I. Backgrounds and preliminaries

Transcription

1 I. Bckgrounds nd preliminries 1.1 Vector 1.2 Mtri nd Liner lger 1.3 Function nd Differentition 1.4 Bsis of mechnics nd Sttics 1.5 Unit nd Dimension 1.6 Bem heor -1-

2 1.1 Vector -2-

3 Definition of vector Vector quntit: Vector quntit is defined s the quntit which hs direction s well s mgnitude. Vector is mthemticl description of vector quntit. Sclr nd rel numer: Sclr quntit hs onl mgnitude of rel numer. Description of vector Vector: Mgnitude: r r uuur,,,, AB r,, Emples of vector: Force, Displcement, Velocit, Accelertion, Het flu, Moment, Momentum, etc. Emples of sclr: Work, Power, emperture, Mss, etc. z Fctors of vector Essentil fctors (Mthemticl requirements) Mgnitude: Distnce etween points A nd B A uuur AB B Direction: Direction of rrow directing from point A to point B Selective fctors Point of ction: Point B Line of ction: Line pssing points A nd B <Definition of vector>

4 Kind of vectors nd mechnics Sliding vector: Essentil fctors of vector + Line of ction Bound vector: Essentil fctors of vector + Point of ction Free vector: Onl essentil fctors of vector Sttics Elsticit igid-od trnsltion F B P u u A Unknowns : A, B P Unknowns : deformtion, etc. Unknowns : displcement, etc. () sliding vector () ound vector (c) free vector <Kind of vectors>

5 Mthemticl description of vector Mthemticl description: Component On 2D plne ow vector:, or,, 2 1 cos Column vector:, 0, 1, 1 () 2D In 3D spce ow vector: Column vector:,, z,, z z Components cn mke us clculte direction s well s mgnitude of vector. Bsicll, vector in mechnics mens row vector,, z,, z 2 3, cos 2 () 3D 1, <Components in the rectngulr coordinte sstem> i z i 2

6 r Zero vector: 0 0= 0, 0, 0 Definition of terminolog Mgnitude of vector : or z Direction of vector : Directionl cosine : Unit vector: A vector with mgnitude of unit u 1 or u cos, cos, cos Unit sis vector : r i i= 1, 0, 0 e 1 r j j = 0, 1, 0 e 2 r k k= 0, 0, 1 e 3 i 1 i cos ( 1, 2, 3) ( 0) i, r i, e1 1 z, 3 r k, e3 r j, e2 <--z coordinte sstem nd unit sis vector>, 2

7 Alger of vector vector ddition Addition of vectors [,, ] Multipliction of sclr nd vector [,, ] Chrcteristics of vector ddition nd sclr-vector multipliction ( ) c ( c) 0+= ( 0= ) +( - )=0 ( +)= + ( + ) = + ( )=( ) 1 = i r z k i r o z j z k r j i+ j 2 r r r r r r r r r <Vector ddition, sclr-vector multipliction> =,, z Prllelogrm rule =,, z + = ( ) i ( ) j ( ) k z z <Appliction of vector ddition, sclr-vector multipliction>

8 Alger of vector Inner product Inner product, dot product, sclr product 3 Geometric mening of inner product , 2, 3 2 = i i = i1 3 cos 0 mens the two vectors re perpendiculr. Chrcteristics of inner product ( c) c ( ) ( ) 0 0implies = 0 Norm of vector : ( ) <Inner product> Miscellneous ij = jk = ki = 0 ii=jj=k k= 1

9 Alger of vector Vector product Vector product, cross product i j k c = = ( 2332) i ( 3113) j ( 1221) k c= Geometric mening of vector product Mgnitude: c=sin (0 ) - Are of prllelogrm constructed the two vectors Direction: Perpendiculr to the plne constructed the two vectors, following the right-hnd rule shown in the figure Chrcteristics ( c) c ( ) c( c) ( ) c ( c) ( ) c c 0 // c <ight-hnd rule> sin <Definition of vector product> Miscellneous i jk, jki, kij ii = jj= kk = 0

