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Models of Ancient Columns and Colonnades

Subjected to Horizontal Base Motions - Study of

their Dynamic and Earthquake Behaviour

G C. Manos, M. Demosthenous

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1. Introduction

Ancient Greek and Roman structurescomposed of large heavy members thatsimply lie on top of each other in a perfect-fit construction without the use ofconnecting mortar, are distinctly differentfrom relatively flexible contemporarystructures. The colonnade (including freestanding monolithic columns or columnswith drums) is the typical structural formof ancient Greek or Roman temples. Thecolumns are connected at the top with theepistyle (entablature), also composed ofmonolithic orthogonal blocks, spanning thedistance between two columns (figure 1). Figure 1 Typical ancient colonnadesThe seismic response mechanisms that develop on this solid block structural systemduring strong ground motions can include sliding and rocking, thus dissipating theseismic energy in a different way from that of conventional contemporary buildingsThis paper presents results and conclusions from an extensive experimental study thatexamines the dynamic response of rigid bodies, representing simple models of ancientcolumns or colonnades. These models are subjected to various types of horizontal basemotions (including sinusoidal as well as earthquake base motions), reproduced by theEarthquake Simulator Facility of Aristotle University. The employed in this study rigidbodies were made of steel and are assumed to be models of prototype structures 20times larger. Two basic configurations are examined here; the first is that of a singlesteel truncate cone assumed to be a model of a monolithic free-standing column whereas

Transactions on the Built Environment vol 26, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

290 Structural Studies, Repairs and Maintenance of Historical Buildings

the second is formed by two steel truncate cones, of the same geometry as the oneexamined individually, supporting at their top a rectangular solid steel, representing inthis way the simplest unit of a colonnade. In the first part, dynamic response results arepresented, from an extensive investigation employing one monolithic free-standing rigidcolumn These are next compared with predictions from numerical simulations thatemployed software developed at Aristotle University. In the second part, the dynamicresponse of two steel columns, supporting at their top a rectangular steel block areinvestigate experimentally under simulated base motions.

2. Investigation of the dynamic response of a solid rigid bodyThe dimensions of the examined rigid body, a monolithic steel truncate cone, are shownin figure 2. This specimen was placed on the earthquake simulator simply free-standingon the horizontal moving platform of the shaking table. A very stiff, light metal framewas built around the studied steel truncate cone in order to carry the displacementtransducers that measured the rocking angle; this metal frame also provided temporarysupport to specimen during excessive rocking displacements indicating overturning.

The dynamic behavior of this specimen isgoverned by its rocking response. Theexperimental investigation of its rockingresponse included periodic sine-wavetests with a chosen range of frequenciesand amplitudes as well as earthquakesimulated tests using the El Centro 1940record. A simple way to estimate thedissipation of energy during rocking is toobtain the value of the coefficient ofrestitution by comparing free vibrationrocking response measurements withthose predicted numerically. From suchcomparisons the coefficient of restitutionis obtained with the value that yields thebest agreement between numerical andmeasured response parameters.

Figure 2. Dimensions of thesteel truncate cone.

The numerical solutions employed were verified for their stability and convergencethrough an extensive parametric study that examined various methods of numericalintegration and time step. This correlation was next extended with measurements fromsinusoidal as well as earthquake simulated tests. The numerical studies in this caseemployed the same base excitations recorded at the Earthquake Simulator during thetests and used as values for the coefficient of restitution the ones determined from thesame experimental sequence and the best numerical solution that was found from theabove mentioned study (Demosthenous 1994)2.1 Free Vibration Tests The sequence of tests for the solid truncate cone includedfree vibrations tests, with the objective of assessing the coefficient of restitution of thetest structure. The following figure (figure 3) depicts the rocking angle response (solidline) from such a free vibration test. Supenmposed on this graph with a dash line is thenumerically predicted rocking response, with a coefficient of restitution value that gavethe best fit to the experimental measurements. The predicted rocking response wasderived from the linearised equations of the rocking motion (Aslam et al 1980). Certain


