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

The analysis of the strains and stresses of the reinforced brick vaults

J. Jasienko & Ł. BednarzWroclaw University of Technology, Wroclaw, Poland

ABSTRACT: The article specifies the aspects of the analysis of strains and stresses of the reinforcement ofthe brick vaults. It introduces the problems of restoring and reconstruction of the vaults and presents variousways of reinforcing these types of brick constructions applied to the dorsal side, minding at the same time thatfrescos and other decorations are normally applied to the intrados side. What is more, the analytical methods andnumerical methods (FEM) used in the analysis of the construction of arches, vaults and domes, are introducedtogether with the findings of the research conducted within the range of technical scale in the laboratory of theEngineering Institute of the Technical University of Wroclaw. In addition, the measurement of the effectivenessof various types of the adopted reinforcements was also made (inserted bar steels, grids and FRP tapes).

1 INTRODUCTION

Keeping and maintaining the monumental construc-tions, particularly the arches and vaults, which wereeither partly or totally devastated, is the problem whichis encountered by more and more people dealing withconstruction reinforcement.

The effective accomplishment of this task requiresan extensive knowledge about the origins of the effectswhich contribute to deteriorating the construction’sproperties, and both the knowledge about the old con-struction methods and modern methods which enableone to prevent those effects and to restore the objectswithin the scope of their former carrying capacity andstiffness.

Implementing the modern methods of restoring andreinforcing the brick constructions or their specificelements (e.g. arches, vaults or domes), leads to theconsiderable decline in costs, allowing at the same timesaving the culture-precious objects.

Reinforcing and restoring the brick constructionsof the arches and vaults are the difficult tasks whichusually require the individual approach to each case.

The analysis of the masonry constructions (espe-cially the monumental arches, vaults and domes) isa very complicated topic because of many problemsresulting from the description and from the simulationof this type of constructions. Moreover, the numericalapproach is also a complex problem (simulation of thematerial which in fact behaves in a nonlinear way), aswell as the limited information about the mechanicalproperties of the materials and the complexity of thegeometrical construction.

2 ANALYSIS OF STRUCTURE

Making the simulation of the masonry construction,there are three major groups of analyses: linear anal-ysis, nonlinear analysis and limit analysis. It is alsoworth mentioning that despite the differences resultingfrom the fact that various types of methods gener-ate different safety coefficients and different finalresults, there are no ample reasons for choosing theone particular method over the others – during theresearches, it was decided that the simplest type ofanalysis will be applied, namely the linear analysis.Such a decision was made because of the applicabil-ity of this type of analysis in the simple constructingsolutions.

2.1 The analytical models of calculation

A mathematical analysis of the curved brick construc-tions placed an important position in the deliberationsof mathematicians and construction theoreticians fora long time. What is more, many instructions sub-stantially based on the engineering practice wereestablished.

Associating the practical knowledge with the theo-retical deliberations, enabled one to the better under-standing of the problem of the static work of arches,vaults and domes. Making the analysis of the currentstate of knowledge about estimating the capacity ofarches, vaults and domes, one can easily notice thatone of the major problems appearing on the level ofthe estimation of carrying capacity is accepting thestatic scheme of the curved brick construction with

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application of the proper method of calculating theinternal forces in constructing elements.

On the way to constructive deliberations aboutthe statistics of the curved masonry constructions,the systematization of those deliberations seems to beintentional. However, this systematization should bedifferent from that which dominates over the histori-cal aspects of the subject. Here, the static scheme of thework of the groups, justifies the division into groupswhich can be attributed to a certain group. The knowl-edge of the static scheme enables the calculation ofinternal forces which are present in elements of thearches, vaults and domes as well it facilitates carryingout the analysis of their work in respect to constructionsecurity and in case of a failure in accepting the propermethod of strengthening.

However, the static calculations are based on vari-ous simplified assumptions which are not truly real.Therefore, such calculations cannot be made withfull precision. Uncertainty of the static calculationsbecomes problematic when weight is taken into con-sideration. The construction weight (especially in rela-tion to monumental buildings) and snow and windburdens, are rather unknown and depend on otherfactors which cannot be precisely calculated. Evenmodel-scale experiment cannot ensure certainty.

The problem which ought to be precisely analysed isthe issue of static calculations of the damaged arches,vaults and domes. It must be specified how the varioustypes of damage (i.e. loosening of squinch from archrib, change of construction geometry) influence theway the internal forces flow through the curved brickconstructions and safety of the entire construction.Although not all the mathematical considerations canbe easily applied, there are a few considerations whichhighly influence the analysis of the curved masonryconstructions’ behaviour. Moreover, further solutionsrelate to various types of the curved brick construc-tions, not only to the one described type of construction(i.e. arches).

The group of major analytical methods com-prises of:

– graphical method of establishing the line of pressurein the arches,

– three-jointed arches calculation method,– Heyman’s non-moment method applied to vaults

and domes.

All of the numerical analyses of the monumentalbuildings related to the static analysis, are encumberedwith a basic error, namely it is very difficult or almostimpossible to describe the resistance and deformabil-ity of materials in present masonry constructions. Ifthis was possible, it would enable taking the correctcalculation model.

