This paper deals with the way of calculating
the load-bearing capacity of masonry arch railway
bridges. It reviews the basic aspects of structural behaviour
of these bridges, such as material non-linearity
of masonry and interaction with the soil. Paper shows,
how to include second order analysis in the calculation,
because in some cases it might have non-negligible influence.
It reminds the requirements of standards and
shows, how to calculate the load-bearing capacity in
accordance with these requirements with influence of
mentioned non-linearities

Vokál, Marek

Drahorád, Michal

English

Stadtarchiv Solingen, Bergische Arbeiterstimme 15. März 1918 Literaturempfehlungen für Säuglings- und Kleinkinderpflege  Für unsere Arbeiterinnen und Arbeiterfrauen  sind zwei wichtige Bücher erscheinen, welche wir den-  selben nicht warm genug empfehlen können, und deshalb  in unserer Buchhandlung vorrätig halten. Das erste ist                                    betitelt:  Hilfsbuch für Säuglingspflege  Von Professor Dr. Rud. Hecker und Schwester Berub     Woerner. – 56 Seiten stark mit 44 A..

Stadtarchiv Solingen

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15. März 1918

Stadtarchiv Solingen, Bergische Arbeiterstimme 15. März 1918 In Solingen diskutierten mehrere gewerkschaftliche Organisationen und Verbände die durchgehende Arbeitszeit bzw. den Achtstundentag.      Um die durchgehende Arbeitszeit.     Gestern abend tagte im Hotel „Monopol“ eine Sitzung der Vor-  stände sämtlicher hier vertretenen kaufmännischen Organisationen,  und von Vertretern des Gewerkschaftskartells, des Metallarbeiter-  und des Industriearbeiter-Verbandes, die sich mit der Einführung ..

Stadtarchiv Solingen, Bergische Arbeiterstimme 15. März 1918 Ratschläge der städtischen Gartenbauberatungsstelle Solingen zur anstehenden Gemüseaussaat     Solingen. Die städtische Gartenbauberatungs-  stelle hat ihre Samenverkaufsstellen mit den jetzt  nötigen Sämereien versehen, so daß die Abnahme er-  folgen kann. Da überall Mangel an Samen herrscht, heißt es spar-  sam wirtschaften. Von allen Kohlarten, Kohlrabi usw., deren  Anzucht am besten von geschulten Gärtnern erfolgt, wird kein  Sa..

Stadtarchiv Solingen, Bergische Arbeiterstimme 15. März 1918 Eine Walder Hausfrau bringt ihren Unwillen über die Qualität und Quantitat des Walder „Wochenspeisezettel“, verglichen mit dem Solinger, zum Ausdruck    Wald. Zur Lebensmittelversorgung. Eine  Hausfrau schreibt uns: Mit Neid sehen unsere Hausfrauen den  Solinger Wochenspeisezettel durch und vergleichen ihn mit dem,  den unser Lebensmittelamt herausgibt. Unsere Frauen stellen  dann jede Woche fest, daß der Walder Speisezettel in der ..

Stadtarchiv Solingen, Bergische Arbeiterstimme 15. März 1918 Die Jungmänner Solingens werden aufgerufen ihre Sachen abzugeben.    Jugendwehr-Solingen. Am Montag den 18. März treten alle  Jungmännern der Kompagnien 285, 286 und 287 (Stammkompagnie,  1. und 2. Kompagnie), am Mittwoch den 20. März alle Jungmannen  der Kompagnien 288, 289 und 290 (3., 4. Kompagnie und Tambour-  korps) vollzählig um 2 Uhr auf dem St. Seb[astianus]-Schützenplatz an zur  Abgabe der Sachen. Die Jungmannen, die wegen ..

