Graphic Statics Research Articles

A Graphical Method for Determining Truss Stability

Graphic statics has been used for over 150 years, having been pioneered by the likes of Maxwell, Cremona, Culmann and Rankine, and has recently seen a resurgence in popularity because of its use in design. However, it is only concerned with equilibrium; as any engineer will testify, whilst equilibrium is necessary, it is not sufficient and stability must also be obtained. This paper develops a novel graphical method for determining the stability and stiffness of prestressable structures. By considering the weighted sum of the Maxwell-Minkowski diagram, the stiffness and stability of the structural mechanisms can be determined. This work extends to cover structures with multiple mechanisms and has been compared to results obtained through experimentation and the finite element method. Furthermore, it extends the work on stiffness to provide a graphical method to estimate the natural frequency of a truss. Whilst this method accurately determines the stiffness of structures, it represents a significant development in the field of graphic statics as it allows an engineer to 'eye-ball' the stability of a given truss. Engineers can also manipulate the form and force diagrams, as desired, to adjust the stiffness of their structure accordingly, whilst being able to visualise the process. Much of the previous work in this area relies heavily upon large matrices, while this method allows a more intimate and hands-on alternative.

Manipulation of topologically optimized structures using graphic statics

A combined topology optimization-graphic static approach is presented in this paper for manipulation and generation of equilibrium lightweight structures. Topology optimization is an effective tool for designing lightweight structures and truss-based designs. Graphic statics is also a powerful platform for truss-based designs and, more importantly, manipulation of such designs to satisfy pre-defined structural/load constraints. This work proposes coupling topology optimization and graphic statics in a sequential framework to take advantage of both methods in generating truss-based designs and offering a way to manipulate such structures without breaking the global equilibrium state and reducing truss forces. After post-processing and structural manipulation, results have shown structures with a uniform load distribution on all members. Optimized structures show an allowable load distribution manipulation and a significant reduction in member loads through form/force algorithms, presenting results with up to 4 and 100 times reduced maximum loads under the same state of global equilibrium. The work is essential in producing structures that avoid critical loads while having arbitrary load/boundary conditions that cannot be solved using existing methods.

When low strength materials meet funicular structures: a sustainable clay floor structure solution for emerging contexts

The inevitable expansion of the built environment due to the rapid growth of the urban population in emerging contexts poses a great challenge for the sustainable development of local communities. This problem could be faced by providing adequate multistorey building solutions while using sustainable and eco-friendly materials, preferably from renewable or upcycled and locally available sources. These materials tend to have lower mechanical properties than those commonly used in conventional structures and therefore not adequate for elements subjected to bending such as floor slabs. This challenge can be met by improving the strength of the locally available materials and/or using geometries specifically designed to lower the structural stresses such as funicular structures. In this study, a novel clay-based cementless material is used. Different mixtures and additives were tested to achieve similar processing advantages as concrete. A parabolic arched floor element is designed based on a parametric analysis using graphic statics to reduce structural stresses and weight. The fabrication system is based on reusable formworks in which locally available earth is used as part of the mould. A four meters span arched floor is built to analyse its structural behaviour and to evaluate the proposed fabrication method.

A form-finding approach for the conceptual design of air-supported structures using 3D graphic statics

This article introduces a novel equilibrium-based form-finding approach for air-supported structures that relies on vector-based 3D graphic statics and the Combinatorial Equilibrium Modelling (CEM). The proposed approach is particularly suitable for the conceptual phase of the design process, as it allows generating a wide range of design options of discrete air-supported structures without membrane material in real-time while taking into account the overall mechanical behaviour and the architectural potentials of these structures. Benchmarking analysis is carried out using Finite Element Method (FEM) models without the membrane material to validate the feasibility of the discrete structures generated using the proposed form-finding approach. The comparison between the models created by means of the proposed approach and the FEM models shows general compatibility in terms of the geometry of the structures and distribution of the internal forces. Moreover, a complete design method for air-supported structures that combines the strengths of vector-based 3D graphic statics and CEM, for the conceptual phase, and FEM, for the advanced phase with models that include the membrane material, is discussed.

