ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES CONTENTS

Transcription

1 CONTENTS 1. EFFECTS OF CURVATURE ON THE STRESS-DEFORMATION EQUATIONS IN THE CROSS SECTION OF A CURVED BEAM. GENERAL EQUATIONS OF DEFLECTIONS IN A CURVED BEAM 3. DIRECTRIX OF THE ARCH 4. DEPTH VARIATION OF THE ARCH 5. TYPICAL CONFIGURATIONS OF ARCH BRIDGES 6. CONSTRUCTION PROCEDURES OF ARCH BRIDGES 7. SECOND ORDER DEFLECTIONS IN ARCHES 8. COMPENSATING TECHNIQUES ON ARCHES 9. ARCH STABILITY 1. TEMPERATURE VARIATION, CONCRETE SHRINKAGE AND SUPPORT DISPLACEMENT EFFECTS 11. GEOMETRY CONTROL OF THE ARCH DURING CONSTRUCTION 1. BIBLIOGRAPHY Conference Theme at the Intituto de la Contrucción y Gerencia: Análii y Dieño de Puente en Arco, June, Concepción Etructural de Puente en Arco, Augut,3 Publihed in the Journal of the ICG, N PT-9 y PT-1, repectively, 3 OSCAR MUROY

2 1. EFFECTS OF CURVATURE ON THE STRESS-DEFORMATION EQUATIONS IN THE CROSS SECTION OF A CURVED BEAM In a traight beam a well a in a curved axi beam, it ha been proved, by model teting and verified by a rigorou analyi uing the Theory of Elaticity, that cro ection initially plane, remain plane after being ubjected to axial force and moment. Fig 1.1 a) Section ubjected to an axial force N: In thi cae we will have the equation Let a egment of a curved beam with AB axi, with a radiu of curvature R and a ubtended angle d (Fig. N 1.1) The total deformation could be divided in an axial deformation d and an angular deformation d. Then, in the fibre a ditance y from the axi, total deformation will be: d yd And the original length i R yd Hence tre will be: d yd E E (1.1) R yd yda M (1.) A da N (1.3) A The Moment of Inertia of the cro ection i defined a: I A And doing: I' A y da, y 1 da, y R Being the value y/r, normally very mall, developing in erie of (y/r) and neglecting larger term to 3th order, we get the following expreion: A 1 1 y R I da A R, 3 OSCAR MUROY 1

3 Uing thee expreion and the Eq. (1.1) in developing the Eq. (1.), we obtain the relation of axial and angular deformation: I d d (1.4) I' R Replacing thi Eq. (1.4) in the Eq. (1.3), and taking into account that: d R d, egment of the curved beam axi Axial deformation will be: d N EA d (1.5) And replacing Eq. (1.5) in the Eq. (1.4), we obtain the angular deformation: d I I' N EAR d (1.6) Replacing Eq. (1.5) and (1.6) in the Eq. (1.1), we obtain the axial tre: N A b) Cro ection ubjected to a bending moment M: In thi cae we have the equation: da N (1.7) A yda M (1.8) A From Eq. (1.7), with the ame aumption a before, we obtain the relation between angular deformation and axial deformation: AR I d d 1 (1.9) I AR R Replacing Eq. (1.9) in the Eq. (1.8), we obtain the axial deformation: d I I' M EAR d (1.1) Replacing thi value, in the Eq. (1.9) we obtain the angular deformation: M I d 1 EI ' AR d (1.11) Latly, replacing thee two value in Eq. (1.1), we obtain the bending tre, in the cro ection: I M M y I M I My 1 I' AR I' y I' AR I' I y, 1 1 R R Hence, in the curved element, we will have axial tree in the neutral axi of the cro ection, due to a bending moment M:, 3 OSCAR MUROY

