Conference Paper A Finite Element Approach for the Elastic-Plastic Behavior of a Steel Pipe Used to Transport Natural Gas

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1 Conference Paers in Energy, Article ID , 10 ages htt://dx.doi.org/ /2013/ Conference Paer A Finite Element Aroach for the Elastic-Plastic Behavior of a Steel Pie Used to Transort Natural Gas Miltiades C. Elliotis Deartment of Mathematics and Statistics, University of Cyrus, P.O. Box 24908, 1305 Nicosia, Cyrus Corresondence should be addressed to Miltiades C. Elliotis; m-elliotis@live.com Received 14 March 2013; Acceted 28 Aril 2013 Academic Editors: P. Demokritou, A. Poullikkas, and C. Sourkounis This Conference Paer is based on a resentation given by Miltiades C. Elliotis at Power Otions for the Eastern Mediterranean Region heldfrom19november2012to21november2012inlimassol,cyrus. Coyright 2013 Miltiades C. Elliotis. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. A finite element technique, with two-dimensional isoarametric elements, is develoed, for the analysis of a steel ie, with a circularcross-section.theieisinstalledabovethegroundandisusedtotransortnaturalgasatveryhighressures.thematerialof the ie is assumed to obey a bilinear elastic-lastic model.double symmetry is considered when setting u the mathematical model roblem and creating the finite element mesh. Ste increments or decrements on the internal ressure are alied (cyclic loading). Yielding is detected by the Von Mises yield criterion. Also, a J 2 flowruleisemloyedtohandlethelasticstraincomonent. A four-oint Gauss-Legendre quadrature is used to numerically erform all necessary integrations. Finally, the so-called shakedown henomenon is studied when cyclic loading is alied after the commencement of lastic deformations on the ie. Numerical results obtained by the method comare favorably with the analytic solution. 1. Introduction The efficient and effective movement of natural gas from roducing regions to consumtion regions requires an extensive and elaborate transortation system. In many instances, natural gas roduced from a articular well will have to travel a great distance to reach its oint of use. The transortation system for natural gas consists of a comlex network of ielines, designedtoquicklyandefficientlytransortnaturalgasfrom its origin to areas of high gas demand. IntheUSAandinEuroetherearethreemajortyesof ielines along the transortation route: the gathering system, the interstate ieline system, and the distribution system. The gathering system consists of low ressure, small diameter ielines that transort raw natural gas from the wellhead to the rocessing lant. Interstate highway system transorts rocessed natural gas from rocessing lants to those areas with high natural gas requirements and can carry natural gas across state or country boundaries at ressures varying between 15 and 200 MPa. This reduces the volume of the natural gas being transorted (by u to 600 times), as well as roelling of the gas through the ieline. The distribution system consists of low ressure ies which distribute the transorted natural gas within any region with high natural gas requirements. Mainline transmission ies, the rincile ielines in a given system, are usually between 200 and 1200 mm in diameter and with a wall thickness between 10 and 100 mm. The original art of a ie, in which roelling of the gas is erformed and ressures u to 200 MPa are often alied, is usually around 1000 mm in its diameter as shown in Figure 1. The actual ieline itself, commonly called line ie, consists of a strong carbon steel material, engineered to meet high standards. Thus, there is usually a need to erform a high quality design of such a ie. The engineering roblem is to find the aroriate external diameter and wall thickness. Usually, the diameter of the ie is chosen according to the needs in natural gas and the desired rate of suly. Therefore, it remains to estimate the wall thickness by considering the roerties of the material used and the internal ressure of the gas.

