Predicting Stresses in Cylindrical Vessels for Complex Loading On Attachments using Finite Element Analysis

Transcription

1 Predicting Stresses in Cylindricl Vessels for Complex Loding On Attchments using Finite Element Anlysis By: Chrles Grey An Engineering Project Submitted to the Grdute Fculty of Rensseler Polytechnic Institute in Prtil Fulfillment of the Requirements for the degree of MASTER OF ENGINEERING IN MECHANICAL ENGINEERING Approved: Prof. Ernesto Gutierrez-Mirvete, Engineering Project Adviser Rensseler Polytechnic Institute Hrtford, Connecticut December, 0

2 Contents LIST OF FIGURES... iii LIST OF SYMBOLS... vi ABSTRACT... vii - INTRODUCTION... - BACKGROUND THEORY & METHODOLOGY RESULTS DISCUSSION CONCLUSION REFERANCES APPENDIX A APPENDIX B... 4 APPENDIX C... 5 ii

3 LIST OF FIGURES Figure Verticl Lug Support (left) nd Guide Lug Support(right) on Piping System Figure - Digrm of Exmple Creted to Illustrte the Results of this Report... Figure 3 - Uniform Lod Plced on the Nozzle Attchment... 5 Figure 4 - Originl Dt vs. Experimentl Extensions [6]... 6 Figure 5 Digrm Showing How Pressure Lod Method is Applied to Moments []... Figure 6 Longitudinl Moment Lod Distribution []... 3 Figure 7 Circumferentil Moment Lod Distribution []... 3 Figure 8 Detil of Lod Distribution []... 5 Figure 9 Detil of Squre Nozzle nd Stress Loctions [6]... 5 Figure 0 - Schoessow nd Kooistr Exmple [4]... 6 Figure Strin guge Lyout for Schoessow nd Kooistr Exmple [4]... 7 Figure - Proof of Concept Mx Von Misses Stress in the Circumferentil Direction. 8 Figure 3 New Exmple Geometry... 9 Figure 4 - Circumferentil nd Longitudinl Moments on Attchment Figure 5 - New Exmple Circumfrentil Loding Only... 3 Figure 6 - New Exmple Longitudinl Loding Only... 3 Figure 7 - Combined Loding Mx Von Mises Stress Figure 8 - Displcement Boundry Condition to Mimic Plte Anchor Figure 9 - Meshing Detil round Nozzle Figure 0 WRC No.07 Figure A Grph to find M for Externl Circumferentil Moment [7] Figure - WRC No.07 Figure A Grph to find M x for Externl Circumferentil Moment [7] Figure - WRC No.07 Figure 3A Grph to find N for Externl Circumferentil Moment [7] Figure 3 - WRC No.07 Figure 4A Grph to find N x for Externl Circumferentil Moment [7] Figure 4 - WRC No.07 Figure B Grph to find M for Externl Longitudinl Moment [7] Figure 5 - WRC No.07 Figure B Grph to find M x for Externl Longitudinl Moment [7] Figure 6 - WRC No.07 Figure 3B Grph to find N for Externl Longitudinl Moment [7] Figure 7 - WRC No.07 Figure 4B Grph to find N x for Externl Longitudinl Moment [7] Figure 8 Digrm Illustrting Boundry Conditions on New Exmple... 5 Figure 9 Digrm Illustrting Meshing For New Exmple Figure 30 - Stresses Presented on New Exmple from Circumferentil Loding Only Figure 3 - Stresses Presented on New Exmple from Longitudinl Loding Only Figure 3 Front View Mx Von Misses Stress for New Exmple Figure 33 Rer View Mx Von Misses Stress for New Exmple Figure 34 - Displcement Vectors for Complex Lod Figure 35 - Displcement Vectors for Complex Loding Inside Shell iii

4 Figure 36 - Principl Stress for Complex Loding Figure 37 - Principl Stress for Complex Loding... 6 Figure 38 - Principl Stresses from Complex Loding... 6 iv

5 LIST OF TABLES Tble Comprison of Results from WRC to FEA Tble WRC No.07 Hnd Clcultion... 4 v

6 LIST OF SYMBOLS Men rdius of cylindricl shell (in.) b Coordinte to locte the center of the nozzle (in.) c Hlf the length of the nozzle in the circumferentil direction (in.) c Hlf the length of the nozzle in the longitudinl direction (in.) l Length of the shell (in.) l Hlf the length of the shell (in.) m,n Integers involved in the Fourier series clcultions p An eqully distributed lod (lbs.) p o Mximum orml lod on the shell cused by Circumferentil or Longitudinl Moments (lbs.) q Internl Pressure for the cylinder shell (lbs / in.) s * t Shell wll thickness (in.) u, v, w Displcements for the X, Y (), Z directions (in.) D E Modulus of Elsticity (lbs / in.) L *π* (in.) M x Bending Moments on the shell for Longitudinl Moment (in.lbs.) M Bending Moments on the shell for Circumferentil Moment (in.lbs.) x Membrne Forces on the shell for Longitudinl Moment (lbs.) Membrne Forces on the shell for Circumferentil Moment (lbs.) Z Rdil loding per unit re (lbs / in.) α Length / α α / β c / β c / γ / t Cylindricl coordinte mn Expression defined by eqution (6) λ λ *λ ν Poisson s Rtio vi

