Modified Stiffness Matrix Method for Macro-Modeling of Infilled Reinforced Concrete Frames. T.C. Nwofor

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1 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 53 Modified Stiffness Matrix Method for Macro-Modeling of Infilled Reinforced Concrete Frames T.C. Nwofor Department of Civil and Environment Engineering University of Port Harcourt, P.M.B. 533 Port Harcourt, Rivers State, Nigeria templenwofor@yahoo.com ABSTRACT In this paper, two kinds of models are used in order to validate a basic stiffness method for the macro-modeling of infilled frames. Previous numerical modeling techniques were faced with several complexities like the existence of plane of weak in the mortar joints and material non-homogeneity, which limited the real non-linear micro-modeling of infilled frames. The new explicit two dimensional finite element method, which is one of the models used in this work is used to study the behaviour of masonry infilled reinforced concrete frames and also considers the effect of the size of openings which is often ignored by most designers. A second model which is basically a macro-modeling technique which uses the stiffness matrix method to analyze an equivalent one-strut model used to replace the infilled panel is also used in this work, and results obtained validated against that of the micro-modeling procedure. It was observed that the stiffness matrix method for macro-modeling of infilled frames can quickly and effectively model the shear strength response of infilled frames with openings up to a failure load. Keywords: 1.0 INTRODUCTION Infilled frame, infill panel, equivalent one strut model, stress and displacement In many countries, situated in seismic regions, reinforced concrete frames are infilled by brick masonry panels. Although the infill panels significantly enhance both the stiffness and strength of the frame, their contribution is often not considered mainly because of the lack of knowledge of the composite behavior of the frame and the infill. However, extensive experimental research [1]-[4] and semi-analytical investigations [5], [6] have been made. Recently, it has been shown that there is a strong interaction between the infill masonry wall and the surrounding frame. Attempts at the analysis of infilled frames since the mid 1950s have yielded several analytical models. For a better understanding of the approach and capabilities of each

2 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 54 model it may be convenient to classify them into macro- and micro- models based on their complexity. The basic characteristic of a macro- (or simplified) model is that they try to encompass the overall (global) behavior of a structural element without modeling all the possible modes of local failure. Micro- (or fundamental) models, on the other hand, model the behavior of a structural element with great detail trying to encompass all the possible modes of failure. Since the first attempts to model the response of the composite infilled frames structures, experimental and conceptual observations have indicated that a diagonal strut with appropriate geometrical and mechanical characteristics could possibly provide a solution to the problem. In 1958, Polyakov [7] suggested the possibility of considering the effect of the infilling in each panel as equivalent to a diagonal bracing and this suggestion was later taken up by Holmes [8] who replaced the infill by an equivalent pin-jointed diagonal strut made of the same material and having the same thickness as the infill panel and a width equal to one third of the infill diagonal length. Another set of researchers [9], [10] related the width of the equivalent diagonal strut to the infill/frame contact lengths using an analytical equation which has been adapted from the equation of the length of contact of a free beam on an elastic foundation subjected to a concentrated load [11]. Based on the frame/infill contact length, alternative proposals for the evaluation of the equivalent strut width have been given [1], [13]. Also efforts were made to stimulate the response of infilled frames under earthquake loading by taking into account stiffness and strength degradation of the infills [14]. They proposed to model each infill panel by six compression-only inclined struts. Three parallel struts are used in each diagonal direction and the off diagonal ones are positioned at critical locations along the frame members. The advantage of this strut configuration over the single diagonal strut is that it allows the modeling of the interaction between the infill and the surrounding frame. Even when it is a well known fact that infill walls have openings, recent research has concentrated on simple cases of infill wall without openings. It is pertinent to note that the direct influence of infill walls to the shear strength of the structural frame is greatly

3 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 55 reduced when the structure is subjected to cyclic or lateral loading as can be seen under real earthquake situations. Useful experimental findings [15], [16] showed considerable reduction in the shear response of infilled frames under cyclic loading. Analytical study of infill frame with opening is limited and has little comparison due to the different materials used and the different type of openings. Experimental investigation by Benjamin and Williams [17] on the lateral stiffness of infilled frames with openings showed a 50% reduction of the ultimate strength in infilled frames having an opening at the center of the infill with dimensions proportional to the infill dimensions by a ratio of one is to three. Also experimental investigation into the effect of opening positions on the behaviour of infilled frames with or without shear connectors was carried out [18]. It was observed that opening at either end of the loaded diagonal of an infilled frame without connectors reduces its shear strength about 75%. For infilled frame with shear connectors the reduction in shear strength was about 60-70% as compared with infilled frame with a solid panel. The reduction of strength in both cases is as a result of the centrally loaded square opening. The main purpose of this research is to model the shear resistance of lateral loaded infilled reinforced concrete frame structure which accounts for the effect of openings in the infill panel using the finite element method as an analytical tool and also to propose a nonlinear macro-model for lateral load analysis of masonry infilled reinforced concrete frame structure. We should note that in most cases, door and window openings are provided in masonry infill panels to make up for functional and ventilation requirements of buildings. Considering these openings which are the true representation of masonry infilled structure adds complexity and difficulties in analysis. The presence of these openings would tend to reduce the lateral strength and stiffness of the infilled frames. However this reduction in strength has not been considered especially in the macromodels mentioned in this review. Hence the models were only applicable to the analysis of solid masonry infilled frames. It has been strongly emphasized by most Emergency Management Agencies [19], that the effect of masonry infilled frames with or without openings should be considered in the estimation of seismic vulnerability of existing framed buildings. In most cases the strength and stiffness of infilled frames with

4 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 56 opening is based on the micro modeling of a composite infilled frame structure. Hence the development of a useful macro-model to predict the shear strength response in terms of the lateral load carrying capacity and component of the internal forces at ultimate load of infilled frames with opening is also necessary..0 FINITE ELEMENT MODELING OF INFILLED FRAME STRUCTURE The developed finite element model would be employed here to simulate the in-plane behaviour of masonry-infilled frames tested by previous researchers. Detailed experimental results of the specimens have been summarized [15]. Among several specimens tested under lateral load in their investigation specimen WC3 would be singled out for this present investigation. The specimen is a single panel of 3600mm long by 800mm high masonry infilled frame with a 0.8 x.m central opening. This particular specimen would correspond to structural model MIP04 used in this work. The basic method was to allow a horizontal load increase up to failure load and applied at the upper Conner of the Reinforced concrete infilled frame. From the foregoing the values obtained from experimental test would be compared with result of finite element analysis on the micro model in order to validate the model..1 Development of Element Stiffness and Stress Displacement Matrix For the purpose of this study the finite element method of analysis for a continuum would be used. Basic triangular elements shall be used and the formulation adopted is the displacement approach. In using this method the model displacements are the basic unknown, while the stresses and strain are assumed to be constant for each element. The finite element method of analysis used in this paper would involve voluminous numerical works which would be considerably simplified by matrix formulation of the whole problem, hence very suitable for computerization. For plane elasticity problem the elastic matrix denoted by [D] can be expressed as

5 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 57 E Exv x yx 0 1 vxy vyx 1 vxy vyx Eyvxy Ey D = (1) vxyvyx vxyvyx Ey 0 0 (1 + vyx) where E x and E y are the modulli of elasticity in the x and y direction respectively, V xy and V yx are the poisson s ratio in the xy and yx plane respectively. The element stiffness matrix [K e ] would be a 6 x 6 matrix for the plane elasticity triangle, because there exist two degree of freedom (DOF) at each node of the triangular element (see Figure ), hence the Nodal force vector [F e ] can be related to the displacement vector as in equation. { F } [ e K e ]{ δ e } = () y V 3 U 3 y Fy 3 Fx V Fy U Fx V 1 1 U 1 Fy 1 1 Fx 1 (a) x (b) x Figure : (a) Nodal displacements vector (b) Nodal force vectors displayed in the Cartesian co-ordinate system. 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

6 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 58 unknown coefficients ( α α ) of a plane triangular element. u = α + α x 1 v = α + α x α 3 y + α y 6 L representing the six degrees of freedom in the case 1, α 6 The triangular element stiffness matrix [K e ] is represented by [ t] e T { K } [ B] [ D][ B] = (4) where matrix [B] constraints constant linear dimensional values, represents area of triangle and t represents the thickness of the triangular elements. It is simpler in practice to perform the matrix multiplications of equation 4 numerically with the computer. To determine the element stresses from the element nodal displacements, the relationship below is considered where { σ ( x y) } = [ D] [ B] { δ e }, (5) Where the stress-displacement matrix [H] equates to the product of matrix [D] and [B] [H] = [D] [B] { ( x, y) } = [ H ] { d e } σ (6) Where σ is the component of normal stress ( σ x, σ y ) and shear stress ( τ xy ). Computer Program Formulation In other to implement the finite element method, a computer programme for two dimensional finite element analysis developed by the author would be used. The computer programme is divided into two parts (subroutines). The first part consists of the routines for the control numbers and data input modulus, the second part consists of routines for tabulated nodal displacements and element stresses. The basic steps to obtain the element stiffness matrix [K e ] and stress matrix [H] have already been discussed in details and would involve voluminous numerical work, hence this processes were well built up in the subroutines to take care of the overall analysis. The input data consists of specifying the geometry of the idealized structure, its mechanical properties, the loading and the support condition. The data also includes certain control numbers that would help the efficiency of the program such as the total number of nodes and elements. (3) THE INPUT data for the micro-model is as follows:

