AN INVESTIGATION 
 
 OF THE 
 
 ONE-HINGED STEEL ARCH 
 
 AND ITS 
 
 COMPARISON WITH OTHER TYPES 
 
 A THESIS 
 
 presented to the 
 
 Faculty of the Graduate School 
 Cornell University 
 
 for the degree of 
 
 Dodtor of Philosophy 
 
 by 
 
 Nee Sun Koo, B.S., M.C.E., 
 
 
 McGraw Fellow in Civil Engineering, 1920-’2 1 
 1921 
 
 Reprinted from the Cornell Civil Engineer, 
 volume 29, pages 110, 129, 150, March, April, May, 1921 
 
Digitized by the Internet Archive 
 in 2017 with funding from 
 
 University of Illinois Urbana-Champaign Alternates 
 
 https://archive.org/details/investigationofoOOkoon 
 
AN INVESTIGATION OF THE ONE-HINGED ARCH AND 
 ITS COMPARISON WITH OTHER TYPES 
 
 By Nee Sun Koo, B. S., M. C. E. (1919). 
 
 McGraw Fellow in Civil Engineering, 1920-21. 
 
 An Abstract of a Thesis to be Presented to the Faculty of the Graduate School of Cornell University for 
 
 the Degree of Doctor of Philosophy. 
 
 PREFACE 
 
 During the last thirty years much study has been 
 given to the design and construction of steel arches 
 by American engineers and investigators. Two- 
 hinged and three-hinged arches seem to have met 
 the greatest favor, while few no-hinged steel arches 
 have been built in this country. One-hinged arches 
 are practically unknown in America, although a few 
 have been successfully constructed on the European 
 continent. After considerable study, the author 
 has been in doubt of the practicability and value 
 of the one-hinged steel arch. He could see no 
 logical or conspicuous reason against its adoption. 
 A number of writers attack it bitterly while enumer- 
 ating a few disadvantages, but fail to demonstrate 
 or justify their statements. Others deem it un- 
 necessary to give a full treatment, because it' has 
 not come into popular use. The deficiency of 
 theoretical knowledge may be the reason why the 
 building of one-hinged steel arches has been at- 
 tempted so rarely. With this idea in mind, the 
 author has undertaken a special investigation of 
 the one-hinged steel arch. 
 
 While this work is undertaken purely for the 
 purpose of discovering and publishing some un- 
 known facts, some contributions made by previous 
 investigators will be mentioned. One of the noblest 
 and most remarkable preliminary designs which 
 have ever been made in bridge engineering was con- 
 tributed by Charles Worthington in competition 
 with other designs for the famous Quebec Bridge. 
 It was a one-hinged steel arch with a span of 1,800 
 feet, more than .twice as long as that of any arch 
 bridge then existing in the world. One of the most 
 noted American bridge engineers. Dr. B. A. L. Wad- 
 dell, in his book called “Bridge Engineering,” 
 praises the work of Mr. Worthington and calls the 
 design ingenious and quite feasible. It is to be 
 regretted that the scheme was not accepted by the 
 Canadian Government and thus the complete treat- 
 ment of the theoretical and practical problems in- 
 volved was prevented. The work gives some indi- 
 cation of what could be done with the one-hinged 
 arch and done economically. Dr. Waddell’s com- 
 ment is valuable on account of his extensive ex- 
 perience and gives promise for the future. In view 
 of these facts, the author felt encouraged to carry 
 on this investigation. 
 
 Special emphasis is laid upon two points, first, 
 
 to make an extensive study of its behavior in carry- 
 ing the load ; and second, to reveal its characteristics 
 by critical comparisons with other types of arches. 
 It is his hope that the work may be of value to the 
 engineering profession in the future, if not at pres- 
 ent, Since so little has been written in engineer- 
 ing literature upon the subject, it is hoped that 
 every bit of the original work in this thesis will 
 appeal to the sympathetic interest of engineers and 
 investigators. Acknowledgment is due to Profes- 
 sor Henry S. Jacoby, chairman of the special com- 
 mittee in charge of the author’s graduate work at 
 Cornell University, for his aid and helpful sugges- 
 tions. 
 
 Reactions Solved by the Author’s Method of 
 Symmetrical Deflection Equations. 
 
 Two kinds of beams, either straight or curved, 
 are used in bridge building; those which are 
 statically determinate, and those statically inde- 
 terminate. For the solution of beams of the first 
 class, three statical equations are available: — 
 
 = O ( a > 
 
 EH- O (b) 
 
 EM O 
 
 (c) 
 
 For the solution of beams of the second class, three 
 elastic conditions are available, besides the statical 
 conditions. These are 
 
 Ay 
 
 -f- 
 
 f 
 
 Pixels 
 
 El 
 
 Plds 
 
 (d) 
 
 (e) 
 
 (f) 
 
 where Ay is the horizontal deflection, Ay the ver- 
 tical deflection, and jo the change of the slope be- 
 tween any two points on the beam, E the modulus 
 of elasticity of its material, I the moment of in- 
 ertia of a section, and .1/ the moment at any point 
 between a and b , where I is taken. By means of 
 these six fundamental equations, all beams can be 
 solved. (For the derivation of formulas (d), (e) 
 and (/) see standard books on' Mechanics). The 
 
author lias used these equations in deriving general 
 formulas of reactions for no-, one-, two-, and three- 
 hinged arches. 
 
 Let an arch-rib with a span l and rise h be fixed 
 at two ends a and b, and hinged at the crown c and 
 be subject to a vertical load P at a distance kl from 
 the left end. (Fig. 1). The load is sustained by 
 the reactions H x , V x and M x at the left support, and 
 by H 2 , V and M 2 at the right support. There are 
 six unknowns and six conditions are required for 
 its solution. The three statical equations furnish 
 three conditions, while the hinge at the crown in- 
 sures that the moment at that point is zero. It 
 remains for us to find two more elastic conditions. 
 In other words, we must choose any two equations 
 from ( d ), (e) and (/) and apply them to the case 
 of the one-hinged arch. 
 
 In order to simplify the algebraic work, the fol- 
 lowing notations are proposed by the author: — 
 
 vertical line at the crown. (Fig. 1). Formulas (d) 
 and (e) are used in finding the vertical and horizon- 
 tal deflections between a and c and between b and c. 
 As the supports are actually fixed, the vertical deflec- 
 tion between a and i must be equal to that between b 
 and c; thus giving the first condition. Also, the horiz- 
 ontal deflections between a and c and between b and c 
 must be equal in magnitude and opposite in di- 
 rection, because what is lengthened in one-half of 
 the rib must be shortened in the other. Thus, we 
 obtain the second condition. For the left half of 
 the rib with the origin at a the moment at any point 
 ■between a and c is 
 
 M = rij -t\£a? ~Hy -P(~c-fd) ••••(!) 
 
 For the right half of the rib with the origin at b 
 the moment at any point between b and c is 
 
 M=Mzi-\t,3c-Hy (2) 
 
 Substituting (1) in ( d ) and ( e ) and replacing the 
 integral forms with the notation above given, we 
 have the horizontal and vertical deflections be- 
 tween a and c respectively, 
 
 EA.ir - i- V x r ~/it -P(r 2 -/ilq 2 )' • (3) 
 JEAy =7^/? i-lfs-Hr -P(p^-kCp^ • • (i 
 
 Substituting (2) in the same formulas and making 
 the same substitutions for the integral forms, we 
 
 P=P, +P? 
 
 q = + <ja 
 
 r = r,+r 2 
 
 have 
 
 Eaot = Pt a q fit — ( 5 ) 
 
 Paz/ - M^p -t- Ur -...(6) 
 
 where the modulus of elasticity is assumed to be 
 constant. Equating (3) to (5) and (4) to (6) as 
 above mentioned, we have 
 
 S£ Jkl 
 
 ^zpc/s 
 
 r ^ 
 
 ~J/d r 
 
 £5 = S, / 
 t = t. £s. 
 
 pr s q 
 
 The hinge at the crown breaks the continuity of 
 the beam, hence a special method must be used 
 in applying the two elastic conditions. M. A. Howe 
 in his book called “A Treatise on Arches” succeeds 
 in reducing the required elastic conditions into one 
 by means of symmetrical loading. But the method 
 is rather lengthy, and the formulas obtained rather 
 complicated. By means of symmetrical deflection 
 equations, the problem is easily solved by the 
 author, with the final formulas expressed in a very 
 simple form. Let us assume that the arch-rib >s 
 divided into two parts at the crown by an imaginary 
 
 y - Ifs -Hr -P(k> -k. ZpP) — -Hr 1 
 
 -h Ifr -Ht ~P(r 3 -ftlq e ) = ~(j% q +l£r-Ht) 
 
 V (8) 
 These two equations together with the three statical 
 equations and the equation furnished by the hinge 
 at the crown as shown below are sufficient to solve 
 all the unknowns. 
 
 M, .... (9) 
 
 / T*L / /l • • • ( 10 ) 
 
 j% -a* - m^ sk ) -° 
 
 V+\£-P=o 
 
 H, -H^ = o ... ( 12 ) 
 
 Thus, by solving (7), (8), (9), (10), (11) and (12;, 
 we have 
 
 4 
 
B'.-iZ-rMlL-ff- ( A) 
 
 v _ „ Ip-kLpx-s -ss (B) 
 
 * Ip -as 
 
 (C) 
 
 Mj = - Vi fj + &(j-zk) (D ) 
 
 M 2 ~ Ml 1-Jfl PI (l-fi ) (E ) 
 
 It must be remembered that these formulas hold 
 good only when the load is on the left of the crown. 
 They are applicable to a one-hinged arch with any 
 form of arch-ribs. 
 
 
 1 
 
 F/gJ 
 v. c 
 0 $ 
 
 \Yz 
 
 
 K- A- f — 
 
 3 
 
 fir 
 
 
 
 Mis 
 
 FO 
 
 t i 
 
 i 
 
 
 
 F/g.2. 
 
 These formulas again hold good only when the 
 load is on the left of the crown with its direction 
 toward the left. They are also applicable to any 
 form of arch-ribs. 
 
