11 th INTERNA TIONAL BRICK!BLOCK MASONRY CONFERENCE TONGJI UNIVERSITY, SHANGHAI, CHINA, 14 - 16 OCTOBER 1997

GROSS DISPLACEMENT MECHANISM ANAL YSIS OF MASONRY BRIDGES AND TUNNEL LININGS

Matthew Gilbere

1. ABSTRACT

Using the mechanism Cor ' kinematic') method of analysis, the failure mo de of a masonry structure comprising of weakly mortared masonry blocks can be calculated directly - no prior assumption as to the internaI stress state is required. However, in the case of many masonry arch bridges designed and constructed in Western Europe, use of backfill material to fill the spandrel void areas means that soil-structure interaction will influence bridge performance. Whilst in the past for mechanism analysis it has normally been assumed that large soil pressures could be mobilized by infinitesimal structural displacements, this c1early does not happen in practice. For this reason the use of a simple iterative, gross displacement, analysis proceáure is investigated. The technique is used to study the behaviour of large-scale model masonry arches and arch bridges, tested in the laboratory. For these bridges the difference in the magnitudes of the failure loads predicted by the standard and gross displacement mechanism approaches are found to be relatively smalL Finally, the capability of the method to assess the stability of multi-ring brickwork tunnel linings in the presence of progressively increasing ground movements is exarnined.

2. INTRODUCTION

2.1 Masonry Arch Bridges

It was pointed out in the 1960's that plastic theory, formulated initialIy for steel structures, could be applied to masonry provided certain assumptions were made. It rnight have been expected that folIowing the work of Heyman 1 and others that the

Keywords: Masonry; Mechanisms; Arches; Tunnels

lLecturer, Department of Civil and Structural Engineering, University of Sheffield, Mappin Street, Sheffield SI 3JD, UK

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mechanism method would by now have completely replaced the MEXE2 method of assessment in the UK, as a more rational routine analysis tool for routine assessment of masonry arch bridges. This has not occurred and the MEXE method is still very widely used, despite its questionable theoretical basis. It is probable that one of the reasons engineers are reluctant to move to mechanism analysis is because of the rather crude way soil-structure interaction problems are treated, using the 'standard' mechanism fcrmulation.

Field and laboratory tests carried out over the last decade in the UK have shown that ultimate bridge strength is considerably enhanced by the presence of horizontal backfill pressures (i.e. so-called 'passive' pressures - e.g. refer to Page3

). However, whilst significant horizontal backfill pressures can only be mobilized by relatively large structural displacements, mechanism analysis programs have generally assumed that peak backfill pressures are mobilized by infinitesimal structural displacements. This assumption will lead to non-conservative results. Indeed the current UK Department of Transport bridge assessment memorandum BA16/93 2 states that:

'the mechanism method ..... may not be appropriate where soil resistance is important, which has found to be the case even for relatively fiat arches .... These (mechanism) programs may therefore produce arbitrary results'

When the mechanism method is applied to multi-span arch bridges, the assumption that peak horizontal soil pressures are mobilized by infinitesimal structural displacements can lead to other problems. It has been found4 that when analysing a three-span brickwork arch bridge loaded in the centre span, application of horizontal backfill pressures to, say, correctly increase the load required to cause barreI movement of span 3, may in fact lead to an erroneous mechanism which also involves span 1 to apparently beco me criticai (using a rigorous solution procedure in which the minimum of the upper bound solutions is sought). In the erroneous mechanism, it is found that the 'passive' type pressures applied to span 3 are in faci being mobilized as 'active' type pressures. This problem may be overcome by subsequently manually adjusting the magnitudes of the fill pressures in span 1 until a correct mechanism not involving that span is found. However, the use of an iterative, gross displacement approach may be a more elegant approach to the problem, .and may lead to a more accurate solution. Using such an approach, soil pressures may be progressively increased according to the increasing structural displacements, without the intervention ofthe analyst.

2.2 Masonry Tunnel Linings

Another situatiun where a gross displacement analysis might be valuable is when examining the stability of masonry tunnel linings subjected to soil movements. Nineteenth century engineers left Westem society a legacy of canal, railway, sewer and water supply tunnels; of these a large proportion were constructed of brickwork. The majority ofthese have proved to be long lived, requiring only minimal maintenance.

