Load carrying capacity of masonry bridges: numerical evaluation of the influence of fill and spandrels

Transcription

1 Load arrying apaiy of masonry bridges: numerial evaluaion of he influene of fill and spandrels A. Cavihi and L. Gambaroa Deparmen of Sruural and Geoehnial Engineering, Universiy of Genova, Ialy ABSTRACT. The onribuion of arh-fill ineraion o he load arrying apaiy of masonry bridges an be large [1]. While a omprehensive desripion of arh-fill-spandrels ineraion requires a hree-dimensional model, a simplified desripion of fill onribuion o he load arrying apaiy of he bridge an be implemened in wo-dimensional models. This is he way followed for insane in Hughes e al.[2] and by Bianić e al. []. In Cavihi and Gambaroa [4] fill is desribed as a oninuum in plane srain. This reduion of fill o a plane model presens a riial aspe whose effes have o be evaluaed: he dependene of he in-plane fill resisane on he onaining apaiy of he spandrels. This aspe is here onsidered in an approximaed way by desribing fill as a oninuum in a plane sae where he ou of plane normal sress is onsrained in an admissible domain represening he limied onaining aion of he spandrels. A saially admissible wo-dimensional model is defined and he Sai Theorem of Limi Analysis is applied. 1. INTRODUCTION One of he mos relevan problem in he assessmen of masonry bridges is he evaluaion of he load arrying apaiy inluding he srenghening effe of fill ineraing wih spandrel walls. This irumsane has been highlighed by experimenal ollapse ess on full sale and model sale bridges [1]. Even if a omprehensive analysis of he ineraion beween arh, fill and spandrels would all for deailed hree-dimensional models [2], in some ases, when ransverse effes an be negleed and spandrel walls are no onsidered, he sruural sysem an be desribed by a plane model. This makes i possible o assume simplified onsiuive models for masonry, hereby reduing he soures of unerainies; moreover, he bridge behaviour an be more synheially desribed. Two-dimensional models based on he no-ension assumpion for masonry, originally proposed in he pioneering works by Casigliano [], Kooharian [4] and Heyman [5], have been exended by various auhors o ake ino aoun fill ([6]-[1]). Cavihi [11] and Cavihi and Gambaroa [12-1] have proposed a model in whih he fill is desribed as a ohesive-friional oninuum ineraing wih he arh-pier sysem represened by no-ension, perfely plasi beams; upper esimaes of he ollapse load and he orresponding ollapse mehanisms have been obained. The unonservaive naure of he resuls from he kinemai approah may represen a problem. Aim of his paper is o overome his limiaion by applying he lower bound heorem of limi analysis o a saially admissible FE disreizaion of he mehanial model defined in Cavihi and Gambaroa [1] and ake ino aoun he effes of a limied ransverse onsraining aion of he spandrel walls on he fill by inroduing a ondiion whih limis he ransverse ompressive sress under an admissible value; his feaure provides a simplified

