ANALYSIS OF FAILURE ASSESSMENT FOR SPHERICAL PRESSURE VESSELS

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1 (ISSN ) VOLUME-5, ISSUE-, March 6 ANALYSIS OF FAILURE ASSESSMENT FOR SPHERICAL PRESSURE VESSELS Sumit Goel, Anil Kumar, Abhishek Kr. Goel M.Tech. Scholar, Department o Mechanical Engineering, Subharti University, Meerut, INDIA Assistant Proessor, Department o Mechanical Engineering, Subharti University, Meerut, INDIA Abstract- The maximum stress and strain criteria are popular because they are operationally simple. The distortional energy and tensor polynomial criteria have built-in generality and mathematical consistency. Experimental methods are useul or veriying the correctness o analytical or computational analyses. Since usually stress cannot be measured directly, most experimental methods serve to measure strains by bonding the gauges to the surace o the structure under test. I nothing is known in advance about the strain ield, a three-element rosette is the best choice or inding the elements o the small strain tensor. A three-element rectangular rosette is used in the present study to obtain the principal strains. A simpliied procedure, which can be easily programmed on a digital computer or elasto-plastic stress analysis, is explained in details in which the Relation between normal strains and stresses are written in the orm i i ( i,, ) where is the stress and is the strain.. Subscripts, & reers to the circumerential, meridional and radial directions o a pressure vessel. KeyWords: Failure assessment, Pressure vessel, Experimental stress, Experimental strain.. INTRODUCTION Dierent types o pressure vessels are being commonly used or both aerospace and ground applications: rocket motor cases, storage tanks or carrying required uel and oxidizer in aerospace vehicles, both high and low pressure storage tanks or ground based applications such as gas cylinders, etc. Most o the pressure vessels in industrial practice basically consist o ew shapes: spherical or cylindrical with hemispherical, ellipsoidal, conical, tori-conical, torispherical, or lat end closures. The shell components are welded together by means o langes, orming a shell with a common rotational axis. Ground based pressure vessel design employs the concept o allowable stresses which are selected to have a actor o saety varying rom to 5. In rocket motor cases/ aerospace pressure vessels, the overall actor o saety varies rom.5 to.4. The allowable stress is either yield or ultimate or buckling strength, whereas the actual stress is the state o the stress in the body due to the applied loads. Factor o saety is deined as the ratio o the allowable stress to the actual stress. Two ailure criteria presented by Williams etal [] are recognized by the rocket industry, yielding and racture. Failure due to yielding occurs when some unctional stress or strain is exceeded and racture occurs when an existing crack extends. Various methods are investigated by many researchers [-6] which are being used or the estimation o the maximum ailure pressure. Various ailure theories were developed to understand the ailure behavior o materials in a biaxial stress ield. Experiments with a variety o materials showed that there is no unique ailure theory, which predicts the ailure load accurately or all materials. Designer has to use/establish a suitable ailure theory or the intended materials. So A three-element rectangular rosette theory is used in the present study to obtain the principal strains. A simpliied procedure, which can be easily programmed on a digital computer or elastoplastic stress analysis, is explained in details in which the Relation between normal strains and stresses are written in the orm. i i ( i,, ) where is the stress and is the strain.. Subscripts, & reers to the circumerential, meridional and radial directions o a pressure vessel.. METHODOLOGY The intensity o distribution depends only on a single actor K, which is called as the stress intensity actor. Unlike the concentration coicient, the stress intensity coicient (actor) is a dimensional quantity (MPa m). The description clearly demonstrates the decisive role o the stress intensity actor (K) in racture mechanics. Hence this stress intensity actor is considered to be the object o analytical as well as experimental investigations. For most o the pressure vessel applications, both leakage and rupture are the modes o ailure and are o primary concern. The cracks and deects in the vessels are initially part-through cracks that are generally open to the surace. Understanding o transitioning (the point at which the part-through crack becomes through crack) is necessary to predict the lie to break-through or leakage (assuming stable crack growth). Once break-through has occurred, the leakage rate may be o concern and detailed description o crack growth and opening becomes necessary. The 64 P a g e

