Transformed Shell Structures Determined by Regular Networks as a Complex Material for Roofing

Abstract

The article presents a comprehensive extension of the proprietary basic method for shaping innovative systems of corrugated shell roof structures by means of a specific complex material that comprises regular transformable shell units limited by spatial quadrangles. The units are made up of nominally plane folded sheets transformed into shell shapes. The similar shell units are regularly and effectively arranged in the three-dimensional space in an orderly manner with a universal regular reference surface, polyhedral network, and polygonal network. The extended method leads to the increase in the variety of the designed complex shell roof forms and plane-walled elevation forms of buildings. For this purpose, the rules governing the creation of the continuous roof shell structures of many shells arranged in different unconventional visually attractive patterns and their discontinuous regular modifications are sought. To obtain several novel groups of similar unconventional parametric roof forms, single division coefficients and double division coefficients are used. The easy and intuitive modifications of the positions of the vertices belonging to the polygonal network on the side edges of the polyhedral network accomplished by means of a parametric algorithm allow one to adjust the geometry of the complete shell units to the geometric and material constraints related to the orthotropic properties of the transformed sheeting by means of these coefficients. The innovative approach to the shaping of the diverse unconventional roof structures requires the solving of many interdisciplinary problems in the field of mathematics, civil engineering, construction, morphology, architecture, mechanics, computer visualization, and programming.

1. Introduction

Small transverse compressive or tensile forces acting on a thin-walled nominally flat folded sheet can substantially alter the width of each fold of a sheet. The thin-walled, folded structure of the sheet allows each fold to change the width at its length in a diversified way. This property enables a strip of shell folds to take several forms of various ruled surfaces with a contraction appearing along the length of the shell fold [1] (Figure 1).

Symmetric elastic shape transformations of the thin-walled folded sheets ought to minimize their effort and allow one to obtain the highest possible degree of the shape changes generating the expected relatively big curvature of the transformed sheet while maintaining the ability to carry the characteristic roof load [2]. The rational forms of the transformed folded roof shells can be provided at the initial step of the process of modeling the buildings roofed with the transformed shells. For this purpose, various sectors of warped surfaces are created so that their contracting line runs halfway along the length of each of the transformed folds. It is not difficult to create shallow doubly-curved rectangular sectors of warped shells with negative Gaussian curvature due to the specific orthotropic geometric and material properties related to the shape transformations. These are mainly central sectors of hyperbolic paraboloids or quarters of these sectors.

In order to obtain a double-curved shell roof structure of medium or high positive Gaussian curvature, for example, an ellipsoid, it is necessary to form an unconventional shell roof structure composed of many similar shell units arranged on an auxiliary regular surface in the three-dimensional Euclidean space [1]. The so-called reference surface represents the general shell shape of a ribbed roof structure. The single shells of such a structure must be properly joined to each other along their edge lines (Figure 2).

In order to develop the rules governing the formation of several novel complex forms of the shell roofs that are impossible to obtain with the help of the single ruled shells due to the transformation constraints, methods based on spatial plane-walled reference networks are used [3,4]. One of such methods was developed and, next, expanded to utilize in elaborating the special rules defining different types of the complex shell roofs [5]. The considered roofs are derivatives of the basic roof shell structures described in the abovementioned articles.

For scientific research and engineering developments, several experimental tests and computer simulations (Figure 3) are carried out to analyze geometric and mechanical properties of thin-walled folded sheeting transformed into shell shapes [6,7].

2. Critical Analysis of the Present Knowledge

The research into the shape transformations of the nominally flat thin-walled corrugated steel sheets was initiated by Nilson in the 1970s. In the initial phase, the studies concerned the geometric and mechanical properties of complete shells transformed into the forms of central hyperbolic paraboloid sectors [8]. In order to increase the stiffness and critical loads of the transformed folded shells, two layers of folded sheets arranged in two orthogonal directions were used by the Winter team [9]. There were only created shallow double-layer shells characterized by a small transformation degree.

The degree can easily be increased by using single-layer sheeting at the expense of reducing the transverse stiffness of the transformed coating. Gioncu and Petcu developed a computer program for the calculation of the critical loads of the transformed shells [10]. The results of these tests point at the fact that the transformed single corrugated shells can be regarded as working in a membrane state.

The diversity, ridge, stiffness, and critical load of the transformed folds can significantly be increased by assembling several single shell units into one continuous ribbed shell roof structure (Figure 4a). The team led by Winter conducted comprehensive studies and published consistent results on the static-strength work of several single shells and shell structures composed of four quarters of the hyperbolic paraboloid central sectors arranged in different configurations (Figure 4b). Parker published the supplementary research results concerning the static-strength work of the complete coatings and [11].

The team led by Gergely presented a uniform description of the static-strength work of the single and complex transformed shells based on a relatively wide range of tests [12]. Simultaneously with the Gergely’s team, similar tests and analysis were conducted by Fisher et al. [13] in the field of static work and critical loads of the complete and complex hyperbolic paraboloid shells. The results of the tests and analysis of the work of the folded coatings were collected by Davis and Bryan to indicate the effective methods for shaping the shells [14].

The abovementioned researchers indicated the great theoretical possibilities of shaping various ruled forms made up of the transformed folded sheets with an open profile. Ultimately, on the basis of the tests and analysis carried out, they found that the encountered significant material and technological limitations drastically reduce the possibility of shaping the diversified shell forms of transformed sheeting to one basic type of the shallow hyperbolic paraboloids called hypars [12].

Therefore, it is reasonable to shape the ribbed structures composed of many single transformed shells by means of complex systems of complete ruled shells separated by sets of planes containing common or mutual displaced sections of the edge lines of these shells. Biswas and Iffland proposed a concept of such a system composed of many congruent transformed single shell units distributed over a sphere by means of a bundle of planes [15]. The transformed folded shells made up of aluminum or PVC “Selchim” corrugated plastic sheets were analyzed by Samyn [16]. The complete shells are shaped as revolved hyperboloids or right hyperbolic paraboloids limited by spatial quadrangles. Pottman proposed a comprehensive method of shaping the systems of planes separating subsequent smooth shell sectors in an arbitrary surface [17].

Reichhart developed a novel method for shaping the complete transformed thin-walled corrugated shells to increase their diversity, ridge, and transformation degree [18]. In accordance with Reichhart’s algorithm, the nominally flat single-layer sheeting is transformed into the position of the rigidly fixed directrices so that a freedom of the transverse width and height changes of all shell folds is assured. The previously mentioned methods did not provide such a freedom. Therefore, the effort of the transformed sheeting designed by means of these methods is significantly higher, and its critical load and ridge are significantly smaller compared to the shells formed by means of Reichhart’s method. Reichhart’s analysis carried out on the basis of the results of the tests allowed him to develop a method for calculating the shape, length, and mutual position of each shell directrix. The designed sheets do not have to be loaded with additional transverse forces to adjust their longitudinal edges or neutral axes to the positions of the selected rulings of an arbitrary ruled surface modeling the transformed roof shell.

On the basis of the results of the performed tests, Abramczyk noticed a significant role played by the contraction of each transformed shell sheeting in creating the effectively transformed thin-walled folded shells [1]. Abramczyk extended the Reichhart method and included the condition related to the central location of the contraction in the effectively transformed folded shells.

