An algorithm for calculating the feasible pre-stress of cable-struts structure

Highlights

• An algorithm to find the feasible pre-stress is proposed.

• Three types of cable-struts structures are selected to demonstrate the algorithm.

• The algorithm can update the geometry for the feasible pre-stress.

• The algorithm can consider loads during the design of the pre-stress.

• The feasible pre-stress of the cable-struts structure is obtained directly.

Abstract

A cable-struts structure consists of tension-only cables and compression-only struts. The structural rigidity comes from the pre-stress. To determine the feasible pre-stress for the cable-struts structure efficiently and accurately, the paper first defines the three states for the cable-struts structure based on the loads applied, develops the fuzzy relationship between the internal force and the pre-stress under different states, and studies various essential conditions for the feasible pre-stress. Second, the paper gives a Newton iteration method to update the pre-stress and a simple method for updating the geometry based on the geometrical difference under different states and subsequently develops a process to find the feasible pre-stress, combining the pre-stress updating method with the geometry updating method. Last, the paper selects three types of cable-struts structures as illustrative examples, a traditional Geiger dome, a new form of Geiger dome and a Kiewitt dome with multiple self-stress modes, and successively calculates their feasible pre-stress using the proposed method. The results indicate that the proposed method is able to search for the feasible pre-stress for cable-struts structures efficiently and accurately; it can calculate not only the pre-stress without loads but also the pre-stress considering loads, and it can change the geometry for the feasible pre-stress. Meanwhile, the method is successful in obtaining the feasible pre-stress for the Kiewitt dome with multiple self-stress modes without the optimal process.

Introduction

Tensegrity structures, which consist of continuous tension elements and discontinuous compression elements, were proposed by Fuller [1], and the first tensegrity structure was designed by Snelson in 1948 [2]. The tensegrity concept has become a basic principle of nature and has been applied to so many fields of science that it is losing its primary meaning [3]. With respect to the basis of the tensegrity concept, Geiger first invented the cable dome, which includes a compressed ring in the boundary of a tensegrity structure [4], the so-called Geiger form. Subsequently, Levy improved the Geiger form and invented the Levy form [5]. Zheng et al. [6] then proposed a rectangular cable dome to enlarge the application of cable domes in practical projects. Kmet and Mojdis [7] improved the traditional Levy form using an action member and built a Levy form adaptive cable dome. The cable dome includes tension-only cables, compression-only struts and a compressed ring. The innovative configuration and lightness of the cable dome have attracted attention from engineers. The first cable dome was designed by Geiger for the Olympics in Seoul (1986), followed by the Redbrid Arena in Illinois (1988), the Florida Suncoast Dome in St. Petersburg (1988), the Taoyuan Arena in Taiwan (1993), and the oval plan Levy form of the cable dome for the Olympics in Georgia (1996) [8].

The tensegrity structures and cable domes (generally called cable-struts structures) have the structural rigidity only when applying the self-equilibrium stress to the cables and struts [9] and is conditioned by its pre-stress state [10], [11]; therefore, calculating the pre-stress is the key step for any cable-struts structure, including cable nets [12], cable-beam structures [13], cable-stiffened arch [14] and the suspended-dome invented by Kawaguchi et al. [15], [16]. Much research has been carried out on the pre-stress design of cable domes. Based on the flexibility method, Hanaor proposed a unified method for the pre-stress design of pre-stressable structures [17]. Pellegrino and Calladine presented a classic singular value decomposition (SVD) technique to obtain the independent self-stress modes and the independent displacement modes of the cable-struts structures [18], [19], [20]. Considering the inherent geometric symmetry of cable domes, Yuan and Dong [21], [22] proposed the concept of feasible integral pre-stress modes and a general method for domes with a single integral self-stress mode. For the cable dome with multiple self-stress modes, Yuan et al. [23] proposed a general method, referred to as double singular value decomposition (DSVD), and then selected a Kiewitt dome to design the pre-stress using DSVD and the optimal method. SVD and DSVD are effective in finding the pre-stress modes when we know the reasonable structural geometry, but those methods cannot consider the structural deformation and loads. Later, Wang et al. [24] proposed a simple approach to design the pre-stress for two Geiger domes with self-weight based on the nodal equilibrium equation after changing the self-weight into the nodal force. It is, however, effective only for cable domes with a single self-stress mode. Moreover, the paper [25] noted that the geometry or topology must be changed if the condition that cables are subjected to tension and struts are subjected to compression is not satisfied during the design of the pre-stress; the detailed method for changing the unreasonable geometry or topology, however, was not introduced.

In practice, the loads always exist in the structure, and we sometimes must change the initial unreasonable structural geometry for the feasible pre-stress, so it seems more meaningful to obtain the feasible pre-stress for cable-struts structures with unfeasible geometry, with loads or with multiple integral pre-stress modes. This paper proposes a Newton iteration method for updating the pre-stress and a simple method for changing the geometry for the feasible pre-stress and gives a process for looking for the feasible pre-stress of cable-struts structures.

The layout of the paper is as follows. Section 2 proposes a Newton iteration method for updating the pre-stress, a simple method for updating the geometry, and the process for designing the feasible pre-stress. Section 3 gives three different numerical models of the cable-struts structures and illustrates the feasible pre-stress design under different conditions; it also compares the results with SVD. The results demonstrate that the process proposed in the paper is effective and accurate for designing the feasible pre-stress for different cable-struts structures, that it can directly consider the possible loads during the design of the pre-stress, and that it can directly update the unreasonable geometry.