A comparative study on the approximate analysis of masonry structures

Transcription

1 Materials and Structures/Matériaux et Constructions, Vol. 3, August-September 998, pp A comparative study on the approximate analysis of masonry structures J. S. Lee, G. N. Pande and B. Kralj 3 () now at KRRI, Korea () Professor (3) Post-Doctorate Research fellow, now at W.S. Atkins Department of Civil Engineering, University of Wales Swansea, Singleton Park, Swansea SA 8PP, U.K. SCIENTIFIC REPORTS Paper received: April 7, 997; Paper accepted: May 8, 997 A B S T R A C T 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 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. R É S U M É Dans cet article, on étudie l analyse numérique d une maçonnerie porteuse soumise à un champ uniforme et plan de contrainte de traction/déformation, au moyen de diverses techniques d homogénéisation. On considère la maçonnerie porteuse comme étant un matériau composite ayant comme constituants des briques, des joints d assise et des joints verticaux. En supposant une adhérence parfaite entre ces constituants, deux techniques d homogénéisation basées sur l approche de l énergie de déformation sont appliquées afin d en déduire les modules d élasticité équivalents de la maçonnerie. Les relations structurelles pour les matériaux constituants sont ensuite déduites afin d établir la relation entre les déformations et contraintes dans les constituants et les déformations et contraintes moyennes dans la maçonnerie. De plus, une conception légèrement différente de la technique d homogénéisation, basée sur la solution d Eshelby au problème de l inclusion ellipsoïdale, est appliquée afin de comparer les résultats avec des méthodes basées sur l énergie. On détermine la résistance à la traction de la maçonnerie sur la base de la rupture de n importe lequel des matériaux constituants. On démonte que la résistance à la traction est fonction des paramètres élastiques briques/mortier, ainsi que de la résistance à la traction du mortier. Ces études montrent également que, bien que la fissuration initiale commence sous des forces de traction horizontales, la résistance ultime du panneau est supérieure dans le sens horizontal à celle dans le sens vertical.. INTRODUCTION The tensile strength of the masonry is an important consideration in the design and assessment of masonry structures since masonry can be subjected to tensile loads due to external loading, e.g. settlement of foundation, thermal loading or movement of moisture/creep. In this paper, the complex state of stresses developed in a masonry panel subjected to a uniform tensile stress/strain are computed, which are in turn used to check tensile failure (cracking) of each constituent. A two-stage homogenisation technique based on a strain energy of the composite material is adopted to compute equivalent orthotropic material properties of the masonry. Based on the structural matrices obtained, stresses in constituents are next computed. In addition, another homogenisation scheme based on the mechanics of composite material is also employed to compare the results with those based on the strain energy rule. For this, the simplified stress path method is utilised to investigate the effect of homogenisation methods. Numerical experiments are performed on masonry Editorial Note Prof. G. N. Pande is a Senior member of RILEM and Chairman of TC MMM (Computer Modelling of Mechanical Behaviour of Masonry Structures) /98 RILEM 473

2 Materials and Structures/Matériaux et Constructions, Vol. 3, August-September 998 panels employing both stress controlled and strain controlled tests. As a result, a semi-analytical approach using a homogenisation technique is proven to be simple to use when analysing composite material, and it is also demonstrated that the material properties calculated are in good agreement with those derived through finite element analysis with very fine meshes.. ORTHOTROPIC MATERIAL PROPERTIES OF MASONRY In this section, orthotropic material properties equivalent to the overall mechanical characteristics of the masonry are derived based on various homogenisation techniques. One technique is based on a strain energy concept introduced in [] where bed and head joints are individually homogenised with brick material. Two homogenisation sequences are possible within this category, i.e. bed joint and brick can be homogenised first and the head joint is considered subsequently; alternatively head joint and brick can be homogenised first, followed by bed joint homogenisation. For simplicity, the former case will be named the bh method and the latter will be denoted as hb method. The material properties obtained will be used to calculate equivalent average stresses in masonry and subsequently, the stresses in the constituent materials, i.e. brick, bed as well as head mortar joints will be computed. The basic assumptions made in the derivations given in [] are,. Brick and mortar are perfectly bonded.. Head mortar joints are assumed to be continuous. The first assumption may appear to be too restrictive since there is considerable research being carried out on the bond strength of mortars. The bond strength is likely to be less than the tensile strength of mortar. One simple and practical way of overcoming this restriction is to assign these lower bond strength properties to the mortar joints, maintaining the assumption of perfect bond between brick and mortar. The second assumption is necessary in the homogenisation procedure, and it will be shown later that the assumption of continuous head joints instead of staggered joints does not have any significant effect on the stress states of the constituent materials. Let the orthotropic material properties of the masonry panel be denoted by E x, E y, E z, v xy, v yz, v zx, G xy, G yz, G zx, with the coordinate system shown in Fig.. If the stress/strain relationship of the equivalent material is represented by σ [D ]ε () or ε [C ]σ () where, σ {σ xx, σ yy, σ zz, τ xy, τ yz, τ zx }T (3a) ε {ε xx, ε yy, ε zz, γ xy, γ yz, γ zx }T (3b) Fig. Masonry subjected to in-plane tensile stress/strain. (a) Geometry of masonry structure; (b) Applied in-plane tensile stress/strain in masonry. then νxy νxz Ex Ex Ex νyx yz ν Ey Ey Ey νzx νzy C Ez Ez E [ ] z Gxy Gyz G and detailed expressions of the orthotropic material properties for masonry in equation (4) are given in [] and will not be reiterated here. Another approximate approach based on the mechanics of composite material can also be used to derive orthotropic material properties of masonry. In this case, zx (4) 474

3 Lee, Pande, Kralj the head joint is considered as an ellipsoidal inclusion within brick, and effective elastic moduli of the composite composed of head joint and brick are derived first, see [] for details. Next, homogenisation of the composite and bed joint is performed to obtain overall orthotropic material properties of the masonry, either using the strain energy concept above or employing the mixture rule introduced by Hill [3]. The homogenisation technique based on the mechanics of composite materials and the mixture rule was first used in [4] to investigate the in-plane behaviour of masonry structures and will be denoted as the eb method for simplicity. Effective elastic moduli of a two-phase composite, in which head joints having a elliptic cylinder shape are assumed to be monotonically aligned within brick, is tedious to derive and full details can be found in []. From the results listed in [] and [], it can shown that the orthotropic material properties are functions of:. Dimensions of the brick. Young s modulus and Poisson s ratio of the brick 3. Young s modulus and Poisson s ratio of the mortar 4. Thickness of the mortar joint. 3. STRESSES IN MASONRY AND CONSTITUENTS The stresses in masonry are considered as notional average stresses as obtained from equation (). These stresses are related to the stresses in the constituent materials through the following equations: b [S b ] bj [S bj ] (5) hj [S hj ] where, subscripts b, bj and hj represent brick, bed joint and head joint, respectively, and [S] is a structural matrix. Details of this matrix employing the hb homogenisation method are also listed in []. Again, the structural matrices are functions of (i) geometry of brick/mortar (ii) material properties of brick/mortar. The structural matrix using the mechanics of composite materials also has the similar form as in equation (5) and details of [S b ] and [S hj ] can be found in [4]. 4. FAILURE OF MASONRY The failure of masonry is detected by the failure of any of the constituents and, in this study, the failure criterion used for each constituent material of masonry will be the well-known Mohr-Coulomb failure criterion accompanied by a tension cut-off. The following stress invariants are used in the derivation: σ σ ij ij δijσkk 3 J σii σ+ σ + σ3 J σ σ ij ij ( σ σ) + ( σ σ3) + 6 σ σ J 3 σσ ij jkσki 3 ( 3 ) (6) where σ ij, δ ij, J, J and J 3 are the deviatoric stress, Kronecker delta, the first stress invariant, the second and the third deviatoric stress invariants, respectively. Here, σ, σ, σ 3 are the principal stresses of a specific constituent material. Finally, the following stress invariants are defined: σm J ; σ J 3 where, σ m and σ can be regarded as the mean stress and deviatoric stress invariant, respectively. If the uniaxial compressive and tensile strength (σ c and σ t ) of the constituent material are known from experiments, the following material parameters for the Mohr-Coulomb failure criterion can be obtained from simple geometry of Mohr circles of failure, Fig. (a), with σ c < 0 and σ t > 0; (7) σc + σt σσ ϕ c sin ; C σ σ c t t (8) Fig. Mohr-Coulomb failure function including tension cutoff. (a) Construction of Mohr-Coulomb failure function; (b) Mohr-Coulomb failure function including tension cut-off in σm 3 σ space. where, ϕ and C are the friction angle and cohesion, respectively. Using the stress invariants defined above, the Mohr- Coulomb failure function can be obtained as follows: 475

