DISPLACEMENT ANALYSIS OF NON-CIRCULAR PLANAR CURVED BEAMS UNDER IN-PLANE IMPULSIVE LOAD A Thesis Submitted to the Graduate School of Engineering and Sciences of İzmir Institute of Technology in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Mechanical Engineering by Ahmet ÇELİK October 2015 İZMİR We approve the thesis of Ahmet ÇELİK name in capital letters Examining Committee Members: ________________________________ Prof. Dr. Bülent YARDIMOĞLU Department of Mechanical Engineering, İzmir Institute of Technology ________________________________ Assist. Prof. Dr. Onursal ÖNEN Department of Mechanical Engineering, İzmir Institute of Technology ________________________________ Assoc. Prof. Dr. Levent MALGACA Department of Mechanical Engineering, Dokuz Eylül University 16 October 2015 _____________________________ Prof. Dr. Bülent YARDIMOĞLU Supervisor, Department of Mechanical Engineering, İzmir Institute of Technology ________________________ _________________________ Prof. Dr. Metin TANOĞLU Prof. Dr. R. Bilge KARAÇALI Head of the Department of Dean of the Graduate School Mechanical Engineering of Engineering and Sciences ACKNOWLEDGEMENTS Foremost, I would like to express my sincere gratitude to my advisor Prof. Bulent YARDIMOGLU for the continuous support of my M. Sc study and research, for his patience, motivation, enthusiasm, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my M. Sc study. ABSTRACT DISPLACEMENT ANALYSIS OF NON-CIRCULAR PLANAR CURVED BEAMS UNDER IN-PLANE IMPULSIVE LOAD In this study, time response of a planar curved beam with variable curvatures under in-plane impact load is analyzed by two numerical methods which are Finite Difference and Finite Element Methods. The solution procedures in both methods are based on solution of eigenvalue and time response problems. Catenary form is selected as the axis of curved beam. A computer program is developed in Mathematica for the solution with Finite Difference Method. Moreover, a computer code is written for the geometric and finite element models of curved beam with variable curvature in ANSYS by using APDL (ANSYS Parametric Design Language). Solutions of the two methods are compared in each other and then good agreement is observed. The effects of impuls and damping properties on the time response are investigated. iv ÖZET DÜZLEM İÇİ İMPULSİF KUVVET ALTINDAKİ DAİRESEL OLMAYAN DÜZLEMSEL EĞRİ ÇUBUKLARIN YERDEĞİŞTİRME ANALİZİ Bu çalışmada, düzlem içi darbe yükü altındaki değişken eğrilikli düzlemsel eğri çubukların zaman cevabı sayılsal yöntemler olan Sonlu Farklar ve Sonlu Elemenlar Yöntemleri ile incelenmiştir. Her iki metoddaki çözüm usulü özdeğer ve zaman cevabı problemlerine dayalıdır. Eğri çubuğun ekseni olarak Katenary biçimi seçilmiştir. Sonlu Farklar Yöntemi ile çözüm için Mathematica’da bir program geliştirilmiştir. Ayrıca, değişken eğrilikli eğri çubuğun geometrik ve sonlu eleman modelleri için ANSYS de APDL (ANSYS Parametrik Tasarım Dili) ile bir bilgisayar kodu yazılmıştır. İki metodun çözümleri birbirleri ile karşılaştırılmış ve iyi bir uyum gözlenmiştir. İmpuls ve sönüm özelliklerinin zaman cavabına etkileri araştırılmıştır. v TABLE OF CONTENTS LIST OF FIGURES.....................................................................................................viii LIST OF TABLES.......................................................................................................ix LIST OF SYMBOLS...................................................................................................x CHAPTER 1. GENERAL INTRODUCTION............................................................1 CHAPTER 2. THEORETICAL BACKGROUND.....................................................8 2.1. Introduction......................................................................................8 2.2. Geometry of Curved Beam..............................................................9 2.3. Derivation of the Equations of Motion............................................10 2.4. Discretization of Continuous Systems.............................................15 2.4.1. Finite Difference Method for Transient Analysis......................15 2.4.2. Finite Element Method for Transient Analysis..........................17 2.4.2.1. Full Solution Method...........................................................18 2.4.2.2. Reduced Solution Method...................................................18 2.4.2.3. Mode Superposition Solution Method.................................19 CHAPTER 3. NUMERICAL RESULTS AND DISCUSSION..................................21 3.1. Introduction......................................................................................21 3.2. Convergence Studies for Natural Frequency...................................22 3.3. Natural Frequencies for Different Models.......................................24 3.4. Proportional Damping Parameters...................................................26 3.5. Comparison Studies for Impact Response.......................................26 3.6. Impact Responses for Different Models..........................................27 CHAPTER 