10 Eucliden spce nd liner comintion k-dimensionl Eucliden spce k Dimension of vector = Numer of components k : k-dimensionl Eucliden, or k-dimensionl rel numer vector spce Liner comintion n c c c Lc i i n n i1 Linerl independent L k l L k i k,,, ; 's re rel,, L, ,, L, k 1 2 l l 1 2 ( ) he cse tht the liner comintion vnishes onl when ll c i 's re zero. Linerl dependent he cse tht the liner comintion vnishes when n is not zero. c i M n k k k

11 Coordinte sstem (C.S) nd coordintes Coordintes Position (components) of point reltive to.c.s. A vector quntit z, 3 (,, ) i i i eference coordinte sstem(.c.s.) nd locl C.S. Orthogonl coordinte sstem ectngulr coordinte sstem Clindricl coordinte sstem Sphericl coordinte sstem z z θ r r cos rsin z z ).C.S. ) C.C.S. c) S.C.S. z <picl orthogonl coordinte sstems> z, θ 1 r i i <Coordinte sstems nd coordintes> rsin cos rsin sin z rcos locl coordinte sstem i reference coordinte sstem, 2

12 1.2 Mtri nd Liner lger -12-

13 Definition of mtri nd terminolog Mtri: ectngulr rr of numers, clled elements m nmtri : erminologies : ow vector : n A [ ] n ij m 1 m2 mn 1 n mti Column vector : m 1 mtri Squre mtri : n n mtri Upper tringulr mtri U n 22 2n nn Lower tringulr mtri L n2 n2 nn Off-digonl term : n n Digonl term : ( i1,2, L, n) for n squre mtri ii ( i j) ij Zero mtri Digonl mtri Unit (identit) mtri D nn 100 I [ ] 010 ij 001 Kronecker delt -13-

14 Definition of terminologies continued Sumtri: A mtri mde deleting some rows or columns from the originl mtri Principl sumtri: A sumtri mde deleting simultneousl some i-th row(s) nd the sme i-th column(s) of squre mtri rnspose of mtri component of mtri A, A, A : A mtri of which (i,j) component is equl to the (j,i) ij ji Smmetric mtri: A A, ij ji Skew-smmetric mtri: A A, ij ji nk: Numer of independent rows = numer of independent columns n Singulr mtri: squre mtri of which rnk is less thn n

15 Mtri nd Vector Epression of mⅹn mtri using vectors n n A ij 2 1, 2, L M M m m1 m2 mn n i M i1 M 1 j i2 2 j j in mj

16 Addition of mtries nd multipliction of rel numer nd mtri Definition of ddition of mtries C A B cij ij ij Multipliction of rel numer nd mtri C A cij ij Properties A+BB+A A+ ( B +C) ( AB ) +C A+0=A A+( A) 0 ( A + B) A B ( ) A A A ( A) ( ) A 1A A ( A B) A B ( A) A

17 Product of mtries Definition of product of mtries A nd B C = A B : Product of m p mtri A [ ] nd q n mtri B [ ] g g p q l c is the essentil requirement for l ij ikkj i j i j k1 Properties of mtri product ( A)B ( AB) A( B) A(BC) ( AB)C ( A B)C AC BC A(B C) AB AC ( AB) B A to e defined. In generl, AB B A nd AB 0 does not lws men A 0 or B 0 IA A, AI A, I Emple: Show (AB) =B A using the following two mtries cij ij 1 j M M M M M L2 j L 1 8 gives (AB) AB 0 7, A2 5 3, B BA herefore (AB) B A j ij j L L i cij i i1 i 2 i3 il KK K K L 3j L M M M M M M L ljl m n mtri ml mtri l n mtri

18 ole of mtri ole of mtri : A = Mthemticl opertor rnsfer function or trnsformtion rnsformtion A Mpping Lw of coordinte trnsformtion of vector : <ole of mtri> F cos sinf cos sin sin cos ' F' sin cos F F cos F F ' ' F cos F F ' ' sin sin F cos sinf' F sin cos F ' rnsformtion mtri =[t1, t ] 2 t cos, sin t sin,cos 2 1 t t 1, t t Orthonorml mtri 1 i cos sin i j sin cos j <Coordinte trnsformtion> -18-

