Structural Analysis of Historic Construction – D’Ayala & Fodde (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5

Load-carrying capacity of multi-span masonry arch bridgeshaving limited ductility

G. de FeliceDepartment of Structures, University Roma Tre, Rome, Italy

ABSTRACT: The assessment of load-carrying capacity of masonry bridges is usually based on limit anal-ysis methods, which implicitly assume that masonry has unlimited ductility capacity in compression; aimingat taking into account the effective ductility capacity of masonry, an approach that makes use of beams withfibre cross is developed. The approach is able to take into account the non-linear behaviour of masonry incompression, including the post-peak softening branch and therefore provides a reliable estimate of the ultimateload. A numerical investigation of one to three span masonry arch bridges under travelling concentrated load,shows that, by parity of constitutive assumptions, the method provides the same estimate of ultimate load givenby limit analysis, as well as the same kinematics at collapse. The influence of limited ductility is then investi-gated and the overestimate provided by classical limit analysis is highlight for the previously examined bridgetypologies.

1 INTRODUCTION

Masonry bridges have been thoroughly studied in thecourse of the last three decades, especially in GreatBritain, in United States and in Italy; several experi-mental investigations have been carried out on existingbridges and on scale models which have revealed someimportant aspects of their structural behaviour underservice loading and under ultimate load conditions;only a few experimental work have been carried outon multi-span arch bridges, among which the experi-mental investigation on scale models by Prentice andPonniah (1994), Melbourne et al. (1995) and Robinsonet al. (1997).

As far as modelling is concerned, a wide literaturederives from Heyman’s studies on the yield designof masonry arches (Heyman, 1966, 1982) amongwhich the contribution by Boothby (1992), Gilbert andMelbourne (1994), Clemente et al. (1995). The greatsimplicity of the model together with its feasibility forpractical applications, makes the yield design basedapproach one of the most suitable methods for anal-ysis of masonry structures; however, the use of suchmethod, point at the question of the limited ductilitycapacity of masonry. It is well known that masonry hasa brittle behaviour in traction and a strain softening incompression, with a decrease of load capacity afterthe peek load. Therefore, the yield design approach,which implicitly assumes a plastic behaviour of thematerial, may lead to an overestimate of the load capac-ity of the structure (Salençon, 1985). The classical

assumption of no- tensile resistance, allows over-coming this problem in traction, that however stillremain in compression, especially for shallow or longspan masonry bridges, which may be subjected tohigh loading with a relatively weak material strength,and therefore the possible failure due to crushingof masonry cannot be neglected. In these cases, thelimited ductility in compression can be handled byapplying a reduction factor to masonry strength, but itscalibration still remains one of the main drawbacks forthe use of yield design-based methods. An other draw-back to the application of yield design-based methods,stands in the lack of control for the displacement orstrain quantities, since the problem is driven by equi-librium and strength conditions and the kinematics isintroduced only through the virtual work equation. Onthe contrary, a displacement-based prediction of thestructural behaviour may be relevant for assessmentpurposes, especially under time-varying loading.

Some recent approaches have succeeded in mod-elling the behaviour of the bridge, such as the inter-action of filling with the arch barrel (Cavicchi andGambarotta 2005) and the contribution of the span-drel walls (Fanning and Boothby 2001), in line withwhat was observed in the experimental tests. Despitethe results achieved, an intrinsic difficulty remains inthree-dimensional finite element modelling, derivingfrom the high dependence on of the results on inputparameters and boundary conditions not easy to deter-mine. In any cases, full 3D non-linear approacheswould generally require a high computational effort,

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especially for multi-span bridges, which pertain to amore refined analysis and is not justified for initial orintermediate-level assessment.

