5.7
Geometry determination of thickness dimensions and of projecting elements.
The ancient documentation
is poor of cross sections and of thickness dimensions (like the thickness
of the bridge) and of dimensions of the projecting elements (like
cornices), moreover there is no data concerning all the "small"
dimensions of details and of variations. This of course constitutes an
important limit to the exact global geometry determination, and these
elements have of course a remarkable esthetical architectural value.
Inaccuracies in the geometry determination of these dimensions may lead to
a different global aspects of the monument.
For what concern the bridge
thickness of the load bearing arch, it can be said that it was most likely
that this measure had small variations and changes in the bridge length
due to usual constructive inaccuracies and imperfections. The 1982 survey
doesn’t allow any evaluation of these variations probably because of the
graphic representation scale, and nowhere else these information are
filed. By summing up all the thickness dimensions of every single stone
row, the results are in many cases far from being reliable, because these
totals are often too much different from the average value of the
thickness, and the differences do not follow a progression. This is also
due to the fact that voussoirs have been measured one by one, but no
global dimension has been taken, therefore there might be an inaccuracy
related to the joint thickness and to the reiteration of the measures for
a number of times equal to the number of arch stones contained in each
row.
Therefore the thickness of
the load bearing arch of the bridge may be evaluated from the 1982 survey
as 395 centimetres, which is also the average value obtained from the
totals of the 1955 survey. During the on site works ordinary constructive
imperfections will lead to the performing of the small differences among
row’s lengths.
For what concern, instead,
the study of the cross sections, and of the dimensions of the cornices,
the phase A work has led to understand that these data may be only
gathered trough the analysis of the recovered stones, and of the remains
of the former bridge. A study of this peculiar aspect has been completed,
and has led to some drawings, which have to be considered the reference
sections towards the determination of the most likely dimensions for all
the architectural finishes. The analysis work of the recovered stones has
been performed either during Phase A either during phase B of the works,
and with the support of technical notes written during the performing of
the ancient restoration works it has been possible to reach a satisfactory
level of accuracy.
5.8
Geometry determination of the spandrels, cornices and parapets
For what concern the other
bridge elements geometry, like spandrels, cornices and parapets, it has
been mostly referred to the 1982 survey. But performing all the necessary
checks (like it has been done for all the work steps) some incoherences
have been found, as it has been already explained in the paragraph §1.2.2
of phase A report.
The most remarkable and
reliable checks have been performed trough the use of the photogrammetric
elaboration of the ancient pictures (refer to paragraph §5.3 of this
report).
All the differences found
have been documented in a sort of superposed representation drawing, on
which importance has been given also to the 1955 survey dimensions. The
above task has been performed in a cad system, where all the data have
been easily compared in the appropriate scale. Data pertaining to the
photogrammetric processing (rectified pictures) have been imported as
raster images and, in overlay, vectorial digital drawings have been
remounted and compared.
It should be underlined
that, for what concern the spandrel stone layout, inaccuracies have been
found mostly on vertical joints, while horizontal ones were correct: this
is probably due to the scarce visibility of details and of short and small
divisions.
Cornices dimensions
constituted a more difficult issue to handle since for small elements the
graphic sign (of the ancient drawings), shadows and low level of
visibility may have led to remarkable inaccuracies. Anyhow it seems that
the size and dimensions of cornices, even if similar, were not constant
and there was a difference among the upper and lower cornices of the
bridge elevations. This matter is strictly linked to the study of the
thickness of the mentioned cornices (see previous paragraph) and has been
better analysed during the phase B for the stone cut (refer to chapter 7
and to SCXX drawings). A procedure has been worked out to make those
elements as much close to the former ones and to manage at the same times
tolerances, joints and slightly variations among one and another.
The above process has
allowed a satisfactory level of accuracy, and what is most important, has
strongly contributed in the determination of the most likely geometry of
the bridge, stone by stone, trough a sharp analysis of the available data
and elaboration. The value of this work is independent from any other
future use of it, may not even be related to this project, because this is
an historical and research contribute for a monument that doesn’t exist
anymore and that is lost forever.
5.9 The
bridge curvature inquiry
Arch voussoirs laying one
next to the other are represented in the 1982 ancient surveys by a
division line that here next will be called "joint line". Joint
lines have different orientation that generally converge in one or more
points which are the arch centres. In the specific case of Mostar Bridge,
this orientation varies depending on irregularities due to craftsman
inaccuracies or due to the bridge settlements and deformations. It has
been verified that, for the following reasons:
 for the final bridge geometry
representation (specially of joint lines)
 for a better understanding of the
intrados shape
 for a better understanding of the
possible cinematic of the bridge
 for a better understanding of the
possible bridge original shape
it has been of use to work
out which are the centres of circles that best fit the points of the arch
curve and where groups of connection lines are oriented to. This
elaboration has produced important reference lines that outspreading form
the centres have been compared to the joint lines of the 1982
photogrammetric survey.
To perform the above task
it has been designed a purpose built software routine that having as input
data the X,Y coordinates of groups of points of the intrados line
(belonging to the final mediated curve), outputs as feedback the centre
coordinates and the radius of a circle that best fits the given points of
the arch.
This inquiry, as first
result, has given the possibility of drawing voussoirs joint lines with a
better control and has avoided the eventual graphical inaccuracies due to
the thickness of 1982 drawn lines.
The arch intrados curves,
of north and south side, have been also mathematically investigated using
the numeric values of all the joints coordinates. But being the curve so
much affected by local irregularities this task hasn’t given meaningful
results: a mathematical equation that could partially fit all the numeric
data may only be represented by a polynomial function of high exponential
level. This curve would be of very little use for the current study or for
the on site work, and anything that is related to these considerations has
been already underlined (see §1.9.1 of the Inception Report).
Going back to the above
mentioned inquiry, it has been found (by geometrical constructions
verifies) that the arch had clearly some discontinuities (cusps) and that
groups of voussoirs could be created being their joint line directions
converging in small areas. As it has been underlined, a purpose built
routine has been made to work out which were the circles and the related
centres that could best fit the groups of points of the stone arch joints.
The routine code is here next listed:
% Calcolo della circonferenza più probabile considerando gli scarti senza segno
clear
%Dati da inserire
a=[………];
b=[………];
passo=50;
Xmin=1330; Xmax=1600;
Ymin=350; Ymax=180;
Rmin=1360; Rmax=1560;
%Processo
intx=floor((XmaxXmin)/passo);
inty=floor((YmaxYmin)/passo);
intr=floor((RmaxRmin)/passo);
for i=0:intx;
Xc=Xmin+i*passo;
for j=0:inty;
Yc=Ymin+j*passo;
for k=0:intr;
Rc=Rmin+k*passo;
R=sqrt(((aXc).*(aXc))+((bYc).*(bYc)));
Er(i+1,j+1,k+1)=sqrt(sum(abs((R.*R)(Rc.*Rc))))/(length(R));
end
end
end
%Visione risultato
for i=0:intx;
for j=0:inty;
for k=0:intr;
U=(Er(i+1,j+1,k+1)>Er);
t=sum(sum(sum(abs(U))))
if t>0
no=1;
else
RC=Rmin+k*passo;
XC=Xmin+i*passo;
YC=Ymin+j*passo;
Scartomin=Er(i+1,j+1,k+1);
end
end
end
end
%Soluzioni: XC=ascissa centro, YC=ordinata centro, RC=raggio, Scartomin=Scarto medio
XC
YC
RC
Scartomin
The routine works by
running different nested loops with a chosen step in a predefined range
for three different variables: X value of the centre, Y value of the
centre and R radius of the circle. The routine tries, this way, many
circles and compare them with the matrix of the coordinates of the
intrados curve a[…] and b[…]; another cubic matrix (3d) is than
created, composed by all the differences resulted, the lowest combination
is the one that best fits all the data.
With the use of this
routine it has been possible to make many attempts that have contributed
to a better understanding of the geometry of the intrados curve; moreover
this research work may be useful during the bridge stone design, the
centering design, on site works and for any future evaluation about the
correct functioning of the arch stones due to their orientations: all this
will be better investigated during the design phase.
An interesting observation about the
results of these inquiries is that there seems to be a relationship among
the cusps of north and south elevations, because of the number, the
location and the distance (refer also to §5.6.9). One of the hypothesis,
(by Mr Pequeux), is that the cusps may have been originated by the joints
of the ancient wooden centering which had sectors that could not be longer
than the available wooden beams.
5.10
Graphic representation of the analysis and of the results
The graphic elaboration
have been detailed planned thinking of the future development of the work.
Graphics and drawings have been all performed in a digital cad system to
allow the following:
 numerical management of drawings with no
graphic inaccuracies
 precise diagnostic and measurements
 constant control and updating of the
designing stage
 longlasting, readytohand and
duplicable data for records
Moreover, as it as already
been explained (see §5.6 of this report), it has been carried out a sort
of relationship among graphic and numerical work trough which it has been
possible to perform verifies and checks, and the system was so strictly
linked that some of the results have been worked out trough the numerical
work and some others trough the graphic work, depending on the type of
researches. The drawings, (apart from all the photogrammetric graphic
outputs and elaboration), may be subdivided in two main categories:
 2d drawings of the bridge elevation
 3d drawings of the whole bridge plus
sections and details
The first group, containing
all the graphic information related to the bridge arch, has been managed
with only one file, and it is the one that is mostly linked to the
numerical analysis file: this choice, of course has led to a complex
organisation of the final charts and of the layers but it has all the
following advantages:
 all the graphic data coming from
different sources may be easily compared
 anything is referred to the same origin
 it can be visualised at the same time
raster and vector information
 any changing or work developing will
take place once and will not have to be repeated in other files with
the risk of making different representation of the same data (we
should remember that a numeric correction may be not graphically
visible)
As already explained in the
Inception report (see §2.5) the chart layout has been organised with many
dimensioning systems (including this way all the numeric data), and with
metric reference of different types, like cross points reference,
graduated lines and graduated goniometer to make the drawings exhaustive
and of easy interpretation.
The second group of
drawings has been carried out trough the use of a set of 3d seed drawings
(see next paragraph) from which have been assembled different cad files to
manage the archivolt and all the other bridge elements like spandrels,
parapets and cornices and for a better study of the joint among bridge and
its abutments.
All the graphic drawings
have been detailed analysed in this report in chapter 10 with a comment
and an explanation on each view.
5.11 The 3d
reference seed files and the bridge springers
The 3d digital drawings of
the bridge are one of the most important graphic elaboration for the
correct management of the delicate design phase. In no other way it is
possible to have a full control of coordinates and of the reliability of
the final result, this because the 3d drawings are the result of 3d
virtual digital model of the bridge and of its stones.
The strategy for managing
all the 3d information is different from the one adopted for the 2d
drawings but the aims are similar: to have the possibility of comparing
and managing all the available data at the same time. The 3d data files
have been divided into seed files or basic files that contain fundamental
and unmoveable objects like springers, topographic points and arch
intrados curves. Some of this 3d seed files are built trough the use of
portions of 2d drawings and remounted in 3d coordinates on the right
positions. All these 3d files have one common peculiarity that links them
one to each other: they are equally referenced in the space, and this
allow their loading one over the other mounting a global view of all the
elements. This different procedure has been followed for the 3d files
because some of the drawings and of the detailed views require large
computer resources, and not all of the drawings require to have all the
work loaded at the same time. Moreover the method of the seed files has
allowed the simultaneous working of more people on the same 3d model
producing different results that could be combined at the end.
The most important 3d file
is the one of the bridge springer position: this file has been already
introduced and described (see §2.6.1 of the Inception Report); here next
the description is enlarged and updated for some values that have been
changed after a more detailed evaluation.
fig.33  A sketch of the
main dimensions of the springers and of the bridge span
This is a very interesting
matter to be investigated, since it can tell us many things on the shape
of the intrados of the vault of about 4 metres thickness. Moreover without
this inquiry it is not even possible to compare north and south elevation.
The results are exposed in previous sketch (fig.33).
The springers have the
following z coordinates (referred to our origin station point):
north east: 3.692 north
west: 3.562 south west: 3.562 south east: 3.682
and referred to the 1982
origin about:
north east: 46.71 north
west: 46.84 south west: 46.84 south east: 46.72
So the differences are
really very low among springers of the same side, and this of course makes
it much easier in comparing the different curves of the north and south
elevation. What it has to be noted is that on the 1982 survey the north
west dimension is reported as 46.92, which is wrong, and that can be also
verified (apart from current instrumental surveys) by measuring the
graphic representation of the 1982 drawing. This way it seems that it was
only a mistake due to drawing editing.