 Reactions Solved by Author’s Cantilever Method 
 
 Another interesting method! is derived by the 
 author in securing the two elastic equations for solv- 
 ing the reactions of a one-hinged arch. It is called 
 the cantilever method, because the main feature lies in 
 separating the arch into two separate cantilevers free 
 at the crown and fixed at the supports. The vertical 
 and horizontal deflections at the free end of each can- 
 tilever are then found and equated 1 so as to furnish 
 the two necessary equations. The method is very 
 simple, because the deflections at the free end's of can- 
 tilevers can be easily obtained. Thus, the following 
 deflection table includes all the data needed for the 
 solution : 
 
 Table 1. Deflection Table for Curved Beams 
 
 Load/ng 
 
 D/agra/7? 
 
 Ax. at c 
 
 Ay at c- 
 
 Vert Lootof of 
 M from <=> 
 
 VLAR. 
 
 - # H- ^ 
 
 
 hor Load at 
 k! from o 
 
 
 
 ( -jr (dkfi-r) 
 
 Vert Loacf of 
 free enof 
 
 _ |C 
 
 -f 
 
 ~ ^ (ip-ri) 
 
 Hor Loo of 
 at free esief 
 
 !/< £ 
 
 o\ i__i 
 
 
 + (hfi- r) 
 
 e-.J.s. 
 
 If instead of a vertical load P on the span, we place 
 a horizontal load P at the same position acting 
 towards the left and at a distance k'h from the line 
 ab, (Fig. 2) the problem can be solved in a similar 
 way. In this case the direction of II. > and V 2 are 
 reversed in comparison with those under the vertical 
 loading. The following formulas are derived for 
 the horizontal load on the span with the same 
 process : 
 
 «• _ d shg-hk'g,-t~t 
 
 fj, 
 
 3 y ^ 
 
 V -knpi 
 
 as> -ip 
 
 m 2 =Mj -jpfih 
 
 ....(F) 
 
 ....(G) 
 
 ....(H) 
 
 .•••(I) 
 
 The expressions in this table can be easily derived 
 
 by using formulas (d) and (e). 
 
 / 
 
 In order to find the elastic conditions, let us sepa- 
 rate the arch at the crown. Since there is no load on 
 the right half of the rib (See Fig. 3) and since the 
 right reaction line must pass through the crown, the 
 two forces H and V 2 must act at the free end of the 
 right cantilever with their directions as shown. 
 Therefore we can consider the left cantilever as sub- 
 jected to two external forces H and V 2 at the free end. 
 On the left half of the rib there is a load P. But 
 since action must equal the reaction, the forces acting 
 at the free end of the left cantilever must be H and 
 V 2 . Let ns find the vertical and horizontal deflections 
 between a and c when the loads H, V 2 , and P are act- 
 ing on the left cantilever. Evidently, the vertical and 
 horizontal deflections at c are contributed by three 
 factors, those due to P, H and V 2 respectively. From 
 table 1 these factors can be summed up in the ex- 
 pressions, 
 
 J2jk.p = -f-H(hp-r) —p(klpi -Sjj -( 13 ) 
 
 =\^(Lq-r) (14) 
 
 Similarly, the vertical and the horizontal deflections 
 at c of the right cantilever are contributed by two fac- 
 tors, those due to H and V 2 . From table 1, we have 
 the expressions, 
 
 5 
 
Table 2 — Reactions and End Moments of a One-Hinged Ar h with Various Forms of Arch-Ribs. 
 
 1 k/ 
 
 P 0-2)1 
 
 Si* 
 / -w 
 /a 
 
 < 
 
 
 r l '' 
 
 One-hinged Arch 
 
 m'V K-K* 
 n* 3z- 3 r* 
 
 0 = 3tt-8 
 
 oc=sin'(zk-/j 
 
 g = r-h 
 
 cosec- ° T7- 
 
 r- +i£±l M 
 th 
 
 1 * I 0 Sec 3 
 
 E * Constant 
 
 P 
 
 r= fiityds 
 
 n-r^yf 6 
 
 S.jfxVs 
 s, 
 
 s A\^ 
 
 K = zarfr- §)(” - oc)+l(f-izr-ahg) 
 d « L2r*l-l 3 -z4r z g(j£ -oc) 
 
 NOTE- 
 
 / Notations as shown above.. 
 
 Z. Assumptions, 1 = 1° Sec e. 
 
 E = constant, shear nea- 
 lected 
 
 3. Formulae under Bending 
 F)omenf ho/d good, when 
 the toad is on tn> 
 of the span. 
 
 CURVE: 
 
 15 
 
 I 
 
 « 
 
 § 
 
 CQ 
 
 4j K 
 
 led 
 
 e left ha If 
 
 b <& 
 
 5 § | 
 
 u> i: ^ 
 <o >u 
 
 H 
 
 Vi 
 
 14 
 
 Mi 
 
 Me 
 
 H, 
 
 Uz 
 
 V, 
 
 Vz 
 
 Mi 
 
 Me 
 
 n, 
 
 Mi 
 
 Hi 
 
 Ml 
 
 GENERAL 
 
 p kiy-'i . 
 2(h 9 -t) 
 
 P bApi-s, 
 Lp-ZS 
 
 -V,±+tih-rQ(,- Z k) 
 
 M,+V,L-Pl(l-k.) 
 
 peho-k’hg.-t-t' 
 
 Z(hq-t) 
 
 pJdhOi-t, 
 
 pJL - . h'fr p, 
 ZS - Lp 
 
 . p n - Ftp , 
 Zs-lp 
 
 vd + h,h-Ph(l-k) 
 
 M,+V,L - Pk'h 
 
 Set" l 
 
 z^-tyin' 
 
 ^ Eet°lh 
 
 if. 
 
 ZLL- 
 
 Z(hq-t)* -ln 2 
 
 - sL 
 
 ZChq-t) 
 
 shL 
 
 2(h ? -t) 
 
 PARABOLIC 
 
 #f (***-**) 
 
 pl [«<i lk ’-4k+i)in+ 
 Tho\^ 3 Q-Z k)(3 tt-zoc) 
 
 P(/-4k 3 ) 
 
 4Pk 3 
 
 -PLQek-i^fJfk) 
 
 ELQsk’-sk*) 
 
 sQ-2ok 3 +4ok?-r6k s ) 
 
 PP(5k-)ok‘)-4k s ) 
 
 lEb(zC- 3k*) 
 
 - ePh(zk 3 -3k‘ > ) 
 
 4Ph(k-k - 7k) /3k? 4k) 
 
 -4Ph(3k-VV-t4K) 
 
 ± iSFe t'lo 
 
 zh*+i5n 3 
 
 zp 15 E<zt°Lh 
 Zh z + l5o* 
 
 _ IS slo 
 Zh z 
 
 L5sIo 
 
 EzLLIPTICAL 
 
 pQ-ok 3 ) 
 
 4-Pk 
 
 -Vl+Hh+%Q-2k) 
 
 -PlQ-k ) 
 
 'o+4k(3-3ki-Zk') 
 
 - 3m(z<c-3n ) 
 
 P- H , 
 
 Ph (4m(akfZkf3)-l 
 
 til ) 
 
 -V, 
 
 ~{iL+ Hh - PhQ-k) 
 
 z 
 
 M.-hVk-Pk'h 
 
 !2 Set's. 
 
 Oh 2 + 120* 
 
 ^ izEet'Lh 
 oh*+ ! 2 .r, z 
 
 12 31° 
 
 oh z 
 
 IZSlo 
 
 oh 
 
 3EGMENTAL 
 
 CIRCULAR 
 
 . — 1 
 
 ZMJ+ZK-Zoc)-Jf’. 
 
 PQ-dk 3 ) 
 
 4Pk 3 
 
 
 M, f til- PlQ-k) 
 
 p-hz 
 
 Zk L [ /2 r 2 - 1\3 -e* i- 4 h 
 
 U6r*fsir>24c'+2(c-*c)]\g+r6tni 
 
 \6lrg(/-4K)sinoc' 
 
 
 pjsiRecV^ sinZoc] 
 [fek z sinoc'--§KJ 
 
 U +d,h-Ph(l-k) 
 
 M,fti,l~Pk'h 
 
 ± /z Eet'll* 
 K+izln* 
 
 T izket°ll°h 
 K + IZlr,* 
 
 ! 2 sl I , 
 
 y. IZslhL 
 H 
 
 Table 3 — General Formulas of Reactions and End Moments of an Arch with the Regular Form of Arch Rib. 
 
 P ,ft*p 
 
 r + 
 
 
 S - ft 4^ds 
 
 Jo l 
 
 p*-ft ¥ 6 
 
 
 <1- find? 
 
 sz Mj ^ 
 
 iMffyds 
 
 t -/j 
 
 q.- £nds 
 
 tpjf'gtds 
 
 
 
 n.Jf^s 
 
 1 * I 0 sec e 
 
 r 2 .fi*qds 
 
 £ ■ ConsEarrb 
 
 NOTE- 
 
 /. Formulae for one and three 
 hinged Arches under the 
 term Bending Moment hold 
 good, when the load is on the 
 left of C. 
 
 2 . B/se or Fall of Temperature 
 has certain effect upon M 
 of the three hinged. The 
 formula is not derived, as 
 its effect upon the truss 
 can best be found bu the 
 graphical method, d — ) 
 
 3. Notations are shown above, 
 r, - radius of Gyration s =■ 
 unit compressive stress in 
 Pib-shortening Formulae. 
 (Average stress) 
 
 ARCH 
 
 NO-HI NGEzD 
 
 ONE-HINGED 
 
 TWO-NINGLD 
 
 THEE L -HINGE D 
 
 £ 
 
 % 
 
 ? 
 
 1 
 
 Q 
 
 § 
 
 -4 
 
 O 
 
 K 
 
 0) 
 
 s 
 
 H 
 
 Bl(brk.l<} 2 )-l<?(k-k') 
 
 2 ir-2q‘L 
 
 e> kloi — n 
 
 Zfrq-t) 
 
 a /7 + k.lQi. 
 z-t 
 
 P>k.l 
 
 ~zK 
 
 V, 
 
 + F-Q-k) 
 
 f=> IfO -kl)0,-S-S* 
 lp-zs 
 
 P(L-k) 
 
 pQ-k) 
 
 Vz 
 
 P-V, 
 
 p kip, -3, 
 lp—2s 
 
 Pk 
 
 Pk 
 
 Mi 
 
 4^0 Qlq - 3 d)-Pkl(/-kf 
 
 -V/L-r Hh + P) Q-Zk) 
 
 o 
 
 o 
 
 Mz 
 
 pr, + Kl -Pl(/-k) 
 
 M, +-V/L- PlQ-k) 
 
 o 
 
 o 
 
 MOEUZON TA L LOAD 
 
 n 
 
 dfa-k'hqj-qlzq,- s 
 
 S\ If k'UQ-te ) 1 > 
 
 2 L it-z<r'/, 1 
 
 p zhg-Fha, -t-ti 
 
 EQ + tt - kings) 
 
 §(*-k) 
 
 Hz 
 
 p-n, 
 
 p k’hat-t, 
 
 Z&q-t) 
 
 P-H, 
 
 Pk.' 
 
 z 
 
 V, 
 
 + Pk'h) 
 
 P n-k'hp, 
 
 2s —Lp 
 
 Pk'h 
 
 L 
 
 Pk'h 
 
 L 
 
 1/a 
 
 -V, 
 
 -P r, - k'hp, 
 
 2 S-Lp 
 
 - Pk'h 
 l 
 
 - Pk'h 
 l 
 
 M, 
 
 %QL<t-3d)Af’])(r,-n)- 
 Zl (2<f-qi)+Pk 'h(l-4k+3k‘) 
 
 -U +H,h-Ph(i-k) 
 z. 
 
 o 
 
 c 
 
 Mz 
 
 rv), + V,L -Pk'h 
 
 M, fV,l -Pk'h 
 
 o 
 
 o 
 
 Qi . jo 
 
 Igg 
 
 JU b k 
 
 K x uj 
 
 H, 
 
 . tzehl 
 
 ± Bei°l 
 
 * Bet°l 
 
 
 * 2 -t - 4 z±L + in' 
 i ~TZ 
 
 Z&q-t) * ifj 
 
 
 
 M, 
 
 -r- 2Be-t 0 n 
 
 
 o 
 
 o 
 
 Sgs 
 
 Q 1 5 $ 
 
 «5 ^ 
 
 h, 
 
 _ sl 
 
 zb- iff. 
 
 — ^ s — 
 
 KM-t) 
 
 -Hr 
 
 o 
 
 M, 
 
 
 + jshl — , — 
 Z(hg-t) 
 
 o 
 
 o 
 
EAy = -%(lp ~s) +H(f?p-r) ’ ‘ ( 15 j 
 
 EJAa? = -V^{Lg-r) -Plf(hg-t) • • • ( 16 ) 
 
 The value of E is assumed to be constant in all of 
 these expressions. Equating the vertical deflections 
 from each half of the rib, we have 
 
 = -IzCkp-'S) -hH(hp-r) ( ' 
 
 from which the formula for V 2 can be obtained di- 
 rectly. Equating the horizontal deflections we have 
 
 +*f(/p-r) -F>(klp, -s,)' 
 
 - v ^kp- r ) -H(hq-t) (18 ) 
 
 from which the formula for H can be obtained di- 
 rectly. From the statical conditions and the condi- 
 tion furnished by the hinge at the crown, we can 
 obtain all necessary formulas for the reactions and the 
 bending moments at the ends. 
 
 In the case of the horizontal loading, the above 
 method applies equally well. The expressions for the 
 vertical and horizontal deflections at the free end of 
 a cantilever under a horizontal load P are also given 
 in table 1. 
 
 Influence of Temperature 
 
 Changes in temperature cause changes in the values 
 of and II i but do not affect the vertical reaction l 7 ^ 
 It. is usually specified that an arch shall be designed to 
 be subject to a certain variation in temperature from 
 a standard value. Let 2 be the coefficient of expan- 
 sion and r the rise of temperature, then the spau 
 length will be increased by %t°l provided one end is 
 free to move. As both ends are fixed in position when 
 the supports do not yield, equal and opposite reactions 
 and end moments are produced. The values of II, 
 and 1/j must be such as to prevent the horizontal dis- 
 placement 2t°£, which is due to the effect of bending 
 and thrust. The horizontal deflection due to bending 
 and thrust are given by the following expressions: 
 
 in which A is the area of cross-section. Equating the 
 deflections, and 
 
 substituting H^h-y) for M and solving for H 1 and M u 
 we have 
 
 ■/. Eet°l 
 
 (K) 
 
 (L) 
 
 where Io is the moment of inertia of the section at the 
 crown and )\ is the radius of gyration of the same sec- 
 tion. For a rise in temperature use the positive sign 
 for II 1 and a negative sign for M\. For a fall in tem- 
 perature, the reverse is true. 
 
 Rib-Shortening 
 
 The direct effect of the thrust along the axis is to 
 shorten the axis of the arch-rib. It would also shorten 
 the span provided one end were free to move, but as 
 this is not the case it will develop equal and opposite 
 negative reactions H 1 and positive moments M\. The 
 effect of displacement due to II 1 and M\ must equal 
 that due to rib-shortening. Let us use S as the aver- 
 age compressive stress on the rib. The shortening of 
 the span is given by 
 
 A. 
 
 Si (21) 
 
 E 
 
 which must be equal to 
 
 ( 22 ) 
 
 Equating (18) to (19), substituting H^h-y) for M in 
 the equation, and solving for H u we have 
 
 r = _ SI 
 
 a(hg - 1) 
 
 
 Shi 
 
 a(hg-t) 
 
 (M) 
 
 (N) 
 
 Application to Parabolic, Segmental Circular, and 
 Elliptical One-Hinged Arches 
 
 The general formulas derived above have been ap- 
 plied to three forms of arch-ribs with parabolic, seg- 
 mental circular and elliptical curves respectively. The 
 equations of these curves all have their origin at the 
 left support a, with the X-axis passing through the 
 supports a and b. The equations of these curves are 
 as follows: 
 
 (y -f-r-hf = ) • ( 24 ) 
 
 An important assumption is made in applying the 
 general formulas, that is I=lo sec e where 7 is the 
 moment of inertia of the cross-section at any point and 
 
 7 
 
o the angle of inclination of the axis of rib at the point 
 where / is taken. This assumption simplifies the work 
 materially and should be dose enough for deriving 
 the formulas to be used in the preliminary designs. 
 Also, ds=sec a dx, while the modulus of elasticity is 
 assumed to be constant in the general formulas. 
 Table 2 gives a comparison of the formulas for the 
 reactions and end moments of the one-hinged arches 
 with parabolic, elliptical and segmental circular arch 
 ribs. These formulas have been derived after a tre- 
 mendous amount of work. Formulas for tempera- 
 ture effects and rib-shortening are also given in the 
 table. 
 
 The Reaction Locus 
 
 Fig. 1 shows a one-hinged arch under a vertical 
 load P. As the ends are fixed, it is unknown where 
 the reaction lines must pass the supports. Let us 
 assume that they pass the left and the right supports 
 at points distant y 1 and y 2 below and above the points 
 of support respectively. The right reaction line must 
 pass through the center hinge. By similar triangles, 
 we have the relation 
 
 in which q is the ordi- 
 
 kl H 
 
 nate to the curve at any point 
 But 
 
 IJt = > therefore 
 
 fj filVi -/-I'll 
 
 ^ III 
 
 ( 0 ) 
 
 which is a general formula of the reaction locus of the 
 one-hinged and the no-hinged arches applicable to 
 any form of arch-rib. It is not applicable when the 
 load is on the right half of the arch rib,. The reaction 
 locus must be symmetrical about the center, so it is 
 not necessary to d'erive the formulas for the load on 
 the right of the span. For parabolic, elliptical and 
 segmental circular arches we have by substitution the 
 following formulas : 
 
 
 h(/4 -/3k) 
 
 9 
 
 h(/+/3k) .(P) 
 
 3(/ik)~ 
 
 for the load on the left and right of the crown hinge 
 respectively. 
 
 Equation Q* 
 
 Equation Rf 
 
 Formulas for Reactions of Three — ,Two — And 
 No-Hinged Arches 
 
 General formulas for reactions of three-, two-, 
 and no-hinged arches have been derived in order to 
 obtain a comparison between them. They have also 
 been applied to these types of arches with parabolic, 
 elliptical, and segmental circular ribs. Table 3 
 shows a comparison of the general formulas of the 
 four types of arches, while those for a comparison 
 of the reactions of parabolic, segmental circular and 
 elliptical forms of arches are not here reproduced. 
 
 The Envelop of Reaction Lines 
 
 The position of live load for the maximum posi- 
 tive and negative moments and the maximum posi- 
 tive and negative shears of the one-hinged arch can 
 be found graphically by means of the reaction locus 
 and the envelop of the reaction lines. Since the 
 ends of the one-hinged arch are fixed, we cannot 
 tell where the reaction lines pass through the sup- 
 port for a load at a certain point of the span. By 
 means of the reaction envelop and the reaction locus, 
 this can be easily done by drawing the tangent line 
 to the reaction envelop from the point where the 
 load line intersects the reaction locus. The equa- 
 tion of the reaction envelop of the one-hinged arch 
 was found by the author to be an equation of the 
 third degree. To plot the curve from the equation 
 involves a tremendous amount of labor. It is easier 
 to draw the reaction envelop from the computed 
 values of ordinates than to plot it from the equation. 
 The envelop passes through the hinge and intersects 
 the Y-axis at a certain distance below the origin. 
 To determine the points of division of the live load 
 for the maximum positive and negative moments, 
 tangent lines are draAvn to the envelop from the 
 point on the arch-rib we wish to investigate. The 
 intersection of the tangent lines with the reaction 
 locus gives the required points of division. To de- 
 termine the points of division for the shear, a 
 tangent line to the envelop is drawn perpendicular 
 to the normal line of the section. The intersection 
 of this tangent line with the reaction locus forms one 
 division point, while the section itself is another 
 division point. 
 
