Plastic vs Elastic Neutral Axis — PNA/ENA Difference & Shape Factor
The elastic neutral axis (ENA) and plastic neutral axis (PNA) are two distinct reference axes within a cross-section, used for different design philosophies. The ENA is used in elastic design (allowable stress) to compute stresses and deflections. The PNA is used in plastic design (LRFD ultimate strength) to compute the plastic moment capacity Mp and the plastic section modulus Z.
Elastic NA: ∫ y·dA = 0 (first moment of area = 0, the centroid)
Plastic NA: A_above = A_below (equal areas above and below)
In a doubly-symmetric section (rectangle, W-shape, I-shape), the ENA and PNA coincide at mid-depth. In any section with asymmetry — T-shapes, angles, unequal-flange plate girders, and composite sections — the two axes diverge, and the difference directly affects moment capacity.
PRELIMINARY — NOT FOR CONSTRUCTION. All content is for educational and reference use only. Must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in any project.
Why the PNA Shifts from the Centroid
Under elastic bending, stress varies linearly: σ = M·y/I. The neutral axis is where σ = 0, which is the centroid (first moment of area zero). As the moment increases and fibers yield, the elastic core shrinks. At full plasticity:
- Every fiber above the PNA is at Fy in compression
- Every fiber below the PNA is at Fy in tension
- Horizontal equilibrium: Fy·A_above = Fy·A_below → A_above = A_below
The PNA shifts to where half the area lies on each side — this is NOT the centroid unless the section is symmetric. For a T-section with a wide flange and thin web, the PNA shifts toward the flange to capture more area in the thinner web region.
Shape Factor Z/S
The shape factor quantifies the reserve between first yield and full plasticity:
f = Z / S = Mp / My
Where Z is computed using the PNA:
Z = Σ(A_i · d_i) sum of area × distance from PNA for all parts above and below
S = I / y_max elastic section modulus
| Section Type | Shape Factor | Z Computation |
|---|---|---|
| Solid rectangle | 1.50 | Z = bd²/4 |
| W-shape (typical) | 1.10–1.18 | Z = Σ A_i·d_i about PNA (at mid-depth) |
| Channel | 1.12–1.20 | PNA slightly above mid-depth (web-heavy top) |
| Solid circle | 1.70 | Z = d³/6 |
| Thin-walled tube | 1.27 | Z ≈ 1.27·S |
| Equal-leg angle | 1.50–1.60 | PNA near leg intersection |
The modest shape factor for W-shapes (1.10–1.18) means most of the section's reserve capacity comes from moment redistribution in indeterminate structures, not from individual section overstrength.
PNA Location — Worked Example (Unequal-Flange I-Section)
Consider a built-up I-section: top flange 12×0.5 in, bottom flange 8×0.5 in, web 20×0.375 in, total depth 21 in.
ENA (centroid) from bottom:
A_bot_flange = 8 × 0.5 = 4.0 in²
A_web = 20 × 0.375 = 7.5 in²
A_top_flange = 12 × 0.5 = 6.0 in²
A_total = 17.5 in²
First moment about bottom:
flange: 4.0 × 0.25 = 1.0
web: 7.5 × 10.5 = 78.75
top: 6.0 × 20.75 = 124.5
Σ = 204.25
ȳ_ena = 204.25 / 17.5 = 11.67 in from bottom
PNA from bottom:
A_total / 2 = 17.5 / 2 = 8.75 in²
Counting from bottom:
Bot flange: 4.0 in² (remaining needed: 4.75 in²)
Web area rate: 0.375 in²/in
Distance into web: 4.75 / 0.375 = 12.67 in from web start
Web starts at 0.5 in (bottom flange thickness)
ȳ_pna = 0.5 + 12.67 = 13.17 in from bottom
The PNA is 13.17 − 11.67 = 1.50 inches above the ENA, shifted toward the larger top flange. The larger top flange requires more area below the PNA to balance forces, pulling the PNA upward.
Practical Significance
| Condition | ENA = PNA | ENA ≠ PNA |
|---|---|---|
| Symmetric sections | Yes | No (not applicable) |
| Asymmetric built-up plate girders | No | PNA shifts toward larger flange |
| T-sections in tension | No | PNA in or near flange |
| Angle sections under biaxial bend | No | PNA depends on principal axes |
| Partially yielded sections | N/A | ENA shifts continuously |
Frequently Asked Questions
Does the PNA always divide the section into equal areas?
Yes, for pure bending at full plasticity. Every fiber is at yield stress, so force equilibrium Fy·A_comp = Fy·A_tens requires equal areas. If an axial load is present, the PNA shifts from the area-equalizing position to accommodate the axial force.
Why is the shape factor for W-shapes so small?
The W-shape concentrates area in the flanges far from the neutral axis, making the elastic section modulus S already close to the plastic modulus Z. The thin web contributes very little area and moment arm — nearly all the capacity comes from the flanges, which yield uniformly.
How does the PNA concept apply to composite beams?
In composite beams (steel beam + concrete slab), the PNA is determined by force equilibrium between the concrete compression block and the steel tension. See the Plastic Neutral Axis in composite beams for specific AISC I3.2d cases.
Related Terms and Pages
- Plastic Neutral Axis (PNA) — Composite Beam Design
- Plastic Modulus (Z) — Definition & Formula
- Elastic Section Modulus (S) — Definition & Formula
- Plastic Moment (Mp) — Definition & Capacity
- Section Classification — Compact, Non-Compact & Slender
- Moment of Inertia — Definition & Calculation
Educational reference only. Section capacity design must be performed per the governing design code (AISC 360 Chapter F, EN 1993-1-1 Clause 6.2, AS 4100 Section 5) by a licensed Professional Engineer.
Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.