Reichhart also developed a simple method of composing of a large number of identical complete transformed shells into a ribbed structure arranged on an oblique plane (Figure 2 and Figure 5) [19]. Abramczyk developed a much more complex method for regular arrangement of many complete shell units in the three-dimensional Euclidean space (Figure 6) based on the so-called reference surface [3]. The method results from the experimental studies and computer simulations of the transformed complete thin-walled folded shells [20]. It is possible to combine different shell units into one regular structure whose general form is similar to the arbitrary regular surface with almost free curvature using in constructions [21,22].

The properties of several thin-walled structures were detailed by Wei-Wen [23]. They can be used in shaping of the elastically transformed roof shells. Marin et al. [24] extended the classical theory of elasticity developed by Green and Lindsay in terms of the theory of thermo-elasticity for dipolar bodies. A novel method for a solution to a dynamical mixed problem was presented using a reciprocal theorem and not very restrictive conditions [25].

It is significant to investigate the space around the designed free-form building, its physical form and cultural patterns appearing in a whole spatial system. The relation between the formation of the urban space and the social experience of the human self was considered by Sharma [26]. The design syntax of urban greenways should also be taken into account. The mathematics-based graph studies of patterns and shapes, thermal based photography, and morphology to perform imagery-derived deductions on the design syntax were carried out by Hasgül [27].

Morphological shaping of buildings using many features specific to architectural, industrial, and structural design must be accomplished. Morphology is the study of the forms taking into account the relationships occurring between the function, structure, internal and external texture, static-strength work, and comfort conditions. Systematic morphology was defined by Eekhout as “the study of the system, rules and principles of form [that] has led to the interpretation of the study of the geometry of regular three-dimensional bodies or forms, usually known as polyhedra” and it plays a significant role in the design process [28]. The analogous universal systems of planes called polyhedral reference networks are utilized in this article.

3. The Aim

The main aim is to present the algorithm of an innovative extended method for shaping diversified unconventional systems of complex roof shell forms. The diversification is targeted at searching for unconventional basic and derivative configurations of the transformed shell roof structures. The research is related to the exploration of the essential dependencies dividing the shell structures into different groups of similar geometric properties and, next, the implementation of the obtained significant relations in the method’s algorithm. The complete transformed shell units characterized by the special orthotropic geometric and mechanical properties are used as a material for creating the shell roof structures. The units are made of flat thin-walled folded steel sheets transformed into similar shapes of ruled shells.

The utilization of the method relies on creating two specific reference networks Γ and B_v and adopting the set of division coefficients of the respective pairs of the vertices belonging to the first reference polyhedral network Γ by the vertices of the second reference polygonal network B_v to determine several general forms of free-form buildings. The network B_v defines the degree of the folding and discontinuity of the roof structure Ω (Figure 7a,b).

The formulas governing the mutual positions of the vertices of the subsequent meshes B_vij (sectors Ω_ij) in the required networks B_v (structures Ω) must be invented. The expected types of the geometric patterns formed by B_vij (Ω_ij) on the network B_v (structure Ω) have to result from the predicted relationships between the mutual positions of the vertices of the subsequent meshes B_vij (sectors Ω_ij). The proposed formulas must relate to the respective mutual displacements of the adjacent meshes B_vij (sectors Ω_ij) and their vertices along the side edges of the created polyhedral structure Γ, which allows one to achieve the appropriate type and form of the final derivative discontinuous roof structures Ω.

4. The Method’s Concept

The following concept of the research was adopted. The previously developed method was significantly extended and includes activities and objects that allow one to search for special rules relating to diversification of the visual patterns combined with many complete shells to achieve unconventional regular roof structures. The formulas governing the patterns result from the method of defining and arranging the single shell units in the three-dimensional space in an orderly and regular manner. For example, a reference surface and a set of division coefficients defining the location of the characteristic vertices of the designed structure can be utilized. The search for the relationships begins with the observation and description of the properties of the continuous ribbed shell roof structures.

For this purpose, a z-symmetric roof structure Ω consisting of two symmetrical and two antisymmetric parts is sought. The search starts with the determination of one symmetrical quarter of Γ. At the beginning, a single central shell Ω₁₁ is created by means of a central tetrahedral mesh Γ₁₁ and a central quadrilateral mesh B_v11 (Figure 8). The subsequent meshes Γ_ij, B_vij and Ω_ij of the nets Γ, B_v and Γ are symmetrically arranged with respect to the z-axis-symmetric Ω₁₁ in the orthogonal and diagonal directions.

The characteristic feature of the reference network Γ is that each single mesh Γ_ij is a specific tetrahedron with four vertices W_ABij, W_CDij, W_ADij and W_BCij, two axes u_ij and v_ij, four side edges a_ij, b_ij, c_ij and d_ij, four triangular side walls (W_ABijW_CDijW_BCij), (W_ABijW_CDijW_ADCij), (W_BCijW_ADijW_ABij) and (W_BCijW_ADijW_CDij) contained in four planes defined by the above vertices. For the first mesh i = j = 1 (Figure 8).

In order to obtain the tetrahedron Γ₁₁, the coordinates of its four vertices W_AB11, W_CD11, W_AD11 and W_BC11 must be defined based on a global coordinate system [x,y,z] [3,21]. The positions of the points S_A11, S_B11, S_C11 and S_D11 are defined with the following division coefficients d_SA11 = (W_AB11, W_AD11)\S_A11, d_SB11 = (W_AB11, W_BC11)\S_B11, d_SC11 = (W_CD11, W_BC11)\S_C11 and d_SD11 = (W_CD11, W_AD11)\S_D11 of the pairs (W_AB11, W_AD11), (W_AB11, W_BC11), (W_CD11, W_BC11) and (W_CD11, W_AD11), where

$$\begin{gathered} (W_{AB11},W_{AD11})\backslash S_{A11}=m\left(\overrightarrow{W_{AB11}{S}_{A11}}\right)/m\left(\overrightarrow{W_{AB11}{W}_{AD11}}\right) \\ (W_{AB11},W_{BC11})\backslash S_{B11}=m\left(\overrightarrow{W_{AB11}{S}_{B11}}\right)/m\left(\overrightarrow{W_{AB11}{W}_{BC11}}\right) \\ (W_{CD11},W_{BC11})\backslash S_{C11}=m\left(\overrightarrow{W_{CD11}{S}_{C11}}\right)/m\left(\overrightarrow{W_{CD11}{W}_{BC11}}\right) \\ (W_{CD11},W_{AD11})\backslash S_{D11}=m\left(\overrightarrow{W_{CD11}{S}_{D11}}\right)/m\left(\overrightarrow{W_{CD11}{W}_{AD11}}\right) \end{gathered}$$

(1)

and $\overrightarrow{W_{AB11}{W}_{AD11}}$ is the vector starting with W_AB11 and ending at W_AD11, m($\overrightarrow{W_{AB11}{W}_{AD11}}$) is the measure of $\overrightarrow{W_{AB11}{W}_{AD11}}$, $\overrightarrow{W_{AB11}{S}_{A11}}$ is a vector with the starting point at W_AB11 and the ending point at S_A11, etc. The points S_A11, S_B11, S_C11 and S_D11 together with the analogous points assigned to the other meshes of Γ define the respective reference surface ω.

