dsn optimization hussain

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Industrial ApplicationsDOI 10.1007/s00158-004-0470-4Struct Multidisc Optim 30, 7688 (2005)

Design optimization of simply supported concrete slabs by finiteelement modelling

K.M.A. Hossain, O.O. Olufemi

Abstract This paper outlines the finite element pre-diction process for the development of charts for accu-rate peak load determination of simply supported, rein-forced concrete slabs under uniformly distributed load-ing. Through a series of parametric studies using a sim-ple concrete model, the simulation of tests on four sim-ply supported slabs was used as a basis for establish-ing a set of optimum parameter values and computa-tional conditions, which guarantees acceptable solution.The reliability of the established parameter values forprediction purposes was verified by the direct simula-tion of 11 other slabs. Following the successful reliabilitycheck, the finite element model was used for analysing270 computer model slabs, from which charts were de-veloped. These charts serve for quick and reliable peakload determination of arbitrary simply supported slabs.

A comparative study of the direct finite element andchart predictions, with values from analytical and de-sign methods, reveals the superiority of the charts overthe latter methods, with accuracy comparable to thatof the optimised finite element model. The chart predic-tion is noted to be accurate to within 4% of test results.A strategy for displacement determination is also estab-lished, with the same degree of success and the paperdiscusses possible practical applications of the developedfinite element system.

Key words mathematical modelling, concrete struc-ture, structure and design

Received: 27 August 2003Revised manuscript received: 6 July 2004Published online: 26 January 2005 Springer-Verlag 2005

K.M.A. Hossain1,u, O.O. Olufemi2

1 Department of Civil Engineering, Ryerson University, 350Victoria St., Toronto, ON, M5B 2K3, Canadae-mail: [email protected] Department of Engineering, University of Aberdeen, Ab-erdeen, AB24 3UE, UK

e-mail: [email protected]

1

Introduction

The use of the finite element method for the stress analy-sis of structures whose materials range from glass, metals,soils to reinforced concrete, is dependent on the choice ofappropriate material models to characterise observed be-haviour, (accounting for material nonlinearity), and onthe inclusion of geometric nonlinearity, accounting forlarge deformation. The increased sophistication of ma-terial models being developed demands that these finiteelement programs be used, not only for the analysis ofexisting structures, but also for predicting the behaviourof proposed ones, especially in the peak load and corres-ponding deflection determination. Work along this linehas been done for reinforced concrete slabs (Famiyesin

et al. 1995; Famiyesin and Hossain 1998a,b).Other methods of peak load determination of rein-

forced concrete slabs include analytical methods such asthe yield line method, and design based methods. Thesemethods have a tendency to grossly under-predict thepeak load of reinforced concrete slabs, mainly becauseof the nonlinear membrane action mobilised at the sup-port due to in-plane forces, which are not accounted forin these methods (Famiyesin and Hossain 1998a,b). Theeffect of membrane action has been recognized since thefirst half of the 20th century. However, it wasnt until 1955when Ockleston (1955) published the results from load

tests on a reinforced concrete building in South Africathat researchers became fully aware of its possible ben-efits. Ockleston (1955) conducted tests on interior floorslabs in the building and found the ultimate load was sig-nificantly greater than both design based methods andyield line predictions. He attributed this enhancement tocompressive membrane action.

Many researchers have looked into compressive mem-brane action since 1955. Some of the more notable workwas done by Park in the 1960s (Park 1964a,b, 1965),while Braestrup (1980) summarizes much of the workdone in this area. Experimental studies by Powell (1956),Wood (1961), Park (1964b), Kirkpatrick et al. (1984) and

Rankin et al. (1991) have shown that slabs in buildings


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and bridge decks, which are restrained against lateral dis-placements at the edges, have ultimate strengths far inexcess of those predicted by analytical methods basedon yield line theory. The increase in strength has beenattributed to membrane action, which is due to the in-plane forces developed at the supports. The two types ofmembrane actions that could be identified from a typical

load-deflection curve are the compressive membrane ac-tion at small deflections and tensile membrane at largedeflections.

