Effects of internal=external pressure on the global buckling of pipelines

Transcription

1 159 Effects of nternal=external pressure on the global bucklng of ppelnes Eduardo N. Dvorkn, Rta G. Toscano * Center for Industral Research, FUDETEC, Av. Córdoba 3, 154, Buenos Ares, Argentna Abstract The global bucklng (Euler bucklng) of slender cylndrcal ppes under nternal=external pressure and axal compresson s analyzed. For perfectly straght elastc ppes an approxmate analytcal expresson for the bfurcaton load s developed. For constructng the nonlnear paths of mperfect (non straght) elasto-plastc ppes a fnte element model s developed. It s demonstrated that the lmt loads evaluated va the nonlnear paths tend to the approxmate analytcal bfurcaton loads when these lmt loads are nsde the elastc range and the mperfectons sze tends to zero. Keywords: Internal pressure; External pressure; Axal compresson; Euler bucklng; Ppelne 1. Introducton When a straght ppe under axal compresson and nternal (external) pressure s slghtly perturbed from ts straght confguraton there s a resultant force, comng from the net nternal (external) pressure, that tends to enlarge (dmnsh) the curvature of the ppe axs. Hence, for a straght ppe under axal compresson, f the nternal pressure s hgher than the external one, there s a destablzng effect due to the resultant pressure load and therefore, the ppe Euler bucklng load s lower than the Euler bucklng load for the same ppe but under equlbrated nternal=external pressures; on the other hand when the external pressure s hgher than the nternal one the resultant pressure load has a stablzng effect and therefore the ppe Euler bucklng load s hgher than the Euler bucklng load for the same ppe but under equlbrated nternal=external pressures. The analyss of the bucklng load of slender cylndrcal ppes under the above descrbed loadng s mportant n many technologcal applcatons; for example, the desgn of ppelnes. In Fg. 1 we present a smple case, for whch the axal compressve load.t / has a constant part.c/ and a part proportonal to the nternal pressure.p /. That s to say, Ł Correspondng author. Tel.: C54 (3489) 435-3; Fax: C54 (3489) ; E-mal: sdrto@sderca.com T D C C kp (1) where k s a constant dependng on the partcular applcaton. In the second secton of ths paper we develop an approxmate analytcal expresson for calculatng the Euler bucklng load for elastc perfectly straght cylndrcal ppes (bfurcaton lmt load) and n the thrd secton we develop a fnte element model to determne the equlbrum paths of mperfect (non straght) elasto-plastc cylndrcal ppes. From the analyss of the nonlnear equlbrum paths t s possble to determne the lmt loads of ppes under axal compresson and nternal=external pressure. Of course, ths lmt loads depend on the ppe mperfectons; however, we show va numercal examples that, for the cases n whch the bfurcaton lmt loads are nsde the elastc range, the ppe lmt loads tend to the bfurcaton lmt loads when the mperfectons sze tends to zero.. Elastc bucklng of perfect cylndrcal ppes.1. Internal pressure In Fg. 1 we represent a perfectly straght slender cylndrcal ppe, n equlbrum under an axal compressve load and nternal pressure; let us assume that we perturb the straght equlbrum confguraton gettng an nfntely close 1 Elsever Scence td. All rghts reserved. Computatonal Flud and Sold Mechancs K.J. Bathe (Edtor)

2 16 E.N. Dvorkn, R.G. Toscano / Frst MIT Conference on Computatonal Flud and Sold Mechancs Fg. 1. Cylndrcal ppe under nternal pressure and axal compresson. confguraton defned by the transversal dsplacement, v.x/, of the ponts on the ppe axs. If for some loadng level, defned by p and by Eq. (1), ths perturbed confguraton s n equlbrum we say that the load level s crtcal (bucklng load) because a bfurcaton of the equlbrum path, n the loads dsplacements space, s possble. Due to the polar symmetry of the problem we consder that all the dsplacements v.x/ are parallel to a plane. For a longtudnal fber defned by the polar coordnates.x; r; / (see Fg. 1) we have, for the case of small strans, " xx D v.x/r cos () where " xx s the axal stran and v.x/ D d v.x/. dx On a dfferental ppe length, the resultant pressure force due to the ppe bendng s normal to the bent axs drecton (follower load) and ts value s, q.x/ dx D Z ³ p cos.1 C " xx /r d dx (3) where r s the ppe nner radus.usng Eqs. () and (3) we get, q.x/ D p ³r v.x/ (4) whch s the resultng force per unt length produced by the nternal pressure actng on the deformed confguraton. Ths load per unt length has horzontal and vertcal components that n our case (v.x/ 1) are, q h.x/ D q.x/ cos ð v.x/ Ł I q v.x/ D q.x/ sn ð v.x/ Ł : (5) Usng a seres expanson of the trgonometrc functons and neglectng hgher order terms, we get: q h.x/ D p ³r v.x/i q v.x/ D : (6) To analyze the equlbrum of the perturbed confguraton, beng ths an elastc problem, we use the Prncple of Mnmum Potental Energy [1,]. When only conservatve loads are actng on the ppe, equlbrum s fulflled f, n the perturbed confguraton, ŽŠ D (7) where Š s the potental energy, Š D U V (8) U: elastc energy stored n the ppe materal, V : potental of the external conservatve loads. In our case we have to consder the dsplacement dependent loads (non-conservatve) gven by Eq. (6), therefore [3]: Ž.U V / and [1], U D EI V D T q h Žv.x/ dx D q h Žv.x/ dx D (9) ð v.x/ Ł dx; ð v.x/ Ł dx; p ³r v.x/žv.x/ dx (1a) (1b) (1c) E: Young s modulus of the ppe materal, I: nerta of the ppe secton wth respect to a dametral axs. Hence, ntroducng the above n Eq. (9) we get for the fulfllment of equlbrum, EI Ž C p ³r ð v.x/ Ł dx T ð v.x/ Ł dx v.x/žv.x/ dx D : (11) We search for an approxmate soluton of the above equaton usng the Rtz Method [1], therefore we try as an approxmate soluton, Qv.x/ D X a n sn n³x : (1) nd1;;:::

