Torsional buckling of thin-walled cylinders under circumferentially varying thermal loads

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1 45th AIAA/ASE/ASCE/AHS/ASC Structures Structurl Dynmics & terils Conference 9 - April 004 Plm Springs Cliforni AIAA Torsionl buckling of thin-lled cylinders under circumferentilly vrying therml lods Kr. Chrles * nd Anthony. Ws University of ichign Ann Arbor ichign Thin-lled cylinders used in vriety of engineering pplictions re often subjected to situtions here the pplied loding is not limited to single loding type. The introduction of multiple lods lters the stbility chrcteristics of the system in mnner tht must be understood to mintin structurl integrity nd bide by sfety regultions. The present study investigtes the elstic buckling response of thin-lled cylindricl shells under combintion of torsionl lods nd circumferentilly-vrying therml lods. E L h m n X X τ r T o δ omenclture Young s modulus Poisson s rtio coefficient of therml epnsion il length of cylinder rdius of cylinder thickness of shell ll number of il hlf sine ves number of circumferentil full sine ves in-plne direct stress resultnts in-plne sher stress resultnt in-plne moment resultnts torsionl lod out of plne displcement il cylindricl coordinte circumferentil cylindricl coordinte rdil cylindricl coordinte men therml lod mimum temperture difference () q first derivtive of quntity ith respect to vrible q I. Introduction hin-lled cylinders re used in vriety of engineering pplictions such s irplne fuselges submrines T building construction nd mnufcturing mchinery. Often the pplictions re in situtions here the loding cnnot be limited to single loding type such s il compression torsion or pressure. The introduction of multiple loding lters the stbility chrcteristics of the system in mnner hich must be understood to mintin structurl integrity nd bide by sfety regultions. The buckling behvior of thin-lled cylindricl shells is understood for simple torsionl loding here clssicl elstic buckling theory uses the potentil energy pproch to describe the stbility of the shell. Donnell equtions described in Refs. nd 4 re frequently used to determine the deformtion of cylindricl shell under combined lods. This study investigtes the influence of therml lods on the buckling behvior of thin-lled cylindricl shells subjected to torsionl lods. * Reserch Assistnt Deprtment of Aerospce Engineering 0 Bel St. Ann Arbor I 4809 AIAA Student ember Professor Deprtment of Aerospce Engineering 0 Bel St. Ann Arbor I 4809 Associte Fello AIAA Americn Institute of Aeronutics nd Astronutics Copyright 004 by the Americn Institute of Aeronutics nd Astronutics Inc. All rights reserved.

2 The motivtion for the present investigtion including therml forces stems from n interest in emining the behvior of pper mill drying cylinders. 67 These cst-iron stem-heted pressure vessels re typiclly 50 to 00 inches in length nd hve rdius of 6 to 9 feet. Since their shell ll thickness is on the order of - inches these dryers cn be nlyzed s thin-lled cylinders. In pper mchines the dryers re positioned ith their center is of rottion horizontl ith the floor. The dryer cylinder rottes t frequency of up to revolutions/second s moist pper is pressed on the surfce to be dried vi evportion. In most dryers n evenly spced condenste removl system inside the structure etrcts condenste from the internl surfce. Typiclly dryers hve four or si loctions here condenste is removed from the dryer. The ction of the condenste removl nd ecess stem pssing through the system cretes bnds of cooler nd rmer tempertures in the shell running illy long the length of the cylinder. A torsionl lod is introduced into the pper drying system by liner lod from crepe blde hich scrpes the dried pper from the cylinder. Since the rottion is driven from only one side of the cylinder the tngentil portion of the liner lod resists the rottion of the system resulting in torsionl lod in the cylinder. On occsion pper dryers hve eperienced sudden nd dngerous instbility. When the instbility occurs the dryer shell distortion increses by n order of mgnitude compromising the opertion of the equipment nd the sfety of the opertors. It is the common belief in the industry tht this is due to the therml vrition round the dryer. Wheres the temperture grdient does crete deflection of the cylinder surfce dryers ith both four- nd si-node therml vritions re found to buckle into five-node distortion pttern indicting tht the therml loding is not the primry driving fctor in the structurl instbility. One eplntion is tht the therml distortion resulting from the norml ction of the condenste removl system increses the torsionl lod produced by the crepe blde ultimtely reching the criticl lod nd producing torsionl buckling of the dryer shell. In the present study the therml lod investigted is pplied s sinusoidl temperture vrition bout the circumference here the medin temperture mplitude of oscilltion nd the number of cycles round the cylinder re mintined s vribles. The individul loding conditions re first considered seprtely then together in the combined loding stte. The system is emined in such y tht the therml lods re ssumed to influence the behvior of the criticl torsionl buckling lod. The present study is crried out under conditions of plne stress ccording to the thin-lled shell ssumption. The influence of therml loding on the pre-bucked stte of the cylinder is emined. Totl strins re found by dding the therml strin component to the mechnicl strin using liner superposition. The second derivtives of the kinemtic reltions re mnipulted to yield reltion beteen the strins nd surfce deflection. An Airy stress function is introduced nd utilized to determine the resultnt lods for the cylinder ith circumferentilly vrying therml loding. With this considered the combined therml nd torsionl loding buckling problem is solved by using Glerkin pproimtion. The results from the nlyses sho the influence of therml lods on the torsionl buckling chrcteristics of thin lled cylinders. When the solutions re specilized to the cse of the stem heted pressure vessels good greement is found beteen the nlysis nd the observed buckling modes. II. Fundmentl equtions for the thin-lled cylinder The nlysis tht is described in the present study considers thin-lled cylinders of isotropic mterils. Directionl properties relting to ply direction of composite lminte cn be integrted into the equtions for n nlysis of cylinders of orthotropic mterils. Hoever for simplicity the isotropic mteril ssumption hs been mde. The folloing figure describes the shell in question here h is the thickness L is the length nd is the undeformed-middle-surfce rdius. For this shell to be considered thin-lled h must be considerbly less thn. The thin-lled ssumption llos the nlysis to be crried out under conditions of plne stress. Wheres this is good pproimtion of the true conditions it lso simplifies the equtions nd the solution significntly. The study of the behvior of thin-lled cylinder under both therml nd torsionl loding begins by emining force nd moment equilibrium of shell ll element hich provides Eqs. (). L r Figure : Circulr cylindricl shell coordinte system reference digrm Θ Americn Institute of Aeronutics nd Astronutics

