Cable-strut Systems Research Articles

Potential Energy Optimization Approach in the Form-Finding of Tensegrities and Cable–Strut Systems

A form-finding process for tensegrities and cable–strut systems is determined as a minimum problem of potential energy that is constructed considering initial estimated pre-stress states. Both the Jacobian and Hessian matrices of a potential energy function are derived explicitly regarding free nodal coordinates. Based on these derivations, both the first-order geodesic dynamic relaxation optimization method and the second-order optimization method of Newton can be applied to solve form-finding problems involving tensegrities and cable–strut structures. It should be noted that both the pre-stress state and the configuration of the systems are adjusted during the optimization process to satisfy self-equilibrium conditions. The geometrical constraints are then included in the proposed work as equality constraint conditions to find the final forms satisfying not only the self-equilibrium states but also the designers’ demands. Two well-known and complex systems are investigated to assess the efficiency of the proposed method, i.e. free-standing truncated icosahedral tensegrity and the Kiewitt Dome.

Comparative study and optimal selection for different cable-stiffened latticed shells

The cable-stiffened latticed shell can be constructed in varied structural forms by diverse types of single-layer latticed shells and cable-strut systems. So, to provide references for model selection, this paper studied the efficient combination of latticed shells and cable-strut systems in different boundary conditions and proposed a performance evaluation method. The general modeling process was illustrated firstly, and several new structural forms with folded curved surfaces were proposed. Then, based on the improved equilibrium matrix theory, the force-finding analysis was carried out for the initial state, which can guarantee actual mechanical properties in the numerical model. Thus, the overall stiffness, the bearing capacity, and the cable reinforcement effect were obtained through the nonlinear stability analysis. And for the optimal selection from multiple structural forms, AHP-TOPSIS was used to judge the latticed shell system for the first time. The comprehensive structural performances were evaluated and ranked by this multi-attribute decision-making method. Finally, the results showed that the complicated cable-strut system could not necessarily lead to higher structural efficiency. Regardless of the cable-strut layouts, the spherical cable-stiffened latticed shell had a high bearing capacity when the ribbed grid was adopted. For the cylindrical structure, the folded latticed shell divided by orthogonal grids maximized the role of in-plane cables and was considered the best choice.

Discussion on cable-strut systems of the suspen-dome structures

According to the architectural requirements, the roof structure of a large-span gymnasium adopts the suspen-dome structure. In the scheme selection stage of the pre-stressed cable-strut system at the bottom part of the suspen-dome structure, a Levy-type scheme and a Loop-free scheme are established. The finite element models are established, and the static analysis under the design loads, the whole process analysis of load-displacement, and the dynamic response analysis after accidental cable break are carried out. The architectural expression of the two schemes are discussed. The component material consumption, the structural stiffness, the tension distribution characteristics, and the static bearing capacity of the two schemes are discussed. The failure mode and the progressive collapse resistance of the two schemes after accidental cable break are also discussed. The results show that the Loop-free scheme requires significantly less in terms of component material consumption than the Levy-type scheme. The static failure mode of the two schemes is strength failure, but the Loop-free scheme has greater bearing capacity. The Loop-free scheme has greater structural stiffness, lower cable forces, and uniformly distributed cable forces in each layer, and lower stress on the top reticulated shell members. Neither of the two schemes experience progressive collapse after accidental cable break. Due to the rupture in the loop cable of the Levy-type scheme, the rigidity of the rear region decreases greatly, and the cable force loss is large. On the contrary, internal force redistribution occurs in the Loop-free cable scheme and the cable force loss is not obvious, hence the progressive collapse resistance is better than that of the Levy-type scheme.

Static Behavior of a Retractable Suspen-Dome Structure

A new design of a radially retractable roof structure based on the concept of the suspen-dome is proposed in this paper. The radially foldable bar structure is strengthened by the lower cable-strut system to obtain a higher structural stiffness. Then the comparison of the static behavior between the retractable suspen-domes and their corresponding foldable bar shell with quadrangular mesh is discussed. Moreover, the effects of different structural and geometric parameters, such as the rise-to-span ratio, the cross-section area of beams, cables and struts, and the pre-stress level of the lower cable-strut system, on the nodal displacements and member forces are investigated systematically. The results show that higher structural stiffness is anticipated with the introduction of cable-strut systems into the hybrid structure. When the rise-to-span ratio is equal to 0.2, the maximal nodal displacement of the suspen-dome reaches the minimal value. The increase of the cross-section area of steel beams contributes an enormous amount to the structural stiffness. Increasing cable and strut sections has little impact on the mechanical behavior of suspen-domes. Moreover, the prestress level of cable-strut systems has a slight influence on the nodal displacements and member forces. Parametric analysis can be regarded as an essential basis for the optimization of the design of a retractable suspen-dome structure.

