The above is inaccurate in neglecting possible bending at the top of the pier. If the top of the pier in Fig. 93, be held against rotation, a bending moment will be produced in section A–B equal to M p = II The actual bending moment in the section is that produced by the eccentricity of the resultant of the thrusts of the arches against the pier, or /If, =Re. In order that no tendency to rotate exist, Re should not be less than The error clue to this cause may usually be made insignificant by careful design.
Methods of analyzing arches with elastic piers may be found in the works of and In a paper by A. C. Janni in the Journal of the Western Society of Engineers, May, 1913, a graphical method of analysis is outlined, by the use of the ellipse of elasticity, which may be applied to a system of arches with elastic piers. These methods are complicated and cannot be discussed here; they all involve assumptions which make it necessary to exercise care in their application.
175. Analysis by Influence Lines.—In important structures, other conditions of loading than those mentioned in the preceding para graphs may be desirable, and a more complete analysis may be ob tained by determining the effect of individual loads at the various points of loading, which is accomplished by using influence lines to determine the effect of a unit load at each load point. In open span drel arches, when the loads are brought upon the arch ring at defi Plain and Reinforced Concrete Arches, by J. Aldan, translated by D. B Steinman, New York, 1015.
2 Reinforced Concrete Construction, Vol. III, by George A. liool, New York, 1915.
nite points, by vertical walls or columns, this method may be easily applied.
Fig. 94 represents an arch SO feet long, 16 feet rise; depth at crown, 2 feet; at springing line, 2.S feet. It is reinforced with 1.6 of steel per foot of arch, placed 2.5 inches from both extrados and intrados. The loads are assumed to be applied through cross walls at points 10 feet apart.
The arch ring is divided into ten parts on each side of the crown, so that the ratio s/I is constant; s being the length of division and I the moment of inertia at the middle of the division. Using the notation of Section 167, the values of x and y for centers of the various divisions are as given in Table XXIII.
The thrusts and moments at any given section of the arch ring, due to each load, may now be found graphically (see Fig. 94). For this purpose, draw the force polygon, laying off 0-K=1.0, the unit load. From K, the value of P, is measured vertically, K-v= Ve, for each load, and horizontally, v-A, v-B, etc. The distance Me/Hc, measured vertically from the middle point of the crown sec tion, gives the point of application of the crown thrust, k-A, k-B, etc. The equilibrium polygon in each case consists of two lines inter
secting on the line of action of the loads and parallel to the correspond ing lines in the force polygon.
The thrusts upon any section of the arch ring due to each unit load may now be taken from the force polygon, while the moment is found by multiplying the value of for the given load by the vertical distance from the center of section to the equilibrium polygon.
Moments and thrusts at any section due to dead or live load at each load point may now he found by multiplying the values found for unit load by the amount of the load. If these be tabulated and combined, the maximum and minimum stresses may be obtained.
176. Analysis Using Arbitrary Divisions.—The method of analysis given in Art. 46 requires that the arch ring be so divided as to make s/I constant for all divisions. This simplifies the formulas used in obtaining values for and Lll,, but makes the lengths of divi sions vary greatly where the thickness of the arch ring increases from crown to springing line, and frequently gives very long divisions near the ends of the arch, which may sometimes introduce considerable error into the results.
A method of analysis based upon the principle of work in dellec tion is sometimes employed. This is demonstrated by Professor Hudson 1 and is applied to the analysis of the stresses in a conduit by Professor French 2 under the name of the method for indetermi nate structures. Practically the same formulas may be produced by the method of Art. 4G by leaving the term s/I as a variable in the for mulas.
If the constant E be eliminated from Formulas (4), (5), and (6) of Section 163, we have Combining these with Equations (10) and (11) of the same section, and solving we find compute the values of IIc, 1', and :1I, for the arch ring given in the example of Art. 47 with the loading employed in Section 1(iS. Fig. 95 shows the arch with divisions of equal length and the loads upon each division. Table XXV gives the coordinates of the centers of divisions, the value of s j I for the mid-section of each division, and combinations of these quantities required in the computations. Table XXVI gives the computations of the moments at centers of divisions, and of the terms in the formulas which include these moments. These computations might be somewhat shortened by expressing the loads in Kips of 1000 pounds and the moments as foot-kips.
These results are preferable to those obtained in Section 168 on account of the better division of the arch axis and the inclusion of a larger portion of the load in the moments. The labor required in the use of this method is not materially greater than that involved in the use of the ordinary method as given in Section 168.