Prestressed concrete beam
A simply supported pretensioned or post-tensioned beam with bonded or unbonded tendons by ACI 318-19: tendon stress limits, losses (lump sums or the Zia et al. expressions), stresses at transfer and at service with the class of Table 24.5.2.1, flexural strength with fps of 20.3.2, the 1.2 Mcr minimum, shear by Vci and Vcw, camber and deflection.
Inputs of the prestressed beam #
- Materials: f'c and f'ci at transfer, the concrete type, Ec; the bar grade for As and the stirrups.
- Section and span: rectangular or T-section, the simple span.
- Prestressing: pretensioned (bonded) or post-tensioned (bonded or unbonded), Aps, the strand diameter (transfer length 50 db), fpu and the steel type (fpy/fpu and γp), the jacking stress, the tendon profile (straight, parabolic, harped at a fraction of the span) by its eccentricities below the centroid at the ends and at midspan, bonded mild steel As and its depth.
- Losses: lump sums at transfer and in total (the supplier's or your values), or the Zia et al. (1979) expressions for elastic shortening, creep, shrinkage and relaxation with the relative humidity, V/S, Ksh and C, plus friction and seating for post-tensioning. These expressions are not in ACI 318 (20.3.2.6 lists what losses include); they are written from knowledge of the paper as the PCI Design Handbook gives them, marked "to confirm".
- Loads: the self-weight, superimposed dead and live load (uniform), the sustained part of the live load; whether bonded reinforcement resists the tension at transfer (24.5.3.2.1).
- Shear and deflection: stirrups; the live load deflection limit; the long-term multipliers (PCI values prefilled, from knowledge).
Auto-design varies the depth (2 in. steps) and the number of strands; the profile, losses and loads stay yours.
Checks performed #
| Check | Basis |
|---|---|
| Jacking stress | ≤ 0.94 fpy and 0.80 fpu; ≤ 0.70 fpu at post-tensioning anchorages after transfer (Table 20.3.2.5.1) |
| Losses | Lump sums or Zia et al. (1979) [to confirm]; fse ≥ 0.5 fpu for the approximate fps (20.3.2.3.1, 20.3.2.4.1) |
| Transfer stresses | 0.70 f'ci / 6√f'ci at the end (transfer length or anchorage), 0.60 f'ci / 3√f'ci at midspan (Tables 24.5.3.1, 24.5.3.2) |
| Class | U (≤ 7.5√f'c) or T (≤ 12√f'c) at service (Table 24.5.2.1); Class C NOT FEASIBLE here (24.5.2.3) |
| Service compression | 0.45 f'c sustained, 0.60 f'c total (Table 24.5.4.1) |
| Flexure | fps by Eq. 20.3.2.3.1 (bonded) or Table 20.3.2.4.1 (unbonded); rectangular or T block; φ by εt with εty = 0.002 |
| Minimum reinforcement | φMn ≥ 1.2 Mcr with bonded tendons (9.6.2.1, unless 9.6.2.2); As ≥ 0.004 Act with unbonded tendons (9.6.2.3) |
| Shear | Vci and Vcw (22.5.6.3) at h/2 and every tenth of the span to midspan, prestress reduced in the transfer length; Vs; Av,min (Table 9.6.3.4); spacing |
| Deflection | Camber and self-weight at transfer (Eci), live load with Ig (Class U) or Ie (Class T, approximation), final by the multipliers |
The drawings show the elevation with the tendon centroid and the midspan section.
Limitations #
- One simple span with uniform loads; continuous members and their secondary moments, composite toppings, point loads, debonding and the anchorage zones (25.9) are not covered. Continuous post-tensioned slabs: the PT slab strip.
- fps by the approximate expressions only (no strain compatibility); for a T-section ρp uses the flange width.
- The Class T deflection uses Ie with the tendons and bars transformed and the axial prestress ignored in Icr; the long-term multipliers are planning values.
- The loss expressions and the multipliers are from knowledge (Zia et al. 1979; PCI Design Handbook), not on file.
Verification examples #
Verification status: 19 values compared in 3 examples (0 published, 3 hand calculations from the cited equations): 19 within tolerance, 0 recorded deviations. No published example: every value is a hand calculation. That checks the program against this reading of the code, not the reading itself. Code basis: partial. Only some provisions were read in the target edition; the rest rests on older editions, examples or summaries.
No published prestressed beam example is on file (the PCI Design Handbook and the PTI manual are not). The values are hand calculations (closed form): the transfer and service fiber stresses, fps of Eq. 20.3.2.3.1, the stress block and Mn, Mu, Mcr, Vcw at midspan, the live load deflection and the camber of a harped tendon for the default 12 × 28 in. beam; the Zia et al. losses as the module writes them; fps of an unbonded tendon and As,min = 0.004 Act.