!fS»»»»«rw»^ -"^ '^' f^ BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Henrs W. Sage 1891 A jz,<. M a.s;3. .:)k.t>^^.v..>L,.. Cornell University Library TG 360.H87 Notes on plate-girder design, 3 1924 015 400 827 TGr; +1 8 n Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924015400827 WORKS OF CLARENCE W. HUDSON PUBLISHED BY JOHN WILEY & SONS Notes on Plate-Girder Deslgrn. 8vo, vii + 75 pages, figures throughout the text and 2 folding plates. Cloth, SI. 50 net. Deflections and Statically Indeterminate Stresses. Small 4to, xiii+ 258 pages, profusely illustrated with figures in the text, and full page plates Cloth, $3.50 net. NOTES ON PLATE-GIRDEK DESIGN BY CLARENCE W. HUDSON, C.E. Mem. Am. Soc. C.E. Professor of Civil Engineering, Polytechnic Institute of Brooklyn, N, Y. Consulting Engineer FIRST EDIT 10 X FIRST THOUSAND NEW YORK: JOHN WILEY & SONS London; CHAPMAN & HALL, Limited 1911 T'Cx- %\'^(o C Hj C^) Copyright, 1911, ET CLARENCE W. HUDSON THE SCIENTIFIC PRESS R08ERT DftUMMOND AND COMPANY BROOKLYN, N. Y. PEEFACE The following Notes on Plate-Girder Design have been used by the author to give his civil engineeTing students the theoretical and practical information nec- essary to enable them to make a design and general detailed drawing for a through plate-girder railway bridge. The theory of the plate girder has been developed 90 that it may be applied to such structures for any duty. The notes may easily be given to the average class in one recitation per week for one semester of the college year. They have greatly facilitated the computation and drawing-room work with my own classes. It is with the hope that they may be useful to other teachers, and engim srs, that they are published. C. W. Hudson. Polytechnic Institute, Brooklyn, N. Y., January, 12, 1911. CONTENTS PAGB Ahticle 1. Stress Distribution and General ^ 1 Theory of stress distribution. Limits of the use of the plate girder. Design and detail drawing for a single-track deck plate- girder bridge. Design and detail drawing for a single-track through plate-girder bridge. The weight of a plate girder. Problems. Article 2. Required Area op Cross-section for the Flanges . Figures showing make up of cross-section for plate girders. Formula for obtaining the required area of the tension flange. Problems. Article 3. The Design of the Cross-section of the Flanges . Composition of the flanges. Minimum pitch of rivets in tension flanges. Formula for a safe unit stress for the compression flange. Thickness of material in compression flanges. Problems. Article 4. Lengths op Cover Plates 14 Formulas for the length of cover plates for girders carrying uniform and concentrated loads by analytic methods. Graphic determination of lengths of cover plates. Examples of the application of the ' formulas for finding lengths of cover plates. Problems. Article 5. Rivet Spacing in Girder Flanges 20 Rivet spacing a function of the shear. Formulas for spacing \ rivets connecting flanges to the web. Formulas for spacing the rivets connecting the component parts of the flanges. Formula for rivet spacing at the ends of cover plates. Problems. V vi CONTENTS PAGE A.BTICLE 6. Web Plates 28 Stresses on the faces of an element of the web. Stresses of maxi- mum intensity and the planes on which they act. Computation of the stress intensity at various points in the web for a definite case. Problems. Article 7. Area op Web Plates 37 Discussion of stress distribution throughout the web plate. Rela- lation of shearing and compression unit stresses. Unit stresses for shear and compression in web plates. Problems. Article 8. Stiffeners for Web Plates 42 Functions of stiffeners. Intermediate stiffeners. Concentrated load stiffeners. Formulas for size of stiffeners. Spacing of stiffeners. Problems. Article 9. Splices for Web Plates 46 Arrangement of splices. Duty of a web splice. Thickness of web splice plates. Number and arrangement of rivets in the splice. Maximum stresses on the rivets of a web spUce. Problems. Article 10. Splices for the Component Parts of the Flanges 51 Location of splices in flange material. Disposition of flange splicing material. Rivets for splicing the component parts of the flanges. Problem. Article 11. Connecting One Girder to Another 55 Size and arrangement of the material for the connection. Rivets for the various parts of the connection. Article 12. End Bearings 59 The duty of end bearings. Area of bearing on masonry. Proper length and width of bearings. Thickness of bearing plates. Differ- ent forms of end bearings. Problem. CONTENTS vii PAGE Article 13. Positions of Loading fob Maximum Shear and Moment 64 Maximum end shear. Maximum siiear at any point. Maximum moment at any definite point. Absolute maximum moment and the point at which it occurs. Problems. Article 14. Preparation of a Table of Bending Moments, Shears, and Concentrated Loads for Cooper's E50 Loading 69 Advantages of tables of moments and shears. Equations for maximum moments. Preparation of a table of moments. Table of, and sketches showing positions of loads for maximum moments. Problems. Article 15. Table of Moments, Shears, and Concentrated Loads FOR Cooper's E50 74 NOTES ON PLATE-GIRDER DESIGN ART. 1, STRESS DISTRIBUTION AND GENERAL The theory of stress distribution for plate girders is the same as that for solid beams of uniform material, it being generally assumed that the common theory of flexure applies with sufficient accuracy to the built-up girder. This assump- tion is certainly not strictly true. The rivets connecting the various parts together cause a local and small irregular- ity of stress distribution; splices of the various parts, and non-prismatic form, produce quite marked irregularity of this distribution. The " Theory of Flexure " properly applied to plate-girder design, however, leads to a thoroughly good engineering structure, as an immense tonnage of such work constructed during the past forty years and doing excel- lent duty witnesses. This wide use of the plate girder for service under greatly varying conditions has increased the knowledge of its capacity and given valuable practical information as to the design of certain of its features. The requirements for its fabrication, shipment and erection have also affected its design. These requirements are not fixed in their nature, although in certain respects, 2 NOTES ON PLATE-GIRDER DESIGN such as minimum tliickness of material in web plates, and minimum edge distances for punching, they fix a limit to the size of certain parts. The correct design of plate girders therefore requires theoretical knowledge and either practical experience or formulated rules, based on practical experience, and so carefully and closely drawn as to prevent a poor design. Plate girders are used for bridges anywhere from 18" deep and 15' long to 126" deep and 130' long, and even beyond these limits; they are also used in buildings and other important engineering work to a great extent. In general, the lower limit of their use is the I beam, which will furnish the proper strength at a less cost per pound of material; their upper limit is the truss, whose total cost is less than the heavier but cheaper per pound girder. In order to design properly any structure it is necessary to understand the composition and relation of each to each of the various parts. The following drawings are meant to illustrate some of the most essential features of plate-girder construction. They are therefore not meant to be casually inspected, but thoroughly studied, and the function of the various parts in carrying a load to the supports, the makeup of the parts and their connection, each to each clearly understood. Fig. la shows a single-track deck plate-girder bridge. Fig. lb shows a single-track through plate-girder bridge. The computation of the stresses in any structure is the preliminary step in the design. The stresses themselves are a function of the weight of the structure, and hence an early estimate of the weight of each portion of the » I gi^m 16 I IX Bolt — >; Estimatea WeigM 66,200 °#° Dead Load 1800°^p.l.f, Live Load Diagram per Traak Fi( "0 H* 4 0M > End Floor beams Sti-s. & Fl. bms. flush on bottom n'l:^ Inter. Floor beam Stringers Bending Moment 1,056,000 inch lbs. End Shear •46,400 lbs. m -0 o a i: ) Q 0=^ i'gtiQ o o o o Q o =e^ ~i j||j9 ^c ^ c c O 1 o 1 1 t' 1 o I <' ! I ' ' 1 1 ! 1 1 : • : 4 il ioi ^ ioi "^i j. 1 t^^ i*l 5l'l'l>et. Back Walls 1 ) ) 1 ) CI o o () ) C) o ) C) (1 Y 1 o () ' o o ) 9 1; -1 o > I 11 1 (> 04 = ag^o"Q fc O O _&: S (P-^e e e e e e — M^ ^ L Girder Chord Stress 272,800^ ■ Load on end shoe 158,600# STANDARD NO. 12 50'-0" THROUGH PLATE GIRDER SPAN. IHE NATIONAL LINES OF MEXICO Scale y - I'-O" April, 1907 BoLLER & Hodge, Consulting Engineers, New York STRESS DISTRIBUTION AND GENERAL 3 structure is necessary. Many valuable compilations of weights of structures are to be had, but they are not usually accompanied by full data. Every engineer must be sure of the quantities which go into his structure, and they therefore should be estimated as the computations are carried along and all computations based on wrong assumed dead loads corrected. The design of a structure should always begin with the part receiving the load and follow with each successive part in the path of the load as it travels to the support. For example, in an ordinary through plate-girder rail- road bridge, the design of the various parts should be made in the following order: Rails, ties, stringers, floor- beams, lateral bracing, and main girders. In making a first estimate of the weight of any girder, compute the live-load stress at the center of the bottom flange and increase this for impact if any is to be added, add to this sum 15% for the stress due to the weight of the girder, then determine the net area of this member in square inches. This net area in square inches multiplied by 14 will give a weight per foot of girder which may be used as a first estimate. This estimate is based on equal gross areas 15% greater than the net area of the bottom flange, for top flange, bottom flange, and web, and their sum increased 15% for details. An error of 1% in the total stress due to a wrong first assumed dead load should require a corrected dead load to be used. It is very difficult to give a good general rule for esti- mating the weight of girders, as the weight obviously is a 4 NOTES ON PLATE-GIRDER DESIGN function of the depth, loading, unit stresses, and many other things. PROBLEMS la. Sketch the cross-section of a plate girder with and without cover plates, and write the name of each part on the sketches. Ifi. Sketch the plan of single-track railroad deck plate-girder bridge and write the names of the various parts on the sketch. Ic. Sketch the plan of single-track railroad through plate-girder bridge, and write the names of the various parts in the sketch. ART. 2. REQUIRED AREA OF CROSS SECTION FOR THE FLANGES The plate girder is a built-up structure, and may be made with a variety of cross-sections depending on the require- ments of the case. The forms of cross-section most fre- quently used are those of a, b and c of Fig. 2. Top Elanges ni~ Bottom Elange = _l L ~]r =U= nr =U= 6 Fig. The object of these forms being to take advantage of the known law of stress distribution in flexure. The relation between the moment of the outer forces and inner stresses for homogeneous bodies subject to flexure is given from the formula