INTRODUCTION TO THE FINITE ELEMENT METHOD

Transcription

1 INTRODUCTION TO THE FINITE ELEMENT METHOD Evgen Barkanov Institute of Materias and Structures Facut of Civi Engineering Riga Technica Universit Riga,

2 Preface Toda the finite eement method (FEM) is considered as one of the we estabished and convenient technique for the computer soution of compe probems in different fieds of engineering: civi engineering, mechanica engineering, nucear engineering, biomedica engineering, hdrodnamics, heat conduction, geo-mechanics, etc. From other side, FEM can be eamined as a powerfu too for the approimate soution of differentia equations describing different phsica processes. The success of FEM is based arge on the basic finite eement procedures used: the formuation of the probem in variationa form, the finite eement dicretization of this formuation and the effective soution of the resuting finite eement equations. These basic steps are the same whichever probem is considered and together with the use of the digita computer present a quite natura approach to engineering anasis. The objective of this course is to present brief each of the above aspects of the finite eement anasis and thus to provide a basis for the understanding of the compete soution process. According to three basic areas in which knowedge is required, the course is divided into three parts. The first part of the course comprises the formuation of FEM and the numerica procedures used to evauate the eement matrices and the matrices of the compete eement assembage. In the second part, methods for the efficient soution of the finite eement equiibrium equations in static and dnamic anases wi be discussed. In the third part of the course, some modeing aspects and genera features of some Finite Eement Programs (ANSYS, NISA, LS-DYNA) wi be brief eamined. To acquaint more cose with the finite eement method, some eceent books, ike [-4], can be used. Evgen Barkanov Riga,

3 Contents PREFACE. PART I THE FINITE ELEMENT METHOD.. 5 Chapter Introduction Historica background...5. Comparison of FEM with other methods...5. Probem statement on the eampe of shaft under tensie oad.6.4 Variationa formuation of the probem Ritz method..6 Soution of differentia equation (anatica soution).7 FEM.... Chapter Finite eement of bending beam. Chapter Quadriatera finite eement under pane stress... PART II SOLUTION OF FINITE ELEMENT EQUILIBRIUM EQUATIONS. Chapter 4 Soution of equiibrium equations in static anasis. 4. Introduction Gaussian eimination method.. 4. Generaisation of Gauss method. 4.4 Simpe vector iterations Introduction to noninear anases Convergence criteria...7 Chapter 5 Soution of eigenprobems Introduction 9 5. Transformation methods.4 5. Jacobi method Vector iteration methods Subspace iteration method..4 Chapter 6 Soution of equiibrium equations in dnamic anasis Introduction Direct integration methods The Newmark method Mode superposition Change of basis to moda generaised dispacements Anasis with damping negected Anasis with damping incuded.5

4 PART III EMPLOYMENT OF THE FINITE ELEMENT METHOD...5 Chapter 7 Some modeing considerations Introduction.5 7. Tpe of eements.5 7. Size of eements Location of nodes Number of eements Simpifications afforded b the phsica configuration of the bod Finite representation of infinite bod Node numbering scheme Automatic mesh generation 59 Chapter 8 Finite eement program packages Introduction.6 8. Buid the mode App oads and obtain the soution Review the resuts...6 LITERATURE 6 APPENDIX A tpica ANSYS static anasis

5 PART I THE FINITE ELEMENT METHOD Chapter Introduction. Historica background In 99 Ritz deveoped an effective method [5] for the approimate soution of probems in the mechanics of deformabe soids. It incudes an approimation of energ functiona b the known functions with unknown coefficients. Minimisation of functiona in reation to each unknown eads to the sstem of equations from which the unknown coefficients ma be determined. One from the main restrictions in the Ritz method is that functions used shoud satisf to the boundar conditions of the probem. In 94 Courant considerab increased possibiities of the Ritz method b introduction of the specia inear functions defined over trianguar regions and appied the method for the soution of torsion probems [6]. As unknowns, the vaues of functions in the node points of trianguar regions were chosen. Thus, the main restriction of the Ritz functions a satisfaction to the boundar conditions was eiminated. The Ritz method together with the Courant modification is simiar with FEM proposed independent b Cough man ears ater introducing for the first time in 96 the term finite eement in the paper The finite eement method in pane stress anasis [7]. The main reason of wide spreading of FEM in 96 is the possibiit to use computers for the big voume of computations required b FEM. However, Courant did not have such possibiit in 94. An important contribution was brought into FEM deveopment b the papers of Argris [8], Turner [9], Martin [9], Hrennikov [] and man others. The first book on FEM, which can be eamined as tetbook, was pubished in 967 b Zienkiewicz and Cheung [] and caed The finite eement method in structura and continuum mechanics. This book presents the broad interpretation of the method and its appicabiit to an genera fied probems. Athough the method has been etensive used previous in the fied of structura mechanics, it has been successfu appied now for the soution of severa other tpes of engineering probems ike heat conduction, fuid dnamics, eectric and magnetic fieds, and others.. Comparison of FEM with other methods The common methods avaiabe for the soution of genera fied probems, ike easticit, fuid fow, heat transfer probems, etc., can be cassified as presented in Fig... Beow FEM wi be compared with anatica soution of differentia equation and Ritz method considering the shaft under tensie oad (Fig..). 5

6 Methods Anatica Numerica Eact Approimate Numerica soution FEM (e.g. separation of variabes and Lapace transformation methods) (e.g. Raeigh-Ritz and Gaerkin methods) Numerica integration Finite differences Fig.. Cassification of common methods.. Probem statement on the eampe of shaft under tensie oad The main task of the course Strength of Materias is determination of dimensions of a shaft cross section under known eterna oads. Apping the genera pan for the soution of probems in the fied of mechanics of deformabe soids, tree group of equations shoud be written: ) equiibrium equations (statics) The equiibrium equation for the separate eement with the ength d has the foowing form X or σf ( σ dσ ) F qd After some transformations we have dσ d F q du Taking into account that σ εe E we obtain the static equiibrium equation d d u EF q d ) geometric equations du ε d ) phsica equations σ εe From this sstem of equations it is possibe to determine a necessar vaues. 6

7 σ d q d q σdσ F z Fig.. Shaft under tensie oad. Another approach for the soution of the probem eamined eists aso. This is utiisation of the principe of minimum of the potentia energ which means: a sstem is in the state of equiibrium on in the case when it potentia energ is minima. Correctness of this principe ma be observed on the foowing simpe eampes: - a ba is in the state of equiibrium on in the ower point of surface (Fig..), - a water on the rough surface takes the equiibrium state in the ower position, - a student tries to take eamination with the minimum ependitures of abour. From the condition that the potentia energ takes the minimum, it is possibe to determine the unknown vaues. The genera agorithm of soution in this case is foowing: ) an epression for the potentia energ of eastic sstem under eterna oads is written, ) conditions of minimum of the potentia energ are written, ) unknown vaues are determined from the condition of minimum, 4) a strength probem is soved. Π P R P R Π min Fig.. Principe of minimum of the potentia energ. 7

8 Compete potentia energ of the deformabe sstem consists from the strain energ U stored in the sstem and energ W ost b the eterna forces (Fig..4). That is wh the work of the eterna forces W is negative vaue Π U W Since the tension of a shaft is eamined, U is the potentia energ of tension. Then for the tension we have Π U W P P P The force oses energ P, but the shaft acquires the tensie energ ( P ). The second part goes on overcoming the friction forces, interna heat, changes into kinetic energ, etc. After remova of oad, the sstem can gives back on the energ equa to the potentia energ of tension U P P P - Π U - W -P H P H- P(H - ) - PH -P W fina initia Fig..4 Energ baance. Π δu() u() Π(uδu) Π(u) δπ Fig..5 Variationa formuation. 8

9 .4 Variationa formuation of the probem A numerica vaue of the potentia energ of tension Π U W is dependent from the function u( ) to be used. Because Π is a functiona, since a functiona is a vaue dependent from the choice of function. This can be epained b the hep of Fig..5. In the ower point, an infinitesima change of the function u ( ) equaed to δ u( ) wi not give an increase of the functiona δ Π. In the point of minimum: δ Π. Free changes of δ u,δπ are caed the variations. The mathematica condition of the minimum of potentia energ can be written as δπ. How it can be seen, variation in the case of functiona investigation has the same meaning as differentia in the case of function investigation. Let s investigate the functiona Π of a tensie shaft under distributed oad q Π ( u( )) U W du EF d qud d U N σ F ε E F EF EF EF du d d d d EFε d EF d Let s determine the variation δ Π as a difference of two vaues the potentia energ with and without increment δ u du du δπ Π ( u δu ) Π ( u ) EF δ d qδud d d du dδu du d u EF d qδud EF δu δu d o qδ ud d d d d du t ; s δu d Integration b parts: s' td st st' d The potentia energ wi has the minimum vaue, if δπ, or b other words, if a items equa to zero in the ast epression. Boundar conditions for our probem are: ) : u du ) : d At a ength : δ u. Apping these boundar conditions to the variation of the functiona Π, we obtain d u EF d δ u d qδud δu EF q d d u d 9

10 d u This equation can be soved if EF q. Moreover, this condition presents the d static equiibrium equation. Epressions obtained show that the potentia energ of sstem has the minimum, if: ) the equiibrium equations wi be reaised ) the boundar conditions wi be reaised The second boundar conditions, so caed as natura boundar conditions for the functiona Π, since the are obtained from the minimum of functiona, reaise automatica. But it is necessar to satisf without fai to the first boundar conditions. Otherwise, these conditions are not taken into account anwhere. These boundar conditions are caed principa. In the case of beam bending: - natura conditions are forces, - principa conditions are dispacements. The probem of determination of u( ) can be soved b two was: ) b soution of the differentia equation, ) b minimising the functiona Π. Soving the probem b the approimate methods using computers, the second wa is more suitabe..5 Ritz method B the Ritz method it is possibe to determine an approimate min Π. An unknown function of dispacements u ( ) is found in the form u( ) ak ϕ k ( ) k where a k are coefficients to be determined, ϕ k ( ) are coordinate functions given so that the satisf to the principa boundar conditions. B insertion u ( ) into functiona Π and then to integrate, it is possibe the probem of the functiona minimisation to come to the probem of determination the function minimum Π Π ( ak ) from unknowns ak. To minimise the function of the potentia energ obtained, it is necessar to equate to zero the derivatives on a k Π a, Π a, Π a,... After this operation, the sstem of agebraic equations is obtained and soved to find the unknowns a k. In the Ritz method, the choice of function u ( ) cosed enough to a truth is a compicated probem requiring a good idea of the resut epected.

