THREE DIMENSIONALLY ANALYSIS OF FLAT PLATE STRUCTURES BY EQUIVALENT GRID METHOD

IIham Nurhuda*, Universitas Diponegoro, Indonesia

Han Ay Lie, Universitas Diponegoro, Indonesia

29th Conference on OUR WORLD IN CONCRETE & STRUCTURES: 25 - 26 August 2004, Singapore

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29!h Conference on OUR WORLD IN CONCRETE & STRUCTURES: 25 - 26 August 2004, Singapore

THREE DIMENSIONALLY ANALYSIS OF FLAT PLATE STRUCTURES BY EQUIVALENT GRID METHOD

IIham Nurhuda*, Universitas Diponegoro, Indonesia Han Ay Lie, Universitas Diponegoro, Indonesia

Abstract

The use of the effective beam width model to predict respond of flat plate structure subjected to lateral loading is a common practice. Nevertheless, usage of this model is still limited to two-dimension model. This paper extends the application of equivalent grid model to analyze flat plate structures three dimensionally. The effective grid width is analyzed empirically from experimental result. Structure analysis conducted by using both linear and nonlinear analysis. The result is discussed in this paper.

Keywords: flat-plate, model, reinforced concrete

1. Introduction

The structure generally used in reinforced concrete floor is a plate fixed 0 n beams 0 n a II four sides. This system is the eldest system which is its analysis widely known. Nevertheless, the plate­beam system has some weaknesses especially from efficiency facet and economics aspects. High cost and more time needed to set the formwork, large story height due to beam depth, are some of the lacking of this system. As time progressed and technology evolved, the column line beams gradually began to disappear. The resulting slab system consisting of solid slabs supported directly on column is called the flat plate [1] . This kind of structure is developed based on understanding that the slab not only distribute load to beam but also function as a unity of structure carrying load.

Difficulty faced in usage of this flat-plate system is in its behavior analysis, especially relate to the effect of lateral loading when the structure is designed to resist strong wind or earthquake. Static analysis with classical mechanics is not adequate because of the complex stress distribution that occurred in plate. Fast growth of computer technology brings big progress to solve the physic problem numerically. Numerical method that is often used in structure analysis is the finite element method.

Structure model that widely used ins tructure analysis is frame element model 0 r bar element. This model is often chosen because it is relatively simple and easy to apply. In the flat-plate structure, an analysis by using frame element is generally carried 0 ut bye ffective beam width model. In this method, the plate is modeled as equivalent beam with the certain effective width. Analysis in this way generally uses two-dimension analysis. Usage this analysis in three-dimension is not commonly used.

Based on the understanding that analysis of flat-plate have to model plate as a unity supporting load into two directions, the use of grid model to models flat-plate structure will be studied. Using this model in flat-plate is relatively simple and easy to apply. This model of analysis is used with the assumption that slab consists of grid formation that can distribute load to any direction. Effective distance of grid will be examined by varying the distance between grid and compare the result of analysis to experiment result. Effect of number of grid to slab deflection is also observed by comparing numerical analysis to experiment.

2. Grid Model Of Flat Plate Structures

2.1 General Behavior Of Flat Plate Structures

Flat-plate structure is a structure system where slab supported directly on column without existence of beam. Flat-plate system is analyzed as two-way slab system but has different behavior

305

from the wo-way slab system in slab-beam structure. In slab-beam system, the positive elastic moment of plate occurred in the middle of plate, and the negative moment occurred in the middle of plate edge on beam. On the contrary, in flat-plate structure, the maximum positive and negative moment occurred at column line.

In the rectangular panel, slabs on rigid beams span predominantly beween the long beams, and for an aspect ratio of 2 the central region is curved practically only in short span direction. In a flat­plate, the central section spans in the longer direction, and the transverse moments are confined to a limited width on either side of the columns as indicated by fig 1.