19 Appliction of the role of mtri Mtri in mechnics Displcement-lod reltion L 1 1 Q u AE P u AF=U L L AE Qu, AE u Q 2 2L 1 1 u P KU=F AE Pu, Displcement-lod reltion z f 0 z z f 0 z z z zz f z 0 z Approimte pproch to differentil eqution + Finite element pproimtion KU F K M U Stiffness mtri Displcement vecor Lod vector 2 =, ( ) =0-19-

20 Determinnt of 2 2 mtri: D A det Determinnt of 3 3 mtri : Determinnt of mtri D det A c c c c c1 det A c2 Determinnt of n n n mtri : n n n det A n n D 3 11 n 12 L n1 n2 nn n2 n3 nn n1 n3 nn n n i j ji ji ij ij ij ij ij i1 i1 D C C ( j 1, 2,, n), C ( 1) M ( C : cofctor) M M C C L L D( 1) n n i j i j ( 1) j im ji ijmij i1 i1 M ij : Minor, Determinnt of the (n-1) (n-1) sumtri, i-row nd j-column were removed. -20-

21 Chrcteristics Chrcteristics of determinnt of mtri 1 2 A A AB = BA = A B 3 If row or column of mtri is multiplied c, determinnt of the newl formed mtri is c-multiple of the originl vlue. 4 When two rows (or columns) re chnged, the resulting determinnt ecomes negtive of the originl vlue. 5 When row (or column) is dded n other row (or column) multiplied constnt, the resulting determinnt does not chnge. 6 When row vectors (or column vectors) of mtri re linerl dependent, its determinnt vnishes. Applictions 1 D det A D det A c21 c22 c23 cdet A D det A det A

22 Inverse mtri Inverse mtri of n n mtri A: A -1 Inverse of mtri A multiplied B ( AB) B A -1-1 AA I or A A I ( ABCD) D C B A * A [ Cij ] [ Aij ] det A det A Liner eqution * [ Aij ] [ Cij ] : [ A ] ij C C C C C C n1 * C12 C22 Cn2 Adjoint of mtri A 1n 2n nn Orthonoml mtri nd trnsformtion mtri A A A, I A, A -1 If AA D, i.e., digonl mtri, the mtri A is orthogonl mtri. If AA I, i j, the mtri A ij is orthonoml mtri, i.e., A -1 A. rnsformtion mtri is orthonoml, i.e., I. herefore, -1. A 1 Kronecker delt if i j ij 0 if i j 1 0 L 0 01L 0 I ij MMLM 00L1-22-

23 Similrit trnsformtion Wht is similrit trnsformtion? %A A Chrcteristics of the similrit trnsformtion Eigenvlues of A % nd A re identicl. 1 eltionship of eigenvectors : % = ( : eigenvector of mtri A, % : eigenvector of mtri A % ) Appliction of trnsformtion mtri i' j' i' p j' q pq ip ' pq jq ' ip ' ' ip ' pq jq ' %

24 Eigenvlue prolem Homogeneous liner eqution: = 0 A = 0 A I A = 0 : rivil solution, meningless solution IF, 0ut A0. Eigenvlue prolem : A = or ( A I ) = 0 ows or columns of the n n mtri ( A I ) should e linerl dependent. : Eigenvlue or chrcteristic vlue : Eigenvector or chrcteristic vector Chrcteristic eqution: A I 0 equirement tht rnk of is less thn n, or tht is singulr. Non-liner eqution of order n. If the mtri is smmetric, the n rel-vlue solutions eist, i.e.,,, L,. A Orthogonlit of eigenvectors : A, A (i) (i) ( i ) ( j ) ( j ) ( j ) ( A A ) ( ) () j () i () i ( j) () i ( j) () i ( j) = 0 () ( ) () ( ) () ( ) If A A 0, ( i j ) i j 0 i j 0 A I j j i A 1 2 i i 0 A, A () i () i () i ( j ) ( j ) () i ( j ) n -24-

