STIFFNESS AND DAMPING OFNINE CYCLICALLY LOADEDPRESTRESSED CONCRETEMEMBERS

Cyclic loading tests to measure member stiffness and dampingproperties were made on nine prestressed concrete members.End rotations were applied which produced either uniform shearor uniform moment loading up to failure. Test results were foundfrom end moment vs. end rotation hysteresis loops. Stiffness anddamping values were dependent on the type of loading,but were found to be independent of frequency and the numberof cycles of load applied.

R. A. SpencerUniversity of British ColumbiaVancouver, B.C., Canada

Although prestressed concretestructures can be designed to with-stand strong earthquakes ( "2,3 ' 4> , thenecessary techniques are not yetwell established for all types ofstructures. Traditional earthquake-resistant design methods for framestructures have been developed overa period of years using a semi-empi-rical approach (51 . These methods as-sume that the frame members candissipate energy by yielding (or de-veloping plastic hinges). However,some prestressed concrete framemembers dissipate relatively littleenergy under earthquake loadingconditions because they recover withonly small residual deflections afterbeing loaded well beyond crack-ing( ).

There is little information aboutthe earthquake response of tallframe structures in which pre-June 1969

stressed concrete members of thistype are used to resist earthquakeforces. The design of such a struc-ture should therefore include a ra-tional dynamic analysis of its re-sponse to strong earthquake motions.Thus, this paper describes tests madeto measure the stiffness and dampingproperties of nine prestressed con-crete members. The results can beused in a non-linear, dynamic analy-sis of earthquake response.

NOTATIONDt = damping energy or ener-

gy dissipated per cycleEl = apparent elastic modulus

of concrete in a memberin which all the cracks areclosed

Il = second moment of areafor a full section of amember

39

Table 1. Details of the test members

Test 28-day Nominal Cross-member concrete Steel No. of prestress Stressing section and

no. strength cables force steel location(psi) (Ib.)

1,2 8700 3/s-in. dia. 1 11,000 Pretensioned7-wire (no anchorages) a^strand

3, 4, 5 8700 2 22,000 " •

6 8700 3 33,000 • •

7 8700 " 4 44,000 •

8 8700 0.276-in. 2 18,000 Post-tensioneddia. wire from 15-in. •

blocks

9 10,500 0.200-in. 4 18,000 Pretensioned •dia. wire (anchorages)

a•'

L = loaded length of testmember

LWe = equivalent loop width, amoment value describinghysteresis loop width

LWF = loop width factor, a di-mensionless hysteresisloop width parameter

Me = end moment applied to amember

M, = nominal cracking mo-ment, defined from a hy-hysteresis loop

S = stiffness, the slope of ahysteresis loop

Sl = primary stiffness, formember with cracksclosed

S2 = secondary stiffness, formember with cracks open

Be = end rotation applied to amember

Oet = double amplitude end ro-tation

TEST MEMBERS

Nine prestressed concrete mem-bers were tested, each 6 x 4 in. insection and 158 in. long. The uni-formly prestressed members werefactory made, except test specimenNo. 9 which was made under care-fully controlled laboratory condi-tions. Prestress forces shown in Table1 are nominal values allowing forprestress losses. Member No. 8 waspost-tensioned from 15-in, long pre-cast blocks, separated by ¼-in. Ci-ment Fondu pads. The rest of thefactory-made members had no ex-

40 PCI Journal

MeDEFLECTEDSHAPE

ee Me

MOMENTDISTRIBUTION

SHEAR FORCEDISTRIBUTION

Fig. 1. Uniform shear loading

ternal anchorages, and memberswith the same number of cableswere cast together in a line. Mem-ber No. 9 had external anchoragesapplied at each end before it wastested. None of the members hadany stirrups or shear reinforcement.

TEST LOADING

The non-linear load-deflectionproperties of a prestressed concretemember loaded past cracking showclearly that its behaviour is not thatof a simple elastic solid. Aside fromthe tension cracks that develop, amember loaded past cracking willexhibit regions of concentrated curv-ature, slip between the concrete andthe tension steel, variations in thedepth of the neutral axis, and dif-ferent strain distributions at variouscross sections. These cause variationsin the effective value of the elastic

modulus and the second moment ofarea along the member, and causeit to behave like a mechanism. Sincethe system used to apply loads to aprestressed concrete member willaffect the location and depth of thetension cracks, the loading systemmust be chosen carefully if the testis to provide meaningful results.

In the tests described in this pa-per, the members were loaded byapplying equal rotations, in the samedirection, at each end. Each rotationwas about a vertical axis passingthrough the centerline of the mem-ber, causing it to deflect horizontally(see Fig. 1). This produces uniformshear loading which is similar to theloading of members in a laterallyloaded frame structure.

The loading was zero-mean cyclic,because the end rotation variedabout a mean of zero in an approxi-

DEFLECTEDaSHAPE

Fig. 2. Uniform moment loadingJune 1969 41

Fig. 3. Testing machine used to apply zero-mean cyclic loading, shownwith a test member clamped in place

mately simple harmonic manner.This gives different results from acyclic loading which varies betweenzero and a maximum value whichis either always positive or alwaysnegative. The frequency was variedbetween 0.5 and 2.0 cycles per second(cps) and the members were loadedto failure whenever possible. Thistype of test loading ensured that thestiffness and damping data obtainedcould be used in non-linear responseanalyses.

Uniform moment loading also wasdeveloped on the members by ap-plying equal and opposite end rota-tions (see Fig. 2). This loading wasused only for comparative purposes.

TESTING MACHINE

The testing machine used to loadthe members is shown in Fig. 3 witha test member in place. Each end ofthe member was clamped inside arigid steel frame by bolts spaced as

shown in Fig. 4. A mechanical link-age system rotated each framearound a vertical axis and, in sodoing, applied end rotations to themember. One of the frames wasmounted so that small variations inthe center-to-center distance be-,tween the frames could be accom-modated freely. The frames weremounted on a supporting frameworkthat resisted the shear forces devel-oped during loading. The support-ing framework provided attachmentpoints for the linkage system thatcould be set to give either uniformshear or uniform moment loading.

The zero-mean cyclic loading wasproduced by a variable amplitudeunit in which two eccentrics of fixedthrow were geared together andcoupled to the linkage system. Rota-tion of the eccentrics produced asimple harmonic motion. The ampli-tude of this motion could be varied,without changing the speed of the

42 PCI Journal

eccentrics, by an adjustment to thegearing system. The eccentrics weredriven by variable speed hydraulicmotors that controlled the frequencyof the motion. Two flywheels wereused to reduce unwanted frequencyfluctuations.

The machine produced memberend rotations anywhere in the rangebetween ±0.0005 and -!-0.04 radians,and end moments of up to -125,000in.-lb. The frequency could be variedbetween 0.25 and 5.0 cps.

It was also possible to apply eitheruniform moment or uniform shear toa member as a static load. MemberNo. 5 was loaded this way.

TEST PROCEDURE

The test procedure called for mak-ing one or more test runs on eachmember. A test run was a sequenceof steady-state cyclic tests, the firstat small amplitude, followed by

others at successively greater ampli-tudes until the member failed or thedesired maximum load was applied.After each steady-state test the am-plitude was increased by a prede-termined amount, and the frequencywas readjusted before the nextsteady-state test was made. Duringeach test run there was no interrup-tion of the zero-mean cyclic loading,and no change was made in the typeof loading applied. If the memberdid not fail when the planned maxi-mum load was applied, the ampli-tude was reduced to zero beforethe zero-mean cyclic loading wasstopped. The test frequency wasnormally 1.0 cps throughout eachtest run, but this was varied to 0.5and 2.0 cps at selected amplitudesto investigate the frequency depen-dence of the results.

The test runs made on each mem-ber are summarized in Table 2.

Table 2. Test runs made on each member

Test Maximum Maximummember Run Cycles Type of Me Bet Failure

No. No. applied loading in.-lb. radians1 1 5000 Moment 28,500 0.074 No2 1 3000 Shear 52,500 0.078 Yes3 1 2000 Moment 53,000 0.075 No

2 5000 51,000 0.070 "4 1 700 Shear 89,000 0.068 Yes6 1 1000 Moment 57,000 0.021 No

2 1000 80,000 0.0653 1000 80,000 0.0654 500 Shear 125,000 0.055 Yes

7 1 2500 Shear 120,000 0.026 No2 500 55,000 0.0093 5000 Moment 93,000 0.062 "

