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|STONE CUT - REQUIREMENTS AND SPECIFICATIONS||
7.4 Stone cut procedure
Stone cut procedure is here globally referred to the work of preparation of the stones for the bridge construction, actually it could be correctly subdivided in different categories related to time and to types as here next explained:
The above classification terms will be used in following paragraphs to allow a better understanding of all the different issue which are often referable to more phases at the time. A preliminary analysis of the main peculiarities of the arch geometry analysis is here next exposed in order to gather the reasons of the technical approach.
7.5 Main irregularities of the archivolt - SMS-A
The most likely survey of the arch bridge, result of different ancient survey comparing, and called here survey 2000, has revealed all the slightly imperfections that were a proper characteristic of the monument. Geometrically speaking these variations may be resumed as follows:
The above listed irregularities have been moreover studied, by the use of numeric electronic charts and co-ordinates with special enquiries about range, highest and lowest values, areas in the arch where these anomalies were more present and by detail digital graphic comparing among north and south elevation. A more detailed comment about each type is here next reported.
7.5.1 Arch curvature irregularities
arch intrados and extrados curvatures have some local irregularities and discontinuities that make the arch of an irregular shape definable only trough co-ordinate points (for more details refer to geometry determination studies); moreover north elevation curvature and south elevation one do not follow same deflections, same height and do not even have the same length and span.
The relationship between north and south elevation curvature points is mainly linked to the relative position of the origins of the co-ordinate system of the curves; in other words, most of the observed anomalies, and their possible quantification is all based on the position of the springer level points in the space. The survey work has chosen, as working reference, both for south and north elevation the east side as the origin (coherently with the 1955 survey). In the three dimensional co-ordinate system the origin has been set on the north-east springer, and the south east has brought the following:
where the height is represented by the Y co-ordinate, and the thickness by the Z co-ordinate.
In the following figure it is shown a correct projection of north side over south one to have a better view of the differences that we have in the thickness of the archivolt.
fig.1 - north and south elevation correctly projected to underline differences
What it is important to observe is that, (apart from local deflections), from the springer level to the arch reins, and also further, there is a fairly good matching among north and south side: rows are almost regular and deviations are low. It is mostly in the central portion of the arch where north and south elevation do not match. It is probable that, during the original construction, they had some settling of the wooden centering by the time they were loading it (this could be the reason why we have highest differences on top); and probably they tried to correct the settling of the centering by using wedges under the stone voussoirs (traces of which have been noted by Mr. Bessac because of small steps among adjacent arch stones). All the above matters anyhow are better analysed in the chapter about the original design and construction method.
In order to complete this analysis of the main anomalies of the intrados curves of the archivolt it is important to point out that the position of north elevation and south elevation have been carefully analysed during the geometrical study (refer to the proper chapter for detailed notes) and have been determined to be parallel either for the 1982 survey results, either for a scarcely amount of other notes about the eventual thickness variations of the bridge, which anyhow was most likely to be quite regular for this peculiar matter.
Already from this preliminary overview it should be clear that the archivolt is not a regular solid figure and that, for the stone cut, a complex dimensioning system is required; this may be also confirmed by the fact that some of the above variations may reach about 20 centimetres for homologous points along an arch row. On the other side the requirements for a practical way of proceeding for the works has led to design a simplified system of dimensioning which will allow, with a small amount of dimensions, the performing of this irregular solid figure.
7.5.2 Front face shape variation
Front faces of the arch voussoirs (elevation side) are commonly of a regular trapezoidal shape and all alike; it is not at all the case of the Old Bridge of Mostar, where front faces of the voussoirs vary in shape, size, dimensions and angles: only an average size is maintained along the archivolt as shown in following figure.
fig.2 - type of variations in front faces of voussoirs
7.5.3 Variation of height above sea level
Each row of voussoirs, going from north to south elevation, varies each height above sea level, of a quantity which is in a range of cm 0-10 on about cm 395 of thickness. This irregularity is one of the most difficult to be managed during the SC-final (final stone cut) and SC-adjusting (stone finishing and adjusting) and requires to be analysed with observations held on the way the bridge was originally assembled (see next paragraphs).
