tain in virtue of its being an equilibrium the center and distributed in the same polygon, and they would induce no bend- manner as the segments of uuʼ: for it ing moments if applied to the arch. is such a distribution of loads or presThe actual loads in general are different- sures which the rib can sustain or proly distributed. By Prop. VI the bending duce. A similar set of moments can be moments induced in the truss are those induced in the stiffening truss by lengthdue to the difference between the weight ening the posts between the rib and actually resting on the arch at each truss. point, and the weight of the same total amount distributed as shown by the segments of the line uu'.

When this deflection and the value of EI in the truss are known, these moments can be at once constructed by Now lay off a load line vv' made up methods like those already employed. of weights which are these differences A judicious amount of cambering of this of the segments of uu' and ww', taking kind is of great use in giving the struccare to observe the signs of these dif- ture what may be called "initial stiffferences. The algebraic sum of all the ness. The St. Louis Arch is wanting in weights vv' vanishes when the weights initial stiffness to such an extent that which rest on the piers are included, as the weight of a single person is sufficient appears from inspection of the construc- to cause a considerable tremor over an tion in the lower part of Fig. 10. The entire span. This would not have been construction above described will differ possible had the bridge consisted of an from that in Fig. 10 in one particular. arch stiffened by a truss which was anThe rib will not in general be parabolic, chored to the piers in such a state of and the loads which it will sustain in bending tension as to exert considerable virtue of its being an equilibrium poly-pressure upon the arch. This tension of gon will not be uniformly distributed, the truss would be relieved to some exhence the differences which are found as tent during the passage of a live load. the loading of the stiffening truss do not generally constitute a uniformly distributed load.

The horizontal thrust of the arch is the distance of uu' from b measured on the scale on which the loads are laid off, and the thrust along the arch at any point is length of the corresponding ray of the pencil between b and uu'. These thrusts depend only on the total weight sustained, while the bending moments of the stiffening truss depend on the manner in which it is distributed, and on the shape of the arch.

Having determined thus the weights applied to the stiffening truss, it is to be treated as a straight girder, by methods previously explained according to the way in which it is supported at the piers.

The effect of variations of temperature is to make the crown of the arch rise and fall by an amount which can be readily determined with sufficient exactness, (see Rankine's Applied Mechanics Art. 169). This rise or fall of the arch produces bending moments in the stiffening truss, which is fastened to the tops of the piers, which are the same as would be produced by a positive or negative loading, causing the same deflection at

The arch rib with stiffening truss, is a form of which many wooden bridges were erected in Pennsylvania in the earlier days of American railroad building, but its theory does not seem to have been well understood by all who erected them, as the stiffening truss was itself usually made strong enough to bear the applied weights, and the arch was added for additional security and stiffness, while instead of anchoring the truss to the piers and causing it to exert a pressure on the arch, a far different distribution of pressures was adopted. Quite a number of bridges of this pattern are figured by Haupt* from the designs of the builders, but most of them show by the manner of bracing near the piers that the engineers who designed them did not know how to take advantage of the peculiarities of this combination. This further appears from the fact, that the trussing is not usually continuous.

A good example, however, of this combination constructed on correct principles is very fully described by Haupt on pages 169 et seq. of his treatise. It is a wooden bridge over the Susquehanna River, 5 miles from Harrisburg on the

*Theory of Bridge Construction. Herman Haupt, A.M.

New York. 1853.

of the extrados, and from any convenient point on bb, as b, draw lines to d and q. These will enable us to find the ordinates bd of the ellipse of the extrados, from those of the circle, by decreasing the latter in the ratio of bg to bd. By this means, as many points as may be desired, can be found upon the intrados and extrados; and these curves may then be drawn with a curved ruler. We can use the arch ring so obtained for our construction, or multiply the ordinates by any convenient number, in case the arch is too flat for convenient work. Indeed we can use the semicircular ring itself if desirable. We shall in this construction employ the arch ring ad which has just been obtained.

We shall suppose that the material of the surcharge between the extrados and a horizontal line tangent at d causes by its weight a vertical pressure upon the arch. That this assumption is nearly correct in case this part of the masonry is made in the usual manner, cannot well be doubted. Rankine, however, in his Applied Mechanics assumes that the pressures are of an amount and in a direction due to the conjugate stresses of an homogeneous, elastic material, or of a material which like earth has an angle of slope due to internal friction. While this is a correct assumption, in case of the arch of a tunnel sustaining earth, it is incorrect for the case in hand, for the masonry of the surcharge needs only a vertical resistance to support it, and will of itself produce no active thrust, having a horizontal component.

