M5' Algorithm for Shear Strength Prediction of HSC Slender Beams without Web Reinforcement

Transcription

1 International Journal of Modeling and Optimization, Vol. 7, No. 1, February 217 M5' Algorithm for Shear Strength Predition of HSC Slender Beams without Web Reinforement Ali Kaveh, Seyyed Mahmoud Hamze-Ziabari, and Taha Bakhshpoori Abstrat Aurate predition of the shear behavior of Reinfored Conrete (RC beams unlike to its flexural behavior whih an generally be well predited is a hallenging problem due to the omplexity of the shear transfer mehanism. This issue an be more ritial for High-Strength Conrete (HSC beams ompared with the beams with normal strength onrete. The present study adopts an effiient rule based data mining approah: M5' algorithm to potential development of a new formulation for prediting the shear strength of HSC slender RC beams without stirrup. A omprehensive database ontaining several effetive parameters that onsider geometrial and mehanial properties of onrete, aggregate and reinforement are involved in predition of the shear strength. Comparison between developed model and the most ommon design odes demonstrates the superiority of the developed model in term of auray. Furthermore, the safety analysis based on Demerit Points Classifiation sale also onfirms the reliability of the proposed formulation. Index Terms High strength onrete, M5' algorithm, predition shear strength, slender beams. I. INTRODUCTION The high-strength onrete (HSC has been emerged as an inreasingly used effiient strutural material in buildings and ivil/strutural infrastrutures due to its physial and mehanial harateristis, strutural effiienies, and eonomi and aestheti advantages. HSC an be defined as a onrete to have strength signifiantly beyond what is used in normal strength onrete (NSC. Different odes and researhers speify different limits for ompressive strength of onrete for demaration between NSC and HSC [1]. However, the definition by ACI 363R-1 [2] whih onsiders the value of 4 MPa as the demaration limit is followed here. Shear arrying apaity of reinfored onrete members has been investigated experimentally and analytially by several researhers. It has been found that the shear failure mehanism in onrete members under bending whih are reinfored longitudinally but have no transverse reinforement, varies signifiantly and size of member and shear span to-depth ratio are two main parameters whih Manusript reeived Otober 7, 216; revised Deember 12, 216. Ali Kaveh is with the Iran University of Siene and Tehnology as professor of Strutural Engineering and also as the hair of the Centre of Exellene for Fundamental Studies in Strutural Engineering, Tehran, Iran ( alikaveh@ iust.a.ir. Seyyed Mahmood Hamze-Ziabari is with Coastal Engineering with Iran University of siene and tehnology, Tehran, Iran ( ziabari@ivileng.iust.a.ir. Taha Bakhshpoori is with the Department of Engineering, East of Guilan University of Guilan, Roudsar, Guilan, Iran, He is a orresponding author. ( tbakhshpoori@guilan.a.ir. influene the shear failure. Beams are divided to deep beams (<2.5 and slender beams (>2.5 on the basis of the shear span-to-depth ratio. Aurate predition of the shear behavior of reinfored onrete (RC beams unlike to their flexural behavior whih an generally be well predited, is a hallenging problem due to the omplexity of the shear transfer mehanism. Based on the available literature [4]-[8] differenes are obvious between NSC and HSC beams without web reinforing. Shear strength predition of HSC slender beams without web reinforement is thus still ontentious. Elsanadedy et al. [1] very reently applied regression models and neural networks for prediting the shear strength of HSC slender RC beams without stirrups. The main objetive of this study is the predition of shear arrying apaity of slender high strength onrete beams without transverse reinforement. The M5' as one of the model trees algorithms is used for developing preditive and simple formulas for estimation of the maximum pull out fore. Unlike most of the SC algorithms suh as ANN and ANFIS, the M5' algorithm an present transparent formulas that are physially sound and interpretable. A omprehensive existing database inluding 25 experimental tests [1] from multiple soures in literature is employed here. Various influential parameters that affet the shear strength suh as the longitudinal steel ratio, the shear span-to-depth ratio, ompressive strength of onrete, the size of the beam speimens, and the size of oarse aggregate are onsidered. The auray and safety of preditions of the derived M5' model are ompared to those of the existing shear strength relationships of the most ommon design odes. The omparison results showed that the proposed model is more aurate and reliable than other design odes. Furthermore, the robustness of the proposed models is onfirmed through sensitivity and parametri analyses. Remaining of the paper is organized as follows: the following setion presents some limitations of urrent design odes. The M5' algorithm is outlined at the third setion. After presenting the used database the fourth setion develops the preditive models by using M5' algorithms. Penult setion disusses the results based on the developed model and also makes a safety analyses to evaluate the reliability and unertainty of the model. The last setion summarizes the findings of this study. II. LIMITATIONS AND DESIGN CODES In the present study, the preditive apability of the most ommon design odes in predition of shear strength of the HSC slender beams inluding ACI [9], CSA A [1], fib Model Code [11], Euroode-2 [12], CEB-FIP Model DOI: /IJMO.217.V

2 International Journal of Modeling and Optimization, Vol. 7, No. 1, February 217 [13], AS [14], JSCE Guidelines [15], and Cladera and Mari [16] are investigated. Table I summarizes the main design parameters inluded in the mentioned odes and also the statistial error parameters related to eah ode. 25 reorded ases for HSC beams existing in different soures in literature are used to evaluate the performanes of these design odes. The average (μ, standard deviation (σ and oeffiient of variation (COV of disrepany ratio between predited and measured values of the shear strength are presented in Table I. It is lear from this table that the performanes of these equations for slender beams made of HSC are remarkably limited. This an be attributed to this fat that all the design equations onsider a limitation value of 65 MPa for ompressive strength of the onrete (f' exept CEB-FIP model. In addition, some influential parameters suh as the nominal maximum size of the oarse aggregate is not inluded in most of the design odes (exept A [1] and level II approximation of the fib Model [11]. The dependeny of the shear strength of the HSC beams to this parameter is investigated and onfirmed by Muttoni [17]. On the other hand, the ontribution of different preditive parameters suh as ompressive strength of onrete and longitudinal steel ratio is inorporated in a different way. For example, the ontribution of f' in predition of the shear strength is proportional to f' 1/3 in Euroode-2, CEB-FIP Model, AS 36, and JSCE while the other odes (ACI , CSA A23.3-4, fib model onsider this ontribution as f' 1/2. This proportion was also reported as f'.2 by Cladera and Mari [15]. In general, it an be onluded from these observations that new preditive equations should be presented to take full advantage of the positive features of the RC beams made of the HSC. Furthermore, the preditive equation must onsider all effetive parameters in predition of the shear strength. TABLE I: DETAILS OF EXISTING DESIGNS EQUATIONS AND THEIR PERFORMANCES IN PREDICTING SHEAR STRENGTH OF SLENDER HSC BEAMS III. M5' ALGORITHM The important advantages of model trees (MTs to other soft omputational approahes an be onsidered as being more aurate than regression trees, more transparent and understandable than Artifiial Neural Networks (ANN, easy to train, and robust when dealing with missing data [19]. Among the soft omputing algorithms, M5' Model Tree is one of the newly developed algorithms that has not been widely applied in strutural engineering. Very reently Kaveh et al. [18] utilized this algorithm for predition of the prinipal ground-motion parameters in whih the effiieny of M5' is demonstrated against many other algorithms. In the following M5' algorithm is outlined [18]. M5 model was developed by Quinlan [2] in 1992 and improved later in 1997 as a system alled M5' by Wang and Witten [21]. M5 model trees are more aurate and understandable than regression trees and ANNs. It an handle large number of attributes and high dimensions [22]. The algorithm onsists of three main steps: building tree, pruning the tree and smoothing. By using the splitting riterion, the basi tree is formed. This splitting riterion is defined as the standard deviation of the lass values that reah a node as a measure of the error at the node, and alulates the expeted redution in error as a result of testing eah attribute at the node. Then, the attribute that maximizes the expeted error redution is seleted. The standard deviation redution (SDR for M5 is alulated using the following formula: T SDR sd T sd T T i i i where T is the set of example that reah the node, T i is resulted set from splitting the node aording to the seleted attribute and sd is the standard variation [21]. The splitting proess eases when the lass values of all the instanes that reah a node vary by less than 5% of the standard deviation of the original instane set, or when only a few instanes remain. An over-fitting problem an our during the MT onstrution based on the training data. In order to redue this problem, a method is termed pruning has been used. The pruning proedure uses an estimate of the expeted error that will be experiened at eah node for the test data. First, the absolute differene between the predited value and the atual output value is averaged for eah of the training examples that reah the node. Sine the trees have been built expressly for this dataset, this average will underestimate the expeted error for new ases. To ompensate for this, the output value is multiplied by the fator (n+v/(n-v, where n is the number of training examples that reah the node and v is the number of attributes in the model that represents the output value at that node. Therefore, this multipliation is performed to avoid underestimating the error for new data, rather than the data against whih it is trained. If the estimated error is lower at the parent, the leaf node an be dropped [19]. Smoothing proess is onsidered to redue and solve the problem of sharp disontinuousness at leaves of the pruned tree. The smoothing proess, as desribed by Quinlan [2], uses the leaf model to ompute the predited value. The value is then filtered along the path bak to the root, smoothing it at eah node by ombining it with the value predited by the linear model for that node. This involves the alulation as follows: (1 np kq P n k (2 49

3 International Journal of Modeling and Optimization, Vol. 7, No. 1, February 217 where P' is predition whih exeed to higher node, P is predition pass to urrent node from the below, q is the predited value by model at the node, n is number of training instanes reah to previous node, and k is Wang & Witten onstant [21]. IV. MODEL DEVELOPMENT D ' E B C ag a F H v u( MPa Ab d f or d d D ' E B C ag a F H V (kn Ab d f d d where A', B',, and H' are onstant values. The developed rules and model tree were as follows: (3 A. Model Inputs Based on the previous experimental studies and the existing design odes, six independent variables inluding the beam width (b, the effetive depth (d, the shear span-to-depth ratio (a/d, the ompressive strength of onrete (f', the aggregate size-to-depth ratio (a g /d, and the longitudinal steel ratio (ρ were onsidered as preditive variables. The single model output is onrete shear apaity, v u. B. Database Used To investigate shear strength of the HSC slender beams and develop new predition formulas, a very reently olleted omprehensive data set by Elsanadedy et al. [1] from 33 experimental studies performed between 1957 and 213 is used. Detail of data set used an be found in [1]. The entire dataset inludes 25 reorded experimental data samples for different ranges of f' between 42.5 MPa to 183 MPa. The data set is randomly divided into two independent parts (i training (2 (ii testing (5. The M5' algorithm is trained using the training dataset and then evaluated with testing dataset. The range of input and output variables for training and testing datasets is presented in Table II. It should be noted that slender beams with a/d 2.5 in whih the failure mode was shear, are seleted. The speimens in mentioned experiment studies are monotonially loaded by either one or two onentrated loads. TABLE II: RANGES OF INPUT AND OUTPUT PARAMETERS FOR TRAINING AND TESTING SETS LM 1 LM 2 LM a ag v 69.2b d f d d a ag v b f d d a ag v 1.796b f d d In Fig. 