IC-NRLF LIBRARY UNIVERSITY OF CALIFORNIA. Class SECONDARY STRESSES IN BRIDGE TRUSSES BY C. R. GRIMM, C.E. MEMBER OF THE AMERICAN SOCIETY OF CIVIL ENGINEERS, MEMBER OF THE AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE, MEMBER OF THE AMERICAN GEOGRAPHIC SOCIETY FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS LONDON: CHAPMAN & HALL, LIMITED 1908 OF THE UNIVERSITY ) OF / "TC-V COPYRIGHT, 1908, BY C. R. GRIMM Stanbope fl>res& F. H. GILSON COMPANY BOSTON. U.S.A. PREFACE. THIS book owes its existence to Dr. J. A. L. Waddell, Member of the American Society of Civil Engineers, who proposed to the writer the treatment of secondary stresses as a timely undertaking. A comprehensive treatise on secondary stresses with numerous numerical examples involves an extraordinary amount of time and labor in its preparation, and considering the very limited amount of time at the writer's disposal for such an attempt, he thought it best to confine himself to a narrower field, as otherwise the publication would surely have been unduly delayed. It is owing to these circumstances that we discuss principally the most important secondary stresses, namely, those which are due to riveted joints in trusses. We give in substance four principal methods of calculation of secondary stresses, which the reader may study separately and apply, The principle of least work leads also to the desired results, but it is far removed from shortening the labor. The nature of the subject forbids a very quick determination of the stresses, and definite rules or formulas cannot be given. The problem of secondary stresses has been put in the proper light and the theories worked out entirely by German authors who have furnished also the greatest and best part of the contri butions to the literature of this subject. In preparing these notes the writer consulted a great many papers and books most of which are noted in Chapter XI. He has endeavored to be clear and precise in his statements, but if he has failed in this he begs the reader to be indulgent with his shortcomings. Should he succeed in interesting his readers in this new field for American bridge literature, and in rendering some service to his colleagues, he would consider himself richly repaid for his labors. C. R. G. GREATER NEW YORK, 1908. 173028 CONTENTS. CHAPTER I. PAGE GENERAL AND HISTORICAL NOTES 1-5 Distinction between Primary, Principal, Additional and Secondary Stress i Deformation of Trusses 2 The Most Fruitful Sources of Secondary Stresses 2 Displacement of Stresses 2 Secondary Stresses as Higher Functions of the Exterior Loads . . 3 Origin of the Expression " Secondary Stress " 3 Solution of the Problem by Manderla 3 Difficulties of the Problem 4 Further Contributions to the Subject by Engesser, Winkler, Lands- berg, Miiller-Breslau, Ritter and Mohr 4 Cases of Examination of Secondary Stresses 5 CHAPTER II. NATURE OF THE PROBLEM AND MEANS FOR ITS SOLUTION . . 6-19 Deflection of a Pin-Connected and a Riveted Truss 6 Manderla's Supposition 6 Deformation of Riveted Members, Resembling Usually the Letter S 7 The Angle of Deflection 7 The Conditions of Equilibrium for a Deformed Bar 7 Solution Effected by Means of the Equation of the Elastic Line ... 8 Simultaneous Performance of the Calculation for the Entire Truss Load 8 Simplified Assumptions the Method of Influence Lines .... 9 Disturbing Influences 9 Determination of the Angle-Alterations A 10 Relations between the Angle of Deflection and the End Moments 12 Determination of the Angle Included Between Successive Positions of the Axis of a Bar, Belonging to a Truss with Frictionless Pins, which is Acted Upon by Exterior Forces 16 Work Done by a Couple if The Elastic Line of a Straight Beam Represented as an Equilibrium Curve after Mohr 17 v vi CONTENTS. CHAPTER III. PAGE MANDERLA'S METHOD 20-34 Assumptions Made for the Solution of the Problem of Secondary Stresses Caused by Riveted Joints 20 Other Elements of Influence than Riveted Joints on Secondary Stresses 20 Manderla's Course of Investigation 20 Differences between Compression- and Tension-Members .... 21 The Deflection Angles 24 Compression-Members 25 Tension-Members 27 Determination of the Deflection-Angles from the Conditions of Equilibrium 30 Secondary Stresses Found by Trial -Computations 53 CHAPTER IV. MULLER-BRESLAU'S METHOD 35-43 Derivation of the Fundamental Equations , 35 Assumed Deformation of Triangles 35 Values of the Deflection Angles 36 Determination of the Quantities U for a Truss 38 Method of Influence Lines 40 CHAPTER V. RITTER'S METHOD 44-51 Notation of Quantities 44 Fundamental Equations 46 Graphic Solution 46 More Exact Method 49 CHAPTER VI. MOHR'S METHOD 52-57 Determination of the Unknown Quantities 52 Determination of the Angles if/, Pratt Truss as Example .... 53 The Bending Moments Expressed as Functions of the Angles and ^ 54 Resulting Equations and their Solution 57 CONTENTS. vii CHAPTER VII. PAGE METHOD OF LEAST WORK . . . 58-67 A Riveted Triangle as an Example 58 The Unknowns of the Problem 59 Influence of the Sectional Areas on the Stresses 59 Application of the Principle of Least Work 60 Determination of the Bending Moments and the Secondary Stresses 63 Displacement of the Stresses 64 Test of Accuracy by a Check-Calculation 64 CHAPTER VIII. OTHER CAUSES OF SECONDARY STRESSES THAN RIVETED JOINTS IN MAIN TRUSSES 68-84 Eccentricities 68 Loads Between Panel -Points in the Plane of the Truss 69 Effects of Dead Load 69 Loads Between and at the Panel-Points of a Member Supposed to Turn Freely Around a Pin 69 Bottom-Chord Eyebar as an Example 70 Changes in Temperature 73 Misfits 74 Brackets on Posts 75 Unsymmetrical Connections 75 Curved Members 75 Pin-Joints 75 Greatest Diameter of a Pin, if an Eyebar Shall Turn Freely ... 76 Amount of Secondary Stress in an Eyebar Due to Frictional Resist- ance 77 Friction at Supports 78 Cross-frames 78 Amounts of Secondary Stresses in the Suspenders of a Two-Track Railway Through Bridge 79 Amounts of Secondary Stresses in the Posts of a Four-Track Rail- way Through Bridge 79 Yielding of Foundations and Settlements of Masonry 80 Trusses Affected by a Displacement of Their Supports 80 Continuous Truss or Girder Over Three Supports 81 Two-Hinged Arch 81 Horizontal Thrust of a Two-Hinged Arch Caused by Vertical Load- ing, Changes in Temperature and Horizontal Yielding of the Sup- ports 83 Bridge Across the Emperor William Canal at Griinenthal, Germany, as an example 83 viii CONTENTS. CHAPTER IX. PAGE IMPACT 85-9! The Propagation of Stresses in Elastic Bodies and the Laws of Sound 86 Ritter, Mach, Radinger and Steiner on the Propagation of Impulses and Vibrations 86 Radinger on Bridge-Stresses by Fast Running Trains 87 Zimmermann's Rigid Solution of the Problem of Impact in the Simplest Case 87 Zimmermann's Impact Formula 90 CHAPTER X. EXAMPLES AND CONCLUDING REMARKS 92-137 Amounts of Secondary Stresses in Three Warren Trusses without Verticals 94-99 Amounts of Secondary Stresses in Two Warren Trusses with Verticals 100-104 Discussion of Secondary Stresses 105-106 Amounts of Secondary Stresses in Three Pratt Trusses 107-115 Amounts of Secondary Stresses in Two Double Intersection Warren Trusses without Verticals 116-119 Amounts of Secondary Stresses in One Double Intersection Warren Truss with Verticals 120-122 Discussion of Secondary Stresses continued 123 Amounts of Secondary Stresses in a Parabola Truss 124-125 Amounts of Secondary Stresses in a Continuous Truss 126-127 Discussion of Secondary Stresses continued 128 Points to be Observed for the Reduction of Secondary Stresses in Trusses Due to Static Loads 128-130 Secondary Stresses Due to Riveted Connections Between Floor- beams and Main Trusses 130-131 Remarks on Floorbeams 132 Points to be Observed for the Reduction of Impact and Vibrations 132-133 Bridge Collisions I 33~ I 35 Remarks on Calculations and their Use I 35~ I 37 CHAPTER XI. LITERATURE 138-140 Concerning Secondary Stresses 138-140 Concerning Impact and Vibrations 140 SECONDARY STRESSES IN BRIDGE TRUSSES. CHAPTER I. GENERAL AND HISTORICAL NOTES. BEFORE entering upon a discussion of secondary stresses, a subject which has been very obscure up to about twenty-five years ago, it is proper to show to the reader the distinction the Germans make between different stresses, so that he can see at the outset the meaning they attribute to the expression, "Secondary Stress." A bridge may be exposed to the influence of a number of causes, dead load, live load, wind pressure, centrifugal and braking forces, changes of temperature, yielding of masonry (as, for instance, in statically indeterminate arches, continuous beams and trusses), impact, etc. Any one of these causes produces primary stresses, which are separated into two classes. Such that are due entirely to dead load and live load are called by German writers Hauptspannungen (principal stresses), while those due to any other cause are called Zusatzspannungen (additional stresses). The resultant of the primary stresses passes through the center of gravity of the section and acts along the axis of the member, producing either an elongation or a shortening. The remainder of the stresses are bending, shearing and torsional stresses; they bend, displace and twist, and are comprised under the expression, secondary stresses (German Nebenspannungen, Sekunddrspannungen; French - efforts secondaires). 2 SECONDARY STRESSES IN BRIDGE TRUSSES In other words, in a truss with frictionless pins, representing an ideal truss, the axes of the truss members remain straight during deformation, while in a riveted truss these axes are subjected to deformations accompanied by secondary stresses. In the computations of bridge stresses we always proceed under the supposition that the joints are provided with frictionless pins; a supposition, which is either not at all realized in the structure, or at best in an imperfect manner, so that each truss member is bound to be deformed, that is to say, each member during the action of the load is subjected to bending moments in the plane of the truss, and its axis cannot remain straight. Not only the members composing a riveted truss, but all those that go into the riveted cross frames and the bracings are subjected to secondary stresses. The riveted joints of the main trusses in particular, as also the stiff connections between floor system and trusses form the most fruitful source of secondary stresses. For this reason, we will give for the present some general remarks about the second- ary stresses in riveted trusses and discuss later those stresses which arise from the riveting between main trusses and floorbeams, as also several other groups of secondary stresses. The deflection of an ideal truss with frictionless pins, due to external forces, causes alterations of leverarms, which the computer of primary stresses completely ignores, and rightfully so, since these alterations have no practical significance whatever. The original leverarms are very great in comparison with the altera- tions that have taken place after deflection and, therefore, these changes are neglected. The lines of stresses of the different mem- bers around a panelpoint still pass through a common center. On the other hand, when we attempt the computation of secondary stresses in a riveted truss, we have to deal with small leverarms, because each member of the truss has been bent and the altera- tions of these leverarms cannot be ignored without some sufficient reason. The resultant stress in each bar does not act any longer parallel to the axis of the bar, on the contrary, it forms an angle with it, which makes the leverarm and the lines of stresses of the GENERAL AND HISTORICAL NOTES 3 different members around a panelpoint pass no longer through one and the same mathematical point, that is, through a panelpoint of the truss. By taking into account the influence of the deformations on the alterations of leverarms, the secondary stresses appear as higher functions of the exterior forces, while in neglecting these influences they appear as linear functions of the exterior forces, and one of the difficulties is removed. It is natural that the deformation of a compression member plays a greater role than that of a tension member, but if ample provision against buckling is made in the design of compression members, the neglect of the deformations in the calculations is justified. The expression, Sekunddrspannung (secondary stress), origi- nated with Professor Asimont of the polytechnic school in Munich, Bavaria. In a paper, " Hauptspannung und Sekundarspannung" (primary and secondary stress), which was published in 1880 in Zeitschrijt jur Baukunde, Asimont discusses the effects of eccentric loading on a column, and here he made the distinction between primary or direct stress and that due to a couple, producing a bending stress, which he called Sekundarspannung. But long before Asimont coined that expression, engineers considered secondary stresses in their designs. The subject of secondary stresses being one of importance, the polytechnic school in Munich, in the year 1877, offered a prize on the solution of the problem of how to calculate these stresses in riveted trusses. In formulating the problem, Asimont suggested that, owing to the fact that the lines of the resulting stresses no longer pass through the centers of the paneljoints, its solution might be effected by the employment of Euler's equation of the elastic line. The prize was awarded to Manderla, in 1879, whose excellent solution is found in a highly scientific and mathematical paper, published in 1880, in Allgemeine Bauzeitung, under the title, "Die Berechnung der Sekundarspannungen, welche im einfachen Fachwerke infolge starrer Knotenverbindungen enstehen" (The calculation of secondary stresses which occur in simple trusses as a 4 SECONDARY STRESSES IN BRIDGE TRUSSES consequence of rigid joints). The nature of the problem of secondary stresses in riveted trusses is such that it offered great obstacles to a solution; in fact, this problem is one of the most difficult in technical mechanics, and although Manderla's solution is a very great step forward, the problem in all its aspects has not yet been completely solved. Before the appearance of Manderla's solution, Engesser had published an approximate method. In the year 1881, the late Professor Winkler gave a lecture on secondary stresses before an organization of engineers and architects in Berlin, in which he said that for some years past he had paid attention to the subject. This lecture is published in Deutsche Bauzeitung in 1881, under the title, "Die Sekundarspannungen in Eisenkonstruktionen " (Secondary stresses in iron constructions).* In 1885 Professor Landsberg contributed a graphical solution under the assumption that the chords alone are riveted; and in 1886 Professor Miiller- Breslau made an analytical contribution. Professor Ritter, in 1890, gave a graphical solution, and in the years 1892 and 1893 Professor Engesser published a book on secondary and additional stresses, the latter expression to be taken in the sense already explained. In this book a systematic representation of the whole subject is given; the treatment is analytical throughout. Pro- fessor Mohr contributed in 1892 an analytical method in " Der Civil Ingenieur," which can also be found in his very valuable book, " Abhandlungen aud dem Gebiete der technischen Mechanik," published in 1906, in which he treats in a masterly manner subjects appertaining to technical mechanics. The fundamental truths and the methods of calculations that came to light in studying the problem that is here under consider- ation, we owe to German scientists. The results they obtained were soon made use of by German bridge constructors, as is noticeable in their designs, but this is a point to which we will return at another place. * Very extensive investigations can be found in Winkler's " Theorie der Briicken," II Teil. GENERAL AND HISTORICAL NOTES 5 The calculations of secondary stresses are very extensive and require therefore much time; but under simplified assumptions, and with the requirement to find these stresses for only one given load system, the calculations can be performed with comparative speed. It is hardly in the nature of the problem to give empirical rules and formulas. Although in common cases there is no necessity for such calculations, yet in particular cases secondary stresses should be investigated; for instance, in cases where we can expect them to be of great magnitude, or where a bridge has to carry much greater loads than those for which it has been designed, which is true of old bridges.* The margin of safety provided for in our specifications must cover the secondary stresses without impairing the safety of the structure. The nature of a truss, its details, as well as the dimensions of its individual members, are of great importance for the reduction of secondary stresses, but in order to obtain the best results even attention must be paid to the manufacture, erection and main- tenance of bridges. * W. J. Watson, Concerning the investigation of Overloaded Bridges, Pro- ceedings Am. Soc. of C. E., April, 1906. CHAPTER II. NATURE OF THE PROBLEM AND MEANS FOR ITS SOLUTION. IN the following discussion it is assumed that all exterior forces, to whose influence a truss is exposed, are acting in the plane of the truss. The deflection of a truss with supposed frictionless pins is, strictly speaking, not the same as that of a riveted truss, all other conditions being the same for both trusses. While in the former case the problem is geometrical, since the bars can turn freely around the pins and remain straight, in a riveted truss the axes of the bars become deformed under the action of the load, a fact which must be considered, together with the moments of inertia of the sections, in determining the deflection. But the difference in the deflection between these two cases is so small that its consideration has no practical value. Manderla's solution is based on the supposition that the posi- tions of the panelpoints in a riveted truss under the action of outer forces are the same as if the truss were provided with fric- tionless pins. In Fig. i, representing a fragment of a truss, the angle formed by the two bars 01 and 02 before any deformation takes place is 4- 102 = a. Under the supposition of frictionless pins this angle a will be changed as soon as the truss is subjected to the influence of exterior forces. Let this change be &a, so that the angle included between the two bars 01 and 02 after deformation equals a + Aa. If we now conceive the truss to have joints riveted in such a manner as to keep the ends of the bars absolutely fixed, then under the influence of the outer forces the originally straight bars will be deformed and the angle included between the two end 6 THE PROBLEM AND ITS SOLUTION 7 tangents o7\ and oT 2 , drawn to the elastic lines, which are the deformed axes of the bars, will remain unchanged during deform- ation, that is to say, the angle 7\or 2 = a or = the original angle before any deformation took place. Each bar will be subjected to bending moments, and its deform- ation generally, but not always, resembles the letter S. The bars Fig. i. must be considered fixed at the ends and under the influence of an axial load, while the effect of the bending moments can be conceived as consisting in the reduction of the changed angle a + \a to its original magnitude a. The angle r, included between a chord of a deformed bar and its corresponding end tangent, as, for instance, o/ and or, we will call the angle of deflection. We will now consider a single bar all by itself, for instance, bar 01 (see Fig. 2), by passing sections close to the panelpoints and applying the resultant P of all the interior forces, which inter- sects the chord of the deformed bar under the angle &. This resultant is resolved into the force 5 , acting along the axis of the deformed bar, the transverse force Q and the moment M . The equilibrium requires that -S = s,, G. = G,, ^~- 1 - G, = o. M + M The magnitude of the transverse force Q = is princi- 8 SECONDARY STRESSES IN BRIDGE TRUSSES pally dependent on the signs of the two moments; the greater values of Q correspond to equal signs of M and consequently to a double curvature of the bar, while opposite signs of the moments mean smaller values of Q and a single curvature. The transverse force Q is constant for the entire length of the bar, as its ends are the only points of application of the outer forces, and the bending moment for any point between the ends of the bar is variable under the influence of Q and S. I Manderla's excellent original solution considers all of the forces represented in Fig. 2. He proceeds from the equation of the elastic line, employs for the integrations hyperbolic functions, and after a relation between the bending moments and the angle of deflection ~ is established, expressing the latter in terms of a, he obtains the desired result by trial. Other scientists, who have occupied themselves with a solution of this problem, proceed from the assumption that the influence of the deformation on the leverarm y is a negligible quantity, or in other words, for well designed compression members tension members are not so important in this respect the leverarm y, and consequently also the moment Sy, is so small that it does not need to be considered. The transverse forces Q, too, are in most cases small enough to be neglected. These suppositions lead to simplified calculations. In Manderla's solution the secondary stresses appear as higher Fig 2. functions of the exterior forces, which means that the calculations must be performed simultaneously for the entire load system. THE PROBLEM AND ITS SOLUTION 9 If, for instance, the secondary stresses in a railroad truss have been determined for a certain position of a train, we have no means of knowing what the magnitude of these secondary stresses amount to as soon as the train is moved to some other position. The method of influence lines does not hold true in this case, while under the simplified assumption, that is, if the effect of the moment Sy on the final result is neglected, the secondary stresses appear as linear functions of the exterior forces and the investigator may employ the method of influence lines if he so desires. The fact that secondary stresses do not increase, or decrease in direct pro- portion to the exterior loads makes the so-called " factor of safety''* appear as the "factor of uncertainty," and uncertainties must be covered by the margin of safety in our specifications. The assumptions which we have made are never strictly realized in a structure, but this is, of course, equally true for the calculations of other bridge stresses. It is within the range of possibilities that the deformation of some truss members will take place outside of the plane of the truss and they may even be sub- jected to the influence of torsional moments. It must also be borne in mind that the desire, of the bridge engineer to save some weight often results in sizes of gusset plates so small that the state of fixity of the bars is complied with in a very imperfect manner. A variation in the value of the modulus of elasticity and possible erection stresses play also a role in making the results somewhat uncertain. In the first place we will deduce some fundamental conclusions from technical mechanics, which are indispensable for a clear understanding of the different methods of calculations of second- ary stresses. * (i) "Factors of safety," Eng. News, Sept. 6, 1906. (2) "The Investigation of Old Bridges a Phase of Maintenance Engineering," Eng. News, Sept. 13, 1906. 