2 2 Conference Paers in Energy σ L t σ H P L d Figure 1: An examle of a steel ie, used to carry natural gas, with a very large external diameter. σ H Figure 2: Hoo stresses on the cross-section of a ie in equilibrium with internal ressure (radial and longitudinal stresses also exist). σ H For the arts of the ieline installed above ground we are interested in solving the roblem of a steel ie with internal ressure thousands of times greater than the external atmosheric ressure. Solutions for the elastic behavior (i.e., before the initiation of yielding) were given by many theoreticians like Timoshenko and Goodier [1], Beer and Johnston [2], and Fung [3],to mention some of them.but of great interest is the elastic-lastic case. The engineers who design such ies consider that lastic deformations aear first on the internal surface and as the ressure increases yielding of the material rogresses until it reaches the outer surface. They choose dimensions and material roerties which allow either artial yielding of the cross-section or no yielding at all. The elastic-lastic roblem is an old roblem. It was first analytically aroached by Prantdl (1925) and Reuss (1930). First they assumed that the changes of strain are to be divisible into static and lastic comonents. The elastic strain increment was exressed in terms of the deviatoric and hydrostatic stress comonents while the lastic strain comonentwasassumedtoberoortionaltothestress gradient of the lastic otential. Also, by using the Von Mises yield criterion and simle equilibrium of stresses (Figure 2), one obtains a hyerbolic system of three semilinear differential equations with first-order artial derivatives, in which the rincial unknown is the radius of the elastic-lastic boundary. These equations were further simlified by Hencky who roosed the so-called stress-deformation law. But the resulting system of differential equations is still much comlicated and cannot be solved theoretically to give a closed form of an exact solution. They can be solved numerically by relacing the artial derivatives with finite differences and obtaining a system of linear algebraic equations. Unfortunately, the numerical result is of oor accuracy. So, instead of numerically solving the original system of differential equations, researchers solved the system of equations for several values of the radius of the elastic lastic boundary and created a large number of grahs of the normalized radial stress σ r /k in the wall of the tube (k being the yield stress) versus the normalized distance r/a from the center (a being the internal radius). But the necessity to aly a numerical scheme to estimate the ultimate loads, as well as the stresses, strains, and dislacements at any osition and with adequate accuracy, was still there. It was the finite element method (FEM) which satisfied this need. This owerful method was develoed in arallel with the mathematical theory of lasticity mainly during the second half of the 20th century. The tremendous rogress in the field of comuter engineering alongside with the develoment of the FEM by Argyris [4], Zienkiewicz and Taylor [5], and most recently by Babuška et al. [6] andbathe[7] andothers(tomention some of them) gave to the engineers a owerful tool to solve many elastic-lastic roblems by running codes on very fast comuting machines. At the same time a more systematic aroach and formulation of the mathematical theory of lasticity was develoed by Hill [8], Prager [9] andothers.later on, in the 70s and 80s, the works of Bath and Wilson [10], of Hinton and Owen [11], and of others inaugurate a new era in the treatment of engineering roblems concerning materials with elastic-lastic behavior. In this work the roblem of a thick ie carrying natural gasissolved.itissubjectedtoagraduallyincreasinginternal ressure, followed by a decrease in ressure. It is also subjected to a reeated increase and decrease in the value of internal ressure (cyclic loading and unloading), with lane strain conditions being assumed in any cross section along the axis of the ie. In Section 2, a brief resentation of the mathematical theory of lasticity is made which includes a resentation of the Von Mises yield criterion, the mathematical models of strain hardening behavior, and the elasticlastic stress/strain relation. In Section 3 a general resentation of the FEM is made, and in Section 4, the model roblem is resented and numerical results are given together with the results of the analytical solution. This aer ends with conclusions and references which are given in Section 5 and References section, resectively. 2. The Mathematical Theory of Plasticity The object of the mathematical theory of lasticity is to rovide a theoretical descrition of the relationshi between stress and strain for a material which exhibits an elastic-lastic resonse. Plastic behavior is characterized by

