THE UNIVERSITY OF MICHIGAN INDUSTRY PFOGRAM OF THE COLLEGE OF ENGINEERING AN ANALYTICAL AND EXPERIMENTAL STUDY OF THE PRESTRESSED BOWSTRING ARCH Movses J. Kaldjian A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1959 September, 1959 IP-385

ACKNOWLEDGMENT The investigation described in this study has been carried on under the direction of Professor Lawrence C. Maugh, chairman of the doctoral committee, to whom I am indebted for his valuable advice, for the time he has generously given for consultation, and for reviewing the manuscript and offering numerous helpful suggestions. I also wish to thank the other members of my committee for their assistance: Professor Robert C. F Bartels, Assistant Professor Glen V, Berg, Associate Professor Samuel K. Clark, and Professor Leo M. Legatski. The Department of Civil Engineering supplied the funds for the "?model study." The Statistical Research Laboratory made available the highspeed digital computer, without which the computation could not have been made. The typing, drafting, and reproduction were done by the staff of the Industry Program of the College of Engineering. I wish to express my thanks to each of these persons for their assistance. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENT.,eo... o o...... o ii LIST OF TABLES.,,,.. *.* *..............*.*.. iv LIST OF FIGURES.......... o a............V..... ~ ~. NOMENCLATURE.. oae...,.... a..a*..****.*@a****S**-* vii CHAPTER I. INTRODUCTION............ **...................... 1 II. THE STRAIN-ENERGY METHOD.......................... 6 III. THE MEMBRANE-ANALOGY METHOD........................ 19 IV. EXPERIMENTAL STUDY.................. 33 V. SUMMARY AND CONCLUSIONS............................. 42 APPENDIX A. RELATION BETWEEN THE VERTICAL COMPONENTS OF THE DEFLECTION OF THE ARCH RIB AND THE TIE-GIRDER..... 47 APPENDIX B. SAMPLE CALCULATION........................... 50 APPENDIX C. INFLUENCE LINE DIAGRAMS AND GRAPHS FOR PRELIMINARY DESIGN WORK......................* 56 REFERENCES......,..... o.o. *.....,................ 81 iii

LIST OF TABLES Table Page I VALUES OF EQUATION (2.4) (ARCH RIB CENTERLINE) AND ITS DERIVATIVES............................. 10 II BENDING MOMENTS AND AXIAL FORCES WITH PARTIAL DERIVATIVES - IN ARCH.......................... 11 III BENDING MOMENTS AND AXIAL FORCES WITH PARTIAL DERIVATIVES - IN TIE-GIRDER.....,...,.......*e....e 12 IV AXIAL FORCES WITH PARTIAL DERIVATIVES - IN SUSPENSION RODS......o.,... o *..*.....*.... a........ 13 V ANALYTICAL AND EXPERIMENTAL VALUES OF HORIZONTAL FORCE X1 COMPARED.......................a....... 36 VI VALUES OF sin (inck) FOR DIFFERENT "i" AND "k"........ 54 VII VALUES OF 10/in (k4 - 2k5 + k) FOR DIFFERENT "i".AND f"k".~..o......iv 55 iv

LIST OF FIGURES Figure Page 1 Analytical and Experimental Values of Bending Moment Compared for mI = 2.62..,o...,......*...o.. 37 2 Analytical and Experimental Values of Bending Moment Compared for mI = 5*.30*........e...........* 38 3 Bending Moment Along Arch and Girder for Different mg and mr Values oo.o................... 57 4 Influence Lines for Bending Moment at Panel Point X4 for Different mg and mr Values.................... 58 For Vertical Load 5 Influence Lines for Horizontal Force X1... o..... a. 59 6 Influence Lines for Suspension-Rod Force X2.o.. 0.... 60 7 Influence Lines for Suspension-Rod Force X3......... 61 8 Influence Lines for Suspension-Rod Force X4.......... 62 9 Influence Lines for Bending Moment at Panel Point X? for Arch..O.O.o....... OO.ooo.OO.oO.Oee 63 10 Influence Lines for Bending Moment at Panel Point X3 for Arch.... oo............................ 64 11 Influence Lines for Bending Moment at Panel Point X4 for Arch.......Oa..oa.....oa.....a.o. oo0oo...o..... 65 12 Influence Lines for Bending Moment at Panel Point X2 for Tie-Girder 6........... 66 13 Influence Lines for Bending Moment at Panel Point X3 for Tie-Girder.. o.. o...................... 67 14 Influence Lines for Bending Moment at Panel Point X4 for Tie-Girder.............o o................ 68 For Horizontal Load 15 Bending Moment at Panel Point X4 for Different Values of L/rc and mI for Tie-Girder................. 69 v

LIEST OF FIGURES (CON'P'D) Figure Page 16 Bending Moment at Crown for Different Values of L/rc and mI for Arch................... o............. 70 17 Horizontal Force X1i for Different L/r0 and mI Values.. 71 18 Suspension-Rod Force X2 for Different L/rc and mI 'Values............................................ 72 19 Suspension-Rod Force X3 for Different L/rc and mI Value s..O O. o ooeoe e e eses................................ 73 20 Suspension-Rod Force X4 for Different L/rc and mI Va luS e o..................c o. e........ o......o e 74 For Horizontal Gap A 21 Bending Moment at Panel Point X4 for Different Values of L/r and mI for Tie Girder..... o o q o o o o o. o o o o 75 22 Bending Moment at C row for Diff Serent Values of L/rc and mI for Arch.o.oooooooo.ooooo oooo. o oe. o... 76 25 Horizontal Force Xi for D'iferent L/r and mi Values.o 77 24 Suspension-Rod Force X2 for Diffrent /r ad mI al. es Q................................................ 78 25 Suspension-Rod Force X3 for Different L/rc and mI 26 Suspension-Rod Force X*4 or D ifferent L rc and mT Values o *..................................0....... 80 27 Plexiglas Modeo o o o o o o o o o o o o o o o o39 28 End Detail of A.luminu.ni M odel 'with Gap o............... 39 29 Altuiinumi Model with Vertical Load o o............. o. 40 50 Aluminum Model and Interchangeable Tie Girders........ 41 vi

NOMENCLATURE The letter symbol.s in this article are defined where they first appearo Those which appear frequent+ly are listed below for reference o A a'AcAg Ar cross-sectional areas EE,Ea EgEr modulii of elasticity F horizontal force (external) H,Ho horizontal components of arch thrust *I 9 I, IC9Ig rrmoments of inertia K a constant L span length between supports MYM YMa M bending moment;s CL g, N' N T axial forces PP1 loads (external.) U strain energy X, ^X i nknown forces or moments (internal) ai, a Fourier constants h rise at crown hm height of arch at i.th suspension rod k x L mg AgEg/AcEa mI IgEg/IcEa mr AArEr/AcEa qg membrane force along the span length r~crg radii of gyration vii

NOMEWCLATIRm (CONT 'D) cpQcOY Cp constants F =constant 8 J6QH horizontal movements of arch at support Cq> a~slope along centerline of arch rib A,4G initial horizontal gaps between arch and girder hAg, ~ vertical deflection of arch due to membrane force q As length along centerline of arch Ax length along girder or horizontal proJection of AX1, AXi relative deflections or rotations at X1, Xi Sub cripts a arch rib c crown g tie girder r suspension rod viii

I. INTRODUCTION The purpose of this study is the analysis of the prestressed bowstring arch of which the structural system is highly indeterminate. With the ever-increasing use of prestressed concrete in construction, such an investigation is important to the structural designer. Bowstring arches are often used when the abutments are not reliable for thrust, and when maximum clearance under the structure is desired. As used in this study, the term "bowstring arch" denotes a combination of an arch rib with a tie girder. These elements are fastened to one another at the supports and connected to each other through equally spaced vertical suspension rods as shown in the diagram below. ~ ~ L~_ _ ~ x L Any load on the tie girder is resisted jointly both by the tie girder and the arch rib. In addition to its flexural action, the tie girder also resists the horizontal thrust of the arch. -1 -

-2 -This type of interaction which involves the flexural resistance of the girder makes the structure highly indeterminate. The exact redundancy will depend on the type of end connections and also on the number of suspension rods. Because of the extensibility of the suspension rods and the deflection of the arch rib, the tie girder resembles a continuous beam on elastic supports to a considerable degree. The general practice is to simplify the above conditions by assuming that the suspension rods are inextensible, that the moment of inertia of the arch rib is small compared to that of the tie girder, i.e., the arch provides no bending resistance at all, and that the arch and the tie girder are pin-connected, although the latter may or may not be the case. Thus no matter how many suspension rods are used, the structure has only one redundant, that is, the horizontal reaction XL of the arch rib. When the rigidity of the arch is considered, there is an additional degree of indeterminacy for every suspension rod. Hence, a bowstring arch with six suspension rods will have seven redundants for pin-connected ends and nine redundants when. the ends are fixed. The bowstring-arch structure differs from the usual tied arch in that the moment of inertia of the tie girder is many times greater than that of the arch rib. As a result, the total bending moment taken by the rib and girder together is divided between these two elements, the major part being resisted by the girder. According to J M. Garrelts(l) his design of St. Georges Tied Arch Span at St. Georges, Delaware, is the first to introduce

