Post-Tensioned Slab Design and Analysis: Tendon Layout Optimization, Loss Calculations, and Long-Term Deflection Prediction

Post-tensioned (PT) concrete slabs enable column-free spans exceeding 12 meters, reduce slab thickness by 25-40%, and eliminate problematic crack patterns—but only when designed correctly. Improper tendon layouts cause excessive camber, cracking, and even edge lift-off failures.

This comprehensive guide covers the advanced analysis techniques required for reliable PT slab design, from first principles through software implementation and field verification.

Why Post-Tensioning?

Conventional RC Slab (8m span): Thickness: 250mm, Deflection: L/250 (borderline), Cracking: Moderate (>0.3mm width), Weight: 6.0 kN/m².

PT Slab (8m span): Thickness: 180mm (-28%), Deflection: L/500 (excellent), Cracking: None (precompression prevents), Weight: 4.3 kN/m² (-28%).

Economic Breakeven: Spans greater than 7.5m make PT cost-effective. Spans greater than 10m make PT significantly cheaper. Complex geometries often make PT the only viable solution.

Fundamental Theory - Equivalent Load Method

Post-tensioning creates internal forces that can be replaced by equivalent external loads for analysis.

Parabolic Tendon Profile: The upward equivalent load (kN/m) is calculated as w_eq = 8 × P × e / L², where P = Prestress force (kN), e = Eccentricity at midspan (mm), and L = Span length (mm).

Example Calculation: Given Span = 9000mm, Prestress = 1200 kN (4 × 12.7mm strands @ 300kN each), Eccentricity = 60mm (tendon at bottom, CGC near mid-depth). Therefore w_eq = 8 × 1200 × 60 / 9000² = 7.1 kN/m (upward). This upward load counteracts dead and live loads, resulting in minimal net deflection.

Tendon Layout Design

Primary Tendons (Spanning Direction): Continuous over multiple spans for efficiency. The optimal profile (moment-balanced) follows e(x) = e_max × [4x(L-x)] / L², which results in parabolic drape matching moment diagram.

Secondary Tendons (Perpendicular): Typically unbonded, spaced 1200-1500mm. They prevent shrinkage cracking, distribute concentrated loads, and control edge lift-off.

Tendon Spacing Rules (PTI DC10.5):

• Maximum spacing in interior zones: 8 × slab thickness
• Maximum spacing at edge/column zones: 4 × slab thickness
• Absolute maximum: 1800mm
• Minimum spacing: 3 × duct diameter (typically 100mm min)
• Sufficient for concrete placement

Critical Zones:

1. Column strips: High shear, need more tendons
2. Middle strips: Flexure-controlled, standard spacing
3. Cantilevers: Draped tendons in top, straight in bottom
4. Edges: Additional tendons to prevent lift-off

Prestress Loss Calculations

Total long-term losses can reach 30-35% of jacking force. Accurate prediction is critical.

Immediate Losses (at stressing)

1. Friction Loss: The prestress at distance x from the jack is calculated as P(x) = P_jack × e^{(-μα - Kx)}, where μ = Wobble coefficient (0.0015-0.0020 per meter for plastic ducts), K = Curvature coefficient (0.15-0.25 per radian), α = Cumulative angular change (radians), and x = Distance from jack (meters).

Example: 30m long tendon, parabolic profile, 4° total angle change. α = 4° × π/180 = 0.070 radians, μ = 0.0020/m, K = 0.20/radian. Loss at far end: P(30) / P_jack = e^{(-(0.0020×30 + 0.20×0.070))} = e^(-0.074) = 0.929. Therefore friction loss = 7.1% at far end.

2. Elastic Shortening (Bonded Tendons Only): For a single tendon in a slab, n = 1, so ΔP_ES = 0 (no interaction). For multiple tendons stressed sequentially, elastic shortening is typically 2-3% of P_jack. The formula is ΔP_ES = (n-1)/(2n) × (E_s/E_c) × f_cgp.

Long-Term Losses (over years)

3. Concrete Creep: ΔP_CR = (E_s/E_c) × φ × f_cgp, where φ = Creep coefficient (1.5-2.5 for normal concrete) and f_cgp = Stress at CGS due to prestress + permanent loads. Typical value: 6-8% of P_jack after 10 years.

