Phân tích ứng xử động của tấm fgm chịu tải trọng di chuyển có xét đến ảnh hưởng của nhiệt độ sử dụng phương pháp phần tử chuyển động mem

I H C QU C GIA TP H IH TR N QUANG S NG X NG C A T M FGM CH U T I TR NG DI CHUY C A NHI S N D PH N T CHUY K thu ng : 8580201 LU TP H NG - MEM NG I: IH I H C QU C GIA TP H h ng d n khoa h c: TS Tr n Minh Thi b ch m nh ch m nh Lu n v n th TS Nguy ng TS Nguy n T ng c b ov t ih m 2021 (tr c n) nH PGS.TS H m: c Duy - Ch t ch - TS Tr n Minh Thi TS Nguy ng .- Ph n bi n TS Nguy n T ng .- Ph n bi n CH T CH H NG NG KHOA K THU NG I H C QU C GIA TP.HCM C I CH IH C H T NAM c L p - T Do - H : TR N QUANG S MSHV: 1970467 m sinh: 10/02/1996 thu I N i sinh: Qu ng : : 8580201 ng x n ng c a nhi II NHI M V I DUNG ng c a t m FGM ch u t i tr ng di chuy n s d n t chuy ng MEM Thi t l n kh ng, ma tr c k t c u t m FGM s d n thu ng t ng th c Ki n t chuy tr n ng x III nt ng gi i h tin c y c i k t qu c hi Ti nc t qu c o s nh m kh ng c ng c a k t c u t m, t MV IV t lu quan n ngh : 22/02/2021 M V : 05/08/2021 V H H NG D N: TS Tr n Minh Thi m 2021 H NG D N CH NHI M B (H (H TS TR N MINH THI NG KHOA K THU (H i NG - ii t li c t ng h nh m m hai hay nhi u lo i v t li t v t li u m t li V t li t li u bi Graded Materials - t lo i ch c bi ng nhi aly ng nh c theo chi ho V t li ct nhi c n nhi t r t th i t li u ch i nhi n kim lo t li u ch c o dai, kh c ph c s r n n t n ng nhi cao ag m c a v t li u ch c u theo nh nc c bi t is d u ki cc c ng d n n ng, Trong m t kho ng th i gian ng t nhi u v t c bi tr tc u ng x c a t t ch u t i ng l c h m ng Lu ng x ng c a k t c u t m -Mindlin s d n t chuy n n ng ch ng c a nhi t c u t m ch u t gi ng s d nt h uh n t chuy n y u t nhi c a Lu ng ng m i n i nhi ph n t t m s c t ng c n ch u nhi t, va ch n pasternak r c hai ng m v s n ng c s d ng ph bi n nh t n: g th u i tr n y u t nhi t c a t m Lu uv n ng x ng c a nhi iii ng l c h c c a t m v t li u ch ng s ABSTRACT A composite is a material that is synthesized from two or more different materials in order to create a new material They are superior and more durable than the original materials Functional Graded Materials (FGM) is a special, heterogeneous, isotropic composite They have mechanical properties that vary continuously with the thickness of the plate and they are most commonly used in high temper ature environments Typical FGM materials are made up of two components: ceramic and metal Ceramics have high elastic modulus, a very small coefficient of thermal expansion and a heat transfer coefficient It makes the functional material highly rigid and very inert to heat In addition, the metal makes the functional material tough, overcoming the cracking that occurs due to the brittleness of the ceramic in the high temperature environment The advantage of functional