This is further evident from Moseley's principle of least resistance, which is stated and proved by Rankine in the following terms:

by them; and will, therefore, not increase beyond the least amount capable of balancing the active forces."

A surcharge of masonry can be sustained by vertical resistance alone, and therefore will exert of itself a pressure in no other direction upon the haunches of the arch. Nevertheless this surcharge will afford a resistance to horizontal pressure if produced by the arch itself. So that when we assume the pressures due to the surcharge to be vertical alone, we are assuming that the arch does not avail itself of one element of stability which may possibly be employed, but which the engineer will hesitate to rely upon, by reason of the inferior character of the masonry usually found in the surcharge. The difficulty is usually avoided, as in that beautiful structure, the London Bridge, by forming a reversed arch over the piers which can exert any needed horizontal pressure upon the haunches. This in effect increases by so much the thickness of the arch ring at and near the piers.

The pressure of earth will be treated in connection with the construction for the Retaining Wall. On combining the pressures there obtained with the weight, the load which a tunnel arch sustains, may be at once found, after which the equilibrium polygon may be drawn and a construction executed, similar in its general features to that about to be employed in the case before us.

Let us assume that the arch is loaded with a live load extending over the left half of the span, and having an intensity which when reduced to masonry of the same specific gravity as that of which the viaduct is built, would add a depth df to the surcharge. Now if the number of parts into which the span is divided be considerable, the weights which may be supposed to be concentrated at the points of division vary very approximately as the quantities of the type af. This approximation will be found to be sufficiently exact for ordinary cases; but should it be desired to make the construction exact, and also to take account of the effect of the obliquity of the joints in the arch ring, the reader will find the For the passive forces being caused by method for obtaining the centers of the application of the active forces to gravity, and constructing the weights, in the body or structure, will not increase Woodbury's Treatise on the Stability of after the active forces have been balanced the Arch pp. 405 et seq. in which is

"If the forces which balance each other in or upon a given body or structure, be distinguished into two systems, called respectively, active and passive, which stand to each other in the relation of cause and effect, then will the passive forces be the least which are capable of balancing the active forces, consistently with the physical condition of the body or structure.

1

given Poncelet's graphical solution of the arch.

19

scribed limits near the crown and near the haunches. Let us assume e at the With any convenient pole distance, as middle of the crown, e' at the middle of one half the span, lay off the weights. a'd', and e, near the lower limit on ad We have used b as the pole and made This last is taken near the lower limit, bw, the weight at the crown because the curvature of the left half of (af+ad) = b'w', w ̧w2 = a1f12 ww the polygon is more considerable than af, etc. Several of the weights near the other, and so at some point between the ends of the span are omitted in the it and the crown it may possibly rise to Figure; viz., ww, etc. From the force the upper limit. The same consideration polygon so obtained, draw the equili- would have induced us to raise e' to the brium polygon c as previously explained. upper limit, were it not likely that such The equilibrium polygon which ex- a procedure would cause the polygon to presses the real relations between the rise above the upper limit on the right loading and the thrust along the arch, is of e'. evidently one whose ordinates are proportional to the ordinates of the polygon

C.

Draw the closing line kk through ee', and the corresponding closing line hh through cc,', and decrease all the ordi

3

It has been shown by Rankine, Wood-nates of the type he in the ratio of hb to bury and others, that for perfect stability, -i.e, in case no joint of the arch begins to open, and every joint bears over its entire surface, that the point of application of the resultant pressure must everywhere fall within the middle third of the arch ring. For if at any joint the pressure reaches the limit zero, at the intrados or extrados, and uniformly increases to the edge farthest from that, the resultant pressure is applied at one third of the depth of the joint from the farther edge.

The locus of this point of application of the resultant pressure has been called the "curve of pressure," and is evidently the equilibrium curve due to the weights and to the actual thrust in the arch. If then it be possible to use such a pole distance, and such a position of the pole, that the equilibrium polygon can be inscribed within the inner third of the thickness of the arch ring, the arch is stable. It may readily occur that this is impossible, but in order to ensure sufficient stability, no distribution of live load should be possible, in which this condition is not fulfilled.

