NEW STRAIN-BASED TRIANGULAR AND RECTANGULAR FINITE ELEMENTS FOR PLANE ELASTICITY PROBLEMS

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1 The Islmic Universit Gz Postgrdute Denship Fult of Engineering Structurl Engineering Deprtment الجامعة الا سلامیة غزة عمادة الدراسات العلیا كلیة الھندسة قسم الا نشاءات NEW STRAIN-BASED TRIANGULAR AND RECTANGULAR FINITE ELEMENTS FOR PLANE ELASTICITY PROBLEMS SUBMITTED BY ENG. SALAH M. TAYEH SUPERVISED BY DR. ATTIA I. MOUSA Associte Professor, Den of Fcult of Engineering Islmic Universit Gz A Thesis submitted in prtil fulfillment of the requirements for the Degree of Mster of Science in Civil / Structurl Engineering JULY 003

2 سورة البقرة الا ية ٢٨٦ - صدق االله العظيم

3 Abstrct ABSTRACT In this thesis, the Finite Element Method of structurl nlsis is used to investigte nd compre the performnce of severl strin-bsed elements. Two new strin-bsed tringulr nd rectngulr finite elements in Crtesin coordintes sstem, for two dimensionl elsticit problems re developed. Ech of these elements hs three degrees of freedom per node. The Strin Bsed Approch is used to develop nd formulte these two dimensionl finite elements. In this pproch, finite elements re formulted bsed on ssumed polnomil strins rther thn displcements. Two min computer progrms re developed to nlze the new finite elements. To test the performnce of these elements, the re used to solve two common plne elsticit problems. The problems considered included re: the problem of plne deep cntilever bem fied t one nd loded b point lod t the free ; nd the problem of simpl supported bem loded t the mid-spn b point lod. The finite element solutions obtined for these problems re compred with the nlticl vlues given b the elsticit solutions. Results obtined using the new tringulr nd rectngulr elements re lso compred to those of the well-known constnt strin tringulr element (CST) nd the biliner rectngulr element (BRE) respectivel. In ll cses, convergence curves for deflection t specific points within ech problem re plotted to show tht cceptble levels of ccurc. Furthermore, convergence curves for bing stress t points on the upper surfce nd sher stress t points on the neutrl is re plotted; gin convergence is ensured. I

4 TO MY PARENTS AND MY WIFE I DEDICATE THIS WORK II

5 Acknowledgments ACKNOWLEDGMENTS The work presented in this thesis ws crried out in the Deprtment of Civil nd Structurl Engineering t the Islmic Universit of Gz, under the supervision of Dr. Atti Mous, Den of Fcult of Civil Engineering. The resercher wishes to tke this opportunit to epress his cknowledgments nd grtitude to Dr. Mous for his continued guidnce nd encourgement throughout the course of this work. M thnks et to the lecturers t the Civil Engineering Deprtment for their effort nd their role in lunching the M. Sc. progrm in the field of Structurl Engineering. I m prticulrl grteful to m prents, whom, without their support I would hve not been ble to complete this work. Finll, I present m thnks for ll people who helped me in completing this work.. III

6 List of Abbrevitions (Nottions) LIST OF ABBREVIATIONS (NOTATIONS) Smbol Nottion, Crtesin coordintes T E ν U V φ ε, ε γ σ, σ τ [ C ] [ B ] [ D ] e [ K ] [ Q ] { δ e } Π DoF CST BRE SBTREIR SBREIR Thickness of the element Young s Modulus of Elsticit Poisson s rtio Displcement in the direction Displcement in the direction In-plne rottion Direct strins in the nd directions, respectivel Sher strin Norml stresses in the nd directions, respectivel Sher stress Displcement trnsformtion mtri Strin mtri Rigidit mtri Element stiffness mtri Strin energ mtri Element nodl displcement vector Totl Potentil Energ Degree of Freedom Constnt Strin Tringle Biliner Rectngulr Element Strin-Bsed Tringulr Element with In-plne Rottion Strin-Bsed Rectngulr Element with In-plne Rottion IV

7 Tble of Contents TABLE OF CONTENTS ABSTRACT...I ACKNOWLEDGMENTS...III LIST OF ABBREVIATIONS (NOTATIONS)...IV LIST OF FIGURES...VIII LIST OF TABLES...X. CHAPTER ONE: INTRODUCTION.... INTRODUCTION.... HISTORY OF FINITE ELEMENT METHOD PROCEDURE OF THE FINITE ELEMENT METHOD Ideliztion of the Structure Formultion of Element Stiffness Mtri Generl Procedure for Derivtion of Element Stiffness Mtri Assembl of the Overll Structurl Mtri nd the Solution Routine Convergence Criteri...9 A- Rigid-Bod Modes...0 B- Constnt Strin...0 C- Inter-element Comptibiit Mesh Size Design... A- Number of Degree of Freedom... B- Aspect Rtio... C- Element Distortions (Skewing)....4 FINITE ELEMENT APPLICATIONS OF TWO-DIMENSIONAL STRUCTURES Constnt Strin Tringulr Element (CST) Liner Strin Tringulr Element (LST) Biliner Rectngulr Element (BRE) Qudrtic Isoprmetric Rectngle Element SCOPE OF THE CURRENT THESIS...6. CHAPTER TWO: STRAIN BASED APPROACH...7. INTRODUCTION...7. REASONS FOR DEVELOPMENT OF STRAIN BASED ELEMENTS HISTORY OF STRAIN BASED APPROACH Strin Bsed Curved Elements Strin Bsed Shell Elements Strin Bsed Two Dimensionl Elements....4 GENERAL PROCEDURE FOR DERIVING DISPLACEMENT FIELDS FOR STRAIN-BASED ELEMENTS...3 V

8 Tble of Contents 3. CHAPTER THREE: COMPUTER PROGRAMS USED COMPUTER SOFTWARE MthCAD Softwre MtLb Softwre IMPLEMENTATION OF THE FEM ANALYSIS Outline of the Mjor Progrm Steps Progrms Flow Chrt CHAPTER FOUR: DEVELOPMENT OF NEW STRAIN-BASED TRIANGULAR ELEMENT INTRODUCTION DERIVATION OF DISPLACEMENT FIELDS OF NEW TRIANGULAR ELEMENT WITH IN-PLANE ROTATION PROBLEMS CONSIDERD DEEP CANTILEVER BEAM PROBLEM Used Mesh Size Anlticl Solution Convergence Results SIMPLY SUPPORTED BEAM PROBLEM Used Mesh Size Anlticl Solution Convergence Results CHAPTER FIVE: DEVELOPMENT OF NEW STRAIN-BASED RECTANGULAR ELEMENT INTRODUCTION DERIVATION OF DISPLACEMENT FIELDS FOR NEW RECTANGULAR ELEMENT WITH IN-PLANE ROTATION PROBLEMS CONSIDERD DEEP CANTILEVER BEAM PROBLEM Used Mesh Size Convergence Results SIMPLY SUPPORTED BEAM PROBLEM Used Mesh Size Convergence Results CHAPTER SIX: COMPARISON OF THE NEW TRIANGULAR AND RECTANGULAR ELEMENTS SUMMARY AND DISCUSSION OF RESULTS FOR THE DEEP CANTILEVER BEAM SUMMARY AND DISCUSSION OF RESULTS FOR THE SIMPLY SUPPORTED BEAM CHAPTER SEVEN: CONCLUSIONS AND RECOMMENDATIONS CONCLUSIONS RECOMMENDATIONS...88 VI

9 Tble of Contents REFERENCES...89 APPENDIX A: COMPUTER CODES...93 A. GENERATION OF TRIANGULAR MESH IN RECTANGULAR DOMAIN (USING MATLAB)...94 A. Smple Tringulr Mesh...96 A. GENERATION OF RECTANGULAR MESH IN RECTANGULAR DOMAIN (USING MATLAB)...97 A. Smple Rectngulr Mesh...99 A.3 MATLAB CODE FOR THE STRAIN BASED TRIANGULAR ELEMENT WITH IN-PLANE ROTATION (SBTREIR)...00 A.4 MATLAB CODE FOR THE STRAIN BASED RECTANGULAR ELEMENT WITH IN-PLANE ROTATION (SBREIR)...06 A4. Smple Input File for New Strin Bsed Rectngulr Element...3 A4. Smple Output File for New Strin Bsed Rectngulr Element...6 VII

