Transactons of JWRI, Vol.36 (2007), No.1 Influental Factors Affectng Inherent Deformaton durng Plate Formng by Lne Heatng (Report 1) The Effect of Plate Sze and Edge Effect VEGA Adan*, RASHED Sherf**, SERIZAWA Hsash*** and MURAKAWA Hdekazu**** Abstract The authors am to propose an accurate and practcal method to predct the deformaton of actual plates of large sze such as those used n shpbuldng. Ths method s to be used to mprove the accuracy of an automatc plate formng process. In ths frst report, the nfluental factors affectng nherent deformaton are outlnes. Numercal nvestgatons of the nfluence of plate length and plate wdth of large plates are presented. The edge effect on nherent deformaton of such large plates s also nvestgated. The nvestgatons have shown that the nfluence of plate length and plate wdth on nherent deformaton, ncludng edge effect, of large plates s small and can be neglected. Therefore, when the dstrbuton of nherent deformaton of a large plate s known along the heatng lne, the nherent deformaton of smaller plates, wth the same thckness and formed under the same heatng and coolng condtons, can be accurately and easly predcted. KEY WORDS: (Lne Heatng) (Inherent Deformaton) (Resdual Stress) (Plate Length Effects) (Plate Wdth Effects) (Edge Effect) 1. Introducton Plate formng by lne heatng s a process routnely performed n many shpyards. Formng takes place n a metal whenever t s subjected to stresses greater than the yeld stress so that the deformaton moves from the elastc to the plastc range. When a plate s beng formed by heatng, plastc deformaton s produced by the thermal stresses generated durng localzed heatng and subsequent coolng of the plate. Durng ths process, only one sde of the plate s heated whle the other sde remans cooler. The temperature gradent across the thckness causes a gradent of thermal expanson and the plate bends n one drecton, convex at the heated surface. In the mean tme, the expanded metal s constraned by the surroundng cooler metal leadng to compressve plastc stran gradent that s larger n hotter metal at and near the heated surface. When the heatng source s removed, the plate cools down and the metal contracts. Compressve plastc stran gradent causes the plate to bend and assume an equlbrum state n the drecton opposte to that when t was heated, that s concave at the heated surface. Receved on June 22, 2007 * Graduate Student ** Specally Apponted Professor *** Assocate Professor Formng by lne heatng has been an actve research topc n manufacturng, especally n shpbuldng. Theoretcal research on the mechansm of lne heatng process amed to predct the fnal shape of metal plates when gven the heatng condtons and mechancal propertes of the plate materal (see references 1-4) for examples). The fnte element method or smplfed beam or plate theory are usually appled. Research on desgn of the proper heatng and/or coolng process (see references 5-9) for examples) has been based on the experence of formng small rectangular plates nto smple geometry. Stran or curvature analyss has been employed to determne the locatons of heatng lnes. The nherent stran method 5,6) s very promsng as an accurate and practcal method to predct the deformaton. However t requres accurate nherent stran data bases n order to get accurate predctons. The relatonshps between bendng deformaton and heatng parameters, materal propertes, and plate thckness have been developed n emprcal models and nherent stran data bases. Addtonal nformaton, such as nfluence of stran hardenng, stran rate effects, edge effect, and sze effects, have also been reported n expermental and numercal **** Professor Transactons of JWRI s publshed by Jonng and Weldng Research Insttute of Osaka Unversty, Ibarak, Osaka 567-0047, Japan. 57
