STRUCTURAL SWQGRTS FOR REF-SEER? VESSELS Thesis for fho Degree of M. S. MiCHiGAN STATE COLLEGE Zigurds Janis Mécheisans 3953 l‘ This is to certify that the thesis entitled Structural Support: for Refinery Vessels presented by Zigurds Michelson: has been accepted towards fulfillment of the requirements for ._l;_8°_ degree in _C_,_§_,___ Major professor + $ iii}. .3 I . I‘hQ'IE- ' {rt- '49‘ ,3 WWW fl STRUCTURAL SUPPORTS FOR REFINERY VESSBLS by 21 girds Janis Michelscns A THESIS Submitted to the School of Graduate Studies of llichian State college of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil maneering 1953 gums TABLE OF CONTENTS Introductory Statement . . . . . . . . . . . . Acknowledgements . . . . . . . ; . . . . . . . Part One: [general Discussion mad. 0 O O 0 O O O O O O O O O 0 loading Combinations. . . . waaelaseoeeoos 0 Horizontal Vessel Supports . Vertical Vessel Supports . . Anchor Bolts . . . . . . . . conclusion . . . . . .'. . . Part Two: Momth Qalxsis 93; g m S or 000000 900000 IntraduOtion O O O O O NOtat 1°11. Uaed . O O O O O O O O O O O O O Moment Sign Convention . . . . . . . . . . General Considerations . . . . . . . . . . Fixed End moments . . . . . . . . . . . Moment Distribution without Sidesway . . . Moment Distribution for Sidesway . . . . . values for the Infinite Series . . . . . . Discussion of Charts . . . . . . . . . . . numerical Examples . . . . . . . . . . . . con°1u.1°n O O O O O O O O O O O 0 O O 0 List of References . . . . . . . . O INTRODUCTORY STATEMENT The subject matter of this thesis was chosen in order to become acquainted with the applications of structural engineering to the petroleum industry and to determine some of the special problems encountered byé/struotural engineer in the desigi of a petroleum refinery. While the theory used in the design of structural parts of an oil refinery is not basically different from structural design in general, there are design requirements and loadingconditions that lave created some distinct types of structures and details best adapted to the needs of the chemical and petroleum industry. ‘ As there are no books published on the subject of structural supports for refinery vessels, all the information had to be obtained from the oil industry and from articles published in periodicals. Only a limited number of articles can be found in periodicals of the petroleum industry, obtainable from specialized libraries; because the subject is too specialized for discussion in general civil engineering magazines and also is not the ‘direct concern of chemical engineers. Thus, a major part of the effort in writing this discussion had to be spent in finding possible sources of information. After obtaining and evaluating the information, some insight in the requirements and problems of the oil industry has been @ined, conditioned, of course, by the fact that the writer has no actual experience in the industry. The first part of the thesis is a general discuss- ion of some aspects of the desigi of structural supports for vessels of a refinery. The second part is a moment analysis of a simple frame that supports a vertical vessel. ACKNOWIEDGEMENTS The assistance by numerous organizations in providing informative materials for this thesis is hereby gratefully acknowledged. Names of these com- panies, who have helped by sending pamphlets, reprints of articles, illustrative plans and suggestions, are listed alphabetically on the following page. A special acknowledgement is given to Dr. Richard H. J. Plan of Michigan State College under whose super- vision this thesis has been prepared, and to Dr. F. E. Wolosewick, an authority on structural design in the oil industry, whose desim charts have been the starting point for the mathematical analysis, using the Moment Distribution method, in the second part of this thesis. - ACKNOWIIEDGEMENTS American Petroleum Institute The Atlantic Refining Germany Continental Oil Company The Dow Chemical Company The H. K. Fergison Company Gulf Oil Corporation an: Publishing Company The M. H. Kellogg Company Leonard Refineries, Inc. The Lincoln Electric Company The Lummus Company - Midwest Refining Co. (Ire Be P018011) The Ohio Oil Company Richfield Oil Corporation Sargent a Lundy Engineers (Mr. F. I. 'olosewick) Sinclair Refining Company Skelly Oil Germany Standard Oil Company of California Standard Oil Company (Indiana) Standard Oil DevelOpment Company The Texas Germany Tide Water Associated Oil Company Union 011 Conpany universal Oil Products Company (Mr. Uebele) PART ONE QENERAL DISCUSSIQN Supports of vessels in refineries range from simple pedestals with a footing to huge open-type frameworks of skysnraper proportions, supporting numerous vessels, emipment and a mass of piping, as well as operating and inspection platforms, stairways and ladders. LOADS The most important part in the design of a structural support is the determination and evaluat- ion of loads and loadingcombinations to be supported. The following loads and forces met be considered: dead loads, live loads, impact, vibration, thermal forces, test loads, erection loads, wind and earth- quake. Dead loads include the weight of the structure, empty eeiipment and piping, and its fireproofing and insulation. For piping under one foot in diameter the structure is often designed by assuming a specified uniformly distributed loading, in order to simplify calculations. Points of support for larger pipes mist be predetermined and considered as concentrated loads including the weight of the pipe, the fittings, valves, insulation and the weight of the fluid. Live loads consist of movable loads, including personnel, portable machinery and equipment, and gravity forces caused by the fluid in the equipment and piping under normal operation. But what forces are created by the liquid, solid or viscous materials in the equipment must oftentimes be determined from observ- ations of the performance of similar equipment in the past. . Because provisions must be made for mechanical handling of materialsyeqiipment and parts, under operating and replacement conditions, the structural framing must support, in many cases, also elevators, trolleys, beams, and hoists in convenient locations. The frame mist be designed to resist not only the weight of these handling devices and the wei ghtcarried by them, but also the vertical and horizontal impact forces due to their operation. The disturbing forces produced by emipment Inving a tendency to vibrateland by surging fluids met be considered in the desim of the supports. Possible vibration caused by nearby vibrating equipment must also be investiated. Dcause many of the vessels operate at high temp- eratures, large thermal expansion forces are set up by the vessel and the piping, that must be absorbed by the structural frame. The tenmerature differential is minimized by proper insulation of the vessels, and piping or by facilitating heat transfer to the supports by continuous welds or heating coils in the supports. Nevertheless, therml expansion cannot be avoided and must be taken care of by arranging the supports to permit clearances for expansion of the vessels. To minimize the thrust of the expanding vessels and piping on the supports, sliding bearings and expansion Joints are provided. . _ Before putting the completed structure into operation, it is usually tested by filling the vessel “4353px“ with water. in most cases water is heavier than the operating fluid, therefore this is considered as a separate loading condition. Temporary loads and forces may be caused under erection conditions and met be considered in the desigi of the structural supports. Iind forces are computed assuming a uniformly distributed load on the vertical projection of the vessel, piping and framing. The magnitude of this uniform load is computed by considering the desigi wind velocity and the shape and height of the exposed surfaces. Commonly used wind design velocity is 100 M.P.H., giving a wind pressure of about 30 lbs. per square foot. Ihere necessary, an earthqiake force is considered. The most commonly assumed seismic force is 0.1 or 0,2 of the weight of the mass acting horizontally at the center of gravity. LOADING COMBINAT IONS After determination and evaluation of all these loads and forces, the structure has to be analyzed for the critical loading conditions. Ordinarily the loads to be considered in each loading condition are