3D Finite Element Analysis Of The Tube Plate Stress-Strain State

Alexey I. Borovkov, Victor S. Modestov, Dmitry B. Birukov,

Sergey O. Gostevskich, Evgeny V. Pereyaslavets

Computational Mechanics Laboratory St. Petersburg State Polytechnical University, Russia

Abstract

The choice of optimal tube plate thickness of high-pressure preheaters of chamber-type (HPCP) by modern computation methods is an actual engineer problem. In the paper the scheme, which allows defining the tube plate thickness of the HPCP by finite element (FE) modeling with application of the composite structure mechanics methods is presented.

Tube plate thickness was chosen with the use of axisymmetric modeling in accordance with the R.F. atomic norms and regulations. The axisymmetric FE model of the HPCP with the spherical bottom and the U-tube bunch was researched and the stress-strain state for a plane plate was defined.

FE research of the tube plate 3D stress-strain state was carried out on basis of the HPCP model accounting the design features in the shape of rigid inclusions and technological fillets. In the last part of the work, were defined the effective properties of the periodicity cell of the material by means of the homogenization method. There was carried out FE simulation of the HPCP with the substitution of a homogeneous material for the tube plate material and with the application of the step-by-step heterogenezation in the areas of the connections of solid and holed material as well as in the maximum stress area.

Account of the new design features influence and application of the composite structure mechanics methods made it possible to determine more exactly the stress-strain state of the HPCP in the jointing places between the uniform and holed materials as well as in the maximum stress area.

Introduction On the way of utilizing high-power energy units of thermoelectric power stations (TEPS) and nuclear power stations (NPS) stand the problems of creating combined capital and auxiliary equipment as well as increasing its quality and dependability. More than urgent stands the problem of achievement of high technical level, high quality and reliability of high-pressure preheaters (HPP). Concerning this problem, it should be singled out the search of a new design of apparatuses with heat exchange surface 2500 m2 and even more (in one structure), compact, bound to the operating conditions of hoisting-and-conveying machinery, considerably exceeding in reliability tube systems of collector-spiral-type HPP and distinguished by its highly active maintenance adaptability.

Utilization of manufacturing large-size forgings and drop forgings as well as tubes of nickel-free steel, grade 0.8C-14Cr-0.6Mo-0.3V, by metallurgical industry made it possible to come to real constructive solutions of HPP designing. There were developed new high-pressure preheaters of chamber-type (HPCP) with U-shaped heat-exchange surface from tubes 161.4 mm in size from the above-mentioned steel (or in the case of its absence from steel, grade 0.12C-18Cr-10Ni-0.6Ti), with lower arrangement of a feeding-water chamber.

By the complex of decisive factors: thermal efficiency, metal consumption, compactness, technological effectiveness, dependability, maintenance adaptability and manufacturers possibilities the preference for HPP of NPS turbine sets is given to a chamber design of a vertical preheater, with lower location of the feeding water chamber. This conclusion is confirmed by the experience of many companies (e.g. Weir, England; BBC, Switzerland; Balcke-Drr, Germany; Westinghuse, USA; Delas-Alsthorm, France; Skoda, Chech Rep.; Bergman-Borsig, Germany).

The present paper considers a multivariant finite element analysis of HPCP and is aimed at the search of new design solutions for lessening the tube plate thickness.

Chapter 1. Some provisions of the composite structure mechanics

1.1.General relations of the heterogeneous anisotropic medium elastic theory

1.1.1. Equilibrium differential equations. Boundary conditions A quasi-stationary elastic theory problem in displacements, for heterogeneous anisotropic medium includes the solution of three differential equations of equilibrium in relation to displacement vector component:

( )( ) 04 =+ vfurC (1.1), where: r radius-vector of a considered point; u displacement vector; Hamiltons nabla-operator; C4 elasticity modulus tensor, 4th rank tensor; vf vector of volume forces.

The system of equations (1.1) determining a bodys behavior in the points of its volume V is complemented by kinematic, static and mixed boundary conditions on its surfaces:

ss uu = (1.2) ( )( ) ss uurCn = 4 (1.3)

ss uu =1 , ( )( ) ss furCn = 4 (1.3), 21 SSS = (1.4) Where su displacement vector prescribed on the boundary; sf superficial load prescribed on the boundary; n isolated vector of outer normal to body surface.