10 SECONDARY STRESSES IN BRIDGE TRUSSES i. Determination of the Angle Alterations Aa. An important problem in the theory of trusses, forming the foundation of Manderla's solution, consists in the determination of the alterations Aa of the angles a between any two adjacent pin-connected bars of a truss when the latter is acted upon by outer forces. Aa will be expressed in terms of the primary stresses, which are known quantities. The deformation of the angle a, in the triangle ABC of Fig. 3, is due to the elastic changes of all three sides, and this deformation is obtained by determining the influence of each bar on the deformation successively, whereupon the results are summed up. This is done in assuming that each bar in turn is elastic, the other two being non-elastic. If the side AC is alone elastic, experienc- ing a contraction to the amount of AL 3 , it will be forced to revolve around the apex A , while the side BC revolves around B, the side AB being supposed to be held fast. From the figure we have Aa 13 = a l /?!, where the index i refers to the angle and the index 3 to the side. Further: ! == 180 - a, -a 3 and /?! = 180 - a, - a 3 - e + 7 consequently : Aa ls = - 7 + s. The angle 7 is found from the equation BCy X cos a l =-- a or a a AZ, 3 7 =: = T = ; on account of the similarity of the BC cos a i b h triangles BCF and CDE. We have also r X = ~T-> and o L 3 L 3 h a &AZ, since the angle e == , we find by substitution: e = . ^3 LJl THE PROBLEM AND ITS SOLUTION II With these values for 7 and e the deformation becomes A 13 = ^ + 3 If 5 3 denotes the total stress in bar AC and A 3 its cross-section, then -~ = s 3 , which is the stress per unit of area. Calling E -4 3 the modulus of elasticity, the deformation of the angle a lt due only to the alteration in the length of the bar AC, is expressed by 5 3 L 3 - b _ _Sy_ ^ 1 3 TT* A i 77* A 2 T-I 2 * &A 3 tl LA 3 JtL If the bar BC alone is elastic, we find in a similar manner, A 12 = -J? cot ci' 3 , and under the supposition that AB alone is subjected to a change in its length, we have : AT 0*7" C 1 ( f } ( ~ = ~^~r~ = ^rr - ] - + - [ = IT } cot r Eyltr EA< r r r\ El Aa, cot 12 SECONDARY STRESSES IN BRIDGE TRUSSES The total deformation of angle a t is now found by summing up the three results, just found, so that we can write: Ao^ = Aa n + Aa 12 + Aa 13 . In substituting the values in this equation and treating the deformations of the angles a 2 and 3 in like manner, we finally arrive at the three equations: (i) If we attribute to a tensile stress the positive and to a com- pressive stress the negative sign, we can obtain, for instance, the positive maximum deformation of angle a 1 by supposing s 1 a tension and s 2 and s 3 a compression, while the negative maximum deformation of the same angle results from a negative ^ and a positive s 2 and s y 2. Relations Between the Angle of Deflection and the End Moments. The Italian engineer Alberto Castigliano in his book, "The'orie de Pequilibre des Systemes elastiques," demonstrated and intro- duced the principle of the derivative of work and the principle of least work.* The first quoted principle means, in our case, that if we express the work of deformation of a bar as a function of the outer forces, its first derivative with respect to a moment equals the angle of revolution of the bar. If the first derivative is taken with respect to a force, we obtain the displacement in the direction of the force of its point of application. In Fig. 4, let AB represent a beam fixed at B and under the influence at its end A of a vertical force Q and a moment M Q . * W. Cain : Determination of the Stresses in Elastic Systems by the Method of Least Work. Transactions Am. Soc. C. E., Vol. XXIV, 1891. THE PROBLEM AND ITS SOLUTION The angle r included between a horizontal line and the end tan- gent drawn to the elastic line of the beam is the angle sought, ._-. Mo Fig. 4 . measured by the length of a circular arc (of unit radius) and expressed by the equation: T M 2 , - dx > dM o 2 El r = r M_ Jo EI dx. EIdM M = Moment around any point of the axis of the beam, E = Modulus of elasticity, /. = Moment of inertia. We will now consider a bar of double curvature, Fig. 5, but with Fig. 5- Fig. 6. the express understanding that the end moments only will be considered in determining the deflection angles T. 14 SECONDARY STRESSES IN BRIDGE TRUSSES In Fig. 6, the moments are represented graphically, M = oa and M 1 = i b, and the moment area appears as the difference of two triangles, so that we can write: Area oiba = Aoafr Aoi5. The moment in reference to any point of the axis of the bar or and M = ~{M + M,} y + M 6M __* dM I Applying now the principle of the derivative of work for the determination of the angle T as expressed in the above equation, we can write: , _ ~o EIdM which gives: T ' = JTr i El ( - + 3 , - - M Q - - M - 3 2 2 - + 2 or T = {2 M Q - M,} (2) TJ is found by exchanging M for M r The deformation shows a single curvature and with no point of contraflexure when a bar is acted upon by two end moments of opposite sign, which case will hereafter be exclusively con- sidered, Figs. 7 and 8. THE PROBLEM AND ITS SOLUTION 15 In regard to Fig. 8, M = oa and M^= i b, and the moment area appears as a trapezoid. The moment at any point of the axis of the bar is equal to and M= {M, -M (7 + M l dM x Fig. 7 consequently we have: C 1 M. T ~ Jo E dM EIdM dx = or T = and also (3) 16 SECONDARY STRESSES IN BRIDGE TRUSSES 3. Determination of the Angle Included Between Two Successive Positions of the Axis of a Bar, Belonging to a Truss with Frictionless Pins, which is Acted Upon by Exterior Forces. We assume OA and OA l are the directions of the axis of the bar "before" and " after" deformation, so that the angle included between OA and OA i is the angle sought; see Fig. 9. The equation states that the work done by the exterior forces equals the work done by the interior forces in case of equilibrium. In this equa- tion P designates an outer force and S its corresponding stress, while A/ is an alteration of the length of a bar and d a displacement in the direction of the force P of its point of application. The work done by the reactive forces is not considered, as we assume that the points of sup- port are immovable in the direction of these forces. The above equation holds true for any possible displacements and alterations A/ and for any values of P which are independent of the d and A/. If we wish to calculate any particular d r in the direction of P r , which is caused by a given load system, it is only necessary to let all of the forces P vanish, except P r , which is put equal to unity. In doing this the new equation results: in which the stresses 5 are caused by the force unity and the A/ are caused by the given exterior forces, r corresponding to the A/. The force unity may represent a single force or a moment, in which latter case the angle of revolution ^', measured by the length of the circular arc, must be substituted for the displace- ment d r and our equation becomes: (4) THE PROBLEM AND ITS SOLUTION 17 We will now show that the expression M X 4> represents work as it should do. In Fig. 10, OA represents a bar, having its center of revolution at O and acted upon by a moment M = Q X a. The angle (p is L= &__ 4 a Fig. 10. measured downward from OA in agreement with the direction of revolution of the moment Q X a, which is the same as that of the hands of a clock. From the figure we have: Work = Q X {a + b\ $ - Q X b = Q x a^ = M X 0. 4. The Elastic Line of a Straight Beam Represented as an Equilibrium Curve after Mohr.* This problem was first solved by Mohr nearly forty years ago dx dx Fig. n. by comparing the equation of the elastic line with that of an equilibrium curve. * Mohr: " Abhandlungen aus dem Gebiete der technischen Mechanik," 1906. 1 8 SECONDARY STRESSES IN BRIDGE TRUSSES In Fig. n, the beam is subjected to a vertical load, which is continuous but non-uniform. The hatched area is the load area of the beam and p designates the variable load per unit of length of the horizontal projection of the curve. Figure 12 is a fragment of the force polygon and Fig. 13 Fig. 12. shows the infinitely small sides mn and no of the equilibrium polygon (curve) with horizontal and vertical projections in an _^ *' ~ *-- dx s \d pdx Fig. 13. exaggerated scale, p doc being the load at point n. As the hatched triangles are similar in Figs. 12 and 13, we can write: ^2 - P-fa dx H Dividing by dx we have the differential equation of the equi- librium curve for vertical forces, that is, dx 2 H THE PROBLEM AND ITS SOLUTION 19 The differential equation of the elastic line d?y = M_ doc 2 ~ El is of the same form, where M = any moment, E = modulus of elasticity and 7 = moment of inertia. Consequently, if we put M = p and El = H, or ' = p and E = H , or j - p and i H, we see that the elastic line can be conceived as an equilibrium curve. CHAPTER III. MANDERLA'S METHOD. THE problem of secondary stresses is solved under the assump- tion that the deformations of a riveted truss, caused by the exterior loading, are accomplished in the plane of the truss and that no torsion exists. It is further assumed that every load is applied at the panelpoints of the truss, which is composed of members having uniform sections. This last assumption is not quite true, since the truss members are connected together by means of gusset plates, which make a sudden change in the sectional areas at the ends of the members. There are other elements which are of influence on the stresses, but which can be examined separately. One-sided connections hardly need any consideration, as they are condemned by our specifications ; but the effect of details not centrally designed, as also the effect of loads, applied at any point between the panel- points, as dead and live load, wind pressure and centrifugal force, and the influence of a change in temperature on the secondary stresses, may be calculated. Of considerable importance is the effect of the deformations of floorbeams riveted to the trusses on the stresses in truss members, but this is a problem whose analytical discussion is entirely outside the range of our considerations. The progress of the investigation is as follows: Proceeding from the supposition that the positions of the panelpoints of a riveted truss under the influence of loads are the same as if the truss had frictionless pins, the alterations of all the angles in the triangles, composing the truss, are calculated according to equa- tions (i) in Chapter II, for that system of loads for which we intend to determine the secondary stresses. But, as the ends of our truss members are supposed to be rigidly fixed, any alterations MANDERLA'S METHOD 21 in the angles of the triangles are impossible, in consequence of which the truss members become deformed under the action of the exterior loads and are subjected to bending moments, which must reduce the supposed changed angles to their original magnitudes. The end bending moments are now introduced into the calcu- lations as the unknowns, and relations are determined between them and the angles of deflection, which latter angles are included between the chords of the deformed bars and the end tangents drawn to the elastic lines of the bars. Hereupon the angles of deflection are expressed in terms of the alterations of the angles in the triangles calculated by means of equations (i) for the desired load system under the assumption of frictionless pins. After the unknown bending moments have been determined, it is easy to find the secondary stresses according to known formulas. Man- derla finds the desired stresses not directly, but by a few trial computations, which are easily performed. Before we proceed with the analysis of the stresses, it will be necessary to give some remarks on the action of the forces to which the truss members are subjected, and to point out the differ- ences which exist between the compression and tension members, as shown in Figs. 14, 15, 16, 17, 18, and 19. If we pass a cut close to a panelpoint and apply the inner forces to establish the equilibrium, then these forces can be represented by their resultant P, which in turn can be resolved into an axial force S, a transverse force Q, and a moment M. We refer the reader here, to what has been said on this subject in Chapter II. The two moments for each bar either act in the same or in the opposite sense. If in the same sense, the deformed bar is either on one side of its chord or partly on one and partly on the other side of the chord, and in both cases that place where the line of the resultant P intersects with the elastic line is a point of inflection. Should the two moments act in opposite directions, then the deformed bar is on one side only of its chord, and it has no point of inflection. The resultant P intersects with the prolongation of the chord of the bar ab. 22 SECONDARY STRESSES IN BRIDGE TRUSSES Fig. 14. Fig. 15- S-. Fig. 16. COMPRESSION MEMBERS. MANDERLA'S METHOD Fig. 17. Fig. 18. Fig. 19. TENSION MEMBERS. 24 SECONDARY STRESSES IN BRIDGE TRUSSES For tension members, the convex side of the elastic line is turned towards the line of the resultant P, and consequently no moment between the end points a and b of the bar can be the maximum moment. For compression members, the concave side of the elastic line V Fig. 20. is turned towards the line of the resultant P, and a maximum moment between the ends a and b of the bar is possible. We will now turn our attention to the deflection angles. Figure 20 represents a panelpoint K where four bars intersect. The straight bars are shown in full lines, and any angle included between two adjacent bars KS, which had the value before deformation, becomes < + A< after deformation. The assumption is that during the action of the actual exterior loads upon the truss each bar is free to turn around a frictionless pin. But in reality the bar ends are fixed by riveting, consequently they must be deformed while the truss deflects, and the angles formed by two adjacent end tangents KT of the deformed bars, shown in dashed lines, must remain constant during deflection, or, in other words, these angles are the angles (/>, which existed before the exterior loads began to act. MANDERLA'S METHOD 25 In Fig. 20, the angles of deflection r are produced by clockwise revolution and are to be taken positive. From the figure we have e + p -= & + p + T, + <" + <' = " + A" + <' + A<' + r", + 0" + "+ ' - <'" + A"-f A" + c// + Ac/,' + T///> or r' = - A0', T" := 6- {A^ + A^i, i 7 " == e - {A^ + A" + A^}, and generally t - ?.- X A ^- (5) This shows that if we knew only one deflection angle , then from it and the A< all others around a panelpoint could be readily calculated. i. Compression Members. Our next step consists in seeking a relation between the moments M and the angles of deflection T, and for this purpose we employ the equation of the elastic line, which we write, bearing in mind the directions of the axes of the coordinates as shown in Figs. 14 to 19, d 2 y _ M x doc 2 El where E = modulus of elasticity, / = moment of inertia and M x a moment at any point of the axis of the bar, that is, M x = + Sy - Qjc + M,. 5 Substituting and putting the three constants = T 2 , El wehave ^ = - Ty + and by integration (6) 26 SECONDARY STRESSES IN BRIDGE TRUSSES The constant B is determined from the condition that y becomes o for x = o, consequently The constant A is found by putting y = o for x = I, that is, M p i A =-'- tan - / tan to - S 2 sin p cu . If / S In this last equation p = IT -= I y , and tan we put in equation (6) the values for the constants A and B, we get by differentiation, dy M t r(, p . ) ( p ) < tan cos xl sin xl } +tan oj < i - cos xl > (7) dx S ( 2 ) ( sin p y The equation (7) gives the angles of deflection dy , , dy , -- == r l for x = o and -- = r 2 for rv = /. These angles are : M T tan - tan 6; 2 sn /o - i T, = - tan - tan 2 P ) - - cos p - i sm p ) (8) If we now put for brevity's sake, 2 + cot - 2 ?M r and p p cot = 2 n, 2 p cot 2 or 2 /? COt L 2 MANDERLA'S METHOD 27 we obtain by addition of the equations (8), f ( . f tan co - {-, d T 2 }, (9) and by subtraction, S l == - X-~i n e T 2 } y (10) and by exchange of the angles r, M 2 =|{ m c T 2 + n,-,}. (ID For practical calculations it is best to express m c and w c in series as follows: 1 P P 3 cot p = - - P 3 45 cot-.2-- -f 2/06 360 2 - p COt 2 6 ( 60 2520 6 i i a - - p- , p- 10 1400 42 II W, ; = - p - - p 3 p 15 6300 n c = ~ + P + -2- p P 30 12600 2. Tension Members. The moment M x at any point of the axis of a tension member is M x - -Sy- and consequently / 7 28 SECONDARY STRESSES IN BRIDGE TRUSSES By integrating we find M -*-' y = A xT + Be xT tan "~ j 5 < p - ~ p ) (f ~ ~ p (12) Of course we have here again T 2 EI = Sj ~ 1 = tan o> and /T 1 = p. After substituting the values of the constants A and B in the equation (12), we obtain by differentiation the angle of deflection dy_ dx = &L _ ^xT J i -- p ^~ 5 ? p - - p ' + + tan & rrxp - i (13) Further, -T = TI for 5f = o and rfjc r 2 for x = I, or : p + ~ p \ Jf t 1 2 e TI 6 1 " p - ~ P p - - p 2 2 M I 2 2 T, - - 2 p - ~ p p - ~ p ^ 2 2 . ' (14) MANDERLA'S METHOD 29 It is suitable for our purpose to replace the exponential func- tions by hyperbolic functions, which latter we express in series. The hyperbolic sine and the hyperbolic cosine are written: . . - , sm hp = - and cos hp = 2 We have further, 7 p i + cos hp , p i cos hp cot h - = - J and tan h - = - ; ; , 2 sm hp 2 sm hp and consequently the equations (14) are transformed into (15) J T tan h ?- + tan co \ 9 - - i 5 2 ( sin hp r 2 = - T tan A + tan cu -r~~ X cos hp- i S 2 sin /zo If we write + cot h = 2 m t . j P 2 p COt fl 2 2 p - cot h = 2 n t , i P 2 p cot h - -2 2 we find by addition, i m + n p cot h - - 2 2 But 7 * x\ ^v^>j /i/r' j. , sm hp sm /z/o sm P n = /O COt /Z-- - 2 = 2 m t + n t 30 SECONDARY STRESSES IN BRIDGE TRUSSES and in adding the equations (15) and using this last expression we obtain tan^=^^{r 1 T 2 }, (16) P and in subtracting them, M l =- {m t TI , r 2 }, (17) and by exchanging the angles r, 5 M 2 = { w, r 2 + , TJ}. (18) We express now cot hp in a series that is cot/2/0= I + _^ + 2 -L ^ 3 45 945 and consequently we have in a similar manner as that for com- pression members, T p p 10 1400 p cot h L - 2 r 2 15 6300 p 30 12600* 3. Determination of the Deflection Angles from the Conditions of Equilibrium. The final step consists in writing down the equations from which the unknown deflection angles are to be computed, and in showing the manner in which these calculations are effected. With the knowledge of the values of the deflection angles the problem is really solved, as the remainder of the computations MANDERLA'S METHOD refers to simple operations with equations already known. With this end in view r , we consider the equations: As M l = \m c r l n c r 2 S M 2 = -- Im c r 2 + n c M 2 = |j ( r 2 + 5 _ s VEI T" VS for compression members. for tension members. j we write 3 12600 ^4 2 II m t -=K= } *- + --p - ^ r (p 15 6300 r n t - = L= ] - T (p 30 12600 r tor com- \ pression ' V F T 9 r members. |V/5 for ten- sion I VEIS members. The general equation for the moment M 1} which is the moment at the end a of any of the bars in Figs. 14-19, is S =-- < mr, nr 2 I = Kr, Lr 2) (19) in which the letter c or /, pointing to compression or tension, has been omitted, in consequence of which m, n may refer to either a compression or tension member; it depends on what bar is under consideration, and with this understanding the equation will be used. We now consider that in Fig. 21, the bar ends, formerly called o, and 4 in this particular case, all meet in the panelpoint c. We SECONDARY STRESSES IN BRIDGE TRUSSES give further to each moment the positive sign, when it turns in the same direction as that of a hand of a clock, as is shown by the arrows, which indicates that the angles of deflection are produced by clockwise revolution. If the result of the computation shows a positive sign for a moment, then it turns as shown in the figure; and if a negative sign, it turns in the opposite direction. The equilibrium of panelpoint o requires that 4 %M = o- (20) i If we substitute in equation (20) the 4 moments of the form of the equation (19) by using the notation as given in Fig. 21, where the angles of deflection r appear either in the form + a or + /?, then we have 01 # #02^0 # 03 U #04^0 ft! ft} = o. (21) Calling X X X X #04 X 4 = X ft = ""Ol X Si ~T -^02 X S2 ~T~ -^03 X S3 I -^04 X 4 [ UNIVLKoiTY OF MANDERLA'S METHOD 33 the four equations (21) are written, = o, or . Each panelpoint gives an equation for a deflection angle , and as we have just as many unknown as panelpoints, the deflection angles can be determined. But the computation of the angle from the equation (22) has a difficulty, which forbids its direct solution, and this difficulty consists in the fact that the third mem- ber of the numerator contains the four unknown quantities t , 2 , 3 , and c 4. In order to remove the difficulty, we resort to trial computa- tions, and put tentatively ^ = 2 = 3 = 4 = o, and compute the angle under this supposition, which is in so far justified, as the SZ, is very small compared with the sum of the others. In a similar way we proceed with the determination of the deflection angles of the other panelpoints, and consequently also with that of l} 2 , 3 , and 4 , and substitute these latter values in equation (22), which now yields a more precise value of , and in this way we continue the operations until satisfactory results are obtained. After these values for c are known they are sub- stituted in equation (5), and then the deflection angles r are com- puted, and finally the values of r thus found are substituted in equation (19), whereupon the moments M are obtained. If d is the distance between the neutral axis and the extreme fiber, we find the secondary stress Md T The position of the lines of direct stresses in the bars is an easy matter to determine after the values M are known. Considering that the angle at, which is included between the resulting force P 34 SECONDARY STRESSES IN BRIDGE TRUSSES and the chord of the deformed bar, is always very small, we commit no material error in putting P = S, so that we have, M l A , M 2 !