Load Bearing Capacity of Masonry Arch Bridges v Imp Printed

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Load-Bearing Capacity ofMasonry Arch BridgesStrengthened with FibreReinforced PolymerCompositesJ. F. Chen

School of the Built Environm ent, Th e University of Nottingharn, University Park, Nottingham NG7 2RD, UKE-mail: [email protected]

Abstract: This paper presents a new mechanism analysis method to calculate theultimate load-bearing capacity of masonry arch bridges strengthened by externallybonding fibre reinforced polymer (FRP) composites on the intrados. The M-Nrelationship of a strengthened arch ring cross section is analysed and fully consideredin the mechanism analysis. The Complex Method of nonlinear programming is used toperform a constrained ininimisation of the load function to obtain the minimumcollapse load and its corresponding hinge positions. This has advantages because thearch ring does not need division into many discrete blocks as in traditional m echanismanalysis and the solution process is fully automatic. Som e initial results are presented,which show that significant strength increases can be achieved by bonding a smallamount of FRP composites to the intrados of masonry arch bridges.

Key words: mason ry arch bridge, strengthening, fibre reinforced polymer co mpo sites, FRP, mechanism analysis,the complex method

1. INTRODUC'TIONMasonry arch bridges make a significant contribution tothe UK's road and rail infrastructure. The majority ofthese were constructed in the 19th century, or evenearlier (Page 1993). It is clear therefore, that they werenot originally designed to carry the extent of loadsimposed on them by current vehicular traffic. Manyexisting masonry arch bridges need reassessment oftheir 'fitness for purpose' and upgrading where, and as,appropriate. This situation has encouraged research inthis domain in the last decade (e.g. Hughes 1995,Hu_ghes and B lackle r 1997, -Hughes et al. 199 8,

Melbourne e t al. 199 7, Peaston and Choo 1996, Prentice1996, Ponniah and Prentice 1998).Many methods are available for strengtheningmasonry arch bridges. Comm only used metho ds includesaddling, sprayed concrete and relieving arch etc.(Department of Transport 1993, Page 1993). Based onthe method established by the Military EngineeringExperimental Establishment (MEXE), the load bearingcapacity is a direct function of the sum of the thicknessof the arch barrel and the depth of fill at the crown(Peas ton and Choo 1996). When the depth of f il l U h ecrown s large, a significant amoun t of strengthening of

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Load-Bearir~gCapac ity of hfaso riry Arch Br idges Str-engtheized with Fibre R einforced Polymer Conzposites-

- - -- -- .- -the arch barrel--i~-Ffien -- equiredincrease of the loadlb-earing-capam:- - --. - This paperlg,po~ts n initiakheore~cal-studyon the -~ t r ~ r i g t h e n i n gf maSOnryarChTdges by bondingfibre-reinforced (FRP) co m~ -si tes o the archintrados. FRP composites are very light and provide

ease of site handling, so the retrofit construction workcan be performed quickly and without specialist orheavy machinery. This means minimal disturbance totraffic; an important consideration in bridge repair orrehabilitation work. Neither will the dimensions of abridge be changed significantly, so the original appear-ance is maintained and, minimal loss of headroom isexperienced where traffic goes under the bridge.Corrosion is not a problem in that FRP is virtuallycorrosion resistant. The use of FRP sheets can alsoprevent the development of longitudinal cracks in bridgemasonry.A new mechanism analysis method is presented inthis paper, which duly considers the M-N relationshipwithin the bridge's arch ring, to calculate the ultimateload-bearing capacity of bridges strengthened in thisway, The minimal collapse load and its correspondinghinge positions are obtained by using the ComplexMethod to perform a constrained minimisation of thelo ad function. This has advantages because the arch ringdoes not need division into many discrete blocks as intraditional mechanism analysis and the solution processis fi~ lly utomatic. Som e initial results are presented,which show that significant strength increases can beachieved by bonding a small amount of FRP compositeto the intrados of masonry arch bridges.