8 1 2500 Moment 48,000 0.070 No2 2500 44,000 0.052 'f3 2000 Shear 55,000 0.072 Yes

9 1 2000 Shear 78,000 0.066 Yes

June 1969 43

FREE _FREELENGTH 13 LOADED LENGTH r13'—r'LENGTH

AXIS OF CLAMP BOLTS AXIS OFROTATION

LMEMBER ROTATION

90"158"

Fig. 4. Spacing of clamp bolts used to apply loads to the testmembers

TEST MEASUREMENTS

Hysteresis loops showing the vari-ation of end moment, Me, plottedagainst the end rotation 0e, were ob-tained for eaoh end of each testmember every time a steady-statetest was made.

End rotations were measured bytransducers coupled to the mem-ber through the rigid steel framesclamped at each end. End momentswere measured by strain gages fixedto the round steel bars which passthroughi each frame. An X-Y oscillo-scope and a camera were used todisplay and record the hysteresisloops. The oscilloscope spot tracedone complete, closed loop duringeach cycle of steady-state loading.The oscilloscope records were re-produced as 8 x 10-in. photographs.Stiffness and damping values werefound by measuring these photo-graphs.

The testing machine and record-ing equipment were checked byloading a steel I-beam within itselastic range. The Me vs. 0, plot ap-peared as a straight line of zero loopwidth.

RESULTS

The M0 vs. 0e hysteresis loopsshown in Figs. 5 and 6 for members6 and 8 are typical of all the loopsobtained for steady-state loadingpast cracking. All such loops showedthree approximately linear portions,a small variation in loop width, and

rI

steeper slopes and greater loopwidth for uniform shear loading thanfor uniform moment loading. Forany given steady-state loading case,the loops recorded were all stableand repeatable, unless the memberwas failing. However, loops were notrecorded while ee was actually beingincreased, so changes in behaviourdirectly following the formation ofnew tension cracks could not bestudied.

To check on these changes, thefirst test run on member No. 5 wasmade by applying the zero-meancyclic loading as a series of staticloads. Separate loops were obtainedfor every cycle. In this test there wasa marked fall in loop width duringthe first two or three cycles immedi-ately after the formation of new ten-

n-3 -2 -1 0 1 7 3

0 (RADIANS X 10"2)

Fig. 5. Typical hysteresis loops formember No. 6

PCI Journal

U. SMEAR

U. MOMENT

-3 •2 •1 0 1 2 3B (RADIANS X1011

Fig. 6. Typical hysteresis loops formember No. 8

sion cracks. Additionally, there werethe expected reductions in stiffnessand strength resulting from tensilefailure in the concrete. However, allsubsequent cycles at the same endrotation gave stable, repeatableloops. The discussion that follows is

limited to the loops recorded forsteady-state loading.Stiffness. The stiffness, S, of a mem-ber at some point during a loadingcycle is defined as the slope of themoment-rotation curve at that pointgiven by:

dMeS dee

Values of S for uniform shear load-ing are then directly applicable inframe analysis.Stiffness parameters. Two specialstiffness parameters can be definedfor any given hysteresis loop. Theprimary stiffness, S1, is the gradientof the line L1, drawn as shown inFig. 7. Primary stiffness is the stiff-

Fig. 7. Stiffness and strength parameters from a typical hysteresisloop

June 1969 45

C UNIFORM MOMENT'° -Cry- •• SHEAR

C TENSION CRACKS OPENC F FAILURE BEGINS

RUN*1

CZ

3

F

4

3 4 5get (RADIANS X 10-2)

Fig. 8. Fall in apparent elastic modulus, member No. 6

•00

XNZ

ld

ness of the member when all thetension cracks are closed by the ac-tion of the prestress force, and thefull section is available to resistbending everywhere along the mem-ber. For the member in this condi-tion the second moment of area happlies along its whole length. Thedefinition of S l implies that simplebending theory can be used to finda value of E l , the apparent elasticmodulus of the concrete in the crack-closed condition, given by

E1=61L

for uniform shear loading, and by

E1 -S

for uniform moment loading, whereL is the loaded length of the mem-ber. Values of E l , calculated using1, = 72 in.4 and L = 90 in., areplotted against get, the double ampli-tude end rotation, in Fig. 8 for eachtest run made on member No. 6.The results are typical of those for

the other beams.Test run No. 1 (Fig. 8) shows

that the value of El for memberNo. 6 fell from an initial value ofapproximately 6 x 106 psi at the startof uniform loading to approximately5 x 106 psi even before the first ten-sion cracks, point "C", were seen.The fall in El continued as get wasfurther increased. Recovery, asshown by the start of run No. 2 wassmall. Runs Nos. 2 and 3 show afurther decrease in El with addi-tional uniform moment loading. Uni-form shear loading was applied intest run No. 4 and El fell to lessthan 50 percent of its initial valuebefore any evidence of failure wasseen. Failure was indicated by con-crete falling away from the cracksand this is noted as point "F".

