Seismic Response and Stability Analysis of Single Hinged Articulated Tower

doi:10.4236/ojce.2013.34028

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Open Journal of Civil Engineering, 2013, 3, 234-241

Published Online December 2013 (http://www.scirp.org/journal/ojce)

http://dx.doi.org/10.4236/ojce.2013.34028

Open Access OJCE

Seismic Response and Stability Analysis of Single Hinged

Articulated Tower

Prashant Atreya1, Nazrul Islam2, Mehtab Alam1, Syed Danish Hasan3

1Department of Civil Engineering, Jamia Millia Islamia University, New Delhi, India

2Civil Engineering Department, Islamic University, Madina, KSA

3University Polytechnic, Aligarh Muslim University, Aligarh, India

Email: prashant[email protected]

Received November 3, 2013; revised December 3, 2013; accepted December 10, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

Study of dynamic stability phenomenon in transient systems has always created interest amongst the researchers be-

cause of its inherent non-linearities. Offshore structures subjected to wave, earthquake or wind loads or a combination

of these loads show non-linear transient behaviour. As oceanic waves are better modelled as stochastic process, there

is a need to investigate the stochastic stability of flexible offshore structures as well. Present study has been carried out

to determine seismic response of Single Hinged Articulated Tower (SHAT) under different categories of wave loads

and earthquake followed by its dynamic stability analysis. Different phases of wave/earthquake loading on SHAT have

been explored to investigate dynamic instabilities existing during each phase. Two dimensional phase plots have been

used to identify phases of dynamic instability existing within the responses of SHAT under various conditions of load-

ing.

Keywords: Single Hinged Articulated Tower; Earthquake; Time History; Phase Plot

1. Introduction

Compliant offshore structures are favoured for deep sea

water operations since they avoid unacceptably high hy-

dro-dynamic loads by yielding to wave and current ac-

tions leading to economic designs. These structures have

large displacements with inherent non-linearities, so pre-

diction of behaviour of these structures in oceanic envi-

ronment is difficult and is met with many challenges.

Efforts are made to use simplified realistic mathematical

models to gain important insight into the response be-

haviour of these structures and to explore the possibility

of their dynamic instability and chaotic motion. The pres-

ence of strong geometric non-linearity and non-linearity

arising due to fluid structure interaction leads to the pos-

sibility of dynamic instability of the systems. On account

of these non-linearities, numerical investigations of com-

pliant offshore structures have revealed complex behav-

iour involving sub-harmonic, super-harmonic and aperi-

odic solutions (Banik and others) [1-3]. Although, during

last more than a decade, researchers have carried out

seismic response of compliant offshore structures (Lina,

H., Youngang, T. and Cong, Y.I., (2006) [4], Chandra-

sekaran et al. 2008 [5]; Hasan, S.D., 2011 [6]), more

efforts are required for stability analysis in non-linear

environment. The stability analysis may consist of per-

turbating an approximate solution. Various methods of

dynamic stability analysis of non-linear system in closed

form by using analytical, semi-analytical and numerical

techniques have been developed (Friedmann et al., 1977

[7]; Chua and Ushida, 1981 [8]; Burton, 1982 [9], Cai,

G.Q. (1995) [10], Lin, Y.K. and Cai, G.Q., 1995 [11]).

Application of these techniques covers a wide range of

application problems including standard problems of

Van-Der-Pol oscillator, Duffing Oscillator, Double Pen-

dulum etc. (Hamdan and Burton, 1993 [12]; Ravindra

and Mallik, 1994 [13]; Blair et al., 1997 [14]; Yu and Bi,

1998 [15]). The main focus of application problems was

to study and investigate capabilities of the methods to

bring out all possible instability phenomenon latent in the

system. Islam Saiful, A.B.M (2013) [16] used two di-

mensional Phase Plots to determine dynamic stability

phenomenon in SPAR platforms.

Single Hinged Articulated Tower Platform is one of

the compliant structures (Figure 1) which is economi-

cally attractive especially as loading and mooring termi-

P. ATREYA ET AL. 235

nal in deep waters. These platforms are comparatively

light compared to the conventional fixed platforms. The

tower itself is a linear structure, flexibly connected to the

sea bed through a cardon/universal joint and held verti-

cally by the buoyancy force acting on it. The part of the

tower emerging from the water supports the super struc-

ture designed to suit the particular application e.g. a

tanker to be loaded, flaring of waste gases, etc. As the

connection to the sea bed is through the articulation, the

structure is free to oscillate in any direction and does not

transfer any bending moment to the base. The articulated

tower which can be used at larger water depth may also

have one or more number of joints at the intermediate

level. Such towers having joints at the intermediate level

are called multi hinged articulated tower.

As the articulated tower is compliant in nature, it

moves with the waves and thus the wave force and

bending moment along the tower will be less compared

to a fixed structure. Usually the natural period of the

towers is of the order of 40 to 90 seconds and hence dy-

namic amplification-factor is small.

2. Scope of Present Study

Effects of seismic excitation/stochastic response of tower

in the presence of regular/random wave forces have been

examined.

Figure 1. Single hinged articulated tow e r.

Dynamic Stability has been examined/analysed for ef-

fect of regular/random waves and earthquake duration on

serviceability and survivability of tower.

Concepts of minimum potential energy and Phase

Plots have been used to determine the regions/range of

dynamically stable periods during the affected motion of

the tower.

Variation of stabilizing and destabilizing forces and

their impacts on Shear Force, Axial Force, Bending mo-

ments along the axis of Tower and stress on hinge have

been calculated.

3. Methodology

In the present work firstly a nonlinear dynamic analysis

of the said structure has been carried out for its time do-

main responses using Langrangian approach. The ran-

dom waves have been simulated by Monte-Carlo tech-

nique represented by Modified PM Spectra. Modified

Morison’s equation has been used for estimation of hy-

dro-dynamic loading. Water particle kinematics has been

governed by Airy’s linear wave theory. To incorporate

variable submergence, Chakraborty’s correction [17,18]

has been applied. Seismic inputs have been applied using

Northridge spectra. Stability assessment has been carried

out using two dimensional phase plots.

Assumptions and Structural Idealizations

In the present study, the solution has been obtained using

Finite Element approach, so the following assumptions

and structural idealizations have been made for formula-

tion of the problem in respect of Single Hinged Articu-

lated Tower (Figure 2):

1) Articulated tower is modelled as a stick with masses

lumped at the nodes. The universal joint at base is

modelled as mass-less rotational spring of zero stiff-

ness. Flexural deformations of the tower have been

assumed to be negligible as compared to its dis-

placements as a rigid body.

2) The entire tower has been discretized into “np” num-

ber of elements of uniform length for the estima-

tion of conservative and non-conservative forces,

while diameter, mass and buoyancy may vary. The

submerged elements of the tower have been subjected

to time dependent hydro-dynamic loading. Due to

non-linear forces acting on the tower, the number of

submerged elements shall also vary with respect to

time.

3) Drag force is assumed to be proportional to the rela-

tive water particle velocity w.r.t. the structure, oscil-

lating under wave and ground motion. The structural

damping of the system is specified as a fraction of the

critical damping corresponding to the un-deflected

coration of the tower. nfigu

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P. ATREYA ET AL.

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236

Figure 2. SHAT model.

4) Earthquake is assumed to be a broad band random

stationary process described with the help of an acce-

logram. The behaviour of the fluid surrounding the

structure shall not be affected by the slow motion of

the compliant tower.

5) Analysis due to earthquake excitation and due to

wave forces are carried out independently, and there-

fore water particle kinematics is taken to be negligible

for the seismic forces. Only two dimensional motion

of the tower in the plane of the environment loading

have been considered in the analysis.

(Geometrical and Mechanical Properties of SHAT

Model are given in Table 1 and Solution Flow chart is

given at Figure 3).

4. Estimation of Load on Structure

4.1. Wave Loads

A variety regular waves as suggested by Jameel & Ah-

mad (2011) [19] have been considered. In order to inves-

tigate the combined effect of current and earthquake,

current velocities of 1.0 m/s, uniform throughout the

depth have also been considered. The influence of vari-

ous parameters such as variable buoyancy, added mass,

instantaneous towers orientation, variable submergence

and effect of current, on the response of SHAT and its

stability aspects has been investigated in detail. To cal-

culate the wave force on latticed articulated tower, the

latticed tower has been replaced with cylindrical shaft of

equivalent diameter. Wave forces on the submerged part

of the latticed tower (cylindrical shaft) have been esti-

mated by the modified Morison’s equations, which duly

takes into account the relative motion of the structure and

water. The water particle velocities and accelerations

have been stipulated by Airy’s wave theory. To incorpo-

rate the effect of variable submergence, Chakrabarti’s

approach has been adopted. The transformation matrix

has been used to compute the normal and tangential

component of the hydrodynamic forces on each element

of the tower corresponding to instantaneous deformed

configuration of the tower. The updated mass-moment of

inertia of the tower has been incorporated in the consis-

tent mass and damping matrices. Newmark’s Beta inte-

gration scheme has been deployed to solve the equation

of motion taking into account all non-linearities involved

in the system.

4.2. Seismic Loading

Northridge accelogram has been used to provide input

for 39.98 seconds for ground acceleration time history

(Figure 4) for calculating seismic response of the tower

by random vibration analysis. The wave loading are not

correlated with seismic loadings. The two analysis are

carried out independently. The analysis under earthquake

alone is carried out using water particle kinematics as

zero. Therefore, it is assumed that in the seismic analysis

alone, the water particle kinematics are zero.

To observe the behaviour due to the combined wave

and earthquake forces, Numerical studies are conducted

to investigate the effects of initial conditions, current and

wave on the seismic response of the tower. Without wave,

the tower is assumed to have zero displacement and zero

velocity at time t = 0. When wave and earthquake are

considered to act together, different initial conditions of

the tower are assumed depending upon the oscillating

state of the tower at the inst nt when the structure en- a

P. ATREYA ET AL. 237

Figure 3. Flow chart for problem solution.

Table 1. Geometrical and mechanical properties of SHAT under study.

Geometric characteristics Mechanical properties

Height of tower (l) 400 m Deck mass (MD) 2.5 × 106 Kg

Water depth (d) 350 m Structural mass (SMT) of tower 2.0 × 104 Kg

Height of ballast (HBL) 120 m Mass of ballast (MBT) 44,840 Kg

Height of buoyancy chamber (H) 70 m Mechanical oscillations

Position of buoyancy chamber (PBC) 310 m Time period 38 sec.

For chamber Hydrodynamic specifications

Effective diameter for buoyancy (DB) 20 m Drag coefficient (CD) 0.6

Effective diameter for added mass 7.5 m Inertia coefficient (CM) 2.0

Effective diameter for drag 14.5 m Mass density of sea water 1024 Kg/cu-m

Effective diameter for inertia 7.5 m Random sea spectrum PM SPECTRUM

For tower shaft

Effective diameter for buoyancy (DB) 7.5 m

Effective diameter for added mass 4.5 m

Effective diameter for drag 13 m

Effective diameter for inertia 4.5 m

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P. ATREYA ET AL.

238

Figure 4. Northridge eart hquake .

counters the earthquake. Further, it is assumed that the

earthquake forces act on the structure when it oscillates

in a steady state under the regular sea.

The responses induced due to earthquake are further

compared with the responses due to strong sea-state/

waves/waves in order to establish relative severity of the

two independent events. The commencement of the

earthquake has been considered such that the first prom-

inent peak of accelogram matches with the crest of the

wave in order to get the maximum impact of wave and

Earthquake load. The seismic responses of tower are

further compared with the response due to strong sea

states.

5. Discussion of Results on Seismic Response

and Dynamic Stability

Single Hinged Articulated Tower in sea exhibits non-

linear behaviour because of inherent non-linearities in the

system. These non-linearties are produced by non-linear

excitation and restoring forces, damping non-linearity,

etc. which may lead to complex response behaviour of

SHAT. Non-linear restoring force is from the geometric

non-linearity of the structure. Non-linear responses of the

system are analysed to investigate different kinds of dy-

namic instability phenomenon. During the present study,

SHAT was subjected to a variety of regular wave load

along with Northridge Earthquake excitation at different

time.

The responses are obtained in terms of heel angle rota-

tion, tip displacement, shear force, bending moment, ax-

ial force, base shear force, Stabilizing/Destabilizing mo-

ments etc. Here, we shall discuss in details a case depict-

ing behaviour of Tower under regular wave load and

earthquake. The response and dynamic stability analysis

has been done using Time history and two dimensional

Phase Plots (Mallik, A.K. and Bhattacharjee, J.K. (2005)

[20].

Strong Regular Wave (H-17.15 m, T-13.26 s)

with Northridge Earthquake at 308.7 Second

The analysis was performed to evaluate the response of

Strong Regular Wave having wave height as 17.15 m,

Time period as 13.26 sec. and Northridge Earthquake

applied at 308.7 second. Wave load was initially applied

at 0 second. Upto 100 seconds the motion was unstable,

then due to hydrodynamic dampening, the impact of reg-

ular wave on Tower got gradually reduced and the Tower

motion became stable by 250 second. There after, North-

ridge Earthquake was applied at 308.7 second i.e. at the

crest of wave. Time history plot for Hinge angle rotation,

is given at Figure 5, which provides information about

magnitude of hinge angle responses generated due to

Wave and Earthquake loads. The phase plots as shown

from Figures 6-10 have been generated for the com-

‐1.0E‐01

‐8.0E‐02

‐6.0E‐02

‐4.0E‐02

‐2.0E‐02

0.0E+00

2.0E‐02

4.0E‐02

6.0E‐02

8.0E‐02

0100 200 300 400 500 600 700 800

TIME(sec.)

ANGULARANGULARROTATION(rad.)

Time HistoryforRegularWaveH‐17.15m,T‐13.26s,EQ‐HingeAngleR otation

Figure 5. Time history for hinge rotation.

‐1.5E‐02

‐1.0E‐02

‐5.0E‐03

0.0E+00

5.0E‐03

1.0E‐02

1.5E‐02

2.0E‐02

‐1.00E‐01‐8.00E‐02 ‐6.00E‐02 ‐4.00E‐02 ‐2.00E‐02 0.00E+002.00E‐02 4.00E‐02 6.00E‐02 8.00E‐02

PHASEPLOTFORCOMPLETEDURATIONOF2000s

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

Figure 6. Phase plot for complete duration of 2000s.

‐1.2E‐02

‐1.0E‐02

‐8.0E‐03

‐6.0E‐03

‐4.0E‐03

‐2.0E‐03

0.0E+00

2.0E‐03

4.0E‐03

6.0E‐03

8.0E‐03

‐1.00E‐01 ‐8.00E‐02‐6.00E‐02 ‐4.00E‐02‐2.00E‐02 0.00E+002.00E‐02 4.00E‐02 6.00E‐028.00E‐02

PHASEPLOTFROM308.7to348.68sDURINGEQ

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

Figure 7. Phase plot from 308.7 - 348.68 s during North-

ridge earthquake.

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P. ATREYA ET AL. 239

‐1.0E‐02

‐5.0E‐03

0.0E+00

5.0E‐03

1.0E‐02

1.5E‐02

‐6.00E‐02 ‐4.00E‐02 ‐2.00E‐02 0.00E+002.00E‐02 4.00E‐02 6.00E‐0

PHASEPLOTFROM348.68to449sINTERVAL

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY(rad/s)

Figure 8. Phase plot from 348.68 to 449 s interval after

Earthquake.

‐8.0E‐03

‐6.0E‐03

‐4.0E‐03

‐2.0E‐03

0.0E+00

2.0E‐03

4.0E‐03

6.0E‐03

8.0E‐03

‐2.00E‐02 ‐1.50E‐02 ‐1.00E‐02 ‐5.00E‐03 0.00E+005.00E‐03 1.00E‐02 1.50E‐02 2.00E‐02

PHASEPLOTFROM449to650sINTERVAL

HEELANGLE(rad.)

ANGULARVELOCITY

(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY

(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY

(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY

(rad/s)

Figure 9. Phase plot from 449 - 650 s interval depicting sta-

bilization process.

‐8.0E‐03

‐6.0E‐03

‐4.0E‐03

‐2.0E‐03

0.0E+00

2.0E‐03

4.0E‐03

6.0E‐03

8.0E‐03

‐1.50E‐02 ‐1.00E‐02‐5.00E‐03 0.00E+005.00E‐03 1.00E‐02 1.50E‐02

PHASEPLOTFROM650to2000s

HEELANGLE(rad.)