2.1 Basic Equ ationsThe ultimate strength of an FR P strengthened rectangu-lar masonry cross-section under eccentric compression,which closely represents the cross-section of an archring in a masonry arch bridge strengthened by bondingFRPs on its intrados, is analysed in this section.A masonry cross-section of width B and thickness t isstrengthened with an FRP sheet or plate with thicknesst,, and cross- section al area A,, (Figu re la ). Th isstrengthened cross-section is under a bending momentM and a compressive axial force N (i.e. under a com-pression load N with eccentricity e = M/N) (Figurelb ). It is assumed h ere that th, is much smaller than t(i.e. tf,/t


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Becau se the stress-strain relationship is linear untilfailure for most types of FRP composites, the stressdistribution on the cross-section may be approximatelyrepresented by Figure 1d. Integrating the stresses on the being the normalised FRP fraction, bending momentcross-section gives the axial force N and the bending and axial force respectively. Eqns 4a and b becomemoment M - XN = 0 . 8 - - O f , - -t ( 1)where E f r p is the young's th e FRP c On l~ Os i t e If the compressive force is known, the height of thean d fm is the 'ltimate strength of the compressive area of the masonry x can be found froinmasonry. Eqn 6a:The cross-section may fail due to either rupture of theFRP composite or compression failure of the masonry.Because FRP composites are brittle materials so the -

- o f , + d(OfipN)l + 3.20f,] (7 )latter failure mode is preferred. These different failure t 1.6mod es are considered separately as follows.I The M-N relationship can thus be determined from

2.2. Maso n ry C ompress ion Fa il u re Eqns 6 and 7.If the maximum strain in the masonry reaches its ulti- The above equation s apply when the stress in the FRPmate strain E,,,, whilst the strain in the FRP composite is tensile, i.e. x / t < 1. When x / t 2 1 (i.e. 2 2 0.8),E~, , is smaller than its ultimate (rupture) strain Eqns 6b and 7 may be replaced by ignoring thecompressive strength of the FRP (setting wfT = 0):E ~ ~ , ~ ~h m / E f r p ,he cross-section fa ils due to compres-sion -failure o f the masonry. Here fh , is the ultimatestrength of the FRP composite. Substituting r , = E , ,into Eqn 1 gives the strain in the FRP composites atfailure:

Equation 8 also applies for all un-strengthened cross-Substituting Eqn 3 into Eqns 2a and b gives sections, independent of the value of N.

N- 0.8- - 2.3. FRP Ruptu re Fai lurefmB t t fmBt (4a) If the FRP composite reaches its ultimate strengthbefore the maximum strain in the masonry reaches itsM ultimate strain E,,,,, the cross-section fails due to ruptureof the FRP. The strain in the FRP composite at failure(4b) is known in this case. Substituting sf, = E ~ , , ~ , , ~ intoEqns 2a and b givesLet

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Load-Bearing Capacity ofh las on gi Ai-clz Bridges Strengthened with Fibre R e i i f o ~ e d ob,iner Coi~zposites i- -p- - ---.- - -- - - .-- -- -- - . -- ---. -- --

-- RC~C-; -s theu ltima te strain ratio of FR P-to masorlry -- mall axial forces and small ultimate strain -=- -.. --.- to-masonry. The ultimate strain-ratio of -- ----

- - --- - f rn ruP_ asonry < has-a very sign ifican f3T ectto n : -5 = --- - (10) the critica l FRP fraction. If 5 = T t h e cross-sec tion wi ll -- Em u always fail due to crushing of masonry when the nor-malised axial force N s greater than 0.4. If < increasesto 3, FRP rupture will not occur when N > 0.2.

2.4. Balanced Failure For commonly used FRP composites and masonry, < isTh e boundary between the masonry compression failure usually g reater than 3.and the FR P rupture failure modes lies at when the FR P Figure 3 shows the effect of the FRP fraction wo n the

and the masonry both reach their ultimate normalised moment M for various axial forces. As wstrains simultaneously, i.e. increases from zero, M increases very fast initially butremains almost unchanged when w becomes large. This-

indicates that it is most economical to strengthen theEm - Em u and Efrp = Efrp,rup (11) structure when only a small amoun t of FRP composite

is needed. Further increase of the load bearing capacitySubstituting Eqn 11 into Eqns 2a and 3 gives the critical by using more FRP composite is uneco nom ical. TheFRP fraction for this balanced failure mode: figure also shows that the strengthening is most effec-tive if the axial force is small.

