# Symmetric Shape Transformations of Folded Shell Roofs Determining Creative and Rational Shaping of Building Free Forms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art

## 3. Aims

## 4. Concept

_{0}])

- [σ] = $\left[\begin{array}{c}{\sigma}_{x}\\ {\sigma}_{y}\\ {\sigma}_{z}\\ {\tau}_{xy}\\ {\tau}_{yz}\\ {\tau}_{zx}\end{array}\right]$—Vector of stresses, [ε] = $\left[\begin{array}{c}{\epsilon}_{x}\\ {\epsilon}_{y}\\ {\epsilon}_{z}\\ {\gamma}_{xy}\\ {\gamma}_{yz}\\ {\gamma}_{zx}\end{array}\right]$—Vector of strains,
- [ε] = $\left[\begin{array}{ccc}\frac{\partial}{\partial x}& 0& 0\\ 0& \frac{\partial}{\partial y}& 0\\ 0& 0& \frac{\partial}{\partial z}\\ \frac{\partial}{\partial y}& \frac{\partial}{\partial x}& 0\\ 0& \frac{\partial}{\partial z}& \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z}& 0& \frac{\partial}{\partial x}\end{array}\right]\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$, $\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$—Vector of nodal displacements,
- [E] = $\left[\begin{array}{cc}\begin{array}{c}\left(1-v\right)c\\ v\xb7c\\ v\xb7c\\ 0\\ 0\\ 0\end{array}& \begin{array}{cc}\begin{array}{c}v\xb7c\\ \left(1-v\right)c\\ v\xb7c\\ 0\\ 0\\ 0\end{array}& \begin{array}{cc}\begin{array}{c}v\xb7c\\ v\xb7c\\ \left(1-v\right)c\\ 0\\ 0\\ 0\end{array}& \begin{array}{ccc}\begin{array}{c}0\\ 0\\ 0\\ G\\ 0\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\\ 0\\ G\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ G\end{array}\end{array}\end{array}\end{array}\end{array}\right]$; c = E/[$\left(1+v\right)\left(1-2v\right)$], G = E/2$\left(1+v\right)$,
- [ε
_{0}] = $\left[\begin{array}{c}a\Delta T\\ a\Delta T\\ a\Delta T\\ 0\\ 0\\ 0\end{array}\right]$—Vector of initial strains produced by the increment $\Delta T$ of temperature, here [ε_{0}] = $\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right]$

_{z}= τ

_{zx}= τ

_{xz}= τ

_{yx}= τ

_{yz}$\cong $ 0, [E] is the matrix of elastic stiffness, and v is Poisson’s ratio.

_{e}] + [G

_{e}]) ∙ [D] − [R]

_{e}] is the symmetrical mechanical element stiffness matrix, and [G

_{e}] is the symmetrical geometrical element stiffness matrix.

^{(i)}] in the nodal point displacement, a new modified total displacement vector the incremental solution

^{t}

^{+Δt}[D]

^{(i)}at time t + Δt in iteration i, instead of the previous one at time [D]

^{(i)}, and then t calculated during iteration i − 1. The two equations that accomplished the Newton–Raphson iteration are as follows [26]:

^{t}

^{+Δ}

^{t}[K]

^{(i−1)}[ΔD

^{(i)}] =

^{t}

^{+}

^{Δ}

^{t}[R] −

^{t}

^{+}

^{Δ}

^{t}[F]

^{(i−1)}

^{t}

^{+Δ}

^{t}[D]

^{(i)}=

^{t}

^{+}

^{Δ}

^{t}[D]

^{(i−1)}+ [ΔD

^{(i)}]

- (a)
- In static analysis:
^{t}[K] [D] =^{t}^{+Δt}[R] −^{t}[F] - (b)
- In dynamic analysis:[M]
^{t}^{+Δt}[D^{″}] +^{t}[K] [D] =^{t}^{+Δt}[R] −^{t}[F]

- (a)
- In static analysis:(
^{t}[K_{L}] +^{t}[K_{NL}]) [D] =^{t}^{+Δt}[R] −^{t}[F] - (b)
- In dynamic analysis:[M]
^{t}^{+Δt}[D^{′′}] + (^{t}[K_{L}] +^{t}[K_{NL}]) =^{t}^{+Δt}[R] −^{t}[F]

^{t}[K] is the linear strain incremental stiffness matrix,

^{t}[K

_{L}] is the linear strain incremental stiffness matrix,

^{t}[K

_{NL}] is the nonlinear strain incremental stiffness matrix,

^{t}

^{+Δt}[R] is the vector of externally applied nodal point loads at time t + Δt,

^{t}[F] is the vector of nodal point forces referring to the element stresses at time t, [D] is the vector of increments in the nodal points displacements, and

^{t}

^{+Δt}[D

^{″}] is the vector of nodal point accelerations at times t and t + Δt.