-3 -the bowstring-arch design to Americaa In his designr he assumes that the rib and the girder are hinged at the supports) and that the hangers are inextensible and resist axial forces onlyo For his first approximation he considers the arch rib to resist no bending momrrent, and subjected only to compression. Hence, the horizontal component H of the arch rib compression is the only redu.indant, and using the principle of virtual work, he determines its val'e a He 'then proceeds with his design and obtains prel1imrinary sections to be used as a basis for his more accurate analysis, Since the total bending moment (simple beam moment minus H o y) at a section is actually taken by both the rib and the girder, Garrelts sets the second derivative of the vert;ical component of t:he arch-rib defllection equal to the second derivative of the girder deflection to find the right proportion for eacho Thu.s he obtains the proper proportion of each and a mnore accu-x.rate formula for the value of Ho A partial summary of his.work is found in the Appendix, Some seven years later, in 1948, in Budapest, Professor Viktor Haviar came out with his designo (2) Were it not for the introduction of the extensibility of t-he suspension rods into his derivation his work would be identical with Garre.ltso Unforturiately, however, his model analysis was not suitable for stu;Ldying this pointo In 1954 Drso So Chandrangsu and So Ro Sparkes described a procedure called "The Method of Influence Coefficients"(3) to design the bowstring arch. Here the relative displacement of the sides of every "cut" is expressed in terms of each force or moment applied, and as

many elastic equations are obtained as there are redundants in the system. To determine the coefficients in these equations, the principle of virtual work is used. A ":nll" —system bowstring arch. is introduced to avoid seriously affecting the accuracy of the moment calculations~ By transforming the elastic equations and calculating the differences, the bending moments are obtained directly. Chandrangsu and Sparkes also describe the membrane-analogy method for a fixed-end bowstring arch by considering only the flexural strain energy in the system. However, a more complete and more accurate analysis must also consider the axial strain energy. This is especially important for low values of the slenderness ratio as well as prestressing forces. By slenderness ratio here is meant the span length of the bowstring arch divided by the radius of gyration of the arch rib at the crown around the horizontal:is. The present study utilizes the strain-energy method to analyze the bowstring arch, and then applies t;his analysis to a bowstring arch which is pin-connected at each end and has six suspension rods equally spaced along its length. Five different tie-girder-to-arch-stiffness ratios have been used to each of ten different slenderness ratios of the arch. In addition, the effect of varying the cross-sectional area of the tie girder to the arch and the ch:ange in the suspension rod area were studied. The results of the above calculations have been used to plot influence line diagrams and graphs with a view to using them in preliminary design work. Were it not for the electronic computing facilities of The University of Michigan, it would have been next to impossible to

. 5 -obtain all these results; there were some 420 sets of seven equations with seven unknowns to be solved, To make the analysis sufficiently general so that it may apply to any desired number of suspension rods and to all rise-to-span ratios, the membrane-analogy method has also been introduced here, (See Chapter III), This assumes that the arch and the tie girder are connected by an inextensible membrane, and the vertical force in the membrane is expressed as a single continuous function, To find the force in any suspension rod, one has only to integrate this force between the proper limits along the length. The accuracy of this method will depend on the number of suspension rods used. However, even with six suspension rods, the results are so accurate that its use for preliminary design work is recommended, To verify the correctness of the various theories used in this study, an experiment was performed on an aluminum model of 49 in. length with two different tie girders. The model was properly instrumented with strain gauges, and gauges to measure the deflection. The experimental results thus obtained were in good harmony with those of the theories, In case the reader has been wondering why the word prestressing has not occurred thus far, it should be emphasized that the analysis is one of determining redundant forces even when the forces applied are those of prestressing. This point will be clarified in the detailed study of the analytical and experimental results presented later. In this discussion, prestressing is regarded asa condition rather than the analysis itself,

II. THE STRAIN-ENERGY METHOD Derivation of the Equation for Bowstring Arch Having Any Number of Suspension Rods i 1 * -^ --- L - - - The above diagram shows a fixed-end bowstring arch with equally spaced vertical suspension rods. To analyze this bowstring arch, "cuts" or "hinges", as the case may be, are imagined to be inserted at suitable places in the structure to make it statically determinate (see diagram below). Next, moments and forces are applied to both sides of these "cuts" or "hinges" to restore the structure to its initial condition. The total strain energy U, which is a function of all the forces and moments acting on the structure, can now be expressed in terms of all these forces and moments. The derivative of this energy U with respect to any one force (or moment) will give the deflection (or the rotation) in the direction of that force (or moment). Hence one can get as many elastic continuity equations from U as there are unknowns in the structure. Since the relative deflection or rotation between the two faces of a "cut" is zero or a predetermined quantity, these equations can be solved simultaneously and the desired results are obtained. -6 -

-7 -2 XX Xn P The above is in accordance with Castigliano's theorem which states that, when a structure is acted upon by an equilibrated force system, the derivative of the total strain energy U in the structure with respect to any force gives the displacement in the direction of that force. The unknown forces and the bending moments to restore the structure to its original condition are X1, X2, X3,...Xn as seen in the above diagram. Let M and N denote, respectively, the bending moment and the axial force on the "cut" structure produced by the externally applied loads, say P; and let M' and N' be the bending moment and the axial force caused by the unknown forces X1, X2, X3,.. Xn. The total strain energy (neglecting shear and torsion) is L L L L(Ma + M-)2 L(Na + N )2 + (M + Ml)2 U = J -a-al ds + ds + (g +g) d ~E IT~ ^ P"E A ~2E I o 2EaIa o 2EaAa o 2EIg L (n-2) + (N + N) dx \ (Xi) h (2.1) 0 2E gAg 2Err

-8 -where subscripts "a", "g" and "r" stand, respectively, for the arch rib, tie girder, and suspension rod. The relative deflection or rotation at the i-th "cut" is au ax- = xi which, by means of Equation (2.1), can be expressed in the form L 6M L ___ L (M a + Ma) i (N + N) ZiX......xi ds + o EaIa o EaAa aMt an N (n-2) + (Mg + Mg) d (Ng + )x + El9 9) ~6X dx.+ 9 9 dx + - I~ (2,2) EIg EgAg ErAr i=2 If, now the summation is used for the integration, the expression for -Xi becomes: L aM' L aN L (Ma + M a) + (Na + N a + ) X (Mg + M)?-~6 a (Xie + ( Mgs +.. Ea- I EaAa EaEg L aN' (n-2) (N + N>)x +E (2X3) + EA Ax X1h1 (2,35) /o-0 EgAg i=2 ErAr Equation (23.) gives "n" independent equations as "i" assumes values of 1, 2, 3, o.n, respectively, i.e., one equation per "cut". Thus one gets as many equations as there are unknowns in the system. If there is no initial lack of fit of members, (LXi)s are equal to zero, These "n" equations now can be solved simultaneously and the values of (Xi)s are obtained. Once the unknown forces are determined, finding the actual bending moment, the axial forces, and the shear at any section in. the structure is a simple matter,

-9 -Solution of Hinged-End Bowstring Arch Having Six Suspension Rods The arch being studied is parabolic in form and has the dimensions shown in the diagram below. For a 1 to 4 rise-to-span ratio, Y 4/I I '"^c 1 L/4 - --- 7 @ L/7 = Lthe equation of the center line of the arch with its origin at mid point of the tie-girder is x2 L = - + (2.4) further, the cross-sectional area of the arch Aa and the length along the center line of arch As are assumed, respectively, to very according to equations A A= sec cp and } (2.5) ASa = Asc sec p = Ax sec cp where the subscript c denotes crown; and taking the width of the arch to be constant throughout, the moment of inertia of the arch is given by Ia = Ic sec3 cp (2.6) To evaluate Equation (2.3), four tables have been prepared. Table I is the tabulated result of Equation (2.4) and its derivative. Tables II, III, and IV show all the forces, acting on the structure,

-10 -TABLE I VALUES OF EQUATION (2.4), (ARCH RIB CENTERLINE) AND ITS DERIVATIVE of Symm,- y = _-t_ + L/4 b b - 7 @ L -'I L Pin xb i ---— @ — Point x y = tan qp Cp cos p d 0 0.25000 L 0 0 1 1 L/14 0.24490. L 0.14286 8~ 08'.98994 2 ~ L/14 0.22959 *L 0.28571 15~ 57'.96150 3 * L/14 0.20408. L 0.42857 23~ 12'.91914 bt 4 * L/14 0.16837 L 0.57143 29~ 45'.86820 5 * L/14 0.12245 L 0.71429 35~ 32'.81378 a' 6 L/14 0.066633 L 0.85714 40~ 36.75927 7 a L/14 0 1 45~ 00'.70711

-11 -o H H H H H H H 0 Lz6L- l66o 60 8 9T' o68 L L89TL' tr899t 0oL91 039989 6901 oLo 9tlT LL93T' gS9 9 S3' 5ti 6o0Lo - 051OT96- 96gz6o T98Lo ) LLLTT' COAT T9Lo- 960o-| s 10 0 0 0 0 0 0 * 0gT96- 936o 0-tT90Lo - ' -oLT * LLLTT. Trgl9 9gz60o * 0e898- 680Loo- 9tt *- j 95g93,J LgqzT 9 LTyT1 J 680Lo Lz65L' 0o9L 1 9 t89r' U L9TL o6983 C69T' [ L66o H C W ~ | ~ l^ - |1 " | > '3> s3> |t - | - o M o... H j r H 3 3 2 > C s1 l g l z.~ 5 ^o 0fl o-^l 5'ry o~zA 'zA o~ ^ i g 5Z08 *TT 5. t* S t 5t 019 fyL 50 5 | ' | S@t | * 5.ft oJLt |/ ^n y S'at o~L tl Io 0 v 58e8 t 5.t 5e1\ 09 \t Itt o.0tl fTt z 58T s t o 00T o6 1 C 9oT gz'I \o 0^\ 9 *T ^ O'T^ *0 3m + / o 00 0 ~ 7 x i -^ "^ ^~~~~~~~~~~~~~~~~~~~~~Q

-12 -0 oh ~o ) X 0 V* o 8 -+o H H 0H H 0, + N- O N L 1 1 1 I I I I mM ~ ~...... I I 'o h - - f. o1- |- | '' <:C U/ hD 1 il i li iSJ~rg O3 - 0O O 8 0- u -|0t 1 5 ~ S ~ l \ 5 t -al0q 1~ g L 9 0- 5l~6 H g $ |k 5 61 |t I tl-=t t 1 0. t 5 k+ 3 5' 6 'o; 1. |Q 55 J50 [ O'T I S'TL 09 t WC t 5.1 t ~ ~ ~s + oo ^ t V o l t 5o f B H X 1 o ' O'T 0 ~6 x LL + ' <'l <'.~k o z <i~~ or ot so