1, the lassifiation of data set by using M5' model tree for prediting shear strength of HSC slender beams are shown. Fig. 1 Classifiation of dataset based on M5' algorithm. Finally, the following statistial error parameters were used to evaluate the performane of the developed M5' algorithm for prediting shear apaity: mean absolute error (MAE, root mean square error (RMSE, sattering index (SI and orrelation oeffiient (R. The readers are enouraged to the speialized literature for the mathematial statement of these parameters. The orrelation oeffiient R is a measure of the relative orrelation between the predited and measured values. The R values ranged between -1 and 1. If the R value is lose to 1, it indiates that there is diret linear relationship between measured and predited values. However, R sometimes may not neessarily indiate better model performane, partiularly when data range is very wide and the data points distributed about their mean. Therefore, the oeffiient of determination R 2, an be used as an unbiased estimate and an be a better measure for evaluating model performane. The MAE and RMSE measure the differene between predited and measured values and values near to zero indiate lose math. (4 (5 (6 C. Developed M5' Model M5' model an be used to generate simple and meaningful rules that an be easily applied in shear apaity alulations. However, it ould just propose a linear relation between the input and the output parameters. In order to overome this limitation, model was developed by log (inputs and log (output. Then, the final developed model an be written as: V. RESULTS AND DISCUSSIONS A. Performane Analysis Performane of the M5 algorithm for training and testing datasets shown in Fig. 2 demonstrates that there is a little sattering around the optimal line between measured and predited values by M5 algorithm for both datasets. 5

4 International Journal of Modeling and Optimization, Vol. 7, No. 1, February predited shear strength. v predited Training Frequeny Euroode2 Frequeny CEB-FIP v predited v measured Testing (a v measured (b Fig. 2. Graphial performane of developed M5 algorithm for (a training (b testing datasets. The statistial error parameters related to training, testing and the whole dataset are presented in Table III. The results indiate that the proposed model performs well in both training and testing datasets. TABLE III: THE PERFORMANCE OF DEVELOPED M5 ALGORITHM Subset MAE RMSE R R 2 Training Testing Total Table IV tabulates statistial error parameters of the M5 and the other shear design equations. As shown, all the error measures of the proposed model show improvement. The performanes of the Euroode2, CEB-FIP, and AS36 are more reasonable than other design odes. However, the proposed model remarkably outperforms the others. In this respet, the RMSE and R 2 values of M5 model shows improvement respetively by 37.8% and 6% ompared to the AS 36 model as the most preise model among the design equations. TABLE IV: COMPARISON OF THE M5 MODEL AND OTHER AVAILABLE METHODS Model MAE RMSE R R 2 ACI CSAA Fib Euroode CEB-FIP AS JSCE Cladera and Mari M5 (proposed The histogram of log (measured/predited of shear apaity for M5 model and the three best design equations are shown in Fig. 3. The normal distribution is also fitted to logarithm of disrepany ratio between measured and Frequeny AS Frequeny M5' Fig. 3. Histograms of log (v u,mes values for different approahes and their fitted normal distribution funtions. It an be seen that the values of log (Measured/Predited for the M5 model are lose to zero. Furthermore, the distribution of log (measured/predited values is nearly symmetrial and approximately onentrates in zero value. The distribution of Euroode 2 model predition is wider than other models. This indiates that the unertainty of this model is high. Furthermore, the most of CEB-FIP and AS 36 model preditions overestimate shear apaity. The log (Measured/Predited values of AS 36 model are loser to zero than other design equations. However, the M5 has remarkably better performane than other design equations in this aspet and also has narrower normal distribution (less unertainty. As stated in previous setions, some design odes did not onsider some important parameters suh as a g /d in their models or onsider different ontributions for some parameters suh as f