LATERAL STIFFNESS CHARACTERISTICS OF TALL.pdf

8/10/2019 LATERAL STIFFNESS CHARACTERISTICS OF TALL.pdf

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buildings. In contrast with the member analysis, no methodology is recommended in these codes of practice to consider the effects of cracking in the analysis of large multistorey reinforced concretebuilding structures. Analyses of reinforced concrete structures are essentially carried out on the basisof linear elastic theory by either neglecting the nonlinear effects of concrete cracking or considering itby arbitrarily reducing the stiffness of cracked members. Consequently, it is quite probable that there

are tall reinforced concrete buildings associated with these designs that cannot meet the serviceabilityrequirement of either the top lateral deection or inter-storey drift, or alternatively are excessivelyconservative. Hence it would be benecial to develop an analytical method for accurately assessing thelateral deections and stiffness of tall buildings, accounting for the effect of nonlinearity due toconcrete cracking.

Researchers and engineers have made signicant advancements in nite element procedures forstructural analysis in the past several decades. Two approaches have been gradually formed forreinforced concrete, namely the microscopic element and the macroscopic element approach (Polak,1996). The microscopic element approach considers the structures to be divided into many small niteelements and the stressstrain relationships of concrete and steel are adopted to model the concrete andthe steel elements, respectively (ASCE Stat-of-the-Art Report, 1982). The discrete crack model (Muftiet al. ,1972; Ingraffea and Gerstle, 1985) and the smeared crack model (Rashid, 1968; Barzegar, 1989)are applied to simulate crack propagation. Thediscrete crack model appears to be signicantly restrictedwhen applied to tall reinforced concrete buildings, since complicated mesh renements becomenecessary to accommodate the propagation of only a few discretecracks. The application of the smearedcrack model encountered a number of numerical difculties that were later solved by introducing theshear retention factor. The selection of the shear retention factor, however, appears to be dependent onthe experience of analysts and the requirements for numerical stability with no signicant validation of experimental data. Tension stiffening introduced by Scanlon and Murray (1974) was modelledindirectly by inclusion of the descending branch in the tensile stressstrain curve, so called tensionsoftening behaviour despite some confusion between the two terminologies (Kotsovos and Pavelovic,1995). On the other hand, the macroscopic element approach incorporates such factors as the crackingeffect, bonding behaviour and mechanical aggregate interlock into a comprehensive constitutive modelof reinforced concrete. Several material property models have been derived directly from experimentalresults, such as the modied compression eld theory of Vecchio and Collins (1986) and the effectivestiffness relationship of Branson (1963)which is, however, restricted to application on simple beam andslab structures.

In this paper, a general effective stiffness model is proposed to determine the relationship betweenexural stiffness reductions and various moments due to loading applied to the members. The mostsignicant feature of the proposed model is its extensive applicability to members that are subjected tovarious load types. This effective stiffness model can be considered as a general constitutive model formembers in tall reinforced concrete buildings. If the force redistribution and ensuing coupled stiffnessreduction among members in structures are considered by using an iterative algorithm, a practicalcracking analysis system can be established by integrating the proposed effective stiffness model andthose iterative algorithms with commercial packages of linear nite element analysis. The advantageof this system is the ability to predict the lateral deection and stiffness of reinforced concrete buildingstructures under service load conditions explicitly and quantitatively.

2. ANALYSIS OF CRACKING EFFECTS

Since the monolithic system is a distinct characteristic of reinforced concrete structures, and thesestructures behave as indeterminate structures, appropriate equations that describe the forcedeformation relationships, equilibrium and deformation compatibility can be fully satised by the

366 C.-M. CHAN ET AL.

Copyright 2000 John Wiley & Sons, Ltd. Struct. Design Tall Build . 9 , 365383 (2000)


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linear nite element method (LFEM) when the cracking effects are not signicant. Using thesemethods, internal element forces such as moment, shear, and axial force can readily be obtained. TheseLFEM analyses are commonly used in todays structural engineering design ofces.

For tall reinforced concrete buildings the lateral drift, affected signicantly by cracking effects,becomes one of the important design criteria, and therefore cracking effects need to be considered inthe analyses. The major difculty with nonlinear cracking analyses is that the stiffness of the crackedmembers needs to be reduced according to the extent of crack formation occurring in the members.Moreover, such stiffness reduction may result in internal force redistribution, which will further affectthe occurrence of cracking. To cope with this difculty, a load incremental procedure has beendeveloped. In this procedure, the cracked members are rst identied at a specic service load leveland their element stiffnesses are then reduced accordingly using the following probability-basedeffective stiffness model. The effective stiffness model provides the nonlinear forcedeformationrelationships for the cracked members.

