Numerical derivation of homogenized dynamic masonry material properties with strain rate effects

Masonry is a composite material composed of bricks and mortar disposed in a regular arrangement. It is commonly used as load bearing or partition walls in building structures. Owing to limitations of computer power, detailed distinctive modelling of brick and mortar of a realistic masonry structure or a structure with masonry infilled walls is usually not possible. Moreover, no dynamic masonry material model can be found in the open literature. Dynamic masonry material properties are important for an accurate prediction of masonry failure and fragmentation under dynamic loads. In this paper, a continuum damage model with strain rate effect is developed for masonry materials based on the homogenization method. The equivalent elastic properties, strength envelope and dynamic increase factors (DIFs) of strength and moduli for the homogenized masonry material are numerically derived from the simulated responses of a representative volume element (RVE). A numerical model of an RVE is analyzed with detailed distinctive modelling of brick and mortar with their respective dynamic material properties obtained from laboratory tests. The homogenized material model can be used to analyse large-scale masonry structures subjected to dynamic loading.     

Numerical modeling of the in-plane behavior of historical brick masonry walls

This paper addresses numerical modeling of historical brick masonry subjected to in-plane loads. Based on experimentally determined material parameters for the masonry constituents brick and mortar numerical simulations of masonry specimens under various loading conditions are performed. These simulations aim at calibrating and validating parameter sets for historical brick masonry on the meso-scale. For computations on a larger scale, a macro-model in the framework of multi-surface plasticity theory is implemented in a finite element program, which captures the fundamental failure modes of in-plane loaded masonry. Material parameters for the macro-model are derived in a numerical homogenization procedure yielding the transition from the meso-scale to the macro-scale. Results from comparative simulations on discrete and homogenized models show both efficiency and limitations of the proposed methodology.     

Nonlinear homogenization techniques to solve masonry structures problems

The behaviour of masonry material subjected to different in-plane loading combination is studied in this work. The masonry is considered as a periodic composite material composed by a regular distribution of brick and mortar and it is analyzed using a homogenization technique. The mechanical properties of the masonry, as an orthotropic homogeneous material, depend on the geometrical and mechanical properties of the components based on the study of the equilibrium and compatibility of a basic cell. The masonry is a frictional material and its behaviour depends on the loading direction, for these reasons, a unilateral damage model is chosen for the analysis. This model describes the behaviour of brittle materials subjected to tension-compression cyclic loads based on the introduction of two damage variables and it assumes that the damage is due to the beginning and growth of cracks only in the mortar joints. It is considered that the bricks have a linear elastic constitutive relationship. Numerical applications are performed with a nonlinear finite element code in order to test the proposed procedure by comparing the results with those available in the literature and also with experimental data.     

Prediction of fragment size and ejection distance of masonry wall under blast load using homogenized masonry material properties

The fragment hazard resulting from a nearby explosion is a major concern in the design of structures which may be subjected to blast loads. This paper presents a predictive method based on the theories of continuum damage mechanics and mechanics of micro-crack development, and numerical simulation to determine the probabilistic fragment size distribution and the launch distances. Theoretical derivations are presented to calculate fragment distribution. The fragmentation process is modeled according to the crack initiation and propagation, which depend on the material damage levels and are estimated using continuum damage mechanics theory. The proposed method involves two steps. First a finite element model is developed to estimate the material damage, fragment distribution and the ejection velocity. Then a simple algorithm is used to predict the fragment trajectory and the launch distance based on the fragment size and the ejection velocity. A masonry wall is used as an example in this study. The wall is modeled with both the distinctive consideration of the brick and mortar material properties and the homogenized masonry material properties. The reliability and efficiency of using the homogenized masonry material model in predicting the masonry wall damage and fragmentation are proven. The program AUTODYN is used in this study to conduct the numerical simulations with the proposed models linked to it as user subroutines. The numerical results indicate that the masonry fragments approximately follow the Weibull distribution, which is consistent with some empirical fragment distributions. The proposed method avoids using erosion technique, which inevitably results in a loss of fragment mass, and avoids discretizing the structure into particles or predefining the failure planes, which may lead to unrealistic prediction of damage propagation and evolution and therefore inaccurate fragmentation process and fragment size distributions.     

