88. The Truss Element or Triangle of Bracing.—A number of the preceding formulæ find their applications to bridge-trusses, as well as to beams; hence it is necessary to give attention at least to some simple forms of those trusses.

The skeleton of every bridge-truss properly designed to carry its load is an assemblage of triangles. In other words, the truss element, i.e., the simplest possible truss, is the triangular frame, such as is shown in skeleton in Figs. 18 and 18a. These simple triangular frames are sometimes called the King-post Truss. The action of such a triangular frame in carrying a vertical load is extremely simple. In Fig. 18 let the weight W be suspended from the apex C of the triangle. The line CF represents that weight, and if the latter be resolved into its two components parallel to the two upper members of the triangular frame, the two component forces CG and CD will result. If from D and G the horizontal lines DH and GO be drawn, those two lines will represent the horizontal components of the forces or stresses in the two bars CA and CB. The force HD will act to the left at the point A, and the force CG will act to the right at B, and as these two forces are equal and opposite to each other, equilibrium will result. Either of the horizontal forces will represent the magnitude of the tension in AB. Both AC and CB will be in compression, the former being compressed by the force CD, and the latter by the force CG. The manner of drawing a parallelogram of forces makes the triangle COG similar to CNB, and CHD similar to CNA; hence HW divided by CH will be equal to AN divided by NB. But HW is the vertical component of the stress in CB, while CH is the vertical component of the stress in AC, the latter being represented by the reaction R and the former by the reaction R′. It is seen, therefore, that the weight W is carried by the frame to the two abutment supports A and B, precisely as if it were a solid beam. In other words, the important principle is established that when weights rest upon a simple truss supported at each end they will produce reactions at the ends in accordance with the principle of the lever, precisely as in the case of a solid beam. In engineering parlance it is stated that the weight W is divided according to the principle of the lever, and that each portion travels to its proper abutment through the members of the triangular frame. If the two inclined members of the triangular frame are equally inclined to a vertical, the case of Fig. 18a results, in which one half of the weight goes to each abutment.

The triangular frame, with equally inclined sides, shown in Fig. 18a, is evidently the simplest form of roof-truss, constituting two equally inclined members with a horizontal tie.

89. Simple Trusses.—The simplest forms of trussing used for bridge purposes are those shown in Figs. 19, 20, and 21. There are many other forms which are exhibited in complete treatises on bridge structures, but these three are as simple as any, and they have been far more used than any other types. The horizontal members af and AB are called the “chords,” the former being the upper chord and the latter the lower chord. The vertical and inclined members connecting the two chords are called the web members or braces. When a bridge is loaded, either by its own weight only, or by its own weight added to that of a moving train of cars, the upper chord will evidently be in compression, while the lower chord is in tension. A portion, which may be called a half, of the web members will be in tension and the other portion, or half, will be in compression.

The function of the upper and lower chords is to take up or resist the horizontal tension and compression which correspond to the direct stresses of tension and compression existing in the longitudinal fibres of a loaded solid or flanged beam. The metal designed to take these so-called direct stresses is concentrated in the chords of trusses, whereas it is distributed throughout the entire section of a beam, whether that beam be solid or flanged. The function of the web members of a truss is to resist the transverse or vertical shear which is represented by the algebraic sum of the reactions and loads. The total section of a solid beam resists these vertical shears, while the web only of a flanged beam is estimated to perform that duty. The horizontal shears, which have already been recognized as existing along the horizontal planes in a bent beam, are resisted by the inclined web members of a truss, the horizontal stress components being the horizontal shears, whereas the vertical shears are resisted by the vertical web members of a truss. If the web members are all inclined, as shown in Fig. 21, each web member resists both horizontal and vertical shear. It is thus seen that the members of a truss perform precisely the same duties as the various portions of either solid or flanged beams. Inasmuch as the chords of bridge-trusses resist the direct or horizontal stresses of tension and compression produced by the bending in the truss, it is obvious that the greatest chord stresses will be found at the centre of the span, and that they will be the smallest at the ends of the span. In the web members, on the contrary, since the vertical shear is the greatest at the ends of the span and equal to the reactions at those points, decreasing towards the centre precisely as in solid beams, the greatest web stresses will be found at the ends of the span and the least near the centre. It is obvious that the areas of cross-sections of either chords or web members must be proportioned to the stresses which they carry. Hence the distribution of stresses just described tends to a uniform distribution of the truss weights over the span.