Even for simply supported slabs, (where membraneforces should be relatively low), results from tests showedhigher strengths than those obtained from yield line the-ory (Taylor et al. 1996; Brotchie and Holley 1971; Roz-vany 1976). Simply supported square reinforced concreteslabs with isotropic (uniformly distributed) as well aswith optimised reinforcements (with various types of re-inforcement layouts) were tested experimentally by Roz-vany (1967) and his research group at Monash Univer-sity (Rozvany and Adidam 1971; Rozvany and Charrett1971). Testing of slabs was facilitated by an efficient ex-perimental method which involved bolting a heavy frameto a 500-ton point load capacity reaction floor. The slabswere subjected to upward uniformly distributed load bypumping compressed air into the air bags placed under-neath the slabs. Using this layout, the tensile surfaceof the slabs was observed continuously for the develop-ment of cracks. Other tests carried out at Monash Uni-versity were concentrated on simply supported circularslabs including circular footing slabs, square slabs withclamped condition as well as supported on three sidesand slabs with rectangular openings. For the simply sup-

ported condition, it was found that (a) slabs with conven-tional isotropic reinforcement required about 80% moresteel than those with an optimum reinforcement for thesame ultimate load, (b) after the development of the col-lapse mechanism, optimised slabs exhibited membraneaction to a lesser extent than isotropic slab and (c) theultimate load of both isotopic and optimised slabs werehigher than the theoretical load predicted by yield linetheory.

Over the years, the theories developed by researchershave largely been based on plastic flow theories and haverequired gross assumptions to be made (e.g., assuming

the concrete to be a rigid-plastic material). The equa-tions derived by these methods are generally unsuitablefor design engineers to use and as a result, the benefitsof membrane action are usually not taken into accountin design or assessment methods (Alan et al. 2001; Das2001). While the existence of membrane action is com-monly acknowledged, its use in practical situations is hin-dered by a lack of knowledge of the stiffness of the ho-rizontal restraints and how this effects the developmentof membrane forces. This surround stiffness is critical tothe development of membrane action. Eyre (1997) hasdeveloped a method to directly assess the strength ofreinforced concrete slabs under membrane action. The

method requires knowledge of the surround stiffness that

the slab is exposed to and determines a safe load that isalways less than the ultimate load.

The finite element method has been used to modelthe membrane action in reinforced concrete slabs (Huanget al. 2003a,b; Salami 1994; Famiyesin and Hossain1998a,b). Huang et al. (2003a,b) used the non-linear lay-ered finite element procedure to model the membrane

action of concrete slabs in composite buildings under fireconditions. This research was concentrated on solid re-inforced concrete slabs with simply supported edges atambient temperature under uniform loading. This wasfollowed by a simulation of a full-scale fire test on a solidreinforced concrete slab floor at the Cardington Labo-ratory in the UK. It is evident that the proposed modelcan predict structural behaviour of reinforced concreteslabs and their influence on composite steel-framed build-ings in fire with good accuracy. In all cases the devel-opment of membrane actions is demonstrated, and thestructural behaviour differs compared with the geometri-cally linear case. These studies provide evidence that atvery high temperatures the floor slab becomes the mainload-bearing element, and the floor loads above the firecompartment are carried largely by tensile membraneforces developed mainly in the steel anti-cracking mesh orreinforcing bars.

The finite element method is capable of analysingmembrane action in slabs due to its ability to incorpo-rate both geometric and material nonlinearities in its for-mulation as opposed to yield line theory or code baseddesign procedures. Implicit in finite element materialmodels are the model parameters whose values controlthe simulation process and hence the ability to use the

model for prediction purposes. Optimisation of the modelparameters for a class of structural problem is there-fore essential in the use of the finite element methodfor accurate peak load, (and displacement), predictionwithin that class. Salami (1994) has applied this prin-ciple for fully clamped slabs and used the subsequentpredictions to develop equations for strength determin-ation for similar slabs. Work done by Famiyesin and Hos-sain (1998a,b) has focused on slabs with variable edgerestraints. The case of simply supported reinforced con-crete slabs under uniformly distributed loading is focusedon in this paper.

