A Graphical Method for Swing-bridges: A Rational and Easy Graphical Analysis of the Stresses in ...

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Preface

In the analysis of swing-bridges given in the following pages, no acknowledgment of indebtedness, or other reference, to any work which has appeared upon the subject, is therein made, as no book has been consulted in its prepai-ation, although much of the ground has been previously well explored; except that equation (1), of Case III, was taken without material change from Professor TV. II. Burr's able work on the " Stresses in Bridge and Roof Trusses." Otherwise the solution of continuity herein given is entirely the result of independent investigation.

In the Introduction no originality whatever has been attempted. The principles given are well known and established. The texts of other writers have been liberally drawn upon, where principles were found so well expressed as to warrant such quotation; and due credit has been given in each case.

It may be noticed that among the fundamental principles stated, no mention is made of the " Parallelogram of Forces." This term is thought to be inelegant in both its logic and its application. The Triangle of Forces concisely covers the same ground, without the use of superfluous lines.

The article on the analysis of the stresses in swing-bridges, of which the greater portion of these pages consists, was originally intended for publication in some engineering journal, and for such purpose to be complete in itself ; which fact explains why it contains matter which otherwise might have been more consistently embodied in the Introduction^ which was not thought of until later.

In the hope that this humble effort may contribute something of value to the constantly increasing knowledge of graphical methods of analysis, it is submitted to the perusal of those interested in the subject.

Jackson, Mich., 1893.

Introduction

The General Theory of Graphical

Statics.

Before taking up the analysis of the stresses in swing-bridges, which is the principal feature of this work, the general principles of Graphical Statics, or the graphical investigation of static force, will be briefly noticed, in order that those who are not familiar with the subject may understand the solution as given without being obliged to first study the general theory elsewhere. Those readers to whom the subject of graphical statics is familiar may omit these introductory pages and pass directly to the analysis for swing-bridges.

At the beginning it will be well to repeat a few well-expressed principles of equivalents in

Static Force.

" The resultant of two or more forces is a force which singly will produce the same mechanical effect as the forces themselves jointly.

'^ The original forces are called components.

" In all statical investigations the components may be replaced by their result ant, and vice versa,

^'The resultant of two unequal forces acting in opposite directions is a single force equal to their difference and acting in the direction of the larger.

" The resultant of any number of forces acting in the same right line is their algebraic sum.

" If three forces are in equilibrium, each will be equal to and opposite in direction to the resultant of the obhier two.

''If three concurring forces are in equilibrium, and a triangle be formed by lines drawn in their directions, the sides of the triangle, taken in order, will represent the forces.

'^ If any number of concurring forces be represented in magnitude and direction by the sides of a polygon, taken in order, they will be in equilibrium/^*

This last-expressed principle brings us directly to the subject of

Graphical Statics.

As lines have but one magnitude, length, they may be appropriately taken to represent forces by the distinct properties of magnitude and direction, and the laws of equilibrium of the forces so represented may be determined by applying the principles of Analytical Geometry.

Such geometrical analysis is called Graphical Statics.

It can be applied very successfully, even elegantly, to the investigation of the stresses in framed structures which

* Smith's Mechanics.

are designed to carry specific loads, such as bridge and roof trusses. The subject will here be treated rather toward such end, and not at all exhaustively.

The following principles will be taken as a basis :

1. If two forces which meet at a point be represented, in magnitude and direction, by straight lines drawn end to end consecutively, and respectively parallel to the actions of the forces which they represent, a third line drawn from the beginning of the first line to the unconnected end of the second will represent, in magnitude and direction, the resultant of the two original forces.

2. If the third line be drawn in the same position as before, but in the opposite direction, following around the triangle, it will represent a force in equilibrium with the two first-mentioned forces.

3. When any number of forces act at a point in any direction in the same plane, the force which will be in equilibrium with them can be obtained by drawing lines which represent the original forces in magnitude and direction, to follow each other consecutively end to end, as the sides of a rectilinear polygon; the remaining side to complete the polygon drawn to the starting-point will represent the force sought.

Note. — This is true in whatever order the forces are taken, but it is usually best to take them in regular order passing in either direction around the point. This practice will generally give the plainest figure and avoid confusion from crossing of lines.

4. If all the forces which meet at a point are known except two and the directiofis of those are known, their magnitudes can be found by first drawing the lines representing the known forces, then completing the polygon by drawing those rspresenting the unknown forces in their proper directions.

