CC BY
17
MECHANICS: Mechanics of Deformable Solids
D0l.org/10.5281/zenodo.1119147 УДК 697.972
Samui P., Bocharova A.
PIJUSH SAMUI, Associate Professor, Department of Civil Engineering, e-mail: [email protected] National Institute of Technology Patna Patna, Bihar India, 80005
ANNA BOCHAROVA, Chief of Mechanics and Mathematical Modeling Department, e-mail: [email protected] Far Eastern Federal University 8 Sukhanova St., Vladivostok, Russia, 690091
Modeling of compressive strength of masonry structure using relevance vector machine and minimax probability machine regression
Abstract: Compressive Strength (f) of masonry structure is a key parameter for designing masonry structure. This article employs Relevance Vector Machine (RVM) and Minimax Probability Machine Regression (MPMR) for estimation of compressive strength (f) of a masonry structure. RVM is a Bayesian model. MPMR is constructed based on the concept of Minimax Probability Machine Classification. Mortar compressive strength (fm), and brick compression strength (fb) are used as inputs of the RVM and MPMR. The output of RVM and MPMR is f. The results of RVM and MPMR have been compared with the other models. The results show that the developed RVM and MPMR are useful tools for estimation of f of masonry structure.
Key words: compressive strength, masonry structure, relevance vector machine, minimax probability machine regression.
Compressive strength (f) of a masonry structure is a key parameter for designing masonry structure. Researchers gave different analytical models for estimation of f [2, 3, 8, 10, 13]. The available analytical models are not so reliable [20]. Different empirical models are also available for determination of f [1, 5, 7, 16]. Artificial Neural Network (ANN) and fuzzy logic were examined for prediction of f of a masonry structure [9]. However, ANN is not a perfect model [12, 18].
This article adopts Relevance Vector Machine (RVM) and Minimax Probability Machine Regression (MPMR) have used for estimation of f of a masonry structure. RVM was developed based on Bayesian concept [22, 23]. Many problems have been solved by RVM [24-26]. MPMR was developed based on the kernel formulation [14, 17, 19]. The predicted output from the MPMR will be within some bound of the true regression function. RVM and MPMR use the database of Roca et al. [9]. The dataset consist the value of mortar compressive strength (fm), brick compression strength (fb) and f. The results of RVM and MPMR have been compared with the ANN and fuzzy logic models. In RVM, the basic equation is given below for estimation of output(y):
N
у = £ wK (xt, x), (1)
i=1
© Samui P., Bocharova A., 2017
About the article: Received: 30.10.2017; financing: budgets: National Institute of Technology, India and Far Eastern Federal University, Russia.
where N represents the number of samples, wi is weight, x denotes input variable and K(x;,x) is kernel function. This article uses fm and fb as inputs of the RVM. The output of RVM is f.
So, -x = \fm, fb ] and y = [f ].
In RVM, a Gaussian prior is assumed on wi. The Gaussian prior has zero mean and hyperparameters (ar1) variance. RVM uses iterative formulae for hyperparameter estimation [15]. In RVM, nonzero weights
are called relevant vectors. The details of RVM have been obtained from Tipping (2000, 2001) [22, 23].
To develop the above mentioned RVM, 76 out of 96 dataset have been taken as training dataset. The remaining 20 datasets have been used to examine the capability of model. These datasets are called testing dataset. This article adopts normalization between 0 and 1. This study employs Radial basis func-
tion ( K (xt, x) = exp <j -
(xt - - x)
2a
2
a is the width of radial basis function) as kernel function. Dif-
ferent kernel functions (radial basis function, polynomial and spline) have been examined to get best performance. However, radial basis function gives best performance. The program of RVM has been implemented in MATLAB.
A brief description of MPMR for prediction of f of a masonry structure. In MPMR model, the basic formulation is given below
y =
N
(x,, x) + b
(2)
where y denotes output, x denotes input, K(xi,x) represents kernel function, N represents number of data, pi and b are output of the MPMR algorithm. MPMR uses fm and fb as inputs. f is the output of MPMR.
Hence x = f, fb ] and y = \f].
Strohmann and Grudic [20] gave the procedure for estimation of pi and b. In MPMR, it determines a bound on the probability that the regression model is within ±s of the true regression function. MPMR has been developed based on mean and covariance matrix statistics of the regression data. MPMR employs the same training dataset, testing dataset, normalization technique and kernel function as adopted by RVM. The implementation of MPMR model has done by using MATLAB.
For RVM, the best performance is obtained at the design value of a is 0.7. The value of f of training and testing dataset has been determined by using the design value of a. Fig. 1 shows the plot between actual f and predicted f for the training dataset.
1
0.9
s«0.8 ■s
80.7
ARVM(R=0.987) °MPMR(R=0.998) Actual=Predicted
50.6
I0.5
£0.4
u
'■§0.3 ^0.2
0.1 0
0.2 0.4 0.6
Actual Normalized f
0.8
Fig. 1. Plot between Actual f and predicted f for training dataset.
