Justifying feasibility calculations

The proof of feasibility of the imagined velum necessarily involves the sizing of the cables and the efforts to be made. To prove that hemp ropes of reasonable diameters are suitable and that the installation and the maneuvers of the velum require only moderate efforts is therefore the objective of the pages of this tab.

Imagined method

It was chosen to calculate the elliptical velum of an ancient amphitheater by extrapolating the calculations applied to a virtual circular amphitheater having a radius of the same value as the largest radius of curvature of said amphitheater and with a number of masts providing a distance from one to the other very close to that measured on the backed up supports (3m in Nîmes, 2m in the Colisée). It is in fact at the right of this greater radius of curvature that the centripetal tensions of the arms and halyards generate the most important tangential reactions on the central ring.

The method consists of first determining the forces at the ends of a connecting line of two elementary rectangles of fabric and then transferring them in the opposite direction to the other members with which it is in contact (action = reaction) and and so on. The equilibrium is acquired when at the same time zero are the algebraic sum of the forces in X, the algebraic sum of the forces in Y, the algebraic sum of the moments of the forces with respect to any point.

The equation of efforts concerns a subset consisting of:

• --- an arm (in brown)
• --- an elementary rectangle of canvas (in red)
• --- a meeting thread of two neighboring rectangles of canvas
• --- the central ring share associated with the canvas rectangle
• --- a deployment halyard, folding
• --- a set of halyard pulley.

Special Conventions

The efforts will be represented by their projections on the usual orthogonal axes X and Y, oriented in their real sense, here detectable without any ambiguity because the tensions exerted at each end of a flexible rope can only be in the same direction as the tangents to those. This arrangement makes it possible to relate to the equations only the absolute values of the forces involved. It will only be necessary to endow with a certain sign the moments causing such direction of rotation and of a sign opposite those causing a direction of rotation. opposite.

Spare the right rules? to simplify the scripts it was chosen to operate with the system of units MKpS, that allows it to express the weights with the same values as the masses that engender them. In this old system a mass of one kilogram weighs one kilogram force while it weighs 9.81 newton in the legal system of today…

The moment with respect to a point of the uniformly distributed weight of the wire or of a cable being of the same value as that of the resultant will be expressed with the said resultant. The lengths of the canvas and the cables will be considered as those of the lines joining their two ends on the ground that the difference is small (3% for an arrow of 10% of the length of "chain").

Geometric Data

According to Jean Claude Golvin and Christian Landes « Amphitheatres and gladiators » (1990 to CNRS Presses) he perimeters of the ancient amphitheatres were most often obtained by four circular arcs whose centers are the vertices of a rhombus built with four triangles of Pythagoras rectangle contiguous. In the calculations which follow the amphitheatres will be considered with external perimeters defined according to this rule.

 By designating u a basic building module length of the amphitheater we have : I O1 = 3.u ; I O3 = 4.u ; O1 O3 = 5.u A and B being the length and width of the amphitheater we have : (1) A = D2 + 6u (2) B = D1- 8u (3) D1/2 – D2/2 = 5u The 3 equations (1), (2) and (3) calculate D1 and D2 from the length A and width B of the amphitheater D1= 2.A – B and D2 = (3.B - A)/2 The angles of construction α and β in radian are : α = arc sin(3/5) et β = arc sin(4/5)

As previously described, calculations are made for virtual circular velums having radii of the same value as the arcs of large radii of the amphitheater concerned ... and with numbers of polesN1 et N2 procurant des écartements très proches de celui que l'on peut encore mesurer sur les ruines (2,25m pour le Colisée, 3,10 m pour les arènes de Nîmes). Cette disposition fait que les velums virtuels pour calcul seront constitués avec des rectangles élémentaires de toile de mêmes dimensions que le rectangle élémentaire du velum d'allure elliptique à N mâts.

The perimeter of the amphitheater is Pe = 2. α .D1 + 2. β. D2

The spacing between masts is lm = Pe/N

The perimeters of the big and the small circle of calculation are:
Pe1 = .D1 et Pe2 = .D2

The number of masts in the large circle of calculation must therefore be N1 such as
Pe /N = .D1/N1 from where N1 = N. .D1/ Pe

The number of masts in the small circle of calculation must therefore be N2 such as
Pe /N = .D2/N2 from where N2 = N. .D2/ Pe

The values given by the calculations will be rounded to even integers.