4. Concrete Shrinkage: ΔP_SH = E_s × ε_sh, where ε_sh = Shrinkage strain (typically 500-800 microstrain) and E_s = Steel modulus (195,000 MPa). Therefore ΔP_SH = 195,000 × 0.0006 = 117 MPa, which is 6-7% of typical jacking stress (1400 MPa).

5. Steel Relaxation: For low-relaxation strand, ΔP_RE = 0.05 × P_jack (at 1000 hours), which equals 3-5% total after 10 years.

Total Long-Term Losses (Typical unbonded PT slab):

• Friction: 5-8%
• Creep: 6-8%
• Shrinkage: 6-7%
• Relaxation: 3-5%
• Total: 20-28% at 10 years
• Effective prestress: P_eff = 0.75 × P_jack

Deflection Analysis - Critical Procedure

Stage 1 - At Transfer (tendons stressed, no live load): Deflection components include Self-weight: Δ_DL = 5wL⁴/(384EI) (downward) and PT equivalent load: Δ_PT = 5w_eqL⁴/(384EI) (upward). Net: Δ_net = Δ_PT - Δ_DL. Target: Small upward camber (L/1000 to L/2000).

Stage 2 - Long-Term (full prestress losses, sustained loads): Additional deflections include Prestress loss (reduces upward component), Creep (amplifies all deflections by factor (1 + φ_creep)), and Shrinkage (additional camber reduction). The final deflection is predicted as Δ_final = (Δ_DL + Δ_SDL) × (1 + φ) - Δ_PT × (1 - loss%) × (1 + 0.5φ).

Stage 3 - Under Live Load: Instantaneous live load deflection: Δ_LL = 5 × w_LL × L⁴ / (384 × E_c × I_eff), where I_eff = I_gross for precompressed sections (accounts for cracking if any), or reduced I if tension exceeds modulus of rupture.

Case Study - 9m × 12m Flat Slab: Design parameters: Slab thickness = 200mm, Concrete = 35 MPa, Tendons = 12.7mm dia at 0.6 ultimate load, Average prestress = 2.5 MPa. Calculated deflections: At transfer: +12mm (upward camber), Long-term DL+SDL: -8mm (slight sag), DL+SDL+LL: -18mm (L/667 - acceptable).

Punching Shear at Columns - PT-Specific Considerations

Post-tensioning significantly improves punching capacity.

ACI 318-19 Approach: V_c for PT slabs = min of: (1) (2 + 4/β) × √f'_c × b₀ × d × (1 + V_p/V_u), (2) (α_sd/b₀ + 2) × √f'_c × b₀ × d × (1 + V_p/V_u), and (3) 4 × √f'_c × b₀ × d × (1 + V_p/V_u), where V_p = Vertical component of tendons crossing critical section and V_u = Factored shear from gravity loads.

Tendon Contribution: V_p = Σ (P_i × sin θ_i). For parabolic profile through column: θ ≈ 8e/(L × 1000) radians (small angle), therefore V_p = P × 8e/L.

Example: Interior column 400×400mm, Critical section d/2 away gives b₀ = 4(400+200) = 2400mm, d = 160mm (effective depth). Tendons crossing section: 8 tendons @ 1200kN each, Drape: 60mm over 9000mm span. sin θ = 8×60/9000 = 0.053, V_p = 8 × 1200 × 0.053 = 510 kN. Enhancement factor: (1 + 510/800) = 1.64. Therefore punching capacity increases 64% over non-PT slab.

When PT Alone Is Not Enough:

• Very high live loads (>7.5 kPa)
• Slab thickness < 200mm
• Large column spacing (>9m)

Solutions:

1. Column capitals (drop panels)
2. Shear reinforcement (studs, stirrups)
3. Increase slab thickness locally
4. Add additional tendons in column strips

Design Workflow - Step by Step

Step 1 - Preliminary Sizing: Span-to-depth ratios: Flat plates = L/45 to L/50, Flat slabs (drop panels) = L/40 to L/45, Beams = L/20 to L/25. Example: 10m span gives 200-225mm slab thickness.

Step 2 - Load Analysis: Dead load = Self-weight + SDL (flooring, partitions, MEP). Live load per code (offices: 2.5-4.0 kPa). Load factors: 1.2D + 1.6L (ACI 318).

