MATLAB Examples

Harmonic excitation Response spectrum

Contents

Initial definitions

The following initial definitions are made (in the order presented below):

Set the time step of the input acceleration time history.

dt=0.01;

Set the input acceleration time history ($$\mathrm{\alpha_g}$). A sinusoidal acceleration time history is defined with maximum input acceleration equal to $3m/{s}^{2}$ and period $$T_g=\pi$.

t=(0:dt:100)';
xgtt=3*sin(2*t);

Set the eigenperiod range for which the response spectra will be calculated.

T=(0.01:0.01:10)';

Set the critical damping ratio ($$\mathrm{\xi}$) of the response spectra to be calculated.

ksi=0.05;

Set the minimum absolute value of the eigenvalues of the amplification matrix.

rinf=1;

Set the algorithm to be used for the integration.

AlgID='U0-V0-Opt';

Set the initial displacement of all SDOF oscillators analysed.

u0=0;

Set the initial velocity of all SDOF oscillators analysed.

ut0=0;

Plot the input harmonic excitation.

figure('Name','Harmonic excitation','NumberTitle','off')
plot(t,xgtt,'LineWidth',1.)
xlabel('t','FontSize',13);
ylabel('a_g','FontSize',13);
title('Harmonic excitation','FontSize',13)

Processing

Find the spectral displacement of a SDOF system with eigenperiod equal to 2 and critical damping ratio equal to 5% using default parameters.

[~,~,Sd,~,~]=LERS(dt,xgtt,2,0.05)
Sd =

    0.7163

Calculation of elastic response spectra and pseudospectra using nondefault parameters.

[PSa,PSv,Sd,Sv,Sa]=LERS(dt,xgtt,T,ksi);

Post processing

Plot displacement spectrum.

figure('Name','Spectral Displacement','NumberTitle','off')
plot(T,Sd,'LineWidth',2.)
grid on
xlabel('T_n','FontSize',13);
ylabel('S_D','FontSize',13);
title('Displacement Spectrum','FontSize',13)

Plot velocity spectrum.

figure('Name','Spectral Velocity','NumberTitle','off')
plot(T,Sv,'LineWidth',2.)
grid on
xlabel('T_n','FontSize',13);
ylabel('S_V','FontSize',13);
title('Velocity Spectrum','FontSize',13)

Plot acceleration spectrum.

figure('Name','Spectral Acceleration','NumberTitle','off')
plot(T,Sa,'LineWidth',2.)
grid on
xlabel('T_n','FontSize',13);
ylabel('S_A','FontSize',13);
title('Acceleration Spectrum','FontSize',13)

Plot pseudo-velocity spectrum.

figure('Name','Pseudo Velocity Spectrum','NumberTitle','off')
plot(T,PSv,'LineWidth',2.)
grid on
xlabel('T_n','FontSize',13);
ylabel('PS_V','FontSize',13);
title('Pseudo Velocity Spectrum','FontSize',13)

Plot pseudo-acceleration spectrum.

figure('Name','Pseudo Acceleration Spectrum','NumberTitle','off')
plot(T,PSa,'LineWidth',2.)
grid on
xlabel('T_n','FontSize',13);
ylabel('PS_A','FontSize',13);
title('Pseudo Acceleration Spectrum','FontSize',13)

Validation

  • Spectral Displacement:
  1. It is zero for rigid structures ($$S_D->0$ for $$T_n->0$)
  2. For $$T_n=\pi$ it is equal to $$\frac{1}{2\xi}max(\mathrm{\alpha_g})(\frac{T_g}{2\pi})^{2}=7.5$
  • Spectral Velocity
  1. It is zero for rigid structures ($$S_V->0$ for $$T_n->0$)
  2. For $$T_n=\pi$ it is equal to $$\frac{1}{2\xi}max(\mathrm{\alpha_g})\frac{T_g}{2\pi}=15$
  • Spectral Acceleration
  1. For rigid structures $$S_A->max(\mathrm{\alpha_g})=3$ for $$T_n->0$
  2. For $$T_n=\pi$ it is equal to $$\frac{1}{2\xi}max(\mathrm{\alpha_g})=30$
  • Pseudo Velocity Spectrum
  1. For rigid structures $$PS_V->0$ for $$T_n->0$
  2. For $$T_n=\pi$ it is equal to $$\frac{1}{2\xi}max(\mathrm{\alpha_g})\frac{T_g}{2\pi}=15$
  • Pseudo Acceleration Spectrum
  1. For rigid structures $$PS_A->max(\mathrm{\alpha_g})=3$ for $$T_n->0$
  2. For $$T_n=\pi$ it is equal to $$\frac{1}{2\xi}max(\mathrm{\alpha_g})=30$

Copyright

Copyright (c) 13-Sep-2015 by George Papazafeiropoulos