Finite Element Analysis of Structures

Transcription

1 KAIT OE5 Finie Elemen Analysis of rucures Mid-erm Exam, Fall 9 (p) m. As shown in Fig., we model a russ srucure of uniform area (lengh, Area Am ) subjeced o a uniform body force ( f B e x N / m ) using a -node russ finie elemen. Assume ha all DOFs for y- and z-direcional ranslaions are prescribed o be zero. f B x U U U m m Figure. A russ problem modeled by a -node russ elemen Find (a) Inerpolaion of displacemen, u( x) HU. (p) (b) Relaion beween srain and nodal displacemens, (c) Equilibrium equaion, K U R. (5p) (d) Displacemen and sress a x. 5m. (5p) xx BU. (p) Noe ha u(x) is a quadraic polynomial, U { U U } denoes Young s modulus. T is in (a), (b) and (c), and E

2 KAIT OE5. We have a plane sress problem ( 4 m4 m ) subjeced o a poin load, p 4e N. The displacemen BC is given as shown in Fig.. The -D problem is modeled by he uniform x mesh of plane sress finie elemens. y hickness, E C E N / m E / p U U 9.5m 4m U 7 y x 4m Figure. A plane sress problem subjeced o a poin load Calculae (a) Diagonal componen of he oal siffness marix corresponding o K U 7 U 7. (5p) (b) Componens of he load vecor, (c) Displacemen R U9 and R U. (5p) U and srain energy sored in he srucure. (p) Assuming a dynamic analysis of he srucure (mass densiy ), find (d) Componen of he oal mass marix, M U 9 U. (p) U 7,

3 KAIT OE5 Finie Elemen Analysis of rucures Final Exam, Fall 9 (4 p). Using wo -node riangular plane sress elemens and wo -node russ elemens, we model a srucure subjeced o an X-direcional ip loading ( P N ), see Fig.. The properies are given 6 - E N / m, and hickness = for he plane sress elemens, 6 - E N / m and secional area = each for he russ elemens. v..5m.m Plane sress elemens m Truss elemens.m Truss elemens P N U m Y Plane sress elemens Y X Z m Fig.. A srucural model in he XY and YZ planes Considering "linear elasic analysis", calculae (a) he Jacobian marix J and he deerminan de J of he shaded riangular elemen (5 p) (b) he relaion beween srain and nodal displacemens B ( r, s) for he shaded riangular elemen (5 p) (c) he oal siffness K corresponding o (5 p) U U (d) he displacemen (5 p) U Noe ha he -node riangular plane sress elemen has he inerpolaion funcions: h r s, h r, h s, and he maerial law is v E C v N / m. v v U

4 KAIT OE5. Consider he single degree of freedom sysem shown in Fig.. F U CU U R In he figure, U R Fig.. A nonlinear spring ( C.) is he ip displacemen, F is he inernal force of he spring and is he exernal loading. Recall ha he siffness of he sysem is F K U (a) Wrie incremenal equilibrium equaion o calculae he response of he sysem. (5 p) (b) We wan o find he ip displacemen U corresponding o Using he full Newon-Raphson mehod, perform ieraions unil he ( i) soluion is converged R F. (5 p). R... Consider he 4-node plane sress elemen shown, where. Using he oal Lagrangian formulaion, calculae he nonlinear srain incremenal siffness marix K. ( p) NL m a ime x m m a ime x Fig.. A 4-node plane sress elemen under roaion ** If you hink ha he calculaion is oo heavy, calculae any componen of K NL. m

5 KAIT OE5 Finie Elemen Analysis of rucures Mid-erm Exam, Fall (p) We have a plane sress problem subjeced o poin loads, and, and he displacemen BC as shown in Fig. The D problem is modeled by a 4-node plane sress finie elemen, P P hickness=.m, E N / m E / C. E U P U P 4m Y m m X 4m Figure. A plane sress problem subjeced o poin loads. Considering he isoparameric procedure, evaluae (a) Global coordinae (X,Y) a r=s= (p) (b) Jacobian marix a r=s= (p) (c) B-marix a r=s= (5p) (d) B T CB a r=s= corresponding o K (5p) U U and calculae (e) Componens of he load vecor, R and U R U (5p).. Le's assume ha we obain he displacemen U U. m, calculae (f) Displacemens a r=s= (p) (g) rains and sresses a r=s= (5p) (h) rain energy sored in he srucure (p)

6 KAIT Ocean ysems Engineering, OE5 Finie Elemen Analysis of rucures Final exam, -- (4p.). Le us consider a -node iso-beam (Timoshenko beam) elemen as shown in Fig.. Y v s v Y θ u X r θ u Z a=.m node L=m node.m Fig.. A -node iso-beam elemen in D (. r. and. s.) The geomery and displacemen inerpolaion funcions and he nodal displacemen vecor are given: s ( x y ) = ( h ix i s a ), (u v ) = ( h iu i a h iθ i ) and U = [u u v v θ θ ] T. h i v i The maerial law used is { τ xx τ xy } = C { ϵ xx γ xy } wih C = [ E G ]. Considering he sandard isoparameric procedure for linear elasic problems, calculae (a) Jacobian marix (5p.) (b) u, u, v, v x y x y (c) B-marix (5p.) in erms of he nodal displacemen vecor (5p.) (d) Componens of he K-marix, K u u and K θ v (5p.)

