role of masonry infill in seismic resistance of rc structures - CiteSeerX

‫‪Salah El-Din Fahmy Taher and Hamdy Mohy El-Din Afefy‬‬

‫‪ROLE OF MASONRY INFILL IN‬‬ ‫‪SEISMIC RESISTANCE OF RC STRUCTURES‬‬ ‫*‪Salah El-Din Fahmy Taher‬‬ ‫‪Professor of Concrete Structures, Faculty of Engineering, Tanta University,‬‬ ‫& ‪Vice-Dean, Post-Graduate and Research Affairs, Faculty of Engineering, Tanta University,‬‬ ‫‪Director, Higher Education Enhancement Project Fund (HEEPF), Ministry of Higher‬‬ ‫‪Education, Egypt.‬‬

‫‪and Hamdy Mohy El-Din Afefy‬‬ ‫‪Lecturer Assistant, Structural Engineering Department, Faculty of Engineering, Tanta‬‬ ‫‪University,Egypt.‬‬

‫اﻟﺨﻼﺻﺔ‪:‬‬ ‫ﺘﻡ ‪ -‬ﻓﻲ ﻫﺫﺍ ﺍﻟﺒﺤﺙ ‪ -‬ﺍﻗﺘﺭﺍﺡ ﺃﻜﺜﺭ ﺍﻟﻨﻤﺎﺫﺝ ﺒﺴﺎﻁﺔ ﺒﺩﺭﺠﺎﺕ ﺤﺭﻴﺔ ﻤﺨﻔﻀﺔ ﻹﻤﻜﺎﻥ ﺍﺴﺘﺨﺩﺍﻤﻬﺎ ﻓﻲ ﺘﺤﻠﻴل‬ ‫ﺍﻟﻬﻴﺎﻜل ﺍﻟﻤﺤﺸﻭﺓ ﻤﺘﻌﺩﺩﺓ ﺍﻟﻁﻭﺍﺒﻕ ﻭﻤﺘﻌﺩﺩﺓ ﺍﻟﺒﻭﺍﻜﻰ‪ .‬ﻴﺘﻜﻭﻥ ﺍﻷﻨﻤﻭﺫﺝ ﻤﻥ ﻭﺴﻁ ﻤﺘﺠﺎﻨﺱ ﻤﻥ ﺍﻟﺨﺭﺴﺎﻨﺔ ﺍﻟﻤﺴﻠﺤﺔ‬ ‫ﻤﻘﻴ ‪‬ﺩ ﻓﻲ ﻜل ﺒﺎﻜﻴﺔ ﺒﺸﻜﺎﻻﺕ ﻗﻁﺭﻴﺔ ﺃﺤﺎﺩﻴﺔ ﺍﻟﺘﺄﺜﻴﺭ ﺘﻜﻭﻥ ﻓﻌﺎﻟﺔ ﻓﻘﻁ ﻓﻲ ﺍﻟﻀﻐﻁ‪ .