. _ . _ . _ -'-0-'­ ' ­ ' ­ ' - ' - ' ­ ' ­ ' ­ . _0-'-'- , ._._ ., j­ ' - ' -'­ ' -'­ ' -'­ ' -'-'­ ' - ' ­ ' -'-'­ ' - ' ­ ' 1'-'-" I

I I

I : : ~ I ~--~----------------{J--~

: t ( ) t: I I I I

~--[}----------------{]--jI I . I I i

._._._ . _._._ . _._ . _ . _ . _._._._ . _ _ . _ . _f ~ . - . - . ~ . - . - . - . -.-.- . - . -.- , - . - . - . -.-.-~- . - , -j

Fig 1. Behavior of Elongated Panels of Continuous Slabs

When the wo-way slab is supported on wall or beam, hence shear in slab often do not become critical factor in design. Generally, this is because the factored shear force still smaller than concrete shear capacity. On the contrary, in wo-way slab system where slab is supported directly to column, slab carry entire load and distribute it directly to column, hence shear around column being a critical matter. Thereby design of structure to shear effect has to get special attention from designer [5].

Two type of shear require to be considered in designing of flat-plate structure are beam shear type and punching shear type. The beam shear type or one way shear become critical in thin and long plate. While punching shear, which is often the commonest mode of failure, is due to relatively small shear area of slab around column that is unable to detain point loading of column. Critical area of the punching shear failure is at distance d/2 from column face, where d is the effective depth of slab.

2.2 Effective Beam Width Model

Regarding the slab-column connection, there are not all portion of slab width work effectively as a flexural element framing into the column. This behavior is due to slab do not experience the same rotation along its width. To overcome this matter, it is common to model the slab in a simple model at the design phase. The simplest way is by modeling the slab in bar element that call slab-beam element. This slab-beam element rotates uniformly across its transverse width. The beam depth generally is taken equal to slab thickness. Whereas the effective beam width has been studied and proposed by many researchers.

Banchik(1987), as quoted by Hwang and Moehle {3], presents effective beam width solutions developed using finite element technique. The solutions apply to interior, edge, and corner connections, for column with square cross sections, and for combinations of c1/11 and 12/11, defined as follows

c1fl1 = 0.06, 0.09, 0.12 12/11 = 0.67, 1.00, 1.50 Where: c1 is a column width

11 is column distance in lateral load direction 1 2 is column distance in transverse direction.

Banchik presented his result using c1/11 as the main geometric variables. The results divide into wo distinct groups: one for the interior frame, and another for the exterior frame. The variations of the effective beam width b for an interior frame, which includes interior connections and edge connections with bending perpendicular to the edge, can be represented as

..... . .. .. ..... .. . ... .. . ... .. .... ... .. ... ...... . .... .. .. . ..... .......... ... (1)

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The effective beam width for an exterior frame, which includes corner connections and edge connections with bending parallel to the edge, can be represented as

I,b =c, +- ....... ... .............. ...... .................... .... .............. . .. . ... (2)

6 According to Hwang and Moehle [3], the influence of crack section is accommodated by giving a reduction factor to the element stiffness, which is generally taken by 1/3.

2.3 Improving Model By Using Grid Model For 3D Analysis

Based on the study conducted by Hwang and Moehle [3], the model proposed by banchik is good enough to model structure two dimensionally. However, this model cannot give a comprehensive understanding of behavior of structure. Behavior of structure such as: plate deformation, internal force distribution, influence of structure and loading configuration can be observed only by using three­dimension analysis. Moreover, by using nonlinear procedure in three-dimension analysis, inelastic behavior of structure can be examined. In order to simplify the analysis, including in nonlinear procedure, the grid model is proposed. In grid model, the slab is replaced by arrangement of equivalent beam with certain width.

The questions are how wide the appropriate effective grid width is, and how the number of grid influence the analysis result. To answer these questions, some numerical experimental are conducted. Details of the numerical experimental are explained in the following sections.

