THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF John S.Prell WORKS OF PROF. M. A. HOWE PUBLISHED BY JOHN WILEY & SONS. The Design of Simple Roof-trusses in Wood and Steel. With an Introduction to the Elements of Graphic Statics. Second edition, revised and enlarged. 8vo, vi + 159 pages, 87 figures and 3 folding plates. Cloth, $2.00. Retaining-walls for Earth. Including the Theory of Earth-pressure as Developed from the Ellipse of Stress. With a Short Treatise on Foundations. Illustrated with Examples from Prac- tice. Fourth edition, revised and enlarged. lamo, cloth, $1.85. A Treatise on Arches. Designed for the use of Engineers and Students in Technical Schools. Second edition, revised and en- larged. 8vo, xxv -f 369 pages, 74 figures. Cloth, $4.00. Symmetrical Masonry Arches. Including Natural Stone, Plain-concrete, and Rein- forced-concrete Arches. For the use of Technical Schools, Engineers, and Computers in Designing Arches according to the Elastic Theory. 8vo, x + 170 pages, many illustrations. Cloth, $2.50. A TREATISE ON ARCHES. DESIGNED FOR THE USE OF ENGINEERS AND STUDENTS IN TECHNICAL SCHOOLS. MALVERD A. HOWE, C.E., Professor of Civil Engineering, Rose Polytechnic Institutes Member of American Society of Civil Engineers. SECOND EDITION, REVISED AND ENLARGED. SECOND THOUSAND. JOHN S. PR,_L Gvil & Mechanical Engineer. SAN FBANCISCO, CAL. NEW YORK: JOHN WILEY & SONS. LONDON: CHAPMAN & HALL, LIMITED. 1911 JOHJM S. PRELL Qvil & Mechanical Engineer. SAN FRANCISCO, CAU Copyright, 1897, 1906, BY MALVERD A. HOWE. THE SCIENTIFIC PRESS IERT DHUMMONO BROOKLVN, Library PREFACE. THE theory of the elastic arch as developed in the follow- ing pages is based upon four fundamental equations demon- strated by Weyrauch in 1879. From these equations have been deduced formulas similar to those commonly given in American text-books, but in a simplified form for practical use. In addition to these a large number of general formulas have been introduced, many of which are new. In Chapter V an attempt has been made to give a set of general formulas which can be applied to any symmetrical arch either fixed or hinged, and subjected to either vertical or horizontal loads. These formulas readily reduce to the com- mon forms, and can be applied in their integral form to any symmetrical arch when the equation of the axis and the law of variation of the moments of inertia of the cross-sections are known. In many cases the reduction of the integrals to a simple form for a given case would be complicated and perhaps impossible ; for such cases these formulas are given in their summation form when they apply to any symmetrical arch subjected to any loading. The effect of the axial stress, which is usually neglected by American authors, is thoroughly discussed, exact as well as approximate formulas being given for all cases likely to iii 737388 ngineering Library iv PREFA CE. occur in practice. It is shown that in flat arches fixed at the ends the effect of this stress should not be neglected if economy of material is considered. Formulas for vertical and horizontal loads are deduced for each case considered, making it possible easily to treat loads making any angle with the axis of the arch. The effect of a couple is discussed, and general as well as special formulas given. Changes of temperature and of shape have been con- sidered, and when not too complicated, formulas for special cases are given. Masonry arches are considered, and the many difficulties and inaccuracies of the common methods of treatment pointed out. With a little good judgment it is easy to so design a masonry arch that the stresses will practically follow the laws demonstrated for the elastic arch. This has been experimentally shown by the " Austrian Experiments" and by many large arches designed and erected by European engineers. Alexander and Thomson's method for designing seg- mental masonry arches has been given as being the most consistent of the many methods which assume all loading due to material to act as vertical forces upon the arch. It is hoped that the practising engineer, who has, as a rule, little time to study mathematical demonstrations or to search through several pages of transformations for a desired formula, will appreciate the collection in simple form (Chapter II) of all of the necessary formulas likely to be needed in practice, and also the ease and celerity with which they can be applied, with the aid of the tables, to the case in hand. A fair trial of the summation formulas given in the same chapter will, it is believed, lead to the adoption of metal arches more artistic in form than the usual American type. PREFA CE. V These summation formulas are readily applied in the design- ing of masonry arches. Nearly all of the formulas given have been deduced for this treatise by two radically different methods. Many of these formulas are old, and while it was desired to give full credit in every particular, it was not found either expedient or possible to do so for each form. The tables were carefully computed, and when possible by the method of differences, each tenth value being checked by direct computation. The demonstrations are believed to be sufficiently simple to be easily followed by senior students in Technical schools. With the aid of the tables, class problems can be solved which otherwise would be impossible on account of the time required where direct computation of the various terms must be resorted to. The author will esteem it a favor if any errors that may be found are at once brought to his notice. M. A. H. TERRE HAUTE, May 1897. NOTE. IN this second and enlarged edition the errors in the for- mulas and example which were discovered in the first edition have been corrected, and it is believed that very few if any errors of importance remain. Three appendices have been added which consider the summation formulas in a simplified form and also the summation formulas as applied to unsym- metrical arches. The tables of arch data have been rearranged and brought up to date, and in addition one reference has been made, for each item, to a publication where a more complete description may be found. M. A. H. TERSE HAUTE, IND., July, 1906. NOTE TO THE SECOND THOUSAND OF THE SECOND EDITION. IN this edition Appendix I has been rewritten, and Table XXX brought up to date. M. A. H TERRE HAUTE, IND., June, 1911. vi JOHfl S. PRELL Civil & Mechanical Engineer. SAN FRANCISCO, CAL. TABLE OF CONTENTS. CHAPTER I. GENERAL PRELIMINARY FORMULAS. PAGB Deformation Formulas Axial-stress Terms Distribution of Stress upon any Radial Section of the Elastic Arch Extreme Fibre-stresses Distribution of Stress when Arch has Two Flanges connected by Web- bracing Location of Resultant Pressure for Like Stresses over Entire Section General Relations between the External Forces Equilibrium Polygons for Vertical and Horizontal Loads I CHAPTER II. FORMULAS FOR PRACTICAL USE. Symmetrical Parabolic Arches -with Two Hinges Vertical Loads with Effect of Axial Stress neglected Vertical Loads with Effect of Axial Stress included Horizontal Loads with Effect of Axial Stress neglected Horizontal Loads with Effect of Axial Stress included Temperature Change of Length in Span. Symmetrical Parabolic Arches -without Hinges Vertical Loads with Effect of Axial Stress neglected Vertical Loads with Effect of Axial Stress included Horizontal Loads with Effect of Axial Stress neglected Horizontal Loads with Effect of Axial Stress included Temperature Change of Length in Span.- Sym- metrical Circular Arches with Two Hinges Vertical Loads with Effect of Axial Stress included Vertical Loads with Effect of Axial Stress neglected Horizontal Loads with Effect of Axial Stress neglected Horizontal Loads with Effect of Axial Stress included Temperature Change of Length in Span. Symmetrical Circular Arches without Hinges Vertical Loads with Effect of Axial Stress neglected Hori- zontal Loads with Effect of Axial Stress neglected Temperature with Effect of Axial Stress neglected. Summation Formulas for Sym* vii Vlll TABLE OF CONTENTS. PACK metrical Arches of any Regular Shape and Cross-section Vertical Loads Horizontal Loads Temperature Effect of Axial Stress. . . . . . 2O CHAPTER III. PARABOLIC ARCHES WITH THE MOMENTS OF INERTIA VARYING ACCORDING TO THE RELA TION Ed COS = A CONSTANT. General Relations General Formulas for Symmetrical Arches Symmetri- cal Arch with Two Hinges Vertical Loads Change of Shape due to the Action of Vertical Loads Horizontal Loads Change of Shape due to Horizontal Loads Temperature Change of Length in Span Uni- form Loads Sinking of Supports. Symmetrical Arch -without Hinges Vertical Loads Change of Shape due to Vertical Loads Horizontal Loads Change of Shape due to Horizontal Loads Tem- perature Change of Length in Span Sinking of Supports Uniform Loads. Formulas for HI, MI, Vi , x , xi , x t , y<> , y\ , y* , etc 52 CHAPTER IV. General Relations Symmetrical Arches Symmetrical Arch with Two Hinges Vertical Loads Change of Shape due to Vertical Loads Horizontal Loads Temperature Change in Length of Span Sinking of Supports. Symmetrical Arch without Hinges Vertical Loads Horizontal Loads Temperature Effect of Axial Stress. Formulas Pi, .to , etc CHAPTER V. SYMMETRICAL ARCHES HAVING A VARIABLE MOMENT OF INERTIA. Symmetrical Arch without Hinges Demonstration of General Formulas for MI and HI Vertical Loads Horizontal Loads Temperature. Symmetrical Arch with Two Hinges Demonstration of General For- mulas for Hi Vertical Loads Horizontal Loads Temperature. Summation Formulas for Arches with and without Hinges Vertical Loads Horizontal Loads Temperature. Symmetrical Arch with a Hinge at the Crown Vertical Loads Horizontal Loads Temperature Parabolic Arch with a Hinge at the Crown. Symmetrical Arch with Three Hinges Vertical Loads Horizontal Loads no TABLE OF CONTENTS. ix CHAPTER VI. COMPARISON OF FOUR TYPES OF ARCHES. PAGE Comparison of the Values of Hi, V\ , Mi, etc., for Four Types of the Parabolic Arch Relative Values of V x and M x for Symmetrical Parabolic Arches with and without Hinges Comparison of Tempera- ture Effects upon Four Types of Symmetrical Parabolic Arches Comparison of Maximum Stresses for Three Types of Parabolic Arches Comparison of Weights 145 CHAPTER VII. ) APPLICATIONS. Point of Application of Vertical and Horizontal Loads Wind Loads- Maximum Stresses Character of Reactions Co-ordinates of the Equilibrium Polygon Bending-moments at Supports Series of Examples illustrating the Applications of Formulas given in Chapter II Effect of Axial Stress 159 CHAPTER VIII. APPLICATION OF GENERAL SUMMATION FORMULAS TO ARCHES HAVING A HINGE AT EACH SUPPORT. Bridge over the Douro in Portugal Data Computation of HI for Vertical Loads Comparison of Values of H\ with those obtained by Seyrig Values of H\ tor Several Distributions of Moving Loads Stress Di- agram for Load over all 182 CHAPTER IX. APPLICA TION OF GENERAL SUMMA TION FORMULAS TO ARCHES WITHOUT HINGES. Parabolic Arch with cos

{ = the angular distance of the right support from the crown. AV A v etc., = value of A, will be found in Tables I, II, etc., respectively. Al = small finite change in /. A(f> = small finite change in 0. A(j) a = small finite change in . A(f>i = small finite change in 0/. Ax a finite value of dx. Ay = a finite value of dy. As = a finite value of ds. X 2 = algebraic sum up to the section x. MASONRY ARCHES. NOMENCLATURE USED IN ALEXANDER AND THOMSON'S METHOD. b = distance of directrix to centre of described circle. 2c = clear span of arch. d = -distance of directrix from soffit of arch at cfowri. ' * e = base of Naperian system of logarithms. k = the clear rise of arch. m = the parameter of catenary. XXl'i NOMENCLA TURE. t o = depth of arch-ring at the crown. t s = depth of arch-ring at the skew-backs. w = weight of a unit mass of masonry. r = ratio of transformation = Vs. R l = the radius of the described circle. R, = the radius of the three-point circle. x and y = general co-ordinates. x l and y l = co-ordinates of the nose of a two-nosed catenary. x, and 7, = the co-ordinates of the point where the two- nosed catenary is cut by the three-point circle. y, = the ordinate of the two-nosed catenary at the crown. K e = the ordinate of the described circle at the crown. *.=Jo-F . a ......... (40) N x = V x sin + H x cos ......... (42) *). (41) . (49) ^). (47) (50) M, M, ^- and j, = -r + r ........ (51) , a = r and ^ = r .......... (54) A TREATISE ON ARCHES. CHAPTER I. GENERAL PRELIMINARY FORMULAS. DEFORMATION FORMULAS.* LET Fig. i represent a portion of an elastic arch; then the relation between the length of any fibre between two adjacent radial sections can be expressed in terms of the length of the neutral fibre (limited by the same radial sections) by. the equation ds n = ds -f- n sin ( d<$) ds nd. . . . (i) * The formulas and demonstrations in this article are essentially the same as given by Prof Weyrauch in " Theorie der Elastigen Bogentr^ger " (Milncben, 1879). v A TREATISE ON ARCHES. Now suppose some circumstance, as the application of a load, changes the lengths of these fibres, and let s become s + As, s n become s n -f- As n , etc., as shown in Fig. 2. Then we have for the new condition Combining (i) and (2), dAs H = dAs ndA$ t or dAs n ~dAs dAs (2) (3) (4) where dAs M represents the change in magnitude of ds n and dAs that of ds. Let F' represent the intensity of the force necessary to change the magnitude of ds n by the amount dAs H , and let E represent the modulus of elasticity of the material. Then -^r = 2r (5) If the force F' is due to a change of temperature, and e rep- GENERAL PRELIMINARY FORMULAS. 3 resents the coefficient of expansion per degree of change, then for an increase of temperature of t we have Let N n be the intensity of any normal force acting upon the fibre s n ; assuming that this force acts at the same time with the change of temperature but of opposite effect, then we have from (5) and (6) From (i) and (2), dAs n dAs ndAtf) ds n ~ ds nd(f> Substituting (8) in (7) and solving for N H , we obtain (8) ds- or ~dAs (9) + Eet. . . (10) H ~ds~ d(f> i Now ds = R sin ( defy = Rd, or -r- = -r,; hence dAs R Let /' represent the area of the fibre s n , and N x the resultant normal force acting upon the section. Then 4 A TREATISE ON ARCHES. If we take the centre of moments on the axis of the arch at the section x, the moment of the radial force upon the section will be zero. If x is the distance of the point of application of the force N x from the axis, we have =themoment of the external forces acting uponthe section ; then from (12) we can write f'n^R dAsf'nR 'n.. . . (13) Let R2~- = Wand 2f = F x ; then 2f'n = o; hence in (12) f>nR V W N Substituting these values in (12) and solving ior~ t we obtain W or N x ( dA . i dAs W (13) reduces to M x ('dA n . dAs ) W (15) GENERAL PRELIMINARY FORMULAS. From (15) ( dA . i dAs \ M x \~~ds~^~ Tt~ds \ ~~ EW Substituting this in (14), N * - _ M* __ dAs or Substituting (16) in (15) and reducing, we have \ N >M X \ i 4 _M x _et_ N *- J r- -- Substituting (16) and (17) in (11), and reducing, we have M x M x nR From Fig. 2, d(x 4- Ax) - d(s 4- As) cos (0 4- J0) ; d(y 4~ Ay) = ^ 4~ ^) sl " n (0 4~ ^0) but ^ xf^ cos (04-J0)=cos cos A(b sin sm J0 = A) ; or, from (16), Ydx -vAQdy. . . (21) A TREA TISE ON ARCHES. But / YA(f)dy can be neglected in comparison with the terms preceding. Hence ...... (22) In a similar manner, (23) From (16) and (17), , ........... (24) and A$=fxds ............ (25) These four equations, (22), (23), (24), and (25), completely determine the effect of any change of position of any point in the axis of the arch when X, Y, and the equation of the axis of the arch are known. The expressions for X, Y, and N H can be simplified by replacing Wby 6 = ~2f'n* = the moment of inertia of the cross- section. Since in the expression W = R2 " , R is usually n -\- K very large in comparison with n, no material error results from the change. In (16) is a very small quantity, and consequently we can neglect and " ( ! 7) 02 A- anc * i^ can be omitted. K rLr x K. In (18) -^ can be neglected and - - be assumed to Kr x n -j- K equal n. GENERAL PRELIMINARY FORMULAS. 7 Making these modifications and collecting our formulas, we have finally (d) The term containing N x shows the effect of the axial stress, which is usually neglected in the common investigation of the problem. In many cases the influence of this stress is of little or no importance, but in very flat arches it should not be neglected. Omitting the terms containing N x greatly simplifies the deduction of equations for special forms of arches, and also the solution of these equations in the determination of the reactions, bending-moments, etc. Omitting the terms containing N x , we have Ax= J* AQdy + effdx ; .... (aa) Ay =f A &,, N l and N x have the same sign. ^ o< _|_^ u N^ and N x " " " " 6 FIG. 4. Hence when x 9 > ^ and < ^ the entire section is sub- jected to the same kind of stress. To illustrate : Suppose the section to be rectangular ; then F = bh and & - ^bh\ and IO A TREA TISE ON ARCHES. N = o when I 4- - o, or x^ - = k~ ; or the k 6 resultant stress acting upon the section must cut the sec- tion at a point distant from the axis one sixth the depth of the section and below the axis (see Fig. 4). Evidently, if N x were applied above the axis and x^ = -, N^ = o and N l = 2p . Hence, in order that all parts of a rectangular section be sub- jected to the same kind of stress, the resultant stress N x must be applied within the middle third of the section. Adding (31) and (32), &> + &, = 2/ -i- A*o( g i-*). If a, = a v Returning to (e), From Fig. 3, letting Q represent the force whose intensity is uniformly varying, we have But M x 7V> = QA t ', therefore e e 2f'n (35) (36) which completely determines the arm of the couple whose moment is Q/i . Now since the intensities of the force Q vary directly with n, the intensity at the axis of the arch must be GENERAL PRELIMINARY FORMULAS. II zero, and the application of Q be // from the axis as indicated in Fig. 3. ARCHES HAVING TWO FLANGES OR CHORDS. In case the arch is composed of two flanges connected by a thin web or by struts and ties, it is customary to consider the material of each flange concentrated at its center of gravity, and that the flanges resist all stresses excepting radial stresses From Fig. 5, or _ . xt (37) ft! Also and FIG. 5. ,, _ NJi, - x n N x _ h,N x - M x ' (38) Q and Q" will be of the same kind as long as -f- x is less than h^ and x^ less than //,,, or N x must lie between Q and VALUES OF X FOR VARIOUS SECTIONS. The following table contains the maximum values which x can have when the entire cross-section is subjected to the same kind of stress for the various sections shown. A TREA TISE ON ARCHES. *=* = '. = 4 , = #* = \b, = a. = a.= 1.414 t-2py F = 2.598^, *, = *,= o.866, = 0.5413^, r' = 0.2083^, F = 2.598^', a, = a t = b, 9 = 0.5413^, r 1 = 0.2083^*, ** = 0.2083^. d!, = tf, = O.924^, 0.638^*, r' = 0.2256^', x a = 0.2446. F =&&-&&, e h 12 bk - i M- - bfc o - 1 - Z7 ~T7 TT~' o// M b.h. GENERAL PRELIMINARY FORMULAS. 9 F- bh + bh - a - h * -r . i #t #, 2 > 12 //-[- ^/^ ' "*" 6^ bh -\- bji^ ' Si p - bh (I b U h 12 bh -(b- b,)h, ' (-& 4 6/z bh-(b- b^/i, ' n F=-d* = 0.7854^', a, = a,= -, 6 = 0.049 1 = 0.062 5 r * = 0.062 5(^'+^,*), SB F = 0.78 5 4#, ^, = ., = -, ^ = 0.049 1 b/i 3 , r 2 = 0.062 5 //', X = O.\2$k. F = 0.78 54(/* *//,), a t =a t = -, _,_ 0.125^' ^,^ 1 3 -t- -L > / / / ll Oil 0/2, A TREATISE ON ARCHES. GENERAL RELATIONS BETWEEN THE EXTERNAL FORCES. In Fig. 6 let ABC represent the axis of any elastic arch, and P and Q the vertical and horizontal components respec- FIG. 6. tively, of any load applied at a point having the co-ordinates a and b ; then for equilibrium we have x = l -2- x>a V x - V, -i/ 7 ; x>a M x = M, + F^ - Hj - P& - a) Referring to Fig. 7, A 7 ; = V x sin + H x cos ; . , = ^ cos //, sin ; . V x tan ? = - (39) (40) (42) (43) (44) ; 3 . ( 45 ) GENERAL PRELIMINARY FORMULAS. Differentiating (41), tan 0. FIG. 7. Since dx ds cos 0, we have from (43) Also, from (40), V x = V, - 2P. Hence =V X -H, tan + i" tan .# = T; + H n tan - (^i - 10 tan Therefore dM, ds = T, (46) Making x = / and j = in (41), Af, becomes M 9t and we have 1 6 A TREATISE ON AKCffES. Solving this for J 7 ,, we obtain Making x = I in (40), and combining with (47), we have Collecting the equations which will be employed in the investigation of special cases, we have r M = V, - 2P\ x>a. . . . . N x V x sin + H x cos 0. ... M l = M t VJ + Hf + 2P(t a) V, j{ M t M, + H,c + 2P(l a) (39) (40) (42) (40 (49) (47) -*)}. (48) ORDINATES LOCATING THE EQUILIBRIUM POLYGONS. (a) Vertical Components. In Fig. 8 let ABC represent the axis of any elastic arch and let a single vertical load (corresponding to the vertical component of any load) be applied at B. Thas load causes GENERAL PRELIMINARY FORMULAS. I/ the reactions R^ and R t and the moments M l and M t at A and C respectively. This condition can be represented graphically by the equilibrium polygon GEK, which must be so situated V V that Hji = M, , H^(y^ c) = M,, tan /?, = -j~, and tan /3 t = -~. From Fig. 8, taking moments about E, or which locates the point E when M lt V lt and H l are known. Taking moments about D, M,-H,y l =Q, or 7, = -. . . . (51) Taking moments about F, M 9 -H t (^-c) = o or ? t = <:+^-.. (52) From the triangles DGA and CKF, tan = 5- and tan fr = ^-. . . . (53) /7j " From the triangles GAD and GLE, and 3 (54) We also have x - , . . . (55) * 77 %**/ K a The above equations completely determine the locations of GDEF and K, and hence the equilibrium polygon GEK i8 A TREATISE ON ARCHES. can be drawn in its true position and the values of R 1 and R t at once determined. Having determined R t and R, in magnitude and position, the distribution of pressure over the section at A can be found, and then the stresses in other portions of the arch determined. When the arch is solid in section the stresses are best deter- mined by equations (39), (40), etc. If, however, the arch is composed of two flanges connected by a thin web or by bracing, the graphical method is the more expeditious. The methods of determining the stresses, etc., at different points of the arch will be fully illustrated by problems, but a brief outline of one method of procedure after R t has been determined will be given here. In Fig. 9 let AB be any radial section of a solid elastic arch. Suppose I, 2, ... 5 represent the positions and FIG. 9. magnitudes of the resultants for five vertical loads. Then the position of their resultant and its magnitude can be found graphically as shown. The distance x n can now be scaled, the force R resolved into the components N x and T x , and the stresses upon the section AB completely determined. (See page 8.) If the arch is composed of flanges, the method is practi- cally the same, with the exception that each flange is assumed GENERAL PRELIMINARY FORMULAS. 19 to have a uniform stress over its entire section, as explained above. (Seepage n.) (b] Horizontal Components. An examination of Fig. 10 shows that we can locate the equilibrium polygon GEK for the horizontal load Q in a !- * --"--4--- OJs- * FIG. 10. manner similar to that employed for the vertical component ; in fact, all of the equations will be the same, with the exception of that for j , which in this case becomes x . We have then M, M, * and **-<= Also, *i = 77 and * = y,- From the figure (56) (57) or (58) CHAPTER II. FORMULAS FOR PRACTICAL USE. IN this chapter all of the important formulas for parabolic and circular arches have been collected and arranged for ready application. The demonstrations of these formulas are given in chapters which follow. (A) SYMMETRICAL PARABOLIC ARCHES. A = Ed cos = a constant, or 6 varies inversely as cos = A a constant; X59*)> (59) a m = ~~ (radius of gyration) 1 ; X^) (6) r I* p parameter of parabola = ; . . . . (61) o/ b = 4/(i k}f = y for x = a = kl\ . (62) f)/. . (63) J, = function given in Table I, J, = function given in Table II, J, = etc. etc. ARCH WITH TWO HINGES, ONE AT EACH SUPPORT. (a) Vertical Loads, with Effect of Axial Stress neglected Common Method. H^\ l -^P\k(i-2k^k^-\ . . p(gi) . . (64) o/ */(59) indicates that this equation is taken from the chapter on Parabolic Arches (Chap. Ill), its number in that place being />(59). FORMULAS FOR PRACTICAL USE. 21 or \-L-kpA ^A^ = function given in Table I). . (640) y X93) . (65) /(95) (66) FIG. ii. or (66*) T x =(V l - 2P) cos - H, sin 0. From (42), N x = (V, - i\P) sin + //; cos 0. From (41), M x = V,x -Hj- - a). (42) /(97) (67) . (68) . (69) The application of the above formulas to either the solid or open arch rib is quite simple. The formulas are exact, of course, for the solid rib alone, and then only when the depth of the rib is small and the loading is applied upon the centre line; yet for practical purposes they can be applied to open ribs. A TREA TISE ON ARCHES. SOLID RIB. For the solid rib we compute the values of H l and V lt and then determine the values of M x , N x , and T x for each section of the arch, the sections being taken at convenient distances apart. The values of M x , T x , and N x can be found from a graphi- cal construction as shown in Fig. 12. Draw the locus line S after computing j (formula 66), and then draw FA and FC for the load being considered. By reso- lution of forces R^ R^ //,, H t , V lt and V^ can be determined. FIG. 12. Let ab be any radial section where M x , N x , and T x are desired. Then N x equals the normal component of R^ upon ab, T x equals the tangential component of ^ upon ab, and M x equals the ordinate (o) multiplied by N* Maximum Value of M x . Let ab, Fig. 14, be any section where the maximum moment is desired. Draw the lines Ao and Co until they cut the locus line S. Then since the loads at these points produce no moment at the section ab, these points separate the loadings which cause moments of different signs. The shaded sections in the figure clearly indicate the loadings which cause maximum M x . Maximum Value of T x . At any section ab, Fig. 13, draw AD perpendicular to ab. Then the loading causing positive FORMULAS FOR PRACTICAL USE. 2$ and negative shear is distributed as shown by the shaded por- tions in the figure ; or, for maximum upward shear the arch must be loaded on the left up to the section ab and on the right between D and E. FIG. 14. Uniform Loading. Thus far only concentrations have been considered. If, however, the loading is uniformly distributed horizontally, we have n f/l and w = load per unit length of span. (70 24 A TREATISE ON ARCHES. V _ wlr ( y T x = *(2 - *) cos - -1- - (wl(k f - k"} cos 0, where ' ...... (41) . . (90) In case of the open rib arch the stresses in the individual members can be found by graphics after H l has been deter- mined. (f) Change of Length in Span. By replacing ef by in the above equations they may be applied to any change in length of span. ARCH WITHOUT HINGES. (a) Vertical Loads, neglecting Effect of Axial Stress Common Method. (91) or r = -2/>J M , where n =///. to**) 4* 3 A TREATISE ON ARCHES. M, = - reading (i - k) for k. ( 92(Z ) FIG. 21. ^=2j\i-k)\l+2k) or or (93) (93) -/ (positive upwards) /(ISO (94) ^ Q( l -t)S = S J reading (i - k) for & /(i 53 ) (96) (97) (97*) FORMULAS FOR PRACTICAL USE. 3! < X.55) - (93) , = - J 9 , reading (I- ) for ........ (980) T x = ( V, - V) cos - H, sin ..... (43) . (99) - a). . . . (41) . (100) As in the case of the two-hinged arch, it is necessary only to compute //",, F,, and j a to determine all the outer forces acting upon the arch, and then the stresses ; but as a check it is advisable to compute x^ x y , j, , and/,. The methods of determining the fields of loading which cause maximum values of M x and 7 x are the same as for the two-hinged arch, only the resultants R t and R t do not neces- sarily pass through the supports, but must have their locations fixed by the ordinates x^ , y l , x tJ and y^ (b) Vertical Loads, with Effect of Axial Stress included. H, = C\ &Pk\i - ^ - 2/( j^ 2/) i^(i - k) } , . (101) /(i62) where Approximately, where ^ = // t in (91). *(i-*) = 4 ....... (105) k\i - kj - J lt ....... (106) * See Appendix C. 32 A TREA TISE ON ARCHES, Note that all quantities in tJie above equations excepting those given by the tables are constant for any given arch. - lD2Pk(i - 2k* + . (107) ^ =I + ^_rzpL (I0 8) where //, is to be found from (101). k(i 2k? + /') = //, (109) 2k 3/P -f k 3 = J 10 . ...... (no) (i /) = z/ 6 (in) * To determine J/, replace k by (i ) in (107), or compute r a from (107) and substitute the value in (so * Note that only the terms containing 2k $& -f- & and zk i change in magnitude when I k is used in place of kc FORMULAS FOR PRACTICAL USE. 33 Having computed H l , M l , M t , F, , and y t , all the other outside forces can be found as follows(Fig. 22) : Lay off H, and F, at A and complete the parallelogram of forces, thereby de- termining the direction and magnitude of R^. Then lay off y l FIG. 22. at A above or below, according to the sign, and draw R l in its proper position, extending its direction until it cuts the load being considered. By parallelogram of forces F, and R^ are readily found. As a check, //,, F,, and y t should be com- puted. (c) Horizontal Loads, with Effect of Axial Stress neglected Common Method. or or . (H5) (n6) (118) M, = /2Q4 19 , entering table with I k. (120) 34 A TREATISE ON AXCHES. V,= !2n2Qk\i - k)* ........ /(i 76) (121) or (1210) or or (122) 2 k(i k}\2 or 7 S =/^ J4 reading I for ......... (124^) or *t = ^i. reading i /^ for /^ ......... (126^) ^. = ^(3 - 12/fc + 24^ a - i6#) .... Xi83) (127) or FORMULAS FOR PRACTICAL USE. 35 (d) Horizontal Loads, with Effect of Axial Stress included. if-\- + ^o) } ' X*87) (128) where ^) ....... (,30) - 15 + 5o -60*" + 24*') = 4,- (130 -k- where H l is given by (128). (I33) # = ^. (134) =z/ 3 . . ( I35) 3 A TREATISE ON ARCHES. k(l-k) = A b ......... (136) For M, read (i k) for k, in (132). ^ = 1(^-^ + 20*) . (47) . (137) y* = jj t ....... (50 ('38) Having the values of //,, J/i, V lt and /,, the remaining outside forces are readily determined in the manner outlined for vertical loads on page 33. ( ; cos = n f(56) . (158) tan 0=^ +j>/ ^(67) . (159) Since the general method of treating circular arches is the same as for parabolic arches, it will be necessary to only give the equations. ARCHES HAVING TWO HINGES, ONE AT EACH ABUTMENT. (a) Vertical Loads, with Effect of Axial Stress neglected Common Method. i^(sin" sin* a) \ -f cos (cos #, (*73) sin a a cos a = A J 18 (174) 3 sin cos -f 20 cos' = J Jg . . . (175) r=-r. = i<2r. (176) or sin a cos a 2 cos (sin a a cos a) - - - - ' -T - : - 3 3 sin cos 20. COS* or (177) (178) Horizontal Loads, including Effect of Axial Stress. f 00 3 sin cos n + 20 cos* -f ~\ r j sin a cos a 2 cos (sin a. a cos a) { + w(0 n + sin cos 0o + a -f sin a cos or) i- ^(125) (180) I 2(0 3 sin n cos n -(- 20 cos* ) I -f 2w(0 + s i n 0o cos 0o) J 42 A TREATISE ON ARCHES. or . . (181) (182) _:#, (f) Temperature. efA sm0, A.

