# Form-Finding in RFEM

### Technical Article

The RF-FORM-FINDING add-on module determines equilibrium shapes of membrane and cable elements in RFEM. In this calculation process, the program searches for such a geometric position in which the surface stress/prestress of membranes and cables is in equilibrium with natural and geometric boundary conditions. This process is called form-finding (hereinafter referred to as FF). The FF calculation can be activated in RFEM globally in the "General Data" of a model, "Options" tab. After selecting the corresponding option, a new load case or a calculation process called RF-FORM FINDING is created in RFEM. An additional FF parameter is available for defining surface stress and prestress when entering cables and membranes. By activating the FF option, the program always starts the form-finding process before the pure structural calculation of internal forces, deformation, eigenvalues, etc., and generates a corresponding prestressed model for further analysis.

#### Input

While defining a model of lightweight structures, you might realize the geometric position of the membranes and ropes is unclear. It is exactly the task of the FF process to find this position and fixate it. In the first instance, RFEM requires the initial input of the FF - elements. This entry provides the program with the information for where, between points, is a cable, and where, between line polygons, there is a membrane included. Furthermore, the initial input requires a determination of surface stress value in the warp and weft direction of the membranes, including their application method (tension or projection), and the prestress level or sag dimension of the cable elements that should act according to the FF calculation. It should be noted that the initial shape of the FF elements is irrelevant. When entering initial data of FF elements, you should only be sure that all required connecting nodes and lines are integrated in the surfaces/members and that the meshing process can generate a mesh for all elements. If the meshing process fails, the operation is ended directly before the calculation.

#### Form-Finding

After a successful meshing, the program starts the FF process. This process adopts the mesh geometry and surface stress/prestress initially entered, and displaces the position of the mesh elements until the surface stress on the FE element is in equilibrium with the boundary conditions. The description of the surface stress on the membrane mesh elements can be defined in two ways. The tension method describes a surface stress vector, which can move freely in space until it reaches the target position. In contrast, the projection method describes a surface stress vector that can move partially in space and is fixed to its XY coordinates. Especially for rotationally symmetric models with conical shapes, it can happen that in the case of prestress vectors freely movable in space, the tangential vectors can contract to a point in the center. You can counteract this reaction by fixating the surface stress vectors in the XY plane when using the projection method.

This displacement step is performed iteratively according to the URS method (Updated Reference Strategy, see https://www.st.bgu.tum.de/forschung/alte-forschungsthemen/form-finding-of-membrane-structures) by Prof. Dr.-Ing. K.-U. Bletzinger and E. Ramm. To control the iteration process, there is the "Form-Finding" tab in the Calculation Parameters dialog box. The following options are available:

Maximum number of iterations

Generally, the FF calculation should come to an end before reaching this limit while meeting all tolerance limits. If the tolerance limits are not met after reaching the maximum number of iterations, the program displays a warning message with the option to further use the intermediate result.

Number of iterations for loading prestress

This number specifies in how many iterations the FF calculation should apply the prestress to the elements with the previously defined value. When exceeding this limit, the program stops repeatedly applying the prestress with the start value during the FF calculation. By increasing the value in the case of isotropic surface stress with the tension method or isotropic/orthotropic surface stress with the projection method, the program converges to a stable solution. Due to the biaxial curvature, it is only possible to find an approximate solution for orthotropic surface stress with the tension method.

Consider self-weight from load case

This load case assignment allows you to use self-weight as a constraint for the FF calculation, in addition to the firmly defined surface stress/prestress.

Integrate preliminary form-finding

This option speeds up the global FF process in most cases. The preliminary form-finding displaces the FE surface elements, assuming rigid edges at a position that is close to the target solution. After this step, the actual iterative FF process has started. Since the way between the initial position and the target position is usually reduced due to the preliminary analysis, the actual iterative calculation should cover a small way path to the target position and thus save a certain amount of the computing time.

Generate NURBS surfaces/lines from form-finding results and regenerate form-finding results

This is used for determination of a new model input. In general, the program shows the shifted mesh generation applying the surface stress/prestress after the FF calculation. This mesh geometry can be displayed in the program, but cannot be edited and modified. All entries and analyses (consequent loads, result evaluation, etc.) can only be entered initially.

In case the FF mesh geometry is shifted very far from the initial geometry, the NURBS transformation can help you. This option transforms the FF geometry (membrane surface, membrane boundary lines, and cable lines) in the determined FF geometry. Since the FF geometry usually has a multiple-curved shape and the corresponding line geometries cannot be edited with other lines, arcs, splines, or the surface geometry with planes, cylindrical or quadrangle surfaces any longer, this option transforms the new element in non-uniform rational B- splines (NURBS) with Order 9. These NURBS elements represent the corresponding lines and surface definitions, which approximately match the previously determined FF geometries.

In RFEM, the entry of NURBS surfaces is fixed to a surface type with four boundary lines. This means that the program can only distribute the position of the necessary matrix nodes on surfaces with four boundary lines uniformly depending on the edge in the middle of the surface and assess them accordingly. Additionally, a special case with three boundary lines is possible as this calculation model - in contrast to a quadrangular surface - considers the boundary line with a length of 0. Therefore, the matrix node distribution on the corner with the zero line is strongly compressed.

After the transformation, the program creates a new FE mesh using the NURBS surfaces on the basis of the previous FF geometry without additional distortions, and starts the FF calculation. Since NURBS elements are very close to the previously found FF geometry, the calculation process usually finds a solution within just a few iterations. As expected, an approximate zero deformation perpendicular to the membrane plane with the intended surface stress/prestress results from the FF calculation in the case of these NURBS transformations. However, an FF deformation may occur in the membrane plane in some cases. Nevertheless, this does not contradict the assumptions and so it can be accepted.

Tolerance for form-finding convergence criteria

This option specifies the solution precision. The value modifies the internally adjusted precision of the FF calculation. Thus, a value lower than 1 increases the precision and forces the program to perform the iterative calculations until the reduced tolerance limit is reached. The FF calculation as a criterion between the iterations verifies the deformations and the equilibrium between the element forces and reactions.

Speed of convergence

This option controls the calculation stability. The pure FF calculation applies the absolute stiffness to the membrane surfaces. This value can be modified with a set value. A value lower than 1 increases the stiffness and thus provides a slower convergence but higher calculation stability. In this way, you can avoid any instability during the FF calculation.

Figure 03 - Calculation Parameters

#### Results

After the FF calculation, the results are displayed under the "RF-FORM-FINDING" load case. The result navigator is the same as in the case of an usual structural design, only without the FF analysis. Deformation results describe the deformation between the initial input and the determined equilibrium shape. Member and surface results show the force or stress conditions for the equilibrium shape, considering the defined FF parameters. The "RF-FORM-FINDING" load case represents a new model configuration with the surface stress/prestress. Then, a subsequent calculation with certain surface load entries such as wind load, for example, uses a model such as the "RF-FORM-FINDING" load case with all corresponding effects as the initial configuration. In the case of these subsequent load cases, the deformation applies to the previously determined equilibrium shape.

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