Steel Tanks for Liquid Storage Revised Edition - 2011
The material presented in this publication is for general information only and should not be used without first securing competent advice with respect to its suitability for any given application. The publication of the material contained herein is not intended as a representation or warranty on the part of the Steel Market Development Institute—or of any other person named herein—that this information is suitable for any general or particular use or of freedom from infringement of any patents. Anyone making use of this information assumes all liability arising from such use.
Published by STEEL MARKET DEVELOPMENT INSTITUTE, A business unit of the American Iron and Steel Institute
In cooperation with and editorial collaboration by STEEL PLATE FABRICATORS ASSOCIATION, Div. of STI/SPFA
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cknowledgement is given to the important and valuable contribution made by members of the Steel Plate Fabricators Association in reviewing and updating the material for publication in this current edition. A special note of appreciation is given to Stephen W. Meier, P.E., S.E. of Tank Industry Consultants for his effort in updating this publication. The Steel Market Development Institute gratefully acknowledges the continued investment of its investor steel-producing companies in the steel pipe and tank markets.
Copyright Steel Market Development Institute 2011
STEEL MARKET DEVELOPMENT INSTITUTE 25 Massachusetts Avenue NW, Suite 800 Washington D.C. 20001 ii
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he purpose of this publication is to provide a design reference for the usual design of tanks for liquid storage. Volume 1, "Steel Tanks for Liquid Storage,” deals with the design of flatbottom, cylindrical tanks for storage of liquids at essentially atmospheric pressure. Volume 2, "Useful Information on the Design of Plate Structures,” provides information to aid in design of such structures. For unusual applications, involving materials or liquids not covered within these pages, nor referenced herein, designers should consult more complete treatments of the subject material. Part I contains general information pertaining to carbon plate steels. This section is most helpful to readers who are not intimately familiar with steel industry terminology, practice and classification. Part II deals with the particular carbon steels applicable to tanks for liquid storage. Part III covers the design of carbon steel tanks for liquid storage. Part IV covers materials, design, and fabrication of stainless steel tanks for liquid storage.
Inquiries for further information on the design of steel tanks should be directed to: Steel Plate Fabricators Association Division of STI/SPFA 944 Donata Court Lake Zurich, IL 60047 www.steeltank.com iii
Part I Part II Part III Part IV
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Materials—General ..................................... .................. ..................................... ..................................... ....................... .... 1 Materials—Carbon Steel Tanks for Liquid Storage ......................... ................... ...... 7 Carbon Steel Tank Design................................... ................. ..................................... .............................. ........... 11 Stainless Steel Tanks for Liquid Storage ................................... ................ ......................... ...... 33
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certain elements increases in the liquid portion of the ingot. The resulting product, known as RIMMED STEEL, has marked differences in characteristics across the section and from top to bottom of the ingot. Control of the amount of gas evolved during solidification is accomplished by the addition of a deoxidizing agent, silicon being the most commonly used. If practically no gas evolved, the result is KILLED STEEL, so called because it lies quietly in the ingot. Killed steel is characterized by more uniform chemical composition and properties than other types. Although killed steel is a quality item, the end result is often not so specified by name, but rather by chemical analysis. Other deoxidizing elements are used, but in general, a specified minimum silicon content of 0.10% on heat analysis indicates that a steel is "fully killed." The term SEMIKILLED designates an intermediate type of steel in which a smaller amount of deoxidizer is added. Gas evolution is sufficiently reduced to prevent rimming action, but not sufficiently reduced to obtain the same degree of uniformity as attained in fully killed steels. This controlled evolution of gas during solidification tends to offset shrinkage, resulting in a higher yield of usable material from the ingot. As a practical matter, therefore, plates originating from ingots are usually furnished as semikilled steel unless a minimum silicon content of 0.10% on heat analysis is specified. The steels with which we are concerned are either continuous cast or cast into ingots. The ingots may be hot rolled to a convenient size for further processing, or they may be rolled directly into plates. The current practice is mostly to use continuous casting of the steel. The steel used for continuous casting is fully deoxidized.
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ost of the steel specifications referred to in this manual can be obtained from the American Society for Testing and Materials (ASTM). Each ASTM specification has a number such as A283, and within each specification there may be one or more grades or qualities. Thus an example of a proper reference would be “ASTM designation A283 grade C." In the interest of simplicity, such a reference will be abbreviated to "A283C." ASTM standards are issued periodically to report new specifications and changes to existing ones having a suffix indicating the year of issue such as "A283-C-03." Thus a summary such as is provided here may gradually become incomplete, and it is important that the designer of steel plate structures have the latest edition of ASTM standards available for reference. At least a nodding acquaintance with the terminology of the steel industry is essential to an understanding of steel specifications. This is especially true because, in common with many other industries, a number of shop and trade terms have become so thoroughly implanted in the language that they are used instead of more precise and descriptive technical terms. The following discussions may be of assistance. Practically all steel is made by the electric furnace process or the basic oxygen process. ASTM specifications for the different steels specify which processes are permissible in each case. In most steelmaking processes, the principal chemical reaction is the combination of carbon and oxygen to form a gas. If the oxygen available for this reaction is not removed, the gaseous products continue to evolve during solidification in the ingot. Cooling and solidification progress from the outer rim of the ingot to the center, and during the solidification of the rim, the concentration of
A discussion of the effects of the many elements added to steels would involve a metallurgical treatise far beyond the scope of this work. However, certain elements are common to all steels, and it may be of help to briefly outline the effects of carbon, manganese, phosphorus, and sulfur on the properties of steel. 1
There are some exceptions to these rules in High Strength Low Alloy (HSLA) steels.
CARBON is the principal hardening element in steel, and as carbon increases, hardness increases. Tensile strength increases, and ductility, notch toughness and weldability generally decrease with increasing carbon content. MANGANESE contributes to strength and hardness, but to a lesser degree than carbon. Increasing the manganese content generally decreases ductility and weldability, but to a lesser degree than carbon. Because of the more moderate effects of manganese, carbon steels, which attain part of their strength through the addition of manganese, exhibit greater ductility and improved toughness than steels of similar strength achieved through the use of carbon alone. PHOSPHORUS. Phosphorus can result in noticeably higher yield strength and decreases in ductility, toughness, and weldability. In the steels under discussion here, it is generally kept below a limit of 0.04% on o n heat analysis. SULFUR decreases ductility, toughness, and weldability, and is generally kept below a limit of 0.05% on heat analysis. HEAT ANALYSIS is the term applied to the chemical analysis representative of a heat of steel and is the analysis reported to the purchaser. It is usually determined by analyzing, for such elements as have been specified, a test ingot sample obtained from the front or middle part of the heat during the pouring of the steel from the ladle. PRODUCT ANALYSIS is a supplementary chemical analysis of the steel in the semifinished or finished product form. It is not, as the term might imply, a duplicate determination to confirm a previous result.
Steel is usually considered to be alloy when either: 1. A definite range or definite minimum quantity is required for any of the elements listed above in (1) under carbon steels, or 2. The maximum of the range for alloying elements exceeds one or more of the limits listed in (2) under carbon steels. Again, the HSLA steels demonstrate some exceptions to these general rules. These steels, generally with specified yield point of 50 ksi or higher and containing small amounts of alloying elements, are often employed where high strength or light weight is desired. Mechanical testing of steel plates includes tension, hardness, and toughness tests. The test specimens and the tests are described in ASTM specifications A6, A20, A370, and A673. From the tension tests are determined the TENSILE STRENGTH and YIELD POINT or YIELD STRENGTH, both of which are factors in selecting an allowable design stress, and the elongation over either a 2" or 8" gauge length. Elongation is a measure of ductility and workability. Toughness is a measure of ability to resist brittle fracture. Toughness tests are generally not required unless specified, and then usually because of a low service temperature and/or a relatively high design stress. Conditions under which impact tests are required or suggested will be discussed in connection with specific structures. A number of tests have been developed to demonstrate toughness, and each has its ardent proponents. The test most generally accepted currently, however, is the test using the Charpy V Notch specimen. Details of this specimen and method of testing can be found in ASTM-A370, "Mechanical Testing of Steel Products," and in A20 and A673. Briefly described, an impact test is a dynamic test in which a machined, notched specimen is struck and broken by a single blow in a specially designed testing machine. The energy expressed in foot-pounds required to break the
Steel is usually considered to be carbon steel when: 1. No minimum content is specified or required for chromium, cobalt, columbium, molybdenum, nickel, titanium, tungsten, vanadium, zirconium, or any other element added to obtain desired alloying effect; 2. When the maximum content specified for any of the following elements does not exceed the percentages noted: manganese 1.65, copper 0.60, silicon 0.60; 3. When the specified minimum for copper does not exceed 0.40%. 2
specimen is a measure of toughness. Toughness decreases at lower temperatures. Hence, when impact tests are required, they are usually performed near temperatures anticipated in service.
Referring to Figure 1-1, if the designer has selected a Charpy V Notch value of "x” ft.-lbs, as desirable under special service conditions, it will be noted that the steel illustrated would not be acceptable at temperatures lower than about +35F in the as-rolled condition. In the normalized condition, the same steel would be acceptable down to about -55F, and if quenched and tempered, to about -80F together with an increase in carbon, manganese, or other hardening elements. Note, however, that heat treatment adds to the cost and is indicated only when service conditions indicate the necessity for increased toughness and/or increased strength.
Grain size is affected by both rolling practice and deoxidizing practice. For example, the use of aluminum as a deoxidizer tends to produce finer grains. Unless included in the ASTM specification, or unless otherwise specified, steels may be furnished to either coarse grain or fine grain practice at the producer's option. Fine grain steel is considered to have greater toughness than coarse grain steels. Heattreated fine grain steels will have greater toughness than as-rolled fine grain steels. The designer is concerned only with the question of under what conditions it is justifiable to pay the extra cost of specifying fine grain practice with or without heat treatment in order to obtain improved toughness. Guidelines will be discussed in later sections.
Plate steels are generally defined or classified in two ways. The first classification, which has already been discussed, is based on differences in chemical composition between CARBON STEELS, ALLOY STEELS and HIGH STRENGTH LOW ALLOY STEELS. The second classification is based primarily on the differences in extent of testing between STRUCTURAL QUALITY STEELS and PRESSURE VESSEL QUALITY STEELS 1. It should not be construed that these terms limit the use of a particular steel. Pressure vessel steels are often used in structures other than pressure vessels. The distinction between structural and pressure vessel qualities is best understood by a comparison of the governing ASTM specifications. ASTM designation A6, General Requirements for Rolled Steel Plates for Structural Use, covers a group of common requirements and tolerances for the steels listed therein, the chemical composition and special requirements for which are outlined under separate specification numbers such as A36, A283, A514, etc. Similarly, ASTM designation A20, General Requirements for Steel Plates for Pressure Vessels, covers a group of common requirements and tolerances which apply to a list of about 35 steels, the chemical composition and special requirements for which are outlined under separate ASTM specification numbers.
POST-WELD HEAT TREATMENT consists of heating the steel to a temperature between 1100F and 1250F, furnace cooling until the temperature has reduced to about 600F and then cooling in air. Residual stresses will be reduced by this procedure. NORMALIZING consists of heating the steel to between 1600F and 1700F, holding for a sufficient time to allow transformation, and cooling in air, primarily to affect grain refinement. QUENCHING consists of rapid cooling in a suitable medium from the normalizing temperature. This treatment hardens and strengthens the steel and is normally followed by tempering. TEMPERING consists of reheating the steel to a relatively low temperature (which varies with the particular steel and the properties desired). This temperature normally lies between 1000F and 1250F. Through the quenching and tempering treatment, many steels can attain excellent toughness, and at the same time high strength and good ductility. To illustrate the effect of heat treatment on toughness and strength, refer to Figure 1-1. The numerical values shown apply only to the specific steel described. For other steels, other values would apply, but the trends would be similar.
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Pressure vessel quality steels were previously known as FLANGE and FIRE-BOX qualities, historically inherited terms used to define differences in the extent of testing, but which have no present-day significance insofar as the end use of the steel is concerned.
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Both A6 and A20 define tolerances for thickness, width, length, and flatness, but for the designer the important difference is in the quality of the finished product as influenced by the difference in the extent of testing. A general comparison of the two qualities follows: 1. Chemical Analysis—The requirements for phosphorus and sulfur are more stringent for pressure vessel quality than for structural quality. Both A6 and A20 require one analysis per heat plus the option of product analysis. Product analysis tolerances for structural steels are given in A6. 2. Testing for mechanical properties— a. In general, all specifications for structural quality require two tension tests per heat, size bracket and strength gradation. A6 specifies the general location of the specimens. b. In general, all specifications for pressure vessel quality require either one or two transverse tension tests, depending on heat treatment, from each plate as rolled 2 (and as heattreated, if any). This affords a check on uniformity within a heat. Specification A20 also specifies the location from which the specimens are to be taken. 3. Repair of surface imperfections and the limitations on repair of surface imperfections are more restrictive in A20 than A6.
The arc is formed between the work to be welded and a metal wire which is called the electrode. The electrode may be consumable and add metal to the molten pool, or it may be nonconsumable and of a relatively inert metal, in which case no metal is added to the workpiece. In the welding of steel plate structures, we are concerned principally with five variations of arc welding: 1. Shielded metal arc process (SMAW) 2. Gas metal arc process (GMAW) 3. Flux-cored arc process (FCAW) 4. Electrogas or Electroslag welding 5. Submerged arc process (SAW) In the early days of arc welding, the consumable electrode consisted of a bare wire. The pool of molten metal was exposed to and adversely affected by the gases in the atmosphere. It became obvious that to produce welds with adequate ductility, the molten metal must be protected or shielded from the atmosphere. This led to the development of the shielded metal arc process, in which the electrode is coated with materials that produce a gas as the electrode is consumed which shields the arc from the atmosphere. The coating also performs other functions, including the possible adding of alloying elements as well as slag-forming materials which float to the top and protect the metal during solidification and cooling. In practice, the process is limited primarily to manual manipulation of the electrode. Not too many years ago, this process was almost universally used for practically all welding. It is still widely used for position welding, i.e., welding other than in the down flat position. For the down flat position some of the later processes described below are much faster and hence less costly.
Many plate structures are fabricated by welding. A brief discussion of welding processes follows. Welding consists of joining two pieces of metal by establishing a metallurgical bond between them. There are many different types of welding, but we are concerned only with arc welding. Arc welding is a fusion process in which the bond between the metals is produced by reducing the surfaces to be joined to a liquid state and then allowing the liquid to solidify. The heat required to reduce the metal to liquid state is produced by an electric arc.
In the gas-shielded arc welding process, the molten pool of metal is protected by an externally supplied gas, or gas mixture, fed through the electrode holder rather than by decomposition of the electrode coating. The electrode is a continuous filler-metal (consumable) bare wire and the gases used include helium, argon, and carbon dioxide. In some cases, a tubular electrode is used to
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The term “Plate as rolled” refers to the unit plate rolled from a slab or directly from an ingot in relation to the number and location of specimens, not to its condition.
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facilitate the addition of fluxes or addition of alloys and slag-forming materials. Some methods of this process are called MIG and CO2 welding. The gas-shielded process lends itself to high rates of deposition and high welding speeds. It can be used manually, semi-automatically, or automatically.
It will be observed from the above that all arc welding processes result in rapid heating of the parent metal near the joint to a very high temperature followed by chilling as the relatively large mass of parent plate conducts heat away from the heat-affected zone. This rapid cooling of the weld metal and heataffected zone causes local shrinkage relative to the parent plate and resultant residual stresses. Depending on the chemical composition of the steel, plate thickness and external conditions, special welding precautions may be indicated. In very cold weather, or in the case of a highly hardenable material, preheating a band on either side of the joint will slow down the cooling rate. In some cases post-heat or stress relief as described earlier in this section is employed to reduce residual stresses to a level approaching the yield strength of the material at the post heat temperature. With respect to chemical composition, carbon is the single most important element because of its contribution to hardness, with other elements contributing to hardness but to lesser degrees. It is beyond our scope to provide a definitive discussion on when special welding precautions are indicated. In general, the necessity is dictated on the basis of practical experience or test programs.
This is an arc welding process wherein coalescence is produced by heating with an arc between a continuous filler-material (consumable) electrode and the work. Shielding is obtained from a flux contained within the electrode. Additional shielding may or may not be obtained from an externally supplied gas or gas mixture. This process is a method of gas metal arc welding or flux-cored arc welding wherein molding shoes confine the molten weld metal for vertical position welding. Submerged arc welding is essentially an automatic process, although semi-automatic applications have been used. The arc between a bare electrode and the work is covered and shielded by a blanket of granular, fusible material deposited on the work ahead of the electrode as it moves relative to the work. Filler metal is obtained either from the electrode or a supplementary welding rod. The fusible shielding material is known as melt or flux. In submerged arc welding, there is no visible evidence of the arc. The tip of the electrode and the molten weld pool are completely covered by the flux throughout the actual welding operation. High welding speeds are achieved. It will be obvious that the necessity of depositing a granular flux ahead of the electrode lends itself best to welding on work in the down flat position. Nevertheless, ingenious devices have been developed for keeping flux in place, so that the process has been applied to almost all positions except overhead welding.
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Part II Materials—Carbon Materials—Carbon Steel Tanks For Liquid Storage
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he intent of this publication is to provide information that may be useful in the design of flat-bottom, vertical cylindrical tanks for the storage of liquids at essentially atmospheric pressure. Considerable attention has been directed to tanks storing petroleum-based liquids or water, which constitute most of the tanks built. However, suggestions have been included for storage of liquids meriting special attention, such as acid storage tanks. There are two principal standards in general use in the U.S.: the American Petroleum Institute (API) covering welded steel tanks for petroleum storage, and the American Water Works Association (AWWA) covering tanks for water storage. The abbreviations API and AWWA will be used for the sake of convenience. While API has developed and maintains numerous standards related to the construction, operation and inspection of tanks, the API 650 Standard, Welded Tanks for Oil Storage and the API 620 Standard, Design and Construction of Large, Welded, LowPressure Storage Tanks are the commonly used tank design basis standards. AWWA has also developed and maintains numerous tank-related standards for concrete and steel tanks, included bolted and welded. The most commonly applied and used in this publication is AWWA D100- Welded Carbon Steel Tanks for Water Storage. Both API and AWWA permit the use of a relatively large number of different steel plate materials. In addition, the basic API Standard 650 and AWWA Standard D100 Section 14 provide refined design, construction and inspection rules for tanks designed at higher stresses in which the selection of steel is intimately related to stress level, thickness and service temperature, as well as the type and degree of inspection. As a result, knowledge of available materials and their limitations is equally as important as familiarity with design principles. Useful information concerning plate steel in general has been covered in Part I. It is the purpose of this section to assist in the selection of the proper steel or steels in the construction of tanks for liquid storage.
As you will learn in more detail in Part III of the publication, both the AWWA and the API offer optional methods of shell design. The AWWA basic procedures apply simplified rules which use conservative allowable stress levels. The optional design methods are based on refined procedures that take into account plate grade, service temperature, thickness and higher standards of inspection. It will be obvious that inasmuch as the simplified design provisions of both standards allow identical design stresses for any of the permissible steels, economic considerations will lead to the selection of the least expensive steel that will be satisfactory for the intended service. Steel selection is not so simple and straightforward in the case of tanks built in accordance with either the API or the AWWA Section 14 design provisions. Unstressed portions of such tanks, including bottoms and roofs, will probably be furnished as A36 unless the purchaser specifies otherwise. The selection of material for the shell demands further attention. The design provisions AWWA Section 14 resulted from a desire to utilize newer and improved steels and modern welding and inspection techniques to build tanks of higher quality. The use of higher stresses demanded attention to other properties of steel, primarily toughness. An exhaustive discussion of toughness is beyond the scope of this work, but it can be pointed out that as the stress level increases and temperature decreases, toughness becomes more important. At the stress level existing in API and AWWA basic design criteria tanks, experience has demonstrated that the steels used in combination with the specific welding and inspection rules have proven adequate for the service temperatures involved. Operating at the field of higher stress levels of the optional design methods requires steels having greater toughness. Thanks to research in metals, such steels are readily available. A number of factors enter into making a proper selection. For example, for any given steel, toughness generally decreases as thickness increases. The 7
Although both the API and AWWA standards permit the ordering of plates for certain parts of the tank on a weight rather than thickness basis, there is no longer any economic advantage in doing so.
toughness of carbon steels is improved if part of the hardness and strength is obtained by a higher manganese content and lower carbon at the same strength level. Fine-grained steels exhibit greater toughness than coarse-grained steels; this can be accomplished in the deoxidizing process, and in heat treatment. Thus as thickness increases and service temperature decreases, more stringent attention must be paid to toughness from the standpoint of materials selection and fabrication. The steels permitted by API and AWWA Section 14 for use at these higher stress levels have statistically demonstrated adequate toughness for the thickness and temperature ranges shown. The API standard includes an Impact Exemption chart which establishes requirements for impact testing, based on thickness, temperature and type of material. In the final analysis, the goal is to design the most cost-effective but acceptable tank for a given set of conditions. API and AWWA rules permitting higher design stresses afford a fairly wide selection of steels and stress levels from which to choose. A definitive treatment of economics is beyond the scope of this work. Basically, the factors involved are: 1. Cost of material 2. Weight of material as it affects freight and handling 3. Fabrication, erection and welding costs 4. Inspection and QA/QC costs None of these factors is necessarily conclusive in itself. In any given case, the lightest weight or lowest material cost may or may not be the least expensive overall depending on the relative importance of the factors listed above. The tank fabricator is usually in the best position to judge which steel or combination of steels will permit construction of the most economical, safe tank based on current market conditions. It is generally unwise to specify a more expensive steel than can be justified by the application. There are material costs not associated with quality. The cost of plates will vary according to both width and thickness, and from this consideration tank shell plate approximately 8' wide will commonly be used. Particular situations may dictate the use of wider or narrower plates for all or part of a tank shell. Plate widths of 10 ft to 12 ft are not uncommon.
To this point, only those steels specifically permitted by API or AWWA have been discussed. Other steels have been used to a minor extent by those thoroughly familiar with the problems involved. Among these are the materials referred to in Part I as high strength low alloy steels, manufactured either as proprietary, trade-named steels, or to ASTM specifications. Some of these steels offer the additional attraction of improved atmospheric corrosion resistance, thus eliminating the necessity for painting outside surfaces. As is the case with all high-strength materials, the designer and user must assure themselves that factors other than strength (toughness for example) are properly allowed for in design and construction. For obvious reasons, construction codes often lag behind technical progress. The extensive research facilities of individual steel producers are constantly searching for ways to better serve the needs of our modern economy. But before any construction standard such as those of API and AWWA can accept and permit a new material, it must have been established that it is suitable for the structure in which it will be used. Usually, but not always, acceptance by API and AWWA implies prior acceptance by ASTM. Primarily this is because ASTM specifications clearly delineate the materials to be furnished, whereas any departure from ASTM requires that the standards involved spell out the requirements in corresponding detail. New ASTM steels may or may not eventually find their way into the construction standards, depending on economics and the proven properties of the materials. It should be left to those who have acquired the necessary experience in tank design and construction to pioneer in the use of materials not approved by API or AWWA. The designer, the user, and the fabricator assume added responsibilities in working outside of recognized industry standards. On the other hand, such pioneering by qualified organizations in the past led to the progress represented by the refined procedures of Section 14 of AWWA D100 and API-650. API -650. 8
As in the case of steels already approved by API and AWWA, time and experience will eventually lead to recognition of the steel or combination of steels that will yield the highest quality tank at least cost.
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Part III Carbon Steel Tank Design
excellent. Very few tank failures have been recorded under even abnormal conditions and properly maintained steel tanks have endured long past their original design lives. Before applying them to tanks storing liquids other than water or oil, the designer should consider which philosophy best fits his circumstances. In either case the design standards provide minimum requirements for safe construction and should not be construed as a design manual covering all possible service conditions.
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art III will consider the design of flat bottom, vertical, cylindrical, carbon steel tanks for the storage of liquids at essentially atmospheric pressure and near ambient temperatures. Practically all tanks in the United States within the scope of this part are constructed in accordance with API 650 covering welded steel tanks for oil storage or AWWA D100 covering welded steel tanks for water storage. Tanks of other shapes and subject to gas pressure in addition to liquid head, and tanks subject to extreme low or high temperatures present radically different problems. Consult ASME Section VIII, API 650 Appendices F and M, and API 620 for further information. API 650 and AWWA D100 contain detailed minimum requirements covering inspection. Any attempt to summarize the inspection requirements of either standard would be voluminous and dangerously misleading. It will be the purpose of Part III to discuss only those portions necessary to understand the various design bases. Anyone concerned with fabrication, erection, or inspection must obtain copies of the complete standards. There are basic differences between the standards of API and AWWA. API 650 is an industry standard especially designed to fit the needs of the petroleum industry. The petroleum tank is usually located in isolated areas, or in areas zoned for industry where the probable consequences of mishap are limited to the owner’s property. The owner is conscious of safety, environmental concerns and potential losses in his operations, and will adjust the minimum requirements to suit more severe service conditions. AWWA D100 is a public standard to be used for the storage of water. The water storage tank is often located in the midst of a heavily populated area, often on the highest elevation available. The consequence of catastrophic mishap could not be tolerated in the public interest. The API 650 and AWWA D100 standards have been in existence for many decades (since the 1930s). The performance of the tank population throughout the U.S. has been
Membrane shell theory, as it applies to cylindrical tanks of large diameter, is elementary and needs no explanation here. Starting with the basic premise that circumferential load in a cylinder equals the pressure times the radius, then expressing H and D in feet for convenience, the circumferential load at any level in a vertical cylinder containing water weighing 62.4 #/ft3, can be expressed as: T = 2.6 HD
where T = H
=
D
=
(3-1)
the circumferential load per inch of shell height depth in feet below maximum liquid level tank diameter in feet
Then the minimum design thickness can be expressed as: 2.6 HDG + C (3-2) t (inches ) = SE where G = contained liquid specific gravity S = allowable design stress in psi E = joint factor C = corrosion allowance in inches Obviously the ideal situation would be to vary the thickness uniformly from bottom to top, but since steel plates are rolled to a uniform thickness, any given course of plates is uniform throughout its width. Thus a course designed for the stress at its lower edge will have excess thickness at the top, which will help carry part 11
asymmetric loading criteria depending on the design standard applied.
of the load in the lower portion of the course above. API takes advantage of this and designs each course of plates for the stress existing one foot above the bottom of the course in question. AWWA designs on the basis of stress existing at the lower edge of each course. Application of other methods of shell design, such as the variable point method, is permitted and explained in API 650.
Live—The minimum roof live load shall be 15 psf to account for future maintenance loads and possible accidental vacuum. Seismic—Because of their flexibility and ductility, flat-bottomed cylindrical steel tanks have had an excellent safety record in earthquakes. Steel has the ability to absorb large amounts of energy without fracture. Prior to the Alaskan earthquake of 1964, flat bottom ground storage tanks had an excellent record of surviving western hemisphere earthquakes with essentially no effects other than broken pipe connections or minor buckles. In the Alaskan quake, the horizontal oscillations of the tank contents caused vertical shell stresses of sufficient magnitude to permanently deform the shell in a peripheral accordion-like buckle near the bottom (exaggerated elephant foot buckling). But again, the properties of steel were sufficient to accommodate this deformation without fracture of the shell plates.4 AWWA D100 and API 650 contain recommendations for the seismic design of tanks. The seismic design requirements in both API and AWWA were recently updated to follow the ASCE 7-05 and IBC 2006 requirements.
As outlined in the preceding section, the thickness of the shell is determined by the weight of the product stored. However, there are other loads or forces which a tank may have to resist and which are common to both oil and water tanks. Wind—Historically for tank design , wind pressure has been assumed to be 30 psf on vertical plane surfaces which, when applying shape factors of 0.6 and 0.5 respectively, becomes 18 psf on the projected area of a cylindrical surface, and 15 psf on the projected area of a cone or surface of double curvature as in the case of tank roofs. These loads are considered to be the pressure caused by a wind velocity of 100 MPH. For higher or lower wind velocity, these loads are increased or decreased in proportion to the square of the velocity ratio, 2 (V / 100) , where V is expected wind velocity expressed in miles per hour. In recent years, the ASCE 7 has been the basis of loads for the U.S. buildings codes. This document is more advanced and includes effects of escalation of wind speed with height, increased wind speed along coastal regions and other factors not considered in the original simplified approach of the tank standards. This newer method was adopted; but, for AWWA the historical wind pressures were retained as minimum design pressures.
Negative Pressure— (such as partial vacuum) — Most tanks of this nature at some time will be subject to a negative pressure (partial vacuum) by design or otherwise. Approximately one-half oz. per square inch negative pressure is built into the shell stability formulae in AWWA D100 and API 650. AWWA D100 tanks are not usually designed for negative pressure but negative pressure due to the evacuation of water is considered in the venting requirements. Occasionally API 650 tanks are specified to resist a certain negative pressure, usually expressed in inches of water column. To meet these requirements the shell and roof must be designed to resist the specified negative pressure. See API 650 Appendix V for the current design methods applicable to flat bottom tanks for external pressure.
Snow—Snow load is assumed to be 25 psf on the horizontal projected area of the roof that has a slope of 30 degrees or less with the horizontal plane. If the roof slope is greater than 30 30 degrees, then the snow load may be zero. Snow loads reduction may also be made in regions where the lowest one day mean temperature is 5F or warmer. Fixed steel roofs on tanks are not usually designed for nonsymmetrical loads but if such load conditions are anticipated, these should be considered by the designer. Aluminum geodesic dome roofs may have an
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should remain flexible to facilitate plate seams, nozzles and other interferences. For example, for a shell plate that is 10π feet long, it would be advantageous to use three anchors per plate and space the anchors at approximately 10.5 feet. Obviously the anchor bolt circle must be larger than the tank diameter, but care should be taken so interference will not occur between the anchor bolts and foundation reinforcing. Volume 2 Part V provides design rules for anchor bolt chairs.
Open top tanks require stiffening rings at or near the top of the shell to resist distortion or buckling due to wind. These stiffening rings are referred to as wind girders. In addition, some tank shells of open top and fixed roof tanks require intermediate wind girders to prevent buckling due to wind. API 650 and AWWA D100 provide differing design requirements for intermediate wind girders and are explained in the examples of Appendix A. The formula for maximum height of unstiffened shell is based on the MODIFIED MODEL BASIN FORMULA for the critical uniform external pressure on thin-wall tubes free from end loadings.
As a minimum for all tanks, bottom plates should be l/4" in thickness and lap welded top side only. If corrosion allowance is required for bottom plates, the as-furnished thickness (including corrosion allowance) should be specified. The thickness of annular ring or sketch plates beneath the tank shell may be required to be thicker than the remainder of the bottom plates and any corrosion allowance should be specified as applicable to the calculated thickness or the the minimum thickness. API 650 and AWWA D100 specify minimum shell plate thicknesses based on tank diameter for construction purposes. If corrosion allowance is necessary, it should be added in accordance with the respective standard. A required minimum above those stated in the standards may also be specified, but it should be made clear if this minimum includes the necessary corrosion allowance. As a minimum for all tanks, roof plates should be 3/16" in thickness and lap welded top side only. If corrosion allowance is necessary, it should be added in accordance with the respective standard. A required minimum greater than 3/16" in thickness may be specified, but it should be made clear if this minimum includes the necessary corrosion allowance. If corrosion allowance is necessary for roofsupporting structural members, it should be added in accordance with the respective standard. If a corrosion allowance requirement different from the standards is necessary, it should be made clear what parts of the structure require the additional thickness (flange or web, one side or both sides) and/or the minimum thickness necessary.
The normal proportions of petroleum tanks are such (diameter greater than height) that anchor bolts are rarely needed. It is quite common, however, for the height of water tanks to be greater than the diameter. There is a limit beyond which there is danger that any empty tank will overturn when subjected to the maximum wind velocity. As a good rule of thumb, if C in the following formula exceeds 0.66, anchor bolts are required: C = M
=
d
=
w
=
2 M dw
where
(3-3)
overturning moment due to wind, ft. lb. diameter of shell in feet weight of shell and and portion of roof supported by shell, lb. 4 M W
Design tension load per bolt = ND − N
(3-4)
Where M and and w are as above and N = number of anchor bolts diameter of anchor bolt circle, feet D = The diameter of the anchor bolts shall be determined by an allowable stress of 15000 psi on the net section at the root of the thread with appropriate stress increase for wind or earthquake loading. Because of proportionately large loss of section by corrosion on small areas, it is recommended that no anchor bolt be less than 1.25" in diameter. Maximum desirable spacing of anchors as suggested by API 650 and AWWA D100 is 10'0.This spacing is a matter of judgment and 13
construction difficulties in order to perform the work in a safe manner. The following information is based on API 650. Anyone dealing with tanks should obtain a copy of the complete standard.
Except for open-top tanks and the special requirements applying to self-supporting roofs, tank shells shall be provided with top angles of not less than the following sizes:
API requires that all joints between shell plates shall be butt-welded. Lap joints are permitted only in the roof and bottom and in attaching the top angle to the shell. API 650 offers optional shell design procedures. The refined design procedures permit higher design stresses in return for a more refined engineering design, more rigorous inspection, and the use of shell plate steels which demonstrate improved toughness. The probability of detrimental notches is higher at discontinuities such as shell penetrations. The basic requirements pertaining to welding, stress relief, and inspection relative to the design procedures are important. Tank shells designed in accordance with refined procedures will be thinner than the simplified procedure, and thus will have reduced resistance to buckling under wind load when empty. The shell may or may not need to be stiffened, but must be checked. This is discussed in the section on wind girders.