-13 -x Hl r H H H H H H 14 k E-4 W; lo o o o o Pg4 o o o o - o H ~H ~ | ||OOO PO O O O C13 0 0N 11 H O O O O O 3 t l K — @ - / - C~~~~~~~-O H O <^ 0 co PC\ P4 CQ H O O O O, -.4 —0 / ^ ^ -~<-^.,~~~~~~~~~Lr f r^ t I t ^ ** **{^' 5~~~Ile \ ^-^-^ ~C ^ ^^^^~>

-14 -and the bending moments successively produced in the arch and in the tie girder by all the redundants XI, X2,.> X7 and the external forces Pi, P2, P3, and P4. With the help of these tables, and making the following substitutions, i.e, letting AZ g = AcEa A E AcEa IcEa c' a and by recognizing that (L/rc)-2 is identical to Ic/L2Ac, and that lX' ZX2 *2 ~.2X7 are all equal to zero, a set of seven equations [Equation (2.7)]* is obtained for solving the values of Xi, X2, a... X70 The numerical values of X1 X2,... X7 of Equation (2~7) depend no doubt on the external load, L/rc, mg, Tmr and mio Once these parameters are known, X 1 X2,. o. X7 can be determined, and one can now proceed to find the bending moments as well as the shearing and the axial forces along the entire stircture, and thus design the bowstring arch. For additional prestressing of the tie girder, whenever desirable, a specified initial (vertically restrained) horizontal gap ' is provided between the ends of the arch and the tie girdero A horizontal force X1, just large enough to close this gap completely, is applied next. Since there is a relative motion now between the ends of the arch and the tie girder, a different set of equations must be obtained. This is done by satisfying the new geometric boundary conditions and by setting all. the The detailed calculation of S1C as a sample may be found in Appendix Bo

-15 -AX1 = 0 (i) [498.8567 + 13228.3095 (r )2 + 16807 x. x (L )-2]X + [165.7657 - 1186.3341 L )- x2 + [303.2085 - 2220.6849 (L )'-]X re mg re rb rh +[382.4850 - 2855.2692 ( )-2]X4 + [382.4850 - 2855.2692 ( ) -2X5 + [303.2085 - 2220.6849 (r) 22]X6 + [165.7657 - 1186.3341 (L )2]X7 + 16807 () 0 mg re aX2 0 (ii) [165.7656 - 1186.3341 ( 2]X + [66.5788 + 799.2929 ( )2 + 80.50 - + 2058.8575 m (r ) X2 + [112.5944 + 642.7477 (r + 151.25 re rb Ml Mre c Ml +1129.5393 - 544.5468 (-)2 + 147.00 i]X4 + 119.6737 + 472.2767 (-) + 14.75 m]X5 + [89.4689 + 374.0758 (- ) + 101-50 IX [ 47.2175 + 217.5306 (L )2 + 54.25 -]X7 - [80.50 P + 131.25 + 147.00 ] 0 X3 " 0 1-2 1 0 1 -2 1 1 (iii) [303.2085 -2220.6849 (r ) X1 + [112.5944 + 642.7477 (L)- + 11.25 mI]X + [200.3580 + 981.2887 ( 227.50 + 3430.3087 (r )X3 + [239.2140++ 2 0 1 ]X4 + + 8 [ 62.3175 (L )8 2 + 248.50 1IX5 + [169.0053 + 626.6610 (L ) 2 + 189.00 X6 L)-12 + 1 01 ] 2 P 27 P=4 *+ [ 89.4689 + 74.0758 ( + 101.50 X - [11.25 + 227.50 + 266.0 'X4 = 0 (iv) [382.2850 - 2855.2692 (L) 2']X + [129.5393 + 544.5468 (L) 2 + 147.00 mI]X + [239.2140 + 845.8723 (L )-2 + 266.oo00 (2.7) + [300.7027 + 912.8602 ()-2 + 329.00 L + 4116.0343 1 (L)-2X4 + [293.0846 + 876.3650 (L )-2 + 320.25 1 + [224.4488 + 762.3175 (L)-2 + 248.50 1-]X6 + [119.2637 + 472.2767 (L )2 + 134.75 ]X7 - [147.00 o5 + 266.oo ( + 329.00 0 ] 0 Ax = 0 -Xi 0 (v) [382.4850 - 2855.2692 (L ) 2]X1 + [119.6737 + 472.2767 (L )-2 + 14.75 LIX2 + [224.4488 + 762.3175 (L )-2 + 248.50 m1X3 + [293.0846 + 876.3650 ( L)-2 + 320.25 -1 X4 + [300.7027 + 912.8602 ()-2 + 329.00 m + 416.0343 - ( )-2X5 ( L )-2 )-2 P- + 248.40 r mI ic mI mr c + [239.2140 + 845.8723 ()-2 + 266.00 1]X6 + [129.5393 + 544.5468 ( + 147.00 X - [134.75 + 248.50 + 320.25 AX6 = 0 (vi) [303.2085 - 2220.6849 (#-)2Xl + [89.4689 + 374.0758 (L)-2 + 101.50 l]X2 + [169.0053 + 626.6610 (1-)2 + 189.00 ] + [224.4488 + 762.3175 (L-2 + 248.50 m[]X4 + [239.2140 + 845.8723 (L)2 + 266.0 m]X5 + [200.3580 + 981.2887 ( )2 227.50 + 3430.3087 X6 + [112.5944 + 642.7477 (-L)2 + 131.25 -x7 - [101.50 + 2 ] = 0 AX7 = 0 (vii) [165.7656 - 1186.41 (L)-]X1 + [47.2175 + 217.5306 ( L )-2 + 54.251 2 + [89.4689 + 374.0758 (L ) + 101.50 1[X3 re ri Ml rd MI (L)-2 1+ 134 ()2 [112544.6 1 427477 (L -)2 + [119.6737 + 472.2767 (r) + 154.75 ]X4 + [129.5393 + 544.5468 (L)- + 147.00 1 + [115944 + 1.25 ]X6 6688 1 mI 2 2058.8575 ( - [5425 + 150 [657(-9.21 + 80.50 - + 2058.8575 - -).]x7 - [54.25 (_L-2 + 11 + 13475 1[ i[+.575 + 8799292m1 mi- m MM M Z~6=Om m m m

-l6 -external forces Pi, P2, P3- and P4 equal to zero, Thus for the case with a gap, Equation (2.7) becomes Equation (2.8). Influence Line Diagrams and Graphs Bending-moment and axial-force diagrams have been prepared from Equations (2~7) and (2.8) for various parameters and are shown in Appendix C, (See Figure 3 through 26)o These will facilitate the preliminary design because the graphs can be read directly and bending moments and axial forces can be obtained at a glance, In all these diagrams the parameters mg and mr were held constant at 2.0 and Ool, respectively, while L/rc varied from 25 to 300 and mI from 2,5 to 20, Thus they cover a wide range of values. The bending moments and the axial forces, for the vertical loads, are drawn as influence lines. For the horizontal loads the bending moments and the axial forces are drawn as a function of mI for different values of L/rc or vice versa. To use Figures 21 through 26, one must first find the particular value of A for which the graphs are prepared and then. by direct proportion obtain. the values wanted, To find the particular gap A, set A = KL where K - 5000/EaA,. Thus, once EaAc is known, A can easily be determined, The effects of varying the parameters mg and mr qn the bending moments are shown in Figures 3 and 4

-17 -AX1 = A (i) 2L -2 L -2 L -2 (i) [498.8567 + 13228.5095 (Tc) ]xi + [165.7657 - 1186.3341 (c) IX2 + [303.2085 - 2220.6849 (jr-) ]X + [82.4850 - 2855.2692 (r -) 4 - -2 L )-2 X2+ (303.2085 -C IX3 + [382.4850 - 2855.2692 (?Fc) ]X4 (iL )-2]X5+ 1( )-2] + (.)-0 2 75 Ic + [382.4850 - 2855.2692 ~) ]X5 + [303.2085 - 2220.6849 (r) ]X + [165.7657 - 1186.3341 ()2 = A AX2 0 (ii) [165.7656 - 1186.5541 ()-2 X1 + [ 66.5788 + 799.2929 ()-2 + 80.50 + 2058.8575 ()]X2 + [112.5944 + 642.7477 ( )- + 11.25 r- r_) + 80-50 F,- + 2058.852 E]xl L )-2 1 1-2 -2 +:129.5595 + 544.5468 ( -) + 147.00 ~1]x4 + [119.6757 + 472.2767 (rc)2 + 134.75 5]X5 + [89.4689 + 374.0758 (L-) + 101-50 ]X6 + 1 47.2175 +_217,5506 (L ) + 54.25 1 x7 - 0 AX3 = 0 (iii) [305.2085 - 2220.6849 (L )-2]X + [112.5944 + 642.7477 ( )-2 + 131.25 e]X2 + [200.5580 + 981.2887 ()-2 + 227.50 1 + 3430.3087 (L )-2X3 1 ]2 + [200.58 + 981.2887 + L -2 1 2 1 2 + [239.2140 + 845.8725 (-)2 + 266.o00 1]x4 + [224.4488 + 762.3175 (r)2 + 248.50 m]X5 + [169.0053 + 626.6610 ( )-2 + 189.00 i] L )-2 +01 + [ 89.4689 + 574.0758 (L ) + 101.50 -]x7 o AX4 = 0 (iv) [582.4850 - 2855.2692 ( )]X1 + [129.5393 + 544.5468 ( )2 + 147.00 []X2 + [259.I240 + 845.872 ()-2 + 266.00 mI]X3 [500.7027 + 912.8602 (L )-2 + 429.00 6. (L)-2]X4 + [293.0846 + 876.3650 (L )-2 + 320.25 1X5 re mI M re r MI (T ~-2 IL-2 (2.8) + [224.4488 + 762.3175 (L)2 + 248.50 1]x6 + [119.6737 + 472.2767 (L)-2 + 134.75 ]7 = AX5 - 0 (v) [382.4850 - 2855.2692 (L)-2]X + [119.6737 + 472.2767 (L)-2 + 134.75 L]X2 + [224.4488 + 762.3176 (L)-2 + 248.50 re + 8 mr]X3 + [293.0846 + 876.3650 ()- + 320.25 1]X4 + [300.7037 + 912.8602 (L )-2 + 329.00 + 4116.0343 L )2X. Mi a, MI 1=2+266.001 L-2 1 + [239.2140 + 845.8723 (r-) + 266.00 i]X6 + [129.5393 + 544.5468 ( )- + 147.00 X7 = 0 M16 = 0 -2 2 re)-"]X1+ [89.4689+ 974.0758 (L)-2 + -2 189.00 1 (vi) [305.2085 - 2220.6849 (L ) ]Xl + [89.4689 + 374.0758 (L) + 101.50 Li2 + [169.0053 + 626.6610 (- + 189.00 l-] + [224.4488 + 762.3175 (L ) + 248.50 1 ]X4 + [239.2140 + 845.8723 (L )2 + 266.00 X5 + [200.3580 + 981.2887 (L )2 + 227.50 + 430.3087 X6 + [112.5944 + 642.7477 (L )-2 + 151.25 1 1X7 = O rg mI AX7 0 (vii) [165.7656 - 1167.5541 ()-2X1 + [47.2175 + 217.5306 ()-2 + 54.25 i]X2 89. + [ 74.0758 ( 2 + 101.50 ]X ir~~ i [O3112a-.59re mI a. MI + [119.6757 + 472.2767 (L )-2 + 134.75 Lj]X4 + [129.5393 + 544.5468 ( )-2 + 147.00 + [1X5 + +642.7477 (L) 2 + 131.25 L]X6 +re mI + 28.85 M a + [ 66.5788 + 799.2929 (L )-2 + 8050 1 + 20588575 1L()-2]X7 re MI Mr 0.=

