Analysis and Experimental Verification of the Strength of Telescopic Booms for Construction Machinery

Transcription

1 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC Anlss nd Eperentl Verfcton of the Strength of Telescopc Boos for Constructon Mchner Zhe CUI, Wengung JIANG*, Le CHENG School of Mechncl Engneerng, Ynshn Unverst, Qnhungdo, Hee, , Chn XCMG Xuzhou Envronentl Technolog Co.,Ltd, Xuzhou, Jngsu, 000, Chn Astrct Telescopc oos re ke eleents for plod lftng n constructon chneres. The usull consst of thn-wlled hollow segents tht cn retrct, etend or elevte durng operton. The echncl propertes of the telescopc oos drectl deterne the lftng cpct of these chnes. The overlppng contct regons etween the djcent oos re crtcl loctons where hgh stresses occur, thus the stress level n these regons s the ke fctor representng the strength of the telescopc oo structures. Accordng to the prncple of superposton, thetcl odel hs een proposed for nlss of stresses n the contct zones of telescopc oo structure. The proposed odel clcultes the glol endng stress ccordng to the e endng theor, then uses prtl telescopc oo geoetr solted fro the contct zones to estlsh locl nltcl odel to deterne the locl stresses ccordng to the plte endng theor, nd fnll superposes the two sttes of stresses to otn the fnl stress stte whch represents the strength of the telescopc oos. The proposed nltcl odel hs een vldted oth fnte eleent nlss nd eperentl test conducted on telescopc erl pltfor. Kewords - telescopc oo; structurl strength; prncple of superposton; endng stress; nltcl odel; eperentl verfcton. I. INTRODUCTION Telescopc oos n constructon chneres conduct plod lftng tsk retrctng, etendng or elevtng. The re usull the drect lodng erng prts n the whole sste of the chner, s shown n Fg.. Reducng the selfweght of the telescopc oos could help ncrese the effectve plod lftng cpct, nd ncrese the lftng nd etenson operton speeds of the oos s well. In recent ers, nufcturers hve pd ore nd ore ttenton on the sectonl shpe selecton nd optzton of the telescopc oos. The consequence of usng these optzed secton desgn cn not onl reduce the weght of the telescopc oos, ut lso ncrese the endng stffness nd torson stffness n the se te[,]. Durng ntennce or repr, plstc deforton nd crcks could often e oserved n the vcnt of the overlp contct regons etween two djcent telescopc oos. Ths ndctes tht the stresses wthn these regons re uch hgher thn other regons. work slewng se lftng frst second thrd Fgure. Structure of the telescopc oo The studes of the stress dstrutons wthn the overlppng contct regon of telescopc oos hve een reported severl reserchers. Bsed on the results of sttc strength eperent on the rectngulr cross-sectonl telescopc oo overlppng structure, Lu nd Wng [3] proposed stress epresson usng superposton of three nltcl solutons of spl supported plte odels under dfferent lodng condtons. The predcted stresses fro ths odel were vldted eperentl results. Djelosevc et l.[4] studed the effect of locl stress on the crrng cpct nd optu desgn of o grders. Svkovc et l. [5] studed the locl stress ncreses t the overlppng contct regons of rectngulr telescopc oos of truck crnes. Wth the developent of odern coputng technolog, the fnte eleent ethods re wdel used to perfor nuercl nlses for the telescopc oo structure. Nuercl solutons of the stress dstrutons could e otned nd structurl optztons could e crred out sed on the stress nlss results [6-8]. Soe of the used the nodl degrees of freedo couplng technolog to connect ech oos. Soe used contct nlss cplt of softwre to solve stresses on the contct res of the telescopc oos. As there est n cople contct regons on the telescopc oo structures nd t s well know prole tht the contct stress nlss processes re hghl nonlner, fnte eleent odellng of these structures re norll ver te-consung, nd n n cses, convergence dffcultes re often encountered durng nlses. These fctors lt the wde pplcton of the fnte eleent nlses to the telescopc oo structures. Copre to fnte eleent odels, nltcl odels re ver effcent to run. However ost of the estng nltcl odels for telescopc oo structures hve poor stress predcton ccurc due to the use of the oversplfed DOI 0.503/ IJSSST ISSN: onlne, prnt