Tank Tank Diam iameter eter 35 feet and less over 35 to 60 ft. incl. over 60 feet
The selection of roof type depends on many factors. In the oil industry, many roofs are selected to minimize evaporation losses. Inasmuch as the ordinary oil tank is designed to withstand pressures only slightly above atmospheric, it must be vented against pressure and vacuum. The space above the liquid is filled with an air-vapor mixture. When a nearly empty tank is filled with liquid, this air-vapor mixture expands in the heat of the day and the resulting increase in pressure causes venting. During the cool of the night, the remaining airvapor mixture contracts, more fresh air is drawn in, more vapor evaporates to saturate the air-vapor mixture, and the next day the cycle is repeated. Either the loss of valuable "light ends" to the atmosphere from filling, or the breathing loss due to the expansion contraction cycle, is a very substantial loss and has led to the development of many roof types designed to minimize such losses. The floating roof is probably the most popular of all conservation devices and is included as Appendices to API Standard 650. The principle of the floating roof is simple. It floats on the liquid surface; therefore, there is no vapor either to be expelled on filling or to expand or contract from day to night. Inasmuch as all such conservation devices are represented by proprietary and often patented designs, they are beyond the scope of this discussion, which will be limited to the fixed roofs covered by API Standards. API 650 provides rules for the design of several types of fixed roofs. The most common fixed roof is the column-supported cone roof, except for relatively small diameters where the added cost of a self-supporting roof is more than offset by saving the cost of structural framing. The dividing line cannot be accurately defined
Tank bottoms are usually lap welded plates having a minimum nominal thickness of 1/4". After trimming, bottom plates shall extend a minimum of 1 inch beyond the outside edge of the weld attaching the bottom to the shell plates. The attachment weld shall be a continuous fillet inside and out as shown in the following table of sizes: Maximum of Shell Plate
Minimum Size of Fillet Weld*
3/16 over 3/16 to 3/4 over 3/4 to 1-1/4 over 1-1/4 to 1-3/4
3/16 1/4 5/16 3/8
Mini Minimu mum m Size ize of of Top Top Ang Angle le 2 x 2 x 3/16 2 x 2 x 1/4 3 x 3 x 3/8
* Maximum size Fillet 1/2"
Butt-welded bottoms are permissible, but because of cost, are seldom used except in special services. Butt-welded bottoms are usually welded from the top side only using backing strips attached to the underside. Welding from both sides presents significant 14
because different practices and available equipment may affect the decision in any given case. If economy is the only consideration, the purchaser would be well advised to specify the size of tank and let the manufacturer decide whether or not to use a self-supporting roof. A self-supporting roof is sometimes desirable for special service conditions such as an internal floating roof, or where cleanliness and ease of cleaning are especially important. All roofs and supporting structures shall be designed to support dead load plus a live load of not less than 15 psf. Roof plates shall have a minimum nominal thickness of 3/16 inch. Structural members shall have a minimum thickness of 0.17 inch. Roof plates shall be attached to the top angle with a continuous fillet weld on the top side only: 1. If the continuous fillet weld between the roof plates and the top angle does not exceed 3/16 inch and the slope of the roof at the top-angle attachment does not exceed 2 inches in 12 inches, and when the cross-sectional area of the roof-to-shell junction does not exceed W A= (3-5) 201,000 tan θ
connection for the device and the drawings should reflect the need for such a device to be supplied by the customer. The top angle may be smaller than previously noted when a frangible joint is specified. 3. Tanks less than 50 ft. diameter may not be considered to have frangible roof joints even when the provisions of item 1 are satisfied. Supported Cone Roofs — Supported cone roofs are usually lap welded from the top side only with continuous full fillet welds. Plates shall not be attached to supporting members, and shall be attached to the top angle by a continuous 3/16" fillet weld or smaller on the top side if specified by purchaser. The usual slope of supported cone roofs is 3/4" in 12". Increased slopes should be used with caution. The columns transmit their loads directly to the supporting soil through bases resting on but not attached to the bottom plates. Some differential settlement can be expected. A relatively flat roof will follow such variations without difficulty. As pitch increases, a cone acquires stiffness, and instead of smoothly following a revised contour, unsightly local buckles may develop. In general, slopes exceeding 1-1/2" in 12" may be undesirable. Rafters in direct contact with the roof plates may be considered to receive adequate lateral support from friction, but this does not apply to truss chord members, rafters deeper than 15", or roof slopes greater than 2" in 12". Rafters are spaced so that, in the outer ring, their centers are not more than 6.28 feet apart at the shell. Spacing on inner rings does not exceed 5.5 feet. All parts of the supporting structure shall be so proportioned that the sum of the maximum calculated stresses shall not exceed the allowable stresses as stated in the appropriate section of API 650
where W =
total weight of the shell and roof framing supported by the shell in pounds = angle between the roof and a θ horizontal plane at the roofto-shell juncture in degrees, the joint may be considered to be frangible and, in case of excessive internal pressure, will fail before failure occurs in the tank shell joints or the shell-to-bottom joint. Failure of the roof-to-shell joint is usually initiated by buckling of the top angle and followed by tearing of the 3/16 inch continuous weld at the periphery of the roof plates. 2. Where the weld size exceeds 3/16 inch, or where the slope of the roof at the topangle attachment is greater than 2 inches in 12 inches, or when the cross-sectional area of the roof-to-shell junction exceeds the value calculated per Equation 3-5, or where fillet welding from both sides is specified, emergency venting devices in accordance with API Standard 2000 shall be provided by the purchaser. The manufacturer shall provide a suitable tank
Self-Supporting Roofs — Self-supporting cone, dome or umbrella roofs shall conform to the appropriate requirements of API 650 unless otherwise specified by the purchaser.
15
such storage are referred Appendices R and Q. API 650 contains specific designs for approved accessories which include all dimensions, thicknesses, and welding details. For all cases, OSHA requirements must be satisfied. No details are shown, but specifications are included for stairways, walkways and platforms. All such structures are designed to support a moving concentrated load of 1000 Ibs. and the handrail shall be capable of withstanding a load of 200 lbs. applied in any direction at any point on the top rail. Normally all pipe connections enter the tank through the lower part of the shell. Historically tank diameters and design stress levels have been such that the elastic movement of the tank shell under load has not been difficult to accommodate. With the trend to larger tanks and higher stresses, the elastic movement of the shell can become an important factor. Steel being an elastic material, the tank shell increases in diameter when subjected to internal pressure. The flat bottom acts as a diaphragm and restrains outward movement of the shell. As a result, the shell is greater in diameter several feet above the bottom than at the bottom. Openings near the bottom of the tank shell will tend to rotate with vertical bending of the shell under hydrostatic loading. Shell openings in this area, having attached piping or other external loads, should be reinforced not only for the static conditions but also for any loads imposed on the shell connections by the restraint of the attached piping to the shell rotations. Preferably the external loads should be minimized or the shell connections relocated outside the rotation area.
to
API
620
Molasses Tanks — Molasses presents no unusual problems other than the fact that its specific gravity is about 1.48, and the shell design must, of course, take this into account. It is quite common to require such tanks to be built in accordance with API 650. It must be remembered that the API Appendix A design stress of 21,000 psi at 85% joint factor is predicated on the tank being full of water during test, and that the actual stress in petroleum service is usually considerably less. Because molasses is heavier than water, the full design stress is present in service. Thus if the designer is depending on the long and successful record of tanks designed in accordance with API 650 Appendix A design, it would be more consistent with the true situation to use a somewhat lower design stress. On the other hand, on tanks built to the basic design of API 650 this difference between usual petroleum service stress and design stress does not exist. However, the addition of a corrosion allowance is required when warranted by service conditions. Acid and Caustic Tanks — To attempt a comprehensive discussion of the subject of storing acids and caustic solutions is beyond the scope of this work. While stainless steel or other high alloy materials are often required, some acids and caustic solutions can be stored successfully in carbon steel tanks, and the following discussion will be limited to such application. In the absence of personal experience, information concerning the corrosive properties of many common solutions can be found in chemistry and chemical engineers' handbooks or in the publications of the National Association of Corrosion Engineers. However, it should be noted that very small differences in content (such as slight impurities) or conditions can influence the corrosive effect of many chemicals. As an example, concentrated sulfuric acid does not attack carbon steel, whereas dilute sulfuric acid is extremely corrosive. Thus concentrated sulfuric acid can often be safely stored in carbon steel tanks provided proper precautions are taken to cope with dilute acid that may form in the upper portions of the tank
There are many applications for steel tanks other than the storage of oil or water. Since most such applications are industrial in nature for which no industry standard has been developed, it is quite common to use API Standard 650 as a basis for design and construction. This is a logical approach provided that problems peculiar to the contents stored are taken into account. Tanks designed to store liquefied gases at or near atmospheric pressure are beyond the scope of this document. However, those interested in 16
when acid fumes and water condensation meet in the vapor space. Thus one fundamental requirement for an acid tank is that the interior of the tank be smooth without crevices or pockets where dilute acid condensation can collect. Selfsupporting roofs are good practice. If the design of the roof or size of tank requires structural stiffeners, it is desirable that they be placed on the outside. If the roof is lap welded, it should be welded underneath as well as the top. The connection of the roof to the shell should eliminate any pocket which might exist at the top of a standard API tank. When using Appendix A design basis of API 650, a lower design stress should be considered for the same reasons as given under "Molasses Tanks." The tank user should specify the amount of corrosion allowance, if any is required, for his particular purpose. In the case of carbon steel tanks storing caustic solutions, both the concentration and temperature are important. Carbon steel tanks should not be used if the combination of concentration and temperature exceeds the following values and may in some cases be unsatisfactory below these limits: 50% and 120F 25% and 150F 5% and 200F It is most important to make sure that the specified design conditions are not exceeded in service. Automatic temperature controls are recommended. In addition to ordinary corrosion, the principal problem in caustic tanks is one referred to as "caustic embrittlement" or "stress corrosion cracking." In the presence of high local stresses this type of corrosion can rapidly result in cracks and leaks. Local stress concentrations approaching the yield point can exist at shell penetrations, in the vicinity of welds and at other details. In caustic service these are the points where stress corrosion cracking can occur. Thus, in the case of caustic storage tanks, all fittings penetrating the shell or bottom, or any permanent attachments welded to the interior surface thereof, should be installed in a plate in the shop and the entire assembly thermally stress relieved. Essentially, this leaves only main seam welding to be performed in the field. Self-supporting roofs without structural members immersed in the tank contents are advisable. It is not necessary, however, to
eliminate crevices and pockets as is recommended for acid tanks. For caustic tanks, a standard API roof is acceptable. Certain additional precautions in welding should be taken for both acid and caustic tanks. Lap welds in the bottom and the inside bottomto-shell fillet should be made in at least two passes. Since the bottom-to-shell weld usually consists of a fillet inside and out, it is advisable to provide a water stop (complete penetration) at each vertical shell joint so that if a leak does occur in the inside fillet, channeling will be limited to one plate length. All other shell joints should be designed for complete penetration and fusion. The inside passes should be made first. The later welding of outside passes will partially heat treat and reduce residual stresses in the inside weld. If anticipated corrosion indicates a bottom plate thickness greater than 3/8", the bottom should be butt welded and the same sequence followed; i.e. weld the inside passes first. Inasmuch as all welds create locally high residual stresses, all brackets, welding lugs, etc. should be kept to a minimum, be located on the outside, and attached with small-diameter electrodes to limit the heat input and consequently the effect on the inside surface. When the corrosive attack is considered sufficiently severe to admit the possibility of local penetration, but not severe enough to warrant the expense of high alloy or clad steel plates, the tank is sometimes supported on a structural grillage to permit inspection from the underside.
The following information is based on the AWWA Standard D100. Anyone dealing with tanks should obtain a copy of the complete standard. With the exception of shells, roofs and accessories, many of the comments made in connection with API tanks also apply to AWWA tanks and will not be repeated here in detail. Bottoms may be either lap or butt-welded with a minimum thickness of 1/4 inch. AWWA does not specify top angle sizes, but the rules of API represent good practice.
17
the high water level will extend up into the roof itself. The resultant upward pressure on the roof is resisted by the combination of the roof dead load and the weld joint between the roof and shell. AWWA requires that for all roof plate surfaces in contact with water, the minimum metal thickness shall be 1/4". Roof plate surfaces not in contact with water may be 3/16". As applied to rolled shapes for roof framing, the foregoing minimum thicknesses shall apply to the mean thickness of the flanges regardless of web thickness. Roof supports or stiffeners, if used, shall be in accordance with current specifications of the American Institute of Steel Construction covering structural steel for buildings, with the following exceptions: 1. Roof plates plates are considered to provide provide the necessary lateral support by friction between roof plates and rafters to eliminate reduction in the basic allowable compressive stress, except where trusses and open web joists are used for rafters, or rafters having nominal depth greater than 15 in. or rafters having a slope greater than 2” in 12.” 2. The roof, rafter and purlin depth may be be
AWWA D100 offers two different design bases – the standard or basic design and the alternate design basis as outlined in Section 14. The alternate design basis permits higher design stresses in return for a more refined engineering design, more rigorous inspection, and the use of shell plate steels with improved toughness. AWWA D100 Section 14 includes steels of significantly higher strength levels and correspondingly higher design stress levels. This introduces new design problems. For example, for A517 steels, the permissible design stress of 38,333 psi will result in reaching the minimum required nominal thickness several courses below the tank top. It would be uneconomical to continue the relatively expensive steel into courses of plates not determined by stress. The obvious answer is to use less expensive steels in the upper rings. To govern this transition, Section 14 adds the following requirements: “In the interest of economy, upper courses may be of weaker material than used in the lower courses of shell plates, but in no instance shall the calculated stress at the bottom of any course be greater than permitted for the material in that course. A plate course may be thicker than the course below it provided the extra thickness is not used in any stress or wind stability calculation. ”
Compliance with this requirement will probably result in the course or courses immediately below the transition point being somewhat heavier than required by stress. Using a steel of intermediate strength level as a transition between A517 steel and carbon steel may help the situation. In any event, the use of two or more steels will result in plates of the same thickness made of different steels. Careful attention to plain marking for positive identification becomes very important. Consideration might be given to varying plate widths for different materials of the same thickness to aid in identification in the event that markings are lost.
less than
f b
600,000
times the span length in inches where f b is the maximum bending stress in psi, providing slope of the roof is ¾” to 12” or greater. 3. The maximum slenderness ratio ( L / r ) for roof support columns shall be 175. 4. Roof support columns shall be designed designed as secondary members. 5. Roof trusses, if any, shall be placed above the maximum water level in climates where ice may form. Roof trusses are not recommended due to the high degree of maintenance required over the life of the tank roof. 6. Roof rafters and connections shall be placed above maximum water level.
Whereas oil tanks are strictly utilitarian, a pleasing appearance is often an important consideration in the case of water tanks. Since the roof line has an important effect on appearance, this striving for beauty has led to a wide variety of roof designs. Often a self-supporting roof, such as an ellipsoid, will extend a considerable distance above the cylindrical portion of the shell, and 18
AWWA does not provide detailed designs of tank fittings and accessories, but specifies the following: 1. Compliance with OSHA and other regulations. 2. Two manholes shall be provided in the first ring of the tank shell. Manholes shall be either a 24" diameter or at least 18" x 22" when elliptical manholes are used. 30-inch diameter manholes are often recommended for safe recovery of personnel. 3. The purchaser shall specify pipe connections, sizes, and locations. Due to freezing hazard these connections are normally made through the tank bottom and as near to the shell as practical. A concrete valve box may be provided to permit access to piping. This valve box must be designed as a part of the ringwall. 4. If a removable silt stop is required, it shall be at least 4" high. If not required, then the connecting pipe shall extend at least 4" above the tank bottom. 5. The purchaser shall specify the overflow size and type. If an overflow to ground is required, it should be brought down the outside of the tank and discharged onto a splash block or other appropriate drainage structure with an air break. Inside overflows are not recommended. They are easily damaged by ice, and a failure in the overflow will empty the tank to the level of the break. 6. An outside vertical ladder shall begin 8 feet (or as specified) above the tank bottom and afford access to the roof. Need for access to AWWA tanks is infrequent and a conscious effort is made to render access difficult for unauthorized personnel. 7. The contractor shall provide access to the roof hatches and vents. The access must be reached from the outside tank ladder and fulfill the AWWA D100 requirements consistent with the roof slope or as specified by the purchaser. 8. A roof door or hatch whose least dimensions are 24" x 15", with a curb 4" high, provided with a hinged door and clasp for locking shall be placed near the outside tank ladder. A second opening of
9.
at least 20" in diameter and with a 4" neck must be provided near the center of the tank. Additional openings may be required for ventilation during painting. Adequate venting shall be provided to accommodate the maximum filling and emptying rates. A frost resistance pressure–vacuum relief device is required. These rates should be specified by the purchaser.
Venting for outflow (partial vacuum condition) is based upon the unrestricted vent area and the pressure differential that can safely be allowed between the outside and inside of the tank. This differential is established by quantifying the strength of the roof and shell above and beyond other structural requirements; for example, the margin of extra strength of the shell against buckling with respect to the design wind load. Venting for inflow (pressure condition) is again based upon the restricted vent area and the pressure differential that can safely be allowed before lifting the roof plates. For example, if 3/16" roof plates are used, the pressure differential would be 7.65 psf, 0.053 psi, or 1.47 inches water column. If the differential is limited to the weight of the roof, the shell/roof juncture does not become involved. The overstress in the shell would be minimal. The equation for outflow vent capacity is:
⎡⎛ Pa ⎞0.286 ⎤ − 1⎥ Q = 0.5 A × 110 × T × ⎢⎜ ⎟ Pi ⎝ ⎠ ⎥⎦ ⎣⎢
1/ 2
(3-6)
where Q =
vent capacity in cubic feet per second A = minimum clear vent open area in square feet T = air temperature in degrees Rankine Pa = atmospheric pressure in psia Pi = pressure in tank during withdrawal in psia The equation for inflow vent capacity is: 1/ 2
0.286 ⎧⎪ ⎡ ⎤ ⎫⎪ 6 ⎛ Pi ⎞ Q = 0.5 A⎨6.25 × 10 ⎢⎜⎜ ⎟⎟ − 1⎥ ⎬ P ⎢ ⎥⎦ ⎪⎭ ⎪⎩ ⎣⎝ a ⎠
19
(3-7)
APPENDIX A
The following design example covers the AWWA D100 tank. Calculate shell thickness using the basic equation: 2.6h p DG t = = 0.1547" (3-8) sE
Design Example
For typical examples of tank design consider two tanks 150 feet in diameter by 40 feet nominal height with flat cone supported roofs. Consider one tank per AWWA D100 and the other tank per API 650. See Figure 3A-1 for tank dimensions. These examples are for illustration only and are not to be used for an actual design or construction. Design of similar tanks should be accomplished by competent people experienced in the design of like structures and the use of applicable standards. For the AWWA tank consider Section 14, shell design and a site with mapped seismic ground motion values per ASCE 7-05 and AWWA D100-05 values of S s = 0.5 and S 1 = 0.15. Assume the Seismic Use Group is III. The Site Classification is “C” and TL = 8 sec. For the API 650 tank consider the standard, shell design by the variable point method, 1/16 inch corrosion allowance on the shell only. The seismic design procedures for API 650, API 620 and AWWA D100 are similar and are not repeated. Consider design metal temperature (DMT) of 20°F, standard 100 mph wind loads, standard 25 PSF roof loads, a maximum liquid content height of 39'-6,” and a design specific gravity of 1.0 for both tanks. The economics of plate selection with respect to width and grade and structural selection will differ with location and construction capabilities. Factors to consider are plate width and grade availability in a particular locality and structural rolling schedules. Also the availability of plate and structural stock in a particular locality will sometimes influence the selection of material. Further discussion of material selection will be beyond the scope of this document.
All nomenclature in the above and following equations is defined in the AWWA D100 standard. Notice that h p in the above equation is the full liquid height above the design point rather than h - 1 as used in API 650. The calculation for ring five (top ring) is:
=
2.6 × 7.66 ×150 × 1.0
= 0.1547" 19,330 × 1.0 The thicknesses for the remaining rings calculate: t 5
hp =15.63’ hp=23.58’ hp=31.54’ hp=39.50’
S=19,330 psi S=23,330 psi S=23,330 psi S=23,330 psi
t 4=0.3152” t 3=0.3942” t 2=0.5273” t 1=0.6603”
using A36 steel for rings 4 and 5 and A573 GR70 for rings 1, 2 and 3. Ring 5 will be increased to 0.3125” because of minimum thickness requirements in AWWA D100. Shell stability is calculated using the basic equation. H =
10.625 × 106 × t
(3-9)
1.5
Pw ( D / t )
The calculation for ring five (top ring) is: H 5
=
10.625 × 10 6 × 0.3125 1.5
18 × (150 / 0.3125 )
= 17.54' > 7.96'
For each ring the “h” calculated is compared to the actual height of shell above the design point. When “h” calculates less than the height of shell above, the shell is unstable. This may be corrected by thickening the shell or adding a stiffening ring. For this example we will consider only thickening the shell. h4 = 17.73’ > 15.92’ h3 = 21.76’ < 23.87’ Recalculate the thickness of ring 3 by using a lower strength steel (A36). t3 = 0.4758”
20
Using an inner support radius of 2.38 ft, which is dependent upon the method of supporting the inner rafters, the maximum design length of the inner rafters is 39.33 ft, as indicated in Figure 3A-2. The maximum design moment calculates to be 27,580 ft-Ibs. Using an AISC allowable stress of 0.66 × Fy, a section modulus of 13.93 in 3 is required. A W12 × 14 section with a section modulus of 14.9 in3 is chosen. See Figure 3A-3 for a typical rafter loading. The maximum design length for the outer rafters is 35.33 ft, as indicated in Figure 3A-2. The maximum design moment calculates to be 27,890 ft-lbs. A section modulus of 14.09 in 3 is required and again we will choose a W12 × 14 section. The rafter reactions are placed on the girder at the locations as determined by the roof framing layout. The outer rafter reactions are 3480 Ibs.; the inner rafter reactions are 2840 lbs.; and the girder design length is 29.07 ft. The maximum design moment calculates to be 150,440 ft-lbs. Again using AISC allowable stresses, a section modulus of 75.98 in 3 is required. AW18 × 46 sections with a section modulus of 78.80 in 3 is chosen. See Figure 3A-4 for a typical girder loading. For the center column a design load of 74,900 lbs. is calculated from the accumulated reactions of the inner rafters. Using AISC design procedures an allowable compressive stress is determined based upon the unsupported column length of 486.5 inches and a calculated slenderness ratio of 131. A 10" diameter schedule 20 pipe will meet the design criteria. See Figure 3A-5 for typical center column detail. For the outer columns we have chosen an 8” diameter schedule 20 pipe based upon a design load of 41,400 lbs, an unsupported column length of 470.6 inches, and a slenderness ratio of 159; using the same design criteria as the center column. See Figure 3A-6 for a typical outer column detail. Seismic design requirements are given in Section 13 of AWWA D100-05 and follow ASCE 7-05. The weights of the tank and liquid are computed to be: Weight of the product = 43,556,000 lbs Weight of the tank’s shell = 340,000 lbs Weight of the roof/framing = 354,000 lbs Weight of roof acting on shell = 205,000 lbs
Recalculate: h3 = 26.37’ > 23.87’ The shell is now stable above ring 3; continuing; h2 = 34.10’ > 31.83’ h1 = 45.67’ > 39.79’ The entire shell is now stable for a design wind velocity of 100 mph. See Table 3A-1 for shell thicknesses before and after minimum thickness and wind stability adjustments. For 90 mph wind load, minimum design loads are 18 psf on projected areas of cylindrical surfaces (shell) and 15 psf on projected areas of double curved surfaces (roof). Based upon the tank geometry and the design loading, the wind shear is calculated: Shell = 150 × 40.04 × 18 = 108,113 lbs Roof = 150 × 4.69 × 0.5 × 15 = 5,273 lbs Total = 113,386 lbs The minimum required coefficient of friction against sliding is: Wind Shear = 113,386 = 0.154 Tank Weight 734,250
(3-10)
This coefficient is well below established values which range as high as 0.4 to 0.5. The wind moment at the base of the shell is calculated: Shell = 108,113× 108,113× 20.02 = 2,164,421 2,164,421 ft-lbs. Roof = 5,273 × 41.60 = 219,357 Total = 2,383,778 ft-lbs. The ratio, C = 2M/dw, calculates to be 0.076 < 0.666; therefore, no anchors are required to resist overturning due to wind. Roof framing concepts, layout and detail vary among tank designers and suppliers. Rafter spacing is dependent upon roof loading and plate thickness. For reasons of plate strength and construction a maximum rafter spacing of approximately 7.00 feet is desirable. For this example consider nine girders and outer columns, 36 inner rafters and 72 outer rafters (see Figure 3A-2). The outer columns will be located on a 42'-6" radius. The rafter spacing is 6.54 feet at the shell and 6.92 feet at the girder. Consider 25 psf snow load and 7.65 psf (3/16" roof plate) dead load. 21
Weight of tank bottom = 181,000 lbs Center of gravity of roof above shell = 3 ft Center of gravity of shell above fdn = 16.7 ft H/D ratio = 0.263
Sc = 416 psi < 4570 psi
OK
The additional hydrodynamic hoop stresses are calculated by Section 13 (Equations 13-43 through 13-46). These hydrodynamic hoop stresses are added directly to the hydrostatic hoop stresses.
Given Ss = 0.5 and S 1 = 0.15. From Table Table 26 and 27 of AWWA D100 and Site class C, Fa and Fv can be determined as Fa = 1.2 and Fv = 1.65. Substituting into the equations using k=1.5, U = 2/3, I=1.5 Ri= 2.5 for self-anchored and R c = 1,5, the values for Ai and Ac are computed as:
Hydrodynamic Hoop Stresses
Y
Ai = 0.172 Ac = 0.022 and Av = 0.056 The sloshing period, Tc is 8.18 sec. Using Section 13.5, the impulsive weight, Wi, and convective weight, Wc, are calculated:
Y/D
Eqn 13-43
Eqn 13-44
1
7.5 .050
790
692
2
15.5 .103
1450
3
23.5 .157
4 5
Eqn Ni 13-45 5388
Eqn 13-46 Nc
790
432
1378
5388 1450
384
1922
2009
5388 1922
351
31.5 .210
2207
2586
5388 2207
332
39.5 .263
2303
3108
5388 2303
326
The addition of the stresses is usually done on a force/per unit length of circumference basis. In the longitudinal or vertical membrane (phi) direction, the forces from dead load, snow load, live load, and overturning are added together in the appropriate load combinations and compared to the applicable allowable stress. Similarly, in the circumferential (hoop or theta) membrane direction the hydrostatic and hydrodynamic forces are added. The results of this calculation are summarized in the table on page 23. Finally, the freeboard must be evaluated per Section 13. Since this was identified as a SUG III tank, and the sloshing period, Tc > 4 sec. Af = 0.03g, and the calculated wave height is 2.2 ft. The freeboard provided is 0 ft (0.5 ft from roof plate was there for rafter ends to project into the tank). Thus, the design must be modified. Either the liquid level must be reduced or the shell height increased by 2.2 ft to provide the required freeboard.
Wi = 13,208,000 13,208,000 lbs Wc = 28,423,000 lbs Similarly, substituting into the equations for the moment arms of the lateral forces, Xi, Xc can be computed: Xi = 14.81 14.81 ft Xc = 21.16 ft Substituting into Equation 13-23 of AWWA D100, the ringwall moment is Ms = 37,354,000 ft lbs Using this value of Ms in Equation 13-36, calculate the “J” ratio. Less than 0.785, so there is no net uplift for the design overturning moment and the tank is self-anchored if the maximum shell compression calculated by Equation 13-39 is met. Substituting into Equation 13-39, 13-39,
22
Summary of Shell Stresses (AWWA) Rin g No Material Des ig n allowa ble ten sile Ht of Rin g Thickness of Ring Y RoofDL, including equ ipm en t Shell Weight Cum mu lative DL, Self Weight Nphi DL(me (metal) tal) Nphi , L L Nphi HEQ , im puls ive Nph Nphi HEQ, HEQ, co conve nvective ctive Nphi HEQ (direct sum) Nphi HE HEQ (sr (srss) ss) Nphi VE VEQ Ntheta, hydrostatic Nthe Ntheta ta HEQ HEQ,, Nimpu Nimpuls lsiv ive e Ntheta HEQ , Nconvective Ntheta, HEQ (direct sum) Nthet Ntheta a VEQ, VEQ, Nthe Ntheta ta x EQ EQ vert vert factor factor Ntheta EQ (s rs s) Nth et eta t ot ot al al Check Load Combinations DL+LL Nphi Sphi Sphi al allowable
Roof ps i ft inc hes ft lbs/in kips lbs/in lbs/ lbs/in in lbs/in lbs/in lbs/ lbs/in in lbs/in lbs/ lbs/in in lbs/in lbs/in lbs/ lbs/in in lbs/in lbs/in lbs/in lbs/in lbs/in l bs bs /i/in
lbs/in psi psi
Ntheta Stheta Stheta allowable
lbs/in psi psi
Nphi Sphi Sphi allowable
lbs/in psi psi
Ntheta Stheta Stheta allowable
lbs/in psi psi
To p A36M 1933 0 8 0.312 5 7.5 36.25 36.3 48 .1 36.3 44 .8 36.3 44 .8 0.0 0 .0 0.5 3 .4 0.0 2 .6 0.5 5.9 0.5 4 .2 2.0 2 .5 292 5 79 0 43 2 122 1 16 5 91 5 384 0 T op
36.3 1 16 6 11 OK
44 .8 14 3 61 1
2 3 4 5 A36M A573Gr70 573Gr70 573Gr70 19330 2 3330 233 30 2 3330 8 8 8 8 0.3152 0 .3942 0.5273 0.6604 15.5 23 .5 31.5 39 .5 36.3 36 .3 3 6.3 36 .3 48.5 60 .6 81.1 101 .6 53.3 64 .0 78.4 96 .4 53.3 64 .0 78.4 96 .4 0.0 0 .0 0.0 0 .0 12.9 34 .5 73.5 135 .4 11.0 25 .3 45.5 71 .6 23.9 59 .9 11 9.1 207 .0 17.0 42 .8 86.5 153 .2 3.0 3 .6 4.4 5 .4 6045 9165 12285 1 5405 1450 1922 2207 2303 384 351 3 32 326 1834 2274 25 39 2629 340 516 691 867 1538 2021 23 36 2482 7583 11186 14621 1 7887 2 3 4 5
OK 292 5 936 0 1933 0 OK
53.3 169 617 OK 6045 19178 19330 OK
64 .0 162 774 OK 9165 23250 2 3330 OK
78.4 1 49 10 43 OK 12285 23298 233 30 OK
96 .4 146 1319 OK 1 5405 2 3327 2 3330 OK
49 .0 15 7 209 6 OK 384 0 1228 8 2577 3 OK
70.4 223 2312 OK 7583 24058 25773 OK
106 .9 271 2895 OK 11186 28376 3 1107 OK
164.9 3 13 37 50 OK 146 21 277 28 311 07 OK
249 .6 378 4570 OK 1 7887 2 7086 3 1107 OK
DL+EQ DL+EQ (srss , default AWWA) 36.8 1 18 8 15 OK
23
APPENDIX B - TANK FOUNDATIONS
Soils Investigation
The subgrade of a potential tank site must be capable of supporting the weight of the tank and contained fluid. A qualified geotechnical engineer should be retained to conduct the subsurface exploration and to make specific recommendations concerning: the type of foundation required, anticipated settlements, allowable soil bearing and specific construction requirements. The ultimate soil bearing capacity should be determined using sound principles of geotechnical engineering. The following minimum minimum factors of safety should be applied to the ultimate bearing capacity when determining the allowable soil bearing: 1. A factor of safety of 3.0 for normal operating conditions. 2. A factor of safety of 2.25 during hydrotest. 3. A factor of safety of 2.25 for operating conditions plus the maximum effect of wind or seismic forces. An allowable soil bearing based solely on the above factors of safety may result in excessive total settlements. If required, these factors of safety should be increased in order to limit the anticipated total settlements to acceptable values. Factors of safety larger than the above minimums are also required by certain codes and standards, such as AWWA D100.
24
3/16” ROOF PLATE LAP WELDED TOP SIDE ONLY
T OP SIDE ONLY ¼” BOTTOM PLATE LAP WELDED TOP Figure 3A-1 – Flat Bottom Tank
Table 3A-1 – Shell Plate Thicknesses
a.) CALCULATED SHELL THICKNESSES FROM STATIC HEAD ONLY (AWWA DESIGN) RING # 5 4 3 2 1
THICKNESS 0.1547” 0.3152” 0.3942” 0.5273” 0.6603”
b.) ADJUSTED FINAL THICKNESSES FOR STATIC HEAD AND WIND STABILITY (AWWA DESIGN) RING# 5 4 3 2 1
MATERIAL A36 A36 A573GR70 A573GR70 A573GR70
25
THICKNESS 0.3125” 0.3152” 0.4758” 0.5273” 0.6603”
MATERIAL A36 A36 A36 A573GR70 A573GR70
Figure 3A-2 – Framing Layout -- AWWA
Figure 3A-3 – Typical Rafter Loading
Figure 3A-4 – Typical Girder Loading
26
7 2
VARIABLE POINT POINT DESIGN: API API 650 RING NO. 1
DESIGN: D = 150.000 H = 39.500 G = 1.000 S = 28000. CA = 0.0625 Td = 2.6*D*(H - 1) * G/S + CA = 0.5362 + CA = 0.5987 T1d = [1.06- (0.463*D/H)* SQRT(H*G/S)] *2.6*D*H*G/S + CA = T1d = 0.5469 + CA = 0.6094 HYDROTEST: D = 150.000 H = 39.500 G = 1.000 S = 30000. TT= 2.6*D* (H-1) * G/S = 0.5005 T1T = [1.06-(0.463* D/H) * SQRT(H* G/S)] * 2.6* D* H* G/S= 0.5115 USE: 0.599 IN. A573 70 L/H = SQRT(6.0* D* T)/H3 = 0.5929 <= 2.0 OK RING NO. 2
DESIGN: D = 150.000
H = 31.542
G = 1.000
S = 28000.
CA = 0.0625
Td = 2.6* D* (H - 1) * G/S + CA = 0.4254 + CA = 0.4879 TX = 2.6* D* (H -X/12) *G/S TU 0.4254 0.4116 0.4121
TL K C 0.5362 1.2606 0.1211 0.5362 1.3028 0.1390 0.5362 1.3013 0.1384 TX = 0.4121 + CA = 0.4746
X2 45.847 52.595 52.369
X1 26.607 28.571 28.506
X3 23.872 23.482 23.495
X 23.872 23.482 23.495
TX 0.4116 0.4121 0.4121
X3 23.062 22.711 22.723
X 23.062 22.711 22.723
TX 0.3851 0.3854 0.3854
DESIGN: PARA. 3.6.4.5 RATIO = 95.500/[SQRT (6 * D * 0.5362) ] = 4.3471 T2 = Tx + (T1 - Tx) * (2.1 - 4.3471/1.25) = 0.2410 T2D = 0.4121 + CA HYDROTEST: D = 150.000 H = 31.542 G = 1.000 TT = 2.6 * D * (H - l) * G/S = 0.3970 TX = 2.6 * D * (H - X/l2) * G/S TU 0.3970 0.3851 0.3854
TL K 0.5005 1.2606 0.5005 1.2998 0.5005 1.2985 TX = 0.3854
C 0.1211 0.1377 0.1372
X2 45.846 52.127 51.924
S = 30000.
