Artificial neural network technology as a method to evaluate the failure bending moment of a pipe with a circumferential crack

Int. J. fks. Vex & t’/pipin,g 68 (1996) l-6 Copyright @ 1YY6 Elsevirr Science LimIted Printed in Northern Ireland. All rights reserved 030x-o I6 I /‘)I?/$ IS.00

0308-0161(95)00033-X

Artificial neural network technology as a method to evaluate the failure bending moment of a pipe with a circumferential crack Mechanical

Engineering

YIPLin Han, Shi-Ming Shen & Shu-Ho Dai Department, Nanjing University of Chemical Technology, Nanjing

210009,

P.R. China

(Received 20 April 1995: accepted 3 May 1995) The artificial neural network (ANN) method is introduced to evaluate the failure bending moment of pipes with circumferential cracks. In particular examples, it is used to evaluate the failure bending moment of full-scale pipes under internal pressure and external bending moment. The examples show the feasibility that the neural network method can be used to predict the failure bending moment of pipes with circumferential cracks. The neural network method is also used to assess the acceptability of flawed pipes in plant. A comparison of the assessment results of the neural network method with those of the ASME Boiler and Pressure Vessel Code Section XI IWB-3650 is also presented. Copyright 0 1996 Elsevier Science Ltd.

1 INTRODUCTION To predict the failure external bending moment of a pipe with a circumferential crack under internal pressure and external bending moment, the necessary knowledge has to be obtained first. In the present paper the method known as neural network technology is introduced to obtain the knowledge about the failure bending moment of pipes with circumferential cracks under internal pressure and external bending moment. Examples of this procedure are also given. The neural network method is selected to assessthe acceptability of flawed pipes in plant, and the comparison of the assessment results determined by the neural network method with those determined by ASME Boiler and Pressure Vessel Code Section XI IWB-3650’ is also presented. Reference 2 has given some simple engineering calculating methods to calculate the failure bending moment of pipes with circumferential crack under internal pressure and external bending moment. The calculating failure moments of pipes determined by the methods of Ref. 2 are compared with those determined by the ANN method in the present paper.

2 ARTIFICIAL NEURAL NETWORK ARCHITECTURE AND ALGORITHM From the point of its material structure, the human brain consists of a large amount of neurons. The human brain has three basic abilities that are: learning ability; information processing ability on the basis of knowledge; and ability of modifying and accumulating the knowledge in the information processing process. The ANN method consists of a large amount of artificial neurons that imitate the information processing process of neurons. The whole network imitates the structure and abilities of the human brain. 2.1 Artificial neurons The artificial neuron that is selected in the present paper is shown in Fig. 1.’ In Fig. 1, the circle means the jth artificial neuron u,; x, is the information from the ith artificial neural network 14, (j f i); w,, is the linking strength between u, and u,; z, is threshold of this artificial neuron, y is the output information of u,: ,I

neti = C W,iX,- Z, ,=I Y, = 1/[1 + exp(-net,)]

(2)

Yu-Lin

2

Han, Shi-Ming Shen, Shu-Ho Dai

yi

*

The artificial neural network also could learn from experiences-the samples that are given to the neural network. There are many kinds of learning algorithms. The back-propagation algorithm (B-P algorithm), which is used in the present paper, is a famous learning algorithm. The B-P algorithm is described as follows. Step 2. Initiate all neuron thresholds 2, and w,; to

Fig. 1. Artificial neuron

2.2 Architecture

of artificial

neural

networks

Three-layer feed-forward neural networks (Fig. 2) are investigated in the present paper. The input layer receives the outside information and transfers the information to the hidden layer. The hidden layer handles the information from the input layer and gives out its own output information to the output layer. The output layer handles the information and sends out the output of the whole neural network. Every neuron in each layer is working on the rules shown in eqns (1) and (2). 2.3 Learning

algorithm

of neural

network

We readily deduce that if it is getting dark with many black clouds and it is getting cold, that it must be going to rain. We do not have this knowledge at birth, but we can obtain this knowledge from many occurences of the same experience. In the beginning, we can not give the correct answer about weather changes, but we can adjust our knowledge after many times of correct and wrong answering until we can give out the correct deduction. We learn from experiences.

small random values. Step 2. Get the learning set of experiences that includes the learning input set and target output set. 3. Calculate the output of the neural network according to eqns (1) and (2). Step 4. Calculate the error function E according to eqns (3) and (4). If E is less than or equal to &resi,o,d(&,resim,c,is the threshold for E), go to step 8. Step

E,, = l/22

E = l/(2& I’

c

(4)

(t,,k

a,,,

=

(t,,k

-

o,,k>“,>k(l

-

O,lk)

(6)

=

77x

(7)

6,>k

(9)

I’

Step 6. Calculate A& and AZ, for hidden layer.

aE

ay.;

layer

=

v”,Jio,Jj(l

-

opj)c

spk

wkj

(11)

k

layer

A W/; = c A,, I$,

Fig. 2. Three-layer neural network

WV

A,,~.i = -77 p

‘/?yi

i

o,>k>’

A,, Wkj = Topj ‘,k

(Wji) Input

-

k

layer

(Wkj) ) Hidden

(3)

where E,, is the difference between the actual output o,,k and the target output t,,k of the pth sample, E is the mean difference of all the samples. Step 5. Calculate Awkj and AZ, for the output layer.

A&

) Output

(t,>k- O,d’ k

(12) (13)

Artificial

neural network technology and pipe bending moments

where n is the learning rate. Step 7. Calculate the modified Wk,,Zk, H$ and Z, according to eqns (U-15), then return to Step 3. W”‘“’ = WOld+ aJ,,j/ XI hi k/ .,I,34= Z”/‘/ + *z & h W”‘W I’

=

W”/‘/ 11

+

aw

Z”‘“’ I

=

Z”“’

+

*z

I

I’ I

(14) (15) (16)

(17)

step 8. output. Step 9. Stop. The neural network learns according to the learning algorithm until it has obtained the knowledge or law that is supposed to be able to describe the action of the things that belong to the same set as the samples do. In other words, if the input data of the learning samples are put into the neural network, the neural network will continue to learn until its outputs (O,,k) are close enough to the target outputs (t,,h) of the same learning samples.

3 ARTIFICIAL NEURAL NETWORK TECHNOLOGY AS A NEW METHOD TO EVALUATE THE FAILURE EXTERNAL BENDING MOMENT OF PIPES WITH CIRCUMFERENTIAL CRACKS A set of failure experiment data of pipes, which are flawed pipes with circumferential cracks under internal pressure and external bending moment, are prepared in this paper. Parts of these data are the learning samples of the neural network. The other parts of the prepared data are for testing the learning result. In the learning stage, the neural network will obtain the knowledge about the failure bending moment of pipes with circumferential cracks under internal pressure and external bending moment. In the testing stage, the neural network will use the prepared data and the knowledge, which has been obtained in the learning stage, to predict the failure bending moment of pipes with circumferential cracks. Then the predicted failure moments will be compared with the experimental failure moments. The comparison of results will say if the neural network has acquired the knowledge in the learning stage. If the predicted

3

failure bending moments are close enough to the experimental ones, then the neural network must have gained the knowledge in the learning stage. 3.1 Experimental data Full-scale pipes with circumferential cracks were tested under internal pressure and external bending moment.2 Reference 2 gives two sets of equations to calculate the failure bending moment of pipes under internal pressure and external bending moment. One is on the basis of moment theory, and the ultimate strength criterion is applied. The other is on the basis of plastic limit load theory with the flow stress criterion. The experimental data (from Ref. 2) are shown in Table 1. Parts of the data (with asterisk mark) are selected as learning samples of the neural network. The neural network uses the other part of the data to test its learning results. There are some data of flawed pipes that are in use in chemical plant. The data are obtained by measurement or nondestructive testing (such as ultrasonic inspection and X-ray inspection). Of course there are no experimental failure bending moments for these pipes. 3.2 Learning of a neural network The neural network learning samples are marked with an asterisk in Table 1. The architecture of the neural network is shown as Fig. 2. The algorithm has been mentioned in Section 2. In the learning of the neural network, the data of pipe No. BVZ091, for example, are put into the neural network. This means the data of the pipe dimension, flaw dimension, material parameters, internal pressure and experimental bending moment are put into the neural network. The neural network will give an output according to eqns (1) and (2) and will calculate the difference between the output and the experiment failure bending moment of pipe No. BVZ091 (see Table 1) according to eqn (3). The data of other pipes are put into the neural network too, and the neural network will give the outputs and the difference of these pipes. The neural network will adjust the WA,, Zh, H$ and Z, in the way discussed in Section 2 until the outputs of the neural network are close enough to the relative experiment external bending moments. To assessthe acceptability of the flawed pipes in Section 4, a special sample (with asterisks in

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Table Material (kNNo;n)

*BVZ161 *BVZ160 *BVZ091 *BVZlOO BVZllO *BVZ120 BVZ130 BVZl50 *BVZ140 BVS102 *BVSllO “BZS060 BVS070 BVS080

A A A A A A A A A B B B B B

Han, Shi-Ming 1. Experimental

Pipe outside diameter (mm)

Pipe wall thickness (mm)

Flaw type

797.9 797.9 797.9 797.9 797.9 797.9 797.9 797.9 797.9 793.9 793.9 793.9 793.9 793.9

47.2 47,2 47,2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2

b E b C

: C C C

E C C

Shen, Shu-Ho Dai data and results’ Flaw dimension Depth (mm)

Length (deg.)