10 List of Figures LIST OF FIGURES Figure ١-: Skewed And Wrped Elements (Poor Shped)...3 Figure ٣-: Finite Element Anlsis Flow Chrt...30 Figure ٤-: Coordintes nd Node Numbering for Tringulr Element with 3 DOF per node 3 Figure ٤-: Dimensions nd Loctions of Considered Points for the Deep Cntilever Bem..37 Figure ٤-3: Smple Tringulr Mesh of the Deep Cntilever Bem Problem...38 Figure ٤-4: Bing Stress Pttern in the Deep Cntilever Bem Using CST...39 Figure ٤-5: Bing Stress Pttern in the Deep Cntilever Bem Using SBTREIR...40 Figure ٤-6: Verticl Deflection t A, (mm) in the Deep Cntilever Bem Using Tringulr Elements...4 Figure ٤-7: Bing stress t B, (KP) in the Deep Cntilever Bem Using Tringulr Elements...43 Figure ٤-8: Shering Stress t C, (KP) in the Deep Cntilever Bem Using Tringulr Elements...44 Figure ٤-9: In-Plne Rottion t A, (Rd) in the Deep Cntilever Bem Using Tringulr Elements...45 Figure ٤-0: Dimensions nd Considered Points for the Simpl Supported Bem...46 Figure ٤-: Smple Tringulr Mesh of the Simpl Supported Bem Problem...47 Figure ٤-: Bing Stress Pttern in the Simpl Supported Bem Using CST...48 Figure ٤-3: Bing Stress Pttern in the Simpl Supported Bem Using SBTREIR...49 Figure ٤-4: Verticl Deflection t "A", (mm) in the Simpl Supported Bem Using Tringulr Elements...5 Figure ٤-5: Bing Stress t "B", (KP) in the Simpl Supported Bem Using Tringulr Elements...5 Figure ٤-6: Shering Stress t "C", (KP) in the Simpl Supported Bem Using Tringulr Elements...53 Figure ٥-: Coordintes nd Node Numbering for the Rectngulr Element with 3-DOF per node...55 Figure ٥-: Smple Rectngulr Mesh of the Deep Cntilever Bem Problem...60 Figure ٥-3: Bing Stress Pttern in the Deep Cntilever Bem Using BRE...6 Figure ٥-4: Bing Stress Pttern in the Deep Cntilever Bem Using SBREIR...6 Figure ٥-5: Verticl Deflection t A, (mm) in the Deep Cntilever Bem Using Rectngulr Elements...64 Figure ٥-6: Bing Stress t B, (KP) in the Deep Cntilever Bem Using Rectngulr Elements...65 VIII

11 List of Figures Figure ٥-7: Shering Stress t C, (KP) in the Deep Cntilever Bem Using Rectngulr Elements...66 Figure ٥-8: In-Plne Rottion t A, (mm) in the Deep Cntilever Bem Using Rectngulr Elements...67 Figure ٥-9: Bing Stress Pttern in the Simpl Supported Bem Using BRE...69 Figure ٥-0: Bing Stress Pttern in the Simpl Supported Bem Using SBREIR...70 Figure ٥-: Verticl Deflection t A, (mm) in the Simpl Supported Bem Using Rectngulr Elements...7 Figure ٥-: Bing Stress t B, (KP) in the Simpl Supported Bem Using Rectngulr Elements...73 Figure ٥-3: Shering Stress t C, (KP) in the Simpl Supported Bem Using Rectngulr Elements...74 Figure ٦-: Verticl Deflection t A (mm) in the Deep Cntilever Bem from Vrious Elements...78 Figure ٦-: Bing Stress t B (KP) in the Deep Cntilever Bem from Vrious Elements79 Figure ٦-3: Shering Stress t C, (KP) in the Deep Cntilever Bem from Vrious Elements...80 Figure ٦-4: In-Plne Rottion t A, (Rd) in the Deep Cntilever Bem from the new Elements...8 Figure ٦-5: Verticl Deflection t Point A (mm) in the Simpl Supported Bem from Vrious Elements...84 Figure ٦-6: Bing Stress t B (KP) in the Simpl Supported Bem From Vrious Elements...85 Figure ٦-7: Shering Stress t C (KP) in the Simpl Supported Bem From Vrious Elements...86 IX

12 List of Tbles LIST OF TABLES Tble ٤-: Mesh Size nd Aspect Rtio of the Cntilever Bem Using Tringulr Elements.37 Tble ٤-: Results of the Cntilever Bem Problem Using Tringulr Elements (CST nd SBTREIR)...4 Tble ٤-3: Mesh Size nd Aspect Rtio of the Simpl Supported Bem Using Tringulr Elements...46 Tble ٤-4: Results of the Simpl Supported Bem Problem Using Tringulr Elements (CST nd SBTREIR)...50 Tble ٥-: Mesh Size nd Aspect Rtio of the Cntilever Bem Using Rectngulr Elements60 Tble ٥-: Results of the Cntilever Bem Problem Using Rectngulr Elements (BRE nd SBREIR)...63 Tble ٥-3: Mesh Size nd Aspect Rtio of the Simpl Supported Bem Using Rectngulr Elements...68 Tble ٥-4: Results of the Simpl Supported Bem Problem Using Rectngulr Elements (BRE nd SBREIR)...7 Tble ٦-: Verticl Deflection t A (mm) in the Cntilever Bem from Vrious Elements.75 Tble ٦-: Bing Stress t B (KP) in the Cntilever Bem from Vrious Elements...76 Tble ٦-3: Shering Stress t C, (KP) in the Cntilever Bem from Vrious Elements...76 Tble ٦-4: In-Plne Rottion t A, (KP) in the Cntilever Bem from New Elements...76 Tble ٦-5: Verticl Deflection t A (mm) in the Simpl Supported Bem from Vrious Elements...8 Tble ٦-6: Bing Stress t B (KP) in the Simpl Supported Bem from Vrious Elements...8 Tble ٦-7: Shering Stress t C, (KP) in the Simpl Supported Bem from Vrious Elements...83 X

13 Chpter : Introduction. CHAPTER ONE: INTRODUCTION. INTRODUCTION The conventionl nlticl pproches to solution of plne elsticit problems re bsed on determining the necessr equtions governing the behvior of the structure b tking into ccount equiibrium nd comptibiit within the structure s in the methods developed in the theor of elsticit [4], [3] & [35]. The solution to these equtions eists onl for specil cses of loding nd boundr conditions. Due to the comple nture of the shpe of the structure, loding pttern, irregulrities in geometr or mteril, the nlticl solution norml becomes difficult nd even impossible to solve such problems for displcements, stress or strins within the structure. The need for some other technique, such s suitble numericl methods for tckling the more comple structures with rbitrr shpes, loding nd boundr conditions, is then essentil. Severl pproimte numericl methods hve evolved over the ers. One of the common methods is the Finite Difference scheme in which n pproimtion to the governing equtions is used. The solution is formed b writing difference equtions for grid points. The solution is improved s more points re used. With this technique, some firl difficult problems cn be treted, but for emple, for problems of irregulr geometries or unusul specifiction of boundr conditions, the solution becomes more comple nd difficult to obtin. On the other hnd, the Finite Element Method (FEM), cn tke cre of ll these comple problems, nd hence hs become more widespred in finding solutions to comple structurl nd non-structurl problems.

14 Chpter : Introduction The Finite Element Method (FEM), or Finite Element Anlsis (FEA), is bsed on the ide of building complicted object with simple blocks, or, dividing complicted object into smll nd mngeble pieces. Appliction of this simple ide cn be found everwhere in everd life s well s in engineering. FEM is powerful method for the nlsis of continuous structures including comple geometricl configurtions, mteril properties, or loding. These structures re idelized s consisting of one, two or three-dimensionl elements connected t the nodl points, common edges, or surfces. An importnt ctegor is the two dimensionl plne elsticit problems. These problems re chrcterized b the following ssumptions: - two dimensions re lrge, the third is smll, - the structure is plne - the lods ct prllel to the plne. These structures re described b the mid plne nd the thickness distribution. Becuse of the specil tpe of loding, the generl three-dimensionl behvior of continuum cn be reduced to two dimensions b the ssumption of constnt distributed stresses or strins throughout the thickness. The English term plte onl reflects the geometr of the structure wheres the Germn term Scheibe dditionll refers to the fct tht onl membrne ction is present with no bing or twisting. The following terms might lterntivel be used: - in-plne loded plte (plne stress) - membrne structure - plne stress / strin structure The two-dimensionl plte elements (tht will be studied in this thesis) re etremel importnt for: () Plne stress nlsis which is defined s the stte of stress in which the norml stress nd the sher stresses directed perpiculr to the plne re ssumed to be zero. This includes problems such s pltes with holes, fillets or other chnges in geometr tht result in stress concentrtions.

