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STUDY OF ENVELOPES OF SEISMICRESPONSE VECTORS

PREPARED BY:

UNDERGRADUATE STUDENTS B.E. CIVIL ENGINEERING

BODHI SUNDER RUDRA

GAURAV KUKREJA

SAURABH GUNECHA

VAIBHAV MITTAL

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INTRODUCTION

In engineering structures subjected to seismic ground motion, the responsespectrum method is commonly used to estimate the maximum values ofresponses. With this method, individual peak modal responses are obtainedusing a prescribed set of response spectra that characterizes the groundmotion expected at the location of the structure. These modal maxima arethen combined ( by ABSBSUM, SRSS, CQC) in an appropriate manner to

estimate the maximum values of responses of interest.

But, , the conventional response spectrum method is ideally suited to thedesign or analysis of structural elements that are controlled by the maximumvalue of a single response quantity, e.g. a beam governed by maximumbending moment. For members in which simultaneous action of multipleseismic responses must be considered e.g. in a column subjected to anaxial load and bending moment, the critical combination of responses maynot coincide with maximum value of any of the responses. For such cases,an envelope that bounds the evolution of vector of seismic responses intime is desirable. This envelope can then be superimposed over thecapacity surface( interaction diagram) of the member to determine thecritical combination of responses.

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In current practice, response spectrum based estimates of the individualresponse maxima to construct a rectangular envelope for this purpose. Thisenvelope provides an upper bound that can be overly conservative formany design situations.

In this report, a response-spectrum-based procedure for predicting theenvelope that bounds a vector of seismic response in a linear structure isdeveloped. When the principal directions along which the ground motion

components are un- correlated are known, the envelope that bounds avector of seismic responses in a linear structure is an ellipsoid that isinscribed within the rectangular envelope described above. Naturally,elliptical envelope provides a tighter bound on a vector response processthan the rectangular envelope.

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RESPONSE VECTOR

For an N-degree-of-freedom linear and classically dampedstructure, x(t) is a time-varying m-vector of responses, whereeach response component is expressed as a linear function ofthe nodal displacements u(t)

When the structure is subjected to three translational

components of ground motion, it satisfies

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MODELLING

For the purpose we choose a singlestory frame and designed it in STAADPro for the loading as per IS 875, andIS 1893:2002 for seismic loading usingI-sections as beams and columns.

The dimension of the frame chosen is4X4X4, with rigid joints and fixed jointrestraint at all four supports.

Although all computations for stiffness,mass, stodal law, etc., are done inMATLAB or MS- EXCEL but for purposeof checking the results obtained fromtime history analysis we are using SAP2000.

SAP 2000 model of THE CHOOSENFRAME showing local axes andFIXED support restraints

http://sap3d.jpg/

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Preliminary Calculations

For the purpose of time history analysis of the model, we have lumped mass of thestructure at 4 upper joints of the frame and the mass source is just the mass of ourstructure, consisting of ISMB 200 as beams and columns.

In the model, 6 degrees of freedom are assumed at each joint location (3 rotationaland 3 translational), making a total of 24 DOFs.

The mass matrix and stiffness matrix for the structure is assembled by firstcomputing mass, stiffness matrices of individual member elements and then

combining these to produce 24 X 24 MASS MATRIX and 24 X 24 STIFFFNESSMATRIX.

Once the mass and stiffness matrices are obtained, mode shape vectors andnatural frequency of the model is computed using the STODLA method to obtain

solution for the Eigen value equation.

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Each eigen value and eigen vectorsatisfies the equation:

When written in matrix form theequation becomes;

The matrix PHI is the modal matrix forthe eigen value problem.

The N (DOFs=24 for present case)eigen values can be assembled into adiagonal matrix which is known asspectral matrix.

.

MODAL MATRIX

SPECTRAL MATRIX

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After obtaining the mass, stiffness matrices the damping matrix is generated.

Once we have all 3 matrices namely, stiffness, mass, and damping matrices we can write theequation of motion for this MDOF system in matrix form.

When the structure is subjected to three component of earthquake ground motion , u(t) satisfies:

Where M = mass matrix, C =Rayleighs damping matrix, and K = stiffness matrix and the terms inthe equation of motion is inertial force, damping force, stiffness force respectively.

The term on the RHS represents the dynamic force on the structure due to ground motion.

I = influence matrix.

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EARTHQUAKE RESPONSEHISTORY

18. The time series used to generate the RHS of the equation of motion is North southcomponent of El-Centro Earthquake. The graph for the function is shown below.

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The equation of motion is solved using MODAL DECOMPOSITION METHOD, which uncouples the Ndifferential equations.

Replacing u(t) by (shown below) in the equation of motion:

and on pre-multiplying the equation by (phi)transpose we get N uncoupled equation.

For i=1 to N the N uncoupled equations are:

where

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DUHAMEL INTEGRAL

The N uncoupled differential equations thus generated can be solved using Duhamel integral.

The final expression for u(t) becomes:

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CQC MODAL COMBINATIONRULE

Complete Quadratic Combination rule was developed by Dr. Der Kiureghian forpredicting the peak value of a single response quantity resulting from all componentsof ground motion, according to which we find the response matrix of thecorresponding member by:

Where ki is the modal participation factor; ij being the correlation coefficient

between the modal responses dki and dkj

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; being the angle of attack and Dki=E[max dki(t)] is the

mean displacement response spectrum ordinate formode i due to the kth principal component of the ground

motion.

Our next step is to plot envelopes for response vectorsin linear structures