Structural Analysis and Calculations
Related to the Tower of Pisa
Work by Vittorio Novelli, a surveyor from Cesena (Forlì)
Tower of Pisa
STATIC CALCULATIONS
1st phase: Anchorage of the Tower by means of steel cables
1) Sizing of the cables:
The sizing of the cables, their number and their position, have been
individually selected to satisfy the following conditions:
a) to make a uniform ground pressure;
b) to stress for size, the different "orders" of rings, in a uniform
way and in terms that are most acceptable.
The first condition is satisfied when the sum of the moments of
all the tension applied to the bands, relative to the center of
the base equal the given moment resulting from the weight of the
Tower for its eccentricity on the base. The second condition is
satisfied when the tension applied to the bands result in a load
proportional to the weights of each single respective ring.
It is stable when each section balances in compression and turnover.
Calculated data is taken or inferred from the publication
"Searches and Studies on the Leaning Tower of Pisa", 1971, from
the Office of the Public Jobs. It is summarized as follows:
- inclination of the plane of support of the Tower on the ground:
5°35'53" (equal to 5.5980° degrees; 0.097704 radians);
- partial and progressive mass of each order:
Partial Progressive
8th order 725t 725t
7th order 1,267t 1,992t
6th order 1,175t 3,167t
5th order 1,192t 4,359t
4th order 1,227t 5,586t
3rd order 1,362t 6,948t
2nd order 1,371t 8,319t
1st order 3,738t 12,057t
at the base 2,396t 14,453t
(In the calculations that follow, at the 6th order, a partial
and progressive mass equal to 3,167 tons will be attributed,
not being the two greatest orders of interest in this same
calculation);
- eccentricity of base (looking along a section normal
to the Tower): 2.24 m.
The position of the characteristic points (Center of Gravity and
centers of the rings) and the dimensions of the structure
(diameters, heights and thickness) are taken again from the
quoted ministerial publication.
How the project data is assembled:
- the position of the center of gravity of each band of cable is
at the point of detachment from the Tower (point of tangency)
and vertically in the center of the respective order. Said
band is in such position that the first band from the top is
respectively the 6th order; the second, the 5th order; the third,
the 4th order; the fourth, the 3rd order; and the fifth,
the 2nd order;
- Cables are embedded in an external anchorage plinth of concrete
set 90m from the center of the Tower in which is embeddedd
an axle (corresponding to the direction of the plane of maximum
inclination) with the points of attachment of the cabless varying,
with respect to the level of the base, from 16.75m to 14.25m.
The calculation of the cables presupposes the use of cables of
braided steel of the type SEALE WORRINGTON, 222 THREADS, diameter
of 24 mm (15/16 inch) whose nominal characteristics are:
weight 2.40 Kg/ m
diameter of the individual threads 1.33 mm
Tensile load guaranteed - minimum 38,400 Kg
resistance 180 Kg/ [mm²] (1770 N/ [mm²])
extension at design load - 2% length (around)
A prudent safety measure restrains the cables to not more than the
50% of their nominal load and therefore with a maximum load of
19.2 Kg each.
The first analysis and the first trial could be made on the tension
and compression on the ground and on the masonry, not forgetting
that the data furnished would mirror transitory conditions and that
they will stop with the stop of the tension of the cables. They
are, in substance, the conditions in which one will find the Tower
at the moment of the execution of the work.
reduction terrestrial masonry
inclination s.max|s.min s.max|s.min
(kg/cm²) (kg/cm²)
0° 5.14 5.14 1.00 8.59
1° 5.13 5.13 10.85 8.63
1° 30' 5.12 5.12 10.77 8.65
2° 5.11 5.11 10.70 8.67
Scheme of positioning anchorage cables
These meaningful data, that alone describe the angle of elevation for
stability of the anchored Tower, coordinate with these other statistical
characteristics:
- eccentricity of void base (and= 0), in all cases;
- maximum eccentricity of 32 cm in the 6° ring;
- solicitation to the cut that doesn't overcome the 0.24 Kg/cm²;
- coefficient of inferior attrition of 0.027 (corresponding to
an inclination of the resultant of 1°32');
- traction in the inferior cables to 8,900 Kg/ [cm²]
(against the 18,000 nominal);
A true and accurate understanding of the conditions of the Tower
is needed in this delicate phase of removal of the underlying
ground and of the sucessive consolidation of the foundation
plinth.
2) Verification of stability of the structures of anchorage of the cables:
Volume and mass of the reinforced concrete and its ballast.
a) Caisson in reinforced concrete
n. length width height volume unit mass mass total eccent
(m) (m) (m) (cu.m.) (tons/cu.m) (tons) (m)
1 20.00 10.00 2.00 400.000
3 20.00 0.80 11.00 528.000
8 0.80 3.80 11.00 267.000
1 20.00 10.00 0.80 160.000
6 5.60 3.80 0.80 102.144
1457.664 2.500 3644.16 0.00
b) Spurs in reinforced concrete
n. length width height volume unit mass mass total eccent
(m) (m) (m) (cu.m.) tons/cu.m (tons) (m)
1 20.00 2.00 2.00 80.000 2.500 200.00 5.00
4 0.80/2 2.00 11.80 37.760 2.500 94.40 4.67
c) Central Void in reinforced concrete (by deduction)
n. length width height volume unit mass mass total eccent
(m) (m) (m) (cu.m.) tons/cu.m (tons) (m)
1 3.80 5.60 0.80 17.024 2.500 42.56 2.30
d) ballast in inert material
n. length width height volume unit mass mass total eccent
(m) (m) (m) (cu.m.) tons/cu.m (tons) (m)
12 3.80 5.60 5.10 1302.336 1.700 2213.97 0.00
e) empty ballast (by deduction)
n. length width height volume unit mass mass total eccent
(m) (m) (m) (cu.m.) tons/cu.m (tons) (m)
2 3.80 5.60 5.10 217.056 1.700 369.00 2.30
Gravitational Center of the total mass
[ Image]
[ Image]
and therefore 6.863 m from the point of the overturning
Plinth for anchorage extracts
Resistant moment to overturning:
Mres= 5740.97 x 6.863= 39400 ton-meters
Tension transmitted from the cables to C, the center of the plinth,
in the more unfavorable hypothesis:
Moment = - 16,699 ton-meters
(with the horizontal component 15.96 m from the theoretical center of
the plinth)
Normal effort (toward the top)= 168 t
Axial effort (horizontal)= 1046 t
Moment transported to the point of tangency (7 m. toward the Tower and 0.80
toward the lower part):
M=- 16699- 1046 x 0,80- 168 x 7,00=- 18703 ton-meters
degree of stability to the tangency:
[ Image]
Position of the resultant as regards the point of turnover:
[ Image]
(therefore with an eccentricity of 2.29 m).