The aim of this paper is to investigate the influenceof a limited ductility capacity of masonry on the load-carrying capacity of arch bridges. To this end, a non-linear incremental analysis tool is developed, along aresearch line that dates back to the well-known study ofCastigliano (1894) for the bridge over the Dora closeto Turin, and includes some recent contributions by(Boothby 1997; Molins and Roca 1998; Brencich andDe Francesco 2004). In the present paper, aiming attaking into account the effective behaviour of masonryin compression, the bridge is modelled using beamswith fibre cross section. The use of beams with fibrecross section has become current for the analysis ofreinforced concrete structures but, at the knowledge ofthe author, has not yet been used for structural analysisof masonry arch bridges.

The paper is organized as follows. In section 2the non-linear beam model with fibre cross sectionis briefly recalled, then, in section 3, a comparison ofthe results provided by the present approach and thosegiven by limit analysis is performed for one to three-span railway arch bridges under travelling concen-trated load; for the comparison, masonry is consideredeither with infinite compressive strength or finite com-pressive strength with unlimited ductility. Finally, insection 4, the influence of a limited ductility capacity isexamined, using the Kent & Park model for describingthe uniaxial constitutive behaviour of masonry, whenvarying the slope of the post-peak descending linearbranch. The numerical results, which provide the ulti-mate travelling load of the bridges, highlight the effectof material post-peak behaviour on the load carryingcapacity, showing to what extent the assumption ofunlimited ductility would lead to an overestimate ofloading capacity.

2 FIBRE BEAM MODEL

Some recent experiments on masonry prisms sub-jected to eccentric compression (Brencich and Gam-barotta 2005; de Felice, 2006) have shown that theclassical hypothesis that plane section remains planeafter deformation, holds true for brick masonry even inthe non-linear range, when stress concentrations andcracks start to develop; besides, the intrinsic difficultyin defining the non-linear constitutive behaviour of thebeam, as for instance the dependence of flexural hys-teretic behaviour on axial load, also due to the lack ofexperimental data, suggests deriving the section force-deformation relation from the uniaxial stress-strainbehaviour of the material. Therefore, looking for acompromise between simplicity and accuracy, the archis discretised into rectilinear beams and each beam

is subdivided into longitudinal fibres, whose stress-strain behaviour is defined according to the materialproperties that can be easily obtained through experi-ments.The non-linear constitutive relation of the beamsection is not given explicitly but is derived by integra-tion of the stress-strain relation of the fibres as follows(Spacone et al. 1996). According to the assumptionthat plane section remain plain and normal to thebeam axis x during the deformation history, referringto figure 1, for a given section generalized defor-mations d(x) = {χz(x) χy(x) ε(x)}, comprising the twocurvatures and the axial strain at the referenceaxis, the strain distribution over the section writes:ε(x, y, z) = a(y, z)d(x), where a(y, z) = {−y z 1}. Oncethe material constitutive relation is specified, thestrain distribution yields the tangent material modu-lus E(x, y, z) and the corresponding stress σ(x, y, z).Therefore, according to the virtual work equality, thesection stiffness matrix k(x) and the resisting forcesD(x) = {Mz(x) My(x) N (x)}, comprising the two bend-ing moment and the axial force, that correspond todeformation d(x) are obtained by integrating over thecross section A(x) as:

The evaluation of the integrals (1,2) is performedby subdividing the generic section x into n(x) fibresand applying the mid-point integration rule:

A flexibility-based formulation is adopted for theFE analysis, which ensures that equilibrium is strictlysatisfied within the element, and allows to overcomethe numerical instabilities arising after the onset ofstrain softening; the latter point is crucial in masonryelements, much more than in reinforced concrete ele-ments, since due to the lack of steel, the masonrysection displays a rapidly descending branch afterthe peak load. According to flexibility approach theelement stiffness matrix is obtained by inverting theflexibility matrix: K = F−1, where

L being the element length, and b(x) the force interpo-lation functions matrix, which is derived form the equi-librium of axial force and bending moments within theelement.

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Figure 1. Fibre beam model.

Figure 2. Two span bridges with deep (up) and shallow(down) arches.

