Paper: ;:,rn1u1 ~t al

... ' :1, .-.. II P~per j. analysis of masonry arches I Three-hinge

l F. W. Smith, BSc, CEng, MICE Wolfson Bridg~ Research Unit, University of Dundee

'~, W. J~ Hal'Vjey, BSc, PhD, CEng, MICE Wolfson Bridgb Research Unit, University of Dundee

~f Professor }\.E. Vardy, BSc, PhD, CEng, MICE, MASCE Wolfson Bridge Research Unit, University of Dundee

synopsis ~lf Brief reviews,~ of the behaviour of masonry arch bridges and of popular methods of analysis highlight the need for a simple, practical and{ysis of arches under working loads. A three-hinge fhodel is proposed and is shown to lead to the same solutio~ as four-hinge mechanism analyses when the structure ap)Jroaches ultimate load. The three-hinge method is extended to ~nable failures by local crushing of the masonry to be simula~ed. It is also shown to permit realistic allowances r: made for movements of abutments. Introduction1J Since 16761, there have been attempts to determine how arches support the loads they[~re called on to carry. In the case of masonry arch bridges, the arch ring rr\ay be greatly stiffened by spandrel walls and by fill between

II the pavement surface and the arch. Nevertheless, these features are by no means decisivJin poorly maintained structures, and engineers usually base judgments onjlthe capacity of the arch .alone (or on the arch and known fill material) ll: . . .

In many areas of structural destgn, attention is focused primarily on the behaviour of ~!structure at ultimate load. Serviceability criteria are evaluated independentlYt1and are assumed to have no direct bearing on the safety of the structure,ieven when they lead to more onerous requirements. With masonry consthiction, however, and with old masonry bridges in particular, this may not ge a sound approach. Certainly, there is a need to know the ultimate cap~~ity of a structure, but this may be strongly influenced by behaviour ~t Jruch smaller loads. Alexander & Thompson2, for example, drew attention to the consequences of the continual opening and closing

f k. '" d' . o crac s between a Jacent voussoirs. In effect, the most pressing need may be for a fuethod of defining and measuring serviceability limits rather than ultimatel

11:limits.

0 rf I ~noadsu ac;e

Masonry arch behaviour ., , There is ample evidence that masonry arches tend to deform when centring is removed34 and that three hinges can form under the action of deid load

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alone (Fig l(a)). Sometimes this is due to shortening of the arch itself under compression, especially in the case of very flat arches. At other tifues; it may be due to abutment spread at the spr_ingings5 W?atever th~:tcau}e, the arch is likely to adopt a statlcally-determmate three-hmge form m whtch the locations of the hinges are the principal unknowns .. Figs l(b) ahd l(c) illustrate two possible configurations for an arch carrying dead lo~d and a concentrated live load. Different lines of thrust are obtained, actording to the load position. The hinges are exaggerated for clarity. One ~r both outer hinges are within the arch itself, and regions of the arch beyor/d these points are effectively rigid. 11

The positions of the hinges can also be influenced by the magnilude of the live l_oad. With successively increasin~ loads, ~inge C in Fig 7:moves progressively further to the left, and the lme of action of the thruSf at the right-hand springing moves upwards. Eventually, the thrust linejreaches the extrados and a fourth hinge is formed. At this stage, the st~ucture becomes a mechanism, and collapse is inevitable in the absence of ~xteinal changes. It .:

Figs I and 2 illustrate the problem of quantifying serviceability limits for masonry arches. Real bridges experience moving traffic load~ a~d a wide range of load magnitudes. The hinge positions vary with time, sometimes quite rapidly. {' :: Masonry arch analysis Early analysis by Castigliano6 and others were based on elastic p;rnciples. The complete voussoir arch was assumed to act as an elastic rib with fixed ends. As loads were applied, tension developed in certain areas. THe cross-sections at these locations were reduced to eliminate the cracked arhs, and successive analyses were obtained, approaching a final soluti~b in an

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(b) One hinge displaced

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0 (c) .t . (d) Fig 2. Influence of applied load magnitude on hinge positions

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l iterative Lanner. The necessary calculations must have been tedious, but the meth~d is potentially sound in principle, provided that proper account is takenilof abutment movements (to which it is highly sensitive). Its disadvantage today lies in the difficulty of incorporating interactions with the fill. Hughes' has. developed a microcomputer-based analysis, treating the fillfs ail elastic medium. - .