Structural Studies, Repairs and Maintenance of Historical Buildings 291

Truncate Cone (Solid)Initial non-dim angle = 0.475

coef. of restitution ( e ) = 0.982

: O 5 TIME (s«c)Figure 3 Measured and Predicted

Free Vibration Rocking

a) max. non dim angle : 0.012 CONE (solid)

discrepancies are apparent at the initial large amplitude rocking stages. This must beattributed to three-dimensional (3-D) rocking (Koh 1990). since rocking in a perfectplane for a 3-D model requires exacting initial conditions and the slightest out-of-planedisturbance will cause rocking to be 3-D

When the specimen tested isprismatic instead of truncatecone the 3-D response duringrocking is minimized and goodcorrelation can be obtainedbetween observed and predicted

10 behavior thus assessing moreaccurately the coefficient ofrestitution (Manos andDemosthenous 1990).2.2 Sinusoidal Tests Duringthese tests the frequency ofmotion was varied from IHz to7Hz in steps of IHz from test totest. This resulted in groups oftests with constant frequency forthe horizontal sinusoidal motionfor each test. In the various testsbelonging to the same group ofconstant frequency, the amplitudeof the excitation was variedprogressively from test to testFigures 4a, 4b and 4c depict therocking response of the solidspecimen during this sequence oftests for three different excitationfrequency cases (3Hz, 4Hz and6Hz). The following points canbe made from the observedbehavior during these tests:

max. base disp. : 4mm FREQ. EXCITATION 3Hz

b) max. non dim. angle : 0.209 CONE (solid)

max. base disp.: 4mm FREQ. EXCITATION 4Hz

c) max. non dim. angle : 0.104 CONE (solid)

max, base disp : 4mm FREQ. EXCITATION 6Hz12O 6 TIME ( sec )

Figure 4 Rocking during sinusoidal baseexcitations.

- For small amplitude tests the rocking behavior is not present; the motion of thespecimen in this case follows that of the base.- As the horizontal base motion is increased in amplitude, rocking is initiated Thisrocking appears to be sub-harmonic in the initial stages and becomes harmonic at thelater stages.- Further increase in the amplitude of the base motion results in excessive rockingresponse, which after certain buildup leads to the overturning of the specimen. At thisstage the rocking response is also accompanied by some significant sliding at the baseas well as by rotation and rocking response out-of-plane of the excitation axisThe above stages are graphically portrayed in the plot of figure 5 (together with thenumerical results as explained below). The ordinates in this plot represent the nondimensional amplitude of the base motion whereas the absiscae represent the non-dimensional frequency given by formulae (1). The following observations summarizethe mam points:



- The non-dimensional stable-unstable limit rocking amplitude increases rapidly with th -non-dimensional frequency.- For small values of the non-dimensional frequency the transition stage from no-rocking to overturning, in terms of non-dimensional amplitude, is very small and itoccurs with minor amplitude increase.A further numerical study has been performed, aimed to simulate the dynamic responseof the solid specimen, using the non-dimensional linear equations for sinusoidalexcitation as given by Spanos and Koh (1984) and with a value for the coefficient ofrestitution as obtained from the free vibration tests. The numerical analyses have beenperformed for various combinations of amplitude and frequency. Figure 5 presents theseresults in a summary form. Fairly good agreement can be seen between numericalpredictions and the corresponding experimental measurements.

A = x / ( 6cr * g ),H = co/p, (1)p2 = W Re / lo

A =Non-dimensional baseacceleration amplitude

x = Actual honz. peak accelerationof the sinusoidal motion

6cr Critical angle(rad) indicatingoverturning of the specimen

g =The gravitational accelerationQ =Non dimensional frequencyto =Actual sinusoidal freq. (Hz)p =The natural rocking frequency

of the specimen (Hz)W =The weight of the blockRc=The distance of the center of

gravity from the rocking polelo =The mass moment of inertia

TRUNCATE CONEe = 0.982

experimental

SINUSOIDAL BASE EXCITATIONLINEAR EQUATIONS

2 3EXCITATION

4 5FREQUENCY

Figure 5. Summary from numerical analysis

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Non Dimentional Rocking A

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6 Correlation of the predicted and measured rocking

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A number of tests were performed during thissequence with progressively increasing intensity, based on the 1940 El CentreEarthquake record. The base acceleration and displacement response was measuredtogether with the rocking response of the tested specimen. The principal objective ofthese tests was to observe again the stable-unstable behavior of the model. The baseexcitations recorded at the Earthquake Simulator were used in the numerical simulationof the earthquake response; the value for the coefficient of restitution was taken fromthe free vibration tests. The full time history of the predicted rocking angle response iscompared with the corresponding measured rocking angle in figure 6. As can be seenfrom this correlation, good agreement was achieved in this case by the employednumerical simulation.