Using dynamic measurements to detect and locate ruptured cables on a tensegrity structure

Tensegrity structures are cable-strut systems held in equilibrium due to self-stress. There is potential for damage tolerance when they are kinematically redundant. In this paper, detection and location of a ruptured cable in a deployable tensegrity footbridge are studied through monitoring changes in dynamic behavior. Position values and axial load values of elements are measured before, during, and after a cable breakage. Free and forced-vibration-induced dynamic behavior of the tensegrity structure are characterized in the state of deployment (one half of the structure) and in-service (full structure). Examination of ambient vibrations for the half structure and forced vibrations for the full structure successfully led to detection of ruptured cables. Exclusion of possible damage cases for location using measurements effectively reduces the number of candidate cases when using nodal displacement measurements. Correlation methods using strain measurements are also successful to locate a ruptured cable. These methods reveal the potential for self-diagnosis of complex sensed structures.

Reliability Model of Consecutive $(2, k)$ -Out-of-( $2, n$ ) :F Systems With Local Load-Sharing

Most previous reliability models of consecutive k-out-of-n system assume that the components are independent or their reliabilities are equal. The system reliability can be obtained according to the configuration of $k$ consecutive failed components. However, for consecutive $(2,k)$ -out-of- $(2,n)$ systems with local load-sharing, workloads on failed components will be equally shared by both vertically and horizontally adjacent components. Consequently, the component failure probability density function (FPDF) may be unequal and dependent on the actual load (both original and sharing load), which is determined by the states of adjacent components. Furthermore, the component FPDF may be time-varying with the failure sequences of adjacent components. Therefore, reliability modeling must consider the component dynamic process determined by the failure sequences of adjacent components, instead of the static configuration determined by failure states of adjacent components in the traditional models. To solve the above issues, this paper develops a new reliability model for consecutive $(2,k)$ -out-of- $(2,n)$ systems with local load-sharing. The system and component dynamic processes are automatically searched and combined into a set of state transition paths. In each path, the component load-varying process can be confirmed, by considering the influence of adjacent failures on its shared load. Based on the state transition paths, the component and system reliability models are further derived, according to the confirmed load processes of each component. Finally, we present graphical illustrations of the reliability evaluation of cable-strut systems in long-span suspension bridges.

Technical Coefficients in Continuum Models of an Anisotropic Tensegrity Module

The presents paper focuses on the application of continuum models for an anisotropic tensegrity module and the determination of its technical coefficients. Tensegrities are cable-strut systems with a special node configuration, which ensures the occurrence of infinitesimal mechanisms balanced with self-stress states. The continuum model of the module is built by assuming that the strain energy of the unsupported tensegrity structure is equivalent to the strain energy of the cube. After the proper validation, the proposed model can be used to determine and interpret physical properties of the module. From the obtained elasticity matrix and the inverse matrix, the technical coefficients such as: Young's moduli, Poisson's ratios and shear moduli can be computed.

Selfstressed Systems Comprising Elements with Unilateral Rigidity: Selfstress States, Mechanisms and Tension Setting

Selfstressed reticulated systems have given rise to many studies in the past. However, the theoretical studies undertaken were based on the assumption that their components had bilateral rigidities when, in fact for systems with elements as cables (tensegrity or cable-strut systems) this hypothesis is not valid any more. We propose here to extend the methods of research of the selfstress states or the mechanisms for such systems. Then we can at the same time: choose the relevant selfstress states verify that cables do not induce unilateral mechanisms optimize the number of active elements for the tension setting of the system.

Nonlinear Analysis of Cable-strut Structures as a Mixed Complementarity Problem

This paper proposes a computation-oriented approach for determining the structural response of cable-strut systems under prescribed, not necessarily monotonic, loading histories. A feature of such structures, and one that is often a source of computational difficulty, is the possible slackening and retensioning of the cables. The methodology used in the present work to deal robustly and efficiently with this behavior is to express the element (cable or strut) constitutive law in a special mathematical format known as “complementarity”. Mathematically, this represents the perpendicularity of two-sign constrained vectors. Mechanically, it can be used to capture precisely the unilateral behavior of cables (viz. they can only support tension) and also the irreversible yielding of struts. We outline how these cable and strut laws can be combined with classical equilibrium and compatibility relations to provide a formulation in rates. Numerical solution of this problem is carried out by converting the rate problem into a finite incremental form leading to a special class of problems known in the mathematical programming literature as a “mixed complementarity problem” (MCP). Two examples are given to illustrate application of the approach proposed.