4 I I' M AR c) Section ubjected to hearing force T: d) Summary: Fig 1. In thi cae we will get a hearing train equal to : T T dy d d GA G A/ T GA Making A 1 A/, equivalent area to hear force, being k, a factor depending on the ectional hape. Then for a ection ubjected to M, N and T, we will get the following deformation: Angular deformation: M d 1 EI I AR d I N I M I d 1 I' EAR I' EI AR ' 1 d I N d I' EAR (1.1) Axial deformation: d N EA I I' Shearing deformation: d y T GA 1 d M EAR d In Eq. (1.1) and (1.13), econd term the deformation of a curved beam I I' N EAR and I I' M EAR (1.13) (1.14), are the effect of curvature in Next we have a table with the value for area, inertia and the parameter I / I, I / I ( 1+ I / AR ), I / I ( I / AR ) y A / A 1 : I I Section A I I /I 1 I' AR Rectangular bxh bh 1 bh h R h 15 R I I I' AR 1 1 A/A 1 h 3/ R Box bxh - b 1 xh 1 Circular h bh b 1 h 1 bh b 1h h h bh b 1h1 1 bh b 1h1 3 R bh b1h * 1R 1 bh b1h R h 1 1 h 16 R 1 16 h R 4/3, 3 OSCAR MUROY 3

5 Tubular h y h h h 1 h h h h h1 h R h h h h1 16R h h1 4 1 h h1 16R * 3 bh b1h1 1 bh b1h R bh b 1h1 1R bh b1h1 Table 1.1 Typical Cro Section Parameter In order to have an idea of the value (h/r), we refer to table N 1. and N 1.3 and we can ee that in concrete Arch Bridge, it i between 1/3 to 1/7 and 1/5 to >1/1 in Steel Arch Bridge. Then (h/r) <1/9~1/1, and therefore very mall comparing to1. N Bridge Name Type Span l(m) Rie f(m) l/f Directrix h (m) h /l h c (m) h c /l Cro Section 1 Nant Ffrwd built-in parab..89 1/ /16.4 rectang. Kimitu built-in / /55 Hollow 3 Mannen Two-hinged / /37.6 rectang. 4 Omokage built-in co hip.1 1/ /56.7 Box 5 Nant Hir built-in parab..91 1/ /14. rectang. 6 Araya built-in / /58.7 Box 7 Yohimi built-in co hip.7 1/ /5.9 Box 8 Miyakawa built-in co hip. 1/ /61.3 rect hollow 9 Taf Fechan built-in parab / /157. rectang. 1 Yumeno built-in / / Taihaku built-in co hip 3.8 1/ /6.4 Box 1 Hokawazu Two-hinged parab 4 th o 3. 1/ /7.8 Box 13 Beppu Myoban built-in / /67.1 Box 14 Río Paraná built-in / /9.6 Box 15 Gladeville built-in / /7.9 Box Table 1. Concrete Arch Bridge N Bridge Name Type Span l(m) Rie f(m) l/f Directrix h (m) h /l h c (m) h c /l Cro Section 1 South Street Two-hinged circular 1. 1/ /58.8 box Northfolk Two-hinged parab..61 1/ /138. box, 3 OSCAR MUROY 4

6 3 New Scotwood Two-hinged / /131.6 box 4 Leavenworth Two-hinged / /15.6 box 5 Smith Av. Two-hinged / /65. box 6 Río Colorado built-in / /78.7 box 7 Cold Spring Two-hinged / /77.9 box 8 Glenfield Two-hinged / /187.4 box 9 Fort Pitt Two-hinged / /139.4 box 1 Lewiton built-in parab 4 th o / /73.8 box 11 Rooevelt built-in parab / /135. box 1 Vltava Valley Two-hinged / /66 box 13 Fremont Two-hinged / /313.7 box Table 1.3 Steel Arch Bridge, 3 OSCAR MUROY 5