>--, and /,=;-' If we make ample provision against buckling in the design of the truss members, that is, if we design them with large moments of inertia, then p = IT = I V ; becomes very small, and the * hi equation (19), assuming a double curvature so that both deflec- tion angles r l and - 2 are positive, is then written and for M 2 we get After some simple transformations we find from these two equations the values of the deflection angles l{2 M , - J/J 6 El 6 El (23) These equations (23) are identical with the equations (2) in Chapter II, which latter we developed by applying the principle of the derivative of work. The derivation of the equations (2) is based on the assumption that the deformation of a bar compared with its dimensions is very small and consequently a negligible quantity, but the foregoing analysis demonstrates that this assumption is only justified for large moments of inertia. Therefore, the equations (23) or (2) should not be employed for slim and flexible members. CHAPTER IV. MULLER-BRESLAU'S METHOD. i. Derivation of the Fundamental Equations. THIS method neglects the influence of the deformation of the bar on the secondary stresses, which is justified, as we have seen in Manderla's method, for sufficiently large moments of inertia. The method proceeds from the assumption that the exterior loads, for which the secondary stresses are to be computed, are exclusively applied at the panelpoints and produce a deformation of the axis of each bar showing no point of inflection, as indicated in Fig. 22. This assumed deformation, made for the sake of determining the character of the signs of the bending moments, does not exist Fig. 22. Fig. 23. in reality, because the sum of the alterations of the angles a, /?, and 7 must vanish, that is to say, we must have Aa- + A/? + AY = o. SECONDARY STRESSES IN BRIDGE TRUSSES For instance, should the moment M^ be found negative, then the deformation of the triangle is that of Fig. 23. The alterations of the angles are to be calculated according to equations (i) in Chapter II, under the assumption of frictionless pins, but, as the bar ends are supposed to be rigidly fixed by rivet- ing, the angle alterations are conceived as reduced by the bending moments to their original values a, /?, and 7, as previously explained in Chapter II. The equations (3) are those for the deflection angles of a bar with single curvature, and they are written with regard to the designations as given in Fig. 22, 2M. El, M -\k' (24) in which l lt 1 2 , 7 13 7 2 , are the lengths and moments of inertia of the bars AC and AB. M denotes a bending moment and E is the modulus of elasticity. We have further, (25) If we substitute the values r 2 , r 3 . . . . from equations (24) in equations (25) we find {M, + 2 M 2 } {2 M} = {M 3 + 2 M,} - + {2 M 5 + M,} - =6 A/?, {M, + 2 MJ - + {2 M! + M 2 } - =6 (26) If we assume for the present that three moments are known, we would then be in a position to calculate the other three bending moments from equations (26). Later illustrations will show the MULLER-BRESLAU'S METHOD 37 manner in which the supposedly known moments are found. Now for the sake of convenience we do not introduce the unknown moments M in the computations, but the expressions M - = U, and put for the bar AC = I, : " " " AB = 1 2 : M 3 == U a and M 4 f- = U 4 . 1 2 1 2 " " " BC - /, : M 5 4 3 - = U. and M e ^ = U r ^3 ^3 With these expressions inserted in equations (26) we obtain U, + 2 {U 2 + Z7J+ Z7 4 = 6EA. (27) J7 3 + 2 {U 4 + t7 5 } + U 9 = 6 A/?. (28) ^ 5 + 2 {t/ 6 + Z7J+ t/ 2 = 6A 7 . (29) We have further the relation U, + U 2 + U 3 + U 4 + U 5 + U, = o, (30) because the sum of the alterations of the angles a, /?, and 7 equals zero, that is, Aa -f A^ + A 7 = o. We now express U 4 , U 5 and t/ 6 as functions of the assumed known quantities U^ U 2 , and Z7 3 , and obtain from equation (27), U 4 = 6 Ea - U, - 2 {Z7, + U 3 }, (31) and from equations (28) and (30), U & = 6 E t 8 + U l + U 2 - U 4 , (32) and from equations (29) and (30), U, = 6 A 7 - Z7, + t/ 3 + ^ (33) As soon as the quantities U are known, the bending moments M are also known, and with them, of course, the secondary stresses; but in order to find these stresses in a truss, it is necessary to apply the foregoing equations successively to the different triangles, SECONDARY STRESSES IN BRIDGE TRUSSES which compose the truss, whereby the triangles are supposed to be alternately deformed according to Fig. 22 or Fig. 24. In order to calculate the alterations of the angles in those tri- angles deformed as in Fig. 24, we must bear in mind to reverse the signs of A a, A/?, and A 7, and to write s l - s 3 \ cot 7 Is, cot a + - **} = o. cot p cot 7 BAA (34) 2. Determination of the Quantities U for a Truss. In calculating the quantities U for a truss, Fig. 25, we assume for the present that the values U^ and U 2 are known, and we then express every other U as a function of U 1 and U 2 . The equilibrium requires that for every panelpoint Slf = o, consequently we have at panelpoint A : M 2 M 3 = o, or U 3 -~ = U z -r-- 1 2 /! After Z7 3 is known, its value is substituted in equation (31) and 7 4 calculated, whereupon U 5 is found from equation (32) after the substitution of Z7 4 , and in a similar way U Q is found from equa- tion (33). MULLER-BRESLAU'S METHOD 39 At panelpoint B the value U 7 is determined from the condition that MQ MI M 7 = o, L or In going over to the triangle called II in Fig. 25, we simply repeat the calculation made for triangle I, that is to say, we write down the equations for U s) U 9 , and U lo in conformity with equa- tions (31), (32), and (33), care being taken that the alterations Aa, U 13 A/? and AY are calculated for triangle / after equations (34) and for triangle II after equations (i). Knowing U 4 , U 5 and t/ 10 , we find Un at panelpoint C from - M 4 - M, + M lo - M u = o, and finally U 12 , U 13 and U 14 are found from equations which again correspond to the fundamental equations (31), (32), and (33). We have now expressed each U as a function of U l and 7 2 , which were supposed to be known, and the next step consists in determining U l and U 2 . To this end we apply the condition 2M = o to the panelpoints D and E, and write LL _ = o. 40 SECONDARY STRESSES IN BRIDGE TRUSSES The solution of these equations gives / x and U 2) and herewith the problem is solved. 3. Method of Influence Lines. The secondary stresses are usually computed for one single position of the loads. A more complete investigation would con- sist in the computation of these stresses under the assumption of full load and one or more partial loadings, but the most complete investigation applies the method of influence lines. By Manderla's method we have seen that the values K and L are higher functions of the primary stresses 5, and as the secondary stresses are dependent on K and L, they are also higher functions of S or of the exterior loads, in consequence of which the employ- ment of the method of influence lines is not to be thought of. It was also shown that if the influence of the deformations of the bars is left out of consideration, then the resulting equations are the more exact the larger the moments of inertia or the stiffer the truss members are, and in this case the secondary stresses are found to be linear functions of the exterior loads, and the method of influence lines is applicable. In using this method, an influence line may be drawn for each bar end under the assumption of a traveling load equal to unity which is successively applied to the panelpoints of the truss. MULLER-BRESLAU'S METHOD 4! Muller-Breslau's method of influence lines is in substance as follows : Figure 26 represents a truss composed of the triangles I to VII, the sides of which show alternating deformations, and whose angle alterations must be calculated alternately in compliance with equations (i) and (34). The values U lt U 2 , U 3 , etc., correspond to the figures i, 2, 3, etc., written near the bar ends in Fig. 26. The first step consists in the calculation of the angle alterations after a load is applied to the nearest panelpoint of the right-hand support in our case panelpoint (4) and of such magnitude that it produces at the left-hand support a reaction = i. For this reaction : = i , each U value from U 3 to U 2T inclusive is expressed as a function of U 1 and U 2 in a manner as has been previously explained. Thereupon the load is shifted from panelpoint (4) to panelpoint (i), and it is given such a magnitude that it produces at the right-hand support B a reaction = i. The calculations must now be repeated. First, all angle altera^ tions are calculated for a reaction = i at B, and each U from U^ to U 4 is expressed as a function of Z7 30 and U 2r If the load is applied at (4) the general expression for U m is U m V m + C m U, + D m U 2J ( 35 ) and if the load is applied at (i) we have U m = W m + F m U 30 + G m U m . (36) V m , W m , C m , D m , F m , and G m are known quantities, and V m and W m alone are functions of the alterations of the angles which are dependent on the assumed exterior load. If we now apply to panelpoint (2) a vertical load equal to unity, then this load produces a reaction at the left-hand support equal to R a = i X j, and the stresses due to this reaction in the sides of the JL/ triangles marked I and II are times greater than those pro- JL/ 42 SECONDARY STRESSES IN BRIDGE TRUSSES duced by the reaction R a = i, in consequence of which the equa- tion (35) is transformed into U, D m U. (37) The value / n is also computed after equation (37) for the simple reason that the condition M = o must be fulfilled, that is to say, we must have - M n + M 10 - M, + M 4 = o, or, by multiplying with i and writing U instead of M, we get i u tl f - u, ^8 / The same reasoning holds also true for the right-hand reaction R b , the stresses and angle alterations in the triangles IV, V, VI, and VII, so that we are in a position to write down the following set of equations : U - +C 8 C7, +D S U S2 U, = V, + C, L\ + D, U, U, U --W U 13 ~ j VV 13 & T1 (38) MULLER-BRESLAU'S METHOD 43 With respect to equations (32) and (30) and the condition M = o, applied to panelpoint (2), we now write down: U 13 + U 14 , U 9 + U 1 (39) After the values U 8 to Z7 15 , as expressed in equations (38), are inserted in equations (39), they can be solved for U ly U 2 , U 30 and t/ 29 . In applying the load = i in succession to the different panelpoints of the truss, we obtain by the same course of treatment the influence lines for U 1} U 2 , t/ 30 and U. M , and with these latter values the influence lines for any U. We caution the reader to observe the fact that Aa' m and A/?,,, in equations (39) require a separate calculation for the reason that, if the load = i is supposed at (2), then the reaction R a = i X - produces - times greater stresses and angle alterations than R a = i in the triangles I and II, but not in triangle III. For each U we have two equations, the use of which depends on the position of the load. If, for instance, the load is applied either at panelpoint (3) or (4), then U a = 2 V 13 + C n U t + D 13 U 2 ; but if the load is at (i) or (2), the equation 77 =-W + F U + G U must be used. CHAPTER V. RITTER'S METHOD. OUR first step will consist in assigning the proper notation for the different quantities with which we have to deal. In Fig. 27, representing a part of a truss, we will for the present exclusively consider panelpoint (5) where four bars intersect, forming three angles. Each of these angles included between any Fig. 27. two adjacent straight bars will be denoted by an index corre- sponding to the two figures of the opposite bar. So, for instance, the 3-5-4 shall have the index 3-4, the ^ 4-5-6 the index 4-6, etc. The two bending moments of the bar 3-5 will be designated M 3 at panelpoint (5) and M 3 ' at panelpoint (3). For the bar 4-5 the moments shall be Af 4 at panelpoint (5) and Mf at panelpoint (4), etc. But in case we consider panelpoint (4) the designations of the two moments of the bar 4-5 are M 5 at panelpoint (4) and M K ' at panelpoint (5). Consequently we can write: M 4 for panelpoint (5) = M & f for panelpoint (4), MI for panelpoint (5) = M 5 for panelpoint (4), 44 RITTER'S METHOD 45 which means that each moment in a truss is characterized in two different ways. A moment will be taken as positive when it deflects a bar in the sense of the hand of a clock. If each bar in our truss could turn around a frictionless pin, then each angle a included between any two adjacent bars would be changed under the influence of the actual loading of the truss; these angles would be either increased or decreased an amount A a corresponding to the alterations in the lengths of the bars. But the bar ends of our truss are riveted, consequently the angle included between any two adjacent end tangents drawn to the curves of the deformed bars, which we assume to be S-shaped, must remain unchanged during deformation, and this unchangeable angle is a. We refer the reader here to Fig. i in Chapter II, and to what has been further said on this subject. Figure 27 shows that or Substituting the values of r 3 and r 4 as given in equations (2) of Chapter II, we have: /3J2M3-M/J / 4 { 2 M 4 -M/j - 4= . 6/ 3 ~67T Similar equations can be written for EAa 4 _ and We put now for the sake of convenience - - = U, U repre- 6 1 senting a force per unit of area and measured with the same unit as s and EAa. The equilibrium at a panelpoint requires that the algebraic sum of the moments vanishes, or that M 3 + M 4 + M 8 + M 7 = o, and the four equations in regard to panelpoint (5) are now as follows : 46 SECONDARY STRESSES IN BRIDGE TRUSSES .< = {2 17.- 17,'} -- \2U t - U t '}, ..- J2t/ 4 - [//} -- J2tf. - 17.'}, ., = J2^ c - f/ 'j -- \2U. ~ U/}. (40) For every panelpoint there are as many equations as bars, which intersect at the panelpoint, or two equations for each bar, so that the total number of equations equals the number of unknown moments. The work of solving a larger set of equations algebraically taxes the patience of a quick and sure computer, experienced in all the short cuts that can be advantageously used, and any means which are calculated to save time are very welcome indeed. For this reason we will now consider a graphical solution of the problem, the object being to find the values U. The EAa are computed by means of equations (i) in Chapter II, and are due to the actual loading of the truss, for which loading we intend to determine the secondary stresses. If we assume for the sake of an illustration that besides the EAa also the values of V were known, we could then easily and quickly find the values U by the simple means of a force and string polygon in the follow- ing manner: We consider the values as forces, lay out a vertical load line, i select an arbitrary pole O 5 with the pole distance H 5 , and draw the rays as in Fig. 28. Hereupon the distances between the forces, which are the known EAa, are laid out horizontally as in Fig. 29, and each of the forces displaced to the left an amount U f and the equilibrium polygon constructed for the forces thus displaced. The double values of U are now equal to the distances of the displaced forces from their resultant, a statement which follows from the equations (40). The distance of a displaced force from the resultant is 2 U for a displacement U', consequently the distance of an undisplaced HITTER'S METHOD 47 force from the resultant equals 2 U-U', and the difference between the distances of two adjacent undisplaced forces must equal E&a. The meaning of this is that the first three of the equations (40) Fig. 29. have been satisfied, and since the string polygon shows that the algebraic sum of the component moments in reference to any point in the direction of the resultant vanishes, the last equation is also satisfied. As a matter of fact, the values of U f are not known at all, and herein of course lies an obstacle to our solution, but this can be removed by trials. Considering the fact that any change in the values of U r has only half the effect on the values of 7, we will for the first trial assume that the quantities U f are non-existent, in which case we obtain roughly approximate values U\ or, in other words, for the first trial we will assume that the forces are not at all displaced and with this understanding we draw a force and equilibrium polygon for every panelpoint. Figures 30-31 show the positions of the resultants of the undis- placed forces for the panelpoints (4) and (5), the equilibrium polygons being omitted. 4 8 SECONDARY STRESSES IN BRIDGE TRUSSES If now the distance between an undisplaced force and the result- ant is designated by F, we have, with reference to Figs. 30-31, 2 U, = U.' + 7,, a U, - U t ' + V t . In compliance with the designations as adopted at the beginning, we have also 77 '_ u[' = u" 6534 3 467 1,111 I 2U4 Panelpoint 4 Fig. 30- Panelpoint 5 Fig. 31. Further, TJ, - > U 5 " or and in a similar way we find The determination of the quantities U would be an easy matter if the positions of the resultant forces were known, in which case they could be found at once ; but since these positions are not known and must be first found, we take for instance U 3 , which equals RITTER'S METHOD 49 and transfer it as // in the equilibrium polygon for panelpoint (5); and in a similar way we transfer any other U as U', which has been obtained by a first trial, into some other and corresponding equilibrium polygon, and continue these correcting operations until the changes in the values are so small that they can be neglected, but we must bear in mind that each equilibrium polygon has to be repeatedly drawn. After the quantities U are known, we obtain the bending moments from the equations: and if d is the distance from the center line of gravity to the extreme fiber, and cr the secondary stress per square unit, then Md 6d T1 ~ ~T U ' Strict attention must be paid to the character of the signs in order to avoid mistakes. The succession of the bars around a panelpoint should be taken in the sense of motion of the hand of a clock, and a Aa, which has a positive sign, should be laid out on the right hand, and a negative A a on the left hand. Any quantity U to be transferred as U' into a corresponding equilibrium polygon should be laid out to the left, if it is situated to the left of the resulting force of the - , and it should be laid out to the right, if on the right side of the resultant. The method we have explained is in so far approximate as the influence of the deformation on the secondary stresses, due to the action of a force along the chord of the deformed bar, has not been taken into account; and if this longitudinal force is also to be con- sidered, it is necessary to develop an expression for the deflection angle r differing from that given in equations (2) in Chapter II. OF TM UNIVERSITY 50 SECONDARY STRESSES IN BRIDGE TRUSSES The equation of the elastic line with the designations as given in Fig. 32 is written doc 2 El M x being the moment at any arbitrary point of the axis of the bar with the coordinates x, y. If it is now supposed that the bar is acted upon not only by the ^ +x two end bending moments M and M', but also by a compressive force of the magnitude S, then M x = Sy + M\l - x} Substituting this value M x in the differential equation, w r e get + M'\ x - Ml tL _ *L i dx> El EIL The integration gives M y-ycos Kx McosK + M' . Kx sin - SsmK + M'}x-Ml SI In this equation the constants have been determined from the condition that y vanishes when x = o and when x = /, and K /SI 2 denotes the expression y for brevity's sake. tLL RITTER'S METHOD 51 The first differential coefficient gives the angle of deflection r for x = o, that is, dy = _ K{McosK + M'\ M + M' dx SI sin K SI Developing cos K and sin K in series gives = l_\2RM - R'M' 6 El and in this last equation 2 K 4 . K" E> J 5 3*5 1575- ^ = I+ 7A 3 + 3^ + -^ c oo 2520 100800 If M and M' are exchanged r' will be found. Should the bar under consideration be a tension member, then the sign of K 2 in SI 2 the equation K 2 = must be reversed and R and R' computed J^sJ. accordingly, that is to say, every second member is negative. For an infinitely great moment of inertia K 2 is reduced to zero and R = R' = i, which leads us back to the equations (2) in Chapter II. From this consideration we conclude that the equations (2) are the more exact the larger the moments of inertia are, or, in other words, they should only be used for stiff truss members which have ample provision against buckling, an assertion pre- viously made and which has now been demonstrated to be true. Should the more exact method be used in connection with graphic statics, it is necessary in constructing the force polygon to lay off the forces and displace them in the equilibrium polygon an amount R'U 1 ', whereupon the values 2 RU are found instead of 2 U. CHAPTER VI. MOHR'S METHOD. i. Determination of the Unknown Quantities. IN this method of calculation the effect of deformation on the leverarms y, as also that of the transverse forces Q, are not con- sidered (see Fig. 2 in Chapter II), for reasons previously given, and only the bending moment at each end of the bar is determined. If for a truss, which* is composed of triangles, p denotes the number of panelpoints and n the number of bars, then n = 2 p - 3; and as 2 unknown moments correspond to each bar, we have as the total number of unknown moments, 2 n = 4 p 6. But instead of introducing the unknown bending moments in the calculation, Mohr introduces two sets of angles on which the bending moments are dependent, and in so doing he reduces the total number of unknowns to 3 p 3. Figure 33 shows the unknown angles (f> , lt and 01 for the bar 01. The lines o a and i b are parallel to each other, and indicate the original direction of the bar 01, which is that before the ex- terior forces began to act. o T and i T l indicate the end tangents drawn to the elastic line of the curved bar after deformation has set in. The angle

is the angle around which the end of each bar revolves during deformation. This fact follows from our assumption that all bar ends are rigidly riveted so that an angle between any two adjacent 52 MOHR'S METHOD 53 bar ends must remain constant while a bar becomes . deformed. The angle ^ 01 included between the original direction o a or i b of the bar and the chord of the curved bar after deformation is the angle around which a bar revolves during deformation under the assumption that the truss is provided with frictionless pins. To each panelpoint of the truss corresponds an angle <, and as the number of the panelpoints is />, we have p as the total number of the angles ; to each bar corresponds an angle ^, and as the number of the bars is 2 p 3, we have 2 p 3 as the total number of the angles ^, consequently the number of angles (f> and ^ taken together is equal to p + 2 p - 3 = 3 # 3, which are now the number of unknowns of our problem. 2. Determination of the Angles (p. These angles $ are calculated by the equation in Chapter II, M([) = 2 s A/, where M == i, so that i X $ = ZsM for the assumption that the points of supports are fixed in the direction of the reactions. This is the case with respect to Fig. 34, which shows a Pratt truss with one fixed end and one roller end. If we wish, for instance, to determine the angle of revolution