11 Eampe. ) u a The principa boundar conditions:, u. du Π [ u( )] EF d qud EFa qa d q const ; du d a Π q q EFa, hence a a EF q du q q u ; ε ; σ εe EF d EF F q q u( ) ; u EF 4EF q EF ) u a a The principa boundar conditions:, u. Π [ u ( )]... Π Π ; a a Then the sstem consisting from inear agebraic equations is soved and unknown coefficients a and a are determined. After this eampe, it is possibe to write the genera agorithm of the Ritz method: ) Presentation of Π. ) Determination of boundar conditions. ) Approimation for a construction. 4) Integration of Π for a construction. 5) Determination of the minimum of Π. Soution of the sstem of inear agebraic equations. 6) Cacuation of dispacements. 7) Cacuation of stresses. For a beam with few areas, it is more easi to guess the defection function for separate areas than for the whoe of beam. Moreover, the function for one area wi be a more simpe than for the whoe of beam. The idea of division of the investigated object is used in FEM. In the Ritz method, the accurac can be increased choosing more terms of the approimated function, but in FEM increasing the quantit of finite eements. To simpif the probem soution b computer, the finite eements and approimated functions are chosen the same.

12 .6 Soution of differentia equation (anatica soution) Anatica soution means determination of the dispacement function u ( ) from the equiibrium equation. Eampe. d u EF q d To sove the probem of u( ) foowing boundar conditions: ) : u du ) : d determination it is necessar to satisf to the d u d q EF After integration we obtain du q q d C C d EF EF q u C C EF The constants are determined using the boundar conditions: q ) C C ; C EF ) q C q ; C EF EF Then we have q q u EF EF q q q u( ) EF EF EF q q q u 8EF EF 8EF du q q σ εe E d F F du q q ε d EF EF

13 .7 FEM FEM was treated previous as a generaisation of the dispacement method for shaft sstems. For a computation of beams, pates, shes, etc. b FEM, a construction is presented in a view of eement assemb. It is assumed that the are connected in a finite number of noda points. Then it is considered that the noda dispacements determine the fied of dispacements of each finite eement. That gives the possibiit to use the principe of virtua dispacements to write the equiibrium equations of eement assemb so, as made for a cacuation of shaft sstems. Let s have a ook the finite eement of tensie shaft (Fig..6). The dispacement function can be chosen in the foowing form u C C Using boundar conditions for a singe finite eement in his oca coordinate sstem, we have u C u C C Now our purpose to epress coefficients through the noda dispacements of the finite eement C u u u C Then the dispacement function for a singe finite eement can be written in the foowing form u u u u u u un( ) un( ) N ( ), N( ) - noda functions. u (i) q L q (i) u Fig..6 Finite eement of tensie shaft.

14 The potentia energ of the finite eement can be epressed as foows e Π du d EF d qud EF u ( ) ( ) N' ( ) d uu N' ( )N' d u N' ( ) d ( qu { K u K u u K u } { u F u F } N( )d qu N( )d ) du d u N' ( ) u N ' ( ) N' ( ) N ' ( ) EF K EF ( N' ( )) d EF EF K EF N' ( )N' ( )d EF EF K EF ( N' ( )) d EF K ij - eements of the stiffness matri. F q q F q N( )d q d q F i - noda forces. N( )d q d q q q Now the potentia energ of the finite eement presents the function of his noda dispacements e e Π Π ( u,u ) 4

15 Let s rewrite the potentia energ of finite eement in the matri form { } { } e et e e et F d d K d F F u u u u K K K K u u e Π d u u e q q F F e F EF K K K K e K Then it is possibe to determine the potentia energ of structure consisting from the separate energies of the finite eements N e F d Kd d T T Π Π where is the stiffness matri of construction as a sum of stiffness matrices of separate finite eements, d is the vector of noda unknowns of construction, F is the vector of given eterna noda forces. B this wa, the potentia energ of structure is epressed in a view of function dependent on unknown noda dispacements d. The condition of the functiona minimum turns into condition the function minimum N e K K e ) ( Π d u i Π Soution of this sstem is unknown dispacements F K d It is necessar to note that soving the present sstem of equations, it is necessar to take into account conditions of structure supports, that is to sa the principa boundar conditions. After determination of the noda dispacements d, the interna forces and stresses are computed. Then the are used for a vauation of the structure s strength. σ 5

16 Eampe. q u ;F I L q II u I F F F II ; () () q q q L - ength of FE u F F II ; () q Fig..7 Finite eement mode of shaft under tensie oad. The potentia energ of shaft under tensie oad can be epressed as foows I II I I I I I Π Π Π ( Ku Kuu Ku ) ( uf uf ) II II II II II ( Ku Kuu Ku ) ( uf uf ) I II II II {( K K ) u Kuu Ku } ( uf uf ) T T { Ku } ( ) d Kd d F K u u K u u F u F I II K K K II K K II K K - in the goba coordinate sstem. K K u F K ; d ; F K K u F EF q K ; F q 6

17 Now it is necessar to determine the noda dispacements of the structure using the principe of minimum of the potentia energ. Π(u,u ) min Fig..8 Possibe soution. u,u These unknowns are determined from the foowing sstem of inear agebraic equations Π u Π u Π Ku Ku F u Π K u Ku F u q ql u EF L u since, then we have 8EF q u ql EF u EF Stresses can be cacuated b the foowing wa u( ) u N( ) un( ) ε du d u N ( ) un ( ) 7

18 I σ Eε ql Eu N' ( ) EuN' ( ) E L 8EF ql 4 F N ( ) ; N ' ( ) L σ II ql ql ql ql EN ' ( )u EN' ( )u E E L 8EF L EF 4 F F ql 4 F N ( ) ; N' ( ) L Eampe.,.,. Let s compare the anatica soution with the soution obtained b FEM and Ritz methods. u σ q F Ritz method anatica soution FEM q 4EF q 8EF anatica soution q 4 F q 4 F FEM Ritz method q EF q F Fig..9 Anatica, FEM and Ritz soutions. 8

19 After this eampe, it is possibe to write the genera agorithm of FEM: ) Presentation of Π. ) Determination of boundar conditions: δ Π. ) Approimation for the finite eement. 4) e K - integration (anatica or numerica). 5) Finite eement meshing. B computer 6) Buiding of K. 7) Determination of the minimum of Π. Soution: Kd F, d K F, where K is smmetrica and banded. 8) Output of dispacements. 9) Computation of stresses. ) Output of stresses. FEM Accurac of FEM: approimate eact, FEM Πapproimate Π eact. 9

20 Chapter Finite eement of bending beam The functiona Π of bending beam oaded b the concentrated forces moments M j and distributed oad qk can be written in the foowing form P i, bending L L dw j Π U W EJ( w ) d Pi wi M j qwd () i j d Then it is necessar to describe the boundar conditions. In our case, the principa boundar conditions are w, dw d and the natura boundar conditions - w ( M EJw ), w ( Q EJw ) w q P M dw d d L w i ' w i i i i ' w i w i w w ' w ' w w i i Fig.. Finite eement of bending beam.

21 Besides, we have additiona conditions - two principa boundar conditions which shoud be reaised at each end of the finite eement. These are conditions of joining of two neighbouring eements w( i ) w( i ), w(i) w( i ) To satisf these four conditions, et s choose the ponom with four coefficients as coordinate function w ( ) a a a a In such view the coordinate function w( ) does not satisf to the boundar conditions et. Therefore, et s change it so, that coefficients a, a,a, a were epressed through unknowns in the noda points of eement ends - w, w, w, w, where and are the numbers of noda points w a (when ) and etc. Then the sstem of equations is soved in reation to a, a,a, a. Substituting these epressions into coordinate function and introducing the noda functions N ( ), N( ), N( ), N4( ), we obtain w( ) w N( ) w N ( ) w N( ) w N 4( ) () N ( ) N ( ) N( ) N 4( ) In such view the coordinate function w() satisfies to the principa boundar conditions. Then we substitute the epression () in () and obtain after integration the potentia energ of the finite eement e Π Π e ( w,w,w,w ) e Π et e e et e d K d d F

22 where is the stiffness matri of the finite eement of bending beam, d is the vector of noda unknowns of the finite eement, F is the vector of given eterna noda efforts, when the eterna oad is presented b the forces and moments in the noda points. e K e e EJ e K, d, w w w w e q q q q M F M F e F Now it is possibe to determine the potentia energ of structure consisting from the separate energies of the finite eements N e Π Π The compete potentia energ is a function of unknowns dispacements and anges of rotations in the noda points. To obtain the minimum of the potentia energ, as in the Ritz method, we take derivatives on unknowns, equate to zero and obtain the sstem of agebraic equations for determination of unknown vaues. Assuming that a beam consists from one finite eement ( Π ), the condition of minimum can be written as e Π w, w, w, w e e e e Π Π Π Π

