Miller, C.D. Shell Structures Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999

Transcription

1 Miller, C.D. Shell Srucures Srucural Engineering Handbook Ed. Chen Wai-Fah Boca Raon: CRC Press LLC, 1999

2 Shell Srucures Clarence D. Miller Consuling Engineer, Bloomingon, IN 11.1 Inroducion Overview ProducionPracice Scope Limiaions Sress ComponensforSabiliyAnalysisandDesign Maerials Geomeries,FailureModes,andLoads BucklingDesignMehod SressFacor Nomenclaure 11.2 AllowableCompressiveSressesforCylindricalShells Uniform Axial Compression Axial Compression Due o BendingMomen ExernalPressure Shear SizingofRings (GeneralInsabiliy) 11.3 AllowableCompressiveSressesForCones UniformAxialCompressionandAxialCompression Due o Bending Exernal Pressure Shear Local Siffener Buckling 11.4 AllowableSressEquaionsForCombinedLoads For Combinaion of Uniform Axial Compression and Hoop Compression For Combinaion of Axial Compression Due obendingmomen,m,andhoopcompression ForCombinaionofHoopCompressionandShear ForCombinaion of Uniform Axial Compression, Axial Compression Due o Bending Momen, M, and Shear, in he Presence of Hoop Compression,(f h 0) ForCombinaionofUniformAxial Compression, Axial Compression Due o Bending Momen, M,andShear,inheAbsenceofHoopCompression,(f h =0) 11.5 TolerancesforCylindricalandConicalShells Shells Subjeced o Uniform Axial Compression and Axial Compression Due o Bending Momen Shells Subjeced o ExernalPressure ShellsSubjecedoShear 11.6 AllowableCompressiveSresses SphericalShells ToroidalandEllipsoidalHeads 11.7 TolerancesforFormedHeads References FurherReading 11.1 Inroducion Overview Manyseelsrucures,suchaselevaedwaeranks,oilandwaersorageanks,offshoresrucures, andpressurevessels,arecomprisedofshellelemenshaaresubjecedocompressionsresses. The shell elemens are subjec o insabiliy resuling from he applied loads. The heoreical buckling srenghbasedonlinearelasicbifurcaionanalysisiswellknownforsiffenedaswellasunsiffened cylindrical and conical shells and unsiffened spherical and orispherical shells. Simple formulas c 1999byCRCPressLLC

3 have been deermined for many geomeries and ypes of loads. Iniial geomeric imperfecions and residual sresses ha resul from he fabricaion process, however, reduce he buckling srengh of fabricaed shells. The amoun of reducion is dependen on he geomery of he shell, ype of loading (axial compression, bending, exernal pressure, ec.), size of imperfecions, and maerial properies Producion Pracice The behavior of a cylindrical shell is influenced o some exen by wheher i is manufacured in a pipe or ubing mill or fabricaed from plae maerial. The wo mehods of producion will be referred o as manufacured cylinders and fabricaed cylinders. The disincion is imporan primarily because of he differences in geomeric imperfecions and residual sress levels ha may resul from he wo differen producion pracices. In general, fabricaed cylinders may be expeced o have considerably larger magniudes of imperfecions (in ou-of-roundness and lack of sraighness) han he mill manufacured producs. Similarly, fabricaed heads are likely o have larger shape imperfecions han hose produced by spinning. Spun heads, however, ypically have a greaer variaion in hickness and greaer residual sresses due o he cold working. The design rules given in his chaper apply o fabricaed seel shells. Fabricaed shells are produced from fla plaes by rolling or pressing he plaes o he desired shape and welding he edges ogeher. Because of he mehod of consrucion, he mechanical properies of he shells will vary along he lengh and around he circumference. Misfi of he edges o be welded ogeher may resul in uninenional eccenriciies a he joins. In addiion, welding ends o inroduce ou-of-roundness and ou-of-sraighness imperfecions ha mus be aken ino accoun in he design rules Scope Rules are given for deermining he allowable compressive sresses for unsiffened and ring siffened circular cylinders and cones and unsiffened spherical, ellipsoidal, and orispherical heads. The allowable sress equaions are based on heoreical buckling equaions ha have been reduced by knockdown facors and by plasiciy reducion facors ha were deermined from ess on fabricaed shells. The research leading o he developmen of he allowable sress equaions is given in [2, 7, 8, 9, 10]. Allowable compressive sress equaions are presened for cylinders and cones subjeced o uniform axial compression, bending momen applied over he enire cross-secion, exernal pressure, loads ha produce in-plane shear sresses, and combinaions of hese loads. Allowable compressive sress equaions are presened for formed heads ha are subjeced o loads ha produce unequal biaxial sresses as well as equal biaxial sresses Limiaions The allowable sress equaions are based on an assumed axisymmeric shell wih uniform hickness for unsiffened cylinders and formed heads and wih uniform hickness beween rings for siffened cylinders and cones. All shell peneraions mus be properly reinforced. The resuls of ess on reinforced openings and some design guidance are given in [6]. The sabiliy crieria of his chaper may be used for cylinders ha are reinforced in accordance wih he recommendaions of his reference if he openings do no exceed 10% of he cylinder diameer or 80% of he ring spacing. Special consideraion mus be given o he effecs of larger peneraions. The proposed rules are applicable o shells wih D/ raios up o 2000 and shell hicknesses of 3/16 in. or greaer. The deviaions from rue circular shape and sraighness mus saisfy he requiremens saed in his chaper.

