Available via license: CC BY 4.0
Content may be subject to copyright.
buildings
Article
Rationalized Algorithmic-Aided Shaping a
Responsive Curvilinear Steel Bar Structure
Jolanta Dzwierzynska
Department of Architectural Design and Engineering Graphics, Faculty of Civil and Environmental Engineering
and Architecture, Rzeszow University of Technology, al. Powstancow Warszawy 12, 35-959 Rzeszow, Poland;
joladz@prz.edu.pl; Tel.: +48-17-865-1507
Received: 22 February 2019; Accepted: 7 March 2019; Published: 11 March 2019
Abstract:
The correlation of the architectural form and the structural system should be the basis
for rational shaping. This paper presents algorithmic-aided shaping curvilinear steel bar structures
for roofs, using modern digital tools, working in the environment of Rhinoceros 3D. The proposed
method consists of placing the structural nodes of the shaped bar structure on the so-called base
surface. As the base surface, the minimal surfaces with favorable mechanical properties were used.
These surfaces were obtained in two optimization methods, due to both the structural and functional
requirements. One of the methods used was the so-called form-finding method. It wasalso analyzed
the amount of shadow produced by the roof and the adjacent building complex, during a certain
research period, to find the roof’s optimal shape. The structure of the optimal shape was then
subjected to structural analysis and its members were dimensioned. The dimensioning was carried
out for two bar cross-sections, and as the optimization criterion, the smallest structure’s mass was
used. The presented research aims to show how it is possible to use generative shaping tools, so as
not to block the creative process, to obtain effective, responsive structural forms, that meet both
architectural and structural requirements.
Keywords:
steel bar structures; structural analysis; parametric design; algorithmic-aided shaping;
responsive architecture; shadows; Grasshopper; Karamba 3D
1. Introduction
Over the years, various methodologies have been developed for shaping building objects in
accordance with the current development of design ideas, building materials, as well as technologies.
The basic principles of composition in architecture and construction were set by Vitruvius in The Ten
Books of Architecture, in the early centuries [
1
]. According to them, architecture should be based
on a combination of three elements: stability, utility and beauty. Classical architecture was mainly
based on the orderly composition of shapes formed mainly on the basis of Platonic solids [
2
]. In turn,
nonlinear and organic architecture was the result of studying biological organisms, the structure
of matter,
and the
application of this knowledge to design and construction. On the other hand,
according to the dominant concepts of modernism, the shaping of structures was based on industrial
technologies, functionalism and universal models, as well as modern ways of construction and the
use of new materials [
2
]. That is why, over the years, the process of shaping building objects has been
changing depending on the contemporary canons of beauty, materials, as well as the technological
conditions. However, irrespective of these conditions, the process of shaping any building object
always has to be adapted to both architectural and structural requirements, which are interrelated.
Generally, the shaping process can be defined as the optimization of the building’s shape and form,
so that it meets the assumed initial criteria to the greatest extent. This means looking for such shapes
and dimensions of the shaped building object, that would allow it to meet the requirements resulting
Buildings 2019,9, 61; doi:10.3390/buildings9030061 www.mdpi.com/journal/buildings
Buildings 2019,9, 61 2 of 11
from its purpose, as well as its future use. Requirements of the contemporary design standards for
structures, as well as requirements of building law, are comprehensive. They are the set of related
conditions regarding reliability, the load-bearing capacity, serviceability, durability, the resistance to
exceptional impacts, and harmful impacts on the natural environment and society, etc. [3–8].
Due to this fact, the shaping of any architectural/structural spatial system can be considered
in various aspects, depending on the designing phase, as well as the nature and complexity of
the design task. However, the shaping criteria always result directly from the starting boundary
conditions for shaping: The structure’s function, safety requirements, as well as the material and
technological solutions.
The shaping phase is the phase preceding all subsequent stages of the design process, which is
why it is the most creative phase on the one hand, and on the other hand, has a significant impact
on the final form of the object [
9
,
10
]. Therefore, it very important that as many aspects as possible
concerning the future project be considered in the initial phase. It can guarantee the creation of a
sensitive, adaptable, and sustainable form, as well as a reliable structure. In addition, the conceptual
phase is related to the multi-variant analysis of initial solutions, most often undertaken within a limited
time. These conditions require the use of comprehensive tools that define the structural model, as well
as accounting for the team and interdisciplinary nature of the designing [11–13].
Due to this fact, another factor influencing the way of shaping is the type and availability of
design tools, which change with the development of technology [
13
–
15
]. Especially, during the last
twenty years, the advancement of digital technologies has influenced the whole field of engineering
design, and digital media have become a convenient means for shaping structures.
Various Computer Aided Design CAD tools enabled not only the creation of two-dimensional
documentation [
16
], but also the creation of three-dimensional models based on two-dimensional
drawings [
17
–
19
]. Moreover, thanks to the widespread use of digital tools in designing, the boundary
between physical and virtual models have blurred. This was mostly caused by the hybridization
of several methods and techniques for acquiring the model’s geometry, as well as the development
of reverse engineering [
20
]. Further, building information modeling (BIM), as a 3D model-based
approach, gave architecture and civil engineering possibilities to streamline the design process by
improving communication between the participants of the design process.
Especially, the development of the concepts and software enabling smooth modeling, which
is digital modeling based on the non-uniform rational B-spline (NURBS) and its application in
architecture, resulted in a significant change of the shaping process [
21
]. This was caused by the
ability to control the geometry of shaped forms on an ongoing basis, enabling the creation of dynamic,
parametric and non-linear forms. In turn, flexible geometry initiated the development of parametric
and respond forms, which can evolve during shaping. Such possibilities created modern software tools
like Computer Aided Three-dimensional Interactive Application CATIA, as well as Rhinoceros 3D
and Dynamo, etc. equipped with visual scripting languages, which have revolutionized the shaping
process. Today, parametric shaping of building structures allows one to create complex shapes, as well
as their fabrication. Virtual spatial models of compound forms are created by means of advanced
algorithms, based on parametric equations that designers can adjust for particular circumstances.
