Investigation Into The Floor Diaphragm Flexibility In Rectangular Reinforced Concrete Buildings And Error Formula

Transcription

1 The th World Conerence on Earthquake Engineering October 7, 008, Beijing, China Investigation Into The Floor Diaphragm Flexibility In Rectangular Reinorced Concrete Buildings And Error Formula M. Moeini, B. Raezy and W. P. Hoson M. Sc o Structural Engineering, The Sahand University o Technology, Tabriz, Iran Assistant Proessor, Dept. o Civil Engineering, The Sahand University o Technology, Tabriz, Iran ABSTRACT : Associate Proessor, Cardi School o Engineering, Cardi University, Cardi, UK Ma.moeini@Gmail.com, Raezyb@sut.ac.ir, Hoson@c.ac.uk Building structures are typically designed on the assumption that the loor systems serve as rigid diaphragms that span beteen the vertical resisting elements. Such an assumption is normally perectly adequate or the seismic analysis o most buildings, but some structural orms, typically those comprising long, thin loor plans and perimeter lateral resisting elements, can exhibit signiicant in plane lexibility in their loor systems. The dynamic behavior o this latter class o structures is dissimilar to the behavior expected o typical structures and can lead to unexpected orce and drit patterns. The main purpose o this paper is to use the inite element method to investigate the inluence o loor diaphragm lexibility on the behavior o concrete structures. Initially a parametric study is undertaken on a variety o reinorced concrete structures ith rectangular plan orm, perimeter shear alls and slabs that contain openings. In the second part o the paper, a number o response spectrum analyses are perormed on the structures assuming both rigid and lexible diaphragm assumptions. A regression analysis is then perormed on the results to obtain an error ormula that can be used to estimate the error involved in using the rigid diaphragm assumption on structures similar to those tested. KEYWORDS: Rigid diaphragm, Flexible diaphragm, Finite element, Responsespectrum analysis.. INTRODUCTION Floor diaphragms are essential structural elements in the lateral orce resisting system o building structures. In the analysis o multistory buildings subjected to lateral loads, a common assumption is that the loor system undergoes no deormation in its on plane [, ]. At the mass center o each rigid loor, there is a master node having three degrees o reedom to represent the to inplane translations and one outoplane rotation o all the other nodes or socalled slave nodes. These slave nodes contain three degrees o reedom, to inplane rotations and one outoplane translation [, ]. Muto [] used a beam ith bending and shear deormation eects to simulate the behavior o lexible loors. Jain [] also used this beam to evaluate the eect o lexible loors on the seismic response o buildings. Saarini and Qudaimat [5] analyzed 7 RC buildings to compare the dierence beteen rigidloor and lexibleloor analyses. They ound that the rigidloor assumption is accurate or buildings ithout shear alls, but it can cause errors or building systems ith shear alls. A quantitative investigation o the dierence beteen lexible and rigid loor analyses o buildings ith shear alls as not given and appears to be largely absent rom the literature. Ju and Lin [6] conirmed the results o [5] and perormed a quantitative investigation. They then proceeded to deduce an error ormula using a regression analysis o the rigidloor and lexibleloor analyses rom 50 rectangular, Ushaped, and Tshaped buildings. The eect o slab openings as not considered and also appears to be absent in the literature. Fleishman and Farro [7] investigated the dynamic behavior o perimeter lateralsystem structures ith lexible diaphragms. Reinorced concrete