15 Chpter : Introduction () Plne strin nlsis which is defined s the stte of strin in which the direct nd sher strins norml to the - plne re ssumed to be zero. This includes problems such s long underground bo culverts subjected to uniform lod cting constntl long its length, or dm subjected to the hdrosttic horizontl loding long its length [5].. HISTORY OF FINITE ELEMENT METHOD The modern development of the finite element method in the field of structurl engineering dtes bck to 94 nd 943, when its ke fetures were published b Cournt [4], Hrenikoff [0] nd McHenr [5]. The work of Cournt is prticulrl significnt becuse of its concern with problems governed b equtions pplicble to structurl mechnics nd other situtions. He proposed setting up the solution of stresses in the vritionl form. Then he introduced piecewise interpoltion functions (shpe functions) over tringulr sub-regions mking the whole region to obtin the pproimte numericl solution. In 947, Lev [3] developed the fleibilit method (force method) nd in 953 he suggested tht the use of the displcement method could be good lterntive for the nlzing stticll redundnt ircrft wings [4]. This method becme populr onl lter fter the invention of the high-speed computers. In 954, Argris nd Kelse [] gve ver generl formultion of the stiffness mtri method bsed on the fundmentl energ principles of elsticit. This illustrted the importnce of the energ principles nd their role in the development of the finite element method. In 956, Turner, Clough, Mrtin nd Topp [34] presented the first tretment of the two dimensionl elements. The derived the stiffness mtrices for tringulr nd rectngulr elements bsed on ssumed displcements nd the outlined the procedure commonl known s the Direct Stiffness Method for ssembling the totl stiffness mtri of the structure. This is regrded s one of the ke contributions in the discover of the finite element method. 3

16 Chpter : Introduction The technolog of finite elements hs dvnced through number of indistinct phses in the period since the mid 950 s. The formultion of the tringulr nd rectngulr elements for plne stress hs motivted the reserchers to continue nd estblish element reltionships for solids, pltes in bing nd thin shells. In 960 s, liner strin tringulr element ws developed [33]. This element hs 6 nodes with degrees of freedom per node. The derivtion of the stiffness mtri for this element ws difficult. The isoprmetric formultion ws then developed [] in which both the element geometr nd displcements re defined b the sme interpoltion functions. This formultion ws then pplied to two nd three dimensionl stress nlsis where higher order tringulr nd rectngulr plne elements were developed. Also, brick elements were developed for three dimensionl stress nlsis. Elements creted cn be non rectngulr nd hve curved sides. B the erl 970 s, this method ws further developed for use in the erospce nd nucler industries where the sfet of the structures is criticl. Since the rpid decline in the cost of computers, FEM hs been developed to n incredible precision. Currentl, there eist commercil finite element pckges tht re cpble of solving the most sophisticted problems for sttic s well s dnmic loding, in wide rnge of structurl s well s non-structurl pplictions. Before reviewing the vilble finite element solutions for two dimensionl structures, brief introduction to the finite element method is presented showing description of the procedure for obtining the stiffness mtri of the generl (tringulr or rectngulr) plne element..3 PROCEDURE OF THE FINITE ELEMENT METHOD In continuum problem of n dimension, the field vrible (whether it is pressure, temperture, displcement, stress or some other quntit) possesses infinite vlues becuse it is function of ech generic point in the bod or 4

17 Chpter : Introduction solution region. Consequent, the problem is one with n infinite number of unknowns. The finite element discretiztion procedure reduces the problem to one of finite numbers of unknowns b dividing the solution region into elements nd b epressing the unknown field vrible in terms of ssumed pproimting function within ech element. The pproimtion functions re defined in terms of the vlues of the field vribles t specified points clled nodes or nodl points. Nodes usull lie on the element boundries where djcent elements re considered to be connected. The nodl vlues of the field vrible nd the pproimtion for the elements completel define the behvior of the field vribles within the elements. For the finite element representtion of problem, the nodl vlues of the field vribles become the new unknowns. Once these unknowns re found, the functions define the field vrible throughout the ssemblge of elements. Clerl, the nture of the solution nd the degree of pproimtion dep not onl on the size nd the number of the elements used, but lso on the selected pproimtion functions. An importnt feture of the finite element method tht sets it prt from other pproimte numericl methods is the bilit to formulte solutions for individul elements before putting them together to represent the entire problem. Another dvntge of the finite element method is the vriet of ws in which one cn formulte the properties of individul elements. The most common pproch to obtining element properties is clled displcement pproch nd the method cn be summrized s described below..3. Ideliztion of the Structure The ideliztion governs the tpe of the element tht must be used in the solution of the structure. A vriet of element shpes cn be used, nd with cre, different element shpes m be emploed in the sme solution. Indeed when nlzing, for emple, n elstic shell tht hs different tpes of 5

18 Chpter : Introduction components such s stiffener bems, it is necessr to use different tpes of elements in the sme solution..3. Formultion of Element Stiffness Mtri The evlution of the stiffness mtri for finite element is the most criticl step in the whole procedure becuse it controls the ccurc of the pproimtion. This step includes the choice of: - The number of nodes nd the number of nodl degrees of freedom tht determines the size of the stiffness mtri. An element m contin corner nodes, side nodes nd/or interior nodes. The degrees of freedom re usul referred to the displcements nd their first-order prtil derivtives t node but ver often include second or higher order prtil derivtives. - The theor tht determines the stress-strin nd strin-displcement reltionships to be used in deriving the element mtrices. - The displcement functions (simple polnomil) or interpoltion functions in terms of the coordinte vrible nd number of constnts (equl to the totl number of degrees of freedom in the element). The displcement functions re then chosen to represent the vrition of the displcements within ech element. B using the principle of virtul work or the principle of minimum potentil energ, stiffness mtri relting the nodl forces to the nodl displcements cn be derived. Hence, the choice of suitble displcement functions is the mjor fctor to be considered in deriving element stiffness mtrices..3.3 Generl Procedure for Derivtion of Element Stiffness Mtri A stiffness mtri epresses the reltion between the nodl lods pplied to the element, nd the nodl displcements. Such reltion cn be derived from considertion of geometr, reltions in the theor of elsticit nd the 6

19 Chpter : Introduction conditions of equilibrium []. Thus, if Δ is the vector contining the displcement functions of the element, σ is the vector contining the stresses nd ε is the corresponding strins, nd if: { } [ f ] { A} Δ = Eq. - where [ f ] is the mtri contining the coordinte vribles (,, etc.) nd A is the vector of constnt terms (,, etc.) of the displcement function Δ respectivel, we cn ppl eqution (-) to the nodes of the element to relte the displcement within the element to its nodl displcements then we obtin: { } [ C ] { A} δ = Eq. - From which A [ C ] where = δ Eq. -3 δ is the vector contining the nodl degrees of freedom of the element nd [ C ] is the trnsformtion mtri resulting from substitution the coordintes of ech nodl point into the [ f ] mtri. Therefore, b substituting eqution (- 3) into eqution (-), the ltter becomes: Δ [ f ] [ C ] = δ Eq. -4 Now, we cn epress the strins b using the fct tht the strins re the derivtives of the displcements nd b using eqution (-4): { ε} [ B ] [ C ] { δ} = Eq. -5 where [ B ] is clled the strin mtri nd it contins the necessr derivtives of [ f ] corresponding to the strin-displcement reltionship. Also from Hook's l w, the stresses within the element cn be epressed s { } [ D ] { ε} σ = Eq. -6 Thus from (-6) nd (-5) { σ} [ D ] [ B ] [ C ] { δ} = Eq. -7 where [ D ] is clled the rigidit mtri tht contins the mteril properties. 7