3. Description Of Test Model

3.1 Configuration and Properties

The Test slab configuration and geometry are adapted from the experiment conducted by Hwang and Moehle [2]. This test slab is an idealized of a flat-plate floor at an intermediate level of a multistory office building. It has three bays in each direction with the center to center span are 4.6 m and 6.9 m for each direction. Slab thickness is 203 mm and story height is 3.0 m. Gravity loading consists of self­weight and live load. DeSign of structure subjected to lateral load is due to wind.

For experimental purposed the structure is scale to 0,4 from the original that is as follOWS: center to center spans in the two principal directions are 1.8 and 2.7m. Slab thickness is 81 mm, the columns extend 305 mm above the slab and 1220 mm below the slab. The column is pin supported to model inflection point of moment around column midheight. Configuration and dimension of test slab in cm are as shown in figure 2.

< 274.

30.5 cmt I

I I

~}.XJ·~~-·-·O~~·X:·~~-·-·-6·-·- ·-·-·- 122cmIi I

. -.-.-.-.- . -.-~-.-.-.-.-.-.-.~ . -.-.-.-.-.-.- · €L-· I I

i i

· -·-·-·-·-·9·-·-·;;~;:;&·-·24AXli~ I

I 24.4xI2.2 i

Fig 2. Configuration and Dimension of Test Slab

Structure has the following properties: ultimate compressive strength of concrete of 21.8 MPa, Splitting tension of 2.6 MPa, and secant modulus of 17860 Pa. Slab reinforcement was No.2 deformed bars, with the average yield strength 450 MPa.

307

~ I

[ II

II

L '- ­~

3.2 Loadings

The test slab is loaded with a combination of vertical and lateral load. The vertical load consists of self-weight load and live load of 5650 Pa. Lateral loading was applied to north-south and east-west direction separately and applied until the structure collapse. In north-south direction, the lateral load is applied to slab edge at two points, one midway between frame lines a and b, and the other between lines c and d. In east-west direction, the lateral load is applied to slab edge at midway between frame Lines 1 and 2, and Lines 3 and 4, as shown in fig 3.

a b c d l-

2 --

-~~~===r~======~r-===~=r~

I I

----------i::t-------- - --~----------! ! I I-------------j--------------j-------------- -f'f-. ­I I

3 -­ ----------i:Jr-----------r::J---------­I

Fig 3. Lateral Loading Direction in Test Slab

3.3 Modeling structure by Grid Model

For the numerical investigation, four grid models with different grid space and its dimension are made. Column is modeled as a frame element with its section properties not changes for all the four grid models. The illustration of the four grid models can be seen on fig 4 (a, b, c, d):

a b c d a b c d I1 r= -.e­

2 r 1

3 r !I

L ~ -4

2

3

4

(a) (b)

a b c d a b c d ~

2 r

3 J

4 ~l..! L­

r

1­ '-:1

2 r

3 r

4 I -(c) (d)

Fig. 4 Illustration of Grid Models (a) Model 1 ; (b) Model 2; (c) Model 3; (d) Model 4

308

The distance between grids of each grid model is as shown in tables 1.

Table1. G ridSipace

Model Distance Between Column

(1 1)

Grid Space (lg)

N-S E-W N-S E-W 1 182.8 cm 274.3 cm 182.8 cm 274.3 cm 2 182.8 cm 274.3 cm 91.4 cm 137.2 cm 3 182.8 cm 274.3 cm 91.4 cm 68.6 cm 4 182.8 cm 274.3 cm 45.7 cm 68.6 cm

The variation of the effective grid width will be derived empirically from the four grid models above. The effective grid width will be the function of grid space (lg) and distance between column in grid direction (11), or b=f (11, 19).