The Jlost popular methods of analysis in use today are # . -the MEXE method -the m~chanism method

-finite1~Iemeni methods ,, "' The K.fEXE method is based on Pippard's8 use of Castigliano's theorenl~. It has a sound theoretical basis, but it is unduly sensitive to parame~ers whose values cannot be determined accurately in many cases, especiaiiy with existing structures. The fuechanism method, .revived by Heyman\ is elegant and easily underst'i>od. The method enables the locations of the four hinges shown in Fig 2(d) to be determined and gives the load at which these will be formed.

Heyma~ did not claim that the method would provide an accurate prediction n of the 1failure load, although this would naturally be possible if all the necessJ\-y parameters were known with sufficient accuracy. Instead, he

propos~d that the safe load should be deemed to be the ultimate load for an arcij of the same shape, but half the thickness. Harvey9 proposed an alternalive interpretation of the same analysis, i.e. determining the minimum

thickn~ss of arch rlng capable of supporting the known (factored) applied ~~~ ' '

With finite element methods, account is taken of strains as well as stresses and s8 more information is obtained than is possible with mechanism anal;Js. Crisfield10 has included both geometric and elastic non-linearities in his ihodel, and he can m!lke quite accurate predictions of bridge collapse loads ~~hen appropriate-~ material characteristics are known. The main disad~antages of such techniques are the large demand on computing

resoui~es and the difficulty in obtaining and inputting adequate data about ,. ' ', ' ' '' ' the g~ometry and the material properties. Although the speed and the capacity of readily available computers are increasing rapidly, these remain a major limitation for engineers wishing to use extensive finite element

progt~ms for interactive design ... Th~ obvious gap in 'the available theoretical methods is for a simple meth~d of simulating the behaviour of arches in the three-hinge state that predJminates for most of their working lives. It is the purpose of this paper . ' to prfsent such a method.

Thr~~e-hinge. analysis In it~- simplest form, the proposed method of analysis closely resembles the J\echanism analysis described by Heyman4:Consider the voussoir arch in Fi~ 3(a) under the action of dead load and a single live load. The analysis belri~s by guessing the locations of the three hinges - typically, at the

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intrados at the springings and at the extrados between the crbwn''and the load, but anywhere in principle. The vertical and horizonta!Jomponents of the thrust at any hinge can be determined by static equilibrium. For example, by taking moments about each of the hinges A and ~in Fig 3(a), the components at C are found to satisfy :r :!

~{v(x-x~)}+ :::{h(y-yA)}- V,(x,-xA)-H,({-;yA)=O il .... (I) ~{v(x-x8 )}+ :::{h(y-y8 )}- V(x -x8 )-H(y -y8 )=0

' ' ' :1' . . . . (2) in which hand v denote the horizontal (L-+ R) and vertica; do~nwards) components of all forces acting on the arch ring. These ar imposed by the fill and are determined from the known live loading i~ the manner described by Harvey5 Briefly, they are due to (i) the weight of the fill, (ii) forces arising from live loads, suitably distributed by the! fill, and (iii) active, passive or intermediate pressure forces arising froml:mov,ement of the ring. The predicted load distribution is displayed in Fig 3 and in subsequent figures by vectors on the arch extrados. j ,;

Knowing the magnitude and direction of the thrust at each hinge, the magnitude and the line of action of the thrust can be det~~miried at any other location around the ring. This enables a so-called 'li~~ of. thrust' to be drawn for the whole ring, giving results such as those i~ Figs 3(b) and 3(c). In the former case, the line of thrust strays outside tile ring, and so the solution is incorrect. Alternative solutions must be soughl with different (assumed) hinge positions until a result such as Fig 3(c) i! obtained.

~ven when ~n acceptable solution has be~n o_btained,;~it may no~ be umque. Sometimes, there are several combmatwns of assumed hmge positions for which a static solution is possible, particular!* in the case of shallow arches. In such cases, as may be inferred from Fig 3(c), hinge C in the real structure is likely to be difficult to locate precisel1. Indeed, there may be many voussoirs showing signs of rotation. Any efror introduced by choosing a particular position will be negligible becausJ th~1 choice has so little influence on the assumed position of the line of th~ust. In the authors' computer program, the case yielding the minimum horirontal thrust

;:,:,~:~;:'"''"" ;, '""'"' th: coc but the spandrels had separated from the arch barrel. ~he road surface was level, and the depth of fill above the crown was 23~f~~- Inspection

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Paper: Smith! et al ~t

(b) Invalid hinge positions

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(c) Possitlle hinge positions

n 3. LinJ of thrust in the arch ring ''f

after collale reveale~ some additional backing (haunching) near the springings.ii

A 750mm-wide line load was applied at a single quarter point across the roadwa~ (between the spandrels) by means of hydraulic jacks acting on steel rods anchored in the. bed of the canal. Deflections and cracking were record~d at successive load increments until failure occurred at a load

-lh ' . ' .: . " ' of 140kN!J:h width across the roadway. Only the latter parameter can be compared directly with the proposed analysis.

