Pressure vessels design

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- 1. chapter one Overview of pressure vessels Contents 1.1 Introduction................................................................................................. 1.2 Development of pressure vessel construction codes........................... References ............................................................................................................. 1.1 Introduction Vessels, tanks, and pipelines that carry, store, or receive fluids are called pressure vessels. A pressure vessel is defined as a container with a pressure differential between inside and outside. The inside pressure is usually higher than the outside, except for some isolated situations. The fluid inside the vessel may undergo a change in state as in the case of steam boilers, or may combine with other reagents as in the case of a chemical reactor. Pressure vessels often have a combination of high pressures together with high temperatures, and in some cases flammable fluids or highly radio-active materials. Because of such hazards it is imperative that the design be such that no leakage can occur. In addition these vessels have to be designed carefully to cope with the operating temperature and pressure. It should be borne in mind that the rupture of a pressure vessel has a potential to cause extensive physical injury and property damage. Plant safety and integrity are of fundamental concern in pressure vessel design and these of course depend on the adequacy of design codes. When discussing pressure vessels we must also consider tanks. Pressure vessels and tanks are significantly different in both design and construction: tanks, unlike pressure vessels, are limited to atmospheric pressure; and pressure vessels often have internals while most tanks do not (and those that do are limited to heating coils or mixers). Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 2. Pressure vessels are used in a number of industries; for example, the power generation industry for fossil and nuclear power, the petrochemical industry for storing and processing crude petroleum oil in tank farms as well as storing gasoline in service stations, and the chemical industry (in chemical reactors) to name but a few. Their use has expanded throughout the world. Pressure vessels and tanks are, in fact, essential to the chemical, petroleum, petrochemical and nuclear industries. It is in this class of equipment that the reactions, separations, and storage of raw materials occur. Generally speaking, pressurized equipment is required for a wide range of industrial plant for storage and manufacturing purposes. The size and geometric form of pressure vessels vary greatly from the large cylindrical vessels used for high-pressure gas storage to the small size used as hydraulic units for aircraft. Some are buried in the ground or deep in the ocean, but most are positioned on ground or supported in platforms. Pressure vessels are usually spherical or cylindrical, with domed ends. The cylindrical vessels are generally preferred, since they present simpler manufacturing problems and make better use of the available space. Boiler drums, heat exchangers, chemical reactors, and so on, are generally cylindrical. Spherical vessels have the advantage of requiring thinner walls for a given pressure and diameter than the equivalent cylinder. Therefore they are used for large gas or liquid containers, gas-cooled nuclear reactors, containment buildings for nuclear plant, and so on. Containment vessels for liquids at very low pressures are sometimes in the form of lobed spheroids or in the shape of a drop. This has the advantage of providing the best possible stress distribution when the tank is full. The construction of a typical pressure vessel is shown in Figure 1.1. A spherical pressure vessel is shown in Figure 1.2. This is a special pressure Figure 1.1 Typical pressure vessel. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 3. Figure 1.2 Spherical pressure vessel. vessel and is really a storage sphere. Functionally it acts as a tank because its purpose is to store a fluid. However since it does so at pressures above atmospheric, it can be classified as a pressure vessel. This however does not have internals and operates at atmospheric temperatures. A horizontally supported cylindrical pressure vessel with a hemispherical head and conical transition is shown in Figure 1.3. This consists of a cylindrical main shell, with hemispherical headers and several nozzle connections. The vessel geometries can be broadly divided into plate- and shell-type configurations. The plate-type construction used in flat covers (closures for pressure vessels and heat exchangers) resists pressure in bending, while the shell-type’s membrane action operates in a fashion analogous to what happens in balloons under pressure. Generally speaking the shell-type construction is the preferred form because it requires less thickness (as can be demonstrated analytically) and therefore less material is required for its manufacture. Shell-type pressure components such as pressure vessel and Figure 1.3 Horizontally supported pressure vessel. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 4. heat exchanger shells, heads of different geometric configurations, and nozzles resist pressure primarily by membrane action. Pressure vessels are made in all shapes and sizes, from a few centimeters (cm) in diameter to 50 meters (m) or more in diameter. The pressure may be as low as 0.25 kilopascals (kPa) to as high as 2000 megapascals (MPa). The American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code, Section VIII, Division 1,1 specifies a range of internal pressures from 0.1 MPa to 30 MPa. Pressure equipment, such as the American Petroleum Institute (API) storage tanks are designed to restrict internal pressure to no more than that generated by the static head of the fluid contained in the tank. A few more examples are provided in this chapter. In the area of nuclear power generation a number of coolant systems are used. The two plant cycles most often found in nuclear power plants are the pressurized water reactor and the boiling water reactor. The pressurized water reactor inside the reactor pressure vessel is subjected to a high coolant water pressure. The pressurized water is heated and the pump circulates the water through a heat exchanger (steam generator) where the steam for the turbine is generated. The part of the nuclear power plant containing the reactor coolant is called the primary circuit. Included in the primary circuit is an important vessel called the pressurizer. The coolant volume varies when the load changes require reactor coolant temperature changes, and when this occurs, the pressurizer serves as the expansion tank in the primary system, which allows the water to undergo thermal expansion and contraction keeping the primary circuit pressure nearly constant. If the pressures are allowed to fluctuate too far, steam bubbles might form at the reactor heating surfaces; these bubbles or voids if formed inside the reactor core greatly alter reactor power output. The pressurizer has electric heating elements located low inside to provide the vapor needed to cushion the flowing liquid coolant. All of these items are included in the primary circuit. Figure 1.4 shows a pressurized water reactor (PWR) vessel. A PWR steam generator and a PWR pressurizer are indicated in Figures 1.5 and 1.6, respectively. The rest of the plant is called the secondary circuit. The steam generator produces the steam that passes through the turbine, condenser, condensate pumps, feed pump, feed water heaters and back to the steam generator. Pressure vessels as components of a complete plant are designed to meet various requirements as determined by the designers and analysts responsible for the overall design. The first step in the design procedure is to select the necessary relevant information, establishing in this way a body of design requirements, as shown in Figure 1.7. Once the design requirements have been established, suitable materials are selected and the specified design code will give an allowable design or nominal stress that is used to dimension the main pressure vessel thickness. Additional code rules cover the design of various vessel components such as nozzles, flanges, and so on. Following these rules an arrangement of the various components are finalized and analyzed for failure. Most of the types of Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 5. Figure 1.4 Pressurized water reactor (PWR) pressure vessel. (Courtesy of Westinghouse Electric Company, Pittsburgh, PA.) failure relevant to pressure vessel design are stress dependent and therefore it is necessary to ensure the adequacy of the stress distribution and check against different types of postulated failure modes. The proposed design is finally iterated until the most economical and reliable product is obtained. The functional requirements cover the geometrical design parameters such as size and shape, location of the penetrations, and so on. Some of these parameters may have to be fixed in collaboration with the overall design team, but in a majority of situations the pressure vessel designer acts freely on the basis of his or her experience. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 6. Figure 1.5 Pressurized water reactor (PWR) steam generator. (Courtesy of Westinghouse Electric Company, Pittsburgh, PA.) In the design of pressure vessels safety is the primary consideration, especially for nuclear reactor pressure vessels, due the potential impact of a possible severe accident. In general however, the design is a compromise between consideration of economics and safety. The possible risks of a given mode of failure and its consequences are balanced against the effort required for its prevention; the resulting design should achieve an adequate standard of safety at minimum cost. Safety cannot be absolutely assured for two reasons. First, the actual form of loading during service may be more severe than was anticipated at Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 7. Figure 1.6 Pressurized water reactor (PWR) pressurizer. (Courtesy of Westinghouse Electric Company, Pittsburgh, PA.) the design stage: abnormal, unpredictable loads inevitably occur during the pressure vessel’s lifetime. Second, our knowledge is seldom adequate to provide a qualified answer to the fracture of materials, state of stress under certain conditions, and so on. It is true that although the fundamental mechanism of failure is not sufficiently understood, it is possible to establish preventive measures based on semiempirical methods. Following this line of thinking, the pressure vessels could be classified according to the severity of their operations since this will affect both the possibility of failure and its Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 8. Figure 1.7 Design procedure. consequences. These considerations lead to the classification of vessels ranging from nuclear reactor pressure vessels at one end to underground water tanks at the other. The design factor used in the ASME Boiler and Pressure Vessel Code1 is intended to account for unknown factors associated with the design and construction of the equipment. The design formulas and the stress analysis methods are generally approximate and have built-in assumptions. Typically it is assumed that the material is homogeneous and isotropic. In the real world the material has flaws and discontinuities, which tend to deviate from this assumption. In 1925, the rules for construction of power boilers was written using a design factor of 5 which was subsequently reduced to 4 in 1942, presumably to help conserve steel. In 1955, new processes in the petrochemical industry were requiring significant design pressures requiring wall thickness in vessels to be between 150 and 200 millimeters (mm). The ASME Pressure Vessel and Code Committee decided to form a task group with the allowable stresses based on a design factor of 3. The purpose was to reduce fabrication costs, with the implied assumptions that this could be applied to limited materials, with the addition of fracture toughness rules along with design rules for cyclic operation (fatigue) and that detailed stress analysis was used for most loading conditions. The committee felt that the nuclear code for pressure vessels would be easier to write than the code for pressure vessels used in petrochemical processes. This is because the nuclear pressure vessels only contained steam and water, and the maximum temperature was 800F (427C). Many nuclear plant design specifications identified that the design cycles for fatigue evaluation should be based on a 40-year life expectancy of the plant. The 40 years was based on nuclear plants being able to last twice as long as fossil plants (which usually lasted Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 9. 