7 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 59 (1) Nodal point coordinates in direction x and y for each node. () Element properties: This includes the mechanical properties E and v for masonry and concrete in two directions x and y, the thickness (t) of the structural model and any other data defining each element and the structures as a whole. (3) Boundary conditions: These consist of the restraints of the nodes of the supports and the stiffness of the elastic supports (4) Loading: Consists of the component of the lateral load placed at the top Conner of the structure. THE OUTPUT consists mainly of (1) The components of displacements { δ } at each node in the directions x and y and the maximum displacement (δ max ) for a the model () The stresses in each element as follows (a) Component of normal stresses (σ) in the directions X and Y (b) The shearing stress (τ xy ) (c) The maximum shear stresses τ max The bulk of the input data for the finite element micro-modelling of masonry infilled structure will consist mainly of coordinates of nodal points and element properties. The typical structural model for the validation of micro-model would consist of 16 elements and 18 nodal points. Note that manual development of the mesh would entail considerable expenditure of time and labour, hence a mesh generation subsidiary program would aid the generation of this mesh and the corresponding coordinates of the nodal points of the elements automatically. The output from this subsidiary programme consists mainly of Coordinates x and y of all the nodal points numbered in a consecutive order. Element properties for all the triangular elements generated and numbered in a specific order. These outputs are then used as input data for the main programme.

8 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: Consideration of Size of Openings on the Shear Strength of Infilled Frame Structure In order to investigate the effect of the size of openings on the lateral strength and stiffness of infilled reinforced concrete frames, a parametric study would be conducted using the finite element analysis on the infilled brick masonry. The effect of the opening size on the shear strength would be studied for values of parameters denoted by β and λ m which is defined as percentage of the opened area of the solid infill panel area and ratio of the infill panel strength with openings to that without openings respectively. Hence a number of one-story one-bay infilled structure with varying size of opening would be analyzed using finite element method aided by the suitable computer programme code. A typical structural micro model for the analysis is shown below (figure and 3) with a 30kN horizontal load acting at the top corner of the infilled reinforced concrete frame structure. To conduct properly this investigation the central opening of a one-bay infilled structure is varied, but with particular interest on opening ratio of 0-5%. Structural models tagged M1P01-M1P05 would be considered with each model having a particular percentage opening in the infill panel. 30kN Figure : Infilled reinforced concrete frame structure with central opening

9 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: kN o o o Figure 3: Triangularly meshed micro-model ready for finite element analysis Shear load-lateral displacement path obtained from finite element analytical modeling of MIP04 model which is similar to the WC3 model can be compared against that obtained from results of experimental data obtained by Dawe and Seah [15] as shown in figure 4 in order to validate the finite element programme for the micro-modelling of masonry infilled concrete frame structure with openings Lateral Load (KN) This Research Dawe and Seah (1989) Lateral Displacement (mm) Figure 4: Lateral load-displacement curves

10 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 6 There was reasonable agreement between the experimental collapse loads of 85kN and the numerical result of 95kN and a good correlation coefficient of c r = 0.9 was obtain between the experimental and the numerical results. The maximum value for shear stress τ max for the different models considered can be obtained from the result of finite element analysis. These results show that the shear stress τ xy is a function of factors such as the applied lateral force, dimensions and the elastic properties of the structure. Hence, a more general formula for shear strength and ultimate shear load will consist of one which includes such variables. In other to investigate the effect of opening size on the shear strength of masonry infilled frames, a study was conducted for various values of the parameters denote β and λ m as defined previously. To this end the infilled frame structural models with central openings denoted as MIP01, MIP0, MIP03, MIP04, MIP05 corresponding to 0-5% were analyzed. The structural models are subjected to lateral loads which could be the result of seismic forces, and a finite element analysis of the models carried out to determine the effect of opening sizes on the lateral strength of masonry infilled frames. Here the estimated shear strength factor λ m (defined as the ratio of infilled panel strength with opening to that without opening) is used for comparison of the numerical data obtained. Figure 5 shows the variation of the shear strength reduction factor λ m to the opening ratio β. The results shows that an increase in the opening percent leads to a decrease in the infilled frame shear strength. The shear strength decreased to about 75% for a bare frame. It was also noticed that the shear strength reduction factor λ m was practically constant for an opening percentage of more that 55%. A comparison is made between the results of this investigation with previous analytical results [0]. The favorably agreement of this work with previous analytical study has further assisted in the validation of the finite element model used in this work.

11 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: Shear Strength reduction factor ( λm) Opening ratio (β) Figure 5: Variation of shear strength reduction factor of infilled frame with opening ratio for a case of central opening. From the foregoing, the values of shear strength reduction factor λ m obtained from the analysis can be plotted against the values of the opening percentage β. Hence a consideration can be made using the shear strength factor λ m to stimulate the equivalent width of the compressed diagonal strut, for the macro-modeling of infilled frame structure. A reasonable regression equation can be obtained relating λ m to β for a case of central opening of the compressed diagonal as λ m = 0.95 e 0.03β (7) 3.0 MACRO-MODELLING OF INFILLED FRAMES A typical macro-model, which consists of a modified, one-structural model proposed by the author, would be used to carry out this investigation. Here, the infill is replaced with an equivalent pin jointed diagonal strut with mechanical property correlated from that of the infill material. A three-strut model which consists of two off diagonal struts which can be used for the nonlinear analysis of actual infilled frames failing in corner crushing mode had been proposed [1]. However, this model was not used to analyze infilled frames with openings. In order to consider openings in the proposed macro model for this investigation, the equivalent diagonal strut area

12 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 64 is modified to account for the variation in these openings. Hence an equivalent structure would be obtained by comparing with available numerical results obtained in the previous section. 3.1 Modeling of Infilled Frames Adopting the One-Strut Model (OSM) The analysis of the proposed model would be carried out using the stiffness matrix method for pin-jointed bar elements. Where the stiffness matrix K for a bar element is represented by e [ k ] = AE L l l m m symmetric l lm l l m m where F e represents the force vector and δ e represents the displacement vector considering two degrees of freedom at each end of the bar, the force vector and the displacement vector can similarly be related in equation () { F } [ e K e ]{ δ e } lm m (8) = () An equivalent one-strut model for macro-modeling of infilled frame is shown in figure 6. 30kN Equivalent diagonal strut.65m L = 3.30m Figure 6: One-strut model for masonry macro-modelling of infilled frame structure

13 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 65 According to Saneinejad and Hobbs [] the total equivalent diagonal region area is simplified as ( α ) C α Cht A = 1 (9) Cos θ Considering the resistance of the infilled panel (R) and the reinforced concrete frame to the collapse load of the structure H = R = M Pj RCos θ + (10) h hh M h cosθ Pj, A = R f ' The equation can be modified to account for opening as R o = hh o M h cos θ pj, A m = R f ' m o m From the foregoing the dimensionless parameter λm relating to the effect of opening size on shear strength capacity of infill panels can be defined as follows R A hh M o m o pj λ m = = = (13) R A hh M pj Hence in order to modify the equivalent diagonal area to account for openings, it is expected that the regression equation relating the shear strength reduction factor (λ) to the opening area ratio (β) be utilized (11) (1) The modified equivalent diagonal region area in the infilled frames with a central opening would given by A m = λ m A d (14) Also carrying out this analysis it would be necessary to note the geometric properties of the diagonal struts are functions of the length of contact between the wall and the column α h and between the wall and beam α L [3]. Hence assuming a beam on elastic foundation as proposed [9], [4] the following relationships have been obtained. α π 4E l h f c 4 h = (15) EmtSin θ

14 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 66 and α π 4E l h f b 4 L = (16) E tsinθ m Where E m, E f = elastic moduli of the masonry wall and frame material respectively. t, h, l = thickness, height and length of the infill wall, respectively. l c, l b = moments of inertia of the column and the beam of the frame respectively. θ = tan -1 h L Hendry [3] also proposed the following equation to determine the equivalent or effective strut width w, where the strut is assumed to be subjected to a uniform stress ω 1 α c α h = + (17) Once the geometric and material properties of the struts are calculated, the stiffness matrix method for bar elements can be employed to determine the stiffness of the infilled frame, the internal forces and the deflections. Considering the one strut macro-model in figure 6, the following geometric and material properties can be deduced. Infill wall: Thickness t = 106mm Elastic modulus E m = x 10 3 N/mm Frame: Area of beam A b = 90,000mm Area of column A c = 90,000mm Moment of inertia of beam and column = I b = I c = 6.75 x 10 8 mm 4 Elastic modulus E f =.9 x 10 4 N/mm 1 h 1.5 Where θ = tan = tan = l 3.0