i = ~ (sin 0. 0o cos ) .............. (211) i __ ^Q^ _ \ (cos a -cos )(sin cos - +2 cos a sin <) ) 2(sin cos - ) / sin (sin sin* a) J ) [ . . . . ) sin 0, cos 0, = JJ, ...... (213) sin , 0o cos 0o + sin a a cos a . . . . ^(143) (212) ) FORMULAS FOR PRACTICAL USE. 45 sin a cos a = ft lt ..... (214) sin a a cos a = JJ 19 ..... (215) sin a a cos a AA^ ..... (216) Tie magnitudes of //, and M, can be found from (207) and (212) by replacing a by / a, etc. r i = l -(M,-M l + 2Qb). c( 144) (21 7) () Temperature, with Effect of Axial Stress neglected. i (2I8) 0." + 0o sin 0o cos a 2 sin 8 = J M . . (219) It will be noticed that all of our tables, with one exception, have been computed for whole degrees. In case the loads do not fall at even-degree points, it will be found advisable to make all computations for .//,, M lt V lt etc., for the even- degree points, and then obtain the values corresponding to the true positions of the loads by reading their values from a dia- gram constructed from the calculations thus made. The effect of the axial stress has been omitted here, as the equations are long ; these are given complete in Chapter IV. When the rise of the circular arch is not greater than two tenths the span, the formulas for parabolic arches can be applied in the determination of the external forces without sensible error. Another approximate method may also be used for arches where ///> 0.30, viz.: Substitute a parabolic arch of the same span which has an area equal to the area of the given circular arch and determine the external forces, and then apply these forces to the given circular arch. For arches approaching a semicircle this method is but a few per cent in error. 46 A TREATISE ON ARCHES. C. SUMMATION FORMULAS FOR SYMMETRICAL ARCHES OF ANY REGULAR SHAPE AND ANY CROSS-SECTION. ARCH WITHOUT HINGES. (a) Vertical Loads, with Effect of Axial Stress considered. ^ ^K'yAs ^N X A X o ft, yAs 7 ^ h 7 /> ^AS_ f T (220) s V N x Ax p ~B^ */ yAs where f), is the horizontal thrust due to two equal and sym- metrically placed loads. For a single load where ' (22I) and AT* A T a. /f sin 0. . . . . (224) FORMULAS FOR PRACTICAL USE. 47. KxAs where ? = ^^ cos (approximately), . . (228) 1 x o ** and /^j is given by (221). r l = l(M t -M t +p(i-W). . . . (229) M y, =jj ............ (230) Having //,, M lt F,, and /,, the remaining outside forces can be found by the method explained on page 22. (b) Vertical Loads, with Effect of Axial Stress neglected. If the effect of the axial stress is to be neglected, we have merely to drop the terms containing N x and F x in (221) and (225), and proceed as before. (c) Horizontal Loads, with Effect of Axial Stress included. H, = -. + <2). . . , v.r A TREATISE ON ARCHES. (232) where COS0. ./-... (235) cos S. =. (236) M 9 for a load situated a distance a from the origin = M l for a load situated (/ a) from the origin. (237) (238) Having determined //",, J/,, ^/ a , F",, and 7,, the other outer forces are readily found, as explained on page 22. FORMULAS FOR PRACTICAL USE. 49 (d) Horizontal Loads, with Effect of Axial Stress neglected. If the effect of the axial stress is to be neglected, we have merely to omit all the terms which contain F, in (236) and (232), and proceed as before. (e) Temperature. Eefl cos ^ (239) yAs . 240) r t = o (241) If the effect of the axial stress is to be neglected, omit the terms containing F x in (239) and (240). ARCH WITH A HINGE AT EACH SUPPORT. (a) Vertical Loads, with Effect of Axial Stress included. As \i As R As \ ~" (242) (243) 50 A TREA TISE ON ARCHES. If the axial stress is neglected, omit the terms containing F x in the above equations. (li) Horizontal Loads, with Effect of Axial Stress included. where (245) (c) Temperature. Eetl When the effect of the axial stress is neglected the terms containing F x are to be omitted. DEFLECTION OF ARCH. In considering the deflection we will assume that the axial stress is neglected and that no change takes place in the rela- tive positions of the several points of the arch other than that produced by the loading. ARCH WITHOUT HINGES OR WITH TWO HINGES (SYMMETRICAL LOADING). JT+"*- e(&) (247) \-ety.- . . g(6i) (248) FORMULAS FOR PRACTICAL USE. 5 1 The above summation formulas are sufficiently accurate for practical purposes, and are quite simple in their application. They apply to any regular arch figure, such as circular, par- abolic, oval, elliptic, gothic, spandrel-braced, etc. They are especially useful in the solution of the spandrel-braced arch, and all arches which have variable or constant moments of inertia not following the laws upon which the formulas of Chapters III and IVare based. CHAPTER III. PARABOLIC ARCHES, WITH THE MOMENTS OF INERTIA VARYING ACCORDING TO THE RELATION A = 6 cos = a constant. GENERAL RELATIONS. IN large arches it is convenient often to arrange the sections so that their moments of inertia vary according to the relation A = EB cos = a constant. This assumption enables us to deduce quite simple formulas for the determination of the reactions, bending-moments, etc. The nomenclature used in this chapter will be the same as heretofore employed, and any new symbols appearing will be found clearly represented in Fig. 24. FIG. 24. We have then A E6 cos = a constant. X59) Let e = -~ = (radius of gyration)* . . . /(6o) 52 PARABOLIC ARCHES. 53 and p the parameter of the parabola. The equation of the parabolic curve referred to its vertex is or For ^ = 0,^ = 0, and g = 2//, X 6 3) From/(62), y ~ zp therefore and dy=*^d X or g = ^ = tan0. . . /(6 5 ) From & I + j-i { Qf\y - b}d x | . /(68) Performing the integrations indicated, factoring, and col- lecting, we obtain . - 2A From (#) the expression for J^r from o up to the section X becomes d X . . . /( 7 o) Substituting the value of ^0 obtained above, integrating and reducing, /(/o) becomes (see Appendix A) PARABOLIC ARCHES. 55 /( 79 ) From (^) the expression for Ay from o up to the section x becomes Ay= f Adx + ef f*dy- f^y* - . - ^/O t/O i/O * which reduces to (see Appendix B) $6 A TREATISE ON ARCHES. Equations p(6g), ^(79), and />(84) are general expressions for the elastic parabolic arch, symmetrical or non-symmetrical, when EO cos is constant. They can be employed in the determination of temperature stresses and stresses caused by concentrated loads acting in any direction in the plane of the arch. Those terms containing the factor m show the influence of the " axial stress." SYMMETRICAL ARCHES GENERAL FORMULAS. For symmetrical arches these equations become somewhat more simple, as g = \l and

= ffif = and y = c = o. Making these changes in /(6o,), we have - J* - H - 2P(l - a)* From (47), F/' = MJ-MJ+:SP(l-a)l + 2Qbl. . /(86) Substituting /(86) in/>(85) and reducing, ; B j*; + Z I jr, + jr. - H !/+ 1 M/ - )* 2yi ( / From /(79) we obtain PARABOLIC ARCHES. $7 /(88) Equation />(84) reduces to At = /J0. + ^+1^,- ^,/+ $P\a(l- SYMMETRICAL PARABOLIC ARCH WITH A HINGE AT EACH ABUTMENT. FIG. 25. In Fig. 25 let ABA' represent a symmetrical parabolic arch having a hinge at A and A' ; then there can be no bend- $8 A TREATISE ON ARCHES. ing-moments at these points; hence M t and M t = o, and the resultants R v and R t will pass through the hinges. For convenience, the effects of vertical loads, horizontal loads, and a change in temperature, with and without omitting the effect of N x , will be considered independently. In all that follows, k = . (a) Vertical Loads, with the Effect of N x omitted Common Method. Assuming that /re-mains constant, J/=o, and by remem- bering that M t and M, = o, and also that all terms containing Q and m do not appear, we have at once from /(88), by solving for.fr,, or Values of k(i 2k* +') are given in Table I. Since all loads are vertical, the horizontal thrust is the same throughout the arch. From (39), H X = H, ........ (39) making x = /, H x = ff t , and we have //, //, = o, or H l and //, are equal in magnitude, but act in opposite directions. From /(9 1 ) the values of ff, for each load can be very quickly found with the aid of Table I, which gives the values of the expression k(i 2k* -\- k*) for values of k from o to i.oo. From (47), PARABOLIC ARCHES. 59 or from which the value of F, for each vertical load is readily obtained. The value of I k can be taken directly from Tables I or V, for k = o to k = i.o. Having H, and F, , the direction of R s for any particular load is found from /(94) When the stresses are to be determined by graphics, we need only to use ^(94) and determine tan /?, for each load ; then since R t for each load must pass through the left hinge, Rj can be drawn in its proper position at once. Since R J} the load, and R t meet in a point, R 9 must pass through the point of intersection of R^ and the vertical force (load) ; it must also pass through the right hinge, and hence its direction is com- pletely determined. The values of //, , //,, F, , and V^ can now be found by simple resolution of forces. The interme- diate stresses can be found by Clerk Maxwell's method of graphics when the arch is trussed. To facilitate the calculation of tan /?, the values of - . , T have been tabulated in Table II. 5 v 1 i * & ) From (50) we have for each load which locates the point of intersection of R t and R 9 , making the application of graphics still easier than the method using The values of -- , _ are given in Table II for values of k from o to i.oo. 60 A TREATISE ON ARCHES. From (39), (40), and (43) we obtain T x = (F, 2P) cos H, sin From (41), M x = V,x - Hj - ZP(x -a) ..... /(97) By means of p(gi), /(93), ^(96), and />(97) tne stresses at any point of the arch can be completely determined by com- putation. Change in Shape Due to the Action of Vertical Loads (N x omitted). From /( 8 9)> J0 = 7 + The term -~ shows the effect of any change in the eleva- tions of the hinges. This does not mean a slight difference of level in the hinges before the arch is in place, but any change which may take place afterwards. In construction an attempt is made to so design the abut- ments, etc., that Ac will be zero. />(98) may be written (Ac assumed zero) . A99 ) From /(69), remembering that g = /, From /(79), PARABOLIC ARCHES. 6 1 From ^(84) Ay = xJt.+fc { V lX -H^(2l- X )-2P( x -aY \ , /(iO2) in which //, = |-i/%(i-2' + ') ......... /(90 and V i = 2P(i -k) .............. /(93) (b} Vertical Loads, Effect of the Axial Stress included. From X 88 ) H = 1 3 . - _ - _ 8// 3 + 30^0.1 3 2(/+2/) or in which j is the value of //, given by/(9l). Let/= /; then Substituting /(iO4) i The values of are given in Table XXV. * HI = ^(i e), (approximately) /(io6) where ^ = //", as given by/(9i). * See Appendix C. 62 A TREAl^ISE ON ARCHES. For a brief discussion of the effect of the 'axial stress, see Appendix C. The expression for V t is not affected by the axial stress ; hence k). . ....... /( 93 ) For any load, from (50) we have y> = jj a = fj- kl - From (39), V.= V,-2P, ........ (40) N x = V x sin -\- H x cos 0, . . . . (42) (c) Horizontal Loads (N x omitted}. FIG. 26 In Fig. 26 is represented a single horizontal load Q acting from the right to the left, which produces a horizontal reaction H^ similar in character to that produced by a vertical load acting downward. PARABOLIC ARCHES. 63 From/(88), for any number of horizontal loads, p(no) .- . /(in) The values of the quantity in brackets, are given in Table III for values of k from o to i.oo. For any load Q, From (47), From Fig. 26, for a single load, Vs = HJ or *. = %& ..... /(1 15) From (47), for a single load, Therefore 64 A TREA T1SE ON ARCHES. or The coefficient of / is given in Table III. Having the value of x t for any load Q, the values of ff lt F. , , etc., are readily determined by graphics. From (39), From (40), V x Vi />(ii8) From (41), * From (43), T x = V x cos H x sin ; (43) (a ) Change of Shape Due to Horizontal Loads (N x neglected). From ^(89, we have o= y + l^,- i(2[i - 2^(2 - 5*+ 5^) + 3*]}- The coefficient of 2Q is tabulated in Table IV. From /(/9), 15(7-- 30/^-5^] I ....... /(I22) PARABOLIC ARCHES. 65 From X 8 4) in which and V, = k(i -k)n (e) Horizontal Loads, Effect of the Axial Stress included. From X88), 15 5 *vn J where i 15 B 8// + 30^/0; Here we see that the effect of the axial stress is small. If np(f> a is neglected in -, the first of/>(i25) at once reduces to/>(m)- is neglected in -^, the first term of the second member 66 A TREATISE ON ARCHES. The expression J2 $k(i k 2k* -f- ^/P) + 8/ 6 } may be / k \ written 2\\ -[5(1 2& -{- 4^) - 8 4 ]J, and hence its value quickly determined from Table III. The value of F, is not affected by N x ; hence V^ = 4 k(i-k}n ...... p(i 13) For any particular load, from (58), (58) The values of H x , V x , N xt and T x are given by (39), (40), (42), and (43). M x = V,x - (/") Change of Shape due to the Action of Horizontal Loads, with Effect of Axial Stress included. The values of ^0 , Ax, and Ay can be found from /(89), />(79), and ^(84) respectively. (^") Temperature. A change in temperature is equivalent to applying a certain horizontal load at the hinges ; or, if the axial stress is neglected, and if the effect of the axial stress is included. PARABOLIC ARCHES. 6f A rise in temperature creates a reaction H^ acting from the left towards the right. The values of H x , V x , M x , N x , and T x can be found from (39), (40), (41), (42), and (43). (/z) Change of Length in the Span. From X 88 )> neglecting the axial stress ; or TT 6oA f - if the axial stress is included. If the span is shortened, //", acts from the left towards the right. The values of H xy V XJ etc., can be found from (39), (40), etc. (t) Sinking of a Support. In case one of the supports changes its elevation after the arch is in place, a slight change in the stresses may result from the effect of the change in the length of the span ; but any change likely to occur may be neglected in the calcula- tion of stresses. (/) Uniform Loads. Thus far we have considered only concentrated loads. If the load is uniformly distributed (horizontally), 2P =fwda = wlfdk, where w represents the load per unit length of the span. 68 A TREATISE ON ARCHES. Let Fig. 27 represent an arch having a partial uniform load; then, from/(90 From /(97), = T[ I ^ (2 - k) - ^ a(5 - From/(96), , where ^ = . /(i3S) PARABOLIC ARCHES. 69 The above equations enable us to determine all the stresses in the arch when the axial stress is neglected. (k) Uniform Load Over All. In case the load is distributed horizontally and uniformly over the entire span, then k" = o and k' = i or x/l, and we have, If n = o, //i = o. From/(i33), V From /( 1 34), * ~ 2 24* If ft = o, then jj/ = o, and the expression for the bending - moment in the ordinary straight girder. From /(i 3 5), T- = cos -zivl sin wx cos 0. /(i39) 2 8 If = o, becomes zero, since our radius is now infinity; hence ^ wl T x = - wx, the expression for shear in the ordinary straight girder. A TREATISE ON ARCHES. SYMMETRICAL PARABOLIC ARCH WITHOUT HINGES. LocuslLine 3 FlG. 28. In this case we have several conditions which must be sat- isfied. As in the case of the arch with two hinges, we shall consider the various loadings, etc., independently. (a) Vertical Loads (N x neglected). From/(87), by transposition, - a)', and from /( 88 ) Now if there are no hinges the ends of the arch must be fixed in direction, and = <& cannot change under any condition of loading. If the length of the span be assumed unchanged and the effect of temperature omitted, we have, by combining ^(140) and/(i4i) and reducing, PARABOLIC ARCHES, or H> = l sP#(i-Kr ....... X'43) The values of k?(i Kf are given in Table XI for values of k from o to i. oo inclusive. From ^(89), assuming that Ac and J0 are zero, M, + \M % = HJ - P(l - a)(2l - a)a. . /(i44) Combining /(i4i) and />(i44), assuming Al and etlto be zero, we obtain by reduction or Substituting the value of H l from /( 143), we have The values of k\i ^)($ 5/) are given in Table VI for values of k from o to I.OO inclusive. Substituting (i k} for k in/( J 47) we have M v =- The values of k(i k)\e>k 2) are given in Table VI for values of k from o to i.oo inclusive, reading (i k) for k. 72 A TREATISE ON ARCHES. From (47), Substituting /( ! 47) and ^(148) in /(I49), we obtain by reduction The values of (i - k?(i + 2/) are given in Table VII for values of k from o to i.oo inclusive. The above equations completely determine all of the ex- ternal forces. The stresses at any section of the arch can be determined from equations (39) to (43) inclusive, remembering that the terms containing Q disappear, as we are not consider- ing the horizontal components or loads. The values of //,, V lt and R t can be found graphically after the ordinates y^ , j , and ^ are determined. From (50), (51), and (52), M, + V,a M t M t y. = . ' y> = jr> and ^ == Substituting the values of ff t , V^, etc., given above, we have (positive upward) . . PARABOLIC ARCHES. 73 which completely determines the position of the equilibrium polygon for any vertical load. When j, orj/ 2 become very large it is more convenient to use the abscissas x^ and x^. From (54) and (55), x l is negative when measured towards the left. 2(3 - 2*) x^ is negative ivhen measured towards the right. The coefficients in/(i52) and /( 153) are tabulated in Table VIII, and those in/(i54) and/(i55) in Table IX. (b) Change of Shape due to the Action of Vertical Loads (N x neglected). From /(8 9 ), where M^,M^, and H l are to be found from /(i48), /(i47), and/>(i43) respectively. The values of k(2 *$k + P) = 2k 3^ -f- ^' are given in Tab)e X for values of k from o to i.oo inclusive. in which M l , V,, and H, are to be found from /(i48),/(i5o), and /( 143) respectively. 74 A TREA TISE ON ARCHES. From /(/p), where Yl/, , F, , and //", are to be found from ^(148), and/>(i43) respectively. From ^(84), - a) 3 , where y)/, , F", , and //, are to be found from /(i48), /(i5o), and /( 143) respectively. These four equations completely determine the change of shape due to the action of any vertical load. (c) Vertical Loads, with Effect of Axial Stress included. a). . ...... /(i6o) From />( 88 )' > = Htf - ,2P(l - a)(r + al- a^a PARABOLIC ARCHES. Subtracting p(i 60) from /( l6 and solving for H lt we have, by reduction, where c= The values of &(i )* and k(i k) are given in Tables XI and V. We have, approximately, where ^ = //, in/(i63). From/(89), -- The first member of /(l63) may be written or . f multiplying /(i6i) by D, we have * Appendix C. A TREA TISE ON ARCHES. M.D + MJ) = #; / - ~ Eliminating M, from X 1 ^) and in which H l is to be found from ^(162) and Z> from (m). For values of k(\ - 2/fe a + /^ 3 ), see Table I ; 2k $& + /P, see Table X ; and for /&(i - K) see Table V. It is to be noticed that D and the coefficients containing m are constant for any particular arch, hence by the aid of the tables, /(i68) can be evaluated very rapidly. The value of M 1 can be obtained from ^(164) by taking everywhere (i k) for k, or by first computing the value of M t from/(i64) and substituting in/(i6i). The value of V l can be found from The stresses at any point of the arch can now be deter- mined from (39) to (43) inclusive, remembering that all terms containing Q disappear. The values of y<>, y^ and y^ can be found from (50), (51), and (52), if graphics is employed in determining the intermediate moments and shears. PARABOLIC ARCHES. 77 (d} Change of Shape due to Vertical Loads, including Effect of Axial Stress. A(f>, Ax, and Ay can be found from /( 6 9)> P(79\ and /( 8 4> respectively, remembering that all terms containing Q dis- appear, and that Ac, Al, and J0 are zero. (e) Horizontal Loads (N x neglected}. From X 8 7) V - 6at* - 40?) From /(88), + M, = Hf - Qa(l - a) s Equating /( l6 5) and/( l66 )> and solving for H lt we have or in which the quantity in [ ] is tabulated in Table XII. Eliminating M, from /( 88 ) and A 8 9)' and solving for M, we obtain A TREA TISE ON ARCHES. /' or and where the expression in j \ is tabulated in Table XIII. Substituting (i k) for k in/(i7i) and changing sign, where the expression in { } is tabulated in Table XIII, reading (I - K) for & From (47), . . . /(I73) Substituting /(i 74), X 1 72), and /(i 70 *)'^ .... /(i75) or The values of (/& ')' can be found from Table XI. The above equations completely determine the external PARABOLIC ARCHES. 79 forces. The stresses at any point of the arch can be found with the aid of (39) to (43) inclusive. The method of graphics may be employed after we have found the values of y l , y^ , x l , x^ , and x^ in determining the external forces. From (51) and (52), y^ _| \_i\ ~~ _i_ A~ f^p i /.'i^ P{ 1 ^} and _y, and y^ are always measured upward, The coefficients of /in/(i78) and ^(179) are tabulated in Table XIV. From (54) and (55), ^r, w always measured towards the left and x^ towards the right. The coefficients of / in /( 1 80) and /( 1 8 1 ) are given in Table XV. From Fig. 29, or ;r = 8O A TREA TISE ON ARCHES. Substituting the values of H l , F, , and b from /(i68), /(i76), and />( 1 74), we have ). . . . /(i 83) w always measured from left to right. FIG. 29. The values of 3 \2k-\- 24^' i6/^ 3 are given in Table XVI. '/) Change of Shape due to Horizontal Loads (N x neglected). From ^(69), since J0 = o, From PARABOLIC ARCHES. 8 1 From ^(84), where M t , V, , and //, are given by/(i7i),/(i76), and/(i68) respectively. (g) Horizontal Loads, with Effect of Axial Stress included. From ^(87) and /(88), in a manner similar to that employed for vertical loads, we have \- 1 5 + &k where The values of are given in Table XII. 82 A TREATISE ON ARCHES. From/(88) and/(89), from which the value of J/, can be found. (i - 4/fc + io/& a - io/6 3 + 3^) = i - 2k(2 - and the values of this expression are given in Table IV. and the values of this expression are tabulated in Table III. and /fj is given by/(i87). The values of k(i k) are given in Table V. It is to be noticed that D and the coefficients containing m are constant for any particular arch ; hence by the aid of the tables, />(i8g) can be readily evaluated. PARABOLIC ARCHES. 83 The value of M l can be found from/(i89) by taking every- where (i k) for k. The value of V l is found from /(I49) The stresses at any point of the arch can now be determined from (39) to (43) inclusive, remembering that all terms con- taining 2P disappear. The values of y 9 ,y lt etc., can be found from (50), (51), and (52), if graphics is employed in determining the intermediate stresses (A) Change of Shape due to Horizontal Loads, including Effect of Axial Stress. A<(>, Ax, and Ay can be found from, ^(69), ^(79) and ^(84) respectively, remembering that all terms containing 2P disap- pear, and that Ac, A I, and A^ = o. (i) Temperature. From/(8/) and/^88), where the term containing m shows the effect of the axial stress. If this be omitted, A*A H l = ^--et (axial stress neglected). . . ^(191) 47 Substituting /(icp) in/(88) and/(89), and solving for M t , letting /> =I+ 84 A TREATISE ON ARCHES. If the axial stress be neglected, D becomes unity, and the terms containing m disappear. M t = ~-et = MI (axial stress neglected) . /093) or and The intermediate stresses, etc., can be found from the general equations (39) to (43) inclusive. (/) Effect of a Change Al in the Lrngth of the Span. An inspection of the general equations /(87), /(88), and ^(89) shows that -- follows the same law as -j- et. Hence and HI = ^TTi^A (neglecting axial stress) PARABOLIC ARCHES. 8$ also or. if the axial stress be neglected, M, = - l ^Al = M t = \HJ, ..... /(200) where r The intermediate stresses can be found from (39) to (43) inclusive. () Effect of any Change in , <&, and the Relative Positions of the Supports in Elevation. From/(8;) and/(88), or, if the axial stress be neglected, ^ = -^(^0.-^0o) X202) From/(88) and/(89), by substituting /(2Oi), 86 A TREATISE ON ARCHES If the effect of the axial stress be neglected, and (/) Uniform Loads. Using the same nomenclature as employed in discussing this case for the two-hinged arch, we have, from/(i43), From p( 147), k> From /( 1 48), From /( 1 50), yf r.^jJfc-aJP-T,*) ...... X209) >i" From (41), PARABOLIC ARCHES. 8/ (m) Uniform Load Over AIL Here k" o and kf = I or x/l. From/(2o6), From/>(207), ~Sn M, o. . From/(2o8), M o. . . . . . . From ^(209), V.-' From/(2io), "wl ivl wx* /(2I2) CHAPTER IV, CIRCULAR ARCHES HAVING ~ = A CONSTANT. GENERAL RELATIONS. Let FIG. 30. _ -pa A -7j = a constant ; ....... ^(59) k' = R-f. ........... d(j) *H t R cos R sin - - cos a)d Substituting the above values in ^(71) and ^(68), *T2 From (), After substituting the value of A$ from R sin 0^0 - a)\(a - sin a sin cosor^sin 0+a cosasin 0X0 r* -AT- Integration of j -^dx i/o ^f* r^*= r J, EF ' Jo /4 A* # >4 T and = . But R = -j=\ hence we have, after substituting j . -j= the value of N M as given by (42), 92 A TREATISE ON ARCHES But from (39) and (40), //, = #,- i<2 and hence the second member of ^(82) becomes* where . - 0) cos 1 x)dx =2PR* sin cos )1 ....... ,( 86 ) CIRCULAR ARCHES, Therefore (a) becomes 93 - *\ 4- k'x + gy- ^ (P. 0)] + (0 - x-gy + From (3), . .(87) -dy. . ,(88) The following integrals are employed in reducing ,(88): = - {^(0o - 0) - ^(0o - 0) - *!: 94 TREATISE ON ARCHES. (g- d)(y - b)\; kf)dx or + b)\(g- X }(a - 0)_ ( , _ ? sin cos in which the following integrals occur after substituting the values of V x and H x from (39) and (40): sin cos CIRCULAR ARCHES. 95 sin - ) + #(* - ) - ^O - J) + xy - *}; / 2QR* si sn cos i = Using the above integrals, ^(88) become Ay = efy + xA

t -la-2b)\ 2 *" a 2^ - / - a) CIRCULAR ARCHES. 97 from which / - 2k '0. + 2P\a(l-a-2k'a)-k'(l,-l - la-2b)\ . 1 ~ ~^ 7 0, + 2*V -3V or, since a = R(sin (1 ^ ~ lU ~ 2 ^ } . . H = 1 1 -m\2Pa(l-a)\ J -k'l) or 2 B -j- 2w(0 -f- sin cos ) or I r(sin* sin* ) ' I 4- (0 + sin cos ) See Appendix C. CIRCULAR ARCHES. IOI in which 1} is to be found from ^(109) and A = \ (sin 2 0o sin 2 ) -j- cos (cos a + a sin a cos sin 0^ B = 2 cos 2 3 sin cos + . Since the value of B is constant for any particular arch, the values of A can be very easily found from Table XVII by multiplying the tabular quantities by B. The denominator of c(\i^) is constant for any particular arch, and hence the value of //, can be found with but little labor. The value of F, can be found from c(m). From (50), where V, and H^ are to be found from c(m) and c(\i?) re- spectively. For practical purposes it will be sufficient to compute but a few values of y^ and then draw the curve D, Fig. 31, by means of a curved ruler. The change in shape of the arch can be found by means of 487), and c( 99 ). (c} Horizontal Loads (N x neglected}. From ^(87), t t (^ + 2 ^)(0 o + ) + ^(2-/) ) ,^-\-^Q\ - &'(! -a) + 2bk'ot __ I ( 2R\

> + d -1} which reduces to ; ,f 2 sin [cos a -f- sin a] H l = ^P-( sin [2 cos fl -f- sin ] e sin* or [-,(133) 00* H~ 0o si n 0o cos 0o ~ which is easily evaluated by the aid of Tables XX, XXI, and XXII. Substituting {[.. *.cos * - W [_ 7.L%.] U -)-0o sin (cos a sin +a) sin a(0 cos sin +0o s ) [ _ 0o(sin 0o cos 0o 0o) 1 2 sin 0o[cos a -f- a sin a] - sin U [2 cos + sin ] sin 2 a J By the aid of Tables XX, XXII, XXIII, and XXIV ^(138) can be readily evaluated. Evidently y t can be obtained from (138) by making (a) equal (/ a). From (50), . sin 0. sin a, \ \ P sin 2 ^o sm2 a 77^^ change in shape can be found from ^(77), ^(87), and ^(99)- (^) Horizontal Loads (N x neglected). From ^(102) and ((104), _//- 4_ T4- 2 r 2 sin 2 - sin cos 2 ) 4 ' which can be easily evaluated by means of Tables XIX and XX. Substituting c(iO2) in ^(105), and eliminating M^ between ^(102) and ^(105), we have CIRCULAR ARCHES. or IT n M l l {sin 0, cos

\Ai'A} ^ d _ /) ^(HS) or _ - - - , 1 . sin cos - 2 sin 3 ) ' ^ 4 where the value of the parenthesis in the denominator can be obtained from Table XX. 108 A TREA TISE OX ARCHES. From c(iO2) and ^(105), A rr r> A sin cos 0o _ 0f) f -{-^(20 sm0 -|-sin0 cos0 ) \ 4" ft( 2< Po si" ^o si" ^0 cos *A) + ^o) j . ( sin which can be evaluated by the aid of Table XIX. From (47), .(149) The stresses can be now found from (39) to (41). The change in shape can be found from ^(77), ^(87), and <99). (d) Effect of the Axial Stress. In order to economize space, the expressions for H^ and M^ will be given which are perfectly general, applying to cases of vertical loads, horizontal loads, change in temperature, etc. From 4102) and 1(104), - 1(1- 2m 9 2Pa(l - a) + or) - k'l %bl+ a(b + k')\ CIRCULAR ARCHES. 109 The terms containing m show the effect of the axia Istress. From c(iO2) and ^(105), + 2^V0 + (k f d}(i + m\2b 2aa /0, + /a) - m\$t(l - 2a)(b + d)- 2a R*] } _ -LsQ\l -a- tec- k'(a+ 0.)} 20, The terms containing m show the effect of the axial stress. By the application of c(\$Q), f dy- f^dy. i/t) / * SYMMETRICAL ARCHES. Assume a single load placed at any point upon the arch; then, since the arch is fixed at the ends and symmetrical, J0 ; = J0 , Al O, and Ac = o. If x = /, ^(59), ^(60), and g(6\) become, neglecting temperature for the present, J- / ~^s= and ^__j. C 1 M^ Now, from (41), where Substituting ^-(65) in g(&2) and ^-(64), we have and r i ds . . c l x ds, c i Kd S , r+r t t - e -+ ~r = o i/O * i/O * Jo V * Fxch r**3 C l K X ds C 1 N * I -^+r* I + I -Q -- / jr t/O * i/O * t/0 * /J r * From (47), where 112 A TREATISE ON ARCHES. Substituting the value of V l in ^(67) and ^(68), and elimi- nating M y , we have ( rK*ds__ f l jv i c l xds r i Kds r i x *ds \ I IF y \ I ~Q ' I 01 ~ r O~ Ti/r . v*/0 x / x JI/Q y x y * in which sin + ^ cos 0, - - />, ^ > a. From (40) ,-Q. x>a. From (39) (42) ^(73) Then in g(ji) we have two unknown quantities,//", and V x . But V x occurs in N x only, which contains the effect of the axial stress ; hence for the common method of arch treatment we can neglect the term containing N x . A method will be given, however, which will enable us to very nearly obtain the actual effect of the axial stress. Value of //,. The vr.lue of ff t can be found as follows: FIG. 33. Assume the arch free to slide longitudinally upon the sup- ports, and that two equal and symmetrical loads are applied ; also assume that there are equal and symmetrical moments Hz applied at the supports; then /J0, = ^0, the same as if SYMMETRICAL ARCHES. 113 the arch were fixed at the ends, since our loading is symmetri- cal. From (62) we have But from (41), where K' = F^ - 2P(x -a) + 2Q(y - b), . . ^76) Hj being zero, since the arch is free to slide upon the sup- ports. Substituting -(75) in or C l Mds C 1 M+K' C lds C 1 K' I -7T = I '-*- ds=M l &-+ -$-<** = JQ * J i/o i/o * /K' -x-ds u * The change in length of the span due to the action of our loading can be found by the aid of ^63). Let A'l be the change in the length of the span ; then Substituting the value cf M l from ^(78), C l K'ds j v = ii-t A*-2 r^^- J A TREA TISE ON ARCHES where K' = V t x - - a] = F, sin + /f, cos ; (42) (40) (39) All of which are known quantities ; hence the value of A' I can be accurately determined from ^(80) for any symmetrical loading. FIG. 34. Now suppose the arch unloaded and free to slide as before, and let two equal and symmetrical moments Q'z be applied at the supports ; then J0 ; = ^0 , and we have from ^(62) /I Q* o. But hence SYMMETRICAL ARCHES. 115 or r l ya J yds g(3) The corresponding change in the length of the span is given by (63), or where M x = Q'(z + y) and N x = + H x cos = + Q cos 0; hence Substituting ^83) in (85), r Let ^j be the horizontal thrust at the support necessary to cause a change in the length of the span of ^7; then we have ^/:J7::>:^ = ^~ Therefore V, I ds Jo J* A TREATISE ON AKCHES. where K' = V,x - 2P(x a x=a. - b) ; For two equal and symmet- rical loads. (a) Vertical Loads only. If the loads are vertical, = Px - For two equal and symmetrical vertical loads, - a) - a) ; V x = V, -2P= P - 2P-, N x = V, sin 2P sin = P sin 2P sin 0. Since our loads are equal and symmetrically placed, and K' is the moment at any point x, considering the arch as an unconfined girder, the value of K' due to one load will have a corresponding equal value due to the other load. Then, since there are symmetrical values of -^-, the value of / K'y-r^ for one load must be equal to that for the other load. Therefore for a single vertical load we have K' = P(i - k}x - [/>(* - a) when x > a~\ SYMMETRICAL ARCHES. 117 and N x = P(i k) sin (f> [P sin when x > a]. For x = o to ^ = a, K' = />(i % and N x P(i - k) sin 0. . ^(88) For x tf to ;tr = /, AT' = Pk(l-x) and ^ = - / sin ; . . ^(89) and we have From ^(90), the horizontal thrust due to any vertical load can be found when the relation between x and y is known. The equation applies equally well to the parabolic, circular, or elliptic arch. Horizontal Loads only. Here we have K' = and For two equal and sym- metrical horizontal loads, x > a. = - 2Q cos 0. 