STRESS ANALYSIS OF THICK WALLED CYLINDER
iSTRESS ANALYSIS OF THICK WALLED CYLINDERA thesisSubmitted bySUSANTA CHOUDHURY (109ME0365)In partial fulfillment of the requirementsFor the award of the degree ofBACHELOR OF TECHNOLOGYinMECHANICAL ENGINEERINGUnder the guidance ofDr. H. ROYDepartment of Mechanical EngineeringNational Institute of Technology RourkelaOdisha -769008, India
CERTIFICATEThis is to certify that this report entitled, “Stress analysis of thick walledcylinder” submitted by Susanta Choudhury (109ME0365) in partial fulfillmentof the requirement for the award of Bachelor of Technology Degree in MechanicalEngineering at National Institute of Technology, Rourkela is an authentic workcarried out by them under my supervision.To the best of my knowledge, the matter embodied in this report has not beensubmitted to any other university/institute for the award of any degree or diplomaDate:Dr. H RoyDepartment of Mechanical Engineering(Research Guide)ii
ACKNOWLWDGEMENTI would like to give our deepest appreciation and gratitude to Prof. H Roy, for hisinvaluable guidance, constructive criticism and encouragement during the courseof this project.Grateful acknowledgement is made to all the staff and faculty members ofMechanical Engineering Department, National Institute of Technology, Rourkelafor their encouragement. In spite of numerous citations above, the author acceptsfull responsibility for the content that follows.Susanta Choudhuryiii
ABSTRACTIt is proposed to conduct stress analysis of thick walled cylinder and compositetubes (Shrink fits) subjected to internal and external pressure. Many problems ofpractical importance are concerned with solids of revolution which are deformedsymmetrically with respect to the axis of revolution. The examples of such solidsare: circular cylinders subjected to uniform external and internal pressure. Thestress analysis of thick walled cylinders with variable internal and external pressureis predicted from lame’s formulae.Different case in lame’s formula arethick walledcylinder having both (a) External and Internal pressure (b) Only Internal Pressure(c) Only External Pressure. In case of Composite tubes (Shrink Fit) the contactpressure between the two cylinders is determined then stress analysis is done byapplying external and internal pressure in tube by lame’s formulae.Theoreticalformulae based results are obtained from MATLAB programs. The results arerepresented in form of graphs.iv
TABLE OF CONTENTSTITLEPAGE NO.CERTIFICATEiiACKNOWLWDGEMENTiiiABSTRACTivTABLE OF CONTENTSvLIST OF FIGURESviiNOTATIONSviiiCHAPTER 1: INTRODUCTION1.1 Problem statement11.2 Literature Review2CHAPTER 2:MATHEMATICAL MODELLING2.1Lame’s Problem32.1. a. Plane Stress2.1. a. i. Cylinder subjected to internal pressure only2.1. a. ii. Cylinder subjected to external pressure only2.2. b. Plane Strain778v
TITLEPAGE NO.CHAPTER 3:RESULTS AND DISCUSSIONS3.1 Matlab Programmes3.1.1 Thick Walled Cylinder113.1.2 Shrink Fit23CHAPTER 4: SUMMARY AND CONCLUSION4.1 Summary274.2 Future Scope of Work27REFERENCES28vi
LIST OF FIGURESTITLEPAGE NO.Fig. 3.1: Graph between radial stress and radius for thick walled cylinder subjectedto internal and external pressure12Fig. 3.2: Graph between hoop stress and radius for thick walled cylinder subjectedto internal and external pressure14Fig 3.3:Graph between radial stress and radius for thick walled cylinder subjectedto internal pressure only.16Fig. 3.4:Graph between hoop stress and radius for thick walled cylinder subjectedto internal pressure only18Fig. 3.5: Graph between radial stress and radius for thick walled cylinder subjectedto external pressure only20Fig. 3.6:Graph between hoop stress and radius for thick walled cylinder subjectedto external pressure only22Fig. 3.7: Graph between radial stress and radius in case of shrink fit24Fig. 3.8:Graph between hoop stress and radius in case of shrink fit26vii
NOTATIONSPlane stress in z-axisRadial StressHoop StressShear Stress in rx-planeShear Stress in ry-planeShear Stress in rz-planeStrain in z-directionCircumferential strainRadial strainEYoung’s modulusνPoission ratioInternal PressureExternal PressureContact Pressureviii
