Form-Finding in RFEM

General Data

Input

When defining a model of lightweight structures, you may realize that the geometric position of the membranes and ropes is unclear. It is precisely the task of the FF process to find this position and fixate it. Therefore, RFEM requires an initial input of the FF elements in the first instance. The initial input provides the program with the information as to between which points a cable is enclosed, and between which line polygon a membrane is enclosed. 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. From a program perspective, when initially entering the FF elements, you only have to make sure that all required connection nodes and lines are integrated into the surfaces/members and that the meshing process can generate a mesh for all the elements. If the meshing process fails, the operation is ended directly before the calculation.

Menu

Form-finding

After a successful meshing, the program starts the FF process. This process adopts the mesh geometry and surface stress/prestress from the initial input and shifts the location 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 which can move partially in space and is fixed to its XY coordinates. 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, especially for rotationally symmetric models with conical shapes. You can counteract this reaction by fixing the surface stress vectors in the XY plane when using the projection method.

This shifting step is done iteratively using the URS method (Updated Reference Strategy, https://mediatum.ub.tum.de/node?id=1095271) by Prof. Dr.-Ing. K.-U. Bletzinger and E. Ramm. K.-U. Bletzinger and E. Ramm. To control the iteration process, there is a separate "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 controls in how many iterations the FF calculation should reapply the prestress with the originally defined value to the elements. After exceeding this limit, the program stops repeatedly applying the prestress with the initial value during the FF calculation. The program converges to a stable solution by increasing the value for an isotropic surface stress using the tension method or for an isotropic/orthotropic surface stress using the projection method. 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 the 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 is started. Since the distance between the initial position and the target position is usually greatly reduced due to the preliminary consideration, the actual iterative calculation only has to cover a small distance to the target position and can thus save a certain amount of calculation time.

Generate NURBS Surfaces/Lines from Form-Finding Results and Regenerate Form-Finding Results
This is used for determining a new model input. After the FF calculation, the program generally outputs the shifted mesh generation under the applied surface stress/prestress. This mesh geometry can be displayed in the program, but cannot be edited or modified. All entries and analyses (consequent loads, result evaluation, and so on) can only be entered initially.
In cases where 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 FF geometry usually takes a multi-curved form and the corresponding line geometries can no longer be realized with straight lines/arcs/spline curves, and surface geometries can no longer be realized with planes/cylindrical surfaces/quadrangle surfaces, the function rewrites the new elements into non-uniform rational B-splines (NURBS) with an order of 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 uniformly distribute the position of the necessary matrix nodes for surfaces with four boundary lines depending on the edge in the middle of the surface and evaluate them appropriately. 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 places a new FE mesh over the NURBS surfaces based on the previous FF geometry without additional distortions, and starts an FF calculation. Since NURBS elements are very close to the previously found FF geometry, the calculation process usually finds a solution within a few iterations. As might be expected, the FF calculation for these NURBS transformations results in near-zero deformation perpendicular to the membrane plane with the intended surface stress/prestress. However, an FF deformation may occur in the membrane plane in some cases. Nevertheless, this does not contradict the assumptions, 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. As a criterion between the iterations, the FF calculation balances the deformations and equilibrium between the element forces and reactions.

Speed of Convergence
This option controls the calculation stability. The pure FF calculation applies the membrane surfaces with the absolute stiffness. 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. This way, you can avoid any instability during the FF calculation.

Calculation Parameters

Results

After the FF calculation, the results are displayed under the "RF‑FORM‑FINDING" load case. The Results Navigator is the same as in the case of a regular structural design, only without the FF analysis. Deformation results describe the deformation between the initial input and the equilibrium shape found. 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. A subsequent calculation with a specific surface load input, such as wind load, then uses the model described under the load case "RF‑FORM‑FINDING" as the initial configuration, with all the associated effects. In these subsequent load cases, the deformation is then related to the previously determined equilibrium shape.

Model