118 A TREA TISE ON ARCHES. For x o to x = a,, K' o and N x = o ; g(9 l ) for x = a t to x = ^ a , JST' = 0(j ^) and N x = Q cos (j> ; . ^(92) for x a^ to -r = /, 7T' = o and N x = o ^(93) Let /f, = the thrust due to the load on the left ; H t = the thrust due to the right load. Then but H^+l hence 2/7, = or Q Therefore r 2 + or -I- / : i i z? /<* cos />". (y - b}ds * & * r'yfo SYMMETRICAL ARCHES. 119 is general, and can be applied to parabolic, circular, and elliptic arches with equal facility. (c) Moments, Vertical Loads only. In -(71), for a single vertical load, K l =-Hj-P(x-a] x = a, and N x F, sin <{> P sin -f H^ cos 0, #;;, where //, can be found from ^(90). There remains then only the term F, sin 0, which is as yet unknown. In case there are equal and symmetrical loads V, becomes known, as it is equal to one half the total loading. If, however, the loading is not symmetrical, the values of M l and M t can be computed with the term F, sin neglected, and the corresponding value of F, found from (47), and then a second calculation made and this value introduced. Generally the value of the expression containing F, is very small, and is omitted entirely by nearly all American authors. Neglecting the term containing F, , for x = o to x a, and N x H^ cos (approximately); ..... for x = a to x /, K=-Hj-P( X -a} ........ and N x = H l cos Psin (approximately). . Therefore -(71) becomes / A " U 120 A TREATISE ON ARCHE3. where the value of H^ is to be found from ^90). J The value of M t can be found from g(ioi) by replacing a by (/-). (d) Moments, Horizontal Loads only. In the case of horizontal loads only, N x = V x sin + H l cos -2<2 cos 0. Then for x = o to x = a, and N x = /^, cos (approximately); for x a to x /, and jV* = /^, cos <2 cos (approximately). Therefore ^(71) becomes where //; is to be found from ^(95). SYMMETRICAL ARCHES. 121 The value of M t can be found from ^(106) by replacing a by (I -a). (e) Effect of a Change in Temperature. Assuming that the span does not change in length, = etl = o; or, if the arch is free to slide upon the supports, Let H t be the horizontal thrust necessary to cause a change in the length of the span of A'l-, then, referring to ^(86), or Eefl From ^(83), /i yds r- - - ~^~ t/ But M, = H& 122 hence M 1 = -. A TREATISE ON ARCHES. / /xyy r^co-0 ^ /ds / v x H ^ c /I . Q* SYMMETRICAL ARCH WITH A HINGE AT EACH SUPPORT. In this case we have no morrtents at the points of support. Assume that the arch is free to slide upon the supports then, from ,^(63), FIG. 35. Let Q be any horizontal load at the hinges ; then and N x = H x cos = Q cos Then ^(113) becomes S I 'MME TRICA L AR CHES. 123 Now suppose the horizontal loads Q' removed and two equal and symmetrical vertical loads applied to the arch ; then E t/o x M x = V,x 2P(x a) and F, = i/'Ci - ); hence N x V x sin + //^ cos (p = V x sin But F x = F, - 2P; therefore N x = 2P(i k) sin 2P sin 0. Then for x = o to x = a, M x = and sn 0; 124 A TREATISE ON ARCHES. for x = a 1 to x = a t , and N, = 2P(i k) sin Ps'm 0; . . . ^(123) for x = #, to x = /, and A^ = -2/\i /&) sin 2Psin 0. . . . Evidently the change in the length of the span due to the left load will equal that due to the right load ; hence we have, for x = o to x = #, and for x = a to ;r = /, J/. = />(! - k)x - P(x - a) . . . . ^(128) and N x P(i k] sin Psin 0. . . ^(129) Therefore ^"(117) becomes dx P r t (x--jfyds P f l sin d x ~E J - --- ^EJ ~~F - ' ' ' *< /ja v */ a r x SYMMETRICAL ARCHES. 125 Let f^, represent the horizontal thrust necessary to cause a change in the length of the span of A" I] then and we have, since for vertical loads the horizontal thrust is constant and //, = ^, , dx COS From g(i$i] the horizontal thrust due to any vertical load can be found with comparatively little labor. i rT 1 I 01 FIG. 37. For two equal and symmetrical horizontal loads we have from (41). assuming the arch free to slide, M x = V.x - but V l = o, hence M x = 2Q(y - b\ 126 A TREATISE ON ARCHES. N x V x sin -\- H x cos = H x cos ; but //, = //;- i& hence N x = - 2Q cos 0. ...... . . Then for ;r = o to ;r = #, , and for x = a t to x = /, ^, = ............ ^-(134) and for x = rt, to JT = #,, and Hence the corresponding change in the length of the span is Then since SYMMETRICAL ARCHES. For a single load, //, = ), + %Q. Hence -COS an equation quite simple in its application. * If the moment of inertia is assumed to vary according to the laws assumed by most writers upon the theory of arches, their equations can be very easily obtained from our general forms. For example, let the horizontal thrust //, for a single vertical load placed upon a parabolic arch having no hinges be required. Assuming cos (f> = A = a constant, and that the terms contain- ing the effect of the axial stress are neglected, and remembering that ds cos = dx, we have, from "(90), + (I _ - x]dx f I/O dx / yd* Jo dx where, using the nomenclature employed in Chapter III, - x)ydx = * For several examples illustrating the application of these general formulas to special cases, see Appendix E. 128 A TREATISE ON ARCHES /*** = /, t/O and we have \fl\k - 2k' + f) - \fl\k - ff) ffl = ~- or which is the same as obtained by the method employed in Chapter III. (See equation X J 43) P a g e 7 1 -) In a similar manner any of the equations usually employed can be quickly deduced from our general formulas, which have the advantage of being general to the extent that they can be employed for any arch when the relation between x and y can be represented by a linear equation. SUMMATION FORMULAS. In many cases it is preferable to replace the sign of integra- tion by that of summation. This is particularly true in arches where the moments of inertia do not change according to some law which permits of readily reducing the above equations to fit the paTficurar case. As examples of such structures may be mentioned the Douro Arch and the Washington Bridge. The summation formulas are as follows : SYMMETRICAL ARCHES. 12$ (A) ARCH WITHOUT HINGES. Vertical Load only. . . A* Horizontal Loaa only. * "^7"^r ~ v f 130 A TREATISE ON ARCHES. 0, P* Temperature. Eefl ARCH WITH TWO HINGES (ONE AT EACH SUPPORT). Vertical Load only. ^ &- sin

i. From ^(63) we have where C l M x yds = \ ^- = o, . i/U When x = , M x = o, since there can be no bending moment at the hinge ; hence . (151) See pages 229 and 230. 132 A TREATISE ON ARCHES. But since our loads are equal and symmetrically pkced, F, = P, and Substituting this value in "(150) and then the value of M x in ^(149), we have or 'yds C^_ J, ^ y. T'^ {'yds , -. ' 'J. . - r- /yg .Tr- J. . " J J, * where a t = / a t . Value of F, and F, / 7 ) But Hence Then if ^,/be the vertical displacement of the crown due to the action of these two loads, //* , /J/a f- For a single vertical load, -V^ + HJ+'^P^-^ ....... g(i6i) Therefore, If /J,/be the vertical displacement of the crown due to a single load, we have 134 A TREATISE <9A' ARCHES. .pT ( ,- a f*-v,L SO x I/O Since the vertical deflection of the crown due to one of two equal and symmetrical loads must be one half that due to both loads, J.,/= 2^,/. Equating these two values and solving for V l , we obtain ///' /// ~// *--P (*-af*-P ^1 v = _ ff * J* * Jo #* ,;.*., 1 2 / /'' /1 / W/ * _ I xds _ / art 2. A -T Jr. T tw equation is to be employed for all loads on the left of the crown. These equations enable us to find the values of V^ and V t for all loads. Values of M^ and M t for Vertical Loads. From (41), making x = - and solving for J/,, we have, for a single load, ' in which the values of V, and 77, are given by ^164) and ,^(154). This equation gives the values of M l for any load on the left of the crown. From (49), M = M V -Pl-a SYMMETRICAL ARCHES. HORIZONTAL LOADS. Value of H^ for a Single Horizontal Load. '35 FIG. 39. Let two equal and symmetrically placed horizontal loads act upon the arch ; then V l = o and (41) becomes If x = -, then M x = o; hence and ^ = 9,/- J-Q(f- From -(63), Substituting the value of M x and solving for I},, we have, & /*/ ^y ds f* l ys riy'ds J ~^~J Q ~e^ I3 6 A TREA TISE ON ARCHES. Value of F, for a Single Horizontal Load. This case will be treated in a manner similar to that em ployed for vertical loads. From "(156), From (41), for two equal and symmetrical loads, But hence The vertical displacement due to two equal and symmetrical loads is I ( fixds fi *f=Mf T x ~^ ( Jo i/o xyds Q(f- *) For a single load, -Q(f-b}+Q(y-b}-V?- 2 SYMMETRICAL ARCHES. 137 B "t *f>=m + Q>\ ..... ^(180) hence and Equating the two values of AJ and solving for F, , we obtain - Qf + Q *i x*ds _ /_ rixds * " Vo ' which reduces to /'^rVj _ /_ ^7^' *m "~ * I 9* J o which holds good for #// /^^r =. a constant and show graphically the relations between the values of the outer forces for the different types. Let Type i = arch with no hinges ; " 2 = " " one hinge ; " 3= " " two hinges; " 4 = " " three hinges. (a) VERTICAL LOADS. Comparison of H^. The formulas* are : P I See pages 29, 139, 20, and 143, respectively. MS i 4 6 A TREA TISE ON ARCHES. These values are represented graphically in Fig. 43, from which we see that the 2 type differs quite considerably from the others, particularly for loads near the crown and those near the springing. 0.5 HI z 0.4 / _1 2 I S - I \- g COMPARISON THE VALUES C OF F H, ft 0.3 L s _l | 1 2 cc J o / z S 4f 09 UI 1 0.1 < */ m / CO > ^ / o & / 0.1 0.2 0.3 V 0.4 U.UE 0.5 SOP 0.6 k 0.7 0.8 0.9 1. FIG. 43 Comparison of V^. Formulas* Type i. -W = (i >) s 2 ^ = (i - 4^) See pages 30, 139, 21, and 143, respectively. COMPARISON OF FOUR TYPES OF ARCHES. 147 These values are represented graphically in Fig. 44. 1.0 NT - 0.9\ ^N \ > M_ >"l 0-6 CO 0.5^ ; N 3 >, 1 'XjV \ COR 1PARISON OF VALUE qp v, THE i A s 0.4 > V 0.3 \ \\ 0.2 N\ \ 0.1 \ \ \ ^ ^^ Ov 1 2 0.3 v 4050 ALUES OF 6 k 8 0*9 1 FIG. 44. The equation for intermediate vertical shear is V x =V l or V x = KP. We give below the values of V x for the arch without hinges and that with a hinge at each support. The span is divided into twenty equal divisions, and the load P assumed to occupy each point of division. The values of V x are given for each division. These tables and those given later for maximum bending- moments are principally useful in preliminary computations unless is assumed to vary as the secant of 0, when of course they very materially decrease the labor of calculation. 148 A TREATISE ON AKCHES. c C-. ---------------- e Division Number. H Q +++++++++ 1 1 1 1 1 1 1 1 1 1 O O o" " *- cTtnco^-^cor'-iN ^-'o ET O O = 1 1 1 I ! 1 1 1 1 ++++++++++ ++++++++ 1 I 1 1 1 1 1 1 1 l 1 53"o*~^m < %S'SS~~oooo8 2 i l l i i i l l -H-+++++++++ 00 fft!isiiss*iij?ioi <2 i i i i i i i +++++++++ i i ++++++ l i i i i l i ++++++ K 2 i i i i i i +++++++ i i i i i i w O co-l-OOco w m w O "- 1 "^0*^^00 w Of^^- N oorrO^m-^ IS l l i +++++ l i l i l l l i i i i CO ++ Mill +++++++++++ co O^O oo co co -C u-> co Tj-o ^ O O* Tf w coco eo I l +++++ l I 1 l 1 I I 1 I il M ||^i|4|||||||tt|tt 2 1 +++++ + 1 i I 1 1 1 1 1 1 1 1 1 _ iHssilltllHIlslll o -f ++44+ i i i i i i i M i i i i |J c jaqimiN fi S 001SIAIQ COMPARISON OF FOUR TYPES OF ARCHES. 149 Division Number. ?%: ill ..... i +++-1-++++++ I l I l I I l I l I I I ++++++ 1 1 1 1 1 1 1 1 1 1 1 1 1 c^,oo N mr^io O OO O OO r^vnoo * * r- O^OO I^\O T C4 O TOO WOO Tf Q r^u^c*^N - O I i l I I I I I I ++++ 2 ? S , 3 s^ i* ? 2 >< i"f > 5 o q S i 1 r r r r f++ +++++++ r r f r i"ti"tioJ> ' i o iitt i" t^T^TrvriO C^cJr^cTQ rnuito Nc^-*-Tt^cj~-ooqoqq 1 f l' l" ++++ -f +-f l' \ 1 f 1 l -f ++ Mill! ++++++++++ ^S ?^g > S5' v g, < 2 ^'g XoT ? - = 8 5" 3 -pcn-^c<->NoOOO<-.--'OOO r f r+++4++i' f r r r r r f \ \ >O Ooo m t^ r^oo O T co n'i-c-i->i-i.qq r f ++++++ r r r f f f f f r \ \ + I M ^1^^ !'++++++ I I I I I I i i iooo^iocnr^c^r^ TOO ~N l-oo ++++++ f f f r r r f f f r r r r 150 A TREATISE ON ARCHES Comparison of 'the Maximum Values of M x . In each of the four types of arches which we are consider- ing, if the values of M l , F, , and H l be substituted in (41), we find that M x = l -f\K-](Z\ where K depends upon k = j and Z upon z == j , showing that for parabolic arches the value of M x varies with the span alone for given values of k and z. Then we may write If the values of J be computed for each load for every value of x and tabulated, the maximum values of M x are readily found by taking the sum of the values of J having like signs. We give below the values of J for types i and 3 which are most common in practice. It will be noticed that the positive and negative moments are approximately equal, although the arch is divided into but twenty equal divisions. For a uniform horizontal load cover- ing the entire structure the positive and negative moments would be equal, since the equilibrium polygon would be a pa- rabola coinciding with the axis of the rib. In Fig. 45 * is shown relatively the maximum values of M x for the four types. It appears from this diagram that type i has moments which vary more nearly according to the variation of the sec- tion of the rib than either of the others. The second type has very large moments near the springing, which rapidly decrease until about the quarter-point, and then * This diagram is from a note by M. Souleyre: "Note sur 1'emploi de quatre types d'arcs dans les Fonts, Viaducts et Fermes Metalliques de grande portee." Annales des Fonts et Chaussees, mai, 1896. COMPARISON OF FOUR TYPES OF ARCHES. I$I after increasing slightly, decrease rapidly, becoming zero at the crown. A crescent-shaped rib corresponds more nearly with the variation of the maximum moments in the third type. FIG. 45. As the formulas of the fourth type do not depend upon the values of 0. the rib can be designed to correspond with the variation in the moments. Thus far we have considered only the live or moving load effects. 152 A TREATISE ON AXCHES. SYMMETRICAL PARABOLIC ARCH WITHOUT HINGES.* PI M x = _/o values of Jo. Point of Divi- sion. ' Crown 10 Pom -.0789 +., + .0,35 + .0102 +.0073 + 0284 +.0047 +.0189 +.0025 + .0106 +.0005 + 0036 .0010 - .0023 -.003, " 3 4 HJ354 -.0647 - -749 + .0085 .OJ79 + .084- + .0429 +.0622 +.1075 +.0427 +.0760 + .0256 +.0484 !: +.0245 .0011 + .0045 -.0109 .0,16 - 0239 '" I -'54 -734 -.0712 -0579 -.03,6 0357 .0069 +0633 + .1186 + .0705 +.0793 + .119, + .0452 + -744 + .0,64 + -0363 -.007, + .0047 .0254 " 7 -.0369 -.0388 .0330 -.0,94 + .0019 +.03,, +.0679 + .,126 + .0650 + .0252 .0069 " 8 || 9 + .0340 -.0173 + .004, -.0,73 .0303 -.0234 +.0259 -.0074 + .0605 + .0,79 + .1037 + .0524 + .0555 + .0965 +.0,60 + 0494 + .0625 + .0234 .0062 -!o266 -0375 -.0390 -.03,3 -.0,4, + .0,25 + .0484 + -937 * ii + .0836 + .0388 + .0032 .0232 -.0404 -.0485 -0473 .0370 .0174 + .0494 ** 12 + 0959 + .0994 +.0490 + 0538 + .0,08 + .0160 -.0188 - .0397 -.0363 .052, -.0509 - -0557 -.0576 -.0508 -.0566 --0372 -.0478 -.0,49 + .0160 -.0069 ** 14 + .0946 + -534 + .0,87 -.0093 - .0307 - -0455 - -0537 - .0552 - .0501 --0385 - .0202 " J 5 + . 0820 + 0475 + .0,83 -.0056 -.0242 -.0376 - -0457 -.0485 -.046, -.0384 .0254 ** 16 + .0640 + 0,57 -.0,73 .0280 -.0348 -.0379 .037, .0324 -.0239 " 7 + 043 1-0259 + .01,3 -.0009 .0,07 .0,80 -.0229 -.0254 - -0254 .0230 .0182 " 18 + .0225 .0,37 + . 0062 .0001 .0052 -.009, .01,7 -.0132 -0134 -.0124 .0103 " 9 + .0065 + .0040 + .00,9 + .000, -.00,4 .0025 -.0033 -.0038 -.0039 -.0037 -.0031 First published by Prof. Greene in Engineering News, vol. iv. SYMMETRICAL PARABOLIC ARCH WITH Two HINGES.* PI MX = /a values of / a . Point of Divi- sion. ' 2 3 4 5 6 7 8 9 Crown 10 ^pni + .0832 , .0676 + 0523 + .0402 +.0289 +.0,78 +.0084 + .0003 - 0066 .0122 " 3 " 4 " I + .0667 J.0509 359 + .022, + .0097 +1359 +.,05, +.0765 + .0498 + .0257 + .1075 + .1634 t32 +.0480 + .08,5 + .,250 + .1715 +.12,9 + .0767 + .0580 + .0902 + .,260 + .1663 + .11,8 + .0370 4-. 0590 +.085, + .1,62 +.1532 + .0784 t' 3 o 5 +.0489 +.07,7 + .,o,6 + 0075 +.0173 + .0328 +055, -.0097 - .0006 +.0155 .0226 - .0297 -.0320 .0283 .0,76 7 .0013 +.0043 + .0,70 +.0366 + .0632 + .0968 + .1374 + .0849 +0394 + .00,0 8 .0,07 -.0,39 .0097 + .0019 + .02,0 +0475 + .0815 +.1229 +.07,7 +.0280 9 -.0,83 - .0289 -.03.8 .0270 -.0,45 + .0058 + .338 + .0695 + "29 +.064, 10 -.0242 -.0492 -.0500 -.0430 -.0281 -.0055 + .0250 +.0633 +.,094 ii .0283 -lots. -.06,8 -.0670 -.0644 .0542 .0362 .0105 +.0229 +.0641 * 12 .0307 - -0539 -.0697 -.0780 .0790 -.0725 - -0585 - .037, -.0083 +.2080 13 -.0313 -.0556 - .0730 -.0834 - .0868 -.0832 .0726 -0551 - -0305 + .0010 J 4 - .0304 -545 .0720 -.0833 -.0882 -.0868 - .0791 -.0649 -C445 .0176 '5 -.0279 -.0502 -.0670 -.078, .0837 -.0838 - -0783 - .0672 -.0505 .0283 .024, - -0435 -.0583 -.0685 - .0740 -.0749 -.07,1 .0627 - -0497 - .0320 *7 -.0191 .0347 - .0467 -.0550 -.0598 -.0609 - -0585 - -0525 - .0429 - .0297 18 -.0133 -.024, - -0325 .0385 - .0420 -.0430 .04,6 -.0365 -0314 " 19 -.0068 .0167 .0198 .0210 - .0197 - .0,66 .0,22 First published by Prof. Greene in Engineering News, vol. iv. COMPARISON OF FOUR TYPES OF ARCHES. 153 The dead load is very nearly a uniform horizontally distrib- uted load, and hence the moments due to this load are prac- tically zero in the four types. If in the i and 2 types the dead-load stresses are com- puted as if the ribs were hinged at the springing and then built with hinges at the springing, when the falseworks are removed the rib will settle into position and the dead-load stresses will be practically those computed. From Fig. 45 we see that the live-load flange-stresses are a minimum for the i type, or the arch without hinges. A rib con- structed with pins at the springing is very easily made into a rib with fixed ends by arranging the details so that the flanges may be rigidly connected with the piers or abutments after the falseworks are removed. This method is followed by French and German engineers in many cases, especially for masonry arches and metal arches with solid webs. The arch without hinges, or type i, appears to be the most economical of the four for the dead and live loads. There remains to be considered the effect of temperature. Comparison of Temperature Effects. Typei . H^Acf. M, 2. H, = Aff- M, = HJ. 3. H,= l ^Aet\ M>=o. 4. H, = o. M, = o. Hence for Type i. M x = //,(/- y) = Aet\^f - 6y}. 154 A TREATISE ON ARCHES. Type 2. M x = Htf - y) = From which we see that the effect of temperature is great- est in the i type and least in the 4 type; also that the effect in the 2 type is greater than that in the 3 type. For structures carrying moving loads the second and fourth types are not desirable on account of vertical vibration of the MAXIMUM MOMENTS DUE TO THE WIND BLOWING AGAINST ONE SIDE OF THE ARCH WITH A FORCE OFpPER UNIT OF THE RISE/ FIG. 46. structure, leaving the first and third types to be selected from. Complete calculations show that for large structures the first type is more economical as well as mor? rigid, and practice proves this type well adapted to the work required for a railway bridge. COMPARISON OF FOUR TYPES OF ARCHES. 155 (V) HORIZONTAL LOADS. Horizontal loads, being usually due to wind, may be consid- ered as a dead load, covering the arch on one side from the crown to the springing. For a uniform load/ per unit of height of the arch,* Fig. 46 shows the relative values of max M x for the four types. Here we see that the first type more nearly agrees with the variation of in the variation of the moments, so that the conclusion drawn above remains unchanged for structures carrying a moving load. In case there is no moving load, as in roof-trusses, the fourth type appears to be most economi- cal. This type is almost always employed by American engineers for large roof-trusses. Comparison of Types i, 3, and 4 designed for a Single-track Railway Bridge having a Span 0/416 Feet.\ To more clearly show the relation between the three types 1, 3, and 4, a comparison of the maximum stresses in the individual members of a trussed parabolic arch rib are shown in Figs. 47, 48, 49, and 50. The diagrams show the maximum stresses due to dead load, live load, wind, and changes in temperature. Figs. 47 and 48 clearly indicate the superiority of the arch without hinges for economy in the flanges. Figs. 49 and 50 show that there is little choice between the types as far as the web is concerned, there being a remarkably close agreement between the stresses for the three types. The principal data employed are as follows (see Fig. 51) : Span 416' o" Rise 67' o" *See note under Fig. 45. f The computations for this comparison were made by Messrs. Crockwell, Wiggins, and Shaneberger in connection with their theses for graduation from the Rose Polytechnic Institute. T 5 6 A TREATISE ON ARCHES. CO , Q \ oc V O X o o I- o 5JL NOISN31 COMPARISON OF FOUR TYPES OF ARCHES. 1 57 15 A TREATISE ON ARCHE$. Batter of arch planes. I in 3 Depth of rib at crown 6' o" " " " " skewbacks 10' O 77 Moving load per lineal foot ot span 4000 Ibs. Dead " " " " " superstructure 1500 " " " " " " " arch' 1000 " Wind " " " " " span (live). 300 " " " " (dead) 600 " Range of temperature 80 F. FIG. 51. Relative Weights of Steel in One Arch Rib, including Gusset- plates, Rivets, etc. Type i , without hinges l.oo " i.2i 3 i two 4, three 30 CHAPTER VII. APPLICATIONS. IN the preceding pages we have deduced formulas for deter- mining the various reactions and moments which result from the application of vertical and horizontal forces to the linear elastic arch, that is, we have assumed that the forces were applied upon the central line or neutral axis of the arch rib. In practice this evidently is not always the case, especially where a super- structure is supported by arch ribs having considerable depth. The weight of the arch rib alone may without serious error be assumed as applied to the centre line or neutral axis. Vertical Loads. Vertical forces due to the superstructure and moving loads may be assumed to act where they intersect the neutral axis in flat ribs and in trussed ribs where one system of the web bracing is vertical* The same assumption may be made for plate-girder ribs, as they are either very shal- low, as in bridges of short spans, or the forces due to the superstructure are applied to the rib quite close together. For the condition where a vertical force does not intersect the neutral axis of the rib, as in the case of a large semicircular rib near the supports, the following method may be employed. In Fig. 52 let P be a vertical force applied at B which does not intersect the neutral axis. At the centre of the strut BD place the two equal and opposite forces P\ then we have for the equivalent of the force P applied at B the force P applied at C and the couple Pd. The reactions can now be found by apply- ing the formula for a vertical load and that for a couple. In passing we may say that this method is general and can be ap- * See Fig. 51. 159 100 A TREA TISE OX ARCHES. plied for any load whether its direction intersects the neutral axis or not. Horizontal Loads. Horizontal forces in the plane of the arch-rib seldom occur in practice, excepting in the case where the arch is employed for supporting a large roof. In this case the horizontal force is the horizontal component of the wind load. If the stresses due to the wind are small in comparison with those caused by the total dead weight of the structure, the wind forces may be assumed to act upon the neutral axis where the normal components intersect it in the determination of re- actions, etc. If greater accuracy is desired, then the force may be replaced by an equal force and a couple, as explained above for vertical forces. Wind Loads. We have just explained how to consider wind loads in the plane of the arch. There remains to be discussed the action of the wind against the arch and super- structure perpendicular to their plane. The superstructure is usually composed of a roadway sup- ported by columns or towers according to the magnitude and design of the structure. The action of the wind against the roadway creates a hori- zontal reaction at the top of each post or tower. This reaction is transmitted to the arch-rib in the form of an equal horizontal force at the foot of the column or tower, and a couple which is equivalent to a vertical force acting upward on the wind side of the structure and an equal vertical force acting downward on the opposite side as illustrated in Fig. 53. The vertical T -T- forces are treated as explained above. The horizontal force with that due to the direct action of the wind against the rib must be considered differently. The actual action of these forces is very com- plex unless we make the assumption that the arch-ribs act a? the chords of a canti- levered beam having a length equal to one naif the length of the axis of the arch. Under this assumption the lateral systems may be 1 APPLICATIONS. l6l developed and the stresses in the different members found by ordinary methods. Although this method is not correct, yet its simplicity and probable safety commend its use. Maximum Stresses. We have explained in Chapter II the methods for selecting those forces which cause the maximum shears and moments at any point. Another method may be employed which has many features in its favor. The values of the reactions, etc., may be found for each load, and then the stresses in each member of the rib due to each individual load. These stresses being tabulated, the maximum positive and negative stresses are readily determined by simple addi- tion. This method is long, but has the advantage of being free from errors, and if each load is taken as unity, the stresses obtained for the individual loads will be coefficients which can be applied to any load. The latter feature is of considerable im- portance, as very often the magnitudes of the loads are changed before the final computation is made. In very large structures where the moving load is small in comparison with the dead load it is customary to make but two computations for the mov- ing load : one for the moving load covering the entire structure, and a second for the load covering one half of the span. Character of Reactions. In the hinged or fixed arch the -vertical reactions (F, and F,) always act upivard when the verti- cal forces which are applied to the arch act downward, and the horizontal reactions (//, and H^) act from the supports towards the centre of the span. In case the vertical forces act upward, F, and F, act downward and //, and ff t act away from the centre of the span. In the case of horizontal loads, if the load acts from the left towards the right, Fj acts downward and F, acts upward. Both of the horizontal reactions act from the right towards the left. Co-ordinates y 9 , x^y^x^y^x % . Vertical Loads. The ordinate y^ is always measured upward from the long chord of the arch. The ordinate y l is always measured upward at the left 162 A TREA TISE ON ARCHES. support for loads on the right of the crown. For loads adja- cent to the left support y l is measured dowmvard. y 1 is zero for a load near a point which is four tenths the span from the left support, this distance varying with different arches. The abscissa x l is measured to the left of the left support when y l is measured ztpward, and to the right when y 1 is meas- ured downward. x l and y l are zero when the arch has a hinge at the left sup- port. The directions of x, and y, are easily determined from what has been said concerning x^ and_y,. Horizontal Loads. x a is always measured towards the right from the left support for loads on the left of the crown. y l and j/, are always measured upward at the left and right support respectively. x l is always measured to the left of the left support, and x^ to the right of the right support. Bending Moments at the Supports. For arches with hinges at the supports M l and M t are zero. When j/, is zero M l is also zero. When the extremity of jy, lies between the flanges of the arch-rib both flanges have the same kind of stress ; for vertical loads acting downward this stress is compression. When the extremity of j, lies above the rib the upper flange is in compression and the lower in tension for vertical loads act- ing downward, or M i is positive. When y l is measured down- ward M l is negative and the upper flange is in tension and the lower in compression unless the extremity of y l falls between the flanges, when both are in compression for vertical loads acting downward. To illustrate the application of our formulas we will now solve various examples in detail. i. Given a parabolic arch, with a hinge at each support, having a span of 100 and a rise of 25, determine H, for a load P placed at a distance 25 from the left support. Here / = 100, 7=25, and = 0.25. A P PLICA TIONS. 1 63 From (64.0) we have The vertical reaction V l is found from (65), or F, = (i - o.25)/> = 0.75^. From (66#) we have 7. = 25(1.3474) = 33-68. In a like manner the values of H l , F, , and j can be found for any other vertical load. The method employed above was the common method neglecting the effect of the axial stress. Although this is of little consequence. in this case (see Appendix C), we will, how- ever, give the solution which includes the axial stress. For this we need the values of /* m = (the radius of gyration)*, / parameter = :, and . Let m be assumed = 4 (an average value), p = 50 and = 0.7854. Then, from (74), H 15 \ 8 X 100(25)* 1 8 X 100(25)' + 30 X 4 X 50 X 0.7854! 15 '' 4(100)' ) 2(50+ so)" 1 ?.P or H, = 0.0000297133333$, - 2QOPk(l - k)\, which is general for this particular arch. 164 A TREA TISE ON ARCHES. Substituting the values of ^, and k, we have //, = 0.000297! 18559.8 - 37.5} = 0.550P. From the approximate equation (75), HI = 0.5568(0.9885) = 0.5 $oP, the difference in results being in the fourth decimal place. The value of V^ remains unaffected by the axial stress. From (76), By the common method y n = 33.68, which is but 0.41 less than obtained above. In a similar manner any other vertical load may be treated. 2. Let a horizontal load Q be applied in place of the ver- tical load P. Then, by the common method from (77) or (77#), H, - 0.57420. Note that the values of ff t are given by Table III when Q = unity. From (780), V, = 4 X 0.25 X 0.18750 = 0.1875^. From (79), * = o.5742/ = 57.42. If the axial stress is included in our calculations we have to apply (83), which contains the factor -. But APPL1CA TJONS. 165 = 0.4949; mp(a + 0.) _ 4 X 50(0463 + 0-785) _ 0.0074. Hence //, = 0.4949! 2(0.5742)} (2 + 0.00740 = 0.57570, which is but a very small amount larger than the result found by the common method. F, = 0.18750, as before. From (85), 3. In place of the loads P and Q, suppose the arch-rib constructed of metal having a modulus of elasticity E = 28,000,000, and let the temperature rise 50. What will be the value of //, if the coefficient of expansion of the metal is 0.0000055? From (86), , = 15^^(0.0000055,50. If 6 is taken at the crown, cos = i. Let = 4 ; then H, = 92.4. From (87), which includes the axial stress, Since a rise in temperature tends to lengthen the arch rib, 166 A TREATISE ON ARCHES. the span will tend to increase, hence H l must act from left towards the right. 4. Let the arch be assumed parabolic in shape and fixed at the ends. Let a load P be applied at the quarter-point and determine the reactions, etc. The following data will be used : / = 100, /= 25, = 0.7854, a= 0.463. For the value of H l we have, from (91) or (910), HI = -V 5 - W(o-0350^= 0.5265^. From (92) or (92^) we have M, = i* ( - o.io54)/> = - 5.27^. From (93) or (930), V, = o.8437/>. From (92) or (920), M t = -4*( + 0.0820)7* ^ +4.IO/ 5 . From (93) or (93^), letting k i k 0.75, From (94), }'o = 7 2 S 3> measured up. From (95), y t = 0.4(25) = 10, measured down. From (96), 5= +7777. measured up. APPLICATIONS. 167 From (97) or (970), x, = \(4( 0.625) = +6.2 5, measured to the right. From (98) or (98^), * a = ^V - ( 2 - 62 5) = 26.25, measured to the right. A good check upon the above work is to lay off the ordi nates and see if the two reaction lines meet on the load line P as indicated in the figure below. Thus far the formulas of the common method have been employed. We will now consider the effect of the axial stress. For this case we apply (101) to obtain the value of ff l , letting m = 4 and/ = 50. From (102), 15 X ioo X 25 4 X ioo X 625 + 90 X 4 X 50 X 0.7854 = 0.1419; 2/1/+2/) 50(100) ^ Q Then , = 0.1419} 100(0.0351) 0.24(0. 1875) * H, = 0.49 1 P. By the approximate formula (103), HI 0.935(0.5265)7' = 0.492/1 168 A TREA TISE ON ARCHES. For the bending-moment J/ a we employ (107), in which there are several coefficients which are constant for this arch. We will first compute these: / - = = 0.7854. at = 0.463. Then, from (107) ^,(0.4805) = ^|i4.98l - 99-025^(1 + 0.2377/^(1 - K)k - o.6/ ? {o.7854(2/^ - + 0.463}. In the case we are considering //, = o.49i/'and ^ = 0.25. Making the proper substitutions, we have ^(0.4805) = + 7-355^'- 22.O53/ 3 + 16.405^ + 0.044^ 0.043/ 3 or 4 P PLICA T10NS 1 69 To determine the value of M, we substitute i k for k in the above formula, or ^1(0-4805) = + 7.355^ 22.053^ + n.7i5^ + o.044/> - 0.043 P or As a check we will apply (112) to this case ; then M> = - 3-555^ - 22.270/ > 4- O.I + 0.047/ 3 or M, = - 5.953^ (6.1 16 5.953)^ = o.i6P, or an error of about 2#, caused by neglecting decimals. From (113), or V, = 0.8467/ 5 , which differs but o.oo^P from the value obtained by the com mon method. From (51), or -6.116 y l = = 12.46, measured downward, 0.491 A TREATISE ON ARCHES. and y = ' 3 ' 555 = _|_ 7.24, measured upward. 0.491 From (54), Therefore 6.116 0.8467 = 7.2, measured to the right, and 3-555 i 0.8467 = 23.2, measured to the right. From (50), MA. y a 6.116 + 21.18 * = ffT = 5^T- - = 3- 6 ' To show the effect of the axial stress upon each of the quantities we will tabulate our results : COMPARISON OF RESULTS. Function. Common Method. Exact Method. Difference. Percentage of Common Method. H, 0.5265^ 0.491^ 0.0 35J P 6.6 /A 0.5265 P 0.49I/' o.035/> 6.6 r, 0.8437P 0.8467 P 0.003 P 0-3 T, O.l&lP O.I533-P 0.003 / 2.0 Mi 5.27-P 6.H6P 0.84/ 1 16.0 M, 4.ioP 3-555^ 0.55^ 15.5 y 1419(3 X EO cos (f>et}l = 525. From (141), ^(0.4805) = Ji 14.9$} Aet - o.uSSAet . APPLICATIONS. 175 Now Aet = 28000000 X 4 X 50 X 0.0000055 or Aet = 30800. Therefore From (51), H, 525 8744 ^ = IO.O. 7. Let the arch be loaded from the left support to the crown with a uniform horizontally distributed load; then k" = o and k' = 0.5. From (147), From (148), From (149)* From (150), The location of the point where the true equilibrium polygon starts can be found from (51). 176 A TREATISE ON ARCHES. 1562 y. = - = 6.2. measured downward * 25 1562 y^ = -- = 6.2, measured upward. 8. What will be the vertical deflection of the arch at the crown when there are two equal and symmetrical loads placed at the quarter points? Let E 28000000 and cos = 4; Since the arch is fixed at the ends, J0 = o; then ^(84), page 55, becomes, making x = 1/2, (We have neglected the effect of the axial stress.) -T = 0.0000037 about. 3. 5 iP + 50.000^ - 19.74^ A P PLICA 7VOJVS. 177 Therefore 6y = 0.0000037(0.76)0 = 0.00000280. Suppose Q = 30000, then dy = 0.084 J and if our span is measured in feet Sy = 1.008 inches. The sign being positive indicates that the crown rises under the action of these two loads placed at the quarter points. Thus far we have considered only parabolic arches. We will now solve a few similar problems for a circular arch having a span of 100 and a rise of 25. The following data will be employed : /= 100, /=25, ^=62.5, r\ k> = 37-5 0o = 53 7i' ** = j^ 0.00102, = 25, a = 23 35'. 9. Determine //, , V l , etc., by the common method, as- suming the arch to have a hinge at each support. Vertical load P. From (i6oz), In order to use Table XVII we must determine the values , 2(/> , a of and . ^ = 0.590, ~ = 0.443. it . Having now determined the values of H l for each load, the stresses in the arch can be found graphically for any given value of P. Since the arch is hinged at the ends, the values of Fi will be the same as for a straight unconfined beam. The following table shows the values of H l obtained above with those given by Seyrig: Load at ff, (I) ff, (Seyrig) (2) Diff. (3) Formula (91), page 29 (4) Diff. ( i) and (4) (5) A B C D E o.37i3/> o.59i4/> o.6304/> 0.6495/ 1 O.6543/* o.37o/> o.592/> 0.63I/* o.bsoP O.OOI3/ 1 o.ooobP o.ooo6/ > o.ooos/' o.35S/> 0.653/ 1 O-7I2/" 0.742 p o 746/* O.OI33/' 0.0616; o.o8i6/ > 0.0925^ O.OQljP * As given by Seyrig this is 0.637; but as he gives it as the quotient of 2048.36 H- 3246.84, which is 0.6309, it is evidently a typographical error. ARCHES HAVING A HINGE AT EACH SUPPORT. l8/ The differences in column (3) are very small and unim- portant. To show the error in applying the common formula to arches where the moments of inertia do not vary according to the law making 6 cos

