Prestressed Concrete - Cheat Sheet
Your 30-minute revision companion for prestressed concrete design, covering pre- vs post-tensioning, service stresses, losses, and load balancing. Master the force equations and sign conventions to crush this topic on the PRC exam.
Fundamental Concepts & Mechanisms
Common values
Value
15–20%
Symbol
ΔP
Quantity
Typical total prestress losses (post-tensioned)
Value
18–25%
Symbol
ΔP
Quantity
Typical total prestress losses (pre-tensioned)
Value
0.80–0.85
Symbol
R
Quantity
Effectiveness ratio (typical range)
Important facts
- Pre-tensioning transfers force by bond friction over the transfer length (plant-controlled, typically 40–60 strand diameters).
- Post-tensioning uses mechanical end anchorages and is more flexible for field application.
- Grouting in post-tensioned ducts provides corrosion protection and increases long-term effectiveness.
- Eccentric prestress (tendon below centroid) applies additional compression at the bottom fiber and relieves the top fiber.
- A draped tendon exerts an upward distributed load on the concrete ('load balancing').
- Critical stresses must be checked at both transfer (Pᵢ, minimal load) and in service (Pₑ, full load).
- The critical fiber reverses between transfer and service due to growth of applied moment.
Key definitions
Prestressing
Example
Parabolic tendons in a post-tensioned bridge beam exert upward distributed force to offset gravity bending.
Introduction of internal compressive force into concrete before service loads, to keep the member in compression or low tension under load.
Pre-tensioning
Example
Precast hollow-core slabs, pile caps, and railway sleepers.
Strands tensioned against external abutments before casting; force transferred by bond after concrete cures.
Post-tensioning
Example
Cast-in-place bridge girders, transfer beams, and post-tensioned flat slabs.
Strands in ducts tensioned after concrete cures; force anchored at ends, ducts typically grouted.
Effective Prestress (Pₑ)
Example
If Pᵢ = 1200 kN and losses = 16.7%, then Pₑ = 0.833 × 1200 = 1000 kN.
Jacking force minus all losses; the actual force available for design in service.
Effectiveness (R)
Example
R = 0.83 means 17% total loss.
Ratio of effective to initial prestress: R = Pₑ / Pᵢ; typically 0.80–0.85.
Diagrams to know
- Stress block at transfer vs service (showing shift from top to bottom as load grows).
- Parabolic tendon profile with sag e and resulting upward load w_bal.
- Equivalent load on concrete from eccentric tendon (axial force + couple).
Service Stress Analysis (Core Design Formulas)
Formulas
fₜₒₚ = P/A − (Pec_top)/I + (Mc_top)/I
Meaning
P = prestress force (kN); A = section area (mm²); e = eccentricity below centroid (mm); c_top = distance from centroid to top fiber (mm); I = second moment of area (mm⁴); M = external bending moment (kN⋅m); compression is positive.
Watch out
Sign of Pec term: eccentric prestress below centroid adds compression at bottom, relieves top. If e is above centroid, flip the sign. Always convert M to N⋅mm.
When to use
When finding the stress in the top fiber at any section; check at transfer and service.
f_bot = P/A + (Pec_bot)/I − (Mc_bot)/I
Meaning
c_bot = distance from centroid to bottom fiber (mm); all other terms as above.
Watch out
The Pec term adds compression at the bottom (where the tendon is), so it enters with a plus sign. The applied moment M reduces bottom compression (sagging load creates tension at bottom).
When to use
When finding the stress in the bottom fiber at any section; check at transfer and service.
f = P/A ± (Pec)/I ∓ (Mc)/I
Meaning
Compact form: compression positive; ± depends on whether c is measured to top (−) or bottom (+) fiber.
Watch out
This is the superposition of three effects: (1) axial prestress P/A, (2) moment from eccentricity Pec/I, (3) external applied moment Mc/I. Do not omit any term.
