Pressure Vessels Thin and Thick-Walled Stress Analysis

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1 Pessue Vessels Thin and Thick-Walled Sess Analysis y James Doane, PhD, PE

2 Conens 1.0 Couse Oveview Thin-Walled Pessue Vessels Inoducion Sesses in Cylindical Conaines Hoop Sess Longiudinal Sess Sesses in Spheical Conaines Thick-Walled Pessue Vessels Inoducion Inenal and Exenal Pessue Inenal Pessue Only Exenal Pessue Only Failue Cieia Bile Maeials Maximum Sess Theoy Moh s Failue Cieion Ducile Maeials Maximum Shea Theoy Von Mises Yield Theoy Copyigh 018 James Doane Page of 31

3 1.0 Couse Oveview This couse coves he asic conceps of he analysis of vessels holding a fluid o gas unde pessue. Some common examples of pessuized vessels include pipes, wae owes, hydaulic cylindes, and oiles. This couse will focus on cylindical and spheical shaped vessels ecause hey ae common in indusial applicaions and saighfowad o analyze. Sess calculaions in his couse ae sepaaed ino wo caegoies. The fis secion will cove asic sesses in hin-walled pessue vessels. The second secion coves sesses in hick-walled pessue vessels. The majoiy of he couse coves opics fo hick-walled pessue vessels due o he inceased complexiy. Noe ha his couse only coves sess caused y pessue loads. Hanges o suppo ackes can e used o suppo anks o pessue vessels. The suppos will cause addiional sess, which when comined wih he woking sesses of he vessel canno exceed an allowale limi fo he maeial. I is also impoan ha he suppo sill allows flexiiliy of he vessel and no cause i o ecome oo igid. Though analyical mehods exis fo deemining addiional sesses caused y suppos, i is ecommended o use FEA [no coveed in his couse] o analyze suppo aeas. This couse coves he asic sess analysis of pessue vessels and does no cove specific design codes fo pessue vessels due o he vas ypes of applicaions. The eade should consul any appopiae codes, such as ASME Code Secion VIII, fo moe deails..0 Thin-Walled Pessue Vessels.1 Inoducion Pessue vessels will fall ino one of wo main caegoies: hin-walled pessue vessels and hickwalled pessue vessels. A hin-walled pessue vessel has a small plae hickness compaed o he oveall diamee of he vessel. Conside a pessuized cylindical vessel wih an inne adius and a wall hickness. Fo cylindes having a wall hickness less han o equal o 10% of he 10, he pessue vessel is consideed hin-walled. I will e seen ha hoop inne adius [ ] sess is assumed o e unifomly disiued ove he wall hickness in hin-walled pessue vessels (wih minimal eo), whee he disiuion is non-unifom in hick-walled pessue vessels. Copyigh 018 James Doane Page 3 of 31

4 . Sesses in Cylindical Conaines We will egin wih cylindical shaped vessels wih hin walls sujeced o inenal pessue. The inenal pessue will cause sesses in he cylinde walls. Fo hin-walled pessue vessels, a poin on he vessel (sufficienly fa fom he ends) will have hoop sess and longiudinal sess as illusaed wih he fee-ody diagam shown in Figue 1. Figue 1 Illusaion of hoop sess h and longiudinal sess L..1 Hoop Sess Fo an inenal pessue in he vessel we can calculae he longiudinal and hoop sess. We will egin wih a deivaion of hoop sess, which is a cicumfeenial sess. Conside he cylindical vessel shown in Figue (a). The vessel conains a fluid unde a pessue p [psi]. A secion wih a lengh L is cu fom he cylinde and is shown isolaed in Figue (). Copyigh 018 James Doane Page 4 of 31

5 Figue (a) Thin-walled pessue vessel () Isolaed secion The secion shown in Figue () is now cu in half o deemine inenal foces. The half secion is shown in Figue 3 (a) and has a lengh L, an inne diamee D, and he wall hickness is. The inenal pessue mus e conveed ino a foce, which is epesened in Figue 3 (). The esulan fluid foce is equal o he fluid pessue muliplied y he aea. Fo his case, he pessue is p and he aea is he pojeced aea of he diamee imes he lengh. F pdl Equaion 1 The inenal foce in he walls can e deemined fom saics. Refeing o Figue 3 () F P pdl P pdl P Equaion Copyigh 018 James Doane Page 5 of 31