The locations of the vertices A₁₁, B₁₁, C₁₁ and D₁₁ of Bv₁₁ (Figure 8) are defined by means of the vertices of Γ₁₁ and the following proportions:

$$\begin{gathered} {\mathrm{d}}_{A11}=(W_{AB11},W_{AD11})\backslash S_{A11}=m\left(\overrightarrow{W_{AB11}{S}_{A11}}\right)/m\left(\overrightarrow{W_{AB11}{W}_{AD11}}\right) \\ {\mathrm{d}}_{B11}=(W_{AB11},W_{BC11})\backslash S_{B11}=m\left(\overrightarrow{W_{AB11}{S}_{B11}}\right)/m\left(\overrightarrow{W_{AB11}{W}_{BC11}}\right) \\ {\mathrm{d}}_{C11}=(W_{CD11},W_{BC11})\backslash S_{C11}=m\left(\overrightarrow{W_{CD11}{S}_{C11}}\right)/m\left(\overrightarrow{W_{CD11}{W}_{BC11}}\right) \\ {\mathrm{d}}_{D11}=(W_{CD11},W_{AD11})\backslash S_{D11}=m\left(\overrightarrow{W_{CD11}{S}_{D11}}\right)/m\left(\overrightarrow{W_{CD11}{W}_{AD11}}\right) \end{gathered}$$

(2)

where $\overrightarrow{W_{AB11}{A}_{11}}$ is the vector with the starting point at W_AB11 and the ending point at A₁₁, etc. The points A₁₁, B₁₁, C₁₁ and D₁₁ determine the spatial quadrangle B_v11 constituting the eaves of a single smooth shell segment Ω₁₁ modeling a single shell of a complex roof structure.

The process of shaping of the reference network Γ consists in creating subsequent tetrahedrons characterized by common planes intersecting each other in axes and side edges. To define the networks Γ and B_v, a set of the respective independent variables must be adopted and specific values have to be assigned to these variables. For all dependent variables, appropriate functions must be defined to determine the vertices of Γ and B_v.

In order to obtain the tetrahedron Γ₁₂ its four vertices W_AB12, W_CD12, W_AD12 and W_BC12 must be defined [3,29]. The vertex W_AB12 = W_CD12. Two next vertices can be calculated by means of two division coefficients as follows

$$\begin{gathered} (W_{CD11},W_{BC11})\backslash W_{BC12}=m\left(\overrightarrow{W_{CD11}{W}_{BC12}}\right)/m\left(\overrightarrow{W_{CD11}{W}_{BC11}}\right) \\ (W_{CD11},W_{AD11})\backslash W_{AD12}=m\left(\overrightarrow{W_{CD11}{W}_{AD12}}\right)/m\left(\overrightarrow{W_{CD11}{W}_{AD11}}\right) \end{gathered}$$

(3)

where m($\overrightarrow{W_{CD11}{W}_{BC12}}$) is the measure of the vector $\overrightarrow{W_{CD11}{W}_{BC12}}$ with the starting point at W_CD11 and the ending point at W_BC12, and m($\overrightarrow{W_{CD11}{W}_{BC11}}$) is the measure of the vector $\overrightarrow{W_{CD11}{W}_{BC11}}$, etc.

The vertices of the other meshes B_vij of B_v should be defined in the same way as B_v11 using formulas analogous to Equations (1)–(3). Each pair of two adjacent meshes Γ_ij and Γ_ij+1 or Γ_ij and Γ_i+1j have one common side edge contained in a plane of Γ, for example, Γ₁₁ and Γ₂₁ has two common side edges b₁₁ = a₂₁ and c₁₁ = d₂₁, and three common vertices W_BC11 = W_AD21, W_CD11 = W_CD21 and W_AB11 = W_AB22 (Figure 9). Four adjacent meshes Γ_ij, Γ_ij+1, Γ_i+1j and Γ_i+1j+1 have one common side edge a_i+1j+1 = b_i+1j = c_ij = d_ij+1 of Γ, for example, a₂₂ = b₁₂ = c₁₁ = d₂₁ for i = j = 1.

The subsequent quadrilateral meshes B_vij of the targeted networks B_v modeling the eaves of the examined shell roof structures are constructed based on the side edges of the auxiliary network Γ. Following to the method’s algorithm, the subsequent adjacent quadrilateral meshes of B_v must have common vertices A_ij, B_ij, C_ij or D_ij determined on the edges a_ij, b_ij, c_ij and d_ij (Figure 9). The specific sum of all individual shells Ω_ij, determined by means of A_ij, B_ij, C_ij and D_ij, constitutes a continuous ribbed roof structure Ω (Figure 10). Finally, a free-form Σ of many Σ_ij is the simplified model of an entire complex building.

The complete tetrahedrons Γ_ij can be regarded as an universal material for creating the spatial polyhedral networks modeling free-form building systems. Similarly, the single spatial quadrilateral meshes B_vij can be accepted as a material for shaping the eaves systems B_v of complex free-form roofs. In addition, the complete shell sectors Ω_ij are used as a universal material for creating the free-form shell roof structures Ω. After all, the abovementioned systems create three subsequent layers producing one complex material used for shaping unconventional building free forms roofed with complex shell structures.

The article primarily presents the results of the research on the geometric properties of the eaves layer B_v determining the form of the shell layer Ω. The observed properties are described with the help of the developed mathematical rules governing the systems of different patterns of the complete meshes B_vij and sectors Ω_ij arranged on ω. The results of these studies can be used in the design of several diversified unconventional free forms of buildings roofed with attractive and rational systems of many regular roof shells made up of transformed corrugated sheets.

In order to carry out the research, the following algorithm was developed. It forbids two adjacent meshes of B_v or segments of Ω to have common vertices and sides. Thus, a discontinuous net B_v and a discontinuous structure Ω can be created as a result of a rotation of each pair of the respective eaves segments belonging to two adjacent shell units Ω_ij and Ω_i+1j or Ω_ij+1 (Figure 11). Four straight segments of each spatial quadrangle B_vij are displaced in the respective planes of the network Γ. Thus, the basic continuous configuration CB and the modified discontinuous configuration CP1 of the structure Ω use the same network Γ.

However, such a separation of the positions of the vertices belonging to the selected pairs of two adjacent meshes of B_v (Ω) requires appropriate changes in the values of the division coefficients assigned to the respective vertices of Γ of each B_vij. The goal of the research is therefore to develop a method of modifying the values of the coefficients so that different groups of discontinuous shell roof structures can be achieved. This activity requires defining some uniform conditions and adopting mathematical formulas describing the modified net B_v and structure Ω.

The designed discontinuous nets B_v and structures Ω can also be created as a result of translations of two respective eaves segments belonging to two adjacent shell units Ω_ij and Ω_i+1j or Ω_ij+1 (Figure 12). Four straight segments of each quadrangle B_vij can be displaced in the respective planes of Γ.