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SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 18 | NUMBER: 2 | 2018 | DECEMBERThe Load Bearing Capacity of Railway MasonryArch BridgesMarek VOKAL, Michal DRAHORADDepartment of Concrete and Masonry Structures, Faculty of Civil Engineering, CTU in Prague, Thakurova 7,166 29 Praha 6, Czech Republicmarek.vokal@fsv.cvut.cz, michal.drahorad@fsv.cvut.czDOI: 10.31490/tces-2018-0019Abstract. This paper deals with the way of calculat-ing the load-bearing capacity of masonry arch railwaybridges. It reviews the basic aspects of structural be-haviour of these bridges, such as material non-linearityof masonry and interaction with the soil. Paper shows,how to include second order analysis in the calculation,because in some cases it might have non-negligible in-fluence. It reminds the requirements of standards andshows, how to calculate the load-bearing capacity inaccordance with these requirements with influence ofmentioned non-linearities.KeywordsLoad-bearing capacity, masonry, material non-linearity, vault.1. IntroductionWhen evaluating existing bridges, the requirement forsufficient mechanical resistance and stability is ex-pressed by the maximum load that the structure is ableto transfer safely - the parameters of the rail vehiclewhich can go across the bridge under the defined con-ditions, ie the load bearing capacity. Determining theload bearing capacity of the existing bridge is governedby similar principles as the design of new structures(see applicable technical standards and regulations EN,DIN, TKP SŽDC and MVL). In the case of masonryvault structures this task is complicated by the struc-tural behaviour and by the typical property of the ma-terial - a very small tensile strength. Due to these facts,the procedures for determining the load bearing capac-ity leads to non-linear analysis, which must consider alarge number of frequently variable parameters. It istherefore very difficult to set up a simple analyticalmodel and, for modelling of usual conventional struc-tures, therefore, special programs developed directlyfor vault structures are often used.The normal load bearing capacity of the bridgeZLM71 (hereinafter referred to as "load bearing capac-ity") is a dimensionless variable which expresses theratio of the vertical effects of the variable load by railtransport (in terms of the relevant ultimate or service-ability limit state) to the effects caused by the loadmodel 71 on the bridge object.2. Structural behavior2.1. Behaviour of masonrystructuresFor performing an analysis and design of the structure,the material - mortar and masonry elements - is homog-enized in a suitable manner in such a way as to preserveits properties in relation to the real behaviour of thestructure or its part. It is assumed that the dimensionsof the masonry elements and joints between them donot significantly affect the distribution of stress in thebrick element. The real stress-strain diagram of themasonry shows non-linear behaviour, particularly dueto weak tensile strength - see [1]. In this work, it isconsidered that the material acts only in compressionand when the tensile stress occurs, cracks occur andopen up. If, subsequently, (e.g. in another load combi-nation) the tension stresses in the cross section disap-pear, the cracks close and the cross-section acts againas full. The stress-strain diagram is considered linearin compression, the introduction of a non-linear dia-gram is not decisive from the engineering point of viewat this stage of the research.c© 2018 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 66SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 18 | NUMBER: 2 | 2018 | DECEMBERFig. 1: Mechanism of change of cross-sectional characteristicsand geometry of structure due to loading.Fig. 2: Scheme of the actions of the backfill on the vault (i -distribution of the loading load, ii and iv - the weight ofthe backfill material, iii - the active earth pressure at thedeformation point of the vault away from the backfill,v - the passive pressure at the point of deformations ofthe vault into the backfill).2.2. Backfill of the vault and itsinteraction with the vaultThe backfill of the vault is an integral part of the struc-ture. The effects of backfill on a vault can be dividedinto several basic kinds (see also Fig. 2 and [4] ):• the distribution of concentrated loads from axlesof load model into the back side of the vault, whichreduces the local loads of the vault structure,• self-weight,• active ground pressure acting at points, where thevault deforms away from the backfill; the pres-sure decreases with the increasing separation ofthe vault from the backfill,• passive earth pressure acting at the points of theconstruction, where the vault moves to the backfilland is therefore stabilizing structure; the pressureis increased