-l8 -Sign Convention Bending-moments producing compression on the upper fibers of the arch rib and the tie girder are considered positive throughout the text.,

III. THE MEMBRANE-ANALOGY METHOD For a Concentrated Vertical Force on the Girder To simplify the analysis of the hinged-end bowstring arch by the membrane theory, the membrane has been assumed to be inextensible so that the deflections of the arch rib and the tie girder are the same. The arch is assumed to be of parabolic form y = 4h - (L] (5.1) Moments of inertia and the cross-sectional area of the arch vary, respectively, as!a = Ic sec 2) (3.2) Aa = Ac seccp Now separate the arch from the girder as shown below. h H1 q H-. H0 (a) H0 ] Ho I 'p (b) P above is any vertical load applied on the girder. The membrane connecting the arch and the girder is assumed to function in the same -19 -

-20 -way as an infinite number of suspension rods, and carries a vertical force q of varying intensity along the span length. The vertical deflection of the arch due to membrane force q, in the preceding diagram may be obtained from the equation: 4 q = EIa cos 0 (5.5) dx4 Mq =EIa cos c (3.4) a dX2 where A denotes the vertical deflection and Mq the bending moment due.q~~~ a to the membrane force only. This deflection can be represented by the Fourier series; 30 oo ao n a~ n_ bn nCIx "i -I + a- sin i - + L I cos L (3 5) EI ~ EIc L EIc L n=l n -here ao, an, and bn denote constants to be determined. Considering the end conditions of the arch, Aq = 0 at x = 0 and x = L; also d2 A/dx2 = 0 at x = 0 and x = L; it is convenient to use the simplified form: A = an si n1x (3.6) nq EIc L By successive differentiation of q and replacing Ia by Ic sec, the membrane force q and the bending moment Mq are found: 00 Mq an (nII)2 sin L (3.7) n=l and Co = nH14 nIIx q = an ( I) sin AL (3.8) n=l

Equation (358) indicates that the membrane force can be regarded as being.composed of an infinite nrumber of distributed vertical loads whose magnitudes vary sinusoidally along the span length. It can be expressed to any desired degree of accuracy depending upon the number of terms consideredo The vertical deflection of any point on the arch is equal to that of the point on the girder immediately below it, and in the arch it is the sum of tbe deflections produced by the horizontal force Ho0 and the membrane force q, that is~ 4 +At q (3~9) The deflection due to HI is given'byj d_. H~y Eh X Xo2. `ha..y. dx2 EI Ec LL L. integrating and satisfying the end conditions ' that. -0 gf x - O and x =L Hence from Equations (3.6) and (53.0) Ao hF( Y.lrY, p (,)5 ~ TE 5 y x., ^ 5EIc l^. L EI L ( and when x KL, the deflection p aunder t:he load P is Hh2 00 EIc k E+k)+ I (12) n=1a H and the constants a. a remain to be determined O.and the constants al.O..an remain to 'be det ermined~

-22 -Since the relative horizontal movement on each side of the cut section at Ho is zero, H = OH + Ho8HH = O (3.13) from which Ho = - (3.14) 6HH 0OH and 5HH denote, respectively, the horizontal movement of the arch at Ho due to the membrane load force q and to the load Ho taken as unity. 00 OH =EIC6OH = -16 an () =Tm- E - 8 h2L ~IH = EcrHH= 1-5hL Substituting the above in Equation (3.14) and reducing, we obtain: H0 I Lt 7 (5.3-15) Thus, Ho can be determined when the Fourier constants are known, and these in turn can be calculated by consideration of the energy in the system. If a small increment dai is given to a Fourier coefficient ai there will be a small change in the internal stresses and also a small consequent deformation of the whole system including the load point. To satisfy equilibrium, the change in potential energy in the system must equal the change in work done by the external load. Thus aU dai = P ( iai (5.16) 6ai 6ai

-23 -The total potential energy in the system iso L L L 2 L MU.. fLo dx + dx - ox + dx (5317) 2EIe c 2JEogIg J 21Ar gA2EA 0 0......0 00oo Ma oy -\ an (:l) 2 sin.T (3i-8) Mg P(lik)x - (rI sin l for 0 < x < L (3 (31i9) n n 2: Pk(L-x) - a 3 sir - o < x < L From Equation (1) 2 r h l6Jh2 64h x 64;2 x2t OS +( 6Ct.r 2 o2 *2 -%() - lj j and for L/. -4 L 2 H oscp. x L (3,20) 0 Substituting these values in Equation. (317). a:.d. integrating t and reducing gives UEss [ L 5 Vo L - L an 'n-I+ ~ ':,..~'.nII?' 4an.. 2l] i ~5 rp/2- — L + 2m,.EII5 -"' [P \ /i'2L ar 13 vak 00 2 + a (nlT)4 + HO:r (5321) 2 3

-24 -Substituting for Ho in the above equation and differentiating with respect to ai, we obtain: +U 1 ai 4 rc o a —= 2E"'c L-3 (i-(i) 2 [960o - 1800 ( ).2] (3.22) + E { 2P sir (ia ) + i (il) + (2 r2 2mIEIc L3 (-il7) h The change in. potential energy in the bowstring arch due to the increment dai is: 6{,ai 4 1 80 r 2 dai 1! - f(il)4 [960 o- 1800 I(rC)2 da i 21c \i4 h ai ( 8oo r + - 2P sin (imi) + [(i1I) + (800 ] da (3.23) Substituting for Ho in Equation (3512) and differentiating, we obtain: ~a P E.10 (k4 - 2k3 + k) + sin ( (524) ~ai AP EIC [ (3,24) and the work done by P is: P ( Ap) dai = [ 10 (k4 - 2k3 + k) + sin (ilk.)]dai (3.25) oai EIc i The charnge in work done by P is equal to the change in the potential energy of the system, so equating Equations (3523) and (3,25) and simplifying gives: m-T + sin (ik) - -10 (k4 - 2k3 + k) ai = 2PL3 -—. --- —------- (3,26) mI + 1 4 120 - (ill)- w( ee where p s = 8 - 15 ( (c)2 + 1 (5;)2 (3.27) L h mi J

-25 -Note that even values of "i" have no effect on Ho. The Fourier constants from Equation (3.26) can now be substituted in Equation (3.15) to give Ho; Equations (3.7) and (3.8) can be used to give Mq and q, and the bending moments in the arch and tie girder are then given by: 00 Ma = HoY- an ()2 sin L (3.28) n=l 00 M = M - an () sin nL (3.29) r -- g pnL( n=l The suspension-rod forces are given by integrating Equation (53.8) between the proper limits. For a Prestressing Force F as Shown in the Diagram Below Since all assumptions for this case are the same as before, Equations (3.1) through (3.8) are valid here as well. The difference is in the application of the external load. y i h HoI, HI L (a) '_F-H__ _ ) f(F-H0) (b)

~26 -Likewise, the relative horizontal movement on each side of the cut section at Ho is zero: H = O6H + H HH + (F - Ho) 6GH = (3.30) from which a + F. H = OH+ FGH (35.531) GH - 5HH denotes the horizontal movement of the arch at Ho due to a unit load on the tie girder. 60H and &HH denote the same as before and 6 = -1L GH EAg Substituting the values of 5, I and 6G into Equation (3.31) and reducing, we obtain: oo F ()2 + 16h \,L 1 g L L, L3 nlT '1-5' Ho - — = - where UF (5.52) 8 ()2 (1 )2 aF - )-+ f () 15 L mL ( L Thus Ho can be determined when the Fourier constants are known; and with the same reasoning as before: a dai F( Ai ) dai (3.33) where oF is the horizontal. movement of force F. The total potential energy in the system is: U-.E:2.. M.. 2 ( O cos )2 dx Ma 1. (i[I.o c o c 2EgIg o 2E A 2EgAg o g

