1 o Experiments and Simulation of Line Heating of Plates by Roger J. Anderson B.S. in Naval Architecture and Marine Engineering Webb Institute of Naval Architecture, 1998 Submitted to the Department of Ocean Engineering in partial fulfillment of the requirements for the degree of Master of Science in Ocean Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 1999 @ Massachusetts Institute of Technology 1999. All rights reserved. Author........................ , ......... ........... .. Department of Ocean Engineering September 7, 1999 Certified by.......... ............................................ Nicholas M. Patrikalakis, Kawasaki Professor of Engineering Thesis Co-Supervisor Certified by........... ...................... y- - .. - - -------- -.-.-.-- Takashi Maekawa, Lecturer and Principal Research Scientist Thesis Co-Supervisor A ccepted by ......... . .. . . . . .. . . . ............ ............ . .--- Nicholas M. Patrikalakis, Kawasaki Professor of Engineering Chairman, Departmental Committee on Graduate Studies ENG Experiments and Simulation of Line Heating of Plates by Roger J. Anderson Submitted to the Department of Ocean Engineering on September 7, 1999, in partial fulfillment of the requirements for the degree of Master of Science in Ocean Engineering Abstract In order to reduce the long computation times normally associated with three dimen- sional non-linear finite element models of the line heating process, a simplified model for the prediction of angular deformations of plates due to line heating is presented. The complete model is composed of thermal and mechanical subsidiary models. The thermal model utilizes an analytically determined temperature distribution, which has been modified from an existing solution to increase accuracy by incorporating the effects of convective heat losses and a distributed heat source in lieu of a point heat source. Based on user-defined values of heat source strength, plate thickness, and heat source speed, the size and shape of isotherms can be calculated. Using the thermal model, the dimensions of the isotherm corresponding to a critical tem- perature, which defines the size of the inherent strain zone, are found and used in the mechanical model. The mechanical model uses the dimensions of the inherent strain zone to calculate estimates of angular deformation and equivalent nodal forces, which can be used in a linear finite element model. Line heating experiments on mild steel plates were performed with a variety of heating conditions. Based on the temperatures measured during these experiments, the resulting temperature field is matched with the model and used to find the angular deformation analytically and with a linear finite element model. These values are compared with corresponding experimental results and a relation between the heating condition and the amount of deformation is established. Thesis Co-Supervisor: Nicholas M. Patrikalakis, Ph.D. Title: Kawasaki Professor of Engineering Thesis Co-Supervisor: Takashi Maekawa, Ph.D. Title: Lecturer and Principal Research Scientist 2 Acknowledgments I would like to express my appreciation to Professor Nicholas M. Patrikalakis, Dr. Takashi Maekawa, and Guoxin Yu, for their advice, assistance, and encouragement. I am also grateful to the American Bureau of Shipping and the MIT Department of Ocean Engineering for their generous funding. I would like to acknowledge Dr. Hidekazu Murakawa, from Osaka University, for his invaluable insights on the line heating process, and Kawasaki Heavy Industries, whose donation of the welding robot made this project possible. Finally, I would like to thank Alice Jensen, who served as a source of inspiration for this work. 3 Contents 1 INTRODUCTION 12 2 SIMPLIFIED MECHANICAL MODEL 15 2.1 INTRODUCTION ....... ............................ 15 2.2 ASSUMPTIONS ....... ............................. 15 2.3 DERIVATION .............................. 18 2.3.1 RESIDUAL STRAIN ...................... . 18 2.3.2 INHERENT STRAIN ZONE ....................... 23 2.3.3 ANGULAR DEFORMATION ....................... 25 2.3.4 EQUIVALENT FORCES .................... 30 3 ROSENTHAL'S SOLUTION FOR MOVING HEAT SOURCES 35 3.1 INTRODUCTION ................................... 35 3.2 GENERAL SOLUTION OF A QUASI- STATIONARY HEAT SOURCE .......................... 37 3.3 LINEAR SOURCE, CONSTANT STRENGTH HEAT SOURCE ... 38 3.3.1 SOLUTION DERIVATION .................. . 39 3.3.2 SOLUTION BEHAVIOR .......................... 