Bearing on the ground:
a) excluding the resistance to traction:
[ Image]
b) admitting the resistance to traction of the poles:
[ Image]
smax (compression)= 2.32 + 2.74 = 5.06 kg/cm²
smax (traction)= 2.32 - 2.74 = 0.42 kg/cm²
with the point of flex 0.92 m from the tension edge.
To guarantee the traction on the ground, whose intensity has given from:
N'= ½ (20.00 m x 0.92 m x 4.2 t/m²) = 38.64 t
20 piles with resistance by friction of 2 tons are foreseen each reinforced
with 4 or 14 (6,16 cm²).
[ Image]
that, obviously, is a limit of total safety.
In the compressed zone are installed cast in place concrete piles for
a total load data taken from:
N'= ½ (20,00 x11,08 x 51.6 t/m²)= 11435 t
200 piles will be installed, therefore, with load of 58 t each, obviously,
one will provide for the yielding of the ground to reach, in the
extreme, a resistance to compression of 5.16 kg/cm².
2nd phase: amplification of the foundation
Dimensions of the structure of reinforcement:
width of the circular crown about the base: 4 m
external height of the crown: 1.5 m
inside height (that is in contact with the existing foundation) 3 m
diameter of the central nucleus 4.5 m
height of the central nucleus 3 m
weight of all the additional complex 1818 t
height of the center of gravity (on the plan of the base) 1.23 m
Since the additional plinth influences only the base of the Tower, in the
sense that the structure standing above is not influenced, from the
static point of view, by the new strength agent (weight of the plinth)
the new response of the base is determined, that is as it concerns
the unloaded Tower, by the action of the cables, but with reduced
inclination, respectively of 1°, of 1.5° and 2° as regards the current
one.
Reduction of the inclination to 1°:
Moment (M) -26,700.35 - 1818 x tan 0.080709 = - 26,847.4 ton-meters
Normal effort (N) -14.406 - 1818 x cos 0.080709 = - 16,218 tons
Eccentricity of base (e) M/N = 1.66 meters
Tangential Force (T) -1165 - 1818 x sen 0.080709 = - 1,312 tons
Area 1/4 * pi * (27.58)^2 = 597.42 m^2
Modulus of resistance (W) 1/32 * pi * 27.58^3 = 2.059.6 m^3
Tension to compression smax = 4.02 kg/cm2
smin = 1.41 kg/cm2
Tension at the split t = 0.22 kg/cm2
Reduction of the inclination to 1° 30':
Moment (M) -23,857.79 - 1818 x tan 0.071982 = - 23,988.9 ton-meters
Normal effort (N) -14.416 - 1818 x cos 0.071982 = - 16,229 tons
Eccentricity of base (e) M/N = 1.48 meters
Tangential Force (T) -1039 - 1818 x sen 0.080709 = - 1,170 tons
Area 1/4* pi * D^2 = 597.42 m^2
Modulus of resistance (W) 1/32* pi * D^3 = 2,059.6 m^3
Tension to compression smax = 3.88 kg/cm2
smin = 1.55 kg/cm2
Tension at the split t = 0.20 kg/cm2
Reduction of the inclination to 2°:
Moment (M) -21,013.41 - 1818 x tan 0.063255 = - 21,128.6 ton-meters
Normal effort (N) -14.424 - 1818 x cos 0.063255 = - 16,238 tons
Eccentricity of base (e) M/N = 1.41 meters
Tangential Force (T) -914 - 1818 x sen 0.063255 = - 1,029 tons
Area 1/4* pi * 27.58 ^ 2 = 597.42 m^2
Modulus of resistance (W) 1/32* pi * 27.58 ^ 3 = 2,059.6 m^3
Tension to compression smax = 3.74 kg/cm2
smin = 1.69 kg/cm2
Tension at the split t = 0.17 kg/cm2
Solicitations on the ground transmitted from the foundations
Complete solidarity between the existing structure and the
supporting one is insured by the application of 18 post-tension
cables of the DYFORM type 6 x 36mm to 38mm (nominal
tensile capacity of 118 tons) with applied tension equal to 70 tons.
View Together
Plan of intervention elaborated by Vittorio Novelli, Nazzareno Paccaloni, Marco Crescentini,
George Crescentini which application of the brief industrial No. 12001A/90 titled "Procedure
to correct the tilt of towers and buildings generally" by Vittorio Novelli ...
Translated by Gary Feuerstein, 25 March 1998, with permission from Mr. Novelli
Personal e-mail to:
gary@endex.com