AREA OF CROSS-SECTION FOR FLANGES 5 in which M iw the moment of the outer forces; r >S is the unit stress in the extreme fiber of the cross-section under consideration; / ihi the moment of inertia of the cross-section; (■ is the distance from the neutral axis to the extreme fiber of the cross-section. Any existing straight prismatic girder may be examined for intensity of flexural stress in a plane normal to the axis by means of this formula. The plate girder of prac- li J h, Fig. 2d. tice is, however, far from the ideal prismatic bar of the Theory of Flexure. The application of the preceding for- mula to the design of girders is very tedious, and while it could be greatly facilitated by the use of tables of - for a great variety of cases, the labor of its application to problems of design is not warranted. The almost universal method of plate-girder flange design will now be developed, with the aid of the following nomenclature and Fig. 2d. NOTES ON PLATE-GIRDER DESIGN Let A be the net area of each flange; h be the distance c. to c. of gravity of flanges; S be the allowable unit stress for tension or com- pression produced by flexure; Til be the depth of the web plate; t be the thickness of the web plate; ^2 be the depth out to out of flanges; If be moment of inertia of each flange about its own horizontal axis. Then the flexure formula becomes L /l2 6 /l2 /l2J The right half of this equation represents the moment of the internal stresses; of the three terms which make up this half it may be said : 47/ The term -t— is small in most cases, but for shallow girders it is relatively quite large, its omission from the expression requires the other terms to be larger. h . The value of j- is always less than unity, and in shallow girders considerably less, frequently as much as 10%. h The value of t- is usually less than unity, for very shallow girders without cover plates it is often equal to unity, and for girders with two or more cover plates it may be as much as 10% less than unity. AREA OF CROSS-SECTION FOR FLANGES 7 If h, hi, and h2 be taken equal, and the term -r- be tl2 dropped the formula may be written: That is, the approximate moment of the internal stresses equals the unit stress times the depth times the net area of one flange plus \ of the area of the web plate. Many engineers drop the -p- from the expression, as its value is small and it is on the side of safety to do so. Others include it with the idea that as the web takes a part of the horizontal stresses these horizontal stresses should be considered in designing the flanges and web splices. Where stiffeners or spHces are used on the web it is impossible to maintain the full web section, and the web available th as flange is usually taken as -5-, which corresponds to a vertical rivet pitch of about 4". The preceding formula should be written for the purpose of design: M . In this formula -r- is known as the flange stress, just as in the case of trusses the moment divided by the depth is the chord stress. 8 NOTES ON PLATE-GIRDER DESIGN PROBLEMS 2a. Compute the net area required in the flanges of a plate girder 30' long and 48" deep out to out of flange angles, when carrying two loads, of 160,000 lbs. each, spaced 5' from the center of the girder and a uniform load of 300 lbs. per Unear foot. Assume a web plate of 48 Xf". 26. Make up a section for the bottom flange of the girder of Prob. 2a, using only two angles. 2c. Make up a section for the bottom flange of the girder of Prob. 2a, using two cover plates and two angles. ART. 3. THE DESIGN OF THE CROSS-SECTION OF THE FLANGES Flanges of plate girders are generally composed of angles in pairs or angles in pairs and plates, as is shown in Fig. 2a, 2b, and 2c. The several parts are connected by rivets. The holes for the rivets are generally punched to a diam- eter ^" greater than that of the rivet, or to a diameter of ^" less and subsequently reamed to ^" greater, or, as in the best class of railway work, drilled to a diameter ^" greater than that of the rivet. It is customary in design- ing tension members to allow for a- hole J" greater in diameter than that of the rivet, that is, for a |" rivet a hole 1" in diameter should be deducted. The number of holes to be deducted from any tension flange depends on the number of rows of rivets and the spacing of the rivets in the rows. Much might be written on this point, but here only little will be given. For the flange shown in a in section, the views of c and d are longitudinal developments of one of the angles showing common methods of grouping the rivets. DESIGN OF THE CROSS-SECTION OF FLANGES 9 It is clear that a symmetrical arrangement of rivets such as shown in c is better than that of d, for the center of gravity of the net area through x-x and y-y of c lies in the root of the angle, while for the corresponding sec- Web PI. Mi %" L8«'i6"xK" 2 Coy. Pis. IJ lij" ^^ ^ <>- -© e- <^ -e- w iy -e- P^ ^- ^ ^ 4" ^5- -e e- ^>- -e- L. Fig. 3. tions of d it is first to the right of the root and then to the left, as indicated. The arrangement at d is sometimes chosen because it permits a little more freedom in driving one of the inner lines of rivets. Care must be taken in locating the sections x and y so that 2D + H^ V; for the case selected the longitudinal pitch p cannot be less than log 1 V3.252 -2.252= JYg = ^\/88 = 2.35". In fact the stresses 10 NOTES ON PLATE-GIRDER DESIGN passing through the space marked D are considerably- bent and thereby increased, and therefore the distance p for a good stagger for this case should be 3". It should be noted that most girders have an excess of strength except at the point of maximum bending moment and at the ends of cover plates, and hence care in staggering the rivets need only be exercised at these points. The entire flange stress is developed in small incre- ments by the web and transmitted to the flange by the rows of rivets connecting the vertical legs of the angles to the web. It is clear that these angles then should comprise a considerable part of the total flange area — some engineers require 50%, others permit as little as 33^%.. The girder which has a flange stress developed in a short distance requires heavier angles than one in which the stress is developed in a great distance. The foregoing provision for maintaining net section applies par- ticularly to the tension flange. The compression flange is usually made equal in gross area to the tension flange. The compression flange is in somewhat the condition of a column as far as liability to failure in a sidewise direction is concerned. For girders with a constant top flange section the maximum unit stress occurs only at the point of maximum moment, for a flange with cover plates; that is, for a flange which varies closely as the flange stress, the unit stress is nearly constant throughout. The web stiffeners, if any are used, give considerable lateral stiff- ness to the top flange, reaching and connecting, as they do for half their length to material in tension. The sketch of Fig. 3e will help make this point clear. The tension flange DESIGN OF THE CROSS-SECTION OF FLANGES 11 is held in line by virtue of its stress; any tendency to side- wise deflection of the compression flange is resisted by the tension flange if stiffeners are used. If the unit stress for static loads in tension is 16,000 lbs. per square inch, then 16,000 —70- is a corresponding unit stress for a compression member, in which the quantities need no definition. If it be assumed that a girder flange J Compression Flange ij Tension Elange Fig. 3e. is similar to a column in its action and that its section is a rectangle whose width = 6, then (l6,000 -70-^) =(l6,000 -70 -^J^) =16,000 -242^, or 16,000 -240^, (2) gives a compressive unit stress for a girder flange cor- responding to 16,000 lbs. per square inch in tension. As the stress in a girder flange is only a maximum for a 12 NOTES ON PLATE-GIRDER DESIGN small part of its length a formula for the safe compressive and unit stress P = 16,000 -20o|-, (2') may be used. Experience has shown that no material should be used which is less in thickness than ^ of the distance between the rivets in the direction of the action of the stress, J of the distance from the center of a rivet to the edge of the piece at right angles to the line of stress, or that any angle leg when used alone in a girder flange shall be longer than 12 times its thickness, otherwise the girder flange may fail in detail rather than as a whole. For simplicity and ease of construction the compression flange should not be made of greater section than the tension flange, therefore the compression flange must be supported in a sidewise direction at frequent intervals, which, from the preceding formula, will be about every ten times the flange width. It is customary to consider that the rivets completely fill the rivet holes in the compression flange and that the full gross section of the flange may be assumed to act to resist compression. This assump- tion with regard to the rivets always filling the holes is open to serious question, and if they do fill the holes they do not offer the same resistance to lateral deforma- tion which takes place under compression as the unpunched material. However, it is fair to assume that they partially make up the punched-out material. The formula for the unit stress for the compression flange is believed to be severe enough to allow it to be applied to the gross area DESIGN OF THE CROSS-SECTION OF FLANGES 13 of the flange. The depth h (called the effective depth) of formula (1) is obtained by taking the gross area of both flanges in computing the location of the center of gravity of the flanges. It should be borne in mind that if there is no lateral deflection of the top flange that its maximum unit stress depends entirely on -, the section modulus, as it is some- c times called. While rivet holes affect the position of the neutral axis for the sections in which they occur, probably two-thirds or one-half of the length of a girder will be undiminished by holes, hence in applying the formula M = — to check the results of formula (1) it will be best to determine the position of the neutral axis from the gross section, and find the moment of inertia of the net section of the entire girder section for / in the formula. PROBLEMS 3a. Compute -the probable maximum unit stress in the tension flange Si/ of the girder of Prob. 26, by means of the formula M= — . 36. Compute the probable maximum unit stress in the tension flange of the girder of Prob. 2c, by means of the formula M = — . 3c. Compute the probable maximum unit stress in the compression flanges of Probs. 3a and 36, by means of the formula M= — . 