= a con- stant, by the summation method and by the formulas demon- strated in Chapter III. DATA. Span =7=190; Rise =7=25; Load = a concentration P or Q = unity at points designated. VERTICAL LOADS. (P = unity). (a) Determination of H l by Summation. 10 o FIG. 57. Let the semi-arch be divided into ten parts as shown in the figure, and the quantities shown in the tables determined. k\f and tan = yr (/ IQO ARCHES WITHOUT HINGES. The moments of inertia are determined for the section at the middle points of As or points having the abscissas x. Only the relative values need be determined now, as we propose to neglect the effect of the axial stress. DATA. Point. k - y AJ A.? ~o* Approximate I 0.026 5 2-5 II. 2 .12 IO.O 26 30' 2 .079 15 7-3 IO.g .09 23 54' 3 .132 25 "5 10.7 .07 21 12* 4 .184 35 15-0 10.5 05 18 23' 5 .237 45 18.1 10.3 .03 15 29' 6 .289 55 20.5 IO.2 .02 12 30' 7 342 65 22.5 IO.I .01 9 26' 8 395 75 23-9 IO. I 1. 01 6 20' 9 447 85 24-7 IO.O .00 3 10' 10 .500 95 25.0 5-o(i) .00 5.0 0' 95.0 SUMMATION TERMS. AJ A* AJ AJ Point. fa* ''T *6^ ** I 25 62.5 50 125 2 73 532-9 150 1095 3 "5 1322.5 250 2875 4 ISO 225O.O 350 5250 5 181 3276.1 450 8i45 6 205 4202.5 550 "275 7 225 5062.5 650 14625 8 239 57I2.I 750 17925 9 247 610O.9 8 5 20995 IO 125 3125.0 475 "875 1585 31647.0 4525 94185 From (221), page 46, remembering that the terms contain- ing N x and F x are to be omitted, since we propose to neglect the axial stress, we have A TREATISE ON ARCHES. VK'As "K'yAs f B x VyJs T~JT ~PT ' * off. H, = where K' = Vjc 2P(x a) ; or for the left half of the arch, K' = Px - P(x - X a\ But P = unity, and hence x> a. K' = x - (x - a). Then and * ( 3) We will first determine the constants in our expression for The denominator becomes 95 ARCHES WITHOUT HINGES. 193 and we have 10406 The following table contains the quantities to be substi- tuted in this equation PARTIAL SUMS. (1) (2) (3) (4) a 9 "I a 6 aT I 5 94060 7800 4475 425 2 15 92965 22305 4325 1125 3 25 90090 34300 4075 1625 4 35 84840 42770 3725 1925 5 45 76695 46845 3275 2025 6 55 65420 45980 2725 1925 7 65 50795 39715 2075 1625 8 75 32870 27900 1325 1125 9 85 II875 10625 475 425 10 95 O o In this table the summation is actually taken between / and (a -\- i), for when x = a the combination of columns I and 2 and 3 and 4 respectively equal zero. For a load at (i), our equation gives us H. = 94185 -(94060-7800) -{4525 (4475 425)116.68. 10406 H l = O.O002.* In like manner the values of H, can be found for all the points from I to 10 inclusive. These values are given in the annexed table. *This value of H l should be zero when all mathematical work is correct. 194 A TREA TISE ON ARCHES, VALUES OF H l FOR A LOAD UNITY AT Point. i Hi I .026 o.ooo 2 3 4 I 7 8 9 .079 .132 .184 -237 .289 -342 395 447 O.I4T 0-357 0.640 0.932 .212 454 .641 757 In case any load other than unity is placed at any point, the corresponding value of //] is found by mul- tiplying the load by the cor- responding coefficient HI in this table. 10 .500 797 (b} Determination of H l by Integration. From (91), page 29, we have , 4/1 which becomes for our arch with load unity H, = 28.6^(1 - )' or H, = 28.64, if the tables are employed. The values of ff l are as follows : VALUES OF HI FOR A LOAD UNITY AT Point. 4 //, I .026 .018 2 .079 .152 3 .132 377 In case any load other 4 .184 643 than unity is placed at any 5 237 938 point the value of H\ is 6 .289 .201 found by multiplying the 7 342 447 load by the corresponding 8 395 633 value of Hi in this table. 9 447 744 10 500 .787 ARCHES WITHOUT HINGES. 195 (c) Comparison of Results. VALUES OF H\ FOR LOAD UNITY AT Point. by Summation. by Integration. Difference. Relative Diff. in Per Cent. j O.OOO .Ol8 r 2 O.I4I .152 + .011 1 1 I 7 3 0-357 377 + .O2O a* 1 3 4 0.640 643 + .003 2.8 | 0.4 5 0.932 938 + .006 I r ' [0.6 6 .212 .201 .Oil Iri fi-o 7 454 447 -.007 "& 1 0.5 8 .641 633 -.008 (frj 1 0.5 9 757 744 -.013 | 0.8 10 797 .787 .OIO H [0.5 Let a load of one ton per horizontal foot of the arch be assumed, and determine the value of //", for a load over all. Then, by summation, H l = 10(9.0325)2 = 180.65 tons; by integration (for concentrations), //i = 10(9.0475)2 = 180.95 tons; 180.95 180.65 = 0.30; and relative error equals We will now determine the values of M l by both methods. (d) DETERMINATION OF Af t . From (225), page 47, iKxAs^xAs tKAs'SAs ,* _ in which ft, 196 A TREA TISE ON ARCHES. The following table contains all the constants entering the above equation. Substituting these constants and the values of K, we obtain an equation quite simple in its application. CONSTANTS. Point. JT y 4 >'t 9 -4 I 5 2.5 25 62.5 50 250 125 2 15 7-3 73 532-9 150 2250 1095 3 25 "5 "5 1322.5 25O 6250 2875 4 35 15-0 150 2250.0 350 12250 5250 5 45 18.1 181 3276.1 450 20250 8145 6 55 20.5 205 4202.5 55 30250 11275 7 65 22.5 225 5062.5 650 42250 14625 8 75 23 9 239 5712.1 750 56250 17925 9 85 24-7 247 6100.9 850 72250 20995 10 95 25.0 250 6250.0 950 90250 23750 9' 105 24.7 247 6100.9 1050 110250 25935 8' "5 23-9 239 5712.1 1150 132250 27485 t 125 135 22.5 20.5 225 205 5062.5 4202.5 1250 1350 156250 182250 28125 27675 5' 145 18.1 181 3276.1 1450 210250 26245 4' 155 15.0 150 2250.0 1550 240250 23250 3' 165 "5 1322.5 1650 272250 18975 2' 175 7-3 73 532.9 1750 306250 12775 I* 185 2-5 25 62.5 1850 3-12250 4625 3170 63294. 18050 2.284750 301150 1585 31647 Determination of Constant Factors. 'As'x*As 2-J-2- = 190(2284750) = 434,102,500; U x " x K A C\ * l a = 325,802,500. Hence Denominator = 108,300,000; ; = 18050 and ^jj 2,284,750; 18,050 108,300,000 2,284,750 108,300,000 = O.2II. ARCHES WITHOUT HINGES. I 9 7 Therefore jurii'Jsr,^F + ' a A ~ *^- As We are now prepared to determine the" value of M l for a load at any of the points I to 10 inclusive. The substitutions in the above formula are quite simple as illustrated by the detailed deduction of M l for a load at point 6 (see table of Partial Sums). PARTIAL SUMS. Point. ; . Y a ~o~ 2** Jt a y*4* a ft t+ a I 18000 2284500 301025 1 80 3M5 2 17850 2282250 299930 170 3072 3 17600 2276000 297055 160 2957 4 17250 2263750 291805 150 2807 5 16800 2243500 283660 140 2626 6 16250 2213250 272385 130 2421 7 15600 2I7IOOO 25776O 1 20 2196 8 14850 2II4750 239835 no 1957 9 14000 2O425OO 218840 100 I71O 10 13050 1952250 195090 90 1460 9' 12000 I842OOO 169155 80 1213 8' 10850 1709750 141670 70 974 7' 9600 1553500 "3545 60 749 6' 8250 I37I250 85870 50 544 5' 6800 n6ioco 59625 40 363 4' 52CO 920750 36375 30 213 3' 36OO 648500 17400 20 98 2' 1850 342250 4625 10 25 l' o o M. = -1 Load at Point 6. a = 55- - .oooi6f|i.2i(30ii50) + 2213250 55(16250)} + .0211 {l.2l(3I70)+ 16250 - 55(130)} .000161(1684494) = 280. 749 + .0211(12942) =+ 273.076 )* i 9 8 Hence A TREATISE ON ARCHES. M, = 273.076 - 280.749 = - 7-673- In like manner the values of M 1 for loads at any other points are determined. We have tabulated below the values obtained by this method. Load at Value of M, . Load at Value of MI. I - 4-683 tf 4- 8.256 2 10.380 8' + 9-490 3 12.923 7' 4- 9-603 4 12.641 6' + 8.927 5 10.687 5' + 7.363 6 7-603 4 + 5-410 7 - 3.887 3' 4- 3-037 8 O.2OO 2' 4- I. 210 9 + 3.317 l' 4- 0.123 10 4 6.200 The corresponding values of M lt as obtained from Table VI, are as follows : Load at k ^6 MI = 95J 8 , page 30. if It by Summation. out. I .026 -.046 - 4-370 - 4-683 + 0.313 2 .079 -.108 IO.26O 10.380 + 0.120 3 .132 .133 - 12.635 12.023 + .288 4 .184 -.132 - 12.540 12.641 -j- .101 5 237 .112 10.640 10.687 + -047 6 .289 -.O8l - 7.695 - 7-603 - .032 7 342 .043 - 4-085 - 3-887 - .198 8 395 -.003 0.285 O.2OO - .085 9 .447 +.032 + 3-040 + 3.317 + -277 10 .500 -J-.062 + 5-890 + 6.200 + -310 9' 553 + .084 + 7.980 + 8.256 + -276 8' .605 + .096 + 9.120 + 9-490 + -370 7' .658 +099 + 9-405 + 9.603 4- -198 6' .711 +092 + 8.740 + 8.927 4- .187 5' .763 +.0 7 8 + 7410 + 7.363 - .047 4' .816 + 057 + 5-415 + 5-410 .005 3' .868 +035 + 3-325 + 3-037 - .288 2' .921 + 015 + 1.425 + I. 210 .215 i' 974 +.002 + 0.190 + 0.123 -3*3 62.510 63.004 + 61.940 + 62.936 - 0.570 0.068 ARCHES WITHOUT HINGES. 199 If our points had been taken closer together, the positive and negative moments would have been practically equal for a load over all. The values of V l can now be found from the formula For a load at 6, Fi = 8.927 + 7-673 The values of y^ are determined from , , (50), page 17. /* 10406 0. From the tables computed for vertical loads the partia sums required above are readily found. PARTIAL SUMS. Point. vW y*A* 8 x *\1 yds ^VJ a 6 x I 31584-5 1560 85 2 31051.6 1487 75 3 29729.1 1372 65 4 27479.1 1222 55 5 24203.0 IO4I 45 6 20000.5 8 3 6 35 7 14938.0 611 25 8 9225.9 372 15 9 3125.0 125 5 10 o o APPLICATION OF THE GENERAL SUMMATION. 2OI To illustrate the application of the formula we will deter- mine the value of //", for a load at point 7. Load at 7, b 22.5 Then _ CJi - ff= /+ [14938 -22.5(61 1)] \ _ \- i6.68[6i i -22. 5 (2 5 )1/ - - ~ 10406 - 16.68(756.6)] 10406 = 0.536. 73 ' By Table XII, H, = A, = 0.537. The following table contains the values of H l as found by two methods for loads at points I to 10 inclusive. Point. HI , by Summation. by Table XII. Diff. I .026 1. 000 0.991 .009 2 .079 0.935 0.930 .005 3 .132 0.839 0.836 .003 4 .184 0.742 0.740 .OO2 5 237 0.652 0.651 .OOI 6 .289 0.585 0.584 .001 7 342 0.336 0-537 .005 8 395 0.511 0.511 .002 9 447 0.501 0.502 .001 10 .500 0.500 0.500 .000 For loads on the right of the point 10 we have merely to subtract the value of 77, for the corresponding load on the left of the crown from unity. The above values of H l are for loads acting from the right towards the left, and hence they are positive and the same in character as for loads acting vertically downward. For bending-moments M l we have from (236), page 48, introducing the constants already found, 202 A TREATISE ON AKCHES. ' The partial sums required above are given on page 197. As the application of this formula is precisely the same in method as that for vertical loads, we will only illustrate its application in a few cases. Load at Point 10. : M _ ( 0.000162(281735) = 46.956, ) +0.0211(2375)= + 50113; or M, = + 3-I57- By the use of Table XIII, M l = + 0.1250(25) = + 3.125. . Load at Point 5 . b = 18.1. M t = jfl _ ( 0.000l6|(2l6770) = 36.128, { + 0.021 1(1975) =+ 41.672. Hence W = + 5.544. By the use of Table XIII, M l = + 5.450. The above results indicate a close agreement in the two methods. To determine M 3 it is necessary to merely consider a as /- *. The method of procedure is now parallel with that outlined for vertical loads. Fig. 59 shows graphically the results obtained by the two methods somewhat exaggerated. APPLICATION OF THE GENERAL SUMMATION. 2O$ The close agreement of the curves shows clearly that the approximate method of summation is quite accurate enough for practical purposes. This method requires considerable more \vork, but it has the advantage of being approximately correct for any form of arch and any values of 0, the circular or elliptical arch requiring no more labor in calculating the values of H t , M t , etc., than the parabolic arch. CHAPTER X. THE ST. LOUIS ARCH.* To further show the accuracy of the results obtained by the use of the summation formulas we will compute the values of //, and M } for the well-known St. Louis or Eads Bridge, using the data given in the History * of the bridge. The results given by Prof. Woodward were computed with great care from formulas deduced ,to fit the peculiarities of the arch-rib. 6 has but two values throughout the rib. For a distance equal to one twelfth of the span from each support 9 has a constant value, and between these two sections another value which is uniform throughout that section ; thus the use of the formulas of Chapter IV is prohibited. DATA. Span = / = 519.2328 ft. Rise =/= 47-31 ft. Radius = R = 736.0 ft. = 20 39' i7 // .92. Area of each flange for -fa the span at the ends = F = 67 sq. in. Area of each flange in centre section = F = 100.5 sq. in. Depth centre to centre of flanges = 12 ft. Dead load = I ton per running foot horizontal. Live " =0.8" " In applying our formulas the linear arch will be assumed to lie midway between the flanges of the rib. We will divide this linear arch into fifty-one divisions, as shown in the first table. The coordinates x and y will be computed for the centre points * See "A History of the St. Louis Bridge," by C. M. Woodward (St. Louis, G. J. Jones & Co., 1881). 204 THE ST. LOUIS ARCH. 205 of these divisions, and the moments of inertia taken at the same points. Since the areas of the flanges are 67 and 100.5 s q- in- and the distance centre to centre of the flanges 12 ft. throughout, the moments of inertia will be in the ratio of two to three. As we propose to neglect the influence of the axial stress as was done by the computers for the structure as built we need not concern ourselves about the actual values of 0, but use relative values. The following data will be used throughout in the computation of H l and M l : TABLE OF CO-ORDINATES, ETC. Point. X y A y 4* 4 Relative 0. ' 6.6 2-33 4.83 13.3 14.14 3 2 18.3 6.52 3.46 IO.O 10.58 3 3 28.3 9.98 3-31 10.53 3 4 38.3 13-22 3.16 10.48 3 5 48.3 16.31 3.02 10.44 2 6 58.3 19.18 2.87 10.40 2 7 68.3 21.98 .65 10.34 2 8 78-3 24-55 50 10.31 2 9 88.3 27.06 43 10.29 2 10 98-3 29-34 .21 10.24 2 ii 108.3 31-55 .14 IO.22 2 12 118.3 33-61 .98 10.19 2 13 128.3 35-45 77 10.15 2 14 138.3 37-21 .69 IO.I4 2 15 148.3 38.76 -55 IO.I2 2 16 158.3 40.23 .40 IO.IO 2 17 168.3 41.56 25 10.08 2 18 178.3 42-73 .10 10.06 19 188.3 43-76 .96 10.04 20 198.3 44-72 .81 10.03 21 208.3 45-46 0-74 10.03 22 218.3 46. 12 0.51 10.01 23 228.3 46-63 0.44 10.01 24 238.3 46.93 0.40 10.007 25 251.4 47-22 0.12 16.3 16.30 2 . Determination cf //,. From (220), page 46, remembering that the terms contain- ing N x and F x are to be omitted, we have for vertical loads 206 A TREATISE ON A K CUES. in which for a load P = unity and ~ *- (223) We note that only the quantities enclosed in the parentheses in (222) and (223) vary with a change in the location of the load. We will first compute the terms which are constant. 7 |y^ = 156,868.7; 2-j- -= 125.0. Combining these quantities and multiplying the product by 2, we have for the value of the denominator 43324.1. ^3 As ~*y-7T = 66 9403-3 r-S- = 16,863.4; THE ST. LOUIS ARCH. 207 125.0 _ 3 * For our purpose it will not be necessary to compute //, for a load at each point of division. We have selected points 2, 4, 6, 10, 15, 20, 23, and 25. The following tables show the method of procedure in the determination of H l for each point designated. FIRST TERM OF NUMERATOR. First Term of Load at j. Numerator: Point ^*yds // A 1 1 A I/ K'v A No. o e * ^ s a~ y 2r~ s a * 2 669,403.3 668,910.9 74,606.5 75,099-1 4 666,152.5 153,035.0 156,285.8 6 " 656,225.8 222,170.7 235,348.2 10 " 611,473.5 322,5I4-9 380,444.7 15 " 495,454-3 353.462.5 527,411.6 20 " 303,881.1 260,155.3 625,677.5 23 " 152,699.9 141,462.2 658,165.5 25 " o O 669,403.3 SECOND TERM OF NUMERATOR. Second Term of Load at Numerator: Point ll Fa 160 180 200 220 240 260 FIG. 60. paper and a smooth curve drawn through them, the value of //, for any value of a can be readily and quite accurately deter- mined. THE ST. LOUIS ARCH. 209 For a uniform load of w per horizontal unit of span the total value of //, will be twice the area included between the above curve (extending from the support to the crown) and the axis of abscissas multiplied by w. Such a curve is shown in Fig. 60. The full line represents the curve located by the above values of H . The broken line is located by values of H l which were obtained by another computation in which only one decimal place was employed in the data. In the computations for the St. Louis arch uniform loads were assumed as follows : For dead load i.o ton per lineal foot. " live " 0.8 " " " " In the history of the bridge the values of H^ are given for a load extending from the support up to each of eight points of division. The corresponding points are marked I, II, III, etc., in Fig. 60. The following table shows the relation between the values of HI given in the history of the bridge and those obtained from Fig. 60. MOVING LOAD OF 0.8 TON PER LINEAL FOOT. VALUES OF H\. Load up to History. Fig. 60. Difference. Remarks. I Tons. 8.IO Tons. 8.04 O.O6 The values in the II 56.20 55-86 0-34 third column were III 155-20 154.78 0.42 obtained from Fig. IV 286.60 286.56 0.04 60 by taking T 8 ff the V 418.10 413.34 0.24 area between the VI 517.00 517-26 0.26 full line and the VII 565.10 565.08 0.02 axis of a. over all 573-30 573-12 o.i 8 1 The above table shows almost perfect agreement between the exact and approximate methods. The errors are of no practical importance. They exist only in the decimal figures, which are quite likely to be in error by either method. For a load over all with w = O.8 ton the area between the 2IO A TREATISE ON ARCHED. broken line and the axis of a is 559-3, being in error 14 tons, or a little over 2 per cent. Even this is of no practical importance. ' We will now show that the effect of the axial stress, which was neglected in the calculations made for the St. Louis Bridge, is very much larger than any error which is likely to be made by using the summation formula. In (221) we have in the numerator A* sin o X F X o F x Since we used only relative values for B, it will be necessary to introduce a factor in the above expression. For the area 67, 6 = 2 . ( ) =2(16.75) approximately. Therefore 144 \ 2 / 2 2 = 2(8.37) = the factor required ; then 7/ a Ax a Ax sin d> r xjr = 2^8.37 g This is very small in comparison with the remaining terms in the numerator, and hence can be neglected without serious error. In the denominator of the same equation we have l !*Ax cos d> '/ Ax cos + 2 p ^ or 2^8.37 ^- r - We may replace Ax cos by As nearly, and have v*As 16.7 '4^-p- = 8300, about. Then the denominator, when the effect of the axial stress is considered, becomes 43,324+2(8300) = 599 2 4. THE ST. LOUIS ARCH. 211 = 1.38, or the values of H l obtained above are too large, and should be divided by 1.38 to obtain the values which include the effect of the axial stress (see page 283). Deduction o Neglecting the axial stress term, (225), page 47, becomes, where The values of the constant terms are as follows : 1 As l x*As 2 - = 250 ^-^ = 22,035,617. Vx V x ^ = 64,887. "* Hence the denominator = 1,298,530,726. * ^ ^ ^ = 0.0169696; D -2 = 0.000050046. o V* Then our equation becomes M, = 0.000050046^-^^ 0.0169696^-^. The following tables contain the necessary quantities for substitution in the above equation, using the values of H l found above and the same points for the location of the loads. 212 A TREA T1SE ON ARCHES. FIRST TERM. Load at Point No. ^'f^~ JgA * * t.*4* a ^ tjEfJt o * First Term. 2 83,226.4 22,034,233.5 1,185,691.6 20,931,768 1047.551 4 352,111-5 22,026,302.9 2,472,605.5 19,905,809 996-205 6 772,5ii-3 21,996,451.0 3,731,410.1 19.037,552 952.750 10 1,985,268.8 21,851,139.9 6,123,034.1 17,713,374 886.483 15 3,664,093.9 21,427,952.6 8,754,026.2 16,338,020 817.652 20 4,999,983-7 20,623,541.6 10,816,260.6 14,807,264 741.044 23 5,473,733-7 19,906,568.2 11,703,816.8 13,676,485 684.453 25 ' 5,644,454.5 19,107,367.4 12,073,221.3 12,678,600 634.513 25' 5,644,454.5 18,522,875.2 12,276,321.8 11,891,007 595.096 23' 5,473.733-7 17,704,577.3 12,502,906.4 10,675,404 534.26I 20' 4,999,983.7 16,250,248.4 12,292,268.3 8,957.963 448.310 15' 3,664,093.9 13,141,610.2 10,927,427-6 5.878,276 294.184 10' 1,985,268.8 9,041,875.9 8,102,179.8 2,924,964 146.382 6' 772,5"-3 4,928,737.8 4,623,750.2 1,077,498 53-924 4' 352,in.5 2,964,105.3 2,835,176.7 481,040 24.074 2' 83,226.4 1,237,593.8 1,209,345.9 "1,514 5.580 SECOND TERM. Load at Point No. *& o o* & '& IK*, 0* Second Term 2 320.6 64,791.9 4,423.8 60,688 1029.864 4 1,356.5 64,558.9 8,990.4 56,925 965.994 6 2,976.2 64,003.6 13,077.6 53,902 914.697 10 7,648.5 62.289.3 20,026.3 49,9" 846.978 15 14.116.4 59,029.2 26,444.2 46,701 792.502 20 19,263.1 54,544-9 30,371.8 43,436 737-094 23 21,088.3 51,265.1 31,536.4 40,816 692.646 25 21,726.0 48,023.9 31,420.7 38,329 650.431 25' 21,726.0 45,841.4 31,287.9 36,279 615.648 23' 21,088.3 42,980.1 31,075.4 32,993 559-8/7 20' 19,263.1 38,305-6 29,455.4 28,113 477.069 15' 14,116.4 29,461.9 24,698.2 18,880 320.386 10' 7.648.5 19,249.6 7.307.4 9-590 162.750 6' 2,976.2 10,032.0 9,425.4 3,582 60.798 4' 1.356.5 5,895.6 5,645.8 1, 606 27-257 2' 320.6 2,414.3 2,359-2 375 6-375 In making the computations above three decimal places were used throughout. These have not been given, hence the last figures may not exactly check THE ST. LOUIS ARCH. ALUES OF MI. 213 Load at Computed Values Load at Computed Values Point No. of M* . Load Unity. Point No. ot Af t Load Unity. 2 - 17-7 25' + 20 6 4 - 30-2 23' + 2 5 .6 6 -38.1 20' + 28.8 IO - 39-5 15' + 26.2 15 - 25.2 10' -4- 16. 4 20 4.0 6' + 6-9 23 + .8.2 4' + 3-2 25 + 15-9 2' + 0.8 With the values of a as abscissas and those of M l as ordi- nates the curve shown in Fig. 61, page 215, can be located. The following table shows the agreement between the values given in the History and those obtained from Fig. 61. COMPARISON OF VALUES OF M\. w = 0.8 TON. Load Upto- Values Given in History of Bridge. Values from Fig. 61. Difference. Percentage of Computed Values. (*) '(2) .(3) (4) I 1206 - 1244 + 38 3-0 11 - 3"4 - 3224 4- no 3-4 III - 4034 - 4226 4- 192 4-5 IV -3588 - 3848 -f 260 6-7 V - 2235 2529 + 294 It. 6 VI - 782 1123 -f 341 30.3 VII + 70 282 + 352 124.8 VIII + 232 - 128 + 36o 281.2 COMPARISON OF VALUES OF M t . I -f 161 -f 154 - 7 4-5 II -f 1013 + 985 - 28 2.8 III -f-2 4 66 -f- 2401 - 65 2.7 IV + 3821 + 3720 101 2.7 V + 4266 + 4008 -258 6.3 VI -f 3346 + 3096 250 8.0 VII + M38 + 1116 - 322 28.7 VIII + 232 - 128 + 360 281.2 214 A TREATISE ON ARCHES, In column (3) the positive sign indicates that the values in column (2) are too large. Here we see that the agreement in values is not as close as in the values of //",, but we also note that the greatest discrep- ancies occur in the small and non-important values. The maximum negative value of M t is in error but 4.5 per cent and the maximum positive value 6.3 per cent errors which are of little importance. The negative area in Fig. 61 is about 4.5 per cent too large and the positive about 6.3 per cent too small. Now since in this particular case the difference between these areas is small, we readily see why our discrepancy for a load over all is so large. The heavy broken line in Fig. 61 represents the correct curve. For practical purposes our curve is quite exact, and will give results as near the truth as any of the common methods in their special cases. Of course the particular advantage in the summation method is its adaptability to any case of the symmetrical arch. Temperature. Data.et = 0.000527, where t = 80, / = 520, E = 1944000 tons per square foot. Value of H t . From (239), _ Eefl _ 532734 ~~ In this case the actual values of d x must be employed in the denominator, or THE ST. LOUIS ARCH. 215 [ 5 rf 1 7 1 1 P s 7 1 / i =/ -~l f 1 i > -1 \ 1 \ 1 - \ 1 H3I N30 V 1 1 \ g \ N * 8 \ 8 \ g / =_ 3 y Q ^x ^ s?T B l N o 2 3AIIS(Dd 2l6 A TREATISE ON A ACHES. From the history of the bridge, H t = 204.9. 205.9204.9 = i, or an error of about one half of I per cent. Value o/M,. From (240), page 49, we see that M, =#; VT = 205.9(32-88) = 6769.9. i|f o V x From the history of the bridge, M, = 6747. 6769.9 6747 = 22.9, or an error of about 3.4 per cent. CHAPTER XI. THE SPANDREL-BRACED ARCH. THE so-called spandrel-braced arch usually consists of an arched bottom chord and a horizontal top chord connected by a system of web-bracing. Evidently the formulas of Chapters III and IV cannot be applied even approximately to this form of arch. The summation formulas, however, enable us to consider this type of arch either with or without hinges with comparatively little more labor than required for the ordinary form having a variable 6. To illustrate the method to be pursued we will take the case of a proposed design for a bridge over the river Douro by Mr. Max Am Ende and Messrs. Handyside & Co.* The form and general dimensions of the bridge are given in Fig. 62. c p ffi All dimensions in meters .86,? i. FIG. 62. When the general form of the structure has been decided upon, the first step is to approximately determine the sections * Design for a bridge over the river Douro by Mr. Max Am Ende and Messrs. Handyside & Co.; Engineering, London, 1881. 21 218 A TREATISE ON ARCHES. of the various members by the formulas of Chapter III or IV, using for the linear arch the parabola or circle which lies approximately midway between the two chords. For the application of the summation formulas the linear arch is assumed to pass through the centres of gravity of each vertical section. (Of course in both cases mentioned above the linear arch must pass through the supports.) The method of procedure is now the same as already explained for the arch with a hinge at each support and the arch without hinges. In computing the values of 6 for each section the moments of inertia of the flange sections about an axis passing through their centres of gravity may be neglected and the moment of each flange be taken as , where h is the distance centre to 4 centre of the flanges. Douro Spandrel-braced Arch, Let ABODE, Fig. 62, represent one half of the bridge, and suppose the approximate dimensions of members and the linear arch have been determined. We will divide the linear arch FIG. 63. into twenty equal parts, measure the co-ordinates at the centre of each division, and take the moments of inertia at the same points. Following are the data required for the determination of THE SPANDREL-BRACED ARCH. 219 DATA. Divi- sion. *AJ *Ajr * by X y *sin_ = 47265 _ | 2I83 _ 7I .2(26. 9 )| =4458.7 and ^-=121 {46.4 - 71.2(0.572)} = 115.2. . 4458.7-115.^25.25) 2000 //5> In like manner the values of H l for loads at the other points are obtained. The following table contains the values THE SPANDREL-BRACED ARCH. 221 of .//, for each division, the interpolated values for the ends of the divisions, and the values given by Mr. Max Am Ende, who took his origin of co-ordinates at the crown and measured the xs and /s to the extremities of the divisions. He then sub- stituted the proper quantities in * three equations, which he demonstrates, and eliminated all unknowns but H t . COMPARISON OF THE VALUES OF Hi Divis;on No. #i ffi at End of Divisions, Fig. 64. ii Max Am Ende. I 0.018 0.032 2 0.037 0.060 0.109 3 0.117 0.167 O.2O4 4 0.218 0.274 0.302 5 0.323 0.385 0.402 6 0.440 0.500 0.509 7 0.550 0.617 0616 8 0.673 0.725 0.701 9 0.775 O.SlO 0.762 10 0.835 0.840 0.792 (?) 0.8 ^~ 0,7 / ~ ^~ 0.6 / il'J 0.5 ^ / 0.4? f 0.3> j V 0.2 A ^ 0.1 4! ' .*** ^ r 3 LOAD 4 AT " 5 =>OIN' 6 r NO. 7 8 9 10 FIG. 64. * As far as known by the author, Mr. Max Am Ende was the first to suc- cessfully treat the fixed arch with variable 9, using the summation formulas By some manipulation his three formulas can be reduced to our general forms. 222 A TREATISE ON ARCHES. We see from Fig. 64 that our values lie above and below those given by Mr. Max Am Ende, and that the areas between the curves located by both series of values and the axis of a are very nearly equal, that is, for a uniform load over all the values of H l would be practically equal. We could not expect any closer agreement in values, considering the difference in method and the very approximate values of x and y which we used It will not be necessary to take up the deduction of J/,, V^ etc., as the method of procedure is precisely the same as that employed for the St. Louis arch. The more common form of the spandrel-braced arch is hinged at each support. The method of treatment is prac- tically the same as outlined above ; only the formulas for the hinged arch are, of course, used. CHAPTER XII. THE MASONRY ARCH. UNDER this heading we will include arches constructed of stone, brick, and concrete having spans of at least twenty-five feet. Before considering the many types of masonry arches we will first consider a type which is amenable for calculation by the formulas deduced for the elastic arch. This type consists of an arch-rib of masonry, with joints carefully made and as thin as practicable. At regular intervals this arch supports thin lateral walls, which in turn carry small arches or slabs which support the roadway. At the abutments the arch is protected from any horizontal pressures by retaining walls. The general features of this type are shown in Fig. 65.* FIG. 65. The dead weight of this style of bridge consists (i) of the weight of the masonry in the rib proper, and (2) the weight of * See "Bericht des Gewolbe-Ausschusses. Sonderabdruck aus der Zeit- schrift des Osterr. Ingenieur- und Architekten-Vereines," No. 20-34, 1895. 