Chen, Jian-Fei

Ooi, Jin

Stratford, Tim

Tao, Y.

Chen, Jian Fei

Ooi, Jin Y.

Queen's University Belfast Research Portal

Numerical modelling of a large scale model masonry arch bridge

The behaviour of masonry arch bridges with sand backfills is extremely complex because it involves not only complex materials including sand, brick units and mortar, but also many highly nonlinear interfaces between these materials. This paper summarises several FE modelling techniques for such a complex bridge at different levels of sophistication. Their accuracy and suitability are assessed by comparing their predictions with a large scale model bridge test results

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     Edinburgh Research Explorer                                      Numerical Modelling of a Large Scale Model Masonry ArchBridgeCitation for published version:Tao, Y, Chen, J-F, Stratford, T & Ooi, J 2012, 'Numerical Modelling of a Large Scale Model Masonry ArchBridge' Paper presented at Structural Faults and Repair 2012, Edinburgh, United Kingdom, 3/07/12 -5/07/12, .Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher's PDF, also known as Version of recordGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 28. Apr. 2017NUMERICAL MODELLING OF A LARGE SCALE  MODEL MASONRY ARCH BRIDGE  Yi Tao, J.F. Chen, T.J. Stratford & J.Y. Ooi The University of Edinburgh Institute for Infrastructure and Environment School of Engineering Edinburgh EH9 3JL, Scotland, UK y.tao@ed.ac.uk, j.f.chen@ed.ac.uk, t.stratford@ed.ac.uk, j.ooi@ed.ac.uk  KEYWORDS: Masonry arch, finite element model, interface element, backfill, model strategies.   ABSTRACT The behaviour of masonry arch bridges with sand backfills is extremely complex because it involves not only complex materials including sand, brick units and mortar, but also many highly nonlinear interfaces between these materials. This paper summarises several FE modelling techniques for such a complex bridge at different levels of sophistication. Their accuracy and suitability are assessed by comparing their predictions with a large scale model bridge test results.  INTRODUCTION Masonry arch bridges form a significant part of the transport network from ancient times to modern days. A masonry arch usually fails in a hinge mechanism. The presence of the backfill can significantly affect the arch behaviour in a number of ways. The fill material increases the stability of the arch, either directly by inducing additional compression area in the arch, or indirectly by allowing a distribution of concentrated forces over larger areas. The passive pressure in the structure also provides additional lateral restraint to the arch. This paper presents a summary of an extensive numerical study on the behaviour of masonry arch bridges using four FE models of increasing complexity.  The goal was to develop a deeper understanding of the loading and failure mechanism of masonry arch bridges and to select an advanced model suitable for modelling of fibre reinforced polymer (FRP) strengthened masonry arch bridges in a follow-on study.  THE TEST BRIDGE A two-span one-third scale model masonry arch bridge was tested at the University of Edinburgh to investigate its behaviour with and without FRP strengthening. The two semi-circular single-ring arches were constructed from concrete bricks and cement mortar, with dry sand backfill to a height of 350 mm above the crowns (Figure 1). Initially, each of the spans was individually loaded at its quarter span location (Figure 1) near to collapse to investigate the loading response and to establish a four-hinge mechanism. FRP strengthening was then applied to the two arches, and each of the spans was again tested separately until the failure of the strengthened system. Further details of the experiments can be found in Tao et al. (2011).  FINITE ELEMENT MODELS Four finite element (FE) models were developed by adopting both macro and micro-macro modelling strategies with various constitutive laws for the masonry arch. The details of each model are summarized in Table 1 accompanied by sketches in Figure 2. The model using the macro modelling strategy is referred as the homogeneous model in this paper because this approach treats the masonry work as a homogeneous isotropic continuum without distinguishing the masonry units, mortar and interfaces between them (Figure 2b). A traction-opening model and a damaged plasticity constitutive model were used to simulate the mortar interface for two meso-interface models in which the units are expanded to include the mortar layer thickness and the interfaces between units are modelled (Figure 2c). A detailed solid model, which models the actual units and mortar using solid elements (Figure 2d), was finally developed to overcome the shortcomings of both the homogeneous and meso-interface models.  