6 Figue 3 (a) Half secion dimension () Foces on half secion The inenal foce in he walls given in Equaion can e used o deemine he nomal sess in he walls. Knowing ha he sess will equal he foce in he wall divided y he aea of he wall, he sess is Hoop Sess h P A pdl 1 L pd h Equaion 3 The hoop sess h given in Equaion 3 acs in he vessel s cicumfeenial diecion. In he equaion p is he inenal pessue in psi, D is he inenal diamee of he cylinde in inches, and is he wall hickness in inches. The esuling hoop sess will have unis of psi. Copyigh 018 James Doane Page 6 of 31

7 Example 1 A cylindical ank has an inenal diamee of 18 inches and a wall hickness of ¼ inch. Deemine he hoop sess if he inenal pessue is 150 psi. Soluion: The hoop sess is deemined fom Equaion 3. h pd l 150 in ( 18in) ( 0.5in) h 5400 in l.. Longiudinal Sess Conside he cylindical vessel wih closed ends shown in Figue 4 (a). The inenal pessue will end o push he ends of he cylinde ouwad causing a longiudinal ensile foce in he walls. Figue 4 () shows a secion of he cylinde wih he pessue foce and esuling wall foces. The esulan foce fom he fluid pessue will equal he pessue imes he pojeced aea. Fo an inenal diamee D, he foce will equal πd F p Equaion Copyigh 018 James Doane Page 7 of 31

8 Figue 4 (a) Thin-walled pessue vessel wih closed ends () Isolaed secion The foce in he cylinde wall mus equal he foce in Equaion 4, and he inenal sess will equal he foce divided y he aea. Fo a wall hickness, he aea will equal π D. Longiudinal Sess L πd p 4 1 πd pd L Equaion 5 4 The longiudinal sess acs in he axial diecion. In he equaion p is he inenal pessue in psi, D is he inenal diamee of he cylinde in inches, and is he wall hickness in inches. The esuling longiudinal sess will have unis of psi. Copyigh 018 James Doane Page 8 of 31

9 Example A cylindical ank has an inenal diamee of 6 inches and a wall hickness of ¼ inch. Deemine he longiudinal sess if he inenal pessue is 00 psi. Soluion: The longiudinal sess is deemined fom Equaion 5. L pd 4 l 00 in 4 ( 6in) ( 0.5in) L 500 in l.3 Sesses in Spheical Conaines Deiving sesses in he walls of a spheical conaine is vey simila o cylindical conaines. Conside he half secion of he spheical conaine shown in Figue 5. The conaine will have an inenal pessue p in psi. Fo his case, he esulan foce is he pessue imes he aea, which is a cicula aea. Fo a sphee inside diamee D, he foce will e πd F p Equaion Copyigh 018 James Doane Page 9 of 31

10 Figue 5 Half secion of a spheical conaine The foce in he walls mus equal he foce fom he fluid, and he inenal sess will equal he foce divided y he aea. Fo a wall hickness, he aea will equal π D. πd p 4 1 πd pd Equaion 7 4 Theefoe, he equaion fo sess in spheical vessel walls is he same as longiudinal sess in cylindical vessels. Copyigh 018 James Doane Page 10 of 31

11 Example 3 A spheical pessue vessel is consuced fom ½ hick plae and has an inne diamee of 3 fee. Wha is he maximum inenal pessue if he sess canno exceed 0 ksi? Wha would e he maximum inenal pessue fo a similaly sized cylindical vessel? Soluion: The maximum pessue fo he spheical conaine can e deemined using Equaion 7. pd p D ( 0.5in) l 0000 in 36in l p 1,111 in Fo a similaly sized cylindical conaine, he maximum sess occus in he cicumfeenial diecion. Fom Equaion 3 h pd h p D ( 0.5in) l 0000 in 36in p 556 in l Theefoe, a spheical pessue vessel will cay wice he inenal pessue. Copyigh 018 James Doane Page 11 of 31

12 3.0 Thick-Walled Pessue Vessels 3.1 Inoducion The equaions used o deemine sess in hin-walled pessue vessels ae ased on he assumpion ha he sess disiuion is unifom houghou he hickness of he cylinde walls. In ealiy, he sess vaies ove he hickness. If he walls ae hin, he vaiaion in sess is small so he sess can e assumed unifom wih minimal eo. Fo hicke walls, he vaiaion in sess ecomes moe impoan and canno e negleced. This secion on hick-walled vessels will e sepaaed ased on he ype of loading. The fis secion will include pessues on he inside and ouside comined, which is he mos geneal case. The following secions will simplify equaions y only consideing one pessue, eihe inenal o exenal. Conside a pessue vessel wih an ouside adius and an inside adius a as shown in Figue 6 (a). Fo he deivaions of equaions fo sess and sain, we will use cylindical coodinaes. Fom Figue 6 (a) we have he axial coodinae z aligned wih he cenal axis of he cylinde. Figue 6 () shows he adial and cicumfeenial coodinaes and θ. Copyigh 018 James Doane Page 1 of 31

