ONTINUOUS, 
 
 WBRIDGES 
 
 HERSCHEL 
 
\JP* LIBRARY 
 
 or TIIF. 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Received 
 Accessions No. 
 
CONTINUOUS, 
 
 EEVOLVING DRAWBRIDGES 
 
 THE PRINCIPLES OF THEIR CONSTRUCTION AND THE 
 CALCULATION OF THE STRAINS IN THEM. 
 
 WITH MORE ESPECIAL REFERENCE TO THE DESIGNING OF 
 CONTINUOUS PANEL GIRDERS OF THIS DESCRIPTION. 
 
 BY CLEMENS HERSCHEL, 
 
 CIVIL ENGINEER, MEMBER OF THE AMERICAN SOCIETY OF CIVIL ENGINEERS. 
 
 WITH NINETEEN WOODCUTS AND TEN HELIOTYPE PLATES. 
 
 BOSTON: 
 LITTLE, BROWN, AND COMPANY. 
 
 1875. 
 
Cambridge : 
 Press of John Wilson & Son. 
 
 -~7 f 
 
TO 
 
 JOS. P. DAVIS, ESQ., 
 
 CITY ENGINEER OF BOSTON, MASS., 
 AS THE INNOCENT CAUSE OF ITS HAVING BEEN WRITTEN, 
 
 THIS CONTRIBUTION TO THE ADVANCEMENT OF BRIDGE BUILDING 
 
 IS RESPECTFULLY DEDICATED, 
 WITH CHEAT APPRECIATION OF HIS TALENTS AS A CIVIL ENGINEER, 
 
 BY 
 
 THE AUTHOR. 
 
PREFACE. 
 
 THIS little book, in its original shape, was a paper written for the 
 AMERICAN SOCIETY OF CIVIL ENGINEERS, and was prompted by the 
 author s design of the Quinnipiac Bridge, at New Haven, Conn., and by 
 his employment in testing a design for the Eastern Avenue bridge, in 
 Boston, Mass. It has been thought that in its present shape it may 
 be of some use to students in engineering construction ; to experts also, 
 on the subject of continuous girders, at home and abroad, so much of the 
 work as relates to subjects, that, it is believed, are here for the first time 
 treated, it may not be without interest. In this connection, the author will 
 refer only to his method of first calculating continuous girders for given 
 reactions, and then weighing off these same reactions to act under the 
 girders in setting them up on their supports ; to those parts of the book 
 that treat of the extension of this idea to the calculation of continuous 
 girders, by first dividing up the total load into parts that make certain 
 reactions =0, and those that act on the girder as though it were contin 
 uous over all its supports ; and to the cases in which two supports are 
 themselves carried and pivoted upon a third support. 
 
 CLEMENS HERSCHEL. 
 
 BOSTON, January 27, 1875. 
 
CONTENTS. 
 
 PAGE 
 INTRODUCTORY 1 
 
 PART I. 
 
 ARTICLE 
 
 I. HISTORY OP THE THEORY OF CONTINUOUS GIRDERS 2 
 
 II. THE PRACTICE AND THEORY OF THE ELASTIC LINE 8 
 
 III. SOME PROPERTIES OF CONTINUOUS GIRDERS 7 
 
 IV. LOADS AND REACTIONS OF CONTINUOUS DRAWBRIDGES 10 
 
 V. PRACTICAL CONSIDERATIONS IN SELECTING THE KIND OF SUPPORTS 
 
 FOR DRAWBRIDGES 20 
 
 VI. STRAINS IN THE GIRDERS 23 
 
 VII. PRACTICAL CONSIDERATIONS AS TO CHOICE OF FORM AND SYSTEM 
 OF TRUSS, TURN-TABLES, AND CENTRE PIVOTS ; AND IN CON 
 CLUSION 26 
 
 PART II. 
 
 VIII. INTRODUCTORY TO THE MATHEMATICAL INVESTIGATIONS 30 
 
 IX. EQUATIONS FOR THE REACTIONS OF A GIRDER CONTINUOUS OVER 
 
 THREE SUPPORTS 32 
 
 X. EQUATIONS FOR THE REACTIONS OP A GIRDER CONTINUOUS OVER 
 
 FOUR SUPPORTS 34 
 
 XI. EQUATIONS FOR THE REACTIONS OF A GIRDER CONTINUOUS OVER 
 FOUR SUPPORTS, THE Two CENTRE ONES RESTING UPON AN 
 INFLEXIBLE BODY, WHICH is PIVOTED AT THE CENTRE .... 38 
 XII. EQUATIONS FOR THE REACTIONS OF A GIRDER CONTINUOUS OVER 
 THREE SUPPORTS, THE Two RIGHT HAND ONES RESTING UPON 
 AN INFLEXIBLE BODY, WHICH is PIVOTED AT THE CENTRE . . 40 
 
 XIII. IN A GIRDER CONTINUOUS OVER FOUR LEVEL SUPPORTS, THE Two 
 
 CENTRE ONES PIVOTED (SAME AS IN 11), TO FIND WHAT 
 LOADING UNIFORMLY DISTRIBUTED, ADDED TO A GIVEN LOAD 
 ING UNIFORMLY DISTRIBUTED OVER THE SPAN A B, WILL 
 MAKE D = 40 
 
 XIV. IN A GIRDER CONTINUOUS OVER Two UNEQUAL SPANS, TO FIND 
 
 WHAT LOADING UNIFORMLY DISTRIBUTED, ADDED TO A GIVEN 
 LOAD UNIFORMLY DISTRIBUTED OVER THE SPAN A B, WILL 
 
 MAKE C = Q 41 
 
 PART III. 
 XV. NUMERICAL EXAMPLE 42 
 
 APPENDIX 
 
CONTINUOUS, 
 
 REVOLVING DRAWBRIDGES. 
 
 First principles, then rules. 
 
 In order that the human intellect may approach the perfect answer in any physical investi 
 gation, there is necessary the combined application, each in its proper proportion, of 
 three instruments: mathematics, experiment, and a well-balanced, trained, common- 
 sense judgment. 
 
 INTRODUCTORY. The choice of subject above presented probably needs 
 no apology, before an audience of American engineers. After having neg 
 lected the construction of continuous fixed spans, with probably good reason 
 in the majority of cases, as we shall see, but on the other hand carried the 
 construction of certain kinds of single fixed spans (link-bridges) to a degree 
 of perfection and size hitherto unknown, American engineers have suddenly 
 been brought face to face with the problem of continuous spans, by the 
 abandonment of the old form of drawbridge, whose ends are suspended 
 from a central tower by chains or rods, and the adoption of the continuous 
 form of truss for such structures. The number of drawbridges of this latter 
 class is steadily increasing, and some of them have already attained a mag 
 nitude of span never before attempted ; yet it is a painful fact that, in the 
 
 * The word " drawbridge" will, in this paper, be used only in the sense of one of that species 
 of drawbridges that are made to revolve around a vertical axis situated between their ends. 
 
calculation of the strains in continuous drawbridges, unsupported opinion 
 and "good enough" approximations have often had a controlling voice; 
 whereas such guides are evidently out of place in engineering or in any other 
 physical science. And it is, perhaps, only a natural consequence of this that 
 there are many unpleasant rumors (facts are difficult to arrive at in such 
 cases) as to the unsatisfactory performances, wear, and life of ncary of the 
 great drawbridges of the country, both of the old and new styles. 
 
 The subject is therefore submitted as one of practical and ample pecun 
 iary interest to the profession and to their employers. 
 
 PART I. 
 
 1. HlSTOHY OF THE TllEORY OF CONTINUOUS GlRDERS. The CalcU- 
 
 lation of continuous girders commences with Navier, about 1830, who, it is 
 believed, first propounded the theory of the 4t elastic line ; " that is, the curve 
 or form which the neutral axis of a body, following the laws of elasticity, 
 would assume upon being supported by any number of fixed supports, which 
 were first taken as being all on the same level, and then acted upon by a 
 series of vertical forces or loads. Navier showed how, from the equation of 
 this curve, could be deduced the reactions at each support, the value of the 
 moments over the supports and generally along the line, &c. Still, his 
 method led to somewhat laborious calculations, and was therefore but little 
 used, until in December, 1857, a French engineer, Clapeyron, published, in 
 the "Comptes Rendus," an improvement on the method of Navier, which 
 consisted in finding primarily the moments over each support, and then de 
 ducing, from the moments so found, the reactions sought for. Clapeyron s 
 equations required that : 1. The load be uniformly distributed on each span. 
 (The several spans may, however, have different loads.) 2. The supports 
 must all be situated on one and the same level. 3. The moment of inertia 
 of the cross-section of the girder, referred to the neutral axis, must be con 
 stant. Other assumptions made by Clapeyron, but which are common to 
 all the theories of the elastic line, before and since his day, will be spoken 
 of further on. 
 
 The publication of Clapeyron s article seems to have given a great im 
 petus, in France and Germany at least, to the investigation of the properties 
 of continuous girders, and speedily led to the extension of the theory of the 
 elastic line to the cases where the supports are not on a level,* to investiga- 
 
 First noticed by C. KOpke in " Zeitachrlft de Archt. & Ing Vereins zu Hannover," 1856. 
 
3 
 
 tions as to the effect produced by such a change of level in the supports, &c. ; 
 and much has since been written in French, German, and English* technical 
 journals, and other publications, with some books partly or wholly on this 
 subject. [See the list given in the Appendix.] 
 
 It is characteristic, however, of all the books and articles that appeared 
 on this subject, between 1860 and 1873, that they severally treat of special 
 cases only, at least what in the light of the most recent workf appear as 
 only special cases. Thus Mohr develops the properties of girders, contin- 
 iious over two or three spans of any relative length, for uniformly distributed 
 loads, with the supports on a level, or raised or lowered certain known 
 quantities, and for girders of varying section. Laissle and Schuebler do 
 nearly the same ; while Quensel gives the equations for concentrated loads, in 
 girders continuous over two spans, when both spans are equal. 
 
 It was the first intention of the writer to add to these cases, by develop 
 ing the equations for concentrated loads, on a girder, continuous over two 
 spans, when the spans are unequal and the supports either in or out of level ; 
 which then seemed the only set of equations needed to meet the general case 
 here treated of, a continuous girder drawbridge, loaded and supported in 
 any manner. J All such partial investigations have been set at rest, how 
 ever, by the recent work of Weyrauch, previously referred to, which, in the 
 language of the preface, presents a complete " theory of straight girders, 
 continuous over any number of openings from 1 to co, and for any kind of 
 concentrated or regularly or irregularly distributed loads." It even treats 
 the single span girder as but a special case of the general subject above pre 
 sented. Following a work as exhaustive as this, there remains but little to 
 be done in the way of generalization ; and it has been freely used in deduc 
 ing the equations needed to calculate the reactions at the several points 
 of support of a drawbridge. 
 
 The general equations thus taken from Dr. Weyrauch s book, the special 
 equations then derived from them, and their mode of derivation, will be 
 found in Part II. 
 
 2. THE PRACTICE AND THEORY OF THE ELASTIC LINE. It is time 
 now to speak of the value of all these investigations, of the assumptions 
 upon which they rest, and in how far we can judge or know that the results 
 obtained represent the exact truth. 
 
 * The writer cannot state as to works in other than these three languages. 
 
 t The work by Dr. Jacob I Weyrauch cited in the Appendix. 
 
 J However, this would not have met the whole case, as will presently appear; for usually 
 the drawbridge, owing to its being supported at two points at the centre, becomes a girder con. 
 tinuous over four supports, or of three spans instead of two. 
 
The theory of the " elastic line," together with the long train of investi 
 gations that has grown from it, rest upon certain assumptions to be found 
 in most theories of flexure, and very fully set forth in the work to which 
 reference has just been made. In order that a bridge structure should 
 exactly conform to these assumptions: 1. Every member and every part of 
 every member would have to extend or compress strictly equal distances by 
 the application of equal strains per unit of section. 2. If the girder were laid 
 over on its side, that is, not be acted on by what ordinarily are its vertical 
 forces, every part of it would have to be without any strain whatever. 
 3. When set up, no part of it must in the slightest yield otherwise than 
 according to 1. -4. Should it act as a continuous girder on supports that 
 are on a level, its actual supports must, when the bridge is set up, conform 
 strictly to the profile of the bottom of the bridge when it is in condi 
 tion 2. This is a point of some importance, as will appear; and it is well 
 perhaps at once to clearly catch the idea of what is meant by supports on 
 a level" and "supports so and so much out of level," speaking after the 
 manner of continuous girders. In point of absolute level, the supports may 
 all be on a different grade and yet react as supports that are on a level, if 
 the girder, when without strain or when it is acted upon by the calculated 
 theoretical reactions, has an under profile fitting to this broken grade line 
 of the supports ; and, if mention should be made of such a support being 
 lowered so much out of level, it would mean, lowered so much from this 
 primary grade line, &c. 5. In the theories generally given it is assumed 
 that the moment of inertia of the cross section of the girder referred to the 
 neutral axis is a constant. There are, to be sure, methods for calculating 
 the reactions, moments, and strains in girders whose cross sections, and 
 therefore moments of inertia, vary by steps, or according to known gradual 
 increments or decrements, accurately as well as approximately, but they are 
 both so laborious as to make their use unwarranted, save in rare cases; and 
 even then the uncertainty and approximations inherent to the theory in 
 general are not removed by this additional accuracy of taking into account 
 these variations. It is safe to affirm, namely, that in actual practice none of 
 the conditions 1, 2, 3, and 4 ever strictly obtain. Under 1, different pieces 
 of iron do not extend and compress strictly as the strains upon them ; each 
 separate plate, angle, Ti or channel will vary more or less in this respect 
 within its own length and each with the other. 2 is a condition which must 
 be dWTercnt in different forms of construction, link or riveted, or nut and 
 rod, and is probably never absolutely attained in practice. 3 depends on 2 
 in the first place, and even starting with 2 perfectly attained it is just as 
 
^^ 
 
 5 
 
 "\^ 
 
 reasonable to suppose that more or less members will yield somc"\ 
 matter how little, other than by extension or compression of (he nmi"ruil 
 they are composed of, as that they will not at all, especially on being first*^ 
 brought into action; and, again, that different forms of construction will act 
 differently in this respect. 4 is a condition of which, ordinarily, really 
 nothing is absolutely known; it is of the more importance, as very small 
 changes of level produce a great increase or diminution of strains. How 
 this condition may be regulated will be shown farther on. 5 has already 
 been discussed. 
 
 There would seem to be, therefore, no particular reason why we should 
 give implicit faith to the results of the theories based on that of the elastic 
 line, any more than we do to those derived from theoretical hydraulics; we 
 can, on the contrary, as a matter of judgment, gravely question their 
 accuracy, and, so far as known to the writer, there are yet wanting any 
 experiments on finished structures from which coefficients of correction, 
 such as are indispensable in the science of hydraulics, could be deduced to 
 apply to the science of bridge-building, more particularly to that of con 
 tinuous girders. 
 
