Complex Building Forms Roofed with Transformed Shell Units and Defined by Saddle Surfaces

Abstract

A novel method and description of creating diversified complex original building forms roofed with a number of transformed folded shell units developed on the basis of a novel reference polyhedral network and arranged according to a reference surface with the negative Gaussian curvature is presented. For that purpose, specific reference polyhedral networks is are defined as a complex material deliberately composed of many regular tetrahedrons that are arranged regularly to obtain original attractive complex general building forms. The proposed method is a significant extension of the previous method for shaping roof structures with the positive Gaussian curvature and fills existing gaps in current scientific knowledge. The extended method enables the designer to significantly increase the variety of the created complex shell roof forms and plane-walled folded elevation forms of buildings and to define the shapes of their rod structural systems. It allows one to overcome the existing significant geometric and material limitations related to shape transformations of nominally flat rectangular folded steel sheets into different shell forms. The developed extension is based on formation of a set of properly connected tetrahedra as a material determining different (a) inclination of elevation walls to the vertical, and (b) distribution of many individual warped roof shells in accordance with the properties of a regular surface with negative Gaussian curvature. A number of the adopted specific sets of division coefficients (parameters) is used for determining the entire network and its complete tetrahedra. The presented description makes it possible to adopt appropriate assumptions and data and then employ the innovative method to obtain the expected characteristics of the unconventional building form shaped. The presented three different special forms created with the help of the novel method and the appropriately selected diversified values of the division coefficients of pairs of the vertices of a polyhedral reference network, a polygonal eaves network and points of a reference surface confirm the innovative scientific nature of the obtained results. The method has to be computationally aided due to the complexity of mathematical operations and the need to visualize the designed forms.

1. Introduction

Nominally flat folded rectangular steel sheets are characterized by relatively low transverse bending and longitudinal torsional stiffness [1]. This property enables the designer to exploit their diversified elastic transformations to shape diversified ruled shell forms [2] (Figure 1). However, there are significant geometric and material limitations in the formation of the shells due to a significant influence of these and other stiffnesses on the final shape of the transformed shells [3]. The limitations very often result in significant values of stresses and, for the cases of high transformation degrees, significant values of strains. The intentional spatial shape deformations are called initial transformations [2].

Further limitations are the straightness of all transformed shell folds and the rectangular shape of the sheets before transformation. On the one hand, these limitations simplify the process of designing and assembling the transformed roof shells. On the other hand, they induce the edge lines of the developed building models to be shaped in the form of various ruled surface sectors [4] each of which is limited by a closed spatial quadrangle with vertex angles close to right [2]. Due to the limited length of complete folded sheets used for roofing, single shell sheeting, being the result of joining the individual sheets with their adjacent longitudinal edges, can cover buildings with a span of up to 15–17 m. Therefore, for medium and long span roofs, edge roof structures composed of a few single folded transformed shells must be accomplished. The simplest way to shape the complex building structures is to determine special sums of many single shells transformed into parabolic-hyperbolic folded sectors, the edge lines of which are closed spatial quadrangles [4,5]. Moreover, all facade wall pieces of the structures have to be flat polygons (Figure 2). In order to obtain a large variety of the designed building forms, the façade walls are inclined to the vertical.

Bearing in mind the above-mentioned assumptions and limitations, a novel polyhedral network called a reference network was developed [6]. The network is used as a geometric material for modeling various unconventional forms of buildings characterized by plane-walled folded facades and edge roof structures composed of many transformed single folded shells. The structure of the layers used in the corrugated sheeting and their temperature affect the mechanical and geometrical properties of these coating segments [7,8].

Each configuration of the network is defined by an appropriate arrangement of many tetrads of vertices defining typical tetrahedral meshes connected by common triangular walls and adequately filling the three-dimensional space. These vertices, planes and side edges of the tetrahedra define the planes and edges of all facade walls of each modeled building. In the above planes, straight segments forming closed spatial quadrangles defining single shell roof are determined. The quadrangles constitute a polygonal network of the roof eaves structure and delimit all individual single roof shells. By varying the geometric properties of the polyhedral reference network and the polygonal network, in particular their subsequent meshes, some variety of building forms can be obtained, including multi-wall folded facades and multi-shell ribbed roofs [9].

2. Critical Analysis of the Present Knowledge

Many diversified unconventional building forms and structural system intended for these forms are presented by Abdel and Mungan [10]. The comprehensive classification of different types of a large number of unconventional structural systems is published by Saitoh [11]. Most of these systems can be employed for structural systems supporting complex building forms characterized by folded elevation walls and multi-segment shell structures [9].

Great theoretical possibilities in the field of shaping various ruled forms made up of the transformed folded sheets with an open profiles have resulted in extensive research development in the field of single shell sheeting transformations and creation of multi-segment shell structures [2]. On the basis of the tests and analysis carried out, Bryan and Davis found that some significant material and technological limitations drastically reduce the diversification of the shell forms made up of transformed sheeting to a basic type of shallow hyperbolic paraboloids called hypars [12].

However, the developed novel Reichhart’s method enables one to increase the diversity of shaping the unconventional roof shells and their structures composed of the nominally flat single-layer sheeting transformed into the position of the rigid skew directrices generating folded shell shapes [3].