40 3.4 CONSTANT STRENGTH HEAT SOURCE, SURFACE LOSSES . . 42 3.4.1 SOLUTION DERIVATION .................. . 42 3.4.2 SOLUTION BEHAVIOR .......................... 42 3.5 VARIABLE STRENGTH SOURCE, NO SURFACE LOSSES .... 43 3.5.1 SOLUTION DERIVATION .................. . 45 4 3.5.2 SOLUTION BEHAVIOR ..................... 49 3.5.3 CALCULATION OF An COEFFICIENTS ........... 52 4 MODIFIED MODELS 57 4.1 INTRODUCTION .................................. 57 4.2 MODIFICATIONS TO ROSENTHAL'S SOLUTION ......... 57 4.2.1 VARIABLE STRENGTH HEAT SOURCE WITH SURFACE LOSSES ........ ................ 58 4.2.2 DISTRIBUTED HEAT SOURCES ....... 70 4.3 INHERENT STRAIN ZONE DIMENSIONS ..... 76 4.3.1 MAXIMUM BREADTH ............... 78 4.3.2 MAXIMUM DEPTH ................. . 81 . . 4.3.3 INHERENT STRAIN ZONE DIMENSIONS WIT A DIS- TRIBUTED HEAT SOURCE ......... . . . . . . 82 4.4 MODIFICATION OF SIMPLIFIED MECHANICAL MOD EL.... 85 4.4.1 ANGULAR DEFORMATION ......... . . . . . . 85 4.4.2 EQUIVALENT FORCES ............. . . . . . . 85 4.4.3 FINITE ELEMENT MODELING ........ . . . . . . 86 5 EXPERIMENTS 87 5.1 INTRODUCTION ................... . . . . . . 87 5.2 EXPERIMENTAL PROCEDURE ........... . . . . . . 87 5.2.1 CALIBRATION PROCEDURE ......... . . . . . . 88 5.2.2 PLATES ..................... . . . . . . 88 5.2.3 TESTING ........................ . . . . . . 89 6 RESULTS AND ANALYSIS 101 6.1 INTRODUCTION .................................. 101 6.2 R ESU LTS . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 101 6.3 ANALYSIS ....................................... 105 5 6.3.1 CRITICAL TEMPERATURE AND RESIDUAL STRAIN .... .... .. . . ....... .. . 106 6.3.2 HEAT FLUX SCALING ................ 108 6.3.3 VARIABLE MATERIAL PROPERTIES .... 108 6.3.4 BOUNDARY CONDITIONS ............ 110 6.3.5 LOCAL EFFECTS ................ 113 7 CONCLUSIONS 116 7.1 INTRODUCTION .................... . . . . . . . 116 7.2 MODEL ATTRIBUTES .................... . . . . . . . 116 7.3 MODEL LIMITATIONS AND APPLICATIONS ..................... 117 8 RECOMMENDATIONS 119 8.1 INTRODUCTION .................... 119 8.2 CALIBRATION PROCEDURE ................ 119 8.3 CONTROLLABLE HEAT SOURCE ............ 120 8.4 LOCAL BEHAVIOR OF MODEL ............ 120 8.5 EDGE EFFECTS ..................... 121 8.6 MULTIPLE HEATING LINES ............. 121 6 List of Figures 2-1 (a) Assumed isothermal region (b) Actual isothermal region. . . . . . 16 2-2 (a) Model of plastic region (b) Model of elastic region. . . . . . . . . 17 2-3 Idealized stress-strain loading diagram for the line heating process. . . 22 2-4 Equivalent gap of plastic region due to deformation. . . . . . . . . . . 23 2-5 Assumed elliptical distribution of critical isothermal region and corre- sponding dimensions. Adapted from [7]. . . . . . . . . . . . . . . . . 23 2-6 (a) Forces in a differential element of the plate (b) Normal stresses in a slice through the plate thickness (c) Plate curvature. . . . . . . . . 26 2-7 Angular deformation J in the y-z plane. Adapted from [7]. . . . . . . 28 2-8 Angular deformation as a function of heating conditions with E* = - .007674. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2-9 Distribution of equivalent nodal forces due to line heating. Adapted from [7]. . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 31 2-10 Effect of depth on the solution of (a) transverse moment, (b) transverse force, (c) longitudinal moment, and (d) longitudinal force. . . . . . . 34 3-1 (a) Plate-fixed coordinate system (b) Heat source-fixed coordinate sys- tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3-2 Temperature time history of a point on the upper surface on the heating line with a constant strength heat source. . . . . . . . . . . . . . . . . 41 3-3 Temperature time history for a point on the upper surface on the heat- ing line with a constant strength heat source (a) without heat loss and (b) with heat loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 3-4 Temperature distribution in the (-y plane on the upper surface with a constant strength heat source (a) without heat loss and (b) with heat loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3-5 Temperature time history of a point on the heating line with a variable strength heat source (a) on the upper surface and (b) on the lower surface. 50 3-6 Temperature distribution in the (-y plane with a variable strength heat source (a) on the upper surface and (b) on the lower surface. . . . . . 