2 The th World Conerence on Earthquake Engineering October 7, 008, Beijing, China Modern structural systems oten employ isolated or perimeter lateralsystems ith long loor spans in hich the rigid diaphragm treatment is not accurate. For these structures, diaphragm lexibility can modiy dynamic behavior. The dynamic behavior o such structures is dissimilar to the behavior expected o typical structures. This dierence can lead to unexpected orce and drit patterns [7]. Due to the idespread use o the rigidloor building analysis assumption, the quantitative study o the error caused rom this scheme is a very important topic. Hoever, it is extremely diicult to obtain an accurate error ormula, since there is no general theoretical solution or the structural analysis o buildings. Thus, a statistical method is adopted to obtain an approximate error ormula. In this paper, the inite element method is used to analyze a number o buildings ith and ithout shear alls. From the results o a number o responsespectrum analyses, the rigidloor model is ound to be suiciently accurate or regular and irregular buildings ithout shear alls. Hoever, the dierence beteen rigidloor and lexibleloor analyses can be large or buildings ith shear alls. In order to estimate the dierence beteen those to types o analyses or buildings ith perimeter shear all and internal symmetric slabopening, an error ormula is developed statistically. Using this ormula, one can estimate the error in the result hen the rigidloor assumption is used. Furthermore, it is easy to use, since only the geometric data o the shear all and the slab are required.. STRUCTURAL MODELLING AND ANALYSIS FRAMEWORK The total number o degrees o reedom in a threedimensional (D) building analysis is equal to three times the total number o slave nodes and master nodes in the mesh. For the equivalent static lateral orce method, the horizontal orces are oten applied to the master nodes in a rigidloor analysis. Hoever, it is diicult to add these horizontal orces to the nodes o a building ith the lexibleloor assumption. For example, adding these horizontal orces only to the node at the mass center o each loor ill cause a stress concentration near the mass center. Thus, to compare the results o the rigid and lexibleloor analyses, dynamic analysis is probably a better choice since the earthquake loading can be applied to the base o the building ithout any dierentiation beteen the rigid and lexibleloor analyses. Forced dynamic analyses include timehistory and responsespectrum analyses. For timehistory analysis, it is not easy to compare the complex analysis results beteen the rigid and lexibleloor analyses. For example, the to results may dier due to a signiicant time shit, so comparing them at a certain time ill cause errors. The responsespectrum analysis does not have this problem, since only the maximum responses are calculated. Hence, this method ith the response spectrum o the 800 code (Fig. ) is used to perorm the to types o building analyses. For the dynamic analysis, the mode superposition method is used, and the irst 0 modes are used to develop the input data or the responsespectrum analysis. The eective masses in the xtranslation, ytranslation and zrotation or these 0 modes are alays larger than 95% o their total masses in the building analyses. Figure. Response Spectrum o 800 Code

3 The th World Conerence on Earthquake Engineering October 7, 008, Beijing, China. CHARACTERISTICS OF THE BUILDINGS CONSIDERED The RC structures considered all have a rectangular plan orm and perimeter shear alls. The nesting o the parameters in the parametric study is given in Fig., hile the slabs are shon in Fig.. For the purpose o analysis, the slabs are divided into the olloing groups according to the size o the opening, as ollos Group, no opening. Group, opening area o approximately %. Group, opening area o approximately 0%. Group, opening area o approximately 50%. In each group there are three dierent aspects ratios (:, : and :5); or each aspect ratio there are our slab thicknesses (5, 0, 5 and 0 cm); or each slab thickness there are our dierent story numbers (, 5, 0 and 0story) and each building is analyzed according to the rigid and lexible diaphragm assumption. Fig. is only dran or irst group. Figure. Style o Buildings Grouping Figure. Plan shape, opening area and shear all positions Table. Member sizes, story height and shear all thickness Number o stories Beam size Column size Story height Shear all thickness (cm) (cm) (m) (cm) 50X80 50X X80 80X X80 00X X80 0X0. 0

4 The th World Conerence on Earthquake Engineering October 7, 008, Beijing, China Table. Properties o buildings in all groups Aspect ratio Plan dimension Opening dimension Approximate opening (m) (m) area % : 8X8 Group : 8X5 :5 8X90 : 8X8 6X6 % Group : 8X5 6X8 % :5 8X90 6X0 % : 8X8 8X8 0% Group : 8X5 6X0 9% :5 8X90 6X5 0% : 8X8 X 5% Group : 8X5 0X6 8% :5 8X90 0X8 5% For simplicity, buildings are labeled according to the olloing procedure. A building ith properties, Group, aspect ratio :, slab thickness 5cm and 5story, ould be labeled It is clear that or the number o groups, aspect ratio, slab thickness and number o stories, the total number o buildings analyzed is (***=9) 9. Since each building is analyzed ith rigid and lexible diaphragms, the total number o analyses is 8. Fig. indicates the plan shape, opening in slab and the position o perimeter shear alls. For the analytical modelling and dynamic analyses o the structures considered, the computer program SAP000 [8] as used. The loor diaphragms and shear alls ere modelled ith shell elements.. ANALYSIS OF THE PARAMETRIC STUDY The eects o the various parameters are no shon in the olloing graphs. In these graphs, the error beteen the results rom the rigid and lexible loor analyses is deined by equation () and plotted along the Ycoordinate. lexible story Error % = di % = 00 () lexible Where lexible = lateral deormation o the diaphragm in the top story under the lexible loor assumption, story = lateral deormation o the diaphragm in the top story according to the rigid loor assumption (average story drit)... Number o Stories Fig. shos the variation o the error ith the number o stories. It is clear that increasing the number o stories reduces the lateral stiness o the building and hence the eect o diaphragm lexibility. Fig. shos the results or buildings 00, 005, 00 and Floor thickness Fig. 5 shos the variation o the error ith the loor thickness. It can be seen that increasing the loor thickness increases the inplane loor stiness and thus reduces the eect o diaphragm lexibility. Fig. 5 shos the results or buildings 5050, 500, 550 and Aspect ratio Fig. 6 shos the variation o the error ith aspect ratio. Again, it can be seen that increasing the aspect ratio reduces the inplane loor stiness and thus increases the importance o diaphragm lexibility. Fig. 6 shos the results or buildings 050, 050 and 5050.