and ρ. The model errors should be independent of the input parameters or less sensitive to them, otherwise it an be interpreted those input parameters did not orretly inorporated in that model or should be added to the model if it has not been onsidered. The variations of disrepany ratio between measure and predited shear strength by Euroode 2, CEB-FIP, AS 36, and M5 models with a/d, a g /d, f, and ρ are shown for testing dataset in Fig 4. The lines in this figure represent the best linear regression between the values of v measured /v predited and input variables. The errors of design odes are remarkably sensitive to variations of a/d and ρ parameters. It may be interpreted that these parameters have not been onsidered orretly in the mentioned models. This sensitivity is less for f and a g /d. In general, the errors of developed M5 model are ompletely insensitive to variation of effetive parameters. B. Safety Analysis Most design odes onsider a marginal auray in trading of safety, therefore, the proposed model should also remain in a suffiient level of safety. To evaluate the safety of the proposed model and existing design odes, a new sale introdued by Collins [23] is employed. This sale is known as Demerit Points Classifiation (DPC, whih onsider the safety, auray and sattering of the design odes as a 51

5 International Journal of Modeling and Optimization, Vol. 7, No. 1, February 217 funtion of the ratio between the ultimate resistanes in experimental tests and the estimated theoretially shear apaity. In the present study, this ratio has been shown as DR parameter. The safety lassifiation based on the Collins sale is presented in Table V. extremely onservative lasses. It should be noted that the M5 algorithm has the lowest demerit penalty and are safer than other design odes. Approximately, 65 perent of the preditions of M5 algorithm are lassified in the safe and appropriate region based on Collins sale. TABLE VI: CLASSIFICATION OF DEVELOPED AND DESIGN EQUATIONS ACCORDING TO THE CRITERIA OF COLLINS Fig. 4. Variation of disrepany ratio (DR between measured and predited shear strength as a funtion of a/d, a g/d, f, and ρ. TABLE V: CLASSIFICATION BY DEMERIT POINTS v u,exp/ v u,predited Classifiation Demerit points <.5 Extremely dangerous 1 [ Dangerous 5 [ Appropriate and safe [ Conservative 1 2. Extremely onservative 2 To quantitative evaluation of the safety of the proposed model and the existing design odes, first a demerit point is attributed to eah predition of these equations for total 25 data points based on Table VI. Then, the total values of demerit of eah formula are alulated by the sum of the produts of the number of speimens in eah interval and their orresponding demerit penalty. The lower the value of total sum indiates that the onsidered formula is safer. The results of developed models and existing equations are presented in Table VI. As shown, the CEB-FIP model is the safest design odes and has the lowest demerit penalty. However, nearly 6 perent of its preditions are ategorized in onservative and VI. CONCLUSION In the present study, new formulation for predition of shear strength of the HSC slender RC beams without web reinforement using a rule based method namely M5' algorithm is proposed. A omprehensive database of 25 available experimental data samples ontaining several effetive parameters that onsider geometrial and mehanial properties of onrete, aggregate and reinforement are is employed to develop the model. Comparison between the developed model and the most ommon design odes are performed to evaluate the auray of the proposed model. The RMSE and R 2 values of M5 model shows improvement respetively by 37.8% and 6% ompared to the AS 36 model as the most preise model among the several design equations. Furthermore, the safety analysis based on Demerit Points Classifiation sale also onsidered to evaluate the safety of proposed formulation. Results indiate the proposed model safer than the CEB-FIP model whih is determined as the safest design ode among others. REFERENCES [1] H.M. Elsanadedy et al., ''Shear strength predition of HSC slender beams without web reinforement,'' Materials and Strutures, vol. 49, no. 9, pp. 1-24, 215 [1] A. C. I., ''Report on high-strength onrete, ACI 363R-1,'' Amerian Conrete Institute: Detroit, MI. 21 [2] H. P. J. Taylor, ''The fundamental behavior of reinfored onrete beams in bending and shear,''speial Publiation, vol. 42, 1974 [3] M. Hamrat et al., ''Shear behaviour of RC beams without stirrups made of normal strength and high strength onretes,'' Advanes in Strutural Engineering, vol. 13, no. 1, pp , 21. [4] K. Maekawa et al., ''Diret path-integral sheme for fatigue simulation of reinfored onrete in shea,'' J Adv Con Teh, vol. 4, no. 1, pp , 26. [5] P. C. Mihael and K. Daniel, ''How safe are our large, lightly reinfored onrete beams, slabs, and footings?,'' Strutural Journal, vol. 96, no. 4, pp ,

