Seismic Response Analysis-3

8/3/2019 Seismic Response Analysis-3

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SEISMIC RESPONSE ANALYSISOF

STRUCTUTRES

KAUSHIK CHATTERJEE

DCE (CIVIL), NPCIL


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Uncoupled Equations of motionUncoupled Equations of motion

unun--dampeddampedEquation of motion of an un-damped MDOF

system is

Substituting from eq. (1), we get,

Pre-multiplying by the transpose of the nth mode

shape vector, Tn ,we get,


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unun--dampeddampedNow let us introduce new symbols as,

These are called the normal coordinate

generalized mass

, generalized stiffness and

generalized loadfor mode n respectively.


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unun--dampeddamped

Thus the eqn. of motion of the MDOF system

can be written as,

Which is a SDOF system equation of motion for

mode n.

We have learnt earlier that,

Multiplying both sides by we get,


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unun--dampeddamped

Hence, a set of N simultaneous differentialequations which are coupled by the off-diagonalterms in the mass and stiffness matrices, gets

converted into a set of N independent normal-coordinates equations.

The dynamic response can be obtained bysolving separately for the response of eachnormal (modal) coordinate and thensuperposing these by eq. 1 to obtain theresponse in the original coordinates. This is

called ModalSuperposition Method.


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dampeddamped

The damped eqn. of motion is,

Normal coordinates transformation can be used to

uncouple the damped equation of motion.

Using eqn. (1) and pre-multiplying by the transpose

of the nth mode shape vector, Tn ,we get,

The orthogonality property causes all componentsexcept the nth mode term in the mass and stiffness

expressions of the eqn. to vanish.


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dampeddampedIf it is assumed that the corresponding

orthogonality condition applies to the damping

matrix also, then the above eqn. can be written

as,

Or,


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dampeddamped

Where,

n is the generalized damping ratio for mode n.

n = cn/cc = cn/2Mn n

The normal coordinate generalized mass,

stiffness and load are identical to those for the

un-damped system.


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TIME HISTORY ANALYSISTIME HISTORY ANALYSIS

Required to : Obtain time varying response of structures

due to dynamic excitation. (Example Turbo-

generator foundation subjected to dynamic

machine load)

Generate floor time history & corresponding

floor response spectra at various floor levels

of multistoried buildings, which are used toseismically qualify equipments located on that

particular floor.


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TIME HISTORY ANALYSISTIME HISTORY ANALYSIS.

Ug U


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In case of seismic excitation of a MDOF system

(say a multistoried building) the forces

generated by the seismic excitation within the

structural members are due to relativedisplacementbetween the stories.

No external force is applied on the structure assuch.

.


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Hence.

Substituting this in the equation of motion, we

get

As the negative sign on the RHS has no real

significance in case of seismic analysis, it is

omitted in the subsequent equations.


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Applying the concept of generalized coordinatesu=z and pre multiplying by

.

we get the equation of motion in generalized

coordinates as

Where, is called theModal earthquake excitation factor


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If the natural mode of vibration isthe natural frequency of vibration is

and the damping ratio is

The equation of motion for the nth

mode ofvibration of the idealized multi-storey building

can be expressed as

.

The ratio Ln/Mn is called

Modal participation factor


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The solution to this equation can be written interms of the Duhamel integral (damped system)

Thus the modal displacement is

The contribution of the nth mode to the

displacement at jth floor is given by

, j=1, 2, 3, ., N


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The deformation or drift in storey j is given by

the difference of displacements of the floors

above and below

Now find out the equivalent lateral force fn(t),

which, if applied as static forces, would cause

structural displacements un(t). Thus

i.e.,


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If we express the forces in terms of the massmatrix, then

Thus, the equivalent force at jth floor, which

causes lateral displacement ujn(t), is

The inertial force in the structure can be

determined by a static analysis, applying these

equivalent lateral forces.


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Then the earthquake response of the structureis obtained by combining the modal responses

in all the modes of vibration.

The total value of any response r(t), is thecombination of the contribution of all the

vibration modes to that response quantity


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RESPONSE SPECTRUM ANALYSISRESPONSE SPECTRUM ANALYSIS

Response spectra is a plot of the maximumresponse of a series SDOF system having

different natural frequencies, for a particular

value of damping.


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As the response in each vibration mode can bemodeled by the response of a SDOF system, the

maximum response in the mode can be directly

computed from the response spectra.

In design Max. response, rather than response

history is required.


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To evaluate the earthquake response of a MDOF

system at any time:

Evaluate the earthquake response integral at

that time for each significant response mode.

To evaluate the maximum response:

Compute the modal response as before, for

each time during the earthquake history,

compare them and find the maximumresponse. ---- Time consuming calculation

involved.


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Analysis based on ground motion responsespectra comes as an alternative handy tool.

For each individual mode of the structure, themaximum response can be obtained directly

from the response spectra.The maximum response in nth mode can beexpressed in terms ofSdn, Svn, and San which arethe ordinates of the displacement, pseudo

velocity and pseudo acceleration responsespectra respectively, corresponding to thevibration period and damping ratio of the mode.


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The maximum modal displacement is

The maximum displacement at the jth floor is

The maximum drift in the jth storey is

The maximum lateral force at the j-th floor is


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The base shear is given by

Substituting the value of fjn, we get

The relation between Sdn, Svn, and San is


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Steps involved: Determine the response spectrum for the

ground motion if not already available.

Compute mass and stiffness matrices m and k. Estimate modal damping ratios.

Solve the eigen value problem to determine

the natural frequencies and mode shapes.


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Compute the maximum response in individualmodes of vibration by repeating the following

steps:

Corresponding to the period and damping

ratio, read the ordinates Sdn, and San of the

displacement and pseudo-acceleration

response spectra of the ground motion.

Compute floor displacements.

Compute storey drifts from floor

displacements.


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RESPONSERESPONSE SPECTRUM ANALYSISSPECTRUM ANALYSIS

Compute equivalent lateral forces.Compute internal forces storey shears and

moments by static analysis of the structuresubjected to equivalent lateral forces.

Determine an estimate of the maximum ofany response (displacement of a floor,deformation in storey, shear or moment in astorey, etc.) by combining the modal maximafor the response quantity

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Seismic Response Analysis-3