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Earthquake analysis of multidrum columns

D. Konstantinidis1 & N. Makris2 1Department of Civil and Env. Engineering, University of California, Berkeley, U.S.A. 2Department of Civil Engineering, University of Patras, Greece

Abstract

This paper presents a numerical investigation on the seismic response of multidrum classical columns. The motivation for this study originates from the need to understand: (a) the level of ground shaking that classical multidrum columns can survive; and (b) the possible advantages or disadvantages of retrofitting multidrum columns with metallic shear links that replace the wooden poles that were installed in ancient times. The study reveals that relative sliding between drums happens even when the g-value of the ground acceleration is less than the coefficient of friction, µ, of the sliding interfaces and concludes that: (a) typical multidrum classical columns can survive the ground shaking from strong ground motions recorded near the causative faults of earthquakes with magnitudes Mw=6.0 to 7.4; (b) in most cases multidrum classical columns free to dislocate at the drum interfaces exhibit more controlled seismic response than the monolithic columns with same size and slenderness; (c) the shear strength of the wooden poles has a marginal effect on the sliding response of the drums; and (d) stiff metallic shear links in-between column drums may have an undesirable role on the seismic stability of classical columns and should be avoided. This paper is an abbreviated version of an extended paper published in the Journal of Earthquake Engineering and Structural Dynamics, Vol. 34, 2005. Keywords: multidrum classical columns, sliding, rocking, seismic response, earthquake engineering.

1 Introduction

During the last two decades, the earthquake response of ancient structures has received increasing attention. Of particular interest are the classical temples that

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 115

are found in Greece and southern Italy and consist of multidrum columns, occasionally connected with each other at the top by epistyles (architraves) as shown in Figure 1. The dynamic response of these rigid-body assemblies involves primarily sliding and rocking, and it has very little in common with the earthquake response of modern structures which respond inelastically by exhibiting ductile behaviour. The fundamental differences between the response of multidrum columns and ductile structures have been stressed by several researchers [1-7, and references reported therein].

Figure 1: Views of the Temple of Zeus at Nemea, Greece. Left: The three columns that have always stood since 330 B.C. Right: The same three columns together with two additional columns at the back that have been recently reconstructed.

More recently, there has been a comprehensive effort by a team of researchers at the National Technical University of Athens to investigate the seismic behaviour of a part of the Parthenon Pronaos which is subject to major restorations [5,6,7]. In this paper, we present selected results of a comprehensive dynamic analysis on the seismic response of multidrum columns in order to examine the observations and conclusions reported by Sinopoli [1] and Psycharis et al. [4,7] and to provide recommendations on the reconstruction of classical columns which are based on sound engineering analysis. The aim of this paper is not to produce another isolated study but rather to build on the work of others. Our results confirm the findings of past investigations and conclude that the use of stiff metallic shear studs in-between drums is an unjustified intervention which may degrade the seismic stability of classical columns.


116 Earthquake Resistant Engineering Structures V

2 Numerical analysis software

Working Model 2D [8] is a software that combines robust numerical techniques with sophisticated editing capabilities. Its main attraction is its capability to compute the motion of mechanically interacting bodies under a variety of constraints and the action of time-varying forces. The drums of the columns analyzed in this study were modelled in Working Model as rigid blocks. This idealization is consistent with the findings of Psycharis et al. [7] who concluded that modelling columns drums as rigid blocks provided a sufficient approximation while reducing significantly the computational cost. Column drums that are engaged in rocking motion can impact onto each other or the base of the column regardless of whether there is sliding or not. During an integration step, two colliding bodies may overlap by a small amount. In Working Model, collisions are detected by finding intersections between the geometries of bodies. Since the bodies are assumed rigid, for any two points of body B, 1 2 1 2− = −x x X X , for all time t, where x is the one-to-one mapping

( , )t=x Xχ , and X is the position vector in some reference configuration. This implies that the position and orientation of the edges of a rigid body are known for any time by tracking a master node. When intersection between edges is detected, Working Model computes forces sufficient to “repel” the bodies. Working Model employs an impulse-based collision model in which the coefficient of restitution is used [8]. The numerical integration of the equations of motion in conjunction with the satisfaction of the constraint conditions (friction and restitution) is done using a robust Kutta-Merson method (5th order Runge-Kutta). Integration error as well as model assembly and collision overlap tolerances can be set to achieve the desired precision. With the available variable-timestep Kutta-Merson scheme, near collision, the timestep is reduced appropriately to restrict the overlap between bodies from exceeding the specified overlap tolerance [8]. For all the simulations presented in this paper, the overlap error tolerance was set to 10-7 m. The Working Model user literature does not provide information on how the software handles multiple simultaneous impacts. A proper formulation of impact laws and how to deal with multiple simultaneous impacts are aspects of rigid body dynamics that remain controversial and unresolved. Nevertheless, these do not affect the internal consistency of rigid body models, but rather they address the issue of how accurately these laws approximate experimentally observed behaviour [9]. Working Model was validated for various limiting cases in the paper by Konstantinidis and Makris [10].