14 NOTES ON PLATE-GIRDER DESIGN ART. 4. LENGTHS OF COVER PLATES The plate girder, being a composite structure, may easily be constructed so that its cross-section may vary approxi- mately as the moments and shears require. The full flange section being required only where the moment is a maxi- mum, a method of determining where the parts may be omitted when no longer required is necessary to economi- cally design the girder. In general the cover plates are the only parts of the flange which do not extend the full length. Two methods will be developed for finding where cover plates (or any other part of the flange) may be omitted. The first: For girders which carry a uniform load, or a load which may be closely represented by a uniform load. Deck plate girders for railway bridges may be included in this classification. In Art. 2 the approximate moment of the internal stresses at any section=(S/i x( A +-^) in which (Ah — ~) is the net flange area, designating this area by a for simplicity, the moment of the internal stresses becomes Sha. Let w= the uniform load per foot of girder, or for locomotive loading the uniform load which would pro- duce the same end shear, then the bending moment at VdI/X wx^ any point distant x from the end=-^ —, and this must equal the moment of the internal stresses. ivlx wx^ ~2 2' = ^ha, LENGTHS OF COVER PLATES 15 if it is desired to find the location of the point n, where the first cover plate must begin, substitute for S, h, and a their value for the portion of the girder between the end and n and solve for x. It is more cnovenient in practice to have the formula in such form as to give the length of cover plate direct. For this puprose Let c = the theoretical length of any cover plate =1 —2x; C = the practical length of any cover plate =c+2/ (2/ = from 2 to 5 feet) the additional length y being required for locating a few rivets, so that the plate may be capable of taking stress where it is theoretically required. Fig. 4a. From the foregoing relation between the moments of the outer and inner forces may be written , P P 2Sha , , , . 11^ 2Sha x!^-lx+j=j — ^ and from this ^ = 2~\A — ^• C = y+c = y+l-2x , (I IP 2Sha\ „ /i 'P 2Sha w (3) To use the formula to find the length of the second plate, take a and h for the portion of the girder between n and 0. 16 NOTES ON PLATE-GIRDER DESIGN Another simple formula by the author for finding the lengths of cover plates for this class of loading is given in Eng. News, XXXII, page 278, the issue of Oct. 4, 1894. The second: For girders carrying loadings which may not be represented by a uniform load. Girders of this class receive their loads through other girders or columns at definite points. The main girders of through plate girder spans with floorbeams are common examples of this class. As before the resisting moraent = Sha, where a = area of flange at end of any cover; /i = depth c. to c. of gravity of flanges at end of any cover; /S = unit stress. The bending moment M at any point along the length of the beam and distant x from the left end, where Pi, P2, P3, etc., of Fig. 4& are the concentrated loads, w the r — — 1 n .1 — U — P3 Pi 1- P^ N < 1 ^ Fig. 46. uniform load per foot of length, and Ri the left hand reac- tion, is given from M = Rix -m-| -Pi(x -0) -Pzix -b) - etc. LENGTHS OF COVER PLATES 17 Since the moment of the external forces = moment of internal stresses Rix — ^-Piix-0)-P2{x-b)+etc.=^Sha, . (4) in which every quantity is known except x. To apply this to any point n between the first con- centrated and the end the above becomes 7/1 r-s RiX—K- = Sha, (4'> in which J?i and Sha are known, and the solution of which is a very simple matter. It should be noted that the proper position for the live load is that which makes Ri a maximum. To apply it to any point q between two loading points the formula becomes, R^o + iR,-Pi){x-o)—^=Sha. . . .(4"> The position of the live load must be taken, first, sO' as to make the bending moment at Pi a maximum; second, so as to make the bending moment at P2 a maximum. This gives two equations of the form of (4") each of which must be solved for x. The value of .r, which is least, i.e., the one requiring the longest cover, is to be taken. The lengths of cover plates are readily found by a graphic method which needs no explanation beyond the following sketch: 4c. The full line is drawn to represent 18 NOTES ON PLATE-GIRDER DESIGN the resisting moment of the various portions-, and the dotted line the moments of the outer forces. The appUcation of formulas (3) and (4) will now be made to finding the lengths of cover plates; for this pur- pose assume a girder 43' long c. to c. end bearings, 60i" ,^r Fig. 4c. deep out to out of flange angles and of the following compo- sition : 1 web plate 60 Xf" =22.50 sq.in.X^ = 2.81 sq.ins. 2 top angles 6 X6 XlQ.li = 11,52 +2.81 =14.33 sq.ins. 1 top plate 14 Xi =7.00 +14.33= =21.33 sq.ins. 1 top plate 14x4 = 7.00+21.33= =28.33 sq.ins. 2 bottom angles 6x6x19.6 = 11.52 -2.00= 9.52+2.81 = 12.33 sq.ins. 1 bottom plate 14 X J = 7.00 -1.00 = 6.00 +9.52 = 18.33 sq.ins. 1 bottom plate 14x^ = 7.00-1.00 = 6.00+15.52 = 24.33 sq.ins. The girder will first be assumed to act as in a single- track deck railway bridge, using formula (3), Maximum bending moment = 1,938,410 ft.lbs. Maximum end shear = 193,500 lbs. (w= .^ ' =9000). a = 12.33, 18.33, and 24.33 sq.in. h = 4.77, 4.85, and 4.93 ft. LENGTHS OF COVER PLATES 19 For first cover Ci =1/ +2 (48)2 2X12.33X4.77X16,000 4 9000 = 2/ +2V4()2.2-209.1=?/ +2x^253.1 = 2+2X16=34' long. For second cover C2 = y +2 J462.2- 2 X-^^ X 4.85 X 16,000 = t/X2\/462.2'-316.1=2/+2\/r463 = 2+24 = 26'. The girder will now be assumed to have floorbeams attached to it as indicated in Fig. 4d. The reaction Ri, when the moment at Pi is a maximum, is 136,500 lbs. and the own weight of the girder 300 lbs. per linear foot. R( -10.75— Fig. id. Then for the first cover 136,500.T -300^ = 12.33 X 4.77 X 16,000 = 941,000. 910x-.r2 = 6273 or x^ -910x +207025= -628 +207025 .T = 45.5 -38 = 7.5. The length of the first cover (7i =2 +(43 -15) =30'. For the second cover 136,500x -150x2 == 18.33 X 4.85 X 16,000 = 1,422,400. a;2-910x = 9483; x = 455 -445 = 10. The length of the second cover C2 = 2 + (43 -20) =25'. 20 NOTES ON PLATE-GIRDER DESIGN PROBLEMS 4a. Let each of the flanges of the girder of this Art. consist of 2 angles 6X6X17.2 lbs. = 10.12 sq.ins. 1 cover plate 14 X I " =5.25 1 cover plate 14 X I " =5.25 1 cover plate 14X I " =5.25 and I of the web plate 60 X |. Find the lengths of the cover plates when it acts as a part of a deck railway bridge. 4b. Find the lengths of the cover plates when the girder of Prob. 4o acts as a part of a through bridge. The concentrated load P, = 105,000 lbs. The loads being spaced as shown in Fig. 4d. ART. 6. RIVET SPACING IN GIRDER FLANGES The connection between the web and flanges of girders, as in other composite structures, is made by means of rivets. These rivets are spaced with reference to the horizontal component of the stress in the flange, for at the extreme fiber the direction of the stress is horizontal and the maximum shearing unit stresses in the flanges are very small, and hence the horizontal component is the only stress of importance. The one important exception to the foregoing is where the girder load is applied to one of the flanges, here the rivets have to transmit the loading which the girder carries together with the horizontal stress increments between the flanges and web. The exceptional case will receive special consideration. The moment of the external forces at any point in a girder equals the moment of the internal stresses, there- RIVET SPACING IN GIRDER FLANGES 21 fore between any two points it is essential that there be enough rivets to properly develop the stresses produced by the maximum increase in moment between the points. The increment in the moment is constantly varying throughout the girder length. The general equation for moment for a simple girder as given in Art. 4 is M = Rix —^ -Pi{x -0) -P2(x ~b) -etc., The derivative of M with respect to x is, —j— = Ri —vox —Pi —P2 —etc., and is the shear. It is therefore seen that the greatest increase in the bending moment occurs when the shear has the greatest possible value. The increase in bending moment between two points so close together that the load between them may be neglected, is the shear multiplied by the distance between the points; the increase in flange stress is this increase in moment divided by the depth. ■Let F = the maximum shear at any point on the girder; fc = the effective depth, i.e., the depth c. to c. of gravity of flange, in inches; i? = the least value of the rivet to resist either crushing or shear, and p=the space between two adjacent rivets in inches. Fxl Then — r — =the maximum increase in flange stress in a '■: space of 1 inch (a) — = stress per inch of girder length carried by the rivets (6) 22 NOTES ON PLATE-GIRDER DESIGN For the proper degree of strength (a) should equal (b). V R h p or p = Rh " V (5) In order to show the application of (5) to determining rivet pitch: Let Fig. 6a represent a portion of the girder 2Cov. Pis. «xM ZLVxc'i 19.6 f- ■Web PI. 60 xH 2L=6x6x 19..6# 2COV..P1B. u'iji" Fig. 5a. used to illustrate the method of finding lengths of cover plates in Art. 4. The shear at a = 193,500, at 6 = 153,000, and ate = 118,000. The value of a rivet in bearing = 24,000, and in shear 12,000 lbs. per sq.in. The pitch for |" rivets, at (a) = p- 4.77X12X7876 193,500 = 2.34" provided the flange angles are designed to carry the entire flange stress at this point. If, however, as was assumed in finding the length of the cover plates, the portion of the web used as flange area is 2.81 sq.in., then .^ , 2.34X12.33 2.34X12.33 „„„ _ the pitch = ^^^ 33 _^g Y)== 9:52 = ^-0^ • The pitch of 2.34" would require that two lines of rivets be used, as good construction requires that the rivets be not closer RIVET SPACING IN GIRDER FLANGES 23 than 3". It is also customary to use two rows of rivets to connect a 6" angle leg to the web. If the load is applied directly to the girder flange then the previous computation needs modification for the rivet pitch in the vertical legs of the angles of the loaded flange. Let Fig. 56 represent a part of the top flange of the girder of Fig. 5a and assume that it carries one-half a railway track on the top flange. One engine wheel weighing 50,000 lbs., including impact, will then be carried in a space of about Fig. 5. 42", it being the custom to asssume that three tics carry one axle load. Let Hr and Vr of Fig. 5c represent the resultant horizontal and vertical forces on the rivet as applied to the rivet by the girder web; the resultant of these two forces R should not exceed the ability of the rivet to safely resist either shear or crushing. To deter- mine the pitch for this case, Let IF = the wheel load including impact (or other con- centrated load); a = total flange area ai=total flange area with the part of web used for flange deducted; .s = three times the distance c. to c. of ties in inches, /i = effective depth of the girder in inches; 24 NOTES ON PLATE-GIRDER DESIGN /ir = horizontal increment of stress in a length of one inch carried by Ui] j;r = vertical load per inch of length of flange affected by the concentrated load, rr = resultant of hr and Vr, =Vli^r + v^r 4 V ai\2 /WY h a) \ s Then \i R= the value of the rivet, as before, R R P = rr l/V-aiY (Wy h-a / \ s Let this be applied to finding the pitch for connecting the top flange angles to the web for the end of the girder of Fig. 5a. Fai 193,500X9.52 193,500x9.52 ^a ~ 57.24X12.33 ~ 705.77 -2bl0; F^SMOO^ s 42 ^^^"' ^26102 +1190^ 2710 For the heavier concentrated loads, such as columns bring when carried on the flanges of the girders, the flange angles will often not contain enough rivets for security, and stiffeners must be used, as will be explained later. The foregoing gives a simple method for finding the max- imum permissible pitch of the rivets connecting the vertical legs of the flange angles to the web. The determination of the pitch of the rivets for con- RIVET SPACING IN GIRDER FLANGES 25 nc'cting the cover plates to the horizontal legs of the flange angles is not a simple matter if theoretical accuracy is desired. At b, Fig. 5a, the point where the first cover plate begins the flange angles haA^e in them all the stress they can carry. It. is clear, therefore, that from 6 to c with the rivets connecting the first cover to the flange angles spaced to take only the flange stress increment, the flange angles simply transmit the increments of stress to the first cover plate. At c the first cover plate and angles have all the stress they can carry, and with rivets spaced as before the increments of flange stress from c to the point of maximum flange stress are simply transferred through the angles and first cover to the second cover. Rivets con- necting a cover plate to a flange are generally spaced to take the increments of flange stress which occur from the end to the point of maximum stress in the cover. If n be the number of lines of rivets connecting the cover to the flange then nRh P = -V'> (5') Assuming two lines of rivets in the cover plates the pitch at b for connecting the first cover to the flange angles 2X7216X57.24 .