Figure 1: Schematic arch model (dimensions in mm)   Figure 2: Modelling strategies for masonry arch: (a) local details of masonry work;  (b) homogeneous model; (c) meso-interface model; (d) detailed solid model  Table 1: Summary of FE models for the masonry arch bridge Model Brick unit Mortar and interface Unit size Constitutive model Representation Constitutive model Homogeneous model NA Concrete damaged plasticity model Not modelled explicitly Meso interface models Traction-opening  (TO) interface model Actual size + mortar thickness Zero thickness interface element Traction-opening Damaged plasticity (DP) interface model Damaged plasticity Detailed solid model Actual size Solid element with actual mortar thickness Concrete damaged plasticity The test arch bridge was modelled as a 2D plane strain problem using the commercial FE software ABAQUS (2007). The whole two-span bridge was modelled because the external load was only applied to a single span in each test so the problem was unsymmetrical. In the FE models, the timber retaining walls and concrete abutments were modelled as linear elastic materials. The sand backfill was modelled using the Mohr-Coulomb plasticity model with a small cohesion c=100Pa and an internal friction angle Φ=33° which was obtained from laboratory tests. The interaction between the sand and the arches, and between the sand and the timber retaining walls were modelled using the contact interface elements in all the FE models with a friction angle between the contact pairs being that of the internal friction angle of sand (i.e. 33°).  The abutments were assumed to be fully restrained in all directions because they were connected to the strong floor using steel bolts as shown in Figure 1. The timber walls were restrained at the top and the bottom laterally using spring boundary elements (Spring 1 element in Abaqus) to model the partial restraint in the test setup. The stiffness of the springs was determined from an analysis of the elastic timber wall (see Tao, 2012). The FE mesh for detailed solid model is shown in Figure 3 and that for other models can be found in Tao (2012). The material properties involved are summarized in Table 2.   Figure 3: The FE mesh adopted for the detailed solid model  Table 2: Material properties of the masonry arches  Parameter Value Masonry work Compressive strength fcm  25 MPa Young’s modulus Em  16 GPa Brick Unit Compressive strength fcb 58.2 MPa Tensile strength ftb 4.1 MPa Mortar interface Tensile strength 0tif   0.1 MPa Mode-I fracture energy 0cnG  0.066 N/mm Initial shear strength τ0   0.39 MPa Mode-II fracture energy 0ctG 0.3 N/mm  Homogeneous model The critical feature of the homogeneous model is that the weak mortar and brick-mortar interfaces are smeared so that the whole arch is represented by a homogeneous isotropic material (Figure 2(b)). All the components including the arches, timber retaining wall, sand backfill and abutments were modelled using the four node plane strain element with four integration points (CPE4).  The masonry work was modelled using the concrete damaged plasticity (CDP) model available in ABAQUS (2007), because masonry work behaves similar to concrete under both compression and tension (Lourenço et al. 1995). The uniaxial compressive constitutive behaviour of masonry was modelled using the relationship proposed by Saenz (1964) for plain concrete:  20021 ⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−+=pcpcppccEEεεεεσεεσ  (1)  where σc and εc are respectively the compressive stress and strain, σp and εp are the maximum stress and its corresponding strain which are taken to be the compressive strength of the masonry fcm and 0.0022 respectively, and E0 is 16000MPa. Tensile failure at the mortar-interface is brittle in nature. Therefore, it is difficult to test the full nonlinear behaviour of masonry under tension. In Pluijm (1997) it was shown that the descending branch of masonry under tension can be described by the model developed by Hordijk (1991) for plain concrete:  ( ) ( )22 3131 11ccrtwwccrttt ecwwewwcfcrt−⎟⎟⎠⎞⎜⎜⎝⎛−+−⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+=σ   ,  tFcr fGw 14.5=  (2)  where wt is the crack opening displacement, wcr is the crack opening displacement at the complete loss of tensile stress, σt is the tensile stress, ft is the tensile strength, and c1=3.0 and c2=6.93 are constants. In addition to the material properties of the masonry work listed in Table 2, the tensile strength of the mortar interface was adopted as the tensile strength of the masonry work.  Meso interface models A critical feature of the two meso-interface models is the use of zero thickness interface elements to simulate the brick to brick interfaces as shown in Figure 2(c), without modelling the actual mortar layer. The thickness of the mortar layer was added to the size of the brick units so that the geometry of the bridge remained unaffected. The brick units were modelled using the CPE4 plane strain element and the interfaces were modelled using the COH2D4 cohesive element available in ABAQUS (2007). The concrete brick units were modelled using the same CDP model as described in the homogeneous model above but with the brick properties listed in Table 2.  Traction-opening interface model In the traction-opening interface model, the interface was modelled by defining the behaviour of both mode-I and mode-II. A linear softening law was adopted to model the two failure modes as shown in Figure 4. A damage initiation law (Eq. 3) and a damage evolution law (Eq. 4) were used to describe the interaction between the two modes.   1,max 00 =⎭⎬⎫⎩⎨⎧ττσtif (3)  12020 =⎭⎬⎫⎩⎨⎧+⎭⎬⎫⎩⎨⎧cttcnnGGGG  (4)  where σ and τ are the normal and shear stresses respectively, and Gn and Gt are the work done by the traction and its conjugate relative displacement in the normal and shear directions respectively. Damaged plasticity interface model One disadvantage of the traction-opening interface model is that the shear stress does not increase when the interface is under compression as shown in Eq. 3, which does not reflect the actual behaviour of the inter-brick interface. To overcome this drawback, the CDP model available in ABAQUS was used instead here. Although this model was originally developed for concrete, it is appropriate for modelling the mortar interface since it was evolved from the original Drucker-Prager plasticity model by considering the damage effects on the yield conditions and the flow rule. The constitutive law of the mortar interface was thus defined by the compressive and tensile behaviour and a damage law. The compressive strength of the interface was taken as that for the mortar at 12 MPa. The mode-I behaviour in Figure 4(a) was used to model the tensile behaviour including the damage model.   