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Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 ISSN: 2251 - 0478 © Wilolud Journals, 2012 http://www.wiloludjournal.com Printed in Nigeria NUMERICAL MODELING OF BRICK-MORTAR COUPLET T.C. Nwofor Department of Civil and Environmental Engineering, University of Port Harcourt, P.M.B. 5323, Port Harcourt, Nigeria. e-mail: templenwofor@yahoo.com ABSTRACT This paper is a presentation of the finite element modeling of stress in a brick-mortar couplet. The brick-mortar couplet here serves as a meaningful substitute for a brick wall structure. In this work the basic steps in derivation of the element stiffness matrix for a plane stress problem using the displacement approach is presented. The main objective in the work is to determine the nodal displacements, which in turn would be used to determine the stress in each element, from which the general stress pattern for the overall structural model can be evaluated. The process involves voluminous numerical work and a computer code is used to do the analysis. The result in this work shows that the stresses obtained are directly affected by the value of the modulus of elasticity for mortar (Em) assuming that the modulus of elasticity for the brick region (EB) remains constant. Hence the usual failure theory formulated by previous investigators by considering a homogeneous structure is limited as it is well noticed that the variation of modulus of elasticity for the mortar region is a significant factor that should not be ignored. KEYWORDS: Brick, mortar, masonry, compression, stress, and strain. INTRODUCTION Brickwork construction has come to be the most commonly used in the construction industry. They are mostly used as infill in framed structures or load bearing elements (compression elements) in unframed structures. The importance of bricks in construction lies in the fact of its numerous advantages, such as durability, appearance, economy, fire protection, heat insulator and workability. In these modern times where we have tall buildings, much load comes on the bricks in form of compression and that is when we consider that the bricks act as load bearing elements (unframed structure) and as a result there is higher stress on the brick wall. Even in framed structure, the brick which serves as infill is also subjected to little stiffening effects which have long been neglected in the design of framed constructions. Brick walls subjected to axial compression and in plane shear has been investigated (Hendry, 1985 and Borchelt, 1970). The state of the art review of seismic shear strength of masonry has also been published (Mayes and Clough, 1975). The first category of researchers (Borchelt, 1970; Mayes and Clough, 1975; Blume and Proulx, 1968) obtained theoretical and empirical expressions for shear strength in terms of principal tensile stress on the assumption that diagonal tensile failure was initiated when the principal tensile stress reached a critical value. The second category (Sinha and Hendry, 1967; Pieper and Trantsch, 1970) showed that shear strength is dependent on bond and friction between the mortar and bricks. Another set of researchers in their paper which is also a proceeding of the second international conference on structural Engineering Analysis and Modeling worked on the plane of weakness theory for masonry Brick Elements (Graham and Andam, 1990). The paper is an experimental verification of the fundamental behaviour of brick elements subjected to in-plane uniaxial compressive stress forces. In their work they saw a brick masonry assemblage as a two – phase composite material. The phases do not only have different strengths but also different deformational characteristics. The mortar phase which acts as a binding medium for the assemblage, introduces two distinct planes of weakness, orthogonal to each other. Hence, the behaviour of brick masonry panels in response to uniaxial compressive stress applications was investigated in relation to the plane of weakness theory. Another work on single plane of weakness theory was also carried out (Orr, 1981). The biaxial compressive strength of brick masonry was studied (Page, 1981; Dhanasekar et al., 1985). Studies were also done on sandcrete block wall subjected to uniaxial stress (Perera, 1993). 1 T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 The condition of masonry in framed structure where the bricks serves as infill has also received attention (Madan et al., 1997; Ephraim, 1990). In these studies several failure mechanisms were identified. Also the failure of masonry due to stress released under compression load have been studied (Stroven, 2002) while the failure of brick wall due to acid rain from radioactive emission has been investigated (Adedeji, 2002). Other comprehensive studies on the resistance of masonry wall to compressive and transverse loads were conducted (Chinwah, 1973, 1985; Abrams, 1996). Brick-mortar elements which are under various stress conditions are primarily vertical load bearing elements in which the resistance to compressive stress is the primary factor in design. Most of the design values at the present have been obtained by empirical test on walls and smaller specimen, without considering the influence of the varying values of the mechanical properties of mortar region. Hence in this present work, it is necessary to consider the effect of the varying mechanical properties of the mortar region on the stress state of a brick- mortar couplet which in this case serves as meaningful substitute for a brickwork structure by using the finite element method of analysis. NUMERICAL ANALYSIS This investigation shall be carried out using the constant strain three node triangular elements. Hence the brick and mortar couplet continuum shall be divided into constant strain triangular elements which shall include elements in the mortar joints. By the use of this method the distribution of stress in the brick-mortar continuum can be obtained easily, and with the result obtained, areas of critical stress states will be obvious. The basic principle of this method is that the continuum is divided into a finite number of elements interconnected at node points situated on their boundaries. The structure thus idealized can be analysed by any of the standard method of structural analysis. The development and application of the finite element method are well published. For the purpose of this brick-mortar analysis, the formulation used will be the displacement approach. In using this method the nodal displacements are the basic unknown, while the stresses and strains are assumed constant for each element. Finite Element Method The basic concept of the finite element method of analysis is that the structure can be considered to be an assemblage of individual structural elements. Hence the idea of the finite element method is the use of two or three dimensional elements for the idealization of a continuum, where accuracy of the solution increases with the number of elements taken. The basic approaches to obtain the triangular element stiffness matrix for a plane stress problem are well documented. I will only present some essential features in this paper. Figure 1, represents the coordinates and node numbering for a plane elasticity triangular element Fy3, v3 y Fx3, u3 3 v1 Fy2, Fy1 22 1 Fx2, u2 x1 Fy1, u1 y1 x Figure 1: Nodal forces and displacements displayed in the Cartesian co-ordinate system 2 T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 A typical node will have two components of displacements u {δ } i = i vi and would produce corresponding force vector f {F } i = xi f yi Therefore, the complete displacement and force vector for a particular triangular element can be represented as u1 v 1 {δ e } = u2 v (1) 2 u3 v3 Fx1 F y1 F {F e } = Fx 2 (2) y2 Fx 3 Fy 3 Hence the element stiffness matrix [Ke] would be a 6 x 6 matrix for this plane elasticity triangle. {F } e = [K e ] { e } δ (3) Note that equation (1) and (2) shows contributions from the three nodes shown in figure 1. A suitable displacement function is chosen to define the displacement at any point in the element. This is simply represented by two linear polynomials functions containing six unknown coefficients (α , α 1 2 Lα 6 ) representing the six degrees of freedom in the case of a plane triangular element. u = α1 + α 2 x + α 3 y (4) v = α 4 + α 5 x + α 6 y For plane elasticity problems the [D] matrix which represent the contribution of modulus of elasticity E and poisons ratio v can be expressed as d11 d12 0 [D] = d 21 d 22 0 0 0 d 33 3 T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 where for plane stress problem d11 = d22 = E (1 − v 2 ) d12 = d21 = vE (1 − v 2 ) d33 = d22 = E 2(1 + v ) and for plane strain problem d11 = d22 = E (1 − v ) (1 + v )(1 − 2v ) d12 = d21 = vE (1 + v )(1 − 2v ) d33 = d22 = E 2(1 + v ) For a plane elasticity problem, the state of stress {σ (x, y )} at any point within the element related to the strain {ε ( x, y )} can be represented by three stress-strain components as follows: σ x 1 v 0 εx E {σ (x, y )} = σ y = v 1 0 εy (5) τ 1−V 2 1− v xy 0 0 σ xy 2 which can be represented simply as {σ (x, y )} = [D ] {ε ( x, y )} (6) where {ε ( x, y )} = [B ] { e } δ (7) Hence the relationship between element stresses and the nodal displacements is obtained as {σ (x, y )} = [D ][B ]{ e } δ (8) where matrix [B] contains constant linear dimensional values. The statically equivalent nodal forces {Fe} related to the nodal displacements {δ } and hence the element e stiffness matrix [Ke] can be obtained as {F } = [∫ [B] [D][B]d (vol )] {δ } e T e (9) Carrying out the integration and noting that ∫d(vol) can be represented by the area of the triangle ∆ multiplied by constant thickness t {F } = [[B] [D][B]∆t ] {δ } e T e (9a) e Hence the triangular element stiffness matrix [K ] is represented by {K } = [[B] [D][B]∆t ] e T (10) Through rigorous calculations the nodal displacements for a whole continuum is obtained from equation 9, from which the element stresses and strains are deduced from equation 8 and 7, hence a suitable computer programme code will aid this analysis. Analytical Model The analytical brick-mortar model consists of the full size brick and mortar couplet. The couplet consists of brick panels of average dimensions 215mm x 103mm x 65mm, fabricated from high quality perforated bricks of standard dimensions. A