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 structures composed of large heavy members that simply lie on top of each other in a perfect- fit construction without the use of connecting mortar, are distinctly different from relatively flexible contemporary structures. The colonnade (including free standing monolithic columns or columns with drums) is the typical structural form of ancient Greek or Roman temples. The columns are connected at the top with the epistyle (entablature), also composed of monolithic orthogonal blocks, spanning the distance between two columns (figure 1). Figure 1 Typical ancient colonnades The seismic response mechanisms that develop on this solid block structural system during strong ground motions can include sliding and rocking, thus dissipating the seismic energy in a different way from that of conventional contemporary buildings This paper presents results and conclusions from an extensive experimental study that examines the dynamic response of rigid bodies, representing simple models of ancient columns or colonnades. These models are subjected to various types of horizontal base motions (including sinusoidal as well as earthquake base motions), reproduced by the Earthquake Simulator Facility of Aristotle University. The employed in this study rigid bodies were made of steel and are assumed to be models of prototype structures 20 times larger. Two basic configurations are examined here; the first is that of a single steel 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 one

examined individually, supporting at their top a rectangular solid steel, representing in this way the simplest unit of a colonnade. In the firstpart , dynamic response results are presented, from an extensive investigation employing one monolithic free-standing rigid column These are next compared with predictions from numerical simulations that employed software developed at Aristotle University. In the second part, the dynamic

response of two steel columns, supporting at their top a rectangular steel block are investigate experimentally under simulated base motions.

2. Investigation of the dynamic response of a solid rigid body The dimensions of the examined rigid body, a monolithic steel truncate cone, are shown

in figure 2. This specimen was placed on the earthquake simulator simply free-standing on the horizontal moving platform of the shaking table. A very stiff, light metal frame was built around the studied steel truncate cone in order to carry the displacement transducers that measured the rocking angle; this metal frame also provided temporary support to specimen during excessive rocking displacements indicating overturning. The dynamic behavior of this specimen is

governed by its rocking response. The experimental investigation of its rocking response included periodic sine-wave tests with a chosen range of frequencies and amplitudes as well as earthquake

simulated tests using the El Centro 1940 record. A simple way to estimate the dissipation of energy during rocking is to obtain the value of the coefficient of restitution by comparing free vibration rocking response measurements with

those predicted numerically. From such comparisons the coefficient of restitution is obtained with the value that yields the Figure 2. Dimensions of the best agreement between numerical and steel truncate cone. measured response parameters. The numerical solutions employed were verified for their stability and convergence

through an extensive parametric study that examined various methods of numerical integration and time step. This correlation was next extended with measurements from sinusoidal as well as earthquake simulated tests. The numerical studies in this case employed the same base excitations recorded at the Earthquake Simulator during the tests and used as values for the coefficient of restitution the ones determined from the same experimental sequence and the best numerical solution that was found from the above mentioned study (Demosthenous 1994)

2.1 Free Vibration Tests The sequence of tests for the solid truncate cone included free vibrations tests, with the objective of assessing the coefficient of restitution of the test structure. The following figure (figure 3) depicts the rocking angle response (solid line) from such a free vibration test. Supenmposed on this graph with a dash line is the numerically predicted rocking response, with a coefficient of restitution value that gave the best fit to the experimental measurements. The predicted rocking response was

derived from the linearised equations of the rocking motion (Aslam et al 1980). Certain

discrepancies are apparent at the initial large amplitude rocking stages. This must be

attributed to three-dimensional (3-D) rocking (Koh 1990). since rocking in a perfect plane for a 3-D model requires exacting initial conditions and the slightest out-of-plane disturbance will cause rocking to be 3-D Truncate Cone (Solid) When the specimen tested is Initial non-dim angle = 0.475 prismatic instead of truncate cone the 3-D response during rocking is minimized and good correlation can be obtained coef. of restitution ( e ) = 0.982 between observed and predicted : O 5 TIME (s«c) 10 behavior thus assessing more Figure 3 Measured and Predicted accurately the coefficient of Free Vibration Rocking restitution (Manos and Demosthenous 1990). a) max. non dim angle : 0.012 CONE (solid) 2.2 Sinusoidal Tests During these tests the frequency of

motion was varied from IHz to 7Hz in steps of IHz from test to max. base disp. : 4mm FREQ. EXCITATION 3Hz test. This resulted in groups of tests with constant frequency for b) max. non dim. angle : 0.209 CONE (solid) the horizontal sinusoidal motion for each test. In the various tests

