Modeling of the Fluid-Structure Interaction in Inelastic Piping Systems

Modeling of the Fluid-Structure Interaction in Inelastic Piping Systems

F. B. de F. Rachid

Laboratory of Theoretical and Applied Mechanics (LMTA)

Department of Mechanical Engineering

Universidade Federal Fluminense

Rua Passo da Pátria, 156

24210-240 Niterói, RJ. Brazil

[e-mail]

H. S. da C. Mattos

[e-mail]

In many engineering applications, compliant piping systems conveying liquids are subjected to inelastic deformations due to severe pressure surges such as plastic tubes in modern water supply transmission lines and metallic pipings in nuclear power plants. In these cases the design of such systems may require an adequate modeling of the interactions between the fluid dynamics and the inelastic structural pipe motions. The reliability of the prediction of fluid-pipe behavior depends mainly on the adequacy of the constitutive equations employed in the analysis. In this paper it is proposed a systematic and general approach to consistently incorporate different kinds of inelastic behaviors of the pipe material in a fluid-structure interaction analysis. The main feature of the constitutive equations considered in this work is that a very simple numerical technique can be used for solving the coupled equations describing the dynamics of the fluid and pipe wall. Numerical examples concerning the analysis of polyethylene and stainless steel pipe networks are presented to illustrate the versatility of the proposed approach.

Keywords: Fluid-structure interaction, viscoelastic piping systems, elasto-viscoplastic piping systems, pressure surges

Introduction

To ensure proper design and consequently an adequate operation of compliant piping systems conveying liquids it is necessary to predict transient pressures and flow the system might be subjected to. Since for many practical situations pressure pulses induce pipe vibrations and vice-versa, it has been shown that an accurate prediction of pressure transients should take into account an approach which couples the dynamics of both fluid and pipe motion. Such kind of approach is referred to in the literature as fluid-structure interaction (fsi) analysis and has been the subject of extensive research in the past years. For a comprehensive review and bibliography of the various methodologies employed on the fsi analysis in liquid-filled compliant piping systems, the reader is referred to the works of Hatfield et al. (1982) and Wiggert et al. (1987).

Because of the inherent complexity of the problem, the majority of these works has mainly been concerned with the appropriate description of the coupling mechanisms between fluid and structural motions. As a consequence, a linear elastic pipewall mechanical behavior has been considered as a simplifying assumption.

In many engineering applications, however, the pipewalls are most likely subjected to inelastic deformations rather than elastic ones, due to adverse operating conditions. Severe pressure surges, piping stiffness, high temperature environments and pipe materials used in the installations are the most prominent factors responsible for the development of deformations in the inelastic range.

In fact, the inelastic character of the pipewalls has long been appreciated in the analyzes of pressure pulse propagation in compliant piping systems (Gally et al., 1979; Rieutord, 1982; Fox and Stepnewski, 1974; Youngdahl and Kot, 1975 and Chohan, 1988). Nevertheless, these models neither have accounted for fluid-structure interaction phenomena nor have made use of constitutive relations capable to describe the inelastic behavior properly. Moreover, they have not attempted for the possibility of treating different types of inelastic pipewall materials within a same context.

From the viewpoint of applications, this feature is quite important since inelastic behavior encompasses a great variety of mechanical responses. According to each specific application, a different mechanical response might occur. For instance, plastic tubes used in modern water supply transmission lines are known to behave as a viscoelastic material (Gally et al, 1979). On the other hand, metallic pipings of nuclear and thermohydraulic installations are likely subjected to viscoplastic deformations (Youngdahl and Kot, 1975).

In the present work, a fluid-structure interaction analysis is extended to cope with different types of inelastic behavior of the pipewalls. The main goal of this paper is to present one-dimensional model equations that combine enough mathematical simplicity to allow their usage in engineering problems along with the capability of describing complex non-linear mechanical behaviors. The description of the mechanical responses of the pipe materials relies upon a general constitutive theory with strong thermodynamical basis. It comprises and generalizes a great number of rate-dependent theories of plasticity with and without yield surfaces. Thus elastic, plastic and viscous behaviors can be treated within the same abstract framework. Despite the possibility of treating strong nonlinear material responses, the theory assumes that the elastic behavior is always present and, in addition, that it is linear. This particular feature permits the use of a very simple numerical technique (based on the method of characteristics) for solving the set of nonlinear differential equations describing the dynamics of both liquid and pipe motion, for any kind of constitutive equations developed within this context.

