CELE Reinforced & Prestressed Concrete — Prestressed ConcreteStudy Notes

Prestressing is the intentional application of an internal compressive stress in a concrete member to counteract or reduce the tensile stresses that will be induced by service loads. The core principle is simple: if you precompress a material that is weak in tension (like concrete), you can reduce or eliminate tension when external loads are applied. Consider an analogy common in Philippine engineering textbooks: imagine a masonry column made of stacked bricks (analogous to concrete). Stacked dry, it collapses under its own weight because bricks cannot handle tension at the mortar joints. But if you clamp the column from the top and bottom (applying compressive force), the bricks stay tightly bound and can carry load far better. Prestressing works the same way—the "clamp" is provided by high-strength steel tendons. The advantages of prestressed concrete are substantial: 1. **Crack Control:** Precompression keeps concrete in compression (or low tension) under service load, virtually eliminating flexural cracks. This greatly improves durability—cracks are the primary pathway for water and chloride ingress (critical in Philippine coastal environments). 2. **Reduced Deflection:** Precompression acts as an upward load (load balancing), reducing the net bending moment and deflection. NSCP 2015 and ACI 318 typically allow larger span-to-depth ratios for prestressed members. 3. **Longer Spans:** Because cracks and large deflections are suppressed, prestressed girders can span 50+ meters (versus 20–30 m for RC), making them ideal for long-span bridges and viaducts. 4. **Efficient Use of Material:** High-strength steel (tendons at 1,700–1,900 MPa) is fully stressed, and concrete is kept in a favorable stress state throughout—reducing concrete section and steel quantity. 5. **Factory Quality Control:** Pre-tensioned members (common in Philippine precast plants) benefit from controlled plant production, ensuring consistent quality and durability. However, prestressed concrete has costs and constraints: - Higher initial material and labor cost (tensioning equipment, specialised labor). - More complex analysis and design. - Less ductility than RC (though still adequate under proper design). - Sensitive to mishandling during production and installation (sudden loss of support can cause cracking).

Prestress is applied to concrete by two fundamentally different methods, each suited to different applications: ### 2.1 Pre-tensioning In pre-tensioning, high-strength steel strands (typically seven-wire strands conforming to ASTM A416) are tensioned **before** the concrete is cast. The process is: 1. **Stressing:** Strands are pulled between two fixed abutments (at a precast plant) using hydraulic jacks, typically to 75% of the ultimate strand strength. For a common 12.7 mm (1/2") strand with f_pu = 1,860 MPa, the jacking stress is ~1,395 MPa. 2. **Casting:** Concrete is cast around the tensioned strands. 3. **Curing:** The concrete cures (steam or normal time). 4. **Release:** Once the concrete reaches sufficient strength (typically f'_ci ≥ 24.5 MPa, per ACI 318), the strands are released from the abutments. The concrete, now bonded to the strands via friction and mechanical interlock, prevents the strands from relaxing fully, retaining the precompressive stress. The transfer of prestress from strand to concrete is by **bond** (friction and adhesion) along the strand length. Near the ends, the strands slip slightly as the load is transferred—the "transfer length" (typically 40–50 strand diameters, or ~600–800 mm for 12.7 mm strands) is the distance over which full prestress is developed. **Advantages of pre-tensioning:** - Excellent bond (no unbonded length). - Long-term durability (no ducts or grout). - Ideal for repetitive members (hollow-core slabs, piles, standard girders)—amortizes plant cost. - Excellent quality control in a factory environment. **Disadvantages:** - Fixed geometry (tendons cannot be draped)—all members have the same eccentricity profile. - Requires large abutments and tensioning beds. - Elastic shortening losses occur before release. - Not practical for unique or curved members (e.g., irregular bridge decks). **Common pre-tensioned members in the Philippines:** - Precast hollow-core slabs (widely used in residential and commercial buildings). - Precast piles (for deep-water and soft-soil foundations). - Standard I-girders and box girders (DPWH bridge standards). - Architectural precast panels. ### 2.2 Post-tensioning In post-tensioning, tendons (cables or individual strands) are routed through ducts in the concrete member and tensioned **after** the concrete has cured to sufficient strength. The process is: 1. **Ducting:** Metal or plastic ducts are cast into the member, running along the intended tendon path (which can be straight, draped parabolic, or polygonal). 2. **Placement:** Strands or cables are threaded through the ducts after casting. 3. **Tensioning:** After concrete reaches sufficient strength, the tendons are tensioned using anchoring devices (typically fixed anchors at one end, movable jacks at the other). Tension is applied in stages, monitored by load cells and extension measurements, to minimize shock and ensure uniform stress. 4. **Anchoring:** Once tensioned, the tendons are anchored against fixed devices at each end (e.g., wedge-cone anchors). 5. **Grouting (optional but recommended):** The duct is filled with cement grout to protect the tendons from corrosion, restore structural continuity, and improve long-term performance. Grouting is **essential** in Philippine climate (high humidity, salt spray in coastal zones). **Advantages of post-tensioning:** - **Draped tendons:** Can be curved (parabolic, polygonal) to follow moment diagrams, improving load balancing and efficiency. - **Flexible geometry:** Suited to unique structures (curved bridges, irregular transfer beams, long-span slabs). - **Reduced dead load during construction:** Prestress is applied after concrete cures, allowing temporary supports to be removed earlier. - **Adaptable:** Tendon layout and force can be tailored to the specific structure. **Disadvantages:** - Friction losses occur during tensioning (especially significant for long or highly draped tendons). - Requires skilled labor and specialized equipment (jacking, grouting). - Ducts must be grouted to prevent corrosion and ensure durability—poor grouting is a common failure mode. - Slightly higher cost per member (but competitive for large, unique structures). **Common post-tensioned structures in the Philippines:** - Bridge decks and box girders (especially curved or long-span). - Transfer beams (supporting columns in high-rise buildings—common in Manila developments). - Long-span slabs (parking structures, auditoriums). - Marine/port structures (subject to corrosion—grouting is critical). - Building facades and special architectural elements. ### 2.3 Comparison Table | Aspect | Pre-tensioning | Post-tensioning | |--------|---|---| | **Timing** | Strand tensioned before casting | Tensioned after curing | | **Bond** | Bond over full length | Unbonded before grouting, bonded after | | **Geometry** | Straight or standard profile | Can be fully draped/curved | | **Transfer** | Bond friction (~600–800 mm) | Mechanical anchors (localized) | | **Applications** | Repetitive, standard shapes | Unique, long-span, variable sections | | **Friction loss** | Minimal (no duct) | Significant for draped tendons | | **Corrosion protection** | No duct (but strand quality critical) | Requires grouting (essential in Philippines) | | **Plant cost** | High (abutments, beds) | Lower per member | | **Philippine use** | Precast hollow-core, piles, standard girders | Bridges, transfer beams, special structures | For PRC exam purposes, be prepared to identify the appropriate method for a given structure and to explain the load transfer mechanism in each case.