3 ( ) 0 0 p () here p is the rdil pressure defined positive outrd nd the forces nd moments re described by the folloing constitutive Eqs. () involving the in-plne force nd moment resultnts (integrted over the thickness of the shell) mid-plne strins nd curvtures of the shell. ( ) ( ) ( ) ( ) ( ) D D D C C C () Equtions () utilize the common structurl rigidity constnts C /(- ) nd D /[(- )]. The formuls for the strins nd curvtures of the circulr cylindricl shell re defined by the folloing Donnell kinemtic reltions Eqs. (). v u v u () here the s re ngles of rottion of the norml to the shell mid-surfce nd re ssumed to be smll. (4) III. Effect of therml loding The therml loding considered in the present study is pplied s sinusoidl temperture vrition round the circumference of the cylinder such s is introduced in the pper dryer shell by the condenste removl system. The medin temperture mplitude of vrition nd the number of cycles round the cylinder re mintined s vribles to fcilitte modeling multiple therml loding scenrios. The ppliction of the therml lod profile is relevnt to mny other situtions involving structures. For instnce ircrft fuselges undergo ide rnge of solr therml loding hile ithstnding torsionl lods from ind nd control surfces. Also the rdition to hich spce Americn Institute of Aeronutics nd Astronutics

4 stellite structures re eposed cn led to lrge temperture vritions through the system tht hen combined ith forces from propulsion my led to structurl instbilities. The totl strin on the surfce hen therml lod is introduced ill be the sum of the mechnicl strin nd the therml strin in both the nd directions. The sher strin ill not be effected by the therml loding nd ill remin solely the mechnicl strin in the - direction. The mechnicl strin is determined by evluting the simultneous equtions for the norml lod from Eqs. () nd solving for the strin components. mech mech mech (5) The totl strins cn then be found by dding the therml strin component T() to the mechnicl strins in the il nd circumferentil directions. T T ) ( ) ( (6) The second derivtives of the kinemtic reltions Eqs. () re found to be v u v u (7) hich cn be mnipulted by substitution of the first to equtions into the third eqution to yield reltion beteen the strins nd the mid-surfce deflection. (8) To simplify the solution n Airy stress function f() is introduced hich is defined to stisfy Eqs. () identiclly. The function reltes to the in-plne stress resultnts such tht f f f (9) Americn Institute of Aeronutics nd Astronutics 4

