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Dynamic analysis of pipe stress : Mathematical foundation andapplications

BY : SH. M. G. CHOUDHURY : VICE PRESIDENT ( PIPING)E-MAIL : ghosh_chowdry@ril.com

ANINDYA BHATTACHARYA (ASSTT. MANAGER)E-MAIL : anindya_bhattacharya@ril.com

SH. HIMANSHU VARSHNEY (ASSTT. MANAGER)E-MAIL :himanshu_varshney@ril.com

RELIANCE ENGINEERING ASSOCIATES LTD. JAMNAGAR.

Dynamic analysis of pipe stress : Mathematical foundation andapplications

1.Introduction :1.1 Importance of dynamic analysis in piping systems :

Piping is the backbone of any process industry and safety and reliability of plants is to a large extentdependent on the accuracy of stress analysis of critical piping systems . The loads on piping systems arecategorised as “ sustained “ comprising of pipe weight, along with fittings, fluid weight, insulation etc. aswell as “operating “ in which thermal loads as a result of expansion and contraction of the system are takeninto account. Besides there will be occasional loads on the piping systems like seismic, impact loads due toslug flow, relief valve opening, fluid hammer etc. Although static analysis for the occasional loads ispossible using values of dynamic load factors or static earthquake factors, an exact simulation of thebehaviour of piping systems under transient loading conditions is not possible and calls for rigorousdynamic analysis. Fundamental differences between the two methods can be explained as :

a) Static loads are applied slowly in a system, as a result of which the system gets enough time to reactand internally distribute the loads, thus maintaining equilibrium and the pipe does not move

(summation of forces and moments both equal to zero)

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b) With a dynamic load, ( which changes quickly with time), the system may or may not have enoughtime for load distribution ,resulting in unbalanced forces and moments and therefore pipe movement.Since the sum of forces and moments in such a system need not be zero, the internally induced loadsmay be different, either higher or lower than the applied loads.

There are two basic approaches to perform dynamic analysis of a piping system, one to perform beforecommissioning of an equipment in a piping system another to evaluate the safety of a piping system againstsome continued vibration problem. Many times the first approach is not feasible as vibration measurement( as with lines connected to rotary equipments ) may be required . However with lines connected toreciprocating equipment’s and dynamic analysis of seismic loading can be done before commissioning.

1.2 Dynamic analysis types :

Three typical force vs. time profiles are encountered in piping systems, namely random ( say earthquake ),harmonic( say a rotary pump ) and impulse types ( say fluid hammer ). The usual methodologies used in theanalysis of these different load profiles are :a) Spectrumb) Harmonicc) Time History

However , loading with random profile can also be solved using the Time History approach ( as withEarthquake loading ) and impulse type loading can also be solved using spectral method. Thesemethodologies will be discussed in detail in the following sections . In this paper the analysis proceduresfor the different force vs. loading profiles, emphasising the mathematical foundations behind them will behighlighted with some specific application guidelines keeping in view the commercially available pipestress programmes.

2. Harmonic analysis :2.1 The harmonic equation :

In Dynamic analysis , the no. of Degrees of freedom ( DOF) of a system is defined as the no. ofindependent co ordinates required to describe the displaced position of the system . The basic equation ofdynamic equilibrium for a multiple degree of freedom or DOF (say an N degree of freedom system ) can bewritten as :

M d2x(t)/dt2 + C dx(t)/dt + Kx(t) = F(t) … … … … … … … … (1)

Where :

M= System mass matrix.K= System stiffness matrixC= System damping matrixx(t) = displacement vector as a function of time.dx(t)/dt = velocity vector as a function of time.d2x(t)/dt2 = acceleration vector as a function of time.

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In Harmonic analysis the right hand side of eqn. 1 is of the form :F(t) = Fo sin (ωt + Φ i) , where the symbol Φ i stands for the input phase angles reflecting the timingbetween the applied forces.