7 ABSTRACT This report contins results of nlysis tht illustrte shortcomings in the Welding Reserch Council Bulletin No.07 for predicting mximum stress vlues from pplying complex loding to nozzle or ttchment on piping system. Since 965, the Welding Reserch Council s Bulletin No. 07 hs been the go to document for finding mximum stress vlues for pplying lod to n externl nozzle or ttchment on n imperforted shell. This work ws bsed off Professor P.P. Bijlrd s theoreticl nd experimentl work on the topic. The issues with his work rise from ssumptions tht were mde. The focus of this report is on the ssumption tht the methods ccurtely clculte the mximum stresses on the four mjor xis loctions. This ssumption does not tke into ccount the re prllel to the mjor xis points. In some cses of multiple complex loding, these stress vlues cn be un-conservtive. To find more ccurte to life prediction of stress vlues, finite element nlysis of the loded ttchment is required. By minimizing ssumptions in the finite element nlysis, more ccurte mximum stress vlues re predicted. The gol of this report is to show how complex loding ffects the nlysis of the piping system round the nozzle nd how the results differ from the stress predictions clculted from Welding Reserch Council Bulletin No. 07. vii

8 - INTRODUCTION Pipe supports re lrge prt of designing piping system. If piping system cnnot tke the point lod t the support loction, then the system must be re-evluted with new loctions to blnce nd even out lods nd movements pprent on the pipe. Incresing the number of supports on the piping system is nother option. Ech of these options decreses the lod t ech support point nd would increse the likelihood tht nozzle ttchment would pss stress nlysis. These support ttchments re nlyzed using Welding Reserch Council Bulletin No.07 [6]. Pipes tht re to be hung in the verticl direction hve sher lugs tht re welded to the outer shell of the pipe s shown in Figure. This is type of support is verticl sher lug support nd would wrrnt the use of the Welding Reserch Council Bulletin No.07 method.

9 Figure Verticl Lug Support (left) nd Guide Lug Support(right) on Piping System In Figure guide lugs re shown on piping system. Guide lugs or nozzles will imprt moments or internl pressure bck into the shell of the pipe. The lugs portryed re tking pressure lod s well s circumferentil lods simultneously. The support shown in Figure is very common style used in the piping industry. This reports min focus will be this style of support. Figure shows the exmple tht hs been creted for the Finite Element Anlysis performed in this report. The geometry will hve both circumferentil nd longitudinl lods plced on the fces of the ttchment simultneously, s would be seen in the field. Figure - Digrm of Exmple Creted to Illustrte the Results of this Report WRC No.07 hs been widely used by the petrochemicl nd boiler industry to engineer nozzles/ ttchments for their vessels. Mny issues from using WRC No.07 re tht engineers often violte the ssumptions tht the bulletin tkes into ccount. Bijlrd s initil clcultions were only vlid for smll grouping of exmples. These ssumptions excluded certin terms in formuls when computing results, limiting the types of shells it

10 could be used to nlyze. Much of wht is used in the industry now exceeds those initil vlues nd the Pressure Vessel Reserch Committee relized this limittion. Mny industry leders pushed the PVRC to extend the curves out to the best of their bility while still trying to keep those curves conservtive. The curves were extended. However they my not be ccurte to the ctul stresses tht would be present in the vessel shell. The curve extensions were creted from very miniml experimentl dt. 3

11 - BACKGROUND The Pressure Vessel Reserch Committee sponsored progrm in the mid 960s tht ws tsked to find, through experimenttion nd nlysis, method of determining the stresses present in the shell of n imperforted pressure vessel when externl loding in the form of sher forces or moments re plced upon ttchments joined to it. The min portion of the reserch ws completed by Prof. P.P. Bijlrd during his time t Cornell University. He published few ppers ([], [], nd [3]) defining his methodology before it ws dopted by the PVRC nd mde into bulletin for the Welding Reserch Council. The reserch involved looking t sphericl s well s cylindricl pressure vessels. For the purpose of this report, only cylindricl vessels will be delt with. In the originl nlysis of cylindricl vessels, shllow shell theory ws used to derive formule, curves, nd dt. This ws one of the ssumptions behind P. P. Bijlrd s clcultions. Severl more ssumptions were mde in the cretion of these formuls tht will be mentioned in the ltter sections nd some of these could led to inccurcies. The initil dt Bijlrd clculted ws minly for smller dimeters of pipe. Severl lrge compnies in the piping industry requested lrger piping rtios. Bijlrd used limited experimentl dt to extend these curves to ccommodte severl of these lrger compnies. He included wrnings to the compnies tht would inform them of the un-conservtiveness of his method if not used properly. Loding ws lwys considered to be in the middle of the ttchment nd uniform in nture. This style of loding is shown by the pink rrows in the digrm from the FEA model used in this report in Figure 3. 4

