pseudo-dynamic testing method (pdtm) * effective force method … · 2014. 5. 16. · that must be...

A new seismic testing method

E. Kausel

Professor of Civil and Environmental Engineering, Massachusetts

7-277, OamWd e, 027 JP,

Introduction

The bulleted enumeration that follows shows five experimental strategies totest the effects of earthquakes on structural components, structural models, orfull scale structures; the first three are commonly used, the fourth one hasrecently been proposed but not yet implemented, and the last one is thesubject of this paper. These five strategies are:

• Quasi-static Testing Method (QSTM)• Shaking Table Testing Method (STTM)• Pseudo-dynamic testing method (PDTM)* Effective Force Method (EFTM)• Hybrid Shaking Table Testing method (HSTM)

In the quasi-static testing method, a test specimen is subjected to slowlychanging prescribed forces or deformations by means of hydraulic actuators—without considering inertia forces within the structure. The purpose of thiselementary test is to observe the material behavior of structural elements,components, or junctions when they are subjected to cycles of loading andunloading.

A shaking table, on the other hand, allows ample opportunities to observe theeffects of dynamic forces on structural test models. However, it is difficult todesign a table capable of reproducing actual ground motions, particularlywhen simulating multi-axial earthquakes. Among the reasons are thedeformability and inertia of the table, its characteristic modes of vibration, the

Transactions on the Built Environment vol 20, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509

4 Earthquake Resistant Engineering Structures

friction of the bearings, the physical capabilities of the hydraulic actuators,and (to a lesser extent) the limitations in the control devices. For large scalestructural models, additional problems develop because the table must be ableto carry both the large dead weight of the specimen and the overturningmoment caused by the inertia forces —all of this without detriment to thefunctioning of the table.

The pseudo-dynamic testing method resembles the quasi-static testing methodin that it also consists in applying slowly varying forces to a structural model.During testing, however, the motions and deformations observed in the testspecimen are used to divine the inertia forces that the model would have beenexposed to during the actual earthquake; this information is then fed back intoa control engine so as to determine and adjust the effective dynamic forcesthat must be applied onto the structure. These pseudo-dynamic forces aretypically accomplished by means of actuators pushing against a large reactionwall.

The effective force testing method is based on applying dynamic forces to atest specimen that is rigidly anchored to an immobile ground; these forces areproportional to the prescribed ground acceleration and the local structuralmasses. The deflections measured in this test correspond to the motions of thestructural points relative to the ground that would have been observed had thespecimen been subjected at its base to the actual test earthquake. While thetheoretical basis for the effective force method is very well known, a proposalto develop and implement an actual testing device based on this principle hasonly recently been made [Leon et al, 1993]. As of this writing, however,there do not exist as yet exist any experimental facilities implementing thisstrategy.

Finally, the hybrid shaking table testing method presented herein is a newtechnique which seeks to combine the advantages of shaking tables withthose of the effective force method while avoiding some of the pitfalls in eachof them. In this method, the test specimen is subjected to both a motion at itsbase by means of a shaking table, and to dynamic forces applied onto themasses by means of force actuators (reacting against either a stationary wallor against a support system moving with the table), or by thrust engines. Theidea is to separate the ground motion into two arbitrary components, and toassign one component to the table and the other to the actuators. In fact, thecharacteristics of the components can be chosen so as to optimize the designof the testing system, as will be seen. While it is possible to show this methodto be valid for complex multiple-degree of freedom systems, includinginelastic deformations and large displacements, for the sake of brevity (and toavoid clouding the fundamental ideas behind a mass of obscure equations andconcepts) only the essential aspects of the method will be presented and


Earthquake Resistant Engineering Structures 5

illustrated through a simple one degree-of-freedom system. A full proof ofconcept for multi-degree of freedom systems undergoing large displacementsand/or deformations will be presented elsewhere (Kausel & Clark, 1996).

Testing methods applied to a one degree-of-freedom (1-dof) system

Consider a 1-dof (perhaps inelastic) system subjected to an arbitrary supportmotion. Let y be the relative displacement and u the absolute acceleration, asmeasured in the inertial system with respect to which the ground is movingwith acceleration ii^ , so that u = y + u^ . The equation of motion is then

m% + /(}',J/) = 0 (1)

where f(y,y)is the restoring force in the member connecting the mass andthe support. In general, this force is not only a function of the instantaneousvalues of the deformation y and the rate of deformation y , but the entireprevious deformation history (i.e. the deformation path). For a linear system,the restoring force is given by the well-known expression f(y,y) = ky + cy ,where k and c are the spring and dashpot constants.

