free, etc.) This determines the effective length factor (K).
- Cross Section: The shape (I-beam, Tube, Solid Round). This determines the Radius of Gyration (r).
- Material: Determines Stiffness (Modulus E) and Strength (Yield Sy).
- Design Factor (N): The safety margin.
- Final Dimensions: The actual width/thickness required.
Because the cross-section (Item 2) determines the slenderness ratio, but you can't pick the cross-section until you know the allowable stress, column design is inherently iterative.
The Iterative Design Loop:
1. Assume a dimension (guess).
2. Calculate the Slenderness Ratio (KL/r).
3. Select the correct formula (Euler vs. Johnson).
4. Solve for allowable load.
5. Compare with required load. If not safe (or too heavy), repeat.
Design software or Excel spreadsheets significantly speed up this process.
1. Assume a dimension (guess).
2. Calculate the Slenderness Ratio (KL/r).
3. Select the correct formula (Euler vs. Johnson).
4. Solve for allowable load.
5. Compare with required load. If not safe (or too heavy), repeat.
Design software or Excel spreadsheets significantly speed up this process.
Long vs. Short Columns (The Formulas)
Before analyzing, we must classify the column based on its Slenderness Ratio (KL/r) compared to the Column Constant (Cc).
Figure 2: Long columns fail by elastic instability (Euler). Short columns fail by yielding (Johnson).
1. Long Columns (Euler Formula)
If KL/r > Cc, the column is "Long." It will buckle elastically before the material yields. The geometry (Stiffness E and Inertia I) controls failure, not the material strength.
Critical Load (Pcr)
Pcr = (π² · E · I) / (K · L)²
Pcr = (π² · E · I) / (K · L)²
Or expressed as critical stress:
Pcr / A = (π² · E) / (K · L / r)²
Advertisement
2. Short Columns (J.B. Johnson Formula)
If KL/r < Cc, the column is "Intermediate" or "Short." It will fail due to a combination of crushing and buckling. The material Yield Strength (Sy) matters here.
Pcr = A · Sy · [ 1 - ( Sy · (KL/r)² ) / ( 4 · π² · E ) ]
Design Factor Guidelines:
- N = 3.0: Typical industrial conditions.
- N = 2.0: Smooth, well-known loading.
- N = 4.0+: Shock or impact loads.
- N = 3.0: Typical industrial conditions.
- N = 2.0: Smooth, well-known loading.
- N = 4.0+: Shock or impact loads.
Eccentrically Loaded Columns
An eccentric load is one applied away from the centroidal axis by a distance e. This creates a bending moment in addition to compression.
Figure 3: Eccentricity 'e' induces bending, significantly reducing the load capacity.
To analyze this, we use the Secant Formula. It calculates the maximum stress in the outermost fibers (at distance c from the neutral axis):
Max Stress (σmax):
σmax = (P/A) · [ 1 + (e·c/r²) · sec( (Le/2r) · √(P / E·A) ) ]
σmax = (P/A) · [ 1 + (e·c/r²) · sec( (Le/2r) · √(P / E·A) ) ]
Note: The term inside the sec() function is in radians. Since standard calculators don't have a "secant" button, remember that sec(x) = 1 / cos(x).
The Calculation Challenge
The Secant formula is transcendental—you cannot simply rearrange it to solve for Load (P). To find the allowable load for a specific Design Factor (N), you must perform an iterative search to find the load P that makes σmax = Sy / N.
Comments