12 Figure 3 - Uniform Lod Plced on the ozzle Attchment Any other type of loding such s point lod would be un-conservtive, forcing uneven loding nd uneven torques to be put into the nlysis. Becuse of the complexity of Bijlrd s work, it becme pprent tht n esier methodology ws needed to help engineers crry out the correct clcultions. The Pressure Vessel Reserch Committee summrized ll of Bijlrd s work into cook book type formt, with n esy to follow clcultion methodology. This method is presented in the Welding Reserch Councils Bulletin No. 07. As mentioned previously, the curves were extended to try nd mtch some of the experimentl dt tht did not mtch Bijlrd s originl clcultions for lrger dimeters nd rtios. These curves cn be seen Figure 4 s well s in the grphs in Appendix B shown with dshed lines. 5

13 Figure 4 - Originl Dt vs. Experimentl Extensions [6] The nlyticl portions tht re directly from Bijlrd s originl work re shown s solid line. All of these curves were discussed in the committees s conservtive nd sfe to implement from the vilble experimentl dt. The purpose of this report is to identify some of the limittions of the bove methods, specificlly for structures subjected to complex loding. A crefully clculted nd vlidted finite element nlysis is used to find the mximum stresses in the shell of the cylinder. 6

14 7 3 - THEORY & METHODOLOGY The method tht Professor P. P. Bijlrd uses consists in developing the lods nd displcements into double Fourier series for numericl evlution. The lodings considered re those of uniformly distributed lod within rectngulr nozzle. The types of loding vilble for nlysis using this method re s follows: pressure lod pushing towrd the center of the shell, moment in the longitudinl direction uniformly distributed over short distnce in the circumferentil direction, nd moment in the circumferentil direction uniformly distributed over short distnce in the longitudinl direction. For the purposes of this report only the ltter two types will be considered. In this cse, n eight order differentil eqution is derived in terms of the rdil displcement nd tngentil lod. Using this method the displcements, bending moments, nd membrne forces re found. Three prtil differentil equtions from the thin shell theory [] re used to strt the derivtions, 0 x w x v u x u υ υ υ () ( ) υ υ υ v dx v t w x w t w v x v x u () ( ) Z Et v x v t w t w v x u υ υ υ (3) u, v, nd w re denoted s displcements of X, Y (), nd Z directions respectively while is considered the men rdius of the cylinder nd t the thickness of the cylinder. ν is the Poisson s rtio nd

15 4 x Since we re deling with thin shells the terms contining cn be discrded. (4) Equtions (), (), nd (3) cn be simplified s follows. Applying the opertors nd to eqution () yields, nd, (5) (6) respectively. Ech of these is to then be solved for ltter is to hve () which results in the formul, nd fter which the pplied to it. Both equtions re then inserted bck into eqution w w υ t w w u υ x x υ x x. (7) Then by tking eqution () nd pplying the terms nd s before it will result in, to it in the sme mnner (8) nd (9) 8

16 9 respectively. Ech of these is to then be solved for nd fter which the ltter is to hve pplied to it nd then both re to be inserted bck into eqution (). Doing this yields the eqution, ( ) υ υ υ υ w x w x w t w x w v (0). For equtions (7) nd (0) one pplies the opertors nd respectively nd then pplying to the ltter eqution nd both of these re inserted bck into eqution (3) resulting in the eqution, ( ) ( ) ( ) Z D x w x w w x w t w υ υ υ υ () The new terms included in order to simplify eqution (0) re the flexurl rigidity of the shell: ( ) 3 υ Et D () nd w w (3) When evluting deflections from eqution () it hs been found tht some engineers hve left off the third term for shells with lrger length to rdius rtios nd lrger thickness to rdius rtios. In some cses it ws found tht the clculted displcement vlues were seen s up to twenty five percent too low when mtching ginst experimentl dt. This will cuse problems when using this method to evlute shells with those chrcteristics. To void those issues this term will remin in the nlysis.

17 When looking t differentil equtions (), (), nd (3) there is certin lck of ttention to the products of the resulting forces nd moments plced on the shell. If these were to be ccounted for they would cuse the differentil equtions to become nonliner nd would increse the difficulty of the computtions. This is discussed in more depth in ltter sections of the report. The internl pressure inside pipe nd the membrne forces tht re prt of this clcultion re esily included. However, Prof. Bijlrd did not include them in his method nd, they will not be including hence s these differences would eventully skew the results being exmined for this report. This is one of the ssumptions tht Bijlrd mkes in his nlysis tht sometimes re not relized by engineers when mking clcultions. Eqution (7) contins only derivtives of w; therefore it cn be solved by developing the w deflections nd the externl lods into double Fourier series. λ w wmn cos ( m ) sin x λ Z Zmn cos ( m) sin x (4) (5) Inserting equtions (4) nd (5) into eqution () yields, 8 4 D 6 4 ( ) ( υ ) λ m λ m λ m ( 6 υ υ ) ( 7 υ) ( λ m ) t Z mn λ 4 m 4 w mn cos λ ( m) sin x 0 (6). Solving eqution (6) for the displcements produces the coefficients, 0