2 (ISSN ) VOLUME-5, ISSUE-, March 6 third concern is stable verses unstable crack growth within the transition region. It has generally been assumed that i the subsequent through crack is stable, unstable crack growth in the transition need not be considered. This criterion is the oundation o all widely used leak-beore-break (LBB) criteria. For most o the racture mechanics problems involving surace laws, transition and break through criteria are not addressed explicitly. Rather, a LBB criterion is employed as an upper bound on transition behavior. The maximum stress intensity is calculated or a through crack o length t, and i it is less than K IC, the LBB criterion is satisied. This approach presumes that the initial surace crack length (c) is less than twice the shell wall thickness (t) and that the maximum K I does not exceed K IC at any point prior to the through crack condition. Where and are the stress components in hoop (circumerential) and meridional directions o the thinwalled vessels. The two principal radii o curvature R and R or any axisymmetric shape in radial (r) axial (z) coordinate system are deined as R r dr dz R dr dz d r dz For to be positive (tension), R R, which is necessary condition or a membrane to avoid buckling. One can impose the more restrictive condition on equation (.) that R R or. pr t Fig.: Plastic region around the crack-tip Fig. : Free-body diagram o end section o spherical thin-walled pressure vessel showing pressure (p) and hoop and axial stresses ( and ).. EXPERIMENTAL ANALYSIS The design ormula or thin-walled vessels in the shape o a body o revolution under internal pressure (p) can be derived rom pr R t R pr t Fig. : Free-body diagram o end section o right circular cylindrical thin-walled pressure vessel showing pressure (p) and hoop and axial stresses and ). ( 65 P a g e

3 (ISSN ) VOLUME-5, ISSUE-, March 6 pr t pr t It should be noted that or the spherical pressure vessel, the hoop and axial stresses are equal and are one hal o the hoop stress in the cylindrical pressure vessel. This makes the spherical pressure vessel more icient pressure vessel geometry. Experimental methods are useul or veriying the correctness o analytical or computational analyses. Since usually stress cannot be measured directly, most experimental methods serve to measure strains by bonding the gauges to the surace o the structure under test. I nothing is known in advance about the strain ield, a three-element rosette is the best choice or inding the elements o the small strain tensor. Dally and Riley [5] presented expressions or various types o rosettes to obtain the magnitude and directions or the principal strains. A three-element rectangular rosette is used in the present study to obtain the principal strains. Keil and Benning [6] described the analytical procedure or the computation o stresses rom the measured strains. A simpliied procedure, which can be easily programmed on a digital computer or elasto-plastic stress analysis, is given below Table-: Stresses and strains at the maximum stressed location in the burst ANFOR 5CDV6 steel rocket motor case. The ective stress ( ) and the ective strain ( ) rom the distortion energy principle are [8]: Fig. 4: Photograph o the ailed AFNOR 5CDV6 steel rocket motor case ater the hydro-burst pressure test [7] ( ) one can ind a relation: Pressure (MPa) E S Strains X -6 Stresses (MPa) The inner radius (R i ) and the thickness (t i ) o the cylindrical shell portion o the rocket motor case are.mm and.6mm respectively. The material constants in the constitutive (stress versus strain) relation (.9) rom the achieved strength properties o the material are: Young s modulus, E =.558 GPa; =.5; n =.85; the Poisson s ratio, =.; ultimate tensile strength, ult = 6MPa; and.% proo stress or yield strength, strain hardening exponent, n =.5. ys = 95MPa. The Experimental stress analysis has been carried out considering the recorded strain data during burst testing o several rocket motor cases made o dierent materials. Table-. gives the recorded strain data at the maximum strained location o AFNOR 5CDV6 steel rocket motor case. 66 P a g e

4 /u IJRREST (ISSN ) VOLUME-5, ISSUE-, March 6 Thickness (mm) Failure Pressure, P ( MPa ) Test [6] Equation (.6) max Equation (.7) Equation (.8) Table-: Burst pressure estimations or the 9mm diameter low ormed maraging steel motor case. ( E 86. GPa, MPa, ult ys 8 55MPa, n.7) Table-. gives the details o the dimensions, mechanical properties and ailure pressure data o 9mm diameter low ormed maraging steel motor cases. Failure pressure estimates are ound to be within % o the test results[6]. Pressu re (MPa) Strains X -6 Stresses (MPa) Fig. 5: A mm diameter maraging steel motor case ater burst []. Experimental stress analysis is perormed considering the burst test data [] on a mm diameter and 5.6mm thick, M5 grade maraging steel motor case. Table-. gives the stresses and strains at the maximum stressed mismatch location. The strain-hardening exponent (n) o the material is ound to be.484. The maximum pressure estimated rom equation (.6) or this motor case is.77 MPa, which is 5% higher than the recorded ailure pressure o.5 MPa. Figure-.5 shows the racture occurrence along the long-seam joint in the cylindrical portion o the motor case at the pressure level o.5 MPa. The relationship between K max and can be o the orm [-6]: K max K F m m u u 4. RESULTS & DISCUSSIONS p m= m=.5 m= Table-.: Stresses and strains at the maximum stressed mismatch location in the burst-tested mm diameter maraging steel motor case c [( u /K F ) ] Fig.6: Tensile racture strength curve generated o a very large (ininite) panel having a centre through crack. 67 P a g e