and d>. In Chapter II, it was shown that the elastic line, whose equation d 2 y M is -r^ 2 = , can be conceived as an equilibrium curve, if we put, CLOC JLH. for instance, El = H y and M = p\ that is to say, by considering first the term El as a force acting parallel along the chord of the deformed bar, and considering secondly the bending moment M as a force per unit of length of the equilibrium curve and at right angles to the chord of the deformed bar. MOHR'S METHOD 55 In order to obtain at once the proper signs for the moments M and the angles and ^ are very small, we find, BC-i O il 3 il 3 The forces (J^ and Q 2 are Q l = Af ia i and -i ( < - I 3 M - Substituting the values of yl.D, 5C, Q l and Q 2 in the equations (42), we have . P 21 6 = > I 2 -Q. (43) Dividing each equation by and putting- =N 12 , the equa- tions (43) are then written, In order to find the moments Jlf it is necessary to determine next the unknown angles (f> in equations (44) ; the values TV are known, and the angles ^ have previously been calculated. As the alge- braic sum of the moments with respect to any point must be equal MOHR'S METHOD 57 to zero, we take in succession the panelpoints of a truss as the centers of moments, and wrile down as many equations as there are panelpoints in the truss, which number equals the number of unknown angles . With reference to Fig. 34, these equations are as follows: Panelpoint i : Mj_ 2 + M^_ z = o. 2 : M 2 __, + M 2 _ s + M 2 _, + M 2 _ 4 = o. 3 : M 3 _, + M 3 _ 2 + M s _ s = o. 4 : Mt_ 2 + M 4 _ 5 + M 4 _ 7 + M 4 _ 6 = o. 5 : M 5 _ 3 + M,_ 2 + M 5 _ 4 + M 5 _ 7 = o. 6:lf c _ 4 +M 6 _ r + M 6 _ 8 = o. I 7 : M^, + M 7 _ 4 + M 7 _t + M T _, + M 7 _, = o. | 8 : M 8 _ 6 + M 8 _. + M 8 _, + M 8 _ 10 = o. 9 : M 9 _ 7 + M 9 _ 8 + Af _ 10 + M 9 _ u = o. 10 :^f 10 _ 8 + M 10 _ 9 +M 10 _ U + M ] _ = o. 11 :M n _ 9 + M U _ 10 + M U _ 12 = o. _+ M 12 _ n = o. The values of the bending moments M as expressed in equations (44) are now substituted in equations (45), and the values of the angles < are ascertained by solving the latter equations. The last step consists in substituting the values < in the equations (44), from which now the moments M can be calculated, &nd here- with our problem is solved. Mohr suggests a short cut in regard to the solution of equations (45), provided the truss to be examined is symmetrical, in which case the conditions of equilibrium appear in symmetrical form. For a symmetrical truss the work of solving the equations is now greatly facilitated by determining first the sums and the differences of the angles <, whose positions are symmetrical in respect to 'the center line of the truss. In our particular case we would determine first { 3 equations available for the deter- mination of the 6 p 9 unknowns. The remainder of the equa- tions, that is, { 6 p 9 j - - {3^ 3} ; = 3 p 6, must therefore be obtained from some other source than statics. From the fore- going remarks we see that a triangular riveted truss is threefold statically indeterminate, a fact which can also be arrived at in a way different from the above. The influence of the sectional areas on the stresses, which always exists in a statically indeterminate structure, can easily be detected in our truss by going to extremes. Let us suppose that the truss is loaded at its apex with a finite load, but that the moments of inertia of the two compression members are infinite, then any deformation of these members is excluded, and as the angle at the apex the truss being riveted remains unchanged, it follows that the length of the horizontal bar is not affected at all by the 60 SECONDARY STRESSES IN BRIDGE TRUSSES exterior load. This is merely another way of stating that the horizontal bar is under no stress. If we now suppose each moment of inertia of the two compression members to be equal to zero, in which case each of these two members is represented by a very thin bar, hinged at its ends, then the horizontal bar receives a pull of 125 tons, but no bending moment. As neither of these conditions can be fulfilled, that is, as the moments of inertia of the members must be between o and x , we draw the conclusion that the pull in the horizontal bar is less than 125 tons. The circumstances that the truss is symmetrical in form and also symmetrically loaded reduces the three unknowns to two unknowns. Generally speaking, we are free to select the unknowns of the problem, and in our case we take the pull of the horizontal bar and the bending moment at its middle as the two unknown quantities to be determined. The only stresses we need to con- sider in our case are direct and bending stresses; the shearing stresses, which may be taken into account, are in so far of no con- sequence, as their influence on the final result is small enough to be neglected, and the effect of the moments Sy, owing to the small deformation of the bars, we will also leave out of consideration. The principle of least work requires the work of deformation of the truss to be a minimum, which means that the partial differ- ential coefficients with respect to the unknowns must be placed each equal to zero. Therefore, we write the equations of condi- tion for our particular case, M 9M C N w iTu dx + J-RA-W**- In this equation M = moment with respect to any point of the axis of any bar, N = direct stress in any bar, / = moment of inertia of any bar, A = sectional area of any bar, E = constant modulus of elasticity, U = any unknown. METHOD OF LEAST WORK 6l As the truss is symmetrical in form and symmetrically loaded, it suffices to extend the work of deformation over one half of the truss, instead of over the entire truss. If we now pass a cut through its middle, Fig. 37, apply the inner forces as outer forces in order to establish the equilibrium, take A as the origin of the abscissae x, coincident with the axis AC and of the abscissae v, coincident with the axis of the horizontal bar, resolve P and H in components parallel and at right angles to AC t we can then write, H and M being the unknowns, M dM dH P cos a X x H sin a X x M , dM .-sinaX*; -^rr = - I, and N = P sin a + H cos a, dH = cos dN ; -- = o, dM. for bar AC. M = - M, = o; dM I, and N = H, W 5 3Mn = o, for bar AB. 62 SECONDARY STRESSES IN BRIDGE TRUSSES By substituting these values in the equation of condition, we obtain, with respect to Fig. 36 and 37 two equations from which the two unknowns can be found. These equations are, C L P cos a sin a 2 , r L H sin a 2 X x 2 , r L M ft sin a X x , - ~r ~ dx + ~ dx+ - dx Jo -LI Jo *! Jo 2 i s* L P cos a sin a , r L H cos a 2 r l H , -f / dx + / ax + I dv = o J A i J ^i Jo ^2 *^ ' T 7" yj J 1\ Jo J l */0 ^ 1 t/0 ^2 Computing these integrals and solving for H and M , we get H = P X j^T^ S ( *" / and M = P- <" ^ c ~ d where a = - -r-, 6 / 1 i cos aL each sin a has been replaced by cos a. If we assume 7 X = oo, we obtain H = o, and M = o; and if we assume 7\ = o, we get H = P, and Af = o, METHOD OF LEAST WORK 63 which are the same results as stated before. But if we substitute the values as given in Table I, we find the horizontal pull: H = 124.55 tons > and the moment M = M 3 = - 32.8 inch- tons = 65,600 inch-pounds, M 3 = M 2 = M 4 = M y See Fig. 39. M 3 has the same direction of revolution as indicated in Fig. 37, and M 2 has the opposite direction of M 3 . The moment at the apex is M l = PI - H sin a X L - 32.8, or M l = -f 94.47 inch-tons = 188,940 inch-pounds. Fig. 38. This moment corresponds to a stress of 1020 pounds per square inch in the outer fiber, and is not more than 8.5 per cent of the primary stress. The stress in each compression member is R = ^124. 55* = 176.45 tons. 64 SECONDARY STRESSES IN BRIDGE TRUSSES This stress must pass through the point of inflection, a point where no moment exists. The fact that the horizontal pull is less than 125 tons points to the displacements of the directions of the compressive stresses as indicated in Fig. 38. Passing a section through each end point of the bar AC and one through the point of inflection, and applying the stresses R as exterior forces, we can then replace R by S = R cos co and Q = R sin oj, a moment M^ at point C and a moment M 2 at A. The equilibrium requires the identity in magnitude of the stresses 5, acting along the chord of the deformed bar and the transverse forces Q at right angles to this chord ; further, we must have QL 2 = M l and QL, = M 2 , or QL = M l + M 2 . The location of the point of inflection is found from or LM, T , . 1 == As /! = - 1 , we find the angle w from K M We can also, as in Fig. 38 c, and without disturbing the equilib- rium, add at each of the points A and C two forces, each equal to R in magnitude, but acting in opposite directions. If now one force R is resolved in two components, one parallel to the chord of the deformed bar, and the other at right angles to it, we obtain the same result as before. In order to test the accuracy of our calculation, we apply now Miiller-Breslau's method as a check. As 2M = o with respect to any panelpoint, and on account of symmetry we must have in reference to Fig. 39, M = M 6 and METHOD OF LEAST WORK 65 M 2 = M 3 = M 4 = M 5 , consequently there remain only two unknown moments to be determined. We select M x and M 2 as the two unknowns, and begin with the 250,Tons X I Fig- 39- calculation of the 6 E fold alterations of the angles in the triangle according to equations (i). These values are, 6 Aa = 6 { (- 6 - 7) i + (- 6 + 6) 0} = - 78, 6EA/? = 6 {(-6 + 6)0 + (-6 + 7)i| = -78,- 6 A7 - 6 { (7 + 6) i + (7 + 6) i j = = + 156. The stresses are given in net tons per square inch, Table I, consequently the modulus of elasticity E must be reduced to the net ton and the square inch as the units, and it is assumed to be 14,500. A a, A/9, and A7 are measured on the arc for a radius = i. If, for instance, we suppose that the bars AC and BC are non- elastic, but the bar AB elastic, then, according to Chapter II, 6 {7(1 + i)} 6 X 14500 = 0.000965 inches; 66 SECONDARY STRESSES IN BRIDGE TRUSSES and if measured for a radius equal to the height of the truss, Fig. 40, we have h&y = 282.84 X 0.000965 = 0.273 inches. The value 0.273 inches is the elongation in the bar AB, for the elongation oW Fig. 40. S 2 I 125 X 2 X 282.84 A2/= r=-r- = =0.273 inches. EA 14500 X 17.86 Considering the identity of the moments at the supports and apex, we write: U -* - TI -1 ? 77 -* U i 1/2 L 3 /, " 4 /, ' ' and using the data from Table I, we get U 2 = U s and U 3 = U 4 = 1.885 U 2 . With reference to equations (31) and (32), we have 1.885*7, -^-5-770^-78, U 2 U,+ U 2 - 1.885 U 2 - 78, Solving for U i and C7 2) we find U l = + 47.18, and U 2 = - 16.35; and the bending moments M, = U l -f - = + 94. 36 inch-tons, or 188,720 inch-pounds. JLi M 2 = U 2 = - 32.70 inch-tons, or 65,400 inch-pounds. METHOD OF LEAST WORK 6/ Comparing these results with those previously obtained, we find the greatest difference not more than three tenths of one per cent. The finding of a positive M l means that the bar AC is deformed at the point C as shown, in Fig. 39, but M 2 being found negative, the deformation of the bar at the support is contrary to that shown, and as M 3 = U 3 2 = U 2 = 16.35 X 2= 32.70 inch-tons, the elastic line of the bar AB is also curved contrarily to what is shown in Fig. 39. The deformations are indicated in Fig. 41. Fig. 41. The displacements of the lines of direct stresses are, /i = f: = 5^ =o - 534inches - = 0.185 inches. *' /w = 0.262 inches. I2 4-55 The foregoing investigation shows that the lines of the dis- placed stresses form a diagram which is not identical with the truss diagram, consisting of the center lines of the truss members. Equilibrium exists, but the lines of the stresses do not intersect any more in the panelpoints of the truss. CHAPTER VIII. OTHER CAUSES OF SECONDARY STRESSES THAN RIVETED JOINTS IN MAIN TRUSSES. i. Eccentricities. WHEN the stress in a truss member does not pass through the center of a panelpoint, then its eccentricity not only brings about a change in the secondary stresses, but it itself is a source of a new stress. These stresses have usually different signs, which means that a stress due to an eccentricity, is in part counterbalanced by the rigid connection. In the execution of calculations, proper attention must be paid to the character of the signs of the moments due to an eccentric connection. So, for instance, we may call a moment positive if it turns in the direction of the hand of a clock, and negative for a counter-revolution. If a bar with the stress S has an eccentricity at each end c and c it then M = Sc and M t = Sc ly and the angle of deflection - c on account of the eccentricity is, according to equations (2), C 6 El 6 El The angles between the different bars of a truss are now sub- jected to a change, not only owing to alterations in their lengths, but also on account of eccentricities; and after having taken care of the deflection angles T C with their proper signs in the determi- nation of the EAce, the calculations proceed then as previously explained. After the calculations are finished, the direct effects of the eccentric connections must be added to the secondary stresses. 68 OTHER CAUSES OF SECONDARY STRESSES 69 2. Loads between Panelpoints in the Plane of the Truss. These loads are dead and live load and braking forces. The live load between panelpoints could be avoided by designing a bridge with floorbeams and stringers; and in case a floor is riveted to the posts at any point between top and bottom of the main trusses so that the posts are subjected to bending by the braking of trains, the insertion of extra members will transmit the braking forces to the panelpoints without causing bending in the posts. The calculation of the effects of loads applied between panel- points is about the same as that for eccentricities. If we have to deal with dead load, for instance, we consider each member individually, calculate the stress S M due to its own weight, deter- mine further the deflection angle r at both ends caused by it, and proceed then similarly as shown for eccentricities. Finally, we add the stresses s^ and the secondary stresses. 3. Loads between and at the Panelpoints of a Member supposed to turn freely around a Pin. Under this head comes an eyebar whose secondary stresses are of particular interest, as has been shown by the discussion they caused among engineers. The exact solution of the problem requires the simultaneous consideration of the action of its dead weight, which consists in a deflection, and that of a pull reducing in part this deflection. A separate consideration of dead weight and pull leads to approximate results. We assume an eyebar 70 SECONDARY STRESSES IN BRIDGE TRUSSES under tension, the centers of the two heads in a horizontal line and one half of the bar walled in, which circumstance will not in any way disturb the equilibrium. Further, we call P 2 the pull, P t the reaction from the dead weight of the bar, supposed to be uniformly distributed over its length, and take A as the origin of coordinates, Fig. 42. The problem consists now in the representation of the ordinate y as a function of oc. The relation between these two variables is the equation of the elastic line; and as soon as the latter is found, it is easy to calculate the deflection, the moment, and the stress for any section of the bar. With respect to Fig. 42 the equation of the elastic line is written, ii> dx 2 2 L since the negative ordinates are below the axis of the abscissae. P P Calling the constants ^- = n and ~ = q, El ILL (Py ( x 2 > , we have ^ = n j - * + - j + qy. To facilitate the investigation, we write / instead of -f- and doc d?y instead of -r^ 2 , and our given equation becomes now nx 2 y" = qy - nx + Differentiating twice, we obtain --+" By integrating this last equation twice, we obtain /' as a func- tion of x\ and if this function of x is substituted in the given OTHER CAUSES OF SECONDARY STRESSES 71 equation, we find then at once y as a function of x, which is the object sought. Multiplying the last equation with dy", we have dy" X 9r - d -f X d\ *f I = gyW + 5 rf/ . dx 2 doc I dx ) L ' / ) 2 Since Xd is the differential of dx ' , dx ( dx ) 2 therefore - is the integral of -f- X d 2 dx and we get by integration, !!'-* /W + **fdf + A- & + I* y" + A, or ^y j = = 0#. Integrating again, we obtain i ( n A and 5 are arbitrary constants; and as these can be changed, we write the last equation, -- " + A B or _ + y/V0 + \/ ^}' //2 + y" + ~B~ ?2 SECONDARY STRESSES IN BRIDGE TRUSSES is the base of the system of natural logarithms. From this equation we obtain the value of y" as a function of x, and by sub- nx 2 stitution of this value in the given equation y" = qy nx + - - , 2 L we get after some simple transformations, A _,y- n { nx nx* ^ 2 qVq I 2 L 2 q 2 V / q XB 2 q\/q X B ) L(f q 2 Lq The next step consists in the determination of the constants. Putting 2 L 2 q 2 Vq XB 2 qVq XB 2 qVq C and D being two ne\v constants, \ve have _ 2 Lq 2 q 2 Lq and by differentiation, dx q Lq dy From the conditions that y = o for x = o, and ~ = o for x = L, the two constants C and D are found. They are C_ __^___ t ^ */ ~ r+i~\y and if C and D in the equation for y are replaced by these values, then OTHER CAUSES OF SECONDARY STRESSES 73 EXAMPLE. An eyebar 15 inches wide by 2 inches thick and 55 feet long is subjected to a pull P 2 = 600,000 pounds, or 20,000 pounds per square inch. Required the maximum bending stress. Let the modulus of elasticity be 29,000,000 pounds per square inch, P l = 2800 pounds, / =562.5, L = 330 inches, The maximum deflection D is equal to D =- < - + - q ( 2 Lq - i j > = o. 488 inches. The maximum bending moment is equal to M = - 2800 X 330 + 2800 X 165 + 0.488 X 600,000, or M 169,200 inch-pounds, consequently the maximum bending stress equals - - = 2256 pounds per square inch. This bending stress amounts to about 11.3 per cent of the direct stress, and is compression in the top fiber and tension in the bottom fiber. 4. Changes in Temperature. Any rise or fall in temperature affects the lengths of truss mem- bers. These alterations in the lengths of bars produce deforma- tions of a truss, which may or may not be connected with stresses. A statically determinate truss whose movable end is free from any frictional resistances and whose members can turn freely around pins, is not subjected to any temperature stresses, not even when its members are unequally affected by temperature. But the case is different with statically indeterminate trusses, no matter whether the indeterminateness refers to the outer or 74 SECONDARY STRESSES IN BRIDGE TRUSSES inner forces. The safety of the structure requires an examination of its temperature stresses, and, if necessary, an inquiry into the secondary stresses arising from them. Statically determinate trusses, which are riveted, can also be affected by temperature stresses. So, for instance, can a top chord have a higher tempera- ture than a bottom chord, if the latter is protected from the rays of the sun by a floor. In this case the difference in temperature produces an upward deflection of the truss, and consequently deformations and stresses in the bars. The calculation of temperature stresses is as follows: If c is the coefficient for expansion or contraction due to i change in temperature, and / the total change of temperature in degrees, then Sl >i EA ^ C > or A This value of s t is now used, for instance, -in equations (i), - in order to calculate the angle alterations, whereupon the suc- ceeding operations are executed as formerly explained. 5. Misfits. It is very essential that the lengths of members for a riveted truss are exact; if they are not, they are a source of secondary stresses. We speak here in particular of statically indeterminate trusses where a small shop mistake may lead to a considerable change in the stresses. In order to determine the effects of these misfits on the stresses, we assume that these misfits are produced by stresses s m , and if we call A/ the amount a truss member is either too long or too short, we can write or s m = X E. OTHER CAUSES OF SECONDARY STRESSES 75 This unit stress s m is now used in the same way as the unit stress s t due to a change in temperature. 6. Brackets on Posts. Eccentric loads on posts caused by brackets give rise to very lengthy calculations, and as these loads affect the entire cross- frame of a bridge we cannot consider them here. It is best to avoid such brackets wherever possible. 7. Unsymmetrical Connections. Good practice does not allow unsymmetrical connections in the design of main trusses; and at such places where they are tolerated, the secondary stresses caused by them are of minor importance. 8. Curved Members. The secondary stresses due to curved members can be computed from the suggestions given for eccentric connections. Bridge trusses with curved members do exist, but, in the opinion of the writer, they are utterly out of their proper place. The defense of such members, even from an aesthetic point of view, is weak. 9. Pin Joints. We believe it to be a safe statement that many an engineer lays too great a stress on the value of a pin joint as a means to reduce secondary stresses. Of course no pin joint is perfect, and in some cases the frictional resistance may be so small that we do not need to pay any attention to it. On the other hand, if a pin is designed with no consideration whatever for a reduction of secondary stresses, there is at least a chance that it will be in- effective, so that a riveted joint could just as well have been built. If a bar does turn around a pin, it is certain that the stress in the bar will be displaced out of its former axial position, Fig. 43. In this case the displacement r is such that the moment Sr over- 76 SECONDARY STRESSES IN BRIDGE TRUSSES comes the frictional moment F X R, F being the frictional resist- ance and R the radius of the pin. The frictional resistance is found by resolving the stress S at its point of application on the pin periphery into two components, the line of one component coinciding with the tangent on the pin periphery, and the other normal to this tangent. The frictional Fig- 43- resistance is now equal to the tangential stress 5 X sin and also equal to the normal pressure S X cos < multiplied by the coeffi- cient of friction. The angle $ included between the line of the stress S and the normal is the angle of friction. We have then, FxR = SXsin(t>xR = SXr, and r = R X sin . . This shows the displacement r is independent of the stress S. If this displacement is smaller than r, then no turning of the bar around the pin is possible ; and if it equals r, a turning takes place. Let us now go back to the triangular riveted truss, calculated in the preceding chapter, in order to find for a given coefficient of friction the greatest diameter of the pin at the apex, which must not be exceeded if a turning of the bars around the pin is to be realized. As the displacement of the stress in this case was OTHER CAUSES OF SECONDARY STRESSES 77 found = 0.534 inches, and assuming 0.2 as the value of the coefficient of friction, we have 0.534 = 0.2 R, or R = 2.67 inches, which means that if we design this pin with a greater diameter than 2 X 2.67 = 5.34 inches, it would be ineffective, and with a greater frictional coefficient the diameter of the pin should of course be still smaller. From the above considerations it follows that if we wish to reduce the secondary stresses to a minimum in hinged members, it is very essential to keep the size of the pins down as much as the strength and safety of a structure will permit. The writer is not aware that experiments have been made with a view to determine the frictional coefficient for cases that are here under consideration, and therefore he is not in a position to fur- nish any reliable data. Some specifications require that the diameter of a pin shall not be less than three quarters of the width of any eyebar which they connect. Calling S the direct stress of an eyebar, w and / its width and thickness, 0.2 the frictional coefficient, then we have for the displacement of the stress, r = 0.375 w X 0.2 = 0.075 w - $ The direct unit stress equals , and the bending stress is equal wt to the moment divided by the section modulus, or equal to S X 0.075 X w X 6 = S_ inH wt The sum of the direct and bending stress equals X } i + 0.45}'. which means that the secondary stress amounts to 45 per cent of the primary stress. In regard to these stresses it should be noted that probably the vibrations due to a passing train cause the eyebars to adjust them- selves to their original positions; they probably turn around their pins, whereby the angle of friction is momentarily decreased from what it would be for static loads. 