Transversal Deformations and Creep Effects in Columns - a Mathematical Research Into the Entasis of Greek Doric Temples

Transversal deformations and creep effects in columns - a mathematical research into the entasis of Greek doric temples

J.M* Rodriguez Ortiz, C.L. Martin

Polytechnical University of Madrid, School of Architecture, Spain

Abstract

The observed bending deformations of marble slabs, as well as the known Poisson effect in compressed cylindrical specimens has led to a research into the possibilities of the entasis curve in Greek temples to be an effect of delayed transversal deformations of viscoelastic type. However, the results of axysimetric finite element computations support the traditional theory of a built-in entasis (a delicate work indeed !) as the computed shape does not match the actual one and, on the other hand, the vertical compression of the column should be of decimeters in order to allow 2-3 cm of "bulging". The comparison does not improve by using more elaborated models as a linear decreasing modulus from the axis to the outer surface or, inversely, dilatant properties clue to development of microcracks in weathered marble.

1 Introduction

As it is well known doric columns show a slight bulge, called entasis, that results in a more pleasant look as compared with a straigth tapered shaft. This "refinement" has been copied in aftercoming styles with varied success and following known design schemes. No written testimony has reached us about the method used by ancient Greek architects to lay out this delicate line and this has given way to some controversy. Early work by Cockrel and Jenkin in 1820 concluded in the entasis been formed

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by hiperbolic and elliptic arches. Penrose (1888) "demostrated" after careful measurements and computations that the curves were hyperbolas. However historic evidence shows that conical curves were mathematically defined not before 340 b.C. More recently Mertens (1988) has suggested that entasis curves are catenaries, as this curve is easily delineated hanging a string from two nails fixed to a wall. This hypothesis seems quite verosimil but it is difficult to understand the setting of that curve in all and any one of the flutes of each column in a temple. Our most recent theory is that the entasis derives from a normal polishing procedure aiming to smoothing up the broken shape conecting a lower almost vertical cylinder with an upper truncated cone, both easily obtained in situ through a rough chiselling process from the stacked drums carried from the quarry. According to this, the best fit to actual measurements can be achieved by two straight lines departing from both column ends. This is illustrated in fig 1, were a catenary, a hyperbola, a circle and two lines are fitted to the profile of a doric column.

5.5 5.0 ww» 4.0 CATENJ\RY «*•**~~^~^ 3.0 ^f*+• zs*-1,71x10"* 2.0 ^^^ 1.0 ^'\ 1 O.b2 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20.22 0 Offset from vertical (m) 5.5 ^ 5.0 & 4.0 +* 3.0 "§» 2.0 1 1.0 ~£L 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 Offset from vertical (m) 5.5 5.0 4.0 » 1.69x10"

0.0.022 0.04 0.00.066 0.08 0.10 0.12 0.14 0.1O.'l86 0.20.10 0.22 Offset from vertical (m)

5.0 , — — 4.0 BILINEAR _ 3.0 .— •— • — 2.0 ^^s^*~^* 2S'o1,28x10"

1.0 ^^ 0.62 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20.202 Offset from vertical (m)

Fig. 1: Curve fitting to the profile of the columns of Paestum "Basilica"

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Although this can be a plausible explanation, the origin and intention of the entasis is far from clear. A lot of hypotheses have been suggested: the bulging of former wooden columns, improvement of visual appearance against straight shafts, aesthetic reasons, etc. But many doric temples have not entasis and in some cases, as the Parthenon, its value is so small that it is difficult to imagine so a refined construction procedure unless one admit an advanced polishing process, leading to a limit beyond wich the danger of strangling the column was very high. However some questions arise, as a straight column may be easier to construct that a column with entasis, but the amount of material removed by polishing is higher. This has lead us to contemplate the possibility that small entasis could be due to transversal deformation of a former straight shaft, instead of deliberatelly constructed. Evidently this should not apply to exagerated entasis as it can be observed in Paestum and Sicilian temples. In these cases the shape was dictated by the poorer materials (travertine, shell sandstones, etc.), were the strength heterogeneities increased the risk of deviations from the desired profile.

2 Deformation of a cylindrical column

The problem of a cylindrical specimen compressed between two rough plates was firstly solved by Filon (1902). The problem was reviewed by Pickett (1944) who corrected the result of Filon at the ends of the cylinder by using the Fourier method. A mathematical expansion of Filon's equations was applied by Kotte et al. (1969) to rupture testing of rock samples. All theoretical solutions yield a deformed shape of barrel type, in accordance to the observed behavior of compressed specimens in a testing machine. However, tests performed by Seldenrath and Gramberg (1958) show a deformed shape of irregular profile, with some divergence from Filon's predictions.