Specialized computer software can generate original and atypical spatial structures, with mathematical
precision and optimization [
22
–
25
]. Along with this line of thought, the paper discusses a novel
algorithmic-aided approach to shaping curvilinear steel bar roof structures with the application of
design tools, working in the environment of Rhinoceros 3D.
Curvilinear steel bar structures are defined as spatial structures made of slender members, directly
connected bars, which carry loads. Historically, curvilinear steel bar structures, mostly in the form
of lattice cylindrical structures, began to be applied in the mid-nineteenth century. However, due to
serious difficulties in their calculation and construction because of the repeatable elements, they began
to be used on a larger scale only in the 1960s.
Buildings 2019,9, 61 3 of 11
However, the inherent properties of steel, such as its high strength-to-weight ratio, make the
range of the use of steel bar structures very wide. They can be used as load-bearing structures of
various roofs, as well as vertical and horizontal partitions of different topologies, in order to separate
the space of various buildings, which also minimizes the visual mass of each structure.
Nowadays, their use is increasing, thanks to advanced technologies. Therefore, more and more
curvilinear steel bar structures are created in a great variety of geometric forms and technical solutions,
that constitute original and sometimes amazing examples of engineering inventions.
The research presented in this manuscript is focused on the rational parametric shaping of
responsive curvilinear steel bar structures of a building, covering over a marketplace. This process is
recognized as a process, where the aim is to shape a structure that satisfies a predetermined geometry
and functionality, with respect to structural demands, as well as environmental conditions. In this
research algorithmicaided shaping curvilinear steel bar structures is done by application of structural
model making algorithms.
The algorithmic-aided structural shaping presented in the paper, where both the geometric model
and structural analysis are realized by means of algorithms, is a rather new field of research. However,
some researchers explore the concept of algorithmic-aided shaping in architecture, engineering and
construction [26–29].
2. Materials and Methods
The task was to shape a covering of a marketplace, between two buildings, the orthogonal
projection of which, is a rectangular trapeze (Figure 1). The dimensions of the trapeze were as follows,
the lengths of the trapezoid bases: 14 m, 20 m, the length of the trapezoid height: 14 m.
Buildings 2019, 9, x FOR PEER REVIEW 3 of 11
various roofs, as well as vertical and horizontal partitions of different topologies, in order to separate
the space of various buildings, which also minimizes the visual mass of each structure.
Nowadays, their use is increasing, thanks to advanced technologies. Therefore, more and more
curvilinear steel bar structures are created in a great variety of geometric forms and technical
solutions, that constitute original and sometimes amazing examples of engineering inventions.
The research presented in this manuscript is focused on the rational parametric shaping of
responsive curvilinear steel bar structures of a building, covering over a marketplace. This process is
recognized as a process, where the aim is to shape a structure that satisfies a predetermined
geometry and functionality, with respect to structural demands, as well as environmental
conditions. In this research algorithmicaided shaping curvilinear steel bar structures is done by
application of structural model making algorithms.
The algorithmic-aided structural shaping presented in the paper, where both the geometric
model and structural analysis are realized by means of algorithms, is a rather new field of research.
However, some researchers explore the concept of algorithmic-aided shaping in architecture,
engineering and construction [26–29].
2. Materials and Methods
The task was to shape a covering of a marketplace, between two buildings, the orthogonal
projection of which, is a rectangular trapeze (Figure 1). The dimensions of the trapeze were as
follows, the lengths of the trapezoid bases: 14 m, 20 m, the length of the trapezoid height: 14 m.
Figure 1. Horizontal, vertical and axonometric view of the considered square with the adjacent
buildings.
The roof structure was shaped as a curvilinear steel two-layered bar structure. As a roofing
material, polycarbonate plastic sheets were used, as well as tubes for structural elements.
In order to obtain the optimal roof structure, the roof shaping was carried out using modern
tools for algorithmic-aided design, working in the environment of Rhinoceros 3D (Robert McNeel &
Associates). The reason for which the above-mentioned tools were selected, is that they gave the
possibility to optimize the model with the assumptions of the various initial criteria.
The approach to algorithmic-aided shaping of curvilinear steel bar structures of the roof
proposed in this research, was realized by application of versatile tools: Grasshopper, Karamba 3D,
and Ladybug, working in the mentioned modeling software, Rhinoceros 3D. We also used advanced
software for structural analysis in order to optimize the bars’ cross sections: Autodesk Robot
Structural Analysis Professional 2019 [30].
In order to generate a parametric model, the educational version of Rhinoceros 5.0 in
combination with the parametric tool Grasshopper, was used. Rhinoceros 3D is three-dimensional
modeling software, which enables creation of various free form shapes, based on NURBS surfaces.
Figure 1.
Horizontal, vertical and axonometric view of the considered square with the
adjacent buildings.
The roof structure was shaped as a curvilinear steel two-layered bar structure. As a roofing
material, polycarbonate plastic sheets were used, as well as tubes for structural elements.
In order to obtain the optimal roof structure, the roof shaping was carried out using modern
tools for algorithmic-aided design, working in the environment of Rhinoceros 3D (Robert McNeel
& Associates). The reason for which the above-mentioned tools were selected, is that they gave the
possibility to optimize the model with the assumptions of the various initial criteria.
