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Castel del Monte |
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Castel del Monte → Elevation Design → Tas-de-Charge and Rib Support System Analysis |
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Note: Figures referenced in text are located in a separate file. Follow link to see figures as slides in a separate browser tab:  Tas-de-Charge Study Slides
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1. Introduction |
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Part of the elevation design is the vault itself. At the height of the Middle Ages, the construction of ribbed cross vault was an established practice, a mechanical procedure based on methodology rather than design. The process was to erect cross vaults using diagonal and transverse ribs as foundation, see the “Cross Vault Technology” section for a detailed description. The resulting cross vault with its four webs have characteristics and dimensions that depend on the accurateness of the ribs construction, which takes place high above the floor, on top of the column support system. The fact that a rib is made up of two different structural sections makes the erection of the ribs more complex and the precision of the rib form more challenging. The complete rib system includes tas-de-charges and free rib voussoirs.
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2. Tas-de-Charge and Ribs |
The tas-de-charge is a section of the vault ribbed system in Gothic architecture. It is the part where the diagonal and transverse ribs merge into a funnel-shaped structure that terminates on a support system, Fig. 1. A square cross vault requires four pillars for supporting the ribs, which in the case of Castel del Monte are the columns , one at each corner of the vault base square. Consequently four tas-de-charges are present in the ribbed square cross vault.
[The support system is composed of a semi-column attached to the wall, which makes the wall a functional part of the support.]The half-conical form of the tas-de-charge at Castel del Monte has a rotational symmetry that matches the semi-circular perimeter of the supporting elements, semi-columns with semi-octagonal capitals and semi-octagonal bases, Fig. 2. The face of the tas-de-charge is shaped as a set of ribs that start as ribs completely fused together at the base and then start separating until they are free of each other, at the top of the tas-de-charge where the freed rib section starts. The rib arches are made of stone blocks mounted in a single circular file, Fig. 3, 4 and 5. The stone blocks are special voussoirs, curved lengthwise but with the same cross sectional form and dimension and the same radial curvature. These voussoirs can be of different length but their end faces are normal to the lengthwise curvature, Fig. 6. The similarity of the curved voussoirs is such that they can be mounted at any location along the rib arc. The exception, of course, is the keystone at the pinnacle of the rib. The full rib is composed of a central section made of a row of free rib voussoirs and two side sections that are the tas-de-charges. Therefore, sculptured in the tas-de-charge for a square cross vault are the forms of two transverse ribs at right angle to each other and a diagonal rib evenly spaced in between them, Fig. 1. The ribs have the same form and dimension on the respective floors at Castel del Monte. The ribs on the upper floor have a more ornate form, Fig. 6. The diagonal ribs have a slimmer profile compared to the transverse ribs on both floor, Fig. 3, 4 and 5. The transverse ribs that separate the square cross vault from the side vaults are completely free and visible. The other two transverse ribs, respectively on the façade and on the courtyard sides, are partially embedded into the wall stonework; only part of these ribs is free and visible. |
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3. Fake Ribs |
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There are side vaults on each side of the square cross vault at Castel del Monte, because of the trapezoidal shape of the rooms. Accordingly, the façade and courtyard walls extend into the side vault areas; there are no walls separating the side vaults from the square cross vault. Accordingly the columns, which have the purpose to serve the ribs within the square area of the cross vault, physically take some space in the side vault areas. The columns actually have a semi-circular envelope with the flat back seemingly plastered to the wall and the center of their back plane set exactly at the corners of the cross vault base square. The tas-de-charge take on the semi-circular form of the columns as mentioned above and they too spread into the side vault areas, Fig. 1.
The ribs are part of the square cross vault area of the room; the side vaults do not have any ribs. Consequently the section of the tas-de-charge in correspondence of the side vaults does not require sculptured curved ribs. Nevertheless, the body of the tas-de-charge on this side is also sculptured with the form of ribs that give the tas-de-charge a rotational continuity in the forms along its perimeter at the base. These are however straight ribs that rise vertically to the top of the tas-de-charge, gradually disappearing into the curving surface of the side vaults, Fig. 7. These are fake ribs that have no structural function, but serve to fill a space, providing some symmetry around the semi-conical form of the tas-de-charge. |
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4. Tas-de-Charge Characteristics |
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Physically the tas-de-charge is made of a layered stack of flat stones gradually extending in front of the wall, thus creating the semi-conical form as the tas-de-charge rises. The stonework has effectively the structural function of corbels. The stone layers are often referred to as drums and are typically single stone pieces, especially in the bottom area of the tas-de-charge where the drums are smaller. The visible sides of the drums are sculptured in the shape of ribs with a curvature that matches the circular form of the ribs.
