Band 8, Heft 1

This paper presents the solution of stress distribution around elliptical cutout in an infinite laminated composite plate. Analysis is done for in plane loading under hygrothermal environment. The formulation to obtain stresses around elliptical hole is based on Muskhelishvili’s complex variable method. The effect of fibre angle, type of in plane loading, volume fraction of fibre, change in temperature, fibre materials, stacking sequence and environmental conditions on stress distribution around elliptical hole is presented. The study revealed, these factors have significant effect on stress concentration in hygrothermal environment and stress concentration changes are significant with change in temperature.

Robustness analyses are very well referenced for concrete or steel frame structures but less for spatial structures; in particular for truss roofs. Here, we present a robustness analysis of an existing airport space structure. A finite element model was implemented based on the original design documents, where the structure is modeled as a spatial truss composed of elastic, perfectly hinged bars. With respect to five main design loading conditions, the most stressed bars among the top layer, the bottom layer and the diagonals, were alternately removed, and the stress variations in the remaining bars monitored. A total of fifteen analyses with removal of either the most stretched or the most compressed bar were run. Also, reductions of the most stressed bars’ axial stiffness were considered to investigate the effects of such local reductions on the global structural flexibility. Linearized global buckling analyses were also conducted to point out again the effect of a global loss of stiffness, due to local losses. The study gives basic information about the general behavior of the structure in case of failure, or damage, of a key element. Results show that this kind of lightweight and efficient structures are very sensitive to local losses, since their redistribution capabilities are not large.

The new paradigms of parametric modelling have been proving promising on the advance of systems for analysis and design of taut (or tensile) structures. With this premise, the presented work consist on the development with a form-finding tool for Computer Aided Design(CAE) and Computer Aided Engineering (CAE) integration using VPL (Visual Programming Language), in the context of parametric modelling. The methods used in the implementation are the Force Density Method (FDM) and the Natural Force Density Method (NFDM), taking advantage of the linear solution approach provided, suitable for fast form-finding computational procedures. The program is implemented as a Grasshopper plug-in and it is named BATS (Basic Analysis of Taut Structures), which enables parametric definition of boundary conditions for the form-finding. The program structure and benchmarks with other available Grasshopper plug-ins for taut structures form-finding are presented, showing considerably superior performance using BATS.

This paper deals with the use of the shells in the architecture of concert halls through research on the historical development of the concert building’s space. These elements are widely applied in civil engineering and architecture due to the good operational qualities, very rich spectrum of forms, the simplicity of production and its technical efficiency. The article considers two types of shells: conoid and truncated cone to choose the most optimal form for covering the concert hall.

The form-finding analysis is a crucial step for determining the stable self-equilibrated states for tensegrity structures, in the absence of external loads. This form-finding problem leads to the evaluation of both the self-stress in the elements and the shape of the tensegrity structure. This paper presents a novel method for determining feasible integral self-stress states for tensegrity structures, that is self-equilibrated states consistent with the unilateral behaviour of the elements, struts in compression and cables in tension, and with the symmetry properties of the structure. In particular, once defined the connectivity between the elements and the nodal coordinates, the feasible self-stress states are determined by suitably investigating the Distributed Static Indeterminacy (DSI). The proposed method allows for obtaining feasible integral self-stress solutions by a unique Singular Value Decomposition (SVD) of the equilibrium matrix, whereas other approaches in the literature require two SVD. Moreover, the proposed approach allows for effectively determining the Force Denstiy matrix, whose properties are strictly related to the super-stability of the tensegrity structures. Three tensegrity structures were studied in order to assess and discuss the efficiency and accuracy of the proposed innovative method.

This is a conceptual work about the form-finding of a hybrid tensegrity structure. The structure was obtained from the combination of arch-supported membrane systems and diamond-type tensegrity systems. By combining these two types of structures, the resulting system features the “tensile-integrity” property of cables and membrane together with what we call “floating-bending” of the arches, a term which is intended to recall the words “floating-compression” introduced by Kenneth Snelson, the father of tensegrities. Two approaches in the form-finding calculations were followed, the Matlab implementation of a simple model comprising standard constant-stress membrane/cable elements together with the so-called stick-and-spring elements for the arches, and the analysis with the commercial software WinTess, used in conjunction with Rhino and Grasshopper. The case study of a T3 floating-bending tensile-integrity structure was explored, a structure that features a much larger enclosed volume in comparison to conventional tensegrity prisms. The structural design of an outdoor pavilion of 6 m in height was carried out considering ultimate and service limit states. This study shows that floating-bending structures are feasible, opening the way to the introduction of suitable analysis and optimization procedures for this type of structures.

