Realization of historical Persian ornamental and geometric patterns as architectural components in innovative reciprocal frame barrel vaults

Abstract

Barrel vaults are one of the most widely used forms in Persian architecture. While they are mainly built with heavy and compressive materials such as brick, today’s advances in construction techniques has led the architectural industry to utilize lightweight structural systems including reciprocal frame structures (RF). The purpose of this paper is to generate a barrel vault form using RFs through a revival of historical Persian ornamental and geometric patterns known as girih. This research was carried out in three phases. After extracting the essential criteria necessary to produce reciprocal configuration, four geometric girih that were compatible with those criteria were selected. The selected patterns were then modified to localize their reciprocal configurations following the Persian ornamental and geometric patterns. A structural analysis was performed using the Karamba Plugin in order to make a structural comparison between a barrel vault constructed with RFs and one made with bricks. The results showed that the use of RFs can significantly reduce the structural weight while using a minimum of material in covering the specified span. In addition, it was concluded that the vault with Hasht-e-moraba and Chahar-lenge ghenas patterns behaved in a more optimal way for the transmission of axial forces with less displacement and deformation.

Keywords

Revival Persian ornamental and geometric patterns barrel vault reciprocal frame structure structural analysis

Introduction

Vaults are among the most important and valuable structural elements of Persian architecture. With their potential to create a variety of forms containing both spatial quality and beauty, vaulted forms historically had a meaningful presence in Iran. The continuity of such forms into contemporary architecture has been overshadowed, however, by complicating factors such as the high weight of the materials used, difficulties in reusing the materials, long construction and erection times, and structural constraints.

The barrel vault—a semi-cylindrical hollow structure that is formed by moving an arc along a line—is widely used in Iran. Because of its shape, this vault has very high thrust forces. Due to this compressive behavior, the materials available to the builders of historical vaults were mostly heavy such as bricks. The amount of used was very high due to the level of adhesion needed, which caused the manufacturing process to last longer. The transmission of forces in this system of pressure vaulting creates limitations to the opening of its surface. Additionally, the inclusion of decorative elements after the completion of the structure requires the employment of separate materials and techniques. This research aims to develop a new type of barrel vault by utilizing Reciprocal frame structures (RF) as lightweight materials containing both structural and decorative functions. RFs, due to the use of repetitive units, their ability to diversify the form as a whole, and to create diverse configurations within a specific form, offer suitable structures for modernizing vaulted forms. In addition, these structures can be made with a variety of linear and planar elements for different purposes.

Reciprocal frame structures (RF) are a kind of three-dimensional structure, requiring at least three beams. They consist of units namely the “fan.” A reciprocal configuration is formed by combining several fans. Because these structures are inherently variable in their configuration, they can be used for creating numerous kinds of exhibition spaces and protective structures with both permanent and temporary functions. Although, generating the form of reciprocal structure is rather more intricate than other grid space structures because any pair of elements contact at their surfaces and it causes the pair of elements are not on the same plane in geometric point of view,¹ the use of limited-length elements that have the ability to cover long spans is one of the special advantages of these structures. In addition, because in RF structures the large spans are covered without any internal supports, the architect can design sub-spaces more easily. In an RF structure, an overall configuration in which the elements play structural and decorative roles simultaneously can be reached by altering the arrangement of individual elements by using a bottom-up design process. This configuration is similar to the Persian geometric pattern known as a girih. This research paper examines the construction possibilities of RF structures extracted from the geometric girih found in historic Persian architecture.

Literature review

A vault is a three-dimensional arch that transfers loads only through pressure to the supports while performing purely in compression. Compressive vaults are divided into two main kinds: cylindrical vaults (with curvature in one direction) and domes (with curvature in two directions). The barrel vault is one kind of cylindrical vault in which the load distribution occurs at a 45° angle in each direction (Figure 1).

Figure 1.