 Study On Designs 
 
 The design of a one-hinged arch can only be made 
 with a series of approximations. Stresses due to 
 temperature and rib-shortening play an important 
 
 q=h^/+8(37r-3) 
 
 (/—2ft) ft 3 "1 
 
 4(4 k e -3- ft -P3)-)k-ff-h3(/-2K)(37r -3a ) J 
 
 in which 
 
 Ct^cos 
 
 t 
 
 1-h 
 
 4 Ik *(/-2./t) 
 
 h[sr 3 &i7z 3 c*'j-3r s l(/-£fi)(stvi £o<'-/- 2 ci- 2 c(')- 8 g 3 ^3l^(j -3 ft +4ft 2 ^ 
 
 in which Cf = r~h tx'— siii~ I (c±k.-/') 
 
 (Q) 
 
 (R) 
 
 8 
 
part. These can not be exactly determined without 
 knowing the area of the cross-section of the arch- 
 rib. Yet the latter again depends upon the total 
 stresses to be carried. The method of trial is the 
 only way to secure the right section for the one- 
 hinged arch-rib. Three comparative designs were 
 made by the author in order to study their compara- 
 tive merits: (1) The moment of inertia at any 
 section of the arch-rib is assumed to vary as the 
 secant of the angle of inclination of the section, 
 with the effect of rib-shortening neglected; (2) 
 Same assumption as in the first case, with the effect 
 of rib-shortening included; and (3) the true moment 
 of inertia of the section is used. 
 
 A one-hinged arch of twenty panels, with a span 
 of 258 feet and a rise of 26' was adopted for investi- 
 gation. The dead load is assumed 59 kips and the 
 live load 18.5 kips per panel load. The depth of the 
 rib at the center is assumed 5 feet. An allowance 
 of 75° F is made for the rise or fall of temperature. 
 The Modulus of elasticity is taken as 26,000 kips and 
 the unit-stress for the gross section of the flanges, 
 15 kips. The arch chosen for investigation has been 
 greatly studied by former graduate students of Pro- 
 fessor H. S. Jacoby when containing three, two or 
 no hinges. The reason for using this arch by the 
 author is, of course, to compare the design of the 
 one-hinged arch with that of the other types. 
 
 Study On Deflections 
 
 The deflections of the one-hinged arch w T ere stud- 
 ied in three ways: (1) Vertical and horizontal de- 
 flections under the vertical loading ; (2) vertical 
 and horizontal deflection under the horizontal load- 
 ing; and (3) maximum and minimum deflections. 
 The same arch used in the design was used for the 
 study of deflections. The deflections of this arch 
 having three, two and no hinges respectively were 
 investigated by Dr. P. H. Chen, a graduate of Cor- 
 nell University. It gives a good opportunity for 
 critical comparisons. (See The Cornell Civil Engi- 
 neer, Vol. 26, Page 229). 
 
 RESULTS OF INVESTIGATION 
 Discussion On Formulas 
 
 So far as the formulas for the reactions of the one- 
 hinged arch are. concerned, those for the parabolic 
 ribs are the simplest in form, while those for the 
 segmental circular form require a vast amount of 
 labor for their derivation and application. Formu- 
 las for the elliptical form are intermediate in these 
 respects. 
 
 Neglecting the effect of the axial thrust and using 
 a ratio of the rise to the span for a segmental cir- 
 cular arch equal to 1-8, the relative effect of tem- 
 perature upon the three forms of the one-hinged 
 arch is as follows : 
 
 Seg. 
 
 Parabolic Elliptical Circular 
 Coefficient of Eet°Io /; 2 7.5 8.4 6.0 
 
 Ratio 125 140 100 
 
 The temperature effect in the parabolic form is 
 125% as great, and that in the elliptical form 140% 
 as great as that for the segmental circular form. 
 The effect upon the elliptical form is 112% as great 
 as that upon the parabolic form. Thus, the tem- 
 perature effect upon the elliptical form is the larg- 
 est, while that upon the circular form is the small- 
 est. It is to be noted that in the segmental form 
 the ratio of the rise to the span is assumed as one- 
 eightli. This is only a particular case. If we as- 
 sumed another ratio, the results would be different. 
 
 The relative effects of rib-shortening upon the 
 horizontal reaction and the end moments of the 
 three forms of one-hinged arches have the same re- 
 lations as the temperature effects. 
 
 There is an interesting fact about the number of 
 equations required to analyze no-, one-, two-, 
 and three-hinged arches. Three-hinged arches re- 
 quire three conditions; two-hinged arches, four; one- 
 hinged arches, five ; and no-hinged arches, six. Thus, 
 we see that the more hinges an arch has, the less 
 are the number of the conditions required to solve 
 its unknowns. In addition to the three statical con- 
 ditions, the three-hinged arch requires no-elastic 
 condition for its solution ; the two-hinged arch re- 
 quires one ; the one-hinged arch, two ; while the ne- 
 hinged arch requires three. Therefore the three- 
 hinged arch is said to be statically determinate. 
 
 The formulas for reactions of three-hinged arches 
 are exact, while those for no-, one-, and two- 
 hinged arches are subject to many imperfections 
 and assumptions. First of all, the elastic limit is 
 not assumed to be passed. If very large loads 
 should ever be applied which cause the stresses to 
 exceed the elastic limit of the material in any mem- 
 ber, the theory of elasticity fails and it is impossible 
 to predict the degree of safety of the structure. The 
 moment of inertia of any section of the rib is as- 
 sumed to vary as the secant of the angle of the 
 inclination of the arch-rib. The modulus of elastic- 
 ity is assumed to be constant throughout the span. 
 Shear is neglected in the derivation of the formulas. 
 
 No- and one-hinged arches are similar in one re- 
 spect; that is, they are both subject to vertical and 
 horizontal reactions and have end moments under 
 either vertical or horizontal loads. Two- and three- 
 hinged arches are all subject to vertical and horizon- 
 tal reactions, the vertical reactions being the same 
 for those two types of arches under either vertical or 
 horizontal loads. Temperature and rib-shortening 
 cause horizontal reactions and the end moments in 
 both the one and no-hinged arches, while in two- 
 hinged arches they cause horizontal reactions but 
 no end moments. It is generally supposed that a 
 three-hinged arch is not subject to stresses due to a 
 change in temperature. Strictly, however, such 
 stresses will occur, for a fall in temperature causes 
 a deflection of the crown hinge, and as the span 
 does not change, the horizontal thrust will be in- 
 
 9 
 
creased. The stresses produced, however, are very 
 small. 
 
 Neglecting the effect of axial thrust upon the rise 
 and fall of temperature, the relative importance of 
 temperature effects upon the horizontal reactions 
 and end moments of no, one-, tw r o-, and three- 
 hinged arches are as follows : 
 
 Arch 
 
 No- 
 
 One- 
 
 Two- 
 
 Three- 
 
 
 Hinged 
 
 Hinged 
 
 Hinged 
 
 Hinged 
 
 ELLIPTICAL 
 
 
 160/8 
 
 67.2/8 
 
 12/8 
 
 0 
 
 M x 
 
 125.6/8 
 
 67.2/8 
 
 0 
 
 0 
 
 PARABOLIC 
 
 H x 
 
 90/8 
 
 60/8 
 
 15/8 
 
 
 M x 
 
 60/8 
 
 60/8 
 
 0 
 
 0 
 
 Thus, the temperature effect upon the no-hinged 
 arch is found to be the greatest of all. Referring 
 to H in the parabolic form, it is one and one-half 
 times as large as for the one-hinged arch, and six 
 times as large as for the two-hinged arch. In the 
 elliptical type it is 2.38 times as large as that for the 
 one-hinged arch, and 13.33 times as large as for the 
 two-hinged arch. The temperature effect upon H of 
 the one-hinged arch of the parabolic form is four 
 times as large as for the two-hinged arch, while for 
 the elliptical form it is 5.6 times as large. The tem- 
 perature effect upon M of the no- and one-hinged 
 arches is the same in the parabolic form. In the 
 
 elliptical form, the former is 18.7 times as great as 
 the latter. In the segmental circular form, the tem- 
 perature effect upon H in the one-hinged arch is 
 three times as great as that for the two-hinged 
 arch. 
 
 With respect to the effect of rib-shortening the re- 
 lative ratios above stated apply to different types 
 of arches equally well. 
 
 So far as the formulas for the reactions are con- 
 cerned, there is a decided advantage gained by those 
 for the parabolic form over all other types of arches 
 because of their simplicity. 
 
 Discussion on Reaction Influence Lines 
 
 In order to study the variation of the reactions 
 and the end moments as a single load moves over 
 the span, reaction influence lines are drawn and 
 compared. (See diagrams 2, 3, 4, 8, 9, and 10). 
 Only a few of the numerous diagrams are repro- 
 duced in the abstract of the thesis. 
 
 Under vertical loading, the elliptical form of the 
 one-hinged arch has greater horizontal reactions and 
 end moments than the parabolic and segemental 
 circular forms. The last two forms cause less dif- 
 ference in the H t and M x . The vertical reactions 
 are all the same in the one-hinged arch for the three 
 forms of ribs. There is not much difference shown 
 under the horizontal loading. The circular form 
 seems to have the greatest advantage for the one- 
 
 10 
 
hinged arch judging from a comparison of the 
 curves. However, the curves for the circular form 
 are only drawn for a single case with a ratio of rise 
 to span equal to 1-8. If other ratios were used, the 
 results may be different. On the other hand, the 
 parabolic form has the advantage of low reactions 
 besides that of simplicity in formulas. Therefore, 
 the author comes to the conclusion that the para- 
 bolic curve is the best form for the neutral axis of 
 the arch-rib to be used for a one-hinged arch. 
 