materials is the ability to fabricat e structures according to desired properties under specific working conditions, such as in environments of high temperature, impact, abrasion, etc In construction, the plate problem on the pasternak foundation has high applicability, such as the foundatio n of civil, industrial, traffic, reactors, In a short period of time, many problems related to the structural analysis of FGM plates have been studied, especially the behavior of the plate on the viscoelastic foundation when subjected to dynamic loads To analyze the dynamics of the plate problem on the Pasternak foundation, the plate model on the Paternak foundation was applied This study focuses on analyzing the dynamic behavior of FGM sheet structures according to the Reissner-Mindlin model using the Moving Element Method (MEM) and considering the influence of temperature Previous studies usually only model the plate structure under moving load by analytical method, finite element method -FEM, and moving element method-MEM, but have not considered the temperature factor in MEM Therefore, the new idea of this study is to develop MEM with more advantages, where the plate elements will be considered as moving and the load can be considered stationary Besides, this study considers the temperature factor on the sides of the plate This study is expected to contribute to the study of the dynamic behavior of FGM plates on Pasternak foundations in a temperature environment iv c hi is ng d n c a th y TS Tr n Minh Thi M t s k t qu Lu t a T ng h ng Vi ng (ISSN -2021 s 355&356 XXXIV Moving element analysis of FGM plate considering effect of temperature t t i H i ngh qu c t l n th hai v ki n ng b n v K t qu Lu hi th mv c th c hi n c Tr n Quang S v i M CL C .i ii iii v GI I THI U .1 1.1 Gi i thi 1.1.1 cv tv 1.1.2 .1 p thi t c 1.1.3 d ng v t li u bi i ch gi i 1.1.4 u .3 1.1.5 M u 1.1.5.1 M u 1.1.5.2 u 1.1.5.3 Ph 1.2 u Gi i thi u c a lu T .7 2.1 m ch u t i tr ng di chuy n 2.1.1 2.1.2 t t m bi Bi n d ng c a t 2.1.3 2.1.4 i ch .7 i qu n h gi a ng su t Pasternak 10 T t Pasternak 11 2.2 2.2.1 bi n d ng m ch u t i tr ng di chuy n 13 Ph n t ng tham s 13 2.2.2 t Pasternak ch u t i tr ng di chuy n 17 2.3 ng c a nhi n ng x ng t n Pasternak ch u t i tr ng di chuy n 24 2.4 29 N U 32 3.1 d ng 32 vi 3.1.1 32 3.1.2 Gi ng gia t c 33 3.1.3 Gi ng chuy n v 33 3.1.4 3.2 it c thu 33 35 36 4.1 Ki m ch 36 4.1.1 tr ng x c a t m FGM ch ng c a t i 36 4.1.2 ng c a t m FGM 38 4.1.3 d ng c a t i tr 4.2 t i tr 4.2.1 t ng l c h c t n Pasternak ch ng 43 ng l c h c t n ng c a nhi n Pasternak ch ng c a 44 h i t c a chuy n v c l p th i gian 44 4.2.2 Pasternak ch u t i tr ng x ng l c h c c a t ng nhi n i 46 4.2.3 Kh ng x ng l c h c c a t n Pasternak ch u t i tr ng nhi h s t l th i 50 4.2.4 Kh ng x ng l c h c c a t n Pasternak ch u t i tr ng nhi chi i 52 4.2.5 Kh ng x ng l c h c c a t n Pasternak ch u t i tr ng nhi h s c ng k wf i 54 4.2.6 Kh ng x ng l c h c c a t n Pasternak ch u t i tr ng nhi s t k sf i 56 4.2.7 Kh ng x ng l c h c c a t n Pasternak ch u t i tr ng nhi h s c n n n cf i 57 4.2.8 Kh ng x ng l c h c c a t n Pasternak ch u t i tr ng nhi l c di chuy n i 60 vii 4.2.9 Kh ng x ng l c h c c a t n Pasternak ch u t i tr ng nhi n c ut ot i 62 K T LU N NGH 66 5.1 K t lu n 66 5.2 Ki n ngh 66 68 U THAM KH O 88 M TS 94 108 viii M TS ng x n n nh c a nhi ng c a t b clear all clc format long syms z Lx=20;% length of x direction (m) Ly=10;% length of y direction (m) nx=20;% (columns) number of element along x direction ny=10;% (rows) number of element along y direction lx=Lx/nx;% side length of x direction ly=Ly/ny;% side length of y direction ndof=3;%numder of DOFs per node nnel=9;%number of nodes per element nel=nx*ny;% total element nnode=(2*nx+1)*(2*ny+1);% total number of nodes in total elements edof=nnel*ndof;% DOFs per element sdof=nnode*ndof;% total of Plates DOFs %FGM plate parameters dac trung cua tam FGM nuy=0.3;%poison's ratio ro_c=2370; %kg/m3 ro_m=8166; %kg/m3 t=0.1;% thickness of the plate (m) k=1 % he so ty le the tich %Temperature-dependent coefficients E0c=348430000000;%Pa Eac=0;%Pa E1c=-0.000307;%Pa E2c=0.000000216;%Pa E3c=-0.00000000008946;%Pa E0m=201040000000;%Pa Eam=0;%Pa E1m=0.0003079;%Pa E2m=-0.0000006534;%Pa E3m=0;%Pa a0c=0.0000058723;%1/K aac=0;%1/K a1c=0.0009095;%1/K a2c=0;%1/K a3c=0;%1/K a0m=0.00001233;%1/K aam=0;%1/K a1m=0.0008086;%1/K a2m=0;%1/K 94 a3m=0;%1/K Tc=400;% K Tm=400;% K Kc=9.19; Km=12.04; Kcm=Kc-Km; CC=1-Kcm/((k+1)*Km)+(Kcm^2)/((2*k+1)*(Km^2))(Kcm^3)/((3*k+1)*(Km^3))+(Kcm^4)/((4*k+1)*(Km^4))(Kcm^5)/((5*k+1)*(Km^5)); L=(z/t+1/2); n=1/CC*(LKcm/((k+1)*Km)*(L^(k+1))+(Kcm^2)/((2*k+1)*(Km^2))*(L^(2*k+1)) (Kcm^3)/((3*k+1)*(Km^3))*(L^(3*k+1))+(Kcm^4)/((4*k +1)*(Km^4))* (L^(4*k+1))-(Kcm^5)/((5*k+1)*(Km^5))*(L^(5*k+1))); Tz=Tm+(Tc-Tm)*n; deltaT=Tz-300; Ec=E0c*(Eac*Tc^-1+1+E1c*Tc+E2c*Tc^2+E3c*Tc^3);%Pa Em=E0m*(Eam*Tm^-1+1+E1m*Tm+E2m*Tm^2+E3m*Tm^3);%Pa Ez =(Ec - Em)*((z/t+1/2)^k)+Em; ac=a0c*(aac*Tc^-1+1+a1c*Tc+a2c*Tc^2+a3c*Tc^3);%1/K am=a0m*(aam*Tm^-1+1+a1m*Tm+a2m*Tm^2+a3m*Tm^3);%1/K az =(ac - am)*((z/t+1/2)^k)+am;%1/K %internal forces generated by temperature Mt=int(Ez*z*az*deltaT,z,-t/2,t/2)/((1-nuy^2))*[1 nuy 0; nuy 0; 0 (1-nuy)/2]*[1;1;1]; % E=int(Ez,z,-t/2,t/2); % E=eval(E) ro_z =(ro_c - ro_m)*((z/t+1/2)^k) + ro_m; % ro=int(ro_z,z,-t/2,t/2); % ro=eval(ro) kapa=5/6; %shear correction factor % G=Emodule/2/(1+nuy); %flexural rigidity of the plate % D=Emodule*t^3/12/(1-nuy^2); %shear modulus %Load parameters -f=1000000;%load vo=10;% initial velocity of load(m/s) v=10;% velocity of