The results of the tests performed by Abramczyk [13] indicate an important role played by a contraction of each transformed shell sheeting in shaping unconventional forms of buildings roofed with thin-walled folded sheeting transformed elastically into different ruled surfaces [4]. Abramczyk has added the condition related to the central location of the contraction in all effectively transformed folded shells to the Reichhart’s algorithm.

The contraction of a roof shell can be modeled with a line of striction of a ruled surface. Thus, the Reichhart’s method was extended with the boundary condition related to the line of striction. This action should result in a rational shaping of the shell roof forms so that they could be characterized by the relatively high visual attractiveness and the minimum effort [14]. If athe central location of the contraction, halfway along the length of each shell fold, is achieved by means of the arbitrary roof directrices, then the static-strength work of the transformed sheeting is stabilized and rationalized [4]. In this way, a freedom of designing of free-form roof and elevation structures should be achieved [2].

Biswas and Iffland [15] elaborated two trivial systems of many congruent rectilinear shell units distributed over a sphere with the help of bundles of planes, where the complete transformed shells are made of revolved hyperboloids or right hyperbolic paraboloids restricted by spatial straight quadrangles. The proposed method for shaping ribbed structures composed of many congruent shells can be extended to create complex systems of many complete ruled shells separated by sets of planes containing their edge lines.

Prokopska and Abramczyk have carried out some simple systems of reference tetrahedrons to model complete free forms with oblique and folded plane elevations, and roofed with many transformed complete shell sectors [16,17]. The presented building structures are often composed of quarters of hyperbolic paraboloids arranged symmetrically in different configurations. The proposed diversified free form building structures roofed with complex corrugated shells are characterized by medium spans.

For the engineering developments, Abramczyk has proposed that each single shell segment is to be modeled with a sector of a warped surface [2] limited by a closed spatial line composed of four straight segments contained in four planes of a polyhedral reference network [6]. The planes of the reference network are a specific system dividing the designed roof structure into many complete shell segments arranged regularly in the three-dimensional space. In this way, all directrices of the shell sectors can be easily defined in these planes [9].

Pottman has proposed a few comprehensive methods for shaping the systems of planes separating subsequent plane and smooth shell sectors arranged on different regular surfaces [18]. If we want to shape transformed folded shell sheeting by means of the methods, their significant modifications taking into account the geometric and material constraints of the folded sheets must be performed. A few methods carried out by Attard [19], Vlasow [20] or Vasiri [21] can also be implemented to design thin-walled sheeting subjected to large displacements. Samyn has described a method for creating the transformed folded shells made up of aluminum or PVC [22].

The procedure for shaping complex building structures covered with transformed shell roof structures, the general form of which is characterized by a positive Gaussian curvature (Figure 3), has been described by Abramczyk [6] in a fairly accurate manner. This procedure has also been implemented in computer procedures [23]. However, there is no analogous procedure and implementation where the overall form of the shell roof structure complies with the negative Gaussian curvature. This issue is analyzed in detail in this article.

The analogous universal systems of planes called polyhedral reference networks are utilized in this article. On the basis of these systems, some derivative systems, including polygonal or shell sector networks can be defined to achieve different free-form buildings characterized by complex folded elevation walls and multi-sector shell roof structures (Figure 4) [24].

To increase the attractiveness of the building edge shell structures and the whole urban system, Prokopska has proposed green plant gardens located on the shell roof segments and specific communication routes between the segments [25]. The space around the designed building form, its physical form, urban greenways and cultural patterns of a whole spatial system have to be investigated. Sharma has described [26] the relation appearing between the formation of the urban space and the social experience of the human self. Hasgül [27] presents very important and interesting mathematics-based graph studies of patterns and shapes, thermal based photography and morphology related to the design syntax. Eekhout [28] provides the results of his research in the field of forms taking account of the relationships occurring between the function, structure, internal and external texture, static-strength work and comfort conditions. The systematic morphology leads to the rationalization of each design process.

A very interesting approach to exploit the mechanical properties of new materials making it possible to shape original forms of elevation and roof structures is used by Marin et al. [29], where the classical theory of elasticity developed by Green and Lindsay is extended in terms of the thermo-elasticity theory for dipolar bodies. The proposed novel method is based on a reciprocal theorem and not restrictive boundary conditions [30].

A method for designing complex architectonic forms and engineering rational systems has been developed by Ręebielak [31]. A number of attractive and effective structures adapted to human needs and the built environment has been provided by Tarczewski at al. [32].

The methods for shaping single and complex shell roofs presented in the above literature should be directed towards making each roof from many nominally flat rectangular thin-walled corrugated steel sheets. In order to use these sheets for shell roof covering, shape transformations are required. It is advisable to shape the edge line of each shell segment in the form of a spatial quadrangle with corner angles close to straight and take into account the shape changes and limitations resulting from the shape transformations.

The given references refer to single shells or simple ribbed structures composed of several single shells. The innovative method developed by Abramczyk allows one for a significant increase in the variety of general building forms. However, the method is limited only to structures, where the arrangement of all individual shell segments is consistent with the properties of surfaces characterized by the positive Gaussian curvature (saddle surfaces). In some articles published by the researcher, the possibility of using various structures composed of many single shells distributed on regular surfaces with the negative Gaussian curvature (saddle surfaces) is identified.