Load distribution in (a) independent arches and (b) a vault.²

In 1908 in Europe and 1925 in the United States of America, Zollinger patented an arched-type structure called the Lamella system which consisted of cross-diagonal arches (Figure 2). A Lamella arch is a cylindrical arch that is particularly suited to the use of relatively small-sized members to span large wooden, steel, or prefabricated concrete spaces.²

Figure 2.

Lamella vault.

Scientific research on reciprocal frame structures began in the late 20th century. In 1989, the term “reciprocal frame” was first introduced by Graham Brown, an English designer.³ A number of studies were later conducted regarding reciprocal structures, all based on primary research presented by Chilton in 1992, 1994, and 1995.³ In these studies, an accurate definition was suggested for this type of structural system and its principles were investigated. Larsen⁴ focused on the geometrical, structural, and construction considerations of RFs in her 1996 thesis. The Ph.D. thesis written by Baverel⁵ was undoubtedly a major step toward the identification of reciprocal frame structures in the modern age. In this thesis, Baverel⁵ explained the geometry of three-dimensional reciprocal grids and focused on finding forms using genetic algorithms. Douthe and Baverel⁶ compared the structural behavior of reciprocal domes with conventional triangular grid structures and examined the design process of these structures using “dynamic relaxation.” Gelez et al.⁷ analyzed the behavior of planar reciprocal frames based on a 4-nexor fan. Parigi and Pugnale⁸ compared the behavior of two- and three-dimensional reciprocal frames in order to study the impact of geometric parameters on the distribution of internal forces. In addition, Parigi and Sassone⁹ conducted research on pinned connections used to fabricate kinetic reciprocal frame structures. Rizzuto¹⁰ carried out a research on structural behavior of an RF honeycomb configuration dome under different conditions including varying applied loading, boundary support conditions and connection stiffnesses. RF structures have a very extensive research history. While at least 14 different sub-topics have been published for these types of structures, the most important of these are the morphology and geometry of reciprocal spatial structures, form-finding using interactive tools, and polygon configurations.³ Table 1 shows these three essential research fields of RF structures and their investigations.

Table 1.

Three important research fields for RF structures.

Three important fields of research in RF structures
Morphology and geometry		Form-finding		Polygon configurations (various pattern-planar element-kinetic RFs)
1989	Graham Brown: introduced the term “reciprocal frame”³	2000	Baverel⁵: form finding using genetic algorithms	2008	Roelofs¹¹: interwoven patterns
199219941995	Chilton: accurate definition of RF and its principles³	2009	Douthe and Baverel⁶: form finding using “dynamic relaxation”	2011	Parigi and Sassone⁸: kinetic reciprocal frame structures
1996	Larsen⁴: geometrical, structural, and construction considerations of RFs	2014	Parigi and Kirkegaard: developed a computational tool called a “Reciprocalizer”³	2011	Gelez et al.⁷: behavior of planar reciprocal frames based on a 4-nexor fan
1998	Saidani et al.: multi-reciprocal grid (MRG)³	2019	Su et al.¹: form finding using a new numerical method based on contact point and contact vector as two new parameters	2013	Song et al.¹²: introduced a tool for the production of reciprocal regular networks

With emerging computational methods, more RF projects and prototypes have been designed and built (Figure 3). The KREOD pavilions¹³ is a triple pavilion formed with double-curved wooden shells. Each shell formed with wooden members in hexagonal RF configuration.^4,12 Mass Imperfections¹⁴ is a 3 m high arch built of 552 mutually supported olive-wood pieces with 1 cm thickness. Each panel has six vertices, three of which are supported by three neighboring panels while the remaining three support three other neighboring panels. Future Tree is the most recent RF project in free-form geometry that manufactured and assembled using an industrial robot in 2019.¹⁴

Figure 3.

