MODELING THE OUT-OF-PLANE BENDING BEHAVIOR OF RETROFITTED URM WALLS. Abstract. Introduction

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1 MOELING THE OUT-OF-PLANE ENING EHAVIOR OF RETROFITTE URM WALLS J. I. Velázuez imas, Professor, Facultad de Ingeniería, Universidad Autonoma de Sinaloa, Mexico M.R. Ehsani, Professor CEEM epartment, The University of Arizona, Tucson, AZ J. H. Castorena González, Professor, Esc. de Ing. Mochis, Universidad Autonoma de Sinaloa, Mexico A. Reyes Salazar, Professor, Facultad de Ingeniería, Universidad Autonoma de Sinaloa, Mexico Astract The classical laminated plate theory has een used to develop a simplified analytical model for predicting the out-of-plane ehavior of URM wall retrofitted with glass fier composites. The predictions of the model were compared with experimental results from reduced-scale wall tests. The correlation etween the analytical model and experimental results was found to e very good. Introduction URM uildings constitute the main part of the uildings inventory worldwide. When sujected to seismic forces, the walls forming the structural system of this type of uildings experience the action of in-plane and out-of-plane forces. Failure modes related to oth types of forces are characterized y a diagonal crack pattern and horizontal cracks along ed-joints as documented y many researchers [,]. Failure due to out-of-plane ending is recognized y the earthuake engineering community as the more dangerous [,], and attriutes to structural damage and loss of life. ue to the advantages of composites materials with respect to conventional construction materials such as steel, they have een used in many applications to strength URM uildings to increase their in-plane and out-of-plane loadcarrying capacity [,, 6]. This paper presents the mathematical modeling of the flexural ehavior of URM walls retrofitted with fier composites using the Classical Laminated Plate Theory (CLPT). The walls analyzed are those sujected to uniform load and with oundary conditions defined y two sides free and the others two simple supported. ackground There have een reported many studies aout modeling the ehavior of URM walls sujected to in-plane and out-of-plane loading [7, 8, 9]. However, little research has een done on mathematical modeling of URM walls retrofitted with composites and sujected to out-of-plane ending. Some such studies are discussed next. Schwegler [], was the first to report a study where caron fier composite strips were used to retrofit one URM uilding. He tested wall panels under in-plane and out-of-plane loading and found that composite materials are a viale alternative to retrofit URM uildings to increase their strength and stiffness. He also developed an analytical model for in-plane ehavior. Triantafillou [], conducted an experimental work on eams constructed with hollow clay ricks strengthened with composites strips. The main ojective in that study was to investigate the effect of the reinforcement on the strength and the mode of failure of eams. From experimental results he

2 concluded that it is etter to use composites strips onded to the wall surface with spaces in etween instead of covering the whole wall surface. He also developed an analytical model for eams retrofitted with composite strips and suggested the use of laminated plate theory for walls having composite reinforcement covering the whole surface. Velázuez-imas and Ehsani [], ased on experimental results of seven half-scale rick masonry walls externally strengthened with vertical glass-faric composite strips, and sujected to outof-plane loading, developed an analytical model for predicting the flexural ehavior of tested walls. The main parameters investigated were the amount of composites, the height-to-thickness ratio (h/t), the tensile strain in composites, and the mode of failure. According to the otained results, they concluded that the ehavior of the walls is est predicted with a linear elastic approach. Modeling with CLPT In this part, the Classical Laminated Plate Theory (CLPT) is used to develop the models for the investigated masonry walls. The ehavior of the walls is divided into two stages that are: the stage corresponding to the first ed-joint crack and the stage corresponding to the ultimate load, for details see []. In order to differentiate oth stages more clearly, the first one is called the symmetrical case and the second non-symmetrical. Therefore the mathematical work is done in two parts that are developed next. Theoretical Concepts of CLPT In this part, general concepts of CLPT that are important for this study are developed. First of all, the oundary conditions of the cases to e investigated are shown in Figure. It is important to say that only plates with two sides free and the other two simple supported will e analysed. This is ecause the availale experimental data is for walls having such oundary conditions. In addition, a uniformly distriuted load ( ) is assumed to act on the surface of the wall parallel to the Z-axis. Figure. Reference system, oundary and loading conditions of the discussed model The corresponding laminar arrangement of the studied walls is depicted is Figure. With this layer arrangement the theoretical work is simplified due to the symmetrical distriution of the layers. It is assumed that similar layers having the same thickness and fier distriution are aove and elow the middle plane of the plate. Therefore, according to the CLPT the components ij of the constitutive euations are zero.