ANGULARVELOCITY

(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY

(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY

(rad/s)

HEELANGLE(rad.)

ANGULARVELOCITY

(rad/s)

Figure 10. Phase plot from 650 - 2000 s duration showing

Stabilized SHAT motion.

plete period of 2000 seconds, from 308.7 to 348.68 sec-

onds, from 348.68 to 449 seconds, from 449 to 650 sec-

onds and from 650 to 2000 seconds. From phase plots, it

was observed that motion from 308.7 to 348.68 seconds

was under earthquake impact and showed bifurcations

and chaos. During Earthquake, absolute Maximum heel

angle was 5.0 degrees, which was well outside the per-

mitted serviceable limits of the Tower. As the earthquake

was applied for 39.98 seconds duration, after 348.68 sec-

onds the hydrodynamic dampening started reducing re-

sponses and trajectories started moving towards stable

limit. However, from 348.68 to 449 seconds period, the

motion has been non-harmonic, aperiodic and asymmet-

ric. It took another 200 seconds for responses to settle

down and stabilize. By 650 seconds, the responses be-

came harmonic, symmetric and periodic and perfectly

stable. From above discussion it is clear that with the

onset of wave load, although the tower is in serviceable

limits but shows slight instability for period of approx.

250 second. After onset of earthquake at 308.7 seconds it

becomes unserviceable as well as unstable for a period of

approximately another 350 seconds. Then it showed the

stable behaviour. The hydrodynamic dampening effects

are more prominent in stronger waves and the system

reaches stability much early. During the entire motion,

the tower has been safe due to net positive stabilizing

moments acting on the tower. The Tower during initial

wave period of 100 seconds can be used with some dis-

comfort. Use of tower should be further avoided during

Earthquake period, especially from 308.7 to 650 seconds.

6. Conclusions

SHAT model was subjected to a variety of waves with

and without earthquake loads. The observations for seis-

mic responses and stability determination were analyzed

and following conclusions have been drawn:

6.1. Conclusive Remarks on Seismic Response of

SHAT

1) The initial condition described by the instant of time

of the steady state tower motion at which the earth-

quake strikes has significant effects on the tower re-

sponse. Peak values differed upto 7% - 8% due to

change in initial conditions i.e. when the earthquake

is applied at axis or applied at crest of wave in regular

sea. These values may further differ appreciably if

earthquake occurs at the time corresponding to trough

or crest of the regular wave.

2) Responses induced by earthquake in regular/random

sea are further compared with the responses due to

independent Strong large size wave in regular/random

sea to obtain the relative severity. Hinge angle rota-

tion of SHAT due to strong regular wave (H-30m,

T-15 sec.) alone is 4.21 × 10−2 radian while it in-

creased to 1.43 × 10−1 radian (3.39 times) when the

tower is hit by only Northridge earthquake. Similarly

in random sea, hinge angle rotation with only Strong

large size wave (H-30m, T-15s) is 3.72 × 10−2 radian,

while it increased to 1.28 × 10−1 radian (3.44 times)

with small wave (H-2.15, T-4.69s) and Northridge

Earthquake. This proves that even small duration

earthquake gives big jolt to the structure leading to

higher values of hinge angle rotation.

3) Maximum Hinge angle rotation due to small size reg-

ular wave (H-2.15 m, T-4.69s) and Northridge earth-

quake is 8.73 times more than that due to regular

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P. ATREYA ET AL.

240

wave alone. Maximum Hinge angle rotation for large

size wave (H-17.15m, T-13.26s) and earthquake is

3.17 times more than that due to regular wave. This is

because larger size waves produce sufficient magni-

tude of hydrodynamic dampening so as to attenuate

peaks in responses.

4) The maximum tower response for Northridge earth-

quake alone in regular sea has been observed to be

2.68 times more than that due to the combined load-

ing of wave (H-2.15m, T-4.69s) and Northridge

earthquake. This again indicates that introduction

wave increases hydrodynamic dampening and thus

size of peak generated due to Earthquake impact is

reduced.

5) With the inclusion of the current in random/regular

sea wave, there is 2.58% - 2.82% increase in Hinge

angle response.