The approach to algorithmic-aided shaping of curvilinear steel bar structures of the roof proposed
in this research, was realized by application of versatile tools: Grasshopper, Karamba 3D, and Ladybug,
working in the mentioned modeling software, Rhinoceros 3D. We also used advanced software for
Buildings 2019,9, 61 4 of 11
structural analysis in order to optimize the bars’ cross sections: Autodesk Robot Structural Analysis
Professional 2019 [30].
In order to generate a parametric model, the educational version of Rhinoceros 5.0 in combination
with the parametric tool Grasshopper, was used. Rhinoceros 3D is three-dimensional modeling
software, which enables creation of various free form shapes, based on NURBS surfaces. Whereas,
its plug-in, Grasshopper, enables the creation of scripts to describe parametric models, which can be
modified and visualized in Rhino’s viewport and carefully analyzed. However, Grasshopper’s plug-in,
Ladybug, was used for the shadow range analysis.
Integration of geometric shaping and structural analysis was carried out using Karamba 3D.
This plugin
made it possible to combine parameterized complex geometric forms of various topologies,
load calculations and perform finite element analysis, according to Eurocode 3. Karamba 3D is
intended to be a fast tool for structural optimization [
31
]. It has been used mostly in the early stage
of design, although it is not limited to it. However, in the presented research, in order to perform
structural analysis of complex curvilinear steel bar structures, the advanced professional software
Robot Structural Analysis Professional, was used.
The validation of the functionality of the shaping tools mentioned above, as well as the benefits
of their application, were accomplished by implementing them for the shaping and analysis of the
responsive curvilinear steel bar structure.
The shaping strategy applied in the research consisted of the shaping of curvilinear steel bar
structures by placing the structural nodes on the so-called base surface [
29
]. The base surface was
a minimal surface obtained as a result of the optimization, or a surface received by the method
of form-finding.
3. Results of Shaping a Roof Based on a Minimal Surface
According to mathematical definition, a minimal surface is a surface that locally minimizes its
area, which is equivalent to having zero mean curvature. It is a surface with the smallest possible area
among all surfaces stretched on given lines.
The application of minimal surfaces in architecture is not new. They are used mostly in the
construction of light roof structures. Moreover, due to their characteristics, there is a greater and
greater interest in the study of minimal surfaces and their wider use to create innovative architectural
objects. Moreover, the application of minimal surfaces brings measurable benefits in structural design
since the minimal surfaces exhibit an optimal system of forces and stresses. It is also an important task
from an architectural and economic point of view, as the application of minimal surfaces means the use
of a minimal area of claddings and the minimization of costs. Due to this fact, the parametric study and
optimization were conducted on the examples of spatial lattices, created based on minimal surfaces.
3.1. Shaping Free-Form Roof Due to Optimization of the Minimal Surface
The first step in shaping the roof was formation of a minimal base surface. The minimal surface
was determined as a surface stretched on four curved lines lying in vertical planes passing through the
sides of the trapeze. Each line was established parametrically by three characteristic points, which
could change their positions. Two of the points were the arches’ ends and the third point was an inner
point of the arch. The rectangular projections of these points on the horizontal plane were, respectively,
the square’s vertices and the centers of the square’s sides (Figure 2).
Buildings 2019,9, 61 5 of 11
Buildings 2019, 9, x FOR PEER REVIEW 4 of 11
Whereas, its plug-in, Grasshopper, enables the creation of scripts to describe parametric models,
which can be modified and visualized in Rhino’s viewport and carefully analyzed. However,
Grasshopper’s plug-in, Ladybug, was used for the shadow range analysis.
Integration of geometric shaping and structural analysis was carried out using Karamba 3D.
This plugin made it possible to combine parameterized complex geometric forms of various
topologies, load calculations and perform finite element analysis, according to Eurocode 3. Karamba
3D is intended to be a fast tool for structural optimization [31]. It has been used mostly in the early
stage of design, although it is not limited to it. However, in the presented research, in order to
perform structural analysis of complex curvilinear steel bar structures, the advanced professional
software Robot Structural Analysis Professional, was used.
The validation of the functionality of the shaping tools mentioned above, as well as the benefits
of their application, were accomplished by implementing them for the shaping and analysis of the
responsive curvilinear steel bar structure.
The shaping strategy applied in the research consisted of the shaping of curvilinear steel bar
structures by placing the structural nodes on the so-called base surface [29]. The base surface was a
minimal surface obtained as a result of the optimization , or a surface received by the method of
form-finding.
3. Results of shaping a roof based on a minimal surface
According to mathematical definition, a minimal surface is a surface that locally minimizes its
area, which is equivalent to having zero mean curvature. It is a surface with the smallest possible
area among all surfaces stretched on given lines.
The application of minimal surfaces in architecture is not new. They are used mostly in the
construction of light roof structures. Moreover, due to their characteristics, there is a greater and
greater interest in the study of minimal surfaces and their wider use to create innovative
architectural objects. Moreover, the application of minimal surfaces brings measurable benefits in
structural design since the minimal surfaces exhibit an optimal system of forces and stresses. It is
also an important task from an architectural and economic point of view, as the application of
minimal surfaces means the use of a minimal area of claddings and the minimization of costs. Due to
this fact, the parametric study and optimization were conducted on the examples of spatial lattices,
created based on minimal surfaces.
3.1. Shaping free-form roof due to optimization of the minimal surface
The first step in shaping the roof was formation of a minimal base surface. The minimal surface
was determined as a surface stretched on four curved lines lying in vertical planes passing through
the sides of the trapeze. Each line was established parametrically by three characteristic points,
which could change their positions. Two of the points were the arches’ ends and the third point was
an inner point of the arch. The rectangular projections of these points on the horizontal plane were,
respectively, the square’s vertices and the centers of the square’s sides (Figure 2).