Results obtained by another re-searcher( 7 ) for cyclic tests', in whichthe members were loaded in onedirection only, show very muchsmaller reductions in El.

There was no frequency depend-ence of the results observed, norany fall in E1 due to prolonged

46 PCI Journal

UNIFORM MOMENT—o----O-- SHEAR

F FAILURE BEGINS

U,0

U,NN

0

N#1

2 3 4 5eft (RADIANS X 10'2)

Fig. 9. Stiffness ratios, member No. 6

steady-state loading at any givenamplitude. The fall in El generallyappeared to be less for the memberswith higher prestress forces, possiblybecause of the reduction in concen-trated curvature in such members.

However, in every case, the zero-mean cyclic loading caused the pri-mary stiffness (stiffness of the crack-closed member) to fall by at least40 percent before the first damageor signs of failure appeared.

The second special stiffness param-eter is the stiffness ratio S 2/S1 . Foreach steady-state hysteresis loop forloading past cracking, the secondarystiffness, S 2 , is defined as having twovalues, equal to the gradients of thelines L2 and L2 respectively (see Fig.7). These are used for each loop, to-gether with the primary stiffness S1,

to define two stiffness ratios, S2/S1.Each is the ratio of the stiffness ofthe member loaded in one direction,so that the tension cracks are open,to the stiffness of the member duringthe same cycle when the cracks are

closed by the prestress force.Although Sl falls as Bet is in-

creased, it might be expected thatSz/S 1 would remain constant, ornearly so. This is not the case, as isshown in Fig. 9, where the stiffnessratios for member No. 6 are plottedagainst get . During run No. 1 underuniform moment loading, S 2 /S 1 fellabout 50 percent and reached anapparently stable value, as shown byrun No. 2. Throughout run No. 4,when uniform shear loading was ap-plied, the stiffness ratios were higherthan at the end of run No. 2. Theseratios probably would have stabi-lized at higher values had failurenot begun at "F".

The general conclusion for allmembers was the same: the testloading caused the stiffness ratios tofall, and higher minimum valueswere recorded for uniform shearthan for uniform moment loading.Stable minimum values of S 2 /S 1 un-der uniform shear loading variedfrom approximately 0.4 for memberNo. 7, which had the most prestress,

June 1969 47

F UNIF. MOMENT-O---o- UNIF. SHEAR

7 MEMBER NO.

F F FAILURE BEGINS6

z8 7 6

4 87

L 3 9 7 ie X 10.2)

Fig. 10. Values of nominal cracking moment

to 0.2 for member No. 2, which hadthe least.Strength. A nominal cracking mo-ment, Ml, was defined as havingpositive and negative values givenby the points of intersection of Lland L2 and Ll and L'2 respectivelyin Fig. 7. Values of M l for a numberof members are plotted against Betin Fig. 10. The actual end momentat which existing tension cracks firstre-open somewhere in a membercannot be calculated unless the pre-stress force is known accurately.However, for any member it shouldbe no greater than the least value ofMz recorded under uniform momentloading. The higher values of Mlrecorded for uniform shear loadingare more significant for design pur-poses because they indicate the endmoment at which cracking caused amajor reduction in member stiffness.

In all members tested, the value ofM l was independent of frequencyand of the number of applied cyclesof steady-state loading, except forthe first few cycles of loading afterthe opening of new tension cracks.

The Ml values did not fall by asignificant amount until evidence offailure was seen, point "F" in Fig. 10.