2 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC ssuptons tht the contcts occur over the entre sldng surfce of the sldng locks. In the current stud, t hs een found fro the elortel conducted fnte eleent nlss results tht the trnsfer of lods etween the overlppng contct oos re ctull relzed through ver nrrow contctng res long the edges of the sldng locks, rther thn the whole sldng surfce of the sldng locks. Bsed on the oservton, we could ke rtonl ssupton tht the dstruton of contct lod trnsferred etween contctng oos s long the nrrow edge res of the slders. Usng ths ore relstc ssupton nd the prncple of superposton for elstc structures, n ths pper, new thetcl odel for the deternton of crtcl stresses n the contct regons hs een proposed superposng the structurl glol endng norl stress wth the locl trnsverse endng stress n the contct zone. Fgure 3. Sldng locks loctons on the out nd nner oos. II. STRESS ANALYSIS IN THE BOOM CONTACT REGIONS The telescopc oo structure s usull coposed of two or ore hollow-sectoned oos nestng together one fter nother. The nner oos cn slde wthn the outer neghorng oos. The lods trnsferred etween the neghorng oos re relzed through the sldng contcts etween the sldng locks nd the sldng surfces of the oos. The upper sldng locks re fed on top of the outsurfce t the end of the nner oo, nd the lower sldng locks re fed on the lower nner surfce of the out oo, see Fg.3. The lod trnsferrng echns s llustrted n Fg.4. In ths pper, the nltcl soluton of the crtcl stress wthn the contct regons of the telescopc oo sste wth nestng hegonl cross-sectons re studed n detl. Fg. shows the pretrc densons of the hegonl oo cross-sectons. H s the heght of the oo cross secton, W the wdth of the oo cross secton, T the plte thckness, H the heght of the we plte, W the wdth of the otto flnge plte, nd α the nclned ngle of the we plte. Fgure 4. Lod trnsfer v sldng locks Wthn the rege of elstc deforton, the stress n the contct re cn e consdered s the superposton of two sttes of stresses. The frst prt s the norl endng stress cused the glol endng deforton of the oo. The second prt s the locl endng stress cused the trnsverse locl endng deforton. The forer s shown n Fg.5, nd the ltter s shown n Fg.6. σ' σ' z z z Fgure 5. Glol endng norl stress Fgure. Cross-secton of the telescopc oo DOI 0.503/ IJSSST ISSN: onlne, prnt

3 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC σ'' σ'' z τ'' τ'' σ'' σ'' l s the stretch-out length of the th oo. s the overlppng length etween the neghorng th oo nd the (+)th oo. G s the self weght of th oo. Assung the self weght of ech oo s unforl dstruted long ts length, the locton of center of self weght should e on the dpont of ech oo. The plod P N s eerted on the tp of the nth oo, refer to Fg.7. The two contct cton ponts on th oo re desgnted s A nd B, where A ndctes the left sde pont, B s the rght sde pont, nd s the oo nuer. Consderng equlru of oth force nd oent, the lnce forces trnsferred fro the (-)th oo to the th oo t the contct ponts re gven s: Fgure 6. Locl endng stress The fnl resultnt stress coponents σ, σ, nd τ n the contct regon cn e clculted s the suton of the correspondng stress coponents of the glol endng stress nd the locl trnsverse endng stress: () () (3) ' '' '' '' where s the drecton long the oo length, the trnsverse drecton of the plte s shown n Fg.5 nd Fg.6. A. The Glol Bendng Norl Stress Provded tht n s the totl nuer of oos n the whole telescopc oo sste. The dgr of forces on porton of the telescopc oo sste fro th oo to the nth oo s shown n Fg.7. Fgure 7. Force dgr of porton of the telescopc oo. n n j- A [ j ( j - j ) j k j j k n PN lj]/ j n n j- B [ j ( j j ) j k j j k n PN lj]/ PN j N G l G l N G l G l Usng equtons (4) nd (5), the lncng forces on ll oos cn e clculted, the endng oent M on n cross-secton of the oos cn e otned, nd the endng norl stress σ cn lso e clculted usng the followng equton: ' M Z (6) I where I s the oent of nert of the th oo crosssecton wth respect to the -s. B. The Locl Bendng Stress As the thckness of the pltes of the oo s uch sller thn the wdth nd length of the pltes, the theor of elstc thn wlled plte could e ppled to solve the locl '' endng prole of the pltes. The stress coponents z, nd re ver sll nd could thus e sfel gnored, '' z '' z copred to the other three jor stress coponents. The contct stress hd een usull ssued to e unforl dstruted on the sldng contctng surfces of the sldng lock when developng nltcl solutons to the prole. However, fro our stud usng ccurte full three densonl fnte eleent nlss, contcts were found to occur onl long the nrrow edge regons of the sldng lock, nd the wdth of the contctng regons s out twce the thckness of the contctng pltes. The reson for ths s nl due to the locl endng deforton s llustrted n Fg.8. Bsed on these fndngs, the nltcl theor developed n ths pper usng ore relstc hpotheses whch ssue tht contcts etween the neghourng oos occur long the nrrow edge regons of the sldng locks, (4) (5) DOI 0.503/ IJSSST ISSN: onlne, prnt