Influental Factors Affectng Inherent Deformaton durng Plate Formng by Lne Heatng (Report 1) y z x h Wdth (W) Length (L) Fg.1 Schematc of the plate model. nvestgatons 10). Expermental observatons have shown that many factors nfluence the formng process. However, only a few nvestgatons have explored these factors other than those nduced by heat nput and plate thckness. Most of the nvestgatons have focused on the deformaton produced by sngle heatng lnes on small plates. Inherent stran data bases derved from sngle heatng lnes on small plates are avalable. No emprcal methods or nherent stran data bases has been developed for actual sze plates, takng nto account the nfluence of multple heatng lnes. At present a practcal method to accurately predct the deformaton of a plate formed by lne heatng does not exst. The authors am to propose a practcal and accurate method to predct deformaton of actual sze plates takng nto account factors such as plate geometry, heatng condton, locaton of heatng lnes and multple heatng lnes. In ths frst report, the man factors affectng the deformaton of plates formed by lne heatng are outlned. Through a 3D thermal-elastcplastc fnte element analyss usng an teratve substructure method 11), the effect of plate length, plate wdth and proxmty to the edge (edge effect) on nherent deformaton are nvestgated and clarfed. Followng reports wll nvestgate the effect of other nfluental factors of the lne heatng process and clarfy ther effect on nherent deformaton. Ths wll help to better understand the phenomenon, quantfy the effect of these factors on nherent deformaton and fnally present an accurate and practcal method to predct the deformaton of actual shp plates when formed by lne heatng. 2. Thermal Elastc-Plastc Fnte Element Analyss An outlne of the procedure appled to compute plate deformaton usng 3D thermal-elastc-plastc fnte element analyss s presented n ths secton. 2.1 Geometry, fnte element models and analyss procedure All analyss cases are carred out usng rectangular flat plates as shown n Fg.1. Fnte element models as Materal Propertes 15.0 10.0 5.0 Fg.2 Example of the fnte element model. Densty(x10-3 g/mm 3 ) Thermal conductvty (x10-2 J/mm/sec/ o C) Yeld stress (x10 2 MPa) Young's modulus (x10 2 GPa) Thermal expantonrato (x10 Heat transfer coeffcent (x10-6 J/mm 2 /sec/ o C) Heat capacty (x10-1 J/G/ o C) Heatng Area 6 1/C o ) 0.0 0 200 0 600 1000 1200 10 1600 Temperature( o C) Fg.3 Mechancal propertes of mld steel. shown Fg.2 are employed. Heatng s appled over a movng zone that conssts of two elements n the longtudnal (movng) drecton, 4 elements n the transverse drecton, and one element n the thckness drecton (80xx10 mm). In each case, the transent temperature dstrbuton resultng from the movng heat source s computed usng a 3D thermal FE analyses. Heat nput per unt length of the heatng lne s adjusted such that the hghest temperature on the surface n the heatng zone s kept at C. Coolng s defned correspondng to natural coolng n ar. Mld steel thermal propertes wth temperature dependency as shown n Fg.3 are used. The plate s heated along the X drecton on the top surface. No meltng s nvolved n the heatng and coolng process. Thermal analyss s followed by a transent mechancal analyss. Necessary constrants are added to elmnate rgd body moton. Mechancal materal propertes wth temperature dependency as shown n Fg.3 are used. The transent temperature dstrbuton hstory obtaned from the thermal analyss s appled and stresses, strans and dsplacements are evaluated. To clarfy the nfluence of plate length, plate wdth and proxmty to the edge (edge effect) on nherent deformaton, the analyss cases shown n Table 1 are used. As shown n Table 1, cases from 1 to 4 are used to 58
Transactons of JWRI, Vol.36 (2007), No.1 Heat Input (J/mm) Table 1 Heatng condtons and plates models. Heatng Condtons Travelng Surface Speed Temperature (mm/s) ( C) Back Surface Temperature ( C ) 5,000 3.0 260 Plate Models Used to Study the Plate Length Effect CASE L (mm) W (mm) h (mm) 1 2 3 4 0 10 3000 Plate Models Used to Study the Plate Wdth Effect CASE L (mm) W (mm) h (mm) 5 6 0 7 10 8 3000 (b) (a) study the nfluence of plate length. Cases 5 to 8 are used to analyze the nfluence of plate wdth. 2.2 Method of analyss An n-house three dmensonal thermal elastcplastc fnte element code based on an teratve substructure method 11) s employed. The teratve substructure method (ISM) takes full advantage of the fact that the regon under plastc-loadng whch exhbts strong nonlnearty s lmted to a very small area compared to the sze of the model to be analyzed and the remanng part s mostly lnear. Therefore a model s dvded nto a non-lnear regon and the remanng lnear regon. Contnuty of tracton on the boundary between the lnear and the nonlnear regons s mantaned throughout an teratve procedure. 