4 Special consideraion mus be given o ends of members or areas of load applicaion where sress disribuion may be nonlinear and localized sresses may exceed hose prediced by linear heory. When he localized sresses exend over a disance equal o one half he wave lengh of he buckling mode, hey should be considered as a uniform sress around he full circumference. Addiional hickness or siffening may be required. Failure due o maerial fracure or faigue and failures caused by dens resuling from accidenal loads are no considered. The rules do no apply o emperaures where creep may occur Sress Componens for Sabiliy Analysis and Design The inernal sress field ha conrols he buckling of a cylindrical shell consiss of he longiudinal, circumferenial, and in-plane shear membrane sresses. The sresses resuling from a dynamic analysis should be reaed as equivalen saic sresses Maerials Seel The allowable sress equaions apply direcly o shells fabricaed from carbon and low alloy seel plae maerials such as hose given in Table 11.1 or Table UCS-23 of [3]. The seel maerials in Table 11.1 are designaed by group and class. Seels are grouped according o srengh level and welding characerisics. Group I designaes mild seels wih specified minimum yield sresses 40 ksi and hese seels may be welded by any of he processes as described in [5]. Group II designaes inermediae srengh seels wih specified minimum yield sresses > 40 ksi and 52 ksi. These seels require he use of low hydrogen welding processes. Group III designaes high srengh seels wih specified minimum yield sresses > 52 ksi. These seels may be used provided ha each applicaion is invesigaed wih respec o weldabiliy and special welding procedures ha may be required. Consideraion should be given o faigue problems ha may resul from he use of higher working sresses, and noch oughness in relaion o oher elemens of fracure conrol such as fabricaion, inspecion procedures, service sress, and emperaure environmen. TheseelsinTable11.1 have been classified according o heir noch oughness characerisics. Class C seels are hose ha have a hisory of successful applicaion in welded srucures a service emperaures above freezing. Impac ess are no specified. Class B seels are suiable for use where hickness, cold work, resrain, sress concenraion, and impac loading indicae he need for improved noch oughness. When impac ess are specified, Class B seels should exhibi Charpy V-noch energy of 15 f-lbs for Group 1 and 25 f-lbs for Group II a he lowes service emperaure. TheClassBseelsgiveninTable11.1 can generally mee he Charpy requiremens a emperaures ranging from 50 o 32 F. Class A seels are suiable for use a subfreezing emperaures and for criical applicaions involving adverse combinaions of he facors cied above. The seels given in Table 11.1 can generally mee he Charpy requiremens for Class B seels a emperaures ranging from 4 o 40 F. Oher Maerials The design equaions can also be applied o oher maerials for which a char or able is provided in Subpar 3 of [4] by subsiuing he angen modulus E for he elasic modulus E in he allowable sress equaions. The mehod for finding he allowable sresses for shells consruced from hese maerials is deermined by he following procedure.

5 TABLE 11.1 Seel Plae Maerials Specified Specified minimum minimum yield sress ensile sress Group Class Specificaion (ksi) a (ksi) a I C ASTM A36 (o 2 in. hick) ASTM A131 Grade A (o 1/2 in. hick) ASTM A285 Grade C (o 3/4 in. hick) I B ASTM A131 Grades B, D ASTM A516 Grade ASTM A573 Grade ASTM A709 Grade 36T I A ASTM A131 Grades CS, E II C ASTM A572 Grade 42 (o 2 in. hick) ASTM A591 required over 1/2 in. hick ASTM A572 Grade 50 (o 2 in. hick) ASTM A591 required over 1/2 in. hick II B ASTM A709 Grades 50T2, 50T ASTM A131 Grade AH ASTM A131 Grade AH II A API Spec 2H Grade API Spec 2H Grade 50 (o 2 1/2 in. hick) API Spec 2H Grade 50 (over 2 1/2 in. hick) API Spec 2W Grade 42 (o 1 in. hick) API Spec 2W Grade 42 (over 1 in. hick) API Spec 2W Grade 50 (o 1 in. hick) API Spec 2W Grade 50 (over 1 in. hick) API Spec 2W Grade 50T (o 1 in. hick) API Spec 2W Grade 50T (over 1 in. hick) API Spec 2Y Grade 42 (o 1 in. hick) API Spec 2Y Grade 42 (over 1 in. hick) API Spec 2Y Grade 50 (o 1 in. hick) API Spec 2Y Grade 50 (over 1 in. hick) API Spec 2Y Grade 50T (o 1 in. hick) API Spec 2Y Grade 50T (over 1 in. hick) ASTM A131 Grades DH32, EH ASTM A131 Grades DH36, EH ASTM A537 Class I (o 2 1/2 in. hick) ASTM A633 Grade A ASTM A633 Grades C, D ASTM A678 Grade A III A ASTM A537 Class II (o 2 1/2 in. hick) ASTM A678 Grade B API Spec 2W Grade 60 (o 1 in. hick) API Spec 2W Grade 60 (over 1 in. hick) ASTM A710 Grade A Class 3 (o 2 in. hick) ASTM A710 Grade A Class 3 (2 in. o 4 in. hick) ASTM A710 Grade A Class 3 (over 4 in. hick) a 1 ksi = MPa Sep 1. Calculae he value of facor A using he following equaions. The erms F xe,f he, and F ve are defined in subsequen paragraphs. A = F xe E A = F he E A = F ve E Sep 2. Using he value of A calculaed in Sep 1, ener he applicable maerial char in Subpar 3 of[4] for he maerial under consideraion. Move verically o an inersecion wih he maerial emperaure line for he design emperaure. Use inerpolaion for inermediae emperaure values. Sep 3. From he inersecion obained in Sep 2, move horizonally o he righ o obain he value of B. E is given by he following equaion: E = 2B A When values of A fall o he lef of he applicable maerial/emperaure line in Sep 2, E = E.