The failure and ultimate strengthbehaviour of these members was notstudied because only a small num-ber of members were tested andthey did not contain any shear re-inforcement. However, all membersdid have end rotations at failure atleast four times greater than thoseat cracking.Damping energy. Following the sug-gestions of Lazan( 8> for a uniformdamping nomenclature, the maindamping parameter is the totaldamping energy, Dt, defined as theenergy dissipated by a member dur-ing one cycle of steady-state loading.Because one hysteresis loop was re-corded at each end of a member foreach steady-state test, Dt was thesum of the measured areas of thesetwo loops. The use of Dt has severaladvantages: it is found without refer-ence to the mass or stiffness of themember; in some cases values of Dtfor individual members can beadded to give an overall damping

48 PCI Journal

value for a frame; and Dt can beused in calculating equivalent vis-cous damping coefficients if theseare required.

Values of Dt for all members, ex-cept No. 5, are shown plotted againstthe steady-state double amplitudeend rotation, Bet , in Fig. 11. Pointsat which cracking was first observedfor various members are marked "C",and points at which failure began,as shown by the falling away ofconcrete, are marked "F". Althoughthe numerical values apply only tothese particular tests, they showsome features of wider interest.

In every case, D t increased withBet, at least until failure began.Uniform shear and uniform momenttests on the same or similar mem-bers (e.g., No. 6) show that for any

given value of Bet, D t is higher foruniform shear loading.

The beginning of failure markedthe maximum value of D t for mem-ber No. 6, but D t continued to in-crease after failure began for mem-bers 2, 4 and 9, and doubled in thecase of member No. 9 before it final-ly fell. The author believes thatbinding of the concrete in zones ofmaximum curvature would have en-sured that Dt either maintained orincreased its value after the begin-ning of failure. At least 1000 cyclesof steady-state loading, at any ampli-tude that did not cause failure, couldbe applied with no reduction in D.Thus, the damping mechanism doesnot appear to "wear

Various tests were at 0.5,1.0 and 2.0 cps, and there was no

Za_

9

UNIFORM MOMENT--o---o— " 1• SHEAR

8 MEMBER NO.C TENSION CRACKS OPENF FAILURE BEGINS

4

F 2

F6 F9

F 8

4

77 6

83

8J 2 1C

2 4O. (RADIANS X 10.2)

Fig. 11. Values of damping energyJune 1969 49

z

A,t (RADIANS X 10-A)

Fig. 12. Values of equivalent loop width

measurable change in D. From thesetests it appeared that there wasno significant frequency dependentdamping, at least within this rangeof frequencies. (The calibration testmade on a steel beam, that had toolittle damping to give a measurableloop width, showed that a smallamount of velocity dependent damp-ing might have been present in thetest members.)

The first tests of each member,made at amplitudes too small tocause cracking, gave negligible val-ues of D. Measurable damping be-gan as soon as the first tension cracksdeveloped, and was observed duringall subsequent tests including thosewhen the amplitude was too smallto causo any of the tension cracksto re-open. This suggests that low-amplitude vibration tests of a realstructure, before it is subjected toits first major earthquake, mightgive a much lower damping valuethan tests after the earthquake, eventhough the structure does not appearto have been damaged.

It appeared from the tests that

the two major sources of damping inthe members might be: 1) inelasticbehaviour of the concrete, especiallyin those regions where cracking re-sults in high stress and curvatureconcentration; and 2) the slip be-tween steel and concrete associatedwith tension cracking. This wouldexplain why the uniform shear load-ing, with more severe cracking andcurvature concentration for any giv-en end rotation, gives the higher Dtvalues shown in Fig. 11.

It also suggests why increasing thesteel area and prestress force in-creased Dt for all cases of uniformmoment loading, but not for all casesof uniform shear loading. In the caseof uniform moment, the increase insteel area apparently results in in-creased friction due to slip. But, inthe case of uniform shear, the ex-tent to which the increased prestressforce tends to reduce the concen-tration of curvature near the cracksmeans that the increase in steel areatends to be balanced by a reductionin slip and perhaps in inelastic con-crete behaviour as well.