Finally, the following algorithm based on four general steps should be adopted. The main characteristic of the algorithm is the usage of the division coefficients of the vertices of the reference polyhedral network Γ by the vertices of a reference polygonal network B_v (Figure 7) and the selected points of a reference surface ω. At the first step of the algorithm, the division coefficients are used to obtain the reference polyhedral network Γ and the repeatability of its tetrahedral meshes Γ_ij, especially in the orthogonal directions. At the second step of the algorithm, the division coefficients are used to determine several points defining the reference surface ω. The expected curvature and general form of the shell roof structure are designated in an easy and intuitive way by means of the relations defined by means of the arbitrary division coefficients.

The division coefficients are also used at the third step of the method’s algorithm to determine the vertices of the reference polygonal network B_v. The vertices are located on the side edges of the polyhedral reference network Γ and defined with respect to the reference surface ω. The obtained vertices define the polygonal network B_v composed of quadrilateral spatial meshes B_vij. Four segments of B_vij determine one complete shell unit Ω_ij and constitute its eaves line. At this step, a basic configuration CB of the sought-after shell roof structure Ω is created. The configuration is composed of many individual shells Ω_ij so that the roof structure Ω is continuous (Figure 4b, Figure 5a,b and Figure 10). Each pair of two adjacent complete shells of the structure shares one edge.

The network Γ and the network B_v determine the curvature and folding of the roof structure Ω. The curvature of Ω can be determined using the properties of ω. The folding of Ω results from the ribs existing between the adjacent complete smooth shell units Ω_ij.

The fourth step of the method’s algorithm allows one to create several derivative configurations of the examined continuous roof structure. The derivative configurations are most often characterized by a mutual translation or rotation of the edge lines of the adjacent shells (Figure 6a,b, Figure 11 and Figure 12) in the planes of Γ. The discussion of the activities and their effects provided for in this step of the algorithm is the essential part of the research presented in this article.

The accuracy of the algorithm is comparable with the accuracy of the adopted data, and it is equal to 1 mm. This results from the 0.5 mm accuracy of the performed experimental tests and the 1 mm accuracy of the calculations of the ruling’s position of each complete shell Ω_ij of Ω.

5. Results

The search for the rules governing the systems of the diversified patterns of the complete transformed shells on the roof structures is presented in three examples of networks B_v using the investigated algorithm. At the first step of the algorithm, the axes u_ij and v_ij, side edges a_ij, b_ij, c_ij and d_ij, and the entire Γ are determined on the basis of the adopted vertices W_ABij, W_CDij, W_ADij and W_BCij [3,21]. This step is divided into two sub-steps requiring the designer to create: (1) the mesh Γ₁₁, meshes Γ_i1 and Γ_1j (for i > 1 and j > 1) located orthogonally in relation to Γ₁₁, (2) meshes Γ_ij located diagonally relative to Γ₁₁, Γ_i1 or Γ_1j. The sub-steps differ from each other in terms of the initial data and the actions necessary to build the subsequent single meshes Γ_ij. The values of the coordinates of the vertices belonging to the Γ₁s (Figure 13a) are presented in

Table A1. The coordinates of the vertices W_ABij, W_CDij, W_ADij, W_BCij (for i, j = 1, 2, 3) of the polyhedral reference network Γ₁.

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
WAB13	3254.3	0.0	468.2
WCD13	5572.7	0.0	1487.4
WAD13	−435.4	−1054.6	12,007.1
WBC13	−435.4	1054.6	12,007.1
WAB21	−1100.0	−87.2	−996.2
WCD21	1100.0	−87.2	−996.2
WAD21	−4500.0	−4880.7	55,786.9
WBC21	0.0	2574.0	9677.1
WAB31	−1210.0	−353.3	−2063.5
WCD31	1210.0	353.3	−2063.5
WAD31	0.0	2574.0	9677.1
WBC31	0.0	4374.6	9074.6
WAB23	3589.7	−95.9	−580.8
WCD23	6173.6	−105.5	435.5
WAD23	−435.4	1054.6	12,007.1
WBC23	−480.0	3133.7	11,876.9
WAB32	1210.0	353.3	−2063.5
WCD32	3959.7	−389.5	−1713.3
WAD32	−110.0	2840.1	10,744.4
WBC32	−121.0	4847.4	10,188.4
WAB33	3959.7	−389.5	−1713.3
WCD33	6838.9	−429.4	−708.6
WAD33	−480.0	3133.7	11,876.9
WBC33	−529.1	5371.0	11,378.6

in Appendix A.

At the second step of the algorithm, all points defining a reference surface ω are determined. In this way, the size of Γ and the curvature of ω are defined. For this purpose, the values of the selected division coefficients are adopted and the coordinates of the points S_Aij, S_Bij, S_Cij and S_Dij defining the reference surface ω are calculated using Equation (1) and the initial data are published in

Table A2. The initial data defining the positions of the points S_Aij, S_Bij, S_Cij, S_Dij for i, j = 1, 2.

Ratio	Value
dSA11 = (WAB11,WAD11)\SA11	5.5000
dSB11 = (WAB11,WBC11)\SB11	5.5000
dSC11 = (WCD11,WAD11)\SC11	5.5000
dSD11 = (WCD11,WAD11)\SD11	5.5000
dSA12 = (WAB12,WAD12)\SA12	5.0000
dSB12 = (WAB12, WBC12)\SB12	5.0000
dSC12 = (WCD12, WBC12)\SC12	5.0000
dSD12 = (WCD12, WAD12)\SD12	5.0000
dSA22 = (WAB22,WAD22)\SA22	4.6667
dSB22 = (WAB22, WBC22)\SB22	4.6281
dSC22 = (WCD22, WAD22)\SC22	4.6667
dSD22 = (WCD22, WAD22)\SD22	4.6364

in Appendix A. The coordinates of these points are published in

Table A3. The coordinates of the points defining the reference surface ω.

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
SA11	4500.0	−4793.6	54,790.7
SB11	4500.0	4880.7	55,786.9
SC11	−4500.0	4793.6	54,790.7
SD11	−4500.0	−4880.7	55,786.9
SA12	−4500.0	−4880.7	55,786.9
SB12	−4500.0	4793.6	54,790.7
SC12	−13,517.1	4793.6	52,918.0
SD12	−13,517.1	−4793.6	52,918.0
SA13	−13,517.1	−4793.6	52,918.0
SB13	−13,517.1	4793.6	52,918.0
SC13	−21,737.1	4793.6	49,304.2
SD13	−21,737.1	−4793.6	49,304.2
SA21	4500.0	4880.7	55,786.9
SB21	4500.0	13,460.5	53,340.4
SC21	−4500.0	13,460.5	53,340.4
SD21	−4500.0	4793.6	54,790.7
SA31	4500.0	13,460.5	53,340.4
SB31	4500.0	21,957.5	50,497.3
SC31	−4500.0	21,957.5	50,497.3
SD31	−4500.0	13,460.5	53,340.4
SA22	−4500.0	4793.6	54,790.7
SB22	−4500.0	13,460.5	53,340.4
SC22	−13,675.6	13,605.3	52,270.2
SD22	−13,517.1	4793.6	52,918.0
SA23	−13,517.1	4793.6	52,918.0
SB23	−13,675.6	13,605.3	52,270.2
SC23	−22,104.0	13,660.9	49,061.5
SD23	−21,737.1	4793.6	49,304.2
SA32	−4500.0	13,460.5	53,340.4
SB32	−4500.0	21,957.5	50,497.3
SC32	−13,692.4	22,263.8	49,770.7
SD32	−13,675.6	13,605.3	52,270.2
SA33	−13,675.6	13,605.3	52,270.2
SB33	−13,692.4	22,263.8	49,770.7
SC33	−22,428.4	22,611.2	47,304.5
SD33	−22,104.0	13,660.9	49,061.5

in Appendix A.