belonging to the same group of constant frequency, the amplitude max. base disp.: 4mm FREQ. EXCITATION 4Hz of the excitation was varied

c) max. non dim. angle : 0.104 CONE (solid) progressively from test to test Figures 4a, 4b and 4c depict the

rocking response of the solid specimen during this sequence of tests for three different excitation max, base disp : 4mm FREQ. EXCITATION 6Hz frequency cases (3Hz, 4Hz and O 6 TIME ( sec ) 12 6Hz). The following points can Figure 4 Rocking during sinusoidal base be made from the observed excitations. behavior during these tests: - For small amplitude tests the rocking behavior is not present; the motion of the specimen in this case follows that of the base. - As the horizontal base motion is increased in amplitude, rocking is initiated This rocking appears to be sub-harmonic in the initial stages and becomes harmonic at the later stages.

- Further increase in the amplitude of the base motion results in excessive rocking response, which after certain buildup leads to the overturning of the specimen. At this stage the rocking response is also accompanied by some significant sliding at the base as well as by rotation and rocking response out-of-plane of the excitation axis The above stages are graphically portrayed in the plot of figure 5 (together with the numerical results as explained below). The ordinates in this plot represent the non dimensional amplitude of the base motion whereas the absiscae represent the non- dimensional frequency given by formulae (1). The following observations summarize the 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 it occurs with minor amplitude increase. A further numerical study has been performed, aimed to simulate the dynamic response

of the solid specimen, using the non-dimensional linear equations for sinusoidal excitation as given by Spanos and Koh (1984) and with a value for the coefficient of restitution as obtained from the free vibration tests. The numerical analyses have been performed for various combinations of amplitude and frequency. Figure 5 presents these results in a summary form. Fairly good agreement can be seen between numerical predictions and the corresponding experimental measurements.

A = x / ( 6cr * g ), H = co/p, (1) p2 = W Re / lo SINUSOIDAL BASE EXCITATION A =Non-dimensional base LINEAR EQUATIONS TRUNCATE CONE acceleration amplitude e = 0.982 experimental x = Actual honz. peak acceleration of the sinusoidal motion 6cr ^Critical angle(rad) indicating overturning of the specimen g =The gravitational acceleration Q =Non dimensional frequency to =Actual sinusoidal freq. (Hz) p =The natural rocking frequency of the specimen (Hz) W =The weight of the block 23 45 EXCITATION FREQUENCY Rc=The distance of the center of gravity from the rocking pole lo =The mass moment of inertia Figure 5. Summary from numerical analysis

: EXPCI~i mental Max value : 8.589 Min value : -8 575 : Hunei*ical Max value : 8.685 Min value : -8.597

Specimen : COME /' M.F.D. e : 8.973

Method : RUNGE KUTTA Non 1 inear equat ions A Constant time stc:p 1 ;'; .': (DT=.8888 1 ) J I! fJS !/| i\f\ .ft ftft'j>_fl'J V f'1'Yi VjVA'riiijV^AlLf^ViWjO/A'SJM^iviiiiaW

f: EL CEMTRO 1948 Span : 2.5 max base accel = 8.568 g r J8 16.81 TIME ( sec )

6 Correlation of the predicted and measured rocking angle response (j q c o No n Dimentiona l Rockin g Angl e

A number of tests were performed during this sequence with progressively increasing intensity, based on the 1940 El Centre Earthquake record. The base acceleration and displacement response was measured together with the rocking response of the tested specimen. The principal objective of these tests was to observe again the stable-unstable behavior of the model. The base

excitations recorded at the Earthquake Simulator were used in the numerical simulation of the earthquake response; the value for the coefficient of restitution was taken from the free vibration tests. The full time history of the predicted rocking angle response is compared with the corresponding measured rocking angle in figure 6. As can be seen from this correlation, good agreement was achieved in this case by the employed numerical simulation.

3. Dynamic Response of two model column with a top-block 17 Genera/ D2jrr%?fzbM The research effort presented before for individual free standing rigid bodies (models of free-standing individual columns) was complemented by this study aiming to investigate the dynamic response of two individual columns

supporting at their top a rectangular prismatic rigid block (representing part of the entablature). This formation, shown in figures 7 and 8. is the simplest possible structural unit that combines more than one free standing column. In this investigation the examined structural unit was formed of two model columns with the axis of symmetry form of solid truncate cones with the same lower part, having dimensions identical to the single column presented Instrumentation in section 2. This lower part was complemented at the top by an upper » Horiontal displ. part formed by a small truncate cone

and a prism (see figures 1. 7 and X). 522mm Vertical displ. These lower and upper parts for each column were not constructed CONE CONE Accelerometer 1 2 separately; instead they were shaped in one piece from a single steel block SHAKING TABLE resulting in two identical solid rigid I BASE -215mni bodies. Each free-standing column was 522.5mm high and had a circular base Figure 7. Studied structural formation and of 97mm diameter and a 103mm by employed instrumentation 100mm almost square top.