In terms of the relative displacement, the equation of motion is

/) = -/%% (2)

which describes the essence of the effective force method: apply fictitiousforces -mu^ directly onto the structure while holding the base stationary.

The left hand side contains the terms associated with the natural reaction ofthe system (i.e. that part which we do not control), while the right hand sidecontains the prescribed excitation (which we do control). Analogously, theleft hand side of equation (1) represents the natural reaction of the systemwhen mounted on a shaking table. By comparison, the equation of motionunderlying the pseudo-static method is of the form

/(;,,;,) = -m( + JP) (3)

with the excitation term y obtained from the previous history of y by whatamounts to a forward difference. To carry out the test, this equation is"slowed down" by appropriate time scaling (which requires neglecting thedependence of / on y ), and sacrifices are made in the estimation of y frompast values of y to ensure dynamic stability.



Hybrid testing method

The basic idea behind the hybrid testing method is best explained by meansof a thought experiment: assume the structure to be resting on a shaking tablethat is in turn supported by another shaking table, as shown in Fig. 1 . Thebottom table 5, moves with acceleration u^ with respect to the inertial

reference frame 5, while the top table moves with acceleration u^ relative

to the bottom table. Clearly, since the motions of the two tables are additive,their combined effect is equivalent to that of a single table with addedmotions u^ = u^ + u^ . If one of the two tables, say the bottom one, is

replaced by actuators exerting equivalent forces directly onto the structure,the response — as far as structural deformations is concerned — remainsunaltered. In other words, as long as both excitations effectively add to theoriginal ground motion, the structural response is indistinguishable from thatobtained solely with a single shaking table, or solely with actuators.

Let w, be the apparent absolute motion of the mass as seen from the

(moving) reference frame 5, , and >>, = y^ = y the actual deformation of themember in either system. The true and apparent absolute motions are relatedby the equation u = u\ + u^ = y + u^ + u^ • From equation (1), it then follows

that

which states that the response of the system can be obtained by applying asupport motion u underneath the (now single) top table simultaneously

with fictitious forces -mii^ exerted by actuators. This equation reduces to

either equation (1) or (2) when w^,or u^ , respectively, go to zero (i.e. the

HSTM method reduces then to either the STTM or the EFTM methods).

Why hybrid testing?

The proposed method could offer several potential advantages, including thepossibility of optimizing the ground motion components from the point ofview of the power or force demands on the table and actuators, or themechanical characteristics of these devices. For example, one of thefollowing criteria could be used:

• Separate the ground motion into low and high frequency components.Assign the low frequency components to the shaking table, and the high



frequencies to the actuators. In fact, the separation need not strictly followa Fourier decomposition. For instance, one could drive the shaking tablewith a quasi-sinusoidal component, modulated by an appropriate bell-shaped window so as to simulate the ground displacement; the remainderwould then be driven with actuators. Conceivably, this alternative couldsimplify considerably the design of the driving and bearing mechanismsof the table. At the same time, one could avoid the problems associatedwith low-amplitude table motions, which are often impeded by friction ofthe bearings.

• Separate again the ground motion into low and high frequencycomponents. This time, however, simulate only the intermediate or highrange of frequencies with the table. The remainder could be accomplishedwith actuators, including perhaps jet engines attached to the structure tosimulate the low frequency components. This could have the advantage oflimiting the stroke of the actuators and the maximum displacement thatthe table would have to accommodate; in addition, it could also reducethe total power requirements of the system.

• Expand by means of actuators the capabilities of an already existing table(or even a planned one) —able to move in only one or two directions—and use it to simulate fully three-dimensional earthquake forces. Forexample, model the strong translational components with the table, andthe remaining components through actuators (particularly rotationalground motion components!).

• Split the components so as to minimize the instantaneous power demandand/or the total energy dissipated. As we shall see in the next section,shaking tables and actuators exhibit different mechanical behaviors andone could, in principle, find an optimal combination.

Whatever the criterion used to separate the ground motion into additivecomponents, this hybrid testing device would normally be supplemented withsophisticated control devices, which would measure the displacements of thetable and the forces exerted by the actuators, and provide instantaneousfeedback and correction to ensure a reliable simulation of the seismicexcitation.

Testing Systems and Power Demand

The instantaneous power demanded by the experimental devices, whiletesting the 1-dof systems considered previously, is simply the product of the



net external force and the velocity of the point of application of this force.This power varies form system to system, as will be seen. From the previoussection, and with reference to Fig. 2, the instantaneous mechanical power(excluding the power required to move the driving assembly, such as the tableitself, or the actuators) is as follows.

Shaking table:The net external driving force is p = mu, and the velocity is that of the

ground. Hence,

W = miiu (5)

Force actuators:The force applied by the actuators is p = -mu^ , and the velocity of the

application point is y . Hence

W = -mu^y (6)

Hybrid system:The force exerted by the actuator is /?, = -mu^ , which acts on the mass

moving with velocity M, ; the driving force acting on the table, moving with

velocity u^ , is p^ = m(u\+u^) = mu (which is independent of how the

ground motion is apportioned between the shaking table and the actuator).The total mechanical power delivered by the system is then W = /?, w, + p^u^ ,

that is

W = -mUgM + muiigi = -mu^y + m(u - u^ )ii (7)

In particular, if the earthquake is simulated only with the shaking table(u^ = 0, u^ =Mg, Wg2 =6,,), or only with actuators (u^ =0, u^ = 0, u^ = u^) ,

then equation (7) reduces to equations (5) and (6), respectively. These twoquantities are not equal. This is illustrated in Figs. 3, 4, which show acomparison of the two expressions for oscillators with unit mass, 5%structural damping, and resonant frequencies of 0.5 and 5 Hz, respectively,when subjected to the 1940 El Centro earthquake (SE component, normalizedto 1 g). A can be seen, the shaking table requires less power for the softspring-mass system, but much more for the stiff system. The converse is truefor the force actuator system.

Fig. 5, on the other hand, shows the power demand for a 1-Hz hybrid systemin which the motion is represented 50% with a shaking table and 50% with



actuators. Remarkably, the peak power demand is less then either systemacting alone.

We next split the motion into low and high frequency components by meansof a low-pass filter that rolls off smoothly in the band from 0.5 to 1.5 Hz.; thetime histories for the two motion components are shown in Figs. 6 and 7. Fig.8 gives the total power demand for a 1 Hz structure for the hybrid device,with the low components going into the shaking table, and the highcomponents into the actuators. As can be seen, the hybrid system's peakpower demand is half as large as that of the shaking table or the actuatorsystem acting alone.

Conclusion

The new hybrid shaking testing concept presented in this paper is based onseparating the ground motion components into two arbitrary additive parts;the simulation of the first part is accomplished with a conventional shakingtable, while the second is carried out with actuators acting directly onto thestructure. The concept was demonstrated here by means of a simple onedegree-of-freedom system, but it can also be shown to be valid for systemwith many degrees of freedom, including those that exhibit material and/orgeometric non-linearities. This strategy could offer several advantages in thedesign of a seismic testing device, particularly when the action of multipleground motion components must be considered simultaneously. Among thepotential benefits one could find:

• considerable reduction in the total power and energy requirements• substantial decrease in stroke and thrust of the actuators needed to

move the table or exert forces on the structural masses.• simplification in the support mechanisms needed to shore up the

structure and resist overturning moments when vertical and rotationalcomponents are simulated.

• possibility of using novel, low frequency force actuators (such as jetengines), which need not have the capacity of rapidly changing thethrust.


JO Earthquake Resistant Engineering Structures

References

Kausel, E. and Clark, A (1996): "A new hybrid method for earthquaketesting", submitted for possible publication in Earthquake Spectra.

Leon, R.T., Clark, A.J., French, C.W., and Bailey, F.N. (1993): "Developmentof an effective force technique for earthquake testing", presented at the U.S.-Japan Seminar on Development and Future Dimensions of Structural TestingTechniques", Ilikai Hotel Nikko Waikiki, June 28 to July 1

u

Fig. 1



a) Shaking Table

S'

b) Force Actuators

c) Hybrid Testing System

Fig. 2



Instantaneous power, 0.5 Hz test structure (5% damping)

El Centro earthquake normalized to Ig

10

-5

-10

Shaking tableForce actuators

M> r. 4--W-\//k/\-- /'f

/ 1/ \' I \,/: v\ r

\ I\l

5 10

Time (sec)

15

Fig. 3



GO

Instantaneous power, 5 Hz test structure (5% damping)

El centro earthquake normalized to Ig

Fig. 4



oo

Instantaneous power, 1 Hz test structure

50% allocation between shaking table and force actuators

-5

Hybrid (50%)Shaking tableForce actuators

-15'

Time (sec)

Fig. 5



Motion component 1

coV\/\

_J

Time (sec)

Fig. 6



Motion component 2

e/5 04s

Time (sec)

Fig. 7



OJD

Instantaneous power, 1 Hz test structure

Ground motion separated into low and high frequencies

15

Si-

Hybrid (0.5 Hz lowpass)Force actuatorsShaking table

Fig. 8


pseudo-dynamic testing method (pdtm) * effective force method … · 2014. 5. 16. · that must be...

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