However, a comprehensive novel method is presented in the present article and fills the gap in the current knowledge. This area is also important because torsional transformations of a single folded sheet lead to its various shell forms with the negative Gaussian curvature (saddle surfaces). After filling this gap, the authors intend to propose some innovative structural systems supporting the transformed folded shells and extend laboratory tests in the field of thin-walled folded sheeting transformed elastically into various forms of saddle surfaces.

3. The Aim

The aim is to present an innovative procedure leading to the creation of some building models characterized by unconventional folded forms of their façades and roofs, where the façade walls are inclined both inside and outside the designed building, and individual roof whose shell sectors are arranged in a three-dimensional space according to the properties of a surface with the negative Gaussian curvature. In order to achieve the above-mentioned goal, a novel algorithm for creating a specific type of the polyhedral reference networks has to be used. The algorithm is implemented into a new procedure for shaping complex building forms in such a way that the created models are located between the vertices of the reference network. However, for the case of the previous building structures with the positive Gaussian curvature, the vertices are positioned on the same side of a reference surface.

The presented procedure allows for arranging the facade and roof elements in relation to the vertices of a reference network using appropriate coefficients expressing some proportions between the distances of the appropriate pairs of the adjacent vertices, and the distances of the characteristic points of the created models from the respective vertices of the reference network. The differentiation of the values of the coefficients adopted as independent variables leads to the differentiation of the obtained building forms. The degree of differentiation of the created models depends on the number of the assumed independent variables. In particular, all the coefficients used may depend on one parameter, which leads to adopting one independent variable and obtaining a single set of building forms with little differentiation.

It is decided that a detailed presentation of the proposed procedure is going to be made using some specific examples. A few different forms obtainable by the procedure are given to observe the possibilities of the procedure. A detailed description of the whole set of the examined diversified complex building forms goes beyond the scope of this article. There are only presented the main relationships governing the position of each mesh B_vij (each shell unit Ω_ij) in the eaves network B_v (the shell structure Ω) and three sets of the values of the partition coefficients assigned to the vertices of the same basic reference polyhedral network to analyze three specific types of the general building forms that are different to each other.

4. The Method’s Concept

At first, a network Γ composed of a number appropriately arranged tetrahedra Γ_ij must be defined. On the basis Γ, a model Σ of a complex folded building form can be created s as follows (Figure 3). The walls of Σ should be included in the planes of Γ. The elevation edges of Σ should be included in the side edges of Γ. The eaves lines of each single shell of the roof structure must be contained in the planes of Γ. The vertices of each eaves line must belong to the respective side edges of Γ. In the presented procedure, these vertices have to be located between the vertices of Γ.

The horizontal base plane P_b of the whole structure Σ must intersect all side edges of Γ. The intersecting points should also belong to the side edges and lie between the vertices of Γ. The division coefficients of the side edges of Γ by the vertices of the base and Σ must ensure appropriate positions of these vertices.

Construction of Σ begins with the determination of the first central tetrahedral mesh Γ₁₁ of Γ. Γ₁₁ is created on the basis of two arbitrary skew straight lines u₁₁ and v₁₁ perpendicular to each other, called the axes of Γ₁₁ (Figure 5a). An arbitrary distance d_uv11 between u₁₁ and v₁₁ has to be adopted. It is measured along the straight line n₁₁ perpendicular to u₁₁ and v₁₁ between the points O_u11 and O_v11 of intersecting n₁₁ with u₁₁ and v₁₁.

In order to determine the positions of four vertices W_AB11, W_CD11, W_AD11 and W_BC11 of Γ₁₁, it is necessary to adopt the distance duv₁₁ between the above skew axes, the distance d_u11 between W_BC11 and W_AD11 and the distance d_v11 between W_AB11 and W_CD11. Based on the above assumptions, it is possible to calculate the coordinates of the above vertices in the orthogonal coordinate system [x,y,z] and define four sides of Γ₁₁ (Figure 5a).

A few subsequent meshes are located in orthogonal and diagonal directions of Γ₁₁ (Figure 6) as follows. For the case of the mesh Γ₁₂, three vertices W_AB12, W_CD12 and W_AD12 are adopted to be identical to the vertices W_AB11, W_CD11 and W_BC11 of Γ₁₁, respectively. The axis u₁₂ of Γ₁₂ is assumed to be identical to u₁₁.

The location of the vertex W_BC12 is determined so that it is contained in the plane (y, z), distant from W_BC11 by the adopted value of dv₁₂ and distant from the axis u₁₁ by the height d_BC12 of the triangle W_BC12W_CD12W_AB12 measured from W_BC12, where W_BC12W_CD12W_AB12 should be congruent to W_BC11W_CD11W_AB11. The above activities lead to the creation of the form presented in Figure 7a.

It is more convenient to control the shape of Σ when the proportions dd_BC12 = d_v12/d_v11 and dd_BC12 = d_BC12/d_BC11 are used instead of the above-mentioned distances. D_BC12 is the distance of W_BC12 from u₁₂. Similarly, d_BC11 is the distance of W_BC11 from u₁₁. However, the distances can be calculated with the help of these coefficients. Adoption of the proportions between all meshes makes it possible to parametrize and control the shape of Σ, and, consequently, create diversified forms Γ and Σ. For the mesh Γ₁₂ presented in Figure 7a, dd_v12 = 1 and dd_BC12 = 1. The parametrization should lead to a division of the forms Σ into different groups having similar geometric properties to predict the expected properties of the designed forms Σ. The subsequent tetrahedra Γ_1j (j = 1 to N, where N—the arbitrary natural number) belonging to the first orthogonal strip of Γ are constructed similar to Γ₁₂.