At the third step, vertices of all quadrilateral meshes of the designed B_v for the basic configuration CB are defined on the side edges of Γ relative to ω (Figure 13b). At this step, the complete shell segments Ω_ij of an entire roof structure Ω constituting the basic configuration CB are defined on the basis of B_vij. The adopted values of the division coefficients d_Aij, d_Bij, d_Cij and d_Dij used for achieving the B_v vertices are presented in

Table A4. The division coefficients employed to calculate the coordinates of the Bv and Ω vertices for i, j = 1, 2.

Ratio	Value
dA11 = (WAB11,WAD11)\A11	5.4000
dB11 = (WAB11,WBC11)\B11	5.6000
dC11 = (WCD11,WAD11)\C11	5.4000
dD11 = (WCD11,WAD11)\D11	5.6000
dA12 = (WAB12,WAD12)\A12	5.0910
dB12 = (WAB12, WBC12)\B12	4.9090
dC12 = (WCD12, WBC12)\C12	5.0910
dD12 = (WCD12, WAD12)\D12	4.9090
dA22 = (WAB22,WAD22)\A22	4.5834
dB22 = (WAB22, WBC22)\B22	4.7114
dC22 = (WCD22, WAD22)\C22	4.5834
dD22 = (WCD22, WAD22)\D22	4.7190

in Appendix A. The coordinates of the Bv and Ω vertices calculated with the help of Equation (2) are presented in

Table A5. The values of the division coefficients (for i, j = 1, 2) used to estimate the folding of Ω (B_v).

Ratio	Value
dS11A11 = (WAB11,WAD11)\(SA11,A11)	−0.1000
dS11B11 = (WAB11,WBC11)\(SB11,B11)	0.1000
dS11C11 = (WCD11,WBC11)\(SC11,C11)	−0.1000
dS11D11 = (WCD11,WAD11)\(SD11,D11)	0.1000
dS12A12 = (WAB12,WAD12)\(SA12,A12)	0.0910
dS12B12 = (WAB12, WBC12)\(SA12,B12)	−0.0910
dS12C12 = (WCD12, WBC12)\(SA12,C12)	0.0910
dS12D12= (WCD12, WAD12)\(SA12,D12)	−0.0910
dS22A22 = (WAB22,WAD22)\(SA21,A22)	−0.0833
dS22B22 = (WAB22, WBC22)\(SA21,B22)	0.0826
dS22C22 = (WCD22, WBC22)\(SA22,C22)	−0.0833
dS22D22 = (WCD22, WAD22)\(SA22,D22)	0.0826

in Appendix A.

The values of four division coefficients d_SijAij = (W_ABij,W_ADij)\(S_Aij,A_ij), d_SijBij = (W_ABij,W_BCij)\(S_Bij,B_ij), d_SijCij = (W_CDij,W_BCij)\(S_Cij,C_ij) and d_SijDij = (W_CDij,W_ADij)\(S_Dij,D_ij) are calculated to estimate the folding of Ω (B_v) related to the diversification of the locations of the vertices of B_v relative to ω. The coefficients are used with the positive or negative sign depending on whether the points A_ij, B_ij, C_ij and D_ij lie above or below ω defined by means of the respective quadrangle S_AijS_BijS_CijS_Dij. The calculated coordinates of the vertices A_ij, B_ij, C_ij and D_ij are presented in

Table A6. The coordinates of the vertices A_ij, B_ij, C_ij, D_ij (for i, j = 1, 2, 3) of the basic configuration CB of the eaves edge net B_v1.

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
A11	4400.0	−4706.4	53,794.5
B11	4600.0	4880.7	55,786.9
C11	−4400.0	4706.4	53,794.5
D11	−4600.0	−4880.7	55,786.9
A12	xD11	yD11	zD11
B12	xC11	yC11	zC11
C12	−13,822.1	4880.7	53,871.7
D12	−13,212.2	−4706.4	51,964.4
A13	xD12	yD11	zD11
B13	xC12	yC11	zC11
C13	−21,240.6	4706.4	48,434.8
D13	−22,233.7	−4880.7	50,173.6
A21	xB11	yB11	zB11
B21	4400.0	13,218.5	52,370.0
C21	−4600.0	13,702.4	54,310.7
D21	xC11	yC11	zC11
A31	xB21	yB21	zB21
B31	4600.0	22,348.2	51,417.8
C31	−4400.0	21,566.7	49,576.8
D31	xC21	yC21	zC21
A22	xD21	yD21	zD21
B22	xC21	yC21	zC21
C22	−13,367.3	13,360.7	51,326.4
D22	xC12	yC12	zC12
A23	xD22	yD22	zD22
B23	xC22	yC22	zC22
C23	−22,608.0	13,906.3	49,928.3
D23	xC13	yC13	zC13
A32	xD31	yD31	zD31
B32	xC31	yC31	zC31
C32	−14,001.5	22,660.5	50,672.4
D32	xC22	yC22	zC22
A33	xC22	yC22	zC22
B33	xC32	yC32	zC32
C33	−21916.7	22,208.4	46,465.1
D33	xC23	yC23	zC23

in Appendix A.

The ratios d_S11A11 = (W_AB11,W_AD11)\(S_A11,A₁₁), d_S11B11 = (W_AB11,W_BC11)\(S_B11,B₁₁), d_S11C11 = (W_CD11,W_BC11)\(S_C11,C₁₁) and d_S11D11 = (W_CD11, W_AD11)\(S_D11,D₁₁) are defined for B_v11 as follows:

$$\begin{gathered} {\mathrm{d}}_{S11A11}=m\left(\overrightarrow{S_{A11}{A}_{11}}\right)/m\left(\overrightarrow{W_{AB11}{W}_{AD11}}\right)={\mathrm{d}}_{A11}-{\mathrm{d}}_{SA11} \\ {\mathrm{d}}_{S11B11}=m\left(\overrightarrow{S_{B11}{B}_{11}}\right)/m\left(\overrightarrow{W_{AB11}{W}_{BC11}}\right)={\mathrm{d}}_{B11}-{\mathrm{d}}_{SB11} \\ {\mathrm{d}}_{S11C11}=m\left(\overrightarrow{S_{C11}{C}_{11}}\right)/m\left(\overrightarrow{W_{CD11}{W}_{BC11}}\right)={\mathrm{d}}_{C11}-{\mathrm{d}}_{SC11} \\ {\mathrm{d}}_{S11D11}=m\left(\overrightarrow{S_{D11}{D}_{11}}\right)/m\left(\overrightarrow{W_{CD11}{W}_{AD11}}\right)={\mathrm{d}}_{D11}-{\mathrm{d}}_{SD11}. \end{gathered}$$

(4)

The above division coefficients allow one to observe the absolute differences in the mutual positions of the points A₁₁, B₁₁, C₁₁ and D₁₁ (Figure 13c). The obtained values can easily be converted into the distances of these points from ω. The coordinates of the B_v vertices calculated with the help of the equations analogous to Equation (4) are presented in Table A5 in Appendix A.