7 . GENERAL EQUATIONS OF DEFLECTIONS IN CURVED BEAMS Fig.1 The Navier-Bree Equation for the diplacement in curved beam are given by: Angular Diplacement: M w w d (.1) EI Horizontal Diplacement: 1 M N T u u w y y y d co d in d (.) EI EA GA Vertical Diplacement: 1 M N T v v w x x x d in d co d (.3) EI EA GA where: d d dy M EI N EA T GA d, i the angular deformation in a egment d of the arch d, axial deformation 1 d, hear deformation In thee equation it i not conidered the effect of curvature of the arch Then, to take into account thee effect, we hould ubtitute thee value, with the value: M I I I N d 1 d EI I AR I EAR (.4) ' ' d N EA I I' M EAR d (.5), 3 OSCAR MUROY 6

8 T dy d GA (.6) 1 The axi curvature, i obtained from the equation: 1 y" y" 3 y"co 3/ (.7) R 1 y' 3/ 1 tg Developing firt the Navier-Bree equation, we have: M w w d (.8) EI 1 M N T u u w y wy d co d in d (.9) EI EA GA 1 M N T v v w x wx d in d co d (.1) EI EA GA Replacing Eq. (.4) in Eq. (.8): w w M I 1 EI I' I AR I I' N EAR d Thi Eq. could be written a follow: w w M EI I 1 I' I AR I I' I AR NR d M (.11) Replacing Eq. (.4) and (.5) in Eq. (.9): M I I I N u u w y wy d 1 EI I AR I EAR ' ' Thi Eq. could be written a follow: u u w y wy N I M co d EA I EAR M I EI 1 I' Replacing Eq. (.4) and (.5) in Eq. (.1): I AR ' 1 I I' I AR Rco d N I co 1 d EA I' Rco T in d GA T in d (.1) GA 1, 3 OSCAR MUROY 7

9 M I I I N v v w x wx d 1 EI I AR I EAR ' ' Thi Eq. could be written a follow: Doing: v v w x wx I I 1 I' AR I I I' AR R co R in Latly, we have the equation: w w u u v v M I EI 1 I' N I M in d EA I EAR I AR ' 1 I I' I AR Rin d N I in 1 d EA I' R in T co d GA T co d (.13) GA M NR d EI M (.14) w y w x wy wx M d EI M d EI N co 1 I d EA I' 1 T in d (.15) GA N in 1 I d EA I' T co d (.16) GA 1 1, 3 OSCAR MUROY 8

10 It ha been examined the value of parameter,, and, for two typical cae: built-in arch of 65 m pan and two-hinged arch of 9 m pan, whence the following concluion could be drawn for thee parameter ( I ). I 1 ( I ) 1. I Latly the Navier equation could be written, taking into account the curvature effect, in the following way: M w w d (.17) EI u u v v w y wy M N T d 1. co d in d (.18) EI EA GA 1 w x wx M N T d. in d co d (.19) EI EA GA 1 That i, the effect of curvature could be incorporated in the normal Navier equation, modifying the value of the ection area for the axial force, with a reduction factor of,8 and,5 in the equation of horizontal and vertical deflection, repectively. In the practice, thi could be done, calculating the reult for each of thee value of reduced area to axial force, with the final reult lying in between thee two value, 3 OSCAR MUROY 9

11 Arch, parabolic axi, two-hinged, parabolic variation of h x y h co in y" R I/I Span=9.m, rie=17.m, hc=.m, ha=1.5m +/ (+/) -/ (-/) 1+(I/I ) 1- (I/I ) , 3 OSCAR MUROY 1

12 Arch, parabolic axi 4th order, two-hinged, parabolic variation of h x y h co in y" R I/I Span=9.m, rie=17.m, hc=.m, ha=1.5m +/ (+/) -/ (-/) 1+(I/I ) 1- (I/I ) , 3 OSCAR MUROY 11

13 Arch, coine axi, two-hinged, parabolic variation of h m=qa/qc=.7 x y h co in y" R I/I Span=9.m, rie=17.m, hc=.m, ha=1.5m +/ (+/) -/ (-/) 1+(I/I ) 1- (I/I ) , 3 OSCAR MUROY 1

14 Arch, circular axi, two-hinged, parabolic variation of h x y h co in y" R I/I Span=9.m, rie=17.m, hc=.m, ha=1.5m +/ (+/) -/ (-/) 1+(I/I ) 1- (I/I ) , 3 OSCAR MUROY 13