This is really the statement of a conclusion rather than a principle; two forces of known directions and unknown magnitudes are determined by substantially the same treatment as one force of which the magnitude and direction are both unknown; each case contains two unknown quantities, which may be determined from a figure having two dimensions, i.e., length and breadth — a plane figure. It is, however, stated as a principle, as it applies with especial value to the determination of the stresses in the members of bridge- trusses and similar structures having a singie system of bracing. Each force or stress is considered to act in a straight line between the centres of the connections of each member, and hence its direction is known.

It will be attempted to illustrate the foregoing principles sufficiently to give a clear understanding of them.

Let a-h and h-c (Fig. la) represent two forces which meet at a common point; their magnitudes are represented by the respective lengths of the lines, and the direction in which each is exerted is indicated by an arrow. As they cannot be made to take the form of a closed figure, i.e., as they cannot be so drawn that the end c of the line, b-Cy will be at the starting-point, a, they are not in equilibrium. The resultant of their combined force is in the direction a-c and of the amount represented by a line drawn between those points; and this resultant, a-6*, or the combined force of its components, will just be held in equilibrium by a force equal to it in magnitude and of the opposite direction, c-a. If the line representing this latter force be combined with the lines a-b and b'C, they will together form a closed triangle, as a-b'C (Fig. Id), Notice that the lines simply represent the forces in magnitude and direction; the point at which they meet is not shown.

The cord A-B-G (Fig. 2a) is attached and supported at A and C, and a weight, Wy is suspended from it at B, This weight is a force exerted vertically downward upon the point B and, with the forces exerted by the cord in the directions B'A and B-C, is in equilibrium at this point. The amount of the force W and its direction B- W are known, and the directions of the forces B-A and B-C are known. They must act in the directions of the cords in which they exist and away from the point By as the weight, Wy causes a tension or " pull '^ in the cord W-B, tending to draw the point B toward the weight, W, which can only be held in equilibrium by tensions in ^-^ and B'C, the amount of which forces may be determined by the application of principles 2 and 4. In what follows it will be noticed that the forces B-A and B-C must act in directions away from the point B in order to close the triangle. Draw a line which in magnitude and direction represents the force W, as a-c (Fig. 2d), then without removing the pencil draw lines in the directions in which the two unknown forces, ^-Cand B-Af act respectively with reference to the point ^, to just close at the starting-point, a; C'h and h-a are such lines and represent the forces sought, and n-h-c is the triangle of the forces which meet at^.

The forces in B-A and B-C, the amounts of which have just been found, are finally resisted at ^ and C by being attached to immovable objects at those points. The forces in B-A and B-O tend to draw the points A and C toward B, and those forces at A and C which resist this tendency are called the reactions.

Now as, practically, it is not always expedient to support a force or system of forces by inclined reactions, a member which is so designed as to resist compression may be introduced between A and C which will receive the horizontal* components of the forces in B-A and B'C so that their, vertical components only will be carried by the reactions, and the latter may be designed to simply resist a downward pressure. This compression member is represented by the line A-C, The modified figure represents a very simple form of truss.

* This is not a strictly correct expression, as the components will act along the direction of the line A-C,

By drawing the triangle of forces as before, the compressive force in A-C may be ascertained. Its direction is known; at A it exerts a pressure upon this point in the direction C-A. Then at A three forces meet: this compressive force in C'A, the tensile force in A-B, and the vertical reaction which is exerted upward. But the force in A-B has been found ; it is a-b of the triangle of forces (Fig. 2d), in which a-c, which represents the weight W, must also represent the sum of the vertical reactions at A and C. Now consider the forces acting at A: pass over a-b, in the triangle of forces, in the direction which the force in A'B acts upon A, then draw b-d, in the direction which the force in C-A acts upon this point, to a point from which a line, when drawn vertically upward, will close at a; b-d will represent the force in C-A, and d-a the reaction at A] c-d will represent W minus the reaction at A {d-a), or the reaction at C. Thus we have found all the forces acting in and upon the frame

A'B-C, The internal forces which exist in the members of a frame, as A-B-C, are called stresses, and the geometrical figure, the lines of which represent the stresses, as a-b-c-d, is called the stress diagram.

Fig. da represents substantially the same truss as Fig. 2a, except that in the former the load IF is transferred to the joint i? by a member under compression instead of tension.

