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By George Papazafeiropoulos

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Chopra (2012): DRSA for shear buiding subjected to the El Centro earthquake motion

Contents

Statement of the problem

  • Chopra (2012), Section 12.8: Consider the structure of Fig. 12.8.1: a uniform five-story shear building (i.e., flexurally rigid floor beams and slabs) with lumped mass m at each floor, and same story stiffness k for all stories.
  • Chopra (2012), Section 13.2.6: The structure is subjected to the El Centro ground motion (Chopra (2012), Fig. 6.1.4). The lumped mass $$m_j = m = 100 kips/g$ at each floor, the lateral stiffness of each story is $$k_j = k = 31.54 kips/in.$, and the height of each story is 12 ft. The damping ratio for all natural modes is $$\mathrm{\zeta_n} = 0.05$.

Initialization of structural input data

Set the storey height of the structure in ft.

h=12;

Set the number of eigenmodes of the structure, which is equal to the number of its storeys.

neig=5;

Set the lateral stiffness of each storey in kips/inch.

k=31.54;

Set the lumped mass at each floor (g=386.4 inch/sec^2).

m=100/9.81*0.0254;

Calculation of structural properties

Calculate the stiffness matrix of the structure in kips/inch.

K=k*(diag([2*ones(neig-1,1);1])+diag(-ones(neig-1,1),1)+diag(-ones(neig-1,1),-1));

Calculate the mass matrix of the structure.

M=m*eye(neig);

Set the spatial distribution of the effective earthquake forces. Earthquake forces are applied at all dofs of the structure.

r=ones(5,1);

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 inch/sec^2.

xgtt=9.81/0.0254*text{1,2};

Set the critical damping ratio of the response spectra to be calculated ($$\mathrm{\xi}=0.05$)

ksi=0.05;

Dynamic Response Spectrum Analysis

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

Plot the natural modes of vibration of the uniform five-story shear building

FigHandle=figure('Name','Natural Modes','NumberTitle','off');
set(FigHandle,'Position',[50, 50, 1000, 500]);
for i=1:neig
    subplot(1,neig,i)
    plot([0;Eigvec(:,i)],(0:h:h*neig)','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

Compare the eigenmodes with those shown in Figure 12.8.2. in Chopra (2012).

Plot the peak modal displacement response.

FigHandle=figure('Name','Displacements','NumberTitle','off');
set(FigHandle, 'Position', [50, 50, 1000, 500]);
for i=1:neig
    subplot(1,neig,i)
    plot([0;U(:,i)],(0:h:h*neig)','LineWidth',2.,'Marker','.',...
        'MarkerSize',20,'Color',[0 1 0],'markeredgecolor','k')
    xlim([-max(abs(U(:,i))) max(abs(U(:,i)))])
    grid on
    xlabel('Displacement','FontSize',13);
    ylabel('Height','FontSize',13);
    title(['Mode ',num2str(i)],'FontSize',13)
end

Plot the peak modal equivalent static force response.

FigHandle=figure('Name','Equivalent static forces','NumberTitle','off');
set(FigHandle, 'Position', [50, 50, 1000, 500]);
for i=1:neig
    subplot(1,neig,i)
    plot([0;f(:,i)],(0:h:h*neig)','LineWidth',2.,'Marker','.',...
        'MarkerSize',20,'Color',[1 0 0],'markeredgecolor','k')
    xlim([-max(abs(f(:,i))) max(abs(f(:,i)))])
    grid on
    xlabel('Static force','FontSize',13);
    ylabel('Height','FontSize',13);
    title(['Mode ',num2str(i)],'FontSize',13)
end

Calculate the peak modal base shear in kips.

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

Vb =

   60.4052   24.2322    9.8689    2.8754    0.5834

Calculate the peak modal base overturning moment in kips-ft.

Mb=zeros(1,neig);
for i=1:neig
    Mb(i)=sum(f(:,i).*(h:h:5*h)');
end
Mb

Mb =

   1.0e+03 *

    2.5467   -0.3500    0.0904   -0.0205    0.0036

Verification of figure 13.8.3 of Chopra (Dynamics of Structures, 2012)

Compare the peak modal responses with those shown in figure 13.8.3 of Chopra (2012).

Modal combination with the ABSolute SUM (ABSSUM) method.

Peak base shear.

VbAbsSum=ABSSUM(Vb);

Peak top-story shear.

V5AbsSum=ABSSUM(f(5,:));

Peak base overturning moment.

MbAbsSum=ABSSUM(Mb);

Peak top-story displacement.

u5AbsSum=ABSSUM(U(5,:));

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

Peak base shear.

VbSRSS=SRSS(Vb);

Peak top-story shear.

V5SRSS=SRSS(f(5,:));

Peak base overturning moment.

MbSRSS=SRSS(Mb);

Peak top-story displacement.

u5SRSS=SRSS(U(5,:));

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

Peak base shear.

VbCQC=CQC(Vb,omega,ksi);

Peak top-story shear.

V5CQC=CQC(f(5,:),omega,ksi);

Peak base overturning moment.

MbCQC=CQC(Mb,omega,ksi);

Peak top-story displacement.

u5CQC=CQC(U(5,:),omega,ksi);

Assemble values of peak response in a table.

Assemble values of peak response in a cell.

C{1,2}='Vb (kips)';
C{1,3}='V5 (kips)';
C{1,4}='Mb (kip-ft)';
C{1,5}='u5 (in)';
C{2,1}='ABSSUM';
C{2,2}=VbAbsSum;
C{2,3}=V5AbsSum;
C{2,4}=MbAbsSum;
C{2,5}=u5AbsSum;
C{3,1}='SRSS';
C{3,2}=VbSRSS;
C{3,3}=V5SRSS;
C{3,4}=MbSRSS;
C{3,5}=u5SRSS;
C{4,1}='CQC';
C{4,2}=VbCQC;
C{4,3}=V5CQC;
C{4,4}=MbCQC;
C{4,5}=u5CQC;
C

C = 

          []    'Vb (kips)'    'V5 (kips)'    'Mb (kip-ft)'    'u5 (in)'
    'ABSSUM'    [  97.9651]    [  56.2087]    [ 3.0113e+03]    [ 7.9562]
    'SRSS'      [  65.8938]    [  29.8774]    [ 2.5723e+03]    [ 6.7964]
    'CQC'       [  66.3294]    [  29.1493]    [ 2.5695e+03]    [ 6.7891]

Verification of table 13.8.5 of Chopra (Dynamics of Structures, 2012)

Compare the peak responses with those shown in table 13.8.5 of Chopra (2012).

Copyright

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


Published with MATLAB® R2012b

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