 Another difficulty lies in the form of the equations presented to the 
 engineer for calculating the dimensions of a desired girder. If the supports 
 are not " on a level," or if it is desired to apply the more exact formulae, 
 which make allowance for the variations in the cross section of the chords, 
 it will be found that these very equations already contain the desired dimen 
 sions, algebraically of course. There is then nothing left to do but to apply 
 the method of successive approximations; that is, first to find the dimensions 
 as well as may be, introducing the value of the dimensions sought in the 
 equation used to find them, according to the best judgment of the calculator, 
 then from the structure so designed to start anew. There is no desire here 
 to overrate the deficiencies of the system as it stands developed to-day, nor 
 to conceal the fact that multitudes of bridges have been built in accordance 
 therewith and are doing good service. The question that remains is: inas 
 much as the formulae do not give exact mathematical results, how near are 
 they to the truth for various kinds of bridges or what are the coefficients of 
 correction? Happily, much uncertainty could be eliminated, and the two 
 last described calculations dispensed with, were the method of weighing off 
 the actual reactions of the finished bridge once substituted for that of calcu 
 lating the proper position or level of the supports, in order that certain 
 reactions might be obtained. To arrive at this method in the development 
 of the proper manner of calculating drawbridges seems very natural, but it 
 
6 
 
 is nevertheless somewhat puzzling that, with the great care and expense 
 applied to the erection of the numerous continuous girders that have been 
 built in France and Germany, this simple method should not have been 
 thought of and practised rather than the uncertain one of calculating 
 minute differences of level of the supports, and then attaining them as 
 nearly as possible. 
 
 To illustrate, take a girder of uniform section, uniformly loaded, of two 
 equal spans, supported on three supports all on a level. 
 
 L_ LJ 
 
 *^ i 
 
 A Fig. 
 
 A, B, C, are the reactions, q the weight per unit of length, therefore 
 from 49 and 50,* A = C = | ql, B= \^ ql, when all the conditions of a 
 continuous girder are fulfilled. But if A and C are each lifted up by a lever 
 or set of levers (the levers ultimately to be kept in position by, say a spring 
 dynamometer) or by any other weighing apparatus more or less approxi 
 mate,! until these scale beams read f ql, and permanent wedges are then 
 driven under A and C until the spring dynamometers or temporary supports 
 return to indicate 0, we shall be certain that we have left the girder acting 
 as a theoretically perfect continuous girder, under that load at least. Again, 
 if the reactions left under the girder were only those for which it had 
 been calculated, it would be a correctly designed structure, no matter 
 whether these reactions were those theoretically due to a continuous girder 
 on level supports or on supports to a certain degree out of level. For a load 
 q l on one span, and q t on the other, the theoretical reactions on level sup 
 ports would be (46 48) 
 
 and further experiment would show whether the same girder, reacting 
 theoretically perfect under the one kind of load, did so under the other 
 kind also, or, if not, what were the actual reactions. From a series of such 
 tests, valuable practical data would speedily be obtained; and in the mean 
 
 * See equations in Tart II. 
 
 t A good hydraulic press that ha<l previously been experimented with and duly rated would 
 make a convenient and direct weighing machine. 
 
while girders would have to be so supported that they should receive the 
 proper reactions when under the loads which produce the most frequently 
 occurring maximum strains. 
 
 3. SOME PROPERTIES OF CONTINUOUS GIRDERS. Before proceeding 
 to speak specifically of only continuous drawbridges, it may be well to 
 become more fully acquainted with some of the properties of continuous 
 girders in general. One of their most remarkable characteristics is the 
 effect they undergo by being supported on supports which are not " on a 
 level." The equations that give the reactions, maximum moments, &c., 
 under these conditions, all contain the moment of inertia of the girder cross 
 section and the ordinates below a level line of the several points of support. 
 Likewise, with given sections of girder, spans, and loads, may be found the 
 ordinates of supports that will produce desired strains and moments. See 
 equations 32-34 containing the term Y, or 59-62 containing Y l and F 2 , and 
 others. A large part of most treatises is taken up with the calculation of these 
 special ordinates; that is, ordinates producing special strains and moments, 
 and the result of some of these calculations it will be instructive to look over. 
 It is proposed in this paper to make no use of any formulas containing the 
 terms Y that is, the ordinates of the support and the moment of inertia of 
 the girder cross section for several reasons : 1 . to abbreviate calculation ; 
 2. because these ordinates cannot be measured practically with the nicety that 
 the equations demand, thousandths of a foot in difference of level usually 
 make tons of difference of reaction and many tons difference of strain; 3. be 
 cause the introduction of these terms, as has been stated, fits the equations 
 indeed for finding the strains in a designed girder, but not for finding the 
 correct dimensions of a girder about to be designed; .4. because by directing 
 our energies toward a different object the weighing of reactions more can 
 be achieved in the investigation of the same subject. In the present state of 
 the science of continuous girders, certainly the value of the deductions 
 about to be given lies rather in the warning they offer as to the effects 
 of accidental or unknown differences of level of the supports, than in 
 the use to be made of them and the formulae they are derived from, in the 
 designing of new works. The following tables, I. for two spans and II. for 
 three spans, are taken from the work of Laissle and Schuebler, and will 
 explain themselves. The object aimed at is to make the cross section of the 
 chords as nearly uniform as possible; that is, equal in maximum in each 
 span to what it is over the centre support in the case of two spans, and 
 equal in all three spans to what it is over the centre supports in the case of 
 three spans. 
 
TABLE I. (for 2 Span,,). 
 
 Lenptfh in ft. of each I 
 span ) 
 
 23. 
 
 32.8 
 
 492 
 
 G5.6 
 
 98.4 
 
 131.2 
 
 196.8 
 
 328. 
 
 Kuti< 
 
 ) . . 
 
 * 
 
 
 
 1 
 5.3 
 
 1 
 4.5 
 
 1 
 3.6 
 
 I 
 
 1 
 2.5 
 
 1 
 1.9 
 
 </ 
 
 WMax. moment 
 
 
 
 
 
 
 
 
 
 in the 2 spans 
 
 0.0927 
 
 00919 
 
 0.0906 
 
 0.0897 
 
 0.0882 
 
 0.0868 
 
 0.0851 
 
 0819 ;/ 2 
 
 
 Max. moment 
 
 
 
 
 
 
 
 
 
 o~ 
 
 over the cen 
 
 
 
 
 
 
 
 
 
 H 
 
 tre pier .... 
 
 0.125 
 
 0.123 
 
 0.125 
 
 0.125 
 
 0.125 
 
 0.125 
 
 0.125 
 
 0.125?P 
 
 i-lc ( 
 
 
 
 
 
 
 
 
 
 
 Max. moment 
 
 
 
 
 
 
 
 
 
 oc *- Z+ 
 
 at centre sup 
 
 
 
 
 
 
 
 
 
 t S 8 
 
 port and in 
 
 
 
 
 
 
 
 
 
 8,8. 
 pc a 
 
 tho two spans 
 
 0.1026 
 
 0.1019 
 
 0.1011 
 
 0.1004 
 
 0.0994 
 
 0.0983 
 
 0.0970 
 
 0.09467/ 5 
 
 *r| 
 
 Depression in 
 
 
 
 
 
 
 
 
 
 ir x ^ 
 
 ft. of centre 
 
 
 
 
 
 
 
 
 
 
 support = s . 
 
 0.0111 
 
 0.0164 
 
 0.0247 
 
 0.0357 
 
 0.0563 
 
 0.0815 
 
 0.1262 
 
 0.2468 
 
 TABLE II. (for 3 Spans). 
 
 Length of 
 middle 
 span in ft. 
 
 = /...,. 
 
 / 
 , 
 
 i 
 
 I 
 
 1 
 
 1 
 
 i 
 
 
 15" 
 
 1 
 
 
 c j^e 
 
 
 
 
 
 
 
 
 
 
 1 
 
 8 = 
 
 0.00042/ 2 
 
 0.00119/a 
 
 0.00148/a 
 
 0.00171/., 
 
 0.00205/ 2 
 
 0.00182/, 
 
 0.00092/, 
 
 
 1 
 
 - 
 
 0.00039 
 
 0.00100 
 
 0.00123 
 
 0.00142 
 
 0.001C9 
 
 0.00155 
 
 0.00083/, 
 
 574 
 
 I 
 
 - 
 
 O.OOO.T7 
 .2123 
 
 0.00093 
 .5338 
 
 0.00114 
 .6543 
 
 0.00132 
 .7576 
 
 0.001 53 
 .8782 
 
 0.00116 
 .8380 
 
 0.00078/.OP 
 .4477 ft. 
 
 328 
 
 * 
 
 
 
 O.OOO.T5 
 .1148 
 
 O.OOOS2 
 
 0.00100 
 .3280 
 
 0.00115 
 .3772 
 
 0.00133 
 
 . 1 ;. _ 
 
 0.00132 
 .4329 
 
 0.00072/, or 
 .2379 ft. 
 
 230 
 
 1 
 
 
 
 0.00033 
 .0739 
 
 0.00076 
 .1748 
 
 0.00092 
 .2116 
 
 0.00106 
 .2438 
 
 0.00123 
 .2829 
 
 0.00123 
 .2829 
 
 0.00008/3 or 
 .1564 ft. 
 
 118 
 
 i 
 
 - 
 
 0.000.32 
 .0473 
 
 0.00072 
 .1005 
 
 0.00087 
 .1287 
 
 0.00100 
 .1480 
 
 0.00115 
 .1702 
 
 0.00119 
 .1761 
 
 0.00004/3 or 
 .0947 ft. 
 
 82 
 
 i 
 
 
 
 0.00031 
 .0254 
 
 0.00006 
 .0541 
 
 0.00080 
 .0656 
 
 0.00092 
 .0754 
 
 0.001 00 
 .0869 
 
 0.00111 
 .0910 
 
 O.OOOO.V.,or 
 .0517 ft. 
 
 GO 
 
 i 
 
 
 
 0.00031 
 .0186 
 
 0.00004 
 .0384 
 
 0.00078 
 .0168 
 
 0.00089 
 .0534 
 
 0.00102 
 .0612 
 
 0.00108 
 .0648 
 
 O.OOOO.V.jOr 
 .0378 ft. 
 
 40 
 
 1 
 
 
 
 0.00030 
 .0120 
 
 o.ooor.i 
 
 .0244 
 
 0.00073 
 .0292 
 
 0.00082 
 .0328 
 
 0.00099 
 .0396 
 
 0.00104 
 .0416 
 
 0.000r>9/,or 
 .0236 ft. 
 
 25 
 
 i 
 
 
 
 0.00030 
 .0075 
 
 0.000.7) 
 .0147 
 
 0.00071 
 .0177 
 
 0.00079 
 .0197 
 
 0.00093 
 .0242 
 
 0.00101 
 .0252 
 
 0.000r>3/ 3 or 
 .0132 ft. 
 
9 
 
 In these tables : 
 
 p own weight per f t. , q = live load per ft. , 
 
 / = span in ft. , both spans equal in Table I. , two outside ones equal in 
 Table II. 
 
 1. 2 span in ft. of middle span in Table II. 
 
 The height of the truss is supposed to be ^ of I and X " /" * for 
 
 o 
 
 Tables I. and II. respectively, with a strain of about 8500 Ibs. per Q in 
 the chords. 
 
 Mr. Mohr, in his article, I860, finds for a strain of about 8,400 Ibs. 
 per n inch, two spans, height -fa of span, modulus of elasticity = about 
 29 million Ibs. per Q inch: 1 + z = 1 -|- 51 -, where z is percentage of 
 
 strain in the chords at centre, due to lowering of centre support, y is 
 amount of such lowering, and h is the height of girder, both expressed 
 in same unit of length. We find, therefore, that for a lowering of the 
 centre support equal to J r part of the height of the girder, the strain in the 
 chord at the centre is already doubled. 
 
 Similarly, for three equal continuous spans, the same admirable writer 
 
 finds 1 + -Hi. = 1 + 09 f , where r = 8ht each . unit of , ft 
 7 r h live load 
 
 Such a girder would therefore require that the central supports should be 
 only Jg- of the height of the girder lower than the outside ones, or that the 
 bridge, when laid on its side and every member without strain, should have 
 a crown at the points where it will rest on the centre supports, above a 
 straight line connecting its two end-supported points of ^ of the height of 
 the truss, in order that it receive a strain in the chords over the centre sup 
 ports of 2 X 8,400 = 10,800 Ibs. per Q inch. Only half the described devi 
 ation of level in the piers, taken in conjunction with half the described crown 
 of the bridge, will produce a like effect. 
 
 For small bridges, especially, the measurements of level required to properly 
 mount a continuous girder are practically unattainable,* and the advantage 
 to be derived from lowering the centre supports is completely swallowed 
 up in the disadvantages resulting from accidental or unknown changes 
 of level. The most advantageous level for the central supports of a three- 
 span continuous girder, as compared with three single spans, will result in 
 a saving of some 18 30% of chord section ; but an equal disadvantage 
 
 * It may not bo out of place to call attention to the fact that this difficulty decreases for 
 largo spans, aud again to note that it can always be entirely avoided by weighing the reactions 
 instead of measuring the relative level of the supports. 
 
10 
 
 arises from a variation of level in the points of support of ^V* to TTo*jnr 
 that is, for small bridges of T ^ 7 and for larger ones of ^J 5 of their own 
 height. 
 
 An endeavor to calculate continuous girders, without having recourse to 
 the measurement of the distances the several supports are " out of level," 
 would therefore seem to be well warranted. The same remark is less true 
 as regards omitting the consideration of variations in the moments of iner 
 tia of the chord sections. The time for this additional approximation will 
 come, when we shall have obtained coefficients of correction from experi 
 ments with many kinds and qualities of finished bridges. It will be to 
 these last, most nearly correct formulae, that the coefficients will have to be 
 applied. At present, the methods to which this paper has been limited can 
 gain but little additional accuracy, by having regard to variations of the 
 moments of inertia, so long as they contain the relatively equal inaccuracy of 
 being without practical coefficients of correction. In the mean while it is 
 gratifying to find that calculating without regard to moments of inertia of 
 the chord sections is of the less importance the more nearly the actual sections 
 are made exactly proportional to the strains in them. For the mathematical 
 case, where the above proportion is exactly obtained, Mr. Mohr finds for two 
 equal continuous spans, uniformly loaded: 
 
 3/ a = 0.146 7^ = moment over the centre pier, 
 
 A = 0.354 ql = reactions at ends, and 
 
 Af = 0.06215 qP = maximum moment in each span, when the change of 
 moment of inertia is taken into account; and, as is well known, 
 
 M a = 0.125^, 
 
 A r=0.375<7/, 
 and M = 070> ql 2 , neglecting the same. 
 
 Similarly, for three equal continuous spans, uniformly loaded, the mo 
 ment over the central piers, tho reactions at the ends, and the maximum 
 moments in the first and third and middle spans are found for the two cases 
 of varying and of uniform moments of inertia : 
 
 f., = 0.100 ?/*, 
 
 A = 
 
 M = 0.0773 qP, 
 M" = 0.0183 qP, 
 
 A =0.4000?, 
 M r=O.OS0^ 2 , 
 M" = 0.025 0/V 
 
 4. LOADS AND REACTIONS OF CONTINUOUS DRAWBRIDGES. 
 Pivot drawbridges, when constructed as link-bridges, usually have two 
 
 See also the article by J. W. Schwodler, 1862, p. 277, cited in Appendix. 
 