^,„ _,,. . ,, . ^, ^ = :nr^r-?^7^ = 5.41". Ihis is the maximum that mav 153,000 be used; the actual pitch would be made considerably less and usually a multiple of h, f , or 1" and at the same time such a pitch as would stagger well with the rivets in the vertical logs of the flange angles. If the maximum 26 NOTES ON PLATE-GIRDER DESIGN pitch permissible were used the stress per square inch in the first cover plate would be zero at b and increase to 16,000 at c. This is highly undesirable, as there would exist in the girder in juxtaposition the angles with a unit stress of 16,000 and a cover with an average of 8000 lbs. The girder flange material cannot act in any such way with- out undue bending stresses on the rivets. For this important reason cover plates should be made longer than a mere consideration of their relation to the moment polygon would require. The additional length, designated by y in formula (3) required, is a function of the number of rivets necessary to equalize the flange unit stress in all the flange material, and may be deter- mined as follows: Let n = number of rows of rivets in the cover plates; a = area of flange without the cover plate under consideration, with unit stress s; ai =area of flange, including the cover plate under consideration, with unit stress Si; 2? = pitch in inches of the cover plate rivets in each line; R = value of one rivet connecting the cover to the flange ; 1/1 = additional length of cover plate required at each end m ft. = ;j. yi = — -— p p. {a\ — a),Si S-a-p ^ RIVET SPACING IN GIRDER FLANGES 27 The value of y for the first cover of the first example used to illustrate the method of finding the lengths is: 16,000X12.33X6 p_3 p y~ 6X18.33X7216 'n~2'n If the rivets be placed 3" apart in two rows 2/ = 2.25. If enough rivets are used at the end of a cover plate as at h of Fig. 5a to transmit to it its full proportion of the flange stress, the rivet pitch required to connect the cover plates to the flange at any point will be the following: Ai n-Rh ^,,. In which Ai is the total flange area, and A the area of all the cover plates, at the point under consideration. The number of rivets as determined by (5') and (5") should be increased by 20% to allow for the bending, stresses. To still further limit these bending stresses no rivet passing through angles and covers should have a grip more than five times the diameter of the rivet. PROBLEMS 6a. Using unit stresses of 10,000 and 20,000 lbs. for rivets in shear and crushing, compute the required rivet pitch for connecting the bottom flange angles to the web at c of Fig. 5a. 5b. For the data of Prob. .5a compute the rivet pitch for connecting the top flange angle to the web at a point vertically over c, including the effect of a locomotive wheel load. 5c. Using the unit stresses of 5o compute the value of y for the second cover plate. bd. Compute the pitch of the rivets in the cover plate of 5c just adjacent to the portion y. 28 NOTES OX PLATE-GIRDER DESIGN ART. 6. WEB PLATES The maximum stresses that the web of a beam or girder must resist are of varying intensity and direction throughout its area. For any elementary parallelepiped cut from the body of a beam in equilibrium subject to flexure, the forces on the faces of the element perpendicular to the plane of the drawing may be represented for the most general case -I 'l 11 1 W Fig. 6a. as shown. The weight of the elementary particle may he neglected, as it is an infinitesimal of the third order as compared with the amount of stress on the faces, which are infinitesimals of the second order. As the length of the sides of the particle approach zero as a limit, the intensities of the oppositely directed forces py are equal, and also the forces px. If the oppositely directed normal forces are of equal intensity the tan- gential or shearing forces q are also of equal intensity. The intensity of the shearing stresses at any point in a homogeneous prismatic body subject to flexure is given WEB PLATES 29 from q=-T-j-, a well-known formula of Mechanics of Mate- rials, in which q = ihc unit shearing stress; 7 = the total shear; 6=the width of the body at the point under con- sideration; Q = statical moment of the portion of the body above the point under consideration with reference to the neutral axis of the body, and 7 = moment of inertia of the body with reference to the neutral axis. The intensity of stress p^ at any point is given from Mil . . the flexure formula Vx=-j-' ^"^ which the quantities are too well known to need definition. Y Y . ■ .■UJ=Uiiiform load per incJx /^^Mn. t J. .1— 10 000 lbs. p. l.f. I ^ Fig. 6e. This problem is so important for a proper understanding of the stress distribution on web plates that it is com- pletely solved for points Nos. 3, 7, 9, 11 and 15, and much of the computation made for the remaining points, in what follows in this article. In Fig. 6e arrows acting toward a point indicate compressiqn and away from the point tension. The computations for -p^, Vv ^^d g should be made about as follows: ^=-,Vx^x (24)3 .. = 576.00 5.56X4 = 22.24 4X3.75X10.822 = 1756.08 = 2354.32 M^ = Ms = 2,205,000 in. -lbs Mc = 2,940,000 in. -lbs At points Nos. 1,2 3, 4 and 5, Px = 1 X X 1 1 - ^ Ll^ 1.18 Fig. 6/. 34 NOTES ON PLATE-GIRDER DESIGN • XT ^ _, n 2,205,000X4 ,„^^ At points Nos. 7 and 9, fx= — 9354 32 — ^ 3,740 At points Nos. 11 and 15, p^ = ' ^tk^ 32 — = 10'000 Qii =2X3.75X10.82= 81.15 +2.00X10.00= 20.00 101.15 Q7 = 2X3.75X10.82= 81.15 2.00X10 = 20.00 2.00X6 = 12 113.15 ^3 = 113.15+2x2 =117.15 No. 3 = F-^^^m^ = 70,000 X. 0993 = 6960 No. 7 = 7 -^^^^22 =35,000 X. 0960 = 3360 No. 11 =F- 5x2354 3 '2^ 0000 X .0853 = 0000 From formula (7) there is found 8334- 8331 X. 5 ' at No. 3, Vy= 5-^ = 830 833i- (8334) .304 at No. 7, p„ = — -^ = 1240 .M 11 833i-(416f+416f).121 733 ,._^ at No. 11, Pj, = ? =—^ = 1470 Values of Q, jj, and -ry should be tabulated as follows : 12".. 11" 10 .. 9 8 .. 7 6 5 4 .. 3 2 .. 1 .. WEB PLATES Q vxfj = 00.000 V .0000 |— = 58.375 T' nifi.'; for 6 =!?<"'— = 74.125 r .0210 " = 88.375 r .0250 " = 101.125 T' .0853 - to'b=}i" = 104.875 F .0883 = 108.125 T .0919 = 110.875 T .0936 = 113.125 T' .0960 = 114.875 V .0976 - = 116.125 T' .0986 = 116.875 T .0992 = 117.125 T .0993 35 f\ Fig. 6g. Let Ai =the area above division No. 11 =4.25 + .75 = 5.00 sq.ins. A2=the area above division No. 10 = 5.00 + 1.50 = 6.50 sq. ins., etc. Values of q X arc as: qXAi = Fx.017 ?XAi3 = T'x.550 A2 = FX.046 Ai4 = Fx.599 As = FX.081 >4i5 = Fx.648 Ai = Fx.l21 Ai6 = Fx.696 A, = Fx.l65 Ai7 = Fx.743 Ae = FX.210 Ai8 = Fx.790 At = FX.257 Ai9 = Fx.835 As = FX.304 A2o = Fx.879 As = Fx.352 A2i = Fx.919 Aio = F X .401 A22 = Fx.954 An =-Fx.450 A23 = Fx.983 Al2 = Fx.500 A24 = Fx.l000 36 I NOTES ON PLATE-GIRDER DESIGN At No. 3: tan 2d = 2X6960 13,920 830 -0 830 16.771 2^ = = 86° -36" =43 -18" = i[0+830±V(830)2 + 4(6960)2] = 4[830 ±13,940] = +7380-6550 At No . 7: tan 20- 2X3360 6720 " 1240 -3740 2500 = 2.683 26 = = 68° -10' 6- = 34° -05' ■p = i-[3740 + 1240 ± V (2500)2 +4(3360)2] = J[4980±7170] = +6080, -1100. At No. 9: 'p^= -3740 833 J -833i X .696 ^,^ Vv = — — — ^^ 510 3 = 3360 „ 2X3360 6720 ^^'^^^^ 510 +3740 "4260 ^^^ 25 = 57°-50' (9 = 28°-55' p = |[_3740 +510 ±'/(4250)^ +4 X3360'] = i[- 3230 ±7950] = -5590, +2360. AREA OF WEB PLATES 37 At No. 11: 2 xO tan 26= ._„ _ „„„ = an infinitesimal p = M10,000 + 1470 ± V(8530)2+4X0] = i[ll,470±8530] = +10,000, + 1470. At No. 15: p^= -10,000 833i -(4161+4161) __^ 100 _. py = ^ 879 =--g- = 200 > _d_ Fig. 7a. Fig. 75. In order to determine whether a web needs stiffening an investigation which will lead to a safe structure can be made on the following basis : Formulas (8) and (9), of Art. 6, show that at the neutral axis of the girder the maximum resultant com- pressive stress intensity approximately equals the maximum shearing stress intensity and that the line of its action makes an angle of about 45° with the neutral axis. A compressive unit stress equal to the maximum shearing 40 NOTES ON PLATE-GIRDER DESIGN stress intensity is therefore assumed to act along the element of the web having a length of I in Fig. 76, and this must be less than the permissible unit stress for a column ot this length. Assuming that a safe unit stress for a column is given from the formula P = 16,000 -70-, the notation for which needs no definition, the 16,000 should be reduced to 12,000 if the average shearing unit stress is used, as the maximum unit compression. The formula then becomes P = 12,000 -52.5-. r Substituting for I and r their values, l = dV2, and r = .29t, where i = the thickness of the web. The formula then is P = 12,000 -52. 5^^ = 12,000 -257.5j . . (a) As the tension in the web below the neutral axis will prevent buckling, the value of d should be k", and as the tension in the upper half of the web helps to prevent buckling the constant may be divided again by 2, the formula then being P = 12,000 -65^ (10) AREA OF WEB PLATES 41 It should be noted that d equals either the clear distance between flange angles or stiffeners. To show the method of applying this formula to see if web stiffeners are necessary let the girder of Fig. 5a be examined. The average shearing unit stress = — „ ' - =8580 lbs. per square inch. 65 X48 5 The allowable unit stress = 12,000 gy^ = 12,000 -8420 = 3580 lbs. per square inch. As the actual stress exceeds the allowable, the web must be made thicker or the stiffeners placed closer together or else both combined. If the web be made J" thick 193 500 The average shearing unit stress =- — stj — = 6450 lbs. per square inch. 65—48 5 The allowable shearing unit stress = 12,000 ^ — ~ = 12,000-6300 = 5700 lbs. per square inch, which shows that the web must be further thickened or the stiffeners placed closer together. If the web be made ^" thick and stiffeners placed 30" apart in the clear the average shearing unit stress 193,500 ,oonlu • u = o^ OF =7380 lbs. per square mch. Jo. 25 65 X30 The allowable shearing unit stress =12,000 r-; — = 12,000 -4420 = 7580 lbs. per sq.in. 42 NOTES ON PLATE-GIRDER DESIGN PROBLEMS 7a. The accompanying sketch shows the cross-section of the girder of Prob. 6a and the curve of unit shear distribution. Construct the Fig. 7c. figure to three times the size of the sketch shown and write in the value of the ordinates to the curve at points 1" apart from the top to the bottom of the section. 