15 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 67 From equation 15 and 16 4E I h f c α 4 h = mm Em + Sin = π 4E I h f b α 4 L = π = 01.80mm Em + Sin Using equation 17 w 1 = α h + α L = 115mm w Area of diagonal strut A d = x 106mm = 119,05mm hence modified area of diagonal strut to account for the cases of opening A m = A d λ m Where λ m = 0.95e 0.03β Analyzing the frame using a classical methods of structural analysis in the stiffness matrix for a two dimensional structure, maximum unknown horizontal deflection would be obtained from the solution of the global structural matrix. 4.0 DISCUSSION OF RESULTS The equivalent one strut system was used for the macro-modelling of infilled frames using a classical method of structural analysis in the stiffness matrix method. Using this model, the non-linear static behaviour of masonry-infilled frames was studied by analyzing structural models MIP01-MIP05. The maximum horizontal displacement in the frame was analyzed for by using a modified area for the equivalent strut from equation 14 and the effective width of strut from equation 17. The value of displacement obtained using this model is compared with that obtained previously with the micro model. It can be seen from Figure 7a to 7e that the equivalent strut model was able to model the ultimate load capacity of the masonry infilled frames with openings up to failure as a very close agreement was seen between the micro and macro models with a general corresponding average magnification factor of 1.1.

16 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: Micro Model Macro Model L ateral L oad (KN) L ateral Displacement δ (mm) (a) Lateral Load (KN) Model Micro Macro Model L ateral Displacement δ (mm) (b)

17 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: Lateral Load (KN) Micro Model Macro Model L ateral Displacement δ (mm) (c) Lateral Load (KN) Micro Model Macro Model Lateral Displacement δ (mm) (d)

18 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: Lateral Load (KN) Micro Model Macro Model Lateral Displacement d (mm) Figure 7: (e) Lateral load displacement relations for different size of opening on the infill panel (a) MIP01 (b) MIP0 (c) MIP03 (d) MIP04 (e) MIP05 CONCLUSION The macro-modeling technique can be used for the design of infilled frame with opening by utilizing a modified area for the equivalent strut. Noting that in this work two kinds of numerical modeling strategies were used to stimulate the in-plane non-linear behaviour of infilled frames with openings, where the two dimensional finite element micro-model developed for the inelastic non-linear analysis of masonry-infilled structure was validated and used for the study of the effect of openings on the shear strength of the structure. Furthermore application of this model may be require a lot of computational skill especially for individuals that may not have useful analytical program software hence, an equivalent one strut model was adopted and modified to investigate the nonlinear behaviour of infilled frames with a central openings. Here a modified diagonal area related against the shear strength reduction factor obtained from the different case of opening sizes can be use improve the estimation of the equivalent strut to account for the effects of openings when utilizing the macro-modeling technique. This model was used

19 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 71 in the study of one storey-one bay infilled frame structures, and the results obtained compared favorably with that obtained from the finite element micro modeling technique. Valuable extension of this study would include but not limited to the following, (a) The utilization of the concept of equivalent strut to model the behavior of multistorey one-bay full and partial infilled frames. (b) The macro-modeling technique should be extended to accommodate the effect of position of openings on the non-linear analysis of infilled frames. (c) Finally the extension of the macro-modeling technique to the response of infilled frame to dynamic shear loads is also necessary in order to obtain results that would be utilized by designers in real dynamic regimes. REFERENCES 1. Mehrabi, A.B., Shing, P.B., Schuller, M. and Noland, J. (1996) Experimental evaluation of masonry-infilled RC frames", J. Strut. Eng., 1(3), Page, A.W., Kleeman, P.W. and Dhanasekar, M. (1985) An in-plane finite element model for brick masonry, New Analysis Techniques for Structural Masonry, Proc. of a session held in conjunction with Structures Congress, Chicago, Illinois, ASCE, Buonopane, S.G., and White, R.N. (1999) "Pseudodynamic testing of masonryinfilled reinforced concrete frame", J. Strut. Eng., vol. 15(6), Santhi MH, Knight GMS, Muthumani K. (005) Evaluation of Seismic Performance of Gravity Load Designed Reinforced Concrete Frames, Journal of Performance of Constructed Facilities, Vol. 19, No.4, pp Dhanasekar, M. and Page, A.W. (1986) Influence of brick masonry infill properties on the behavior of infilled frames", Proc., Instn. Civ. Engrs., London, Part,81, Moghaddam, H.A. (004) Lateral Load Behavior of Masonry Infilled Steel Frames with Repair and Retrofit, J. Strut. Eng., 130 (I), Polyakov, S.V. (1960) On the interaction between masonry filler walls and enclosing frame when loading in the plane of the wall, Translation in earthquake engineering, Earth quake Engineering Research Institute, San Francisco, Holmes, M. (1961) Steel frames with brickwork and concrete infilling, Proc., Instn. Civ. Engrs., London, Part, 19,

