We simulate the prestressing of the concrete shell of nuclear power plants

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Dear Khabrovites! Three months have passed

since the start of the Masters of Simulation project , and we already have results that we would like to share with you.

We received a lot of applications and were very pleased that this initiative of ours found a lively response in the minds and hearts of engineers and designers, young graduate students and already seasoned specialists. The tasks that the project participants sent to us in their profiles turned out to be interesting and, at times, very extraordinary. There was here the calculation of processes during the destruction of structures, and the calculation of composite materials, and highly non-linear processes, and much more.

The greatest progress in the solution was achieved in the following three profiles:
1. Objective: to simulate the prestressing of the “piece” of the concrete shell with friction and sliding losses. Author: Markevich Maxim Alexandrovich
2. Task: Modeling the rigidity of linear guides and rolling bearings. To analyze the rigidity of individual components (mechatronic modules) of machine tools Author: Nail Khamitovich Yusupov
3. Task: It is necessary to obtain temperature fields and stresses in a multilayer structure when exposed to a temperature source and under the influence of a power load. Author: Dolgopolova Natalya Vladimirovna

Below in this publication we provide a detailed description of the solution to the problem of Maxim Aleksandrovich Markevich mentioned above.

So ...
Imperfect techniques - the path to disaster

Designing nuclear power plants (NPPs) is an extremely important task, because cases of negligence are too expensive for all of humanity. Traditionally, elements of nuclear power plants, such as the inner protective shell of the reactor compartment, are calculated using the theory of plates and shells. This is a technique that allows you to get an accurate analytical calculation. But the word "exact", in fact, would be nice to put in quotation marks, because this calculation was performed after a large number of simplifications and assumptions. So, very often the anisotropy of concrete behavior due to reinforcement is not taken into account, the fact of pretensioning and physical non-linearity in behavior are simplified. Not to mention the fact that many features of the geometry of the containment are simply ignored: for example, the influence of local features on the uneven distribution of stresses over the shell and its thickness. All this is usually corrected by a set of coefficients, with the help of which excess strength from operational loads is achieved. And in most cases this is enough. However, such a technique may not be enough, and this can lead to disaster. And here the need comes to the forefront to significantly increase the detail of calculations, to take into account previously discarded factors, to use more accurate methods to complete the task, with which you can make less simplifications and idealizations in the design. As a result, a more adequate technique is required to account for such effects. And in most cases this is enough. However, such a technique may not be enough, and this can lead to disaster. And here the need comes to the forefront to significantly increase the detail of calculations, to take into account previously discarded factors, to use more accurate methods to complete the task, with which you can make less simplifications and idealizations in the design. As a result, a more adequate technique is required to account for such effects. And in most cases this is enough. However, such a technique may not be enough, and this can lead to disaster. And here the need comes to the forefront to significantly increase the detail of calculations, to take into account previously discarded factors, to use more accurate methods to complete the task, with which you can make less simplifications and idealizations in the design. As a result, a more adequate technique is required to account for such effects.

Naturally, within the framework of the Masters of Simulation project, no one takes responsibility for the preparation of a new method for calculating and designing the protective shell of nuclear power plants, which will immediately go to work. But in one of the applications received, we found questions about the fundamental possibility of taking into account some effects of such design using the Autodesk Simulation line. And it was precisely these questions that we tried to find more accurate answers. In order not only to say “yes, of course”, but also to say “how exactly”.

Initial data

The initial statement was described as follows: To
model the prestresses of the “piece” of the shell with friction and slip losses - Maxim Aleksandrovich Markevich, engineer of the RUE Belnipienergoprom company.

Fig. 1. Image provided by the author of the task for its explanation

I would like to thank Maxim Alexandrovich for prompt answers to questions and interest in solving his own problem. This allowed us to work out the task much deeper.
Now let's move on to a more detailed description of the problem. And start with a description of the structure under study.
The inner protective shell of the reactor compartment of a nuclear power plant is made of prestressed monolithic reinforced concrete in the form of a cylinder, covered by a dome in the form of a hemisphere (Fig. 2).

Fig. 2. The protective shell (1 - the outer reinforced concrete shell; 2 - the inner metal shell)

The protective shell pre-stressing system (SPS) consists of reinforcing beams, support anchor blocks, channel formers, equipment for mounting reinforcing beams, their tension, and a diagnostic system for the reinforcing beam (Fig. 3, Fig. 4).

