CELE Reinforced & Prestressed Concrete — Prestressed ConcreteRevision Notes
Revision notes for CELE Reinforced & Prestressed Concrete Prestressed Concrete — designed for time-pressed reviewers. These notes skip the basics and focus on what Professional Regulation Commission (PRC) — Board of Civil Engineering consistently tests, so you spend your revision hours on the content most likely to appear on exam day.
Exam context
The Civil Engineer Licensure Examination is conducted by Professional Regulation Commission (PRC) — Board of Civil Engineering and is scheduled for May and November 2026. The Reinforced & Prestressed Concrete subtest is marked as "Core" in the official pattern, and Prestressed Concrete appears in position 7th of 7 in the CELE Reinforced & Prestressed Concrete review rotation. Passing mark: 70% weighted average, no sub-test below 50%. Recent CELE 2026 papers have drawn roughly a meaningful share of questions from this subject.
Prestressed Concrete - Revision Notes
Prestressed concrete is one of the most frequently tested topics in the PRC Civil Engineer Licensure Examination under Reinforced and Prestressed Concrete Design. The core idea is elegantly simple: introduce a controlled internal compression into the concrete before service loads arrive, so that the tensile stresses caused by those loads are either eliminated or kept within allowable limits. This approach allows concrete to span longer distances with shallower sections, control cracking, and reduce long-term deflections — all critical in modern Philippine infrastructure such as post-tensioned bridge girders, prestressed piles for coastal reclamation projects, and flat-plate post-tensioned slabs for high-rise buildings in Metro Manila. These notes cover the four pillars tested in board exams: (1) pre-tensioning vs post-tensioning, (2) service stress analysis using the elastic formula, (3) prestress losses, and (4) the load-balancing concept.
Sections
Formulas
Example
For a 12.7 mm diameter strand: l_t ≈ 50 × 12.7 = 635 mm ≈ 640 mm from the end.
Formula
l_t ≈ 50 d_b (transfer length, pre-tensioning)
Variables
l_t = transfer length (mm); d_b = nominal strand diameter (mm)
Application
Determines the minimum distance from the end of a pre-tensioned member where full prestress is available. Critical for checking stresses near the beam ends.
Example
Used when checking whether a prestressed pile or girder has adequate strand development at the support region.
Formula
l_d = l_t + l_f (development length)
Variables
l_d = development length (mm); l_t = transfer length (mm); l_f = flexural bond length (mm); l_f = (f_ps - f_se) d_b / (per ACI 318-19 Eq. 25.8.2.1a)
Application
Ensures the strand can develop its full design stress f_ps at the critical section for flexure.
Exam Tips
- Board questions often ask you to IDENTIFY which loss category applies to which system — memorize: Friction & Anchorage Seating = post-tensioning only; Bond failure = pre-tensioning concern.
- When a question says 'jacked to P_i = X kN and total losses = Y%,' the effective prestress is P_e = (1 - Y/100) × P_i. This is the most common loss-related computation pattern.
- Pre-tensioned members are ALWAYS precast; post-tensioned can be precast OR cast-in-place. This detail sometimes appears as a multiple-choice distractor.
- Remember 'PRE = BEFORE casting, POST = AFTER casting' — a simple memory anchor.
Key Points
- Pre-tensioning: High-strength strands (typically 12.7 mm or 15.2 mm dia, Grade 1860 MPa) are tensioned against external abutments BEFORE concrete is cast. After the concrete reaches a specified transfer strength (commonly f'ci ≥ 0.75 f'c or as specified), the strands are released and the prestress is transferred to the concrete by BOND over a transfer length.
- Post-tensioning: Concrete is cast first with ducts (metal or plastic sheathing) containing un-bonded or bonded strands/tendons. After the concrete cures to the required strength (commonly f'ci ≥ 28 MPa for most bridge applications), the tendons are tensioned against the hardened concrete ends using hydraulic jacks and anchored with wedge-type or nut-type anchorage devices.
- Bonded post-tensioning: ducts are grouted with cementitious grout after stressing — provides corrosion protection and structural redundancy. Required by DPWH specifications for bridge girders.
- Unbonded post-tensioning: tendons coated with corrosion-inhibiting grease and wrapped in plastic sheathing; used for slabs and minor beams. More flexible but less redundant.
- Pre-tensioned members are typically precast plant products: prestressed concrete piles (common in Philippine ports), hollow-core slabs, double-tee girders, and AASHTO/PCPCI girders.
- Post-tensioned members are typically cast-in-place: bridge decks, transfer beams in high-rise buildings, post-tensioned flat slabs, and segmental bridge construction.
- Transfer length (pre-tensioning): the distance over which the strand develops its full prestress by bond, approximately 50 strand diameters (50db) per ACI 318-19 Section 25.8.1.
- Development length (pre-tensioning): total length for full flexural strength, ld = transfer length + flexural bond length.
Definitions
Term
Prestressing
Definition
The intentional introduction of a predetermined internal stress (compressive) into a structural member to counteract, partially or fully, the tensile stresses induced by service loads.
Importance
Fundamental concept — appears in virtually every board exam question on this topic.
Term
Transfer (Pre-tensioning)
Definition
The act of releasing the tensioned strands from the external abutments so that the prestress force is transferred to the concrete through bond. Occurs when concrete reaches the required transfer strength f'ci.
Importance
Defines the 'transfer' loading stage — a critical check for top-fiber tension and bottom-fiber over-compression in pre-tensioned beams.
Term
Anchorage (Post-tensioning)
Definition
The mechanical device (wedge plates, barrel-and-wedge, or threaded nuts) at the ends of post-tensioned tendons that permanently maintains the tension force after the jack is removed.
Importance
Anchorage seating loss is a key immediate loss in post-tensioning problems.
Term
Grouting
Definition
Injection of cementitious grout into post-tensioning ducts after stressing to bond the strand to the duct and provide corrosion protection.
Importance
Distinguishes bonded from unbonded post-tensioning; affects structural behavior after cracking.
Term
Jacking Force (P_j)
Definition
The force applied by the hydraulic jack to the tendon during stressing. This is the maximum force in the tendon and is used as the starting point for loss calculations.
Importance
Board problems often give P_j and ask for P_e after losses; or give P_e and ask for required P_j.
Section Title
1. Pre-tensioning vs Post-tensioning
Common Mistakes
- Confusing which system transfers by BOND (pre-tensioning) vs by END ANCHORAGE (post-tensioning). Bond = pre; Anchorage = post.
- Applying friction losses to pre-tensioned members — friction losses apply ONLY to post-tensioned members with curved or straight tendons in ducts.
- Applying anchorage seating loss to pre-tensioned members — seating loss applies ONLY to post-tensioning.
- Forgetting that elastic shortening loss occurs in BOTH systems, but for post-tensioning with multiple tendons, it affects only previously stressed tendons (the last tendon stressed has zero elastic shortening loss).
- Assuming transfer occurs at full design strength f'c — transfer occurs at the TRANSFER strength f'ci which is typically less than f'c.
Formulas
Example
300×600 mm beam, P=900 kN, e=150 mm, M=120 kN·m: P/A=5.0 MPa; Pec/I=900000×150×300/(5.4×10⁹)=7.5 MPa; Mc/I=120×10⁶×300/(5.4×10⁹)=6.67 MPa. f_top = 5.0 − 7.5 + 6.67 = +4.17 MPa (compression). ✓
Formula
f_top = P/A − (P·e·c_top)/I + (M·c_top)/I
Variables
f_top = stress at top fiber (MPa, compression positive); P = effective prestress force (N); A = gross cross-sectional area (mm²); e = eccentricity of CGS below centroid (mm); c_top = distance from centroid to top fiber (mm); I = moment of inertia of gross section (mm⁴); M = applied bending moment (N·mm, sagging positive)
Application
Used at EVERY board exam problem involving service or transfer stresses. Compute for both transfer (P = P_i, M = M_sw) and service (P = P_e, M = M_total).
Example
Continuing above: f_bot = 5.0 + 7.5 − 6.67 = +5.83 MPa (compression). Both fibers in compression — no cracking. ✓
Formula
f_bot = P/A + (P·e·c_bot)/I − (M·c_bot)/I
Variables
f_bot = stress at bottom fiber (MPa, compression positive); all variables same as above; c_bot = distance from centroid to bottom fiber (mm)
Application
Critical for checking bottom-fiber tension in service and top-fiber over-compression at transfer. The governing stress controls the design.
Example
For the same 300×600 beam: S = 300×600²/6 = 18×10⁶ mm³. Pe/S = 900000×150/(18×10⁶) = 7.5 MPa. M/S = 120×10⁶/(18×10⁶) = 6.67 MPa. Same answers as above — confirming formula equivalence.
Formula
f = P/A ± P·e/S ∓ M/S (using section modulus form)
Variables
S = I/c = section modulus (mm³); top fiber: f_top = P/A − Pe/S_top + M/S_top; bottom fiber: f_bot = P/A + Pe/S_bot − M/S_bot
Application
Faster computation for rectangular and symmetric I-sections where S_top = S_bot = S. Preferred in timed board exam conditions.
Example
300×600 mm: I = 300×600³/12 = 5.4×10⁹ mm⁴. This appears directly in most board problems.
Formula
I = b·h³/12 (rectangular section)
Variables
b = width (mm); h = total depth (mm)
Application
Standard moment of inertia for the most common cross-section in board problems. For T-sections and I-sections, use the parallel-axis theorem.
Exam Tips
- Always write out all three stress terms (P/A, Pec/I, Mc/I) separately before combining — this reduces sign errors dramatically.
- For a rectangular beam, memorize: A = bh, I = bh³/12, S = bh²/6, c = h/2. You can compute everything from these four formulas.
- A quick sanity check: under prestress alone (M=0), the bottom fiber should be MORE compressed than the top (for below-centroid steel). If not, you have a sign error.
- When the problem asks for 'stresses at transfer,' use P_i and only the self-weight moment. When it asks 'stresses in service,' use P_e and full moment.
- Board problems often test: given the allowable stresses, find the MINIMUM required prestress P or MAXIMUM allowable eccentricity e. Set up the stress equations as inequalities and solve — same formula, different unknown.
- Check both fibers at both stages — four stress values total for a typical simply supported beam. The design is governed by the most critical one.
Key Points
- Prestressed concrete at service is analyzed elastically (uncracked section) — all cross-section properties (A, I, c_top, c_bot) are computed for the gross or transformed section.
- Three stress components superpose at any fiber: (a) uniform axial precompression P/A, (b) bending stress due to eccentricity ±Pec/I, and (c) bending stress due to applied moment ∓Mc/I.
