Journal of Constructional Steel Rearch64(2008)
624–633
/locate/jcsr
Neural networks for modelling ultimate pure bending of steel circular tubes
Mohamed Shahin a,∗,Mohamed Elchalakani b,1
a Department of Civil Engineering,Curtin University of Technology,Perth,WA6845,Australia
周密简介b School of Architectural,Civil and Mechanical Engineering,Victoria University,Melbourne,VIC8001,Aust
ralia
Received12July2007;accepted8December2007
Abstract
The behaviour of steel circular tubes under pure bending is complex and highly nonlinear.The literature has a number of solutions to predict the respon of steel circular tubes under pure bending;however,most of the solutions are complicated and difficult to u in routine design practice.In this paper,the feasibility of using artificial neural networks(ANNs)for developing more accurate and simple-to-u models for predicting the ultimate pure bending of steel circular tubes is investigated.The data ud to calibrate and validate the ANN models are obtained from the literature and compri a ries of49pure bending tests conducted on fabricated steel circular tubes and55tests carried out on cold-formed tubes.Multilayer feed-forward neural networks that are trained with the back-propagation algorithm are constructed using four design ube thickness,tube diameter,yield strength of steel and modulus of elasticity of steel)as network inputs and the ultimate pure bending as the only output.A nsitivity analysis is conducted on the ANN models to investigate the generalization ability(robustness)of the developed models,and predictions f
rom the ANN models are compared with tho obtained from most available codes and standards.To facilitate the u of the developed ANN models,they are translated into design equations suitable for spreadsheet programming or hand calculations.The results indicate that ANNs are capable of predicting the ultimate bending capacity of steel circular tubes with a high degree of accuracy,and outperform most available codes and standards.
Crown Copyright c 2007Published by Elvier Ltd.All rights rerved.
Keywords:Neural networks;Back-propagation;Steel circular tubes;Pure bending
1.Introduction
Circular hollow steel tubes have good energy absorption characteristics under pure bending,thus,have been ud in veral large-scale engineering applications such as offshore pipelines and platforms;chemical and nuclear power plants; and land-bad pipelines.The deformation of circular tubes under bending exhibits significant changes to their cross ction profile along the tube length through what is known as ovalization[1,2].This phenomenon is highly non-linear and makes the behaviour of circular tubes under bending very complex.An accurate simulation of the behaviour of circular tubes under bending using the conventional analytical solutions requires
rigorous mathematical procedures that are difficult to achieve from the pragmatic point of view.Most available ∗Corresponding address:Department of Civil Engineering,Curtin Univer-sity of Technology,GPO Box U1987,Perth,W A6845,Australia.Tel.:+618 92661822;fax:+61892662681.
E-mail address:m.shahin@curtin.edu.au(M.Shahin),
mohamed.elchalakani@vu.edu.au(M.Elchalakani).
1Tel.:+61399194727;fax:+hods for predicting the pure bending of circular tubes (e.g.[3–7])incorporate veral assumptions to simplify the problem and to make it amenable to solution,which in turn, affects the modelling accuracy.In this respect,the artificial neural networks(ANNs),which do not need incorporation of any assumptions or simplifications,are more efficient.
In recent times,ANNs have been applied successfully to different structural engineering [8–11]). ANNs are numerical modelling techniques inspired by the functioning of the human brain and nervous system.The ANNs modelling philosophy is similar to that ud in the development of more conventional statistical models.In both cas,the purpo of the model is to capture the relationship between a historical t of model inputs and corresponding outputs. Howeve
r,unlike most available statistical methods,ANNs do not need predefined mathematical equations of the relationship between the model inputs and corresponding outputs and rather u the data alone to determine the structure of the model and unknown model parameters.This enables ANNs to overcome the limitations of existing modelling methods.The
0143-974X/$-e front matter Crown Copyright c 2007Published by Elvier Ltd.All rights rerved. doi:10.1016/j.jcsr.2007.12.001
M.Shahin,M.Elchalakani/Journal of Constructional Steel Rearch64(2008)624–633625新学期黑板报
main objective of this paper is to explore the feasibility of using ANNs for developing accurate models to prediction of ultimate pure bending of fabricated and cold-formed steel circular tubes.The predictive ability of the ANN models is examined by comparing their results with experimental data,and with tho obtained from most available codes and standards.The robustness of the ANN models is further investigated in a nsitivity analysis.