The main problem of such approach is the accep-tance of the most precise and efficient model of

Figure 1. Research model of an arch.

Figure 2. Real view of one of the examined arches –arch A2.

material. After a profound study of the subject’sliterary sources, it was decided that the masonrymaterial will be treated as homogenous and isotropicmaterial as it has got the same physical properties inall directions.

3 DESCRIPTION OF TEST MODELS

Laboratory tests were conducted on 7 brick arches(0.12 m thick, 0.77 m wide, 4 m span and 2 m radiuseach – as in Fig. 1 and Fig. 2). Barrel arches built ofmaterials of characteristics similar to those of mate-rials in historical objects (full ceramic bricks of class100 and pure lime mortar). Static pattern of supportand loading realized in the same way in all the archestested.

All of the arches models underwent monotonicallyincreasing static loading which was realized on theentire width of the arch, 1/3 by an actuator until thedestruction of the arch.

Each arch was reinforced in a different way:

– A1 model – the arch without any reinforcement –witness arch,

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Figure 3. Model and arch A1.

Figure 4. Model and arch A2.

– A2 model – the arch reinforced with FRCM –Ruredil X Mesh C10 carbon fibre screen set in theRuredil X Mesh M 25 mortar on the side of theridge,

– A3 model – the arch reinforced with 2 CFRP S&PLamelle CFK 150/2000 carbon tapes (100/1.4 inter-section) placed in a 40 cm axial spacing and withFRCM material – Ruredil X Mesh C10 carbon fibrescreen set in the Ruredil X Mesh M 25 mortar onthe side of the ridge,

– A4 model – the arch reinforced with FRCM –Ruredil X Mesh C10 carbon fibre screen set in theRuredil X Mesh M 25 mortar on the side of the ridgeand on the intrados side,

– A5 model – the arch reinforced with ø 8 steel bars,set in furrows made in the ridge surface of brickarch, put ins with glue made on the basis of Epidian5 epoxide resin,

– A6 model – the arch reinforced with 2 CFRP S&PLamelle CFK 150/2000 carbon tapes (100/1.4 inter-section) glued in a 40 cm axial spacing on the sideof the ridge,

– A7 model – the arch reinforced with a girder madeof GL24 glued wood, joined with the brick arch bythe use of ø 8 steel anchors glued on the side of theridge.

Figures 3–5 present a diagrammatic way of enforc-ing compared with a real view of another arch.

4 LABORATORY FINDINGS

In the course of the researches, the tracks of the staticbalance for all the arches were marked out by mea-suring the intensity of the force loading the laboratorymodels. Figure 7 presents those tracks.

The reinforcing materials were supposed to beset by the dorsal side of the arches or inside thearches’ section. This type of reinforcement is domi-nant in case of the historical arches and vaults whichare vividly decorated on the intrados side. In thesubject’s literary source, one cannot find any simi-lar cross-sectional researches with so many methodsof reinforcing the curved brick constructions whichare acceptable with respect to the doctrine in theconstructional conservation of the historical brickarches, especially with respect to FRCM reinforce-ments (with the use of screens e.g. carbon screensplaced in a mineral die-block). The laboratory find-ings showed that the reinforcement favorably affectsthe carrying capacity and stiffness of the examinedarches.

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Figure 5. Model and arch A5.

Figure 6. Model and arch A6.

Figure 7. Track of the static balance for arches A1, A2, A5 and A6.

Figure 8 presents the laboratory findings of all the7 models of arches with specification of the ana-lyzed arches: A1, A2, A5, and A6. One can alsonotice the considerable growth of the destructiveforce in all the reinforced arches comparing to the

model arch. Table 1 shows the proportional growth offorce.

The displacements of the arches at the applicationforce points with the different levels of loading werealso marked (see Figure 9).

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Figure 8. Frontier amts of destructive force Fmax in the specific laboratory models.

Table 1. Growth of destructive force Fmax.

Failure force Fmax Growth of failure forceArch [kN] [%]

A1 2.882 –A2 10.866 377.03A5 19.314 670.16A6 16.588 575.57

Figures 10–13 show the real schemes and modelsof destruction of the examined arches. One shouldalso notice the transformation of those models ofdestruction. Starting from the typical destruction tak-ing place in the arch A1 and described in literature(the model with 4 articulated joints), the next archesthe forming articulated joints change their place (theyare formed within bearings and under the place ofapplication force. One can also observe the shearingof the arches on the bearings’ areas. In the practicaluse of constructions, there are also very massive wallsat the places of bearings, which are much higher thanarches and which have the additional loadings (e.g.loadings of pinnacles). Those walls prevent the archesfrom a thrust so as the case of shearing in bearingcannot occur.

The reinforced models gain much bigger deflectionsthan the witness arch, even with an additional loading.It relates to especially those models of reinforcementwhich involve using the carbon fibre screens wherethe deformation and destruction proceeds slowly andsmoothly.