3 Figure. Coordinate system and layer arrangement [9 / / /9 ] for a typical wall Analysis of symmetrical laminated plate For analysis of this type of plates, many theories have een developed and they can e linear and nonlinear. Also Finite Element Method has een widely used for analysis of laminated plates. According to the geometry properties, the plates can e thin or thick ones, and symmetrical and nonsymmetrical. However, the approach ased on thin plates is the most common for doing analysis since the shear and transverse normal strains are not considered. Therefore, in this study the CLPT is used for a thin plate. In order to apply this approach the following assumptions are taken into account [].. Each plate is composed of an aritrary numer of layers perfectly onded such that no relative displacement takes place.. Each layer is homogeneous, orthotropic and ehaves elastically.. The plate thickness is small with respect to its other dimensions.. The displacement components u, v and w are small compared with the plate thickness.. Each ply is of uniform thickness. 6. Transverse shear strains are zero. 7. Transverse normal strain is zero. 8. ody forces are neglected. 9. The displacement u and v are linear function of z. Assumptions 6 and 7 are called Kirchhoff assumptions and the prolem can e reduced to a two dimensional study of the middle plane. Comining euilirium, displacements and strain euations with the latter assumptions, the constitutive relationships for laminated plates can e developed. Such euations are shown next:

4 N x N y N xy M x M y M xy A A A 6 6 A A A 6 6 A 6 A 6 A x y xy κ x κ y κ xy () Where A ij is called extensional stiffness, ij the ending stiffness, and ij the endingextensional coupling stiffness [], which are defined in terms of the lamina stiffness Q (k) ij and the plate thickness h, and the coordinate z as: h / ( A,, ) (,, ) k ij ij ij z z Q dz i, j,,6 () h / ij Taking into account the layer arrangement shown in Figure, the force vector of () is divided in two parts which are the stress resultant and moment resultant euation. It is important to say that a noncoupling system is otained where the extension is independent of ending. Furthermore, the differential euation for a symmetric and orthotropic laminated plate with no inertial terms is given as []: w w w ( 66 ) () x x y y Analytical Solution In order to solve E. (), many solutions are availale. However, in this study an approach suggested y Levy [, ] is used. This approach was chosen ecause allows solutions for plates having oundary conditions with two sides free and the other two simple supported, i.e. similar to the walls tested in the availale experimental studies. For solving E. () Levy suggested the following displacement function. w (x,y) w o(y) w (x,y) () Where each term is given y:

5 w ( y ) w (x, y) π n n X sin n n ( x ) n π y sin n π y () (6) Ashton and Whitney[], have solution for plates with oundary conditions denoted y two sides simple supported and two fixed. However, for the case discussed in this study no reference was found for a solution. The full development of such solution can e found in []. ue to the fact that the solution reuires many time consuming steps its development is not shown in this study. According to the solution given y E. () and the oundary conditions to e satisfied, the following euation can e otained: '''' n '' n ( ),,,. X ( ) 66 X n ( ) X n x n x n( x) π π nπ sen (7) where the prime denotes differentiation with respect to x. Thus the function X n (x) must satisfy the homogeneous euation. '''' nπ '' nπ ( ) (8 ) X ( ) 66 X ( ) X n x n x n( x) The general solution of E. (8) can e written in terms of four aritrary constants A n, n, C n and n. The particular form of the solution depends upon the roots of the characteristic E. (9): s ( 66 ) s (9) Three solutions exist for E. (9), which are: Case : The roots are real and uneual and denoted y ±s and ±s (s, s >). This solution can e written as: w [ n,,,... π n A n cos α cosh γ n sin α sinh γ ]sin nπy w [ n,,,... π n π n π n cos α cosh γ sin α sinh γ ]sin nπy ()

