Hyperstatic Moments in Nonlinear Analyses for Post-Tensioned Elements

 

In the field of prestressed concrete structural design, particularly for civil engineering structures such as bridges, the accurate determination of internal forces is essential to ensure structural safety and durability. One of the key concepts in this analysis is that of hyperstatic moments, which result from the restraint imposed by support continuity and by the stiffness of the structural system against the deformation induced by prestressing. Their proper evaluation is imperative, since these moments are superimposed on the effects of the remaining actions, directly influencing the design.

 

Traditionally, in linear static analyses, determining hyperstatic moments is a relatively straightforward process. Structural analysis software such as SAP2000, ETABS, SAFE, and CSiBridge provides specific load case types (Load Cases) designated as "Hyperstatic," which automatically isolate these effects. These load cases compute the forces resulting from deformation incompatibility imposed by the prestressing tendon profile in a statically indeterminate structure.

 

However, the conceptual and practical challenge arises when the analysis is carried out using a Staged Construction method. By its nature, this type of analysis is intrinsically nonlinear. The nonlinearity stems from multiple factors, such as the evolution of boundary conditions throughout the stages, the activation and deactivation of structural elements, and the consideration of concrete rheological effects such as shrinkage and creep. This nonlinear nature prevents the direct application of the "Hyperstatic" load case, which assumes linear elastic system behavior for its formulation.

 

To overcome this limitation and extract hyperstatic moments at any stage of a nonlinear analysis, a robust and versatile methodology is to use the Section Cuts tool. This functionality enables the integration of all forces and moments acting on a user-defined cutting plane, providing a complete "snapshot" at that location.

 

The underlying principle is as follows: at a given point in the construction process, the total moment acting on a section is equal to the sum of the moment due to self-weight and the hyperstatic moment due to prestressing.

 

To implement this technique, it is essential to define a cutting plane that intercepts all structural elements contributing to equilibrium and stiffness at the section under analysis. This includes not only the main beam or slab elements (Frame or Shell), but also—critically—the prestressing tendons themselves (Tendon). For this procedure to be valid, the tendons must also be modeled as elements (As Elements). By including all these elements in the cutting plane, the software computes the summation of internal forces and moments at that section and for that specific construction stage.

 

Thus, by isolating the hyperstatic moment from the equation, the following expression is obtained:

 

Although this procedure requires more detailed intervention by the engineer, it provides the flexibility needed to rigorously quantify the “secondary” effects of prestressing in complex nonlinear analysis scenarios. Applying this methodology to real-world cases demonstrates its effectiveness and general applicability, and it is recommended as standard practice in projects involving staged construction.

 

Next, a practical example of a beam-plus-slab prestressed system constructed in stages is presented. The objective of this example is to demonstrate that hyperstatic moments can be obtained in a simple and efficient manner at any stage of construction.

 

In the video below, it is possible to understand the procedure that should be followed to obtain the hyperstatic moment value.

 

In the example presented in the video, the values obtained from the Section Cut for the construction stage under analysis are as follows: the total moment read by the Section Cut is Msd(Section Cut)= 1195.13 kN·m, and the moment due to self-weight is Mg= −1252.076 kN·m (negative according to the model sign convention). By applying the expression, the following is obtained: