Evaluation of Local and Global Mode Shapes Using RSBUCK for Determination of Equivalent Member Length

Technical Article

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When performing the stability analysis of members according to the equivalent member method, considering internal forces according to the linear static analysis, it is very important to determine the governing equivalent member lengths.

Theoretical Background

The slendernesses and the reduction factors determined from them are determined in the flexural buckling analysis according to EN 1993-1-1, Chapter 6.3, taking into account the ideal critical load Ncr . This critical load is determined analytically in the STEEL EC3 add-on module using the governing effective length. For simple systems, for example, the four Euler cases are known here.

Figure 01 - 1 - Euler Cases

For more complex systems, the estimation of the effective length is no longer so trivial. Here, the user can use RSBUCK.

A critical load factor is determined for the system. This is multiplied by the normal forces of the members to obtain the critical loads. With the converted buckling formula Ncr = E ∙ I ∙ π²/Lcr , the corresponding buckling lengths for buckling about both axes are determined. Finally, the buckling length coefficients can be determined from the relation kcr = Lcr/L.

Global and Local Mode Shapes in RSBUCK

Using a simple frame, the determination of the mode shapes and the correct evaluation will be explained.

Figure 02 - 2 - Frame

When calculating the buckling shape and thus the buckling lengths, the loading plays a crucial role: The buckling values depend not only on the structural model but also on the ratio of the axial forces to the total critical load Ncr. Effective lengths can only be calculated for members with compressive forces. The distribution of loads in the entire system also affects the determination of the critical factors. By using the graphical evaluation of the individual mode shapes, you can see if the mode shape is global or local. If the most unfavorable critical load is the critical load of a single member, this is clearly shown in the graphic. For this failure case, the results are useless for all other members and must not be evaluated.

In the example, the first mode shape with a critical load factor of 5.32 describes the global deflection of the frame in the frame plane. The second mode shape with a critical load factor of 11.42 describes the local deflection of the left column from the frame plane (buckling about the minor axis z).

Figure 03 - 3 - Eigenvalues

Divided members

When evaluating the effective lengths and effective length coefficients, a division of members must be considered. In the example, the left column of the frame consists of two single members. For technical reasons, the column was divided in the middle. If we now consider the local eigenmode number 2, this is actually the typical Euler case no. 2 and a buckling length factor kcr, z = 1.0 is expected in the results. However, the result window 2.1 of the add-on module shows a buckling length factor kcr, z = 2.0 for the two "partial members" of the column.

This can be easily explained by the relations given under "Theoretical Background". For the total column, the buckling length in this case is equal to the column length and thus the buckling length factor of 1. However, since single members are evaluated in RSBUCK, kcr = Lcr/L with L = 0.5 ∙ column length results in a buckling length factor of 2.0 .

The buckling length coefficients for continuous members cannot be determined directly with RSBUCK. Here you can evaluate the results of the individual members. The member for which the smallest buckling load Ncr is output can be considered as governing for the continuous member tension. The kcr values can then be determined from the buckling length of this member and the total length of the member tension.

Figure 04 - 4 - Buckling Length

Keywords

Stability analysis equivalent member method Eigenvector Critical load

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