13 Figue 6 Thick walled pessue vessel and cylindical coodinae sysem 3. Inenal and Exenal Pessue We will egin wih he mos geneal case whee he vessel is sujeced o oh inenal and exenal pessue. The inside pessue is p i and an ouside pessue p o as shown in Figue 7. The hick-walled cylinde can e eaed as seveal layes of hin-walled vessels, one is shown as he dashed lines in Figue 7. One segmen of a hin walled shell can e isolaed, as shown in Figue 8. The figue shows a ypical segmen wih a adius and hickness d. Copyigh 018 James Doane Page 13 of 31

14 Figue 7 Inenal and exenal pessue on a hick-walled cylinde Figue 8 Sesses on he half shell Copyigh 018 James Doane Page 14 of 31

15 The equaion of equiliium can e developed using cylindical coodinaes [noe ha cylindical coodinae sysems will use θ in place of fo he angenial coodinae, u will e used hee o say consisen wih he hin-walled pessue vessel equaions]. Wihou poof, he equaion of equiliium educes o d d ( ) Equaion 8 Similaly, he sain compaiiliy condiions ae given y d ( ε ) ε εθ Equaion 9 d Addiional equaions come fom Hook s law fo iaxial sess. The common fom of Hooke s law would e in ecangula coodinaes. [ ν ( )] ε x 1 x y + E [ ν ( )] ε y 1 y z + E [ ν ( )] ε z 1 z x + E z x y whee is sess, ε is sain, E is modulus of elasiciy, and ν is Poisson s aio. Hooke s law fo iaxial sess can also e wien in ems of cylindical coodinaes o give Copyigh 018 James Doane Page 15 of 31

16 [ ν ( )] ε 1 + z Equaion 10 E [ ν ( )] ε 1 + z Equaion 11 E [ ν ( )] ε z 1 z + Equaion 1 E Fom his poin, hee ae muliple ways o develop he geneal equaions fo sess a any poin. The pocess can ecome vey edious, and I will only povide a geneal appoach o no ovecomplicae he issue. As a asic appoach, we can ake Equaion 1 and ealize ha E and ν ae all consan maeial popeies. Defomaion a a coss-secion sufficienly fa fom he ends will no vay [o vay insignificanly] in he z diecion, so ε z and z ae consideed consan. Theefoe, + will also e consan. I is common o call he consan A (he faco of is fo simplificaion of fuue equaions) giving + A Equaion 13 Adding Equaion 8 and Equaion 13 gives d d d d ( ) + A ( ) ( A ) Equaion 14 Equaion 14 is a fis ode diffeenial equaion wih a single vaiale, which can e solved using sepaaion of vaiales o give Copyigh 018 James Doane Page 16 of 31

17 B A Equaion 15 whee A and B ae consans. Susiuing Equaion 15 ino Equaion 13 gives B A + Equaion 16 The consans A and B can e deemined y using ounday condiions fo adial sess, which mus equal he inenal pessue a he inne adius and he exenal pessue a he oue adius (oh negaive). Deemining he consans will give he final fom of he equaions shown in Equaion 17 and Equaion 18. Radial Sess a any poin Tangenial Sess a any poin ( p p ) a pi po a o Equaion 17 a i ( a ) ( p p ) a pi po a o + Equaion 18 a i ( a ) Fo he case of inenal pessue and exenal pessue, he adial displacemen a any adial locaion is given y ( ) ( )( ) ( 1+ ν ) 1 pia po + a a ν Equaion 19 ( p p ) u i o E Copyigh 018 James Doane Page 17 of 31

18 Example 4 A hick-walled cylinde has an inside adius of 1.5 inches and an ouside adius of 3.5 inches. The ouside pessue is 100 psi and he inenal pessue is 1300 psi. Plo he disiuion of adial sess ove he hickness. Soluion: The adial sess can e calculaed using Equaion a p i p a o a ( p p ) i ( a ) ( 1300) 3.5 ( 100) 1.5 ( 3.5 )( ) o ( ) Ploing he equaion fo values of eween 1.5 and 3.5 gives Noe ha he adial sess equals he inenal pessue a a and he exenal pessue a (all negaive). Copyigh 018 James Doane Page 18 of 31