The service stress state in a prestressed beam is found by superposing three components: (1) the axial prestress, (2) the moment due to eccentricity of the prestress (secondary moment), and (3) the bending moment due to applied loads. This superposition is the cornerstone of prestressed concrete design. ### 3.1 Stress Components For a simply supported beam with uniform cross-section, mid-span section, consider: - **Prestress force:** P (in kN or N), acting at eccentricity e below the centroid (positive e means below = toward bottom fiber). - **Applied moment:** M (in kN·m), due to sustained and live loads, causing sagging (compressing top, tensioning bottom). The stress at the top fiber (y = c_top from centroid) is: $$f_{\text{top}} = \frac{P}{A} - \frac{Pec_{\text{top}}}{I} + \frac{Mc_{\text{top}}}{I}$$ The stress at the bottom fiber (y = c_bot = d − c_top) is: $$f_{\text{bot}} = \frac{P}{A} + \frac{Pec_{\text{bot}}}{I} - \frac{Mc_{\text{bot}}}{I}$$ Where: - P/A = axial prestress (compression, positive). - Pec/I = flexural effect of eccentric prestress; note the **sign difference** between top and bottom. - Mc/I = flexural effect of applied load. **Sign Convention (Critical for PRC exams):** - **Compression** is taken as **positive** (common in prestress analysis). - An eccentric prestress **below** the centroid (e > 0 downward) causes **extra compression at the bottom** and **relief (less compression or tension) at the top**. - Applied sagging moment causes **compression at the top** and **tension at the bottom**. The design goal is to balance these stresses so that: - **At transfer (initial, Pi applied):** concrete is just barely at the edge of tension (or safe compression) at the top fiber, and safely in compression at the bottom. - **In service (effective, Pe applied + full load):** both fibers are within allowable stresses (typically no tension, or very low allowable tension per ACI 318 Section 24.4.2). ### 3.2 Allowable Service Stresses (NSCP 2015 and ACI 318) NSCP 2015 adopts ACI 318-19 for prestressed concrete design. Key allowable stresses (for service load conditions) are: **Immediately after transfer (before time-dependent losses):** - **At top fiber:** f'_ci is the compressive strength at transfer (typically 24.5–27.6 MPa for rapid curing). Allowable compression = 0.6 f'_ci ≈ 14.7–16.6 MPa. Allowable tension (limited, to control cracking) = 0.5√(f'_ci) ≈ 2.5–2.8 MPa (if bonded reinforcement provided). - **At bottom fiber:** Allowable compression = 0.6 f'_ci (same as top). **In service (sustained load + live load, after losses):** - **Compressive stress:** 0.45 f'_c (where f'_c is 28-day concrete strength, typically 35–40 MPa for prestressed members). Allowable ≈ 15.75–18 MPa. - **Tensile stress:** 0 (zero; no tension in service, or very limited per ACI 318 Table 24.4.2). Some designers allow 0.5√(f'_c) ≈ 3–3.3 MPa if longitudinal bonded reinforcement is provided to control cracking. **Design approach (typical NSCP/ACI process):** 1. **At transfer:** Check that the top fiber does not exceed allowable tension (or compression), using Pi and initial moment M_i (if applicable, e.g., self-weight on temporary supports). 2. **In service:** Check that both fibers remain within allowable stresses under Pe and full applied moment (dead load + live load). Often, the **critical stress** is the **top fiber at transfer** (risk of cracking/delamination) or the **bottom fiber in service** (risk of over-compression or yielding of reinforcement if present). ### 3.3 Practical Example: Rectangular Beam Consider the example from the reference material: **Given:** - Section: 300 mm wide × 600 mm deep. - A = 180,000 mm², I = 5.4 × 10⁹ mm⁴, c_top = c_bot = 300 mm. - Prestress: P_e = 900 kN at e = 150 mm (below centroid). - Applied moment: M = 120 kN·m (due to sustained + live load). **Calculate section properties:** $$\frac{P}{A} = \frac{900 \times 10^3 \text{ N}}{180 \times 10^3 \text{ mm}^2} = 5.0 \text{ MPa}$$ $$\frac{Pec}{I} = \frac{900 \times 10^3 \text{ N} \times 150 \text{ mm} \times 300 \text{ mm}}{5.4 \times 10^9 \text{ mm}^4} = \frac{40.5 \times 10^9}{5.4 \times 10^9} = 7.5 \text{ MPa}$$ $$\frac{Mc}{I} = \frac{120 \times 10^6 \text{ N·mm} \times 300 \text{ mm}}{5.4 \times 10^9 \text{ mm}^4} = \frac{36 \times 10^9}{5.4 \times 10^9} = 6.67 \text{ MPa}$$ **Top fiber stress:** $$f_{\text{top}} = 5.0 - 7.5 + 6.67 = 4.17 \text{ MPa (compression)}$$ **Bottom fiber stress:** $$f_{\text{bot}} = 5.0 + 7.5 - 6.67 = 5.83 \text{ MPa (compression)}$$ **Interpretation:** - Both fibers are in **compression**—no cracking risk. - The top fiber has lower stress (4.17 MPa) because the eccentric prestress (below centroid) relieves the top, but the applied moment compresses it back down. - The bottom fiber has higher stress (5.83 MPa) because both the eccentric prestress and the applied load moment compress it. - Assuming f'_c = 35 MPa, allowable service compression = 0.45 × 35 = 15.75 MPa. Both actual stresses are **well within** the limit—indicating efficient, safe design with room for live-load growth or cost optimization (e.g., reducing P or section). ### 3.4 Transfer vs. Service: The Critical Fiber Flip One of the most common pitfalls in PRC exams is forgetting that the **critical fiber changes** between transfer and service: - **At transfer (immediately after release, with Pi and minimal load):** The top fiber is at maximum risk of **tension** (or over-compression if prestress is excessive). The bottom fiber is safe. **Check top fiber first.** - **In service (with Pe and full load):** The applied moment compresses the top and tensions the bottom. Now the **bottom fiber is at risk** (over-compression or insufficient precompression). **Check bottom fiber first.** Example scenario: - At transfer, top fiber: f = P_i/A − P_i e c_top/I = 8 − 10 = −2 MPa (tension). **Unsafe if allowable tension < 2 MPa.** - In service, bottom fiber: f = P_e/A + P_e e c_bot/I − M c_bot/I = 4 + 8 − 12 = 0 MPa (neutral). **Safe, but only just.** Designers must check **both stages** and iterate on section and prestress level to satisfy both. ### 3.5 Load Case Analysis NSCP 2015 and ACI 318 require stresses to be checked at multiple load cases: 1. **At transfer (immediately after release):** - Prestress force: P_i (initial, before losses). - Load: typically self-weight (if the member is propped) or zero (if simply supported during release). 2. **Service—sustained load (dead load only):** - Prestress force: P_s (after immediate/elastic losses). - Load: self-weight + other permanent loads. 3. **Service—total load (dead + live):** - Prestress force: P_e (effective, after all losses). - Load: self-weight + live load (and other temporary loads). Stress limits vary slightly between cases. For a typical exam problem, assume: - **At transfer:** Top fiber limited to ~0.5√(f'_ci) tension or 0.6 f'_ci compression. - **Service (sustained):** Slightly relaxed limits (some provisions allow 0.25 f'_c tension with bonded reinforcement). - **Service (total):** Most restrictive—typically zero tension unless bonded reinforcement is present. The correct approach is to **read the problem carefully** and **identify which load case is being asked about**, then apply the correct stress limits.

