The results above have shown how various parameters like pipe geometry, dent geometry and pipe material can affect the total strain in the dent. These parameters are very important in predicting the total strain. From the discussions presented above, the below conclusions can be drawn
1. Circumferential strains are higher in circumferential dents compared to longitudinal strains and reduce as the diameter to thickness ratio increases
0.00 0.05 0.10 0.15 0.20 0.25 0 20 40 60 80 100 120 max strai n D/t
FEA predicted strain ASME B31.8 strain Noronha et al
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2. Longitudinal strains are higher in longitudinal dent compared to circumferential strains and reduce as D/t increases
3. The difference between the circumferential and longitudinal strain is higher in circumferential dent compared to that of the longitudinal dent.
4. The total strain in a dent is higher in a longitudinal dent compared to a circumferential dent
5. As the dent depth increases, there is a corresponding increase in the strain for both dent models
6. Pipes with higher material grades exhibits higher strain and the difference in strain level gets smaller as the Diameter to thickness (D/t) increases
7. The ASME and the FEA predicted strain shows similarity in patterns in regards to how the parameters affect the strain, however, the ASME equation under- predicted the strain
8. The proposed formula by Noronha el al[21] gave a good correlation to the FEA results
9. ASME needs to revise equation to consider the radial components of the strain and also consider plane strain state
The objective of this chapter is to study the effect of these parameters on the strain prediction. The data extracted from the FEA study is eventually used to train an artificial neural network (ANN) in chapter 6 to be able to predict the maximum strain level in the pipeline. The ANN-based formula will present a more comprehensive method for evaluating the total strain in the dent as it includes the effect of pipe grade which was not considered in the ASME equation.
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CHAPTER 6
ARTIFICIAL NEURAL NETWORK APPLICATION FOR
PREDICTING REROUNDING, SCF AND STRAIN IN
DENTED PIPELINE
6.1 Introduction
Artificial neural network is a method than can be used to predict data through learning. It derives its origin from the human nervous system which consist of a large parallel interconnection of a large number of neuron. These neurons are interconnected in a very complex way between each other to create a network. The artificial neural network mimics a small part of the human brain to perform some specific task such as data classification or pattern recognition through a learning process.
An artificial neural network consists of a set of neurons or processing element (PE) that are connected by links of certain numeric weights. Each neuron has
a set of input links from other neurons a set of output links from other neurons a current activation function
an activation function to compute the activation level in the next time step
A typical neuron is seen in figure 2-7 in the literature review. From the figure x1, x2…xn
are the inputs, w1j, w2j….wnj are the weights. The total weighted input is the sum of the input activation multiplied by their respective weights as shown by equation 6-1
𝐺 = ∑ 𝑊𝑖𝑗𝑥𝑖
𝑖=0
(6-1)
A typical neural network architecture will consist of an input layer, one or more hidden layer, and an output layer. The input layer represents the information that is being fed into the system. The hidden layer is dictated by the activities of the input units and the
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weights between the input and hidden layer. The output layer relies on the action of the hidden layer and weights between the input and output layer.
For this study, only one hidden layer is used because its efficiency is enough for this application.
The number of neurons in the input layer is equal to the number of input variables, similarly, the number of output layer is equal to the number of output variables. The accuracy of the model is determined by the number of neurons in the hidden layer. The relationship between the input xj and output yp of a single perceptron is given by the
expression below 𝑦𝑝= 𝑓 (∑ 𝑤𝑝𝑗𝑥𝑗+ 𝑏𝑝 𝑁 𝑗=1 ) (6-2)
Where 𝑤𝑝𝑗 are the weights, bp is a constant usually referred to as bias and 𝑓( ) is the
activation function [6]. The number of weights and biases to be determined depends on the network architecture itself. It can be determined by the equation (6-3).
(ni x nh) + (nh x no) + (nh +no) (6-3)
Where ni is the number of inputs, nh is the number of processing elements in the hidden
layer and no is the number of output.
There are several algorithms that can be used to train a neural network. Some of them include back-propagation, genetic algorithm, and particle swam optimisation. For this study, the back propagation algorithm is used as it is the fastest and the most common technique. It also gives insight on how the weights and biases can affect the behaviour of the network.
Suppose we have an input layer X and an output layer Y with the transpose of the vector of the input variables is XT = (x1, x2, x3,… xn )and the transpose of the output
variable as YT = (y1, y2, y3,… yl).The mathematical expression between the input and
105 Y =𝑓{𝑊2𝑇x[ 𝐼 𝑓(𝑊1 x 〈 𝐼 𝑋〉)]} (6-4)
Where I is a unit one by one unit matrix and the expressions of matrices W1 and 𝑊2𝑇
are W1 = [ 𝑏1 . 𝑤𝑖ℎ1 1 . 𝑤𝑖ℎ1 2 ⋯ . . 𝑤𝑖ℎ1 𝑛 . 𝑏𝑚 𝑤𝑖ℎ𝑚1 𝑤𝑖ℎ𝑚2 … 𝑤𝑖ℎ𝑚𝑛 ] (6-5) 𝑊2𝑇=[ 𝑐1 . 𝑤ℎ𝑜1 1 . 𝑤ℎ𝑜1 2 ⋯ . . 𝑤ℎ𝑜1 𝑚 . 𝑐𝐿 𝑤ℎ𝑜𝐿1 𝑤ℎ𝑜 𝐿2 … 𝑤ℎ𝑜𝐿𝑚 ] (6-6)
The behaviour of an ANN greatly depends on the activation function, weights, and biases. There are different types of activation functions. The most common ones are the logistic sigmoid and the hyperbolic tangent transfer functions. Both functions are compared to determine the one that best predicts the SCF. The expressions for both the logsig and hyperbolic tangent functions are given below
Logsig=Y= 1 1+ 𝑒−𝑥 (6-7) Hyperbolic tangent=Y=𝑒 𝑥−𝑒−𝑥 𝑒𝑥+𝑒−𝑥 (6-8)
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In this Study, ANN is used to predict the total rerounding, stress concentration factor and the maximum strain in the dent. A commercial software MATLAB by MATHWORKS is used to develop the model. The ANN architecture constitute of an input layer with 4 input variable for the strain and rerounding model and 5 input variables for the SCF models. It consist of one hidden layer and one output layer with one output variable. The performance of the network is measured by alternating the number of neurons in the hidden layer, the transfer functions, and the weights. The ranges of the input variables are seen in table 6-1.
D/t d/D L/D σy Pmean
Dome Bar Dome Bar Dome Bar Dome Bar Dome Bar
Minimum 18.6 18.6 1.3 1.5 0.2163 2.9046 317 317 3.55 3.55 Maximum 96 96 11.3 11.4 3.6154 6.1183 690 690 48.07 48.07
Table 6-1 input variable ranges for ANN models