Indian Journal of Engineering & Materi als Sciences Vo l. 9, April 2002, pp. 109- 11 4

Prediction of strength and workability of high performance concrete composites using artificial neural networks

S Rajasekaran & R Amalraj

Department of Civil Engineerin g. PSG Co llege of Technology. Coimbatore 64 1 004. Ind ia

Received 3 1 May 2001; accepled 4 l alll/w y 2002

A model based on an Art ific ial Neura l Network for predicting the compress ive strength and workabil ity for various mixes of High Performance Concrete (HPC) from M 80 to M 150 is presented. It is felt th at the conventional methods or design or concrete mixes may not be suitable for HPC because, w ith the addition of sili ca fume, more water content is needed that can be eliminated by using high range water reducing admi xtures for ach iev ing the same workabi lity . Further. the compress ive strength and workability properties or HPC mixes are greatly influenced by several parameters, viz. fine aggregatc. coarse aggregate, water-cement ratios, and percentage of silicafume and super-plasticizer. Consequentl y. develop ing a standard mi x design procedure for HPC mixes requires an ex tensive understanding of relati on between these parameters and the propert ies of the result ing mi x. Developing an empirical formula for pred icting the compressi ve stre ngth and workab ilit y of HPC is ex tremely difficult becau se of non-linear interacti on among the vari ous parameters.

Hi gh Performance Concrete (HPC) consists of all ingredi ents of Conventional Cement Concrete (CCC) with chemica l admi xtures as superpl asti cisers and mineral admi xtures like fl yash and sili ca fume. Super­pl asticisers and mineral ad mi xtures improve the utility of HPC. The perform ance of cement concrete can be improved fo r both strength and durabi I i ty considering impermeabili ty charac teri stics, which is achieved by adopting lower water-cement rati o and using pozzo lanic admi xtures such as fl yash and sili ca fum e. It shoul d be noted that the HPC has a very hi gh degree of du rab ility because of very low permeability and stronger-denser transiti on zone between aggregate and cement paste in concrete. Though, studies on HPC have been carri ed out ex tensively by researchers 1-.1 , there is no standard procedure for the design of HPC mi xes . It is felt that the conventi onal methods of des ign of concrete mi xes may not be suitabl e for HPC, because with the additi on of silica fum e, more water content is needed that can be eliminated by using hi gh range water reducing admi xtures fo r achiev ing the same workability. Hence, a new water-cement rati o law has to be es tabli shed fo r HPC. Further, the compress ive strength and workability properties of HPC mi xes are greatl y influenced by several parameters vi z., fine aggregate, coarse aggregate, water-cement ratios, and percentages of sili ca fume and super-plasti cizers. Consequentl y, deve loping a standard mi x design procedure for HPC mi xes requires an extensive

understanding of the relati on between these parameters and the properti es of the resul ting mix . Developing empirical and semi -empirical fo rmul ae for predi cting the strength and workabi Ii ty properties of HPC mi xes is ex tremely di fficult because of non­linear interaction among the vari ous parameters. The material s used for HPC are cement-43 grade OPC, fin e aggregate-naturally ava il abl e sand , coarse aggregate-locally available blue grani te, minera l admi xtures-silica fume imported fro m Australia, chemi cal admi xture-superplasti ciser (Co nplas t SP337) and potable water.

In thi s paper, it is proposed to use a new computer paradigm, called Artifi cial eural Network (ANN) in which 'Back-propagati on architecture' is used to achi eve the above-menti oned goa l. The ANN model will be of great help to the designer to est imate the properti es of the HPC mi xes even before they are prepared in the laboratory and thus save a lot of ex pensive ex perimentati on. The advantage of A N is that it can be trained using ex perimental da ta avail able from publi shed research to recogni ze the relationship between these influencing fac tors. After training, the ANN is applied to pred ict the compress ive strength and workability of concrete. The network is trained to deduce likely results from specified input data. Once trained, the network can be used indefinitely to make predicti ons from data no t previously known.

11 0 INDI AN J. ENG. MATER. SCI, APRIL 2002

Theory

Basic concepts of ANN

The network models fo r predicting the strength , slump and th e materi als lIsed in HPC are shown in Figs I and 2. The 'Back-propagation network' shown in Fig. 2, is a widely lIsed type of network archi tec ture, based on a multi layered feed forward topology, w ith supervised learning. Thi s network consists of an inpu t layer, a hidden layer, an output layer and two levels of adapti ve connecti ons. It is also fully interconnec ted , i .e. each neuron is connected to all th e neurons in the nex t level. The overall idea behi nd back propagation4 is to make large change to a particular weigh t, IV i f the change leacls to a large reduction in thc errors observed at the output nodes. Let y be a smooth function of several vari ab les Xi. We want to know how to make incremental changes to in i tial values of each Xi, so as to increase the value of y as fast as possible. T he change to each initial Xi

va lue should be in proportion to the parti al deri vati ve of y \·"i th respect to that parti cular X i ' I n other words,

uv ~x. DC-'-, ax, ... ( I )

Suppose that y is a function of a several intermediate variables Xi and that each X i is a function

Fmc Aggregate

IC()ar'~ !\ggrt.!galt:

Water Cement ratio

Percentage of silica fume

Percentage orl superplas tl cl ser

I

OJ

E o 7.