4. Analysis Method

Numerical analysis was governed by two methods of analysis that is linear analysis and material­nonlinearity analysis. Linear analysis was done to investigate the effective grid width. Nonlinear analyses that count for the material nonlinearity, was governed to observe inelastic behavior of structure after yield occur, how the models meet the experimental result. Behaviors of structure evaluated here are the relation of the load to lateral displacement, vertical deflection of slab, and shear stress distribution around column. The analysis was conducted by using MASTAN 2 program [4,6].

5. Results And Discussion

5.1 Effective Beam Width

Analysis result of effective grid width for each model is present in table 2. The result is divide into two groups that is for interior and exterior grid.

61. 45. 36. 36. 91. 68. 68. 40.

30. 22. 18. 18. 45. 34. 34. 20.

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Result of analysis are plotted in figure 5, in terms of 19l 11. The effective grid width derived by curve fitting .

1 0.9 0.8 0.7 0.6 • Interior GridE' 0.5 • Exterior Grid

.0 0.4 0.3 0.2 0.1

0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

Fig 5 Plot of Effective Grid Width

From the graph above, the variations of the effective beam width b for an interior grid, which includes interior connections and edge connections with bending perpendicular to the edge, can be represented as

Lg/b::= 0,9.Lg.e - 7Ll

Whereas the effective beam width for an exterior grid, which includes comer connections and edge connections with bending parallel to the edge, can be represented as

Lg/ b::= 0,45.Lg.e - 7L1

According to these expressions, the effective grid width in exterior grid is half the value of the interior grid.

5.2. Analysis of Vertical Deflection of slab

Vertical deflection is an important factor in the structure design, especially in service load level. Thereby estimation of deflection that may occur, play an essential rule in design process. In the fig 6 below, the vertical deflection resulted from numerical analysis of the three models is plotted together with the experimental result. Due to the limited number of data, hence the deflection only controlled at the middle of a3~ column line.

According to the analysis result, it is seen (fig 6) that model 4 has the best prediction compared to other models. Model 2 and model 3 seen overestimate in the lateral drift ratio of 1/50. The increasing of grid number in the model will improve prediction of prevailing vertical deflection. Large number of grid will make the stiffness not change immediately. In model 2, due to its few number of grid, yielding in one element cause the global stiffness drop and make the vertical deflection increase very fast. Another result derived from the analysis is the three models perform good prediction until the lateral drift ratio of 1/100.

310

III

25 -l---=+==ExDei- - ­I • Experiment - - • - - Model 2 - -.- model 3 ~------------~.

)( model 4 c:: o 15 ·r---- ----------~--0------~--------- --~-----;:

~ c;: III Q 10 ~------------------------------_,~-.ii . ".

~ ~.......~~.....~; :..~

:: I.)

'I o -'--~---

1/800 1/400 1/200 1/100 1/50

Lateral Drift Ratio

Fig 6. Calculated vertical deflection and experimental result

5.3. Shear in Slab

Effect of lateral loading to shear force in slab is analyzed by grid model. The shear force around column is got from shear force at the end of grid element that connected to column joint. Shear strength is calculated in one side of column where the shear width (bo) is equal to column width(c) added by effective slab height (h). Shear strength is calculated using an empirical formula [1] as follows:

Vc =i* JfC *bo*d .. .... ............................................................... ... (3)

Result of analysis of the shear force in three locations that is B4, 82, A3, to its corresponding lateral drift ratio are shown in figure 7(a, b, c, d)

: ~Veast .............·Vwest .-4-V northI ~Veast --- Vwest II

~Varound ---7f.- V south20.000 1 -6-~ north ~V around

~ 15.000 t K==rc===x===Qj' )t)( IE ~

~ 10.000 I :. 5.000 k~F==;::::::=~~===~~ ~ 0.000 +-,--------.-------r-----=r=-==--- ,

o 0.5 1.5 2

Late ral Drift Ratio(%)

(a)

I ~Veast .........- V north _ ~Varound -=-*== V south

z .~~.======================~i 20:000 E :~ ~~:~ -= : : :: A

• ca 1! 5.000 '"' w • .. f1) 0.000 '. - ,

o 0.5 1.5 2

Late ral Drift Ratio(%)