Figure 2. Establishing assumed arches for creation of a roof’s base surface.
Figure 2. Establishing assumed arches for creation of a roof’s base surface.
The position of characteristic points: A, B, C, D, E, F, G, H, were variables that could take different
values, and were defined as local increments of the co-ordinates x,y,z. The intervals of these variables
were defined for individual points as follows:
•Point A, B, C, D – z: 0.0 m–2.5 m.
•Point F, E, G – z: 4.0 m–5.0 m.
•Point E: x: 1.0 m–2.0 m (according to direction of axis x).
•Point G: x: 1.0 m–2.0 m (in the opposite direction than the axis x), Figure 2.
Depending on the adopted variables, we could get different shapes of arcs, and therefore, different
shapes of the minimal surfaces defined by them. Consequently, this led to obtaining many alternative
base surfaces. Some of them are presented in Figure 3.
Buildings 2019, 9, x FOR PEER REVIEW 5 of 11
The position of characteristic points: A, B, C, D, E, F, G, H, were variables that could take
different values, and were defined as local increments of the co-ordinates x, y, z. The intervals of
these variables were defined for individual points as follows:
• Point A, B, C, D – z: 0.0 m - 2.5 m.
• Point F, E, G – z: 4.0 m - 5.0 m.
• Point E: x: 1.0 m - 2.0 m (according to direction of axis x).
• Point G: x: 1.0 m - 2.0 m (in the opposite direction than the axis x), Figure 2.
Depending on the adopted variables, we could get different shapes of arcs, and therefore,
different shapes of the minimal surfaces defined by them. Consequently, this led to obtaining many
alternative base surfaces. Some of them are presented in Figure 3.
Figure 3. Generations of possible solutions of covering surfaces.
The optimal surface selected from the possible alternatives, was the surface with the smallest
area, as it could guarantee the minimum amount of materials used. In order to find such a surface,
the optimization process was performed in Grasshopper environment.. As the optimization
criterion, the minimum surface area was assumed, while the variable parameters were the
individual positions of the points. Due to the optimization carried out, a minimal surface area equal
to 263.042 m2 was obtained for the surface, as well as characteristic heights of the points (coordinate
z), presented in Figure 4. Additionally, both points, E and G, were moved horizontally one meter
outside the trapeze bases, according to axis x.
(a)
(b)
Figure 4. The best result of the roof surface optimization: (a) Perspective view of the roof’s base
surface; (b) The heights in meters of the characteristic roof’s points.
This optimal surface constituted a base surface in order to create the roof’s steel bar structure.
The multi-layered grid on this surface was applied in such a way that the structural nodes of the top
layer were included in the surface. In order to distribute the nodes, the surface was divided into the
same number of segments, equal to ten in the x and y directions (Figure 4-5).
Figure 3. Generations of possible solutions of covering surfaces.
The optimal surface selected from the possible alternatives, was the surface with the smallest
area, as it could guarantee the minimum amount of materials used. In order to find such a surface,
the optimization process was performed in Grasshopper environment. As the optimization criterion,
the minimum surface area was assumed, while the variable parameters were the individual positions
of the points. Due to the optimization carried out, a minimal surface area equal to 263.042 m
2
was
obtained for the surface, as well as characteristic heights of the points (coordinate z), presented in
Figure 4. Additionally, both points, E and G, were moved horizontally one meter outside the trapeze
bases, according to axis x.
Buildings 2019, 9, x FOR PEER REVIEW 5 of 11
The position of characteristic points: A, B, C, D, E, F, G, H, were variables that could take
different values, and were defined as local increments of the co-ordinates x, y, z. The intervals of
these variables were defined for individual points as follows:
• Point A, B, C, D – z: 0.0 m - 2.5 m.
• Point F, E, G – z: 4.0 m - 5.0 m.
• Point E: x: 1.0 m - 2.0 m (according to direction of axis x).
• Point G: x: 1.0 m - 2.0 m (in the opposite direction than the axis x), Figure 2.
Depending on the adopted variables, we could get different shapes of arcs, and therefore,
different shapes of the minimal surfaces defined by them. Consequently, this led to obtaining many
alternative base surfaces. Some of them are presented in Figure 3.
Figure 3. Generations of possible solutions of covering surfaces.
The optimal surface selected from the possible alternatives, was the surface with the smallest
area, as it could guarantee the minimum amount of materials used. In order to find such a surface,
the optimization process was performed in Grasshopper environment.. As the optimization
criterion, the minimum surface area was assumed, while the variable parameters were the
individual positions of the points. Due to the optimization carried out, a minimal surface area equal
to 263.042 m2 was obtained for the surface, as well as characteristic heights of the points (coordinate
z), presented in Figure 4. Additionally, both points, E and G, were moved horizontally one meter
outside the trapeze bases, according to axis x.
(a)
(b)
Figure 4. The best result of the roof surface optimization: (a) Perspective view of the roof’s base
surface; (b) The heights in meters of the characteristic roof’s points.
This optimal surface constituted a base surface in order to create the roof’s steel bar structure.
The multi-layered grid on this surface was applied in such a way that the structural nodes of the top
layer were included in the surface. In order to distribute the nodes, the surface was divided into the
same number of segments, equal to ten in the x and y directions (Figure 4-5).
Figure 4.
The best result of the roof surface optimization: (
a
) Perspective view of the roof’s base surface;
(b) The heights in meters of the characteristic roof’s points.
This optimal surface constituted a base surface in order to create the roof’s steel bar structure.
The multi-layered
grid on this surface was applied in such a way that the structural nodes of the top
Buildings 2019,9, 61 6 of 11
layer were included in the surface. In order to distribute the nodes, the surface was divided into the
same number of segments, equal to ten in the xand ydirections (Figures 4and 5).