Figure 4: Normal (a) and shear (b) traction-opening displacement relationships  of the brick-mortar interface  Detailed solid model In the above meso-interface models, the mortar layer was neglected. This may not only reduce the accuracy of the models, but also lead to numerical difficulties when modelling more complex systems where the zero thickness interface elements intersect such as in multiple ring arches and in FRP strengthened masonry arch bridges. To overcome these drawbacks, a detailed solid model was developed to represent the mortar interface using the CPE4 solid element with the actual mortar thickness as shown in Figure 2d. The unit-mortar interfaces are smeared into the mortar elements. The mortar was modelled using the CDP model as that in the damaged plasticity interface model, with compressive properties of the mortar and tensile properties of the interfaces. The brick units were modelled in the same way as those in the meso-interface models. 0510152025303540455055-30 -25 -20 -15 -10 -5 0Radial displacement (mm)Load (kN)N2 (Test)N2 (Homogeneous model)N2 (T-O interface model)N2 (DP-Interface model)N2 (Detailed solid model)Unloading in the test Figure 5: Comparison between FE predictions and experiment  FE PREDICTIONS AND COMPARISON WITH TEST DATA Figure 5 shows the comparison between the test load-displacement curves and the predictions from the four FE models. The comparison of the four hinge mechanism between the detailed solid model and the test results at the peak load is shown in Figure 6. Note that the location of hinge D reported in Tao et al. (2011) was incorrect and it has been corrected here by having carefully checked the photographic records. The loading capacity and locations of the hinges from the four FE models together with that of a mechanism analysis are summarised in Table 3. It is evident that all four models are capable of producing a comparable prediction; however each model has its own characteristics. The homogeneous model was the simplest one which gave the most conservative prediction because the weaker tensile strength was smeared in the whole arch so that cracking can occur inside the bricks rather than at the interfaces in the other models and practice. The traction-opening interface model did not capture the mortar interface behaviour correctly because it did not consider the enhanced shear strength under compression. Although the damaged plasticity interface model delivered a good prediction, it is not suitable for modelling FRP strengthened masonry arch bridge because it can lead to modelling difficulties (Tao 2012). The detailed solid model produced a good agreement with the test and is also suitable for modelling FRP strengthened arch bridges which is one of the purposes of this study.  Table 3: Comparison of main results  Loading capacity (kN) Locations of  four hinges Test North 46.2 -68°, -20°, 20°, 90° South 49.7 -68°, -20°, 20°, 90° Homogeneous model 42.5 -65°, -20°, 29°, 90° TO interface model 50.0 -65°, -20°, 25°, 90° DP interface Model 47.5 -65°, -20°, 20°, 90° Detailed solid model 45.0 -60°, -20°, 20°, 90° Mechanism analysis 41.9 -58°, -13°, 26°, 90°   Figure 6: Comparison of the four hinge mechanism between the detailed  solid model prediction (upper) and test (lower)  CONCLUSIONS This paper has presented a numerical study on a masonry arch bridge with sand backfill. The arch bridge was modelled using four different FE models. They all produced reasonably good predictions of the loading behaviour and the associated hinge failure mechanism, with the homogeneous model which does not model the unit-mortar interfaces predicted the lowest loading capacity. As the failure of masonry arch bridges was dominated by the behaviour of the brick-mortar interfaces, an accurate interface model is essential for modelling more complex systems but the interface models pose many numerical challenges. The proposed detailed solid model smears the brick-mortar interfaces into the thin mortar layers, leading to an advantage of being able to accurately modelling the interface behaviour without the need of modelling the actual interfaces and thus avoiding numerical difficulties in complex systems.  A B CDCrack opening HingeREFERENCES ABAQUS (2007). ABAQUS Analysis user's manual, version 6.7. Hordijk, D. A. (1991). "Local Approach to Fatigue of Concrete." PhD, Delft University of Technology. Lourenço, P. B., J.G. Rots and J. Blaauwendraad (1995). "Two Approaches for the Analysis of Masonry Structures: Micro and Macro-Modeling." HERON, 40(4), 313-340.  Pluijm, R. v. d. (1997). "Non-linear Behaviour of Masonry under tension." HERON, 42(1), 25-55. Saenz, L. P. (1964). "Discussion of equation for the stress-strain curve of concrete-by Desayi, P. and Krishan, S." ACI Journal 61(9), 1229-1235. Tao, Y., Stratford, T. J. and Chen, J. F. (2011). "Behaviour of a masonry arch bridge repaired using fibre-reinforced polymer composites." Engineering Structures, 33(5), 1594-1606.  Tao, Y. (2012). Behaviour of Masonry Arch Bridges Strengthened with FRP. PhD thesis, The University of Edinburgh. 

Numerical Modelling of a Large Scale Model Masonry Arch Bridge

Numerical Modelling of a Large Scale Model Masonry Arch Bridge

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