6 Case : The roots are real and eual and denoted y ±s (s>). This solution can e written as: nπy w [ A cosh cosh ]sin,,,... n α C n α n π n [ sinh cosh K w α α n,,,... π n π n n sinh cosh ]sin K nπy α α π n n () Case : The roots are complex and denoted y s±ti and -s±ti (s, t>), and written as: nπy w [ A cosh sinh ]sin,,,... n α n x α n π n [ ( ) cosh nπ ξ w α n,,,... π n π n ξ ξ ξ ξ ( ) sinh ]sin nπ ξ x α π n ξ ξ ξ ξ nπy () The last three euations are solutions ased on an approach suggested y Levy. The Levy approach assumes a displacement function consisting of two parts which are: cylindrical flexure on a unit strip parallel to the Y axis and the other is the flexure parallel to X. Since the developed solutions are not easy for practical purposes, a simplified solution is developed next and this is the main ojective of this study. The simplified solution is otained y considering only the cylindrical ending. With such models, displacements and load carrying capacity of URM retrofitted with fier composites, and modeled with CLPT, can e predicted. Symmetrical Case This case corresponds to a wall that has no cracks. As pointed out at the eginning of this study, the ehavior of the investigated walls is divided in two stages corresponding to the uncracked and cracked ones. Since many steps are involved in to develop the simplified model, in this paper only the final model is presented. However, it is important to say that the model is derived y keeping the first term of Es. () and (), and y simplifying the euations due the symmetrical arrangement of the plate, and y assuming that E m.e m [6] and E f E f. Thus, the resulting simplified euation for displacements is given y E. (). 6

7 w ( y y y ) 8 K T K T K K re inf wall n W E f f h h K ( m re inf a ν 8 8 ), K wall E m ν. h m 8 () In addition, an expression for calculating the load corresponding to the rupture is developed and given y E. () E m h m ( y ) y σ y σ rup () ν [ K K ] reinf wall. where K reinf and K wall are defined in E. (). In order to get, the value of modulus of rupture f r σ rup f m (psi) (UC-97) is given in E. (). So the point corresponding to maximum tensile stress is otained y using Es. () and () Nonsymmetrical Case As done in the previous case, a simplified mathematical model for a cracked ehavior was developed. For this situation many mathematical developments were made. ue to space limitations, only general steps are herein descried. The layer distriution showing the cracked and uncracked areas is depicted in Figure a, and the stress distriution is shown in Fig.. In addition, a linear strain distriution across the wall section is assumed. a) ) Figure. Cracked cross sectional and stress distriution 7