23 Chapter Quadriatera finite eement under pane stress Since genera reations of pane strain and pain stress differ on b the eastic constants, a soution of the pane probem in the theor of easticit we eamine on the base of pane stress. For the cacuation of pates oaded in their pane, the functiona of compete potentia energ for the pane stress is written in the foowing form: L dl v ) p u p ( )d ( W U Ω Ω γ τ ε σ ε σ Π () where are the norma and tangentia stresses, ε are the inear and ange strains, u are the inear dispacements of the points on the midde pane of pate in reation to aes and, are the vector components of eterna oading in reation to aes and, are infinite sma eement of two-dimensiona area and outine. τ σ σ,,, v dω,γ ε, p p, dl, For the pane probem in the theor of easticit we have,,, () d v u p p F τ σ σ σ γ ε ε ε For the isotropic materia, the genera reations of the pane stress can be presented as ) ( E E E E E γ ε ε τ σ σ σ, () v u γ ε ε ε (4) or in the matri form:

24 σ Eε, ε Dd (5) where E is the matri of easticit, D is the matri of differentiation. Now the functiona of compete potentia energ of the pate oaded in it pane can be written in the compact form: T T Π ε σ d Ω F d dl (6) Ω L For the buiding of stiffness matri, it is necessar to set the dispacement approimation for the finite eement area and to connect it with the degrees of freedom. For an eistence of the functiona of compete potentia energ, the approimation functions of dispacements shoud contain terms not ower than first order. The inear ponomia from two variabes contains three terms. To connect four nodes of the quadriatera finite eement with necessar quantit of constant coefficients of the approimation functions, the foowing form of these ponomias are taken u(, ) a a a a4 v(, ) a5 a6 a7 a8 This mode corresponds to the inear distribution of dispacements aong coordinate aes. The number of inear independent coefficients is twice more than the number of finite eement nodes. On this reason, for each node it is possibe to give two degrees of freedom. Thus, the finite eement has eight degrees of freedom (Fig..). The vectors of noda dispacements and noda reactions have the foowing form: u R v R u R v R d, R (8) u R v R u 4 R 4 v4 R4 (7) u v v u b u 4 4 u v 4 v a Fig.. Quadriatera finite eement. 4

25 The stiffness matri K with dimension 88 connects these vectors b the foowing wa R Kd (9) Let s epress the inear independent constant coefficients of approimation functions b the noda dispacements. For this purpose coordinates (, ) for the first node are substituted to the epression (7) and we have u a a a a4 The same operation is repeated for the second and other nodes. After that we have the sstem of four inear agebraic equations in reation to the constant coefficients a i ( i,,, 4) : u u u u4 4 4 a a a 4 4 a4 or in the compact form: d Ca () Soving the sstem (), the constant coefficients a i ( i,,, 4) are determined a C d Then approimation functions can be written in the form: u(, ) v(, ) 4 d i N i i 8 d i N i i 5 where d i ( i,,..., 8) are the degrees of freedom of the finite eement, N i ( i,,..., 8) are the noda functions. As it is seen, on coefficients for the function u (, ) were determined, since the coefficients for the function v (, ) have the same form. In the detaied form, the functions can be epressed as u(, ) v(, ) () () ( a )( b )u ( b )u u ( a )u4 () ab ab ab ab ( a ab )( b )v ( b )v v ab ab From the principe of possibe dispacements we have ( kij ) r σ jεidω r Ω ( a ab )v4 where σ j are the stresses on the finite eement area from dispacement d j, ε i are the strains on the finite eement area from dispacement (4) d i. If degrees of freedom have the 5

26 phsica meaning of dispacements, ( k ij ) r, as an eement of the stiffness matri, is an effort arising aong i-th degrees of freedom from j-th unit dispacement under condition that a others ( i j ) degrees of freedom d i. Thus, we can obtain now the stiffness coefficients using epression (4) and taking into account the pane stress state and epression (5) kij ab h ( Eε i ) T ε jdd where h is the thickness of pate, εi ( i,,..., 8 ) is the strain vector (4) on the finite eement area in the case, when the node dispacement with number i is equa to unit but a other dispacements are zero, ε j ( j,,..., 8 ) is the strain vector (4) on the finite eement area in the case, when the node dispacement with number j is equa to unit but a other dispacements are zero. As an eampe, et s epress the stiffness matri eement k k u presenting the u reaction arising in the node aong ais from the unit dispacement of the node in the same direction. The numbering of degrees of freedom is given in reation of their recording in the coumn (8). At the beginning we buid the strain vector ε εu corresponding to the deformation state on the finite eement area from the unit dispacement u, when a other noda dispacements are zero. In this case the vector of approimate functions is formed from the epression () taking into account u and u u u4 v v v v4 : u (, ) ( a )( b ) N (6) ab v(, ) The strain vector ε is formed in reation with the epression (4): ε ( b ) ) ab u(, ε ε (7) v(, ) γ ( a ) ab B the same wa, the strain vector ε ε u (5) is buid. This vector corresponds to the deformation state on the finite eement area from the unit dispacement noda dispacements are zero. u, when a other ( b ) ab ε (8) ab 6

27 Substituting the vectors (7) and (8) into epression (5), we have dd ab ) a ( ab ) ( E ab b ) ( b ab E h k ab 6 b a b a b a ab Eh, or after introduction of a b m : m m Eh k To obtain remaining eements of the stiffness matri of quadriatera finite eement under pane stress, the same procedure shoud be appied and we have m m Eh k 6, 8 Eh k, Eh k, m m Eh k 6 5, 8 6 Eh k, m Eh k, Eh k, m m Eh k 6, 8 8 Eh k, m m Eh k 6 6 4, 8 5 Eh k, m m Eh 6 6 k, Eh k, m m Eh k 8, m m Eh k 6, 8 4 Eh k, m m Eh k 6 6 5, Eh k, m m Eh k 6 7, 8 8 Eh k, m m Eh k 6 44, Eh k, m m Eh k 46, 8 47 Eh k, m m Eh k 6 48, m m Eh k 6 55, 8 56 Eh k, m m Eh k 57, Eh k, m m Eh 6 66 k, Eh k, m m Eh k , m m Eh k 6 77, 7

28 8 78 Eh k, m m Eh k 6 88 Based on epressions (5), stresses on the finite eement area are determined using known nodes dispacements of the sstem. The strain vector with three components is epressed b the vector of approimate functions b the foowing wa: (9) EDd Eε σ where E is the matri of easticit, D is the matri of differentiation, d is the vector of approimate functions consisting of two components and. The approimate functions () are written in the matri form: ), ( u ), v( 4 4 T v u v u v u v u ) a ( ) ( b ) )( b a ( ) a ( ) ( b ) )( b a ( ab ), v( ), u( () After substituting epressions (), (4) and () into epression (9), we have 4 4 v u v u v u v u a ) ( a ) ( a ) ( b ) ( b ) ( b ) ( b ) ( a ) ( a ) ( a ) ( b ) ( b ) ( )ab ( E T τ σ σ () where and are coordinates of the points on the finite eement area. As it is seen from the epression (), stresses on the finite eement area are the inear functions of coordinates. In the centre of gravit of the finite eement ( b, a ), the stress vector has the foowing form: 8

29 v u v u v u v u b a a a b b b a a a b b b a a a b b b a a a b b )ab ( E T τ σ σ 9

30 PART II SOLUTION OF FINITE ELEMENT EQUILIBRIUM EQUATIONS Chapter 4 Soution of equiibrium equations in static anasis 4. Introduction When FEM is used for a soution of static probems, we get a set of simutaneous inear equations, which can be stated in the form KX F where K is the stiffness matri of a structure, X is the dispacement vector and F is the oad vector. This equation can be epressed in the scaar form as k k Kkn n f k k Kkn n f M kn kn K knn n fn where the coefficients kij and the constants fi are given. The probem is to find the vaues of i, if the eist. In the matri form k k K n n n M kn k k kn K K K kn k, X, n M knn n f f F n M fn It is necessar to note that in the finite eement anasis, the order of the matri K is ver arge. The methods avaiabe for soving of the sstems of inear equations can be divided into two tpes: direct or iterative. Direct methods are those, which, in the absence of round-off and other errors, wi ied the eact soution in a finite number of eementar arithmetic operations. In practice, because a computer works with a finite word ength, sometimes the direct methods do not give good soutions. Indeed the errors arising from round-off and truncation ma ead to etreme poor or even useess resuts. The fundamenta method used for direct soutions is Gaussian eimination, but even within this cass there are a variet of choices of methods, which var in computationa efficienc and accurac. Iterative methods are those, which start with an initia approimation, and which b apping a suitab chosen agorithm, ead to successive better approimations. When the process converges, we can epect to get a good approimate soution. The accurac and

31 the rate of convergence of iterative methods var with the agorithm chosen. The main advantages of iterative methods are the simpicit and uniformit of the operations to be performed, which make them we suited for use on computers and their reative in sensitivit to the growth of round-off errors. Matrices associated with inear sstems in the finite eement anasis are cassified as sparse and have ver few nonzero eements. Fortunate, in most finite eement appications, the matrices invoved are positive definited, smmetric and banded. Hence soution techniques, which take advantage of the specia character of such sstems of equations, have aso been deveoped. 4. Gaussian eimination method The basic objective of this method is to transform the given sstem into an equivaent trianguar sstem, whose soution can be more easi obtained. We sha consider the foowing sstem of three equations to iustrate the process To eiminate the terms from equations () and (), we mutip equation () b - and - 4 and add respective to equation () and equation () iving the first equation unchanged. We wi have then 5 6 To eiminate the term equation (). We wi have the trianguar sstem 5 from equation (), we mutip equation () b and add to 5 This trianguar sstem can be soved now b the back substitution. From equation () we find. Substituting this vaue for into equation () and soving for, we obtain 4. Fina, knowing and, we can sove equation () for, obtaining. 4. Generaisation of Gauss method Let s the given sstem of equations be written as