2.3 Obtaned results In all cases, nodal dsplacements, element stress components and elastc and plastc stran components are obtaned. The deformaton of the plate s expressed usng the nherent deformaton method whch conssts of the ntegraton of the plastc stran over the cross secton of the plate. The nherent deformaton can be dvded nto four components; longtudnal shrnkage ( δ x ), transverse shrnkage ( δ y ), longtudnal bendng ( θ x ), and transverse bendng ( θ y ). These four components are defned by the followng equatons: δ = ε dydz h (1) x x / δ = ε dydz h (2) y y / 3 θ = ε ( z h / 2) /( h /12 dydz (3) x x ) 3 θ = ε ( z h / 2) /( h /12 dydz (4) y y ) Inherent Transverse Shrnkage (mm) 0.20 0.10-0.10-0.20 (c) Fg.4 Schematc of (a) the plate deformaton, (b) resdual stress n x-drecton and (c) resdual stress n y-drecton. Entrance edge δy max Ext edge -0.30 0 100 200 300 0 500 600 700 L 1 L 2 L 3 Fg.5 Schematc of nherent transverse shrnkage dstrbuton. 3. Influental Factors Affectng Inherent Deformaton Heat-nduced deformaton s affected by many complex and uncertan factors that make t dffcult to obtan accurate predctons requred by automatc formng systems. In order to dentfy mportant factors affectng nherent deformaton, let us consder a formed plate such as that shown n Fg.4(a), formed by applyng a heatng lne along x-drecton n the mddle of the plate. An example of analyss results s shown n Fg.5 whch presents the nherent transverse shrnkage dstrbuton along the x-drecton (heatng drecton). 59
Influental Factors Affectng Inherent Deformaton durng Plate Formng by Lne Heatng (Report 1) A prmary factor nfluencng nherent deformaton s the heatng method. The four components of nherent deformaton are strongly related to varables such as the heat nput (Q), sze of the heatng zone and the heatng source speed. Inherent deformaton almost proportonally ncreases wth the heat nput and decreases wth the heatng source speed. Coolng method s another mportant factor. Inherent deformaton s drectly dependent on the rate of coolng. Rate of coolng can be ncreased by usng water leadng to an ncrease of transverse shrnkage and a slght decrease of longtudnal shrnkage (tendon force). The locaton and area of applcaton of water coolng affects all 4 components of nherent stran. Also, ncreasng the rate of coolng, the varaton on nherent deformaton along the heatng lne decreases. As shown Fg.5, nherent transverse shrnkage vares along the heatng lne. At and near both the entrance and the ext edges (regons L 1 and L 3 ), nherent deformaton s smaller than that n the mddle regon of the plate (L 2 ). The same tendency can be observed n the other three components of nherent deformaton. The varaton of nherent deformaton at and near the edges from the maxmum nherent deformaton near the mddle of the plate s another mportant factor known as the edge effect. When the same heatng lne s appled n the same x-drecton but close to one sde of the plate, the nherent deformaton becomes smaller. Ths varaton of nherent deformaton s called the sde effect. The regon (L 2 ) showed n Fg.5, ncreases n the case of a larger plate whle L 1 and L 3 only slghtly changes. In the case of short plates, L 2 almost dsappears and the nherent deformaton at center becomes smaller as the plate becomes shorter. Ths decrease of nherent deformaton s called the plate length effect. In large plates, L 2 s large and ths effect becomes neglgble. In a smlar way, n narrow plates the nherent deformaton decreases wth plate wdth. Ths decrease of nherent deformaton s called the plate wdth effect. Both, plate length effect and plate wdth effect depend on plate edge restrant and are observed only n small plates. The transverse shrnkage dstrbuton shown n Fg.5 s obtaned from the analyss of a mm thck plate. Wth a smaller plate thckness, the nherent deformaton ncreases. The edge effect also ncreases whle the mddle regon L 2 decreases. These varatons are called the plate thckness effect. Fgures 4(b) and 4(c) show the resdual stress