Three recent RF projects. From left to right: Kreod pavilion,^4,13 Mass imperfection,¹⁴ and Future tree.¹⁵

As can be seen in Table 1, one of the areas of research in RF structures is their various configurations based on different polygons. The placement of reciprocal units (called “fans”) together will create a network of repetitive patterns. These patterns can have various geometries depending on the primary modules and joints. Most studies have been done on square, rectangular, triangular, and hexagonal patterns.¹¹ The plane patterns of these structures were studied based on the theory of symmetry groups and vertex transitivity developed by Grünbaum and Shephard.¹⁶ In other words, such patterns are made by congruent regular polygons with every point in the pattern surrounded by the same sequence of polygons.¹¹ Roelofs¹¹ achieved different configurations through his research into two- and three-dimensional RF patterns (Figure 4, left). Roelofs¹¹ also used interwoven patterns to transform a flat pattern into a spherical construction (Figure 4, right).

Figure 4.

Three interwoven patterns (left) and interwoven sphere (right).¹⁰

This results in a non-planar unlimited construction that has also kinetic properties. The elements can slide between particular boundaries and the total construction can be pressed together or stretched (Figure 5).¹⁰ In other words, he used 3D pattern configuration of reciprocal elements.

Figure 5.

Three dimensional reciprocal.¹⁰

In 2013, Song et al.¹² introduced a tool for the production of reciprocal networks, but this tool only manages the modeling of regular networks that have symmetric rotational rods. These parametric tools have only been applied to simple square, triangular, and hexagonal patterns. Less research on unconventional patterns with different tessellations has been undertaken.

In the end, vaults created by using the Lamella system have only one diagonal configuration. In RF structures, not only can the combination of a number of small-sized elements cover large spans, but they can also be conceived of in different patterns and configurations. In this paper, the authors try to make reciprocal patterns based on Persian girih to revive barrel-vaulted forms using the patterns as structural elements in the field of polygon configurations.

Methodology

In reciprocal frame structures, all elements are mutually supported by each other and can take a variety of forms and configurations. This research attempted to create reciprocal configurations by employing Persian girih patterns. As mentioned earlier, a barrel vault configuration was used as the main form, and the Persian girih were changed according to the principles of reciprocity in order to create the required geometry. In this research, after drawing a barrel vault geometry in which the rise is equal to half of the span’s length, three following stages were carried out:

(1) Modification of a Persian geometric girih on a 2D surface (the girih was selected according to the design requirements and selection criteria as shown in Figure 5).

(2) Simulation of the reciprocal grid from Phase 1 on the barrel vault (as a 3D surface).

(3) Structural analysis and optimization.

To do this research, it was first necessary to modify the selected girih. These modifications were done in order to construct the girih based on the principles of reciprocity. Simulation operations were then performed to map the modified geometry onto the 3D surface of the barrel vault. In the simulation phase, different Plugins were used, according to the objectives and structural requirements, as indicated in Figure 6.

Figure 6.

RF structures’ design process using non-conventional patterns.

For parametric simulation, the Grasshopper and Kangaroo Plug-ins were used. Form-finding of the reciprocal structure was done by using a physics-based particle-spring system engine called Kangaroo, in which the elements’ positioning is simulated in an equilibrium state.¹⁷ Structural analysis was done using the Karamba Plugin, a parametric structural engineering tool for predicting structural behavior to obtain numerical values.¹⁸ This plug-in makes it possible to integrate data between the structural and geometrical models.¹⁹ Finally, quantitative and qualitative analysis was done to compare the simulated structures.

Phase 1: Modification of Persian geometric girih according to reciprocity principles on a 2D surface

There are various ways to divide a flat plane using a grid of lines in a regular or irregular pattern, such that each of them produces considerable variation in the length of the lines and the angles between them. In the past, it was normally considered advantageous if, in any particular space structure, the number of different member lengths can be limited and connection angles at the joints standardized. However, nowadays with advanced machining methods and types of equipment, this standardization is not considered as an advantage and it is now almost as simple to fabricate members with various lengths and nodes with many different connection angles in an inexpensive way. There are only three regular polygons (i.e. polygons with all sides of equal length and equal interior angles) that can be used exclusively to completely fill a plane. These are the equilateral triangle, square, and hexagon, which are common configurations in reciprocal frame structures as well as space frame structures.²⁰ Geometry has always been the primary factor in the interaction between structure and architecture²¹ and plays an undeniable role as the linking agent of form and structure. One of the geometric factors affecting RF structures is the different configurations of the reciprocal modules. The first phase includes two steps: (1) Girih selection according to our defined criteria. (2) Modification of the selected girih based on reciprocity principles.