5 The th World Conerence on Earthquake Engineering October 7, 008, Beijing, China.. Opening in slab Fig. 7 shos the variation o the error ith the opening in slab. Once more it can be seen that increasing the size o the opening in the slab reduces the inplane loor stiness and thus increases the importance o diaphragm lexibility. Fig. 7 shos the results or buildings 505, 505, 505 and 505. Figure. Variation o the error ith number o stories Figure 5. Variation o the error ith loor thickness Figure 6. Variation o the error ith aspect ratio Figure 7. Variation o the error ith opening in the slab 5. ERROR FORMULA A statistical method is no used to ind a regression curve or estimating the error o the rigidloor building analysis. First, the error value beteen the rigidloor and lexibleloor analyses is deined to be the Ycoordinate, hile the sample input value, deined as the displacement dierence ratio o the lexibleloor and the rigidloor analyses, is the Xcoordinate. Each pair o rigidloor and lexibleloor building analyses produces one error value and one sample input value. Regression analysis is then used to determine the error ormula rom the 8 building analyses. The error value deined in () is used to calculate the dierence in analysis results beteen the rigidloor and the lexibleloor models or shearall buildings. All the our column end moments are compared in this equation. The eightings o these column moments are also used in () to magniy the importance o large moments and suppress the eect o small moments. Error % = 00 n i= j= ( M rij M n i= j= ij M / M ij ij ) M ij = 00 n i= j= n M i= j= Where n = total number o columns in the building; index j = index o the bending moments o the to axes at the to column ends; M rij = moments o column i using the rigidloor building analysis; and M ij= moments o column i using the lexibleloor building analysis. An appropriate deinition or the sample input value is very important, since the variation o the error rij M M ij ij () 5

6 The th World Conerence on Earthquake Engineering October 7, 008, Beijing, China ormula is highly dependent on this deinition. Hoever, a complex deinition should be avoided or practical purposes. The error calculated rom () should be proportional to the displacement dierence ratio (R) beteen the lexibleloor and rigidloor analyses. In this paper, the olloing steps are used to deine this displacement dierence ratio (R):. The loor is assumed to be a simply supported beam subjected to a unit uniorm load (load/length) along the length direction (Fig. 8). Since the shear alls are symmetric, only one o the to symmetric shear alls is chosen. This shear all is assumed to be a cantilever beam subjected to a concentrated load at the all top (Fig. 8). The magnitude o the concentrated load is L/, hich is transerred rom the unit uniorm load o the loor, here L is the loor length shon in Fig. 8. [ Figure 8. Model and Dimensions o Floor and Shear Walls. The approximate averaged displacements o the rigidloor and lexibleloor analysis are assumed as ollos () () Where = approximate averaged displacement o the loor according to the rigidloor assumption; = approximate averaged displacement o the loor according to the lexibleloor assumption; = displacement at the top o the cantilever beam (all) according to the assumption o step ; and = displacement at the center o the simply supported beam (loor) according to the assumption o step.. From () and (), the displacement dierence ratio is deined as: lexible rigid R = = + (5) lexible For rectangular slabs ith internal symmetric slabopening, and can be obtained rom (6) and (7) hich ere derived using an energy method. I D= D and L= 0 in (7), o a rectangular slab (ithout opening) is obtained rom (8). (LL = Et t H L αhl = + (6) ED GD t α( LL L + Gt D L ) L[( L / ) + D / 6Et D ) α[( LL + / ) LL ( L Gt D / )] L ] [( L / ) L ] / 8Et D / 5L αl Re c = + (8) ED t 8GD t (7) 6