5 After the norml lods in the strin equtions re replced by the reltions in Eqs. (6) nd Eqs. (9) the strindeflection reltion Eq. (8) is determined. f ( ) f f T T (0) This study is specific to therml loding hich vries only round the circumference. Tht is T() T(). The temperture distribution of interest cn vry sinusoidlly round the cylinder s follos: δ T( ) To sin k () here the integer k represents the number of full sine ves in the therml loding round the circumference nd δ represents the mplitude of the therml vrition. It cn be ssumed tht the stress function ill tke the form of the temperture loding tht is pplied to the shell. f ) f o f sin k ( () Substituting these formuls into the strin-deflection reltion Eq. (0) leds to the simplifiction 4 f k δk sin k sin k 0 () Solving for the mplitude constnt in the stress function f leds to δ f ( ) f o sin k k (4) This function cn then be used to determine the resultnt lods for the cylinder ith circumferentilly-vrying therml loding. To complete the model uniform il loding must be included to ccount for the compressive rection force t the boundries of the shell. The cylinder is ssumed to be sufficiently long such tht boundry effects my be neglected nd this uniform loding ill not pper in either the circumferentil or sher force terms. δ o sin k 0 0 (5) IV. The torsionl buckling problem With the therml loding fully defined the net step is to pply this informtion to the torsionl buckling problem. The Donnell stbility eqution Eq. (6) hich is described in terms of the lterl displcement ill be used to solve for the criticl buckling lod. 8 D 0 4 (6) 5 Americn Institute of Aeronutics nd Astronutics

6 The il norml force is tht determined from the therml loding Eqs. (5). Also incorported is mechnicl torsion loding τ such tht τh. The solution to this eqution cn be obtined by pplying the Glerkin method. In the cse of cylindricl shell the equtions needed re obtined s follos: π L 0 0 L ( ) ϕ dd m 0 (7) here L() is in this cse the Donnell stbility opertor nd is n ssumed solution for the displcement of the buckled configurtion. To ccommodte the ves in the deflection tht re coupled ith tisting behvior n pproprite solution ould be of the form ( m n ) kn sin (8) here m mπ/l ith m being the integer defining the number of hlf sine ves long the length of the cylinder nd n defining the number of full sine ves round the circumference. This leds to n epression for the criticl torsionl buckling lod. This pproch my be verified by compring the solution for the cse here there is only uniform torsionl lod ith the ccepted result from Ref. []. ( m 4) D m (9) 4m 4 o ( m 4) Here the circumferentil ve number n is set to to cpture the loest buckling lod. If hoever the term is left in vrible form the solution for the torsionl lod becomes: ( m n ) D m o nm n n ( m ) (0) V. Discussion of results The Glerkin pproimtion described in the previous section s incorported into computer code using themtic. With geometric nd mteril dt entered the code solves for the criticl torsionl lods for mtri of mode shpes such tht m nd n re lloed to vry s integers. From this informtion the loest torsionl buckling lod cn be identified s the criticl torsionl lod for the criticl mode shpe. The torsionl shpe function given in Eq. (8) s investigted for suitbility in combined torsionl nd il loding stte. The concern s tht the torsionl shpe function ould not llo considertion for the effect of il lods on the torsionl buckling. Hoever the nlysis s performed for il loding only using both the torsionl shpe function nd the clssicl il shpe function Eq. (). ( m) ( n ) kn sin sin () The clculted il buckling lod s the sme for both shpe functions indicting the sufficiency of the torsionl shpe function for the combined loding cse. So s to dhere to boundry conditions the shpe function used in the Glerkin pproimtion s ltered from the form given by Eq. (8). The folloing eqution is consistent ith the boundry conditions hich require the ends to be pinned into the originl circulr shpe. kn sin L L ( m n ) sin ( ml n ) ( n ) sin () 6 Americn Institute of Aeronutics nd Astronutics

