STRESS ANALYSIS OF THE MULTI-LAYERED THICK CYLINDERS

Transcription

1 Yea STRESS ANALYSIS OF THE MULTI-LAYERED THICK CYLINDERS Assist Lectue Abdul Munium Razoki Majeed Algboy Institute of Medical Technology (Baghdad) Foundation of Technical Education ABSTRACT In this study, the stess analsis of the steel-aluminum compound thick cylindes unde the effects of intenal pessue, themal loading and otational loading has been caied out using the finite element method The stuctue is teated as axisymmetic body, because each of the geomety and applied loads ae symmetic about the longitudinal axis The stesses vaiations (Hoop, Axial, Radial, Equivalent) though the walls thickness ae detemine hee and the esults wee checked using two theoies of elastic failues (Tesca and Von-Misses) The esults showed that, the Hoop stesses at the inne suface is about (6 MPa) due to effect of Intenal pessue, (-5 MPa) due to themal load, (57 MPa) due to effect of otational speed while about (5 MPa) due to the effect of the total loading It can be seen that the max hoop stess concentated at the contact suface between the two cylindes Also the tempeatue distibution though the cylinde thickness has been detemined Key wod: thick cylinde, compound, tempeatue distibution, themal stess /( ) (Equivalent, Radial and Hoop) (Tesca, Von-Misses) (6 MPa) (Hoop Stesses) (57 MPa) (57 MPa) (-5 MPa) 5

2 Yea INTRODUCTION The compound mould of the centifugal casting technique, epesent the most common of the applications of the compound thick cylinde which subject to intenal pessue, otational speed and themal loading Theefoe, an accuate and eliable technique is essential in ode to obtain the stesses The themal load and otational speed ae one of the most impotant paametes that effects on the behavio of the stess though the thickness of the compound thick cylinde and this behavio depends on the mechanical and themal popeties which, in fact, vay with the tempeatue vaiation [] The effect of tempeatue on the defomation field is not one-way phenomena When the mechanical and themal aspects ae coupled, and insepaable In geneal, themal stess geneation can be attibuted to two main causes [, 3] - When the tempeatue distibution in the body is unifom two paticula causes that poduces stesses: Existence of extenal constaints, as occus, if when the ends of a beam ae fastened within a wall And non homogeneity of the body such as discontinuity of sufaces - When the tempeatue distibution in the body is not unifom which gives a non-unifom defomation, and a system of stesses within the body is developed when the body undegoes a themal tansient state M Jabbai et al pesented an analytical solution fo the calculation of the axisymmetic themal and mechanical stesses in thick hollow cylinde made of FGM by using navie equation [4] Nejad et al shows the stess vaiation along the adial diection of otating FGM cylinde subjected to intenal pessue [5] Jahomi et al investigated the esidual stesses induced by autofettage pocess in layeed and functionally gaded composite vessels This study showed that the induced esidual stess at the inne suface of composite vessels much highe values compaed to a metal vessel counte pat depending on the popeties of composite constituents [6] Noda 987 studied the tansient themal stesses in a tansvesely isotopic, semi-infinite solid cicula cylinde subjected to a convection heat loss on the end suface The theoetical analysis consideed the effect of the themal and elastic isotopic of the mateial popeties on themal stesses in a tansvesely isotopic semi-infinite cicula cylinde due to cylindical suface heat geneation [7] THEORY It is vey impotant to study the tempeatue distibution and stess analysis of the compound thick cylinde subjected to diffeent loading simultaneously such as intenal pessue, otational speed and themal loading to know the behavio of stess between the single cylinde and compound cylinde Thick Cylinde Subjected to Intenal Pessue Only [8] Fom the follow equilibium equation fo thick cylinde d d () The Lame s equation can be deived A at i B, A p pi 5 B