The present paper presents a review of the works developed in the last ten years by the authors, not only encompassing but also generalizing some results on the subject (Freitas Rachid et al., 1990, 1991a, 1991b) by presenting an abstract, systematic and general approach to consistently incorporate the inelastic behavior in the fluid-structure interaction analysis.

Nomenclature

A =

a = coefficient of the elato-viscoplastic constitutive equation, N/m²

b = coefficient of the elato-viscoplastic constitutive equation, N/m²

B^b = thermodynamical force associated to b , N/m²

B^p = thermodynamical force associated to p, N/m²

B^c = thermodynamical force associated to c, N/m²

C = elasticity tensor, N/m²

c = internal variable associated to the kinematical hardening, dimensionless

d = coefficient of the elato-viscoplastic constitutive equation, dimensionless

E = Young's modulus, N/m²

e = pipewall thickness, m

F = inelastic potential, Ns/m²

J = Von-Mises'equivalent stress, N/m²

K = isentropic bulk modulus of liquid, N/m²

K^*= constant, Eq.(4)

k = coefficient of the elato-viscoplastic constitutive equation, Ns/m²

L = pipe length, m

n = coefficient of the elato-viscoplastic constitutive equation, dimensionless

p = cumulated plastic strain, dimesionless

P = pressure, N/m²

R = internal radius of the pipe, m

t = time, s

u = axial displacement of the pipewall, m

V = axial fluid velocity, m/s

W_e= elastic strain energy density, N/m²

W_a= inelastic strain energy density, N/m²

Greek Symbols

b

= general internal variable, dimensionless

Dt = time step, s

Ds = axial grid spacing, m

e = strain tensor, dimensionless

e^a = inelastic strain tensor, dimensionless

g = constant, Eq.(5)

l = wavespeed, m/s

n = Poisson's ratio, dimensionless

F = dissipation potential, N/sm²

Y = Helmholtz free energy, Nm/kg

r = density, kg/m³

s = stress tensor, N/m²

s^y= coefficient of the elato-viscoplastic constitutive equation, N/m²

j = coefficient of the elato-viscoplastic constitutive equation, N/m²

Subscripts

a

= relative to inelastic

e = relative to elastic

f = relative to fluid

s = relative to axial coordinate

q = relative to circumferential coordinate

t = relative to structure (pipe)

1 = relative to a point in the pipe

2 = relative to a point in the pipe

Constitutive Theory

The basic assumption of this theory is that the state of the body at a given material point and at a given time instant is completely defined by a set of state variables. Within the general framework of Thermodynamics of Irreversible Processes, one internal variable is introduced per each dissipative mechanism. To each internal variable it is associated one evolution law in such way that the Second Law of Thermodynamics must be satisfied. In this theory two thermodynamical potentials, the Helmholtz free energy and a dissipation potential, are sufficient to define a complete set of constitutive equations (Germain and Muller, 1995; Lemaitre and Chaboche, 1990).

For the isothermal evolution of an inelastic viscous solid of constant density r_t, the local state is supposed to be characterized by the total strain tensor e, the inelastic strain tensor e^a and by a

E

²

e = pipewall thickness, m

F = inelastic potential, Ns/m2

J = Von-Mises'equivalent stress, N/m²

set of internal variables, noted abstractedly as b. The variables b are associated with irreversible changes of the internal state of the material (such as the strain hardening due to plasticity). The choice of the variables b will depend upon the degree of detail desired in the modeling and so they can vary from problem to problem. The particular families of viscoelastic and elasto-viscoplastic equations considered ahead in the section labeled applications will make the role of these variables clearer. Meanwhile, to simplify the presentation of the theory, b will be considered a scalar variable.

The Helmholtz free energy y is assumed to be a differentiable scalar function of the state variables with the following form:

where is the elastic strain energy density and W_a(b) is the inelastic strain energy density related to the internal variable b. C is the classical symmetric positive-definite fourth-order tensor of elasticity. If the elastic behavior is isotropic, then:

where E and n stand for the Young's modulus and Poisson's ratio whereas 1₂and 1₄represent, respectively, the rank two and rank four identity tensors.