3 Seismic response of multidrum columns

In the case of a multidrum column where the drums are allowed to slide and rotate, the degrees of freedom are many. Assuming that the drums do not liftoff and that the motion is planar, the number of degrees of freedom is two times the number of drums. Each of the perimeter columns of the Temple of Zeus at Nemea has 13 drums and a capital. In this analysis we adopted a four-drum-and-



capital column in an effort to identify trends in the response of multidrum columns while preserving a high fidelity in the accuracy of the numerical solution. Figure 2 (top) illustrates the relative horizontal displacements and total rotations of a four-drum column with a capital.

wooden polede 40 mm ≈

woodenempolium

Figure 2: Top: Profile of relative displacements and total rotations of the four drums of a classical column with overall dimensions equal to those of the columns of the Temple of Zeus at Nemea, Greece. Bottom: Direction of sliding at the instant of impact between two adjacent drums (left). Sketch of two adjacent drums together with the wooden pole and the two wooden empolia (right).

3.1 The role of poles/empolia

From the end of the archaic period and throughout the classical period in ancient Greece, the drums of columns were connected with a wooden system that consisted of the pole and the two empolia. Figure 2 (bottom-right) shows the two wooden empolia that were installed flush with the interface of the adjacent drums together with the cylindrical pole that was made of harder wood. The



dimensions of poles and empolia vary according to the scale of the column. Orlandos [11] offers details on the dimensions of poles and empolia installed in columns of various ancient temples in Greece. The diameter of the poles varies from 27d = mm at the Temple of Poseidon in Cape Sounion to 47d = mm at the Erectheion on the Acropolis; up to 55d = mm at the Temple of Athena Nike, also on the Acropolis. The pole was usually fixed at the top drum which upon erection was rotated with respect to the bottom drum in an effort to achieve the best possible contact [11]. The role of the wooden poles was primarily for rotating the drums—not to provide a shear link between them.

3.2 Yielding mechanism at the interface of drums

The mechanical behaviour along the drum interfaces is governed by friction. For limestone, the coefficients of static, sµ , and kinetic, kµ , friction may vary appreciably depending on the type of the limestone and the prevailing environmental conditions. Typical values of the friction coefficients for limestone rock are s 0.8µ = and k 0.7µ = [4,6].

In an effort to assess the resisting contribution of the wooden poles when appreciable dislocation occurs, the shear strength of the pole, q, is superimposed to the static friction force along the drum interface. According to Orlandos [11], the poles were made of hard pine wood, which has a typical shear strength of

w 7τ = MPa. Adopting an average value for the diameter of the pole 40d = mm, the strength of the pole is 2

w( / 4) 9q d= π τ ≈ kN. Accordingly, because of the presence of the pole, the apparent coefficient of static friction, sµ̂ , at each drum interface is

ss

( )ˆ( )

w z qw z

µ +µ = (1)

where w(z) is the weight of the column above the interface of interest. Table 1 indicates the height of the drum interfaces and the coefficients of friction used in this study. The fourth column lists the values of the apparent coefficients of static friction, sµ̂ , as determined by Equation (1).

Table 1: Values of the friction coefficients at the drum interfaces of a four-drum-and-capital column (Figure 2) with overall dimensions equal to those of the columns of the Temple of Zeus at Nemea, Greece.

z [m]

w(z) [kN]

sµ sµ̂ kµ

0 306 0.80 0.83 0.70 2.425 232 0.80 0.84 0.70 4.850 149 0.80 0.86 0.70 7.275 82 0.80 0.91 0.70 9.700 22 0.80 1.20 0.70



3.3 Seismic response analysis

Figure 3 plots with a solid line the relative displacements (left) and the total rotations (right) of the four drums and the capital of a column with the same size and slenderness as the column of the Temple of Zeus at Nemea when subjected to the Cholame 02 motion recorded during the 1966 Parkfield, California, earthquake. The plots on the left of Figure 3 indicate that the drums undergo some minor sliding (of the order of 0.5 cm) while the plots on the right indicate that most of the rotation happens at the base. This study reveals that there is a small relative sliding between the drums even when the ground acceleration is less than µg. This minor sliding between the drums is induced by the small-duration but high-acceleration spikes that develop during impacts when the rocking column alternates pivot points. Figure 2 (bottom-left) illustrates schematically the direction of sliding at the instant of the impact at each of the two pivot points.