3 Conference Paers in Energy 3 an irreversible straining which is indeendent of time and which can only be sustained once a certain level of stress has been reached. In this section we outline the basic assumtions and associated theoretical exressions for a general continuum. An extensive treatment of this theory can be found in the works of Hill [8], Prager [9], and others. In order to formulate a theory which models elastolastic material deformation, three requirements must be met: (i) An exlicit relationshi between stress and strain must be formulated to describe material behavior under elastic conditions, that is, before the onset of lastic deformation. (ii) A yield criterion, indicating the stress level at which lastic flow commences, must be ostulated. (iii) A relationshi between stress and strain must be develoed for ostyield behavior, that is, at the stage where the deformation consists of both elastic and lastic comonents. Before the onset of lastic yielding the relationshi between stress and strain is given by the standard linear elastic exression σ ij =C ijkl ε kl, (1) where σ ij and ε kl are the stress and strain comonents resectively, and C ijkl isthetensorofelasticconstantswhich,for an isotroic material, has the exlicit form C ijkl =λδ ij δ kl +μ(δ ik δ jl +δ il δ jk ), (2) where λ and μ are the Lamé constants and δ ij is the so-called Kronecker delta, a useful symbol in mechanics defined by δ ij ={ 1, if i=j, 0, if i =j. Equation (1) willsufficeintheidealcaseofanelasticand homogeneous material. But, in general, this is not the case for the metals and esecially for carbon steel and other steel alloys which exhibit both elastic and elastic-lastic behaviors under certain combinations of loadings. The steel ies used for the transortation of natural gas are not an excetion The Yield Criterion. The yield criterion determines the stress level at which lastic deformation begins and can be written in the general form (3) f(σ ij )=Y(k), (4) where f is a function of stresses σ ij and Y exresses the yield surface of a material (Figure 3). It has the meaning that a state of stress which gives a oint inside the surface reresents elastic behavior. In (4) arameter k is a material arameter to be determined exerimentally. It is the so-called yield stress and usually is denoted as σ y. On hysical grounds, any yield criterion should be indeendent of the orientation of the coordinate system Tresca yield curve Von Mises yield surface Von Mises yield surface σ 1 σ 2 Hydrostatic axis σ 3 σ 1 =σ 2 =σ 3 Tresca yield surface π-lane (deviatoric lane) σ 1 +σ 2 +σ 3 =0 Figure 3: The mathematical reresentation of the Von Mises and Tresca yield surfaces, suitable for three-dimensional stress states. emloyed, and therefore it should be a function of the three stress invariants only J 1 =σ ii, J 2 = 1 2 σ ijσ ij, J 3 = 1 3 σ ijσ ik σ ki. Exerimental observation, notably by Bridgman [12], indicate that lastic deformation of metals is essentially indeendent of hydrostatic ressure. Consequently, the yield function can only be of the form (5) f(j 2,J 3 )=Y(k), (6) where J 2 and J 3 are the second and third invariants where J 2 = 1 2 σ ij σ ij, J 3 = 1 3 σ ij σ ik σ ki, (7) σ ij =σ ij 1 3 δ ijσ kk. (8) Most of the various yield criteria that have been suggested for metals are now only of historic interest, since they conflict with exerimental redictions. The two most simle criteria which do not have this fault are the Tresca criterion and the Von Mises criterion. InthisworkweadotedtheVonMisesyieldcriterion becauseithasbeenroved,sofar,toworkverywellformetals. It was first roosed by Von Mises [13] in According to this criterion yielding occurs when J 2 reaches a critical value as J 2 =Y(k), (9)