A peculiar feature that separates the sculptured ribs of the tas-de-charge from the rest of the ribs is that the grout line between the various rib segments, Fig. 8 and 9. The grout lines in the tas-de-charge sections are horizontal; they are not normal to the curvature of the rib arc as it is the case along the central segment of the rib where the rib voussoirs are free. The exception is the top drum of the tas-de-charge. The top surface of the sculptured rib portion of the top drum is cut downward, at a downward angle just like the free rib voussoirs, Fig. 8, thus providing a matching surface of the proper angle for the first free rib voussoir. These features are readily visible on the ground floor of Castel del Monte where the tas-de-charges are mostly in good shape, Fig. 9. The tas-de-charges on the ground floor are made of five stone layers, likely five stone drums, with four horizontal grout lines in-between them. The top drum is recognized by the grout line that changes from horizontal at the bottom surface of the drum to a downward angle at the top of the drum. This feature of the grout line is not distinguishable on the upper floor, Fig. 10, because most of the tas-de-charges are defaced or gone altogether in a few cases, with the exception of room 7 where they are in a good state of preservation. |
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5. Structural Bonding to the Wall |
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The half-conical form of the tas-de-charge would have a flat back-surface where it is attached to the wall. However, the tas de-charge is not just plastered to the wall; the stone drums extend into the wall, thus providing a structural bond between these elements, Fig. 8. Besides being buried and not visible, the back-surface is neither a physical stone feature of the tas -de-charge; it is just a geometric design reference.
The bonding to the wall extends to the column support system. The back of the semi-columns and semi-octagonal capitals lines up with the finished surface of the walls, Fig. 8; their stonework also extends into the wall, thus providing a similar structural bond. Accordingly, the back-surface of the semi-columns system (base, shaft, and capital) is also a geometric reference, not an identifiable physical stone feature. Notable on both floors is that the stone-work of the columns has ear-like extensions along the intended wall finished surface, Fig. 11. The ear-like extensions were intended to provide a butting edge for the finishing wall panels, but also show how the column stone-work extends into the wall itself. |
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6. Alignment |
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The back plane of the tas-de-charge and column support system lines up with the finished surface of the wall, not the stone surface, Fig. 12. The room dimensions and the important diagonal of the cross vault base square that also sets the height of the room at the keystone and the radius of the rib arcs are referenced to the finished surfaces of the walls.
The tas-de-charge and the column are aligned vertically; the geometric alignment is along the centerline on the back plane of both the semi-column system and the semi-conical form of the tas-de-charge, Fig. 12. This vertical center-line is again a geometric reference, not an identifiable masonry feature. I is located on the finished surface of the walls at the corners of the square cross vault. This center-line would be the geometric marker for the corners of a square room if there were no side-vaults. The back-plane centerline crosses the top surface of the column capital at the center of the back border. This is where the arcs of the cross vault ribs geometrically merge into a single point, at the corners of the cross vault square, on top of the column capitals. Of course, the rib arcs are not just lines; the ribs are made of stone blocks. The rib arcs that merge geometrically are at the center of the extrados of the ribs. This geometry then leads to the next item, which is the footprint of the tas-de-charge. |
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7. Tas-de-Charge Footprint |
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Merging on top of the column capital are not just geometric arcs but ribs made of actual stone blocks. Since stone voussoirs cannot physically merge into each other, the rib forms are sculptured into a common stone block; this is the pragmatic reason for the tas-de-charge. However, the sculpturing has to start at the base, with the tas-de-charge footprint. Geometry is the tool that serves to define the footprint form and dimension. Rectangles representing the cross sectional areas of individual ribs are drawn in a common drawing, aligned along the directions of the ribs, Fig. 13.