_{j,a}are the edges of these three folds (j = n = 1 to 3), and u

_{i,a}represent transverse measuring lines (i = 1 to 7). The scheme of the arrangement of the above edge and measuring lines is presented in Figure 9.

_{wpi}of its selected central section limited by its two rulings as follows [9]:

- AA = $\left[\begin{array}{cc}\frac{\partial \psi}{\partial u}& \frac{\partial \chi}{\partial u}\\ \frac{\partial \psi}{\partial v}& \frac{\partial \chi}{\partial u}\end{array}\right]$, BB = $\left[\begin{array}{cc}\frac{\partial \chi}{\partial u}& \frac{\partial \phi}{\partial u}\\ \frac{\partial \chi}{\partial v}& \frac{\partial \phi}{\partial u}\end{array}\right]$, CC = $\left[\begin{array}{cc}\frac{\partial \phi}{\partial u}& \frac{\partial \psi}{\partial u}\\ \frac{\partial \phi}{\partial v}& \frac{\partial \psi}{\partial u}\end{array}\right]$

## 5. Method for Geometrical Shaping Transformed Folded Shells Based on Lines of Striction

_{i}of ω (Figure 11), and p(u) is the unit director vector of ruling t

_{i}. All vectors p(u) have a common origin at point O

_{L}that determines the spherical indicatrix p contained in sphere f of the unit radius and center O

_{L}, and u, v are two independent variables that are well-known to be curvilinear coordinates of ω.

_{i}of ω, and v is the parameter describing the position of any point of s(u) on respective ruling t

_{i}in relation to directrix e(u). Therefore, if the line s(u) is to be the directrix e(u), the following condition must be satisfied:

_{1}of the contraction of Ω.

_{j}of a single fold is enough to build the whole shell roof because all its folds are identically transformed. A parametric equation of this surface can be given as:

_{o}is the radius, b

_{s}is a coefficient referring to the spiral lead, and u is the selected parameter of the helix of striction of the helicoid. The helix s of contraction and ruling t

_{j}of the examined helicoid are shown in Figure 13. The ruling t

_{j}is a binormal of the line s, i.e., it is perpendicular to the osculating plane of s.

_{j}

_{−1}, t

_{j}and t

_{j}

_{+1}of the considered helicoid w, modeling the longitudinal edges of two adjacent shell folds Ω

_{j}and Ω

_{j}

_{+1}, are shown in Figure 14. The position of these rulings on the surface ω can be found from the condition that the surface area of each Ω

_{j}segment must be equal to the surface area of a rectangle modeling the respective shell fold before the transformation. The method of shaping the transformed shells using the lines of striction of warped surfaces will be presented in detail in one of the authors’ subsequent publications using a specific exemplary architectural free form roofed with a transformed roof shell.

_{1}to H

_{4}. The axis of symmetry adopted for the reference tetrahedron is also taken as the axis of symmetry of the transformed roof shell. The way of creating axis-symmetric entire free forms roofed with transformed corrugated shells is presented in the next section by a detailed example based on two border plane directrices of a transformed roof shell.

## 6. Method for Geometrical Shaping Transformed Folded Shells Based on Border Directrices

_{1}limited by a horizontal base plane, and (b) a roof form Σ

_{2}bounded by two shells: upper Ω

_{g}and lower Ω

_{d}. The coordinates of the characteristic vertices of Σ are given in Table 1. A visualization of Σ is shown in Figure 16.

