MATLAB Examples

Dynamic Response Spectrum Analysis (RSA) for an industrial structure subjected to the El Centro earthquake motion

Contents

Statement of the problem

An industrial structure is modeled as 2-DOF system as shown in the following figure. Determine the lateral displacement, base shear and base moment of the structure due to El-Centro, 1940 earthquake ground motion using the response spectrum method. Take $$EI=80000 Nm^2$, $$L=2m$, $$m_1=100kg$ and $$m_2=200kg$. The damping ratio for all natural modes is $$\mathrm{\xi} = 0.02$.

Initialization of structural input data

Set the length of the structure members in m.

L=2;

Set the flexural stiffness in Nm^2.

EI=80000;

Set the lumped masses in kg.

m1=100;
m2=200;

Calculation of structural properties

Calculate the stiffness matrix of the structure in N/m.

K=6*EI/(7*L^3)*[8,-3;-3,2];

Calculate the mass matrix of the structure in kg.

M=[m1+m2,0;0,m2];

Set the spatial distribution of the effective earthquake forces. Earthquake forces are applied only at dof No 1 of the structure.

r=[1;0];

Calculation of response spectrum ordinates

Open file elcentro.dat.

fid=fopen('elcentro.dat','r');

Read the text contained in the file elcentro.dat.

text=textscan(fid,'%f %f');

Close file elcentro.dat.

fclose(fid);

Set the time step of the input acceleration time history.

time=text{1,1};
dt=time(2)-time(1);

Set the input acceleration time history ($$\mathrm{\alpha_g}$) in m/sec^2.

xgtt=9.81*text{1,2};

Set the critical damping ratio of the response spectra to be calculated ($$\mathrm{\xi}=0.02$)

ksi=0.02;

Dynamic Response Spectrum Analysis

[U,V,A,f,omega,Eigvec] = DRSA(K,M,r,dt,xgtt,ksi);

Set the number of eigenmodes of the structure.

neig=numel(omega);

Plot the natural modes of vibration of the industrial structure

FigHandle=figure('Name','Natural Modes','NumberTitle','off');
set(FigHandle,'Position',[50, 50, 1000, 500]);
for i=1:neig
    subplot(neig,1,i)
    plot([0;10*Eigvec(1,i);L],[0;L;10*Eigvec(2,i)+L],'LineWidth',2.,...
        'Marker','.','MarkerSize',20,'Color',[0 0 1],'markeredgecolor','k')
    grid on
    xlabel('Displacement','FontSize',13);
    ylabel('Height','FontSize',13);
    title(['Mode ',num2str(i)],'FontSize',13)
end

Calculate the peak modal base shear (N).

Vb=zeros(1,neig);
for i=1:neig
    Vb(i)=f(1,i);
end
Vb
Vb =

   1.0e+03 *

    0.2120    2.3148

Calculate the peak modal base overturning moment (Nm).

Mb=zeros(1,neig);
for i=1:neig
    Mb(i)=L*sum(f(:,i));
end
Mb
Mb =

   1.0e+03 *

    1.0783    2.6296

Modal combination with the ABSolute SUM (ABSSUM) method.

Calculation of peak base shear.

VbAbsSum=ABSSUM(Vb);

Calculation of peak base overturning moment.

MbAbsSum=ABSSUM(Mb);

Calculation of peak lateral displacement.

u1AbsSum=ABSSUM(U(1,:));

Modal combination with the Square Root of Sum of Squares (SRSS) method.

Calculation of peak base shear.

VbSRSS=SRSS(Vb);

Calculation of peak base overturning moment.

MbSRSS=SRSS(Mb);

Calculation of peak lateral displacement.

u1SRSS=SRSS(U(1,:));

Modal combination with the Complete Quadratic Combination (CQC) method.

Calculation of peak base shear.

VbCQC=CQC(Vb,omega,ksi);

Calculation of peak base overturning moment.

MbCQC=CQC(Mb,omega,ksi);

Calculation of peak lateral displacement.

u1CQC=CQC(U,omega,ksi);

Assemble values of peak response in a table.

C{1,2}='Lateral Displ. (m)';
C{1,3}='Base Shear (N)';
C{1,4}='Base Moment (Nm)';
C{2,1}='ABSSUM';
C{2,2}=u1AbsSum;
C{2,3}=VbAbsSum;
C{2,4}=MbAbsSum;
C{3,1}='SRSS';
C{3,2}=u1SRSS;
C{3,3}=VbSRSS;
C{3,4}=MbSRSS;
C{4,1}='CQC';
C{4,2}=u1CQC;
C{4,3}=VbCQC;
C{4,4}=MbCQC;
C
C = 

          []    'Lateral Displ. (m)'    'Base Shear (N)'    'Base Moment (Nm)'
    'ABSSUM'    [            0.0506]    [    2.5269e+03]    [      3.7079e+03]
    'SRSS'      [            0.0359]    [    2.3245e+03]    [      2.8421e+03]
    'CQC'       [            0.0359]    [    2.3247e+03]    [      2.8431e+03]

Copyright

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