The thermodynamical forces s and B^b related to the state variables (e,e^a,b) are defined from the free energy potential by taking its partial derivatives. The relations between the state variables and the thermodynamical forces are the so-called state laws:

To complete the constitutive equations, evolution laws are required for the internal variables. These are obtained by introducing a differentiable scalar function F = F(s,B^b;b,e^a) , named dual of the pseudo-potential of dissipation, in which the internal variables are regarded as independent parameters. When F is differentiated with respect to the arguments s and B^b, the evolution laws are obtained:

where superimposed dot represents partial derivative with respect to time. To assign a convenient and general character to the theory, we have taken eq. (2.b) into account and expressed the evolution laws in terms of generic functions g and h of the arguments s, e^a and b. Equations (2) and (3) define a complete set of inelastic constitutive equations.

Governing Equations

Once the context in which constitutive relations are derived has been established, the desired extension of the fsi continuum models to incorporate inelastic behaviors of the pipewall is straightforward. Here a coupled fsi model based on the work of Walker and Phillips (1977) is used. It is a one-dimensional wave equation formulation which accounts for the interaction of plane axial stress waves in the pipe wall and plane pressure waves in the liquid.

Basically, the model considers an inviscid transient axisymmetrical slightly compressible flow of a liquid contained in a thin-walled pipe (inside radius R and wall thickness e) for which both liquid and pipe wall motion are relevant. For the fluid motion, it is assumed that the moderately long pulse approximation holds and that the liquid bulk modulus K is constant. On the other hand, small deformations, axisymmetrical plane-stress distribution and inelastic behavior given in accordance to the preceding section are the underlying assumptions for the pipe.

By considering an isotropic elastic behavior in the constitutive equations presented in the previous section, neglecting the radial inertia of the pipe and using the one-dimensional continuity and momentum equations proposed for both liquid and pipe, the following set of equations result:

where g_s, g_q and h are functions of (P, s_s , , and .

In the above equations, P, V and u represent the pressure, liquid axial velocity and pipe wall displacement in the axial direction, while s_s and s_q P designate the only non-vanishing stress components in the axial and circumferential directions, respectively. and are the inelastic strains in the axial and circumferential directions. All of these variables are functions of the axial coordinate s along the pipe and the time t. r_f and r_t stand for fluid and pipe densities.

The first four equations in (4) are similar to the four-equation model (Stuckenbruck et al., 1985) for elastic pipe materials, except to the terms and that appear in (4.a) and (4.d). With the introduction of these new terms and the three last equations, one can handle not only the elastic behavior but also several types of inelastic behaviors of the piping. To do so, it suffices to choose the internal variables and to specify the expressions of the evolution laws (4.e).

No matter how these expressions look like, the system of equations (4) can be solved by using a simple numerical technique based on the method of characteristics.

Numerical Method

Equations (4) define a first-order, quasilinear, hyperbolic system of partial differential equations in the dependent variables P, V, , s_s, , and b. Due to its hyperbolic nature, this system can be transformed into a simultaneous set of ordinary differential equations (classically named compatibility and characteristics equations) by means of the method of characteristics.

When this transformation is carried out, four families of propagating characteristics lines (with slopes ds/dt = ± l_f , ± l_f,) and other three stationary characteristics (ds/dt = 0) are found.

The values of ds/dt, which themselves define the characteristics, are obtained by solving the eigenvalue problem associated with equations (4). They represent the wavespeeds with which disturbances propagate in the system. The non-vanishing wavespeeds are found to be and for the pressure waves in the liquid and for the axial stress waves in the pipe walls, respectively; with and where:

After the system of equations (4) has been transformed by the method of characteristics, the following set of compatibility equations, which are valid along the corresponding characteristics lines, are obtained:

where

At this point, it should be emphasized that the internal variables represented by b do not alter the mathematical structure of the problem since their evolution take place along non-propagating characteristics. This particular feature allows the treatment of many kinds of pipe material behaviors within a same mathematical framework, regardless the number of internal variables used in the formulation.

To derive the finite difference equations, each compatibility equation is integrated along its appropriate characteristic line represented on the computational time-space grid shown in Figure 1. The grid is set by dividing the pipe into equal reaches of Ds in length. Since l_t > l_f , the time step is chosen equal to D_t = D_s / l_t , so that the Courant-Friedrichs-Levy (Leveque, R. J., 1990) criterion for convergence and stability is always satisfied.

In order to avoid numerical errors caused by interpolation, in the present work the ratio between wavespeeds is taken as an integer by adjusting slightly the liquid and pipe densities. With this procedure, interpolation becomes unnecessary since every point in the time-space grid is intercepted by all characteristic lines.