The Colosseum measures188 x 156 m and is equipped with 240 masts. To take into account that its velum is hung on the edge of a 10 m wide platform, we will retain for it a large axis of 168 m (188 – 2.10) and a small axis of 136 m (156- 2.10), diameters of calculation circles D1 = 200 m and D2 = 120 m, numbers of poles N1= 314 and N2= 188

The arenas of Nîmes measure 134 x 101 m and are equipped with 120 masts.

 D'où : D1 = 167 m D2 = 85 m N1 = 170 N2 = 86

if L is the projection of the width of the cover velum the covered area is : Sc = 2.L.(α.D1 + β. D2) –2.L2.(α + β)

Definition of the main symbols used

D1 : diameter in m of the large circle of calculations.
D2 : diameter in m of the small circle of calculations.
N1 : number of virtual masts on D1
N2 : number of virtual masts on D2
mt : unit mass of the fabric in Kg / m2
L : projection length of an elementary rectangle of canvas
a : distance in m between the ends of the fabric and the central ring-top overhanging the arena
p : slope at the low point of the rope inserted into the canvas.
Hf : vertical drop in m of the rope inserted into the canvas.
hd : Distance in m between top of rope in canvas and upper strand of halyard.
hbr : Distance in m between top of rope in canvas and arms.
R : breaking strength in Kg / mm2 of all the cables used.
d : density in Kg/dcm3 of all the cables used.
Kf : desired breaking coefficient of safety for the rope inserted into the fabric.
Ka :desired breaking coefficient of safety for the central ring cable.
mp : mass in kg of a set of halyard pulley
z : Ratio of halyard cable and rope inserted in the fabric.
y : Ratio of arm and halyard cable diameters.
l+ : extra width in m width of canvas rectangle, for assembly.
N : Number of masts of the amphitheater.
E : modulus of elasticity of the wood of the masts.
Rm : Resistance to breaking wood of masts in Kgf / mm2.
Km : Desired breaking factor for masts.
øtr : Winch diameter in mm
L': projection length of the halyard and the arm: L' = L + a
Lt : approximate length of a canvas rectangle, in m.
Ld : approximate length of the upper strand of the halyard, in m.
Lb : approximate length of an arm, in m.
lm1 : distance in m between two masts of the virtual diameter velum D1.
lm2 : distance in m between two masts of the virtual diameter velum D2.
lt : width in m of a rectangle of canvas of the velum. lt = lm1 + l+
Hd : vertical drop in m of the upper strand of halyard = Hf + hd
mf : unit mass of the rope inserted into the fabric in Kg / m.
øf : diameter of the rope inserted into the fabric in mm
ød :diameter of the halyard cable in mm
øa : diameter of the ring cable in mm
øbr :diameter of the arm cable in mm
øm :mast foot diameter in mm
Kd : safety coefficient of halyard cable.
Kbr et Kbr' : safety coefficient of the arm cable stretched and at the beginning of folding.
Cu : useful torque on winch in Kgf m m

A/ Equilibrium of a rope between two rectangles of canvas when the velum is stretched (in red)

The equilibrium studied is that of the drawing above, that of a carrier rope inserted between two elementary rectangles of canvas. In addition to its own weight it carries two half rectangles of canvas and the part of the rope of halyard relaxed. Having a null component the two sets of forces perpendicular to the plane of the drawing, exerted by the neighboring rectangles of canvas, are ignored.

The length of the canvas is Lt = ( L2+ Hf2)1/2 and its width lt is calculated based on the size of the amphitheater and the number of masts (see "Definition of Symbols Used"). The halyard cable and the linen rope are made of the same material and have diameters in a ratio. z.

Equilibrium forces are horizontal and vertical actions :

--- reactions XA, XB, YA, YB at the extremities.
--- halfway between A and B the weight of the canvasmt.lt.Lt, the weight mf.Lt of the rope between two rectangles and the weight z2.mf.Lt loosened halyard cable, the tension of the fabric being then ensured by the inserted rope.