32 k k k k k k k4 k k k k4 k5 k44 k44 k444 k544 The wide of matri band is. After eimination of k55 k455 k555, we have f f f f4 f5 k k ( ) k ( ) k ( ) k4 k ( ) k ( ) k ( ) k4 ( ) k5 ( ) k4 ( ) k4 ( ) k44 ( ) k54 ( ) k 5 ; ( ) k 45 ( ) k55 ( ) f ( ) f ( ) f ( ) f4 ( ) f5 where new coefficients are epressed b the foowing wa ( ) k k k k k ( ) k k k k k ( ) k k k k k ( ) k k k k k ( ) f f f k k ( ) f f f k k The upper inde () has been used to denote the first eimination. The genera reation of an arbitrar coefficient after first eimination has the foowing form ( ) k j kij kij ki, i, j > k To eimination with number n corresponds the foowing genera reation ( n) k ( n ) ( n) ( n) nj kij kij kin, i, j > n ( n) knn Anaogous formuas are obtained for a vector of oad

33 f ( n ) i ( n ) ( n ) ( n ) f n fi kin, i > n ( n ) knn The smmetr in coefficients after eimination operation maintains. Thus the matri decomposition ma be carried out using on coefficients situated on the main diagona and above it. Therefore, it is not necessar to store the fu matri. Another positive feature of the Gaussian eimination, which can be used, is that matri eements situated out of band do not infuence on the eimination process. The equa to zero. Hence, it is not necessar to store them. That gives the possibiit to store the goba stiffness matri as a rectanguar arra with wide equa to the wide of the matri band. After apping the above procedure ( n ) times, the origina sstem of equations reduces to the foowing singe equation ( n ) ( n ) knn n f n from which we can obtain n ( n ) f n ( n ) knn The vaues of the remaining unknowns can be found b the back substitution. Note: If zero or negative diagona eement occurs in the Gauss eimination the structure is not stabe or, b other words, the goba stiffness matri is not positive definite. The Gauss eimination scheme fas under the categor of direct methods. This categor incudes aso Choeski method (a direct method for soving a inear sstem which makes use of the fact that an square matri can be epressed as the product of an upper and ower trianguar matrices), the Givens factorization (rotation matrices are used to reduce the goba stiffness matri into upper trianguar form), the Househoder factorization (refection matrices are used). 4.4 Simpe vector iterations The power and inverse iteration methods are the methods not used wide now, but the shoud be eamined, since hep to understand more compe modern agorithms. Agorithm of the power method: The unit vector X is chosen. Then, for k,,,... ) To form F k KXk Fk ) To set the rate of X k Fk ) To subject X on the convergence test. k

34 The inverse iteration method is a power method appied to K. It is not necessar to make an inverse matri K, instead of that we change F k KXk in the power method on ) To sove in reation of F k the foowing equation KF k X k. In the cass of iterative, the Gauss-Seide method is we known. The conjugate gradient and Newton s methods are other iterative methods based on the principe of unconstrained minimisation of a function. It is to be noted that the indirect methods are ess popuar than the direct methods in soving arge sstems of inear equations. 4.5 Introduction to noninear anases In the inear anasis we assumed that dispacements of the finite eement assembage are infinitesima sma and that a materia is inear eastic. In addition we aso assumed that a nature of boundar conditions remains unchanged during appication of oads on the finite eement assembage. Figure 4. gives a cassification that is used ver convenient in practica noninear anasis because this cassification considers separate materia noninear effects and kinematic noninear effects. The basic probem in a genera noninear anasis is to find the state of equiibrium of a bod corresponding to the appied oad. Assuming that the eterna appied oads are described as a function of time, the equiibrium conditions of a sstem of finite eements representing the bod under consideration can be epressed as t F t R t where the vector F stores the eterna appied noda oads and t R is the vector of noda point forces that are equivaent to the eement stresses. This reation must epress the equiibrium of the sstem in the current deformed geometr taking due account of a noninearities. Considering the soution of the noninear response, we recognise that the equiibrium reation eamined must be satisfied throughout the compete histor of oad appication, i.e. the time variabe ma take on an vaue from zero to the maimum time of interest. The basic approach in the incrementa step-b-step soution is to assume that the soution for the discrete time t is known, and that the soution for the discrete time t t is required, where t is a suitab chosen time increment. Hence, considering the previous reation at time t t we have t t t F t R Since the soution is known at time t, we can write t t t R R R where R is the increment in noda point forces corresponding to the increment in eement dispacements and stresses from time t to time t t. This vector can be approimated using a tangent stiffness matri t K which corresponds to the geometric and materia conditions at time t () () () 4

35 σ L σ,ε P / E L P / σ P / A ε σ / E ε L ε <.4 ε (a) Linear eastic (infinitesima dispacements). σ P / A L L P / P / σ P / A σ ε E ε <.4 σ Y σ σ E T Y E E T ε (b) Materia-noninear-on (infinitesima dispacements, but noninear stress-strain reation). ' ' ε' ' L L ε' <.4 ' ε' L (c) Large dispacements and arge rotations but sma strains. Linear or noninear materia behaviour. Fig. 4. Cassification of anases. 5

36 (d) Large dispacements, arge rotations and arge strains. Linear or noninear materia behaviour. P / P / (e) Change in boundar condition at dispacement. Fig. 4. (continuation) R t KX (4) where X is a vector of incrementa noda point dispacements. Substituting two ast epressions () and (4) into previous, we obtain t t t t KX F R and soving for X, we can cacuate an approimation to the dispacements at time t t t t X X X The eact dispacements at time t t are those that correspond to the appied oads t t F. We cacuate in equation (6) on an approimation to these dispacements because equation (4) was used. Having evauated an approimation to the dispacements corresponding to time t t, we can now sove for an approimation to the stresses and corresponding noda point forces at time t t, and coud then proceed to the net time increment cacuations. However, because of the assumption in equation (4) such a soution ma be subject to ver significant errors and, depending on the time or oad step sizes used, ma indeed be t (5) (6) 6

37 unstabe. In practice, it is therefore frequent necessar to iterate unti the soution of equation () is obtained to sufficient accurac. A wide used iteration procedure is the modified Newton iteration, in which we sove for i,,,... (i) t t t t (i) F F R (7) t (i) (i) K F (8) t t (i) t t (i) (i) X X X with the initia conditions t t ( ) t t t ( ) t X X ; R R These equations were obtained b inearizing the response of the finite eement sstem about the conditions at time t. In each iteration we cacuate in equation (7) an out-ofbaance oad vector, which ieds an increment in dispacements obtained in equation (8), (i) and we continue the iteration unti the out-of-baance oad vector F or the (i) dispacement increments X are sufficient sma. The most frequent used iteration schemes for the soution of noninear finite eement equations are some forms of Newton-Raphson iteration, when the equation (7), (8), (9) are a specia case. In the Newton-Raphson iteration, in genera the major computationa cost per iteration ies in the cacuation and factorisation of the tangent stiffness matri. As an aternative to forms of Newton iteration, a cass of methods known as matri update methods or quasi-newton methods has been deveoped for iteration on noninear sstems of equations. These methods invove updating the stiffness matri to provide a secant approimation to the matri from iteration ( i ) to i. (9) 4.6 Convergence criteria If a soution strateg based on iterative methods is to effective, reaistic criteria shoud be used for the termination of the iteration. At the end of each iteration, the soution obtained shoud be checked to see a convergence. If the convergence toerances are too oose, inaccurate resuts are obtained, and if the toerances are too tight, much computationa effort is spent to obtain needess accurac. Since we are seeking the dispacement corresponding to time t t, it is natura to require that the dispacements at the end of each iteration be within a certain toerance of the true dispacement soution. Hence, a reaistic convergence criterion is (i) X t t X ed / n where e D is a dispacement convergence toerance, X k is the Eucedean k norm. The vector t t X is not known and must be approimated. 7

38 A second convergence criterion is obtained b measuring the out-of-baance oad vector. For eampe, we ma require that the norm of the out-of-baance oad vector be within a preset toerance of the origina oad increment t t t t (i) F R t t t F R e F ef In the case, when both the dispacements and the forces are near their equiibrium vaues, a third convergence criterion ma be usefu, in which the iteration (i.e. the amount of work done b the out-of-baance oads on the dispacement increments) is compared to the initia interna energ increment. Convergence is assumed to be reached when, with e E a preset energ toerance, (i)t t t t t (i) X ( F R ) e ( )T t t t E X ( F R) 8

39 Chapter 5 Soution of eigenprobems 5. Introduction The forced vibration equation after the finite eement discretization of structure can be epressed as foows M X & KX F () where M and K are the mass and stiffness matrices of structure; F is the eterna oad vector; X and X & are the dispacement and acceeration vectors. In the free vibration anasis, the eterna oad vector is zero and the dispacements are harmonic as i t X Xe ω () After substitution of Equation () into Equation (), we obtain [ M] X K ω or KX λmx () where X represents the ampitudes of the dispacements X caed the mode shape or eigenvector, ω denotes the natura frequenc of vibration and ω λ is caed eigenvaue. Equation () is the generaised eigenvaue probem. It has a nonzero soution for K ω M is zero, i.e. X, when the determinant of the foowing matri [ ] K ω M (4) The structure with n degrees of freedom has n natura frequencies. It was assumed that the rigid bod degrees of freedom were eiminated in Equation (). If rigid bod degrees of freedom are not eiminated in deriving the matrices K and M, some of the natura frequencies ω woud be zero. In such case, for a genera three-dimensiona structure, there wi be si rigid bod degrees of freedom and hence si zero frequencies. In most of the numerica methods used for the soution of Equation (), the generaised eigenvaue probem is first converted into the form of a standard eigenvaue probem, which can be stated as HX λx or [ H λ I] X (5) Mutiping Equation () b M, we obtain Equation (5), where H M K (6) However, in this form the matri H is in genera nonsmmetric, athough M and K are both smmetric. Since a smmetric matri is desirabe from the points of view of storage 9

