Why does the Endless Column seem everlasting?

Structural Analysis of Historical Constructions - Modena, Lourenço & Roca (eds)
© 2005 Taylor & Francis Group, London, ISBN 04 1536 379 9
Why does the Endless Column seem everlasting?
C.A. Safta
Univers ity Politehnica 01 Bucharesr, Romania
ABSTRACT: The paper firstly presents the geometrieal data used for shaping the Endless Column, loeated in
Targu-Jiu, Romania, as well as the physical data about site and environment. Recent investigations in the subsonic
wind tunnel have shown the influenee of Colurnn shape on wind flow around it. The influence of Geometry on
Kármán 's alternate vortiees and wind gusts was further analytically searehed. Finally, the virtual analysis was
devoted to wind flow around horizontal and vertical cross-seetions. The answer to title question is affirmative,
indeed.
(Sofronie 200 I, 2003). Brancusi succeeded this performance with the aid of Geometry. By combining
the geometrical properties of his modules he bridged
according to his own statement the earth with the sky.
Particularly, with his masterpiece it seems Brancusi
either defined the wind or on the contrary succeeded
to tame and control its force. The paper tries to disclose
the influence of Colurnn shape on wind power.
INTRODUCTION
The dynamic parameters ofthe Endless Colurnn were
measured in situ atTargu Jiu only in 1984 after4 7 years
since its erection, with the participation of Colurnn
builder Stefan Georgescu-Gorjan. It was found that
the period ofnatural oscillations in fundamental mode
is near 2 seconds (Georgescu 2004). This large value
is typical for very flexible structures. The History of
Science is mentioning many damages caused to similar constructions like submarine periscopes, industrial
smoke stacks and gas pipelines. The failure ofTacoma
suspension bridge near Seattle in 1940 is also due
to its high flexibility (Yamada 2004). However, the
Endless Column by its motionless standing, has never
drawn attention on any danger coming from wind or
earthquakes. Only after the maintenance works carried
out between 1996 and 2000 some questions about its
stability to wind were raised. On the basis of some
laboratory tests performed in Italy it was established
that the Column has a reliable behaviour under wind
actions (Lungu 2002). On the other hand by a com parative study and theoretical analysis it was found that
the Colurnn is aeroelastically stable (Safta 2002a, b).
Subsequent studies have shown that the Colurnn is not
only stable but even aeroelastically indifferent (Safta
2003). Finally, after some investigations on a model
in wind tunnel , new data have been brought for supporting the aeroelastic stability of the Colurnn (Safta
2004).
There are many colurnn-like monuments but the
Colurnn in Targu Jiu is the only endless in the world.
lts infinity is not an adjective resulting from a mystic
rich imagination, a legend or a fairy tale. The endlessness of the Column is a reality based on the laws of
automorphism as it was topologically demonstrated
2
GEOMETRY AND SHAPE
The Endless Colurnn in Targu Jiu is composed by cast
iron modules threaded along a central steel pillar. Each
module is shaped as a decahedron with eight identical
isosceles trapeziums as equally inclined faces and two
identical squares as bases (Fig. I).
The decahedron has five symmetry plans, four vertical and one horizontal. Since the eight faces are sl ightly
swelled out the inc1ined sides of the hexagons in the
four vertical plans of symmetry are a little eurved.
Due to the existing horizontal plan ofsymmetry at the
middle of module height, the decahedron shaped as
above can be regarded as composed by two congruent
hexahedra with eommon squared bases. Both types of
polyhedral shapes satisfy the topological relationship
F+V=E+C
(1)
where F = number of faces; V = number of vertices; E = number of edges; and C = 2 is Euler's
characteristie.
As metrical relations, the sides of the two base
squares have 45 em, the sides of middle square have
90 cm while the height of each hexahedron is 90 em
that means the total height of decahedral module is
667
180 em. From the above data for eaeh hexahedron
results the proportion 1: 1: 1 while for deeahedron is
1:2:4. The Column eonsists of one basie module with
a height of 136 em, 15 identieal modules of 180 em
eaeh, and one semi-module at the topo Its total height
is 29.35 m, and total weight 29, 173 kg.