- SIGN CONVENTION (compression positive, as is standard in Philippine board problems): Axial term P/A is always compressive (+). The eccentric prestress (e below centroid for sagging beams) adds compression at the BOTTOM (+) and relieves the top (−). The applied sagging moment adds tension at the BOTTOM (−) and compression at the TOP (+).
- TWO critical loading stages must be checked: (i) AT TRANSFER — use P_i (initial prestress, before time-dependent losses), with only the self-weight moment M_sw acting. The top fiber is most critical for tension. (ii) IN SERVICE — use P_e (effective prestress after all losses), with full factored or service moment M_total. The bottom fiber is most critical for tension.
- Allowable stresses per ACI 318 / NSCP 2015 Section 418: At transfer — compression ≤ 0.60 f'ci; tension ≤ 0.25√f'ci (MPa) for members without bonded reinforcement at the tension zone. In service — compression ≤ 0.45 f'c (sustained loads) or 0.60 f'c (total loads); tension in precompressed tensile zone ≤ 0.50√f'c (MPa) for Class C (cracked) or zero for Class U (uncracked).
- Section moduli: S_top = I/c_top; S_bot = I/c_bot. For a rectangular section b×h: A = bh, I = bh³/12, c_top = c_bot = h/2, S = bh²/6.
- The eccentricity e is measured from the centroid of the gross section to the centroid of the prestressing steel (CGS). Positive e = steel below centroid (typical for sagging beams).
- For a rectangular section, the stress formula simplifies using section modulus: f = P/A ± Pe/S ∓ M/S.
Definitions
Term
Eccentricity (e)
Definition
The perpendicular distance from the centroidal axis of the cross-section (CGC — centroid of gross concrete) to the centroid of the prestressing steel group (CGS). Measured as positive when the steel is BELOW the centroid.
Importance
This single parameter controls the magnitude of the flexural prestress component. Misidentifying e is the most common source of computational error.
Term
Transfer Stage
Definition
The loading condition immediately after prestress is applied (released for pre-tensioning, or after jacking for post-tensioning) when P = P_i (initial prestress) and the only load is the member self-weight. Typically governs the TOP fiber for tension.
Importance
Board problems frequently ask for the CRITICAL fiber at transfer — almost always the top fiber for a simply supported beam with below-centroid steel.
Term
Service Stage
Definition
The long-term loading condition with P = P_e (effective prestress after all losses) and full service or factored loads applied. Typically governs the BOTTOM fiber for tension.
Importance
P_e must be used — using P_i in service calculations overestimates prestress and is a critical error.
Term
Kern (Middle-Third / Core)
Definition
The central region of a cross-section within which the resultant force must lie to ensure no tension occurs anywhere in the section. For a rectangle, the kern is the middle-third of the section (h/6 from centroid each way).
Importance
Kern concept is sometimes tested directly: 'Find the minimum eccentricity that avoids tension at the top fiber under P only.' Answer: e ≤ h/6 for a rectangular section.
Term
Cracking Moment (M_cr)
Definition
The applied moment at which the tensile stress at the extreme tension fiber reaches the modulus of rupture f_r = 0.62√f'c (MPa) per ACI 318-19 Section 19.2.3.
Importance
Required for computing minimum steel and deflection calculations in partially prestressed beams.
Section Title
2. Service Stress Analysis — The Elastic Formula
Common Mistakes
- SIGN ERROR on the Pe term: Eccentric steel BELOW the centroid puts the BOTTOM in extra compression (+) and RELIEVES the top (−). Many students reverse this.
- SIGN ERROR on the M term: Sagging applied moment puts the BOTTOM in tension (−) and the TOP in compression (+). This is opposite to what the eccentric prestress does — they partially cancel, which is the whole point.
- Using P_i (initial prestress) for service stress calculations instead of P_e (effective after losses) — this overestimates the prestress benefit.
- Using P_e for transfer stress calculations instead of P_i — this underestimates the prestress at transfer and misses potential over-compression.
- Forgetting to include self-weight moment M_sw at transfer — at transfer the beam is already loaded by its own weight, which relieves the top-fiber tension somewhat.
- Using h/2 for c_top and c_bot in non-rectangular (T or I) sections — always compute c from the centroid of the actual section.
- Mixing units: P must be in Newtons (not kN) when I is in mm⁴, c is in mm, and the result is in MPa (N/mm²). Convert P to N by multiplying kN × 1000.
Formulas
Example
P_i = 1200 kN, losses = 16.7%: P_e = (1 − 0.167) × 1200 = 0.833 × 1200 = 1000 kN.
Formula
P_e = P_i − ΔP_total = (1 − ξ) P_i = R · P_i
Variables
P_e = effective prestress after all losses (kN or N); P_i = initial prestress at transfer (kN or N); ΔP_total = total prestress loss (kN or N); ξ = total loss ratio (fraction, 0.15 to 0.25); R = effectiveness ratio (0.75 to 0.85)
Application
The most basic loss formula — used in almost every board problem involving losses. Given P_i and total loss %, find P_e. Or given P_e and R, find P_i.
Example
Δ=6 mm, L=15000 mm, A_ps=1000 mm², E_ps=195000 MPa: ΔP_anc = (6/15000)×195000×1000 = 78,000 N = 78 kN.
Formula
ΔP_anc = (Δ / L) × E_ps × A_ps
Variables
ΔP_anc = anchorage seating loss (N); Δ = anchor slip (mm, typically 6 mm); L = tendon free length (mm); E_ps ≈ 195,000 MPa; A_ps = total area of prestressing steel (mm²)
Application
Post-tensioning only. Used when the problem gives anchor slip and asks for the immediate loss or the stress drop in the tendon.
Example
P_j=1500 kN, μ=0.20, α=0.30 rad, K=0.002/m, x=20 m: ΔP_FR = 1500[1−e^(−0.20×0.30−0.002×20)] = 1500[1−e^(−0.10)] ≈ 1500×0.0952 = 142.8 kN.
Formula
ΔP_FR = P_j [1 − e^(−μα − Kx)]
Variables
ΔP_FR = friction loss (N); P_j = jacking force (N); μ = curvature friction coefficient (typically 0.15–0.25 for metal ducts); α = total angular change of tendon path (rad); K = wobble coefficient per mm (typically 0.001–0.003 per m = 0.000001–0.000003 per mm); x = tendon length from jack to point considered (mm)
Application
Post-tensioning only. For small μα + Kx values (<0.3), approximate: ΔP_FR ≈ P_j (μα + Kx).
Example
f'ci=28 MPa, E_ci=4700√28=24,870 MPa, E_ps=195,000 MPa, f_cgs=8.5 MPa, A_ps=1000 mm²: ΔP_ES = (195000/24870)×8.5×1000 = 7.84×8.5×1000 = 66,640 N ≈ 66.6 kN.
Formula
ΔP_ES = (E_ps / E_ci) × f_cgs × A_ps (pre-tensioning, single-stage)
Variables
ΔP_ES = elastic shortening loss (N); E_ps = modulus of prestressing steel ≈ 195,000 MPa; E_ci = modulus of concrete at transfer = 4700√f'ci (MPa); f_cgs = compressive stress in concrete at the CGS level due to P_i
Application
Pre-tensioning: all tendons stressed at once before concrete is cast, so 100% elastic shortening loss applies. Post-tensioning: if n tendons stressed sequentially, average ES = ES × (n−1)/(2n), approaching zero for the last tendon.
Exam Tips
- Memorize the IMMEDIATE vs TIME-DEPENDENT loss categories and which systems they apply to — this is tested directly in multiple-choice questions.
- Key association: FRICTION + ANCHORAGE SEATING → POST-TENSIONING ONLY. BOND DEVELOPMENT → PRE-TENSIONING CONCERN. ELASTIC SHORTENING, CREEP, SHRINKAGE, RELAXATION → BOTH.
- Typical total loss ranges are testable facts: 15–20% (post-tensioned), 18–25% (pre-tensioned). These numbers appear in matching and fill-in board questions.
- When a problem gives jacking stress f_pj and asks for effective prestress f_pe after losses, use: f_pe = (1 − loss ratio) × f_pj. Maximum allowable jacking stress per ACI 318-19 = 0.94 f_py or 0.80 f_pu, whichever is less.
- For the anchorage seating loss formula, always check units: Δ in mm, L in mm, E_ps in MPa (N/mm²), A_ps in mm² → result in Newtons. Convert to kN for practical use.
Key Points
- Prestress losses are the reductions in tendon force from the jacking force P_j to the effective prestress force P_e. They are classified as IMMEDIATE (occur during or shortly after stressing) and TIME-DEPENDENT (develop over months and years).
- IMMEDIATE LOSSES: (a) Elastic Shortening (ES) — as the concrete is compressed, it shortens, reducing the tendon strain and hence force. (b) Anchorage Seating Loss (ANC) — post-tensioning only; the wedge or nut slips slightly (typically 6 mm) as the jack is released, reducing tendon strain. (c) Friction Loss (FR) — post-tensioning only; friction between the tendon and duct as the tendon is pulled, especially at curves.
- TIME-DEPENDENT LOSSES: (a) Creep of Concrete (CR) — the concrete shortens further under sustained stress over time. (b) Shrinkage of Concrete (SH) — the concrete shortens as moisture evaporates during curing and afterward. (c) Steel Relaxation (RE) — high-strength prestressing steel loses stress over time under sustained high strain.
- Friction loss formula: ΔP_f = P_j (1 − e^(−μα − Kx)) where μ = wobble-curvature friction coefficient, α = total angle change of tendon (radians), K = wobble friction coefficient per unit length, x = tendon length from jack.
- Anchorage seating loss: ΔP_anc = (Δ/L) × E_ps × A_ps where Δ = seating slip (mm), L = tendon length (mm), E_ps = modulus of elasticity of prestressing steel ≈ 195,000 MPa, A_ps = area of prestressing steel (mm²).
- Elastic shortening loss (pre-tensioning, single stage): ΔP_ES = (E_ps/E_ci) × f_cgs × A_ps where f_cgs = concrete stress at the centroid of the steel due to P_i, and E_ci = modulus of concrete at transfer = 4700√f'ci (MPa).
- Typical TOTAL losses: 15–20% for post-tensioned members; 18–25% for pre-tensioned members. These ranges are commonly tested as multiple-choice questions.
- The effectiveness ratio R = P_e/P_i relates the effective to the initial prestress. Common board values: R = 0.80 to 0.85.
- ACI 318-19 Section 26.10.2 requires that losses be considered in design. NSCP 2015 Section 418 follows the same provisions for Philippine practice.