7b. Are stiffeners required on the web of the girder of Prob. 6o? If not, show it by computation. 7c. What is the minimum thickness permissible for the web of the girder in Prob. 6a? ART. 8. STIFFENERS FOR WEB PLATES Stiffeners have several important functions, the chief of which are : (a) They keep the web from buckling, due to the com- pressive stresses in it. (b) They help hold the compression flange from lateral failure as a whole and from failure in detail in any direction. (c) They are used to relieve the rivets connecting the STIFFENERS FOR WEB PLATES 43 loaded flange to the web, by transferring the load directly to the web. (d) They are used to reduce to pro.per amount the vertical stresses, on horizontal planes, in the web brought by local concentrated loads. (e) They help hold the web true to shape during man- ufacture and erection. According to their principal functions they should be divided into two general classes, intermediate stiffeners, which are chiefly used for the purposes of (a), (6) and (e); concentrated load stiffeners, for (c) and (d), although each class performs all the functions of course. Intermediate stiffeners act as a beam standing vertically to resist lateral displacement. They are generally made of angles in pairs and riveted to the girder web as shown in Fig. 8a. The a .6 :0» Fig. 8o. legs of the stiffening angles which are against the fillers are near the neutral axis of the pair and therefore should be only large enough to receive the connecting rivets; a 3i" leg for |" rivet and 3" leg for f" are enough. The width h of the outstanding legs is the principal 44 NOTES ON PLATE-GIRDER DESIGN element of efficiency for resistance to transverse displace- ment. A common rule is to make the width of the outstand- ing leg (0) equal to the depth of the girder (d) in inches divided by 30 plus 2", or expressed as a formula = — + 2" (11) The following is a common method of locating inter- mediate stiffeners. Intermediate stiffeners should be used at points as required by (10) or wherever the unsupported depth between flange angles is more than 160 times the web thickness. Where intermediate stiffeners are required they should never be further apart in the clear than the clear distance between the flange angles, with a maximum limit of 5'. Concentrated load stiffeners act to transfer a concen- trated load into the web or to transfer the girder load to a support. The sketch of Fig. 86 will illustrate both cases. h^ I Web PlM'k }i' 1/8 6x6'x^'l° m rh \Xl: gii'xn -^t<- FiG. 8b. The girder here shown being loaded with three con- centrated loads of 180,000, and its own weight of 250 lbs. per linear foot. The lines of maximum web stress STIFFENERS FOR WEB PLATES . 45 for a girder of this character are very different from those of the girder of problem 6a. If such a girder were con- structed without stiffeners, and formula (7) were applied to finding the intensity of p„, this intensity would be found to be very high. The heavy vertical stifleners are added to transfer the point of application of these loads from the outer edges of the flanges to an average position, of the center of the web, and thereby decrease the inten- sity of vertical stress on horizontal planes just at the inner edge of the flange angles. These stiffeners cannot fully accomplish their object, as they are of elastic material and they are shortened by compression, and, as they are fastened to the web, the web and stiffeners move together except for a variation due to the deformation of the rivets. The stiffeners also serve to relieve the flange rivets from the component due to the vertical load. Stiffeners supporting concentrated loads may be designed as columns with the formula P = 16,000 —70— , in which i should be taken as one-half the girder depth and r as the radius of gyration about an axis in the longitudinal center line of the girder. Wherever the combined stress intensity on a web plate exceeds the allowable unit stress the stresses may be reduced by means of side plates, as shown by dotted lines. Where the concentrated load stiffeners are at the end of a girder they should have their outstanding leg about equal to the horizontal leg of the flange for the sake of good appearance, and where it is desired to face the ends of such girders the stiffening angles must be made thicker 46 NOTES ON PLATE-GIRDER DESIGN by I" than theoretical considerations determine, to allow for material removed by such facing. PROBLEMS 8a. Determine the size of the end stiffeners for the girder of Fig. 5a, using two pairs of angles, one on the outer and one on the inner edge of the sole plate. 8b. Assuming the wall plate to be 18" long for Fig. 5a, determine the size and spacing of the intermediate stiffeners for the girder. 8c. Determine the sizes of the stiffeners at the points of concen- trated loading for the girder of Fig. 86. 8d. Determine the sizes and spacing of the intermediate stiffeners for the girder of Fig. 8b. ART. 9. SPLICES FOR WEB PLATES Webs of plate girders should not be spliced unless neces- sary, but as wide plates cannot be procured in one piece, in general, for girders over 60 ft. long, splicing for long girders is unavoidable. The manufacturers furnish lists of the maximum dimensions of their plates. The number and location of the web splices should be determined by a combined consideration of the cost and resulting efficiency of the possible arrangements. The splice should be de- signed with the idea of transferring the stresses in the web across the cut in the most direct manner possible, just as in plate girder design as a whole the effort should be made to have the stresses developed as indicated by the theory of flexure for solid beams. Let it be required to splice the web at the center of the girder which has an effective length of 80', and a composition as shown in Fig. 9a. The girder has been designed on the assumption that SPLICES FOR WEB PLATES 47 ^ the gross area of the web is available as flange area. The maximum bending moment = 6,300,000 ft.-lbs. The center shear simultaneous with this moment = 0. The maximum ceJiter shear = 75,000 lbs., and the bending moment simultaneous with this = 3,400,000 ft.-lbs. Fig. 9a. The flange area is made up of: ^ web = 96Xi;% Xi= 5.25 sq.in. 2 angles 8x8x32.7 = 19.23-2.50 = 16.73 1 cover plate 20 Xi = 10 -1.00= 9.00 1 cover plate 20X^ = 10-1.00= 9.00 a total of 39.98 sq.ins. The simplest form of web splice consists of two ver- tical plates terminated at the top and bottom by the vertical legs of the flange angles, but this makes a normal distribution of the web stresses impossible in the vicinity of the splice. The resultant of all the forces on any beam cross-section may be represented by a shear and a couple 48 NOTES ON PLATE-GIRDER DESIGN as is well known from mechanics, but the proper form of splice should be designed with reference to the distri- bution of these forces. The portion of the web between the flange angles is so near the extreme fiber that the stresses may be assumed to be horizontal throughout. The flange unit stress has been taken at 16,000 lbs.; the bearing and shearing values for rivets will be taken to correspond to this at 24,000 and 12,000 lbs. per square inch respect- ively, or at 50% more and 25% less than for tension. The thickness of the splice plates should be sufficient to transmit the maximum possible stresses which can occur at the point of maximum stress in the splice. The total stress on the portion of the web between the flange angles = (8" -1") X^ X 14,667 = 44,800 lbs.; the least that can be used for this part of the splice will be 2 — 7x| flats for each flange. The net area of these two plates = 4.5 sq.in., while that of the strip of the web = 3.06 sq.in. The bearing value of the rivets is less than that for double shear, hence one rivet is worth 9190 lbs. at 43.75 the edge of the web plate and 9190 X ."n =8350, an average for this part of the splice. The number of rivets on each side of the splice for splicing the strip = ooka = 5.37 or 6. For the condition of maximum moment there is no shear at the splice and the web stresses are horizontal. The number of vertical rows of rivets is evidently a SPLICE FOR WEB PLATES 49 function of the vertical pitch, the more vertical rows the greater the vertical pitch. Let w=th.e pitch of the rivets in a vertical direction; n = number of vertical rows of rivets; d = distance from the neutral axis to horizontal row of rivets. For three vertical rows or or (w -1)^ Xl6,000Xd/48 = 3 X9190 Xd/48 w = 34.570/7000 = 4.94", say 4|". For two vertical rows {w -1)^ X 16,000 Xd/48 = 2 X9190 Xd/48 «) = 25,380/7000 = 3.62, say 3". For this splice 3 vertical rows on each side spaced 4J will be used, as it gives a web plate with less reduction of strength at the splice. The splice should now be examined for the case of 75,000 shear and simultaneous moment of 3,400,000 ft.-lbs. 9190 X96 From (5) the flange pitch at the center p=— _- „„„ = 11.6" for no part of the web acting as flange. The flange as designed uses 5.25 sq.ins. of the web for flange area; the pitch of the rivets connecting the flange angles , . 9190X96 „ ^ 39.98 ,„^,, to the web IS ^.^^ = ll.bX ^. „^ = 16.6 . 75,000 Xgg^gg The 7xf flats on the vertical legs of the flange angles will therefore be made to take about 6 or 8 rivets on each side of the splice. 50 NOTES ON PLATE-GIRDER DESIGN For a web splice at some point other than the center of the girder, the number of rivets in the flange plates on the side toward the center will be the number required for splicing the web; on the side toward the end the number will be the sum of those, for connecting flange to web plus those for web splice. The extreme fiber stress on the web for a moment of 6,300,000 ft.-lbs. was 16,000; for a moment of 3,400,000 ft.-lbs. it is 8630. For the strip of web between the flange angles the previous determination is evidently ample. A study of the stress intensity at different points in the web for this condition of loading would show that the stress intensity was nowhere as great as for the case for which the splice was designed. A safe resultant rivet bearing of 9190 should not be exceeded. The vertical component on a rivet at either edge of the main splice plates will not exceed 75,000/54 = 1390 lbs., the horizontal component =3.5 X^ X8630 x38/48xi = 3480 per rivet. The resultant = ^ 1390 +3480 =3800. PROBLEMS 9a. What is the relative efficiency of the unriveted plate of Fig. 9a to that of the section of the plate through the first vertical line of rivets in the splice. 9b. Compute and compare the resisting moment of the rivets in the splice of Fig. 9a with that of the net section of the web plate through the first vertical line of rivets. 