Fig 4 shors computer simulations at successively increased loads, based on the abo~e analysis. With dead load only (Fig 4(a)), the hinges are at the crown ~hd the springings. At a relatively small load of lOkN/m width (Fig 4(b )), t'he central hinge has moved towards the load, thus highlighting the concerl expressed by Alexander & Thompson2 about the successive opening an'U closing of joints between adjacent voussoirs. At a load of 32 kN/m width (noi shown), the hinge has reached the load point, and this configurati~n obtains until the right-hand hinge begins t~ move along the

' ' arch, away)from the springing at loads in excess of about 40 kN/m width. As the loa~;increases, the hinge moves successively further to the left and the line of thrust at the right-hand springing moves away from the intrados \7"'' '1 ~wulru. Fig 4() how typkol ~mpk duri"' thi p

'I' lh Papen Smith et al I'

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'II l1 (a} lniinite stress

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------=

f r 'r, -::-----r---=x_----~ 1_'[----- . I :1.

(b) 1jinite stress Fig 5. In]luence of material strength on the zone of thrust

:II foliowin~ section. Secondly, the lines of thrust are much more curved at low load~ than at high loads. In case (a), the curvature is due to the dead load of the arch and the fill alone. As successively greater live load is applied,

0he thrusf lines tend to straighten and the magnitude of the thrust increases. . ~I .

- Influence of matenal strength In the sidtple line of thrust approach, no account is taken of the strength of the rrl~sonry. As a consequence, hinges are assumed to form exactly at the exlrados or at the intrados of the arch ring (Fig 5(a)). In practice,

however,)~ this is not possible because the material cannot transmit infinite stresses. 'II

A moFe realistic representation of a hinge is as a region of masonry at the crusffing stress of the voussoirs (or mortar, if present). In Fig 5(b), the line ofthhist may be assumed to act at the mid-depth ofa region of depth h where 'I[ . ______ _

1F h =-

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;_ .... (3) ou

in whicnlt denotes the local thrust and a" is the assumed crushing str~s~: This is J considerable simplification of the true state of stress, but it is much bdtter than assuming infinite stresses.

The v~lue of a" used will depend on the needs of the analyst. In most cases, it ~}viii be sufficient to accept the characteristic values derived from graphs in the Code of Practice for the bridge assessment BD2l/8412, and apply sJltable partial factors of safety.

The dbuble lines of thrust in Fig 4 and in subsequent figures depict the ,--... ,, t depth ofilmaterial which would be required locally to form a hinge capable _ -' of transrhitting the local thrust. They provide a visual means of interpreting

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~~ .lr (a) 50 .k/Nm width 11,

(b) 100 k/Nm width

j[ 200 .!

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370

(c) 200 k/Nm width (d) 370 k/Nm width II

Fig 6. Roredicted behaviour of Preston-upon-the- Weald Moors Bridge

;[ il the behaviour of the structure under increasing load (or varying loads). By inspection, one effect is to decrease the effective ring thickpess; :stability exists only when both lines are contained wholly''within the ring:i This reduces the load at which a fourth hinge forms, causing the structure to become a mechanism. t.

Example 2: Preston-upon-the- Weald Moors Bridge 1 Another of the full-scale tests organised by TRRL 11 was at Preston-upon-the-Weald Moors Bridge in Shropshire. This semi-elliptical!:fanal bridge with a skew of 17 was built around 1834 and had a span of 4950mm and a rise of 1636mm. The 360mm-thick sandstone arch ring apbeared to be in reasonable condition, and there was no separation from the ~rick,spandrel walls. The road surface was 375 mm above the crown and the fill was mainly granular. Haunching consisting of 'large coursed sandstone blJcks; possibly cemented together' was present behind the arch above the sp~~-~ngings, thus effectively raising the height of the springings. ,, ..