20 Chpter : Introduction In order to relte the pplied nodl lods to the nodl displcements, the nodl lods should first be relted to the internl stresses, using the conditions of equilibrium, nd then to the nodl displcements using equtions (-5) & (-7). Using the principle of minimiztion of the potentil energ, we cn derive the stiffness of the element. The totl potentil energ, Π, of n element is the difference between the strin energ stored b the internl stresses, U, nd the potentil energ of the pplied lods Ω, Π = U Ω Eq. -8 An epression for the strin energ m be written s: T U = ε σ dvol Eq. -9 vol Using equtions -5 nd -6, we get = T T [ C ] [ B] [ D][ B][ C] T δ δ dvol Eq. -0 while the potentil energ of the pplied lods Ω (including the contribution of bod forces nd surfce trction forces) is written s T { δ} { P} Ω = Eq. - So we hve Π = T T T T {} δ [ C ] [ B] [ D][ B][ C] dvol {} δ {}{} δ P Eq. - where δ hs been tken out of the integrl, s it is indepent of the generl - coordintes. Now, b differentiting Π nd equting it to zero, we get T T [ C ] [ B] [ D][ B][ C] dvol {} δ {} P = 0 dπ = dδ e or, { P} = [ K ] { δ} where [ dvol ][ C] e T [ K ] [ C ] [ B] T [ D][ B] = Eq. -3 is the element stiffness mtri. 8

21 Chpter : Introduction.3.4 Assembl of the Overll Structurl Mtri nd the Solution Routine To obtin solution for the overll sstem modeled b network of elements, we must first ssemble ll the elements stiffness mtrices. In other words, we must combine the mtri equtions epressing the behvior of the entire structure. This is done using the principle of superposition; it is lso clled the direct stiffness method. The bsis for the ssembl procedure stems from the fcts tht: - At node where elements re interconnected, the vlue of the field vrible (generted displcements) is the sme for ech element shring tht node so tht the structure remins together nd no tering or overlp occur nwhere in the structure. - Equilibrium is stisfied t ech node. i.e. the sum of ll the internl nodl forces meeting t node must be equl to the eternll pplied forces t tht node. Therefore, to implement these two fcts simple computer progrm cn be written nd used for the ssembl of n number of elements. The resulting stiffness mtri of the element (nd hence tht of the totl structures) is smmetricl nd singulr mtri. So, the resulting simultneous equtions cn be solved, fter the introduction of the boundr conditions to the specific problem to obtin the nodl displcements nd these re then used for the clcultion of the stresses..3.5 Convergence Criteri A chrcteristic of the finite element method is tht the results should pproch the ect vlues s more nd more elements re used. With good displcement functions, convergence towrds the ect vlue will be much fster thn with poor functions, thus resulting in reduction of the modeling nd computing time nd effort. In order to chieve the convergence towrds the ect vlue of the required vribles, the displcement functions chosen should tr to 9

22 Chpter : Introduction represent the true displcement distribution s close s possible nd should hve certin properties, known s convergence criteri. These re conditions to gurntee tht ect nswers, within the pproimtion mde, will be pproched s more nd more elements re used to model the rbitrr structure. These criteri re discussed net. A- Rigid-Bod Modes The pproimtion function should llow for rigid-bod displcement nd for stte of constnt strin within the element. For emple, the one-dimensionl displcement function u = stisfies this criterion becuse the constnt ( ) llows for rigid bod displcement (constnt motion of the bod without strining) while the term ( ) llows for stte of constnt strin ( ε = du / d =). This simple polnomil is then sid to be complete nd is used for the onedimensionl br element. Completeness of the chosen displcement function is necessr condition for convergence to the ect vlues of displcements nd stresses [5], [35]. The inclusion of rigid bod modes is necessr for equilibrium of the nodl forces nd moments nd hence the stisfction of globl equilibrium in the structure being nlzed []. B- Constnt Strin In order for the solution to converge to the ctul stte of strins, the pproimtion function should lso llow for stte of constnt strin within the element. The stte of the constnt strin in the element cn occur if the elements re chosen smll enough. This requirement is obvious for structures subjected to constnt strins becuse s elements get smller, nerl constnt strin conditions previl in them. As the mesh becomes finer, the element strins re simplified, nd in the limit the will pproch their constnt vlues. 0

23 Chpter : Introduction C- Inter-element Comptibiit The concept of comptibilit mens tht the displcements within the elements nd cross the boundries re continuous. It hs been shown tht the use of complete nd comptible displcement functions is bsic gurntee to obtin converging solutions s more elements nd nodes re used (the mesh size is refined). Note tht the bove criteri re mthemticll included in the sttement of Functionl Completeness..3.6 Mesh Size Design The finite element nlsis (FEA) uses comple sstem of points clled nodes tht mke up grid clled the mesh. Nodes re ssigned t certin densit throughout the mteril deping on the nticipted stress levels t prticulr res. Regions tht will receive lrger mounts of stress usull hve higher node densit thn those with little or no stresses. In order to conduct finite element nlsis, the structure must be first idelized into some form of mesh. Meshing is the procedure of ppling finite number of elements to the FEA model. The rt of successful ppliction of the meshing tsk lies in the combined choice of element tpes nd the ssocited mesh size. If the mesh is too corse, then the solution will not give correct results. Alterntivel, if the mesh is too fine, the computing time nd effort cn be out of proportion of the results obtined. A corse mesh is sufficient in res where the stress is reltivel constnt while fine mesh is required where there re high rtes of chnges of stress nd strin. Locl mesh refinement m be used t re of mimum stress sttes. To ensure tht convergent results of the FEA solution re obtined, the following modeling guidelines should be considered in the design of the mesh:

24 Chpter : Introduction A- Number of Degree of Freedom It is the essence of the finite element method tht incresing the totl number of degree of freedom (clculted s the number of nodes times the number of DoF per node) hs the bsic influence on the convergence of the FEA solution towrds true solution. B- Aspect Rtio The spect rtio is defined s the rtio between the longest nd shortest element dimensions. Acceptble rnges for spect rtio re element nd problem depent. It is generll known tht the ccurc of the solution deteriortes s lrger spect rtio is used. The limit of the spect rtio is ffected b the order of the element displcement function, the numericl integrtion pttern for stiffness, the mteril behvior (liner of nonliner) nd the nticipted solution pttern for stresses or displcements. Elements with higher-order displcement functions nd higher-order numericl integrtion re less sensitive to lrge spect rtios. Elements in regions of mteril nonlinerit re more sensitive to chnges in spect rtio thn those in liner regions. However, fied numericl limits re given such s 3: for stresses nd 0: for deflections [9]. C- Element Distortions (Skewing) Distortions of elements or their out-of-plne wrping re importnt considertions. Skewing is usull defined s the vrition of element verte ngles from 90º for qudrilterls nd from 60º for tringles. Wrping occurs when ll the nodes of three-dimensionl pltes or shells do not lie on the sme plne or when the nodes on single fce of solid devite from single plne [9]. These concepts re illustrted in Figure - below.

25 Chpter : Introduction Figure -: Skewed And Wrped Elements (Poor Shped).4 FINITE ELEMENT APPLICATIONS OF TWO-DIMENSIONAL STRUCTURES The finite element solution of two-dimensionl structures is mde b dividing the structure into mesh of elements, minl tringulr nd/or rectngulr elements. There re two min tpes of elements; the bsic elements re those hving onl corner nodes such s the constnts strin tringulr element (CST) nd the biliner rectngulr element (BRE). The more dvnced elements re those who hve dditionl mid-side nodes such s the liner strin tringulr element (LST) nd the (qudrtic iso-prmetric element). The use of mid-side nodes llows qudrtic vrition of strins nd hence fster rte of convergence. Another dvntge of the use of higher order elements is tht curved boundries of irregulrl shped structures cn be pproimted more closel thn b the use of simple stright-sided elements. The gol of developing ll these elements is to introduce more degrees of freedom into the solution nd hence get more ccurte results s well s better modeling of vrious structure boundries. 3

26 Chpter : Introduction The following is brief description of the bsic fetures of the commonl used elements for two-dimensionl problems [5]..4. Constnt Strin Tringulr Element (CST) The constnt strin element is the bsic nd most common element used for the solution of plne elsticit problems. This element hs two degrees of freedom (U nd V displcements) t ech of its three corner nodes, thus it hs totl of si degrees of freedom. The element is bsed on indepent displcement fields in the nd directions, i.e. the constnts ppering in the epression for the displcement in one direction do not pper in the epression for the other direction s follows: U = V = The ssocited strins for this element re given b ε ε γ U = = V = = 6 U V = = 3 5 It is obvious tht the vlues of strin components, nd so the stresses, re constnt nd do not vr throughout the element, hence cme the nme. This mens tht stisfctor convergence towrds the ect solution cn onl be ensured b using lrge number of elements [9], [5] & [35]..4. Liner Strin Tringulr Element (LST) This element hs si nodes, i.e. three corner nodes s well s three mid-side nodes. This is considered s development over the CST [9], [5] & [35]. It is higher order tringulr element tht hs twelve degrees of freedom. Its strins 4

27 Chpter : Introduction 5 vr linerl with nd coordintes. The element is bsed on indepent displcement fields in the nd directions s follows: V U = = The ssocited strins for this element re given b ) ( ) ( ) ( V U γ V ε U ε = = = = = =.4.3 Biliner Rectngulr Element (BRE) This rectngulr element is lso bsic nd common element used for the solution of plne elsticit problems. This element hs two degrees of freedom (U nd V displcements) t ech of its four corner nodes, thus it hs totl of eight degrees of freedom. The element is bsed on indepent displcement fields in the nd directions s follows: V U = = The ssocited strins for this element re given b V U γ V ε U ε = = = = = = The strin components, nd hence the stresses, vr linerl in both the nd directions, hence come its nme. This mens tht stisfctor convergence towrds the ect solution cn be chieved better thn the CST mentioned bove [9], [5] & [35].