51 3-7 Temperature distribution in the (-z plane along the heating line (a) for a constant strength heat source and (b) a variable strength heat source. 51 3-8 Temperature distribution in the y-z plane under the heat source (a) for a constant strength heat source and (b) a variable strength heat source........ .................................... 52 3-9 Triangular heat flux distribution. . . . . . . . . . . . . . . . . . . . . 52 4-1 Plot of the two functions in equation 4.9 used to solve for c, numerically. 60 4-2 Heat flux representations (a) with 20 coefficients and (b) with 100 coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4-3 Temperature time history for a point on the upper surface on the heat- ing line with a variable strength heat source (a) without heat loss and (b) with heat loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4-4 Temperature distribution in the (-y plane on the upper surface with a variable strength heat source (a) without heat loss and (b) with heat loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4-5 Temperature distribution in the (-z plane on the upper surface with a variable strength heat source (a) without heat loss and (b) with heat loss. ......... .................................... 69 4-6 Temperature distribution in the y-z plane on the upper surface with a variable strength heat source (a) without heat loss and (b) with heat loss. ......... .................................... 69 8 4-7 Temperature distribution in the (-y plane on the upper surface (a) with one point heat source and (b) multiple point heat sources. . . . 71 4-8 Distribution of multiple point heat sources within a concentrated heat sp ot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4-9 Temperature distribution in the (-z plane on the upper surface (a) with one point heat source and (b) multiple point heat sources. . . . . . . 72 4-10 Temperature distribution in the y-z plane on the upper surface (a) with one point heat source and (b) multiple point heat sources. . . . 72 4-11 Gaussian distribution of a heat source. . . . . . . . . . . . . . . . . . 74 4-12 Temperature distribution in the (-y plane on the upper surface (a) with one point heat source and (b) continuously distributed heat source. 76 4-13 Temperature distribution in the (-y plane on the upper surface from a non-linear finite element model. . . . . . . . . . . . . . . . . . . . . . 77 4-14 Temperature distribution in the (-z plane on the upper surface (a) with one point heat source and (b) continuously distributed heat source. . 78 4-15 Temperature distribution in the y-z plane on the upper surface (a) with one point heat source and (b) continuously distributed heat source. 78 5-1 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5-2 Line heating process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5-3 Line heating process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5-4 Typical distribution of thermocouples used in experiments. . . . . . . 93 5-5 Temperature time history at x=0.0508[m], y=0.0508[m], and z=0[m]. 93 5-6 Temperature time history at x=0.0508[m], y=0.0508[m], and z=0.00635[m]. 94 5-7 Temperature time history at x=0.15[m], y=0[m], and z=0.00635[m]. . 94 5-8 Temperature time history at x=0.15[m], y=0.0254[m], and z=0[m]. 95 . 5-9 Temperature time history at (=0.15[m], y=0.0762[m], and z=0[m]. 95 . 5-10 Temperature time history at (=0.15[m], y=0.127[m], and z=0[m]. . . 96 5-11 Temperature time history at x=0.254[m], y=0.0508[m], and z=0.00635[m]. 96 5-12 Temperature time history at x=0.254[m], y=0.0508[m], and z=0[m]. . 97 9 5-13 Temperature time history at x=0.254[m], y=-0.0508[m], and z=0[m]. 97 5-14 Temperature time history at (=0.0508[m], y=-0.0508[m], and z=0[m]. 98 5-15 Comparison of experimental and matched analytic time temperature distributions (a) on lower surface of the plate on heating line and (b) on upper surface of the plate, 1 inch from the heating line. . . . . . . 100 6-1 Deformed geometry of plate after equivalent nodal force loading. . . . 103 6-2 Comparison of experimental and analytic solutions of angular deflection. 104 6-3 Effect of temperature on the tensile strength of mild steel. Adapted from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6-4 Temperature time history for a point on the upper surface of a lead plate on the heating line (a) without heat loss and (b) with heat loss. 114 6-5 Temperature distribution in the (-y plane on the upper surface of a lead plate (a) without heat loss and (b) with heat loss. . . . . . . . . 114 10