X1 26.202 28.036 27.977
HYDROSTATIC: PARA. 3.6.4.5 RATIO = 95.500/[SQRT (6 * D * 0.5005) ] = 4.4997 T2 = TX + (T1 - TX) * (2.1 – 4.4997/1.25) = 0.2128 T2T = 0.3854 MINIMUM DESIGN THICKNESS = 0.4746 USE: 0.475 IN. A573 70 Figure 3A-7 – API 650 VARIABLE POINT CALCULATIONS
28
TABLE 3A-2 — ADJUSTED FINAL THICKNESSES (API 650 DESIGN)
RING # 5 4 3 2 1
THICKNESS 0.3125” 0.3125" 0.3750” 0.4750” 0.5990”
Wtr (ring 5) Wtr (ring 4) Wtr (ring 3) Wtr (ring 2) Wtr (ring 1) SUM OF Wtr
= = = = = = =
MATERIAL A36 A36 A573GR70 A573GR70 A573G R70
95.50 inches 95.50 60.54 33.61 18.79 303.94 inches 25.33 feet
Figure 3A-8 — TRANSPOSED SHELL HEIGHT (API 650 DESIGN)
top. If this is not done, the sand will percolate down through the voids in the coarser rock. An excellent tank grade can also be obtained by substituting about 1½ inches of asphalt road paving mix for the sand cushion. This material is available from ready mix plants in many sections of the country. It is very important that the paved tank grade be constructed level and to the proper profile, particularly near the shell. Once the asphalt has set up, it is extremely difficult for the tank builder to correct inaccuracies by taking down the high and filling in the low spots. Drainage is important both from the standpoint of soil stability and bottom corrosion. Good drainage should be provided under the tank itself and in the general area around the tank. Where the terrain does not afford natural drainage, proper ditching around the tank may help to correct the deficiency.
Factors of safety lower than the above minimums may be considered when actual experience with similar tanks and foundations at a particular site indicates that satisfactory performance can be expected. Tank Grade
The tank grade (surface which supports the tank bottom) can be constructed of earth materials provided the subgrade beneath the tank bottom is capable of supporting the weight of the contained fluid. The tank grade usually consists of a 4" sand cushion placed over properly compacted fill or soil. It is recommended that the finished tank grade be constructed at least 6 inches above the surrounding ground surface and be crowned from its outer periphery to its center. A slope of 1 inch to 10 feet is suggested. The sand should be clean and free of corrosive elements. Care should be taken to exclude lumps of earth or other deleterious materials from coming into contact with the bottom. These materials can cause electrolytic action that will result in pitting of the bottom plate. If the sand cushion is placed on top of crushed rock fill, the rock should be carefully graded from coarse at the bottom to fine at the
Foundations
The shell of a flat bottom tank can be supported on a compacted granular berm, concrete ringwall or concrete slab foundation. Local soil conditions, tank loads and the intended use of the tank will determine which 29
of these foundations is suitable s uitable for a particular site. Tanks that require anchor bolts must be supported by ringwall or slab foundations.
on the inside horizontal projection of the ringwall. 3. Bending, torsion and shear resulting from lateral, wind or seismic, loads. A rational analysis, which includes the effect of the foundation stiffness, should be used to determine the soil-bearing stress distribution and the above internal design forces. The area of reinforcement provided must be sufficient to resist the above forces and should not be less than the following minimums. These minimums are intended to prevent excessive cracking due to shrinkage and temperature. 1. For wall-like ringwalls, the area of vertical reinforcement provided should not be less than 0.0015 times the horizontal cross-sectional area of the ringwall. 2. The area of hoop reinforcement provided should not be less than 0.0025 times the vertical cross-sectional area of the ringwall. The detailed design of the ringwall and requirements for the materials, construction and testing should be in accordance with the American Concrete Institute's Building Code Requirements for Reinforced Concrete (ANSI/ACI 318). Recesses shall be provided in the concrete ringwall for flush type cleanouts, drain off sumps and any other appurtenances that require recessing. Refer to API 650 for details of recesses at flush type cleanouts. The top of the ringwall should be smooth and level to within ± 1/8 inch in any 30 feet of circumference. No point on the total circumference should vary more than ± l/4 inch from the specified finish elevation. Slab Foundation - When the subgrade beneath the tank bottom cannot adequately support the weight of the contained fluid, a slab foundation is required. The area of the slab must be sufficient to produce a soil bearing (due to the total weight of the tank, foundation and contained product) less than the allowable soil bearing. The depth to the bottom of the slab s lab will depend on local conditions and must be sufficient to place the bottom of the slab below anticipated frost penetration and within the specified bearing strata.
Granular Berm Foundation - When a qualified geotechnical evaluation concludes that it is unnecessary to construct a ringwall or slab foundation, the shell can be supported by a granular berm foundation. The berm should be constructed of well-graded and properly compacted stone or gravel. The berm should extend a minimum of 3 feet beyond beyon d and 2 feet inside the tank shell as shown in Figure 3B-1. The berm should be level to within ± 1/8 inch in any 10 feet of circumference and to within ± 1/2 inch in the total circumference. Adequate drainage away from the berm must be provided to prevent erosion of the berm under the shell. Alternatively, a welded or bolted steel grade band can be used to retain the outer portion of the berm. Concrete Ringwall Foundation - When suitable bearing is not available at the surface, s urface, but is available at a reasonable depth below the surface, a ringwall foundation should be considered. The depth of the ringwall will depend on local conditions and must be sufficient to place the bottom of the ringwall below anticipated frost penetration and within the specified bearing strata. As a minimum, the bottom of the ringwall should be located 2 feet below the lowest adjacent finish grade. The width of the ringwall must be sufficient to produce a soil bearing less than the specified allowable soil bearing. As a minimum, the ringwall width should be 1 foot. The inside horizontal projection (inside the tank shell) should be no less than 4 inches. The ringwall must be reinforced to resist the following forces: 1. Direct hoop tension resulting from the lateral earth pressure on the inside face of the ringwall. Unless substantiated by proper geotechnical analysis, the lateral earth pressure should be assumed to be 30% of the vertical pressure due to the contained fluid and the soil weight. 2. Bending moment resulting from the uniform moment load. The uniform moment load is due to the eccentricities of the shell and pressure loads relative to the centroid of the soil bearing stress. The pressure load is due to the fluid pressure
30
The detailed design of the slab and requirements for the materials, construction and testing should be in accordance with the American Concrete Institute's Building Code Requirements for Reinforced Concrete (ANSI/ACI 318). References, References, Part Ill
1. API Standard 650 Welded Steel Tanks for Oil Storage, Division of Refining, American Petroleum Institute. 2. AWWA Standard D100 Welded Steel Tanks for Water Storage, American Water Works Association. 3. Manual of Steel Construction, American Institute of Steel Construction, Inc. 4. "Oil Storage Tanks," The Prince William Sound, Alaska, Earthquake of 1964 and Aftershocks, Volume II, Part A, U.S. Dept. of Commerce, 1967. 5. "Fluid Mechanics," Dodge and Thompson.
31
Figure 3B-1 — Example of Foundation with Crushed Stone Ringwall from API 650
Note:
Bottom of excavation should be level. Remove any unsuitable material and replace with suitable fill, thoroughly compacted.
Figure 3B-2 – Example of Concrete Ringwall Foundation Foundation
32
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Part IV Stainless Steel Tanks For Liquid Storage
Introduction
Type 304
A
t the present time, the only rules for stainless steel storage tanks are given in Appendix Q of API Standard 620(l) which covers low-pressure tanks for liquefied hydrocarbon gases, particularly liquefied ethane, ethylene, and methane, at a minimum temperature of —270F. Rules for the design and construction of, pressure vessels — including stainless steel vessels— are given in the ASME (2) Boiler and Pressure Vessel Code, Section VIII, Division 1 and Division 2, Pressure Vessels. In the following discussion rules are presented for design and construction of stainless steel tanks at atmospheric pressures. These rules are not intended to cover storage tanks which are to be erected in areas subject to regulations more stringent than specified in the following pages. These rules are recommended only insofar as they do not conflict with local requirements.
Possessing corrosion resistance, strength and fabricability, this is the general purpose stainless steel, long known as "18-8”. Type 304 is the most widely used type and is extensively specified for food handling and storage, dairy equipment, nuclear fluids, and in general most applications where even small amounts of corrosion product would be intolerable. Type 316
Containing higher nickel than Type 304, and 2-3% molybdenum, Type 316 possesses greatly improved resistance to corrosion by pitting. It is used under conditions too severe for Type 304, such as mineral acids (phosphoric acid, sulfuric acid), strong organic acids (oxalic, formic, etc.) and halides in various dilutions. Types 304L and 316L
Containing 0.03% maximum carbon, these are the low carbon counterparts of Types 304 and 316. The lower the carbon content, the less the chromium carbide that can be formed. Chromium-nickel stainless steels form a grain boundary chromium-carbide precipitate when heated in the 800-1650F temperature range for sufficient time (see Figure 4-1) (5). If the degree of precipitation is severe — i.e. the grains are completely surrounded — there may be a loss of corrosion resistance in aggressive media such as hot, oxidizing acids (e.g. strong nitric acid), iron or copper sulfates in hot dilute sulfuric acid, and air-saturated hot sulfuric acid. Such aggressive corrosion conditions do not normally exist in storage tanks. Intergranular corrosion attack used to be a common occurrence when the stainless steels contained up to 0.12% carbon (as in Type 302, for example). This was enough carbon to remove considerable chromium from solution during welding cycles, causing mild to heavy carbide precipitation in the weld heat-affected zone. Corrosive attack would be evident in this zone, if the environment was severe. This situation resulted in widespread specifying of low carbon (0.03% maximum) stainless steels, but it should be understood that there are relatively few situations where the L grades are actually required for storage vessels. Even these should be carefully investigated to establish
FACTORS AFFECTING SELECTION OF STAINLESS STEEL
There are a total of 62 stainless steel compositions that are recognized as standard by the American Iron and Steel Institute (3), as well as commercially available proprietary compositions. The five stainless steels most generally used as plate material for construction of liquid storage tanks are Types 304, 304L, 316, 316L and 410S. The last is not recognized as standard by the American Iron and Steel Institute. The chemical compositions of these types are listed in Table 4-1 and their mechanical properties are listed in Table 4-2. The selection of a particular type of stainless steel for a given corrosive environment often follows extensive study of comparative data, and sometimes even pilot or service testing. However, a general understanding of the corrosion resistance capabilities of the five stainless steels, in terms of their relative resistance to various common media, is shown in Table 4-3. The five types fall within two categories: namely, Types 304, 304L, 316 and 316L are in the chromium-nickel group, while Type 410S is in the straight chromium group.
33
such a need before the additional expense of the L grades is incurred. Types 304 and 316 (0.08% maximum carbon) can, in many cases, be welded free of carbide precipitation. If a small amount does develop, it may be unaffected by the liquid being stored, except possibly as indicated above. It should be noted here that galvanized material or other zinc products welded to stainless steel will cause intergranular cracking. In general, the L grades should be used when — and only when — it is ascertained that conditions will be present, which are conducive to intergranular attack on as-welded 0.08% maximum carbon stainless steel (see Figure 4-1). The general corrosion and pitting resistance of the L grades is not better than their higher carbon counterparts in the annealed condition; nor is there any advantage in weldability or fabricability.
2. 3.
4.
5.
Type 410S
This straight-chromium stainless steel is not subject to the above form of carbide precipitation. It finds use where moderate corrosion resistance is needed, and slight product contamination is not critical (see Table 4-3). The low carbon (0.08% maximum) results in a tough plate product which avoids formation of the less-tough metallurgical structures possible in high strength, low alloy steels. Stress Corrosion Cracking
Another phenomenon associated with the chromium nickel stainless steels Types 304, 304L, 316 and 316L, is stress corrosion cracking. By definition, stress corrosion cracking involves the combined action of a tensile stress and a corrosive medium. Aside from some ultra-pure metals, most commercial metals are subject to this phenomenon in certain specific environments. While the initial reaction may be one of great concern, it should be emphasized that throughout industry there are numerous applications of stainless steel in environments where stress corrosion cracking does not occur or which have been engineered to avoid stress corrosion cracking. Although stress corrosion cracking is not fully understood, there are some general guidelines that can be related to stainless steels: 1. The form of stress corrosion usually found in the AISI 300 stainless steels is
related to exposure to hot chloridecontaining corrosive media. At temperature much below 160°F, stress corrosion failures are not very likely to occur. At temperatures exceeding 160°F, the time to failure of life of the vessel is very dependent on temperature, concentration of chloride and stress. Increased values in each case will shorten the life of the vessel. While Types 304 and 316 are both susceptible to hot chloride stress corrosion cracking, Type 316 under similar service conditions tends to give better life than Type 304. Because of variation in fabrication and service stresses, it is frequently difficult to predict the life of an austenitic stainless steel vessel in hot chloride-containing media.
STAINLESS STEEL PLATE Manufacture
Stainless steel plate is defined as a flat rolled or forged product, 3/16 inch (4.76 mm) and over in thickness, and 10 inches (254 mm) and over in width (3). It is formed in the same type of equipment as utilized for carbon steel plate, although production allowances must accommodate the much greater hot strength of stainless steel over carbon and low alloy steels. In producing plate, care is taken to attain the good surface condition that is essential to corrosion resistance. The first of several surface cleaning operations occurs at the slab stage, where the 4 to 10-inch-thick bloomed slab is ground or scarfed overall to remove not only the scale but some of the underlying base metal as well. The slab is then cut to size, yielding the ordered plate size, reheated and hot rolled. The plate is then annealed, and again cleaned of scale by either a chemical solution or mechanical means, or both. During the hot rolling, high pressure water jets and other mechanical devices are employed to assure that the refractory scale is not rolled into the surface. Light gauge plate (3/16 inch and 1/4 inch thick) can be rolled in coils up to 60 inches wide on continuous mills. This product normally has
34
improved surface, gauge accuracy and offers greater flexibility in length.
is clean edges (and near-edge areas) prior to welding. Contaminants, whether organic or inorganic, can cause problems when they decompose in the arc heat. The oxide film on stainless steel surfaces is very refractory and reduces the wetting action between parent and weld metal. The filler metals for welding stainless steels are listed in Table 4-4. When stainless steel is welded to carbon or low alloy steels, Type 309 electrodes are normally used.
Forming
Press brake flanging or bending, and roll bending are the most widely used cold forming operations performed on stainless steel plate. Type 410S has cold forming characteristics similar to carbon and low alloy steels in the 35,000-50,000 psi yield strength range. The chromium-nickel stainless steels (Types 304, 304L, 316, 316L), on the other hand, workharden quite rapidly with increasing plastic deformation. As the steel increases in strength with increased deformation, the bending forces exerted by the forming equipment rise commensurately. The most noticeable effect of work hardening is the greater degree of springback, compared with carbon steel. Dies for brake bending and rolls for roll bending must overbend the stainless steel to compensate for the springback. In brake bending, bending to a smaller radius can compensate for the greater springback.
Cleaning and Passivation
These two terms are actually synonymous if the word clean is strictly interpreted to mean "the complete removal of all contaminating materials from the stainless steel surface.” If the surface is clean, it will self passivate. Iron particles and/or oil are the contaminants most generally encountered, and to the extent that they are present some surface staining or rusting may occur. Whether such an occurrence is serious or not depends on the requirements of the application which can range from "food quality surfaces" to simply "retention of structural strength." Prevention of contamination is to be preferred over removal of contaminants. A main source of contamination due to iron particles results from comingling plate fabrication operations involving carbon steel and stainless steel. Another source is the existence of weld scale or weld spatter, which can only be removed by energetic chemical or mechanical means. In extremely demanding requirements, finishing may include a final polishing to produce the brightest surface possible. Various cleaning practices are summarized in ASTM A380 (6) and more comprehensively described in ASTM Special Technical Publication 538 (7).
Cutting
Most stainless steel plates are cut by tank fabricators with the plasma arc process. Thin plates can be sheared. Thicker plates can be cut by saw cutting or abrasive wheel cutting. Gasoxygen (oxy-gas) is also applicable if used in conjunction with iron powder. Stainless steel cannot be cut by conventional flame cutting, but in some cases may be cut and beveled with the carbon arc gouge if the cut edges are ground to remove oxides. Welding
Gas metal arc and submerged arc welding are high-production methods and are usually used in the downhand position, fully automated. Both give deep penetration and, for high volume welding that can be positioned, are the lowest cost methods for joining plates. A modification of gas metal arc called interrupted (or pulse) arc welding is useful for butt, fillet, and lap welding. Shielded metal arc welding is widely used for all types of stainless steel welding, particularly where automatic welding is impractical. Advantages are low cost equipment and mobility. Disadvantages are slow speeds and high labor cost for skilled operators. An essential requirement for any welding method
FACTORS AFFECTING TANK DESIGN
The design rules and details of API Standards 650 (8) and 620 (1) are applicable for stainless steel tanks at atmospheric pressure with certain exceptions. In the following discussion, the design stresses recognize the increased toughness of stainless steels over carbon steels and the low yield/tensile ratios of Types 304, 304L, 316 and 316L. The increased toughness permits designing to a higher proportion of 35
yield strength, but the lower yield strength introduces the problem of permanent strain. Section VIII of the ASME Code (2) recognizes the strain by stating two allowable design stresses for the austenitic stainless steels. The higher stresses are related to the following footnote to the ASME table of stresses: "Due to the relatively low yield strength of these materials, these higher stress values were established at temperatures where the short time tensile properties govern to permit the use of these alloys where slightly greater deformation is acceptable. These higher stress values exceed 62X percent but do not exceed 90 percent of the yield strength at temperature. Use of these stress values may result in dimensional changes due to permanent strain. These stress values are not recommended for flanges of gasketed joints or other applications where slight amounts of distortion can cause leakage or malfunction.” After consideration of the allowable design stresses of the ASME Code and the API Standards, the following stress basis is suggested for stainless steel tanks at atmospheric pressure: a. The design basis for shells where permanent strain ≤ .05% is acceptable is the lesser of: Sd = .8 x Fy or .3 x Ft see Table 4-5(a) b. The design basis for shells where permanent strain ≤ .1% is acceptable is the lesser of: Sd = .9 x Fy or .3 x Ft see Table 4-5(b) where Fy = min. yield strength Ft = min. tensile strength Sd = design stress c. Because the lower carbon grades (Types 304L, 316L) usually exhibit yield strengths at room temperature greater than the specified minimum value, the allowable stress for 100oF has been based only on the tensile factor. d. Where a lower level of permanent strain is desirable such as mentioned above for gasketed joints or other applications where slight amounts of distortion can cause leakage or mechanical malfunction see Table 4-5(c) for values.
e. The yield values at temperature can be obtained from table AHA2 of ASME Section VIII, Div. 2. The allowable stresses listed in Table 4-5, a, b, and c, result from these bases, with the higher stresses for the austenitic grades taking into account the greater deformation of item b above. Table 4-5(b) should be the default basis unless directed otherwise for shells and Table 45(c) should be used for flanges. The design thickness for each shell course can be calculated by the formula in API Standard 650, modified by the allowable stress and joint efficiency: (2.6)( D )( H − 1)(G ) t = + C ( E )(1000 S d ) where: = t C =
minimum thickness, in inches an additional thickness required for corrosion allowance (rarely if ever required) D = nominal diameter of tank, in feet. This shall be the centerline diameter of the shell plates, unless otherwise specified by the purchaser H = height, in feet, from bottom of course under consideration to top of top angle, or to bottom of any overflow which limits tank filling height G = specific gravity of liquid to be stored, but in no case less than 1.0 Sd = maximum allowable stress in ksi. Values tabulated in Table 4-5, a or b 1.0 if tank is radiographed in E = accordance with section 6 = 0.85 if tank is radiographed in accordance with A.5.3 = 0.7 if tank is not radiographed. This value shall be given as part of the buyers' specifications Stainless steel separation pads (or poison pads) should be considered at points where carbon steel is welded to the stainless steel tank to avoid carbide precipitation. Typical areas for this would be anchor bolts and support brackets. The recommended nominal thickness of shell plates should not be less than the following based on construction minimums:
36
Nominal Tank Diameter Smaller than 50' 50' to 120' excl. 120' to 200' incl. Over 200'
FACTORS AFFECTING FABRICATION AND CONSTRUCTION
Nominal Plate Thickness 3/16" 1/4" 5/16" 3/8"
Before proceeding with any fabrication or construction of a stainless steel tank, satisfactory weld procedure qualifications should be performed in accordance with all the essential variables of Section IX, “Welding Qualifications,” of ASME Boiler and Pressure Vessel Code, including conditions of postweld heat treatment or the omission of postweld heat treatment. Requirements and restrictions for postweld heat treatment are described in ASME Section VIII. Materials that would require postweld heat treatment should not be used for storage tanks designed by the rules in this manual. All austenitic chromium-nickel alloy steel welds, both butt and fillet, between plates exceeding 3/4 inch nominal thickness, shall be examined for detection of cracks by the liquid penetrant method, before the hydrostatic test of the tank. All cracks shall be eliminated. Butt-welded joints in Type 410S welded with electrodes that produce an austenitic chromium-nickel weld deposit shall be radiographed when the thinner plate at the welded joint exceeds 1 ½ inches. Referring to Part AM of ASME Section VIII Division 2 it will be noted that both values of thermal conductivity (TC) and thermal diffusivity (TD) (given in Btu/hr ft °F and ft2/hr respectively) are considerably lower (about 2 to 1) for stainless compared to carbon steel. This indicates that heat (from welding) is not conducted away or diffused as rapidly with stainless steels and therefore distortion is likely unless design steps are taken to ensure that nozzle location (with respect to vertical and horizontal seams) should be thought out. Also, the tolerances given in API 650 for banding and peaking may not be achievable for stainless steel shells. For the design of stainless steel tanks at refrigerated temperatures, the designer is referred to Appendix Q, "Low-Pressure Storage Tanks for Liquefied Hydrocarbon Gases" of API Standard 620. This subject is beyond the intended scope of this discussion.
Throughout this design procedure it shall be remembered that Young's Modulus for stainless steel is less than that of carbon steel. Therefore designs for compression and stability should consider this fact. Normally the tank shell should be designed to resist the design wind velocity given in the customer's specifications. API Standard 650 provides rules for stiffening tank shells. The background for the API rules was given in a paper, "Stability of API Standard 650 Tank Shells,” presented in May 1963, to the API Division of Refining. For the design of shells under external load (small negative pressures) the designer should refer to ASME Section VIII or Part III of "Design of Plate Structures" (11). For the design of structural members, the designer should refer to the Stainless Steel Cold-Formed Structural Design Manual (10). NOTE: Roof designs for stainless steel tanks may be done in a similar manner as that outlined in Part III for carbon steel tanks, but normally all structural units will need to be fabricated in custom shapes. A lighter gauge lighter than .17 allowed in API 650 for carbon steel structural units, may be used to accommodate forming.
37
Figure
Time required for formation of carbide precipitation in stainless steels with various carbon contents. Carbide precipitation forms in the areas to the right of the various carbon-content curves. Within time-periods applicable to welding, chromium-nickel stainless steels with 0.05% carbon would be quite free from grain boundary precipitation. (5)
38
Table 4-1 — STAINLESS STAINLESS STEELS COMMONLY USED FOR CONSTRUCTION OF LIQUID STORAGE TANKS (4) COMPOSITION, PERCENT Carbon Manganese Phosphorus Sulfur Silicon max. max. max. max. max. Chromium
ASTM Type
UNS No.a
304
(S30400)
0.08
2.00
0.045
0.030
1.00
18.0020.00
8.0010.50
N 0.10 Max
304L
(S30403)
0.03
2.00
0.045
0.030
1.00
18.0020.00
8.0012.00
N 0.10 Max.
316
(S31600)
0.08
2.00
0.045
0.030
1.00
16.0018.00
10.0014.00
2.00-3.00 Molybdenum N 0.10 Max.
316L
(S31603)
0.03
2.00
0.045
0.030
1.00
16.0018.00
10.0014.00
2.00-3.00 Molybdenum N 0.10 Max.
410S
(S41008)
0.08
1.00
0.040
0.030
1.00
11.5013.50
0.60 (max)
Nickel
Other Elements
Unified Numbering System, originated by ASTM and SAE, developed to provide a single orderly system for designating commercial metals and alloys.
a
Table 4-2 — MECHANICAL MECHANICAL PROPERTIES OF STAINLESS STEELS COMMONLY USED FOR CONSTRUCTION OF LIQUID STORAGE TANKS (4)
ASTM Type
UNS No.
Tensile Strength, min ksi MPa
Yield Strength, min ksi MPa
Elongation, a min Percent
Hardness, max b Bhn RBc
304
(S30400)
75
515
30
205
40
202
92
304L
(S30403)
70
485
25
170
40
183
88
316
(S31600)
75
515
30
205
40
217
95
316L
(S31603)
70
485
25
170
40
217
95
410S
(S41008)
60
415
30
205
22.0
183
88
a Elongation
in 2 inches (50.8 min)
b Brinell c Rockwell-B
39
Table 4-3 — RELATIVE RELATIVE CORROSION RESISTANCE OF STAINLESS STEELS COMMONLY USED FOR CONSTRUCTION OF LIQUID STORAGE TANKS
ASTM
UNS
Mild Atmospheric
Type
No.
and Fresh Water
304
(S30400)
X
X
-
304L
(S30403)
X
X
316
(S31600)
X
316L
(S31603)
410S
(S41008)
Atmospheric
Salt
Industrial Marine Water
Chemical Mild
Oxidizing
Reducing
-
X
-
-
X
-
X
X
-
X
-
X
X
-
X
X
X
X
X
X
X
X
X
-
-
-
X
-
-
Note: X’s indicate environments environments to which the various stainless steels may be considered resistant.
Table 4-4 — TYPICAL TYPICAL FILLER METALS FOR WELDING STAINLESS STEELS
Base Metal
Electrodes (AWS)
Type 304
E308-15 or 16; ER308; E308T-2
Type 304L
E308L-15 or 16; ER308L; E308T-2
Type 316
E316-15 or 16; ER316; E316T-2
Type 316L
E316L-15 or 16; ER316L; E316T-2
Type 410S**
E410*-15; ER410*; E410T*-2
* Type 410 electrodes must be specified to 0.08% maximum carbon in all cases. ** It is permissible (and often desirable) to weld Type 410S with austenitic (chromium-nickel) (chromium-nickel) electrodes.
40
Table 4-5 — ALLOWABLE ALLOWABLE STRESSES FOR TANK SHELLS OF STAINLESS STEEL
Minimum Yield, KSI
Minimum Tensile, KSI
304a
30.0
304b
Type
100°F
For Metal Temperatures Not Exceeding 200°F 300°F 400°F 500°F
600°F
75.0
22.5
20.0
18.0
16.6
15.5
14.6
30.0
75.0
22.5
22.5
20.3
18.6
17.5
16.4
304La
25.0
70.0
21.0 c
17.0
15.3
14.0
13.0
12.4
304Lb
25.0
70.0
21.0 c
19.2
17.2
15.8
14.7
14.0
316a
30.0
75.0
22.5
20.6
18.6
17.1
15.9
15.0
316b
30.0
75.0
22.5
22.5
21.0
19.3
17.9
16.8
316La
25.0
70.0
21.0 c
16.9
15.1
13.8
12.7
12.0
316Lb
25.0
70.0
21.0 c
19.0
17.0
15.5
14.3
13.5
410Sa
30.0
60.0
18.0
18.0
18.0
18.0
18.0
18.0
Note: a, b, and c explained on page 39.
41
Table 4-5(a) — ALLOWABLE STRESSES FOR TANK SHELLS OF STAINLESS STEEL Limiting % Strain = 0.05%
Minimum Yield, KSI
Minimum Tensile, KSI
100°F
304
30
75
22.5
20
18
16.6
15.5
14.6
304L
25
70
21
17
15.3
14
13
12.4
316
30
75
22.5
20.6
18.6
17.1
15.9
15
316L
25
70
21
16.9
15.1
13.8
12.7
12
410S
30
60
18
18
18
18
18
18
Type
For Metal Temperatures Not Exceeding 200°F 300°F 400°F 500°F
600°F
Table 4-5(b) — ALLOWABLE ALLOWABLE STRESSES FOR TANK SHELLS OF STAINLESS STEEL Limiting % Strain = 0.1%
Minimum
Minimum
Yield,
Tensile,
KSI
KSI
100°F
200°F
300°F
400°F
500°F
600°F
304
30
75
22.5
22.5
20.3
18.6
17.5
16.4
304L
25
70
21.0
19.2
17.2
15.8
14.7
14.0
316
30
75
22.5
22.5
21.0
19.3
17.9
16.8
316L
25
70
21.0
19.0
17.0
15.5
14.3
13.5
Type
For Metal Temperatures Not Exceeding
Table 4-5(c) — ALLOWABLE ALLOWABLE STRESSES FOR FLANGES OR GASKETED JOINTS OF STAINLESS STEEL Limiting % Strain = 0.01% per 62.5% Fy Yield Strength Limit
Minimum
Minimum
Yield,
Tensile,
KSI
KSI
304
30
75
304L
25
316
Type
For Metal Temperatures Not Exceeding 100°F
200°F
300°F
400°F
500°F
600°F
20.0
16.7
15.0
13.9
12.9
11.5
70
16.7
14.3
12.8
11.7
10.9
10.3
30
75
20.0
16.7
15.0
13.9
12.9
11.5
316L
25
70
16.7
14.3
12.8
11.7
10.9
10.3
410S
30
60
18
18
18
18
18
18
42
5. Svetsaren English edition 1-2; 1969, p. 5. 6. ASTM Designation A380-78 (ANS G81.16) Standard Recommended Practice for Cleaning and Descaling Stainless Steel Parts, Equipment and Systems. 7. ASTM S.T.P. 538 "Cleaning Stainless Steel" includes ASTM A380 and 22 papers presented at a symposium. 8. API Standard 650 – Welded Steel Tanks for Oil Storage; Division of Refining, American Petroleum Institute, Eighth Edition. 9. ASTM Designation A370-77 (ANS G60.1) —Standard Methods and Definitions for Mechanical Testing of Steel Products. 10. "Stainless Steel Cold-Formed Structural Design Manual," American Iron and Steel Institute. 11. "Design of Plate Structures," Vol. 2, AISI/SPFA. 12. Steel Products Manual - Plates; Rolled Floor Plates: Carbon, High Strength Low Alloy, and Alloy Steel.
Table 4-6 FACTORS FOR LIMITING PERMANENT STRAIN IN HIGH-ALLOY STEELS1
Limiting Permanent Strain, % 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
Factors 0.90 0.89 0.88 0.86 0.83 0.80 0.77 0.73 0.69 0.63
NOTE: (1) Table 4-6 lists multiplying multiplying factors factors which, when applied to the yield strength values shown on Table AHA-2, will give a value that will result in lower levels of permanent strain. If this value is less than the design stress intensity value listed in Table AHA-1, the lower value shall be used.
References, Part IV
Other Information on Corrosion
1. API Standard 620 — Recommended Rules for Design and Construction of Large, Welded, Low-Pressure Storage Tanks; Division of Refining, American Petroleum Institute. 2. American Society of Mechanical Engineers. 3. "Steel Products Manual — Stainless and Heat Resisting Steels," American Iron and Steel Institute. 4. ASTM Designation A240-80b (ANS G81.4) — Standard Specification for HeatResisting Chromium and ChromiumNickel Stainless Steel Plate, Sheet and Strip for Fusion-Welded Unfired Pressure Vessels.
"Corrosion Resistance of the Austenitic Chromium-Nickel Stainless Steels in Chemical Environments," The International Nickel Co., April 1970, 16 pages. "Selection of Stainless Steels," American Society for Metals, 1968, 82 pages. “Corrosion Engineering," G. Fontana and N.D. Greene, McGraw-Hill Book Co., 1967. "The Possibility of Service Failure of Stainless Steels by Stress Corrosion Cracking," J.E. Truman and H.W. Kirkby, Metallurgia, August 1965.
43
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Useful Information on the Design of Plate Structures Revised Edition – 2011
The material presented in this publication is for general information only and should not be used without first securing competent advice with respect to its suitability for any given application. The publication of the material contained herein is not intended as a representation or warranty on the part of the Steel Market Development Institute—or of any other person named herein—that this information is suitable for any general or particular use or of freedom from infringement of any patents. Anyone making use of this information assumes all liability arising from such use.
Published by STEEL MARKET DEVELOPMENT INSTITUTE, A business unit of the American Iron and Steel Institute
In cooperation with and editorial collaboration by STEEL PLATE FABRICATORS ASSOCIATION, Div. of STI/SPFA
Copyright Steel Market Development Institute 2011
This page has been left blank intentionally.
Introduction
T
he purpose of this publication is to provide a design reference for the usual design of tanks for liquid storage. Volume 1, "Steel Tanks for Liquid Storage,” deals with the design of flatbottom, cylindrical tanks for storage of liquids at essentially atmospheric pressure. Volume 2, "Useful Information on the Design of Plate Structures,” provides information to aid in design of such structures. Scope
Volume 2, "Useful Information on the Design of Plate Structures,” covers many facets of plate design that are generally applicable to many types of structures. Information on these is now conveniently collected in one source. Drawing on many sources, this volume offers such information and discusses some of the more commonly encountered problems. Included is an outline of membrane theory, data for weld design, commonly used details, plus data and mathematical tables useful in design of steel plate structures. The intent is to include information principally pertinent to plate structures. For convenience of users of this volume, some data readily available elsewhere, particularly in mathematical tables, has been incorporated.