A non‐linear interface element for 3D mesoscale analysis of brick‐masonry structures

This paper presents a novel interface element for the geometric and material non‐linear analysis of unreinforced brick‐masonry structures. In the proposed modelling approach, the blocks are modelled using 3D continuum solid elements, whereas the mortar and brick–mortar interfaces are modelled by means of the 2D non‐linear interface element. This enables the representation of any 3D arrangement for brick‐masonry, accounting for the in‐plane stacking mode and the through‐thickness geometry, and importantly it allows the investigation of both the in‐plane and the out‐of‐plane responses of unreinforced masonry panels. A co‐rotational approach is employed for the interface element, which shifts the treatment of geometric non‐linearity to the level of discrete entities, and enables the consideration of material non‐linearity within a simplified local framework employing first‐order kinematics. In this respect, the internal interface forces are modelled by means of elasto‐plastic material laws based on work‐softening plasticity and employing multi‐surface plasticity concepts. Following the presentation of the interface element formulation details, several experimental–numerical comparisons are provided for the in‐plane and out‐of‐plane static behaviours of brick‐masonry panels. The favourable results achieved demonstrate the accuracy and the significant potential of using the developed interface element for the non‐linear analysis of brick‐masonry structures under extreme loading conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

A comparative study on the approximate analysis of masonry structures

In this paper, numerical analysis of structural masonry subject to a uniform in-plane tensile stress/strain field is investigated employing various homogenisation techniques. Here, structural masonry is regarded as a composite material with brick, bed joints and head joints as its constituents. Assuming a perfect bonding between constituents. Assuming a perfect bonding between constituent materials, two homogenisation techniques based on the strain energy approach are applied to derive equivalent elastic moduli of masonry. Structural relationships for the constituent materials are next derived to relate strains and stresses in constituents to the average strains and stresses in the masonry. In addition, a slightly different concept of the homogenisation technique based on Eshelby's solution of the ellipsoidal inclusion problem is also applied to compare the results with the energy based methods. The tensile strength of the masonry is found on the basis of the failure of any of the constituent materials. It is shown that tensile strength is a function of the elastic parameters of brick/mortar as well as the tensile strength of mortar. These studies also show that, although initial cracking occurs under horizontal tensile forces, the ultimate strength of the panel is higher in this direction than in the vertical direction.     

FE homogenized limit analysis model for masonry strengthened by near surface bed joint FRP bars

A homogenized limit analysis model for the prediction of collapse loads and failure mechanisms of masonry walls reinforced with near surface bed joint GFRP bars is presented. Reinforced masonry homogenized failure surfaces are obtained by means of a compatible identification procedure, where each brick is supposed interacting with its six neighbors by means of finite thickness mortar joints, filler epoxy resin and FRP rods.In the framework of the kinematic theorem of limit analysis, a simple constrained minimization problem is obtained on the unit cell, suitable to estimate – with a very limited computational effort – reinforced masonry homogenized failure surfaces.A FE strategy is adopted to solve the homogenization problem at a cell level, modeling joints, bricks, filler and FRP rods by means of eight-noded infinitely resistant parallelepiped elements. A possible jump of velocities is assumed at the interfaces between contiguous elements, where plastic dissipation occurs. For mortar and bricks interfaces, a frictional behavior with possible limited tensile and compressive strength is assumed, whereas for epoxy resin and FRP bars some formulas available in the literature are adopted in order to take into account in an approximate but effective way, the delamination of the bar from the epoxy and the failure of the filler at the interface with the joint.In order to validate the model proposed, two meaningful examples are critically analyzed. The first relies on a reinforced masonry beam in four-point bending, whereas the second is a full scale wall constrained at three edges and loaded until failure with a distributed out-of-plane pressure. While the first example is useful to test the model at a cell level, since only horizontal ultimate bending moment is involved in the failure mechanism, the second provides a full assessment of the procedure proposed at a structural level. In both cases, very good agreement is found with literature data, meaning that the model proposed may provide useful information for all practitioners interested in the design of masonry walls reinforced with bed joint FRP bars.     

Strength and elasticity of brick masonry prisms and wallettes under compression

The characteristics of brick masonry are influenced by the properties of bricks and mortar. This paper attempts at studying the properties of brick masonry using table moulded bricks and wire-cut bricks of India with various types of mortars. The strength and elastic modulus of brick masonry under compression have been evaluated for strong-brick soft-mortar and soft-brick strong-mortar combinations. Various sizes of prisms and wallettes have been tested during these experiments to study the size effect and different bonding arrangements. The failure mechanisms of such specimens have been studied. Attempts are also made to derive empirical relationships for masonry strength as a function of brick and mortar strength in the Indian context.     

Approximate limit analysis of full scale FRP-reinforced masonry buildings through a 3D homogenized FE package

A 3D homogenized FE limit analysis software for the numerical prediction of collapse loads and failure mechanisms of entire masonry buildings reinforced with FRP strips is presented. In particular, a two steps approach is adopted: in step I, masonry homogenized failure surfaces are obtained through an admissible kinematic FE approach in the representative element of volume (REV), constituted by a brick interconnected with its six neighbors with finite thickness mortar joints. 8-Noded rigid infinitely resistant parallelepiped elements interconnected with interfaces with frictional behavior and limited tensile and compressive strength are utilized to model the REV. A simple linear programming problem in few variables is obtained, suitable to recover numerically masonry failure surfaces when loaded in- and out-of-plane. In step II, homogenized failure surfaces are implemented in the novel FE kinematic limit analysis software for an inexpensive evaluation of collapse loads of entire buildings. Delamination is considered in the model imposing to FRP–masonry interfaces a limited resistance in agreement with Italian code CNR-DT-200. 6-Noded rigid infinitely resistant 3D wedge-shaped elements are used to model homogenized masonry, whereas FRP strips are modeled by means of triangular 3-noded rigid elements.     