- Figure 4 shows the M-N relationship for various0.8 N (12) w values, which again shows that the moment capacity@frp,cr 5 + 1 2 5 can be most effectively increased when the compressionforce is small. This implies that this strengthening

0.52.5. Numerical Results 5 .45When calculating the moment capacity for a given 2 0.4 NIB^^,normalised axial force N, he critical FRP fraction 0.35 -of,,,,,eeds to be calculated using Eqn 12 first. If the 0.3actual FRP fraction is greater than w,,,c,, the cross- 2 0.25section fails due to masonry compression failure and Z 0.2.c 0.15Eqns 6 an d 7 should be used. Otherwise the cross-sec- -2 0.1tion fails due to FRP rupture and Eqn 9 should be usedinstead. $ 0.055igure 2 shows the critical FRP fraction for o I 2 3 4 5various FRP to masonry ultimate strain ratios. Clearly, FRP fraction a,,,mo re FRP composites are needed to prevent FRP rupture Figure 3 . Normalised moment versus FR P fraction

0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1. Norm alised axial force NlBtf, Norm alised axial force NIBtf,

Figure 2. Minimum FRP fraction to prevent FRP rupture failure Figure 4. hl-N relationship for various FRP fractions

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techniqu e may be mo st effectively used for shallow archbridges.It may be noted that a similar analysis for masonrystructures strengthened using bonded FRPs waspresented by Triantafillou (1998a). However, this wasincomplete because no analysis for w less than thecritical value was presented. It also contains errors, evenafter a correction in Triantafillou (1 998b) because he pre-dicted that the moment capacity decreases as w increaseswhen the axial force exceeds certain values. The workhere thus represents a fuller and more rigorous analysis.

3. A NE W M ECHA NISM ANALYSISME TH OD FOR STRENGTHENEDMAS ONR Y ARCH BRIDGES3. I. Virtual Work RelationshipFor a plastic hinge subjected to a virtual rotation 19(Figure 5), the virtual work done by the internal forcesM and N s:

For the load distribution and a failure mechanism ofan arch bridge as shown in Fig. 6, the virtual workrelationship m ay be expressed as

in which W, s the dead weight for each of the blocks; Hfis the total horizontal forces provided by the passiveresistance of the fill; the virtual displaceinents Ap atloading places, Aj at the gravity centre of the jth blockand AHf at the centre of horizontal fill resistance force,and the v irtual rotation 19, f each hinge can be obtainedfrom basic kinematics. Forces and displacements aredownwards an d rightwards positive here.

-- --

figure S F V ~ a l - d e f m m a t i o n s t a h in ge- - --- -- -

Figure 6 . Load distribution and mechanism

The internal forces at each of the hinges Mi nd Nican be found from th e equilibrium for the given mecha-nism, together with the appropriate M-N elationship,using an iterative procedu re. Depending on whether thevalue of wfi, is greater than either Eqns 6 and7 or 9 should be used as appro priate. If there is no FRP,or the bending moment is opposite to the direction asshown in Figure 5, Eqn 8 should be used. An unstrength-ened masonry a rch structure is thus a special case here.The collapse load for a given mechanism can thusbe determined by using the virtual work relationship(Eqn 14) and the static equilibrium. It may be notedthat the eccentric compression force at a hinge can beoutside the ring if th e FR P is effective (i.e. in tension).

3.2. Determination of Minimum CollapseLoad and Its Corresponding HingePositionsIn traditional mechanism analysis, the arch ring isdivided into a set of (typically 50) elements. The mini-mum collapse load P,,, and its corresponding hingepositions have to be obtained by trial and error (Page1993). This is a tediou s process. A new method is thusdevised here to obtain the minimum collapse load andits correspond ing hinge positions.In this new method, the arch ring is not divided intoblocks, so that all the hing e positioils can vary continu-ously within the ring. For simplicity, Hinge B isassumed to be located right beneath the centre of theloading here. However, there should be no difficulties inthe following analysis to allow the position of Hinge Bto vary by removing this assumption.