2 61 ARCH 7 5h Inernaional Conferene on Arh Bridges means o evaluae he dependene of he ollapse load on he ransverse effes and release and disuss he plane srain assumpion [14]. 2. PLANE EQUILIBRIUM MODEL OF THE ARCH BRIDGE A longiudinal ross seion of a ypial masonry bridge is diagrammaially desribed in Figure 1.a; he sruural model desribed in Figure 1.b is assumed, in whih he spandrel walls are no onsidered. The volume oupied by fill is represened by he wo-dimensional domain Ω f, orresponding o a longiudinal seion of fill, while arh barrels and piers are desribed by plane urved beams. The piers a he base and he arh barrels a springings are onsidered buil in and he onneions beween he arh springings and he op of he piers are assumed rigid. The fill is resrained a he opposie ends of he bridge. p i I Fill Haunhing i L Ω f b Ω f Abumen (a) Arh barrel Pier O Fig. 1. (a) Longiudinal ross seion and wo-dimensional model of he bridge. The exernal fores, represened by self weigh b and he line raions p over he upper boundary of he fill, are assumed parallel o he plane of he longiudinal ross seion of he bridge and uniform aross he widh.. The load veor p is deomposed as p=p + sp, where p is fixed and sp represens he live load, s being he muliplier of he referene live load p. Among he hree-dimensional sress fields in equilibrium wih he presribed loads, only plane fields are onsidered. The ou-of-plane sress omponen σ is indued by smooh boundaries whih represen he effes of eiher he spandrel walls or he ie rods insered in he fill and onneing he opposie spandrel walls o srenghen masonry bridges [15]. A simple desripion of he sress field in arh barrels and piers is obained by ignoring he membrane and shear fores and he bending momen aing on planes parallel o he longiudinal ross seion and he wising momen as well. Boh fill and masonry arhes and piers are assumed made of no-ension rigid perfely plasi maerials. Fill is assumed as a Mohr-Coulomb ohesive-friional no-ension maerial. A onsrain on ransverse sress is assumed σ σ%, where he limiing value σ% ( ) an represen eiher he presribed maximum allowable pressure on he spandrel walls or he maximum allowable ensile srengh of he ie rods per uni area. The resuling ondiions of plasi admissibiliy [14] are expressed in he form σ1 σ2 + ( σ1 + σ2) sinϕ 2 osϕ, σα, σ α +σ rσ%, α = 12,, (1)(2)() σ = 2osϕ ( 1-sinϕ ) and σ = 2osϕ ( 1+sinϕ ) being he uniaxial ompressive and ensile srengh, respeively, and r = σ σ = ( 1+sinϕ) ( 1-sinϕ ). The admissible range of σ is obained a poseriori [14] as a funion of σ 1 and σ 2 ; he orresponding minimum admissible ompressive value is (a) if min ( σ α, α = 1, 2 ) σ, hen σ = ; else σ =σ + 1 r min ( σ α, α= 1, 2). (4)(5) The yield funion of he ross seions of arh barrels and piers is obained by assuming vanishing ensile srengh aross he morar joins orhogonal o he enre line and rigid-ideal plasi response under ompression. Sliding failure is negleed. x 2 x 1 b b b b i Ωb

3 A. Cavihi and L. Gambaroa 611. FINITE ELEMENT EQUILIBRIUM MODEL The esimaes of he ollapse load muliplier s of he model are obained by a finie elemen appliaion of he Lower Bound Theorem of Limi Analysis. The fill is approximaed by hreenode riangular elemens and sress disoninuiies are allowed a heir shared edges [16]; arhes and piers are approximaed by sraigh beam elemens (Fig. 2). The presene of he disoninuiies allows for enrihing he sress field. l h k - β α α β α x 2 x 1 O β (a) Fig. 2. (a) Finie elemen model of he bridge; he finie elemen equilibrium model: arh-fill ineraion raions. The sress field in riangular elemens is assumed linearly dependen on he nodal sresses, so ha he in-plane differenial equilibrium equaions of he -h elemen are expressed as linear h h k k l l equaions of he nodal sresses in he marix form Cσ + Cσ + Cσ + =, where he veor depends on he applied load. In order o preserve he lower bound propery of he resul, he ondiions of plasi admissibiliy (1)-() are pieewise linearized in he spae of sress omponens by an inernal polyhedron obained as inerseion of hree polyopes wih an even number p of faes. The orresponding pf = p inequaliies are olleed in marix form and imposed a nodes, r T r f (σ )=N σ r, r= h, k, l. Arhes and piers are disreized by wo-node sraigh beam elemens having onsan seion heigh h and self-weigh per uni lengh b b. Eah beam elemen disreizing he arhes ineras wih he edge of a riangular elemen of fill (Fig. 2b); he generalized fores aing on he beam axis are hen obained by simply aking ino aoun he effe of he beam heigh on he bending momen. The lineariy of he aions oming from he fill allows for wriing he overall i i j j h h k k equilibrium equaions of he beam in he marix form Cs b b + Cs b b + C b σ + C b σ + b =, h k where σ and σ are he nodal sresses of he edge of he riangular elemen and veor b depends on he applied body fore b b (see Fig. 2b). To obain a linear formulaion, he plasi limi envelope in he ( NM, ) plane is pieewise linearised by an inernal polygon and imposed a a disree number p s (ps 2) of seions. The laer approximaion allows a violaion of equilibrium beween he disree seions ha an be heked a poseriori; however, wih he mesh refinemen neessary for a good desripion of fill, he effe of his approximaion was negligible even assuming ps 2 in he examples shown in he nex seion. The generalized nodal sresses in he riangular and beam elemens, olleed in veor σ, and he load muliplier s are he disree variables whih define he finie elemen equilibrium model. The equilibrium equaions are olleed in he linear marix equaion Cσ =. The boundary ondiions on he nodes where aive and reaive fores are applied are expressed in he linear form Qσ sq= q, where q and sq are he nodal sress veors depending on he applied dead and live loads, respeively. Finally, he ondiions of plasi admissibiliy are olleed in he marix inequaliy f = N σ r. The larges lower bound s T on he ollapse load muliplier is obained as he soluion of he Linear Programming problem. β β i j