2.1. Probability-based effective stiffness model

Figure 1 shows the effect of cracking in a exural beam member. Where the applied moment, M (x), issmaller than the cracked moment M cr dened as

M cr v f r I uncr yt 1where f r = exural tensile strength of concrete = 0 6 f c MPa, yt = distance from centroid of grosssection to extreme bre in tension, v = axial compressive stress, the uncracked regions have amoment of inertia represented by I uncr , equivalent to the gross uncracked moment of inertia whichaccounts for the contribution of reinforcing steel to the stiffness. In the region where the appliedmoment, M (x), is larger than or equal to the cracked moment, M cr , the exural tensile cracks develop in

Figure 1. Cracking in a reinforced concrete beam

RC BUILDING CHARACTERISTICS UNDER SERVICE LOAD 367

Copyright 2000 John Wiley & Sons, Ltd. Struct. Design Tall Build. 9 , 365383 (2000)


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the tensile bres of the exural member and propagate gradually towards the neutral axis causing themoment of inertia of the cracked section to be reduced to the value I cr .

The occurrence of cracking at a section is dependent in some manner on the value of the moment,which in turn is determined by the shape and magnitude of the moment diagram due to the externalload. The larger the magnitude of moment, the greater is the probability that a crack will occur. The

shape and the magnitude of the moment diagram are the main factors affecting the probability of crack formation of a reinforced concrete member.

The probability density function determining the occurrence of the random variable when themoment is larger than a certain value, can be expressed as

p x M x

S 2where M ( x) is the moment distribution function and S is the total area of the moment diagram. Hencethe probability that the moment value is larger than M cr can be obtained by integrating the probabilitydensity function p( x) in the region where M ( x) M cr ,

P M x M cr M x M cr M x

S dxS crS 3

in which S cr = the area of the moment diagram segment over which the working moment exceeds thecracking moment M cr . The probability of the moment value being less than M cr is

P M x M cr M x M cr

M xS

d x

1 M x M cr

M xS

d x

1 P M x M cr 4The probability of occurrence of cracked sections P cr associated with the outcome I cr , the cracked

moment of inertia, is determined by equation (3):

P cr P M x M cr 5while the probability of occurrence of the uncracked section Puncr with outcome I uncr , is derived inequation (4) as

P uncr P M x M cr1 P cr 6

Hence, the expected value of moment of inertia, the so-called effective moment of inertia, of thereinforced beam subjected to a certain type of loading is

I e P uncr I uncr P cr I cr1 P cr I uncr P cr I cr 7




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The advantage of the proposed model, represented in equation (7), is that the effective stiffness I e isdetermined from the probability of the occurrence of cracking. This probability is dened as the ratioof the area of the moment diagram segment over which the working moment exceeds the crackingmoment M cr , to the total area of the moment diagram. The proposed model has been veried by a seriesof experimental tests taking into consideration different loading congurations (Ning, 1998) and byother recent publications on the subject (Al-Zaid et al. , 1991). This concept can also be extended fromthe analysis of the beams and slabs to columns and shear walls with dominant exural crackingbehaviour (Mickleborough et al. , 1999).

2.2. Iterative Integrated cracking analysis method

The analysis method presented in Figure 2 provides the history of the nonlinear behaviour of reinforced concrete structures due to cracking, by applying the external loads in an incrementalmanner. The force redistribution and the cracked moment M cr affected by the change of axial force areupdated in every load increment, which in turn determines the stiffness reduction of reinforcedconcrete members for the subsequent increment. These coupled variations can be adjusted to trace thephysical propagation of cracking in the reinforced concrete structures from iteration to iteration. Thesteps for the proposed method are described as follows and shown in Figure 2.

Step 1. For n = 1, where n is the incremental number, all members of the reinforced concrete structure

Figure 2. Flowchart of iterative analysis procedureload incremental method




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are considered to be initially uncracked. The full lateral service load is divided into N loadincrements, such that each load increment has a value equal to F . Initially the applied load isset as F l = F .

Step 2. The load F (n ) is applied to the structure, and the internal moment M ni ,axial force An

i andlateral deection D(n ) are computed using linear nite element software.

Step 3. The element cracking moment M crn

i is updated by considering the element axial force A(n )

,according to equation (1).

Step 4. If M ni M crn

i ,then the ith member is deemed to be a cracked member; the exuralstiffness, ( I unc )i , is then reduced to the effective exural stiffness, I e

ni ,using equations (3)

and (7) based on the probability of cracking.Step 5. If M ni M cr

ni ,cracking is considered not to occur in the ith member, hence the exural

stiffness maintains the uncracked values ( I uncr )i .Step 6. If n N , the iteration proceeds to Step 7; otherwise F (n 1) = F (n) F and n = n 1, and

the analysis proceeds to Step 2 to repeat the incremental load analysis and to determine thecurrent effective stiffness of the structure.