6) The absolute maximum values of Hinge angle rota-

tions have been found to be in line with nature/size of

waves. For each load case, values of RMS and Stan-

dard Deviation for Hinge angle rotation are in the

same range.

7) The base shear in case of only Northridge earthquake

is 19.69 times the base shear value due to Strong

large size regular wave. This shows that Base shear

value in this load case has increased more rapidly as

compared to increase in heel angle values. In each

load case, values of RMS and Standard Deviation for

Base Shear are in the same range.

8) When current is introduced in regular sea with or

without Northridge Earthquake, the Maximum value

of Base shear reduces.

9) For different loadings, Base Axial Force values,

Mean and RMS values have common coefficient as

108. The changes in these values due to different

combinations of loadings are very small.

10) Maximum bending moment due to small size regular

wave (H-2.15m, T-4.69s) and Northridge earthquake

is 2.46 times more than that due to regular wave alone.

Maximum Bending moment for large size wave

(H-17.15m, T-13.26s) and earthquake is 1.51 times

more than that due to regular wave. This is because

larger size waves provide hydrodynamic damping and

thus reduce the seismic response.

6.2. Conclusive Remarks on Dynamic Stability

Analysis of SHAT

1) During the earthquake, the tower tends to vibrate at

its own natural frequency while the steady state re-

sponse again takes place in wave frequency when the

earthquake is over.

2) The time required to achieve the steady state response

after the duration of earthquake depends upon the sea

environment at that time. High sea state dampens the

seismic response quickly.

3) Although short duration intensive earthquake load

gives a big jolt to the tower but it survives due to in-

herent restoring capacity. The responses due to short

duration Earthquake die out quickly when the waves

are also present. In the absence of earthquake and

other environmental loads tower oscillations are

checked by hydrodynamic dampening due to waves

and due to tower oscillation and inherent tower

buoyancy.

4) Subsequent to application of Wave/Earthquake loads,

the time taken for Stabilization depends upon the na-

ture/Size of the wave. The larger the size of wave, the

smaller is the Stabilization period. For Small size

wave (H-2.15, T-4.69s) with or without Earthquake,

the time taken to achieve dynamic stability was 610

seconds. For middle sized wave (H-11.15, T-10.69s),

the time taken for dynamic stability is 350 seconds.

For another higher size wave (H-17.15, T-13.26s), the

time taken for dynamic stability is 250 seconds. For

Strong large size wave (H-30m, T-15s), the time tak-

en for dynamic stability is 200 seconds.

5) When only Northridge Earthquake is applied in calm

sea (with minimal size regular wave), the time taken

for dynamic Stability was 1000 sec. Waves/current

provide hydrodynamic dampening to the structure, in

the absence of dampening the structure takes more

time to stabilize.

6) With the introduction of current in wave load with or

without Earthquake, the time taken for dynamic sta-

bility slightly reduces which shows that the current

adds up to the hydrodynamic damping effects.

7) SHAT model subjected to small regular wave

(H-4.8m, T-10.4s) and 4 varieties of Earthquake

loadings (1994 Northridge-NWH360, 1979 Imperial

Valley H-E11140, 1979 Imperial Valley H-E13140 &

1999 Duzce, Turkey-1062-N) showed that absolute

Maximum values of responses arising out of impact

of all the four types of Earthquakes are having similar

magnitude(except for some minor variation). North-

ridge Earthquake is found to be severest of all four.

The stabilization period after each earthquake is 640

sec., which once again emphasize the fact that stabi-

lization period is a property of nature/size of wave.

Nature of Earthquake, however governed the shape of

Time History/Phase plot of responses.

8) In all the regular wave cases analysed during study, it

was observed that during the initial period pertaining

to onset of waves or period pertaining to Earthquake,

the dynamic instability is visible in Phase plots. The

motion is non-harmonic, aperiodic and asymmetric.

Bifurcations are easily visible in the phase plots con-

firming dynamic instability. With the passage of time,

the hydrodynamic dampening effects reduce responses.

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P. ATREYA ET AL.

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241

The trajectories start moving towards the stable limit

cycle and the motion gradually becomes harmonic,

periodic and symmetric. No bifurcations are visible

on the phase plots after longer duration loadings and

the structure show dynamic stability.

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