Buildings 2019, 9, x FOR PEER REVIEW 6 of 11
(a)
(b)
Figure 5. The roof structure received due to optimization: (a) View from above; (b) view from below.
The orthogonal projections with dimensions of the considered structure are shown in Figure 6.
Figure 6. The views of the considered curvilinear steel bar structure.
3.2. Finding the roof’s shape by the form-finding method
Another way to shape the minimal base surface and the roof structure is using the so-called
form-finding method. Historically, it is an quite an old process, although the tools for the realization
of it have been developing over the years. There are two form-finding techniques: the first one it is a
method, which uses hanging models to simulate compressive forces; and another one is the method
of the minimal energy shapes of soap films. The hanging model method can be applied to simulate
the behavior of a family of structures known as funicular structures. Thus, finding the shape of the
structure based on the hanging model method means that the form found is adjusted to magnitudes
and positions of the forces acting on it. This method is mostly used in order to shape structures,
which work mainly in tension and compression.
The form-finding method has become very popular in recent years, mainly as a method for the
shaping of free-form structures. Nowadays, due to the development of modern computerized
modeling and calculating tools, which give the freedom to explore the design space, it can be
realized automatically in various ways [32]. In the case of parametric shaping structures,
form-finding is determined as the process in which parameters are directly controlled in order to
find an optimal geometry of a structure, which is to be in static equilibrium with assumed loads. The
Figure 5. The roof structure received due to optimization: (a) View from above; (b) view from below.
The orthogonal projections with dimensions of the considered structure are shown in Figure 6.
Buildings 2019, 9, x FOR PEER REVIEW 6 of 11
(a)
(b)
Figure 5. The roof structure received due to optimization: (a) View from above; (b) view from below.
The orthogonal projections with dimensions of the considered structure are shown in Figure 6.
Figure 6. The views of the considered curvilinear steel bar structure.
3.2. Finding the roof’s shape by the form-finding method
Another way to shape the minimal base surface and the roof structure is using the so-called
form-finding method. Historically, it is an quite an old process, although the tools for the realization
of it have been developing over the years. There are two form-finding techniques: the first one it is a
method, which uses hanging models to simulate compressive forces; and another one is the method
of the minimal energy shapes of soap films. The hanging model method can be applied to simulate
the behavior of a family of structures known as funicular structures. Thus, finding the shape of the
structure based on the hanging model method means that the form found is adjusted to magnitudes
and positions of the forces acting on it. This method is mostly used in order to shape structures,
which work mainly in tension and compression.
The form-finding method has become very popular in recent years, mainly as a method for the
shaping of free-form structures. Nowadays, due to the development of modern computerized
modeling and calculating tools, which give the freedom to explore the design space, it can be
realized automatically in various ways [32]. In the case of parametric shaping structures,
form-finding is determined as the process in which parameters are directly controlled in order to
find an optimal geometry of a structure, which is to be in static equilibrium with assumed loads. The
Figure 6. The views of the considered curvilinear steel bar structure.
3.2. Finding the Roof’s Shape by the Form-Finding Method
Another way to shape the minimal base surface and the roof structure is using the so-called
form-finding method. Historically, it is an quite an old process, although the tools for the realization
of it have been developing over the years. There are two form-finding techniques: the first one it is a
method, which uses hanging models to simulate compressive forces; and another one is the method
of the minimal energy shapes of soap films. The hanging model method can be applied to simulate
the behavior of a family of structures known as funicular structures. Thus, finding the shape of the
structure based on the hanging model method means that the form found is adjusted to magnitudes
and positions of the forces acting on it. This method is mostly used in order to shape structures, which
work mainly in tension and compression.
The form-finding method has become very popular in recent years, mainly as a method for
the shaping of free-form structures. Nowadays, due to the development of modern computerized
modeling and calculating tools, which give the freedom to explore the design space, it can be realized
Buildings 2019,9, 61 7 of 11
automatically in various ways [
32
]. In the case of parametric shaping structures, form-finding is
determined as the process in which parameters are directly controlled in order to find an optimal
geometry of a structure, which is to be in static equilibrium with assumed loads. The load used for
form-finding is usually the structure’s self-weight, however, they can also be other external loads.
In the considered case, the roof’s base surface was generated as the form found over the
trapeze-shaped place, by means of plug-ins working in environment of Rhinoceros 3D. During such
a form-finding process, the structural load direction had been inverted in order to achieve a proper
arrangement of the shaped structure. The height of the roof during simulation was assumed to be
the same as in the previous case, which was 1.5 m. On the generated surface, a two-layered grid
structure with the same division along axes x and y, like in the previous case, was applied (Figure 7).
The structure was placed at the same height as the structure presented in Section 3.1.
Buildings 2019, 9, x FOR PEER REVIEW 7 of 11
load used for form-finding is usually the structure’s self-weight, however, they can also be other
external loads.
In the considered case, the roof’s base surface was generated as the form found over the
trapeze-shaped place, by means of plug-ins working in environment of Rhinoceros 3D. During such
a form-finding process, the structural load direction had been inverted in order to achieve a proper
arrangement of the shaped structure. The height of the roof during simulation was assumed to be
the same as in the previous case, which was 1.5 m. On the generated surface, a two-layered grid
structure with the same division along axes x and y, like in the previous case, was applied (Figure 7).
The structure was placed at the same height as the structure presented in section 3.1.
Figure 7. The roof structure generated due to form-finding process.
The orthogonal projections with dimensions of the obtained structure are shown in Figure 8.
Figure 8. The views of the considered curvilinear steel bar structure.