For the class of simply supported concrete slabs, testscarried out by Taylor et al. (1996) on four slabs are usedas a basis for the basic simulation process and parametricstudies, to identify a set of conditions and model parame-ter values. The reliability of the identified values and com-putational conditions for prediction purposes is furtherverified by the direct simulation of 11 other simply sup-ported slabs tested by Taylor et al. (1996) and Brotchieand Holley (1971). Following this, a series numerical pre-dictions are carried out on a total of 270 numericalmodel slabs, varying geometric and strength properties.The data generated are used as a basis for developingcharts for arbitrary simply supported slabs, which can

be used for the determination of the peak strength and


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Fig. 2 Tension stiffening in cracked concrete

tensile strength ft. A smeared representation for crackedconcrete is assumed, where cracks are distributed acrossa region of the finite element. As the overall structural be-haviour is of primary concern in this study and the size ofelements used for analysis are relatively large, the prob-lem of strain localisation that such assumption may causein a fine mesh situation does not arise. To take account ofthe effect of tension stiffening due to the presence of re-inforcement, a gradual release of the concrete stress com-ponent normal to the cracked plane and shown in Fig. 2,is adopted. Unloading and reloading of cracked concreteis assumed to be linear, with a fictitious elasticity modu-lus Ei defined as:

Ei = f

t (1 i/m) /i t i m , (3)

where and m, are tension stiffening parameters and iis the maximum value of tensile strain at the point consid-ered. The normal stress 1 is obtained by the expressions:

1 = f

t (1 1/m) , t 1 m or by:

1 = i1/i if 1 < i , (4)

where 1 is the current tensile strain in material direc-tion 1. The value offt is taken as the modulus of rupture,fr of the concrete and can be related to the uniaxial com-pression strength (in MPa) by : fr = 0.62(f

c)0.5. An ap-

propriate value of cracked shear modulus based on the ap-

Table 1 Details of Taylors simply supported slabs (Taylor et al. 1996)

Slab Dimensions Ly/Lx Lx/h fy# fc

# Design Test load

no. LyLxh (mm) load, kN N/mm2(kN)

S1 1830183050.8 1.00 36 375.5 27.9 80 0.043 (144)

S4 375.5 25.9 80 0.042 (141)

S5 420.3 26.8 77 0.036 (120)

S10 1830183076.2 1.00 24 375.5 26.35 80 0.038 (128)

Mean value of dry and wet strengths of concrete, #Unit in N/mm2,

Design based on:

Yield line theory,

Strip method

proach of Cedolin and Deipoli (1977), which is a functionof the current tensile strain, is employed to account for ag-gregate interlock and dowel action in the smeared crackmodel. This allows for shear transfer across the rough sur-face of the cracked concrete.

2.3Modelling of steel behaviour

The reinforcement bars are modelled as layers of equiva-lent thickness (Owen and Figueiras 1984) and followsan elastoplastic material behaviour, with the Von Misesyield criterion defining the yield surface (Owen and Hin-ton 1980).

3

Basic finite element simulations

Four of the ten slabs tested by Taylor et al. (1996) areused as basis for the preliminary finite element simulationand subsequent parametric studies. All the slabs weresimply supported along their edges and a uniformly dis-tributed load was applied on the surface. The successfulsimulation of the behaviour of these slabs in the para-metric studies, would lead to the identification of a setof optimised parameter values and computational con-ditions, which would guarantees reliable predictions insubsequent analyses.

3.1Slab details

Taylor et al. (1996) carried out tests on ten two-way span-ning, square, simply supported reinforced concrete slabs,under uniformly distributed loading. All slabs were de-signed for the same ultimate load of 80 kN on the as-sumption of a uniformly distributed load, although thearrangement of reinforcements was varied. The methodof design used for proportioning the reinforcement waseither Johansens yield-line theory (Johansen 1962) orHillerborgs strip method (Hillerborg 1960). Four of the


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Fig. 3 Reinforcement details in Taylors slabs (Taylor et al. 1996)

slabs, (S1, S4, S5 and S10), have been selected for thebasic finite element simulations and parametric studies.Details of the slabs are summarised in Table 1 and thereinforcement patterns are as shown in Fig. 3.