,=i
0
1
FEFU: SCHOOL OF ENGINEERING BULLETIN. 2017. N 4/33
The value of Coefficient of Correlation (R) has been determined to assess the performance of RVM and MPMR. For a good model, the value of R is close to one. Fig. 2 depicts the performance of testing dataset.
1
0.9
^-0.8 -o
.§0-7 1*6 |0.5 ■o
«0.4 u
|0.3
0.1
0
•RVM(R=0.986) A MPMR(R=0.996) Actual=Predicted
0.2 0.4 0.6
Actual Normalized f
0.8
Fig. 2. Plot between Actual f and predicted f for testing dataset.
It is observed from fig. 1 and 2 that the value of R is close to 1 for training as well as testing dataset. The constructed RVM employs the following equation for estimation of f.
f = £ exp <
-(Xj - .x X-X - x) 0.98
(3)
where xi is the inputs of training dataset, x is the inputs of unknown output dataset and T is transpose.
User can obtain the values of w from fig. 3. The value of w will be obtained based on the serial number of training datasets. For example, the value of w will be zero for serial number 1 of training dataset. There are 10 training datasets of non zero w in the training dataset (see fig. 3). Hence, the developed RVM produces 10 relevance vectors for estimation of f.
0.35 0.3 0.25 0.2 0.15 £ 0.1 0.05 0
-0.05 -0.1 -0.15
O
I
<XXX)0000000000000«X>
o
I
I
<xx»
<»0006
11
61
21 31 41 51 Training Dataset Fig. 3. Variation of w with the training dataset.
71
0
1
!=1
For MPMR, the best performance is obtained at a = 0.45 and s = 0.002 . The value of f of training and testing datasets has been determined by using a = 0.45 and s=0.002. Fig. 1 illustrates plot between actual f and predicted f for training dataset. Fig. 2 shows the plot between actual f and predicted f for testing dataset. It is clear from figures 1 and 2 that the developed MPMR predicts f reasonable well for training as well as testing datasets. Therefore, the developed MPMR shows his ability for estimation of f.
The results of constructed RVM, MPMR have been compared with the other models [7-9, 13,
f
24-26]. Table depicts the value of mean and standard deviation of the proposed. The performance of de-
freal
veloped MPMR and RVM is almost identical with the ANN. However, the developed MPMR and RVM outperform the other methods. ANN uses many design parameters compare to the RVM (design parameter = a) and MPMR (design parameter = s and a). RVM has no control on the predicted f for future datasets. However, the developed MPMR has control on the predicted f for future datasets. RVM uses only some parts of training data (called relevance vector) for estimation of f. However, ANN and MPMR adopt all training dataset for determination of f.
Values of mean and standard deviation of the different models
Method Mean Standard Deviation
Mann (1982) 1.19 0.22
Dayaratnam (1987) 0.32 0.22
Kaushik et al. (2007) 0.48 0.34
Dymiotis et al. (2007) 0.79 0.37
Eurocode 6(1998) 0.57 0.34
ACI 530.99 (1999) 0.29 0.32
ANN (Roca et al., 2013) 1.02 0.35
Fuzzy Logic(Roca et al., 2013) 1.09 0.34
Linear Regression (Roca et al., 2013) 1.22 1.52
RVM 1.03 0.37
MPMR 1.01 0.40
This study describes RVM and MPMR models for estimation of f of a masonry structure. The developed RVM and MPMR give reasonable performance. MPMR employs convex optimization for prediction of f. Advantage of RVM is that it produces sparse solution. The results of the developed models (RVM and MPMR) have been compared with the other models. In summary, it can be concluded that the developed RVM and MPMR are new reliable tools for estimation of f of a masonry structure.
REFERENCES
1. ACI committee 530. Building code requirements for masonry structure. American Concrete Institute, Farmington Hills, MI; 1999.
2. Atkinson R.H., Noland J.L., Abrams D.P. A deformation theory for stack bonded masonry prisms in compression. Proc., 7th int. brick masonry conf., Melbourne, Melbourne Univ., 1982, p. 565-576.
3. Binda L., Fontana A., Frigerio G. Mechanical behaviour of brick masonries derived from unit and mortar characteristics. Proc. of the 8th international brick/block masonry conf, Dublin, 1988, pp. 205-216.
4. Dayaratnam P. Brick and reinforced brick structures. Oxford and IBH, New Delhi, 1987.
5. Department of transport. The assessment of highway bridges and structures. British Standard BD 21/93, Her Majesty's Stationery Ofc., London, 1993.
6. Dymiotis C., Gutlederer B.M. Allowing for uncertainties in the modeling of masonry compressive strength. Constr Build Mater, 2007;16(7):1385-1393.
7. ENV 1996-1-1. Eurocode no. 6 Design of masonry structures, Part 1-1: general rules for buildings-rules for reinforced and un-reinforced masonry, 1998.
8. Francis A.J., Horman C.B., Jerrems L.E. The effect of joint thickness and other factors on the compressive strength of brickwork. Proc. of the 2nd int. brick masonry conf. Stoke-on-Trent, UK, British Ceramic Research Association, West HWH, Speed KH editors, 1971, p. 31-37.