typical representation of a brick-mortar couplet under uniform loading is shown in Figure 2. Two types of analytical models are shown in Figures 3 and 4. While Model 1 is divided into 100 triangular elements, model 2 is a more typical representation of a practical couplet with 216 elements. It is also interesting 4 T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 to note that the elements within the mortar joint are made smaller in two rows. The idea in the smaller mesh size seen in model 2 is to as much as possible keep the aspect ratio as near unity as possible, i.e. to avoid long thin triangles since their use reduces accuracy. The geometry and the loading condition of the structure made in this analysis in order to formulate a finite element model for the analysis of the stress-strain in the brick-mortar couplet is shown in the figures below. Other features of the analytical model are as follows. (i) The thickness of the mortar joint is 19mm (ii) A compressive force consisting of a unit of uniformly distributed load (udl) acts on it. (iii) The support reaction of the couplet will consist of a pin support of one end and a roller support on the other end and this is to ensure a determinate structure. The elastic properties of brick and mortar component are given in Table 1. Table 1: Average values for Modulus of Elasticity and Poisson’s Ratio E(N/mm) x 103 ν Brick 8.83 0.060 Mortar See comment* 0.170 *Comment:- The value of modulus of elasticity E for mortar varies with the actual water/cement ratio or cube crushing strength fcu while the Poisson’s ratio ν for mortar can be taken as a average of 0.170. The value for Em from literature varies from 1.24 x 103N/mm2 to about 37.23 x 103N/mm2 (Page, 1981). Figure 2: Typical representation of a Brick-mortar couplet under uniform loading. 5 T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 Figure 3: Analytical model of the Brick-mortar couplet for finite element analysis Figure 4: Analytical model of the Brick-Mortar couplet for finite element analysis 6 T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 Using the mechanical property from Table 1 and a computer programme code, a finite element analysis of the two models are carried out, varying the modulus of elasticity of the mortar joint. From the results, only the maximum vertical compressive stress and strain in the models have been tabulated and shown in Table 2 and 3. Table 2: Finite element results for models idealized with 100 elements Modulus of Elasticity for Maximum Compressive Stress Maximum Compressive Mortar (Em) (N/mm2) x 103 (N/mm2) Strain (mm/mm) 1.24 6.10 0.00290 3.72 6.90 0.00320 13.10 7.85 0.00370 24.82 9.01 0.00400 31.03 9.85 0.00405 37.23 10.35 0.00415 Table 3: Finite element results for models idealized with 216 elements Modulus of Elasticity for Maximum Compressive Stress Maximum Compressive Mortar (Em) (N/mm2) x 103 (N/mm2) Strain (mm/mm) 1.24 5.75 0.00285 3.72 6.45 0.00315 13.10 7.23 0.00330 24.82 8.75 0.00340 31.03 9.45 0.00402 37.23 9.98 0.00413 DISCUSSION OF RESULTS The comparison of stress-modulus of elasticity for mortar (Em) curves for both models, and experimental study are shown in Figure 5. It should be pointed out that the numerical results presented above were obtained after collection of the mechanical property data for brick and mortar. The first FEM calculation with the mortar, masonry, and brick's mechanical properties taken from the experimental study results led to big discrepancies between the results from FEM, and testing of the brick mortar couplet with different elasticity property for mortar joint. This confirms that the properties of mortar inside the joints of the couplet were by far different from those exhibited by the mortar samples, which were tested separately. Therefore, in the next FEM, the values of the mechanical properties were increased significantly and according and the results obtained were much closer to the experimental results. The stress strain curve obtained from both analyses shows that the maximum compressive strength of brickwork was slightly higher for the case of higher value of elastic property of the mortar region. The mortar joint having lower strength than the brick strength, will cause a reduction in the compressive strength of masonry. The actual compressive strength of the masonry determined by experimental method was much higher than the strength obtained by numerical method. However, in reality, the brickwork was constructed from two layered materials, namely brick and mortar, therefore, the idealization of typical brick work material for the analysis should be adopted. Models with 216 elements show about 4% higher strength than the model with 100 elements which shows a typical convergence of the finite element result. From the foregoing in order to get maximum compressive strength in brickworks, it may be recommended that the average value of elasticity modulus greater than 13.10 x 103(N/mm2) for the mortar region should be use in brick work construction. The finite element result compared favorably with the experimental result (Page, 1981; Bakhteri and Sambasivam, 2003) as they both show increase strength for corresponding increase in the elastic property of the mortar region. However, comparison of experimental and numerical results indicates that the compressive strength of masonry obtained by experimental method was 40% higher than those obtained by finite element method. Therefore, to get the actual compressive design strength of brick masonry, the finite element analysis results should be enhanced by a factor of 1:4. 