‬ﻭﻗﺩ ﺘﻡ ﺘﺤﺩﻴﺩ ﺨﺼﺎﺌﺹ ﺍﻟﻨﻅﺎﻡ‬ ‫ﺍﻟﻤﻜﺎﻓﺊ ﻭﺨﻭﺍﺹ ﺍﻟﻤﻭﺍﺩ ﻏﻴﺭ ﺍﻟﺨﻁﻴﺔ ﺒﺎﺴﺘﺨﺩﺍﻡ ﻤﻔﺎﻫﻴﻡ ﺃﺴﻠﻭﺏ ﺍﻟﺘﺤﻠﻴل ﺍﻟﻤﻌﺎﻜﺱ ‪ ،‬ﺇﻀﺎﻓﺔ ﺇﻟﻰ ﺍﺨﺘﺒﺎﺭﺍﺕ‬ ‫ﺍﻟﻔﺭﻭﺽ ﺍﻹﺤﺼﺎﺌﻴﺔ ﻟﻠﺤﺼﻭل ﻋﻠﻰ ﺃﻨﺴﺏ ﺃﺴﻠﻭﺏ ﺘﻨﻘﻴﺢ ﺒﺎﻟﺩﻗﺔ ﺍﻟﻤﻼﺌﻤﺔ ﻓﻰ ﺍﻟﺘﻔﺎﻭﺕ ﺍﻟﻤﺴﻤﻭﺡ‪ .‬ﻭﻴﺴﻤﺢ ﺍﻟﻨﻅﺎﻡ‬ ‫ﺍﻟﻤﻘﺘﺭﺡ ﺒﺎﻟﺘﺤﻠﻴل ﺍﻻﺴﺘﺎﺘﻴﻜﻰ ﻭﺍﻟﺩﻴﻨﺎﻤﻴﻜﻲ ﺒﻁﺭﻴﻘﺔ ﺍﻟﻌﻨﺎﺼﺭ ﺍﻟﻤﺤﺩﺩﺓ ﻏﻴﺭ ﺍﻟﺨﻁﻴﺔ ﻟﻠﻬﻴﺎﻜل ﺍﻟﺨﺭﺴﺎﻨﻴﺔ ﺍﻟﻤﺴﻠﺤﺔ‬ ‫ﺍﻟﻤﻌﻘﺩﺓ‪ .‬ﻭﻗﺩ ﺘﻡ ﺇﺠﺭﺍﺀ ﻓﺤﺹ ﻟﺤﺴﺎﺴﻴﺔ ﺩﻗﺔ ﺍﻟﻨﻅﺎﻡ ﺍﻟﻤﻘﺘﺭﺡ ﻟﻠﺘﺤﻘﻕ ﻤﻥ ﻤﻼﺌﻤﺘﻪ ﻓﻲ ﻤﻌﺎﻟﺠﺔ ﺍﻟﺘﻁﺒﻴﻘﺎﺕ ﺍﻹﻨﺸﺎﺌﻴﺔ‬ ‫ﺍﻟﻤﺨﺘﻠﻔﺔ‪.‬‬ ‫ﻭﺒﻌﺩ ﺍﻟﺘﺤﻘﻕ ﻤﻥ ﺩﻗﺔ ﺍﻷﻨﻤﻭﺫﺝ ﺍﻟﻤﻘﺘﺭﺡ ﺘﻤﺕ ﺩﺭﺍﺴﺔ ﺍﻟﺴﻠﻭﻙ ﺍﻟﺯﻟﺯﺍﻟﻲ ﻟﻺﻁﺎﺭﺍﺕ ﺍﻟﺨﺭﺴﺎﻨﻴﺔ ﺍﻟﻤﺴﻠﺤﺔ ﺘﺤﺕ‬ ‫ﺘﺄﺜﻴﺭ ﺍﻟﺤﺸﻭ ﺍﻟﺠﺯﺌﻲ ﻤﻥ ﺍﻟﻁﻭﺏ ﻓﻲ ﺃﺴﻔل ﻫﺫﻩ ﺍﻹﻁﺎﺭﺍﺕ ﻭﻭﺴﻁﻬﺎ ﻭﺃﻋﻼﻫﺎ‪ .‬ﻭﻗﺩ ﺍﺨﺫﺕ ﺍﻟﺩﺭﺍﺴﺔ ﻓﻲ ﺍﻻﻋﺘﺒﺎﺭ‬ ‫ﺘﺄﺜﻴﺭ ﻋﺩﺩ ﺍﻷﺩﻭﺍﺭ ﻭﻋﺩﺩ ﺍﻟﺒﺎﻜﻴﺎﺕ ﻟﻺﻁﺎﺭ ‪ ،‬ﻭﻜﺫﺍ ﻨﺴﺒﺔ ﺍﻟﺤﺸﻭ ﻭﻤﻜﺎﻨﻪ‪ .‬ﻭﻗﺩ ﺍﺴﺘﺨﺩﻤﺕ ﻁﺭﻴﻘﺔ ﺭﺍﻴﻠﻰ ﻟﻠﻁﺎﻗﺔ‬ ‫ﻟﺘﺤﺩﻴﺩ ﺍﻟﻤﻌﺎﻤﻼﺕ ﺍﻟﻤﺨﺘﻠﻔﺔ ﻟﻠﺴﻠﻭﻙ ﺘﺤﺕ ﺍﻟﺘﺄﺜﻴﺭ ﺍﻟﺩﻴﻨﺎﻤﻴﻜﻲ ﺍﻟﻌﺭﻀﻲ‪ .‬ﻭﻗﺩ ﻋﻜﺴﺕ ﺍﻟﻨﺘﺎﺌﺞ ﺘﺄﺜﻴﺭ ﺤﺸﻭ ﺍﻹﻁﺎﺭﺍﺕ‬ ‫ﺍﻟﺨﺭﺴﺎﻨﻴﺔ ﺍﻟﻤﺴﻠﺤﺔ ﻓﻲ ﺯﻴﺎﺩﺓ ﺍﻟﻤﻘﺎﻭﻤﺔ ﺍﻟﺠﺴﺎﺀﺓ ﻭﺍﻟﺘﺭﺩﺩ ﻟﻠﻨﻅﺎﻡ ﺍﻹﻨﺸﺎﺌﻲ ﻜﻜل ﻁﺒﻘ ﹰﺎ ﻟﻨﺴﺒﺔ ﺍﻟﺤﺸﻭ ﻭﻭﻀﻌﻪ ‪.‬‬ ‫ﻭﻗﺩ ﺃﻅﻬﺭﺕ ﺍﻟﺩﺭﺍﺴﺔ ﺃﻥ ﻭﺠﻭﺩ ﺍﻟﺤﺸﻭ ﻓﻲ ﺍﻷﺠﺯﺍﺀ ﺍﻟﺴﻔﻠﻴﺔ ﻴﻌﻁﻲ ﺠﺴﺎﺀﺓ ﺃﻜﺒﺭ ﻟﻠﻤﻨﺸﺄ ﺒﺎﻟﻤﻘﺎﺭﻨﺔ ﺒﻭﻀﻌﻪ ﻓﻲ‬ ‫ﺍﻷﺠﺯﺍﺀ ﺍﻟﻌﻠﻭﻴﺔ ﻟﻺﻁﺎﺭ ﺍﻟﺨﺭﺴﺎﻨﻲ‪.‬‬

