Response of a uniaxial bar for a ramp loading

Calcul dynamic

% FETHALLAH AMARA
% central difference method
% problem parameters
Area=10e-5;E=100e9;rho=4000;L=1/3;Fmax=0.001*Area*E;tmax=3e-3;
Dt=21e-6;Nmax=round(tmax/Dt)+1;
t=linspace(0,tmax,Nmax);
% Global Matrix
K=(Area*E/L)*[2,-1,0;-1,2,-1;0,-1,1];
M=(rho*Area*L/6)*[4,1,0;1,4,1;0,1,2];
F=[zeros(2,Nmax);linspace(0,Fmax,round(Nmax/2)), Fmax*ones(1,Nmax-round(Nmax/2))];
Q=zeros(3,Nmax);
% Time integration
Minv=(Dt^2)*(inv(M));
Q(:,2)=Q(:,1)+Dt*zeros(3,1);
for n=2:Nmax-1
 Q(:,n+1)=2*Q(:,n)-Q(:,n-1)+Minv*(F(:,n)-K*Q(:,n));
end
% Static solution
Qs=K\F;
plot(t,Q(3,:),t,Qs(3,:),'--','LineWidth',2);
xlabel('Time(sec)','FontSize',13,'Color','k')
ylabel('Displacement(m)','FontSize',13,'Color','k')
hleg1 = legend('Central finite difference',' Static solution','Location','Best');