4 4 Conference Paers in Energy where, again, k istheyieldstress.theseconddeviatoricstress invariant, J 2, can be exlicitly written as J 2 = 1 2 σ ij σ ij σ y σ 1 Von Mises yield surface Tresca (maximum shear) yield surface = 1 6 [(σ 1 σ 2 ) 2 +(σ 2 σ 3 ) 2 +(σ 3 σ 1 )] (10) σ y σ y σ2 = 1 2 [(σ 11 )2 +(σ 22 )2 +(σ 33 )2 ] σ y +σ σ2 23 +σ2 13. Yield criterion (9) may be further written as Figure 4: The mathematical reresentation of the Von Mises and Tresca yield surfaces, suitable for two-dimensional stress states. σ eff = 3J 2 = 3Y (k). (11) Quantity σ eff is termed effective or equivalent stress. Figure 3 shows the geometrical interretation of the Von Mises yield surface to be a circular cylinder whose rojection onto the π-lane is a circle of radius k 2. The two-dimensional lot of this yield surface has an ellitic form (Figure 4). Now, a hysical meaning of the constant k can be obtained by considering the yielding of materials under simle stress states. So, the case of uniaxial tension (σ 2 =σ 3 = 0) requiresthat valuek 3 is the uniaxial yield stress. Initial yield surface σ 1 f 1 (σ (1) y ) Current yield surface f 3 (σ (3) y ) f 2 (σ (2) y ) σ Mathematical Models of Strain Hardening Behavior. After initial yielding, the stress level at which further lastic deformation occurs may be deendent on the current degree of lastic straining. Such a henomenon is termed work hardening or strain hardening. Thus the yield surface will vary at each stage of the lastic deformation, with the subsequent yield surfaces being deendent on the lastic strains in some way. Some alternative models, which describe strain hardening are illustrated in Figures 5 and 6. In the case where the yield surfaces are a uniform exansion of the original yield curve, without translation, as shown in Figure 5, the strain hardening model is said to be isotroic. On the other hand if the subsequent yield surfaces reserve their shae and orientation but translate in the stress sace as a rigid body, as shown in Figure 6, kinematic hardening is said to take lace. Such a hardening model gives rise to the exerimentally observed Bauschinger effect on cyclic loading. Therogressivedevelomentoftheyieldsurfacecan be defined by relating the yield stress Y, to the lastic deformation by means of the hardening arameter k.this can be done in two ways. Firstly, the degree of work hardening can be ostulated to be a function of the total lastic work, W,only.Then, k=w = σ ij d(ε ij ), (12) where (dε ij ) are the lastic comonents of strain occurring during a strain increment. Alternatively, k can be related to Figure 5: Mathematical model for the reresentation of isotroic strain hardening. Initial yield surface O σ 2 Current yield surface σ 1 Loading Figure 6: Mathematical model for the reresentation of kinematic strain hardening. a measure of the total lastic deformation termed the effective or equivalent lastic strain which is defined incrementally as dε (eff) = 2 3 (dε ij) (dε ij ). (13)

5 Conference Paers in Energy 5 For situations where the assumtion that yielding is indeendent of any hydrostatic stress is valid, we have (dε ij ) = (dε ij ).Consequently,(13)canberewrittenas dε (eff) = 2 3 (dε ij ) (dε ij ). (14) Then, the hardening arameter k is defined as k=ε (eff), (15) where ε (eff) istheresultofintegratingdε (eff) over the strain ath. This behavior is termed as strain hardening. By considering an isotroic hardening model, all the oints on the yield surface f=y(k)reresent lastic states, while elastic behavior is characterized by f<y(k).atlastic state, f=y(k), the incremental change in the yield function due to an incremental stress change is df = f dσ σ ij. (16) ij Then, if df < 0, elastic unloading occurs (stress oint is inside the yield surface); df = 0, neutral loading takes lace (stress oint remainsontheyieldsurface); df > 0, lastic loading occurs (stress oint remains on the exanding yield surface). Itcanalsobeshown[8] that, for a stable material, the initial and all subsequent yield surfaces must be convex. According to Drucker s definition [14], strain hardening in a material, which already exhibits lastic deformations, occurs when the following imortant inequality is true: uon loading, and when a comleted cycle of loading is indicated. dσ ij dε ij >0, (17) dσ ij (dε ij ) 0, (18) 2.3. Elastic-Plastic Stress/Strain Relation. After initial yielding thematerialbehaviorwillbeartlyelasticandartlylastic. During any increment of stress, the changes of strain are assumed to be divisible into elastic and lastic comonents, so that dε ij = (dε ij ) e + (dε ij ). (19) The elastic strain increment is related to the stress increment by (1), and by decomosing the stress terms into their deviatoric and hydrostatic comonents the exression for d(ε ij ) e is as