7 KAIT Ocean ysems Engineering, OE5. Considering he oal Lagrangian formulaion, we analyze an elasic bar problem by using one -node russ elemen. Assume ha he cross secion does no change during deformaion and he maerial law is xx = E ϵ xx. ime ime R.m X L =m ΔL=m load.m u u R node node ime Fig.. An elasic bar problem For he configuraion a ime, calculae (a) Tangenial siffness, K u u (5p.) For he configuraion a ime, calculae (b) Linear par of he angenial siffness, ( K L ) u u (5p.) (c) Nonlinear par of he angenial siffness, ( K NL ) u u (5p.) (d) Componen of he force vecor ( F ) corresponding o u (5p.)

8 KAIT OE5 Finie Elemen Analysis of rucures Mid-erm Exam, Fall (4p.). (5p.) Le us consider a D 5-node plane sress elemen as shown in Figure. p V 5 (-.5,) s 5 V (,5) 5 V (,4) s (4,4) U (,) r r (-.5,-.5) Naural coordinae y (,) 4 x Global coordinae 4 (4,) (a) Find he shape funcion h 5 when h = 4 r( + r)( + s), h = 4 r( r)( + s), h = ( r)( s),h 4 4 = ( + r)( s). 4 (b) Find (x p, y p ) corresponding o r =.5 and s =. (c) Calculae he Jacobian marix J and de J a r = s =. (d) Calculae he column vecor of he srain-displacemen marix B a r = s = corresponding o U. Noe ha ε = {ε xx ε yy γ xy } T. (e) Calculae he componens R V, R V and R V5 of he load vecor for he poin load P = pe y applied a (x p, y p ).. (5p.) We sudy he convergence of finie elemen soluions in poenial energy. The poenial energy is defined by π( v ) = a( v, v ) ( f, v ), in which v is he displacemen vecor. Prove π( u ) π( u h ), where u is he exac soluion and u h is he finie elemen soluion, and explain he meaning of he inequaliy.

9 KAIT Ocean ysems Engineering, OE5 Finie Elemen Analysis of rucures Final Exam, Fall, -- (5p.). Le us consider a russ srucure shown in Fig.. The srucure consiss of wo bar members conneced by a pin and he pin connecion is suppored by a spring. Each bar member of lengh L is modeled by a -node bar elemen. x U, R U, R, x r=- r= r u u L L Figure. Lef: A linear elasic russ problem ( L = 5, EA =, spring consan = k ). Righ: -node bar elemen in he naural coordinae sysem. (a) (5p.) Find he B-marix of he bar elemen. (b) (p.) Find he siffness marix of he russ problem, U = [U U ] T. (c) (5p.) Find he ension forces of he bar members when k = and U = U =.. (d) (p.) When k = and R =, wha happens in he linear analysis? Plo he nonlinear load-displacemen curve ha you expec (beween R and U ) as he load increases.

10 KAIT Ocean ysems Engineering, OE5. Le us consider he 4-node plane sress elemen shown in Fig.. U ime = ime = x U x.5.5 Figure. A -D plane sress problem. A ime =, = and all oher sresses = are given in he elemen. Assume ha he maerial law wih Young s modulus E and Poisson s raio (v = ) relaes he incremenal nd Piola-Kirchhoff sresses o he incremenal Green-Lagrange srains and assume hickness =. a ime =. Using he oal Lagrangian formulaion, calculae he following. (a) (5p.) The componens of he linear siffness marices K L and K L corresponding o δu and U. (b) (p.) The componen of he nonlinear siffness marix K NL corresponding o δu and U. (c) (5p.) The componen of he force vecor F corresponding o U.

11 KAIT OE5 Finie Elemen Analysis of rucures Mid-erm Exam, Fall (4p.). (5p.) As shown in Fig., a 8-node D solid elemen is subjeced o a uniformly disribued normal pressure q (force per area). Calculae he nodal poin consisen loads a nodes,, and 4. f s q e 4.. X Z Y Fig.. A 8-node D solid elemen wih a pressure loading.

