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1 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS A - INTRODUCTION AND OVERVIEW INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
2 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Course Content: A INTRODUCTION AND OVERVIEW Numericl method nd Computer-Aided Engineering; Physicl problems; Mthemticl models; Finite element method;. B REVIEW OF -D FORMULATIONS Elements nd nodes, nturl coordintes, interpoltion function, br elements, constitutive equtions, stiffness mtri, boundry conditions, pplied lods, theory of minimum potentil energy; Plne truss elements; Emples. C PLANE ELASTICITY PROBLEM FORMULATIONS Constnt-strin tringulr (CST) elements; Plne stress, plne strin; Aisymmetric elements; Stress clcultions; Progrmming structure; Numericl emples. INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
3 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS COMPUTER-AIDED ENGINEERING (CAE) The use of computers to nlyze nd simulte the function (structurl, motion, etc.) of mechnicl, electronic or electromechnicl systems. Computer-Aided Design (CAD) - Drfting - Solid modeling - Animtion & visuliztion - Dimensioning & tolerncing Engineering Anlyses - Anlyticl & numericl methods INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
4 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS The Process of FE nlysis Ref: K.J. Bthe, Finite Element Procedures, Prentice-Hll, 996 INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 4
5 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS y EI P d y d Bucling of Euler Column PHYSICAL PROBLEM AND MATHEMATICAL MODEL INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 5 m m ω Eigenvlue problem
6 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS EXAMPLE OF A MODEL Stedy-stte het conduction through thic wll INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 6 ) ( : T Solution Q d dt d d ht R R R T T T h L L LL L L L L M M L M L L
7 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS REVIEW OF MATRIX ALGEBRA INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 7
8 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Mtri Algebr In this course, we need to solve system of liner equtions in the form n n n n nn n n n b b b n (-) where,,, n re the unnowns. Eqn. (-) cn be written in mtri form s [ ]{ } { b} A (-) where [A] is (n n) squre mtri, {} nd {b} re (n ) vectors. INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 8
9 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS [ ] { } { } n n b b b : b nd, :,... : : : : A The squre mtri [A] nd the {} nd {b} vectors re is given by, INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM (-) Note: Element locted t i th row nd j th column of mtri [A] is denoted by ij. For emple, element t the nd row nd nd column is. n n nn n n b... 9
10 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Mtri Multipliction The product of mtri [A] of size (m n) nd mtri [B] of size (n p) will results in mtri [C], with size (m p). [ A ] [ B] [ C] ( m n) ( n p) ( m p) (-4) Note: The (ij) th component of [C], i.e. c ij, is obtined by ting the DOT product, c ij ( ith row of [ A]) ( jth column of [ B]) (-5) Emple: ( ) 4 5 ( ) 7 - ( ) 5 7 INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
11 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Mtri Trnsposition If mtri [A] [ ij ], then trnspose of [A], denoted by [A] T, is given by [A] T [ ji ]. Thus, the rows of [A] becomes the columns of [A] T. Emple: [A] Then, [ A] T Note: In generl, if [A] is of dimension (m n), then [A] T hs the dimension of (nm). INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
12 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Trnspose of Product The trnsposeof product of mtrices is given by the product of the trnsposes of ech mtrices, in reverse order, i.e. T T T T ([ A ][ B][ C]) [ C] [ B] [ A] (-6) Determinnt of Mtri Consider squre mtri [], [ ] The determinntof this mtri is give by, det [ ] (-7) INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
13 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS EXAMPLE Given tht: } { 4 ] [ ] [ 4 ] [ E D C A INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM Find the product for the following cses: ) [A][C] b) [D]{E} c) [C] T [A]
14 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Solution of System of Liner Equtions System of liner lgebric equtions cn be solved for the unnown using the following methods: ) Crmer s Rule b) Inversion of Coefficient Mtri c) Gussin Elimintion d) Guss-Seidel Itertion Emple: Solve the following SLEs using Guss elimintion method (i) (ii) (iii) INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 4
15 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Guss Elimintion Method Reducing set of n equtions in n unnowns to n equivlent tringulr form (forwrd elimintion). The solution is determined by bc substitution process. Bsic pproch -Any eqution cn be multiplied (or divided) by nonzero sclr -Any eqution cn be dded to (or subtrcted from) nother eqution -The position of ny two equtions in the set cn be interchnged INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 5