All tetrahedra Γ_i1 of the second orthogonal strip are formed in the same way as Γ_1j belonging to the first strip. The vertices W_AD21, W_BC21 and W_AB21 of the first Γ₂₁ tetrahedron, Figure 6, are assumed to be identical to the vertices of W_AD11, W_BC11 and W_CD11 of Γ₁₁, respectively, when creating the network Γ.

The fourth W_CD21 vertex of Γ₂₁ is constructed in the (x, z) plane in two arbitrary distances d_u21 from W_CD11 and d_CD21 from v₁₁. The method employed for parameterizing Γ consists in defining a certain number of coefficients expressing the proportions between the distances of the vertices of the subsequently created Γ_ij’s axes. In the case of the Γ₂₁ tetrahedron, these proportions can be given as follows: dd_u21 = d_u21/d_u11 and dd_CD21 = d_CD21/d_CD11. The above activities lead to the construction of the form presented in Figure 7b for which dd_u21 = dd_CD21 = 1.

In the presented algorithm, there is no freedom in determining the vertices of Γ₂₂ and other Γ_ij tetrahedra arranged in diagonal strips (for i ≠ 1 or j ≠ 1). For the Γ₂₂ tetrahedron, it is assumed that W_AD22 = W_BC11, W_BC22 = W_AD12, W_CD22 = W_CD21, W_AB22 = W_CD11. The above activities lead to the creation of the form presented in Figure 8.

Generally, for diagonal tetrahedra, W_ADij = W_BCi−1j−1, W_BCij = W_{AD i−1j}, W_CDij = W_CDij−1, W_ABij = W_CDi−1j−1. A slight modification of the described procedure allows one to set W_ADij, W_BCij, W_CDij and W_ABij at any points of the respective network side edges, not only at the vertices of the previously created tetrahedra. This reduction of the limitations in relation to the Γ network leads to fundamental changes in the proportions between the overall dimensions and the size of the elements of the building model shaped. The description of the modified procedure for shaping the Γ network goes beyond the scope of this work.

Each network Γ₁ (Figure 8) formed according to the proposed algorithm is composed of four tetrahedra Γ_ij (i, j = 1, 2) and can be extended to a symmetrical forms Γ (Figure 9a,b) for which (x, z) and (y, z) are the planes of symmetry. Thus, Γ consists of four symmetrical parts Γ_i (i = 1 to 4). Based on the symmetrical reference network Γ, a polygonal B_v network is created to define a multi-shell roof structure. B_v is created so that some respective relationships between the location of the vertices of the roof structure and the vertices of Γ are adopted. The relationships are defined by means of the division coefficients of the pairs {W_ABij, W_ADij}, {W_ABij, W_BCij}, {W_ADij, W_CDij}, {W_ADij, W_BCij} of each Γ_ij tetrahedron by the vertices of the eaves polygonal network B_v, reference surface and the base of the form Σ.

The first group {dd_SAij, dd_SBij, dd_SCij and dd_SDij} of the division coefficients is used to determine the points S_Aij, S_Bij, S_Cij and S_Dij on the side edges of Γ defining the reference surface ω_r to search for the network B_v. The second group {dd_Aij, dd_Bij, dd_Cij and dd_Dij} of the division coefficients is used to determine the points A_ij, B_ij, C_ij and D_ij constituting the vertices of the polygonal network B_v. For the networks under consideration: (a) all acceptable values of the above-mentioned two types of the coefficients are in the range (0, 1), (b) the reference surfaces defined by the points S_Aij, S_Bij, S_Cij and S_Dij have negative Gaussian curvature. These limitations results from the geometric properties of the network Γ made. They proves the innovative nature of the performed analysis and the proposed procedure. In the case of the topics discussed so far in other articles, the values of these coefficients are within the range (1, +∞), and the reference surface is characterized by the positive Gaussian curvature. The case in which the values of the above-mentioned coefficients are within the range (−∞, 0) is similar to that with the range (1, +∞) but requires some modifications.

As it is convenient to define the positions of the vertices A_ij, B_ij, C_ij and D_ij with respect to the reference surface ω_r, it is rational to use coefficients taking account of the difference in the levels of the vertices belonging to ω_r and B_v. For this purpose, the quotient of: (a) the respective values of the division coefficients of pairs {W_ABij, W_ADij}, {W_ABij, W_BCij}, {W_ADij, W_CDij} and {W_ADij, W_BCij} by: (a) points S_Aij, S_Bij, S_Cij and S_Dij, and (b) points A_ij, B_ij, C_ij and D_ij can be calculated. However, the exact description of this problem is presented by Abramczyk [24] and goes beyond the scope of the present article.