4 Materials and Structures/Matériaux et Constructions, Vol. 3, August-September 998 or 3σ+ Msσm Fs 0 where, (0) 3sin ϕ Ms 3 cosθ sin θsin ϕ ; 3C cosϕ () Fs 3 cosθ sin θsin ϕ and sin θ J ; () 3 θ σ 6 6 The tension cut-off, i.e. the condition that no principal stress exceeds the uniaxial tensile strength, t, can be incorporated with equation (0) by considering the following expressions: or (9) F eral constraints are imposed so that the same strain energies are stored both in masonry cell and in equivalent σ m sin ϕ + σ C 0 cosθ 3 sin θsin φ cosϕ orthotropic material. F m t 3σ+ Mtσm Ft 0 where, M σ σ + sin θ σ σ t 3 sin θ+ 3 (3) (4) (5) In this section, the results of numerical experiments on masonry panels in horizontal and vertical tension are reported. The size of the panels are sufficiently large so that the assumption of equivalent material behaviour can be justified. Two types of tests were performed, viz, stress controlled and strain controlled. At first, the effect of homogenisation methods on overall orthotropic material properties are investigated. For this, material parameters and geometry shown in Table and Fig., respectively, are considered. The material are identical to those used in [5] where the orthotropic material properties derived from finite element analysis of the masonry are also described. In [5], to evaluate Young s modulus in a specific direction of the representative masonry cell composed of brick and mortar, uniform loading in that direction as well as lat- Fig. (b) illustrates the failure function in σm 3σ space which is the combination of equations (0) and (4) where the failure mode is changed from tensile cracking to shearing at the transition point. It is assumed here that the failure criterion shown in Fig. (b) can be used for brick bed as well as head mortar joints by substituting known compressive/tensile strength values. 5. NUMERICAL EXAMPLES ; F s 3σt sin θ Brick Dimension l h t b Young s modulus E b N 0,000 Poisson s ratio ν b 0.5 Compressive strength σ c,b N Tensile strength σ t,b N.85 Mortar Table Material properties of masonry Thickness t bj, t hj mm 0 Young s modulus E m (E bj, E hj ) N 0 to 0,000 Poisson s ratio ν m (ν bj, ν hj ) 0.5 Compressive strength σ c,bj, σ c,hj N -0.0 Tensile strength σ t,bj, σ t,hj N 0.5 Fig. 3 shows the orthotropic material properties evaluated from 3 different methods, as well as finite element modelling of a representative masonry. Here, bh represents the energy based homogenisation method with brick and bed joint homogenisation first, while hb means the energy based method where brick and head joint are homogenised first. Also, eb denotes the material model derived from the homogenisation technique coupled with Eshelby s solution of a medium having elliptical inclusions. Finally, the results of finite element analysis are adopted from the figures illustrated in [5]. As is clear from the figure, the orthotropic material moduli calculated from the bh and hb models are almost the same regardless of orthotropic moduli plotted. However, E x, the value calculated from the eb model, is quite different from energy based models as the stiffness ratio E b / E m becomes large. It is also noted that the F.E.M. model gives moduli almost identical to those obtained by the energy based model, especially the hb method. The discrepancy of the eb method can be explained by the fact that in this method the moduli are the functions of aspect ratio, representing the ratio of minor to major axes of the embedded ellipse and, in the case of masonry, the ratio of thickness to height of the head joint. Since the eb model has been developed for the case of elliptical inclusions, it is not quite applicable to the embedded head joint as the stiffness ratio becomes large. This is further verified in Fig. 4 where various aspect ratios, including that of head joint thickness and height, 0.9, are plotted. It is clear from the figure that as the aspect ratio is reduced, the E x value is close to the F.E.M. value, and a very small aspect ratio rather than direct calculation from the thickness and height of head joint is necessary to obtain better results. Similar discussion has been made in [4] where

5 Lee, Pande, Kralj Fig. 3 Orthotropic material properties of masonry according to various homogenisation techniques, E b 0,000 N/ and aspect ratio of eb method 0.9. (a) E x, (b) E y, (c) E z, (d) G yz Fig. 4 Variation of E x according to aspect ratio. Homogenisation is based on mechanics of composite materials (eb) method. small values of the aspect ratio are interpreted as a continuous head joint within masonry. It is also noted that the results listed in [5] show much better approximation when the hb method is adopted, while both bh and hb methods considered in this study give almost the same moduli, although the hb method shows slightly better results when compared with those calculated by finite element analysis. The influence of the homogenisation techniques can be further investigated by utilising the simplified stress path method. For this, stress/strain control tests are performed to find the failure mode of masonry subject to uniform horizontal or vertical tensile loading. The material properties including tensile/compressive strength of masonry used are the same as in Table and the stiffness ratio, E b / E m, has been taken as 0. Figs. 5(b) and (d) show the stress paths of the bed as well as head joints along with the incremental tensile strains in a strain controlled test. Also shown in the figures are the threshold values of applied strain, ε v,t and ε h,t, at which tensile cracking begins to develop. At the same time, the stress path of the brick is such that it does not reach failure function and, therefore, will not be shown in this study. Furthermore, stress control tests in both the vertical and horizontal directions are performed. Figs. 5(a) and (c) show the stress paths of the head and bed joints subjected to incremental tensile stresses. Clearly, for the eb method with aspect ratio of head joint as 0.9 in the case of horizontal loading, erroneous stresses in the head joint are obtained and, therefore, this method cannot be used in the analysis of masonry structures. It is also noted from Fig. 5 that, when horizontal tensile stress/strains are applied to masonry, significant stress concentrations occur within the head joint and, therefore, tensile cracking is initiated at a relatively small 477

