Where Can the Cable Go?

1. The question left open

At the end of ÖNG-01 we saw this: with a constant eccentricity the governing section is the support, and the limit the support imposes leaves the section under-used at midspan. In the corrected design the midspan utilisations came out at 0.21 and 0.64.

The natural question follows: if the cable need not run straight, where may it sit at each section?

2. Two states, four conditions

A prestressed beam is checked at two separate moments, and the two push in opposite directions:

• At transfer. The tendons have just been stressed; the prestressing force is at its largest (P_i) and the external load at its smallest (usually self-weight only). The danger is the prestress over-stressing the section.
• In service. Losses have occurred; the force has fallen (P_e) and the external load is at its largest. The danger is the prestress being insufficient.

Two fibres are checked in each state; four conditions in total:

3. Why transfer gives an upper and service a lower limit

This symmetry is not accidental but physically necessary.

At transfer there is no load to balance the prestress. As the eccentricity grows, the prestress moment grows and drives the top fibre into tension and the bottom into excessive compression. So a larger e makes things worse → an upper limit.

In service the external moment drives the bottom fibre into tension, and what balances it is the prestress moment. If the eccentricity is too small, the balance fails. So a smaller e makes things worse → a lower limit.

4. At the support: the same number as ÖNG-01

At the support M_min = M_max = 0. Computing the zone:

This means two separate functions of the engine — one computing the constant-eccentricity limit, the other the cable zone — arrive at the same number by independent derivations. It is kept as a separate case in the test suite.

5. At midspan: the zone is empty

At midspan the moments enter: the self-weight moment M₀ at transfer, and the total moment in service.

One detail deserves attention: the upper limit found here is precisely the 46.3 cm discussed in ÖNG-01. In the original solution that value was found as "the eccentricity optimised for midspan", and it was correct — but only for the transfer condition. The service condition was never checked, and at this section it cannot be satisfied.

6. What does an empty zone mean?

What can be done is limited, and none of it concerns cable placement:

• Enlarge the section (increase the section modulus)
• Raise the concrete grade (increase the allowable stresses)
• Change the prestressing force — but this is a double-edged sword: raising it pushes the transfer limit harder
• Add passive reinforcement and move to partial prestressing, i.e. accept cracking and control it

7. An independent cross-check

Now the interesting part. We can compute the section modulus that would bring the zone to exactly zero width — that is, "how large would the section have to be for the zone to be only just closing?"

The two routes are entirely independent. A direct section check finds the required modulus in one step from the stress limits. The cable zone approach sets up four inequalities, writes the zone width and sets it to zero. That they land on the same number is evidence that both are correctly constructed.

8. Behaviour along the span

As you move along the beam and the moment grows, how does the zone change?

Both limits rise with the moment — and that is the direct answer to why tendons are laid out parabolically: the cable must follow the moment diagram, because the permissible zone follows it too.

But the lower limit rises faster than the upper one, because the service moment is larger than the transfer moment and the service force smaller than the transfer force. So the zone narrows towards midspan, and at this section it closes.

9. What the zone does not tell you

In this case the zone is empty anyway; but even if it were not, whether the eccentricity found is less than half the section depth would have to be checked separately. In another problem mentioned in ÖNG-01 the required eccentricity came to 89.74 cm while the section geometry allowed at most 62.5 cm — physically impossible. The stress calculation does not say so on its own.

10. What to do in practice

1. Plot the zone along the beam, not at one section. An eccentricity satisfied at one section may fail at another — that was the whole subject of ÖNG-01.
2. Check both states together. Looking only at transfer produces exactly the 46.3 cm error in that case.
3. Treat a narrowing zone as a warning. A width approaching zero means construction tolerances alone could break the design.
4. If the zone is empty, stop working on cable placement. The problem is in the section or in the force, not in where the cable sits.
5. Check the physical layout separately. The stress zone may not fit inside the section.
6. Verify by two independent routes. A direct section check and the cable zone must agree; if they do not, one of them is wrong.

11. Test yourself

1. Why do transfer conditions give an upper limit for e and service conditions a lower one?
2. Why does the upper limit of the cable zone at the support coincide with the constant-eccentricity limit?
3. Which four interventions can close an empty zone? Which of them is double-edged?
4. Why are tendons laid out parabolically? Answer in terms of the zone.
5. Why does the zone narrow towards midspan? Which two effects produce that together?
6. Why might an eccentricity that is valid in stress terms still be unusable?
7. What does it mean that two independent routes give the same required section modulus? What if they had not?