 Under vertical loading the reaction influence line 
 of H for a three-hinged arch consists of straight 
 lines ; for a two-hinged arch, a curve resembling a 
 parabola ; for a no- and a one-hinged arch, curves 
 symmetrical at the center. Curves for one- and 
 three-hinged arches break at the crown while those 
 for no — and two-hinged arches do not. This is be- 
 cause the hinge at the crown of the one- and three- 
 hinged arches breaks the continuity of the rib. 
 Curves of no-, two-, and three-hinged arches are 
 very close to each other, while that for the one- 
 hinged arch has a great variation, being lower at 
 the supports and higher near the crown. This brings 
 out one disadvantageous fact for the one-hinged 
 arch. When the load is at the crown, the value of 
 H is about two times as much as that for the other 
 types. This means that the supports have to resist 
 a greater horizontal thrust, and extra care has to be 
 taken in making the end supports secure. The end 
 
 moment of the one-hinged arch has a greater varia- 
 tion than that of the no-hinged arch. 
 
 of fk&cfar? Zoc/ 
 S//s/?Acr&/ & fnx/Arr 
 
 The one-hinged arch has an advantage over the 
 no-hinged arch with regard to the effect of tempera- 
 ture and thrust, but it is not so favourable with re- 
 gard to the reactions and the end moments under 
 vertical loads. For reactions due to temperature, 
 thrust and vertical loading, the one-hinged arch is 
 not as advantageous as the two — and three-hinged 
 arches. So far as the simplicity of the formulas for 
 
 fc‘srr / a?/-*sas? of~ Ahfefao Zoc/ 
 
 Os?&, /mo < 5 * 
 
 Ztsae /? styes' asx/er & Mr*/ /&&/ 
 
 Ctsrya SA&e/ 2 / 
 
 
 the reactions is concerned, the same statement holds 
 true. However, attention will be called subsequent- 
 ly to some decided advantages of the one-hinged 
 arch. 
 
 Discussion On Reaction Loci 
 
 In order to compare the reaction loci of different 
 types of arches, the equations of the reaction loci 
 for no-, one-, two-, and three-hinged arches are 
 plotted for different forms of arch ribs. They are 
 shown in diagrams 20 and 21. 
 
 All reaction loci for the one-hinged arch with dif- 
 ferent forms of arch-ribs pass through the crown 
 hinge. When the load is at the crown, the reaction 
 lines must pass through the crown hinge, for other- 
 wise there would be rotation at the crown. All re- 
 action loci break their continuity at the crown but 
 the two halves are symmetrical. The parabolic one- 
 hinged arch has the highest ordinate when the load 
 
 11 
 
is near the support, while the segmental circular 
 form has the least ordinate. 
 
 For a parabolic arch-rib, the reaction locus of the 
 three-hinged arch is a straight line for each half of 
 the span; that for the two-hinged arch, a curve like 
 a parabola; that for a no-hinged arch, a horizontal 
 straight line at an ordinate of 1,2 h; and that for 
 the one-hinged arch, a curve breaking at the center. 
 Curves for no- and two-hinged arches are contin- 
 uous, while those for the other two break at the 
 center. Curves for no- and two-hinged arches do not 
 pass through the crown, while those for the one- and 
 three-hinged arches must do so. At the support, the 
 three-hinged arch has a maximum- ordinate of 2.00h; 
 the two-hinged arch, 1.60 h; the one-hinged arch, 
 1.40 h; and the no-hinged arch,. 1.20 h. At the 
 crown the curve for the two-hinged arch has the 
 highest ordinate. 
 
 Similar statements can be drawn for the reaction 
 loci for the elliptical forms of arches. The ordi- 
 nates are highest in certain cases and lower in 
 others, while the general forms are about the same. 
 
 Discussion On Reaction Envelop 
 
 By means of the reaction locus and the reaction 
 lines, the position of live loading for the maximum 
 positive and negative moments and maximum posi- 
 tive and negative shears for the one-, two-, three, 
 and no-hinged arches can be found graphically. In 
 the three- and two-hinged arches, the reaction lines 
 can easily be drawn as soon as the position of the 
 live load is given; because the hinges at each sup- 
 port fix the direction of the reaction lines. In the 
 one- and no-hinged arches, the reaction lines cannot 
 be so easily drawn, if the position of the load is only 
 given ; for the fixing of the supports makes the di- 
 rection of the reaction lines uncertain. In order to 
 avoid the difficulty, reaction envelops are required 
 for both the one- and no-hinged arches. 
 
 The reaction envelop of the no-hinged arch was 
 found by A. V. Saph, a graduate of Cornell Uni- 
 versity, to be two hyperbolas symmetrical and tang- 
 ent to each other at the center at a point 2-3 h above 
 the level of the supports. The reaction envelop of 
 the one-hinged arch was found by the author to 
 consist of two curves very similar to those for the 
 no-hinged arch in form yet quite different in their 
 equations. The curves are symmetrical and meet 
 each other at the crown hinge. The reaction en- 
 velop is a plane curve of the third degree ; because 
 the formulas of the reactions of the one-hinged arch 
 involves the k’s in the fourth power. In the case 
 of the no-hinged arch, the k's in the formulas of the 
 reactions are of third power; hence a second-degree 
 curve is obtained for the envelop. 
 
 The method of finding the positions of live loading 
 is the same for the four types of arches, except 
 that the reaction envelops of the no- and one-hinged 
 arches have the function of the end hinges in the 
 
 case of the three- and two-hinged arches. A com- 
 parison of the reaction envelops of the no- and one- 
 hinged arches is given in diagram 28. 
 
 Discussion On Design With the Moment of Inertia 
 Varying As the Angle of Inclination of the Arch-Rib 
 
 There is no marked difference in the procedure of 
 designing the arch-rib for the two-, one-, and no- 
 hinged steel arches with the assumption that I 
 varies as sec b . The procedure may be generalized 
 in the following heads: (1) calculation of reactions 
 and end moments from the formulas for a unit load 
 at each panel point; (2) to find the position of live 
 loading for maximum positive and negative mo- 
 ments, by either algebraic or graphical methods; 
 (3) calculation of dead load and live load moments 
 for each section; (4) calculation of dead load and 
 live load thrust; (5) calculation of moments and 
 thrust due to temperature effect by assuming a cer- 
 tain moment of inertia at the crown; (6) calculation 
 of moments and thrust due to effect of rib-shorten- 
 ing by means of the assumed moment of inertia ; 
 (7) design of the flange area of the crown section 
 using the maximum moments and thrusts so obtain- 
 ed; (8) test the moment of inertia of the crown 
 section, and see if the assumed value of the moment 
 of inertia at the crown section agrees with the value 
 obtained; (9) after the right value of I 0 is obtained 
 the flange areas at other sections can be calculated 
 by using the relation that / = I t sec Q ; (10) the 
 position of live load for maximum positive and nega- 
 tive shear is obtained by either graphical or alge- 
 braic method, the maximum shear at each section is 
 secured by combining the dead load, live load, tem- 
 perature and rib-shortening shears, and the webs are 
 designed accordingly. The design of the three- 
 hinged arch may be carried out in the similar or- 
 der with the exception that no trial is required in 
 securing the sections. 
 
 There is an interesting fact found in combining 
 the moments and thrusts caused by the dead load, 
 live load, temperature and rib-shortening to produce 
 the maximum stress on the sections of the one- 
 hinged arch. The live load may be considered in 
 three ways; (1) loading for maximum positive mo- 
 ments; (2) loading for the maximum negative mo- 
 ments; (3) loading for maximum thrust. In the 
 design with the effect of rib-shortening neglected, 
 case (1) controls for the section 0-4 and case (3) 
 controls for the section 5-10. In the design with 
 the effect of the rib-shortening included, case (2) 
 controls for the section 0-4, while case (3) still con- 
 trols for the sections 5-10. The reason is quite 
 clear; for in the sections near the center, the mo- 
 ments are comparatively small, while the thrusts 
 are large. This means that a full live load is re- 
 quired to design the sections near the center. As 
 there exist large bending moments in the sections 
 near the support, naturally the moments control 
 
 12 
 
the design for those sections. The effect of rib- 
 shortening is to produce the negative moments and 
 thrusts. This is why the negative moments control 
 the end sections when the effect of the rib-shorten- 
 ing is included. The design of the one-hinged arch 
 is different from that of the two-, and no-hinged 
 arch in two respects. The design of the center sec- 
 tions of the one-hinged arch is controlled by the 
 thrust while that of the two- and no-hinged arches 
 is controlled by the moments. The reason is be- 
 cause the no-hinged and two-hinged arches have 
 large moments at the middle sections, while the 
 one-hinged arch has large thrusts. The signs of 
 the moments and thrusts produced by the effect of 
 temperature and rib-shortening are the same for all 
 sections of the one-hinged arch, while those for the 
 no-hinged arch are the same for the sections below 
 2-3 h of the rib, and opposite each other for the 
 sections above 2-3 h of the rib. The signs of the 
 moments and thrusts produced by the effect of tem- 
 perature and rib-shortening are all opposite for the 
 two-hinged arches. 
 
 In designing the sections at the crown, only thrust 
 is used for the one-hinged arch. This makes the 
 design of the one-liinged arch easier, because the 
 right value of the moment of inertia can be secured 
 immediately. 
 
 Positive shear governs the design of the web for 
 all types of arches, whether the effect of rib-short- 
 ening is included or not. The latter which tends to 
 produce negative thrust, tends to increase the posi- 
 tive shear in all sections for the one-hinged arch. 
 
 Under maximum loading the comparative effects 
 of dead load, live load, temperature and rib-short- 
 ening on the flanges are shown in the following 
 table and also on curve sheet 22 : 
 
 
 Dead 
 
 Live 
 
 Tempera- 
 
 Rib- 
 
 Section 
 
 load 
 
 load 
 
 ture 
 
 shortening 
 
 
 Per Cent. Per Cent. 
 
 Per Cent. 
 
 Per Cent. 
 