3.3. Analysis of the shadow created by a set of connected buildings
Due to the fact that the function of the shaped roof was not only to provide protection against
atmospheric precipitation but also against the sun in the summer period, the shadow produced by
the roof, together with the adjacent complex of buildings, was analyzed. The aspect of the
construction of the shadow cast by the building complex is presented in [33]. In our research, the
analysis was carried out for the period from March to September, during the hours 10 am to 5 pm by
Grasshopper’s plug-in, Ladybug. For the analysis, the city of Warsaw was assumed as the location of
Figure 7. The roof structure generated due to form-finding process.
The orthogonal projections with dimensions of the obtained structure are shown in Figure 8.
Buildings 2019, 9, x FOR PEER REVIEW 7 of 11
load used for form-finding is usually the structure’s self-weight, however, they can also be other
external loads.
In the considered case, the roof’s base surface was generated as the form found over the
trapeze-shaped place, by means of plug-ins working in environment of Rhinoceros 3D. During such
a form-finding process, the structural load direction had been inverted in order to achieve a proper
arrangement of the shaped structure. The height of the roof during simulation was assumed to be
the same as in the previous case, which was 1.5 m. On the generated surface, a two-layered grid
structure with the same division along axes x and y, like in the previous case, was applied (Figure 7).
The structure was placed at the same height as the structure presented in section 3.1.
Figure 7. The roof structure generated due to form-finding process.
The orthogonal projections with dimensions of the obtained structure are shown in Figure 8.
Figure 8. The views of the considered curvilinear steel bar structure.
3.3. Analysis of the shadow created by a set of connected buildings
Due to the fact that the function of the shaped roof was not only to provide protection against
atmospheric precipitation but also against the sun in the summer period, the shadow produced by
the roof, together with the adjacent complex of buildings, was analyzed. The aspect of the
construction of the shadow cast by the building complex is presented in [33]. In our research, the
analysis was carried out for the period from March to September, during the hours 10 am to 5 pm by
Grasshopper’s plug-in, Ladybug. For the analysis, the city of Warsaw was assumed as the location of
Figure 8. The views of the considered curvilinear steel bar structure.
3.3. Analysis of the Shadow Created by a Set of Connected Buildings
Due to the fact that the function of the shaped roof was not only to provide protection against
atmospheric precipitation but also against the sun in the summer period, the shadow produced by the
roof, together with the adjacent complex of buildings, was analyzed. The aspect of the construction of
Buildings 2019,9, 61 8 of 11
the shadow cast by the building complex is presented in [
33
]. In our research, the analysis was carried
out for the period from March to September, during the hours 10 am to 5 pm by Grasshopper’s plug-in,
Ladybug. For the analysis, the city of Warsaw was assumed as the location of the building complex
and the complex’s position was with respect to the north direction, as shown in Figure 9. The analysis
showed that the complex of buildings with the shaped roof, presented in Figure 5, produced much
more shadow during the considered period, than the complex presented in Figure 7. The amount of
shadow generated in the analyzed period is shown graphically in Figure 9.
Buildings 2019, 9, x FOR PEER REVIEW 8 of 11
the building complex and the complex’s position was with respect to the north direction, as shown
in Figure 9. The analysis showed that the complex of buildings with the shaped roof, presented in
Figure 5, produced much more shadow during the considered period, than the complex presented in
Figure 7. The amount of shadow generated in the analyzed period is shown graphically in Figure 9.
(a)
(b)
Figure 9. The analysis of shadows: (a) Location of the buildings with respect to the north direction;
(b) the amount of shadow generated in the analyzed period.
3.4. Optimization of the structural members’ cross sections
We carried out structural optimization using the Autodesk Robot Structural Analysis
Professional 2019 software. The border conditions regarding the wind and snow zones were
assumed according to the location of the building objects. Due to the shape of the roof, that is, its
mostly flat nature, the snow load was assumed to be the same as for the shed roof. However, due to
the roof’s installation, that is, its adherence to adjacent buildings, the possibility of a snow drift has
been included in the calculations [34]. The maps on bars showing the distribution of the axial force
Fx are presented in Figure 10.
Figure 10. Maps on bars showing the distribution of axial force Fx.
The bars of the structure were divided into three groups for the dimensioning: The top truss
bars, the bottom truss bars, and the truss diagonal bars. The structure was optimized assuming as
the optimization criterion the mass of the structure. Moreover, the structure was optimized for two
cross-sections: circular and square. The results of the optimization are presented in Table 1.
Table 1. The results of the structural optimization of the considered structure.
Kind of member
Cross section
Cross section
Circular hollow
Figure 9.
The analysis of shadows: (
a
) Location of the buildings with respect to the north direction;
(b) the amount of shadow generated in the analyzed period.
3.4. Optimization of the Structural Members’ Cross Sections
We carried out structural optimization using the Autodesk Robot Structural Analysis Professional
2019 software. The border conditions regarding the wind and snow zones were assumed according
to the location of the building objects. Due to the shape of the roof, that is, its mostly flat nature,
the snow load was assumed to be the same as for the shed roof. However, due to the roof’s installation,
that is, its adherence to adjacent buildings, the possibility of a snow drift has been included in the
calculations [
34
]. The maps on bars showing the distribution of the axial force Fx are presented in
Figure 10.
Buildings 2019, 9, x FOR PEER REVIEW 8 of 11
the building complex and the complex’s position was with respect to the north direction, as shown
in Figure 9. The analysis showed that the complex of buildings with the shaped roof, presented in
Figure 5, produced much more shadow during the considered period, than the complex presented in
Figure 7. The amount of shadow generated in the analyzed period is shown graphically in Figure 9.
(a)
(b)
Figure 9. The analysis of shadows: (a) Location of the buildings with respect to the north direction;
(b) the amount of shadow generated in the analyzed period.
