TfCH DAO DONG

DlJN PAN KHOA HOC CONG NGHE

PHAN TfCH DAO DONG TLT NHIEN VA MAT ON DINH CUA D A M SUf DUNG MO HINH VI KET CAU

FREE VIBRATION AND BUCKLING ANALYSIS OF BEAMS BY USING MICROSTRUCTURE MODEL

PHAM XUAN T O N G , L l / d N G VAN HAI, TRAN M I N H THI, N G U Y I N H L / U PHU

TOM TAT: | Trong bi\ bio ndy, si/ mit Sn (Jjnh vd dao ddng tt/ do cua dim dUdc phSn tfch dt/a tren md hinh vi \ik clu. D^m vi kit c l i

ddQc clu thanh bdi ht^ han cdc pliln tCr gi6ng nhau lign kgt v6\ nhau bing cdc id xo xoay. Ma tran dd ciJng cOa md hinh dlnr vi kit cau dUdc thilt i^p di/a trgn phi/dng phdp ndng iUdng. Cdc cdng thij'c tinh todn th^ ndng dan hdi, dOng ndng vd ndng Iddng todn phln di/dc thigt lap. Nguyfin ly Hamilton dUdc ap dung d^ xkc dinh ma trdn dd ci?ng cua dim vi kit clu, Cdc vl dij s6 phan tich dao ddng td nhign vd mat 6t\ djnh cDa dam vi kit cdu vdi cdc dilu kign bign vd tai trgng khdc nhau ddgc thi/c hi$n.

Kit qud cQa mo hinh vi ket clu d l xult dddc so sdnh vdi_ phi/dng phdp phln tif hull h^n (FEW) vdi si/ h6 tn? cua phln mim SAP2000. Cdc kit qud s6 ^ u di/dc cho Uiiy tinh dilng ddn va higu qud cua vi$c phdn tfch i^ig xijf mat d^n dinh vd dao ddng td do cua dim sijr dung phi/dng phdp vi l<lt clu.

T^ kh6a: cdng nghe nano, ly thuyit phi cgc bd, vi kit clu, phdn tich mit 6n i^nh va dao ddng tf; do.

ABSTRACT

)n tills paper, tiie buckling and free vibration analysis of beam based on microsb^uctfjre model will be carried out.

Microstf uctfjre beams are composed of finite elements which are connected by rotation^ spnngs. Stiffness matrix of the beam is obtained by using ttie energy metiiod. Formulations of elastic potential energy, kinetic energy and total energy are estab- lished. Hamilton's principle is applied to determine the stiffness matrix. Numerical examples are carried out to understand the vibration and buckling of microstf-ucture beams against various boundary conditions and load cases. The obtained results are compared with tiiose produced by the traditional finite element method (FEM). The numerical results cieariy illustrated that the proposed approach is accurate as well as computationally efficient and Is more suited generally for the study of buckling and free vibration of beams as compared to the FEM.

Tjir khda: nano technology, nonlocal theory, microsti'uctfjre, buckling and free vibration analysis.

_ a^SESia 1 . G i 6 i t h i g u

Ly thuyit ccr hpc phi cue bo dupc gi61 thiSu bcfi nha khoa hoc My Eringen (1972) [1] v& {1983) [2], trong do g\k thik r i n g trgng th^i ufng suSt tai mot d i l m ehju Snh hufing bcrt sir bidn dang cua xix ca cic diem lien tuc.

0i§u nky hokn to^n kh^c h4n v6i ly thuyit moi tru&ng li§n tyc c6 dien cho rSng ijfng suat tai m6t diem chi phy thu^c vao b i l n dgng tai d i l m d6. Tren cff scf ly' thuyit d^n h6i d6o phi cue bO, Peddieson (2003) [3]

da phdt t r i l n mpt mo hinh d§m Euler-Bernoulli phi cue bO ehju u6n. Zhang vk cpng sir (2004) [4] da phan ti'ch su m i t 6n dinh cua 6ng nano carbon n h i l u Icfp ehju Snh hu&ng cua hieu Lfng ti le ehilu dai nho. Zhang va c^ng s a (2005) [5], Ece vk Aydogdu (2007) [6] da phSn ti'ch dao dpng t u do cua I n g nano carbon ldp k6p dua v^o ly thuyit phi cue bp. Wang va cong s u (2006) [7], (2007) [8] da phan tich s u m i t 6n (Snh va dao dOng cua I n g nano tren mo hinh d i m Timoshenko phi cue b5.