Merging in a quadrant at the base of the tas-de-charge are two transverse ribs and a diagonal rib in-between them; on the adjacent quadrant is a fake transverse rib and a fake diagonal rib. A rectangle is positioned for each rib so that the center of one of its sides, the one that correspond to the extrados, is at the geometric merging point of the rib arcs, Fig. 13. The rectangles overlap each other at the side close to this central merging point, but are free at the opposing sides. The footprint of the tas-de-charge is the outside perimeter formed by the overlapping rectangles. The stonecutters would start sculpturing the rib profiles from this perimetric boundary, curving forward with the correct curvature in the case of the actual ribs; the fake ribs are sculptured along straight vertical lines. The tas-de-charge footprint is wide laterally two measures of the transverse rib rectangles, and deep one length of the transverse rib rectangle. Not surprisingly, the tas-de-charge footprint profile fits within the form of a semi-octagon, a match for the semi-octagonal form of the column capital. The matching is a structural requirement for a proper stonework support of the tas-de-charge by the column capital. |
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8. Dimension of the Rib Cross Section |
The actual cross-sectional profile of the ribs on the ground floor is essentially that of a vertical rectangle with beveled corners , Fig. 2 and 4. Schirmer indicates that the depth of the ribs cross-sectional rectangle is 0.50 m for the diagonal ribs and 0.52 m for the transverse ribs on the ground floor (Abb. 27, pg. 39), Fig. 14. Schirmer also indicates that the capital and base of the columns have a semi-octagonal form extending 1.024 m from side-to-side against the wall (Abb. 20, pg. 27), Fig. 15. As delineated above, the lateral extension of the column capital matches the length of two transverse ribs side by side. These measures point therefore to a common design measure for the cross-sectional depth of the ribs, which is 0.51 m for both diagonal and transverse ribs. The ribs on the upper floor, which have a larger curvature radius, have likely similar dimensions for the rectangular frame of their cross-section.
[The diagonal ribs on the upper floor have an ornate circular face on the bottom portion of the rectangular cross-sectional frame.]Missing for this analysis is a measure for the width of the rib cross-sectional rectangle. It is clear from a visual observation that the diagonal ribs have a smaller cross-sectional width than the transverse ribs, Fig. 3, 4 and 5. A measure for the missing data on the width of the ribs is theorized based on a geometric model with a tie to the established depth of the rib, and consistent with visual observation on the width of these ribs. The form of the tas-de-charge footprint is constructed by overlapping rib cross-sectional rectangles, Fig. 16. Half the measure of the lateral width of the column capital, 0.51 m, is assigned to the depth of the ribs, which is the length of the rib rectangular frame, as indicated above. It is posited that the width of the transverse rib rectangle might be the width of the full side of the semi-octagonal perimeter of the column capital, which is 0.43 m. The two transverse rib rectangles at right angle to each other overlap close to the geometric merging center, Fig. 16. The overlapping area is a square with a measure of the side that is half the width of the transverse rib rectangle. It can be shown from geometry that the diagonal of this square has the measure of the sides of the equilateral triangles (sides ”a” of triangle “T” in Fig. 16) formed at the corners of a square that circumscribes an octagon and shares four of its sides. This is the octagon-squared figure that has been identified as the model that the medieval designers used to define the measure of the semi-octagon for the column base and capital and involve the Castel-del-Monte foot used as a unit of measurement. The diagonal of this overlapping square has therefore the measure of exactly one CdM-foot, 0.302 m (the measure “a” of the equilateral triangle “T”). The diagonal rib rectangle has the same depth as the transverse rib, 0.51 m, spanning from the geometric merging center of the rib arcs to the side of the capital semi-octagon, along the line bisecting the 90° angle between the transverse ribs. The width of the diagonal rib rectangle can be set geometrically as the diagonal of the square formed by the overlapping transverse ribs, Fig. 16. The width of the diagonal rib rectangle would then be exactly one CdM-foot, 0.302 m. This makes the width of the diagonal rib 70% the width of the transverse rib. This difference contributes to the separation of the ribs in the form of a notch, which starts at the footprint of the tas-de-charge. Based on the posited measures for the width of the rib rectangles, this notch between the transverse and diagonal ribs is deep 6.5 cm at the footprint of the tas-de-charge . [The notch depth of 6.5 cm is measured from the corner of the diagonal rib rectangle; the actual ribs have beveled corners.]Such choices for the width measures for the rib cross-sectional rectangular frames are very plausible. They are consistent with visual observation and are grounded on geometry involving the linear unit of measurement at Castel del Monte. This follows the geometric theme in the design at Castel del Monte, and was likely the only “scientific” and rational basis that the medieval architects had to guide their creative planning. |
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9. Height of the Tas-de-Charge |
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A final item of interest is the height of the tas-de-charge. It is a key dimension to achieve the correct round arcs. It is a key measure for the stonecutters preparing the stone elements of the tas-de-charge. It is also a critical measurement for coordination with the portion of the rib arc made of free voussoirs. The planners of Castel del Monte would have likely determined this height from geometric consideration.