specif- ied in standards of the cranization designing the structure“. In general, the structure is investi'ted for the erection, testing and operating condition. The erection condition gives the most critical case for stability of the structure due to the over- turning moment of wind or earthquake, with the least stabilizing moment of the supports and the empty equipment. Under erection condition the specified minim stability factor is ordinarily less than under the operating condition. Also the allowable stresses may be increased by one third when uncluding wind or earthgake forces in the design; this, however, does not necessarily permit increase of the safe soil bear- ing values. The testing condition includes forces due to the weight of the completed structure and equipment plus the weight of the test fluid filling the vessel and pipes. Ho wind or earthquake force is included in this case or, at least, the forces are reduced in magnitude. A In analyzing the structure for the normal operatixg condition, full wind or earthquake force, whichever is pester, is considered, in addition to the weight of the completed structure and equipment, the operating weight of fluid, applicable live loads from platforms and walkways, thermal forces, vibration and impact. Only live loads from permanent storey are included when checking the stability aginst horizontal over- turning forces or when designing anchor bolts. VBSSEIB The vessels to be supported are of cylindrical form with elliptical, hemispherical or conical heads. The rounded outline of the vessels is necessitated by the fact that many of the vessels operate under high pressure, and the sphere and the cylinder, of course, are best suited as containers of pressure. In addi- tion to the internal pressure, the cylindrical shell of the vessel has to resist external forces at points where the vessel is attached to its supports and it Ins to act as a hollow beam in resisting the bending moment due to wind or other horizontal forces. In most cases, ladders, platforms, cranes or other equip- ment are supported from the vessel shell, ‘as well as internal brackets and beams supporting internal parts of the vessel. Complex stresses of highly indeter- ‘ minate nature are caused in the vessel shell at these points of support:6 The stress analysis of the vessel shell is made more difficult by the high temperatures at which many of the vessels have to Operate, requiring the vessel to be built of special material, becoming a problem partly in metallurg. Investigtions aimed at determining more exactly the stresses in the vessel shell due to the various factors and finding a practical desigi method are very much welcomed by the oil ind- ustry. The design and construction of the vessels is larply governed by the 'API-ASlm Code for the Desim, Construction, Inspection, and Repair of Unfired Pressure Vessels for Petroleum Liquids and Gases. that specifies the allowable stresses for different con- ditions. Different strength theories have been used for computing the required thickness of the shell. The Maximum Shear theory seems to be the most widely used; others are Maximum Stress theory, Maximum Strain theory, and the Modified Strain-Enery theory”. The Code specifies a minimum permissible shell thickness as a function of the diameter of the vessel. The required shell thickness is computed to resist the internal pressure, the weight of the vessel, its l .l..|| flat. 6.4.F1F.Il.eufilll. lr! ID x..- contents and superimposed loads, and bending moments. from wind or other forces” The bending stresses are computed by the flexure formula, considering the vessel a; a tzhin-walled hollowflbeam with a section modulus Eli ‘\ . Thepheight of thin-walled tall towers may be limited by its strength ayinst buckling, the maximum height being a function of the allowable compressive stress in the shell”. To the comuted required plate thickness a corrosion allowance (usually 1/8“) is added, to insure longer life. The inner parts of the vessel, being exposed to chemical erosion, may require the protection of weak- plates in addition to a corrosion allowance.‘?’2 Most vessels are of welded construction because high temperatures and pressure preclude the use of riveted construction, also a smooth inside is required.29 Welded construction permits highest efficiency of Joints and effects savings in materials and weight.18 when considering their supports, cylindrical vessels may be divided into vertical and horizontal vessels. Vertical pressure vessels vary in height and diameter. Also the ratio of height to diameter varies from a drum with proportions of a barrel to slender fractionating columns in excess of 160 feet in height and only about 10' diameter. 'na 6 II 0 III. P \l II mRIZONTAL VESSEL SUPPORTS _ Horizontal vessels usually rest on saddle supports (Fig. l). Welded saddle supports insure a more uniform distribution of bearing stresses in the vessel shell than other types of supports. The supports should be set at such a distance from the ends of the vessel, so that the stresses in the shell are the same at the center of the span as over the supports, or else the supports should be located near the ends where the head acts as a stiffener for, the cylindrical shell.21 when necessary, stiffening rings and internal struts may be installed in the vessel at thehports to take care of the bending stresses/3‘ . Usually only two piers are used for supporting horizontal vessels. Using separate footing for more than two piers, differential settlement my distribute the bearing stresses at the supports unequally. Also the vessel may be lifted off the middle support due to a possible greater expansion at the top of the vessel which is more exposed to sun 's heat and less cooled by the liquids contained in the vessel.21 Generally, a slide plate is provided at the free end of the vessel to take care of therml expansion. Rollers may be required for expansion of horizontal vessels operating under high temperatures. In that r?" e ‘tfi J2 t‘: T‘W ° '- c--v"‘.' ‘fi-‘TW - O * v - u- '77 ."T‘ a 0“. 36-4 I a ' ".7 I‘R’i‘ :c " R l e t 3W$t§;ekéfimw FIG. 4 FIG. 2. FIG. 3 v . ‘gfi‘ ‘r‘ o ‘ “—- l > a ... “". ' ~ ft egi'J-fii.,fi,. . ’ 90......zcg! ‘ Q, ‘ a -> I . . , [2. case the fixed-end pier may have to be designed to resist the total seismic force parallel to the longitud- inal axis of the vessel. VERTICAL VESSEL SUPPORTS Smaller vessels may be supported by legs welded directly to the vessel shell. This, however, intro- duces high localized stresses in the vessel shell at the supports. The same applies to vessels. supported by lugs bracketed to the shell atpthe lower portion of the r vessel. The strength ofthe shell at the support may be increased by circumferential stiffening rings. Most large vessels rest on an extension skirt of steel with a base ring (Fig. 2). Tapered skirts(Fig. S) are used where it is required to increase the diameter of the base ring to give larger leverage to the anchor belts in resisting the overturning moment due to wind or other forces. The simplest foundation for a vertical vessel is a pedestal, resting on a footing, to which the skirt base ring is fastened by bolts (Fig. 4). A commonly used shape of the pedestal and the footing is octagonal. The most customary arrangement of reinforcing steel in the footing is shown in Fig. 5. ‘By necessity, the steel must be placed in four planes, requiring a flqfl/ A6 5\ \ FIG ‘7 13 greater footing depth. Another disadvantage of this steel .at'tangement is the unnecessary}; close grid underneath the pedestal, making placing of. concrete difficult. An alternate arrangement is obtained using bent bars radiating from the center (Fig. 6). These bars can be placed in one plane, thus providing a greater effective footing depth.