In the considered examples, the values of the partition coefficients used for the meshes B_vij lead to small or big folding of the examined roof structure covered with complete transformed shells separated by ribs. On the other hand, the base of the modeled building is flat and horizontal, so the z-coordinates of all base points, belonging to the P_b plane, are the same. The arbitrary level of the P_b plane is adopted as constant z_P from the interval <0, dd_uv11>. The coordinates of all points P_Aij, P_Bij, P_Cij and P_Dij of the base belonging to the edges of the facade walls are obtained as a result of the intersection of the plane P_b with all Γ’s side edges.

The proportions taken into account, inter alia, the length, width and height of the entire Σ model and its fragments depend on the mutual position of the vertices of the reference polyhedral Γ network, the vertices of the polygonal B_v network, the base plane P_b and the reference surface used. These proportions are determined by the assumed values of the independent variables and the relationships between the values of the dependent and independent variables. In order to illustrate the impact of adopting different sets of values of the above-mentioned variables on the form of the Γ model, three examples of complex folded building forms covered with different shell roof structures characterized by the negative Gaussian curvature are presented in the next section.

5. Results

There are presented three examples showing the way of using the developed procedure in the process of shaping various unconventional complex forms of buildings, including the possibility of parameterizing these forms. In order to create the first tetrahedron Γ₁₁ of a polyhedral network Γ, the values of the following parameters have to be adopted: (1) the distances d_v11 of two vertices W_BC11 and W_AD11 belonging to the first axis v₁₁ (Figure 6), (2) the ratio dd_uv11 of the above distance d_uv11 to the distance d_v11 between the oblique axes u₁₁ and v_11, (3) the ratio dd_u11 of the distance d_u11 of two vertices W_AB12 and W_CD12 belonging to the second axis u₁₁ to d_v11. The values employed are given in

Table 1. The values of the initial parameters of Γ₁.

Initial Parameter	dv11 [mm]	dduv11	ddu11
	20,000	5	1

.

Identical values are adopted for the remaining congruent tetrahedra located in the orthogonal directions of the developed Γ₁ reference network. As a result of the respective composing of Γ₁₁, Γ₂₁ and Γ₁₂ a network Γ₁ can be created. The vertices of Γ₂₂ are lain at four vertices of the above tetrahedrons obtained previously. The co-ordinates of these vertices are given in

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

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
WAB11	−10,000	0	0
WCD11	10,000	0	0
WAD11	0	10,000	50,000
WBC11	0	−10,000	50,000
WAB12	−10,000	0	0
WCD12	28,793	0	6840
WAD12	0	10,000	50,000
WBC12	0	−10,000	50,000
WAB21	0	−28,793	43,160
WCD21	0	−10,000	50,000
WAD21	−10,000	0	0
WBC21	10,000	0	0
WAB22	28,793	0	6840
WCD22	10,000	0	0
WAD22	0	−28,793	43,160
WBC21	0	−10,000	50,000

in Appendix.

Then, to determine the positions of the vertices S_A11, S_B11, S_C11 and S_D11 of the first mesh of the reference surface ω_r lying in the side edges of Γ₁₁, the coefficients dd_SA11, dd_SB11, dd_SC11 and dd_SD11 constituting the division coefficients of the vertices of the Γ₁₁ tetrahedron by S_A11, S_B11, S_C11 and S_D11 should be adopted. The arbitrary values of these coefficients are presented in

Table A2. The initial division coefficients defining the positions of the points SA_ij, SB_ij, SC_ij, SD_ij and the B_v’s vertices of the first structure Σ₁ for i, j = 1, 2.

Ratio	Value
ddSA11	0.465
ddSB11	0.478
ddSC11	0.489
ddSD11	0.489
ddSA12	0.478
ddSB12	0.600
ddSC12	0.617
ddSD12	0.489
ddSA22	0.327
ddSB22	0.450
ddSC22	0.600
ddSD22	0.478
ddA11	0.480
ddB11	0.462
ddC11	0.500
ddD11	0.474
ddA12	0.462
ddB12	0.600
ddC12	0.600
ddD12	0.500
ddA22	0.343
ddB22	0.435
ddC22	0.600
ddD22	0.462

in the Appendix. The calculated values of the coordinates of these vertices are presented in

Table A3. The coordinates of the points defining the reference surface ω₁ of the first structure Σ₁.

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
SA11	−5354.1	4645.9	23,228.7
SB11	−5221.6	−4778.5	23,892.5
SC11	5107.3	−4892.7	24,463.6
SD11	5107.3	4892.7	24,463.6
SA12	−5221.6	−4778.5	23,892.5
SB12	−3748.6	−18,000.2	26,980.8
SC12	3826.3	−17,776.5	26,645.6
SD12	5107.3	−4892.7	24,463.6
SA21	−19,413.7	3257.6	20,899.9
SB21	−19,366.8	−3273.9	20,970.3
SC21	−5221.6	−4778.5	23,892.5
SD21	−5354.1	4645.9	23,228.7
SA22	−19,366.8	−3273.9	20,970.3
SB22	−15,836.3	−12,957.2	23,184.0
SC22	−3748.6	−18,000.2	26,980.8
SD22	−5221.6	−4778.5	23,892

in the Appendix.