dstrbuton of one half of the deformed plate model shown n Fg.4(a). Hgh tensle and compressve resdual stresses appear n and close to the heatng area whle far from the center of the plate, resdual stresses are small. δ y(mm) 0.24 0.20 0.16 0.12 0.08 0.04-0.04-0.08-0.12-0.16-0.20 0 300 600 900 1200 1500 1 2100 20 2700 3000 Fg.6 Effect of the plate length on transverse shrnkage. Now, let us consder a second heatng lne appled parallel to the frst heatng lne. The total nherent deformaton after the two heatng lnes s not a smple addton of that produced by each heatng lne when appled alone 12). The same s observed when the second heatng s appled at the same poston of the frst heatng lne (overlapped heatng) or when the second heatng lne s appled n the transverse drecton of the frst heatng lne (crossed heatng). In these three cases, the nherent deformaton produced by the second heatng lne s greatly affected by the resdual stresses produced by the frst heatng lne. In ths paper, the dfference of the fnal nherent deformaton from the smple summaton of the nherent deformaton of two heatng lnes n each of these three cases s called, the effect of parallel heatng lnes, the effect of overlapped heatng lnes and the effect of crossed heatng lnes respectvely. In the case of multple heatng lnes, the heatng sequence also nfluences the nherent deformaton. In addtonal to the above mentoned nfluental factors, the nherent deformaton s affected by the resdual stresses produced by the cuttng process and by the ntal curvature of the plate. 4. Effects of Plate Sze and Edge Effect In the prevous secton a bref ntroducton of the nfluental factors that affect nherent deformaton s presented. In ths secton the effect of plate length, the effect of plate wdth and the edge effect are dscussed and clarfed. Other factors are dscussed and clarfed n followng reports. 4.1 Effect of plate length Fgure 6 shows the varaton of nherent transverse shrnkage wth plate length whle plate wdth and thckness are kept constant. Heat condtons and heat 60
Transactons of JWRI, Vol.36 (2007), No.1 δ x (mm) δ x(mm) θ y(mm) 0.03 0.02-0.02-0.03-0.05-0.08-0.09-0.11-0.12 0 300 600 900 1200 1500 1 2100 20 2700 3000 Fg.7 Effect of the plate length on longtudnal shrnkage. 0.030 0.025 0.020 0.015 5-5 - -0.015-0.020-0.025-0.030 0 300 600 900 1200 1500 1 2100 20 2700 3000 Fg.8 Effect of the plate length on transverse bendng. 0.018 0.016 8 6 4 2-2 -4 0 300 600 900 1200 1500 1 2100 20 2700 3000 Fg.9 Effect of the plate length on longtudnal bendng. source speed are also kept constant. It may be seen that the maxmum nherent transverse shrnkage (descrbed n Fg.5) n small plates, changes wth plate length. However, for plates larger than about, maxmum nherent transverse shrnkage s almost the same. The central regon of the plate (L 2 ) ncreases wth δ y(mm) 0.21 0.18 0.15 0.12 0.09 0.06 0.03-0.03-0.09-0.12-0.15-0.18 0 100 200 300 0 500 600 700 Fg.10 Effect of the plate wdth on transverse shrnkage. plate length, and the dstrbuton of nherent transverse shrnkage s almost flat. The same characterstcs are also observed n Fgs.7, 8 and 9 whch present the varaton of dstrbutons of nherent longtudnal shrnkage, nherent transverse bendng and nherent longtudnal bendng respectvely. Although, the nherent deformaton vares along plate length, only n the case of small plates, a consderable decrease of maxmum nherent deformaton s observed. In shpbuldng, such small plates are not usually formed by lne heatng. Therefore, n actual shpbuldng plates, t may be sad that the nherent deformaton and ts dstrbuton along the heatng lne n regon L 2 s not a functon of plate length. 4.2 Effect of the plate wdth Fgure 10 shows the varaton of nherent transverse shrnkage wth plate wdth whle plate length and thckness are kept constant. It s observed that n the case of narrow plates, there s a small reducton of the maxmum nherent transverse shrnkage compared wth that obtaned from wde plates. From the fgure t may be seen that ths reducton s small and can be neglected. Ths reducton s also observed n Fgs.11, 12 and 13 whch present the varaton of nherent longtudnal shrnkage, nherent transverse bendng and nherent longtudnal bendng along plate length wth dfferent plate wdths. Therefore t may be sad that the dstrbuton of nherent deformaton along the heatng lne for real shp plates s not a functon of plate wdth. 4.3 Edge effect In prevous sectons t may be seen that when the heatng lne s long enough, the dstrbuton of the nherent deformaton n the nteror of the plate (regon L 2 ) s almost unform. It may also be seen that n all 61
