Understanding Secondary Effects in Post-Tensioning : A Physical Explanation

We mostly recall that M = M_primary + M2, but here is a simpler explanation that can be made, so we don't just memorize the math.

1. Moment Decomposition

For a prestressed member, at any section:

M = M_{primary} + M_{secondary}

• Primary:

  M_{primary} = F*e

  (prestressing force × eccentricity, as if the member were determinate and free to bend)

• Secondary:

  M_{secondary} = M - F*e

  This is the “extra” moment you only get because restraints prevent the tendon from producing its natural deflected shape.

What can cause this? Yes, the restraint reaction!

M_{secondary} = R_{restraint}*x

where R_{restraint} is the additional reaction caused by compatibility and x is the lever arm.

That’s the essence: secondary moments are just restrained reaction effects redistributed as moments.

2. Reaction Decomposition

Similarly, at supports:

R = R_{static} + R_{restraint}

• Static part R_{static}: comes from external loads only (dead, live, wind, seismic, etc.).

• Restraint part R_{restraint}: arises from prestressing when the structure can’t freely deform.

That's our conclusion

If restraint is absent (say, a simply supported single-span girder, free to take the tendon profile without continuity):

• R_{restraint} = 0

• So M_{secondary} = 0

• Meaning: prestress only contributes through F*e.

3. The Unifying Interpretation

• Secondary = effect of restraint = “compatibility correction”.

• Without restraint, prestressing acts like a free body load (pure F*e).

• With restraint, structure fights back → extra reactions → extra moments/shears.

• If R_{restraint} = 0, then

  M_{secondary} = M - F*e = 0

• If R_{restraint} ~= 0, all secondary actions are just “reaction forces from restraint”.

Why is post-tensioning a special case?

Restraint and secondary effects mostly occur in post-tensioning because of structural continuity. In pre-tensioning, the member is freely deformed, so that the restraints and hence secondary effects are not present.

How to use this in the ACI318-19 code?

In ACI, the secondary effects from post-tensioning must be included by 1.0P, i.e.

1.2D + 1.6L +1.0P

where P is not the total prestressing load as we might be misunderstood by its notation P (it shall be replaced by R in my opinion), but only this secondary (restraint) effect.

The systematic way to get this P can be done as following:

1. Analyse the structure under external load without considering prestressing effect, i.e. to get 1.2D + 1.6L

2. Apply equivalent prestressing load (some software might allow you to input the physical tendon inside the member).

3. From step 2, we get {R_restraint} which only occurs if the structure is indeterminate.

4. In load case P, compute member forces, mostly M & V, due to this R at the design section (at distance x from support). This can be easily done by hand calculation or a dedicated post-processing script.

   
   

   SAP2000 can perform these post-processing steps when you define the load pattern (of prestressing) as HYPERSTATIC. In fact, this can be applied to other similar load patterns like shrinkage and temperature.

AGAIN, load case P is not the member forces directly caused by any equivalent loads from prestressing. They are caused by restraint reactions only!