Another operation provided for in the algorithm of the procedure is to determine the positions of the vertices A₁₁, B₁₁, C₁₁ and D₁₁ of the first mesh B_v11 of the polygonal net B_v1. For the form shaped (Figure 10) the appropriate values of the coefficients dd_A11, dd_B11, dd_C11 and dd_D11 of the vertices of the Γ₁₁ mesh by A₁₁, B₁₁, C₁₁ and D₁₁ are adopted. These values are given in

Table A4. The coordinates of the vertices A_ij, B_ij, C_ij, D_ij (for i, j = 1, 2) of the eaves edge net B_v1 (and the first structure Σ₁).

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
A11	−5200.3	4799.7	23,998.5
B11	−5375.4	−4624.6	23,122.8
C11	5000	−5000	25,000
D11	5261.2	4738.8	23,693.8
A12	xB11	yB11	zB11
B12	−3748.6	−18,000.2	26,980.8
C12	4002.5	−17,2693	25,885.2
D12	xC11	yC11	zC11
A21	−19,849.7	3106.2	20,246.4
B21	−18,930.8	−3425.3	21,623.8
C21	xB11	yB11	zB11
D21	xA11	yA11	zA11
A22	xB21	yB21	zB21
B22	−16,258.5	−12,535.1	22,651.6
C22	xB12	yB12	zB12
D22	xB11	yB11	zB11

in Appendix. The p_z level of the base plane of the considered form is equal to 15,920 mm in [x,y,z]. The individual vertices P_A11, P_B11, P_C11 and P_D11 of this base can be constructed as the points of intersection of the horizontal plane p_z with the side edges of Γ₁.

The following meshes of the Γ₁, B_v1 nets and the reference surface are created in the same way. Alternatively, some vertices must be adopted at the positions of the corresponding vertices belonging to the previously constructed meshes.

A characteristic feature of the folded forms created by the procedure is the different inclination of the adjacent facade walls to the vertical. The form presented in Figure 10a–c is characterized by two opposite façade walls directed in the x-axis direction and inclined with their bases inwards, while the other two façade walls are inclined with their bases outwards.

On the other hand, a specific feature distinguishing this form from two following forms is the fact that its elevation walls with their bases inwards are much longer than two other elevation walls with bases inclined outwards. As a result, the form has an elongated elliptical shape when projected onto a horizontal plane, and the x axis can be considered as the principal axis of the imaginary ellipse (Figure 10c).

Three exemplary forms Σ were created based on the same reference network Γ. For these forms, appropriate and different values of three sets of the partition coefficients related to the points of the created reference surfaces, eaves networks and the bases were adopted. The first set was adopted so that the form Σ has an elongated form in the direction of the axis x in the projection on the plane (x, y) and the base plane is closer to the lower vertices of the reference network Γ (Figure 10).

The second set of the partition coefficients was adopted so that the second form Σ has an elongated form in the direction of the y axis in the projection on the (x, y) plane, the plane of the base is halfway between the upper and lower vertices of the network Γ, and the points of the reference surface ω_r lie close to the base (Figure 11). The third set of the partition coefficients was adopted so that the third form Σ has a square form when projected onto the (x, y) plane, the base plane is halfway between the upper and lower vertices of the network Γ, and the points of the reference surface ω_r lie further from the base than in the previous case (Figure 12). The differences between the essential dimensions of the above forms are noticeable when comparing the respective views of each of the above-mentioned three forms.

The values of the relevant parameters and coordinates of these forms are given in Table A2, Table A3, Table A4,

Table A5. The initial division coefficients defining the positions of the points S_Aij, S_Bij, S_Cij, S_Dij and the B_v’s vertices of the second structure Σ₂ for i, j = 1, 2.

Ratio	Value
ddSA11	0.682
ddSB11	0.682
ddSC11	0.682
ddSD11	0.682
ddSA12	0.682
ddSB12	0.869
ddSC12	0.882
ddSD12	0.682
ddSA22	0.564
ddSB22	0.750
ddSC22	0.869
ddSD22	0.682
ddA11	0.624
ddB11	0.739
ddC11	0.624
ddD11	0.739
ddA12	0.739
ddB12	0.819
ddC12	0.919
ddD12	0.624
ddA22	0.510
ddB22	0.805
ddC22	0.819
ddD22	0.739

,

Table A6. The coordinates of the points defining the reference surface ω₂ of the second structure Σ₂.

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
SA11	−3184	6816	34,080
SB11	−3184	−6816	34,080
SC11	3184	−6816	34,080
SD11	3184	6816	34,080
SA12	−3184	−6816	34,080
SB12	−1814.9	−24,600.6	37,870.4
SC12	1868.1	−24,539.6	37,793.5
SD12	3184	−6816	34,080
SA21	−12,549.5	56,415	31,189
SB21	−12,546.6	−5642.5	31,193.4
SC21	−3184	−6816	34,080
SD21	−3184	6816	34,080
SA22	−12,546.6	−5642.5	31,193.4
SB22	−71,984	−21,595.4	34,080
SC22	−1814.9	−24,600.6	37,870.4
SD22	−3184	−6816	34,080

,

Table A7. The coordinates of the vertices A_ij, B_ij, C_ij, D_ij (for i, j = 1, 2) of the eaves edge net B_v1 (and the second structure Σ₂).