2.Development of ANN models
The type of neural networks ud in this study are multilayer feed-forward that are trained with the b
ack-propagation algorithm[12].A comprehensive description of this type of neural networks is beyond the scope of this paper and can be found in many [13,14]).The typical structure of a multilayer feed-forward neural network consists of a number of processing elements(also called nodes or neurons) that are fully or partially linked via connection weights.The processing elements are usually arranged in layers:an input layer;an output layer;and one or more layer in between called hidden layers.Training of a multilayer feed-forward neural network commences at the input layer,where the network is prented with an actual measured t of he training t)and the output of the network is obtained by utilizing a learning rule.The network output is compared with the desired output from which an error is calculated.This error is then ud to adjust the connection weights so that the best input/output mapping is obtained.Once training has been successfully accomplished,the performance of the trained model has to be verified using an independent validation t.
In this work,two ANN models(one for fabricated steel tubes and another for cold-formed tubes)are developed us-ing the computer-bad software package NEUFRAME Ver-sion4.0[15].The data ud to calibrate and validate the ANN models are obtained from the literature and include a ries of 49ultimate pure bending tests that were conducted on fabri-cated steel circular tubes,and55tests that
were carried out on cold-formed tubes.The49tests of fabricated tubes compri a number of27tests reported by Sherman[2,16],ten tests by Schilling[17],four tests by Jirsa et al.[18]and eight tests by Korol and Huboda[19].The55tests of cold-formed tubes were reported by Elchalakani et al.[20–23].The databa ud for the fabricated and cold-formed tubes are given in Tables1and 2,respectively.Tables1and2also include the predicted ulti-mate bending by ANN M u(ANN)(column7)and ratio of mea-sured to predicted ultimate bending M u/M u(ANN)(column8).
2.1.Model inputs and outputs
Four variables are prented to the ANN models as inputs. The include the tube thickness,t,tube diameter,d,yield strength of steel,f y,and modulus of elasticity of steel,E.The single model output is the ultimate pure bending,M u.
2.2.Data division and preprocessing
For each ANN fabricated model and cold-formed model),the available data are randomly divided into two statistically consistent ts:training and validation,as recommended by Masters[24]and detailed by Shahin et al.
[25].The statistics of the data ud for training and validation ts are given in Table3,which include the mean;standard deviation;minimum;maximum;and range.In total,80%of the data records are ud for training and20%for validation.It should be noted that,like all empirical models,ANNs perform best when they do not extrapolate beyond the range of the data ud for model training[26]and conquently,the extreme values of the available data should be included in the training t,as shown in Table3.Once the available data have been divided into their subts,the input and output variables are preprocesd by scaling them between0.0and1.0,to eliminate their dimension and to ensure that all variables receive equal attention during training.The simple linear mapping of the variables’practical extremes to the neural network’s practical extremes is adopted for scaling,as it is the most common method for this purpo[24].
2.3.Model architecture,weight optimization and stopping criterion
Following the data division and preprocessing,the optimum models he number of hidden layers and corresponding number of hidden nodes)and weight lection of the optimal learning rate and momentum term that control the training process)must be determined.It should be noted that a network with one hidden layer can approximate any continuous function provided that sufficient connection weights are ud[27,28].Therefore,one hidden layer is ud in the
current study.The optimal number of hidden nodes is obtained by a trial-and-error approach in which the networks are trained with a t of random initial weights and hidden layer nodes of2,3,...,and(2I+1), respectively,where I is the number of input variables.It should be noted that(2I+1)is the upper limit for the number of hidden layer nodes needed to map any continuous function for a network with I inputs,as discusd by Caudill[29].For each number of hidden layer node from the previous step,different combinations of learning 0.05,0.1,0.2,0.4,0.6,0.8, 0.9and0.95)and momentum 0.01,0.05,0.1,0.2, 0.4,0.6,0.8and0.9)are ud with a tan h transfer function in the hidden layer nodes and a sigmoidal transfer function in the output layer nodes.To determine the criterion that should be ud to terminate the training process,the scaled mean squared error(MSE)between the actual and predicted values on the validation t is monitored until no significant improvement in the error occurs.This is achieved at approximately5000 training cycles(epochs).