The prestress force in a member decreases from the jacking value (P_i) to a smaller effective value (P_e) due to various loss mechanisms. These losses must be predicted and accounted for in design; underestimating losses can lead to insufficient precompression in service. ### 4.1 Classification of Losses Losses are classified as **immediate** (occurring during and shortly after tensioning) or **time-dependent** (ongoing over the life of the structure). #### 4.1.1 Immediate Losses **1. Elastic Shortening (ΔP_es)** When the prestress force is applied to the concrete member, the concrete itself undergoes elastic compression. The tendon, bonded to the concrete, follows this shortening and thus loses stress proportional to the shortening strain. **For pre-tensioning:** Elastic shortening occurs during concrete curing (before release), so the loss is subtracted from P_i to find the stress just after release. The loss depends on the stress in the concrete at the centroid of the prestressing steel: $$\Delta P_{\text{es}} = \frac{A_p}{A_c} \cdot E_p \cdot \epsilon_{\text{cc}} = \frac{A_p}{A_c} \cdot \frac{f_{\text{cir}}}{E_c}$$ where: - A_p = area of prestressing steel. - A_c = area of concrete. - E_p = modulus of elasticity of steel (~197 GPa for strands). - f_cir = stress in concrete due to prestress (P_i / A_c, approximately). - E_c = modulus of concrete (typically ~30–40 GPa for 35 MPa concrete). Typically, ΔP_es ≈ 2–4% of P_i for pre-tensioned members. **For post-tensioning:** Elastic shortening occurs slightly differently. As the tendon is tensioned, the concrete pushes back, and the anchorage point (jack) must move slightly to maintain the target stress. The loss is often negligible in post-tensioning because the jacking stage is brief and the member is already (mostly) at full load. However, if **sequential tensioning** is used (tensioning multiple tendons in stages), each earlier tendon loses stress due to elastic shortening caused by the later ones. This can be significant and is calculated as: $$\Delta P_{\text{es, sequential}} = \frac{A_p}{A_c} \cdot E_p \cdot \frac{\sum P_i}{E_c A_c}$$ For typical post-tensioned buildings with multiple tendons, sequential loss ≈ 3–5% per tendon. **2. Anchorage Seating Loss (ΔP_as)** **Post-tensioning only:** When a tendon is anchored (e.g., by driving wedges into a cone-shaped anchor head), the wedges slip slightly as they seat into their conical cavity. This slip—typically 5–10 mm for common anchors—causes the tendon to relax slightly near the anchor, losing stress over a distance ("seating distance") of typically 1–3 m. The loss is: $$\Delta P_{\text{as}} \approx \frac{\Delta L_{\text{seat}} \cdot E_p \cdot A_p}{L_{\text{tendon}}}$$ For a 20 m tendon with 6 mm seat slip, E_p = 197 GPa, A_p = 140 mm² (one strand): $$\Delta P_{\text{as}} \approx \frac{6 \times 197 \times 140}{20 \times 10^3} \approx 82.6 \text{ kN per strand} \approx 3–5\% \text{ (typical)}$$ Anchorage seating loss is reduced or eliminated by carefully selecting anchorage devices and monitoring the stressing process. **3. Friction Loss in Post-tensioning Ducts (ΔP_f)** As the tendon is pulled through a curved duct, friction between the tendon and the duct wall resists motion. The relationship is given by the **Capstan equation** (also called the Eytelwein formula): $$P(x) = P_0 e^{-\mu(\theta + Kx)}$$ where: - P(x) = tension at distance x along the tendon. - P_0 = initial (jacking) tension at the active (tensioning) end. - μ = coefficient of friction between tendon and duct (typically 0.20–0.30 for steel strand in metal or plastic duct). - θ = total angular change (in radians) of the tendon along the duct. - K = wobble coefficient, accounting for unintended curvature per unit length (typically 0.004–0.005 per meter for common practice). Rewritten in terms of force loss: $$\frac{\Delta P_f}{P_0} = 1 - e^{-\mu(\theta + KL)} \approx \mu(\theta + KL) \quad (\text{for small exponent})$$ **Example:** A parabolic tendon drapes over a 24 m span with sag e = 1.5 m. Angular change: θ = 8e/L = 8 × 1.5 / 24 = 0.5 rad. Wobble loss: KL = 0.004 × 24 = 0.096 rad. Total: 0.5 + 0.096 = 0.596 rad. With μ = 0.25: $$\Delta P_f / P_0 \approx 0.25 × 0.596 \approx 0.149 \text{ (14.9% loss)}$$ Friction loss can be large for heavily draped tendons and is a key design consideration. NSCP 2015 and ACI 318 provide friction coefficients; designers must use values appropriate to the actual materials and conditions. #### 4.1.2 Time-Dependent Losses **1. Concrete Creep (ΔP_cr)** Concrete continues to deform under sustained stress, gradually (over months to years). This creep increases concrete strain, which (if the tendon is bonded) pulls the tendon along, reducing its stress. The creep strain is related to the sustained stress and the creep coefficient C_c: $$\epsilon_{\text{creep}} = C_c \cdot \frac{f_c}{E_c}$$ where f_c is the sustained concrete stress and C_c ≈ 2–3 for typical long-term creep (10+ years) in tropical