The influence of a wide temperature range in the glass transition region of a shape memory polymer (SMP) matrix on micro-buckling of the fiber reinforcements in shape memory polymer composites (SMPC) under large bending deformation is described. Analytical expressions to estimate the strain energy, neutral strain surface, critical buckling surface and half wavelength of the buckled fibers in the SMPC are presented based on the minimum energy method. This study considers the reinforced fibers as three-dimensional elastic bodies and the matrix as a temperature stimulated flat plate. A comprehensive study was performed to understand the dynamic temperature behavior of the micro-buckled fibers and corresponding results were validated by previous works in the literature. The effects of fiber volume fraction and thickness of the SMPC plates on the half wavelength are also discussed along with the simultaneous influence of temperature on the parameters computed in the minimum energy analysis.

In the present paper, the acoustic emission (AE) device is used with an innovative approach, based on the calculation of P-wave propagation velocity ( v p ), to detect the stiffness characteristics and the diffused damage of in-service old concrete structures. The paper presents the result of a recent testing campaign carried out on the slant pillars composing the vertical bearing structures designed by Pier Luigi Nervi in one of his most iconic buildings: the Hall B of Torino Esposizioni. In order to investigate the properties of these inclined pillars, localizations of artificial sources (hammer impacts), by the triangulation procedure, were performed on three different inclined elements characterized by stiffness discrepancies due to different causes: the casting procedures, executed in different stages, and the enlargement of the hall happened a few years later the beginning of the construction. In the present work, the relationship between the velocity of AE signals and the elastic characteristics (principally elastic modulus, E) is evaluated in order to discriminate the stiffness level of the slanted pillars. The procedure presented made it possible to develop an innovative investigation method able to estimate, by means of AE, the state of conservation and the elastic properties and the damage level of the monitored concrete and reinforced concrete structures.

Free-edge gridshells represent the majority of built gridshells. Indeed, the gridshell reference geometry usually needs to be trimmed in order to provide building access or to insert the gridshell within an existing building, giving rise to one or more elastic boundaries. Despite the current design practice, so far a very limited number of scientific studies has been devoted to investigate the influence of elastic boundaries on the overall structural behaviour of gridshells. This paper focuses on the effects of the orientation of the boundary structure with respect to the grid direction. This is done by studying the buckling behaviour of an ideal single-layer steel gridshell, for different grid layout (quadrangular, hybrid, triangular) and orientation. The results of the parametric study demonstrate that the sensitivity of free-edge single-layer gridshells to the free-edge orientation strongly depends on the grid pattern. In particular, isotropic gridshells have shown an almost negligible influence of the free-edge orientation in terms of buckling load, in opposition to orthotropic gridshells. Moreover, the change in free-edge orientation induces significant variations of the global structural stiffness for all the layouts, resulting in possibly unacceptable displacements in service conditions.

The stability check of masonry structures is a debated problem in Italy that poses serious problems for its extensive use. Indeed, the danger of out of plane collapse of masonry walls, which is one of the more challenging to evaluate, is traditionally addressed not using finite element models (FEM). The power of FEM is not properly used and some simplified method are preferred. In this paper the use of the thrust surface is suggested. This concept allows to to evaluate the eccentricity of the membrane stresses using the FEM method. For this purpose a sophisticated, layered, finite element with a no-tension material is used. To model a no-tension material we used the smeared crack method as it is not mesh-dependent and it is well known since the early ’80 in an ASCE Report [1]. The described element has been implemented by the author in the program Nòlian by Softing.