2.4.Model performance and robustness
The performance of the optimum ANN models for fabricated and cold-formed tubes is shown in Figs.1and2, respectively.It can be en from Fig.1that the predictions of fabricated tubes model has a high coefficient of correlation,r, of0.99in both the training and testing ts,and Fig.2shows
626M.Shahin,M.Elchalakani/Journal of Constructional Steel Rearch64(2008)624–633
Table1
Summary of the data ud for the fabricated tubes ANN model development
Reference t(mm)d(mm)f y(MPa)E(MPa)M u(kN m)M u(ANN)(kN m)Mu,M u(ANN) [2]26.64572791996301442.11473.00.98冬瓜玉米汤
[2]18.74572991998991237.01175.6 1.05
[2]16.54583382003591198.11176.9 1.02
[2]13.1458299200353830.9855.10.97
[2]9.9458294200590562.9594.20.95
[2] 6.9458325199433381.5387.70.98
[2] 6.1456314200909346.6324.9 1.07
[2] 6.3456309201066358.6332.4 1.08
迷人远足
[2]12.96103142000801490.11528.10.98社区活动有哪些
[2] 6.8610373201196810.4797.0 1.02
[2]25.44573741999401892.71743.3 1.09
[2]19.64583901998591408.61493.40.94
[2]18.84553671989431391.71402.10.99
[2]16.44584241999201302.91308.4 1.00
[2]13.34584111999311111.51024.5 1.08
[2]10.0458410200485783.4735.7 1.06
[2] 6.8458434199501538.8494.0 1.09
[2]13.66104051991381729.71724.6 1.00
[2]13.76083781997201828.21682.1 1.09
[2]7.0609429200617918.3934.90.98
[2]9.96084012002511317.21338.90.98
[16]14.9273290210000306.1195.8 1.56
[16]7.8273304210000160.094.0 1.70
[16] 5.6273405210000150.9125.7 1.20
[16] 4.9273419210000139.7127.3 1.10
[16] 3.527328721000064.735.6 1.82
[16] 2.527331121000048.833.9 1.44
[17] 1.91092692100007.1 6.2 1.15
[17] 1.4103270210000 4.3 5.60.77
[17] 1.1105270210000 3.2 5.40.59
[17]0.94103245210000 2.3 4.80.48
[17]0.76104267210000 1.6 5.00.32
[17] 2.910035821000010.312.30.84
[17] 2.312535921000012.213.50.90
[17] 1.61123572100007.410.60.70
[17]0.9989370210000 2.79.20.29
[17] 1.3117394210000 6.215.50.40
[18] 5.9273380210000167.2110.7 1.51
[18]8.9273334210000232.1130.4 1.78
[18] 6.6406342210000385.7286.8 1.34
[18] 6.5508375210000593.1511.7 1.16
[19] 3.911430821000015.411.5 1.34
[19] 3.916830521000033.217.1 1.94
财富的近义词
[19] 4.816836821000048.332.1 1.50
[19] 5.6273306210000126.762.7 2.02
[19] 6.4324377210000248.9176.5 1.41
[19] 6.4356297210000231.3150.4 1.54
[19] 6.3406309210000297237.8 1.25
[19] 6.4508362210000509.5487.5 1.05
that the predictions of cold-formed tubes model has also high r of0.99and0.95in the training and testing ts,respectively. Figs.1and2also demonstrate that there is a little scatter around the line of equality between the measured and predicted values of ultimate bending capacity in the training and testing ts for both fabricated and cold-formed tubes models.
To further examine the generalization ability(robustness) of the ANN models,a nsitivity analysis is carried out that demonstrates the respon of predicted model ultimate bending to a t of hypothetical input data that lie within the range of the data ud for model training.For example,the effect of one input variable,such as tube thickness,t,is investigated by allowing it to change while all other input variables are t to lected constant values.The inputs are then accommodated in the ANN model,and the predicted ultimate pure bending is calculated.This process is repeated for the next input variable and so on,until the model respon has been examined for all inputs.The robustness of the ANN model is determined by examining how well the predictions compare with available structural knowledge and experimental data.The results of the nsitivity analysis for fabricated and cold-formed tubes models are shown in Fig.3.It should be noted that the range of the
M.Shahin,M.Elchalakani/Journal of Constructional Steel Rearch64(2008)624–633627 Table2
Summary of the data ud for the cold-formed tubes ANN model development
Reference t(mm)d(mm)f y(MPa)E(MPa)M u(kN m)M u(ANN)(kN m)M u/M u(ANN) [20] 2.53101.83651998008.88.880.99
[20] 2.6088.64322095008.07.74 1.03
[20] 2.4576.3415217100 5.1 4.88 1.05
[20] 3.3589.34122179009.910.090.98
[20] 2.4460.6433211100 3.1 3.200.97
[20] 3.2476.24562111007.67.790.98
[20] 3.0160.6408204700 4.2 4.05 1.04
[20] 1.9833.64422042000.8 1.130.71
[20] 2.6333.8460207100 1.1 1.08 1.02
[21] 1.10110.1408190900 3.9 4.100.95
[21] 1.00109.9408190900 3.7 3.59 1.03
[21]0.90109.7408190900 3.4 3.13 1.09
[21] 1.25110.4408190900 4.5 4.990.90
[21] 1.7098.6410212300 5.8 5.66 1.02
[21] 1.2098.8404191200 4.3 3.62 1.19
[21] 1.4099.2404191200 4.9 4.35 1.13
[21] 1.6099.6365199800 5.4 5.19 1.04
[21] 1.80100.0365199800 5.3 5.780.92
[21] 2.3099.84102123008.98.52 1.04
[21] 2.4087.3412217900 5.7 6.270.91
[21] 2.10100.64041912007.57.580.99
[22] 2.44101.83652000008.78.52 1.02
[22] 2.5289.3378182000 6.4 6.590.97
[22] 2.1776.3415217000 4.7 4.52 1.04
[22] 3.1089.34122180009.49.450.99
[22] 2.2360.7433211000 3.0 3.090.97
[22] 3.0776.24562110007.7 6.99 1.10
[22] 2.9060.7408205000 3.7 3.970.93
[22] 2.4033.8460207000 1.1 1.08 1.02
[22] 2.44101.83652000008.48.520.99
[22] 2.44101.83652000008.78.52 1.02
[22] 2.5289.3378182000 6.7 6.59 1.02
[22] 3.0889.147320100010.010.20.98
[22] 2.2960.2407211000 3.3 3.470.95
[22] 3.0776.24562110007.4 6.99 1.06
[22] 2.9560.37413196000 4.0 4.200.95
[22] 2.54101.140019000010.59.51 1.10
[22] 2.5289.33781820007.2 6.58 1.09
[22] 2.3576.1370202000 4.6 4.990.92
[22] 3.0889.147320100010.610.23 1.04
[22] 2.2960.2407211000 3.3 3.470.95
[22] 3.1375.9402198000 6.7 6.54 1.02
[22] 2.9560.4413196000 4.3 4.21 1.02
[22] 2.5289.33781820007.3 6.59 1.11
[22] 2.2960.23407211000 3.8 3.47 1.10
[22] 2.9560.4413196000 4.6 4.21 1.09
[23] 2.54101.14001900008.79.510.91
[23] 2.5289.3378182000 6.4 6.590.97
[23] 2.3576.1370202000 4.3 4.990.86
[23] 3.0889.14732010009.810.230.96
[23] 2.2960.2407211000 3.3 3.470.95
[23] 3.1375.9402198000 5.9 6.530.90
[23] 2.9560.4413196000 4.1 4.200.98
[23] 2.5289.3378182000 6.2 6.590.94
[23] 2.9560.4413196000 4.0 4.200.95
hypothetical data ud for fabricated tubes model is different from that lected for cold-formed tubes model as each model is trained with different range of model inputs,as shown in Table3.It can be en from Fig.3that predictions of the behaviour of ultimate bending from the ANN models agree