Let us write that the sums of the projections of the forces on the traditional axes X and Y are null :
(1) XA - XB = 0 et (2) YB - YA + mt.lt.Lt+ mf.Lt.(1+z2) = 0

Let us write that the sum of the moments of the forces with respect to the point B is null :
(3) YA.L – XA.Hf - mt.lt.Lt.L /2 - mf.Lt.L.(1+ z2)/2 = 0

The requirement of a slope p at the low point B of the thread inserted into the canvas is written :
(4) YB = p.XB

A help (sesame) for further calculations :

If R is the breaking strength of a section cable S and K the desired safety factor its permissible voltage is :
T = R.S /K

If L is its length and d its density its mass M is : M = S.L.d
By eliminating S between these two equations it comes : T = R.M /(K.L.d)
is T = R.m /(K.d) by designating m mass per unit length.
When we express T in Kgf, R in Kgf/mm2, m in Kg/m and d in Kg/dcm3 the last formula is written :
T = 103. R.m /(K.d)
The maximum tension of the rope inserted in the fabric is developed A. It is worth Tf = (XA2+YA2)1/2.

So we have:

(5) XA2+YA2 = ((103. R . mf)/(Kf .d))2

The system of 5 equations (1), (2), (3), (4) and (5) gives a second degree equation to unknown mf of shape :
A.mf2 + B.mf + C = 0

(103.R)/(Kf .d) ; Z1 = - ((L.Lt.(1 + z2))/(2.(Hf – p.L)))

Z2 = - (
(mt . lt .Lt.L)/(2.(Hf – p.L))) ; Z3 = p. Z1 + Lt.(1 + z2) ; Z4 = p. Z2 + mt.lt .Lt

We have : A = Z12 + Z32 - G2 B = 2.
( Z1.Z2 + Z3.Z4) C = Z22 + Z42

The determinant of the second degree equation is : = B2 – 4.A.C with which one calculates mf then :
XA =XB = Z1.mf + Z2 ; YA = Z3.mf + Z4 ; YB = p. XB

De mf in Kg we deduce the diameter Øf in mm rope inserted into the canvas.

Øf = ((4.103.mf)/(Pi.d))1/2 and the diameter of the halyard cable : Ød = z . Øf

B/ Balance of the upper strand of the halyard :

The studied equilibrium is that of drawing 2. When the velum is deployed on each side of the pulley the halyard cable is subjected to the same tension as is inB the rope inserted into the canvas, is (XB2 + YB2)1/2.

hd being the height between the high point of the rope and the top of the halyard, the unevenness of that is almost Hd = (Hf +hd). The low altitude difference induced by the distance a between central ring and canvas is neglected.

By designating L' = (L + a) the length of the projection of the upper strand of the halyard and Ld the approximate length of the halyard as : Ld = (L'2 + Hd2)1/2 the balance is written :

(6) YD2 + XD2 = XB2 + YB2 (7) YD + (md .Ld) – YC = 0

(8) XD.Hd -((md .Ld.L')/2)– YD.L' = 0 (9) XC - XD = 0

The system of the four equations above gives an equation of the second degree to unknown XD of shape :
A1.XD2 + B1.XD + C1 = 0 with :

A1 = 1+
(Hd /L)2 ; B1 = -(Hd .md .Ld/ L ) ; C1 = (md2.Ld2/4) - XB2 -YB2

From the positive root of XD we calculate :
YD = XD .(Hd /L)(md.Ld /2) puis YC = YD + md.Ld

The maximum tension TC on point C in a strand of halyard is TC = (XC2 + YC2)1/2

The coefficient of safety at break of the halyard cable : Kd =((. Ød2/4). R) /((YC2 + XC2)1/2)

PRELIMINARY NOTE : the tension of the fabric is ensured by the weights of the central ring, pulleys and various additive masses (which can be decorative pendants) attached to them.

It is agreed that when the canvas is stretched the arm retains a residual tension with components XE et YE such as the arm cable has a diameter øb = y.ød

The equilibrium studied is that of the drawing above representing in section the short connection between the end B from the canvas and the topE of the central ring.

The point E,on an arch of greater curvature, (see further drawing seen from above) is subjected to a horizontal centripetal effort
Q1 = XB + (1+ w).XD which induces in the ring cable a tension Ta as :

Ta = Q1/(2.sin(/N1)) = (103.R.ma)/(Ka .d) from where ma = (Ta .Ka .d)/(103.R)
ma being the mass per m of the ring cable the diameter of that is : øa = ((4.103.ma)/(.d))1/2

On the central ring the arms are separated from la1 = (D1-(2. L')).sin (/N1) on the arcs of large radius and of
la2 = (D2-(2. L')).sin(/N2) on small radius arcs.