25 Complements of eigenvlue prolem Emple of homogeneous liner eqution Prolem: 2 0 c c Solution: c Chrcteristic euqtion 0 c( 2) 0 c 2 : 2:1 Emple of eigenvlue prolem Prolem: z n 0 z n 0 z z zz n z 0 n n z n z n n z z zz z z n Solution: z 3 2 z I1 I2 I 3 0 1, 2, 3 z z zz Chrcteristic euqtion -25-

26 1.3 Function nd Differentition -26-

27 Nture nd function -27-

28 Quntifiction of function in 2D Height of mountin Atmospheric pressure h h(, ) p p(, ) -28-

29 Mechnics nd unknown functions 1-Dimensionl 2-Dimensionl 3-Dimensionl

30 Ordinr differentition nd slope Bem deflection Averge rte of chnge v ( ) A..C. = v ( ) v ( ) Differentition v ( ) v ( ) dv 0 v( ) lim d v ( ) in em theor EIv( ) M ( ) M ( ) V( ) V( ) ( EIv( )) Instntneous rte of chnge v ( ) v ( ) v( ) lim v ( ) v ( ) lim 0

31 Ordinr differentil eqution Bem deflection M ( ) V( ) dv ( ) q ( ) d M ( ) ( v ) EI EIv( ) M ( ) ( EIv( )) V ( ) Boundr conditions: Essentil BC: Nturl BC: v(0) 0, vl ( ) 0, v(0) 0, v( L) 0 V(0) P, M (0) M, V( L) P, L ( EIv( )) q( ) M () EIv () V () M() ( EIv()) v(0) 0 (0) 0 M v(0) 0 v(0) 0 V( L) W M ( L) 0 vl ( ) 0 M ( L) 0 v(0) 0 v(0) 0 V( L) 0 M ( L) 0

32 Prtil differentition nd grdient Height of mountin h h(, ) h h(30,10) h(20,10) h h(20,15) h(20,10) h h(, ) h(, ) h lim lim 0 0 fied (20,15) Y O (20,10) H (30,10) Stress in 2D (, ) (, ) (, ) lim lim 0 0 fied (, ) (, ) (, ) (, ) -32-

33 Prtil difference equtions Continuit eqution of incompressile mteril v (, ) v v 0 Eq n of conduction under sted-stte condition q q 0 k k 0 q k, q k Eq n of equilirium 0, Miscellneous u lim 0 fied u 0 Infinitesiml re v (, ) Infinitesiml re v (, ) v (, ) q (, ) Infinitesiml re q (, ) q (, ) (, ) (, ) q (, ) q q(, ) q(, ) q q q -33-

34 1.4 Bsis of mechnics nd Sttics -34-

35 Bsis of mechnics he mjor fctors Force Displcement Mteril Mteril nd Continuum, Deformtion of mteril Solid <Grin> <Grin nd oundr> <Continuum> igid-od : No chnge in distnce etween prticles occurs under eternl force. Elstic deformtion: Deformtion due to eternl force disppering when it removes. Plstic deformtion: Deformtion due to eternl force remining when it removes. Fluid including gs -35-

36 Some detils of the 3 mjor fctors Force Force Internl Eternl Action nd rection force rction Eerted lod, ection force Bod force Grvit, Mgnetic force Displcement or motion igid-od motion Deformtion Mteril: Set of prticles continuousl distriuted Continuum Solid igid-od: No reltionship etween force nd displcement is needed. Elsticll deformle od: Liner elstic(hooke s lw) nd Non-elstic, Isotropic(Common mteril) nd Anisotropic (Composite mteril) Plstic od: Yield criterion, Isotropic hrdening, Kinemtic hrdening Fluid including gs -36-