When to use
General service stress formula for any fiber; substitute c and M with correct signs.
Common values
Value
√(f'c) MPa
Symbol
f_t,allow
Quantity
Allow tensile stress, top fiber, transfer (ACI/NSCP)
Value
0.5 f'c MPa
Symbol
f_c,allow
Quantity
Allow compression, top fiber, transfer
Value
0 MPa (no cracking)
Symbol
f_t,service
Quantity
Allow tension, service (exposed member)
Important facts
- Compression is positive, tension is negative in the stress formulas.
- The critical fiber at transfer is usually the top (smallest precompression opposes least applied load).
- The critical fiber in service often flips to the bottom as applied moment grows.
- A typical design goal: keep both top and bottom fibers in compression throughout the member's life.
- If the top fiber goes into tension at transfer, the member will crack unless grouted post-tensioned tendons provide restraint.
- The moment term (Mc/I) opposes the eccentricity moment (Pec/I), so the net stress depends on which is larger.
Key definitions
Eccentricity (e)
Example
A 300 × 600 mm beam with centroid at 300 mm height; tendon at 150 mm above base gives e = (300 − 150) = 150 mm below centroid.
Distance of tendon from the section centroid; positive if below centroid (usual).
Transfer Stress Limit (Top Fiber)
Example
For f'c = 40 MPa: allow ≤ 20 MPa compression; allow ≤ 6.3 MPa tension.
At release, top fiber must not exceed 0.5 f'c in compression or √(f'c) in tension (ACI 318, NSCP 2015).
Service Stress Limit (Top Fiber)
Example
For f'c = 40 MPa and f_top = −0.15 f'c = −6 MPa, the fiber is in 6 MPa tension (undesirable unless mitigated).
In service, top fiber typically kept in compression (0 to 0.45 f'c) to prevent cracking; zero tension allowed if member is exposed to weather.
Diagrams to know
- Stress distribution at transfer (heavily compressed at bottom where tendon is).
- Stress distribution in service (compression may flip sign at one or both fibers).
- Stress envelope over the span of a simply-supported prestressed beam (maximum at mid-span).
Prestress Losses
Formulas
P_e = P_i − ΔP_total = R × P_i
Meaning
Pₑ = effective prestress (kN); Pᵢ = jacking (initial) force (kN); ΔP_total = sum of all losses (kN); R = effectiveness ratio.
Watch out
Do not use Pᵢ for service calculations; always use Pₑ. Losses accumulate: elastic + anchorage seating + friction + creep + shrinkage + relaxation.
When to use
Convert jacking force to service force after accounting for all losses; this Pₑ is used in service stress equations.
ΔP_ES = (E_s / E_c) × (P_i / A_c) × A_c = E_s × (P_i / A_c) / E_c × A_c
Meaning
Elastic shortening loss; Eₛ ≈ 200 GPa (strand), Eₑ = concrete modulus; Aᶜ = concrete area; loss is proportional to initial stress on concrete.
Watch out
This is an immediate loss; it happens as soon as strands are released or tensioned. Approximate loss: 2–4% of Pᵢ.
When to use
Estimate immediate loss at release (pre-tensioning) or right after tensioning (post-tensioning).
ΔP_friction ≈ μ × P_i × (k × x + f_rec)
Meaning
μ = coefficient of friction (0.20–0.25 for strand in metal duct); k = curvature coefficient (0.001–0.002 rad⁻¹); x = path length along duct (m); f_rec = wobble coefficient (0.0005–0.001 m⁻¹); applies only to post-tensioning.
Watch out
Friction loss increases with curvature (k⋅x term) and with total path length (f_rec⋅x term). Typical loss: 5–10% for a curved duct.
When to use
Calculate friction loss in post-tensioned members with parabolic or other curved tendon paths.
ΔP_anchorage = (Δ_slip × E_s × A_s) / L_transfer
Meaning
Δ_slip = anchorage seating loss (mm, typically 5–10 mm); Eₛ = strand modulus; Aₛ = strand area; L_transfer = transfer length over which force regains.