_{gi}(for i = 1 to 4), whose coordinates are the entered initial data. Segments e

_{i}and f

_{i}of directrices e and f, shown in Figure 17, are the determined auxiliary short lines modeling the supporting lines of subsequent folds of the shell roof being sought. The lengths of these segments were calculated to develop the simplified smooth shell model of each roof shell fold as a central sector Ω

_{i}of a warped surface limited by two rulings t

_{i}

_{−1}and t

_{i}. Each pair of these segments e

_{i}and f

_{i}was determined on the basis of two points E

_{i}and F

_{i}displaced on e and f.

_{i}on e and F

_{i}on f were changed in relation to points E

_{i}

_{−1}and F

_{i}

_{−1}, and the shape of Ω

_{I}was decided. This change makes it possible to satisfy the two conditions discussed in the concept of the paper, and relate them to the surface areas of simplified models of transformed folds and the optimal position of a contraction line along the length of each fold.

_{i}

_{−1}and t

_{i}.

_{g}and Ω

_{d}(see Figure 16). The upper one of which is the sought-after symmetrical model of the transformed folded shell sheeting. Half of this model is presented in Figure 19. It was determined using the innovative application built by one of the authors in the Rhino/Grasshopper program.

_{i}and f

_{i}, the lengths and unit twist angles α

_{j}of the consequent folds of the investigated shell, are tabulated in Table 2. The properties of the other symmetrical part can be obtained by using the z-axis symmetry of the set of the two directrices e and f.

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Foraboschi, P. The central role played by structural design in enabling the construction of buildings that advanced and revolutionized architecture. Constr. Build. Mater.
**2016**, 114, 956–976. [Google Scholar] [CrossRef] - Foraboschi, P. Structural layout that takes full advantage of the capabilities and opportunities afforded by two-way RC floors, coupled with the selection of the best technique, to avoid serviceability failures. Eng. Fail. Anal.
**2016**, 70, 377–418. [Google Scholar] [CrossRef] - Abel, J.F.; Mungan, I. Fifty Years of Progress for Shell and Spatial Structures; International Association for Shell and Spatial Structures Publishing: Madrid, Spain, 2011. [Google Scholar]
- Medwadowski, S.J. Symposium on Shell and Spatial Structures: The Development of Form. Bull. IASS
**1979**, 70, 3–10. [Google Scholar] - Saitoh, M. Recent Spatial Structures in Japan; J. JASS: Madrid, Spain, 2001. [Google Scholar]
- Foraboschi, P. Optimal design of glass plates loaded transversally. Mater. Des.
**2014**, 62, 443–458. [Google Scholar] [CrossRef] - Liu, Y.; Zwingmann, B. Carbon Fiber Reinforced Polymer for Cable Structures—A Review. Polymers
**2015**, 7, 2078–2099. [Google Scholar] [CrossRef] - Makowski, Z.S. Analysis, Design and Construction of Double-Layer Grids; Applied Science Publishers: London, UK, 1981. [Google Scholar]
- Abramczyk, J. Influence of the Shape of Flat Folded Sheets and Curved Directrices on the Geometrical Forms of Transformed Shells; Rzeszow University of Technology: Rzeszów, Poland, 2011. (In Polish) [Google Scholar]
- Abramczyk, J. Shell Free Forms of Buildings Roofed with Transformed Corrugated Sheeting; Rzeszow University of Technology: Rzeszów, Poland, 2017. [Google Scholar]
- Obrębski, J.B. Observations on Rational Designing of Space Structures. In Proceedings of the Symposium Montpellier Shell and Spatial Structures for Models to Realization IASS, Montpellier, France, 20–24 September 2004; pp. 24–25. [Google Scholar]
- Rębielak, J. Review of Some Structural Systems Developed Recently by help of Application of Numerical Models. In Proceedings of the XVIII International Conference on Lightweight Structures in Civil Engineering, Łódź, Poland, 7 December 2012; pp. 59–64. [Google Scholar]
- Reichhart, A. Geometrical and Structural Shaping Building Shells Made up of Transformed Flat Folded Sheets; Rzeszow University of Technology: Rzeszów, Poland, 2002. (In Polish) [Google Scholar]
- Grey, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 4th ed.; Champman & Hall: New York, NY, USA, 2006. [Google Scholar]
- Abramczyk, J. Principles of geometrical shaping effective shell structures forms. JCEEA
**2014**, XXXI, 5–21. [Google Scholar] [CrossRef] - McDermott, J.F. Single layer corrugated steel sheet hypars. Proc. ASCE J. Struct. Div.
**1968**, 94, 1279–1294. [Google Scholar] - Egger, H.; Fischer, M.; Resinger, F. Hyperschale aus Profilblechen. Stahlbau
**1971**, 12, 353–361. [Google Scholar] - Gergely, P.; Banavalkar, P.V.; Parker, J.E. The analysis and behavior of thin-steel hyperbolic paraboloid shells. In Review in A Research Project Sponsored by the America Iron and Steel Institute; Report 338; Ithaca: New York, NY, USA, 1971. [Google Scholar]
- Davis, J.M.; Bryan, E.R. Manual of Stressed Skin Diaphragm Design; London: Granada, Spain; London, UK, 1982. [Google Scholar]
- Petcu, V.; Gioncu, D. Corrugated hypar structures. In Proceedings of the I International Conference on Lightweight Structures in Civil Engineering, Warsaw, Poland, 1 December 1995; pp. 637–644. [Google Scholar]
- Abramczyk, J. Shape transformations of folded sheets providing shell free forms for roofing. In Proceedings of the 11th Conference on Shell Structures Theory and Applications, Gdańsk, Poland, 11–13 October 2017; Pietraszkiewicz, W., Witkowski, W., Eds.; CRC Press Taylor and Francis Group: Boca Raton, FL, USA, 2017; pp. 409–412. [Google Scholar]
- Abramczyk, J. Transformed Shell Roof Structures as the Main Determinant in Creative Shaping Building Free Forms Sensitive to Man-Made and Natural Environments. Buildings
**2019**, 9, 74. [Google Scholar] [CrossRef] [Green Version] - Prokopska, A.; Abramczyk, J. Parametric Creative Design of Building Free Forms Roofed with Transformed Shells Introducing Architect’s and Civil Engineer’s Responsible Artistic Concepts. Buildings
**2019**, 9, 58. [Google Scholar] - Kidmann, R.; Kraus, M. Steel Structures. Design Using FEM; Ernst & Sohn: Berlin, Germany, 2011. [Google Scholar]
- Cook, R.D.; Malkus, D.S.; Plesha, M.E.; Witt, R.J. Concepts and Applications of Finite Element Analysis. Design Using FEM; John Wiley & Sons Inc.: New York, NY, USA, 2002. [Google Scholar]
- Bathe, K.J. Finite Element Procedures in Engineering Analysis; Prentice-Hall Inc.: Englewood Cliffs, NJ, USA, 1982. [Google Scholar]
- Abramczyk, J.; Prokopska, A. Responsive Parametric Building Free Forms Determined by Their Elastically Transformed Steel Shell Roofs Sheeting. Buildings
**2019**, 9, 4. [Google Scholar]