The left-hand sides of the compatibility equations are integrated exactly while the integration of the right-hand sides is evaluated by employing a second-order approximation (the trapezoidal rule). As a result, a set of seven non-linear algebraic equations is obtained for the unknowns (P, V, , s_s ,, at any interior point (i,t) as a function of these variables at the previous time steps. Once initial and boundary conditions are given, the non-linear system is solved at any point in the time-space grid by using the Newton-Raphson method. As an initial estimative for the Newton's method, the solution of the correspondent elastic problem was investigated and found to give satisfactory results. For the examples run in this work, no more than five interactions were necessary to achieve convergence.

Applications

This section presents some particular constitutive equations for viscoelastic and elasto-viscoplastic behaviors which do fall within the general context of the constitutive theory presented before. As stated earlier, these constitutive relations will be derived by choosing an appropriate set of internal variables b and by specifying particular forms for W_a and F. Later on the next section some numerical results obtained by using these constitutive models are presented in order to illustrate the applicability and versatility of the theory. Particular forms of the constitutive relations for viscoelastic and elasto-viscoplastic behaviors are presented next.

Viscoelasticity Within the field of linear viscoelasticity theory, analogic models composed of the assemblage of mechanical components (such as dashpots and springs) constitute an useful way to derive constitutive equations. In the particular case of solids in which the dilatational response can be assumed to be purely elastic, the generalized Kelvin-Voigt model is well-suited for describing the viscoelastic behavior which in turn is restricted to pure shear.

In this model, the viscoelastic shear response is described by coupling a number of n Kelvin-Voigt units in series. To each Kelvin-Voigt unit it is associated a second-order strain tensor eⁱ, for i = 1,K ,n, so that the inelastic strain tensor e^ais equal to the summation over i of eⁱ.

By referring to the previous section, the state variables associated to this viscoelastic model are:

If one restricts the analysis to isotropic viscoelastic behavior, the inelastic strain energy density is chosen as:

where E_i is the Young's modulus of the i-th Kelvin-Voigt unit.

So, by considering equation (2) the state laws related to eⁱ are given by:

In this case, the dissipation potential assumes the following form:

where X_dev is the deviatoric part of X and t_i is the retardation time of the i-th Kelvin-Voigt unit. In view of equations (3), the evolution laws for this viscoelastic behavior are:

The viscoelastic constitutive equations (Eq. (8) and Eqs. (10)) presented above are suitable for polymeric materials such as polyethylene and polyvinyl chloride (PVC) at moderate temperatures. For detailed information about this Kelvin-Voigt model and the material parameters one should see the works of Freitas Rachid and Stuckenbruck (1990) and Gally et al. (1979).

Elasto-viscoplasticy The particular set of elasto-viscoplastic constitutive equations shown ahead is used to describe the mechanical behavior of metallic materials submitted to non-monotonic loadings. In this case, the set of state variables is defined as:

where p is a scalar internal variable associated with the isotropic hardening and c is a tensorial internal variable related to the kinematic hardening. The variable e^ais usually called the plastic strain and the variable p is known as the cumulated plastic strain.

The inelastic part of the free energy is chosen such that

and, from (2), the state laws associated to p and c are:

where a, b, and d are material parameters.

For this class of materials, the potential F is supposed to have the following form:

where k and n are positive material parameters and áxñ = max{0,x}.

In the above equation, F stands for the yield function which is given by:

where f (s,B^p,B^c) = J(S) + B^p - s^y and J(S) = [ ] [S]_dev^. [S]_dev]^½. S = s + B^c and [S]_dev is the deviatoric part of S. s^y is the initial yield stress in traction and j is a material parameter. From (3), the evolution laws for this elasto-viscoplastic behavior are:

From equations (15) one can note that the yield function F is strictly responsible for the evolution of the inelastic behavior. Moreover, by virtue of equations (13), it can be verified that F<0 if and only if f<0. Thus, if f<0 then and, consequently, the material will behave elastically. When B^p = 0 and B^c = 0, the condition f<0 is nothing else than the classical Von-Mises criterion. If B^p = 0 and B^c = 0 at time t = 0, the evolution of the elastic domain (the set of the stresses s^* such that f(s^*,B^p,B^c)<0) will be characterized by an homothetical expansion (due to B^p(t)) and by a translation (due to B^c(t)) of the initial elastic domain (defined by the Von-Mises criterion). Equations (13) and (15) describe adequately the mechanical phenomena of elasticity, plasticity, creep and relaxation observed in many metallic materials at room and high temperatures. For further details about this kind of elasto-viscoplastic constitutive equations, such as experimental procedures to identify the material parameters a, b, d, s^y, k, n and j, one should report to the works of Lemaitre and Chaboche (1990).

