verification Energy time histories of nonlinear SDOF oscillator

Calculate the time histories of the energy dissipated by damping, the energy dissipated by yielding, as well as the energy that is input to two SDOF oscillators with two different eigenperiods.

Reference
Description
Load earthquake data
Setup parameters for NLIDABLKIN function for nonlinear SDOF with T=0.2s
Calculate dynamic response of the nonlinear SDOF with T=0.2s
Plot the energy time histories of the nonlinear SDOF with T=0.2s
Setup parameters for NLIDABLKIN function for nonlinear SDOF with T=0.5s
Calculate dynamic response of the nonlinear SDOF with T=0.5s
Plot the energy time histories of the nonlinear SDOF with T=0.5s
Copyright

Reference

Uang, C. M., & Bertero, V. V. (1990). Evaluation of seismic energy in structures. Earthquake engineering & structural dynamics, 19(1), 77-90.

Description

Figure 3(b) of the above reference is reproduced in this example, for both linear elastic-perfectly plastic SDOF systems (short period and long period). Thetwo eigenperiods considered are T=0.2 sec and T=5.0 sec. The damping ratio is 5% of the critical value and the ductility ratio achieved is equal to 5 for both SDOF systems. Figure 3(b) is verified in terms of damping energy (noted as E_ksi in the Figure), yielding energy (noted as E_h in the Figure) and input energy (noted as Ei in the Figure).

Load earthquake data

Earthquake acceleration time history of the 1986 San Salvador earthquake will be used (San Salvador, 10/10/1986, Geotech Investigation Center, component 90)

fid=fopen('SanSalvador1986GIC090.txt','r');
text=textscan(fid,'%f %f');
fclose(fid);
t=text{1,1};
dt=t(2)-t(1);
xgtt=text{1,2};

Setup parameters for NLIDABLKIN function for nonlinear SDOF with T=0.2s

Mass

m=1;

Eigenperiod

T=0.2;

Calculate the small-strain stiffness matrix

omega=2*pi/T;
k_hi=m*omega^2;

Assign post-yield stiffness

k_lo=0.0001*k_hi;

Assign yield limit

uy=0.004875;

Critical damping ratio

ksi=0.05;

Initial displacement

u0=0;

Initial velocity

ut0=0;

Algorithm to be used for the time integration

AlgID='U0-V0-Opt';

Minimum absolute value of the eigenvalues of the amplification matrix

rinf=1;

Maximum tolerance of convergence for time integration algorithm

maxtol=0.01;

Maximum number of iterations per integration time step

jmax=200;

Infinitesimal acceleration

dak=eps;

Calculate dynamic response of the nonlinear SDOF with T=0.2s

Apply NLIDABLKIN

[u,ut,utt,Fs,Ey,Es,Ed,jiter] = NLIDABLKIN(dt,xgtt,m,k_hi,k_lo,uy,...
    ksi,AlgID,u0,ut0,rinf,maxtol,jmax,dak);

Calculate the kinetic energy of SDOF

Ek=1/2*m*ut.^2;

Plot the energy time histories of the nonlinear SDOF with T=0.2s

Plot the damping energy, yielding energy and input energy

figure()
plot(t',cumsum(Ed),'k','LineWidth',1)
hold on
plot(t',cumsum(Ed)+cumsum(Ey)-Es,'b','LineWidth',1)
plot(t',cumsum(Ed)+cumsum(Ey)+Ek,'r','LineWidth',1)
hold off
xlim([0,10])
ylim([0,0.6])
xlabel('Time (sec)','FontSize',10);
ylabel('Energy/unit mass (N*m/kg)','FontSize',10);
%title('(a)','FontSize',10)
grid on
legend('Damping energy','Damping+yielding energy','Input energy')
drawnow;
pause(0.1)

Show the achieved ductility factor (must be equal to 5 according to the above reference)

max(abs(u))/uy

ans =

          5.00074829936011

Setup parameters for NLIDABLKIN function for nonlinear SDOF with T=0.5s