For the given- four-hinge mechasism, th ec ol la ps eload -P can he found+om-the-virtual work an ist at ic -- - ----- -- .-Adva nces in Structui-a1 Engi11eer.ing E l . 5 No . 1 2002


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Load-Bearing Capacity of M asonr y Arch Bridges Strengthened lvith Fibre Reinforced Po lyii~erCo~nposites I

-.mgdibfium for each sethinge ~&ions. As Hinge-a-function of& e positions ofthe explicit expression -of thTherefore, the problem is to find a set ofadmissible hinge positions so that th e cominiinised.If the hinge positions are expressed in terms of angles author for more information./3 (can also be any other geometrical measures), theproblem can be mathematically expressed as

4. NUM ERICAL ANALYSIS OF ANMinimise P(P> (15a) EXAMPLE BRIDGE4. I. The Unst rengthened Br idgewhere The B olton full scale mo del segm ental bridge tested by

P = {PA,PC,P D ) ~ Melbourne and Walker (1989) was used here to applythe devised analysis method. Details of the bridge wereobtained from Page (19 93). The square span of the archSubject to PC> PD ( 1 5 ~ ) ring was 6 m and the total width was 6 m. A schematicdrawing of the bridge geometry is shown in Figure 7.The ring was constructed using concrete bricks. ThePmn = {PBo, DO, 5 Pmax = {PAo, BO, (15d) brickwork had a density o f 2 1952 kN/m3 and a com-

pressive strength of 1 1.2 M Pa. The fill was limestonekinematically admissible with a density of 21 3 6 4 W m 3 and internal f ic t ionangle 4 = 54". The ultimate strain of the concrete brickin which PA,,&,, an d hore angle positions of the two was assumed to be 4000 p~ in the analyses in thespringings and the load (Figure 6) and represent the following two gections.limits to which Hinges A, C and D may vary (Eqn 15d). A 'line' load with 750 m m width was applied. TheThe constraint Eqn 15c is used here to keep the hinges bridge failed with a 4 hinge mechanism and the maxi-in the right order. Constraint Eqns 15c and d together m um measured load was 1170 kN. The prediction of theensur e that the solution will be in the order o f above analysis method was 1144kN , representing anPA > PB > PC> PD.The constraint Eqn 15e is used error of less than 2.5%. However, the error of predic-to ensure that the obtained mechanism is kinematically tion is expected to be much bigger than this for deeperadmissible. It need s to be quantified in numerical bridges because of the inherited limitation of mecha-analysis and this was done here by using an indicator nism analysis.KA which equals 1 if a set of the hinge positions islunematically admissible and -1 if inadmissible. Theconstraint Eqn 15e thus becomes

K A > oThis is a constrained nonlinear optimisation problem(or constrained nonlinear programming problem): tominimise the objective function P subject to constraintsof Eqns 15c*. Man y constrained nonlinear prograin-ming methods are available. Detailed descriptions ofnonlinear programming m ethods can be found in manytextbooks (e.g. Himmelblau 1972, Avriel 1976). TheComplex Method proposed by Box (1965) was usedin this study. Detailed description of the ComplexMethod is beyond the scope of this paper. Interestedreaders may consult Box (1965) or textbooks innonlinear programming (e.g. Himmelblau 1972, Figure 7. Geometry of the Bolton full scale model bridgeAvriel 1976). (Melbourne and Walker 1989)

---I I _A,.-YX- -_- _ _-._--- -- -.- . - - -- ----

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4.2. Effect o f Strengthening o n Col lapse predicted collapse load increases from 1144 kN for theLoad unstrengthened bridge, to 1906kN (67% increase) ifThe above bridge is now assumed to be strengthened by only Hinge B is strengthened and 2 181 kN (9 1%bonding an FR P sheet on the intrados. It is assumed that increase) if both Hinges B and D are strengthened. Bya carbon fibre reinforced polymer (CFRP) composite bonding a 1 nlrn thic k FRP, the collapse load is furtherwith strength &,= 2400MPa and Young's modulus increased to 2932kN (156% increase) and 3424kNEfm= 2 X 105 MP a sused.Two options are considered: (199% increase) for the two strengthening schemesa) The F RP is bonded on the full intrados, but notanchored in the abutments ( i t . not effective for

Hinge D ). Because both the loading positioil an d theposition of Hinge B were fixed in this analysis, theeffect will be minimal whether the full iiltrados isstrengthened given that Hinge B is always strength-ened. The only effect which may arise is that whenthe predicted position of Hinge D is not at the spring-ing. However, the intrados should be fully strength-ened in practice because of moving loads.b) The FRP is bonded on the full intrados and fullyanchored in the abutments so that it is effective forHinge D as well.