11 
 
 centre posts at equal distances each side of the pivot or centre. Indeed, 
 whatever the construction of the girder, two supports at the centre will be 
 the rule, and one only, exactly at the pivot, the exception. In the latter 
 event, the formulas for two equal and those for two unequal spans, for the 
 several kinds of loading, will give the reactions for any case that may arise. 
 It is proposed to discuss in full only the more difficult case of two centre 
 supports, in the course of which the other and less usual case cannot fail to 
 be likewise sufficiently elucidated, the more so as the two have many points 
 in common. Take, first, the supposition of symmetrical loading only; that 
 is, loading such that no tendency to pivot at the centre exists, and let us see 
 the effect on the truss, of the several possible levels of the end supports as 
 referred to that of the two centre ones. The former couple can always, and 
 the latter two can for all cases of symmetrical loading, be considered as 
 though on a level. 
 
 Case I. The bridge open and loaded with snow. In Southern climates, 
 and in those cases of railroad bridges that have no flooring, the snow load 
 may of course be omitted. In this latitude (40 and above) and for highway 
 bridges it cannot be overlooked. The end supports are wanting, the bridge 
 being open, and we have a girder resting on two supports only; and the 
 reactions at each must necessarily be equal to half its own weight, plus the 
 weight of the snow on half its length. 
 
 Case II. The bridge shut and loaded with snow plus the maximum live 
 load. It will be evident that the reactions at the four points of support 
 depend now very much on their relative levels. If the end supports are 
 wanting, one extreme case, the girder would be in the same condition as 
 when open, as regards support, but subjected to the maximum loading; if 
 the end supports were lifted so high as to lift the ends of the girder higher 
 than the two central supports, by an amount exceeding or just equal to the 
 deflection of the girder, acting as a single span-bridge between the two end 
 supports when subjected to the moving load, the other extreme case, it 
 would of course act as such a single span girder alone, and undergo a strain 
 at the centre equal to what it would in Case /., under an equal load and were 
 there only one central support, but with a general reversal of the directions 
 of all strains. Both those cases are, however, practically out of question, 
 and exist only as logical possibilities ; * but between them lie an infinity of 
 
 * Tlio first is too outrageous a case of wrong for even an unconscientious draw-tender to 
 permit; the other could be prevented by not furnishing said draw-tender with any apparatus 
 to lift the bridge off its centre supports. In the case of railroad drawbridges whose ends ave 
 intended to be lifted, the passage of trains before such lifting has taken place may effectually 
 
12 
 
 sapposable cases. Ilaving eliminated from the discussion the consideration 
 of measured differences of level of the supports, and substituted therefor that 
 of the actual weights of the reactions, we next eliminate this latter by con 
 sidering only the several cases when the end reactions arc just equal to 0.* 
 It will readily be seen, therefore, that for the case in hand, a bridge loaded 
 with snow and a live load, the reactions at the ends may be made=:0: when 
 the bridge is empty, when loaded with snow, or when loaded at its maxi 
 mum, though this last brings us again to one of our suppositions that may 
 practically be neglected. Or any given number of tons or Ibs. per unit of length 
 or panel mat/ be considered as acting on (he girder continuously, and the balance 
 will be supported at the central points, which is equivalent to saying that, when 
 loaded with this balance, the skeleton outline of the girder will endure no reac 
 tions at the ends. It is evident that, by thus dividing up the loading into two 
 parts, one acting on the girder continuously, the other supported at the two 
 central points alone, the total reactions at the four several points can in all 
 cases be obtained. 
 
 Another way to explain this process of finding the total reactions by first 
 finding the reactions due to certain elements of the loading is this; and the 
 remarks above, relative to what constitute " level supports," and what 
 "supports out of level," speaking after the manner of the elastic line 
 theory, must be borne in mind in this connection. Take, first, the girder 
 loaded with any part of the total load ?, and the end supports then brought 
 just into contact with the ends of the girder. For all future loads, the reac 
 tions will be the same as those due a straight girder on level supports; in point 
 of fact, it is a curved girder on supports just fitting it, which amounts to the 
 same thing. Sometimes the strains in each member have been calculated in 
 parts, due to different loads, and these then summed up; the plan sub 
 mitted of summing up only the reactions in parts, and then getting the 
 strains of members at one operation, is generally to be preferred. 
 
 A practical choice, for the present, will be to suppose the end reactions 
 equal to or end supports barely in contact, when the bridge is empty. It 
 will shut a little hard in that case, when loaded with snow; but, as the draw 
 is opened least about the time that heavy snow-storms occur, this will only 
 be taking advantage of a convenient provision of nature, and not working 
 
 bo prevented by connecting the signal that the bridge is ready for travel with the bolt that bolts 
 tho draw ends into the fixed bridge, and then arranging this bolt so that it cannot be shut until 
 the draw ends are lifted. Still, tho ingenuity with which such labor-involving contrivances 
 are sometimes circumvented is fully equal to that which produced them. 
 
 * Attention i* here called to the three widely different cases of having a reaction r 0, &plua 
 or a in in a* quantity 
 
13 
 
 against one. Certain drawbridges, as will appear, have the ends raised 
 before being subjected to travel ; but even for such it is proper, unless the 
 draw-tenders are under rules no whit less strong than military rules, to 
 allow for the case where travel goes upon the bridge before the draw-tender 
 has lilted or sufficiently lifted the ends of the draw; in other words, always 
 to take the reactions at the ends for the empty bridge just equal to in 
 calculating Case II. We shall have, then, the reactions at the ends due to 
 a load equal to the live load of the bridge -j- snow load acting on a girder 
 continuous over four supports, or to be derived from equations 51-76, 
 for the several kinds of loading and proportions of length of spans one to 
 the other. To the centre reactions derived from these equations, add the 
 weight of each half girder, to get the total reactions at the two central 
 supports. 
 
 These two, I. and //. , constitute all the cases of symmetrical loading, and 
 we next take up the more numerous cases of unsymmetrical loading. This 
 leads at once to a discussion of the manner of support of the two central 
 supports (of the nature of the turtle, that is under the elephant, that sup 
 ports the world). We shall distinguish two sub-cases : (a) when the two 
 central supports remain firm, the same as though they rested upon a solid, 
 inelastic, stone pier, or as though the draw took bearing upon the circular 
 girder, which in turn rests upon the "live ring"; and (b) where the live 
 ring is not considered as offering support, and the whole weight is thrown, 
 by suitable framing, upon the centre pivot.* It will be readily seen, also, 
 that any deflection that may occur in this " framing " maybe neglected: we 
 have but to lower the end supports an amount equal to this deflection, to 
 render it entirely nugatory. 
 
 For equal outside spans, which alone it will be necessary to discuss, 
 (unequal spans are treated just like equal ones, only that the short span is 
 loaded with concentrated loads sufficient to balance its deficiency of own 
 weight), there will arise twelve different cases of unsymmetrical loading, 
 belonging under (a) and (b) to three different styles of drawbridge, and 
 according as the snow load is considered or not. We find, namely, that for 
 unsymmetrical loading, load on Am (in is point at centre of bridge) whether 
 sub (a) or (&), we very often get the reaction at D first = 0, then a minus 
 quantity. That is, in the latter event, in order that the elastic line shall 
 remain in contact with the end support opposite to the loaded side, the light 
 end must be weighted or force of some sort must be applied to keep it in 
 
 * In some drawbridges the centre pivot is used only as an axis of rotation, never as a sup 
 port; in such, of course, the cases (b) cannot arise. 
 
14 
 
 position. In the absence of such provision, it will rise or "kick up," espe 
 cially if the loaded end is not very firmly supported. Now it requires but a 
 little thought to see that so and so many tons, suddenly rising \ inch, 1 inch, 
 or 1^ inches, &c., many times a day, and as suddenly falling again, and with 
 nothing else to spend their work upon, are only so many foot-pounds assid 
 uously laboring to destroy the bridge or the piers. Such a structure par 
 takes too much of the nature of a tilt-hammer to be entirely satisfactory as 
 a permanent drawbridge. Upon taking up therefore the question as to how 
 to prevent this tipping, we find two methods in practice, the first long in 
 use, the second apparently for the first time used by Mr. Charles Mucdonald 
 of New York. They may be likened to the two methods at command for 
 preventing a horse or an ox or a man from kicking. One way would be to 
 fasten his heels to the floor, the other to lift them up so high that any 
 attempt to kick would indeed diminish the strain upon the chords that held 
 said heels up, but would never result iu their going any higher. In the same 
 way may be managed this restive end of unsymmetrically loaded drawbridges: 
 it may either be lifted up so high that the reaction D shall under no case of 
 loading become either equal to or a minus quantity, or it may be fastened 
 down, this last being Mr. Macdonald s way. 
 
 As previously remarked, and as often found, the ends may also be left 
 free; and we have therefore the above-stated twelve cases (two of each kind, 
 according as we consider the snow load or not) of unsymmetrical loading as 
 follows : 
 
 Case III. (a) ends free, on firm central supports. 
 
 ,, ///. (b) ,, ,, pivoted ,, ,, 
 
 ,, IV. (a) ,, latched, ,, firm ,, 
 
 ,, IV. (1} ,, ,, ,, pivoted ,, ,, 
 
 ,, V. (a) ,, lifted, ,, firm ,, 
 
 ,, V. (6) ,, ,, pivoted ,, ,, 
 
 Caie III. (a). Let us again so place the end supports that when the bridge 
 is empty they shall be just in contact with the bridge, no more and no less; 
 that is, their reactions shall equal 0. Imagine now a girder so supported, 
 no part latched down, to be loaded on length Am (m being point at centre) 
 with dead -f-snow-f live, and on length mD with dead -|- snow. This will 
 generally (it may be proven in each case) give the maximum absolute strains 
 in the chords at least. It would be rdatirrbj more severe for the two main 
 spans of the girder, to suppose one side loaded with dead -f- snow -j- live, 
 and the other with dead only ; but there is a little of the absurd in thus 
 
15 
 
 supposing a bridge only half covered with snow, and at the same time it 
 docs not furnish absolute maxima. We take therefore, in all the unsym- 
 metrical cases of loading, for maximum strains, dead-|-snow-|-h ve on Am, 
 and dead -J- snow on mD. The snow load has, however, an important effect 
 in forming the nature of the end reactions and of some of the webbing in 
 some cases; and it is necessary carefully to consider the bridge with and 
 without snow upon it, as it may easily happen that a bridge that is firm under 
 snow -]- live on one side, and snow on the other, becomes tipping and unsatis 
 factory with the snow load removed. To determine the nature or positions 
 of the end supports and in some cases certain members of the webbing, we 
 therefore calculate all the unsymmetrical cases for dead -[- live on one half 
 and dead only on the other, as well as for snow added to each. In the case of 
 panel bridges having only one panel over the centre, half the centre panel load 
 can be supposed to be applied at the inner extremities each of spans A B and 
 CD : if there should be more panels, the formulae will easily admit of being 
 extended to cover those cases also; and, for plate girders and uniformly 
 distributed loads, the load on centre span can likewise, if desired, be taken 
 into account, so as to make in all the unsymmetrical cases the load on just 
 half the length of the girder. 
 
 To resume, then, the special case under consideration, we have a.girder 
 on four supports, unequally loaded (Fig. 2), and B and C higher than A and 
 D by an amount just equal to the. deflection of the bridge under its own 
 weight when resting on B and C alone, no matter what the exact depth of 
 such deflection be in decimals of a foot or inches. 
 
 Fig. 2 
 
 First, take the action of the bridge under its dead load only. Reactions 
 at B and C are each equal to half of dead on whole girder. Now place 
 end supports in contact. For all future loadings the bridge is equivalent 
 to a straight girder on level piers. Add snow -f- live on Am and snow 
 only on mD. Find reactions for this unsymmetrical load from 63-66 
 (or 69-72), four supports on a level, and to B and C previously found add 
 (algebraically) the new B and C for the final B and C. A and D are found 
 at once and directly. Now see whether C and D, one or both, are plus or 
 minus. We can have both plus (a),* C minus and D plus (/3), (D minus 
 
 * Equivalent, if D is large enough, to Case V. (a). 
 
16 
 
 and C plus is impossible), and both minus (y). If (.,), the above calculation 
 was correct; if (3), it is a case of two unequal spans AD and J5D. Start 
 anew (Fig. 3) this time with the whole of the dead load of the whole girder 
 on B. 
 
 Fig. 3, 
 
 But, if put in that position, it would need still another force or better 
 couple to keep it there, since BD is longer than AB, and the dead load is 
 supposed to be symmetrical about m. Such a balancing couple could be 
 applied at A downwards like a load, and at B upwards like a reaction, that 
 is, diminishing any A to be found hereafter, and as much increasing 11; or at 
 D upwards and B downwards, that is, increasing any D to be found here 
 after, and as much diminishing B ; or finally at both A and D, with the same 
 respective characteristics. Such force is without doubt applied at both A and 
 D; and, to find it, we must first see how much of the unsymmetrical load on 
 Am, when the supports are as in III. (a) (a), will just make (7 = 0, which 
 can be done from 63-66. For this much of the unsymmetrical load and 
 dead -|- snow load, the reactions are as in ///. (a) (a); for the balance of the 
 unsymmetrical load, they are according to 32-34 (or 43-45), for two un 
 equal spans AB and BD, This is evidently a rare case, and moreover moves 
 between very narrow limits of possibility. It would require a proportionally 
 very large live load, and peculiar ratios of loads and lengths, to produce such 
 a case : it is given, however, for the sake of completeness, and well illustrates 
 the general adaptability of the method of separating the effect of combina 
 tions of loads into the effects of certain elements of the same. 
 
 Cane HI. (a) (y). If both C and D are minus (Fig. 4), it is a case of 
 one span and an overhanging arm BD. We now have again the whole of 
 
 0-0 
 
 the dead load upon 72, and the reactions required to balance it about B can 
 be applied at A and B only. These will tend to diminish any future A, and 
 
17 
 
 will equally increase B. The snow will now act precisely as does the dead, 
 and the live will be distributed between A and B according to the simple and 
 well-known laws of the lever. Necessarily A and B are the only reactions. 
 This, again, is a rare case, and requires still greater live loads in proportion 
 to the dead load and snow Equations to meet ///. (a) (/3 and -y) could be 
 evolved by introducing the conditions C, or C and Z>, = 0, and the known 
 values of moments that that implies, in the equations of the elastic line. 
 See book of Wcyrauch, often cited. 
 
 Case III. (7>). Fig. 5. The end supports are so placed, that for the 
 empty bridge, they shall again be just in contact: dead -f- snow -f- live on 
 Am; dead -f- snow on mD. For the dead load, .Band C again each equal 
 half dead load of whole girder. Now find A, J5, C, and D from 89-91 for 
 live on Am, nothing on mD ; also, A, B, C, and D for snow over whole 
 
 bridge from 63-66 (four supports on a level).* C cannot now be a minus 
 quantity: it must equal B, in order that there shall be equilibrium, and 
 there remains only to see whether D is or not. If plus (a), the reactions 
 found are correct; if minus (/3) (Fig. G), take first again B and C each 
 equal half dead load of girder. Then, snow over whole girder from 63-66 
 (four level supports), then find (from 89-91) what part of live will just 
 
 Fig.6, 
 
 QOOO 
 
 = 
 
 make D equal to 0. The reactions for that part are found from 89-91; 
 those for the balance of live from 93-94. Or, a shorter way, take first 
 dead on B and C. If now, in such a pivoted girder, forces 2P on AB 
 
 * The two operations just described with different formulae could be done in one operation 
 and by only one set of formulae by introducing into 89 91 the terms for loads on mD as well 
 as on Am only. 
 