Numerical Examples

To illustrate the versatility of the theory, numerical results concerned with the transient response of a liquid-filled piping are presented by considering the two different types of mechanical behaviors for the pipewall presented before.

Firstly, to check the effectiveness of the theory, the result of a simulation is compared to some existing data. Gally et al. (1979) have carried out an experimental study in a polyethylene pipe conveying water (K=2.2 GN/m²) at 13.8^ºC. The pipe is 43.1 m long with an internal diameter of 45.8 mm and a wall thickness of 4.2 mm. Hydraulic transients were generated in a simple reservoir-pipe-valve installation by rapid valve closure (12 ms for an initial flow velocity of 0.57 m/s). The pipe is anchored at the extremities and is made of low density polyethylene (r_t= 920 kg/m³) which presents a viscoelastic behavior. The pipe material coefficients at 13.8^ºC are: E= 0.874 GN/m², v= 0.39, E₁= 1.938 GN/m², E₂= 1.569 GN/m², E₃= 1.148 GN/m², t₁= 0.56' 10^-4 s, t₂= 0.0166 s, t₃= 1.747 s.

The pressure history close to the valve is depicted in Fig. 2. The solid curve represents the experimental results and the dashed curve corresponds to the computed result obtained using the model proposed herein with ^Dt= 2.3 ms. Frictional losses in the fluid were not taken into account. The comparison indicates that the model may provide reliable results since both curves are in good agreement.

In the next examples, the schematic piping system shown in Figure 3 is considered. It represents a typical reservoir-pipe-valve installation where fast transients are generated by valve slam. The upstream end of the piping (location D) is attached to a constant pressure reservoir, and a valve rigidly fixed to the ground is positioned at the downstream end (location A). Rigid supports are installed at the elbows, located at B and C, to prevent their motions. However, these supports can be removed and the elbows move freely in the plane of the structure. Thus, different constraint conditions can be considered by unrestraining the motion of the elbow located either at B or at C. The piping is always anchored at the reservoir and at the valve and the pipe lengths L₁, L₂, and L₃ are equal to 7, 3 and 5 m, respectively. Water (K=2.2 GN/m²) flows initially on steady-state from the reservoir towards the valve. The transient regime is generated at t = 0 by rapid valve closure.

Boundary conditions for this piping arrangement are imposed at the reservoir, elbows and valve. For a constant pressure reservoir with a pipe rigidly connected to it, the following conditions hold:

where as those associated to an instantaneous closure of a valve fixed to the ground are:

Different boundary conditions apply to the elbow according to its constraints. For a moving massless elbow, the relationship between the liquid flow and the elbow motion is derived from the continuity and momentum equations applied over a translating control volume. The appropriate relations for this boundary condition, neglecting the bending stiffness of the pipes, are (Otwell et al., 1985):

where A_f and A_t represent the cross-sectional area of the fluid and pipe, respectively. Here, the subscripts 1 and 2 refer to the conditions at the pipe's cross section at the entrance and exit of the elbow, respectively. For instance, if the elbow B is considered (see Fig. 3), then the subscripts 1 and 2 refer to the conditions at the pipes L₂ and L₃ close to the elbow. However, if the elbow is rigid, the last two equations in (18) are replaced by .

Besides the boundary conditions, initial conditions must be provided at the time t = 0 in order to solve the problem. As initial conditions for the examples run in this work, we consider the steady state solution of the system of equations (4). In addition, it is also assumed that the piping has never been subjected to inelastic deformations, so that inelastic strains are set equal to zero before the transient regime takes place.

In the sequel, the dynamic response of the piping system shown in Fig. 3 is analyzed for two kinds of pipe materials under different constraint conditions imposed at the bends. These examples show the effects of pipe motion and pipewall mechanical behavior on the system response.

One piping is made of low density polyethylene which presents a viscoelastic behavior characterized by the same pipe material coefficients presented earlier in this section. The initial flow velocity is 0.5 m/s and the pipe has an internal diameter of 26.5 mm and a wall thickness of 1.7 mm.