37 Newton s lw of motion Newton s lw 2 nd lw 3 rd lw Sum of ll the forces eerting on prticle ( f ) is equl to the ccelertion ( ) multiplied the mss ( m ), i.e., f = m. Action nd rection lw: wo internl forces eerting etween two prticles hve the sme mgnitude nd line of ction nd the opposite direction. i ij F 0 j i, j 1,2,, ij ( F i ij ) 0 F i ij 0 i j i i j ij ij 0 i j 20 (O) 30 (X) 50 A mteril is set of infinite numer of prticles i F : Sum of eternl force eerting on prticle i ij : Internl force eering on prticle j from prticle i All forces re ounding vectors. F A r r r o r F F. F F 5 F F equirement on equilirium i j r i F 0 i ij i ij ri F 0 ri F ri 0 i i j i i j i r F 0 i i ij 0 F i 0 or F 0 i i r F 0 i he ove requirement of equilirium should stisf for ll susstem s well s the whole sstem. Leding to differentil equtions, for emple, equtions of equilirium. or M A 0-37-

38 Derivtion of requirement on equilirium F A r F r F using si-prticle od F F F F = F = F = F = F = F =0 i ij ji F =0 Q = r1f +r1 +r1 +r1 +r1 +r1 = r2f +r2 +r2 +r2 +r2 +r2 = r3f +r3 +r3 +r3 +r3 +r3 = r4f +r4 +r4 +r4 +r4 +r4 = r5f +r5 +r5 +r5 +r5 +r5 = r F +r +r +r +r +r = i ij ij ri F =0 Qri =-rj Actul numer of prticles is infinite. -38-

39 Susstem he requirement of equilirium should stisf for n susstem. his sttement is the sme with the following sttement: he requrement of equilirium should stisf for ritrr infinitesiml re in 2D or volume in 3D, leding to differentil eqution. Note tht we should define stress, i.e., force per unit re to define the force eerting on oundr of the infinitesiml re in 2D or volume in 3D. -39-

40 1.5 Unit nd Dimension -40-

41 Unit nd dimension Sstem of Units hree sis units chrcterize the sstem of units. Conventionl unit: Bsis units re length, weight, time, current, temperture, etc. Science unit: Bsis units re length, mss, time, current, temperture, etc. Dimension [Length] [L], [Mss] [M], [ime] [], [Weight] [F] Bsis units of sstem of science units: [L], [M], [] Bsis units of sstem of conventionl units : [L], [F], [] -41-

42 British unit nd SI unit British unit Bsis unit: ft, l, sec British unit elongs to conventionl unit, i.e, l mens sicll l f (pound force). Unit for mss : slug = l sec 2 /ft SI unit (he Interntionl Sstem of Units) Bsis unit: m, kg, s In SI unit, kg mens sicll kg m (kilogrm mss). Unit for force: N = kg m/s 2 Entngled or mied use of conventionl nd science units Mn countries elonging to the Commonwelth of Ntions re using British unit. he other countries re using SI unit. However, most people re using the SI unit like British unit, i.e, the re using kg (i.e., kg force) insted of N (Newton). Sometimes mss is epressed in kg s 2 /m, where kg mens kg f. -42-

43 Bsis unit, Complement unit, Assemled unit Bsis unit Length : m, ft Current : A Complement unit rd Mss : kg, slug emperture : K ime : s, sec Voltge : V Assemled unit Velocit : m/s, ft/sec 1N = 1kg m/s 2, l, kg Pressure : 1P = 1N/m 2, psi = l/in 2, psf = l/ft 2, kg/m 2, kg/mm 2 Work : 1J = 1N m, l ft, kg m Moment : N m, l ft, kg m Power : 1W = 1J/s, l ft/sec, kg m/s, PS, hp Fctors for unit (10 12 ), G(10 9 ), M(10 6 ), k(10 3 ), c(10-2 ), m(10-3 ), (10-6 ), n(10-9 ), p(10-12 ) eltionship etween the two sstems of unit 1ft = m, 1 in = 25.4 mm, 1l = kg -43-

44 1.6 Bem heor -44-

45 Bem nd eternl lod q ( ) = Lod intensit function = Lod/length q ( ) q ( ) Concentrted lod q ( ) q ( ) Concentrted moment

46 Sher force nd ending moment Definition of es Ais of smmetr Definition of cross-section Positive -fce Negtive -fce Neutrl is z 1 z Internl forces eerting on cross-section z M z M 3D M : twisting moment M, M : ending moment F z : il force F, F : sher force z F F F z M Sign convention 2D M F V First suscript: Direction of plne Second suscript: Direction of force M F V F F, V F, M M z Newton s 3 rd lw should e stisfied.

