3D Finite Element S tructural Analysis of Attachments of Steam Turbine Last Stage Blades

Alexey I. Borovkov Alexander V. Gaev

Computational Mechanics Laboratory, St.Petersburg State Polytechnical University, Russia

Abstract

The work presents the results of finite element (FE) simulation and study of 3-D stress-strain state of attachments of steam turbine last stages blades performed with the use of the ANSYS FE software. The attachments of the steam turbine last stages blades consist of a great deal of structural elements (e.g. rivets, forks) that have complicated geometrical shape and are in contact interaction with each other.

To obtain the FE results, correctly describing local stresses and to reduce the computation time, a series of 3-D finite element models has been built and a method of multi-level submodeling has been used. In the process of the finite element study of the 3-D stress-strain state of the blades attachments the following models have been used:

1. Initial model – a macro-model incorporating a disc sector and a blade with a shroud flange, damper links, and the attachment;

2. First-level submodel comprising a blade attachment and a disc sector;

3. Second-level submodel intended for correct determination of local stress fields in rivets.

With the use of the method of multi-level submodeling on a submodel 717675 degrees of freedom, fields of local stresses and strains for the rivets have been obtained.

To calculate the complete model without applying the method of multi-level submodeling, it would be necessary to solve a problem of 3-D contact interaction with as many degrees of freedom as 1102743.

As a result of the finite element simulation performed with due regard to the 3-D multiple contact interaction of the blade tail and the disc, there have been revealed the areas where the equivalent, Mises stresses exceeded the yield strength. The practical recommendations have been worked out allowing reduction of the equivalent stresses in the mentioned areas.

The results obtained have been used in modernizing the existing constructions as well as for designing the new blade attachments.

Introduction In spite of a great variety of existing and worth-while thermal power station, a steam turbine plant is their integral part, therefore development, construction and improvement of steam turbines for present-day and promising thermal power stations is an important field of development of power industry.

Growth in power and more complicated design of turbomachines are accompanied by higher requirements for their reliability. To increase operational life of turbomachines is also one of the main tasks of quality improvement.

In this connection at present, when developing and mastering the steam turbines, modern computational and experimental methods are used to determine strength and reliability characteristics.

The present work is dealing with a study of stress-strain state of attachment of powerful steam turbine last stage blade. Special attention is paid to the analysis of the stress-strain state of the most highly loaded attachment components – rivets. It is the first time for this particular design that the study is performed with due regard for 3-D contact interaction with an assumption of elasto-plastic properties of material.

The object under study – an attachment – incorporates the following elements:

• Fork-type tang;

• Rivets that fix the blade;

• Disc.

The fork-type joints are characterized by high supporting power. The choice of the number of forks in a tang depends on a value of the centrifugal force acting upon the blade. For slightly loaded blades – a tang design with 2–3 forks is used, while for heavy-loaded blades – the number of forks may be as high as 5–7.

To the disc collar the blades are fixed with the rivets which are installed with reaming, their edges being rolled. The rivets can be installed between the two neighboring blades (a tang with side rivets) or along the tang center line (a tang with center rivets). The advantage of the fork-type tangs is first of all the absence of bending stresses in disc rim caused by centrifugal forces. By increasing the number of forks and making the section stepwise an attachment with high carrying capacity can be obtained. Moreover such connection does not need installation of special locking blades and allows for partial replacement of individual damaged blades.

Multilevel Submodeling Method To analyze local 3-D stress-strain state of attachments of steam turbine last stages blades with the use of the ANSYS FE software it is necessary to apply multilevel submodeling method that is based on locality principle in composite structures mechanics, developed in details in Computational Mechanics Laboratory of SPbSPU and successfully evaluated by solution of various complex problems of structural mechanics.

