I. Current Analysis Methods for Membrane Structure Engineering

Three major issues exist in the design analysis of membrane structures: shape determination (form-finding), load analysis, and cutting analysis. Among them, the shape determination issue is fundamental, serving as the basis for the analysis of the latter two problems.

Currently, the main methods applied to determine the shape of membrane structure engineering include the force density method, the dynamic relaxation method, and the nonlinear finite element method. Among them, the nonlinear finite element method is the most widely used and effective. The force density method is developed by...Inkwitz and Schek propose a shape-finding method for cable net structures, which can also be applied to membrane structure engineering by discretizing the membrane into equivalent cable nets. Force density refers to the ratio of the internal force of a cable segment to its length. Treat the cable net or equivalent membrane structure as composed of cable segments connected by nodes. During shape-finding, boundary points are constrained, while intermediate points are free. By establishing and solving the equilibrium equations of nodes using the force density of specific cable segments, the coordinates of each free node can be obtained, representing the shape of the cable net. Different force density values correspond to different shapes. Once the shape meets the requirements, the corresponding prestress distribution can be calculated from the force density. Dynamic relaxation is a numerical method for solving nonlinear problems, having been applied to shape-finding in cable and membrane structures since the 1970s. Dynamic relaxation discretizes the structural system in both space and time. Spatially, it divides the system into elements and nodes, assuming mass is concentrated at the nodes. If an excitation force is applied at the nodes, they will vibrate, and due to the presence of damping, the vibration will gradually diminish until static equilibrium is reached. Temporal discretization refers to the vibration process of the nodes. Dynamic relaxation does not require forming the overall stiffness matrix of the structure, allowing for modifications to the structure's topology and boundary conditions during shape-finding. Calculations can continue, and a new equilibrium state can be obtained, which is used to solve the equilibrium surface under given boundary conditions.

Nonlinear finite element method is a technique that applies the theory of geometric nonlinear finite element methods to establish and solve a set of nonlinear equations. It is commonly used in membrane structure engineering analysis. There are two basic algorithms: starting from the initial geometry and starting from the plane state. The former involves first establishing an initial geometric shape that meets boundary conditions and shape control, assuming a set of prestress distributions. Generally, the initial structural system does not satisfy the equilibrium conditions and is in an unbalanced state. At this point, an appropriate method is used to solve a set of nonlinear equations to determine the equilibrium state of the system. The latter assumes that the material's elastic modulus is very small, allowing the elements to deform freely. The initial shape is a plane, and the support points of the system are gradually raised to a specific position. Since the elements can deform freely, the internal forces of the system remain unchanged. When the equilibrium state is reached, the internal forces of the system are at predetermined values; to ensure the stability of the calculation, the supports need to be raised in sections.

The algorithm achieves a satisfactory solution by avoiding grid distortion, ensuring computational convergence, and employing an appropriate method for solving the nonlinear equations.

Issue with Current Analytical Methods in Membrane Structure Engineering

The density method only requires solving linear equations, and for simple structures, it can even be calculated manually. However, its accuracy is not as good as that of the finite element method, which degrades with increased complexity of the structure. The dynamic relaxation method has significantly more iterations than the general finite element method and is not applicable to cases with undefined boundary conditions, such as analyzing the process of a membrane material being stretched from a flat to a spatial state. Moreover, even if the shape-finding problem is resolved using these two methods, load analysis and trimming analysis still require the finite element method. Consequently, a change in calculation methods is necessary, impacting calculation efficiency.

At present, the finite element method remains the primary approach for solving the form-finding problem in membrane structures. However, this method also encounters some challenging issues in form-finding solutions. For instance, slight inaccuracies in meshing can lead to mesh distortion, preventing computation; the lifting of supports must be done in sections, with the number of sections significantly affecting convergence; the choice of solution methods for the selected nonlinear equations also impacts the accuracy of the solutions.