by pushing the construction into thebackfill.The material used in the backfill is important, be-cause it determines the behaviour of the entire struc-ture. Usually, local materials were used to backfill ofthe arch, incoherent materials were preferred.Fig. 3: Stress distribution of masonry cross-section at the ulti-mate limit state.2.3. Ultimate limit stateThe ultimate limit state (ULS) describes the behaviourof the structure in the state just before collapse, i.e. onthe bearing capacity (load factor equals to 1.35 for deadload and 1.45 for live load). It is generally assumed thatthe structure is in a plastic state, there is a significantmaterial damage and the full development of plasticjoints. Conditions of resistance of the structure undera given applied load are then expressed according to[2] as:NRd = fdb (h− 2eu) (1)VRd = (fvk0 + 0.4σd) b (h− 2eu) /γM . (2)fd is the design strength of masonry in compression,fvk0 is the characteristic value of initial shear strengthat normal stress equal to 0, b, h is width and height,respectively, eu is the eccentricity of the resultant pres-sure force in cross-section in the ultimate limit state,σd is the compressive stress in the compressed area atthe ultimate limit state, m is the coefficient of frictionin the masonry joint, γM is the factor of the material.The distribution of stress in the cross-section is consid-ered constant, see Fig. 3.2.4. Serviceability limit stateThe serviceability limit state (SLS) describes the be-haviour of the structure under usual operating condi-tions. Fulfilling the conditions of serviceability limitstate provides the required properties of the structurethroughout its lifetime. In terms of serviceability limitstate, crack width and structural stress under usualload are verified (load factor equals to 1.0). In terms ofverification of vault structures, it is necessary to ver-ify the maximal stresses in the cross-section and theheight of the compressed area of the cross-section ac-cording to ČSN EN 1996-1-1. The behaviour of thecross-section is considered elastic with a linear distri-bution of stress in the compressed part of the crosssection and excluded tensioned part (see Fig. 4).c© 2018 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 67SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 18 | NUMBER: 2 | 2018 | DECEMBERFig. 4: Stress distribution of masonry cross-section at the ser-viceability limit state.σn,max =NEk3b (h− 2e) ≤ 0.45fk (3)hc ≥ h2−→ e ≤ h3(4)e =MEkNEk+ einit =MEkNEk+LCr450(5)LCr is critical buckling length, MEk, is momentcaused by load, NEk is normal force caused by load.3. Ways of modelling3.1. General assumptions forcalculation of programLimitState: RING and thisstudy• The maximum span of the bridge L is 20 m.• Minimal arch sagitta f is L/6 (not suitable forvery flat arches and especially their middle parts).• The structure must have a backfill.• The thickness of the backfill is less than L/2(providing the dominant influence of the variableload).• The masonry must have at least approximatelyperpendicular joints.• The impact of parapet and spandrel walls is ne-glected.3.2. Content of this studyThe load bearing capacity of the bridges with spansranging from 2.5 to 10 m has been calculated in the cri-terions of ULS and SLS. Three different relevant vaultthicknesses, masonry strengths 2, 4, 6, 8 and 10 MPa,the sagitta L/2 and L/4 and height of the backfill 0.5,1.0 and 1.5 m were considered. These properties areFig. 5: Plot of load bearing capacity versus compressivestrength of masonry in ULS.common in these stuctures according to authors’ ex-perience (authors have already calculated tens of loadbearing capacities of existing arch bridges). For thestudy of structures in ULS, the vaults were modelledin the LimitState: RING program and for the studyof SLS the vaults were modelled in program Scia Engi-neer. In addition, a Matlab script was created, whichcalculates the deformations and internal forces on thevault structure including the effect of mentioned non-linearities. The calculation is iterative, and at theoccurrence of tensile stresses, the cross-sectional areais reduced, and thus the geometry of the structurechanges in the next calculation step. The algorithmiterates until the calculation converges to the equilib-rium position and the geometry of the structure doesnot change. See chapter 3.5. Since the modelling inthe program LimitState: RING is less time-consumingthan other methods, it was the task of this study todetermine which of the limit states is decisive and todetermine the value of the load bearing capacity ac-cording to the requirements of ULS, in which the loadfactor according to the requirements of SLS (Z1) isequal at least one.c© 2018 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 68SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 18 | NUMBER: 2 | 2018 | DECEMBER3.3. ULSLimitState: RING is a program that checks the loadbearing capacity of the vault in the plane of the longitu-dinal section of the bridge structure, including the loaddistribution by the backfill. It uses equilibrium condi-tions on the parts of vault as a rigid bodies, the struc-ture is divided into rigid bodies depending on creatingthe plastic hinges in locations with the lowest heightof compressed area. It addresses also other influenceswhich are not taken into account in this study: possibledisplacements of supports, true geometry, weakeningof joints by the extraction of mortar and multi-spanvaults. The disadvantage of the program is, that itdoes not consider the strength failure of the arch. Theresult of study using LimitState: RING can be seen inFig. 5. For detailed results see [3].3.4. SLS – Scia EngineerTo write a study on checking the serviceability limitstate with different parameters of vaults, separate mod-els for each vault was created in program Scia Engineer.The individual masonry elements were modelled by 2Delements – plane stress/plane strain, joints were mod-elled as beam members - 10 elements across the thick-ness of the vault. These beam members were entereda non-linearity "compression only". This has resultedin the masonry behaviour according to chapter 2.1. Toconsider the behaviour of the soil, the backfill was mod-elled also by the 2D element, with the behaviour con-sidered linear. Since the program Scia Engineer cannotperform a non-linear calculation together with the cal-culation of the moving load, manual loading by forceand continuous load of the model 71 in several (about10) loading states with different load positions were al-ways entered. Then the load factor was changed man-ually until either the limiting stress was reached in oneof the bars representing the joint - Eq. 3, or there wasa tension in the eleventh control beam which is in themiddle of the height of each joint - this would meanbreaking the requirement according to Eq. 4. Thisfactor was recorded in the graph - see Fig. 8 and 9.3.5. SLS – MatlabThe script in Matlab model the arch with beam ele-ments. The vault support was for this study chosenat both ends of the arch pinned. The backfill is mod-elled as a media, in which the load is distributed. Justas in the LimitState: RING program, the Bousinesqdistribution is considered – see Fig. 11.According to the distribution angle in the soil, it isdetermined which elements are loaded by the axle ofload model and the vertical load intensity at the par-Fig. 6: Global model of the structure.Fig. 7: Detail of the joint – ten beam elements across the heightof the joint and eleventh in the middle.Fig. 8: Plot of load bearing capacity versus compressivestrength of masonry in SLS for f = L/2.c© 2018 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 69SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 18 | NUMBER: 2 | 2018 | DECEMBERFig. 9: Plot of load bearing capacity versus compressivestrength of masonry in SLS for f = L/4.Fig. 10: Distribution of load.Fig. 11: Vertical load intensity in X location.ticular node of the structure is determined accordingto the following relationship:σz =qpi[α+ sinα cos (α+ 2β)] . (6)q is line continuous load on the surface and the anglesα and β see in the Fig. 11.The backfill is also modelled by horizontal springs toprevent the structure from the deforming to the back-fill, thus transmit only the pressure. Their stiffness ischosen as the size of Edef – modulus of deformation ofthe soil multiplied by the horizontal projection of theelement. The magnitude of the force they can transferis in reality limited by the passive pressure, for the pur-poses of this script it is not limited for simplicity, (inthe serviceability limit state there must never be anyincreased deformations in which the passive pressureis activated). Given the nature of the problem (abovementioned) solving is performed by method of inverseiterations. This method is explained in [5]. The systemof equations of the following form is solved:(K−KG) r = 0. (7)K is global stiffness matrix considering the boundaryconditions (regular matrix),KG is a geometric stiffnessmatrix and r is a vector of displacement. Geometricstiffness matrix for a beam hinged on both sides is de-rived from a simple equilibrium conditions – see Fig.12.KG =Nl0 0 0 0 0 00 1 0 0 −1 00 0 0 0 0 00 0 0 0 0 00 −1 0 0 1 00 0 0 0 0 0 . (8)N is normal force in the beam and l is a beam length.c© 2018 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 70SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 18 | NUMBER: 2 | 2018 | DECEMBERFig. 