19 3.3 Inenal Pessue Only The equaions can e simplified fo specific loading cases. If he pessue vessel only has p 0, Equaion 17 and Equaion 18 educe o inenal pessue ( ) o Radial Sess Tangenial Sess a p i 1 Equaion 0 a a p i 1 + Equaion 1 a Figue 9 povides a gaphical epesenaion of he sesses. The adial sess disiuion calculaed fom Equaion 0 is shown in Figue 9 (a). The adial sesses ae negaive (compession) wih a maximum value a he inne suface. Figue 9 () shows he angenial sess disiuion calculaed fom Equaion 1. The sesses ae posiive (ensile) wih a maximum value a he inne suface. Figue 9 (a) Radial sess disiuion and () angenial sess disiuion fo inenal pessue only Copyigh 018 James Doane Page 19 of 31

20 The adial sess is always compessive and he angenial sess is always a ensile sess, wih he angenial sess lage han he adial. The maximum adial sess occus a he inside a giving suface ( ) a p 1 max a ( ) i a a a ( ) pi max Maximum Radial Sess ( ) i p Equaion max Similaly, he maximum angenial sess occus a he inne suface. a p 1 max + a ( ) i a Maximum Tangenial Sess a + max a Equaion 3 ( ) pi Fo he case of inenal pessue only, he adial displacemen a he inne suface is ap i a + ua + ν Equaion 4 E a Copyigh 018 James Doane Page 0 of 31

21 Example 5 A hick-walled cylinde has an inside adius of 6 inches and an ouside adius of 10 inches. Plo he disiuion of angenial sess fo an inside pessue of 9 ksi. Wha is he maximum angenial sess? Soluion: The angenial sess is calculaed using Equaion 1. a p i a ( 9000) [ psi] Ploing he equaion fo values of anging fom 6 inches o 10 inches gives Copyigh 018 James Doane Page 1 of 31

22 The maximum angenial sess can e calculaed using Equaion 3 (o fom he aove equaion wih 6 inches). a + a ( ) pi max ( ) 9000 max ( ) psi 19, 15 max Example 6 The pessue vessel in he pevious example is made fom seel wih a modulus of elasiciy of 9x10 6 psi and has a Poisson s aio of 0.3. Find he adial displacemen a he inne suface in inches. Soluion: The adial displacemen is deemined using Equaion 4. u a u a ap i a + + ν E a ( 9000) u a in Copyigh 018 James Doane Page of 31

23 3.4 Exenal Pessue Only If he pessue vessel only has exenal pessue ( p 0) i, Equaion 17 and Equaion 18 educe o p a o 1 Equaion 5 a p a o 1 + Equaion 6 a Equaion 5 and Equaion 6 ae gaphically epesened in Figue 10 (a) and Figue 10 () especively. The adial sesses ae negaive (compession) wih a maximum value, equal o he magniude of he exenal pessue, a he oue suface. The angenial sesses ae compession wih he maximum value a he inne suface. Figue 10 (a) Radial sess disiuion and () angenial sess disiuion fo exenal pessue only Copyigh 018 James Doane Page 3 of 31

24 Copyigh 018 James Doane Page 4 of 31 The maximum adial sess occus a he oue suface ( ). 1 a a p o a p a a p o o + Maximum Radial Sess o p Equaion 7 The maximum angenial sess occus a he inne suface ( ) a giving + 1 a a a p o Maximum Tangenial Sess a p o Equaion 8 Fo he case of exenal pessue only, he adial displacemen a he oue suface is + + ν a a E p u o Equaion 9

25 Example 7 A hick-walled pessue vessel has an ouside diamee of 16 inches and an inside diamee of 1 inches. The vessel is sujeced o an exenal pessue only, and he maximum adial sess has a magniude of 4 ksi. The maeial popeies ae 6 E 9 10 psi and ν Wha is he maximum angenial sess and adial displacemen a he oue suface? Soluion: Based on Equaion 7, he exenal pessue will equal he magniude of he maximum adial sess. Theefoe, po 4, 000 psi Equaion 8 gives p a o 8 8 ( )( 4000) 6 18, 86 psi Equaion 9 gives he adial displacemen. u u p o a + + ν E a ( 4000) u in Copyigh 018 James Doane Page 5 of 31

26 4.0 Failue Cieia A complee coveage of failue cieia is eyond he scope of his couse. Howeve, a geneal coveage will e povided wih he focus on failue of hick-walled pessue vessels. Many diffeen failue heoies exis, u hey can e sepaaed ino wo majo caegoies ased on maeial ype. The caegoies ae failue heoies fo ile maeials and failue heoies fo ducile maeials. Many heoies exis fo each maeial caegoy, u he mos common will e discussed hee. 4.1 Bile Maeials Fo ile maeials, failue is specified y facue. The wo mos common heoies of failue fo ile maeials ae descied elow. The maximum sess heoy should e used if he maeial has simila ehavio in ension and compession. If he maeial ehavio is diffeen in ension and compession, hen Moh s failue cieion should e used Maximum Sess Theoy The maximum sess heoy is ased on he assumpion ha failue occus when he maximum pincipal sess eaches a limi value such as ulimae sengh. 1 ul ul Equaion 30 whee 1 and ae he pincipal sesses and ul is he ulimae sengh. Equaion 30 is shown gaphically in Figue 11 (a). If he sess a a poin ( ) o on he ounday, he maeial will facue. 1, falls ouside he shaded egion Copyigh 018 James Doane Page 6 of 31

