15.4 Prestressed Concrete Fundamentals and Prestress Losses

15.4 Prestressed Concrete Fundamentals and Prestress Losses

Prestressed concrete is an advanced structural system wherein high-strength steel tendons are tensioned against high-strength concrete to introduce controlled internal compressive stresses. In conventional reinforced concrete, concrete carries compression while internal steel bars carry tension only after the concrete cracks. Prestressed concrete transforms concrete into an active elastic uncracked material: internal pre-compression neutralizes tensile bending stresses caused by service gravity loads, eliminating cracking, reducing deflections, permitting longer spans with shallower depths, and substantially improving durability against marine corrosion and environmental degradation.

In Philippine civil engineering practice, prestressed concrete dominates bridge superstructures (AASHTO girders, segmental box girders), long-span precast building floors (double-tee slabs, hollow-core planks), and large liquid-retaining tanks under NSCP 2015 Chapter 4 (ACI 318).

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1. Fundamentals: Pretensioning vs. Post-Tensioning

Prestressed concrete members are constructed using one of two primary mechanical methods:

1. Pretensioning:

   • Tendons are stressed between rigid external bulkheads on a casting bed before concrete is poured.
   • Concrete is placed around the stressed tendons and allowed to cure until reaching required transfer strength ($f'_{ci} \ge 25 - 30\text{ MPa}$).
   • The tendons are flame-cut or gradually released; as the steel attempts to contract elastically, compressive force is transferred to the concrete through mechanical bond and friction along the transfer length ($l_t \approx 50 - 60 d_b$).
   • Widely utilized in precast manufacturing plants for standard bridge girders, piles, and hollow-core slabs.

2. Post-Tensioning:

   • Hollow metal or plastic ducts are cast inside the concrete formwork alongside mild reinforcement.
   • Tendons are threaded through the ducts after the concrete has hardened and achieved adequate compressive strength.
   • Tendons are jacked using hydraulic rams bearing directly against the hardened concrete ends and anchored with mechanical split-wedge assemblies.
   • Ducts are subsequently injected with cementitious grout (bonded post-tensioning) to protect steel from corrosion and ensure bond, or tendons are coated with grease inside plastic sheaths (unbonded post-tensioning).
   • Standard for cast-in-place commercial floor slabs, transfer girders, and segmental bridge construction.

High-Strength Materials

• Prestressing Tendons: High-strength 7-wire low-relaxation steel strands conforming to ASTM A416 Grade 270 ($f_{pu} = 1,860\text{ MPa}$, yield stress $f_{py} = 0.90 f_{pu} \approx 1,674\text{ MPa}$) with elastic modulus $E_p \approx 195,000 - 200,000\text{ MPa}$. Mild steel ($f_y = 275 - 415\text{ MPa}$) cannot be used because prestress losses would completely eliminate the initial strain.
• Concrete: Compressive cylinder strength $f'_c \ge 35 - 50\text{ MPa}$ to withstand high local anchor bearing stresses, minimize creep deformation, and provide high shear capacity.

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2. Core Concepts: Analytical Methods for Prestressing

Engineers analyze prestressed concrete beams using three conceptually distinct, mathematically equivalent approaches:

1. Stress Superposition Method (Combined Direct & Bending Stress)

The prestressing force $P$ acting at eccentricity $e$ below the section centroid produces an axial compression force $P$ and a hogging internal bending moment $M_p = P \cdot e$. Combining these with external gravity bending moments ($M = M_D + M_L$):

$\sigma = -\frac{P}{A} \pm \frac{P e y}{I} \mp \frac{M y}{I}$

Using the standard civil engineering sign convention (compression negative, tension positive):

$\sigma_{\text{top}} = -\frac{P}{A} + \frac{P e}{S_{\text{top}}} - \frac{M}{S_{\text{top}}}$

$\sigma_{\text{bot}} = -\frac{P}{A} - \frac{P e}{S_{\text{bot}}} + \frac{M}{S_{\text{bot}}}$

Where:

• $A$ = cross-sectional area of concrete.
• $S_{\text{top}} = I / y_{\text{top}}$, $S_{\text{bot}} = I / y_{\text{bot}}$ = section moduli for top and bottom fibers.
• $e$ = tendon eccentricity measured from the cross-sectional centroid (positive downward).