3. The eigenvalue problem3.1 The problem formulation :

In general one of the fundamental objective of dynamic analysis is to extract the natural frequencies of thesystem. In doing so usually the damping of the system is neglected

Neglecting damping the free vibration equation can be written as :

M d2x(t)/dt2 + Kx(t) = 0 … … … … … … … … … … … … … … … (2)

Considering sinusoidal response , we get the matrix eigenvalue problem

(K- M ωn 2 ) φn =0 … … … … … … … … … … … … … … … … … … … … … … … … … … . (3)

The non trivial solution of which requires det(K- M ωn 2 )=0The solution of this eigenvalue problem gives the N ( for a N degree of freedom system ) naturalfrequencies and the N mode shapes matrix where each column represents the mode shape for aparticular natural mode of vibration. Solution methods for the eigenvalue problem include numericalmethods like inverse vector iteration with or without spectral shifts , Gram- Schmidt orthogonalization,subspace iteration, etc. A detail discussion on these methods can be found in reference (5) :

The eigen value problem determines the natural modes to only within a multiplicative factor. Scalefactors are sometimes applied to natural modes to standardise their elements associated withamplitudes in various DOFs. This process is called normalization. Sometimes it is convenient tonormalize each mode so that its largest element is unity. In most computer programs, it is commonto normalize modes so that Mn ( mass normalization ) have unit values , where:

Mn = φn T

M φn = 1 … … … … … … … … … … … … … … … … … (4)

Or φ T M φ = II is the identity matrix.φ is the mode shape matrix ( N x N ) = [φin]where j denotes dofs and n denotes modes.

One important parameter in structural dynamics is the orthogonality of normal modes.

φnTK φr = 0 and … … … … … … … … … … … … … … … … … … (5)

φnT M φr = 0 … … … … … … … … … … … … … … … … … … … … ..(6)

for r not equal to n.where ωn is the natural frequency of nth mode. ωr is the natural frequency of rth mode. φn is the mode shape vector for the nth mode φr is the mode shape vector for the rth mode

One physical implication of the modal orthogonality is that the work done by the nth modeinertia forces in going through the rth mode displacements is zero. In dynamic analysis of pipingsystems the mathematical model usually comprises of lumped mass at node points , connected bymassless springs in between. The lumped mass at a node is determined from the portion of the

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weight that can reasonably be assigned to the node. The lumped mass at a node of the system is

the sum of the mass contributions of all the elements connected to the node.The mass matrix formed by this idealization is known as the lumped mass matrix. Since therotational inertia has negligible influence on the dynamics of practical systems hence in generalfor a lumped mass idealization, the mass matrix is diagonal:

Mij = 0 for i ≠ j Mij = Mj or 0

Where Mj is the lumped mass associated with the jth translational DOF and Mij = 0 for arotational DOF.

4.0 Damping matrix :4.1 The significance of the damping matrix :

One might think that it should be possible to determine the damping matrix for the structure fromthe damping properties of individual structural elements just as the structural stiffness matrix isdetermined . However it is impractical to determine the damping matrix in this manner, becauseunlike the elastic modulus , which enters into the computation of stiffness the damping properties ofmaterials are not well estimated.

The damping matrix for a system should be determined from its modal damping ratios whichaccounts for all energy dissipating mechanism. In this paper only on the classical damping matrix.(classical damping is an appropriate idealisation if similar damping mechanism are distributedthroughout the structure) will be highlighted . In computer programmes for stress analysis, the dampingmatrix is typically used in Harmonic analysis.

A classicial damping matrix consistent with experimental data is the Rayleigh damping.

C = a0 M+ a1 K … … … … … … … … … … … … … … … … … (7)

The damping ratio for the nth mode of such a system is

ξn = a0 / 2ωn + (a1 / 2) ωn … … … … … … … … … … … … … … .(8)The constants a0 and a1 can be determined from specified damping ratios ξi and ξj for the ith andjth mode respectively.If the modes are assumed to have the damping ratio ξ which is reasonably based on experimentaldata :

a0 = ξ 2ω I ω j / ( ωI + ω j ) … … … … … … … … … … … … … (9)a1 = 2 /( ωI + ω j ) … … … … … … … … … … … … … … … .(10)

For practical problems a0 is extremely small and can be neglected. In this form only usually thedamping matrix is used for mathematical modelling of a harmonic system. If the modal frequencies arenot known prior to generation of the damping matrix , the forcing frequency ω can be used inplace of the natural frequency of a mode. When the forcing frequency of the load is in thevicinity of a modal frequency this gives a good estimation of the true damping. Typical values ofdamping ratio for piping systems , as recommended in USNRC Regulatory guide 1.61 and ASMEcode case N-411 range from 0.01 to 0.05 based upon pipe size, earthquake severity and thesystems natural frequencies.