follows: (dε ij ) e = dσ ij 2μ + 1 2V E δ ij (dσ kk ), (20) where E and V are, resectively, the elastic modulus and Poisson s ratio of the material. In order to derive the relationshi between the lastic strain comonent and the stress increment, a further assumtion on the material behavior must be made. In articular, it will be assumed that the lastic strain increment is roortional to the stress gradient of a quantity termed the lastic otential Q,sothat (dε ij ) =dλ Q, (21) σ ij where dλ is a roortionality constant termed the lastic multilier. A theoretical basis for this assumtion was develoed by Hill [8]. Equation (21) is termed the flow rule. It governs the lastic flow after yielding. The otential Q must be a function of J 2 and J 3 but as yet cannot be determined in its most general form. However, the relation f=qhas a secial significance in the mathematical theory of lasticity, since for this case certain variational rinciles and uniqueness theorems can be formulated. The identity f Qis a valid one since it has been ostulated that both are functions of J 2 and J 3 and such an assumtion gives rise to an associated theory of lasticity. In this case (21)becomes (dε ij ) =dλ f (22) σ ij and is termed the normality rule. The derivative f / σ ij is a vector directed normal to the yield surface at the stress oint under consideration, as shown in Figure 7.Itisseenthatthe comonents of the lastic strain increment are required to combine vectorially in an n-dimensional sace to give a vector which is normal to the yield surface. For the articular case of f=j 2,wehave Then, (22)becomes f σ ij = J 2 σ ij =σ ij. (23) (dε ij ) =dλσ ij (24) which is the so-called set of the Prandtl-Reuss equations [8]. Exerimental observations indicate that the normality condition is an accetable assumtion for metals and esecially for carbon steel with which ielines transorting natural gas aremadeof.now,onuseof(19), (20), and (22), the comlete incremental relationshi between stresses and strains for elastic-lastic deformation is found to be dε ij = dσ ij 2μ + 1 2V E δ ij (dσ kk ) +dλ f. (25) σ ij The hardening law Y=Y(k)couldjustbeeasilyexressed in terms of the effective stress σ eff (since it is roortional to J 2 ) to give, for the strain hardening hyothesis (15), σ eff =H(ε (eff) ), (26)

6 6 Conference Paers in Energy σ y σ 1 σ y σ ij Von Mises yield surface Tresca (maximum shear) yield surface (dε ij ) σ dσ y ij σ 2 σ y By differentiating F(σ,k), we have which is exressed as with the definitions df = F F dσ + dk=0, (31) σ k α T dσ Adλ = 0, (32) Figure 7: Geometrical reresentation of the normality rule of lasticity. Stress, σ dσ σ y O Sloe E T elastic-lastic tangent modulus dε e dε dε Sloe E elastic modulus Strain, ε Figure 8: Elastic-lastic strain hardening behavior for the uniaxial case. or differentiating, dσ eff =H (ε (eff) dε (eff) ). (27) In (27) the derivative H is the hardening function and it is the sloe of the uniaxial stress/lastic strain curve for which σ eff =σ 11 and dε (eff) =dε.thus,itcanbefurtherexressed as follows: H (ε (eff) )= dσ 11 = dσ 11 E = = T dε dε dε e 1 (E T /E), (28) where E T is the elastolastic tangent modulus (Figure 8). The theoretical exressions develoed so far can now be converted to matrix form. The yield function, first defined in (4), can be rewritten as f (σ) =Y(k), (29) where σ is the stress vector and k is the hardening arameter which governs the exansion of the yield surface. In articular, from (12), we have dk = σ T dε for the work hardening hyothesis. Rearranging (29), we get F (σ, k) =f(σ) Y(k) =0. (30) α T = F F =[, σ σ 11 F σ 22, F σ 33, A= 1 F dλ k dk. F σ 23, F σ 31, F σ 12 ], (33) Zienkiewicz et al. [15] give an elegant roof of the following result: A=H. (34) Thus, arameter A is obtained to be the local sloe of the uniaxial stress/lastic strain curve and can be determined exerimentally from (28). 