2. Internal Couple Method ($C-T$ Concept)

The prestressing tendon acts as an internal tension tie ($T = P$), while the concrete acts as a compressive block ($C = P$). As external bending moment increases, the internal compressive resultant $C$ shifts upward by distance $z = M / P$. The internal resisting couple is $M = C \cdot a$, where $a$ is the internal lever arm between $C$ and $T$.

3. Load Balancing Method (T. Y. Lin)

Formulated by Prof. T. Y. Lin in 1963, this method models the curved tendon as applying an equivalent transverse upward force on the concrete beam due to cable curvature. For a parabolic draped tendon with mid-span sag $e$ and span length $L$:

$w_{\text{bal}} = \frac{8 P e}{L^2}$

• The upward balancing load ($w_{\text{bal}}$) directly counteracts the downward distributed gravity load ($w_{\text{gravity}}$).
• The net effective transverse load acting on the concrete member is: $w_{\text{net}} = w_{\text{gravity}} - w_{\text{bal}}$
• If the prestressing is designed such that $w_{\text{bal}} = w_{\text{dead}}$, the dead load bending moment is completely eliminated ($M_{\text{net}} = 0$). Under dead load, the beam experiences zero deflection and pure uniform axial compression: $\sigma = -\frac{P}{A}$
• Live load stresses are then simply calculated on the uncracked transformed section as $\pm M_L / S$.

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3. Kern of Section and Limiting Eccentricities

The kern (or core) of a structural cross-section defines the geometric boundary within which an axial compressive force $P$ must be applied to ensure that no tensile stress develops anywhere across the cross-section under prestress alone ($M = 0$).

Setting bottom fiber stress $\sigma_{\text{bot}} = -\frac{P}{A} + \frac{P e}{S_{\text{bot}}} \le 0$:

$e \le \frac{S_{\text{bot}}}{A} = \frac{I}{A y_{\text{bot}}} = \frac{r^2}{y_{\text{bot}}} = k_{\text{top}}$

For a solid rectangular cross-section of width $b$ and total depth $h$:

• $r^2 = \frac{h^2}{12}$, $y_{\text{top}} = y_{\text{bot}} = \frac{h}{2}$.
• Upper kern limit: $k_t = \frac{h^2 / 12}{h / 2} = \frac{h}{6}$.
• Lower kern limit: $k_b = \frac{h}{6}$.
• This forms the famous middle-third rule: as long as the tendon remains within the middle third of the beam depth ($e \le h/6$), the entire section remains in compression under prestress alone.

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4. Prestress Losses: Immediate and Long-Term

The prestress force in a tendon decreases continuously over time from the initial jacking force ($P_j$) to the initial transfer force ($P_i$), and ultimately to the effective service force ($P_e$). Total prestress losses typically range from $15\% \text{ to } 22\%$ of the initial jacking stress.

Prestress losses are categorized into two stages:

$\Delta f_p = \underbrace{\Delta f_{pES} + \Delta f_{pANC} + \Delta f_{pF}}_{\text{Immediate Losses}} + \underbrace{\Delta f_{pSH} + \Delta f_{pCR} + \Delta f_{pRE}}_{\text{Time-Dependent Long-Term Losses}}$

Immediate Losses (Occur during jacking and transfer)

1. Elastic Shortening of Concrete ($ES$): As prestress is transferred to the concrete, the concrete shortens elastically, causing the bonded steel tendons to shorten simultaneously:

   • Pretensioned Members: $\Delta f_{pES} = K_{es} \frac{E_p}{E_{ci}} f_{cs}$ Where $n = E_p / E_{ci}$ is the modular ratio at transfer, and $f_{cs}$ is the concrete compressive stress at the tendon centroid immediately after transfer ($f_{cs} = \frac{P_i}{A} + \frac{P_i e^2}{I} - \frac{M_D e}{I}$).
   • Post-Tensioned Members (Tensioned sequentially): Tendon 1 shortens when Tendons 2, 3, and 4 are jacked. For $N$ sequentially jacked tendons: $\Delta f_{pES} = \frac{N - 1}{2 N} \left(\frac{E_p}{E_{ci}}\right) f_{cs}$ (If all post-tensioned tendons are jacked simultaneously, $\Delta f_{pES} = 0$).