Step 7. The iterative procedure terminates with the nal effective stiffness of the structure and thelateral displacement of the structure determined under the specied lateral load.

A disadvantage of this method maybe that the best value of the load increment, F is not readilyobvious and this load increment may become a critical parameter affecting the accuracy of calculation.Generally a larger value of F may cause a larger error, while a smaller F lead to more accurateresults. A relatively large incremental load step may not detect the initial cracking sequence of variousmembers, which in turn may cause some discrepancies at the onset of cracking. The magnitude of theerrors from the actual solution is dependent on the discrepancies in the actual effective stiffnesses inthe current iteration and the equivalent effective stiffnesses determined in the previous iteration, whichin turn are dependent on the value of the load increment, F .

3. EXPERIMENTAL VALIDATION

Two full-size structural sub-assemblage tests were conducted to investigate the behaviour and stiffnesscharacteristics of indeterminate reinforced concrete structures under service load. Two types of planarstructures were involved, namely: a rigid frame and a wall-frame structure. These structures are typicalof the lateral load resisting systems in high-rise buildings, and it is considered appropriate to studythese systems for the proposed iterative integrated nonlinear analysis accounting for cracking effects.The purpose of these tests is to validate the applicability and determine the limitations of the proposedanalytical method on these types of structures. The wall-frame test has been used to expand the rangeof its application to more general tall building systems containing shear walls as components to resistthe lateral load due to either wind or earthquake. Detailed experimental programs are presented. Thisexperimental work is followed by the comparison of the analytical predictions and experimentalresults.

3.1. Frame: structure and material detailsThe moment resisting frame was fabricated with a centre-to-centre span of 3000mm, a rst storeyheight of 1170 mm (including a steel pin joint height of 170 mm forming the pin joined base), a secondstorey height of 2000mm, and an overall height of 3670 mm. The section dimensions of columns were250mm wide by 375mm deep and the section dimensions of beams were 250mm wide by 350mmdeep. The frame was connected to the rigid ground of the laboratory strong oor by two fabricatedsteel pin joints. This form of boundary condition assumes that, in the rigid frame analysis, points of




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inection are located at approximately the mid-heights of the columns of every storey. Figure 3 showsthe elevation and lateral view of the frame structure and details of the pin joints. This frame structurecan be considered as a sub-assemblage in tall reinforced concrete rigid frame buildings.

The columns of the frame were reinforced with three No. 20 deformed bars each as top and bottomreinforcement, as well two No. 20 bars placed at the neutral axis and No. 10 closed stirrups with150mm spacing used as shear reinforcement. The beams were designed with three No. 20 deformedbars as bottom reinforcement, three No. 20 bars as top reinforcement and similar shear reinforcementas the columns. Commercially produced and supplied concrete was used, and the concrete had acompressive strength of 29MPa and a Youngs modulus of 15700MPa. Table I presents detail

properties, dimensions and reinforcement of the frame specimen.

3.2. Wall-frame: structure and material details

The wall-frame structure chosen for investigation was a single-span, two-storey reinforced concretestructure. The difference between the frame and wall-frame was in the inclusion of a wall(750mm 180mm in cross-section) to replace a column in the frame structure. The boundaryconditions were also changed from the pin-joint to a xed base connection, since it was assumed that

Figure 3. Layout and dimensions of the test frame

Table I. Detail properties, dimensions and reinforcement of rigid-frame Specimen

Member b (mm) h (mm) d c (mm) i (%) c (%) I umcr

(10 6 mm 4) I cr

(10 6 mm 4)

B1-6 250 350 50 1 26 1 26 1254 585B2-5 250 350 50 1 26 1 26 1254 585C1-2 250 375 40 1 88 1 13 1601 804C2-3 250 375 40 1 88 1 13 1601 804C4-5 250 375 40 1 88 1 13 1601 804C5-6 250 375 40 1 88 1 13 1601 804




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there was no longer an inection point occurring in the shear wall. The beam span between the columncentreline and the wall centreline was 2100 mm and the typical storey height was 1500mm. The framewas supported on a reinforced concrete base with dimensions 400mm wide, 400mm thick and3400mm long. The frame and the base were built integrally. The overall height of the wall-framestructure was 3575mm. The beams and columns had the same rectangular cross-section as those usedin the frame structure. Figure 4 shows the overall geometry of the wall-frame structure.