47.2 26 47.2 47.2 39.4 20 20 20 20 20 36 47.2 20 20

20 20 60 60 60 60 60 90 120 20 20 60 60 120

Material A: 20 MnMoNi 5 5 in Basis Safe Quality (upper shelf impact energy Material B: NiMoCr Special Melt (upper shelf impact energy -50 J). Flaw Type: b-through-wall crack c-part-through flaw (outside) d-part-through flaw (inside).

Table 3) with target output of the mean calculated failure bending moment by methods used in Ref. 2. Because the target output of this sample is not the experiment failure bending moment, the relative output of neural network in learning need not be as close to the target output as the outputs of those samples with target outputs in the experiment data. The learning results of the neural network are shown in Table 2. Reference 2 also gives two sets Table No.

Material

2. Comparison

Exp. bending moment max. value [kN m (Ref. 2)

(Me)1 *BVZ161 *BVZ160 *BVZ091 BVZlOO BVZllO *BVZ120 BVZ130 BVZISO “BVZ140 BVS102 “BVSllO *BVS060 BVS070 BVSOSO

A A A A A A A A A B B B B B

14 200 13 400 9650 8800 7400 10500 10 600 9000 8650 11000 85.50 5500 6500 5700

of experimental

MPA method moment [kN m (Ref. 2)

(Mm)1 10 151 9870 7620 6450 6920 8770 8720 7860 6990 10400 9510 6450 8960 7150

Temperature (“C)

130 250 20 20 250 250 250 270 250 230 250 150 250 280

-

Internal pressure WW

External bending moment (max.) W m)

0 15.5 0 15 1.5 15 15 15 15 15 15 1.5 15 15

14 200 13 400 9 650 8800 7 400 10 500 10600 9 000 8 650 11000 8 550 5 500 6 500 5 700

>150 J).

of equations to calculate the failure external bending moment. The calculated results by the methods in Ref. 2 are listed in Table 2. 3.3 Examples of predicted failure external bending moment of pipes with circumferential cracks by neural network

After learning, the neural network is supposed to have learned the knowledge about the failure and calculated

Plastic limit load concept moment [kN m (Ref. 2)

WP)I 13 267 13 222 10 988 10 273 10 288 12 220 12 220 11231 10639 13 339 12 838 9624 12 141 10424

bending

moments

Neural network method moment [kN m @WI

13 143 12 519 10418 7945 8573 11 187 11 179 9513 8407 10442 8303 5804 8145 5892

Mm/Me

Mp/Me

Mn/Me

0.7149 0.7366 0.7896 0.733 0.9351 0.8352 0.8226 0.8733 0.8081 0.9455 1.1123 1.1727 1.3785 1.2544

0.9343 0.9867 1.1387 1.1674 1.3903 1.1638 1.1528 1.2479 1.2299 1.2126 1.5015 1.7498 1.8678 1.8288

0.926 0.934 1.08 0.903 1.159 1.065 1.055 1.057 0.972 0.949 0.971 1.055 1.253 1.034

Artificial

neural network technology and pipe bending moments

external bending moment of pipe with circumferential crack from the learning samples; it has memorized them in the form of Wxi, Z,, l4$ and Z,. When new inputs are sent in, the neural network uses this knowledge to give an answer. In this section, the data in Table 1 without an asterisk will be put into the neural network. For example pipe No. BVZlOO, the data of the pipe dimension, flaw dimension, material property parameters and internal pressure are put into the neural network. The neural network treats the input information with the knowledge which has been accumulated in the learning stage of the neural network, to give out an output that is the calculated failure external bending moment of pipe No. BVZlOO. Since the experiment failure moment is known, so by comparing the calculated failure moment and the experiment failure moment we can see how well the neural network has gained the knowledge in the learning process. This is the testing stage of the neural network (the testing stage of the neural network has been explained in Section 3.1). The test results are shown in Table 2. Reference 2 also gives two sets of equations to calculate the failure external bending moment. The calculated results by the methods given in Ref. 2 are also listed in Table 2.

which has learned the knowledge as in Section 3, is used to assessthe acceptability of those flawed pipes. The finite-element method (FEM) is introduced to calculate the load on these pipes. The internal pressure, self-weight of the pipes and thermal change are considered in the FEM process. The output of the FEM is axial mechanical load, axial thermal load and bending moment. The axial mechanical load (8,) and axial thermal load (P,) are conservatively combined as pressure (p) in the way described in eqn (18).

where R, is the inside radius of the pipe. Because of the constraint of the flawed pipes, p is not always larger than internal pressure. If p is smaller than internal pressure, p is conservatively assumed to be equal to internal pressure. Then the pressure p, the material property data, the radius and thickness of pipe, and the size and location of flaw are put into the neural network, which has supposedly learned the essential knowledge, as in Section 3, to give out the neural network calculated failure bending moment (Table 3). The acceptability assessment results of these flawed pipes are determined by way of Ref. 1, and the allowable bending moments that are calculated from the allowable bending stress obtained by way of Ref. 1 are listed in Table 3. In addition, the calculated failure bending moments determined by way of Ref. 2 are also listed in Table 3.

4 APPLICATION OF NEURAL NETWORK METHOD IN PLANT-ACCEPTABILITY ASSESSMENT OF FLAWED PIPES IN PLANT Some pipes in plant are found by X-ray inspection to be flawed. The neural network, Table NO.

Pl P2 P3 “P4 P5 P6

Pipe outside diameter (mm)

Pipe wall thickness (mm)

457 457 356 356 356 356

9.5 9.5 12 12.7 12.7 12.7

3. Acceptability

Flaw dimension Depth (mm) 5 2 6 6 6 I .s

Length (mm) 18 51 20 80 10 5x

5

assessment of flawed pipes

Temperature w

M(NN) W ml

0.75 7 3.14 3.14 3.14 3.14

26 15 6 4 4 6

712 697 440 304 447 422

M(MPA) W ml

691 6Y3 576 4x7 526 526

M( PLL) (kN ml

738 737 518 550 580 578

M(FEM)-moment calculated from the finite element method. M(NN)-moment calculated from the neural network method. M(MPA)-moment calculated from the MPA method (see Ref. 2). M(PLL)-moment calculated from the plastic limit load method (see Ref. 2). M(lWB)-allowable moment calculated from the method of Ref. 1. Material is ASTM A53 GR.B. Because X-ray inspection could not determine where the flaw resides in the wall of the pipe, all flaws are supposed part-through cracks. M(NN), M(MPA). M(PLL) and M(IWB) > M(FEM). All flawed pipes are accepted.

M(IWB) W m)

167 167 113 108 114 172

as outside

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Han, Shi-Ming

5 CONCLUSION The neural network method is introduced to evaluate the failure external bending moment of pipes with circumferential cracks under internal pressure and external bending moment. The calculated moments are compared with those determined by the methods in Ref. 2 and/or Ref. 1. For the pipes in Table 1, the error analysis results of calculated moment obtained by neural network method and Ref. 2 methods are listed in Table 4. From the error analysis results in Table 4, it is found that the neural network method can predicate the failure external bending moment more accurately. For flawed pipes in services, the acceptability assessmentresults determined by neural network are the same with those determined by IWB3650’ and the methods of Ref. 2. Because of the safety margin of the ASME code,’ the moment obtained by the neural network method is much larger than the moment calculated by the ASME code. There are still some problems about the scope of the knowledge that the neural network learned from a few samples. If we cannot get enough samples to assure the universality of the knowledge, what can we do then? In the present

Shen, Shu-Ho Dai

paper it is suggested that the knowledge, which we have gained in other ways such as the knowledge in Ref. 2, can be used to deal with the problem. The suggested method is to make samples according to the knowledge such as the knowledge in Ref. 2. Table 4. Error

analysis of Table 2

Mm/Me Mean Standard Error Standard Deviation Variance Minimum Maximum

0.936 555 0.056 401 0.211 034 0.044 535 0.714 859 1.378 462

Mp/Me 1.326 599 0.080 322 0.300 539 0.090 324 0.934 296 I.867 846

Mn/Me 1.029 438 0.025 846 0.096 707 0.009 352 0.902 841 1,253 077

REFERENCES Boiler and Pressure Vessel Code Section XI IWB-3650 and Appendix H. Herter, K.-H., Julisch, P., Stoppier, W., & Sturm, D., Behavior of pipes under internal pressure and external bending moment-comparison between experiment and calculation, Fracture Mechanics Vertification by LargeScale Testing, EGF/ESIS8 (Edited by K. Kussmaul), (1991), Mechanical Engineering Publications, London, 223-241. Yin, Q. Y., Yang, Z. K., & Tan, Z., Pattern Recognition and Neural Network, Machinery Industry Press, P. R. China, 1992. ASME