5.0 Method of solving the dynamic equilibrium problem in Harmonic analysis as well as the generalmodal analysis principle :

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Usually computer programs solve the dynamic equilibrium problem ( as in Harmonic analysis ) by directsolution of the equation using the damping matrix C as linearly dependent on the mass and the stiffnessmatrix.( The Rayleigh damping as already discussed)This is possible as the force vector is well defined in Harmonic analysis. The more general approach, themodal combination approach can be described as :The dynamic response of a system can be expressed as :

x(t)=Σ φr qr(t) summed for r=1 to N … … … … … … … … (11)

for a physical representation of modal superposition using lumped mass , refer to fig. (2)

Where qr = modal co ordinate for the r th mode.

qr = φr T M x / φr

T Mφr = φr T M x /Mr … … … … … … … … .(12)

The resulting differential equation ( dynamic equilibrium eqn. ) can now be written as :

Mn qn'' (t)+∑Cnrqr

' (t)+Kn qn (t) = Pn (t) … … … … … … … … … … … .. (13)

(Using the modal orthogonality relations. )whereMn = φn

TMφn

Kn = φn TKφn

Cnr = φn TCφr

The modal equations will be uncoupled for classical damping and linear systems.

For a Harmonic system, Pn (t) will be of the form Pno sin (ωt + φi n ) where φi n are the input phase angles.Accordingly the solution will be of the form

x(t) = A Sin (ωt + φr1 ) +B Sin (ωt + φr2) + … … … … … . (14)

Where φr1, φr2 etc. are “response phase angles “, indicating the time lag between imposed and responseparameters.

So far computer programmes are concerned infinite values of x(t) for infinite time durationcannot be shown as output. So some programmes ( Caesar II ) show the output for different valuesof t ( different values of angle ωt ) where ωt symbolises the infinite angles between 0 and 360 deg.The range 0 – 360 deg. is usually divided into 18 intervals of 20 deg. each and results aretabulated for each value of wt ranging from 0, 20,40 … ..degs. etc. Once the displacements arecomputed forces and moments can be computed easily.

6.0 A typical Harmonic analysis problem :6.1 Excitation of a piping system connected to a rotary equipment, (say a centrifugal pump.)

Usually in standard computer programmes, the user is has to give Force and or displacement as an input.The objective of giving the displacement input is to convert it into force input. Since it is difficult tocorrectly quantify Harmonic forces, the inputs in the computer programmes can best be given asdisplacements at the point (s) where the piping is connected to the source of excitation ( in this casethe connection to the pump nozzle) The displacement will be converted to a force by multiplyingwith the stiffness.

Many times the user makes a mistake of giving the displacement inputs at other points (where thepipe is not connected to the source of excitement ) and try to check the displacement output fromthe computer programme at those nodes with the measured values to check the correctness of the

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modelling. This approach is wrong as in this case what we are measuring is the “responsedisplacement” , which the programme has to calculate and not the “imposed displacement” whichshould be the input. The difference between the “imposed and the response” displacements can beunderstood from this analogy :

When we crack a whip, we move the end in our hand a couple of inches ( the imposeddisplacement ) and the other end of the whip slings across the room ( the response displacement ).To summarise the imposed displacement is imposed on the piping system from an external source,some thing stronger than the pipe which moves and moves the piping system with it. The excitingfrequencies are the pump speed and multiples thereof.

Force vs. time profiles which are random in nature are best solved by response spectrum method. (However time history analysis can also be done). We will highlight the response spectrum method ,particularly with reference to dynamic analysis of earthquake.