47 Cutting method for sher force nd ending moment 0 ection force q ( ) 0 V M V( ) 0 M ( ) 0 0 L 2 L 0L 8 2 L L 0L 2 L 0 0L 0 2 L 2 Clcultion of V( ) nd M ( ) 0L C V M 0L F 0; V( ) 0 2 L 0; ( ) M C M

48 q ( ) q ( ) V( ) F 0; V( ) V( ) q( ) 0 M ( ) V( ) C M ( ) V( ) V( ) lim q ( ) ( ) dv q ( ) d 0 MC 0; M( ) M( ) V( ) q( ) 0 2 M( ) M( ) dm ( ) lim V( ) V( ) 0 d dm ( ) V( ), M ( ) V( ) 0 d dv ( ) q ( ), V( ) q ( ) 0 d M ( ) q ( )

49 V M P V( ) M ( ) P ection force P L P P Pure ending L P L Lod intensit function q () V () V () M() L V( ) P V( ) PP 0 M ( ) P 3 LL V( ) PPPP M ( ) PPPPPLP PLP M () P P PL PL P P PL PL Clcultion of sher force nd ending moment digrm P P PL PL PP PL PP PL 1 1 Drwing of sher force digrm nd ending moment digrm M ( ) PPP P

50 Distriution of internl force Bending moment nd ending stress = Sher force nd sher stress z = M

51 Strin Pure ending deflection curve-strin reltion A B C P D E F Q S Plne of smmetr Neutrl is M P S PQ Q S M PQPQ ( ) v( ) PQ v z v 0, 0, 0 z z z z z z Anticlstic curvture

52 Hooke s lw of n isotropic mteril Pure ending - stress-strin reltion zz zz zz zz z z z z z z z z 1 E 1 E 1 E 2 1 E 1 2G 2 1 E 1 2G E 2G Stress z z z z z ij ji z fce z z z z zz z fce fce Appliction of Hooke s lw Assumption: 0, z 0 E E, ν zz 0, 0, 0( 0, 0, 0) z z z z zz z zz zz Flnge z 0 We 0

53 Pure ending Force equilirium Appliction of requirement of force equilirium M F 0; da 0 A 1 E ( A), A EdA 0 If E constnt A da 0 M 0; dam z A E, ( A) 1 EdA 2 A M If E constnt EI zz : Fleurl rigidit EI zz M E M 0; z da zda 0 A A I zz A 2 da Automticll stisfied for smmetric em 2 nd moment of inerti of the cross-section

54 ( ) ( ) ( ) Summr of pure ending em theor q v M Curvture-deflection curve : Curvture-strin : Constitutive lw : E 1 v( ) E Ail force : da 0 da 0 A A Bending moment : 1 M da M v( ) A EI EIv( ) M ( ) ( EIv( )) M q( ) M I zz Boundr conditions dv ( ) q ( ) d dm ( ) V( ) d 2 dm 2 ( ) d q ( ) Geometric : v(0) 0, v(0) 0 vl ( ) 0, v( L) 0 Mechnicl : V(0) P, M (0) M, V( L) P, L M M (0) EIv(0) M v(0) EI P V M ( ) V(0) ( EIv(0)) v(0) EI

55 Purpose of engineering em theor Engineering em theor When sher force eists, z - ending moment vries from position to position. -sher stress s well s ending stress eist. M ( ) Purpose of engineering em theor is to clculte the sher stress M ( ) z Assumption of engineering em theor he following reltionships otined the pure ending em theor re vlid even though sher force does not vnish. M, I zz 1 M v( ) EI esults of engineering em theor V zz Q I 1 zz Q A da

56 Summr of em theor P P / << 1 P F F P = F, F / P = /, F >> P z Assume E 0 he sme with uniil loding in rod Assume A z c da 0 F 0 M z 0 M c Izz M Izz v( ) EI v( ) M ( ) zz I zz A 2 da