Origami folding structures can find significant applications in the general area of building design, as they can be lightweight and deployable. An inherent property of folded surfaces, which is related to the degrees of freedom of each origami crease pattern, is form flexibility. Therefore, when building-scale applications are considered, in many instances, the folded surfaces, in order to become stiff and load-bearing, need to be constrained. A study of different types of deployable structures has led to the observation that in planar scissor linkages, hinges and pivots follow the same deployment path in space, as sets of vertices in certain origami structures. Due to the similarities in their kinematic behavior, selected origami patterns and scissor linkages can function as effective kinematic pairs, leading to structures able to transform in a controlled manner through a wide range of possible spatial configurations. A few examples of combining these two types of structures already exist [1–3]. In this paper a systematic approach for coupling origami crease patterns characterized by biaxial and rotational symmetry, with translational, polar and angulated scissor linkages, towards the development of novel forms of deployable structures, has been attempted. For the design and evaluation of the kinematic performance of the developed new structures, existing geometric modeling and calculation methods, parametric and simulation processes, as well as testing with physical models have been used. It is anticipated that research in this direction will lead to promising novel hybrid types of deployable structures.

Shell Eco-Marathon (SEM) is an international competition among university students that involves designing, building, and driving energy-efficient cars. The car frame is the most crucial aspect influencing the strength of the car. This research aims to obtain maximum torsional strength with variations in the material and thickness of the frame. Calculation and testing are done using the simulation method to obtain a strong car frame. This simulation method is calculated by a series of finite element analyses. Then, data from the simulation method are obtained in the form of deformation and safety factors. By comparing the moment received with its deformation, torsional stiffness is then obtained. Furthermore, the torsional stiffness is divided by the weight to produce a value ratio. It is known that the factor which has the most significant influence on the difference in torsional stiffness of each variation is the shear modulus of the material used. In contrast, the weight of the chassis is influenced by the density of the material and the thickness of the chassis. Additionally, the safety factor of each variation is strongly influenced by the strength of the chassis structure itself. The results of this study will demonstrate the car frame design with the best performance.

A research subject in structural engineering is the problem of vibration under a loading object. The two-dimensional (2D) model of a structure under loading is an example. In general, this case uses an object that is given a random frequency, which then causes various changes in shape depending on the frequency model. To determine the difference in performance by looking at the different forms of each mode, modal analysis with ANSYS was used. The samples to be simulated were metal plates with three variations of the model, namely, a virgin metal plate without any holes or stiffness, plates with given holes, and metal plates with stiffness on one side. The model was simulated with modal analysis, so that 20 natural frequencies were recorded. The sample also used different materials: low-carbon steel materials (AISI 304), marine materials (AISI 1090), and ice-class materials (AR 235). Several random-frequency models proved the deformation of different objects. Variations of sheet-metal designs were applied, such as pure sheet metal, giving holes to the sides, and stiffening the simulated metal sheet.

In the field of logistics, containers are indispensable for shipments of large quantities of goods, particularly for exports and imports distributed by land, sea, or air. Therefore, a container must be able to withstand external loads so that goods can safely reach their destination. In this study, seven different models of container skins were developed: general honeycomb, cross honeycomb, square honeycomb, corrugated wall, flat, flat with a single stiffener, and flat with a cross stiffener. Testing was performed using the finite element method. In the static simulation, the best results were obtained by the model with corrugated walls. As the main element and the content of the sandwich panel structure, the core plays a role in increasing the ability of the structure to absorb force, thereby increasing the strength of the material. In the thermal simulation, the best results were obtained by the general honeycomb walls. Vibration simulations also showed that the square honeycomb design was better at absorbing vibration than the other models. Finally, the corrugated model had the best critical load value in the buckling simulation.

In this paper, an original analytical formulation to evaluate the natural frequencies and mode shapes of high-rise buildings is proposed. The methodology is intended to be used by engineers in the preliminary design phases as it allows the evaluation of the dynamic response of high-rise buildings consisting of thin-walled closed- or open-section shear walls, frames, framed tubes, and dia-grid systems. If thin-walled open-section shear walls are present, the stiffness matrix of the element is evaluated considering Vlasov’s theory. Using the procedure called General Algorithm, which allows to assemble the stiffness matrices of the individual vertical bracing elements, it is possible to model the structure as a single equivalent cantilever beam. Furthermore, the degrees of freedom of the structural system are reduced to only three per floor: two translations in the x and y directions and a rigid rotation of the floor around the vertical axis of the building. This results in a drastic reduction in calculation times compared to those necessary to carry out the same analysis using commercial software that implements Finite Element models. The potential of the proposed method is confirmed by a numerical example, which demonstrates the benefits of this procedure.