7 The th World Conerence on Earthquake Engineering October 7, 008, Beijing, China Where E = Young s modulus; G = shear modulus; = thickness o the loor slab; = thickness o the shear all; α= ratio o the eective shear area o the slab, hich can be approximated to 6/5; H = total shearall height; and L, L, L, D, D, D, and = dimensions o the loor slab and shear all as shon in Fig. 8. Fig. 9 indicates the results rom 8 building analyses, here each points represents the error evaluated rom a rigidloor and a lexibleloor building analysis. From this igure and the values o errors, in order to obtain appropriate accuracy in the linear regression analysis, the analysis should be perormed in the olloing to regions o R, 0 < R < 0. 85and 0.85 R <, as shon in Fig. 0. Figure 9. The results o analyses Figure 0. Regression data points and curve (9) Fig.0 indicates that all the data points are concentrated along the banded lines ithout too much variation and the straight lines are shon in (9) are a good regression model or the regions ( 0 < R < 0. 85) and ( 0.85 R < ). The average and maximum variations o equation (9) are shon in Table, here the variation is deined as the absolute value o the error percentage rom (9) minus the real value. The values o this table indicate satisactory accuracy in the regression analysis. Thus, using (5) as a regression base is acceptable. Equations (9) are useul or estimating the error in columns moments, hen the idespread rigidloor method is used to simulate buildings ith continuous symmetric shear alls and internal slab openings. Furthermore, it is easy to use, since only the geometric data o the shear alls and slab are required. In the irst region ( 0 < R < 0. 85), hen R < 0., the percentage error obtained rom (9) is less than 0%, hich indicates that the diaphragm is eectively behaving rigidly. In the remainder o the region, the percentage error is alays less than 0% and the diaphragm can be classed as moderately lexible. In the second region ( 0.85 R < ), the percentage o error obtained rom (9) is alays more than 0% and the diaphragm can be classed as highly lexible. Thus, in general, it is recommended that or rectangular RC structures ith continuous symmetric shear alls and internal slabopening, the behavior o the diaphragm should be considered to be suiciently lexible that the lexibleloor analysis be used in place o the rigidloor analysis hen R 0.. 7

8 The th World Conerence on Earthquake Engineering October 7, 008, Beijing, China Table. Average and Maximum Variations o Eq. (9) R Variation o Eq. (9) Average Maximum 0.00<R< <R< <R< <R< <R< <R< <R< <R< <R< <R< <R< <R< <R< <R< SUMMARY AND CONCLUSIONS For buildings ithout shear alls, the rigidloor model is as accurate as the lexible model even or irregular loor systems. This is due to the act that the inplane stiness o the slab is much larger than the outoplane column stiness. For buildings ith shear alls, the rigidloor and lexibleloor analyses can dier greatly due to the very large lateral stiness o the shear all system. In such cases, the inplane stiness o the slab is relatively insigniicant, and the slab inplane deormation cannot be ignored. An error ormula has been generated using a regression analysis on the results o rigidloor and lexibleloor analyses rom 96 rectangular reinorced concrete buildings ith internal symmetric slabopenings. Using this ormula, one can estimate the error involved in applying the rigidloor assumption or this class o building. Furthermore, the ormula is easy to use, since only the geometric data o the shear all and the slab are required. REFERENCES [] MacLeod, I. A. Shear all rame interaction: A design aid, Engrg. Bull. No., Portland Cement Association, Skokie, IL, 970. [] Wilson, E. L., Hollings, J. P., and Dovey, H. H. Three dimensional analysis o Building systems (expanded version), Rep. No. EERC 75, University o Caliornia, Berkeley, cali, 97. [] Muto,K. A seismic design analysis o buildings, Marunzen Co., Ltd., Tokyo 60, Japan, 97. [] Jain, S. K. Seismic response o buildings ith lexible loors, J. Engrg. Mech. ASCE0, 59, 98. [5] Saarini, H.; Qudaimat,M., Inplane loor deormations in RC structures, Journal o Structural Engineering, vol. 8, No., 089 0, 99. [6] Ju, S.H. ; Lin,M.C. Comparison o building analyses assuming rigid or lexible loors, J. Struct. Engrg. ASCE 5, 59, 999. [7] R.B.Fleischman; K.T.Farro, Dynamic behavior o perimeter lateralsystem structures ith lexible diaphragms, Earthquake engineering and structural dynamics 00; 0:7576. [8] SAP000, Advanced 9.0., Structural Analysis Program User Manual, Copyright Computer and Structural, Inc. 8