4 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC the wdth of the nrrow contct regons s twce the thckness of the pltes nd the contct lods re unforl dstruted wthn these nrrow regons. M M q () q M M (6) () M 6 M 3 Fgure 8. Locl endng deforton of the cross- secton t the contct zone of the telescopc oos: out oo (left) nd nner oo (rght). Consder sngled-out porton of the oos wth length of ( = H, s uch greter thn the length of the contct slder, see Fg.9). For the convenence of nlss, the oo portons could e further dssseled to s sller suportons,.e. () top flnge, () rght we, (3) rght nclned plte, (4) otto flnge, (5) left nclned plte, nd (6) left we, see Fg. 0. These dssseled pltes re nlzed s spl supported pltes sujected to endng oents, nd the se endng oents re ppled to oth sdes of the dssseled edges, see Fg. 0. (5) (3) (4) M 5 M 4 M () M M M (6) () M 6 M 3 (5) (3) q 5 (4) q 5 q 3 q 3 M 5 M 5 M 4 Fgure 0. Model for nlss: out oo (top) nd nner oo (otto). X M M u q Y q v Fgure. Superposton of the lods for the top flnge plte / Fgure 9. The porton of oo loded v sldng locks: out oo (top) nd nner oo (otto) The dfferentl equton for the trnsversel loded plte hs the for [] w w w q(, ) 4 4 D (7) where: 3 E D s the endng stffness of the plte. ( ) DOI 0.503/ IJSSST ISSN: onlne, prnt

5 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC E s Young s odulus. μ s Posson rto. δ s the thckness of the plte. The deflecton of spl supported plte due to unfor contnuous lod q s: n sn sn 6q wq ( ) 6 D n n n( ) (8) u nv n sn sn sn sn The nclnton of ths plte due to the lod q s gven the epresson: n sn sn wq ( ) 6q 5 D n n ( ) (9) u nv n sn sn sn cos The deflecton of spl supported plte due to the cton of oents M nd M + s: E E( ) w( M, M ) [ 4D cosh ( tnh cosh - snh ) (0) E( ) - E ( coth snh - snh cosh )]sn The oent M, M + nd α re deterned ccordng to the followng seres: sn M E sn M E( ) where E nd E (+) re unknown coeffcents. The correspondng nclntons of the plte t Y=±/ re deterned usng the epressons: w( M, M ) - [ - ( E 4D () E( ) ) F ( E( ) - E) H]sn w( M, M ) [( E 4D () E( ) ) F ( E( ) - E) H]sn The coeffcents F nd H re gven through the followng epressons: F - ( tnh ) cosh H ( - coth ) snh As shown n Fg. 0 (),the top flnge plte of the out oo s utull sujected to the locl dstruted lods nd the endng oents. B usng the prncple of superposton, the nlss of the top flnge of the out oo cn e consdered s the superposton of the two spler lodng cses s shown n Fg.. For the lodng cse of the top flnge plte eng consdered s spl supported plte sujected to the locl unfor contnuous lod q, the deflecton cn e otned fro forul (8): n sn sn 6q w ( q) 6 D n n n( ) (3) u nv n sn sn sn sn For the lodng cse of the top flnge eng consdered s spl supported plte sujected to dstruted endng oent M nd M t Y= ± /, the deflecton cn e otned fro forul (0): E E w( M, M) [ 4D cosh ( tnhcosh - snh ) (4) E- E ( cothsnh - snh cosh )]sn B superposng Eq. (3) nd Eq. (4), we hve the deflecton equton of the upper wng plte: w w( q) w( M, M) (5) The rest of the pltes cn e consdered s spl supported pltes t ll four edges, sujected to dstruted endng oent M nd M(+) t Y= ± / sdes usng the Eq. (0). The deflecton of equtons re: w w( M, M ) (6) In order to ensure the contnut of the deforton of the oo, the followng oundr condtons ust e stsfed: w6( M6, M) wq () (7) w ( M, M) - DOI 0.503/ IJSSST ISSN: onlne, prnt