The other piping is made of an AISI 316L stainless steel (r_t= 7142 kg/m³) which exhibits elasto-viscoplastic behavior and whose constitutive coefficients at 20^ºC are (Lemaitre and Chaboche, 1990): E= 185 GN/m², v= 0.3, s^y= 82 MN/m², k= 151 MNs/m², b= 60 MN/m², d= 8, n=24, a= 108 GN/m², j= 2800 GN/m². The pipe has an internal diameter of 73.2 mm with a wall thickness of 1.65 mm and the initial flow velocity is of 6 m/s.

The dynamic response of both pipings to valve slam is considered for different constraint conditions imposed at the bends.

Firstly, the simulation is carried out by considering both elbows immobile. The results obtained for this situation are depicted in Fig. 4 for the viscoelastic and the elasto-viscoplastic pipings. Figure 4a shows the pressure history at the valve for the viscoelastic piping and Fig. 4b displays the pressure at the valve for the elasto-viscoplastic piping.

By comparing the pressure curves in Fig. 4 one can see that the system response is strongly dependent on pipewall mechanical behavior. It is observed that the viscoelastic behavior introduces much more mechanical damping than the elasto-viscoplastic behavior does. Besides attenuation, the viscoelastic behavior introduces dispersive effects which, on the other hand, are not observed in the viscoplastic behavior (at least to that extent).

In a second simulation, the flexibility of the structure is increased by removing the supports of the elbow either located at B or at C, so that in each case one of the elbows is free to move in the x-y plane. The system responses are shown in Fig. 5 for the viscoelastic piping when the elbow located at B is allowed to move and in Fig. 6 for the elasto-viscoplastic piping when now the elbow at C is left free.

An examination of Figs. 5a and 6a reveals that, for both pipe materials, pressure responses are significantly altered when compared to the case where both elbows are rigid. Due to pipe motion, pressure values up to 25% and 28% higher than those predicted in the fixed-elbow setup are observed for the viscoelastic and elasto-viscoplastic pipings, respectively. Different from the elasto-viscoplastic behavior, where high frequencies disturbances generated by the fsi mechanisms remain visible throughout the response, in the viscoelastic behavior such disturbances are virtually eliminated after the first fluid cycle.

Finally, elbow displacements in the x and y direction, driven by axial stress waves in the pipe wall and pressure waves in the liquid, are displayed in Figs. 5b and 6b for both pipe materials. The results show that the piping may be subjected to significant displacement amplitudes if no constraints are imposed on critical points of the system, such as bends and variations of pipe diameters.

Concluding Remarks

It has been presented in this work an extension of a fluid-structure interaction analysis in order to handle different types of inelastic behaviors of the pipewall. Besides its capability of treating several kinds of inelastic behaviors within a same abstract framework, the model developed herein allows the use of a very simple numerical technique based on the method of characteristics for solving the resulting problems.

This reasonably simple theory makes possible the analysis of quite different piping systems, such as those employed in water supply transmission lines and in thermohydraulic installations. Examples of some complex effects caused by the mechanical couplings described by the proposed theory were presented and analyzed for viscoelastic and elasto-viscoplastic pipes. It is shown that the system response is strongly dependent on pipewall mechanical behavior. For fast transients generated by valve slam, the elasto-viscoplastic behavior of a typical stainless steel pipe acts by limiting the pressure oscillations without presenting dispersion. For polyethylene tubes, however, the viscoelastic response introduces not only considerable attenuation but also dispersive effects on the system response. It has also been observed that for both pipe materials pipe motion induced by the coupling mechanisms can cause pressure, and plastic deformation, higher than those predicted by the rigid set up model.

Although it has not been addressed directly in this work, the theory (mechanical model and numerical technique) adopted here can also be used as a promising tool for the design of safer and reliable piping systems through a structural failure prediction analysis. To do so, it suffices to introduce an additional internal variable related with the degradation (damage induced by inelastic deformation) of the pipe material into the constitutive equations as proposed and analyzed in detail in Freitas Rachid et al. (1994a, 1994b, 1998a, 1998b, 1999). With such a procedure it is possible to perform a cumulative damage analysis in the pipe and, consequently, to predict the remaining life of operating piping systems which had been subjected to severe pressure transients.

Acknowledgment

The author F. B. Freitas Rachid gratefully acknowledges the partial financial support provided by the agency CNPq through grant n^º 301323/94-1.

Manuscript received: July, 2001. Technical Editor: José Roberto de F. Arruda.

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Publication Dates

• Publication in this collection
  18 Sept 2002
• Date of issue
  Mar 2002

History

• Received
  July 2001