3. The Finite Element Method The finite element method (FEM) is a numerical technique for finding aroximate solutions to artial differential equationsandtheirsystems,aswellasintegralequations.thebasic concet of this method is the discretization of the domain oftheroblemintofiniteelements.thereareseveraltricks about how to create an efficient finite element mesh in order tohaveresultsasaccurateasossibleandsendacomutational time as small as ossible. Several tyes of elements have been develoed of which theisoarametricelementswerefoundtobethemostsuitable for the work resented in this aer. So, in a finite element reresentation with isoarametric elements, the dislacements and strains and their virtual counterarts may be exressed by the relationshis u= n i=1 ε= N i d i =N d, δu = n i=1 B i d i =B d, δε = n i=1 n i=1 N i δd i =N δd, B i δd i =B δd, (35) where for node i, vectord isthevectorofnodalvariables, δd is the vector of virtual nodal variables, N is the matrix of global shae functions, and B is the global strain-dislacement matrix. For a 4-node isoarametric element shae function, N i is exressed in the local system Oξζ of the element as N i (ξ, ζ) = (1/4) (1+ξξ i ) (1+ζζ i ). The total number of nodes on the whole mesh is n. The basic exressions required for solution can be obtained by use of the rincile of virtual work. So, the virtual

7 Conference Paers in Energy 7 work exression for the whole domain, reresented by the finite element mesh, is as follows: n i=1 [δd i ]T { [B i ] T σdv [N i ] T bdv [N i ] T tds} =0, (δd ) T (B T σ N T b) dv (δd )f=0, (36) where t aretheboundarytractionswhicharealsodenotedby f (external alied forces) and b are the distributed internal loads er unit volume. Since (36)mustbetrueforanyarbitrary δd value, we have B T σdv N T bdv f = 0. (37) Here, for the lane strain roblem, the discretized elemental volume dv (e) reresents an element area which on the local coordinate system of the element is given as dv (e) = J(e) dξ dζ, (38) where J (e) is the Jacobian matrix of the transformation from the local system of the element to the global Cartesian system of the roblem s domain. For the solution of nonlinear roblems, with elastic-lastic behavior and strain hardening, (38) will not generally be satisfied at any stage of the comutation and ψ= B T σdv ( N T bdv + f) =0, (39) where vector ψ is the residual force vector of the nodal forces. For an elastic-lastic situation the material stiffness is continually varying and instantaneously the incremental stress/ strain relationshi is given by dσ = D e dε. Fortheurose of evaluating the global tangential stiffness matrix K T at any stage, the incremental form of (4) mustbeemloyed. Thus, within an increment of load, we have where Δψ = B T (Δσ) dv (Δf + N T (Δb) dv) =K T d (Δf+ N T (Δb) dv), (40) K T = B T D e BdV. (41) The rocedure for evaluating Δψ in (40), after the initiation of yielding, involves the imlementation of suitable iteration methods (refined rocess) in order to reduce a stress oint to the current yield surface, at ositions where loading in the lastic region occurs. Therefore, vector ψ of the nodal forces must be equivalent to the stress field satisfying elastolastic conditions. During the alication of an increment of load, an element, or art of an element, may yield. All stress and strain quantities are monitored at each Gaussian integration oint, and therefore it is ossible to determine whether or not lastic deformation has occurred at such oints. Consequently an element can behave artly elastically and artly elastolastically if some, but not all, Gauss oints indicate lastic yielding. For any load increment it is necessary to determine what roortion is elastic and which art roduces lastic deformation and then adjust the stress and strain terms until the yield criterion and the constitutive laws are satisfied. 4. Model Problem and Numerical Results The roblem studied in this work is that of a thick cylinder subjected to a gradually increasing internal ressure, with lane strain conditions being assumed in the radial direction of the cross section. In fact it is a art of a ieline carrying natural gas which must be designed for a given variation of the internal gas ressure. This means that having considered the self-weight of the ie, which is installed horizontally abovethegroundandhavingestimatedthebendingmoments and the normal stresses on the cross sections, at several ositions along the ie, one then examines the stress state at selected Gaussian oints on the surface of the isoarametric elements, to check if yielding occurs. It must be mentioned that the normal stresses due to bending moments are negligible comared to the values of the radial and of the hoo stresses in the lane of the cross section. The Von Mises yield criterion is assumed and the numerical solutions, obtained by imlementing the rocedure resented in the revious sections, are comared with the theoretical results of the elastic solution and the elastic-lastic solution develoed by Prandtl and Reuss. A 4-oint Gauss-Legendre