6 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC w ( q) w ( M, M ) w ( M, M ) 3 - wmm (, ) w ( M, M ) - (8) (9) (=,3,4,5, when =5,+=) Eqs.(7)-(9) re equvlent to Eqs.(0)-() E6 ( F6- H6 ) E [ ( F6H6 ) D6 D6 ( F H)] E ( F - H ) D D (0) 4 w () q ( ) 0 sn E ( F - H ) E[ ( F H ) D D ( F H)] E3 ( F - H) D D () 4 w ( q) - ( ) sn E ( F - H ) E( ) [ ( F H ) D D ( F( ) H( ) )] () D E( ) ( F( ) - H( ) ) 0 D (=,3,4,5,when =5, +=) Eqs. (0), () nd () for sets of equtons. In ech set of equtons, there re 6 equtons nd there re lso 6 unknown vrles (E, E E 6), therefore these sets of equtons hve unque solutons. And the endng oents (M, M M 6 ) of ech pltes could lso e otned. Susequentl, the deflecton equtons of ech pltes could e derved. We cn then clculte the surfce stress of ech plte fro the known deflecton equtons usng the followng equtons: '' E w w ( ) (3) ( - ) '' E w w ( ) (4) ( - ) '' E w (5) ( ) Susttutng Eqs. (3), (4) nd (5) nd (6) nto the Eqs. (), () nd (3), the stress coponents (σ, σ nd τ ) t n loctons of the out surfce of the out oos cn e derved. Both the theor nd the ethod of clculton of the locl endng stress for the nner oos re dentcl to those of the out oos. Therefore the correspondng equtons re not gven n ths pper for the ske of revt. III. VALIDATION OF THE ANALYTICAL MODEL A. Preters of the Boo Geoetr nd Mterls To vldte the ccurc of the nltcl odel developed n ths pper, oth eperents on rel telescopc oo structure nd fnte nlses hve een crred out. The structure used to perfor the eperent nd nlses s the telescopc oo on tpcl erl work pltfor. The geoetrc dt of the telescopc oo sste re gven n Tle. The etenson lengths nd the lengths of the overlppng contct regons of the telescopc oos re: l 3 =540, l =4890, l =4975, 3=800, =050, =00. The plod P N=3Kg. The length of the sldng lock u =50, Young s odulus of the terl E=. 05(MP), Posson rto μ=0.3, nd the denst of the steel terl ρ= (Kg/ 3 ). TABLE. THE DIMENSIONS OF THE CROSS-SECTION Preter of Cross Secton Frst Boo Second Boo Thrd Boo H/ W/ H/ W/ T/ α/ B. Eperent The erl work pltfor used for the test s shown n Fg.. The pctures showng the strn guges on the surfce of the tested oo re gven n Fg.3, nd the dstruton of the esureent ponts re llustrted n Fg.4. The se pont of the esureent,, s locted on the setrc lne of the surfce of top flnge of the frst oo, nd just ove the upper sldng lock of the second oo, see Fg.4. The loctons of the rest nne strn guges re deterned the reltve dstnces to ths se pont. Durng testng, the telescopc oo ws kept t horzontl poston nd ll ts coponent oos were full etended. Ths s one of the ost crtcl lodng cses. When the telescopc oo ws operted under ths DOI 0.503/ IJSSST ISSN: onlne, prnt