Fig. 3. The cross-section of the containment along the axis

Fig. 4. The cross-section of the protective shell perpendicular to the axis

For prestressing, strands of cable reinforcement in a polyethylene sheath are used. Each reinforcing rope has a polyethylene sheath and is located in a channel former filled with a special injection solution (Fig. 5). Thus, there is no connection between the reinforcing beam and the concrete of the containment. If necessary, this allows the possibility of tightening or replacing the rope. Channel formers are metal pipes and flexible metal hoses with an outer diameter of not more than 219 mm and a thickness of about 3-5 mm.

Fig. 5. Section of the channel former and the reinforcing rope

Description of the fittings:Prestressed reinforcement (Fig. 5) - bundles consisting of 55 ropes with a diameter of 15.7, 17.4-19.4 mm, type NRE with lubrication. Strength class - 1860 MPa. Such reinforcement is produced by Trefileurope (France.

Actually, the solution.

Now you can smoothly proceed to the solution of the problem. Let's start with some comments and conclusions that can be made on the basis of the available information:
1. Channel formers are built into the concrete of the main part of the containment. Their main purpose : Separate the ropes from concrete and allow these ropes to be tensioned, creating a prestressed state, and replace the ropes if necessary.
2. The channel former does not have particular rigidity and, therefore, as a first approximation, it most likely can be ignored. But due to the difference in the stiffness of the materials of the channel former and concrete, it can work as a pressure redistributor from the ropes.
3. The aforementioned (in clause 2) fact needs to be verified.
4. The channel former has an extremely small thickness compared to the main dimensions of the structure, and its modeling in a volume formulation can lead to a very high dimensional problem. Therefore, the channel former will be executed in Autodesk Inventor as a surface, and during FE analysis it will be modeled by plate elements in those tasks in which it will be taken into account (Fig. 6).

Fig. 6. Canal formers built by surfaces

5. It is also necessary to separately consider a small part of the structure near the channel former to see the local behavior of the structure. Based on this, it will be possible to draw conclusions about the need for modeling channel formers (Fig. 7).

Fig. 7. A detailed volumetric model of the local segment of the sheath and the channel former

6. It also makes sense to solve the problem of the contact interaction of the rope and the channel former within this local piece (Fig. 7). This will allow you to better choose the type of contact and solve the question of the possibility of replacing the prestressed rope with its power analogue.
7. Thus, the basic geometric model for the basic calculation began to look as follows (Fig. 8). This is a solid-state volumetric geometric model of the shell (or rather part of it) with voids under the ropes. Ropes are also made in the form of volumetric elements. The channel formers, as mentioned earlier, are made in the form of surfaces.

Fig. 8. The original version of the part of the investigated design in Inventor


To solve this problem, from the arsenal of Autodesk calculation programs, you can use: Autodesk Simulation Mechanical (ASM), Autodesk Nastran In-CAD, Autodesk Sim 360. Part of the problem can also be solved using the Inventor Professional functionality, or rather, its Inventor Simulation module. In the case of a positive solution to the first part of the problem, it is supposed to develop in the direction of solving the conjugate thermo-strength problem, so ASM was chosen as the basic tool for solving the problem. A detailed step-by-step description of the solution of the problem will be published a little later, now we will proceed to the statement of the problem.
8. For the first calculations, the default grid was selected, and global condensation was up to 30%. A general view of the FE grid is shown in Fig. 9. In the task tree, you can see volumetric parts and a number of parts, divided into plate-shell elements (selected in the tree).

Fig. 9. FE-grid of the initial object under investigation

9. After conducting various types of contacts for the current task, the most suitable ones in ASM were identified: “Sliding / No Separation” and “Surface”. The “Bonded” contact type, which is offered by default by ASM, in this case leads to a fundamentally incorrect design behavior, because the ropes cannot slide freely along the channel former (as it should be), but are rigidly glued over the entire surface to the channel former and concrete.
10. To speed up the counting, the contact type “Sliding / No Separation” will be used as a first approximation, hereinafter - “Surface”.
11. In the cylindrical part of the sheath, the vertical ropes practically do not interact with the channel, and in fact simply create a compressive force at the ends of the sheath along the boundary of the holes of the channel former. In this case, the presence of vertical ropes in the model makes a very small correction, and at the first steps they can be excluded from the calculation model.
12. The cement reinforcing ropes themselves need a separate description and calculation of their mechanical properties.
Due to their structure and production method, steel ropes (Fig. 10) work very well in tension, but practically do not work in compression. Due to the fact that in the current case the ropes are bonded with an injectable cement mortar, the resulting reinforcing-cement composite gains some resistance to compressive loads.

