The Constant Eccentricity Trap

1. Why does this article exist?

A worked design was reviewed and four separate errors were found in it. One of them is a structural safety error: a section that would crack in service was declared adequate. The other three are subtler but each would distort a real design.

The point is not to criticise a particular solution. All four errors are of a kind that anyone can make, and each has the same shape: a check performed in the right way at the wrong place, or a rule applied correctly from the wrong system.

2. The governing equation

The fibre stresses in a prestressed section are the superposition of three effects:

Everything in this article follows from this one relation. There is nothing code-specific in it.

3. Variable and constant eccentricity

In a post-tensioned beam the tendon is usually laid out on a curve: deep at midspan, rising towards the supports. In a pretensioned member it is often straight — that is, the eccentricity is constant along the beam, because a straight bed is far simpler to build.

4. Error 1: forgetting the support

The worked design chose e = 46.3 cm, and that value is correct — for midspan, at transfer. But the check was carried out only there. Applying the same eccentricity at the support gives:

5. Deriving the correct limit

With a constant eccentricity the support governs, so the limit must be derived there. Setting M = 0 in the governing equation and solving for e:

6. Which fibre governs?

7. The corrected design

Notice the cost. At midspan the utilisations are now well below 1 — the section is being used inefficiently there, because the support dictated the eccentricity. That is the price of a straight tendon, and it is exactly the question ÖNG-02 takes up.

8. Error 2: ignoring the horizontal restraint force

At a dapped end the beam bears on a corbel or bearing pad. As the beam shortens under prestress, creep and shrinkage, friction at the bearing resists that movement and generates a horizontal force H_d. The original solution took it as zero.

The horizontal reinforcement row deserves attention: with H_d = 0 it does not exist at all. Ignoring the force does not merely under-size a bar; it removes a reinforcement layer from the design entirely.

9. Error 3: mixing two code philosophies

This error is the most instructive, because it is conservative — and therefore easy to defend and hard to notice.

The original solution used the design strength of philosophy A together with the reduction factor of philosophy B. The reinforcement comes out — larger than it should be.

10. Error 4: reading the geometry

The fourth error is the plainest: section properties were taken as given rather than derived from the dimensions. When A, I and W are entered by hand, an arithmetic slip propagates silently through every subsequent check.

Our engine does not accept manual section properties. It takes the dimensions and computes A, I, W and the self-weight itself. That is not a convenience — it removes a whole class of error.

11. How the engine makes these impossible

One design decision is worth stating. The midspan bottom fibre in the corrected design comes to 19.2004 MPa against a limit of 19.2. A strict comparison would fail it. We use a relative tolerance of 0.1% — but the utilisation is always reported, so a section sitting on the limit is visible rather than hidden behind a pass.

12. Test yourself

1. With a constant eccentricity, why is the support the critical section rather than midspan?
2. Both limits in section 5 are upper limits. Why must both be computed?
3. Where does the horizontal force at a dapped end come from, and what does ignoring it remove from the design?
4. Mixing two safety philosophies gives a conservative result. Why is that still an error?
5. In the corrected design the midspan utilisations are far below 1. What is the cost of that, and what would you do about it?
6. Why should section properties be derived rather than entered?