18 
 
 cause D to become =0, that part of the symmetrically distributed load 
 which is on CD will just balance the part on All, about the pivot under 
 B and O, which will leave the 2P acting only according to 93-94 on 
 A,B, and C, and the symmetrical loads must all rest in equal parts on 
 B and C. 
 
 Case IV. (a). Fig. 7. The end supports are so placed that for the 
 empty bridge they are just in contact. Centre supports firm. Ends latched 
 down. Loading as before. We have, first, half of dead on B and C respec 
 tively. From 63-66 find .1, B, C, and D for snow -f live on An, snow 
 
 Fig. 7. 
 
 only on mD, and add middle terms to B and C already found. It matters 
 not now whether the resultant D be plus, minus, or : in either event there 
 is a body at hand to produce the needed reaction. If C is plus (a), the 
 reactions are correct ; if C is minus (/3), Fig. 8, it. is again a case of two 
 unequal spans, AB and BD. Start again with B and C each equal to 
 
 half of dead on whole girder. Find and sum up A, B, (7, and D for snow 
 on whole girder, from 63-66; now find what part of live on Am will make 
 C = from 63 -66 ; that part will produce reactions, also to be found from 
 63-66, the balance will act on A and B only, according to equations 
 32 - 34, for loads on two unequal spans. This also may be considered as a 
 rare case in actual practice. The minus values of D will show the necessary 
 strength of the latching apparatus.* 
 
 The latching apparatus must be constructed to resist, also, a strain that will arise, due to 
 R difference of temperature in tlie two chords, tending to raise the bridge ends, i.e. top chord 
 warmer than bottom chord. See C. Shaler Smith s paper cited below. The effect of general 
 changes of temperature on the strains and shapes of trusses is very lucidly treated in Hitter s 
 b<x>k. cited in the appendix; and a study of the principles of the same would, no doubt, soon 
 lead to the method to bo followed in computing the strains and changes produced by a differ 
 ence of temperature between tho two chords of a bridge. 
 
19 
 
 Case IV. (&). Fig. 9. The end supports as before. Centre supports 
 pivoted. Ends latched down. Loads as before. "VVe have, first, half of 
 dead on B and C respectively ; then, for snow on girder, A, B, C, and D, 
 
 Fig.9, 
 
 to be found from 63-66; also live on Am, with nothing on mD, A, B, 
 C, and D, to be found from 89-91; sum up, and we have the final 
 A, B, C, and D. 
 
 The relative merits of IV. and V. it is proposed to discuss further on. 
 
 Case V. (a). End supports on such a level, relatively to B and C, that 
 they are more than high enough to prevent either A or D from ever being 
 r= 0. Centre supports firm. Loading as hitherto. Now as to the amount 
 the ends should be lifted. The only object of lifting them can be either to 
 prevent what we have called the " kicking " of the light end or to regulate 
 the strains over the centre. We have to deal with the first only at this 
 time. Find, therefore, with what loading of the whole girder the light end 
 reaction is just equal to 0, when the heavy side is loaded with its maximum 
 moving or live load, and then make the end reactions = at a loading 
 greater than this, so as to attain a certain factor of safety against such an 
 occurrence. This subject is treated of, for both the case of the pivoted 
 draw and of two unequal spans, in 13 and 14 of Part II. To illustrate, 
 in the numerical example V. (&), if the ends are raised (see 13), so that 
 2.24 tons* per panel react on A, B, C, and Z>, as a girder continuous over 
 four supports, with B and C pivoted, it will make D equal to 0, when Am 
 is in addition loaded with 12. G tons per panel. In such a case we would 
 take, say 3 tons per panel, as the uniform load, of which the end supports 
 must carry their due share, in order that the live load on Am shall never 
 cause D to equal 0. And it may be noted here that railroad bridges, sub 
 ject to having trains move over them at a high rate of speed, would need a 
 larger factor of safety against this occurrence than roadway bridges. 
 
 In our calculations we therefore take the end reactions equal to 0, when 
 the skeleton lines of the girder are loaded with the own weight, minus the 
 
 * Wherever the word " ton" is used in this paper, it means 2,000 Ibs. 
 
20 
 
 weight selected as above, which we will call 5 (for stability), and proceed.* 
 Just as 5 must now always make D plus, it can and should be required to 
 always keep C plus. The case is therefore analogous to ///. (a) (), only 
 substituting dead s i or dead, and snow -[- live -f- a for snow -j- live, and 
 similarly. 
 
 Case V. (b). End supports lifted a certain amount higher than for 
 /. - / V. Centre supports pivoted. Loading as hitherto. $ is found from 
 1:2, Tart II., which reduces this case to ///. (b) (). Or mid A,B, C, 
 and D from 63-66 (two central supports pivoted), for snow -f- live -\- s on 
 Am and snow -}- s on m D. D will surely be plus, if s has been taken right, 
 and after adding half (dead s) on whole girder, to B and C each, we have 
 the final A,B, C, and D.-\ 
 
 5. PRACTICAL CONSIDERATIONS IN SELECTING THE KIND OF SUP 
 PORTS FOR DRAWBRIDGES. The subject, as left in the last section, is 
 perhaps seemingly complex, but seemingly only. We had in all: /., //., 
 ///. (a)(a,,&y), III. (i) (&/S), IV. (a) (a &/3), IV. (&), V. (a) (a&/3), 
 and V. (i), that is, fourteen cases, or, adding those without snow, twelve 
 more, in all twenty-six cases ; but this is only because of the generality of 
 the treatment of the subject, which adapts it to all cases of double centre 
 post drawbridges that may occur. All those marked ///., in all ten cases, 
 need never, and it is advised should not, occur, or at least reduce to V. ; that 
 is, have D equal plus. For any particular loading and form of truss, only 
 one of a, /3, or y can ever prevail, and the choice of end support determines 
 whether to use IV. or V. For any one drawbridge, therefore, we shall 
 have, generally, only four cases of loading to consider ; and if it is desired 
 to calculate for both, firm centre supports, and for the two central supports 
 pivoted, this will add two more cases, making in all six. 
 
 To calculate for partial loadings of any one span will not be necessary for 
 the chords. This can be proved mathematically: numerical proofs can be 
 derived from the equations given, for the construction of the formula? render 
 the introduction of a single load at any point or distributed loads of various 
 
 s may be taken 0, or even a minus quantity, as far as tipping is concerned, if the dead 
 load is BO great, in comparison to the live, and the lengt.li of spans are so proportioned, that the 
 dead alone prevents D from being equal to 0. This weight * should also be such that the 
 raising of the ends due to it will make I) plus, when the draw chords are unequally affected by 
 temperature, and the draw ends tend to rise on that account, that is, where the bottom chord 
 is warmer than top chord See C Shaler Smith s paper cited below. 
 
 t It is proper to remark that the snow load, or, more exactly, the fact that the snow load 
 reactions are not equal to 0, when the bridge is shut, have introduced a triple element into all 
 the unsymmetrical loadings. By leaving out the snow load, as in the second series of cases of 
 unsymmetrical loading, or by making the reactions equal with the snow load on, the several 
 deductions are simplified. 
 
21 
 
 values per unit over any length in any position an easy matter. Instead of 
 2P substitute for distributed loads f qdx between the limits that the load q 
 is distributed. In case of the webbing, it may occasionally be necessary to 
 consider the spans as partially loaded. 
 
 Generally speaking, Case I. rules the strains at each side of a point about 
 |- the length of the two end spans, measured in either direction from the 
 centre supports; Case II. gives maximum strains over the centre; and the 
 unsymmetrical cases rule the dimensions of the two main spans, from where 
 Case I. leaves off to the two ends respectively, and the strength of the latch 
 ing or position of end supports in IV. and V. There may be some question 
 whether it is wise to make the end reactions = for the empty bridge in 
 Case //., the reason for which has been explained in discussing that case. 
 By making them due to s, or still more, for Case II., we decrease the strains 
 over the centre, and, if certainty existed that the bridge would, before loading, 
 be always supported properly at either end, and Case II. were then calculated 
 for reactions at ends due to 5, Case V. , which follows from the same arrange 
 ment, would have an advantage over Case IV. in producing less chord and 
 web over the centre. If this certainty does not exist, and //. be calculated 
 for end reactions equal to 0, IV. and V. are placed on an equal footing, 
 as far as centre sections are concerned. A great point in favor of IV. is 
 the little time and power required to latch the ends, as compared with 
 that needed to lift them ; though this last should not be overestimated. 
 The power required to lift grows from 0, and can increase as high as 
 half the weight of the girder, if the bridge be lifted off its centre sup 
 ports. 
 
 We shall see, however, in a numerical example, that in point of fact 
 comparatively little, sometimes no lift, is required to keep the bridge steady 
 under the unsymmetrical load, according as the centre supports are pivoted 
 or firm and snow on or off. In the numerical example, when pivoted and 
 with snow off, it is necessary to have the end reactions due to a distributed 
 load of about 4.5 tons per panel, or, roughly calculating for a 200 ft. draw 
 end to end, about f of 25. tons, or about 9.3 tons, and that only at the 
 end of the lift.* By means of eccentrics, toggle-joints, or any other of the 
 mechanisms that have increasing power, such a final pressure is easily pro 
 duced. Such an increasing and rising pressure can also be produced by 
 a constant, descending counter-weight. See " Zeitschrift fiir Bauwesen," 
 
 * In the numerical calculations, s = 4.8 per panel, and A = D, when bridge is empty = 2.9520 
 4-1.7712 + 3.1980 + 1.9188 = 9.84. See table of reactions. 
 
Berlin, 1371, plates 41 and 42, or also "Engineering," Vol. 12 (1871), 
 page 151.* 
 
 In bridges subject to frequent opening and shutting, whatever arrangement 
 is used at the ends, latch or lift, should be operated from the centre of the 
 drawbridge to save time. 
 
 If / V. is to be used for railroad bridges, it must be carefully constructed, 
 and the bolt that serves to latch the draw should be made to lit as exactly as 
 possible in its socket, or all "lost motion" be taken up by the mechanism; 
 if there be any play-room above or below, it would allow of appreciable 
 destructive power being exercised respectively at the light end or over the 
 centre every time an engine runs on the bridge. 
 
 A description of Mr. Macdonald s drawbridge at Point Street, Providence, 
 R. I., 2oO ft. long, and fitted with a latching apparatus, may be found in 
 11 Engineering," March 21, 1873, p. 202. 
 
 Whether the centre supports should, in the calculation, be considered as 
 firm or pivoted (a or &), or the calculation made for both, the construction 
 of the bridge must determine. Drawbridges will be differently constructed 
 in this respect and in that of the manner of end supports, among other rea 
 sons, accordingly as they are more or less frequently opened. It would be 
 an* approximation, when the draw rested on the pivot, to consider it in 
 effect as converted into a girder of two spans only, the pivot forming the 
 middle support, and the framing over the pivot part of the truss, but an 
 approximation only, as may be proven. In a case of symmetrical loading 
 of A H and C-D, for example, (Fig. 10) the truss is evidently a case of three 
 
 Fig-. 10. 
 
 continuous spans. If, now, the reactions A and D are calculated, they will 
 be found less than if calculated supposing the truss to consist of two spans 
 only, with exactly the same loading. 
 
 And the reason for this is not far to seek. The plastic line would take 
 the shape as drawn in Fig. 11, rising higher than the level of the supports 
 
 Fig. II. 
 
 * The same apparatus, together with the peculiar, and in some situations admirable, kind of 
 drawbridge to which It is in this instance applied, is fully and clearly descril>ed and calculated 
 by Hiisfter, Civil Kngineer, in " Zcitschrift des Archt. and Ing. Yen-ins zu Hannover," 18f>9, 
 p. 412. The form of draw referred to is made without any "live ring," and the pivot is used 
 at a support only when the bridge is being opened. 
 
23 
 
 in the middle, or the case would be equivalent to the one of two spans, as 
 sketched in Fig. 12, where the centre support is liiyher than the end supports; 
 
 Fig. 12. 
 
 but this, we knoio, decreases the end reactions, showing why, for a girder of 
 three spans, the end reactions are less than for the same girder considered as 
 composed of only two spans, but with the three supports on a level. Both 
 cannot be right, and from the above three spans evidently is. Making the 
 end reactions less involves making the centre ones greater, and the demon 
 stration just given is as applicable for unsymmetrical loads as it is for sym 
 metrical. To convert the girder into one of two spans only, it would be 
 necessary to give it only one centre post, or arrangement of struts equivalent 
 thereto, these bearing on the centre pivot, or on a single arm framing resting 
 upon the same. 
 
 6. STRAINS IN THE GIRDERS. It may seem strange to some readers 
 that no word has yet been said about any thing but finding the reactions under 
 the girder, as though that constituted the whole problem of the calculation 
 of the strains in continuous drawbridges ; and with but little exception it 
 does. The reactions once found, any structure, of just the proper degree of 
 strength to meet them and the loads that caused them, can be interpolated 
 between the two, without the slightest difficulty or labor other than that 
 inherent to the calculation of any framed structure, if the right way be only 
 followed. And, as this has been written for the purpose of contributing to 
 the elucidation of the construction, more especially, of panel drawbridges, 
 the writer ventures to remark that he shall not consider his labors to have 
 been in vain if, achieving nothing else, he will awaken in the minds of some 
 of his readers a realizing sense of the beauty that seems to be the cor 
 rect appellation to apply to them of the principles used by Aug. Hitter * 
 in the calculation of all link structures, and which it is now intended to 
 describe. 
 
 To understand this thoroughly, let us return to first principles and see 
 what a framed link structure really is. It may be described as a properly 
 designed skeleton system of lines, each line endowed with the faculty of 
 exercising muscular or elastic force in both directions, each line pivoted 
 
 * See Hitter s book, cited in Appendix. In Van Nostrand s " Eclectic Engineering Maga 
 zine" for 1871, pp. 136 and 332, Ritter s method may be found translated and described for two 
 example?, in a way, however, that to the author seems devoid of appropriate emphasis. 
 
24 
 
 at its extremities, and then acted upon in space by generally only vertical 
 forces, or, including roof -trusses, by vertical and horizontal forces, in short, 
 by forces in the plane of the structure; these forces are, in the first case, 
 the reactions and loads, in the second the reactions, loads, and force of 
 the wind. We have given the skeleton lines, the loads, and points of appli 
 cation, and the points of application of the reactions. We must find what 
 has been called above the muscular force required to keep every thing in 
 equilibrium, and the only way, in cases of continuous girders, to do this, is 
 by first finding the reactions; and, going farther, there is much in favor 
 of the statement that, when on account of there being only two supports, 
 as in case of single spans, the laws of the lever (which, however, are 
 only a special case of the elastic line) suffice, and strains, as this mus 
 cular force is more frequently called, can be found without starting from 
 the reactions, the latter method is the best, simplest, and easiest remem 
 bered. 
 