Fig. 3d is the stress diagram for the same; it will be readily understood from its similarity to Fig. 2d, which has been explained.

Fig. 4a is the same truss inverted. It is the same as the truss shown in Fig. 3a, except that those members which in that truss were under a compressive stress, or in compression, are in tension in Fig. 4a, and vice versa. This figure represents a form of truss very commonly used for short-span bridges; and as this is a practical case, the stress diagram will be followed through in detail. The members of the truss are designated by figures, which in the diagram of the truss (Fig. 4a) are placed in the spaces, and at the angles in the stress diagram (Fig. 4^). In the latter the line between any two figures represents the stress in that member of the truss included between the same two characters in the diagram of the truss. The load W is situated at midspan, and by the principle of the lever we know that it is received in equal parts by each reaction. We will start with the reaction at A^ which equals one half the load and is the extel*nal force exerted upwards against the truss at this point. Consider, then, the forces acting upon the point A : lay off upwards on a vertical line the amount of this reaction, represented to any convenient scale, as 1-2 (Fig. 46?); draw 2-3 parallel to that member of the truss (^4-6), and in the direction which its stress (tension) is exerted upon the point A^ also 3-1 parallel to that member of the truss (B-A) and in the direction which its stress (compression) acts upon this point, to close at 1, the starting-point. The tension in A-b will be represented to the same scale by 2-3, and 3-1 will represent the compression in A-B.

Next consider the forces acting at the point B: retrace 1-3 in the opposite direction from which it was drawn, as the compression in A-B acts against B in the opposite direction from which it acts against A; draw 3-4 parallel to that member of the truss and in the direction which its stress (tension) '^ pulls ^' upon

B, and 4-1 parallel to that member of the truss and in the direction which the stress (compression) in it is exerted against this point, to close again at 1. In the stress diagram 3-4 represents the stress in that member of the truss; it is evidently equal to the load W and will scale that amount by the scale used in laying off 1-2. The stress represented by 1-4 equals that represented by 1-3. Next, considering the forces acting at

C, we retrace 1-4 and draw 4-5 and 5-1, which in a sense complete the stress diagram. The line 5-4 equals the line

2-3; this is correct, as tliey each represent the stress iu what is really the same member, A-C, The line 5-1 represents the reaction at C; it equals the reaction at A which at the beginning was found to be one half the load W and was represented by the line 1-2; evidently the line 5-2, which represents the sum of the reactions, equals the line 3-4, which represents the load. Each separate triangle in the stress diagram, the sides of which represent the forces acting at some point, is completed, or closed, by a line which returns to the starting-point. One of the many checks to graphics is here given by the fact that it must always be possible to trace each complete polygon, beginning and finishiTig at the starting-point, without removing the pencil. As many of the lines represent forces which act at two points, i.e., the two ends of a member, and hence belong to two triangles, they must be passed over twice in opposite directions.

This is true of all lines which represent the internal forces (i.e., the stresses in the members of the truss), when the complete stress diagram is drawn. It is also true of the line which represents the external forces (i.e., loads and reactions) if those forces are parallel, which is generally the case in a truss. The lines representing the resultants of the external forces, whatever may be the directions of the original forces, will always coincide, but that representing the resultant of the loads will be drawn in the opposite direction from that representing the resultant of the reactions.

As has been mentioned, the stress diagram has been in a sense completed; yet for the theoretically complete diagram we have still to consider the joint b. The forces in ^-Cand b-B are perpendicular to each other at this point and hence have really no influence upon each other; but it is a point at which forces meet, and it must be taken into consideration in the stress diagram if the latter be in every sense complete.

Now consider the forces which meet at the point b: in the stress diagram pass downward 2-5, the amount of the weight W, and in the direction which its force is exerted upon this point, then pass over 5-4, 4-3, and 3-2, the amounts aud directions which the forces in those members are exerted upon the same, closing on 2. Each line in the stress diagram has now been passed over twice in opposite directions and the graphical work is complete in every sense. To be thoroughly consistent with good logic the points should have been considered before the point B, in order to obtain the stress in the member B-b; but as it is evident that this stress is simply a tension equal to the load PT, and as also it is obtained in the diagram by drawing the polygon (in this case a triangle) for the point B, the point b was omitted till later. It could have been omitted altogether.

In fact, in this and most ordinary trusses of one span, the two portions between each end and the center are symmetrical and it is only necessary to draw the stress diagram for one half, although by drawing the complete diagram a convenient check is obtained. In this case the only portion of the stress diagram actually necessary to be drawn is the triangle 1-2-3.