CHAPTER-1INTRODUCTION1.1 Problem statement:Thick walled cylinders are widely used in chemical, petroleum, military industriesas well as in nuclear power plants. They are usually subjected to high pressure &temperatures which may be constant or cycling. Industrial problems often witnessductile fracture of materials due to some discontinuity in geometry or materialcharacteristics. The conventional elastic analysis of thick walled cylinders to finalradial & hoop stresses is applicable for the internal pressure up to yield strength ofmaterial.General applicationn of Thick- Walled cylinders include, high pressure reactorvessels used in mettalurgical operations, process plants, air compressor units,pneumatic reservoirs, hydraulic tanks, storage for gases like butane LPG etc.In this Project we are going to analyze effect of internal and External Pressure onThick walled cylinder , How radial stress & hoop Stress will vary with change ofradius. Contact pressure in shrink Fit and it’s affect on hoop stress and radial stressin analysed.1
LITERATURE REVIEWXu& Yu [1] carried down shakedown analysis of internally pressurized thickwalled cylinders, with material strength differences. Through elasto-plasticanalysis, the solutions for loading stresses, residual stresses, elastic limit, plasticlimit & shakedown limit of cylinder are derived.Hojjati&Hossaini [2] studied the optimum auto frittage pressure & optimumradius of the elastic plastic boundary of strain hardening cylinders in planestrain and plane stress conditions. They used both theoretical and & Finiteelement modelling. Equivalent von-Mises stress is used as yield criterion.M. Imanijed& G. Subhash[3] developed a generalized solution for small plasticdeformation of thick- walled cylinders subjected to internal pressure andproportional loading.Y.Z. Chen & X.Y. Lin [4] gave an alternative numerical solution of thickwalled cylinder and spheres made of functionally graded materials.Li &Anbertin [5] presented analytical solution for evaluation of stresses arounda cylinder excavation in an elastoplastic medium defined by closed yieldsurface.2
CHAPTER 2MATHEMATICAL MODELLING2.1 LAME’S PROBLEM-Thick walled cylinder subjected to internal andexternal pressureConsider a cylinder of inner radius a and outer radius b. Let the cylinder to besubjected to internal pressureand external pressure . It will have two casesplane stress case ( 0)or as a plain strain case ( 0)2.1. a. Plane StressLet the ends of the cylinder be free to expand. We shall assume that 0 oursresults just justify this assumption . Owing to uniform radial deformation 0, Neglecting body forces we can write r r 0 rrSince r is the only independent variable the aboveequation can be written asd r r 0dr----------------eq(1)From Hooke’s Law:3
1 r E1 r E r Stresses in terms of strainE r 1 2E r 1 2 r After putting valuesandur E dur 1 2 drr dur E ur 1 2 rdr r bstituting above values in equation (1), we will getdur d dur urr u r 0dr drdr rdur d 2urdu udu r 2 r r r 0drdrdr rdrd 2ur 1 dur ur 2 0dr 2r drrd 1 d ur r 0 dr r dr ur can be found from this equation asur C1r C2r4
Substituting this values in Eq. (2)1 C1 (1 ) C2 (1 ) r 2 E 1 r C (1 ) C2 (1 ) 2 2 11 r r E1 2C1 and C2 are constants of integration and can be found out by applyingboundary conditions.When r a,σr -paWhen r b,σr -pbso1 C(1 ) C(1 ) pa2 1r 2 E 1 C (1 ) C2 (1 ) 2 pb2 11 r E1 2On Solving,C1 1 pa a 2 pbb 2Eb2 a 21 a 2 b 2C2 pa pb E b2 a 2On substituting these values we get, r pa a 2 pbb 2 a 2b 2 pa pb 2 2b2 a 2r b a2pa a 2 pbb 2 a 2b 2 pa pb 2 2b2 a 2r b a25
2.1. a. i. Cylinder Subjected to Internal Pressure onlyIn this case pb 0 and pa p.Hence, r p a 2 b2 1 b2 a 2 r 2 p a 2 b2 2 1 b a2 r 2 These equations show that σr is always a compressive stress and σθ is a tensilestress.2.1. a. ii. Cylinder subjected to external pressure onlyIn this case pa 0 and pb pHence,p b2 a 2 r 2 1 b a2 r 2 p b2 a 2 2 1 2 b a2 r 2.1. b. Plain StrainFor long cylinder stresses are calculated as sate of plane strain.presumed does not vary along the z axis.d r r 0dr------------------eq(3)From Hooke’s law1 r ( z ) E1 ( r z ) E1 z r ( r ) E r 6
As 0 z ( r )1 r (1 ) r E1 (1 ) r EWhile solving r andE r (1 ) E (1 ) r 1 2 1 1 2 1 Putting values ofand u dur (1 ) r r 1 2 1 dr r durur (1 ) drr 1 2 1 ----------------------------eq(4)EESubstituiting these in the equation of equilibriumduduud 1 r r ur r 1 r 0dr drdrr dur d 2ur ur r 2 0drdrrd du ur dr dr r 0 We can writeur C1r C2rPutting these values in equation(4)7
C C1 (1 2 ) 22 r 1 2 1 r C2 C (1 2 )1r 2 1 2 1 EEBoundary conditionsWhen r a,σr -paWhen r b,σr -pb,We can getE 1 2 (1 )C2 C (1 2 ) pa1 a 2 E(1 2 )(1 )C2 C1 (1 2 ) b 2 pb and solving we can find out 1 2 1 pb b 2 pa a 2Ea 2 b22 21 pb pa a bC2 Ea 2 b2C1 Substituiting thse values we can find out that r pa a 2 pbb 2 a 2b 2 pa pb 2 2b2 a 2r b a2pa a 2 pbb 2 a 2b 2 pa pb 2 2b2 a 2r b a28
RESULTS AND DISCUSSIONSProgram for plotting graph between radial stress and radius for thickwalled cylinder subjected to internal and external pressureclose allclear allpa 17000*10 3;pb 1000*10 3;a 0.04;b 0.08;r [a:(b-a)/1000:b];n 1;while(n 1001)sigma r(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) - (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));sigma t(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));n n 1;endplot(r,sigma t)xlabel('r')ylabel('sigma t')9