7.0 Response spectrum method7.1 The underlying theory :

A Response Spectrum for an earthquake load can be developed by placing a series of single DOFoscillators on a mechanical shake table and feeding a typical ( typical for a specific site )earthquake time history through it , measuring the maximum response ( displacement , velocity oracceleration ) of each oscillator . The Response spectra can be plotted for different damping valuesof a system. The methodology used in Response spectrum analysis to calculate the systemresponses can be summarised as :a) Determining the mass matrix M stiffness matrix K and modal damping ratios ξn of the system.(

which is typically considered to be same for all modes in a piping system )b) Determining the natural frequencies and natural modes of the system.c) Computing the peak response of the nth mode of the system by the following steps to be

repeated for all modes. n = 1,2,3,4,… … … .1) Corresponding to the natural period Tn and damping ratio ξn (which can be considered samefor all modes ) , reading the values of Dn and An, the deformation and pseudo-acceleration from theresponse spectrum, where the pseudo acceleration is defined as the product of the maximum relativedisplacement ( relative w.r.t ground motion ) times square of the angular natural frequency i.e. An = ωn 2.Dnwhere Dn= u0 the maximum relative displacement.

Computing the displacement at DOFs in mode n by the equation :

xjn = Γnφjn Dn … … … (15)

Where Γn is the modal participation factor, φjn is the mode shape

Γn is defined as = φn TMυ / φn

TMφn where υ is the influence vector represents the displacement of themasses resulting from static application of a unit ground displacement.

b) Computing the equivalent static lateral forces from

fjn = Γn mj φjn An … .. (16)

c) By static analysis of the system subjected to forces fjn the peak value rn of the n th mode contribution to aresponse quantity r is determined.

d) The total response is found by addition of the peak modal responses.

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7.2 Modal combination methods :

The different methods are ( in this paper not all methods are highlighted ) ;

a) Absolute : This method states that the total response is equal to the sum of the absolute values of theindividual modal responses.

The system response can be calculated as :

R= Σ Ri summation from i= 1 to N … … … (17)

This method gives the most conservative result, since it assumes that all modal responses occur at exactlythe same time during the course of the applied load. This is over conservative since the modes withdifferent natural frequencies will probably experience their maxima at different times during the loadprofile.

Square root of sum of squares : (SRSS) This method states that the total response is equal to the squareroot of the sum of squares of the individual modal responses.

R = Σ [ Ri2 ]1/2 summed from i = 1 to N … … … … … … ..(18)

This method is based on statistical assumption that all modal responses are completely independent, withthe maxima following a relatively uniform distribution throughout the duration of the applied load.

This method is non conservative especially if there are modes with very close frequencies, since thosemodes will probably experience their maximas at approximately the same time during the load profile.

Group method :This method requires that modes with “ closely spaced frequencies “ ( typically within10%of each other ) are correlated and should be added absolutely, those sums and all other frequencies arethen considered to be uncorrelated and combined using the SRSS method.

Many times the stress analyst may not be equipped with a response spectrum suitable for his site location.Keeping this in mind, most computer programmes have in built data base for response spectrum.

Some typical available data bases include response spectrum as per United states Nuclear regulatoryguide 1.60 ( based on different damping values like 5%,7% etc )or as per Uniform building codes ( fordifferent soil types.

These response spectras are normalized with respect to peak ground acceleration ( zero periodacceleration or ZPA ) of 1g ( which may be changed for specific site requirement ).As per UBC (1994) thespectrum values have to be scaled by product ZN,( for design basis earthquake ) where Z is the seismiczone factor and N the near field coefficient relating the sites proximity to an active fault.( UBC gives thisrecommendation for “ Seismically isolated structures but not for “ Non Seismically isolated structures “.However it is reasonable to consider the same recommendations for “ Non Seismically Isolated structures“.). The UBC ( 1994) Response Spectra is shown in Fig. (1).

If the piping system is subjected is subjected to multiple support excitement, then different spectras arebeing introduced into the piping system through the vibrations of separate sets of restraints. Thisdifferential displacement gives rise to a quasi static force , the effects of which has to be added with thedynamic analysis results by any combination ( say ABS or SRSS ) methods.