The columns of the wall-frame were reinforced with three No. 20 deformed bars, each as top andbottom reinforcement, and No. 10 closed stirrups at 150mm spacing as shear reinforcement. Thebeams were designed to have similar main reinforcement and shear reinforcement as the framespecimen. The wall was reinforced with two layers of six No. 20 steel bars distributed evenly in thesection and No. 10 @ 150mm stirrups. From the cylinder tests, the concrete had a compressivestrength of 36MPa and Youngs modulus of 19000MPa. Table II presents detail properties,dimensions and reinforcement of the frame specimen.

3.3. Frame: load and instrumentation details

The testing setup involved the application of a total vertical axial load of 200kN to each column andmaintenance of this load in a force-controlled mode throughout the test by a device called a gravityload simulator. This device approximates true gravity load and is used together with tension-loadinghydraulic actuators (Yarimci et al. 1967). The load line of this system is such that when the sidesway isbetween 0mm and approximately 200mm, the load remains vertical and moves with the sidesway of the structure. Since the actual lateral deections are generally less than 100mm, no horizontalcomponent of vertical load was induced. A lateral hydraulic actuator with 400kN capacity wasinstalled on the laboratory reaction wall. Lateral load was then monotonically applied in a stroke-controlled mode at a rate of 0 02kNs 1, until the ultimate capacity of the frame was achieved.

To monitor the behaviour during testing, the frame model was extensively equipped with electronicinstrumentation. Fourteen displacement transducers (LVDTs), positioned at key points around the

Figure 4. Details of geometry of wall-frame specimen




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structures, were used to monitor both lateral deection and vertical deformation. A load cell attachedto the lateral actuator measured the applied lateral load and two load cells monitored the vertical loads.The resolution of LVDTs and loading system were 0 01mm and 0 01kN respectively. Allexperimental data from the applied loading and horizontal and vertical deformations were recordedby a DATA LOG system at regular intervals of ve seconds. Figures 5 and 6 show the detail setup of this reinforced concrete rigid frame and wall frame experiments.

3.4. Experimental and analytical Investigation

Figure 7 compares the analytical prediction for the proposed method with both the elastic predictionand experimental load-deections. The linear nite element method predicts values of 46 and 47% of the measured test values for the lateral deections at the 2nd and 1st storeys, respectively, loaded to70% of the ultimate lateral load level (i.e. 140kN). The proposed method gives predictions of 83 and85% of the experimental values. At a load level of 50% of ultimate (i.e. 100kN), the predictions of the

linear elastic method give 55 and 58% of the experimental deections; however, the predictions of theproposed method, which considers the cracking effects, give 92 and 95% of the test results. Within the

Table II. Detail dimensions and reinforcement of wall-frame specimen

Member b (mm) h (mm) d c mm Ten. bars t (%)Comp.

bars c (%) I uncr

(10 6 mm 4) I cr

(10 6 mm 4)

Beam 250 350 50 3T20 1 26 3C20 1 26 1146 464

Column 250 375 40 3T20 1 12 3C20 1 12 1450 607Base 400 400 50 8T20 1 79 6C20 1 35 2618 908

t (mm) bw (mm) d c (mm) Ver. bars v (%)Hori.bars h (%)

I uncr(10 6 mm 4)

I cr(10 6 mm 4)

W, 2nd 180 750 50 12V20 2 79 20H10 0 58 7685 2983W, 1st 180 750 50 12V20 2 79 20H10 0 58 7685 2983

Figure 5. Rigid-frame test setup




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serviceability loading range, the assumption of linear elastic behaviour is relatively inaccurate sincethe rst abrupt stiffness reduction caused by initial cracking of structural members usually appears at avery low lateral load level. In contrast, the proposed method accounts very well for the cracking effectswith stiffness reduction. This method not only predicts lateral deection with a high degree of accuracy at a value of load equal to approximately half of the ultimate capacity, but also gives anestimation of behaviour at 70% ultimate load with an acceptable degree of accuracy. The proposedmethod is most applicable in the serviceability loading range, beyond which signicant discrepancieswill be found as shown in Figure 7. Such discrepancies are primarily due to the fact that concrete andsteel materials become inelastic and nonlinear near ultimate limit states.