6 Materials and Structures/Matériaux et Constructions, Vol. 3, August-September 998 Fig. 5 Stress paths of masonry (aspect ratio of eb method 0.9): (a) Stresses in head joint with horizontal tensile stress control test, σ h,t 0.43 N/ (b) Stresses in head joint with horizontal tensile strain control test, ε h,t % (c) Stresses in head joint with vertical tensile stress control test, σ v,t 0.48 N/ (d) Stresses in head joint with vertical tensile strain control test, ε v,t % Fig. 6 Stress path of masonry (aspect ratio of eb method 0.000): (a) Stresses in head joint with horizontal tensile stress control test, σ h,t 0.43 N/ (b) Stresses in head joint with horizontal tensile strain control test, ε h,t % load intensity compared with the case of vertical stress/strain loading. As discussed above, if the aspect ratio of the eb method is reduced to 0.000, much improved results can be obtained, see Fig. 6, and the stress path as well as the tensile failure mode are almost the same with the eb method. It can be also verified from the conventional finite element analysis that the initial cracking of the head joint occurs when σ h 0.4 N/ due to local stress concentrations. However, since the brick and bed joints are still within the elastic limit, increments of tensile stress can be further applied after initial cracking in head joints. The structural failure of the panel, in this case, occurs when σ h is increased upto 0.86 N/. As for the vertical tensile stress/strain control tests, the threshold value for the cracking of the bed joint, σ vt, is 0.49 N/ which can be also calculated from finite element analysis involving the elastic-brittle behaviour of masonry. In this case, the stress at which initial cracking takes place is also the ultimate tensile load that can be applied to a masonry panel. Finally, the influence of mortar thickness is considered. For this, the stiffness ratio, E b / E m, has been taken as 0 again and the thickness of bed as well as head joint is changed from to 30 mm although too thin or thick mortar is not practical in real situation. Fig. 7 shows the variation of the E x value according to the thickness of joint and homogenisation methods. As in the above, the aspect ratio of the eb method plays a major role when deriving E x and a small value is again necessary to obtain proper results. Another interesting point is that, when 478

7 Lee, Pande, Kralj Fig. 7 E x of masonry according to various mortar thicknesses, E b / E m 0. (a) Aspect ratio 0.9; (b) Aspect ratio the mortar joints are very thin, the E x value is the same regardless of homogenisation methods and, as the mortar joint becomes relatively thick, the E x value varies significantly depending on the homogenisation method adopted. This is basically due to the fact that the homogenisation method used is an approximation procedure and, therefore, errors in orthotropic material properties will be significant if the volume fraction of weak material is increased. 6. CONCLUSIONS Homogenisation techniques based on the strain energy, as well as based on Eshelby s solution for elliptical inclusions, have been applied to masonry to evaluate orthotropic material properties and, from these, the tensile strength of masonry panels subject to incremental in-plane tensile stresses/strains has been investigated. These have been compared with the results of an accurate finite element analysis where each brick and joint is discretised. Employing the equivalent orthotropic material model and elastic-plastic behaviour of the brick/mortar, the following conclusions are made: The assumption of continuous head joints coupled with the hb method in which head joint and brick are homogenised first using the strain energy concept gives results which are close to the finite element results. To use the eb method in which the head joint is regarded as an embedded ellipse, the aspect ratio has to be artificially reduced, although the aspect ratio represents the ratio of the thickness to the height of the head joint. By using a reduced aspect ratio, the head joint can be interpreted as continuous discontinuity rather than embedded ellipse. Although initial cracking occurs at a lower level of horizontal tensile forces, the ultimate strength of the panel is higher in this direction than the vertical direction. REFERENCES [] Pande, G. N., Kralj, B. and Middleton, J., Analysis of the compressive strength of masonry given by the equation f k K(f b ) α (f m ) β, The Structural Engineer 7 (994) 7-. [] Zhao, Y. H. and Weng, G. J., Effective elastic moduli of ribbonreinforced composites, J. Appl. Mech. 57 (990) [3] Hill, R., Elastic properties of reinforced solids: some theoretical principles, J. Mech. Phys. Solids (963) [4] Pietruszczak, S. and Niu, X., A mathematical description of macroscopic behavior of brick masonry, I. J. Solids Str. 9 (99) [5] Lourenço, P. B., Computational strategies for masonry structures, PhD thesis, Dept. Civil Eng., Delft Univ. of Technology, The Netherlands,