223 224 A TREA TISE ON ARCHES. the material above the arch which is transmitted to the rib through the thin lateral walls. The forces acting upon the arch-ring are evidently vertical. Now since any rectangular masonry joint will have the same kind of stress at all points when the resultant pressure upon the joint is applied within the middle third, our arch-ring will be in compression throughout if the equilibrium polygon lies within the middle third of each section. Then if the effect of the mortar joints be neglected the masonry rib will behave quite similarly to an elastic rib, and hence we may consider the formulas already demonstrated as applicable in this case. If the skew-backs are well fitted and the abutments or piers supporting the arch practically immovable, then the masonry rib is fixed at the ends, or at least more nearly fixed than hinged, as long as the equilibrium polygon remains within the middle third of the section. Since the arch-ring is necessarily made up of many pieces where either stone or brick is employed, it is practically impossible to so construct the arch-ring that there will not be more or less change in the position of the axis when the false- works or centring is removed. As a consequence the true position of the equilibrium polygon in the arch as constructed is somewhat uncertain. To avoid this uncertainty in the location of the equilibrium polygon, it is advisable to place in three or more joints which divide the ring symmetrically some material, as lead, covering the middle third of the joint. This locates the polygon within the limits of the area of the lead plates, and hence the maximum possible thrusts at these joints can be determined. After the falseworks are removed and the arch with its spandrels, etc., completed, these joints can be filled with cement, and become fixed at the ends for any additional loads.* This method is successfully followed by German engineers. For arches of the above type all loads are considered vertical, aud the arch-rib is assumed to be without hinges for moving loads. * See page 229. THE MASONRY ARCH. 22$ As in all arch designs, the general dimensions must be* assumed, and then the corresponding loads computed and the equilibrium polygons f drawn to determine if they lie within the middle third of the arch-ring assumed, and further, to be sure that the intensity of the pressure at any point in the rib does not exceed the safe strength of the material and that frictional stability is not exceeded. Having decided upon the shape of the arch, the span and rise of the axis being assumed, the next dimensions required are the thickness of the rib at the crown and that at the skew- backs. The assumption of these dimensions can be made with the aid of Table XXX. THICKNESS OF ARCH-RING AT THE SKEW-BACK. Theory (except in hinged arches), practice, and appearances demand that the depth of the arch-ring at the skew-back should be somewhat greater than at the crown. For vertical forces the horizontal thrust is constant throughout the arch, and hence the axial thrust increases as the secant of the angle of inclination of the axis. The thickness of the rib, however, should increase more rapidly than the secant of this angle, since it is seldom that the equilibrium polygons follow the centre of the arch-ring. As the polygon departs from the centre of the ring the maximum intensity of the pressure upon the joint changes quite rapidly, being twice the average intensity when the polygon passes through the third point of the joint. Having decided upon the depths of the crown and the skew-backs, the arch-ring can be drawn to scale. * See Alexander and Thomson's direct method for proportioning masonry arches, page 234. \ The graphic method is preferred for the preliminary investigations, being much shorter than the algebraic methods, and quite accurate enough. 226 A TREATISE ON ARCHES. EQUILIBRIUM POLYGON FOLLOWING THE AXIS OF THE ARCH-RING. The ideal arch would be one in which the pressure over the area of each joint is uniform, or the resultant pressure would pass through the centre of each joint of the arch-ring. This, of course, is impossible when the loading is movable ; but for the dead load of the structure the various parts can be so located that the equilibrium polygon will very nearly pass along the axis of the arch-ring. Now since the dead load is usually much greater than the live load, if the arch be designed so that the equilibrium polygon follows the axis for the dead load and a live load over all, the ring will be safe usually for a variable moving load. The loading necessary to make the equilibrium polygon follow the axis can be obtained approximately as follows : Assume the dimensions of the arch-ring and draw it to scale as shown in Fig. 66. Determine the distance mp and the location of the points a, b, c, etc., where the lateral walls rest upon the arch-ring. Then in Fig. 66 let abc, etc., be these points of division ; connect them by the straight lines ab, be, cd, etc. ; then abcde is one half of the equilibrium polygon which follows (nearly) the axis of the arch. We have now to deter- mine the relative and actual magnitudes of P,/ 5 ,/ 5 ,, etc., so that the points a, b, c, etc., will not be changed in position. The load at the crown can be determined at once from the assumed dimensions and weights. Lay off one half of this load as shown in Fig. 66, and draw 5 6 parallel to ge until it cuts the horizontal at P\ draw S t S 3 , etc., parallel to ed, dc, etc., respectively : then the distances P t P 4 P 3 P t , etc., cut off on the vertical are the required values of the loads at e, d, c, b, etc. A few trials will place the material above the ring so that these values will very nearly obtain. We have now all of the general dimensions of the structure from which the actual loads at abc, etc., can be computed. THE MASONRY ARCH. 227 Taking abcde as the axis of the arch, and assuming the above loads applied at the points abcde, the actual values of //,, V lt and M l can be found by means of the formulas already demon- strated, and the true equilibrium polygon drawn. FIG. 66. If lead joints are employed at the skew-backs and the crown, the values of //,, V^ etc., can be found under the assumption that the arch has three hinges, trials being made under the assumption that the hinges lie within the middle third of the arch-ring. If an actual hinge is placed at the crown, the starting-point of the equilibrium polygon is fixed. If no hinges are assumed, then the starting-point of the polygon must be determined in the same manner as for the metal arch. Extent of Loading which will cause the Equilibrium Polygon to follow the Axis of the Arch. In Fig. 66 let mg' represent the load at g ; then at the joints e t d, c, etc., lay off upwards from the lower limit of the arch-ring the loads P^P^ etc., and draw the curve jkm. This represents very nearly the upper limit of a homogeneous load corresponding to the polygon abc, etc. If now nop is drawn parallel to jkm at a distance mp below this 228 A TREA TISE ON ARCHES. curve, the shaded portion between the curve nop and the upper limit of the arch-ring represents the relative amount of material to be placed in the lateral walls. If the live load over all is included with the dead load, the point m would be raised an amount proportional to the added live load measured in masonry units. The axis of the arch shown in Fig. 66 is circular. If the angle at the centre had been larger and the curve jkm continued, we would have found the distance between it and the arch-ring increasing quite rapidly beyond an angle of 45 or 50 from the crown and becoming infinite for the semicircular arch. For this reason it is customary to consider the arch-ring to act as an arch for only about 45 or 50 from the crown, the masonry in the abutments or piers being built solid in horizontal courses up to this point. Moving Load. There remains now to be determined the effect of the moving load. If the actual maxima stresses are desired, the best method of procedure is to determine the effect of each load or concentration independently and combine the results. In most cases, however, the effect of the moving load is small, and it is necessary to consider but two cases, namely, moving load over all and moving load extending from one support up to the crown. Change in Dimensions. If after trial it is found that some equilibrium polygon for dead and live load combined departs from the middle third of the ring, the depth of the ring may be changed ; this need not necessitate a new calculation unless a great change is made, for the effect of the added material is likely to be very small, especially if the equilibrium polygon for the dead load follows the axis of the arch-ring. In case the equilibrium polygon lies outside of the middle third at any section, it does not necessarily make the structure unsafe unless the intensity of the pressure is sufficient to crush the material. The joints may open a little on the side farthest away from the polygon, so that it is not good policy to so design the ring that there is any such tendency. Concrete and Brick Arches. Evidently concrete and brick THE MASONRY ARCH. 229 arches can be designed in the manner outlined for the stone arch, using proper judgment as to the strengths of the materials. The concrete arch may even be made lighter, since it has considerable strength in tension. ARCHES WITH LEAD IN THE JOINTS AT THE SPRINGING AND THE CROWN.* In order to reduce as much as possible the uncertainty of the location of the equilibrium polygon at the crown and the springing, and also to reduce to a certainty its location within limits, German engineers have placed lead in the middle thirds of the joints specified. Evidently the equilibrium polygon cannot lie far outside of the middle third at these joints, as the lead acts similarly to a hinge. After the false- works have been removed the masonry adjusts itself until every joint is in equilibrium. Nearly all, if not all, this adjustment takes place at the lead joints, which are com- pressed in thickness and expanded around the edges until the pressure per square inch does not exceed about 3500 pounds. German engineers design these lead joints so that the maximum intensity of the pressure does not exceed about 1600 pounds per square inch, and have been very successful in their application of the method. After the structure is about completed and the entire dead weight is in place the joints at the springing are filled with cement and the arch becomes fixed at the ends for any additional loads. *" Fonts en Magonnerie avec Articulations a la clef et au joint de Rup- ture." Par M. G. La Riviere. Annales da Fonts et Chaiisse'es, juin, 1891. Abstract, Engineering News, Oct. 24, 1891. 230 A TREATISE ON ARCHES. ARCHES WITH STEEL OR IRON PINS AT THE CROWN AND THE SKEW-BACKS. Recently there has been constructed in Switzerland a concrete-arch bridge which has articulations at the springing- joints and at the crown composed of convex steel bearings resting in concave steel sockets or grooves. The entire depth of the arch-ring is reinforced with metal and the steel bearings placed at the centres of the joints. This mode of construction definitely fixes the equilibrium polygon at the springing-joints and the crown. EARTH-FILLED SPANDRELS In small arches the spandrels are often filled with earth from the arch-ring up to the roadway. FIG. 67. Assuming the earth to produce only the vertical pressures upon the arch-ring due to its weight, the determination of the equilibrium polygon offers no especial difficulties. But prob- ably the earth causes other than vertical forces, and these are more or less indeterminate. If the earth is assumed to be a homogeneous granular * The Coulouvreniere Concrete-arch Bridge, Geneva, Switzerland. gingering News, Aug. 6, 1896. En- THE MASONRY ARCH. 2$l mass, then the pressure upon the arch-ring at any point can be fairly well determined from the Theory of Earth-pressure.* If the arch is designed for this earth-pressure, it will sup- port a very considerably increased load at the crown, owing to the resistance of the earth over the haunches against heaving. Another feature which places the method of considering the earth-pressure acting against the ring as against a retain ing-wall upon the safe side is that longitudinal side walls must be used to retain the earth in the spandrels. These walls undoubtedly relieve the arch-ring from the direct thrust of the earth. If a retaining-wall is placed over the abutments, then the earth-filling may as well be treated as a vertical weight upon the arch-ring. PART EARTH AND PART MASONRY SPANDREL-FILLING. Under the assumption of vertical loading, it is found often that spandrels filled with earth alone are too light to cause the equilibrium polygon to follow the axis of the arch; then the spandrels are partially filled with masonry, as shown in Fig. 68. EARTH FIG. 68. This masonry is usually concrete or rubble masonry. It is seldom of the same class as the arch-ring masonry. As constructed, the upper limit of this masonry filling * Retaining Walls for Earth, by M. A. Howe ; John Wiley & Sons, N. Y. 232 A TREATISE ON AXCHES. slopes very gradually from the crown towards the skew-backs; hence the horizontal thrust of the earth above is practically eliminated. The exact action of this spandrel-filling upon the arch-ring is indeterminate. The assumption that it acts as vertical forces is on the safe side. MASONRY SPANDRELS WITH LONGITUDINAL VOIDS. Here the haunches are lightened by running longitudinal walls above the arch-rib and connecting them by arches or slabs immediately below the roadway, as shown in Fig. 69. FIG. 69. The amount of space to be left void can be found by the method outlined for lateral voids, but the masonry un- doubtedly exerts a much less pressure upon the arch-ring than under the assumption of vertical loads. Just what the pressure is cannot be determined. Such arches seldom if ever fail at the haunches owing to the resistance offered by the solid longitudinal spandrel-walls. If the arch-ring is designed for vertical loads the crown will not rise, as these walls cannot possibly exert a pressure equivalent to their weight. In fact good masonry can be stepped at an angle of at least 50 from the horizontal and be perfectly stable, provided the weight is balanced over the pier or abutment. THE MASONRY ARCH. 233 ARCHES HAVING SPANS LESS THAN TWENTY-FIVE FEET. These can be proportioned in the manner outlined for larger arches, but usually the ring is made much deeper than necessary owing to the economy in using material of certain dimensions. Stone arches seldom have ring-stones less than one foot deep. We have pointed out some of the difficulties which arise in the consistent designing of masonry arches of the usual type. The principal difficulty appears to be the determina- tion of the magnitudes and directions of the forces due to the dead load. If these forces are assumed as acting vertically and in magnitude the weight of the material included between vertical planes then the arch can be designed by the formulas already deduced for elastic arches, or by the direct and very consistent method proposed by Alexander and Thomson, which we will explain in the following pages. For the assumptions made, this method is the most general and con- sistent which has been advanced up to the present time. CHAPTER XIIL ALEXANDER AND THOMSON'S METHOD FOR DESIGNING SEGMENTAL MASONRY ARCHES.* " THE Transformed Catenary is shown by Rankine (Civil Engineering, Art. 131) to be the form of equilibrium for an ideal linear rib or chain under the uniform-vertical-load area between itself and a horizontal straight line. This curve has received considerable attention from early times because of its importance in designing arches, and is known best, per- haps, by engineers as the equilibrium curve. " It seems to have been assumed that the transformed catenary, like the common catenary and the parabola, had its curvature continuously diminishing from the vertex out- wards. " In the following investigation it is shown that a very close resemblance exists between certain of these equilibrium curves and the circle a fact important to engineers." EQUATION OF THE COMMON CATENARY. From Rankine's Civil Engineering, Art. 128, (I) * Transactions of the Royal Irish Academy, vol. xxix, part ill, 1888. On Two-nosed Catenaries and their Application to the Design of Segmental Arches. By T. Alexander, C.E., Professor of Engineering, Trinity College, Dublin; and A. W. Thomson, B.Sc., Assoc. Mem. Inst. C.E., Lecturer in the Glasgow and West of Scotland Technical College. This is an elaborate paper, contain- ing many interesting things which are omitted here as not being essential for the mechanical method of designing arches. 234 SEGMENTAL MASONRY ARCHES 235 where y = the ordinate of any point ; x = the abscissa of any point having the ordinate^ ; m the parameter ; and e = the base of the Naperian system of logarithms. w J Directrix N K FIG. 70. THE TRANSFORMED CATENARY. The locus of a transformed catenary is obtained by in- creasing or decreasing all the ordinates of a common catenary by a given ratio r. Then for the transformed catenary dy tan = -r- = -\e m where y^ is the value of y when x = o P = <*-y dx* m = rm. my py_ m? . (II) . (Ill) (IV) A TREATISE ON AKCHES. where is the slope at any point and p the radius of curvature. For the crown (iv) becomes and hence (v) becomes sec 8 = -L PoJo (VI) (VII) THE TWO-NOSED CATENARY. An investigation of (iv) for maxima and minima shows that for values of r less -=. there is a maximum radius of curvature p t at the crown and a minimum radius of curvature X Q a;, KO Directrix FIG. 71. p, at a pair of points (-?,/?/) symmetrical about the crown, where (VIII) Such catenaries are called two-nosed. SEGMENTAL MASONRY ARCHES. 2tf If m be assumed as unity and r be given values less than V^, the values of p lt y^ x lt y a , and 0, can be readily com- puted. A large number of these values are given in Table A. As an aid in computing the ordinates, etc., of the two- nosed catenary, the general formulas may be put in the following forms : Let ,==,.-*. = * ..... (ix) f ./a Then, from (vili), From (III), 1 ^; (xi) 2 , -*) ( XII > From Rankine's Civil Engineering, Art. 131, (XIII) It may be noted here that, given a certain value of s, all quantities are directly proportional to m excepting 0, , which is constant for any given value of s, regardless of any change in m or />,. 238 A TREATISE ON ARCHES. THE DESCRIBED CIRCLE. In Fig. 71 if Bi be prolonged, it will cut AQ, in Q t . If Q 1 be taken as a centre and B t Q, as a radius, and a circle described, it will evidently lie wholly above the two-nosed catenary between the points B, and Z?/. This circle will also lie beyond the catenary curve for some distance beyond B^ and B{, cutting it finally in B 4 and B{. Let RI be the radius of the described circle. Then, from Fig. 71, R^ = x^ cosec 0, , (XIV) OQ l = b =y 1 +^j cos0,, .... (XV) and (xvi) The values of j?, , b, and F are given in Table A for the values of r which were used in computing the ordinates, etc., of the two-nosed catenary. An examination of this table shows for r = V%, or s = , that j = F , or the described circle touches the two-nosed catenary at the crown. That is, .#,/?,' and A and K coincide. Also, that between the values of s 0.027 and s = 0.0204, F changes sign, indicating that the described circle cuts the directrix. The distance apart of the described circle and the two- nosed catenary at the crown is XA=*.=s.-Y t ...... (xvii) The values of 6 are given in Table A. THE THREE-POINT CIRCLE. Evidently for the two-nosed catenary there must be a point beyond B l which has the same radius of curvature as at SEGMENTAL MASONRY ARCHES. 239 the crown. There will be a similar point on the opposite side of the crown. A circle passed through these three points will Directrix evidently lie below the catenary between /? A, and /?/. This circle is called the three-point circle. Let R^ be the radius of the three-point circle which passes through the three points of equal curvature of the two-nosed catenary. , (XVIII) y. Vs y^ = y 9 sec 8

6, > 6 y , or the two circles approach each other as they leave the crown. Between s = 0.027 and s =0.0204 (s = T V) ^0=^1= ^,* r the two circles are concentric. Beyond these values of s the circles diverge. Then if through A a symmetrical circular arc is passed concentric with the described circle, the equilibrium polygon or two-nosed catenary will lie between the two out to the points of rupture B, and B. Then much more will it lie be- tween the described circle and one of a less radius than the concentric circle. Table B. In Table A are given the various co-ordinates of points on the described circle and the three-point circle for a modulus m = unity. Suppose now we wish to base all of these quantities upon the radius of the described circle and take its value as unity, then it is necessary to divide each linear quantity in A by the corresponding value of /?,. Table B is the result of such an operation. The object of this will appear from the following: Suppose a circle of radius unity be drawn, and let this circle be taken as a described circle ; then R^\. Since s, 0,, and 0, are independent of R lt it is evident that all of the two-nosed catenaries in Table A can be constructed within this circle of unit-radius merely by changing each ordinate proportional to R t . We have then to the scale unity (R l = unity) an exact representation of the relations between the several curves we have been considering. If R t has any other value than unity, we have only to mul- tiply these quantities by the new value of the radius. In Fig. 73 let XX be the upper limit of masonry to be supported by an arch, and OA the depth at the crown when R l unity; then, from Table B, OA = y a can have values from 0.23 to 0.05, R^ S , 0,, etc/, corresponding values, and 242 A TREATISE 0A' ARCHES. for any particular case the equilibrium polygon wiii never be above B^KB{ between B^ and j5/, and never below B^ABJ between B^ and B, . Then if the portion of the masonry between B^KB^ and B^AB^ were cut into arch-stones the structure would be stable under the assumption that B^AB^ does not sensibly differ from BJB^AB^B^ which is the case within the limits of our values of s. If this arch with R, = unity is in equilibrium, then any other arch of the same proportions would be in equilibrium. But although we have equilibrium, we have not strength probably; and besides, in masonry the equilibrium polygon should not depart at any place from the middle third of the joints, and often it must follow more closely the centre line to obtain intensities of pressure consistent with the strength of the material employed. The arch-ring specified above decreases in depth as it leaves the crown while the pressures upon the radial joints increase, indicating that the lower boundary of the ring should be changed so that the depth would increase as the stresses increase. This can be done after the depth at the crown is known. As this depends upon the material, we must deter- mine the permissible intensities of the stresses, etc. Horizontal Thrust. According to Rankine's Civil En- gineering, Art. 131, ff i = H x = H wm\ . . . (xxix) where w is the weight per unit volume of the material taken as solid from the directrix down to the two-nosed catenary or the equilibrium polygon, or approximately the three-point circle. If d is the depth at the crown from the directrix to the soffit, H u< m* = wpji. .... (xxx) SEGMENTAL MASONRY ARCHES. 243 If T is the thrust at any point, (xxxi) Intensity of Pressure. In Chapter I it was shown that if the resultant pressure on any rectangular joint was applied at the third point, the maximum intensity was twice the average intensity and the minimum intensity was zero. Assuming, then, that the equilibrium polygon at the crown is to be applied at the lower third point of the key- stone, if t^ is the depth of the key, / = 3 sec # a . By means of the values obtained for Po <^> ^j the thrust at crown and joint of rupture and the centre of stress at joint of rupture are calculated, as in pre- ceding example. A tangent from this last point enables a suitable abutment to be designed. CHAPTER XV. TESTS OF ARCHES. RECENTLY the Austrian Society of Engineers and Archi- tects have published a report of a series of tests made upon full-size arches. The publication contains 131 folio pages with 27 plates.* The experiments are minutely described and thoroughly discussed, and a comparison made between the results and those theoretically obtained. The tests of greatest interest were those made upon five arches having a span of 75. 4 feet, a clear rise of 15.1 feet, and a width of 6.56 feet. These arches were i. Rough quarry-stone; 2. Brick; 3. Concrete; 4. Concrete Monier type; 5. Steel. Rough Quarry-stone Arch. This arch was constructed of rough quarry-stone laid in Portland cement-mortar composed of i part cement and 2.6 parts sand, the test being made 51 days after its completion. The thickness at the crown was 23.6 inches and at the skew-backs 43.3 inches. The loading was applied vertically at five points, dividing the half-span into five equal parts. The ultimate load causing rupture was 660 pounds per square foot over one half of the span. * Bericht des Gewolbe-Ausschusses des Oesterreichischen Ingenieur- und Architekten-Vereins. Vienna, 1895. See also Eng. News, Nov. 21, 1895, p. 351, and April 9, 1896. 253 254 A TREATISE ON ARCHES. The arch failed by radial cracks appearing on the extrados near the skew-backs on the loaded side and over the haunches on the unloaded side. Brick Arch. This arch was identical in dimensions with the stone arch, and failed in a similar manner under a load of 602 pounds per square foot over one half of the span. Concrete Arch. The thickness at the crown was 27.6 inches, and at the skew-backs 27.6 inches. This arch was made up of segments of concrete composed of mixtures of different proportions, and at the skew-backs the joints between the arch-ring and the abutments were filled with asphalt about \ inch thick. The arch failed under a load of 742 pounds per square foot over one half of the span. Monier Arch. Here the general dimensions were the same as before, but the thickness at the crown was only 13.8 inches and at the skew-backs 23.6 inches. The arch failed under a load of 1300 pounds per square foot over one half of the span, failing by cracking as follows: 1st. On the loaded side at the skew-back; 2d. On the unloaded side at the haunches; and, 3d. On the loaded side at the haunches. Steel Arch. Failure took place under a load of 1564 pounds per square foot over one half of the span, by the buckling of the unloaded portion near the haunches. Deformations. Throughout the tests careful measure- ments were made of all deformations caused by removing the falseworks, temperature changes, and the changes in loading. The appearance of the first crack was noted, with the magnitude of the load causing it. The arches were finally tested to destruction and the load causing failure carefully determined. From these records a comparison was made with theoreti- cal results. Comparison with Theory. It was found for the stone and brick arches that failure occurred in the joints, the mortar TESTS OF ARCHES. 2$$ separating from the stone or brick. The adhesive strength of the mortar for the stone arch was found to be about 120 pounds per square inch and the value of E about 960,000. In the brick arch the adhesive strength of the mortar was about 70 pounds per square inch, and the values of E varied from 340,000 to 470,000. From the results of the tests of the concrete arch the average ultimate strength of the concrete was placed at 290 pounds per square inch, and the value of E at 1,430,000. The Monier arch cannot be discussed theoretically, owing to the use of metal imbedded in the concrete. The value of E as determined from the tests of the steel arch was about 26,000,000, which is a little smaller than the value obtained from the tests of small specimens. Even in the masonry arches the deformations were pro- portional to the loads up to a certain point, showing that the material behaved the same in the arch as in small specimens for testing. The measuring devices placed near the skew-backs indi- cated that on the loaded side the arch was practically fixed at the ends and on the unloaded side very nearly so. Of course the concrete arch with asphalt plates at the skew-backs must be excepted. This . arch behaved neither as fixed nor hinged, and the theoretical results were taken as the mean of those obtained by considering the arch as fixed and then as hinged. In all cases the arches failed at points which theory pre- dicted, CONCLUSIONS DRAWN FROM THE RESULTS OF THE FIVE EXPERIMENTS. The very important conclusion drawn from these experi- ments was that the masonry arches behaved very nearly as elastic arches fixed at the ends, and hence the formulas for elastic arches were the only formulas which should be em- ployed in designing such structures. 256 A TREATISE ON ARCHES. The close agreement with theory under the method of applying the loading employed in these experiments is a very strong argument in favor of the type of spandrel construction advocated in Chapter XII. A few very old bridges and some modern bridges have been constructed after this form. The only argument against this method is that in bridges of long span the effect of changes in temperature sometimes cracks the masonry above the small arches; but this can be avoided by making a vertical joint near the skew-backs, as was done in the Coulouvreniere bridge. SPECIFICATIONS. The following specifications are advocated in Chapter VII of the Austrian report. All Large Arches must be designed according to the Elastic Theory. Two cases of live loading may be considered: (i) load covering entire span, and (2) load covering but one half of the span. For railway bridges the rails should be at least 3.28 feet above the crown of the arch, and this space filled with some cushioning material. Brick and stone arches, where the ratio of the rise to the span lies between one half and one fifth, may have depths at the crown as specified below. For spans of 30 metres, thickness of crown = i.i m. 40 " " " 1.4 " it 55 " 120 " " " 4.1 " For segmental arches the thickness at the skew-backs may be i the thickness at the crown, and for semicircular arches 1.7 the thickness at the crown. TESTS OF ARCHES. 2 57 The width of the bridge at the crown should never be less than the following: For spans of 30 metres, width = 2.4 m. 40 " " 3.0 " " 65 " 4.5 80 " " 5.6 " IO o " " 7.0 " I20 dy = yA^ + -LF* dy 2 M,x + Vj? - H, Substituting the value of dy S - - dx [from/(65)], P ; '01 i -L- - 6pb( X - )].( - x} 263 264 A TREATISE ON ARCHES which reduces to From (42), ^, = F, sin + 7/, cos 0. From/(59)> Then APPENDIX A. 265 n From./(6o), m~\ hence ~ I _V* = ^j / */9 x ( t/0 From (39) and (40), and V * ^ ~ 2P ' /(40) Substituting these values in the above equation, - f ^dx=" 1 ^ /Tocos' EJo F * A (Jo L rr. * -,) + / //, cos dx > c/O L J) - ^\_ P f a cos ' ^ J - i [^jf^ 08 ' i ^i* anc ^ ^ = pdz ; hence dy zpdz, and we have 266 A TREATISE ON ARCHES. Substituting the value of 2, s' 0rfy = - l -p log f I 2V In practice seldom exceeds -; then for this ratio -^- > = ; when y < /, - - < . Then without sensible error we may take and /(74) / cos 2 <}>dx. or * As before, let z = tan = ; then cos* = and ^ = pds, and we have * * /'cos' 0dfor = / T^dz p\ tan" 1 z = p\ . I/O I j 2 \_ *. Therefore ., s ll 0^=-/(0-0 )=X < / > o-0). . . . /(75) /'cos* 0^/7. . . * From demoastration by Prof. Weyrauch. APPENDIX A. 267 Therefore - / cos* <>dx. - jT'cos' $dx = [~= -/ftan- 1 ar = -/F0 = /(0 ) a a a Therefore jTcos' 0d^ = X - 0) ..... /(77) Substituting /(74). X75). /(7 6 ). and/(77) in /(73). reducing and factoring, we obtain - 2P(y -b}- Substituting /(7i),/(7 2 ) and/(?8) n/(7o), we have 268 A TREATISE ON ARCHES. - a)' -f 3 5 - 15 X79) APPENDIX B. INTEGRALS EMPLOYED IN THE DEDUCTION OF Ay FOR PARABOLIC ARCHES. EQUATION ^(84). ,,0 P d t/O et f'dy = et t/O t/O -)] or 269 A TREATISE ON ARCHES. e, dx dx dy( V* sm +H x cos 0) = : \ V f i sin* dx -j- //^ I cos 1 ]dx = x /(0 0); /* ,-* sin* (fidx = / (i cos* (f>}dx = x a t>(a (b\ l'a r py I COS 0 = 0.13607'; 0.1360 - = 0.069 = relative error. 272 APPENDIX C, 273 2. Assume a single load acting at the crown of the arch, and let the rise be 25. Then = 45 = 0.785, p = 50, and * = f From (640), From (74), H< = -^ {26041 - (0.7813 o.7724)/ ) = 0.0089 = o.oi 14 = relative error. 2a. The same as 2, with k \. From (640), From (74), { 18560 - 37}^ = 0.5503^ ; (0.5568 -o. 0.0065 0.5568 = o.oi 1 6 = relative error, which is practically the same relative error which was found in 2. 3. Assume a single load acting at the crown of the arch, and let the rise be 50. Then 0, = 63 26' = i.n, p = 25, and 274 A TREA TISE ON ARCHES. From (640), H l = I (0.3125)7'= O.39O4/ 3 . From (74), //" = 152O53 40}^ = o.3894/ > ; 2003330* 3 (0.3904-0.3894)^ = O.OOI 0.3904 = 0.0025 = relative error. 