An iteration scheme based on Newton-Raphsonmethod is used for solving the non-linear equation atthe element level, while at the structural level, an archlength method with Riks iteration scheme and adaptivesolution strategy is adopted.

3 LOAD-CARRYING CAPACITY OFMULTI-SPAN MASONRY ARCH BRIDGES:COMPARISON WITH LIMIT ANALYSIS

In this section, the load carrying capacity of one tothree-span arch bridges representative of Europeanrailway network (figures 2–3) has been estimatedaccording to the present approach that makes use offibre beam elements, and the results compared withthose provided by limit analysis. The same bridgetypologies studied in (Brencich and De Francesco2004) have been considered, having 15 mt span and0.75 mt arch depth; the bridges are made either withshallow (rise/span ratio equal to 0.25) or deep arches

Figure 3. Three span bridges.

Table 1. Bridge typologies characteristics.

Shallow DeepGeometrical and Mechanical arch archproperties bridges bridges

Span (at the intrados) [mm] 15000 15000Arch depth [mm] 750 750Fill in crown [mm] 750 750Masonry spec. weight [KN/m3] 22 22Fill specific weight [KN/m3] 24 24Rise (at the intrados) 3750 5250Pier height [mm] 7500 7500Pier depth [mm] 2200 2200

Figure 4. Load diffusion through the fill.

(rise/span ratio equal to 0.35); their geometrical andmechanical characteristics are reported in table 1.

The ultimate value of a concentrated travelling loadhas been evaluated for the classical hypothesis ofno-tensile resistance and either infinite compressivestrength, or 5 MPa compressive strength with unlim-ited ductility. As regards the application of travellingload, a diffusion angle trough the fill 2θ = 40◦ + 40◦was assumed, and the effective loading on the arch wasschematised as uniformly distributed (figure 4).

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Figure 5. Load carrying capacity of a single-span shallowarch bridge, under concentrated travelling load.

Figure 6. Load carrying capacity of a two-span deep archbridge, under concentrated travelling load.

Each arch was modelled by 100 non-linear beamswith cross section discretised in 100 fibres. The resultsobtained with the Fibre Beam (FB) model have beencompared to those obtained with Limit Analysis (LA)making use of the program Ring (Gilbert 2005). Thefigures 5–8, give the load carrying capacity of thebridge (one meter depth) as a function of load posi-tion for the different bridge typologies under study.All the analyses show an excellent agreement betweenthe results provided by the two approaches, by parity ofconstitutive assumptions, with differences that neverexceed 4%. When neglecting the compressive strengthof masonry, an overestimate of the ultimate load of

Figure 7. Load carrying capacity of a two-span shallow archbridge under concentrated travelling load.

Figure 8. Load carrying capacity of a three-span shallowarch bridge, under concentrated travelling load in the centralspan.

about 25% and 40% is obtained for deep and shallowarch bridges examined typologies respectively.

As regards the kinematics of the bridge close tocollapse, the formation of sufficient number of plastichinges that turn the bridge into a mechanism takesplace.

As shown in figure 9, the collapse mechanism of thebridges provided by FB approach is similar to the oneprovided by LA in terms of “hinges” location, apartfrom a different treatment of the kinematics of joint’sopening. In fact, while in LA approach no overlappingof the blocks in contact across a joint is allowed, in

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Figure 9. Collapse mechanisms of two and three-span archbridges obtained with Limit Analysis (up) and Fibre Beam(down) approaches respectively.

FB approach the arch deforms, also in compression,especially in plastic hinges regions, according to thestiffness of masonry.

It appears that for shallow arch bridges the col-lapse involves the pier and the adjacent span, while fortwo span deep bridges, only the loaded arch collapses,according to the four hinges mechanism, that does notinvolve the adjacent spans. This result is clearly influ-enced by the slenderness (the height-to-depth ratio)of the pier, as well as by the presence of the backfill,that acts as a restraint to the single arch mechanism,allowing a multiple arch mechanism to take place. Theinfluence of the backfill is highlight in figure 10, wherea three span arch bridge with deep arches is analysedwhen varying the height of the backfill form 1 to 3meters. in the case of fibre beam approach, the back-fill is modelled trough non-linear horizontal trussesable to carry compression only. the results also showa good agreement between LA and FB approaches.