40 and computer time, we adopt the foowing procedure to derive a standard eigenvaue probem with smmetric H matri. Assuming that M is smmetric and positive definite, we use Choeski decomposition and epress M as M U T U, where U is the upper trianguar matri. B substitution of M into Equation (), we obtain T KX λu UX i.e. i.e. T U KX λux T U KU UX λux Defining a new vector Y as probem as [ H λ I] Y where the matri H T H U KU Y UX, Equation (7) can be written as a standard eigenvaue is now smmetric and is given b Then we app the inverse transformation to obtain the desired eigenvectors Xi U Y i corresponding to the eigenvaues λ i. Two genera tpes of methods, name, transformation methods and iterative methods, are avaiabe for soving eigenvaue probems. The transformation methods such as Jacobi, Givens and Househoder schemes are preferabe, when a the eigenvaues and eigenvectors are required and the dimension of the eigenvaue probem is sma. The iterative methods such as the power method, subspace iteration and Lanczos methods are preferabe, when few eigenvaues and eigenvectors are required on and the eigenvaue probem has a arge dimension. (7) 5. Transformation methods X, Transformation methods empo the basic properties of eigenvectors in the matri T X KX T X MX Λ I Since the matri X, of order n n, which diagonaizes K and M in the wa given in Equations (8) and (9) is unique, we can tr to construct it b iteration. The basic scheme is to reduce K and M into diagona form using successive pre- and post-mutipication b T matrices P k and Pk, respective, where k,,... Specifica, if we define K K and M M, we form (8) (9) 4

41 T K P KP T K P K P M T K k P k K k Pk M T M P MP T M P M P () M () T M k P k M k Pk M where the matrices P k are seected to bring K k and M k coser to diagona form. Then K k Λ and M k I as k and is eamined as the ast iteration, X P P...P. In practice, it is not necessar that M k converges to I and K k to diagona form. Name, if K k diag( K r ) and M k diag( M r ) as k then with indicating the ast iteration ( ) K Λ diag r ( ) M r X P P... P diag ( ) M r The basic idea described above is used in Jacobi and Househoder-QR methods appied effective in the finite eement anasis. 5. Jacobi method The basic Jacobi method has been deveoped over a centur ago for the soution of standard eigenprobems. A major advantage of the procedure is its simpicit and stabiit. Considering the standard eigenprobem KX λx, the k iteration step defined in Equation () reduces to T K k Pk K k Pk () where P k is an orthogona matri, i.e. Equation () gives T P k Pk I In Jacobi soution the matri P k is a rotation matri, which is seected in such wa that an off-diagona eement in K k becomes zero. If eement ( i, j) is to be reduced to zero, the corresponding orthogona matri P k is 4

42 i coumn j coumn P k O M M M cosθ O M M M sinθ L i row sinθ cosθ L L L j row O where θ is seected from the condition that eement ( i, j) in K k be zero. Denoting ( k ) eement ( i, j) in K k b k ij, we use ( k ) kij tan θ for ( k ) ( k ) kii k jj π θ for 4 k L ( k ) ii ( k ) ii L k k k Since K k is smmetric for a k, the upper (or ower) trianguar part of the matri, incuding its diagona eements, is used. It is necessar to note that athough the transformation in Equation () reduces an off-diagona eement in K k to zero, this eement wi again become nonzero during the transformations that foow. Therefore for the design of an actua agorithm, we have to decide which eement to reduce to zero. One choice is to awas zero the argest offdiagona eement in K k. However, the search for the argest eement is time consuming and it ma be preferabe to simp carr out the Jacobi transformations sstematica, row-b-row or coumn-b-coumn, which is known as the ccic Jacobi procedure. The disadvantage of this procedure is that the eement ma aread be near zero and a rotation is sti appied. A procedure that has been used ver effective is the threshod Jacobi method, in which the off-diagona eements are tested consequent, name rowb-row (or coumn-b-coumn), and a rotation is on appied if the eement is arger than the threshod. ( k ) jj ( k ) jj 5.4 Vector iteration methods In the vector iteration methods the basic reation is KX λmx () 4

43 If assume a vector for X, sa X, and assume a vaue for λ, sa λ, we can evauate then the right-hand side of Equation (), i.e. we ma cacuate R ( ) MX Since X is an arbitrari assumed vector, we do not have, in genera, that KX R. Instead, we have the foowing equation KX R, X X We ma assume that X ma be a better approimation to an eigenvector than was X. B repeating the cce we obtain an increasing better approimation to an eigenvector. The procedure described above is the basic of inverse iteration. We wi see that other vector iteration techniques work in a simiar wa. Specifica, in forward iteration, in the first step we evauate R KX and then obtain the improved approimation X to the eigenvector soving MX. R 5.5 Subspace iteration method This method is ver effective in finding the first few eigenvaues and the corresponding eigenvectors of arge eigenvaue probems. The various steps of this method are given beow brief. Agorithm: ) Start with q initia iteration vectors X, X,..., Xq, q> p, where p is the number of eigenvaues to be cacuated. Bathe and Wison suggested a vaue of q min( p, p 8 ) for good convergence. Define the initia moda matri X as X [ XX...X q ] and set the iteration number as k. ) Use the foowing subspace iteration procedure to generate an improve moda matri X k : ~ a) Find X k from the reation K X ~ k MXk b) Compute ~ T ~ K k Xk KXk ~ T ~ M k Xk MXk c) Sove for the eigenvaues and eigenvectors of the reduced sstem K k Qk M k Q k Λk and obtain Λ k and Q k d) Find an improved approimation to the eigenvectors of the origina sstem as 4

44 ~ X k Xk Qk Note: () It is assumed that the iteration vectors converging to the eact eigenvectors eact eact X, X,... are stored as the coumns of the matri X k. () It is assumed that the vectors in X are not orthogona to one of the required eigenvectors. ( k ) ( k ) () If λ i and λ i denote the approimations to the i eigenvaue in the iterations k and k respective, we assume convergence of the process whenever the foowing criteria is satisfied: ( k ) ( k ) λi λi ε, i,,..., p ( k ) λi where ε 6. 44

45 Chapter 6 Soution of equiibrium equations in dnamic anasis 6. Introduction The dnamic equation of motion of a structure can be written as M X && CX & KX F where M,C and K are the mass, damping and stiffness matrices of structure, F is the eterna oad vector, X, X & and X & are the dispacement, veocit and acceeration vectors of a finite eement assembage. It shoud be noted that Equation () is derived from considerations of static at time t, i.e. Equation () ma be written as F (t) F (t) F (t) F(t) () I D E where FI (t) are the inertia forces, F MX& I (t) ; FD (t) are the damping forces, F D (t) C X & ; FE (t) are the eastic forces, F E(t) KX, a of them are time-dependent. Mathematica, Equation () represents a sstem of inear differentia equations of second order and, in principe, the soution of the equations can be obtained b standard procedures for the soution of differentia equations with constant coefficients. However, these procedures can be ver epensive if the order of the matrices is arge. Therefore few effective methods have been eaborated to app them in practica finite eement anasis. These methods are divided into direct integration and mode superposition. Athough these two techniques ma at first sight appear to be quite different, in fact, the are cose reated, and the choice for one method or the other is determined on b their numerica effectiveness. We wi consider on soution of the inear equiibrium equations (Equation ()). () 6. Direct integration methods In direct integration the equations in () are integrated using a numerica step-b-step procedure, the term direct meaning that prior to the numerica integration, no transformation of the equations into a different form is carried out. In essence, direct numerica integration is based on two ideas. First, instead of tring to satisf Equations () at an time t, it is assumed to satisf Equations () on at discrete time intervas t apart. Therefore it appears that a soution techniques empoed in static anasis can probab aso be used effective in direct integration. The second idea on which a direct integration method is based is that a variation of dispacements, veocities and acceerations within each time interva t is assumed. This assumption determines the accurac, stabiit and cost of the soution procedure. 45

46 In the soution the time span under consideration T is subdivided into n equa time T intervas t (i.e. t ) and the integration scheme empoed estabishes an n approimate soution at times, t, t, t,...,t,t t,...,t. Since an agorithm cacuates the soution at the previous times considered, we derive the agorithms b assuming that the soutions at times, t, t, t,...,t are known and that the soution at time t t is required net. Let s have a ook now a more effective and wide used in the genera purpose finite eement programs (incuding ANSYS), the Newmark method. 6. The Newmark method The Newmark integration scheme can be eamined as an etension of the inear acceeration method. The foowing assumptions are used t t t t t t X & X& [( δ) X&& δ X& ] t () t t t t t t t X X X& t α X&& α X& t (4) where α and δ are parameters that can be determined to obtain integration accurac and stabiit. The inear acceeration method: a inear variation of acceeration from time t to time t t is assumed. When δ and α 6, reations () and (4) correspond to the inear acceeration method. Newmark origina proposed, as an unconditiona stabe scheme, the constantaverage-acceeration method (aso caed trapezoida rue), in which case δ and α 4. In addition to () and (4), for soution of the dispacements, veocities and acceerations at time t t, the equiibrium Equations () at time t t are aso considered t t M X&& t t t t t t C X& K X F (5) t X & t ( X & t t X& ) t t X & t t t Fig. 6. Newmark s constant-average-acceeration scheme. 46