0 5 10 15−0.5−0.25

00.25

0.5

a g [g]

time [sec]

1966 Parkfield, C02−065

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time [sec]

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]0 5 10 15

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]

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θ 4 [rad

]

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θ c/α

Total Drum Rotations

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θ c [rad

]

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]

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]

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Relative Drum Displacements

−202

u c [cm

]

Figure 3: Histories of relative displacements and total rotations of the four drums and capital of a multidrum column (solid lines: without poles; dashed lines: with wooden poles) and of a monolithic column (heavy lines) when subjected to the Cholame 02 motion recorded during the 1966 Parkfield, California, earthquake.

The solid lines in Figure 3 and in all subsequent figures plot the response of the drums without accounting for the shearing resistance of the poles ( s 0.8µ = and k 0.7µ = ) while the dashed lines plot the response of the drums after including the resistance of the poles. The enhanced shearing strength across the drum interfaces has been approximated by increasing the static coefficient of friction according to Equation (1). The negligible differences in the sliding



response of the drums when the shear-resistance contribution of the wooden poles is included or neglected confirm that the poles were installed originally to achieve the best possible fit between drums—not to enhance the shearing resistance of the column. The heavy lines shown in the second-from-the-bottom windows plot the displacement (left) and the rotation (right) of a monolithic column with the same size and slenderness as the five-block assembly. The analysis shows that while the multidrum column and the monolithic column exhibit approximately the same peak rotations (20% of the column slenderness) the rocking motion of the five-block column decays faster due to the additional energy dissipation that takes place at the block interfaces. It should be noted that the decay of the free-vibration response of a rocking block is very sensitive to the actual value of the coefficient of restitution. In this analysis, the maximum values for the coefficient of restitution (minimum damping) have been used. In reality, the actual values (which are not known) may be smaller (more damping), and the decay of the rocking motion may be even faster.

0 2 4 6 8 10−0.6−0.3

00.30.6

a g [g]

time [sec]

1973 Lefkada, OTE NS

0 2 4 6 8 10−0.6−0.3

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a g [g]

time [sec]

0 2 4 6 8 10−0.2−0.1

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−0.0200.02

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]0 2 4 6 8 10

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]

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]

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]

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]

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]

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]

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−202

u c [cm

]

Figure 4: Histories of relative displacements and total rotations of the four drums and capital of a multidrum column (solid lines: without poles; dashed lines: with wooden poles) and of a monolithic column (heavy lines) when subjected to the OTE motion recorded during the 1973 Lefkada, Greece, earthquake.

Figure 4 (left) shows that the dislocations of the interfaces of the five-block column when excited by the 1973 Lefkada, Greece, earthquake are of the order of few millimetres while Figure 4 (right) shows that the rotations from the second drum-up are appreciably different than the rotation at the base of the column. Again, the solid lines plot the drum response without poles whereas the dashed lines plot the drum response with wooden poles. The two responses are



almost identical. The heavy line which plots the response of the monolithic column shows that the rotations of the monolithic column at the base are larger than those of the multidrum column. Figure 5 indicates that when the column is subjected to the 1995 Aigion, Greece, earthquake, the rotations of the multidrum column are comparable to the rotations of the monolithic column—a result that is in agreement with the conclusion reached by Psycharis et al. [4].

0 2 4 6 8 10−0.6−0.3

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a g [g]

time [sec]

1995 Aigion, OTE FP

0 2 4 6 8 10−0.6−0.3

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a g [g]

time [sec]

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]

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u 1 [cm

]

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]

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−202

u c [cm

]

Figure 5: Histories of relative displacements and total rotations of the four drums and capital of a multidrum column (solid lines: without poles; dashed lines: with wooden poles) and of a monolithic column (heavy lines) when subjected to the OTE motion recorded during the 1995 Aigion, Greece, earthquake.

Figure 6 presents the response of the multidrum column when subjected to the 1977 Bucharest, Romania, earthquake record. While this record has a

0.2PGA g≈ , the long duration of the motion’s main pulse ( 2≈ sec) is responsible for large rotations that reach 55% of the column slenderness. In this case, the multidrum column exhibits slightly larger rotations than the monolithic column. During the impacts, the drums slide appreciably and reach dislocations of the order of 2.0 cm. The values of drum dislocations shown in Figures 3 and 6 are consistent with what one observes on column K-31, one of the three columns of the Temple of Zeus that have always stood up since antiquity. Some of the drums of column K-31 exhibit dislocations of 5 mm while between drums 7 and 8, the dislocation reach 1.5 cm.



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time [sec]

1977 Bucharest, NS

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]

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u c/bRelative Drum Displacements

−202

u c [cm

]

Figure 6: Histories of relative displacements and total rotations of the four drums and capital of a multidrum column (solid lines: without poles; dashed lines: with wooden poles) and of a monolithic column (heavy lines) when subjected to the 1977 Bucharest, Romania, earthquake.

4 Conclusions

This paper presents a numerical investigation on the seismic response of multidrum classical columns. The configuration examined is a column with size and slenderness equal to those of the columns of the Temple of Zeus at Nemea, Greece, yet it is divided only in four drums and the capital rather than 13 drums that each column has in reality. The aim of this study was to understand the beneficial effects on the stability of the column that result from the ability of the drums to dislocate (relative sliding movements).

The numerical study presented in this paper is conducted with the commercially available software Working Model which was validated in great length for pure sliding and pure rocking response. The seismic response analysis presented in the paper uses twelve strong ground motions, seven of which have been recorded near the causative faults of earthquakes with magnitudes ranging from Mw=6.0 to 7.4. In this analysis, the maximum values for the coefficient of restitution (minimum damping) have been used. In reality, the actual values for the coefficient of restitution may be smaller (more damping), and the decay of the rocking motion may be faster.

The most important conclusion of the paper is that relative sliding between the drums happens even when the g-value of the peak ground acceleration is less than the coefficient of friction along the sliding interfaces. This relative sliding between drums (which in most occasions is of the order of only a few



millimetres) is induced by the small-duration but high-acceleration spikes that develop during impacts when the rocking column alternates pivot points. The effects of these small dislocations are shown to be beneficial since they result in a rocking response of the multidrum column that in most cases is more controlled than the response of the equal-size-and-slenderness monolithic column. This finding suggests the installation of stiff metallic shear links in-between drums may have an undesirable role since they will oppose these small but so beneficial drum dislocations. The dynamic response analysis presented in this paper also examined the role of wooden poles installed in ancient times, and it is concluded that their effect in modifying the response of the freestanding drums is marginal.

Acknowledgments

Partial financial support for this study was provided by the National Science Foundation under Grant CMS-0116354. The authors have benefited greatly through discussions with Professor Stephen G. Miller, director of the Nemea excavations.

References

[1] Sinopoli, A., Kinematic approach in the impact problem of rigid bodies. Applied Mechanics Reviews, ASME, 42(11), pp. S233-S244, 1989.

[2] Psycharis, I.N., Dynamic behaviour of rocking two-block assemblies. Earthquake Engineering and Structural Dynamics, 19, pp. 555-575, 1990.

[3] Papantonopoulos, C.L., The articulated structural system: studying the earthquake response of a classical temple. Structural Repair and Maintenance of Historical Buildings, STREMA, Bath, U.K., pp. 483-489, 1993.

[4] Psycharis, I.N., Papastamatiou, D.Y. & Alexandris, A.P., Parametric investigation of the stability of classical columns under harmonic and earthquake excitations. Earthquake Engineering and Structural Dynamics, 29(8), pp. 1093-1109, 2000.

[5] Mouzakis, H.P., Psycharis, I.N., Papastamatiou, D.Y., Carydis, P.G., Papantonopoulos, C. & Zambas, C., Experimental investigation of the earthquake response of a model of a marble classical column. Earthquake Engineering and Structural Dynamics, 31(9), pp. 1681-1698, 2002.

[6] Papantonopoulos, C., Psycharis, I.N., Papastamatiou, D.Y., Lemos, J.V. & Mouzakis, H., Numerical prediction of the earthquake response of classical columns using the distinct element method. Earthquake Engineering and Structural Dynamics, 31(9), pp. 1699-1717, 2002.

[7] Psycharis, I.N., Lemos, J.V., Papastamatiou, D.Y. & Zambas, C., Numerical study of the seismic behaviour or a part of the Parthenon Pronaos. Earthquake Engineering and Structural Dynamics, 32(13), pp. 2063-2084, 2003.



[8] Working Model. User’s Manual. MSC.Software Corporation: San Mateo, California, 2000.

[9] Stewart, D., Rigid-body dynamics with friction and impact. SIAM Review. Society for Industrial and Applied Mathematics, 42(1), pp. 3-39, 2000.

[10] Konstantinidis, D. & Makris, N., Seismic response analysis of multidrum classical columns. Earthquake Engineering and Structural Dynamics, 35, 2003 (in press).

[11] Orlandos, A.K., Construction Materials of the Ancient Greeks. Vol. 2. Archaeological Society of Athens, Greece, 1958 (in Greek).



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Earthquake analysis of multidrum columns - WIT Press · Earthquake analysis of multidrum columns ... The motivation for this study originates from the ... three columns together with