First step: Criteria and girih selection

RF networks is a modular system consisting of a combination of modules named fan. Each fan requires at least three beams known as “nexor.” They are designed to act hierarchically in the 2D surface. First, the fan is defined and then a series of them are gathered to form a larger network.¹¹ To form a single fan, nexors are arranged in a closed circuit, forming a three-dimensional geometric shape such as a dome. To create a free-form shape, some non-circular arrangements may be required.¹¹ Reciprocal structures are self-similar and form very symmetrical patterns that create various architectural spaces.²²

The position of the nexors in an RF structure create various grids based on the style of the fan that is defined by the number of used nexors in each fan (Figure 7).These grids (configuration) have features that provide continuity and coherence for the structure. In fact, any flat pattern of straight lines can form a suitable reciprocal pattern. However, some configurations of special polygons are preferable due to their repeatability and tendency to distribute loads more uniformly.¹⁰

Figure 7.

Different types of linear configurations (patterns) based on the style of the fan (N), N = the number of used nexors (beams) in fan.¹¹ Hybrid fan is a kind of fan that combines two or three fans of different styles.

What was essential at this stage of the research was answering the following question: what are the features of reciprocal patterns and is it possible to use any pattern as a structural-functional pattern according to the principles of reciprocity? One of the main differences between reciprocal patterns and other similar structures (such as space frame structures) is the discontinuity of the elements. Additionally, in the RF pattern there are no node systems (like MERO^I in spatial structures²⁰), therefore the number of elements connected at each joint is limited. Patterns with identical reciprocal modules have elements with equal length, while in hybrid fans, the length of the elements can be varied. According to the principles of the reciprocal frame and through review of the common RF geometries, four significant criteria necessary to produce a suitable reciprocal pattern were extracted as follows (Table 2).

Table 2.

Essential criteria for RF patterns.

1	Repetitive geometrical configuration
2	Discontinuity of elements
3	Limitation to the number of elements at connection points
4	Identical rotational direction of elements in each fan

In Persian architecture, interest in surface decoration holds a similar value to building design and construction. In other words, structural, architectural, and decorative elements are designed simultaneously and in close connection to each other.²³ In the Encyclopedia of Arts, “Girih tiling” is defined as a method to organize geometric patterns in a coherent and balanced design.²⁴ Persian girih patterns can be developed and expanded through the x, y, and z-axes based on their geometric constraints.²¹ The most prominent features of girih patterns are geometric order, balance, generative ability, and variability. There are over 300 different types of Persian girih in a wide range of geometrical complexities.

The selection process is carried out in two stages. First, the essential criteria for RF patterns have been extracted (Table 2). These criteria exclude many patterns from our study but still, there are other patterns that remain in the competition and have the potential for conversion and use in reciprocal structures. In the second stage, we chose girih which has a similarity in their basic pattern with the conventional RF patterns. In other words, you can find the square, triangle, and hexagonal geometry in their configuration whether directly or indirectly. Therefore, the four girih patterns known as Sorme-dan moraba (A), Shesh-e-hasiri (B), Hasht-e-moraba (C), and Chahar-lenge ghenas (D) which were considered for performing this research (Figure 8). As the name of selected girih are long and repeating along with the paper, we use (A), (B), (C), and (D) instead of their original name.

Figure 8.

Selected girih to create non-conventional reciprocal patterns in order from left to right: (A) Sorme-dan moraba, (B) Shesh-e-hasiri, (C) Hasht-e-moraba, and (D) Chahar-lenge ghenas.^25,26

However, it still remained to determine if these girih could be used in the creation of reciprocal networks. Moreover, was it possible to extract different reciprocal configurations from the selected girih?