1.1.2. Defining relations

In the case of small strains ( 1

For composite materials, elasticity moduli tensor C4 and elastic yielding tensor S4 are breaking functions of coordinates. For i -phase of a composite, in the case of general-type anisotropy, elastic moduli (flexibilities) tensor contains 21 independent components, for orthotropic body 9 independent components, for transverse-isotropic 5, for isotropic 2 independent components.

In straight-line orthogonal system of coordinates x1x2x3, the equations of the generalized Hookes law for orthotropic body may be written in the following form:

E111 = 11-1222-1333

E222 = -2111+22-2333 (1.7)

E333 = -3111-3222+33

G1212 = 12, G2323 = 23, G3131 = 31

Youngs moduli and Poissons ratios are bound with each other by the following relations:

E121 = E212; E232 = E323, E313 = E231 (1.8)

Defining relationships may also be written in the following way:

( )

= W , ( )

= W (1.9)

1.2 Some definitions and assumptions Materials consisting of two or several components (phases) with distinctly expressed conjugation surfaces and different physical and mechanical properties are named composite materials or composites.

Generally, composite materials are classified by the shape of inclusions. A composite which inclusions present lengthened cylinders is called fibrous composite. If these cylinders (fibers) are parallel to each other, fibrous composite is called unidirectional.

While studying mechanical behavior of fibrous elastic composites, the following suppositions are generally put forward:

groundmass and inclusions are linearly elastic, homogeneous, isotropic: relationship between stresses and strains in composite components are described by the Hookes law;

separate composite components are ideally bound with each other the conditions of continuity of displacement vectors and stress vectors are realized on conjugation surfaces of components;

fibrous composite presents linear-elastic macrohomogenous material with no initial stresses. Representative volume element (RVE) is volume element with geometrical and physic-mechanical properties of a considered heterogeneous medium, in which volume-averaged strain tensors and stress tensors are equal to corresponding tensors calculated for heterogeneous medium as a whole.

If geometrical characteristics (shape, size, disposition) and mechanical properties of components are periodically repeated in space, such composite is named periodical structure composite or simply periodical composite.

If a medium has doubly periodical structure, then:

it is possible to take account of interaction between a singled-out volume element and surrounding space if only one periodicity (or periodic) cell (CP) is singled out;

all strain and stress tensors, CP-volume-averaged, are equal to corresponding average tensors calculated for heterogeneous medium as a whole.

1.3 Efficient defining relationships. Multiscale direct homogenization The general algorithm [1] for the analysis of complex composite structures is fully based on the implementation of the finite element method and consists of three steps [2]:

1. Computation of the effective elastic characteristic of the lamina. In the consideration of the special boundary conditions on the outer boundary of the RVE V we shall determine: the microscopic fields of the displacements U(r), strains (r) and stresses (r). The basic results of the solution of the boundary-value problem of elasticity theory are the effective elastic (C*) characteristics of the microheterogeneous anisotropic medium (macroscopic effective properties of the equivalent homogeneous medium) and tensors of effective surface strength (2F*, 4F*) on the base of the prescribed microstructure of the RVE and known properties of composite components, that is formulation of the effective constitutive equation:

=C*; =1/V V

dV... (1.10)

and formulation of the effective tensor-polynomial strength TsaiWu criterion:

f = 2F** + *4F**+=1, (1.11)

where * - tensor of effective (macroscopic) stresses. The dimensions of the RVE are: d=10-6 m, L=210-6 m (see Fig 1).

Figure 1. Representative volume element The direct homogenization method [2] procedure for described composite lamina is following:

The calculation of the effective Youngs modulus *3E by using the volume concentrations is:

)1(33*3

fmff VEVEE += (1.12)

Two problems (See Fig. 2 for the Problem 1 and Fig. 3 for the Problem 2) for the transverse tension of composite element must be solved to calculate the effective Youngs moduli *1E ,

*2E and Poissons ratios.

Figure 2. Problem 1

Figure 3. Problem 2

Boundary conditions for the Problem 1 are following: 01111 :5.0 uuhx == , 012 = ; 0:0 11 == ux , 012 = ; 0:5.0 222 == uhx , 012 = ; 0:0 22 == ux , 012 = ;

Boundary conditions for the Problem 2 are following:

0:5.0 111 == uhx , 012 = ; 0:0 11 == ux , 012 = ; 02222 :5.0 uuhx == , 012 = ; 0:0 22 == ux , 012 = ;

As a result, we obtain for the problem statement:

0)( = uC

lkjiijkl eeeeCC =

11011

)1( ee >=< , >>==< , >>===

>>===