‫‪* Address for correspondence:‬‬ ‫‪Prof. Dr. Salah El-Din M. Fahmy Taher‬‬ ‫‪23a Anas Ibn Malik St.‬‬ ‫‪Al-Mohandseen 12411,‬‬ ‫‪Giza, Egypt‬‬ ‫‪Cell #: (+20) 10 1692682 Tel./Fax: (+20) 2 37491056‬‬ ‫‪* E–mail: [email protected]‬‬ ‫‪Paper Received 16 April 2007; Revised 4 September 2007; Accepted 28 November 2008‬‬

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‫‪The Arabian Journal for Science and Engineering, Volume 33, Number 2B‬‬

‫‪October 2008‬‬

Salah El-Din Fahmy Taher and Hamdy Mohy El-Din Afefy

ABSTRACT The influence of partial masonry infilling on the seismic lateral behavior of low, medium, and high rise buildings is addressed. The most simple equivalent frame system with reduced degrees of freedom is proposed for handling multi-story multi-bay infilled frames. The system is composed of a homogenized continuum for the reinforced concrete members braced with unilateral diagonal struts for each bay, which are only activated in compression. Identification of the equivalent system characteristics and nonlinear material properties is accomplished from the concepts of inverse analysis, along with statistical tests of the hypotheses, employed to establish the appropriate filtering scheme and the proper accuracy tolerance. The suggested system allows for nonlinear finite element static and dynamic analysis of sophisticated infilled reinforced concrete frames. Sensitivity analysis is undertaken to check the suitability of the proposed system to manipulate various structural applications. The effect of number of stories, number of bays, infill proportioning, and infill locations are investigated. Geometric and material nonlinearity of both infill panel and reinforced concrete frame are considered in the nonlinear finite element analysis. Energy consideration using modified Rayleigh’s method is employed to figure out the response parameters under lateral dynamic excitations. The results reflect the significance of infill in increasing the strength, stiffness, and frequency of the entire system depending on the position and amount of infilling. Lower infilling is noted to provide more stiffness for the system as compared with upper locations. Key words: infilled reinforced concrete frames, damage mechanics, nonlinear finite element modeling, equivalent frame, statistics, inverse problem, back analysis, dynamic analysis, masonry

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ROLE OF MASONRY INFILL IN SEISMIC RESISTANCE OF RC STRUCTURES

1. INTRODUCTION Nonlinear dynamic analysis of high rise infilled reinforced concrete framed systems involves several intricate aspects [1]. These comprise the number of parameters characterizing the composite material nonlinearity of the various components constituting the entire structural system including concrete, steel reinforcement, interface elements, masonry, mortar, joints, fixtures, and connections whenever applicable. Even though mesh generation capabilities for media discretization in finite element framework or for internal cell development in boundary element scheme are provided, the input phase might still be very sophisticated for real life applications. In addition, analysis of the output results might be formidable, especially where time history analysis is required or when the frequency domain has to be conceived. Moreover, computational limitations in commercial packages through the built-in dimensioning of arrays or through convergence restrictions in nonlinear schemes may encumber the whole process. In turn, these drawbacks have provided the incentive for establishing the various equivalence approaches developed to date [2]. As categorized originally by Whittman in 1983 and modified later by other investigators [3], nano-, micro-, meso-, macro-, and structural-scales are the different levels that can be considered for tackling the problem. Albeit approximate, an equivalent system at the structural level with a satisfactory degree of accuracy is basically required to handle structural problems, especially under dynamic excitations. The efficiency of the equivalent system resides in its capability for simulating the real behavior. Once a cost-effective, reliable, efficient, and accurate model is achieved, extrapolation of existing experimental results may be carried out and minute details on deformations, strains, internal stresses, mode shapes, frequencies, and time-history can be determined. Nonlinearity of the behavior is evident, and an incremental–iterative finite element computational scheme should, therefore, be adopted. Besides, a reliable equivalence has to take into account the following features: (i)

The orthotropic nature of planar infilled structures requires the use of very sophisticated constitutive relations and complex elements to represent the various components.

(ii)

The highly nonlinear response of infilled frames, even at low load levels, makes irrelevant the use of linear elastic elements in most cases.

(iii)

The simulation of certain brittle infill materials may create serious numerical problems.

(iv)

The softening behavior, tension stiffening, shear retention, interface slippage, anisotropic, or orthotropic nature of the constituent materials.

(v)

The unilateral features of the behavior due to non-uniform contact and separation between the frame and the infill and the development of interfacial stresses.