3.4. Optimization of the structural members’ cross sections
We carried out structural optimization using the Autodesk Robot Structural Analysis
Professional 2019 software. The border conditions regarding the wind and snow zones were
assumed according to the location of the building objects. Due to the shape of the roof, that is, its
mostly flat nature, the snow load was assumed to be the same as for the shed roof. However, due to
the roof’s installation, that is, its adherence to adjacent buildings, the possibility of a snow drift has
been included in the calculations [34]. The maps on bars showing the distribution of the axial force
Fx are presented in Figure 10.
Figure 10. Maps on bars showing the distribution of axial force Fx.
The bars of the structure were divided into three groups for the dimensioning: The top truss
bars, the bottom truss bars, and the truss diagonal bars. The structure was optimized assuming as
the optimization criterion the mass of the structure. Moreover, the structure was optimized for two
cross-sections: circular and square. The results of the optimization are presented in Table 1.
Table 1. The results of the structural optimization of the considered structure.
Kind of member
Cross section
Cross section
Circular hollow
Figure 10. Maps on bars showing the distribution of axial force Fx.
The bars of the structure were divided into three groups for the dimensioning: The top truss
bars, the bottom truss bars, and the truss diagonal bars. The structure was optimized assuming as
the optimization criterion the mass of the structure. Moreover, the structure was optimized for two
cross-sections: circular and square. The results of the optimization are presented in Table 1.
Buildings 2019,9, 61 9 of 11
Table 1. The results of the structural optimization of the considered structure.
Kind of Member Cross Section
Box Hollow
Cross Section
Circular Hollow
Top lattice’s bars 40 ×40 ×4 38 ×3.6
Bottom lattice’s bars 40 ×40 ×4 38 ×3.6
Diagonal bars 40 ×40 ×4 31.8 ×45
The considered structure was composed of 221 nodes and 800 members. However, the total mass
of the shaped structure was equal to 186,525 kg in the case of members with box hollow cross-sections
and 185,621.286, in the case of round tube members. Due to this fact as the most efficient structure,
it has been chosen the structure with circular hollow cross-sections as the lighter structure than the
structuer with box hollow cross sections.
4. Discussion
The use of Rhino and Grasshopper gave much creative freedom and flexibility in shaping. Due to
this fact, it was very convenient to use during the initial design process. In addition, to the advantages
of the possibility of the creation of many alternatives of geometric forms by Grasshopper, there was a
huge potential for combining parametric shaping with interactive evolutionary optimization.
The conducted research on shaping the roof over the market square, using algorithmic-aided
shaping, has shown that a comprehensiveapproach to shaping is possible. Therefore, as early as
possible, we could take into account many aspects that affect the future form and work of the structure.
These aspects can be conditions regarding the planned function, reliability, load-bearing capacity,
resistance to exceptional impacts and natural environment.
Moreover, defining the geometry of the shaped structure by means of the algorithms, gave the
possibility of various modifications and obtaining a virtually unlimited number of geometric forms.
Especially, in the case of curvilinear steel bar structures, which are characterized by a variety of
topologies, this is of great importance. In turn, the creation of a structural model using the Karamba
3D plug-in gave the possibility of an initial verification of the geometry, in relation to the structural
requirements, which led to effective shaping.
5. Conclusions
The form of the roof received by algorithmic-aided shaping has been appropriately adapted to
environmental conditions, whereas external conditions have caused the adaptability of the structural
members of the roof. Moreover, the choice of the minimal surface as the base surface for the design of
the roof form brings great benefits. First of all, the amount of cladding needed to make the roof could
be reduced. In addition, the use of a minimal surface as the surface for the location of the structural
nodes can influence the beneficial arrangement of forces in the structure. Thanks to this, it was possible
to reduce the cross-sections of the bars, and hence the mass of the structure. Dimensioning was carried
out for two bar cross-sections, which also gave the opportunity to choose the optimal cross section,
due to the mass of the structure.
The research presented in the paper shows how using the tools for generative design, the process
of shaping curvilinear steel bar structures, could be potentially improved in order to create responsive
structural forms that meet both architectural and structural requirements.
Due to the fact that, in the case of bar structures, the unification of structural members is very
important, the optimization of this issue will be the subject of further considerations of the author.
Funding: This research was funded by Rzeszow University of Technology.
Conflicts of Interest: The author declares no conflict of interest.
Buildings 2019,9, 61 10 of 11
References
1.
Vitruvius, P. The Then Books on Architecture, 1st ed.; Morgan, M.H., Ed.; Harvard University Press: Cambridge,
UK, 1914; pp. 13–31.
2.
Biermann, V.; Borngasser, B.; Evers, B.; Freigang, C.; Gronert, A.; Jobst, C.; Kremeier, J.; Lupfer, G.; Paul, J.;
Ruhl, C.; et al. Architectural Theory from the Renaissance to the Present, 1st ed.; Taschen: Koln, Germany, 2003;
pp. 6–20, ISBN 978-3-8365-5746-7.
3. Woli ´nski, S. On the criteria of shaping structures. Sci. Papers Rzeszow Univ. Technol. 2011,276, 399–408.
4.
Ku´s, S. General principles of shaping the structure. In General Building Construction, Building Elements and
Design Basics; Arkady: Warsaw, Poland, 2011; Volume 3, pp. 12–71. (In Polish)
5. PN-EN 1990:2004. Eurocode. Basis of Structural Design; PKN: Warsaw, Poland, 2004. (In Polish)
6.
PN-EN 1991-1-1:2004. Eurocode 1. Actions on Structures. Part 1-1: General Actions—Densities, Self-Weight,
Imposed Loads for Buildings; PKN: Warsaw, Poland, 2004. (In Polish)
7.