Ngay ca khi cSc mo hinh phi cue b6 dang ducft:

nghidn cCru n h i l u thi v l n d l tinh todn he s6 anh hLrang cua quy m6 nho v l n c6n r i t kh6 khan. Eringen (1987) [9] da xdc dinh he s6 knh hudng cua ti \i ehilu d§ii nh6 eO b i n g each so sanh vdi mfi hinh song phan tdn cua Born-Karman. Duan v^ c6ng s i i (2007) [10] vk

Narendar vk cong s u (2011) [11 ] da so sdnh m6 hinh dan h l l phi cue b6 vdi mo phong p h l n tCrd^ng d l tinh todn h^ s6 ti le c h i l u dai nho. Trong nghidn ciJru gin day cua Challamel et al. (2013) [12], d i m vi k i t c l u duoc m6 hinh b6i cac p h l n tir lap bao g6m cde dogn cUng v^ cdc Id xo xoay ddn hdi dupc sir dung d l thilt ldp hd sa ti 1^ c h i l u dai nh6. Hd s6 e^ ndy duoc tim t h i y c6 nhijng gid trj khdc nhau tCiy thuOc vdo I09I phan tich, Cu the la e^.

mdt on (^nh ddm 2 ^ ' Euler

; 0.289 cho phdn tich phi cue b6 vd eg = - ^ = 0.408 cho phan tfch dao ddng ciia d i m

V 6

Euler phi cue b$. Wang va cpng s u (2013) [13] bdng each so sdnh m6 hinh d i m vi k i t c l u duoc Idm lifin tuc hod vdi md hinh ddm phi cue bd da tfnh todn dui?c hd s6 ti Id chidu ddi nho b i l n thidn phu thude vdo Ong s u i t ban ddu tU 0.289 ddn 0.408. Wang vd cdng si/

(2013) [13] eung ehi ra hd sd dnh hudng ciia tl l§

c h i l u ddi nhd eb khdng phy thude vdo logi phdn tfch md phy thude vdo ilng s u i t dpc true ban d i u trong dao ddng cua ddm. S u m i t 6n dinh dupc tim thdy id

NGl/dl XAY Dl/NG s 6 THANG 3 & 4 - 2015

PHAN TICH DAO OONG TU" NHIEN VA MAT dN DINH..

trufing hop dac bidt eiJa d i m dao dpng co itng s u i t dpc tryc khi tdn sd dao ddng t u nhien bdng O.Dilu ndy giai thi'eh ly do khi phan tich dao ddng hay m i t I n dirih thi he sd anh hudng t i Id c h i l u dai nho dupc tim t h i y ed gid trj khdc nhau.

Trong bai bao nay, md hinh vi k i t cdu dupe dp dyng 6i phdn tieh dao ddng vd mdt on dinh cua k i t cdu d i m thdng thu&ng vdi cde d i l u kidn bien khde nhau vd so sdnh vdi k i t qua cua phuong p h d p phdn tir hiJU han. Md hinh vi k i t cdu s§ eho thay tfnh hOfu hidu khi md phdng cac bdi todn cd d i l u kidn bidn phijrc tgp thdng qua vide md hinh bdng cdc Id xo xoay. Cdc ket qud se cho thdy md hlnh ddm vi k i t cdu khdng nhufng cd t h i dp dyng phdn tfch d i m nano ma cdn ed t h i sir dyng cho k i t cau d i m thdng thudng.

2. Ccr s d ly t h u y i t

Xet m$t ddm don gian hai ddu khdp, c h i l u ddi L dupc md hinh bdi huu hgn cdc phdn tif vdi edc Id xo xoay ddn hdi cd dO cUng la C vd dupc the hidn d . Ddm ndy duoc gid thidt chiu mdt ifng s u i t dpc tryc ban ddu Id Go.

nbi&id«ag

w,-0 Mi w, H u y * ^

Hinh 1. M6 hlnh dim vi kit ciu n doan dddi tie dung cua Ong suit ban diu ue vdi hai diu khdp.