 0 
 
 31.5 
 
 29.7 
 
 24.2 
 
 14.6 
 
 1 
 
 38.8 
 
 24.2 
 
 23.0 
 
 14.0 
 
 2 
 
 47.4 
 
 17.7 
 
 21.7 
 
 13.2 
 
 3 
 
 58.2 
 
 10.6 
 
 19.3 
 
 11.9 
 
 4 
 
 72.7 
 
 1.3 
 
 16.1 
 
 9.9 
 
 5 
 
 75.8 
 
 23.7 
 
 20.6 
 
 —20.1 ' 
 
 6 
 
 75.9 
 
 23.8 
 
 15.5 
 
 —15.1 
 
 7 
 
 76.0 
 
 23.8 
 
 11.4 
 
 —11.1 
 
 8 
 
 76.0 
 
 23.9 
 
 8.5 
 
 — 8.4 
 
 9 
 
 76.0 
 
 23.8 
 
 6.6 
 
 — 6.5 
 
 10 
 
 75.8 
 
 23.8 
 
 6.3 
 
 — 5.9 
 
 It is 
 
 seen from the 
 
 curve that the dead load has 
 
 the greatest effect of 
 
 all, and 
 
 its effect 
 
 on the sec- 
 
 tions near the center is greater than those near the 
 ends. In the first place, this is because the dead 
 panel load is comparatively great, and naturally it 
 takes greater stress. In the second place, the effect 
 of temperature and rib-shortening near the center 
 sections is greater than their effect on the sections 
 
 near the ends. This makes the percentage of the 
 stress carried by the dead load gradually decrease 
 in the sections near the ends. The live load is the 
 next important factor in causing flange stresses. 
 The effect of temperature and rib-shortening can 
 never be omitted. They have about the same effect. 
 Temperature has the greater effect in the sections 
 near the ends than in those near the center. The 
 same is true for the rib-shortening. The negative 
 signs of the rib-shortening in the sections near the. 
 ends are explained by the fact that both negative 
 thrusts and moments are used. 
 
 Under maximum loading the comparative effects 
 of moments and thrusts on the flanges is shown in 
 the following table and also shown on curve sheet 
 
 > : — 
 
 Section 
 
 Moment 
 
 Thrust 
 
 
 Per Cent. 
 
 Per Cent. 
 
 0 
 
 73.1 
 
 26.9 
 
 1 
 
 67.4 
 
 32.6 
 
 2 
 
 60.3 
 
 39.7 
 
 3 
 
 51.8 
 
 <M # 
 
 GO 
 
 4 
 
 43.1 
 
 56.9 
 
 5 
 
 2.5 
 
 97.5 
 
 6 
 
 1.5 
 
 98.5 
 
 7 
 
 .7 
 
 99.3 
 
 8 
 
 .4 
 
 99.6 
 
 9 
 
 .1 
 
 99.9 
 
 10 
 
 .0 
 
 100.0 
 
 One apparent conclusion 
 
 can be drawn 
 
 from the 
 
 curve ; that is, the moment has far greater effect on 
 the sections near the ends than the thrust, while 
 the thrust has far larger effect on the sections near 
 the middle than the moment. The effect of moment 
 on the sections 5-10 is practically nothing. As a 
 whole, the thrust is more important than the mo- 
 ment. This peculiar fact is a special feature of the 
 one-hinged arch. 
 
 The comparative areas of the designs with the as- 
 sumption / 5 sect* with rib-shortening included and 
 neglected are shown in the following table and on 
 curve sheet 24 : — 
 
 
 Rib-Shortening 
 
 Rib-Shortening 
 
 Section. 
 
 Neglected. 
 
 Inc. 
 
 
 Area in sq. in. 
 
 Area in sq. in. 
 
 0 
 
 166.69 
 
 153.70 
 
 1 
 
 142.77 
 
 127.00 
 
 2 
 
 121.39 
 
 103.60 
 
 3 
 
 105.00 
 
 84.00 
 
 4 
 
 88.84 
 
 67.75 
 
 5 
 
 82.15 
 
 67.70 
 
 6 
 
 77.39 
 
 66.80 
 
 7 
 
 73.72 
 
 66.00 
 
 8 
 
 71.10 
 
 65.45 
 
 9 
 
 69.38 
 
 64.95 
 
 10 
 
 68.71 
 
 64.71 
 
 It is seen 
 
 from the curve 
 
 that the one-hinged 
 
 arch requires far larger areas in the sections near 
 the ends than the sections near the crown. This is 
 
 13 
 
due to the presence of the large end moments. It 
 is also seen that the effect of rib-shortening is to 
 decrease the areas of sections. We thus see that for 
 this particular rise of span, the effect of rib-shorten- 
 ing can be safely but not economically neglected in 
 the design. The author believes that this is equally 
 true for other ratios of rise to span, because in any 
 
 So far as the areas of sections in the flanges 
 are concerned, the three-hinged arch is the most 
 favorable of all, while the no-hinged and one-hinged 
 arch require the largest areas. The areas of the 
 two-hinged arch are intermediate between them. 
 The three-hinged arch requires the largest area at 
 the quarter points; for the moments at those sec- 
 
 V 60 
 .V 40 
 
 V 20- 
 
 r 
 
 0-2Or 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 -t- 
 | 
 
 i 
 
 
 
 (r) 
 
 i 
 
 i — 
 
 r- 
 
 /o(Sedions) 
 
 ~ /0 (Sections) 
 
 Comparative Effects of D.L. L.L. 
 77 <3 R.S. on the Flanges 
 
 ISO 
 
 %40- 
 
 
 ij; o- 
 
 
 
 r- — 
 1 
 
 1 
 
 
 T~ 
 
 
 ■ 
 
 1 
 
 ! 
 
 
 © 
 
 
 L_ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 Comparative Web Frees of 0-R, 
 /-H, 2-H, <&■ 3-ti Arches 
 
 * 
 
 | 
 
 -,o (Sections; 
 
 Comparative Effects of Moment 
 and Thrust On the Flanges 
 
 — 
 
 © 
 
 
 
 
 
 y 
 
 
 
 
 1 
 
 
 
 
 
 — t n 
 
 
 
 "~C- 
 
 —l.. 
 
 *-»- 
 
 m*ml 
 
 2_ 
 
 0 
 
 
 
 2 
 
 
 ___r — 
 
 
 4 
 
 0 
 
 8 K> 
 
 to (Sections) 
 
 X 
 '/-to 
 
 .%/oo 
 
 $ 
 
 P(Sections) 
 
 Comparative Design of F/ange 
 Ft re as 
 
 Comparative Flange Freas of 
 O-tt, /-H, 2-H, & 3-H Fashes 
 
 Curves Nos. 22-26 
 
 Dl -Dead Load, LL = Live Load, T= Temperature 
 PS = Rib -Shortening RN= Rib-shortening 
 neglected , R I = Rib-shortening included. 
 
 fvO/npa. 
 
 Envelops 
 
 •relive /Reaction 
 of No- end One-h/’nged 
 S/ee/ ftrches 
 
 O-M /St SC) ’* dtrzf-tdhtx *ht* - O 
 /-A. soco-ufA * r- ^-A $7 * robra-o 
 ScbB^rOCbi *S-BJ J r Sc — O 
 
 A— - # - fen it* pj * a-£ S-f 
 S- aFc r / -dbcrdb’^F c-saAb -7 
 
 @ Reaction locus of o-A 
 ® Reeclon locus of /-A 
 © Reeclon Envelop oF o-A 
 
 @ Reaction Envelop oF t-A 
 20/agram 20 
 
 case the moments and thrust produced by the rib- 
 shortening always have the same negative sign in 
 the case of a one-hinged arch. This naturally tends 
 to decrease the area of section required, if the area 
 of section is controlled by the positive moments and 
 thrust, in the design, as is usually the case in the 
 one-hinged arch. 
 
 The comparative flange areas required for the 
 three-, two-, one-, and no-hinged arches for the same 
 form of the arch-rib under the same specifications 
 are shown in the following table and on curve sheet 
 25. The areas given for the three-hinged arch are 
 exact, while those for the two-, one-, and no-hinged 
 arches are obtained from a design with the as- 
 
 sumption 
 
 / 5 seed 
 
 with the 
 
 effect of r 
 
 ib-shortening 
 
 included. 
 
 Three- 
 
 Two- 
 
 One- 
 
 No- 
 
 Section. 
 
 IIinged. 
 
 Hinged. 
 
 Hinged. 
 
 Hinged. 
 
 
 sq. in. 
 
 sq. in. 
 
 sq. in. 
 
 sq. in. 
 
 0 
 
 68.0 
 
 68.4 
 
 153.7 
 
 119.5 
 
 1 
 
 73.0 
 
 72.8 
 
 127.0 
 
 93.1 
 
 2 
 
 78.2 
 
 78.1 
 
 103.6 
 
 75.9 
 
 3 
 
 82.4 
 
 85.0 
 
 84.0 
 
 72.3 
 
 4 
 
 84.4 
 
 90.7 
 
 67.7 
 
 65.0 
 
 5 
 
 84.1 
 
 93.1 
 
 67.7 
 
 67.9 
 
 6 
 
 81.2 
 
 94.9 
 
 66.8 
 
 72.8 
 
 7 
 
 77.7 
 
 94.8 
 
 66.0 
 
 76.9 
 
 8 
 
 71.0 
 
 92.3 
 
 65.4 
 
 78.4 
 
 9 
 
 65.0 
 
 89.3 
 
 64.9 
 
 78.3 
 
 10 
 
 64.0 
 
 89.0 
 
 64.7 
 
 77.3 
 
 tions are the largest of all. The two-hinged arch 
 requires the largest area in the sections near the 
 middle for the same reason. The one-hinged and 
 no-hinged arches require the largest areas in the 
 sections near the ends, because the end moment is 
 the dominating factor in those sections. In com- 
 paring the areas in flanges of the one- and no-hinged 
 arches, we see that the one-hinged arch requires 
 comparatively large areas in ihe sections near the 
 ends and small areas in the sections near the crown. 
 The areas near the crown of the one-hinged arch 
 are rather uniform, because the thrusts at these 
 points are large and about equal in magnitude. 
 