4 612 ARCH 7 5h Inernaional Conferene on Arh Bridges T T T ( α1 q α2 r α ) s = min + +, T T C α + + =, 1 Q α2 Nα T q α2 = 1, α. (6) 4. EXAMPLE OF APPLICATION:PRESTWOOD BRIDGE The firs example refers o Preswood bridge, a single span bridge esed up o ollapse [17] wihin he experimenal researh on masonry bridges suppored by he Transpor Researh Laboraory (TRL). The geomery of he bridge and he posiion of he live load are desribed in Figure.a. The experimenal arh ollapse mehanism exhibis four hinges; he mehanism akes plae wih negligible maerial rushing and he experimenal ollapse load is P = 228kN. exp b = P a= 1 4l hr = 165 h = s Loaion of piles Vl Lef plinh 545 Righ plinh 25 Vr Loaion of piles f =1428 (a) l = 2 l = 655 l = 2 Fig.. (a) Preswood bridge model (mm). Posiion (mm) of he plinhs of he loading sysem and of he piles a abumens. Figure 4 shows he numerial resuls obained by assuming plane srain fill and he mehanial parameers given in Table 1; he values of he angle of inernal friion ϕ and he ompressive masonry srengh σ ome from experimenal evaluaions [17], while daa abou ohesion furnished in [17] are no suffiien and he assumed value is suh ha he experimenal ollapse load Pexp = 228kN is approximaely he average beween he equilibrium ( Psrain = 29kN ) ub and he ompaible ( Psrain = 254kN ) soluions. An analysis of he sensiiviy of he resuls on he mehanial parameers has been arried ou and will be disussed shorly. P η (a) () 1 1 max max 1 2 max 1 max 1 4 max () Fig. 4. (a) Collapse mehanism wih non resisan fill ( P nrf = 46.2kN ); in-plane prinipal sress field and line of hrus (plane srain fill) ( Psrain = 28.8kN ); () ollapse mehanism and onour plo of he maximum shear srain rae from ompaible model (plane srain fill) ( P = 254.5kN ). ub srain