5 (ISSN ) VOLUME-5, ISSUE-, March 6 Fig. 7: Tensile racture strength design curve o a very large (ininite) panel having a centre through crack. 5.ANALYSIS OF FIGURES WITH LABELLLING Fracture analyses have been carried out on the throughthickness and surace cracked structural conigurations. When the crack size is negligibly small, tends to the ultimate strength o the material. Since the stress intensity actor, K I is a unction o load, geometry and crack size, it is more appropriate to have a relationship between stress intensity actor at ailure (K max ) and the ailure stress ( ) rom the racture data o cracked specimens useul or racture strength evaluation o lawed conigurations. Value o ratio racture stress to ultimate stress is decreased with increase in value constant o ck.it is suggested rom this studies that, i one draws a tangent rom the ultimate tensile strength (i.e., ) to the theoretical racture u strength curve, the experimental data or the structural components having smaller cracks are marginally above the tangent line. 6. CONCLUSIONS The achieved strength properties o the material are: ultimate tensile strength, = 469MPa; and.% ult proo stress or yield strength, maximum pressure ( P max ys = 8MPa. The ) estimated are 47.6 and 45.6MPa respectively, which are ound to be 4 to 9% higher than the actual burst pressure o 44MPa. Burst pressure estimations are ound to be reasonably in good agreement with the existing test results. Nomenclature: a u. D A B Crack depth (mm) m=.5 DBC-Design curve c [ ( u /K )] F C b Minor axis o the elliptic hole (mm) c Major axis o the elliptic hole or surace/through crack length (mm) D i andd o E and E S Kc Inner and outer diameters (mm) Young s modulus and secant modulus (MPa) Plane stress racture toughness (MPa m) K F, m, p Fracture parameters K IC K max n n Plane strain racture toughness (MPa m) Stress intensity actor at ailure (MPa m) Strain hardening exponent Parameter deining the shape o the nonlinear stress-strain curve REFERENCES [] F.A. Williams, M. Barrere and N.C. Huang, Fundamental aspects o solid propellant rockets, (AGARD No.6), (969) The Advisory Group or Aerospace and Development, Technical Services, Slough, UK. [] B.H. James, Structural integrity analysis o solid rocket motors, Conerence on Stress and Strain in Engineering, Brisbane, National Committee on Applied Mechanics, (97) The Institution o Engineers, Australia. [] B. NageswaraRao, Fracture o solid rocket propellant grains, Engineering Fracture Mechanics, Vol.4, pp (99). [4] J.H. Faupel, Yielding and bursting characteristics o heavy walled cylinders, Trans ASME, Vol.78, pp.-64 (956). [5] N.L. Svensson, Bursting pressure o cylindrical and spherical vessels, Trans ASME, Vol.8, pp (958). [6] J. Margetson, Burst pressure predictions o rocket motors, AIAA Paper No AIAA/SAE 4th Joint Propulsion Conerence, Las Vegas, NV, USA (July 978). [7] A.P. Beena, M.K. Sundaresan and B. Nageswara Rao, "Destructive tests o 5CDV6 steel rocket motor cases and their application to lightweight design", International Journal o Pressure Vessels & Piping, Vol.6, pp.- (995). [8] B.S.V. Rama Sarma, P.K. Govindan Potti and B. Nageswara Rao, Failure behaviour o an ultra high strength low alloy steel, Materials Science and Technology, Vol.8, pp (). [9] T. Christopher, B.S.V. Rama Sarma, P.K. Govindan Potti, B. Nageswara Rao and K. Sankarnarayan Samy, A comparative study on ailure pressure estimations o unlawed cylindrical vessels, International Journal o Pressure Vessels and Piping, Vol.79, pp.5-66 (). [] A. Subhananda Rao, G. Venkata Rao and B. Nageswara Rao, Eect o long-seam mismatch on the burst pressure o maraging steel rocket motor cases, International Journal o Engineering Failure Analysis, Vol., pp.5-6 (5). [] U. Gamer, The expansion o the elastic-plastic spherical shell with nonlinear hardening, International Journal o Mechanical Sciences, Vol., No.6, pp (988). [] M.M. Megahed, Elastic-plastic behavior o spherical shells with nonlinear hardening properties, International Journal o Solids and Structures, Vol.7, No., pp (99). 68 P a g e

6 (ISSN ) VOLUME-5, ISSUE-, March 6 [] A. Nayebi, R. El Abdi, Cyclic plastic and creep behavior o pressure vessels under thermo-mechanical loading, Computational Materials Science, Vol.5, pp (). [4] X.L. Gao, Strain gradient plasticity solution or an internally pressurized thick-walled spherical shell o an elastic-plastic material, Mechanics o Research Communications, Vol., pp.4-4 (). 69 P a g e