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

October 2008 The Arabian Journal for Science and Engineering, Volume 33, Number 2B 291

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.

:

ف استخدامهامكااقتراح أكثر النماذج بساطة بدرجات حرية مخفضة-في هذا البح-تم تحلي

المحشوة متعددة الطوابق ومتعددة البواكى ون األنموذج من وسط متجانس من الخرسانة المسلحةيت.الهياك

ديقمفةيداحأ ةيرطق تالاكشب ةيكاب وقد تم تحديد خصائص النظام. الضغفط فعالة فالتأثير تكو ك

المعاكس أسلوب التحلي مفاهيم باستخدام الخطية غير إلى اختبارات إضافالمكافئ وخواص المواد

على أنسب أسلوب تنقيح بالدقة المالئمة فى التفاوت المسموح ويسمح النظام.الفروض اإلحصائية للحصو

االستاتيكى الخرسانية المسلحةناميكوالدالمقترح بالتحلي غير الخطية للهياك بطريقة العناصر المحددة

معالجة التطبيقات اإلنشائيةفجراء فحص لحساسية دقة النظام المقترح للتحقق من مالئمتهوقد تم.المعقدة

.المختلفة

األنموذج

دقة

من

التحقق

تموبعد

الخرساالمقترح

لإلطارات

الزلزالي

السلوك

دراسة

تحت

المسلحة

ية

هذه اإلطارافتأثير الحشو الجزئي من الطوب االعتبارفوقد اخذت الدراسة.ووسطها وأعالهسف

نسبة الحشو ومكان تأثير عدد األدوار وعدد الباكيات لإلطار وقد استخدمت طريقة رايلى للطاقة.وكذا

وقد عكست النتائج تأثير حشو اإلطارات. العرضميكالدينلتحديد المعامالت المختلفة للسلوك تحت التأثير

لنسبة الحشفالخرسانية المسلحة ًقبط . ووضعه زيادة المقاومة الجساءة والتردد للنظام اإلنشائي كك

بالمقارنة بوضعهكب جساءة األجزاء السفلية يعفن وجود الحشووقد أظهرت الدراسة ف للمنشأ

. لإلطار الخرسااألجزاء العلوية

* 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




The Arabian Journal for Science and Engineering, Volume 33, Number 2B October 2008 292

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, whichare 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





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 itscapability 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 orthotropicnature 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.





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, whichdepict 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.

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)). Theentire 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.

(a) Real problem configuration

(b) Idealized macro-scale idealization

(c) Idealized structural-scale idealization

a

b c

Interface material

Continuum idealization

for the infill panel

Equivalent unilateral

strut idealization

for the infill panel

and the interface

Continuum idealization

for reinforced concrete





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 todamage, 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 C ij expressed by Von-Karman (Voigt) notations, as functions of brick and mortar Young’s moduli E b and

E m in MPa , and a prescribed damage variable for mortar as follows

C 1 1 = ½ (0.3 E m +1.775 E b) – D (0.12 E m – 0.05 E b), (1)

C 2 2 = ½ (0.55 E m +1.525 E b) (1– Dn)1/2 , (2)

C 1 2 = 0.2 C 2 2 , (3)

C 3 3 = 0.4 C 2 2 (4)

with n = 1+( E m / 15 000) ( E m / E b)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 equivalentlyreplaced 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– d ai) Aoi εi (5)





= (1– d ei) E oi εie (6)

= (1– d pi) Poi εi p (7)

in which d ci (c = a, e, p) are the damage variables associated with the pseudo initial moduli C oi (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 β1 = 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.,

C oi = C oi (β1 , εi), (8)

d ci = d ci (β1 , εi , εi) (9)

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

where (.) is the time derivative while εi, εi are meant for c = a in εic and εi

c, respectively. Additionally, for unloading in

the j th direction d ej = 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 = ( J 2)

0.5, as follows

where σ y 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 applicationthrough commercial nonlinear software packages, Drucker–Prager criterion is used as follows

where I 1 is the first stress invariant of stress tensor σij, J 2 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 C c, are related to the compressive strength, f c and tensile strength, f t of concrete as follows

The model parameters for the homogenized media Φ and C are to be assessed through the inverse analysis. Eight-

noded 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

)11(03 =σ−= yk f

(10)

)1(

.

..