27 Figue 11 Bile maeial heoies (a) Maximum sess heoy () Moh s failue cieion 4.1. Moh s Failue Cieion If he maeial exhiis diffeen popeies in ension and compession, a modified plo will exis as shown in Figue 11 (). The only diffeence is ha he ulimae sengh in ension is no he same as ha in compession. The same pinciple applies, ha sess inside he shaded egion is consideed safe. 4. Ducile Maeials Fo ducile maeials, failue is specified y he iniiaion of yielding. Failue will occu due o shea sess. The wo mos common heoies fo ducile maeials ae descied elow, wih he maximum shea heoy eing moe consevaive. Copyigh 018 James Doane Page 7 of 31

28 4..1 Maximum Shea Theoy The maximum shea heoy assumes ha yielding will egin when he maximum sheaing sess is equal o he ha of simple ension. The maximum shea sess can e deemined fom equaions of Moh s cicle o e one half he diffeence eween he pincipal sesses. τ max max min Equaion 31 Le s look a he case of a hick-walled pessue vessel wih inenal pessue only and z is equal o zeo. The maximum shea sess will occu a he inne suface and will have a magniude of τ max ( ) ( ) max max Equaion 3 Fom Equaion and Equaion 3, noing ha angenial sess is posiive and adial sess is negaive, he maximum shea sess ecomes τ max a + a a p a i p i Maximum Shea Sess: Inenal Pessue Only τ p i a max Equaion 33 Based on he idea ha ducile maeials fail in shea, he maximum shea heoy of failue can e wien as Copyigh 018 James Doane Page 8 of 31

29 1 Y If 1 and have he same sign Y 1 If 1 and have he opposie signs Y Equaion 34 whee 1 and ae he pincipal sesses and Y is he yield sengh. Equaion 34 is shown gaphically in Figue 1 as he solid line shaded egion. If he sess a a poin (, 1 ) falls ouside he shaded egion o on he ounday, he maeial will yield. Figue 1 Ducile maeial heoies: Maximum shea heoy (solid line shaded egion) and Von Mises yield heoy (dashed line) Copyigh 018 James Doane Page 9 of 31

30 4.. Von Mises Yield Theoy The las failue heoy is ased on disoional enegy. The Von Mises yield heoy is fomulaed ased on disoions caused y sain enegy. Wihou poof, he heoy saes ha failue occus when + Equaion Y Equaion 35 is shown gaphically in Figue 1 as he dashed line ellipical cuve egion. If he falls ouside he shaded egion o on he ounday, he maeial will sess a a poin ( ) 1, yield. I can e seen ha he maximum shea heoy is moe consevaive han he Von Mises yield heoy. Example 8 A hick-walled pessue vessel has an inne adius of 5 inches and an oue adius of 6 inches. The vessel is sujeced o an inenal pessue of 3500 psi. Based on he Von Mises yield heoy, deemine if failue occus using a maeial yield sengh of 36 ksi. Soluion: The maximum adial sess and maximum angenial sess oh occu a he inne adius, so ha will e he locaion of he maximum Von Mises sess. Fom Equaion Fom Equaion 3 ( ) p max i 3500 in l Copyigh 018 James Doane Page 30 of 31

31 a + a 5 6 ( ) p ( 3500) 19, 409 max i Because he sess elemen on he inne suface will no have any shea sess loading, he adial and angenial sess values will also e he pincipal sess values. l 1 3,500 in l 19,409 in Based on Equaion 35, failue occus if Y ( 3500 ) ( 3500)( 19409) < Theefoe, failue does no occu ased on he Von Mises yield heoy l in Copyigh 018 James Doane Page 31 of 31

MECHANICS OF MATERIALS Poisson s Ratio

MECHANICS OF MATERIALS Poisson s Ratio Fouh diion MCHANICS OF MATRIALS Poisson s Raio Bee Johnson DeWolf Fo a slende ba subjeced o aial loading: 0 The elongaion in he -diecion is accompanied b a conacion in he ohe diecions. Assuming ha he maeial