Second step: Modification of the reciprocal grid on a 2D surface

The appearance of a reciprocal pattern is determined by three factors: (1) Geometric parameters of the RF units (fans); (2) Connection points of the RF units; and (3) the geometry of the guiding surface.¹¹ The first two factors play a decisive role in creating an RF constructive pattern. The first step in the modification of the girih is to identify its constitutive unit (fan). Fans can be formed by a variety of elements based on their shape (linear or planar). The second factor is the connection points of the RF units. Although there is a certain degree of freedom in the position of the connections leading to variable grid forms, these connection points must take into consideration the location of the adjacent elements and the three following factors: (a) that the distance of one element is not such that the element is faced with shear and weakness; (b) the position of the connecting points is such that the elements have the required overlap; and (c) there is no interconnection with more than two elements at any one junction point.

Therefore, the selected patterns need to be changed by modifying the geometrical shape of the elements and the location of elements’ connections (Figure 9). The barrel vault is also used as a guide to the surface geometry.

Figure 9.

Methods of pattern modification.

For instance, in the (C) girih, the modification and generation of the pattern was done with the following process (Figures 10 and 11). In the (C) pattern, the geometry of the element can be considered as both a linear and planar form. The elements’ positions are modified by a radial transmission toward the center of the polygon containing one side of each square in a single fan (Figure 10).

Figure 10.

Modification and generation process of the (C) girih by using appropriate location with the radial transmission (left to right).

Figure 11.

Script of modification and generation of the (C) girih.

By completing this process for all fans and applying these changes to the girih network, a modified reciprocal network is created. In addition to its decorative role, this network can be used as a structural grid. These steps and processes are briefly summarized in Table 3 for the other selected girih (Table 3).

Table 3.

Steps in converting the selected girih patterns to the reciprocal pattern ((1): supported points (2): supporting points).

Type of girih	Refined grid		Extracted fan	Final grid	Element’s geometry
Type of girih	Shape’s geometry	Fitting location	Extracted fan	Final grid	Element’s geometry
(A) Sorme-dan moraba	Linear element	-
(A) Sorme-dan moraba	Planar element	-
(B) Shesh-e-hasiri	Linear element	-
	Linear element	-
	Planar element	-
	Planar element	-
(C) Hasht-e-moraba	Linear element	Radial Transmission
(C) Hasht-e-moraba	Planar element	Radial Transmission
(D) Chahar-lenge ghenas	Linear element	Axial Transmission
(D) Chahar-lenge ghenas	Planar element	Scale up

Phase 2: Simulation of a reciprocal grid derived from Phase 1 on a barrel vault (3D surface)

In RF structures, unlike other space structures, with removing knot joints in the intersections, the grid lines are not continuous anymore. In other words, the members are directly connected to each other without any intermediate joints. Therefore, because of RFs’ non-hierarchical nature, conventional CAD tools have failed in the modeling of reciprocal geometry. This geometry must be acquired from complicated relationships between various factors such as the shape, geometry, and position of all members. So, a numerical solution is required based on the geometrical compatibility of the elements.²⁷ In recent years, Grasshopper (a plug-in for Rhinoceros) has remarkably simplified the modeling of such structures.

The design process of reciprocal structures has many geometric restrictions that prevent the estimation of the final form by changing the parameters discretely. Form finding is an indispensable step toward defining the basic configuration and parameters of the structure. Recently, genetic algorithms and dynamic relaxation method have been applied in RFs’ form-finding process. The logic of the dynamic relaxation is based on the alteration of the primitive configuration through optimization of the engagement length between two elements.⁵

Three dimensional reciprocal structures can be obtained by introducing eccentricity between the neutral axes of the elements in a fan. However, the amount of overlap between the elements depends on the isoparametric curve (or isocurve) of basic surface and the element’s effective length. Therefore, it is highly challenging to predict the final form of a 3D reciprocal structure based on its constitutive member’s shape, particularly for RFs with free-form surfaces.²⁸