 Taking now a girder as above described, acted upon by its forces (loads 
 and reactions), and considering all parts in. equilibrium, we have in any 
 section: 
 
 1. The sum of all the horizontal components each side the section 
 must = 0, or 2/7=0. 
 
 2. The sum of all the vertical components each side the section 
 must=rO, or 2 F=0. 
 
 3. The sum of all tendencies to rotate about any point in the plane of the 
 girder that is, all moments must=:0, or 2 j\I=Q. 
 
 Hitter s simplification consists in using equation 3 onfy, \vhich can always 
 be done by merely no choosing the points of rotation that they shall be situ 
 ated at the intersection of the direction of two out of three forces which are 
 being investigated. The moments of two thus become 0, and there is left a 
 simple equation for the value of the third. The general enunciation of the 
 principle is this: Conceive a section of the girder, taken in such manner 
 that it shall cut, if possible, only three members, and imagine the forces 
 A , 1", and Z applied at the section, in the line of these members, and rep 
 resenting the strains in them. For that part of the structure which lies at 
 either side of the section write the equation of moments, and so choose the 
 point of rotation that, in determining X, the point of intersection of the 
 directions Y and Z is taken ; in determining F, that of A and Z, and in 
 determining Z that of X and Y. 
 
25 
 
 For example, take a roof truss (Fig. 13): 
 
 D = reaction, P and Q are loads, X, Y, Z strains required, or, y, and z 
 lever arms. Moments with the motion of the hands of a clock, -[-; against 
 it,. 
 
 1. About E, intersection of Z and Y, 
 
 XxP. CE + D.AE = 1 
 
 _ P.CE D.AE 
 
 or X = - 
 
 
 
 2. About -4, intersection of X and Z", 
 
 " Q. ZE = 0, 
 
 or _ 
 
 y 
 
 3. About .H", intersection of X and 1", 
 Zz Q EL P. 
 
 D.AL Q.EL P.CL 
 or Zi = - 
 
 Where a member cannot be cut by a section cutting only two others, it 
 will always be possible (unless there is a redundance of members, and that 
 means indeterminate strains and bad construction) to cut, if there are four 
 together, some one of them, if five, two of them, &c., by a section taking in 
 only three members ; and, their strains once determined, they can be used as 
 known forces in getting the others of the same group. Or, any number all 
 meeting in one point and one more, may be cut, and the strain of this odd 
 one determined ; it being remembered that the section may be taken in any 
 direction, straight or curvilinear. For trusses with parallel chords the 
 demonstration is still correct; we have but to introduce oo as the distance at 
 which the chords meet, and it will be found that the cos all cancel out, leav 
 ing easily remembered equations and forms of equations. 
 
2G 
 
 This constitutes, in fact, the whole method, which needs but to be tried 
 to be appreciated. Its advantages are, primarily, that it throws aside all 
 burdens upon the memory, in the shape of special rules for special cases, rules 
 for each different kind of roof or bridge truss, not to mention a set of rules 
 for variations of one and the same kind of girder.* The advantage of this it 
 is difficult to overestimate; it exceeds, if any thing, the value of the well-known 
 Napier s rules in Spherical Trigonometry. Another value lies in the readi 
 ness with which the strain on any one member can be calculated independently 
 from that on any other; and, finally, in simple trusses, in being able to 
 recognize at a glance, from the form of the equation for the strain on each 
 member, what loads diminish and what loads increase the same ; that is, 
 under what loading it is maximum or minimum. The numerical examples 
 have all been worked according to liitter, and the reader is referred to them 
 (and more especially to Hitter s book) for more on this branch of the sub 
 ject. In the cases of finding the strains on the diagonals, it is frequently 
 simpler to choose the first of the equations of equilibrium written above, 
 instead of Hitter s method ; that is, after having found the chord strains, 
 by remembering that the horizontal components must equal 0, the diagonal 
 strain becomes merely the horizontal component of the strain in one chord 
 minus that of the strain in the other, multiplied by the secant of the angle 
 the diagonal that connects them makes with the horizon, and it is of the 
 kind of strain, tension or compression, according as needed for equilibrium. 
 If the two chords are unlike, it always has the same quality, t. or c. , as the 
 smallest; if both chords are alike (happens in Warren girder), the quality 
 is opposite to that of the chords, and must equal their sum instead of 
 difference. 
 
 Plate girders are not specially treated of in this paper; nothing new is to 
 be said of their calculation, and, by getting equations for moments from the 
 reactions, the curves of moments (and lines of shear from reactions) are 
 easily drawn, and calculations of sizes of parts made as well known and long 
 used. The value of plate girders for drawbridges is seriously diminished 
 by the large surface they expose to the force of the wind. Cases are not 
 wanting, where such bridges have been blown off the centre pier or have 
 shut with such force as to break the centre pivot, &c. For large spans 
 especially, their use must be decidedly condemned. 
 
 7. PRACTICAL CONSIDERATIONS AS TO CHOICE OF FORM AND SYSTKM 
 OF TRUSS, TURN-TABLES, AND CKNTIIE PIVOTS; AND IN CONCLUSION. 
 
 Most of these rules are, moreover, of tho perplexing kind, sometimes known as "sink or 
 swim " rules; i.e , the answer sought Is found In one of two ways, but you cannot tell which. 
 
27 
 
 Lnissle and Schuebler, in their work of 1870, give at the close a digest of 
 16 typical iron bridges that were built between 1855 and 1870, of all con 
 ceivable shapes and forms, both as regards chord, web, position of platform, 
 number of trusses and corresponding number of tracks ; two of them contin 
 uous over three spans; in fact no two alike, reducing them all to the same 
 strain per unit of section, under the same load per unit of length. The 
 result, if intended as an exhibition of the superiority of one style over 
 another, is entirely nugatory, with the single exception that it is clearly 
 better to support two tracks on two trusses only, than it is to support them 
 on three or four. The authors arrive at the conclusion that the economy of 
 each bridge depends, not on the choice of any one from among the dozens 
 (hundreds it might almost be said) of good styles of skeleton outlines of 
 chord and web that are fit to be used in any given locality, but on the skill 
 of the constructor in each particular case; but this again is resolved, almost 
 wholly, into that of reducing to a minimum that part of the weight of the 
 bridge which is literally "dead" weight, that is, carries no load, does not 
 exercise muscular energy, and is only so much apoplectic, adipose tissue. 
 Again, every member necessarily brings with it into the bridge more or less of 
 this very dead weight ; the less members, therefore, other things being equal, 
 the better the bridge, the more load it will carry with the same total weight 
 of material in it, or, conversely, the less material it will require to carry a 
 given load. The rules for a choice of system for drawbridges will not be 
 different from those for single fixed spans, with some few exceptions, due to 
 reversal of strains from c. to /., and vice versa. 
 
 At first sight, it might seem as though the height of the truss should be 
 in a very different proportion to the span in draw from what it is in fixed 
 bridges; because, when the draw is open, the span, of which each side rep 
 resents half, is in fact double that of the doubly supported single span, of 
 equal moment. It must not be forgotten, however, that when it is in this 
 position the bridge is not loaded. Taken altogether, therefore, about the 
 same proportions of height to span will obtain in continuous as in single 
 spans, though with a leaning towards greater height in drawbridge spans 
 than in single spans of equal length. In a recent competition for a draw 
 200 ft. over all and for two fixed spans each 121 ft. long, the same designer 
 made only $000 difference in favor of 21 ft. trusses as against trusses 15 ft. 
 high, on a bid for the latter height of $50,560. Perhaps Mr. Baker s* in- 
 
 * On the Strengths of Beams, Columns, and Arches. B. Baker, London. E. and F. N. 
 Spon, 1870. 
 
28 
 
 vestigation as to the proper height of single span trusses and plate girders 
 could be extended to the consideration of continuous spans also. 
 
 In continuous drawbridges, more than in single spans, the defects of sep 
 arate counter-bracing become apparent. It is very much a question with 
 the writer, whether separate counter braces and ties have not been used far 
 too often in all structures, wooden or iron, but especially in the latter. A 
 member that will resist both tension and compression can nearly always be 
 made to weigh less than one to resist compression or tension running in one 
 direction, added to its counter of like quality running opposite to it, especially 
 when we have regard in the two cases to the useless dead weight above spoken 
 of. In the way indicated, that of introducing members capable of resist 
 ing both t. and c., wherever they are liable to exchange of strain, any kind 
 of truss can be made to dispense with the customary counters and their extra 
 weight.* This is recommended in all continuous spans, drawbridges in 
 cluded. In case of wooden trusses also, an iron rod parallel and beside the 
 wooden strut, so as to make a counter- tie instead of counter-brace, would 
 probably make a good wooden continuous truss. 
 
 A point to be careful about also is that of either superfluous or deficient 
 members. The numerical example is one sent in in a recent competition for 
 a drawbridge, slightly altered. As first designed, the end panels were as in 
 Fig. 14, the dotted line representing a tie rod. Now when such a truss is 
 
 Fig. 14 
 
 latched down, which is equivalent to putting a concentrated load on A, it 
 will exert compression on the piece T. T cannot take, however, even the 
 compression due to the loading on its own system, being only a rod, and 
 should be made so as not to suffer from such compression, by haring a link 
 in it, or by being passed loosely through the top or bottom chord, &c. But, 
 when thus cleared of compression, the minus reaction at A, which should 
 
 * It is curious sometimes to sec counters introduced where, in point of fact, tliey appear like 
 "cats in a strange garret." A "Post" truss, for example, is nothing but a Warren girder 
 whose struts and tics arc at an unequal angle with the horizon; and this* web, ins:. -a. I of being 
 made, like all other Warren girders, to take cither /. or c. at the centre of the truss, is fitted out 
 with counters in the shape of tension rods. They are, however, more proper here than in some 
 other cnses, owing to the great length of the regular tension members, which without them 
 would have to be made to resist compression also. 
 
29 
 
 pervade the whole webbing, can act on one system only, the one drawn in 
 double lines, which must be characterized as undesirable. For this reason 
 the end has been changed as in Fig. 15. Each system can now take any load 
 
 Fitf. 15. 
 
 or reaction due to its own loading, and the end post belongs to the second 
 system, the one drawn in single lines.* 
 
 A valuable paper on turn-tables and pivot centres by C. Shaler Smith will 
 be found in the July number, 1874, of the " Transactions of the American 
 Society of Civil Engineers." The data given there are probably unexcelled 
 by those to be found scattered in the descriptions of various drawbridges. 
 It is more such data that the profession is sadly in need of. 
 
 In the same number, Mr. Macdonald cites an equation from the (** Journal 
 of the Society of German Engineers") Zeitschrift des Vereins Deutscher In- 
 genieure, Berlin, April, 1874, for the safe pressure on bearing rollers. The 
 same subject may also be found analytically treated by Jules Gaudard, in 
 "Etudes comparatives de Divers Systemes de Fonts en fer," quoted in 
 Spon s Dictionary of Civil Engineering, p. 789, and in Winkler s book, 
 cited in the Appendix. 
 
 The writer makes no apology for the length of this paper or of any 
 formulae to be found in it. There is no " royal road " to the calculation 
 of the strains in continuous girders, and the methods described in this 
 paper are believed to be an improvement upon and are shorter than any of 
 the exact modes of calculation that have come under the author s notice. 
 It has been to him a pleasant task, though task it was. But there is no 
 branch of engineering so cleanly cut and finely moulded in its results as 
 this of finding the strains in framed structures. The reactions must first 
 be equal in algebraic sum to that of the total loads; then, around any 
 point of meeting of the several members in the structure the vertical com 
 ponents up must just balance those down ; those to the right must balance 
 those to the left; and any member cut in two must reveal in its interior 
 an elastic or muscular force that will prevent revolution about that point 
 
 * There is still in this arrangement some indeterminateness; viz., the absolute direction that 
 the reaction at A will take, whether up the end post or the first strut or what part on each. 
 This is a defect common to all multiple systems of web, but is not of serious importance. 
 
30 
 
 around which the parts of the structure would rotate, but for the duty by it 
 performed. And the members and reactions must do all this, if need be, 
 to any desired place of decimals, just as in the ledgers of some stately 
 mercantile house the separate accounts and the trial balances must all foot 
 up equally on both sides to the fraction of a cent. The principles involved 
 in the one case are no more certain to produce the desired results than those 
 upon which hinges the other; and, if those clear results are not forthcoming, 
 it is but a case of human error or imperfection. The principles remain, 
 perfect, immutable, and eternal. 
 
 PART II. 
 
 8. INTRODUCTORY TO THE MATHEMATICAL INVESTIGATIONS. The 
 arrangement of this Essay is intended to be such that, while the argument 
 is all to be found in Part I., and its application illustrated in Part III., 
 Part II. shall contain only the mathematical demonstrations, or the deriva 
 tion of the formula? presented for use, and used in Parts I. and III. The 
 reader that is prepared to accept these formula? on faith, therefore, need not 
 read Part II. at all, but will, it is expected, find in the remainder of the 
 paper sufficient to enable him to design any revolving drawbridge. Those 
 that seek for mathematical proof will find the basis of Part II. in the work 
 of Weyrauch, to which reference has many times been made ; and the equa 
 tions, taken from it, that are needed for the development of the formula? it 
 is now proposed to find, are the following : 
 
 17. M,_, u 2 .i/ r (u + /,) + j/ r+1 1, = - ~ + -^ 6 EI 
 
 _Ji I p a (I a) (/ + ) ^ Pa (I a) (21 a). 
 
 20. -V.= A =j [j/ J/+ 2P (I a)] , and 
 
 21. V l = A =- l [j/ M> + 2 Pa] , 
 
 in which the several characters have the meanings indicated by Fig. 1C, and 
 the schedule given below. 
 
31 
 
 Fig. IB. 
 
 M r , M r+l &c., are moments over the supports, at left hand of the spans 
 
 4, JN-. &c. 
 M & AT are the moments over the left and right hand supports at either 
 
 end of the span I. 
 
 l> Z r Z r+1 &c., are the lengths of spans. 
 c r , c r+1 &c., are the ordinates of the points of support at left hand of spans 
 
 Z r , Z r+1 &c., referred to the axis of abscissas. 
 E represents the modulus of elasticity. 
 / represents the moment of inertia of the beam, referred to its neutral axis. 
 
 P represents a load, or element of load, within the span /. 
 
 i 
 
 2 P indicates the summation of all loads P between and I. 
 
 2P, 2P, &c., indicate the summation of all the loads P, in the spans 
 
 * /H-I &C - 
 
 a is the distance of point of application in the span I of the load P, from 
 
 the left hand support. 
 <?r ?r+i &c., are the amounts of uniformly distributed load, per unit of length, 
 
 in the spans / r , / r+1 &c. 
 
 V a & Vi are the vertical shearing forces at and at I. 
 A & A are the parts of the reactions at the left and right hand extremities 
 
 of a span Z, which are in equilibrium with the loads on that 
 
 span. 
 