The coefficients d_SijAij, d_SijBij, d_SijCij and d_SijDij do not give the precise information about the folding of the network B_v (structure Ω) because they express the proportions in relation to the distance of two adjacent vertices of Γ. Therefore, the proportions signifying the position of the points A_ij, B_ij, C_ij and D_ij in relation to the position of the points S_Aij, S_Bij, S_Cij, S_Dij and the position of the vertices of Γ must be preferred to present the folding more precisely.

Thus, the following double division coefficients play an important role in describing the geometrical properties of the roof structures:

d_ASij = (W_ABij,W_ADij)\(A_ij/S_Aij) = d_Aij/d_SAij
d_BSij = (W_ABij,W_BCij)\(B_ij/S_Bij) = d_Bij/d_SBij
d_CSij = (W_CDij,W_BCij)\(C_ij/S_Cij) = d_Cij/d_SCij
d_DSij = (W_CDij,W_ADij)\(D_ij/S_Dij) = d_Dij/d_SDij.

(5)

If the values calculated by means of Equation (5) are greater than one, then the respective points lie above the surface ω. If the values are less than one, then the points lie below ω.

The following double division coefficients are a much more convenient and intuitive variable describing the folding of a shell roof structure:

d_A\Sij = (d_Aij − d_SAij)/d_SAij
d_B\Sij = (d_Bij − d_SBij)/d_SBij
d_C\Sij = (d_Cij − d_SCij)/d_SCij
d_D\Sij = (d_Dij − d_SDij)/d_SDij.

(6)

The new division coefficients show the relative proportions of the folding of B_v (Ω) in relation to the positions of ω and the respective vertices of Γ. The values of these coefficients were calculated by means of Equation (6) for the selected meshes of B_v. They are presented in

Table A7. The values of the division coefficients (for i, j = 1, 2) used to precisely define the folding of Ω (B_v).

Ratio	Value
dA\S11	−0.0182
dB\S11	0.0182
dC\S11	−0.0182
dC\S11	0.0182
dA\S12	0.0182
dB\S12	−0.0182
dC\S12	0.0182
dD\S12	−0.0182
dA\S22	−0.0178
dB\S22	0.0180
dC\S22	−0.0178
dD\S22	0.0178

in Appendix A.

The polyhedral reference networks Γ, reference surfaces ω, polygonal networks B_v and shell roof structures Ω are the main geometric objects created by means of the investigated method in the process for shaping building free forms roofed with complex transformed corrugated shell structures. The network B_v is the most important because the edge lines of all its complete meshes B_vij determine all individual shell segments Ω_ij of the resultant roof structure Ω. The B_v sides are the eaves of Ω_ij. To create a network B_v characterized by the expected properties, the method introduces the auxiliary reference polyhedral network Γ assisting and facilitating the determination of the positions of the vertices A_ij, B_ij, C_ij and D_ij of B_v.

The main property of each reference network Γ is that each pair of its adjacent side edges must intersect. The intersecting points of the respective pairs of two adjacent side edges denoted as a_ij, b_ij, c_ij or d_ij (Figure 8 and Figure 9) are called vertices W_ABij, W_CDij, W_BCij and W_ADij of Γ. The edges define all planes of Γ. It is very important that two adjacent meshes of B_v have one common segment of their edge lines (for continuous structures Ω) or two different segments (for discontinuous structures Ω) contained in the same plane of Γ, so the segments are coplanar. This property significantly simplifies the processes for shaping of the regular basic continuous configurations of Ω and their derivative configurations.

The research on searching for the rules governing the locations of the B_v vertices and the patterns of the complete Ω_ij shells starts with the analysis of the properties of the basic configuration CB of the shell structures shown in Figure 10 and Figure 13. The configuration CB allows one to create a so-called continuous roof structure Ω. This configuration is characterized by the fact that each four adjacent quadrangles B_vij, B_vi+1j, B_{vij +1} and B_vi+1j+1 have one common vertex C_ij = B_ij+1 = D_i+1j = A_i+1j+1. The process of the creation of the new polygonal networks B_v derivative of the basic configuration CB is relatively simple because the sides of B_vij are displaced in the abovementioned planes of Γ. Similarly, the B_v vertices belong to the side edges of Γ during these displacements.

The goal of this article is to focus on the next step of the method’s algorithm related to some modifications of the basic nets B_v. In particular, the activities forcing a diversification of the positions of the vertices of four adjacent B_v meshes corresponding to each other are analyzed. The change of the positions of the B_v vertices consists in varying their positions on the side edges of Γ in relation to ω. These modifications lead to diversified and original patterns of B_vij and Ω_ij on B_v and Ω (Figure 14 and Figure 15).

The first derivative configuration CP1 defines a discontinuous shell structure Ω and a discontinuous polygonal network B_v characterized by flat areas of discontinuity between the subsequent shell sectors Ω_ij of Ω (Figure 11 and Figure 14a,b). The discontinuous areas occurring between each pair of two adjacent meshes B_vij and B_vi+1j (Ω_ij and Ω_i+1j) or B_vij and B_{vij +1} (Ω_ij and Ω_ij+1) are the combinations of various triangles contained in the Γ planes. For this configuration, all points A_ij and C_ij are located below ω and all points B_ij and D_ij are located above ω at the distances resulting from the respective values of the division coefficients d_Aij, d_Bij, d_Cij and d_Dij. The configuration CP1 is characterized by tetrads of the adjacent meshes B_vij, B_vi+1j, B_vij+1 and B_vi+1j+1 with two pairs of common vertices: C_ij = A_i+1j+1 and B_ij+1 = D_i+1j located independently on the same side edge c_ij = b_ij+1 = d_i+1j = a_i+1j+1 of Γ.

The first spatial quadrangle B_v11 is defined so that the points A₁₁ and C₁₁ lie on the side edges a₁₁ and c₁₁ below ω and the points B₁₁ and D₁₁ above ω at the distances used for the basic configuration CB and resulting from the adopted values of the following division coefficients: d_S11A11 = d_S11C11 = −0.1 and d_S11B11 = d_S11D11 = 0.1. The positions of two next meshes B_v12 and B_v21 are obtained by moving the points A₁₂, B₁₂, C₁₂, D₁₂, A₂₁, B₂₁, C₂₁ and D₂₁ of the configuration CB along the respective side edges of the Γ network to their new positions A₁₂, B₁₂, C₁₂, D₁₂, A₂₁, B₂₁, C₂₁ and D₂₁ of the new configuration CP1 at the distances resulting from the values of the respective division coefficients adopted for B_v and Γ. The values of the coordinates of the vertices belonging to the quarter Γ₁ of the z-axis-symmetric network Γ calculated for CP1 are presented in

Table A8. The coordinates of the vertices A_ij, B_ij, C_ij, D_ij (for i, j = 1, 2, 3) of the first modified configuration CP1 of the eaves edge net B_v1.