15 Arch, parabolic axi, built-in, Straner variation of h x y h co in y" R I/I Span=65.m, rie=13.m, ha=1.65m, hc=1.1m +/ (+/) -/ (-/) 1+(I/I ) 1-(I/I ) , 3 OSCAR MUROY 14

16 Arch, parabolic axi 4th order, built-in, Straner variation of h x y h co in y" R I/I Span=65.m, rie=13.m, ha=1.65m, hc=1.1m +/ (+/) -/ (-/) 1+(I/I ) 1-(I/I ) , 3 OSCAR MUROY 15

17 Arch, hyperbolic coine axi, built-in, Straner variation of h x y h co in y" R I/I Span=65.m, rie=13.m, ha=1.65m, hc=1.1m +/ (+/) -/ (-/) 1+(I/I ) 1-(I/I ) , 3 OSCAR MUROY 16

18 Arch, circular axi, built-in, Straner variation of h x y h co in y" R I/I Span=65.m, rie=13.m, ha=1.65m, hc=1.1m +/ (+/) -/ (-/) 1+(I/I ) 1-(I/I ) , 3 OSCAR MUROY 17

19 3. DIRECTRIX OF THE ARCH The directrix of an Arch hould be the more approximate poible to the funicular curve for the loading applied to the arch. A the funicular curve i derived, graphically, for an tatically determined tructure (a threehinged arch) and will give an approximate hape of the funicular configuration, a it not conidered, tree produced by the elatic deformation of the tructure. Fig 3.1 Alo equally, we have to take into account, that the funicular correpond to a certain loading cae. A the bridge i ubjected to varying condition of traffic live load, we have to chooe the tate of loading more repreentative or critical to find a funicular, uch that in the other cae of loading, the funicular divert the leat poible from the directrix choen. To find the locu of the funicular, we aumed the arch of Fig. N 3.1 The triangle of force correponding to load in the egment of arch x: dy Q OA dx dy Q dx d dx dy x OA' dx The difference of thee value OA and OA, give a reultant of load q x x Then: Q d dx dy x OA' OA qxx dx Latly, the equation of the funicular i: d y Q dx q x a) Funicular for an uniform load q: Being q x = q, uniform along the length of the arch: Then:, 3 OSCAR MUROY 18

20 d y Q q dx Solving the differential equation and the condition, x =, y = and y = : qx y Q We have alo, for x = l/ and y = f: ql f The iotatic thrut being: 8Q Then, the funicular equation i a parabola: 4 f x l y, ql Q, 8 f b) Funicular for a load varying parabolically: Let q x the parabolic varying load: q x q c x l / q q a c q c q q a c 4 x l Then, in the equation of the locu: d y Q dx q c q q a c 4 x l Solving the differential equation and the condition, x =, y = and y = : qc qa q y Q 3Ql c x x Fig 3. From the condition, x = l/ and y = f: f qc qa q Q 3Ql l l 4 4 c l the iotatic thrut i: Q 5 48 f q a q c Replacing thi value of Q in the funicular equation: y q a 8 f 5q c 3q c q a q c x l x l Thee equation are equally valid for the cae of load diminihing to the pringing, 3 OSCAR MUROY 19

21 c) Funicular for a load varying imilarly to the directrix: qa When the load increae toward the pringing: 1, q c Let the load be imilar to the directrix curve: q x q c q y a q f Then, in the locu equation: d y Q q dx c q y a c q f c Fig. 3.3 or: d dx y q a q fq c qc y Q The general olution of thi differential equation without the econd term i: y C coh qa q Qf c x And the particular olution of the equation with the econd term: y C 1 Then the general olution of the differential equation with the econd term i: qa qc y C coh x C 1 Qf From the condition: x =, y = and y =, differentiating and replacing in the differential equation, we obtain: C C1 Doing: qc qa qc qc qa qc y f coh x 1 qa qc Qf f qa m 1, and q c k qa q Qf c l From the condition: x = l/, y = f, we obtain: 1 f f coh k 1, the relation between m and k being: m coh k m 1, 3 OSCAR MUROY