The tas-de-charge would have to rise to the point where the merged ribs sculptured on the side of the drums are completely free from each other and can continue as separate stonework. This free-point is reached when the curve at the extrados of the ribs reaches a specific distance from the vertical centerline at the back plane of the tas-de-charge. This specific distance is the measure of rib cross-sectional depth, Fig. 15. In the simple case where all the ribs have the same width as the side of the capital semi-octagon, the rib free point is reached when the form of the semi-octagonal footprint is replicated at the extrados, at the top of the tas-de-charge. This free-point is calculated to be 2.09 m at the extrados and 1.85 m at the intrados for ribs that are 0.51 m deep, Fig. 17. Because the diagonal ribs are slimmer than the transverse ribs, a notch opens up between ribs at the footprint, and the rib free point would be somewhat lower. For the rib widths theorized above, the notch is deep 6.5 cm; the rib free point is 1.96 m at the extrados and 1.77 m at the intrados. |
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10. Complexity in the Design and Assembly of Ribs |
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The tas-de-charge with a height of 2.09 m at the extrados on the ground floor covers a portion of the rib arc subtending an angle of 27.5° along the diagonal rib; the tas-de-charge with a height of 1.96 m at the extrados subtends instead an angle of 25.6°. The rest of the diagonal rib semi-circle that is made of free voussoirs subtends the remaining angle, which is 180° minus the angles covered by two tas-de-charges, 125.0° and 128.8° respectively. On the ground floor, the portion of the vault height at the keystone made by the free rib voussoirs of the diagonal ribs is theorized to be 2.44 m for a tas-de-charge height at the extrados of 2.09 m, and 2.57 m for a tas-de-charge height of only 1.96 m.
The height of the rib portion made by the free voussoir is complementary to the height of the tas-de-charge. Coordinating these heights is critical to achieve a perfect semi-circle. Small discrepancies in the measures of the tas-de-charge height versus the arc of free voussoirs can create slight inflections in the curvature where the free rib arc rests against the tas-de-charges, resulting in a diagonal rib profile that diverges from a perfect semi-circle. This criticality is heightened further by the different construction methods. The tas-de-charge is assembled more simply by overlaying horizontal flat stone drums. The arc of free ribs requires a complex centering frame to be raised to the correct height in-between the corbelling forms of the tas-de-charges. The centering frame includes two circular crossing frames for the diagonal ribs and the shorter and pointed frames for the transverse ribs that converge but do not join at the tas-de-charges. While imperfections in the curvature of the ribs may be so slight to be imperceptible, issues with the complex erection of the ribs likely results into two possible outcomes. One is the height of the vault that may deviate from the intended design. Another is defacing of the ribs, especially in the tas-de-charge area, Fig. 10. The defacing is likely caused, at last in part, by excessive shearing stress created at the edge of the rib forms due changes in the curvature of rib segments. |
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11. Conclusions |
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Stone, cut and sculptured in different forms and sizes depending on the application, was the main construction material for permanent edifices in ancient times. The tas-de-charge was the inevitable solutions to resolve the stone construction where the ribs converge in the ribbed cross vault style of Gothic architecture. The full rib construction is therefore made up of tas-de-charges and free rib sections; the tas-de-charge covers about 45% of the vault height on the ground floor of Castel del Monte. Assembling such complex structures involved completely different stone components and erection techniques. Horizontal stone drums of increasing sizes are stacked vertically to make up the tas-de-charge, with ribs sculptured on the side. The drums and the whole tas-de-charge are discernible by the horizontal grout lines between the drums. Curved voussoirs make up the sections of ribs freed from the tas-de-charges and require elaborate supporting frames for their erection. The diversity of stone forms and erection processes presented challenges in coordinating the material and erection process for the medieval builders and the resulting construction. The most significant result of this is the frequent defacing of the ribs noted nowadays especially with the tas-de-charge on the upper floor.
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12. Documentation Notes |
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File: “cdm-154-130=tas-de-charge-study-en”
Castel del Monte — Study of the Elevations Tas-de-Charge and the Rib Support System: Analysis: “-154-130=tas-de-charge-study-en” Figures: “cdm-154-130=tas-de-charge-figs-en” Room and Vault Heights: Data Analysis: “-152-130=elev-measurement-analysis-en” Figures: “cdm-152-130=elev-figs-en” Findings: “cdm-152-130=elev-study-findings-en” Domenico Lanera domenico.lanera@gmail.com www.lanera.com January 2021 |
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