climates. Prestress loss due to creep: $$\Delta P_{\text{cr}} \approx A_p \cdot E_p \cdot C_c \cdot \frac{f_c}{E_c}$$ Typically, creep loss ≈ 5–8% of P_i for members in moderate conditions, and **higher in tropical/humid conditions** (like the Philippines), where concrete creep is accelerated. NSCP 2015 includes provisions for this. **2. Concrete Shrinkage (ΔP_sh)** Concrete shrinks as it loses moisture to the environment. Shrinkage occurs most rapidly in the first few weeks but continues for months. If the tendon is bonded (pre-tensioned or grouted post-tensioned), it resists the shrinkage, resulting in stress loss. Shrinkage strain (ultimate, after months to years): $$\epsilon_{\text{sh, ultimate}} \approx 400–800 \times 10^{-6} \quad (\text{depends on w/c, curing, ambient conditions})$$ Loss: $$\Delta P_{\text{sh}} = A_p \cdot E_p \cdot \epsilon_{\text{sh, ultimate}}$$ For a typical strand: $$\Delta P_{\text{sh}} \approx 140 \text{ mm}^2 \times 197 \text{ GPa} \times 500 \times 10^{-6} \approx 13.8 \text{ kN} \approx 2–4\% \text{ (typical)}$$ Shrinkage loss is roughly **proportional to surface-area-to-volume ratio**; thin members (slabs) shrink more than massive members (girders). In the Philippines, low humidity (coastal dry seasons) can exacerbate shrinkage loss. **3. Steel Relaxation (ΔP_rel)** High-strength steel (strands and wires) exhibits **stress relaxation**—gradual stress loss under sustained high stress, even if strain is held constant. This is a material property of the steel, independent of concrete behavior. Relaxation loss depends on: - **Initial stress level:** f_p / f_pu (higher stress → faster relaxation). - **Steel grade:** Low-relaxation strand (common, especially post-1970s) loses ~2–3%; stress-relieved wire loses ~10–15% over 1000 hours. - **Temperature:** Relaxation accelerates at higher temperatures (Philippine tropical climate slightly increases relaxation rate). For low-relaxation strand at f_p / f_pu = 0.75 (typical jacking), relaxation at 1000 hours ≈ 2.5% (one-day relaxation ~1%). At 10,000 hours: ~3.5–4%. Over 10+ years (approximately 87,600 hours), relaxation approaches 5–7% for low-relaxation strand. Typically, **relaxation loss ≈ 3–5%** of P_i for low-relaxation strands. ### 4.2 Total Losses and Effectiveness Factor **Approximate total losses (experienced in practice):** | Condition | Pre-tensioned | Post-tensioned | |-----------|---------------|----------------| | **Immediate** | ES: 2–4% | ES: 1–2% (minor), AS: 3–5%, Friction: 5–20% (depending on geometry) | | **Time-dependent (long-term)** | Creep: 5–8%, Shrinkage: 2–4%, Relaxation: 3–5% | Creep: 5–8%, Shrinkage: 2–4%, Relaxation: 3–5% (tendon only, concrete gains from grouting) | | **Total** | ~15–25% (typical 18–22%) | ~15–35% (typical 20–28%, higher if heavily draped) | **Effectiveness factor (loss ratio):** $$R = \frac{P_e}{P_i} = 1 - \frac{\sum \Delta P}{P_i}$$ For design, R ≈ **0.80–0.85** is commonly used. NSCP 2015 and ACI 318 provide detailed methods and tables for more precise calculations. ### 4.3 Design Practice: Using Losses The design process typically follows: 1. **Estimate total loss percentage** (using code methods or experience tables). 2. **Calculate effective prestress:** P_e = (1 − loss%) × P_i. 3. **Check stresses at transfer** using P_i (before losses occur) to ensure top fiber does not go into excessive tension. 4. **Check stresses in service** using P_e (after all losses) to ensure both fibers remain within allowable limits. 5. **Iterate** section and P_i if either stage violates limits. **A critical insight for exam candidates:** When a problem states "effective prestress P_e = 1,000 kN," this **already includes losses**. Do not apply losses twice. Conversely, if the problem states "jacking stress 1,400 MPa" or "initial prestress P_i = 1,200 kN," you must **calculate or apply losses** to find P_e for the service stage. Read the problem carefully. ### 4.4 Reducing Losses: Engineering Practice For critical structures or high-performance requirements, losses can be minimized: 1. **Use low-relaxation strand** (modern standard in most of the world, including Philippines). 2. **Limit jacking stress** (e.g., 0.75–0.80 f_pu instead of 0.85), reducing creep, shrinkage, and relaxation (but requiring larger section/more strands). 3. **Control concrete shrinkage** (low w/c ratio, proper curing, extended initial moist curing). 4. **Post-grouting** in post-tensioned members stabilizes concrete and can recover some stiffness (though not prestress). 5. **Optimize tendon routing** (minimize friction by avoiding sharp bends). 6. **Two-stage stressing** (partial stress immediately, final stress after some concrete curing)—reduces elastic shortening effects—not common but possible for special projects. In the Philippines, where tropical climate increases creep and shrinkage rates, careful attention to concrete mix design and curing (especially humidity control during the first week) is essential.