90. The Pratt Truss Type.—In the discussion of these three simple types of trusses, the simplest possible loading of a perfectly uniform train will be assumed. The portions into which the trusses are divided by the vertical or inclined bracing are called panels. In Fig. 19, for instance, the points 1, 2, 3, 4, 5, and 6 of the lower chord and a, b, c, d, e, and f of the upper chord are called panel-points. The distance between each consecutive two of these points is called a panel length. The uniform train-load which is to be assumed will be represented by the weight W at each panel-point. This is called the “moving load” or “live load.” The own weight of the structure is called the “dead load” or the “fixed load.” The dead load per upper-chord panel will be taken as Wʹ, and W₁ for the lower chord. The loads to be used will, therefore, be as follows:

There will also be used the length of panel and depth of truss as follows:

• Panel length = p;
• Depth of truss = d.

In these simple trusses with horizontal upper and lower chords the stress in any inclined web members is equal to the shear multiplied by the secant of the inclination of the members to a vertical line. Also, at each panel-point every inclined web member, in passing from the end to the centre of the span, adds to either chord stress at that point an amount represented by the horizontal component of the stress which it carries; or, what is the same thing, an amount equal to the shear at the panel in question multiplied by the tangent of its angle of inclination to a vertical line.

It has already been shown in discussing solid beams that the greatest shear at any section will be found when the uniform moving load covers one of the segments of the span. This principle holds equally true for trusses carrying uniform panel-loads like those under consideration. In determining the stresses in these trusses, therefore, the inclined web members will take their greatest stresses when the moving train or load extends from the farthest end of the span up to the foot of the member in question. In this connection it is to be observed also that any two web members meeting in the chord which does not carry the moving load take their greatest stresses for the same position of the latter. The so-called “counter web members” take no stresses from the dead load.

Inasmuch as every load placed upon a truss will produce compression in the upper chord and tension in the lower, the greatest chord stresses will obviously exist when the moving load covers the entire span, and that condition of loading is to be used for the stresses in the following cases.

Bearing these general observations in mind, the ordinary simple method of truss analysis yields the tabulated statement of stresses given below for the three types selected for consideration. The first case to be treated is that of Fig. 19, which represents the Pratt truss type. The moving load is supposed to pass across the bridge from right to left. The plus sign indicates tension and the minus sign compression.

It is easy to check any of the chord stresses by the method of moments. As an example, let moments first be taken about the panel-point 5 in the lower chord, and then about the panel-point c in the upper chord. The following expressions for the chord members U₁ and L₄ will be found, and it will be noticed that they are identical with the stresses for the same members given in the preceding tabulation, the counter-members, shown in broken lines, being omitted from consideration as they are not needed.

91. The Howe Truss Type.—The truss shown in Fig. 20 is the skeleton of the Howe truss, to which reference has already been made. The inclined web members are all in compression, while the vertical web members are all in tension. In the Howe truss all compression members are composed of timber. It has the disadvantage of subjecting the longest web members to compression. It thus makes the truss, if built all in iron or steel, heavier and more expensive than the trusses of the Pratt type. As in the preceding case, the moving train or load is supposed to pass across the bridge from B to A. Also, as before, the + sign indicates tension and the - sign compression. The greatest stresses, given in the tabulated statement below, can be computed or checked by the method of moments in this case, precisely as in the preceding.

It will be noticed in the cases of Figs. 19 and 20 that upper and lower chord panels in the same lozenge or oblique panel have identically the same stresses, but with opposite signs. For instance, in Fig. 20 the stress in U₂ is equal in amount to that in L₂; and the same observation can be made in reference to the stresses in U₂ and L₄ of Fig. 19. This must necessarily always be the case in trusses having vertical web members.

In making computations for these forms of trusses it is very essential to observe where the first counter-member, as c₁, must be used. These counter-members may be omitted if the proper main web members near the centre of the span are designed to take both tension and compression.

92. The Simple Triangular Truss.—The truss shown in Fig. 21, in which all the web members have equal inclination to a vertical line, is sometimes called the Warren Truss, although that term has also been applied specially to this type of truss so proportioned as to make the depth just equal to the panel length. As before, the moving train is supposed to pass over the bridge from B toward A, while the + sign represents tension and the - sign compression. The greatest stresses are the following.

The chord stresses may be checked or found by the method of moments, precisely as in the case of Fig. 19. If, for instance, it is desired to determine the stresses in the upper chord member U₂, moments must be taken about the lower-chord panel-point 5, and about the upper-chord panel-point d for the lower-chord stress in L₄. Taking moments about those points, results given in equations (31) and (32) will at once follow, which it will be observed are identical with the values previously found for the same members.

93. Through- and Deck-Bridges.—These simple trusses have all been taken as belonging to the “through” type, i.e., the moving load passes along their lower chords. It is quite common to have the moving load pass along the upper chords, in which cases the bridges are said to be “deck” structures. The general methods of computation are precisely the same whether the trusses be deck or through. It is only necessary carefully to observe that the application of the methods of analysis depends upon the position of each panel-load as it passes across the structure.