Apart from Liauw’s idea [4] of using an equivalent frame of the same stiffness and strength through a transformed composite section of the infilled frame, most other idealizations were directed towards proposing an appropriate strut system, originally proposed by Polyakov [5] and subsequently developed by Smith [6], rather than the relatively cumbersome analytical solution using the polynomial stress function [7]. Micromechanical and macromechanical approaches have been widely used in previous work [8–11]. Micro-modeling was found to be relatively time-consuming for analysis of large structures where existence of mortar joints is taken into account [12]. For example, Mosalam [13] and Dhanasekar and Page [14] used a nonlinear orthotropic model, while Liauw and Lo [15] a employed smeared crack model and Mehrabi and Shing [16] utilized a dilatant interface constitutive model to simulate the infill behavior. On the other hand, the infill panel was macromechanically treated as homogeneous material and the effect of mortar joints between masonry units was smeared over the whole panel and taken on an average sense [17, 18, 19]. Because of the sophistication of the problem description, most of the numerical investigations were restricted to frames of limited number of bays and stories [8, 12, 14, 17, 19, 20–32]. Discarding nonlinear nature of the behavior, Sayed [32] studied the free vibration of multi-bay multi-story infilled frames through skeletal idealization of the structure. The infill was modeled using Mainstone’s representation [33] and infinite (continuous) treatment for the stiffness and mass of the frame members was considered to investigate the effect of location and percentage of the infill. Therefore, it can be concluded that the analysis of high rise infilled frames still requires a simple, yet rigorous, finite element idealization of the problem. Afefy [20] carried out a more elaborate nonlinear finite element analysis of multi-bay multi-story infilled reinforced concrete frames under dynamic loading.

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In the present work, a new nonlinear equivalent frame is proposed in which the inverse analysis for system parameter identification with the appropriate filtering technique is employed. Statistical testing of the hypotheses is applied to judge the accuracy of the equivalent system from the energy absorption standpoint. The suggested equivalent system is represented by continuum idealization for the reinforced concrete members, while the infill panel and the interface are idealized by a equivalent unilateral diagonal strut. The equivalent system is thus suitable for nonlinear finite element analysis with reduced degrees of freedom that is capable of capturing most of the salient features of the response. Static as well as dynamic verification and validation of the equivalent system are carried out for several study cases, which depict its reasonable accuracy. 2. METHODOLOGY The basic idea of deriving the equivalent system is to reduce the total number of degrees of freedom while attaining the physical description of the problem almost unchanged [20]. This intention is motivated by incorporating problem nonlinearity in the static and dynamic analysis of high rise buildings with masonry infill. Figure 1 illustrates conceptually the successive system reduction from the micro-scale to macro-scale then to structural scale idealization in order to achieve the required equivalent system.

Continuum idealization for reinforced concrete

Equivalent unilateral strut idealization for the infill panel and the interface

Interface material

Continuum idealization for the infill panel

a

b (a) (b) (c)

c

Real problem configuration Idealized macro-scale idealization Idealized structural-scale idealization Figure 1. Methodology of establishing the system equivalence

The first step is a homogenization phase to replace the concrete and steel reinforcement with its intricate features by equivalent media with mutually equivalent responses. In addition, the brick and the mortar in joints and beds are replaced correspondingly by an equivalent masonry panel. The interface between the two equivalent homogenized materials is kept unchanged from the actual problem configuration because of its importance in delineating the actual behavior. The outcome of the substitution process of either or all, (Figure 1(b)), individual components (R.C. and/or infill panel) by equivalent homogenized media represents the micro- to macro-modeling reduction. The final step is to replace the infill and its frame-interface by equivalent diagonal struts with compression bracing an by, as shown in (Figure 1(c)). The entire process is formulated through computational modeling by the finite element method. The well-established approaches that can serve for the proposed methodology are inverse analysis [34], back analysis [35], and advanced statistical approaches by semi-variogram and Kriging estimation [36]. Recent applications of inverse analysis focused on structural applications, while back analysis had many advances in tunneling and geotechnical projects. The latter approach is widely used for geological, mining, oil production, and in-situ testing to determine the connatural system parameters. However, the three approaches are not contradictory and the fundamental concepts can be combined for broader applications. In the present work system identification on the combined bases of inverse analysis and statistical considerations is followed.