“ A too large footing for high towers may be avoided by mintaining stability by means of gay wires, fastened to the vessel, instead of by a self supporting found- ation. Ordinarily, the vessel ms to be held a certain distance above grade to accomodate piping below the vessel. The length of the skirt is increased accord- ingly. Holes have to be provided in the skirt for piping and additional opening my be necessary for servicing and ventilation. A structural steel or reinforced concrete frame- work is used if it is required to supportthe vessel at a higher elevation with more space available under- neath the vessel. Such a frame in its simplest form, a table top support, is shown in Fig. '7. This frame has been chosen for a Moment Distribution analysis in the second part of this thesis. This support is soon- omical for various sizes of vessel diameters and heights of support up to 30 feet. For supporting larger vessels, [4 additional columns and horizontal bracing may have to be added to avoid a bulky appearance of the support. Banding in the colums due to wind shears may be de- creased by struts to adjacent structures or by giying the vessels. ‘ . When additional equipment and platforms are necessary that cannot be supported by a simple frame or from the vessel itself, the vessel is enclosed by a malty-level framework on which the vessel, the servicing and inspection platforms, auxiliary equip- ment, and piping are supported. Due (to the high fire hazard in the oil industry, all main structural members are enclosed by concrete. Gunite is commonly used for fireproofing structural steel members. The most outstanding example of a refinery struct- ure, because of its large size and complicated desigi, is the intricate framework of a catalytic cracker, supporting numerous large vessels, piping and other equipment. The carefully calculated lay-out, determ- ined by process requirements, results in a compact structure that permits replacements with minimum loss of production time. The required structural framework is an irregular structure with many strength consider- ations subordinated to the needs of the chemical process. The structure has to behnalysed separately for various loading conditions. has [5 The catalytic cracking structure may be about 20 stories high, narrow and with its heaviest load - the re generator close to the top. This presents special structural design problems. The structural frame being an open type structure, it has to withstand wind pressures without the aid of masonry walls or floor slabs.’ Cross or 'K' bracing cannot always be used to resist wind moments because it interferes with the piping, expansion loops and the operating platforms. In such cases knee braces (Fig. 8) are desigied8 to give the needed stiffening to the framework to resist the effects of sidethrust due to pipe stresses, wind, and equipment therml expansion or contraction forces. Sometimes also special wind trusses may be required underneath the girders. Moment resistant beam-to-colum con- neetiont28 may be undesirable because requiring large columns to resist bending, when no advantage can be taken of continuity due to irregular framing of the beam. This discontinuity of the framework is caused by the my large opening required for equipment and piping, often demanding the use of diagonal beams. Great care has to be taken in leveling the main girders supporting the larg vessels, in order to insure a uniform bearing surface for the skirt base ring. Special bearing plates may be fitted to the tops of FIG-9 I6 girders and planed level-for skirt bearing to avoid unequal bearing stresses. In desigxing the girders, the vessel lead is often assumed to be uniformly dis- tributed over the length of the girder. This, how- ever, is not in accordance with the actual conditions because when the vessel is set onto the supporting girders, it usually rests on shims placed between the vessel skirt plate and the girder. Ihen later grout of the unexpanding type is poured underneath the skirt base ring between the shimmingpoints, this still does not produce a continuous bearing on the girder because of shrinkage of the concrete. To insure eqial reactions at all points of support, the supporting girders must be designed for equal deflections as the loading points. AAcommon type of girder framing used to support a vertical vessel at 8 points with opening for the head of vessel is shown in Fig. 9. The short diapnal beams should have the same stiffness as the min girders, otherwise each point of support will not take its proportionate share of the total load. On the other hand, in case of a reinforced concrete support where the whole top of the support is poured monolitically, the assumption of concentrated loads may have less meaning and a uniform load distribution may be the best assumption. In case of hot vessels resting on the supporting frame directly through lugs or base ring angles that are integral with the hot shell, radial expansion forces 17 from the vessel are transferred to the supporting horizontal bent. .. Before the base of the hotmtower is able to expand, it has to overcome the frictional resistance of the support. In case of friction between the base plate of the tower and a smooth concrete sur- face of the support, the radial expansion force, trans- ferred to the supporting frame, will be about 20% of the supported weight at each point of support. When using fitted sole plates, the coefficient of friction may be reduced to 13-15%. Still the thermal expansion forces may be high, causing ring tension, in case of an integrally poured concrete top of the support. in critical cases special low friction bearing plates are sandwiched in between, permitting radial expansion of the vessel on these sliding bearings. A bearing plate made of bronze with trepanned holes filled with graphite sticks and both faces greased with a graphite lubricant reduces the coefficient of friction to fin-"7%.5 In case of a large vessel, whose stability is not overcome by wind or other horizontal forces, rollers in boxes anchored to the concrete foundation ring have been used to permit radial expansion but not side slippage of the vessel..22 In general, there are two methods of reducing the destructive force of the thermal expansion and con- traction, either by designing rigid piping that displaces FIG. IO ~: '9 92. Wr: ---~~ [8 the mJor equipment (on sliding bearings) through the full amount of the expansion, or by keeping the vessels stationary and absorbing the expansions by bendsor expansion Joints in the interconnecting piping)“ Differential thermal expansion may also be decreased by connecting various process equipment together (Fig.10). Desigung expansion connections for a refinery is a large field of enterprise for research investigiions. ANCHOR $12.15 The providing of adequate bolts at the base of the tower,anchored to the foundation to give stability to the vessel aainst overturning moments,is an important part of the structural design. The anchor bolts are located on a belt circle in the base plate outside the skirt of the vessel, pro- viding at least 3-3;“ clearance from bolt circle to skirt. Ordinarily, at least 8 bolts or more are used for larger vessels to give a more even distribution of stresses along the circumference of the vessel and to minimise the danger in case of a loose bolt. The anchor bolts should be developed to their full tensile strength by embedment into the foundation. This requirement limits the practical size of the bolts to 3" diameter, usually, though, below 24-“ diameter. The bolt stresses are transferred to the tower by lugs TOPOF ,PEDEETAL '.o -¢§ oLee e we. Her "9’ I I .. ' n ‘ i /9 welded to the skirt of the vessel. Thelength of the lugs should be long enough to allow space for sufficient weld of the required strength. It is safer to design the lugs for only one weld for every lug because the welds may be difficult to make on- the inner faces of the lugs.6 For larger bolts two nuts are used. Also an initial tension is introduced into the bolts by tightening them after erection of the tower. The ends of the anchor bolts, protruding from the foundation, may be bent or their threads dama ged when setting the tower in place, therefore a sleeve nut is sometimes attach to the end ofthe bolt below the‘top of the concrete (Pig. ll), into which a stud bolt is inserted from the top through the lugs of. the tower.2 The force resisted~ by each bolt is proportional to its distance from the neutral axis of the base circle. The neutral axis is, usually, assumed to be located at the center of the bolt circle and bolt stresses com- puted by the flexurs formula, using the section modulus of the whole bolt coup. Using another procedure, the stresses per foot of circumference of the skirt are computed and the bolts desigzed to take care of tensile forces within their segnent of the base ring.2 Although the assumption that the rotation occurs about the center line of the base ring is not exactly correct, it is convenient from desigi sallpoint and it is practical to design the anchor bolts for most towers by thivs .ll H I s . s . a. V I c '(‘11‘" in ‘ . .' .3 a. -0 id. 0" . A . . V4 . . I Greeli’t Q 0.1, . . .