18 where, mn w mn mn Z mn 4 l D ( m α n π ) ( m α n π ) ( υ ) n π α γ m α [ m α ( 6 υ υ ) n π ( 7 υ) m α n π ] α, nd γ t. Combining ll equtions nd simplifying produces the displcement eqution, l (7) (8) 4 l λ w mnzmn cos( m) sin x D The u nd v displcements of the piping system my be expressed s λ u umn cos ( m) cos x λ v vmn sin ( m) sin x By inserting equtions (9), (0), nd () bck into equtions (7) nd (0), the (9) (0) () displcement coefficients become, u mn λ ( λ m ) m t υλ υ m υ ( λ m ) wmn () v mn m t 4 3 υ 4 ( υ) λ m λ m m w mn λ λ υ υ (3) ( m ) The equtions pictured bove contin the term ( systems tht hve thick shell nd high lmbd vlues. For the report exmples, only thin shells re being considered mking the vlues for these terms insignificnt nd thus they cn therefore be ignored. This trnsforms the equtions () nd (3) into equtions, ). This term is only importnt for

19 ( m υλ ) w ( λ m ) [( υ) λ m ] ( λ m ) λ λ u mn cos( m)cos x m λ v wmn sin( m)sin x where w mn is tken from eqution (7). (4) (5) Equtions (9), (4), nd (5) cn now be used to find the three directionl displcements. To properly nlyze the lod plced on the shell, the method of externl pressure loding on cylindricl shell will be considered for it is the bsis in how the longitudinl nd circumferentil moments re clculted. A visuliztion of the technique cn be seen in Figure 5. Figure 5 Digrm Showing How Pressure Lod Method is Applied to Moments [] The externl pressure lod is considered equl nd opposite from the centrl xis. P. P. Bijlrd exmined the longitudinl moment nd postulted tht it is the sme thing s uniform pressure lod considering tht the lod is vrying from the center of the ttchment. The lrgest pressure lod is considered s being pplied t the outermost edge of the nozzle. This is lso true for the circumferentil moment. Bijlrd considered the circumferentil moment to be vrying lod with the highest pressure lod being

20 pprent from the outermost edge of the ttchment. Both of these re shown in Figure 6 nd Figure 7. Figure 6 Longitudinl Moment Lod Distribution [] Figure 7 Circumferentil Moment Lod Distribution [] The moments of system tht re the key to Bijlrd s pproch re represented by equtions [7], nd where, nd ( X ) Mx D x υx ( X υ ) M D X X x w w (6) (7) (8) 3

21 4 x w X x. (9) Combining the two momentums these equtions with the given terms yields, υ w w x w D M x (30) nd x w w w D M υ (3) The membrne forces cn lso be exmined the sme wy s the moments by using the following equtions s considered from reference [7] s well. w v x u Et x υ υ (3) x u w v Et υ υ (33) The previously obtined displcement equtions ((9), (4), nd (5)) cn now be inserted into equtions (30) through (3) to give ( ) ( ) x m m n Z l M mn mn x λ υ α π α sin cos (34) ( ) x m n m Z l M mn mn λ α π υ α sin cos (35) ( ) ( ) ( ) x m n m n m Z mn mn x λ π α γ α υ π sin cos 6 5 (36)

22 ( υ ) n 6π α γ mnzmn cos ( m α n π ) λ ( m) sin x (37) These four equtions re the bsic equtions for the computtion of the moment nd membrne forces on shell when given n outside lod. It will now be shown how the externl loding must be modified so tht it will ccommodte either circumferentil or longitudinl moment pplied to the system. Figure 8 Detil of Lod Distribution [] The lod is exmined s being contined in the squre tht is the ttchment s seen in Figure 8. The externl lod will be developed into Fourier series, i.e. where p( s) o o / L L L / m πms cos L p( s) ds (38) (39) 5

23 nd m / 4 L L 0 πms p( s)cos ds L Combining these three equtions yields the eqution, (40) p( s) pc l p π πmc sin m l πms cos l (4) which then simplifies to β p p( ) p sin π π m ( mβ ) cos( m ) (4), This eqution is used to represent uniformly distributed lod cross n ttchment connected to cylinder. A circumferentil or longitudinl moment lod is not considered n eqully distributed lod in ll directions like n externl pressure lod. The differences cn be seen in Figure 6 nd Figure 7. Over the length of the ttchment, the lod is highest t the outermost point nd will decrese to zero s it psses the centroid of the loded ttchment. The lod will continue to increse in the opposite direction s it moves wy from the centroid in the opposite direction on the shell. The lod on one side of the shell is therefore in tension while the opposite side of the shell is equl in mgnitude but in compression. This lod must be conformed to meet these conditions. By looking t the uniformly distributed loding one cn then turn the loding of the ttchment into n odd function of x, i.e. by writing the eqution, where, p( x, ) b n b c nπx bn sin l n x 4 p dx l ( )sin π l b c (43) (44) 6

24 7 nd l b n l c n n p b n π π π sin sin ) ( 4 (45) Therefore the finl loding distribution looks like, l x n l b n l c n n n p x p π π π π sin sin sin ) ( 4 ), ( (46) With longitudinl moment, the uniformly distributed lod will vry with the vlue of x. This is considered to be negtively mirrored from the centroid of the ttchment. This will be represented by p o c x p l x x l l x ' ' '. This rtio cn then be inserted into (46) to produce, ( ) ( ) cos sin ' ' ) ( β π π β m m m x c p x c p p o o. (47) By then combining ll of these terms the conclusion cn be mde tht Z x p ), (. (48) Forming the eqution, x m Z x p Z mn ' )sin cos( ), ( λ (49) where l n l n π π λ λ ' ' leds to