**Figure 1.**Two asymmetric experimental corrugated shells supported by: (

**a**) curvilinear; (

**b**) straight skew directrices.

**Figure 3.**Two symmetrical experimental hyperbolic paraboloid shells: (

**a**) with stiffen edges; (

**b**) without stiffen edges.

**Figure 4.**Symmetrically arranged hyperbolic paraboloid shell units: (

**a**) an erected corrugated shed; (

**b**) geometrical smooth models.

**Figure 5.**Folded mechanical thin-walled model of nominally plane folded sheet transformed elastically and initially into a shell shape and the graphical expression of the “effective” stresses in MPa on its top surface. (

**a**) The direction of the view according to the longitudinal axis of the edge fold. (

**b**) The direction of the view skew to the longitudinal axis of any fold. (

**c**) An initially transformed and uniformly loaded sheeting.

**Figure 6.**Linear dependence between the normal stresses σ

_{yy}= σ

_{y}acting perpendicularly to the longitudinal axes of shell folds and the corresponding strains ε

_{yy}= ε

_{y}of an effectively transformed thin-walled corrugated shell.

**Figure 7.**Symmetrical character of the relative width increments db

_{w}of three folds (n = 1 to 3) of the same sheet measured as compatible with seven (i = 1 to 7) measuring lines passing perpendicularly to the longitudinal fold’s axes in relation to the longitudinal axes of all folds of the length L = 4900 [mm] and the unit twist angle α

_{jed}= 6928 [

^{o}]. (

**a**) The width increments presented along transverse measuring lines. (

**b**) The width increments presented along longitudinal axes of these folds.

**Figure 9.**Scheme of the arrangement of measuring points Pi, j, α located at the intersection of longitudinal lines (n = 1 to 3) separating single folds and transverse measuring lines (i = 1 to 7) on a rectangular folded sheet.