.

d

d -1C

d C C d

c

icic

i

cicioi

cicioi

c

ioicii

ε⎥⎥⎦

⎤

⎢⎢⎣

⎡ε

ε∂

∂−=

ε−ε−=σ

)14(sin

)13(5.0

1

t c

t cc

t cc

f f

f f

f f C

+

−=Φ

=

−

)12(0),( 2121 =−+α= k J I J I f





8-noded element 6-noded elementInterface

Reinforced Concrete

3-noded element

8-noded element

Concrete Infill

Steel Diagonal Strut

2-noded element

8-noded element

300

0mm

8φ163φ16

Frame Panel; Geometry and Reinforcement

Girder 250 Column

4000 mm

600

3φ16600

250

Applied Load

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 reinforcementwhile 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].

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





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.

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.

Figure 4. Equivalence validation for various frame topologies

0

100

200

300

400

500

600

0 5 10 15 20Top Drift, mm

L a t e r a l L o a d ,

k N

Original Bare Frame

Equivalent Bare Frame

Original Infilled Frame

Equivalent Infilled Frame

N a t u r a l F r e q u e n c y

( H z )

0

10

20

30

40

50

60

1 2 3 4 5

Original system

Equivalent System

1 - Bare frame 2 - Red brick with 12 0mm thickness3 - Red brick with 25

4 - Loam perforated brick with 12

5 - Loam perforated brick with 25

0mm thickness

0mm thickness

0mm thickness





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 theoriginal 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. Isotropichardening 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.

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 theequivalent 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]

∑

∑

=

==ω N

iii

N

iii

xW

x f g

1

2

1 (15)

0

2

4

6

8

10

12

% E n e r g y o f O r i g i n a l

E n e r g y o f E q u i v a l e n t S y s t e m





Typical

floor height

= 3 m

Typical bay width = 4 m

Reinforced concrete300 x 900 mm modeled by 300 x

300 serendipity element

Unilateral strut equivalent to half brick

wall with section 120 x 1000 mm

Single-bay 5-story frame fully infilled

σus

ε

Es = 5 kN/mm2

σus= 105 N/mm2

ρs = 3.839 E-12 kN/mm3

τ

σceq

φeq

σI σ

II

σIII

π− plane

Drucker Prager envelopEeq = 208 kN/mm2 ET = 0.1 Eeq

ρeq = 2.501 E-12 kN/mm3

ceq = 38 N/mm2

φeq = 56o

Figure (6-a) Problem idealization

Figure (6-b) Load-time history

Load

Time

10 kN

Loading envelope

2 sec

5-Stories10-Stories

20-Stories

No. of Stories

Single-Bay

Double-Bay

No. of Bays

Parameters

Infill

Location

Lower MiddleUpper

Lower Middle Upper

Percentages

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

0% 20% 40% 60% 80% 100%

Lower Infilling

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





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 f 1 , f 2 , …, f N producing story drifts x1 , x2 ,…, x N .

The gravitation loads at each floor are W 1 , W 2 , …, W N as shown in Figure 7.

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 aremonotonically 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

∑ ∫

∑ ∫

=

=

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ =ω

N

i t ii

N

i t ii

dxW

dx f g

1

2

1 (16)

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.

f 1

f 2

f 3

f 4

f 5

f N

f N-1

x N





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





0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4Relative Story Drift at Failure

R e l a t i v e S t o

r y H e i g h t

Bare Frame

20 % Infilling

40 % Infilling

60 % Infilling

80 % Infilling

Full Infilling

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Relative Time

R e l a t i v e T o p V e l o c i t y Bare Frame

20 % Infilling

40 % Infilling

60 % Infilling

80 % Infilling

Full Infilling

Figure 11. Story drift for lower infilling relative to bare frame

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.





-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Relative Time

R e l a t i v e

T o p A c c e l e r a t i o n

Bare Frame20 % Infilling40 % Infilling60 % Infilling80 % InfillingFull Infilling

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

20-story

10-story

5-story

Infill Location

Percentage

140

90

86

Bare Frame

140

100

75

Lower Infilling

150

90

71

Middle Infilling

140

110

100

Upper Infilling

20 %

Infilling

140 180 100 Lower Infilling

140

90

56

Middle Infilling

140

110

100

Upper Infilling

40 %

Infilling


140

100

75

Middle Infilling

140

90

86

Upper Infilling

60 %

Infilling


140

133

111

Middle Infilling

140

90

75

Upper Infilling

80%

Infilling

140 167 178 100 % Infilling

6. CONCLUSIONS

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.





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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role of infill wall

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