28 Chpter : Introduction.4.4 Qudrtic Isoprmetric Rectngle Element This element hs eight nodes, i.e., four corner nodes s well s four mid-side nodes. This is considered s development over the BRE. It is higher order rectngulr element tht hs siteen degrees of freedom [9], [5] & [35]. All of the bove-mentioned elements re bsed on ssumed polnomil displcements. Another pproch clled the Strin Bsed Approch eists where ssumed polnomil strins re used for deriving the displcement fields. This new pproch is pplied in the present work for deriving new displcement fields of tringulr nd rectngulr elements with three degrees of freedom pre node, which re the two trnsltions nd the in-plne rottion..5 SCOPE OF THE CURRENT THESIS The purpose of the work presented in this thesis is to derive two new strinbsed elements for two dimensionl elsticit problems. In Chpter 4, new strin bsed tringulr element hving three degrees of freedom per node is derived. The element is then pplied to solve two problems in plne elsticit, i.e., deep cntilever bem problem nd simpl supported bem problem. Convergence of the solutions for deflection nd stresses is studied with mesh refinement. The performnce of this element is compred to tht of the vilble displcement bsed Constnt Strin Tringle element, CST nd the ect elsticit solutions. In Chpter 5, new strin bsed rectngulr element hving three degrees of freedom per node is derived. The element is then pplied to solve the sme problems mentioned bove. Also, convergence of the solutions using this element is studied. The performnce of this element is compred to tht of the vilble displcement bsed Biliner Rectngulr Element, BRE nd the ect elsticit solutions. In Chpter 6, the performnce of these elements is compred for the results of deflection nd stresses. 6

29 Chpter : Strin Bsed Approch. CHAPTER TWO: STRAIN BASED APPROACH. INTRODUCTION A new pproch to develop the stiffness mtri of finite elements ws proposed b Ashwell, Sbir nd Roberts [] & [3]. In this pproch, polnomil strin components re ssumed. Then the displcement fields re obtined b integrtion of strin components ccording to the relevnt strin/displcement reltions. The displcement fields hve two essentil components. The first component reltes to the zero-strin rigid bod mode of displcement while the second is due to the strining of the element, which cn be represented b ssuming indepent polnomil terms of strins in so fr s it is llowed b the comptibilit equtions governing the chnges in the direct stresses nd sher stresses. A min feture of the strin-bsed pproch is tht the resulting components of displcements re not indepent s in the usul displcement pproch but re linked. This linking is present in the ect terms representing rigid bod modes nd the pproimte terms within the contet of the finite element method representing the strining of the element. Another feture of this pproch is tht the method llows the in-plne components of the displcements to be presented b higher order terms without incresing the number of degrees of freedom beond the essentil eternl degrees of freedom. 7

30 Chpter : Strin Bsed Approch. REASONS FOR DEVELOPMENT OF STRAIN BASED ELEMENTS The resons for seeking n element bsed on generlized strin functions rther thn displcements were given b the uthors who developed this pproch [3]. The wrote tht: Firstl if we wish to minimize the contribution of strin energ to the potentil energ of n element we should seek vritions of strin which re s smooth s possible. This considertion follows from the fct tht the strin energ is clculted from squres nd products of the strin, nd imposing locl vritions on n initill smooth distribution without ltering the locl men vlues increses the vlue of the squres when the re integrted over the element. Secondl the equtions relting displcements nd generlized strins re coupled in such w tht some strins re functions of more thn one displcement thus mking displcements indepent of one nother will not mke the strins indepent of ech other. In ddition, since two rules in the convergence criteri nd directl concerned with strins the would be esier to stisf with ssumed strins rther thn displcement functions..3 HISTORY OF STRAIN BASED APPROACH.3. Strin Bsed Curved Elements The development of displcement fields b the use of the Strin Bsed Approch ws first pplied to curved elements. It ws reveled tht to obtin stisfctor converged results, the finite elements bsed on indepent polnomil displcement functions require the curved structures to be divided into lrge number of elements. 8

31 Chpter : Strin Bsed Approch Ashwell, Sbir nd Roberts [] & [3] showed tht when ssumed indepent polnomil displcement fields re used in the nlsis of curved elements, the structure needs to be divided into lrge number of elements in order to get stisfctor converged results. However, when ssumed indepent polnomil strin fields re used, converged results re obtined even when less divisions of the structure re used, i.e., the strin bsed elements showed fster convergence. Therefore, the continued to develop new clss of simple nd efficient finite elements for vrious tpes of problems bsed on ssumed indepent strins rther thn indepent displcement fields..3. Strin Bsed Shell Elements Sbir nd Ashwell [] presented strin-bsed rectngulr clindricl element tht hs twent degrees of freedom. It uses onl eternl geometricl nodl displcements (three liner displcement nd two rottions). It includes ll rigid bod displcements ectl nd stisfies the constnt strin condition in so fr s tht condition pplies to clindricl shells. Becuse the rectngulr elements which hve been developed cn not be used for modeling shells hving irregulr curved boundries, Sbir nd Chrchfchi [5] used the strin pproch to develop qudrilterl element nd Sbir used this element to investigte the problem of stress concentrtion in clinders with circulr nd ellipticl holes [4] nd the problem of normll intersecting clinders [8]. Sbir et l lso used the strin pproch to develop element for rches deforming in the plne [] nd out of the plne contining the curvture [7] nd took the opportunit to show tht higher order strin bsed elements cn be obtined nd lso cn be condensed to the onl essentil eternl degrees of freedom. This stticl condenstion t the element level ws shown to produce further improvement in the convergence of the result. 9

32 Chpter : Strin Bsed Approch In 985, Sbir nd Rmdnhi [6] developed simple curved finite element bsed on shllow shell equtions. The element is rectngulr in plne nd hs three principl curve lines nd hs the onl essentil eternl degrees of freedom. The element presented is bsed on ssumed strins rther thn displcement fields. The element ws tested b ppling it to nlsis of clindricl [6], sphericl [9] nd hperbolic prbolic shell nd it gve high degree of ccurc. Convergence of the results for deflection s well s stresses ws more rpid when compred with other finite elements bsed on ssumed displcements. Sbir nd El-Erris [8] lso developed new curved strin-bsed hperbolic prbolic shell element similr to tht developed b Sbir nd Rmdnhi but hving the in plne rottion s sith degree of freedom. The results obtined b the use of this element were shown to converge more rpidl for vriet of problems even when compred with high order elements. Sbir nd El-Erris developed curved conicl shell finite element suitble for generl bing nlsis of conicl shells. This element is simple nd possesses ll the necessr requirements for less computtionl effort. The element hs 0 degrees of freedom nd stisfies the ect representtion of rigid bod modes of displcement. The convergence chrcteristics of the element were tested b ppling to the bing nlsis of conicl shells nd it ws shown tht results of cceptble level of ccurc re obtined when few elements re used. Djoudi [6] developed curved tringulr shllow shell element. The element hs onl the five essentil degrees of freedom t ech node nd is bsed on ssumed strins. Severl emples of shells with different loding nd boundr conditions were considered nd the results obtined were shown to be stisfctor for most problems. The strin pproch ws lso used in developing severl strin bsed shell elements were developed b Mous, A. [7] to []. These elements include conicl, clindricl nd sphericl shell elements. Also, two groups of doubl curved tringulr elements were developed, the first group included 0

33 Chpter : Strin Bsed Approch three elements tht hve five degrees of freedom t ech node while the second group included two elements tht hve si degrees of freedom t ech node. These elements gve high ccurc of results for displcements nd components of stresses when the were used in complete nlsis of vriet of comple structures such s: - Comple tpe of shell roof referred to s fluted roof, with studing the effect of using stiffening bems on deflections. - Clindricl storge tnk (mde up of clindricl nd conicl shell components) tht ehibits lrge concentrtion of stresses t the junction of the two components with studing the effect of using stiffening ring bems on stresses t the junction. - Doubl curved hperbolic prbolic dm with constnt or vrible thickness, - Doubl curved sphericl shell roof in the form of four-corner str with studing the effect of using stiffening bems on deflections..3.3 Strin Bsed Two Dimensionl Elements The strin-bsed pproch ws further eted b Sbir [3] to the two dimensionl plne elsticit problems. A new fmil of such elements ws developed. These elements stisf the requirements of strin-free rigid bod mode of displcement nd the comptibilit within the element. A tringulr in-plne element ws developed hving the two bsic degrees of freedom. The resulting displcement fields of this element re given b U = V = Eq. - However, this element gve no improvement since its performnce ws found to be ectl the sme tht of the displcement bsed Constnt Strin Tringulr element.