2. Anchorage Slip / Seating Loss ($ANC$): When the hydraulic jack releases the tendon, the anchoring wedges slip inward into the conical anchor head before gripping the strand. If anchorage seating slip is $\Delta L_{\text{slip}}$ (typically $6 - 10\text{ mm}$): $\Delta f_{pANC} = \frac{\Delta L_{\text{slip}}}{L} E_p$

3. Friction Along Post-Tensioning Tendons ($F$): Friction between the tendon and the duct wall consists of two components: the curvature effect (intentional angular change $\alpha$) and the wobble effect (unintentional misalignment along length $x$): $P_x = P_0 e^{-(\mu \alpha + K x)} \approx P_0 [1 - (\mu \alpha + K x)]$ Where $\mu$ = curvature friction coefficient ($0.15 - 0.25$), $\alpha$ = total angular change (radians), $K$ = wobble coefficient ($0.0003 - 0.0020\text{ per meter}$).

Long-Term Time-Dependent Losses

1. Concrete Creep ($CR$): Progressive deformation of concrete under sustained compressive stress over years. In the Zia et al. method, $\Delta f_{pCR} = K_{cr} (E_p / E_c)(f_{cir} - f_{cds})$, with $K_{cr} = 2.0$ for pretensioned and $1.6$ for post-tensioned members.
2. Concrete Shrinkage ($SH$): Volume loss as the concrete dries. In the same method, $\Delta f_{pSH} = 8.2 \times 10^{-6} K_{sh} E_p (1 - 0.06 V/S)(100 - RH)$, with the volume-to-surface ratio $V/S$ in inches. In SI units with $V/S$ in mm, the term becomes $(1 - 0.0024\,V/S)$.
3. Steel Relaxation ($RE$): Loss of stress in high-strength steel held under constant high strain over time. For low-relaxation strands, relaxation loss is small ($20 - 30\text{ MPa}$).

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5. Serviceability Stress Limits (NSCP 2015 / ACI 318)

Prestressed concrete sections are classified based on the extreme fiber tensile stress ($f_t$) at service loads:

• Class U (Uncracked): $f_t \le 0.62 \sqrt{f'_c}\text{ MPa}$. Gross uncracked section properties govern.
• Class T (Transition): $0.62 \sqrt{f'_c} < f_t \le 1.0 \sqrt{f'_c}\text{ MPa}$.
• Class C (Cracked): $f_t > 1.0 \sqrt{f'_c}\text{ MPa}$. Cracked section properties must be analyzed.

Allowable Concrete Stresses at Transfer (Before Losses)

• Extreme fiber compression: $f_{ci} \le 0.60 f'_{ci}$.
• Extreme fiber tension (except ends): $f_{ti} \le 0.25 \sqrt{f'_{ci}}\text{ MPa}$.

Allowable Concrete Stresses at Service (After Losses)

• Extreme fiber compression under sustained load: $f_c \le 0.45 f'_c$.
• Extreme fiber compression under total load: $f_c \le 0.60 f'_c$.

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6. Comprehensive Worked Examples

Worked Example 1: Extreme Fiber Stresses at Transfer and Service

Problem: A simply supported pretensioned beam of span $L = 10.0\text{ m}$ has a rectangular cross-section $300\text{ mm} \times 600\text{ mm}$. A straight tendon is positioned at an eccentricity $e = 120\text{ mm}$ below the neutral axis. The initial prestress force at transfer is $P_i = 900\text{ kN}$. Long-term prestress losses are $18\%$ ($P_e = 0.82 P_i = 738\text{ kN}$). Concrete unit weight is $24\text{ kN/m}^3$. Service superimposed live load produces a mid-span moment of $M_L = 90\text{ kN}\cdot\text{m}$. Calculate extreme fiber stresses at mid-span: (a) at initial transfer (prestress + self-weight), and (b) at service stage (effective prestress + total dead and live load).