78 SECONDARY STRESSES IN BRIDGE TRUSSES 10. Friction at Supports. The frictional resistances at the supports of trusses are depend- ent on the coefficient of friction, the vertical loads, and the length of span. These resistances would -be very great indeed for long trusses, if sliding friction were allowed; but as for such trusses only rolling friction is allowed, the resistances are very con- siderably reduced, and need not to be considered. But it is, of course, important that the roller ends be kept in proper working order. ii. Cross-Frames. The analytical discussion of cross-frames, if a complete solution of the problem is attempted, leads to very extensive investigations and is outside of our province. Nevertheless, we will give a few remarks. A thorough examination of cross-frames for either a through or deck railway bridge consisting of verticals, floorbeams, and cross- constructions of any description, would consider the bending effects on the frame of the following forces: dead load of floor- beams, dead and live loads transferred from the stringers to the floorbeam, centrifugal force, impact, wind pressure against the train, wind pressure against the structure concentrated at top and bottom and uniformly distributed against the posts, unequal deflection of the main trusses, and unequal change of temperature for different parts of the frame. The influence of these forces is felt in the entire cross-frame, causing also bending and twisting in the members of the main trusses. The secondary stresses in cross-frames are next in importance to those in the main trusses ; and concerning cross-frames with no diagonals, it may be said that they are predisposed to higher stresses. For the purpose of finding out to what extent the secondary stresses are affected in the verticals by dead and live load, and exclusively by changing the dimensions of the verticals, the writer OTHER CAUSES OF SECONDARY STRESSES 79 examined a riveted cross-frame of a two-track railway bridge, consisting of a 6-foot-deep floorbeam, a lattice strut, and two sus- penders, each of the latter composed of 4-8-inch bulb angles with a total area of 22.5 square inches. Deep floorbeams with large moments of inertia tend toward a reduction of secondary stresses. The trusses were 30 feet from center to center, the panel length 27 feet, and the live load carried by each of the suspenders 187,000 pounds. The lengths of the verticals and their depths parallel to the web of the floorbeam were the only dimensions changed; and if we express the secondary stresses in percentages of the unit stress due to dead + live load, the results are as follows: Per cent. Verticals 36 feet long and 14^ inches deep I 3-S Verticals 36 feet long and 29 inches deep 2 3-3 Verticals 18 feet long and 14^ inches deep 25.2 Verticals 1 8 feet long and 29 inches deep 40 .9 A cross-frame as described 36 feet deep may belong to a through Warren truss bridge, the verticals being suspenders; and a cross- frame 1 8 feet deep may belong to a Baltimore deck bridge with the top chord projecting above the floor, the verticals being short posts. For the analytical treatment it does not make any differ- ence whether the floorbeam is at the bottom or top. In another example, the writer selected a cross-frame of a four- track railway through bridge, with two main trusses, symmetri- cally loaded, gave purposely the post the unusual transverse depth of 52 inches, designed it under three prominent specifications, and found in each case that the secondary stress amounted to closely 55 per cent of the primary stress. But by reducing the depth parallel to the floorbeam of this comparatively short post to 18 inches with ample provision against buckling, the secondary stress was reduced to 22.3 per cent. These high stresses in the verticals are verified by Winkler, Querkonstruktionen, pp. 179-182; by Jebens, fiie Spannungen in 8o SECONDARY STRESSES IN BRIDGE TRUSSES den V erticalstandern der eisernen Br'ucken, Zeitschrijt des Vereins deutscher Ingenieure, 1880, p. 127; and by that standard work of engineering science, Handbuch der Ingenieurwissenschaften, vol. II. From the above we can draw the conclusion that a small ratio between the length and the transverse depth of a vertical should be avoided, and that the transverse depth should be restricted to a proper limit, otherwise we run the risk of exceeding by far the unit 'stresses as prescribed by our specifications. It is of interest to note that investigations of riveted cross-frames disclose the assumption of the fixity of floorbeams at the posts as quite erroneous; the floorbeams cannot even be approximately so considered. 12. Yielding of Foundations and Settlements of Masonry. The influence on the stresses of a truss caused by yielding of the foundations or settlements of masonry could be calculated if these displacements were known. As a matter of fact, they can only be estimated or judged from uncertain or incomplete evidence. In regard to these displacements we must distinguish between elastic and non-elastic deformations. Elastic deformations vanish as soon as the load is removed, while the non-elastic are perma- nent. A structure should not be built if the computations prove that it is very sensitive to assumed displacements of its supports, pro- vided we cannot give it supports which are almost as good as fixed. It is also essential in such a case that the supports are placed with the utmost care in those positions as assumed in the computations. Of course not every truss is affected by a displacement of its supports. So, for instance, a truss, resting on two supports and with one movable end, a cantilever truss or a three-hinged arch are free from this influence. On the other hand, a continuous truss is very susceptible to the influence of a yielding of the founda- tions or a settlement of the piers (particularly so if very massive OTHER CAUSES OF SECONDARY STRESSES 8l and deep); also the one-hinged, the two-hinged, and the hingeless arch and others are affected by these causes. Continuous truss over three supports. We will suppose that the center pier of a continuous truss over three supports yields verti- cally to the amount of D inches, Fig. 44. Such a displacement reduces the center reaction an amount X, which, when found, enables us to compute the stresses in the truss members, and con- sequently also the secondary stresses. In order to find X, we place a load equal to unity at the center of the truss, the latter assumed to be resting on its end supports only, and determine the deflection D l at the center due to this load = i. If we now apply a load X at the center of the truss, then the deflection will be XD^ but as this deflection must be equal to the supposed deflection D, we have XD l =D,orX == - The examination of a continuous plate girder is quite similar to that of a truss. Two-hinged arch. Of all forms of arches, the two-hinged arch has probably found the widest application, and from this reason we will take it as an example. This kind of arch is statically indeterminate with one unknown quantity, which is the hori- zontal thrust, provided the truss has no redundant members. The principle of the derivative of work furnishes us with convenient means to determine the thrust. For this purpose we first remove the statical indeterminateness by giving the truss one movable end, the two supports being supposed in this case in 82 SECONDARY STRESSES IN BRIDGE TRUSSES a horizontal line. The stress S in any of the truss members, due to vertical loading, can now be found by statics. There- upon we apply a force = i at the movable end, acting toward the fixed hinge, and determine the stress v either analytically or graphically in every member of the truss. If now the real hori- zontal thrust = H, and if 5 denotes the stress in any of the bars of the statically indeterminate truss, we have 5 = 5 + vH. The stress S is a function of the exterior forces, and is independ- ent as well as the stress v of the thrust H. The work of deforma- tion W is expressed by ^-y^l M -* aE A' where I = length of any bar, A = sectional area of any bar, E = modulus of elasticity. The principle of the derivative of work states that, if we express the work of deformation of the bars as a function of the exterior forces, then the displacement of the point of application of a force (in our case H) equals the partial derivative of the work of defor- mation with respect to that force. Differentiating the work of deformation with respect to H } we get ITT ~ ' ' r> A ^ ITT ~~~ ^*<*J L being the length of the span from center to center of end pins, and AL the supposed horizontal outward yielding of the masonry supports under the action of the vertical loading. In case temperature stresses are to be considered, w r e call the SI SI alteration in the length of a bar: + etl instead of -=rr; being EA k,A the coefficient for extension or contraction due to a change in temperature of i degree and for a unit of length, and / the total change in temperature. OTHER CAUSES OF SECONDARY STRESSES 83 But 6" = S + vH> and - = ^, consequently we obtain by, substitution, or TT The derivation of this formula for H assumes that no initial stresses exist, that is to say, with the removal of the outer loading all stresses must vanish. The three different causes which influence the thrust of the arch may also be considered separately. The thrust caused by the sole action of the vertical load is and by a change in temperature, X v v 2 l [t being positive for an increase] and by a change in the length from center to center of end hinges, being positive for an increase]. The expressions ^TrV X v and ^TTT~ designate horizontal * zZ/^T. jC/.ri. displacements of the hinges, which are supposed to move freely; the former is the displacement caused by the vertical load, and the latter is that due to a horizontal thrust equal to unity and applied at the hinge. 84 SECONDARY STRESSES IN BRIDGE TRUSSES The length of the distance between the hinges can be increased by the thrust of the arch in pushing the supports bodily outward or by crushing or compressing the masonry. Such an action naturally decreases the horizontal thrust and consequently exer- cises an influence on the stresses in every member of the arch truss with corresponding changes in the secondary stresses. As an example of a thorough examination of the effects of the yielding of masonry supports on the stresses in a truss, can be mentioned the bridge across the Emperor William canal at Gruen- enthal, described by Fiilscher in " Zeitschrift fiir Bauwesen," 1898. This bridge is built for a single-track railway and highway traffic, has two crescent-shaped arch ribs, projecting above the floor system, and measures 513.3 feet from center to center of end pins. The wind bracing in the plane of the floor consists of a wind chord and diagonals, and the place of the struts is taken by the floorbeams. It is divided into three sections: a central section, lying between the intersection points of the arch ribs and the floor; and two end sections, each extending from these intersection points to the abutments. The central section of the bracing transfers the wind pressure to the arch, and the end sections partly to the arch ribs and partly to the abutments. In the central section the floor is suspended, and in the end sections it is supported by posts resting on the arch ribs. Apart from the effects of the dead load, live load, wind pressure, and temperature changes, the stresses in each truss member have been calculated under the supposition that, before riveting up the wind chords, but after the arch carried its own weight, a hori- zontal yielding of each of the masonry supports of it of an inch would take place, and a further yielding of the supports of iV of an inch under the influence of the live load. While the two-hinged arch is affected only by horizontal dis- placements of the masonry, the one-hinged and hingeless arch are susceptible to horizontal and vertical displacements, and, moreover, to a possible turning of the masonry in the plane of the truss. CHAPTER IX. IMPACT. IT is outside the scope of this book to take the reader over the field of mathematical investigations of dynamical effects on bridges, or to discuss the many suggestions that have been made, how impact and vibrations could be covered in our specifications. If it had been the intention, it would have been proper to extend the discussion of secondary stresses under static loads to the cross- sections of bridges, floors, and wind bracings first, before giving some notes on the secondary stresses under moving loads. But a few words on this subject are not out of place. While the progress in the theory of bridges has been gigantic, the same cannot be said of the theory of dynamical effects on bridges, and the reason for this is not far to seek. If every element had to be considered which has some connection with the effect produced on a bridge by a fast-moving train, then the problem of impact would, of course, be insoluble, and even in the simplest case. But barring such elements, as, for instance, a defective track, or inequalities of the rail ends at rail splices, etc., whose influence is naturally outside the province of a calculation, the mathematical difficulties presented by the problem are almost unsurmountable ; and, in fact, they have been only overcome for the case of a single load moving over a beam. A train in passing over a bridge causes the latter to deflect, whereby the pressure or centrifugal force exerted by the train against the bridge is influenced by the deflection and the velocity of the moving masses, and this pressure in turn exercises an influ- ence on the form of the deflection curve. The mutual relations between the quantities which enter into consideration are compli- 85 86 SECONDARY STRESSES IN BRIDGE TRUSSES cated, and made more so by the counterweights of the locomotive drivers, which affect the values of the pressures. In consequence of the great velocity with which a train enters a bridge, of the variable loads produced by the counterweights of the locomotive, of defective rail splices and unround wheels, the bridge is subjected to vibrations. Not only does the bridge as a whole vibrate ver- tically and horizontally, but also the different bars perform rapid oscillations longitudinally and transversely. The view held by some engineers, that a fast-moving train does not give a truss the necessary time to offer its full resisting power, does not harmonize with the fact that the propagation of stresses in elastic bodies follows the laws governing the velocity of sound. The velocities of sound vary greatly in different mediums; in liquids the velocity is greater than in air, and in solids the range is rather wide. In caoutchouc the velocity is from 100 feet to 200 feet per second, while in steel wire, wrought iron, and steel it amounts in round figures to 16,000 feet, or about 3 miles per second. A telegraph wire furnishes a good illustration of the propagation of sound in solids. Filing at one end of the wire can be heard at a distance of several miles by placing the other end in the ear. The subject in question has been treated from various points of view. In the year 1890, Professor Ritter, of Aachen, published calculations as a result of theoretical considerations 'which place the velocity of the propagation of impulses in wrought iron as high as three miles per second, and Professor Mach showed optically the propagation of longitudinal vibrations. Professor Radinger treats the subject in his book on steam engines with high piston velocities, published in Vienna, 1892. In the same year appeared in " Zeitschrift des osterreichischen Ingenieur- und Architekten-Vereins," a paper of great interest to engineers on " Metal Constructions of the Future" by the late Professor Steiner. In this paper Steiner shows how impulses can be made visible to the eye by the construction of a model of a bridge truss, each bar of the truss to be provided with a groove. If such a model is placed in a horizontal position and the grooves filled with quick- IMPACT 87 silver, it is then possible to follow the propagation of an impulse, which has been made by the finger at any point of the truss. However, it should be remarked that the propagation of stresses from section to section in the members of a bridge truss experiences a delay, firstly, because the line of progress must be changed, and secondly, on account of the imperfections of the joints. A pin- connected truss appears to be at a disadvantage compared with a riveted steel truss, as the latter resembles more closely a con- tinuous mass. But, whatever may be left of the velocity of propa- gation of stresses, it appears to be of sufficient magnitude to be looked upon as instantaneous. Under the assumption that the stresses, in traversing a truss, encounter so many difficulties which reduce their velocity, Radinger arrives at the conclusion that a truss may be subjected at the ends to particular high stresses by fast-running trains. He arrives at this conclusion in this way: Let us suppose for an illustration that the velocity of stresses is reduced from 16,000 feet per second to 4000 feet per second, then the time required by a span of 400 feet in length to act as a structure on two supports would be 2 X 400 i ,. - = or a second. A train running over the bridge with a 4000 5 speed of 90 feet per second would cover a length of 18 feet in ^ of a second, which means that for this time the bridge would have only one support for the live load. The solution of the problem of impact, but only in the simplest case, namely for a single constant load, moving over a weightless beam of uniform sectional area and resting on two rigid supports, has been rigidly effected by Dr. H. Zimmermann, whose brilliant researches are contained in his paper, "Die Schwingungen Eines Tragers Mit Bewegter Last," Berlin, 1896. Many investigators have attacked the problem without success on account of the mathematical difficulties. But in this respect it should be remarked that the general integral of the differential equation of the curve described by the moving mass is of a kind that was unknown up to the publication of Zimmermann's writing, although he knew it 88 SECONDARY STRESSES IN BRIDGE TRUSSES as early as 1892. Zimmermann tells us that he hesitated to pub- lish the purely mathematical solution of the problem, as he had wished to work out a practical method of calculation, but the great amount of time and labor spent on this undertaking caused a considerable delay of his publication. Zimmermann's extensive paper is naturally of a highly mathe- matical character, and his penetration into the subject is deep. Therefore, we will give only the results of his investigations, which may be applied in practice. As has been said before, a weightless beam of uniform sectional Fig. 58. area and resting on two rigid supports is assumed, over which moves a mass with any constant velocity. If the mass moves over the beam so slowly that the pressure against the beam is invariable, we have the familiar equation, which expresses the form of the deflection curve, Fig. 58. The object is now to find for greater velocities the path described by the moving mass, in which case the pressure against the beam is variable. In other words, if the curve shall be found, it is necessary to consider the effect of the centrifugal force of the mov- ing mass. The differential equation expressing the desired curve is IMPACT ,89 and the meaning of the symbols is as follows : The time / required by the mass to cover one half of the span length / with the velocity c is t = - ; and if we suppose the mass were to descend from the G height h inside the same time /, then 2 h = gt 2 = g , g being the acceleration of gravity. The ratio 2h C* P -=W_ <=,and- 6 El E and / are modulus of elasticity and moment of inertia of the beam, and P equals the weight W of the mass m + the load W l transferred to the wheel by the spring. x Zimmermann puts further = c, and takes the fall 2 h as a unit / of measure for y by writing rj = -2 . 2 h a and /? are constants, if P and c are assumed to be invariable, but Zimmermann's method of integration can also be employed in case ft is variable, representing an arbitrary function of . The form of the path described by the moving mass is dependent on a in a high degree. This curve is known for static loads where a = oo for c = o. The total pressure P does not influence a. If P changes, then the ordinates of deflection for both the moving mass and the static load change also, but in the same proportion. The stresses of the beam are naturally greatest for the maximal values of W and P. In order to find a value for a, which can be used in practice, we determine first two limiting values. If there are no springs assumed, we put W = P, and W' 6 El 90 SECONDARY STRESSES IN BRIDGE TRUSSES PI The moment M c at the center of the beam is M c = ; and if 2 s is the stress at the extreme fiber, and D the depth of the beam, we have D and 2C 2 S If we put W = j P for a perfect spring, then 6gDE c~s As the wheel load is neither rigidly supported nor a spring per- fect, we will assume a -- -^ . C 2 S Zimmermann has conclusively shown that for velocities up to 62 miles per hour, or 91 feet per second, the path of the moving mass can be considered as symmetrical about the center line of the beam, so that the stress of the beam can be determined from the deflection at the center by the sufficiently accurate equation a 12 Consequently the greatest proportionate increase of the deflection, or of the bending moment, or of the stress, is expressed by T Increase = i ; - 3 In the equation for , the quantities g, D, and c are expressed in feet, and E and s in tons per square inch, a is independent of the length of the span, nevertheless the equations can only be used for IMPACT 91 very short spans on account of the assumption that only one load moves over the beam. Assuming, for example, D = i foot, c = 91 feet per second, s =8 tons per square inch, E = 14,500 tons per square inch, we find _ 3,X 3 2.i7X i X 14590 _ 9 i 2 X8 which gives about 44 per cent impact. For D = 1.5 foot, c =91 feet per second, 5 =4 tons per square inch, E == 14,500 tons per square inch, 3 X 32.17 X 1.5 X 14500 _ 6 VL U < . 