There are several methods for RF geometric modeling that can be classified into two main groups: the first method generates a two-dimensional reciprocal pattern and then projects it on a 3D surface using conformal mapping.¹¹ This method isn’t useful for geometries with multiple curvatures in different directions. In second method, after discretizing of a surface into polygonal form meshes, the reciprocal pattern is created by moving or rotating the mesh edges. This is depicted in Figure 12.⁵ To create the common reciprocal pattern (namely four-, three-, and six-sided grids) on each continuous surface, first, the surface must be prepared to create an RF grid taking into consideration the three methods of translation, rotation, and extended translation by converting the surface to mesh and dividing it into smaller portions. In this study, the modeling of two reciprocal patterns derived from the (A) and (D) girih was performed through the second method because it allows for the possibility of converting these patterns into equal square divisions (Figure 13). The modeling of the (B) and (C) girih was done by the first method (Figure 14).

Figure 12.

The second method of simulation process (left to right).

Figure 13.

Script of samples with (A) and (D) girih.

Figure 14.

Script of samples with (B) and (C) girih.

After modeling, an RF vault’s relaxation using Kangaroo was done to facilitate form-finding. The elements’ positioning was then simulated in an equilibrium state. This is clearly shown before and after relaxation in the first design sample, a barrel vault using the (A) pattern, in Figure 15. Before the relaxation process (left), the beams are non-linear and connect together in how they intersect completely and slice each other in sub-beams while they must have a partial intersect and still remain in their basic length. After carrying out the relaxation (right), the elements’ positioning has been changed in an equilibrium state and you can see the linear beams with a partial connection as notch with other members.

Figure 15.

Before (left) and after (right) relaxation using the Kangaroo plugin.

In the simulation process, in order to minimize the cost of construction, all elements were considered to be linear and flat without any bending. The materials, cross-sections, and joints were considered the same in all elements. Modeling and form-finding was done on a barrel vault with a 10 m span, 5 m rise, and 10 m length. (The dimensions were hypothetically selected according to the proportional dimensions of a barrel vault in which the structure’s rise is equal to half of the span’s length.) This kind of structure has less thrust with more height. Elements were considered to have a rectangular cross-section of 5 cm × 3 cm for the first stage of simulation. The type of joint used was a notch which was defined as a hinge through parametric simulation by Karamba. According to the type of joint, the fabrication process is based on subtractive manufacturing that involves removing sections of material by machining or cutting it away such as CNC. Wood was selected as the main material with the following physical properties (Table 4).

Table 4.

Wood’s physical properties.

Material	Young’s modulus (kN/cm²)	Shear modulus (kN/cm²)	Density (kg/cm³)	Thermal expansion (1/°C)	Yield strength (kN/cm²)
Wood	1050	360	600	5/0E-1	1.3

The linear samples of the four-selected girih are shown in Figure 16.

Figure 16.

Linear samples formed by patterns derived from Persian girih: (A), (B), (C), and (D).

Phase 3: Structural analysis and optimization

Considering the influence of the dimensions, geometry, and material properties of the elements in addition to analysis of the forces imposed on the structure, the change and influence of each parameter can be observed during the design process using the Karamba Plugin. The modeling process of these structures follows a reciprocating design process. Finally, steps were taken to optimize the structures using the Galapagos Plugin. In this case, by taking into account the least amount of displacement as fitness,^II a cross-section of the members can be obtained in their optimum state (Figures 17 and 18).

Figure 17.

Structural analysis of the (C) sample using the Karamba plugin: (a) position of the elements, (b) displacement, (c) axial forces, and (d) bending moment.

Figure 18.

Script of structural analysis and optimization process of the (C) sample using the Karamba plugin and Galapagos.