9c. Make a sketch of a splice for the girder, using only two vertical rows of rivets on each side of the splice. SPLICES FOR THE FLAXGES 51 ART. 10. SPLICES FOR THE COMPONENT PARTS OF THE FLANGES SjDlicing any of the component parts of a girder flange should be avoided, as it is never necessary except to meet some emergency, or reduce the cost slightly. The flange angles and cover plates, for girdei's of the maximum length now used, may be obtained in one i)iece. The cost of a few very long pieces, however, may be so high, due to the expense of shipment in less than proper amounts, or the need of the completed structure may be so urgent, as to make it advisable to permit splicing of component parts of the flange. Flange angle splices should be located when jjossible as follows: a. Where there is an excess of flange section over the required amount; b. Between adjacent pairs of stiffeners, where these pairs of stiffeners are far enough apart to jjcrmit a splice of proper length to be made; c. So that not more than two flange angles shall be spliced in any cross-section of the girder; d. So that the flange angles on opposite sides of the girder flange will be spliced on opposite sides of the center of the girder. The splicing material should be disposed with reference to the shape of the section spliced. The splice for an angle with equal legs should be made by a cover angle with two equal legs. The net area of each leg of the splice 52 NOTES ON PLATE-GIRDER DESIGN should equal the net area of the leg sphced. In general in splicing or connecting material under stress the splice or connection should be arranged to avoid any redistri- bution of stress. This requires that angles with unequal legs be spliced by cover angles with unequal legs, and that the net area of each leg of the one be equal to the net area of each leg of the other. The number of rivets in the splice on each side of the point where the angles are cut should be determined with care. The duty of the rivets in developing the flange stress as well as in connecting the spliced parts should be clearly understood. For the purpose of illustration let it be supposed that for the girder of Fig. 5a the only material available for the flange angles is 4 angles 6 X 6 X J -16' 4i" long and 4 angles 6 X6 X J -28' 4 J" long. This makes it necessary to splice the bottom flange at a point 6' from the center of the girder. Fig. 10 shows the splice in considerable detail. The net area of the two angles = 9.52 sq. ins., and this at 16,000 = 152,320 lbs. as the strength which must be developed by both the rivets and angles of the splice. Two 6x6xA angles with the legs cut to 5V' each and the corner of the angle planed so to fit the fillet of the flange angles will be used for the splicing material. The shear, at the point of splicing, simultaneous with maximum flange stress, is 50,000 lbs. The pitch of the rivets cannot be less than 6" in each line without reducing the flange section (see Art. 3). The horizontal flange increment / per rivet from Formula (5)= — ^Q — X3 = 2540 when none of the web is assumed SPLICES FOR THE FLANGES 53 to act as flange area; for this girder, however, 2.81 sq.in. of the web was assumed to act as flange area. The value of / therefore is 2540 -2540 X 2.81 24.33 " 2540-290 = 2250 lbs. Fig. 10 (a and 6). Fig. 10 (c) shows the forces on a rivet through the wcl) and vertical legs of the splice and flange angles due to increment /. Spl. L" Fig. 10 (c, d and e). Fig. 10 {d) shows the forces on a rivet through the web and vertical legs of the sphce and flange angles due to splicing, for the left portion of the splice. Fig. 10 (e) shows the forces on a rivet through the 54 NOTES ON PLATE-GIRDER DESIGN web and vertical legs of the splice and flange angles due to splicing, for the right portion of the splice. The unit stresses on the rivet will be taken at 12,000 lbs. per square inch for shear, and 24,000 lbs. per square inch for bearing. Consideration of the forces on the rivets and the thick- ness of flange and splice material shows that the shearing value of the rivets is the determining one along the plane AA. 152 320 The number required in the vertical legs =7 — joTfi = 5.3 or 6. Ctx. g h Fig. 10 (/, g and h). ^^SpI.L Fig. 10 (/), (g) and (h) show the forces on the rivets through the horizontal legs of the flange angles of the top flange splice. It is readily seen that the number of rivets . , 1 52,320 ^„ „ required ^J^T^oVq ^ ^-^ ^^ ^■ Fig. lOi. Cover plate splices should be located so that a simple lengthening of an outer cover plate will form the splice. For example if a splice were required in the first cover CONNECTING ONE GIRDER TO ANOTHER 55 plate of the girder of Art. 4, it should be spliced by extending the second cover a distance x, which must be enough to permit the location of a proper number of rivets, shown by the dotted lines in Fig. 10 (i). PROBLEM 10. Design a splice for the flange angles of the girder of Prob. Aa^ which shall be located at the same point in the length of the girder as was the one used to illustrate this article. ART. 11. CONNECTING ONE GIRDER TO ANOTHER The problem of framing one girder into another is one so frequently met with that a connection for two adjacent stringers and the floorbeam between them will be designed to illustrate the method of making the computations for such connections. Fig. 11a. Fig. 11a is a line drawing of a plan showing the relative- position of girders, floorbeams, and stringers. A load on any stringer is carried first to the floorbeams at its ends. The floorbeam carries it to the girders and the girders to the abutments or other end supports. 56 NOTES ON PLATE-GIRDER DESIGN The stringer is composed of 1 web plate 27x1; 4 angles 5X3JX13.6 (|"). The floorbeam is composed of 1 web plate 38 X J; 4 angles 5 X3ix 19.8 (f") The maximum stringer reaction is 97,800 lbs. (ZZ = 48,800 Imp. = 46,600 and dZ = 2400). The maximum reaction for two stringers is 131,700 lbs. (ZZ = 66,300, Imp. = 60,600 and dZ = 4800). The requirements of construction will be assumed to demand that the bottom flange of the stringer be 4J" above the bottom flange of the floorbeam. The connection between the stringer and floorbeam is made by two ver- tical angles which are riveted onto the stringer in the bridge shop. The rivets through the outstanding legs of the connection angles and the floorbeam web must be field driven. The unit stresses are 12,000 and 24,000 for shear and bearing for shop rivets and 20% less than these values for field-driven rivets; |" rivets will be used and the connection angles will be made J" thick so that after they are faced on the back of their outstanding legs they will not be less than f " thick on these legs. Fig. 11 (d) shows how the load W is applied to each rivet and carried to the stiff eners. The rivet should be examined for crushing by the web for W and for shear for^. 97 800 The number required for crushing = .. ^'i-^,^ = 9.3 or 10. 97 800 For shear the number = Jjo- Tq = ^-^ or 7. The upper and lower rivet in the lines xx and zz of (b) cannot be CONNECTING ONE GIRDER TO ANOTHER 57 counted for this connection, as they have other duty to perform. Therefore, in order that the rivets be not closer than 3", a leg for the connection angles which will contain 2 rows of rivets must be used. As 10 rivets are required, a symmetrical arrangement demands 11. These rivets pass through fillers and a good requirement is to extend the fillers for such cases and put 50% additional rivets through {b) Cc) Fig. 11. them. Many good designers consider it allowable to assume that fillers connected to web plates by a line of rivets such as yy increase the bearing value of rivets in lines XX and zz by the entire bearing value of the rivets in yy. Consideration of (c) of Fig. 11 shows that it will be possible to get 16 rivets in the two lines oo and have them a little over 3" apart, and 58 NOTES ON PLATE-GIRDER DESIGN 18 rivets in the two lines oo and have part of them 2| and part 3" apart. Fig. 11 (e) shows how the forces act on a rivet con- necting the outstanding legs of the connection angles to the floorbeam web. The bearing value of a field rivet in the f (net thickness) stifTener leg is 6300. The shearing value of a field rivet is 5770. For the maximum stringer load the number of rivets 97 800 required for shear = ^.i-,„ = 16.95 or 18, which would enable two lines of rivets to be used, if part of the number are 23" apart. The rivets must also be examined for bearing on the floorbeam web when both stringers are bringing their com- bined maximum loads at this connection. The bearing value of a field rivet in the i" floorbeam web =8400 Ibs'. The number of rivets required =—n~-j— = 15.6 or 16, which is one less than the number which are required for maximum shear on the rivets for one stringer loaded. The number of rivets by the second computation is usually, though not in this case, the larger and hence the governing one. The rivets should be located on the lines 00 in (c) of Fig. 11. The two fillers under the connection angles and on the stringer web must be Y' thick and 3" wider than the angle leg. They should therefore be 9" xY' flats. The requirement of two lines of rivets for the leg of the connection angle against the stringer web demands a END BEARINGS 59 6" leg. One line between each leg of the connection angle and the floorbeam web fixes the width of the out- standing leg at SJ or 4". The connection angles cannot be less than |" in thick- ness. Examination of this net section through lines xx and 00 shows them to be ample to resist shear. Some- times they must be increased in thickness for this. The connection angles are therefore made 6 X4 X16.2 (J). PROBLEM H. Design the connection between two adjacent stringers and their floorbeam. The stringer consisting of 1 web24XA; 4 angles 5 X3iX 15.2 (A) The floorbeam consisting of 1 web34Xi; 4 angles 5X3^X13.6 (i); 2 covers 11 xi. The shears being the same as for the example solved in this article. ART. 12. END BEARINGS The end bearings of a girder must receive the load brought to the end of the girder and distribute the same over the masonry or other support. For the single-track girder of Art. 4, which was taken to illustrate the method of finding lengths of cover plates an end reaction of 193,500 lbs. was assumed. For girders less in length than 60' one end of the girder is bolted to the masonry and the other 60 NOTES ON PLATE-GIRDER DESIGN allowed to slide on a plate. This arrangement makes the determination of the reactions possible. The area of the bearing on the wall, assuming the bridge seats to be of granite and 600 lbs. per square inch as a unit stress, should be — ^' =323 sq.in. bOO It is not advisable to make an end bearing of the simple nature indicated in Fig. 12 too long. The tendency of Fig. 12. such a bearing is to overstress the bearing along its inner edge. The upper plate of the two shown at each end is called the sole plate, and is connected by rivets, counter- sunk on the bottom, to the girder. The bottom of the sole plate at (6), the expansion end, should be planed. END BEARINGS 61 The lower plate of the two shown at each end is called the wall plate. The upper surface of the wall plate at (6), the expan- sion end, should be planed. The wall plates are held in position by the two anchor bolts at each end. The bearings at the ends of the girder, when of two simple plates, should not be too wide, as the tendency of the bearing is to overload the masonry along the portion covered by the bottom flange of the girder. The bearing for this case will be made 18 X18". The two plates which project beyond the bottom flange angles must be strong enough in flexure to distribute the load on the bridge seat. Each inch of length of the plates may be considered acted on in a transverse direction by the forces shown in (d). The bending moment then = 5400(9/2 -6.18/2) =5400X1.41=7614 in.-lbs. If the sole and wall plates be made of equal thickness, then the resisting moment of each plate must be 3807 in.