Fig. 3.1: Variation of radial stress along the radius subjected to internaland external pressureThe graph above shows the variation of radial stress along the radius of thickwalled cylinder subjected to internal and external pressure. The graph showsthat radial stress in this case is a compressive stress as its magnitude is negativethroughout the graph.10
Program for plotting graph between hoop stress and radius for thickwalled cylinder subjected to internal and external pressureclose allclear allpa 17000*10 3;pb 1000*10 3;a 0.04;b 0.08;r [a:(b-a)/1000:b];n 1;while(n 1001)sigma r(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) - (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));sigma t(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));n n 1;endplot(r,sigma t)xlabel('r')ylabel('sigma t')11
Fig. 3.2: variation of hoop stress along radius subjected to internal andexternal pressureThe graph above shows the variation of hoop stress along the radius of thickwalled cylinder subjected to internal and external pressure. The graph showsthat hoop stress in this case is a tensile stress as its magnitude is positivethroughout the graph.12
Program for plotting graph between radial stress and radius for thickwalled cylinder subjected to internal pressure onlyclose allclear allpa 17000*10 3;pb 0;a 0.04;b 0.08;r [a:(b-a)/1000:b];n 1;while(n 1001)sigma r(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) - (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));sigma t(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));n n 1;endplot(r,sigma r)xlabel('r')ylabel('sigma r')13
Fig 3.3: Variation of radial stress along radius subjected to internalpressure only.The graph above shows the variation of radial stress along the radius of thickwalled cylinder subjected to internal pressure. The graph shows that radialstress in this case is a compressive stress as its magnitude is negativethroughout the graph.14
Program for plotting graph between hoop stress and radius for thickwalled cylinder subjected to internal pressure onlyclose allclear allpa 17000*10 3;pb 0;a 0.04;b 0.08;r [a:(b-a)/1000:b];n 1;while(n 1001)sigma r(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) - (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));sigma t(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));n n 1;endplot(r,sigma r)xlabel('r')ylabel('sigma r')15
Fig 3.4:Variation of hoop stress along radius subjected to internal pressureonly.The graph above shows the variation of hoop stress along the radius of thickwalled cylinder subjected to internal pressure. The graph shows that hoop stressin this case is a tensile stress as its magnitude is positive throughout the graph.16
Program for plotting graph between radial stress and radius for thickwalled cylinder subjected to external pressure onlyclose allclear allpa 0;pb 1000*10 3;a 0.04;b 0.08;r [a:(b-a)/1000:b];n 1;while(n 1001)sigma r(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) - (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));sigma t(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));n n 1;endplot(r,sigma t)xlabel('r')ylabel('sigma t')17
Fig 3.5: Variation of radial stress along radius subjected to externalpressure only.The graph above shows the variation of radial stress along the radius of thickwalled cylinder subjected to external pressure. The graph shows that radialstress in this case is a compressive stress as its magnitude is negativethroughout the graph.18
Program for plotting graph between hoop stress and radius for thickwalled cylinder subjected to external pressure onlyclose allclear allpa 0;pb 1000*10 3;a 0.04;b 0.08;r [a:(b-a)/1000:b];n 1;while(n 1001)sigma r(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) - (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));sigma t(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));n n 1;endplot(r,sigma r)xlabel('r')ylabel('sigma r')19
Fig 3.5:Variation of hoop stress along radius subjected to external pressureonly.The graph above shows the variation of hoop stress along the radius of thickwalled cylinder subjected to external pressure. The graph shows that hoop stressin this case is a compressive stress as its magnitude is negative throughout thegraph.20