The major advantage of the proposed incremental load method is that the historical variation in theexural stiffness reduction for every member in the reinforced concrete structure can be shownexplicitly. This feature can provide design engineers with signicant information on the consequences

Figure 6. Wall-frame test setup

Figure 7. Loaddeection comparison among analysis results of proposed method, elastic method andexperimental results




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of cracking in the concrete members and therefore also minimize the uncertainty of stiffness reductionin the lateral load resisting systems. Figure 8 illustrates the quantitative analytical results of theexural stiffness reduction of the six members involved in the test frame. As indicated in Figure 8, thebeams at the 1st and 2nd storeys crack initially at the 25 and 35kN lateral load level, respectively, andtheir exural stiffnesses reduce quite rapidly after the initial cracking. At the lateral load of 70kN, bothcolumns on the lateral loading side, C4-5 and C5-6, start to crack, while the exural stiffness reductionratios of the beams at the 1st and 2nd storeys have already reduced to 0 54 and 0 60, respectively. Asignicant amount of the load carried by the beams has been transferred to columns C4-5 and C5-6 andas a consequence cracking has occurred in those columns. The columns on the opposite side of thelateral loading, namely C1-2 and C2-3, crack at very different lateral load levels with the upper storeycolumn cracking at the 75kN load level, and the lower storey column cracking at the 95kN load. Thisbehaviour is due to the axial compression forces in these columns being greater than those of thecolumns on the loading side, and maximum axial force occurring in the column at the lower storey onthe opposite loading side.

3.5. Wall-frame: experimental and analytical Investigation

Figure 9 presents the analytical prediction using the proposed method together with experimentalresults for the loaddeection response of the 1st and 2nd storey levels; 5kN is chosen as anappropriate load step. At the three lateral load levels of 175, 210 and 245kN (equivalent to 50, 60 and70% of the ultimate load of 350kN) there are generally only insignicant differences between thepredictions of the proposed method and the test deections. These differences are 2, 2 and 4% at the2nd storey level and 4, 10 and 18% at 1st storey level, respectively. Such discrepancies are consideredacceptable in a practical engineering sense even though larger errors were found when the lateral loadwas larger than 70% of the ultimate load. The comparison presented indicates that the proposedmethod can be applied to wall-frame structures to capture the lateral deection with consideration of the cracking effects in the serviceability loading range.

The quantitative variation of the exural stiffness reduction of each member, including the walls inthe wall-frame structure, is also presented explicitly in Figure 10. This gure indicates that the beamsat the 1st and 2nd storey start to crack at lateral loads of 55 and 65kN and their exural stiffness

Figure 8. Flexural stiffness reduction of members in the rigid-frame test




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reduces rapidly after initial cracking. At 110kN (approximate 30% of the ultimate load), the 1stcolumn starts to crack and the exural stiffness reduction ratios of beams at the 1st and 2nd storeyshave already decreased to a value of 0 64 and 0 76. The 2nd column cracks at a lateral load level of 175kN (approximate 50% of the ultimate load). When the lateral load level reaches 205kN(approximate 59% of the ultimate load), initial cracks start to occur in the shear walls. In consideringthe results of the two respective frame and wall-frame tests, the beams are the rst members in thesestructures to experience cracking and to have their stiffness reduced to a value below 50% of theirgross moment of inertia.

4. EXAMPLE OF A 40-STOREY REINFORCED CONCRETE BUILDING

A 40-storey, 140m high, wall-frame building, with storey height 3 5m, is considered in this analytical

Figure 9. LoadDeection comparison between analysis results of proposed method and experimental results

Figure 10. Flexural stiffness reduction of members in the wall-frame test




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study. Reinforced concrete is used as the structural material, in which the design strength and Youngsmodulus of the concrete are assumed to be 30 and 24 800MPa, respectively, and the Youngs modulusof the steel is assumed to be 200000MPa. The typical oor live load is 1 5kPa. The design windpressures in accordance with the Code of Practice on Wind Effects Hong Kong (1983) are taken as1 2kPa within the lower 30m of the boundary height, and 1 9, 2 4 and 3 0kPa in the ranges 3050m,50100 and 100150m, respectively. The maximum allowable lateral deection is taken to be H /500where H is the building height. Idealized xed boundary conditions are assumed at the supports.

Figure 11 shows the typical oor plan with columns located on a grid of 5300 7000mm. Thebeam span is 7000mm in the y-direction and 5300mm in the x-direction. The typical section size of thebeams is 625mm deep by 300mm wide, while the lintel beams between two shear walls at every storeyhave cross-section 800mm deep by 400mm wide. Two kinds of column are chosen, one with a cross-section of 800mm by 800mm being used for the 1st to 20th storeys, and 750mm by 750mm beingused for the 21st to 40th storeys. The typical thickness of oor slab is 150mm. In addition, foursections of wall are used at every storey to strengthen the lateral stiffness in the short direction of thebuilding. Two kinds of cross-section of the walls are designed. At the 1st to 20th storeys, the dimensionof the wall is 7800mm wide by 400mm thick. The cross-section of the wall is 7750mm by 350mmfrom the 21st to 40th storeys. The details of reinforcement ratios and design properties are presented inTables IIIV.