6 Sep 4. Calculae he allowable sresses from he following equaions: F xa = F xe E FS E F ba = F xa F ha = F he E FS E F va = F ve E FS E Geomeries, Failure Modes, and Loads Allowable sress equaions are given for he following geomeries and load condiions. Geomeries 1. Unsiffened Cylindrical, Conical, and Spherical Shells 2. Ring Siffened Cylindrical and Conical Shells 3. Unsiffened Spherical, Ellipsoidal, and Torispherical Heads The cylinder and cone geomeries are illusraed in Figures 11.1 and 11.3 and he siffener geomeries are illusraed in Figure The effecive secions for ring siffeners are shown in Figure The maximum cone angle α shall no exceed 60. FIGURE 11.1: Geomery of cylinders.

7 FIGURE 11.2: Secions hrough rings. FIGURE 11.3: Geomery of conical secions. Failure Modes Buckling sress equaions are given herein for four failure modes ha are defined below. The buckling paerns are boh load and geomery dependen.

8 FIGURE 11.4: Siffener geomery. 1. Local Shell Buckling This mode of failure is characerized by he buckling of he shell in a radial direcion. One or more waves will form in he longiudinal and circumferenial direcions. The number of waves and he shape of he waves are dependen on he geomery of he shell and he ype of load applied. For ring siffened shells, he siffening rings are presumed o remain round prior o buckling. 2. General Insabiliy This mode of failure is characerized by he buckling of one or more rings ogeher wih he shell ino a circumferenial wave paern wih wo or more waves. 3. Column Buckling This mode of failure is characerized by ou-of-plane buckling of he cylinder wih he shell remaining circular prior o column buckling. The ineracion beween shell buckling and column buckling is aken ino accoun by subsiuing he shell buckling sress for he yield sress in he column buckling formula. 4. Local Buckling of Rings This mode of failure relaes o he buckling of he siffener elemens such as he web and flange of a ee ype siffener. Mos design rules specify requiremens for compac secions o preclude his mode of failure. Very lile analyical or experimenal work has been done for his mode of failure in associaion wih shell buckling. Loads and Load Combinaions Allowable sress equaions are given for he following ypes of sresses. a. Cylinders and Cones 1. Uniform longiudinal compressive sresses 2. Longiudinal compressive sresses due o a bending momen acing across he full circular cross-secion 3. Circumferenial compressive sresses due o exernal pressure or oher applied loads 4. In-plane shear sresses 5. Any combinaion of 1, 2, 3, and 4 b. Spherical Shells and Formed Heads 1. Equal biaxial sresses boh sresses are compressive 2. Unequal biaxial sresses boh sresses are compressive 3. Unequal biaxial sresses one sress is ensile and he oher is compressive

9 Buckling Design Mehod The buckling srengh formulaions presened in his repor are based on classical linear heory which is modified by reducion facors ha accoun for he effecs of imperfecions, boundary condiions, nonlineariy of maerial properies, and residual sresses. The reducion facors are deermined from approximae lower bound values of es daa of shells wih iniial imperfecions represenaive of he olerance limis specified in his chaper. The validaion of he knockdown facors is given in [7], [8], [9], and [10] Sress Facor The allowable sresses are deermined by applying a sress facor, FS, o he prediced buckling sresses. The recommended values of FS are 2.0 when he buckling sress is elasic and 5/3 when he buckling sress equals he yield sress. A linear variaion shall be used beween hese limis. The equaions for FS are given below. FS = 2.0 if F ic 0.55 (11.1a) FS = F ic / if 0.55 <F ic < (11.1b) FS = if F ic = (11.1c) F ic is he prediced buckling sress, which is deermined by leing FS = 1 in he allowable sress equaions. For combinaions of earhquake load or wind load wih oher loads, he allowable sresses may be increased by a facor of 4/ Nomenclaure Noe: The erms no defined here are uniquely defined in he secions in which hey are firs used. A = cross-secional area of cylinder A = π(d o ), in. 2 A S = cross-secional area of a ring siffener, in. 2 A F = cross-secional area of a large ring siffener which acs as a bulkhead, in. 2 D i = inside diameer of cylinder, in. D o = ouside diameer of cylinder, in. D L = ouside diameer a large end of cone, in. D S = ouside diameer a small end of cone, in. E = modulus of elasiciy of maerial a design emperaure, ksi E = angen modulus of maerial a design emperaure, ksi f a = axial compressive membrane sress resuling from applied axial load, Q, ksi f b = axial compressive membrane sress resuling from applied bending momen, M, ksi f h = hoop compressive membrane sress resuling from applied exernal pressure, P, ksi f q = axial compressive membrane sress resuling from pressure load, Q p,onheendofa cylinder, ksi. f v = shear sress from applied loads, ksi f x = f a + f q, ksi F ba = allowable axial compressive membrane sress of a cylinder due o bending momen, M,in he absence of oher loads, ksi F ca = allowable compressive membrane sress of a cylinder due o axial compression load wih λ c > 0.15, ksi F bha = allowable axial compressive membrane sress of a cylinder due o bending in he presence of hoop compression, ksi