6 International Journal of Modeling and Optimization, Vol. 7, No. 1, February 217 [6] J. Sagaseta and R. Vollum, ''Influene of aggregate frature on shear transfer through raks in reinfored onrete,'' Magazine of Conrete Researh, vol. 63, no. 2: pp , 211. [7] S. Sarkar, O. Adwan, and B. Bose, ''Shear stress ontributions and failure mehanisms of high strengh reinfored onrete beams,'' Materials and Strutures, vol. 32, no. 2, pp , 1999 [8] A. A. Standard, Building Code Requirements for Strutural Conrete Amerian Conrete Institute, 211. [9] C. Standard, Design of Conrete Strutures, Canadian Standard Assoiation, 24. [1] F. I. D. Béton, Fib Model Code 21 Final Draft, 212, p. 37. [11] Institution, B.S., Euroode 2: Design of Conrete Strutures: Part 1-1: General Rules and Rules for Buildings. 24: British Standards Institution. [12] C. MC9, Design of Conrete Strutures, CEB-FIP Model Code 199, 1993, Thomas Telford. [13] S. Australia, Conrete Strutures (AS 36-29, 29. [14] K. Sakai and K. Kawai, JSCE Guidelines for Conrete No. 7: Reommendation of Environmental Performane Verifiation for Conrete Strutures, Japan Soiety of Civil Engineering, 26. [15] A. Cladera and A. Mari, ''Shear design proedure for reinfored normal and high-strength onrete beams using artifiial neural networks. Part I: beams without stirrups,'' Engineering Strutures, vol. 26, no. 7, pp , 24. [16] A. Muttoni, ''Punhing shear strength of reinfored onrete slabs without transverse reinforemen,'' ACI strutural Journal,. vol. 15, no. EPFL-ARTICLE , pp [17] A. Kaveh, T. Bakhshpoori, and S. M. Hamze-Ziabari, ''Derivation of New Equations for Predition of Prinipal Ground-Motion Parameters using M5 Algorithm,'' Journal of Earthquake Engineering, vol. 2, no.6, pp , 216. [18] I. H. Witten and E. Frank, Data Mining: Pratial Mahine Learning Tools and Tehniques, 25: Morgan Kaufmann. [19] J. R. Quinlan, Learning with Continuous Classes, in Por. 5th Australian Joint Conferene on Artifiial Intelligene, Singapore. [2] Y. Wang and I. H. Witten, Indution of Model Trees for Prediting Continuous Classes,1996. [21] D. P. Solomatine and A. Ostfeld, ''Data-driven modelling: some past experienes and new approahes,'' Journal of hydroinformatis, vol. 1, no. 1, pp. 3-22, 28. [22] M. P. Collins and D. Kuhma, ''How safe are our large, lightly reinfored onrete beams, slabs, and footings?,'' Strutural Journal, vol. 96, no. 4, pp , Ali Kaveh was born in 1948 in Tabriz, Iran. After graduation from the Department of Civil Engineering at the University of Tabriz in 1969, He ontinued his studies on strutures at imperial College of Siene and Tehnology at London University and reeived his MS, DIC, and a PhD degrees in 197 and 1974, respetively. He then joined the Iran University of Siene and Tehnology in Tehran where he is presently a professor of Strutural Engineering. Professor Kaveh is the author of 54 papers published in international journals and 15 papers presented at international onferenes. He has authored 23 books in Farsi and nine books in English published by Wiley, the Amerian Mehanial Soiety, Researh Studies Press, and Springer Verlag. Taha Bakhshpoori was born and grew up in Astara. He earned his B.S. in ivil engineering from University of Guilan (29, M.S. (211 and Ph.D. (215 both in strutural engineering from Iran University of Siene and Tehnology. He joined the University of Guilan, department of Engineering in Eastern Guilan as Assistant Professor in Deember 215. His researh interests inlude the following: Artifiial Intelligene and Appliations to Civil Engineering Problems, Meta-heuristi Modeling and Optimization, Multi Objetive Optimization, Strutural optimization, Reliability, Robust, and Risk Based Design Optimization, andstrutural Health Monitoring and Damage Detetion. 53