There is no any rapid movement related to separa-tion of the reinforced material as it is with the models

reinforced by the CFRP tapes of steel bars alone. Thechange of the schemes of destruction and growth ofthe carrying capacity of the particular models of archesprove the usefulness of the applied reinforcements andthe good quality of their workship.

In the course of the researches, also the defor-mations were measured (arches A5 and A6) and thenormal stresses as well as tangential stresses in theadhesive-bonded joints, carbon tapes (A6) and steelbars (A5) were assigned. The results of those measuresare shown on the Figures 14 and 15 (normal stresses)and on Figures 16 and 17 (tangential stresses).

5 NUMERICAL FINDINGS

It is generally acknowledged that for the detaileddescription of the functioning and destruction ofmasonry constructions, it is necessary to adopt thecalculative models with respect to nonlinearity andbrittleness of walls. However, in the case the analy-sis of the static work of the examined arches, it wasdecided to adopt much less complicated linear-elasticmodels. This was mainly the result of the fact thatone of the assumptions was to enable the analysis ofthe effectiveness of the reinforcements in the commonengineering papers.

On the bases of the created numerical models of thearches, the series of numerical calculations in the areaof flexibility were made with respect to resistance andstrength. Basing on those calculations, the contour lineof deformations and level of stresses for the particularcalculative models was received. Figure 18 shows theexamples of the deformations in the arch A5 whicharose as a result of unloading on 1 kN level. Figure 19

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Figure 9. Relation: force “F” – displacement “u” at the loading application point.

Figure 10. Model and real scheme of the destruction of the arch A1(force: 2,882 kN).

Figure 11. Model and Real scheme of the destruction of the arch A2 (force: 10,866 kN).

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Figure 12. Model and Real scheme of the destruction of the arch A5 (force: 19,314 kN).

Figure 13. Model and Real scheme of the destruction of the arch A6 (force: 16,588 kN).

Figure 14. Normal stresses “σ” in the steel bar at points 1–10 in the sequent amts of the loading with “F” force in the arch A5.

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Figure 15. Normal stresses “σ” in the carbon tape at points 10–13 in the sequent amts of the loading with “F” force in thearch A6.

Figure 16. Tangential stresses “τ” in the adhesive-bonded joint between points 1–10 in the sequent amts of the loading with“F” force in the arch A5.

Figure 17. Tangential stresses “σ” in the carbon tape between points 10–13 in the sequent amts of the loading with “F” forcein the arch A6.

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Figure 18. Deformations in the numerical model of the arch A5 – the loading level – 1 kN.

Figure 19. Comparison of the deformations at 5 rodman points in the arch A2 (for the real model – a full line and for thenumerical model – dashed line.

shows the comparison of the curves of deformationsat rodman points for the real model (a full line) andthe numerical model (dashed line).

6 CONCLUSIONS

It is easy to notice that the schemes of destruc-tion in all of the reinforced arch models changed incomparison with the A1 arch which was not reinforced

(Figure 10–13). It shows that the static work of the rein-forced arches changed and it also proves the need forthe applied reinforcements. One should also considerthe schemes of destruction of the joints between anenforcing material and an arch:

– within the A2 arch it was noticed that the carbonfibre screen loosened from masonry constructionwith brick elements – it shows that destruction(discerption) took place inside the brick,

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– within the A5 arch it was noticed that steel barsloosened from masonry construction with brick ele-ments – it shows that destruction (discerption) tookplace inside the brick and glue composition fulfilledits role,

– within the A6 arch it was noticed that the FRP tapeloosened from the masonry construction it showsthat destruction (discerption) took place inside thebrick.

All reinforced models of arches considerablyincreased their carrying capacity. As it is presentedin the picture 8, the increase of carrying capacity isvery huge, ranging from 3,77 (A2 arch) through 5,76(A6 arch) up to 6,70 (A5 arch).

The conclusion of the performed research on themodels in technical scale shows, it may be stated that:

– all research models were performed with refer-ence to the practical use of analysed techniques inconservation of the masonry constructions,

– the research proved the possibility and usabilityof FRCM enforcement application, that is the car-bon fibre screens submerged in mineral mortarin order to enforce the historical curved brickconstructions,

– the serving facility of FRCM systems in arches andvaults constructions was confirmed,

– the research proved possibility and usability of jointapplication of CFRP tapes and FRCM system glu-ing techniques and, in result use of the positiveproperties of both aforementioned systems,

– the research proved the possibility and usabilityof the application of steel bars glued inside theconstruction (however, it seems to be very time-consuming and labour-consuming),

– the research proved the possibility and usabilityof the application of glued wood elements (joinedwith the construction by the glued steel anchors)in order to stiffen the arch layers (it should benoticed that there are some difficulties with match-ing of the curvature of enforcing elements made ofglued wood to the curvatures of arches and vaultsconstructions),

– the intersections of CFRP tapes and steel bars arevery rarely used; the best solution of this problemseems to be their preliminary conjunction whichenables the substantial use of their carrying capacitywhat further results in increasing the effectivenessof reinforcement. However, such solution is barelyfeasible in this type of constructions.

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