3.2

Finite element idealisation

The slabs are idealised with the 3D degenerated, 9-nodedHeterosis elements, with each node having five degrees offreedom. A symmetric quarter of the slab is discretizedinto nine elements as shown in Fig. 4. For slabs with irreg-ular reinforcement spacing, the averaged spacing is usedto calculate the idealised amount of reinforcement in theelements. The concrete depth is discretized into 10 layers

of equal thickness as shown in Fig. 5.

3.3

Load control strategy

The finite element simulations are carried out based onthe load control (LC) strategy. The suitability of LCsimulation has been discussed (Hossain and Famiyesin1998a,b). For the LC simulation, uniformly distributedloads are directly applied to the slabs in increments.A typical finite element load-deflection response underLC is superimposed to that from a typical experiment

in Fig. 6. In most of the cases, the peak load from the

finite element simulation can be taken as the load as-sociated with a sudden large displacement, although insome cases a negative pivot may indicate the ultimateload. A relationship can be established between the finite

element peak deflection (dfe) and experimental peak de-flection (dexp), via a displacement factor (df) such that;dexp = dfedf.

Fig. 4 Typical finite element mesh


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Fig. 5 Dicretization of steel and concrete layers

Fig. 6 Qualitative load-deflection responses

3.4

Parametric studies

A series of comprehensive parametric studies have beencarried out to establish the sensitivity of various param-eters and computational conditions involved in the fi-nite element modelling. These are the convergence cri-teria, integration rules, non-linear solution techniques,

ultimate compressive strain of concrete (cu), tensionstiffening parameters ( and m), modulus of elastic-ity of concrete (Ec) and the elastoplastic modulus ofsteel (Es).

The single parameter to which the slab simulation wasmost sensitive, and can be adjusted to achieve accuratesimulation while the other parameters are fixed, is theultimate concrete crushing strain (cu). The other com-putational conditions established from the parametricstudies and are adopted for the simulation are the selec-tive integration scheme (SI), modified NewtonRaphsonmethod (MNR), and convergence criterion based on dis-

placement norm (DN). Actual test values were chosenfor material properties such as the Youngs moduli forsteel and concrete, (Es & Ec), cylinder strength of con-crete, fc and the yield strength of steel fy, while thevalues chosen for other parameters are, for tension stiffen-ing, ( : 0.5, m : 0.002) and the elastoplastic modulus forsteel, Es is taken as Es/15, where Es is the steel Youngsmodulus. The Poisson ratios for concrete and steel arechosen as 0.18 and 0.25, respectively. The uniaxial ten-sile strength of concrete is assumed to be one tenth of thecompressive strength.

Typical load-deflection responses from the finite elem-ent simulations, for various values of cu are super-

imposed on those from experiments as shown in

Fig. 7 (a) Effect ofcu on the load-deflection response of slabS1 (Taylor et al. 1996). (b) Effect ofcu on the load-deflectionresponse of slab S4 (Taylor et al. 1996). (c) Effect ofcu on the

load-deflection response of slab S5 (Taylor et al. 1996)


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Table 2 Comparison of experimental and numerical simulations (Taylors Slabs)

Experiments Finite element LC simulationSlab Pred. load Pred. defl. Pred. load Ratio Pred. defl. Ratio

no. N/mm2 mm N/mm2 mm

S1 0.043 77.32 0.040 1.07 65.70 1.18S4 0.042 64.00 0.044 0.955 66.10 0.97

S5 0.036 64.00 0.036 1.00 64.00 0.93S10 0.0385 82.55 0.050 0.77 75.10 1.10

Average ratio values 0.95 1.05

cu = 0.005 df = 1.05

Experimental to finite element values

Fig. 7ac. From these figures it can be concluded that thebehaviour of the slabs can be closely simulated by usingan cu value of 0.005. The results obtained by using thiscu value in the simulation of the four slabs, are comparedwith experimental results and summarised in Table 2.From the table it is noted that the ratios of experimen-tal to finite element predicted loads and deflections areaveraged at 0.95 and 1.05, respectively, showing a rea-sonably good agreement. The consistency of the finiteelement results, (in comparison with experiment), ob-tained by the use of the identified parameter values andcomputational conditions, in the simulation of the fourslabs, suggests that predictions carried out for similarlysimply supported slabs with the established parameterswill yield reliable and accurate results. This should bethe case for both the peak load and peak displacementpredictions.