D i m dupc c l u thanh XU n p h l n tir gidng nhau cd chilu ddi m i l p h l n tir a = L/n. C h i l u ddi phdn tir a lidn quan ddn s u r&i rac vi k i t c l u eua md hinh vdt iy.

De phdn tfch dao ddng vd mdt I n djnh eua md hinh ddm vi k i t edu, the nang ddn h6i U, t h i nang do ufng s u i t dpc tryc ban ddu V vd ddng nang do dao ddng T i l n lupt dupe tfnh toan. B d n g thdi, nguyen ly- Hamilton duac dp dung d l tfnh todn ma tran dO cUng K ciia d i m . TU dd, tan sd dao ddng va iimg s u i t mdt I n djnh ciia ddm thu dupc bang cdch gidi phuong trinh ddc tamg detK = 0.

Thd nang ddn hoi U sinh ra do b i l n dang eua cdc Id xo xoay trong md hinh d i m vi k i t cdu dupc cho bdi:

duong Id Ung sudt nen vd ifng s u i t ao dm Id Ung sudt keo.

Bdng ndng T do dao ddng cua ddm vi k i t c i u dupc bieu d i l n nhu sau:

2,.2 L

⁽¹⁾

trong 46 dO cimg C = El/a.

T h i nang V do umg s u i t doc true ban dSu trong m6 hinh dSm vi l<St c j u dupc viSt nhcr sau:

(2)

trong d6 A 1^ di6n tich t i l t di$n dilm, iJng su4l OQ

r^-S^

⁽³⁾

trong dd m, la khdi lupng tai niit /. Tong khdi luong M c i i a ddm vi k i t cdu duoc phdn bd n h u sau: cho cdc nut giijfa m^ =/W/n vdiy = 2,3,...,/7, v a c h o h a i n u t & d d u Id m, = m^^, = JW/(2n) bdi vi niit d hai ddu chi cd mdt doan cimg tham gia vao khdi lupng nut. Ap dyng nguydn ly Hamilton, thu dupc phucmg trinh sau:

S^'{U + V-T)dt = (i (4) trong dd t, vd tg Id thdi gian ban ddu vd thdi gian

k i t thuc. Bdng cdch t h i phuong trinh (1), (2) vd (3) vdo phuong trinh (4) vd xdt chuyen ddng d i l u hda, tuc Id Wj x,t =Wj X e'"^ vdi i = V - l vd 0} Id t i n sd gdc ciia dao ddng:

+ V 6 l ; = 2 :

- 2S + ^ = 0 (5)

(6)

(7)

— w.— 4w + 5w, —

+ va y = 3 -^ n - 1 :

+ Vdi y = n : - 5w„-4H'^_, + w„_2 —

~y -2\v^ + w^_| + 0w^ = 0

trong 66 fi = Mto^a' I nC vk y = a^AaIC.Mdi n = 3 phdn tir thi chiing ta ehi c6 hai phuong trinh (6)vd (7). Trong tru&ng hop nay, d i m hai ddu khdp cho w, = 0 vd w„+, = 0 .

€ ) l xdc djnh tdn sd t u nhidn cua dao dpng ddm vi kdt cdu dudi tdc dung eiia img s u i t ban ddu OQ, dinh thifc CLia ma t r | n K phai bdng CTQ, nghTa Id:

detK = 0 (8)

Gidi phuong trinh dgc trung ta thu dupe n h i l u nghidm , vd moi nghidm tuong umg vdrt mpt t i n sd dao ddng t u nhien eiJa d i m vi ket c l u . Bi xdc dinh Ung s u i t gay mdt I n djnh eua d i m vi kdt c l u , t h i l t ldp ra = 0 vd sau dd gidi phuong trinh ddc trung .

3. Cac vf du s o

D I chimg minh su tin cay ciia phuong phdp dupe d l xudl cung nhu cac y i u td anh hu&ng d i n Ung xii eua d i m , edc vi du sd v l phdn tfch tdn sd dao ddng t u nhidn vd mdt on dinh ciia ddm se ldn luot dupc thuc hien thdng qua vide so sdnh vdi phuong phdp phdn tir hiju han FEM (Finite element method) truyin thdng. Cdc vf du sd se dupc t i l n hdnh cho k i t c l u ddm vdi cdc didu kien bien vd tai trpng khdc nhau nhdm the hi^n tfnh uu vidt vd dung ddn ciia phuong phdp de nghi.