10 F hba = allowable hoop compressive membrane sress of a cylinder in he presence of longiudinal compression due o a bending momen, ksi F he = elasic hoop compressive membrane failure sress of a cylinder or formed head under exernal pressure alone, ksi F ha = allowable hoop compressive membrane sress of a cylinder or formed head under exernal pressure alone, ksi F hva = F hxa = allowable hoop compressive membrane sress in he presence of shear sress, ksi allowable hoop compressive membrane sress of a cylinder in he presence of axial compression, ksi F a = allowable ension sress, ksi F va = allowable shear sress of a cylinder subjeced only o shear sress, ksi F ve = elasic shear buckling sress of a cylinder subjeced only o shear sress, ksi F vha = allowable shear sress of a cylinder subjeced o shear sress in he presence of hoop compression, ksi F xa = allowable compressive membrane sress of a cylinder due o axial compression load wih λ c 0.15, ksi F xc = inelasic axial compressive membrane failure (local buckling) sress of a cylinder in he absence of oher loads, ksi F xe = elasic axial compressive membrane failure (local buckling) sress of a cylinder in he absence of oher loads, ksi F xha = allowable axial compressive membrane sress of a cylinder in he presence of hoop compression, ksi = minimum specified yield sress of maerial, ksi F u = minimum specified ensile sress of maerial, ksi FS = sress facor I s = momen of ineria of ring siffener plus effecive lengh of shell abou cenroidal axis of combined secion, in. 4 I s = I s + A s Z 2 s L e A s + L e + L e 3 12 K = effecive lengh facor for column buckling I s = momen of ineria of ring siffener abou is cenroidal axis, in. 4 L = design lengh of a vessel secion beween lines of suppor, in. A line of suppor is: 1. a circumferenial line on a head (excluding conical heads) a one-hird he deph of he head from he head angen line as shown in Figure a siffening ring ha mees he requiremens of Equaion L B = lengh of cylinder beween bulkheads or large rings designed o ac as bulkheads, in. L c = unbraced lengh of member, in. L e = effecive lengh of shell, in. (see Figure 11.2) L F = one-half of he sum of he disances, L B, from he cener line of a large ring o he nex large ring or head line of suppor on eiher side of he large ring, in. (see Figure 11.1) L s = one-half of he sum of he disances from he cener line of a siffening ring o he nex line of suppor on eiher side of he ring, measured parallel o he axis of he cylinder, in. A line of suppor is described in he definiion for L (see Figure 11.1). L = overall lengh of vessel as shown in Figure 11.1, in. M = applied bending momen across he vessel cross-secion, in.-kips M s = L s / R o M x = L/ R o P = applied exernal pressure, ksi P a = allowable exernal pressure in he absence of oher loads, ksi

11 Q = applied axial compression load, kips Q p = axial compression load on end of cylinder resuling from applied exernal pressure, kips R = radius o cenerline of shell, in. R c = radius o cenroid of combined ring siffener and effecive lengh of shell, in. R c = R + Z c R o = radius o ouside of shell, in. = hickness of shell, less corrosion allowance, in. c = hickness of cone, less corrosion allowance, in. Z c = radial disance from cenerline of shell o cenroid of combined secion of ring and effecive lengh of shell, in. Z c = A sz s A s +L e Z s = radial disance from cener line of shell o cenroid of ring siffener (posiive for ouside rings), in. S = elasic secion modulus of full shell cross-secion, in. 3 r = radius of gyraion of cylinder, in. λ c = slenderness facor S = π ( Do 4 ) D4 i 32D o r = λ c = KL c πr ( D 2 o + Di 2 ) 1/2 4 ( Fxa FS E ) 1/ Allowable Compressive Sresses for Cylindrical Shells The maximum allowable sresses for cylindrical shells subjeced o loads ha produce compressive sresses are given by he following equaions Uniform Axial Compression Allowable longiudinal sress for a cylindrical shell under uniform axial compression is given by F xa for values of λ c 0.15 and by F ca for values of λ c > F xa is he smaller of he values given by Equaions 11.3 and Equaion λ c = KL c πr ( ) Fxa FS 1/2 (11.2) where KL c is he effecive lengh. L c is he unbraced lengh. Recommended values for K [1] are 2.1 for members wih one end free and he oher end fixed, 1.0 for members wih boh ends pinned, 0.8 for members wih one end pinned and he oher end fixed, and 0.65 for members wih boh ends fixed. Local Buckling (For λ c 0.15) E

12 135 F xa = FS for D o 466 F xa = ( ) D o FS F xa = 0.5 FS for D o for 135 < D o 600 (11.3a) < 600 (11.3b) (11.3c) or F xa = F xe FS (11.4) where F xe = C xe D o (11.5) C x = 409 c D o no o exceed 0.9 for D o C x = 0.25 c for D o 1247 c = 2.64 for M x 1.5 c = 3.13 for 1.5 <M x < 15 Mx 0.42 c = 1.0 for M x 15 < 1247 M x = L (R o ) 1/2 (11.6) Column Buckling (For λ c > 0.15) F ca = F xa for λ c 0.15 (11.7a) F ca = F xa [ (λ c 0.15)] 0.3 for 0.15 <λ c < 2 (11.7b) F ca = 0.88F xa λ 2 c for λ c 2 (11.7c) Axial Compression Due o Bending Momen Allowable longiudinal sress for a cylinder subjeced o a bending momen acing across he full circular cross-secion is given by F ba.