Homogenization of Periodic Masonry Using Self-Consistent Scheme and Finite Element Method

Masonry is a heterogeneous anisotropic continuum, made up of the brick and mortar arranged in a periodic manner. Obtaining the effective elastic stiffness of the masonry structures has been a challenging task. In this study, the homogenization theory for periodic media is implemented in a very generic manner to derive the anisotropic global behavior of the masonry, through rigorous application of the homogenization theory in one step and through a full three-dimensional behavior. We have considered the periodic Eshelby self-consistent method and the finite element method. Two representative unit cells that represent the microstructure of the masonry wall exactly are considered for calibration and numerical application of the theory.     

Masonry infilled reinforced concrete frames under horizontal loading / Stahlbetonrahmen mit Ausfachungen aus Mauerwerk unter horizontalen Belastungen

The behaviour of infilled reinforced concrete frames under horizontal load has been widely investigated, both experimentally and numerically. Since experimental tests represent large investments, numerical simulations offer an efficient approach for a more comprehensive analysis. When RC frames with masonry infill walls are subjected to horizontal loading, their behaviour is highly non‐linear after a certain limit, which makes their analysis quite difficult. The non‐linear behaviour results from the complex inelastic material properties of the concrete, infill wall and conditions at the wall‐frame interface. In order to investigate this non‐linear behaviour in detail, a finite element model using a micro modelling approach is developed, which is able to predict the complex non‐linear behaviour resulting from the different materials and their interaction. Concrete and bricks are represented by a non‐linear material model, while each reinforcement bar is represented as an individual part installed in the concrete part and behaving elasto‐plastically. Each brick is modelled individually and connected taking into account the non‐linearity of a brick mortar interface. The same approach is followed using two finite element software packages and the results are compared with the experimental results. The numerical models show a good agreement with the experiments in predicting the overall behaviour, but also very good matching for strength capacity and drift. The results emphasize the quality and the valuable contribution of the numerical models for use in parametric studies, which are needed for the derivation of design recommendations for infilled frame structures.     

Model validation and parametric study on the blast response of unreinforced brick masonry walls

Numerical simulations are carried out to estimate the response and damage of unreinforced brick masonry walls subjected to explosive blast loading based on the transient dynamic finite element program LS-DYNA. A previously developed dynamic plastic damage model was used for brick and mortar. A new model for strain rate effects of bricks and mortar is included in the numerical analysis. The results obtained from the numerical models are compared with field test data and good agreement can be found. Parametric studies are conducted to evaluate the effect of material strength, boundary conditions, and thickness of the wall on the blast response of unreinforced brick masonry walls. It was found that boundary conditions and wall thickness significantly affect the blast response, while the effect of material strength is relatively small.     

A micromechanical model for linear homogenization of brick masonry

A micromechanical model, originally developed for long-fiber composites, is applied to determination of the overall linear-elastic mechanical properties of simple-texture brick masonry. The model relies upon exact solution after Eshelby and describes brickwork as a mortar matrix with insertions of elliptic cylinder-shaped bricks. The macroscopic elastic constants are derived from the mechanical properties of the constituent materials and the phase volume ratios. The ability of the suggested model to predict the behavior of real brickwork has been checked by performing uniaxial compression tests on brick masonry panels of two types, with cement mortar and lime mortar. The results obtained through the proposed model fit experimental data more closely than other models selected from the literature for the sake of comparison.     

Determination of masonry crack evolution due to differential displacements: a numerical study*

Brick walls of ceramics without any mortar covering or paint are used extensively in building façades in Spain. One of the most used masonry wall systems is based on non‐bearing panels partially supported, about two‐thirds of the brick width, over the edge beams of the structural skeleton. The edge beam is veneered with special thinner bricks to achieve the visual continuity of the façade. A considerable number of these walls show cracking. In a previous work, finite element simulations were performed in order to gain insight into the causes of cracking. A special finite element, based on the strong discontinuity analysis and the cohesive crack theory, is used in the numerical simulations. The results agree with the overall cracking patterns observed but if an imposed displacement is applied in the range allowed by the standards, extensive cracking occurs. This implies that the design displacements are not the actual ones. In this work, an elastic study using the principle of superposition is used to determine the effective deflections under service loading. Then, these deflections are applied to the structure and the evolution of cracking is studied. This study shows that the masonry panels of the first and last store have the major probability of cracking. Another parametric study is carried out changing the elastic and tensile properties of the masonry. This study shows that although the cracking of the masonry panels starts at different loads for different tensile properties, the crack patterns are similar for a given panel geometry and loading. This numerical study provides a method of design to determine the crack width for different geometries, loadings and fracture properties.     