25 8,,3,... 0 ' cos ' ' sin ) ( ' 3 n m n n n n p Z n o mn β α π β α π β α π β π β α (50),,3,...,,3,... sin( ' cos ' ' sin ) ( ' 4 ) 3 n m m n n n mn p Z n o mn β β α π β α π β α π β π α (5) where ' ' l α α The vlue of Z mn will be used in conjunction with some of the previous equtions to find the displcements t ech point. By using equtions (50) nd (5) in conjunction with (8) nd inserting tht into eqution (34) nd (35), one cn find the moments M x nd M. If equtions (50), (5), nd (8) re used in conjunction with equtions (36) nd (37) the vlues for membrne forces N x nd N will result. When computtion of displcements is desired, equtions (50), (5), nd (8) cn be inserted bck into the eqution (). By tking eqution (7) nd using it to solve (9), (4), nd (5) ll directionl displcement vlues result. The sme procedure is now completed for circumferentil moment with chnges mde to some of the equtions. The lod is considered to be uniformly plced on the shell while still being proportionl to the ngle of wy from the zero ngle on the centroid of the ttchment. This will men tht the forces nd moments t from the centroid of the ttchment re equl nd opposite t the from the centroid. The generic eqution for the lod induced into the system is given by,

26 4p( ) nπc nπb nπx p( x, ) sin sin sin nπ n l l l (5) Then, proceeding s before this needs to be turned into n odd function of. where This extrpoltes out to L bm π m when c p( ) b m b m / 4 l s po l c 0 mπc posin L mπs sin L πms sin ds L mπc L mπc cos L (53) (54) (55) Combining ll these terms together gives,, sin cos sin (56) Then, using the sme methodology described erlier we cn ssume where, sinsin (57) Therefore, the finl result is, (sin cos sin,,3 (58),3,5 We cn then tke equtions (8) nd (58) nd combine them in the eqution, sinsin (59) The displcement equtions below re different to the displcement equtions for longitudinl moments. The chnge in force clcultions drive the terms to become sin 9

27 (m) rther thn cos (m). This does not cuse sign chnge in the results of the equtions put in. Therefore, sinsin (60) sincos (6) cossin (6) For Bijlrd s finl method in the cretion of Welding Reserch Council Bulletin No.07 guideline, the following equtions re formulted to use equtions (9) nd (58) to find the moments nd membrne forces s found in the bulletin, i.e. sinsin (63) sinsin (64) x Et υ u v w υ x (65) Et υ v w u υ x (66) These terms must be broken down by using the equtions nd sincos (67) cossin (68) Derivtives re tken of these equtions nd then plced bck into the originl equtions given ((36) nd (37) to obtin the equtions, 0

28 6 sinsin (69) 6 sinsin (70) Equtions (8) nd (58) re used to find the vlues used by Bijlrd. To complete the guideline (WRC No. 07), Bijlrd took ech of these equtions nd solved them for chnges in the vlue β, grphing them ccording to ech type of lod induced into the system. This is wht ws used in this project to clculte his results nd compre them to the results from the finite element nlysis. The next section of the nlysis explins hnd clcultion of the Welding Reserch Council method s it is trnscribed in the bulletin. There re few prmeters tht must be found in order to use the method properly. The shell prmeter γ is the rtio of the shell s men rdius to the thickness of the shell. This prmeter is used to red ech curve off the chrt in which to cpture the correct dt for input in the clcultion sheet. The second prmeter tht is needed is the β term. This term will vry depending on the type of ttchment used nd its orienttion. The method hs the possibility of using round ttchment, squre ttchment, or rectngulr ttchment. Ech of these types hs different formul to clculte β. For the purposes of this report, the squre ttchment will be considered. For this we will use the formul,

29 After finding ech vlue in the chrts, clcultions involving some of these initil terms will tke plce. Ech of the chrts re derived from the originl equtions tht were described bove. When looking for stresses resulting from circumferentil moment pplied to the ttchment, these re the steps tht should be followed. First, to find the circumferentil stresses in the shell Figure in Appendix B will be used. Reding the vlue from the chrt for /, if the vlue does not fll directly on specific γ vlue then one must interpolte between the vlues of the upper nd lower limits surrounding the desired. The next step is to find, in Figure 0 in Appendix B, the vlues for /. Once these vlues re found, the initil conditions re used to solve for the membrne stress nd the circumferentil bending stress by using the following equtions And M / Rmβ R M c * β T c m * M 6* Mc Mc / Rmβ Rm * β * T (7) (7) Once ech of these is solved for N /T nd 6M /T then they cn be combined bck into generl stress eqution of the form, ± (73) Where K n nd K b re stress concentrtion fctors to be considered in cses where there is brittle mteril or ftigue nlysis is to be completed on the ttchment nd pipe.