In Multilevel submodeling method after obtaining solution for macro model (on the macro level – i = 0) at every next step i (i >1) 3D problems for i-level submodels should be solved. These problems can be problems for elastic, elasto-plastic media, contact problems, and composite structures or, at last, elasto-plastic contact interaction of composite structures. It depends on the peculiarities of local stresses in the zone where the correct FE solution is to be obtained. Deriving submodel of i level from (i-1)-level submodel, it is necessary to apply kinematical boundary conditions on the submodel cut-boundary S(i-1, i). These boundary conditions are interpolated displacements obtained from the solution for (i-1)-level submodel. Such boundary conditions bring about some edge-effect-like perturbation into stressed state of i-level submodel [6], but this perturbation appears only close to the cut-boundary. If the cut-boundary is far enough from stress concentration zone of interest there will be obligatory zone of results correlation for i-level and (i-1)-level submodels U(i) ≈ U(i-1). In this way with use of multilevel submodeling method correct results can be obtained even for the problems with singular stresses (e.g. stress intensity factors in solutions of fracture mechanics problems [7, 8]).

Computation of the stress-strain state of the blade with the use of the full model of the blade and the attachment

The present work is aimed at the analysis of the three-dimensional stress-strain state of the blade with account of three-dimensional contact interactions between all possible contact surfaces of the attachment.

For solving the stated problem, a full model, comprising a blade airfoil with a shroud flange and holes intended for damping links, a fork-type tang, rivets and a disc, was constructed.

At this stage of computations is determined the three-dimensional, stress-strain state of the blade airfoil with account of contact interactions in the blade attachment, not taking into consideration its contact interactions with damping links, shroud flanges and the disc.

Full-scale model of the moving blade and the attachment Figure 1 demonstrates a full-scale model of the last stage moving blade and its attachment to the disc.

Figure 1. 3D FE full model of a blade.

The model consists of 3-D “SOLID45” elements. The finite-element grid is built with due regard for possible concentration of stresses in places of root section fillet and holes for damping wire. The study of the stressed state of the blade at this stage is performed for the number of degrees of freedom NDF=126282.

The model takes into account all presumable contact interactions. Figure 2 presents the contact surfaces of the attachment. The contact surfaces are assigned by 12129 contact elements “TARGE 170” and “CONTA 174”with friction coefficient µ = 0.15. For numerical solution of the contact problem, the Lagrange’s multiplier method [4, 9] was chosen.

Figure 2. The contact surfaces of the attachment.

Figure 3 presents a scheme of boundary conditions specified for the blade. The conditions of cyclic symmetry, i.e. degrees of freedom of the attachments located on the surfaces 1 and 2 (Figure 3) are connected with each other by the following relationships:

Ur1 = Ur2

UΘ1 = UΘ2 = 0

Figure 3. Scheme of boundary conditions assigned for full model.

For the present FE formulation of a problem interaction of the blade with the damping wires is not taken into account. As soon as the working temperature of the steam turbine last stage moving blades is close to 40° C, according to [2], temperature strains are not taken into account in the computation.

The bending stresses arising as a result of steam acting upon the blade airfoil are not taken into account because they are by an order of magnitude lower that stresses caused by centrifugal forces [2].

The blade is in the area of action of centrifugal forces arising as a result of rotor rotation at 3000 rpm.

Figure 4 demonstrates the results of stress computations for the upper and the lower rivets with account of elasto-plastic properties of material (kinematic hardening model, [3,9]), namely Von Mises stress intensity.

Figure 4. Von Mises stress intensity of the full model.

The results of stress-strain computations for the upper and the lower rivets are presented in Figure 5. It should be noted that the stress state of rivets, obtained for this model is only available for approximate estimation, and it should be defined more accurately with the use of a model with the greater number of degrees of freedom or by applying submodeling.

Figure 5. Von Mises stress intensity in the rivets.

Time required for solving the full model using PC with the processor Intel P4 1.7GHz, 1024MB RAM comes to about four hours. This length of time is not acceptable for engineering analysis.

Necessity of using the multi-level submodeling for creating a method of efficient engineering computations of the stress-strain state of rivets In industrial production conditions, time required for solving a problem is one of the fundamental characteristics of a computation method. Therefore, for creating effective computation procedures, it is necessary to apply those methods which would make it possible to substantially shorten time required for fulfillment of complex structure computations without significant accuracy losses of the results obtained. A method of multi-level submodeling may be related to such methods. The application of the submodeling method for the analysis of the stress-strain state of the attachment of gas turbine moving blades is considered in the paper [5].

In the present work, a multi-level submodelling method is used for the first time in computations of the stress-strain state of an attachment of a moving blade of the high-power vapor turbine.