3 E.N. Dvorkn, R.G. Toscano / Frst MIT Conference on Computatonal Flud and Sold Mechancs 161 An example of ths case s the hydraulc testng of a ppe. In ths case: C D, k D ³ re r Ð. Hence, usng Eq. (14b) we get, p cr D EI³ : re Obvously, f there are.n 1/ ntermedate supports we have, p cr D n EI³ : re.. External pressure For the cases n whch the ppe s submtted to external pressure we rewrte Eq. (6) as, q h.x/ D p e ³r v.x/i q v.x/ D : (15) Fg.. Smply supported ppe open on both ends under nternal pressure. Introducng the proposed approxmate soluton n Eq. (11) and takng nto account that the a n are arbtrary constants we get for equlbrum, EIn 4 ³ 4 Tn ³ p 3 r n ³ 3 a n D n D 1; ;::: (13) The above equatons have two possble soluton sets: ž a n D ; whch corresponds to the unperturbed straght confguraton. h EIn 4 ³ 4 Tn ³ 3 n ³ 3 D ; whch corresponds ž p r to an equlbrum confguraton dfferent from the straght one. The second soluton gves the locaton of the bfurcaton pont (crtcal loadng), T cr C p cr ³r D n EI³ (14a) C cr C kp cr C p cr ³r D n EI³ : (14b) It s nterestng to realze that the above equatons predct that there s a crtcal (bucklng) pressure also f there s no axal compresson.t D / and even f there s axal tenson on the ppe.t < /. et us consder the followng cases: ž Smply supported ppe, closed on both ends, under nternal pressure. In ths case, C D andk D ³r ; hence, from Eq. (14b) t s obvous that the only possble soluton s the straght confguraton and no bfurcaton s possble. ž Smply supported ppe, open on both ends, under nternal pressure (Fg. ). Hence, after some algebra we get for the equlbrum of the perturbed confguraton, EI Ž p e ³r ð v.x/ Ł dx T ð v.x/ Ł dx v.x/žv.x/ dx D (16) usng as an approxmaton for the equlbrum confguraton the one wrtten n Eq. (1), we fnally get, EIn 4 ³ 4 Tn ³ C p 3 e r n ³ 3 a n D n D 1; ;::: (17) therefore, for the nontrval soluton, T cr p ecr ³r D n EI³ (18a) C cr C kp ecr p ecr ³r D n EI³ : (18b) From the above equatons t s obvous that the external pressure has a stablzng effect on the ppe; that s to say, the axal compressve load that makes the ppe buckle s hgher than the Euler load of the ppe under equlbrated nternal=external pressures. et us consder the followng case: ž Smply supported ppe, closed on both ends, under external pressure. For ths case C D andk D ³re therefore from Eq. (18b) we get, EI³ p ecr D.re r / and f the ppe has.n 1/ ntermedate supports, p ecr D n EI³.r e r / :