load(m/s) a=0;%acceleration %Foundation parameters -kwf=1*10^7; %(N/m3) ksf=1*10^5;%(N/m3 cf=1*10^4;%(N.s/m3) %Newmark tolerance -tole=10^(-6); %tolerance to=1;%total analysis time (s) deltat=0.0025;%time step 95 %Matrix containing the density of the material and thickness -[ C11, C22, C12, C21, C66, M11, M22, M33 ]= Coefficient(t,Ez,ro_z,nuy,kapa) m=[M11 0; M22 0; 0 M33]; %Material matrix related to bending deformation and shear deformation -Db=int(Ez*(z^2),z,-t/2, t/2)/((1-nuy^2))*[1 nuy 0; nuy 0; 0 (1-nuy)/2]; Ds=int(Ez,z,-t/2, t/2)*kapa/2/(1+nuy)*[1 0;0 1]; %Mindlin Plate meshing -[gcoord,ele]=mesh2d_rectq9(Ly,nx,ny,lx,ly); %Sampling points and weights nglx=3; ngly=3;%3x3 Gauss-Legendre quadrature nglxy=nglx*ngly;%number of sampling points per element [point2,weight2]=memglqd2(nglx,ngly); %Loop for the total number of elements -for iel=1:nel for i=1:4 nd_corner(i)=ele(iel,i); % extract connected node for (iel)-th element xc(i)=gcoord(nd_corner(i),1); % extract x value of the node yc(i)=gcoord(nd_corner(i),2); % extract y value of the node end xcoord=[xc (xc(1)+xc(2))/2 (xc(2)+xc(3))/2 (xc(3)+xc(4))/2 (xc(4)+xc(1))/2 (xc(1)+xc(2)+xc(3)+xc(4))/4]; ycoord=[yc (yc(1)+yc(2))/2 (yc(2)+yc(3))/2 (yc(3)+yc(4))/2 (yc(4)+yc(1))/2 (yc(1)+yc(2)+yc(3)+yc(4))/4]; end K1=zeros(edof,edof); K=zeros(edof,edof); M=zeros(edof,edof); C=zeros(edof,edof); FT=zeros(edof,1); %Numerical integration -for intx=1:nglx x=point2(intx,1); % sampling point in x axis wtx=weight2(intx,1); % weight in x-axis for inty=1:ngly y=point2(inty,2); % sampling point in yaxis wty=weight2(inty,2) ; % weight in y-axis 96 [N,dNdr,dNds,d2Ndr2,d2Ndrds,d2Ndsdr,d2Nds2]=memisoq9(x,y); %Compute shape functions and derivatives at sampling point [jacob2]=memjacob2(nnel,dNdr,dNds,xcoord,ycoord); % compute Jacobian detjacob=det(jacob2); % determinant of Jacobian invjacob=jacob2\eye(2,2); % inverse of Jacobian matrix [dNdx,dNdy,d2Ndx2,d2Ndxdy,d2Ndydx,d2Ndy2]=memderiv2(nnel ,dNdr, dNds,d2Ndr2,d2Nds2,d2Ndrds,d2Ndsdr,invjacob);%derivatures in physic coordinate [Bb,Bs,Nw, dNwdr, d2Nwdr2, d2Nwds2, N, dNdr, d2Ndr2]=memkine2d(dNdx,dNdy,d2Ndx2,d2Ndxdy,d2Ndydx,d2Ndy2,N); K1=K1+(Bb'*Db*Bb+Bs'*Ds*Bs+vo^2*N'*m*d2Ndr2a*N'*m*dNdr-cf*vo*Nw'*dNwdr+kwf*Nw'*Nwksf*(Nw'*d2Nwdr2+Nw'*d2Nwds2))*wtx*wty*detjacob;% element stiffness matrix at initial time K=K+(Bb'*Db*Bb+Bs'*Ds*Bs+v^2*N'*m*d2Ndr2-a*N'*m*dNdrcf*v*Nw'*dNwdr+kwf*Nw'*Nwksf*(Nw'*d2Nwdr2+Nw'*d2Nwds2))*wtx*wty*detjacob;% element stiffness matrix M=M+(N'*m*N)*wtx*wty*detjacob;% element mass matrix C=C+(-2*v*N'*m*dNdr+cf*Nw'*Nw)*wtx*wty*detjacob;% element damping matrix FT=FT+(Bb'*Mt)*wtx*wty*detjacob; end end %Stiffness, mass, damping matrix of plate KOS1=zeros(sdof,sdof); KOS=zeros(sdof,sdof); MOS=zeros(sdof,sdof); COS=zeros(sdof,sdof); FOST=zeros(sdof,1); for