Influental Factors Affectng Inherent Deformaton durng Plate Formng by Lne Heatng (Report 1) δx(mm) θ y(mm) θ x (mm) 0.03 0.02 0.01-0.01-0.02-0.03-0.04-0.05-0.07-0.08-0.09-0.10-0.11 0 100 200 300 0 500 600 700 Fg.11 Effect of the plate wdth on longtudnal shrnkage. 0.030 0.025 0.020 0.015 5-5 - -0.015-0.020-0.025-0.030 0 100 200 300 0 500 600 700 Fg.12 Effect of the plate wdth on transverse bendng. 0.016 8 6 4 2-2 -4 0 100 200 300 0 500 600 700 Fg.13 Effect of the plate wdth on longtudnal bendng. cases, the dstrbuton of nherent deformaton at the entrance edge (regon L 1 ) does not dffer whle at the ext edge (regon L 3 ), slghtly changes. In ths paper, the decrease of nherent deformaton at both, the entrance and the ext edge of the plate s referred to as the edge effect. δ y(mm) δ x (m m) -0.02-0.04-0.08-0.10-0.12-0.14-0.16 0 20 60 80 100 120 1 160 180 200 Fg.14 Entrance edge effect on transverse shrnkage. 0.03 0.01-0.01-0.02-0.03-0.04-0.05-0.07-0.09 0 20 60 80 100 120 1 160 180 200 Fg.15 Entrance edge effect on longtudnal shrnkage. Due to the fact that nherent deformaton does not sgnfcantly change wth plate wdth, n ths secton, we only examned the relaton between edge effect and plate length. Fgure 14 shows the dstrbuton of nherent transverse shrnkage at the entrance edge (regon L 1 ). In ths fgure t s observed that the edge effect at the entrance edge does not depend on the plate length. The same tendency s observed n Fgs.15, 16 and 17 whch present the dstrbuton of nherent longtudnal shrnkage, nherent transverse bendng and nherent longtudnal bendng, respectvely. Fgure 18 shows the dstrbuton of nherent transverse shrnkage at the ext edge of the plates (regon L 3 ). These dstrbutons follow the same tendency. However, there are dfferences between these dstrbutons for dfferent plate lengths, especally n the case of plate length equals to. For plates about or larger, these dfferences become very small and may be neglected. The same tendency s observed n Fgs.19, 20 and 21 whch present the dstrbuton of nherent longtudnal shrnkage, nherent transverse 62
Transactons of JWRI, Vol.36 (2007), No.1 θ y(mm) 0.024 0.021 0.018 0.015 9 6 3 0 20 60 80 100 120 1 160 180 200 Fg.16 Entrance edge effect on transverse bendng. δ x (m m) -9-0.018-0.027-0.036-0.045-0.054 3-0.072-0.081-0.090-0.099-0.108 0 20 60 80 100 120 1 160 180 200 220 2 260 280 300 Fg.19 Ext edge effect on longtudnal shrnkage. 0.028 0.021 8 7 θ x (m m) 6 4 2 θ y(mm) -7 - -0.021-2 -4 0 20 60 80 100 120 1 160 180 200 Fg.17 Entrance edge effect on longtudnal bendng. -0.028-0.035 0 20 60 80 100 120 1 160 180 200 220 2 260 280 300 Fg.20 Ext edge effect on transverse bendng. δ y(mm) 0.210 0.175 0.1 0.105 0.070 0.035-0.035-0.070-0.105-0.1-0.175 0 20 60 80 100 120 1 160 180 200 220 2 260 280 300 Fg.18 Ext edge effect on transverse shrnkage. bendng and nherent longtudnal bendng at the ext edge, respectvely. Consderng the smlarty of nherent deformaton at the entrance edge and at the ext edge of plates wth dfferent lengths larger than about, the dstrbuton of nherent deformaton for a shorter plate θ x (m m) 0.015 9 7 5 3 2 0 20 60 80 100 120 1 160 180 200 220 2 260 280 300 Fg.21 Ext edge effect on longtudnal bendng. length (same thckness) can be obtaned by superposng the dstrbuton of nherent deformaton of both, the entrance and the ext edges (L 1 and L 3 ), obtaned from the analyss of a larger plate. The value of the nherent stran n the central regon (L 2 ) stays almost unchanged. 63