12: Derivation of geometric stiffness matrix for a beamhinged on both sides.Geometric stiffness matrix for a beam fixed on bothsides:KG =Nl0 0 0 0 0 00 36 3l 0 −36 3l0 3l 4l2 0 −3l −l20 0 0 0 0 00 −36 −3l 0 36 −3l0 3l l2 0 −3l 4l2 . (9)Equation 7 can be modified for the purpose of cal-culation to:Kr = f + fekv. (10)f is a load vector, and fekv is vector of equivalentload calculated as follows:fekv = KGr. (11)In the (i+1)–th step of iteration we can calculate:Kri+1 = f +KiGri. (12)i is number of step. It is used in upper indices. Anal-ysis uses following algorithm:Kr0 = f (13)Kri+1 = f +KiGri (14)||ri+1 − ri)|| < . (15)Equation 13 is solved first to obtain r0, from whichN0,K0G are calculated. Then, from Eq. 14 we obtainri+1 – vector of displacement of next step. This stepis repeated until the criterion of Eq. 15 is fulfilled. InMatlab algorithm, error is chosen to be smaller than = 10−8 m. As an example, the vault with the thick-ness 0.5 m, diameter 5 m and sagitta 1.9 m was cal-culated. It was loaded by self-weight and axles of loadmodel LM71. The height of backfill at the top of vaultis considered 2 m. The result is shown by Fig. 13, 14and 15.The calculation converged within eight steps.Fig. 13: The resulting thickness of the compressed part of struc-ture when loaded by the self-weight of the structureand the backfill.Fig. 14: The resulting thickness of the compressed part of struc-ture under the load of the axles of the LM71 in the lefthalf of the vault.Fig. 15: The resulting bending moment on the three elementsapproximately in the middle of the arch in the individ-ual calculation steps.c© 2018 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 71SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 18 | NUMBER: 2 | 2018 | DECEMBERFig. 16: Plot of load bearing capacity versus height of backfillin SLS.4. Discussion of the result4.1. Scia EngineerFor the various selected values of sagitta f , height ofthe backfill h and strength of masonry fk, a plot ofload bearing capacity versus height of backfill was con-structed – see Fig. 16.Generally, it can be read out from the graphs (butonly in the range of parameters mentioned in chapter3.1.), that:• at lower sagitta, the vault has a higher load bear-ing capacity,• for a higher backfill, the vault has a higher loadbearing capacity,• for a higher vault thickness, the vault has a higherload bearing capacity.The total number of Z1 coefficients that are largerthan 1 in this study is 32. The value varies from 1to 3.22. From Fig. 8 and 9 it can be seen that themost tough of all the criteria of limit states consideredis the criterion according to Eq. 3 for the low-strengthof the masonry (2-4 MPa), which determines the max-imum admissible pressures in the masonry, while forthe vaults with a higher strength is the load bearingcapacity limited by the criterion according to the Eq.4 determining the maximum eccentricity of the resul-tant force in the cross section.4.2. MatlabOne vault with one axle position was verified in theMatlab script. To obtain the results of a study ofmultiple vaults of different parameters, another Matlabscript must be created, that will call the original scriptmany times with different parameters and store the re-sults different load positions and different parametersof vaults. So far, we can say that the results obtainedso far are real and that it is easy to distinguish whetherthe loaded structure fulfils the requirements after fin-ishing the calculation. Only the thickness, diameterand arch of the vault, the characteristic strength of themasonry and the height of the backfill at the top ofvault are specified by user in the script. Entering thisdata is therefore very operative and fast. Until now,only circular arches hinged on both sides were verified,the author can modify the shape of the vault, for ex-ample on an ellipse or other shape, or to modify thesupport to fixed or elastic.5. ConclusionA study has been carried out showing how masonryvault structures behave and how they should be anal-ysed. It compares which of the limit states – ULSor SLS – is more tough and therefore results in lowerload bearing capacity of masonry vault railway bridges.It has revealed that the requirements set by SLS aretougher than ULS requirements. It also showed thegeneral rules, how the differing parameters (thickness,sagitta, height of the backfill) influence the load bear-ing capacity.References[1] DRAHORAD, M. Load–Bearing Capacity of Ma-sonry Arch Bridges. International Conference En-gineering Mechanics . 2013, vol. 19, iss. 1, pp. 33–34. ISSN 1802-1484.[2] EN 1996–1–1. Design of masonry structures.ÚNMZ, 2005.[3] DRAHORAD, M. The guide to use of softwareLimitState: Ring for employees of SŽDC.[4] LimitState:RING Manual. Available at:http://www.limitstate.com/files/pdf/ring/RING_User_Manual.pdf[5] Linear stability and theory of II. Order. Avail-able at: http://mech.fsv.cvut.cz/~leps/teaching/ankc/cviceni_08.pdfc© 2018 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 72

Transactions of the VŠB - Technical University of Ostrava Civil Engineering Series