—A Superadobe dome is a parameterization in the sense that given some initial parameters, the structure can be completely determined by them [1]. While a set of such initial parameters can produce a safe and functional structure, a diff erent set can result in an unsafe , unstable, or even geometrically senseless structure. It is necessary for a Superadobe designer to predict the structure’s final dimensions and mechanical behavior from the initia l parameters of his or her choice, so as to be able to choose them wisely. These initial independent numbers must satisfy some equations containing variables of both geometrical and mechanical nature, yet the parameters’ compliance with the single, purely -geometrical equation showcased in this article, is a necessary condition f or all functional Superadobe domes. Keywords — Superadobe , Sand -bags, Self -construction, eco -friendly, emergency shelter, parametric analysis Figure 1. Section of a Superadobe dome with adequate set of initial parameter (𝒔𝒘, 𝒉𝒔,𝑬𝒙𝒕𝒇 , 𝒓𝒃, 𝒓𝒓 𝒂𝒏𝒅 𝒕𝒓 resulting in a geometrically sound structure Rewriting (1) we have: 𝑭=1−(√(𝑡𝑟+𝑟𝑟−𝑟𝑏)2+2(√𝑟𝑟2−(𝑡𝑟+𝑟𝑟−𝑟𝑏)2)(ℎ𝑠)−ℎ𝑠2+𝑟𝑏−𝑟𝑟−𝑡𝑟) 𝑠𝑤−𝐸𝑥𝑡𝑓 (1’) For some combinations of values ( 𝒔𝒘, 𝒉𝒔,𝑬𝒙𝒕𝒇 , 𝒓𝒃, 𝒓𝒓 𝐚𝐧𝐝 𝒕𝒓 ), F can result in a very small number, or even a negative one, and the respective dome would be geometrically ‘ unsound ’: Figure 2. Section of a Superadobe dome with inadequate set of initial parameters 𝐬𝐰, 𝐡𝐬,𝐄𝐱𝐭𝐟 , 𝐫𝐛, 𝐫𝐫 𝒂𝒏𝒅 𝐭𝐫 resulting in a geometrically unsound structure The previous figures illustrate that the set 𝒔𝒘, 𝒉𝒔,𝑬𝒙𝒕𝒇 , 𝒓𝒃, 𝒓𝒓 𝐚𝐧𝐝 𝒕𝒓 must be correctly set in the design of a dome . 3 Applications of a “geometrical soundness equation” for Superadobe domes ETSEM Polytechnic University of Madrid Applications of Equation (1): Equation (1) depends upon a comprehensive set of Superadobe dome initial geometric parameters , and relates them with a plausible indicator of behavior and stability F, geometrically observable as well . Hence , it alone constitutes a potent filter that will rule out inadequate designs. When all variables save one in (1 ’) are defined, this equation can be solved implicitly. A simple code, which solves the implicit equation resulting from defining all parameters in (1) , except for one, was written in Python programming Language. This allows us to solve the equation for any variable, not only for F, but also for parameters such as 𝑡𝑟,𝑟𝑟 𝑜𝑟 𝑟𝑏, which cannot be isolated, and for which only implicit functions exist in subdomains determined by (1). With the aid of this tool, even F, which is the observable, isolated func tion in (1 ’), can be treated as another independent parameter . Examples of these cases will be discussed in the upcoming section. Example 1 : What is the minimum degree of contact between rows that occurs in a conventional Superadobe dome ? If a Superado be structure has geometric conventional initial parameters: 𝒓𝒓=𝟓.𝟐𝒎,𝒓𝒃=𝟐.𝟒𝒎,𝒔𝒘=𝟎.𝟒𝒎 , satisfying equations (3,4,5 ) in [1] , and hs=0.2m , 𝑬𝒙𝒕𝒃 =𝟎.𝟏𝒎, and we set the radius of opening at Max height H, to be for example ,𝟎.𝟓𝒎, then we can simply calculate F , using (1’), or equivalently solve for F in (1), discovering the minimum length of contact between the transversal sections of any two consecutive blocks for this design. In the following figure, we see the program interfac e that solves the implicit equation (1) for parameter F. The result (0.234), means that if we take the conventional Superadobe sack, base radius and roof radius values, we have a minimum fraction of contact between transversal sack sections of 0.234 times 0.3m, (roughly 7 cm of contact between the top sections), up to H(0.5m) =4.019m of height . The program solves the equation behind the scenes, and the code for this program is attached in the appendix. Figure 3. Console Interface for solving implicitly eq (1) for F Observation: F in the console is represented by ‘ frac’. As a second example of the many applications of (1), we can test the maximum height up to which a conventional dome is geometrically stable. Example 2 : What is the maximum height of a given geometrically stable dome with set parameters ? (This is proportional the maximum travel of the external compass arch [1], marked in red in the figure ) Figure 4. Travel of the external compass If for example a Superadobe structure has parameters 𝒓𝒓,𝒔𝒘 𝒂𝒏𝒅 𝒓𝒃 satisfying equations ( 3,4,5 ) of [1] : hs=0.2m , 𝒓𝒓= 𝟓.𝟐𝒎, 𝒓𝒃=𝟐.𝟒𝒎, 𝒔𝒘=𝟎.𝟒𝒎, 𝑬𝒙𝒕𝒃 =𝟎.𝟏𝒎, and we set a condition for stability that minimum f raction of contact between top rows, 𝑭= 𝟎.