Inquiries for further information on design of steel tanks should be directed to: Steel Plate Fabricators Association Division of STI/SPFA 944 Donata Court Lake Zurich, IL 60047 www.steeltank.com
ii
Contents
Part I ― Part II ― Part III ― Part IV ― Part V ― Part VI ― Part VII ― Part VIII ―
Flat Plates .................................. ................. ................................... ................................... ................................1 ...............1 Membrane Theory .................................. ................ .................................... ...................................7 .................7 Self-Supported Stacks ...................................... ................... ...................................... ........................ .....17 Supports for Horizontal Tanks Tanks and Pipe Lines ................... ................ ...25 25 Anchor Bolt Chairs .................................... .................. ..................................... .............................. ........... 42 Design of Fillet Welds .................................... .................. ..................................... ......................... ......45 45 Inspection and Testing of Welded Vessels ........................... ................. ..........57 57 Appendices .................................... .................. .................................... ................................... ......................... ........ 59
Part I Flat Plates
F
elastic limit. In general it must not be expected that these formulas will yield stresses accurate to better than 5%. The mode of support and manner of loading specified must be complied with if the stresses are to be realized. No commercial edge fastening will correspond exactly with the theoretical conditions. The exact restraint of the edge, where bending is of prime importance, will depend on the rigidity of the support, the flexibility of any gaskets used, the position of the bolting circle and the spacing of the bolts therein, as well as the tightness with which the joint is bolted up. When membrane action is of importance, the degree of bolting up and the ability of the reinforced opening to resist slight deformations under radial tensions will largely determine the exact stress in the plate and the corresponding deformation. The bending moment at the edge is of less importance than at points where plate resistance depends primarily on bending. In view of these remarks, the conditions "Fixed" and "Supported" serve as guides to the possible range of stress and deflection. Notation a = length, in. of semi-minor axis of supporting ellipse for elliptical plates A = length, in., of semi-major axis of supporting ellipse for elliptical plates b = length, in., of short side of rectangular plate at supports B = length, in., of long side of rectangular plate or side of square at supports B1 = factor for stress in uniformly loaded, fixed edge, rectangular plates (Tables 1-1A and 1-1B) B2 = factor for stress in uniformly loaded, simply supported, rectangular plate (see Tables 1-1A and 1-1B) E = modulus of elasticity, psi f = maximum fiber stress in bending, psi F y = specified minimum yield strength, psi H = uniform load, ft. of water LS = stiffener spacing, in. n = a/A or b/B p = uniform load or pressure, psi
lat plates are used in many conventional structural forms, such as plate girders, built-up columns, or component parts of trusses. Such uses are well covered in standard texts or handbooks and are not discussed in this volume. Instead, Part I will cover applications in steel tanks. Bending Stresses and Deflections
Used as a membrane, as in the shell of a tank, a steel plate is a very efficient member. In contrast, a flat plate in bending normal to its plane is inefficient. Circumstances, nevertheless, sometimes dictate the use of a flatwalled tank because of space limitations, or the storage of a corrosive liquid may dictate use of a grillage-supported bottom to facilitate inspection. In such cases, a stiffened flat surface is indicated. On the next page, formulas are given for calculating the maximum bending stresses and maximum center deflections of certain flat plates. These formulas have been derived from various sources, the most important being based on an analytical derivation from elastic theory. However, those relating to three classes of elliptical plates and to certain others with a central applied load are less rigid in their derivation though sufficiently reliable for the use of the designer. It must be remembered that all formulas apply to materials such as steel, for which Poisson's ratio is 0.30. The inherent limitations of these formulas must be kept in mind. It is assumed that tensions in the plane of the plate appropriate to membrane action are small or negligible compared with the stresses due to bending. In general, the deflection must be small compared with the plate thickness if this is to be true. For greater deflections, other more complicated formulas must be used in whose derivation both membrane and bending action are considered. The formulas given may yield reliable working stresses yet be absolutely unreliable in calculating the load at failure and the corresponding deflection, particularly in the case of materials which elongate materially before failure, or which assume a dished form under load through initial stressing beyond the 1
P = r = r =
concentrated load, lb radius, in., of central loaded area inside knuckle radius, in., for flat, unstayed, circular plates R = radius, in., to support for circular plates S = spacing, in., of adjacent staybolts at corners of square plates t = plate thickness, in. Δ = center deflection, in., of plate relative to supports Φ = factor for stress in circular flanged plate (see Table 1-1A) for deflection of uniformly Φ1 = factor loaded, fixed-edge, rectangular plates (see Tables 1-1A and 1-1B) for deflection of uniformly Φ2 = factor loaded, simply supported rectangular plates (see Tables 1-1A and 1-1B) Φ3 = factor for deflection of fixed-edge, rectangular plates subjected to central concentrated load (see Tables 1-1A and 1-1B) One of the most commonly encountered conditions is a uniformly loaded flat plate supported on uniformly spaced parallel stiffeners. In the absence of any code or specification requirement, assume an allowable bending stress equal to 3/4 of the specified minimum yield stress value in the plate for determination of stiffener spacing L , s, in. The plate stress can be obtained from the formula in Table 1-1A for the case of a rectangle b x B, where B = ∞ and b is taken as Ls. Thus, for the fixed condition (continuous over the supports), the maximum permissible spacing of stiffeners becomes:
taken into account. An alternative solution, therefore, is to assume that yielding does occur at the support and the plate acts as a catenary between supports. At intermediate supports, the tension in the plate will be balanced; but at the outside support, restraint must be provided to resist that tension. This is not always easily accomplished. When the span is such that the profile of the plate approaches a catenary between supports, the support spacing is given approximately by the following formula:
′
1/ 2
tf ⎛ 24 f ⎞
Ls
=
Ls
= 900
(1-3) ⎜ ⎟ ⎝ F ⎠ Because of the approximate nature of the solution, a conservative value for f is indicated. Assuming f = 10,000t 10,000t and E = 29,000,000 psi for mild carbon steel, the equation becomes: p
t p
= 2.076
t H
(1-4)
Figure 1-2 gives graphical solutions for Equations 1-3 and 1-4. For the catenary approach, it is essential that a lateral force of 10,000t be resisted at the peripheral support. Since this is not always feasible, application of the catenary approach is limited. Similarly, it should not be used where pressure is reversible or where deflection is objectionable. In the above discussion, only plate stresses have been considered, and it is assumed that any welded plate joints will develop the full strength of the plate including appropriate joint efficiencies. Also, the stiffener system should be in accordance with accepted structural design principles. Protection against brittle failure of a structure should be considered at the time of design. Since environmental extremes, design detail, material selection, fabrication methods and inspection adequacy are all interrelated in protecting a structure from such failure, these factors should be evaluated.
1/ 2
⎛ 54,000t 2 ⎞ ⎟ Ls = ⎜ (1-1) ⎜ p ⎟ ⎝ ⎠ For convenience in connection with tank bottoms, the load can be expressed in feet of water, rather than psi, in which case: 1/ 2
⎛ 124,615t 2 ⎞ ⎟ (1-2) Ls = ⎜ ⎜ H ⎟ ⎝ ⎠ Figure 1-1 gives graphically stiffener spacing determined from Equations 1-1 and 1-2 for an allowable bending stress of 27,000 psi (i.e. F y = 36,000 psi). If deflection exceeds t /2, the plate will tend to act as a membrane in tension and exert a lateral pull on the outside support that must be 2
Figure 1-1. Stiffener Spacing for Flat Plate Acting as Continuous Beam.
Figure 1-2. Stiffener Spacing for Flat Plate with Catenary Action.
3
Table 1-1A. Flat Plate Formulas Poisson’s Ratio = 0.30 SHAPE
Loading
Edge Fixation Fixed
0.75 p
Supported
1.24 p
⎡
Central concentrated P on r
Fixed
Supported
Fixed Uniform
1 . 43 ⎢ log 10
⎢ ⎣
1 . 43
⎡ ⎢ log ⎢⎣
10
⎛ R ⎞ ⎜ ⎟+ ⎝ r ⎠
⎛ R ⎞ ⎜⎜ ⎟⎟ + ⎝ ⎠
Ellipse
Central
Fixed
R 2 t 2
⎛ r ⎞ ⎟ ⎝ R ⎠
+
6 3n 4
+ 2n 2 + 3
0 . 06
p
0.42n 4 3n
⎤ ⎥ ⎥ ⎦
P / t 2
⎛ r ⎞ ⎜ ⎟ ⎝ R ⎠
2
⎤ ⎥ ⎥⎦
0.17
⎛ p ⎞ ⎜ ⎟ ⎝ E ⎠
R 4
0.695
⎛ p ⎞ ⎜ ⎟ ⎝ E ⎠
R 4
0.22
⎛ p ⎞ ⎜ ⎟ ⎝ E ⎠
R 2
⎛ p ⎞ ⎜ ⎟ ⎝ E ⎠
R 2
P
0.55 t 2
a
+ n2 + 1
t 3 3
t
t 3
1.365
t 2
p
t 3
⎛ p ⎞ a 4 ⎜⎜ ⎟⎟ 3n 4 + 2n 4 + 3 ⎝ E ⎠ t 3
2
3
4
2
0 . 11 ⎜
2A X 2a a
t
0 . 334
p Supported
R 2 t
Uniform p Circle Radius R
Center Deflection Δ In.
Maximum Fiber Stress psi f
a
2
2
t
50
P
2
2
13.1
P
+ 2n + 12.5 t
Supported
0.42n 4
Fixed
B1 p
Supported
B2 p
+ n 2 + 2.5 t 2
Rectangle BXb b
n=a/A Approximate Fits n=1, load over 0.01% of area
t
b2
Central concentrated P
Supported
Fixed
1 + 2n 5.3
Square X B B X Central concentrated P
Flat Stayed Plate
Circular Flanged * † ‡
Uniform p
Uniform p
B
2
‡
t
P
1 + 2.4n
Supported
0.287 p
Fixed
1.32
Supported
1.58
Staybolts spaced at corners of square of side S
Fastened to shell
0.228 p
Φ2 and B2 depend on B/b.
2
See Table 1B.
⎛ p ⎞ b 2 ⎟ ⎝ E ⎠ t 3
b/B = n Approximate Fits n =1 and n = = 0
φ 3 ⎜
b/B = n Approximate
‡
Fits n =1 and n = = 0
t
4
0.0138
⎛ p ⎞ B ⎜ ⎟ ⎝ E ⎠ t 3
ƒ max. center of side
0.0443
⎛ p ⎞ B 4 ⎜ ⎟ ⎝ E ⎠ t 3
ƒ max. of center
2
2
t
Uniform P
See Table 1B.
⎛ p ⎞ b 4 ⎟ ⎝ E ⎠ t 3
t 2
0.308 p
Φ1 and B1 depend on B/b.
φ 2 ⎜
2
B 2 t 2 P
As above. Deflection nearly exact.
t 2 P
0.0125
⎛ p ⎞ B 2 ⎜ ⎟ ⎝ E ⎠ t 3
Approximately for ƒ; Area of contact not too small.
0.0284
⎛ p ⎞ S 4 ⎜ ⎟ ⎝ E ⎠ t 3
If plate as a whole de-forms, superimpose the stresses and deflections on those for plate flat when loaded.
t 2 s2 t 2
2 ⎡ ⎛ R r 1 + r ⎞ ⎤ ⎜ ⎟ ⎥ ⎢ r p ⎢ + Φ⎜ 2 R ⎟ ⎥ t ⎟⎟ ⎥ ⎢ 2t ⎜⎜ ⎢⎣ ⎝ ⎠ ⎥⎦
n=a/A Exact Solution
†
⎛ p ⎞ b 4 ⎟ ⎝ E ⎠ t 3
P
As above Center Stress
Fits n=0 and n=1
φ 1⎜
2
P uniform over circle, radius r., Center Stress
n=a/A Approximate Fits n=0 and n=1 Load over 0.01% of area
2
4.00 Fixed
ƒ max. at center
†
b2
Uniform p
ƒ max. at edge
n=a/A Approximate *
Concentrated P
Remarks
Φ varies with shell and joint
stiffness from 0.33 to 0.38 Knuckle Radius, r′
Formula of proper form to fit circle and infinite rectangle as n varies n varies from 1 to 0 Formulas for load distributed distributed over 0.0001 plate area to match circle circle when n = = 1. They give reasonable values for stress when n = = 0. Stress is lower for larger area subject to load. Formulas of empirical form to fit Hutte values for square when n = = 1. They give reasonable values when n = = 0. Assume load on 0.01 of area. Apparent stresses only considered. These formulas are not to be used in determining failure.
4
Table 1-1 B. Flat-Plate Coefficients
Stress Coefficients – Circle with Concentrated Center Load 1.0
0.10
0.09
0.08
0.07
0.06
0.05 0.05
0.04
0.03
0.02
0.01
Fixed1
0.157
1.43
1.50
1.57
1.65
1.75
1.86
2.00
2.18
2.43
2.86
Supported2
0.563
1.91
1.97
2.05
2.13
2.23
2.34
2.48
2.66
2.91
3.34
4.0
5.0
∞
1.91 0.435
1.95 0.442
2.00 0.455
r/R
Stress and Deflection Coefficients - Ellipse 1.0
1.2
1.4
1.6
1.8
2.0
2.5
Uniform Load Fixed. Stress3 Deflection4
0.75 0.171
1.03 0.234
1.25 1.284
1.42 0.322
1.54 0.350
1.63 0.370
1.77 0.402
1.84 0.419
Uniform Load Supported5
1.24
1.58
1.85
2.06
2.22
2.35
2.56
2.69
2.82
2.88
Central Load Fixed6 Supported7
2.86 3.34
3.26 3.86
3.50 4.20
3.64 4.43
3.73 4.60
3.79 4.72
3.88 4.90
3.92 5.01
3.96 5.11
3.97 5.16
A/a A/ a
3.0
3.00
4.00 5.24
Stress and Deflection Coefficients - Rectangle B/b
1.0
1.25
1.5
Stress B1
0.308
0.399
0.454
Stress B 2
0.287
0.376
0.452
1.75
2.0
0.490
0.497
0.517
0.569
0.610
1.33
1.75
2.12
2.25
2.42
2.67
1.56
2.09
2.56
2.74
2.97
3.31
Deflection Φ1
0.0138
0.0199
0.0240
0.0264
0.0277
Deflection Φ2
0.0443
0.0616
0.0770
0.1017
0.1106
0.1336
0.1400
Deflection Φ3
0.1261
0.1802
0.1843
0.1848
4 1 + 2n 2 5.3 1 + 2.4n 2
1.6
0.0906
0.1671
(r /R )2 ] Values of 1.43 [ log 10 R /r / r + 0.11 (r
1
2.5
3.0
0.741
0.748
0.750
3.03
3.27
3.56
3.70
4.00
3.83
4.18
4.61
4.84
5.30 0.0284
of 3/[0.42n 3/[0.42n 4 + n 2 + 1)
of 1.43 [ log 10 R / r + (r/R )2 ] + 0.334 + 0.06 (r/R
6Values
of 50/(3n 50/(3n 4+ 2n 2n 2 + 12.5)
3Values
of 6/[ 3n 3n 4 + 2n 2n 2 + 3)
7Values
of 13.1/[0.42n 13.1/[0.42n 4 + n 2 + 2.5)
5
∞
0.713
5Values
3n 4 + 2n 2n 2 + 3) Values of 1.365/[ 3n
5.0
0.500
2Values
4
4.0
0.1416
0.1422 0.1849
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Part II Membrane Theory
M
ost vessels storing liquid or gas are surfaces of revolution, formed by rotation of one or more continuous plane curves about a straight line in their plane. The line is called the axis of revolution. All sections of a shell of revolution perpendicular to the axis of revolution are circles. Usually the axis of revolution of a storage vessel is vertical, in which case all horizontal sections are circles.
Note: Radii R1 and R2 lie in the same line, but have different lengths except for a sphere where R1 = R2. T 1 and T 2 are loads per inch and will give the membrane stress in the plate when divided by the thickness of the plate. General Equation for Membrane Forces
Consider an element of a spherical section of unit length in each direction. Figure 2-1 indicates the radii and forces T1 and T2 acting on the element. Figures 2-2 and 2-3 indicate the pressure on the element and the components of the membrane unit forces in the latitudinal and meridional planes. For equilibrium, the summation of forces must be equal to zero.
Notation
P = The internal pressure on shell. It may be be due to gas alone (PO ), ), liquid alone (PL ), ), or both together (P + P) (psi). T 1 = The meridional force (sometimes called longitudinal force). This is force in vertical planes, but on horizontal sections (pounds per inch). T 1 is positive when in tension. T 2 = The latitudinal force (sometimes called hoop or ring force). This is force in horizontal planes, but on vertical section (pounds per inch). T 2 is positive when in tension. R = Horizontal radius at plane under consideration from axis of revolution (in). R1 = Radius of curvature in vertical (meridional) plane at level under consideration (in). Generally Rl is negative if it is on the opposite side of the shell from RP. RP. R2 = Length of the normal to the shell at the plane under consideration, measured from the shell to its axis of revolution (in). Generally R2 is positive unless the plane results in more than one circle. W = = Total weight of that portion of the vessel and its content, either above or below the plane under consideration, which is treated as a free body in computations for such plane (pounds). W has the same sign as P when acting in the same direction as the pressure on the plane of the free body, and the opposite sign from P when acting in the opposite direction. AT = Cross-sectional area of the interior of the vessel at the plane under consideration (square inches). γ = Density of product (pounds per cubic inch).
Σ Outward force = PR2 Φ2 R1 Φ1 Φ Φ Σ Inward Force = 2T 1 2 R2 Φ 2 + 2T 2 2 R1Φ 1 2
2
Equating the two: PR2 Φ2 R1 Φ1 = 2T 1
Φ1 2
R 2 Φ 2
+ 2T 2
Φ2
R1 Φ 1 2 ∴ PR1 R2 = T 1 R2 + T 2 R1
∴P =
T 1 R1
+
T 2 R 2
(2-1)
Equation 2-1 is the general equation for membrane forces. This equation considers membrane forces primarily produced by the product contained within the vessel. The weight of the vessel itself may add to these forces and should be considered in the analysis. Modified Equations for Membrane Forces
In general, the meridional force is the unit force in the wall of the vessel required to support the weight of the product, internal pressure and plate weights at the plane under consideration. In the free body diagram (Figure 2-5), consider the forces acting at Plane 1-1. The total forces acting at Plane 1-1 from above the Plane = P Π R2. 7
General Equation for Membrane Forces
Figure 2-1
Figure 2-2
Figure 2-3
8
Modified Equations for Membrane Forces
F
Figure 2-4
Figure 2-5
9
Total forces acting at Plane 1-1 from below the plane = W . Total vertical downward force= P.Π .R2 + W Vertical force required along circumference at Plane 1-1 to support the downward forces: P.Π R . 2 + W T VI = 2Π R Membrane force T VI P.Π. R 2 + W T 1 = = Sin Φ 2 Π R Sin Φ or
PR
=
=
⎡ W ⎤ ⎥ ⎢P + 2 Sin Φ ⎢⎣ Π R 2 ⎥⎦
Sin Φ
2 Π R Sin Φ
= R2 and Π R2 = AT
The equations for membrane forces can be further simplified for some of the shapes.
⎡ W ⎤ P+ (2-2) ⎢ ⎥ 2 ⎣ AT ⎦ From Equation 2-1 ⎡ T ⎤ T 2 = R2 ⎢ P − 1 ⎥ ⎣ R1 ⎦ These are the equations used in API 620. T 1
=
= PR2
The sign of R1, R2, P, W, and AT are shown in Table 2-1 and must be included in computing the forces. For any other vessel configuration, a free body diagram can be drawn and the forces T1 and T2 calculated in a similar way.
R
R
Since
+
T 2
W
T 1
2 Sin Φ
⎡ T ⎤ = R2 ⎢ P − 1 ⎥ ⎣ R1 ⎦ For Figures 2-10, 2-11, 2-12, and 2-13 where R1 = ∞, the equations for membrane forces reduce to: R ⎡ W ⎤ T 1 = 2 ⎢ P + ⎥ AT ⎦ 2 ⎣ T 2
R2
a. Spheres For spheres with no product (gas pressure only), the equations reduce to: T 1
=
PG R2
2
⎡ PR ⎤ = R2 ⎢ P − 2 ⎥ 2 R1 ⎦ ⎣ Since R1 = R2 = R T 2
Simplified Equations for Commonly Used Shapes
Figures 2-6 to 2-14 show the common vessel shapes used and the direction and magnitude of the radii, pressure, and weights acting on the free body diagram. Table 2-1 indicates the sign for each variable. The figures use the following notations:
T 1
= T 2 =
PR
2
where R = radius of sphere. b. Cylinders If the weight of the plate is neglected and there is no internal pressure in the vessel and since R2 = R : R :
Level of product in the vessel. Volume of product to be used in calculating the weight of product above or below the free body diagram.
T 1
Area of plate to be used us ed in calculating the weight of plate above or below the free body diagram. For all figures: P = PG + γ H H
=
R ⎡
2
⎢ P L ⎣⎢
Π R 2γ H ⎤ − ⎥ ΠR 2 ⎦⎥
Since γH = = PL T 1 = 0 T 2 = PL.R where R = radius of cylinder.
AT = = ΠR2 For Figures 2-6, 2-7, 2-8, 2-9, and 2-14, the equations for membrane forces are: 10
Figure 2-6 Spherical Vessel or Segment. Plane below line of s upport.
Figure 2-7 Spherical Vessel or Segment. Plane above line of s upport.
11
Figure 2-8 Spheroidal Vessel or Segment. Plane below line of support.
Figure 2-9 Spheroidal Vessel or Segment. Plane above line of support.
12
Figure 2-10 Conical Vessel or Segment. Plane below line of support.
Figure 2-11 Conical Vessel or Segment. Plane above line of support.
13
Figure 2-12 Conical Vessel or Segment. Pressure on convex side. Conical Vessel or Segment. Pressure on convex side. Plane above line of support.
Figure 2-13 Cylindrical Vessel. Plane above line of support .
14
Figure 2-14 Curved Segment. Pressure on convex side. Plane above line of support.
TABLE 2-1
Figure 2-6 2-7 2-8 2-9
R1 + + + +
2-10 2-11 2-12 2-13 2-14
-
R 2 + + + + + + + + +
P + + + + + + + +
15
W + + + + -
AT + + + + + + + + +
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Part III Self-Supported Self-Supported Stacks
If investigation shows that stresses due to vibrations are not within safe limits, the only solutions are to change the stack diameter or to add a damping device. Such devices might consist of a gunite or similar lining or so-called “wind spoilers” on the exterior of the stack. The subject is quite complex. To attempt a brief summarization could be dangerously misleading. Instead, a bibliography of references is appended at the end of this part for the benefit of those who wish to explore the subject more thoroughly.
Scope
T
he scope defined for this Volume stated that stacks would not be discussed in detail because of the complicated problem of resonant vibrations. Apart from this phase, however, there are purely structural facets that may be of interest. For the benefit of those not familiar with the problem, a brief explanation of stack vibration follows: Wind-Induced Vibrations
When a steady wind blows on an unsheltered, unguyed stack, formation and shedding of air vortices on each side of the stack can apply alternating lateral forces that cause movement of the stack perpendicular to the direction of the wind. The frequency of vortex shedding is a function of wind velocity and stack diameter. The term critical velocity denotes the wind velocity at which the frequency of vortex shedding equals the natural frequency of the stack. Under such conditions, resonance occurs. Excessive lateral dynamic deflection and vibration of the stack from vortex shedding may occur at wind velocities considerably below the maximum wind velocity expected in the area. One way to avoid resonance and consequent damage to the stack is to proportion the stack so that the critical wind velocity exceeds the highest sustained wind velocity that is likely to occur. In most areas, for example, it is unlikely that a steady wind of more than 75 mph will occur. Hence, a stack having a critical velocity of 75 mph is probably safe in those regions, though gusts of greater velocity might occur. There may be reasons, however, why a stack of such proportions will not serve the purpose. If so, the effects of dynamic vibrations must be thoroughly investigated. If the critical wind velocity is low enough, it may be that the stresses due to dynamic deflections are within design limits. In that case, the stack is structurally adequate if noticeable movement of the stack is not objectionable.
Minimum Thickness and Corrosion
In view of the corrosive nature innate to stack operation, it is wise to add a corrosion allowance to the calculated shell thickness. The nature of the flue gasses and moisture content in the area are some important parameters in determining the amount of corrosion for which to allow. Erection requirements usually dictate minimum plate thicknesses and the stress formulae in this part are not considered valid for thicknesses less than 1/4". Therefore, the minimum thickness for shell plate is taken to be 1/4" nominal.
A
α As
β C C’c C L D Do E EL F a
17
= Cross-sectional area of base ring, in.2 = Vertical angle of cone to cylinder, deg. = Required area for stack stiffeners, in.2 = Critical damping ratio of stack = See Figure 3-4 (Sec. A-A) = Euler Factor = Lift coefficient (0.2 for circular cylinder) = Outside diameter of stack, in. = Outside diameter of cylindrical portion of stack, ft. = Modulus of elasticity, psi at design temperature = Joint efficiency for base plate design = Allowable compressive stress for circumferential stiffeners, 12,000 psi (unless otherwise noted)
F b F c F cr cr F L F s F y FS H H 1 I s K Φ K Ls L Ls1 M N Pd R1 Ro S Ss T V V cr1 cr1 V cr2 cr2 V o W W’ W s do f c f o f t g h p qcr r t
w
= Allowable bending stress, 0.6 F4, psi for stiffeners = Allowable compressive stress, ksi = Critical buckling stress, ksi = Equivalent static force, lb/ft of height = Allowable compressive stress, psi (in cone cylinder junction area) = Yield point of stack material, ksi = Factor of safety = Overall height of stack, ft. = Overall height of stack, in. = Required moment of inertia for stack stiffeners, in.4 = Effective length factor = Slenderness reduction factor = Stiffener spacing, ft. = Length for KL/ KL/r = Stiffener spacing, in. = Moment at any design point, inchpounds = Number of anchor bolts = Wind load, psi = Outside conical radius, in. = Outside radius of cylinder portion of stack, in. = Strouhal number (0.2 for steel stack) = Required section modulus for stack stiffeners, in.3 = Load per bolt, lb. = Total direct load at any design point, lb. = Critical wind velocity, mph = Critical wind velocity, ft/sec. = Resonance velocity, ft/sec. = Chord for arc W’, W’, in. = Arc length of breeching opening, in. = Unit weight of stack shell, lb./in.3 = Outside diameter of belled stack base, ft. = Compression stress, ksi = Frequency of the lowest mode of ovaling vibration, cps = Natural frequency, cps = Acceleration of gravity, 386 in./sec. = Height of stack bell, ft. = Wind load, psf = Dynamic wind pressure, psf = Radius of gyration, in. = Thickness of stack, in.
= Uniform load over breeching opening, lb./in.
Static Design Criteria
In the suggested static design criteria below, the proportions indicated are those desirable from a structural standpoint. Independent calculations are needed to determine sizes to satisfy draft or capacity requirements. In general, stacks proportioned as suggested will probably have a high critical wind velocity, but a dynamic check should be made to verify this. Short stacks (less than 100 ft. high) may be straight cylinders without a belled base. Minimum base diameter do = H / 10 10 (3-1) Minimum bell height h = 0.3H (3-2) Minimum diameter of cylinder, do = H / 13 13 (3-3)
Figure 3-1. Cylindrical Stack with Belled Base.
Stacks are likely to be subjected at least to the following loads: 1. Metal Weight. 2. Lining Weight. 3. Wind: Wind load provisions may be found in ASCE 7-95. Local building codes should also be consulted. 4. Icing (if required). 5. Seismic (if required). 6. Thermal cycling (vertical and circumferential). 7. Possible negative pressures. 8. Other requirements of local building codes. Dynamic Wind Criteria
The dynamic influence of wind may be approximated by assuming an equivalent static force, FL, in pounds per foot of height, acting in the direction of oscillations, given by: F L = C L Do qcr /2 β (3-4) 18
NOTE: NOTE: β = Critical damping factor which varies from 1% for an unlined steel stack of small diameter to 5% for concrete.
Stack Stresses
The stresses associated with buckling have four ranges into which they can fall depending on the t/R ratio. They in turn may be affected by the Euler effect or slenderness ratio reduction factor. The stresses calculated in this manner are not to be increased for wind or earthquake stresses.
The dynamic wind pressure, qCR, in psf, is given by: *q *qCR = 0.00119 V CR CR2. The critical wind 2 velocity, V CR CR in fps, for resonant transverse vibration is given by: 2
V CR ( ft / sec ) =
f t Do S
(3-5) Fcr
The natural frequency, f frequency, f t (cps), of vibration of a stack of constant diameter and thickness is given by: 3.52 D ⎡ Eg ⎤ f t = (3-6) ⎥ 2 ⎢ 4π H 1 ⎣ 2W s ⎦ Critical velocity for a steel stack with an S value of 0.2 is given by: V cr 1 (mph ) = 3.41 Do f t
(3-7)
C ' c
Values of effective diameters and effective height for stacks of varying diameter and thickness may be determined by methods found in reference number 19.
t/Ro Range
5.8 x 103 t/Ro
0.017 < t/Ro ≤ Fy/11600 Fy/11600
Fy [0.35 + 50t 50 t / Ro]
Fy/11600 Fy/11600 ≤ t/R, t/R, ≤ 0.01
5 t /Ro] Fy [0.8 Fy [0.8 + 5t
0.01 ≤ t/Ro ≤ .04
Fy
t/Ro > .04
=
2π 2 E Fcr
FS = 2.0
Fc = Fc = K ΦFcr/FS
⎡ C ' c ⎤ If C’c < C’c < KL/ KL/r K Φ = 0.5 ⎢ ⎣ KL / r ⎥⎦
* Reference number 14(b)
(3-10) 2
Critical Wind Velocity for Ovaling Vibrations
In addition to transverse swaying oscillations, stacks experience flexural vibration in the cross-sectional plan as a result of vortex shedding. This frequency of the lowest mode of ovaling vibration in a circular shell is: f o
=
678.5t Do
2
2
⎡ KL / r ⎤ If C’c ≥ KL/ KL/r K Φ = 1 – 0.5 ⎢ ⎣ C ' c ⎥⎦ Tables 3-1, 3-2 and 3-3 have been developed using ASTM A36 steel with a yield of 36 ksi. The value of K is is taken as 2 in view of the fact that a stack is normally a cantilever. These allowable stresses will also be used for tapered or belled base stacks using the equivalent cylindrical radius approach as shown below. In order to arrive at allowable stresses in the conical section one would substitute R1 into the above formulae for Ro.
(3-8)
Resonance occurs when frequency of the lowest mode of ovaling vibration is twice the vortex shedding frequency; thus, the critical wind velocity for ovaling frequency is: V o
=
f o Do
2S
= ( ft / sec)
(3-9)
Unlined stacks are subject to ovaling vibrations. In order to prevent this phenomenon, the thickness of the stack should not be less than D/250 or D/250 or intermediate stiffeners are required to raise the resonant velocity above 60 mph. Care should be exercised in coastal areas to give special attention to high winds as outlined in the aforementioned ASCE 7-95. Figure 3-2. Loads on Cylinder-Cone Junction
19
resistance to negative draft. Spacing of intermediate stiffener Ls is:
Cylinder-Cone Junction
In many applications of tubular columns, it is desirable to use a base cone to provide a broader base for anchorage. At the junction of the cone and cylinder (Figure 3-2), it is necessary to provide reinforcement to resist the maximum vertical force. P
V =
2 Ro
+
M Ro
2
and the radial thrust H = = V tan α The ring compression to be resisted is C = HRo = VRo tan α The area of reinforcement required is A s
=
VR o tan F s
Ls = 60
E
(3-11) (3-12)
S s
=
(3-13)
I s
=
As
=
(3-14)
(3-17)
pLs1 D 2
1100F b Pd Ls1 D 3
8 E Pd Ls1 D 2 F a
(in3)
(3-18)
(in4)
(3-19)
(in2)
(3-20)
To satisfy the requirements of the above intermediate stiffener design formulae a portion of the stack equal to 1.1t Dot may be included. Breeching Opening
The breeching opening should be as small as consistent with operating requirements with a maximum width of 2Do/3. The opening must be reinforced vertically to replace the area of material removed increased by the radio of Do/ Do/C. Therefore, C. Therefore, each vertical stiffener on each side of the opening should have a cross-sectional area of: As
3
p
To insure a nominal size of intermediate stiffener, the spacing is limited within 1.5 times the stack diameter. Intermediate stiffeners should meet the following minimum requirements:
Under load, the junction reinforcement, or stiffener, will move elastically inward. This will induce secondary vertical bending stresses on each side of the junction. For that reason, it is desirable to keep allowable stress F s relatively low. If F s is in the range of 8,000 psi, the secondary stresses can usually be ignored if Do is not greater than about 15 ft. For greater diameters or higher values of F s it would be advisable to evaluate the secondary stresses. Note that V is is the maximum value resulting from both vertical load and bending moment in the cylinder at the junction level. The moment of inertia I s of the stiffener section should not be less than: HRo
Do t
(3-15)
=
W d Ls1 D
2C
(3-21)
Each vertical stiffener in conjunction with a portion of the liner shell would be designed as a column. Each stiffener should extend far enough above and below the opening to develop its strength. Horizontal reinforcement should be provided by a ring girder above and below the opening. These girders should be designed as fixed-end beams to carry the load across the opening above and below. The span in bending is the width W between between the side column, but the girders should encircle the stack to preserve circularity at the opening. To form each ring girder, stiffener rings should be placed to act as top and bottom flanges. The shell of the stack will serve as the web. Each ring girder must be capable of carrying a uniform distributed load, in terms of pounds per inch of arch W’, W’, of:
based on a factor of safety of 3 for critical buckling. The area of reinforcement and computation of I s provided by a stiffener may include an area of cylinder and cone plate equal to: 0.78(t Rot + t 1 R1t ) (3-16) Where R1 = Ro /cos α This approach can be used in designing the junction of two cones having different slopes, except that H would would be the difference between the horizontal components of the axial loads in the two cones. Circumferential Stiffeners
A stiffener is required at the top of the stack; also intermediate ring stiffeners are required to prevent deformation of the stack shell under wind pressure and to provide structural 20
w=
V
π Do
+
4 M π Do
2
(3-22)
The bending moment in the girder is: Mq =
mW '2
12
(3-23)
Allowable bending stresses may be chosen using AISC rules. Base Plates
In addition to bending stresses due to bending loads, the stack base plate must resist ring tension due to the horizontal component of the base cone if one is used. Maximum ring tension should be limited to 10,000 psi to account for secondary bending stresses in the base cone. This value may be varied upward depending upon the extent of secondary stress evaluation. Tension should be checked at the minimum cross-section occurring at the anchor bolt holes or at a weld joint where 85% or 100% efficiency may be assumed. A base plate area may be calculated by the following equation: A =
VDo tan α
20,000 E 1
Figure 3-4. Horizontal Section Through Opening.