- Refined (detailed) loss calculations use the PCI or AASHTO LRFD methods for actual design; simplified (lump-sum percentage) is accepted for board exam problems unless the problem specifically gives friction and relaxation data.
Definitions
Term
Elastic Shortening Loss (ES)
Definition
The reduction in tendon force caused by the instantaneous elastic shortening of the concrete member when the prestress force is applied. The concrete compresses, the tendon shortens with it, and the tendon stress drops.
Importance
Applies to BOTH pre- and post-tensioned systems but is handled differently. A very common board exam topic.
Term
Anchorage Seating Loss
Definition
The loss of prestress in a post-tensioned tendon due to the small inward movement (slip) of the wedges or anchors as the hydraulic jack pressure is released. Typically 3–10 mm depending on the anchorage system.
Importance
Post-tensioning ONLY. The formula is straightforward and board problems love it.
Term
Creep Loss
Definition
The long-term shortening of concrete under sustained compressive stress causes additional tendon shortening and stress loss. Creep is proportional to the sustained concrete stress at the steel level.
Importance
The largest single time-dependent loss in most prestressed members. Affected by humidity, concrete age at loading, and mix proportions.
Term
Shrinkage Loss
Definition
The shortening of the concrete member as moisture evaporates from the concrete over time. This shortening reduces the tendon elongation and hence the prestress force.
Importance
Significant in Philippine conditions where high ambient temperature accelerates drying shrinkage.
Term
Steel Relaxation
Definition
The reduction in stress in a high-strength prestressing strand or wire held at constant strain over time. Relaxation is a material property of the steel itself (not of the concrete). Low-relaxation strands (preferred) have about 1/3 the relaxation of normal-relaxation strands.
Importance
Board problems sometimes ask to identify which loss is a STEEL property (relaxation) vs a concrete property (creep, shrinkage) — a classic multiple-choice test.
Term
Wobble Effect (Friction)
Definition
Unintended angular deviations of the tendon duct from the intended profile, causing additional friction losses even in 'straight' tendons. Quantified by the wobble coefficient K.
Importance
Distinguishes wobble friction (unintentional, always present) from curvature friction (intentional curvature, characterized by μ).
Section Title
3. Prestress Losses
Common Mistakes
- Applying friction and anchorage seating losses to PRE-TENSIONED members — these apply only to POST-TENSIONED systems.
- Forgetting that elastic shortening applies to the LAST tendon stressed in post-tensioning with zero loss, while the FIRST tendon stressed suffers maximum ES loss (total ES if only one stage).
- Using f'c instead of f'ci in the elastic modulus formula at transfer: E_ci = 4700√f'ci, NOT 4700√f'c.
- Adding all percentage losses arithmetically without recognizing they are cumulative on P_j — technically, each loss should be applied sequentially, but for board exam purposes, lump-sum percentages are added and applied to P_i.
- Forgetting that time-dependent losses (creep, shrinkage, relaxation) occur AFTER the immediate losses — the concrete stress that drives creep is already reduced by ES.
- Confusing the effectiveness ratio: R = P_e/P_i (not P_e/P_j). Some problems give P_j (jacking force) which is larger than P_i (force after immediate losses at transfer).
Formulas
Example
P=900 kN, e=150 mm=0.15 m, L=10 m: w_bal = 8×900×0.15/10² = 1080/100 = 10.8 kN/m. This tendon balances 10.8 kN/m of gravity load.
Formula
w_bal = 8 · P · e / L²
Variables
w_bal = upward equivalent load balanced by the tendon (kN/m); P = effective prestress force (kN); e = midspan sag of the parabolic tendon from the chord line (m); L = span of the beam (m). Keep P in kN, e and L in meters for result in kN/m.
Application
The single most important load-balancing formula. Use it to: (a) find how much load is balanced by a given tendon, (b) find the required P for a given balanced load, or (c) find the required sag e.
Example
Balance w=14 kN/m, L=12 m, e=250 mm=0.25 m: P = 14×12²/(8×0.25) = 14×144/2 = 2016/2 = 1008 kN.
Formula
P = w_bal · L² / (8 · e)
Variables
Required tendon force (kN) to balance a specified uniform load; w_bal in kN/m, L in m, e in m.
Application
Design formula: given the balanced load and geometry, find the required prestress force.
Example
P=900 kN=900,000 N, A=300×600=180,000 mm²: f = 900,000/180,000 = 5.0 MPa compression uniformly throughout the depth.
Formula
f_uniform = P / A (stress at balanced condition)
Variables
f_uniform = uniform axial precompression (MPa) throughout the section when w = w_bal; P = effective prestress (N); A = gross cross-sectional area (mm²)
Application
At balanced load, bending stress is zero; the only stress is this uniform compression. This stress must be checked against the allowable compressive stress.
Example
Balance 10.8 kN/m, L=10 m, P=900 kN: e = 10.8×10²/(8×900) = 1080/7200 = 0.15 m = 150 mm.
Formula
e = w_bal · L² / (8 · P)
Variables
Required tendon sag (m) to balance a specified uniform load; w_bal in kN/m, L in m, P in kN.
Application
Used when the tendon force is fixed (e.g., by strand selection) and you need to determine the drape profile.
Exam Tips
- The formula w_bal = 8Pe/L² is a direct manipulation of the catenary/parabola geometry — derive it once from moment equilibrium to memorize it permanently.