- ‘l - v....a.. . . .. . fiery... -. V I n,e \ 0'. ‘ I .. . ‘.o-‘ . -uw1t '. . J ewW'r- ?-I'L}W\‘u (I 9 20 method with a minimum of effort. When designing tall towers, the use of a method, based on the actual location of the neutral axis, my be Justified by resulting in a base design with smaller anchor bolts, lugs, and base plate. A theoretical approach to the problem is to consider the section between the base ring and the concrete pedestal as a hollow cylindrical reinforced concrete cantilever beam of balanced desigi, with an axial load equal to the weight of the tower and an external bending moment equal to the overturning moment.“ After finding the neutral axis of the base section; by the transformed area method (Fig. 12), the tensile stresses in the bolts and the concrete bearing stress may be calculated in accordance with the theory of flexurs. This, of course, requires the base plate to be stiffened, in order that it may be considered rigid, with negligible deflection. In this analysis the usual initial tension in the bolts may be neglected. Tokind the most economical arrange- ment of anchor bolts, it may be necessary to repeat these calculations several times, trying out different number and sizes of anchor bolts and some variation of the bolt circle diameter. For the purpose of finding the minimum anchor bolt area that is consistent with a given base ring area and a given working stress in . steel, nomograms have been devised that produce the solution without the need of numerous cumbersome trials.15 2! CONCLUS ION This summary of applications of structural engineer- ing to the refinery industry should give a general _ survey to an engineer outside the industry, about some of the problems encountered in the structural design of a refinery. A more detailed information on this subject may be obtained from articles listed as refer- ences at the end of this thesis. This list is believed -t-e-to include the under part of articles on the struct- ural design of refineries, published in american periodicals within the last ten years preceding 1952. PART T'O MOMENT ANALYSIS A 8 MP E 8 P RT F1617 22 INTRODUC T ION Dr. F. l. 'olosewick has prepared charts for the functions of bending moments in the beams, columns, and foundation girders, as well as vertical reactions, end shears in girders, and unit soil pressures for the simple frame of the type shown in Fig. '7, and has found tm functions to haves straight-line variation.” Following his examle, end-moments of the members of such a frame are analysed here by moment distribution, extending the study by taking into consideration the effects of varying amount of end-restraint of the columns, varying loading conditions, haunching of the members and side-sway of the frame. First, expressions are derived for the end moments in the members, then, values for the end-moments are computed for different ratios of beam stiffness to column stiffness and various loading conditions, and, finally, the functions are plotted in charts, for easier interpretation. A,B,C,D,E,F c 0c: z r re! d a r m r+0.'75 F H h 1: er 3- a kc 1 L l “AB, MBA, etc. P Q k 1-! if 25 N O TAT IONS USED corner points on the centerline of the: members of the bent _' Carry-lover factor for the beam Carry-over‘factor for the column from the column-beam Joint to the colum end at the footing Moment distribution factor for the beam at the Joint with a column, assumed fixed at the footing Moment distri‘mtion factor for the beam at the Joint with a bolumn, assuming columns pin-connected Fixed end moment Horizontal thrust, applied at top of bent Height of bent, from top of footing to centerline of beam Relative stiffness factor of the beam Relative stiffness factor of the colum Length of beam span, center to center of colums Applied moment at top of frame due to the horizontal force I End moments in members of a bent caused by vertical loads without sidesway Concentrated vertical load on beam concentric Wei ght supported by frame Ratio of beam stiffness factor to column stiffness factor; also relative stiffness factor of the beam, with kc equal to one GL4 l-(l-c)d-(c-c2)d2-(cz-c3)-. . . -(cn-cn’1)dn‘1-. . . 2 2 s 4 4 4 5 aldal-cafld2nel. l-deczd -c‘d sc d -c d +...+c Cd ~cd2¢c.d3-c3d‘+. . . ‘caeld2n+1_c&eld2ne2. . 2 3 4 5 t- e + do - <§> we) -...-<-13‘I%)?.. Horizontal force applied on vessel with moment arm a, causing a moment M about the top of frame Coefficient for the fixed end moments in the beam, caused by the loads due to q Coefficient for the fixed end moment at the left end of the beam, caused by the loads due to M. Coefficient for the fixed end moment at the right end of the beam, caused by the loads due to M. 25 MOMENT SIGN CONVENTION _ The Moment Distribution sign convention is used in the computations, adapting the following criterion in determining the algebraic signs of the resisting moments in the ends of the structural members acting on the Joint: End moments of the member are positive when they act in a clockwise direction on the Joint, and negative when they act in the opposite direction. This sign convention gives positive fixed end moments at the left end of a beam with.downward loading and negative moments at the right end. ‘ Sidesway causes positive fixed end moments when the line Joining the ends of the member rotates in a clockwise direction during relative lateral displacement and negtive - otherwise. 26 GENERAL CONS IDERAT IONS The tableetop supports may be either of reinforced concrete or structural steel. Ihen making the frame. of structural steel, it is also encased in concrete, for fireproofing purposes, developing end-restraints at the Joints. The structure will be treated as a rigid frame, using centerline distances of the members for the Moment Distribution analysis. Only frames with equal length of beams, giving a square outline of the frame in top view, will be con- sidered in this analysis, because these table-top supports will be designed to carry only one vessels at the center. Because the vessel load will be assumed to act symmetrically about its center, and the support-will be desi med for horizontal forces from either direction, all vertical at W bouts of the frame will be alike and symmetrical about their vertical centerlines. The frame is analyzed as a plane structure. This is much simpler than a three-dimensional analysis; besides, analysis of a rectangilar frame as a three- dimensional structure gives» values for end-moments that differ only by a few per cent,in most cases, from 35 This is because those obtained by plane analysis. a member, framing into a Joint perpendicular to the plane of the bending moment, does not add much to the stiffness of the Joint because the torsional stiffness 27 of structural members as much smaller thantheir bending stiffness. This small difference in results Justifies the analysis of the frame as divided into two-dimensional bents. Only the vertical (bents mm and mo (Fig. 19) are considered, the thermal or other forces acting on the horizontal bent Am at the top of the support being neglected for purposes of this investiption. The frame is acted upon by forces cue to the weight of the vessel q, and due to a horizontal force I from wind, earthquake, or other causes, that creates a bending moment I at the top of the support (Fig. 13). The bent DAB: supports vertical loads due to. Q and II that may either be considered as concentrated at several points or as uniformly distributed along the length of the beam A3. The bent FRAD resists a horizontal thrust Elli/2, in addition 'te vertical forces, A typical bent DAR: on the leeward side of the frame, supporting the largest downward leads, will be analysed for end moments due to vertical loads without sidesway. Two kinds of vertical loading on the horizontal beam A) will be considered: a uniformly distributed load along the entire length I. of the beam, or con- centrated loads from bolts at 8 points, with support top framing as shown in Fig. 9. For computing loads on the beams due to M, the simplified assumption of rotation about the centerline of the vessel will be made. 28 FIXED END MOMENTS The familiar formulae: 2 nus : E?