We can assume any three points at will, within this inner third, and cause a projection of the polygon c to pass through them, and then determine by inspection whether the entire projection lies within the prescribed limits. In order to so assume the points that a new trial may most likely be unnecessary, we take note of the well known fact, that in arches of this character, the curve of pressure is likely to fall without the pre

ke, by help of the lines bn and bl, in a manner like that previously explained. For example h,c,=n ̧o,, and 10,=k ̧€ ̧• By this means we obtain the polygon e which is found to lie within the required limits. The arch is then stable: but is the polygon e the actual curve of pressures? Might not a different assumption respecting the three points through which it is to pass lead to a different polygon, which would also lie within the limits? It certainly might. Which of all the possible curves of pressure fulfilling the required condition, is to be chosen, is determined by Moseley's principle of least resistance, which applied to the case in hand, would oblige us to choose that curve of all those lying within the required limits, which has the least horizontal thrust, i.e. the smallest pole distance. It appears necessary to direct particular attention to this, as a recent publication on this subject asserts that the true pressure line is that which approaches nearest to the middle of the arch ring, so that the pressure on the most compressed joint edge is a minimum; a statement at variance with the theorem of least resistance as proved by Rankine.

Now to find the particular curve which has the least pole distance, it is evidently necessary that the curve should have its ordinates as large as possible. This may be accomplished very exactly, thus: above e, where the polygon approaches the upper limit more closely than at any other point near the crown, assume a new position of e, at the upper limit; and be

low e where it approaches the lower at the most exposed edge a factor of only limit most nearly on the right, assume a 3 instead of 5. new position of e, at the lower limit. At the left e, may be retained. Now on passing the polygon through these points it will fulfill the second condition, which is imposed by the principle of least resist

ance.

A more direct method for making the polygon fulfill the required condition will be given in Fig. 18.

It is seen in the case before us, the changes are so minute that it is useless to find this new position of the polygon, and its horizontal thrust. The thrust obtained from the polygon e in its present position is sufficiently exact. The hori

zontal thrust in this case is found from

the lines bn and bl. Since 2vv, is the horizontal thrust, i.e. pole distance of the polygon c, 2vv, is the horizontal thrust of the polygon e.

By using this pole distance and a pole properly placed, we might have drawn the polygon e with perhaps greater accuracy than by the process employed, but that being the process employed in Figs. 2, 3, etc., we have given this as an example of another process.

the

The joints in the arch ring should be approximately perpendicular direction of the pressure, i.e. normal to the curve of pressures.

It may be desirable in a case like that under consideration, to discuss the changes occuring during the movement of the live load, and that this may be effected more readily, it is convenient to draw the equilibrium polygons due to the live and dead loads separately. The latter can be drawn once for all, while the former being due to a uniformly distributed load can be obtained with

facility for different positions of the load. The polygon can be at once combined into a single polygon by adding the ordinates of the two together. Care must be taken, however, to add together only such as have the same pole distance. In

case the construction which has been

given should show that the arch is unstable, having no projection of the equilibrium polygon which can be inscribed within the middle third of the arch ring, it is possible either to change the shape of the arch slightly, or increase its thickness, or change the distribution of the loading. The last alternative is usually the best one, for the shape has been chosen from reasons of utility and taste, and the thickness from consideration of the factor of safety. If the center line of the arch ring (or any other be considered to be an equilibrium polyline inscribed within the middle third) gon, and from a pole, lines be drawn With regard to what factor of safety parallel to the segments of this polygon, is proper in structures of this kind, all a weight line can be found which will engineers would agree that the material represent the loading needed to make at the most exposed edge should never the arch stable. If this load line be be subjected to a pressure greater than compared with that previously obtained, one fifth of its ultimate strength. Owing to the manner in which the pressure is assumed to be distributed in those foints where the point of application of the resultant is at one third the depth of the joint from the edge, its intensity at this edge is double the average intensity of the pressure over the entire joint. We are then led to the following conclusion, that the total horizontal thrust (or pressure on any joint) when divided by the area of the joint where this pressure is sustained ought to give a quotient at least ten times the ultimate strength of the material. The brick viaduct which we have treated is remarkable in using perhaps the smallest factor of safety in any known structure of this class, having

it will be readily seen where a slight additional load must be placed, or else a hollow place made in the surcharge, such as will render the arch stable. In general, it may be remarked, that an additional load renders the curvature of the line of pressures sharper under it, while the removal of any load renders the curve straighter under it.

The foregoing construction is unrestricted, and applies to all unsymmetrical forms of arches or of loading, or both. As previously mentioned, a similar construction applies to the case of an arch sustaining the pressure of water or earth; in that case, however, the load is not applied vertically and the weight line becomes a polygon.