-lbs. The depth (or thickness) of plate required may be obtained from M=^ = ^^ = 3807 or ^2=^^^^X6 = 1.43" and d = 1.25" about. These plates should neither of them be less than f"" thick, even if a less thickness would furnish proper strength. For small spans generally only two pairs of stiffeners, one over the outer and the other on the inner edge of the bearing plates, are used. 62 NOTES ON PLATE-GIRDER DESIGN The end stiffencrs for this case should be 5X3J angles and their radius of gyration in a direction transverse to the web is 2.78" approximately. The required area for 1 r IS" 1 18 xOOO 10800 " 'It ' , , . w A^f -i-a/" pn' (c) (d) Fig. 12. ■(e) these stiffencrs =t^^ = 12.7 sq.in. [P = 16,000 -707^7^ = 16,000-750 = 15250]. Therefore 4 angles 5 X3i X12(^) =14.11 sq.in. will be ample. The addition of another pair of stiffencrs over the center of the bearings would help greatly in distributing the pressure over the wall plates, as their outstanding -43'OEfE: — i ton (ff) Fig. 12. legs would prevent an upward deflection of the plates and angles between the pairs of stifTeners. The additional stiffencrs are shown by dotted lines in Fig. 12 (a) and (&). END BEARINGS 63 A better bearing for the ends of this girder could be made by making it shorter in the direction of length of the girder and using either a cast or built-up pedestal to distribute the load in a transverse direction as is indicated in (/) and (g) of Fig. 12. For a length of 12" the width should be 27". The sizes for either case (/) or (g) should be determined from the laws of flexure, and direct stress. For longer girders with greater end reactions to secure proper distribution of the load on the masonry and proper application of the reaction to the girder, a bearing should be used which by means of its form and action will insure this result. This is generally accomplished by means of an upper and lower shoe, both of which may freely rotate about a pin. These upper and lower shoes should have ■EfE. Length- Fig. 12h. proper bending strength and proper bearing on the pin and masonry. These shoes may be either built up of rolled material or cast in one piece. Their detailed design will not be undertaken here, as no new principles are to be developed. PROBLEM 12. Design an end bearing for a girder of a single-track railway bridge of 36' effective length, the cross-section of the girder at the 64 NOTES ON PLATE-GIRDER DESIGN end being 1 web plate 48XA and 4 angles 6X6X19.2(J), the end reaction being 180,000 lbs. APPENDIX ART. 13. POSITIONS OF LOADING FOR MAXIMUM SHEAR AND MOMENT The shears and moments, which are necessary for the design of plate girder bridges, are required for such a great variety of span lengths within the limiting lengths of span for such structures, that special methods of computing and tabulating them are advisable. In the principal Railway Engineers' Bridge Works and Consulting Engineers' offices the major part of the structures designed will be made for some standard loading. The loading known as Cooper E50 is perhaps more widely used than any other for railway bridges. The quantities given in Table No. 1 are for Cooper's E50 and are similar to those which should be determined and kept on record for any standard loading to enable design to be made with proper facility. Shears In computing shears it should be noted that for deck bridges the maximum end shear is given for the engine located so that either the first or last driving wheel stands at the end of the span. The maximum shear at an interior point on the left of the center of the span is given when the load extends from the right and up to the point and perhaps a little beyond. Every load that passes from just to the right LOADING FOR MAXIMUM SHEAR AND MOMENT 65 of the point to just the left of the point under consideration causes a decrease in shear by the amount of that load. Any further movement of the system to the left increases the shear until another load passes from the right to the left of the point. The maximum shear must therefore be determined by trial and generally will occur for the first or second engine wheel just to the right of the given point. For through bridges the maximum shear in any panel W is given when the well-known criterion P= — is satisfied, in which P is the load in the panel, W the total load on the bridge and m the number of equal panels in the span. Sometimes two or even three positions of the load satisfy the criterion, in which case the maximum shear is determined by comupting the shear for all positions which satisfy the criterion. For maximum concentrated load brought to a floor beam or trestle bent by two adjacent stringers or girders: nW The criterion for position is P= — ; — , in which n and m are the lengths of the adjacent spans and P the load on the span of length n, and W the load on both spans. Bending Moments The bending moment, due to a certain number of moving loads at any definite point in a girder either deck nW or through, is a maximum when P=— — , in which P is the load on the left of the point, W the total load on the structure and n the distance from the point to the left 66 NOTES ON PLATE-GIRDER DESIGN end and m the span length. This criterion enables any group of loads to be placed to produce a maximum moment. The criterion will sometimes be satisfied by more than one position of the live load, generally that position which has the heaviest load at the point under consideration and in addition the greater load on the structure is the one giving maximum moment. The moment for all posi- tions of load which satisfies the criterion must be com- puted for a certain determination of the maximum. For a deck structure the point of maximum bending moment occurs at or near the center, generally a little distance away from the center, the location of this point of maximum moment being different for different systems of loading. To aid in determining the position of the loading for absolute maximum it should be remembered that: (a) The maximum moment must occur where the shear passes through zero; (6) For a system of concentrated loads the shear must pass through zero at one of the loads; (c) The amount of load on either segment, into which the point of maximum moment divides the span, is to the total load on the span as the length of the segment is to the span length. These requirements for practical cases fix the point of maximum bending moment under one of the two wheels adjacent to the resultant of the system. The criterion for the exact position -^f the loading to produce maximum bending moment follows : LOADING FOR MAXIMUM SHEAR AND MOMENT 67 For any given system of loading, the loads should be so placed that the center of the span is half loay between the resultant of the system and one of the two loads which are nearest to the resultant. This criterion may be established as follows: Let Fig. 13a show any girder carrying the system of moving loads shown. i=! w o OOOCP I oooo o «^ -(- Fig. 13a. Let a be the center of the span; be the point of maximum moment which occurs under o or n; the resultant of the system = W; c be the distance of W from the right end; the resultant of the loads to the left of o be P; b be the distance from o to P; X be the distance from o to a; Then the bending moment at o = M^R^{\-x) -Pb-^{{-x) -Pb Wc Wcx "^~ I -Ph. («) 68 NOTES ON PLATE-GIRDER DESIGN Now suppose the loading to advance a small distance to the left of dx, then h = h, x = x+dx, and c = c+dx, and the bending moment M' = Ki(^-[x + dx]) -P& as R^=-^^~^mB neglecting terms containing dx^, M'=^—^ ^+^ 1 P^- ■ C«) Wc Wcx Wcdx . Wdx Wxdx ^ r ' Subtracting (a) from (/3) Wdx Wxds Wcdx M -M' = dm= 2~ +~^ +—[— dm_ _W Wx^ Wc^ dx~ 2 ^ I '^ I ' and for a maximum this must equal 0. TV Wx Wc ^ -T+-r+-r-'^ I which establishes the criterion as stated. PROBLEMS 13a. Compute the maximum bending moment for a deck plate girder of 18' effective span length for Cooper's E50. Assume the girder to carry one-half the loading. 13b. Compute the maximum bending moment for a 2.5' span, same loading as 13a. 13c. Compute the maximum center moment for a 25' span, same loading as 13a. TABLE OF BENDING MOMENTS, ETC. 60 ART. 14. PREPARATION OF A TABLE OF BENDING MOMENTS, SHEARS, AND CONCENTRATED LOADS FOR COOPER'S E^ LOADING The great advantage of a table of moments and shears is that it may be prepared in a few days for all spans for which it is at all likely to be needed. Computations for a system of quantities made at one time show by the law of the increase or decrease of the quantities any error, and the similarity of the computations enables them to be very rapidly made. When a system of many loads moves over spans of varying length it is evident that one load produces maximum moment for spans up to a certain length, two loads for a certain other length, and so on. s , I X 5 Fig. 14a. For one load on a span see Fig. 14a. M = Ril -.) ^'-^il -X) =50,000x -^^ I or M = a max., when 50,000Z M- , which, as is well known, is the general expression for max. moment for any span with one load. 70 NOTES ON PLATE-GIRDER DESIGN The general equation for moment for two loads spaced 5' apart is 250,000 +100,000x. M = R{l-5-x)=- l -{l-x-5) X x^ 1 250 000 = 250,000 Xl00,000x -750,000-^ -100,000-y — ' — ^ ' dM _^^^^ 750,000 200,000a; ^ ^ n nr -— = 100,000 ^ -j^ =0. .-. x = 2-3.75. ^l-s-x ^ ^_ 6' . . t t '1 Fig. 146. Substituting for x its vlaue, we have for M = I 250,000 + 100,0001 2 -3.75 I ^ ^ I = 250,000^-125,000 — ^[z-^ +3.75 -5] 156,250 By plotting the curves of moments for different lengths of spans wc see that somewhere between 8' and 9' one load and two loads produce the same moment. To find this point exactly we make the equations for moments simultaneous and solve to find the value of I for which the moments are equal. For this case ^^^ = 250,000Z - 125,000 + i^, whence Z = 8.54'. The next step is to get the general equation of moments for three loads. Make it simultaneous with that for two TABLE OF BENDING MOMENTS, ETC. 7L loads, and solve for Z, which gives the upper limit for two- loads as 11.125'. The equation for 4 loads is -17 = 50,000/ -500,000 + 312,500 — r 1 ^ i t. —^ ^ ^ ~ t [ = — 1 — ^ 7^ .g 1 c -r^ a t' , — ■ ■ ■ ^^ ^kX~ _^^ s-^^ , ^ - 5,- , ^ --" 1 1 S 3 1 6 6 7 8 9 10 11 12 13 U 16 16 17, Spans in Feet* Fig. 14c. The equation for five loads will now be written r. . f • Kf 1 1 2,075,00 dist. of e.g. from right load ^^o^c^qoo" ""^-^-^ • Fig. 14d. The point of max. moment will be under the second driver. Reaction at R\ = 2,075,000 +225,000.T I 72 NOTES ON PLATE-GIRDER DESIGN Maximum bending moment 2,075,000 +225,000x „ „„ , „, .«.„„„ =~ J '- X (l -X -23+13) -575,000 _ 2,075,000+225,000. ^ ^^ _^ _^„, __^,^ „^„ -2,075,000 +225,000. - MZiiOOOx +225,000. I 20,750,000 +2,250,000.T -575,000 ^^ 99r;nnn ^,075,00 450,000:^ 2,250,00 ^ ^ = 225,000 ^ ^ ^ = _ 450,000x ^ _225 QQQ ^2, 075,000+2,250,000 450,000x = 225,000Z - 4,325,000 a; = 2 -9.61. Substituting this value for x in the following simplified expression for the moment 1,500,000 +225,000x - '-^^^f^ - ?^'- - MZ^O there is given ilf = l,500,000+225,000(|-9.6l) -i^E^ (|- -9.6l) 225,000 / Z Y 20,750,000 -g.eiV r~V2-^-"V 1 = 1,500,000 +11 2,500Z -2,162,500 -2,162,500 +^i^^^- 20,783,925 20,750,000 -56,250Z +2,162,500 I I 34 131 ^4. 1 "^i = -662,500+-^^ — +56,250Z = 56,250Z -662,500+^^—!