Watch out
This is a localized loss near the anchor; total loss over the entire span is small but critical at the anchor end.
When to use
Account for tendon slip at the anchor during post-tensioning; significant near the anchor, negligible away.
ΔP_creep + ΔP_shrinkage + ΔP_relaxation ≈ 8–12% × P_i (total time-dependent)
Meaning
These three losses occur over weeks to months; creep depends on sustained stress, shrinkage on humidity, relaxation on strand stress and time.
Watch out
Time-dependent losses dominate total loss (~10% alone); relaxation loss increases if the strand is kept at high stress (> 0.7 f_pu). Use 2–3 year or 5-year loss tables from design codes (ACI 318, NSCP 2015).
When to use
Estimate the long-term loss; use in service stress calculations and deformation predictions.
Common values
Value
0.20–0.25
Symbol
μ
Quantity
Coefficient of friction (strand in metal duct)
Value
0.001–0.002 rad/m
Symbol
k
Quantity
Curvature coefficient (typical parabolic duct)
Value
0.0005–0.001 m⁻¹
Symbol
f_rec
Quantity
Wobble coefficient (typical duct)
Value
5–10 mm
Symbol
Δ_slip
Quantity
Anchorage seating loss (typical)
Value
200 GPa
Symbol
Eₛ
Quantity
Strand modulus
Important facts
- Immediate losses (elastic shortening, anchorage seating, friction) occur within hours to days.
- Time-dependent losses (creep, shrinkage, relaxation) develop over months to years; use code tables (ACI 318 Appendix B, NSCP 2015 equivalent).
- Total loss for post-tensioned: ~15–20% of Pᵢ.
- Total loss for pre-tensioned: ~18–25% of Pᵢ (higher due to different creep environment).
- Effective prestress Pₑ = (1 − ΔP_total/Pᵢ) × Pᵢ must be used for all service calculations.
- Friction loss depends on tendon profile (straight ducts have zero friction loss); draped tendons lose more.
- At the anchorage end of a post-tensioned member, friction loss starts at zero and grows along the duct; this creates a stress gradient near the anchor.
- Relaxation loss is reduced if the tendon is stress-relieved (low-relaxation strand); always specify low-relaxation in design.
Key definitions
Jacking Force (Pᵢ)
Example
For f_pu = 1860 MPa, typical Pᵢ = 0.75 × 1860 = 1395 MPa ≈ 75% f_pu per strand.
Initial prestress force applied; typically 75–80% of strand ultimate strength f_pu to avoid over-stress.
Elastic Shortening
Example
Concrete stress = 12 MPa, Eₛ/Eᶜ = 5 → loss ≈ 5 × 12 = 60 kPa in strand (order of magnitude).
Immediate reduction in prestress as concrete compresses under the applied force; occurs at transfer or tensioning.
Anchorage Seating
Example
5 mm slip over a 2 m transfer length; effective strain ≈ 5/2000 = 0.0025 → loss ≈ 0.0025 × 200 000 = 500 MPa in a small region.
Slip of the strand at the anchor as the wedge or cone seats; loss is localized near the anchor.
Friction Loss (Post-tensioning)
Example
Parabolic duct with curvature → 7% loss at the far end, 0% at the jack end.
Loss of tendon force due to friction between strand and duct as the jack tensions the strand.
Diagrams to know
- Loss diagram along the span of a post-tensioned beam (friction decreases from anchor end; ends at Pₑ after all losses).
- Timeline of loss accumulation (immediate spike at transfer/tensioning, then gradual creep/shrinkage/relaxation curve).
- Stress–time plot showing relaxation loss as a curve (higher initial stress = higher relaxation loss).
Reactions or equations
ΔP_total = ΔP_ES + ΔP_friction + ΔP_anchorage + ΔP_creep + ΔP_shrinkage + ΔP_relaxation
Note
The order of magnitude: elastic shortening (1–3%), friction (5–10%), anchorage seating (small, localized), time-dependent (8–12%). Total: 15–25%.
Conditions
Sum of all loss components; friction applies only to post-tensioning.
Load Balancing (Design Shortcut)
Formulas
w_bal = (8 × P × e) / L²
Meaning
w_bal = uniformly distributed load balanced by the tendon (kN/m); P = prestress force (kN); e = sag of parabolic tendon at mid-span (m); L = span (m).
Watch out
This formula assumes a parabolic tendon profile; straight tendons produce zero load balancing. All units must be consistent: P in kN, e and L in m, result in kN/m. If L is in mm, divide by 10⁶ to convert.
When to use
Estimate how much uniform gravity load is 'carried' by the draped tendon as axial precompression (no bending).
P × e = (w_bal × L²) / 8
Meaning
Rearranged: product of prestress and sag equals one-eighth times load times span squared.
Watch out
This is the moment equivalent of the balanced load; use it to size the tendon profile and force.
When to use
Design the tendon: given allowable load w and span L, find the required P×e product.
Common values
Value
L/10 to L/8
Symbol
e
Quantity
Typical sag, simply-supported beam
Value
50–80% of total applied load
Symbol
w_bal / w_total
Quantity
Load balancing ratio (typical design)
Important facts
- A parabolic tendon (sag e at mid-span) exerts an upward distributed load on the concrete; the magnitude is w_bal = 8Pe/L².
- A straight tendon produces no load balancing (e = 0).
- The balanced load acts as a distributed upward force along the entire span; it is not a point load at mid-span.
- If applied load < w_bal, the member has negative bending moment (compression throughout); deflection is near zero.
- If applied load = w_bal, bending moment is zero, and the member carries only axial precompression P/A (ideal design).
- If applied load > w_bal, the excess load causes positive bending (conventional sagging); the precompressed section resists this as an ordinary RC beam would.
- The load balancing concept is the key reason prestressed members are shallow and long-span friendly: most of the gravity load is 'pre-balanced' by the tendon, so deflection and cracking are minimal.
- In service, the effective force is Pₑ (after losses), so the actual balanced load is w_bal,e = 8Pₑe/L².
Key definitions
Load Balancing
Example
A 15 m span beam with 20 kN/m self-weight: if P = 1000 kN and sag e = 300 mm, then w_bal = 8(1000)(0.3)/(15²) = 10.7 kN/m ≈ half the load is 'balanced'.
Selection of tendon force P and sag profile e such that the upward distributed load from the eccentric tendon cancels a chosen portion of gravity load; the result is pure axial precompression with minimal bending.
Equivalent Load (from Draped Tendon)
Example
w_bal = 10.7 kN/m upward; acts to lift the beam, canceling 10.7 kN/m of downward weight.
Upward distributed load exerted on the concrete by an eccentric, parabolic tendon; acts like an inverted load.
Diagrams to know
- Parabolic tendon profile with sag e; annotate that it exerts upward distributed force w_bal.
- Shear and bending moment diagrams for: (a) applied load alone, (b) balanced load alone, (c) net (showing cancellation if loads are equal).
- Load-balancing design flowchart: choose w_bal → calculate P×e → select tendon profile and force.
Reactions or equations
M_applied − M_tendon = Net moment → bending stress
Note
If w = w_bal, then M_net = 0 and bending stress = 0; the member carries only axial precompression.
Conditions
M_tendon = P×e = (w_bal × L²)/8 (moment from eccentric prestress); M_applied = (w × L²)/8 (moment from applied load).
Design Steps & Stress Checks
Formulas
Check at Transfer: f_i = P_i / A ± (P_i × e × c) / I ± (M_i × c) / I
Meaning
At the moment of release; Pᵢ = jacking force; Mᵢ = moment from self-weight and formwork (usually small); use allowable stress limits per code.