20 years). The design for cyclic operation was based on the estimated cycles for 40 years, with the severity of the cycles based on estimated worst conditions. This method of design was a rough attempt to ensure freedom from fatigue cracking during the 40-year period. Using the code fatigue curves, a cumulative usage factor was calculated that was arbitrarily required to be equal to less than unity, which is based on the estimated number of cycles for the postulated 40-year period. The methodology has many conservative design factors in it, namely a factor of 2 for stress and a 20 for cycles. 1.2 Development of pressure vessel construction codes Numerous boiler explosions took place through the late 1800s and early 1900s. This led to the enactment of the first code for construction of steam boilers by the Commonwealth of Massachusetts in 1907. This subsequently resulted in the development and publication of the ASME Boiler and Pressure Vessel Code in 1914, which sought to standardize the design, manufacturing, and inspection of boilers and pressure vessels. In 1921 the National Board of Boiler and Pressure Vessel Inspectors was organized to promote consistent inspection and testing. The publication of the section on locomotive boilers also appeared in 1921. The ASME and the ASTM (American Society for Testing and Materials) material specification merged in 1924. The first publication of Section VIII ‘‘Unfired Pressure Vessels,’’ appeared in 1925. This document was referred to as one of a theoretical factor of safety of 5. The petroleum industry did not consider it to be adequate for their purposes and also desired better utilization of available materials. The year 1928 saw the advent of welded pressure vessels. For higher pressures the welded shells were made thicker than 70 mm. These required nondestructive examination (NDE) before service. In 1934, a joint API–ASME Committee published the first edition of an unfired pressure vessel code specifically for the petroleum industry. In 1952 these two separate codes merged into a single code – the ASME Unfired Pressure Vessel Code, Section VIII. The ASME Pressure Vessel Code, Section VIII Division 2: ‘‘Alternative Rules for Pressure Vessels,’’ was published in 1968 and the original code became Section VIII Division 1: ‘‘Pressure Vessels.’’ A considerable boost was provided to the understanding of the basic behavior of pressure vessel components following the development of the nuclear power program in the U.S. and Europe in the late 1950s and early 1960s. Similar developments can be found in the British, French, German and Japanese codes, to name but a few. By 1960 the need for a code for pressure vessels for commercial nuclear plants became imperative. This resulted in publication of the 1963 Edition, Section III: ‘‘Nuclear Pressure Vessels.’’ This was a design by analysis code with a theoretical safety factor of 3. After the publication of Section III: ‘‘Nuclear Pressure Vessels’’ in 1963, it was necessary to modify Section VIII for general pressure vessels. ASME Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 10. Code Section VIII Division 2: ‘‘Alternate Rules for Pressure Vessels’’ appeared as a result and provided a theoretical factor of safety of 3. In 1971, Section III: ‘‘Nuclear Power Components’’ were classified as (a) pumps, (b) valves, and (c) piping. The stress limits for emergency and faulted conditions were introduced. In addition, the addenda of 1971 added storage tanks. The addenda of summer 1972 introduced Appendix G on nonductile failure. The Appendix F on evaluation of faulted conditions was included in the addenda of winter 1972. The design of component supports and core support structures appeared in the addenda of winter 1973. ASME Section III Division 1 is devoted entirely to nuclear power components and also contains the rules for the design of nuclear pumps and valves. The recognition of concrete reactor and containment vessels led to the publication of the Section II Division 2 code in 1975. Three subsections (NB, NC and ND) of ASME Section III Division 1 cover the design and construction of equipment of Classes 1, 2, and 3, respectively. The most stringent is Class 1, which requires design by analysis. Class 2 permits design by analysis as well as the use of formulas. Class 3 prescribes design by formula, and is equivalent to Section VIII Division 1. The designer evaluates the safety function of each pressure vessel and applies the appropriate code class. Design of supports for Section III Division 1 vessels are not prescribed in the ASME Code. Section III has a subsection NF, which prescribes the design of supports for Class 1, 2, and 3 pressure vessels. The addenda of winter 1976 changed the nomenclature of design, normal, upset, testing and faulted conditions to level A, B, C and D service conditions. In the 1982 addenda, the fatigue curves were extended to 1011 cycles. In the 1996 addenda, the design rules for high-temperature service were incorporated. In 1976, Division 3 was published which contained rules on transport of irradiated materials. The need for uniform rules for in-service inspection of nuclear power plants led to the issuance of the 1970 edition of Section XI: ‘‘Rules for In-service Inspection of Nuclear Plant Components.’’ The organization of the ASME Boiler and Pressure Vessel Code is as follows: 1. Section I: Power Boilers 2. Section II: Material Specification: i. Ferrous Material Specifications – Part A ii. Non-ferrous Material Specifications – Part B iii. Specifications for Welding Rods, Electrodes, and Filler Metals – Part C iv. Properties – Part D 3. Section III Subsection NCA: General Requirements for Division 1 and Division 2 i. Section III Division 1: a. Subsection NA: General Requirements b. Subsection NB: Class 1 Components Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 11. c. Subsection NC: Class 2 Components d. Subsection ND: Class 3 Components e. Subsection NE: Class MC Components f. Subsection NF: Component Supports g. Subsection NG: Core Support Structures h. Appendices: Code Case N-47 Class 1: Components in Elevated Temperature Service ii. Section III, Division 2: Codes for Concrete Reactor Vessel and Containment 4. Section IV: Rules for Construction of Heating Boilers 5. Section V: Nondestructive Examinations 6. Section VI: Recommended Rules for the Care and Operation of Heating Boilers 7. Section VII: Recommended Guidelines for Care of Power Boilers 8. Section VIII i. Division 1: Pressure Vessels – Rules for Construction ii. Division 2: Pressure Vessels – Alternative Rules 9. Section IX: Welding and Brazing Qualifications 10. Section X: Fiberglass-Reinforced Plastic Pressure Vessels 11. Section XI: Rules for In-Service Inspection of Nuclear Power Plant Components The rules for design, fabrication and inspection of pressure vessels are provided by codes that have been developed by industry and government in various countries and are indicated in Table 1.1. The design and construction codes all have established rules of safety governing design, fabrication and inspection of boilers, pressure vessels and nuclear components. These codes are intended to provide reasonable protection of life and property and also provide for margin for deterioration in service. Table 1.1 also includes the ASME Boiler and Pressure Vessel Code. Some of the significant features of the latest version of the ASME Code Section III are: Explicit consideration of thermal stress Recognition of fatigue as a possible mode of failure The use of plastic limit analysis Reliable prediction of ductile failure after some plastic action. In addition there is a continuous attempt to understand all failure modes, and provide rational margins of safety against each type of failure. These margins are generally consistent with the consequence of the specific mode of failure. A word or two about the impact of technological advances in pressure vessel design should be mentioned. The last three decades have seen great strides made in the improvement of digital computations. In the 1960s the use of computers began to make an impact on design and analysis of Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 12. Table 1.1 Design and Construction Codes for Pressure Vessels Country Code Issuing authority U.S. ASME Boiler Pressure Vessel Code ASME U.K. BS 1515 Fusion Welded Pressure Vessels BS 5500 Unfired Fusion Welded Pressure Vessels British Standard Institute Germany AD Merblatter Arbeitsgemeinschaft Druckbehalter Italy ANCC Associazione Nationale Per Il Controllo Peula Combustione Netherlands Regeis Voor Toestellen Dienst voor het Stoomvezen Sweden Tryckkarls kommissionen Swedish Pressure Vessel Commission Australia AS 1200:SAA Boiler Code AS 1210 Unfired Pressure Vessels Standards Association of Australia Belgium IBN Construction Code for Pressure Vessels Belgian Standards Institute Japan MITI Code Ministry of International Trade and Industry France SNCT Construction Code for Unfired Pressure Vessels Syndicat National de la Chaudronnerie et de la Tuyauterie Industrielle pressure vessels. The rapid development of finite-element software has remarkably impacted the detailed design of pressure vessel components. These developments along with continuing increase in computing speed and storage capacity of the computer have really made the design process extremely quick and at the same time have led to very accurate design assessment. Initially in the early to mid-1970s, detailed finite-element analyses were generally performed for confirmatory analyses. Today these tasks are routinely accomplished in an interactive mode. The three-dimensional finite-element analysis programs using solid elements are rapidly replacing plate, shell, and two-dimensional programs for routine structural design analysis of pressure vessels. In addition the concepts of computer-aided design (CAD) and computer-aided manufacturing (CAM) are being integrated. In spite of some of the most rigorous, well-conceived safety rules and procedures ever put together, boiler and pressure vessel accidents continue to occur. In 1980, for example, the National Board of Boiler and Pressure Vessel Inspectors reported 1972 boiler and pressure vessel accidents, 108 injuries and 22 deaths.2 The pressure vessel explosions are of course rare nowadays and are often caused by incorrect operation or poorly monitored corrosion. Safety in boiler and pressure vessels can be achieved by: Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 13. Proper design and construction Proper maintenance and inspection Proper operator performance and vessel operation. The design and construction cures are dependent upon the formulation and adoption of good construction and installation codes and standards. Thus the ASME Pressure Vessel Code requires that all pressure vessels be designed for the most severe coincident pressure and temperature expected during the intended service. There can be no deviation from this requirement, even if the severe condition is short term and occurring only occasionally. Bush has presented statistics of pressure vessels and piping failures in the U.S., Germany and the UK.3 He has concluded that a 99 percent confidence upper boundary for the probability of disruptive failure to be less than 1 10–5 per vessel year in the U.S. and Germany. According to his study, periodic inspection is believed to be a significant factor in enhancing pressure vessel reliability, and successful applications of ASME Boiler and Pressure Vessel Codes (Sections I and VIII) are responsible for the relatively low incidence of noncritical failures early in life. Pierre and Baylac authored an international perspective of the design of pressure vessels in 1992.4 They recommend that the governing authorities be vigilant by constantly monitoring accident statistics. They also insist that the authorities be prudent and maintain a flexible attitude in enforcing regulations. References 1. American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, ASME, New York. 2. Jagger, R., Causes of boiler and pressure vessel accidents, Prof. Saf., 29, 39–42, 1984. 