47 t t Soving from (4) for X & t t t t in terms of X and then substituting for X & into (), t t t we obtain equations for X & t and X &, each in terms of the unknown dispacements t t t t on. These two reations for X & and t t X X & are substituted into (5) to sove for t t t t, after which, using () and (4), X & and t t X X & can aso be cacuated. Step-b-step soution using Newmark integration method: A. Initia cacuations:. Form stiffness matri K, mass matri M and damping matri C.. Initiaise X, X & and X.. Seect time step size t, parameters α and δ, and cacuate integration constants: δ.5 ; α. 5(. 5 δ ) δ a ; a ; a α t α t α t a ; α δ t δ a 4 ; a 5 ; α a 6 t( δ ) ; a7 δ t 4. Form effective stiffness matri K : K K a M ac. 5. T Trianguarize K : K LDL B. For each time step:. Cacuate effective oads at time t t : t t t t t t t t t t F F M(a X a X& a X&& ) C(a X a X& a X& 4 5 ). Sove for dispacements at time t t : T LDK t t t t X F. Cacuate acceerations and veocities at time t t : t t X & t t t t ( X X) X& t a a a X& t t X & t X& a6 t X&& a7 t t X& 6.4 Mode superposition The numbers of operations required in the direct integration are direct proportiona to the number of time steps used in the anasis. Therefore, in genera, the use of direct integration can be epected to be effective, when the response for a reative short duration is required. However, if the integration must be carried out for man time steps, it ma be more effective to first transform the equiibrium equations (Equation ()) into a form in which the step-b-step soution is ess cost. In particuar, since the number of operations required is direct proportiona to the haf-bandwidth m k of the stiffness matri, a reduction in m k woud decrease proportiona the cost of the step-b-step soution. 47

48 6.5 Change of basis to moda generaised dispacements We propose to transform the equiibrium equations into a more effective form for direct integration b using the foowing transformation on the finite eement noda point dispacements X X (t) PU(t) where P is a square matri and U(t) is a time-dependent vector of order n. The transformation matri P is sti unknown and wi have to be determined. The components of U are refered to as generaised dispacements. Substituting (6) into () and T premutiping b P, we obtain M U & (t) CU& (t) KU(t) F(t) (7) T T T T M P MP, C P CP, K P KP, F P F (8) The objective of the transformation is to obtain new sstem stiffness, mass and damping matrices, K, M and C, which have a smaer bandwidth than the origina sstem matrices, and the transformation matri P shoud be seected according. In theor, there can be man different transformation matrices P, which woud reduce the bandwidth of the sstem matrices. However, in practice, an effective transformation matri is estabished using the dispacement soutions of the free vibration equiibrium equations with damping negected, M X & KX (9) B substitution i t X Φe ω, Equation (9) becomes KΦ ω MΦ () where Φ is an eigenvector and ω is an eigenvaue, and Equation () is caed the generaised eigenvaue probem with unknowns Φ and ω. The eigenprobem () eads the n eigensoutions ( ) ( ) ω, Φ, ( ) ω, Φ,, ω n,φ n, where the eigenvectors are M- orthonormaized, i.e. T, i j Φ i MΦi (), i j ω ω... ωn () The vector Φ i is caed the i mode shape vector and ω i is the corresponding frequenc of vibration. Defining a matri Φ, whose coumns are the eigenvectors Φ i and a diagona matri Ω, which stores the eigenvaues ω i on its diagona, i.e. (6) 48

49 ω Φ [ Φ, Φ,..., Φn ], ω Ω () O ω n we can write the n soutions to () as KΦ MΦΩ (4) Since the eigenvectors are M orthogona, we have T Φ KΦ Ω T, Φ MΦ I (5) It is now apparent that the matri Φ woud be a suitabe transformation matri P in (6). Using X (t) ΦU(t) we obtain equiibrium equations that correspond to the moda generaised dispacements T T U & (t) Φ CΦU& (t) Ω U(t) Φ F(t) (7) The initia conditions on U(t) are obtained using (6) and the M orthonormait of Φ, i.e. at time we have T U Φ M X, U & T Φ M X& (8) The equations in (7) show that if a damping matri is not incuded in the anasis, the finite eement equiibrium equations are decouped, when using in the transformation matri P the free vibration mode shapes of the finite eement sstem. Since the derivation of the damping matri can in man cases not be carried out epicit, but the damping effects can on be incuded approimate, it is reasonabe to use a damping matri that incudes a required effects, but at the same time aows an effective soution of the equiibrium equations. (6) 6.6 Anasis with damping negected If veocit-dependent damping effects are not incuded in the anasis, each equation presents the equiibrium equation of a singe degree of freedom sstem with unit mass and stiffness ω i. In summar, the response anasis b mode superposition requires first, the soution of the eigenvaues and eigenvectors of the probem, then the soution of the decouped equiibrium equations and, fina, the superposition of the response in each eigenvector. The choice of whether to use direct integration or mode superposition wi be decided b considerations of effectiveness on. The essence of a mode superposition soution of a dnamic response is that frequent on a sma fraction of the tota number of decouped equations need be considered, in order to obtain a good approimate soution to the actua response of the sstem. However, the finite eement mesh shoud be chosen such that a important eact frequencies and vibration mode shapes are we approimated. In this case, the mode superposition procedure can be much more effective than direct integration. 49

50 For earthquake oading, in some cases on the owest modes need be considered, athough the order of the sstem n ma be arger than. On the other hand, for bast or shock oading, man more modes need genera be incuded, and number of eigenmodes required p ma be as arge as n /. Fina, in vibration ecitation anasis, on a few intermediate frequencies ma be ecited, such as a frequencies between the ower and upper frequenc imits ω and ω u, respective. Considering the probem of seecting the number of modes to be incuded in the mode superposition anasis, it shoud awas be kept in mind that an approimate soution to the dnamic equiibrium equations is sought. In summar, assuming that the decouped equations have been soved accurate, the errors in a mode superposition anasis using p < n are due to the fact that not enough modes have been used, whereas the errors in a direct integration anasis arise because too arge a time step is empoed. 6.7 Anasis with damping incuded Considering the anasis of sstem in which damping effects can not be negected, we sti woud ike to dea with decouped equiibrium equations, mere to be abe to use essentia the same computationa procedure whether damping effects are incuded or negected. In genera, the damping matri C can not be constructed from eement damping matrices, such as the mass and stiffness matrices of the eement assembage, and its purpose is to approimate the overa energ dissipation during the sstem response. The mode superposition anasis is particuar effective if it can be assumed that damping is proportiona, in which case T Φ i CΦ j ω iξiδ ij (9) where ξ i is a moda damping parameter and δ ij is the Kronecker deta (δ ij for i j, δ ij for i j ). Therefore, using Equation (9), it is assumed that the eigenvectors Φ i, i,,...,n are aso C orthogona and the equations in (7) reduce to n equations of the form u&& i ( t ) ω iξiu& i( t ) ωi ui( t ) fi( t ) where f i ( t ) and the initia conditions on u i ( t ) are defined as T fi( t ) Φi F( t ), i,,..., n () T ui Φ t i & T ui Φ t i M X M X& We note that Equation () is the equiibrium equation governing motion of the singe degree of freedom sstem. In considering the impications of using (9) to take account of damping effects, the foowing observations are made. First, the assumption in (9) means that the tota damping in the structure is the sum of individua damping in each mode. The damping in one mode coud be observed, for eampe, b improving initia conditions corresponding () () 5

51 to that mode on (i.e. X Φi for mode i ) and measuring the ampitude deca during the free damped vibration. A second observation is that in the numerica soution we do not cacuate the damping matri C, but on the stiffness and mass matrices K and M. However, assume that it woud be numerica more effective to use the direct stepb-step integration and that the reaistic damping ratios ξ i, i,,..., p are known. In that case, it is necessar to evauate the matri C epicit, which when substituted into (9) ieds the estabished damping ratios ξ i. If p, Raeigh damping can be assumed, which is of the form C α M βk () where α and β are constants to be determined from two given damping ratios that correspond to two unequa frequencies of vibration b the foowing wa T Φi ( αm βk ) Φi ω iξi α βωi ω iξi In actua anasis it ma we be that the damping ratios are known for man more than two frequencies. In that case two average vaues, sa ξ and ξ, are used to evauate α and β. If more than on two damping ratios are used to estabish C, a more compicated damping matri ma be suggested. Assume that the p damping ratios ξ i, i,,..., p are given to define C. Then a damping matri that satisfies the reation in (9) is obtained using the Caughe series, [ M K] p k C M ak (5) k where the coefficients a k, k,,..., p are cacuated from the p simutaneous equations a p ξ i aω i aωi... a pω i (6) ωi We shoud note that with p, Equation (5) reduces to Raeigh damping, as presented in (). An important observation is that if p >, the damping matri C in (5) is, in genera, a fu matri that considerab increases the cost of anasis. Therefore Raeigh damping is assumed in most practica anases using direct integration. A disadvantage of Reeigh damping is that the higher modes are considerab more damped than the ower modes, for which the Raeigh constants have been seected. In the case of nonproportiona damping (anasis of structures with wide varing materia properties), it ma be resonabe to assign in the construction of the damping matri different Raeigh coefficient α and β to different parts of the structure, which resuts into a damping matri that does not satisf the reation in (9). Another case of nonproportiona damping is encountered, when concentrated dampers corresponding to specific degrees of freedom (e.g. at the support points of a structure) are specified. The soution of the finite eement sstem equiibrium equatoins with nonproportiona damping can be obtained using the direct integration agorithms without modifications, because the propert of the damping matri did not enter the derivation of the soution (4) 5

52 procedures. In the mode superposition method for the case of nonproportiona damping, the equiibrium equations in the basis of mode shape vectors are no onger decouped. An eact mathematica formuation ma be presented in an aternative anasis procedure, where the decouping of the finite eement equiibrium equations is achieved b soving a quadratic eigenprobem, in which case compe frequencies and vibration mode shapes are cacuated. 5