5. DISCUSSION OF ANALYSIS RESULTS

Figure 11 presents the typical oor layout and the two loading conditions acting on the tall building inthe x- and y-directions. Figure 12 presents the lateral loadtop deection relationships with variousload increments under y-direction loading. In this gure, F represents the full service wind loaddetermined from the design wind pressure along the height of the building, N stands for the loadincrement number used in the incremental analysis. The dashed vertical line represents the limiting

Figure 11. Typical oor plan




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lateral drift ratio of 1/500 for this building. A total of ve different load increments are selected in thecomparisons. N = 1 is used to represent elastic analysis without reducing the stiffness of the membersdue to cracking. N = 3, 10, 40, 200, are used to represent load increments equivalent to 100/3 = 33%,100/10 = 10%, 100/40 = 2 5% and 100/200 = 0 5% of the full service load. Using linear elasticanalysis, the lateral top deection is found to be 0 24m, indicating that the lateral drift ratio has a valueof 1/584, which satises the serviceability limit of 1/500. However, when fully accounting for crackingeffects, the proposed method with 40 load increments predicts that the value of top deection is0 312m, representing a lateral drift ratio of 1/449, which is 30% more than that of the elasticprediction. Such a result implies that, if concrete cracking is taken into account in the analysis, thelateral serviceability performance of the building becomes unsatisfactory and this particular structuredoes not meet the code criteria for lateral drift.

Loadtop deection curves resulting from the use of various load increments are presented in Figure12. By means of a number of incremental load analyses, accelerated convergent behaviour is found.The top deection calculated with three load increments is found to be 22% more than the linear elasticprediction, while the prediction of 10 increments is 30%. In addition, the curves of 40 and 200 loadincrements are very close, and these predictions of the top deection at full service load do not showsignicant deviation from the values calculated by 10 load increments. This feature of convergence

Table III. Section dimension and properties of columns ( A = area; I uncr I cr = gross, cracked-moment of inertia)

Column at storeySection (mm 2)

Cover (mm) (%)

Steel A (m 2) I uncr (m4) I cr (m

4)

120 800 800 3% 0 64 0 04626 0 02062

50 24 322140 750 750 2 1% 0 56 0 03297 0 0116950 24 25

Table IV. Section dimension and properties of beams ( A = area; I uncr I cr = gross, cracked-moment of inertia)

Beam at storeySection (mm 2)

Cover (mm)l (%) (top)

(%) (bottom) A (m 2) I uncr (m4) I cr (m

4)

B4 4036 400 800 50 1, 6 25 1, 6 25 0 32 0 02132 0 00775B4 3531 400 800 50 1 3, 8 25 1 3, 8 25 0 32 0 02274 0 00983B4 3026 400 800 50 1 6, 8 28 1 6, 8 28 0 32 0 02419 0 01185B4 251 400 800 50 2 1, 8 32 2 1, 8 32 0 32 0 02637 0 01475

Other 40-1 300 625 50 0 8, 4 28 0 8, 4 28 0 19 0 00848 0 00395

Table V. Section dimension and properties of shear walls ( A = area; I uncr I cr = gross, cracked-moment of inertia)

Wall at storeyFlange (mm)Web (mm)

(%) (Ver. Steel) (%) (Hori. steel) A m2 I uncr m

4 I cr m4

120 800 800 3 0, 24 32 3 76 27 00 6 63400 0 8, 42 25

2140 750 750 2 1, 24 25 3 31 22 80 4 18350 0 22




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has an instructive signicance for application of the method in practical design. Generally, this methodwith 10 load increments, where each load increment is equal to approximately 10% of the full serviceload, can provide very accurate prediction as well as efciency in terms of computational time.

Figure 13 presents the exural stiffness of individual cracked members at the 40th storey calculatedusing 200 load increments. The coupling beams B4 are found to be critical members where cracks arerst caused by lateral load. The sudden stiffness reductions in B5, B7, B8 and B10 at the verybeginning of the iteration process are mainly due to the initial applied dead and live gravity loads.These reduced beam stiffnesses remain basically constant during the entire application of lateral loads.Several beams at the interior span, such as B2, B4, B6 and B9, exhibit greater stiffness reductions sincea larger shear force is taken by the beams of the interior span.