The load-balancing method is an elegant and intuitive design approach for prestressed concrete, particularly for post-tensioned members with draped tendons. It shifts the design perspective from stresses to loads, offering insight into how a well-designed tendon profile can significantly reduce bending moments and deflections. ### 5.1 Conceptual Basis: Upward Distributed Force Consider a simply supported beam with a parabolic tendon that drapes (sags) from the ends toward midspan. As the tendon curves downward, the internal tension in the tendon creates a distributed upward force on the concrete (by Newton's third law—the tendon pulls the concrete down, so the concrete pulls the tendon up, and the tension transmits a distributed upward load to the concrete). For a parabolic tendon with constant tension P (ignoring friction and losses momentarily), the distributed upward load w_bal is: $$w_{\text{bal}} = \frac{8Pe}{L^2}$$ where: - P = tendon force (kN). - e = sag (maximum deflection of tendon below the chord connecting end points, in meters). - L = span (meters). - w_bal = uniform distributed upward load (kN/m), equivalent to the load "balanced" by the tendon. **Derivation (brief):** A parabolic tendon has curvature κ = 8e/L². At any point, the vertical component of tension creates an upward force; integrating over the span yields w = P × κ = P × 8e/L² = (8Pe)/L². ### 5.2 Practical Implications: Moment Reduction If the actual gravity load on the beam equals w_bal, the internal moment in the concrete is approximately zero (or only the axial precompression remains). If the actual load is less than w_bal, the beam has a net upward load and a negative (hogging) moment. If the actual load exceeds w_bal, there is a sagging moment as usual, but reduced. **Example:** A simply supported beam, span L = 12 m, carries uniform load w = 18 kN/m. Tendon with sag e = 0.75 m and force P = 450 kN gives: $$w_{\text{bal}} = \frac{8 \times 450 \times 0.75}{12^2} = \frac{2700}{144} = 18.75 \text{ kN/m}$$ Since w ≈ w_bal, the applied load is nearly balanced. The internal moment is nearly zero, so the beam experiences primarily **axial precompression** P/A and minimal bending stress. The deflection is tiny. This is why prestressed beams with well-designed tendon profiles can span 50+ m with depth-to-span ratios of 1:50 or better, whereas RC beams at the same span might need depth ratios closer to 1:25. ### 5.3 Load-Balancing Design Procedure A common design approach is: 1. **Choose target load to balance:** Typically, balance a fraction of the sustained load (e.g., 60–80% of dead load + sustained live load), leaving the remainder (and temporary loads) to be resisted by bending of the section. 2. **Select sag e:** Limited by the geometry of the structure (e.g., available headroom, architectural constraints). For bridges, e/L ≈ 1/20 to 1/50 is typical. 3. **Solve for required tendon force:** P = w_bal L² / (8e). 4. **Select strands/tendons:** Determine the number and size of tendons to provide the required force, given material properties and jacking percentages. 5. **Verify stresses:** Check that the section (with chosen P and eccentricity profile) satisfies stress limits at transfer and service, accounting for unbalanced loads and other load cases. The elegance of load balancing is that it **separates concerns**: the tendon provides a known upward load (geometric + force property), and the remaining (unbalanced) load is analyzed using ordinary bending theory on a precompressed section. ### 5.4 Limitations and Refinements **1. Non-uniform loading:** The formula w_bal = 8Pe/L² applies to **uniform load**. For concentrated loads or non-uniform distributions, the equivalent "balancing load" varies along the span. A tendon profile can be optimized to balance a non-uniform load by adjusting the curvature profile (e.g., flatter near the supports, more sag at midspan), but this requires more complex tendon routing. **2. Friction and eccentric anchorage:** In post-tensioning, friction loss and the distance from anchorage points affect the actual stress profile and, therefore, the effective w_bal. The formula above assumes constant tension along the tendon; in practice, tension varies. **3. Deflection:** Even with perfect load balancing (w = w_bal), the beam is not perfectly rigid. The axial precompression P/A prevents bending deflection, but axial elastic strain (P × L / (E_c × A)) remains. For long-span structures, this axial strain, though small, can be significant. **4. Multiple load cases:** A single tendon profile balances **one specific load**. In practice, the structure experiences variable loads (e.g., asymmetric live load, wind, earthquake). Designers often balance the **permanent load** (dead load + essential live load) and accept that other load cases will produce some bending stress. ### 5.5 Example: Load-Balancing Design **Given:** Simply supported beam, L = 20 m, carries uniform dead load w_d = 12 kN/m and uniform live load w_l = 8 kN/m. Total w = 20 kN/m. Design to balance 100% of dead load + 50% of live load = 12 + 4 = 16 kN/m. Available sag: e = 1.2 m (architectural limit). **Find:** Required prestress force P. **Solution:** $$w_{\text{bal}} = 16 \text{ kN/m} = \frac{8Pe}{L^2} = \frac{8P \times 1.2}{20^2}$$ $$16 = \frac{9.6P}{400}$$ $$P = \frac{16 \times 400}{9.6} = \frac{6400}{9.6} \approx 666.7 \text{ kN}$$ **Verification of load balancing:** $$w_{\text{bal}} = \frac{8 \times 666.7 \times 1.2}{400} = \frac{6,400}{400} = 16 \text{ kN/m}$$ ✓ **Unbalanced load:** w_unbal = 20 − 16 = 4 kN/m (the remaining live load). This 4 kN/m is analyzed as ordinary bending on a precompressed section. The moment due to 4 kN/m is: $$M_{\text{unbal}} = \frac{w_{\text{unbal}} L^2}{8} = \frac{4 \times 20^2}{8} = \frac{1600}{8} = 200 \text{ kN·m}$$ This is **much smaller** than the total moment if the entire 20 kN/m load were resisted by bending: M_total = (20 × 400) / 8 = 1,000 kN·m. The prestress has reduced the design moment by 80%. **Benefit:** A section optimized for M ≈ 200 kN·m plus axial P = 666.7 kN is much smaller and lighter than a section for M = 1,000 kN·m with minimal axial stress. This is why load balancing yields efficient designs. ### 5.6 Practical Tendon Profile: Parabolic vs. Polygonal **Parabolic profile:** Smooth curve, suited for simply supported spans. Provides uniform w_bal over the span. **Polygonal (two-point) profile:** Two straight segments meeting at a point (often at midspan or near support). Easier to construct (fewer anchorages) but produces concentrated shear at the inflection point. Common in older designs. **Stepped or multiple-segment profile:** Used for complex loading (e.g., cantilevers + spans). Tendon follows the moment envelope more closely. For **Philippine infrastructure**, parabolic profiles are standard in modern bridge design (per DPWH guidelines), while polygonal profiles are common in building slabs (simpler formwork and construction). ### 5.7 Interaction with Structural Actions Load balancing is most effective when the **dominant load** is gravity (dead load + service live load). For structures subject to significant **dynamic** or **lateral** loads (e.g., seismic, wind, vibration from traffic), the full structural analysis must account for all load cases. Prestress helps with gravity-induced cracking but does not directly resist lateral load; lateral resistance requires adequate concrete strength, reinforcement, and section properties. In the Philippines, seismic design is mandatory (per NSCP 2015, which adopts IBC provisions). A structure must be designed for both gravity loads (with load balancing benefits for prestressed members) and seismic actions. The two must be checked separately (envelope approach).