Influental Factors Affectng Inherent Deformaton durng Plate Formng by Lne Heatng (Report 1) In a prevous report 13) the authors demonstrated that the edge effect manly depend on heatng condton, heat source speed and plate thckness. The authors also examned methods to reduce the edge effect. They concluded that trangular heatng and edge heatng are the most effectve methods to ncrease the nherent deformaton at plate edge. 5. Conclusons In the above, nfluental factors affectng nherent deformaton n plate formng by lne heatng are outlned. Investgatons of plate length effect, plate wdth effect and edge effect are presented. Through these nvestgatons the followng conclusons are drawn. (1) In small plates, nherent deformaton ncreases wth both, plate length and plate wdth. However, when plate dmensons become larger than about (for a mm thck plate), the effects of plate length and plate wdth on nherent deformaton become neglgble. Therefore, for real shp plates, both plate length and plate wdth do not apprecably nfluence the nherent deformaton. (2) Edge affect at entrance edge (regon L 1 ) does not change wth nether plate length nor wth plate wdth. (3) The edge effect at the ext edge (regon L 3 ) slghtly changes wth the plate length. However, for plates larger than about, the change s small and may be neglected. (4) For plates longer than about, the dstrbuton of nherent deformaton for a shorter plate length (same thckness) can be obtaned by superposng the dstrbuton of nherent deformaton of both, the entrance and the ext edge (L 1 and L 3 ), obtaned from the analyss of a larger plate. The value of the nherent stran n the central regon (L 2 ) stays almost unchanged. Acknowledgements The authors would lke to thank IHI marne department, especally Dr. Mornobu Ishyama and hs research team for sharng ther nvaluable experences. References 1) A. Moshaov and W. S. Vorus, The Mechancs of the Flame Bendng Process: Theory and Applcatons. Journal of Shp Research, 31(4):269-281, 1987 2) J. G. Shn and A. Moshaov, Modfed Strp Model for Analyzng the Lne Heatng Method-part 1: Elastc Plates. Journal of Shp Research, 35(2):172-182, 1991. 3) C. D. Jang and S. C. Moon, An Algorthm to Determne Heatng Lnes for Plate Formng by Lne Heatng Method. Journal of Shp Producton, 14(4):238-245, November 1998. 4) A. K. Kyrsand, T. B. Kermands, and S. G. Pantelaks, Numercal and Expermental Investgaton of the Laser Formng Process. Journal of Materals Processng Technology, 87:281-290, 1999. 5) Y. Ueda, H. Murakawa, A. M. Rashwan, Y. Okumoto and R. Kamc: Development of Computer-Aded Process Plannng System for Plate Bendng by Lne Heatng (Report 1) - Relaton between Fnal Form of Plate and Inherent Stran. Trans. SNAME, J. of Shp Producton, Vol.10, No.1 (Feb. 1994), 59-67. 6) Y. Ueda, H. Murakawa, A. M. Rashwan, Y. Okumoto and R. Kamchka: Development of Computer-Aded Process Plannng System for Plate Bendng by Lne Heatng, (Report 2) - Practce for Plate Bendng n Shpyard Vewed from Aspect of Inherent Stran. Trans. SNAME, J. of Shp Producton Vol.10, No.4 (Nov. 1994), 239-247. 7) Y. Ueda, H. Murakawa, A. M. Rashwan, Y. Okumoto and R. Kamchka: Development of Computer-Aded Process Plannng System for Plate Bendng by Lne Heatng, (Report 3) relaton between heatng condton and deformaton. Journal of Shp Producton 10(4):248-257, 1994 8) Y. Ueda, H. Murakawa, A. M. Rashwan, Y. Okumoto and R. Kamchka: Development of Computer-Aded Process Plannng System for Plate Bendng by Lne Heatng, (Report 4) Transactons of Japan Weldng Research Insttute, 22(2):305-313, 1993. 9) C. D. Jang, S. Seo, and D. E. Ko: A Study on the Predcton of Deformaton of Plates due to Lne Heatng usng a Smplfed Thermal Elasto-Plastc Analyss. Journal of Shp Producton, 13(1):22-27, 1997. 10) J. Bao and Y. L. Yao: Analyss and Predcton of Edge Effects n Laser Bendng. Journal of Manufacturng Scence and Engneerng, ASME Vol. 123:53-61, 2001. 11) H. Nshkawa, H. Serzawa and H. Murakawa: Development of a Large-scale FEM for Analyss Mechancal Problems n Weldng, Journal of the Japan Socety of Naval Archtects, 2(2005), pp.379. 12) A. Vega, Y. Tajma, S. Rashed and H. Murakawa: Numercal Study on Inherent Deformaton of Thck Plates Undergong Lne heatng. Transactons of Internatonal Socety of Offshore and Polar Engneers (ISOPE), Vol. 4(pp. 3472-3479) 2007. 13) A. Vega, W. Lang, H. Serzawa and H. Murakawa: Numercal Study on Edge Effect n Transverse Shrnkage under Lne Heatng. Transactons of Asan Techncal Exchanges and Advsory Meetng on Marne Structure (TEAM), (pp.283-290), 2006. 64