C14. More about prestressed concrete

The proper modeling of prestressing requires using specific software that can manage the cable layouts, the calculation of tensions (calculation of losses), and consider the phasing and creep laws.

Example of complex cabling

However, it is always possible to perform a prestressing calculation or to verify a complex calculation to model the prestressing in a simplified way.

Beams (and shells) must be described at their center of gravity to ensure the correct positioning of the ropes in the cross-section.

The next two sub-sections present respectively the simplified modeling of a rope within a concrete section and a rope outside of a concrete section. It assumes that the rope path is known and in constant tension (after instantaneous or long-term losses, for example). It is useful to specify that the modeling of prestressing losses would follow the same logic but with a sign opposite to the action of the initial prestressing.

C.14.1 Cable inside of a concrete section

The external forces method makes it possible to understand the effects of ropes by modeling them as concentrated forces at the ends and as pressures (thrusts) along the rope.

End anchors or embedded anchors, we will have:

a horizontal force H_A = P.cos(α),

a vertical force V_A = P.sin(α),

a bending moment M_A = H_A.e.

(with the sign convention adapted to the software)

Along the beam, a cable exerts radial thrusts of pi≈P/Ri, which in the general case can be assumed to be vertical. This is the thrust exerted by the totality of the cables. They are applied as classical distributed loads.

Straight segments do not produce thrust (R=∞).

H_A and P are frequently confused, as the cos(α) is often close to 1.00.

Example of a simplified manual definition of a rope.

We analyze in this example a two-span beam of 25m each. The section is symmetrical and 1.25m tall (to set the limits of the cable).

Input data of the cable.

One must ensure that the cable remains inside the beam (in this case the limit is set as 10cm away from the upper and lower faces) and that the connections at the inflection points have the same slope.

Converted to loading on the beam:

The resulting shear efforts, i.e. the shear efforts considering the hyperstaticity of the system:

The bending moment:

For control purposes, we can:

always return to an isostatic system (hereby removing the central support),

calculate the isostatic bending moment at the middle of the span (or in any section), by summing the forces on the left or the right. The differences cannot be greater than the %,

in this case, verify that the results are symmetrical since the structure and the prestressing are symmetrical.

“Isostatic" shear effort diagram:

“Isostatic" bending moment diagram (the curve divided by HA is the rope path):

It should be ensured that the support reactions in the "isostatic" prestressed load case are equal to zero:

Calculation using specific software:

For this example, a software capable of modeling directly the prestressing ropes was used. The comparison of the results is available in the document. Example of prestressing and eccentricity.

C.14.2 Cable outside the concrete section: forces at anchors and deflectors

Just as before, assuming a uniform tension for the entire cable, the external forces method allows apprehending the effects of a prestressing rope by modeling it as a sequence of concentrated forces.

At the anchor A, the rope applies to node n1 of the model the loads (H_A, V_A, M_A), M_A being the moment produced by H_A at node 1. At each deflection point, the rope applies the force FS to the bar n1-n2. This is done for all the deflections along the rope up to the last anchor.

C.14.3 Modelling prestressed slabs

The study of prestressed slabs is carried out according to the same principles used for beams but applied to shell elements.

The use of specific software is desirable, maybe even essential. It will be necessary to make sure that the elements are modeled at their center of gravity, and that the sum of the support reactions of the prestressed load case is zero.

C.14.4 Tensioning cables (side, order)

Be aware that the forces brought by the prestressing, after losses due to friction and anchor recoil, are strongly dependent on the tensioning method (from one side only, from both sides). For exceedingly long ropes, the error can be significant.

Also, for highly prestressed structures, the order of tensioning might have an impact and it is important to analyze the structure at certain intermediate phases of tensioning.