Watch out
The critical fiber at transfer is usually the top (least precompressed); if top fiber goes into tension > √(f'c), the member will crack. Top fiber is compressed least by eccentricity (since tendon is below centroid).
When to use
Immediately after releasing pre-tensioned strands or after tensioning a post-tensioned tendon; check that concrete does not over-compress or excessively crack.
Check in Service: f_service = P_e / A ± (P_e × e × c) / I ∓ (M_service × c) / I
Meaning
After all losses have occurred and full service load is applied; use Pₑ and all applied loads (dead + live); check against service stress limits (typically 0 to 0.45f'c for top fiber, no tension if exposed).
Watch out
The critical fiber often flips to the bottom in service due to growth of applied moment. Bottom fiber must also be checked and usually remains in compression. If service top fiber goes into tension, provide auxiliary reinforcement or adjust P and e.
When to use
Final design check; must satisfy both top and bottom fiber limits to prevent cracking and excessive compression.
f_allowable (top, transfer) = min(0.5 × f'c, 0.6 × f'y)
Meaning
Conservative allowable compression at top during transfer; f'c = concrete strength (MPa), f'y = yield stress of regular reinforcement (typically 275 or 415 MPa in Philippines).
Watch out
This is a code limit (ACI 318, NSCP 2015); exceeding it may cause micro-cracking or spalling. The 0.5f'c rule is standard.
When to use
Set the upper limit for top fiber stress at transfer to prevent over-compression and concrete crushing.
f_allowable (top, tension, transfer) = √(f'c) MPa
Meaning
Maximum tensile stress allowed at top fiber during transfer (per ACI, NSCP); f'c in MPa.
Watch out
If initial top fiber tension > √(f'c), the design fails; increase P, increase e, or reduce formwork loads. For f'c = 40 MPa, limit ≈ 6.3 MPa tension.
When to use
Ensure top fiber does not crack during handling or transfer; if exceeded, the member is damaged and must be re-cured or rejected.
Common values
Value
0.5 f'c
Symbol
f_c,allow,xfer
Quantity
Allowable top fiber compression (transfer, NSCP/ACI)
Value
√(f'c) MPa
Symbol
f_t,allow,xfer
Quantity
Allowable top fiber tension (transfer, NSCP/ACI)
Value
0.45 f'c
Symbol
f_c,allow,service
Quantity
Allowable top fiber compression (service, exposed)
Value
0 to 0.1√(f'c) (typically 0)
Symbol
f_t,allow,service
Quantity
Allowable top fiber tension (service, exposed, protected from weather)
Important facts
- Design always requires two stress checks: (1) at transfer with Pᵢ, (2) in service with Pₑ.
- At transfer, the applied load is minimal (self-weight or formwork only); the critical constraint is preventing over-compression or cracking of the top fiber.
- In service, the full dead and live load are applied; the critical constraint is usually the bottom fiber (it must not be overly compressed, and the top must not crack).
- The ratio of applied moment to eccentric moment changes between transfer and service, causing the critical fiber to shift.
- If the design fails at transfer (top fiber in excessive tension), increase P or e. If it fails in service (top fiber in tension or bottom over-compressed), reduce applied load or redesign the section.
- Intermediate checks (at different stages of loading) are sometimes required for complex member shapes (T-beams, etc.).
- A typical 'balanced' design is one where top and bottom fibers just meet their limits at transfer and service, respectively.
Key definitions
Transfer (Release)
Example
Precast beam in the yard: jacking ram is retracted, strands slip and bond to concrete; stresses jump instantly.
The moment when pre-tensioned strands are released from the abutments; prestress is suddenly applied to the concrete.
Service State
Example
Bridge carrying traffic, 6 months after post-tensioning; use Pₑ and full design load w.
The long-term condition under full dead and live load, after all losses have occurred (weeks to months after construction).