NGUdi XAY DIA4G SO THANG 3 & 4 • 2015

PHAN TiCH DAO fldNG Ttf NHIFM "A MAT 6 N DINH...

3.1 KiSm chititg cht/ang tr'mh

Tiin hanh l(hao s i t luc t6i han cua mo hinh dkm vi l<6t c j u hai d i u l<hap nhu vfli c4c thong s6 d4u vao cua d i m cho trong . Mo hinh d i m se duoc chia tang d i n tir 4 p h l n to d i n 14 p h l n ti> vfli mdi bubc tang 1 p h l n tir. K i t quS duoc so sanh vfli iuc tfli han tfnh toan bfli Challamel (2013) [12] vS duoc t h i hien fl.

Bang 1 . Thflng s i cua d i m trong cac v i du

thong qua vide sir dung p h l n m i m SAP20(K). Kit qua i l n iuot duoc the hi#n fl Hinh 3 v4 Hinh 4.

4(m) ^ 2x10Mnfl ^ 7850(lso/ni=)

.

- • - S i i bao -o- O"!**™*' (2013)

,

•

sAphlntv

Hinh 2. So sinh life tdi h^n cfia bii bio vdi Challamel (2013) [12].

LUC tdi hgn cua phuong phdp vi k i t c l u thu dupc cho thdy hodn todn trung khdp vdi k i t qua cua Challamel (2013) [12], Do dd, cd t h i kdt luan rdng chuong trinh tfnh ed dd tin cay cao vd cd t h i dp dung duoc cho cdc bdi todn phdn tfch tidp theo.

3.2 Khao sit stf hgi tfj cua dam vi kit ciu Khdo sdt dup'c thyc hidn v&i ddm vi kdt c l u hai d i u khdp nhu vdi thdng sd d i u vdo eho trong Bang 1.

Ddm dupc chia phdn til tang d i n tU 5 p h l n ti> d i n 50 p h l n tir vdi mdi bude tdng Id 5 phdn tir. Lue tdi han vd t i n sd dao dpng rieng sir dyng md hlnh vi k i t c l u dupc so sdnh v&i l&i gidi cho b&i phuong phdp FEM

Hinh 3. Khio sit dd h^ tu cua ldc tdi ban cOa dim vi kit du hai (^u khdp.

S^liiintft

Hinh 4. Idiio sit di hdi tu cua tin sd ndng dim vi kit dau hai diu l^dp Hlnh 3 vd Hinh 4 cho thdy khi tang sd p h l n tif thi Iyc tdi han yd t i n sd dao dpng ridng eung tang theo, tuy nhien ddn khi sd p h l n t& xdp xi 50 thi k i t qua 6a hdi tu. Do dd, cdc khao sat sau se duoc thyc hidn vdi ddm dupc chia 50 phdn tif.

3.3 l^ng xt} dam vi Itet ciu hai dau ngam Khao sdt d i m hai d i u ngdm dupc t h i hign trong Hinh 5. D i m dupc chia thdnh n phdn tif ndi vdi nhau bdng cdc Id xo xoay vd mdi phdn tif cd c h i l u ddi a.

D i m ehju mat Iyc dpc tn,ic id P vd Iyc phdn bd dfiu P

theo phuong thdng dimg g = . Trong md hinh lOOOL

vi k i t cdu, cdc lien k i t ngdm se dupc md hinh bdng Id xo xoay cd dp cUng Id C, = 2C theo d l x u l t ciia Challamel (2014) [14]. Cdc thdng sd ddm duPC cho trong Bang 1.

i. 1

-M

[^ — i^ H Hinh 5. Md hlnh dim vi kit ciu hai diu ngim.

T i l n hdnh plidn tfch mdt on dinh vd dao dpng rieng CLia md hlnh d i m hai d i u ngdm. Ddng thdi, cac ket qua phan tfeh dupe so sdnh vdi p h l n m i m SAP2000.

C h u y i n vj & giOfa nhjp d i m trudc khi mdt on djnh v4 dang dao ddng dmjc the hien tuong iimg trong Hlnh 6 va Hinh 7.