(Section A-A, Figure 3-3) Base plate thickness may be determined by using AISC formulae and allowable bending stresses. Anchor Bolts
Minimum diameter = 1 ½” Maximum spacing of anchor bolts = 5’-6’ Maximum tension at root of threads = 15,000 psi. Each bolt should be made to resist a total tension in pounds of:
(3-24)
T =
4 M ND
−
V N
(3-25)
A suggested design procedure for anchor bolt brackets is covered in Part V.
Figure 3-3. Elevation of Stack.
(See Figure 3-4 for Section A-A)
21
Table 3-1 Fc Allowable Compressive Stress (Fy = 36 ksi) For t/Ro from .0017 through Fy/11600 t/Ro KL/r 0 17.5 35 52.5 70 87.5 105 122.5 140 157.5 175
.0017
.00192
.00214
.00236
.00258
.0028
.00302
4930 4917 4878 4813 4722 4605 4462 4293 4097 3877 3630
5568 5551 5502 5419 5303 5154 4971 4755 4507 4225 3909
6206 6185 6124 6071 5876 5691 5414 5196 4887 4537 4145
6844 6819 6744 6618 6443 6217 5942 5616 5240 4814 4338
7482 7452 7362 7212 7003 6733 6404 6015 5565 5056 4487
8120 8085 7979 7803 7556 7238 6850 6392 5862 5263 4593
8758 8717 8594 8389 8101 7732 7281 6747 6132 5434 4655
Table 3-2 F c Allowable Compressive Stress (F ( F y = = 36 ksi) For t / Ro from F y /11600 /11600 to .01 t/Ro KL/r 0 17.5 35 52.5 70 87.5 105 122.5 140 157.5 175
.003104
.00425
.0054
.00655
.0077
.00885
.00999
9094 9049 8917 8695 8386 7988 7501 6926 6262 5510 4670
10128 10073 9908 9634 9250 8756 8152 7439 6616 5683 4673
11162 11095 10895 10562 10095 9496 8762 7896 6896 5763 4673
12196 12116 11888 11480 10928 10207 9331 8297 7103 5769 4673
13230 13136 12855 12387 11732 10889 9859 8642 7237 5769 4673
14264 14155 13829 13284 12523 11543 10345 8930 7298 5769 4673
15298 15173 14797 14171 13295 12168 10791 9163 7301 5769 4673
Table 3-3. F c Allowable Compressive Stress (F y = 36 ksi) For t / Ro from .01 to .04 t/Ro KL/r 0 17.5 35 52.5 70 87.5 105 122.5 140 157.5 175
.01
.015
.02
.025
.03
.035
.04
15300 15175 14798 14173 13296 12169 10792 9163 7302 5769 4673
15750 15617 15219 14556 13627 12432 10972 9247 7302 5769 4673
16200 16060 15638 14936 13954 12690 11146 9320 7302 5769 4673
16650 16502 16057 15315 14277 12942 11311 9383 7302 5769 4673
17100 16944 16474 15692 14597 13189 11468 9435 7302 5769 4673
17550 17385 16891 16067 14914 13431 11618 9476 7302 5769 4673
18000 17827 17307 16440 15227 13666 11760 9305 7302 5769 4673
F c = .5 x F y x KΦ
If t/Ro > .04
Underlined numbers are an indicator at which point C’c > C’c > KL/ KL/r 22
14. F.B. Farquaharson, "Wind Forces Structures: Structures Subject Oscillations,” Proceedings of ASCE, Vol. 84, ST 4, Paper 1712, 1958, p.13. ASCE Transaction Paper #3269 ("Wind Forces on Structure"). 15. C.F. Cowdrey and J.A. Lewes, "Drag Measurements at High Reynolds Numbers of a Circular Cylinder Fitted with Three Helical Strakes,” NPLIAeroI384, July 1959. 16. L. Woodgate and J. Maybrey, "Further Experiments on the Use of Helical Strakes for Avoiding Wind-Excited Oscillations of Structures with Circular or Near Circular Cross-Section," NPL/Aero/381, July 1959. 17. A. Roshko, "On the Wake and Drag Bluff Bodies," presented at Aerodynamics Sessions, Twenty- Second Annual Meeting, IAS, New York, NY, January, 1954. 18. J.O. Smith and J.H. McCarthy, "Wind Versus Tall Stacks," Mechanical Engineering, Vol. 87, January, 1965, pp. 38-41. 19. Gaylord and Gaylord, "Structural Engineering Handbook." 2nd Edition, Chapter 26. 20. R. Stuart III, A.R. Fugini, A. DeVaul, Pittsburgh-Des Moines Corp. Research Report #98528, "Design of Allowable Compressive Stress Cylindrical or Conical Plates, AWWA DlOO," May 1981. 21. Roger L. Brockenbrough, Pittsburgh-Des Moines Corp. Research Report 98030, "Determination of The Critical Buckling Stress of Cylindrical Plates Having Low t/R Values," October 5, 1960. 22. Tom Buckwalter, Pittsburgh-Des Moines Corp. Supplement to RP 98030, "Determination of the Critical Buckling Stress in a Cylinder Having a t/R of 0.00426," December 20, 1960. 23. AlSC "Specification for Structural Steel Buildings – Allowable Stress Design and Plastic Design." 24. ASCE 7 “Minimum Design Loads for Buildings and Other Structures.”
References
1. M.S. Ozker and J.O. Smith, "Factors Influencing the Dynamic Behavior of Tall Stacks Under the Action of Winds,” Trans. ASME Vol. 78, 1956, pp. 1381-1391. 2. P. Price, "Suppression of the Fluid-Induced Vibration of Circular Cylinders," Proceedings of ASCE, Vol. 82, EM3, Paper No. 1030, 1956, p. 22. 3. W.L. Dickey and G.B. Woodruff, "The Vibration of Steel Stacks,” Proceedings of ASCE, Vol. 80, 1954, p. 20. 4. T. Sarpkaya and C.J. Garison, "Vortex Formation and Resistance in Unsteady Flow,” Journal of Applied Mechanics, Vol. 30, Trans. ASME, Vol.85, Series E, 1963, pp. 16-24. 5. A.W. Marris, "A Review on Vortex Streets, Periodic Wakes, and Induced Vibration Phenomena," Journal of Basic Engineering,Trans. ASME, Series D, Vol. 86, 1964, pp. 185-l 96. 6. J. Penzien, "Wind Induced Vibration of Cylindrical Structures,” Proceedings of ASCE,Vol. 83, EM 1 Paper No. 1141, January, 1957, p. 17. 7. W. Weaver,"Wind-Induced Vibrations in Antenna Members,” Transactions of ASCE, Vol. 127, Part 1, 1962, pp. 679-704. 8. C. Scruton and D. Walshe, "A Means of Avoiding Wind-Excited Oscillations of Structures with Circular or Nearly Circular Cross-Section," NPL/Aero/335, October 1957. 9. C. Scruton, D. Walshe and L. Woodgate, "The Aerodynamic Investigation for the East Chimney Stack of the Rugeley Generating Station," NPL/Aero/352. 10. A. Roshko, "On the Development of Turbulent Wakes from Vortex Streets,” NACA Report 1191, 1954. 11. A. Roshko, “On The Drag and Shedding Frequency of Two-Dimensional Bluff Bodies,” NACA Technical Note 3169, July 1954. 12. N. Delany and N. Sorensen, “Low-Speed Drag of Cylinders of Various Shapes,” NCA Technical Note 3038, November, 1953. 13. G.B. Woodruff and J. Kozok, "Wind Forces on Structures: Fundamental Considerations," Proceedings of ASCE, Vol. 84, ST 4, Paper No. 1709, 1958, p. 13. 23
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Part IV Supports for Horizontal Tanks and Pipe Lines
T
the saddle, which result in a different distribution of stress in the pipe or vessel wall from that encountered with a full ring support, are discussed in the following paper by L.P. Zick. It includes some revisions of and additions to the original paper published in “The Welding Journal Research Supplement,” September, 1951, and reprinted in “Pressure Vessel and Piping Design Collected Papers 1927-1959,” published by ASME in 1960.
here is considerable information available on design of supports for horizontal cylindrical shells where a complete ring girder is used. There are many installations where a horizontal tank, pressure vessel, or pipeline is supported by a saddle extending less than 180 ° around the lower part of the cylinder. The effects of vertical deflection of the cylinder and the concentration of stress around the horn of
Original paper published in September 1951 “THE WELDING JOURNAL RESEARCH SUPPLEMENT.” This paper contains revisions and additions to the original paper based upon questions raised as to intent and coverage.
Stresses in Large Horizontal Cylindrical Pressure Vessels on Two Saddle Supports Approximate stresses that exist in cylindrical vessels supported on two saddles at various conditions and design of stiffening for vessels which require it. By L.P. Zick* pressure vessels made of mild steel for storage of liquid weighing 42 lb. per cu. ft
INTRODUCTION The design of horizontal cylindrical vessels with dished heads to resist internal pressure is covered by existing codes. However, the method of support is left pretty much up to the designer. In general the cylindrical shell is made a uniform thickness which is determined by the maximum circumferential stress due to the internal pressure. Since the longitudinal stress is only one-half of this circumferential stress, these vessels have available a beam strength which makes the two-saddle support system ideal for a wide range of proportions. However certain limitations are necessary to make designs consistent with the intent of the code. The purpose of this paper is to indicate the approximate stresses that exist in cylindrical vessels supported on two saddles at various locations. Knowing these stresses, it is possible to determine which vessels may be designed for internal pressure alone, and to design structurally adequate and economical stiffening for the vessels which require it. Formulas are developed to cover various conditions, and a chart is given which covers support designs for
HISTORY In a paper1 published in 1933, Herman Schorer pointed out that a length of cylindrical shell supported by tangential end shears varying proportionately to the sine of the central angle measured from the top of the vessel can support its own metal weight and the full contained liquid weight without circumferential bending moments in the shell. To complete this analysis, rings around the entire circumference are required at the supporting points to transfer these shears to the foundation without distorting the cylindrical shell. Discussions of Schorer’s paper by H.C. Boardman and others gave approximate solutions for the half-full condition. When a ring of uniform cross-section is supported on two vertical posts, the full condition governs the design of the ring if the central angle * L.P. Zick is a former Chief Engineer for the Chicago Bridge and Iron Co., Oak Brook, Illinois. 25
Figure 4-1. Strain gauge test set up on 30,000 gal. propane tank. ............................................................................
had actually slumped down over the horns of the saddles while being filled with liquid, but had rounded up again when internal pressure was applied. Testing done by others,2,3 gave very useful results in the ranges of their respective tests, but the investigators concluded that analysis was highly indeterminate. In recent years the author has participated in strain gauge surveys of several large vessels. 4 A typical test setup is shown in Figure 4-1. In this paper an attempt has been made to produce an approximate analysis involving certain empirical assumptions which make the theoretical analysis closely approximate the test results.
between the post intersections with the ring is less than 126°, and the half-full condition governs if this angle is more than 126 °. However, the full condition governs the design of rings supported directly in or adjacent to saddles. Mr. Boardman’s discussion also pointed out that the heads may substitute for the rings provided the supports are near the heads. His unpublished paper has been used successfully since 1941 for vessels supported on saddles near the heads. His method of analysis covering supports near the heads is included in this paper in a slightly s lightly modified form. Discussions of Mr. Schorer’s paper also gave successful and semi-successful examples of unstiffened cylindrical shells supported on saddles, but an analysis is lacking. The semisuccessful examples indicated that the shells
Table 4-1 – Values of Coefficients in Formula for Various Support Conditions Saddle Angle,
θ
Maximum long. bending stress, Min. K 1
Tangent shear, K 2
120° 150°
0.63 ( A/L = 0.11) 0.55 ( A/L = 0.11)
1.171 0.799
120° 150°
1.0 ( A/L = 0) 1.0 ( A/L = 0)
0.880 0.485
120° 150°
0.23 ( A/L = 0.193) 0.23 ( A/L = 0.193)
0.319 0.319
120° 150°
0.23 ( A/L = 0.193) 0.23 ( A/L = 0.193)
1.171 0.799
Circumf. stress top of saddle, K 3j
Additional Ring head compres. stress, in shell, K 4 K 5 Shell unstiffened 0.0528 … 0.760 0.0316 … 0.673 Shell stiffened by head, A ≤ R/2 0.0132 0.401 0.760 0.0079 0.297 0.673 Shell stiffened by ring in plane of saddle … … … … … … Shell stiffened by rings adjacent of saddle 0.0132 … 0.760 0.0079 … 0.673
Ring Stiffeners Circumf. Direct bending, stress, K 6 K 7 7 … …
… …
0.204 0.260
… …
… …
0.204 0.260
0.0528 0.0316
0.340 0.340 0.303 0.303
0.204 0.260
0.0577 0.0353
0.263 0.228
0.204 0.260
against A/L,, for values of K 1, corresponding to values of A/L of A/L not not listed in table. *See Figure 4-5, which plots K 1 against A/L jSee Figure 4-7
26
Tension across saddle, K 8
Figure 4-2. Location and type of s upport for horizontal pressure vessels on two supports.
27
liquids usually associated with pressure vessels. Where liquids of different weights are to be stored or where different materials are to be used, a rough design may be obtained from the chart and this design should be checked by the applicable formulas outlined in the following sections. Table 4-1 outlines the coefficients to be used with the applicable formulas for various support types and locations. The notation used is listed at the end of the paper under the heading Nomenclature.
SELECTION OF SUPPORTS
When a cylindrical vessel acts as its own carrying beam across two symmetrically placed saddle supports, one-half of the total load will be carried by each support. This would be true even if one support should settle more than the other. This would also be true if a differential in temperature or if the axial restraint of the supports should cause the vessel acting as a beam to bow up or down at the center. This fact alone gives the twosupport system preference over a multiplesupporting system. The most economical location and type of support generally depend upon the strength of the vessel to be supported and the cost of the supports, or of the supports and additional stiffening, if required. In a few cases the advantage of placing fittings and piping in the bottom of the vessel beyond the saddle will govern the location of the saddle. The pressure-vessel codes limit the contact angle of each saddle to a minimum of 120 ° except for very small vessels. In certain cases a larger contact angle should be used. Generally the saddle width is not a controlling factor, so a nominal width of 12 in. for steel or 15 in. for concrete may be used. This width should be increased for extremely heavy vessels, and in certain cases it may be desirable to reduce this width for small vessels. Thin-wall vessels of large diameter are best supported near the heads provided they can support their own weight and contents between supports and provided the heads are stiff enough to transfer the load to the saddles. Thick-wall vessels too long to act as simple beams are best supported where the maximum longitudinal bending stress in the shell at the saddles is nearly equal to the maximum longitudinal bending stress at mid-span, provided the shell is stiff enough to resist this bending and to transfer the load to the saddles. Where the stiffness required is not available in the shell alone, ring stiffeners must be added at or near the saddles. Vessels must also be rigid enough to support normal external loads such as wind. Figure 4-2 indicates the most economical locations and types of supports for large steel horizontal pressure vessels on two supports. A liquid weight of 42 lb. per cu. ft. was used because it is representative of the volatile
MAXIMUM LONGITUDINAL STRESS
The cylindrical shell acts as a beam over the two supports to resist by bending the uniform load of the vessel and its contents. The equivalent length of the vessel (see Figures 4-2 and 4-3) equals L + 4H/3, H/3, closely, and the total weight of the vessel and its contents equals 2Q 2 Q. However, it can be shown that the liquid weight in a hemispherical head adds only a shear load at its junction with the cylinder. This can be approximated for heads where H ≤ R by representing the pressure on the head and the longitudinal stress as a clockwise couple on the head shown at the left of Figure 4-3. Therefore the vessel may be taken as a beam loaded as shown in Figure 4-3; the moment diagram determined by statics is also shown. Maximum moments occur at the midspan and over the supports. Tests have shown that, except near the saddles, a cylindrical shell just full of liquid has practically no circumferential bending moments and therefore behaves as a beam with a section modulus I/c = I/c = πr 2t. However, in the region above each saddle, circumferential bending moments are introduced, allowing the unstiffened upper portion of the shell to deflect, thus making it ineffective as a beam. beam.
28
Figure 4-3. Cylindrical shell acting as beam over supports.
Figure 4-4. Load transfer to saddle by tangential shear stresses in cylindrical shell.
29
This reduces the effective cross section acting as a beam just as though the shell were split along a horizontal line at a level above the saddle [See Figure 4-4 (a)]. If this effective arc is represented by 2A (A in radians) it can be shown that the section modulus becomes: 2 ⎞ ⎛ ⎜ Δ + sin Δ cos Δ − 2 sin Δ ⎟ ⎜ Δ ⎟ I / c = π r 2 t ⎜ ⎟ ⎛ sin Δ ⎞ ⎜⎜ ⎟⎟ π ⎜ − cos Δ ⎟ Δ ⎝ ⎠ ⎠ ⎝ Strain gauge studies indicate that this effective arc is approximately equal to the contact angle plus one-sixth of the unstiffened shell as indicated in Section A-A of Figure 4-4. Of course, if the shell is stiffened by a head or complete ring stiffener near the saddle the effective arc, 2 Δ, equals the entire cross section, and I/c = π r r 2 t. Since most vessels are of uniform shell thickness, the design formula involves only the maximum value of the longitudinal bending stress. Dividing the maximum moment by the section modulus gives the maximum axial stress in lb. per sq. in. in the shell due to bending as a beam, or: 3K 1 QL S 1 = ± π r 2 t
Figure 4-5. Plot of longitudinal bendingmoment constant, K1.
This maximum bending stress, S1, may be either tension or compression. The tension stress when combined with the axial stress due to internal pressure should not exceed the allowable tension stress of the material times the efficiency of the girth joints. The compression stress should not exceed one half of the compression yield point of the material or the value given by: ⎛ E ⎞ S 1 ≤ ⎜ ⎟ (t / r )[2 − (2 / 3) (100) (t / r )] ⎝ 29 ⎠ which is based upon the accepted formula for buckling of short steel cylindrical columns.* The compression stress is not a factor in a steel vessel where t/r >0.005 and the vessel is designed to be fully stressed under internal pressure. *See also par UG-23(b) ASME Code Section VIII Div. 1.
K 1 is a constant for a given set of conditions, but actually varies with the ratios A/L ratios A/L and and H/L R/L for different saddle angles. For ≤ R/L convenience, K 1 is plotted in Figure 4-5 against A/L A/L for various types of saddle supports, assuming conservative values of H = 0 when the mid- span governs and H = R when the shell section at the saddle governs. A maximum value of R/L R/L = 0.09 was assumed because other factors govern the design for larger values of this ratio. As in a beam the mid-span governs for the smaller values of A/L of A/L and the shell section at the saddle governs for the larger values of A/L; A/L; however, the point where the bending stress in the shell is equal at mid-span and at the saddle varies with the saddle angle because of the reduced effective cross section. Figure 4-10 on page 39 gives acceptable values of K 1.
TANGENTIAL SHEAR STRESS
Figure 4-4(d) shows the total shear diagram for vessels supported in saddles away from the heads. Where the shell is held round, the tangential shearing stresses vary directly with the sine of the central angle Φ, as shown in Section B-B of Figure 4-4, and the maximum occurs at the equator. However, if the shell is free to deform above the saddle, the tangential shearing stresses act on a reduced effective cross section and the maximum occurs at the horn of the saddle. This is approximated by assuming the shears 30
continue to vary as the sin Φ but only act on twice the arc given by( Θ/2 + (β/20) or (π-α) as shown in Section A-A of Figure 4-4. The summation of the vertical components of these assumed shears must equal the maximum total shear. The maximum tangential shear stress will occur on the center side of the saddle provided the saddle is beyond the influence of the head but not past the quarter point of the vessel. Then with saddles away from the heads, the maximum shear stress in lb. per sq. in. is given by: ⎛ K Q ⎞ L − 2 A S 2 = ⎜⎜ 2 ⎟⎟ ⎝ rt ⎠ L + 4 H
CIRCUMFERENTIAL SADDLE
values of K 2 listed in Table 4-1 for various types of supports are obtained from the expressions given for the maximum shears in Figure 4-4, and the Appendix. Figure 4-4(f) indicates the total shear diagram for vessels supported on saddles near the heads. In this case the head stiffens the shell in the region of the saddle. This causes most of the tangential shearing stress to be carried across the saddle to the head, and then the load is transferred back to the head side of the saddle by tangential shearing stresses applied to an arc slightly larger than the contact angle of the saddle. Section C-C of Figure 4-4 indicates this shear distribution; that is, the shears vary as the sin Φ and act downward above angle α and act upward below angle α. The summation of the downward vertical components must balance the summation of the upward vertical components. Then with saddles at the heads the maximum shear stress in lb. per sq. in. is given by: S 2 =
S 2 =
K 2Q
AT
HORN
OF
In the plane of the saddle the load must be transferred from the cylindrical shell to the saddle. As was pointed out in the previous section the tangential shears adjust their distribution in order to make this transfer with a minimum amount of circumferential bending and distortion. The evaluation of these shears was quite empirical except for the case of the ring stiffener in the plane of the saddle. Evaluation of the circumferential bending stresses is even more difficult. Starting with a ring in the plane of the saddle, the shear distribution is known. The bending moment at spy point above the saddle may be computed by any of the methods of indeterminate structures.
3
K 2 Q
STRESS
Figure 4-6. Circumferential bending-moment bending-moment diagram, ring in plane of saddle.
If the ring is assumed uniform in cross section and fixed at the horns of the saddles, the moment, M Φ, in in.-lb. at any point A is given by: ⎧ 3 sin β cos β 1 ⎛ Φ + − ⎜⎜ cos Φ − ⎪cos Φ + sin Φ − 2 2 β 2 4 ⎝ ⎪ ⎪ ⎡ 2 ⎤ Qr ⎪ ⎛ sin β ⎞ M Φ = ⎟⎟ + 2 cos2 β ⎥ 4 − 6 ⎜⎜ ⎨ ⎢ π ⎪ ⎢ ⎥ ⎝ β ⎠ ⎪× ⎢9 − 2⎥ ⎛ sin β ⎞ ⎥ sin β ⎪ ⎢ ⎟ ⎥ cos β + 1 − 2 ⎜⎜ ⎪ ⎢ β β ⎠⎟ ⎦ ⎝ ⎣ ⎩
rt
in the shell, or rt h
in the head. Values of K 2 given in Table 4-1 for different size saddles at the heads are obtained from the expression given for the maximum shear stress in Section C-C of Figure 4-4 and the Appendix. The tangential shear stress should not exceed 0.8 of the allowable tension stress.
⎫ ⎟⎟ ⎪ ⎠ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭
sin β ⎞ β
This is shown schematically in Figure 4-6. Note that β must be in radians in the formula. The maximum moment occurs when Φ = β. Substituting β for Φ and K 6 for the expression in the brackets divided by π, the maximum circumferential bending moment in in.-lb. is: M β = K 6 Qr 31
When the shell is supported on a saddle and there is no ring stiffener the shears tend to bunch up near the horn of the saddle, so that the actual maximum circumferential bending moment in the shell is considerably less than M β, as calculated above for a ring stiffener in the plane of the saddle. The exact analysis is not known; however, stresses calculated on the assumption that a wide width of shell is effective in resisting the hypothetical moment, Mβ, agree conservatively with the results of strain gauge surveys.
assumption that this moment is divided by four when A/R when A/R is less than 0.5. The change in shear distribution also reduces the direct load at the horns of the saddle; this is assumed to be Q/4 for shells without added stiffeners. However, since this load exists, the effective width of the shell which resists this direct load is limited to that portion which is stiffened by the contact of the saddle. It is assumed that 5t 5 t each side of the saddle acts with the portion directly over the saddle. See Appendix B. Internal pressure stresses do not add directly to the local bending stresses, because the shell rounds up under pressure. Therefore the maximum circumferential combined stress in the shell is compressive, occurs at the horn of the saddle, and is due to local bending and direct stress. This maximum combined stress in lb. per sq. in. is given by: S 3
=−
Q
4t (b +10t )
−
3K 3 Q
−
12 K 3 QR
2t 2
if L
≥
8R
or S 3 =
−
Q
4t (b + 10t )
L t 2
if L * < 8R
* Note: For multiple supports: L = Twice the length of portion of shell carried by saddle. If L ≥ 8R use 1st formula.
Figure 4-7. Plot of circumferential bendingmoment constant, K3.
It was found that this effective width of shell should be equal to 4 times the shell radius or equal to one-half the length of the vessel, whichever is smaller. It should be kept in mind that use of this seemingly excessive width of shell is an artifice whereby the hypothetical moment Mβ, is made to render calculated stresses in reasonable accord with actual stresses. When the saddles are near the heads, the shears carry to the head and are then transferred back to the saddle. Again the shears tend to concentrate near the horn of the saddle. Because of the relatively short stiff members this transfer reduces the circumferential bending moment still more. To introduce the effect of the head the maximum moment is taken as:
It seems reasonable to allow this combined stress to be equal to 1.50 times the tension allowable provided the compressive strength of the material equals the tensile strength. In the first place when the region at the horn of the saddle yields, it acts as a hinge, and the upper portion of the shell continues to resist the loads as a two-hinged arch. There would be little distortion until a second point near the equator started to yield. Secondly, if rings are added to reduce this local stress, a local longitudinal bending stress occurs at the edge of the ring under pressure. 5 This local stress would be 1.8 times the design ring stress if the rings were infinitely rigid. Weld seams in the shell should not be located near the horn of the saddle where the maximum moment occurs. EXTERNAL LOADS
M β = K 3 Qr
Long vessels with very small f/r values are susceptible to distortion from unsymmetrical external loads such as wind. It is assumed that vacuum relief valves will be provided where required, so it is not necessary to design
where K 3 equals K s when A/R when A/R is is greater than 1. Values of K 3 are plotted in Figure 4-7 using the 32
against a full vacuum. However, experience indicates that vessels designed to withstand 1 lb. per sq. in. external pressure can successfully resist external loads encountered in normal service. Assume the external pressure is 1 lb. per sq. in. in the formulas used to determine the sloping portion of the external pressure chart in the current A.S.M.E. Unfired Pressure Vessel Code. Then when the vessel is unstiffened between the heads, the maximum length in feet between stiffeners (the heads) is given approximately by: L + 2 / 3 H =
E rt ⎛ t ⎞
combined stress be allowed to be 25% greater than the allowable tension stress because of the nature of the stress and because of the method of analysis.
2
⎜ ⎟ ⎝ r ⎠ When ring stiffeners are added to the vessel at the supports, the maximum length in feet between stiffeners is given by: L − 2 A =
52.2
E rt ⎛ t ⎞
52.2
Figure 4-8. Loads and reactions on saddles.
RING COMPRESSION IN SHELL OVER SADDLE
2
⎜ ⎟ ⎝ r ⎠
Figure 4-8 indicates the saddle reactions, assuming the surfaces of the shell and saddle are in frictionless contact without attachment. The sum of the assumed tangential shears on both edges of the saddle at any point A is also shown in Figure 4-8. These forces acting on the shell band directly over the saddle cause ring compression in the shell band. Since the saddle reactions are radial, they pass through the center O. Taking moments about point O indicates that the ring compression at any point A is given by the summation of the tangential shears between and Φ. This ring compression is maximum at the bottom, where = π. Again, a width of shell equal to 5t each side of the saddle plus the width of the saddle is assumed to resist this force. See Appendix B. Then the stress in lb. per sq. in. due to ring compression is given by:
ADDITIONAL STRESS IN HEAD USED AS A STIFFENER
When the head stiffness is utilized by placing the saddle close to the heads, the tangential shear stresses cause an additional stress in the head which is additive to the pressure stress. Referring to Section C-C of Figure 4-4, it can be seen that the tangential shearing stresses have horizontal components which would cause varying horizontal tension stresses across the entire height of the head if the head were a flat disk. The real action in a dished head would be a combination of ring action and direct stress; however, for simplicity the action on a flat disk is considered reasonable for design purposes. Assume that the summation of the horizontal components of the tangential shears is resisted by the vertical cross section of the flat head at the center line, and assume that the maximum stress is 1.5 times the average stress. Then the maximum additional stress in the head in lb. per sq. in. is given by: S 4
=
S 5 =
or S 5 =
K 5 Q t (b +10t )
The ring compression stress should not exceed one-half of the compression yield point of the material.
⎛ ⎞ sin 2 α ⎜ ⎟ ⎜ 8r t h ⎝ π − α + sin α cos α ⎠⎟ 3Q
S 4 =
⎛ ⎞ 1 + cosα ⎜⎜ ⎟ t (b + 10t ) ⎝ π − α + sin α cos α ⎠⎟ Q
WEAR PLATES
K 4 Q r t h
The stress may be reduced by attaching a wear plate somewhat larger than the surface of the saddle to the shell directly over the saddle. The thickness t used in the formulas for the assumed cylindrical shell thickness may be
This stress should be combined with the stress in the head due to internal pressure. However, it is recommended that this 33
taken as (t (t1 + t2) for S5 (where t1 = shell thickness and t 2 = wear plate thickness), provided the width of the added plate equals at least (b (b + 10 t1) (see Appendix B). The thickness t may be taken as (t ( t1 + t2) in the formula for S2 provided the plate extends r/10 r/10 inches above the horn of the saddle near the head, and provided the plate extends between the saddle and an adjacent stiffener ring. (Also check for S 2 stress in the shell at the equator.) The thickness t may be taken as (t (t1 + t2) in the first term of the formula for S 3 , provided the plate extends r/10 inches r/10 inches above the horn of the saddle near the head. However, (t 1 2 + t 2 2 ) should be substituted for t 2 in the second term. The combined circumferential stress ( S 3 ) at the top edge of the wear plate should also be checked using the shell plate thickness t 1 and the width of the wear plate. When checking at this point, the value of K 3 should be reduced by extrapolation in Figure 4-7 assuming θ equal to the central angle of the wear plate but not more than the saddle angle plus 12°.
found by statics. Then the direct load at the horn of the saddle is given in pounds by n P β =
Q
π
⎡ β sin β ⎤ cos β ( M β − M 1 ) − cos β ⎥ + ⎢ ⎣ 2 (1− cos β ) ⎦ r (1− cos β )
or P β = K 7
Q n
If n stiffeners are added adjacent to the saddle as shown in Figure 4-4 (b), the rings will act together and each will be loaded with shears distributed as in Section B-B on one side but will be supported on the saddle side by a shear distribution similar to that shown in Section A-A. A-A. Conservatively, the support may be assumed to be tangential and concentrated at the horn of the saddle.
DESIGN OF RING STIFFENERS
When the saddles must be located away from the heads and when the shell alone cannot resist the circumferential bending, ring stiffeners should be added at or near the supports. Because the size of rings involved does not warrant further refinement, the formulas developed in this paper assume that the added rings are continuous with a uniform cross-section. The ring stiffener must be attached to the shell, and the portion of the shell reinforced by the stiffener plus a width of shell equal to 5t 5t each side may be assumed to act with each stiffener. The ring radius is assumed equal to r. When n stiffeners are added directly over the saddle as shown in Figure 4-4 ( c ) , the tangential shear distribution is known. The equation for the resulting bending moment at any point was developed previously, and the resulting moment diagram is shown in Figure 4-6. The maximum moment occurs at the horn of the saddle and is given in in.-lb. for each stiffener by: M β = K 6
Figure 4-9. Circumferential bending-moment bending-moment diagram stiffeners adjacent to saddle.
This is shown schematically in Figure 4-9; the resulting bending moment diagram is also indicated. This bending moment in in.-lb at any point A A above the horn of the saddle is given by: M Φ
=
Q r ⎧ π − β
⎨
2π n ⎩ sin β
⎫ − Φ sin Φ − cos Φ [3 / 2 + (π − β ) cot β ]⎬ ⎭
For the range of saddle angles considered, M ϕ is maximum near the equator where Φ = ρ. This moment and the direct stress may be found using a procedure similar to that used for the stiffener in the plane of the saddle. Substituting ρ for Φ and K 6 for the expression in the brackets divided by 2 π, the maximum moment in each ring adjacent to the saddle is given in in.-lb. by:
Q r n
Knowing the maximum moment M moment M β and the moment at the top of the vessel, M t, the direct load at the point of maximum moment may be
M ρ = K 6
34
Qr n
Knowing the moments M ρ and M t, the direct load at ρ may be found by statics and is given by: P ρ =
circumferential bending in the shell at the horn of the saddle has been neglected. The upper and lower flanges of a steel saddle should be designed to resist bending over the web(s), and the web(s) should be stiffened according to the AISC. Specifications against buckling. The contact area between the shell and concrete saddle or between the metal saddle and the concrete foundation should be adequate to support the bearing loads. Where extreme movements are anticipated or where the saddles are welded to the shell, bearings or rockers should be provided at one saddle. Under normal conditions a sheet of elastic waterproof material at least ¼ in. thick between the shell and a concrete saddle will suffice.
⎡ ρ sin ρ ⎤ cos ρ ( M + M t ) − cos ρ ⎥ − ⎢ ( π n ⎣ 2(1 − cos ρ ) 1 r − cos ρ ) ρ ⎦ Q
Then the maximum combined stress due to liquid load in each ring used to stiffen the shell at or near the saddle is given in lb. per sq. in. by: S 6 = −
K 7Q na
±
K 6QR nl / c
where a = the area and l/c l/c = the section modulus of the cross section of the composite ring stiffener. When a ring is attached to the inside surface of the shell directly over the saddle or to the outside surface of the shell adjacent to the saddle, the maximum combined stress is compression at the shell. However, if the ring is attached to the opposite surface, the maximum combined stress may be either compression in the outer flange due to liquid or tension at the shell due to liquid and internal pressure. The maximum combined compression stress due to liquid should not exceed one-half of the compression yield point of the material. The maximum combined tension stress due to liquid and pressure should not exceed the allowable tension stress of the material.
Nomenclature
Q = load on one saddle, lb. Total load = 2Q. L = tangent length of the vessel, ft. A = distance from center line of saddle to tangent line, ft. H = depth of head, ft. R = radius of cylindrical shell, ft. r = radius of cylindrical shell, in. t = thickness of cylindrical shell, in. th = thickness of head, in. b = width of saddle, in. F = force across bottom of saddle, s addle, lb. S1 , S2 , etc. = calculated stresses, lb. per sq. in. K 1 , K 2 , etc. = dimensionless constants for various support conditions M Φ , etc. = circumferential bending moment Φ , M β β due to tangential shears, in.-lb. θ = angle of contact of saddle with shell, degrees. β = (180 - θ/2) = central angle from vertical to horn of saddle, in degrees (except as noted) Δ = π/180 (θ/2 + β/6) = π/180 (50/12 + 30) ⋅ 2Δ = arc, in radians, of unstiffened shell in plane of saddle effective against bending. α = π- π/180 (θ/2 + β/20) = the central angle, in radians, from the vertical to the assumed point of maximum shear in unstiffened shell at saddle. Φ = any central angle measured from the vertical, in radians.