8.0 The Force spectrum method8.1 Fundamentals of the method :

A slight variation of the seismic response spectrum method can be used to solve problems pertaining toloading such as opening of relief valve, fluid hammer, slug flow etc. Here, instead of displacement, velocityor acceleration spectrum used for the seismic problem, a dynamic load factor is used. A dynamic load

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factor can be defined as ratio of maximum dynamic displacement to the maximum static displacement ).Just as for the earthquake, the time history of the loading can be applied to a shake table of SDOF bodieswith a response spectrum ( in this case DLF vs. natural frequency ) by dividing the maximum oscillatordisplacements by the static displacements expected under the same load.

9.0 The non extracted modes :9.1 Method of inclusion of these modes in the final computation

In dynamic analysis of a piping system usually the analyst does not go in for extraction of all the modes ofvibration . Also the higher modes usually do not cause considerable differences in the final result. Howeverin order to find out the effects of the non extracted modes the following procedure is followed :

The mode shape martix can be partitioned as :

φ = [ φe φr ] … … … … … … … … … … … … … … … … … … … (19)φe = mode shapes extracted for dynamic analysis ( i.e. lower frequency modes )φr = residual ( non extracted ) mode shapes .

The displacement components can be expressed as linear combinations of mode shapes as per eqn. (11)

Let qe = partition of q matrix corresponding to extracted modes. qr = partition of q matrix corresponding to residual modes.

The dynamic load vector can be expressed as :

F= Kφq = Kφe qe + Kφr qr … … … … … … … … … … … .. (20)

Multiplying both sides by φeT and using modal orthogonality relations as per eqns. ( 5 ) & (6) the residual

force comes out to be Fr = F- φeT Mφe F

The missing load is applied to the structure as a static load. This static structural response can then becombined with the dynamically amplified modal responses by either ABS or SRSS method.

The ABS obviously produce the more conservative result and is based upon the assumption that thedynamic amplification is going to occur simultaneously with the maximum ground acceleration. Sincemodal and rigid responses are statistically independent, so that an SRSS combination is more accuraterepresentation of reality.

10. Modal and spatial combination philosophy :

When performing a spectrum analysis, each of the modal responses must be summed. In addition ifmultiple shocks have been applied to the structure in more than one direction, the results from the differentdirections also must be summed. For example, if we apply shocks in X, Y and Z directions and saycalculate the force Fx ( force in X direction ) at a particular node, Fx can be calculated as the sum of all theFx at the node in all modes caused by the X component of the shock. However since shocks are applied inY and Z directions also besides the X shock, Fx responses will be there at that node due to thesecomponents and the sum will be the modal combinations of all the Fx values caused individually by the Yand Z components respectively. To answer the question whether , we should sum the Fx caused by shocksin X , Y and Z directions in one mode and then go on adding the values (calculated likewise) in all themodes by the modal summation method ( spatial combination first )or we sum the modal responses causedby the X component first, repeat for Y and Z components and then add all the values of Fx ( modalcombination first ) , we need to consider the following :

A difference in the final results arises whenever different methods are used for the spatial and modalcombinations.

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A combination of spatial components first implies that the shock loads are dependent, while thecombination of modal components first implies that the shock loads are independent.

Dependent and independent refer to the time relationship between the X ,Y and Z components of theearthquake. With a dependent shock case, X,Y and Z components of the earthquake have a directrelationship, a change in the shock along one direction produces a corresponding change in the otherdirections. This would be the case when the earthquake acts along a specific direction having componentsin more than one axis, such as when a fault runs at a 30 degree angle between the X and Z axes.

An independent shock is one where the X, Y and Z time histories produce related frequency spectra buthave completely unrelated time histories. It is the independent type of earthquake that is far more commonand thus in most cases the modal components should be combined first. As a reference : IEEE 344-1975 (IEEE recommended practices for seismic qualification of class 1E equipment for Nuclear power generatingstations ) states :

“ Earthquakes produce random ground motions which are characterised by simultaneous but statisticallyindependent horizontal and vertical components “ This is usually less of an issue for Force spectrum combinations since normally there are no spatialcombinations to combine i.e. there are no X, Y and Z components acting simultaneously. However in theevent that there is more than one potential force load ( such as when there is a bank of relief valves that canfire individually or in combination ), If two relief valves may or may not fire simultaneously ( i.e. they areindependent )and the combination method should be modal before spatial. If under certain circumstancesthe two valves will open simultaneously ( i.e. the loadings are dependent, ) the combination method shouldbe spatial before modal. Since directional forces are usually combined vectorially , the SRSS method is themost appropriate method for spatial combination. However if two or more spectras are applied in the samedirection ( as in multiple support excitation system ), the directional combination has to be done first inpreference to modal combination.