94. Multiple Systems of Triangulation.—Figs. 19, 20, and 21 exhibit what are called single systems of triangulation or single systems of bracing, but in each of those types the system of web members may be double or triple; in other words, they may be manifold. There have been many bridges built in which two or more systems of bracing are employed. Fig. 22 represents a truss with a double system of triangulation, known at one time as the Whipple truss. Fig. 23, again, exhibits a quadruple system of triangulation with all inclined web members. The method of computation for such manifold systems is precisely the same as for a single system, each system in the compound truss being treated as carrying those loads only which rest at its panel-points. This procedure is not quite accurate. The complete consideration of an exact method of computation would take the treatment into a region of rather complicated analysis beyond the purposes of these lectures, but its outlines will be set forth on a later page. The exact method of treatment of two or more web systems involves the elastic properties of the material of which the trusses are composed. In the best modern bridge practice engineers prefer to design trusses of all lengths with single web systems, although the panels are frequently subdivided to avoid stringers and floor-beams of too great weight.

95. Influence of Mill and Shop Capacity on Length of Span.—In the early years of iron and steel bridge-building the sizes of individual members were limited by the shop capacity for handling and manufacturing, and by the relatively small dimensions of bars of various shapes, and of plates which could be produced by rolling-mills. As both mill and shop processes have advanced and their capacities increased, corresponding progress has been made in bridge design. Civil engineers have availed themselves of those advances, so that at the present time single system trusses with depths as great as 85 feet or more and spans of over 550 feet are not considered specially remarkable.

96. Trusses with Broken or Inclined Chords.—As the lengths of spans have increased certain substantial advantages have been gained in design by no longer making the upper chords horizontal in the case of long through-spans, or indeed in the cases of through-spans of moderate length. The greatest bending moments and the greatest chord stresses have been shown to exist at the centre of the span, while the greatest web stresses are found near the ends. Trusses may be lightened in view of those considerations by making their depths less at the ends than at the centre. This not only decreases the sectional areas of the heaviest web members near the ends of the truss, but also shortens them. It adds somewhat to the sectional area of the end upper-chord members, but the resultant effect is a decrease in total weight of material and increased stability against wind pressure by the decreased height and less exposure near the ends. It has therefore come to be the ruling practice at the present time to make through-trusses with inclined upper chords for practically all spans from about 200 feet upward. A skeleton diagram of such a truss is given in Fig. 24.

97. Position of any Moving Load for Greatest Web Stress.—In the preceding treatment of bridge-trusses with parallel and horizontal chords a moving or live load has been taken as a series of uniform weights concentrated at the panel-points. This simple procedure was formerly generally used, and at the present time it is occasionally employed, but it is now almost universal practice to assume for railroad bridges a moving load consisting of a series of concentrations, which represent both in amount and distribution the weights on the axles of an actual railroad train. If a bridge is supposed to be traversed by such a train, it becomes necessary to determine a method for ascertaining the positions of the train causing the greatest stresses in the various members of the bridge-truss. The mathematical demonstration of the formulæ determining those positions of loading need not be given here, but it can be found in almost any standard work on bridges.

In order to show concisely the results of such a demonstration let it be desired to find the position of a moving load which will give the greatest stress to any web member, as S in Fig. 24. Let the point of intersection of GK and DC be found in the point O, then let CK be extended, and on its extension let the perpendicular h be dropped from O. The distance of the point O from A, the end of the span, is i, while m is the distance AD. Using the same notation which has been employed in the discussion of beams, together with that shown in Fig. 24, equation (33) expresses the condition to be fulfilled by the train-loads in order that S shall have its greatest stress. The first parenthesis in the second member of that equation represents the load between the panel p and the left end of the span, while the second parenthesis represents the load in panel p itself.

It will be noticed in equation (33) that the quantity m shows in what panel the inclined web member whose greatest stress is desired is located, and it is important to observe that panel carefully. If, for instance, the vertical member KD were in question, the point O would be located at the intersection of the panel NK and the lower chord of the bridge. In other words, the point O must be at the intersection of the two chord members belonging to the same panel in which the web member is located.

98. Application of Criterions for both Chord and Web Stresses.—The criterion, equation (33), belongs to web members only. If it is desired to find the position of moving load which will give the greatest chord stresses in any panel, equation (27), already established for beams, is to be used precisely as it stands, the quantity l′ representing the distance from one end of the span to the panel-point about which moments are taken.

If the desired positions of the moving load for greatest stresses have been found by equations (27) and (33), those stresses themselves are readily found by taking moments about panel-points for chord members and about the intersection-points O, Fig. 24, for web members. These operations are simple in character and are performed with great facility. Tabulations and diagrams are made for given systems of loading by which these computations are much shortened and which enable the numerical work of any special case to be performed quickly and with little liability to error. These tabulations and diagrams and other shortening processes may be found set forth in detail in many publications and works on bridge structures. They constitute a part of the office outfit of civil engineers engaged in structural work.