9i 2 X 4 which reduces the impact to about 8 per cent. A further result of Zimmermann's investigation can be summed up in the statement that the end of the girder and its support in the direction of the motion of the load is subjected to a particular impact, which increases in intensity with the rigidity of the support, the rails, and the tires. CHAPTER X. EXAMPLES AND CONCLUDING REMARKS. A FEW years ago Professor E. Patton in Kiew, Russia, published a book on secondary stresses in the Russian language, whose title, translated into English is, "Calculation of Trusses with Stiff Joints," Moscow, 1901. This book contains a number of examples of bridge trusses, calculated by German authors, which Professor Patton collected from various books and periodicals, besides enriching this collection by a number of examples of his own. An abstract of this book is published in "Zeitschrift fur Architektur und Ingenieurwesen," Hannover, 1902. Through the courtesy of Professor Patton the writer is enabled to republish some examples of the above collection, adding three trusses of American design. It is hardly necessary to remark that although a truss is of foreign origin it cannot fail to be serviceable in the study of secondary stresses. The calculation of secondary stresses is naturally always pre- ceded by that of primary stresses, which is executed under the assumption that the bars can turn around frictionless pins and in this case only do the lines of stresses coincide with the axes of the bars. But this assumption is never fulfilled in our trusses. Either we have a riveted joint or a pin joint with frictional resist- ances of more or less severity. As we have seen in previous discussions it is owing to the nature of joints that the bars become deformed under the action of loads upon the truss and the lines of stresses displaced, and moreover, the primary stresses in a riveted truss for a given load are not identical with those found under the assumption of frictionless pins. But the differences in the primary stresses between a 92 EXAMPLES AND CONCLUDING REMARKS 93 riveted and a pin-connected truss, all other conditions being equal, is inconsiderable and consequently can be neglected for all practical purposes. We have also explained that the calculation of secondary stresses does riot need to take into consideration the deformations of the bars, provided sufficient provision against buckling has been made. In this case the stresses in any given section of a bar are dependent only on the bending moment with respect to that section, and these bending moments generally reach their greatest values at the ends of the bars. After the bending moments have been found we can then calculate two different stresses in the extreme fibers at each end of each bar according to known formulas. Intersection points of diagonals, which are riveted at these points, are treated in exactly the same manner as any other panel points. The designation of the letters in the tables is as follows: / is the moment of inertia of a bar, / is the length of a bar, b is the width of a bar, e is the distance from the neutral axis of a bar to the extreme fiber. The primary stresses are given per unit of area, and the secondary stresses, which are tension on one side of the bars and compression on the other side, are expressed in percentages of the primary stresses. The maximum total stresses are of course obtained by adding primary and secondary stresses of the same signs. A study of secondary stresses proves to be instructive, for it discloses the weaknesses in our designs, but at the same time we are also taught how to minimize the defects resulting in stronger and therefore safer bridges. Besides the character of the truss, which plays an important role in regard to the magnitude of secondary stresses, we must also point out the marked influence on these stresses exercised 94 SECONDARY STRESSES IN BRIDGE TRUSSES TABLE 2 AND FIG. 45. Span 15 m. Wrought Iron. Gross I Most Stresses. 1 Member. area, sq. gross 1 cm. b=2C cm. danger- ous cm. Kg. Sections in mm. cm- cm. 4 fiber. sq. cm % e 1 2-4 60 1000 500 19 Top 9-5 -567 14 53 o f U 0, 4-6 60 IOOO 500 19 Top 9-5 -467 18 53 i nn H 90 z 90 z 8 U 1-3 5-7 5 5 75 75 500 500 17 17 Bottom Bottom 8-5 8-5 + 360 + 300 28 21 59 59 LjH 80 z 80 z 9 PQ 3-5 70 1125 500 17 Bottom 8-5 -457 18 59 1 =]M ! i ji 80 x 80 z 12 l b JL 1-2 5 79 350 17 Top 8-5 -520 6 21 nn 6-7 5 790 35 17 Top 8-5 420 4 21 | HMO | H- 170- -w 80 z 80 x 9 1 3-4 30 440 35 17 8-5 + 100 65 21 Mb PS 5 4-5 3 440 35 17 8-5 200 34 21 r--170 } 80 i 40 x.? 2-3 5-6 40 40 610 610 350 35 16 16 ... 8 8 + 575 + 45 12 19 22 22 JL HQ I -] [*10 i Tfi i 75 z 8 Secondary stresses are calculated for gross moments of inertia. SECONDARY STRESSES IN BRIDGE TRUSSES 95 a O f '<5 ^ VI P fr ^ . 3 S J3 t/i li (/i 3 PL' C. ' P g g- O C3 Q ^ f H -2.6m 1 9 6 SECONDARY STRESSES IN BRIDGE TRUSSES TABLE 3 AND FIG. 46. Span 20 m. Wrought Iron. , Gross I Most Stresses. Member. area, sq. gross 4 cm. b= 26 cm. danger- ous cm. Sections. Kg. cm. fiber. sq. cm. % 1 65 4000 400 24 Top 12 -49 22 33 9 ' 911 o 5'-6' 138 35 42 Bottom 7-6 + 35 II 46 > PQ 6-8 178 350 43 Top 35-8 + 450 III 9.8 }~$ 1 Web 40 x 1 6'-8' 178 35 43 Bottom 6.8 + 400 10 3 2 C. PI. 40 x 0.8 f * 1C. PI. 40x1 S-v 178 35 43 Top 35-8 + 45 45 9.8 -ib * 2^-9x9*1 8'-9' 178 35 43 - Top 35-8 + 400 13 9.8 2-4 106 14600 700 4i Bottom 3i-8 -480 21 22 ) ~ K~ I Web 40 x 1 \ ^ 1 C. PI. 40 x 0.8 2>- 4 ' 106 14600 700 4i Top 9.0 -300 13 78 ) *? -i l_f 9x9x1 1 4-7 162 18100 700 42.2 Top 6-9 -480 8 101 ) - n__-_ 1 Web 40 x 1 ( 1 C. PI.- 40x1.4 U a, 4'-7' 162 18100 700 42.2 Top 6-9 -35 9 101 ( g 1 C. PI. 40xO. ^ 2 Ij- 9 x 9 x 1 H 7-7' 182 18450 700 43-2 Top 7- 1 -400 10 99 *^^^* 1 Web 40 x 1 r ic. PI. 40 x 1.4 -T 1 C. PI. -40x0.8 b == BI = 1 g 1C. Pl.-lOxl 2 e e 2 b Jf 2(_!-9x9xl 2 -3 2'-3' 93-6 93-6 712 712 35 35 17-5 + 460 + 320 ii 17 20 20 |_ 1 Web 35 x 1 J^ e* 2 C. PI 17. 5 x 1 \ \_ 2 [f- 7 x 7 xO.9 4-6 4 '-6' 58.8 58.8 712 712 26 26 13.0 13.0 + 340 + 400 18 15 2 7 27 ni Web - 2G x 1 2^-8x8x1.1 M 7-9 38.8 712 J 7 8-5 260 27 42 fT_ ,1 2^-8x8*1.1 1 7'^ 38.8 712 i7 8-5 + 53 ii 42 "3 1 8P Q -2 114.8 712 40 20.0 -520 44 18 ) ' 1 Web 40 x 1 L. J, 2C. PI 20-xl 0'~2' 114.8 712 40 2O. O -300 30 18 ) ' 2|J-x8xl.l 3-4 87.6 712 3 2 16.0 -380 26 22 Jl Web 32 x 1 3 '-4' 87.6 712 3 2 16.0 -310 39 22 2 (J 7 x 7 x 0.9 6-7 52-1 712 21 10.5 -150 47 34 T Si 21" 10x10x1.4 6'- 7 ' 5 2 - I 712 21 10.5 420 T 4 34 T. A L I -2 21 . I 600 15 8.0 + 590 12 40 JM 4-5 21 . I 600 J 5 8.0 + 420 17 40 ' 7-8 7 '-8' 21 . I 21 . I 600 600 15 12 8.0 8.0 + 575 + 100 5 90 40 40 I-"~| 2 Ij- 7 x 7 x 0.3 4'-5' 21 . I 600 IS 8.0 + 100 60 40 I '-2' 21 . I 600 15 8.0 + 100 65 40 Secondary stresses are calculated for gross moments of inertia. SECONDARY STRESSES IN BRIDGE TRUSSES IOI IO2 SECONDARY STRESSES IN BRIDGE TRUSSES Verticals Diagonals Bottom Chord On Oo M ^ I 4x M 4^ K) O ON 4^- to to to to to to to 00 00 00 Oo 4*. -J to Oo 4* ON 0? ^J ^ Oo Oo Oo ON ON ON C_/l Ol Cy\ 000 M Oo to Oo NO 00 ON 00 On Oo Oo On 00 CO O 00 00 00 00 00 Oo Oo Oo to to to On On 4- O Oo Oo Oo to to to 4- 4-4-. p ^ Cfl M M M M tO 00 4^ K) to to 4- 4- 4* On Cn en O O O p p n* 00 Co 00 Oo Oo Oo 00 00 00 M to to to t-n. O\ to to to to to to Ol C-f\ O*l ulated for : : : gross moment + 1 + M tO M tO O N> O O Ol M O O O On On > 5' to O M M NO to M to 4- to Oo Oo O to p' to to to ON ON ON to to to to Oo 00 OJ ^'"^ L J L. _l l_ J L J L J 11 N r n v-i ec 4 * r -i r- -| r n !i It n l'L "M, N '] a SECONDARY STRESSES IN BRIDGE TRUSSES 103 L Top Chord L U?5 IM I I End Post *' M M i *.r B .s M. W g A 1 8" SB * o> e > l *H ^rl S M co J i IO4 SECONDARY STRESSES IN BRIDGE TRUSSES - -36.0- SECONDARY STRESSES IN BRIDGE TRUSSES 105 by the dimensions of each bar and the distribution of its material. It is for this reason that each table contains a column, which gives the ratio between length and width of each bar having a symmetrical section, and for bars with unsymmetrical sections the ratio between length and distance of outer fiber from the neu- tral axis is given, and with a few exceptions the tables show also the distribution of material in the sketches of the sectional areas. We will now point out some facts which indicate how different trusses are affected by secondary stresses. An inspection of the Tables II, III and IV, with Figs. 45, 46 and 47, shows that the secondary stresses of the chords in single Warren trusses without verticals increase from the center of the span toward the end, while for the web members these stresses increase in the opposite direc- tion. This circumstance is in so far fortunate as the web mem- bers in the neighborhood of the center of the span have often a surplus in section and the end members of both the top and bottom chords show this surplus still oftener. The secondary stresses in Warren trusses with verticals, Fig. 48, Table V, and Fig. 49, Table VI, do not entirely bear out the statements made in reference to the Warren trusses without verticals. It is the bottom chord which shows in both cases irregularities not observed in Warren trusses without verticals. The run of the secondary stresses in Fig. 48 and Table V, can easily be accounted for by the fact that the live load covers only one-half of the span. If verticals carry a heavy concentrated load, then they elongate a good deal. These elongations are the cause of considerable deformations in the bottom chord members accompanied by severe secondary stresses. The bottom chord member 4-6 in Fig. 49, shows the greatest secondary stress to be 43 per cent of the primary stress, while in the members 1-3 of Fig. 48, it is as high as 143 per cent. In com- paring these secondary stresses with respect to the unit primary and total stresses, we find that the secondary stress in the member 1-3 of Fig. 48 amounts to 5690 pounds per square inch, which 106 EXAMPLES AND CONCLUDING REMARKS with the primary stress of 3980 pounds per square inch, results in 9670 pounds per square inch, while in the member 4-6 of Fig. 49, the secondary stress is 5160 pounds per square inch, making a total of 12,000 + 5160 = 17,160 pounds per square inch. But the bottom chord section 6-8 of Fig. 48, with an increase of stress of in per cent or 7100 pounds per square inch shows a primary stress of 6400 pounds per square inch, which gives a total of 13,500 pounds per square inch or a stress considerably higher than in the member 1-3 of the same truss. Of course the idea presents itself as natural that the reduction of these high secondary stresses is easily affected by providing the verticals with ample sectional areas, because in so doing we reduce their elongations and consequently the deformations of the bottom chord. Excluding the truss Fig. 51 and Table VIII from a comparison on account of the incompleteness of the data, we find the run of the secondary stresses in the two Pratt trusses Fig. 50, Table VII, and Fig. 52, Table IX, in a rather close agreement, although these stresses have been determined under varying conditions and such differences as do exist can be explained. The secondary stresses for the truss Fig. 50 and Table VII, have been calculated for one single position of the live load, while those for the truss Fig. 52 and Table IX, have been found by the method of influence lines. Sometimes the writer reversed the direction of the train, but he always placed the second driver of the first engine at any one of the panel points, so that the stresses given are not the absolute greatest. The secondary stresses from the center of the span along the top chord and the end post in Fig. 50 first increase and then decrease, and this is also true for the truss, Fig. 52, up to the inter- section point between end post and collision strut, the maximum stress being reached in the lower fragment of the end post. This maximum is due to the collision strut, which by its division of the end post into two fragments decreases the ratio - and has, there- SECONDARY STRESSES IN BRIDGE TRUSSES 107 io8 SECONDARY STRESSES IN BRIDGE TRUSSES Verticals Diagonals Jo 00 OO I I, I HH . 1 .-sj 7 -_J u 27-^j |*~W_^ UL _JL JU U nr ni ii r~ SECONDARY STRESSES IN BRIDGE TRUSSES 109 Bottom Chord 7 CO M Top Chord O w Co Co 00 co Cn Cn Cn Cn W td o" o" 33 O O H B - l 3 > I | | i I i : 8 ft 8 g II il pi- i i ft 8 or T no. iv'ERSJTY 110 SECONDARY STRESSES IN BRIDGE TRUSSES Verticals SECONDARY STRESSES IN BRIDGE TRUSSES Diagonals III -vj I Oo -BrS-6-E-B r* 22 K 13 S^? c 3 ~ 03 112 SECONDARY STRESSES IN BRIDGE TRUSSES K p << p X rt 8 o" & 3. H ^ s. p' en 1 r i p I3_ crq p EL 09 c/T T3 !? 1' o' H p 3 *< P 3 CL ublished b 3 d, P H '< C/I K a p ^ P cr p < w iiij o' Q c o w o 00 n> x 3 a 3. B- o 3 p c D- 9' $ <' 3 3 ^ Cfi P '-1 i i o 5" *~, to 5 CD 3 P O- N * D- c o p 3 w w ~ 1 O "8 fa c R _ K" en ^ O g O ? TO O ;^r 1 1 1 3 r& w C 5 "s 3 " M ^d s ^ 3. w ^o B ' R S. ^ o H w P tr 5f - a 3 w P P H3 ^^3. ft. n 3 g ^ ^ B. o " ^ p . a ? p s "8 g 3 S ^^^ ? g' R- S 3 3 o 3. * C". H o' SECONDARY STRESSES IN BRIDGE TRUSSES 113 Ei. O P [3* 5* ^ hj P to i " 3 3 ~ l ~ 1 "* C/a i_3 r-f- O " - ^ e. g ^ ? &. p; P p fC C/J W M- 5" 3*^^ n- P o c 5-1 On i 3 fa 3 n> 3 P P t J o" ^ e. B 8.3 w * P t3- CL O ? . *d f-* UJ CD <-! o 1 f ! 3 5' 3- 3- 3 40000 10000 23000 23000 r 114 SECONDARY STRESSES IN BRIDGE TRUSSES Collision Strut Verticals Diagonals to C^l OO M OO M 1 1 1 | | P ON -fi. to o 4-4- Oo 4- to 00 CO ^J '-TI O 00 00 to to M 00 Go 10 00 oo to Oo g 8" K 3 Oo Oo C>o 4^ 4- c 8* 8" 8" ON ON O- Oo to to to to to 3 ^, ,V, y. Ln 0~l i + CN ON ON ON ON n Oo Oo ~ Cn ^ c p c c' era <~/\ to Oo ^ Oo ^r g 8" 8 Oo O g o "-'l ^ ON c ta *'U g- O ~0 4^ O M to Cn Oo Oo to O O 00 00 Oo 00 00 3' T" ill! T LJ L-J M> 10 * - 1 | r - E K. 1 ^ * c- c- " E> t> i 1 * 8^ . g *. i % 5k K- E^ 2,4x CW*^ SECONDARY STRESSES IN BRIDGE TRUSSES 115 Bottom Chord Top Chord End Post f 4^ to O OJ M Ji OJ i 2 1 O CO 00 i 8 8 OJ to to CO o oo OJ OJ Ol Ol ir OJ M Ol 4- 4- o o CO O o o Ol Ol 1" to to to to to CO 00 tO M OO ^J 4^ to 5' OJ 10 to Ol Ol Ol Ol 4- 4- 4- 4- Ol Ol 5' cr f p [S, ON ON ON Oi OJ OJ Ol Ol Ol VO 4- O 01 oi vb D A ABLE Feet. S i 3 td o o T) S O 3 1 % a. I I i ^* M 4 Oo Oi ^ 8 8 00 00 ON ON Oi 4^ OJ OJ o ~o tr | S 01 c to 10 l-i ON M O M tO o *o K> OJ OJ M S9 B | tO KJ tO 10 to to 4^ to O 00 OJ to o o n 1 - JL , 1 1 . i_J ji Ml I 1 1 i r 1 1 Ml i ii i i i i 1 1 1 I i a t M ' ^M 1 1 1 f "E" ' <* " 2 c 3 5i if if > ^"^ c si TC-* 5- g K-^K. t- 1 t- 1 S.K) g.10 EE fO K) U ffi r{? S. ^ S" Il6 SECONDARY STRESSES IN BRIDGE TRUSSES TABLE 10 AND PIG. 53. Span 27 m. Wrought Iron. Gross 3TC3. I 1 b Most Stresses. 1 Case I. Case H. Member. sq. gross cm 4 . cm. cm. danger- ous fiber. cm. Kg. Max. e Sections. cm. sq. cm. % Kg. % sq. cm. i-3 60 2800 45 2 3 Bottom 15.8 o -12.5 5 28 8 1 u 3-5 80 3700 45 24 Bottom 18.7 -18.8 83 -28.1 56 24 ex 5-7 IOO 4500 45 25 Bottom 21 . I -15.0 177 -30.0 88 21 T Sections con- "2 0-2 60 2800 45 23 Top 15-8 + 18.8 41 + 18.8 41 28 sist of i web, 2 angles and U i or 2 or no g 2-4 80 3700 45 24 Top l8. 7 + 14.1 164 4-32.8 70 24 cover plates o 4-6 IOO 4500 45 2 5 Top 21 . I 4-18.8 127 + 33-8 7! 21 b = 2 e b 1-2 36 2100 75 26 13 + 34-7 12 29 3-4 28 9OO 75 20 10 + 22.3 23 4-22.3 23 37 "I 5-6 43 400 75 14 7 + 9-7 17 54 13 1 1 | .2 Q o-3 1 06 2500 75 24 12 -17.7 12 -18.3 12 3i 1 2-5 96 1700 75 20 ... 10 o -13-0 22 37 4-7 80 I IOO 75 18 9 -7.8 33 -10.4 25 42 II o-i 96 1400 600 16 8 -10.4 24 37 Secondary stresses are calculated for gross moments of inertia. SECONDARY STRESSES IN BRIDGE TRUSSES 117 3 S"tS3s 3 tr n> CD CD o e | g* i. & I* ^ B i-1 J S 1 3 y fiT P E tf P W 00 i-, K*, Ji l-i'I 8- ? a. * 3- ri' g I S I s O- 1-1 i" 1 /is >-r l 3 r* g- f I o w g ^ & o 5 c n p oq w OQ e-s.5:- p co 3 HH w O w 3 3' ^ g O ^* p-t- K-( (/> " o 5. crq LO K- i S.i"B 5 5|l p ^ p B C3 3 3 M n. I 3 8 g q CL g. P !F a o Il8 SECONDARY STRESSES IN BRIDGE TRUSSES TABLE 11 AND FIG 54. Span 27 m. Wrought Iron. Stresses. Gross area, I 1 b Most e 1 Case I. Case II. Member. sq. cm gross cm 4 - cm cm danger- ous fiber. cm. Kg. Max. Kg e Sections. i /O /O sq. cm. sq. cm. "g 1-3 60 2800 45 2 3 Top 7-2 -12.5 27 62 U OH 3-5 5-7 80 IOO 3700 4500 45 45 24 25 Bottom Bottom 18.7 21 . I -18.8 -15.0 55 227 -28.1 30.0 36 24 21 T "g 0-2 60 2800 45 2 3 Top 15-8 + 18.8 73 + 18.8 73 29 Sections con- sist of i web, I 2 angles and U i or 2 or no o 2-4 80 3700 45 24 Top l8. 7 + 14.1 IO2 + 32.8 44 24 cover plates I 4-6 IOO 4500 450 2 5 Top 21 . I + 18.8 115 + 33-8 64 21 I b= 2 e . ej = e 2 1 b 12 36 2100 375 26 13 ... + 34-7 28 14 3-4 28 900 375 20 10 + 22.3 56 + 22.3 56 19 "3 1 5-6 43 400 375 14 7 + 9-7 9 27 tj fi 5 0-3 1 06 2500 375 24 12 -17.7 29 -18.3 28 16 1 g CO 2-5 96 1700 375 20 10 o -13.0 5' 19 4-7 80 I IOO 375 18 9 -7-8 IOO -10.4 75 21 "c O-I 96 1400 600 16 8 10.4 2 3 37 Secondary stresses are calculated for gross moments of inertia. SECONDARY STRESSES IN BRIDGE TRUSSES 119 ? s. p 3 3 OQ F |3- ^^ 2. 3 J 6.0m. 120 SECONDARY STRESSES IN BRIDGE TRUSSES TABLE 12 AND FIG 55. Span 40 m. Wrought Iron. Case I. Case II. Sections in cm, Stresses, 1 b Most e when diagonals are 1 Member. cm. cm. danger- ous fiber cm. riveted. not riveted. e Kg. Kg. sq. cm. /o sq. cm. /o --3 400 46.2 Bottom 36.5 1501 117 -So 120 ii i i 3-5 400 46.2 Top 9-7 ~39 12 -39c 12 41 i 1 Web 45 x 1.5 1 C. P . 48 x 1.2 2L S 11.8 x 11.8 E 1.3 1 5-7 400 47-4 Top 8.6 -460 12 -460 12 46 r "I U a, 8 o h -_ >_ 1 We ). 45 x 1.5 2 C.I M 48 x 1.2 2 L 8 11.8 x 11.8 s.1.3 7-9 400 48.6 Top 8.0 -500 7 -500 7 5 "| 1 9-1 1 400 48.6 Top 8.0 -540 8 -540 8 5 1 L 1 Web - 45 x 1.5 3 C. P . - 48 x 1.2 se- 11.8 x 11.8 x l.a 12 0-2 400 40.2 Top 36.5 +210 145 + 210 117 1 1 same as lor 13 1 2-4 400 46.2 Bottom 9-7 + 480 7 + 480 8 4i and 4-5 u | 4-6 400 47-4 Bottom 8.6 + 530 8 + 53 10 46 same as for 5-7 pq 6-8 8-10 400 400 48.6 48.6 Bottom Bottom 8.0 8.0 + 560 + 590 7 7 + 560 6 6 5 .same as for 7-9 and o-n u 18 --* w o-i 400 48 -105 133 -105 124 l b 8 t* 1 -5 .43 x 1.5 .y a PU- 48 x 1.3 "Tj SH. 11 x 1 I> 4L 8 _10.5 x 10.5 x 1.8 2-3 4-5 6-7 400 400 400 i7 .... -3 "1 60 483 220 141 -bo 220 141 24 24 24 fi 8-9 400 17 -70 50 -70 24 i 10-11 400 17 -90 o 90 24 4 ,?_R x 8 x 1 Secondary stresses are calculated for gross moments of inertia. SECONDARY STRESSES IN BRIDGE TRUSSES 121 TABLE 13 AND FIG. 55. Span 40 m. Wrought Iron. Case I. Case II. ! i Stresses, Gross ; when diagonals are Member. area, | 1 sq. cm. b ie t = e 2 cm. cm. riveted. not riveted. Sections in cm. cm. Kg. 07 1 Kg- L l sq. cm. /O b " " i /C sq. cm. b 3 ''- 2 84 56630 15 + 543 23 '9. 4 + 543 II .0 L so *1 SF1 ' 30 1 - 4 bJD c -5 C *j 3-4 73 566 28 14 + 500 22 10 + 5 20 2C j^__ 28 __ i 2 Fl. 28 x 1.3 UO .IJP C J^ 5-6 60 566 25 I2 -5 + 420 23 ii + 420 17 2 3 _ - 2ru25xl . a IT 7-8 60 566 2 5 12-5 + 235 34 1 1 + 235 25 23 U-- 25-J 5 9-10 49 566 18.2 9.1 + 90 78 16 + 90 33 3 ] nr 1^^ o-3 95 566 21 10.5 -580 16 13 -580 9 27 ! 1 I ] 4 L s _10.5 at 10.5 xf.2 nr W---19--*) 2 2-5 85 566 19 9-5 -500 14 15 500 14 So | || ! 4 L 8 9.5 x.9.5 it 1.2 o nr 'C c 4-7 72 566 19 9-5 -440 16 15 -440, 1 1 ,y< *--- 19--*i S c/j 6^ 72 566 I 9 9-5 -275! 24 15 -275 18 3 4Cf-9.Sx9.5il 1 o _ i nr % i 5 ' w T8.2-l 8-1 1 49 566 18.2 9-i -155 40 16 -^ 16 S' 1 Fl. 0.5 x'1.2 2 L 3 -8.& i 8.5 x 1.2 122 SECONDARY STRESSES IN BRIDGE TRUSSES a o EXAMPLES AND CONCLUDING REMARKS 123 fore, no beneficial influence on the end post as far as secondary stresses are concerned. The run of the secondary stresses in the web members for both trusses is very nearly the same, increasing from the end toward the center of the span. The vertical 5-6 in Fig. 52 has been assumed as carrying no dead weight, otherwise the percentage of secondary stress would be very high. Comparing the secondary stresses in the bottom chords of Fig. 50 and Fig. 52, we see that they decrease in Fig. 50 from the end toward the center of the span, while they increase in Fig. 52 in the same direction. This phenomenon is attributable to the collision strut, which is wanting in Fig. 50. This strut resists the hip vertical in its elongation and consequently lessens the defor- mations of the bottom chord sections 0-2 and 0-4. The middle vertical 5-6 in Fig. 52, and the collision strut experi- ence both a reversal of stresses and this is quite natural, as these members are supposed to carry neither dead nor live load stresses. The double intersection Warren truss without verticals, Fig. 53 and Table X, as also Fig. 54 and Table XI, shows rather severe secondary stresses, which increase from the end of the span towards the center and this direction is contrary to that shown by the single Warren truss. The reason for these high secondary stresses is that for unequal loading of the single systems which compose the truss, the chords are subjected to rather large defor- mations. These latter,- and consequently the secondary stresses, can be reduced by the insertion of verticals, which effectively connect the two single systems. The truss shown in Fig. 55 and Tables XII and XIII proves this clearly. The secondary stresses in the chords of these trusses increase from the center of the span toward its end and those of the diagonals increase from the end toward the center of the span, which is the same run as found in single Warren trusses. The stresses in the diagonals also increase when they are riveted at their intersection points. 124 SECONDARY STRESSES IN BRIDGE TRUSSES TABLE 14 AND FIG. 56. Span 27.02 m. Steel. Case I. Case II. Stresses. Mem- ber. Gross area, sq. I gross cm. 4 1 cm- b cm. Most danger- ous fiber . e cm. 1 e cm. For each uniform total load- For the most unfa- vorable po- sitions of Sections. cm. | ing. the loads. % Max. % I I Kg. I i net. gross. sq. cm. net. o-i 239 43736 386 47-6 Top 10.6 37 33 28 -591 27 -t p f 1 1-3 239 43736 386 47-6 Top 10. 6 37 29 25 -582 tz i CJ (X 3-5 239 43736 386 47-6 Top 10.6 37 7 6 -559 7 1 Li 2 WeBte 18' x % H 5-5' 239 43736 386 47-6 Top 10.6 37 7 6 -557 5 iC. P1.18,"* %*. 1 " 0-2 216 40757 213 47 Bottom ii 19 40 33 + 658 42 10_ j 5 2-4 216 40757 410 47 Bottom ii 37 36 30 + 7i4 38 1 4-6 216 40757 395 47 Bottom ii 45 8 7 + 651 6 J L ? 1 6-8 216 40757 387 47 Bottom ii 35 8 7 + 632 7 Z \Veba 18* x % 1 C. PI. 20J4* i %* 4 L'_;3*;x *'. H 1 e 2 b 1-2 55 688 213 16 Bottom 8 J 3 60 47 -332 54 s 1-4 55 688 298 16 Top 8 19 76 59 -432 86 J L a a -~i r 1 3-4 55 688 298 16 Bottom 8 i9 80 62 -34o 5 2 i 4L'_3*, 3*i % Q '3-6 55 688 365 16 Top 8 23 28 22 -459 23 '5-6 55 688 365 16 Bottom 8 23 35 27 -396 18 5-8 55 688 389 16 Top 8 24 49 38 -436 27 ' The direct stresses and the sums of the direct and secondary stresses have been each calculated for the most unfavorable positions of the live load. SECONDARY STRESSES IN BRIDGE TRUSSES 125 i? g.w "^ HI 9- 9-4 H e-S 9* I i T i 1 126 SECONDARY STRESSES IN BRIDGE TRUSSES TABLE 15 AND FIG. 57. Draw Span 202 Feet. Single Track. Steel. Gross Most Stresses. 1 Sections. Member. area, danger- Ibs sq. gross. in. in. in. erf e in. sq. in. /o "O w o-i 2 3-45 500 452 12.44 4. 12 Top -6080 g log 1 C. PL 20" x 5^f 1-3 23-45 500 303 12.44 8.32 Bottom 4300 22 36 ~ l| _ 3-5 23.45 500 12 . 44 4. 12 Top 4300 78 72 js> 00 U 1 1 ^ - 12 U i,, T b "" 5-7 20.58 359 3i8 12 6 + 7210 80 26 .1 O 2 17.64 3 2 3 33 12 6 + 61^0 4 2 5 .-rfB-, 0* U PQ 2-4 4-6 6-8 17.64 17.64 17.64 323 323 323 33 33 33 12 12 12 6 6 6 ... + 6120 + 1020 + IO2O 1 6 1186 868 25 2 5 2 5 LI 2 12"l!l 25it- 1-4 14.70 288 452 12 6 + 610 190 38 j c: .i B Diagonal 4~5 17.64 323 452 12 6 ... + 8120 60 38 1 II e 1-3 r 5-8 26.40 55 452 12.44 4-38 Top -7900 40 103 !12"lT-30ff.. 1 I C. Plr 20"x Jfi 1 b 1-2 13.68 120 33612.38 6.19 + 8200 3 27 (/3 3-4 13.68 120 336 12.38 6.19 . . . 