12 KAIT OE5. (5p.) Le us consider a riangular canilever problem modeled by a -node plane sress elemen as shown Fig.. The force and displacemen BCs are presened in Fig., hickness is., ε XX YY XY and E C ( ) ( ) / wih. (,) (,) Y P s (, ) X v (, ) r (, ) (, ) < Global coordinae > < Naural coordinae > u Fig.. A riangular canilever subjeced o a poin load P The shape funcions of he -node elemen are given by Calculae he followings h r, h s and h r s. (a) Global coordinaes (X,Y) corresponding o r=s=.5 (b) Jacobian marix J (c) B -marix [ by ] corresponding o u and v (d) iffness marix K [ by ] corresponding o u and v (e) Tip displacemens u and v, and sress a X=Y=

13 KAIT OE5 Finie Elemen Analysis of rucures Final Exam, Fall (45p.). We here wan o exend our experience on finie elemen analysis of srucures ino hea ransfer problems. For D hea ransfer problems, he governing equaions are given k q x i B in V, (differenial hea flow equilibrium equaion) on, (essenial boundary condiion) k ni q on, (naural boundary condiion) x i q where k and are he hermal conduciviy and he emperaure of he body, rae of hea generaed per uni volume, hea flux inpu on he surface q, see Fig.. is he surface emperaure on, and B q q is he is he q x V q x n Fig.. A D body subjeced o hea ransfer (, q q ) (a) (p.) Derive he principle of virual emperaures given as follows V k dv x x i i q V B dv q d, q in which is he virual emperaure disribuion ( on ). f (Hin) Divergence heorem: dv f nid V x wih a scalar funcion f. i

14 (b) (5p.) Assume ha he inerpolaion of emperaure is T and he nodal emperaure vecor x x elemen formulaion for D hea ransfer problems, H B R K is. KAIT OE5, and he relaion beween. Derive he finie (c) (p.) Considering he 4-noede finie elemen shown in Fig., find he componen of siffness marix ( and. 6 (Hin) ( x ) dxdy, ( x)( y) dxdy ( x)( y) dxdy 4 K ) corresponding o x x 4 Fig.. Four-node finie elemen for D hea ransfer problems. The configuraions of a body a ime, and and he second Piola-Kirchhoff sresses for he plane srain four-node elemen are shown in Fig Roaed by oime from ime x Configuraion a ime x Configuraion a ime Fig.. Four-node finie elemen subjeced o sreching and roaion

15 KAIT OE5 (Uni hickness a all ime seps) Calculae he followings (a) (5p.) Deformaion gradiens (b) (5p.) Cauchy sresses a ime X, and τ (c) (5p.) econd Piola-Kirchoff sresses a ime (d) (5p.) Cauchy sresses a ime τ X X (Hin) T, X τ,

16 KAIT OE5 Finie Elemen Analysis of rucures Mid-erm Exam, Fall (4p.). ( p.) Le us consider a D finie elemen model shown in Fig.. Fig.. A D canilever plae problem. (a) Finie elemen model (x mesh), (b) A 4-node plane sress elemen, (c) iffness marix of he 4-node plane sress elemen. Find he componens of he global siffness marix, componens of (m) K e in Fig. (c). K, U 7 U 7 K and U 6 U 7 K U U, in erms of he

17 KAIT OE5. (5 p.) Le us consider a apered bar problem and is finie elemen model in Fig.. Fig. A apered bar problem (E=Young s modulus). (a) Problem descripion, (b) Finie elemen model ( R qa ). The exac soluion of his problem is given by RL u e ( x) ln. Ea x / L Using a -node bar elemen, he following FE soluion is obained u h ( x) h ( x) u h ( x) u wih h ( x) x / L, h ( x) x / L, u and LR u ae. The principle of virual work specialized o his bar problem is given by L du dx du EA dx dx Ru xl u, u x, wih x (a) For he following 4 cases, check wheher he principle of virual work is saisfied or no. - u ue(x), u ah( x), and u ue(x), u a x - u uh(x), u ah( x), and u uh(x), u a x (Hin) du e dx R Ea ( x / L) (b) Discuss he resuls.

18 KAIT OE5. (5 p.) A -node bar finie elemen is shown in Fig.. Fig. A -node bar finie elemen (E=Young s modulus, A=area, =densiy). Considering he isoparameric procedure, he shape funcions for he -node bar finie elemen are given by h r( r ), h r( r ), h ( r)( r). Calculae he followings (a) Jacobian (b) B-marix, B (r) when ( r) B( r) u (c) Componen of he siffness marix, (d) Componen of he mass marix, wih u u T xx K. u u M. u u u. u

19 KAIT OE5 Finie Elemen Analysis of rucures Final Exam, Fall (4p.). (5 p.) A -node riangular plane sress elemen shown in Fig. is subjeced o he prescribed displacemen. Fig.. A -node riangular plane sress elemen (a) in he Caresian coordinae sysem, (b) in he naural coordinae sysem. Thickness =., hape funcions: v E C v, E = and v =. v ( v) / h r s, h r, h s Calculae he followings (a) rain-displacemen marix, B (b) by siffness marix where he boundary condiion is imposed, K (c) Displacemens a nodes and (d) Reacion force corresponding o he prescribed displacemen