quadrature rule is used in order to estimate the values of the elements of the stiffness matrix of an element of the structure +1 ij = 1 K (e) [B (e) i ] T D (e) B (e) j dv(e), (42) where dv (e) is the elemental volume (here is an elemental surface) given by (38). Using double symmetry of the roblem we work only on a quartet of the cross section, and the solid wall is discretized into N E 4-oint isoarametric elements having all together n nodes. The roblem was solved for an inner radius r i equalto100mmandanouterradiusr o which was varied from 140 to 200 mm. The internal ressure P i was alsovariedfrom0mpato192mpa.theatmoshericressure (about 0.1 MPa) is neglected being extremely small comared with the internal ressure on the ie. In Figure 9 afinite element mesh is resented for a quartet of the cross section with an outer radius equal to 140 mm. But the solution resented here is for r o = 200 mm. The material of the ieline is carbon steel with Young s modulus E = 210 MPa, Poisson s ratio ] = 0.3,andauniaxial yield stress σ y = 240 MPa. The strain hardening arameter is H = 0MPa; that is, there is an elastic-erfectly lastic

8 8 Conference Paers in Energy P i (MPa) Figure 9: A finite element mesh for a quarter of the cross section δ (mm) behavior for the material. A FORTRAN 77 code was used to solve the roblem on the comuter. Only the nonzero elements of the global stiffness matrix are stored in the comuter smemory.theelasticroblemwassolvedfortwovalues of the internal ressure: P i =80and 100 MPa. For the elasticlastic case the values of the internal ressure were P i = 120, 140, 160, 180, and 190 MPa. Also, the following cycling loadings were examined: (i) (10 cycles) (ii) (10 cycles). The ressure/radial dislacement characteristics for the elastic and for the elastic-lastic loading are shown in Figure 10. A good agreement between the numerical and analytical solution, given by Prandtl-Reuss, is evident. Indeed, in using the Von Mises criterion the analytic aroach gives the value of MPa as the value for the initiation of yielding. The corresonding value on the grah of Figure 10 is MPa. Thus, a very good agreement is observed between the numerical value and the Prandtl-Reuss analytic solution. In fact, the numerical and analytic curves ractically fall one uon the other in Figure 10. In the numerical studies, collase was deemed to have occurred if the iterative rocedure diverged for an incremental load increase. For the secific roblem the collase load given by the analysis reviously described was found to be equal to 192 MPa. Indeed, for this value of internal ressure or for greater values, there was no convergence in the numerical rocedure and the rogram on the comuter had to be stoed. In Table 1 the numerical results obtained for P i =80MPa (elastic case) are comared with the analytical solution. One may observe that in almost all ositions the theoretical results coincide with those obtained by the FEM (isoarametric element aroach). In Table 2 we resent the results obtained by running the rogram for P i = MPa. On the same table the analytical results given by the theoretical grahs of the Prandtl- Reuss method, for the same value of the internal ressure, are also resented. Comarison between the two sets of values indicates, in general, a good agreement between numerical Figure 10: Plot of internal ressure P i versus δ (dislacement of the internal surface of the ie). Table 1: Numerical and analytical results for the elastic case. r (mm) σ r (MPa) σ θ (MPa) FEM Analytic solution FEM Analytic solution Table 2: Numerical and analytical results for the elastic-lastic case for an internal ressure P i =169.36MPa. r (mm) σ r (MPa) σ θ (MPa) FEM Prandtl-Reuss FEM Prandtl-Reuss and theoretical results. Some deviations from the analytical resultsaremainlyduetoerrorsinreadingthevaluesonthe Prandtl-Reuss theoretical grahs and the assumtion of linear interolation for intermediate values. Since we have a lane strain condition the axial stress σ z is found from σ z = ](σ r +σ θ ).Itisalwayslessthanσ θ. Figure 11 shows the grah of σ r versus r for an internal ressure P i = 180 MPa, and Figure 12 resents the lot of σ θ versus r for the same internal ressure. On the σ θ versus r grahonemayclearlyseetheositionoftheelastic-lastic boundary, that is, at a distance 165 mm from the center of the ie. Figure 13 shows the distribution of the tangential