8 It is important to say that ending stresses in rick masonry through the plate thickness (h) were calculated ased on a model proposed y [7], and the factors γ and α r in model proposed y [8]. y using a statistical study of experimental availale data, an averages values of α r.6 and γ.6, were determined []. Thus, y taking into account the effect of delamination of composite strips on reducing the ultimate load-carrying capacity and to increase the deflection capacity, two models for ultimate load ( ult ) and ultimate deflection (W ult ) were developed. The corresponding mathematical expressions are:.67 max (. ω ) ult w max ult ( y y y) w w max ult.9ln[ ω( ) ].7 h Where composite reinforcement index,ω, and max are given y: () (6) (7) E ω. f max 8 f ' m M a resist ρ v 8 ' g g ω g fmh.6( )[.7( )] ( ) h h g ( ) h h (8) (9) Application of the Proposed Euations The mathematical models developed were used to estimate the experimental results of seven half-scale URM walls retrofitted with glass-fier composite strip reported y [9, and ]. The set of tested walls was divided into two sets which are corresponding to short walls and slender walls. All specimens were sujected to the same standard history pattern of static cyclic out-of-plane loading applied with an air-ag system. The specimens were simple supported at top and ottom and remained free along the vertical edges. The walls were mm (8 in.) wide and mm (.9 in.) thick. The small-scale ricks were cut from solid clay ricks and had a dimension of x9x8 mm (.x.9x. in.). Type N mortar similar to that used in old masonry uildings was used y the mason to construct the walls. Each set of walls had a different height. The slender walls were mm (6 in.) high (i.e. h/t8) and the short walls were 7 mm (8 in.) high (i.e. h/t). Two walls were selected for the purposes of validating the proposed simplified model for predicting their ehavior. One wall is short one having aspect ratio h/t and reinforced with composite strips euivalent to 7% of the corresponding alanced condition(ρ ). The other wall is a slender one with h/t 8 and retrofitted with a composite reinforcement ratio eual to % the corresponding alanced condition (ρ ). The results are shown in Figs. and. As can e oserved from oth graphs, the predicted ones are very close to the experimental curves. 8

9 Load (KPa) Experimental Theory eflection (mm) Figure. Theoretical and experimental graphs of a wall with h/t and ρ.7ρ Load (KPa) 8 6 Experimental Theory 6 7 eflection (mm) Figure. Theoretical and experimental graphs of a wall with h/t 8 and ρ ρ 9

10 Conclusions ased on Classical Laminated Plate Theory and using an approach suggested y Levy, a simplified mathematical model that allows the estimation of points corresponding to the cracking and ultimate points of the load-deflection curves for URM walls retrofitted with FRP materials was developed. From calculated values and comparison to availale experimental data, it can e concluded that good correlation exists etween oth results References [] runeau, M. (99), State-of-the-art report on seismic performance of unreinforced masonry uildings. ASCE Journal of Struc. Engineering., Vol., No. :- [] Kehoe,.E. (996), Performance of retrofitted unreinforced masonry uildings. paper No. 7, Eleventh World Conference on Earthuake Engineering, Acapulco, Mexico. [] Prawel, S.P. and Reinhorn, A.M. (98), Seismic retrofit of structural masonry using a ferrocement overlay. Proceedings Third North-American Masonry Conference. University of Texas at Arlington, [] Saadatmanesh, H and Ehsani, M.R. (998). Fier Composites in Infrastructure, Proceedings of the Second International Conference on Composites in Infrastructure, Tucson, Arizona, Vols. I and II, USA [] Schwgler, G.(99), Masonry Construction Strengthened with Fier Composites in Seismically Endangered Zones. th European Conference In Earthuake Engineering. [6] El-adry, M.M. (996), Advanced Composites Materials in ridges and Structures, Proceedings ACMS II, Montreal, CA, 7 pp. [7] Sinha,.P. (978). A simplified ultimate load analysis of laterally loaded model orthotropic rickwork panels of low tensile strength, The Structural Engineer, (6), pp [8] awe, J. L. and Seah, C.K. (989). Out-of-plane resistance of concrete masonry infilled panels, Canadian Journal of Civil Engineering, Vol.6, pp [9] Arams,.P., Angel, R. and Uzarski (99). Transverse strength of damaged URM infills, 6 th North American Masonry Conference, rexel University, PA. [] Triantafillou, T.C.(998), Strengthening of Masonry Structures Using Epoxy-onded FRP Laminates. Journal of Composites for Construction, Vol., No.. [] Velazuez-imas, J.I, Ehsani, M. R., and Saadatmanesh, H. (). Modeling Out-of-Plane ehavior of URM Walls Retrofitted with Fier Composites, ASCE Journal of Composites for Construction, (), 7-8.