However an increased understanding of the structural behaviour of masonry tunnels may be required when new construction work is planned elose to existing brickwork tunnels, a particularly common scenario in built up areas. On the basis that excavations invariably cause ground movements, there is a need for engineers to assess the effect of

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the ground movements on the stability of a given tunnel lining. This information could be used either to reassure tunnel owners that problems are unlikely to be encountered, or be used to help plan preventative/remedial measures, such as compensation grouting. In the case of tunnel linings which are in generally good condition, overall geometry in relation to the surrounding soil pressures is of p;ime importance when considering their inherent stability. Also possibly important is the bonding pattern used for the brickwork. Very often the soffits of old brickwork linings were built up using individual rings ofbrickwork, using stretcher bond.

3. FORMULATION OF GROSS DISPLACEMENT MECHANISM METHOD

3.1 Standard Mechanism method

The mechanism method of analysis as applied to masopry arches 1 has traditionally relied on a number of assumptions, some ofwhich have subsequently been modified or discarded5 as required to make the solution more general. The assumptions used here are that (a) the masonry (in the arch) has zero tensile strength, (b) strains in the masonry are small, and so cause negligible changes in the global geometry of the arch, and (c) the blocks initially fit perfectly together. Sliding using an associated flow rule is perrnitted in the formulation adopted here.

e.g. the principIe ofvirtual work may be used in conjunction with a linear prograrnrning solution procedure to obtain the collapse load W. i.e.

tvlinirnize W, where:

wdT - W= O (1)

and where: W={W1' w2 •..... wn } , the selfweights ofthe blocks.

d = {di' c4 . .... . dn }, the vertical displacements ofthe blocks.

Subject to appropriate constraints (e.g specifying abutment fixity, no overlap of blocks in adjacent rings etc). Note that to make the problem linear, small displacement theory must be used. Additional terms may be added tJ expression (1) in order to allow inclusion of horizontal soil pressures, material crushing at hinge positions, etc. More details ofthe method are provided elsewhere6

Following an analysis the dual variables in the linear prograrnrning tableau correspond to the internaI forces in the structure. These may be used to calculate the extent of zones of crushing due to finite material strength (this applies both to simple arch ribs and to more complex assemblages of blocks, for example multi-ring arches) . Once initial crushed depths are calculated the analysis is repeated. Normally only two or three iterations are required to obtain a converged solution.

3.2 Gross displacement analysis

The suggested procedure involves carrying out 'standard' mechanism analyses sequentially. i.e:

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(i) Perform 'standard' mechanism analysis ofstructure (ia) Iterate to determine the effect ofmaterial crushing at the lunges (ii) Magnify 'infinitesimal' displacements to determine new positions of blocks (iii) Modify soil-structure pressures as necessary (iv) Repeat trom step 1 [step (ia) not required if displacement increment small]

Clearly to obtain accurate results the above procedure relies on the use of relatively smal! displacement increments in step (ii).

4. APPLICATIONS

4.1 Masonry Arch Bridges

4.1.1 Laboratory Tests - 3m single span arch ribs

In order to check that the gross displacement arialysis method, as implemented in a computer program, was functioning correctly, the load vs. displacement responses of two 3m span bare arch ribs, tested in the laboratory were considered. As no backfill material was present, there was no soil-structure interaction.

Figure I shows an elevation of the ribs, which were 215mm wide. The two rings of arch I were mortar bonded, the two rings of arch 2 were unmortared.

Fig. 1. Elevation of arch ribs tested in the laboratory

Figure 2 shows the experimentally recorded load vs displacement plots for both the arch ribs, together with the predicted gross displacement analysis results (radial displacement measured on intrados at quarter span). It is evident that except for the initial 'elastic' deformation there is reasonable agreement between the experimental and predicted load vs displacement plots. This provides some confidence that the gross displacement method is performing as required. It is also evident that these bare arch ribs are fairly sensitive to smal! displacements. For example 20mm radial displacement at quarter span (= span/150) reduces strength by some 48 percent in the case of the mortar bonded rib . The rib built with ring separation is somewhat less sensitive to the gross displacements.

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5.0 4.5 4.0 3.5

z 3.0 2:S ""O 2.5 co 2.0 o

....J 1.5 1.0 0.5 0.0

O

Gross displacement . . . . f mechanism solutions -. . .