i=1:ny for j=1:nx ie=nx*(i-1)+j; ele(ie,1)=2*ie-1+(i-1)*(nx+1)*2; ele(ie,2)=2*ie+1+(i-1)*(nx+1)*2; ele(ie,3)=2*ie-1+(i+1)*(nx+1)*2; ele(ie,4)=2*ie-3+(i+1)*(nx+1)*2; ele(ie,5)=2*ie+(i-1)*(nx+1)*2; ele(ie,6)=2*ie+(i)*(nx+1)*2; ele(ie,7)=2*ie-2+(i+1)*(nx+1)*2; ele(ie,8)=2*ie-2+(i)*(nx+1)*2; ele(ie,9)=2*ie-1+(i)*(nx+1)*2; ix=memindexos(ele(ie,:),nnel,ndof); [KOS1]=hpsystemmatrix(KOS1,K1,ix); [KOS,MOS,COS]=hpmatrix(KOS,MOS,COS,K,M,C,ix); 97 [FOST]=hpsystemmatrixf(FOST,FT,ix); end end %Load vector -FOS=zeros(sdof,1); FOS(3*((2*nx+1)*ny+nx+1)-2,1)=-f; %load's position at the middle of the center line of the plate %FOS(3*((2*nx+1)*ny+nx/2+1)-2,1)=-f; %load's position at 1/4 of the center line of the plate %FOS(3*((2*nx+1)*ny+3*nx/2+1)-2,1)=-f; %load's position at 3/4 of the center line of the plate FOS=FOS-FOST;%Load by temperature STEP =0; FOS1=zeros(sdof,1); FOS1(3*((2*nx+1)*ny+nx+1)-2,1)=-f; %load's position at the middle of the center line of the plate %FOS1(3*((2*nx+1)*ny+nx/2+1)-2,1)=-f; %load's position at 1/4 of the center line of the plate %FOS1(3*((2*nx+1)*ny+3*nx/2+1)-2,1)=-f; %load's position at 3/4 of the center line of the plate FOS1=FOS1-FOST;%Load by temperature %Boudary condition -option='F-SS-F-SS';%maping to infinity for clamped edge [ bcdof ] = boundary_condition( nx,ny,option ); [ KOS1, FOS1 ] = apply_condition( KOS1,FOS1,bcdof ); %Displacement at initial time yini1=KOS1\FOS1; y=zeros(sdof,to/deltat); y1d=zeros(sdof,to/deltat); y2d=zeros(sdof,to/deltat); yini=zeros(sdof,1);% the initial displacement of the system for i=1:sdof yini(i)=yini1(i); end y(:,1)=yini; % : denotes an entire row or column %Newmark constant beta=1/4; alpha=1/2; a0=1/(beta*deltat^2); a1=alpha/(beta*deltat); a2=1/(beta*deltat); a3=1/(2*beta)-1; a4=alpha/beta-1; a5=deltat/2*(alpha/beta-2); a6=deltat*(1-alpha); a7=alpha*deltat; tt=0:deltat:to-deltat; 98 h=0; step=0; for i=1:(to-deltat)/deltat fprintf('STEP=%d/%d',i,(to-deltat)/deltat); y(:,i+1)=y(:,i); y1d(:,i+1)=y1d(:,i); y2d(:,i+1)=y2d(:,i); h=h+deltat; for j=1:10000000 d1=y(3*((2*nx+1)*ny+nx+1)-2,i+1); d2=y1d(3*((2*nx+1)*ny+nx+1)-2,i+1); d3=y2d(3*((2*nx+1)*ny+nx+1)-2,i+1); FOS=zeros(sdof,1); FOS(3*((2*nx+1)*ny+nx+1)-2,1)=-f; %load's position at the middle of the center line of the plate with changeable intensity KK=KOS+a0*MOS+a1*COS; FF=FOS+MOS*(a0*y(:,i)+a2*y1d(:,i)+a3*y2d(:,i))+COS*(a1*y(:,i)+ a4*y1d(:,i)+a5*y2d(:,i)); [ KK, FF ] = apply_condition( KK,FF,bcdof );%apply boundary for clamped edge mapping to infinity y(:,i+1)=KK\FF; y2d(:,i+1)=a0*(y(:,i+1)-y(:,i))-a2*y1d(:,i)a3*y2d(:,i); y1d(:,i+1)=y1d(:,i)+a6*y2d(:,i)+a7*y2d(:,i+1); e1=abs((y(3*((2*nx+1)*ny+nx+1)-2,i+1)-d1)/d1); e2=abs((y1d(3*((2*nx+1)*ny+nx+1)-2,i+1)-d2)/d2); e3=abs((y2d(3*((2*nx+1)*ny+nx+1)-2,i+1)-d3)/d3); step=step+1; if e1