The Endless Column is far from having an aerodynamie shape and its reed slenderness allows to be
suspeeted of instability. However, due to repeti tive
modules, the shape of Column has a perfeet regularity
and the eonvex edges are rigorously alternating with
the eoneave ones. As a eonsequenee if a uniformly
flow meets sueh an obstacle it beeomes turbulent, but
with a high degree of uniformity. On the other side
a non-uniformly flow is expeeted to be transformed
by the Column into less non-uniformly one. Probably
Braneusi was aware ofthis quality ofhis Column when
he stated: "fts shapes are the samefrom earth to sky;
it is standing by its own power, motionless and well
inserted in earth; everything should startfrom earth ".
3
Figure I.
The Endless Column was ereeted in the first fortnight of November 1937 (Georgeseu-Gorjan 2004).
lt is loeated on a large open platform that rose up a
little over the town Targu Jiu giving a good view of its
surroundings. The former hey market is now arranged
as a publie park, but without trees or other vegetation,
just grass and flowers along some wide alleys. From
aerodynamie point of view that means the ground
roughness is redueed and there are no obstacles around
the site able to ereate turbulenee or disturb in any
way the air eurrents. Aetually, looked from the ground
leveI, the Column has always a motionless appearanee,
that being very eosy. There is no any testimony about
the slightest movement during its long serviee.
The whole plateau near the Carpathian Mountains,
where the Column is loeated, enjoys a rather mi ld
weather: warm but not hot summers and eool but
not eold winters, both being dry seasons. The rainy
seasons of spring and autumn are poor enough in preeipitations . Aeeording to the records received from the
National lnstitute of Meteorology and Hydrology in
Bueharest, between 196 1 and 2000 the medium frequeney of calm days was 64.1 %. In the same period,
the medium annual wind velocity was 3.6 mls in NV
direction and 2.1 mls in SE direction (Fig. 2). However,
the maximum annual wind velocity reaehed 34 mls on
SV direetion (Table 1).
Oecahedral module.
N
4
SE
SV
Figure 2.
WIND TUNNEL INVESTIGATIONS
The wind tunnel 'Elie Carafoli' from the National
Institute for Aerospace Research in Bueharest was
established in 1968. It was eertified by Aerospatiale in
1994 using a powered ATR 42 model. The test ehamber
in elosed circuit, with a maximum velocity of 100 m/s
Frequency and wind velocity.
Table l.
SITE AND ENVIRONMENT
Maximum w ind velocities (\96 1- 2000).
Month
Velocity (m/s)
Oirection
20
N
Il
111
IV
V
VI
VII
VIII
IX
X
34
SV
24
NV
18
N
24
NV
26
N
24
N
16
SV
20
N
668
Xl
XII
Max
17
16
N
12
N
34
SV
SV
is equipped with six components, pyramidal type, and
an externai mechanical balance. The measuring process is automatically controlled and data are displayed
on the control panel and handled by a built in data
acquisition system. The model is mounted directly on
the balance or by using 1,2 or 3 pods. The model position is automatically set by the aid of the pods and by
rotating the balance around the vertical axe. Capabilities: maximum load 10,000 N; accuracy 0.0 I %. Apart
from aircraft models, the tunnel usually performs aerodynamic tests on car, building, antenna, tower or other
civil structure models.
For smoke and wire tests, a wooden model of the
Endless Colurnn, scaled at 1:22, was used. The model
was painted in black before testing to ensure a good
visibility and provide smooth surfaces. 80th principal directions ofthe cross section, at 0° and 45°, were
investigated. However, due to the reduced model scale
the test had only a qualitative character. Its aim was
to observe the influence of Column geometry on the
f10w appearance. For this purpose, the velocity of air
stream was gradually increased from about I m/s up to
Figure 3.
View ofthe tunnel and its control pane!.
Figure 4.
View ofthe model on its base in wind tunne!.
30 m/s. A special attention was devoted to the crosssections where the modules are convex, assuming a
maximum thickness and to the cross-sections where
the modules are concave, assuming a minimum thickness. Two phenomena have been observed: (1) At all
velocities behind the model, the f10w beca me strongly
turbulent but never occurred alternate vortices. (2) The
dispersion of streamlines behind the model produced
in all directions without any preferences while f10w
Figure 5.
Lateral view ofthe model in wind tunne!.
Figure 6.
View from above ofthe model in wind tunne!.
