Load Determination in RF-/STEEL Warping Torsion

Technical Article

This article explains how to determine loads on the basis of the internal force situations defined in the RF-/STEEL Warping Torsion extension of the RF-/STEEL EC3 add-on module. Since this new program allows you to also analyze extracted chain-like beam structures in addition to entire chain-like beam structures, it is necessary to determine the loads of the partial structure separately. For this, a special transformation function has been developed, which determines new loads of all partial structures (depending on the internal forces calculated in RFEM/RSTAB) according to each load situation for geometrically nonlinear warping torsion analysis with seven degrees of freedom.

Introduction

You can calculate deformations and forces for statically determined and overdetermined structures using the finite element method. In order to get the solutions for the background equation system, which depends on the selected member cross-sections, lengths and rotation, the geometric (for example, support) and load-technical (for example, structural load) constraints are required:
[K] ∙ {u} = {F}
where
[K] is the stiffness matrix
{u} is the vector of nodal displacement
{F} is the vector of nodal point loads

Example: A spring with the spring constant K = 3 N/m is extended due to the force F um u = 0.5 m. Thus, the force F is 3 N/m ∙ 0.5 m = 1.5 N.

RF-/STEEL Warping Torsion determines forces and deformations of the specified sets of members again in a new calculation with seven degrees of freedom. However, this means that the extracted beam structures without boundary conditions are not computable. A calculation requires the corresponding geometric constraints in the form of a support definition and the load-technical constraints in the form of a member load.

Since the definitions of geometric supports are usually the same for various load situations, you can define nodal supports in Window 1.7 and elastic line supports in Window 1.13 on each node of the extracted beam structure. The program obtains the load-technical constraint from the load situations (load cases, load combinations, and result combinations) selected in Window 1.1. Since the load situations only include the loads for the entire structure in RFEM/RSTAB and not for the partial beam structure, it is necessary to define each load situation and set of members (parts of the structure) for the calculation of the partial structure in RF-/STEEL Warping Torsion. These loads are determined at the beginning of the calculation in the module, using the internal forces of the global calculation in RFEM/RSTAB. These new member loads of the partial structure and the nodal supports already defined in the module are then used to determine the new forces and deformations according to the warping torsion analysis.

Determining Member Loads for Partial Structure

The design software performs the following described examination for each partial structure and the associated load situation.
In order to determine the loads for the advanced analysis, the program uses the differential equation of the bending line:
$$\mathrm w''(\mathrm x)\;=\;\frac{-\;\mathrm M(\mathrm x)}{\mathrm{EI}(\mathrm x)}$$
where
w(x) is the function of the displacement
M(x) is the function of the bending moment distribution
EI(x) is the function of the bending stiffness using longitudinal member axis (elastic modulus ∙ moment of inertia)

From the relationship between the bending line and the loading (Schwedler theorem), the program can derive load distributions qy(x), qz(x) using the bending moment My(x), Mz(x):
Bending moment M(x) = - EI(x) ∙ w''(x)
Shear force Q(x) = - (EI(x) ∙ w''(x))'
Load q(x) = (EI(x) ∙ w''(x))''

Figure 01 - Bending Moment Distribution on Entire Structure and Extracted Structure

The transfer function determines the corresponding line loads for the extracted structure and the nodal loads in the distribution steps. The internal forces from the axial force and torsion are converted similarly and applied to the partial structure as loads. The shear forces should not be further considered in this analysis as they result directly from the derivation of the bending moments and arise indirectly again from the new equivalent loads.

By using this procedure, the final partial structure will be loaded by the internal forces similar to those resulting from the global calculation of the entire structure in RFEM/RSTAB, provided that the user-defined geometric constraints (supports) for the partial structure are applied to the partial structure as affine to the global effects of the structure. The following rules for defining supports must be respected:

  1. The support must be applied as affine to the effect in the entire structure.
  2. The partial structure must be statically determined or overdetermined.
  3. In the case of the partial structures conforming to the entire structure, it is necessary to specify the supports in the same way as for the entire structure.
  4. Intermediate supports in the partial structure should always be defined with the same stiffness as for the entire structure.
  5. For the extracted partial structures, the supports should be open on cutting points relating to the transferring bending moments about the respective rotational direction. In order to represent the distributions of axial and torsional forces caused by external loads, you have to open any edge support in and about the respective direction. Internal restraint forces on the partial structure are only considered partially (as an external load by the transfer function).

Figure 02 - Normal Force Distribution on Entire Structure and Extracted Structure with Open Support

This transfer function can be used for load cases LC, load combinations CO, and result combinations RC.

Summary

The new transfer function is a complex tool for determining loads on partial structures. Full integration in RF-/STEEL Warping Torsion allows you to realize the full potential of this feature. Thus, the load determination for the calculation according to seven degrees of freedom only depends on the selection of load situations to be analyzed.

Figure 03 - RF-/STEEL EC3 Add-on Module with Extension RF-/STEEL Warping Torsion

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