- 22.3 MFBA : — L for a concentrated load, and MFAD = - llFBA s ‘%3 for uniform load (Fig. 1e) are used to cormute the fixed end moments for beams of uniform moment of inertia along their entire length. The computations give a result that may be expressed in general terms as RF 3 XPI. where the value of x is determined by the location of the load 1?. As the vertical loads, in the present analysis, are a combination of effects from Q and M, the fixed end moments will be expressed in the form MFAB 8 XLQ e Yll, where x and Y are functions both of the location of the load P and of its expression in terms of Q or M. The weight of the tower Q is assumed to be distributed equally to all pointsof loading which are symmetrically located, and therefore gives equal fixed end moments in the beams at both ends. A general expression for the fixed end moment at the ri git side of the beam is use = - (no, '4» zu). When assuming a uniformly distributed load on the beam, the load per unit length will be taken as _q_+4n "u. «m.2 ,.293L . cm. me. 9 R Pa .‘zealizov Famine ’ A T 5 L L. of ' .Lc FIGJ5 29 where the second part of the expression is the maximum stress in a unit segment of the skirt shell, computed by the flexurs formula using theapproximate expression for s section modulus of a thin cylindrical-shell LE3 , transferred to a unit length of the bolt circle with a diameter 1.. Computed fixed end moments in the beam A3 for uniformly distributed load with wind direction perpend- icular to the beam: - .1. une- nms '431'4‘g'1?‘ SI“ 5F” xs-fia- Is For wind direction perpendicular to the beam AB, and framing arrangement of Fig. 9 with 8 bolts, the beam AB carries loads located as shown in lig. ls. Computations by the flexurs formla, using L2 as moment of inertia for the group of 8 equally spaced bolts about the centerline of its bolt circle, gives the following values for the vertical loads: P2 s ‘8’ + gr - O 767 _ P1 - P5 2 £3 4- ML P1 and P3 is half of the load carried by the short diagonal beams, transferred to the beam AB. Superposition of the effects of the three loads gives the following expression for fixed end moments of R P. - .2951. .207 .201 .293 . I a A I FIGJ9 . OJ: FIGJ5 30 the beam A3 for 8-point poading on the frame at perpend-e icular wind: . 1 RFAB a -MFBA : 0.00818198LQ +0.028382’63M Fixed end moments due to another 8-point loading arrangement, used in the practice (Fig. 16), were investiated and found to be smaller than in the other 8-point leading case, thereilore fixed-end moment values aswabove will be used as representing the more signific- ant concentrated loading. Assuming wind from a diagnal direction (Fig. 1‘7), with rotation about the diagonal, the following mamitude of the concentrated loads (Fig. 15) has been computed: P1 ,, 9% 1,2 . g4, 0.3535! P3 : f3» 4- If . The fixed end moments in the beams caused by these leads due to 8-point loading on the frame at diagonal wind are as follows; [FAB 0.02857194Lq, + 0.059362! NEDA : -(O. 02857194LQ + 0.0808141!) X : 0.02857194 Y 3 0.059362 2 8 0.0808014 A wind parallel to the plane of the bent FEAD (Fig. 19) causes the following fixed end moments in the beams due to concentrated loads; 3" 2h 0.2L 0.6L ' 925, L. HAUNCHED BEAM kes‘ “an“ 7.8! Cos ' can '065'9 FIG. l8 3! j5 - 0.028.571.9414 - 0.015158171! FFAB : -(o.028571941.q + 0.0151581711) For the haunched beam shown in Fig. 18, the fixed end moments due to a uniformly distributed load along its entire length, neglecting the effect of the haunch loads, area": ‘ me =- - 1mm - 0.0993sz a 0.0993 g + ah : 2 0.126411 + 0.024825LQ Wind is assumed to be perpendicular to the beam. MOMENT DISTRIEITION OF FIXED END MOMENTS CAUSED BY VERTICAL LOADS ON THE M WITHOUT SIDESWAY OF THE MT A general case of moment distribution has been carried out on the following sheet for a symmetrical beam. The beam is supported on columns, fixed at the footing, forming a bent symmetrical about the midspan of the beam. “(1" is the distribution factor for the beam at the beam-column Joint. The distribution factor for the column is (l - d). ."d', in terms of the ratio 'r" r of the beam stiffness to column stiffness, is d I -—- ; rel 1' - ke/ke 'c' is the carry-over factor for the beans. ”cc“ is the carry-over factor for the column, from the beam-column Joint to the base of the column. Because no moments are carried over from the fixed end of the colum at the footing to the beam- colum Joint, the moment distribution procedure is shown only for the beam end-moments. The final moment in the column at the beam-colunm Joint is of the same magnitude as the end-moment in the beam at the same Joint. Rcause there are no initial fixed end moments in the column, the final end-moment in the column at the footing is equal in magiitude to the moment at the upper end, multiplied by ca. 32 35 Moment distribution of fixed end moments of a beam caused by vertical loads on the beam without sidesway of the bent, carried out in general terms. A B 4- O -> ‘U F.E.MLL +XLQ +YM -XI.Q .211! J. _m____-d -am M +chLQ. +chM -chLQ ~chM -cd 29,9,- -cd‘ZM gog‘gg, god‘YM 4c:d XLQ+c¢d YM ac‘d‘ XLQ-c‘d‘zn a -c d’ QQ- -c“d‘YM go'd" Qgtc c‘d’ZM d' u ec’d XLQec’d’ZM -c d XLQ-c d’YM * oc’ gag-e’d‘zm +9 “$31,943; 3 A etc. etc. D l C Moment distribution results in the following end-moments: 11‘;XLQLi-Jlnc)d--(c-ca )d' -(c"-c’)d' -, ..-(c"- o“)"d."f'. + +21![1-d+c”d"-¢"d’ ec‘dI-cud +...+c ”ah-332%.. + “Efficducdz +c “d -c ”Mendez?" e‘h’fiiJ Designating the infinite series by 81,Sz,and S , respectively, gives final end-moments 3 in an abbreviated form: a“? Xquslj+ mfiaz‘ja- zufsa] Mme: -{xLQ [$131+ HESS] + zu [32]} M A=-MAB Mec=-MBA MDABCcMAD': --i_'-.c OMAB MCB = “’ch 54 It can be seen that the moment distrihition results in end-moment expressions that contain the infinite series 81, 82, and 83. If the loading is symmetrical, the fixed end moments in the beam are equal ('1 a Z). Their moment distribution gives a simplified expression for the final moments; “AB = dim 8 (ZLQ + Yll)81 , 8 plus 83 being equal to 8 2 1' For a beam with a uniform moment of inertial c is v} and the infinite series are simplified as follows; . 2 3 o G d d d d s - 1 - C - . -...- — -... 1 3 22 '23 '21 2n - e3 43 d‘ d5 ' ‘ 42h dad) ‘ 32'1"!*?"?.I'2T';T""*;E'Th” 3 : -- + - +1~1+ + - + 3 ‘5 5 2:5 25 2 2 ‘7' 2 “ 4(5‘“ 7* The same formulas can be used also to find final end-moments in a bent with colums hinged at the bottom, only d in the series 81, 82, and 83 has to be substituted by a modified distribution factor dm. For a bent with members having a uniform moment of inertia dll is equal to Efiv , the relativ being modified multiflying by 3/4. The end moments at the bottom of the column are, of course, equal to zero. e stiffness factor of the colum 35 MOMENT DISTRIBUTION FOB SIDES‘AY If the bent is not restrained from swaying in a lateral direction, horizontal forces and unsymmetrical vertical leading cause lateral translation of the Joints, inducing end-moments, in addition to those caused by the vertical loads. In case of a symmetrical bent with members having a uniform moment of inertia, the fixed end moments caused by the sidesway are equal in mamitude at all. Joints. These fixed end moments have been denoted by F and a moment distribution carried out on a following page, for a bent with fixed-end columns. The moment distribution gives end moments expressed in a formula containing the infinite series 84. The magnitude of F can be computed from an equation of equilibrium satisfying the condition that the sum of the sidesway end-moments resists the horizontal shears caused by the horizontal forces and the asymmetry of the vertical forces. To find an algebraic expression for the end- moments in the members, with consideration of the effects of the sidesway, it would be necessary to express F in terms of the horizontal thrust eff?» height of the bent h. In order to keep the expressions for end momentslin the same simpler terms as the previous end meet rpm». cw he- mew 4....) Home the formula for the sidesway-moments can be used to. add, . the effects of "sidesway to numerical values (of moments computed from the previously, developed end-moment . formlas or their graphical expression in form of charts, when applying equations of equilibrium to the bent. 37 Moment distribution for sidesway moments for a bent with members of uniform moment of inertia, carried out in a general form. ' 0 ates :5 "1 Ft. eoeohof + 0 e '7 e :. :1 .... O u o O'clk. .3... 