7 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC confgurton, the recton forces on the sldng locks re u. The plod of P N=KN ws lfted fro the ground. After the oo ws held n ths testng poston for out 0s, the vrton of the oo ws found to e not ovous nd then the strn guge vlues were red nd recorded. The stress coponents n the nd drectons were esured seprtel. O r s q J t p 60 X Z Y K P Second oo 70 Fgure. A tpcl erl work pltfor Fgure 3. Mesurng ponts for testng the erl work pltfor: out oo (left) nd nner oo (rght). Frst oo B e d h D c E f g C A Z X Y Fgure 4. Lout of esurng ponts: out oo (top) nd nner oo (otto). C. Fnte Eleent Anlss In order to further verf the ccurc of the nltcl odel developed, n ore ccurte fnte eleent odel hs een develop. The fnte eleent esh s shown n Fg.5. To vod the odelng error cused the structurl geoetrc splfcton, the whole upper crrge structure ws ncluded n the fnte eleent odels. To ncrese the nlss ccurc, sold rck eleents were used throughout. Contct eleents were used to sulte the relstc contct ehvor t the overlppng regons etween neghorng oos. The Mses equvlent stress dstruton t the overlppng contct zone of the frst nd second telescopc oo coponents fro the fnte eleent nlss s gven n Fg.6. The contour plots of stress coponents n the vcnt of the locl contct regon s gven n Fg. 7. j F 050 plod 66 fed constrnt Fgure 5. Fnte eleent esh for the telescopc oo DOI 0.503/ IJSSST ISSN: onlne, prnt

8 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC e Fgure 6. Mses equvlent stress n the contct zone: out oo (top) nd nner oo (otto). f Fgure 7. Stress n drecton of frst oo, Stress n drecton of frst oo, c Stress n z drecton of frst oo, d Stress n drecton of second oo, e Stress n drecton of second oo, f Stress n z drecton of second oo. c d X D. Coprson of Models nd Eperentl Vldtons The stress nlss results fro two theoretcl odels, sold fnte eleent odel (FEM) nd eperent dt re ll copred n detl n ths secton. The frst theoretcl odel presented s the trdtonl nltcl odel (leled s Trdtonl nlss n the grphs) whch uses the ssupton tht the contct occur on the whole surfce of the contct sldng locks. The second theoretcl odel s the one newl proposed n ths pper (leled s New odel n the grphs), whch usng the ssupton tht the contcts occur onl on the nrrow edge res of the sldng locks. Fg.8 shows stress dstruton long trnsverse pth lne AB (s rked n Fg.(4)) on the top flnge plte of the frst oo. Fg.9 shows stress dstruton long longtudnl pth lne CD (see Fg.(4)) on the top flnge plte of the frst oo. Fg.0 shows stress dstruton long trnsverse pth lne EF (see Fg.4()) on the vertcl we plte of the frst oo. Fg. shows equvlent stress dstruton long trnsverse pth lne JK (see Fg.4()) on the nclned we plte of the second oo. Fg. shows equvlent stress dstruton long longtudnl pth lne OP (see Fg.4()) on the nclned we plte of the second oo. Fro ll these coprson fgures, t cn e seen tht the results fro the new odel correlte ecellent well wth those fro oth the ccurte sold fnte eleent nlss nd the eperentl dt. However the trdtonl nlss shows sgnfcnt dscrepnces, whch ens tht the wdel used ssupton tht the contct pressures re evenl dstruted on the entre surfce of the sldng lock s not rtonl nd the use of t led to sgnfcnt errors n predctng stress n the telescopc oo structures. The new ssupton tht the contcts occur onl on the nrrow edge DOI 0.503/ IJSSST ISSN: onlne, prnt

9 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC res of the sldng locks, derved fro the oservton fro the ccurte FEM nlss results s rtonl nd the newl proposed nltcl odel sed on t cn eld ver ccurte stress predcton. The u equvlent stress ppers on the top flnge plte of the frst telescopc oo, the u equvlent stress vlue clculted fro the new odel s σ e =50.8 (MP). The equvlent stress t ths locton fro eperentl testng nd FEM nlss re σ T e=6.8 (MP), nd σ E e=46.5 (MP), respectvel. The u reltve dfferences of equvlent stress t ths locton etween the newl proposed theor nd oth of the eperentl results nd ccurte FE nlss re less thn 4.6%. Fgure 9. Coprson of stresses long pth CD on the frst oo: (top), (d) nd eqv (otto). Fgure 8. Coprson of stresses long pth AB on the frst oo: (top), (d) nd eqv (otto) Fgure 0. Coprson of z stress long pth EF on the frst oo DOI 0.503/ IJSSST ISSN: onlne, prnt