3. Bush, S.H., Statistics of pressure vessel and piping failures, Am. Soc. Mech. Eng. J. Pressure Vessel Technol., 110, 225–233, 1988. 4. Pierre, D., and Baylac, G., French pressure vessel regulations within the European context, J. Pressure Technol., 114, 486–488, 1992. Additional Readings Cepluch, R.J., The ASME Boiler and Pressure Vessel Code Committee – challenges: past and future, ASME J. Pressure Vessel Technol., 109, 256–259, 1987. Merrick, R.D., Design of pressure vessels and tanks to minimize corrosion, Mater, Performance, 29–37, 1987. Nasman, G.D., Pressure vessels: total design specification, Chem. Eng., 72–76, 1990. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 14. Pai, D.H., Pressure vessel and piping technology: two decades of progress and future challenges, J. Pressure Vessel Technol., 112, 319–322, 1990. Windenberg, D.F., and Trilling, C., Pressure Vessel and Piping Design: Collected Papers, American Society of Mechanical Engineers, New York, 1960. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 15. chapter two Pressure vessel design philosophy Contents 2.1 General overview....................................................................................... 2.2 Structural and material considerations .................................................. 2.3 Factor of safety ........................................................................................... 2.4 Design by rule ............................................................................................ 2.5 Design by analysis ..................................................................................... References ............................................................................................................. 2.1 General overview Engineering design is an activity to ensure fitness for service. Within the context of pressure vessel design, this primarily involves strength considerations. The ‘‘total design’’ is a topic with far-reaching ramifications. It might include aspects of fuel system design, reactor design, or thermal hydraulic design. In our subsequent discussions, the underlying philoso-phy, decisions and calculations related solely to the strength design are referred to the ‘‘pressure vessel design.’’ For certain pressure vessels and related equipment, preliminary design may still be governed by heat transfer and fluid flow requirements. Although the aspect of thermal hydraulic design is intricately related to the structural design, especially for thermal transient loadings, we will not be discussing them in any detail. It will be assumed that the temperature distribution associated with a particular thermal transient has already been evaluated in a typical design application. However, in these cases the designer still has to consider how the desired configurations of the vessel are to be designed from a structural standpoint and how these designs will perform their intended service. The role of engineering mechanics in the pressure vessel design process is to provide descriptions of the pressure vessel parts and materials in terms Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 16. of mathematical models, which can be analyzed in closed form in a limited number of situations and mostly have to be solved numerically. Even the so-called simple models that can be solved in closed form might involve fairly complex mathematics. In a few isolated instances, intelligent applications of well-known principles have led to simplifying concepts. These concepts have generally eased the designer’s task. However, in a majority of cases, especially when advanced materials and alloys are at a premium, there is a need to make the optimum use of the materials necessitating application of advanced structural analysis. As the complexity of the analysis increases, the aspect of interpretation of the results of the analysis becomes increasingly extensive. Furthermore, a large number of these models approximate the material behavior along with the extent of yielding. As we understand material behavior more and more, the uncertainties and omitted factors in design become more apparent. The improvement will continue as knowledge and cognizance of influencing design and material parameters increase and are put to engineering and economic use. The safety demands within the nuclear industry have accelerated studies on pressure vessel material behavior and advanced the state of the art of stress analysis. For instance, the nuclear reactor, with its extremely large heavy section cover flanges and nozzle reinforcement operating under severe thermal transients in a neutron irradiation environment, has focused considerable attention on research in this area which has been directly responsible for improved materials, knowledge of their behavior in specific environments, and new stress analysis methods. High-strength materials created by alloying elements, manufacturing processes, or heat treatments, are developed to satisfy economic or engineering demands such as reduced vessel thickness. They are con-tinually being tested to establish design limits consistent with their higher strength and adapted to vessel design as experimental and fabrication knowledge justifies their use. There is no one perfect material for pressure vessels suitable for all environments, but material selection must match application and environment. This has become especially important in chemical reactors because of the embrittlement effects of gaseous absorp-tion, and in nuclear reactors because of the irradiation damage from neutron bombardment. Major improvements, extensions and developments in analytical and experimental stress analysis are permitting fuller utilization of material properties with confidence and justification. Many previously insoluble equations of elasticity are now being solved numerically. These together with experimental techniques are being used to study the structural discontinuities at nozzle openings, attachments, and so on. This is significant because 80 percent of all pressure vessel failures are caused by highly localized stresses associated with these ‘‘weak link’’ construction details. It is therefore apparent that the stress concentrations at vessel nozzle openings, attachments, and weldments are of prime importance, and Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 17. methods for minimizing them through better designs and analyses are the keys to long pressure vessel life. Control of proper construction details results in a vessel of balanced design and maximum integrity. In the area of pressure vessel design there are important roles played by the disciplines of structural mechanics as well as material science. As mentioned earlier, we try to provide a description of pressure vessel components in terms of mathematical models that are amenable to closed-form solutions, as well as numerical solutions. The development of computer methods (sometimes referred to as computer-aided design, or CAD) has had a profound impact on the stress and deflection analysis of pressure vessel components. Their use has been extended to include the evaluation criteria as well, by a suitable combination of postprocessing of the solutions and visual representation of numerical results. In a number of cases advanced software systems are dedicated to present animation that aids the visualization and subsequent appreciation of the analysis. A number of design and analysis codes have been developed that proceed from the conceptual design through the analysis, sometimes modeling the nonlinear geometric and material behavior. Results such as temperatures, deflections and stresses are routinely obtained, but the analysis often extends to further evaluations covering creep, fatigue, and fracture mechanics. With the advent of three-dimensional CAD software and their parametric, feature-driven automated design technology, it is now possible to ensure the integrity of designs by capturing changes anywhere in the product development process, and updating the model and all engineering deliverables automatically. Pressure vessel designs that once averaged 24 hours to finish are completed in about 2 hours. Such productivity gains translate into substantial savings in engineering labor associated with each new pressure vessel design. The typical design of a pressure vessel component would entail looking at the geometry and manufacturing construction details, and subsequently at the loads experienced by the component. The load experienced by the vessel is related to factors such as design pressure, design temperature, and mechanical loads (due to dead weight and piping thermal expansion) along with the postulated transients (typically those due to temperature and pressure) that are anticipated during the life of the plant. These transients generally reflect the fluid temperature and pressure excursions of the mode of operation of the equipment. The type of fluid that will be contained in the pressure vessel of course is an important design parameter, especially if it is radioactive or toxic. Also included is the information on site location that would provide loads due to earthquake (seismic), and other postulated accident loads. In assessing the structural integrity of the pressure vessel and associated equipment, an elastic analysis, an inelastic analysis (elastic–plastic or plastic) or a limit analysis may be invoked. The design philosophy then is to determine the stresses for the purpose of identifying the stress concentration, the proximity to the yield strength, or to determine the Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 18. shakedown limit load. The stress concentration effects are then employed for detailed fatigue evaluation to assess structural integrity under cyclic loading. In some situations a crack growth analysis may be warranted, while in other situations, stability or buckling issues may be critical. For demonstrating adequacy for cyclic operation, the specific cycles and the associated loadings must be known a priori. In this context, it is important for a pressure vessel designer to understand the nature of loading and the structural response to the loading. This generally decides what type of analysis needs to be performed, as well as what would be the magnitude of the allowable stresses or strains. Generally the loads acting on a structure can be classified as sustained, deformation controlled, or thermal. These three load types may be applied in a steady or a cyclic manner. The structure under the action of these loads may respond in a number of ways:2 When the response is elastic, the structure is safe from collapse when the applied loading is steady. When the load is applied cyclically a failure due to fatigue is likely; this is termed failure due to high cycle fatigue. When the response is elastic in some regions of the structure and plastic in others, there is the potential to have an unacceptably large deformation produced by both sustained and deformation-con-trolled loads. Cyclic loads or cyclic temperature distributions can produce plastic deformations that alternate in tension and compres-sion and cause fatigue failure, termed low cycle fatigue. Such distribution of loads could be of such a magnitude that it produces plastic deformations in some regions when initially applied, but upon removal these deformations become elastic, and subsequent loading results in predominantly elastic action. This is termed shakedown. Under cyclic loading fatigue failure is likely and because of elastic action, this would be termed as low cycle fatigue. When the sustained loading (due to bending or tension) is such that the entire cross-section becomes plastic, gross collapse of the structure takes place. Ratcheting is produced by a combination of a sustained extensional load and either a strain-controlled cyclic load or a cyclic temperature distribution that is alternately applied and removed. This produces cycling straining of the material which in turn produces incremental growth (cyclic) leading to what is called an incremental collapse. This can also lead to low cycle fatigue. Sustained loads in brittle materials or in ductile materials at low temperatures could result in brittle fracture, which is a form of structural collapse. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 19. 2.2 Structural and material considerations The continued and prolonged use of pressure vessels for power generation, nuclear or chemical reactions, industrial processing, and storage requires them to withstand severe conditions of pressure, temperature, and other environments. Such environmental conditions include corrosion, neutron irradiation, hydrogen embrittlement, and so on. Pressure vessels are required to operate at a temperature range from as high as 600C to as low as –20C, with design pressures as high as 140 MPa. Some vessels are designed to carry noncorrosive fluids; while others are designed to withstand harsh corrosive and highly radioactive environments. The type of service, whether steady or cyclic, may also vary considerably. For each set of operating parameters, the pressure vessel material may be required to have certain properties. For example, operation at very low temperatures would require the use of materials with high notch toughness, while operation at high temperatures would require materials with high creep strength. Apart from the mechanical properties, considerations on manu-facturability, commercial availability, as well as cost, has to be accounted for in the selection process. The materials that are used in pressure vessel construction are: Steels Nonferrous materials such as aluminum and copper Specialty metals such as titanium and zirconium Nonmetallic materials, such as, plastic, composites and concrete Metallic and nonmetallic protective coatings The mechanical properties that generally are of interest are: Yield strength Ultimate strength Reduction of area (a measure of ductility) Fracture toughness Resistance to corrosion The failures that the pressure vessels are to be designed against are generally stress dependent. For this reason it becomes necessary to obtain the stress distribution in the pressure vessels. There is a need to evaluate the operating stresses due to the imposed conditions by analytical methods and sometimes by experimental means. Furthermore we also need to under-stand the significance of these stresses on the structural integrity of the pressure vessel by considering the material properties of the vessel. Developments in aerospace, nuclear, chemical, and petrochemical indus-tries have put demands on pressure vessel materials to sustain thermal shock, dynamics, and cyclic operation (fatigue). Knowledge of the material behavior is necessary not only to ensure that the vessel can withstand the Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 20. loading but also to make sure that the material has been chosen and utilized in an optimum manner. The requirements that are imposed on the design of a pressure vessel by the mode of operation specified for the overall plant are divided into two groups. The first group includes those resulting from the operation at maintained loading either under maximum or normal conditions. For this group the operating pressure (internal and external) existing during the normal operation is required. The second group includes the transient conditions that exist during start-up or shutdown or during a general change in loading. For this group it is necessary to know the maximum maintained pressure that may be anticipated. The fluid temperature is another fundamental requirement. The maximum and minimum values as well as the history of temperature variation need to be known. The material selection is dictated to some extent by this requirement. Further requirements might involve environ-mental characteristics such as corrosion, erosion, and irradiation. Mechanical loads on the pressure vessel include those due to: Pressure Dead weight Seismic factors Piping In addition, snow and wind loadings should be considered wherever applicable. Other loads due to various postulated accidents must also be considered. Pressure vessels are designed for a postulated or expected design life. In addition the possibility of periodic inspections is of importance. Thus it is required to provide inspection ports in terms of handholes or manholes as necessary. The detailed description of the mode of operation, the definition of the rate of change of fluid temperatures as well as the number of occurrences of various transient events need to be specified. The vessels need to be designed according to the severity of operation. For example, pressure vessels for nuclear applications have to be designed according to postulated accidents and associated possible risks of failure, including the release of radioactive materials. This is also the case for vessels with corrosive fluids at high pressure. The energy released in the event of a catastrophic failure is an important consideration in the design of vessels. These considerations lead to a classification of vessels varying from nuclear reactor vessels at one end of the scale to underground water tanks at the other. The designer uses his or her own discretion as to the position of the particular design in the scale of the severity. The stress level is maintained below the allowable level, which is based on consideration of many failures; for example, plastic collapse, fatigue, brittle fracture, or buckling. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 21. Stress analysis involves determining the relationship between the applied loads on the vessel and the associated response in terms of deflections, stresses, and strains. When a bar is subjected to a tensile test we can obtain the stress–strain curve for the material. A common tensile test specimen is shown in Figure 2.1, and a typical stress–strain curve is shown in Figure 2.2.3 The curve generally contains a linear portion depicting the material’s elastic behavior, the modulus of elasticity being the ratio of stress to strain in this region. Beyond the linear portion, or beyond the elastic limit, the stress–strain characteristic is usually nonlinear, and the material is subject to nonrecov-erable plastic deformation. The 0.2 percent offset yield strength refers to the stress which when removed leaves a permanent strain of 0.2 percent in the material. This is the practical definition of the yield strength of a material. When the specimen breaks, the nominal level of stress is the ultimate strength of the material. The older design procedures of pressure vessels were based on sustained loading and on the concepts of the static strength of materials. These were mostly appropriate and adequate, because repetitive loadings were uncommon and parts were designed with ample factors of safety. In recent years, with the development and use of power machinery and equipment, inexplicable failures of ductile materials at stresses below the ultimate strength and sometimes even below the yield strength have taken place. These have been attributed to fatigue, since these failures tended to appear after a period of service. It has been established that the important factor is the repetition of stress rather than the duration of time at a particular stress level. The modern view of the fatigue process is characterized by three main stages: 1. Fatigue crack initiation 2. Fatigue crack growth to a critical size 3. Failure of the net section. The crack is generally believed to initiate at a surface flaw and to spread from this location during the stress cycling until the section is reduced sufficiently for an eventual tensile fracture to take place. Since fatigue Figure 2.1 Tensile test specimen. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 22. failure involves the combined effect of a number of small-scale events taking place over many stress cycles, it is fairly difficult to predict the fatigue life. Some aspects of fatigue, however, can be addressed in a semiempirical way. Cyclic testing may be performed for direct compression and tension, bending, torsion, and in some cases a combination of these factors. The simplest and most frequently used method is the R.R. Moore rotating reversed beam bending test. Here, the beam specimen is subjected to bending by a load applied at its center while being rotated at a constant speed, thus creating a completely reversed bending stress with each revolution. Data from such tests are termed S–N curves – the abscissa indicating the stress level and the ordinate representing the number of cycles to failure. A typical S–N curve for mild steel is shown in Figure 2.3. Initially the stress level S decreases with increase in the number of cycles N, then the curve is shown to approach asymptotically a constant stress value beyond which no further reduction in S takes place with increasing N. This is called the endurance limit of the material. This is not a universal property for all materials; only for some ferritic steels this endurance limit is realized between 106 to 107 cycles. For other materials S is seen to drop, albeit at a small rate, with the number of cycles. Actual service conditions are often characterized by a number of cycles of stress of different magnitudes. One method of assessing this failure from repetitive stresses involves the concept of cumulative damage and posits that fatigue failure will take place when the cumulative damage (the summation of incremental damages) equals unity. This is represented as: Xm 1 ni Ni ¼ 1 ð2:1Þ Figure 2.2 Typical stress–strain curve. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 23. Figure 2.3 S–N curve. where ni is the number of cycles at stress level i, and Ni is the number of cycles to failure at the same stress i. The ratio ni/Ni is the incremental damage or the cycle ratio, and it represents the fraction of the total life that each stress ratio uses up. If the sum of all the different stress cycles (m) is less than unity the vessel is presumed safe. 2.3 Factor of safety The design equations in the various codes of construction always contain factors of safety. Realistically this factor is intended to account for the uncertainties in load, the dimensions, and the material properties. The approach taken in pressure vessel design, however, is to incorporate the types of material properties relevant to different modes of failure. These safety factors are strongly dependent on the modes of failure, as indicated in the design equations. The safety factors are generally applied to the pressure vessel materials so that significant assurances exist that the component can safely perform in the operating environment. Because of the complexity and the multiplicity of demands placed on the material of construction, the allowable stresses (hence the safety factors) are not based on a single material property, but on a combination of a number of properties. These properties could be the tensile strength, the yield strength, elongation, and so on. For example, the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code, Section VIII Division 1: ‘‘Rules for the Construction of Pressure Vessels,’’ used for establishing allowable stress values, advocates using the lesser of the following: 25 percent of the specified minimum tensile strength at room temperature 25 percent of the tensile strength at design temperature Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 24. 