7 Tble depicts section of the results output by the themtic code for cylinder ith length of 50 inches rdius of 08 inches nd shell thickness of inches. The mteril properties re those for luminum nd the therml distribution is tht of si-noded sinusoidl ve hose verge temperture of 400±F vries by 00±F. The cse corresponding to three circumferentil ve (independent of the number of il hlf-sine ves) leds to zero denomintor nd is therefore lbeled Indeterminnt. This occurred only for circumferentil mode vlues equl to hlf of the therml nodl shpe number. It cn be seen tht the loest torsionl lod listed is lb-in for the mode shpe ith seven full sinusoidl ves in the circumferentil direction nd to hlf sinusoidl ves in the il direction. It should be noted tht the ve number corresponding ith the temperture distribution does not in generl correspond to tht of the buckled mode shpe. It is noted tht the number of circumferentil ves corresponding to the buckling instbility s predicted here correltes ith field mesurements s reported in References 6 nd 7. Tble : Torsionl buckling lod (0 lb-in) for cylinder under vrying therml lod Circumferentil ode - Full Sine Wve Ail ode - Hlf Sine Wve Indeterminnt Indeterminnt Indeterminnt Indeterminnt The question tht nturlly follos is to ht degree the therml loding influences the torsionl buckling of the cylinder. The conditions bove ere repeted ith the therml loding set to zero. Wheres the buckling mode is not ltered by the removl of therml loding the criticl torsionl lod is incresed by. 0 lb-in. This result is eplined by ppeling to the pre-buckling il loding introduced by the therml lod hich contributes to the totl strin energy of the system. Tble : Torsionl buckling lod (0 lb-in) for cylinder under no therml lod Circumferentil ode - Full Sine Wve Ail ode - Hlf Sine Wve It is cler tht lthough the buckling mode remins unchnged in this instnce the differences beteen djcent criticl lods is ltered. This indictes tht the mode shpe ill chnge for sufficiently high therml loding. For the cylinder in question the mode shpe drops to si circumferentil ves nd to hlf il ves hen the 7 Americn Institute of Aeronutics nd Astronutics

8 constnt therml lod ners 75±F. Further increses in temperture bove 50±F chnges the mode shpe gin to five circumferentil ves nd one hlf il ve. The therml vrition hs less effect on the chnging mode shpe since tht prt of the resultnt lod term is multiplied by sine fctor. Vlues for the vrition hich do lter the buckling re so high tht mterils ith resonble therml conductivity ould not mintin the given therml stte nd re not eperienced in prcticl pplictions. The mode shpe into hich the cylinder buckles is dependent not only on loding conditions but lso on the cylinder geometry. To illustrte this effect the bseline thermlly-loded configurtion described bove s ltered geometriclly. Considertion s first given to the rtio beteen the length nd the rdius of the cylinder. It s found tht scling this rtio did not result in comprble buckling lods or modes. For the sme thickness rdilly lrger cylinders demonstrted less rigidity nd the criticl torsionl lods ere reduced hile the number of ves in the mode shpe incresed. A visul representtion of the buckled shpe of cylinder under torsionl lod is shon in Fig.. The illustrtion is from the computtionl progrm ple bsed on the boundry condition shpe function Figure : Thin-lled cylinder deformtion under torsionl lod ith clmped boundry conditions used in the Glerkin pproimtion Eq. (). For clrity the mode shpe ith five full sinusoidl circumferentil ves nd one hlf sinusoidl il ve hs been chosen nd the displcements hve been eggerted. VI. Conclusion The nlysis performed on the torsionl stbility of thin-lled cylinder shos tht lthough therml lod distribution does lter the torsionl buckling lod the therml contribution to the criticl lod is mrginl. Therml loding increses the strin energy nd therefore reduces the criticl torsionl buckling lod. Wheres the direct influence is slight reness of the role of the therml strins in torsionl buckling is importnt for understnding the behvior of cylindricl shells subjected to multiple-loding situtions. A mthemticl model tht ccurtely predicts the buckling behvior of thin-lled cylindricl shells under combined therml (circumferentil temperture distribution) nd torsionl loding hs been presented. Further development of this model ill incorporte dditionl loding sttes such s due to forces generted on ccount of cylinder rottion. Acknoledgments The uthors ould like to thnk Dr. John A. Chrles for describing this problem nd for his ssistnce ith editing the mnuscript. References Lennon R. F. nd Ds P. K. Torsionl buckling behviour of stiffened cylinders under combined loding Thin-Wlled Structures Vol. 8 o. 000 pp Brush D. O. nd Almroth B. O. Buckling of Brs Pltes nd Shells cgr-hill Inc. e York 975 Chps Donnell L. H. Stbility of Thin-lled Tubes under Torsion ACA Rep Americn Institute of Aeronutics nd Astronutics

9 4 Btdorf S. B. A Simplified ethod of Elstic Stbility Anlysis for Thin Cylindricl Shells ACA Rep Btdorf S. B. Stein. nd Schildcrout. Criticl Stress of Thin-Wlled Cylinders in Torsion ACA T Chrles J. A. An Investigtion of the Ynkee Dryer Out of Round Condition Proceedings of the Tissue World Conference ice Frnce rch 005 (submitted for presenttion). 7 Chrles J. A. Precision Infrred Imging for Ynkee Dryers Proceedings of the Tissue World Conference ice Frnce rch Americn Institute of Aeronutics nd Astronutics

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description