5

. . .

---

10

Displacement (mm)

15

...

20

Fig. 2. Gross displacement mechanism & experimentalload-displacement curves

4.1.2 Laboratory Tests - 3m single span arch bridge

A number of single-span brickwork arch bridges have been tested in the laboratory by Melbourne and Gilbere. In order to maintain comparability with the arch ribs described in section 41.1, a 3m span brickwork arch bridge with detached wa\1s is selected for further study (designated as bridge no . 3-1). The geometry of the arch barrei of this bridge was identical to that of the arch ribs except that the width was much greater at 2880mm. The bridge was backfi\1ed with a crushed limestone filI material.

BackfilI pressures were carefu\1y monitored in the case oftwo similar 3m span bridges built with attached spandrels, and Fig 3 shows the horizontal backfi\1 pressure vs. barrei rotation relationships fo! these bridges (also shown on this figure is the horizontal backfi\1 pressure vs. net c10sing barrei rotation plot recorded above the South pier of multi-span bridge no . 2. This plot wi\1 be referred to later). The values given for backfi\1 pressure are averaged over depth. This was justified by the fact that there was no evidence of a consistent variation of horizontal soil pressure with depth, for the model bridges tested. Note that becâuse the pressure ce\1s used to obtain the plots shown were positioned vertica\1y above the arch spl1ngings, the magnitudes of the pressures shown in Fig. 3 are likely to underestimate those pressures actually exerted on the arch barrei (because of dispers;.:ln) . Thus, for the purposes of analysis, a horizontal pressure of uniform magnitude 55kN/m2 was applied to restrain sections of the arch barrei rotating into the backfi\1 material for the 'standard' mechanism method analyses. The predicted bridge carrying capacity was 553kN (cf. 540kN recorded experimenta\1y).

For the gross displacement mechanism analyses a maximum pressure of 55kN/m2 was maintained, but it was assumed that this pressure would be mobilized gradua\1y. The experimenta\1y recorded build up in pressure was found to be non-linear, approximately described in this case by the empirical relationship : pressure P = Ppe•k(1 - (l - 2R)\ where R is the net barrei rotation, in degrees. According to the relationship, peak pressures wi\1 be mobilized when barrei rotation equals 0.5°. Figure 4 shows the

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Fig. 6. Predicted displaced structure, showing hinge locations

It is of interest to note that when using the gross displacement mechanism method some movement of the North pier of the bridge is predicted to occur, although the magnitude of the predicted movement at the pierhead of l.4mm is significantly lower than the recorded movement of approximately 5mm.

4.2 Masonry Tunnel Linings

Nineteenth century engi neers adopted many different geometries for their tunnel linings. Normally geometry was governed by end use rather than to provi de the optimum shape to carry soi! pressures (for example some sewer tunnels were tall and narrow specifically to allow a person to walk through the tunneI8

). By the end oi the century however, engineers had a c1ear understanding of the importance of tunnel shape9

For tunnels at shallow depths 'elastic' deformation ofthe masonry might reasonably be considered small, allowing the mechanism method of analysis to be used. Knowledge of the surrounding soil pressures is c1ear1y required in order for an analysis to be carried out. Additionally, in practice when ground movements lead to structural movements, the distribution of pressures will often change accordingly; this points to the need for a complex linked soil-structure analysis scheme. However, it has been suggested (e.g. refer to Hansmire lO

) that the ratio of horizontal to vertical soil pressures in c1ay will often tend towards 1.0, as movements of the structure commence. This latter value is therefore used throughout for the simplified analyses presented here.

As an initial assumption, vertical and horizontal pressures which were ·both equal in magnitude and uniform in distribution were applied to the exterior of the sample tunnel lining considered here. The sample lining was of circular cross section, with internai diameter 2356mm. The wall thickness was taken as 222mm. The lining was considered in turn as comprising ofa single ring ofblocks, and oftwo separate rings.