30 The sme process is then used to find vlues for from Figure 3 in / Appendix B nd from Figure. The vlues re then input into equtions, / And M / Rmβ R M c * β T c m * M 6* Mc Mc / Rmβ Rm * β * T (74) (75) Once ech of these re solved for N x /T nd 6M x /T then they cn be combined bck into generl stress eqution of, ± (76) When looking for stresses resulting from longitudinl moment pplied to the ttchment, the sme steps re followed using different chrts. First, to find the circumferentil stresses, Figure 6 in Appendix B is used. Reding the vlue from the chrt for for /, the next step is to find in Figure 4 in Appendix B the vlues / from the grphs into. As mentioned previously the steps remin the sme by inputting the results M / Rmβ R M c * β T c m * (77) nd M 6* Mc Mc / Rmβ Rm * β * T Then putting the solved vlues into the finl stress eqution ± (78) (79) 3

31 The next term needed is from Figure 7 in Appendix B nd / from Figure 5. They re then input into the equtions, / nd M / Rmβ R M c * β T c m * M 6* Mc Mc / Rmβ Rm * β * T (80) (8) nd they cn then be combined bck into generl stress eqution of, ± (8) The clcultion hs been simplified down to single sheet for Bijlrd s computtion scheme. This clcultion pge cn be seen in Tble in Appendix A. The curves tht re present in Appendix B re directly tken from WRC Bulletin No.07. 4

32 4 RESULTS In the previous chpter Prof. Bijlrd s methodology ws presented. In the computtions, some of the sme ssumptions tht he mde were continued through the report s clcultions to mke sure tht the derivtions mtch exctly with wht Bijlrd used to crete the method incorported in WRC bulletin No.07. Ech of these ssumptions will hve wekening effect on his methodology in finding the true to life stress distribution for nozzle ttched to cylindricl pipe. These ssumptions re s follows:.) All stresses re computed t (when looking in pln view) the up, down, right, nd left midpoints t ech side of the ttchment. Ech of these loctions both interior nd exterior of the shell re shown in Figure 9 below. Figure 9 Detil of Squre ozzle nd Stress Loctions [6].) When initilly creting the stress equtions, the terms for were removed from the eqution. 5

33 3.) After finding the derivtions of the equtions the term were ignored. 4.) Stresses t the men rdius were to be considered to be zero. 5.) Internl pressure is ignored from the initil sets of equtions. 6.) Stresses presented from this method re considered to be equl nd opposite for the stresses considered in the sme xis plne. The effects of ech of these ssumptions will be explined in the discussion section of this report. For proof of concept n exmple from one of Bijlrd s ppers ws used. In this exmple, he took experimentl dt to compre his methods to. The work of Schoessow nd Kooistr describes test cylinder 7 inches in length nd 56 inches t the men dimeter. This cylinder ws.3 inches in thickness nd hd n.75 inch pipe ttched to the side. The cylinder ws fixed on the end by steel plte welded round the dimeter, s shown below in Figure 0. Figure 0 - Schoessow nd Kooistr Exmple [4] 6

34 This system ws subjected to 40,000 in. lbs. in both the circumferentil nd the longitudinl directions. The strin ws recorded from guges ttched t vrious distnces from the welded ttchment on the pipe; these loctions re illustrted in Figure. Figure Strin guge Lyout for Schoessow nd Kooistr Exmple [4] The computed vlue of the circumferentil moment using Bijlrd s method is equl to 3,80 psi. Stress vlues equl to 5,000 psi were recorded from the extrpolted dt obtined from the strin guges. The difference is bout 6.8%. This is n cceptble devition. The sme exmple ws simulted using Finite Element Anlysis, s shown in Figure. 7

35 Figure - Proof of Concept Mx Von Misses Stress in the Circumferentil Direction The mximum stress vlues clculted with this model re exctly where Bijlrd predicted them to be. For circumferentil moment the mximum stresses re locted on the mjor xis on either side of the ttchment. The mximum stress vlues equl,80 psi, giving vrition of 6.3%. The devition shows proof of concept mong ll three nlyses. The longitudinl moment nlysis ws performed with the sme vlues. The resulting stress ws 3,90 psi using Bijlrd s method. This must be compred to 3,500 psi extrpolted from the experimentl dt. The difference in the clcultions ws only.9%. When using the finite element model difference of 9.3% ws chieved. This shows proof of concept for the clcultions tht were performed during the nlysis using the finite element method. Using the sme methods tht hve been used for proof of concept, seprte exmple ws creted in order to fcilitte the modeling of combined longitudinl nd circumferentil moments on nozzle. The geometry is shown in Figure 3: 8

36 Figure 3 ew Exmple Geometry The men rdius of the cylinder is 5 inches with thickness of 0.3 inches. These dimensions give γ vlue of 50. The ttchment chrcteristics re squre ttchment hving length of 7.5 inches nd height of 0 inches from the men rdius of the shell. This gives the nozzle β vlue of.5. The loding of the ttchment will be in both the circumferentil nd the longitudinl direction. The loding of the nozzle consisted of moments equling 5,000 in. lbs shown in Figure 4. 9

37 Figure 4 - Circumferentil nd Longitudinl Moments on Attchment In the hnd clcultion from the Welding Reserch Councils method in Appendix A, the clculted vlues for the combined stress re shown in Tble. Ech of these vlues were clculted from numbers extrpolted from curves found in WRC bulletin No.07. The curves were creted directly from the method derived in the theory section. Ech of the vlues In Tble mtch directly from the loctions tht would be found on the xis round the nozzle s defined from WRC 07 (see Figure 9 ). The vlues for ech column represent the loctions where the stress ws mesured in the experiment. Tble Comprison of Results from WRC to FEA AU AL BU BL CU CL DU DL WRC Method FEA Difference 39.9% 39.90% 6.8% 7.8% 30