0 'I' F D Moment distribution results in the following final end-moments: MAE-HEAaFE-d Angel .5 a g at... a =F|'_I.3(a/2).z(d/2) -3(d/2)+ ...+(-1)'3(d/2)" -...]= = 3r -a/2.(d/2)‘ -(d/2)’ + .. .+(-1)" (d/2) e. . .] MFE-Mm=F{l-§-(lod)+d/2'(l-d)-d /2 (1 -d)+d”/2‘(1-d)-... ...-d ”/2 (l-d)+ d "/2 (1-d)-...}.. -r{1+(1-d)(-e+d/2 - a “/2 , d /2 -...-d"‘/2”‘ ed” 2"“3...}= =F{-§--3d/2 -34 /2 -34 /2 -...-3d /2 -3d /2 -...}.. 8F {4.3/25/244/2) +(d/2)’ -...-(-1)" (e/z)" -...]} In abbreviated form: new ma ar [4.] mm a MD,- fled/2&1} 38 VALUES FOR THE INFINITE SERIES A systemtic computation of approximate values of the four infinite series is given in the tables #1, #2, #3, #4, #3, for r values from 1 to 10. The computations have been carried out to the seventh decimal place and to the d8 member of the series. Conputations of the series for bents with uniform moment of inertia beams with c equal to i are given in tables #1, #2, #3, and #4. Table #3 contains values for 81 for a haunched beam with c equal to 0.659. In compiling values for 81 with c equal to f, it was realized that the exact value of 8 is 37%. , for l bents with fixed end columns. A proof of this is given on the next sheet by expanding the function .35. into re a liaclaurin's series, resulting in 81 . .. d d " 351' 1' ' r21: rez 1' 2:d f(d) : .3533 f(O) ma) : {5:32-72 1"(0) " = “'4‘ f"(0) 1(4) (“)3 ma) : 43—; mm (2-4) f~(d) a 45-8—5- 17%” (2-4) r'm) a ""9'6"24 1"(0) (2—4) ma) .4322, r"- -- ,- .- e . . - -. r ‘ s q . \ o . . a '- _,_,,. .. - -... -..--. g . . . \._l '< . l f ' ' l ’ ‘ . . , . . — , on _ — m __ *.. “ -,- ._ 7,- _ M C v .1 ’ L - . - << 5 ‘ - —o~—o-n. 10 :9 g-n- £9» to so so see: so :9- .5 .5714285 .6666667 .7272727 .75 .8 .8 .8421052 .8335333 .8695652 .8571428 .8888889 .8750000 .9052258 .8888889 .9142857 .9 .9230769 .9090909 .9302325 (12 .25 .3265305 .4444445 .5289256 .5625 .64 .64 .7091412 .6944444 .7561456 .7546937 .7901235 .7656250 .8158168 .7901235 .8559183 .81 .8520710 .8264463 .8655325 d .125 .188: .296: .3841 .421! .512 . 512 . 5971 .5781 .657! .6291 .7022 .6699 .7368 .7023 .7642 .729 .7865 .7513 .8049 TABEB 1 40 -v - v ‘0 ' ~ . . - c J . r b. e O I .— .4- —-o~. - fl - p D u d- . ‘ I ". -o , 1 ‘ - 2 '- s- ‘ - - ’7 - do. - . a- - . H . A ~— ' . - H' Any-*x e... _ ‘ s . . . ‘ g - - \ —- ~ . ‘ .- - - -- y c . ol- no . s . - a. ' . I . — O . .‘ _ ' *- ~“M-v-- _ w I ‘ \ - e I —- _ 1 .fl “-5- ..-.--so - I I - ‘ - '. r I an. ' , - n “. u-oo. I-‘H h, .- e . . 10 .5 .5714285 .6666667 .7272727 .75 .8 .8 .8421052 .8535355 .8695652 .8571428 .8888889 .8750000 .9032258 .8888889 .9142857 .9 .9250769 .9090909 .9502525 d2 .25 .5265305 .4444445 .5289256 .5625 .64 .64 .7091412 .6944444 .7561436 .7546957 .7901255 .7656250 .8158168 .7901235 .8559185 .81 .8520710 .8264465 .8655525 d5 .125 .1865888 .2962965 .3846751 .421875 .512 .512 .5971715 .5787057 .6575162 .6297374 .7025520 .6699219 .7568668 .7025320 .7642681 .729 .7865271 .7513148 .8049604 TABLE #1 5 d4 d .0625 .03125 .1066222 .0609269 .1975509 .1316873 .2797622 .2034634 .3164063 .2373047 .4096 .32768 .4096 .32768 .5028812 .4234789 .4822531 .4018776 .5717532 .4971767 .5397749 .4626642 .6242951 .5549290 .5861817 .5129090 .6655571 .6011483 .6242951 .5549290 .6987594 .6388657 .6561 .59049 .7260250 .6701769 .6830135 .6204214 .7488003 .6965584 4() .010625 .054815% .0877915 .1479754 .1779785 .262144 .262144 .3566158 .5548980 .4523276 .5965695 .4952702 .4487954 .5429727 .4932702 .5841058 .531441 .6186248 .5644740 .6479615 d7 .0055125 .0198945 .0585277 .1076170 .1534859 .2097152 .2097152 .3003063 .2790816 .5759370 .3399165 .4584624 .4487954 .4904270 .4384624 .5540596 .4782969 .5710583 .5151582 .6027547 d8 .0026563 .0115685 .0390185 .0782691 .1001129 .1677722 .1677722 .2528895 .2325680 .5269017 .2913570 .3897444 .5926960 .4429663 .3897444 .4876888 .4504672 .5271125 .4665074 .5607020 TABLE #2 COMPUMTN’NSL FOR 8‘ 4] 808 1 4 2 5 4 5 6 7 a r g’ 9" is 1" 95 L 9'7 16 2 2 2 2 2 2 26 2 2 1 d .25 .0625 .015625 .0039063 .0009766 .0001660 .0000415 .0000104 dm .2857145 .0816326 .0233236 .0066639 .0019040 .0005440 .0001554 .0000444 2 d .5555553 .1111111 .0570570 .0125457 .0041152 .0015717 .0004572 .0001524 dm .5636564 .1522514 .0460841 .0174851 .0063582 .0025121 .0008408 .0005057 5 a .375 .140625 .0527544 .0197754 .0074158 .0027809 .0010428 .0005911 dm .4 .16 .064 .0256 .01024 .004096 .0016384 .0006554 4 d .4 .16 .064 .0256 .01024 .004096 .0016384 .0006554 dm .4210526 .1772853 .0746464 .0314501 .0132557 .0055721 .0023461 .0009878 5 d .4166667 .1736111 .0723380 .0301408 .0125567 .0052328 .0021803 .0009085 dm .4547826 .1890359 .0821895 .0557545 .0155368 .0067551 .0029570 .0012770 6 d .4265714 .1836734 .0767172 .0337360 .0144583 .0061964 .0026556 .0011381 dm .4444444 .1975309 .0877915 .0390184 .0175415 0077073 .0054255 .0015224 7 d .4575000 .1914063 .0837402 .0366364 .0160284 . .0070124 .0030679 .0015422 dm .4516129 .2059542 .0921084 .0415975 .0167859 .0084839 .0038315 .0017303 8 d .4444444 .1975509 .0877915 .0390184 .0175415 .0077075 .0054255 .0015224 dm .4571429 .2089796 .0955555 .045725 .0199646 .0041722 .0041722 .0019050 9 d .45 .2025 .091125 .0410063 .0184528 .0083038 .0037369 .0016815 dm .4615585 .2130178 .0983159 .0453766 .0209450 .0096660 .0044612 .0020590 10 d .4545455 .2066116 .0959144 .0426883 .0194055 .0088199 .0040090 .0018223 dm .4651163 .2163331 .1006201 .0468000 .0217675 .0101244 .0047092 .0021902 -Q o: co m. o: no *4 (I) 10 .5 .6666667 .75 .8 .8335555 .8571428 .8750000 .8888889 .9 .9090909 1:"; 2 .125 .2222222 .28125 .52 .3472222 .5673469 .5828125 .5950618 .405 .4132232 TABLE #5 42 COMPUTATIONS FOR 82 & 83 d5 .2 2 .05125 .0740741 .1054688 .128 .1446759 .1574546 .1674805 .1755850 .18225 .1878287 9.; 2 .0078125 .0246914 .0595508 .0512 .0602816 .0674719 .0752727 .0780369 .0820125 .0855767 d5 ._Z 2 .0019551 .0082305 .0148515 .02048 .0251174 .0289165 .0520568 .0546851 .0569056 .0588076 d6 ‘3; 2 .0005520 .0027455 .0055618 .008192 .0104656 .0125928 .0140249 . 01 541 47 .0166075 .0176598 d7 35 .0000830 .0009145 .0020857 .0052768 .0045607 .0055112 .0081559 .0068510 LOO74754 .0080181 4:; 2 .0000208 .0005048 .0007821 .0015107 .0018169 .0022762 .0026844 .0030449 .0033630 .0036446 43 TABLE-#4 FROM TABLE #5 FROM COMPUTED ACTUAL FROM TABLE #2 TABLE #2 (b) ( ) (b)-(C) 01) (e) l+(°)-(d) (b)-(e) 2 c O (a) 12(1) “'2 2114-1 2n+2 S4 dzm'l Z-—-—-d2n+~ 82 r d S 8 221-01, m 2 3 Z 31-]- l 1 2 2 .- 1 d .3332258 .6667742 2/5 :.6666667 .2666431 .0665827 .2000604 .5332861 .1551555 .5332966 .1334778 dm .5999822 .6000178 .5110975 .0888849 .2222124 . 2 d .4999236 .5000764 1/2 =.5 .5749427 .1249809 .2499618 .7498858 .2499619 .5750951 .1249808 dm .5712538 .4287462 .4189195 .1523545 .2665852 5 d .5997654 .4002346 2/5 :.4 .4361950 .1635724 .2726206 .8723860 .5271447 .2911864 .1090483 dm .6662298 .5557702 .4758784 .1905514 .2855270 4 d .6662298 .5557702 1/5 =.5555555 .4758784 .1905514 .2855270 .9517568 .5807027 .2385946 .0951757 dm .7265541 .2755449 .5112788 .2152755 .2960035 5 d .7136369 .2863631 2/7 :.2857155 .5057457 .2098952 .2958505 1.0074873 .4197863 .2024059 .0839574 dm .7682484 .2317416 .5554459 .2328025 .3026434 6 d .7491464 .2508536 1/4 :.25 .5244025 .2247439 .2996586 1.0488051 .4494878 .1759388 .0749147 dm .7987819 .2012181 .5550029 .2457790 .3072259 7 d .7767558 .2232662 2/9 :.2222222 .5403365 .2363973 .5059592 1.0806732 .4727945 .1557241 .0675420 dm .8221044 .1778956 .5663387 .2557657 -3105730 8 d .7987819 .2012181 1/5 :.2 .5550029 .2457790 .5072259 1.1060060 .4915583 .1597750 .0614446 dm .8404970 .1595050 .5769132 .2636838 .3131294 9 d .8168063 .1831937 2/11:.1818182 .5633147 .2534916 .3098231 1.1266290 .5069830 .1268626 .0564317 dm .8553780 .1446220 .5852586 .2701194 .5151592 ;. 