- Board problems often give: 'A prestressed beam with P=X kN and e=Y mm on a Z m span. What uniform load does the tendon balance?' → Direct substitution into w_bal = 8Pe/L².
- Reverse problems: 'Find P to balance 15 kN/m on a 10 m span with e=200 mm.' → P = w_bal × L²/(8e) = 15×10²/(8×0.20) = 15×100/1.6 = 937.5 kN.
- After finding w_bal, always check the RESIDUAL stress: f = P/A at the balanced condition, and check it against the allowable compression.
- In post-tensioned flat plates, the design percentage of balanced load is typically 60–80% of the sustained dead load — this factual statement appears in theory-type board questions.
Key Points
- Proposed by T.Y. Lin, the load-balancing method views the draped prestressing tendon as exerting an UPWARD distributed load on the concrete beam, which balances (cancels) the gravity downward load. This is an elegant and powerful design tool.
- For a parabolic tendon (the most common profile for simply supported beams) with sag e (vertical distance from the chord connecting the end anchorage points to the lowest point of the tendon at midspan), the equivalent upward uniform load is: w_bal = 8Pe/L².
- When the applied load equals exactly w_bal, the only remaining stress in the beam is a UNIFORM axial precompression P/A — no bending at all. This is the 'balanced' condition: deflection is theoretically zero (ignoring self-weight contributions outside the balanced load), and fiber stresses are uniform = P/A throughout the depth.
- Any load ABOVE the balanced load (w_net = w_total − w_bal) must be carried by ordinary flexural action of the precompressed section. The precompression P/A provides a stress reserve before cracking.
- The load balancing concept is especially useful for FLAT PLATE AND TWO-WAY SLAB design with post-tensioned tendons running in both directions, where you balance a chosen percentage (typically 60–80%) of the dead load.
- For a harped (single-point draped) tendon in a simply supported beam, the equivalent upward force is a CONCENTRATED force at the harping point: F = Pe × (2/L) × (L_1 + L_2)/(L_1 × L_2) — but the parabolic formula w = 8Pe/L² is far more common in board problems.
- Tendon force P in the load balancing formula is the EFFECTIVE prestress P_e (after losses), since it is used for service-load analysis.
- The parabolic tendon profile approximates the shape of the bending moment diagram for a uniformly distributed load — this geometric harmony is why it is so effective for balancing UDLs.
Definitions
Term
Balanced Load (w_bal)
Definition
The upward equivalent distributed force exerted by a parabolic prestressing tendon on the concrete member, equal in magnitude to the gravity load component it is designed to cancel.
Importance
Central concept of the load-balancing method. A well-balanced prestressed beam has near-zero deflection under the balanced load.
Term
Tendon Sag (e)
Definition
For a simply supported beam with a parabolic tendon anchored at the centroid at both ends and passing below the centroid at midspan, the sag e is the vertical distance from the centroidal axis (chord line in general) to the lowest point of the tendon (at midspan). Sag drives the load-balancing capacity.
Importance
Maximum e is limited by the beam geometry (cover requirements) and the need to keep the tendon within the kern at the ends to avoid tensile stresses.
Term
Balanced Condition
Definition
The loading state when the applied load exactly equals w_bal. At this condition, the beam has only uniform axial precompression P/A, zero bending stresses, and (ideally) zero net deflection.
Importance
Explains why prestressed slabs are so valued for deflection control — by balancing most of the sustained dead load, long-term deflection is virtually eliminated.
Term
Equivalent Load (Tendon Equivalent Force)
Definition
The set of forces (distributed, concentrated, or moments) exerted by the tendon ON the concrete, representing the net effect of the curved tendon force on the member. For a parabolic tendon: upward UDL = 8Pe/L². For end anchorages: horizontal force P and moment Pe.
Importance
The load-balancing method replaces the tendon with its equivalent loads, then analyzes the concrete structure as an ordinary loaded member — powerful simplification.
Section Title
4. Load Balancing Concept
Common Mistakes
- Using e in mm while L is in m in the formula w_bal = 8Pe/L² — keep ALL length units the same (both in meters, or P in N and e, L in mm for result in N/mm = MPa, which doesn't make direct sense for UDL). The safest approach: P in kN, e and L in METERS, result in kN/m.
- Using P_i instead of P_e in the load-balancing formula — load balancing is a SERVICE condition analysis, so effective prestress P_e must be used.
- Thinking that the balanced load equals the TOTAL design load — it only equals the portion chosen to be balanced. The remaining unbalanced load still causes bending.
- Forgetting the end moments at fixed supports or continuous beams — the equivalent tendon load includes end moments = Pe at anchorage points, which affect continuous beam analysis.
- Assuming the sag e equals the eccentricity at midspan for all cases — e equals the eccentricity only when the tendon passes through the centroid at the beam ends. If the tendon has eccentricity at the ends, the sag is measured from the chord between the end anchor points, not from the centroid.
Connections
- ORDINARY RC vs PSC: In ordinary RC (NSCP 2015 Chapter 4 / ACI 318 Chapter 9), tension is resisted by mild steel rebar — the concrete is assumed cracked. In PSC, the pre-compression keeps the concrete uncracked under service loads. Both use the same concrete materials (ASTM C150 cement, RA 6541-compliant aggregates) but PSC requires high-strength steel (Grade 1860 strands, ASTM A416) and is governed by NSCP 2015 Chapter 18 / ACI 318 Chapter 26.