13 F ba = F xa for D o F ba = FS F ba = FS ( D o ) for 100 for D o F ba = ( γ) FS D o (11.8a) < 135 (11.8b) < 100 and γ 0.11 (11.8c) for D o < 100 and γ < 0.11 (11.8d) where F xa is he smaller of he values given by Equaions 11.3 and 11.4 and γ = D o E Exernal Pressure The allowable circumferenial compressive sress for a cylinder under exernal pressure is given by F ha and he allowable exernal pressure is given by he following equaions: P a = 2F ha (11.9) D o F ha = FS for F he F ha = 0.7 FS (11.10a) ( ) 0.4 Fhe for < F he < (11.10b) F ha = F he FS for F he (11.10c) where F he = 1.6C he D o (11.11) C h = 0.55 D o for M x 2 ( ) 0.94 Do C h = 1.12M x for 13 <M x < C h = M x for 1.5 <M x 13 C h = 1.0 for M x 1.5 ( ) 0.94 Do Shear Allowable in-plane shear sress for a cylindrical shell is given by F va. F va = η vf ve FS (11.12)

14 where F ve = α vc v E D o (11.13) C v = for M x 1.5 (11.14a) ( ) 9.64 ( ) 1/2 C v = Mx 3 for 1.5 <Mx < 26 (11.14b) M 2 x C v = Mx 1/2 for 26 M x < D o ( ) 1/2 C v = for M x D o D o (11.14c) (11.14d) α v = 0.8 for D o 500 ( ) Do α v = log 10 η v = 1.0 for F ve 0.48 for 500 < D o η v = 0.43 F ve for 0.48 < F ve < 1.7 η v = 0.6 F ve for F ve Sizing of Rings (General Insabiliy) Uniform Axial Compression and Axial Compression Due o Bending When ring siffeners are used o increase he allowable longiudinal compressive sress, he following equaions mus be saisfied. If M x 15, siffener spacing is oo large o be effecive. A s [ M 0.6 s ] L s and A s 0.06L s (11.15) also I s 5.33L s 3 M 1.8 s (11.16) Exernal Pressure (a) Small Rings F he = sress deermined from Equaion wih M x = M s. I s 1.5F hel s R 2 c E ( n 2 1 ) (11.17) n 2 = 2D3/2 o 3L B 1/2 and 4 n2 100

15 (b) Large Rings Which Ac As Bulkheads I s I F where I F = F hef L F R 2 c 2E (11.18) I F = he value of I s which makes a large siffener ac as a bulkhead. The effecive lengh of shell is L e = 1.1 D o (A 1 /A 2 ) A 1 = cross-secional area of small ring plus shell area equal o L s, in. 2 A 2 = cross-secional area of large ring plus shell area equal o L s, in. 2 R c = radius o cenroid of combined large ring and effecive widh of shell, in. F hef = average value of he hoop buckling sresses, F he, over lengh L F where F he is deermined from Equaion 11.11, ksi Shear C v = value deermined from Equaion wih M x = M s. Local Siffener Buckling I s 0.184C vm 0.8 s 3 L s (11.19) To preclude local buckling of he siffener prior o shell buckling, he following siffener properies shall be me. See Figure11.4 for siffener geomery. (a) Fla Bar Siffener, Flange of a Tee Siffener, and Ousanding Leg of an Angle Siffener ( ) h 1 E 1/ (11.20) 1 where h 1 is he full widh of a fla bar siffener or ousanding leg of an angle siffener and one-half of he full widh of he flange of a ee siffener and 1 is he hickness of he bar, leg of angle, or flange of ee. (b) Web of Tee Siffener or Leg of Angle Siffener Aached o Shell ( ) h 2 E 1/2 1.0 (11.21) 2 where h 2 is he full deph of a ee secion or full widh of an angle leg and 2 is he hickness of he web or angle leg Allowable Compressive Sresses For Cones Unsiffened conical ransiions or cone secions beween rings of siffened cones wih an angle α 60 shall be designed for local buckling as an equivalen cylinder according o he following procedure. See Figure11.3 for cone geomery Uniform Axial Compression and Axial Compression Due o Bending Allowable Longiudinal and Bending Sresses Assume an equivalen cylinder wih diameer D e = D/ cos α,whered is he ouside diameer of he cone a he cross-secion under consideraion and lengh equal o L c. D e is subsiued for

16 D o in Equaions 11.3 o Equaions 11.8 o find F xa and F ba and L c for L in Equaion The allowable sress mus be saisfied a all cross-secions along he lengh of he cone. Unsiffened Cone-Cylinder Juncions Cone-cylinder juncions are subjec o unbalanced radial forces (due o axial load and bending momen) and o localized bending sresses caused by he angle change. The longiudinal and hoop sresses a he juncion may be evaluaed as follows: Longiudinal Sress In lieu of deailed analysis, he localized bending sress a an unsiffened cone-cylinder juncion may be esimaed by he following equaion. f b = 0.6 D ( + c ) 2 e (f x + f b ) an α (11.22) where D = ouside diameer of cylinder a juncion o cone = hickness of cylinder c = hickness of cone e = o find sress in cylinder secion e = c o find sress in cone secion α = cone angle as defined in Figure 11.3 f x = uniform longiudinal sress in cylinder secion a juncion resuling from axial loads f b = longiudinal sress in cylinder secion a juncion resuling from bending momen For srengh requiremens, he oal sress (f x + f b + f b ) shall be limied o he minimum ensile srengh given in Table 11.1 ortableu,subpar1of[4] for he cone and cylinder maerial and f x +f b shall be less han he allowable ensile sress F,whereF is he smaller of 0.6 or F u /3. Hoop Sress The hoop sress caused by he unbalanced radial line load may be esimaed from: f h = 0.45 D/ (f x + f b ) an α (11.23) For hoop ension, f h shall be limied o he ensile allowable. For hoop compression, f h shall be limied o F ha where F ha is compued from Equaion wih F he = 0.4E/D. A cone-cylinder juncion ha does no saisfy he above crieria may be srenghened eiher by increasing he cylinder and cone wall hicknesses a he juncion, or by providing a siffening ring a he juncion. Cone-Cylinder Juncion Rings If siffening rings are required, he secion properies shall saisfy he following requiremens: A c D (f x + f b ) an α (11.24) I c DD2 c 8E (f x + f b ) an α (11.25) where D = cylinder ouside diameer a juncion D c = diameer o cenroid of composie ring secion for exernal rings D c = D for inernal rings A c = cross-secional area of composie ring secion I c = momen of ineria of composie ring secion