-27 -where 00 M - H y - a. ( sin-, see Eqiuation (3,58) a L0 L L.1 lL n=00 (3~3555 7,nH 2 ntx M - an (L) sin — L see Equation (3o7) n=l and L cos cp dx = L ^ see Equation (3520) Substituting these in Equation (3534), and integrating and reducing gives. 00 00 U=2E 15 h a (,L 4 ]L c 135`5 n,=1 20 42 (3.36) 1 1 V 2 4 ^ T 2 2 (F - Ho) In1 + a (pnI) + H - L + ( o) r L 1 2LL n 40 o g. g Replacing the value of He from. Equation (3532) in (5336) and di:.ferentiating with respect to ai, we obtain~ - < i a 96 0 h+ 512;| + 5^ n ~ X j ai 2EIC L3S i (il)2Q La 1.5 L2 m 2 ' 4 L2 1241 mM +1 8 r2 h 1 F -1 52 11 P M } IT L2 L (iii) F '~mF- (15 L2 + 4 4-8 (3037) e c e L2 The hange in poent energy in the bowsing ach e to en dai so

-28* _ da 1/a f ha 1 [1 (4096 h 512 r 2.7L 2 m+l 4 8r h i F 1 52 h2 1024] +I + (ill) r mI +m'L L (il)' aF L'!' 2 4 r2 II r2 m " |+ 4 - T)."8 d-a The horizontal movement of force F is.% = C FL (5359) Eic where C is a constant, Differentiating.Equation (3539) with respect to a, gives..~. 0 o (3.4o) and the york done by F is: (....,4) di 0 (3.41) The change in work done by F is equal to the change in the potential energy of the systenm Thus, equating Equations (3,38) and (3541) and reducing gives: 8 r 3 MI(L L I-LIF a -"... (3*42) mI+l rITT"' -F 128 (/^ _ ((i L LL where 8. 2 )2 1=5 L mI L (3.43) and BF3 -*-4 2 ()

-29 -As before, even values of "i" do not effect Ho. From Equation (3.42) Fourier constants can now be substituted in Equation (3.32) to give Ho; Equations (3.7) and (3.8) can be used to give Mq and q, and the bending moments in the arch and tie girder are then given by: 00 Ma = HoY - an (I-) sin nIx (3.44) n=l M = - a n (n) sin nEx (3.45) g nL L n=l The suspension-rod forces are given by integrating Equation (3.8) between the proper limits as before. For an Initial Predetermined Gap 4h, as Shown in the Diagram Below All assumptions and Equations (3.1) through (3.8) apply for this condition also. y h L (a) (b) (b)

-30 -The relative horizontal movement on each side of the cut section at the origin "0" tis sG aG - H = H^OHH (3.46) fronm wThich HO -QH b(3.47) ~HH Substituting the values of 50H and 8HH in Equation (3047) and reducing we obtain 00 1' -.a_ + 30 (3.48) 0 8 hL G h L3 nn( As before, H0 can be determined once the Fourier constants are known. With the same reasoning.as in previous cases, LU dai = AG(a ) dai (3.49) The total potential energy in the system iso L. L. U - 1 dx + L~ d MJ+(Ho cos )2 p (3.50) 2EIc 2EgIg 2E 0 o. o where 00 Ma Hoy -, (I) 2 sin r 1 [see Equation (3,18)] n=1 00 Mg - a ( sin. [see Equation (3.7) ] mg L L and n=l L cos Cq dx = L o [see Equation (3.20)] bsit ng thee in Eqatn (350) ad tegatg ad edu ng gves Substituting these in Equation (3550), and integrating and reducing gives:

-31 -00 00 U 1 28 f 2h2 32Hoh a1 l1 2 ( )4 U = 2It {5 ah L- n an + an (nI) ] 2EI~ V5 0 L Ln n 2L- - 1-355 n=l 00 (3.*51) 1 1.2 41 22 2 +mI an (nil) + y HorcL Substituting Ho from Equation (3.48) in Equation (3.51) and differentiating with respect to ai gives: au _ a ai M mI 3 1 (iE)4 - 50 (32 - 60 n r) } (i ~() '(2 1- (il) 6aI 2EIc E- mI (i)2 '4 h2 225 K h + 4(ilT)i 'h3? AG The change in potential energy in the bowstring arch due to the increment dai is: iU da a [i + (i + )4 30 - - 2EI c L3 mI i)2 4 h2 dai '{ * 5 [ T-('il )4 ( 3 -( 60 ~ + 225 H rI }da 4(iH) 4 h 3 dai From Equation (3.48) the horizontal force is: 00 H = 1^7 ' 6G A + 30 h ) Hr o oh2 L 3 h 5 nH Differentiating Ho with respect tol ' gives: aHo 30 1 (3,,54).ai = I hL2 (3054) and the work done due this change in H is: QA (H) dai ^ ^. dai (3.55) AG (Ha) cdam G = The work done by the change in Ho is equal to the change in the potential energy of the system; thus equating Equations (3a53) and (3555) and reducing gives:

-32 -30 a ^G EI6 a = L3 (il) h L2 (3.56) i mI + 1 4 480 mi ) (-i) 2 G where ''i" is odd for H 0 and,G = 2 - 15 (4I). (2)2 (5 G 4 4 h From Equation (3556) the Fourier constants can now be substituted in Equation (3.48) to give Ho; Equations (357), (3,8), (3544), and (3.45) can be used to give, respectively, Mq, q, the bending moments Ma in the arch, and M in the tie girder. The suspension rod forces, as before, are given by integrating.Equation (358) between the proper limits,

IV. EXPERIMENTAL STUDY Preliminary Work To determine the load-deflection relationship in the bowstring arch, prior to any analytical work, a preliminary experiment was carried out on a 21-inch-length double-plane Plexiglas model, shown in Figure 27. The model was loaded in several increments either by a vertical load at a panel point or by a horizontal force at the ends (produced by tightening the bolts on the prestressing rod) and the corresponding deflections on the tie girder were measured at all the panel pointso The results of the tests, when plotted load vs. deflection, indicated the existence of a linear behavior between the applied load and the deflection produced by it, Main Experiment The main experimental work was carried out on an aluminum model of 49 in, c. to c, span, and the results were compared with exact and approximate theoretical study, Dimensions The arch rib, cut from l-inch-rthick aluminum (6061-T6) plate, was 1 in. wide at the crown, making L/rc = 169.74, and varied according to Aa = Ac sec p and Ia = Ic sec3p toward the abutments, The interchangeable tie girders were made of extruded aluminum (6063-T5) rectangular tubing. There were two different sizes, 1 1/2 x 1 1/2 x 1/8 inches for two, and 1 1/2 x 2 x 1/8 inches for one tie girder. One of the former was for the case with an initial gap, This had horizontally slotted holes at each end -33 -

-34 -to permit relative horizontal motion between the ends of the arch and the tie, girder. The suspension-rods were made of 7/32-inch-diameter brass rods. To find the moduli of elasticity for aluminum and brass, samples prepared from the same stock as the model were tested from which the following ratios were obtained: mI - 2~6208 for 1 1/2 x 1 1/2 x 1/8 inch, mI 5~3004 for 1 1/2 x 2 x 1/8 inch, and mr = 0.0637 throughout. Loading and Measurements The model, in vertical position, was loaded in two increments of 100 lb at each panel point, and strains of a total of 25 points (the distribution and location of which are shown in the diagram, below) were measured by SR-4 electric strain gages after the application of each i:ncrement of load. To minimize the errors in strain measurements, two gages in series were used per point. These were glued one on each side of the point in question. The horizontal prestressing load was applied in. increments of 1000 lb to a maximum of 4000 lbo This was done with the help of a hydraulic jack, a calibrated 3/4-inch-diameter load rod, prestressing wire that ran concentrically inside the tie girder and anchor plateso See Figures 28 through 30. As with the panel point loading, strain measurements were taken after every increment of load. For the case with the initial gap, the relative horizontal deflection were varied in different decrements to a nmaximum of 181o7 x 10 3 in, and the correspQoding horizontal force and strain measfurements were

-35 -taken for every decrement of the relative deflection. 9,( Q ----- — 7 @ 7 —" 49 It The theoretical and experimental values of the axial force and the bending moments in the arch and the tie girder are shown compared in Table V and Figures 1 and 2.

-36 -TABLE V ANALYTICAL AND EXPERIMENTAL VALUES OF HORIZONTAL FORCE X1 COMPARED LOAD CASES P2 =l B + P3-1 C nP4=1 Pl=l Pl=l | Ho Ho L &K/2 4/2-4 A = KL & K = 5000 Ea Ac Horizontal Force Load Case mI By Energy Method By Membrane Analogy By Experiment A 2.62 0.3329 0.3296 0.3300 B 0.5999 o.6o64 0.5892 C 0.7482 0.7662 0.7161 D 0.005615 0.005422 E 19.41 18.74 20.14 A 5.30 0.3304 0.3190 0.3304 B 0.5938 0.6000 0.5648 C 0.7390 0.7682 0.7114 D 0.008061.007915 E 33.01 32.41

-37 -Full Lines show bending moments calculated by the energy method with extensible suspension rods. o -- Indicates Experimental Vaues A -- Indicates Membrane Analogy Values Bending Moments are in units of in-lb x 49 m = 2.62 ^> BENDING MOMENT ALONG BENDING MOMENT ALONG w THE GIRDER THE ARCH ~ os.s06 C a (o) '(b! D '4: - 0I I0 0. \ - j.02C a.0. -.002.03 A.o, 1 -8 I I I I ~.0006.6 -.00 1- - - // 2 ' / -.03.04 -.02 -.02 ~Mome.02nts Compared fr m1 = c 0| -.02 - -.01 --.02 -o -004.000 -.000820 -.0004L 14 @ 3 1/2" 49" TYPICAL_ E 0 0 Figure 1. Analytical and Experimental Values of Bending Moments Compared for mI = 2.62

-38 -Full lines show bending moments calculated by the energy method with extensible suspension rods. -Indicates Experimental Values AIndicates Membrane Analogy Values Bending Moments are in Units of in-lb = 49 mI = 5.30,W BENDING MOMENT ALONG THE m 5 BENDING MOMENT ALONG THE ARCH Ca w GIRDER.06 t0.o3.06 D h o - /V1) 1.02r (b) 0 / \ ^ ^:*4~~.04 o o. 06 * Fi 2.l and ~ E r n V e f d Moments 0maefo I 53-.01 -.04.03 -. 0 -._~~~~~031 ~-.03 14 @ 3 1/2" =49" TYPICAL D o o -.0008 - ^ 1 -.0002 -.0016 -.0004 E 0 - 0 -4 -I -8' -2 Figure 2. Analytical and Experimental Values of Bending Moments Compared for mI = 5.30.