A lot of evidence shows that the end conditions, or the restraint imposed to the high shear stresses developed at the contact with the plattens, can significantly influence the deformed shape near the ends. Most of the quoted studies refer to specimens with a slenderness ratio of 2 or smaller, that is not the case of the columns were slendernesses of 5 are quite common.

Some computations have been made with cylindrical columns of varied slenderness, with the results shown in fig. 2. The computations were made through a finite element program, using an axisymetrical modelling of the column and elastic parameters.

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(10 m.)

Fig. 2: Influence of the slenderss on the deformed profile of cylindrical shafts (rough ends)

As it can be observed, with increasing slenderness the barrel shape is not longer maintained and the swelling is progresivelly displaced to the lower part of the column. Slight differences arise from taking into account the self weight or not,

and from considering the lower contact surface rough or smooth, but these do not influence the general shape above the bottom. The resulting profile is far from

pleasant, althought columns with bulging at the lower part were extensivelly used by Aegyptians as a reproduction of a papyrus plant.

3 Deformation of conical columns

As it could be expected the deformed profile does not significantly change when

the shape of the column is conical. In fig. 3 the results of several computations on a Parthenon-type column are shown. In order to check the influence of the loading

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So O) oCM p=130t CM ro m II II II ! Q. CL Q. 7 •// 9 •f Smooth base 9 \ll E Rough base 8 j, _____ Column weightless 8 \ • ji \ II 7 7 6 \\ 6 \\ 1! c 13 5 i 4 4

3 3 \ \ \ \ \ \ 2 \ 2 \ \ \ \ :j \ j ) 1 F 1 x«^«^*"^ A( , N '//M//J///S'/////////7 ^ I\ Ii Ii r>V 0 1 2 (10 m.) 0 1 2 3 (lGT*m.)

a) Under P = 130 t. b) Increased loading

Fig.3: Horizontal displacements of a tapered column under varied load conditions the load on the column has been increased from the actual one of 130 t to 530 t. It is clear that the increased load causes an outward swelling of the upper part of the column, but this effect quickly disipates downwards, without affecting the lower part.

This leads to the conclusion that the barrel type deformation observed in compressed specimens cannot be expected in more slender columns, were maximum outward deformations concentrate near the base. An important question is wether transversal deformation could attain values close to those observed as entasis in doric columns. A preliminary question refers to the elastic parameters to be used in mechanical computations. Greek architects used different materials as Poros and Corinthian limestones, Pentelic arid Parian marbles, etc. Andesite was used in the Assos temple and travertine or sandstone were common in Sicilian temples. Elastic moduli range between 20,000 and 60,000 MPa, with Poisson ratios from 0.12 to 0.20.

Simple elastic computations show that the compressive stresses in actual columns are so low that the expected deformations lay in the order of 10 "\ for elastic

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moduli above 10,000 MPa. These results have been confirmed by finite element computations, showing a ratio of 50:1 between the vertical and radial displacements. The point of maximum radial displacement appears at about 10% of the height, for a friction angle at the base of 70°. A smooth contact results in a bell-shaped deformation with maximum displacements at the base.

Faced with the impossibility of reproducing observed deformations through elastic models, a further attempt was directed towards creep models.

It is well known that marble slabs in wall claddings, grave covers, etc. show important creep deformations in bending or out of plane distortions from thermal effects or compressive loadings. These deformations are logically more apparent in the so called soft marbles. After observation of some burial slabs of medium-strength marble in some

Spanish cemeteries, a typical 216x100x3 cm slab was choosen, which showed a central deflection of 2.5 cm in 18 years. It appeared that this result colud be reproduced by means of a viscoplastic model of the type

e^, = A o"

where the first term is the velocity of plastic deformation and a is the von Mises stress. A and n are constants. This model implies no volume change.

After several trials with n = 1, a value of A = 0,06 M Pa" * years'* yielded the expected value for the 18 years deflection of the slab. Upon application of the above parameters to the case of the column, for a creep period of 1000 years, the results are clearly unrealistic as a radial displacement of 17 cm is obtained, for avertical displacement of near 3 m (a 17:1 ratio).