Cdc k i t qud eho thdy c h u y i n vj tgt giQa ddm vd dang dao ddng ciia ddm cd kfch thude thdng thudng theo md hinh d l x u l t hdu nhu gidng vdi k i t qua cho bdi phuong phdp FEM. Tuy nhien, bdng cdch si^fdyng md hinh vi k i t c l u thi didu kien bien cua bdi todn se dupc gidi quydt dd ddng hon thdng qua vide md hlnh

N G U d i X A Y DLTNG SO T H A N G 3 & 4 > 2 0 1 5

PHAN TICH DAO O O N G Ttf NHIEN VA MAT dN flINH..

! . f

^•

-o-Bii bia -X-SAP2000

"a? .'@j •

H@r 2±j,

Chuy& vi gifia dim (m) x 10^

Hinh 6. So sinh chuyin vi tai giOi dim cOa md hinh vi kit cau hai diu ngim vdi SAP2Q0O.

Hinh 7. So sinh dang dao ddng cOa dim vi kit cau hai diu ngim vdiSAP2000.

bang edc Id xo ed cdc dp cimg xoay tuong img.

3.4 t)iig xif dim vi kit ciu mgt dau ngam vd m^t diu tt/ do

Khdo sdt d i m vi k i t c l u mot d i u ngdm vd m0t ddu t^r do dupc t h i hidn trong Hinh 8. Ddm dupc md hlnh bdi n phdn tir gidng nhau cd c h i l u ddi a vd lidn kdt vdf nhau bdng edc Id xo xoay. Ddm ehju mdt luc dpc tnjc Id P^ vk luc tdp trung dgt tgi d i u t u do theo

p

phuong mang dUng P. = — ^ — . Lidn k i t ngdm sif 10000

dung mOt Id xo xoay cd dd cUng c, = 2C theo d l x u l t cQa Challamel (2014) [14] vd cde thdng sd ddm (Jm?c cho trong Bdng 1.

S I ; m i t I n dinh vk dao dOng ridng ciia md hlnh d i m mdt d i u ngdm vd mOt d i u t i ; do dupc khdo sdt trong vf dy ndy. C h u y i n vj tgi d i u t y do eiia d i m trudc khi m i t I n dinh vk dgng dao d0ng dupc tfnh todn bdng phuong phdp vi k i t edu dd x u l t , dong thdi cde kdt qud dupe so sdnh vdi phdn m i m SAP2000. K i t qud phan tfch s u m i t I n djnh vd dgng dao ddng t u d o dupc t h i

Hinh B. Md hinh dim vi kit ciu mdt diu ngim vi mdt (^u til do.

Cbuy^ vi gitb dim (m)

I So sinh chuyin vi tai diu td do cua dim vi kit ciu mdt diu ngim. mdt diu tddo vdi SAP2000.

Hlnh 10. So sinh dang dao ddng cOa dim vi kit ciu mdt diu ngim vi mdt diu tddo vdi SAP2000.

hien tuong Ung trong Hlnh 9 vd Hlnh 10.

Tif kdt qud d va cho t h i y c h u y i n vj tgi giila ddm tuong img v&i cdc luc tdi hgn vd dang dao ddng khi sir dung md hlnh vi k i t edu eho d i m ed kich thude thdng thu&ng hodn todn gidng vol k i t qua eho bdi FEM.

3.5 lfng xi} dam vi kit ciu mgt diu ngdm va mgt dau khdp

Ap dyng md hlnh vi k i t c l u cho ddm mdt d i u ngdm vd mdt d i u khdp dupc t h i hien trong Hlnh 1 1 . Dam dupe chia thdnh n p h l n ti> ndi vdi nhau bdng edc Id xo xoay vd mdi p h l n tir ed chidu ddi a. Ddm ehju mdt luc dpc tryc P I vd iyc tap trung tgi giOfa d i m theo

N G U d i X A Y D U N G SO T H A N G 3 & 4 • 2 0 1 5

P H A N T i C H DAO OONG T t f NHIFM V A M A T O N O I N H . .

phuong thang dijftig R = — — . Do cthig ifl xo xoay 1000

tai lign k i t ng^m C, = 2C theo d l x u l t cua Chaiiamei (2014) [14] va cdc th6ng s6 d i m dupc cho trong Bang 1.

fp ^ # ^ ' / /

K-

Hinii 11. Ilfo tiinh dkm vi Itet cau mdt dau ngam vd mdt dau khdp.