Program for plotting graph between radial stress and radius for CompositeTubes (Shrink fit)close allclear allp 8000*10 3;a 0.04;b 0.07;c 0.12;r [a:(c-a)/10000:c];n 1;while(r(n) 0.07)pa 0;pb p;sigma r(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) - (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));n n 1;endn n-1;while(n 10001)pa p;pb 0;sigma r(n) ((pa*b 2 - pb*c 2)/(c 2 - b 2)) - (((b 2*c 2)/r(n) 2)*((papb)/(c 2-b 2)));n n 1;endplot(r,sigma r,'k')xlabel('r')ylabel('sigma r')21
Fig. 3.7: Variation of radial stress along radius in case of shrink fitThe graph above shows the variation of radial stress along the radius in case ofshrink fit. The graph shows that radial stress in this case is a compressive stressas its magnitude is negative throughout the graph. It can also be seen that themagnitude of radial stress increases upto the external radius of the inner radiusand then decreases till it reaches zero at the external radius of the outer cylinder.22
Program for plotting graph between stress and radius for CompositeTubes (Shrink fit)close allclear allp 8000*10 3;a 0.04;b 0.07;c 0.12;r [a:(c-a)/10000:c];n 1;while(r(n) 0.07)pa 0;pb p;sigma t(n) ((pa*a 2 - pb*b 2)/(b 2 - a 2)) (((a 2*b 2)/r(n) 2)*((papb)/(b 2-a 2)));n n 1;endn n-1;while(n 10001)pa p;pb 0;sigma t(n) ((pa*b 2 - pb*c 2)/(c 2 - b 2)) (((b 2*c 2)/r(n) 2)*((papb)/(c 2-b 2)));n n 1;endplot(r,sigma t,'k')xlabel('r')ylabel('sigma t')23
Fig. 3.7: Variation of hoop stress along radius in case of shrink fitThe graph above shows the variation of hoop stress along the radius in case ofshrink fit. The graph shows that hoop stress in this case is a compressive stressas its magnitude is negative upto the external radius of the inner cylinder and isa tensile stress from thereon till the external radius of the outer cylinder as itsmagnitude is positive.24
CHAPTER 44.1 SummaryAn attempt has been made to know the load capacity of a thick walled cylinder .Classical bookwork formulas have been employed to obtain the stressdistribution in cylinder subjected to internal and external pressure. Variation ofstress across the thickness are shown using Matlab.4.2 Future Scope of WorkIn this Project,Stress analysis is done on simple thick walled cylinder. But inindustries there is wide application of thick walled cylinder having holes.Stressanalysis of that should be done in future. Plasticity and yield strength ofmaterial should be analysed so that we can have better understanding whileusing. In case of shrink fit there was no external forces ware applied in analysis.In future it should be analysed with exernal forces25
REFERENCES[1][2][3][4][5]S. Xu & M. Yu, “Shakedown Analysis of thick walled cylinderssubjected to internal pressure with inified strength criterion”,International Journal of Pressure vessels & piping, Vol. 82,pp706712,2005M. H. Hojjati & A. Hossaini, “Theoretical & Finite Element Modellingof autofrettage process in strain hardening thick walled cylinders”,International Journal of Pressure vessels & piping, Vol. 84.pp 310-319,2007M. Imaninejad & G. Subhash, “proportional loading of thick walledcylinders”, International Journal of Pressure vessels & piping, vol. 82,pp129-135, 2005Y. Z. Chen & X. Y. Lin, “An alternative numerical solution of thickwalled cylinders and spheres made of functionally graded materials”,Computational Materials Science, vol. 48, pp640-647, 2010L. Li & M. Anburton, “Anelasto-plastic evaluation of the stress statearound cylindrical openings based on a closed multiaxial yield surface”,International Journal of Pressure vessels & piping, vol. 33, pp193-213,2009.26
Thick walled cylinder , How radial stress & hoop Stress will vary with change of radius. Contact pressure in shrink Fit and it’s affect on hoop stress and radial stress in analysed. 2 LITERATURE REVIEW Xu& Yu [1] carried down shakedown analysis of internally pressurized thick
Internal Pressure, Pi 25 MPa . The thick walled cylinder subjected to only internal pressure at a constant temperature with open-end condition is considered. For a cylinder with open end condition the axial forces are absent hence the axial stress, σ zz 0. The radial and circumferential stress components are given by the equationsCited by: 2Publish Year: 2017Author: Phalguna B N
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2.1 Pressure Vessel Design Model for Cylinders 2.1.1 Thick Wall Theory Thick-wall theory is developed from the Theory of Elasticity which yields the state of stress as a continuous function of radius over the pressure vessel wall . The state of stress is defined relative
This is to certify that the thesis entitled “ Design and Analysis of Thick Walled Cylinder with holes” submitted by Rashmi Ranjan Nath . 4 ANSYS command line code for elastic-plastic analysis 3.4 30 5 Variation of equivalent stresses with internal pressure 3.5 32 . iv
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