11.0The time history analysis :11.1 The mathematical modelling :

Although this method is technically most accurate and can theoretically be applied to any load profile, butit is best used in those applications where the user is absolutely sure of the load time profile i.e. data of loadvariation w.r.t time at small time steps is available. Typically this method is used for impulse type loading.This methodology actually calls for solving the dynamic equilibrium eqn. numerically. Hence it is timeconsuming and requires substantial computational resource.

It has been already mentioned how the dynamic equilibrium equation can be written down as a set ofuncoupled equations in modal co ordinates ( for linear systems with classical damping )

Mn qn'' (t)+ ∑Cnrqr

' (t) + Kn qn (t)= Pn (t) … … … … … … … … … . (21)

This differential equation can be integrated using numerical techniques by slicing the duration of the loadinto many small steps. Based on an assumption of the behaviour of the system between slices ( i.e. that thechange in acceleration between time slices is linear , the system accelerations, velocities, displacements andcorrespondingly the reactions , internal forces and stresses can be calculated at successive time steps. Themost commonly used technique is the Wilson method which is unconditionally stable. By unconditionallystable we indicate procedures that lead to a bounded solution regardless of the time step length. ( for adetail discussion the reader can refer to ref. 1 and 5 )

11.2 Application :

Application of the Time history analysis for stress analysis problems pertaining to earthquake analysis :The dynamic equilibrium for earthquake can be written down as :

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Mx(t) ′′+ Cx(t) ′+ Kx(t)= -Mυxg ′′ (t) … … … . (22)

Where υis the influence vector denoting the displacement of the masses for unit ground displacement andM, C and K are the mass, damping and stiffness matrices respectively.

For a time history analysis of earthquake, the values of ground acceleration at time steps ( say 0.02sec) hasto be given as an input to solve the above differential eqn. by numerical methods ( In Response spectrumanalysis also this eqn. is solved to generate the response spectrum, where first the eqn. is solved fordifferent natural frequencies keeping the same value of the damping ratio and then solved for differentdamping ratios ) The dynamic displacements at different times are converted into equivalent static forcesjust as the way they were in Response Spectrum analysis, the only difference is that now they are not timeindependent.

For stress analysts who use standard software for dynamic analysis ,may find it difficult to analysis aseismic excitation through time history method as many computer programmes do not have the facility totake input of ground motion wherein the other possibility is to convert the motion into forces bymultiplying restraint displacement times restraint stiffness.. Hence time history analysis of earthquake isusually not possible with most commercially available standard computer programmes . Also the outputwill be time dependent unlike the output from response spectrum analysis which is time independent.

Usually Time History analysis should be used where the user is absolutely sure of the force vs. timeprofiles as otherwise solutions won’t be realistic and will require lot of computational resource. In thosesituations, solution using the probabilistic spectrum method is a better option. In fact seismic effects areseldom identical , even at the same location at different time earthquake motions show lot of difference i.e.are not identical thus leaving the analyst with little surety about the force- time profile.

12. Conclusion :

Dynamic analysis of piping systems gives most accurate mathematical models under condition of transientloading. Since proper mathematical modelling of such systems require a good understanding of thefundamentals of structural dynamics so the relevant fundamentals have been discussed along with specificapplication guidelines . Wherever possible references have been given to standard software packageswhich are widely used by piping stress analysts .Hopefully this paper will act as good road map forunderstanding the importance of dynamic analysis and making accurate mathematical models both by thedevelopers and users of pipe stress programmes.

References ;

1. Dynamics of Structures- Theory and applications to Earthquake engineering – Anil. K.Chopra2. Uniform Building Code ( 1994)3. Finite Elements in Engineering- T.R. Chandrupatla and Ashok Belegundu4. Structural Dynamics : Theory and computations – Mario Paz.5. Numerical methods in Finite elements analysis- K.J.Bathe and E.L. Wilson

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