4. CONCLUSIONS..................................................................................32 REFERENCES............................................................................................................33 APPENDIX A. CENTRAL DIFFERENCES..............................................................36 vi LIST OF FIGURES Figure Page Figure 1.1 Archimedes-type spiral spring....................................................................2 Figure 1.2. Arch geometry and loads on an arch element...........................................3 Figure 1.3. A quadrantal circular beam subjected to radial impact in its own plane at its tip by a rigid mass....................................................................5 Figure 1.4. Dimensions of the RC arch beam model...................................................6 Figure 1.5. Pictorial view of the experimental setup...................................................6 Figure 2.1. A planar curved beam with variable radius of curvature under impact loads...............................................................................................8 Figure 2.2. Parameters of catenary beam.....................................................................9 Figure 2.3. A curved beam with internal forces and moments....................................10 Figure 2.4. A curved domain divided into six sub domains........................................15 Figure 3.1. The three geometrical models of the curved beam....................................21 Figure 3.2. Convergence of first natural frequency by FDM and FEM......................23 Figure 3.3. Effects of h and R on first natural frequencies.........................................24 o Figure 3.4. Effects of h and R on second natural frequencies....................................24 o Figure 3.5. Effects of h and R on third natural frequencies........................................25 o Figure 3.6. Effects of h and R on fourth natural frequencies.....................................25 o Figure 3.7. Time response for ξ =0.05 by FDM..........................................................26 b Figure 3.8. Time response for ξ =0.05 by FEM..........................................................26 b Figure 3.9. Time response for Model 1.......................................................................28 Figure 3.10. Time response for Model 2.....................................................................29 Figure 3.11. Time response for Model 3.....................................................................29 Figure 3.12. Time response for Model 4.....................................................................29 Figure 3.13. Time response for Model 5.....................................................................30 Figure 3.14. Time response for Model 6.....................................................................30 Figure 3.15. Time response for Model 7.....................................................................30 Figure 3.16. Time response for Model 8.....................................................................31 vii LIST OF TABLES Table Page Table 3.1. Material properties in the models...............................................................22 Table 3.2. Convergence of natural frequencies based on FDM...................................22 Table 3.3. Convergence of natural frequencies based on FEM...................................23 Table 3.4. Parametric details of the models.................................................................28 viii LIST OF SYMBOLS A cross-sectional area b width of the beam B bending stiffness C [ ] differential operator for damping [C] damping matrix E modulus of elasticity f (s) - f (s) variable coefficients of differential equation 0 6 f ith natural frequency i F external force in normal direction n F external force in tangential direction t F(s, t) force vector in continuous domain {F(t)} force vector in discrete domain {F } initial force vector 0 G shear modulus h depth of the beam KE kinetic energy [K] stiffness matrix L [ ] differential operator for stiffness m mass per unit length M internal moment about y- axis y M [ ] differential operator for mass [M] mass matrix n number of grids N internal normal force, number of element or number of time interval {q } initial displacement vector 0 {q } displacement vector at time t 1 1 {q } displacement vector at time t j+1 j+1 {q(t)} displacement vector R radius parameter of the catenary curve 0 s circumferential coordinate ix s length of the beam L SE strain energy t time T internal axial force T final time f u(s, t) radial displacement w(s, t) tangential displacement W work of the non-conservative forces nc (x ,z ) the tip co-ordinates of the curved beam r r α,α slope of the curve at any point and point r r α,β Rayleigh damping coefficients δ variation operator or dirac delta function ∆t time interval ε tangential strain due to tension ρ density of material of beam κ′ initial curvature 0 κ′ curvature after displacement occurs 1 ρ(z),ρ(α),ρ(s) variable radius of curvature 0 0 0 ω natural frequency ( ` ) derivative with respect to “s” ( . ) derivative with respect to “t” x