20 KAIT OE5. ( p.) Consider a 4-node plane srain elemen in he configuraions a ime and shown in Fig.. Fig.. A 4-node plane srain elemen a ime and Calculae he followings a r r (a) Jacobian marices, (b) Deformaion ensor, J X and J (c) Green-Lagrange srain ensor a ime, ε

21 KAIT OE5. (5 p.) Using he oal Lagrangian formulaion, we consider a russ srucure modeled by wo -node russ elemens in Fig.. Assume ha he cross secion ( no changed during he deformaion and he maerial law is given by A ) is E Fig.. A russ srucure modeled by wo -node russ elemens. When u for he configuraion a ime, evaluae he followings (a) Componen of he linear par of he angen siffness marix ( K ) L corresponding o u and u (b) Componen of he nonlinear par of he angen siffness marix ( K NL ) corresponding o u and u (c) Componen of he inernal force vecor ( F) corresponding o u (Hin) Due o symmery, you may consider a half-symmeric model.

22 KAIT OE5 Finie Elemen Analysis of rucures Mid-erm Exam, Fall 4 (4p.) Le us consider a hin membrane srucure of hickness subjeced o a uniform emperaure variaion C as shown in Figure. The srucure is clamped along lef, righ and boom edges. We model he srucure using a uniform x mesh of four 4-node isoparameric plane sress elemens. Figure. Thermal expansion of a hin membrane srucure. (a) Finie elemen model, (b) A 4-node plane sress elemen. For his hermal expansion problem, we use he sress-srain law given by h τ C( ε ε ) wih in which τ xx τ, yy xy E C E, E / is he sress vecor, ε is he oal srain vecor, xx h ε, yy ε, xy h ε is he hermal srain vecor, C is he maerial law marix wih Young s modulus E, and is he hermal coefficien of expansion. (a) (8p.) pecialize he principle of virual work o his problem considering he given sress-srain law. Wrie down he finie elemen formulaion for he nodal load vecor due o he hermal srain. (b) (8p.) Calculae he siffness componens Noe ha ( ) drds 6 / k v v, r and k v 4 v and 4 (c) (8p.) Consruc he x oal siffness marix K using U U U nodal displacemen vecor is T (d) (8p.) Calculae he nodal load vecor, T k v v of he elemen (), see Figure (b). 4 ( s )( s) dsdr 8/. k v v, k v 4 v and 4 k v v 4. Noe ha he problem is symmeric. R R R.. The corresponding (e) (8p.) Assuming U / and U 8/, calculae he sress jump yy a poin A beween he elemens () and (), see Figure (a).

23 KAIT OE5 Finie Elemen Analysis of rucures Final Exam, Fall 4 (4p.). (p.) Consider a single degree of freedom (DOF) sysem subjeced o he force shown in Figure. p() as m, c, k 6 and p( ) Figure. A single degree of freedom sysem. For his DOF sysem, he linear equaion of moion is given by mu ( ) cu ( ) k u( ) p( ) wih u ( ) and u ( ), in which m, c, k, and displacemen, respecively. u are he mass, damping coefficien, spring consan and Using he Cenral Difference Mehod (CDM) wih ime sep size cr, calculae he response of he sysem from ime o sec. (Fill in he blanks (a), (b) and (c) in Table.) Table. Response of he DOF sysem. Time [sec] u () (a) (b) (c) Noe ha he criical ime sep size / and he free-vibraion period of he sysem cr T n T n is.5. In CDM, he following approximaions are used for he discreizaion of ime. ( ) u ( ) u( ) u( ) and u ( ) u( ) u( ) u( )

24 KAIT OE5. (5p.) Le us consider a 4-node axisymmeric finie elemen as shown in Figure. The finie elemen is clamped along he boom edge. Figure. A 4-node axisymmeric finie elemen. (a) Global DOFs, (b) Local DOFs For linear elasic analysis, he maerial law marix C wih Young s modulus E, E E C and.5e E u v xx, yy, x y xy and he srain vecor ε are given by xx yy ε xy zz u v, and y x u zz x. Using he isoparameric procedure, calculae he siffness componen K. U U Noe ha ( r) ( s) 4 r ( 4 r )( s) drds 64/ dsdr 8/5., ( 4 r )( r) drds 4 and

25 KAIT OE5. (5p.) Le us consider a plane srain elemen as shown in Figure. Figure. A D plane srain elemen. The Cauchy sress a ime, no including zz, is given by τ Pa. Using he Toal Lagrangian Formulaion, compue he componen of he nonlinear siffness K NL corresponding o U and K K L K. U. Noe ha NL