The values assigned to the cross-section (5 cm × 3 cm) confronted the structures with a large amount of displacement. Since the RF structure is considered as one of the spatial structures, the amount of permissible structural displacement can be extracted according to the static regulations of these structures. The vertical displacement on a surface such as a dead load should not exceed 1/200 and for the live load from 1/250 of the span’s shorter length. Therefore, given the length of the span (10 m), the displacement in this case should not exceed 0.05 m. The final stage of the optimization process was performed on samples with linear elements using the Galapagos Plugin. The cross-section of the elements was considered as a genome and the amount of displacement as fitness.

Results

After defining the structure’s geometry in Rhinoceros and Grasshopper, the element’s positions were adjusted with the Kangaroo Plugin. Then, for structural analysis, the extracted geometry from Kangaroo was simulated in the Karamba environment. To generate samples with linear elements, the Kangaroo’s output geometry was directly transmitted into the Karamba Plugin. In sampels with planar elements, geometric operations were performed after the form-finding process to generate plates and then transmitted to the Karamba Plugin. In this paper, only the results of samples with linear elements were considered. Finally, by using Analyze Th1^III (one of the Karamba’s methods of analysis),²⁹ stress results were obtained for each linear sample as shown in Table 5. To validate the results, a similar size structure with the Lamella system is simulated and analyzed as a benchmark for comparison. Its results are added to Tables 5 and 6. However, it is worth to be mentioned the equal dimension of the basic structure does not guarantee the comparison, since the number and length of elements in each sample are totally different as well as their intersection position.

Table 5.

Structural analysis of linear samples using Karamba, before optimization. (C: compression; T: tension; M: bending moment; S: shear force).

	Length (m)	Num.	C (kN)	T (kN)	M (kNm)	S (kN)	Weight (kg)
(A)	1.15	912	0.21	0.21	0.52	0.56	966.64

(B)	0.99	672	0.16	0.16	0.054	0.27	601.56

(C)	0.77	1200	0.54	0.54	0.11	1.19	852.59

(D)	0.95	926	0.67	0.67	0.75	1.17	799.15

Lamella vault	1.097	672	3.38	3.38	1.41	0.89	657.91

Table 6.

Optimization of linear samples using Galapagos.

Optimization	(A)	(B)	(C)	(D)	Lamella
Cross section (cm)	8 × 8	10 × 10	9.0 × 9.7	9.8 × 9.7	10 × 10
Displacement (m)	0.071	0.126	0.008	0.017	0.04
Weight (kg)	6446.29	4010.44	5003.01	5153.17	4337.82
Compression (kN)	237.59	16.80	758.23	883.09	42.95
Tension (kN)	186.81	29.85	758.29	883.09	42.95
Bending moment (kN · m)	33.17	6.92	30.10	71.29	29.17
Shear force (kN)	132.21	17.74	1.97	442.04	14.47

As mentioned earlier, after analysis in the Karamba Plugin, the optimization process for the samples was completed with the Galapagos Plugin. Also the lateral arch-shape supports are added. The results of this stage are illustrated in Table 6 and Figure 19.

Figure 19.

Comparsion of samples’ displacement in three states.

Discussion

Stresses and their influencing factors

Elements in RF structures are affected by axial, shear, and bending forces. In light of the results obtained from analyzing the simulations and optimizations, vaults created with the (D) girih have the highest amount of axial stress, which seems natural due to the angular shape of the elements in this pattern and the necessity of having rigid connections between elements. The vault created with (C) withstands the lowest amount of shear force among the samples because of the lowest amount of connections in every single element. Generally, in RF structures the maximum concentration of stress occurs in the joint connections. At these points, the shear force and bending moment reach their maximum. If the number of connections in each beam increases, the beam is confronted with more stress. In order to prevent the beam’s deflection, it is necessary either to reinforce them at the points of connection or to increase their cross-section. The results of the Galapagos optimization show that vaults generated with the (C) and (D) patterns, in addition to maintaining equilibrium in the axial stresses, maintain the required amount of displacement within the permitted range.

Figure 19 shows the compersion between samples’ displacement in three states: (1) before optimization, (2) after optimization, and (3) adding lateral supports. The acceptable results are those which occurred below the dash line.