34 Chpter : Strin Bsed Approch A bsic rectngulr element ws developed nd tested b ppling it to the two dimensionl nlsis of bem nd plte with holes. Another version of this element stisfing the bove requirements s well s stisfing equilibrium equtions ws lso developed nd tested. These elements hve the two essentil trnsltionl degrees of freedom t ech of the corner nodes. The resulting displcement fields of this element re given b V - U = = Eq. - This element ws pplied to solution of severl plne elsticit problems such s deep cntilever bem nd simpl supported bem nd ws found to give good results nd fst rte of convergence. The displcement fields of the bove mentioned rectngulr element developed b Sbir ws pplied b Sfji [3] to tringulr element with four nodes (three corners nd mid-hpotenuse node) nd using the stticl condenstion of two such tringulr elements. Bsed on the sme rectngulr element, he lso derived two new rectngulr elements stisfing the equilibrium equtions for plne stress nd plne strin respectivel. The displcement fields of these elements re given b ) ν ν - ( ) ( V ) ν ν - ( ) ( U = = Eq. -3 for plne stress nd ) ν - ( ) ν ( V ) ν - ( ) (ν U = = Eq. -4 for plne strin. A sector finite element ws developed in polr coordintes b Djoudi [6]. This element hs three degrees of freedom t ech node (two essentil

35 Chpter : Strin Bsed Approch degrees of freedom nd the in-plne rottion). It ws pplied to the two nlsis of rottionll smmetric curved bem subject to sher..4 GENERAL PROCEDURE FOR DERIVING DISPLACEMENT FIELDS FOR STRAIN-BASED ELEMENTS The following re generl guidelines of deriving displcement fields for strinbsed elements. The displcement fields re required to stisf the requirement of strin-free rigid bod mode of displcement nd strining of the element. To get the first prt of the displcement fields corresponding to rigid bod movement of the element, we begin b writing the governing strin/displcement reltionships to the considered element tpe ( dimensionl, 3 dimensionl, flt, curved, etc) nd mke them equl to zero. The resulting equtions re pplicble to n tpe of finite elements of tht tpe regrdless of the totl number of degrees of freedom per node. Deping on the number of nodes nd the number of degrees of freedom per node in the considered element, it is generll essentil tht the totl number of degrees of freedom in the element (nd hence the number of constnts used in defining the displcement fields within the element) equls the number of nodes times the number of degrees of freedom per node. Some of the required constnts will hve lred been used to describe the strin-free rigid bod mode of displcement s described bove. To get the remining prt of the displcement field, corresponding to the strining of the element, n epression is ssumed for ech of the strin components utilizing the remining number of constnts. The ssumed epression should be checked to ensure tht if the re twice differentited, the stisf the generl comptibilit eqution of strins. These epressions should include constnt terms corresponding to the stte 3

36 Chpter : Strin Bsed Approch of constnt strin tht ensures the convergence of the solution with mesh refinement. The prts of the strin/displcement reltionships involving direct strins re integrted to give epressions of direct displcements (for emple U nd V) tht will include integrtion constnts. These epressions re substituted in the equtions tht link them to other prts of the strin/displcement reltionships (for emple, the sher strin equls the derivtives of U nd V). The terms corresponding to ech of the coordinte vribles (for emple nd ) re collected nd seprted. This will give the vlues of integrtion constnts. B combining the displcement functions due to the rigid bod motion nd those due to the strining of the element, we get the finl epression of the displcement fields. To compre the strin-bsed elements with the commonl used displcement-bsed elements, it is noted tht the displcement fields of the strin-bsed element re linked through the terms representing both the rigid bod mode s well s the strining of the element, i.e. most of the constnt terms pper in epression for ech of the displcement fields. After clculting the displcement fields, the element stiffness mtri cn be clculted using the generl epression e T T [ K ] [ C ]. [ B].[ D ][. B ].d(vol).[ C ] = Eq. -5 where the trnsformtion mtri, [C] is clculted b substituting the vlue of displcement vribles (,, etc.) t ech node. For emple, for elements with two degrees of freedom per node, [C] is clculted s: [ C] (, ) (, ) (, ) ( 3, ) = - - -, where n is the totl number of nodes U ( n, n ) ( n, n ) 4

37 Chpter : Strin Bsed Approch Mtri [ B ] is the strin mtri involving the coordinte vribles ttched to ech of the constnt terms in the epression for the vrious strin components. And [ D ] is the rigidit mtri relting the strins to stress nd using the mteril properties (minl modulus of elsticit nd Poisson s rtio in structurl solid mechnics nd plne elsticit problems). 5

38 Chpter 4: Development of New Strin-Bsed Tringulr Element 3. CHAPTER THREE: COMPUTER PROGRAMS 3. USED COMPUTER SOFTWARE Two min progrms were used for the derivtion of the new finite elements nd computer implementtion of the nlsis. A description of ech progrm is given below. 3.. MthCAD Softwre MthCAD is powerful mthemtics softwre for technicl clcultions. The progrm is used to implement lmost ll kinds of mthemticl opertions in n es w. The progrm interfce is just like n open pge so the user cn write nwhere in this pge. The bsic feture of the progrm is tht it implements smbolic mthemtics such s simplifing epressions, differentition, integrtion, collecting vribles, fctoring terms, etc. In this thesis, MthCAD is used to derive the displcement fields for tringulr nd rectngulr elements s will be detiled in the net chpters. In this regrd, the strins re ssumed then the remining smbolic nlsis re mde until the complete epressions for displcement fields re completed nd then the dequc of the resulting trnsformtion mtri is checked b ppling it to n rbitrril oriented element. The use of MthCAD mkes it es to chnge the ssumptions for strins nd see the resulting displcement fields immeditel. 6

39 Chpter 4: Development of New Strin-Bsed Tringulr Element 3.. MtLb Softwre MtLb is n interctive sstem for doing numericl computtions. The progrm got its nme from the fct tht it is Mtri Lbortor becuse it is bsed on mtri mnipultions, which produce high efficienc in clcultions. Ech entr into the MtLb is treted s mtri, even if it is single number. Computtion speed of the progrm is incredible so it is ver suitble to be used to solve the lrge number of simultneous equtions resulting from finite element nlses. For emple, the inverse of ver lrge stiffness mtri is clculted using one commnd in contrst to the old progrmming lnguges where tht tsk usull requires lrge mount of progrmming effort. 3. IMPLEMENTATION OF THE FEM ANALYSIS Specil purpose computer progrms were developed using MtLb to generte the mesh dt including node coordintes nd element connectivit for the cses of tringulr nd rectngulr elements within ech problem domin. Furthermore, nother progrm ws developed to solve the problems using ech element tpe, i.e. the eisting CST nd BRE elements s well s the new tringulr nd rectngulr elements, which will be developed in the following chpter of this thesis. Input dt ws written in seprte file to orgnize the dt nd fcilitte the process of generting the nodl coordintes nd element connectivit (mesh definition). Output of the progrm ws lso received into seprte file. The output included the input dt, nodl forces, nodl displcements nd element stresses t the nodes of ech element. Smples of the used computer progrms re shown in Appi A including: A-: A-: A-3: MtLb Code for Genertion of Tringulr Mesh in Rectngulr Domin MtLb Code for Genertion of Rectngulr Mesh in Rectngulr Domin MtLb Code for the Strin Bsed Tringulr Element with In-plne Rottion (SBTREIR) 7