^. TABLE OF BENDING MOMENTS, ETC. 73 Makin;^ the moments for four and fivo loads equal, we have: 50,000^ -500,000 + 312,500 I Q4 1 01 -662,500+^^ +5G,250i 278,369 -6250Z= -162,500 6250^2 -162,500^= +278,369 Z2-26Z = 44.54 Z2 -26^+169 = 44.54 + 169 / -13= ±213.54= ±14.61 Z = 13 ±14.61 =27.61' as the limiting length of span for which four loads pro- duce maximum moments. The tabulation following shows the moments and the jjositions of the loads that produce then for the spans from 9 to 27' for intervals of 1'. Span" 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Mom.= 117,360 140,625 164,200 200,000 237,500 275,000 312,500 350,000 387,500 425,000 466,4.50 515,625 564,880 614,210 663,500 713,020 762,500 812,020 861,570 Limits: / o -rc^ 8.536' a:=2--3.75 t° !'','■> 000 11.124' 7J=50,OOO-^^— M= 25,000^-125,000+ mm' 1.56,2 .50 I ^111 m I Limits: 11.125' to fi= 75,000 18.66' M=37, 500^-250,000 x = --o' iWJ^ M.11U.1 <-6*-5'< Limits: 18.67' to 27.61' .T=:^-8.75' .B= 100,000- 2.50,0 00 I mm §LlllIlll Af=50,000i-.50,000 + 312,5 00 T ■«-5'*«-5^5'->Ki 74 NOTES ON PLATE-GIRDER DESIGN PROBLEMS 14a. Compute the foregoing table up to 38'. 14&. Compute the maximum II end shear for one girder of a 38' effective single-track deck plate-girder bridge. Two girders carry one track. Cooper's E50. ART. 15. TABLE OF MOMENTS, SHEARS AND CONCENTRATED LOADS FOR COOPER'S E,„ The method of writing equations for bending moments of Art. 14 need not be followed for spans over 30' long. For spans over 30' in length the maximum center moment is only a small fraction of one per cent, less than the absolute maximum moment. The following table gives a convenient arrangement of the quantities for spans up to 75' effective length. For through spans tables of moments at the panel points and shears in all the panels for varying number of panels and panel length arc readily prepared and should be made for any loading as much used as Cooper's systems. PROBLEMS. Loading Cooper's E^o. 15a. Compute the live load concentralion for a trestle bent which carries adjacent spans of 30' and 60'. Shuw by a sketch the position of loads. 15b. Compute the maximum center moment for a 60' span. Show by sketch the position of loads. 15c. C'ompute the maximum shear ;it the center of a 60' span. Show by sketch the position of loads. 15d. Compute the maximum shear at a point 15' from one end of a 60' span. Show by sketch the position of the loads. TABLE OF MOMEXTS, SHEARS, ETC. Span End Q. Pt. C. C. Cone. Span End Q. Pt. C. c. in ft. Sh. Sh. Sh. Morn. Fl. Bm. in ft. Sh. Sh. Sh. Mom. 10' 75.0 50.0 25.0 140.6 100.0 to 43' 198.4 122.1 57.3 1856.5 11' 81.9 52.3 27.3 164.3 109.1 CO 44' 201.4 123,8 58.0 1929.0 12' 87.5 54.1 29.1 200.0 116.6 \\ 45' 204.3 125.4 58.6 2001.5 13' 92.3 55.8 30.8 237.5 123.1 o 46' 207.0 126.8 59.3 2074.0 14' 96.4 58.9 32.1 275.0 130.4 CO 47' 209.9 128.5 59.9 2146.5 15' 100.0 62.5 33.4 312.5 136.6 48' 212.6 130.3 60.4 2219.0 16' 106.2 65.6 34.4 350.0 142.3 o 49' 215.4 132.0 61.3 2297.0 17' 111.8 68.4 35.3 387.5 147.0 Tl< 50' 218.1 133.6 62.1 2377.3 18' 116.6 70.9 36.1 425.0 151.6 to 51' 220.9 135.3 63.0 2457.6 19' 121.0 73.0 36.0 466.5 157.3 CO .52' 223.5 136.9 63.8 2.5.3S.0 20' 125.0 75.0 35.9 515.6 163.9 11 53' 226.3 138.4 64.5 2618 4 21' 128.6 78.6 37.0 564.9 169.9 o 54' 229.0 140.1 65.3 2702.6 22' 131.9 81.9 38.0 614.3 175.4 to t3 55' 231.8 142.0 66.0 2791.5 23' 134.8 84.7 38.9 663.6 180.4 56' 234.4 143.8 66.6 28S0 , 1 24' 137.5 87.5 39.8 713.0 184.9 o 57' 237.1 145 5 67.3 2968.9 25' 142.0 90.0 40.5 762.5 189.1 « 58' 239.8 147 1 67.9 3057.6 26' 145.3 92.3 41.4 812.0 194.3 g 59' 242.4 148.8 68.8 3150.5 27' 148.1 94.5 42.1 861.6 200.3 o o 60' 245.0 150.3 69.6 3247 . 1 28' 151.1 96.4 42.9 913.8 205.8 a 61' 247.8 151.8 70.5 3343.9 29' 153.9 98.3 43.5 969.9 210.9 m 62' 250.4 153.3 71.3 3440 . 5 30' 157.6 161.1 100.0 101.9 44.1 45.5 1026.1 1082.3 215.6 E 63' 64' 253.6 256.8 154.8 156.3 72.0 72.8 3537.3 31' "a 2 ■ ,3638.8 32' 164.3 103.9 46.9 1138.6 65' 259.5 157.8 73 . 5 3743.8 33' 167.4 105.9 48.1 1194.9 66' 262.4 159.3 74.3 3848. 8 34' 170.3 107.8 49.3 1251.0 S d s -H 67' 266.0 160.8 74.9 .3963 , S 35' 172.9 109.5 50.4 1307.4 68' 269.5 162.3 75.6 40.58 S 36' 176.4 111.1 51.4 1371.7 -1 § 69' 273.0 163.8 76.2 4163.8 37' 179.6 112.5 52.4 1435.9 70' 276.3 165.3 76.9 4268. S 38' 182.8 113.8 53.3 1500.0 g g _^ £ 71' 279.5 166.8 77.7 4376 . 39' 185.8 115.0 54.1 1566.6 g _ 1" m 72' 283.4 168.3 78.5 4481.3 40' 188.5 116.9 55.0 1639.1 -5 a ° « CO y CD 73' 287.0 169.9 79.2 4588.1 41' 192.0 118.8 55.8 1711.6 ^^.|s 74' 290.6 171.3 80.0 4700.0 42' 195.3 120.5 56.5 1784.1 » S S l2 75' 294.3 172.6 80.7 4813.8 Short-title Catalogue OF THE PUBLICATIONS OF JOHN WILEY & SONS New York London: CHAPMAN & HALL, Limited ARRANGED UNDER SUBJECTS Descriptive circulars sent on application. Books marked with an asterisk (*) are sold at net prices only. All books are bound in cloth unless otherwise stated. AGRICULTURE— HORTICULTURE— FORESTRY. Armsby 's Principles of Animal Nutrition 8vo, S4 00 * Bowman's Forest Physiography 8vo, 5 00 Budd and Hansen's American Horticultural Manual : Part I. Propagation, Culture, and Improvement 12mo, 1 50 Part II. Systematic Pomology 12mo, 1 50 Elliott's Engineering for Land Drainage 12mo, 1 50 Practical Farm Drainage. 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(Hall and Hayward.) 8vo, Burgess and Le Cliatelier's Measurement of High Temperatures. Third Edition. (In Press.) Douglas's Untechnical Addresses on Technical Subjects 12mo, Goesel's Minerals and Metals: A Reference Book .' 16mo, mor. * Iles's Lead-smelting 12mo, Johnson's Rapid Methods for the Chemical Analysis of Special Steels, Steel-making Alloys and Graphite Large 12mo, Keep's Cast Iron 8vo, Mstcg-lf's Steel. A Manual for Steel-users 12mo[ Minet's Production of Aluminum and its Industrial Use. (Waldo.). . 12mo, Palmer's Foundry Practice. (In Press.) * Price and Meade's Technical Analysis of Brass 12mo, * Ruer's Elements of Metallography. (Mathewson.) 8vo, Smith's Materials of Machines 12mo, Tate and Stone's Foundry Practice 12mo, Thurston's Materials of Engineering. In Three Parts 8vo, Part I. Non-metallic Materials of Engineering, see Civil Engineering, page 9. Part II. Iron and Steel 8vo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 Ulke's Modem Electrolytic Copper Refining Svo, 3 00 West's American Foundry Practice 12mo, 2 50 Moulders' Text Book 12mo. 2 50 MINERALOGY. BaskerviUe's Chemical Elements. (In Preparation.) * Browning's Introduction to the Rarer Elements Svo, Brush's Manual of Determinative Mineralogy. (Penfield.) Svo, Butler's Pocket Hand-book of Minerals 16mo, mor. Chester's Catalogue of Minerals Svo, paper, Cloth, * Crane's'Gold and Silver Svo, Dana's First Appendix to Dana's New "System of Mineralogy". .Large Svo, Dana's Second Appendix to Dana's New " System of Mineralogy." Large Svo, Manual of Mineralogy and Petrography 12mo, Minerals and How to Study Them /. 12mo, System of Mineralogy Large Svo, half leather. Text-book of Mineralogy Svo, Douglas's Untechnical Addresses on Technical Subjects 12mo, Eakle's Mineral Tables Svo, Eckel's Building Stones and Clays. (In Press.) Goesel's Minerals and Metals: A Reference Book 16mo, mor. * Groth's The Optical Properties of Crystals. (Jackson.) Svo, Groth's Introduction to Chemical Crystallography (Marshall) 12mo, * Hayes's Handbook for Field Geologists 16mo, mor. Iddings's Igneous Rocks Svo, Rock Minerals Svo, Johannsen's Determination of Rock-forming Minerals in Thin Sections. Svo, With Thumb Index * Martin's Laboratory Guide to Qualitative Analysis with the Blow- pipe c 12mo, Merrill's Non-metallic Minerals: Their Occurrence and Uses Svo, Stones for Building and Decoration Svo, * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. Svo, paper, Tables of Minerals, Including the Use of Minerals and Statistics of Domestic Production Svo, * Pirsson's Rocks and Rock Minerals 12mo, * Richards's Synopsis of Mineral Characters 12mo, mor, * Ries's Clays: Their Occurrence, Properties and Uses Svo, 17 1 50 4 00 3 00 1 00 1 25 5 00 1 00 1 50 2 00 1 50 L2 50 4 00 1 00 1 25 3 00 3 50 1 25 1 50 5 00 5 00 5 00 60 4 00 5 00 50 1 00 2 50 1 25 5 00 * Ries and Leighton's History of the Clay-working Inaustry of the United States 8vo, $3 SO * Rowe's Practical Mineralogy Simplified 12m,o, 1 25 * Tillman's Text-book of Important Minerals and Rocks 8vo, 2 00 Washington's Manual of the Chemical Analysis of Rocks Svo, 2 00 MINING. * Beard's Mine Gases and Explosions Large 12mo, 3 00 * Crane's Gold and Silver Svo, 5 00 * Index of Mining Engineering Literature Svo, 4 00 * Svo, mor. 5 00 * Ore Mining Methods Svo, 3 00 * Dana and Saunders's Rock Drilling Svo, 4 00 Douglas's Untechnical Addresses on Technical Subjects 12mo, 1 00 Eissler's Modern High Explosives Svo, 4 00 Goesel's Minerals and Meials: A Reference Book 16mo, mor. 3 00 Ihlseng's Manual of Mining. Svo, 5 00 * Iles's Lead Smelting 12mo, 2 50 * Peele's Compressed Air Plant Svo, 3 50 Riemer's Shaft Sinking Under Difficult Conditions. (Coming and Peele.)8vo, 3 00 * Weaver's Military Explosives Svo, 3 00 Wilson's Hydraulic and Placer Mining. 2d edition, rewritten 12mo, 2 50 Treatise on Practical and Theoretical Mine Ventilation 12mo, 1 25 SANITARY SCIENCE. Association of State and National Food and Dairy Departments, Hartford Meeting, 1906 /. Svo, Jamestown Meeting, 1907 Svo, * Bashore's Outlines of Practical Sanitation 12mo, Sanitation of a Country House i 12mo, Sanitation of Recreation Camps and Parks 12mo, * Chapin's The Sources and Modes of Infection Large 12mo, Folwell's Sewerage. (Designing, Construction, and Maintenance.) Svo, Water-supply Engineering Svo, Fowler's Sewage Works Analyses 12mo, Fuertes's Water -filtration Works 12mo, Water and Public Health 12mo, Gerhard's Guide to Sanitary Inspections 12mo, * Modern Baths and Bath Houses Svo, Sanitation of Public Buildings 12mo, J* The Water Supply, Sewerage, and Plumbing of Modem City Buildings. Svo, Hazen's Clean Water and How to Get It Large 12mo, Filtration of Public Water-supplies Svo, * Kinnicutt, Winslow and Pratt's Sewage Disposal Svo, Leach's Inspection and Analysis of Food with Special Reference to State Control Svo, Mason's Examination of Water, (Chemical and Bacteriological) 12mo, Water-supply. (Considered principally from a Sanitary Standpoint). Svo, * Mast's Light and the Behavior of Organisms Large 12mo, * Merriman's Elements of Sanitary Engineering Svo, Ogden's Sewer Construction Svo, Sewer Design , 12mo, Parsons's Disposal of Municipal Refuse Svo, Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysis 12mo, * Price's Handbook on Sanitation 12mo, Richards's Conservation by Sanitation Svo, Cost of Cleanness 12mo, Cost of Food. A Study in Dietaries 12mo, Cost of Living as Modified by Sanitary Science 12mo, Cost of Shelter 12mo, * Richards and Williams's Dietary Computer Svo, 18 3 00 3 00 1 25 1 00 1 00 3 00 3 00 4 00 2 00 2 50 1 50 1 50 3 00 1 SO J. 00 1 50 3 00 3 00 7 50 1 25 4 00 2 50 2 00 3 00 2 00 2 00 1 50 1 50 2 50 1 00 1 00 1 00 1 00 1 50 r^-- < 3 i- — > T _ ^-■jH i^^^ 1 1 ! l' 2 3X Rollers — Expansion slioe other end Estimated AVeight 96,800 #= Dead Load 1700 #= p. l.f. Live Load Diagram per T o S -JJTL- Full Lungth of Qii-dor X Hook bolt every ibird tie o'i 8"Guai'd Rail ^ Notched yi over lowest Cover plate LJI^L Fio. lo. Bending, Live 3,340,900 Dead ..... 658.100 3,990,000-^8.5 = 470,500 at 10,000=47.05 aq. in. End shear. Live 196,000 Dead 34.000 230,000 STANDARD NO. 17 80'-0" OVER ALL. DECK PLATE GIRDER, 8.T. THE NATIONAL LINES OP MEXICO Scale i"=l'-0". April, 1907 BoLLER & Hodge, Consulting Engineers, New York Richards and Woodman's Air, Water, and Food from