Table 3 Details of Taylors (Taylor et al. 1996) simply supported slabs

Slab Dimensions Ly/Lx Lx/h fy# fc

# Test load Design

no. LyLxh (mm) N/mm2(kN) load, kN

S2 1830183050.8 1.00 36 375.5 29.4 0.040 (134) 80

S6 420.3 27.8 0.040 (133) 77

S7 1830183044.45 1.00 41 375.5 30.84 0.039 (131) 80

S8 375.5 30.62 0.039 (131) 80

S9 1830183076.2 1.00 24 375.5 25.25 0.038 (128) 80

Mean value of dry and wet strengths of concrete, #Unit N/mm2,Design based on: Yield line theory, Strip method

Table 4 Slabs tested by Brotchie and Holley (1971) (Aspect ratio Ly/Lx=1.0)

Slab Dimensions Lx/h d % of steel fy f

c

no. LyLxh (mm) mm x y N/mm2

Slab 8 38138119.05 20 14.22 1.00 1.00 413.4 30.41Slab 9 20 14.22 3.00 3.00 413.4 27.86Slab 12 38138138.1 10 31 1.00 1.00 379.0 29.10Slab 15 10 31 3.00 3.00 379.0 23.93Slab 19 38138176.2 5 65.8 1.00 1.00 365.0 20.15

Slab 23 5 65.8 3.00 3.00 365.0 28.41

4

Direct simulation of previous tests

To test the reliability of the parameter values and compu-tational conditions established from the basic finite elem-ent simulation for prediction purposes, the direct finiteelement simulation of eleven other simply supported slabsare carried out, five slabs tested by Taylor et al. (1996)and six slabs tested by Brotchie and Holley (1971). Thisdirect simulation process is based on load control strat-egy and adopts an cu value of 0.005, as identified fromthe basic simulation of four of Taylors slabs, with fixedconditions and parameter values indicated earlier.

The details of the slabs are presented in Tables 3 & 4.The five slabs of Taylor et al. (1996) (Table 3) have vari-able spacing as shown in Fig. 8, while those of Brotchieand Holley (1971) have uniform reinforcement (Table 4).


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Fig. 8 Layout of reinforcement in Taylors slabs (Taylor et al. 1996)

The finite element idealisation is similar to that of basicsimulation shown in Fig. 4.

4.1

Comparison of peak loads

Table 5 compares the peak loads obtained from the ex-periments with those obtained from direct finite elementpredictions. The average values of the ratios of experi-mental to finite element predicted loads, are found to be

0.91 for Taylor et al. (1996), and 0.92 for Brotchie andHolley (1971), thus showing a reasonably good agreement

Table 5 Comparison of experimental and direct FE predicted loads

Slab no. Experimental load; Direct finite element Ratio of experimental

N/mm2 predicted load; N/mm2 to predicted load

Taylor et al. (1996)

S2 0.040 0.046 0.87

S6 0.040 0.038 1.05

S7 0.039 0.042 0.93

S8 0.039 0.042 0.93

S9 0.0385 0.050 0.78

Mean: 0.91

Brotchie and Holley (1971)

Slab 8 0.12 0.14 0.86

Slab 9 0.289 0.28 1.03

Slab 12 0.503 0.58 0.87

Slab 15 1.07 1.08 0.99

Slab 19 2.03 2.24 0.91

Slab 23 4.13 4.88 0.85

Mean: 0.92

with the 0.95 ratio value established from the basic simu-lation.

4.2

Comparison of central deflections

The experimental and direct finite element predictedvalues of the central deflections for typical slabs are sum-marised in Table 6. The average ratio of the experimentalto predicted deflections is found to be 1.06, which is

consistent with the value identified from the basic simu-lations in Table 2 (averaging 1.05 = df).