4 LOAD-CARRYING CAPACITY OFMULTI-SPAN MASONRY ARCH BRIDGES:EFFECT OF A LIMITED DUCTILITY

The other curves in figures 5–8 provide the loadcarrying capacity of the bridge according to the FBmodel in the case of limited ductility; the results

Figure 10. Load carrying capacity of three-span deep archbridges, under concentrated travelling load, when varyingbackfill height.

have been obtained having made recourse to theKent and Park (1971) relationship for modelling theuniaxial constitutive behaviour of masonry in com-pression. Originally formulated for confined concrete,the Kent & Park (K&P) relationship consists of anascending branch represented by a second-degreeparabolic curve, a subsequent linear descending soft-ening branch and a final constant branch. The modelis defined by three parameters, namely the peak stressfc, the corresponding strain εc, the ultimate strain εu.Denoting by σ̃ = σ/fc, and ε̃ = ε/εc the normalizedstress and strain respectively, the three branches takethe form:

where η = εu/εc. As shown in a previous paper (deFelice, 2006) the K&P model is able to accuratelyreproduce the effective behaviour of masonry incompression.

Aiming at evaluating the influence of the post-peakbehaviour, a parametric study is performed, wherethe different curves in figures 5–8 correspond to dif-ferent slopes of the descending branch, from plasticbehaviour with unlimited ductility (α = 0◦, η = ∞), topost-peak brittle behaviour (α = 90◦, η = 1).

A significant reduction in loading capacity isobtained when increasing the slope of descendingbranch, more pronounced for shallow arches. Forthe two span arch bridges under study, a decrease inthe capacity of about 17% and 24% is found when the

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material behaviour turns from perfectly plastic to post-peak brittle, in the case of deep and shallow archesrespectively. An even stronger dependence is foundfor three-span shallow arch bridges, with a reduc-tion in load capacity of about 32%, from 385 kN/mto 260 KN/m.

These results can be compared with those presentedin (Brencich & de Francesco 2004) since at sectionlevel, the post-peak slope of softening branch can beassociated with an equivalent available ductility; how-ever, as far as the overall behaviour of the bridge closeto collapse is concerned, the differences in the constitu-tive assumptions would imply a different redistributionof the normal force and bending moment through thearches. It is worth noting that, while in (Brencich & deFrancesco 2004) the hinge mechanism does not takesplace since the arch collapses when the ultimate duc-tility capacity in the critical section is reached, in thepresent approach, thanks to a more accurate descrip-tion of the behaviour of masonry in compression, theanalysis can follows the progressive flexural strengthdecrease until the hinge mechanism forms.

5 CONCLUSIONS

An approach for estimating the load carrying capac-ity of multi-span masonry arch bridges is proposedthat makes use of a beam model with fibre cross sec-tion. A numerical investigation of one to three spanmasonry arch bridges under travelling concentratedload, shows that, by parity of constitutive assumptions(infinite strength in compression, or finite strengthwith unlimited ductility) the method provides the sameestimate of the ultimate load given by limit analysis,as well as a similar kinematics at collapse. How-ever, while limit analysis method implicitly assumesa perfectly ductile behaviour, the proposed approachis able to take into account the effective behaviour ofmasonry in compression, including the post-peak soft-ening branch. Once the limited ductility capacity istaken into account, a significant decrease in load car-rying capacity is found, which depends on the slope ofsoftening branch: a reduction between 15% and 30%has been estimated for the bridges under study, whenthe material behavior turns from perfectly plastic topost-peak brittle. Now, since masonry cannot be con-sidered as perfectly ductile in compression, it appearsthat the use of fibre beam method provides a signif-icant improvement in the evaluation of load carryingcapacity of masonry arch bridges.

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