62.5 percent of the specified minimum yield strength at room temperature 62.5 percent of the yield strength at design temperature Stress to produce 1 percent creep strain in 100,000 hours at design temperature 80 percent of the minimum stress required to produce material rupture, at the end of 100,000 hours at design temperature For a simple environment a criterion based entirely on yield strength seems appropriate, therefore European pressure vessel construction codes typically employ a factor of safety of 1.5 for the yield strength. 2.4 Design by rule By following design-by-rule methods, the designer simply follows the rules laid out in the procedures for components such as nozzles, heads, flanges, and so on. This procedure has the great advantage of simplicity and consistency but has several limitations. For example, there are cases when the loadings and geometries are such that the procedure cannot be applied effectively. Some of the rules are based on elastic stress analysis with some limitations on maximum stress. Some are based on shakedown concepts without specifically considering stress ranges, while others are based on limit load concepts with suitable shape factors. Design-by-rule methods were used in earlier ASME design codes (Sections I and VIII). Generally speaking, design-by-rule methods of design are based on experience and tests. This process requires the determination of design loads, the choice of a design formula and the selection of an appropriate stress allowable for the material used. The procedure provides the information on required vessel wall thickness as well as the rules of fabrication and details of construction. These rules do not typically address thermal stresses and fatigue. The fatigue issues are considered covered by the factors of safety. 2.5 Design by analysis This philosophy originated in the 1960s and was motivated by the sophisticated design work performed in the nuclear industry at the time. It effectively integrates design and stress analysis efforts and recognizes that different stress states have different degrees of importance. Furthermore, this process accounts for most failure modes and provides rational margins of safety against each mode of failure. The process involves detailed evaluation of actual stress including thermal stresses and fatigue. This design approach provides a rational safety margin (not unduly excessive) based on the actual stress profile and optimizes design to conserve material, leading to consistent reliability and safety. This Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 25. philosophy is appropriate for pressure vessels involving cyclic operation and requiring superior reliability and safety, and is suitable for pressure vessels for which periodic inspection is deemed difficult (e.g., nuclear vessels). This viewpoint was first incorporated into the ASME Boiler and Pressure Vessel Code Section III and Section VIII Division 2 in 1968.1 References 1. American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, ASME, New York. 2. Burgreen, D., Design Methods for Power Plant Structures, C.P. Press, New York, 1975. 3. Harvey, J.F., Theory and Design of Pressure Vessels, Van Nostrand Reinhold, New York, 1985. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 26. chapter three Structural design criteria Contents 3.1 Modes of failure ......................................................................................... 3.2 Theories of failure...................................................................................... 3.3 Theories of failure used in ASME Boiler and Pressure Vessel Code.............................................................................................................. 3.4 Allowable stress limits in the ASME Boiler and Pressure Vessel Code.............................................................................................................. 3.5 Service limits............................................................................................... 3.6 Design for cyclic loading.......................................................................... 3.7 Protection against fracture........................................................................ References ............................................................................................................. Problems................................................................................................................ 3.1 Modes of failure Two basic modes of failure are assumed for the design of pressure vessels. These are: (a) elastic failure, governed by the theory of elasticity; and (b) plastic failure, governed by the theory of plasticity. Except for thick-walled pressure vessels, elastic failure is assumed. When the material is stretched beyond the elastic limit, excessive plastic deformation or rupture is expected. The relevant material properties are the yield strength and ultimate strength. In real vessels we have a multiaxial stress situation, where the failure is not governed by the individual components of stress but by some combination of all stress components. 3.2 Theories of failure The most commonly used theories of failure are: Maximum principal stress theory Maximum shear stress theory Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 27. Maximum distortion energy theory According to the maximum principal stress theory, failure occurs when one of the three principal stresses reaches a stress value of elastic limit as determined from a uniaxial tension test. This theory is meaningful for brittle fracture situations. According to the maximum shear stress theory, the maximum shear equals the shear stress at the elastic limit as determined from the uniaxial tension test. Here the maximum shear stress is one half the difference between the largest (say 1) and the smallest (say 3) principal stresses. This is also known as the Tresca criterion, which states that yielding takes place when ð1 3Þ 2 ¼ y 2 ð3:1Þ The distortion energy theory considers failure to have occurred when the distortion energy accumulated in the component under stress reaches the elastic limit as determined by the distortion energy in a uniaxial tension test. This is also known as the von Mises criterion, which states that yielding will take place when p ð1 2Þ2þð2 3Þ2þð3 1Þ2 1ffiffiffi2 ¼ y ð3:2Þ To understand the essential differences between the Tresca and von Mises criteria let us consider the simplified case of a biaxial stress state, where we assume that the principal stress, 3 is zero. Let us first consider the case of Tresca criterion. We further assume that 1 and 2 have the same sign. Then, following Eq. (3.1), we have 1 3 ¼ y ð3:3aÞ or 2 3 ¼ y ð3:3bÞ This gives 1 ¼ y; 1 ¼ y; 2 ¼ y; 2 ¼ y ð3:4Þ Next we that 1 and 2 are of the opposite sign. The yielding will then take place when 1 2 ¼ y ð3:5Þ Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 28. This implies that 1 2 ¼ y ð3:6aÞ or 2 1 ¼ y ð3:6bÞ If Eqs. (3.4) and (3.6) are plotted with 1 as abscissa and 2 as the ordinate, then we get six straight lines (shown as dashed hexagon in Figure 3.1). The values of 1 and 2 falling on the hexagon and outside would cause yielding. We have of course assumed that the material yield strength is equal in magnitude when in tension or in compression. Next we consider the von Mises criterion. With the assumption that 3 ¼ 0, Eq. (3.2) gives 2 1 12 þ 2 2 ¼ 2 y ð3:7Þ This equation is plotted in the 1 – 2 plot as shown in the solid lines (forming an ellipse) in Figure 3.1. According to the von Mises criterion, the points falling on or outside of the ellipse would cause yielding. Figure 3.1 Tresca and von Mises theories of failure. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 29. 3.3 Theories of failure used in ASME Boiler and Pressure Vessel Code Two basic theories of failure are used in the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code, Section I, Section IV, Section III Division 1 (Subsections NC, ND, and NE), and Section VIII Division 1 use the maximum principal stress theory. Section III Division 1 (Subsection NB and the optional part of NC) and Section VIII Division 2 use the maximum shear stress theory or the Tresca criterion. The maximum principal stress theory (sometimes called Rankine theory) is appropriate for materials such as cast iron at room temperature, and for mild steels at temperatures below the nil ductility transition (NDT) temperature (discussed in Section 3.7). Although this theory is used in some design codes (as mentioned previously) the reason is that of simplicity, in that it reduces the amount of analysis, although often necessitating large factors of safety. It is generally agreed that the von Mises criterion is better suited for common pressure vessels, the ASME Code chose to use the Tresca criterion as a framework for the design by analysis procedure for two reasons: (a) it is more conservative, and (b) it is considered easier to apply. However, now that computers are used for the calculations, the von Mises expression is a continuous function and is easily adapted for calculations, whereas the Tresca expression is discontinuous (as can be seen from Figure 3.1). In order to avoid dividing both the calculated and the yield stress by two, the ASME Code defines new terms called stress intensity, and stress difference. The stress differences (Sij) are simply the algebraic differences of the principal stresses, 1, 2, and 3, so that S1;2 ¼ 1 2;S2;3 ¼ 2 3;S3;1 ¼ 3 1 ð3:8Þ The stress intensity, S, is the maximum absolute value of the stress difference S ¼ max S1;2 ; S2;3 ; S3;1 ð3:9Þ In terms of the stress intensity, S, Tresca criterion then reduces to S ¼ y ð3:10Þ Throughout the design by analysis procedure in the ASME Code stress intensities are used. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 30. 3.4 Allowable stress limits in the ASME Boiler and Pressure Vessel Code The overall objective in determining the allowable stress limits is to ensure that a pressure vessel does not fail within its established design life. The modes that are most likely to cause a failure, as identified by the ASME Code, are as follows:2 Excessive elastic deformation including elastic instability Excessive plastic deformation Brittle fracture Stress rupture or creep deformation (inelastic) Plastic instability and incremental collapse High strain and low cycle fatigue Stress corrosion Corrosion fatigue The first failure mode, namely that of excessive elastic deformation, is generally related to functional requirements. The aspect of elastic instability deals with the propensity of buckling in thin shells. The aspect of excessive plastic deformation could lead to complete collapse as outlined in the previous chapter. This failure mode requires that the analysis be addressed from the standpoint of bursting and gross distortion from a single load application. The failure mode associated with brittle fracture is related to the fracture toughness and is addressed later in this chapter. The failure mode associated with stress rupture or creep is appropriate for pressure vessels operating at high temperatures and as such will not be discussed here. The failure associate with plastic instability and incremental collapse was identified in the previous chapter as ratcheting causing progressive growth due to cyclic load application and should be addressed at the analysis stage. The high strain and low cycle fatigue is an important consideration for cyclic thermal loads. The crack initiation from fatigue damage should be addressed in the analysis. The failure modes associated with stress corrosion and corrosion fatigue are related to the environmental considerations as well as mode of operation. The allowable stress limits in the ASME Code are established on two modes of failure and are characterized as: Avoidance of gross distortion or bursting Avoidance of ratcheting In order for sustained loads to produce collapse in a structure, it is necessary that the loads produce full plasticity over the cross-section bearing the load, leading to what is commonly termed as the ‘‘plastic hinge.’’ The stresses they produce are designated primary stresses. The set Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 31. of primary mean stresses (or primary membrane stresses), Pm, represent the sustained load acting on the structure divided by the cross-sectional area resisting the load. In fact Pm is the stress intensity derived from the stress distribution and as such is the difference between the largest and the smallest of the principal stresses. Pm determines the susceptibility of the structure to fail by plastic collapse. In order to avoid gross distortion it is necessary to avoid a significant portion of the wall of the vessel from becoming fully plastic. For an elastic–perfectly plastic stress strain law (Figure 3.2) such a vessel would be fully plastic when the membrane stress reaches the yield stress. A safety factor of 1.5 is provided to avoid this situation (see Figure 3.3 for the design limit for Pm/Sy). The allowable design stress (primary membrane) is therefore limited to a stress limit typically two-thirds of the yield (referred to as material allowable Sm). Large bending moments acting over the full cross-section can also produce structural collapse. The set of bending stresses generated by sustained bending moments are termed primary bending stresses, Pb, and at any particular point in the structure, being the stress intensities, they represent the differences between the largest and the smallest values of the principal stresses. The mode of collapse is bending, as opposed to extension, and the collapse will take place only when there is complete plastic yielding of the net cross-section. The pattern of plasticity in this plastic hinge so formed, consists of part of the cross-section becoming plastic in tension and the remainder of the section becoming plastic in compression. When there are both direct (membrane) as well as bending stresses, the avoidance of gross distortion or bursting in a vessel is treated in the same way as direct and bending stresses in a rectangular beam. If such a beam is loaded in bending, collapse does not occur until the load has been increased by a factor known as the ‘‘shape factor’’ of the cross-section when a plastic hinge is formed. The shape factor of a rectangular section in bending is 1.5. Figure 3.2 Strain–strain characteristics for an elastic–perfectly plastic material. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 32. Figure 3.3 Membrane plus bending versus membrane stress for a rectangular beam. When the primary stress in a rectangular section consists of a combination of bending and axial tension, the value of the limit load depends on the ratio between the tensile and bending loads. Figure 3.3 shows the value of the maximum calculated stress at the outer fiber of a rectangular section required to produce a plastic hinge plotted against the average tensile stress across the section, with both values expressed as multiples of the yield stress Sy. When the average tensile stress Pm is zero, the failure stress for bending is 1.5 Sy. The ASME Code limits the combination of the membrane and bending to the yield stress Sy. It can be seen from Figure 3.3 that there are variable margins depending on the particular combination of stresses, but it was decided to keep the design limits simple. The repeated plastic straining or ratcheting is sometimes termed incremental collapse. If a structure is repeatedly loaded to progressively higher levels, one can imagine that at some highly stressed region a stage will be reached when the plastic strain will accumulate during each cycle of load, a situation that must be avoided. However, some initial plastic deformation is judged permissible during the first few cycles of load provided the structure shakes down to elastic behavior for subsequent loading cycles. Consider, for example, the outer fiber of a beam strained in tension to a value 1, somewhat beyond the yield strain as shown in Figure Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 33. Figure 3.4 Ratcheting behavior. 3.4(a) by the path OAB. The calculated elastic stress would be S ¼ S1 ¼ E1. When the beam is returned to its undeformed position O, the outer fiber has a residual compressive stress of magnitude S1 – Sy. On any subsequent loading, the residual compression must be removed before the stress goes into tension and thus the elastic stress range has been increased by the quantity S1 – Sy. If S1 ¼ 2Sy, the elastic range becomes 2Sy, but if S1 2Sy, the fiber yields in compression, as shown by the line EF in Figure 3.4(b) and all subsequent cycles produce plastic strain. Therefore the limit of 2Sy could be regarded as a threshold beyond which some plasticity action would progress. 3.5 Service limits The loading conditions that are generally considered for the design of pressure vessels include pressure, dead weight, piping reaction, seismic, thermal expansion and loadings due to wind and snow. The ASME Boiler and Pressure Vessel Code delineates the various loads in terms of the following conditions: Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 34. 1. Design 2. Testing 3. Level A 4. Level B 5. Level C 6. Level D Test conditions refer to the hydrostatic tests that are performed on the pressure vessel during its operating life. Level A service limits correspond to those of normal operating conditions. Level B service limits are sometimes referred to as ‘‘upset’’ conditions, and are those for which the component must withstand without sustaining damage requiring repair. Typically this includes the operating basis earthquake (OBE) and thermal transients for which the power level changes are on the order of 10 to 20 percent. Level C service limits constitute the emergency conditions in which large deformations in the area of discontinuity are created. Level D service limits are so called faulted conditions, for which gross deformation with a loss of dimensional stability is permitted. The component may require repair or removal. Examples are safe shutdown earthquake (SSE), pipe break or a combination of such events. Specifically for the ASME Code, the primary membrane stress intensity, Pm, and the combined membrane plus bending stress intensity, Pm þ Pb, (also the local membrane plus bending stress intensity, PL þ Pb in some cases) for the various loading conditions are shown below. 1. Design condition: Pm Sm Pm þ Pb 1:5Sm ð3:11Þ 2. Testing condition: Pm 0:9Sy Pm þ Pb 1:35Sy; for Pm 0:67Sy Pm þ Pb ð2:15Sy 1:2Pm Þ for 0:67Sy Pm 0:9Sy ð3:12Þ 3. Level C condition (emergency): Pm Sy PL þ Pb 1:5Sy; for Pm 0:67Sy PL þ Pb ð2:5Sy 1:5PLÞ for PL 0:67Sy ð3:13Þ Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 35. 4. Level D condition (faulted): Pm lesser of 0:7Su and 2:4Sm Pm þ Pb lesser of 1:05Su and 3:6Sm ð3:14Þ In the following chapter, the above limits have been critically appraised by introducing the shape factor of the cross-section. The above limits are strictly applicable for rectangular cross-sections. The new limits have been proposed and discussed. 3.6 Design for cyclic loading Due to loads that are applied in a cyclic fashion the material can fail by fatigue when sufficient cycles of loading are applied. The number of cycles that will cause fatigue failure depends on the magnitude of strain that is incurred during each cycle of loading. Fatigue data are generally obtained at room temperature and plotted in the form of nominal stress amplitude (one half of stress range) versus number of cycles to failure. The stress range is obtained by multiplying the strain range from the fatigue test by the modulus of elasticity. The endurance limit is defined as the cyclic stress amplitude, which will not cause fatigue failure regardless of the number of applied cycles of stress. However, for pressure vessels sometimes the endurance limit and one-million cycle fatigue limit are used interchange-ably. Pressure vessel codes commonly use a factor of safety of 2 on the fatigue stress and a safety factor of 20 on fatigue life (number of cycles to failure). The design for cyclic loading is performed to check whether a pressure vessel designed statically will not fail due to multiple stress cycling. The process entails: 1. Identifying design details which introduce stress concentrations and therefore potential sites for fatigue failure 2. Identifying cyclic (or repeated) stresses experienced during service 3. Using appropriate S–N curves and deducing design life. The concept of cumulative damage factor is a simple yet reliable method to determine the factor of safety against fatigue failure. If Ni denotes the allowable number of cycles corresponding to a stress range Si, then the usage factor Ui at the material point due to ni applied number of cycles of stress range Si is Ui ¼ ni Ni ð3:15Þ Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 36. If the material is subjected to m different cycles of frequency ni and corresponding to stress ranges Si (I ¼ 1, 2, . . .m), then the cumulative damage factor, U, is given by U ¼ Xm i¼1 Ui ¼ Xm i1 ni Ni ð3:16Þ Safety from fatigue failure requires U 1 ð3:17Þ The ASME design fatigue curves are based on strain controlled data in which the best fit curves are constructed by a factor of 2 on stress or a factor of 20 on cycles to account for environment, size effect, and data scatter. 3.7 Protection against fracture Pressure vessel materials are primarily steels, and the main point of concern is the effect of temperature on the fracture toughness of steel. Steels are generally ductile, but their resistance to brittle fracture diminishes as the temperature is lowered. The lower limit of the operating temperature is therefore determined by the transition point at which there is a change from ductile to brittle fracture. The value of the stress at fracture under those situations can be considerably lower than the yield strength. The fracture properties including the transition temperature depend on the composition, heat treatment, prior cold work, and the size of the flaws that may be present. As the carbon content is increased from 0.1 to 0.8 percent, the NDT (nil ductility transition) temperature increases from –45C to þ50C. Small amounts of manganese or niobium can produce large decrease in transition temperature. The four design criteria for mild steels can be summarized as follows: 1. NDT design criterion: The maximum principal stress should not exceed 34.5 MPa, to assure fracture arrest at temperatures below NDT temperature. 2. NDT þ17C design criterion: The temperature of operation must be maintained above an NDT of þ17C, to assure that brittle fracture will not take place at stress levels up to one half the yield strength. 3. NDT þ33C design criterion: The temperature of operation must be maintained above an NDT of þ33 C, to assure that brittle fracture will not take place at stress levels up to the yield strength. 4. NDT þ67C design criterion: The temperature of operation must be maintained above an NDT of þ67C, to assure that brittle fracture will not take place at any stress level. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 37. The margin of safety from brittle fracture is therefore dependent on the stress level as well as the expected minimum temperature of operation. Some design codes use a single margin of safety criterion based on energy absorption in a Charpy test conducted at the minimum expected temperature of operation.1 References 1. Burgreen, D., Design Methods for Power Plant Structures, C. P. Press, 1975. 