38 The finite element method mimics Bijlrd s methods nd theories closely when the circumferentil moment is nlyzed on its own s cn be seen in Figure 5. The stresses creted from the circumferentil lod crete stress concentrtion round the mjor xis of the ttchment. This cse contining the loding in the circumferentil direction is testment s to the results shown in Tble only hving devition equl to 6.8% nd 7.8% t loctions CU nd DU respectively. Figure 5 - ew Exmple Circumfrentil Loding Only The finite element method vries considerbly from the WRC methodology in the cse of loding long the longitudinl direction. The FE nlysis with the longitudinl moment produces results inconsistent with Bijlrd s method. Figure 6 shows tht the mximum stress is not locted t the mjor xis of the nozzle. The mximum stress is concentrted t ech corner of the ttchment. The results hve stggering difference of 39.9%, lmost four times the ccepted vrition. 3

39 Figure 6 - ew Exmple Longitudinl Loding Only The results from finite element clcultions for the combined loding model re shown in Figure 7. The mjor on xis stresses were not the highest recorded stress vlue in the finite element model. There is stress concentrtion t the corner of the nozzle (seen in Figure 7) tht equls 3,680 psi. This vlue hs difference of 67.% when compred to the mximum clculted stress from Bijlrd s method. This is not cceptble for mximum vlue when looking for mximum stress results from the WRC method. How these types of stress re ignored in Bijlrd s guideline will be discussed in the ltter sections. 3

40 Figure 7 - Combined Loding Mx Von Mises Stress 33

41 5 DISCUSSION Bijlrd mde few ssumptions in his nlysis in order to simplify the clcultions nd produce finl method esy enough for cookbook-type computtion..) The ssumption ws mde tht ech of the midpoints on the exterior nd the interior of the shell would reflect the mximum stress vlues for lods tht re plced onto shell from n ttched nozzle. This is very resonble for simple systems nd it hs been shown in the report s proof of concept. However becuse of the neglect of the surrounding prllel plnes in n nlysis involving complex multiple lods, the stresses presented on these xes my not be representtive of the mximum stress for the overll system. There re higher stress concentrtions in res from prllel elements occurring from longitudinl loding nd circumferentil loding. As shown in the finite element results nd the digrms in Appendix C, there is stress concentrtion in the corner of the loded shell. This stress concentrtion will not be represented when looking for mximum llowed stress on the shell using the simplified pproch. The bsence of this stress concentrtion cn cuse the results found from WRC 07 to be unconservtive..) When first deriving formuls to crete the curves for WRC 07, the terms for were neglected, since the thickness to shell rdius rtio ws low. This is pproximtely true for the thin shells used in the exmples considered. Leving 34

42 ny terms out of clcultion however, will decrese its overll true to life ccurcy. 3.) The terms contining becuse of the very smll vlues this term would produce. This will hve little effect on the finl clcultions in this report. 4.) The men rdius of the shell is considered to hve zero stress becuse of the equl nd opposite nture of the loding put into the shell. This ssumption would be invlidted if there ws lrge enough deflection in the shell tht would cuse the centrl xis to move. This occurrence cn be seen in Figure 6. A high displcement vlue ws chieved with longitudinl loding lone. The results become skewed nd stress concentrtions move from their predicted loctions t the mjor xes. The sme finite element conditions were used in the proof of concept; therefore the ssumption cn be mde tht the filure is in the WRC No.07 s method nd not in the FEA nlysis. 5.) Internl pressure is ignored from the clcultion nd derivtion of vlues in Bijlrd s methods for the ddition of such force cn mke the stress clcultion equtions become non-liner nd would increse the difficulty of the clcultion by hnd. The finite element model is not subject to this limittion. 6.) Stresses re considered to be equl nd opposite on either side of the nozzle ttchment. This is good ssumption tht cn be tken nd is proven in the stresses tht were clculted using the finite element nlysis method. The stresses in Tble show tht there is little vrition between the stresses clculted on either side of the ttchment. 35

43 In the exmple from Schoessow nd Kooistr the Stresses clculted re within 0% cceptble devition. When creting the exmple I used mster/ slve physicl constrint for the ttchment of the long pipe to the shell of the test pipe, this locked the pipe nd the ttchment together in mnner tht would be ccurte to the welding shown in Figure 0. A full displcement restrint of the ends of the pipe ws mde to imitte the pltes tht re welded to the end of the pipe to lock the rig in plce during testing in the originl experiment (Figure 8). Figure 8 - Displcement Boundry Condition to Mimic Plte Anchor The sme mteril used in the test exmple is used for properties throughout both finite element nlyses. The modulus of elsticity considered to be 30E06 psi nd the Poisson s rtio to equl 0.3. The geometry of the shell ws creted s shell elements. These elements use the thin shell theory s their bse clcultion. Shell elements produced the best results when compred ginst experimentl dt. The geometry of the ttchment ws modeled s hex elements. Solid structures like the ttchment produce more ccurte results thn tetrhedrl elements when modeling the exmples used in this report. The mesh (shown in Figure 9) ws concentrted in the re round the nozzle to increse ccurcy. 36