 The comparative theoretical web areas required 
 for the three-, two-, one- and no-hinged arches with 
 the same form of the rib and designed under the 
 same specifications are shown in the following table 
 
 and also 
 
 in curve 
 
 sheet 26 : — 
 
 
 
 
 Three- 
 
 Two- 
 
 One- 
 
 No- 
 
 Section. 
 
 Hinged. 
 
 Hinged. 
 
 Hinged. 
 
 Hinged 
 
 1 
 
 sq. in. 
 
 7.91 
 
 sq. in. 
 
 10.04 
 
 sq. in. 
 
 19.59 
 
 sq. in. 
 
 22.83 
 
 2 
 
 6.95 
 
 8.98 
 
 16.76 
 
 19.88 
 
 3 
 
 6.31 
 
 8.20 
 
 14.95 
 
 18.22 
 
 4 
 
 5.77 
 
 7.67 
 
 13.03 
 
 15.51 
 
 5 
 
 5.45 
 
 7.45 
 
 11.10 
 
 13.58 
 
 6 
 
 6.10 
 
 7.61 
 
 9.55 
 
 12.08 
 
 7 
 
 6.95 
 
 7.92 
 
 7.95 
 
 10.87 
 
 8 
 
 7.61 
 
 8.11 
 
 6.89 
 
 9.70 
 
 9 
 
 7.97 
 
 8.12 
 
 6.45 
 
 9.18 
 
 10 
 
 8.13 
 
 7.96 
 
 5.75 
 
 10.46 
 
 14 
 
It is seen that the no-hinged arch requires the 
 largest web area, while the three-hinged arch, the 
 least of all. The one-hinged arch requires smaller 
 web areas than the no-hinged arch, while the two- 
 liinged arch requires greater areas than the three- 
 hinged arch. The no-hinged and one-hinged arches 
 require larger web areas in the sections near the 
 ends and smaller areas in the sections near the 
 crown. This is due to the fact that the shear in 
 the no-hinged and one-hinged arches is far greater 
 near the ends than in the sections near the crown. 
 On the other hand, the web areas of the three-, and 
 two-hinged arches are rather uniform, since the 
 shears in the sections of the three- and two-hinged 
 arches are rather uniform. The marked difference 
 is caused by the fixing of the supports in the one 
 case, and the hinging of the supports in the other. 
 
 The following table shows the results of the de- 
 sign based upon the assumption / 5 sec 0 
 
 Section. 
 
 Composition 
 
 Moment of 
 Inertia. 
 
 Ratio of I. 
 
 See 0 . 
 
 Proposed 
 Assumption, 
 of I. 
 
 0-2 
 
 6z.s 6" X 6” X 9/16" 
 
 3 Pis 14" X 9/16" 
 
 5 Pis 18" X 7/8" 
 
 1 Pis 18" X 3/4" 
 
 286,000 
 
 2.49 
 
 1.07 
 
 250% 
 
 2-4 
 
 6z.s 6" X 6" X 9/16" 
 
 3 Pis 14" X 9/16" 
 
 2 Pis 18" X 3/4" 
 
 1 Pis 18" X 13/16" 
 
 190,600 
 
 1.59 
 
 1.05 
 
 160% 
 
 4-6 
 
 6z.s 6" X 6" X 9/16" 
 
 3 Pis 14" X 9/16" 
 
 3 Pis 18" X 5/16" 
 
 123,800 
 
 1.03 
 
 1.03 
 
 103% 
 
 6-8 
 
 6z.s 6" X 6” X 9/16" 
 
 3 Pis 14" X 9/16" 
 
 1 Pis 18" X 1/4" 
 
 121,700 
 
 1.02 
 
 1.01 
 
 101% 
 
 8-10 
 
 6z_s 6" X6" X 9/16" 
 
 1 Pis 14" X 9/16" 
 
 1 Pis 18" X 3/16" 
 
 119,700 
 
 1.00 
 
 1.00 
 
 100% 
 
 It is seen that the assumption I ' sec Q is about 
 right for the sections near the crown, and is in great 
 error for the sections near the ends. 
 
 DISCUSSION ON TRUE DESIGN 
 So far as the reactions are concerned, the effect 
 of final design as compared with that of the pre- 
 liminary design, assuming I z sec. e, is to increase 
 the horizontal reactions when the load is near the 
 center of the span and to decrease the same when 
 the load is near the ends. There is no appreciable 
 change in the vertical reactions for the two de- 
 signs. The end moments have been affected so 
 greatly that under the full loading, a negative end 
 
 moment is produced in this design, which was not 
 the case in the preliminary design. The latter has 
 a serious effect in designing the sections, because 
 the sign of the dead load moment is changed to 
 negative, and the maximum moments at various 
 sections are thereby changed. 
 
 The preliminary design by assuming I z sec. o is 
 far from being correct. It makes the flange areas 
 at the various sections far different from the true 
 values. The error is on the unsafe side. It is 
 necessary that a revised design be made in de- 
 signing a one-hinged arch. 
 
 The preliminary design by assuming / ; sec. e is 
 nearest to the true value for the two-hinged arch, 
 while it differs most in the one-hinged arch. The 
 relative error of the preliminary design from the 
 true design of the no-, one-, and two-hinged arches 
 with same dimensions and designed to carry the 
 same load is shown below : 
 
 Section 
 
 No-hinged 
 
 One-hinged 
 
 Two-liinged 
 
 
 per cent 
 
 per cent 
 
 per cent 
 
 0 
 
 —15.5 
 
 —30.7 
 
 5.2 
 
 1 
 
 —13.4 
 
 —31.6 
 
 —3.7 
 
 2 
 
 — 7.6 
 
 —33.1 
 
 —4.7 
 
 3 
 
 — 2.4 
 
 —34.4 
 
 —2.5 
 
 4 
 
 — 3.6 
 
 —40.5 
 
 2.4 
 
 5 
 
 8.7 
 
 —33.3 
 
 3.3 
 
 6 
 
 5.6 
 
 —27.0 
 
 7.6 
 
 7 
 
 4.1 
 
 —21.0 
 
 12.8 
 
 8 
 
 3.7 
 
 —16.1 
 
 12.1 
 
 9 
 
 2.5 
 
 —12.8 
 
 11.6 
 
 10 
 
 2.1 
 
 —11.7 
 
 13.7 
 
 Thus, we see that the error in the one-hinged arch 
 is the largest of all. This is due to the presence of 
 the large moments in the sections near the ends and 
 the large thrust in the sections near the crown. The 
 variation of the moments and thrusts in the sections 
 is such that the moment of inertia in the various sec- 
 tions required is not in accordance with the relation 
 jz sece. 
 
 As the assumption / s sec . o is far from being true, 
 the I-Curve for a one-hinged arch with 1/10 rise is 
 recommended by the author, by means of which a 
 closer result can be secured in the approximate de- 
 sign. A similar curve was recommended for the no- 
 hinged arch by P. H. Chen in his thesis above named. 
 The comparative values are shown below : 
 
 Section 
 
 One-hinged 
 
 N 
 
 o-h 
 
 inged 
 
 0 
 
 Io 
 
 + 
 
 185% 
 
 Io 
 
 1 
 
 ~r 
 
 120% 
 
 1 
 
 Io 
 
 + 
 
 135% 
 
 Io 
 
 
 60% 
 
 2 
 
 Io 
 
 + 
 
 100% 
 
 Io 
 
 _L 
 
 30% 
 
 3 
 
 Io 
 
 + 
 
 70% 
 
 Io 
 
 1 
 
 ~r 
 
 15% 
 
 4 
 
 Io 
 
 + 
 
 45% 
 
 Io 
 
 
 
 6 
 
 Io 
 
 + 
 
 15% 
 
 Io 
 
 + 
 
 10% 
 
 8 
 
 Io 
 
 + 
 
 5% 
 
 Io 
 
 — 
 
 10% 
 
 10 In I 
 
 15 
 
{OQOOO 
 
 V 
 
 \ 
 
 \ 
 
 
 — i — 
 
 
 i 
 
 • - y — 
 
 X 
 
 Ch/ 
 
 vrres 
 
 :Ju 
 
 
 
 ■ 
 
 
 
 0/7e 
 
 
 */\ /9 
 
 re/? 
 
 . .. . ( . 
 
 7/4 
 
 % 
 
 5(5 
 
 
 \ 
 
 k' ' 
 
 \ 
 
 \ 
 
 \ 
 
 
 -- 
 
 j 
 
 c7 
 
 1 
 
 ~ I * • 
 
 
 
 
 
 
 
 
 
 Fr&d/c 
 
 
 
 - h 
 
 . : | ' 
 
 
 J 
 
 ©> 
 
 -<> \ 
 
 1 
 
 
 Z + 7o?. 
 
 s-Cvrre 
 
 1 ' 1 
 
 f . 
 
 ■i- 
 
 
 L 
 
 
 
 
 ! 
 
 
 : i 
 
 
 i..| 
 
 U_, 
 
 
 
 
 
 Zj. 
 
 sec_ 
 
 
 
 
 1 
 
 1 
 
 . 
 
 1 1 
 
 '-t|— 
 
 . j. 
 
 
 Cos7sfe. 
 
 
 
 
 
 -t- 
 
 
 
 
 
 
 
 
 ... _L - 
 
 
 
 
 
 *. _ J 
 
 
 
 
 
 
 
 ■! - 
 
 
 • i - ; 
 
 o 
 
 r . 
 
 5 . . V " ^ 
 
 \ 
 
 t : -i 
 
 ’ fri-lrlrtirhiS 
 
 
 - ; 1 
 
 z. i a 
 
 f 
 
 
 It is seen from the curves that the assumptions of 
 I s sec.® and 1= constant are nearly correct in the 
 sections near the center, while they are too small in 
 the sections near the ends. The exceedingly large 
 moments near the ends both in the no-hinged and 
 one-hinged arches necessitate the use of the large 
 sections and accordingly the values of the large 
 moments of inertia. 
 