The mass Ma1 of a length la1 of this cable is Ma1 = (10-3).((a2)/4).la1.d

The mass Ma2 of a length la2 the same cable is Ma2 = (10-3).((a2)/4).la2.d

Equilibrium equations of the arm (according to drawing 4 ) when the canvas is stretched are :

(10) XF - w.XD = 0 et (11) XF .(Hf+hbr) + y2.md.Lb.L’/2 -YF .L’ = 0

(12) YF – y2.md.Lb -YE = 0 d’où YE = YF - y2.md.Lb

They allow to express XF et YF

XF = w.XD YF = ( XF .(Hf+hbr))/L’ ) + y2.md.Lb / 2

The maximum tension of the arm is then Tbr = (XF2 + YF2)1/2

Inducing a factor of safety Kbr = R /((XF2 + YF2)1/2/((π. y2. ød2)/4))

The balance of vertical forces of the short bond between canvas and ring according to the drawing 3 writes :

YB + YD +YE - mP - Ma1 - Madd1 - md.a = 0
is YB + YD + (YF - y2.md.Lb) - mP - Ma1 - Madd1 - md.a = 0
from where : Madd1 = YB + YD + (YF - y2.md.Lb) - mP –Ma1 - md.a

Madd2 = YB + YD + (YF - y2.md.Lb) - mP –Ma2 - md.a

D/ Checking the safety factor of the arm at the beginning of folding

It is at the beginning of folding (or in the final phase of deployment) that the arm is the most tense. In this configuration the inserted rope and the halyard no longer participating in the portage of the ring is the arm that ensures only the portage of the various constituents. Drawing 5 shows the vertical forces at play at these times. Equilibrium equations are then:

(13) YE' = mt.lt .L'/2 + mf.L'/2 + (2.md.Ld'/2) + md.a + mP+ Ma1 + Madd1

(14) YF'- YE'- y2.md.Lb' = 0

(15) (YF'.Ld )(y2.md.Ld.Lb'/2) - (XF'.(Hf+hbr)) = 0

From where: YF'= YE'+ y2.md.Lb' et XF'= Q2 = ((YF'.Ld)(y2.md.Ld.Lb')/2)/(Hf+hbr)

The new tension of the ring cable is then : Ta'= XF'/(2.sin(/N1)) inducing a new factor of safety of the ring cable : Ka'= R /(Ta'/(a2/4))

According to the convention the arm cable diameter is y times that of halyard cable : øbr = y.ød The new safety factor of the arm cable at the beginning of aliasing is therefore :

Kbr' = R /((XF'2 + YF'2)1/2/((. y2d2)/4))

E/ Verification of the balance of a sub-assembly of the velum (stretched canvas)

The sum of the masses of all the cables, the web, the halyard pulley and the additional mass must have the same value as the sum of the vertical reactions at the mast.

For a subassembly located on an arch of greater curvature the sum of the masses is :

For a subset located on a smaller curvature arc the sum of the masses is :
MT2 = (mt.lt.L')+(mf.L')+(y2.md.Lb')+(2.md.Ld')+ Ma2+Madd2 + mp

The sum of the vertical reactions to the mast right is in both cases : RT = YA+YC + YF

Computer-assisted numerical calculations indicate that equality MT1 = MT2 =RT is respected regardless of the combinations of input parameters.

F/ Dimensioning of the masts, of circular section of diameter øm realized with a wood of resistance to the rupture Rm

The maximum bending moment applied to a mast is in Kgf.mm:
Mf = 103. XF'.(Hf +Hbr) et I /v = ( /32)m3

To respect the factor of safety Krm desired with respect to breaking strength Rm used wood one must have :
Rm/Krm = Mf/(I /v) = 103. XF'.(Hf +hbr)/(( /32)m3)

From where øm = (103. XF'.(Hf +hbr).Krm /( /32). Rm))1/3

G/ Various calculations

Total length of the central ring : LTa = 2.α.(D1-2.L) + 2.β.(D2-2.L)