1280$ 27 H sq.in. ^y 5-6 13.68 120 336 12.38 6.19 + 8200 IIO 2 7 o 2 Vi eb.-l9S/ * ^ 16 > rr=_ 7-8 28.54 1475 432 20 10 -2750 VI t=T- Secondary stresses are calculated for gross moments of inertia. SECONDARY STRESSES IN BRIDGE TRUSSES 127 ct> 2 S it g 3 -o = a " a qq Oi -j % * I* ^___Xr-36-0-c-^_-/^| ---23-0-c-c J 128 SECONDARY STRESSES IN BRIDGE TRUSSES The parabola truss, Fig. 56 and Table XIV is a Russian railroad bridge with the wooden ties resting directly on the top chords. The ratios between total stress and primary stress remain constant for any uniformly distributed load. The single Warren truss with verticals, Fig. 57 and Table XV, is continuous over three supports and intended for a drawbridge. The writer determined the secondary stresses for " bridge closed," the truss resting on three supports and covered with the live load from end to end. The center reaction has been taken as the unknown quantity and calculated by means of the principle of the derivative of work. After this reaction was found the primary stresses were calculated. Owing to the fact that this truss is continuous over three sup- ports, the run of the secondary stresses is contrary to that in single Warren trusses on two supports. We see that they increase in the chords from the end of the span toward the center and decrease in the diagonals in the same direction. High secondary stresses can be expected at the center of the span and in its neigh- borhood, no matter whether the bridge is closed or open, as it is at these places where the deformations are the greatest, and they are the smallest in the end panel with inconsiderable bending stresses when the bridge is open. An increase in the stress of the bottom chord section 4-6 of 1 1 86 per cent is very severe, but we must not forget that its primary stress is very small, amounting only to 1020 pounds per square inch. The primary stress in the end bottom chord sections is six times greater than that in the bars 4-6 and 6-8 and conse- quently offers a greater resistance to the deformations at the panel point 2. The total stresses per square inch in the bottom chord sections from the end of the span to the center are 6360, 9790, 13,120 and 9870 pounds, while the primary stresses amount to 6120 and 1020 pounds per square inch. We will now enumerate some points which are guiding for a designer to minimize secondary stresses due to static loads. The effects of impact, vibrations, derailments and collisions are beyond EXAMPLES AND CONCLUDING REMARKS 129 an analysis, but they should not be overlooked because they too are of a secondary nature. It has already been mentioned that secondary stresses decrease if the ratio between the length of a bar and its width increases. This means that members of great circumference and shortness are more susceptible to secondary stresses than long and slender members, a fact already noted when we spoke of secondary stresses in cross frames. In this respect a truss with a curved chord would be a good selection, as, for instance, the parabolic truss or the Schwedler truss where the web members are particularly of light sections. The double intersection Warren truss is unfavorable, as we have "seen, but it can be improved by the insertion of verticals, which connect the two single systems in an effective manner. If in multiple intersection trusses the single trusses act more or less independent of each other, we may predict high secondary stresses, because in such cases the wave like deformations of the chords are rather large. The secondary stresses in continuous trusses over three supports are very severe at the center and in its neighborhood and for this reason pins at these points appear to be desirable, in order to reduce the stresses. The width of a member in the plane of the truss should not be greater than buckling and good connections dictate. The transverse width of verticals to which floorbeams are riveted, should also be kept inside proper limits, as otherwise the secondary stresses caused by the floorbeams may prove to be very high. The removal of the material from the axis of the member to its periphery gives the required moment of inertia in the most economical way. Suspenders should be liberally proportioned to avoid great elon- gations and consequently large deformations of the bottom chords. Strict attention should be paid to good detailing of the panel points ; the axes of the bars must lie in the plane of the truss and eccentricities should be avoided as far as practicable. Curved members should not be tolerated under anv circum- 130 SECONDARY STRESSES IN BRIDGE TRUSSES stances, and brackets on posts may be used only when nothing better can take their place. The movable ends of bridges must be kept in proper working order, otherwise they will be the cause of unnecessary stresses. The use of collision struts cannot be recommended as far as secondary stresses are concerned, because they divide the end posts into two fragments, decreasing the ratio between length and width of these members and increasing the secondary stresses. The collision struts are, as a rule, rather weak members and it is a question whether it would not be better to use the metal on the end posts instead of on these struts, increasing their strength in the direction of the plane of the truss as well as also at right angles to this plane. It is very essential to connect the end posts by a substantial bracing, designing the connections in the best manner possible, and moreover, if a weak end bracing is used the money spent on the horizontal top bracing is wasted. The position of the horizontal bracings should be selected with a view of reducing eccentricities. On account of the riveted connections between the floorbeams and the main trusses, the secondary stresses in the vertical posts of the trusses very often exceed by far the limit of stress set by the specifications. Generally speaking this question has not yet been thoroughly settled. A number of suggestions have been made to remedy the defect and also partly carried out. Deep floorbeams tend to reduce the secondary stresses, but they cannot always be made deep enough to be effective from want of depth in the floor, and besides, if they are very deep, they may exceed the limit of economy. The suggestion to give the floorbeams a down- ward camber, as shown in Fig. 59, originated with Professor Engessor. It is clear that, if a floorbeam is given a deformation corresponding to the load it has to carry and afterwards riveted between the main trusses, it will be quite effective in reducing the secondary stresses in the posts. Another way to avoid these stresses in the verticals would be to give the floorbeams on the main trusses free supports, provided care is taken to properly 11 EXAMPLES AND CONCLUDING REMARKS 131 transmit the wind and braking forces to the main trusses. An important example to accomplish the object in question is fur- nished by the double track railroad arch bridge across the Rhine river at the city of Worms, Germany. The river spans consist of two shore spans, each about 351 feet long and one central span of about 388 feet. The main trusses or arch ribs produce vertical reactions only owing to a tension member running along the floor, which intersects with the main trusses. The floor is carried by the main trusses by means of stiff suspenders to which the floorbeams are pin-connected. The design of the floor is such that it is fixed transversely, not by riveting, but by abutting ends, and longitudinally it is fixed exclusively at the center of the span. The main trusses and all of the bracings are riveted work. The intermediate cross bracings are attached to the main trusses by means of plates, in such positions that they offer only a very ; " small resistance to the deformations of the trusses in vertical planes. These arrangements of the details allow a cross-section for unequal loading of the bridge to take the shape of a rhomboid. Here we see then that the reduction of secondary stresses in the suspenders is aimed at by the use of pins and a skillful design of the bracings and their attachments.* * Our practice of stiff connections between floorbeams and main trusses seems to be well founded, as loose-jointed cross constructions may prove to be more injurious to a bridge than severe secondary stresses. But this does not mean that we should leave the secondary stresses to take care of themselves; on the contrary, proper attention should be paid to these with a view to their reduction. Deep floorbeams, substantial gussets and a generous number of rivets to connect the floorbeams to the main trusses are desirable, as also posts, whose width longi- tudinally and transversely is kept within proper limits. Although these points are well known to experienced bridge engineers never- theless the German government called attention to them in printed circulars, issued 1904, for the observance of those engineers who are charged with the de- signs of engineering structures. It appears from these circulars that the German government is inclined toward stiff connections between floorbeams and main trusses and that it discourages the designs of hinged floor systems in so far as it requests a proof of their advantages. 132 SECONDARY STRESSES IN BRIDGE TRUSSES It is faulty to design the top chord flanges of the floorbeams exclusively for vertical loads, if at the same time they are also charged with the duty of transmitting the braking forces to the main trusses, which they can only do by being bent in a horizontal plane. In such cases the lloorbeam flanges should be provided with brackets, as is shown in Fig. 60, or any other suitable con- struction may be designed to prevent the bending of these flanges. We believe the reduction of secondary stresses by the use of Main Truss Main Truss Fig. 60. pins has been overestimated and that in general the diameters of the pins are too great. Secondary stresses require the pins to be as small in diameter as is consistent with the strength and safety of a bridge. Attention should be paid to have the sur- faces, which are in contact with each other, very smooth to facili- tate turning. It has been suggested that light colored paints would be pref- erable to dark ones, as they absorb less heat and consequently reduce the temperature stresses. For the reduction of the effects of impact and vibrations the following points deserve consideration. Since the mass of a body which receives a blow must be great if the effects of a blow shall be small, it is therefore beneficial to use heavy floor construction and a heavy track. Full webbed floorbeams and trackstringers and a ballasted track diminish the range of vibrations. From economical reasons the ties are often placed directly on. EXAMPLES AND CONCLUDING REMARKS 133 the top chords, when the use of a steel floor would be much better. Tension members should be built stiff, as it is within the range of possibilities that such members, for instance suspenders, are subjected to compression in consequence of vibrations. Riveted connections and stiff members throughout the bridge are in so far of advantage as they tend to decrease the kinetic energy of vibrations. The riveting of the members at their inter- section points, the introduction of secondary and even redundant members, all tend to diminish the time and the amplitude of vibrations. To further decrease the effects of moving loads, it is necessary to keep the track in perfect working order. The passage from the road bed to the bridge should be smooth. Great attention must be paid to the rail ends at the splices. If these rail ends are not of the same height they will be the cause of severe shocks, and particular stress should be laid on the tightness of the bolts at the rail splices. Long rails are naturally of advantage, since they reduce the number of splices, but on a bridge of short span there should be no rail splice at all. In reference to derailments and collisions we will mention a few points which are deserving of consideration from the side of the designer. Of course, it cannot be his object to design a bridge so that it is proof against accidents, but he can do a great deal to reduce the effects of accidents. A ballasted track will prevent the derailed wheels from breaking through the floor. A floor which permits this may cause the collapse of the entire span. A derailed train at the end of a bridge is likely to strike the end post. Consequently, it is of great importance to provide these posts in through spans, and also the portals, with a very robust constitution. The use of inner and outer guard rails on bridges, firmly fastened to ties embedded in ballast, is good practice, as also inside guard rails, flaring guards and rerailing frogs on the bridge 134 SECONDARY STRESSES IN BRIDGE TRUSSES approaches. If these guard rails are higher than the track rails and spaced so that the wheels cannot drop into the space between guard rail and track rail, the train runs practically in a groove. Bridges which are exposed to collisions with floating objects deserve particular attention in their designs. If such collisions are due to causes other than ships, as for example, tree trunks, blocks of ice, etc., it may be sufficient to raise the main trusses above the floorbeams, allowing the latter to take the blows, which would be principally delivered in the direction parallel to their length. Then the stringers and their connections may be so designed that they are self-supporting, preventing the collapse of the bridge in spite of a disabled main truss.* While the stringers could be charged with this duty for short spans, it is hardly feasible to resort to such means for longer spans. Drawbridges which are in great danger of coming into collision with ships are likely to have their bottom chord members struck and possibly ruptured so that a collapse would appear to be cer- tain. In such cases it should be the principal aim of the designer to prevent a rupture of a bridge member and this we believe, at least in the case of a bottom chord section, can be done with a high degree of success and at no great expense. Indeed, the extra cost should not play any role whatever if all the evil conse- quences of a collapse of a bridge are considered. The writer would design the stiff web and bottom chord members with the object of giving them increased resisting power against blows, using rather solid plates instead of latticing and lacing, and paying of course great attention to the detailing of the panelpoints. As a further precaution he would brace the bottom chords in a horizontal plane against the track, but without the use of stiff connections. The track could be designed in a manner that it acts like a cushion to dissipate the effects of an impact. Should the track be above the bottom chord, a collision bracing may extend over the entire width of the bridge, and in this case it * See Eng. News, August 10, 1906. EXAMPLES AND CONCLUDING REMARKS 135 would take at the same time the functions of the bottom lateral bracing. If a bridge has sidewalks, so much the better. These can easily be constructed to form an effective protection for the main trusses. It would be out of place to go here over the manifold details that are possible. It suffices to say that the conditions which govern each individual case must be thoroughly studied in order to properly solve the problem at hand. Before we conclude we will say a few words in regard to the methods of calculations and their use. The method of influence lines cannot be used if it is required to consider the effects of deformations of the bars on the result, as in such cases the secondary stresses appear as higher functions of the exterior loads. Where these deformations can be neglected, there is no doubt that the method of influence lines is the best that can be used in so far as it gives the maximum stresses required. But the trouble with this method lies in the fact that it involves an exceedingly great amount of time and labor, which is best appreciated by the one who undertakes an investigation of this sort. This must be the reason why the calculations published so far are nearly all made on the assumption of only one position of the live load. When the writer undertook the examination of that i4o-foot span after Muller-Breslau's method, which is among the examples, he made use of every facility he could think of to shorten his labors. He never went over the same operations the second time unless there was a necessity to do so. He also prepared tables intended to facilitate the overlooking of the situation as much as the nature of the subject allowed him to do. But in spite of all the precautions taken, the computation proved to be a very labori- ous task indeed. But this is not all. It would be a mistake to believe that that position of the live load which gives the maxi- mum primary stresses is at the same time the one corresponding with the maximum secondary stresses. This may be more or 136 SECONDARY STRESSES IN BRIDGE TRUSSES less true for the chords of a truss resting on two supports, but it is certainly not true for the web members. The writer was not able to foretell either the approximate amount of stress or the deformation of a bar for a given position of the live load. There was only one way to find out something about these points and that was to perform the various operations from beginning to end. The aspect of this subject changes considerably in case we have to deal with the examination of a truss for just one fixed position of the live load. All methods of calculation of secondary stresses require more time than those for primary stresses, but this should not be a reason to avoid them in cases where the knowledge of these stresses appears to be very desirable. Much depends also on the computer himself in point of time. One man may find it a real hardship to compute the secondary stresses in a 2oo-foot span for one fixed position of the live load; while another, obtain- ing his results with ease and rapidity, does not think much of it. There is no doubt that cases may occur where the safety of the structure imposes the duty on the designer to give an account of the secondary stresses. For instance, old trusses designed for loads which are much lighter than those they actually carry are very good subjects for examinations, because it is here in particu- lar within the range of possibilities that the stresses are raised to a dangerous point. Then there are trusses whose examination appears to be desirable on account of peculiarities in their con- struction, because they lead us to the expectation of high secondary stresses. As far as we are aware secondary stresses in trusses of great length, as cantilever trusses, arch ribs, etc., are not known. Such trusses have members whose stresses are subjected to a reversal, which is aggravated by secondary stresses. The secondary stresses are taken into account by the common practice of lowering the unit stresses, but this is a matter of experi- ence and hardly feasible by theoretical considerations. Our knowledge of secondary stresses could be improved either EXAMPLES AND CONCLUDING REMARKS 137 in measuring these stresses by the use of suitable instruments, or by analytical investigations, or by both. The writer suggests that readers who take a particular interest in this subject and have the time to do so, examine trusses and publish their results, which cannot fail to be instructive as they would show us where we have failed. CHAPTER XI. LITERATURE. CONCERNING SECONDARY STRESSES. Asimont. Hauptspannung und Sekundarspannung. Zeitschrift fiir Baukunde, 1880. Manderla. Die Berechnung der Sekundarspannungen, welche im einfachen Fachwerke infolge starrer Knotenverbindungen entstehen. Allgemeine Bauzei- tung, 1880, p. 34. Jebens. Die Spannungen in den Vertikalstandern der eisernen Briicken. Zeitschrift des Vereins deutscher Ingenieure, 1880, p. 127. Manderla. Formanderung des Fachwerks bei wechselnder Belastung. Allge- meine Bauzeitung, 1884, p. 81, 89. Engesser. Die Sicherung offener Briicken gegen Ausknicken. Centralblatt der Bauverwaltung, 1884, p. 415; 1885, p. 71. Ritter. Uber die Druckfestigkeit stabformiger Korper mit besonderer Riicksicht auf die im steifen Fachwerk auftretenden Nebenspannungen. Schwei- zerische Bauzeitung, 1884, I, p. 37, 43, 47. Mliller-Breslau. Uber Biegungsspannungen in Fachwerken. Allgemeine Bauzeitung, 1885, p. 85, 89. Landsberg. Ebene Fachwerkssysteme mit festen Knotenpunkten und das Princip der Deformationsarbeit. Centralblatt der Bauverwaltung, 1885, p. 165. Landsberg. Beitrag zur Theorie der Fachwerke (graphische Ermittelung der Sekundarspannungen infolge fester Knotenverbindungen der Gurtstabe). Zeit- schrift des Architekten-und Ingenieur-Vereins zu Hannover, 1885, p. 361. Miiller-Breslau. Beitrag zur Theorie des Fachwerks. Zeitschrift des Archi- tekten-und Ingenieur-Vereins zu Hannover, 1885, p. 417. Weyrauch. Aufgaben zur Theorie elastischer Korper. Leipzig, 1885, p. 269. Manderla. Uber die Wirkungsweise gelenkformiger Knotenverbindungen. Allgemeine Bauzeitung, 1886, p. 9, 20, 32, 37. Landsberg. Beitrag zur Theorie der Fachwerke. Zeitschrift des Architekten- und Ingenieur-Vereins zu Hannover, 1886, p. 195. Miiller-Breslau. Zur Theorie der Biegungsspannungen in Fachwerktragern. Zeitschrift des Architekten-und Ingenieur-Vereins zu Hannover, 1886, p. 399. Winkler. Aussere Krafte gerader Trager, 1886, p. 166 and 1875, p. 169, 170. Winkler. Querkonstruktionen, p. 179-182. Landsberg. Beitrag zur Theorie des ebenen Fachwerks. Festschrift der technischen Hochschule zu Darmstadt, 1886. 138 LITERATURE 139 Considere. Note sur les effets anormaux dans les ouvrages metalliques. Annales des ponts et chaussees, 1887, I, p. 372. Frankel und Kniger. Spannungs-und Formanderungsmessungen an dem eisernen Pendelpfeiler Viadukte iiber das Oschiitzthal bei Weida. Civil Ingenieur 1887, p. 439. Nebenspanmmgen der Pfeiler, p. 484. Allievi. Equilibrio internio delle pile metalliche. Roma, 1882. (Translated into German by Totz, Wien, 1888.) Hacker. Uber Biegungspannungen in Schwedler'schen Kuppeln. Zeitschrift des Architekten-und Ingenieur-Vereins zu Hannover, 1888, p. 223. Miiller-Breslau. Beitrag zur Theorie der ebenen elastischen Trager. Zeit- schrift des Architekten-und Ingenieur-Vereins zu Hannover, 1888, p. 605. Ritter. Anwendungen der graphischen Statik. II, Das Fachwerk. Zurich, 1890, p. 171. Handbuch der Ingenieurwissenschaften. Vol. II. 1890. Brick. Fachwissenschaftliche Erorterungen zu dem Berichte des Briicken- materialkomites iiber die durchgefiihrten Versuche mit genieteten Tragern aus Flusseisen und Schweisseisen. Zeitschrift des Osterreichischen Ingenieur-und Architekten-Vereins, 1891, p. 76. Jebens. Die seitliche Standsicherheit von eisernen Briicken ohne oberen Querverband. Centralblatt der Bauverwaltung, 1892, p. 148. Engesser. Die seitliche Standfestigkeit offener Briicken. Centralblatt der Bauverwaltung, 1892, p. 349. Engesser. Die Zusatzkriifte und Nebenspannungen eiserner Fachwerk- briicken. I, Die Zusatzkrafte. Berlin, 1892. II, Die Nebenspannungen. Berlin, 1893. Barkhausen. Der Steifrahmen im Wind-und Querverbande geschlossener Trogbriicken. Zeitschrift des Vereins deutscher Ingenieure, 1892, p. 421, 492. Barkhausen. Biegungsspannungen in Blechen und Bandern infolge von einseitiger Verlaschung oder von Uberlappungsverbindungen. Zeitschrift des Vereins deutscher Ingenieure, 1892, p. 553. Mohr. Die Berechnung der Fachwerke mit starren Knotenverbindungen. Der Civil Ingenieur: Organ des Sachsischen Ingenieur-und Architekten-Vereins, 1892, p. 577; 1893, p. 67. Jaquier. Note sur les efforts secondaires qui peuvent se produire dans les systemes articules a attaches rigides. Annales des ponts et chaussees, 1893, I, p. 1142. Engesser. Die zusatzlichen Beanspruchungen durchgehender (kontinuir- licher) Briickenkonstruktionen. Zeitschrift fur Bauwesen, 1894, p. 305. Engesser. Uber die Verringerung der Nebenspannungen von Fachwerk- briicken durch die Art der Aufstellung. Centralblatt der Bauverwaltung, 1895, P- 3i7- Rapport sur les epreuves de charge jusqu' a rupture de 1'ancien pont sur 1'Emme a Wolhusen. Berne, 1895. Haseler. Berechnung der auf Verdrehung beanspruchten Briickentrager. Zeit- schrift des Vereins deutscher Ingenieure, 1896, p. 761. 