DESIGN OF SADDLES
Each saddle should be rigid enough to prevent the separation of the horns of the saddle; therefore, the saddle should be designed for a full water load. The horn of the saddle should be taken at the intersection of the outer edge of the web with the top flange of a steel saddle. The minimum section at the low point of either a steel or concrete saddle must resist a total force, F, in pounds, equal to the summation of the horizontal components of the reactions on one-half of the saddle. Then: ⎡ 1+ cos β −1 / 2 sin 2 β ⎤ ⎥ = K 8Q ⎣⎢ π − β + sin β cos β ⎦⎥
F = Q ⎢
the effective section resisting this load should be limited to the metal cross section within a distance equal to r /3 /3 below the shell. This cross section should be limited to the reinforcing steel within the distance r /3 /3 in concrete saddles. The average stress should not exceed two-thirds of the tension allowable of the material. A low allowable stress is recommended because the effect of the 35
ρ = central angle from the upper vertical to
any central angle measured from the bottom, the moment of inertia is:
the point of maximum moment in ring located adjacent to saddle, in radians. per Ε = modulus of elasticity of material, lb. per sq. in. l/c = section modulus, in.3 n = number of stiffeners at each saddle. a = cross-sectional area of each composite stiffener, sq. in. Pρ, Pβ = the direct load in lb. at the point of maximum moment in a stiffening ring.
⎛ ⎜ ⎝
Λ⎜ 2r 3 t ∫ο cos 2 δ − 2 cos δ
2. 3.
4.
5.
⎡
r 3t ⎢sin Δ cos Δ + Δ − 2
sin 2 Δ ⎞⎟
d δ = Δ2 ⎠⎟ Δ
= ο
sin 2 Δ ⎤
⎥ Δ ⎥⎦
⎣⎢
The section modulus for the tension side of the equivalent beam is: 2 ⎡ sin Δ ⎤ ⎢ Δ + sin Δ cos Δ − 2 ⎥ Δ ⎥ r 2t ⎢ sin Δ ⎢ ⎥ − cos δ ⎢ ⎥ Δ ⎣ ⎦ Then the stress in the shell at the saddle in lb. per sq. in. is given by:
Schorer, Herman, "Design of Large Pipe Lines," A. Lines," A.S. S.C. C.E. E. Tran Tr ans. s.,, 98, 101 (1933) and discussions of this paper by Boardman, H.C., and others. Wilson, Wilbur M., and Olson, Emery D., "Test of Cylindrical Shells," Univ. 111. Bull. No. 331. Hartenberg, R.S., "The Strength and Stiffness of Thin Cylindrical Shells on Saddle Supports," Doctorate Thesis, University of Wisconsin, 1941. Zick, L.P., and Carlson, C.E., “Strain Gauge Technique Employed in Studying Propane Tank Stresses Under Service Conditions,” Steel, Steel, 86-88 (Apr. 12, 1948). U.S. Bureau of Reclamation, Penstock Analysis and Stiffener Design, Design, Boulder Canyon Project Final Reports, Part V. Technical Investigations, Bulletin 5.
⎡ ⎛ A R 2 H 2 ⎢ ⎜ 1− + − 3Q L ⎢ 4 A ⎜ L 2 A L S 1 = ⎜1 − 4 H π r 2 t ⎢ L ⎜ 1+ ⎢ ⎜ 3 L ⎢⎣ ⎝
⎞ ⎛ sin Δ ⎞ ⎤ ⎟ π ⎜ − cos Δ ⎟ ⎥ ⎟ ⎝ Δ ⎠ ⎥ ⎟× 2 sin Δ ⎥⎥ ⎟ Δ + sin Δ cos Δ − 2 ⎟ Δ ⎥⎦ ⎠
or S 1 =
3K 1QL π r 2t
where ⎡ ⎛ sin Δ ⎞ ⎤ ⎡ ⎛ A R 2 − H 2 ⎞⎟⎤ ⎥ ⎢ π ⎜ Δ − cos Δ ⎟ ⎥ ⎢ 4 A ⎜ 1− + ⎝ ⎠ ⎥ × ⎢ ⎜1− L 2 AL ⎟⎥ K 1 = ⎢ ⎟⎥ 4 H ⎢ sin 2 Δ ⎥ ⎢ L ⎜ 1+ ⎟⎟⎥ ⎢ Δ + sin Δ cos Δ − 2 ⎥ ⎢ ⎜⎜ 3 L Δ ⎦ ⎣ ⎝ ⎠⎦ ⎣
The bending moment in ft.-lb. at the mid-span is:
APPENDIX A
⎛ R 2 − H 2 ⎞ ⎜ 1+ 2 ⎟ 2 2 2 2⎤ ⎡ 2 ( L − 2 A) 2 HA A R − H QL ⎜ 2Q A ⎟ L 4 − − + = − ⎢ ⎥ ⎜ 4 H ⎢ L⎟ 3 2 4 ⎥⎦ 4 ⎜ 1+ 4 H ⎣ 8 L + ⎟⎟ ⎜ 3 3 L ⎝ ⎠
The formulas developed by outline in the text are developed mathematically here under headings corresponding to those of the text. The pertinent assumptions and statements appearing in the text have not been repeated.
The section modulus = π r 2 t, and S 1 =
Maximum Longitudinal Stress
Referring to Figure 4-3, the bending moment in ft.-lb. at the saddle is: ⎡ A R 2 − H 2 1− + 2 2 2⎤ ⎢ ⎡ 2Q 2 HA A R − H 2 AL + − ⎢ ⎥ = QA ⎢1− L 4 H 4 H ⎢ 3 2 4 ⎢ ⎣ ⎦⎥ 1+ L + ⎢ 3 3 L ⎣
Δ
+
⎡ δ 2 sin δ sin Δ sin 2 Δ ⎤ 2r 3t ⎢ 1 sin δ cos δ + − + 2 δ ⎥ 2 Δ Δ ⎢⎣ 2 ⎥⎦
Bibliography
1.
sin Λ
3K 1 Q L π r 2t
where
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
⎛ R 2 − H 2 ⎞ ⎜ 1+ 2 ⎟ 2 A ⎜ L K 1 = ⎜ − 4 ⎟⎟ 4 H L ⎜⎜ 1+ ⎟⎟ 3 L ⎝ ⎠
Referring to Section A-A of Figure 4-4 the centroid of the effective arc = r sin Δ . If δ equals Δ
36
Tangential Shear Stress
Circumferential Stress at Horn of Saddle
Section B-B of Figure 4-4 indicates the plot of the shears adjacent to a stiffener. The summation of the vertical components of the shears on each side of the stiffener must equal the load on the saddle Q. Referring to Figure 4-4 (d (d), the sum of the shears on both sides of the stiffener at any point is Q sin Φ/πr . Then the summation of the vertical components is given by:
See under the heading Design of Ring Stiffeners.
2 ∫π o
Q sin 2
Φ
π r
r d Φ
=
2Q ⎡ Φ
⎢2 ⎣
π
−
sin Φ cos Φ ⎤ 2
Additional Stress In Head Used as Stiffener
Referring to Section tangential shears have which cause tension summation of these vertical axis is:
π
π ∫α
the maximum shear stress occurs at the equator when sin Φ = 1 and K 2 = l/π = 0.319 Section A-A of Figure 4-4 indicates the plot of the shears in an unstiffened shell. Again this summation of the vertical components of the shears on each side of the saddle must equal the load on the saddle. Then the total shear at any point is: Q sin Φ r (π − α + sin α cosα ) and the summation of the vertical components is given by: 2
⎡
Q
π r
⎤ ⎥ r d Φ 2 ⎣ π − α + sin α cosα ⎦
sin Φ 2 cos Φ 2 ⎢
S 4 =
K 4Q rt h
where K 2
=
3 8
∫α o
Q sin 2 Φ1
π − α + sin α cos α
π r d Φ1 = 2 ∫α
π r
The ring compression at any point in the shell over the saddle is given by the summation of the tangential shears over the arc = (Φ - α) shown in Section A-A or C-C of Figure 4-4 or in Figure 4-8. Then ⎞ ⎜⎜ ⎟ r d Φ 2 − π r ⎝ π −α + sin α cosα ⎠⎟ Q sin Φ1 Q sin Φ 2 d Φ 2 Φ r d Φ1 = − ∫α ∫α Φ = (π −α + sin α cosα ) π r ⎡ ⎤ Q Φ −⎢ ⎥ [cos Φ 2 ]α = π α sin α cos α − + ⎣ ⎦ Φ − ∫α
Q sin 2 Φ ⎡ α − sin α cosα
π r
⎤ ⎢ ⎥ r d Φ ⎣ π −α + sin α cosα ⎦
or
Q
2Q ⎡ Φ1 π
⎢ ⎣
2
α
−
sin1 Φ cos Φ1 ⎤
⎥ = ⎦o
2
π
Q sin Φ 2 ⎛ α − sin α cosα
⎡ − cos Φ + cosα ⎤ ⎢ π −α + sin α cosα ⎥ ⎣ ⎦
the ring compression becomes a maximum in the shell at the bottom of the saddle. Or if Φ = π this expression becomes:
π
2Q ⎡ α − sin α cos α
⎤ ⎡ Φ 2 sin 2 Φ cos Φ 2 ⎤ ⎢ π − α + sin α cosα ⎥ ⎢ 2 − ⎥ 2 ⎦α ⎣ ⎦⎣
⎡ ⎤ 1+ cos α ⎢ π α sin α cos α ⎥ ⎣ − + ⎦ ⎡ ⎤ then K 5 = ⎢ 1+ cosα ⎥ ⎣ π −α + sin α cos α ⎦
Finally:
Q
Q
π
(α − sin α cosα )= Q (α − sin α cosα ) π
The maximum shear occurs when Φ2 = α and K 2 =
sin α ⎡ α − sin α cos α π
⎛ ⎞ sin 2 α ⎜ ⎟ ⎜ ⎟ ⎝ π −α + sin α cos α ⎠
Wear Plates
sin α
Section C-C of of Figure 4-4 indicates the shear transfer across the saddle to the head and back to the head side of the saddle. Here the summation of the vertical components of the shears on arc α acting downward must equal the summation of the vertical component of the shears on the lower arc (π - α) acting upward. Then 2
α − sin α cosα
Then assuming this load is resisted by 2r th and that the maximum stress is 1.5 times the average:
the maximum shear occurs where Φ = α and K 2 =
sin Φ1 cos Φ1 r d Φ 2 −
π ⎫ ⎧⎡ 2 ⎤α ⎡ ⎪ sin Φ1 − α − sin α cosα ⎤ ⎡ sin 2 Φ ⎤ ⎪ = ⎥ ⎥ ⎬ ⎨⎢ ⎢ ⎥⎢ π r ⎪⎢⎣ 2 ⎥⎦ o ⎣ π − α + sin α cosα ⎦ ⎢⎣ π r ⎥⎦α ⎪ ⎩ ⎭ 2 ⎛ ⎞ Q⎜ sin α ⎟ ⎜ π − α + sin α cos α ⎟ 2 ⎝ ⎠
π
Φ
π r
Q
⎡ Φ − sin Φ cos Φ ⎤ r d Φ = Q ⎢ 2∫ ⎥ =Q r (π − α + sin α cosα ) ⎣ π − α + sin α cosα ⎦α Q sin
π α
Q
∫α o
⎥ =Q ⎦o
C-C of Figure 4-4, the horizontal components across the head. The components on the
⎤ ⎢ π α sin α cos α ⎥ − + ⎣ ⎦
37
Design of Ring Stiffeners; Stiffener in Plane of Saddle
−
Referring to Figure 4-6, the arch above the horns of the saddle resists the tangential shear load. Assuming this arch fixed at the top of the saddles, the bending moment may be found using column analogy. If the arch is cut at the top, the static moment at any point A point A is: is: π r
β
∫ o (sin Φ1 – sin Φ1 cos Φ1 cos Φ - sin2Φ1 sin Φ)
Then the indeterminate moment is:
dΦ1
= =
Q
⎡
π r ⎢⎣
cos Φ1
⎡ 1 π r ⎢⎣ Q
cos Φ 2
Φ
cos
2
sin 2 Φ1 +
1 2
M 1 =
Φ
sin Φ sin Φ1 cos Φ1
Φ1 sin Φ ⎤ ⎥ 2 ⎦o
r 3 El
⎢ 4r ⎢⎣
⎤ ⎦
sin ⎥
d =
β o
El
2
sin β ⎞ r 3 β
⎟⎟ ⎠
El
d =
2r 3 El β
⎡1 1 2 sin Φ sin β Φ sin 2 β ⎤ + ⎢ sin Φ cos Φ + Φ − ⎥ = 2 β β 2 ⎦⎥ ⎣⎢ 2 o ⎡ 2 sin 2 β ⎤ ⎢sin β cos β + β − ⎥ El ⎣⎢ β 2 ⎦⎥ r 3
The load on the analogous column is: q=2
∫ β o
M s El
r d Φ =
q
=
2Q r 2
2Q r 2 π El
Q r 2
π El
Φ ⎛ ⎞ ∫ β o ⎜1 − cos Φ − sin Φ ⎟ d Φ 2 ⎝ ⎠
π El
β
sin Φ Φ cos Φ ⎤ ⎡ ⎢Φ − sin Φ − 2 + 2 ⎥ = ⎣ ⎦o
[2 β − 3sin β + β cos β ]
The moment about the horizontal axis is: M h = − 2
∫
M s El
M h y I h
=
Q r ⎧ 2 β − 3 sin β + β cos β ⎫
π
⎨ ⎩
2 β
β sin β cos β + β 2 − 2 sin 2 β
⎛ sin β ⎞ ⎟ r Y = ⎜⎜ cos Φ − β ⎠⎟ ⎝
2 β r
⎛ ⎜⎜ cos ⎝
−
⎬− ⎭ ⎤ ⎥ ⎥⎦
The distance from the neutral axis to point A is given by:
The centroid is sin/r, and the moment of inertia about the horizontal axis is: lh = 2
q
α 1
Y ⎡ 9 β sin β cos β + 3 β 2 −12 sin 2 β + 2 β 2 sin 2 β
then the M s /El /El diagram is the load on the analogous column. The area of this analogous column is: a1 = 2 ∫ β o
π El
⎡2 cos Φ − 2 cos2 Φ − Φ sin Φ cos Φ −⎤ ⎢ ⎥ ∫ ⎢ sin β ⎥ d Φ = ( ) 2 2 cos sin − Φ − Φ Φ ⎢ β ⎥ ⎣ ⎦ β o
sin Φ cos Φ Φ ⎡ ⎤ 2 sin Φ − cos Φ sin Φ − Φ + − ⎥ 3 ⎢ 4 4 Q r ⎢ ⎥ − ⎥ π El ⎢ Φ sin 2 Φ sin β (2 Φ − 2 sin Φ − sin Φ + Φ cos Φ )⎥ − ⎢ β ⎣ 2 ⎦o Q r 2 ⎡ 9 3 3 sin 2 β β sin 2 β ⎤ = + ⎢ sin β cos β + β − ⎥ π El ⎣⎢ 4 4 β 2 ⎦⎥
Q Φ
M s =
Q r 3
⎛ sin β ⎞ 2 ⎜⎜ cos Φ − ⎟ r d Φ = β ⎠⎟ ⎝
38
Figure 4-10. Plot of longitudinal bending-moment bending-moment constant K1
Finally, the combined moment is given by:
Finally:
⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 sin β cos β ⎪ Φ ⎪cos Φ + sin Φ − + −⎪ 2 2 β 2 ⎪ ⎪ ⎪ ⎪ sin β ⎞ Qr ⎪ 1 ⎛ ⎪ ⎟⎟ × M Φ = − M s + M l = ⎨ 4 ⎜⎜ cos Φ − ⎬ π ⎪ β ⎠ ⎝ ⎪ ⎪⎡ ⎪ 2 ⎤ ⎛ sin β ⎞ ⎪⎢ ⎪ 2 ⎥ ⎟⎟ + 2 cos β ⎪ 4 − 6 ⎜⎜ ⎪⎢ ⎥ ⎪ ⎝ β ⎠ ⎪ ⎢9 − 2⎥ ⎪ ⎪ ⎢ sin β ⎛ sin β ⎞ ⎥ ⎪ ⎪⎢ ⎟⎟ ⎥ cos β +1− 2 ⎜⎜ ⎪⎩ ⎣ β ⎝ β ⎠ ⎦ ⎪⎭
M β = K 6Qr
Because of the symmetry the shear stress is zero at the top of the vessel; therefore, the direct load in the ring at the top of the vessel, Pt, may be found by taking moments on the arc β about the horn of the saddle. Then (1− cos β )r Pt = Q r ⎡⎢1 − cos β − β π
Pt =
Q
π
⎣
2
⎤ ⎦
(
)
sin β ⎥ − M β − M t
⎡ β sin β ⎤ 1 ⎢1 − 2 (1− cos β ) ⎥ − r (1− cos β ) ( M β − M t ) ⎣ ⎦
The direct load, P β , at Φ = β, the point of maximum moment may be found by taking moments about the center. Then
This is the maximum when Φ = β; then:
(
)= Qr (1− cos β )− ( M β − M t )
r P β + Pt
⎧ sin β ⎞ ⎫ ⎛ ⎜ cos β − ⎟⎪ ⎪ β sin β 3 3 sin β β ⎟ ⎪ ⎜ ⎪ − cos β + +⎜ ⎟⎪ ⎪ 2 4 4 β 4 ⎜ ⎟⎪ ⎪ ⎝ ⎠ ⎪ ⎪⎪ Qr ⎪ M β = 2 ⎨⎡ ⎬ ⎤ ⎛ sin β ⎞ π ⎪ ⎢ ⎪ ⎟⎟ + 2 cos 2 β ⎥ 4 − 6 ⎜⎜ ⎪⎢ ⎪ ⎥ β ⎠ ⎝ ⎪⎢ ⎪ ⎥ 2 ⎪ ⎢ sin β cos β ⎪ ⎛ sin β ⎞ ⎥ ⎟⎟ ⎥ + 1 − 2 ⎜⎜ ⎪⎢ ⎪ β ⎪⎩ ⎣ ⎪⎭ ⎝ β ⎠ ⎦
π
Substituting the value above for Pt, and solving for P β , gives: P β =
39
Q ⎡ β sin β
− ⎢ π ⎣ 2 (1− cos β )
⎤
cos β
⎦
r (1 − cos β )
cos β ⎥ +
( M β − M t )
or
Pβ = K 7 Q
APPENDIX B
After the article had been published, certain refinements seemed desirable; therefore, the following has been added to take greater advantage of the inherent stiffness of these vessels. The methods outlined in the paper will give conservative results. The effective width of shell has been limited to lot in order to prepare the chart of Figure 4-2. It has been shown 5 that this effective width may be taken as 1.56 rt . That is, where 5t each side of the saddle or stiffener has been used, the more liberal value of 0.78 rt each side could be used. The values plotted in Figure 4-5 for K 1 cover conservatively all types of heads between H = H = O and H = H = R. More liberal values are given in Figure 4-10 for hemispherical and 2 to 1 ellipsoidal heads for values of H/L between 0 and 0.1. The minimum values of K 1 given in Table 4-1 have not been listed for specific values of R/L R/L and H/L; H/L; so they are conservative. Specific minimum values of K 1 may be read from Figure 4-10.
where K 7 =
⎡ β sin β ⎤ cos β ( M β − M t ) − cos β ⎥ + ⎢ π ⎣ 2 (1− cos β ) ⎦ Q r (1− cos β ) 1
If the rings are adjacent to the saddle, K 6 and K 7 7 may be found in a similar manner, except that the static structure would become the entire ring split at the top and loaded as indicated in Figure 4-9. Design of Saddles
The summation of the horizontal components of the radial reactions on one-half of the saddle shown in Figure 4-8 must be resisted by the saddle at Φ = π. Then this horizontal force is given by: F =
π ∫ β
Q ( − cos Φ sin Φ + cos β sin Φ r (π − β + sin β cos β )
r d Φ =
⎡ − 1 sin 2 Φ − cos Φ cos β ⎤ 2 ⎥ Q⎢ ⎢ π − β + sin β cos β ⎥ ⎣ ⎦ ⎡ 1+ cos β − 1 sin 2 β ⎤ 2 ⎥ Q⎢ ⎢ π − β + sin β cos β ⎥ ⎣ ⎦
Then :
1+ cos β − 1 sin 2 β 2 K 8 = π − β + sin β cos β
The bending at the horn would change the saddle reaction distribution, and increase this horizontal force.
40
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Part V Anchor Bolt Chairs
W
hen anchor bolts are required at supports for a shell, chairs are necessary to distribute the load to the shell. Small tubular columns (less than 4 ft in diameter) may be an exception if the base plate is adequate to resist bending. Otherwise, chairs are always needed to minimize secondary bending in the shell. For flat-bottom tanks, choose a bolt circle to just barely clear the bottom without notching it. For other structures, follow the minimum clearances shown in Figure 5-1(a). The designer must evaluate anchor bolt location for interference with base or bottom plate.
W = total load on weld, kips per lin. in. of weld W H = horizontal load, kips per lin. in. of weld H = W V = vertical load, kips per lin. in. of weld V = θ = cone angle, degrees, measured from axis of cone Z = reduction factor Top Plate
Critical stress in the top plate occurs between the hole and the free edge of the plate. For convenience we can consider this portion of the top plate as a beam with partially fixed ends, with a portion of the total anchor bolt load distributed along part of the span. See Figure 5-2. P (0.375 g − 0.22d ) (5-1) S = 2
Notation
a = top-plate width, in., along shell b = top-plate length, in., in radial direction c = top-plate thickness, in. d = anchor-bolt diameter, in. e = anchor-bolt eccentricity, in. emin = 0.8864 + 0.572, based on heavy hex nut clearing shell by ½ in. See Table 5-1. f = distance, in., from outside of top plate to edge of hole f min min = d/ 2 + 1/8 vert ical plates g = distance, in., between vertical (preferred g (preferred g = = d + 1) [Additional distance may be required for maintenance.] h = chair height, in. j = vertical-plate thickness, in. k = vertical-plate width, in. (average width for tapered plate) L = column length, in. m = bottom or base plate thickness, in. P = design load, kips; or maximum allowable anchor-bolt load or 1.5 times actual bolt load, whichever is less r = least radius of gyration, in. R = nominal shell radius, radius, in., either to inside or centerline of plate (radius normal to cone at bottom end for conical shells) S = stress at point, ksi t = shell or column thickness, in. w = weld size (leg dimension), in.
fc
or ⎡P c=⎢ (0.375 g − ⎣ Sf
⎤ 0.22d )⎥ ⎦
1
2
(5-2)
Top plate may project radially beyond vertical plates as in Figure 5-1(d), but no more than ½”. Chair Height
Chair must be high enough to distribute anchor bolt load to shell or column without overstressing it. If the anchor bolt were in line with the shell the problem would be simple – the difficulty lies in the bending caused by eccentricity of the anchor bolt with respect to the shell. Except for the case where a continuous ring is used at the top of chairs, maximum stress occurs in the vertical direction and is a combination of bending plus direct stress. Formulas which follow are approximations, based on the work of o f Bjilaard. S =
42
Pe t 2
⎡ ⎢ 1.32 Z ⎢ + ⎢ 1.43 ah2 2 ..333 ( 4 ) + ah ⎢ ⎣ Rt
⎤ ⎥ ⎥ Rt ⎥ ⎥ ⎦
.031
(5-3)
Minimum chair height h = 6”, except use h = 12” when base plate or bottom plate is 3/8” or thinner and where earthquake or winds over 100 mph must be considered. Maximum recommended chair height h = 3a. Table 5-1. Top-Plate Dimensions
Based on anchor-bolt stresses up to 12 ksi for 1 ½-in.-diameter bolts and 15 ksi for bolts 1 ¾ in. in diameter or larger; higher anchor bolt stresses may be used subject to designer’s decision.
g=d+1
a
e min
c min
Bolt Load, kips P
2½
4½
1.87
0.734
19.4
Top Plate Dimensions, in.
d
f
1½
Figure 5-2. Assumed Top-Plate Beam. Z =
Rt
2¾ 3
4¾ 5
2.09 2.30
0.919 1.025
32.7 43.1
2¼
1¼
3¼
5¼
2.52
1.145
56.6
Vertical Side Plates
1.0 .177 am ⎛ m ⎞
1 1
If chair height calculated is excessive, reduce eccentricity e, if possible, or use more anchor bolts of a smaller diameter. Another solution is to use a continuous ring at top of chairs. If continuous ring is used, check for maximum stress in circumferential direction, considering the ring as though it were loaded with equally spaced concentrated loads equal to Pe/h. Pe/h. Portion of shell within 16t 16t either side of the attachment may be counted as part of the ring. (Refer to Figure 5-3) Note that the base plate or bottom is also subjected to this same horizontal force, except inward instead of outward. This is true even if a continuous ring is not used around the top of the chairs – but it should never cause any very high stresses in the base, so we do not normally check it. However, it is a good thing to keep in mind in case you have a very light base ring.
Figure 5-1. Anchor-Bolt Chairs.
Where:
1¾ 2
2
⎜ ⎟ + 1.0 ⎝ t ⎠
(5-4)
Be sure top plate does not overhang side plate (as in Figure 5-1d) by more than ½” radially. Vertical-plate thickness should be at least j least jmin = ½” or 0.04 (h (h – c ), whichever is greater. Another requirement is jk is jk ≥ P/ 25, 25, where k is the average width if plate is tapered. These limits assure a maximum L/r of 86.6 and a maximum average stress in the side
Maximum recommended stress is 25 ksi. This is a local stress occurring just above the top of the chair. Since it diminishes rapidly away from the chair, a higher than normal stress is justified but an increase for temporary loads, such as earthquake or wind is not recommended. The following general guidelines are recommended. 43
plates of 12.5 ksi, even assuming no load was transmitted into the shell through the welds.
Figure 5-5. Typical Welding, Base or Bottom Field Attached. Figure 5-3. Chair with Continuous Ring at Top. Assembly of Chair
For field erected structures, ship either the top plate or the entire chair loose for installation after the structure is sitting over the anchor bolts. Where base plate is welded to skirt or column in shop, attach side plates in the shop and ship top plate loose for field assembly. (See Figure 5-4) Where base or bottom plate is not welded to shell in the shop, as for flat-bottom tanks and single pedestal tanks, shop attach side plates to top plates and then ship the assembly for field installation. When you do this, weld both sides at top of side plates so shrinkage will not pull side plate out of square. s quare. (See Figure 5-5) Welds between chair and shell must be strong enough to transmit load to shell. ¼” minimum fillet welds as shown in Figures 5-4 and 5-5 are nearly always adequate, but you should check them if you have a large anchor bolt with a low chair height. Seal welding may be desired for application in corrosive environments. Assume a stress distribution as shown in Figure 5-6 as though there were a hinge at bottom of chair. For the purpose of figuring weld size, the base or bottom plate is assumed to take horizontal thrust only, not moment. Note that loads are in terms of kips per inch of weld length, not in terms of kips per square inch stress. Critical stress occurs across the top of the chair. The total load per inch on the weld is the resultant of the vertical and horizontal loads.
Figure 5-6. Loads on Welds .
Formulas may also be used for cones, although this underrates the vertical welds some. W V = W V =
P a + 2h Pe
ah + 0.667h 2 2
W = W
v
+W 2 H
(5-5) (5-6) (5-7)
For an allowable stress of 13.6 ksi on a fillet weld, the allowable load per lin. in. is 13.6 x 0.707 = 9.6 kips per in. of weld size. For weld size w, in., the allowable load therefore is: 9.5w > W
(5-8)
Design References
H. Bednar, “Pressure Vessel Design Handbook,” 1981, pp. 72-93. MS. Troitsky, “Tubular Steel Structures,” 1982, pp. 5-10 to 5-16. P.P. Bjilaard, “Stresses From Local Loadings in Cylindrical Pressure Vessels,” ASME Transactions, Vol. 77, No. 6, 1955. P. Buthod, “Pressure Vessel Handbook,” 7 th Edition, pp. 75-82.
Figure 5-4. Typical Welding, Base Plate Shop Attached .
44
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Part VI Design of Fillet Welds
D
should check to see what code, if any, applies to the work. In these papers, however, the fillet weld throat dimension for an equal-leg fillet is assumed to be the leg length times 0.707 (i.e. cos 45°).
esign of butt welds is closely controlled by weld details and joint efficiencies clearly specified in various codes and specifications. Design of fillet welds, however, is not so clearly outlined. The following pages are intended to fill the gap. While referring to the following pages and designing fillet welds, the designer is encouraged to keep in mind actual shop and field welding practice and the quality of fillet welds that can consistently be expected. The size and length of the weld as well as the allowable stresses used in their design should reflect the actual shop and field welding and not necessarily the value used here. Size of an equal-leg fillet weld is the leg width W of of the largest 45° right triangle which fits in its cross section. They are referred to by their leg sizes, such as a ¼ in. fillet weld.
Economy of Welding
Economical design of fillet welds includes the following: 1. Use of 450 (equal leg) fillet welds whenever possible 2. Minimum size of fillet 3. Lower cost of down welding position 4. Locate weld to eliminate eccentricity 5. Balanced welds to control distortion 6. Avoid locating welds in highly stressed areas 7. Readily accessible Use the smallest size of fillet permitted (see Fillet Weld Limitations). Flat fillets 5/16” and smaller are normally made in one pass and are more economical than larger fillets. Generally, the fillet with the least cross-sectional area is the most economical. Increasing the size of a fillet weld from 1/4” to 3/8” more than doubles the amount of filler metal, but the strength only increases 50%. A gap also requires additional filler metal.
Figure 6-1. Fillet-Weld Sizes (Leg Dimensions).
Size of an unequal length fillet weld is described by the leg lengths of the largest right triangle which fits in its cross section, such as a 3/8” by 1/2” fillet weld. The strength of a fillet weld is assumed to equal the allowable shearing stress times the throat area of the weld. The throat area of a weld is the length of weld times the theoretical throat distance, which is the shortest distance from the root of the weld to the theoretical weld’s surface. Some codes, however, define the throat distance differently. AWWA defines the throat as .707 times the length of the shorter leg of the fillet weld. AISC distinguishes between welding processes to be used when determining throat distances (eg. AISC ASD J2.2). The designer
Figure 6-2. Volumes of 1-in. Long Welds .
45
Parallel welds have forces applied parallel
Flat welding position is the most economical and overhead the least. For example, the relative costs of 3/8” fillets for different positions are: lap flat flat fillet vertical fillet overhead fillet
to their axis. Fillet weld throat is stressed only in shear. Parallel welds may also be called longitudinal welds.
100% 110% 240% 250%
Figure 6-6. Parallel Weld.
The costs can vary according to weld procedure used. Specify shop welding whenever practical. The fitted-up material can normally be repositioned easier in the shop.
Transverse welds have forces applied at right angles to their axis. Fillet weld throat has both shear and normal (tensile or compressive) stresses. Transverse welds are about 33% stronger than parallel welds.
Type of Fillet Welded Joints
Single-fillet welded joints Strength depends on size of fillet. Do not use when tension due to bending is concentrated at root of weld. Do not use for fatigue or impact loading. Difficult to control distortion.
Figure 6-7. Transverse Weld Allowable Loads on Fillet Welds
Stress in a fillet weld is assumed as shear on the throat area, for any direction of applied load. Many codes express the allowable shear stress for fillet welds in psi on the throat area. It is more convenient, however, to express the strength of fillet welds as allowable load f load f , kips per lin. in. for 1” fillet. The following formula may be used to convert allowable shear stress on throat area to allowable load for 1” fillet with equal leg lengths.
Figure 6-3. Types of Single Fillet Welds.
Double-fillet welded joints Used for static loads. Economical when fillet size is ½” or less. Lap joint maximum strength in tension when length of lap equals at least 5 times the thickness of thinner material.
F = F = 0.707 x allowable shear stress, ksi
Figure 6-4. Types of Double Fillet Welds .
(6-1)
Since transverse welds are stronger than parallel (or longitudinal) welds, some codes permit different allowable stresses for them. API 620 9th Edition and AWWA D100-96 are two codes that have different allowable stresses for the two types of welds. API 650 9th Edition and AISC 9th Edition, however, make no distinction between transverse welds and parallel welds and use the same allowable stress for both. The designer is cautioned to check which code applies to the work at hand as well as the most recent edition of the code
Double-fillet welded corner joint - Complete penetration and fusion. Used for all types of loads. Economical on moderate thickness.
Figure 6-5. Corner Joint.
Welds transmit forces from one member to another. They may be named according to the direction of the applied forces. 46
= polar moment of inertia, in.4, of member transmitting load to weld J w = polar moment of inertia, in.3, of weld lines subjected to torque L = column length, in. M = bending moment, in.-kips n = number of plate sides welded or number of welds loaded P = allowable concentrated axial load, kips Q = statical moment of area, in. 3, above or below a point in cross section, about neutral axis r = least radius of gyration, in. S = section modulus, in.3, of member transmitting load to weld or of weld subjected to moment Sw = section modulus, in.2, of weld lines subjected to bending moment t = plate thickness, in., or thickness, in., of thinnest plate at weld T = torque, in.-kips V = vertical shear, kips w = fillet weld size (leg dimension), in. W = total load on fillet weld, weld, kips per lin. in. of weld W b = bending force on weld, kips per lin. In. of weld W h = horizontal component co mponent of torsional force on weld kips per lin. in of weld W q = longitudinal shear on fillet weld, kips per lin. in. of weld W s = average vertical shear on fillet weld, kips per lin. in. of weld W sa sa = actual shear on fillet weld, kips per lin. in. of weld W t = torsional load on fillet weld, kips per lin. in. of weld W v = vertical component of torsional force on weld, kips per lin. in. x = distance from y axis to vertical weld y = distance from x axis to horizontal weld J
to see if their approach to these types of stresses has changed. In the following pages, however, for the sake of completeness, a distinction will be made between the two types of stresses, f stresses, f p and f and f t. When a joint has only transverse forces applied to the weld, use the allowable transverse load f t. If only parallel forces are applied to the weld, use the allowable parallel load f p. If one of the forces is parallel and the other forces are transverse, use the allowable transverse load when the resultant force is found from Equation 6-1. Specifications on allowable stress for fillet welds are given in Section 2.4 of AWS Structural Welding Code, D1.1-94. Current AISC specifications also refer to: 1. allowable stress at at weld for both weld metal and base metal 2. minimum length of fillet weld 3. minimum size of fillet weld 4. maximum size of fillet weld 5. end returns or “boxing of welds” 6. spacing of welds 7. fatigue loading of welds Notation
A = cross-section area, sq. in., of member transmitting load to weld Aw = length, in., of weld b = length, in., of horizontal weld c = distance, in., from neutral axis axis to outer parallel surface or outer point ch = horizontal component of c, in. cv = vertical component of c, in. d = depth, in., of vertical weld f = allowable load on fillet weld, kips per lin. in. per in. of weld size f b = bending stress, ksi f p = allowable parallel load on fillet weld, kips per lin. in. per in. of weld size f t = allowable transverse load load on fillet fillet weld, kips per lin. in. per in. of weld size f toto = torsional stress, ksi I = moment of inertia, in. 4, of member transmitting load to weld or of weld subjected to torque I o = moment of inertia about o axis, in.4 I x = moment of inertia about x axis, in.4 I y = moment of inertia about y axis, in.4
It is necessary to designate the size and length of fillet welds. Since neither are known, it is usually simpler to assume the length and then calculate the size. Fillet weld size w, in., is found by dividing the force W , kips per lineal inch, on the weld by the allowable load f (kips (kips per lin. in. for 1” fillet) for the weld. w=
47
W f
(6-2)
Table 6-1. Formulas for Force on Weld
Type of Loading
Tension or Compression
Vertical Shear
Bending
Torsion
Common Design Formulas for Stress, psi P A
V A
M S
Tc J
Formula for Force on Weld K/Kips per in W = =
W s
W b
P
Figure 6-8. Forces on Weld Combined.