22 Latly, uing thee relation and: 1 y f m 1 coh k 1 x, we have: l / And the iotatic thrut: Q l k q a f q c d) Funicular for a load varying imilarly to the directrix: qa When load diminihe to the pringing: 1, q c Let it be the load imilar to the directrix: qc qa qx qc y f Then, for the locu equation: d y Q q dx c q y c q f a Fig. 3.4 or: d y qc qa dx fq qc y Q General olution of thi differential equation without econd member i: y C co qc q Qf a x And the particular olution of the equation with the econd member i: y C 1 Then the general olution of thi differential equation with econd member i: qc qa y C co x C 1 Qf From the condition: x =, y = and y =, differential and replacing in the differential equation, we obtain Doing: C C1 qc q q c a f q c qc qa y f co x 1 qc qa Qf, 3 OSCAR MUROY 1

23 qa m 1, and q c k qc q Qf a l From the condition: x = l/, y = f, we obtain: 1 f f co k 1, the relation between m and k i: m co k m 1 Latly, uing thee relation and: 1 y 1 m f 1 co k x, we have: l / And the iotatic thrut: Q l k q e) Circular funicular: c q f a The circular funicular or circular egment correpond to a tate of uniform radial preure q: Then, in thi cae, correpond to load tate with vertical load: q v = q in and horizontal load: q h = q in Fig 3.5 Next it i hown fig. 3.6 and 3.7, where it ha been obtained for arch bridge, the directrixe for a parabolic, circular, 4th order parabolic and the hyperbolic coine or trigonometric coine a the cae maybe. x y (co hip) y (parab 4 th o) y (parab) y (circular) , 3 OSCAR MUROY

24 Y ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES Table 3.1 Built-in arch 65m pam and 13m rie Built-in arch 65m pam and 13m rie co hip parab 4 g parab circular X Fig. 3.6 x y (parab) y (parab 4th o) y (coeno) y (circular) Table 3. Two-hinged Arch, 9m Span and 17m Rie, 3 OSCAR MUROY 3

25 Y ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES Two-hinged Arch, 9m pan and 17m rie parab parab 4 g coeno circular X Fig. 3.7, 3 OSCAR MUROY 4

26 4. VARIATION OF THE ARCH DEPTH The hape and the depth of the cro ection of the arch are determined by the ection deign, in uch a way that the tree produced by the mot unfavourable combination of bending moment and axial force acting on the ection, would not exceed the permiible ervice load tree or atified the factor of afety at ultimate load. The maximum and minimum bending moment due to tranit live load and the correponding axial force are the condition which in mot cae define the dimenion and other parameter of the cro ection. Two of the mot important and critical ection of the arch, are the pringing and the crown ection. Then normally we tart determining the parameter of the ection at thee location, and for thee purpoe it i ueful to take reference to already built bridge, a it lited in table 1. and 1.3. Having been defined thee two ection, we can aume a progreive variation of the ection between thee two point, graphically, or by mean of equation, to complete the geometry of the arch a) Two-hinged Arch: Among the different propoal to define the depth variation h, it could be mentioned the following: 4 Parabolic variation: hx hc px, being: p h c h a l I c Variation according to Chalo from the Ecole de Pont et Chauée: I x 5 x 1 k l I c being: k 1 I a Variation proportional to the inertia of the arch ection I: I c being: k 1 I a Variation proportional to the depth of the arch ection h: hc being: k 1 h a I h x x x I a k co 1 l x ha k co 1 l, 3 OSCAR MUROY 5

27 h ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES x h (parab) h (coine Ix) h(coine hx) h(chalo, n=5) Table 4.1 Two-hinged Arch, 9m Span and 17m Rie.1 Variation of h parab coeno Ix coeno hx Chalo, n= X Fig. 4.1, 3 OSCAR MUROY 6