Unified geometrical framework for the plastic design of reinforced concrete structures

AbstractAlthough the analysis and design of structures in static equilibrium can be intuitively carried out using simple equilibrium‐based methods such as graphic statics, the application of these methods to engineering problems that take into consideration specific material properties is generally limited. Within the domain of reinforced concrete, existing geometric approaches for developing stress fields and yield lines based on the theory of plasticity are especially useful. However, these approaches usually rely on iterative constructions and are generally limited to two‐dimensional cases. By taking advantage of graphic statics, this article introduces the theoretical basis for an entirely geometrical method to generate discrete stress fields and yield line patterns in two‐ and three‐dimensional reinforced concrete structures. The proposed approach is based on the use of reciprocal stress functions and the relationship between form and force diagrams.

Shape optimization of no-tension arches subjected to in-plane loading

In this paper a novel method for the shape optimization of tapered arches subjected to in-plane gravity (self-weight) and horizontal loading through compressive internal loading is presented. The arch is discretized into beam elements, and axial deformation is assumed to be small. The curved shape of the tapered arch is discretized into a centroidal B-spline curve with beam elements. Constraints are imposed for allowable axial force and bending moment in the arch so that only compressive stress exists in the section. The computational cost for optimization is reduced, and the convergence property is improved by considering the locations of the control points of B-spline curves as design variables. The height of section is also modeled using a B-spline function. A section update algorithm is introduced in the optimization procedure to account for the contact and separation phenomena and to further speed up the computation process. The objective function to minimize the total strain energy of the arch under self-weight and horizontal loading. Numerical examples are presented that demonstrate the effectiveness of the proposed method. To validate the findings, the properties of the obtained optimal shapes are compared to shapes obtained by a graphic statics approach.

Graphic statics in a continuum: Strut-and-tie models for reinforced concrete

This research bridges the gap between the numerical layout optimization method and the geometry-based analysis and design method of graphic statics. The study connects the two methods for the application of strut-and-tie models in reinforced concrete design. It suggests a new algorithm for the algebraic graphic statics of indeterminate trusses inspired by the layout optimization method. This research also contributes to the field of graphic statics by providing a formulation that generates form and force diagrams in a continuum, where the topology for none of the diagrams is initially provided.The generation of strut-and-tie models for reinforced concrete has extensively relied on optimization methods, which are helpful techniques for the initiation of a load path inside a continuous domain. However, the resulting truss model is a single answer, and there exists limited control to methodically modify the topology or the force distribution of the model. Furthermore, the minimized-weight truss does not guarantee a practical strut-and-tie model or an optimized performance of a reinforcement design.In contrast to the conventional optimization techniques with a single solution, the intuitive method of graphic statics provides the designer with a vast design space. It also offers explicit control over the geometry and force distribution of the generated truss models. The algorithm provided in this paper is applied to various continuous domains to systematically generate a variety of strut-and-tie models, their force diagrams, and constant stress fields. The production of the optimized truss model and its force distribution allows the direct interactive manipulations of the design while observing the changes in the stress fields and the reinforcement arrangement. The open-source repository of the implemented integrated algorithm and the examples used in this paper are also provided.

Machine learning assisted evaluations in structural design and construction

This paper proposes a new design approach based on an iterative machine learning algorithm to speed up the topological design exploration of compression-only shell structures with planar faces, considering both structural performance and construction constraints. In this paper, we show that building neural networks allows one to train a surrogate model to accelerate the structural performance assessment of various possible structural forms without going through a significantly slower process of geometric form-finding. The geometric form-finding methods of 3D graphic statics are used as the primary structural design tool to generate a single-layer, compression-only shell with planar faces. Subdividing the force diagram and its polyhedral cells using various rules results in a variety of topologically different compression-only structures with different load-bearing capacities for the same boundary conditions. The solution space for all possible compression-only forms for a given boundary condition is vast, which makes iterating through all forms to find the ideal solutions practically impossible. After training with an iterative active sampling method, the surrogate model can evaluate the input data, including the subdivision rules, and predict the value of the structural performance and the construction constraints of the planar faces within milliseconds. As a result, one can then evaluate the nonlinear relations among all the subdivision rules and the chosen structural performance measures, and then, visualize the entire solution space. Consequently, multiple solutions with customized thresholds of the evaluation criteria are found that show the strength of this method of form-finding in generating design solutions. Besides, considering the total training time of the neural network model, the proposed framework is still faster than a traditional optimization method, such as the genetic algorithm that can find only the optimum values. This process will result in interactive sampling methods in which the machine learning models assist the designer in choosing and controlling different design strategies by providing real-time feedback on the effects of the selected parameters on the design outputs.