53 PART III EMPLOYMENT OF THE FINITE ELEMENT METHOD Chapter 7 Some modeing considerations 7. Introduction An estabishment of appropriate finite eement mode for an actua practica probem depends to a arge degree on the foowing factors: understanding of the phsica probem incuding a quaitative knowedge of the structura response to be predicted, knowedge of the basic principes of mechanics and good understanding of the finite eement procedures avaiabe for anasis. Discretization of the domain into finite eements is the first step in the finite eement method. This is equivaent to repacing the domain having an infinite number of degrees of freedom b a sstem having finite number of degrees of freedom. The shape, size, number and configuration of eements have to be chosen carefu so that the origina bod or domain is simuated as cose as possibe without increasing the computationa effort needed for the soution. The various considerations taken in the discretization process are: tpe of eements size of eements ocation of nodes number of eements simpifications afforded b the phsica configuration of the bod finite representation of infinite bodies node numbering scheme automatic node generation After meshing of the bod it is necessar to add the materia properties, eterna oads, and app the boundar conditions. Before start of the probem, on parameters of the cacuation regime shoud be added to the input fie. 7. Tpe of eements Often the tpe of eements to be used wi be evident from the phsica probem itsef and geometr of the bod. Let s consider brief various tpes of finite eements, which are subject to certain static and kinematic assumptions. Truss and beam eements Truss and beam eements are ver wide used in structura engineering (Fig. 7., 7.) to mode for eampe buiding frames and bridges. 5

54 FE Across section A-A: σ is uniform, A a other stress components are zero. A P P Fig. 7. Uniaia stress condition: frame under concentrated oads. FE A M P σ Across section A-A:, σ τ, A Fig. 7. Beam under concentrated oad and moment. q FE structura -D soid triange rectange M q σ, σ, τ are uniform across the thickness. A other stress components are zero. P Fig. 7. Pane stress conditions: membrane and beam under in-pane actions. 54

55 z z u, v, w εz γ z γ z σ z Fig. 7.4 Pane strain conditions: ong dam subjected to water pressure and ong retaining wa subjected to soi pressure. Pane stress and pane strain eements Pane stress eements are empoed to mode membranes, the in-pane action of beams and pates and so on (Fig. 7.). In each of these cases a two-dimensiona stress situation eists in - pane with the stresses σ z, τ z, τ z equa to zero. Pane strain eements are used to represent a sice (of unit thickness) of a structure in which the strain components ε z, γ z, γ z are zero. This situation arises in the anasis of ong dam, retaining wa and so on (Fig. 7.4). Pate and she eements The basic proposition in pate and she anases is that the structure (Fig. 7.5) is thin in one dimension and therefore the foowing assumptions can be made: ) The stress through the thickness ( σ z ) of the pate/she is zero. ) Materia partices that are origina on a straight ine perpendicuar to the mid-surface of the pate/she remain on a straight ine during deformations. In the Kirchhoff theor, shear deformations are negected and the straight ine remains perpendicuar to the mid-surface during deformations. In the Mindin theor, shear deformations are incuded and therefore the ine origina norma to the mid-surface does in genera not remain perpendicuar to the mid-surface during the deformations. In certain probems, the given bod can not be represented as an assembage of on one tpe of eements. In such cases, we ma have to use two or more tpes of eements for ideaisation. Eampes of this woud be the anasis of an aircraft wing or the anasis of a car bod. 7. Size of eements Size of eements infuences the convergence of the soution direct and hence it has to be chosen with care. If the size of eements is sma, the fina soution is epected to be more accurate. However, we have to remember that the use of eements of smaer size wi aso mean more computationa time. Sometimes, we ma have to use eements of different sizes in the same bod. For eampe, in the case of stress anasis of a pate with a hoe, eements of different sizes have to be used. Size of eements has to be ver sma near the 55

56 z mid-surface FE h h z z mid-surface σ z A other stress components are nonzero. Fig. 7.5 Pate and she structures. hoe (where stress concentration is epected) compared to far awa paces. In genera, whenever steep gradients of the fied variabe are epected, we have to use a finer mesh in those regions. Another characteristic reated to the size of eements, which affects the finite eement soution, is the aspect ratio of eements. The aspect ratio describes the shape of eement in the assembage of eements. For two-dimensiona eements, the aspect ratio is taken as the ratio of the argest dimension of the eement to the smaest dimension. Eements with an aspect ratio of near unit genera ied best resuts. 7.4 Location of nodes If the bod has no abrupt changes in geometr, materia properties and eterna conditions (ike oad, temperature, etc.), the bod can be divided into equa subdivisions and hence the spacing of nodes can be uniform. On the other hand, if there are an discontinuities in the probem, nodes have to be introduced obvious at these discontinuities, as shown in Figure 7.5, where (a) and (b) - discontinuit in oading, (c) - discontinuit in geometr, (d) - discontinuit in materia properties, (e) - discontinuit in materia. 7.5 Number of eements The number of eements to be chosen for ideaisation is reated to the accurac desired, size of eements and number of degrees of freedom invoved (dimension of probem). An increase in the number of eements genera means more accurate resuts (Fig. 7.7). However, the use of arge number of eements invoves arge number of degrees of freedom and we ma not be abe to store the resuting matrices in the avaiabe computer memor. 56

57 q P auminium stee node noda ine a) Concentrated oad on a beam. d) Discontinuit in materia properties. q q nodes node q q b) Abrupt change in the distributed oad. e) A cracked pate under oading. node c) Abrupt change in the cross section of a beam. Fig. 7.6 Location of nodes at discontinuities. eact soution FEM no significant improvement beond N N number of eements Fig. 7.7 Convergence in resuts. 57

58 (v) (u) v v u Fig. 7.8 A pate with a hoe with smmetric geometr and oading. 7.6 Simpifications afforded b the phsica configuration of the bod If configuration of the bod, as we as the eterna conditions, is smmetric, we ma consider on haf or quarter of the bod for finite eement ideaisation (Fig. 7.8). The smmetr conditions, however, have to be incorporated in the soution procedure. 7.7 Finite representation of infinite bodies In some cases, ike in the case of anasis of foundations and semi-infinite bodies, the boundaries are not cear defined. Fortunate it is not necessar to ideaise the infinite bod. So, in the case of anasis of foundation, the effect of oading decreases gradua with increasing distance from the point of oading and we can consider on the continuum in which the oading is epected to have significant effect. In this case, the boundar conditions for this finite bod have to be incorporated in the soution. In the present eampe, the semi-infinite soi has been simuated considering on a finite portion of the soi (Fig. 7.9). In some appications, the determination of size of the finite domain ma pose a probem. In such cases, one can use infinite eements for the modeing. P u v or v Footing P Semi-infinite soi (v) u u (u) Fig. 7.9 A foundation under concentrated oad. 58

59 7.8 Node numbering scheme Since most of the matrices invoved in the finite eement anasis are smmetric and banded, the required computer storage can be considerab reduced storing on the eements invoved in the haf bandwidth instead of storing the whoe matri. The bandwidth of the assembage matri depends on the node numbering scheme and the number of degrees of freedom considered per node. If we can minimise the bandwidth, the storage requirements, as we as soution time, can aso be minimised. Since the number of degrees of freedom per node is genera fied for an given tpe of probem, the bandwidth can be minimised using a proper node numbering scheme. As an eampe, consider a three-ba frame with rigid joints, stores high (Fig. 7.). A shorter bandwidth (B) can be obtained numbering the nodes across the shortest dimension of the bod. 7.9 Automatic mesh generation For arge sstems, the procedure of node numbering becomes near impossibe. Hence automatic mesh generation agorithms capabe of discretizing an geometr into an efficient finite eement mesh without user intervention are appied. Most of the automatic bandwidth renumbering schemes permits an arbitrar numbering scheme initia. Then the nodes are renumbered through an agorithm to reduce the bandwidth of the sstem equations. After the sstem equations are soved, the node numbers are often converted back into the origina ones B5 B6 a) aong the shorter dimension b) aong the onger dimension Fig. 7. Node numbering scheme. 59

60 Chapter 8 Finite eement program packages 8. Introduction The genera appicabiit of the finite eement method makes it a powerfu and universa too for a wide range of probems. Hence a number of computer program packages have been deveoped for the soution of a variet of structura and soid mechanics probems. Among more wide used packages are ANSYS, NASTRAN, ADINA, LS-DYNA, MARC, SAP, COSMOS, ABAQUS, NISA. Each finite eement program package consists from three parts: programs for preparation and contro of the initia data, programs for soution of the finite eement probem, programs for processing of the resuts. Let s consider now some genera features of a more wide appied finite eement program - ANSYS. The ANSYS program is a computer program for the finite eement anasis and design. The ANSYS program is a genera-purpose program, meaning that ou can use it for amost an tpe of finite eement anasis in virtua and industr - automobies, aerospace, raiwas, machiner, eectronics, sporting goods, power generation, power transmission and biomechanics, to mention just a few. Genera purpose aso refers to the fact that the program can be used in a discipines of engineering - structura, mechanica, eectrica, eectromagnetic, eectronic, therma, fuid and biomedica. The ANSYS program is aso used as an educationa too at universities. ANSYS software is avaiabe on man tpes of computers incuding PC and workstations. Severa operating sstems are supported. The procedure for a tpica ANSYS anasis can be divided into three distinct steps: buid the mode, app oads and obtain the soution, review the resuts. 8. Buid the mode Buid the mode is probab the most time-consuming portion of the anasis. In this step, ou specif the job-name and anasis tite and then use pre-processor (PREP 7) to define the eement tpes, eement rea constants, materia properties, and the mode geometr. The ANSYS eement ibrar contains over 8 different eement tpes. Each eement tpe is identified b unique number and prefi that identifies the eement categor: 6