**Figure 10.**Symmetrical character of the relative width increments db

_{w}of shell folds along seven (i = 1 to 7) measuring lines passing perpendicularly to the longitudinal fold’s axes obtained for: (

**a**) four various sheets twisted by the same unit angle α

_{jed}= 6928 [

^{o}] and characterized by various lengths; (

**b**) the same sheet of length L = 4900 [mm] and twisted by various unit angles.

**Figure 11.**A central sector Ω determined on the basis of the line of striction s and spherical indicatrix p of a warped surface.

**Figure 12.**Z-axis-symmetrical sector Ω of oblique (a differs from b) hyperbolic paraboloid ω adopted as a model for a corrugated transformed shell.

**Figure 13.**Line of contraction s and binormal t

_{i}of the respective Frenet’s frame (s

_{j}, n

_{j}, t

_{j}) of s used as a ruling of the designed helicoid.

**Figure 14.**Geometrical shaping of consequent folds of a transformed corrugated shell with sectors Ω

_{j}limited by pairs of skew rulings t

_{j}

_{−1}and t

_{j}distinguished on a helicoid.

**Figure 15.**Shaping of an axial-symmetric unconventional building free form by means of a reference tetrahedron Γ.

**Figure 16.**(

**a**) Simplified model Σ = Σ

_{1}∪ Σ

_{2}of the discussed building free forms taking account of the thickness and overhang of the transformed shell roof Σ

_{2}and the spatial shape Σ

_{1}limited by elevation walls. (

**b**) Visualization of the examined building free form Σ.

**Figure 17.**Narrow smooth longitudinal shell strip Ω

_{i}modeling a complete shell fold, created by means of a Loft component and limited by two rulings t

_{i}

_{−1}(E

_{i}

_{−1}, F

_{i}

_{−1}) and t

_{i}(E

_{i}, F

_{i}) as well as two curves e

_{i}and f

_{i}.

**Figure 18.**Two basic green components representing two basic conditions related to the fold’s surface areas and line of contraction.

**Figure 20.**The architectural stadium of two free form buildings roofed with various transformed shells determined on the basis of straight and curved directrices.

Vertex | X-Coordinate | Y-Coordinate | Z-Coordinate |
---|---|---|---|

P_{1} | 5000 | −5000.0 | 0 |

P_{2} | 5000 | 5000 | 0 |

P_{3} | −5000.0 | 5000 | 0 |

P_{4} | −5000.0 | −5000.0 | 0 |

B_{1} | 10,000.00 | −7500.0 | 10,000.00 |

B_{2} | 7777.7 | 6388.9 | 5555.6 |

B_{3} | −10,000.0 | 7500 | 10,000.00 |

B_{4} | −7777.7 | −6388.9 | 5555.6 |

D_{g}_{1} | 10,650.60 | −8016.9 | 11,382.60 |

D_{g}_{2} | 8697 | 7448.5 | 5996.4 |

D_{g}_{3} | −10,650.6 | 8016.9 | 11,382.60 |

D_{g}_{4} | −8697.0 | −7448.5 | 5996.4 |

D_{d}_{1} | 11,129.50 | −8585.8 | 9526 |

D_{d}_{2} | 8156.6 | 6739.5 | 4206.1 |

D_{d}_{3} | −11,129.5 | 8585.8 | 9526 |

D_{d}_{4} | −8156.6 | −6739.5 | 4206.1 |

H_{1} | −5000 | 0 | −20,000.0 |

H_{2} | 5000 | 0 | −20,000.0 |

H_{3} | 0 | −2500.0 | −10,000.0 |

H_{4} | 0 | 2500 | −10,000.0 |

_{g}surface was modeled in the Rhino/Grasshopper program in the following way. The directrices e and f of Ω

_{g}were adopted in the form of straight sections having ends in points D

_{gi}(i = 1 to 4) at the beginning of the calculations. The straight line (D

_{g}

_{3}, D

_{g}

_{4}) was adopted as an initial ruling t

_{p}of Ω

_{g}. Selected rulings ti of the surface Ω

_{g}modeling the longitudinal edges of the subsequent folds of this shell are sought. The distance between two adjacent rulings t

_{j}

_{−1}and t

_{j}, which are the longitudinal edges of the same smooth strip modeling a single shell roof fold, was determined on the basis of two conditions related to the contraction of each transformed shell fold and the equality of the surface areas of two smooth models of this fold before and after transformation.