5 A. Cavihi and L. Gambaroa 61 Table 1 Preswood bridge: onsiuive parameers; experimenal and numerial ollapse loads. Masonry densiy Masonry ompr. srengh Fill densiy Angle of in. friion m 2 kn/m Exp. ollapse load P exp 228kN σ 4.5MPa Non res. fill oll. load P nrf 46kN f 2 kn/m Plane srain P srain 29kN o ϕ 7 Plane srain ub P 254kN Figure 4.a shows he ollapse mehanism obained by assuming heavy bu non resisan fill o and disribuing he live load on he arh by assuming η= ; he dark irles in he arh are he generalised hinges. The orresponding ollapse load is Pnrf = 46kN and agrees wih he resul obained by Crisfield [9]. The limi sress sae and he ollapse mehanism [1] obained by assuming plane srain fill are shown in Figure 4.b and 4., respeively. In Figure 4.b he hik line in he deph of he arh represens he hrus line, while he sraigh lines in he fill represen he prinipal direions of he sress field; he onour plo in he fill in Figure 4. shows he maximum shear srain rae field. A good qualiaive agreemen beween he resuls from he equilibrium and ompaible models is obained. The limi sae desribed by he numerial resuls appears o well reprodue he experimenal behaviour; in pariular, he loaion of he hinges from he analysis are a good approximaion of he experimenal posiions. The onour plo of he minimum ompressive sress field σ is shown in Figure 5. The minimum values are reahed in small regions loaed below he live load and a springings; he in-plane sress field in he regions in whie is ompaible wih plane sress fill. ub srain (kpa) (kn) σ% σ (kpa) Fig. 5. (a) Conour plo of he minimum ompressive sress σ. Influene of he allowable ransverse sress σ% on he lower esimaes of ollapse load (bold line); esimaes obained by imposing plain srain under he punh (hin line); esimaes obained by assuming heavy non resisan fill (dashed line). The effes of he maximum allowable ransverse sress σ% on he lower esimaes of he ollapse load are shown in he graph of Figure 5b (bold line). For σ% 6kPa he plane srain limi sress sae is obained; for σ % < 6kPa he ou of plane limi ondiion affes he resul unil he value P = 57kN obained by assuming σ % = kpa (plane sress). sress P u (kn) UB Exp LB Non resisan fill (kpa) o σ = 4.5MPa, ϕ= 7 ( ) P u (kn) Exp UB LB Non resisan fill ϕ ( σ = 4.5MPa, = 1kPa) P u (kn) Exp UB LB Non resisan fill σ (MPa) o ( = 1kPa, ϕ= 7 ) Fig. 6. Effes (a) of he ohesion, of he angle of inernal friion ϕ and () of masonry ompressive srengh σ.

6 614 ARCH 7 5h Inernaional Conferene on Arh Bridges The equilibrium soluion orresponding o σ % = 2kPa is Pu = 149kN and he sress field, no shown, is similar o he ase of plane srain fill, wih lower values of σ. The limi sae due o he limiaion of he ransverse sress σ is firs reahed jus under he live load, so ha he load is equilibraed by a uniform disribuion of he maximum ompressive sress σ r σ%. The dereased esimae of he ollapse load is herefore due o a loal effe; his aspe is well shown by he resuls obained by keeping he one under he live load in plane srain and varying σ% hrough he res of he fill; he lower bound on he ollapse load (Figure 5b, hin line) dereases only for very low values of σ%. The effes of he ohesion and he angle of friion ϕ of he fill on he lower and upper bound esimaes of ollapse load by assuming plane srain are shown in he graphs of Figure 6.a and 6.b, respeively; he effe of he masonry ompressive srengh σ is shown in Figure 6.. The onribuion of he fill resisane o he ollapse load esimaes is imporan also for low values of ohesion and friion. The effes of a derease in he ohesion are limied, and he lower esimae of he ollapse load is sill higher han 15 kn by assuming = 5kPa. Finally, he effes of he masonry ompressive srengh σ are negligible only for σ 6MPa. The pariular proedure followed o apply he load during he experimenal es [17] makes i neessary o evaluae is possible effes on he resuls. Dead load o provide reaion for he jaks was provided by onree bloks suppored above he bridge by hree beams resing on onree bolk plinhs a eah abumen (Fig..b). The verial fores V l and V r (Figure.b) applied on he fill depend on he weigh of he onree bloks (8kN) and he supporing sruure and on he fore exered by he jaks. The resuls obained by inroduing hese fores in he numerial simulaion have shown ha heir effe is negligible. The analysis has shown ha only large inremens of he weigh of he loading sysem an signifianly affe he resuls. Finally, an analysis of he effe of he fill onsrains a he abumens has been arried ou by onsidering a redued domain of he fill, orresponding o he shaded region in Figure.b. Also in his ase, he differenes in he ompaible and equilibrium soluions provided by he model in plain srain ondiion wih respe o he resuls obained by he referene one are less han 1%. This resuls shows ha, for his pariular load ondiion, he inroduion in he simulaion of he piles presen a he abumens would no have affeed he resuls. 5. CONCLUSION A wo-dimensional saially admissible finie elemen model and a numerial proedure have been presened o analyse he ollapse behavior of masonry bridges aking ino aoun he ineraion beween he arh-pier sruural sysem and fill.. The effes of a limied ransverse srengh of spandrel walls on fill and, as a onsequene, on ollapse load are aken ino aoun by inroduing a simplified ondiion ha limis he ransverse ompressive sress in he fill. The proedure has been applied o simulae an experimenal ollapse es [17] ha allowed for a omparison beween he numerial and experimenal resuls. The resuls obained by varying he admissible ransverse sress have highlighed he imporan influene of his parameer on he ollapse load and he validiy limis of he plane srain assumpion. AKNOWLEDGEMENTS The auhors aknowledge finanial suppor of he (MURST) Ialian Deparmen for Universiy and Sienifi and Tehnologial Researh in he frame of he researh proje PRIN 2/ 24 Proje Safey, mainenane and managemen of Masonry Bridges. REFERENCES [1] Page J., Masonry Arh Bridges - TRL Sae of he Ar Review, HMSO, 199. [2] Fanning P. J., Boohby T.E., Three-dimensional modelling and full-sale esing of sone arh bridges, Compuers and Sruures, 79, , 21.