Graphical method for the construction of pairs of 3D skew funiculars in equilibrium with coplanar loads and controlled supports' positions

AbstractTwo‐dimensional (2D) funicular structures can be easily generated and controlled by means of simple geometric constructions. These geometric constructions derive from reciprocal relationships between the form and force diagrams of a structure, which form the basis of graphic statics. Unlike in 2D space, in 3D space, similar geometric constructions cannot always be established for the design of three‐dimensional (3D) funicular structures. In some specific cases though, purely geometric procedures requiring no numerical approximations can be defined. In particular, this contribution explores the geometric transformation of 2D funicular structures as a means to graphically generate 3D funicular structures in equilibrium with coplanar loads. A fully graphical method for the construction of such pairs of 3D skew funiculars in equilibrium with coplanar loads and with control over the position of the supports is presented. It is based on geometrical linear relationships between the original and the transformed structures. Ultimately, it is shown how such pairs of 3D skew funiculars can be applied to the design of bridge and roof structures for instance.

A Computer-Aided Approach to Equilibrium Design Based on Graphic Statics and Combinatorial Variations

This article introduces a series of significant enhancements to the Combinatorial Equilibrium Modelling (CEM), a computer-aided form-finding approach to generate and control three-dimensional structures in static equilibrium based on topological considerations. In comparison to the initial formulation of the CEM, this article presents several procedures that extend the applicability of the CEM to several structural topologies and load cases, while allowing the designer to engage with different types of constraint-based optimisation problems. As described in the article, these new features of the CEM have been implemented in a digital tool for form-finding, which has been applied to a series of design examples and case studies to demonstrate the usability of the proposed approach for conceptual structural design.

Pravčasta geometrija i prostorna grafostatika

Pravčasta geometrija i prostorna grafostatika

Vector-based 3D graphic statics: A framework for the design of spatial structures based on the relation between form and forces

This article develops a vector-based 3D graphic statics framework that uses synthetic and intuitive graphical means for the analysis and design of spatial structures such as networks of bar elements in static equilibrium. It is intended to support the collaborative work of structural engineers and architects from the conceptual phase of the design process. Several procedures for the construction of a vector-based 3D force diagram for any given 3D form diagram with an underlying planar or non-planar graph are identified and described. In the non-planar case, the proposed procedures rely on the preliminary topological planarization of the graph by cutting the crossing edges and reconnecting them to one or more newly inserted auxiliary vertices. Resulting planar graphs can be then used as a base for the assembly of 3D force diagrams, without altering the static equilibrium of the structure. An implementation of the proposed framework to real design scenarios is presented through two case studies. These examples demonstrate the benefits of bi-directional manipulations of form and force diagrams in the structural design process.

Form Generation of Cable-Stayed Structures considering Structural Art - by using Graphic Statics -

Form Generation of Cable-Stayed Structures considering Structural Art - by using Graphic Statics -

Form finding of corrugated shell structures for seismic design and validation using non-linear pushover analysis

The geometry of shell structures plays an essential role in their capacity to withstand earthquakes. However seismic loading is rarely considered when determining the overall geometry of shells. This paper presents a novel form finding methodology for the conceptual seismic design of corrugated shells. The method ensures that a compression load path exists to carry lateral earthquake accelerations by deriving shell geometries from a series of funicular polygons obtained through a graphic statics procedure for combined gravity and horizontal loads. While the method can be applied to any material that resists compressive stresses, it is employed in this paper to find the shapes of corrugated thin-tile masonry shells. Non-linear pushover analysis is then used to quantify lateral capacity and evaluate form finding results in terms of material efficiency to resist lateral loads. The analysis furthermore provides insights regarding the collapse mechanisms and flow of forces. It is demonstrated that the lateral capacity before cracking in the corrugated shell shapes is up to 79% higher than the capacity of a non-form-found reference shell shapes considering identical material use. All form-found shells were found to fail through a similar collapse mechanism which is defined by four crack zones. The location of these crack zones can be manipulated through the form finding process and identify the locations where reinforcement could be most efficiently introduced. Finally, the flow of forces within the form-found shells is used to propose alternative designs that provide additional openings in the shell surface while maintaining similar seismic capacity. Thus, the paper provides a new approach for the conceptual design of safe corrugated shell structures in earthquake prone areas.