17 In compuing A c and I c he effecive lengh of he shell wall acing as a flange for he composie ring secion shall be compued from: ( ) b e = 0.55 D/ + Dc / cos α (11.26) Exernal Pressure Allowable Circumferenial Compression Sresses Assume an equivalen cylinder wih diameer D e = 0.5(D L +D S ) and lengh L e = L c / cos α. This lengh and diameer shall be subsiued ino Equaions and o deermine F ha. Inermediae Siffening Rings If required, circumferenial siffening rings wihin cone ransiions shall be sized using Equaion wih R c = D/2 where D is he cone diameer a he ring, is he cone hickness, L s is he average disance o adjacen rings along he cone axis, and F he is he average of he elasic hoop buckling sress values compued for he wo adjacen bays by he mehod given in he preceding paragraph. Cone-Cylinder Juncion Rings A juncion ring is no required for buckling due o exernal pressure if f h <F ha where F ha is deermined from Equaion wih F he compued using C h equal o 0.55 (cos α)(/d) in Equaion D is he cylinder diameer a he juncion. Circumferenial siffening rings required a he cone-cylinder juncions shall be sized such ha he momen of ineria of he composie ring secion saisfies he following equaion: { I c D2 L 1 F he + } cl c F hec 16E cos 2 (11.27) α where D = cylinder ouside diameer a juncion L c = disance o firs siffening ring in cone secion along cone axis L 1 = disance o firs siffening ring in cylinder secion or line of suppor F he = elasic hoop buckling sress for cylinder (see Equaion 11.11) F hec = F he for cone secion reaed as an equivalen cylinder = cylinder hickness c = cone hickness Shear Allowable In-Plane Shear Sress Assume an equivalen cylinder wih a lengh equal o he slan lengh of he cone beween rings (L c / cos α) and a diameer D e = D/ cos α,whered is he ouside diameer of he cone a he crosssecion under consideraion. This lengh and diameer shall be subsiued ino Equaions o o deermine F va. Inermediae Siffening Rings If required, circumferenial siffening rings wihin cone ransiion shall be sized using Equaion where L s is he average disance o adjacen rings along he cone axis.

18 Local Siffener Buckling To preclude local buckling of a siffener, he requiremens of Equaions and mus be me Allowable Sress Equaions For Unsiffened and Ring-Siffened Cylinders and Cones Under Combined Loads For Combinaion of Uniform Axial Compression and Hoop Compression For λ c 0.15 The allowable sress in he longiudinal direcion is given by F xha and he allowable sress in he circumferenial direcion is given by F hxa. ( ) C 1 F xha = Fxa C 2 F xa F ha C2 2F ha 2 (11.28) where C 1 = F xa FS + F ha FS f x = f a + f q = Q A + Q p A 1.0 and C 2 = f x f h and f h = PD o 2 F xa FS is given by he smaller of Equaion 11.3 or 11.4, and F ha FS is given by Equaion F hxa = F xha C 2 (11.29) For 0.15 <λ c < 1.2 F xha is he smaller of F ah1 and F ah2 where F ah1 = F xha given by Equaion wih f x = f a and F ah2 is given by he following equaion. ( F ah2 = F ca 1 f ) q (11.30) F ca is given by Equaion For Combinaion of Axial Compression Due o Bending Momen, M, and Hoop Compression The allowable sress in he longiudinal direcion is given by F bha and he allowable sress in he circumferenial direcion is given by F hba. F bha = C 3 C 4 F ba (11.31) where C 3 and C 4 are given by he following equaions and F ba is given by Equaion C 4 = f b F ha f h C 2 3 F ba ( C C 4 f b = M S ) + C 2n 3 1 = 0 (11.32) f h = PD o 2 n = 5 4 F ha FS

19 Solve for C3 from Equaion by ieraion. F ha FS is given by Equaion F hba = F bha f h f b (11.33) For Combinaion of Hoop Compression and Shear The allowable shear sress is given by F vha and he allowable circumferenial sress is given by F hva. [ ( F 2 ) 2 ] 1/2 F vha = va + Fva 2 F va 2 (11.34) 2C 5 F ha 2C 5 F ha where C 5 = f v f h and F va is given by Equaion and F ha is given by Equaion F hva = F vha C 5 (11.35) For Combinaion of Uniform Axial Compression, Axial Compression Due o Bending Momen, M, and Shear, in he Presence of Hoop Compression, (f h = 0) For λ c 0.15 Le K s = 1 ( fv F va ) 2 (11.36) ( fa ) f b 1.0 (11.37) K s F xha K s F bha F xha is given by Equaion 11.28, F bha is given by Equaion and F va is given by Equaion For 0.15 <λ c < 1.2 f a + 8 f b 1.0 for F xha 9 F bha f a F xha 0.2 (11.38) where C m = F e = π 2 E 1 f a FS/F e (KL c /r) 2 See Equaion 11.2 for KL c and Equaion for F xha. F bha is given by Equaion FS is deermined from Equaion 11.1 where F ic = F xa FS (see Equaions 11.3 and 11.4). C m is a coefficien whose value shall be aken as follows [1]: 1. For compression members in frames subjec o join ranslaion (sidesway), C m = For roaionally resrained compression members in frames braced agains join ranslaion and no subjec o ransverse loading beween heir suppors in he plane of bending, C m = (M 1 /M 2 ) where M 1 /M 2 is he raio of he smaller o larger momens a he ends of ha porion of he member ha is unbraced in he plane of bending under consideraion. M 1 /M 2 is posiive when he member is ben in reverse curvaure and negaive when ben in single curvaure.

