Plastic Neutral Axis (PNA) — Composite Beam Design
The Plastic Neutral Axis (PNA) is the axis within a cross-section where the stress reverses from compression to tension at the fully plastic moment. In composite beam design — a steel beam acting compositely with a concrete slab — the PNA location fundamentally determines the moment capacity, ductility, and shear connection requirements.
At full plasticity: ΣF_compression = ΣF_tension
The PNA is the dividing line where F_comp = F_tension at full yield stress
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.
Elastic vs Plastic Neutral Axis
| Property | Elastic NA (ENA) | Plastic NA (PNA) |
|---|---|---|
| Basis | Transformed section centroid | Force equilibrium (C = T) |
| Material behavior | Linear-elastic | Fully plastic |
| Shifts with loading? | No — constant for given section | Yes — moves as plasticity develops |
| Used for | Deflection, elastic stress | Ultimate strength, moment capacity |
| Location (same section) | Geometric centroid | Not at centroid if strength asymmetry |
For a homogeneous steel section in pure bending, ENA = PNA (at mid-depth). For a composite section with concrete above steel, the PNA shifts upward toward the slab because concrete's compressive strength is lower than steel's tensile strength — more concrete area is needed to balance the steel's tension force.
AISC 360 — Three PNA Cases for Composite Beams
Per AISC I3.2d, the PNA location determines the design case:
Case 1: PNA in Concrete Slab (Most Efficient)
Condition: a ≤ t_slab where a = (As × Fy) / (0.85 × f'c × b_eff)
The entire steel section is in tension. The compressive force is carried entirely by the concrete slab. This is the most efficient configuration — the steel's full yield strength is utilized, and the moment arm is maximized.
Mn = As × Fy × (d/2 + t_slab − a/2)
Practical example: A W18x55 (As = 16.2 in², Fy = 50 ksi, d = 18.1 in) with a 4.5 in slab (f'c = 4 ksi, b_eff = 90 in):
C_max_concrete = 0.85 × 4 × 90 × 4.5 = 1,377 kips
T_steel = 16.2 × 50 = 810 kips
Since 810 < 1,377, PNA is in the slab.
a = 810 / (0.85 × 4 × 90) = 2.65 in
Moment arm = 18.1/2 + 4.5 − 2.65/2 = 9.05 + 4.5 − 1.33 = 12.22 in
Mn = 810 × 12.22 = 9,898 kip-in = 825 kip-ft
Case 2: PNA in Steel Top Flange
Condition: a > t_slab but C_slab + C_flange ≥ T_web + T_bot_flange
Partial yielding of the top flange. The PNA divides the top flange into compression and tension zones. Less efficient than Case 1 because part of the steel is in compression rather than providing additional tensile capacity.
Case 3: PNA in Steel Web
Condition: C_slab + C_top_flange < T_web + T_bot_flange
The PNA enters the web. The steel section's top portion (top flange + part of web) is in compression, reducing the net tensile contribution. This is the least efficient case — common with very shallow beams or thick slabs where the concrete compression block is small relative to the steel tension capacity.
Stress Distribution at Full Plastic Moment
Compression
┌─────────────────────────────┐
│ Concrete slab │ ← 0.85 f'c (Whitney stress block)
│ ██████████████████████████ │ ← depth = a (stress block depth)
╞═════════════════════════════╡ ← steel-concrete interface
│ Steel top flange │ ← Fy in compression (if PNA below flange)
│ ───────────────────────── │
│ Steel web │ ← Fy compression above PNA
╞═════════════════════════════╡ ← PNA (stress = 0 at the dividing line)
│ Steel web │ ← Fy tension below PNA
│ ───────────────────────── │
│ Steel bottom flange │ ← Fy in tension
└─────────────────────────────┘
Tension
Full vs Partial Shear Connection
The PNA is also influenced by the degree of shear connection:
| Connection Type | ΣQn / C_max | PNA Behavior |
|---|---|---|
| Full shear connection | ≥ 1.0 | PNA determined by section equilibrium only |
| Partial (≥25%) | 0.25-1.0 | PNA shifts — compression in slab is limited by shear stud capacity |
| No composite action | 0 | PNA at mid-depth of steel section alone |
With partial shear connection, the shear studs cannot transfer the full force required for Case 1 PNA location. The PNA shifts downward into the steel, reducing capacity. AISC I3.2d(2) provides the interpolation for partial composite action.
EN 1994-1-1 — European Composite PNA
EN 1994 uses the same fundamental C = T equilibrium but with different material factors:
Concrete compression: C_c = (0.85 × fck / γC) × b_eff × 0.8x (instead of 0.85f'c × b_eff × a)
Steel tension: T_s = As × fyd (where fyd = fy / γM0)
The 0.8x factor accounts for the rectangular stress block approximation in Eurocode (where x is the neutral axis depth in cracked concrete analysis). The effective width b_eff in EN 1994 may differ from AISC — typically L/8 on each side of the beam.
Significance in Design
| PNA Location | Ductility | Moment Capacity | Shear Studs Required |
|---|---|---|---|
| In concrete slab | Highest | Highest | Highest (full C_max) |
| In steel top flange | Moderate | Slightly reduced | Full or partial |
| In steel web | Lower | Reduced | Partial may be sufficient |
| In steel (non-composite) | N/A | Lowest (steel alone) | None |
The designer's goal is typically Case 1 — PNA in the slab — because it maximizes both capacity and ductility. This requires sufficient slab thickness, effective width, and concrete strength to develop the full tensile capacity of the steel beam.
Frequently Asked Questions
How does the PNA differ from the centroid of the section? The centroid is the geometric center of area (elastic neutral axis). The PNA is the force equilibrium point at full plasticity. In a symmetric homogeneous steel section, they coincide at mid-depth. In a composite section, they differ because concrete and steel have different yield stresses — the PNA shifts upward toward the slab to balance a larger concrete area (lower stress) against a smaller steel area (higher stress).
What happens if the PNA is very close to the top of the steel beam? This is the transition zone between Case 1 and Case 2. The capacity calculation switches equations, and the behavior changes qualitatively — the top flange moves from full tension to partial compression. At the exact transition point (PNA at the steel-concrete interface), both sets of equations should give the same Mn. The designer should verify that the chosen shear stud layout provides adequate horizontal shear transfer at this critical interface.
Can a composite beam have the PNA below the steel beam entirely? No — this would mean the entire composite section is in compression, which requires no bending moment (pure axial compression). In flexure with positive moment (sagging), the bottom fiber is in tension and the top fiber in compression. The PNA must lie between them. For negative moment (hogging) regions over supports, the slab is in tension and cracked — typically, only the steel reinforcement in the slab contributes to tension capacity.
Related Terms and Pages
- Compact Section — Lambda Limits & AISC Table B4.1
- Effective Width — Composite Beam Design
- Plastic Modulus — Definition & Formula
- Shear Lag — Definition & Design Effect
- Beam Capacity Calculator — Free Online Tool
- Composite Beam Design — Complete Guide
Educational reference only. Composite beam design must be performed per the governing design code (AISC 360 Chapter I, EN 1994-1-1, AS 2327.1) by a licensed Professional Engineer. All designs must be independently verified.
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.