𝟐𝟓, then we can solve (1) for Tr, obtaining a top opening (skylight) radius of 0.541m , and we can know the building’s height by calculating the corresponding H when Tr=0.541 m. [1], equation (3’) Since 𝑯(𝑻𝒓)=√𝑟𝑟2−(𝑇𝑟+𝑟𝑟−𝑟𝑏)2 [1], then H (0.541 ) = 3.98m . The geometrically stable building with this character istics will not surpass 3.98+0.2 m in height. The point of the arch which’s distance from the y axis is 0.541 m is 3.98m high, and does not coincide with the building’s top sack section midpoin t (by a measure of less than 0.2 m), becaus e the resulting height is not divisible by hs. The following is an illustration of what was just said: Figure 5. Verification of example 2 : Max height for a Conventional Superadobe Structure, geometrically stable to a degree of 0.25 * 0.3m (The top sack surpasses the limit) 5 Applications of a “geometrical soundness equation” for Superadobe domes ETSEM Polytechnic University of Madrid According to the illustration above, and as predicted by equation (1), a sack’s sect ion midpoint above 3.98m, will have a shared contact with the previous section of just less than 0.25*0.3m=0.075m , (0.059m) , and a sack’s section midpoint up to a height of 3.98m , will have a shared contact with the previous section of at least 0.25*0.3m=0 .075m , (0.08571m). Observation s:  Equation (1) optimizes a dome structure: It sets the parameters ( 𝒔𝒘,𝒓𝒓,𝑻𝒓 ) at a minimum stable value when the equation is solved for them. When 𝒔𝒘 is minimized, the structure is optimal in economy and weight, when 𝒓𝒓 is minimized, the structure is the lowest possible stable structure (also optimal in economy and weight), and optimal in response to stresses from wind load and also excess mass coming from excess height and a steep roof radius. When 𝑻𝒓 is minimized, the building will be as high as possible within this minimal 𝒓𝒓.  Lastly, equation (1) maximizes 𝒓𝒃 ,𝑬𝒙𝒕𝒃 , and hs when the equation is solved for them. When 𝒓𝒃 is maximized, t he resulting building is geometrically stable with maximum possible base area. When 𝑬𝒙𝒕𝒃 is maximized, the bulk s of the lateral extremes of the sack’s transversal section are allowed to be as great as possible while the building maintains stablility , and when hs is maximized, the sack is allowed to be as high as possible while the structure also remains stable, thus having to use less layers or courses to build it. Example 3 : Given the structure’s width, what is the optimal height? (What is the optimal roof curvature / smallest 𝒓𝒓 for a Superadobe dome given its base area?) If the builders have a predefined base surface area, minimum factor of contact , skylight radius and bu ilding sack characteristics, solving (1) for 𝒓𝒓 will produce the lightest and lowest (optimal) geometrically sound structure . Figure 6. Console interface for solving implic itly eq (1) for the parameter ,𝒓𝒓. Calculation of o ptimal height given the structures width The resulting structure in this experiment: Figure 7. Solid of revolution. The resulting dome from the above experiment has 27 superblocks, an optimal height of 5.4m, a top openin g of just over 0.5m, and all of its courses’s transversal sections have at least 10cm of contact. Example 4 . What is the minimum S uperblock width needed to build a stable Superadobe structure of a given area and height? When builders have predefined the size of all three compasses: base area, the building’s final height and skylight radius, they must know the minimum sack dimensions needed to guarantee a geometrically stable dome with this guidelines . Since the height of the dome H depends upon the three compasses (𝒓𝒃 ,𝒓𝒓,𝑻𝒓), the situation described in example 4 is similar to a predefined base radius 𝒓𝒃 (since base area i s dependent on this) , skylight radius 𝑻𝒓 and final dome height H [1], together with presetting minimum factor of contact F. solving (1) for 𝒔𝒘 will produce the thinnest (optimal) geometrically stable structure , with the minimum width of sack needed for the building to be geometrically stable. Note in the figure below of the execution of the program that H is set as a preset requisite by answering yes to the question: “Is final height one of your pre -defined parameters?” Figure 8. Minimum required Superblock width given the structure's size 7 Applications of a “geometrical soundness equation” for Superadobe domes ETSEM Polytechnic University of Madrid The resulting structure: Figure 9. Solid of revolution. The resulting dome from the experiment above has 35 superblocks, a height of 7m, as required in the conso le, a top opening of just over 0.5m, it is built with sacks of 0.641m wide, and all of its course’s transversal sections have at least 0.135 m of their straight sections resting on the previous rows. (0.135m is 0.25*0.64 -Extb) Observation : The final height of the structure, as we have seen in [1] , depends on three parameters: Tr, 𝒓𝒓,𝒂𝒏𝒅 𝒓𝒃. It is clear why H depends on Tr and 𝒓𝒓, but it can be less clear why H also depends on 𝒓𝒃. The reason is H's original dependence on a. This dependence on a means that, although one may know the length of the compass opening to trace the roof radius and the radius of the opening at the top (up to where the roof curve is traced), one must al so know the location of the end point of the compass with respect to t he dome's central axis to define H, and this is where 𝒓𝒃 is needed, because 𝒓𝒓-𝒓𝒃=a, which states the distance between the central axis and the center of the roof compass . Equivalently, 𝒓𝒃 is the distance between the central axis and the roof comp ass endpoint . When a designer has H pre-determi ned, he must also know two of 𝒓𝒓, 𝒓𝒃 or Tr, to know the missing compass . This means that knowing H and two other radiuses rules out the need for the stability equation, unless this equation is solved for a parameter other than 𝒓𝒓, 𝒓𝒃 or Tr, which is the case in the last example , where (1) was solved for 𝒔𝒘. Example 5 : What is the optimal (greatest) base area for a structure of a given roof curvature? In this situation, designers have defined building’s roof radius (external compass size: a pattern for gradual closing of the roof is already defined by pre conceived elements, for example) and skylight radius , with the rest of the sack -eleme nt characteristics also defined. They want to know the maximum base area permitted (maximum distance between central axis and roof compass endpoint) for a ge ometrically stable dome with these parameters and the defined degree of contact. If the builders have a predefined roof radius and top opening radius , with determined sack characteristics and minimum facto r of contact, solving (1) for 𝒓𝒃 will produce the biggest (widest ) geometrically st able structure, with the optimal base area permitted for the bui lding to be geometrically sound up to the desired 𝑭 level [1]. Note that as in Example 3, here the builder does not have the building height H predetermined, and it is a variable depending on 𝒓𝒃. All that is known is that the larger 𝒓𝒃, the higher the building will be. Conversely, the longer the predetermined 𝒓𝒓 is, the wider the building can be, while remaining geometrically stable. Figure 10. Calc ulation of maximum building internal compass given the curvature of the roof The resulting structure: Figure 11. The resulting structure has 30 superblock levels, measuring 6 meters of height, and having a skylight radius of just over 0.6 m, with sack width 0.5m and minimum degree of contact F of 0.25 (10cm) Example 6: Unknown 𝑬𝒙𝒕𝒃 : Solving (1) for 𝑬𝒙𝒕𝒃 will yield the maximum round (non -supportive) element parts , so that the building w ith given characteristics remains stable up to the desired 𝑭𝒓𝒂𝒄 level. This case can be associated with maximizing possible sack undesired behavior while remaining with a stable building. 9 Applications of a “geometrical soundness equation” for Superadobe domes ETSEM Polytechnic University of Madrid Example 7: Unknown 𝒉𝒔: How does the height of the sacks affect dome stability or geometrical sondness ? It is perhaps evident , that the taller the sack, the greater the displacement between consecutive courses in a Superadobe dome is needed. Given two different sack heights, a formal demonstration could be that if h is the height of the lowest one , the displacement between the last two sacks of greater height kh, which is given by: cos(sin−1(0.5𝒌𝒉)) -cos(sin−1(1.5𝒌𝒉)), Will be always greater than the displacement between the two last smaller sacks at the same building´s height, given by cos(sin−1(1.5𝒌𝒉)) -cos(sin−1(0.5𝒉(3𝒌+2)))). Proving the inequality: cos(sin−1(1.5𝒌𝒉)) -cos(sin−1(0.5𝒉(3𝒌+2)))) ≤ cos(sin−1(0.5𝒌𝒉)) -cos(sin−1(1.5𝒌𝒉)) For any positive h and any k greater than 1, can prove the intuition that given two identical domes except for hs, that the one with lower hs will be more stable (have greater F), than its analogous and thus, solving (1) for hs will produce the greatest possible hs with which a given dome will remain geometrically stable up to the desired level F. Example 8: Evaluation of (1) within a range of values . We can keep all parameters identical to the parameters of a given building, solving equation (1) different times for a chosen parameter (changed defined parameter) , but each time giving (forcing) a specific parameter within a range, as an input to the equation . This allows for comparisons betwe en analogous buildings. This is equivalent to not knowing two parameters, feeding (1) with a range of values of one parameter, and solving for the other unknown, to devise graphs and patterns of how one parameter induces another .