The above mentioned irregularity may have been caused by a settling and a consequent lowering of the centering during the construction stage, and should not be confused with local deflections that may have been caused by arch springers settling or other similar happenings that have slightly changed the geometry of the arch in defined spots.
fig.3 - variation of height of a single row in the bridge thickness
7.5.4 Different raising of the rows
During the bridge construction, one of the parameter that was more difficult to be monitored was the raising of the rows. Arch rows of voussoir have been assembled in a way that they do not lie perfectly horizontal and parallel one to each other, and by the time that we get on top (key stone level) this imperfection increases progressively mostly on one side.
fig.4 - different raising of the arch voussoirs rows on north and south elevation (plan view of the arch extrados) This irregularity of the rows may be caused by the fact that during the assembling procedure was not so easy for a worker, standing over the centering, to gather the correct progression at the same level either form north and south side. Another explanation of this could have been that this irregularity easily increases by the time it is repeated due to ordinary construction inaccuracies of stone sizes (see chapter about hypothesis on the original construction method).
The range of this anomaly is the widest: cm 1-19 on about cm 395 of thickness.
7.5.5 Size and rotation of front face
Undoubtedly one of the most interesting findings of the geometrical observations has been the fact that this kind of irregularity, (size of front face variation), is extremely low compared to the others and we can say that voussoirs belonging to the same row were almost of the same intrados dimension from north to south and in the bridge thickness: the gradual shift is in a range of only cm 1-3.5 on about cm 395 of thickness, there are only a few cases around the key stone in which exceptionally cm 11 are reached.
fig.5 - intrados size variation in a single row in the bridge thickness and front face rotation
All the above reveals interesting notes about the construction technique used at the time: it is most likely in fact that one of the construction rule that was followed during the load bearing arch assembling was to choose for each row similar voussoir of same intrados length; so even if adjacent rows could be different due to the natural availability of tenelija each row was almost constant in its transversal thickness even if affected to a different raising (as explained in previous paragraph). And it is only because of the raising anomaly that at the key stone they were compelled to shape the last three rows gradually with a strong size variation (until cm 11 in only one row) in order to recover the raising inaccuracy and to close correctly the archivolt (see for more details chapter about hypothesis on the original construction method). In the next paragraphs it will be explained how all these observations will influence the SC-final and the SC-adjusting procedures.
Rotation of front face is also very low, but this irregularity is much more difficult to be managed during the stone cut procedure and assembling, and even small values may lead to very high incoherence. Rotation of front faces in the thickness of the bridge archivolt is mostly a matter related to the intrados surface (extrados had more construction imperfections and was not even smoothed), this is why rotation is analysed in the following paragraph as a deviation from the row plan.
7.5.6 Deviation from row plan
The geometrical analysis held on the ancient surveys (see related paragraphs) was mostly aimed at finding all the co-ordinates of the connections joints of the arch curves on both north and south elevations. Once the results of this inquiry have been related in 3d co-ordinates each intrados row was delimited by four points: two of a voussoir of north elevation, and the other two of the homologous voussoir belonging to the south elevation. But from geometrical rules we know that one plan is defined by three different points in the space, and four points may be not lie together on the same plan. So the model of the archivolt had to be worked out facing this irregularity mostly for what concern the intrados surface. Numerically it has been monitored for each intrados row plan how far it was the fourth point from a plan that was perfectly matching three of the four points. This anomaly has been defined as "deviation from row plan" and represents the non planarity of every single intrados row, the range found is from cm 0 to cm 3.5 out of cm 395 of thickness, luckily in most of the cases this anomaly is lower than cm 1. And even if this type of monitoring gives the error to one point only, we should not forget that even small values of this parameter may lead to a rotation of the stone face that if not managed and planned will cause a very bad matching of adjacent rows.
fig.6 - deviation from intrados row plan (equal to rotation of front faces)
7.6 Rough stone cut - (SC-rough)
After the quarry of blocks (QY-blocks) an intermediate cut is required in order to have smaller blocks: the closest as possible to the final ones. This cut is here next called "rough stone cut" (SC-rough) as before defined. As already pointed out, in the warnings of this chapter, it has been agreed with PCU TA that a minimum tolerance of cm 3 was required in order to have a margin, large enough, to perform the final stone cut (SC-final) and the possibility of adjusting and finishing the surfaces (SF-adjusting).
It has to be pointed out clearly that the quarrying tolerance (qt) has to be considered as an additional quantity to be added for each side of a length, which means that globally the examined length will get six centimetres longer than its final size once the stone cut (SC-final) has been performed.