Diagrams to know
- Stress diagram at transfer: mostly compression, concentrated at bottom (tendon location).
- Stress diagram in service: more uniform compression or even tension at top (if not well designed).
- Stress envelope over the length of a simply-supported beam (maximum stress at mid-span, zero at supports).
Worked Numerical Examples — PRC Exam Style
Important facts
- Example 1 — Service Stresses: A 300 × 600 mm rectangular beam (A = 180,000 mm², I = 5.4 × 10⁹ mm⁴, c = 300 mm) has P_e = 900 kN at e = 150 mm below centroid, plus applied moment M = 120 kN⋅m. Top stress: f_top = 900,000 / 180,000 − 900,000(150)(300) / (5.4 × 10⁹) + 120 × 10⁶ (300) / (5.4 × 10⁹) = 5.0 − 7.5 + 6.67 = 4.17 MPa compression. Bottom stress: f_bot = 5.0 + 7.5 − 6.67 = 5.83 MPa compression. Both in compression → OK.
- Example 2 — Load Balancing: Parabolic tendon P = 900 kN, sag e = 150 mm, span L = 10 m. Balanced load: w_bal = 8(900)(0.15) / (10)² = 1080 / 100 = 10.8 kN/m. If self-weight = 10 kN/m, then 0.8 kN/m of the load causes bending; stress from that is minimal.
- Example 3 — Losses: Jacking Pᵢ = 1200 kN. Losses: elastic 2%, friction 5%, anchorage 1%, creep/shrinkage/relaxation 8%. Total = 16%. Effective P_e = (1 − 0.16) × 1200 = 1008 kN. Use P_e for service calculations.
- Example 4 — Transfer vs Service: f'c = 40 MPa. At transfer: top fiber = − 3 MPa (tension) → 3 < √40 ≈ 6.3 MPa → OK. In service: top fiber = + 2 MPa (compression) → 2 < 0.45 × 40 = 18 MPa → OK. Design passes both checks.
- Example 5 — Effective Stress: P_i = 1500 kN, losses 20%, A = 200,000 mm². At transfer: axial = 1500 / 200,000 × 1000 = 7.5 MPa. In service: axial = 1200 / 200,000 × 1000 = 6.0 MPa. The drop (1.5 MPa) is from losses.
Philippine Standards & Code References
Important facts
- NSCP 2015 Section 424.1 defines two types: precast (pre-tensioned) and cast-in-place (post-tensioned); applies same stress limits to both.
- NSCP 2015 Section 424.4 specifies transfer stress limits: top compression ≤ 0.5f'c, top tension ≤ √(f'c).
- NSCP 2015 Section 424.5 specifies service stress limits: varies by member exposure and cracking control.
- ACI 318 Appendix B provides creep and shrinkage coefficients for loss calculation; NSCP references these tables.
- Both NSCP and ACI allow two design methods: (1) Allowable Stress Design (ASD, most common for existing structures), (2) Ultimate Strength Design (USD, more modern, not often on the PRC exam).
- For the PRC exam, focus on ASD with allowable stresses (traditional method) and load balancing (practical design shortcut).
- Post-tensioned members must have grouted ducts after tensioning (per NSCP 2015, Section 424.11) unless specifically allowed otherwise; grouting ensures corrosion protection and long-term durability.
Key definitions
NSCP 2015 (National Structural Code of the Philippines)
Example
NSCP 2015, Section 424 specifies allowable stresses, loss estimation, and design checks for prestressed members.
The official design code for structures in the Philippines; Chapter 4 covers prestressed concrete (precast and post-tensioned).
ACI 318-19 (Building Code Requirements for Structural Concrete)
Example
ACI 318 Section 27 (Prestressed Concrete) is the basis for NSCP Chapter 4.
The primary reference code used by Philippines; provides detailed provisions for prestressed concrete, loss equations, and design limits.