4 16 E.N. Dvorkn, R.G. Toscano / Frst MIT Conference on Computatonal Flud and Sold Mechancs Comparng ths result wth the one correspondng to the ppe under nternal pressure t s obvous that the ppe under external pressure can wthstand a hgher pressure wthout reachng the bfurcaton load; hence, t s obvous the stablzng effect of the external pressure. 3. Nonlnear equlbrum paths for non-straght elasto-plastc cylndrcal ppes An actual ppe s not perfectly straght, and ts random mperfectons wll have a projecton on the bucklng mode of the perfect ppe; hence, when analyzng the equlbrum path of a non-perfect ppe we shall encounter a lmt pont rather than a bfurcaton pont [4]. The load level of ths lmt pont shall depend on the ppe mperfectons, wll be lower than the bfurcaton load of the perfect ppe and wll tend to ths value when the mperfectons sze tends to zero. In order to analyze the nonlnear equlbrum paths of mperfect ppes we developed a fnte element model usng the general purpose fnte element code ADINA [5]. Some basc features of the developed fnte element model are: ž The ppe behavor s modelled usng Hermtan (Bernoull) beam elements [6]. ž The ppe model s developed usng an Updated agrangan formulaton wth an elasto-plastc (assocated von Mses) materal model (fnte dsplacements and rotatons but nfntesmal strans) [6]. ž Actng on the beam elements we consder a conservatve load (T ) and a deformaton dependent load normal to the ppe axs, that for the case of nternal pressure s (see Eq. (6)), q h D p ³r [v.x/ C.x/] where.x/ s the ntal mperfecton of the ppe axs. We smply calculate, n our fnte element mplementaton,the second dervatves usng a fnte dfferences scheme. To provde a numercal example, we use the fnte element model to analyze the followng case: Ppe outsde dameter 6.3 mm Ppe wall thckness 3.9 mm Ppe length 1, mm Intermedate grps 4 Ppe yeld strength 38.7 kg=mm Hardenng modulus. under the loadng defned by an nternal pressure and, C D, k D ³ re r Ð No clearance between the ppe and the grp We consder the followng ntal mperfecton for the ppe axs,.x/ D Þ : 5³x 1 sn (19) whch s obvously zero at the grps and s concdent wth the frst bucklng mode predcted usng the Rtz method (Eq. (1)). In Fg. 3 we plot the load dsplacement equlbrum path for varous values of Þ andnthesamegraphweplot the bfurcaton lmt load obtaned usng Eq. (14b). Fg. 3. Grps wth no clearance. oad dsplacement curves.

5 E.N. Dvorkn, R.G. Toscano / Frst MIT Conference on Computatonal Flud and Sold Mechancs 163 Fg. 4. Clearance between grps and ppe body. oad dsplacement curves. We can verfy from ths fgure that the lmt load ncreases when the sze of the mperfecton (Þ) dmnshes, and that t tends to the bfurcaton lmt load when Þ!. 3.. Clearance between ppe and grps Ths s a more realstc case because, unless the grps are welded to the ppe body, there s usually some clearance between the ppe and the grps. We analyze the same case that was consdered n the prevous subsecton but allowng for a clearance between the grp and the ppe body of 5 mm. We consder the followng ntal mperfecton for the ppe axs,.x/ D : 5³x 1 sn C : : 1 1 sn ³x () and between the rgd grp and the ppe we ntroduce a contact condton. In Fg. 4 we plot the nonlnear equlbrum paths correspondng to the cases: ž Clearance between grps and ppe body (ntal mperfecton as per Eq. ()). ž No clearance between grps and ppe body (ntal mperfecton as per Eq. (19) wth Þ D 1:). From the results plotted n Fg. 4 t s obvous that the only mperfecton that has an nfluence on the ppe crtcal load s the mperfecton that s concdent wth the frst ppe bucklng mode. 4. Conclusons We derved an approxmate analytcal expresson for calculatng the Euler bucklng load of a ppe under axal compresson and nternal=external pressure. Ths expresson ncorporates the destablzng=stablzng effect of the nternal=external pressure. We constructed a fnte element model to determne the nonlnear equlbrum paths, n the loads dsplacements space, of mperfect (non-straght) elasto plastc ppes. From the analyss of the nonlnear equlbrum paths t s possble to determne the lmt loads of ppes under axal compresson and nternal=external pressure. Of course, these lmt loads depend on the ppe mperfectons; however, we showed va numercal examples that, for the cases n whch the bfurcaton lmt loads are nsde the elastc range, the ppe lmt loads tend to the bfurcaton lmt loads when the mperfectons sze tends to zero. Acknowledgements We gratefully acknowledge the fnancal support from SIDERCA (Campana, Argentna). References [1] Hoff NJ. The Analyss of Structures. John Wley and Sons, New York, NY: [] Washzu K. Varatonal Methods n Elastcty and Plastcty. New York, NY: Pergamon Press, 198.

6 164 E.N. Dvorkn, R.G. Toscano / Frst MIT Conference on Computatonal Flud and Sold Mechancs [3] Crandall SH, Karnopp DC, Kurtz EF, Prdmore-Brown DC. Dynamcs of Mechancal and Electromechancal Systems. McGraw-Hll, New York, NY: [4] Brush DO, Almroth BO. Bucklng of Bars, Plates and Shells. McGraw-Hll, New York, NY: [5] ADINA R&D. The ADINA System. Watertown, MA, USA. [6] Bathe KJ. Fnte Element Procedures. Englewood Clffs, NJ: Prentce Hall, 1996.