7 T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 16 100 elements 216 elements Experimental Maximum compressive stress (N/mm ) 14 2 12 10 8 6 4 2 0 0 5 10 15 20 25 30 35 40 3 Modulus of elasticity for Mortar (Em ) (N/mm ) Figure 5: Graph of modulus of elasticity for mortar versus maximum compressive stress for brick mortar couplet CONCLUSION Numerical and experimental analyses results showed that, by increasing the Em value, the strength of the masonry will increase it was proposed that in idealization of numerical analysis, the ideal brickwork should be assumed, since brick masonry is in reality, a layered material. For the design purposes, the compressive strength of brick masonry obtained from finite element analysis should be factored by a value of 1.4, in order to get the actual strength of the brickwork. The comparison of the results from present study with that of the experimental investigation carried out earlier showed that there is a significant difference in some cases. Since the experimental result in each set is the average value obtained from the testing of four to five couplets, therefore, by testing large number of couplets in each set, having a unique Em may give a better average value. REFERENCES Abrams, D.P. (1996). “Effects of Scale and Loading Rate with Test of Concrete and Masonry structures,” Earthquake spectra, Journal of Earthquake Engineering Research Institute, vol.12, no.1, pp.13-28. Adedeji, A.A. (2002). “Evaluation of Brick-wall mortar joint in Acid Rain from Radioactive Emission,” Proceedings of the fourth International Conference on Structural Engineering, Analysis and Modelling (SEAM 4), University of Science and Technology, Kumasi Ghana, February 26-28, vol.1, pp.67-74. Bakhteri, J. and Sambasivam, S. (2003). “Mechanical Behaviour of Structural Brick Masonry: An Experimental Evaluation,” Proceeding of 5th Asia-Pacitic Structural Engineering and Construction Conference Malaysis. 305- 317. Blume, J. A. and Proulx, J. (1968). “Shear in grouted brick masonry wall elements,” Western state clay product Association, San Franscico, California 8 T.C. Nwofor: Continental J. Environmental Design and Management 2 (1): 1 - 9, 2012 Borchelt, J.G. (1970). “Analysis of brickwalls subjected to axial compression and in-plane shear,” Proceedings of second international brick masonry conference Chinwah, J.G. (1973). “Shear resistance of brick walls,” Ph.D thesis, University of London. Chinwah, .J.G. (1985). “Shear stress path failure criterion for brickwork,” International Journal of masonry construction, No.4, PP, 22-28. Dhanasekar, M., Page, A.W. and Kleeman, P.W. (1985) “The failure of brick masonry under biaxial stresses,” Proc. Inst. Civil Engineers, Part 2, 79. June 195-313. Ephraim, M.E., Chinwah J.G., Orlu I.D. (1990) “Mechanisms Approach to Composite Frame and infill,” Proceedings of the Second International Conference on Structural Engineering Analysis and Modelling (SEAM 2), University of Science and Technology, Kumasi, Ghana, July 17-19, ol.1, pp.13-26. Graham, E.K. Andam, K.A. (1990). “Plane of Weakness theory for masonry Brick,” Proceeding of the second International Conference on structural Engineering and Modelling, pp.27-36. Hendry, A W. (1985). “Recent research on load bearing brickwork,” The British Ceramic Research Association, Special Publ. No.38. Madan, A. Reinhoru, A.M. Mander, J. and Valles, R. (1997). “Modelling of masonry infill panels for structural analysis,” ASCE Journal of Structural Engineering, vol.123, no.10, pp.1295-1302 Mayes, R .C and Clough, R.W. (1975). “State-of-the art in seismic shear strength of masonry -an evaluation and Review,” EERC Report No.75-21 Orr, D.M.F. (1981). “Single plane of weakness theory applied to masonry,” International Journal of Masonry Construction, Vol. 2, No.1, 1981. Page, A.W. (1981). “The biaxial compressive strength of brick masonry,” Proc. lnst. Civil Engineers, Part 2, 71, Sept., 893-906. Perera, A. (1993). “Low cost House Using Cement Stabilized Blocks and Timber Poles,” Department of Civil Engineering, University of Moratuwa, Sri Lanka. Pieper, K and Trantsch,W. (1970). “Shear tests on wall,” Proceedings of the seconding international brick masonry conference. Sinha, B. P. and Hendry, A. W. (1967). “Racking tests on storey height shear wall structures with openings subjected to pre compression,” Proc. 3rd International Brick Masonry Conference, Essen, West Germany, 1967, pp. 141-145. Stroven P. (2002) “Damage in Compressed masonry due to stress release,” Proceedings of the fourth International Conference on Structural Engineering, Analysis and Modelling (SEAM 4), University of Science and Technology, Kumasi Ghana, February 26-28, vol.1, pp.162-171. Received for Publication: 14/02/2012 Accepted for Publication: 20/04/2012 9

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