Many prestressed structures in practice use **composite sections**—two-part members with different materials, properties, or casting times—to optimize cost, construction, and performance. ### 6.1 Composite Beam: Precast Beam + Cast-in-place Slab A common Philippines application (especially in DPWH bridge standards and building slabs) is a **precast prestressed beam supporting a cast-in-place (CIP) concrete slab**. The beam is pre-tensioned at a plant, transported to site, and erected. Then, forms are placed above the beam, reinforcement is tied, and concrete is cast to form the slab. After the slab cures, the composite section acts together. **Stress analysis changes:** 1. **Transfer stage (at plant, just after releasing strands):** Only the precast beam section resists moment. Stress = P_i / A_beam ± P_i e c / I_beam ± M_sw,beam c / I_beam, where M_sw,beam is the self-weight moment of the beam alone. 2. **Composite stage (after slab cures and bonds to the beam):** The moment due to slab self-weight and superimposed load is resisted by the **composite section** (beam + slab acting together). This is the long-term, in-service condition. - The neutral axis shifts (typically toward the slab due to its large area). - The moment of inertia increases significantly (composite I >> beam I). - Stresses in the beam at a given fiber reduce because M/I decreases (larger I). - Stress in the slab (newly formed from the composite action) is different from the beam. **Design procedure:** 1. Check beam stresses at transfer (using beam properties only) for the beam's self-weight moment. 2. Check composite-section stresses in service (using composite I and P_e) for the full applied moment (slab + superimposed load). 3. Ensure the interface between beam and slab is adequate (reinforcement, friction) to transfer shear and maintain composite action. **Practical detail:** In the Philippines, DPWH bridge designs often specify `600 mm` precast I-beams with a `200 mm` cast-in-place deck slab, creating a composite T-beam or box-beam. The precast beam carries its self-weight; the slab moment is shared by the composite section. ### 6.2 Hybrid Sections: Combining Prestressed and Reinforced Concrete Some members use both prestressing and conventional reinforcement (bonded longitudinal bars). This can be optimal when: - Prestress handles sustained loads (gravity, creep, shrinkage). - Reinforcement handles temporary or accidental loads (earthquake, impact, fatigue). - Cost is minimized by using less prestress and more conventional steel. Example: A building column subjected to both sustained axial load and occasional seismic moment. Prestressing reduces the sustained stress; reinforcement handles the bending. Stress analysis is similar (superposition), but the reinforcement yield and strain-hardening behavior must be verified (typically via section capacity analysis under combined action). ### 6.3 Grouted vs. Ungrouted Tendons in Post-tensioning **Grouted post-tensioned members** (ducts filled with cement grout after stressing): - Tendons are protected from corrosion (critical in the Philippine coastal environment and tropical humidity). - Concrete and steel are bonded along the tendon—more monolithic behavior, similar to pre-tensioned members. - Duct grouting ensures continuity; voids or incomplete grouting can fail catastrophically (water reaches tendon). **Ungrouted (unbonded) tendons:** - Tendons are protected by plastic sheathing or a thin grease coating (less effective, especially in saltwater). - Tendon is not bonded to concrete—stress redistribution after cracking is limited. - Used mainly in temporary structures or where grouting is impractical. - **Not recommended** for critical or long-lived structures in the Philippines due to corrosion risk. NSCP 2015 and ACI 318 emphasize grouting for all post-tensioned structures. Non-grouted members have seen failures in tropical climates (corrosion of strands leading to sudden loss of prestress). ### 6.4 Durability and Maintenance **Crack control via prestressing:** - Prestressed members inherently resist cracking (precompression) if designed correctly. - Even if fine hairline cracks form under overload, they close when load is removed (unlike RC, where cracks remain). - This greatly improves durability in harsh environments (salt, acid, freeze-thaw—though freeze-thaw is uncommon in the Philippines). **Corrosion protection:** - Pre-tensioned members: strands are bonded and surrounded by concrete. If the concrete cover is adequate, corrosion is unlikely over the design life (50–100 years). - Post-tensioned members: **grouting is essential**. Regular inspection of grout quality (via test cores or acoustic scanning) is recommended for critical structures. - In the Philippines, especially in coastal zones (e.g., Manila's waterfront infrastructure, port structures), pre-tensioned members (no grout interface) are often preferred for simplicity and durability. **Monitoring and maintenance:** - Prestressed structures require less maintenance than RC if designed and built correctly. - Periodic visual inspection (cracks, spalling, water stains) is recommended. - For post-tensioned structures, re-stressing or partial stressing (if prestress is found to be inadequate due to unexpected losses) is possible but expensive and should be avoided by good design. ### 6.5 Special Cases: Cantilevers, Continuous Spans, and Indeterminate Structures **Cantilever prestressed members:** - Hogging (negative) moment is the concern. Tendon should be placed **near the top** (above the centroid) to provide compression there. - Load balancing requires the tendon to curve **upward** (increasing curvature from support to free end), opposite to a simply supported span. This is less efficient geometrically but feasible. **Continuous prestressed beams:** - Internal supports create hogging moments at those supports and sagging moments in the spans. Tendon profile must be optimized to balance the moment envelope. - A typical profile (e.g., low at mid-spans, high at internal supports) requires more complex routing and is more difficult to construct than a simple parabolic profile. - Some designs use **selective stressing** (different tendon forces in different spans) to optimize each region. **Indeterminate structures:** - Prestress introduces internal forces (hyperstaticity) that can be difficult to analyze (especially distribution among redundant members). - Linear analysis (superposition) is valid if concrete remains elastic; for large prestress or overload, nonlinear analysis may be required. For PRC exam purposes, focus on **simply supported members** and **basic composite sections**. Complex statically indeterminate structures are less commonly examined at the PE level but may appear in advanced specialist exams. ### 6.6 Construction Sequence and Staged Loading Real-world construction introduces temporary conditions that differ from final design: 1. **Precast beam erected, before slab cast:** Beam carries only its own weight. Must not crack or deflect excessively during handling and erection (temporary support conditions may be needed). 2. **Slab cast, formwork propping:** Beam + formwork/props carry slab weight. Beam experiences temporary higher moment (props reduce but don't eliminate this). 3. **Slab cures, props removed:** Composite section comes into play. Full moment now shared. 4. **Superimposed load (finished flooring, live load):** Final service state. Stress checks at each stage are required (per NSCP 2015 / ACI 318 Section 24.2.2). Missing checks at intermediate stages is a common design error. **Example:** A precast beam, if over-stressed during the slab-casting stage (temporary), might develop cracks that persist and reduce long-term durability, even if final stresses are acceptable. ### 6.7 Connection Details: Bearing Pads, Supports, and Anchorages Prestressed members must be connected carefully to the supporting structure (columns, walls, abutments). Key considerations: 1. **Bearing pads:** Elastomeric or neoprene pads distribute the reaction and allow rotation. Size and material are critical to prevent local crushing of the beam end. 2. **Anchorages for post-tensioned members:** Wedge-cone anchors, barrel-and-wedge, or other mechanical devices must reliably transfer very high tension (often 500+ kN per tendon). Improper anchorage has caused catastrophic failures. 3. **Shear keys and friction considerations:** Horizontal shear (from temperature expansion, earthquake) must be resisted by friction or shear keys. Prestress contributes to friction (normal force), but explicit shear keys or reinforcement may be needed. 4. **Concrete bearing strength:** Local bearing stress at the support must not exceed allowable values (per ACI 318 Section 22.8: bearing stress ≤ 0.85 f'_c or more, depending on bearing area). In the Philippines, DPWH standards and PCA (Portland Cement Association) design guides specify these details; engineers must follow them rigorously.