3 i pi o i i pi o i z o o Yea () (3) (4) Thick Cylinde Subjected to Intenal Pessue and Rotational Speed [8 & 9] A B A 3 v w (5) 8 B w 3 v 8 (6) While Due to Themal Loading [8 & 9] Also the tempeatue ise of the thick cylinde will lead to induce themal stesses E i T d T d v o i i i E i o T d T d T v o i i i o z v ET (7) (8) (9) When the model stuctual components ae otationally symmetic about an axis such as the pessue vessels and solid ings, and if these stuctues ae also subjected to axisymmetic loads, a two-dimensional analysis of a unit adian of the stuctue yields the complete stess and stain distibution as illustated in Fig() [] At fist the shape functions wee developed on a maste element which is defined in ξ, η coodinates fo nomal coodinates [] The Lagange shape function Ni, whee i=,, 8 ae defined such that Ni is equal to unity at node i and is zeo at othe nodes Inpaticula, conside the definition of Ni: N= at node = at node, 3, 8 This element belongs to the seendipity family of elements The element consists of eight nodes Fig (,a) all of which ae located on the bounday Indefining Ni which efe to the maste element shown in Fig (,b) The geneal equation fo shape functions at all cone nodes is 53

4 Ni Yea i i i i 4 Fo midside nodes with ξi = i Ni Fo midside nodes with ηi = i Ni The displacement appoach to the solution of finite-element poblems, a method moe widely used than the stess is illustated by an axially loaded sping; K e e F e Whee: () K e is the element stiffness matix e is the element displacement vecto F e is the element applied load vecto K e B T D B dv () e dv d ddz d da ddz ddz det [ J ] d d K e 6 6 B 6 4 D 4 4 B 4 6 d d T () B 4 6 Stain-displacement elationship matix is given by: N B 4 6 N N y N N N 8 N N N 54 N8 N 8 N 8 N 8 y

5 Yea [D] is the elasticity matix which is in axisymmetic is given by: D 4 4 v E ( v)( v) v e v v ( v) v v u v u v u3 v3 v8 And the load vecto is given by: F e Pd e F e F the whee : (3) Pd e -is the load vecto due to distibution pessue [] F e -Centifugal load unde (inetia) foce pe unit volume F the - Themal load vecto The load vecto due to distibution pessues is given by: Pd e y T P N d y whee N N N N N 55 N 8 (4)

6 Yea But the centifugal load vecto (Inetial) foce pe unit volume depends on assumption that the cente of otation coincides with oigin of the x, y axes Theefoe the adially outwad body foce P on an element aea da is given by: P whee: (5) -is the angula velocity in ad/s ρ-is the mass pe unit volume of the mateial -is the adial distance fom the oigin to the centoid of the element aea P- can be esolved into components paallel to x and y axes such that Px x y Py whee the x and y coespond to the coodinates of the centoid of the element aea x dxdy y T x N det J d d y F N T While the themal load vecto is given by [3] F the B T dv v D whee -Themal stess vecto -Themal stain vecto,,,,,, T, T,, T xx xx yy yy T zz xy T zz xy T ΔT-Unifom incease in tempeatue F e th B D det J d d T 56 (6)

7 Yea MODELING Fo the finite element method analysis of the compound thick cylinde poblem, the ANSYS package pogam is adopted This pogam has vey efficient capabilities to pefom finite element analysis of most engineeing poblems Whee the mateial popeties, dimension and loads, diffe fom the inside and outside cylinde Due to symmety only this poblem teated as axisymmetic poblem ELEMENT SELECTED A single type of element is used thoughout this study namely the paabolic isopeimetic element, a typical two-dimensional vesion of which is illustated in Fig () Paabolic isotopic elements ae extemely vesatile, good pefomes and ae well tied and tested Pactical expeience suggests that, fo a given numbe of total degees of feedom in a stuctue, geate accuacy is achieved by used of fewe complex elements in placed of a lage numbe of simple elements MESH GENRATION The compound thick cylinde as peviously stated is plane-stain poblem o axisymmetic poblem, and the fist step of finite element analysis is to discetize the stuctue into finite elements connected at nodes Fo a stuctue as a compound tubes, it is necessay to discetize it into a sufficient numbe of elements in ode to obtain a easonable accuacy The mesh geneation of the compound thick cylinde as shown in Fig(3) Whee the thickness divided into twenty of 8-node quadilatic (paabola) isopaametic element with total of 85 nodes RESULTS AND DISCUSSIONS The poblem is solved using the finite element method as axisymmetic poblem with longitudinal axis fo compound thick cylinde The finite element solution was based on the following case study Case Study Two type of mateial used hee in the analysis (steel, aluminum) The compound thick cylinde shown in Fig(4) used fo the modeling analysis consists of two cicula cylinde with vaying popeties and have the following specification Geomety Cylinde Cylinde Inne adius = 5 mm Inne adius = 8 mm Oute adius = 8 mm Oute adius = mm The two cylinde ae fitted without any foce (The cylinde fits slightly into the one at oom tempeatue) Popeties [4] Cylinde Cylinde E = 7 Gpa E = 73 Gpa v= 3 v= 33 α = 7*-6 /oc α = 3*-6 /oc K = 35 w/moc K = w/moc ρ = 785 kg/m3 ρ = 3 kg/m3 57