9 Conference Paers in Energy 9 σ r (MPa) r (mm) Figure 11: Plot of σ r versus r for an internal ressure P i = 180 MPa. Notice the discontinuity in entering form the gas to the solid material. σ θ (MPa) r (mm) Figure 12: Plot of σ θ versus r for an internal ressure P i = 180 MPa. residual stresses which aear on the cross section after the endofthelastcycleofthesecondloadingrogramwhich is discussed earlier. At the last two cycles of this loading rogram a solidification of the maximum effective lastic strain ) wasobservedatthevalue This observation is in accord to the theory which has been develoed,thelastthreedecades,aroundtheso-called shakedown henomenon. It is a henomenon which takes lace when cyclic loading and unloading occur elastically (i.e., within the yield surface of the chosen yield criterion) after the initiation of the lastic deformation in the structure. Since this is a dee theory which is much discussed in the literature (e.g., [16]), there will be no further discussion here. Finally, it is of interest to have a look at the total CPU time used in running the FORTRAN code on the multiuser Cyber. Table 3 shows the execution time and the number of iterations required to achieve convergence at each load. Convergence occurs when Δψ becomes less than error. (max ε (eff) 5. Conclusions In this work the roblem of a steel ie carrying natural gas was solved by using the FEM with isoarametric elements. The elastic-lastic behavior of the material of the ie was studiedbyimlementingthevonmisesyieldcriterionand σ θ (MPa) r (mm) 40 Figure 13: Distribution of residual hoo stresses σ θ versus r after cyclic loading. Table 3: Execution time for different values of internal ressure. Internal ressure (MPa) Number of iterations until convergence occurs CPU time (sec) a J 2 flow rule of the mathematical theory of lasticity. The numerical results given by the FEM for both the elastic and the elastic-lastic loadings were comared favorably with the analytical results of the classical theory of elasticity and with the theoretical results of the Prandtl-Reuss solution. For cyclic loading and unloading erformed elastically, after the commencement of yielding in the structure, the FEM was able to redict the shakedown henomenon. Thus, residual stresses in the wall of the ie after a comlete cycle of loading give an accetable state of stress which is within the safety limits. The rediction of this henomenon and of its residual stresses is robably the most imortant caability of the FEM for this class of roblems together with its utility as a valuable tool for the design of such ies to avoid collase under high internal ressures. References [1] S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, NY, USA, [2] F. P. Beer and E. R. Johnston, Mechanics of Materials,McGraw- Hill,NewYork,NY,USA,1981. [3] Y. C. Fung, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, [4] J. Argyris, Energy Theorems and Structural Analysis, Butterworths Sc. Publications, London, UK, [5] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method for Solid and Structural Mechanics, Elsevier Eds, 6th edition, 2005.

10 10 Conference Paers in Energy [6] I. Babuška, U. Banerjee, and J. E. Osborn, Generalized finite element methods: main ideas, results and ersective, International Comutational Methods,vol.1,no.1, , [7] K.J.Bathe,Finite Element Procedures,Prentice-Hall,Englewood Cliffs, NJ, USA, [8] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, Oxford, UK, [9] W. Prager, An Introduction to Plasticity, Addison-Wesley, Amsterdam, The Netherlands, [10] K.J.BathandE.L.Wilson,Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, [11] E. Hinton and D. R. J. Owen, Finite Element Programming, Academic Press, London, UK, [12] P. W. Bridgman, Studies in Large Plastic Flow and Fracture, McGraw-Hill, New York, NY, USA, [13] R. Von Mises, Mechanik der festen Körer im lastischdeformablen Zustand, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, vol. 1, , [14] D. C. Drucker, A more fundamental aroach to lastic stressstrain relations, in Proceedings of the 1st US National Congress of Alied Mechanics, , Chicago, Ill, USA, [15] O. C. Zienkiewicz, S. Valliaan, and I. P. King, Elasto-lastic solutions of engineering roblems; initial stress finite element aroach, International Journal for Numerical Methods in Engineering,vol.1, ,1969. [16] Q. S. Nguyen, On shakedown analysis in hardening lasticity, the Mechanics and Physics of Solids, vol.51,no.1, , 2003.

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