Proportionnally, a maximum entasis of 2 cm would require a vertical displacement of 35 cm. This hypothesis can be rejected even without historic evidence of changes in height of the columns (which probably would result in sagged entablature or cornice profiles), because the point of maximum deflection is at the lower H/10 of the column, not at the observed height. Upon further adjusting of creep parameters the best results were obtained with

A = 5 x 10"* (s~™) m = 0.15 in a model of the type e^ = a A t™. The resultant radial deflection was of 4 mm for a vertical shortening of 110 mm, of course far from the observed values and with a belled deformed shaped (fig. 4). In fig. 5 the development of deflections with time is shown. The evidence of the funerary slab, however, opens the question of a very different

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DEFORMATIONS X 50

Fig. 4: Deformed creep profiles

YEARS

Fig. 5: Creep displacements of point with maximum entasis

behavior of marble in bending (or tension) and under compression. Some insight is provided by Korre and Boura (1983) who observed in the Parthenon significant long terra microscopic loosening of the marble, accompanied by swelling and bulging towards the free surfaces. This phenomenon is impeded by compression or not yielding boundaries. Taking into account the possibility of a dilatant behaviour of marble upon exposure to environmental agents, aging, thermal effects, etc. a new model for

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creep deformations was used, based in a Drucker-Prager potential, developped by Martinez (1993). In this model a velocity of volumetric deformation is considered related to the deviatoric velocity of deformation times the tangent of a certain angle of dilatancy (\|/). The computations with i|/=75° show, as it could be expected, a ratio of 6:1 between vertical and radial displacements (12 cm of vertical deflection for an entasis of 2 cm), but without significant changes in the height of the maximum entasis. Some other models were also used in order to reproduce the expected transversal profile of strength of marble in exposed columns. Very often the columns show an intense surficial weathering leading to the loosening of the material, which can be removed by hand rubbing. As this weathering occurs with swelling and distortion of crystals, the strength and deformability of marble should decrease from the axis of the column to its outer surface. Computations with columns formed by cylindrical annuli of decreasing moduli of deformation did not yield significant radial deformations due to the fact that the core of the column absorbed most of the load, without allowing the transfer of load to the outer parts.

In fact the swelling behaviour could explain some cortical bulging but the weathered material would be easily removed by erosion. The alteration is probably very slow as most of the columns keep a distinct outward profile after more than 2500 years of exposure.

As pointed out by Gram berg (1989) marble specimens show a clear bulging due to multi-shear cataclasis which causes volume increase and creep, but this needs stresses above 50% of the compressive strength, i.e. greater than 20 MPa for common marbles. In doric temples stresses above 0.6 MPa are seldom attained.

Thus, cataclastic models were not used in this research.

4 Conclusions

The research carried out can be summed up in the following conclusions: The well known "barrel" shape of cylindrical specimens under compres-

sion in laboratory of predicted by current elastic theories is highly dependent of the dimensional ratio of the sample (slenderness) as well as

the contact conditions at both ends. Computations with tappered bars with dimensions close to those of doric columns have shown that the point of maximum radial displacement lies

close to the bottom of the column instead the observed entasis at midheight. Furthermore, the value of the radial displacements are far

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below the observed ones.

Numerical reproduction of observed entasis by elastic or viscoplastic models, with or without dilatancy, or transverse anisotropic modeles related

to some profiles of weathering, requires excessive vertical deformations, between 6 and 50 times the radial ones, which is inconsistent with

observed geometry of doric temples. It must be concluded that the entasis of doric temples does not obbey to

a deformational behaviour, at least between perceptible limits. Thus it is neccessary to admit a constructive intervention.

Curve fitting to the profiles of doric columns suggest that the entasis could be the result from smoothing the sharp contact between a conical upper

part of the column and a near cylindrical lower part.

References

1. Filon, L.N.G. "The Elastic Equilibrium of Circular Cylinders Under Certain Practical Systems of Load". Philosophical Transactions of the

Royal Society, series A, vol. 198, 1902, 147-233. 2. Gramberg, J. A Non-Conventional View on Rock Mechanics and Fracture

Mechanics, Balkema, Rotterdam, 1989. 3. Korre, M. and Boura, Ch. " Meleti apokatastaseos tou Partenonos" (in

Greek), Athens, 1983. 4. Kotte, JJ. et al. "Stress-sttain relations and breakage of cylindrical granitic

rock specimens under uniaxial and triaxial loads". Int. J. Rock Mech. Min. Sci. Vol. 6, 1969,581-595. 5. Martinez, F. " Analisis de fendmenos de localizacion de deformaciones en

materiales cohesivo-friccionales" Ph.D. Thesis (In Spanish). Univ. Polyt. of Madrid, 1993.

6. Mertens, D. "Zur Entstehung der Entasis griechischer Saulen". Beitrage zur Architektur und verwandten Ktinsten. Archaologisches Institut des

Universitat des Saarlandes, 1988. 7. Pickett, G. "Application of the Fourier Method to the Solution of Certain

Boundary Problems in the Theory of Elasticity". Journal of Applied Mechanics, Sept. 1944, 176-182.

8. Penrose, F.C. "An investigation of the Principles of Athenian Architec- ture". Society of Dilettanti, London, 2nd edition, MacMillan and Co. 1888.

9. Power, L.D. and Childs, S.B. "Axisymmetric stresses and displacements in a finite circular bar". Int. J. Eng. Sci. ,Vol. 9, 1971,241-255.