Nevertheless, after a preliminary check, it was clear that, as far as the arch stones (SMS-A) were concerned, it wouldn't have been possible to work out the rough stones dimensions just by adding the quarrying tolerance to the final measures of the stone faces and thickness. this mainly for two reasons:
1. Rough stone cut is meant to be an easy cut in parallelepiped blocks with no irregularities and variations which are proper of the stone final cut.
2. Irregularities of height and raising in the rows actually require wider rough block dimensions in order to contained wholly final blocks with the predefined quarrying tolerance.
Of course all the above is referred to the arch stones (SMS-A) and not to the bridge elements (SMS-B), where geometry is much easier and not linked to the vault shape.
Figure n°7 of current page may give an easier explanation of the above mentioned peculiarity:
section A of fig. 7 represents: plan view (bottom) and front view (top) of a block (coloured in grey) in its final shape (after SC-final) which has got a perfectly regular shape. In this hypothetical case no problem arise to determine the shape of the block for the rough cut (SC-rough), which is drawn, as an outer profile, of the previous one, respecting the tolerance (qt).
fig.7 - regular and irregular shaped stone compared with the related rough block (SC-rough) section B of fig. 7 represents: plan view (bottom) and front view (top) of a block (coloured in grey) in its final shape (after SC-final) which has got an irregular shape due to different heights and raising of the row. In this case it is clear, from the shape of the profile, that the rough block has to be much wider compared to the dimensions of the final block (coloured in grey).
section C of fig. 7 represents: the same of section B but with a different profile of the rough stone block which is oriented in the direction of the average axe of the final block. This way the rough block, being always a parallelepiped may minimise either the use of stone either the stone final cut work (SC-final).
section D&E of fig. 7 represents: this two schemes are respectively referred to section B and C and they point out, in an enlarged plan view, the difference of the two approaches in matter of optimisation. Stripes are oriented as the natural bedding of the lime stone and their gradient is compared to the acting forces (arrow on the left). Of course best gradient would be for the one represented in section D (=section B).
Conclusion of all the above notes is that the procedure to work out dimensions of parallelepiped blocks (SC-rough) with the planned tolerance is not short and may be not only based on the final dimensions of the final stones (SC-final).
In other words shape and volume are parameters that are strictly related in this section of the design work, since an irregular shape, which has deviations in all the directions, requires a larger volume, to be contained in it with a planned margin, than a regular shaped parallelepiped (for which it is enough to increase its dimensions of the desired "qt"). Being the irregularities of the arch stones (SMS-A) in a wide range that may reach 10-20 centimetres, it is clear that, even if distributed on three or more stones assembled in the same row, this quantity may get higher than the "qt" with serious consequences and the impossibility of working the final shaped stone out of the rough block.
7.6.1 Design procedure for rough stone cut
Design procedure to determine dimensions of the parallelepiped blocks of the stone rough (SC-rough) has been performed following different steps to better evaluate the most desirable choice to adopt.
First all the dimensions of the stone cut (SC-final) of every single row have been examined and compared in order to get the widest ones for the stone faces. For every single block, this way, it has been determined the rectangular face shape by an X and Y measure (maximum found in the row plus two times the quarrying tolerance). Thickness has been examined for every single block depending on stone cut-final measure and by adding two times the quarrying tolerance. All this has brought to a preliminary calculation of the required volume of rough stone blocks even if it wouldn't have been enough to contain all the final blocks (as before explained): this volume was about 190 cubic metres (for SMS-A only).
Subsequently all the co-ordinates of every single point of every single final block have been calculated in order to manage the design by stone and not by rows (approach by rows was not enough anymore for optimisation reasons). This calculation has been performed trough mathematical calculations using as input data either the co-ordinates of north and south faces, either the thickness dimensions of the different block in a row (previously worked out with ancient surveys analysis). Co-ordinates of a point that subdivides a segment with a predefined ratio are here next reported:
Where x1,y1,z1 are the 3d co-ordinates of a point on the north side, and x2,y2,z2 are the 3d co-ordinates of an homologous point on the south side; k is the thickness of the stone (procedure has to be repeated as many times as the number of arch stones in the examined row and for every front face point). Co-ordinate has been worked out with calculation because no ancient survey data was available for this purpose. This work has led to more than 10.000 new data.