10 ZHE CUI et l: ANALYSIS AND EXPERIMENTAL VERIFICATION OF THE STRENGTH OF TELESCOPIC of unfor stress dstruton on the entre surfce of the sldng lock predct fr too hgh stress level. Whlst the results fro the new odel sed on the nrrow locl contct ssupton re n good greeent wth oth the fnte eleent nlss nd crefull conducted eperentl results on rel telescopc oo structure of n erl work pltfor. ACKNOWLEDGMENTS Fgure. Coprson of Mses stresses long pth of JK on the second oo Fgure. Coprson of Mses stresses long pth of OP on the second oo. IV. CONCLUSIONS A thetcl odel hs een proposed for the effcent nd ccurte nlss of stresses n the contct zones of telescopc oo structure. Accordng to the prncple of superposton, the totl stress cn e clculted s superposton of the glol longtudnl endng stress nd the locl lterl endng stress. For the locl stress nlss odel, new ssupton hs een proposed on the contcts etween the djcent oos. The new odel ssues tht the contcts onl occur on ver nrrow res long the edges of the sldng locks, nd ths ssupton hs een justfed fro the ccurte fnte eleent nlss. Nuercl results show tht the conventonl nlss sed on the ssupton Ths work ws supported the Ntonl Nturl Scence Foundton of Chn (Grnt No ), Hee Nturl Scence Reserch fundng under grnt E REFERENCES [] Z. M. L, Stress nlss of the sldng locks on the telescopc oo nd the reserch of ts chrcter, Mster Thess, Southwest jotong unverst, Chn, 009. [] Y. Z. C, Reserch on cross-sectons nd sze optzton of odern telescopc oo, Mster Thess, Dln unverst of technolog, Chn, 0. [3] B. S. Lu, J. N. Wng, Clculton of locl stress t the sldng shoe of the telescopc oo of truck crne, Hostng nd Conveng Mchner, pp. 4-8, M, 994. [4] M. Djelosevc, V. Gsc nd D. Petrovc, Identfcton of locl stress preters nfluencng the optu desgn of o grders, Engneerng structures, vol. 40, pp , 0. [5] M. Svkovc, M. Gsc nd G. Pvlovc, Stress nlss n contct zone etween the segents of telescopc oos of hdrulc truck crnes, Thn-Wlled Structures, vol. 85, pp , 04. [6] H. Sho, Q. Feng nd S. L. Zhng, Fnte eleent nlss on telescopc oo of n erl work pltfor wth dfferent odelng ethods, Constructon Mchner nd Equpent, vol. 39, pp. 6-, 008. [7] Y. S. Ren, X. J. Yu nd Z. P. Zhou, Fnte eleent clculton nd optl desgn of the telescopc r of crne, Journl of Chngchun Insttute of Technolog(Nturl scences edton), vol. 8, pp , 007. [8] Z. W. Chen, Z. Cu nd W. G. Jng, Structurl optzton of the telescopc oo of certn tpe of truck-ounted crne, Appled Mechncs nd Mterls, vol. 548, pp , 04. [9] X. Ln, Anlss of contct proles nd stud of regulrt on telescopc oo s slders, Mster Thess, Dln unverst of technolog, Chn, 0. [0] J. Z. He, nd Y. Chen, Anlss nd Optl Desgn out knd of Berng Be wth Sleeve Jont Structure, Proced Envronentl Scences, pp , Noveer 0. [] W. Wng, Q. H. Yu nd J. R. Zhng, Applcton of the Theor of Elstc Thn Plte n the Mechncl Anlss of Bo Grder, Journl of Chngsh counctons unverst, vol. 0, pp , Aprl 004. [] S. Toshenko nd S. Wonowsk, Theor of pltes nd shells, Bejng:Scence Press, 997. DOI 0.503/ IJSSST ISSN: onlne, prnt