well with the experimental results in the n that the bending moment increas with the increa of the tube thickness,tube diameter,yield strength of steel and modulus of elasticity of steel for both fabricated and cold-formed tubes.It can also be en that,within the range of the modulus of elasticity ud for ANN models training,the cold-formed tubes em to be less nsitive to the modulus of elasticity of steel than that of the
628M.Shahin,M.Elchalakani/Journal of Constructional Steel Rearch64(2008)624–633
Table3
ANN input and output statistics
Model Variables and data ts Statistical parameters
Mean SD a Min.b Max.c Range Fabricated tubes Tube thickness,t(mm)
Training t8.4 6.60.7626.625.8
Validation t7.8 5.3 1.618.817.2
Tube diameter,d(mm)
自主是什么意思
Training t347.7180.789.0610.0521.0
Validation t343.5132.9112.0458.0346.0
Yield strength of steel,f y(MPa)
Training t340.351.8245.7434.0188.3
Validation t354.846.6294.0411.0117.0
Modulus of elasticity of steel,E(MPa)
Training t205918496919913821000010861姚贝娜
Validation t205101519119894321000011056
Ultimate pure bending,M u(kN m)
Training t564.1608.7 1.611892.71891.1
Validation t469.7484.67.41391.71384.3 Cold-formed tubes Tube thickness,t(mm)
Training t 2.450.610.9 3.35 2.45
Validation t 2.350.59 1.2 3.07 1.87
Tube diameter,d(mm)
Training t82.620.733.7110.476.7
Validation t75.815.760.299.639.4
Yield strength of steel,f y(MPa)
Training t411.329.9365.0473.0108.0
Validation t397.827.8365.0456.091.0
Modulus of elasticity of steel,E(MPa)
Training t2013281108818200021800036000
Validation t201100943618200021100029000
Ultimate pure bending,M u(kN m)
Training t 6.1 2.60.8110.69.8
Validation t 4.7 1.5 3.37.7 4.4
a SD indicates standard deviation.
b Min.indicates minimum.
c Max.indicates maximum.
fabricated tubes.This is in agreement with the results of the tests performed on large size fabricated cylinders reported by Sherman[2,16],where it was found that the buckling strength increas with the elastic modulus of steel tubes.On the other hand,this phenomenon was not obrved in the tests carried out by Elchalakani et al.[21]on small size cold-formed tubes, which was most likely due to the residual stress from rolling and welding of cold-formed tubes.The results of the nsitivity analysis indicate that the developed ANN models are robust and can be ud with confidence.
3.Comparison of ANN models with available codes and standards
In order to examine the accuracy of the ANN models, they are compared with available design codes and standards currently ud in practice.The codes considered are Eurocode 3[30],Australian New Zealand Standards AS/NZS4600[31], Australian Standards AS4100[32]and American Institute of Steel Construction ASIC[33].Details of formulae and definitions of parameters needed for each method are given in Table4.It can be en from Table4that the circular steel tubes are classified as compact,C,noncompact,N,and slender, S,bad on their ctions slendernessλs(for AS4100and AS/NZS4600)andα(for AISC).On the other hand,the tubes are classified as Class1,2,3or4for Eurocode3.It should be noted that compact ctions often achieve their plastic bending capacity,M p,and adequate rotation capacity,R1,in the tests and thus,their full plastic bending capacity is specified in the design codes.Noncompact ctions could achieve M p but do not have adequate rotation capacity,and slender ctions cannot achieve their yield capacity,M y.
The performance of the ANN models and available codes, in relation to the validation t ud for ANN models development,is prented graphically in Figs.4and5,and summarized analytically in Tables5and6.It can be en from Figs.4and5that the predictions obtained from the ANN models exhibit less scatter around the line of equality than tho