 2 P (I ) means the summation of all the loads in the span I, each multi 
 plied by the distance of its point of application from the right 
 
 hand support; and 
 2 Pa, the same sum multiplied by the distance of its point of application 
 
 from the left hand support. 
 
 We pass now to the development of the general formulae required for the 
 calculation of pivot drawbridges. 
 
32 
 
 9. EQUATIONS FOR THE REACTIONS OF A GIRDER CONTINUOUS OVER 
 THREE SUPPORTS. Fig. 17. In 17, above, make r= 2, r 1 = 1, r-f-l = 3, 
 
 B-M B * Fig. 17. 
 
 a in the span ^ = 6^, and a in the span I 2 = l z gl r M^ and 3/ 8 are each 
 = 0, the ends being free (not walled in). Hence 17 becomes 
 
 
 For convenience in writing, put the term ^ ?-j--? ? GEI=Y t 
 
 L LI / 2 J 
 
 and find the value of M 2 : 
 
 I ^ 
 
 Now, from 20 and 21, introducing the new notation as just now in 17, and 
 making M and M 8 = : 
 
 30. B=5 1 +B a =- 
 
 Substituting the value of Af 2 found above : 
 
 V- a w - 2 
 
 oo >i _ _ fi __ fa 
 
 I sPft-f.) 
 
 , 
 
 7+ 1 SPrf, (V-e tf) +1 2 
 -- 
 
33 
 
 34. C = 
 
 I 
 
 Generally 1^ = 1.^=. /, and the points of support are on a level, in which last 
 event Y vanishes. Then after a few changes we get : 
 
 35. 4=a-(4-[5-"]) -2 ^(1_^), 
 
 36. 5 = 
 
 37. 
 
 which are convenient forms for calculation. If further c=g and P=Q 
 [equal spans, supports on a level, equal panel lengths and corresponding 
 panel points equally loaded] : 
 
 . C = 2 [|- (4-0 [5-^])] -2 [-*(! _.*)], 
 
 38. .4 = C = 
 
 39. = 2P[e(3 e 2 )]. 
 
 For uniformly distributed loads, q in span ^ and <? 2 in span Z 2 , for F 0, 
 i.e. the three supports on a level, r 1 = 1, r = 2, and r-{-l = 3, 21/ 1== M S = Q 
 as before, 17a would become : 
 
 Similarly we should have for values of ^4 , ^, and C, derived from 20 and 21, 
 substituting 
 
 a:) for 2 P (/ a) in span ^, 
 
 and I <7 2 rfa:. a; for the same in span Z 2 , 
 
 /* 
 
 also I q l dx. x for 2 Pa in span Z 1? 
 
 o 
 
 ^^ 2 
 and I 5- 2 (fa (/ 2 a:) for the same in span / a : 
 
 ^^ 2 
 
 I 5- 2 (fa (/ 2 
 
34 
 
 41. B = - 
 
 Or substituting value of 
 
 
 These are sometimes found in the forms 
 
 _ l 
 _ 3g aa 
 
 ~ 
 
 If in these ^ / 2 ?, 
 
 46. A = ^<7fc 
 
 47. *=!(7i + fc) 
 
 48. C=(7? a 7i)- 
 
 Finally for q^ = q. 2 (equal spans, supports on a level, both spans equally and 
 uniformly loaded) : 
 
 49. A = C = *ql. 
 
 50. B = i-ql. 
 
 10. EQUATIONS FOR THE REACTIONS OF. A GIRDER CONTINUOUS OVER 
 
 Fig. IB. 
 
 I I 
 
 A B C 
 
 FOUR SUPPORTS. Fig. 18. Making in 17, r first = 2 and then = 3, first J/, 
 and again A/ 4 become = 0, and the equation becomes 
 
35 
 
 designating for brevity [^2 + 8_f] 6 / by F t 
 
 L ^ / 2 J 
 
 and c 2~~ c 3-{- C ^~~ c 8 1 6 / by F 2 , replacing also the general load P and 
 L 1 2 3 -I 
 
 distance a by P, Q, 72 and x, y, 2 in the spans l v L 2 , / 8 , respectively. Solving 
 these equations for M. 2 and J/ 3 , we obtain : 
 
 / 
 52. M 3 = - i- 
 
 + - 2 Px (V /) + y- 2 Qy (/ 2 y) (2 Z 2 
 
 *i h, 
 
 Comparing these,. we find that the first quantity enclosed in [] in the value 
 of J/ 2 is entirely similar with the second quantity enclosed in [], in the 
 value of l/ 3 , and that the same similarity occurs between the second term 
 of M. 2 and the first of M B , so that for convenience we may write, 
 
 - 3 " U~ 
 
 From 20 and 21, making Jl/j = Jf 4 = 0, we obtain 
 55. ^=l 
 
 56. 5 = -3f 2 
 
 57. C=M- 
 
36 
 
 Substituting the values of Af 2 and M 8 given above, and reducing, these 
 become : 
 
 59 x _ 
 
 r-i,ir spft-*) 
 ku i~ 
 
 60. B 
 
 , SQ(/ a y) 
 
 "*"" 
 
 Qy 27? (/ 8 2 ) 
 I r ~; 
 
 4 *3 
 
 -vr "1--V 
 
 Usually F! and F 2 will = 0, the supports being on the same level, 
 Z 2 = mZ, x = el, Z 8 2 = </Z, and Q = 0. Under these conditions 
 
 63 A-* 
 63. ^ _ 2 
 
 0*), 
 
 = r- [m 2 4 (1 -|- m 2 )], and 
 
 e\ fl 20+)a + 
 - [_! -- _ -^ 
 
 ^ 
 
 e, c== 
 
 
 fi R /) sizn ^ fi 2(l+mHL+^l 
 
 66. ^ = 272(1-^7) [I- - 3 - 2 8m1 
 
 . 2 P n -\ _J^ 
 )2 
 
37 
 
 If further the loading be symmetrical, that is, if e = g and P = 7?, a further 
 reduction is possible : 
 
 67. A=D = *P<l-e 1_ 
 
 3 m 2 -f- 8 m -{- 4 J 
 
 68. ^c 
 
 Similarly for uniformly distributed loading, 2 P will become f 
 
 l l 
 
 2 Q ,f q. 2 dx, 2 R ,f <? 3 dx, and we shall have from formulae 59 - 62 : 
 
 69 A - 8 i r * ft V] , i i 
 
 4[ 
 
 70 . B== ft + ^ ( 2/ i + 
 
 7]L c _ 
 
 72 D - ii - zi2 i a , 4 . , 
 
 -- 
 
 When again 1^ = 1^ = 1, I 2 =.ml (in for drawbridges would be a proper frac 
 tion), also q- L = q 3 = q: 
 
 73 ^ = jD= i , i 
 
 ~ 2 2 f 2 " ?2 2 --. 
 
 74. 5=C = 
 
 4 J 
 
 ., 
 2 (o ?n 2 -{- b m 
 
 Finally when q. 2 = q and m = 1, that is ^ = / 2 = J 8 = /, 
 
 75. ^=1) = ^^, 
 
 76. J5=C = iJ^. 
 
 There remain to be developed the equations for the very peculiar cases, 
 when two of the supports are themselves supported on the ends of an inflex- 
 
38 
 
 ible body, which, in turn, rests upon a knife edge or pivot at its centre. 
 (Fig. 19.) It is evident that, for equilibrium, the reactions of such supported 
 
 Fie. 13. 
 
 B-t-C 
 
 supports must be equal. An incidental quality is that, when such supports 
 are not symmetrically loaded, the ordinate of one support, measured from 
 the line connecting them when in a horizontal or symmetrically loaded posi 
 tion, equals the ordinate of the other measured in the opposite direction, or, 
 for a three span girder, c 2 = c s .* The calculation of the equations for the 
 reactions under these conditions is tedious but not difficult, after the method 
 to be followed and the circumstances to be taken into account are once 
 clearly imagined. 
 
 11. EQUATIONS FOR THE REACTIONS OF A GIRDER CONTINUOUS OVER 
 FOUR SUPPORTS, THE Two CENTRE ONES RESTING UPON AN INFLEXIBLE 
 BODY, wincii is PIVOTED AT THE CENTRE. For our cases of unsym- 
 mctrical loading, we must make Q = R = 0. We then have the main con 
 dition of equilibrium, that B=C (see fig. and equations in 10). 
 
 77. B = 4-\ 
 
 or 
 also, 
 
 78. 2 H. 2 
 
 79. AT, L 
 
 80. 
 
 value of 
 
 ?^3 I ^3_ 2P.T 
 
 / ~ / / ~~~ * 
 
 Substituting in 78 and finding 
 
 * This is exactly trnc when the pivot is situated in the line joining B and ( , and cqui-diHtunt 
 from these points. There is no ditliculty about developing the equations whenc, = rtc a , or 
 even/(c a ) = /(t-,). 
 
39 
 81. 3/3 = y / 2/ Ji"7 2 y?w/ 2 \ 5 / v Substituting 80 in 77 will give 
 
 z" v 1 "I 2/ v. 2 ~T~ 8/ 
 
 82. J/ Q = 
 
 Make now ^ = / 3 I, and / 2 = ml, and put 81 = 82, we get, 
 
 83. S R = ml 2 T. Substituting values of S, 11, and T t we get, finally, 
 
 84. Y 2 Y l = l- 
 
 and 
 
 F 2 = P 2 ~ C3 + -^^- 3 1 6^7, also ^ = ^ = 0, and c a = <?,; 
 L / / J 
 
 85. Y 1 = -2 
 
 86. -F 2 = 2 P* (^ _ x 2 -j- 
 
 * 
 
 Substituting values of Y l and F 2 in 78 and 79, and reducing, we get, 
 
 87. OKI 9 ^^ ^ 
 
 2 3 m 2 -{- 8 m -j- 4 
 
 88. ^12 q - - - ^r - 5 j 5 - j -. - - * 
 
 3 m 2 -(- 8 m -f- 4 
 
 These values substituted in the general equations for A, B, C, and D (55-58) 
 [(2 = 72 = 0, / 1 = / 8 = /, / 2 = mZ] give finally,- 
 
 8 , ^ 
 
 i 
 " 
 
 (2 -f m) S Pa; (Z 2 a 2 ) -J- (2 + 3m) ml 2 2 P* 
 a 
 
 and A -f + C + Z> = 2P, as they should. 
 
 To bring these equations to the same known quantities as equations 63-66, 
 we have but to substitute el for x. 
 
 * From the values of Yi and y 2 , in terms of c, m, and /, and the same values in terms of 
 2 Ac, m, and /, the value of c can readily be found, but it will contain the term El. 
 
40 
 
 12. EQUATIONS FOR THE REACTIONS OF A GIKDER CONTINUOUS OVER 
 TIIKEE SUPPORTS, THE Two RIGHT HAND ONES RUSTING UPON AN IN 
 FLEXIBLE BODY, WHICH is PIVOTED AT THE CENTRE. Taking again 
 for our case of unsymmetrical loading Q = 0, ^ = /, / 2 = ml, we have for 
 the main condition of equilibrium B=C, or 
 
 m2 Px 
 
 92. Jf 1 =-Vo- Also, 
 
 77i -4- 2 
 
 93. 2 M. 2 (/+ ml) = Y -f 2 Px (P x 2 ), or 
 
 i 
 
 -r- ISP* (*_*) 
 
 94. llf. 2 =- * From 92 and 93, 
 
 2 / (1 + 777) 772 2 PX -f ?Lt? 2 PX (^ X 2 ) 
 
 95. r= _ _J 
 
 7/1+2 
 
 Substituting in 93, 
 
 96. Af. 2 = -7-7j 2 Px,* and substituting this in the general equations 
 for A, B, and C(29-31). 
 
 97 - ^ = 
 
 98 pr 
 
 -/(m + 2) 
 and /I -f- B -\- C is, as it should be, equal to 2 P. 
 
 Again, substituting el for x, we obtain A, B, and C, in terms of the same 
 known quantities as in equations 35-37. 
 
 13. IN A GIRDER CONTINUOUS OVER FOUR LEVEL SUPPORTS, THE 
 Two CENTRE ONES PIVOTKD (SAME AS IN 11), TO FIND WHAT LOADING 
 
 UNIFORMLY DISTRIBUTED, ADDED TO A GIVEN LOADING UNIFORMLY DIS 
 
 TRIBUTED OVER THE SPAN AB, WILL MAKE D = 0. We had for thp 
 unsymmetrical load (91), 
 
 D= 
 
 m) mP 2 Px 
 
 2/8(3m -|-8m-f-4) 
 
 * Being the same as found (92), which shows that the principles of the lever alone suffice in 
 this instance, and that these principles are but a special case of those of the elastic line. 
 
41 
 
 For uniform loading over the whole span AB, we must SU^CTWJI j^Q j^j j^ 
 for 2 P, which gives, 
 
 ^,,, i+8 
 
 From 70, with q. 2 = q, we have, 
 
 The condition is that unsymmetrical D -|- symmetrical D shall equal 0, or 
 (3 + 8 m) g.- (2 + m ) 
 
 2 / 3 (3 ? 8 + 8 MI -}- 4) * 2(3ro 2 -f-8m 
 
 from which is found, 
 
 101. g = -r- r-? 1 ^- 7r = uniform load per unit of 
 
 * 2 6--15wi ^>w 2m 3 m 4 
 
 length over all three spans, necessary to make Z) = 0, when span /! is 
 loaded with ^ per unit of length in addition to <?, and the supports are in 
 the condition above stated. 
 
 14. IN A GIRDER CONTINUOUS OVER Two UNEQUAL SPANS, TO FIND 
 WHAT LOADING UNIFORMLY DISTRIBUTED, ADDED TO A GIVEN LOAD 
 UNIFORMLY DISTRIBUTED OVER THE SPAN AB, WILL MAKE (7=0. 
 
 We had (45a), 
 
 _ B 
 
 8 / 2 
 For q. 2 =. 0, or unsymmetrical loading, and 1 2 = ml. we have 
 
 102. Unsymmetrical C = - ^^y 
 
 For q 1 = q. 2 = g, or symmetrical loading, 
 
 3 ql m s 4- 4: ql m 12 ql 
 
 103. Symmetrical C= 8m7l~4^n -- r P g 
 
 ^ I . 3 qtf m 8 -4- 4 r// m 2 o/ _ , , . 
 
 -- ^JL -- L _JL__ / i- 0, we obtain, 
 
 8 m (1 + m) 8 m (1 + wi) 
 
 104. O=,T- 
 
 a 8 
 
42 
 
 PART III. 
 
 15. NUMERICAL EXAMPLE. (See Plate I-) Loading. The assumed 
 dead load of each truss (which is in 3 spans of 90, 20, and 90 feet, and made 
 continuous over them all) is taken at 1000 Ibs., the snow load at 200 Ibs., 
 and the live load at 1100 Ibs. per lineal foot. Each panel is 18 feet long, 
 except the centre one, which is 20 feet long. The above loads, concentrated 
 at the panel points and points of support, are as follows, in tons : 
 
 Over each end support 
 
 At each panel point 
 
 Over each centre support 
 
 4.5 tons dead load. 
 0.9 tons snow ,, 
 G.3 tons live ,, 
 
 9.0 tons dead load. 
 