30. The same as 3, with k = . From (64^), From (74), (0.2751 o.2744)/ > = 0.0007/ 3 0.0007 = 0.0025 = relative error. The above results are tabulated below for convenience in comparison. HINGED ARCH WITH VERTICAL LOADS. Load at Crown, * = i. Load at Quarter-point, k =* J. /// Values of H t . Values of ff, (6 4 a) (74) Diff. Rel. error, J{. (6 4 a) (74) Diff. Rel. error, *. O.IO 0.25 0.50 I-953I 0.7813 0.3904 I.8I7I 0.7724 0.3894 0.1360 0.0089 O.OOIO 6.9 I.I O.2 0.5568 0.2751 0.5503 0.2744 0.0065 0.0007 6.9 I.I 0.2 APPENDIX C. 2/5 The results in the above table are probably not correct in the fourth decimal place, but for our purpose they are suffi- ciently exact. The following conclusions may be drawn from the tabu- lated results : i. The position of the load has little or no effect upon the magnitude of the relative error. 2. The common method in general gives results which are too large. 3. To obtain results which are not six or seven per cent too large, the formulas wJiich consider the influence of the axial stress must be employed for flat arches. 4. For arches having a rise equal to one fifth or more of the span the common formulas are sufficiently accurate. (b) Horizontal Loads. A series of computations similar to those made for vertical loads indicated that for arches having a rise of one fifth or more of the span the common formulas can be employed. For flat arches the effect of the axial stress should betaken into account, as the results obtained by the common formulas are from six to ten per cent too small for arches having a rise of about one tenth the span. ARCH WITHOUT HINGES. (a) Vertical Loads. i. Assume a single load on the crown of the arch, and let the rise be 10. Then = 21 48' = 0.38, / = 125, and k = . From (910), // l = H?(o.o625)/> = '2.3437* 4 ;.*.":?* ".:. '.; ?rn>- v'T .:>'..-, From (101), H, =-0.262616.25 0.103 }/> = i.6i$oP. 2/6 A TREA TISE ON ARCHES. (2.3437 - i.6iso)P - = 0.309 relative error. 2-3437 ia. The same as i, with k = . From (910), = 1.3162^. From (101), H l = 0.2626(3.51 o.o77)/> = 0.9007/*. (1.3162 0.9007)^ = 0.41 5 5/>. = o. 3 1 5 = relative error 1.3162 2. Assume a single load on the crown of the arch and let the rise be 25. Then 0. = 45 = 0.785, / = 50, and k = %. From (gia), ^ = ^(0.0625)^=0.9375^ From (101), HI = 0.1419(6.25 o.o6)/ > = o.8784/> (0-9375 0.8784)/ 1 = 0.0591^. 0.0501 ---- = 0.063 = relative error. 2a. The same as 2, with k = \. From (910), ff, = 15(0.035 i)P=o. APPENDIX C. 277 From (101), I 0.1419(3.51 o.04)P = - 0.4910) = 0.0355 0.5265 = 0.067 = relative error. 3. Assume a single load on the crown of the arch and let the rise be 50. Then = i.n,/ = 25, and k = %. From (910), , = ^^(0.0625)^ = 0.4687^. From (101), i = 0.074(6.25 o.O2)P = 0.4610/1 (0.4687 0.46 1 d)P = 0.007 7 P. 0.0077 0.4687 = 0.017 = relative error. Collecting the above results for convenience, we have the following table : ARCH WITHOUT HINGES VERTICAL LOADS. Load at Crown, k = i. /// Values of//,. Values of H l .> doi) Diff. Relative Error, %. , (JOI) Diff. Relative Error, %. 31-5 6.7 o 10 0.25 0.50 2.3437 0-9375 0.4687 1.6150 0.8784 0.4610 0.7287 0.0591 0.0077 30.9 6-3 1-7 1.3162 0.5265 0.9007 0.4910 0.4155 0.0355 Load at Quarter-point, * = }. 2/8 A TREA TISE ON ARCHES. From this table the following conclusions may be drawn : 1. The position of the load has little or no effect upon the magnitude of the relative error. 2. The common method in general gives results which are too large. 3. In arches which do not have a rise equal to at least one fourth the span, the effect of the axial stress is too great to be neglected. It amounts to about thirty per cent for arches having a rise equal to one tenth their span. (b) Horizontal Loads. A series of computations similar to those made for vertical loads indicated that for loads near the crown of the arch the effect of the axial stress can be neglected. For loads near the supports and the quarter-points the effect of the axial stress amounts to at least six per cent for arches having a rise of one tenth their spans, but decreases rapidly as the ratio increases. Since horizontal loads are usually caused by wind, and the ratio of the wind stresses to the live and dead load stresses is small (ordinarily), the common method is probably sufficiently exact for practical purposes. CIRCULAR ARCHES. The above conclusions are based upon examples of par- abolic arches. For flat circular arches (rise less than one fourth the span) we can safely predict that practically the same conclusions will obtain, since the parabola and circle so nearly coincide. We will solve a few examples, which will show the exact effect of the axial stress upon arches of greater rise. APPENDIX C. 279 CIRCULAR ARCH WITH HINGE AT EACH SUPPORT. (a) Vertical Loads. i. Let /= 100 and/ = 25. Then R = 62.5 and k' 62.5 25 = 37.5. tan = ^ = 1.333 ... .'. 0. = 53 71'- 20 n IO6.25 = 5 ^ = 0.590. 7T ISO From Table XVII, for a = o, or a load on the crown : For ' = 0.58, = 0.758. For = 0.60, = 0.726. 7T D Then for = 0.59, = = 0.742. 7T 02 From (160), From (164), I (sin 1 sin* a) ir*--5r- I + g(0o + Sin ^o COS 0.) \vhere ^ = 0.742^. 280 A TREA TISE ON ARCHES. From (153), 44 4 m = 2 - = ^, say ^-, = 0.00102. (62.5) 3906 3900 From Table XVIII. for0 = 53, 6 = 0.1532454 for = 54, B = o. 1 67 1 294 600.0138840 .00023140 =diff. for l' 71 0.00179335 = diff. for 7!' 0.1532454 .% for = 53 7f, B = 0.1550387 g = 0.742. /. A = (0.742X0.155) =0.115. From Table XIX, by interpolation, (0 e + sin cos ) = 1.407 Substituting these values, 0.00102, e i (0.64 o) *, = 0.742-^? O.OOIO2 or ^ = 0.733^; (0.742 o.733)/' = 0.009^; 0.009 = 0.012 = relative error; APPENDIX C. 28l or for a load at the crown the results by the common method formula (160), are about 1.2 per cent TOO LARGE, which is practically the same as found for parabolic arches of the same rise. la. Let a load be placed at the quarter-point in i. Then ... a = 23 35' = 23^.583, = = 0.443. Interpolating in Table XVII, = = 0.570. D From example i, B = 0.155 .-. A = (0.155X0.570) = 0.08835. From (160), From c(i\6), which is (164) in another form, H p 2 A m(s\n* sin' a] 2B + 2w(0 -f- sin cos ) or H = p ai 7 6 7~ 0.00102(0.64-0.16) = 0.31287 ' (0.570 O.563)/ 5 = o.oo7/ ) ; 0.007 - ' = 0.012 = relative error, 0.570 which is the same relative error obtained for a load on the crown. 2. Assume a vertical load on the crown of a semicircular arch. / = 100, / = 50, and k = . Let (radius gyration) 8 = 4, o = 90, and a = o. Then ^ = i and = o. 71 ( t 282 A TREATISE ON ARCHES. From Table XVII, 2d> a. A n for = i and = o, = 0.318. 7t (fi g B From (160), From (153), Then from (164), (0.318 0.317)7'= o.ooiP; = 0.003 = relative error, 0.318 which is too small to be of any practical importance. APPROXIMATE FORMULAS. Very close approximate formulas for parabolic arches can be formed by applying correction factors to the common formulas for vertical loads. Arch with a Pin at Each Support. Let fy = the horizontal thrust as given by the common method ; then where e is the relative error. Computing the value of e for several problems and plotting these results, the following table can be made by means of a curve drawn through the plotted points. APPENDIX C. 28 3 Arch without Hinges. Results obtained by the use of the approximate formulas will be sufficiently accurate for the ordinary problems met with in practice. VALUES OF i e AND i e' IN THE APPROXIMATE FORMULAS FOR HI. //' Hinged. Fixed. I f' I - e* O.IO .0690 .9310 .310 .690 O.II .0570 9430 .265 735 0.12 .0480 .9520 .230 770 0.13 .O42O .9580 .207 793 0.14 .0370 .9620 .186 .814 0.15 .0330 .9670 .170 .820 0.16 .0295 9705 153 .847 0.17 .O265 9735 .140 .860 0.18 .O24O .9760 .126 .874 0.19 .O2I5 .9785 .115 .885 O.2O .0195 .9805 .104 .896 0.21 .0170 .9820 .094 .906 O.22 0153 .9847 .086 .914 0.23 .0140 .9850 .078 .922 0.24 .0125 .9875 .070 930 0.25 .0115 .9885 .065 935 0.26 .0100 .9900 .060 .940 0.27 .0095 -* 49905 .058 .942 0.28 .0085 .9915 053 947 0.2Q .0080 .9920 .050 .950 0.30 .0074 .9926 047 953 0.31 .0068 9932 043 957 0.32 .0063 9937 .040 .960 0.33 .0060 .9940 .038 .962 0-34 .0056 9944 035 .965 0-35 .0052 .9948 034 .966 0.50 .0020 .9980 .020 .980 The above table can be used when the arch rib does not have too great a varia- tion in 6 from the crown to the supports. Increasing 6 at the supports' decreases the effect of the axial stress. This is quite marked in the present form used in concrete arches. APPENDIX D. SPECIAL CASE SEMICIRCULAR ARCH. 2Ed =r- = a constant. J\. SINCE semicircular arches are sometimes employed for large roof-supports, we will give the necessary formulas for determin- ing the outer forces. The effect of the axial stress will be omitted, as its effect can be neglected in practice. See Appendix C. ARCH WITH FIXED ENDS. (a) Vertical Loads. Tt Since = -, sin 0, = I, and cos = o. Then from ^(133), . 7t n . 2(cos a + oi sin a) ---- sin* a and + cos a -f- sin a By making a negative (hence the sin a will be negative) in the value for M^ we have 284 APPENDIX D. 285 It Tt 71 -f cos a -f a sm a _ For any single load we have, from (5 i), M. Substituting the values of M^ and H^ found above, we obtain after reduction ^ >m a I cosaJ a ue cos a -|- sin a. 2 . -j~ 2 cosa-\-2, + -(i sin a)y t + cos 1 at. 286 A TREATISE ON ARCHES. The values of cos a -j- a sin a can be found from Table XXII. (b) Horizontal Loads. From (154), , n \ (sin a cos a a) -j- 2(sin a a cos a) TT ^"(2 I I _2 ' * 1 2 - "4 From ^(156), a cos a sin a I =11 ^ -' x | -| cos cos 3 a By making a negative in the above equations, they become * n f -(a sin a cos a) -f- 2(a cos a sin a) 7? 2 and i^( sm - M it j -| cos a cos* a f From (47), The values of ^, , j, , j, , and ^r, can be found from (51) and (54> APPENDIX D. 287 (c) Effect of a Change in Temperature. From (159), From c(i6i), From (51), , _ Ae? ~ ~~~ ~ ~' M, 2R = = -=0.632^. ARCH WITH A HINGE AT EACH SUPPORT. (a) Vertical Loads. From = a constant. LET 6 cos = A = a constant, and neglect the terms con- taining F x ; then for a single vertical load we have from ^(90), page 117, Value ofH,. x)dx dx P. Substituting the value of y in terms of x, the following values of the respective integrals are easily obtained : * f\l - = -//'(! ~ * The general formulas for the parabola are given on pages 52 and 53. 290 A TREA TISE ON ARCHES. (I - ty - X}dx = \l*k(\ - Substituting these integrations in the expression for //, and reducing, _ \fl\k - 2k* + ^) - \fl\k - V) ~ or flr ......... (90 i- For a single vertical load, from ^(101), page 119, we have M t = f' a)ydx = (i 2 APPENDIX . 293 Hence . (64) Value of ff t . For a single horizontal load we have, from (140), page 127, The value of the integral in the numerator of the fraction is given above in the deduction of (115); the denominator equals -faff; hence . . (77) SYMMETRICAL CIRCULAR ARCH. 2E (a) Arch without Hinges Special Case where ^-6-=.a constant. Value of H t . For a single vertical load we have, from (90), page 117, neglecting the terms containing F xy - x)d 294 A TREA TISE ON ARCHES. Performing the integrations indicated, we have, remember- ing that a kl t - f X yd = - ^70. i/O - (I - k) f t/O xyd = - #70. -f \Kla + - + ' a* ab.. - k f'y(l - c/a Combining these values we obtain \al - i^/0. + \k'la + bk' - ak'a - . - r Combining these two values, we have aa ^la. + f /0 b. // d 20 ; . - 3*7); APPENDIX E. 295 Substituting the values found above for the integrals in- dicated in the expression for //, it becomes, after reduction, _ 2bl - 1(1 - 2a)(& -a)- 2a >. '~ which readily reduces to (192). Value of M^ For a single vertical load we have, from (101), page 119, - pi , n j * nl -I d\- X *d\-\- xd /o I / ) ( /o The values of the integrals not already found above are / *^ = ^J ^. Then the terms containing //", reduce to r where d = ~-. 2 9 6 A TREATISE ON ARCHES. a)d = i/0. 4- */ - - aa + ; Then for the terms containing /*we have (l2d)(b d}t -\- 2R*a

) ds, . r^- (~^W I From integrals already evaluated the denominator of this at once becomes APPENDIX E. 297 2bk'a ; Making the proper substitutions, we obtain _ ral-lk'a-lba-ak'fa -f- tylfaabfa R^.aR^* ~ which reduces to (207). Value of M^ For a single horizontal load we have, from -(106), page 120, 1 ~~ /*' /*' / *' \* JfaJ&W -(/,*<**) From integrals already evaluated the denominator becomes . where f\y - b}xd = - (/' - - (^ + J)K0. + ) + 2*]); = - (I - a - ba - k'a - #0.). The integrals in the terms containing ff, have been evaluated above. 298 A TREATISE ON ARCHES. Substituting the values determined above, we have )!. . . ,(155) which reduces to (212). (b] Arch with a Hinge at each Support Special Case where 2E 6=a constant. J\. Value of H t . For a single vertical load we have, from page 125, />; Therefore Numerator = %a(l a 2k' a) \k\l^ la Hence _ D a(l - a - 2k' a] - k'(l t -let- 2&) *~ - - which reduces to (160). APPENDIX E. 299 Value of HI. For a single horizontal load we have, from ^(140), page 127, = - 2 = K*( + sin a cos a + 2 cos [sin or a cos a]). - ?*$ = ^(^o - 3 c s 0o sin + 2 cos s 0.0.). t/O Hence Q( a sin a cos a* 2 cos (s?n a a cos a \ /.- 2 \ " i 0;-3cos0 sin0 + 2cos s f* V 7 ' APPENDIX F. EFFECT OF A COUPLE UPON A SYMMETRICAL ARCH. (a) Arch with a Hinge at each Support. Value of H,. Let M a be any couple applied at any point a of the arch : then VJ=M. or V, = ^. Evidently V t is numerically equal to V^ but acting in the opposite direction, and //, = H f If another couple equal and symmetrical to M a be placed upon the arch, r i = =r. and fi, = 2ff l= \ th f horizontal thrust at the | left support. If the arch be assumed free to slide upon the supports, the change in the length of the span due to a horizontal load Q applied at each support is given byfn6), page 122, or Now let the loads Q be removed and two equal and sym- metrical moments be applied to the arch; the corresponding change in the length of the span will be where M, is the resultant moment at any section x. 300 APPENDIX F, 301 If ^, represents the magnitude of the horizontal thrust necessary to cause a change in the length of the span of the loaded arch of A" I, we have Substituting the values of A' I and A" I, we have C 1 M X \ -f /0 * X = O and jl/, = J/ a from x = a, to ^r = Then since //, = 1^ , This equation is perfectly general for any symmetrical arch having a pin at each support. (b) Arch without Hinges. Value of MS Let a couple M a be applied at any point a on the arch; then the moment at any point x is M M = M l -\- V,x - Hj + M a . . . x > a. Since the arch is fixed at the ends J0 = A t , and as it is symmetrical Ac = o. Substituting the value of M x in g(62) and ^64), page in, we obtain, neglecting the axial stress term, 3O2 A TREATISE ON ARCHES. and Eliminating V l and solving for M t , we have in which everything is known excepting H lt which can be deter- mined as follows: Value of H^. Let two equal and symmetrical couples act upon the arch, and assume the arch free to slide upon the sup- ports. Also assume that there are equal and symmetrical moments applied at the supports. Then from ^(62) we have C l M x ds = 4/0 * Since the arch is free to slide upon the supports, //, = o ; and since the applied couples are equal and symmetrical, F = O. Therefore where K' is the additional moment at the section x caused by the action of the applied couples. Substituting the value of M x and solving for M l f'Ei J. <>' r' J, . l K'ds M, = - APPENDIX F. 303 The change in the length of the span due to our couples is (see ^(80)) l K'ds C l / .* Now suppose the arch unloaded and free to slide as before. Let two equal and symmetrical moments Q'z be applied at the supports ; then the corresponding change in the length of the span is given by ^86), page 115, /x /",/ /v Q \ I y " s i "X cos j/ =-2rUir+7 ^--- *ds Let ^, be the horizontal thrust necessary to cause a change in the length of span of A' I; then or - C'K'yds r>N x dx B x Jo * Vo F* ' C l dsJ / o* __ ^o ~/ /*/ // \ A/Vc C l dx cos _ Wo ~j Jo e *J F* r* ./o ^ For x = o to ^r = a lt K' o; For ^r = ^ to x = a t , K' = M a ; For ^r = a t to ^r = /,, K' = o. 304 A TREATISE ON ARCHES Therefore since //, for a single couple equals ffy, we have /v/i , ids u.j^ I J, * This equation is perfectly general for any symmetrical arch which has no hinges. PARABOLIC ARCH. cos = a constant. (a) Arch with a Hinge at each Support. Value of H^. Our general expression immediately becomes, neglecting the axial stress term, ydx Therefore H, = M a - (i - 6P APPENDIX F. 305 (J)] Arch without Hinges. Value of H '. Neglecting the term which contains F Xi we have /*/. o Jo dx From Appendix E, page 290, the denominator is found to be^./V. r't / ^r=rt-(i - 2 >& ; v - 3*' + Therefore H, = 2 3O6 A TREA T1SE ON ARCHES. Value of M^ Our general equ" 'on becomes _ The denominator becomes /'. fxdx = -/ 2 ; f l xydx -Tf. t/o 2 i/o -^ 3 J Hence APPENDIX G. SPECIAL CASE WHERE = A CONSTANT. PARABOLIC ARCH WITH A HINGE AT EACH SUPPORT VERTICAL LOAD. In practice it sometimes happens that the arch-rib has a constant moment of inertia, especially in large arches. The formulas already deduced do not apply to such a condition, though they may be considered as approximately correct. This case has been very thoroughly considered in two* papers by M. Belliard in the Annales des Fonts et Chausstes. The principal results are given below. According to the assumption that 6 cos = a constant, the general equation for H l becomes, ( // (*idx pti _ JA -j / xydx -|- 6 cos / -~- sin ( i/O I/O f 1 x ( /*f pidx ) P \ I y(x )^ir -f- ^ cos / -^r- sin 1 # = -^ /y^dx + 6 cos / T=T- cos i/O /%. while for 6 = a constant 'f sin0 } * Note sur L'erreur relative que Ton commet en substituant o&r a <& dans la Formule de Navier. April, 1893. Memoire sur le calcul de la Resistance des arcs paraboliques a grande fleche. November, 1893. 307 308 A TREATISE ON AKCHES. These two equations are the same in form, and their only difference is, in the second ds replaces dx in the first in all terms excepting those containing F x . Although the integration of the second equation offers no serious difficulty, yet the final results are long, and their appli- cation in practice tedious without special tables. The equation for the common method is very simple and easy in application. Since the location of the load does not affect the relation between the results obtained by using the equation containing dx and that containing d&, the relative error between the results can be found, and the results obtained by the common method corrected to correspond with those which would have been obtained by the application of the correct formula containing ds. M. Belliard found that the relative error depended only upon the ratio of the rise to the span. For j = 0.50 he found that the common formula H, = | l -?Pk(i k)(i -f k /P), which oy neglects the axial stress, gave a result 3.3 per cent larger than that given by the exact formula. For ~ = 0.25 the result was 1.7 per cent larger, or practically one half that (per cent) for j = 50. This being the case, it is a very simple matter to find the percentage for any ratio of j by interpolation. The magnitudes of the above errors are too small to be of much practical importance. APPENDIX H. SYMMETRICAL ARCHES HAVING A VARIABLE MOMENT OF INERTIA. SUMMATION FORMULAS. THE summation formulas demonstrated in Chapter V can be simplified by introducing the common moment for loads on a beam supported at the ends. The following formulas, while approximate, can be safely applied in the consideration of con- crete and reinforced-concrete arches, which usually have forms which prevent the use of the integration formulas. SYMMETRICAL ARCH WITHOUT HINGES. In g(87), page 115, K' = V l x-I(x-a} + IQ(y-l)} > (x>a,y>b) in which Then b) = m x , (x>a,y>b) which is the common moment for equal and symmetrical loads on a beam supported at the ends. Neglecting the effect of the axial stress, (87) becomes, in summation form, IvA IA Js in which 2 = the sum between the limits / and o and ^ = z~- 3IO A TREATISE ON ARCHES. For Vertical Loads only. 2ff t - in which m x =R 1 -2P(x-a), (x>a) the common moment for equal and symmetrical loads on a beam supported at the ends. For horizontal loads only, g(95), page 118, becomes f in which m x =2Q(y-b), (y>b) the common moment for equal and symmetrical loads on a curved beam supported at the ends. The two equations for H l given above are quite simple and easily applied. They can be used with equal facility for one or many loads. The determination of Mj and M 2 will now be considered. Ing(7i), page 112, K=-H iy -P(x-a) + Q(y-b). (x>a,y>b) g(66) This equation applies for a single vertical and horizontal load applied at the same point. Let b), (x>a, y>b) APPENDIX H. 3H in which R^ is the common reaction of P. Then K = m x -R l x-H 1 y. Neglecting the effect of the axial stress, (71) becomes l ~ since the arch is symmetrical the values of J will be symmetri- cal and hence IxA becomes -JJ. For similar reasons becomes 2yA. The equation can now be written 7 Wl\ X A[ X -- ^ ~ I V 2x4 ) readily reduces to The value of M l for a load upon the right of the crown is evidently the value of M 2 for a corresponding load upon the left of the crown. The values of m x for the load upon the right will be identical with those for the load upon the left taken in an inverse order. The value of the expression 2m x Ax-\ for the load upon the right can be found by 312 A TKEAT1SE ON ARCHES. using the values of m x for the load upon the left and replacing x by x' = l x\ that is, instead of using the values of m x in an inverse order, the values of x - are used in an inverse order. Then for a load upon the right of the crown, replacing m^ by w 2> ZA The expressions 2 A, 2yA, and 2A( jnr] remain un- changed regardless of the position of the load considered. Im x A for a load upon the right is identical in value for a cor- responding load upon the left. Therefore IA in which -b), (x>a,y>b) For Vertical Loads only. m x = R i x-IP(x-a), (x>a) the common moment for vertical loads on a beam supported at the ends. F, =the horizontal thrust produced by the loads considered. APPENDIX H. 313 For Horizontal Loads only. ff a = the horizontal thrust at the left support produced by the horizontal loads considered. Effect of a Change in Temperature. Neglecting the effect of the axial stress, g(no), page 121, at once becomes and g(ii2), page 122, reduces to Effect of the Axial Stress. Let H! represent the horizontal thrust at the left support produced by vertical loads, when the effect of the axial stress is neglected, and H a = the horizontal thrust at the left support due to the axial stress corresponding to the vertical loads. An inspection of (90), page 117, shows that the axial stress terms in the numerator are comparatively small in effect. If these terms be neglected, the numerator remains the same for the two cases, one when the axial stress is considered, and the other when it is neglected. Therefore, for A TREA TISE ON ARCHES. Vertical Loads only, I 2y4(y7 This may be considered in effect equivalent to a drop in tem- perature which produces the same horizontal thrust and the stresses in the arch rib determined as if such were actually the condition. For Horizontal Loads only. The effect of the axial stress produced by horizontal loads cannot be easily expressed independently. The horizontal thrust at the left support when the effect of the axial stress is included is This expression differs from that which neglects the effect of the axial stress in the denominator of the last term only. For Changes in Temperature. i In all cases APPENDIX H. 315 This method of considering the effect of the axial stress is approximate, but sufficiently exact for practical purposes. Symmetrical Arch with a Hinge at Each Support. In this type of arch the vertical reactions are the same as for loads on a beam supported at the ends. M l =M 2 =o. The expressions for H l are readily determined and become as follows: For Vertical Loads only. From g(n6), page 122, g(ii7), page 123, and the propor- tion at top of page 125, in which m x =R l x-!P(x-a}. (x>a) the common moment for vertical loads on a beam supported at the ends. For Horizontal Loads only. From g(i40), page 127, in which For Changes in Temperature. Eefl 3 I& A TREA TISE ON ARCHES. Effect of the Axial Stress. Neglecting the axial stress terms in the numerators of the general expressions for H lt the above formulas can be used when the axial stress effect is considered by adding 2jr cos < f to ly^A in each formula. Note. A close comparison of the above summation for- mulas with those given on pages 46 to 50 inclusive shows that they are in reality identical. In the new formulas m x has been carried through intact, while in the old formulas it is separated into two parts. The new expression for M t and M 2 is quite superior to the old form, as but one half the loads need be considered. Inasmuch as all summations are between I and o, errors introduced by using the wrong limits are avoided. APPENDIX I. UNSYMMETRICAL ARCHES WITHOUT HINGES SUMMATION FORMULAS. From #(59), g(6o), and (6i), page no, we have, if is assumed constant, o, (i) . , . , . 8s in which 4=-^-. O x For a single vertical load P acting downward and a single horizontal load Q acting toward the left support, we have, from (41), page 16, b). . (41) From (47), page 16, M 2 -M, c l-a c-b Vi= +H IJ +P ~~I Q~T- - (47) Substituting this value of F x in (41) it becomes . . 4 (4) 317 318 A TREATISE ON ARCHES. in which . . (5) The expression for m x is simply that for the static moment of P and Q on a curved beam which is hinged at the right support and on rollers at the left support. Combining (39), (40), and (42), page 16, x>a x>a N x =H 1 cos + ViSmPsm. . . (6) Substituting the value of V t from (47) in (6) and letting f I- a *><> c -b} *><* #=|P-j -- P-Q-pjsin^-^cos^ . . (7) we obtain N^^^^smt+H^osf+jsm^+K. . (8) Substituting the value of M x from (4) and the value of N x from (8) in (i), (2), and (3); eliminating M 2 M^ from (i) and (2), (2) and (3) and then M^ from the two resulting expres- sions, the value of H l is found to be Im x B"IA"-Im x A"IE"- - (IK'-IT) (W'IA"-2B"}-(IK"-ZT'} VIA" 1 ~ '""-'"" - 'A"-IB"} U + WU'IA" When the span is divided in n equal divisions of dx each APPENDIX I. 319 and x=z and a = k the symbols in (9) represent the follow- ing expressions: 2 2 dx F x ds 2 (i5) Expressions (10) to (15) are independent of the loading and are constants for any given arch. ZT=etEl, 2T' = etEc ..... (16) b- 320 A TREATISE ON ARCHES. IT and IT' represent the temperature terms, and equations (17), (18), and (19) are dependent upon the loading. If the effect of the axial thrust is to be neglected all terms containing F x are dropped. If the arch is symmetrical then c becomes zero and / becomes y. The value of M 1 can be readily found from (i) and (3) after MX and NX have been substituted. The general form is 2 IG" IG" ~ IG" dx' ' in which the only new expression is . . (2I) [ IzA F x ds IzA \dx/ j In a similar manner the value of M 2 M x can be found from (i) and (3) and then the value of M 2 , Mt-Mi- in which G=J (z- APPENDIX J. UNSYMMETRICAL ARCH WITH TWO HINGES, ONE AT EACH SUPPORT. Summation Formulas. There will be no bending moments at the supports and the condition that the central angle shall remain constant no longer obtains. The length of the span and the relative positions of the supports, however, must remain fixed. where m x = 2P^-x-I(x-a) + 2Q(y-b}- 2Q C j~x. x>a If the arch is symmetrical, c = o and 321 TABLES. TABLES. (The tables that follow are arranged according to the scheme here given.) A. Tabulated Properties of the Two-nosed Catenary. B. A Series of Two-nosed Catenaries inscribed in the Circle of Radius Unity B,. Arch-rings with the Line of Stress lying within the middle Third. I. k(l - 2k* 8 * III. i - |[ 5 (i - k - 2k* + 4/P) IV. I-2/K2- 5^ + 5^') + 3^ V. k\ k(i - k\ and (i+k VI. ^(1-^(3-5^) = ^. VII. (I -)'(! +2^)= J 7 . VIII. ?5^_2 5 9^ - 2) _ . X. 2/^ - XI. ^ a (i - ^) 2 = XII. I + k\- 1 5 XIII. 2k(l - kj(2 -7k + 8P) = 4. 324 TABLES. 3 2 5 XIV. w 2.K\\ K) 1 If /K-\~ OK ) ^ I + '(-15 + 50*-^ + ^ J\. V . XVI. XVII. 6k 1 -f- cos (cos a (sin* sin" a) -\-asin a cos sin ) 20 COS 2 < ^ 3 sm ^o cos 0o H~ 0o XVIII. XIX. 2* cos 8 * 3 sin x" cos *" -f- *~ = .18 4243 .9706 .6403 25-37 2.1304 2.2446 .4196 S3 17 .4123 1.0163 .6442 26.20 2.1564 2.2910 .4065 .16 .4000 1.0630 .6481 27.01 2.1824 2.3400 .3927 .0 15 .3873 I.IIIO .6519 27.40 2.2084 2.3922 .3783 I .14 3742 i. 1606 6557 28.18 2-2344 2.4477 3631 1 .13 .3606 1. 2122 .6600 28.55 2.26O3 2.5075 3471 J3 .12 .3464 1.2663 .6633 29.30 2.2863 2.5719 .3300 * .1 3333 I.3I70 .6667 30.00 2.3094 2.6339] .3138 1 .11 3317 1.3235 .6671 30.04 2.3123 2.6420 .3117 .IO .3162 1.3843 .6708 30.37 2.3383 2.7188 .2920 C 3 .09 _ .3000 1.4498 6745 31.08 2.3643 2.8037 .2706 la .08 .2828 I.52II .6782 31.39 2.3902 2.8988 .2471 07 .2646 1.5999 .6819 32.09 2.4162 3.0068 .2209 .0625 i .2500 1.6655 .6847 32.31 2.4357 3.0987 .1990 1 .06 .2450 1.6888 .6856 32.38 2.4422 3.1318 .1912 g -05 _ .2236 1.7914 .6892 33.06 2.4682 3.2801 .1569 .04 ^ .2000 1.9141 .6928 33-33 2.4942 3.4628 "57 S 0357 .1889 1-9757 6944 33-45 2.5053 3.556i .0951 a .03 1732 2.0688 .6964 34.00 2.5201 3-6994 .0639 i .027 i .1667 2.1096 6972 34.06 2.5259 3.7631 +.0503 Fora value of s here, sen ibly the next, dir ectrix tou ches descr ibed circ 1 .0204 * .1429 2.2716 6999 34-25 2.5451 4.0191 -.0037 1 g .02 .01 A .1414 .1000 2.2821 2.6391 .7000 7036 34.26 34-51 2.5461 2.5721 4.0360 .0072 4.6178: .1249 .005 .0707 2.9907 7053 35.04 2.5851 5. 2065! -.2394 3 .0048 .0693 3.0II4 7054 35-04 2.5856 5.2410 .2461 3 s For a va ue of t icre, sen sibly the 1 ast, a = i .OO2 .0447 3.4519 .7064 35-11 2.5929 5.9909 -.3881 i .OOI .0316 3-7994 .7068 35-M 2-5955 6.5872 -4994 i .OOO ~ .OOOO CO .7071 35-16 2.5981 00 oo For this table, the modulus of common catenary from which the members are transformed two of the above values being given or assumed, the values of the others may be determined for circular linear arches under vertical and conjugate horizontal loads has been calculated TABLES. OF THE "TWO-NOSED CATENARY." 327 Three-point Circle. j-j )'i t. * /3 Po = Pa ft, tfj- /?, ,;(*, -P) c hanges sign. 5-3 T 9 4.5 6 5I 77-39 80.43 22.3607 5.3895 .6013 .4328 .4628 5659 5.8495 5.4874 79.40 86.01 31.6228 5-8637 -.7235 5310 .5654 .6863 co oo 57.04 90.00 90.00 OO oo is-taken as unity; all quantities except s, r, and angles are directly proportional to m. Any irom the table. Intermediate values can be easily interpolated. Rankine's point of rupture for certain values of s; it is given in the column i . 328 A TREATISE ON ARCHES. TABLE B. A Series of " TWO-NOSED CATENARIES" inscribed in the Circle of Radius (i?i) Unity, and having Parallel Directrices at Graduated Distances (Ki -{- K ) from its Centre from (l + .026) to (i + .234). This Table has for its purpose, in conjunction with supplementary tables, the de- signing of arch-rings, so as to secure the condition of the line of stress lying within the middle third, fifth, seventh, etc., of the arch-ring, as may be required to give strength and stability for every variation of proportion of parts and of the nature and distribution of load. s 2 .230 32 '3 .4890 i .0016 9783 0201 2345 .2340 0007 .0003 .225 21 56 33 oo 4847 i .0017 .9761 .0223 .2299 .2293 .0007 .0003 .220 22 24 33 48 4805 i .0017 9739 0245 2254 2247 .0008 .0004 .2I S 22 50 34 34 .4762 fc t> .0019 .0*71 .2208 .2200 .0009 .0004 23 .6 35 21 4719 a". .0020 .9687 .0296 .2163 2153 .0010 .0005 .205 3 4' 36 06 .4675 w"c .0021 .9658 .0327 2117 2106 .0011 .0005 .195 2406 24 30 3651 37 35 .111 4455 - .0024 .9491 .0501 .,890 .1869 .0021 .0010 '75 .170 .765 .160 2558 26 20 26 40 27 01 40 28 41 ii 41 53 42 3 6 43'9 4273 This cir die third, .OO24 .0024 .0024 .0024 9452 94'3 9370 9327 0543 .0586 .0635 .0684 ,1845 .1800 '754 .1709 .1821 1774 .1726 .1678 .0023 .OO26 .0028 0031 .0011 .0013 .0017 ?55 27 20 43 18 .4226 fS .0023 9279 .0738 .1664 .1629 0034 .0019 .150 27 40 44 oo .4180 ~ .0022 9232 0793 .1619 .0038 .0031 '45 27 59 44 42 4132 p. .OO2I .9180 .0856 '574 1532 .0042 .0023 .140 28 18 45 24 .4085 cfl*O .0020 .9129 .0919 1529 .1484 .0045 .0026 '35 2836 4606 .4036 C.^ .0019 9071 .0990 1483 M34 .0049 .0029 .130 2855 46 49 .3988 "rt ** .0017 .9014 . 06 1 .1438 .1384 .0054 .0032 125 29 12 47 3' 3938 *"" .0014 .8952 142 .1392 1333 .0059 0035 .120 29 30 48 14 .3888 " jg .OOI I .8890 . 224 '347 .1283 .006 4 .0039 .115 29 47 48 57 3836 " u .0007 .8821 3l8 .1301 .1231 .0070 0043 .110 3004 49 4" 3785 U rt .0004 8752 412 1180 .00 7 6 .0048 u^o .100 095 .090 30 37 30 52 31 08 50 25 Si 09 5' 54 52 40 1$ .3622 3567 11 '! 0.9998 9993 9987 .998. .8676 .8600 8517 8433 .1521 iS .,889 .1209 .1070 .1127 1074 .0082 .0089 .0097 o 05 .0053 .085 31 23 53 27 3508 5* .9972 .8340 .2043 .1023 !o9ol .0 14 .0079 .080 3 1 39 54 M 3450 " V 0-5 .9963 .8246 .2197 .0976 0852 o 24 008 7 075- 3i 54 55 03 .3388 So 995' .8,41 2384 .0928 0793 o 34 .0097 .070 .065 .060 32 09 32 23 3" 38 55 53 56 45 57 38 .3326 3259 II .9940 99 2 5 .9910 .8036 as .3036 .0831 .0782 0735 .0673 .0611 : 05 i .0 72 .0108 .0131 0134 055 3 52 58 34 .3121 11 9890 .7661 3335 0732 0544 0188 .0150 .050 33 06 59 3' .3O49 .9870 7524 3634 .0682 '0478 .0204 .0167 .045 33 20 to 34 .2968 1 9842 7363 4037 .0630 .0406 .0223 .0190 .040 035 33 33 3346 61 37 62 40 ! 