50 PCI Journal

Ii-

_l0

OL0

UNIFORM MOMENT^--^ " 11 SHEAR

8 MEMBER NO.

9

o Dv-%–^' 6 8

786

2 3 4 5e, (RADIANS X 10-2)

Fig. 13. Values of the loop width factor

0

The suggested causes of dampingalso make it clear that values of Dtcannot be simply related to the vol-ume of the member or extended di-rectly to other members.Loop width parameters. Two addi-tional damping parameters were alsodefined. Because of the approxi-mately constant width of the meas-ured hysteresis loops, an equivalentloop width, LWe (a moment valuerepresenting the average loop width),was defined for each steady-stateloop by:

LWe =loop area

double amplitude end rotation, Bet

The second parameter is the loopwidth factor, LWF, defined for anysteady-state loop by:

LWF =equivalent loop width, LWe

cracking moment, Ml

The loop width factor is a dimen-sionless parameter and it relates thedamping energy of a member to its

cracking strength for different values'of get.

Values of LWe for members 6 and7 are shown in Fig. 12, and valuesof LWF for a number of membersare shown in Fig. 13. The values ofLWF for members 6, 7 and 8 donot vary greatly with changes in bet,and suggest a range of values withinwhich the values for other similarmembers might be expected to lie.

Idealized loop. The measured loops(Figs. 5 and 6) suggest that the re-sults for primary stiffness S 1 , stiff-ness ratio S2 /S l , cracking momentMl , and the loop width factor LWFmight be used to define an idealizedbi-linear loop with constant loopwidth, like that shown in Fig. 14:The limited number of tests madeindicates that loops of this type—used with a range of stiffness,strength and damping parametersbased on uniform shear loading tests—would be adequate to describemember properties, at least in a pre-liminary dynamic analysis of a pre-stressed concrete frame structure.

June 1969 51

Fig. 14. An idealized moment-rotationhysteresis loop

CONCLUSIONS

The uniform shear loading testsgave stiffness and damping valuessuitable for use in a non-linear dy-namic frame analysis. The two mostimportant features of the loading inthese tests were the application ofend rotations to produce the uni-form shear, and the cyclic variationof these rotations about a mean ofzero. Neither the frequency nor thenumber of cycles of loading ap-peared to affect the stiffness,strength, or damping during anysteady-state test. The fall in stiffnessafter each increase in Bet , and thenegligible damping energies meas-ured for new untracked members,suggest that the first earthquakeloading might significantly changethe response characteristics of a pre-stressed concrete frame. For ana-lytical purposes, the hysteresis loopsobtained after the first two or threecycles of loading past cracking couldbe approximated by bi-linear loopsof constant width.

ACKNOWLEDGMENT

The work described here was car-

ried out at the University of Auck-land School of Engineering, NewZealand, with the support of theUniversity Grants Committee. Theauthor is indebted to Professor N. A.Mowbray and other members of theuniversity staff for their help andencouragement.

REFERENCES

1. Lin, T. Y., "The Design of PrestressedConcrete Buildings for Earthquake Re-sistance," Journal of the PrestressedConcrete Institute, Vol. 9, No. 6, De-cember 1964.

2. Galezewski, S., "Principles of the De-sign and Construction of Earthquake-Resistant Prestressed Concrete Struc-tures," presented at Eleventh AnnualConvention, Prestressed Concrete Insti-tute, 1965.

3. Despeyroux, J., "The Use of PrestressedConcrete in Earthquake-Resistant De-sign," Proceedings, Third World Con-ference on Earthquake Engineering,1965.

4. Sutherland, W. M., "Prestressed Con-crete Earthquake-Resistant Structures,Development, Performance and CurrentResearch," Proceedings, Third WorldConference on Earthquake Engineering,1965.

5. Clough, R. W. and Benuska, K. L.,"FHA Study of Seismic Design Cri-teria for High-Rise Buildings," HUDTS-3, August 1966, pp. v, 1.1.

6. Nakano, K., "Experimental Behaviourof Prestressed Concrete Four StoreyedModel Structure Under Lateral Load,"Proceedings, Third World Conferenceon Earthquake Engineering, 1965.

7. Venuti, W. J., "A Statistical Approachto the Fatigue Failure of PrestressedConcrete Beams," Journal of the Amer-ican Concrete Institute, Proceedings,Vol. 62, No. 11, November 1965.

8. Lazan, B. J., "Energy Dissipation Mech-anisms in Structures, with ParticularReference to Material Damping," Sec-tion 1, Structural Damping, ed. J. E.Ruzicka, Pergamon Press, 1960.

Discussion of this paper is invited. Please forward your discussion to PCI Headquartersby September 1 to permit publication in the December 1969 issue of the PCI JOURNAL.

52 PCI Journal