RA 544 (Regulations for the Practice of Civil Engineering in the Philippines)
Example
Section 34 requires a civil engineer to have knowledge of prestressed concrete design as part of the 'Reinforced and Prestressed Concrete Design' discipline.
Legislation governing professional registration and licensure of civil engineers; PRC administers the exam based on this law.
Must remember
- Stress Formula (Core): f = P/A ± (Pec)/I ∓ (Mc)/I. Compression is positive. Eccentric prestress below centroid: +Pec at bottom (adds compression), −Pec at top (relieves compression).
- Two Design Checks: Transfer (Pᵢ, minimal load) and Service (Pₑ, full load). Critical fiber at transfer is usually top; in service, may flip to bottom. Fail either check = design fails.
- Load Balancing: w_bal = 8Pe/L² (in consistent units). A parabolic tendon creates upward distributed load that cancels gravity; if w_load < w_bal, minimal bending. Straight tendon = zero balancing.
- Prestress Losses: Total ~15–20% (post-tensioned) or 18–25% (pre-tensioned). Always use Pₑ = Pᵢ − ΔP_total for service calculations. Forget this → wrong answer on exam.
- Transfer Stress Limits: Top fiber ≤ 0.5f'c compression and ≤ √(f'c) tension. If exceeded at transfer, member cracks and is rejected. Typical for f'c = 40 MPa: allow 20 MPa compression, 6.3 MPa tension.
- Service Stress Limits: Top fiber 0 to 0.45f'c compression; no tension if exposed to weather (or minimal, ≤ 0.1√f'c). Bottom fiber checked to avoid over-compression. If top goes into tension in service → design fails.
- Pre- vs Post-Tensioning: Pre: strands vs abutments before casting, transfer by bond, factory-controlled, limited profile flexibility, losses 18–25%. Post: ducts after casting, anchors at ends, field-applied, full profile flexibility, losses 15–20%.
- Friction Loss (Post-tensioning Only): ΔP_friction ≈ μ Pᵢ (k⋅x + f_rec⋅x). Depends on curvature k and path length x. Straight duct = zero friction. Typical loss on parabolic duct: 5–10%.
- Effectiveness Ratio: R = Pₑ / Pᵢ typically 0.80–0.85. If total losses = 16%, then R = 0.84. This ratio is the ultimate check: if R < 0.75, the member is inefficient.
- Sign Convention (Critical): Compression positive, tension negative. Eccentric prestress (below centroid) → bottom in compression (+), top in tension relief (−). Applied sagging moment → bottom in tension (−), top in compression (+). The two moments oppose; net stress = superposition.
Last minute tips
- Memorize the stress formula in both forms: f_top = P/A − Pec_top/I ± Mc_top/I and f_bot = P/A + Pec_bot/I ∓ Mc_bot/I. Know when to use + or − signs (eccentricity below centroid → + at bottom, − at top). Draw a sketch on the exam paper to verify signs before calculating.
- Always check BOTH transfer and service stresses. Many students check only one and get the answer wrong. The critical fiber often changes between the two. If transfer is OK but service fails (or vice versa), the entire design is rejected.
- Load balancing formula (w_bal = 8Pe/L²) is a direct gift on the exam.** If the problem gives P, e, and L, calculate w_bal immediately. If it matches the applied load, bending is zero. This shortcut often appears as a 'trick' question: students over-complicate the answer.
- Forget losses = instant fail. Always deduct losses from Pᵢ to get Pₑ before calculating service stresses. A common exam trap: problem gives Pᵢ but asks for service stress; students use Pᵢ directly (wrong). Reduce by 15–20% first.
- Transfer stresses are tighter. At transfer, only self-weight acts; the applied moment is nearly zero. So the critical constraint is usually preventing over-compression or cracking at the top fiber (which gets the least precompression if the tendon is below centroid). This is why pre-tensioned members are carefully released in stages or moved to loading frame early.