 1.8 tons snow ,, 
 12. G tons live ,, 
 
 9.5 tons dead load. 
 
 1.9 tons snow ,, 
 1:5.3 tons live 
 
 LEVER ARMS* FOR THE FIRST SYSTEM (IN FEET). 
 
 ffi 
 
 19.929 
 
 Zi 
 
 18.5 
 
 s t 
 
 172.01 
 
 T* 
 
 181.02 
 
 ^3*4 
 
 22.92 
 
 L 248 
 
 21.5 
 
 s 
 
 211.02 
 
 T * 
 
 222.43 
 
 C7 6 
 
 25.909 
 
 -445 
 
 24.5 
 
 S 6 
 
 251.44 
 
 P 2 
 
 CO 
 
 u< 
 
 20. 
 
 L 
 
 20. 
 
 PI 
 
 222. 
 
 T< 
 
 co sin </> 
 
 LEVER ARMS* FOR THE SECOND SYSTEM (IN FEET). 
 
 c/ 243 
 
 21.425 
 
 ^142 
 
 20. 
 
 s* 
 
 191.77 7\ 
 
 105.01 
 
 u<^ 
 
 24.414 
 
 .1*4 
 
 23. 
 
 s* 
 
 231.:.- 
 
 T* 
 
 203.17 
 
 u* 
 
 20. 
 
 /. 6 &z c 
 
 20. 
 
 
 
 T t 
 
 241.72 
 
 Formula for the Reactions. For our case m = $ . Substituting this in 
 the equations for reactions, and making e = g = , they reduce to the 
 following : 
 
 Sec Plate II. 
 
43 
 
 f 2 P 
 
 Formulae for four level supports ; spans \ A = D = (8 11 k -}- 3 k 
 
 AB and CD symmetrically loaded, \ 
 
 from 67 and 68, IB C = (11 k 3 3 ) . 
 
 Formulae for four level supports ; span 
 AB only loaded, from 63-66, 
 
 A = (80 
 
 00 
 
 ~ 
 
 * 
 
 Formulae for four supports, when B and 
 C are pivoted" and AB only 
 loaded, from 89-91, 
 
 A = (80-91 k + 
 
 2 P 
 
 ^ = T77- 
 
 and C are "pivoted" and AB only 
 loaded, from 97-98, 
 
 9/c). 
 
 In all cases ^4-j-I?+ C-f-Dr=2P, except the first written, when 
 = 22P, as it should. 
 
 Example. Snow load in the first system : 
 
 P = 1.8& = .2; P = 1.8&&=.6; P = 1.9&* = 1 
 . by first formula, 4 D = 1.8 X .084 = 1.7712 
 
 B = C= 1.8 X 1-016 -f 1.9 = 3.7288 
 
 Snow load in the second system : 
 
 .*. by same formula as above : 
 
 A = D = 1.8 X -566 -f 0.9 = 1.9188 
 B = C = 1.8 X 1-434 = 2.5812 
 
 The reactions of the girder, for the several cases given in Parti., and 
 found in a manner similar tc the above, are given in the following Table : 
 

 
 
 
 S2 
 
 s 
 
 |3 
 
 - 
 
 
 
 
 QC CI i*f 
 
 I fl!i 
 
 f 18 1 
 
 
 q 
 
 d d 
 
 d 
 
 . e :" r 
 
 
 eH ?l 
 
 
 c c 
 
 I 
 
 
 
 ss5 5 
 
 . *T r i-? * 
 
 
 
 
 
 
 
 o ^ o 
 
 
 
 
 
 i 
 
 o WT-IO jo 
 
 "1 
 
 STEM. 
 
 
 
 3! 
 
 s 
 
 si^i 
 
 
 T 
 
 1 
 
 p! 
 
 - 
 
 
 
 
 P^ i 
 
 JSiiS 
 
 ^ci|g 
 
 v: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 C 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 S 
 
 ^ 
 
 P 
 
 K 
 
 !? ^ C- 
 
 43.1490 
 
 e^ 
 
 1 
 
 - 
 
 
 *l 
 
 ? 
 
 :: 
 
 
 L0l~ = 
 
 CM 7 ^ 05 co 
 
 S^?5 S 
 
 ^MZ 
 
 i^fSNCi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 ~ 
 
 o d 
 
 d 
 
 2 
 
 1 
 
 " 
 
 cc tc 
 
 1 
 
 . *-] 
 
 e o t- 
 
 ** 
 
 
 
 O THI^ t-^ 
 
 .1111 
 
 Hi! 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 |r- 
 
 
 
 
 
 
 23 
 
 
 
 00 
 
 
 
 
 
 ej |tp 
 
 C ! T 
 
 r? 5 
 
 
 - 
 
 dd 
 
 d 
 
 
 J 
 
 {" ^ 
 
 r 
 
 odd 
 
 d 
 
 
 di^c-i d 
 
 . g I- Vl | 
 
 . c- - ^ -I 
 
 c ;j -i ;i r 
 
 
 
 
 
 
 
 O - O 
 
 N 
 
 
 
 
 1 1 
 
 o c-i 1-1 d 1 i 
 
 1 
 
 w 
 
 o 
 
 S- 
 
 ;; 
 
 ,11 
 
 " ^X . " 
 
 cs " ^ 
 
 
 >C I* Ci 
 
 1 
 
 Pi 
 
 ^ 
 
 
 
 II 
 
 Ji 1 
 
 i~oc 
 
 CC d CO 1- ^5 
 
 ^ec 
 ac d co :t 
 
 
 
 
 
 
 
 
 
 
 
 8 
 I 
 
 
 T 1 ^ 
 
 
 H 
 
 
 
 
 
 
 
 
 
 
 g 
 
 
 ++ 
 
 H- 
 
 
 
 
 
 
 aco 
 
 
 ?? 
 
 | 
 
 
 T " 
 
 5 
 
 8SI 
 
 
 illlii 
 
 r-r. 
 
 c^ 
 
 t-- o 
 
 
 u* f J ^ 
 
 
 i- 
 
 ; 
 
 1- L^ O 
 
 
 
 ^ l . "; * 
 
 C^ "^1 f- ** ** 
 
 c^f r 1 1- o r7 
 
 
 
 
 
 r ro ^2 
 
 CM CM 
 
 . : 
 
 C-4 "" rf 
 
 ;j 
 
 
 
 & 
 
 ^-2 5 
 
 55|8 
 
 S^*"^; |? 
 
 
 
 
 
 
 
 
 
 
 
 rt 
 
 
 
 
 
 ^ 
 
 dd 
 
 d 
 
 e*i 7f 
 
 d^^i 
 
 E 
 
 II 
 
 | 
 
 -- 
 
 1 
 
 - 
 
 e reactions 
 
 c * o 
 
 1111 
 
 d c-i ^ :! 1 d 
 
 Jill 
 
 
 
 
 
 
 
 i- 1 
 
 
 
 
 a 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 rV 
 
 
 ~ : 
 
 
 .. 
 
 
 - i -s 
 
 
 
 - > . 
 
 ---%% 
 
 ... 
 
 rt 
 
 
 li 
 
 HOQ 
 
 
 m 
 
 
 115 
 
 
 
 
 
 115 
 
 ^ yl " 
 
 C - i 
 
 O ^_; 
 
 
 
 
 
 
 c 
 
 
 3. 
 
 
 4 
 
 2. 
 
 I 
 
 
 
 t- ~ 
 
 
 ^ 
 
 
 ^ 
 
 
 ^ 
 
 
 K, 
 
 
 K, 
 
 ***, 
 
 h^ 
 
 U 
 
 c ^ 
 
 
 tt5 
 
 
 QJ 
 
 
 Q- 
 
 
 -. 
 
 
 %9 
 
 . 
 
 ^ 
 
 <i> 
 
 rt s 
 
 
 X 
 
 
 QB 
 
 
 X 
 
 
 &} 
 
 
 v: 
 
 
 BQ 
 
 
 ~ A 
 
 
 
 
 Q 
 
 
 Q 
 
 
 JO 
 
 
 5 
 
 Q 
 
 C 
 
 C! 
 
 
 
 i^j 
 
 
 
 
 
 Qj 
 
 
 O 
 
 
 
 
 
 
 
 
 
 
45 
 
 Strain Sheets. (See Plates III. -IX.) The figures represent resultant 
 strains in tons of 2,030 Ibs. (-J-) means tension, ( ) means compression. 
 To abbreviate calculation, the dead load has been taken as though it were 
 concentrated at the lower panel points ; it is more exact, and in principle 
 just as easy to follow out, to take the own weight as divided into two main 
 parts, (1) that of the truss, and (2) that of the roadway platform, and each 
 as though applied at that part of the skeleton outline of the bridge at which 
 it is applied. A close approximation is to take the platform weight -|- ^ 
 weight of truss, at lower panel points, and 1 weight of truss at upper panel 
 points in " through " bridges ; reverse this arrangement in " deck" bridges, 
 and other combinations will readily suggest themselves in their proper places. 
 
 To illustrate the manner in which these Strain Sheets have been calcu 
 lated, the computation for Case II. is next given, worked throughout sir Icily 
 according to Hitter s method, although for chord members another way may 
 sometimes be shorter, as alluded to in Part I. 
 
 Equations for finding the Strains in the several members of the Draw, loaded 
 and supported (/hat w, reacted on) as in Case II. By making the proper 
 substitutions for loads and reactions, all the other cases can be calculated 
 according to the same formulae.* The centre panel diagonals are made to 
 take tension only; when the resultant strain on one of them comes out minus 
 therefore, neglect that diagonal, and make the calculation over again [for 
 that and for such other members as depend upon such change], supposing now 
 only the other diagonal in place. All other members are made to take either 
 the compression or extension that they may be called upon to withstand. 
 
 FIRST SYSTEM. 
 
 X 10. 929 -f 14.17 X 18 = 0. 
 
 X 22.92 -f 14.17 X (3X18) 23.1 X (2X18) = 0. 
 U 6 X 25.009 + 14.17 X (5 X 18) 23.4 X [(2 + 4) X 18] = 0. 
 7 6 X 20. -f- 14.17 X [(5 X 18) -f- 20] + (57.33 24.7) X 20 
 23.4 X [((4 X 18) +20) + ((2 X 18) + 20)] =0. 
 
 Z x X 18.5 + 14.17x0 = 0. 
 
 -L. 2&3 X 21.5 + 14.17 X (2 X 18) 23,4 X 18 = 0. 
 
 Z 4 &5 X 24.5 + 14.17 X (4 X 18) - 23.4 X [(1 + 3) Xl8] = 0. 
 -Z 6 X2G. +14.17X (5X18) 23.4 X [(2 + 4) X 18] = 0. 
 
 * The several equations have been written out in full, and without the slightest attempt at 
 reduction; on the contrary, all the elements of each term have been conspicuously written and 
 separated, so as to enabb any one to follow the process of evolving the equations for each nirm- 
 ber in any case. In practice, no one, after doing a single day s work according to Ritter, would 
 need to write out any equations as fully as done here, and for most members would not write 
 them out at all, unless indeed "ill his mind s eye." 
 
46 
 
 S 1 X 172.01 11.17X222 = 0. 
 
 7 2 X 184.02 14.17 X 222 + 23.4 X(222 + 18)=0. 
 
 .V 8 " X211.G2 14.17 X 222-|- 23.4 X (222 -[- 18) = 0. 
 
 T 4 X 222.43 11.17 X 222 + 23.4 X [(222+ 18) + (222 + (3 X 18))] = 0. 
 
 S 6 X 251.41 11.17 X 222 + 23.4 X [(222 + 18) + (222 + (3 X 18))] =0. 
 
 I\ X [222+(5Xl8)] (57.38 24.7) X[222+(5X18)] 14.17X222 
 + 23.4X[444+(4X18)]=0. 
 
 r 6 sm<ao 11. 17Xoo + 23.1Xoo + 231Xoo (57.33 21.7)X<=0, 
 where <j> is the angle T 6 makes with the horizontal. 
 
 26 
 .-. Sin </> = so-p,Ti and dividing by cc: 
 
 O .OUo 
 on 
 
 T * X 327^03 -14.17 + 23.1 + 23.1-32.63 = 0. 
 
 SECOND SYSTEM. 
 
 7 2 &3 X 21. 125+ (15.35 11.7) x(2Xl8)=0. 
 0*4*5 X24.414 + 3.65X(X18)-23.4X(2X18)=0. 
 C/ 6 x 20 + 3.05 X [(5X18) +20] +43. 15X20 
 -23.4 [((3X18) +20) + (18 + 20)] = 0. 
 
 (15.35 11.7) X 18 = 0. 
 Z 3 &4 X23 + 3.05 X (3X18) -23.4X18 = 0. 
 _ 5 x OG + 3.05 X (5 X 18) -23.4 [(3 X 18) + 18] = 0. 
 
 L, X 26 + 3.G5 X [(5 X 18) + 20] - 23.1 [((3 X 18) + 20) + (18 + 20)] 
 
 + 13.15 X 20 = 0. 
 
 _ 7 1 X 105.01 3.05 X 222. = [or, revolving about second load point : 
 + 7\ X V/30- 20.907^ + 3.05 X 30 = 0]. 
 
 S a X 101.77 3.05X222 = 0. 
 
 7 8 X 203. 17 3.05 X 222 + 23.4 X [222 + (2 X 18)] = 0. 
 S t X 231.52 3.05 X 222 + 23.1 X [222 + (2 X 18)] = 0. 
 
 - !T 6 X 241.72 3.65x222+23.4x[(222+(2xl8))+(222+(4xl8))] = 0. 
 
 P 2 X [222 + (5 X 18)] 3.05 X 222 + 23.4 [141 + (0 X 18)] 
 
 - T B X 211.72 13.15 X [222 + (5 X 18)] =0. 
 
 T fl sin </> 3.05 + 23.1 + 23.1 13.15 = 0. 
 
 The other half girder must be made symmetrical, or, if desired (in the 
 latched cases), can be calculated, either by continuing to make the supposed 
 sections in the order of procedure from left to right, always introducing 
 reactions as though they were merely minus loads (\vhich is all they are), 
 or by taking sections anew in the order of from right to left. 
 
47 
 
 In this way have been found the strains in Plates III. -IX., and tabulated 
 in Table (vl), p. 50. Observe that the centre panel diagonals present a 
 peculiar feature : the diagonal in service is a different one for the cases (a) 
 and (6), and has a different value for the two, but each diagonal has the 
 same value whether in ///., 7F., or V. A -}- B is constant, in pivoted as 
 well as fixed centre support draws : 
 
 In the first, A -}- B < loads on A B. 
 In the second, A -\- B > loads on AB. 
 