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Table 6 Comparison of experimental and predicted deflections

Slab no. Experimental FE predicted Ratio of experimental

Taylor et al. (1996) central deflection deflection to predicted

(mm) (mm) deflection (df)

S2 51.00 55.55 0.92

S6 81.00 65.20 1.24

S7 76.20 69.40 1.10S8 76.20 65.80 1.16

S9 83.8 95.00 0.88

Mean: 1.06

5

Development of charts by finite element predictions

Establishing a finite element system that generates reli-able solutions for a class of problem has many implica-tions. A possible practical application of such a systemis the development of load and displacement charts, fromthe finite element analysis of many computer modelslabs, varying geometric and strength properties. Suchcharts may serve as quick and accurate strength and dis-placement determination of arbitrary slabs, without theneed for extensive finite element analysis when such in-formation is required, or the physical testing of sample

Table 7 Parameters of theoretical computer-model slabs

Aspect ratio Breadth to depth fc fy % of

Ly/Lx ratio Lx/h N/mm2 N/mm2 steel,

2.0 15 25 250 0.21.5 20 40 460 0.51.0 25 60 0.1

3030

Fig. 9 Comparison of loads from different predictions

slabs which will be time consuming (noting that the slabrequires at least 28 days before testing). The charts mayalso be used for design purposes.

Table 7 shows a summary of the geometric and mate-rial parameters considered in the prediction process, usingvarying aspect ratios, breadth to depth ratios, concretestrengths, reinforcement ratios (in %) and strengths, togenerate the database from which the charts are de-veloped. This involves a total of 270 computer modelslabs. Each slab is assumed to be isotropically reinforcedat the bottom (the percentages of steel in x and y direc-tions are equal) and cover to reinforcement is assumedto be 20% of the slab thickness. The load is uniformlydistributed over the surface of the slabs and applied inincrements and the percentage of steel is calculated onthe basis of the mean effective depth of the slab. Thefinite element discretization is similar to those describedin the basic and direct finite element simulations process.

5.1

Design charts

From the results of the finite element predictions, designcharts, (with a polynomial curve fit of 3rd degree), have

Fig. 10 Design charts showing best fit equations


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Table 8 Comparison of chart loads with experiments and direct FE predictions

Slabs Expt. load Direct FE Chart Expt/direct Expt/Chart

N/mm2 prediction pred. loads FE ratio ratio

N/mm2 N/mm2

Brotchie & Holley (1971)

Slab 8 0.12 0.14 0.138 0.8 0.86Slab 12 0.503 0.58 0.54 0.867 0.93

Taylor et al. (1996)

S1 0.043 0.04 0.0395 1.075 1.09S2 0.040 0.046 0.0365 0.87 1.096S4 0.042 0.044 0.041 0.955 1.02S5 0.036 0.036 0.0365 1.0 0.99S6 0.04 0.038 0.04 1.053 1.00S7 0.039 0.042 0.036 0.93 1.08S8 0.039 0.042 0.035 0.93 1.11S9 0.0385 0.050 0.0343 0.77 1.12S10 0.0385 0.050 0.0346 0.77 1.11

Mean 0.911 1.04

(0.942)#

#Average ratio when values with () are not included

been produced and a typical one is as shown in Fig. 9.This chart has been established for a particular concretecylinder strength (fc), steel yield strength (fy) and as-pect ratio of the slab, and for percentage of reinforcementvarying from 0.2 to 1.0%. The design loads are expressedas a function of Breadth/depth ratios (varying from 15to 35). The estimation of peak load for the slab havinga typical breadth to depth ratios of 20 and percentage

reinforcement of 1.0% is shown by lines with arrows inFig. 9. For a slab with percentage reinforcement of 0.75%,a linear interpolation between 0.5 and 1.0% should beadopted. The best fit chart equations can also be used forprediction purposes (see Fig. 10).

5.2

Comparative study

The developed charts are used to predict peak load valuesfor representative slabs from Taylor et al. (1996) and

Brotchie and Holley (1971) whose reinforcement ratiosfall within the range of computer model predictions car-ried out. The chart-predicted values are compared to re-sults obtained from experiments and direct finite elementsimulation, and is as shown in Table 8. For slabs of thesame aspect ratio, but whose breadth to depth ratio isout of range in the prediction process (e.g., Slabs S7 andS8 from Taylors tests having a breadth to depth ratioof 41, and Slab 12 of Brotchie and Holley having a valueof 10), the equations of best fit identified for the curves(e.g., Fig. 10), could be used for such slabs where directchart predictions are not possible.