The following step has been to analyse every single stone by means of co-ordinates: all the co-ordinates were referred to a reference system which was oriented coherently to the bridge elevation. For every stone all the co-ordinates have been considered and it has been assumed the following:
X rough parallelepiped dimension = Xmax-Xmin+2qt;
Y rough parallelepiped dimension = Ymax-Ymin+2qt;
Z rough parallelepiped dimension = Zmax-Zmin+2qt.
The result of this procedure has brought to the scheme represented in fig.7 (sections B&D) of previous page: rough blocks were big enough to contain the final shape of the stones but were not optimised for construction material saving. The difference was so remarkable that the total volume of rough blocks, worked out this way, was +70% wider than the one previously calculated.
Since optimisation of stone was required and methodology of stone cut would have been atypical compared to the ancient methodologies and to any other ever adopted, design of stone cut has been performed following another mathematical and geometrical device for calculation of rough stone block.
The final geometrical approach is the one represented in fig.7 (sections C&E) of previous page: the axe of the SC-rough parallelepiped has been oriented along the average axe of the SC-final block in order to minimise the quantity of wasted stone locating it mostly at the short edges of the block. This calculation has required the use of a purpose built software routine, (see next paragraph), that, trough rotations of a temporary reference system for each stone, has worked out all the required measures in an automatic process.
The above system has given, as total amount of volume for the rough stones, about 202 cubic metres, which is quite an acceptable value being about +7% compared with the first estimation (that was moreover inadequate to cover the whole vault in many cases). Moreover this design system will allow an easier procedure during the other stages of stone cut with following facilities: smaller quantity of stone to be cut for the SC-final (hand cut), more similar methodological procedure to the ancient followed one. The only disadvantage which is foreseen is that stone bedding will have a small gradient compared to the direction of the acting forces (see fig.7 section E); this gradient will depend mostly on the different raising of the rows in the arch and will be present only in the top portions of the vault: it has been evaluated, (after agreement with PCU TA and stone expert), that it is negligible being not so wide and not a risk for the durability and efficiency of the masonry.
7.6.2 Purpose built software routine for rough stone cut
As explained in previous paragraph, it has been necessary to process all the co-ordinate data of the arch stones in their final design position in order to work out, with a mathematical routine, the dimensions of the rough blocks. This routine works as here next explained:
Each stone has been defined by its eight points in 3d co-ordinates, and a new reference system has been defined to match the average axe of the examined block. This new reference system has been rotated three times (on X, Y and Z) in order to be as closest as possible to the spatial orientation of the blocks. Co-ordinates of the block have been transformed in the new reference system and the dimensions of the rough parallelepiped block as been determined as it has been done in the previous calculation:
X rough parallelepiped dimension = Xmax-Xmin+2qt;
Y rough parallelepiped dimension = Ymax-Ymin+2qt;
Z rough parallelepiped dimension = Zmax-Zmin+2qt.
(The software routine is here not reported)The above described procedure has been determined after many different approaches and tentative; the process is not short and not even of easy management due to the amount of data and compared to the ordinary architectural design procedures, but it is the best that could be carried out to optimise stone wasting and to guarantee dimensions wide enough to face all the geometrical anomalies.
7.6.3 Conclusion about the results of the rough stone cut
Rough stone cut (SC-rough) has given, as final result, three dimensions for each voussoir that represent a parallelepiped block wide enough to contain the final shape of the designed arch stone with a minimum margin of three centimetres. This way it will be possible to perform rough cut easily with no care for any deviation or imperfections and with the use of mechanical instruments for stone cut. A more accurate work during the design stage has given the possibility of an easier process on site.
The above described procedure was required for what concern all the arch stones (SMS-A), all the other bridge elements (SMS-B) had absolutely no similar difficulties and it was possible to determine the SC-rough dimensions by choosing the highest dimensions for each stone block and by adding two times (one per side) the quarrying tolerance.
fig.8 arch voussoir after final cut and related rough cut block with bedding and code dimensions used in final charts
CREDITS:Intellectual property of this report and of the design drawings is owned by General Engineering s.r.l.
author of the text: arch. Manfredo Romeo other contributes have been mentioned in related paragraphs
drawings: arch. Paola Marrone arch. Alessio Talarico - sketches: arch. Manfredo Romeo
database management: arch. Manfredo Romeo - arch. Giovanni Anzani
software development and calculation: ing. Francesco Cenni - ing. Niccolò Baldanzini
yard consultant: Bruno Bonuccelli
© - General Engineering Workgroup -
Final Design Report
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