This paper is an experimental study of the quasi-static mechanical compressive properties of the reinforced closed-cell aluminum alloy foams with different cell orientations at different strain rates. The reinforced foam samples were obtained via the powder metallurgical route. The results of the compression tests revealed that the deformation behavior and mechanical properties of foamed aluminum composites are highly dependent on the orientation of the reinforcing mesh. Differences in the deformation behavior of foams appear to be influenced by the mechanical properties of the matrix material, by foam deformation mechanisms, and by the mechanical properties of the reinforcement. The yield stress, plateau stress, densification stress, and energy absorption capacity of unreinforced foam samples improved linearly with increasing strain rate due to dynamic recrystallization and softening of the foam matrix material. The reinforced foam samples exhibit nonlinear deformation behavior. It was also found that the mechanical properties reduction of transverse reinforced foams was slightly lower compared to foams with longitudinal reinforcement at varying strain rates because of the large contribution of the mechanical properties of the reinforcement. The results of the present study can be employed to modelling and obtain impact-resistant fillers for complex structures in transport construction.

In the specific case of curved cable-stayed bridges, the horizontal component of the load introduced by the stay cables on the deck is variable, concentric and dependent on the connection configuration between the tower and the cables, becoming a challenge in the design of these type of bridges. Hitherto, designers have dealt with this challenge in different ways, either by optimizing the position of the tower and its geometric characteristics, or by modifying the morphology of the stay cable system. This paper proposes the use of funicular and anti-funicular curves of the horizontal concentric load, introduced by the stay cables, to design the curved deck directrix, reducing lateral forces on the deck under the self-weight hypothesis. For the design of the deck directrix, two different formulations are considered: one discrete by means of summations and the other continuous by means of non-linear differential equations. Both formulations study the two possible signs of the axial force which will govern the design (funicular and anti-funicular curves). A least squares approximation is developed to facilitate the implementation of these formulations. The paper introduces a method to liberate the deck from its most important lateral loads, i.e., the concentric loads introduced by the stay cables. This way, it develops a deck dominated by axial forces instead of lateral ones (Bending moment with vertical axis, Mz, and lateral shear force, Vy), which can be critical for its design and decrease the stay-cable system efficiency. It explains, by different methods, how this directrices vary with different design decisions, so that the designer can develop the directrix that suits his design. Finally, two examples of directrices are given as a conclusion.

We exploit the possibility of deforming a shell by assigning a target metric, which, for 2D structures, is decomposed into the first and second target fundamental-forms. As well known, an elastic shell may change its shape under two different kinds of actions: one are the loadings, the other one are the distortions, also known as the pre-strains. Actually, the target fundamental forms prescribe a sought shape for the solid, and the metric effectively realized is the one that minimizes the distance, measured through an elastic energy, between the target and the actual fundamental forms. The proposed method is very effective in deforming shells.

According to Heyman’s safe theorem of the limit analysis of masonry structures, the safety of masonry arches can be verified by finding at least one line of thrust entirely laying within the masonry and in equilibrium with external loads. If such a solution does exist, two extreme configurations of the thrust line can be determined, respectively referred to as solutions of minimum and maximum thrust. In this paper it is presented a numerical procedure for determining both these solutions with reference to masonry arches of general shape, subjected to both vertical and horizontal loads. The algorithm takes advantage of a simplification of the equations underlying the Thrust Network Analysis. Actually, for the case of planar lines of thrust, the horizontal components of the reference thrusts can be computed in closed form at each iteration and for any arbitrary loading condition. The heights of the points of the thrust line are then computed by solving a constrained linear optimization problem by means of the Dual-Simplex algorithm. The MATLAB implementation of presented algorithm is described in detail and made freely available to interested users (https://bit.ly/3krlVxH). Two numerical examples regarding a pointed and a lowered circular arch are presented in order to show the performance of the method.

Curved envelope structural building envelopes have been quite popular in architecture in the past decades, and pose many challenges in their design, manufacturing and planning. In gridshells, a popular structural morphology for curved structure, designers will often strive to orient beams such that their top face is parallel to the envelope surface. However, this tends to induce geometrical torsion along the beam centerline, which complexifies significantly the manufacturing of the connection nodes or of the beams themselves. It is well known that such issue can be avoided by aligning beams with principal curvature directions of the envelope surface, thus yielding a quadrangular paneling. In this article, we study how other types of patterns (non-quadrangular) can be used to design torsion-free grid-shells. Based on asymptotic considerations, we derive a set of geometrical rules which, if fulfilled by a pattern, insure that a surface can be covered by this pattern with negligible torsion and limited deviation of beams from surface normals. A wide variety of patterns fulfill these rules, offering interesting possibilities for the design of curved architectural envelopes (Figure 1) is shown. As these rules are based on first order asymptotic analysis, we perform global validation on case studies. One main application is for structures in which face planarity is not necessary, for example ones cladded with ETFE cushions.