Mr. E. H. Bow, one of the pioneers in graphical statics, calls the complete stress diagram the ^' Keciprocal Diagram of Forces." He states the "chief reciprocal relationship " thus : " Conesponding lines which meet in a point in the one figure form a closed polygon in the other;" and adds: "This is, however, too comprehensive, as it can seldom be altogether complied with." At each angle in the stress diagram meet the lines corresponding to the forces in those which surround or cut off a space in the diagram of the truss, and each space or polvgon in the former is surrounded by lines corresponding to the forces in those lines wiiich in the latter meet at a point. This mutual reciprocity is indicated by the positions of the characters in the system of numerals used to designate the lines in Figs. 4a and 4f/. Corresponding lines are included between like characters.

This system for designating the corresponding lines of the two figures was first used by Mr. Bow, and is described in his "Economics of Construction" in the following language:

" This plan of lettering consists in assigning a particular letter to each inclosed area or space in, and also to each space (enclosed or not) around or bounding the truss, and attaching the same letter to the angle or point of concourse of Imes which represents the area in the diagram of forces. Any linear part of the truss, or any line of action of an external force applied to it, is to be named from the two letters belonging to the two spaces it separates; and the corresponding line in the reciprocal diagram of forces, which represents the force acting in that part or line, will have its extremities defined by the same two letters.'^

In the diagrams which accompany these pages Mr. Bow's system of nomenclature has been used, excepting that Arabic numerals have been used instead of letters.

From this author will also be quoted the following explicit

"Directions for the Construction OF Diagrams of Forces.

'*1. Assign a letter to each enclosed area of the truss, also to each division of the surrounding space as separated by the lines of action of the external forces.

"2. The lines in the diagram are to be drawn parallel to the corresponding lines or parts of the truss figure.

** 3. The forces acting in lines radiating from a point in the truss must in the reciprocal diagram form a closed polygon.

" 4. The sides of any uncrossed space, triangle, or other polygon, around or in the truss, must always be represented by lines radiating from a point in the reciprocal diagram, and that point is to be named by the letter assigned to the space or polygon.

• •••••••

"7. All the external forces acting upon a truss or other erection must, taken together, be represented in the reciprocal diagram by the sides of a closed polygon." *

This method of graphical analysis is usually known as the Clerk Maxwell method, after the name of one of its earliest English investigators. The writer, however, prefers to designate it as the Direct Method. As it deals directly with the action of each force, this name seems to be consistent.

It will be well next to briefly notice that somewhat peculiar figure which has been most appropriately called

The Equilibrium Polygon.

This figure will here be defined as: The graphical representation of the

The closed polygon, however, may be a

\trht linfi

Straight line design of an ideal frame in which the chords are in such perfect equilibrium under the stresses from a given system of external forces that no web members are required.

This definition is, perhaps, unusual; but the writer believes it to be consistent and finds no objection to it. It includes the closing line and, in case of continuity, may include the pier moment, so as to form a complete polygon. The figure is sometimes called a Jointed Frame. It includes the figure quite commonly known as the Funicular Polygon, and also the inverted figure, often called the Linear Arch or Line of Resistance.

It will not be attempted here to consider the Equilibrium Polygon in a manner at all comprehensive, but merely to notice such special class and features of it as may be necessary to intelligently understand the analysis which follows. That figure only will be considered in which the chords of the load curve represent tension and the external forces are vertical. The polygon will be consideied to be composed of imaginary lines having no weight and exerting no influence, except that of resistance to the external forces.

The external forces which will be here considered will be those due to superimposed weights or loads, whose actions are exerted vertically downward, and those vertical reactions which support the actions of the loads when transfeiTcd by means of the internal forces of the frame.

The subject of inclined reactions will be noticed in a general way farther on, though not at any length.