3 8o8 J .9815 .9788 .7203 .7040 4440 4843 .0578 .0526 0334 .0260 0243 .0263 Independent of J?,. Directly proportioned to A',, and subject to any multiplier. For the values of s ending with 5, on this and on the Supplementary Table, the quantities are only interpolated as arithmetical means, and are correct to about i per cent. TABLES, 329 SUPPLEMENTARY TABLE B,. Arch rings with the Two-nosed Catenary or Line of Stress tying within the mid- dle third, and loaded from directrix to a circular soffit which is the three-point circle of another member of the same family of transformed catenaries as the line of stress. Strong Brick. Average W't Ibs. p. cub. ft. Sandstone. Average W't 140 Ibs. p. cub. ft. Granite. Average W't Ibs. p. cub. ft. if II 1<~ 1 g C . >e li o a c 3 1 A 1 Strength 154,000 Ibs. per sq. ft. Strength 576,000 Ibs. per sq. ft. Strength 1,350,000 Ibs. per sq. ft. d from ( crown c jfl JS 54h 1 sf SE ii g c 55*3 11; value of m ic with multiplier. ^ "5 '1 c 1 s! If ss S..t Thickness of to soffit Thickness of to soffit at i Radius of the 1 it i For reference. 1 3 98 |f 1*1 1 1 ~*r d .. ty R k t Feet. Feet. Feet. so 72 8.2 S .141 .019 .026 973 3*5 451 .no 55 81 7-47 137 .021 .029 .970 333 .462 J J 5 61 90 6.70 133 .023 33 .966 34' 473 .110 68 101 6.07 .129 .025 37 .962 349 .482 .105 38 56 76 112 5-45 .125 .027 .041 957 49i .too 42 63 84 126 4-94 .121 .029 045 -952 364 497 .095 46 7 92 140 4-44 .117 .032 .049 .946 .372 504 .090 78 102 I 5 6 4.03 .113 .034 .055 .940 .380 59 .085 9 39 57 86 "4 172 3.62 .110 .037 .062 933 .388 5'4 .080 ai 32 63 95 126 190 3-31 .106 .040 .071 925 395 .516 .075 33 35 69 >5 138 3 oo .103 .044 .081 .917 43 .518 .070 S J 39 43 77 85 116 128 154 170 li 2.71 2.42 .099 .095 .048 .052 .089 .097 .907 .896 .409 .416 5'4 .065 .060 3 1 47 94 '41 _ 2.20 .092 .056 10 .883 .421 506 .055 104 '55 1 .98 .089 .061 23 .869 .427 498 .050 42 I ~ I fa .085 .082 .067 73 41 59 85 431 .436 .480 .462 045 040 46 67 *-44 .078 .078 77 !si2 444 035 One third. - 8 - 1 1 - Directly prop, to tf,. and subject to any multiplier less than given max. - ' Note that * '+ d = \ nearly. 33 A TREA TISE ON ARCHES. TABLE I. VALUES OF k(\ - 2/6* + k 4i k ^i k A k 4i k 4, o .21 0.1934 .42 0.3029 .63 0.2874 .84 0.1525 .01 0.0099 .22 .2010 43 3052 .64 2835 85 .1438 .02 .0199 23 .2085 44 3071 65 2793 .86 1349 03 .0299 .24 .2157 45 .3088 .66 .2748 87 1259 .04 0399 25 .2227 .46 .3101 .67 .2699 .88 .1166 .05 .0498 .26 .2294 47 .3112 .68 .2649 .89 .1075 .06 .0596 27 2359 .48 3"9 .69 2597 .90 .0981 .07 .0693 .28 .2422 49 .3124 70 2541 9i .0886 .08 .0790 .29 .2483 50 3125 71 2483 92 .0790 .09 .0886 30 .2541 Si .3124 72 .2422 93 .0693 .10 .0981 31 2597 52 3119 73 2359 94 .0596 .11 .1070 32 .2649 .53 .3112 74 .2294 95 .0498 .12 .1166 33 .2699 54 3101 75 .2227 .96 0399 13 1259 34 .2748 55 .3088 .76 2157 97 .0299 .14 1349 35 2793 .56 3071 77 .2085 98 .0199 .15 .1438 .36 2835 57 3052 78 .2010 99 .0099 .16 1525 37 2874 58 .3029 79 1934 I.OO o .17 .1610 38 .2911 59 .3004 .80 .1856 .18 .1694 39 2945 .60 .2976 .81 .1776 .19 .17/6 .40 .2976 .61 2945 .82 .1694 .20 .1856 .41 .3004 .62 .2911 83 .1610 VALUES TABLE II. or 8 5 = **. 5 i + k - i k J k A* k A* k //, k A* o .6000 .21 .3723 42 .2866 63 1-2975 .84 1.4104 .01 5843 .22 3657 43 .2850 64 1.3004 85 1.4191 102 5692 23 3593 44 2837 .65 I.3035 .86 1.4280 03 5548 24 3532 45 .2826 .66 1.3068 87 1-4374 .04 .5408 25 3474 .46 .2816 .67 1.3103 .88 1.4472 05 .5274 .26 .3418 47 .2809 .68 1.3141 .89 1-4573 .06 5146 27 3366 .48 .2804 .69 1.3181 .90 I 4679 .07 .5022 .28 .33i6 49 .2801 .70 1.3223 .91 1.4789 .08 4903 29 .3268 50 .2800 7i 1.3268 .92 1.4903 .09 .4789 30 .3223 51 .2801 72 t.33i6 93 1.5022 ,IO .4679 31 3181 52 .2804 73 1.3366 94 1.5146 .11 4573 32 3141 53 .2809 74 1.3418 95 1.5274 .12 .4472 33 3103 54 .2816 .75 1-3474 .96 1.5408 -13 4374 34 .3068 55 .2826 .76 1-3532 97 1.5548 .14 .4280 35 3035 56 2837 77 1-3593 .98 1.5692 .15 .4191 .36 .3004 57 .2850 .78 I.3657 99 1.5843 .16 .4104 -37 2975 58 .2866 79 1.3723 I.OO l,6ooo -17 .4022 38 .2949 59 .2883 .80 1-3793 .18 3942 39 .2925 .60 2903 .81 1.3866 .19 .3866 40 2903 .61 .2925 .82 1.3942 .20 3793 41 2883 .62 .2949 83 1.4022 TABLES. 33* TABLE III. VALUES OF i - H$& - * - 2 * 2 + * .00 I.OOOO .21 -6137 .42 5025 .63 .4892 .84 .3217 .01 9753 .22 .6029 -43 5017 .64 .4865 .85 ,3066 .02 .9510 23 .5927 44 .5011 65 .4834 .86 .2909 .03 .9274 24 5831 45 .5006 ,66 4799 .87 2745 .04 .9043 25 5742 .46 .5003 .67 .4760 .88 2573 05 .8818 .26 .5659 47 .5001 .68 .4716 .89 2395 .06 .8600 .27 -5583 .48 5000 .69 .4667 .90 .2210 -07 .8387 .28 5512 .49 ,5000 .70 .4613 9* .2017 .08 .8182 .29 5447 50 ,5000 7i -4553 .92 .1818 .09 .7983 30 .5387 51 .5000 72 .4488 93 .1613 .10 7790 31 5333 52 .5000 73 .4417 .94 .1400 .11 .7605 32 -5284 53 4999 74 4341 95 .1182 .12 .7427 33 5240 54 4997 -75 .4258 .96 .0957 13 .7255 34 .5201 55 4994 .76 .4169 -97 0726 ,14 .7091 35 .5166 .56 .4989 77 .4073 .98 .0490 .15 .6934 -36 .5135 57 4983 78 397r 99 0247 .16 .6783 37 .5108 ,58 -4975 -79 3863 I.OO .0 .17 .6640 38 5085 59 .4964 -80 3747 ,18 .6504 39 .5066 .60 4950 ,81 .3625 .19 6375 .40 .5050 .61 4934 .82 3496 .20 .6253 .41 5036 .62 4915 83 .3360 TABLE IV. VALUES OF i 2^(2 k J k 4 k //4 k A k 4 .00 I.OOOO ,21 5H2 42 .4365 >63 .4211 ,84 .2626 .01 .9610 .22 .5045 43 4365 .64 .4179 .85 .2498 .02 9239 23 4957 44 .4366 .65 4143 .86 .2365 03 .8887 .24 .4877 45 .4368 .66 .4103 .8? .2227 .04 .8554 25 .4805 .46 4370 .67 4059 .88 .2084 05 .8238 .26 4739 -47 -4372 .68 .4011 .89 .1936 .06 .7939 27 .4681 .48 4373 .69 3959 .90 1783 .07 .7656 .28 .4629 49 4375 .70 .3903 .91 .1625 .08 .7390 .29 .4583 .50 4375 71 3842 .92 .1463 .09 7139 30 4543 51 4374 ,72 3777 93 ,1296 .IO .6903 3 1 .4508 -52 4372 .73 ,3708 94 .1124 .11 .6681 32 .4478 53 .4369 .74 .3634 95 .0948 .12 .6473 33 4452 54 .4365 75 3555 .96 .0767 13 ,6279 34 4431 55 .4358 .76 -3471 -97 .0581 .14 ,6097 >35 .4413 .56 .4349 77 .3383 .98 .0392 .15 .5928 .36 4398 57 .4338 .78 .3289 99 .0198 .16 5770 37 .4387 -58 .4324 79 3I9 1 I.OO .0000 17 .5624 38 -4378 59 .4307 .80 .3088 .18 .5488 39 4372 .60 .4288 .81 .2980 .19 53 6 3 .40 .4368 .61 4266 .82 .2867 .20 .5248 - -4i .4366 .62 .4240 .83 .2749 33 2 A TREATISE ON ARCHES. TABLE V. k & *,*->* + >*'=*,.. k J,, k 4u k J,. k J, k 4,, oo .21 0.7006 42 0.1870 63 0.2024 .84 0.3502 .01 32.1600 .22 .6418 43 .1818 .64 2075 85 3588 .02 15.5266 23 .5892 44 1775 .65 .2128 .86 .3676 03 9.9844 .24 .5422 45 .1740 .66 .2189 .87 .3765 .04 7.2116 25 .5000 .46 1713 67 .2242 .88 .3855 05 5.5666 .26 .4620 47 .1692 .68 .2302 .89 3945 .06 4.4688 .27 4279 .48 .1677 .69 .2364 .90 437 .07 3.6888 .28 3947 49 .1669 .70 .2428 .91 .4126 .08 3.1070 .29 .3698 50 .1666 71 2495 .92 .4223 .09 2.6572 30 3444 51 .1669 72 2563 93 .10 2.3000 31 .3219 52 .1667 73 .2633 94 4413 .11 2.0107 32 .3016 53 .1690 74 .2705 95 4509 .12 .7711 33 .2834 54 .1707 75 2778 .96 .4606 13 .5709 34 .2670 55 .1727 .76 .2853 97 .4703 .14 .4011 35 .2523 56 1752 77 .2929 .98 .4801 15 2555 36 .2392 57 .1781 78 .3007 99 .4900 .16 .1301 37 .2275 58 .1813 79 .3086 I.OO .5000 17 .0208 38 .2171 59 .1849 .80 .3167 .18 0.9251 39 .2080 .60 .1889 .81 3249 .19 .8410 .40 .2000 .61 1931 .82 3332 .20 .7666 .41 .1930 .62 .1976 83 .3416 338 A TREA TISE ON ARCHES. TABLE XVI. VALUES OF 3 izk + 24^ = //, 8 . k A. k AM k A. k 4n k 4 o 3.0000 .21 3902 42 .0081 63 0.9648 .84 0.3711 .01 8823 .22 3512 -43 .0054 .64 .9560 85 .3140 .02 .7694 23 3M9 44 .0034 65 .9460 .86 2535 03 .6611 .24 .2812 45 .0020 .66 9344 .87 .1895 .04 5573 25 2500 .46 .OOIO .67 .9213 .88 .1220 .05 .4580 .26 .2211 47 .0004 .68 .9066 .89 .0508 .06 3629 27 .1946 .48 .OOOI .69 .8902 Negative*. .07 .2721 .28 1703 49 .OOOO .70 .8720 .90 o . 0240 .08 .1854 .29 .1481 50 .0000 7i .8518 .91 .IO27 .09 .1027 .20 .1280 5i 0.9999 72 .8296 .92 .1854 .10 .0240 31 .1097 52 .9998 73 8053 93 .2721 .11 .9491 32 0933 53 9995 74 .7788 94 3629 .12 .8779 33 .0786 54 9989 75 7500 95 .4580 13 .8104 34 .0655 55 .9980 76 .7187 .96 5573 .14 .7464 35 0540 56 99^5 77 .6850 97 .6611 15 .6860 36 0439 57 9945 .78 .6487 .98 .7694 .16 .6288 37 0351 58 .9918 79 .6097 99 .8823 17 5749 38 .0276 59 9883 .80 .5680 1. 00 I. 0000 .18 .5242 39 .0212 .60 .9840 .81 5233 .19 .4766 .40 .OI6O .61 .9787 .82 4757 .20 4320 .41 .OIl6 .62 9723 83 -4250 TABLES. TABLE XVII (BRESSE). VALUES OF =- IN EQUATION c(iog) H^ = ~SP^-, o a 339 2*0 Value " o.oo 0.05 O.IO 0.15 0.2O 0.25 0.30 0-35 0.40 0-45 0.12 4.125 4.112 4-075 4.012 3.926 3.816 3.682 3-526 3-348 3.I49 13 3-804 3-793 3-758 3-700 3-621 3.519 3.396 3-251 3.087 2.903 .14 3-529 3-5i8 3.486 3-432 3-359 3.264 3-150 3.016 2.863 2.692 15 3.291 3.281 3-251 3.200 3-I32 3-043 2.936 2.811 .669 2.509 .16 3.082 3.072 3-044 2.997 2-933 2.862 2-749 2.632 .498 349 17 2.897 2.888 2.862 2.817 2-757 2.679 2-584 474 -348 .207 .18 733 2.725 2.700 2.657 2.600 2.526 2-437 333 .214 .081 .19 .586 2.578 2-554 514 2.460 390 2.305 .206 .094 .968 .20 453 446 2.423 -385 2-334 .267 2.187 093 -985 .866 .21 333 .326 2.304 .268 2.219 .156 2-079 989 .887 774 .22 .224 .217 2.196 .162 2.115 -054 1.981 .895 798 .689 -23 .124 .117 2.098 .064 2.019 .961 1.891 .809 .716 .612 .24 .032 .026 2.007 975 932 .876 1.809 -730 .641 541 25 947 .941 1.923 -893 .851 .798 1-733 .658 572 .476 .26 .869 .863 1.846 .817 777 .725 1-663 -590 .508 .416 .27 797 .791 1-774 .746 .707 .658 1.598 .528 448 .360 .28 .729 .724 1.708 .680 .643 595 1-537 .470 393 .308 .29 .666 .661 1.645 .619 -583 537 1.481 415 341 -259 -30 .607 .602 1-587 .561 527 .482 1.428 .365 .293 .213 31 552 547 1-533 .508 474 .431 1-378 317 .248 .170 32 .500 .496 1.481 457 424 .389 1-332 .272 .205 130 33 452 447 i 433 .410 378 337 1.288 .230 .165 .092 34 .406 .401 1-388 .365 334 294 1.246 .190 .127 1-057 35 .362 358 1-344 .322 .292 254 1.207 153 .091 1.023 36 321 317 1.304 .282 253 .215 1.170 .117 057 0.991 37 .282 .278 1.265 244 .216 .179 I-I35 .083 .025 .960 .38 245 .241 1.228 .208 .180 .144 I.IOI .051 0-994 -931 -39 .209 .205 1.194 .174 .146 .in 1.069 .021 .965 94 .40 .176 .172 1.160 .142 .114 .080 1.039 0.991 937 .877 .42 "3 .109 1.098 .080 054 .022 0.983 937 .885 .828 44 .056 .052 1.042 .024 0.999 0.968 931 .887 .838 783 .46 003 l.OOO 0.990 0.972 949 .919 .883 .841 794 742 .48 0-955 0.951 .942 925 903 .874 839 799 754 .704 50 .910 .907 .897 .881 859 .832 .798 .760 .716 .668 52 .868 .865 .856 .840 .819 793 .760 723 .681 635 54 .829 .826 .817 .802 .782 756 .725 .689 .648 .604 56 793 .790 .781 .767 747 .722 .6 9 tr 657 .618 575 58 758 .756 747 733 .714 .690 .661 .627 589 548 .60 .726 723 715 .702 .683 659 .631 599 .562 .522 .62 .696 .693 .685 .672 654 .631 .603 572 536 497 .64 .667 .665 .657 .644 .626 .607 577 546 .512 474 .68 .614 .612 .604 592 575 554 .528 .499 .467 431 72 .566 .564 557 545 529 .508 .484 .456 .426 392 .76 .522 .520 .516 .502 .486 .467 444 .417 .388 -356 .80 .482 .480 473 .462 447 ! -429 .406 .381 353 323 .84 445 443 .436 .426 .411 393 372 347 .320 .292 .88 .410 .408 .402 391 378 .360 339 .316 .290 .262 .92 .378 376 370 360 346 329 309 .286 .261 235 .96 347 345 -349 -320 .316 .300 .280 .258 234 .209 1. 00 .318 316 -3" 301 .288 .272 253 .231 .208 .184 340 A TREATISE ON ARCHES. TABLE XVII Continued. *. w Values of -. 0-50 0.55 0.60 0.65 0.70 0-75 0.80 0.85 0.90 0-95 O.I2 2.931 2.694 2.441 2.171 1.888 1.592 1.286 0.972 0.651 0.327 13 2.702 2.484 2.250 2.0OI 1.740 1.467 1.185 895 .600 .301 .14 2.506 2.303 2.086 I.S55 i. 612 1.360 1.098 .830 -556 279 15 2-335 2.146 1-943 1.728 1.502 1.266 1.023 772 517 259 .16 2.186 2.008 1.818 I.6I7 1.405 1.184 0.956 .722 .484 .242 .17 2.054 1.887 1.708 1.518 I-3I9 1. 112 .898 .678 454 .227 .18 1.936 1.778 1.610 I-43I 1.243 1.048 845 .638 .427 .214 .19 1.830 1.68r 1.521 1-352 1-175 0.990 799 .603 403 .202 .20 1-735 1-594 1.442 I.28I 1. 112 937 756 571 -382 .191 .21 1.649 I.5I4 1-370 1.217 1-057 .890 .718 -542 .362 .181 .22 I-57I 1.442 1.304 1. 159 1.006 .847 .683 -SIS 344 .172 23 1.499 1.376 1.244 1.105 0.959 .807 .651 .491 .328 .164 .24 1-433 I.3I5 1.189 1.056 .916 771 .621 .468 313 157 .25 1-372 1.259 1.138 I.OIO .876 737 -594 .448 .299 .149 .26 I.3I5 1.207 1.091 0.968 839 .706 569 .428 .286 MS .2? 1.263 1.158 1.047 .929 .805 677 545 .411 .274 137 .28 1.214 1.114 i. 006 .892 -773 .650 523 394 .263 131 .29 1.169 1.072 0.968 .858 744 .625 503 379 253 .126 30 1.126 1.032 932 .826 .716 .601 .484 .364 243 .121 31 1.086 0.995 .899 .796 .690 579 .466 350 234 .116 . -33 1.049 .961 .867 .768 .665 558 449 337 .225 .112 33 1.013 .928 .837 742 .642 539 433 325 .217 .108 -34 0.980 .897 .809 .716 .620 .520 .418 314 .209 .104 35 .948 .868 .782 .693 599 .502 403 303 .202 .IOO .36 .918 .840 757 .670 579 .486 390 .292 .195 097 37 .890 .814 733 .649 .560 .470 377 -283 .188 093 .38 .863 .789 .711 .628 543 454 364 273 .181 .090 39 837 .765 .689 .609 .526 .440 353 .264 175 .087 .40 .812 742 .668 .590 509 .426 341 .256 .170 .084 42 .766 .700 .629 .555 479 .400 .320 .240 159 079 44 .724 .661 594 524 451 377 .301 .225 .149 .074 .46 .685 .625 .561 494 425 345 .283 .211 .140 .069 .48 .650 592 531 .467 .401 334 .266 .198 .131 .065 50 .616 559 .502 442 379 315 .251 .187 .123 .O6l .52 .585 532 .476 .418 358 297 .236 .176 "5 057 .54 556 505 451 396 339 .281 .223 .I6 5 .108 053 .56 .529 .480 .428 375 .320 .265 .210 155 .102 .050 58 .503 456 .406 355 303 .250 .198 .146 .096 047 .60 479 433 .385 .336 .285 .236 .186 .137 .090 .044 .62 456 .412 .366 319 .271 .223 175 .129 .084 .041 .64 434 391 .347 .302 .256 .210 .165 .121 .078 .038 .68 393 354 313 .271 .228 .187 .146 .106 .068 033 .72 356 319 .281 .242 .203 .165 .128 .092 059 .028 .76 .322 .287 .251 .215 .180 MS .in .080 .050 .024 .80 .291 .258 .224 .191 .158 .126 .096 .068 .042 .019 .84 .261 .230 .199 .168 137 .108 .081 .057 035 .Ol6 .88 234 .204 175 .146 .118 .092 .068 .046 .027 .OI2 .92 .208 .180 .152 125 .100 .076 .055 .036 .021 .008 .96 .183 157 .131 .106 .082 .061 .042 .027 .014 .005 .100 159 134 .no .087 .066 047 .030 .017 .008 .002 TABLES. 341 TABLE XVIII. VALUES OF 2x cos' 2 x '3 sin x" cos x" -\-x x 4* x ^.. 11 x \ J,i x" 4,i I 24 0.0033256 47 0.0870414 70 0-5433799 2 25 .0040687 48 .0961634 71 5783850 3 26 .0049329 49 .1059980 72 .6149548 4 27 0059379 50 .1165809 73 .6531228 5 28 .0071012 5i .1279485 74 .6929174 6 29 .0084352 52 .1401378 75 .7343691 7 30 .0099597 53 .1532454 76 .7775070 8 31 .0108680 54 .1671294 77 .8201831 9 32 .0136523 55 .1834638 78 .8689473 10 33 .0158626 56 .1978582 79 .9172998 ii 34 .0183339 57 .2153204 80 .9674381 12 0.0001064 35 .0211197 58 .2336325 Si 1.0193834 13 .0001586 36 .0242131 59 .2508080 82 I.073I55I 14 .0002295 37 .0277103 60 .2717593 83 1.1287710 15 .0003238 38 .0314555 61 .2930503 84 1.18 ? 16 .0004463 39 .0356564 62 3155469 85 i. 25 I 17 .0006032 40 .0402812 63 .339i8o8 86 1-30 o 18 .0008006 4i .0453581 64 .3643046 87 1-37 19 .0010472 42 .0509171 65 .3906428 88 1-433 20 .0013503 43 .0569885 66 .4183343 89 1-50 B 21 .0017190 44 .0636040 67 .4474183 90 1.5707963 22 .0021640 45 .0707964 68 4779307 23 .0026954 46 .0785976 69 .5099063 342 A TREATISE ON ARCHES. TABLE XIX. VALUES OF x + sin x cos x J 19 " " x" sin * cos x fiia x 4. 0,, X* 4 U /3, o 4 6 1.3025470 0.3031560 I 0.0349031 o.oc 00035 47 1.3190868 .3215226 2 .0697848 .0000284 48 .3350190 .3404970 3 .1046243 .0000955 49 .3503454 .3600772 4 .1393998 .0002266 50 .3650685 .3802607 5 .1740906 .0004424 5i >379 I 9i7 .4010441 6 .2086737 .0007659 52 .3927190 .4224234 7 2431339 .0012121 53 4056552 .4443938 8 .2774450 .0018076 54 .4180060 .4669496 9 .3115881 .O0257II 55 -4297774 .4900848 10 3455421 .0035219 56 .4409764 5137924 ii .3792895 .0046829 57 .4516104 .5380650 12 .4128078 .O0607I2 58 .4616881 .5628939 13 .4460784 .OO77O72 59 .4714926 .5879960 14 .4790819 .0096103 60 .4802103 .6141849 15 .5"7994 .0117994 61 .4886749 .6406267 16 .5442123 .0142931 62 .4966229 .66758=3 17 .5763024 .0171096 63 .5040658 .6950490 18 .6080520 .O202666 64 .5110162 .7230052 19 .6394434 .0237818 65 .5174862 .7514418 20 .6704597 .0276721 66 5234897 .7803449 21 .7010844 0319538 67 .5290405 .8097007 22 .7313016 .0366432 68 534I53I .8394947 23 .7610956 .0417558 69 5388425 .8697119 24 .7904514 .0473066 70 .5431243 .9003367 25 8193545 .0533101 7i .5470146 9313530 26 .8477911 .0597801 72 5505298 .9627444 27 .8757473 .0667305 73 .5536868 .9944940 28 .9032110 .0741734 "74 5565032 .0265840 29 .9301696 .0821214 75 5589969 .0589969 30 .9566115 .0905861 76 .5611860 .0917144 31 .9828004 .0993038 77 .5630891 .1247179 32 .0079025 .1091083 78 .5647251 .1579885 33 .0327314 .1191860 79 5661134 .1915068 34 .0570039 .1298199 80 5672735 .2252533 35 .0807115 .1410189 81 .5682252 .2592082 36 .1038467 .1527903 82 .5689887 2933513 37 .1264025 .1651411 83 .5695842 .3276624 38 1513729 .1810773 84 .5700305 .3621227 39 .1697522 .1916046 85 5703540 .3967058 40 .1905356 .2057278 86 .5705698 .4313966 4i .2107191 .22O45O9 87 .5707008 .4661720 42 .2302993 .2357773 88 .5707679 .5010115 43 .2492737 .2517095 89 .5707928 5358932 44 .2676404 .2682494 90 5707963 .5707963 45 .2853982 .2853982 TABLES. 343 TABLE XIX Continued. VALUES OF sin x" x" cos x = x //, x Ju x Jti x ^M o O 23 0.0212167 46 0.1616323 69 O.5O20O59 I 24 .0240716 47 .1719073 70 .5218362 2 25 .0271669 48 1825754 71 .542O8OO 3 26 .0305115 49 .1936404 72 .5627342 4 O.OOOII34 27 .0341145 50 .2051066 73 .5837967 5 .OOO2203 28 .0379820 51 .2169764 74 .6052637 6 .COO3829 29 .0421247 52 .2292541 75 .6271325 7 .OOO6O7I 30 .0465502 53 .2419419 76 .6493983 8 .0009047 3i .0512658 54 .2550423 77 .6720575 9 .0012888 32 .0562797 55 .2685580 78 .6951057 10 .OCI7668 33 .0615995 56 .2824910 79 .7185380 ii .0023501 34 .0672321 57 .2968431 80 .7423494 12 .0030490 35 .0731849 58 .3116145 81 .7665342 13 .0038735 36 .0794651 59 .3268099 82 .7910878 14 .0048339 37 .0860787 60 .3424266 83 .8160034 15 .0059402 38 .0930329 61 .3584668 84 .8412750 16 .0072OI9 39 .1003340 62 .3/49305 85 .8668965 17 .0086304 40 .1079877 63 3919339 86 .8928607 18 .0103238 4t .1159999 64 .4091287 87 .9191605 19 .OI2O222 42 .1243768 65 .4268617 38 .9457892 20 .OI4OO56 43 1331235 66 .4450185 89 .9927379 21 .0161930 44 .1422451 67 4635955 90 I.OOOOOOO 22 .0185936 45 .1517464 68 .4825916 344 A TREATISE ON ARCHES. TABLE XX. VALUES OF x* -\- x sin x" cos x z sin 5 x jr 4 x" // 3 1 x Ao x A** o o.oooo 23 0.0001818 1 46 0.0108522 69 0.1100482 I 24 .0002342 47 .0122966 70 .1191376 2 25 .0002985 48 .0138994 71 .1290242 3 26 .0003766 49 o:-C577 72 .1385871 4 27 .0004688 50 .0175990 73 .1504997 5 28 .0005851 5i .0197320 74 . 16224*42 6 29 .0007206 52 .0220699 75 .1746965 7 30 .0008810 53 . 0246290 76 . 1878885 8 3 1 .0010693 54 .0274221 77 .1971387 9 3 2 .0012902 55 .0304683 78 .2166033 10 33 .0015486 56 0337811 79 .2321884 ii 34 .0018452 57 0373797 80 .2486336 12 35 .0021893 58 .0412871 81 .2659695 13 36 .0025843 59 .0455069 82 2837735 14 37 .0030365 60 .0500725 83 .3034412 15 38 0035517 61 .0549993 84 .3236408 16 39 .0041368 62 .0603087 85 . 3448606 17 40 .0047987 63 .0663576 86 .3671316 18 4i .0055458 64 .0721610 87 .3904870 19 42 .0063850 65 .0787504 88 .4149618 20 43 .0077259 66 .0858036 89 .4405888 21 0.0000776 44 .0083771 67 .0933561 90 .4674012 22 0.0001394 45 .0095494 68 .1014311 TABLE XXI. VALUES OF 2 sin x cos x -\- x sin* x = x 4 n x J 2 , * A n x A n O o 23 0.7806258 46 .4148264 69 7187453 I 0.0349049 24 .8124424 47 .4363274 70 .7216027 2 .0697989 25 .8439761 48 4571858 71 .7235986 3 . 1046722 26 .8752146 49 4773852 72 .7244246 4 .1385129 27 .9061424 50 .4959084 73 .7243722 5 .1743111 28 .9367471 5i 5157396 74 .7233366 6 .2090523 29 .9670128 52 5338617 75 .7213107 7 2437363 30 .9969252 53 5512591 76 .7182896 8 .2783410 31 .0270184 54 .5679161 77 .7142691 9 .3128610 32 0556305 55 .5838162 78 .7092456 10 .3472830 33 .0843930 56 5989435 79 .7032170 ii .3815964 34 .1127420 57 .6132827 80 .6961813 12 . .4157902 35 .1406611 58 .6268207 81 .6885656 13 .4498526 36 1681353 59 . 6400866 82 .6790869 14 4837723 37 I95I479 60 .6514236 83 . 6690300 15 5175372 38 .2216839 61 .6624632 84 6579655 16 5501357 39 .2477263 62 .6726419 85 .6459092 17 5845556 40 2732589 63 .6817446 86 6318529 18 .6177850 4i .2982657 64 .6903668 87 .6188062 19 .6509107 42 .3227295 65 .6978876 88 6037755 20 .6836206 43 .3466341 66 .7044952 89 5877332 21 .7162018 44 3699635 67 .7101816 90 .5707963 22 .7485413 45 3926991 68 7149359 TABLES. 345 TABLE XXII. VALUES OF cos x + x sin x = X* ^* * A,, .r A^ *" A^ O .OOOO 23 0773544 46 1.2721814 69 .4826576 1 .OOOI52I 24 .0839186 47 1.2819313 70 .4900699 2 .0006092 25 .0907097 48 1.2917062 71 .4972383 3 .0013697 26 .0977206 49 1.3014952 72 .5042390 4 .0024339 27 .1049445 50 1.3112876 73 .5107902 5 .0038005 28 .1123748 5i 1.3210720 74 5I7I505 6 .0054680 29 . i 200040 52 1.3308372 75 5232129 7 .0074354 30 .1278248 55 1.3405723 76 .5289699 8 .0097006 31 .1358278 54 1.3502657 77 .5344104 9 .0122609 32 .1440108 55 1.3599061 78 5395194 10 .0151152 33 .1523602 56 1.3694812 79 .5442864 n .0182597 34 .1608693 57 1.3789801 80 .5486991 12 .0216925 35 .1695300 58 1.3883923 81 5527461 13 .0254098 36 1783332 59 1.3977012 82 5564119 14 .0294083 37 .1872707 60 1.4068996 83 5596947 15 .0336845 38 .1963330 61 I.4I59743 84 .5625740 16 .0382339 39 .2055108 62 1.4249128 85 5650397 I? .0430530 40 .2147947 63 1.4337014 86 .5670838 18 .0481371 41 .2241757 64 1.4423340 87 .5686 9I3 !9 .0534812 42 2336435 65 1.4507935 88 .5698536 20 .0590802 43 .2431877 66 1.4590655 89 5705585 21 1.0649291 44 .2527990 67 1.4671423 90 .5707963 22 .0710225 45 .2624672 68 1.4750112 TABLE XXIII VALUES OF x' 2 x sin x cos * An * 4 n x 4 n x A 23 0.0167618 46 0.2433892 69 1.0473746 I 24 .0198158 47 .2637466 70 1.0998688 2 25 .0232609 48 .2852497 71 1.1541174 3 26 .0271274 49 .3079423 72 1.2106700 4 27 .0314472 50 .3318402 73 1.2670749 5 28 .0362481 51 .3569768 74 1.3258782 6 29 .0415654 52 .3833793 75 1.3862235 7 O.OOOI48I 30 .0564302 53 .4110740 76 1.4481051 8 .0002521 31 .0538771 54 .4400895 77 1.5162235 9 . 0004044 32 .0609376 55 .4704481 78 I-576435I 10 .0006148 33 .0686462 56 .5021727 79 1.6528610 ii .0008991 34 .0770368 57 5352871 80 1.7107760 12 .0012617 35 .0861435 58 5698155 81 1.7801637 13 .0017487 36 .0960009 59 .6057683 82 1.8514589 14 .0023472 37 . 1066433 60 .6431729 83 1.9232828 15 .0030891 38 .1181055 61 .6820439 84 1.9969740 16 .0039914 39 .1304212 62 .7223967 85 2.0720558 i? .0050765 40 .1436249 63 .7637878 86 2.1485030 18 .0063670 4i 1577514 64 .8076048 87 2.2262890 19 .0078863 42 .1728338 ' 65 .8424868 88 2.3053880 20 .0096600 43 .1885059 66 .8988928 89 2.3857684 21 .0117398 44 .2060007 67 .9468397 90 2.4674012 22 .0140700 45 .2241512 68 9963335 346 A TREATISE ON AKCHES. TABLE XXIV. VALUES OF x* + x sin x cos x = X* AM JT J jr *** jr J* o 23 0.3055234 4 6 1.0457516 69 .8531934 I 0.0006092 24 .3311036 47 .0820530 70 .8851820 2 .0024360 25 .3575109 48 .1184278 71 .9170352 3 .0054782 26 3847152 49 .1548309 72 .9476041 4 .0097319 27 .4126846 50 .1912472 73 9795371 5 .0151922 28 .4413923 51 .2276436 74 2.0102920 6 .0218524 29 .4708012 52 .2639917 75 2.0407219 7 .0297045 30 .5008810 53 .3002664 76 2.0708359 8 .0387385 31 .5315977 54 .3364391 77 2.0959325 9 .0489436 32 .5629190 55 .3724885 78 2.1301487 10 .0603086 33 .5948106 56 .4083877 79 2.1593724 ii .0728185 34 .6272388 57 .4441165 80 2.1883262 12 .0864681 35 .6601691 58 .4796585 81 2.2170261 13 .1012119 36 6935673 59 .5149785 82 2.2450355 14 .1170628 37 .7273991 60 .5500725 83 2.2737366 15 1339887 38 .7616299 61 .5849191 84 .3017884 16 .1519728 39 .7962252 62 .6195015 85 .3296680 17 .1709923 40 .8311505 63 .6531418 86 3573996 18 .1911252 41 .8663726 64 .6878216 87 .3850088 19 .2120475 42 .9018566 65 .7215380 88 .4125262 20 .2339556 43 .9379693 66 .7549342 89 4399794 21 .2569324 44 9734777 6? .7880145 90 2.4674012 22 .2808096 45 1.0095494 68 .8207711 TABLES. TABLE XXV (WINKLER). 347 - Arc x (Arc *) o O .5707963 O 2.4674012 90 I 0.0174533 .5533430 . 0003046 2.4128739 89 2 .0349066 5358897 .0012185 2-3589571 88 3 .0523599 .5184364 .0027416 2.3056489 87 4 .0698132 .5009832 .0048739 2.2529513 86 5 .0872665 .4835299 .0076154 2.2008619 85 6 .1047198 .4660766 .0109662 2.1493812 84 7 .1221730 .4486233 .0149263 2.0985097 83 8 .1396263 .4311700 .0194953 2.0482472 82 9 .1570796 .4137167 .0246740 1.9985949 81 10 .1745329 .3962634 .0304617 1.9495511 80 ii .1919862 .3788101 .0368588 1.9011167 79 12 .2094395 .3613568 .0438649 1.8532919 78 13 .2268928 3439035 .0514803 I 8060780 77 14 .2443461 .3264502 .0597050 1-7594705 76 15 .2617994 .3089969 .0685389 1.7134727 75 16 .2792527 .2915436 .0779821 1.6680851 74 17 . 2967060 .2740904 .0880344 1.6233060 73 18 .3141593 .2566371 .0986961 1 5791371 72 IQ .3316126 .2391838 . 1099669 I 5355763 7i 20 .3490659 .2217305 .1218476 1.4925254 70 21 .3665191 .2042772 1343361 1.4502840 69 22 .3839724 .1868239 .1474348 1.4085523 68 23 .4014257 .1693706 .161 1426 1.3674271 67 24 .4188790 .1519173 .1754597 1.3269135 66 25 4363323 .1344640 .1903859 1.2870124 65 26 .4537856 .1170107 .2059213 1.2477132 64 27 .4712389 0995574 .2220651 I 2084648 63 28 .4886922 .0821041 .2388202 1.1709491 62 29 .5061455 .0646508 .2561833 1.1334815 61 30 .5235988 .0471976 .2741556 1.0966227 60 31 .5410521 .0297443 .2927374 1.0603734 59 32 .5585054 .0122910 .3119283 1.0247370 58 33 .5759587 0.9948377 .3317284 0.9897018 57 34 .5934119 0.9773844 .3521378 .9*52802 56 35 .6108652 0.9599311 .3731563 .9214683 55 36 .6283185 0.9424778 .3947841 .8882643 54 37 .6457718 0.9250245 .4170212 .8556702 53 38 .6632251 0.9075712 .4398677 .8236855 52 39 .6806784 0.8901179 .4633232 .7923102 5i 40 .6981317 0.8726646 .4873877 .7615437 50 41 .7155850 0.8552113 .5120620 .7313866 49 42 .7330383 0.8377580 .5373452 .7018386 48 43 .7504916 0.8203047 .5632376 .6728998 47 44 .7679449 0.8028515 .5897392 .6445704 46 45 .7853982 0.7853982 .6168503 .6168503 45 Arc JT (Arc *)* JC 348 A TREA TISE ON ARCHES. TABLE XXVI (WINKLER). X Sin* Cos* I COS X o I O 90 Z 0.0174524 0.9998475 O.OOOI525 0.9825476 89 2 .0348995 .9993910 .0006090 .9651005 88 3 0523359 .9986293 .0013707 .9476641 87 4 .0697565 .9975640 .0024360 .9302435 86 % 5 .0871547 .9961947 .0038053 .9128453 85 6 .1045287 .9945218 .0054782 .8954713 84 7 .1218694 .9925462 ..0074538 .8781306 83 8 .1391731 .9902682 .0097318 .8608269 82 9 .1564345 .9876882 .0123118 .8435655 81 10 .1736482 .9848079 .0151921 .8263518 80 ii .1908090 .9816273 .0183727 .8091910 79 12 .2079117 .9781476 .0218524 .7920883 78 13 .2249510 .9743700 .0256300 .7750490 77 14 .2419219 .9702957 .0297043 .7580781 76 J5 .2588190 .9659258 .0340742 .7411810 75 16 .2756374 .9612614 .0387386 .7243626 74 17 .2923717 .9563046 .0436954 .7076283 73 - 18 .3091070 .9510565 .0489435 .6908930 72 19 .3255681 .9455187 .0544813 .6744319 71 20 .3420202 .9396926 .0603074 .6579798 70 21 .3583680 .9335804 .0664196 .6416310 69 22 .3746066 .9271839 .0728161 .6253934 68 23 .39073" .9205049 .0794951 .6092689 67 24 .4067366 9135455 .0864545 .5932634 66 25 .4226183 .9063077 .0936923 .5773817 65 26 .4383712. .8987941 .1012059 .5616288 64 2? 4539905 .8910065 .1089935 .5460095 63 28 .4694717 .8829476 .1170524 .5305283 62 29 .4848096 .8746198 .1253802 .5151904 61 30 .5000000 .8660254 .1339746 .5000000 60 31 .5150380 .8571673 .1428327 .4849620 59 32 .5299192 .8480480 .1519520 .4700808 58 33 .5446391 .8386706 .1613294 .4553609 57 34 .5591929 .8290375 .1709625 .4408071 56 35 .5735764 .8191521 .1808479 .4264236 55 36 .5877853 .8090169 .1909831 .4122147 54 37 .6018150 .7986355 .2013645 .3981850 53 38 .6156615 .7880108 .2119892 .3843385 52 39 .6293204 .7771459 .2228541 .3706796 '51 40 .6427876 . 7660446 2339554 .3572124 50 41 .6560589 .7547096 .2452904 .3439411 49 42 .6691306 .7431449 .2568551 .3308694 48 43 .6819983 .7313537 .2686463 .3180017 47 44 .6946584 .7193398 .2806602 .3053416 46 45 .7071068 .7071068 .2928932 .2928932 45. COS X sin x I COS X - TABLES TABLE XXVII (WINKLER). 349 - sin'* cos" x sin x cos x sin* x COS'* o o I O I 90 I 0.0003046 0.9996953 0.0174498 0.0000053 0.9995428 89 2 .0012180 .9987822 .0348782 .0000425 .9981739 88 3 .0027391 .9972609 .0522644 .0001437 .9958942 87 4 .0048660 .9951340 .0695866 .0003394 .9927100 86 5 .0075961 .9924037 .0868241 .0006620 .9886273 85 6 .0109262 .9890738 .1039539 .0011421 9836552 84 7 .0148521 .9851477 .1209609 .OOlSlOO .9778047 83 8 .0193692 .9806310 .1378187 .0026957 .9710876 82 9 .0244717 .9755283 .1545085 .0038282 .9635178 81 10 .0301537 .9698463 .I7IOIOI .005236! .9551124 80 ii .0364080 .9635920 1873033 .0069470 .9458883 79 12 .0432273 .9567727 .2033683 .0089875 .9358650 78 13 .0506030 .9493969 .2191856 .0113832 .9250638 77 14 .0585262 .9414737 .2347358 .0141588 .9135079 76 15 .0669873 .9330127 .2500000 .0173376 .9012213 75 16 .0759760 .9240239 .2649596 .0209418 .8882288 74 17 .0854812 .9145187 .2795964 .0249923 8745788 73 18 .0954915 .9045085 .2938927 .0295085 .8602386 72 19 .1059947 .8940055 .3078308 .0345085 .8452989 71 20 .1169778 .8830222 .3213938 .0400088 .8297694 70 21 .1284274 .8715726 3345653 .0465573 .8136828 69 22 .1403301 .8596700 .3473292 .0525686 .7970722 68 2 3 .1526708 .8473292 .3596699 .0596532 7799707 67 24 .1654347 .8345653 3715724 .0672884 7624135 66 25 .1786062 .8213938 .3830222 .0754823 7444355 65 26 .1921693 .8078308 .3940055 .0842415 .7260735 64 27 .2061079 .7938921 .4045084 .0935708 .7073636 63 28 .2204036 7795964 .4M5I88 .1034732 .6883427 62 .29 .2350403 .7649599 .4240241 ,1139498 .6690480 61 30 .2500000 .7500000 .4330127 .1250000 .6495189 60 31 .2652642 .7347358 .4417483 .1366211 .6297915 59 32 .2808144 .7191857 .4493971 .1488090 .6099040 58 33 .2966310 .7033684 .4567727 .1615573 .5898943 57 34 .3126968 .6873033 .4635920 .1748579 .5698003 56 35 .3289899 .6710101 .4698463 .1887009 .5496592 55 36 3454915 .6545085 .4755282 .2030748 .5295083 54 37 .3621813 .6378187 .4806307 .2179661 . 5093846 53 38 .3790391 .6209609 .4851478 2333598 .4893237 52 39 .3960442 .6039558 .4890738 .2492387 .4693619 51 40 .4131759 .5868241 .4924039 .2655844 4495335 50 4i .4304134 .5695866 495I34I .2823766 .4298726 49 42 .4477358 .5522642 .4972610 .2995937 .4104124 48 43 .4651217 .5348782 .4987821 .3172122 .3911853 47 44 .4825503 .5174497 .4996955 .3352075 .3722223 46 45 .5000000 .5000000 .5000000 3535534 3535534 45 ' COS 5 X sin'-r sin x cos x cos' x sin x " x 350 A TREATISE ON A A CUES. TABLE XXVIII (WINKLER). sin x X x sia .r X COS X X O 1.5707963 o I 0.6366197 90 I 0.000304' I.553I06I 0.0174506 0.0271096 0.9999486 .6436748 89 2 .OOI2I82 1.534954! .0348853 .0536018 .9997967 .6506919 88 3 .0027404 I.5I63554 .0522881 .0794688 .9995428 .6576697 87 4 .0048699 1.4973273 .0696431 .1047033 .9991875 .6646070 86 5 .0076058 1.4778850 .0869344 .1292982 .9987307 .6715028 85 6 .0109462 1.4580453 .1041458 .1532468 .9981757! .6783559 84 7 .0148892 1.4378255 .1212623 .1765428 .9975147 .6851651 83 8 .0194324 1.4172388 .1382684 .1991804 .9967479 .6919290 82 Q .0245727 1.3963116 .1551457 .2211540 .9958928 .6986465 81 IO .0303073 I.3750509 .1718814 .2424585 9949309 .7053168 80 ii .0366327 1-3534774 .1884589 .2630893 .9938682 .7H9380 79 12 .0435449 1.3316077 .2048627 .2830419 .9927055 .7185100 78 J 3 .0510398 1.3094594 .2210775 .3023125 .9914420 .7250297 77 14 .0591126 1.2870480 .2370880 .3208974 .9900789 .7314980 76 15 .0677587 1.2643939 .2528788 .3387933 .9886157 7379I3I 75 16 .0769725 1.2415131 .2684355 3559977 .9870534 7442735 74 17 .0867484 1.2184185 .2837413 .3/250/9 .9853918 7505785 73 18 .0970806 I.I95I320 .2987832 .3883223 .9833316 .7568267 72 19 .1079625 1.1716702 .3135459 .4034387 .9817725 .7630174 71 20 .1193876 1.1480497 .3280146 .4178564 .9798155 .7691489 70 21 .1313487 1.1242896 .3421750 4315745 .9777607 .7752204 69 22 .1438386 1.1004046 .3560130 .4445923 .9756082 .7812309 68 23 .1568495 1.0764112 .3695144 .4569094 .9733584 .7871798 67 24 .1703731 1.0523289 .3826650 .4685270 .97IOI22 .7930653 66 25 . I 844020 1.0281752 3954514 .4794460 .9685698 .7989851 65 26 .1989265 1.0039628 .4078597 .4896654 .9660318 .8046422 64 27 .2139380 o 9797109 .4198760 .4990726 .9634002 .8105204 63 28 .2294272 95544" .4314897 .5080171 .9606691 8159544 62 29 .2453842 .9311647 .4426849 .5161530 .9578462 .8215086 61 30 .2617994 .9068996 .4534498 .5235988 .9549298 .8269933 60 31 .2786605 .8826632 .4637722 .5303574 .9519194 .8324079 59 32 .2959628 8584731 .4736395 .5364335 .9488167 .8377498 58 33 .3136896 .8343410 .4830396 .5418275 .9456220 .8432167 57 34 .3318318 .8102883 .4919608 5465465 .9424352 .8482206 56 35 .3503779 .7863297 .5003915 .5505941 .9389574 8533443 55 36 .3693163 .7624804 .5083202 .5539746 .9354896 .8583937 54 37 .3886352 .7387573 .5157363 5566936 .9319313 .8633671 53 38 .4083222 7I5I757 .5226286 5587567 .9282843 .8682632 52 39 .4283649 .6917516 .5289864 .5601695 .9245487 .8730820 51 40 .4487501 .6685000 5347999 .5609380 .9207255 .8778223 50 4i .4694661 .6454363 .5400590 .5610692 .9168148 .8824831 49 42 .4904986 .6225756 .5447538 .5605695 .9128181 .8870639 48 43 .5118340 5999330 .5488748 .5594464 .9087354 .8915636 47 44 .5334592 .5775230 5524133 .5577075 .9045682 .8959812 46 45 .5553604 .5553604 5553604 5553604 .9003163 .9003163 45 sin x x sin x jr COS X X JC TABLES. TABLE XXIX (WiNKLER). 