669
velocity strongly decreased comparing with the initial velocity, recorded in fro nt ofthe mode!. A special
attention was given to the non-uniformity ofhorizontal
flow on its thickness due to modules variable geometry. The results of these tests are independent of the
scale effect and were used in the virtual analysis as
well as for preparing an advanced research program in
the wind tunne!.
5
where m = the mass on unit length of cylindrical
obstaele; w = lateral displacement of obstaele perpendicular on the flow direction; k = 3EI/ P spring
constant; and w = natural circular frequency (Petre
(973).
If the expression of Kármán force (2) would be
a harmon ic function with constant coefficients then
equation (7) assumes the solution
(8)
GEOMETRY AND KÁRMÁN VORTICES
It was natural to suspect Kármán vortices able to harm
the Column. History is mentioning many cases when
submarine periscopes and smoke stacks were damaged
or completely destroyed. It seems that the same vortices were responsible for the catastrophe of Tacoma
suspension bridge in 1940.
Indeed, when a uniform flow ofvelocity V meets a
cylindrical obstacle at rest then perpendicular on the
flow direction develops the disturbing Kármán force
(2)
where CK = dimensionless coefficient
q = Newton's dynamic basic pressure,
pv'
q=2
of drag;
w(t) = r========CKqA sinyt
1(
<)2 +(2~r.)2
llJ
(9)
-k-
Cu OJ
It is obvious that when the wind disturbing frequency coincides with natural frequency of obstaele, i.e. y --+ w then the resonance occurs and the
amplitude in (8) indefin itely increases while in (9) it
becomes
(3)
p = fluid density; A = cylinder area submitted to flow
pressure; y = 2nf circular frequency of disturbing
force F K ;
f=S~
or by also considering the structural damping defined
by ratio c/cc r, that can assume values beginning
w ith 3%,
(lO)
In the case ofthe Endless Column the dimension D
is not constant along the height and for a wind with
uniform flow and of constant velocity V from Strouhal
formula (4) results
(4)
D
jD = SV = consto
frequency of vortices; S = Strouhal number; and
D = obstaele dimension submitted to flow action or,
if D is constant, A =D in m 2 /m.
According to d' Alem bert principie the disturb ing
force FK is balanced by the inertia force Fi and the
elastic force Fe of cylindrical obstacle, i.e.
(11 )
Expression (10) defines Strouhal's hyperbola that
demonstrates how small variations of cross-section
dimensions D produce large variations of disturbing
frequencyf (Fig. 7). In these conditions the expression
of disturbing force (2) becomes
(5)
Explicitly, the equation ofmotion becomes
(6)
or
(7)
and the motion described by equation (7) is no longer
harmonic. Since physically the mechanical system
composed by Column and wind is elosed, i.e. there
are no other externai sources to induce energy in the
system; there is no possibility to increase the intensity
of oscillations (Landau & Lifchitz 1966). Practically,
that means behind the Column the flow is strongly
disturbed and becomes turbulent, but the emission of
alternate vortices does not occur any longer. This result
was confirmed by the tests carried out in wind tunnel
as explained in the previous chapter.
670
q(t)
Figure 9.
o
Diameter
Figure 7.
Gust impulse.
where m = W/g mass of oscillating structure;
W = weight of structure; g = gravity acceleration;
k = 3EI/[3 spring constant; w 2 = k / m natural circular
frequency; and q = Q / m gust force per unit ofmass.
Since at the initial moment, 1 = O, the structure
is at rest, i.e. u = 0, but with an initial velocity V o
equation (14) assumes the solution
D(m)
Strouhal's hyperbola.
u = Vo sincol.
co
(15)
To calculate this displacement, the continuous action
ofthe force q is represented by a series of elementary
impulses (Fig. 9).
As a result ofthe unit impulse dvo = qdr occurring
at the instant r , the structure experiences an increment
ofvelocity
du
Figure 8.
Elastic structure.
=dv" sin CO(I- r)
co
or replacing dvo with its value qdr one finds
6 GEOMETRY AND GUST ACTlON
du = i.sin co(t - r)d r
The gust action is a transient phenomenon of short
duration. Assume a gust force Q(t) applied on the top
of Column producing a horizontal displacement u(t),
(Fig. 8).
By using again the dynamic equilibrium ofthe disturbing force Q(t) with the inertia force Fi and the
elastic force Fe, one obtains
F; + F.
