earthquake response spectrum analysis of 4 story shear building

8/13/2019 Earthquake Response Spectrum Analysis of 4 Story Shear Building

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Earthquake Response Spectrum Analysis of 4

Story Shear Building

[ Shear Building ] [ Modal Analysis ] [ Earthquake Response Spectrum ]

[ Modal Spectral Accelerations ] [ Modal Displacements ] [ Floor Displacements ]

[ Story Drifts ] [ Inertia Forces ] [ Shear Forces ] [ Overturning Moments ] [ Input and Output Files ]

In this example we compute the displacements, shear forces, story drifts, and over-turning moments in a 4 story

shear structure subject to moderate earthquake loads.

Loads due to ground motions are represented by a scaled earthquake acceleration response spectrum.

SHEAR BUILDING

Figure 1 is a schematic of the 4 story shear building, and its corresponding mass and stiffness matrices.

Figure 1 : Shear Building with Mass and Stiffness Matrices

The vertical distance between floors is 3m (see computations on overturning moments below).

We assume that all of the building mass is lumped at the floor levels, that the floor beams are rigid, and that the

columns are axially rigid. It follows from these assumptions that floor level displacements may be described by

one degree-of-freedom alone, with only four degrees of freedom being needed to describe total displacements
http://www.isr.umd.edu/~austin/aladdin.html


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throughout the structure.

The mass and stiffness matrices are simply given by

mass = ColumnUnits( 1500*[ 1, 0, 0, 0;

0, 2, 0, 0;

0, 0, 2, 0;

0, 0, 0, 3], [kg] );

stiff = ColumnUnits( 800*[ 1, -1, 0, 0;

-1, 3, -2, 0;

0, -2, 5, -3;

0, 0, -3, 7], [kN/m] );

and generate the matrix ouput we are already familiar with.

MODAL ANALYSIS

Input :The natural periods of vibration and modal shapes are computed are needed by the generalised mass,stiffness, and load vectors, and for the computation of displacements in each mode.

The relevant block of ALADDIN code is:

/* [c] : Calculate natural periods of vibration and mode shapes */

no_eigen = 3;

eigen = Eigen( stiff, mass, [ no_eigen ] );

eigenvalue = Eigenvalue( eigen );

eigenvector = Eigenvector ( eigen );

period = ColumnUnits( Matrix( [ no_eigen,1 ] ), [ sec ] );

for( ii = 1; ii


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Figure 2 : Modal Shapes and Periods of Vibration for Shear Building

The textual output is as follows:

*** SUBSPACE ITERATION CONVERGED IN 10 ITERATIONS

Mode 1 : w^2 = 117.8 1/sec^2 T = 0.5789 sec

Mode 2 : w^2 = 586.5 1/sec^2 T = 0.2595 sec

Mode 3 : w^2 = 1125 1/sec^2 T = 0.1873 sec

MATRIX : "eigenvector"

row/col 1 2 3

units

1 1.00000e+00 1.00000e+00 -9.00864e-01

2 7.79103e-01 -9.96239e-02 1.00000e+00

3 4.96553e-01 -5.39890e-01 -1.60784e-01

4 2.35062e-01 -4.37612e-01 -7.05724e-01

EARTHQUAKE RESPONSE SPECTRUM

Input :The block of ALADDIN code:

/*

* [a] : Setup Matrix for Piece-wise linear approximation to earthquake

* acceleration Spectra

*

* Column 1 : Natural Period (sec)

* Column 2 : Spectral Acceleration (at 2% damping)

*/


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print "*** ACCELERATION SPECTRA FOR MODAL ANALYSIS \n";

print "*** ======================================= \n";

npoints = 18;

spectra = Matrix( [ npoints , 2] );

spectra = ColumnUnits( spectra, [sec], [1]);

spectra = ColumnUnits( spectra, [cm/sec^2], [2]);

spectra [ 1][1] = 0.0 sec; spectra [ 1][2] = 981.0*0.15 cm/sec/sec; spectra [ 2][1] = 0.1 sec; spectra [ 2][2] = 981.0*0.18 cm/sec/sec;

spectra [ 3][1] = 0.2 sec; spectra [ 3][2] = 981.0*0.25 cm/sec/sec;







spectra [10][1] = 1.2 sec; spectra [10][2] = 981.0*0.19 cm/sec/sec;

.... details of spectra removed .....

spectra [18][1] = 3.4 sec; spectra [18][2] = 981.0*0.02 cm/sec/sec;

PrintMatrix( spectra );

defines a (18x2) matrix, spectra, for the acceleration response spectrum. The earthquake ground motions have a

peak ground acceleration of 0.15 g.