44 Figure 9 - Meshing Detil round ozzle The rest of the model ws prtitioned up to hve corser mesh nd to not increse the time nd difficulty of the clcultion. The limits to the finer mesh were determined s where stresses would not devite nd chnge gretly with further refinement. The nozzle ttchment ws determined to hve n equl lod cross the entire width of the ttchment to represent uniform lod, just s is required from Bijlrd s ssumptions. Point lods were not used since they would cuse locl deformtion of the lug tht would skew the finl results. The sme method used on the cretion of the proof of concept ws implemented on the new exmple creted to show the complex loding on nozzle ttchment. The lods in the circumferentil direction gree very well with those obtined from the WRC methodology. However the longitudinl lods were not in good greement. This is cused by some of Bijlrd s ssumptions erlier in this section. The mjority of errors between rel life stresses nd Bijlrd s clcultions come from the ssumption tht ll of the clcultions consider tht the mximum stress vlues re locted t points on the mjor xes of the nozzle. When the prllel loding plnes re not considered nd ccounted for, dverse effects hppen tht cn cuse the finl results to vry from the ctul rel world stresses. 37

45 6 - CONCLUSION The Welding Reserch council gve the tsk of creting simplified clcultion method for piping systems with welded ttchments to Prof. Bijlrd. In creting this method, certin ssumptions were mde. The method ws only to be used for thin shelled cylinders nd spheres. The stresses considered were only to be found for ech side of the ttchment on the mjor xes. These ssumptions re not lwys stisfied in prctice nd they will not give ccurte concentrtions of stresses resulting from complex lods imprted on nozzle ttchment. The exmples presented in this report show the unconservtive nture of the PVRC s clcultion. However it cnnot be sid tht Welding Reserch Council Bulletin No. 07 is un-conservtive for ll sets of geometry with complex loding. There re mny different loding combintions nd only few of them were discussed in this pper, but they could well be the subject of further study using the finite element progrm developed herein. 38

46 REFERANCES. Bijlrd, P. P. "Stresses From Locl Lodings in Cylindricl Pressure Vessels." Trnsctions Of The ASME (August, 955): "Stresses From Rdil Lods nd Eternl Moments in Cylindricl Pressure Vessels." The Welding Journl (December, 955): 608-s - 67-s. 3.. "Stresses from Rdil Lods in Cylindricl Pressure Vessels." The Welding Journl (December, 954): 65-s - 63-s. 4. F., Schoessow G. J. nd Kooist L. "Stresses in Cylindricl Shell Due to Nozzle or Pipe Connection." Trnsctions of The ASME (June, 945): A-07 - A-. 5. Gould, Phillip L. Anlysis of Shells nd Pltes. New York, New York: Springer- Verlg New York Inc., K. R. Wichmn, A. G. Hopper, nd J. L. Mershon. "Locl Stresses in Spericl nd Cylindricl Shells Due to Externl Loding." Welding Reserch Council Bulletin No. 07 (July, 970): ii Timoshenko, S. Theory of Pltes nd Shells. New York, New York: McGrw-Hill Book Compny Inc.,

47 APPENDIX A 40

48 Tble WRC o.07 Hnd Clcultion 4

49 APPENDIX B 4

50 Figure 0 WRC o.07 Figure A Grph to find M for Externl Circumferentil Moment [6] 43

51 Figure - WRC o.07 Figure A Grph to find M x for Externl Circumferentil Moment [6] 44

52 Figure - WRC o.07 Figure 3A Grph to find for Externl Circumferentil Moment [6] 45

53 Figure 3 - WRC o.07 Figure 4A Grph to find x for Externl Circumferentil Moment [6] 46

54 Figure 4 - WRC o.07 Figure B Grph to find M for Externl Longitudinl Moment [6] 47

55 Figure 5 - WRC o.07 Figure B Grph to find M x for Externl Longitudinl Moment [6] 48

56 Figure 6 - WRC o.07 Figure 3B Grph to find for Externl Longitudinl Moment [6] 49

57 Figure 7 - WRC o.07 Figure 4B Grph to find x for Externl Longitudinl Moment [6] 50

58 APPENDIX C 5

59 Figure 8 Digrm Illustrting Boundry Conditions on ew Exmple 5

60 Figure 9 Digrm Illustrting Meshing For ew Exmple 53

61 Figure 30 - Stresses Presented on ew Exmple from Circumferentil Loding Only 54

62 Figure 3 - Stresses Presented on ew Exmple from Longitudinl Loding Only 55

63 Figure 3 Front View Mx Von Misses Stress for ew Exmple 56

64 Figure 33 Rer View Mx Von Misses Stress for ew Exmple 57

65 Figure 34 - Displcement Vectors for Complex Lod 58

66 Figure 35 - Displcement Vectors for Complex Loding Inside Shell 59

67 Figure 36 - Principl Stress for Complex Loding 60

68 Figure 37 - Principl Stress for Complex Loding 6

69 Figure 38 - Principl Stresses from Complex Loding 6