At the time of completion in 1961, the roof of St. Charles Church became the largest unbalanced hyperbolic paraboloid structure in the United States and the only shell structure in Spokane, WA. Situated on an 8-acre site on the north side of the city, St. Charles is a modernist structure designed through partnership of Funk, Molander & Johnson engineers, architect William C. James and in consultation with Professor T.Y. Lin of the Structural Engineering Laboratory at the University of California, Berkeley. This asymmetric structure spans over 33.5 m (110 ft) and utilizes folded edge beams that taper from 1067 mm (42 in) at the base to a 76.2 mm (3 in) thickness at the topmost edge using regular strength reinforcing steel and concrete load carrying components. The novelty of the pre-stressed shell structure serves both architectural and structural design criteria by delivering a large, uninterrupted interior sanctuary space in materially and economically efficient manner. This structural assessment summarizes the roof’s historic design and construction according to the original construction documents, newspaper reports and historic photographs. The FEA is completed using UBC 1955 design loads and ACI 334 Concrete Shell Structures provisions.

An historical overview is presented concerning the theory of shell structures and thin domes. Early conjectures proposed, among others, by French, German, and Russian Authors are discussed. Static and kinematic matrix operator equations are formulated explicitly in the case of shells of revolution and thin domes. It is realized how the static and kinematic matrix operators are one the ad-joint of the other, and, on the other hand, it can be rigorously demonstrated through the definition of stiffness matrix and the application of virtual work principle. In this context, any possible omission present in the previous approaches becomes evident. As regards thin shells of revolution (thin domes), the elastic problem results to be internally statically-determinate, in analogy to the case of curved beams, being characterized by a system of two equilibrium equations in two unknowns. Thus, the elastic solution can be obtained just based on the equilibrium equations and independently of the shape of the membrane itself. The same cannot be affirmed for the unidimensional elements without ‚exural stiffness (ropes). Generally speaking, the static problem of elastic domes is governed by two parameters, the constraint reactions being assumed to be tangential to meridians at the dome edges: the shallowness ratio and the thickness of the dome. On the other hand, when the dome thickness tends to zero, the funicularity emerges and prevails, independently of the shallowness ratio or the shape of the dome. When the thickness is finite, an optimal shape is demonstrated to exist, which minimizes the flexural regime if compared to the membrane one.

Recently, structural monitoring technology invested in methodologies that give direct information on structures’ stress state. Optic fibers, strain gauges, pressure cells give real-time data on the stress condition of a structural element, often determining the area where peak stresses have been reached, with a clear advantage over other less direct monitoring methodologies, such as, e.g. , the use of accelerometers and inverse analysis to estimate internal forces. In addition, stresses can be recorded in a data log for analysis after a loading event, as well as for taking into account the lifelong stress state of the structure. Beams and columns of a reinforced concrete frame can be effectively monitored for flexural loads. Differently, thin shells are most of their lifespan under membrane regime, and, when properly designed, they rarely move to the bending regime. Our proposal is to monitor the stress in thin structures by small-sized low-cost devices able to record the stress history at key locations, sending alerts when necessary, with the aim of ensuring safety against the risk of collapse, or simply to perform maintenance/repairing activities. Such devices are realized with cheap off-the-shelf electronics and traditional strain gauges. The application examples are given as laboratory tests performed on a reinforced concrete plate, a masonry panel, and a steel beam. Results shows that the permanent monitoring control of stresses can be conveniently carried out on new structures using low-cost devices of the type we designed and realized in-house.