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Inverse analysis is generally defined as the process of developing an analytical model for a certain structure, based on the knowledge about its measured input and output signals. If the obtained model depicts an accurate representation of the true system behavior, it can be used for prediction of the system response to any possible future events. Otherwise, the model will have limitations in its applicability to arbitrary problems. Reliability in its application for system identification is dependent on the precision of the adapted model as well as the possibility of obtaining a convergent and accurate value for its parameters. The determination of the state of a system from measurements contaminated with noise is called filtering. The system noise is the difference between the real behavior of the physical system and the one produced by the adopted mathematical model. The filtering problem can be interpreted as a technique to find the best matching between a constructed mathematical model and the actual response of the system behavior. 3. MICRO- TO MACRO-SCALE HOMOGENIZATION 3.1. Infill Panel The work of Papa and co-workers [9, 37] is considered where a homogenized continuum exhibiting elasto-plastic damage behavior was conceived. The damage model was phenomenonlogically adopted and experimentally calibrated for mortar, while bricks were described as brittle–elastic. The homogenization procedures led to orthotropic constitutive law for masonry wall under monotonic and cyclic loading that was validated by experiments on infill masonry panels and entailed differences of no more than 3%. In the micro-scale, bricks were assumed as linear-elastic–brittle, the failure threshold being defined by Grashoff’s criterion of maximum tensile strain. On the other hand, mortar was considered as an elastic material susceptible to damage, understood as degradation of stiffness and sometimes also of strength (softening). The homogenization procedure to macro-scale substantiated a semi-heuristical expression for the non-zero entities for the masonry stress– strain matrix Cij expressed by Von-Karman (Voigt) notations, as functions of brick and mortar Young’s moduli Eb and Em in MPa , and a prescribed damage variable for mortar as follows C1 1 = ½ (0.3 Em +1.775 Eb) –D (0.12 Em – 0.05 Eb),

(1)

C2 2 = ½ (0.55 Em +1.525 Eb) (1–Dn)1/2 ,

(2)

C1 2 = 0.2 C2 2 ,

(3)

C3 3 = 0.4 C2 2

(4)

with n = 1+(Em / 15 000) (Em / Eb)1/2 Such an isotropic representation of the damage variable is acceptable for in-plane loading of infilled frames where the behavior of masonry panel is predominantly characterized by the formation of unilateral diagonal struts with almost unchanged principal directions. 3.2. Reinforced Concrete Homogenization of reinforced concrete members has been a scope of research for several decades [38]. This procedure may not be appropriate for meticulous analysis of members and connections while its suitability may be achieved, on an average sense, for structures where the minute details does not influence the overall behavior significantly [20]. For example, Mehrabi and Shing [16] noted that the bond-slip characteristics between steel and concrete were found insignificant in analysis of infilled frames. Steel reinforcement may be modeled by the smeared approach with distributed properties [39]. However, the most important property is the material nonlinearity of the homogenized media that accounts for the elasto–plastic behavior. For the micro-modeling, the theory of dichotomy [40, 41] that was developed for elasto–plastic damage modeling of concrete is used. The basic idea for an element in any deformable material is that the continuum can be equivalently replaced by an orthogonal nonlinear spring system whose stiffness depends on the ratio of the principal stresses. For each principal direction, the total behavior is dichotomized into elastic–damage and plastic–damage components by decomposing the strain tensor. The comprehensive loading history can be deduced using the appropriate stress–strain spaces. Thus for a material point loaded under biaxial stress states, six stress–strain spaces are deduced. Three damage variables are described through monitoring the degradation of the three moduli depicting the behavior. The constitutive equations can be expressed as σi = (1–dai) Aoi εi

October 2008

(5)


295


= (1–dei) Eoi εie

(6)

(1–dpi) Poi εip

(7)

=

in which dci (c = a, e,p) are the damage variables associated with the pseudo initial moduli Coi (C =A, E, P) in the ith direction. These moduli represent the initial tangents of the stress–total strain (A) space, stress–elastic strain component (E) space and the stress–plastic strain component (P) space in the direction under consideration. In order that the constitutive equations account for the path dependence as prevalent in concrete and other rock-like materials, the forms of the pseudo initial moduli, for plane-stress analysis were functionally dependent on the biaxiality ratio β 1 = σ1 /σ2, and the current state of strain in the ith direction, while the incremental damage variables were expressed additionally in terms of the strain increment εi i. e., Coi = Coi (β 1 , εi),

(8)

dci = dci (β1 , εi , εi)

(9)

The stress increment was, therefore, given in the following form .

.c

.

σ i = (1 − d ci ) C oi ε i − C oi ε ic d ci ⎡ ∂ d ci c ⎤ .c = C oi ⎢1 - d ci − εi ⎥ εi ∂ ε ic ⎥⎦ ⎣⎢

(10)

where (.) is the time derivative while εi, εi are meant for c = a in εic and εic, respectively. Additionally, for unloading in the j th direction dej = 0 , since the process is purely elastic. Equation (10) represented the canonical form of the incremental stress–strain relationships. As far as micro-modeling is concerned, the special steel–concrete interface element and steel boom element are used [39]. Elasto–plastic behavior with isotropic hardening is considered for reinforcement according to von Mises criterion, where the onset of yielding is assumed to take place when the octahedral shearing stress reaches a critical value k, k= (J2) 0.5 , as follows f = 3k − σ y = 0

(11)

where σy is the yield stress from uniaxial tests. For macro-modeling, reinforced concrete is modeled as a unilateral nonlinear isotropic hardening elasto–plastic material, where the behavior in compression and in the tension is different. Aiming at model versatility for application through commercial nonlinear software packages, Drucker–Prager criterion is used as follows f ( I1, J 2 ) = αI1 + J 2 − k = 0