Aw
=
=
W t =
To determine the resultant force for combined forces, use Equation 6-3. If only two forces exist, use 0 for one force.
V
2
W = W 1
Aw
+ W 2 2 + [W 3 ( f t /
f p
)]
2
(6-3)
Refer to Figure 6-8 for explanation of W 1, W 2, and W 3. The total force shall be determined in accordance with the applicable code.
M S w
Simple tension or compression loads. loads. The force W, kips W, kips per inch of weld, is the load P divided by the length Aw of weld. As shown in Table 6-1 the tensile or compressive force on a weld is: P (6-4) W = =
Tc J w
Aw
With this force W , the required fillet weld is calculated from Equation 6-1. Longitudinal Shear
VQ tl
W q =
Example: Find size of fillet welds for the connection shown in Figure 6-9. Assume A Assume Aw = 5 + 5 + 2 ½ = 12 ½”.
VQ nl
Force W on a weld depends on the loading and shape of the weld outline. Table 6-1 shows the basic formulas for determining weld forces for various types of loads.
30,000 lbs
Combining forces: There may be more than one force on the weld, such as bending force and shear force. It is usually easier to determine each force independently and then combine vectorially to obtain a resultant force. All forces which are vectorially added must occur at the same position in the weld. Be sure to find the position on the welded connection where the combination of forces will be maximum. To simplify calculations, increase parallel forces by the ratio f t/ t f / p before combining to account for the lower allowable parallel shear stress specified by some codes.
Figure 6-9. Tension-Member Connection1.
Referring to API 650, the allowable basic shearing stress of an E60 electrode fillet weld is 13.6 ksi. f = (.707) (13.6 ksi) (1 inch weld) = 9.6 kips/inch/1 inch weld.
1
AISC for E60 electrodes would give f = (.707) (.3) (60) = 12.7 ksi shear stress with max shear stress on base metal of .4 yield of base metal.
48
W =
P Aw
w=
=
W f
30 12.5
=
= 2.4 kips per lin. in.
2.4 9.6
W sa at 2 =
nΙ
=
V (0) nΙ
=0
(6-8)
Bending or torsional load load may be applied to the same weld outline.
Use ¼” fillet
= 0.25"
VQ
2 Weld volume = (1/ 4) ×12.5 = 0.39 cu. in.
2
Try A Try Aw2 = 5 + 5 = 10” W 2 = W 2 =
P Aw2 W 2 f
=
=
30 10
3.0 9.6
= 3.0 kips per lin. in.
Use 5/16” fillet
= 0.312
Bending and Shear Load on a Weld
2 Weld volume = (5 / 16) ×10 = 0.49 cu. in.
2
Use ¼” fillet on three sides because of less weld volume. Check fillet size (see Fillet Weld Limitations). Shear load is considered uniformly distributed over the length of weld. Force formula W s = V/Aw from Table 6-1 gives average shear force. Use average shear force when combining with bending force or torsional force. However, if the average shear force about equals or exceeds the bending or torsional force, determine the actual shear force distribution to aid in locating the maximum combined force. The actual shear force per weld at any point can be determined from: VQ (6-5) W sa =
In the figure with the bending load, the weld must transfer the same stress as in the member at the connection. This stress can be determined using the common formula for bending stress. M f b = (6-9) S
In the connection with the torsional load, the weld wants to rotate or twist about the center of gravity of the weld group. The stress in the weld can be found from: Tc (6-10) f to = J
However, before using these formulas, it is necessary to determine the section modulus S or polar moment of inertia J of the weld without knowing its width (size). A simple way to determine the section modulus or polar moment of inertia of the weld is to treat the weld as a line. The property, such as section modulus S, of any thin area is equal to the property of the section when treated as a line Sw times its thickness w. S = S ww (6-11)
For example, the average shear force and actual shear force distribution are compared for a rectangular member in Figure 6-10.
The common formula for bending stress can now be used to find the bending force on the weld. M M W b = w= (6-12)
Figure 6-10. Shear Distribution at Welds.
W sa at 1=
VQ nΙ
V
=
d
2
V
W s =
V Aw
=
V
2d
S
2
=
3V 4d
=1.5W s
S w
Properties of sections treated as lines for typical weld outlines are shown in Table 6-2. The method for determining these properties is given later. When designing welds using the line method, select the weld outline with care.
(6-6)
d
⎛ td 3 ⎞ ⎟ 2⎜ ⎜ ⎟ ⎝ 12 ⎠
Weld Outline
Figure 6-11. Moment and Torque on Weld.
nΙ
Average shear force
Torsional and Shear Load on a Weld
(6-7)
49
Several combinations of line welds will produce the required property Sw or J w. However, select the weld outline where the weld distribution is consistent with the load distribution in the member at the connection. For non-circular members (such as beams, channels, etc.) resisting torsion loads, transverse forces on the weld are present in addition to parallel forces computed from Tc/Jw. Tc/Jw. These transverse forces are the result of the non-circular cross section warping and should not be neglected.
Bending and shear forces on a welded connection are combined vectorially after determining each force independently from Equations 6-11 and 6-5. Determine the combined force W on on the weld using Equation 6-2. Make sure you have found the position on the welded connection where the combination of forces will be maximum. See Figure 6-10 for shear force distribution. Calculate the required weld size from Equation 6-2.
Table 6-2. Properties of Weld Outlines (Treated as a Line) Outline of Welded Joint
Bending (about x – x axis) S w =
d 2
6
S w =
Torsion
in.2
d 2
J w
3
S w = bd
d 3
J w =
12
in.3
(
d 3b 2 + d 2
=
)
Figure 6-12. Bending and Vertical Shear on Welds.
6
J w =
(
b b2 + 3d 2
)
Example: Example: Find size of fillet weld on clip loaded as shown in Figure 6-13. Use f t = 8.9 kips per lin. in. and f p = 6.4 kips per lin. in. from API 620. Assume length of fillet = 10” (5” each side)
6
d (4b + d )
S w (top )=
6
S w (bott )=
d (4b + d ) 2
6 (2b + d )
J w =
(b + d )4 − 6b 2d 2 12 (b + d )
(max force at bott )
S w = bd +
S w (top )=
d 2
J w =
6
(2b + d )3 12
−
Sw from Table 6-2
2
b 2 (b + d )
=
2b + d
3 d (2b + d ) 2
3 (b + d )
J w =
(b + 2d )3 12
−
d 2
3
=
52 3
W b
= 8.33 sq.in =
M
=
S w
4×3 8.33
= 1.44 kips per lin. in.
2
d 2 (b + d ) b + 2d
Avg shear force
(max force at bott )
S w = bd +
3
Bending force
d (2b + d )
S w (bott )=
d 2
W s
=
V Aw
=
4 10
= .40 kips per lin. in. J w =
Figure 6-13. Loaded Clip.
(b + d )3
f t = (.707) (12.6 ksi) (1 inch weld) = 8.9 kips/inch/1 inch weld
6
f p = (.707) (9.0 ksi) (1 inch weld) = 6.4 kips/inch/1 inch weld S w =
π d 2 4
J w
π d 3 4
Revised and expanded outline properties given in Lincoln Electric publication D8 10.17. Solutions to Design of Weldments, Weldments, p. 3.
50
Resultant force W = =
W b
2
⎡ ⎛ f ⎞⎤ + ⎢W s ⎜ t ⎟⎥ ⎜ f p ⎟⎥ ⎢⎣ ⎝ ⎠⎦ ⎡
2
⎛ 8.9 ⎞⎤ ⎟⎥ ⎝ 6.4 ⎠⎦
W v =
= Fillet size w =
W F t
=
1.544 8.9
J w
(6-15)
Equation 6-2 can now be used to find the resultant force on the weld. Increase the forces parallel to the weld at the point considered by f /f t/f p before combining. The required fillet size is calculated from Equation 6-2.
2
1.442 + ⎢0.40 ⎜
⎣
Tch
1.544 kips per lin. in. = .173"
Use 3/16” fillet
Example: Example: Find fillet size for connection shown in Figure 6-15. Use f Use f t = f p = 9.6 kips per lin. in.
Note that the designer is still cautioned to check the shear capacity of the plate. Torsional and shear forces forces on a welded connection are combined vectorially after determining each force independently from Equation 6-3 and the torsional force formula. Tc (6-13) = W t = J w
Maximum torsional force occurs at the most distant weld fiber measured from the center of gravity of the weld outline. This distance to the outer fiber is c in Equation 6-12. The direction of the torsional force W t may be other than horizontal or vertical. By resolving the torsional force into vertical and horizontal components, the problem of combining forces is simplified. Resolve the torsional force into components by using the horizontal and vertical components of dimension c as indicated by Equations 6-13 and 6-14.
Figure 6-15. Loaded Bracket . From Table 6-2. x
b2
=
J w
2b + d
=
=
(2b + d )3 12
32 2×3+ 6
−
= 0.75" 2
b 2 (b + d )
2b + d
3 2 2 = (2 × 3 + 6) − 3 (3 + 6) = 83.25 in.3
2×3+ 6
12
Find components of maximum torsional force at 1. cv = 3"
ch = 3 − x = 2.25"
By Equation 6-13, the horizontal component of torsional force is: W h =
=
Tcv J w
5 (3.75 + 2.25 ) (3) 83.25
= 1.08 kips per lin. in.
Figure 6-14. Torque and Shear on Welds
NOTE: (3.75 + 2.25) is the distance from the point load to the centroid of the weld.
The horizontal torsional force component is: Tc (6-14) W h = v J w
The vertical torsional force component is: 51
Longitudinal shear force may vary along the length of the beam. The vertical shear diagram for the beam can be used as a picture of the amount and location of welds between flange and web.
By Equation 6-14, the vertical component of torsional force is: Tcv
W v =
=
J w
5 (3.75 + 2.25) (2.25) 83.25
= 0.810 kips per lin. in. Find average vertical shear force: W s
=
V Aw
=
5 3+6+3
= 0.416 kips per lin. in. Combine forces using Equation 6-2 W =
⎡ 9.6 ⎤ (0.810 + 0.416)2 + ⎢1.08⎛ ⎜ ⎞⎟⎥ 9.6 ⎝
⎣
Figure 6-17. Shear in Beams.
2
*
Notice there is no shear in the middle portion of beams 1 and 2; therefore, little or no welding is required in this portion. When there is a difference in shear along the length of beam, as in beam 3, the welding could vary in this same ratio along the length of beam. This is why continuous welding is sometimes used at the ends of beams and reduced size or intermittent fillet welds used throughout the rest of the beam.
⎠ ⎦
= 1.635 kips per lin. in. Calculate weld size using Equation 6-1. w=
W f
=
1.635 9.6
= 0.17"
Use 3/16” fillet: Check fillet size (see Fillet Weld Limitations).
Built-up members subject to axial compression: Welds joining the component parts of a built-up compression member, such as a cone roof tank column, are also stressed in longitudinal shear. Determine this longitudinal shear force W q from Equation 6-15 using the shear V at any position along the member as given by Equations 6-16 or 6-17.
Welds in Built-up Members
Built up members subject to bending: Welds attaching the flange to the web are stressed in longitudinal shear and must be adequate to transfer the calculated longitudinal shear force.
Longitudinal shear force at any position along the length of beam is calculated from VQ nΙ
(6-16)
Note that if we had been using API 620 where f t = 8.9 kips per lin. in. and f and f p = 6.4 kips per lin. in. this equation would be
⎡ 8.9 ⎤ (.810 + .416)2 + ⎢1.08 ⎛ ⎜ ⎞⎟⎥ 6 .4 ⎣
V = = 0.01P 0.01P for L/r < 60
(6-18)
w=
*
W =
(6-17)
Also at each end of a built-up compression member, use a total length of continuous fillet weld equal to the maximum width or depth of the member or 4”, whichever is greater. Fillet weld size at any position along the beam or column is determined from Equation 6-1 with the longitudinal shear force W, at W, at the same position.
Figure 6-16. Examples of Built-up Members.
W q =
V = = 0.02P 0.02P for L/r > L/r > 60
⎝
2
⎠⎦
52
W f
=
W q f q
(6-19)
Table 6-3. Length and Spacing of Intermittent Welds
Table 6-4. Maximum Clear Space Between Intermittent Fillet Welds
Continuous Welds %
Length of Intermittent Welds and
60
3-5
Distance Between Centers, In.
57 50
2-4
Tension
3-6
rolled shape flange
4-8
plate flange
4-9
43
Compression
rolled shape flange
3-7 2-5
37 33
plate flange 4-7
44 40
(Carbon Steel Built-up Members)
4-10
30
3-9
4-12
3-10
25
2-8
20
2-10
16
2-12
24” 22t (12” max)* 24”
Maximum clear space between intermittent fillet welds depends on the component parts of the built-up member. The clear space between welds must be close enough to prevent local buckling of the component parts when the loading develops the full strength of the builtup member.
3-8 2-6
24t (12” max)*
3-12
Example: Find size and spacing of fillet weld joining plate and angle of built-up member shown in Figure 6-19. Use f t = 8.9 kips per lin. in., f in., f p = 6.4 kips per lin. in.
Use intermittent fillet welds when the calculated leg size is smaller than the minimum specified in Table 6-5. The calculated size divided by the actual size used, expressed in percent, gives the length of weld to use per unit length: calculated leg size (continuous) (6-20) %= ×100 actual leg size (int ermittent )
Intermittent weld lengths and distances between centers for given percentages of continuous welds are shown in Table 6-3. Figure 6-19. Plate Grinder.
Shear diagram for beam shows that welding for longitudinal shear could be reduced in center portion of beam. Because the vertical shear is small, design the welds for maximum shear throughout the length of beam.
Figure 6-18. Spacing of Intermittent Welds.
The longitudinal shear force is: Minimum length of fillets for intermittent welds is 2” or 4w 4 w, whichever is greater. Selecting the longest fillet possible is usually the most economical. However, do not exceed the maximum clear space between fillets in Table 6-4.
W q
=
VQ nΙ
=
0.623 (0.1875 ) 6 (0.518) 1 (1.094)
= 0.332 kips per lin. in.
*
Many of the built-up members we use have an assumed flange. This flange, usually part of a roof, bottom or shell, may be partially restrained from local buckling when the maximum load is applied. When the built-up member has restraint on the flange, the clear space between fillet welds could be increased to about 32t 32t maximum.
53
Minimum length of fillets for strength welds: 1 ½” or 4w 4w, whichever is greater (Use 2” or 4w 4w for intermittent welds)
The continuous weld size required is: w=
W q f p
=
0.332 6.4
= .052
Spacing of Fillet Welds: 1. When bars or plates plates are are connected only by a set of parallel longitudinal fillets, the length of those welds should not be less than the perpendicular distance between those two welds. 2. When fillet welds are used for end connections, the distance between them must not be greater than 8 inches unless transverse bending is otherwise prevented.
(Use f p because longitudinal shear force is parallel to weld.) Minimum size fillet from Table 6-5 is 3/16”. % continuous weld =
0.052 0.1875
× 100 =
27.7%
Minimum length fillet permitted for intermittent welds is 2”. Maximum clear space between fillets is, from Table 6-4, 22 x 3/16 = 4.1”. Maximum spacing with 2” fillet = 2” + 4.1” = 6.1”. Use 2” – 6” intermittent fillet on one side. This provides 33% (Table 6-3) continuous weld which is more than adequate to transfer the calculated longitudinal shear.
Determining Weld Outline Properties
Properties Sw and J and J w of a weld outline when treated as a line are nearly equal to the section modulus or polar moment of inertia divided by the width w of the weld. When w is small, say 10% of d, the error is usually less than 1%. The properties Sw and J w in Table 6-2 are determined as follows: From handbook
Fillet Weld Limitations
Minimum size fillet: The calculated weld size may be small. To eliminate cracks resulting from rapid cooling, it is best not to put too small a fillet on a thick plate. Follow Table 6-5 for minimum sizes.
I x =
≤1/2” ≤ 3/4” > 3/4”
12
S = I x
Table 6-5. Minimum Size Fllets Thickness3
w3 d
÷
d
2
=
w d 2
6
Treated as a line dividing by w, then
Minimum Leg Size of Fillet 2 3/16” 1/4” 3 1/4” 3, 4
S w
=
S w
d 2
=
6
by
about x axis (6-21)
Figure 6-21. Vertical Weld. 3
From handbook, I w = w d
Maximum size fillet for strength welds:
12
When w is small, let I y = 0 J = I x + I y =
w3d
12
+0
Treated as a line, then J w
Figure 6-20. Weld Size Limited to Plate Thickness.
3
Thickness of thicker part part to be joined.
Leg size of fillet need not not exceed thickness of thinner thinner part to be joined. 2
A minimum fillet of 3/16” 3/16” is acceptable provided 200°F preheat or surface examination of the weld (PT,MT ( PT,MT ) is performed. 3
4
AWS D1.1-82 or AISC require a minimum minimum 5/6” fillet.
54
=
J w
=
d 3
12
(6-22)
⎛ I x ⎞ d 2 2wd 2 (d + 3b ) ⎟ = 6wd ⎝ w ⎠ 3
From handbook, for a horizontal weld I o =
S w = ⎜
w3 d
12
I y = I o
+ Ay 2 =
=
0 + wb y2
I y = w b y 2
J w =
S = I y ÷ y = w b y
=
S
= by about x axis
(6-23)
From handbook I y =
wb3
12 wb3
J = I x + I y = wby 2 +
12
Treated as a line, then J w =
J w
= by 2 +
b3
12
(6-24)
By adding the properties of the two basic lines in Figures 6-21 and 6-22, properties for other straight line outlines may be determined. For example, find Sw and J w for the outline in Figure 6-23.
Figure 6-23. Combination of Welds.
I x
=2
w d 3
12
+ 2 (w b y 2 ) = d
When y = I x =
w d 3
6
I y = 0 + 2
w b d 3
=
2 3
wd
12
6
+ 2w b y 2
,
2
+
w d 3
=
J w b
3
+ bd about x axis
=
I x + I y w 2
= 3
+ 3bd + d 6
(6-25)
w d 2 (d + 3b )+ w b3
6w
(6-26)
Some designers and engineers are not aware of a form of cracking called lamellar tearing, which can occur beneath highly stressed T joints in steel plate. Plate forced to deform plastically in the thruthickness direction by welds which are large, multi-passed, and highly restrained can decohere at a plane of microscopic inclusions. A crack may then progress from plane-toplane in a terrace-like fashion. While lamellar tearing is not frequent, even one incident has the potential of becoming a serious problem. Since there are means to minimize the hazard, it behooves the engineer to take every precaution by optimizing joint design and welding procedure selection. Where these factors cannot be controlled, it may be necessary to use special steels. The reader is referred to the following sources for guidance in designing against lamellar tearing: 1. Engineering Journal, Third Quarter, 1973, Vol. 10, No. 3, pages 61-73. American Institute of Steel Construction, Inc., 1221 Avenue of the Americas, New York, New York 10020. 2. Bibliography on Lamellar Tearing, Welding Research Council Bulletin 232. Welding Research Council, 345 East Forty-Seventh Street, New York, New York 10017.
Treated as a line, then W
3
Cautionary Note
Figure 6-22. Horizontal Weld.
S w =
d 2
w d 2 (d + 3b )
6
3
wd
12
55
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the leak. Since the bottom of a tank is inaccessible from the underside, the differential pressure necessary for the test is accomplished by means of a vacuum box placed on the top side. This box has a glass top and is open on the bottom. The portion of the weld to be inspected is brushed with a soapy solution, the box is fitted over it, and a vacuum created in the box. The weld is inspected through the glass top for leak-indicating bubbles.
A
treatise on the subject of defects in welded vessels and their detection is beyond the scope of this work. But an acquaintance with some of the available inspection and testing tools may serve to dispel the mystery of unfamiliar terms. In the interest of economy, the refinement of inspection and testing must be in tune with the degree of perfection necessary for various classes of work. For example, a pressure vessel storing a lethal substance, or one constructed of a special material known to be crack sensitive, may require as a minimum that 100% of all main joints be radiographed. On the other hand, simple structures such as oil and water tanks, constructed of readily weldable materials, usually require only spot examination. In general, it is safe and wise to follow the inspection requirements of the applicable codes. First, let us distinguish between hydrostatic or overload testing to demonstrate strength or liquid tightness, and inspection to determine weld quality.
Prior to the beginning of any welding, weld qualification and welder certification tests should be performed. These tests insure that the type of welds proposed are adequate for the application and that the welders are capable of applying the required welds. VISUAL INSPECTION is usually the first stage in the inspection of a finished weld, regardless of any other tool that may be employed. Visual inspection can determine conformity with specifications as to dimensional accuracy, extent, etc. It can also reveal noticeable surface flaws, such as obvious cracks, surface porosity, undercutting of parent metal, etc. In some types of work, visual inspection is the only inspection performed; e.g., welds subjected only to compression as in a tubular column, or lowstressed fillet welds. But for most important structures, further inspection is usually required for the main joints, on which the strength of the structure depends. Visual inspection should also verify proper contour, weld preparation and removal of weld spatter and burrs per NACE standards for application of the coating. Some of the more commonly used methods are described below.
Required overload tests are clearly outlined in the various governing codes. Whenever the structure itself, its supports, and foundation conditions will permit, the overload test is usually hydrostatic, i.e., the structure is full of water when the overload, if any, is applied. For the water and oil tanks of Volume 1, no overload can be applied other than that inherent in any difference between the specific gravity of water and that of the product to be stored in service. The normal cone roof will withstand internal pressures only slightly greater than the weight of the roof plates. It will not withstand hydrostatic pressure due to overfilling. Hence, the water test level is limited to the top capacity line. The testing of the flat bottom, however, may warrant brief comment. The liquid tightness of a flat bottom is usually demonstrated by means of a soap bubble test. A soapy liquid is brushed on the weld and a small differential positive pressure created on the opposite side of the plate. Leaks in the weld will be indicated by bubbles as the air passes through
RADIOGRAPHY is an inspection method that shows the presence and nature of macroscopic defects or other discontinuities in the interior of welds. Just as in the case of medical X-rays with which we are all familiar, radiography utilizes the ability of X-rays or gamma rays to penetrate objects opaque to ordinary light. Radiograph films can reveal slag (non-metallic) inclusions, porosity or gas 57
pockets, cracks, lack of fusion, inadequate penetration, and even surface defects, such as undercut. However, welds are rarely perfectly free of all minor defects nor do they need to be. As a result, the inspector must have a good background of experience in reading films, and knowledge of standards. The various construction codes, such as AWS and ASME, define limits of acceptability. MAGNETIC PARTICLE INSPECTION is an aid to visual inspection for surface defects too fine to be detected by the naked eye, plus those that lie slightly below the surface. With special equipment, more deeply seated discontinuities can be detected. The method is applicable only to magnetic materials. It will not function on non-magnetic materials such as the austenitic stainless steels. The basic principle involved is as follows: When a magnetic field is established in a ferromagnetic material containing one or more discontinuities in the path of the magnetic flux, minute poles are set up at the discontinuities. These poles have a stronger attraction for magnetic particles than the surrounding surface of material. Normally the area to be inspected is magnetized between two “prods” by introducing high amperage current or some other convenient means. Then the area is covered with a powder of finely divided magnetic particles. These form a visible pattern of any discontinuity due to the stronger attraction at those points. LIQUID PENETRANT INSPECTION is another method for detecting surface discontinuities too small to be readily seen by the naked eye. It is particularly useful on nonmagnetic materials where the magnetic particle method is ineffective. The method utilizes liquids with unusual penetrating qualities, which, when applied to a previously cleaned surface, will penetrate all surface discontinuities. The surface is then cleaned of all excess penetrant and a developer applied. Penetrant that has entered a crack or other discontinuity will seep out, make contact with the developer and indicate the outline of the defect. There are two principal types of penetrant used. When a FLUORESCENT PENETRANT is used, the indications will fluoresce when exposed to near ultra violet or black light. DYE
PENETRANT utilizes visible instead of fluorescent dyes. As the dye penetrant rises from the flaw by capillary action, it stains the developer (usually a chalky substance) and clearly marks the flaw. ULTRASONIC INSPECTION requires a great deal of explanation for even a rudimentary understanding of how it works. Briefly, ultrasonic testing makes use of an electrically timed wave of the same nature as a sound wave, but of a higher frequency, hence the name ultrasonic. The sound wave or vibrations are propagated in the metal being inspected and are reflected back by any discontinuity or density change. The search unit contains a quartz or similar crystal, which can be moved over the surface much like a doctor’s stethoscope. The search unit applies energy to the metal surface in short bursts of sound waves for a very short, controlled period of time. The crystal then ceases to vibrate for a sufficient period of time to receive the returning echoes. The reflected signals are indicated on a cathode ray tube or oscilloscope. From the reflection or oscilloscope pattern, a trained operator can determine the distance to the discontinuity and some measure of its magnitude. Ultrasonic testing is a valuable tool for certain applications. But it must be used only by an operator skilled in the interpretation of the reflection patterns. In addition to the above methods the following can be used: Eddy Currents, Acoustic Emission, Video Enhancement, Ultrasonic Holography, and Neutron Radiography. Only technically qualified personnel should use these methods.
58
A. B. C. D. E. F. G.
Trigonometry.……………………………………………………………………. A-1 Elements of Sections……………………………………………………………..A-2 Properties of Circles and Ellipses………………………………………………A-7 Surface Areas and Volumes…………………………………………………….A-8 Miscellaneous Formulas………………………………………………………...A-10 Properties of Roof and Bottom Shapes………………………………………...A-12 Conversion Factors………………………………………………………………A-13
59
A-1
A = d 2 c =
d
A = bd
2
c = d
d 4
I = =
I =
12
S = =
d 3 S =
6 d 12
r =
= Z =
= .288675 d
r =
bd 3
3 Bd 2
3 d 3
= .577380
d
d 3
4 A = bd
A = d 2
c
bd
= b
c = d d 4
I =
I =
3
S =
d 3
(
6 b 2 + d 3
)
b 2 d 2
6 b 2 + d 3
d 3
r =
+ d 2
b 2 d 3
S =
3
2
= .577350
d
bd
r =
(
)
6 b 2 + d 3
A = d
2
c
A = bd
d 2
=
= .707107
c
d 4
I = =
12
= .117851 d 3
S =
6 2 r =
Z =
d 12
2c
3
3
= .288675 d =
3
d
= .235702
I =
2
(
2
(
2
bd b sin 2 a + d 2 cos 2 a
bd b sin a + d 2 cos 2 a
6 (b sin a + d cos a )
b 2 sin 3 a + d 3 cos 2 a
12
d 3
A = bd − b1d 1
d
c =
2 bd
I =
12 2
bd
r =
d 12
d
2 3
3
S =
6
bd 3 − b1d 1
12 3
bd
− b1d 13 6d 3
= .288675 d
r =
bd 3
Z =
4
A-2
)
12 3
3 2
S =
Z =
b sin a + d cos a
r =
A = bd c =
=
I =
d 2
S
d
bd 3 − b1d 1
12 A bd 2
4
−
b1d 1
4
2
)
A = b (d − d 1 ) c = I = S =
2
(
− d 13 )
b d 3
12
(
=
I =
S =
6d
(d
b
2
4
− d 12 )
2
1 / 2 bt
12
I
+ bty 2 +
12
S 1 +
c
+ b1t 1 y12
c1
2
32
+
4
I =
= .098175 d 3 = .785398 R 3
2
6
(
2
π d 2 − d 1 4
) = .785398 (d − d ) 2
2 1
d
2
(
4
π d 4 − d 1 64
bd
S =
(
4
π d 4 − d 1 32d
bd 2
) = .049087 (d − d ) 4
) =.098175
4 1
4
d 4 − d 1 d
2
24
d 2 + d 1
r =
= .235702
4
d Z =
3
d
6
bd A
2
=
2
d 1
−
6
π R
2
=
1.570796 R
2
2
⎛ 4 ⎞ ⎟ = .575587 R ⎝ 3π ⎠ ⎛ π 8 ⎞ I = R 4 ⎜ ⎟ = .109757 R 4 ⎝ 8 9π ⎠ c = R ⎜1−
bd 3
12 bd 2
12 d 6
4
d
3
d 18
R 3 π
=
= .049087 d 4 = .785398 R 4
3
c =
36
4
d R
A =
c = d
r =
π d 3
π R 4
=
64
Z = =
3
r =
S = =
π d 4
r =
2
= R
2
S =
I
⎡ ⎛ t + t 1 ⎞⎤ ⎢d − ⎜ 2 ⎟⎥ ⎣ ⎝ ⎠⎦
2d
I =
d
=
I =
A A
)
= π R 2 = .785398 d 2
4
I
c =
A =
c
A bt 3
2
= 3.141593 R 2
+ b1t 1 (d −1 / 2 t 1 ) 3 b1t 1
(
2 b 2 + 4bb1 + b1
6 (b + b1 )
π d 2
A =
36 (b + b1 ) 2
d
r =
+ b1 2)
+ 4 bb1 + b12 ) 12 (2b + b1 )
d b
S =
12 (d − d 1 )
bd
= S =
(
2
3
A =
I = =
(
d 3 b 2 + 4 bb1
d 3 − d 1
= Z =
3 (b + b1 )
I =
− d 13 )
b d 3
d (2b + b1 )
=
c
A = bt + b1t 2 c
2
d
r = Z =
d (b + b1 )
A =
S =
= .408248 d
R3 (9π 2 − 64)
24
r = R
A-3
(3π − 4) 9π 2 − 64 6π
= .190687 R 3
= .264336 R
4
A =
2
m=
5
=
I 2
=
I 3 =
2 2
I 4 =
a 3b
480
105
m
=
n
=
3
=
ab3
a 3b
ab
14 =
10 3
A
a
4b
3
2100 1
1 16
π a3b
1 16
π ab3
⎛ ⎝
ab ⎜⎜1 −
a b n=
ab 3
⎛ ⎝
⎛ ⎝
6 ⎜1 −
t
1 6
I 1 = I 2
=
4 5
⎟
4 ⎠
⎞ ⎟ ⎟ ⎛ π ⎞ ⎟⎟ 36 ⎜⎜1 − ⎟⎟ ⎟ ⎝ 4 ⎠ ⎠ 1
⎛ ⎞ ⎜ ⎟ π 1 1 ⎟ 3⎜ 12 = ab ⎜ − − π ⎞ ⎟ 3 16 ⎛ ⎜⎜ 36 ⎜1− ⎟ ⎟⎟ ⎝ 4 ⎠ ⎠ ⎝
t 3
m =n =
⎟
4 ⎠
π ⎞
⎜ ⎝
t
2 A =
π ⎞
⎜ 1 π − 11 = a 3b ⎜ − ⎜ 3 16
2 2 b =
⎟⎟
4 ⎠
b
⎛ ⎜
a =
π ⎞
a
6 ⎜1 −
37
80
3π
=
m=
b
4
=
4a 3π
⎛ π 4 ⎞ ⎜⎜ − ⎟⎟ ⎝ 16 9π ⎠ ⎛ π 4 ⎞ ⎟⎟ 12 = ab 2 ⎜⎜ − ⎝ 16 9 π ⎠ 13 =
7
π ab
4
11 = a 3b
ab3
15 1
1
A = m=
19
16
=
I 1
1 13 = π a3b 8
n =
2
⎛ π 8 ⎞ ⎜ − ⎟ ⎝ 8 9π ⎠
a 3b
175
A
4a 3π
1 12 = π ab 2 8
8
=
π ab
2
ab3
b
8
1
11 = a 3b
a
5
I 3 =
a3 b
ab
3
3
I 2
I 2
4 15 32
=
I 1
m=
105
m= n=
a
175
=
A
A
16
I 1
=
ab
3
t
11 2100
* To obtain properties of half circle, quarter circle and t 4
circular complement substitute a = b = R.
A-4
n = = Number of sides o
s
=
180
a
=
2 R 2
n
A
2
IRREGULAR PLANE SURFACE
R1
a
R = R1
−
2 sin ϕ
= =
a
2 tan ϕ 1 4
na 2 cot ϕ =
1 2
nR 2
2
sin 2ϕ = nR1 tan ϕ I 1 = I 2 = r 1 = r 2 =
tan 2θ =
(
A 6 R 2 − a 2
24 6 R 2 − a 2 24
) = A(12 R
2 1
+ a2 )
48
2
12 R1
=
+ a2
48
2 K I Y − I X
A = t (b + c ) x =
b 2 + ct
y =
2(b + c )
d 2
+ at
Divide the plane surface into an even number of parallel strips of equal width. The given figure has been divided into ten strips of width, d; the ordinates are h0 to h10. When the ends are curved, h 0 and h10 are zero and cancel out of formulas.