 • 
 
 DISCUSSION OF METHODS OF DEFLECTION 
 COMPUTATIONS 
 
 Deflections in the one-hinged arch-ribs are con- 
 tributed by two factors, that due to thrust and that 
 due to moment. The methods used by the author in 
 finding these deflections are rather interesting and 
 are here given. 
 
 (a) DEFLECTION DUE TO MOMENTS— The 
 formulas for finding the horizontal and vertical de- 
 flections of a curved beam were given in equations 
 (d) and (e) respectively. As the continuity of the 
 one-hinged arch is broken at the crown by the 
 presence of the hinge, the formulas could not be ap- 
 plied directly to the entire span. A method of 
 separating the arch into two cantilevers is intro- 
 duced (Fig. 3). For example, when the load is at 
 a certain point on the left half of the span, the re- 
 actions acting on the left cantilever are H x and V 2 , 
 while those on the right cantilever are H x and V 2 . 
 lever are contributed by three factors, due to P, H x 
 and V 2 , while those in the right cantilever are con- 
 tributed by two factors, H x and V 2 . These factors 
 can be easily found by considering each half in 
 turn as a cantilever with one load on the arch. 
 Thus, under the vertical load P on the left cantilever, 
 we only need to find the deflections in the cantilever 
 due to a horizontal load at the free end, a vertical 
 load at the free end, and a vertical applied load. 
 The deflections due to H x and V 2 in the left half 
 of the arch are just the same in magnitude as those 
 due to H x and V 2 in the right half of the arch. The 
 final value of the vertical or horizontal deflections 
 
 can be easily obtained by taking the algebraic sum 
 of the vertical or horizontal deflections contributed 
 by the various factors. 
 
 If the deflection diagrams are required for a unit 
 load at different positions on the left half of the 
 span, the method is exceedingly simple. We need 
 only to draw the deflection diagrams of the canti- 
 lever with a unit horizontal load at the free end, 
 and a unit vertical load at different positions on the 
 cantilever corresponding to the positions on the 
 arch. These are obtained as unit deflections. The 
 values of H x and V 2 are then found for different 
 positions of the unit applied load. The deflections 
 due to II \ and V 2 are then found by multiplying the 
 unit deflections by the values of H x and V 2 . Using 
 proper signs of deflections contributed by each 
 factor and taking the algebraic sum, the deflections 
 at various points on the arch can be found for dif- 
 ferent positions of the loading. 
 
 The method has several advantages : first, it is 
 simple as well as easy; second, it offers the oppor- 
 tunity of studying the deflections contributed by 
 each factor; third, it is easy to discover mistakes. 
 By means of the graphical method, the deflection on 
 the cantilever due to a unit vertical or horizontal 
 load at any position on the beam can be easily cal- 
 culated. By arranging the work in a systematic 
 way, errors can be easily discovered by comparison. 
 
 The method is applicable in any of the four cases : 
 (1) vertical deflection under the vertical load, (2) 
 horizontal deflection under the vertical load, (3) 
 vertical deflection under the horizontal load, (4) 
 horizontal deflection under the horizontal load. The 
 only difference lies in the magnitude and direction 
 of II x and V 2 and the unit horizontal or vertical de- 
 flections to be found in the cantilevers. 
 
 (b) DEFLECTION DUE TO THRUST— The 
 general formula for the deflection due to thrust is 
 given by 
 
 Deflection = f*Tt 
 
 in which T is the thrust due to the applied load P; 
 while small t is the thrust due to a load unity P" 
 applied at the point whose deflection is sought, the 
 direction of P" being the same as the direction of 
 of the deflection required. Let H' and V' be the hori- 
 zontal and vertical shear (not in the normal section) 
 immediately on the left of the section considered; and 
 H" and V", those due to P" respectively. Then, by 
 substitution, the general formula of deflection due to 
 thrust becomes, 
 
 Deflect /on =J' b H’H"cos 2 e^ +J^‘VV r "sjn *6^ 
 
 which is applicable to the four cases above named, that 
 is, the vertical and horizontal deflections due to the 
 horizontal load. 
 
 16 
 
p 
 
 S 3 
 
 In order to simplify the numerical work( the above 
 formula may be transformed into a simpler form for 
 each of the different cases. For example, let us take 
 the ease of the horizontal deflection under the vertical 
 loading'. Let a unit load P' be applied at t and the 
 horizontal deflection at 5 be found. (Figs. 59 and 
 60). The corresponding points on the other half of 
 the arch are called s' and t' In examining the loading 
 and reactions closely, we find that H' and H" in s-c 
 and s'-c are correspondingly equal. H' in a-s is also 
 equal to H' in b-s' under the load P ' ; while under the 
 load P" , IP in a-s is equal to H 1 and H’ in b-s' is equal 
 to H.,. Also V" in t-c and t'-c are correspondingly 
 equal. V" in a-t is equal to V" in t'-b, while V" in 
 a-t=V 1 and V” in b-t is equal to V 2 . Using the proper 
 signs, the formula is given by 
 
 H'Hcos e Oj{jz —J' (Hr )H coo P Q Ylt 
 +f(v i ’V;') v s<n*Q$ : -2fYVs,n*e& 
 
 It is seen that the terms containing Us can be 
 neglected, because the angle of is not greater 
 than 30 degrees and the squares of sines will be small 
 in value. The net result of the horizontal deflection 
 due to Us is negligible in comparison with the de- 
 flections caused by Hs in the thrusts and that by the 
 moments. By means of transformations the fol- 
 lowing formulas are derived with several approxi- 
 mations : 
 
 For vertical deflection due to thrust under the verti- 
 cal load ; 
 
 D-ef'H'H'cos’e 
 
 for horizontal deflection due to thrust under the 
 horizontal load; 
 
 D - z p rH " C03 ‘ e & ~jJf(HrWl) c o S ‘e j* 
 
 -j'/q&H) 
 
 and for vertical deflection due to thrust under the 
 horizontal load; 
 
 D=2j C H'H"cos*e £§ -ffarH^Hcos^e ^ 
 
 DISCUSSION ON DEFLECTIONS 
 
 The moments and thrusts are equally important 
 in causing the deflections in the one-hinged arch. 
 The latter is especially important in the sections 
 near the center. In considering the deflections 
 caused by the thrust, the vertical forces may be 
 neglected without appreciable error. Shear can be 
 neglected too. 
 
 Under the vertical loading the vertical deflections 
 near the end of the arch are similar to those of the 
 no-hinged arch, while those near the center resemble 
 in the form those of the three-hinged arch. As a 
 whole, the one-hinged arch is subject to greater ver- 
 tical deflection than the no-hinged arch and less 
 vertical deflections than the two-, and three-hinged 
 arches for the vertical loading. Thus, the no-hinged 
 arch is the most favorable in stiffness among the 
 four types of arches, so far as vertical deflections 
 under the vertical load are concerned. The one- 
 hinged arch comes the next, the two-hinged arch 
 comes third, and the three-hinged arch is the most 
 unfavorable of all. Comparative deflection curves 
 for the four types of arches with the same dimen- 
 sions and designed under the same loading are pre- 
 pared by the author for different positions of the 
 loading. They are not here reproduced due to the 
 limit of space. 
 
 The same relation is found for the horizontal 
 deflections under the vertical loading for the four 
 types of arches. The horizontal deflections of the 
 three- and two-hinged arches are much greater than 
 those of the no- and one-hinged arches, while the 
 horizontal deflections of the no- and one-liinged 
 arches are about the same. Comparative curves 
 of horizontal deflections under the vertical load for 
 the four types of arches are also prepared by the 
 author. 
 
 Under the horizontal loading, the vertical and 
 horizontal deflections for the one-hinged arch are 
 found to be very small. This seems to indicate that 
 the vibrations under the moving train are low, and 
 the effect of wind is also rather unimportant for 
 this type of the arch. Comparative curves of de- 
 flections under the horizontal loading for the four 
 types of the arches shows that the one-hinged arch 
 has greater horizontal and vertical deflections than 
 the 110 -hinged arch, and less vertical and horizontal 
 
 17 
 
deflections than the two- and three-hinged arches. 
 
 The comparative maximum and minimum deflec- 
 tions for the four types of arches are shown in 
 Fig. 70. The one-hinged arch is again seen to be 
 stiffer than the two- and three-hinged arches, and 
 not so stiff as the no-hinged arch. 
 
 The yielding of a support has a serious effect on 
 the stresses. With- a horizontal movement of the 
 support of one inch away from the original position, 
 an additional value of 126.8 kips is increased in the 
 horizontal reactions. This increases the thrust at 
 the crown to twice the original value, and thus 
 
 greatly affects the safety of the arch. The yielding 
 of supports has a similar effect on the two- and no- 
 hinged arches. The effect on the one-hinged arch 
 is a little less than that on the no-hinged arch, and 
 much greater than that in the two-hinged arch. 
 For the same amount of horizontal movement, the 
 increase in the horizontal reaction in the two-hinged 
 arch is 29.8 kips, while that in the no-hinged arch 
 is 161.0 kips. Thus, we see that a perfectly firm 
 foundation is required for these three types of 
 arches. 
 
 Remarks 
 
 The main feature of this investigation consists 
 in the discovering new formulas and relations for 
 the one-hinged arch. These formulas and relations 
 are then carefully studied with the aid of numerous 
 curves and diagrams, and their characteristics are 
 revealed by means of critical comparisons. Al- 
 though the main purpose of this thesis consists in 
 searching for new theoretical facts, it is hoped that 
 it may stimulate investigations in a practical treat- 
 ment of the one-hinged arch with regard to its de- 
 sign and construction. The knowledge of the sub- 
 ject, however, is still in its infancy. Notwithstand- 
 ing the large amount of study and the labor de- 
 voted to the subject treated in this thesis, other 
 subjects remain to be investigated, such as the 
 economic ratio of rise to span, the relative merits 
 of solid and open webs, the secondary stresses, etc. 
 

OF , U[;n , ; 
 
 m 2 „ m 
 
 « »«r