If you’ve ever stood on a post-tensioned slab during stressing operations, you know the moment of truth: the jack gauge reads one number, but does the elongation match? Getting the elongation calculation right is the bridge between design intent and field verification — and it’s one of the most nuanced calculations in structural engineering.
Why Elongation Matters
Post-tensioning works by stressing high-strength steel tendons and anchoring them against the concrete. The tendon elongation during stressing is the primary field verification that the correct force has been applied. ACI 318-19 Section 26.10.3.1 requires that measured elongations agree with calculated values within ±7% (or within a tolerance acceptable to the licensed design professional).
If your measured elongation is significantly less than predicted, it could mean:
- Higher-than-expected friction along the tendon path
- The tendon is snagged or kinked
- The concrete has cracked or the duct has collapsed
- Incorrect stressing sequence
If it’s significantly more than predicted, the friction might be lower than assumed, or the tendon profile doesn’t match the design drawings.
The Physics: What Drives Elongation?
At its core, elongation is governed by Hooke’s Law applied along the tendon:
Where f(x) is the tendon force at position x, Eps is the modulus of elasticity of the strand (typically 28,500 ksi for 7-wire strand), and Aps is the strand area. The challenge? f(x) is not constant. It decreases along the tendon due to friction.
Understanding Friction Losses
The force in a post-tensioned tendon decreases from the stressing end (jacking end) toward the dead end due to two friction mechanisms:
1. Curvature Friction (μα)
As the tendon curves through its draped profile, friction between the strand and the duct reduces the effective force. The cumulative angular change (α) at any point determines the curvature loss:
Typical curvature friction coefficient μ:
- Unbonded tendons (greased and sheathed): μ = 0.07
- Bonded tendons in metal duct: μ = 0.15–0.25
- Bonded tendons in plastic duct: μ = 0.19–0.23
2. Wobble Friction (κ)
Even in a “straight” duct, the tendon wobbles due to construction tolerances. This introduces a friction loss proportional to length. The wobble coefficient κ is typically:
- Unbonded tendons: κ = 0.0010–0.0015 per ft
- Bonded tendons: κ = 0.0001–0.0005 per ft
Anchor Set Loss: The Reverse Friction Zone
When you release the jack, the wedges seat and the strand pulls back into the anchor by a small amount — typically 3/8 inch (0.375 in.) for most anchor systems. This creates a “reverse friction” zone near the stressing end where the force drops below the immediate post-stressing value.
The anchor set loss zone extends a distance Lset from the stressing end. Beyond this zone, the force profile is unchanged. Calculating the set loss zone requires iterating — you need to find the point where the “mirror image” of the friction curve intersects the original curve, such that the area between the two curves (which represents the physical set distance) equals the anchor set value.
This is where hand calculations get tedious and error-prone. Most engineers use a bisection method or numerical solver to find the set loss distance.
Numerical Integration: Getting the Elongation Right
Because the force varies continuously along the tendon, the elongation integral is best solved numerically. The most practical approach for engineering applications is Simpson’s Rule, which divides the tendon into segments and approximates the integral:
Process:
- Divide the tendon into segments (finer near the stressing end where friction changes rapidly)
- Calculate the tendon force at each segment using the friction equation
- Apply anchor set loss to modify forces in the set loss zone
- Integrate f(x)/(Eps × Aps) over each segment using Simpson’s Rule
- Sum segment elongations for total elongation
For a typical 60-foot slab tendon stressed from one end with fjack = 0.80fpu (216.8 ksi for 270-ksi strand), you might see:
Jacking force: 33.2 kips (0.153 in² strand)
Friction loss at dead end: ~8–12% (unbonded)
Anchor set loss zone: ~15–25 ft from stressing end
Calculated elongation: ~4.5–5.5 in. (varies with profile and span)
Field Verification Tips
Even a perfect calculation means nothing if the field measurement is wrong. Key practices:
- Mark the strand before stressing. Measure from the mark to the anchor after stressing. This is the gross elongation minus the initial reference.
- Account for live-end seating. The initial “take-up” when the jack engages is not part of the elastic elongation. Use a reference load (typically 10% of jacking force) as your baseline.
- Record both gauge pressure and elongation. They should correlate. If one is right and the other is off, investigate — don’t just average and move on.
- Watch for sequential stressing effects. Stressing adjacent tendons compresses the concrete, which can affect elongation of previously stressed tendons (elastic shortening loss).
Stop Doing This by Hand
The elongation calculation involves friction exponentials, numerical integration, iterative anchor set computation, and multi-tendon batch processing. It’s the kind of calculation that demands a computer — not because the math is hard, but because the iteration count makes hand calculation impractical and error-prone.
The FicientDesign PT Elongation Calculator handles the full calculation: Simpson’s Rule integration, bisection-based anchor set loss, multi-segment tendon profiles, and batch processing for multiple tendons. It’s built by a licensed P.E. who has been in the field watching these calculations get verified.
It’s one of our most powerful engineering tools — and if you’re doing PT work, it’ll save you hours per project.
FicientDesign builds professional engineering tools for practicing engineers. All calculators are developed by licensed Professional Engineers and follow current ACI, AISC, and PTI code provisions.