- FLEXURAL STRENGTH: At ULTIMATE limit state, PSC beams are analyzed similarly to RC — using equilibrium of forces on the cracked section. The design ultimate moment φMn must exceed Mu (factored moment). The stress in the prestressing steel at ultimate f_ps is computed using ACI 318-19 Eq. 20.3.2.4 (bonded tendons) or the simplified formula for unbonded tendons. This connects PSC service design to ultimate strength design.
- SHEAR DESIGN: Prestressed beams have enhanced shear capacity because the precompression raises the principal tensile stress threshold. ACI 318-19 provides Vc formulas for prestressed members (Section 22.5) that are larger than for ordinary RC. The inclined tendon also contributes a vertical shear component V_p = P_e sin(θ). This connects to structural analysis and mechanics of materials.
- DEFLECTION CONTROL: PSC deflections at service are computed by superposing the upward camber (due to prestress, treated as the equivalent balanced load) and the downward deflection due to applied loads. Since load balancing reduces net deflection dramatically, PSC members satisfy deflection limits (NSCP 2015 Table 406.4.1) with shallower sections than RC — connecting PSC to structural serviceability design.
- BRIDGE ENGINEERING (RA 8975 / DPWH): Philippine bridge design uses AASHTO LRFD adapted by DPWH. Most Philippine highway bridges use post-tensioned AASHTO Type IV girders or pre-tensioned PCPCI girders. Understanding PSC losses and service stresses is essential for the Bridge Structural Design component of the CE board exam.
- FOUNDATION ENGINEERING: Prestressed concrete piles are widely used in the Philippines for soft ground conditions (Metro Manila, Cebu reclamations). The pile must resist prestress transfer stresses plus handling stresses (lifting, driving impact). This connects PSC principles to geotechnical and foundation engineering.
- CONSTRUCTION METHODS (RA 544 — Civil Engineering Law): The Civil Engineering Law (RA 544) requires that civil engineers exercise professional judgment in construction supervision. Proper monitoring of prestressing operations — jacking sequence, elongation measurements, and grout injection — falls under the supervisory responsibilities of the licensed civil engineer, connecting this technical topic to professional ethics and practice.
- MATERIALS ENGINEERING: The behavior of PSC depends critically on concrete compressive strength (f'c, f'ci), modulus of elasticity (E_c = 4700√f'c per ACI 318), and creep coefficient — all topics in materials science and concrete technology that connect to PSC loss calculations.
Exam Strategy
For PRC CE Board Exam questions on Prestressed Concrete: (1) IDENTIFY the question type first — is it a service stress problem, a loss problem, or a load-balancing problem? Each has a specific formula set. (2) For SERVICE STRESS problems, always write the three-term formula f = P/A ± Pec/I ∓ Mc/I before plugging in numbers. Determine the signs carefully using the physical reasoning: eccentric-below steel adds compression at the bottom; sagging moment adds tension at the bottom. (3) COMPUTE section properties A, I, c from the given dimensions before starting — do not skip this step under time pressure. (4) CHECK BOTH STAGES (transfer and service) unless the problem specifies only one. At transfer: use P_i, M = M_sw only. In service: use P_e, M = M_total. (5) For LOSS problems, identify which type of loss is asked, which system (pre vs post), and apply the correct formula. For 'total losses given as percent,' simply multiply (1 − loss fraction) × P_i to get P_e. (6) For LOAD BALANCING, the formula w_bal = 8Pe/L² is a one-step computation — be careful with units (P in kN, e and L in meters, result in kN/m). (7) TIME MANAGEMENT: PSC problems are typically 5–10 points each. A typical service stress problem should take 3–4 minutes with systematic computation. If a problem seems complicated, check if it simplifies to the standard formulas with correct signs — most board problems do. (8) AVOID COMMON TRAPS: Do not use P_i for service stresses; do not apply friction losses to pre-tensioned systems; do not mix units in the load-balancing formula. These are the three most common errors that cost marks. (9) For theory-type multiple-choice questions, remember the key associations: bond → pre-tensioning; anchorage → post-tensioning; relaxation → steel property; creep and shrinkage → concrete properties; total losses 15–25%.
Quick Review Questions
A 300 mm × 600 mm simply supported prestressed beam has an effective prestress P = 900 kN applied at an eccentricity of 150 mm below the centroid. The applied midspan moment is M = 120 kN·m. Compute the stress at the TOP fiber. (Take A = 180,000 mm², I = 5.4 × 10⁹ mm⁴, c_top = 300 mm.)
Step 1 — Compute the three stress components: P/A = 900,000/180,000 = 5.0 MPa (compression, +). Pec/I = 900,000 × 150 × 300 / (5.4×10⁹) = 40,500,000,000/5,400,000,000 = 7.5 MPa. Mc/I = 120×10⁶ × 300 / (5.4×10⁹) = 36,000,000,000/5,400,000,000 = 6.67 MPa. Step 2 — Apply signs: f_top = P/A − Pec/I + Mc/I = 5.0 − 7.5 + 6.67 = +4.17 MPa. The top fiber is in compression (+4.17 MPa). No cracking. ✓
For the same beam in Question 1, compute the stress at the BOTTOM fiber.
f_bot = P/A + Pec/I − Mc/I = 5.0 + 7.5 − 6.67 = +5.83 MPa (compression). The eccentric prestress adds extra compression at the bottom (+7.5 MPa) while the applied moment relieves some of it (−6.67 MPa), leaving a net compression. Both fibers remain in compression under these loads — a well-designed prestressed section.