PN-EN 1993-1-1:2006. Eurocode 3. Design of Steel Structures. Part 1-1: General Rules and Rules for Buildings;
PKN: Warsaw, Poland, 2006. (In Polish)
8.
PN-EN 1993-1-10:2005. Eurocode 3: Design of Steel Structures—Part 1-10: Material Toughness and
Through-Thickness Properties; PKN: Warsaw, Poland, 2005. (In Polish)
9.
Dzwierzynska, J.; Prokopska, A. Pre-Rationalized Parametric Designing of Roof Shells Formed by Repetitive
Modules of Catalan Surfaces. Symmetry 2018,10, 105. [CrossRef]
10.
Elango, M.; Devadas, M.D. Multi-Criteria Analysis of the Design Decisions in architectural Design Process
during the Pre-Design Stage. Int. J. Eng. Technol. 2014,6, 1033–1046.
11.
Luo, Y.; Dias, J.M. Development of a Cooperative Integration System for AEC Design. In Cooperative Design,
Visualization, and Engineering, CDVE 2004; Lecture Notes in Computer Science; Luo, Y., Ed.; Springer:
Berlin/Heidelberg, Germany, 2004; Volume 3190.
12.
Wang, J.; Chong, H.-Y.; Shou, W.; Wang, X.; Guo, J. BIM—Enabled design Collaboration for Complex Building.
In Proceedings of the International Conference on Cooperative Design, Visualization and Engineering
(CDVE2013), Mallorca, Spain, 22–25 September 2013; Springer: Berlin/Heindelberg, Germany, 2013.
13.
Kim, D.-Y.; Lee, S.; Kim, S.-A. Interactive decision Making Environment for the Design optimization
of climate Adaptive building Shells. In Proceedings of the International Conference on Cooperative
Design, Visualization and Engineering (CDVE 2013), Mallorca, Spain, 22–25 September 2013; Springer:
Berlin/Heindelberg, Germany, 2013.
14. Oxman, R. Theory and design in the first digital age. Des. Stud. 2006,27, 229–265. [CrossRef]
15.
Oxman, R. Thinking difference: Theories and models of parametric design thinking. Des. Stud.
2017
,52,
4–39. [CrossRef]
16.
Dzwierzynska, J. Descriptive and Computer Aided Drawing Perspective on an Unfolded Polyhedral
Projection Surface. IOP Conf. Ser. 2017,245, 062001. [CrossRef]
17.
Dzwierzynska, J. Single-image-based Modelling Architecture from a Historical Photograph. IOP Conf. Ser.
2017,245, 062015. [CrossRef]
18.
Dzwierzynska, J. Reconstructing Architectural Environment from a Panoramic Image. IOP Conf. Ser.
2016
,
44, 042028. [CrossRef]
19.
Dzwierzynska, J. Direct construction of Inverse Panorama from a Moving View Point. Procedia Eng.
2016
,
161, 1608–1614. [CrossRef]
20.
Biagini, C.; Donato, V. Behind the complexity of a folded paper. In Proceedings of the Conference: Mo. Di.
Phy. Modelling from Digital to Physical, Milano, Italy, 11–12 November 2013; pp. 160–169.
21.
Pottman, H.; Asperl, A.; Hofer, M.; Kilian, A. Architectural Geometry, 1st ed.; Bentley Institute Press: Exton,
PA, USA, 2007; pp. 35–194, ISBN 978-1-934493-04-5.
22.
Kolarevic, B. Architecture in the Digital Age: Design and Manufacturing, 1st ed.; Spon Press: London, UK, 2003;
pp. 20–98, ISBN 0-415-27820-1.
23.
Barlish, K.; Sullivan, K. How to measure the benefits of BIM—A case study approach. Autom. Constr.
2012
,
24, 149–159. [CrossRef]
24.
Wortmann, T.; Tuncer, B. Differentiating parametric design: Digital Workflows in Contemporary Architecture
and Construction. Des. Stud. 2017,52, 173–197. [CrossRef]
25. Hardling, J.E. Meta-parametric Design. Des. Stud. 2017,52, 73–95. [CrossRef]
Buildings 2019,9, 61 11 of 11
26.
Stravic, M.; Marina, O. Parametric Modeling for Advanced Architecture. Int. J. Appl. Math. Inform.
2011
,5,
9–16.
27.
Bhooshan, S. Parametric design thinking: A case study of practice-embedded architectural research. Des. Stud.
2017,52, 115–143. [CrossRef]
28.
Turrin, M.; von Buelow, P.; Stouffs, R. Design explorations of performance driven geometry in architectural
design using parametric modeling and genetic algorithms. Adv. Eng. Inform. 2011,25, 656–675. [CrossRef]
29. D´zwierzynska, J. Shaping curved steel rod structures. Tech. Trans. Civ. Eng. 2018,8, 87–98. [CrossRef]
30. Autodesk. Available online: https://www.autodesk.com/ (accessed on 21 February 2019).
31. Preisinger, C. Linking Structure and Parametric Geometry. Arch. Des. 2013,83, 110–113. [CrossRef]
32.
Veenendaal, D.; Block, P. An overview and comparison of structural form finding methods for general
networks. Int. J. Solids Struct. 2012,49, 3741–3753. [CrossRef]
33.
Dzwierzynska, J. Computer-Aided Panoramic Images Enriched by Shadow Construction on a Prism and
Pyramid Polyhedral Surface. Symmetry 2017,9, 214. [CrossRef]
34.
PN-EN 1991-1-1:2004. Eurocode 1. Actions on Structures. Part 1-3: General Actions—Snow Loads; PKN: Warsaw,
Poland, 2004. (In Polish)
©
2019 by the author. 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/).