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
A11	−3761.4	6238.6	31,193
B11	−2606.7	−7393.3	36,966.7
C11	3761.4	−6238.6	31,193
D11	2606.7	7393.3	36,966.7
A12	xB11	yB11	zB11
B12	−1811.7	−23,577.4	35,340.5
C12	757.6	−26,612.6	39,890
D12	xC11	yC11	zC11
A21	−914.7	6209.3	33,639.4
B21	−14,114.4	−5098	28,843.3
C21	xB11	yB11	zB11
D21	xA11	yA11	zB11
A22	xB21	yB21	zB21
B22	−1811.7	−23,577.4	35,340.5
C22	xB12	yB12	zB12
D22	xB11	yB11	zB11

,

Table A8. The initial division coefficients defining the positions of the points S_Aij, S_Bij, S_Cij, S_Dij and the B_v’s vertices of the third structure Σ₃ for i, j = 1, 2.

Ratio	Value
ddSA11	0.544
ddSB11	0.544
ddSC11	0.552
ddSD11	0.552
ddSA12	0.544
ddSB12	0.715
ddSC12	0.721
ddSD12	0.552
ddSA22	0.460
ddSB22	0.600
ddSC22	0.715
ddSD22	0.544
ddA11	0.545
ddB11	0.531
ddC11	0.567
ddD11	0.536
ddA12	0.531
ddB12	0.730
ddC12	0.706
ddD12	0.567
ddA22	0.475
ddB22	0.585
ddC22	0.730
ddD22	0.531

,

Table A9. The coordinates of the points defining the reference surface ω₃ of the third structure Σ₃.

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
SA11	−4549.4	5450.6	27,252.8
SB11	−4416.7	−5577.5	27,228.5
SC11	4484.2	−5515.9	27,577.8
SD11	4484.2	5515.9	27,577.8
SA12	−4416.7	−5577.5	27,228.5
SB12	−2847.1	−20,596	30,871.7
SC12	2790.6	−20,758.7	31,115.5
SD12	4484.2	−5515.9	27,577.8
SA21	−15,814.4	45,076	26,295.0
SB21	−15,544.2	−4601.5	26,700.1
SC21	−4416.7	−5577.5	27,228.5
SD21	−4549.4	5450.6	27,252.8
SA22	−15,544.2	−4601.5	26,700.1
SB22	−11,517.3	−17,276.3	28,631.9
SC22	−2847.1	−20,596	30,871.7
SD22	4484.2	−5515.9	27,577.8

and

Table A10. The coordinates of the vertices A_ij, B_ij, C_ij, D_ij (for i, j = 1, 2) of the eaves edge net B_v1 (and the third structure Σ₃).

Point	x-Coordinate [mm]	y-Coordinate [mm]	z-Coordinate [mm]
A11	−4401.7	5598.3	27,991.6
B11	−4694.9	−5305.1	26,525.3
C11	4330.5	−5669.5	28,347.7
D11	4638.4	5361.6	26,808.1
A12	xB11	yB11	zB11
B12	−2695.7	−21,031.9	31,525.1
C12	2942	−20,322.7	30,462.1
D12	xC11	yC11	zC11
A21	−16,250.4	4356.2	25,641.5
B21	−15,101.8	−4743.8	27,344.6
C21	xB11	yB11	zB11
D21	xA11	yA11	zA11
A22	xB21	yB21	zB21
B22	−11,939.5	−16,854.1	28,009.4
C22	xB12	yB12	zB12
D22	xB11	yB11	zB11

. It is worth noticing that some geometric properties of two new forms, distinguishing them from the previously presented form are as follows. Two elevation walls of the form shown in Figure 11, whose direction is in line with the y-axis and the bases are tilted outwards, are much longer than the elevation walls titled with their bases inwards and running along the x-axis. The p_z level of the base plane of the considered form is equal to 31,010 mm in [x,y,z].

On the other hand, all façade walls of the form shown in Figure 12 are of similar length regardless of whether their bases are tilted outwards or inwards. As a result, the projection of the third form onto a horizontal plane has a shape similar to a circle or a square. The differentiation of three above-mentioned general forms is generated by the deliberate differentiation of the values of the single division coefficients used. A discussion on this subject is presented in the next section. The p_z level of the base plane of the considered form is equal to 23,194 mm in [x,y,z].

6. Discussion

The following relationships can be noticed from the observation of geometric properties of the orthogonal projections of three different structures presented in the previous section.

The inclination of the facade walls with their bases towards the inside or outside of a considered building form depends on the orthogonal directions of two types of axes v_ij and u_ij of a polyhedral reference network Γ. For example, such a network is presented in Figure 10, Figure 11 and Figure 12. If the considered dimension of the building form is consistent with the axis v₁₁ of the Γ network, located above the base (x, y) of the form, as can be seen in Figure 10a, Figure 11a, Figure 12a the projections show facade walls inclined towards the inside of the form. However, if the dimension of the form is considered orthogonal to the axis v₁₁ located above the base of the building, as it is shown in Figure 10b, Figure 11b, Figure 12b, then the elevation walls tilted with their bases outside the considered form are visible.

The overall dimensions of a building form in all three directions of the axes of its local system [x,y,z] are determined by the shape of the base edge line, the eaves line and their mutual position in the horizontal and vertical directions (Figure 10, Figure 11 and Figure 12).