In this paper the mechanical characteristics of an innovative bioplastic material, the HBP ® -HempBioPlastic ® filament, is investigated. HBP ® was recently patented by an Italian company Kanésis that focused its activity on nature-derived materials. The filaments are the upshot of an original process allowing to reuse the surplus of the agricultural supply chains and transform it into new sustainable materials. At first, the 3D printed HBP ® samples were tested in tensile tests according to the ASTMD638 standard and monitored in term of deformations by the Digital Image Correlation techniques (DIC) in order to evaluate the stress-strain behavior of different HBP ® textures under loading. In addition, using the HBP ® and the results coming from the experimental campaign, the design of an exhibition pavilion was proposed. The pavilion was modelled starting from the geometric construction of the fullerene. The supporting modular structure is combined by HBP ® modular elements, that can be produced by 3D printing or moulding. Finally, in order to demonstrate the feasibility of the proposed pavilion, a linear finite element analysis is presented on the base of the experimentally determined mechanical properties of HBP ® elements, under the effects of wind and seismic environmental actions.

The new serious consideration to masonry and non-metallic structures evidenced their direct prospective to be, even in the present days, advanced architectural and engineering solutions. In the present paper, a form finding for a cement based tessellated pierced vault is studied. The multi-body rope approach (MRA) was used to define compression-only vault optimal shapes. Successively, the thrust network analysis (TNA) was implemented by Rhino-vault for a further validation of the shape and the definition of different tessellation meshes of the surfaces, according to different hole pattern configuration. Different piercing percentage of the vaults were considered and compared for the best solution identification. In addition, the geometrical solutions were analyzed by means of global stability analysis, taking into account the different positions of the holes. Furthermore, 3D printing with a Fuse Deposition Modeling (FDM) technique in polylactide (PLA) material (completely eco-friendly) is used for the construction of the formworks of the cement based blocks (dowels) useful for the assembly of a vault scaled prototype. The prototype of the vault, characterized by a certain piercing percentage was subjected to different loading conditions and monitored by a non-contact device based on the Digital Image Correlation (DIC) technique. The 3D-DIC was performed to recognize the structural behavior during the loading process of the model (prototype). DIC measurements were used to recognize in advance the critical condition of the vault under loading and the displacement measurements were correlated to the different loading phases up to the collapse condition.

Classification and development of the deployable structures is an ongoing process that started at the end of 20 th century and is getting more and more attention throughout 21 st . With the development of the technology, constructive systems and materials, these categorizations changed – adding new typologies and excluding certain ones. This work is giving a critical review of the work done previously and on the change of the categories. The special interest is given to the pantographs (or scissor structures) and the Zeigler’s dome as the form of their application. It is noticeable that after its introduction in 1977, the dome was a part of the initial classification, but with the time it lost its place. The reason for this is the introduction of more efficient scissor dome structures. However, perhaps with the use of data-driven design, this dome can be optimized and become relevant again. The second part of the paper is dedicated to the development of the structural optimization algorithm for panto-graph structures and its application on the example of Zeigler’s dome. Besides the direct analysis, the final part includes the generative optimization algorithm which could help to a decision-maker in the early stages of the design to understand and select the options for the structure.

The research starts from an analogy found between two apparently very different structural solutions: the double spiral pattern of the herringbone brick courses in the domes built by Antonio da Sangallo the Younger (1484-1546) during the Renaissance, and the particular pattern of a wooden floor ‘à la Serlio’, described by Amand Rose Emy in his Treatise at the beginning of 19 th century, made by diagonal beams reciprocally sustained. The diagonal pattern of the floor has a geometrical relationship with the cross-herringbone pattern, so that the latter can be obtained by some geometrical transformations of the former. This pattern was also used in thin shells built by Nervi, from the destroyed airplane hangars in Tuscany to the Palazzetto dello sport in Rome, and even by Piacentini in 1936 and earlier in some neoclassical domes. Thus the construction tool, useful for building domes without expensive scaffolding, could have a structural role at the completed construction stage. Within the research different structures were investigated, in order to observe the relevance of this peculiar structural scheme particularly in the construction of modern domes.

This paper presents the evolution of and influence that the use of a triangular grid has had on the transformation of layered grid structures during the last two centuries. Historical facts, technological advances, economic and social crises among other factors influence the process of change and evolution of the grid systems in architecture to make construction processes more efficient by reducing the consumption of materials and qualified labor. Geometry and structure innovate styles and processes that are perfected over the years, achieving lighter and more stable structures. For the purpose of this paper historical data will be collected that allows the development of a theory that shows the influence of various periods based on technological advances. This research shows four historical periods in the evolution of layered grid structures. Being the triangular grid a widely used methodology in the construction of a diversity of geometric proposals and innovative construction systems for the solution of a diversity of buildings from the first domes built in the early 19th century to the free-form structures built from the late 20th century to the present day.