Prestressed Concrete
Principles of prestressing, pre-tensioning and post-tensioning, stress concept and load balancing, stresses at transfer and service, and losses of prestress with formulas.
Concrete is strong in compression but weak in tension. Prestressing puts a planned compression into the member before loading, so that service loads cause little or no tension.
Methods
- Pre-tensioning: tendons are stretched between abutments, concrete is cast around them, and when it hardens the tendons are released; the force is transferred by bond. Used for factory-made members (sleepers, hollow-core slabs).
- Post-tensioning: concrete is cast with ducts; tendons are threaded, stressed and anchored after hardening. Used for bridges and large members on site.
Three ways to look at it
1. Stress concept. With prestress P at eccentricity e below the centroid, and moment M from loads:
f = P/A ∓ P e y/I ± M y/I
At the bottom fibre: f_bottom = P/A + Pe/Z_b − M/Z_b. Compression is positive here.
2. Strength (internal couple) concept: the tendon and the concrete compression form a couple, like an RCC beam.
3. Load balancing: a parabolic tendon with sag h over span L exerts an upward load w_p = 8Ph/L². If w_p equals the applied UDL, the beam is under pure axial compression P/A with no bending.
Losses of prestress
| Loss | Pre-tensioned | Post-tensioned |
|---|---|---|
| Elastic shortening of concrete | m f_c (f_c = concrete stress at tendon level) | Only if tendons are stressed one after another (about half of m f_c on average) |
| Shrinkage of concrete | Yes | Yes (smaller, concrete is older) |
| Creep of concrete | Yes | Yes |
| Relaxation of steel | Yes | Yes |
| Anchorage slip | No | Δ E_s/L |
| Friction (wobble and curvature) | No | P_x = P₀ e^(−(μα + kx)) |
Total losses are typically 15–25%, so high-strength steel is essential: with mild steel the losses would wipe out most of the prestress.
Worked example
A rectangular beam 300 × 600 mm with a straight tendon at e = 200 mm, P = 1200 kN.
- A = 180,000 mm²; Z = 300 × 600²/6 = 18 × 10⁶ mm³.
- P/A = 6.67 MPa; Pe/Z = 1200 × 10³ × 200/(18 × 10⁶) = 13.33 MPa.
- Top: 6.67 − 13.33 = −6.67 MPa (tension); bottom: 6.67 + 13.33 = 20 MPa (compression), before load.
Practice
A parabolic tendon with P = 1000 kN has a sag of 0.2 m over a 10 m span. What uniform upward load (kN/m) does it apply?
Show answer and explanation
Answer: 16w_p = 8Ph/L² = 8 × 1000 × 0.2/100 = 16 kN/m.
A post-tensioned tendon 20 m long has an anchorage slip of 5 mm (E_s = 200 GPa). What is the loss of stress in MPa?
Show answer and explanation
Answer: 50Loss = ΔE_s/L = 5 × 200,000/20,000 = 50 MPa.
The concrete stress at the tendon level is 8 MPa and the modular ratio is 6. What is the elastic shortening loss in a pre-tensioned member in MPa?
Show answer and explanation
Answer: 48Loss = m f_c = 6 × 8 = 48 MPa.
Which loss does NOT occur in pre-tensioned members?
Show answer and explanation
Answer: CPre-tensioned tendons are straight in open beds with no ducts, so there is no duct friction (and no anchorage slip).