140 SECONDARY STRESSES IN BRIDGE TRUSSES Dupuy. Resistances des barres soumises a des efforts agissant parallelement & leur axe neutre et en dehors de cette axe. Annales des ponts et chausse'es, 1896, II, p. 223. Haseler. Der Briickenbau. I, Die eisernen Briicken. 3. Lief. Braunschweig, . 1897. Luegers. Lexikon der gesammten Technik mit ihren Hiilfswissenschaften im Verein mit Fachgenossen herausgegeben. Franke. Berechnung der Durchbiegung und der Nebenspannungen der Fachwerktrager. Zeitschrift fiir Bauwesen, 1898. Patton. Beitrag zur Berechnung der Nebenspannungen infolge starrer Knotenverbindungen bei Briickentragern. Zeitschrift fiir Architektur-and Ingenieurwesen. Heft 4, 1902. Isami Hiroi. Statically Indeterminate Bridge Stresses. 1905. Mehrtens. Vorlesungen iiber Statik der Baukonstruktionen und Festig- keitslehre. Vol. III. 1905. Mohr. Abhandlungen aus dem Gebicte der technischen Mechanik. 1906. CONCERNING IMPACT AND VIBRATIONS. Resal. Effet des charges roulantes. Annales des ponts et chaussees, 1882, II, p. 337-35 2 ' Resal. Effet des charges roulantes sur les ponts metalliques. Annales des ponts et chaussees, 1883, I, p. 277-299. Robinson. Vibration of bridges. Transactions of the American Society of Civil Engneers, 1887, Vol. XVI. Soulyere. Action dynamiques des charges roulantes sur les poutres rigides qui ne travaillent qu' a la flexion. Annales des ponts et chaussees, 1889, p. 341-441. Glauser. Dynamische Wirkungen bewegter Lasten auf eiserne Briicken. Glaser's Annalen fiir Gewerbe und Bauwesen, 1891, Vol. 29, p. 113; 1892, Vol. 30, p. 61; 1894, Vol. 34, p. 56. Zimmermann. Die Wirkungen bewegter Lasten auf Briicken. Centralblatt der Bauverwaltung, 1891, p. 448; 1892, p. 159, 199, 215. Melan. Uber die dynamische Wirkung bewegter Lasten auf Briicken. Zeit- schrift des osterreichischen Ingenieur-und Architekten-Vereins, 1893, p. 293. Melan. Uber die dynamische Wirkung bewegter Lasten auf eiserne Briicken. Glaser's Annalen fur Gewerbe und Bauwesen, 1894. Deslandres. Note sur les epreuves par charge roulante et 1'action des chocs. Annales des ponts et chaussees, 1894, I, p. 735. Zimmermann. Die Schwingungen eines Tragers mit bewegter Last. Berlin, 1896. Stone. The determination of the safe working stress for railway bridges of wrought iron and steel. Transactions of the'American Society of Civil Engineers, 1889. Vol. XLI. Turneaure. Some experiments on bridges under moving train-loads. Trans- actions of the American Society of Civil Engineers, 1899, Vol. XLI. SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OF JOHN WILEY & SONS, NEW YORK. LONDON: CHAPMAN & HALL, LIMITED. ARRANGED UNDER SUBJECTS. 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(Waldo.). . . . i2mo, Mixter's Elementary Text-book of Chemistry I2mo, Morgan's An Outline of the Theory of Solutions and its Results i2mo, Elements of Physical Chemistry i2mo, * Physical Chemistry for Electrical Engineers i2mo, Morse's Calculations used in Cane-sugar Factories i6mo, morocco, * Muir's History of Chemical Theories and Laws 8vo, Mulliken's General Method for the Identification of Pure Organic Compounds. Vol. I Large 8vo, O'Driscoll's Notes on the Treatment of Gold Ores 8vo, Ostwald's Conversations on Chemistry. Part One. (Ramsey.) i2mo, " " " Part Two. (Turnbull.) i2mo, * Palmer's Practical Test Book of Chemistry 12mo, * Pauli's Physical Chemistry in the Service of Medicine. (Fischer.) .... I2mo, * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8vo, paper, Pictet's The Alkaloids and their Chemical Constitution. (Biddle.) 8vo, Pinner's Introduction to Organic Chemistry. 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(Wood.) 8vo, 5 oo * Descriptive Geometry. 8vo, i 50 Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50 Merriman and Brooks's Handbook for Surveyors i6mo, morocco, 2 oo Nugent's Plane Surveying 8vo, 3 50 Ogden's Sewer Design i2mo, 2 oo Parsons's Disposal of Municipal Refuse 8vo, 2 oo Patton's Treatise on Civil Engineering 8vo half leather, 7 50 Reed's Topographical Drawing and Sketching 4to, 5 oo Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 4 oo Riemer's Shaft-sinking under Difficult Conditions. (Corning and Peele.) . .8vo, 3 oo Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, i 50 Smith's Manual of Topographical Drawing. (McMillan.) 8vo, 2 50 Sondericker's Graphic Statics, with Applications to Trusses, Beams, and Arches. 8vo, 2 oo Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 5 oo Tracy's Plane Surveying I6mo, morocco, 3 oo * Trautwine's Civil Engineer's Pocket-book i6mo, morocco, 5 oo Venable's Garbage Crematories in America 8vo, 2 oo Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo Sheep, 6 50 Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, 5 oo Sheep, 5 50 Law of Contracts 8vo, 3 oo Warren's Stereotomy Problems in Stone-cutting 8vo, 2 50 Webb's Problems in the Use and Adjustment of Engineering Instruments. i6mo, morocco, i 25 Wilson's Topographic Surveying 8vo, 3 50 BRIDGES AND ROOFS. Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 oo Burr and Falk's Influence Lines for Bridge and Roof Computations 8vo, 3 oo Design and Construction of Metallic Bridges 8vo, 5 oo Du Bois's Mechanics of Engineering. Vol. II JTirall 4to, 10 oo Foster's Treatise on Wooden Trestle Bridges 4to, 5 oo Fowler's Ordinary Foundations 8vo, 3 50 Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from an Orifice. (Trautwine.) 8vo, 2 oa Bovey's Treatise on Hydraulics 8vo, 5 oo Church's Mechanics of Engineering 8vo, 6 oo Diagrams of Mean Velocity of Water in Open Channels paper, i 50 Hydraulic Motors 8vo, 2 oo Coffin's Graphical Solution of Hydraulic Problems i6mo, morocco, 2 50 Flather's Dynamometers, and the Measurement of Power. . i2mo, 3 oo Folwell's Water-supply Engineering 8vo, 4 oo Frizell's Water-power , 8vo, 5 oo Fuertes's Water and Public Health i2mo, i 50 Water-filtration Works i2mo, 2 50 Ganguillet and Kutter's General Formula for the Uniform Flow of Water in Rivers and Other Channels. (Hering and Trautwine.) 8vo, 4 oo Hazen's Clean Water and How to Get It Large I2mo, 1 5o Filtration of Public Water-supply 8vo, 3 oo Hazlehurst's Towers and Tanks for Water- works : 8vo, 2 50 Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal Conduits 8vo, 2 oo * Hubbard and Kiersted's Water- works Management and Maintenance.. .8vo, 4 ca Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) 8vo, 4 oo Merriman's Treatise on Hydraulics 8vo, 5 oo * Michie's Elements of Analytical Mechanics. 8vo, 4 oa Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- supply Large 8vo, 5 oo> * Thomas and Watt's Improvement of Rivers 4to, 6 oa Turneaure and Russell's Public Water-supplies 8vo, 5 oo Wegmann's Design and Construction of Dams, sth Edition, enlarged. . . 4to, 6 oa Water-supply of the City of New York from 1658 to 1895 . .4to, 10 oa Whipple's Value of Pure Water Large i2mo, i oo- Williams and Hazen's Hydraulic Tables 8vo, i 50 Wilson's Irrigation Engineering , Small 8vo, 4 oo- Wolff's Windmill as a Prime Mover 8vo, 3 oo> Wood's Turbines 8vo, 2 50 Elements of Analytical Mechanics 8vo, 3 oo MATERIALS OF ENGINEERING. Baker's Treatise on Masonry Construction 8vo, 5 oo Roads and Pavements 8vo, 5 oo Black's United States Public Works Oblong 4to, 5 oo- * Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 50 Byrne's Highway Construction 8vo, 5 oo- Inspection of the Materials and Workmanship Employed in Construction. i6mo, 3 oo Church's Mechanics of Engineering 8vo, 6 oo>- Du Bois's Mechanics of Engineering. Vol. I Small 410. 7 5<> *Eckel's Cements, Limes, and Plasters 8vo, 6 oo Johnson's Materials of Construction Large 8vo, 6 oo Fowler's Ordinary Foundations 8vo, 3 50 Graves's Forest Mensuration 8vo, 4 oo * Greene's Structural Mechanics 8vo, 2 50 Keep's Cast Iron 8vo, 2 50 Lanza's Applied Mechanics 8vo, 7 SO Martens's Handbook on Testing Materials. (Henning.) 2 vols 8vo, 7 SO Maurer's Technical Mechanics 8vo, 4 oo Merrill's Stones for Building and Decoration 8vo, 5 oo Merriman's Mechanics of Materials 8vo, 5 oo * Strength of Materials i2mo, i oo Metcalf's Steel. A Manual for Steel-users i2mo, 2 oo Patton's Practical Treatise on Foundations 8vo, 5 oo Richardson's Modern Asphalt Pavements 8vo, 3 oo Richey's Handbook for Superintendents of Construction i6mo, rnor., 4 oo * Ries's Clays: Their Occurrence, Properties, and Uses 8vo, 5 oo Rockwell's Roads and Pavements in France I2mo, i 25 Sabin's Industrial and Artistic Technology of Paints ard Varnish 8vo, 3 oo *Schwarz's Longleaf Pine in Virgin Forest ... i2tno, i 25 Smith's Materials of Machines i2mo, i oo Snow's Principal Species of Wood 8vo, 3 50 Spalding's Hydraulic Cement i2mo, 2 oo Text-book on Roads and Pavements i2mo, 2 oo Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, . 5 oo Thurston's Materials of Engineering. 3 Parts 8vo, 8 oo Part I. 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Andrew's Handbook for Street Railway Engineers 3x5 inches, morocco, i 25 Berg's Buildings and Structures of American Railroads 4to, 5 oo Brook's Handbook of Street Railroad Location i6mo, morocco, i 50 Butt's Civil Engineer's Field-book i6mo, morocco, 2 50 Crandall's Transition Curve i6mo, morocco, i 50 Railway and Other Earthwork Tables 8vo, i 50 Crockett's Methods for Earthwork Computations. (In Press) Dawson's "Engineering" and Electric Traction Pocket-book . . i6mo, morocco 5 oo Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 oo Fisher's Table of Cubic Yards Cardboard, 25 Godwin's Railroad Engineers' Field-book and Explorers' Guide. . . i6mo, mor., 2 50 Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- bankments 8vo, i oo Molitor and Beard's Manual for Resident Engineers i6mo, i oo Nagle's Field Manual for Railroad Engineers i6mo, morocco, 3 oo Philbrick's Field Manual for Engineers i6mo, morocco, 3 oo Raymond's Elements of Railroad Engineering. (In Press.) 9 Searles's Field Engineering i6mo, morocco, 3 oo Railroad Spiral. i6mo, morocco, x 50 Taylor's Prismoidal Formulae and Earthwork 8vo, x 50 * Trautwine's Method of Calculating the Cube Contents of Excavations and Embankments by the Aid of Diagrams 8vo, 2 oo The Field Practice of Laying Out Circular Curves for Railroads. i2mo, morocco, 2 50 Cross-section Sheet Paper, 25 Webb's Railroad Construction i6mo, morocco, 5 oo Economics of Railroad Construction Large i2mo, 2 50 Wellington's Economic Theory of the Location of Railways Small Svo, 5 oo DRAWING. Barr's Kinematics of Machinery Svo, 2 50 * Bartlett's Mechanical Drawing Svo , 3 oo * " " " Abridged Ed Svo, 150 Coolidge's Manual of Drawing Svo, paper, i oo Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- neers . Oblong 4to, 2 50 Durley's Kinematics of Machines Svo, 4 oo Emch's Introduction to Projective Geometry and its Applications Svo, 2 50 Hill's Text-book on Shades and Shadows, and Perspective Svo, 2 oo Jamison's Elements of Mechanical Drawing Svo, 2 50 Advanced Mechanical Drawing Svo, 2 oo Jones's Machine Design : Part I. Kinematics of Machinery Svo, i 50 Part II. Form, Strength, and Proportions of Parts Svo, 3 oo MacCord's Elements of Descriptive Geometry Svo, 3 oo Kinematics; or, Practical Mechanism Svo, 5 oo Mechanical Drawing 4to, 4 oo Velocity Diagrams Svo, i 50 MacLeod's Descriptive Geometry Small Svo, i 50 * Mahan's Descriptive Geometry and Stone-cutting Svo, i 50 Industrial Drawing. (Thompson.) Svo, 3 50 Moyer's Descriptive Geometry Svo, 2 oo Reed's Topographical Drawing and Sketching 4to, 5 oo Reid's Course in Mechanical Drawing Svo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. Svo, 3 oo Robinson's Principles of Mechanism Svo, 3 oo Schwamb and Merrill's Elements of Mechanism Svo, 3 oo Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) Svo, 2 50 Smith (A. W.) and Marx's Machine Design Svo, 3 oo * Titsworth's Elements of Mechanical Drawing Oblong Svo, i 25 Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. i2mo, i oo Drafting Instruments and Operations i2mo, i 25 Manual of Elementary Projection Drawing i2mo, i 50 Manual of Elementary Problems in the Linear Perspective of Form and Shadow i2mo, r oo Plane Problems in Elementary Geometry i2mo, 125 Elements of Descriptive Geometry, Shadows, and Perspective Svo, 3 50 General Problems of Shades and Shadows Svo, 3 oo Elements of Machine Construction and Drawing Svo, 7 50 Problems, Theorems, and Examples in Descriptive Geometry Svo, 2 50 Weisbach's Kinematics and Power of Transmission. (Hermann and Klein.) 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(Merriam.) i2mo, i 25 Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 oo Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von Ende.) i2mo, 2 50 Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 oo Flather's Dynamometers, and the Measurement of Power i2mo, 3 co Gilbert's De Magnete. (Mottelay.) 8vo, 2 50 Hanchett's Alternating Currents Explained i2mo, i oo Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 Hobart and Ellis's High-speed Dynamo Electric Machinery. (In Press.) Holman's Precision of Measurements 8vo, 2 oo Telescopic Mirror-scale Method, Adjustments, and Tests. . . .Large 8vo, 75 Karapetoff's Experimental Electrical Engineering. (In Press.) Kinzbrunner's Testing of Continuous-current Machines 8vo, 2 oo Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 oo Le Chatelier's High-temperature Measurements. (Boudouard Burgess.) i2mo, 3 oo Lob's Electrochemistry of Organic Compounds. 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Vol. 1 8vo, Thurston's Stationary Steam-engines ' 8vo, * Tillman's Elementary Lessons in Heat. . . . .8vo, Tory and Pitcher's Manual of Laboratory Physics Small 8vo, oo 50 50 50 oo Ulke's Modern Electrolytic Copper Refining 8vo, 3 oo LAW. * Davis's Elements of Law 8vo, 2 50 * Treatise on the Military Law of United States 8vo, 7 oo * Sheep, 7 50 * Dudley's Military Law and the Procedure of Courts-martial . . . 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(Henning.) 8vo, 7 50 Maurer's Technical Mechanics 8vo, 4 oo Merriman's Mechanics of Materials 8vo, 5 oo * Strength of Materials i2mo, i oo Metcalf's Steel. A Manual for Steel-users i2mo, 2 oo Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 oo Smith's Materials of Machines i2mo, i oo Thurston's Materials of Engineering 3 vols., 8vo, 8 oo Part II. Iron and Steel 8vo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 14 Wood's (De V.) Treatise on the Resistance of Materials and an Appendix on the Preservation of Timber 8vo, 2 oo Elements of Analytical Mechanics 8vo, 3 oo Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel 8vo, 4 oo STEAM-ENGINES AND BOILERS. Berry's Temperature-entropy Diagram i2mo, i 25 Carnot's Reflections on the Motive Power of Heat. (Thurston.) i2mo, i 50 Creighton's Steam-engine and other Heat-motors 8vo, 500 Dawson's "Engineering" and Electric Traction Pocket-book. . . .i6mo, mor., 5 oo Ford's Boiler Making for Boiler Makers i8mo, i oo Goss's Locomotive Sparks 8vo, 2 oo Locomotive Performance 8vo, 5 oo Hemenway's Indicator Practice and Steam-engine Economy i2mo, 2 oo Button's Mechanical Engineering of Power Plants 8vo, 5 oo Heat and Heat-engines 8vo 5 oo Kent's Steam boiler Economy 8vo, 4 oo Kneass's Practice and Theory of the Injector 8vo, i 50 MacCord's Slide-valves 8vo, 2 oo Meyer's Modern Locomotive Construction .- 4 to, 10 oo Peabody's Manual of the Steam-engine Indicator 12 mo, T 50 Tables of the Properties of Saturated Steam and Other Vapors 8vo, i oo Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 oo Valve-gears for Steam-engines 8vo, 2 50 Peabody and Miller's Steam-boilers 8vo, 4 oo Pray's Twenty Years with the Indicator Large 8vo, 2 50 Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. (Osterberg.) i2mo, i 25 Reagan's Locomotives: Simple, Compound, and Electric. New Edition. Large i2mo, 3 50 Sinclair's Locomotive Engine Running and Management i2mo, 2 oo Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50 Snow's Steam-boiler Practice 8vo, 3 oo Spangler's Valve-gears 8vo, 2 50 Notes on Thermodynamics i2mo, i oo Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 oo Thomas's Steam-turbines 8vo, 3 50 Thurston's Handy Tables 8vo, i 50 Manual of the Steam-engine 2 vols., 8vo, 10 oo Part I. History, Structure, and Theory. 8vo, 6 oo Part II. Design, Construction, and Operation 8vo, 6 oo Handbook of Engine and Boiler Trials, and the Use of the Indicator and the Prony Brake 8vo, 5 oo Stationary Steam-engines 8vo, 2 50 Steam-boiler Explosions in Theory and in Practice i2mo, i 50 Manual of Steam-boilers, their Designs, Construction, and Operation. 8vo, 5 oo Wehrenfenning's Analysis and Softening of Boiler Feed-water (Patterson) 8vo, 4 oo Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) 8vo, 5 oo Whitham's Steam-engine Design 8vo, 5 oo Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. ..8vo, 4 oo MECHANICS AND MACHINERY. Barr's Kinematics of Machinery 8ro, 2 50 * Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 Chase's The Art of Pattern-making I2mo, 2 50 15 Church's' Mechanics of Engineering 8vo, 6 oo Notes and Examples in Mechanics 8vo, 2 oo Compton's First Lessons in Metal- working izmo, 50 Compton and De Groodt's The Speed Lathe i2mo, 50 Cromwell's Treatise on Toothed Gearing i2mo, 50 Treatise on Belts and Pulleys I2mo, 50 Dana's Text-book of Elementary Mechanics for Colleges and Schools. .i2mo, 50 Dingey's Machinery Pattern Making i2mo, 2 oo Dredge's Record of the Transportation Exhibits Building of the World's Columbian Exposition of 1893 4to half morocco, 3 oo Du Bois's Elementary Principles of Mechanics : Vol. I. Kinematics 8vo f 3 50 Vol. II. Statics 8vo, 4 oo Mechanics of Engineering. Vol. I Small 4to, 7 50 Vol. II Small 4to, 10 oo Durley's Kinematics of Machines 8vo, 4 oo Fitzgerald's Boston Machinist i6mo, i oo Flather's Dynamometers, and the Measurement of Power i2mo, 3 oo Rope Driving i2mo, 2 oo Goss's Locomotive Sparks 8vo, 2 oo Locomotive Performance 8vo, 5 oo * Greene's Structural Mechanics 8vo, 2 50 Hall's Car Lubrication i2mo, i oo Hobart and Ellis 's High-speed Dynamo Electric Machinery. (In Press.) Holly's Art of Saw Filing i8mo, 75 James's Kinematics of a Point and the Rational Mechanics of a Particle. Small 8vo, 2 oo * Johnson's (W. W.) Theoretical Mechanics i2mo, 3 oo Johnson's (L. J.) Statics by Graphic and Algebraic Methods 8vo, 2 oo Jones's Machine Design: Part I. Kinematics of Machinery 8vo, i 50 Part II. Form, Strength, and Proportions of Parts 8vo, 3 oo Kerr's Power and Power Transmission 8vo, 2 oo Lanza's Applied Mechanics 8vo, 7 50 Leonard's Machine Shop, Tools, and Methods 8vo, 4 oo * Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean.) .8vo, 4 oo MacCord's Kinematics; or, Practical Mechanism 8vo, 5 oo Velocity Diagrams 8vo, i 50 * Martin's Text Book on Mechanics, Vol. I, Statics i2mo, i 25 * Vol. 2, Kinematics and Kinetics . .I2mo, 1 50 Maurer's Technical Mechanics 8vo, 4 oo Merriman's Mechanics of Materials 8vo, 5 oo * Elements of Mechanics I2mo, i oo * Michie's Elements of Analytical Mechanics 8vo, 4 oo * Parshall and Hobart's Electric Machine Design 4to, hah* morocco, 12 50 Reagan's Locomotives : Simple, Compound, and Electric. New Edition. Large 12010, 3 5o Reid's Course in Mechanical Drawing 8vo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo Richards's Compressed Air i2mo, i 50 Robinson's Principles of Mechanism 8vo, 3 oo Ryan, Norris, and Hoxie's Electrical Machinery. 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(Herrmann Klein.). 8vo. 5 oo Wood's Elements of Analytical Mechanics 8vo, 3 oo Principles of Elementary Mechanics I2mo, i 25 Turbines. 8vo, 2 50 The World's Columbian Exposition of 1893 4to, I oo MEDICAL. * Bolduan's Immune Sera , 12mo, 1 50 De Fursac's Manual of Psychiatry. (Rosanoff and Collins.). .. .Large i2mo, 2 50 Ehrlich's Collected Studies on Immunity. (Bolduan.) 8vo, 6 oo * Fischer's Physiology of Alimentation Large 12mo, cloth, 2 oo Hammarsten's Text-book on Physiological Chemistry. (Mandel.) 8vo, 4 oo Lassar-Cohn's Practical Urinary Analysis. (Lorenz. ) i2mo, oo * Pauli's Physical Chemistry in the Service of Medicine. (Fischer.) .... i2mo, 25 * Pozzi-Escot's The Toxins and Venoms and their Antibodies. (Cohn.). i2mo, oo Rostoski's Serum Diagnosis. (Bolduan.) 12010, oo Salkowski's Physiological and Pathological Chemistry. 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(Waldo.). . . . i2mo, 2 50 Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, 4 oo Smith's Materials of Machines i2mo, i oo Thurston's Materials of Engineering. In Three Parts 8vo, 8 eo Part II. Iron and Steel 8vo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 Ulke's Modern Electrolytic Copper Refining 8vo, 3 oo MINERALOGY. Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 2 50 Boyd's Resources of Southwest Virginia 8vo, 3 oo 17 Boyd's Map of Southwest Virignia Pocket-book form, a oo * Browning's Introduction to the Rarer Elements Kvo, i 50 Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 4 oo Chester's Catalogue of Minerals 8vo, paper, i oo Cloth, i 25 Dictionary of the Names of Minerals 8vo, 3 50 Dana's System of Mineralogy Large 8vo, half leather, 12 50 First Appendix to Dana's New "System of Mineralogy." Large 8vo, i oo Texi-book of Mineralogy 8vo, 4 oo Minerals and How to Study Them .... I2mo, 50 Catalogue of American Localities of Minerals Large 8vo, oo Manual of Mineralogy and Petrography i2mo oo Douglas's Untechnical Addresses on Technical Subjects i2mo, oo Eakle's Mineral Tables 8vo, 25 Egleston's Catalogue of Minerals and Synonyms 8vo, 2 50 Goesel's Minerals and Metals: A Reference Book ibmo, mor. 300 Groth's Introduction to Chemical Crystallography (Marshall) I2mo, i 25 Iddings's Rock Minerals 8vo, 5 oo Johannsen's Key for the Determination of Rock-forming Minerals in Thin Sections. (In Press.) * Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe. I2tno, 60 Merrill's Non-metallic Minerals. Their Occurrence and Uses 8vo, 4 oo Stones for Building and Decoration .. 8vo, 500 * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8vo, paper, 50 Tables of Minerals 8vo, l 00 * Richards's Synopsis of Mineral Characters i2mo. morocco, i 25 * Ries's Clays. Their Occurrence. 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Small 4to, half morocco 5 oo Letteris's Hebrew Bible 8vo, 2 25 19 ^1 UNIVERSITY Ur THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW NOV 13 UWARYUW APR28 REC'D APR 2 8 1962 RSC'D LD MAY 1 2 1962 laou 30M-V15