A 750mm-wide line load was applied across the roadway, Pilrallel to the springings, at a span third point, the method of application being similar to that at Torksey. As the applied load increased, significa~t horizontal movement was detected at or near the springings. Failure eventifally occurred at a load of 370kN/m width, separation of the arch ba~~el from the spandrels having occurred a:t a much smaller load. i't "

Fig 6 shows computer simulations for this case, with no all0wance being made for skew and with granular fill over the whole arch (i.e. n~ haunching). By inspection, the line of thrust is not contained wholly within the arch eveh at the quite low load 9f lOOkN/m width. Indeed, for all i~ads.in excess of about 60kN/m width, the line of thrust passes through''!he arch' ring

' into the haunching. Effectively, this implies that the haunchrng acts as an abutment, preventing the formation of a fourth hinge. Jt

It would be misleading to suggest that the computer simuf~tions can be .,

used to 'predict' the failure load in this case. The difficulty is ttiat the degree of restraint offered by the haunching is unknown. Unddubtedly, the

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haunching provides far more constraint than the granular fill assumed in the analysis. Nonetheless, the observed horizontal moveme;yts above the springings show that the structure could not be regarded as ~gid. All that can fairly be concluded, therefore, is that the simulations are consistent with th_e observat~ons. _ _ . H .: .

In this context, It must be emphasised that all known methods of analysis are sensitive to the (unknown) properties of haunching, es~rcially where the ring is more vertical than horizontal. The value of three-hinge simulations such as those in Fig 6 is that they provide engineers'twith evidence on which to base judgment. Similar results could'be obtained with a four-hinge (mechanism) analysis for all loads above about 60k~/m.,width in this particular case, but not for loads below this value. 1~ '

A further advantage of the three-hinge approach is that. it provides information about the position and magnitude of the thlust on each abutment. The predicted values are subject to the same limifations as the rest of the analysis, but they are the best estimates available for designing or assessing the abutment requirements. !

i.:~'""'"-~-~., Paper:

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(a) 500 l/Nm width (b) 800 k/Nm width Fig 7. No1~ional behaviour at loads exceeding the true failure load

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Fig 8. Response of a masonry arch to abutment spread

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. I (a) 40 ;k/Nm width

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(b) 65 k/Nm width

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(c) Failure at 70 k/Nm width .. :II Fig 9. Behaviour of Torksey Bridge assuming abutment movement .... -r . ., . . . I~ . . It is ins:tructive to continue the analysis to even greater loads, as illustrated

in Fig 7. Suppose that the haunching had been sufficiently strong to prevent the form~tion of a fourth hinge, even at these loads, and ask the question 'w01ild t?.e structure have survived?'._ According to a simple mechanism analysis, :ithe answer is undoubtedly 'yes', but this is difficult to believe. The matbrial is very highly stressed, particularly close to the hinges. In practice, j~palling of the outer surface of the voussoirs is likely to occur, thus redJ.cing the ring thickness and rendering the arch less stable. It is noteworthy that considerable spalling occurred at the outer hinges of the 11 Preston-~.Jpon-the-Weald Moors Bridge. Also, several observers concluded that faiiJre occurred through crushing at the hinge under the load, not through ~ simple mechanism. The proposed analysis helps an engineer to

. take acc~unt of this effect. In reality, large compressive stresses Imply the existence of large strains

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which will induce shortening of the arch.ring. This phenomenon cannot readily b~ taken into account in three-hinge or mechanism analyses, but engineers%1ssessing a structure should make due allowances, especially when the arch J'1is fairly flat, as is quite often the case. I .

~ ~ Abutme.nt spread It is pos{ible to allow directly for the influence of abutment spread. Fig n.., -lie ---~'~ 'L--~-- L! ___ ---...l!.o.!-- ----L!-1.... --- L- ------1-...1

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as two rigid bars, LA and AR. If the abutments L and R moyhe apart, the new position of the intermediate hinge must lie on the arc AI\ centred on L and on the arc BB centred on R (Fig 8(b). These arcs ha~e a unique inters~ction point an? so thene~ position of the ring can b~tdete.rm. ined by a ~1mple geometnc.constructiOn: 'i!i :_

This effect has been mcorporated mto the authors: computer programs, which allow movement of the abutments in proportion to th~~thr~st. The iterative procedure for each increment is as follows: . 1l ': (i) determine the line of thrust assuming no additional movement; (ii) using the predicted thrust at the springings, evaluate the nbw positions

of the abutments and hence the new position of the interm'~diate hinge; (iii) repeat steps (i) and (ii) as often as necessary (usually dbly once)

In practice, spreading of the abutments usually causes at :feast one of the outer hinges to move away from the springings and into'ihe ring. To simulate this case, it is necessary to know the rotational compbnent of the abutment movement as well as the linear displacements. Thl~ is because the portion of the ring beyond the outer hinge effectively acts':~ompositely with the abutment itself. The programs assume that this portiC:h of the ring undergoes the same linear displacement as the abutment and also rotates about the springing. ' ..