Figi. 27*. Pexiglas Model. Flew. 28. End detal tfr AliWinua Model aith Gap.

C~4~0~~~~~~~~~~~~~~~~~C rd 0 C) a) rd a)::~~~~~~~r 'i~~~~aes~~~, P~~ I~~~~~~~~i ed~~~~~~~~~~~~~'::~-~:~~~~~~~~~~~.:~i~~~~~x~~~: bD~p~

-4l41.~~~~~~~~~~~~~~~~~-t i:~~~~~~~r C) r3 II 0 ':- -::I:::':i:::i:~b~~~i::i~i::::::-:i::~:i~.~:::~~

V. SUMMARY AND CONCLUSIONS In the foregoing chapters, different methods of solving the prestressed bowstring arch for various types of loadings and their experimental verification have been described, In Chapter I, the problem was described and a brief history of previous work on bowstring-arch design was given. In Chapter II, a general expression for solving bowstring arches with extensible suspension rods was obtained from the strain energy of the structure. Then specific equations were derived for a hinged-end bowstring arch with six suspension rods. Lastly, from these equations (with the help of a high-speed digital computer), influence line and other diagrams were prepared for the bending moments and the axial forces to be used in design. In Chapter III, the membrane-analogy method of solving the bowstring arch was introduced. This method is of special value since it covers a wider range of parameters (ie, all rise-to-span ratios, as many suspension rods as desired, and even different curves for the arch rib) with almost no additional work, In Chapter IV, the experimental work carried on 49-inch-span aluminum models was described briefly and the results obtained were compared with those given in Chapters II and III, From the results of the work discussed in this study, the following conclusions can be drawn., -42 -

_43 -(1) The assumption that the total moment at a section (simple bending moment minus H- y) is distributed between the rib and the girder according to their respectire moment of inertia is quite justified. For horizontal loading the error in the maximum moment is within 2% (Figures 1 and 2) and for vertical loading the error is around 10% at the load point at best, but away from the load there is a much larger error (Figures 1 and 2.) (2) A good correspondence between the results of strain-energy and membrane-analogy methods (Table V and Figures 1 and 2) seems to suggest that the assumption of inextensible suspension rods in simplified design is not unreasonable. Suspension rods were assumed to be extensible in the former and inextensible in the latter, (3) The manner in which Ia varied along the arch rib did not affect the results appreciably. (Ia was assumed to be Ic sec5 cp in the strain-energy method and Ic sec cp in the membrane-analogy methods) (4) Omitting the axial energy in the arch rib caused by the suspension-rod force components produced only a maximum error of 0,3% anywhere in the bowstring arch. This item can safely be left out from the design. (5) For horizontal loading an inverse linear variation w noticed to exist between mg and the bending moment in both the girder and the arch (Figure 3). The maximum error in this assumption, as mg varied from 0.5 to 10, was 1.0%. For vertical loads the same variation in mg changes the bending moment by only -3o3% in the arch and by -56% in the girder (Figure 4) (ioe,o the effect of varying mg on vertical loads can be totally neglected). * This can also be due in part to (3)

(6) Varying mr from 0.05 to o, prod-uced a maximiium bending moment change of -2,o4% in the arcPh and -2o7% in the girder for the horizontal loads, and 26o7% in the arch and -9o8% in the girder for the vertical loads (Figu.res 3 and.d 4), This same variation in mr caused in the largest of the su.spensionT rod forces an increase of about, 21%o In practice the percentages mentiorned above cacn be considerably less, for mr may take values a few times larger than 0o0O5 (7) For vertical loads T,he. h.o:rizontal force in the arch and the bending moment along the girder were founrd to be very sensitive to variations in mI when L/rc was small (Figures 5.? 1,2-l4) but for large values of L/rc, changing mI had very little affect on the bending moment and practically none on the horizonral. forceo Suspension-rod ai.d horizontal., forces were direr ctaly proportional to L/rC and inversely to m.- They neve:r chaned sign (Figures 5-8) The bending moment varied with mr, d.r..ectly in the girder and inversely in the arch (Figures 9 —14). As L/rc incre asedy along the girder, the ()) bending moment in.creased. and the (" momerin dec-..Veased For horizontal loadso Suspen.sil.on-od and. hor.izonital forces as well as the bending moments both;) in "the a`rch and in. the g.irde.r were all. found to vary inversely with V-/r, and exC:ept f9or the bendin&g mooment; in the arch, directly with mri. They were very sensitive ';o changes in L/rc when the latter had low values and became insensitive as L/r took on. large values (Figures 15-26)0

-45 -(8) As a whole the results of the experimental work showed very good agreement with those of the analytical methods. The maximum error in the horizontal force, using the strain-energy method as the standard for comparison, was 344% for the membrane-analogy method and 4.9% for the experimental work. (See Table V). For the vertical loading the bending-moment curve of the experimental work showed a small shift upwards along the girder, and downwards along the arch (Figures 1 and 2)o The bending moment along the arch (Figures lb and 2b) was not a smooth curve, while upper and lower-envelop curves, if drawn, tend to be smooths Figure 3 shows upper-envelope curves only. (9) The membrane-analogy method yielded very satisfactory results. The maximum error in the bending moments for horizontal loading did not exceed 2,6% along the girder and 6.6% along the arch (Figures 1 and 2) compared with the results of the strain-energy method. It was further observed that, as mI increased, the error in the bending moment increased in the arch and decreased in the girder in a linear fashion, This can be atrributed to the assumption that the suspension rods were inextensible, In some cases the results from the membrane-analogy method can be obtained by using only the first term of the series, This is especially true for finding the horizontal force for all types of loading and the bending moment along the girder and the arch for horizontal loads. In other cases, say to find the bending moment for a vertical load, since here the series does not converge so rapidly, one must use a few more terms of the series, The latter will no doubt depend on the accuracy

-46 -desired. One can assume (Ia = Ic sec cp)variation without introducing any error which is significant for preliminary design. In conclusion, the use of the Membrane Analogy method for design is highly recommended. (10) When a uniform load is applied over the entire length of the bowstring arch, the bending moment along the arch is found to be positive (Figures 9-11). On the other hand, prestressing the bowstring arch, with or without gap, by a horizontal force always produces a negative bending moment in the arch rib (Figures 3, 16, and 22). Thus prestressing the bowstring arch to counteract the dead-load bending moment is very advantageous for the arch rib. In the tie girder for low L/rc values the bending moment is also positive when mi is around 10 or larger (Figures 12-14). Here again there is an advantage to prestressing for the dead-load bending moment. When L/rc has large values (Figures 12-14), the girder part of the structure does not profit from this combined prestressing; it may even suffer. In this case, the gap prestressing system may be followed and the suspension rods released when the gap is being closed. After prestressing, the suspension rods are fastened back. This will allow one to get the benefit of prestressing without its harmful effect. If the tie girder is to be made of prestressed concrete, the two prestressing operations can be performed in one. Thus, within the limits described above, there are definite advantages to prestressing the bowstring arch. NOTE: It may be of interest also to mention here that for short spans, the roadway floor may be designed as part of the tie girder itself.

APPENDIX A RELATION BETWEEN THE VERTICAL COMPONENTS OF THE DEFLECTION OF THE ARCH RIB AND THE TIE GIRDER(1) Let v represent the component of deflection along a normal to the arch-rib curve at point Q in the diagram below. Q, ~~~~,x~x c — From the differential equation for the deflection of a curved beam: d2v v = M v+ v- — t L (A.la) ds2 p2 Ela in which ds is an element of length of the arch rib and p is the radius of curvature of the rib at point Q. The second term of Equation (A.la) can be neglected without appreciable error, and the equation reduces to 2v M dv. Ma (A.lb) ds EIa Neglecting the tangential component of deflection at Q, the vertical component r of v is: T = v cos p (A.2) -47 -

-48 -The first term of Equation (Aolb) can 'be replaced by an expression involving B, x, and. c as follows: from Equ.ation (Ao2) (i dvS ds ds d d cos c - v ds in c and cos p d- 2-2 dv d sin:p - v (j sp - ( cos (p- 2 sin p +-. - sins (A,5) ds ds p p ds p Neglecting the last three terms of Equation (Ao 3) we obtaino d2v d2 '- =_ sec cp d- (approx.) (A,4) Using dx d s cos cp Equation (Ao4) reduces to d 2 COs q) (A,5a) ds2 'P dx^ or + C oS d(p (approxo) (.A5b) For the girder (see preceding diagranm)o EIg. The total. momenrt M is eq.al to Ma + Mg and r.eglectig the change in lengths of the hangers, we obtain. dx2 d.2X dx2 dx-2

-49 -and from Equations (A.5b) and (A.6)> letting m Ig/Ia, Mg m + cos p (A-7a) and Ma T (m> + cp) (Am7b) Professor Haviar and Drs, Chandrangsu and Sparkes in their respective designs replace cos cp of Equations (Ao7a, b) by one,