351 - x sin 8 x X COS" 4- x sin x cos x o o .5707963 O O O O go I 0.0000053 5528336 0.0174480 O.OOO473I 0.0003046 O.O27IO55 89 2 .0000425 5340191 .0348641 .0018707 .0012175 .0535055 88 3 .0001434 5142774 .0522165 .0041591 .0027366 .0793599 87 4 .0003397 4936797 .0694735 .0073037 .0048580 .1044483 86 5 .0006629 .4722610 .0866036 .0112691 .0075768 .I288o6l 85 6 .0011445 4500577 1035753 .0160187 .0108862 .1524072 84 i .0018145 .4271082 .1203585 .O2I5I52 .0147782 .1752269 83 8 .0027036 .4034495 .1369227 .0277206 .0192432 .1967883 82 9 .0038440 .3795486 .1532356 .0345961 .0242696 .2184312 Si ro .0052628 .354i6ii .1692701 .0421025 .0298469 .2387751 80 ii .0069898 .3286104 .1849964 .0501998 0359597 .2582557 79 12 .0090536 .3025090 .2003859 .0588477 .0426032 .2768568 78 13 .0114814 .2758979 2I54"3 .0669182 .0447316 .2898545 77 14 .0143007 .2488180 .2300454 .0776321 0573578 .3"3654 76 15 0175372 .2213107 .2442622 .0876861 .0654498 ,3272492 75 16 .0212165 .1934174 .2580362 .0981263 .0739907 ,3422069 74 17 .0253628 .1651794 .2713432 .1089108 .0829579 .3562311 73 18 .0299996 .1366392 .2841597 .1199979 .0923291 .3684670 72 19 035I49 1 .1078370 .2964635 .1313468 .1020806 .3814589 71 20 .0408330 .0788151 .3082329 .1429154 .1121876 .3926566 70 21 .0470712 .0496147 3194479 .1546625 .1225963 .4029094 69 22 .0538829 .0202775 .3300896 .16654/2 .1333648 .4122188 68 23 .0612860 0.9908418 .3401397 .1785287 .1443808 .4205874 67 24 .0692976 .9613504 3495819 .1905671 1556439 .42802O7 66 25 0779317 9318432 .3584015 .2026227 .1671250 .4345256 65 26 .0872036 .9023558 .3665819 .2146552 .1787939 .4401084 64 27 0971256 .8727278 .3741122 .2265/43 .1906187 .4446770 63 28 .1077095 .8436043 .3809827 .2384996 .2025721 .4485524 62 29 .1189646 .8144150 .3871810 .2502359 .2146179 .4514376 61 30 .1308998 .7853982 .3926990 .2617994 .2267254 .4534498 60 3i .1435218 .7565900 .3975304 .2731543 .2388603 .4546051 59 32 .1568363 .7280265 .4016691 .2842664 .2509907 .4549215 58 33 .1708476 .6997373 .4051110 .2951004 .2630822 .4544147 57 34 .1855580 6717595 .4078540 .3056249 .2751010 4531075 56 35 .2009685 .6441236 .4098967 .3165358 .2870128 .4510202 55 36 .2170789 .6168597 .4H2396 .3256181 .2987832 .4481748 54 37 2338865 .5899977 .4"8853 .3350265 .3103779 .4445962 53 38 .2513883 .5635661 .4118369 .3440050 .3217622 .4403062 52 39 .2695787 5375920 .4110997 .3525260 .3329020 4353334 5i 40 .2884511 .5121006 .4096806 .3605649 .3437628 .4297035 50 41 3079975 .4871170 .4075877 .3680945 .3543106 .4234443 49 42 .3282075 .4626638 .4048309 .3750942 .3645114 .4165892 48 43 .3490699 4387632 .4014216 .3815415 .3747317 .4091532 47 44 3:05725 .4154354 3973728 .3874163 .3837385 .4011812 46 45 .3926991 .3926991 .3926991 .3926991 .3926991 .3926991 45 x sin 8 x _r cos 8 x .r sin x cos x * 352 A TREATISE ON ARCHES. TABLE XXX. STONE ARCHES.* Name, etc. Date. f Max. Span. Rise. 4>- Reference. 1905 i93 1380 19031 1859 1893 1901 1902 1888 1833 1833 1889 1901 292. 75- Si- r 3. 20. 13- b 09. 00. ?S9-i 101.8 87-8 57-3 59-0 S2.S 21.4 90.2 4-9 4-8 4.0 4.9 tt 6.6 3-4 4-5 B Z-28--04 N. IO-I2-OI 3. l2- 7 -'93 N. io-i 7 -'o 3 V. 7-29-' 99 3. 1 2-7-' 93 SI. 2-l8- ? 02 SI. 1 0-4-' 02 3. 2-27- - 02 3. 5-2-'o2 >.' '92 B. I2-26-'0! A. 6-44 P. '52-294 EL Blue prints B. i2- 7 -'93 F. '52-276 F. '52-290 Luxemburg, Germany H. ' Italy R Cabin John, Washington, D. C. . . . . Aq. Black Forest, Germany R. Bogenhausen, Bavaria H. Lavaur, France R. Grosvenor, Chester, England H. Turin, Italy H. 96. 87. 83- 80. 60. 60. 60. 52.8 SS-8 f6o.o 90.0 t6o.4 65 .0 44.0 S-6 5-9 S-3 t6.o tS-9 l l:l Vieille Brioude, France Ballochmoyle, Scotland R. 1775 1793 1856 1892 1893 1 545 1830 Gignac, France H. Victoria Low Lambton, England. ..R. Main St., Wheeling, W. Va H. Janma, Austria R. Tournon, France H. London, England H. 59- 57- 56. 52. '!f:i '6 S :o' 37.7 1:1 2.8 Berne, Switzerland Gloucester, England H. 1204? 1827 1886 1611 1898? "i*34 1336 SO. B. I2-I9-'9S F. '52-290 54-0 4-5 Near Grenoble, France H. ^ellefield Pittsburg Pa H. S SO 48 48 47 54-4 36.6 49- 5 73^8 3-2 4.0 5-6 4-9 4.6 5-3 4-5 K. '52-276 B. 6-2 2-' 90 I. 5 94 B. i2- 7 -'93 F. '52-276 F. '52-274 B. io-27~'o4 K. 5-i7-'9S L and Q. P. '96-140 Engineer, '04 G. '97-179 G. '88-575 F. '52-280 F. '52-276 P. '96-130 F. '52-296 F. '52-117 B. 5-18-93 B. i2-7-'s> 3 L. B. 6-19-02 B. 7-6-' 9 3 F. '52-276 F. 'q 2-280 Moulins France ... . H. Verone 'italy H. 1354 1904 , T I8SS 1755 1899 . 46 b 44- 44- 35- 35-8 19-3 Outer Maximilian, Bavaria. . , H. Putney Road, England H. Narni, Italy H. Alma, Paris, France H. Pont-y-tu-prydd, South Wales H. 40 40 35-0 2-5 4-0 C 33. Bellows Falls, Vt H. Albula River Viaduct R. Verdun, France H. 189? 1884 1766 1732 1203 i8so1 1849 rS 9 o 1893 1 80 if 3 34-S 34-5 33-8 32.6 32.0 31-2 31. 31- 31- 30.1 t42.6 38.2 66.3 43-7 16.4 16.4 32.8 3-9 5-6 7-7 5-3 5-9 2.6 it Waldi-tobel, Bludenz, Austria R. Vizille, Grenoble France H. Villeneuve, France H. St. Martin, Toledo, Spain H. Scrivia, Italy R. Viaduct de Moret, France R. Boucicault, Verjue, France H. First Worochta, Austria R. Aberdeen, Scotland H. Wan Hsien, China H. North Ave.. Baltimore. Md. H. Echo Bridge, Newton Upper Falls, Mass Aq.&H. 1895 1876 1765 1774 30. 29.0 28.2 28.2 65.0 26.0 42-3 38-5 32.0 5-0 S.o 6.4 <;. * Neuilly. France ... . . . H. * For data for about five hundred masonry arches see "Symmetrical Masonry Arches.' by Malverd A. Howe. John Wiley & Sons. N. Y. t About. J 27 B.C.-I4 A.D. Brick Ring. a. Two hinges. b. Three hinges- TABLES. 353 TABLE XXX (Continued). STONE ARCHES. Name, etc. Date. 6& Max. Span. Rise. to- Reference. Maidenhead, England R. 1838 1785 6 28.0 27 6 24.3 63 8 5-3 K. io-2 S -'9S Near Oisilly, France R. 7 27.0 46.0 6 9 [ ' I2 7 93 Tetes, France H. Devil's Bridge, Lucca, Italy H. Waterloo (new), London H. Hartford, Conn H. Tongueland, Scotland H. Napoleon, Paris, France R. 1732 riooo 1817 ^1904 1806 5 9 8 7 23.6 20.5 20.0 19.0 18 .0 16.0 6r.8 60.3 34-6 29.8 38.0 14.8 4-7 t4.5 4-8 "3.6' F. '52-278 F. '52-288 T. 2-i9-'o 4 L and F. Nantes, France '....' H. St. Esprit, France H. Second Worochta, Austria R. 1765 1309 1893 3 19 iS-4 14.1 13.5 34-0 44-8 56.8 6.4 5-9 T j A 93 i and Q. F. '52-274 Toulouse, France H. Lodi St., Elyria, Ohio H. 1632 1894 7 13-0 38.4 3-7 3ity Engineer Sault, France Winstone. England 1827 1762 3 08.8 Wurtemberg, Germany H. Balersbronn, Germany ?H. 1882 1889 i 08.8 b 08.2 10.8 10.8 3-3 2.O G. '91-903 G. '01-38 Hartford, Conn H. Orleans France . H. 11904 1760 8 L and F Ponthaut, France 1793 06.3 53-i 5-7 F. '52-284 Wissahickon, Philadelphia, Pa H. Potomac Aq., Georgetown, D. C. ..Aq. 1897 7 05.0 B. 9-9-' 9 7 A. '37-148 Prague, Bohemia H. Herault, France Port-de-Piles France R 1878 1847 7 i 04.4 03 8 ti6.o 15-4 4.0 2.7 K. s-io-'78 F. '52-280 Avignon, France ; H. Gere, Vienna, Austria Munich, Bavaria H. Pont-de-la-Concorde, Paris H. Guillotiere, Lyons, France H. Pont-au-Double , Paris, France H. 1187 1781 1814 1792 1265 1847 21 3 02.9 02.7 02.3 02.3 02.3 or.8 Si-5 28.2 '5:5 11 2.4 5-2 4-3 3-7 2. I 5-3 J, L, and F. F. '52-284 F. '52-288 J, F. '5 2-284 F. '52-274 J, F. '52-296 Goltzsch Viad. Bavaria j R. Wellington, Leeds, England H. 1851 1819 6 01.7 00.0 50.9 7-4 Q. Am. Supp. A. '44-128 Blackfriars (old), London, England .H. 1770 9 00.0 43-0 5-0 L, F.' 5 2-280 Bishop Aukland, England H. 1388 1.8 I. Alcantara Aq., Lisbon, Spain Aq. Rutherglen, Scotland H. Minneapolis, Minn. . . . . .R. ti775 1895 1883 35 3 00.0 00.0 88.0 12-5 39.7 4.0 3.0 K K^ls t About. J Brick ring. b Three hinges. 354 A TREATISE ON ARCHES. TABLE XXX (Continued). PLAIN CONCRETE ARCHES. Name, etc. Date. 0) Max. Span. Rise. to- Reference. Ulm, Wiirtemberg, Germany H. Neckarhausen, Germany H. {1903 6187. 6165. bi6s 8.7 1:1 3.6 K ?;-30-04 B o 26 '01 Munderkingen, Germany H. '1893' bi6 4 . 6. 4 3.3 Q Y Connecticut Ave. Bridge, Washing- ton D C H 5 o N _g_' os Vauxhall, London, England H. Inzigkofen Germany. ti849 l8o6 6144. 6141 1 1:? iV ' . N. 2-25- 99 Big Muddy, 111. Cen. Ry R. 140. 5- Coulouvreniere, Geneva, Switz H. 1895 8^2 3 . Y'. *9 I2 ~ 3 Borrodate Burn Viad., Scotland R. 1899 12 . 2. 5 4-0 B. 2-9-' 99 Sixteenth St., Washington, D. C.. . .H. 1905 12 . 9-0 5-0 B. lA^'os Kircheim WUrtemberg H (1898 fc}l 4 6 t 9 .0 2.6 Grand-Maitre, France Aq. Worms Germany H. 1869 1900 fen t 9-3 K. io- ? 6 9 00 Mittenberg, Germany H. Danville 111 R 1899 bn . 6! 4 ' 2.5 B. 7-25-'oi Near Mechanics ville, N. Y E.R. I 95 100. N. 3 3 '06 B. u-5-'o3 Thebes, bridge approach R. Schlitza River, Austria H. ti903 1 100. 0.0 o . o 4.5 2 . 3 B. s-n-'os K. 4 12 '04 Silver Lake, Pittsburg, Pa R. Imnau, German v H. Morar Viaduct, Scotland R. Santa Ana Viaduct, Riverside, Cal. .R. 1905 1896 1904 100. b 98. 90. 86. 0.0 9.8 24.0 43-0 4.0 1.5 3-5 N. 5-6-' 05 G. 'ox-40 B 2-9-99 N. 9-9-05 San Leandro, Cal H. 1901 Piano, 111 R. Rechtenstein, Wiirtemberg 1904 1893 , 75- b 74. ' 'k'.'a 3-0 N. 6- a-' 04 Y. '98 Ashtabula, Ohio R. 1904 74. 37.0 1*6 s T. 2-27-05 Ehingen, Wiirtemberg H. Bridge No. 163, W. Cincinnati, O.. . .R. 1898 1904 b 69. 68. 17.0 3-5 B. 2-9-' 02 N. 3-5-04 Concord, Mass H. 1901 66. I I . O Thebes bridge approach (east) R. Bridge No. 242, W. of Cincinnati, O. R. 1905 1904 V 65- 60. 32-5 26.0 3-3 2.7 B. II-20-'02 N. 3-5-04 t About. * Clear. b Three hinges. SUPPLEMENTARY TABLE. PLAIN CONCRETE ARCHES. Name, etc. Date. ~ J! Max. Span. Rise. to. Refer ece. Nashua to Hudson N H H 6 Scotland, Pa R. B. 4- S-'og Near Guggersbach, Switz H. Wiesen Switz . . R 1906 169 1 80 4 Is 3-7 B. 4-30-'o8 N n-2i-'o8 Lautrach, Bavaria R. ^187.5 B. 5- 2-'o7 Mannheim, Germany H. Near Kempton, So. Germany R. Walnut Lane, Phila., Pa H. Between Teufen and Stein, Switz H. Rocky River, Cleveland, Ohio. .E. & H. Munroe St., Spokane, Wash.. . .E. & H. 1908 1906 1908 1908 1909 19091 4 6 4 6192 6211.5 233 2 2 L 9 ' 3 281 18.1 86.7 70.3 'to.9 us 4.4 9-5 6!o 6.8 B. 6-i8-'o8 j N. 2-23-'o7 IN. s-n-'o? j B. I-3I-'07 IN. 2-i 5 -'o8 B. 12- 3-'o8 N. i-23-'o9 B. 9- 2-'o9 t About. b Three hinges. TABLES. 355 TABLE XXX (Concluded). REINFORCED CONCRETE ARCHES. Name, etc. Date. I* Max. Span. Rise. to- Reference. Munich, Germany. H. 1904 6230.0 B. ii 17 '04 Decize, France H. Bormida River, Italy H. 1902 1899 183.7 167.3 5-3 6.7 1.6 2 .O G. '05-292 G. [04 Playa-del-Re'y Cal H. | ' Vj. oc 377 Schwinmschulbrucke, Steyr H. Park Ave., Newark, N. J H. Kansas Ave., Topeka, Kan H. Zanesville, Ohio, V. Br H. Wildegg Route, Switzerland H. Washington Ave., Lansing, Mich. . .H. Jacaguas River, Porto Rico H. Yellowstone National Park H. Milwaukee Wis H 1905 1898 1902 1890 1901 1903 138.4 132.0 125.0 122.0 122.0 120.0 120.0 Jiaoio 118 o 9.4 16.2 18.9 n-5 11.4 *|i : ? 2.0 2-5 0.6 2.3 2.0 N. 8-i2-'o S B. 4 -2-; 9 6 Cement, 3-'o2 N. 8-3-'oi B. i- I4 -'p 4 Third St., Dayton, Ohio H. Green Island, Niagara Falls, N. Y. . H. Laibach Austria H 1901 IIO.O 14.3 ii . 2. I 3-3 ft 3-4-'o~4 S B. i2-6-'oo Route Francois- Joseph, Austria. . . . H. Stockbridge, Mass H. N. 6th Ave., Des Moines, Iowa H. Wayne St., Peru, Ind H. Bridge 113, near Marshall, 111 R. Yorktown, Ind H. W St Paterson N J H 1900 1895 TI902 1905 tioos 1898 6108. 2 100. 100. 100. 95-0 14.. 20. IS- 40. 1.6 o . 8 4.0 j. '04 B. i2- 7 -'95 Cement, 7-' 02 B 3-29-'o6 Slue prints Main St., Dayton, Ohio H. Grand Rapids, Mich H. Seeley St., Brooklyn, N. Y H. Route Payerbach, Austria H. Fabriano Viaduct, Italy H. Route de 1'Empereur, Bosnie H. Haldu H 1903 1904 1904 1900 1905 1897 88.0 87.0 85-3 85-3 84.9 83.2 t8. "s'. 5- 26. 8. i-3-5 i-S 2.0 I .0 3. 5-19-04 3. i2-i-'o4 5- "-i" 31 ' N. 12-9-05 Soissons, France H. & R. Rock Creek, Washington, D. C H. La Salle St., So. Bend, Ind H. Wishawaka, Ind .H. Hyde Park on Hudson NY... H [1903 1901 1*1906 1807 81.0 80.0 79.0 76.0 8.1 14.0 12.5 15.8 1.7 i.S i. 10-7- 04 1. 10-31-01 Blues Blues Hamilton St., Hartford, Conn H. Grand Rapids, Mich H. Polasky Cal . H 1898 1905 75-0 75-0 7-5 14.0 1-3 3. 3-22-'o6 N 2-2 4 -'o6 Wabash' Ind H. Illinois St., Indianapolis, Ind H. Meridian St., Indianapolis, Ind H. La Salle, 111 H. 1905 1900 1900 1905 0000 18.0 9-5 9-5 7-5 1-3 1-3 '2.0 ^J. 12-2-05 3. 4-n-'oi 3. 9-2i-'os Route de Pa'inperdu, Belgium H. Near Copenhagen, Denmark H. Trinidad, Col H. Eden Park, Cincinnati, Ohio H. Bloomfield Ave., Newark, N. J H. Guayo River, Porto Rico H. 1899 1879 1905 1895 1904 71.8 71 7 70.0 70 .0 7.0 10. 8.5 I .2 G. '04 3. 7-21-98 "f. 2-io-'o6 J. io-3-'95 sT. 8-12-05 N. 8-3-' oi Auch, France H. Jacksonville, Florida H. Herkimer Viaduct, N. Y R. 1899 1904 I 68.9 66.0 66.0 6.8 t?.o 1.0 >. '04 Route Cantal, Italy H. 1902 65.6 65 6 6.6 14 8 i 6* >. '04 G. "04 Route Ebhausen, Wlirtemberg H. Troy, N. Y H. Montgomery St.. Jersey City, N. J. .H. Franklin Bridge, St. Louis H. Eighth Ave., Carbondale, Pa H. 1891 1897 1896 1898 1896 65.6 65.0 61.2 60.0 58.7 8.2 8.5 12.0 '5-5 6.6 I .0 I .0 0.9 G. -04 J. i2-io-'98 t About. t Center to centre. b Three hinges. 356 A TREATISE ON ARCHES. SUPPLEMENTARY TABLE. REINFORCED CONCRETE ARCHES. i/i Name, etc. Date. y Max. Span. Rise. to Reference. Broad St., Bethlehem, Pa E. & H. 1910 5 100 J Cement Era, I 8-'o 9 Near Nashville Term H. 1910' i IOO N. ii- s-'io Mulberry St., Harrisburg, Pa. \p a. TJ Cameron St. approach J K 1909 29? IOO 14 1.5 f N. 8-i S '-o8 \N. 4-3-09' Ross Drive, Washington, D. C H. Vermilion R., Danville, 111 R. 1908 1905 i 3 6100 00 15 40 2-5 4 -0 3. s-2i-'o8 T. 7-i3-'o6 No 113 Marshall 111 R oo 4 Blue prints Highland Boulv., Milwaukee, Wis. . .H. 1909 2 oo 21.7 i 7 N. 6-12-09 Cumberland R., K. & T. Ry R. 1907 5 02 . I T. 3-22-*07 Red Bridge, Huntington, Ind H. 1907 2 05 14 1.8 J Cement Age, \ io-'o8 Pelham, Borough of Bronx, N. Y H. 1908 6 05 16.5 2 .0 N. io-3i-'o8 Factory St.,Canal Dover, Ohio.E. & H. San Diego Cal .... ' H 1906 1910 3 6 06.7 b 07 ii. 8 2 .0 N 2- 9-'o 7 N. 3 25 ? u Paterson, N. J H. 1907 3 08 12 2 .3 N. 3- 7-'o8 Cedar St Mishawaka, Ind H. 1 903 3 10 14 N. 7- 7-'o6 Indianapolis, Ind H. Jefferson St South Bend Ind . . H 1905 5 10 IO 14-3 14 . 2 . I 2 . 3 Blue prints N. 7 28 '06 igoSt B 4 22 'oo Austin, Texas H. 1910 8 15 ll" B! 6-2 3 -'io Paulius Kill, Near Portland, Pa R. 1909! 7 20 60 6 .0 N. 7 16 *io Bridge St., Peoria, 111 H. 1907 4 25 2-5t / Eng. World, 1 2-'o7 St. Paul Minn H 1909 25 _ ft 4 _ ,_- 00 Charles River, Boston, Mass. . .H. & E. Bannock St., Denver, Colo H. 1910 1907 10 i 3 19. 13. 4-5 2 .0 XT' ^ , " N. 12-17- 10 N. 3-2i-'o8 Baltimore, Md H. & E. 1909 4 39 43 . 3-8 N. 6-i9-'o9 Grand Ave., Milwaukee, Wis H. 1910 10 45 28. 2 . 2 N. 2- 9-'o7 Playa del Rey. Cal H. St. Jean La Riviere, Viaduct E. 1906? 1908 46 a 49.2 18. 40. 2 .0 I .9 N. 3-31-06 N. 12- 3~'io Delaware R. near Portland, Pa R. 1910 9 50 B. 12 30 '09 Grand R.. Painsville, Ohio R. 1908 3 60 s&'. 7 -3 N. 5- i-'o9 Tavanasa, Switzerland H. 1908 i b 67.3 18. 0.7 B. 2-i8-'o9 Pyrimont, France H. Guindy River Bridge, France R. Eel River Bridge, Humboldt Co., Cal.H. 1907 1907? igut 3-5 i 7 , 69 b 77 . So 25- 21. 26. s'o B. 4- 2 -'o8 B. 3-26-'o8 B. 3-23-'i i Berlin, Germany H. & R. 1910 3 83.7 N. 8-13-10 Decize, France H. 83.7 IS. "i'.6' G. 4 Tri. 1905 Meadow St., Pittsburgh, Pa H. 1910 7 09 46. 6.2 B. 12- I-'lO Sitter Bridge, Switzerland H. 1908 7 59 87. 3-9 V. 3-13-09 43d St., Philadelphia, Pa H. Grafton Bridge, Auckland, N. Z H. 1909 1910 i i 62 b 3 20 'B: B. s- 2 o-'o9 B. 8- 4 -'io Hudson Memorial Br. (Proposed) i 703 177. 15-0 N. n-i6-'o7 t About. t Clear. a Two hinges b Three hinges. TABLES. 357 TABLE XXXI. CAST-IRON ARCHES. Name, etc. Date. Engineer. "Z, Span. Rise. Reference. Southwark, London, England. . Sunderland, Durham, England. St. Louis, Paris, France Rock Creek, Washington, D. C. El-Kantara, Algeria Pont du Carrousel, Paris 1819 1796 1862 1858 1864 1836 Rennie Burdon Martin Meigs Martin Polenceau 3 3 *2 4 236 210 200 34 26 6 A. '56-261 A. '42-334 Iconographic K. 5-3-67 K. 3-8-' 67 A. '39-81 Blackfriars, London 1869 Cubitt 5 *i86 Chestnut St., Philadelphia Staines, England 1866 1803 Kneass Wilson 185 1 80 ' K. 6-26-' 68 K. 9 13 '95 Galton 1 80 Lendal York England 1862 1826 Page Telford 172 170 \. Vol. 25185 Tewksbury, Gloucester, Engl'd. Battersea, England New North, Halifax, England. . . Hill's, Bristol, England 1890 1869 1809 Bazalgette Fraser 5 2 *i6 3 1 60 1 60 K. 4-i9,-'9S K. 5-7-69 A. 55-111 France 1899 3 "158 ^- 9-5~'o3 Bonar, Scotland 1812 Telford Craigellachie, Scotland Telford 1 150 5 High Bridge, England l83O Potter j 140. " ' 37 151 Buildwas England Telford Barnes, England Victoria, Windsor, England. . .. 1849 1851 Locke Page 3 *I2o! < 5-3l-]9S Westminster (new) England 1863 Page '120. C. 3-8-' 95 Chepstow, England 5 *II2. Pont d'Austerlitz, Paris "1806' Lamand6 5 *I06. >. New Leeds England Steel i IO2. Coalbrookdale, England Lary, Plymouth, England 1779 1827 Darby Rendel I 5 l~ Bristol, England 1806 Jessop *IOO. Nottingham, England Richmond England 1871 1848 Tarbottom Locke 3 100. [' I ~ 6 1~' 71 St. Denis Paris : IOO . 1 1 . * * 95 Ravenswharfe, Dewsbury, Eng. 1848 Grainger 2 100. 12. L. '48-62 W. of Leicester, England New Logan Glasgow, Scotland 1839 1890 Vignoles. 3 4 * 91 is'. ' 3-25-'o4 C. 8-22-',JO Witham, Boston, England Rennie 86. 5- 0- Maximum. 358 A TREATISE ON ARCHES. TABLE XXXII. WROTJGHT-IRON AND STEEL ARCHES. Name, etc. Date. Engineer. "8 I d w Span. Rise. Reference. Clifton, Niagara H. Viaur Viaduct, France R. Bonn, Germany H. Dlisseldorf , Germany H. Douro, Portugal H. & R. Kaiser Wilhelm, Germany. .R. Niagara H. & R. 899' 898 898 885 897 897 884 Krohn Krohn Seyrig Rieppel Buck Eiffel i 3 4 6 j 7 i a 840.0 6*721.6 a*6i 4 .o 0*594-5 566.0 * 557. 6 550.0 76.2 9S! 4 59.5 31.5 24.0 R. n-i-'Sp B. 8-8-' 95 B. 4-2o-'99 K. 7 -2-'86 N. i2-25-'97 B. 8-6-; 96 Bellows'Falls, Vermont H. Levensau, Germany. . .H. & R. 90S 894 877 Worcester Lauter Eiffel & Co i i b 540.0 a 536.0 T90 .0 69.0 N. 4-29-05 K. 8-i6-'9S Eads, St. Louis H. & R. Grlinenthal, Germany. H. & R. Washington Br., New York. . H. Victoria Falls, Africa R. 873 892 889 Eads Eggert Hutton 3 i 2 *520.0 a 513-5 a 510.0 T53-0 77-3 98.3 Br. History K. 8-i6-'95 Br. History Paderno, Italy R. Lake St. Minneapolis H. Costa Rica R. 888 902 Rothlisberger Sewall Cooper 4 492.0 6*456.0 123-0 jgo.o B. 6-iI-; 3 N. 12-7-95 Driving Park, Rochester, N. Y.H. Doran Arch, Richmond, Ind. H. Trisana, Austria R. Worms, Germany R. 889 882' 900 Buck Doran Huss 3 b 428.0 b 408.0 a 393- 6 0*383.1 t48 2 B. 6-22-'99 G. '88xvi-73S Schwarzwasser, Switzerland R. Panther Hollow, Pitts., Pa..H. Kornhaus, Berne, Switz. . . .H. Pont Alexandre III, Paris, 882 ' 898' 896 Probst Schultz Resal 6 373-9 360.0 * 376.7 45-0 103.7 G. '91 N. 6-4-'98 B. i2-i6-'<>7 Austerlitz, Paris, France. . ..R. Worms, Germany H. ? 90St i b 351.6 0*346 4 T40.0 B. 12-7-05 Troitsky, St. Petersburg, Rus- sia H. En neer 'o Stony Crk.. British Columbia R. Palatinus, Mayence, Germany . . Mirabeau. Paris, France. . . .H. Pesth, Austria H. Coblenz, Prussia R. Foot-bridge, Paris, France. .H. Germany Verona, Italy H. 893 885 896 873 866 t 882 885' t 898 Peterson Lauter Resal Gouin Co. Hartwich Moisant Biadego i 4 3 5 3 a 336.0 0*335.0 0*326.0 0*321 .0 315.0 302.3 a 298.5 a 288.6 80.4 19.5 28i6 40.3 J3S.V N. 2-20- ! 04 K. 6-5-96 K. 6-7-67 K. 3-9-83 K. 4-i7-'8s Viad. de Passy Paris, FranceR. a 281 i Arcole, Paris, France H. Main St., Minneapolis, Minn . H. Paris, France H. White Pass. Alaska R. Blaauw-Krautz, Cape Colony. . D'Argenteuil, France. Aq.& H. Versham, Switzerland H. El Cinca, Spain H. 855 888 899? 900 884 894 897 866 Oudray Strobel Lion Wood Max Am Ende Bechmann Berg 2 3 3 262.4 b 258.0 0*246.0 b 240.0 229.7 *229.6 229.6 26.0 49.3 90.0 F97-0 22. 2 B. 4-i4-'88 N. s-io-'90 B. 8-30-00 B. 3-28-01 B. 1-24-85 G. '97 Engineer 97 Lafayette. Lyons, France. . .H. Moraude, Lyons, France . . . H. So. Market St., Youngstown, Ohio H. Fraser River. Can. Pac R. Near Iron Mountain. Mich. . . R. Pont du Midi, Lyons, France. H. Fairmount. Philadelphia, Pa.H. Croton Dam, N. Y H. 889 890 898 893 902 ' 897' 1" 9S Tavernier Tavernier Fowler Peterson Loweth Thayer Smith 3 3 I I I 3 4 *22I . I *22I . I a 210.6 210.0 7 207.0 1 200.0 b 200.0 I 4 .6 14-6 60.0 l"47- o 14.7 40.0 43-4 ' 93 . R. 9-91 N. 2-4-99 K. n-29-195 B. 11-20-02 ^enie Civil '92 Blue prints B. I 2-1-' 04 Noce Chasm, Austria H. Canton Berne, Switzerland. . H. Forbes St.. Pittsburg, Pa H. Cambridge Boston, Mass. . .H. 890 897 90ot Hagen Jackson i i i ir a 196.8 a 196.8 ) 195.0 01*188.5 33-9 J32.8 59.0 26.7 3. 2-I-'90 3. 2-26-'03 N. 4-i5-'5 Maximum. t About. t Clear. a Two hinges. b Three hinges. TABLES. TABLE XXXII (Continued). WROUGHT-IRON AND STEEL ARCHES. 359 Name, etc. Date. Engineer. c r Span. Rise. Reference. Blackfriars, London, EnglandR. 1886 Barry 5 *i8 S -o 18.5 K. 2-i-'9s Fille, France H. 1896 i b 184.8 16.4 Anel River, Sumatra R. Post a 184.0 R. 1 1-' 97 Trevallyn, Launceston, Tas- Doyne 184 o Vienna, Austria R. 1897 Gridl j 183.7 Pimlico, England R. 1860 Fowler 4 a 175-0 17-5 K. 3-22-'9 S New North, Edinburgh, Scot. H. Pont Boiddien Rouen, Fr. ..H. 1899 1888 Blyth 3 ti75.o K. 1 0-6-' 99 Becton, England ' Gas 1870 Evans i fil'.o 3'. Vienna, Austria H. ti897 Pfeuffer * b 174.0 5 N. 8- 1 8-' oo Garibaldi, Rome H. 1888 Vescovali 2 ti6^3 R. I-'93 Cerveyrette, France H. Fall Creek, Ithaca, N. Y. . . .H. 1892 1898 Baldy Landon I a 170.0 137-8 34-0 R. 2-'92 B. 4-28-98 Chagrin River, Bentleyville, Ohio H. 1896 Osborn 1 a 168.8 29.5 Osborn Co. Manhattan Arch, N. Y H. tlQ02 ,Qf\A I a 168.5 B. 2-IO-'03 Brooklyn, Ohio.. . - H. King Charles, Stuttgart, Ger. H. Mill St Watertown, N. Y.. . H. I OQ4 1893 Leibbrand 5 i 0*165 .6 b 165.0 t:? N. 3^-5-'98 Canningtown, England H. Forbes St., Pittsburg, Pa H. 1897 1874 Binnie Pfeifer i Jtso.o 150.0 1*5.1 26.0 N. 7-i S -'99 Cedar Ave., Baltimore, Md. . H. 1891 Latrobe b 150.0 38.0 T. 9-i8-'9i Battersea, England R. 1863 Baker 5 144.0 K Forbes St., Pittsburg, Pa. ... H. Riverside Cemetery, Cleveland, 1899 Brunner a 144.0 24.0 N. 7-i5-'99 Ohio H. 1896 Osborn I a 142.0 27.0 Dsborn Co. Anacostia, Washington, D. C.H. Manhattan Viaduct, N. Y. . .H. 1900 Douglas Williamson 6 23 6tl29. 2 *i 8.6 ti4.5 N. 8-i9-'os B. 6-8-'99 Vienna , Austria R. 1897 Gridl 3 *i 9.4 Albert, Glasgow. Scotland. . H. Victoria, Stockton, England. H. 1870 1887 Bell Neate 3 3 *i 4.0 K. 7-1-70 Michigan Ave., Lansing, Mich.H. Parahyba River, Brazil R. 1895 Landor Ellison Oldfield tio.5 K. 8-2 1-' 68 Mvtao... '.'.'.'.'.'.!".'.'.".'.".". ".".ft Mill Creek, Youngstown, O . . H. 1894 Page Fowler I Jioo.o b 96.0 tio.5 K. 8-7-68 Fowler Carlsburg Viaduct, Denmark H. a* 90.0 J38.'o' ST. 5-i6-'o3 Lake Park, Milwaukee, Wis. H. '1897' Sanne I a 87.0 14.0 Sanne Rock Lane, New Haven, Con- necticut H. 1891 Hill I b 84.3 14-3 B. 8-1 6-V Weston Aq., Southboro, Mas- sachusetts Aq. & H. Oker. Brunswick, Germany. . H. 1903 Stearns Haeseler I t 80.0 b 78.7 Uo V. IO-25-'O2 R.. 2-' 89 Richmond Wier, England. . . H. i892J" More 5 t* 66.0 EC. 6-28-'9S Thirtieth St., Philadelphia, Pa. R. Lake Park. Milwaukee, Wis. H. 1894 Wilson Sanne i i i 64.1 ' 50.0 '0: . 7-2 2-' 70 Sanne t About. t Clear. a Two hinges. SUPPLEMENTARY TABLE. WROUGHT-IRON AND STEEL ARCHES. b Three hinges. Name, etc. Date. Engineer. o'w Max. Span. Rise. Reference. 2 Rio Fiscal Br., Guatemala. .. R. Assopus Viaduct, Greece. . . .R. Fort Snelling, Minn H. Mannheim, Germany H. 1907 1906 1909 1908 Vogue Contractors Shunk 6364^0 38.5 78.0 84.5 22.8 N. 4-4-08 B. 1 1 -4-09 N. 6-26-'o9 B. 6-18-08 Oakland Br., Pittsburgh, Pa.H. 1907 Whited 440 70.0 B. 5-16-07 Tonkin, China R. b 532.8 94.8 B. 5-27-09 a Two hinges. b Three hinf A TREATISE ON ARCHES. TABLE XXXIII. METAL ROOF-ARCHES. Name, etc. Date. Span. Rise. Reference. Liberal Arts Bldg., Col. Ex., Chicago... . b 1892 368.0 206.3 B. 9 -i-' 9 2 Rpof of Main Bldg., Lyons Ex a 1894 361.0 108.0 Train-shed, Philadelphia, Pa., Penn. R.R.b 1893 300.7 100.3 B. 6-i-' 93 "Pa.&R.R.R6 1892 259.0 JS8. 3 B. i-i 3 -'9 3 " Pittsburg, Pa., Penn.R.R. . . b (-1902 255-0 89.0 N. 8-23~'o2 Jersey City, N. J., Penn. R.R. b 1891 252.7 89.8 B. 9~2o-'9i ' ' St. Pancras 1868 240.0 tl2 4 .8 74th Regiment Armory, Buffalo, N. Y. . . b Chicago Coliseum (old) b ti16 221.0 215 .O 94-o 73-o N. 6-^-'oo Train-shed, Cologne, Prussia a 209.0 78-7 B. 10 6 '92 Chicacro Live Stock Pavilion a IOOC 108 o B. 6 28 '06 Dome, West Baden, Ind b A y w j 195.0 42.5 B. 9 4 '02 47th Regiment Armory, Brooklyn, N. Y. b 69* " " New York b 11905 189.8 84.0 N. i2-2 3 -' 99 B. 6-i-'o5 Kansas City Coliseum 187-3 B. 7-5 -'oo Train-shed Frankfort Germany . . b "f"i8oi 184.0 hnr o M" _ Dome, Horticultural Bldg., Col. Ex. 1892 181.6 >yj - w 91.0 i^ . y LZ y I B. '92-1-240 St Louis Coliseum b tiSoo 178 c 8O.O D Q Tf)Jf\>7 22d Regiment Armory, New York i *yy 1889 / - o 176.0 t62.0 97 U S Gov Bldg St Louis Ex b t I 94 172 .0 66 8 B o 20 'o/i i 2th Regiment Armory, New York 1888 171-3 55-6 * y ^y U 4 ist " " Newark, N. J. .. b 1900 163.5 73-3 ST. 5-26-'oo ist " " Chicago 1894 155-5 77-5 B. '94-11-176 Chicago Coliseum (new) b 1802 149.8 T 2O O 66.0 ST. 12 24 '92 Machinery Hall Col Ex b Dancing Hall Lattain Beach b 10y. 1803 'O ' w 118. 7 ^ " 93~ II 379 1 3th Regiment Armory, Scranton, Pa o Vo 112. 49-5 N. 8-2 4 -'oi t About. J Clear. a Two hinges. b Three hinges. TABLES. 361 KEY TO REFERENCES. A. Civil Engineers' and Architects' Journal. B. Engineering News. C. Weale's Bridges. D. Pennsylvania Railway Company's Blues. E. Wm. H. Brown, Chief Engineer, Penna. Ry. Co. F. Construction des Viaducts, Tony Fontenay. G. Annales des Fonts et Chausses. H. Mahan's Civil Engineering. I. Masonry Construction by Baker. J. Spon's Dictionary of Engineering. K. Engineering. L. Edinburgh Encyclopaedia, gth Edition. M. Scientific American Supplement. N. Engineering Record. O. Engineering Magazine. P. Journal of the Association of Engineering Societies. Q. Encyclopaedia Britannica, gth Edkion. R. Railway and Engineering Journal. S. Cresy's Bridges. T. Railway Gazette. U. Murray's Handbook of Northern Italy. V. Le Genie Civil. W. Messrs. Keepers & Thacher. X. The Melan Arch Construction Co. Y. Transactions of Am. Soc. C. E. INDEX. PACK Alexander and Thomson's method 234 Appendices 263 Application of vertical loads 159 Applications, Chapter VII 159 Arch-ring, thickness of, at skew-back 225 Axial stress, effect of 272, 283 Brick arch 228, 254 Catenary, equation of 234 " two-nosed, i . 236 " transformed 235 Circle, the three-point 238 " described 238 Circular arch, - = constant: .A Curve, general equations for 39, 88 A (see general equation) 90 Ax (see general equation) 93 Ay (see general equation) 95 Hi for horizontal load, N x included 42, 102 " " " " neglected 41, 101 " changes of temperature 42, 103 " " in length of span 42, 103 " vertical loads, N x included 40, loo 363 364 INDEX. PAGE ffi for vertical loads, N x neglected 39, 98 V\ ' ' horizontal loads, N x included 42, 103 " " " " neglected 42,102 " vertical loads, N x neglected 39, 99 x a for horizontal loads, N x neglected 41, 102 y a for vertical loads, N x included 40, 101 " " " " neglected 40, 100 Symmetrical circular arch without hinges : A(p (see general equation) 90 Ax (see general equation) 93 v Ay (see general equation) 95 HI for horizontal loads, N x neglected 44, 106 " changes of temperature, TV 7 ! neglected 45, 107 " change in length of span, N x neglected 107 " changes in Z/0o, N x neglected 107 " vertical loads, N x neglected 43, 104 general expression for 108 M\ for horizontal loads, N x neglected 44, 106 " changes of temperature, N x neglected 108 " changes in length of span, N x neglected 108 " vertical loads, N x neglected 43, 104 general expression for 109 V\ for horizontal loads, N x neglected ". 45, 107 " vertical loads, N x neglected 44, 105 y a , y\, and yi, values of, for vertical loads, N x neglected 44, 105 Comparison of four types of arches : If i for vertical loads 144 " changes of temperature 153 MX for horizontal loads 155 " vertical loads 151 M x for symmetrical parabolic arch with two hinges, vertical loads only, table of values 153 M x for arch without hinges, table of values 152 Fi for vertical loads 146 V x for symmetrical parabolic arch with two hinges, vertical loads only, table of values 149 V x for arch without hinges, table of values 148 Stresses, comparison of, for three types 156, 157 Weights, comparison of, for three types 155 Comparison of results of tests with theory 254 " " " Douro spandrel-braced arch 221 " " " fixed parabolic arch 191,198, 201 " " " " St. Louis arch 213,209 " " " " Douro bridge 186 Concrete arch 228 INDEX, Concrete 254 Conclusions drawn from tests 255 Co-ordinates x a , y a , etc 161 Ax, Ay, As, and A = constant : Curve, general equations for ... 52, 53 4n for horizontal loads, N x neglected 64 " " " general 54 " vertical loads, N x neglected 60 Ax for horizontal loads, N x neglected 64 Ay for horizontal loads, JV X neglected 65 " vertical loads, N x neglected 6 1 Hi for horizontal loads, N x included 28, 65 " " " " neglected 27,63 " temperature changes 29, 66 " changes in length of span 29, 67 " uniform loads, N x neglected 23, 68 " " load over all 24,68 " vertical loads, N x included 26, 61 " " " " neglected 20,58 M x for uniform loads, N x neglected 24, 68 " " load over all 69 table of values for vertical loads 153 V x table of values for vertical loads 149 Vi for horizontal loads, N x included 66 INDEX. 367 Fi for horizontal loads, N x neglected 27, 63 " uniform loads, N x neglected 23, 68 " " load over all , 69 " vertical loads 21, 58 #0 for horizontal loads, N x included 28 " " " neglected 27,63 yo for vertical loads, N x included 27, 63 " " " '' neglected 21, 59 Symmetrical parabolic arch without hinges : A

, general expression for 54 Ax for horizontal loads, N x neglected. * 80 " vertical loads, N x neglected 74 general expression for 55 Ay for horizontal loads, N x neglected 81 " vertical loads, N x neglected 74 general expression for 55 HI for horizontal loads, N x included 35, 81 " " " " neglected 33,77 ' ' changes in length of span 37, 84 " " of temperature 36,83 " " in 0o , 0/ , Ac, etc 85 " uniform loads 37, 86 " " load over all 87 " vertical loads, N x included. 31, 75 " " " " neglected 29,71 Mi for horizontal loads, N x included 35, 82 " " " " neglected 33, 78 " changes of temperature 37, 84 " " in 0o, Ac, etc 86 " uniform loads 37, 86 " vertical loads, N x neglected 30, 71 Mt for horizontal loads, N x included 35, 82 " " neglected 33, 78 " vertical loads, N x included 32, 76 " " " " neglected 30,71 MX for uniform loads 86 table of values for vertical loads 152 V x table of values for vertical loads 148 V\ for horizontal loads, N x included 36, 83 " " " " neglected 34, 78 " uniform loads 37, 86 " vertical loads, N x included. ... 76 " neglected 30,72 368 INDEX. x for horizontal loads, N x neglected 34, 80 Xi for horizontal loads, N x neglected 34, 79 ' ' vertical loads, N x neglected , 30, 73 jr a for horizontal loads, N x neglected 34, 79 " vertical loads, N x neglected 30, 73 y y> t and_j2 for horizontal loads 34, 79 ' ' vertical loads 30, 72 Symmetrical parabolic arch with one hinge: HI for a single horizontal load 140 " " " vertical load 139 M\ for a single horizontal load 140 " " " vertical load 139 V\ for a single horizontal load 140 " " " vertical load 139 Pins, steel 230 Reactions, character of 161 " for Douro bridge 187 Resultant, application of 1 1 " " " for several forces 18 Rough quarry-stone arch 253 ^-Semicircular arch X^85-288" Seyrig .777 182 Spandrel-braced arch 218 Spandrels (see Earth, Masonry, etc.). Special formulas, deduction of 289 Specifications, masonry arch, Austrian 256 St. Louis arch 204 Stresses, comparison of, three types 156, 157 Stress diagram, Douro bridge 188 Summation formulas applied to spandrel-braced arch 217 " " fixed parabolic arch 190 " " " " two-hinged arch 182 Summation formulas : Symmetrical arch with two hinges : Hi for a single horizontal load 50, 130 ' " vertical load 49, 130 " changes of temperature 50, 130 Symmetrical arch without hinges ' Hi for a single horizontal load 48, 129 < vertical load 46, 129 " changes of temperature 49, 130 Mi for a single horizontal load 48, 129 " a . vertical load 47, 129 Tests of arches 253 Temperature, St. Louis arch. . _ 214 INDEX. 369 PAGE T x , maximum value of . 22 Unsymmetrical loading, masonry arches 245 Variable moment of inertia : Symmetrical arch with two hinges; Hi for a single horizontal load 50, 127 <> .. vertical load 49, 125 " change of temperature 50 Symmetrical arch -without hinges : H\ for any symmetrical loading 115 " a single horizontal load 48, 118 " " " vertical load 46, 117 " changes of temperature 49, 121 Ali for any loading 112 " a single horizontal load 48, 120 < < vertical load 47, 119 " changes of temperature 49, 121 Symmetrical arch -with one hinge : H l for a single horizontal load 135 " vertical loads 132 MI for a single horizontal load 138 " vertical loads 134 Fi for a single horizontal load 137 " vertical loads 134 Vertical loads (see Parabolic, Circular, etc.). 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