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
A11	4400.0	−4706.4	53,794.5
B11	4600.0	4880.7	55,786.9
C11	−4400.0	4706.4	53,794.5
D11	−4600.0	−4880.7	55,786.9
A12	−4400.0	−4706.4	53,694.9
B12	−4600.0	4880.7	55,786.9
C12	−13,212.2	4706.4	51,964.4
D12	−13,822.1	−4880.7	53,871.7
A13	−13,212.2	−4706.4	51,964.4
B13	−13,822.1	4880.7	53,871.7
C13	−21,240.6	4706.4	48,434.8
D13	−22,233.7	−4880.7	50,173.6
A21	4400.0	4706.4	53,794.5
B21	4600.0	13,702.4	54,310.7
C21	−4400.0	13,218.5	52,370.1
D21	xB12	yB12	zB12
A31	4400.0	13,218.5	52,370.1
B31	4600.0	22,348.2	51,417.8
C31	−4400.0	21,566.7	49,576.8
D31	−4600.0	13,702.4	54,310.7
A22	xC11	yC11	zC11
B22	xD31	yD31	zD31
C22	−13,367.3	13,360.7	51,326.4
D22	xB13	yB13	zB13
A23	xC12	yC12	zC12
B23	−13,983.9	13,850.0	53,213.9
C23	−21,599.9	13,415.6	48,194.7
D23	−22,233.7	4880.7	50,173.6
A32	−4400.0	13,218.5	52,370.1
B32	xC21	yC21	zC21
C32	−13,383.2	21,867.1	48,869.1
D32	xB23	yB23	zB23
A33	xC22	yC22	zC22
B33	−14,001.5	22,660.5	50,672.4
C33	−21,916.7	22,208.4	46,465.1
D33	−22,608.0	13,906.3	49,928.3

in Appendix A.

The second configuration CP2 derivative of CB is characterized by many flat quadrilateral areas of discontinuity between adjacent meshes B_vij, B_vi+1j, B_vij+1 and B_vi+1j+1 (Figure 12 and Figure 15). The locations of its vertices A_ij, B_ij, C_ij and D_ij defining the quadrangular areas of the Ω discontinuity can be found as follows.

The first quadrangle B_v11 is the same as for the basic configuration CB and derivative configuration CP1. The positions of two following quadrangles B_v12 and B_v21 are obtained by moving the points A₁₂, B₁₂, C₁₂, D₁₂, A₂₁, B₂₁, C₂₁ and D₂₁ of the configuration CB along the respective side edges of the Γ network into their new positions A₁₂, B₁₂, C₁₂, D₁₂, A₂₁, B₂₁, C₂₁ and D₂₁ of the new configuration CP2 at the distances resulting from the values of the division coefficients d_S12A12, d_S12B12, d_S12C12 and d_S12D12, etc. The values of the coordinates of the vertices belonging to the quarter Γ₁ are presented in

Table A9. The coordinates of the vertices A_ij, B_ij, C_ij, D_ij (for i, j = 1, 2, 3) of the second modified configuration CP2 of the eaves edge net B_v1.

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
A11	4400.0	−4706.4	53,794.5
B11	4600.0	4880.7	55,786.9
C11	−4400.0	4706.4	53,794.5
D11	−4600.0	−4880.7	55,786.9
A12	−4500.0	−4793.6	54,790.7
B12	−4300.0	4619.2	52,798.3
C12	−13,517.1	4793.6	52,918.0
D12	−12,907.3	−4619.2	51,010.0
A13	−12,602.3	−4532.1	50,057.1
B13	−13,212.2	4706.4	51,964.4
C13	−20,247.5	4532.1	46,696.0
D13	−21,240.6	−4706.4	48,434.8
A21	4500.0	4793.6	54,790.7
B21	4300.0	12,976.6	51,399.8
C21	−4500.0	13,460.5	53,340.4
D21	−4300.0	4619.2	52,798.3
A31	4200.0	12,734.7	50,429.5
B31	4400.0	21,566.7	49,576.8
C31	−4200.0	20,785.3	47,735.8
D31	−4400.0	13,218.5	52,370.1
A22	−4200.0	4532.1	51,802.1
B22	−4400.0	13,218.5	52,370.1
C22	−12,750.7	12,871.3	49,438.9
D22	−13,212.2	4706.4	51,964.4
A23	−12,907.3	4619.2	51,010.8
B23	−12,442.4	12,626.7	48,495.1
C23	−21,095.8	13,170.2	47,328.0
D23	−19,751.0	4444.9	45,826.6
A32	−4300.0	12,976.6	51,399.8
B32	−4100.0	20,394.5	46,815.3
C32	−13,074.1	21,470.3	47,967.4
D32	−12,442.4	12,626.7	48,495.1
A33	−12,134.1	12,382.0	47,551.3
B33	−12,764.9	21,073.6	47,065.8
C33	−19,870.0	20,597.1	43,107.6
D33	−20,591.8	12,924.8	46,461.2

in Appendix A.

The positions of the other vertices belonging to the subsequent new quadrangles B_v13, B_v22 B_v23, B_v32 and B_v31 of CP2 are obtained as a result of moving the points A₁₃, B₁₃, C₁₃, D₁₃, A₂₂, B₂₂, C₂₂, D₂₂, A₂₃, B₂₃, C₂₃, D₂₃, A₃₁, B₃₁, C₃₁, D₃₁, A₃₂, B₃₂, C₃₂ and D₃₂ of CB along the respective side edges of Γ into their new positions of CP2 resulting from the respective values of the division coefficients d_SijAij, d_SijBij, d_SijCij and d_SijDij for i = 1 and j = 3 or i = 3 and j = 1. In particular, the coefficients can be equal to twice or three times the value of the respective coefficient d_S11A11 or d_S11B11 or d_S11C11 or d_S11D11.

6. Discussion

All vertices A_ij, B_ij, C_ij and D_ij of the basic configuration CB are divided into two groups. The first group includes the vertices lying on one side of a reference surface ω, for example, above ω. The second group is composed of the other B_v vertices lying on the opposite side of ω, that is, under ω.

The positions of the sought-after vertices A_ij, B_ij, C_ij and D_ij lying on the side edges a_ij, b_ij, c_ij and d_ij of Γ result from the adopted or calculated values of the division coefficients d_SAij, d_SBij, d_SCij and d_SDij. If we know the values of the above coefficients, the values the division coefficients d_Aij, d_Bij, d_Cij and d_Dij, required to calculate the positions of A_ij, B_ij, C_ij and D_ij, can be calculated as follows:

d_Aij = d_SAij + d_SijAij
d_Bij = d_SBij + d_SijBij
d_Cij = d_SCij + d_SijCij
d_Dij = d_SDij + d_SijDij.

(7)

The values of the division coefficients d_Aij, d_Bij, d_Cij and d_Dij corresponding to the vertices of CB can be calculated with Equation (7) and the values are published in Table A1 and Table A5 in Appendix A.

In the case of the investigated basic configuration CB (Figure 10 and Figure 13), the first subset is composed of the points A_ij and C_ij located under ω, whereas the second subset is composed of the other points A_ij and C_ij positioned above ω. Similarly, some of the points B_ij and D_ij are located above ω and others lie under ω.