Figures 8 and 9 show that the co llapse load can be verysignificantly increased even by bonding a very thinsheet. If a 0.1 m m thick CFR P sheet is used, the

a - -- *Both Hinges B & D strengthened 1-o 2000 - +Only Hlnge B strengthened0 I I1000 a ! I I

0 1 2 3 4 5FRP thickness t,,,, m m

Figure 8. Effect of FR P thickness on the collapse load

respectively. However, it is clear that it is most effectiveto use very thin (e.g. 0.1 mm ) sheets in this case.

It may also be n oted that the current theoretical pre-dictions are the sa me for bonding a 0.1 m m sheet in thefull intrados, and for bonding 1 mmx5Omm str ips @50 0m in centres. The latter is indeed more convenient toapply in practice. However, the former is the preferredchoice for two reasons. Firstly, the continuous sheet canprovide transverse strengtheningtco nstraint so that longi-tudinal cracks may be prevented from developing.Secondly, the analysis has so far assumed that there is aperfect bond between the masonry and the FRP.However, this is unlikely to be true especially for deeparches. A recent study on the bond behaviour betweenFRP com posites an d concrete (Chen and Teng 2000,2001) shows that the use of thin sheets is much moreeffective than thick sheets. Th e Young's modulus has alsoa very significant effect in this respect. More details canbe found in Chen and Teng (2001) and Teng et al. (2001).

Figures 8 and 9 demonstrate clearly that it is a veryeffective measure to anchor the FRP in the abutments,so this should be adopted wherever possible.4.3. Ef fect o f St rengthen ing on HingePositionsThe predicted positions for Hinges A and D are alwaysat the springings in this example. Thi s may be due to thefact that this is a very shallow arc h.The predicted position for Hinge C varies as thethickness of the FRP sheet chang es. Figure 10 showsthis positioil measu red by using the angle from vertical(pC n Figure 7). If only Hinge B is strengthened, the

'0 16g 250% w- 15w rn4 14"000%m- 6 2 3

150% & E 12C .E $0$ 1oo0/o Z E l 1O 9 10m c -2o 50% .g 9.-c- ln0 8

P0% 70 1 2 3 4 5 0 1 2 3 4 5

FRP thickness t rnrn- FRP thickness t m m ----Figure 9. Increase of co ;Effect o f thickn ess on hinge positioa- - ------

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Load-Bearing Capacity of Masonry Arch Bridges Strengtl~enedwith Fibre Rei~ljorced olyrrle~- orilposites I- -- -- - - -.- - . . - ---an gle in creas es - from 73-central -- -- 0.67 m D e p a r t m e n t o f - ~ f i % ~ o r t( L-- --- nent -of highw ay

from the symmetry) forfiz-Tnstrengthened-bridge, to bridges and stniifures", Part 3, BICXI93; --art 4, BA 16/93.1 1.1" (0.98--f rom the sy-mmetwjhen ti, = O.l_mm H i n ~ m e l b l a u ~ .M. (1 972 ) ~ ~ ~ 1 z k d ~ N o a l i ~ e ~ rrogrorn~nitig.-- -a nd c lo se to th e t h r e e - q u m f o r t,, = 5mrn.A similar -- ~ c G r a w - H i l K ~ n c . ,ew York.trend is seen, but the position cKange is much smaller if Hugh es, T.G. (1995) "Analy sis and assessm ent of twin-span nlasonryboth Hinges B an d D are strengthened.If the ring is divided into 50 blocks as in traditionalmechanism analysis, each block represents 1.5" which

will make it impossible to produce smooth curves asshown in Figure 10. This further demo nstrates theadvantage o f the new method devised in this study.

5 . CONCLUSIONSThis paper h as presented an initial study on strengtheningmasonry arch bridges using externally bonded FRPs onthe intrados. The M-N relationship for a rectangularmasonry cross section strengthened with FRP has beendeveloped and it has been used in the mechanism analysisof strengthened bridges. A new mechanism analysismethod has been developed. It finds the minimum col-lapse loa d an d its corresponding hinge positions, by usinga constraint optimisation method. The arch ring is notneeded to be divided into artificial blocks so that the hingepositions can change continuously. The analysis proce-dure is fully automatic and is of high efficiency because itneeds minimal user input. Initial numerical results haveshow11 that it is very effective to strengthen masonry archbridges by bonding a thin FRP sheet on their intrados. Thestrengthening effect can be significantly enhanced byanchoring the FR P in the abutments.