(12)

where I1 is the first stress invariant of stress tensor σij, J2 is the second stress invariant for deviatoric stress tensor, α and k are material constants dependent on is the angle of internal friction, Φ and the cohesion C. These properties for concrete, Φc and Cc, are related to the compressive strength, fc and tensile strength, ft of concrete as follows Cc = 0.5 fc ft Φc = sin−1

f c − ft f c + ft

(13) (14)

The model parameters for the homogenized media Φ and C are to be assessed through the inverse analysis. Eightnoded Serendipity elements are used in the finite element discretization. For both micro- and macro-modeling, the frame-masonry interface is one of the most influencing parameters [1]. The interface is modeled as non-integral continuum material with no tensile capacity and brittle behavior in compression using an appropriate interface element [39]. 4. MACRO- TO STRUCTURAL-SCALE IDEALIZATION For further reduction of the degrees of freedom, the infill panel along with the frame–masonry interface are replaced by diagonal unilateral prismatic strut while the homogenized reinforced concrete is maintain unchanged. The member is postulated to withstand no tensile resistance while linear brittle behavior is assumed in compression. The geometric dimensions are determined according to the aspect ratio of the infill panel after Mainstone’s representation [33]. The

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finite element representation makes use of the 8-noded serendipity element for reinforced concrete while 2-noded link element for the diagonal strut. The statistical filteration scheme suggested by the authors and co-workers [20] is utilized in the present study. Inverse analysis is applied to the study case illustrated in Figure 2. The reinforced concrete bare frame is orthogonal skeleton with prismatic members of 250×600mm with bay width of 4000mm and story height of 3000mm made of ordinary strength concrete 25 MPa. High tensile steel of proof strength 360 MPa is used for the main reinforcement while mild steel of yield stress 240 MPa is used for 8mm @ 150mm stirrups. Following the pre-mentioned methodology micro- to macro-scale homogenization for infill panel consisting of half-red clayey brickwork (120mm panel thickness) and mortar grade 18 MPa provided an average masonry modulus of elasticity 500 MPa. The loading is laterally applied in incremental manner at the centerline of the top girder. The nonlinear finite element analysis is carried out using the package DMGPLSTRS [3]. 4000 mm

Applied Load

3000 mm

3φ16

600

600

250 250

3φ16

Girder

8φ16 Column

Frame Panel; Geometry and Reinforcement Interface

6-noded element

8-noded element

Concrete

8-noded element

8-noded element

Infill

Reinforced Concrete

3-noded element

Steel

2-noded element

Diagonal Strut

Figure 2. Single-bay single-story infilled frame considered in inverse analysis

In the beginning, a detailed “accurate” finite element analysis for the original system is carried out using the mesh shown in Figure 3. Then, an approximate simplified analyses using the equivalent system are made where the filtration process required the execution of 1024 computer run to distinguish the most suitable finite element mesh, schematically illustrated in Figure 2, with appropriate system parameters. This large number of analyses was required because there was no previous knowledge about the most appropriate mesh topology for the equivalent system and a rigorous sensitivity study for mesh choice was binding. This number may be thus reduced in future studies and less restrictive permutations among system parameters can be selected. However, in the presence of automated system similar to that employed in the present work, the process is not that difficult for applications with high statistical confidence limits. Figure 3 depicts the load-top drift of both the bare and infilled frames. It is obvious the close agreement between the predictions of the macro- and structural-scale models with a pronounced saving in the execution time. This is of course is attributable to the less number of equations associated with the equivalent system. These features represent the main

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Lateral Load, kN

advantages of the proposed methodology that facilitate the analysis of multiple-cell high rise buildings which is usually hindered by the limited computer capacity of commercial software packages due to the enormous degrees of freedom. The difference between the equivalent and original systems is about 8% and 6% for the bare and infilled frames, respectively, which satisfies the statistical tolerance. 600 500

400 300 Original Bare Frame

200

Equivalent Bare Frame Original Infilled Frame

100

Equivalent Infilled Frame

0 0

5

10 Top Drift, mm

15

20

Figure 3. Load–deflection characteristics for the original and equivalent systems

Two parameters are considered in the validation phase: (a) effect of frame topology and (b) effect of infill material rather than those considered in the inverse analysis. Single-bay two-story bare, half infilled at both lower and upper locations, fully infilled frames are investigated for the influence of frame topology. On the other hand, clayey and perforated loamy brickwork of 120 and 250 mm thickness are examined for free vibration analysis. The average masonry Young’s modulus of the latter type is almost five times that of the former and thus delineating the relative frame-infill stiffness.