28 b) Built-in arch: Among the different propoal to define the variation of the depth h, it could be mention the following: Parabolic variation: h x hc px, being p h a h c 4 l Variation according to Chalo, family of equation = 1,, 3 ó 4 I x I c x 1 k l n, being k I I c 1 y n Variation according to Straner, the inertia I inverely proportional to coine: I x I c x co 1 l, being 1 I a I c co r Variation according to Straner, the depth h, inverely proportional to coine: a h x hc x co 1 l 1/3, being I c 1 3 I a co h x h (parab) h (Straner1) h (Straner)(Chalo,n=1) Table 4. Built-in for 65m pan and 13m rie, 3 OSCAR MUROY 7

29 h ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES Variation of h parab Straner 1 Straner Chalo, n= X Fig. 4. Variation according to Chalo x h (parab) h (n=) h (n=3) h (n=4) h (n=1) Table 4.3 Two-hinged Arch, 65m pan and 13m rie, 3 OSCAR MUROY 8

30 h ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES 1.7 Variation of h parab Chalo, n= Chalo, n=3 Chalo, n= 4 Chalo, n= X Fig. 4.3, 3 OSCAR MUROY 9

31 5. TYPICAL CONFIGURATIONS OF ARCH BRIDGES Arch Bridge are economically competitive, from the 5 m pan upward in concrete arche and larger for teel bridge, due to a cotlier contruction procedure and the arch in itelf i an element more to build beide the bridge deck, o in lower limit of thee pan, an economical comparion hould be made with beam or frame alternative The typical baic configuration of arch bridge that are contructed nowaday, belong largely to built-in and two-hinged arche and for the relative poition of the bridge deck, in upper, intermediate and lower deck arche. In the next figure ketche of thee configuration are hown. a) Two-hinged Arch b) Built-in Arch Fig 5.1 There i very little number of three-hinged arche, although one of bet known and depicted in any anthological review of Bridge i the Salginatobel Bridge, deigned by the Swi Engineer Robert Maillart, built in 193. hinged Fig 5. One-hinged arche do not repreent any tructural advantage repect to the other type and there are not known bridge of thi type. However, thee two lat configuration, have been ued a a temporary tage of contruction, before a contruction technique called compenation of arche i applied. Thi technique will be explained afterward. In relation of Arch Bridge, it hould be ditinguihed when the arch i of a tru or lattice contruction, which could be conidered a a peudo-arch, becaue although it hape correpond to an arch, tructurally it i analyed more properly a a tru. Fig. 5.3, 3 OSCAR MUROY 3

32 Fig 5.3 With the extraordinary advancement of the Structural of Analyi, which ha broaden the cope of computable tructural type, have emerged a large number of variant of thee baic configuration. Uually, the tructure of the Arch Bridge i compoed of parallel arche in the width of the deck or it a lab type arch, with the width of the deck. A variant, in thi repect, are the configuration with arche, in loped plane approaching or converging at the crown zone. Fig 5.4, 3 OSCAR MUROY 31

33 When for reaon of poor oil or being an intermediate pan over elevated upport, that ha no capacity to take large lateral thrut of the lower deck arch, it i convenient to adopt the tructural cheme of a tied arch. Variant of the tied arch are the ue of lateral emi arche and compreion trut, which reduce the thrut or thi i tranferred far apart to a afer zone c Fig 5.5 Latly, the hanger or column of the arch are in mot cae vertical. Variant in thi apect, are the loped hanger or even interlaced hanger and column with triangular arrangement of the column. Fig 5.6, 3 OSCAR MUROY 3

34 For maller than 4m pan, it ha been deigned arch bridge with filled pandrel, of reinforced concrete arche, although in thee cae an economic comparion hould be made with the frame or beam olution. Fig 5.7, 3 OSCAR MUROY 33

