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

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Transversal Deformations and Creep Effects in Columns - a Mathematical Research Into the Entasis of Greek Doric Temples Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 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 Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 338 Architectural Studies, Materials & Analysis 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" Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 Architectural Studies, Materials & Analysis 339 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. Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 340 Architectural Studies, Materials & Analysis (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 Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 Architectural Studies, Materials & Analysis 341 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 Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509 342 Architectural Studies, Materials & Analysis 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.
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