Design Strategies of Hybrid Bending-Active Systems Based on Graphic Statics and a Constrained Force Density Method

Design Strategies of Hybrid Bending-Active Systems Based on Graphic Statics and a Constrained Force Density Method

The Application of 2d and 3d Graphic Statics in Design

The Application of 2d and 3d Graphic Statics in Design

Algebraic 3D graphic statics: Reciprocal constructions

The recently developed 3D graphic statics (3DGS) lacks a rigorous mathematical definition relating the geometrical and topological properties of the reciprocal polyhedral diagrams as well as a precise method for the geometric construction of these diagrams. This paper provides a fundamental algebraic formulation for 3DGS by developing equilibrium equations around the edges of the primal diagram and satisfying the equations by the closeness of the polygons constructed by the edges of the corresponding faces in the dual/reciprocal diagram. The research provides multiple numerical methods for solving the equilibrium equations and explains the advantage of using each technique. The approach of this paper can be used for compression-and-tension combined form-finding and analysis as it allows constructing both the form and force diagrams based on the interpretation of the input diagram. Besides, the paper expands on the geometric/static degrees of (in)determinacies of the diagrams using the algebraic formulation and shows how these properties can be used for the constrained manipulation of the polyhedrons in an interactive environment without breaking the reciprocity between the two.

Polarities in structural analysis and design: n-dimensional graphic statics and structural transformations

This paper proposes a unified and entirely geometrical methodology for generating 2D and 3D force diagrams for given planar and spatial trusses in static equilibrium within the context of graphic statics. The trusses, regarded as form diagrams, are projections of higher dimensional simply-connected stress functions, can be either self—stressed or loaded with external forces, and need not be tension-only/ compression-only. First, we provide an in-depth overview of fundamental notions of graphic statics and projective geometry related to the construction of reciprocal form and force diagrams. Specifically, we describe a series of polar transformations and discuss them from a geometric and an algebraic standpoint. Moreover, we provide an exegesis and visualization of Maxwell's reciprocal methods while discussing them within the scientific framework of his time. We then develop a contemporary graphic statics framework grounded on projective geometry and higher-dimensional reciprocal stress functions. Within this framework, we provide a general methodology for deriving any type of reciprocal diagrams for the planar and the spatial case. Advantages of this method include: direct control on any of the four reciprocal objects (form diagram, force diagram, corresponding stress functions) and no need for iterative/ procedural reconstruction of any of the diagrams since it is based on direct geometrical transformations. Finally, we highlight how duality, and in particular polarity, can be used for a wide range of applications in structural analysis and design.

Disjointed force polyhedra

This paper presents a new computational framework for 3D graphic statics based on the concept of disjointed force polyhedra. At the core of this framework are the Extended Gaussian Image and area-pursuit algorithms, which allow more precise control of the face areas of force polyhedra, and consequently of the magnitudes and distributions of the forces within the structure. The explicit control of the polyhedral face areas enables designers to implement more quantitative, force-driven constraints and it expands the range of 3D graphic statics applications beyond just shape explorations. The significance and potential of this new computational approach to 3D graphic statics is demonstrated through numerous examples, which illustrate how the disjointed force polyhedra enable force-driven design explorations of new structural typologies that were simply not realisable with previous implementations of 3D graphic statics.

On Structural Behavior of a Funicular Concrete Polyhedral Frame Designed by 3D Graphic Statics

This paper investigates the mechanical behavior of the first built funicular structure designed by 3D graphic statics (3DGS) using reciprocal polyhedral diagrams. Graphic statics methods are unique among other structural design techniques in providing an intuitive control over the form of the structure and its equilibrium of forces for designers. Since graphic statics does not include material properties in the form finding process, further numerical investigations to foresee the behavior of the system under loading scenarios other than the 3DGS design loads are unavoidable. This research reports the structural behavior of Hedracrete which is a prefab, concrete, polyhedral frame with both compression and tension members. A simplified (bar-node) and a detailed volumetric mesh model of the structure are numerically analyzed under eight different loading scenarios. Different combinations of linear and nonlinear material behavior for steel and concrete in addition to linear and quadratic element types are used in those analyses. The analyses are planned to initially verify the equilibrium results of the 3DGS model and to predict the maximum load-bearing capacity of the structure by studying the failure mechanism of the system.