Input Layer

of one vari able z. A lso we want to know the deri vati ve of y with respect to z, using the chain rul e:

... (2)

The standard way of measuring performance is to pi ck a parti cu lar sample input and then sum up the squared error at each of the outputs. We um over all sample inputs and add a minus sign for an overall measurement of performance that peaks at 0:

... (3)

where P is the measured performance, S is an index that ranges over all sample inputs, Z is . n index that ranges over all output nodes, ds: is the desi red output for sample input's' at the / ' node, o s; is the actual outpu t fo r sample input 's' at the /' node. T he performance measure P is a function of the weights5

.

We can deploy the idea of grad ien t ascen t i f we can calcu late the partied deri vati ve of performance wi th respect to each digit. With these partia l deri vati ves in hand , we can cli mb the performance hi ll most rap idly by altering all weights in proportion to the correspondi ng partial deri vative. The performance is given as a sum over all sample inputs. We can

I lidden I.ayer

Output Layer

Co mpressed Strength

Slump

Fig. I - The network Illodel for pred icting the strength ::Ind slulllp

RAJASEKARAN & AMALRAJ: HIGH PERFORMANCE CONCRETE COMPOSITES

Col. 'h'

Input Layer (a)

Co l. 'i'

"'\ '~. ~ ~

Col. 'j'

Hidden Layers (b)

Co l. 'k '

Fig. 2- Typi ca l back-propagation network architecture

or aO j ap

=-----

Col. 'z'

Output Layer (c)

III

... (S a)

... (5 b)

compute the parti al derivative of performance with respect to a parti cular weight by adding up the parti al derivative of performance for each sample input considered separately. Each weight will be adjusted by summing the adjustments derived from each sample input. Let us denote m, n, p as the number of neurons in input, hidden and output layers respectively with the notati ons: W ;-,j Weights of synapses connect ing the neurons in input layer to the neurons in the hidden layer; W j->k Weights of syna­pses connecting the neurons in hidden layer to the neurons in the output layer; 0, Output of input layer. (Since we use linear neuron, O;=I;=lnput of the input

As seen earli er OJ is obtained by apply ing sigmoida l function to the input of hidden neuron as:

11/

layer); 0 Input of the hidden layer = I 0; w; ..... j ; OJ ;= 1

Output of the hidden layer. (S ince we use sigmoidal

function , OJ = f( a j) where f is the sigmoidal

/I

functi on); ('h Input of the output layer = .L 0 j W j ..... k ; Ok ;=1

f (Ch) = Output of the output neuron; and, P Performance value.

Consider the partial derivative:

ap ... (4)

This can be written by apply ing chain rul e of di fferentiation as

Again appl y ing chain rul e of differentiation:

aO j af (a j ) a.f(a j) aa j = =

aa Since ____ J_ = o.

aw; ..... j I

~= af(a j ) 0

aW;-'j aa j I

Substituting Eg. (8b) in Eg . (Sa), we get:

ap af(a j) ap ----0 aw;-, j - ; aa j ao j

... (6)

. . . (7)

... (8a)

.. . (8b)

... (9)

11 2 INDIAN 1. ENG. MATER. SCI, A PRIL 2002

ap Now it is necessary to calculate In Eq. (9).

aO j

Again by app ly ing cha in rul e of differentiation,

ap = ap dOk aer k aOj aOk (Jerk aO j

Since Ok = f (erk), we rewrite Eq. ( IO)as:

(J p = ap a.l(erk) \V j->k

aOj aOk aer k

Combining Eqs (9) and ( II ), we get:

. . . ( 10)

. . . ( I I )

. . . ( 12)

Thus, the two important consequences of the above relat i ons are: ( I ) The partial deri vati ve of perfor­mance wi th respect to a weigh t depends on the partial derivati ve or performance wi th respect to the fol lowing output ; (2) The parti al derivative of performance with respect to one output depends on the partial deriva ti ve of performance wi th respect to the ou tputs in the next layer. The overa ll error o f the network is reduced if the error fo r each trai ni ng pattern is reduced. Th us, at step 's+ I ' of the training process, the weight adj ustment should be proportional to the derivative of the error measure computed on i teration's' . Th is can be written as:

L1 W(s+ I )=- 1]oPloW(s) ... ( 13)

where 11 is a constant learning coefficient, and

There is another poss ible way to improve the rate of convergence by adding some inerti a or momentum to the grad ien t expression, accompli shed by add ing a fraction of the prev ious weight change w ith current weight change. The addit ion of such term helps to smooth out the descent path by preventi ng ex treme changes in the grad ient due to local anomali es . A commonly used update rul e introduced by Rumelhart el al.6

includes such momentum ten . The updated equati on used by Rumelhart is defined as:

ap L'1w(s + I) = - 17 - + a L'1w(s) aw . .. ( 14)

where a is the momentum coeffici ent, should be posit: ve, but less than I . T he typi ca l va lue li es in the range 0.5 to 0.9 but for some prob lems Fahlman 7 used

a value for a==O and showed to be the be t.

Tr'aining the ANN

Traini ng the propagation charac teristics of the ANN takes place in an iterative fashi on in order to minimi ze the errors. Each iterati on cycle involves a forward propagation step followed by an error backward-propagation step to update the connection weights. The error signals are transmitted backward layer by layer to update the connecti on weights,

T:.Jb lc: I - Calculatcd ex pt:r i ll1ental v:.Jl ue vs obtained ANN va lue w ith given input va luc L1 sed for training

51. Fine Coarse Water Percentage Percentage of CO l11pres, i ve Slullip/ 1000

No aggregate/ agg regate/ Cement o f silica Superp last icizer/ strength/ I 000 (M Pal (111m)

(2xCe l11cnt) (-IXCC I11Cllt ) ratio fume/ I 00 1000 Exp. A NN Exp. AN N

I. 0.9-1 0.7 1 0.-15 0.1 0.2 O.OS O.OR 0. 11 0. 11 .., O.S 0.60 040 0. 1 0.2 0.09 0.09 0.09 0.09

3. 0 9 0.67 045 0.2 0. ] 0. 11 0.11 0. 12 0.12

-I. O. 65 0.53 0. 35 0.1 D.: 0. 11 O. 1 I 0.07 0.08

5. 0-18 043 0.30 0.05 0.2 0. 13 0. 13 0.05 0.05

6. 0.5 1 0.-15 0.30 0.15 0.25 0. 13 0. 12 0.06 0.06

7. 0.54 0.48 0.]0 0. 15 0.25 0. 12 0.1 3 0.07 D.07

8. 0.39 0. 3R 0.25 D. I D.2 D.1 4 0. 14 0.14 0. 14

9. 0.82 0.6 1 0.35 0.1 2 0.2 D.OS 0.08 0. 10 0. 10

10. 0. R5 0.64 0.35 0.16 0.25 0.09 0.09 0. 12 0. 12

II . 0.90 0.67 0.35 0.2 0.3 0.09 0.09 0. 13 0.1 3

12. 0.95 0. 7 1 D. 35 0.24 0.35 0.09 0.09 0.1 5 0. 15

13. 099 0.75 0.35 0.28 0.4 0.09 0.09 0. 17 0.1 7

1-1 . 0.69 0.57 0.35 0.075 0.2 0. 10 0.09 o. m~ 0.08

,

'"

RAJASE KARAN & A M A LR AJ: HIGH PERFORMANCE CONCRETE COM POS ITES 11 3

according to certain learning rul es, in order to reduce the difference between the output of the network and the real output. Training iterati ons continue until the network is capable of reproducing the desired output within an acceptable error fo r the particul ar input pattern s. An efficient learning algorithm with two initi al parameters, i.e., learning rate and a momentum determines the step size of the change of weights in the des ired directi on. The momentum is a fract ion

of previous weight change added to the current weight change, this provides an averaging effect which smoothes the trajectory of the movement of the weights. To speed up learning process and improve the predi cti on accuracy based on the previous ex perience, the initial learning rate was set to 0.0 1 and the momentum at 0.90. The training data used in the present study were ex tracted from earli er studies8

.<J.

Fourteen sets of ex perimental data were assembled

Tab le 2-Calculated experimental value vs obtaincd ANN va lue with given input va lue used for in ference

51. Fine Coarse Water

No. aggregate/ aggregate/ Cement

Pcrcentage

of sili ca

Percelllage of

superplastcizer/

Compress i ve strength/ I 000 (M Pal

Ex p. ANN Error

Siump/ I 000 (mm)

Exp. ANN Error

1.

2.

3.

4.

5.

6.

7.

8.

9.