Chuyen vj d giijfa nhjp d i m tnJdc khi m i t on djnh va d^ng dao ddng dupc tinh t o i n b i n g phuong phap vi k i t c l u d l xuat va dupc so sanh vdi p h l n m i m SAP2000. K i t quS i l n iuot duDC the hien 6 Hinh 12 v4 Hinh13.

/ ^

-

-»-BiJbio -Z-SAF2000 Clnty&vigHkidimOid xio"^

2. So Sinh chuyin vj tai gidi dim cua md hlnh vi kit cau mdt diu ngim vi mdt diu khdp vdi SAP2000.

Hinh 13. So sinh dang dao ddng cOa dim vi kit ciu mdt diu ngim vi mdt diu khdp vdi SAP2000.

Hinh 12 vd Hlnh 13 cho thdy c h u y i n yi tgi glQa d i m vd dgng dao dpng cua phuong phdp d l x u l t dp dyng cho tru&ng hpp ddm cd didu kidn bien mi^t d i u ngam vd mdt dau khdp hodn toan gidng vdi kdt qua cho b6i FEM. Qua dd, cd t h i thdy md hlnh d i m vi kit c l u khflng nhung cd the dp dung cho d l r n nano md cdn cd t h i sir dung de phdn tich k i t c l u d i m ed kfch thude thdng thu&ng.

4. K i t ludn

Bai bdo da tidn hdnh phan tfch s u mdt on dinh vd dao dpng t u do cua ode md hinh d i m vi k i t c l u . Trong dd, k i t c l u d i m dupc md hlnh bdng cdch chia nhd thanh cdc phdn tir ndi vdi nhau bdng cdc 16 xo xoay.

Cdc tnjdng hdp phdn tfch bao gdm thay d l i didu ki$n bidn vd tdi trpng tdc dyng. Cac k i t qud dupe so sdnh vdi cac tdi lieu tham khao khdc vd vdi phuong phdp FEM siJr dyng p h l n m i m SAP2000. Qua eae k i t qud phan tfch sd dat dupc, mdt sd k i t ludn quan trpng dupc rut ra nhu sau:

• Trong phdn tfch mdt on (finh vd dao dpng d i ; do, phuong phdp d l x u l t cho kdt qud hodn todn phii hpp vdi kdt qud duoc tim t h i y bdi phuong phdp FEM.

€)dng thdi, phuong phdp vi k i t edu md phdng cdc dilu kidn bidn khde nhau bdng cac Id xo xoay cd t h i giup gidi q u y l t bdi todn nhanh vd vdn dgt s u chfnh xdc nhdt djnh.

• Md hlnh vi ket c l u khdng nhung cd t h i sir dyng khao sdt cdc k i t cdu d i m nano md cdn phdn dnh dung iimg xir mdt on ^ n h vd dao dOng rieng cOa dim ed kfeh thude thdng thudng. •

L d i C A M O N

Nghien cufU ndy duoe tdi trp bdi Bgi lipcJQulcJ gia Thdnh phd H 6 Chf Minh { V N U - f l 6 l ^ r 5 H g 1 khudn k h i d l tdi ma sd C2016-20-xx: "Phan&fciT fing xiir tTnh v d d a o d^ng t u d o cua k l t . p l u i tdm sif dyng md hinh vi ket c d i f .

ThS. Phaim Xuan Tung, KS. NguykHfhj ^ h ^ , ' Wioa K? thuat xay dtfng, tnfang,0ai'hgc B d c V M Qudc gia TRHCM " '-''••'^••^-^*

PGS. TS. LiMngVan H i i

Khoa Ky thuit )Gy diHg, Tni^gDgi hoc 84ch Kto^-, Qu6c gia TRHCM " .^I'fM-

Email: lvhai@haniit.edu.vn Bign thogi: 094^28^99]

NCS. TrUn Minh Thi ^V Khoa Ky thuit xay^d^O va Mdi tn/dng, Dai hoc

Singapore

TAI LI|U THAM KHAO

1. A. C. Eringen, Nonlocal polar elastic continua, Int. J. Eng. Sd (1972).

2. A. C. Eringen, On differential equations of nonloc^ elasticity (Xem Sip trang 78}

N G U d i X A Y D U N G SO T H A N G 3 & 4 • 2 0 1 5