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Advanced Stability

Automatically calculate 2nd order effects on steel elements, considering the element imperfections and the torsional warping effects

 

Advance Design comes with a new FEM engine implemented for a 2nd order analysis with 7 degrees of freedom (DOF). The feature enables the design of a broad variety of cross-sections shapes, even mono-symmetric or asymmetric shapes, which are widely used in building design but are not covered by code rules.

Advanced Stability functionality helps structural engineers obtain more economic structures, by using thin-walled profiles, which are required more and more often in day-to-day project-related activities.

Steel Structures Design using FEM

Source: Stability problems for the buckling of columns and beams - “Steel Structures Design using FEM”, Rolf Kindmann / Matthias Kraus

Workflow

 Advanced Stability feature automatically calculates 2nd order effects on the steel elements. The functionality is available starting with Advance Design 2019.

This enhanced solver will perform a thorough analysis of individual members during the steel design sequence, considering the following:

  • The element imperfections by introducing a scaled eigenmode as initial deformation;
  • The torsional warping effects by considering it as a 7th degree-of-freedom;
  • The eccentricity of the loads by taking into account the application point on the cross-section;
  • The boundary conditions and lateral restrains by introducing centric or eccentric nodal and continuous spring along the member;
  • The effects of the deformed geometry by performing a second-order analysis.
Forces

Process
  • The new solver is fully integrated into the existing workflow of EC3 Steel Design and can be activated on a selection of elements;
  • Each element is extracted/isolated from the 3D model and the equivalent external loads are calculated by first derivation of efforts diagrams, for each load case combination;
  • The same diagrams of efforts and displacements are used to model the equivalent boundary conditions, including elastic supports to simulate the rest of 3D structure;
  • The user can impose an initial imperfection to be taken into account for 2nd order analysis;
  • The cross section of the member can be any of the Advance Design’s standard sections; the member can be with variable section (different sections at extremity, but from the same family), including haunches;
  • A 2nd order analysis is then carried out to compute:
    • 7 displacements and rotations (Tx, Ty, Tz, Rx, Ry, Rz, Rw) per node
      • Where:
        • Tx, Ty and Tz stand for the displacement along the local x, y and z axes respectively.
        • Rx, Ry and Rz stand for the rotation about the local x, y and z axes respectively.
        • Rw is the warping.
    • and 7 forces and moments (Nx, Vy, Vz, Mx, My, Mz, Mw) per node
      • Where:
        • Nx is the normal force.
        • Vy and Vz stand for the shear force in the local y and z directions respectively.
        • Mx is the torsional moment. It includes the 1st order torsional moment (Mxp) as well as the 2nd order torsional moment (Mxs).
        • My and Mz stand for the bending moments about the local y and z axis respectively.
        • Mw is the warping moment.
Forces

Based on these results, cross-section design is performed by computing:

Normal stress σx is based on normal force Nx, bending moments My and Mz as well as warping moment Mw

Shear stress τ is based on shear forces Vy and Vz as well as torsional moment Mx (Mx = Mxp + Mxs)

Von Mises stress σVM is based on normal stress and shear stress

The obtained values are then compared to the limit stresses:

 

Result
  • All calculated internal forces and displacements / rotations are taking into account warping deformations from 7th degree of freedom;
  • Standard results from the 2nd order calculation for bending My, Mz, torsion Mx and warping bimoment Mw, shear Fy, Fz, axial Fx, displacements, rotations and warping rotation q are displayed;
  • with these 2nd order results, stresses are calculated and the cross section is directly checked with the code limit;
  • The deflection verification of EC3 is done with 2nd order displacements.

Theoretical background

The internal forces and moments resulted from the effects of deformed geometry of the structure can be determined using a 2nd order theory that takes into account this influence, according to EN1993-1-1 (§5.2.1). 2nd order effects are also known as P-Δ (non-linear) effects and occur in every structure where elements are axially loaded with compression forces. P-Δ effects are associated with the magnitude of the applied axial compression (P) and displacement (Δ).

Usually, there are no analytical solutions for flexural-torsional problems that include nonlinear deformations. Therefore, a FEM approach is used in order to determine approximate solutions for the differential equations.

The impact from 2nd order effects on a structure must be assessed for every designed structure, since they increase the deflections, moments and forces beyond those calculated by 1st order analysis. In many cases, by calculating the elements according to 2nd order analysis, lighter structures can be designed by using thin-walled steel profiles.

Warping is a deformation that occurs under uniform torsion and causes the section to no longer remain plane. This phenomenon has significant effect on open cross sections such as I, H or channels. For these type of sections, warping results cannot be neglected.

Warping representation on open cross sections

warping representation on open cross sections

Source: Design of steel beams in torsion – SCI Publication P385

In thin­-walled closed cross sections (the most appropriate type of cross section to resist torsion), uniform torsion is predominant. Therefore, in the analysis of thin-­walled closed cross sections subjected to torsion, the warping torsion (Mxs) is normally neglected.

Torsional moment is the internal twisting moment about the beam’s longitudinal axis and it is usually considered in two components: St. Venant torsional moment and warping torsional moment. Warping moment is the bending moment in a flange acting as a result of restraint of warping. The moments in the two flanges are equal and of opposite sign.

In members with thin-walled open cross sections (such as I or H sections), inside of which only the uniform torsion component appears, it is necessary that the supports do not prevent warping and the torsional moment is constant.  Otherwise, if the torsional moment is variable or warping is restrained at some cross sections (usual situation), the member is under non-uniform torsion.

Non-uniform torsion on I section member

Due to the fact that the cross sections also rotate around the longitudinal axis (especially next to the free end),  there also appears uniform torsion. Thus, in this case, the resistance to torsion is given by the sum of both effects (M = Mxp + Mxs), the warping torsion component, Mxs, being significantly larger than the uniform torsion component, Mxp, in sections near the built­-in end.

 

Advance Stability, a brand-new design method of Graitec Advance Design, enables the design of any shape of cross-sections included in the Advance Design library, even mono symmetric or asymmetric shapes which are widely used in building practice but are not covered by code rules. With Advance Design, you can let imagination flow inside your structures, without sacrificing the safety provided by this proven design method.

Bibliography

[1] Rolf Kindmann, Matthias Kraus, Steel Structures Design using FEM
[2] Eurocode 3: Design of steel structures – Part 1-1 : General rules and rules for buildings

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