35 6. CONSTRUCTION PROCEDURES FOR ARCH BRIDGES In a large proportion of cae, arch bridge are built over deep ravine or over permanent water coure, with the additional problem of being a waterway which will make cotlier or even unviable for conventional contruction uing fale work upported on the terrain. Fig 6.1 From thee ituation, it emerge naturally the idea to contruct from above. Thi type of contruction procedure, ha gained a general acceptance for many year, and the mot pread that could be adapted to the national realitie could be mentioned; it i the ue of cable tay to upport the tructure or fale work temporally during the contruction procedure and the ue of contruction traveller, while advancing the contruction tage., 3 OSCAR MUROY 34

36 Fig 6. The ue of thee contruction procedure implie a tight involvement to them with the analyi and deign proce, a the tructure hould be deigned for the different contruction tage and at the ame time the contruction procedure mut be executed o a to agree to the foreeen behaviour for the tructure, in it different tage of the contruction. Fig 6.3, 3 OSCAR MUROY 35

37 7. SECOND ORDER DEFLECTIONS IN ARCHES Deflection w, u and v in the Navier Bree equation, are obtained from an undeformed geometry of the arch, by auming that the deflection are mall and can be neglected, and being a yet unknown the arch deformation. With pan urpaing the 1 m, (at thi time Arch Bridge larger than a 5 m pan have already been contructed), it become neceary to calculate the real deformation, from the deformed hape of the arch, when applying load. Thi i particularly ignificant in arche, a when deforming the arch, rie diminihe and conequently the compenating moment due to horizontal thrut, deriving into larger bending moment Angular Diplacement: M w w d (7.1) EI Horizontal Diplacement: u u w y wy M N T d 1. co d in d (7.) EI EA GA 1 Vertical Diplacement: v v w x wx M N T d. in d co d (7.3) EI EA GA 1 The real deformation determination could be made by ucceive approximation a follow: We hall call x (x), y (x), (x) to the initial geometry, from which we obtain the force N, T y M Applying the equation 7.1 to 7.3, we obtain the elatic deformation, which we will call w 1 (x), u 1 (x) y v 1 (x) Then the firt approximation of the deformed hape, would be: y1( x) y( x) v1( x) x ( x) x( x) u1( 1 x 1( x) ( x) w1 ( x ) ) Deformation in the x direction can be negligible, in comparion to the element dimenion of the arch and it won t be taken into account further. With the arch deformed geometry; we would obtain the corrected value N 1, T 1 and M 1 With thi new geometry and the applied force, we would get a econd et of deformation for the tructure w (x), u (x) and v (x) The econd approximation of the deformed hape would be: y ( x) y( x) v ( x) ( x) ( x) w ( x) With thi new deformed geometry of the arch, we correct again the value N, T and M Proceeding in thi way, we would get after n iteration w n (x), u n (x) and v n (x):, 3 OSCAR MUROY 36

38 The nth time approximation of the deformed hape would be: y ( x) y( x) v ( x) n ( x) ( x) w ( x) n n n So we would get a erie of value y 1 (x), y (x), y 3 (x),..., y n-1 (x), y n (x) and of 1 (x), (x), 3 (x),..., n-1 (x), n (x) In a table tructure, for the loading that i ubjected, thee erie of value are convergent to the final value of the deformed hape. And latly the force, taken into account the deformed hape of the tructure would be N n, T n and M n A an example, we hall examine the cae of an arch of 6m pan, teel with 6cm depth, ubjected to concentrated load of dead weight, a per fig. N 7.1: Fig 7.1 For the cae of two-hinged arch, the ucceive deformation obtained are a hown in table N 7.1 and Fig. N 7. Computation for the final deflection, ha been repeated until relative error of.1, i reached, which ha been obtained after 3 iteration. Final deflection are in thi cae, therefore, larger: In the maximum poitive deflection, at 7.5m from pringing: 5.99/3.338=1.53 In the maximum negative deflection, at crown /1.911=1.46, 3 OSCAR MUROY 37