APPENDIX B SAMPLE CALCULATI ON Calculation for AXI To find AX1, from Table II, III, and IV evaluate all five terms of Equation (3) and add them up together. Thus, a L (Ma + M;) T Z As = (498.8567X1 + 165.7656X2 + 303.2085X3 0 Ea Ia + 3824850X4 + 82.85 82.4850X5 + 303.2085X6 + 165.7656X7) 75 Ea IC (See page 52 for detailed calculations of this first term). a r, (Na + Na) Z EaAa Es -= (5.5095x1 -.4941X2 -.9249X3 - 1.1892X4 0 Ea Aa 1.1892X -.9249X6 -.4941X ) EA (See page 53 for details of this second term). i (Mg"M~ +x L (Mg + Mg xl,- -- Ax = 0 see Table III 0 Eg Ig L (Ng + Ng) (Pi + X1) L Z ---- Ax = i, see Table III 0 gAg EgAg (n-2)Xi hi Z - = 0, see Table IV i=2 Er Ar -50 -

-51 -and after adding and simplifying, AX1 = [498.8567 + 13228.3095( -) + 16807 L ( 2 ]Xi rn re ^c o g..x 2 + [165.7657 - 1186.3341 (L-) ]X2 rC L -2 + [382.4850 - 2855.2692 (rL)2]x4 + [303.2085 - 2220.6849 (r ) ]X6 + [165.7657 - 1186.5534 (-__) JX m7c -2 L3 +16807. 6. ( mg rc 75 E I

-52 -0 ON CYN cmLn~~~~~~~~~ ~\ g3 w _cj| H ^^ ^-^ 'ir^ 'ir^ 'Lf^ 'ir^ 'u^ 0 Cd X P n s o Cc CU CQ Cm CU fO C. 0U C - )C rC 0 C.) t _0 'O OJ OJ * * H t~ t~!~ I~ C KOd OHO H NH H H HvH MOO ' cO H ^ r- H-t x + + + s H as + a + O l H\ l Co0 O CM t^-CM? - -- b — T '- 0)- - LO r L L r LO LO L\ O * * *.. *. * 4) O- H- r- r<! o Co + + + + + + + N O O o o o.. U P FL H U O Lr\ \ 0U U0 0 0 0 O0 + + + + + + + Or UN HH Sn h C E L O\ OL O O C\ + O r; d d d c; +;C + + + + + + + X N X N N N, r o(\ r 0cc 0CN 0 00 + + + + +I + + W C *rl O ~0 -0 l r,l o + + + + + + + 1 11 0 o Un U O U U U + + + + + + O o t 4-> 5- ~ - - ~ - N P.

-53 -r I I 0 O i M I| i O | O O H 1 C1 WS CO HS -1Xf -1 -1 00 0P 0I ~ ~ o x a o cx \o oo oo o o o ^0 0 k 0 0 0c o1 ***** KcK + + + + I + - 0 '! O <1 <<;< M H ~ ~ c o Co o n o.N.... *. (:3 c\ co\ t0 " - 0 0.n W oO N t t- O r- co t IM CM CU CM o 0 o 0, 0- OL 3 \ H K H H + H 0 CO 0- O \0 U\ co Lo\. H H, 0 0C CM O h + + + + + + + ri Lr\ r-I H co MN O O LCn \n, - - 00 H - i -\.- -- -t i - CC H OJ H COJ OJ O\ H + + I + + + + I Q oU oX oU oU U H.0 05 ~ -..* ~'O CO,-'1OC CM 0CC 0\ t C\ '0 - Hr o t 0 o o O 0 0 0 - - I,.- = 1 + + + + + + + H 110 3 t'- 0 0 0 0 CO OH I in - I - I- - t - t - ^ ^ J^ ^' J, ^ ^^C C o 4a ~+~+ ~+ ~+ ~+ + II~L + + s.H oio 'c+ + + + + +*I r g~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0

-4 -'~O 02) ON O' 0 ' G0 0 \ 0\ 00 0 O Kr\ Cj tC- i oj KC\ ONH -4- N C \ - Hn 00 t>- KT\ K0C\ 1l 0 -ON _:f- -t 0 tON \) C~ C A ) ~ 0\ >- rC\ C M CM N h o rI ~ 4 = r| K IX 0\ \0 CO C 'I' 0\ ON '-oKIN C OJ cc <!O O C ) ON\ CO\ co CD.~fK c c C\- C- co W\ co i — - co,, —. co ~ t Q C~ — oN O C —:' ---, '\ Co.O 3ON E CU OJ C- C- CM K\ cc Co - r^ i^ >- cc, ' H C —O ONC) OD co - TX) O C o;i -H —~ I- ~ 0`dd I 3 C0 0 0.0 0 O 00 c ON ON) O O cc CMj -- c\ t — Cj KI cc K^ L1 K(o CM C — C C rco Cc \ 0 O CO 0N 0c CM H -z- r\ C\ HI - ~ O~ i -L- K\ CM CM L co H o r e o cc C K -- C -O 0 s-A N 0,\ ^4' I I 4- 0 -'- C ON ON -- - - C- C — C — H- CH CM rC z L\ kG

. 55 -H H K\ M HN H - 0 0 0 O _o 0 H H 0 0 O0 1 — 0 0 s- O\ \0 V- \ C - \O CC -d- 1- H H t.- -- Lr\ 0N\ CN CJ \ L\ 0 0 H H 0 0 O ** O 0 -0 ~ C -- 0 u'\- t- VHIL\ \L Cu, 0 r4 H 0 CM '^0 H t rc H CO PX 0 H H H H 0 <i H O L\ C(> U, O (2! 0 0-\ OtN ON- CY 0 F-t- 'S O - - KO Lt_^ O0 H H H H 0. 0 c + H CU I0 L- C( N H - C0 C t I0 HL\ 0 Lf0 L\ ~N \C \O ~ \(D K~N ON\ Lf\ C\ CM LO\ ON - H H CM C H H o- 0 CO 040 b —K F( < <~ H *C C CM H* * ~ i ~ I,, ~ > - 0 C0 CM \ Lco -O CTC- ON H - ON ON - H H C -zi- e k~ 0 H CC1 NO O O CU C5 C- OJ K 1 O H CU 00 0 n H9

APPENDIX C Influence Line Diagrams and Graphs for Preliminary Design Work Figures 3 - 26 -56 -

57 -w ci 6 — -&i.nj,\, -,, EE,- \ - ~ " i -— oas — 0 -__.._m c 0#0 00 s W 2 5 \_-_ a 0a. - 0 0 ia. "" a. i E E E EE-. 0' ~~~~~~~~~~~~ 0 0 g, II I I ) I - W 0 --- z 44-^ 04- - I — d 0L 0 to0o -' N Ii o ~ i =| = i - 7 1 - KgO - ___ o -- - _ E _ E I -I_ _ -_ _-_ U' t~ - o.) -I~i: ~ ~ ~ C z

-58 -15 151 7 J " 7 @ L/7 FOR DIFFERENT VALUES OF mg mg - 0.5 XI- X, p. 5 ARCH 0 1 2 3 4 5 75 s50 --.L/rc = 175 mg 0.5- \ = 10 25 mr = 0.1 I o 2___m g = 10 TIE GIRDER -25 2 3 4 5 6 7 x 10 20 _K _______ FOR DIFFERENT VALUES OF mr m -J u-: mr= O.Or = 0.50 o 0 o O0 I2 3 4 5 6 75 mr=O.0I L 175 50r 1mr =.05-s,\ m = 10 mr = 0.507 \Xos / g \\\ mg = 2.0 25 r = ----- m=50 TIE GIRDER ' ~ ~,-,~: -25 -- -50' - --— lI- --— I 0 2 3 4 5 67 DISTANCE ALONG SPAN, FT x I 7 Figure 4. Influence Lines for Bending Moment at Panel Point For Different mg and mr Values.

-59 -mi= e.o.8 - 7 L/7 L — mI 0 L/rc=300.D_.XXL amj = = 20 XI m-2.5 0.7....m1=5 mI =I0 L/rc=75 / =-15 0.6 - -/// - mx=20 0.5 - FROM IO X L 0.4 -L 0.4 L /rc= 25 0 im0. iI 0.3 i^i^~: 0 I 2 3 3.5 HORIZONTAL DISTANCE FROM SUPPORT,. FT xFigure 5. Influence Lines for Horizontal Force X1.

-6o7 @ L/7 =L 0.2 -2 4 0.1 x l 0.21 /S-O — L/r= 25 - - 0O I I I II 0 12 4 5 6 _., 4 /^- ^ ^ ^-3 DISTANCE ALONG SPAN, FT x L 0. I /ap /~~~~ ~ /I I I 75 0 0 12 3 4 5 6 7 uL 0.3 0.2 0. I Li/rc = 300 03 2 3 4 5 6 7 DISTANCE ALONG SPAN, FT x Figure 6. Influence Lines for Suspension-Rod Force Xp.

rn=2.5 7 @ L/7 = L 0.2o 4 4 '"S//-mi=15 X, P=I 03. J /-g~mI=25 0.1..>/ 1 L/rc=25 -. 0 2 3 4 5 6 7 0.3 m i= 2.5 mI=5 0.2 m- m =15 mI=25 >. / a- y ^ DISTANCE ALONG SPAN, FTxL/7 - 0e In e 4 5 0.3 - mI=25 _ mz=10 X, \/ mi=15 - 0.2- mz=20 — - -.... --- - 0.1 ~,,J^~~/'/ ____ L/rc=300 0 2 3 4 5 6 7 DISTANCE ALONG SPAN, FTxL/7

-62 -0.2 -------------? - ----- - -- m l= 2.5 ~~j__ ^~.~~~~~~ // —ml= 5 10 in= 15 Ml= 20 0.3 i _____________ / \ ^~ // — mI= 5 /O / ^X ra- "',l 10 0 I 2 3 4 5 6 7 0.3 | 7 @ L/7:L i= 2.5 - T hN4 L m I=5 0.3 X I p i 5 ---j mI I 20 m/ -^ xx/- m 20 0.2 — _ _ 0.1_ __ ___ ___ L/r..300 0 2 3 4 5 6 7 DISTANCE ALONG SPAN.FT x Figure 8. Influence Lines for Suspension-Rod Force X4.