Total mass of the central ring : MTa = ma .LTa

Total mass of the velum (canvas + cables + pulley) : MTV = N. MT1 = N. MT2

Total mass of canvas : MTT = N. mt.lt.Lt

Covered surface: Sc = 2.L.(α.D1+ β.D2) - (2.L2(α + β))

Total mass of the velum per m2 : M/m2 = MTV/ Sc

Total area of useful canvas : Su = N. lt. Lt

Total cable length for halyard : LTd = 3.Ld.N (about)

NOTA : 3 because there may be a length Ld stored on the winch drum

Total length of arm cable : LTb = Lb.N (off dead strands)

H/ Verification of the lower halyard cable being deployed or folded. (see drawing 5 )

During maneuvers it is the lower strand of halyard that carries the cloth through the rings connected to that one. In this configuration the verification of the halyard cable must be carried out with a finite element calculation. This was done for the velum project on the arena of Puy du Fou with rings spaced 1.5 m. It suffices that the lower halyard strand has a certain slack (of a very acceptable value) so that the efforts of all orders remain inferior to those of the "stretched canvas" configuration. ».

I/ useful power and maneuvering force

Until the final phase of tension of the fabric the energy for the folding of the fabric is, almost, that of the elevation of its center of gravity of the half value of the difference in altitude, ie Hf / 2. For a basic rectangle of canvas the energy in Kgm is therefore: mt.Lt.lt .Hf / 2. If the operation is done in a few minutes it is easy to see that the power per mast is only of the order of fifty watt. This means that one man is amply enough to ensure almost all folding. It is even conceivable that the entire maneuver, including the final tensioning can be done with two or three times less men than masts. At the end of the tension of the fabric the maximum punctual effort to exert on the upper strand of halyard in C est :

TdC = (Xc2 + Yc2)1/2

THE PREVIOUS EQUATIONS WERE VALID BY BUREAU VERITAS at the end of 2004.

J/ Computer-assisted calculations

All the equations numbered above have been translated into computer writing. At 70 years old first exercise of the kind for René Chambon. He discovered that the 3.14 of his 20 years was to be written (Pi ())! From numerical values assigned to twenty or so input parameters the very common Excel software gives almost instantaneously all the efforts involved, the cable diameters and their safety coefficients, the section of the masts. It also verifies that all equilibrium equations used are well respected and that the sum of all the masses in play(MT) is strictly equal to the algebraic sum of the vertical forces applied to the mast (RT). And on the frame (last right column).

K/ Digital applications

Twenty entries and outputs of computer-assisted calculations fit on a single A4 page.

The following two tables are computer-assisted numerical calculations for the Coliseum and for the Nîmes arena. Hemp ropes were retained with a breaking strength of 8,8 Kgf/mm2 and a density of 0,9 Kgf/dcm3, values found in the CORDALPES manufacturer's catalog. The canvas is linen with a realistic weight of 0,3 Kg/m2 as, no doubt, that of the togas of the beautiful Romanes of old. The results show clearly that nearly two thousand years ago the Romans could equip the two amphitheatres with velvets of very ordinary diameters, from 10 to 60 mm. This with security factors of 6 could not be more secure. The weight of the blankets is less than a kilo per square meter: staggering !

the Slope p rope inserted in the fabric was chosen at 5%. A lower value would significantly reduce the masses of the central ring and additive masses (pendants).

In order to protect the first ranks of notables from the oblique rays of the sun, the said ring had to be placed rather low: so much better to reduce the diameters of the ropes and the efforts! It was arbitrarily retained a vertical drop of 14 m for the Colosseum and 10 m for the arena of Nîmes.

The calculations presented take into account central rings made with hemp ropes working with a coefficient at break of 6. It is likely that the Romans made the central ring with an iron chain much stronger than necessary: additive masses (pendants) lighter, high coefficient of safety, insensitivity to hygrometry, ease of fixing halyard pulleys and arms.

 A click on the EXCEL icon opens "calculation program for Roman velums". For Internet users wishing to modify the numerical values of the input parameters to display the effects of this or that on the dimensioning of masts, cables, maneuvering forces ... etc. It can be seen instantly that reducing the height of the Colosseum's velum from 14 to 10 m makes the rope diameter of the central ring of 82 à 105 mm…
Click on the excel icon to download the file before opening it first excel Coliseum second Arenas Nîmes

Case Number 2

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