20 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: Smith, B.S. (1966) Behavior of square infilled frames, J. Strut. Div., ASCE, STI, Smith, B.S. and Carter, C. (1969) A method of analysis for infilled frames, Proc., Instn. Civ. Engrs., 44, Hetenyi, M. (1946) Beams on elastic foundations, University of Michigan Press. 1. Mainstone, RJ. (1971) On the stiffnesses and strengths of in filled frames, Proc., Instn. Civ. Engrs., Supp. (iv), Kadir, M.R.A. (1974) The structural behaviour of masonry in fill panels in framed structures, PhD thesis, University of Edinburgh. 14. Chrysostomou, C.Z. (1991) Effects of degrading infill walls on the nonlinear seismic response of two-dimensional steel frames, PhD thesis, Cornell University. 15. Dawe J.L. and C.K. Seah, (1989), "Behavior of Masonry Infill Frames," in Canadian Journal of Civil Engineering, vol. 16, Vintzeleou, F., and Tassios. T.P. (1989) Seismic behaviour and design of infilled R.C. frames, Proc., J. European Earthquake Eng.,, Benjamin, J.E. and Williams, H.A. (1958) The behaviour of one story brick shear walls, J Struct. Div., ASCE, 84(4). 18. Mallick, D.V. and Garg, R.P. (1971) Effect of openings on the lateral stiffness of infilled frames, Proc. Inst. Civ. Eng., Struct. Build, 49, Federal Emergency Management Agency (FEMA) (000). Pre-standard and Commentary for the Seismic Rehabilitation of Buildings, Report No. FEMA 356, FEMA, Washington, D.C. 0. Giannakas, A., Patronis, D., and Fardis, M., (1987) The influence of the position and size of openings to the elastic rigidity of infill walls. Proc. 8 th Hellenic Concrete Conf. Xanthi, kavala, Greece, El-Dakhakhni, W.W., Elgaaly, M. and Hamid, A.A. (003). Three-Strut Model for Concrete masonry-infilled Steel Frames, J. of Struct. Eng., 19(), Saneinejad, A. and Hobbs, B. (1995) "Inelastic Design of Infilled Frames", Journal of Structural Division, Proceedings of ASCE, Vol. 11, No.4, pp Hendry, A. (1981) Structural Brickwork, Macmillan, London. 4. Amrhein, J.E., Anderson, J. and Robles, V. (1985) "Mexico Earthquake - September 1985," The Masonry Society Journal, Vol. 4, No., G.1-G.17.

21 International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 1 No: 0 73 NOMENCLATURE A = Area of loaded diagonal region of infill panel C r = Correlation coefficient E x, E y = Modulus of elasticity in x and y direction { F e } = Element force vector h = Height of column H = Lateral load carrying capacity of solid masonry infilled frame [H] = Triangular element stresses matrix H o = Lateral load carrying capacity of masonry infilled frame with opening l = Length of beam K = Constant depending upon brick properties and brick-mortar joint configuration; [K e ] = Triangular element/bar element stiffness matrix M p = Plastic moment capacity of frame members n = Total number of data points R = Resistance of solid infill panel R o = Resistance of infill panel with opening t = Thickness of infilled plane { δ e } = Element displacement vector τ = Shearing stress component in rectangular coordinates xy σ x, σ y ε x, ε y β λ m δ c θ = Normal component of stress in the x and y axes = Strain in the x and y directions = Ratio of central window opening to infill panel area = Modification factor of diagonal region area = Ratio of column contact length to height of column = Tan -1 (h/l)