Comparison tables
Pre-Tensioning vs Post-Tensioning
| Characteristic | Pre-Tensioning | Post-Tensioning |
|---|---|---|
| When strands are tensioned | Before casting (against external abutments) | After concrete cures (in ducts) |
| Force transfer mechanism | Bond friction over transfer length (~40–60 diameters) | Mechanical end anchors; ducts usually grouted |
| Typical application | Precast members: hollow-core slabs, beams, piles, sleepers (plant-controlled) | Cast-in-place members: bridges, transfer beams, flat slabs (field-applied) |
| Typical losses | 18–25% (includes higher creep at transfer) | 15–20% (controlled environment) |
| Tendon profile flexibility | Limited to straight or simple curves (dictated by abutment locations) | Highly flexible: can be parabolic, draped, or even straight as desired |
| Cost (relative) | Lower per unit (amortized over many members in factory) | Higher (unique anchoring and grouting per project) |
| Durability (if grouted) | Good (no ducts, strands bonded throughout) | Excellent (if ducts are grouted; poor if ducts not grouted) |
| Load balancing | Limited by abutment geometry | Full control: can achieve any sag e and force P desired |
Stress Limits at Transfer vs Service (NSCP 2015, ACI 318)
| State | Fiber Location | Allowable Compression | Allowable Tension | Common Failure Mode if Exceeded |
|---|---|---|---|---|
| Transfer (minimum load) | Top | ≤ 0.5f'c | ≤ √(f'c) | Micro-cracking (compression); surface cracking (tension) → damaged member |
| Transfer (minimum load) | Bottom | ≤ 0.6–0.7f'c (usually not critical) | Not checked (rarely in tension) | Over-compression, spalling (rare at transfer) |
| Service (full load) | Top | ≤ 0.45f'c (exposed); ≤ 0.5f'c (sheltered) | 0 to 0.1√(f'c) (typically 0 if exposed to weather) | Flexural cracking, water ingress (exposed member) |
| Service (full load) | Bottom | Checked to prevent excessive compression and loss of prestress benefit | Rarely; if present, indicates under-prestressed | Crushing (rare); uneconomical design (over-sized section) |
Loss Components: Magnitude & Timing
| Loss Type | Timing | Typical Magnitude (% of Pᵢ) | Applies To | Formula / Note |
|---|---|---|---|---|
| Elastic shortening | Immediate (at transfer/tensioning) | 2–4% | Both pre- and post- | ΔP = (Eₛ/Eᶜ) × (Pᵢ/Aᶜ) × Aᶜ |
| Anchorage seating | Immediate (first few hours post-tensioning) | 0.5–1% | Post-tensioning only | Localized near anchor; typically 5–10 mm slip over transfer length |
| Friction | Immediate (during jacking) | 5–10% | Post-tensioning only (curved ducts) | ΔP = μ Pᵢ (k⋅x + f_rec⋅x) |
| Creep | Time-dependent (weeks to years) | 4–6% | Both pre- and post- | Depends on sustained stress, humidity, age; use ACI/NSCP tables |
| Shrinkage | Time-dependent (weeks to months) | 2–4% | Both pre- and post- | Depends on exposure and curing; greater if concrete is allowed to dry |
| Relaxation (steel) | Time-dependent (weeks to years) | 2–4% (low-relax strand); 5–8% (regular strand) | Both pre- and post- | Higher if initial stress > 0.7fᵤₚ; minimize by using low-relaxation strand |
Diagrams
Prestressed Concrete: Complete Concept Map
Mind map showing the hierarchical structure of prestressed concrete: definition, methods (pre- vs post-), service stress analysis, losses, load balancing, and design checks. Use this to quickly organize your knowledge during the exam.
Prestressed Concrete Design Flowchart
Iterative design flowchart for prestressed concrete beams. Shows the sequence: select section → choose P and e → check transfer → estimate losses → check service → verify load balancing. Red paths show how to iterate if checks fail. This is the exact procedure used in PRC exams.