11 [] Ashton, J.E., Whitney, J.M. (97) Theory of Laminated Plated. Progress in Materials Science Series, vol.iv. [] Reddy, J.N (997) Mechanics of Laminated Composite Plates: Theory and Applications CRC Press, 77 pp. [] Reddy, J.N., Khdeir, A.A.(987) Lévy Type Solutions for Symmetrically Laminated Rectangular Plates Using First-Order Shear eformation Theory. Journal of Applied Mechanics, No., 7-7, [] Castorena Gonzalez, J.H. () Modelación Mediante la Teoría de Placas Laminadas de Muros de Mampostería Reforzados con Materiales Compuestos y Sujetos a Flexión Tesis de Maestría, Facultad de Ingeniería de la Universidad Autónoma de Sinaloa. [6] Hendry, A. W., (97) The Lateral Strength of Unreinforced rickwork, The Structural Engineer, V, pp. -. [7] Turkstra, C. J. (97) Resistencia de Muros de Mampostería Ante Cargas Verticales Excentricas, Instituto de Ingeniería, Revista No. 7, UNAM, Mexico. [8] Park, R y Paulay, T., (979) Estructuras de Concreto Reforzado, Editorial Limusa, México. [9] Ehsani, M. R., Saadatmanesh, H. and Velazuez-imas, J.I. (999), ehavior of Retrofitted URM Walls Under Simulated Earthuake Loading, ASCE Journal of Composite for Construction, Vol., No. : -. [] Velazuez-imas, J.I., Ehsani, M.R. and Saadatmanesh, H.(), Out-of-Plane havior of rick Masonry Walls Strengthened with Fier Composites. ACI Structural Journal, Vol.9, No E E ν ij G ij κ x κ y κ xy Notation Modulus of elasticity of composite layer in principal () direction; Modulus of elasticity of composite layer in transverse() direction; Poisson ratio in the j-direction when a load is applied in the i-direction; Shear modulus; Plate curvature parallel to the xz-plane; Plate curvature parallel to the yz-plane; Torsion of the plate surface along the x e y-axis; a Side length of the plate parallel to the x-axis; Side length of the plate parallel to the y-axis;. N x Force per unit length parallel to the x-axis (side a); N y Force per unit length parallel to the y-axis (side ) N xy Shear resultant force per unit length; M x Flexural Moment per unit length around Y-axis; Flexural Moment per unit length around X-axis; M y

12 M xy Torsion Moment per unit length; x Strain parallel to the x-axis of the plate middle plane; y Strain parallel to the y-axis of the plate middle plane; xy Shear strain of the plate middle plane. z k istance in z-direction from the middle plate surface to the layer k; s Real root or real part of a complex root; t imaginary part of a complex root; α nπta/ γ nπsa/ α nπs x/ α nπs x/ α nπs a/ α nπs a/ u Middle plane displacement of the plate in the X-direction; v Middle plane displacement of the plate in the Y-direction; E m Modulus of Elasticity perpendicular to ed-joint; E m Modulus of elasticity perpendicular to the rick head joints E m >E m E f Modulus of Elasticity parallel to the fier orientation;. E f Transverse modulus of composite E f >>E f ; E f n W f /a E f.e f n Numer of composite strips onded to the wall surface; W f Wide of the composite strip; h m Wall thickness h m h m / h f Thickness of composite strip; h t h m h f gr epth of the cracked zone; θ Main fier orientation measured with respect to principal axis plate; g Neutral axis depth on cracked walls; h istance from top wall surface to the composite strip;. M resist Nominal moment capacity of reinforced masonry; f m Compression stress on rick masonry; m Strain on rick masonry; f fr Tension stress on composite strip; A f Tensile composite reinforcement area; ρ v A f /ah ω ( m E f /f m )ρ v E Modulus of elasticity calculated on straight part of the σ- curve Strain at maximum stress on masonry; f m Compressive strength of masonry specimens; Strain on fier f

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