REFERENCESAvriel, M. (1976). Nonlinear Progra~nming:Analysis and Methods,

Prentice-Hall, Inc., New Jersey.Box, M.J. (1965) "A new method fo r constraint optirnisation and a com-

parison with other methods", Coinputer Journal, Vol. 8, pp. 42-52.Cllen, J.F. and Teng, J.G. (2000) "A new bond streng th mo del fo r

FRP and steel plates attached to Concrete", P~aceedingsof the6th Intenzational Co ~lje ren ce r1 Structzrral Failure, Dul-abilityand Retrofitting, 14-15 Septem ber Singapore, pp. 229-136.

Chen, J.F. and Teng, J.G. (2001) "Anchorage strength models forFRP and steel plates bonded to concrete", Jounlal ofStructura1Engineerirzg, ASC E, Vol. 1 27, No . 7, pp. 784-791.

arch b ridges", PI-oceedings of the Institution of Civil Engineers:Strttctures and Buildings, Vol. 110, pp. 373-382.

Hugh es, T.G. and B lackler, M.J. (1997) "A review of the UK masonryarch assessment methods", Proceedings ofthe bzstitrltion of CivilEngineers: St~u ctures nd Buildings, Vol. 122 , pp. 305-3 1 5.

Hugh es, T.G., Davies, M.C.R . and Taunton , P.R. (1998) "Small scalemodelling of brickwork arch bridges using a centrifuge",Pi-oceedi~zgs f the I~zstitution f Civil Engineers: Structures andBuildings, Vol. 128, pp. 49-58.

Melbo urne, C. and Walker, P.J. (1989) "Load test to collapse on a fullscale model six meter span brick arch bridge", Depart~~rentfTransport, TRRL contractor Report 189, Transport ResearchLaboratory, Cro\v-thorne.

Melbourne, C., Gilbert, M. and Wagstaff, M. (1997) "The collapsebehaviour of multispan brickwork arch bridges", The Str~tcturalEizgineer, Vol. 75, No. 17, pp. 297-305.

Page, J. (1993) Masonly arclr bridges: state of the art reviews,HMSO publications.

Peaston, C. and Choo, B.S. (1996) "Predicting the capacity ofsprayed concrete strengthened masonry arch bridges: A compar-ison experimental and analytical data", ICCI'96, FibreColizposites in Infiastructur-e (ed by H. Saadatrna~zesh& M.R.Ehsani), Proceedings of the First Inter7zational Conference onCo~npositesll Irljrastructtrre, Arizona, USA, pp. 91-95.

Pomiah, D.A. and Prentice, D.J. (1998) "Load-carrying capacity ofmasonry arch bridges estimated from multi-span model tests",Proceedings of the Institution of Civil Engineels: Structures andBuildiizgs, Vol. 128, pp. 81-90.

Prentice, D.J. (1996) "An appraisal of the geotechn ical aspects of multi-span masonry arch bridges", PhD thesis, Edinburgh University.

Teng, J.G., Chen, J.F., Smith, S.T. and Lam, L. (2002) FRP-Strengthened RC Structures, John Wiley and Sons, Chichester,U.K.

Triantafillou, T.C. (1998a) "Strengthening of masonry structuresusing epoxy-bonded FRP laminates", Journal of Conposites forConstruction, ASCE, Vol. 2, No. 2, pp. 9 6 1 0 4 .

Triantafillou, T.C. (1998b) "Errata", ASCE Journal of Corxpositesfor Construction, Vol. 2 , No . 4, p. 203.

Dr Jian-Fei Chen is a Lecturer in structural engineering in the School of the Bui l tEnvironment, the Universi ty of N ott ingham, UK. He received his BSc and MSc fromZhej iang Universi ty in China and PhD from Edinburgh Universi ty. He has interest inmany areas in c iv i l engineering, including reinforced concrete, masonry and steelstructures; space and shell structures; granular solids flow and wall pressures in silosand inverse problems.

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Load Bearing Capacity of Masonry Arch Bridges v Imp Printed