These model columns were placed on the horizontal moving platform of the shaking

table at a distance of 215mm between their axes of symmetry. A rectangular prismatic rigid block was placed in a symmetric way on top of these columns, as shown in figures 7 and 8; it was also made of the same steel that the two columns were made of and it had a length of 215mm. a height of 68mm and a width of 87mm. This block was not connected in any other way with the two columns but was simply free standing on their top square sections. The longitudinal horizontal axis of this rectangular block coincided

with the vertical plane formed by the two axes of symmetry of the columns; its left and right end sections (figures 7 and 8) were located at the center planes of the left and right column, respectively.

3.2 Expected Modes of Response - Instrumentation

Instrumentation was provided in order to measure the expected complex response of this structural formation during a series of tests that included free rocking vibrations as well as sinusoidal and

simulated earthquake excitations of the base. This instrumentation is depicted in figure 7; it included accelerometers at the top and bottom of each column placed in a way as to coincide with the longitudinal vertical plane of symmetry of this system. Additional accelerometers were placed at the longitudinal

Figure 8. Rigid Block Formation horizontal axis of the rigid top-block at both ends on the Shaking Table again at the longitudinal vertical plane of symmetry. All these accelerometers had the purpose of recording the acceleration response of the various parts of this system. Horizontal base excitations used during this sequence with direction coinciding with this longitudinal vertical plane of symmetry. However, an accelerometer was placed at the mid-side of the rigid top-block perpendicular to this longitudinal plane of symmetry in order to identify out-of-plane response modes of the block. Apart from the accelerometers, displacement sensors were also used, as indicated with the arrows in the same figure, aimed at recording the rocking displacement response of either the columns or the top-block. Finally, the acceleration of the horizontal motion of the shaking ui table moving platform was ui also recorded. Figure 9a depicts the ideal rocking response of this complex a) 1 2 system. This response is 0- derived with the assumption 1 — i If i! that it occurs only in the vertical plane of symmetry IDEAL ROCKING RESPONSE OF THE SYSTEM and that there is no COLONNADE WITH 2 COLUMNS (TRUNCATE CONES) UPLIFT OF THE BLOCK VS ROCKING ANGLE OF THE COLUMN differential sliding of the two b ) Uplift over the 1st Column Uplift over the 2nd column free-standing columns at the base. Moreover, the top- "J U1 ( O < 0 ) block is assumed to rock on U2 ( 0 > 0 ) ..."•" U top of the columns, as shown P in figure 9a. without either L I sliding or losing contact with F U2 (0 < 0 ) the supporting columns As a T U1 ( 0 > 0 ) result from these assumptions (mm) both columns develop the same rocking angle during all -Gcr ROCKING ANGLE +Gcr stages of the response. Figure 9. Ideal Rocking Response of the System. The rocking and uplifting response of the top-block can be obtained as a function of the rocking angles of the supporting cones based on these assumptions. Figure 9b depicts the ideal uplift displacements of the left and right ends of the top-block for rocking angles of the columns from - Ocr to Ocr (where Ocr the critical rocking angle beyond which each one of these two cones studied individually overturns).

TransactionsStructura onl theStudies Built Environment, Repairs vol an 26,d ©Maintenanc 1997 WIT Press,e wwofw.witpress.com, Historical Building ISSN 1743-3509s 295

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

12.88 3.95 TIME ( sec ) 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 TIME