For fast and efficient engineering computation of the stress-strain state of components of the moving blade attachment with the use of the multi-level submodeling method, the following sequence of FE models has been developed:

1. An original model – macromodel comprising the blade with the attachment and the disc;

2. A first-level submodel comprising the blade attachment and the disc;

3. A second-level submodel comprising the components of the blade attachment and the disc, which are considered in contact interaction.

Computation of stressed state of blade airfoil using a macromodel A macromodel is a blade airfoil with shroud flange, holes for damping links and fork-type attachment with disk. The model is intended to determine the displacements in cross-section A (See Figure 6) for further submodeling and also can be used for preliminary estimation of stress-strain state of blade airfoil in the area of root sections.

Figure 6. 3D solid model of an attachment.

At the first stage of the computation the 3D stressed state of the blade airfoil is determined without consideration for contact interactions with damping wires, shroud flanges and disc.

FE model of the last stage moving blade is shown in Figure 7. The model consists of 3-D “SOLID45” elements. The finite-element grid is built with due regard for possible concentration of stresses in places of root section fillet and holes for damping wire. The study of the stressed state of the blade at this stage is performed for the number of degrees of freedom NDF=76800.

Figure 7. 3D FE model of an attachment.

Figure 8 presents a scheme of boundary conditions specified for the blade. The conditions of cyclic symmetry, i.e. degrees of freedom of the attachments located on the surfaces 1 and 2 (Figure 8) are connected with each other by the following relationships:

Ur1 = Ur2

UΘ1 = UΘ2 = 0

Figure 8. Scheme of boundary conditions assigned for macromodel.

For the present FE formulation of a problem interaction of the blade with the damping wires is not taken into account. As soon as the working temperature of the steam turbine last stage moving blades is close to 40° C, according to [2], temperature strains are not taken into account in the computation.

The bending stresses arising as a result of steam acting upon the blade airfoil are not taken into account because they are by an order of magnitude lower that stresses caused by centrifugal forces [2].

The blade is in the area of action of centrifugal forces arising as a result of rotor rotation at 3000 rpm.

Figure 9 shows the result of computation of the stressed state of the blade with due regard for elasto-plastic properties of material (kinematic hardening model, [3, 9]), namely Von Mises stress intensity.

Figure 9. Von Mises stress intensity of the macromodel.

Maximum equivalent stresses are observed in the areas of holes for damping wire as well as in the area of fillet of the blade airfoil root section.

Time required for solving the given problem comes to about 20 minutes while using PC with the processor Intel P4 1.7GHz, 1024MB RAM.

In order to estimate the effect of contact interaction in the attachment on the stress-strain state of the blade airfoil, the comparison of Von Mises equivalent stresses along the line AB (Figure 10) was carried out for the full model and for the macromodel.

Figure 10. Von Mises stress intensity for the full model and for the macromodel.

Analyzing the results obtained (Figure 10), it should be marked that the macromodel brings about excessive values (by 17%) of stress intensities, though acceptable for engineering computations.

Figure 11 presents the relationship between the Von Mises equivalent stresses along the line AB and the number of degrees of freedom in the macromodel. As it may be seen from Figure 11, convergence of the results on Von Mises equivalent stresses is achieved on the FE grid chosen.

Figure 11. Convergence checking for macromodel.

Computation of stress-strain state of blade attachment with the help of the first level submodel At this stage of the computation the 3-D contact task of interaction of the moving blade and the disc is solved, with this contact interactions between all possible contacting surfaces of the rivets, the forks and the disc are taken into account.

First level submodel of moving blade attachment A first level submodel presented in Figure12 consists of the fork-type attachment (1), the rivets (2) and 1/96 disc portion (3). The model is intended for computation of stress-strain state of the moving blade attachment.

Figure 12. Solid first level submodel of the attachment.

The choice of submodel size As a result of the macromodel stress-strain computations, at the distance equal to 0.14 of the blade length from the root cross section, a stress concentration area is revealed on the blade back. For minimization of the influence on the stress-strain state of free edge effects arising in the zone of conjugation of the submodel with the macromodel, the conjugation surface shall not be located within the stress concentration area in the macromodel. Because of this circumstance, and aiming at the decreasing of the number of submodel degrees of freedom, the conjugation area is taken at the distance from the root cross section equal to 0.07 of the blade length outside the stress concentration zone.