To divide the points A_ij and C_ij into two complementary subsets, the following formulas were developed. The points A_ij and C_ij are located under ω if the subscripts i and j meet the following conditions:

i = j + 2 ∙ k_C for i > j

(8)

or

j = i + 2 ∙ k_C for i < j

(9)

or

i = j

(10)

where k_C is the integer constant corresponding to the examined mesh B_vij. If Equations (8)–(10) related to the values of i and j of the respective mesh Ω_ij are fulfilled, the points B_ij and D_ij are located above ω. For example, if k_C = 0 and j = 1, then i = 3 and the points A₃₁ and C₃₁ lie below ω and the points B₃₁ and D₃₁ lie above ω. The same result is achieved when adopting k_c = 0 and i = 1. For this case, j = 3 and the points A₁₃ and C₁₃ lie below the surface ω and the points B₁₃ and D₁₃ lie above ω. If we use Equation (10) and i = j = 3, then we obtain the points A₃₃ and C₃₃ lying below ω.

The points A_ij and C_ij take the positions above ω and the points B_ij and D_ij lie under ω if the following conditions are met:

i = 1 + j + 2 ∙ k_C for i > j

(11)

or

j = 1 + i + 2 ∙ k_C for i < j.

(12)

For example, if k_C = 0 and j = 1, then it follows from Equation (13) that i = 2 and the points A₂₁ and C₂₁ lie above ω, and the points B₂₁ and D₂₁ lie below ω. The same result is achieved when using Equation (12) and adopting k_C = 0 and i = 1. In this case, j = 2 and the points A₁₂ and C₁₂ lie above ω, and the points B₁₂ and D₁₂ lie below ω.

The coefficients express the proportions between the diversity of the positions of the B_v vertices in relation to ω, and the diversity of the locations of the same vertices in relation to the Γ vertices. They allow one to describe and parameterize the form of not only the roof structure, but also the form of the entire building, including the attractiveness and proportions between its basic dimensions. In this case, the double division coefficients must relate to the base level of the designed building. The above issues go beyond the scope of the article.

Thus, each of four vertices of B_vij of each basic configuration CB is shared with three adjacent meshes. In the search for several discontinuous configurations derivative of the base configuration CB, it is advisable to analyze the number of the adjacent B_v meshes possessing a common vertex. Four, three, two or only one vertex may be shared by the mesh with four adjacent meshes. The order of the common vertices of the adjacent meshes also affects the diversification of the derivative configurations.

The first derivative configuration CP1 (Figure 14) examined in the previous section is characterized by the fact that two B_vij and B_vi+1j+1 subsequent meshes of B_v arranged in the diagonal directions have one common vertex. The meshes arranged in the orthogonal directions do not have such a common vertex, and the respective sides of two adjacent meshes are mutually rotated in the planes of Γ.

The conditions determining the positions of the vertices belonging to each tetrad of the adjacent B_v meshes of the configuration CP2 are adopted so that a complete separation of the common vertices of the basic configuration CB is achieved. Thus, no pair of the adjacent meshes of CP2 has common vertices, and one side of each pair of two adjacent B_v meshes is displaced in the respective plane of Γ. As a result, a discontinuous roof structure consisting of many single transformed shells Ω_ij limited by the mutually rotated eaves B_vij is built (Figure 12 and Figure 15). The displacement is accomplished by simultaneously moving two ends of the above side along the respective side edges of Γ either in the directions of the respective Γ vertices or in the opposite directions. This action causes the additional increments of the division coefficients d_SijAij, d_SijBij, d_SijCij and d_SijDij resulting from the movement to be of the same sign.

For the special case when the values of these increments are identical, it is necessary to consider the number of the modifications accomplished for the subsequent pairs of two adjacent B_v meshes when transmitting from B_v11 to the examined B_vij. The transition from B_v11 to B_v23 requires three skips between B_v11 and B_v12, B_v12 and B_v13, and B_v13 and B_v23. Thus, in order to build a mesh B_vij, the number i + j − 2 of the skips is required. If the values of the increments assigned to the division coefficients d_SAij, d_SBij, d_SCij and d_SDij of CP2 are equal to the same constant dd_ij, then the increments dd_Aij, dd_Bij, dd_Cij and dd_Dij used for the vertices A_ij, B_ij, C_ij and D_ij (in relation to the base configuration CB) can be calculated from the formula

dd_Aij = dd_Bij = dd_Cij = dd_Dij = (i + j − 2) ∙ dd_ij

(13)

where dd_ij is the arbitrary constant related to the abovementioned skips.

It should be noted that two selected vertices of each pair of the adjacent meshes of B_v arranged in the diagonal directions (Figure 15), for example, B_v21 and B_v12, have one common vertex. If we change the values of the coefficients d_SijAij, d_SijBij, d_SijCij and d_SijDij of all B_vij to create the configuration CP2 (in relation to the basic configuration CB) using Equation (13), then we should obtain the identical location of the vertices of the adjacent diagonal meshes B_vij and B_vi-1j+1 only for the very specific mutual position of all vertices W_ABij, W_CDij, W_BCij and W_ADij of Γ. In fact, it will not be possible to achieve the abovementioned property of CP2 if we change the positions of these vertices of Γ. This issue goes beyond the scope of the article.

7. Conclusions

The method for creating discontinuous configurations derivative of the specific continuous basic configurations of the transformed shell roof structures defined by means of the systems of planes is proposed due to the necessity to assemble many complete transformed shells into one roof structure Ω resulting from the material limitations of the transformed corrugated sheeting. The method uses two specific reference networks Γ and B_v and the proposed set of the division coefficients of the respective pairs of the vertices belonging to the first reference polyhedral network Γ by the vertices of the second reference polygonal network B_v. The network Γ determines the general form of a building. The network B_v defines the degree of the folding and discontinuity of a roof structure Ω. Both networks Γ and B_v define the particular form and the general curvature of Ω.

The obtained results of the conducted research on the development of the rules governing the formation of the examined continuous roof shell structures and their modifications to the forms of discontinuous regular structures of many complete shells arranged in unconventional visually attractive patterns were implemented into the method’s algorithm. In particular, the relationships governing the position of each mesh B_vij (Ω_ij) in the network B_v (structure Ω) and the values of the partition coefficients assigned to the vertices of each mesh were determined for the basic configuration CB and two derivative configurations CP1 and CP2. The position of each mesh B_vij in B_v defined by the appropriate formula related to the independent variables i and j was discussed. The complete tetrahedra Γ_ij, closed spatial quadrangles B_vij and ruled shell sectors Ω_ij constituted the complex material used for creating the structures.

The invented formulas govern the mutual position of the vertices A_ij, B_ij, C_ij and D_ij of the subsequent meshes B_vij (sectors Ω_ij) in the network B_v (structure Ω). The position results from: (a) the proportions assumed as functions of the positions of these vertices in relation to the respective points of the reference surface and the vertices of the Γ polyhedral network used, (b) the adopted formulas and integer values of the variables i and j related to the sequence of B_vij in B_v. The expected types of the geometric patterns formed by B_vij (Ω_ij) on the network B_v (structure Ω) result from the relationships between the mutual positions of the vertices of the adjacent meshes B_vij. The developed formulas lead to the respective mutual displacements of the adjacent meshes and their vertices along the side edges of the polyhedral structure Γ, which allows one to achieve the appropriate type and form of the final discontinuous roof structure.