3. Dynamic Response of two model column with a top-block

17 Genera/ D2jrr%?fzbM The research effort presented before for individual freestanding rigid bodies (models of free-standing individual columns) was complementedby this study aiming to investigate the dynamic response of two individual columnssupporting at their top a rectangular prismatic rigid block (representing part of theentablature). This formation, shown in figures 7 and 8. is the simplest possiblestructural unit that combines more than one free standing column. In this investigation

the examined structural unit wasformed of two model columns with theform of solid truncate cones with thesame lower part, having dimensionsidentical to the single column presentedin section 2. This lower part wascomplemented at the top by an upperpart formed by a small truncate coneand a prism (see figures 1. 7 and X).These lower and upper parts for eachcolumn were not constructedseparately; instead they were shaped inone piece from a single steel blockresulting in two identical solid rigidbodies. Each free-standing column was522.5mm high and had a circular baseof 97mm diameter and a 103mm by100mm almost square top.

axis ofsymmetry

522mm

CONE1

CONE2

Instrumentation

» Horiontal displ.

Vertical displ.

Accelerometer

SHAKING TABLEI BASE-215mni

Figure 7. Studied structural formation andemployed instrumentation

These model columns were placed on the horizontal moving platform of the shakingtable at a distance of 215mm between their axes of symmetry. A rectangular prismaticrigid block was placed in a symmetric way on top of these columns, as shown in figures7 and 8; it was also made of the same steel that the two columns were made of and ithad a length of 215mm. a height of 68mm and a width of 87mm. This block was notconnected in any other way with the two columns but was simply free standing on theirtop square sections. The longitudinal horizontal axis of this rectangular block coincidedwith the vertical plane formed by the two axes of symmetry of the columns; its leftand right end sections (figures 7 and 8) were located at the center planes of theleft and right column, respectively.



3.2 Expected Modes of Response - InstrumentationInstrumentation was provided in order to measurethe expected complex response of this structuralformation during a series of tests that included freerocking vibrations as well as sinusoidal andsimulated earthquake excitations of the base. Thisinstrumentation is depicted in figure 7; it includedaccelerometers at the top and bottom of each columnplaced in a way as to coincide with the longitudinalvertical plane of symmetry of this system. Additionalaccelerometers were placed at the longitudinalhorizontal axis of the rigid top-block at both endsagain at the longitudinal vertical plane ofsymmetry. All these accelerometers had the purpose

of recording the acceleration response of the various parts of this system. Horizontalbase excitations used during this sequence with direction coinciding with thislongitudinal vertical plane of symmetry. However, an accelerometer was placed at themid-side of the rigid top-block perpendicular to this longitudinal plane of symmetry inorder to identify out-of-plane response modes of the block. Apart from theaccelerometers, displacement sensors were also used, as indicated with the arrows in thesame figure, aimed at recording the rocking displacement response of either the columnsor the top-block. Finally, the acceleration of the horizontal motion of the shaking

Figure 8. Rigid Block Formationon the Shaking Table

table moving platform wasalso recorded. Figure 9adepicts the ideal rockingresponse of this complexsystem. This response isderived with the assumptionthat it occurs only in thevertical plane of symmetryand that there is nodifferential sliding of the two b )free-standing columns at thebase. Moreover, the top- "Jblock is assumed to rock ontop of the columns, as shownin figure 9a. without eithersliding or losing contact withthe supporting columns As aresult from these assumptionsboth columns develop thesame rocking angle during allstages of the response.

ui ui

1 —

10-

2

i If i!

a)

IDEAL ROCKING RESPONSE OF THE SYSTEM

COLONNADE WITH 2 COLUMNS (TRUNCATE CONES)UPLIFT OF THE BLOCK VS ROCKING ANGLE OF THE COLUMN

Uplift over the 1st Column Uplift over the 2nd column

UPLIFT(mm)

U1 ( O < 0 )

U2 (0 < 0 )

U2 ( 0 > 0 ) ..."•"

U1 ( 0 > 0 )

-Gcr R O C K I N G A N G L E +GcrFigure 9. Ideal Rocking Response of the System.