Figure 13 presents a 3D FE model of the last stage moving blade attachment.

Figure 13. 3D FE first level submodel of the attachment.

The model consists of 3-D “SOLID45”elements. Cyclic symmetry of the construction is taken into account, therefore in the computation only 1/96 of the disc is considered. The FE model allows for taking into consideration the presumable concentration of stresses in the places of fillet of the blade root section. The study of the stressed state of the blade at this stage is performed for the number of degrees of freedom NDF=170070.

A scheme of boundary conditions specified for the blade is presented in Figure 14. The conditions of cyclic symmetry, i.e. degrees of freedom of attachment arranged on the surfaces 1 and 2 (Figure 14) are connected with each other by relationships:

Ur1 = Ur2

UΘ1 = UΘ2 = 0

Figure 14. Scheme of boundary conditions assigned for the first-level submodel.

Kinematic boundary conditions are specified for the surface of conjugation with the macromodel (Figure 14). They present interpolated values of displacements (interpolated functions in ANSYS are Shape functions for applied SOLID-45 FE elements [9]), obtained as a result of the solution of the problem for the macromodel.

The model takes into account all presumable contact interactions. Figure 15 presents the contact surfaces of the attachment. The contact surfaces are assigned by 12129 contact elements “TARGE 170” and “CONTA 174”with friction coefficient µ = 0.15. For numerical solution of the contact problem, the Lagrange’s multiplier method [4, 9] was chosen.

Figure 15. The contact surfaces of the attachment.

Stress-strain state of the fixing elements Figure 16 shows the calculated field of equivalent stresses (von Mises stress intensity) in the disc.

Figure 16. Von Mises stress intensity in the disc.

It is worth noting that there is no symmetry of the field of equivalent stresses with reference to the plane, which is perpendicular to the plane of rotation, due to airfoil twisting. It follows from the Figure 16 that the intensity of stresses (von Mises stress intensity) is higher in the lower row of the rivets.

Figure 17 demonstrates the results of stress computations for the upper and the lower rivets with account of elasto-plastic properties of material (kinematic hardening model, [3,9]), i.e. Von Mises stress intensity.

Figure 17. Von Mises stress intensity in the rivets.

The stress states of the upper and the lower rivets obtained for the full model and for the first-level submodel are shown in Figure 18. From the results presented it follows that in the central parts of the upper and the lower rivets the Von Mises’ equivalent stresses conform highly with those computed with the use of the full model (for central parts of rivets the difference does not exceed 15%). There is substantial difference in the values of Von Mises’ equivalent stresses at rivet edges, caused by the influence of the zone of conjugation of the submodel with the macromodel (edges of rivets are gray-colored in Figure 18).

Figure 18. Stress state in rivets for the full model and for the first-level submodel.

The following conclusion may be made from the results of the attachment computations:

Von Mises stress intensity at the rivet edges in the submodel significantly exceeds that obtained with the use of the full model; therefore, it is necessary to carry out more precise computations of local stresses for the upper and lower rivets, taking into account elasto-plastic properties of material.

Solution time for this model comes to about 20 minutes while using PC with the processor Intel P4 1.7GHz, 1024MB RAM.

Computation of stress-strain state of the upper and the lower rivets with the help of the second level submodels At this stage of the computation the 3-D contact task of interaction of the upper and the lower rivets with the disc is solved and the local fields of stresses in the rivets are refined.

Second level submodels of the moving blade attachment The second level submodels given in Figure 19 simulate the upper (а) and the lower (b) rivets as well as the parts of the forks and the disc with which they are in contact interaction. The submodels are intended for computation of stress-strain state of the upper and the lower rivets.

Figure 19. 3D solid second level submodels of the attachment.

The choice of submodel size Aiming at the shortening of the number of degrees of freedom for submodels, and correspondingly – the decrease of the solution time, the upper and the lower rivets are analyzed separately. A possible reciprocal effect of rivets is not taken into account.