-63 -20 -~~~~~20 ~,~ 7 @ L/7 = L 15 /r ~ LL,/^ ^"-.. __...~ ',-\ ~t l,, M, L/rc 25 10 C 2 3 4 - 7 25 -5L _ " J _5 20 -20 _1LL~ 3..X5 15 7^ --- —- I 0'5 m0 I0 225 /. "-T'-' --- ---- /... — -----— 0 mc~ %. ~ITAC5 6A U. 35 -10 ~- 30 0 QDS AO SA to "r N —',o 2 -- -. 2L o/r- 300 5 0 2 4 \ *- 5 6 -I0 -15- L DISTANCE ALONG SPAN, FT x Figure 9. Influence Lines for Bending Moment at Panel Point X2 for Arch.

-64 -25 7 @ L/7 = L 20 Ii I=20 L mI=55 I.0 0 5 L L/r — 25 — ~0 1 2 3 4 ~ 663 7 -5M I 25 I ________ 15 -- / L,/ m<=M20 // -' l /yL^^_..' '^^- '^<___________ _______ 5 LL 25~~~ I',,,r ---'-......_....i 0, 2 3 - ' --- —....... - \ —:.:m=2. 15z " '______/ ______ \-5 w -10 20 m I = 2.5 '5mI=5 15 10 ^^^"'^%mI'0 o, 2 3, 4o... r..... _ - 7 -5 I I \'., 5 6_ 1 DISTANCE ALONG SPAN, FT x -' Figure 10. Influence Lines for Bending Moment at Panel Point X3 for Arch.

-65 -20 —, ~ L ~/-m =2.5 l'ol 15 -------------- ^ --- —--- — ^ - /- ---:5 Ix///-x^i = " - ---- 1n I0 ~ - -- L/r 725 0 2 3 4 5 6 7 20 7 L/7 L 15 --- - X4 I0 -- I --- —m: 20 -10 _....____ 0 I 2 3 4 5 6 7 \ 7 -Lfor Arc h. 0 2 ---- 5 6 - for Arch.

-66 -125 7 _ ' L/7 = L /'. ~ —m! =20 I X, 0O y v -'- I''M 2 3 4 5 6 7I# DISTANCE ALONG S, ", 7 25 100 ___ire12. Influence Lins o d m P i___ __nt X2 - 75 GirderMI26 2.5 50 LL 1 n/ L/ 0 I 2 3 4 5 6 7 50 75 2.3 6mI =20 i,,:-'5 -------------— mI = '-5 I../75~ITAC ALN S ---PA —mN, —=I NF5L 50 — II/ L/rc= 75 0 2 3 4 5 6 7,? / \,,\ =Io 50 =5 —" ---m- -- - 25 Figure 12. Influence Lines for Bending Moment at Panel Point Xp for Tie Girder. for Tie Girder.

-67 -75 7 L./7: L 150.z; L X. L~,/ XI \ / I,00 ------- / /, \ ", V ---.J —m,-_, 125\ // ~~~~~~~~~~~~50 ^ ^^ —~m=20 ~~~~~~~~~~~~~25 ~ ~ ~?A / 0 / \50 75. m-=20 re i-15 50 mI 10 ', mI=5 LL 25. m1=25 I~ ' L/rc=75 '%~ _0 0 -25 0 2 3 4 5 6 7 75 -25 ________ 300 75O~ ~~~ 3, 4. —m7=2 0 Figure 1. Influence Lines for ending Moment at Panel Point for ie irder. Figure 15. Influence Lines for Bending Moment at Panel Point X, for ~te Girder.

-68 -200, 7 @ L/7 =L 175 — ih- L 3 456 175 V' ^ \x x P. I# 150 75 r l\ /m l =20 - 25 5 /2//,\ '".\./ ----m! =>0 27 J 1\00 /2:-/ // 75 m =20., /.', ".5 0; -25 _________ ___________ 0 I 2 3 4 5 67 75 50 1 I = I m e l _.for1~~~ ~ i I =5OO -25 0 I 2 3 4 5 6 7 DISTANCE ALONG SPAN, FT x L -7 -Figure 14. Influence Lines for Bending Moment at Panel Point X4 for Tie Girder.

-69 -7 L/7:-L P-XI P- I1 -35 ---- L /re 25 -30, x 0 -25 ' " L -20 15,-10 LkOm Figure 15. Bending Moment at Panel Point X4 for Different Values of L/r- and mc for Tie Girder. -4 4 L/re 200 - 3 L/,250 o 5 10 15 2o m] Figure 15. Bending Moment at Panel Point X4 for Different Values of L/rc and mI for Tie Girder.

-70 -7 L/7 =L -550 F,__ P-X, P= 1 - 525 -500 -475 x -450_S - 425 - - 00 L/rc.S75 -75 L,125, /r, 175 - 5 L L/rct 200 0 5 10 1 5 20 m] Figure 16. Bending Moment at Crown for Different Values of L/rc and mI for Arch.

-71 -14.... 7 ) L/7:L L 12 L \\ I I I I I PIx P-XI 12 II I0 0._J 7 w 6 F0oFra 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 L rc Figure 17. Horizontal Force X1 for Different L/rc and mT Values.

-72 -35 7? L/7 =L P-X, P | 30 25 20 - 5 3 2 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 L rec Figure 18. Suspension-Rod Force X2 for Different L/rc and mi Values

-73 -7Q@L/7 =L 35 X \ P-XI I 30 25..J w C., 0 I0 5 4 2 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 L rc Figure 19. Suspension-Rod Force X3 for Different L/rc and mI Values

-74 -7 Q L/7 L 351 - --- P-XI P= I 30 25 In r \ \\\\l o. 20 20 _ __ —_ w IiCr~~~~~~~~~~~~~~~(C igure 20. Suspension-Rod orce X4 for Different /r and m Values 5 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 304 rc Figure 20. Suspension-Rod Force X4 for Different L/re and mI Values

-75 --400.. 7 D) L/7 =L Xi X23 X41 4 - 350 ____; ~A/2 A/2-_5000. A:=KL, WHERE K: 5000O - 300 2 -250 UJ -200 4 Co z 0 -150 / ii~~~~~~~~~~~~L /. — 00 i_ Co L/rz=300 0 5 10 15 20 mI Figure 21. Bending Moment at Panel Point X4 for Different Values of L/rc and mT for Tie Girder.

-76 --60 7 ) L/7 ' L -55 \2 /2 I=KL, WHERE K 5000. -50 -J -45 - -40 1 z 0 L/r 50 L /rc = 5 o, __L/rjc=_ 100 __ L/L/r c= 150 --— O ---- - =^-1/^c250...... ======== 0 0 0 5 10 15 20 mI Figure 22. Bending Moment at Crown for Different Values of L/r0 and mi for Arch.

-77 -7 P~ L/7 = L 1600 -,, L KL WH/2ERE K 5000. A=KL, WHERE K=: 0 1400 1200 1000 -J 0 800 600 400 200 - 0 40 80 120 160 200 240 280 L/rc Figure 23. Horizontal Force X1 for Different L/rc and mI Values

-78 -400 _7? ) L/7 = L I 3, X 1 4 4 I..-,,/2,/2- I — 350:=KL, WHERE K 5000. 300 250 u 200 0 150 O50 5 \0 \c\ 0 40 80 120 160 200 240 280 L/rc Figure 24. Suspension-Rod Force X2 for Different L/re and mi Values.

-79 -450 7 P L/7 = L xo 4 X,3 ) _\ A/2 A/2- H400 -- 25000. A=KL, WHERE K= 5000 350 _ 300 250.J,, 200 150 100 50 0 40 80 120 160 200 240 280 L/rc Figure 25. Suspension-Rod Force X3 for Different L/rc and mI Values.

-8o450 7 c L/7 L F~~igure ~ ~ ~ 26x., |p - FX 4a. k \/2 A/2-114 400 A=KL, WHERE K= 5:O. 350 300 250 w o 200 LA!50 100 0 40 80 120 160 200 240 280 L/r, Figure 26. Suspension-Rod Force X4 for Different L/rc and mI Values.

REFERENCES 1. Garrelts, J. M. "Design of St. Georges Tied Arch Span.' Proceedings, ASCE, 67, (December, 1941) No. 10. 2. Haviar, V. "The Arch with Connected Stiffening Girder.' Final Report of Third Congress, International Association for Bridge and Structural Engineering, Liege, Belgium, September, 1948. 3. Chandrangsu, S., and Sparkes, S. R., "A Study of the Bowstring Arch Having Extensible Suspension Rods and Different Ratios of Tie-Beam to Arch-Rib Stiffness." Proceedings of the Institution of Civil Engineers, 3, No. 2, (August, 1954) Paper No. 5966. 4. Maugh, L. C. Statically Indeterminate Structures New Yorko John Wiley and Sons. Inc., 1951. 5. Newmark, N. M. "The Distribution of Moment Between Rib and Girder. " Transactions, ASCE, 1035 (1938) 6. Melan, J., Theory of Arches and Suspension Bridges Chicago: The Myron C. Clark Publishing Co., 1913. 7. Courbon, J. "Bowstring Design." Les Annales des Ponts at Chaussees, (September-October, 1941) 8. Courbon, J. Pont de la Raterie sur la Sarthe Bulletin of the International Association for Bridge and Structural Engineering, August 15, 1958. 9. Chang, J. C L., "Electrqnic Computers in Design of Tied-Arch Bridges.' Civil Engineering, 28 No 1 No. ovember, 1958). 10. Anonymous, "A First in Bridge Construction." Engineering News-Record, 162, No. 22 (June, 1959). 81.

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