60

1 2 3 4 5

- Bare frame - Red brick with 12 0mm thickness - Red brick with 25 0mm thickness 12 0mm thickness - Loam perforated brick with 25 0mm thickness - Loam perforated brick with

Original system Equivalent System

Natural Frequency (Hz)

50 40 30 20 10 0 1

2

3

4

5

Figure 4. Equivalence validation for various frame topologies

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The first fold of validation is carried out by conducting nonlinear analysis of the original and simplified equivalent finite element models under monotonic quasi-static loading up to failure in order to estimate the energy absorption capacity of each system as represented by Figure 4. For two story frame loading is incrementally applied at each floor level with values at the lower story half that of the upper story. The second fold of validation is undertaken through frequency analysis using the initial material properties of the original and simplified equivalent finite element models to determine the natural frequency of each system as depicted in Figure 5. Single-bay single story frames with various infill types and thicknesses have been considered in the analysis. Figures 4 and 5 illustrate the close agreement between the original and equivalent systems for the seven validation cases. The relative energy absorption capacity is conferred for various frame topologies in the first histogram whereas the natural frequency is compared in the second bar-chart. All differences are noted to be minor for practical applications and within the specified confidence limit. 5. ROLE OF INFILL In order to depict the time history of infilled framed structures, lateral excitation is imposed upon the system. Cyclic triangular load of envelope incrementally increasing with time, Figure 6, is laterally applied at each floor level proportional with the story number. Single- and double-bay frames are considered for five, ten and twenty story building with ordinary half-clayey brickwork infill. Full and partial infilling of different percentages of 20, 40, 60, 80% are arranged at the bottom, middle, and top third of the height. In all cases, the infill panel is treated as non-integral to reinforced concrete frame. This equivalence is of extreme importance because of the tremendous number of degrees of freedom involved with the solution of infilled high-rise buildings. The struts are activated only in compression, thus maintaining the unilateral characteristics of infill behavior in contact and separation modes of deformation. Isotropic hardening Drucker–Prager is used to simulate the elasto–plastic behavior of reinforced concrete [38]. Isoparametric eight-noded elements are used to discretize the frame skeleton while link-elements are adapted for the diagonal struts. Figure 6 outlines the main properties of various materials along with the geometric variables and modeling scheme as well as the basic parameters considered hereinafter.

%Energy of Original

Energy of Equivalent System

12 10 8 6 4 2 0 Figure 5. Equivalence validation for different infill material

The time period is one of the major dynamic parameters of structural systems subjected to vibrating actions or liable to seismic movements. Moreover, it constitutes a fundamental quantity, which has to be incorporated in evaluating the equivalent static load given in many code provisions. Since existence of infilling alters both the stiffness and mass distribution, Rayleigh’s approach is best suited for such applications of nonuniform systems. The system frequency is calculated from energy consideration using the modified Rayleigh’s method according to the following formula [32]

N

ω =

g ∑ fi xi i =1

N

∑ W i xi

(15)

2

i =1

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299

Salah El-Din Fahmy Taher and Hamdy Mohy El-Din Afefy σIII

Typical bay width = 4 m

τ Reinforced concrete 300 x 900 mm modeled by 300 x 300 serendipity element

φeq ceq

σ σI Unilateral strut equivalent to half brick wall with section 120 x 1000 mm

Typical floor height =3m

Es = 5 kN/mm2 σus= 105 N/mm2 ρs = 3.839 E-12 kN/mm3

σus

ε

σII π− plane

Drucker Prager envelop Eeq = 208 kN/mm2 ET = 0.1 Eeq ρeq = 2.501 E-12 kN/mm3 ceq = 38 N/mm2

φeq = 56o

Single-bay 5-story frame fully infilled Figure (6-a) Problem idealization

Loading envelope

Load

10 kN 2 sec

Time Figure (6-b) Load-time history

Parameters No. of Stories

No. of Bays

5-Stories 10-Stories 20-Stories

Single-Bay Double-Bay

Infill Percentages 0%, 20%, 40%, 60%, 80%, 100%

0%

20%

40% 60% Lower Infilling

Location Lower Middle Upper

80%

100%

Lower

Middle

Upper

Figure (6-c) The main parameters considered in the analysis

300


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where the indices i and N represent the story number and the total number of floors, g is the gravitational acceleration and ω is the system pseudo frequency estimated for applied floor forces f1, f2, …, fN producing story drifts x1, x2,…, xN. The gravitation loads at each floor are W1, W2, …, WN as shown in Figure 7.

