Post-Tensioned Concrete Slabs: Load Balancing and Tendon Layout
A post-tensioned slab threads high-strength steel tendons through the concrete before stressing them against hardened end anchorages. The lateral force from each curved tendon profile pushes upward against the slab soffit, partially counteracting gravity loads. The result is a thinner, stiffer slab that can span farther without beams — a combination that makes post-tensioned flat plates the predominant choice for parking structures, office buildings, and high-rise residential floors above roughly six stories.
The Load Balancing Concept
T. Y. Lin’s load balancing method, introduced in 1963, reframes prestress design in terms familiar from ordinary structural analysis. A tendon following a parabolic profile exerts a uniformly distributed upward load on the slab equal to:
wb = 8Pe / L2
where P is the effective prestress force (after losses), e is the sag of the tendon below the chord connecting the high points at supports, and L is the span. If wb equals the sustained dead load, the slab is in a state of uniform axial compression P/A with no bending and no deflection under that load. All bending and deflection arises only from loads not balanced by the tendon profile. This insight makes tendon layout decisions physically intuitive: band the tendon concentration where gravity loads are highest to maximize local balancing.
Tendon Layout: Banded and Distributed Directions
Two-way post-tensioned flat plates are typically laid out in a banded-distributed pattern. In one direction (usually the longer span), tendons are concentrated in bands 4 to 6 feet wide centered on column lines. In the other direction, tendons are uniformly distributed across the slab width. This arrangement achieves several goals simultaneously:
- Banded tendons in the column direction provide substantial column strip moments that resist punching shear by increasing the slab’s flexural stiffness at supports.
- Distributed tendons in the other direction provide uniform pre-compression across the full slab width, suppressing cracking.
- The combined system allows each tendon to be stressed from one end only, reducing the stressing labor and cost compared with symmetric layouts.
The minimum average pre-compression required by ACI 318 Section 8.6.1 is 125 psi in both directions. This limit ensures that the prestress is sufficient to suppress tensile cracking under service loads. Most designers target 150 to 200 psi to provide margin above the code minimum and control long-term cracking.
Tendon Profile and High-Point Location
The tendon profile must follow the moment diagram envelope: high points (maximum cover at top) over supports, low points (minimum cover at bottom) at midspan. ACI 318 Chapter 26 specifies minimum cover requirements that indirectly set the maximum available drape e. For a 7-inch slab with 3/4-inch cover to a 0.5-inch diameter tendon duct:
| Position | Tendon centroid from top (in) | Available for profile |
|---|---|---|
| Over support (top) | 1.0 in (minimum cover + half duct) | — |
| Midspan (bottom) | 6.0 in (7 in − 1 in) | — |
| Drape e | — | 5.0 in (127 mm) |
With a drape of 5 inches and a span of 22 feet, the balanced load from a 26-kip prestress force per tendon at 5-foot spacing is wb = 8 × 26 × (5/12) / 222 = 15.5 psf — reasonable for a typical office dead load of 12 to 15 psf without finishes.
Prestress Losses
The jacking stress fpj for low-relaxation strand is typically 0.80fpu = 0.80 × 270 ksi = 216 ksi. By the time the concrete is in service, several loss mechanisms reduce this to the effective prestress fpe:
- Friction and wobble losses: Tendon friction against the duct walls and unintended curvature accumulate over the tendon length. For unbonded monostrand in flat slabs, friction coefficients μ ≈ 0.05 to 0.07 and wobble k ≈ 0.001 per foot are typical, producing 5 to 10 percent loss over a 60-foot tendon length.
- Anchorage seating loss: The wedge anchor slips 0.25 to 0.375 inch as it sets, causing a local loss concentrated near the stressing end.
- Elastic shortening: Stressing the tendon compresses the concrete, which shortens and relaxes the tendon slightly.
- Time-dependent losses: Creep, shrinkage, and relaxation reduce prestress over the service life. Total long-term losses from these three sources typically total 15 to 25 ksi for normal-weight concrete.
After all losses, effective prestress is commonly 145 to 160 ksi, giving an effective force of about 23 to 25 kips per 0.5-inch strand. The design must use the effective force, not the jacking force, for service-level stress checks and load balancing calculations.
Flexural Strength Check
ACI 318 requires that the factored moment demand Mu not exceed the nominal flexural capacity φMn using strand stress at nominal strength fps. For unbonded tendons, fps is estimated by ACI 318 Eq. 20.3.2.4 rather than strain compatibility, because the strand stress is uniform along the entire length between anchors: fps = fse + 10,000 + f′c/(100ρp) ≤ fpy, where ρp = Aps/(bdp) is the prestress reinforcement ratio. Minimum bonded supplemental reinforcement of 0.004Act per ACI 318 Section 8.6.2 is required in each direction to ensure ductile behavior and limit crack widths if the slab is ever stressed beyond its cracking load in the unbalanced moment region.