Analysis of steel‐reinforced masonry walls regarding maximum shear loads / Traglastberechnung von vertikal bewehrten Mauerwerksscheiben

Loadings on masonry for the earthquake case pose particular challenges for the material. In order to improve the load‐bearing and deformation behaviour, masonry building elements can be strengthened with reinforcement. This article presents an analytical model for the calculation of the load‐bearing capacity of vertically reinforced masonry panels. The masonry is modelled as a homogeneous and anisotropic material and failure conditions are based on the plastic theory. Using uniaxially loaded stress fields and considering the structural constraints, a lower load‐bearing threshold can be given. In order to verify the model, three shear tests on reinforced sand‐lime block masonry were recalculated regarding their load‐bearing capacity. The test panels each contained vertical steel reinforcement in the blocks. The blocks were laid in thin bed mortar.     

组合材料加固砖砌体的有限元模拟方法

基于ABAQUS有限元软件,提出了组合材料加固砌体数值建模方法。该方法是在未加固砌体整体式模型的基础上,结合分离式思想建立组合材料加固砌体模型。通过对8片采用粘钢-聚合物砂浆组合材料加固的砖砌体墙体（其中,4片采用粘贴正交钢片,4片采用粘贴斜撑钢片）的拟静力试验结果进行了数值模拟对比分析,结果显示:模拟所得墙体滞回、骨架及刚度退化曲线与试验曲线基本吻合;仿真破坏形态与试验现象一致;计算所得荷载、位移、延性和耗能等全部指标中有81%的误差在20%以内。     

Experimental assessment of an innovative strengthening material for brick masonry infills

The vulnerability of masonry infill walls has been highlighted in recent earthquakes in which severe in-plane damage and out-of-plane collapse developed, justifying the investment in the proposal of strengthening solutions aiming to improve the seismic performance of these construction elements. Therefore, this work presents an innovative strengthening solution to be applied in masonry infill walls, in order to avoid brittle failure and thus minimize the material damage and human losses. The textile-reinforced mortar technique (TRM) has been shown to improve the out-of-plane resistance of masonry and to enhance its ductility, and here an innovative reinforcing mesh composed of braided composite rods is proposed. The external part of the rod is composed of braided polyester whose structure is defined so that the bond adherence with mortar is optimized. The mechanical performance of the strengthening technique to improve the out-of-plane behaviour of brick masonry is assessed based on experimental bending tests. Additionally, a comparison of the mechanical behaviour of the proposed meshes with commercial meshes is provided. The idea is that the proposed meshes are efficient in avoiding brittle collapse and premature disintegration of brick masonry during seismic events.     

Three-dimensional computational micro-mechanical model for woven fabric composites

This work presents a computational material model for plain-woven fabric composite for use in finite element analysis. The material model utilizes the micro-mechanical approach and the homogenization technique. The micro-mechanical model consists of four sub-cells, however, because of the existing anti-symmetry only two sub-cells have to be homogenized for prediction of the elastic material properties. This makes the model computationally very efficient and suitable for large-scale finite element analysis. The model allows the warp and fill yarns not to be orthogonal in the plane of the composite ply. This gives the opportunity to model complex-shaped composite structures with different braid angles. General homogenization procedure is employed with two levels of property homogenization. The model is programmed in MATLAB software and the predicted material properties of different composite materials are compared and presented. The material model shows good capability to predict elastic material properties of composites and very good computational efficiency.     

高延性混凝土加固砖柱轴压性能试验研究

高延性混凝土（HDC）是一种具有高强度、高韧性和高耐损伤能力的新型结构材料。该文提出采用HDC面层加固砖柱,对27个砖柱试件进行了轴压性能试验研究。结果表明:1）HDC作为砌筑砂浆,可对砌体形成一定的约束作用,使砖柱的轴压承载力和变形能力均有所提高;2）HDC面层发挥了较强的套箍作用,使砖柱处于三向受压状态,承载力和变形能力均得到较大幅度提高,且改善了砖柱的脆性破坏特征;3）HDC面层与砖柱具有良好的协调工作能力,对提高砖柱的整体性能具有重要作用。考虑HDC面层对砖柱的约束作用,提出了HDC面层加固砖柱的轴压承载力计算方法,计算结果与试验结果吻合较好。该文研究结果为砌体结构加固提供了一种新方法,具有良好的推广应用前景。