39 Mf (T.m) v (cm) ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES VERTICAL DEFLECTIONS DUE TO DEAD WEIGHT 'v1' 'v' 'v3' 'v4' -18 X (m) Fig 7. BENDING MOMENTS DUE TO DEAD WEIGHT 'MF1' 'MF' 'MF3' 'MF4' -6 X (m) Fig 7.3 Final ucceive Bending Moment are a hown in Table N 7.1 and Fig. N 7.3 Final Bending Moment are in thi cae, therefore, larger: In the maximum negative bending moment at 9.m from pringing: 48.85/34.81=1.4 In the maximum poitive bending moment at the crown: 39.53/8.13=1.41, 3 OSCAR MUROY 38

40 X v1 v v3 v4 X Mz1 Mz Mz3 Mz4 (m) (cm) (cm) (cm) (cm) (m) (T.m) (T.m) (T.m) (T.m) Table 7.1 Vertical Deflection and bending moment in the two-hinged arch For the cae of built-in arch, ucceive deflection obtained are a hown in table N 7. and Fig. N 7.4 Computation for the final deflection, have been repeated until a relative error of.1, which wa obtained with only iteration., 3 OSCAR MUROY 39

41 Mf (T.m) v (cm) ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES VERTICAL DEFLEXIONS DUE TO DEAD WEIGHT 'v1' 'v' 'v3' X (m) Fig 7.4 BENDING MOMENT DUE TO DEAD WEIGHT 'MF1' 'MF' 'MF3' X (m) Fig 7.5 Final deflection are in thi cae, therefore, larger: In the maximum poitive deflection at 9.m from pringing: 1.65/.87=1.57 In the maximum negative deflection at crown 8.61/6.678=1.1 Final ucceive Bending Moment are a it i hown in Table N 7. and Fig. N 7.5 Final Bending Moment are in thi cae, therefore, larger: In the maximum negative bending moment at 9.m at pringing 3.3/.1=1.17 In the maximum poitive bending moment at crown, 3 OSCAR MUROY 4

42 4.83/.75=1. In the maximum poitive bending moment at pringing 47.4/4.87=1.15 X v1 v v3 X Mz1 Mz Mz3 (m) (cm) (cm) (cm) (m) (T.m) (T.m) (T.m) Table 7. Vertical Deflection and bending moment in a built-in arch, 3 OSCAR MUROY 41

43 8. COMPENSATION OF ARCHES Thi i a contruction procedure aimed to incorporate a favourable tate of tree to improve the tructural behaviour of the arch. In the pat it ha been ued to decentre or remove the fale work for it diaembly. For an arch which will become a build-in type, it could be embedded one or two joint. In an arch which will latly become a two-hinged type, it could be embedded a joint at the crown. There are alo two way to execute thee temporary joint: one i to effectively build a hinge, in a tage of contruction and then afterward retore the monolithim of the hinge and o the capacity to withtand the bending moment. The econd way i to inert flat jack in the joint, and jacking up to introduce controlled compreive force, to generate a favourable tate of force for the improved behaviour of the tructure. Fig. 8.1 Applying thee concept for a build-in arch bridge of 65m pan, we hall examine the variation of moment and the eccentricity of the axial force due to permanent load (elfweight + dead weight) a) When it i built temporary hinge in the pringing and we have therefore a two hinged arch temporarily for the permanent load. BENDING MOMENTS AXIAL FORCE X PP PM PP+PM X PP PM PP+PM exc (m) (T.m) (T.m) (T.m) (m) (T) (T) (T) (cm) , 3 OSCAR MUROY 4

44 Table 8.1 Bending Moment and eccentricity of the axial force in a two-hinged arch, 3 OSCAR MUROY 43

45 Mf (T.m) ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES BENDING MOMENT CALIBRATION PP PM CAL X (m) Fig. 8. b) When it i contructed temporary joint: one at crown or two in the fourth pan, inerting hydraulic jack to generate a total horizontal diplacement of.6cm BENDING MOMENTS AXIAL FORCES X PP PM TEMP (*) CALIB X PP PM TEMP (*) CALIB exc (m) (T.m) (T.m) (T.m) (T.m) (m) (T) (T) (T) (T) (cm) , 3 OSCAR MUROY 44