10 6 .8318148 .1681852 1/6 =.1666667 .5718727 .2599421 .3119306 1.1457455 .5198843 .1161968 .0519864 dm .8676608 .1525592 .5922151 .2754477 .5167654 C) ) M O 44 COMPUTATIOHS FOR IIAUNCmD EEAI‘A ,. ’1 L; 0 fl - C f- I ‘I . a ‘rr .2 (I ’4 r CO .659 .4342810 .2861912 .1885000 .1243874 .0819054 . - p ,5 n2 .5 m3 ,* é .3 .P ”6 .341 32247190 31480898 .0975912 .0543126 .7425Q29 Lur-C) Lc-C)d2 (62-671165 65-6%“ (c4-c5)a5 (65-66m6 l .1705 .0551798 .0185112 .0060995 .0020098 .0004504 2 .2273353 .0998751 .0458785 .0192775 .0084692 .0037208 5 .25575 .1264044 .0624754 .0308785 .0152617 .0075431 4 .2728 .1438202 .0758220 .0599754 .0210740 .0111102 5 .2841667 .1560549 .0857001 .0470657 .0258458 .0141056 6 .2922857 .1650995 .0932577 .0526773 .0297551 .0158074 7 .298375 .1720505 .0992085 .0572052 .0329865 .0190208 8 .5031111 .1775558 .1040082 .0609257 .0356889 .0209058 9 .3069 .1820224 .1079575 .0640296 .0579759 .0225255 10 .3100000 .1857182 .1112621 .0566561 .0399551 .0239235 7 8 c 0 .0569757 .0555700 Csucv 07-08 .0279297 .0184057 (gs-c?)d7 (ca-cam?w .0001484 .0000489 .2539480 '.7460520 .0016347 .0007182 .4049071 .5950929 .0037282 .0018426 .5038339 .4961161 .0058573 .0030880 .5755451 .4264549, .0077947 .0042806 .6251001 .3748999 00094958 .0053626 .6647392 .3352608 .0109679 .0063244 .6961399 .5058601 .0122461 .0071755 .7216151 .2783849 .0133587 .0079251 .7426907 .2575095 .0143524 .0085864 .7604118 .2395882 45 DISCUSS ION OF CHARTS The formlas for end-moments obtained by moment distribution have been represented graphically in charts #1, #2, #3, and #4. The appropriate constants x, Y, z and values for 81, 82, 83 from table #4 or #5 where substituted in the formulas and the end-moments IAB and IDA plotted as ordinates in terms of QL, with ll/Qt. as abscissas, using r values from 1 to 10. Chart 1; gives a comparison between end-moments of a fixed colum bent and a hinged column bent for symmetrical loading. The darker lines in chart #1 represent the functions of the end moments of a beam in a bent with fixed-end columns, the lighter lines - for a bent with hinged columns. Lines representing functions for the same r value have been connected by arrows, indicating the possible range of variation of beam end moments in a given bent due to varying end restraint of columns. It can be seen from the charts that the difference between the end moments in beams of fixed column bents and those of hinged column bents is larger for higher r values. As this difference is larger both absolutely and relatively, it indicates the greater necessity of determining the end condition of the columns in analyz- ing a bent with a relatively stiff girder. The difference between the beam end-moments of a fixed-column bent and those of a hinged column bent is 46 caused by the smller end restraint given to the beam by the hinged columns. The smallerend restraint of the beams ”taken care of in the moment distribution pro. cedure by modifying the stiffness factor of the column, resulting into a modified distribution factor. 'r' being a function of the stiffness factors, the modification from a fixed to a hinged column bent can be expressen by means of a modified r. The modified r for a hinged column bent is 4/3 of the r for a fixed column bent. Using this relationship, beam end moments for a hinged colum bent can be obtained from the graphs of moment functions of a fixed column bent by using the modified r. Thus, e'.g., the end moments for a bent with r equal to 6, having hinged-end columns, can beOobtained from the graph for a fixed-end colum bent with r equal to 8. As can be expected, bents with a greater r ratio have relatively greater beam end moments due to a greater end restraint given to the beams by relatively stiffer columns. The colunm stiffness has an increas- ing influence on the beam end moments with a decreasing r ratio. This indicates the need for a closer evaluat- ion of stiffness of the members of a bent with relatively stiff columns, when moments in the beam are computed. Of course, the. beam moments increase with increas- ing load. The intercept of the end-moment line with the vertical axis of the coordinate system (ll/Q1. equal to zero) represents the magiitude of the beam end-moments when only a concentric load Q is applied to the frame. 47 The cmrt gives magnitude of the end moments in the beam AB on the side of the frame where Q and 1! both . cause downward load and, consequently, end moments of the same sign. As the l/QL ratio increases, the influence of M on the end-moments increases. When 11 causes an. end moment greater than that caused by Q (the ordinate for the particular M/QL being more than two times greater than the ordinate at the intercept with.the vertical axis), it may be necessary to investigate end moments in the beam on the opposite side of the frame,_where uplift loads are imposed on the bent with.end-moments of an opposite sign in the beam. _ The slopes of the functions with lower r are steeper, indicating a greater rate of increase of the end-moments with an increase in the loading for a bent with a relatively flexible beam. W represents graphically the influence of uniformly distributed and concentrated loading on the end-moments of the beam. The darker lines are the funct- ions of the end-moments when the frame is loaded at 8 points as shown in the figure. because not less than 8 bolts are used for anchoring a large vertical vessel, and this particular 8-point loading causes the greatest end-momenta“: the area between lines of equal r values may be taken as representing the range of variat- ion in the end-moments due to concentrated or uniform load assumption in practical cases. DJ- .Lc LOADING CONDITION FOR CHART N21 2.02. film . 3.! a . g . It‘s-Ox. . bag it is. 00.. a 0.5' 004 0.3 0-2. 0.! 24$ LOADING CONDITION FOR CHART N2 2. I. o‘ .O .3 Z- ICC: .8 Zia‘nfl.3 III5~JI 102. (in OIKOG :UL‘L ILL-(l, 2“,.‘-‘-\.— \l‘. 3" x. >< +——->k5-——-v— 1/] . A; 481442—389 o 19 F IG.I7 LOADING CONDITION FOR CHART N93 . TI... 1. I6,II¢I .nv , A . .I I v Isl w 1+ 1.. IYl..sI.Iv I: \II \IlI-I.Iv|l. 1.---.IIIIOI _ .I.I f. 0.71.114. i..-I vl .. . . . .. . v. I .I.Iv,tII....IeII+.I . . I I.IF.III..I L -+I. III.. . I m. m .- I I, T-.. Iii _ . _ . 1 Ir III? III I Ila III QIIIII13 -IAIIO . a _ , . I‘III_‘ It. .6 ,7! .II 41 III , . . IvIlvllIItleIIQIIIfI .- .4 . (VIIIIII L IeIIII la I164. Ierlve. . h _ fl .1 A. ~ _ _. I 'n I 5. lm I-IIJI . h I. IIII _ hell ..I.. .I.-+.IA_I.. .I. .I AIVII.II» .IIan . . . 1 I..- .. A IWII-IJ.-I. I .. I". 1! IT: e 17in1 IIIIIVIIT , _ , L..II.I~-: It an. II , _ . IDI1I , II. II Isl III. HIIT IIJILI _ . III ILII v. e. I InalI Illa: _ IOIII 1 C ‘1 1.8011 102. [H1 0F X0— .OU 200N530 uZu03m aunt-.1 11(10 ZUBNFHRO OTOOVO .DZ A\ B UNIFORM “59.91207;— 293'— A" I _..—————— *— ‘I‘I-‘h on D -1 _LC LOADING CONDITION FOR CHART N24 I“ Ear. 0.21: 0.6L 7 .21. L- . -. .4 HAUNCHED BEAM kss z ken: 7.8! Ca; " Cm: 30.659 8‘ .. .3 7: ID(: 102- (Ht ON ION 52 m gives moment functions for the leeward beam A3 of a frame with the wind blowing along a horizontal diagmal of the frame. Bcause the loading is not symmetrical, the end-moments are not emal and have been plotted separately for each end. . This loading condition, with the horizontal force diagmally, gives smaller. end-moments than when the horizontal force is applied perpendicularin to the beam AB, as in the previous charts. This isbecause the heaviest load is applied to the short diagonal beam and only half of it is transferred to the beam AB. Chart is was plotted to determine the influence of haunched beams on the end-mements of the bent. The solid lines represent the functions of the end-moments in haunched beams. The haunched beams have greater end-moments than the beams with a uniform moment of inertia along their length, because of a pester fixed end moment and greater carry-over factor for the haunched beam. This difference is more sipificant for bents with a higher r ratio. Most reinforced concrete table-top supports have haunches at enda tax of the beams, therefore the evaluat- ion of the effects of the haunches on the stresses in the frame is of practical simifdcance. Although chart #4 indicates only the influence of one particular type of haunched beams, the difference in results seems 53 to Justify the consideration of the haunches in frame analysis. The practical significance of the error arising from neglecting the haunches at the Joints is, however, minimized by the fact that the moments at the midspan of the beam are considerably greater than the end- moments and thus the midspan moments are likely to govern the desigz of the beam. Neglecting the effects of the haunches,would result in a safer design of the ham for greater midspan moments, while the haunches at the Joints would take care of the increased end- moments. The practical importance of the moment distribution with consideration of the haunches is also decreased by the difficulty of obtaining actual stiffness factor values for reinforced concrete members with varying reinforcement and thus varying moment of inertia along the length of the member. Because the moment distrib- ution constants cannot be determined exactly for the members, the consideration of the haunches not always is Justified, because the application of the more con- venient uniform member moment distribution procedure is easier. Due to the need of considering additional variables, when analizing the frame for sidesway moments, no charts have been plotted that would include the effect of the sidesway on the end moments. An investigation, using numerical examples, however, indicates that the influence of the sidesway is of chr importance, . even to the extent of reversing stresses in the members due to change of 8191 of the end-moments. In cases where sidesway can occur, its effects should be invest- igted when designing the frame. 