16 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Emple: Solve the following SLEs using Gussin elimintion. (iii) 6 (ii) 8 4 (i) INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM Eliminte from eq.(ii) nd eq.(iii). Multiply eq.(ii) by.5 we get, (iii) 6 (ii)* 4.5 (i) 6
17 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Subtrct eq.(ii)* from eq.(i), we obtin Add eq.(iii) with eq.(i), yields (iii) 6 (ii)** (i) INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM Add eq.(iii) with eq.(i), yields (iii)* 5 4 (ii)** (i) 7
18 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Eliminte from eq.(iii)*. Multiply eq.(ii)** by nd eq.(iii)* by we get (ii)*** (iii)** Subtrct eq.(iii)** from eq.(ii)***, we obtin (i) (i) (ii)** (iii)*** From eq.(iii)*** we determine the vlue of, i.e. 5.5 INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 8
19 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Bc substitute vlue of into eq.(ii)** nd solve for, we get 7 4.5( ) Bc substitute vlue of nd into eq.(i) nd solve for, we get 6 INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 9
20 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Emple Solve the following systems of liner equtions by using the Gussin elimintion method. 4 4 ) INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM b)
21 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Steps in solving continuum problem by FEM Identify nd understnd the problem (This essentil step is not FEM) Select the solution domin Select the solution region for nlysis. Discretize the continuum Divide the solution region into finite number of elements, connected to ech other t specified points / nodes. Select interpoltion functions Choose the type of interpoltion function to represent the vrition of the field vribles over the element. Derive element chrcteristic mtrices nd vectors Employ direct, vritionl, weighted residul or energy blnce pproch. [] (e) {φ} (e) {f} (e) INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
22 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Steps (Continued) Assemble the element chrcteristic mtrices nd vectors Combine the element mtri equtions nd form the mtri equtions epressing the behvior of the entire solution region / system. Modify the system equtions to ccount for the boundry conditions of the problem. Solve the system equtions Solve the set of simultneous equtions to obtin the unnown nodl vlues of the field vribles. Me dditionl computtions, if desired Use the resulting nodl vlues to clculte other importnt prmeters. INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
23 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Wht is the problem? P Solution region of interest Stress concentrtion long shft Other emples: -Scrtches on tensile surfce -Oil groove on shft -Threded connections -Rivet holes under tension Ts: To simulte stress nd strin fields in the vicinity of shrp notch under tensile lod INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM
24 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS FE procedures: FE softwre user steps: Select the solution domin Discretize the continuum Choose inerpoltion functions Derive element chrcteristic mtrices nd vectors Assemble element chrcteristic mtrices nd vectors Solve the system equtions Me dditionl computtions, if desired Drw the model geometry Mesh the model geometry Select element type (The FEA softwre ws written to do this) Input mteril properties (The computer will ssemble it) Input specified lod nd boundry conditions (The compiler will solve it) Request for output Post-process the result file INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 4
25 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS EXAMPLE: Stresses in C(T) specimen Drw the model geometry Mesh the model geometry Select element type [-node Hehedrl elements] Input mteril properties [Elstic modulus, Poisson s rtio] Input specified lod nd BC [Pin loding, displcement rte, zero displcement t lower pin] Request for output [Displcement, strin nd stress components] Post-process the result file INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 5
26 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS EXAMPLE: Stresses in C(T) specimen Chec for obvious mistes/errors Etrct useful informtion Eplin the physics/ mechnics Vlidte the results INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 6
27 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS MODELING CAPABILITIES Microelectronic device relibility - Prediction of sptil distribution of physicl prmeters Het sin Solder ms Silicon Die Substrte Solder joints T 8 5 Motherbord PCB - Prediction of dmge evolution chrcteristics εin..8.6 TD 5-4 Re-flow Time. 4 5 N INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 7.4. TC MT
28 MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS MODELING CAPABILITIES Deflection of composite lmintes bem w w EIy ( 6 L) 4 lyers [/45/9/-45/45/-45/ ] s Lyer thicness.57 mm Al Comp-º Comp-9º E E E 7 GP GP.46 GP INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM 8
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