2(b + c )
K = Pr oduct of inertia about X − X & Y − Y =
abcdt
+ 4(b + c )
⎛ 1 ⎞ = ⎜⎜ t (d − y )3 + by 3 − a( y − t )3 ⎟⎟ ⎝ 3 ⎠ ⎛ 1 3 3 ⎞ I y = ⎜ t (b − x ) + dx 3 − c( x − t ) ⎟ ⎝ 3 ⎠ I x
I z I w
Area=
= I x sin 2 θ + I y cos 2 θ + K sin 2 θ = I x cos
2
2
3
[h0 + h10 + 4(h1 + h3 + h5 + h7 + h9 ) + 2(h2 + h1
+ h6 + h8 )]
Area=
2
θ + I y sin θ − K sin θ
d [0.4(h0
K is negative when heel of angle, with respect to c.g., is in 1st or 3rd quadrant, positive when in 2nd or 4th quadrant.
+ h10 ) + 1.1(h1 + h9 ) + h2 + h3 + h4 + h5 + h6 + h7 + h8 ]
Area=
⎡1 (h0 + h10 ) + h1 + h2 + h3 + h4 + h5 + h6 + h7 + h8 + h9 ⎤⎥ 2 ⎣ ⎦
d ⎢
I 3 = I x sin 2 φ + I y cos 2 φ
When the ends are not curved, but are the straight
I 4 = I x cos 2 φ + I y sin 2 φ
⎛ Y x = M ⎜ sin φ + ⎜ I x I y ⎝
d
lines h1 and h9 then,
⎞ cos φ ⎟ ⎟ ⎠
⎡1 ⎣2
⎤ ⎦
Area= d ⎢ (h1 + h9 ) + h2 + h3 + h4 + h5 + h6 + h7 + h8 ⎥
Where M is bending moment due to force F.
A-5
A = π dt A = π dt I = S =
π d 3t 8 π d 2t 4
r = 0.355d
b = d A = 4dt A =
2d 3t 3
A = 0.408d
d > b A = 2(b + d )t d 2t
(3b + d )
I 1−1
=
S 1−1
=
r 1−1
= 0.289d
6 dt
3
(3b + d ) 3b + d b + d
Sector of thin annulus A = 2α Rt
⎛ sin α ⎞ ⎟ α ⎠ ⎝ ⎛ sin α ⎞ Y 2 = R⎜ − cosα ⎟ ⎝ α ⎠
y1 = R⎜1 −
A-6
Area
Circumference
= 6.28318 r = 3.14159 d
Diameter
= 0.31831 circumference
Area
= ½ (length of arc, arc, m p n x radius, radius, r)
= 3.14159 r 2
= area of circle circle x
Arc
=
a
arc, m p n, in deg rees
360
Angle A c
π r Ab 180 o
=
180 b a π r
= 0.0087266 x square of radius, r , x 2
angle of arc, m p n, in degrees
Area
Radius r =
triangle, m o n =
Rise
(length of arc, m p n, x radius, r) – (radius, r, - rise, b) x chord, c
b
2
= r − 1 / 2 = 2r sin
Area = area of circle – area of segment m n p
Area = square of diameter, d 2, multiplied by the coefficient given opposite the quotient of Intermediate coefficients for values of
b d b d
2 A
y
= b − r +
x
=
r 2
4r 2 − c 2
4
=
c
2
b
r
2
8b
tan
− 2 r sin
A
2
A
4
r 2 − x 2
= r ÷ y −
r 2 − x 2
− (r + y − b )2
not
r = radius of circle
y = angle ncp in degrees
Areas of Sector Sector ncpo=1/2 (length of arc nop nop
Example – Given: rise=2 /16 and diameter=5 /x d
+c
a
Diameter of circle of equal periphery as square = 1.27324 side of square Side of square of equal periphery as circle = 0.78540 diameter of circle Diameter of circle circumscribed about square = 1.41421 side of square Side of square inscribed in circle = 0.70711 diameter of circle
given in tables are obtained by interpolation. 7
4b
2
= 57.29578
Chord C = 2 2br − b 2
= area of sector, m o n p – area of
Given: rise, b, and diameter, d = 2r.
= 0.017483 r Ab
3
×
= Area of Circle
= 2 7 16 ÷ 5 3 32 = 0.478528
r)
×
y
360
= 0.0087266 × r
y
2 ×
Coefficient by interpolation = 0.371233.
r=radius of circle x=chord b=rise
Area =d x coeff.=25.91629 x 0.371233 = 9.6321 2
Area of Segment nop = Area of Sector neopo– Area of triangle ncp
Given: rise, b, and chord c.
=
2 Area of Segment nsp = Area of of Circle – Area of Segment nop
Area = product of rise and chord, b x c, multiplied by the coefficient given opposite the quotient of Intermediate coefficients for values of
b c
( Length of arc nop × r ) − x(r − b )
. b
π=3.14159265359,
c
Log=0.4971499
π2 = 9.8696044, log = 0.9942997
not given in tables are obtained by interpolation. Example – Given: rise = 1.49 and chord = 3.52, b c
=
1.49 3.52
= 0.4233 . Coefficient=0.7542
Area = b x c x coeff. = 1.49 x 3.52 x 0.7542 x 3.9556
1.5028501
1 π
=0.5641896, log = 1.7514251
1 π
2
= 0.1013212, log = 1.0057003
Area = area of circle – (area of segment, t p u +
Area = segment m p n – segment, m s n.
=0.3183099, log =
π3 = 31.0062767,
log=1.4914496
π = 1.7724539, log = 0.2485749
area of segment, v q w).
1 π
180 π
π 180 1 π 3
=0.0174533, log=2.2418774
=0.0322515, log = 2.5085500
=57.2957795, log = 1.7581226
Note: Logs of fractions such as 1.5028501 and 2.5085500 may also be written 9.5028501 - 10 and 8.508550 – 10 respectively.
A-7
D and R = Mean Diameter and Mean Radius, respectively, of Ring d and r = Mean Diameter and Me an Radius, respectively, of Section Surface =
π2 Dd = 4 π2 Rr
Volume = 2 π2 Rr 2 =
π 2 4
Dd 2
End faces are in parallel planes. Volume = i(A + A’ + 4M), where 1 = perpendicular distance between ends A, A’ = areas of ends M = area of midsection, parallel to ends
(As formed by cutting plane oblique to base) I.
Base, abc, less than semicircle; Convex Surface = h [2re ⎯ (d (d x length arc abc)]
÷ (r ⎯ d) d)
⎡2 ⎤ Volume = h ⎢ e 3 − (d × area base abc )⎥ ÷ (r − d ) ⎣3 ⎦ II.
Base, abc, = semicircle; Convex Surface = 2rh
III.
Volume = r 2 h
Base, abc, greater than semicircle (figure); Convex Surface
÷ (r+d) ⎡2 ⎤ Volume = h ⎢ e3 + (d × area base abc )⎥ ÷ (r + d ) ⎣3 ⎦ =h [2re + (d x length arc abc)]
IV.
Base, abc, abc, = circle, oblique plane touching circumference. Convex Surface = πrh
V.
Volume = ½ π r 2 h
Base, abc, = circle, circle, oblique plane plane entirely entirely above above (figure) Convex Surface = 2 π r ½ (h, minimum + H, maximum)
×
Volume = π r 2 x ½ (h, minimum + H, maximum)
Let abcd represent the generating section about axis A-A of solid abef. Let g at distance h from A-A be the center of gravity of abcd. Let a ° be the angular amount of generating revolution. Then Total Surface of solid abef = (2π ha ÷ 360) x perimeter abcd Volume of solid abef = (2 π ha ha ÷ 360) x area abcd For complete revolution (2 π ha ha ÷ 360) = 2 π h
A-8
Convex Surface = π dh
Surface = π d2 = 4π r 2
π d 2
Total Surface = π dh +
2
π
Volume =
d2 h 4 Volume Cylinder, right or oblique = area of section at right angles to sides X length of side. h Center of Gravity above Base = 2 Lateral Surface = h X Base Perimeter Total Surface= Lateral Surface + (2 X Base Area) Volume = h X Base Area
Total Surface =
h
Center of Gravity above Base =
Volume =
2
X Base Perimeter 2 Total Surface = Lateral Surface + Base Area h Volume = X Base Area 3 h
Center of Gravity =
2 3
π r 2h
π 2
Volume =
π h 24
d ds = π d 2 + 4h 2
π
d2h =
24
d2 4 s 2
Volume =
h
4 3
2
4
Total Surface =
π s 2
Volume =
π h 12
Center
(
h d
(
2
4 d
+ 2
(d
2
+
of 2 d d ′
+ + d d ′ +
⎣⎢
)
4 h
2
+ (d −
(d + d ′ ) +
d d ′ + d ′ 2
Gravity 3 d ′ d ′
2
2
)
π 4
d ′
Volume =
6
(d
2
+ d ′
2
Volume =
)
+
4
e
3
π R2r
above
base
=
⎛ 1 + e ⎞⎤ ⎟⎥ ⎝ 1 − e ⎠⎦⎥
log .⎜
Where e=
R 2
− r 2
R
⎡(r 2 + 4h 2 )3 2 − r 3⎤ ⎥⎦ 6h ⎢⎣ Total Surface = Convex Surface + π r 2
Convex Surface =
m
2.303r 2
Use common or base 10, log.
)
)
(l +
+ c′ 2 )
)2
Surface = Sum of surfaces of bounding planes wh
2
π Rr 2
Surface = π ⎢2 R 2
d ′
(d +
(c
4
Convex Surface = π
4
Sin-1e = Angle, in radians, whose sine = e
⎡
)=
π
⎛ sin −1 e ⎞⎤ ⎟⎥ ⎜ e ⎟⎥ ⎝ ⎠⎦
⎢⎣
− d 2
Center of Gravity above base =
d ′
⎞ ⎟ 4 ⎟⎟ ⎠
c2
Surface = 2 π r ⎢r + R⎜
4
(d +
3
⎛ ⎜ ⎜ ⎝
π r 2 ⎜ r − r 2 −
(3c2 + 3c’ 2 + 4h2)
⎡
4
Total Surface = Convex Surface + π d
π s
2
Total Surface = 2 π rh +
2
12
=
(4h + c )
Convex Surface = 2 π rh
⎞ h ⎛ ⎜ 3 a + A + 2 aA ⎟ 4 ⎜ a + A + aA ⎟ ⎝ ⎠
π
2
Spherical Surface= 2 π rh = π(c2 + 4h2) ÷ 4 Total Surface = Spherical Surface + ( πc2 ÷ 4) Volume = π h2 (3r – h) ÷ 3 = π h (3c 2 + 4h2) ÷24 Center of gravity above base of segment = h (4r – h) ÷ 4(3r-h)
Lateral Surface = s (Top + Base Perimeters) ÷ 2 If a = top area and A = base area. Total Surface = Lateral Surface + (a + A) Volume = h (a + A + aA ) ÷ 3 Center of Gravity above base =
Volume =
π r
h ⎞ 3 ⎛ ⎜ r − ⎟ 4 ⎝ 2 ⎠
, above base
4
Convex Surface =
4
=
Center of Gravity above center of sphere =
s
Lateral Surface =
π d 3
π r 3 6 3 Side of an equal cube = diameter of sphere X 0.806 Length of an equal cylinder = diameter of sphere X 0.6667 Center of Gravity of Half Sphere = 3/8r above spherical center
Volume =
+ n)
Volume = base
A-9
π r 2h 2
π r
2
Center of Gravity =
h
3
above
1. Area of Roofs.
7. Heads for Horizontal Cylindrical Tanks :
D = diameter of tank in feet.
3. Expansion in steel pipe = 0.78 inch per 100 lineal feet per 100
Hemi-ellipsoidal Heads have Heads have an ellipsoidal cross section, usually with minor axis equal to one half the major axis-that is, depth = ¼ D, or more. Dished or Basket Heads consist Heads consist of a spherical segment normally dished to a radius equal to the inside diameter’ of the tank cylinder (or within a ran e of 6 inches plus or minus) and connected to the straight cylindrical flange by a “knuckle” whose inside radius is usually not less than 6 per cent of the inside diameter of the cylinder nor less than 3 times the thickness of the head plate. Basket heads closely approximate hemi-ellipsoidal heads. Bumped Heads consist Heads consist of a spherical segment joining the tank cylinder directly without the transition “knuckle”. The radius = D, or less. This type of head is used only for pressures of 10 pounds per square inch or less, excepting where a compression ring is placed at the junction of head and shell. Surface Area of Heads:
degrees Fahrenheit change in temperature = 0.412 inch per mile per degree Fahrenheit temperature change.
(7a) Hemi-ellipsoidal Heads: S=π R2 [1 + K 2 (2-K)]
Umbrella Roofs: Surface area in square feet
= 0.842 D2 (when radius=diameter) = 0.882 D2 (when radius=-0.8 diameter)
Conical Roofs: Surface areas in square feet
= 0.787 D2 (when pitch is ¾ in 12) = 0.792 D2 (when pitch is 1 ½ i n 12)
2. Average weights. Steel 490 pounds per cubic foot, specific gravity 7.85 Wrought iron 485 pounds per cubic foot, foot, specific gravity 7.77 Cast iron 450 pounds per cubic foot, specific gravity 7.21 1 cubic foot air or gas at 32° F., 760 mm. barometer = molecular weight x 0.0027855 pounds.
S=surface area in square feet R=radius of cylinder in feet K=ratio of the depth of the head (not including the straight flange) to the radius of the cylinder The above formula is not exact but is within limits of practical accuracy.
4. Linear coefficients coefficien ts of expansion per degree increase in temperature:
STRUCTURAL STEEL-A-7 70° to 200°C ........................ ........................ 21.1° to 93°C ....................... ....................... STAINLESS STEEL-TYPE 304 32° to 932°C ........................ ........................ 0° to 500°C .......................... .......................... ALUMINUM -76° to 68°F.......................... .......................... -60° to 20°C ......................... .........................
Per Degree Fahrenheit
Per Degree Centigrade
0.0000065 -
0.0000117
(7b) Dished or Basket Heads:
0.0000099 -
0.0000178
(7c) Bumped Heads: S=π R2 [1 + K 2 (2-K)]
0.0000128 -
0.0000231
Formula (7a) gives surface area within practical l imits.
S, R, and K as i n formula (7a) Volume of Heads:
(7d) Hemi-ellipsoidal Heads: V=2/3 π K R3
5. To determine determine the net thickness of shells for horizontal horizontal cylindrical cylindrical pressure tanks: T=
R=radius of cylinder in feet K=ratio of the depth of the head (not including the straight flange) to the radius of the cylinder
6 PD
(7e) Dished or Basket Heads:
S
P = working pressure in pounds per square inch D = diameter of cylinder in feet S = allowable unit working stress in pounds per square inch T = Net thickness in inches Resulting net thickness must be corrected to gross or actual thickness by dividing by joint efficiency.
Formula (7d) gives volume within practical limits.
(7f) Bumped Heads: V=1/2 π K R3 (1 + 1/3 K 2)
V, K and R as in formula (7d) Note:
Dished Heads - K = M -
6. To determine the net thickness thickness of heads heads for cylindrical cylindrical pressure tanks:
⎡
6 PD
MR mR R
S
T, P and D as in formula 5 (6b) Dished or Basket Heads: T=
M=
10.6 P( MR) S
T, S and P as in formula 5 MR = principal radius of head in feet Resulting net thickness of heads is both net and gross thickness if one piece seamless heads are used, otherwise net thickness must be corrected to cross thickness as above. Formulas 5 and 6 must often be modified to comply with various engineering codes, and state and municipal regulations. Calculated gross late thicknesses are sometimes arbitrarily increased to provide an additional allowance for corrosion.’
( M − 1)( M + 1 − 2m) ⎤
Bumped Heads – K = ⎢ M − M 2 − 1 ⎥ ⎣ ⎦
(6a) Ellipsoidal or Bumped Heads: T=
K in above formulas may be determined as follows: Hemi-ellipsoidal heads – K is known
= principal radius of head in feet = radius of knuckle in feet = radius of cylinder in feet MR R
m=
mR R
For bumped heads, m=0
8. Total Volume of a Sphere: V = total volume D = diameter of sphere in feet V = -0.523599 D 3 Cubic Feet V = -0.093257 D 3 Barrels of 42 U.S. Gallons Number of barrels of 42 U.S. Gallons at any inch in a true sphere = (3d-2h) h 2 x .0000539681 where d is diameter of sphere and h is depth of liquid both in inches. The desired volume must include appropriate ullage for the stored liquid.
A-10
9. Total volume or length of shell in cylindrical tank with ellipsoidal ellipsoidal or hemispherical heads: V L
= Total volume = Length of cylindrical shell KD = Depth of head D 2 π
V
=
L
= (V +y)-1’/iKD
4
(L + 1 KD)
10. Volume or contents of partially filled horizontal cylindrical tanks: (10a) Tank cylinder or shell (straight portion only) Q
⎡⎛ πθ ⎞ ⎤ ⎟ − sin θ cos θ ⎥ ⎜ 180 ⎟ ⎢⎣⎝ ⎥⎦ ⎠
= R 2 L ⎢⎜
o
(10c) Dished or Basket Heads: Formula (lob) gives partially filled volume within practical limits, and formula (7d) gives V within practical limits. (10d) Bumped Heads: Formula (lob) gives partially tilled volume within practical limits, and formula (7f) gives V. Note: To obtain the volume or quantity of liquid in partially filled tanks, add the volume per formula (10a) for the cylinder or straight portion to twice (for 2 heads) the volume per formula (10b), (10c) or (10d) for the type of head concerned.
11. Volume or contents of partially filled hemi-ellipsoidal heads with major axis vertical: Q = Partially filled volume or contents in cubic feet V = Total volume of one head per formula (7d) R = Radius of cylinder in feet (11a) Upper Head Q = 1 ½ V Δ (I - Δ2)
Q = partially filled volume or contents in cubic feet R = radius of cylinder in feet L = l ength of straight portion of cylinder in feet The straight portion or flange of the heads must be considered a part of the cylinder. The length of flange depends upon the diameter of tank and thickness of head but ranges usually between 2 and 4 inches. A = Δ R = depth of liquid in feet Δ = i = a ratio Cos θ = 1 - Δ, or θ = degrees (10b) Hemi-ellipsoidal Heads: Q = 3/4V Δ2 (1-1/3 Δ) Q = partially filled volume or contents in cubic feet V = total volume of one head per formula (7d) Δ = k = a ratio a = ΔR = depth of liquid in feet R = radius of cylinder in feet
=
a KR
= a ratio
A = Δ KR = depth of liquid in feet (11b) Lower Head: Q = 1 ½ V Δ2 (1 - Δ)
Δ =
a KR
= a ratio
A = Δ KR = depth of liquid in feet
A-11
3 3
2
2
D D 3 5 4 5 0 2 1 . 1 . 1 0
D 9 6 1 . 0
D D D 0 9 6 1 5 8 8 0 2 0 . 6 . 9 . 0 0 0
D
D D D 2 1 5 7 8 3 2 4 6 3 . 4 . 9 . 0 2 1
D 7 0 7 . 0
D D D 7 2 1 7 7 6 7 7 6 2 . 0 . 6 . 0 2 1
D 5 5 7 1 . 0
3 2 D D 5 7 D 7 6 2 1 3 2 7 5 8 8 0 . . . 0 0 0
D 4 3 1 . 0
3 D D D 8 9 1 3 3 0 1 4 5 . 4 . 8 . 0 0 0
D D 2 6 7 0 4 9 0 . 0 . 1 0
h
) ) h r h r 3 3 ( ( h 2 2 r h h π 2 3 2 7 3 4 8 0 . . 1 7
e l g r o n t a c e e s t a e l t u l a c l u a l c C a C
3 3
h
3 3
2
2
3
3
2
2
D 9 - 3 6 5 . 0 2
D 4 - 6 4 4 . 0 2
D D 0 5 8 9 0 . 1 . 1 1 0 2
∇
D 0 0 0 1 . 0
-
-
-
-
0
0 9
D 2 0 6 6 . 0
-
-
-
-
0
0 9
D 3 4 5 4 . 0
-
-
-
-
0
0 9
D 5 5 7 0 . 0
D 6 9 5 0 . 0
-
-
-
W 8 9 1 . 0
7 6 . 8 3
D 6 5 0 . 0
D 1 5 4 0 . 0
-
-
-
W 6 7 2 . 0
0 3
r p 6
. l o d v n a w ) x e n ( h e s t i a s l u a c b l a n C o
) y l h g u o r (
2
h
5 8
±
) h h h 4 + 3 − H − r ( h r r 2 6 . 8 1 2
∇
h h 2 2 a D D a D 8 4 h 3 h 4 1 8 D 2 - D 2 h 2 3 H h 6 5 π 6 . 2 9 . . 2 0 1
a D p h 6
2
D D D 9 2 4 9 D 4 0 3 7 8 0 1 . 9 . . 0 0 1
2
D D 1 4 1 6 2 . 9 . 1 1 0
D 9 3 4 1 . 0
D 3 6 5 1 . 0
D 4
D 4 D 3 r + + o H H D D p ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ p 6 D D 6 6 . 2
+
6 . 2
+
−
r o D 3 D 4
D D 2 5 6 D 5 4 D 3 4 7 0 2 . 1 . 3 . 1 0 1
2
D D 2 8 2 1 3 . 6 . 1 2 0
D 9 1 9 1 . 0
D 3 8 0 2 . 0
D 3
+
H H 8 ⎜ ⎝ D ⎛ D 5
⎛ ⎜ ⎝ 9 . 1
+
3 3
2 D D D 8 4 1 1 8 D 2 6 5 7 5 2 . 9 . . 0 1 1
2
D D 8 7 0 2 7 9 5 . 3 . 1 0
D 8 7 8 2 . 0
y
D 5 2 1 3 . 0
y
+
D D 2
6 . 2
H
⎛ ⎜ ⎝ 3 . 1
°
°
2
t c a r t b u s
D 4
e t a l u D l c W a 3 C 8 −
r
2
1 3 . 0
D W 6 9 h
e t a l u c l a C
0
0 9
0
0 9
0
0 9
7 0 . 0
2
X 3 4
−
2
D 4 X
°
⎛ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎠ 2
X 4
D D p 4 3 p 3 − 5 . 4 − D 4 + r ⎜ ⎜ ⎝ o ⎛ 2
°
X π
−
⎞ ⎟ ⎟ ⎠
⎞ ⎟ ⎠ +
°
π
⎞ ⎟ ⎠ ⎞ ⎟ ⎠ 3 3
°
⎞ ⎟ ⎟ ⎠
⎞ ⎟ ⎠ ⎞ ⎟ ⎠ 3 3
. e l t o a v l u c ∇ l a . C l o v
°
2
D p 3
X 3 −
2
D 4 X
⎛ ⎜ ⎜ ⎝ π
°
2 1 A
3 1 A
Length
inch foot yard mile(U.S. Statute) millimetre metre metre kilometre
Area
25.4 25.400 00 0.304 800 b 0.914 400 1.609 347
mill millim imet etre re metre metre kilometre
39.370 079 x10-3 3.280 840 1.093 613 0.621 370
in inch foot yard mi m il e
0.645 160 x 10
4.046 873 x 10 3 0.404 687
square milli millime metr tre e mm2 square square meter m 2 square meter m 2 square kilometre km2 square square meter m2 hectare
1.550 003 x 10 -3
squar quare e inc inch in2
10.763 10.763 910 1.195 1.195 990 0.386 0.386 101
square square foot foot ft 2 square square yard yd 2 square square mile mile mi2
b b
square inch
b
square foot square yard square mile (U.S.Statute) acre acre
b
square millimetre square square metre metre square square metre square kilometre square metre hectare
Vol
cubic inch
cubic millimetre cubi cubic c metr metre e cubic cubic metre metre litre
b
-3
0.946 353
61.023759 x 10 6 35.3 35.314 14 662 662 1.307 1.307 951 0.264 0.264 172 1.056 1.056 688
ounce (avoirdupois) pound (avoirdupois) short ton gram
16.387 16.387 06 x 10 3 28.316 85 x 10 0.764 0.764 555 3.785 412
litre
Mass
0.092 0.092 903 0.836 127 2.589 998
in ft yd mi
0.247 104 x 10 -3 acre 2.471 044 acre
cubic foot cubic cubic yard yard gallon(U.S liquid) quart(U.S. liquid)
3
mm m m km
28.349 52 0.453 592 0.907 185 x 10 -3 35.273 966 x 10-
cubic milli millime metr tre e mm3 cubic ubic met meter m3 cubic cubic meter meter m 3 litre l litre
cubic inch
Bending Moment
Pressure Stress
Energy, Work, Heat
in3
cubi cubic c foot foot ft cubic cubic yard yard yd 3 gallon (U.S. (U.S. liquid) liquid) gal quart (U.S. (U.S. liquid) liquid) qt
Power
3
gram
ounce-force pound-force
0.278 014 4.448 222
newton newton
newton newton pound-force-inch
3.596 942 0.224 809 0.112 985
pound-force-foot
1.355 818
newton-metre
8.850 748
newton-metre
0.737 562
pound-force per sq inch foot of water (39.2 F) inch of mercury (32 F) kilopascal
6.894 757
ounce-force pound-force lbf newtonmetre N•m newtonmetre N•m pound-force inch lbf• in pound-force foot lbf• ft kilopascal kPa
2.988 98
kilopascal kPa
3.386 38
kilopascal kPa
0.145 038
kilopascal
0.334 562
kilopascal
0.295 301
pound-force per sq. inch lbf/in2 foot of water (39.2 F) inch of mercury (32 F) joule J joule J
foot-pound-force cBritish thermal unit ccalorie kilowatt hour joule
1.355 818 1.055 056 X 10 3 4. 4 .186 800 3.600.000 X10 6 0.737 562
b b
joule
0.947 817 X 10 10 -3
joule joule foot-pound-force/ second cBritish thermal unit per hour horsepower (550 ft. lbf/s) watt
0.238 846 0.277 778 X 10 10 -6 1.355 818
l
g
kilogram
kg
ki kilogram
kg
ounce oz av (avoirdupois) kilogram 2.204 622 pound lb av (avoirdupois) kilogram 1.102 311 x 10-3 short ton a Refer to ASTM E380-79 for more complete complete information on SI. b Indicates exact exact value. 3
Force
joule J joule J foot-pound force ft •lbf cBritish thermal unit Btu ccalorie kilowatt hourkW • watt W
0.293 071
watt
0.745 700
kilowatt
W kW
foot-poundforce/ sec. ft•lbf/s cBritish watt 3.412 141 thermal unit/hr Btu/h kilowatt 1.341 022 horsepower hp hp (550 ft•lbf/s) Angle degree 17.453 29 X 10 -3 radian rad radian 57.295 788 degree Temper° Fahrenheit t°C=(t°Fx32)/1.8 degree ature Celsius ° Celsius t°F=1.8 x t °C+32 degree Fahrenheit a Refer to ASTM E380-79 for more complete information on SI. b Indicates exact value. c International Table
A-14
0.737 562
N N
SPECIFIC GRAVITY AND WEIGHTS OF VARIOUS LIQUIDS Liquid Acetaldehyde Acetic Acid Acetic Anhydride Acetone Aniline Asphaltum Bromine Carbon Disulfide Carbon Tetrachloride Castor Oil Caustic Soda,66% Solution Chloroform Citric Acid Cocoanut Oil Colza Oil (Rape Seed Oil) Corn Oil Cottonseed Oil Creosote Dimethyl Aniline Ether Ethyl Acetate Ethyl Chloride Ethyl Ether Formaldehyde #1 Fuel Oil #2 Fuel Oil #4 Fuel Oil #5 Fuel Oil #6 Fuel Oil Furfural Gasoline (Motor Fuel) Glucose Glycerin Hydrochloric Acid, 43.4% Sol Kerosene Lactic Acid Lard Oil Linseed Oil – Raw Linseed Oil – Boiled Mercury Molasses Naphthalene Neatsfoot Oil Nitric Acid, 91% Sol. Olive Oil Peanut Oil Phenol Pitch Rosin Oil Soy Bean Oil Sperm Oil Sulfur Dioxide Sulfuric Acid,87% Sol. Tar Tetrachloroethane Trichloroethytene Tung Oil Turpentine Water (Sea) Water (0 °C) Water (20 °C) Whale Oil
At Temp. of °F 64.4 68.0 68.0 68.0 68.0 68.0 68.0 68.0 68.0 59.0 68.0 68.0 68.0 59.0 68.0 59.0 60.8 59.0 68.0 77.0 68.0 42.8 77.0 68.0 60.0 60.0 60.0 60.0 60.0 68.0 60.0 77.0 32.0 60.0 68.0 59.0 59.0 68.0 59.0 68.0 68.0 68.0 59.0 68.0 59.0 59.0 77.0 68.0 68.0 59.0 59.0 80.0 64.4 68.0 68.0 68.0 59.0 68.0 59.0 39.2 68.0 59.0
Specific Gravity 0.783 1.049 1.083 0.792 1.022 1.1-1.5 3.119 1.263 1.595 0.969 1.70 1.489 1.542 0.926 0.915 0.921-0.928 0.926 1.040-1.100 0.956 0.708 0.901 0.917 0.712-0.714 1.139 0.80-0.85 0.81-0.91 0.84-1.00 0.91-1.06 0.92-1.08 1.159 0.70-0.76 1.544 1.260 1.213 0.82 1.249 0.913-0.915 0.93 0.942 13.595 1.47 1.145 0.913-0.918 1.502 0.915-0.920 0.917-0.926 1.071 1.07-1.15 0.98 0.924-0.927 0.878-0.884 1.363 1.834 1.2 1.596 1.464 0.939-0.949 .0.87 1.025 1.00 0.998 0.917-0.924
Weight in Lbs. Per U.S. Gal. 6.52 8.74 9.02 6.60 8.51 9.2-12.5 25.98 10.52 13.28 8.07 14.16 12.40 12.84 7.71 7.62 7.67-7.73 7.71 8.66-9.2 7.96 5.90 7.50 7.64 5.93-5.95 9.49 6.7-7.1 6.7-7.6 7.0-8.3 7.6-8.8 7.7-9.0 9.65 5.8-6.3 12.86 10.49 10.10 6.83 10.40 7.60-7.62 7.8 7.84 113.23 12.2 9.54 7.60-7.65 12.51 7.62-7.66 7.64-7.71 8.92 8.91-9.58 8.61 7.70-7.72 7.31-7.36 11.35 15.27 10.0 13.29 12.19 7.82-7.90 7.25 8.54 8.34 8.32 7.64-7.70
Weight in Lbs. Per Cu. Ft. 49 65 68 49 64 69-94 195 79 100 60 106 93 96 58 57 57-58 58 65-69 60 44 56 57 44-45 71 50-53 51-57 52-62 57-66 57-67 72 44-47 96 79 76 51 78 57 58 59 849 92 71 57 94 57 57 57 73 67-72 61 58 55 55 85 114 75 100 91 59 54 64 62.4 62.3 57
The parameters given are approximate for estimating purposes only. The properties of the stored liquid should be determined analytically and used in the final design.
A-15
A.P.I. AND BAUME´ GRAVITY AND WEIGHT FACTORS The relation of Degrees Baume´ or A.P.I. to Specific Gravity is expressed by the following formulas: For liquids lighter than water: Degrees Baume´ = Degrees A.P.I. =
140
-130, G =
G
141.5 G
-130, G=
140 130 + Degrees Baume′
Formulas are based on the weight of 1 gallon (U.S.) of oil with a volume of 231 cubic inches at 60 degrees Fahrenheit in air at 760 m.m. pressure and 50% humidity. Assumed weight of 1 gallon of water at 60° Fahrenheit in air is 8.32828 pounds. To determine the resulting gravity by mixing oils of different gravities: D
141.5 131.5 + Degrees A.P I . .
145 G
, G=
md 1 + nd 2 m+n
D = Density or Specific Gravity of mixture m = Proportion of oil of d, density n = Proportion of oil of d, density d1 = Specific Gravity of m oil d2 = Specific Gravity of n oil
For liquids heavier than water: Degrees Baume´ = 145 -
=
145 145 − Degrees Baume′
G = Specific Gravity = ratio of the weight of a given volume of oil at 60° Fahrenheit to the weight of the same volume of water at 60° Fahrenheit.
PRESSURE EQUIVALENTS PRESSURE = 2.31 ft. water at 60 °F = 2.04 in. hg at 60 °F
1 lb. per sq. in.
1 ft. water at 60 °F
= 0.433 lb. per sq. in. = 0.884 in. hg at 60 °F
1 in. Hg at 60 °F
= 0.49 lb. per sq. in. = 1.13 ft. water at 60 ° F
lb. per sq. in. Absolute (psia)
= lb. per sq. in. gauge (psig) + 14.7
A-16
Gauge No.
U.S. Standard Gauge for Uncoated Hot and Cold Rolled Sheetsb
Galvanized Sheet Gauge for Hot-Dipped Zinc-Coated Sheets b
USA Steel Wire Gauge
Gauge No.
US Standard Gauge for Uncoated Hot and Cold Rolled Sheets b
Galvanized Sheet Gauge For HotDipped Zinc-Coated Sheets b
USA Steel Wire Gauge
7/0
-
-
.490
13
.0897
.0934
.092 a
6/0
-
-
.462 a
14
.0747
.0785
.080 .080
5/0
-
-
.430 a
15
.0673
.0710
.072 .072
-
-
.394
a
4/0
16
.0598
.0635
.062 a
3/0
-
-
.362 a
17
.0538
.0575
.054 .054
2/0
-
-
.331
18
.0478
.0516
.048 a
1/0
-
-
.306
19
.0418
.0456
.041
1
-
-
.283
20
.0359
.0396
.035 a
2
-
-
.262 a
21
.0329
.0366
-
3
.2391
-
.244 a
22
.0299
.0336
-
4
.2242
-
.225
23
.0269
.0306
-
5
.2092
-
.207
24
.0239
.0276
-
6
.1943
-
.192
25
.0209
.0247
-
7
.1793
-
.177
26
.0179
.0217
-
8
.1681
.1681
.162
27
.0164
.0202
-
9
.1495
.1532
.148
28
.0149
.0187
-
10
.1345
.1382
.135
29
-
.0172
-
11
.1196
.1233 .1233
.120 a
30
-
.0157
-
12
.1046
.1084 .1084
.106 a
a
a
Rounded value. The steel wire gauge has been taken from ASTM A510 “General Requirements for Wire Rods and Coarse Round Wire, Carbon Steel”. Sizes originally quoted IO 4 decimal equivalent places have been rounded to 3 decimal places in accordance with rounding procedures of ASTM “Recommended Practice” E29. b The equivalent thicknesses are for information only. The produce is commonly specified to decimal thickness, not to gauge number. a
A-17