8 Yea Loading PI = 3 Mpa Ti = 5 oc To = 5 oc ω = ad/sec Fig(5) shows the elationship between tempeatue and the cylindes thickness It is clea fom this figue that the tempeatue is decease with incease the adius in nonlinea elationship and in diffeent manne fo steel and aluminum that depend on themal conductivity Fig(6, 7) show the elationship between stesses vaiation (Radial, Hoop, Longitudinal and Equivalent due to Von-Misses and Tesca) and the cylinde thickness due to the themal load It is clea fom Fig(6) that Hoop stess and Longitudinal stess incease with incease cylinde thickness and the ate of incease diffe fom to mateial depend on themal conductivity It can be seen fom this figue that the behavio of adial stess in thickness diffes in thickness While fom Fig(7) fo Equivalent stess, behavio accoding to Tesca is highe than VonMisses and have nonlinea elationship with cylinde thickness Figs(8,9) show the elationship between stesses vaiation and the cylinde thickness due to effect of the otational speed It can be seen fom Fig(8) that the Hoop stess and Longitudinal stess decease with incease the cylinde thickness in diffeent manne fo the two metals used While the Radial stess incease and then decease to zeo value While fom Fig (9) the Equivalent stess decease with the incease the cylinde thickness and the value of Tesca is highe than Von-Misses Figs(,) show the elationship between stesses vaiation and the cylinde thickness due to intenal pessue only It is clea fom this figue that the Hoop stess decease with incease the cylinde thickness while the longitudinal stess appoximately constant and the Radial stess decease to zeo value at oute diamete While fom Fig () the Equivalent stess decease with incease the cylinde thickness and the Equivalent stess due to Von-Misses lowe than Tesca Fig () illustate the stess vaiation though the cylinde thickness due to total load It can be seen fom this figue the stesses incease with incease the cylinde thickness in nonlinea elationship fo and mateial that depend on the mateial popeties Fig(3) show the elationship between equivalent stess vaiation and the cylinde thickness due to total load It can be seen that the Tesca behavio is highe than Von-Misses and have nonlinea elationship CONCLUSIONS Tempeatue deceases with the incease of the adius of thick cylinde, and the behavio of tempeatue in steel wall diffe in that in aluminum wall Stess gadient in steel wall ae highe than in aluminum wall 3 Hoop and Longitudinal stess incease with adius of thick cylinde due to effect of themal loading while deceases with the adius due to effect of intenal pessue and otational speed 4 Behavio of Radial stesses though the wall thickness due to effect of themal load is veses that due to effect of otational speed 5 Equivalent stesses (Von-Misses and Tesca ) due to themal load have behavio diffe than the Equivalent stesses due to otational speed and intenal pessue 58

9 Yea 6 Hoop stess at inne suface due to effect of type of loading as follow: Type of Loading Hoop Stess Themal Load -5 Mpa Rotational Speed 57 Mpa Intenal Pessue 6 Mpa Total Load 57 Mpa 7 Max Hoop stesses concentates at the intemediate adius of the compound thick cylinde REFERENCES [] Dete, G E, Mechanical Metallugy, McGaw Hill Book Company, London, (988) [] Nowinski, J L, Theoy of Themoelasticity with Applications, Sijthoff and Noodhoff Int pub, (978) [3] Boley, B A and Weine, J H, Theoy of Themal Stesses, John Wiley and Sons Inc [4] M Jabbai, S Sohabpou, M R Eslami, Mechanical and Themal Stesses in a Functionally Gaded Hollow Cylinde due to Radially Symmetic Loads, Intenational Jounal of pessue vessels and Piping, Elsevie, () [5] Gholam Hosein Rahimi and Mohammad Zamani Nejad, "Elastic Analysis of FGM Rotating Cylindical Pessue Vessels", Jounal of the Chinese Institute of Enginees, Vol 33, No 4, () [6] B H Jahomi, A Ajdai, H Nayeb-Hashemi and A Vazii, "Autofettage of Layeed and Functionally Gaded Metal-Ceamic Composite Vessels", Jounal of Composite Stuctues, Elsevie, () [7] Noda, N and Ashida, F, Thee dimensional Tansient Themal Stesses of Reacto Gaphite subjected to Intenal Heat Geneation and Asymmetic Suface Heating, Nuc Eng Des, Vol, (987) [8] Hean, E T, Mechanics of Mateials Vol (), pegamon pess Ltd, (977) [9] Anthony C Fische-Cipps, "Intoduction to Contact Mechanics", Spinge, New Yok, Second Edition, (7) [] Klans, J Bathe, Finite Element Pocedues, pentice-hall Intenational, Inc, (996) [] Tiupathi, R C and Ashok, D B Intoduction to Finite Elements in Engineeing, Second Edition Pintice, hall of India, (997) [] Y K, Cheuny and MF Yeo, A pactical Intoduction to Finite Element Analysis, Piiman Publishing Limited, (979) [3] E Hinton and DRJ, Owen, Finite Element Pogamming, Academic Pess, (997) 59

