Academic Open Internet Journal

ISSN 1311-4360

www.acadjournal.com

Volume 21, 2007

 

 

 

 

PREDICTING STRESS VARIATIONS IN A REPETITIVE DESIGN PROCESS USING ARTIFICIAL NEURAL NETWORKS

 

 

C Santosh Kumar 1, Mahendra K V 2

Senior Undergraduate Student and Corresponding Author 1, Head of the Department 2

Department of Mechanical Engineering, New Horizon College of Engineering,

Bangalore, Karnataka 560 087, India

Email: cskjain@gmail.com 

 

 

 

ABSTRACT: Design of a plate with geometrical discontinuity such as a hole or notch requires special attention, as the stress concentration near the hole will be higher than the average stress in the body. Due consideration should be given to stress concentration factor (Kt), as Kt varies phenomenally for the variation in the width (W) of the plate for a constant hole diameter (d). The plate with a hole is studied using finite element method. The results obtained are used to model the Artificial Neural Networks (ANN). In the present investigation, the stress distribution across the plate for varying values of the load & d/W is modeled using ANN. The network is utilized to predict the stress variations with varying load conditions for a constant d/W ratio and the least d/W ratio which provides minimum stress concentration in a plane with a hole.

 

Keywords: Stress Variations, Stress Prediction, FEM, Artificial Neural Networks.

 

 

1. INTRODUCTION

Metal plates are widely used in manufacturing industries. When metal plates are loaded across their ends, a uniform stress distribution exists throughout the metal plate. Many applications require notches or holes to be present in metal plate, under such conditions the stress concentration near the discontinuity will be higher than the average stress in the body and this varies with change in the width of the metal being used and the load acting on it. In these circumstances, an evaluation of the plate behavior should be performed in order to ensure the structural integrity during operation in such demanding loading conditions.

 

2. STRESS CONCENTRATION

 

A geometrical discontinuity in a body, such as a hole or a notch, results in a region of non-uniform stress distribution at the vicinity of the discontinuity. As some region near the discontinuity the stress will be higher than the average stress at distances away from the discontinuity. Hence, the discontinuity acts as a stress raiser.  If the discontinuity were not present, the stress would be uniform through out the cross section of the plate. This condition is usually formulated as

 

                                                              (2.1)

 

With the presence of the hole, the stress distribution is maximum at the edges of the hole and reduces in magnitude moving away from the hole. The severity of stress concentration is often measured by the stress concentration factor, defined as the ratio of stress at the notch-tip (σmax) in the plate to the remotely applied nominal stress (σο) as (Kudari, 2005):

                                                         (2.2)

 

The charts for stress concentration factors were formulated by Peterson (1974). In this investigation, a series of detailed stress distribution analysis were carried on the plate with a hole with different width and for varying load conditions using finite element method.

 

3. ARTIFICIAL NEURAL NETWORK

 

According to Haykin (1994):

“A neural network is a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use.” It resembles the brain in two respects:

 

1.       Knowledge is acquired by the network through a learning process.

2.       Interneuron connection strengths known as synaptic weights are used to store the knowledge.

 

In principle, artificial neural networks can compute any computable function, i.e., they can do everything a normal digital computer can do (Valiant, 1988; Sima and Orponen, 2001), or perhaps even more, under some assumptions of doubtful practicality (see Siegelmann, 1998, but also Hadley, 1999). Practical applications of artificial neural networks most often employ supervised learning. For supervised learning, the training data has to be that which includes both the input and the desired result (the target value). After successful training input data alone can be presented to the NN (that is, input data without the desired result), and the NN will compute an output value that approximates the desired result.

 

A Multi-Perceptron Backpropagation model was used for this work, which intended to compute the output, based on three inputs. Backpropagation refers to the method for computing the gradient of the case-wise error function with respect to the weights for a feedforward network, a straightforward but elegant application of the chain rule of elementary calculus (Werbos 1974/1994).

 

In a feedforward network, the network collects the input data and this information is processed when flowing from one layer to another hence, resulting in an output answer. This output answer is compared with the most approximate required answer, and subsequently the error is calculated. If the result is not satisfactory, then the algorithm of the network goes back to the previous step, thus back propagating, layer by layer and updating the weight needed from the output to the input.

 

4. NOMENCLATURE

 

A = cross – section area of the pipe (mm2)

P = load acting on the plate (MPa)

Kt = stress concentration factor

H = height of the plate (mm)

W = width of the plate (mm)

t = thickness of the plate (mm)

d = diameter of the hole (mm)

σ = stress (MPa)

σmax = maximum stress (MPa)

σο= nominal stress (MPa)

σx= nominal stress in x direction (MPa)

σy= nominal stress in y direction (MPa)

σ1,2= principal stresses (MPa)

τxy= shear stress on the plane perpendicular to the x axis in the direction of y axis (MPa)

σvon= Von-Mises equivalent Stress (MPa)

σint= stress intensity (MPa)

σsut= ultimate tensile strength (MPa)

σsyt =yield tensile strength (MPa)

E =Young’s modulus (GPa)

εp=Poisson’s ratio

 

 

5. FINITE ELEMENT ANALYSIS

 

A number of stress analyses are conducted on plate with a hole using finite element method. Standard finite element structural codes available on the market, e.g. ANSYS, Hyper mesh, MSC Nastran, and ABAQUS, have been introduced to evaluate the plate behavior. In this investigation, the commercial FE analysis package ANSYS was used to predict the plate behavior.  These analyses have been made for loads varying from 50 MPa to the load for which the maximum stress in the plate just crosses the ultimate strength the plate material. The plate material considered here is Steel1030. The Mechanical properties of Steel 1030 is as shown

 

Ultimate tensile strength,

517 MPa

Yield Tensile Strength,

441 MPa

Young’s Modulus,

204 GPa

Poisson’s ratio,

0.29

 

 

 

The analyses were conducted with a fixed diameter of hole and varying width for varying loads. The details of loading and the specimen geometry are as shown in figure 1(a). Due to symmetry of the specimen, only one quarter of the specimen was considered for modeling as shown in figure 1 (b).

 

Figure 1 (a) A plate with a Hole, 2H = 200mm, 2W=100mm, d=10mm

                   (b) Quarter portion of the plate used for Finite Element Analysis
 

 

 

 

 

 

 

Fig. Typical Axial Stress Distribution for the considered geometry

 
 

Figure 2 Geometry of the specimen considered

 
 
 

 

 


A six-nodded triangular two-dimensional element was considered assuming plane stress condition. The mesh is refined at the hole tip for proper stress distribution and is as shown in figure 2.  This indicates the high stress distribution at the discontinuity and the analysis also indicated the presence of bi-axial stresses at the discontinuity. A typical axial stress distribution for d/W = 0.2 is as shown in figure 3.

 

 

Figure 3 Typical Axial Stress Distribution for the considered geometry
 
 

 

 


The stress concentration factor () for different values of d/W as computed by Peterson is compared with the stress concentration factor computed by finite element method.

 

 

6. ANN MODELING AND SIMULATION

 

The ANN used in this investigation was a Feedforward Multilayer Network which is constituted by:

·         One Input Neural Layer (with 3 nodes supplied with varying data of the plate)

·         One Hidden Neural Layer (with 8 nodes)

·         One Output Neural Layer (with 13 nodes)

However, there is no connection between the Neurons in the same layer. The multilayer NN was trained using the back propagation algorithm. The back propagation algorithm is a methodology of decreasing in the gradient related to the quadratic error function. The Neural Network is as shown in figure 4.

 

Text Box: Figure 4 Artificial Neural Network



            The results obtained from the finite element analysis are considered as the training data for the neural network. The NN is supplied with the data and trained to get the average mean error below 0.01 corresponding to the expected output.

To test the ability of the neural network after training with 24 different sets of data obtained from the finite element analysis of the plate, the network is instructed to predict the stresses for varying load conditions within the test range and for varying d/W ratios. The network converged to the average mean error of 0.01after 275 iterations, resulting in sets of suggested stresses. A Plot of Error encountered while training and the number of Learning cycles is as shown in figure5.

 

Figure 5 Plot of Training Error and the number of Learning cycles
 
 

 

 



Text Box: Figure 6 The two boxes show how close the predicted value for each example output is to the true value. They are scaled from (0) minimum to (1) maximum value.

 

 

 

7. RESULTS

 

The modeling and simulation of the NN with the data obtained from finite element analysis in this investigation has produced considerably encouraging results. The results have been analyzed by their percentage errors.

 

 

           

 

 

This indicates that ANN is a powerful tool for modeling and predictive applications. The ability of ANN to learn from actual non-linear data and predict the results effectively for varying loading conditions makes this attractive. The areas examined during this work appear to have good practical applications in the industry.

 

The Trained Neural Network is at last made to predict the minimum d/W ratio for minimum stress, which it identifies as 0.1. Therefore, for minimum stresses to be induced in a plate with a hole, the optimum d/W ratio is 0.1.

 

 

8. CONCLUSION

 

It has been shown that Neural Networks are capable of predicting the stress variations for varying load conditions and d/W ratios by training them suitably. This method of predicting stress distribution involves less running cost and time, as just the input parameters need to be varied for querying the trained network.

 

This method can further be improved as an online stress prediction system, which will lead to further reduction in running costs. Further, to increase the accuracy of prediction, the Neural Networks are to be trained with more data with smaller variations in load for all the d/W ratios.

 

REFERENCES

 

1.                   Peterson, R.E.: Stress Concentration Factors, Wiley Publishing, 1974.

 

2.  Kudari, S. K. and Kodancha, K. G.: Effect of plastic deformation of stress concentration factor, Proceedings of National Conference on Advances in Engineering Design, BIT, Satyamangala, April 28-30, 2005, pp. 29-30.

3.  Haykin, Simon.: Neural Networks A Comprehensive Foundation, IEEE Press, Macmillan, New York, 1994.

4.  Valiant, L.G.: Functionality in Neural Nets, Proc. AAAI88, 7th National Conference on AI, Saint Paul MN, 1988, pp. 629-634.

5.  J. Sima and P. Orponen.: A Computational Taxonomy and Survey of Neural Network Models, submitted for publication (available from http://citeseer.nj.nec.com/ sima01computational.html), 2001.

6.  H. T. Siegelmann.: Neural Networks and Analog Computation: Beyond the Turing Limit, Birkhauser publishers, 1998.

7.  Hadley, R.F., and Cardei, V.C.: Connectionism and novel combinations of skills: implications for cognitive architecture, Minds and Machines, 9 (2), 1999, pp. 217- 235.

8.  Werbos, P.J. Beyond Regression: New Tools for Prediction and Analysis in the Behavioural Sciences, PhD dissertation, Harvard University, 1974.

9.  ANSYS Help System: Release 5.4, SAS IP, Inc. 1997.         

 

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