9. Hilsdorf H.K. Investigation into the failure mechanism of brick masonry loaded in axial compression. Designing, engineering and constructing with masonry products. Gulf Publishing Company, 1969:34-41.
10. Julio Garzón-Roca, Creu Obrer Marco, Jose Adam M. Compressive strength of masonry made of clay bricks and cement mortar: Estimation based on Neural Networks and Fuzzy Logic. Engineering Structures, 2013;48:21-27.
11. Kaushik H.B., Rai D.C., Jain S.K. Stress-strain characteristics of clay brick masonry under uniaxial compression. J. Mater Civil Eng. 2007;19(9):728-39.
12. Kecman V. Learning and Soft Computing: Support Vector Machines, Neural Networks, and Fuzzy Logic Models. Cambridge, Massachusetts, London, England, the MIT Press, 2001.
13. Khoo C.L., Hendry A.W. A failure criterion for brickwork in axial compression. Proc., 3rd int. brick masonry conference. Foertig L., Gobel K., editors, 1973, p. 139-145.
14. Lanckriet G.R.G., Ghaoui L.E., Bhattacharyya C., Jordan M.I. Minimax probability machine. T.G. Diet-terich, S. Becker, and Z. Ghahramani, editors. Advances in Neural Information Processing Systems 14. Cambridge, MA, MIT Press, 2002.
15. MacKay D.J., Bayesian methods for adaptive models. Ph.D. thesis. Pasadena, Calif, Dep. of Comput. and Neural Sysyt., Calif Inst. of Technol., 2001.
16. Mann W. Statistical evaluation of tests on masonry by potential functions. Sixth international brick masonry conference, 1982.
17. Marshall A.W., Olkin W. Multivariate chebyshev inequalities. Annals of Mathematical Statistics. 1960; 31(4): 1001-1014.
18. Park D., Rilett L.R. Forecasting freeway link ravel times with a multi-layer feed forward neural network. Computer Aided Civil and infra Structure Engineering. 1999; 14:358-367.
19. Popescu I., Bertsimas D. Optimal inequalities in probability theory: a convex optimization approach. Technical Report TM62, INSEAD, Dept. Math. O.R., Cambridge, Mass, 2001.
20. Sarangapani G., Venkatarama Reddy B.V., Jagadish K.S. Brick-mortar bond and masonry compressive strength. J. Mater Civil Eng. 2005;17(2):229-237.
21. Strohmann T.R., Grudic G.Z. Minimax probability machine regression. Advances in Neural Information Processing Systems (NIPS). 2003:15.
22. Tipping M.E. The relevance vector machine. Advances in Neural Information Processing Systems. 2000; 12:625-658.
23. Tipping M.E. Sparse Bayesian learning and the relevance vector machine. J. of Machine Learning Research. 2001; 211-244.
24. Wang T., Li Y., Wang H., Wang P. A RVM reliability analysis model for wireless sensor of networked synchronization control system. J. of Information and Computational Science. 2012;9(17):5345-5351.
25. Zhang Z., Cui J. Network intrusion detection based on robust wavelet RVM algorithm. J. of Information and Computational Science. 2011;8(14):2983-2989.
26. Zhou Y., Bao Y. Structural damage detection by integrating non-negative matrix factorization and relevance vector machines. Applied Mechanics and Materials. 2013;249-250:137-146.
THIS ARTICLE IN ENGLISH SEE NEXT PAGE
МЕХАНИКА: Механика деформируемого твердого тела
D0l.org/10.5281/zenodo.1119147 УДК 52-17
Пиюш Самуи, А.А. Бочарова
ПИЮШ САМУИ - доцент, кафедра гражданского строительства, e-mail: [email protected] Национальный институт технологий Патна Патна, штат Бихар, Индия, 80005
БОЧАРОВА АННА АЛЬБЕРТОВНА - заведующая кафедрой механики и математического моделирования Инженерной школы, e-mail: [email protected] Дальневосточный федеральный университет Суханова ул., 8, Владивосток, Россия, 690091
Моделирование прочности на сжатие структуры каменной кладки с использованием методов релевантных векторов и минимакса вероятности регрессии
Аннотация: Прочность на сжатие (f) структуры каменной кладки является ключевым параметром для ее проектирования. В данной статье используется метод релевантных векторов (RVM) и метод минимакса вероятности регрессии (MPMR) для оценки прочности на сжатие (f) структуры каменной кладки. RVM - байесовская модель, MPMR построен на основе концепции классификации минимаксных вероятностей. Прочность на сжатие раствора (fm) и прочность на сжатие кирпича (fb) используются в качестве входов RVM и MPMR моделей, выход RVM и MPMR равен f. Результаты RVM и MPMR были сопоставлены с другими моделями. Результаты показывают, что разработанные RVM и MPMR модели являются полезными инструментами для оценки структуры каменной кладки.
Ключевые слова: прочность на сжатие, каменная кладка, метод релевантных векторов, метод минимакса вероятности регрессии.
О статье: поступила: 30.10.2017; финансирование: бюджеты Национального института технологий, Индия и Дальневосточного федерального университета, Россия.
