Calculating Critical Load Factor for Linear Buckling Analysis

Technical Article

Buckling analysis according to the effective width method or the reduced stress method is based on the determination of the system critical load, hereinafter called LBA (linear buckling analysis). This article explains the analytical calculation of the critical load factor as well as utilization of the finite element method (FEM).

Critical Load Factors Relating to Stress

[1] provides the following equation (Chap. 10, Eq. 10.6) for a pure analytical determination of the critical load factor of a buckling panel:

$$\frac1{{\mathrm\alpha}_\mathrm{cr}}\;=\;\frac{1\;+\;{\mathrm\psi}_\mathrm x}{4\;\cdot\;{\mathrm\alpha}_{\mathrm{cr},\mathrm x}}\;+\;\frac{1\;+\;{\mathrm\psi}_\mathrm z}{4\;\cdot\;{\mathrm\alpha}_{\mathrm{cr},\mathrm z}}\;+\;\left[\left(\frac{1\;+\;{\mathrm\psi}_\mathrm x}{4\;\cdot\;{\mathrm\alpha}_{\mathrm{cr},\mathrm x}}\;+\;\frac{1\;+\;{\mathrm\psi}_\mathrm z}{4\;\cdot\;{\mathrm\alpha}_{\mathrm{cr},\mathrm z}}\right)^2\;+\;\frac{1\;-\;{\mathrm\psi}_\mathrm x}{2\;\cdot\;\mathrm\alpha_{\mathrm{cr},\mathrm x}^2}\;+\;\frac{1\;-\;{\mathrm\psi}_\mathrm z}{2\;\cdot\;\mathrm\alpha_{\mathrm{cr},\mathrm z}^2}\;+\;\frac1{\mathrm\alpha_{\mathrm{cr},\mathrm\tau}^2}\right]^{1/2}$$

As you can see, stress ratios as well as critical load factors are determined separately for the individual stress components or must be known. You can determine the critical load factors by recalculating the critical plate buckling stresses. This determination has already been explained in this technical article.

Thus, the following relations result for the individual stress components:

$$\begin{array}{l}{\mathrm\alpha}_{\mathrm{cr},\mathrm x}\;=\;\frac{{\mathrm\sigma}_{\mathrm{cr},\mathrm p,\mathrm x}}{{\mathrm\sigma}_{\mathrm x,\mathrm{Ed}}}\\{\mathrm\alpha}_{\mathrm{cr},\mathrm z}\;=\;\frac{{\mathrm\sigma}_{\mathrm{cr},\mathrm p,\mathrm z}}{{\mathrm\sigma}_{\mathrm z,\mathrm{Ed}}}\\{\mathrm\alpha}_{\mathrm{cr},\mathrm\tau}\;=\;\frac{{\mathrm\tau}_{\mathrm{cr},\mathrm p}}{{\mathrm\tau}_\mathrm{Ed}}\end{array}$$

This method is particularly suitable for unstiffened or longitudinally stiffened buckling plates that apply the corresponding buckling values from [2] or [3].

Calculation Using Finite Element Analysis

If there is a strongly stiffened buckling plate with longitudinal and transverse stiffeners, the FEM calculation should be used to determine the critical load on the entire structure. As a basis, you should apply a surface model and consider all boundary conditions (for example, supports at the edges, the geometrical position and stiffener loading as well as boundary stresses). For the determination according to LBA, elastic material behavior is applied. The following example shows modeling of a longitudinally stiffened buckling plate in RFEM.

Figure 01 - FE Model of Longitudinally Stiffened Buckling Plate

The RF‑STABILITY add‑on module is used to determine the critical load factor. When selecting a mode shape, the global system failure must be taken into account.

Figure 02 - Results of Mode Shapes

The first mode shape in this example shows global buckling and is therefore to be regarded as governing. However, higher mode shapes may be relevant for design in some cases. Thus, the critical load factor can be calculated for all stress components as well as separately (only one stress component per load case).

The stand‑alone program PLATE‑BUCKLING allows you to perform a complete buckling analysis using the reduced stress method, including automatic determination of eigenvalues for each stress component.

Critical Load Factor in Buckling Analysis

Now, it is possible to either determine the individual critical load factors and then analytically calculate the total critical load factor by using Eq. 10.6 provided in [1], or to use them directly from the FEM calculation. In some cases, the analytical solution may be regarded as conservative. Therefore, PLATE‑BUCKLING provides the following options.

Figure 03 - Calculate Critical Load Factor Analytically or Using Finite Element Analysis

Reference

[1]  Eurocode 3: Design of steel structures - Part 1‑5: Plated structural elements; EN 1993‑1‑5:2006 + AC:2009
[2]  Klöppel, K., & Scheer, J. (1960). Beulwerte ausgesteifter Rechteckplatten, Band 1. Berlin: Wilhelm Ernst & Sohn.
[3]  Klöppel, K., & Scheer, J. (1968). Beulwerte ausgesteifter Rechteckplatten, Band 2. Berlin: Wilhelm Ernst & Sohn.

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