40 Chpter 4: Development of New Strin-Bsed Tringulr Element A-4: MtLb Code for the Strin Bsed Rectngulr Element with In-plne Rottion (SBREIR) A-5: A-6: Smple Input File (Rectngulr Element) Smple Output File (Rectngulr Element) 3.. Outline of the Mjor Progrm Steps The following is summr of the bsic stges tht one hs to go through when implementing the MtLb developed computer progrm. Strt Ask for input file nme to red input dt Ask for output file nme to write results Red input file: o Totl number of nodes in the structure o Totl number of elements in the structure o Node coordintes o Element connectivit dt o Mteril Dt: Modulus of elsticit, Poisson s rtio, thickness o Nodl forces o Nodl fition dt (boundr conditions) Plot the structure to ensure correctness of dt Open output file nd prepre hedings to write results Write the input dt into the output file Prepre smpling points nd weights to be used in numericl integrtion Clculte the rigidit mtri for (plne stress / plne strin) Strt clculting element stiffness mtri: o Red element coordintes o Clculte the trnsformtion mtri [C] o Assign zero mtri for the element stiffness mtri, 8

41 Chpter 4: Development of New Strin-Bsed Tringulr Element o For ech of the smpling points: - Clculte the vlue shpe functions nd their derivtives. - Clculte the strin mtri [B]. - Clculte the vlue of the stiffness mtri nd ccumulte it. Assemble the globl stiffness mtri of the structure [GK] Appl boundr conditions to stiffness mtri nd nodl force vector {F} Solve for globl nodl displcements {GD} = inv [GK].{F} Write nodl displcements to the output file Prepre hedings to write stresses into the output file Strt clculting the element stresses: o Red element coordintes gin o Etrct element nodl displcements {d} from the globl displcements [GD] o Clculte the trnsformtion mtri [C] gin o Clculte the rigidit mtri [D] o Clculte the strin mtri [B] o Clculte the element stresses = [D].[B].inv[C].{d} o Write element stresses to the output file End 3.. Progrms Flow Chrt The following figure shows the usul flow chrt of processes involved in the implementtion of the finite element nlsis. 9

42 Chpter 4: Development of New Strin-Bsed Tringulr Element START Red Input Dt Geometr Loding {F} Mteril Properties Supporting Conditions P re - Processing St g e Stiffness Mtri Requirements Shpe Functions Displcement Fields Trnsformtion Mtri Strin Mtri Rigidit Mtri Evlution Evlute individul element stiffness Mtri, k Assembl Assemble overll stiffness mtri of the structure, GK Processing St g e Boundr Condition Appl boundr Condition to GK nd {F} Solution Solve for globl displcements {F} = [GK].{GD} Evlute Stresses P lot Stress es ( I f required) Write Displcements nd Stresses to Output File P ost - Processing St g e END Figure 3-: Finite Element Anlsis Flow Chrt 30

43 Chpter 4: Development of New Strin-Bsed Tringulr Element 4. CHAPTER FOUR: DEVELOPMENT OF NEW STRAIN-BASED TRIANGULAR ELEMENT 4. INTRODUCTION In this chpter, the strin-bsed pproch is eted to investigte the bilit of deriving displcement fields for new strin bsed tringulr element with the inclusion of the third degree of freedom t ech node, which is the in-plne rottion (lso clled drilling degree of freedom). The performnce of the new tringulr element is investigted b ppling it to the solution of two common plne elsticit problems. These problems re: the problem of plne deep cntilever bem fied t one nd loded b point lod t the free nd the problem of simpl supported bem loded t the mid-spn b point lod. The results obtined b the developed tringulr strin bsed element re compred to those given b the well known Constnt Strin Tringle (CST) nd the nlticl solutions for deflection nd stresses s detiled below. 4. DERIVATION OF DISPLACEMENT FIELDS OF NEW TRIANGULAR ELEMENT WITH IN-PLANE ROTATION The following outlines the ssumptions nd steps to derive the new strinbsed tringulr element. The new tringulr element hs three corner nodes with three degrees of freedom t ech node s shown in the figure below. The displcement fields must stisf the requirement of strin-free rigid bod mode of displcement nd strining of the element. 3

44 Chpter 4: Development of New Strin-Bsed Tringulr Element (,v) (, ) (u,v,f ) 3 ( 3, 3 ) (u 3,v 3,f 3 ) (, ) (u,v,f ) (,u) Figure 4-: Coordintes nd Node Numbering for Tringulr Element with 3 DOF per node To get the first prt of the displcement fields for rigid-bod mode (U nd V ), we begin b writing the strin/displcement reltionships for plne elsticit nd mke them equl to zero. These reltionships re given b: ε U V U V = ε = γ = Eq. 4- where U nd V : re the displcements in the nd directions respectivel. ε nd ε : re the direct il strins in the nd directions respectivel. γ : is the sher strin. Net these stins re set equl to zero nd then integrted, the will give ε ε γ U = = 0 V = = 0 ===> V = U V = = 0 ===> U = f () f () ===> f '() f '() = 0 where f '() nd '() re constnts tht cn be tken s f f '() = nd f '() = 3 hence, f () = 3 nd f () = 3 thus, 3

45 Chpter 4: Development of New Strin-Bsed Tringulr Element U = V = 3 3 Eq. 4- where U nd V : re the displcements in the nd directions respectivel corresponding to rigid-bod mode of displcement. nd : represent the trnsltions of the element in the nd directions respectivel. 3 : represent the rigid-bod in plne rottion of the element. It is noted tht equtions Eq. 4- re pplicble to n tpe of two dimensionl plne finite elements regrdless of the totl number of degrees of freedom per node. Deping on the number of nodes nd the number of degrees of freedom per node in the considered element, it is generll essentil tht the totl number of degrees of freedom in the element (nd hence the number of constnts used in defining the displcement fields within the element) equls the number of nodes times the number of degrees of freedom per node. Three constnts hve lred been defined while the remining si hve to be used to describe the strining of the element. A first ttempt to do so is to ssume tht ε ε γ = = 4 6 = Eq. 4-3 This rrngement of strins does not contin constnt term in the epression for γ nd it is epected tht it wouldn t give good solutions. Also it ws found tht it leds to singulr displcement trnsformtion mtri nd hence it cn t be used to derive stiffness mtri. Severl other rrngements were tried to void this problem. A good rrngement tht gives non-singulr trnsformtion mtri is found to be s follows: 33

46 Chpter 4: Development of New Strin-Bsed Tringulr Element ) ( γ 4 ε 4 ε = = = Eq. 4-4 We observe tht, if the terms of this eqution re twice differentited, the stisf the generl comptibilit eqution of strins, nmel: γ ε ε = Eq. 4-5 The constnts 4, 6 nd 8 re the terms corresponding to stte of constnt strin tht ensures the convergence of the solution with mesh refinement. The constnts 5, 7 nd 9 re the terms corresponding to the strin behvior. To get the second prt of the displcement fields for strining mode (U, V ), we first integrte the fist two equtions s follows. f() 4 V f() 4 U = = Eq. 4-6 To get the functions f() nd f(), we substitute their derivtives in the third eqution then seprte the resulting epressions for nd respectivel s follows: () f' () f' 4 4 ) ( V U γ = = Eq. 4-7 ) ( ]d 4 [ f() ) ( ] d 4 [ f() = = = = Eq. 4-8 The in-plne rottion cn be clculted using the reltion:

47 Chpter 4: Development of New Strin-Bsed Tringulr Element 35 ) U - V ( φ = Eq. 4-9 Now, f() nd f() re substituted in U, V. B dding the epressions for U nd V to U nd V then clculting the in-plne rottion, the complete epressions for the displcement fields re obtined s: )/ ( ) 8 ( ) 8 ( Φ ) 4 ( ) ( V ) 4 ( ) ( U = = = Eq. 4-0 It is noted tht we obtined qudrtic nd cubic terms (, 3, & 3 ) without incresing the number of nodes beond the three corner nodes. This is not chieved in the known constnt strin tringulr element (CST). It is epected tht this increse in the degree of the polnomils will result in more ccurte solutions using this element; s will be shown in the subsequent sections. Hving obtined the displcement fields, the stiffness mtri of the tringulr element cn be evluted using the generl epression [ ] [ ] [ ] [ ][ ] [ ] T T e C B.d(vol). D.. B. C K = Eq. 4- where, the trnsformtion mtri [C] is clculted s [ ] Φ Φ Φ = ( ( V ( U ( ( V ( U ( ( V ( U C the strin mtri [B] for this element is