The acceleration response spectra is represented as piecewise linear segments, whose end-points are defined by

a (time, acceleration) coordinate.

Output :Figure 3 shows the lower- half of the earthquake acceleration response spectrum that we will use forthe analysis.


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Figure 3 : Acceleration Response Spectrum

The textual counterpart to Figure 3 is:

*** ACCELERATION SPECTRA FOR MODAL ANALYSIS

*** =======================================

MATRIX : "spectra"

row/col 1 2units sec m/sec^2

1 0.00000e+00 1.47150e+00

2 1.00000e-01 1.76580e+00

..... details removed ....

17 3.20000e+00 1.96200e-01

18 3.40000e+00 1.96200e-01

MODAL SPECTRAL ACCELERATIONS

Now that we know the systems natural periods, the next step is to compute the spectral accelerations

corresponding to each period.

The block of ALADDIN code:

/* [d] : Find Spectral Accelerations at Modal Periods */

SpectralAccn = ColumnUnits( Matrix( [ no_eigen,1 ] ), [ m/sec^2 ] );

for( ii = 1; ii


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for( ij = 1; ij < npoints; ij = ij + 1 ) {

period1 = spectra [ ij][1];

period2 = spectra [ij+1][1];

if(period [ii][1] >= period1 && period [ii][1] < period2 ) {

dAccn = spectra [ij+1][2] - spectra [ij][2];

dPeriod = (period [ii][1] - period1)/(period2 - period1);

SpectralAccn[ii][1] = spectra [ij][2] + dPeriod*dAccn;

}

}

}

PrintMatrix( SpectralAccn );

walks along the spectra array, and for each element in the matrix ``period'' (shown above), finds the spectral

acceleration corresponding to natural period of the mode. Modal periods that fall between array values are

estimated by linear interpolation.

Output :Figure 4 shows the acceleration response spectra with the modal periods and corresponding spectral

accelerations superimposed.

Figure 4 : Acceleration Response Spectra

The textual output from this computation is:

MATRIX : "SpectralAccn"

row/col 1

units

1 m/sec^2 4.13122e+00

2 m/sec^2 3.21070e+00

3 m/sec^2 2.36550e+00


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A quick comparison of the elements in `period'' and `spectra'' should reveal that these spectral accelerations

match the computed natural periods.

MODAL DISPLACEMENTS

Input :The displacements in each mode are a function of the generalised mass, stiffness and load, and the

spectral displacement (the relevant formulae may be found in standard texts on seismic design/analysis).

The block of ALADDIN code:

/* [e] : Generalised mass, stiffness, and loading matrices */

eigenTrans = Trans (eigenvector);

gmass = eigenTrans*mass*eigenvector;

gstiff = eigenTrans*stiff*eigenvector;

gload = eigenTrans*mass*[ 1; 1; 1; 1 ];

PrintMatrix( gmass, gstiff, gload );

/* [f] : Compute and print floor level displacements */

Y = ColumnUnits( Matrix([no_eigen,no_eigen]), [ m ] );

for( ii = 1; ii


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Figure 5 : Modal Displacements for Shear Building

The textual output is:

MATRIX : "gmass"

row/col 1 2 3

units kg kg kg

1 4.30934e+03 2.84217e-13 -1.13687e-13

2 1.13687e-13 3.26599e+03 -4.54747e-13 3 -2.27374e-13 -2.27374e-13 6.53610e+03

MATRIX : "gstiff"

row/col 1 2 3

units N/m N/m N/m

1 5.07691e+05 -2.18279e-11 1.16415e-10

2 -1.16415e-10 1.91539e+06 -1.16415e-10

3 2.32831e-10 -2.32831e-10 7.35295e+06

MATRIX : "gload"

row/col 1

units

1 kg 6.38475e+03

2 kg -2.38779e+03

3 kg -2.00941e+03

MATRIX : "Y"

row/col 1 2 3

units m m m

1 5.19545e-02 0.00000e+00 0.00000e+00


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2 0.00000e+00 -4.00257e-03 0.00000e+00 3 0.00000e+00 0.00000e+00 -6.46441e-04

MATRIX : "modaldispl"

row/col 1 2 3

units cm cm cm

1 5.19545e+00 -4.00257e-01 5.82355e-02

2 4.04779e+00 3.98752e-02 -6.46441e-02

3 2.57982e+00 2.16095e-01 1.03938e-02 4 1.22125e+00 1.75157e-01 4.56209e-02

Points to note:

As expected, the generalised mass and stiffness matrices, gmass and gstiff, are diagonalized, thereby

indicating the eigenvector tranformation has decoupled the equations of motion.

Notice how we have put the modal displacements "Y" is a (3x3) matrix, thereby simplifying the

subsequent calculations.

FLOOR DISPLACEMENTS

Input :In this section we compute the maximum floor displacements possible by summing the absolute value

floor displacements in each mode.

print "\n";

print "Maximum Possible Floor Displacements (absolute values) \n";

print "====================================================== \n\n";

print " Floor Mode Modes Modes\n";

print " No 1 1 & 2 1, 2 and 3\n";

for( ii = 1; ii


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Clearly, mode 1 makes the greatest contribution to overall floor displacements.

STORY DRIFTS

Input :The story drifts in each mode are defined as the displacement of a floor relative to the floor below it.

In the block of ALADDIN code:

/* [g] : Compute and print story drifts */

print "\n";

print "Maximum Likely Story Drifts (using SRSS) \n";

print "======================================== \n\n";

print " Story Mode Modes Modes\n";

print " No 1 1 & 2 1, 2 and 3\n";

for( ii = 1; ii


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Input :The inertial forces for each mode are simply given by the product of the stiffness matrix times the modal

displacement. The relevant ALADDIN code is:

/* [h] : Compute and print equivalent d.o.f. forces in each mode */

print "\n";

print "Inertia Forces for each mode \n";

print "============================ \n\n";

inertia_forces = stiff*modaldispl; PrintMatrix( inertia_forces );

Ouput :Figure 6 shows the distribution of inertia forces in each of the first three modes of vibration.

Figure 6 : Distribution of Inertia Forces

The textual output is:

Inertia Forces for each mode

============================

MATRIX : "inertia_forces"

row/col 1 2 3

units N N N

1 9.18127e+03 -3.52106e+03 9.83037e+02

2 1.43063e+04 7.01546e+02 -2.18364e+03

3 9.11798e+03 3.80201e+03 3.55155e+02

4 6.47450e+03 4.62254e+03 2.30532e+03


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SHEAR FORCES

Input :The base shear forces in each mode are shown in Figure 6, and correspond to the sum of inertial forces

over the height of the structure.

In the block of ALADDIN code:

/* [i] : Compute and print base shear force */

print "\n";

print "Shear Forces (at base of the structure) \n";

print "======================================= \n\n";

base_shear_forces = [1,1,1,1] * inertia_forces;

PrintMatrix( base_shear_forces );

shear1 = 0.0 N^2; shear2 = 0.0 N;

for ( ii = 1; ii


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/* [j] : Compute and print overturning moments */

print "\n";

print "Overturning Moments (at base of the structure) \n";

print "============================================== \n\n";

floor_heights = [12 m, 9 m, 6 m, 3 m];

base_overturning_moments = floor_heights * inertia_forces;

PrintMatrix( base_overturning_moments );

mom1 = 0.0 (N*m)^2; mom2 = 0.0 N*m;

for ( ii = 1; ii

earthquake response spectrum analysis of 4 story shear building

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