Figure 12. Lateral loaddeections with various load increments in the Y = direction, 40-storey building example(F = full lateral load N = 1 = elastic analysis N = 3, 10, 40 and 200 f = 33%, 10%, 2 5% and 0 5% full lateral

load)

Figure 13. Flexural stiffness reduction of cracked member at 40th Storey of the 40-storey building example




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It is also noted that points of discontinuous reduction rates occur and are shown in the stiffnessreduction curves of the beams at the interior span with increasing lateral loads. This is apparently dueto the action of constant negative moments, which are generated at both ends of those beams from thevertical gravity load such as dead and live loads. On the other hand, increasing positive moment at oneend, and negative moment at the other end, are caused by the lateral load with increasing load

increments. The superposition of these two moment diagrams results in the cracks occurring at the endwhere two negative moments appear together rst, followed by cracking at the other end when the netmoment is larger than the cracked moments of those members. The probability of crack occurrenceincreases abruptly at that moment, leading to discontinuous stiffness reduction rates observed in somecurves shown in Figure 13.

Moreover, the nal stiffness reductions of members B1, B2 and B3 at the full service load arepresented with respect to each storey in Figure 14. The x-axis in these gures contains three ranges of 0100%, to present variations of stiffness of the beams at three locations in the layout of the building.Owing to the characteristics of the wall-frame building, the shear force resisted by the frame is smallerat the lower storeys where the shear walls take a large amount of the shear force. The shear force in thiscase is the main factor generating the moments in the beams. Hence, from this gure, lower stiffnessreductions are found in beams located in the 1st5th storeys. From the 6th to the 30th storeys, the mostsevere stiffness reductions occur in all beams since larger components of the shear force are resisted bythe frame structures.

Not all the beams exhibit the same cracking effects in their stiffness characteristics. The interiorbeam (B2) indicates approximately 50% stiffness reduction above the 5th storey. The beams locatedleeward are categorized into another group that has approximately 60% of the gross moment of inertiabetween the 5th and 30th storeys and 7090% of the uncracked values beyond the 30th storey. Thewindward beams are found to have the least stiffness reductions, but the variations of the stiffnessreductions in those beams at the mid-height of the building are larger than the values at both the topand bottom storeys.

The top deection in the x-direction is calculated by using 10 load increments. Unlike the wall-frame behaviour of the building under Y -direction wind loading, this tall building can be considered asa frame-type structure under X -direction wind loading. Hence, the stiffness of the beams contributes

Figure 14. Reduction of exural stiffness of B1, B2, B3 in the 40-storey building example under y-direction lateralload




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more to the overall lateral stiffness in this type of structure and loading. The uncertainty of the elementstiffness due to cracking effects has been captured by the proposed method. Concrete cracking causes a68% increase in the top deection in the X -direction and a 38% increase in the Y -direction, as indicatedin Table VI. Under frame action in the X -direction of the building, most of the beams are found tocrack severely at the lower storeys and have reduced stiffness to values of approximately 50% of theiroriginal stiffness, as shown in Figure 15. Owing to the large gravity load of the building, the crackedmoments in the vertical members are signicantly increased and only minor cracking is found in thewindward vertical members at the 1st storey of the 40-storey building under respective X and Y -direction lateral loads.

6. CONCLUSIONS

6 1. An effective stiffness model, based on the probability of cracking in the reinforced concretemembers, has been presented and integrated into a linear nite element package in an incrementalmanner to take account explicitly and quantitatively of the cracking effects on the lateral stiffness of tall reinforced concrete buildings.6 2. The proposed method can be used to determine the loaddeection history of reinforced concretebuildings. The advantages of this approach are that it is computationally more efcient and more directthan the typical nonlinear nite element analysis method.

Table VI. Comparison of top deection of elastic and proposed method along x- and y-directions

Top deection analyses along x- and y-direction

Wind direction Elastic Drift Proposed Drift crack

x 0 265 1/529 0 446 1/314 68% y 0 240 1/583 0 312 1/449 31%

Figure 15. Reduction of exural stiffness of B11, B12, B13 in the 40-storey building example under x-directionlateral load




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6 3. Validation of the proposed method has been presented by the results of tests conducted on a rigid-frame and a wall-frame structure.6 4. Within a load level up to about 70% of ultimate, the exural stiffness reduction due to cracking isfound in the tests to be the dominant factor in the resulting nonlinear loaddeformation response of reinforced concrete structures. Beyond the range up to approximately 70% of ultimate, material

nonlinearity becomes a signicant factor in the behaviour of reinforced concrete structures.6 5. Cracks occurring in the beams of tall reinforced concrete buildings are found to be the main factorin the loss of lateral stiffness for lateral load resisting systems.6 6. The effects of cracking on the top lateral deections of structures are dependent on the type of lateral load resisting system. The increase of the lateral top deection of rigid-frame structured due tocracking effects is found to be more than that of wall-frame structures. The greatest reduction in beamstiffness occurs at approximately 1/3 of the height of wall-frame structures, but in the lower storeys of rigid-frame structures.