**Table 2.**Parameters describing the subsequent shell folds in the transformed shell whose simplified smooth model is shown in Figure 19.

Shell Fold [no.] | Length of Supporting Line e _{i} [mm] | Length of Supporting Line f _{i} [mm] | Fold’s Length [mm] | Fold’s Unit Twist Angle α_{j} [^{o}] |
---|---|---|---|---|

1 | 286.8 | 294.1 | 16,467 | 1.809 |

2 | 286.9 | 294 | 16,416 | 1.8199 |

3 | 287 | 293.8 | 16,367 | 1.8306 |

4 | 286.4 | 293.1 | 16,319 | 1.8411 |

5 | 287.1 | 293.6 | 16,273 | 1.8514 |

6 | 287.2 | 293.5 | 16,227 | 1.8614 |

7 | 287.3 | 293.4 | 16,184 | 1.8713 |

8 | 287.4 | 293.2 | 16,142 | 1.8808 |

9 | 287.5 | 293.1 | 16,101 | 1.8902 |

10 | 287.5 | 293 | 16,061 | 1.8992 |

11 | 287.6 | 292.9 | 16,024 | 1.908 |

12 | 287.7 | 292.7 | 15,988 | 1.9165 |

13 | 287.8 | 292.6 | 15,953 | 1.9247 |

14 | 287.9 | 292.5 | 15,919 | 1.9326 |

15 | 288 | 292.4 | 15,887 | 1.9403 |

16 | 288.1 | 292.3 | 15,857 | 1.9476 |

17 | 288.2 | 292.1 | 15,828 | 1.9545 |

18 | 288.3 | 292 | 15,800 | 1.9612 |

19 | 288.3 | 291.8 | 15,774 | 1.9675 |

20 | 288.4 | 291.8 | 15,750 | 1.9735 |

21 | 288.6 | 291.6 | 15,727 | 1.9791 |

22 | 288.6 | 291.5 | 15,705 | 1.9843 |

23 | 288.7 | 291.4 | 15,685 | 1.9892 |

24 | 288.8 | 291.3 | 15,667 | 1.9938 |

25 | 288.9 | 291.2 | 15,650 | 1.998 |

26 | 289.1 | 291.1 | 15,635 | 2.0017 |

27 | 289.1 | 291 | 15,622 | 2.0051 |

28 | 289.3 | 290.8 | 15,610 | 2.0081 |

29 | 289.4 | 290.7 | 15,599 | 2.0108 |

30 | 289.5 | 290.6 | 15,590 | 2.013 |

31 | 289.6 | 290.5 | 15,583 | 2.0149 |

32 | 289.7 | 290.3 | 15,577 | 2.0163 |

33 | 289.8 | 290.3 | 15,573 | 2.0174 |

34 | 289.9 | 290.1 | 15,570 | 2.018 |

35 | 290 | 290 | 15,569 | 2.0183 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abramczyk, J.; Prokopska, A.
Symmetric Shape Transformations of Folded Shell Roofs Determining Creative and Rational Shaping of Building Free Forms. *Symmetry* **2019**, *11*, 1438.
https://doi.org/10.3390/sym11121438

**AMA Style**

Abramczyk J, Prokopska A.
Symmetric Shape Transformations of Folded Shell Roofs Determining Creative and Rational Shaping of Building Free Forms. *Symmetry*. 2019; 11(12):1438.
https://doi.org/10.3390/sym11121438

**Chicago/Turabian Style**

Abramczyk, Jacek, and Aleksandra Prokopska.
2019. "Symmetric Shape Transformations of Folded Shell Roofs Determining Creative and Rational Shaping of Building Free Forms" *Symmetry* 11, no. 12: 1438.
https://doi.org/10.3390/sym11121438