Innovative use of star prism (SP) and di-pyramid (DP) for spatial structures

Abstract Cable–strut structures are constructed from a system of short struts interconnected by high-tensile cables, providing lightweight structures with large-span capability. In this paper, two novel cable–strut systems constructed using star prism (SP) and di-pyramid (DP) grids are proposed. The main innovation of these systems is to extend cable net structures into freestanding grids. The innovation leads to the invention of a number of new structural forms and a wide variety of ornamental forms. Investigation work that led to the discovery of these two systems is presented. Studies on weight efficiency are conducted on square-on-square space grid structures using various modules such as reciprocal prism (RP), SP and DP grids. The SP and DP systems show promise to have lower weight-to-strength ratios than conventional space truss systems. They are also structurally more efficient for long-span spatial applications. Finally, innovative joint details are proposed considering the ease of fabrication and erection of spatial structures.

Cable-strut systems: part I — tensegrity

The concept of cable-strut is extended from that of tensegrity. The broad interpretation of cable-strut systems includes tensegrity systems, RP (Reciprocal Prism) and CP (Crystal-cell Pyramid) system, etc. Its narrow interpretation excludes tensegrity systems. Thus this paper is divided into two parts. Part I gives concept, properties and feasibility studies of tensegrity structures to put forward cable-strut systems. Part II presents the theory and novel concepts concerning the application of cable-strut systems, and concludes that cable-strut systems are revolutionary in space structures. In this part, the essential idea of tensegrity is analyzed and the concept of tensegrity is systematically redefined. Tensegrity grids can be classified into two types of configurations: non-contiguous strut and contiguous strut. Their properties are presented and compared. The properties of the latter are also compared with those of RP grids. The low efficiency of tensegrity grids is analyzed. The feasibility studies, concerning applicable tensegrity forms and their application scale, are also introduced in this paper.

Cable-strut systems: Part II — Cable-strut

In this part, the basic concept of cable-strut systems is introduced. Cable-strut systems, including reciprocal prism (RP) and cell pyramidal (CP) grids invented by the author, as successful attempts to introduce cables into simplexes to form grids, are revolutions in space structures. Their properties are presented in this paper. Cable-strut systems possess self-stressed equilibrium, avoiding reliance on a bulky anchorage system, which is a most important advantage in construction over conventional flexible structures. It has improved greatly the structural properties of tensegrity systems, which also possess self-stressed equilibrium. Its planar form becomes the lightest self-stressed equilibrium space bar systems. The additional advantages over conventional space bar systems are that its joint design can be simplified, and its grid depth and grid length can be adjusted easily to sustain large bar forces and to lower bar forces further. Moreover, its stiffness can be increased by introducing bars to replace connecting cables to form double-layer and triple-layer forms. Super-span domical and cylindrical forms of RP system have also proved to be feasible and economical. The advantage of cable-strut systems in architecture is that the systems are clear in sense of sight, which makes them very attractive.

Linear Complementary Equation Method Applied in the Load Response of Cable-Strut Systems

In this paper, the drawbacks of the Newton iteration method are analyzed. Then the Linear Complementary Equation (LCE) method is introduced into the cable-strut system to cope with the “retiring” problem of elements under load. The corresponding variants of the function of zero-stress and the flow factor of zero-stress are invented to represent the characteristics of cables and struts retiring from functioning. The derivation process is given in detail and its efficiency is proved by case studies. The LCE method has overcome all the drawbacks of the iteration method, because its computation process is non-conditionally stable, and its step length can be determined at about 1/5 of the given load value for all cases to ensure accuracy. It is novel method for studying the load response of cable-strut systems.

Definitions and Feasibility Studies of Tensegrity Systems

In this paper, the developments concerning tensegrity are briefly reviewed. The concept of tensegrity is systematically redefined. The concept is then expanded to cable-strut systems. The feasibility of tensegrity (cable-strut) systems is discussed, including some tools for further research. Finally, work concerning further research is considered.

General Study of Tensegrity Grids of Bar-To-Bar Connection

Tensegrity (TG) grids of bar-to-bar connection are studied in a general way in this paper. Their configurations can be classified into two types: one of TG prisms (anti-prisms), the other of TG truncated-pyramids (anti-truncated-pyramids). The present study shows distinct advantages for the latter type, and its structural properties are studied intensively. The results show that TG grids of bar-to-bar connection are not suitable for conventional usage and integral cable-tension is not applicable. New types of cable-strut systems may be more appropriate. The paper also points out that a low proportion of the number of struts to that of cables is not necessarily advantageous for structural efficiency. Moreover, the drawbacks of the present Newton iteration method are analyzed.