WN

xN

fN

WN

xN

fN-1

W5

x

f5

W4

x

f4

W3

x

f3

W2

x

f2

W1

x

f1

Figure 7. Main parameters in modified Rayleigh’s equation

The previous equation is well applied to linear elastic behavior where both the force and displacement are monotonically increasing with mutual correlation for a system of shear building. For this purpose, the deformational characteristics obtained from the finite element solution should be considered at each load increment. Consequently, this expression can be recast to nonlinear behavior with incremental parameters as follows N

ω =

g ∑ ∫ f i dx i

(16)

i =1 t

⎛ ⎞ ∑ W i ⎜⎜ ∫ dx i ⎟⎟ i =1 ⎝t ⎠ N

2

in which dxi is the incremental drift of the ith floor evaluated at time t. The nonlinear behavior of the system is evident for both bare and infilled frames as shown in Figure 8. Existence of infilling is noted to increase the ultimate lateral resistance of the system while resulting in less ultimate lateral deflection for lower infilling. The effect on both parameters is more pronounced for higher percentages of infilling. Two phenomena arise through the stage of loading and result in the response nonlinearity. First is stiffness degradation of the reinforced concrete with load-induced orthotropy depending on both the applied dynamic load and the inherent deformational characteristics of the frame. Second is the progressive strength reduction of either of the diagonal struts, which is supposed to be sequential according to level of loading. In all next illustrations, dimensionless arguments are utilized relative to the bare frame parameters at failure. The relative time is taken as the ratio of elapsed time during the course of loading for each particular study case to the time at failure of the bare frame. The curves are presented along the loading path envelope. Information presented in these diagrams is related to a single-bay five-story frame whereas the comparative data for other variables are demonstrated in tabulated form.

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301


Figure 8. Base shear-top drift for lower infilling relative to bare frame

Figure 9. Stiffness degradation of the system for lower infilling relative to bare frame

Figure 10. System frequency for lower infilling relative to bare frame

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Relative Story Height

1.0 0.8 0.6

Bare Frame 20 % Infilling 40 % Infilling 60 % Infilling 80 % Infilling Full Infilling

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Relative Story Drift at Failure Figure 11. Story drift for lower infilling relative to bare frame

Relative Top Velocity

1.2 Bare Frame 20 % Infilling 40 % Infilling 60 % Infilling 80 % Infilling Full Infilling

1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Relative Time Figure 12. Top lateral velocity for lower infilling relative to bare frame

Strut failure is considered as a local mechanism that reduces the degree of structural indeterminacy such that does not induce overall collapse of the frame. Such mechanism is noted, however, to take place just prior to the ultimate capacity is reached and the continuous behavior is maintained up till failure. Figure 9 illustrates the stiffness degradation of the system based on monitoring the reduction of the secant modulus obtained from base shear-top drift diagram. Implementation of Equation 16 derived after modified Rayleigh’s method illustrates frequency attenuation associated with the stiffness degradation of the system as depicted in Figure 10. More lower infilling is noted to induce higher initial frequency and hence less time period. Albeit on an average sense attributable to the adopted modeling, the behavioral trend is noted to be almost similar for the considered bare and infilled frames with differences only in magnitude. Racking mode of deformation is nearly dominant for all double-bay frames and even for five- and ten-story frames as shown in Figure 11. Infilling is noted to significantly alter the top lateral velocity and acceleration only in last third of the time up to failure as illustrated in Figures 12 and 13. This, in turn, implies that the system kinematics is related to the progression of stiffness degradation and hence frequency attenuation. October 2008


303

Relative Top Acceleration


1.2 Bare Frame 20 % Infilling 40 % Infilling 60 % Infilling 80 % Infilling Full Infilling

1 0.8 0.6 0.4 0.2 0 -0.2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.4 -0.6 -0.8 Relative Time Figure 13. Top lateral acceleration for lower infilling relative to bare frame

Table 1 demonstrates the various parameters for single-bay frames. Similar observations are noted for double-bay frames but with different values. Lateral strength is considered as the ultimate lateral load capacity of the system. Stiffness and frequency values tabulated hereafter are the foremost highest values for each particular case evaluated at initial conditions. Close results are obtained when estimations are carried out at the onset of failure. Table 1. Strength Percentage Increase of Double-Bay Relative to Single-Bay Frames Percentage

Infill Location

5-story

10-story

20-story

86

90

140

Lower Infilling

75

100

140

Middle Infilling

71

90

150

Upper Infilling Lower Infilling Middle Infilling Upper Infilling Lower Infilling Middle Infilling Upper Infilling Lower Infilling Middle Infilling Upper Infilling

100 100 56 100 178 75 86 178 111 75

110 180 90 110 180 100 90 167 133 90

140 140 140 140 140 140 140 140 140 140

178

167

140

Bare Frame 20 % Infilling

40 % Infilling 60 % Infilling 80% Infilling

100 % Infilling

6. CONCLUSIONS

304

1.

The proposed statistically equivalent system for infilled frames which is represented by nonlinear finite elements with unilateral diagonal strut yields reasonable predictions with considerable reduction in the numerical operations.

2.

Conventional half-brick wall infilling is noted to affect nearly all of the dynamic parameters of reinforced concrete frames.


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3.

Infill influence on the kinetic and kinematic coefficients related to lateral excitation is found to depend on frame features such as number of stories and number of bays as well as infill amount and position.

4.

Lower location yields the higher strength, stiffness, and frequency of the system.

5.

Nonlinearity of the behavior is basically due stiffness degradation, which consequently results in frequency attenuation during the loading regime.

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