The prediction of strength for Slab 8 of Brotchie and

Holley (1971) using the design chart is shown in Fig. 9.

The value of 0.157N/mm2 obtained directly from thechart is corrected by multiplying it with a factor (i.e.,chart-predicted load fy,test/fy,chart), taking into ac-count the steel strength properties used in the chartdevelopment (460 N/mm2) and the actual reinforcementyield strength of test Slab 8 (which is 413.4 N/mm2,see Table 4). The corrected chart-predicted load is0.138N/mm2, and is compared with those from direct

finite element prediction and experiments, in Fig. 9.The average value of the ratios of the experimental to

direct finite element prediction is found from Table 8 tobe 0.911, or 0.942, (when unreliable results are not takeninto consideration), which is consistent with earlier re-sults. There is therefore a general tendency for the directfinite element system to over-predict the peak load (aver-age of 6% error). The experimental to chart-predictedload ratios on the other hand, average at 1.04, whichshows a general tendency to under predict the peak load(4% error). On balance therefore, the performance of thechart is comparable, if not better than the direct finite

element system, from which it was developed.

6

Predictions based on analytical and design approach

Peak load predictions from the yield line method of an-alysis and design calculations based on BS8110 (1985) arecompared in this section with experimental results, for all15 slabs considered in this paper. As most of the slabstested by Brotchie and Holley (1971) have either theirwidth to depth ratios or reinforcement ratios outside therange of values specified in Table 7 used for chart devel-

opment, the chart predicted load values that could be ob-


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Table 9 Comparison of yield line and BS8110-based load predictions with experiments

Slabs Expt. Chart Yield BS8110 Ratio of Ratio of Ratio of loads predicted line based expt. to expt. to yield line

(N/mm2) loads# loads# predicted yield BS8110 to BS8110

loads# line

Brotchie and Holley (1971)

Slab 8 0.12 0.138 0.127 0.0358 0.945 3.32 3.548Slab 9 0.289 0.3057 0.0788 0.945 3.67 3.879Slab 12 0.503 0.54 0.556 0.157 0.905 3.204 3.541Slab 15 1.07 1.30 0.33 0.823 3.242 3.939Slab 19 2.03 2.3335 0.662 0.87 3.067 3.525Slab 23 4.13 6.05 1.596 0.683 2.59 3.791

Taylor etal. (1996)

S1 0.043 0.03944 0.0238 0.008 1.807 5.375 2.975S2 0.040 0.0365 0.0239 0.006725 1.674 5.948 3.554S4 0.042 0.041 0.0239 0.00643 1.757 6.532 3.717S5 0.036 0.04224 0.023 0.010 1.565 3.6 2.3S6 0.04 0.04 0.023 0.00951 1.739 4.206 2.419

S7 0.039 0.036 0.0239 0.00814 1.632 4.791 2.936S8 0.039 0.035 0.0239 0.007365 1.632 5.295 3.245S9 0.038 0.0343 0.0239 0.00766 1.59 4.961 3.120S10 0.038 0.0346 0.0239 0.0074 1.59 5.135 3.230

Average: 1.28 4.33 3.315

#Loads in (N/mm2)Width to depth ratio or reinforcement ratio outside the values used for chart predictions(Slabs 9, 15, 23: 3% reinforcement ratio; slabs 19 & 23: width/depth ratio = 5)

tained for only two of them is considered here. The directfinite element predicted values for all Brotchie and Holley

(1971) slabs have been shown in Table 5. All slabs testedby Taylor et al. (1996) are within the specified range, orclose enough for results to be extrapolated.