Let AhcdE (Fig. 5a) represent a cord suspended from A and E and supporting loads at b, c, and t/; this cord being free to change its form will assume a certain definite position according to the relative magnitudes of the loads. It represents the tension members of the frame, which in this case compose the lower chord. In the example the loads have been taken as equal. This was done merely for convenience; it is never necessary. The line E-A represents the compression chord of the frame and is called the closing lin^, because it is always drawn to the starting point of the load curve and closes the polygon. It is not necessarily nor usually horizontal, though it is so in the present case. This chord receives all the horizontal stress from the tension chord, which has been designated as the load curve, so that the reactions are simply vertical. Vertical ordinates at the points of application of the loads, as h, c, and d, between the load curve and the closing line, will be proportional to the bending moments at the respective points due to the loads; they will, indeed, be definite factors of the bending moments, as will be hereafter shown. A cliange in the length of the cord will affect the lengths of the ordinates, but not their relative lengths. The entire system of loads thus sustained will react vertically at A and E on the principle of the lever.

The graphical analysis of the stresses of this frame is easily performed by drawing the stress diagram in the manner previously shown; and this stress diagram will be of peculiar value. Three forces meet at each joint of the frame, or angle of the polygon, and if, to start with, any one force be known, from the principles which have been given, the other two can each time be found by drawing the triangle of the forces. The loads being known, the reactions can be readily ascertained by simply applying the principle of the lever. The reaction at A will be taken for a beginning; R-1 (Fig. Dcl) represents this reaction, and its force is exerted upward. It may be drawn to any convenient scale. In this case, as the loads are equal and are taken at equal distances from each other and from the reactions — or, in other words, as the loads and panel lengths are equal — each reaction will equal one half the sum of the loads supported, or two and one half loads.

As this force is exerted upward at Ay pass upwards in the stress diagram its amount, R-l, drawn to any convenient scale, then draw 1-0 parallel to the line between the same two characters in the equilibrium polygon and in the direction which the cord " pulls ^' upon ^, then 0-R parallel to O-E of the equilibrium polygon and in the direction which its force is exerted upon A, to close on R; 1-0 and 0-R in the stress diagram will represent the stresses in the corresponding members of the equilibrium polygon; 1-0 will represent tension and O-R compression.

Now consider the joint b : pass over 0-1 of the stress diagram in the opposite direction from that in which it was drawn, as the force in that member acts upon the joint b in the opposite direction from which it is exerted upon A, then pass downward on the vertical line 1-2, the amount of the load at b, which acts downward at this point, and draw 2-0 parallel to that member of the frame and in the direction which it " pulls " upon this point, to close on 0; thus the stress in 2-0 or be is found. It will not be necessa^'y to here follow the stresses through each member of the polygon in detail, as by the aid of the reference characters they can be readily followed through the entire diagram if desired. The lines representing the stresses for the chord AbcclE all radiate from in the stress diagram. The lines in the stress diagram (Fig. bd) represent the stresses in those members of the frame (Fig. 5a) situated between the same characters. In the frame the characters are placed in the spaces, and at the angles in the stress diagram, as lieretofore explained. This completed stress diagram for the equilibrium polygon has been called the Polygon of Forces, or simply the Force Polygon ; 1-4 is usually called the load line- and the pole; those designations will be used hereafter. The amount that is horizontally distant from the load line is usually called the pole distance, and is often designated simply as H] in this case it equals 0-R, If a vertical ordinate of the equilibrium polygon be multiplied by this horizon 'i^nl pole distance, the product will be the bending moment at that point where the ordinate is taken. This fact is apparent when we consider the equilibrium polygon to represent a frame or truss which is in perfect equilibrium under the assumed system of loads. The pole distance, H^ equals the horizontal component of each chord stress, and the ordinates of the equilibrium polygon are the heights of the truss at each panel point. As the entire frame is in equilibrium simply under chord strains and therefore no web strain can exist, it follows that the horizontal component of the chord strains must be uniform throughout the truss; for it can be changed only by the introduction of web members. In a simple truss the bending moment at any point divided by the effective height of truss at that point, which is the lever of the resistance, will equal the horizontal component of the chord strain; and conversely, if we multiply H, the horizontal component of the chord strains in our ideal truss, by the height of truss (ordinate of the equilibrium polygon) at any panel point, the product will be the bending moment at that point due to the assumed loads.

Now instead of following this operation in the order given above we may reverse the process, and, by laying off the loads to some convenient scale on a load line, as 1-4 (Fig 5d), assuming any convenient pole distance and drawing the radiating lines 0-1, 0-2, etc., we can readily construct the force polygon wholly independent of the equilibrium polygon; then, by drawing the chords of the load curve for the latter parallel to the corresponding radial lines in the former, beginning at one reaction and taking them in order through to the second reaction, and finally drawing the closing line to the starting-point, the equilibrium polygon is drawn for the assumed loads.