7 A. Cavihi and L. Gambaroa 615 [] Casigliano C.A.P., Theorie de l equilibre des sysemes elasiques e ses appliaions, Auguso Federio Negro, Torino, 1879; Andrews E.S., Elasi sresses in sruures. Translaion by Andrews, So Greenwood & Son, London, [4] Kooharian A., Limi analysis of voussoir (segmenal) and onree arhes, Pro. Am. Conr. Ins., 89, 17-28, 195. [5] Heyman J., The sone skeleon, In. J. Solids Sruures, 2, , [6] Crisfield M.A., Finie elemen and mehanism mehods for he analysis of masonry and brikwork arhes, Transpor Researh Laboraory, Crowhorne, Researh Repor 19, [7] Choo B. S., Couie M. G., Gong N. G., Finie elemen analysis of masonry arh bridges using apered elemens, Pro. Insn. Civ. Engrs., Par 2, 91, De , [8] Crisfield M.A., Pakam A. J., A mehanism program for ompuing he srengh of masonry arh bridges. Transpor Researh Laboraory, Crowhorne, Researh Repor 124, [9] Hughes T.G., Hee S. C., Soms E., Mehanism analysis of single span masonry arh bridges using a spreadshee, Pro.Ins. Civ. Eng. Sru. Build., 152, 41-5, 22. [1] Bianić N., Ponniah D., Robinson J., Disoninuous deformaion analysis of masonry bridges, Compuaional Modelling of Masonry, Brikwork and Blokwork Sruures, J.W. Bull Ed., , Saxe-Coburg Publiaions, Sirling, 21. [11] Cavihi A., Analisi limie agli elemeni finii di arhi murari inerageni on il riempimeno per la valuazione della apaià porane di poni in muraura, Ph.D. Thesis, Deparmen of Sruural and Geoehnial Engineering, Universiy of Genova, Ialy, 24 (in Ialian). [12] Cavihi A., Gambaroa L, Collapse analysis of masonry bridges aking ino aoun arh-fill ineraion, Engineering Sruures, 27, , 25. [1] Cavihi A., Gambaroa L, Two-dimensional upper bound limi analysis of masonry bridges, Compuers & Sruures, 84, , 26. [14] Cavihi A., Gambaroa L, Lower bound limi analysis of masonry bridges inluding arh-fill ineraion, Engineering Sruures, o appear, 27. [15] Oliveira D.V., Loureno P.B., Repair of sone masonry arh bridges, Pro. Arh Bridges ARCH '4, Barelona, , 24. [16] Sloan S.W., Lower bound limi analysis using finie elemens and linear programming, In. J. Numerial and Analyial Mehods in Geomehanis, 12, 61-77, [17] Page J., Load ess o ollapse on wo arh bridges a Preson, Shropshire and Preswood, Saffordshire, Deparmen of Transpor, TRRL Researh Repor 11, TRL, Crowhorne, England, 1987.

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