The rocking and uplifting response of the top-block can be obtained as a function of therocking angles of the supporting cones based on these assumptions. Figure 9b depictsthe ideal uplift displacements of the left and right ends of the top-block for rockingangles of the columns from - Ocr to Ocr (where Ocr the critical rocking angle beyondwhich each one of these two cones studied individually overturns).



non dim. rocking angle of the 1st col. ( 0 = 8 / Gcr ) / max. 0.281, mln. - 0.236Uplift of the block over the 1st col. (U1) / max. 4.45mm, min. -0.493mm

3.95 T I M E ( sec ) 12.88

Figure lOa Left-cone and top-block response (3Hz)

• non dim. rocking angle of the 1st col. ( 0 = 8 / Gcr ) / max. 0.037, mln. • 0.043-Uplift of the block over the 1st col. (U1) / max. 0.223mm, min. -0.113mm

2.99 T I M EFigure 1 Ob Left-cone and top-block response (7Hz)

-Top acceleration of the 1st column / max accel. 0.32 gHor. accel. of the block over the 1st column / max. accel. 0.35g

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111'* T I M E ( sec ) 11.9

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9Figure lOc Left-cone and top-block response (3Hz)

j. j 5"z/z ozWa/ 7^^ A limited number of results from this experimental sequence ispresented here and discussed Figures lOa to lOc depict the measured response duringthis sinusoidal base excitation sequence. The peak base acceleration during thissequence was kept equal to 160 gals The excitation frequency was varied from 3Hz to



7Hz in steps of IHz from test to test. From the measured response the rocking angle ofthe left column (1) was derived and is depicted with a solid line in the plots of figureslOa (3Hz). 10b(7Hz). In these figures the uplift displacement of the left end of the top-block is also plotted with a dashed line. As can be seen, considerable rocking angle andblock uplift response develops when the excitation frequency is equal to 3Hz (fig. lOa).However, when the excitation frequency becomes equal to 7Hz (fig. lOb) the observedcone and top-block response becomes almost 10 times smaller. This observation agreesvery well with a similar conclusion reached from the extensive study of the rockingresponse of the individual solid rigid columns (see section 2.2 and figure 4). Figure lOcdepicts the acceleration response for the left cone (1) and the left side of the top-block.The base motion is in this case 3Hz sinusoidal excitation with peak base acceleration160gals. As can be seen in this figure, the acceleration response of the top-blockexhibits strong correlation with that of the supporting cone. The peak block uplift.measured during the sinusoidal test rocking stages, is plotted with asterisks in figure 11against the ideal response curves described in section 4.2 (see also figure 9a and 9b). Ascan be seen there is reasonable agreement between measured and expected values.

COLONNADE WITH 2 COLUMNS (TRUNCATE CONES)UPLIFT OF THE BLOCK AS ESTIMATED FROM THE ROCKING ANGLE

Corellation with the Uplift from measurements5.55mm

UpLIFT

(mm)

U1 (O < 0 )

SINUSOIDAL BASE EXCITATIONU2 (0 > 0 )

U2 (G < 0 )U1 ( 0 > 0 )

• 0.3 0 cr R O C K I N G A N G L E + 0.3 G crFigure 11. Observed and expected top-block response

non dim. rocking angle of the P' column (0) / max.. 0.065 min.-0.10Uplift of the block over the 2"** column (U2) / max. 1.23mm

a)EL CENTRO BASE EXCITATION-SPAN 1 - max. base accel. 0.304g

00 T I M E ( sec ) 8.05Figure 12 Left-cone and top-block right displacement response (Span 1)



non dim. rocking angle of the 1" column (0) / max. 0.54 min. - 0.30Uplift of the block over the 2"^ column (U2) / max. 7.3 mm

EL CENTRO BASE EXCITATION-SPAN 3 - max. base accel. 0.935g

00 T I M E (sec) 8.05Figure 13 Left-cone and top-block right displacement response (Span 3)

COLONNADE WITH 2 COLUMNSUPLIFT OF THE BLOCK AS ESTIMATED FROM THE ROCKING ANGLE

Correlation with the Uplift from measurements

EL CENTRO BASE EXCITATION

(mm)