2. Anon., Criteria of the ASME Boiler and Pressure Vessel Code for Design by Analysis in Sections III and VIII, Division 2, American Society of Mechanical Engineers, New York. Problems 1. The in-plane normal stresses in a flat plate are 10 MPa and 60 MPa and the shear stress is 30 MPa. Find the stress intensity and the von Mises equivalent stress. What is the factor of safety corresponding to (a) Tresca criterion, and (b) von Mises criterion if the material yield strength is 150 MPa? 2. The in-plane stresses in a flat plate are –50 MPa and –150 MPa on two perpendicular planes and a shear stress of 40 MPa on those planes. Compute the maximum shear stress, the stress intensity and the von Mises equivalent stress. What is the factor of safety corresponding to (a) Tresca criterion, and (b) von Mises criterion if the material yield strength is 200 MPa? 3. The hoop stress in a cylindrical shell with closed ends is pR/t and the longitudinal stress is pR/(2t), where p is the internal pressure, R the mean radius and t the thickness. If the shell is of diameter 0.5 m and a thickness of 12.5 mm, and is subjected to an internal pressure of 7 MPa, determine the maximum shear stress, the stress intensity and the von Mises equivalent stress. What is the factor of safety corresponding to (a) Tresca criterion, and (b) von Mises criterion if the material yield strength is 160 MPa? 4. A carbon steel pressure vessel is subjected to 1000 pressure cycles at an alternating stress of 300 MPa. At this alternating stress the number of cycles to failure is 7000 from the design fatigue curve. Subsequently the vessel is subjected to 400 temperature cycles at an alternating stress of 700 MPa for which the number of cycles to failure is 600 from the fatigue curve. Is the vessel adequate for the given cyclic loading? Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 38. chapter four Stress categories and stress limits Contents 4.1 Introduction................................................................................................. 4.2 Stress intensity............................................................................................ 4.3 Categorization of stresses ......................................................................... 4.3.1 Primary stress ................................................................................. 4.3.2 Secondary stress ............................................................................. 4.3.3 Peak stress ....................................................................................... 4.4 Stress limits ................................................................................................. 4.5 Special stress limits.................................................................................... 4.6 Practical aspects of stress categorization ............................................... 4.7 Shape factor considerations...................................................................... References ............................................................................................................. Problems................................................................................................................ 4.1 Introduction First of all we need to define the term stress. Stress is a tensor quantity (neither a vector nor a scalar) that depends on the direction of applied load as well as on the plane it acts. Generally speaking, at a given plane there are both normal and shear stresses. However, there are planes within a structural component (that is being subjected to mechanical or thermal loads) that contain no shear stress. Such planes are called principal planes and the directions normal to those planes are called principal directions. The normal stresses (only stresses in those planes) are called principal stresses. For a general three-dimensional stress state there are always three principal planes along which the principal stresses act. In mathematical terms we can say that the problem of principal stresses is an eigenvalue problem, with the magnitudes of the principal stresses being the Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 39. eigenvalues and their directions (normal to the planes on which they act) being the eigenvectors. Principal stress calculations form an essential activity for a general stress analysis problem. 4.2 Stress intensity Let us indicate the principal stresses by 1, 2, and 3. Then we define the stress differences by: S1;2 ¼ 1 2; S2;3 ¼ 2 3; S1;3 ¼ 1 3 ð4:1Þ The stress intensity, SI, is then the largest absolute value of the stress differences, or in other words SI ¼ max S1;2 ; S2;3 ; S1;3 ð4:2Þ The computed stress intensity is then compared with the material allowables taking into consideration the nature of the loading. The material allowables are based on yield and ultimate strength of the material with an implied factor of safety. Within the context of pressure vessel design codes, the comparison of the allowable strength of the material is always done with respect to the stress intensities. This puts the comparison in terms of the appropriate failure theory either the maximum shear stress theory (Tresca criterion) or the maximum distortion energy theory (von Mises criterion). These failure theories have been discussed in some detail in Chapter 3. 4.3 Categorization of stresses Stresses are generally characterized as (a) primary stress, (b) secondary stress, or (c) peak stress. In the following discussion, the primary stresses will be denoted by P, the secondary stress by Q and the peak stress by F. These nomenclatures also apply to the ASME Boiler and Pressure Vessel Code.1 We will now define each of the three categories of stress. 4.3.1 Primary stress Primary stress is any normal stress or a shear stress developed by the imposed loading which is necessary to satisfy equilibrium between external and internal loads. These stresses are not self-limiting. If primary stresses are increased such that yielding through net section occurs, subsequent increase in primary stress would be through strain hardening until failure or gross distortion occurs. Generally primary stresses result from an applied mechanical load, such as a pressure load. The concept of equilibrium is based on a monotonic load and a lower bound limit load Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 40. (for a discussion on limit load and the lower bound theorem see Appendix C). When the limit load is exceeded, gross deformation takes place, hence the qualification ‘‘not self-limiting.’’ A further elaboration of primary stresses is provided in a definition by Pastor and Hechmer where they state: Primary stresses are those that can cause ductile rupture or a complete loss of load-carrying capability due to plastic collapse of the structure upon a single application of load. The purpose of the Code limits on primary stress is to prevent gross plastic deformation and to provide a nominal factor of safety on the ductile burst pressure.2 Primary stresses are further divided into three types: general primary membrane (Pm), local primary membrane (PL), and primary bending (Pb). Quite often the concepts of general primary membrane stress and local primary membrane stress are used interchangeably; the local primary membrane stress representing a general primary membrane stress along a local structural discontinuity. The rigorous definition of the general primary membrane stress is the average primary stress across a solid section produced by mechanical loads, and excludes discontinuities and concen-trations. The local primary membrane stress is defined as also the average stress across any solid section, but includes discontinuities. However, the general primary membrane stress is one that is so distributed in the structure that no redistribution of load occurs as a result of yielding. The failure mode associated with the general primary membrane stress and the local primary membrane stress are meant to be different; the general primary membrane stress leads to gross distortion with no redistribution, and the local primary membrane stress to excessive plastic deformation with redistribution of load. The primary bending is the component of primary stress proportional to the distance from the centroid of the solid section, and is produced by mechanical loads. This definition excludes discontinuities and concentra-tion. The concept of bending stress is akin to the situation of beam bending, with a neutral axis along the center line with regions of tension and compression. The membrane stress is the component having a constant value through the section and represents an average value. 4.3.2 Secondary stress Secondary stress originates through the self-constraint of a structure. This must satisfy the imposed strain or displacement (continuity require-ment) as opposed to being in equilibrium with the external load. Secondary stresses are self-limiting or self-equilibrating. The discontinuity conditions or thermal expansions are satisfied by local yielding and minor distortions. The major characteristic of the secondary stress is that it is a Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 41. strain-controlled condition. Secondary stresses occur at structural dis-continuities and can be caused by mechanical load or differential thermal expansion. The local stress concentrations are not considered for secondary stresses. There is no need for further dividing the secondary stress into membrane and bending categories. In terms of secondary stress we imply secondary membrane and bending in combination. 4.3.3 Peak stress Peak stress is the highest stress in a region produced by a concentration (such as a notch or weld discontinuity) or by certain thermal stresses. Peak stresses do not cause significant distortion but may cause fatigue failure. Some examples of peak stresses include thermal stresses in a bimetallic interface, thermal shock stresses (or stresses due to rapid change in the temperature of the contained fluid), and stresses at a local structural discontinuity. Within the context of local primary membrane stress, PL, as well as secondary stress, Q, the discontinuity effects need not be elaborated. The structural discontinuity can be either gross or local. Gross structural discontinuity is a region where a source of stress and strain intensification affects a relatively large portion of the structure and has a significant effect on the overall stress or strain pattern. Some of the examples are head-to-shell and flange-to-shell junctions, nozzles, and junctions between shells of different diameters or thicknesses. Local structural discontinuity is a region where a source of stress or strain intensification affects a relatively small volume of material and does not have a significant effect on the overall stress or strain pattern or on the structure as a whole. The stress classifications for various parts of a pressure vessel are indicated in Table 4.1 and are reproduced from the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code Sections III and VIII.1 It can be observed that the membrane stress is considered primary for mechanical loads. For a number of geometries and loading situations, the bending stress is considered secondary. The bending stress is considered primary when the net section experiences the applied bending moment. 4.4 Stress limits The allowable stresses (or more correctly the stress intensities) in pressure vessel design codes such as the ASME Boiler and Pressure Vessel Code are not expressed in terms of the yield strength or the ultimate strength but instead as multiples of tabulated design value called the design stress intensity (denoted for example as Sm). This value is typically two-thirds of the yield strength of the material or for other cases one-third of the ultimate Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 42. Classification Pm stress of Type membrane General stress of Origin pressure Internal Location stresses of Classification or shell part 4.1 Table Vessel Cylindrical spherical Q Through thickness gradient Remote from discontinuities Q Q½2 Axial thermal gradient Membrane Bending Junction with head or flange General membrane averaged across full section Bending across full section External load or moment, or internal pressure External load or moment Any shell or head Any section across entire vessel Pm Pm PL Q F Local membrane Bending Peak (fillet or corner) External load or moment, or internal pressure Near nozzle or other opening Q Q Membrane Bending Any location Temperature difference between shell and head Pm Crown Internal pressure Membrane P½3 Pb Bending Dished head or conical head L Q Internal pressure Membrane Bending Knuckle or junction to shell Membrane Bending Membrane Bending Internal pressure Internal pressure Flat head Center region Junction to shell Pm Pb PL Q½2 ½1 Q and F classification of stresses refers to other than design condition. ½2 If the bending moment at the edge is required to maintain the bending stress in the middle to acceptable limits, the edge bending is classified as Pb. Notes: ½3 Consideration should also be given to the possibility of wrinkling and excessive deformation in vessels with large diameter to thickness ratio. Otherwise it is classified as Q. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 43. strength. Therefore we have a factor of safety of 1.5 or 3 in terms of yield strength or ultimate strength, respectively. It is the purpose of the design codes that these multiples of either the yield or the ultimate strength are never exceeded in design. The pressure vessel design codes often make specific recommendations on the limits depending on the conditions (or situations) of design. One typical such classification is in terms of design, normal, and upset (levels A and B), emergency (level C), faulted (level D) and test loadings, and accordingly limits are set appropriately. These stress limits have been discussed in some detail in Chapter 3. As an example, for design conditions the limits for the general primary membrane stress intensity, Pm, the local primary membrane stress intensity, PL, and the combined membrane and bending stress intensity Pm (PL) þ Pb, are typically expressed as: Pm Sm PL Sm PL þ Pb 1:5Sm ð4:3Þ These limits are sometimes higher than the actual operating conditions. It is the intent of the design code that the limit on primary plus secondary stresses be applied to the actual operating conditions. For normal and upset conditions (sometimes indicated as levels A and B), the range of primary and secondary stresses, PL þ Pb þ Q is not allowed to exceed 3Sm, or PL þ Pb þ Q range 3Sm ð4:4Þ A word of caution is needed here. For example, a stress limit on some of the combination of stress categories such as Pm(PL) þ Pb, PL þ Q needs to be carefully understood. The confusion arises because of the tendency to denote the stress intensity in a particular category by the symbol of that category, for example P is the stress intensity for the primary bending stress category. However (PL þ Pb þ Q) is not the sum of the individual components of primary membrane, primary bending and secondary stress intensities. It is in fact the stress intensity evaluated from the principal stresses after the stresses from each category have been added together in the appropriate manner (that is not by adding the stress intensities). The primary plus secondary stress limits are intended to prevent excessive plastic deformation leading to incremental collapse, and to validate the application of elastic analysis when performing the fatigue evaluation. The limits ensure that the cycling of a load range results in elastic response of the material, also referred to as shakedown (when the ratcheting stops). Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 44. 4.5 Special stress limits The theories of failure have been outlined in Chapter 3, the important ones being the von Mises and the Tresca theories. However, none of the theories indicate any limit to the magnitude of the principal stresses, as long as their differences are within the specified limits (stress intensity limits). For uniform triaxial compressive stresses such a position is appropriate. However, for uniform triaxial tensile stresses, failures have been observed to occur, the predictions from the failure theories being that failures would not occur at all. From the available experimental data, it seems reasonable that limiting the mean of the principal stresses to the yield strength would ensure an adequate safety margin against failure, that is 1 þ 2 þ 3 3 Sy ð4:5Þ The ASME Boiler and Pressure Vessel Code takes a somewhat conservative estimate by limiting the mean of the principal stresses to 8/9 Sy, or 1 þ 2 þ 3 3 8 9 Sy ð4:6Þ with an assumed value of the design stress intensity Sm equal to two-thirds Sy, or Sm ¼ 2 3 Sy ð4:7Þ The limit on the sum of the principal stresses becomes equal to 4Sm or 1 þ 2 þ 3 4Sm ð4:8Þ 4.6 Practical aspects of stress categorization Hechmer and Hollinger have proposed ten guidelines on the evaluation of stresses in pressure vessels calculated by finite-element method in the spirit of the ASME Boiler and Pressure Vessel Code.3 1. Guideline 1 establishes the relationship between the membrane and bending stresses and the associated failure modes. The failure modes of concern are collapse/gross distortion (Pm), plastic collapse (Pb), excessive plastic deformation (PL þ Pb), and ratcheting and lack of shakedown (P þ Q). 2. Guideline 2 relates to the first guideline by establishing the finite-element assessment for each failure mode. They maintain that for the Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 45. general primary membrane stress, Pm, finite-element analysis is not necessary, and simple equilibrium equations for net force and net moment are adequate. Finite-element analysis is an appropriate tool for evaluating the local primary membrane stress intensity, PL, the primary membrane plus bending stress intensity, PL þ Pb, and the primary plus secondary stress intensity, P þ Q. For bending of shell cross-sections (as shown in Figure 4.1), simple equations as in Eqs. (4.9) and (4.10) below are proposed Pm ¼ p A ð4:9Þ Pb ¼ 6m t2 ð4:10Þ Here p is the load and m the bending moment per unit length of the shell. 3. Guideline 3 defines stress classification lines (SCL) and stress classification planes (SCP) for the purpose of evaluating membrane and bending stresses. An SCL is shown in Figure 4.1. Figure 4.1 Stress classification line (SCL). Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 46. 4. Guideline 4 establishes the global locations for assessment of stresses, and states that the general primary membrane stress intensity, Pm, should be evaluated remote from a discontinuity; whereas the primary membrane plus bending stress intensity, PL þ Pb, and primary plus secondary stress intensity, P þ Q, should be evaluated at a discontinuity. 5. Guideline 5 establishes the criteria for local locations in terms of the SCL and SCP, and the orientation for the SCL–SCP caused by a discontinuity or blend radius. The blend radii are normally included in the finite-element model. 6. Guideline 6 provides the definition (rewording the code definition) of linearized stress as the stress represented by linear distributions which develop the same net forces and moments on a section as the total stress distribution. 7. Guideline 7 provides procedure for calculating membrane and bending stresses from component stresses, and not principal stresses. Linearized stresses are based on explicit computations by Eqs. (4.9) and (4.10) and not based on arbitrarily linearizing individual stress components. 8. Guideline 8 indicates the method for calculating principal stresses, stress intensities, and stress ranges. All six components of the stresses are linearized individually and then the principal stresses are calculated for the membrane stresses. For the bending stress only two of the component stresses are linearized, leaving the bending stress due to shear and through-thickness components. 9. Guideline 9 provides recommendation on the use of SCL and SCP for various geometries and states that for most axisymmetric situations, SCLs are appropriate. SCPs are recommended for special cases, such as flat plate with penetrations, and are deemed appropriate where the geometry has a well-defined plane that can be directly related to the failure mode. 10. Guideline 10 provides six fundamental recommendations on the evaluation of stresses by finite elements, which are: a. The FEA modeling techniques (mesh refinement, etc.) should be adequate for the level of accuracy needed for structural evalua-tion. b. The finite-element nodes should be such that the location of the SCLs can be readily established. c. An SCL or SCP may start or end at a singularity, because the integration of the loads along the line or the plane may override the effect of the singularity. d. Along discontinuities it is desirable to use an equilibrium type of analysis to obtain more accurate results. e. The failure locations and hence the locations of SCL and SCP should be established based on an overall review of the flow of stresses. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 47. f. The methods recommended above generally apply for shells with R/t ratios greater than 4. For lower ratios of R/t, the level of accuracy is questionable. 4.7 Shape factor considerations The primary stress intensity in the ASME Boiler and Pressure Vessel Code is intended to prevent uncontrolled plastic deformation and to provide a nominal factor of safety on the ductile burst pressure. These limits are based on the principles of limit design. The material is assumed to be elastic–perfectly plastic. For a straight bar in tension, a load producing yield stress, Sy, results in a collapse. If it is loaded in bending, collapse does not occur until the yield moment has been increased by the shape factor of the section. The shape factor, , is defined as the ratio of the load set producing a fully plastic section to the load producing initial yielding of the extreme fibers of the section. The shape factor for a rectangular section in pure bending is 1.5. The current stress intensity limits in the ASME Code rules are based on rectangular cross sections. For combined axial and bending loads, the load set to form a ‘‘plastic hinge’’ depends on the ratio of the tensile and bending loads. An interaction curve as shown in Figure 4.1 is for a rectangular section. Note that when the membrane stress, Pm, is zero, the stress calculated elastically from the collapse moment for bending is 1.5Sy. The factor 1.5 is the shape factor for the rectangular cross section. It should be noted that the current code limits are nonconservative for some sections. Such nonconservatism arises typically for sections with shape factors lower than 1.5 (such as an I-section with a thin web). Chattopadhyay has recommended design equations for different types of loadings to provide adequate safety for all combinations of axial and bending loads.4 The proposed limits in Eqs. (4.11) to (4.13) are intended to replace the existing ones in the ASME Code, and apply to design, level C and testing limits as provided in Eqs (3.6) to (3.8) and shown pictorially in Figure 4.2. The proposed limits are shown below. 1. Design condition: Pm Sm Pm þ Pb Sm ð4:11Þ Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 48. Figure 4.2 ASME Code limits. 2. Testing condition: Pm 0:9Sy Pm þ Pb 0:9Sy; forPm 0:67Sy Pm þ Pb ½ð0:9þ0:1 Þ=ð1 ÞSy ½ð 1Þ=ð1 ÞPm for 0:67Sy Pm 0:9Sy ð4:12Þ 3. Level C condition: Pm ¼ Sy PL þ Pb ¼ S; for PL¼ 0:Sy PL þ Pb ¼ ½ð Þ=ð1 ÞPL þ½ð Þ=ð1 ÞS for Sy ¼ PL¼ 0:Sy ð4:13Þ where S is greater of 1.2Sm or Sy. The value of is 1/ for full sections identified as ones for which no abrupt changes in boundary occur. Copyright 2005 by CRC Press, Inc. All Rights Reserved.
- 49. For non full sections such as the I-sections and T-sections the values of and have been calculated by Chattopadhyay4 and are: I Section: ¼ 1.2, ¼ 0.55 T Section: ¼ 1.7, ¼ 0.75 The procedure for obtaining and will be discussed later. Figures 4.3 to 4.6 give the interaction curves for circle, diamond, I- and T-sections respectively and have been obtained using techniques outlined in Appendix C. These interaction curves represent the upper limits of (Pm þ Pb) as a function of Pm. It is possible to observe certain characteristics in all these figures. At Pm ¼ 0, the limiting value of (Pm þ Pb) is the shape factor of the cross-section times the yield strength (Sy). Then (Pm þ Pb) increases with Pm, reaches a peak and then drops to Sy when Pm reaches Sy. The peak value of (Pm þ Pb) is strongly dependent on the geometry of the Figure 4.3 Interaction curve for a circular cross-section. Copyright 2005 by CRC Press, Inc. All Rights Reserved.