 A careful study of this table will prove instructive also as to choice of kind 
 of supports for drawbridges. The table is not complete, however; there 
 should be added the strains for the cases where there is no snow load, espe 
 cially for Case IV. (ft), and in the span CD. These may be obtained in two 
 ways : (1) we could get the new reactions, by subtracting, for each support, 
 the reactions produced there by the snow load from those already found, and 
 then proceeding in the calculation of strains as before, or (2) by finding first 
 the strains produced in each member by the snow load, and then algebrai 
 cally subtracting these strains from those already found in the corresponding 
 members. The snow strains are, however, of two kinds ; once when the 
 weight of the snow is all carried by the centre supports, and again when the 
 truss carrying the snow loads rests on 4 level supports. The first mentioned 
 snow strains obtain in Cases I. and ///. (&), and must also be subtracted to 
 get the strains without snow in Cases V. (a) and (&), whenever the end re 
 actions without snow are to be the same they were with snow. (In Cases 
 V. (a) and (6), it will be remembered, the end supports are lifted, just 
 enough to prevent the end reactions from ever becoming less than a chosen 
 plus quantity.) For, the end reactions remaining the same, the centre 
 reactions only can have diminished, and must have diminished by just the 
 weight of the snow ; and the strains can have diminished algebraically, 
 only by the strains produced by these two centre reactions, together equal 
 to the snow load, acting as opposed to the several panel snow loads. 
 Another view would be this : we had made s, the part of the own weight 
 that was carried on the 4 supports, instead of on only the central ones, just 
 large enough to securely keep the light end from rising in the event of 
 maximum unsymmetrical loading. If now the snow melts off, it will 
 render the light end insecure, and enough additional s must be taken 
 out of the own weight to restore the original end reactions ; that is, the 
 new s must = old s -j- snow. The process is, the snow is removed from 
 the 4 level supports, then a weight equal to it is taken off of the centre 
 
48 
 
 supports, and is distributed over the 4 level supports again, leaving, as a net 
 result, the centre supports lighter, by the total snow load. Mechanically, 
 this will necessitate a higher position of the end supports , at the end of the 
 lift, in order that the same end reactions may be obtained without, as before 
 were obtained with, the snow. Were, on the contrary, the end nupports to 
 remain in the same position, or at the same level, then, to get tie strains 
 in the trusses without the snow load, from those above with the 3now on, 
 we should have to subtract the snow strains produced by the snow load 
 resting on 4 level supports. 
 
 Figs. 20 and 21, Plate X., show the two kinds of snow strains ; Fig. 20, 
 being those when the snow reactions are produced only at the centre sup 
 ports, and are evidently simply | the strains given for Case I. The others 
 (Fig. 21) are the strains due a load of 200 Ibs. per foot on a bridge skeleton 
 without weight, and of the form under consideration, and supported on 
 4 level supports. Now subtract these strains in the several members, alge 
 braically, from the corresponding strains in the table, remembering which 
 set to use in the several cases, as above given. Fig. 20 has been used in 
 Case V. (/>) and Fig. 21 in Case V. (a), since dead was more than enough 
 for s in this latter case, and D = -{-, even without the snow. 
 
 The resultant strains for all the cases will be found in Table (D) , on 
 p. 51. 
 
 While this book is going through the press, the author notices, in a 
 work* recently issued, the following sentence : " But no book in common 
 use gives us any method for determining the shearing stress under a partial 
 load, a determination which is necessary before the bracing can be correctly 
 proportioned." It is to be hoped, however, that no insuperable objection 
 lies in the way to the attainment of such an object, and that such a book 
 may be written. The foregoing has been an attempt in that direction; for 
 the careful reader will have noticed that, by the method and the formulae 
 herein given, the effect of a load at any point or points, on a continuous 
 girder, upon any of its members, may be calculated. 
 
 The method of strain calculation followed throughout has been what may 
 be termed the " arithmetical " one ; and the author has purposely presented 
 the same purely and alone, and omitting, therefore, considerations of mo- 
 
 * Graphical Method for the Anilysis of Bridge Trusses, &c. Charles E. Greene. New 
 York. 1875. Van Nostrand. 
 
49 
 
 ment-curves, of vertical and horizontal shears, points of contrary flexure, 
 and other paraphernalia which usually accompany treatises on continuous 
 girders. 
 
 To calculate the absolute maxima and minima strains in the members of 
 a continuous panel girder by the method that has been followed, the process 
 would be like this : 
 
 First find the strains in each member, under the own weight of the 
 bridge. 
 
 Next find the strains in each member for the maximum panel live load 
 on the first panel point, all the rest of the bridge being supposed to be 
 without weight and unloaded. 
 
 Next place the panel live load on the second panel point, the rest of the 
 bridge being supposed to be without weight and unloaded, and so on, for 
 each panel point. By tabulating the strains thus found for each member of 
 the bridge, a clear oversight will be gained over them. Then summing up 
 all the minus strains liable to occur in any one member, with the strain due 
 to the own weight, will give the maximum minus strain, and summing up 
 its plus strains with the same will give the maximum plus strain that can 
 come upon it by any possible combination of loads on the bridge. 
 
50 
 
 -.00 h- 08 <O O ^? O <N o -* O -r x cc t-. r: X r? 
 
 sspssi 
 
 777 i? 
 
 II III 
 
 ^;P 
 + + I 
 
 o o n cc t- 
 
 1 1 
 
 o rs o os 
 
 od OW 
 
 
 
 i" OX 
 
 OOriMO C5Ot^C5 GO C5 <-" COCCOr^^ (5 t-t^ 
 
 SS23+ ^^j^gi^ 1 Z <<*%& + ^^223 
 
 1 1 ++ + + + II III + +++ ++ 1 \ \ 
 
 o o 
 
 M M +1 
 
 1 1 II 1 
 
 p o o t^ co 
 
 flf? 
 
 ^SS - N ^3 2^1 ^^ 
 
 , 
 ++II 1 
 
 O O O CN 
 
 t^t^ O C5 CC -J ?< O !O 
 
 
 --o CN 
 
 lTT w 1 
 
 co do ad H 
 
 - 1 
 
 + ^- w c<i 
 
 O c 3 * co r 
 
 co r o 
 
 ii co o op 
 
 II 1 I 
 
 a t -t o c 
 
 I- ^ 
 
 MM 
 
 1 M ++ 
 
 ^s zz adci ?,^^+ cTl^^^?^ 
 
 + + M III + + + + ++ II 1 
 
 CN d CNr^T}5 ir-lcor^ CN cc d -? o 
 
 C^rfO CN 1-1 i- 1^47^ < " i ?1 
 
 1 1 1 TT 1 1 +++++ +i i M 
 
 CN OD yro -f- coddx " 1 ?" co aj ^^Coo "f ^^^W 
 
 : 
 
 M 
 
 > ^-OX)CiO 
 
 d o d co co* oo - co 
 
 co o ri d 
 P*N ea io Op 
 
 fill 
 
 O ^t^ W ^ CO O O . ^COCCCNCN 
 
 w d d r-^ r: >c ri d d d d ?i t-^ ri 
 d> co -< Nt^ -* i-* CN 
 
 MM 
 
51 
 
 O CD O O OS O CO O CO O C^? ^. CN i ^ CD O CD t-~ 35 O CO 
 
 CN O > t^- !~ T I O ^ O O CN 35 CN C5 CO 30 7$ CN i t-^ i CO 
 
 ^- -M Tf< T-I CO O CO 1-HCN T-. T- 
 
 I + + + +o I I I I I* M +++++o +1 I I I 
 
 3? Hi T-J C5 ^ ^ O 00 T}* CC O r- I-H ifl CO O O CD S rj 55 q O O 
 
 I I If 
 
 ++ | | | 
 
 Q 
 
 w 
 
 o 
 
 r-*. 
 
 M 
 
 
 p 
 ^ 
 o 
 (J 
 
 35<717^Ol>- OO - S^P CN^ ^ .30 O r-c p 
 
 1 1+++ TTTfT^ M +++++ 
 
 + 1 I I I 
 
 1 1 1 ++ 
 
 ++I 1 1 
 
 us cs FH GO 00 -^ QO 
 
 1 1 1 1 +1 
 
 eooooo 
 
 11 1++ +++7T 7T i++++ 
 
 ++I 1 1 
 
 c? 
 
 + 1 I I I 
 
 i- r- T-H o r: 
 
 ++I 8 ? 1 ? 77 
 
 -j- O 71 -* CO 
 
 ++77T 
 
 --.!>. -H T-- TJ4 ,-4 T-4 p _O 
 
 N "?^ i77TT 7 
 
 00 fr- f-i i- CO 1O if O O Tf4 rfi t^ 
 
 * TlJ t- O O Ift T-I t-- t-- 30 7-1 O 7-1 71 CO 35 
 
 I I I++ + + + I I I 
 
 + + I I I 
 
 73 00 O O O co co co 
 
 ?o i7nt M 
 
 o o i i <M 
 
 P35^fi?35t ; - T-jCOC^f0^35 i O T-HO35O71 CO r--l--COGOCD 
 
 77+++ ++i77 77-I++++ ++7Ti 
 
 CNCDMOO O " ! t ~;^i3035O ^-O O^lcOt-^35 O O O rt; 
 
 ** O O * "- 71 o 00 i-! t-1 r^ <>i in co co ^ t-1 i~ o O o co ^ co 
 
 77++++ ++i M i 7 1 i++++ ++7 M 
 
 T-I CO T-H r- O 35 t-- b- 03 35 r~- ri 7-1 CO O 
 
 Tjn o -i- t^- r- O O CD" i-i CD O O O O O O -1< CO 
 
 i71 1 7t7 1 +++++ i777 
 
52 
 
 APPENDIX. 
 
 LIST OF BOOKS AND ARTICLES IN TECHNICAL JOURNALS, RE 
 LATING TO THE SUBJECT OF CONTINUOUS GIRDERS (IN PART 
 OR IN WHOLE), IN THE GERMAN, FRENCH, AND ENGLISH 
 LANGUAGES, 1854-1874. 
 
 [Without wishing to presume the part of an adviser, the writer yet offers the opinion that 
 the starred works are especially to be recommended to those desirous of further pursuing the 
 subject of Continuous Girders. The annexed list is doubtless incomplete.] 
 
 *WeyraucJi, Dr. Phil. Jacob /., Ingenieur, &c. Allgemeine Theorie und 
 Berechnung dcr Continuirlichcn und Einfachcn Trager. Leipzig: B. G. 
 Teubncr. 1873. 
 
 *Ilitter, August, Prof. &c. Elemcntare Theorie und Berechnung ciserner 
 Dach- & Briicken-Constructioncn. (Contains, amongst other matters, a 
 peculiar kind of continuous girder.) Hannover: Carl Riimpler. 1873. 
 
 * Lassie fr SchueUcr. Dcr Bau dcr Briickcn-Trfigcr. Stuttgart: Paul Neff. 
 2 vols. 1809 & 1870. (This work has also appeared in a French trans 
 lation.) 
 
 Winkler, Dr. E. Elasticitat und Festigkeit. Frag : II. Dominicus. 1867. 
 
 ZEITSCHRIFT DES ARCIIT. & ING. VEREIN zu HANNOVER. 
 
 Mohr (now Prof. &c. in Stuttgart) I860, 1802, & 1808. 
 
 Kopke, C., Ingr. &c 1850 & 1857. 
 
 ZEITSCHRIFT FUR BAUWESEN, BERLIN. 
 
 Quensel, 1805. 
 
 Schwedler, J. W 1802. 
 
 Winkler, E. 1830. 
 
 ZEITSCHRIFT DES OESTR. ARCIIT. & ING. VEREIN. 
 Winkler, Prof. &c 1872. 
 
 DER CIVIL INGENIEUR. 
 
 Franl-cl, W 1808, p. 271. 
 
 1800, Heft 3 & 4. 
 
 HuUbauer 1858, 5 & 6. 
 
 1858, 2. 
 
53 
 
 ZEITSCHRIFT DES VEREINS DEUTSCHER INGENIEURE. 
 Grashof 1857. 
 
 ZEITSCHRIFT DES BAYERISCHEN ARCIIT. & ING. VEREIX. 
 Gerber. (A Patent Continuous Girder.) 1870. 
 
 ALLGEMEINE BAUZEITUNG. 
 Schmidt, Heinrich 1868. 
 
 De Montdesir. (Strains in Continuous Trusses of Iron Bridges.) 
 
 2dEd 1873. 
 
 *Bresse. Cours de Mecanique Appliquee. l re & 3 ms partie. Paris. 
 
 1859 & 1865. 
 
 ANNALES DES FONTS ET CHAUSEES. 
 
 *Itenaudot 1866, p. 311. 
 
 Pierre 1871, p. 44. 
 
 Albaret 1866, p. 53. 
 
 " COMPTES RENDUS." 
 
 Clapeyron Dec. 1857. 
 
 Molinos fy Pronnier. Construction des Ponts Metalliques. Paris. 1857. 
 
 ANNALES DES TRAVAUX PUBLICS DE BELGIQUE. 
 
 De Clerq 1855-56. 
 
 \_Nuvier. Resume des Lemons donnees a 1 Ecole des P. & C. &c. 
 2d Ed. Paris. 1833.] 
 
 MINUTES OF THE PROCEEDINGS OF THE INST. Civ. ENGRS. 
 
 Bell, W Vol. 32, p. 171. 
 
 *Stoney, E. W ,, 2D, p. 382. 
 
 *PIeppel, J. M. ,, 19, p. 625. 
 
 *Barton, James ,, 14, p. 443. 
 
 Contains a description of the celebrated, and, considering the time it was 
 built, iconderfally perfect, Boyne Bridge at Drogheda, Ireland, 3 contin 
 uous spans of 141, 267, and 141 feet, opened for traffic, double track 
 R.R.,in April, 1855. 
 
 Pole, W Vol. 9, p. 201. 
 
 Remans, G.W. ,, 3, p. 65. 
 
 TITE PHILOSOPHICAL MAGAZINE, LONDON. 
 *Hcppel, J. M. Vol. 40, p. 416. 
 
 Stoney, B. B. Theory of Strains in Girders, &c. London: Long 
 mans, Green, & Co 1873. 
 
 Rankine, W. J. M. Civil Engineering pp. 287-2,02. 
 
 Plumber, William. Strains in Girders. Am. Ed. New York: Van 
 
 Nostrand . . 1870. 
 
54 
 
 Clark, Edwin. Brittania and Conway Tubular Bridges. London. 1850. 
 Moseley, Henry. Mechanical Principles of Engineering and Architecture 
 
 (1840-42). Am. Ed. By D. II. Mahan, Professor at West Point. 
 
 New York: J. Wiley & Son. 1809. (Continuous beams are treated of 
 
 in this work by the name of " Breast summers.") 
 Chanute, #., and G. Morison. Kansas City Bridge, pp. 86, 87, 108-114. 
 
 JOURNAL OF THE FRANKLIN INSTITUTE. 
 Frizell, J.P 3d Series, Vol. G4, 1872. 
 
 TRANSACTIONS ENGINEERS CLUB OF ST. Louis. 
 Smith, Charles A. Girders Continuous over Supports . . Dec. 4, 1872. 
 
 VAN NOSTRAND S ECLECTIC ENGINEERING MAGAZINE. 
 Eddy, C. H p. 552, 1874. 
 
 DUTCH LANGUAGE. 
 
 Tijdschrift van hot Koninklijk Instituut van Ingenieurs.* s Graven- 
 
 hage 1870 & 1871. 
 
 The writer does not know the exact nature of these articles. 
 
UNIVE RSITY OF CALIFORNIA LIBRARY