-0.6 0cr ROCKING A N G L E - 0.6 0crFigure 14 Observed and expected top-block response

3.4 Simulated Earthquake tests A number of tests were performed during thissequence with progressively increasing intensity, indicated by the parameter Span.based on the 1940 El Centro Earthquake record. The larger the value of the parameterSpan the more intense the simulated earthquake motion. Again the principal objectivehere was to observe the top-block response relative to the response of the supportingfree-standing cones (see figures 12 and 13). Measured peak top-block uplift responsevalues are listed in Table 1 together with the peak base acceleration and the peak non-dimensional rocking angular response for the columns. As can be seen from this tablethe more intense simulated earthquake motions result in larger rocking angular responsefor the columns and consequently to larger uplift response for the top-block. Moreover.the columns together with the top-block exhibit rocking response during all the durationof the intense earthquake motion, although this does not lead to unstable response forthis particular earthquake motion. Finally, figure 14 depicts the measured peak top-block uplift (circles) against the ideal response curves described in section 4.2 (see alsofigure 9a and 9b). As can be seen the correlation between measured and predicted top-block response exhibits similar trends with the similar correlation that was observedduring the sinusoidal tests (figure 11)



4. Discussion of the results - conclusionsa. The coefficient of restitution can be defined with sufficient accuracy from the freevibration test results in combination with the performed numerical analysis, despitecomplications from the three dimensional rocking, which was present for both solid aswell as sliced specimens.b. The numerically predicted upper-bounds for the stable-unstable rocking of the solidprismatic specimen, subjected to sinusoidal base motions, agrees well with the observedbehavior.c. The behavior of the examined two columns with the rigid top-block subjected tosinusoidal base excitations exhibits similar trends to those observed for the individualcolumn, with regard to the influence that the excitation frequency exerts on the rockingamplitude. That is, for the same non-dimensional excitation amplitude, the higher thefrequency of the excitation the smaller the rocking amplitude.d. For not very large rocking angles the observed response of the two columns with thetop-block system tends to remain in-plane. It also exhibits good correlation with the onepredicted under the assumptions of no differential sliding of the columns at their baseand no sliding between the top-block and the supporting columns. This observation isvalid for both the sinusoidal as well as the simulated earthquake excitations.

Table 1. Measured peak top-block upliftduring the simulated earthquake sequence.IntensityofBase

MotionSpan 1

Span 2

Span 3

PeakBaseAce.

(8)0.304

0.614

0.935

e/6crmaxmm0.065-0.1000.183-0 1900.540-0.300

Ul(mm)maxnun0.401.181.393.652 JO5.90

U2(mm)maxmin1.231.003.282.437.30397

ACKNOWLEDGEMENTS : Thepartial support of the GreekOrganization for Seismic Planningand Protection (OASP) as well as theEuropean Commission (DIR XII ,ENV.4-CT.96-0106, ISTECH),is gratefully acknowledged.

0 Non Dimensional Cone Rocking AngleUl Top-Block Left Uplift DisplacementU2 Top-Block Right Uplift Displacement

ReferencesAs lam . MM. et.all. 1978. Rocking and Overturning response of rigid bodies toearthquake motions. Report No. LBL-7539, L B Lab. Univ. Of California. Berkeley.

Demosthenous M. 1994. Experimental and numerical study of the dynamic response ofsolid or sliced rigid bodies. Ph. D. Thesis, Dep. Civil Eng . Aristotle University

Koh. A.S. and Mustafa G. 1990. Free rocking of cylindrical structures. J of EngMechanics. Vol. 116. NO 1. pp.35-54.

Manos. G. C. and Demosthenous. M . 1990. The behavior of solid or sliced rigid bodieswhen subjected to horizontal base motion. Proc. 4^ U.S. Nat. Confer. Earth. Eng, Vol.3, pp. 41-50

Manos, G.C.. and Demosthenous, M. 1991. Comparative study of the dynamic responseof solid and sliced rigid bodies, Proc , Florence Modal Analysis Confer., pp.443-449.

Spanos, P.D. and Koh A.S., 1984 Rocking of rigid blocks due to harmonic shaking. J ofEngm. Mech.. ASCE 110 (11), pp. 1627-1642


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