20 3. For compression members in frames braced agains join ranslaion and subjeced o ransverse loading beween heir suppors he following apply: a. for members whose ends are resrained agains roaion in he plane of bending, C m = 0.85 b. for members whose ends are unresrained agains roaion in he plane of bending, C m = For Combinaion of Uniform Axial Compression, Axial Compression Due o Bending Momen, M, and Shear, in he Absence of Hoop Compression, (f h = 0) For λ c 0.15 ( fa K s F xa ) f b K s F ba 1.0 (11.39) F xa is given by he smaller of Equaions 11.3 or 11.4, F ba is given by Equaion 11.8 and K s is given by Equaion For 0.15 <λ c < 1.2 f a + 8 f b f a 1.0 for 0.2 K s F ca 9 K s F ba K s F ca (11.40) f a + f b f a 1.0 for < 0.2 2K s F ca K s F ba K s F ca (11.41) F ca is given by Equaion 11.7, F ba is given by Equaion 11.31, and K s is given by Equaion See Equaion for definiion of Tolerances for Cylindrical and Conical Shells Shells Subjeced o Uniform Axial Compression and Axial Compression Due o Bending Momen The difference beween he maximum and minimum diameers a any cross-secion shall no exceed 1% of he nominal diameer a he cross-secion under consideraion. Addiionally, he local deviaion from a sraigh line, e, measured along a meridian over a gauge lengh L x shall no exceed he maximum permissible deviaion e x given below. e x = 0.002R L x = 4 R bu no greaer han L for cylinders L x = 4 R/ cos α bu no greaer han L c / cos α for cones L x = 25 across circumferenial welds Also L x is no greaer han 95% of he meridianal disance beween circumferenial welds Shells Subjeced o Exernal Pressure The difference beween he maximum and minimum diameers a any cross-secion shall no exceed 1% of he nominal diameer a he cross-secion under consideraion. Addiionally, he maximum

21 deviaion from a rue circular form, e, shall no exceed he value given by Figure 11.5 or by he following equaions. e = (M x ) e R (11.42) FIGURE 11.5: Values of e/ which give a buckling pressure of 80% of he heoreical buckling pressure. Also, e shall no exceed 2. Measuremens o deermine e are made wih a gauge or emplae wih he chord lengh L c given by he following equaion. where L c = 2R sin(π/2n) (11.43) ( ) d R/ n = c 2 n 1.41(R/) 0.5 (11.44) L/R c = 2.28(R/) d = 0.38(R/)

22 Shells Subjeced o Shear The difference beween he maximum and minimum diameers a any cross-secion shall no exceed 1% of he nominal diameer a he cross-secion under consideraion Allowable Compressive Sresses for Spherical Shells and Formed Heads, Wih Pressure on Convex Side Spherical Shells Wih Equal Biaxial Sresses The allowable compressive sress for a spherical shell under uniform exernal pressure is given by F ha and he allowable exernal pressure is given by P a. F ha = FS 1.31 F ha = FS ( F he ) for 1.6 < F ha = 0.18F he FS F ha = F he FS for F he 6.25 (11.45a) F he < 6.25 (11.45b) for 0.55 < F he 1.6 (11.45c) F he for 0.55 (11.45d) F he = 0.075E R o (11.46) P a = 2F ha R o (11.47) where R o is he radius o he ouside of he spherical shell and F ha is given by Equaion Wih Unequal Biaxial Sresses Boh Sresses Are Compressive The allowable compressive sresses for a spherical shell subjeced o unequal biaxial sresses, σ 1 and σ 2, where boh σ 1 and σ 2 are compression sresses resuling from applied loads, are given by he following equaions. F 1a = k F ha (11.48) F 2a = kf 1a (11.49) where k = σ 2 /σ 1 and F ha is given by Equaion F 1a is he allowable sress in he direcion of σ 1 and F 2a is he allowable sress in he direcion of σ 2. The larger of he compression sresses is σ 1. Wih Unequal Biaxial Sresses One Sress Is Compressive and he Oher Is Tensile The allowable compressive sress for a spherical shell subjeced o unequal biaxial sresses σ 1 and σ 2,whereσ 1 is a compression sress and σ 2 is a ensile sress, is given by F 1a where F 1a is he