Figure 1 Ob Left-cone and top-block response (7Hz) -Top acceleration of the 1st column / max accel. 0.32 g Hor. accel. of the block over the 1st column / max. accel. 0.35g c ) j SIN useMDAl. BASE EXCITATIC)N - FREQ. EXC TATICN 3.0 Hz

j /\ * A fl 1 \ ft\i I/ \ 1 f 1 I Plf 1 / V

II 1 y y V V r- V II ft e \? ^ 1 !\ 1)111 111 '* TIME ( sec ) 11.99 Figure lOc Left-cone and top-block response (3Hz) j. j 5"z/z^ozWa/ 7^^ A limited number of results from this experimental sequence is presented here and discussed Figures lOa to lOc depict the measured response during this sinusoidal base excitation sequence. The peak base acceleration during this sequence 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 of the left column (1) was derived and is depicted with a solid line in the plots of figures lOa (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 and block 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 observed cone and top-block response becomes almost 10 times smaller. This observation agrees very well with a similar conclusion reached from the extensive study of the rocking response of the individual solid rigid columns (see section 2.2 and figure 4). Figure lOc depicts 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 acceleration 160gals. As can be seen in this figure, the acceleration response of the top-block exhibits 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 11 against the ideal response curves described in section 4.2 (see also figure 9a and 9b). As can 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 measurements 5.55 mm SINUSOIDAL BASE EXCITATION U2 (0 > 0 ) U U1 (O < 0 ) p L I F T (mm) U2 (G < 0 ) U1 ( 0 > 0 )

• 0.3 0 cr ROCKING ANGLE + 0.3 G cr Figure 11. Observed and expected top-block response

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

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

00 TIME ( sec ) 8.05 Figure 12 Left-cone and top-block right displacement response (Span 1)

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

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

00 TIME (sec) 8.05 Figure 13 Left-cone and top-block right displacement response (Span 3)

COLONNADE WITH 2 COLUMNS UPLIFT OF THE BLOCK AS ESTIMATED FROM THE ROCKING ANGLE Correlation with the Uplift from measurements

EL CENTRO BASE EXCITATION

(mm)

-0.6 0cr - 0.6 0cr ROCKING ANGLE Figure 14 Observed and expected top-block response

3.4 Simulated Earthquake tests A number of tests were performed during this sequence with progressively increasing intensity, indicated by the parameter Span. based on the 1940 El Centro Earthquake record. The larger the value of the parameter Span the more intense the simulated earthquake motion. Again the principal objective here was to observe the top-block response relative to the response of the supporting free-standing cones (see figures 12 and 13). Measured peak top-block uplift response values 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 table the more intense simulated earthquake motions result in larger rocking angular response for 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 duration of the intense earthquake motion, although this does not lead to unstable response for this 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 also figure 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 observed during the sinusoidal tests (figure 11)

298 Structural Studies, Repairs and Maintenance of Historical Buildings

4. Discussion of the results - conclusions a. The coefficient of restitution can be defined with sufficient accuracy from the free vibration test results in combination with the performed numerical analysis, despite

complications from the three dimensional rocking, which was present for both solid as well as sliced specimens. b. The numerically predicted upper-bounds for the stable-unstable rocking of the solid prismatic specimen, subjected to sinusoidal base motions, agrees well with the observed behavior.

c. The behavior of the examined two columns with the rigid top-block subjected to sinusoidal base excitations exhibits similar trends to those observed for the individual column, with regard to the influence that the excitation frequency exerts on the rocking amplitude. That is, for the same non-dimensional excitation amplitude, the higher the frequency of the excitation the smaller the rocking amplitude. d. For not very large rocking angles the observed response of the two columns with the

top-block system tends to remain in-plane. It also exhibits good correlation with the one predicted under the assumptions of no differential sliding of the columns at their base and no sliding between the top-block and the supporting columns. This observation is valid for both the sinusoidal as well as the simulated earthquake excitations.

Table 1. Measured peak top-block uplift during the simulated earthquake sequence. Intensit Peak e/ Ul U2 yof Base 6cr (mm) (mm) ACKNOWLEDGEMENTS : The Base Ace. max max max partial support of the Greek Motion mm nun min (8) Organization for Seismic Planning Span 1 0.304 0.065 0.40 1.23 and Protection (OASP) as well as the -0.100 1.18 1.00 European Commission (DIR XII , Span 2 0.614 0.183 1.39 3.28

-0 190 3.65 2.43 ENV.4-CT.96-0106, ISTECH), is gratefully acknowledged. Span 3 0.935 0.540 2 JO 7.30 -0.300 5.90 397 0 Non Dimensional Cone Rocking Angle Ul Top-Block Left Uplift Displacement U2 Top-Block Right Uplift Displacement

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

Demosthenous M. 1994. Experimental and numerical study of the dynamic response of solid 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 Eng Mechanics. Vol. 116. NO 1. pp.35-54. Manos. G. C. and Demosthenous. M . 1990. The behavior of solid or sliced rigid bodies when 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 response of 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 of Engm. Mech.. ASCE 110 (11), pp. 1627-1642