=Q(I) .
(12)
Explicitly the equation of motion becomes
(13)
co
Applying the superposition principIe over the who le
interval of time t, one obtains the dynamic response
of structure to gust action
1 '
u =- Jqsinco(t -r):tr.
(14)
671
(17)
COo
This is Duhamel's convolution integral, firstly
solved in 1934 by M.A. Biot (Kármán & Biot 1940).
lt contains both free and forced oscillations produced
by the disturbing force Q(t). A realistic representation
of gust force is given by the function
Q(t) = Cte- a,
or
(16)
(18)
where C and a are constants that can be chosen so as
to give the desired strength Qmax of the structure at
0. 1
0.08
E
'5 0.06
c
Ql
E 0.04
Ql
()
co
C.
'"
(5
0.02
O
Figure 10.
0 ---:670---:7'::-0--:!
80
-0.02 0:----I--1:':o--2-:':o:------:<3o:--~40:----.:5::-
Gust function.
Parameters a [1/s1 and c [N/kg s]
the time ta (Timoshenko & Young 1965) (Fig. 10). For
instance, Qmax could be reported to the static pressure
of wind provided by codes.
Setting the time derivative of expression (7) to zero
one obtains
C(I-atVGI =0
Figure 11.
The response ofColumn structure to gust action.
A good approximation to the maximum value of
this displacement can be obtained by evaluating u in
equation (24) for t = T12, where T = 2rrlw. lndeed,
(19)
from which results that Qmax occurs at ta = lia . Substituting this value of tinto equation (7) one finds
that
or
c
Qmax =-.
(20)
ae
Thus, when Qmax and ta are specified, the constants C
and a can be determined.
Adopting the notation for the ratio between gust
constant C and mass m of oscillating structure
C
(21)
e= -
m
the expression of gust force per unit of mass of
structure at instant r becomes
(22)
and the dynamic response of structure is defined by
the equation
u(t) =~ fr e-
al
wo
(25)
sinw (t - r)dr .
(23)
Integrating by parts one finds
ete-I"
2ae
( _
u(t) = - Z - -2 + 2
2 2 e ai
a +w
(a + w )
e(a 2 -w -' ) sinwt
w(a 2 +w 2 )
-
"u
(_!". )
lree--;;
2ae
u(w) =
+
e '" + l .
w(a z + w 2 ) (a 2 + W 2 )2
(26)
It is easily to notice that displacement u is proportional with e and vary exponentially with a as was
represented below for T = 2s, w = 3.14 ç I and e =
1 Nlkg·s (Fig. 11).
One can see that only for a S w the displacement u
assumes significant values. Forthe values of a between
3.14 and 15 the displacement exponentially decreases
and then vanishes, meaning that the structure does not
respond any longer to wind gusts. That explains why
during its service of almost 70 years, the Column did
not experience any response to gust actions. Parameter
c contains the influence of Column's geometry. Paradoxically, due to the divergence of module surfaces,
the dynamic response of Column structure is weaker
than the static response assessed according to code
provisions. lt appears that due to its original shaping
the Column is endowed with a self-damping system.
Wind tunnel investigations to be carried at Yokohama
National University might disclose this puzzle.
)
coswt +
7 VIRTUAL ANALYSIS ON HORIZONTAL
DIRECTION
(24)
The following model examines unsteady, incompressible flow past a long cylinder, having a square section,
672
Flow: [x velocity (u),y velocity (v)]
Flow: [x velocity (u),y velocity (v)]
;
0.225
Figure 12.
45cm.
00.45
Flow around a circular cross-seclion wilh <I> =
Figure 15.
aI 45°.
Flow around lhe same square cross-seclion
Flow: [x velocity (u),y velocity (v)]
Flow: [x velocity (u),y velocity (v)]
o 0.45
00.45
Figure 13.
Flow around a square cross-seclion aI 0°.
Figure 16.
aI 60°.
Flow: [x velocity (u),y velocity (v) ]
00.45
Figure 14.
aI 30°.
Flow around lhe same square cross-seclion
placed in an open field of flow, at right angle to the
incoming fluido The cylinder is offset somewhat from
the centre ofthe flow to make the steady-state symmetrical flow unstable. The simulation time, necessary for
a periodic flow pattern to appear, is difficult to predict.
