Module 13 · Lesson 13.2
Losses, and load balancing
Why the force falls by a fifth, and the mental model that makes prestressed design intuitive.
Why this matters
Two things are worth having from this lesson. The prestress force is not constant — it falls by 20 to 25% over the life of the member, dominated by effects the designer cannot control at the moment of stressing. And there is a way of thinking about prestress that turns a page of algebra into one line: treat the tendon as applying an upward load, and ask what load it balances.
By the end of this lesson you should be able to
- Name the four losses and say which dominate
- Explain why pre-tensioned members lose more to elastic shortening
- Use load balancing to size a prestress force
- Explain why a fully balanced member has no deflection at all
What you should already know
- The two stages (previous lesson)
- Creep and shrinkage (Module 6)
Four losses, and which ones matter
Elastic shortening. As the force is transferred, the concrete shortens, and the tendon bonded to it shortens with it, losing stress. A pre-tensioned member loses the full amount at once. A post-tensioned member with tendons stressed in sequence loses roughly half, because each tendon is stressed against concrete that has already shortened under the earlier ones — and the last tendon stressed loses nothing at all.
Shrinkage. The concrete shrinks as it dries, and the tendon goes with it. This depends on humidity and member size, not on anything structural.
Creep. The concrete goes on shortening under the sustained prestress. This is the largest single loss, and it is proportional to the creep coefficient.
Relaxation. The tendon itself loses stress at constant strain, like any highly stressed steel. Modern low-relaxation strand keeps this small.
For a typical member these come out at roughly 4%, 5%, 7% and 2.5% — a total near 20%, dominated by creep and shrinkage. Both are properties of the concrete and its environment. Nothing the designer does at the moment of stressing changes them.
Load balancing
A parabolic tendon, pulled taut with force P and dropping a distance d between its ends and midspan, is a cable — and from the Structural Analysis Fundamentals course, a cable's shape is the bending moment diagram of the equivalent beam divided by its horizontal force. Turn that round and the tendon applies an upward distributed load to the concrete:
wbalanced = 8 P d / L²
Choose the drape so that this equals the applied load, and the member has no net transverse load at all. It carries pure axial compression, its stress is uniform top to bottom, and its deflection is zero.
That is the most useful mental model in prestressed concrete. It converts a two-stage stress problem into a single question — how much load do I want to balance? — and it explains at a glance why prestressed members deflect so little.
Worked example
Losses and load balancing for the same beam
Given
- The beam of the previous lesson: 350 × 800, span 14.0 m, C45/55
- Initial tendon stress 1300 N/mm²; concrete stress at tendon level 9 N/mm²
- Creep coefficient 2.0; shrinkage strain 300 × 10⁻⁶; relaxation 2.5%
- Final prestress force 1716 kN at a drape of 250 mm
Find
The losses, and how much load the tendon balances.
Assumptions
- Pre-tensioned, so the full elastic shortening applies
- Ep = 195 GPa
- The loss estimate is indicative; a rigorous calculation needs the specification
Practice
A parabolic tendon carries 1500 kN with a drape of 300 mm over a 15 m span. What uniform load does it balance, in kN/m?
Practice
What drape would be needed to balance 24 kN/m with 1500 kN over the same 15 m span, in mm?
Practice
Losses are elastic 4%, shrinkage 5%, creep 7% and relaxation 2.5%. What is α, the ratio of final to initial prestress?
Summary
- Four losses: elastic, shrinkage, creep, relaxation — totalling 20 to 25%
- Creep and shrinkage dominate, and neither is under the designer's control at stressing
- Post-tensioned members lose about half the elastic shortening of pre-tensioned ones
- Losses affect SERVICE only; transfer sees close to the full initial force
- Over-estimating losses is unconservative at transfer
- A parabolic tendon applies an upward load of 8Pd/L²
- Balancing the permanent load exactly gives zero long-term deflection
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint