MATHEMATICAL MODEL AND STATISTICAL ANALYSIS OF THE RESIDUAL STRESSES ( σ

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1 7 th Research/Exert Conference wth Internatonal Partcatons QUALITY 20, Neum, B&H, June 0 04, 20 MATHEMATICAL MODEL AND STATISTICAL ANALYSIS OF THE RESIDUAL STRESSES ( σ resdual ) IN THE CROSS SECTION AREA OF STEEL QUALITY PIPES J55 API 5CT Malush Mjaku Mnstr of Educaton, Scence and Technolog, Prshtna, R. Kosovo E- mal: malushmjaku@ahoo.com malush.mjaku@ks-gov.net Fehm Krasnq Unverst of Prshtna Mechancal Engneerng Facult Prshtna, R. Kosovo E- mal: fehkrasnq@ahoo.com Dmtr Koznakov Mechancal Engneerng Facult Karoš II bb. P.BOX Skoje, R. Macedona E- mal: dmtar@mf.edu.mk Dervsh Elez Poltechnc Unverst of Trana, Mechancal Engneerng Facult, Trana, R. Albana E- mal: dervel@ahoo.com ABSTRACT Object of ths stud s of the steel qualt J55 API 5CT and the rocess of e formng [mm], [mm], and [mm], wth longtudnal seam es-erw Am of ths aer s to stud the mact of lastc deformaton degree n the cold of resdual stresses n the cross secton area of steel qualt es J55 API 5CT[]. For the realzaton of ths stud we have used the lannng method of the exerment wth one-factor. We have bult the mathematcal model for the exerment wth one ndex (resdual stressesσ resdual ) and wth one factor (deformaton degree n the cold) and wth three deformaton levels. The results obtaned n an exermental method are shown n the table and are rocessed n an analtcal wa whle mlementng the one factored exerments [2]. Kewords: One-factor exerments, e, resdual stresses ( σ resdual ).. INTRODUCTION Durng the technologc roducton rocess of the longtudnal seam es sgnfcant factor wth nfluence s the lastc deformaton n the cold whch s realzed accordng to dstortng forces on the curvature durng the rocess of formng and calbraton of es. It s exected that the mact to be much hgher the smaller the e dameter s. To dscover and evaluate ths mact n resdual stresses we made measures for three e dameters: Ø [mm], Ø [mm], and e Ø [mm]. These three e rofles exress three levels (, 2 and 3) of the qualt factor deformaton degree. For each level are erformed four tests [3]. The slttng rngs are taken from the rofles of these es and tests are erformed whle alng the crtera of the chance. The measured ndcator s resdual stress durng formng and calbraton n cross cuttng of the es, marked wth. Gven results (tab. ) of the resdual stresses are calculated accordng to the formula [4]. 533

2 σ res = E t D x 0 + D 0 π () a) b) Fgure. Schematc of the resdual stress dstrbuton n rngs manufactured from tube: a) before and b) after slttng. E - modulus elastct; t - thckness; D0 - ntal dameter; D - dameter after slttng; x - net oenng dslacement [4]. Table.Results of resdual stresses σ res [MPa] Reteratons/Levels R=62[mm] R=22[mm] R=70[mm] Sum = Average values MATHEMATICAL MODEL AND STATISTICAL ANALYSIS 2.. Mathematcal Model Mathematcal model whch s redcted to reflect such a stud s comosed from a sstem b n equatons forms [5]: = m + j a + ε j (2) = (-36.56) + ε ; 2 j = (-7.72) + ε 2 j; 3j = ε 3j j j The formulas for calculaton of round constant n whch are based all observng results of ndex/ndcator ( m ) and effects ( a ) are: m= ++ ; n a = - m (3) + Based on values from table and formulas (2) we wll have: 534

3 2.2. Statstcal Analss m= ++ = = n 2 a= = = a2 = = = a3 = = = Varance Analss Total sum of the squares of dfferences (devatons) of the measured values from the average s comosed b two comonents [2]: S = S + S = = (4) Value of summar of error squares Sg s: g S = - = - = = μ g j + 3,4 3+ = j= = = j= = In smlar method we wll have also the value of devaton of exermental mstake. S = - = - = = u = μ = 2.3. Control of Hothess, uon equalt of the effects For ths s requred control of hothess based on the equalt of the effects a. Accordng to the equaton (2), hothess of equaton of the effects H 0, wll take the form [6]: H : a = a =... = a μ = 0... (5) 0 Alternatve hothess s: H : a 0 (6) Table 2. Summar table of varance analss Reason of change Sum of squares No. of DOF Average square of devatons Processng S = μ - = 2 2 s = Reasons of the case S g = n - μ = 9 2 s = g Sum of devatons S = n - = Value of calculated Fsher s crtera s: 535

4 2 s Fcal = = = (7) 2 sg For level of mortance α = 0.05 lmt value of Fsher s crtera: Ftab ( α );2;9 = (0.05);2;9 = 4.26 F cal = > F = 4.26 Then, wth the level of mortanceα = 0.05 hothess H 0 s rejected and effects a ( =,2,3) are acceted Comarson of the effects Comarson of the effects accordng to mnmal vald dfference To emhasze whch levels are wth mortant changes, frst s requred to calculate mnmal vald dfference Δ ( α) for the level of mortanceα = 0.05 k 2 ( α Δ k ) = s g + ( -) F( ; -, n- ) μ α μ μ = k Based on the crtera (8) levels of effects and k factor, so t comares a and a k : a - ak >Δ k( α) ; (-36.56) = 28.84; > >Δ ( α) ; = 28.84; > 26.4 (8) + k+ k From alcaton of ths crtera result that: tab - = = > between levels 3 and t has mortant mact = = 52 > between levels 3 and 2 t has mortant mact = = > between levels 2 and t has mortant mact Comarson of the effects accordng to collectve crtera of devatons In ths wa frst te of mstake to revoke a true hothess would be: = 0.42 (and no more 0.05). To avod ths ncrement of mstake we should use other crtera, Duncan s collectve crtera of devatons, whch wll be descrbed bellow. In case when number of exerments n ever level s same, standard mstake s calculated [2]: 2 sg S = + = 4 = (9) 536

5 B statstcal tables, for α = 0.05 and number of degrees of freedom f = n μ = 2 3 = 9, are wth row for q = 2,3 vald devaton: r 0.05(2;9) = 3.08 and r 0.05(3;9) = 3.23 Wth vald devatons r α and standard mstakes of levels, calculaton of mnmal vald devatons accordng to the formula: R = r ( q, f) S, q= 2,3,..., μ q α + (0) r = 3.08; r = (2;9) 0.05(3;9) R = = 8.23; R = = Mnmal vald devaton wll be: k Rq... () Now the comarson between levels of averages whch are sstematzed n grous can be done: - = = > 9.2 = + + R ; q = 3- + = = = 52 > 8.23 = + + R2; q = = 2 - = = > 8.23 = + + R ; q = 2- + = PROCESSING DATA WITH SOFTWARE PROGRAM DESIGN EXPERT 7 Resonse Resdual Stress ANOVA for selected factoral model Analss of varance table [Classcal sum squares Te II] Sum of Mean F value Source Squares df Square Value Prob > F Model sgnfcant A- Defor. Degree Pure Error Cor Total The Model F-value of 29.6 mles the model s sgnfcant. There s onl a 0.0% chance that a "Model F-Value" ths large could occur due to nose. Values of "Prob > F" less than ndcate model terms are sgnfcant. In ths case A are sgnfcant model terms. Std. Dev. 5.2 R-Squared Mean Adj R-Squared C.V. % 3.89 Pred R-Squared PRESS Adeq Precson The "Pred R-Squared" of s n reasonable agreement wth the "Adj R-Squared" of "Adeq Precson" measures the sgnal to nose rato. A rato greater than 4 s desrable. Your 537

6 rato of ndcates an adequate sgnal. Ths model can be used to navgate the desgn sace. Treatment Means Mean Standard t for H 0 Treatment Dfference df Error Coeff=0 Prob > t vs vs < vs Values of "Prob > t " less than ndcate the dfference n the two treatment means s sgnfcant. Desgn-Exert Software Resdual Stress Desgn Ponts X = A: Deformaton Degree Resdual Stress [MPa] Resdual Stress 60 R 62 R 22 R 70 A: Def ormaton Degree [R, mm] 4. CONCLUSION In three aled methods (crtera) for results analss, wth degree of decreasng the mstake of the frst te, from 0.42, n 0.05 and n = 0.000, are confrmng the formng of es, the deformaton degree throughout the bendng of sheet and calbraton n the cold nfluences n the ncrease of resdual stresses. The nfluence of the mact s much hgher the smaller the e dameter s. 5. REFERENCES [] Standard: API Secfcaton 5CT, Washngton [2] V. Kedh: Metoda të lanfkmt dhe të analzës së eksermenteve, (Methods of lannng and analss of exerments) Poltechnc Facult, Trana 984. [3] Standard: ASTM-A370, Washngton [4] htt:// [5] I. Pantelć: Uvod u teorju nžnjerskog eksermenta, Radnčk Unverstet Nov Sad 976. [6] Douglas C. Mongomer: Controllo statstco d qualtà, Parte III: (Statstcal controll of qualt), McGraw-Hll, [7] Software: Desgn Exert 7 538