48 Chpter 4: Development of New Strin-Bsed Tringulr Element 36 [ ] = B nd [D] is the rigidit mtri given b [ ] = ν ν 0 ν ) ν ( E D for the stte of plne stress nd [ ] = ν ν - ν 0 ν ν - ν) ν)( ( E D for the stte of plne strin. In the subsequent sections, this element will be clled Strin Bsed Tringulr Element with In-Plne Rottion, (SBTREIR). 4.3 PROBLEMS CONSIDERD The performnce of the new strin bsed tringulr element derived in the previous section is pplied to solve deep cntilever problem nd simpl supported bem problem s detiled below. 4.4 DEEP CANTILEVER BEAM PROBLEM The first problem is deep cntilever bem loded b point lod t the free. The bem hve length L=0m, height H=4m, nd thickness t=0.065m. The mteril properties: modulus of elsticit nd Poisson s rtio re tken s E=00,000 KP nd ν=0.0 respectivel. The point lod t the free of the bem is tken s P =00 KN. In order to chieve full fiit t the built-in of

49 Chpter 4: Development of New Strin-Bsed Tringulr Element the cntilever bem, ll the nodes occurring t tht re ssumed to be restrined in the nd directions s well s the in-plne rottion. Dimension nd loctions of the investigted points within the cntilever bem re shown in Figure 4- below. P B A C H = 4m t L = 0m Figure 4-: Dimensions nd Loctions of Considered Points for the Deep Cntilever Bem 4.4. Used Mesh Size Severl mesh sizes were used in the solution of the problem with incresing the totl number of tringulr elements. The dopted spect rtio is : in lmost ll cses. The following tble shows the number of elements, number of nodes nd the spect rtio of ech mesh size. Tble 4-: Mesh Size nd Aspect Rtio of the Deep Cntilever Bem Using Tringulr Elements Elements in Short Elements in Long Totl no. of Totl Side (L=4m) Side (L=0m) Aspect Mesh Tringulr no. of Dimension Dimension Rtio Size No. No. Elements Nodes (m) (m) : : : : : A smple mesh size is illustrted in the figure below. 37

50 Chpter 4: Development of New Strin-Bsed Tringulr Element Figure 4-3: Smple Tringulr Mesh of the Deep Cntilever Bem Problem 4.4. Anlticl Solution Bsed on the given geometr nd mteril properties, the nlticl solution for deflection nd stresses t the specified points re clculted s follows: Verticl deflection t the free, point A =.05 mm. Bing stress t middle of upper fce, point B = 3000 KP. Sher stress t middle of centerline, point C = 600 KP. In-Plne rottion t the free, point A = 0.56 rd Convergence Results The problem ws solved using the developed computer progrm (described in Chpter 3 nd Anne A) for ech of the mesh sizes listed in Tble 4- bove. Convergence of the overll pttern of bing stress in the cntilever bem is shown below for the nlticl solution s well s the tringulr elements (CST nd SBTREIR) using ech mesh size (Red: tension, Blue: compression). Bing Stress Pttern Anlticl Solution 38

51 Chpter 4: Development of New Strin-Bsed Tringulr Element CST Mesh Size 5 CST Mesh Size 40 CST Mesh Size 5 CST Mesh Size 65 Figure 4-4: Bing Stress Pttern in the Deep Cntilever Bem Using CST 39

52 Chpter 4: Development of New Strin-Bsed Tringulr Element SBTREIR Mesh Size 5 SBTREIR Mesh Size 40 SBTREIR Mesh Size 5 SBTREIR Mesh Size 65 Figure 4-5: Bing Stress Pttern in the Deep Cntilever Bem Using SBTREIR 40

53 Chpter 4: Development of New Strin-Bsed Tringulr Element Tble 4- shows summr of the used meshes nd results for verticl deflection, bing stress nd shering stress t the specified points within the deep cntilever bem s percentge of the ect solutions for the CST nd the new SBTREIR. Tble 4-: Results of the Deep Cntilever Bem Problem Using Tringulr Elements (CST nd SBTREIR) Mesh Size No. of Elements No. of Nodes Verticl Deflection t A Bing Stress t B Shering Stress t C CST SBTREIR CST SBTREIR CST SBTREIR % 7.% 36.53% 70.8% 63.7% 63.% % 89.44% 67.78% 87.48% 85.47% 97.79% 5 * % 9.68% 75.00% 90.65% 86.83% 94.73% 6 5* % 95.56% 80.7% 93.8% 93.8% 99.8% % 98.5% 86.% 96.00% 96.08% 99.59% Anlticl Solutions * Note: In the cse tht n of the required points does not lie on node, (s in the mesh sizes of 5 nd 65 in this problem), results of bing stress nd/or shering stress re verged from the nerest nodes to the loction of the required point. For the deep cntilever bem problem, we notice tht the new tringulr element, SBTREIR gives higher ccurc results thn the CST for the cses of verticl deflection, bing stress nd sher stress. For both elements, the convergence of the solution to the nlticl vlue is ensured s more elements nd nodes re used (mesh refinement). The overll stress pttern converges to the nlticl pttern in the solutions of the two tringulr elements. Figures 4-6 to 4-9 show grphicl comprison between the results obtined b the SBTREIR element, the CST element nd the nlticl solutions. 4

54 Chpter 4: Development of New Strin-Bsed Tringulr Element Verticl Deflection, mm A Anlticl Solution SBTREIR No. of Elements CST Figure 4-6: Verticl Deflection t A, (mm) in the Deep Cntilever Bem Using Tringulr Elements 4

55 Chpter 4: Development of New Strin-Bsed Tringulr Element Verticl Deflection, mm Anlticl Solution SBTREIR No. of Elements Figure 4-7: Bing stress t B, (KP) in the Deep Cntilever Bem Using Tringulr Elements CST B 43

56 Chpter 4: Development of New Strin-Bsed Tringulr Element Shering Stress, KP Anlticl Solution SBTREIR CST No. of Elements C Figure 4-8: Shering Stress t C, (KP) in the Deep Cntilever Bem Using Tringulr Elements 44

57 Chpter 4: Development of New Strin-Bsed Tringulr Element In-plne rottion, Rd A Anlticl Solution SBTREIR No. of Elements Figure 4-9: In-Plne Rottion t A, (Rd) in the Deep Cntilever Bem Using Tringulr Elements 45

58 Chpter 4: Development of New Strin-Bsed Tringulr Element 4.5 SIMPLY SUPPORTED BEAM PROBLEM The second problem used to test the performnce of the new element is tht of simpl supported bem loded b point lod t the middle of the upper surfce of the bem. The bem hve length, L=4m, height H=m, nd thickness t=0.5m. The mteril properties re tken s E=0,000 KP nd ν=0.0. The point lod t the midspn of the bem is tken s P=4. KN. The loctions of the investigted points within the simpl supported bem re shown in Figure 4-7 below. P.0m B C A 0.5m 0.5m H =.0 m t L = 4m Figure 4-0: Dimensions nd Considered Points for the Simpl Supported Bem 4.5. Used Mesh Size The following tble shows the number of elements, number of nodes nd the spect rtio of ech mesh size. Tble 4-3: Mesh Size nd Aspect Rtio of the Simpl Supported Bem Using Tringulr Elements Elements in Short Elements in Long Totl no. of Totl Side (L=m) Side (L=4m) Aspect Mesh Tringulr no. of Dimension Dimension Rtio Size No. No. Elements Nodes (m) (m) : : : : :

59 Chpter 4: Development of New Strin-Bsed Tringulr Element A smple mesh size is illustrted in the figure below. Figure 4-: Smple Tringulr Mesh of the Simpl Supported Bem Problem 4.5. Anlticl Solution Bsed on the given geometr nd mteril properties, the nlticl solution for deflection nd stresses t the specified points re clculted s follows: Verticl deflection t point A = 6.7 mm. Bing Stress t point B = 5. KP. Sher Stress t point C = 6.3 KP Convergence Results The problem ws solved using the developed computer progrm (described in Chpter 3 nd Anne A) for ech of the mesh sizes listed in Tble 4-3 bove. Convergence of the overll pttern of bing stress in the simple bem is shown below for the nlticl solution s well s the tringulr elements (CST nd SBTREIR) using ech mesh size (Red: tension, Blue: compression). Bing Stress Pttern Anlticl Solution 47

60 Chpter 4: Development of New Strin-Bsed Tringulr Element CST Mesh Size 4 CST Mesh Size 8 CST Mesh Size 3 CST Mesh Size 46 CST Mesh Size 50 Figure 4-: Bing Stress Pttern in the Simpl Supported Bem Using CST 48