10 Yea [4] William D Calliste, J, Mateial Science and Engineeing An Intoduction, John Whiley and Sons, Inc, (7) Stuctue and Loading ae Axisymmetic (a) σyy σxx σxy σzz y z x (b) (c) Fig() Thick Cylinde with Symmetic Stuctue and Loading 6

11 Yea 3 η ζ= ζ 6 7 η= 4 8 ζ=- η=- 5 (a) η η= ζ-η= ζ-η= ζ 8 6 ζ=- ζ= ζη= η=- -ζη= 5 (b) Fig(): 8-Node Quadilateal Isopeimetic Element (a) in x,y space and (b) in ζ, η space 6

12 Wall Y Wall Yea X 3 Fig (3) Mesh Geneation of Compound Thick Cylinde St Al ω Pi To Ti i=5 m m=8 m o= m Fig (4) Compound Thick Cylinde Coss-Section

13 Yea Tempeatue ( C) º Cylinde Thickness (m) 9 Fig(5): Tempeatue Distibution Though the Cylinde Thickness 63

14 Yea 3E8 E8 Stess (Pa) E8 E -E8 -E8-3E8-4E8 Hoop Stess Longitudinal Stess Radial Stess -5E Cylinde Thickness (m) Fig(6): Stess Vaiation Thought the Cylinde Thickness Due to Themal Load 6E8 5E8 4E8 3E8 Stess (Pa) E8 E8 E -E8 -E8 Tesca -3E8 Von-Misses -4E8-5E Cylinde Thickness (m) Fig(7): Equivalent Stess Vaiation Though the Cylinde Thickness Due to Themal Load 64

15 6E7 5E7 Hoop Stess Longitudinal Stess Radial Stess Stess (Pa) 4E7 3E7 E7-3E8 E -E Cylinde Thickness (m) Fig(8): Stess Vaiation Thought the Cylinde Thickness Due to Rotational Load 6E7 Tesca 5E7 Von-Misses Stess (Pa) 4E7 3E7 E7 E7 E Cylinde Thickness (m) Fig(9): Equivalent Stess Vaiation Though the Cylinde Thickness Due to Rotational Load 65 Yea

16 6E8 5E8 4E8 Yea Hoop Stess Longitudinal Stess Radial Stess Stess (Pa) 3E8 E8 E8 E -E8 -E8-3E Fig(): Stess Vaiation Thought the Cylinde Thickness Due to Intenal Pessue E9 9E8 8E8 Tesca Von-Misses 7E8 Stess (Pa) 6E8 5E8 4E8 3E8 E8 E8 E -E8 -E8-3E8 5 6 Cylinde 7 Thickness 8 (m) 9 Cylinde Thickness (m) Fig(): Equivalent Stess Vaiation Though the Cylinde Thickness Due to Intenal Pessue 66

17 Yea 7E8 6E8 5E8 4E8 Stess (Pa) 3E8 E8 E8 E -E8 -E8-3E8-4E8-5E8 Hoop Stess Longitudinal Stess Radial Stess -6E Cylinde Thickness (m) Fig(): Stess Vaiation Thought the Cylinde Thickness Due to Total Load 7E8 6E8 Stess (Pa) 5E8 4E8 Tesca Von-Misses 3E8 E8 E Cylinde Thickness (m) Fig(3): Equivalent Stess Vaiation Though the Cylinde Thickness Due to Total Load 67