61 BEAM4, SOLID96, PIPE6, etc. The foowing categories are avaiabe: BEAM, COMBINation, CONTACT, FLUID, HYPEReastic, INFINite, LINK, MASS, MATRIX, PIPE, PLANE, SHELL, SOLID, SOURCe, SURFace, USER, and VISCOeastic (or viscopastic). The eement tpe determines, among other things, the degree-of-freedom set (which impies the discipine - structura, therma, magnetic, eectric, fuid, or couped-fied), the characteristic shape of the eement (ine, quadriatera, brick, etc.), and whether the eement ies in -D space or -D space. BEAM4, for eampe, has 6 structura degrees-of-freedom (UX, UY, UZ, ROTX, ROTY, ROTZ), is a ine eement and can be modeed in -D space. Eement rea constants are properties that are specific to a given eement tpe, such as cross-sectiona properties of a beam eement. For eampe, rea constants for BEAM, the -D beam eement, are area (AREA), moment of inertia (IZZ), height (HEIGHT), shear defection constant (SHEARZ), initia strain (ISTRN), and added mass per unit ength (ADDMAS). Materia properties are required for most eement tpes. Depending on the appication, materia properties ma be inear, noninear, and/or anisotropic. The main objective of the step Creating the mode geometr is to generate a finite eement mode - nodes and eements - that adequate describes the mode geometr. There are two methods to create the finite eement mode: soid modeing and direct generation. With soid modeing, ou describe the geometric boundaries of our mode and then instruct the ANSYS program to automatica mesh the geometr with nodes and eements. You can contro the size and shape of the eements that the program creates. With direct generation, ou manua define the ocation of each node and the connectivit of each eement. Severa convenience operations, such as coping patterns of eisting nodes and eements, smmetr refection, etc. are avaiabe. 8. App oads and obtain the soution In this step, ou use SOLUTION menu to define the anasis tpe and anasis options, app oads, specif oad step options, and initiate the finite eement soution. The anasis tpe is chosen based on the oading conditions and the response ou wish to cacuate. For eampe, if natura frequencies and mode shapes to be cacuated, ou woud choose a moda anasis. The foowing anasis tpes are avaiabe in the ANSYS program: static, transient, harmonic, moda, spectrum, bucking, and substructuring. Not a anasis tpes are vaid for a discipines. Moda anasis, for eampe, is not vaid for a therma mode. Anasis options aow ou to customise the anasis tpe. The word oads as used in the ANSYS program incudes boundar conditions as we as other eterna and interna appied oads. Loads in the ANSYS program are divided into si categories: DOF constraints, forces, surface oads, bod oads, inertia oads, couped-fied oads. Most of these oads can be appied either on the soid mode (kepoints, ines and areas) or the finite eement mode (nodes and eements). 6

62 Load step options are options that can be changed from oad step to oad step, such as number of substeps, time at the end of a oad step, and output contros. A oad step is simp a configuration of oads for which ou obtain a soution. In a structura anasis, for eampe, ou ma app wind oads in one oad step and gravit in a second oad step. Load steps are aso usefu in dividing a transient oad histor curve into severa segments. Substeps are incrementa steps taken within a oad step. The are main used for accurac and convergence purposes in transient and noninear anases. Substeps are aso known as time steps - steps taken over a period of time. After SOLVE command, the ANSYS program takes mode and oading information from the database and cacuates the resuts. Resuts are written to the resuts fie (Jobname.RST, Jobname.RTH, or Jobname.RMG) and aso to the database. The difference is that on one set of resuts can reside in the database at one time, whereas a sets of resuts (for a substeps) can be written to the resuts fie. 8.4 Review the resuts Once the soution has been cacuated, ou can use the ANSYS postprocessors to review the resuts. Two postprocessors are avaiabe: POST and POST 6. POST, the genera postprocessor, is used to review resuts at one substep (time step) over the entire mode. You can obtain contour dispas, deformed shapes, and tabuar istings to review and interpret the resuts of the anasis. Man other capabiities are avaiabe in POST, incuding error estimation, oad case combinations, cacuations among resuts data, and path operations. POST 6, the time histor postprocessor, is used to review resuts at specific points in the mode over a time steps. You can obtain graph pots of resuts data versus time (or frequenc) and tabuar istings. Other POST 6 capabiities incude arithmetic cacuations, and compe agebra. 6

63 Literature. Bathe K.-J. and Wison E. L. Numerica Methods in Finite Eement Anasis. Prentice-Ha, Inc., Sigerind L. J. Appied Finite Eement Anasis. John Wie and Sons, Inc., Варвак П. М., Бузун И. М., Городецкий А. С., Пискунов В. Г., Толокнов Ю. Н. Метод конечных элементов. Головное издательство издательского объединения «Вища школа»: Киев, Rao S. S. The Finite Eement Method in Engineering. Pergamon Press, Ritz W. Über eine Neue Metode zur Lösung gewisser Variationsprobeme der Matematischen Phsik // J. Reine Angew. Math., 99, Vo. 5, P Courant R. Variationa methods for the soution of probems of equiibrium and vibrations // Buetin of the American Mathematica Societ, 94, Vo. 49, P Cough R. W. The finite eement method in pane stress anasis. // Proc. American Societ of Civi Engineers ( nd Conference on Eectronic Computation, Pitsburg, Pennsvania), 96, Vo., P Argris J. H. Energ theorems and structura anasis // Aircraft Engineering, 954, Vo. 6, Part (Oct. Nov.), 955, Vo. 7, Part (Feb. Ma). 9. Turner M. J., Cough R. W., Martin H. C. and Topp L. J. Stiffness and defection anasis of compe structures // Journa of Aeronautica Science, 956, Vo., No. 9, P Hrennikov A. Soution of probems in easticit b the frame work method // Journa of Appied Mechanics, 94, Vo. 8, P Zienkiewicz O. C. and Cheung Y. K. The Finite Eement Method in Structura and Continuum Mechanics. McGraw-Hi: London,

64 APPENDIX A tpica ANSYS static anasis The goa of this eampe is to mode the probem as shown in Figure A. This is a D pate mode where ANSYS genera she eements wi be used to predict the dispacement and stress behaviour of the pate subjected to concentrated oads from one side. The structure is camped at the eft end so that no transations or rotations are aowed there.,5,) (,5,) ( α) (,5,4) (,5,) in in in (,5,) (,5,) (,,) in 45 (,,4) F in F F (,,) F h. in F.5 bs (,,4) 4 in X(R) Fig. A D pate mode. Materia properties: a) Isotropic. b) Young`s moduus, Ee6 psi. c) Poisson`s ratio,.. 64

65 GEOMETRIC MODELLING STEP : To buid geometr of the mode more easi, we change the coordinate sstem - from the defaut decart to cindrica. ANSYS Utiit Menu WorkPane > Change Active CS to > Goba Cindrica STEP : Since the geometr of the mode is D, we change the dispa from top view to isonometric view. ANSYS Utiit Menu Pot Ctrs > Pan,Zoom,Rotate > Iso STEP : Create ten kepoints aocated in corners of the pate b opening the window, defining the kepoint number and coordinate vaues, where X coordinate is radius, Y coordinate is ange, Z coordinate is height. ANSYS Main Menu Preprocessor > -Modeing-Create > Kepoints > In Active CS Kepoint number X,Y,Z Location in active CS App Kepoint number X,Y,Z Location in active CS 4 App etc. OK STEP 4: Create thirteen ines b joining two kepoints. You have to specif start and end of the ine b picking with mouse to kepoints. Kepoints must be joined as shown in Figure A. ANSYS Main Menu Preprocessor > -Modeing-Create > Lines > Straight Line STEP 5: Mode four new areas b picking with mouse corresponding four ines as shown in Figure A. ANSYS Main Menu Preprocessor > -Modeing-Create > -Areas-Arbitrar > B Lines 65

66 9 L 6 5 L6 L7 7 A L A4 L 8 L9 L A L5 L8 L L A L4 4 L Fig. A Kepoints, ines and areas of the geometric mode. DESCRIPTION OF FINITE ELEMENTS STEP 6: For the pate mode we choose four-node she eement. Anss Main Menu Preprocessor > Eement Tpe > Add/Edit/Deete > Add > Structura She Eastic 4 node 6 Ok Cose STEP 7: Pate thickness must be aso definite. Anss Main Menu Preprocessor > Rea Constants... > Add... > OK She thickness at node I, J, K, L. Ok Cose 66

67 MATERIAL MODELLING STEP 8: Add materia data for the isotropic materia: Young`s moduus Ee6 psi and Poisson`s ratio.. Anss Main Menu Preprocessor > Materia Props > Constant-Isotropic... OK Young s moduus Poisson s ratio (major) OK EXe6 PRXY. FINITE ELEMENTS MESHING STEP 9: For smooth mesh we choose a ines divided into three eements ecuding the division of two ong sides divided into eight finite eements. After choosing size of the mesh, we aow computer to perform mesh of a marked areas. The finite eement mesh is presented in Figure A. Anss Main Menu Preprocessor > -Meshing-Size Cntrs > Picked Lines No. of eement divisions No. of eement divisions 8 OK Anss Main Menu Preprocessor > -Meshing-Mesh > -Areas-Free OK APPLICATION OF LOADS AND BOUNDARY CONDITIONS STEP : Mark the right-side edge nodes and input the vaue of appied force -.5 bs. The appied oad is shown in Figure A. Anss Main Menu Soution > -Loads-App > -Structura-Force/Moment > On Nodes Direction of force/mom FY App As Constant vaue Vaue -.5 OK 67

68 STEP : App boundar conditions to the eft-side edge nodes defining no deformation and rotation for a nodes. The boundar conditions are shown in Figure A. Anss Main Menu Soution > -Loads-App > -Structura-Dispacement > On Nodes DOF s to be constrained OK A DOF SOLUTION STEP : Anasis tpe is static. Now soution can be started. When the window Soution is done appears, soution is competed. Anss Main Menu Soution > Anasis tpe-new Anasis... > STATIC Anss Main Menu Soution > -Sove-Current LS OK OK Fig. A Finite eement mesh, appied oad and boundar conditions. 68

69 ANALYSIS OF RESULTS STEP : After static anasis ou can pot a deformation shape of the pate (Fig. A4). Anss Main Menu Genera Postproc > Pot Resuts > Deformed Shape... Def undeformed STEP 4: Besides deformation, stress state of the pate (Fig. A5) can be cacuated b von Mises theor. Anss Main Menu Genera Postproc > Pot Resuts > -Contour Pot-Noda Sou... Item to be contoured Stress von Mises SEQV OK Fig. A4 Deformation shape of the pate. 69

70 Fig. A5 Stress state of the pate. 7