A parabolic tendon with effective prestress P = 900 kN and midspan sag e = 150 mm is used in a simply supported beam of span L = 10 m. What uniform load does this tendon balance?
Using w_bal = 8Pe/L²: Convert e to meters: e = 0.150 m. w_bal = 8 × 900 × 0.150 / 10² = 1080 / 100 = 10.8 kN/m. This tendon exactly balances 10.8 kN/m of downward gravity load. Under this balanced load, the only stress in the beam is uniform axial precompression: P/A = 900,000/180,000 = 5.0 MPa compression.
A tendon is jacked to P_j = 1200 kN. Total prestress losses are estimated at 16.7%. Find the effective prestress P_e.
P_e = (1 − 0.167) × P_j = 0.833 × 1200 = 999.6 ≈ 1000 kN. The effectiveness ratio is R = P_e/P_i = 1000/1200 = 0.833. Design service stresses must use P_e = 1000 kN; transfer stresses (immediately after prestress application) use P_i = P_j (minus only immediate losses, if distinguished from total), typically approximated here as 1200 kN.
Which prestress losses apply ONLY to post-tensioned members (and NOT to pre-tensioned members)?
Friction loss occurs as the tendon is pulled through the duct against friction — this mechanism only exists in post-tensioning where the tendon moves through a duct after the concrete is cast. Anchorage seating loss occurs when the jack releases and the wedge/anchor slips — this also only occurs in post-tensioning. Pre-tensioned members transfer by bond (no duct, no anchorage seating). All other losses — elastic shortening, creep, shrinkage, and relaxation — apply to both systems.
What prestress tendon force P is required to balance a uniform load of 14 kN/m on a simply supported beam of span 12 m, if the maximum available tendon sag is 250 mm?
Rearranging w_bal = 8Pe/L²: P = w_bal × L² / (8e) = 14 × 12² / (8 × 0.250) = 14 × 144 / 2.0 = 2016 / 2.0 = 1008 kN. This is the required effective prestress. The required jacking force (before losses) would be larger: P_j = P_e / R, e.g., if R = 0.83, P_j = 1008/0.83 = 1214 kN.
A pre-tensioned concrete beam is stressed in a plant. At which loading stage does the TOP fiber typically govern for TENSION, and at which stage does the BOTTOM fiber govern for TENSION?
At transfer: P = P_i (high, full initial prestress) and only self-weight moment M_sw (small) acts. The eccentric prestress strongly reduces (or reverses) the top fiber stress: f_top = P_i/A − P_i·e·c_top/I + M_sw·c_top/I. The Pe term dominates and can cause TENSION at the top. In service: P = P_e (reduced by losses) and full service moment M_total (large) acts. The large M term now dominates at the bottom: f_bot = P_e/A + P_e·e·c_bot/I − M_total·c_bot/I, and the bottom can go into tension. Both stages must be checked.
What is the significance of the term w_bal = 8Pe/L² in the context of the deflection of a prestressed beam?
The parabolic tendon exerts an upward UDL = w_bal on the concrete. If the gravity UDL applied to the beam also equals w_bal, these loads cancel perfectly, and the beam experiences no net transverse loading — only the axial compression P/A acts. Without transverse load, there is no bending and no transverse deflection. This is why post-tensioned flat slabs are so effective: by balancing the sustained dead load, long-term creep deflection (which is proportional to sustained bending moment) is nearly eliminated.
For a 350 mm × 700 mm rectangular beam, P_e = 1100 kN at e = 200 mm below the centroid, with an applied moment M = 180 kN·m. Compute both fiber stresses. (Use A = 245,000 mm²; I = b·h³/12; c = 350 mm.)
Step 1 — Section properties: A = 350 × 700 = 245,000 mm²; I = 350 × 700³/12 = 350 × 343,000,000/12 = 10,004,166,667 ≈ 1.0004×10¹⁰ mm⁴; c = 350 mm. Step 2 — Stress components: P/A = 1,100,000/245,000 = 4.490 MPa. Pec/I = 1,100,000 × 200 × 350/(1.0004×10¹⁰) = 77,000,000,000/10,004,166,667 = 7.697 MPa. Mc/I = 180×10⁶ × 350/(1.0004×10¹⁰) = 63,000,000,000/10,004,166,667 = 6.297 MPa. Step 3 — Combine: f_top = 4.490 − 7.697 + 6.297 = +3.09 MPa. f_bot = 4.490 + 7.697 − 6.297 = +5.89 MPa. (Slight rounding variation from I approximation; exact I = 1.00042×10¹⁰ mm⁴ gives approximately these values. Both fibers in compression — acceptable.)
If P_i = 1500 kN and losses are 20%, find P_e and the service axial precompression stress for a section with A = 200,000 mm².
Step 1: P_e = (1 − 0.20) × 1500 = 0.80 × 1500 = 1200 kN. Step 2: Axial precompression = P_e/A = 1,200,000/200,000 = 6.0 MPa compression. This is the stress throughout the section at the balanced condition (when w_applied = w_bal). In service with applied moment, this 6.0 MPa provides a stress reserve before tension develops.
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