Influence of the number of elements and their length on structural density

While the number of elements in the (C) vault is greater than those of other patterns, the (A) and (D) vaults have greater structural density. One of the reasons for the low density of (C) vaults is the smaller length of the elements (77 cm) in comparison to the other patterns. With the (A), not only is the element’s length bigger that of the other patterns (115 cm), there are also three supported points in each element (Table 3). That in turn increases the structural density. The vault generated with (B) has not only the lowest number of elements, but in this pattern, each element has only one supported point, so it shows a lower structural density (Figure 20). In other words, if the number of supported points in each element increases, the structural density will also be increased. In addition, although linear elements lead to a lighter structure, but they provide less shading and structural depth which affect the structural strength. Elements in planar form prepares more shadow if the structure is used as a sunshade.³⁰ Therefore, both the number and type of elements could also affect the shade created by the structure, the amount of incoming light, and the required number and shaping of covering pieces.

Figure 20.

Structural density in different linear patterns (left to right: (A), (B), (C), and (D)).

It is necessary to mention that unlike brick vaulting, it is not possible to cover the surface completely with RF elements (even with planar elements), so the covering arrangements must be considered separately. However, due to the type of force transmission in these kind of structures, architects, and designers have more comprehensive options for locating windows and openings in RF vaults.

Conclusion

Reciprocal frame structures have many potential benefits including efficient force transmission, lightness, ease of construction and erection, the ability to cover a broad span without any internal supports, a combination of structural and decorative elements, form variations, and so forth. They can be used as one of the most effective structures for construction in a variety of architectural configurations including free forms and curved assemblies. The constituting members of these structures can form a variety of networks in addition to the commonly used square or triangular patterns. In Persian historical buildings, compressive structures such as barrel vaults were used mostly to cover spaces due to the availability of brick, stone, and clay as construction materials. As lightweight structures, reciprocal frames not only realize the primary purpose of this group of structures for weight reduction and ease of construction, but also provide the possibility of contemporizing and reviving vaulted forms through new methods and lighter materials. Additionally, this type of structure can be designed using Persian geometric patterns as both architectural and decorative elements.

In this research, four different patterns were examined and simulated by parametric tools with the aim of implementing RF principles and construction techniques in an attempt to contemporize historical Persian brick-vaulted architecture. The RF principles also allowed for the integration of decorative and structural elements by mimicking patterns derived from the historical Persian geometric patterns known as girih. This research was carried out in three phases. In the first phase, some criteria and required factors for the generation of reciprocal patterns were extracted, and four Persian girih, including the Sorme-dan moraba (A), Shesh-e-hasiri (B), Hasht-e-moraba (C), and Chahar-lenge ghenas (D), were selected using these criteria. The required modifications were then performed on sample geometries to adjust them according to reciprocity principles on a 2D surface. In the second phase, by using parametric design tools, reciprocal patterns obtained from the previous phase were simulated onto a barrel vault as a 3D surface. Finally, in the third stage, structural analysis and optimization were carried out on the developed structures.

The simulation results show that the heaviest barrel vault developed with RFs has a weight equal to one quarter of common brick vaults. Of course, it is important to note that the weight of RF vaults will increase with the addition of covering materials but given the possibility of using lightweight materials for this, the total weight of the structure will still be less than the total weight of a conventional brick vault. The issue of weight in RF structures is variable due to the structural geometry of the patterns and their structural density. Among the simulated and optimized samples, the vault designed using the (B) and (A) patterns had the lowest and the highest weight, respectively. A comparison between the simulated vaults showed that the vaults generated with (C) and (D) behave optimally in terms of axial force transmission and they have less displacement in comparison to the other examples. It is worth mentioning that despite all the research carried out on RF structures, their design and construction still have limitations and challenges regarding connections, covering, and drainage systems. Designing the joints in RF structures on a real and long-term scale is now one of the main issues for these structures in terms of future research, especially considering their ease of replacement, covering, and drainage due to a lack of flat plate forms between the elements.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Maziar Asefi

Notes

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