A key predictor is the Reynolds number, which is based
on cylinder diameter. For low values - below 100 - the
Flow around lhe same square cross-section
flow is steady. In this simulation, the Reynolds number
equals 100, which gives a developed Karman path; but
the flow is still not fully turbulent.
The method of computation can be validated at a
lower Reynolds number, using a nonlinear solver.
By comparing the flow patterns for the same flowing parameters, one can find the strong influence of
both the shape of cross-section and the angle of attack
on the phenomena developed behind the obstacle. The
pattem of flow is only of qualitative interest in this
analysis beca use the aim of the study was to catch the
changes caused by the variable geometry of modules.
Around the square cross-section, the streamlines are
much clearly defined than around the circular shape.
The attack angle is defined by direction of flow with
normal to square side.
The configuration of streamlines behind the crosssection is very sensible with any slight change in attack
angles. Even when the flow follows one of square symmetry axes the flow pattem behind cross-section is
not necessarily symmetric. One can conclude that the
flow in the horizontal direction behind square cross
673
Flow: [x velocity (u),y velocity (v)]
Flow: [x velocity (u),y velocity (v)]
•
-
o 0.48
00.45
Figure 17.
at 0°.
Flow around a square w ith the side of 45 em
Figure 20.
at 0°.
Flow around a square w ith the side of 48 em
Flow: [x velocity (u),y velocity (v)]
Flow: [x velocity (u),y velocity (v)]
=-
0.46 ~ •...
O
::::::.
~Ir--::,r:::.~;:::...==. .=...=~,:::.~
=::.._
~_ .
o 0.46
Figure 18.
at 0° .
F low around a square with the side of 46 em
Flow around a square with the side of 49 em
Flow: Lx velocity (u),y velocity (v)J
F low: [x ve\ocity (u),y velocity (v)]
,
-
Figure 21.
at 0°.
:
0 . 47 ~,
O ~<C
.....
~
•
o 0.47
Figure 19.
at 0°.
00.45
Flow around a square with the side of 47 em
sections is not favorab le to develop and preserve steady
phenomena of vortex emission, at least to reduced
velocities. The tests carried out in the wind tunnel at
high velocities have shown a stabilization of flow but
that field of velocities is not of practical interest for
paper purpose.
Figure 22.
at 45 °.
8
Flow around a square with the side of 45 em
VIRTUAL ANALYSIS ON VERTICAL
DIRECTION
Numerical simulations were performed in steadystate, 2-D flow using commercial software MATLAB
with FEMLAB. In order to study the flow in vertical
674
Flow: [x velocity (u),y velocity (v)]
.
Flow: [x velocity (u),y velocity (v)]
-
.< :::::::;::;:>..
- - .:",g;:_-;~>~--
....".;:....~
- -.;. . . -----=-==------=-
,
00.49
00.46
Figure 23.
aI 45°.
Figure 26.
Flow around a square wilh lhe side of 46 em
Flow:
Flow around a square with lhe side of 49 aI 45 °.
Flow: [x velocity (u),y velocity (v)]
Ix velocity (u),y velocity (v)J
~~
.
0.88 ~J\~
o ~~~C . c .
-
-=
~
o
00.47
Figure 24.
aI 45°.
Figure 27.
at 0°.
Flow around a square wilh lhe side of 47 em
,
0.88
Flow around a square with lhe side of 88 em
Flow: [x velocity (u),y velocity (v)]
Flow: [x velocity (u),y velocity (v)]
•
o
00.48
Figure 25.
aI 45°.
Figure 28.
at 0°.
Flow around a square wilh lhe side of 48 em
direclion lhe neighbouring cross-sections were successively considered for two angles of attack, 0° and 45°.
The flowing parameters were lhe same in ali cases.
By comparing the above ten figures one can observe
lhat ali are displaying different pattems. One can
not find two identical configurations. The conclusion
0.89
Flow around a square with the side of 89 em
is that small variations of si de dimensions immediately modify the flow behind the cross-sections. The
changes are not large, but enough to avoid lhe development ofsteady phenomena ofvortex emission what
can harm the stability ofthe Column.
675
Flow: [x velocity (u),y velocity (v)]
Flow: [x velocity (u),y velocity (v)]
. .. . ....... . ..
