Mathematical model and statistical analysis of the elongation of the steel J55 API 5CT before and after the development of the pipes.

Mjaku, Malush ; Krasniqi, Fehmi ; Maksuti, Rrahim 等

1. INTRODUCTION

During technological process of pipe production with rectilinear seam entrance, a factor with significant impact is plastic deformation in the cold which is realized based on the deformation forces in inflexion throughout formation process of pipe calibration. It is more likely that the impact will be bigger as long as diameter of the pipe is smaller. To invent and assess this impact in mechanical attributes, extension in pulling, we have planned the experiment in three conditions of the material: preliminary steel coil, pipe [empty set] 139.7x7.72 mm and pipe [empty set] 219.1x7.72 mm [1]. These three conditions, express three levels (1, 2 and 3) of quality factor"deformation level". For each level there have been conducted 5 experiments in inflexion [3]. Specimens have been taken in direction of pipe's axis and experiments have been conducted based on application of fortuity's criteria. Calculating indicator is percentage of elongation ([A.sub.2]), marked with y.

2. MATHEMATICAL MODEL AND STATISTICAL ANALYSIS

2.1. Mathematical Model

Mathematical model which is predicted to reflect such a study is composed from a system by n equations forms (Pantelic, 1976):

[y.sub.ij] = [bar.m] + [a.sub.i] + [[epsilon].sub.ij] (1)

The formulas for calculation of round constant in which are based all observing results of index/indicator y ([bar.m]) and effects ([[bar.a].sub.i]) are:

[bar.m] = 1/n x [y.sub.++] [[bar.a].sub.i] = 1/p [y.sub.i] + -[bar.m] (2)

Based on values from table 1 and formulas (2) we will have:

[bar.m] = 1/15 437 = 29.13

[[bar.a].sub.1] = 32.20 - 29.13 = 3.07

[[bar.a].sub.2] = 27.20 + (-29.13) = -1.93

[[bar.a].sub.3] = 28 + (-29.13) = -1.13

With replacement of effects values in equations (1) mathematical model will have this form:

[y.sub.1j] = 29.13 + 3.07 + [[epsilon].sub.1j]

[y.sub.2j] = 29.13 - 1.93 + [[epsilon].sub.2j] (3)

[y.sub.3j] = 29.13 - 1.13 + [[epsilon].sub.3j]

2.2. Statistical Analysis

2.2.1. Variance Analysis

Total sum of the squares of differences (deviations) of the measured values from the average is composed by two components (Kedhi, 1984):

S = [S.sub.g] + [S.sub.p] (4)

Value of summary of error squares [S.sub.g] is:

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In similar method we will have also the value of deviation of experimental mistake.

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2.3. Control of Hypothesis, upon equality of the effects

For this is required control of hypothesis based the equality of the effects [a.sub.i]. According to the equation (2), Hypothesis of equation of the effects Ho, will take the form [Douglas & Mongomery, 2000):

[[mu].summation over (i=1)] [[bar.i].sub.i] = 0 [H.sub.0]: [a.sub.1] = [a.sub.2] = ... = [a.sub.[mu]] = 0 (5)

Alternative hypothesis is:

[H.sub.1] : [a.sub.i] [not equal to] 0 (6)

Value of calculated Fisher's criteria is:

[F.sub.c] = [s.sup.2.sub.p]/[s.sup.2.sub.g] (7)

[F.sub.c] = 36.07/3.80 = 9.49

For level of importance [alpha] = 0.05 limit value of Fisher's criteria:

[F.sub.t([alpha])2;12] = [F.sub.t(0.05);2;12] = 3.89; [F.sub.c] = 9.49 > [F.sub.t] = 3.89

Then, with level of importance [alpha] = 0.05 hypothesis [H.sub.0] is rejected and effects [a.sub.i] (i = 1,2,3) are accepted.

2.4. Comparison of the effects

2.4.1. Comparison of the effects according to minimal valid difference

To emphasize which levels are with important changes, first is required to calculate minimal valid difference [DELTA].sub.ik] ([alpha]) for level of importance [alpha] = 0.05.

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Based on the criteria (8) levels of effects "i" and "k" factor, so it compares [a.sub.i] and [a.sub.k]:

[absolute value of [[bar.a].sub.i] - [[bar.a].sub.k]] > [[DELTA].sub.ik]([alpha]) [absolute value of 3.07 - (-1.93)] = 5 > 4.95

[absolute value of [[bar.y].sub.i+] - [[bar.y].sub.k+]] > [[DELTA].sub.ik]([alpha]) [absolute value of 32.20 - 27.20] = 5 > 4.95 (8)

from application of this criteria result that:

[absolute value of [[bar.y].sub.1+] - [[bar.y].sub.2+]] = [absolute value of 32.20 - 27.20] = 5 > 4.95, between levels 1 and 2 it has important impact [absolute value of [[bar.y].sub.1+] - [[bar.y].sub.3+]] = [absolute value of 32.20 - 28 = 4.20 < 4.95, between levels 1 and 3 it has not important impact [absolute value of [[bar.y].sub.3+] - [[bar.y].sub.2+]] = [absolute value of 28 - 27.20] = 0.80 <4.95, between levels 3 and 2 it has not important impact

2.4.2. Comparison of the effects according to collective criteria of deviations

In this way "first type of mistake" to revoke a true hypothesis would be: 1 - 0.857=0.142 (and no more 0.05). To avoid this increment of mistake we should use other criteria, Duncan's collective criteria of deviations, which will be described bellow. For case when number of proves/experiments p in every level is same, standard mistake is calculated (Kedhi, 1984):

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By statistical tables, for [alpha] = 0.05 and number of degrees of freedom f = n-[mu]=15-3=12, are with row for q=2, 3 valid deviation: [r.sub.0.05(2;12)] = 3.08 and [r.sub.0.05(3;12)] = 3.23

With valid deviations [r.sub.[alpha]] and standard mistakes of levels, calculation of minimal valid deviations according to the formula:

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[R.sub.2] = 3.08 x 0.87 = 2.67 and [R.sub.3] = 3.23 x 0.87 = 2.81

Minimal valid deviation will be:

[[bar.y].sub.i] - [[bar.y].sub.k] [greater than or equal to] [R.sub.q] (11)

Now the comparison between levels of averages which are systematized in groups can be done:

[[bar.y].sub.1+] - [[bar.y].sub.2+] = 32.20 - 27.20 = 5> 2.81 = [R.sub.3], q = 3 - 1 + 1 = 3

[[bar.y].sub.1+] - [[bar.y].sub.3] = 32.20 - 28 = 4.20 > 2.67= [R.sub.2], q = 3 - 2 + 1 = 2

[[bar.y].sub.3+] - [[bar.y].sub.2+] = 28 - 27.20 = 0.80 < 2.67 = [R.sub.2], q = 2 - 1 + 1 = 2

3. DISCUTION/ CONCLUSIONS

Due to the plastic deformation, in cold, which is exercised upon the laminated tin, in warm, during the pipe formation and calibration it came to the strain hardening of steel's quality J55 API 5CT as a consequence of dislocations' formation and blockage.

Hypothesis [H.sub.0] of effects equation: [a.sub.1] = [a.sub.2] = [a.sub.3] = ... = [a.sub.1] x [mu] = 0 doesn't exist, while alternative hypothesis [H.sub.1] exist at least for one effect [a.sub.i] [not equal to] 0.

As the experimental calculated values of elongation in percentage ([A.sub.2]) of [F.sub.c] > [F.sub.t], with importance level [alpha] = 0.05 is accepted, effects ([a.sub.i] = 1, 2, 3) are not zero.

Since the effects' difference for elongation in percentage ([A.sub.2]) of levels "i" and "k" of factor's level [[bar.a].sub.i] and [[bar.a].sub.k] is more larger than minimal valid difference [[DELTA].sub.ik] ([alpha]) for importance level [alpha] = 0.05, we have: [absolute value of [[bar.a].sub.i] - [[bar.a].sub.k]] [greater than or equal to] [[DELTA].sub.ik]([alpha]), therefore it is accepted that levels "i" and "k" have important differences based on their impact in the experimental results.

While effects' difference of two pairs (1, 2) and (1, 3), with exception of pair (3, 2), of averages of arithmetical values watched in p levels probations "i" and "k" are larger than minimal valid deviations [R.sub.q], so: [[bar.y].sub.i] - [[bar.y].sub.k] > [R.sub.q]. Therefore, from this analysis we can conclude how important are the differences of level's of the two pairs during the research of elongation in percentage ([A.sub.2]) Results are done in "Laboratori mekaniko-metalografik IMK", Ferizaj-Kosovo.

4. REFERENCES

Standard, API Specification 5CT, Washington 2000.

V. Kedhi, Metoda te planifikimit dhe te analizes se eksperimenteve, (Methods of planning and analysis of experiments) Politeknik Faculty, Tirane 1984.

Standard, ASTM-A370, Washington 2000.

Douglas C. Mongomery, Controllo statistico di qualita, Parte III:(Statistical controll of quality, Part III), McGraw-Hill, 2000.

I. Pantelic, Uvod u teoriju inzinjerskog eksperimenta (Basic theory in engineering experiments), Radnicki Universitet, Novi Sad 1976.

Table 1. Results

 Reiterations / 1 2
 Levels

 1. 29 28
 2. 30 27
 3. 35 27
 4. 31 27
 5. 36 27
Sum 161 136
[y.sub.i+]
Average values 32.20 27.20
[[bar.y].sub.i,+] [[bar.y].sub.1+] [[bar.y].sub.2+]

 Reiterations / 3
 Levels

 1. 27
 2. 27
 3. 28
 4. 28
 5. 30
Sum 140
[y.sub.i+] [y.sub.++] = 437
Average values 28
[[bar.y].sub.i,+] [[bar.y].sub.3+]

Table 2. Summary table of variance analysis

Reason of change Sum of squares No. of DOF

Processing [S.sub.p] = 72.14 [mu] - 1 = 2
Reasons of the case [S.sub.g] = 45.60 n - [mu] = 12
Sum of deviations S = 117.74 n - 1 = 14

 Average square
Reason of change of deviations

Processing [S.sup.2.sub.p] = 36.07
Reasons of the case [S.sup.2.sub.g] = 3.80
Sum of deviations