Abstract
In this paper, three artificial neural networks are presented using the experimental results from bolted moment connections among cold-formed steel members and the software MATLAB in order to predict the rotation at the connections. A common neural network which has a multilayer perceptron along with back propagation learning algorithm is applied in this research. Each of the networks consists of four layers including two hidden ones. The number of neurons in the first hidden layer is changed from 1 to 10 to achieve optimal results. The best results are obtained when the networks had 10, 10, and 9 neurons in the first hidden layer for column base and beam column connections (in positive and negative rotations), and they had the performance of 0.0001371, 0.00044, and 0.00047, respectively, after being trained in the software MATLAB. Thirty percent of the data from each test series were omitted randomly in order to verify the networks. The Mann–Whitney P value tests are 0.9933, 0.9393, and 0.9653 for column base and beam column connections (in positive and negative rotations), respectively.
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Introduction
Cold-formed steel
Lightweight steel structures offer an interesting alternative to more traditional construction technologies, especially for low and medium rise residential and office buildings (Casafont et al. 2006) due to some considerable advantages such as high strength-to-weight ratio, reduced labor costs, fast erection, and economical transportation due to the light weight of cold-formed members (Ebadi Tabrizi and Vosoughifar 2011; Yu and LaBoube 2010; Zaharia and Dubina 2006). Cold-formed steel sections had been considered as secondary structural members such as purlins to support roof cladding (Chung and Lau 1999). Since 1990, there has been a growing tendency to utilize cold-formed steel as structural members in buildings in a variety of countries such as USA, Canada, Australia, and Japan (Ebadi Tabrizi and Vosoughifar 2011; Wong and Chung 2002). The most common cold-formed steel sections are lipped C sections and lipped Z sections, and the thickness typically ranges from 1.2 to 3.2 mm. Common yield strengths are 280 and 350 N/mm2 (Chung and Lau 1999; Wong and Chung 2002). Moreover, there is a whole range of variants of these basic shapes, including sections with single and double lips and sections with internal stiffeners (Chung and Lau 1999). Furthermore, welds, screws, and steel bolts are used as connector for these sections (AISI 2008; Pedreschi et al. 1998).
There are various design recommendations (AISI 19962008; British Standard Institution 1987; Code AS/NZ 4600 2005; Eurocode EN 1993; ECCS Committee TC7, Working Group TWG 7.2 1983) on the design of cold-formed steel structures along with complementary design guides (AISI 19952007; ECCS Committee TC7, Working Group TWG 7.2 1983; Hancock 2007; LaBaube and Yu 2006; Martin and Purkiss 2011; Zaharia and Dubina 2006) and worked examples to assist practicing engineers. ‘For bolted moment connections, the presence of bolts will introduce high localized stresses onto the connected parts of the members which may have adverse effects on the section resistances of the members. There is a lack of general design rules for moment connections among cold formed steel members in the literature particularly for open sections such as lipped C sections’ (Chung and Lau 1999).
In this paper, it is attempted to obtain some nonlinear equations between the results achieved from Chung and Lau’s investigation (Chung and Lau 1999) by means of the artificial neural networks (ANNs).
Artificial neural networks
Neural networks are learning systems based on a simplified model of the biological neuron, which can model the relation between a set of inputs and a set of outputs. In the same way as the biological neural network changes itself in order to perform some cognitive task (such as recognizing faces or learning a concept), artificial neural networks modify their internal parameters in order to perform a given computational task. The typical tasks neural networks that perform efficiently and effectively are as follows: classification (for example deciding which category a given example belongs to or the identification of a disease from some symptoms), recognizing patterns in varied data, and prediction. Artificial neural networks are usually non-parametric approaches represented by connections between a very large number of simple computing processors or elements (neurons). The ANN is trained by supplying it with a large number of numerical observations or the patterns to be learned whose corresponding classifications are known. During training, the final sum-of-squares error over the validation data for the network is calculated. The selection of the optimum number of hidden nodes is made on the basis of this error value. Once the network is trained, a new object is classified by sending its attribute values to the input nodes of the network, applying the weights to those values, and computing the values of the output units.
The multilayer perceptron is generally divided into three layers: the input layer, the hidden layer, and the output layer, where each layer in this order gives the input to the next. The extra layer gives the structure needed to recognize non-linearly separable classes. Figure 1 demonstrates a general architecture of an artificial neural network with two hidden layers.
Applying artificial neural networks for the research
In this study, three diverse ANNs are trained in order to predict the rotation of column base connections and beam column connections under different applied moments using an empirical data (Chung and Lau 1999), and in order to obtain the best possible results, the number of neurons was changed in the first hidden layer from 1 to 10. The network which had the best performance was selected for each test series. The achieved results show the sufficient compatibility of the ANNs with experimental data.
Methods
In this investigation, the empirical data (Chung and Lau 1999) in each test series are divided into two groups as training and testing data. The first one includes 70% of the whole data and the second consists of the remaining 30%. As the names imply, they were used to train and test the networks consecutively. In addition, each group includes two sub-groups known as the input and target data. Using sigmoid function which has a range between 0 and 1, it is necessary to normalize the whole data before applying to the software.
Experimental method
Chung and Lau 1999conducted an experimental investigation on the structural performance of cold-formed steel members with bolted moment connections utilizing lipped C sections. They experienced various specimens with different bolt arrangements to examine the effect of bolt arrangement on the structural performance of column base and beam column connections in order to obtain the optimal connection configuration and evaluate the rational amount of maximum rotational resistance. In the present research, three column base connection specimens and four beam column connection specimens which were configured based on the former results of Chung’s investigation on beam column connections are proposed.
Generation of the input data
Column base connections
In order to generate the input data, an innovative method is applied based on the effective area of each bolt. That is, six general bolt positions (a, b, c, d, e, and f) are defined as shown in Figure 2 which displays all possible bolt positions in column base connections (the position of a column base connection is shown in Figure 3). This method results in six kinds of input data. Additionally, the moment resistance ratio of each specimen, which is defined as equivalent to according to the experimental research article Chung and Lau (1999), and the applied moment (the diverse moments which applied to the connections in the experimental research in any rotation) are considered as two other kinds of input data. Three different bolt arrangements which were experimented in Chung et al.’s experimental investigation are shown in Figures 4, 5, 6 and the related equations to generate the input data are shown next to each figure (Ebadi Tabrizi and Vosoughifar 2011). All lengths and areas in Figures 4, 5, 6 are based on millimeters and square millimeters, respectively, and the title of the specimens are the same as the empirical study (Chung and Lau 1999).
Set 1 equation is related to Figure 4:
In Figure 4, half of the section is considered as the effective area of each bolt. In those positions which there are no bolts, the number 0 is supposed as the related input for the position.
Set 2 equation is related to Figure 5:
In Figure 5, after drawing a triangle by connecting the bolt positions a, c, and e, a perpendicular bisector of each segment was drawn. Ultimately, the effective area of each bolt was determined as shown in the figure. In those positions which there are no bolts, the number 0 is supposed as the related input for the position.
Set 3 equation is related to Figure 6:
In Figure 6, one-fourth of the section is considered as the effective area of each bolt. In those positions which there are no bolts, the number 0 is supposed as the related input for the position.
Table 1 (Ebadi Tabrizi and Vosoughifar 2011) shows some random data related to the test series. The first eight columns in the table are some parts of the 70 × 8 input matrix and the last is a part of the 70 × 1 target matrix.
Beam column connections
In this test series, the innovative method is somewhat different. In these connections, the effective area of each bolt row is considered rather than that of each single bolt to generate input data as shown in Figures 7, 8, 9 and the related equations. This method results in four kinds of input data. Additionally, the moment resistance ratio of each specimen, ψ, defined above, and the moment ratio are considered as two other kinds of input data. It is important to mention that the moment ratio in the empirical study (Chung and Lau 1999) is the applied moment, defined above, divided by 178 kNm. The unit of all lengths and areas in Figures 7, 8, 9 are millimeters and square millimeters, respectively, and the title of the specimens are the same as the empirical study (Chung and Lau 1999).
Set 4 equation is related to Figure 7:
Set 5 equation is related to Figure 8:
Set 6 equation is related to Figure 9:
Table 2 demonstrates some random data related to the positive rotations in beam column connection experiment. The first six columns in the table are some parts of the 88 × 6 input matrix and the last is a part of the 88 × 1 target matrix. For negative rotations, all data are the same, but those related to the moment ratio and rotation are different.
Training the neural network
After generating the input and target data and training the network, a nonlinear relation between input and target data would be established. For this purpose, tansig and pureline functions are utilized as two hidden layers of the neural networks in the software MATLAB. Achieving the desired results is in accordance with the mean squared error (MSE) method which is based on reaching the minimum error (Equations 1 to 3) (Howard and Mark 2006).
In Equation 2, I demonstrates the target function (amount of rotation in normalization space), c represents column matrix of input data in normalization space, w shows the amount of weight, and b exhibits the amount of bias.
In fact, the main equation would be as in Equation 3.
Results
After generating the input and target data as explained in the ‘Generation of the input data’ section for all of the specimens on column base connections and randomly removing 30% of them, a matrix of 70 × 8 as the input and a matrix of 70 × 1 as the target resulted. Likewise, for positive rotations on beam column connections, a matrix of 88 × 6 as the input and a matrix of 88 × 1 as the target and for negative rotations a matrix of 97 × 6 and a matrix 97 × 1 as the input and target matrixes were obtained. In order to reach the best results, the number of neuron in the first hidden layer (tansig) were changed from 1 up to 10, and for the second one, the number remained 1 consistently. For column base connections, the best result was obtained when the first layer had 10 neurons. In the same way, for beam column connections, the best results for positive rotations were obtained when the first hidden layer had 10 neurons, and for the negative ones, when it had 9 neurons.
Obtained weights and bias
Weight and bias matrixes for column base connections
Weight and bias matrixes for positive rotations on beam column connections
Weight and bias matrixes for negative rotations on beam column connections
Ultimately, the nonlinear relation created between both groups of data will be as in Equation 4:
In Equation 4, R represents the amount of rotation in the connection.
After training the networks for each test series and converting those data, which are in normalized amounts, to the real amounts, the inputs of training and testing data were given to the software in order to compare the amounts of real rotations with those calculated by the neural networks. Figures 10, 11, 12 provide these comparisons on the graphs for the training data and Figures 13, 14, 15 do so for the testing ones.
Discussion
In order to make a better verification, the software SPSS was used in order to attain the amounts of P value and compare the amounts of rotation resulting from the ANNs and those obtained from the empirical tests in each test series for both testing and training data. The closer amount of P value to 1 demonstrates a better agreement between the compared data. On the other hand, less than 0.05 exhibits a marked difference between them. The calculated P values, related to each comparison, are shown in Figures 10, 11,12, 13,14, 15.
It is worthwhile to mention that because the current research is based on a number of specimens, a numerical analysis and more empirical studies would be useful to have a better validation of the proposed ANN models.
Conclusions
Based on the study, the following points were obtained:
-
1.
The proposed neural network models have the ability to create appropriate nonlinear relations to evaluate the amount of rotation at bolted moment connections among cold-formed structural members. As a result, using the innovative method, the amount of rotation would be calculated.
-
2.
The achieved results prove that the innovation method applied to generate the input data for bolt arrangements suits the experimental model.
-
3.
Evaluating the rotation at bolted moment connections in cold-formed steel members by the innovative method could be helpful in compiling design recommendation as the rotation would be computed accurately and rapidly.
-
4.
The obtained amounts of P value prove that the achieved results match the empirical results.
Authors' information
BET graduated B.S. Civil Engineering from the Islamic Azad University, South Tehran Branch and started his scientific researches 2 years ago starting with the first part of this paper (ANN for column base connections in CFS) which was presented in the 6th International Conference on Seismology and Earthquake Engineering (SEE6). He has two other full papers. One was recently presented in 15th World Conference on Earthquake Engineering Lisbon, Portugal (15WCEE) entitled A Two-Ring Energy Dissipating Device with Similar Behaviors in Tension and Compression to Create Buckling Resistant Braces. Another full paper is regarding the evaluation of ASCE7-10 which is under review. Dr. MH is an associate professor at the Structural Engineering Research Center, board member of the Center of Excellence on Risk Management, and head of the Lifeline Engineering Department, The International Institute of Earthquake Engineering and Seismology (IIEES). He has more than 250 publications in a wide variety of local and international journals and conferences including the Journal of Structural Design of Tall and Special Buildings.
Abbreviations
- ANNs:
-
Artificial neural networks
- MSE:
-
Mean squared error.
References
AISI: Cold formed steel design manual. AISI, Washington; 2008.
AISI: Load and resistance factor design specification for cold formed steel structural members. LRFD cold-formed steel design manual, part 1. AISI, Washington; 1996.
AISI: Design guide D110–07. 2nd edition. AISI, Washington; 2007.
AISI: Fastening of cold-formed steel framing. AISI, Washington; 1995.
British Standard Institution: Structural use of steelwork in buildings. Part 5: code of practice for the design of cold formed sections. BSI, London; 1987.
Casafont M, Arnedo A, Roure F, Rodriguez-Ferran A: Experimental testing of joints for seismic design of lightweight structures. Part 1. Screwed joints in straps. Int J Thin Walled Struct 2006, 44: 197–210. 10.1016/j.tws.2006.01.002
Chung KF, Lau L: Experimental investigation on bolted moment connections among cold formed steel. Int J Eng Struct 1999, 21: 898–911. 10.1016/S0141-0296(98)00043-1
Code AS/NZ 4600: Cold-formed steel structures. SAI Global Ltd, Sydney; 2005.
Ebadi Tabrizi B, Vosoughifar HR Paper presented at the 6th international conference on seismology and earthquake engineering. In Usage neural network to create relation between bolt arrangements on foundation column connection in Light weight steel structure. Milad Tower Convention Center, Tehran; 2011. 16–18 May 2011 16–18 May 2011
Eurocode EN: Eurocode 3: design of steel structures. Part 1.3: general rules—supplementary rules for cold formed thin gauge members and sheeting. European Committee for Standardization (CEN), Brussels; 1993.
ECCS Committee TC7, Working Group TWG 7.2: European recommendations for steel construction: the design and testing of connections in steel sheeting and sections. Publication no. 21 and 35 ECCS Committee TC7, Working Group TWG 7.2. ECCS, Brussels; 1983.
Hancock GJ: Design of cold-formed steel structures. 4th edition. Australian Steel Institute, Sydney; 2007.
Howard D, Mark B: Neural network toolbox: for use with MATLAB. User’s guide. The MathWorks, Inc., Natick; 2006.
LaBaube RA, Yu WW: Design of cold-formed steel structural members and connections for cyclic loading (fatigue). Missouri S&T, Rolla; 2006.
Martin L, Purkiss JA: Structural design of steelwork to EN 1993 and EN 1994. Elsevier, Butterworth-Heinemann; 2011.
Pedreschi R, Sinha B, Davies R, Lennon R: Factors influencing the strength of mechanical clinching. In: Fourteenth international specialty conference on cold-formed steel structures. University of Missouri-Rolla, St. Louis, Missouri; 1998. 15–16 October 1998 15–16 October 1998
Wong MF, Chung KF: Structural behavior of bolted moment connections in cold-formed steel beam-column sub-frames. Int J Construct Steel Res 2002, 58: 253–274. 10.1016/S0143-974X(01)00044-X
Yu WW, LaBoube RA: Cold-formed steel design, 4th edition (Trans: Mir ghaderi R). Wiley, New York; 2010.
Zaharia R, Dubina D: Stiffness of joints in bolted connected cold-formed steel trusses. Int J Construct Steel Res 2006, 62: 240–249. 10.1016/j.jcsr.2005.07.002
Acknowledgments
The authors wish to acknowledge Mr. Younes Mirzaaghaee, Ms. Azam Dolatshah, and Mr. Ahmad Ghodselahi for their valuable assistance regarding the study.
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Authors' contributions
BET introduced and promote the innovation used in this study and performed the tasks related to the utilized computer programs. MH was the supervisor of the study and the organizer of the article. Both authors read and approved the final manuscript.
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Tabrizi, B.E., Hosseini, M. Creating relation between bolt arrangements at bolted moment connections among cold-formed steel members using artificial neural network. Int J Adv Struct Eng 4, 6 (2012). https://doi.org/10.1186/2008-6695-4-6
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DOI: https://doi.org/10.1186/2008-6695-4-6