The yield line analysis is based on the use of a momentper unit width (m), calculated from the equation first pro-posed by Whitney (1937) and used in Leet et al. (1996),conforming to 1995 ACI code. This is expressed as:

m = fyd2(10.59fy/f

c) (5)

where is the reinforcement ratio, fy is the steel yieldstrength and fc is the concrete uniaxial compressive

strength. In the BS8110-based calculations, the materialstrengths are appropriately factored by the relevant par-tial factors of safety and the calculated factored loads arelisted as the predicted peak load which is used for com-parison. The peak load values obtained from the differentmethods are listed in Table 9, which also shows the ra-tios between the experimental values and values obtainedby yield line and design methods. The ratios of the ex-perimental to yield line calculated loads vary from 0.683to 1.807, averaging at 1.28. For Brotchies slabs, wherethe width to depth ratio is low (ranging from 5 to 20),the yield line method over-predicts the load, while forTaylors slabs, with higher width to depth ratios (rang-

ing from 24 to 41), the method generally under-predicts

the peak loads. As would be expected, the predictionscarried out using the yield line method for simply sup-

ported slabs have lower ratios of experiment to predictedloads, compared to slabs with fully restrained supportconditions, where ratio values as high as 8 have beenrecorded (Famiyesin and Hossain 1998b). This is becausethe in-plane forces causing membrane action is not as sig-nificant in simply supported situation in comparison to

Fig. 11 Comparison of FE chart with yield line method, for

typical values


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Fig. 12 Comparison of FE chart with BS8110-based calcula-tions

Fig. 13 Comparison of chart, yield line and BS8110-basedcalculations for typical values

fully clamped condition and membrane action is not ac-counted for by the yield line method.

Calculations with BS8110-based method results ina general under-prediction of the peak loads, with ra-tios of experiment to design loads varying from 2.59 to6.532, and averaging at 4.33. Predictions based on dir-

ect finite element analysis and the developed charts havebeen shown in Tables 2, 5 and 8, to lead to a more accu-rate peak strength determination. Figures 1113 comparethe performance of the prediction methods, for typicalvalues of material strengths, reinforcement ratios and as-pect ratio.

7

Conclusions

A process of model parameter optimisation has beenundertaken, in which previous test results of simply

supported concrete slabs under uniformly distributed

loading, are used as a basis for finite element simula-tion and parametric study. A set of optimal computa-tional conditions and parameter values were obtainedfrom the study, which was further verified for predic-tion purposes, by the direct finite element analysis ofother simply supported slabs. Results show a good levelof accuracy. The finite element system thus established

was used to carry out peak load (and correspondingdisplacement) predictions, by the analysis of a totalof 270 computer model slabs, with varying geometricand material-strength values. Charts and equations forstrength determination were developed from the predic-tions, whose accuracy was compared with predictionsfrom the yield line method and BS8110-based design cal-culations. From the tests considered, the developed chartsproved to be more accurate than the analytical and de-sign methods of prediction, and are noted to be at leastas accurate as the direct finite element system from whichit was developed. The charts though (within its rangeof applicability), have the advantage of providing easyand quick peak load determination for proposed slabs,before actual construction takes place. The use of finiteelement simulation every time such information is re-quired can be time consuming and the physical testingof sample slabs is also not a viable option, as the slabswill be required to have achieved 28-day strength beforetesting.

The charts may also be used for design purposes (withan appropriate choice of factor of safety). Design based onBS8110 adopts partial factors of safety which to a largeextent limit the structural performance under serviceload to within the elastic range. An average ratio value

of 4.33 of experimental to BS8110-based loads has beenrecorded in Table 9, showing that the actual peak loadsare far in excess of those calculated from design methods.The use of a safety factor of, say 2 on the developedcharts, will result in a safe design and optimise the mate-rials used for construction, thus saving on cost.

A deflection factor df has also been established, re-lating the experimental to finite element predicted dis-placements. This has been demonstrated (Table 6) to givedisplacement values with an acceptable level of accuracy,and can thus be used for reliably predicting displacementsat peak load. As with the load charts, the processed dis-

placement values (i.e., finite element predicted displace-ments multiplied by the deflection factor), can be usedto develop charts for displacement determination. Use ofsuch charts and the corresponding load charts, with ap-propriate safety factors for design purposes, could provideanother basis for ultimate and serviceability limit statesdesign approach.

Apart from the development of charts, another prac-tical application of achieving reliable finite element pre-diction could be in its use to generate a database of peakloads and displacements, which may be used for develop-ing a computer-aided design package or the developmentof a knowledge based system. Application to neural net-

works is also a possibility.


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