- -: - - - - - -
:.~- .
o
Figure 29.
at 0°.
o
0.9
Figure 32.
at 45°.
Flow around a square with the side of 90 cm
0.9
Flow around a square with the side of 90 cm
Flow: [x velocity (u),y velocity (v)]
Figure 30.
at 45 °.
Flow around a square with the side of 88 cm
Apparently, it is a common concept of shaping, but
in fact it has well defined idiomorphic and topological
functions. Standing with such qualities on a platform
in open air for the Wind Engineeri ng as Science, the
Colurnn in Targu Jiu means not only a challenge, but
also a great responsibility. Fortunately, ali the results
obtained by ana lytical studies, experimental investigations and virtual analysis, as presented above in this
pape r, are showing that in normal conditions the wind
can not harm the Colurnn and therefore, from this
standpoint, it can be considered not only endless, but
everlasting too. lt reached the perfection. Actually, ali
of Brancusi masterpieces are fascinating because the
power of its prying questions is much stronger than
the convincing capacity of the answers, sometimes by
chance, the Science could findo
Flow: [x velocity (u),y velocity (v)]
REFERENCES
Figure 3 I.
at 45° .
9
With the side flow around a square of 89 cm
CONCLUSION
The geometry ofthe Endless Colurnn distingu ishes by
the natural and simply shapes ofmodules successively
standing one atop another with perfect regularity.
Georgescu, E.S. 2004. Dynamic parameters of the Endless Column. Proc. ofthe Romanian-Japanese Workshop
" Wind Engineering and Cultural Heritage", UNESCO
Bucharest, Romania, 8 March 2004. Paper #3.
Georgescu-Gorj an, S. 2004 . Assemblage of the Endless
Column at Târgu Jiu in November 1937 Proc. of the
Romanian-Japanese Workshop "Wind Engineering and
Cultura l Heritage", UNESCO Bucharest, Romania, 8
March 2004. Paper #2 .
Kármán, Th. & Biot, M.A. 1940. Mathematical Methods in
Engineering. New York: McGraw Hill Book Co. Inc.
Landau, L. & Lifchitz, E. 1966. Mécanique. Moscou: Éditions Mir.
Lungu, D. et a!. 2002. Reliabi lity under wind loads of the
Brancusi End less Column, Romania. Proc. of lhe 3rd
Easl Europ ean Conference on Wind Engineering. Kiev,
Ukraine: 1- 10.
Petre, A. 1973. TheOly of Aeroelasticity. Dynamic Periodic
Phenomena. Bucharest: Romanian Academy Printinghouse.
676
Safta, C.A. 2002a. Oevice for preventing the oscillations
induced by wind in metallic stacks and gas pipelines.
Bucharest: Romanian Patent Office OSIM No. A 00928.
Safta, C.A. 2002b. The Aeroelastic Stability ofthe Column.
Bul/etin oflhe Technical University ofCivil Engineering
Bucharest, 51(1): 51-55.
Safta, C.A. 2003. Aeroelastic Indifference of the Endless
Column. Proc. of the 28th Congress of the AmericanRomanian Academy. Targu Jiu , Romania, 2- 8 June 2003,
Paper EGAM # I O.
Safta, C.A. 2004. Stability to wind ofthe Endless Column.
Proc. of lhe Romanian-Japanese Workshop "Wind Engineering and Cultural Heritage", UNESCO Bucharest,
Romania, 8 March, 2004. Paper #4.
677
Sofronie, R.A. 200 I . Brancusi and the obsession of gravity. Proc. of lhe lnlernalional Congress of ICOMOS and
UNESCo. Special Session BRANCUSI. Paris, France, 12
September 2001. Paper # 1.
Sofronie, R.A. 2003. ldeomorphic concept of the Endless
Column. Proc. of the 28th Congress of the AmericanRomanian Academy. Targu Jiu, Romania, 2- 8 June 2003,
Paper HAFA #4.
Timoshenko, S.P & Young, O.H. 1965. Theory ofStructures.
New York: McGraw-Hill Book Company Inc.
Yamada, H. 2004. Wind and Long-Span Bridges. Proc.
oflhe Romanian-Japanese Workshop "Wind Engineering
and Cu ltural Heritage", UNESCO Bucharest, Romania, 8
March 2004. Paper # I.