In order to diminish the influence of free edge effects on the stress-strain state of the rivet, which arise in the zone of conjugation of the second-level submodel, the conjugation surface is chosen at the distance equal to 1.5 rivet diameter.

Figure 20 presents the FE models of the the upper (а) and the lower (b) rivets and the adjacent parts of the forks and the disc. A model of each rivet consists of 3-D “SOLID45”elements. The study of the stressed state of the rivets at this stage is performed for the number of degrees of freedom NDF=717675.

Figure 20. 3D FE second level submodels of the attachment.

The rivet models take into account elastic-and-ductile properties of material as well as all presumable contact interactions. The contact surfaces are assigned by 12129 contact elements “TARGE 170” and “CONTA 174”with friction coefficient µ = 0.15. For numerical solution of the contact problem, the Lagrange’s multiplier method [4, 9] was chosen.

The scheme of the boundary conditions assigned to the disc is presented in Figure 21. Some kinematic boundary conditions are applied to the surface of conjugation with the macromodel (Figure 21). They present interpolated displacement values (interpolating functions in ANSYS are Shape functions for SOLID-45 FE elements [9]) obtained by solving the problem for the macromodel.

Figure 21. Scheme of boundary conditions assigned for second level submodels.

Stress-strain state of the upper and the lower rivets Figure 22a presents computed Von Mises stress intensities in the upper and the lower rivets obtained with the use of the second-level submodels. For comparison, the analogous results calculated both with the use of the first-level submodel and the full model are shown in Figure 22b.

Figure 22. Von Mises stress intensity in the rivets.

Solution time for each rivet submodel comes to about 4 hours while using PC with the processor Intel P4 1.7GHz, 1024MB RAM.

In order to analyze the effect of conjugation surfaces on the stress state of rivets, for second-level submodels was solved the problem, in which on the surfaces of submodel conjugation were specified kinematic boundary conditions presenting interpolated values of displacements obtained by solving the problem for the full model. Let this submodels be Submodels 1 Thus, it becomes possible to compare more precisely determined stress state with that computed with the use of the full model and the second-level submodels (Figure 23).

Figure 23. Stress state in rivets for the full model, macromodel, first-level submodel, second-level

submodels and submodels 1.

Computation time for each rivet submodel comes to about 4 hours while using PC with the processor Intel P4 1.7GHz, 1024MB RAM.

Comparing stress-strain states of the upper and the lower rivets determined using the second-level submodels with those obtained with the use of submodels 1, one may conclude that the built-up second-level submodels make it possible to obtain correct local stress fields in the central zone of rivets.

The application of submodels 1 allows obtaining more precise values of stresses (by 15%) in rivets than those computed with the use of the full model (Figure 23). From Figure 23 it also follows that the reciprocal influence of rivets on each other does not take place. Comparing the stress-strain state in rivets determined for the second-level submodels with the stress-strain state in the submodeks 1, one may state that the main error is introduced in the stress state of rivet edges (these areas are gray-colored in Figure 23) when we turn from the macromodel to the first-level submodel.

Conclusions To obtain the FE results, correctly describing local stresses and to reduce the computation time, a series of 3-D finite element models has been built and a method of multi-level submodeling has been used. In the process of the finite element study of the 3-D stress-strain state of the blades attachments the following models have been used:

1. Initial model – a macro-model incorporating a disc sector and a blade with a shroud flange, damper links, and the attachment;

2. First-level submodel comprising a blade attachment and a disc sector;

3. Second-level submodel intended for correct determination of local stress fields in rivets.

With the use of the method of multi-level submodeling on a submodel 717675 degrees of freedom, fields of local stresses and strains for the rivets have been obtained.

To calculate the complete model without applying the method of multi-level submodeling, it would be necessary to solve a problem of 3-D contact interaction with as many degrees of freedom as 1102743.

The procedure developed makes it possible to carry out the full-scale computation of the stress-strain state for each of rivets in 15 hours, for central parts of each of rivets – in 10 hours. Application of the first-level submodel allows calculation of the stress-strain state for central parts of both rivets in 40 minutes.

The practical recommendations have been worked out allowing reduction of the equivalent stresses in the mentioned areas.

The results obtained have been used in modernizing the existing constructions as well as for designing the new blade attachments.

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