The magnitudes ofthe pressures applied corresponded approximately to a depth of 5m in Lonrlon Clay. In order to determine the amount of horizontal movement required in order to cause collapse, the tunnel lining was initially analysed in its undeformed state by applying a loadldisplacement to the crown to obtain a predicted collapse mechanism. The lining was then deformed in accordance with this collapse mechanism, and then re-analysed. This process was repeated as necessary until the lining was

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predicted either to be elose to collapse, or was predicted to have collapsed already. Figure 7 shows the predicted tUl11'el shapes immediately prior to collapse.

limiting horizontal movement =114mm limiting horizontal movement = 87mm

Fig. 7. Predicted Iimiting structural deformations ofbrickwork tunnellinings

In this case it was found that the limiting horizontal deformation of the lining with debonded rings was only 25 percent lower than that of the fully bonded lining. This finding appears to confirm that structures containing debonded rings are remarkably tolerant to gross displacements.

When the pressures on the exterior of the lining were reduced (corresponding to shallower depths), the ability of the tunnel linings to accommodate movement was predicted to diminish. This was because the self weight of the masonry had proportionately more influence on the analysis. However, at shallow depths the idealized distribution of soil pressures becomes increasingly unrepresentative. Investigations are continuing.

5. CONCLUSIONS

For most short span bridges subjected to applied loading, elastic shortening of the masonry is a second order effect, and the mechanism method which does not require assumptions about elastic effects, instead homing in directly on the collapse mechanism, should be regarded as a useful analysis tool. The gross displacement mechanism approach allows one of the perceived shortcomings of the ' standard ' mechanism method to be at least partially addressed, allowing the effects on bridge carrying capacity of the progressive mobilization of horizontal soil pressures to be accounted for. However in this context the method, in common with elastic methods, will only provide reliable results provided the analyst has a sound knowledge of the behaviour of the backfill when subject to structural movements. This point is an important one, and in future more attention should be devoted to the geotechnical aspects of arch bridge behaviour to ensure simplified 'routine' analysis tools provide more rational results.

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On a practical levei, given that the computations required for a standard mechanism analysis rnight typically take on1y a small fraction of a second on a modem PC, the dozen or so steps in an iterative, gross displacement analysis, might be carried out in less than a second or so (i.e. practically instantaneously).

When the gross displacement method was used to predict the magnitudes of the load carrying capacities of laboratory model arch bridges it was found that these were between 8 percent and 12 percent lower than those corresponding to the 'standard' mechanism method. This difference is relatively small, given the nature of other uncertainties. the magnitude of this difference will in general depend on how rapidly structural movements lead to a build up ofhorizontal soil pressures, for a given bridge.

Finally, the gross displacement mechanism method has also been used to examine the stability ofbrickwork tunnellinings subject to ground movements. Indications are that this technique may prove useful in situations where use of more complex and computationally expensive methods of analysis cannot be justified.

6. ACKNOWLEDGEMENTS

The arch bridge tests referred to in the paper were carried out under the overall supervision ofprofessor Clive Melbourne, present1yat the University of Salford, UK.

7. REFERENCES

1. Heyman, J, "The Masonry Arch", Ellis Horwood, Chichester, 1982.

2. Advice note BAI6/93, "The Assessment of Highway Bridges and Structures", Department of Transport, London, 1993.

3. Page,1. (ed), "Masonry Arch Bridges", TRL, HMSO, London, 1993.

4. Melbourne, C. Gilbert, M. and Wagstaff, M., "The Collapse Behaviour of Multi-span Brickwork Arch Bridges", to appear in The Structural Engineer.

5. Harvey, W. F., "The Application of the Mechanism Method to Masonry Arch Bridges", The Structural Engineer, 66, No.S, 1988, pp77-84.

6. Gilbert, M. and Melbourne, C., "Rigid-block Analysis ofMasonry Structures", The Structural Engineer, Vol. 72, No. 21, 1994, pp356-361.

7. Melbourne, C. and Gilbert, M., "The Behaviour ofMulti-ring Brickwork Arch Bridges", The Structural Engineer, Vol.73 No.3, 1995, pp39-47.

8. Reed, G. F., "Sewer Dereliction and Renovation", in Proceedings of International Conference on Restoration of Sewerage Systems, Thomas Telford, London, 1982, pp267-281.

9. Alexander, T. and Thomson, A.W., "Scientific Design ofMasonry Arches", Dublin University Press, 1900.

lO. Hansrnire w., "Example Analysis for Circular Tunnel Lining", in "Tunnelling in Soil and Rock", ASCE 1984.

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