7. NOTATION

Ain = the axial force of the ith member in the nth iteration

D(n ) = the lateral deection of an RC structure at the nth iteration f c = the ultimate strength of concrete f r = the modulus of rupture of concreteF (

n ) = the applied force at the nth iteration I cr = the cracked moment of inertia I e = the effective moment of inertia( I e)i

(n ) = the effective moment of inertia of the ith member in the nth iteration I g, I uncr = the uncracked moment of inertia( I uncr )i

(n ) = the uncracked moment of inertia of the ith member in the nth iteration M , M ( x) = the moment distribution on the member M cr = the cracking moment in a concrete exural member

M i(n )

= the moment distribution of ith member in the nth iteration( M cr )i(n ) = the corresponding cracking moment of the ith member in the nth iteration

N = the total iteration number corresponding to the load increment p( x) = the probability density functionP cr = the probability of occurrence of cracked sections associated with the

outcome I crP uncr = the probability of occurrence of uncracked section with outcome I uncrS = the total area of the moment diagramS cr = the area of the moment diagram segment over which the working moment

exceeds the cracked moment M cr yt = the distance from centroid of gross section to extreme bre in tension

v = the axial compressive stress

F = the load increment

ACKNOWLEDGEMENTS

This work was supported by the Research Grants Council of Hong Kong under Project No. HKUST-543/94E, and was based upon the research conducted by Mr Feng Ning under the supervision of NeilMickleborough and Chun-Man Chan for the Degree of Doctor of Philosophy in the Department of Civil Engineering at the Hong Kong University of Science and Technology, Hong Kong.




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REFERENCES

ACI Committee 318, 1995. Building code requirements for reinforced concrete (ACI 318-95). American ConcreteInstitute, Farmington Hills, MI.

Al-Zaid RZ, Rajeh Z, Al-Shaikh AH, Abdulrahman H, Abu-Hussein MM, Mustafa M. 1991. Effect of loadingtype on the effective moment of inertia of reinforced concrete beams. ACI Structural Journal 88 (2): 184190.

AS 36001994, 1994. SAA concrete structures code, Standards Association of Australia, Sydney.ASCE State-of-the-Art Report, 1982. Finite Element Analysis of Reinforced Concrete. Structural DivisionCommittee, ASCE Special Publication, New York.

Barzegar F. 1989. Analysis of RC membrane elements with anisotropic reinforcement. Journal of Structural Engineering, ASCE 115 (3): 647665.

Branson DE. 1963. Instantaneous and time dependent deection of simple and continuous reinforced concretebeams. Alabama Highway Research Report No. 7, Bureau of Public Roads.

CAN3-A23 3-M94 1994. Design of concrete structures for buildings. Canadian Standards Association, Toronto,Canada.

Ingraffea AR, Gerstle W. 1985. Non-linear fracture models for discrete crack propagation. In Application of Fracture Mechanics to Cementitious Composites , Shah SP. (ed.). The Hague, The Netherlands: Martinus-Nijhoff; 171209.

Kotsovos MD, Pavlovic MN. 1995. Structural ConcreteFinite Element Analysis for Limit-State Design .London: Thomas Telford.

Mickleborough NC, Ning F, Chan CM. 1999. Prediction of the stiffness of Reinforced concrete shear walls underservice loads. ACI Structural Journal 96 (6): 10181026.

Mufti AA, Mirza MS, McCutcheon JO, Houde J. 1972. A study of the nonlinear behavior of structural concreteelements. Proceedings of the spec Conference on Finite Element Method in Civil Engineering , Montreal,Canada.

Ning F. 1998. Lateral stiffness characteristics of tall reinforced concrete buildings under service loads. PhDdissertation, Department of Civil Engineering, Hong Kong University of Science and Technology.

Polak MA. 1996. Effective stiffness model for reinforced concrete slabs. Journal of Structural Engineering, ASCE 122 (9).

Rashid YR. 1968. Analysis of prestressed concrete pressure vessels. Nuclear Engineering and Design 7 (4): 334344.

Scanlon A, Murray DW. 1974. Time dependent reinforced concrete slab deection. Journal of Structural Division, ASCE 100 (9): 19111924.

Vecchio FJ, Collins MP. 1986. The modied compression eld theory for reinforced concrete elements subjectedto shear. ACI Journal Proceedings 83 (6): 219231.

Yarimci E, Yura JA, Lu LW. 1967. Techniques for testing structures permitted to sway. Experimental Mechanics7 (8).



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