Solution:

• Step 1: Section Properties: $A = 300\text{ mm} \times 600\text{ mm} = 180,000\text{ mm}^2$ $S = \frac{b h^2}{6} = \frac{300 \times (600)^2}{6} = 18.0 \times 10^6\text{ mm}^3$ Beam self-weight: $w_D = 0.30\text{ m} \times 0.60\text{ m} \times 24\text{ kN/m}^3 = 4.32\text{ kN/m}$. Self-weight moment: $M_D = \frac{w_D L^2}{8} = \frac{4.32 \times (10)^2}{8} = 54.0\text{ kN}\cdot\text{m} = 54.0 \times 10^6\text{ N}\cdot\text{mm}$. Total service moment: $M_{\text{total}} = M_D + M_L = 54.0 + 90.0 = 144.0\text{ kN}\cdot\text{m} = 144.0 \times 10^6\text{ N}\cdot\text{mm}$.

• Step 2: Stresses at Initial Transfer ($P_i = 900\text{ kN}$, $M_D = 54.0\text{ kN}\cdot\text{m}$):

  • Axial stress: $-\frac{P_i}{A} = -\frac{900,000\text{ N}}{180,000\text{ mm}^2} = -5.00\text{ MPa}$.
  • Eccentricity moment stress: $\frac{P_i e}{S} = \frac{900,000 \times 120}{18.0 \times 10^6} = +6.00\text{ MPa}$.
  • Dead load stress: $\frac{M_D}{S} = \frac{54.0 \times 10^6}{18.0 \times 10^6} = 3.00\text{ MPa}$.
  • Top fiber stress: $\sigma_{\text{top}} = -5.00 + 6.00 - 3.00 = -2.00\text{ MPa} \quad (2.00\text{ MPa Compression})$
  • Bottom fiber stress: $\sigma_{\text{bot}} = -5.00 - 6.00 + 3.00 = -8.00\text{ MPa} \quad (8.00\text{ MPa Compression})$

• Step 3: Stresses at Final Service Stage ($P_e = 738\text{ kN}$, $M_{\text{total}} = 144.0\text{ kN}\cdot\text{m}$):

  • Axial stress: $-\frac{P_e}{A} = -\frac{738,000}{180,000} = -4.10\text{ MPa}$.
  • Eccentricity moment stress: $\frac{P_e e}{S} = \frac{738,000 \times 120}{18.0 \times 10^6} = +4.92\text{ MPa}$.
  • Total moment stress: $\frac{M_{\text{total}}}{S} = \frac{144.0 \times 10^6}{18.0 \times 10^6} = 8.00\text{ MPa}$.
  • Top fiber stress: $\sigma_{\text{top}} = -4.10 + 4.92 - 8.00 = -7.18\text{ MPa} \quad (7.18\text{ MPa Compression})$
  • Bottom fiber stress: $\sigma_{\text{bot}} = -4.10 - 4.92 + 8.00 = -1.02\text{ MPa} \quad (1.02\text{ MPa Compression})$ Result: Both top and bottom fibers remain comfortably in compression throughout full service life; the beam operates fully uncracked.

Worked Example 2: Load Balancing Method

Problem: A post-tensioned beam has a span of $L = 12.0\text{ m}$ and carries a dead load of $w_D = 15.0\text{ kN/m}$ (including self-weight). A parabolic tendon is draped with zero eccentricity at both supports and maximum sag $e$ at mid-span. If the effective post-tensioning force is $P_e = 600\text{ kN}$, what sag $e$ is required to balance $100\%$ of the dead load?

Solution:

• Setting upward balancing load equal to dead load: $w_{\text{bal}} = w_D = 15.0\text{ kN/m}$. $w_{\text{bal}} = \frac{8 P_e e}{L^2} \implies e = \frac{w_{\text{bal}} L^2}{8 P_e}$ $e = \frac{15.0\text{ kN/m} \times (12.0\text{ m})^2}{8 \times 600\text{ kN}} = \frac{15.0 \times 144}{4,800} = \frac{2,160}{4,800} = 0.450\text{ m} = 450\text{ mm}$

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7. Licensure Exam Pitfalls & Review Notes