Fig. 10. Examples of ropes and crossovers

This feature could introduce nonlinearity in the behavior of the object under study, if not for the fact that the ropes are poured with cement mortar in an already pre-stretched form. As a result, even in the presence of some compressive load, the ropes still continue to be in a stretched state, and continue to work exclusively in tension. Therefore, the nonlinear properties of the reinforcing-cement rope can be neglected. However, in view of the fact that the rope works as a whole, it is, in fact, a composite, whose characteristics depend on the materials used, but differ significantly from them. Ideally, it is necessary to conduct a comprehensive study of a part of the rope to work along all directions and calculate the anisotropic characteristics of the final composite material.
1) the properties of the structure will strongly depend on the location of the ropes, which can vary in some way,
2) in classical calculations, the geometry of the ropes and their properties are not taken into account at all, and, therefore, this will be a step forward. In this case, it is proposed to calculate the mechanical characteristics according to the volume fraction.
Nevertheless, in the future we can recommend a more detailed look at the behavior of the ropes (Fig. 11).

Fig. 11. Stress distribution during axial compression of a reinforcing-cement rope

13. Cement-reinforcing ropes operate under tensile stress conditions and are considered isotropic in terms of material characteristics. Their properties are calculated according to the classical formulas for composite materials:

modulus of elasticity (E), Poisson's ratio (v) depending on the volume:

14. To simulate symmetrical boundary conditions, Frictionless type fastening (Fig. 12) will be used on three sides (two sides and a bottom). Thus, for sidewalls, a behavior similar to the presence of a plane of symmetry is modeled. For the lower side, additional research is needed regarding the correctness of the type of fastening, since the lower part should be able to perform plane-parallel movement (in full construction).

Fig. 12. Boundary conditions (fixing)

The level of the preload will be set by force (Force) along the normal to the ends of the rope.

Fig. 13. Power conditions

Thus, there are a number of simplifications and a number of controversial points that it makes sense to check additionally during the test calculations. Some of them were tested on the basis of the initial geometry built on the basis transmitted by the customer. For some tasks, separate models were built that focused on testing this or that effect.
The results of test calculations (various stresses and displacements) with the included scale (10% of the geometry dimensions) are shown below in Figures 14-20.

Fig. 14. The total displacement

Fig. 15. Circumferential movements

Fig. 16. The radial displacement

Fig. 17. The equivalent von Mises stress

Fig. 18. Radial stresses

Fig. 19. Circumferential stresses

Fig. 20. Shear

Analysis of VAT shows that the initial formulation needs to be clarified. This is best demonstrated on radial movements with the option of deformation of the model turned on at a scale of 40% of the dimensions of the structure (Fig. 21):

Fig. 21. The deformation of the object is shown to scale.

When correctly set up, the model should be compressed almost uniformly, and not twisted relative to the center. In view of the fact that the fastenings are absolutely identical on both sides, this may indicate the occurrence of a significant unevenness from the load, which was applied from one side. Also, near the boundary surfaces, one can observe a distortion of the VAT pattern due to the influence of edge effects. Thus, based on the preliminary calculations, the following conclusions can be drawn:

Conclusions on preliminary calculations

- For an adequate solution of the original problem, you should take a larger section of the shell (approximately 3-4 times) (Fig. 22). In this case, edge effects will not appear in the central area.

Fig. 22. Design option proposed for the study
- To determine the detailed SSS and the level of stress concentration, a local concrete section with a rope will be modeled. In this case, the channel former will be modeled in the form of solid-state geometry.
- To compensate for the force skew and more symmetrical deformation of the shell, it makes sense to apply forces in a checkerboard pattern on both sides of the shell (Fig. 23).

Fig. 23. The proposed method of loading

For the sheath, you can use the default setting, for the ropes and the central part it makes sense to set the mesh size (40% of the face value) - this will make the mesh balanced in terms of accuracy / speed.

The results obtained allowed us to proceed to the second stage, within the framework of which the statement will be refined and a solution to the problem will be obtained. With the successful completion of the second stage, it is possible to consider a more complex problem located in the field of multiphysics, namely, solving the problem of the thermally stressed state of the containment taking into account the preload and working pressure.

In one of the following posts we will talk about working on a second project - modeling the rigidity of linear guides and rolling bearings.

We wish you success and recall that the acceptance of applications for participation in the Masters of Simulation program continues.

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