To consolidate the concepts, here is a comprehensive design example, following typical NSCP 2015 / ACI 318 procedures. ### Example: Design of a Simply Supported Prestressed Bridge Girder **Given:** - **Span:** L = 25 m (simply supported). - **Superimposed load:** Uniform dead load (besides self-weight) w_d = 12 kN/m. Live load w_l = 8 kN/m (design per DPWH/AASHTO standards). - **Concrete:** f'_c = 40 MPa (28-day), f'_ci = 32 MPa (at transfer). γ_c = 24 kN/m³. - **Prestressing:** Use low-relaxation 7-wire strands, 12.7 mm (1/2 in), f_pu = 1,860 MPa. Jacking stress f_pj = 0.75 × 1,860 = 1,395 MPa. Effective losses: R = 0.85, so P_e = 0.85 P_i. - **Design constraints:** - At transfer: top fiber stress ≤ 0.5√(f'_ci) = 0.5√32 ≈ 2.83 MPa tension (or just in compression if favorable). - In service: no tension allowed (strict), compression ≤ 0.45 f'_c = 18 MPa. - Deflection: not to exceed span/240 in service. **Design steps:** #### Step 1: Estimate section depth and cross-sectional area For long-span prestressed girders, typical depth-to-span ratio is 1:30 to 1:40. For 25 m, estimate h ≈ 25,000 / 35 ≈ 714 mm. Use **h = 750 mm** (round number, conducive to formwork). Assume a **T-section** (typical for bridge girders): flange width b_f = 3,000 mm (to accommodate a two-lane deck + cantilevers), flange thickness t_f = 250 mm. Web width b_w = 400 mm, web depth = 750 − 250 = 500 mm. (A precast hollow-core or box section might be optimized further, but a solid T is assumed here for simplicity.) Area calculation: - Flange: 3,000 × 250 = 750,000 mm². - Web: 400 × 500 = 200,000 mm². - Total: **A = 950,000 mm²** ≈ 0.95 m². Centroid: Located approximately 320 mm from the top (accounting for the large flange). c_top = 320 mm, c_bot = 750 − 320 = 430 mm. Moment of inertia (approximate, using parallel-axis theorem for component parts): - Flange: I_f = (3,000 × 250³) / 12 + 750,000 × (320 − 125)² ≈ 3.9 × 10⁹ + 22.8 × 10⁹ = 26.7 × 10⁹ mm⁴. - Web: I_w = (400 × 500³) / 12 ≈ 4.17 × 10⁹ mm⁴. - Total: **I ≈ 31 × 10⁹ mm⁴**. (More precise calculation requires detailed geometry; the above is an estimate.) #### Step 2: Calculate moments and loads **Self-weight of girder:** w_sw = γ_c × A = 24 × 0.95 = 22.8 kN/m. **Total loads in service:** w = w_sw + w_d + w_l = 22.8 + 12 + 8 = 42.8 kN/m. **Moments:** - M_sw = (22.8 × 25²) / 8 = 1,781 kN·m. - M_d = (12 × 25²) / 8 = 938 kN·m. - M_l = (8 × 25²) / 8 = 625 kN·m. - M_total = 1,781 + 938 + 625 = 3,344 kN·m. #### Step 3: Choose eccentricity and estimate prestress force To balance the permanent load (w_sw + w_d = 22.8 + 12 = 34.8 kN/m), choose sag e such that: $$w_{\text{bal}} = 34.8 = \frac{8 P e}{L^2}$$ Assume e = 250 mm (available space in a deep girder; not to interfere with ducts or reinforcement): $$P = \frac{34.8 \times 25^2}{8 \times 0.25} = \frac{21,750}{2} = 10,875 \text{ kN}$$ This seems very high (unrealistic for a single girder). Re-examine: perhaps e = 350 mm is more typical: $$P = \frac{34.8 \times 625}{8 \times 0.35} = \frac{21,750}{2.8} ≈ 7,768 \text{ kN}$$ Still quite high. In practice, for a 25 m span, a typical bridge girder might have 30–60 strands, which, with f_pj = 1,395 MPa and area ~140 mm² per strand, gives: $$P_i ≈ 40 \text{ strands} \times 140 \text{ mm}^2 \times 1,395 \text{ MPa} / 1000 ≈ 7,812 \text{ kN}$$ So 40 strands (or 50 for a larger girder) is reasonable. Let's **design for P_i ≈ 7,850 kN** (approximately 56 strands of 12.7 mm each). **Effective prestress:** P_e = 0.85 × 7,850 = 6,672 kN. #### Step 4: Check stresses at transfer **At transfer:** P_i = 7,850 kN, M = M_sw = 1,781 kN·m, e = 350 mm (below centroid). Axial compression: P_i / A = 7,850,000 / 950,000 = 8.26 MPa. Eccentricity moment (M_e = P_i × e = 7,850 × 0.35 = 2,747.5 kN·m): - At top: − (M_e × c_top) / I = − (2,747.5 × 10⁶ × 320) / (31 × 10⁹) = −28.4 MPa (relief/tension). - At bottom: + (M_e × c_bot) / I = + (2,747.5 × 10⁶ × 430) / (31 × 10⁹) = +38.1 MPa (compression). Applied moment (self-weight): - At top: + (M_sw × c_top) / I = + (1,781 × 10⁶ × 320) / (31 × 10⁹) = +18.4 MPa (compression). - At bottom: − (M_sw × c_bot) / I = − (1,781 × 10⁶ × 430) / (31 × 10⁹) = −24.7 MPa (tension). **Top fiber at transfer:** f_top = 8.26 − 28.4 + 18.4 = **−1.74 MPa (tension).** Allowable: 0.5√(32) ≈ 2.83 MPa tension. **1.74 < 2.83, so it passes,** but barely. If the allowable were stricter (e.g., zero tension), we'd need to reduce P_i or increase e, or increase f'_ci (faster curing). **Bottom fiber at transfer:** f_bot = 8.26 + 38.1 − 24.7 = **21.66 MPa (compression).** Allowable: 0.6 f'_ci = 0.6 × 32 = 19.2 MPa. **21.66 > 19.2, so it FAILS.** **Remedial action:** Increase e (place tendons lower, if geometry allows) or reduce P_i (increase the number of strands to a lower jacking stress, or use a different section). Let's reduce P_i to 7,000 kN and increase e slightly to 380 mm. (Iterative process; for brevity, assume this resolves the transfer check.) #### Step 5: Check stresses in service **In service:** P_e = 6,000 kN (after 15% losses), M_total = 3,344 kN·m, e = 380 mm. Eccentricity moment (M_e = 6,000 × 0.38 = 2,280 kN·m): - At top: − (2,280 × 10⁶ × 320) / (31 × 10⁹) = −23.5 MPa. - At bottom: + (2,280 × 10⁶ × 430) / (31 × 10⁹) = +31.6 MPa. Applied moment: - At top: + (3,344 × 10⁶ × 320) / (31 × 10⁹) = +34.5 MPa. - At bottom: − (3,344 × 10⁶ × 430) / (31 × 10⁹) = −46.4 MPa. **Top fiber in service:** f_top = 6.32 − 23.5 + 34.5 = **17.3 MPa (compression).** Allowable: 0.45 f'_c = 18 MPa. **17.3 < 18, passes.** ✓ **Bottom fiber in service:** f_bot = 6.32 + 31.6 − 46.4 = **−8.48 MPa (tension).** Allowable: 0 (strict) or up to ~0.5√(40) ≈ 3.16 MPa if bonded reinforcement is present. **−8.48 < 0, so it FAILS.** Again, design iteration is needed. Perhaps increase P_e or e further, or accept that the bottom fiber will have some non-prestressed (ordinary) reinforcement to carry the tension. In practical design, a few strands of bonded reinforcement in the bottom are often provided as a "safety net." For the purpose of this example, assume a revised design with P_e = 6,500 kN passes all checks (detailed calculations omitted for brevity). #### Step 6: Deflection check The effective prestress provides an "upward" distributed load w_bal = 8 × P_e × e / L² = 8 × 6,500 × 0.38 / 625 ≈ 31.5 kN/m. This is close to the permanent load (22.8 + 12 = 34.8 kN/m), so the beam is nearly load-balanced. Unbalanced load = 34.8 − 31.5 = 3.3 kN/m (plus live load 8 kN/m total unbalanced ≈ 11.3 kN/m). Defection due to unbalanced load (assuming moment of inertia I_eff ≈ 31 × 10⁹ mm⁴): $$\delta = \frac{5 w L^4}{384 E_c I}$$ With w = 11.3 kN/m, L = 25 m, E_c = 33 GPa: $$\delta = \frac{5 \times 11.3 \times 10^3 \times 25^4}{384 \times 33 \times 10^3 \times 31 \times 10^9} ≈ \frac{443 \times 10^9}{4 \times 10^{12}} ≈ 0.11 \text{ m} = 110 \text{ mm}$$ (Simplified; actual deflection is slightly less due to prestress contribution.) Allowable: L / 240 = 25,000 / 240 ≈ 104 mm. **Deflection 110 mm slightly exceeds limit; may need slightly larger I or more prestress.** Iterate. #### Step 7: Reinforcement details Provide: - **Longitudinal reinforcement (non-prestressed):** At least a few bars in the bottom layer to resist the service tension and control any cracking. Use #16 rebars, typically 4–6 bars. - **Transverse reinforcement (stirrups):** Design for shear per NSCP 2015 Section 22.4. At supports, higher shear; stirrup spacing reduces from ~150 mm near support to ~300 mm at mid-span. - **Confinement reinforcement:** At the ends (transfer zone), place helical or spiral reinforcement to prevent splitting of the concrete due to prestress transfer. ### Summary of Design Outcome After iteration (details omitted for brevity), the final design might be: - **Section:** T-beam, 3 m wide flange, 750 mm deep, web 400 mm wide. - **Prestress:** 50 strands, 12.7 mm, initial force P_i ≈ 7,850 kN, effective P_e ≈ 6,700 kN, eccentricity e ≈ 380 mm. - **Reinforcement:** 4 × #16 bars in bottom (non-prestressed), stirrups #10 @ 150 mm near support tapering to #10 @ 300 mm mid-span. - **Performance:** - Stresses at transfer and service: within limits. - Deflection: ~100 mm (at limit). - Crack control: excellent (prestress prevents cracking under service load). - Durability: high (proper cover, pre-tensioned construction, low water/chloride ingress risk). This girder would be manufactured in a precast facility, transported to the bridge site, and erected on bearing pads (elastomeric neoprene pads, ~50 mm thick). A cast-in-place deck slab would then be formed and cast on top, creating a composite section for the final stage. ### Key Lessons from This Example 1. **Iteration is essential:** Initial estimates are refined through multiple checks (transfer, service, deflection, reinforcement). 2. **Multiple limits:** Must satisfy stress limits at **both** transfer and service, **both** top and bottom fibers. Usually, one limit is tight and drives the design. 3. **Load balancing:** Choosing e and P to balance a significant fraction of the permanent load dramatically reduces the unbalanced moment and deflection—a major advantage of prestressing. 4. **Practical reinforcement:** Non-prestressed reinforcement is often added to ensure ductility and to carry any unexpected overload or impact loads. 5. **Composite action:** The design is staged—transfer stage uses beam properties, final stage uses composite properties (if a slab is cast on top). 6. **Code compliance:** NSCP 2015 / ACI 318 govern every step. Always reference the code sections (e.g., ACI 318 Section 24 for prestressed flexural members).