Effects of the Self‐Stress Level on the Instability Behavior of the Tensegrity Barrel Vaults

Tensegrity systems are self-equilibrium systems that contain discontinuous compressive components and continuous tensile components. Main factor for self-equilibrium of tensegrity systems is the initial self-stresses of the components. Therefore, consideration of the effects of the levels of these self-stresses on the instability behavior and collapse mechanisms has significant importance. Moreover, because of the application of these systems in large spans and crowded places, sudden collapse leads to a lot of kills and damages. Thus this paper is presented in two parts: first part: presentating a method to find “feasible” self-stress states based on a mathematical topic called “simplex method” and using artificial variables; and second part: using a self-stress state obtained from previous part, and scale it in three levels. The effects of the self-stress level on the instability behavior and collapse mechanisms of the barrel vault tensegrity system is considered, and based on the analysis results some suggestionsare presented to determine maximum and minimum self-stress level so the structure doesn’t collapse suddenly and have enough rigidity. The initial state of the system is very specific since it is a self-equilibrated state; moreover the rigidity of the tensioned components is unilateral (no rigidity in compression) and the relational structure is very specific: compressed components are inside a continuum of tensioned components [1]. The equation of static equilibrium of an unconstrained reference node i connected to nodes j and k are given by Where any member (A, B), that connects nodes A and B, has an internal force fa,b and a length la,b; and fext is the external force. A simplified linearised notation qa,b= fa,b / la,b known as tension coefficient or force density is used. Let x = [x1…xn]T, y = [y1…yn] T, and z = [z1…zn]T, be the vectors of coordinates for n nodes along the x, y, and z directions, respectively. Let q be a vector of tension coefficients, with one entry for each of the b members. We can write the matrix form of Eq. (1) by factorizing the projected lengths in the equilibrium matrix A and a vector q of tension coefficients [2] Results of the analysis is shown in Figs. 2-4. In these figures, the label StoCx_y shows the structure with the rigidity ratio of the struts to the cables equal x and self-stress level y%. Figures show that the increasing the selfstresslevel may lead to overall collapse mechanism. It can be said that the self-stress level doesn’t have aneffect on the initial stiffness and final load capacity of the structure. Self-stress is the one of the interesting and important properties of the tensegrity systems. In this study, to obtain a feasible self-stress for the tensegrity structures, an extended method using simplex method and artificial variableswas presented. Then the effects of the self-stress level on the barrel vault tensegrity structures was studied. Results showed that the self-stress level doesn’t affect the initial stiffness of the structure; but it can change the collapse mechanism of the structure from a local collapse to overall collapse.