(2xCcment) (4xCement) ratio fume/ IOO

0.82 0.62 0.40 0. 1

0 88 0.66 0.43 0. 15

0.96 0.72 0.45 0. 1

0.68 0.55 0.36 0. 1

0.72 0.59 0.38 0. 15

0.77 0.63 0.40 0.2

0 58 0.52 0.32 .0.1

0.55 0.49 0.30 0.05 0.(,2 0.55 0.34 0.15

Com pressive Streng th o f Tra ine d D a t a

123456789 1011121 3 14

Numbe r of Sa mple s

-- C alc ulate d Strength • Actu al S ireng th

1000

0.2

0.25

0.2

0.2

0.25

0.3

0.2

0.2

0.25

Fig. 3a- Pcrformancc of compressi ve strcngth for trai ned inswnces of data

Worka bil ity o f Tra ined Data

0 .2

f/) 0 .15 '" ::J

'" 0 . 1 > C;;

'" 0 .05 n:: ,

0 I

n of) r0- O> ~ ~ ~

N UIll b er o f ~ a m p ies

--Ca lculated Slu m p _ Ac ttla l S lu mp

Fig. 3b- Pcrfllnnance of workabili ty for trained instances of uat:!

0.08 0.09 0.01 0. 10 0.10 0.00

0. 10 0.09 0.0 1 0. 11 0. 11 0.00

0.08 0.09 0.0 1 0. 12 0.12 0.00

0.1 0 0.10 0.00 0.08 0.08 0.00

0. 10 0. 10 0.00 0.09 0.09 0.00

0. 14 0.10 0.04 0. 11 0.10 0.0 1

0. 11 0.1 3 0.02 0.07 0.08 0.01

0.1 0 0.11 0.01 0.06 0.06 0.00

0.1 2 0.11 0.01 0.09 0.08 0.01

Com pressive Stren gth of Un t r ai ned Data

o 15 -f/) /. -'" ::J 0 .1 - - • -'" - -> '" o 05 '" n::

0 ._---2 3 4 5 6 7 8 9

Num ber of Sam pies

--Ca lcu la te d S tre n gth _ Actua l Stre ng th

Fig. 4a- Perforlllalice of compressive sirength for ulllrained i nst:tJlccs of data

Wo r k abi li t y o f Un trai ned D ata

o 14 0 . 12

f/) o 1 '" ~ 0 .08 '" > 0 .06 C;;

'" 0 .04 n:: 0 .02

0 - ----------2 3 4 5 6 7 8 9

Num b er of S a m pi es

--C al c u lated S lu mp • Actua l S ldmp

Fig. 4b-Per li.mnunce of workability for uJlt rained in tancc~ of data

11 4 INDIAN J. ENG. MATER. SC I, APRIL 2002

(Table I ). The dosage of chemical admixture for slumps of 40- 170 mm are 2-4% by weight of the cement.

Pred iction of sl renglh and slulllp

After the neural network had been fully trained, nine sets of our own experimenta l data, wh ich had not been used i the training, were used to verify the accuracy. These nine sets of data are given in Table 2. In Tab le I , the predict ions generated by ANN are compared wi th these experimental results for compress i ve strength and workabi I i ty of H PC and represented as separate chart s in Fig 3a,b respectively. I t can be seen from Tab le 2 that the average percentage of error fo r compress ive strength is 11 % between the actual experimental results and those predicted by ANN whereas for slump it is only 4%. These va lues are very much permi ssible for HPC mi x des ign. In add ition, the charts plotted in Fig 4a,b demonstrate that the predicted results closely foll ow those from the ex peri ment and the same trend is ev ident. Th is shows th at prediction of strength and slump using an A is not only feasible but also offers promise for the so lution of other types of engi neeri ng problems.

Conclusions The proposed neural network can predict the

strength and slump va lues for HPC. The experimental data reported by others are found to be suitable for training the network, the prediction capability of which is then va lidated successfully Ll sing new experimenta l data.

The neural network is able to lea rn and generali ze the relationship between the strength , slump and water-cement rati o, and percentages of si li ca fume, superp las ti cizers of a high performance concrete. The

back-propagation neural network used in thi s study is an effective network model for predicti ng the strength of HPC. The use of an adaptive learning rate and a momentum has been proved to be efficient with an eITor tolerance of 0.045 173 being reached within 20,000 iterations. The neural network provides accurate prediction of the strength and slump of HPC with an average error of 1 I % for strength and 4% for slump which are permi ss ible fo r HPC mix design. Consequentl y, the NN has great otential for implementation in practice. The mix proporti ons for high grade concretes 80 MPa and 140 MPa obtained in the experiments are given as I : 1.92:2.88 and I : I. 16:2.08 for the same percentages of silicafume and superpl as tici zers.

Acknowledgements Sincere thanks are due to the Management and the

Principal, Dr P Radhakrishnan, of PSG Co llege of Technology, Coimbatore 641 004, for providing necessary faci Ii ti es and constant encouragement to carry out thi s work.

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Wes ley Internation:1I students Ed ition, New Delhi 11 0040), 1999,454-457.

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