(c)

z':l 30.000 )(

. "")l*(==,)~E~1*",E-)(}E----ll*'(-><==-l~IC"'- ­!t'l

~ 20.000 t I J10.000 t:t.t+Q=;;:;Q=:::::::::::==t=~,.~ :s~=E= , ~I 0.000 +-,----"-"T------.,......-------,----=-......,

[ :teral Drift RatiO(~;5' 0 O- 2

---------- - ------------------------' (b)

Colum n Joint bo

She a r S tre n 9 th of Slab

B4 163m m 14630 kN B2 163mm 16575kN A3 163mm 16575kN

(d)

Fig 7. (a) Shear Force in Slab at Joint 84; (b) Shear Force in Slab at Joint 82 (c) Shear Force in Slab at Joint A3; (d) Shear Strength of Slab at one column side

311

B4 Joint (fig. 3) is in south edge, whereas B2 joint (fig. 3) is in the middle of plate, and A3 joint (fig. 3) is in West edge. During the analysis, the vertical loading remain constant while the lateral loading increase. It can be seen from fig . 7 that compared to the shear strength, until 2% lateral drift ratio the calculated shear force still below its shear strength capacity. These results agree with the experimental by Hwang and Moehle [2] where the shear failure occurs at 4%Jateral drift ratio.

6. Conclusion

Based on the result of numerical analysis, the following conclusions are 1. The result of numerical simulation shows that the three-dimension of grid model can be used

to model behavior of flat-plate structures. 2. According to the analysis result, there are relation among the effective grid width, grid space,

and grid length from column to column. This analysis derives the variations of effective grid width as follows:

b =0,9.Lg.e -L~

L1 for the interior grid, and Lg/

b = 0,45.Lg.e - 711 for the exterior,

Where b is the effective grid width, Lg is grid space and L 1 is grid length from column to column.

3. Analysis result shows that until lateral drift ratio of 1 %, the three models (model 2, model 3, and model 4) still perform good prediction. For the elastic analysis purposed, model 2 can be used. Whereas for the inelastic analysis, increasing of grid number is suggested.

4. In this analysis, shear failure is avoided by designing the shear strength capacity almost 3 times of shear from gravitational load. It can be seen that by this way the structure perform good ductility.

5. Analysis shows that grid model able to reckon the shear force and meet the experimental result.

Acknowledgements This research is carried out with the financial support of the Directorate General of Higher

Education of the Republic of Indonesia through Technological and Professional Skills Development Sector Projects (TPSDP), ADB Loan No 1792-INO, under contract No. 568ITPSDPITS/KlXIII03. This paper is prepared by the financial aid of Department of Civil Engineering Diponegoro University.

References [1] Ghosh, S.K., Fanella, D.A., Rabbat, B.G, Notes on ACI318-95, pp. 20.1 - 20.11 , Portlad Cement

Association, 1996 [2] Hwang, S.J., Moehle, J.P., Vertical and Lateral Load Tests of Nine Panel Flat-Plate Frame, ACI

Structural journal, vol 97, no.1, pp. 193-202, American Concrete Institute, 2000. [3] Hwang, S.J., Moehle, J.P., Models for Laterally Loaded Slab-Column Frames, ACI Structural

journal, vol 97, no.2, pp. 345-352, American Concrete Institute, 2000. [4] McGuire, W., Gallagher, RH., Ziemian, RD., Matrix Structural Analysis, t ld Edition, pp. 216-325,

John Willey & Sons Inc, 2000 [5] Regan, P.E. , Behavior of Reinforced Concrete Flat Slab, pp. 11-19, Construction Industry

Research and Information Association, 1981 [6J Ziemian, RD., Mc Guire, W., MASTAN 2, Version 2, John Willey & Sons Inc, 2002.

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