23 value of F ha deermined from Equaion wih F he given by Equaion F he = ( C o + C p ) E R o (11.50) Shear C o = R o / for R o C o = for R o C p = p p = σ 2 R o E < 622 When shear is presen, he principal sresses shall be calculaed and used for σ 1 and σ Toroidal and Ellipsoidal Heads The allowable compressive sresses for formed heads is deermined by he equaions given for spherical shells where R o is defined below. R o = he ouside radius of he crown porion of he head for orispherical heads, in. R o = he equivalen ouside spherical radius aken as K o D o for ellipsoidal heads, in. K o = facor depending on he ellipsoidal head proporions D o /2h o (see Table 11.2) h o = ouside heigh of he ellipsoidal head measured from he angen line (head-bend line), in. TABLE 11.2 Facor K o D o /2h o K o D o /2h o K o Noe: Use inerpolaion for inermediae values Tolerances for Formed Heads The inner surface of a spherical shell or formed head shall no deviae from he specified shape more han 1.25% of he nominal diameer of he vessel. Such deviaions shall be measured perpendicular o he specified shape. Addiionally, he maximum local deviaion from a rue circular form, e, for spherical shells and any spherical porion of a formed head designed for exernal pressure shall no exceed he shell hickness. Measuremens o deermine e are made wih a gauge or emplae wih he chord lengh L c given by he following equaion: L c = 3.72 R

24 References [1] AISC Manual of Seel Consrucion, Allowable Sress Design, 9h ed., Secion C-C2, American Insiue of Seel Consrucion, Chicago, IL. [2] API 2U API Bullein 2U, Bullein on Sabiliy Design of Cylindrical Shells, 1s ed., American Peroleum Insiue, Washingon, D.C. [3] ASME ASME Boiler and Pressure Vessel Code, Secion VIII, Rules for Consrucion of Pressure Vessels, Division 1, American Sociey of Mechanical Engineers, New York. [4] ASME ASME Boiler and Pressure Vessel Code, Secion II, Maerials, Par D-Properies, American Sociey of Mechanical Engineers, New York. [5] AWS. 1992, Srucural Welding Code, AWS D1.1-92, American Welding Sociey. [6] Miller, C.D Experimenal Sudy of he Buckling of Cylindrical Shells Wih Reinforced Openings, ASME/ANS Nuclear Engineering Conference, Porland, OR. [7] Miller, C.D ASME Code Case N-284: Meal Conainmen Shell Buckling Design Mehods, Revision 1, in Code Cases: Nuclear Componens, ASME Boiler and Pressure Vessel Code, American Sociey of Mechanical Engineers, New York, March 14, [8] Miller, C.D An Evaluaion of Codes and Sandards Relaed o Buckling of Cylindrical Shells Subjeced o Axial Compression, Bending and Exernal Pressure, UMI Disseraion Services, Ann Arbor, MI. [9] Miller, C.D. and Saliklis, E.P Analysis of Cylindrical Shell Daabase and Validaion of Design Formulaions, API Projec 90-56, Chicago Bridge & Iron Technical Services Co., Plainfield, IL. [10] Miller, C.D. and Saliklis, E.P Analysis of Cylindrical Shell Daabase and Validaion of Design Formulaions, Phase 2: For D/ Values > 300, API Projec 92-56, Chicago Bridge & Iron Technical Services Co., Plainfield, IL. Furher Reading Addiional informaion on he design of shell srucures can be found in he following references: [1] American Iron and Seel Insiue, Seel Plae Engineering Daa, Volume 1 Seel Tanks for Liquid Sorage and Volume 2 Useful Informaion on he Design of Plae Srucures. [2] Wozniak, R.S Seel Tanks, in Srucural Engineering Handbook, Gaylord, E.H. and Gaylord, C.S. Eds., 3rd ed., McGraw-Hill, New York, 27-1 o The following is a lis of codes, specificaions, and sandards ha provide rules for he design of shell srucures subjec o insabiliy from loads which produce compressive sresses in he shell elemens. A comparison was made by Miller and Saliklis [8, 9, 10] of he prediced failure sresses given by each of hese ses of rules wih he es daa obained from over 600 ess on seel models represenaive of fabricaed shells. The bes fi equaions were deermined for each shell ype and load. These equaions were hen modified o obain a beer fi wih he es daabase. The equaions given in his chaper are he resuls of hese sudies. [3] API BUL 2U Bullein on Sabiliy Design of Cylindrical Shells, 1s ed., American Peroleum Insiue, Washingon, D.C. [4] API RP 2A-LRFD Recommended Pracice for Planning, Designing and Consrucing Fixed Offshore Plaforms Load and Resisance Facor Design, 1s ed., American Peroleum Insiue, Washingon, D.C. [5] API RP 2A-WSD Recommended Pracice for Planning, Designing and Consrucing Fixed Offshore Plaforms Working Sress Design, 20h ed., American Peroleum Insiue, Washingon, D.C.

25 [6] API STD Design and Consrucion of Large, Welded Low-Pressure Sorage Tanks, 8h ed., American Peroleum Insiue, Washingon, D.C. [7] API STD Welded Seel Tanks for Oil Sorage, 9h ed., American Peroleum Insiue, Washingon, D.C. [8] ASME VIII Pressure Vessels, Division 2, ASME Boiler and Pressure Code, American Sociey of Mechanical Engineers, Washingon, D.C. [9] AWWA D AWWA Sandard for Welded Seel Tanks for Waer Sorage, American Waer Works Associaion, Denver, CO. [10] DIN Sabiliy of Shell Type Seel Srucures, German Code DIN 18800, Par 4. [11] ECCS No Buckling of Seel Shells, European Recommendaions, European Convenion for Consrucional Seelwork, Publicaion No. 56, 4h ed., Brussels, Belgium. [12] NPD Buckling Crieria for Cylindrical Shells, Norwegian Peroleum Direcorae, Oslo, Norway.