The criterion, equation (27), for the greatest bending moments in a bridge is applicable to any truss whatever, whether the chords are parallel or inclined, but it is not so with equation (33). If the chords of the trusses are parallel, the quantity i in equation (33) becomes infinitely great, and the equation takes the following form:

Ordinarily the span l divided by the panel length p is equal to the number of panels in the span. Hence equation (34) shows, in the case of parallel or horizontal chords, that when the moving load is placed for the greatest web stress in any panel, the total load on the bridge is equal to the load in that panel multiplied by the total number of panels.

99. Influence Lines.—A graphical method, known as that of “influence lines,” is used for determining the greatest shears and bending moments caused by a train of concentrated weights passing along a beam or bridge-truss. Obviously it must express in essence that which has already been shown by the formulæ which determine positions of moving loads for the greatest shears and bending moments. In reality it is the application of graphical methods which have become so popular to the determination of the greatest stresses in beams and bridges.

100. Influence Lines for Moments both for Beams and Trusses.—It is convenient to construct these influence lines for an arbitrary load which may be considered a unit load; the effect of any other load will then be in proportion to its magnitude. The results determined from influence lines drawn for a load which may be considered a unit can, therefore, be made available for other loads by multiplying the former by the ratio between any desired load and that for which the influence lines are found.

AB in Fig. 25 represents a beam simply supported at each end, so that any load g resting upon it will be divided between the points of support, according to the law of the lever. Let it be desired to determine the bending moment at the section X produced by the load g in all of its positions as it passes across the span from A to B. Two expressions for the bending moment must be written, one for the load g at any point in AX, and the other for the load at any point in BX. The expression for the first bending moment is

and that for the latter

As shown in the figure, z and x, the latter locating the section at which the bending moments are to be found, are measured to the right from A. Equation (a) shows that if the quantity g(l-x) be laid off, by any convenient scale, as BK at right angles to AB, XC will represent the moment M by the same scale when x = z or when z has any value between 0 and x. Similarly will AD be laid off at right angles to AB by the same scale as before, to represent gx. Then when x = z the expression for M′ will have the same value XC as before. Hence if the lines AC and CB be drawn as parts of AK and DB, any vertical intercept between AB and ACB will represent the bending at X produced by the load g when placed at the point from which the intercept is drawn. The lines AC and CB are the influence lines for the bending moments produced by the load g in its passage across the span AB. It is to be observed that the influence lines are continuous only when the positions of the moving load are consecutive. In case those positions are not consecutive the influence lines are polygonal in form.

If there are a number of loads g resting on the span at the same time, the total bending moments produced at X will be found by taking the sum of all the vertical intercepts between AB and ACB, drawn at the various points where those loads rest. The influence lines drawn for a single load, therefore, may be at once used for any number of loads.

The load g is considered as a unit load. If the vertical intercepts representing the bending moments by the scale used are themselves represented by y, and if W represent any load whatever, the general expression for the bending moment at X, produced by any system of loads, will be

If this expression be written as a series, the general value of the bending moment will be the following:

The effect of a moving train upon the bending moment at any given section is thus easily made apparent by means of influence lines. It is obvious that there will be as many influence lines to be drawn as there are sections to be considered. In the case of a truss-bridge there will be such a section at every panel-point.

A slight modification of the preceding results is to be made when the loads are applied to the beam or truss at panel-points only.

In Fig. 25 let 1, 2, 3, 4, 5, 6, and 7 be panel-points at which loads are applied, and let the load g be located at the distance z′ to the right of panel-point 5, also let the panel length be p. The reactions at 5 and 6 will then be

The reactions at A will then be

Hence the moment at any section X in the panel in question will be

Remembering that z-z′ is a constant quantity, it is at once clear that the preceding expression is the equation of a straight line, with M and z or z′ the variables. If z′ = 0, equation (e) becomes identical with equation (a), while if z′ = p, it becomes identical with equation (b). Hence the influence line for the panel in which the load is placed, as 5-6, is the straight line KL. It is manifest that when the load g is in any other panel than that in which the section X is located, the effect of the two reactions at the extremities of that panel will be precisely the same at the section as the weight itself acting along its own line of action. Hence the two portions AK and BL of the influence line are to be constructed as if the load were applied directly to the beam or truss, and in the manner already shown. The complete influence line will then be AKLB, and it shows that the existence of the panel slightly reduces the bending at any section within its limits. The panel 5-6, as treated, is that of a beam in which the bending moment will, in general, vary from point to point. If AB were a truss, however, X would always be taken at a panel-point, and no intercept between panel-points, as 5 and 6, would be considered.

101. Influence Lines for Shears both for Beams and Trusses.—The influence lines for shears in a simple beam, supported at each end, can be drawn in the manner shown in Fig. 25a. In that figure AB represents a non-continuous beam with span l supported from A. The reaction at A will be