The locations of the edge lines of the base and the eaves of the created form depends on the position of their vertices in the side edges of the polyhedral reference network Γ. The more the levels of the eaves line vertices are closer to the level of the axis u₁₁, the more the base is stretched along the axis x and compressed along the axis y. However, in order to obtain similar dimensions of the form (dimensions of the edge lines of the base and the eaves) in two orthogonal horizontal directions, the vertices of the eaves edge and the vertices of the base edge should be placed at levels equidistant (or close to such) from the level of the horizontal plane passing through the midpoint of the straight line normal to u₁₁ and v₁₁, on both sides of this middle plane. All levels are defined by horizontal planes and their distances from the principal plane (x, y).

The shape of the eaves line B_v of a building form may be generated by the division coefficients of the respective pairs of vertices belonging to the polyhedral reference network Γ by the vertices of B_v. On the other hand, the shape of the base edge line can be determined by the coefficient defining their plane level. Control of the mutual position of the eaves line B_v and the base edges can be done by double division coefficients of the vertices of Γ by the eaves vertices and the base vertices. This issue goes beyond the scope of the paper.

To obtain an elongated horizontal projection of a building base along the axis x, the base level coefficient dd_P should be taken significantly lower than 0.5 (Figure 10). To generate a horizontal projection of the building base elongated along the axis y, the coefficient dd_P should be assumed significantly greater than 0.5. However, in order to obtain a horizontal projection of the building base with a comparable length of two overall dimensions measured in the x and y directions, the dd_P coefficient should be adopted close to 0.5.

If the examined polyhedral reference network Γ has properties analogous to the networks presented in Figure 11 and Figure 12, i.e., u₁₁ is identical to x of [x,y,z], v₁₁ is perpendicular to u₁₁ and distant by d_uv11 from z, then the following are true. To obtain an elongated plan view of the designed eaves line along the axis x, the coefficients dd_Aij, dd_Bij, dd_Cij and dd_Dij should be taken significantly less than 0.5. To obtain an elongated plan view of the eaves line along the axis y, the coefficients dd_Aij, dd_Bij, dd_Cij and dd_Dij should be significantly greater than 0.5. To generate a horizontal projection of roof eaves line characterized by comparable overall dimensions measured in the x and y directions, the arbitrary division coefficients close to 0.5 should be taken.

In order to generate an innovative form of a building having two façade walls inclined inwards and another two outwards of the form, and being close to a circle when projected onto a horizontal plane, the absolute values of the differences dd_Aij—0.5, dd_Bij—0.5, dd_Cij—0.5 and dd_Dij—0.5 should be adopted equal or close to an absolute value of the difference dd_P—0.5, especially for all central meshes of the net Γ.

The examined general forms are to be influenced by corrugation of the polygonal eaves network Bv generated by the size of the differences between: (a) the division coefficients of the vertices of polyhedral reference network Γ by the vertices A_ij, B_ij, C_ij and D_ij of B_v, and (b) the division coefficients of the same vertices of Γ by the corresponding points S_Aij, S_Bij, S_Cij and S_Dij of a reference surface. In addition, these forms are to be influenced by the ratio between the size of this corrugation and the height of the entire form, which depends on the division coefficients of the vertices of the polyhedral reference network by points belonging to three groups: (a) base vertex group, (b) reference surface point group, and (c) eaves vertex group. The discussion about the influence of some important proportions between the values of the above-mentioned coefficients and the geometrical properties of the building forms shaped goes beyond the scope of the article and requires definition of the so-called multiple division coefficients.

7. Conclusions

An innovative procedure significantly extending the method for modeling unconventional building forms with complex folded elevations and multi-segment transformed shell roof structure was developed. The specific features of the elaborated procedure is are: (a) an arrangement of many complete transformed shells of a roof structure in the three-dimensional space in accordance with the properties of a regular saddle surface with negative Gaussian curvature, (b) achievement of different inclinations of elevation walls to the vertical, including outside and inside of the same building. Thus, the final innovative method presented in the article fills the important gap existing in the current knowledge.

The algorithm of the procedure was presented on three specific examples of creating complex building forms using appropriate proportions between the distances of all characteristic vertices of their elevation walls and roofs. The procedure enables the designer to take some dependencies determining the respective interrelationships between the distances of the subsequent vertices of the designed reference network deciding on the complex shape of elevation walls. On the basis of the created reference network, the vertices of the polygonal eaves network determining the arrangement and forms of all multi-segment shells of the determined roof structures are defined by means of the division coefficients. The ranges of variability of the above coefficients are also defined so that it is possible to create a few specific types of the building forms under consideration.

A characteristic feature of the presented procedure is that the vertices of each polyhedral reference network are divided into two groups such that the vertices belonging to these groups define two families of the orthogonal axes lying on both sides of the reference surface employed. This leads to a complex shell roof structure the overall shape of which is characterized by the negative Gaussian curvature. This property of the designed roof form is generated by a model whose division coefficients of the vertices belonging to the reference polyhedral network by the roof eaves vertices are in the range (0, 1).

There is a need for searching for further unconventional forms of roofs and elevations as well as their innovative structural systems supporting the transformed folded shell roof sheeting. Some extend laboratory tests in the field of thin-walled folded sheeting transformed elastically into various forms of saddle surfaces are also going to be performed.