55' NUMERICAL EXAMPLES Given: Design weight of vessel, contents and auxiliary equipment and piping supported from the vessel: Q O 10000 lbs. Exterior bending moment about the top of support due to wind earthquake or other horizontal forces acting on the vessel: It I 20000 ft. lbs. Beam span (determined by vessel diameter); 1. a 10 ft. Ratio of beam stiffness to column stiff- ness: r a 6. Finding fixed-end moments in structural members of a table-top support. The M/QL ratio is computed from the above design data to be 0.2 . Assuming the columns to be fixed at the footing and wind perpendic- ular to the beam under consideration, the end-moments in the beam can be obtained from start #1 by reading the ordinate on the moment line for r a 6, correspond- ing to the abscissa 0.2. Result: "AB 3 “BA I 0.0121QI. :3 12.10 ft.1b. The end-moment in the columns at the footing is i- of this value. For hinged-end colum support the beam end moment is obtained from the GE line (which happens to coincide with the moment line of a fixed-colum bent having r 8 8). “AB ' MBA : 1170 ft.1b. When a uniform distribution of the vessel load along the length of the beam is assumed, the end moment is obtained from the 6u line in chart #2 as being MAB : “BA 8 920 ft.1b. FIGZO 56 APPLICATION OF SERIES 31. The algebraic expression 3??? for 81 can be used to find end-moments in symmetrical frames with members of uniform moment of inertia, with the central member loaded symetrically and the other members being fixed or hinged at one end and rigidly connected to the central member at the other end. The end moments of the loaded member can be found directly by the formula I“. ' lAvA 3 5-2;; (F). where 'r' is the ratio of the stiffness factor of the loaded member to the sum of the stiffness factors of the other members at the Joint. F is the fixed end moment in the loaded member due to the load. Example. Find the end moments in the frame shown in Fig. 20. The computed stiffness factors of the mem- bers are indicated on the frame. The fixed end moment in the loaded beam AB is computed to be I 8 120 ft.1b. First, the stiffness factor of the column Ac is modified by 3/4 to account for its hinged endz“ kAc a (3/02 : 1.5. Then r 1- found: r - 1-7373- - z. The end moment in A'A is: n“, a 33-2- (3) a 2-3-2. (120) = so ft.1b. This moment is counteracted by the members Ac and AB at the Joint A in the ratio 1.5/3 8 i. Thus, MAC 8 (1/3)eo -.- 20 ft.1b. and MAB - (2/3) so a 40 ft.1b. nm is t HA3 : 20 ft.1b. M = 0. The other end - CA moments are symmetrical about the center of the span A'A. 57 FIGZO 58 CONCIUS ION m: moment distribution analysis is helpful in recognizing the factors affecting the end moments of members in simple symmetrical bents. Application of infinite series is shown to be a possible means of conveniently solving moment distrib- ution problems. The series Sl'was shown to have. a value that can be expressed in a simple algebraic equationn By use of this relationship, values of end-moments for symmetrical bents having members of uniform moment of inertia caused by symmetrical vertical loading can be obtained directly, without the need of a moment distribution, by substituting the appropriate values into the equation: "as ' In (5%?) ' when the vertical loading is expressed in terms of a load P. This relationship is not confined solely to the simple bents considered in this thesis, but is applic- able to all symmetrical plane frames with members of a uniform moment of inertia and a symmetrical loading on the central member, with the other members having fixed or hinged ends. An example of such a frame is given in Fig. 20. The r value for such a frame is the ratio of the stiffness factor of the loaded member to the sum of the stiffness factors of the other members at the Joint 59 modified to take care of the hinged-end members. It may be possible to find a simple solution also for end moments in frames of a different structure and loading condition when simple expressions, exact solut- ions or convenient approximations can be found to express the moment distribution relationships, The graphs of the end«-moments can be used for desigl purposes, by computing the M/QL ratio from known desigl data and entering the chart with this value to find the end-moment in terms of QL. For practical purposes, it would be necessary to supplement the charts for end moments with charts of maximum positive moments and other necessary desigl function graphs . 1) 2) 3) 5) 6) 9) 10) ll) 12) LIST OF REFERENCES PETROLEUM REFINEB 8e “Vin. Method of Design of Steel Stacks and Foundations April 1943; Vol. 22 v.0. Marshall The Desigl of Foundations for Stacks and Towers Aumat 1943; V01. 3. F. Humerstedt Advantages of Large Diameter Crude Still Towers October 1943; Vol. 22 S. Garnett ' Design of Foundations for Elevated Towers Feb. 1944; Vol. 23 v.3. Blank Sliding Barings for Hot Towers and Tanks March 1945; Vol. 24 Jo Yeakel Desigl ‘of Structural Details for High Towers Dec. 1945; Vol. 24 8.11. Jorpnsen Anchor mlt Calculations Hay 1946; I.H. Blank Analysis of Knee-Raced Rnts Used in the Oil Industry Dec. 1946; Vol. 25 EJ. Stothart Importance of Flexible Pipe Supports July 1947;701. 26 Emunngren Strength against molding, Design of Thin-Walled Towers Nbv. 1948; vol. 27 0.0. Iazaari a 0.3. Sanderson Design of Octagonal Foundations Dec. 1948; vol. 27 8.9. mick Pipe Supports and Pipe Restraints D90. 19‘9; V01.28 13) 14) 15) 16) 17) 18) 19) 20) 21) 21) ' 22) 6! F.E‘. Wolosewick ‘ EXpansion Joints and their Application May 1950; Vol. 29 11.3. Lonngren Pressure Vessel Considerations Vol. 29 A. I. Gartner Nomograms for the Solution of Anchor Dlt Problems July 1951; Vol. 30 FJ. Wolosewick Supports for Vertical Pressure Vessels July, Aug., Oct. at Dec. 1951; Vol. 30 F.E. Iolosewick Supports for Vertical Pressure Vessels 1953 PETIDLEU M PROCESS IN G E. F. Brummerstedt Welding saves 20% in Steel in Cat-Cracker Equipment Feb. 1949; Vol. 4 8.12. McMurray Blilding‘a Modern "Cat-Cracker" Jan. 1950; Vol. 5 PETIDIEUM ENGINEER 0.1L Scudder Modern Pressure Vessel Design and Construction materials May 1944; Vol. 15 J.A. ShJarback Practical Design of Foundations for Vertical and Horizontal Vessels in Refineries July 1944 NAT . PETROLEUM NEWS J .A. ShJarback Practical Desigl of Foundations for Vertical and Horizontal Vessels Oct. 6, 1943 ,E.F. Brummerstedt Desigl and Construction of a 33-ft Diameter Arc- Welded Vacuum Fractionating Tower Feb. 3, 1943 2:5) 24) 25) 26) 27) 28) 29) so) 31) 32) 33) 34) 62 E.F. Hummerstedt Stress Analysis of Tall Towers Nov. 3, 1943; Vol. 35 Pt.2 EJ‘. Brummerstedt Desigl of Anchor Bllts for Stacks and Towers Jan. 5, 1944; Tech. Sect. OIL AND GAS JOURNAL W.L. Nelson Thickness of Welded Towers Nov. 3, 1945; V01. 44 '.L. Nelson Self Supporting Foundations for Towers and Stacks Aug. 4, 1945; Vol. 44 S.W. Iewaren Anchor mlt Design for Tall Refinery Vessels June 14, 1947; Vol. 46 8.1. Lewaren Design of Tall Refinery Frames Feb. 19, 1948 WELDING JOURNAL J .0. Holmberg Welded Pressure Vessels in the Oil Industry April 1945; Vol. 25 H. Greaves Heavy Flexible Beam and Girder Connections Oct. 1948; Vol. 27 False lenmer Field Erected Pressure Vessels Nov. 1946; Vol. 25 API-ASME Code for the Design, Construction, Inspect- ion, and Repair of Unfired Pressure Vessels for Petroleum Liquids and Cases. Standard 011 Development Company's engineering standard: desigl loads; open type structures. Structures; wind and earthquake design requirements Standard Oil Company of California desigl guide: design of stiffening rings and internal struts at supports of horizontal cylindrical vessels. 35) 36) Dean PeaMdy, Jre The Desigi of Reinforced Concrete Structures Chapt. 14 Portland Cement Association Handbook of Frame Constants 65 N ”'TITII‘HQIII1EJEIMEILI' [Mllflfiflilflflfllfflfim'f’ss