Step 3 - Tendon Layout: Primary direction (long span): Spacing = 1200-1500mm, Force per tendon calculated from load balancing as P = w_bal × L² / (8e). Secondary direction: Minimum reinforcement to control shrinkage, typically 0.5-0.7 kg/m² (vs 1.5-2.0 kg/m² primary).

Step 4 - Loss Calculations: Use refined estimates: Friction from actual tendon path geometry, Time-dependent from age at stressing, humidity, concrete mix. Software options: RAM Concept, ADAPT-PT, or manual spreadsheet.

Step 5 - Stress Checks: At transfer (P_jack - immediate losses): Compression f_c < 0.6 f'_ci (initial strength), Tension f_t < 0.25 √f'_ci (generally no tension). At service (P_eff after all losses + service loads): Compression f_c < 0.45 f'_c, Tension f_t < 0.5 √f'_c (Class U, uncracked).

Step 6 - Ultimate Strength: Moment capacity (unbonded PT): M_n = A_ps × f_ps × (d - a/2) + A_s × f_y × (d - a/2). For unbonded tendons per ACI 318-19: f_ps = f_se + 70 + (f'_c/100 × ρ_p) ≤ f_py. Typical: f_ps = 1600-1750 MPa (vs 1860 MPa bonded).

Step 7 - Serviceability: Deflections checked at transfer, long-term, and under live load. Cracking generally none if properly designed. Vibration: Natural frequency > 4 Hz for residential/offices.

Software Comparison

RAM Concept (Bentley): Integrated analysis and design ✓, Sophisticated loss calculations ✓, Automated tendon layout optimization ✓, Expensive ($8K+ per license) ✗.

ADAPT-PT: Specialized for PT design ✓, Excellent tendon balancing algorithms ✓, Separate analysis required for complex geometries ✗.

SAFE (CSI): Integrates with ETABS ✓, Reasonable cost (~$3K) ✓, Less sophisticated tendon optimization ✗.

Midas Civil: Handles complex bridge decks ✓, Time-dependent analysis ✓, Steep learning curve ✗.

Common Design Errors

Insufficient anchorage zone reinforcement - Bursting forces can crack slab at anchorages. Use spiral reinforcement.

Ignoring tendon eccentricity variation - Tendons must have physical space for drape. Check against duct clash.

Underestimating long-term losses - Conservative loss estimates prevent cracking from loss of precompression.

Poor tendon detailing at openings - Deflect tendons around openings with adequate radius (greater than 6m typically).

Neglecting construction sequence - Stressing sequence affects redistribution. Coordinate with contractor.

Field Verification

During Stressing: Monitor jack pressure and elongation. Elongation should match calculated value ±7%. Record actual losses for comparison with design.

After Construction: Survey slab elevations at transfer and 28 days. Compare actual camber vs predicted. Significant deviation (>25%) indicates design/construction issue.

Long-Term Monitoring: Re-survey at 6 months and 1 year. Check for excessive deflection or cracking. Document for future projects (refine loss models).

Performance Comparison Table

Parameter	RC Flat Slab	PT Flat Slab	Improvement
Span (m)	6-8	8-14	+75%
Thickness (mm)	250	180	-28%
Deflection	L/250	L/500	2× better
Cracking	Moderate	None	Eliminated
Self Weight (kN/m²)	6.0	4.3	-28%
Story Height Saving	—	70mm/floor	Significant

Regulatory and Code References

Design Codes: ACI 318-19 (Chapter 22 - PT slabs), PTI DC10.5-12 (Unbonded single strand tendons), AS 3600-2018 (Australian standard), Eurocode 2 EN 1992-1-1 (European approach).

Inspection Standards: PTI DC80.3 (Specification for unbonded tendons), ASME B31.3 (Pressure systems for grouted PT), ACI 423.7 (Specification for bonded tendons).

Post-tensioned slab design requires mastery of prestress mechanics, material behavior, and long-term performance prediction. The techniques in this guide—from tendon optimization through loss calculations and deflection control—represent best practices developed over decades of PT construction worldwide.

The difference between a good PT design and a problematic one often comes down to accurate loss prediction and rigorous serviceability checks. Master these fundamentals and you master PT design.