# Plate Buckling Analysis of Steel Shell Structures Using MNA/LBA Concept

### Technical Article

Shell buckling is considered to be the most recent and least explored stability issue of structural engineering. This is less due to a lack of research activities, but rather due to the complexity of the theory. With the introduction and further development of the finite element method in structural engineering practice, some engineers no longer have to deal with the complicated theory of shell buckling. Evidence of the problems and errors to which this gives rise is very well summarized in [1].

In this article, it is also strongly recommended not to simply create an FE model for each steel shell, apply loads, and then click "Calculate." In most cases, this procedure results in an additional effort because numerous simple analytical methods are available for the design of simple cases that often occur in design practice. These analytical methods, the so-called manual calculation formulas, also have the big advantage of a space-saving and simple documentation. For some containers, it is possible to perform the buckling analysis on a DIN A4 page. Such a space-saving documentation is not possible with an FE analysis.

However, there are also numerous cases where the use of a finite element analysis makes sense and should be preferred to a manual calculation. The following points are just a few exemplary cases in which it makes sense to use an FE calculation:

- Local load application in the shell wall
- Discrete supports (i.e. single supports) of the shell
- Use of nonlinear methods for design verification

In the following, a steel shell against buckling will be determined by means of RFEM according to the MNA/LBA concept. Thus, a nonlinear material behavior of the steel is applied.

Figure 01 - Model of Steel Shell Structure

#### Buckling analysis according to EN 1993-1-6

EN 1993-1-6 shows three ways to perform a buckling analysis for steel shells. In this section, they are briefly listed and evaluated with regard to the requirements for computing technology and the planning engineer.

**Stress-based buckling analysis**

The stress-based buckling safety design is considered the classic design method, which has probably been used by any engineer who has already carried out a shell design. This method can be classified as simple by a competent engineer and the requirements for the computing technology are either very low or nonexistent, because it is often worked with manual calculation formulas.

A major problem of this design method is that economic results are hardly achievable for shell structures with load situations that differ significantly from the classic buckling shapes. Moreover, as a user of this concept, the method itself actually lures you on the wrong track, because it would be easy to think that the buckling safety of the shell structure is only dependent on the occurring stresses. If this were the case, a stiffening of a shell wall, for example by longitudinal ribs, would be of little use because the stresses are not significantly reduced in this way. In reality, however, the buckling resistance of a skilfully stiffened shell is much higher than for an unstiffened shell with the same wall thickness.

**Numerical buckling analysis by means of global MNA/LBA calculation**

This method should be used for the shell to be designed in the following. An MNA/LBA calculation certainly requires more background knowledge about the shell stability than the stress-based design method. The computing technology should also be somewhat more efficient, since a linear elastic branching analysis (LBA) and a material nonlinear calculation (MNA) have to be carried out for a correct application of this method.

In the author's opinion, this design method represents the most sensible type of buckling safety analysis if you want to calculate by means of FE analysis. The reason for this is that in the design by means of the MNA/LBA concept, the computer technology is also consistently used without the user being expended too much effort. If you calculate the internal forces of the shell in a linear elastic manner and then use it for the stress-based buckling analysis, you are actually using the calculation technology too inconsistently, because powerful programs such as RFEM are able to determine the load-bearing capacity of the shell structure.

**Numerical buckling analysis based on global GMNIA calculation**

A GMNIA calculation to determine sufficient shell stability is probably the most consistent method of a buckling safety analysis. In this case, the internal forces are calculated geometrically and materially nonlinearly using imperfections.

This method requires the user excellent background knowledge about shell stability, because, among other things, the correct application of imperfections (buckling pattern) is very difficult. If this background knowledge is not available to the user, it is recommended to omit the design with the GMNIA concept. Considerable requirements are also placed on the computing technology in this method. Thus, the program system used must be able to perform a critical analysis for each load increment of the nonlinear calculation in order to detect, if necessary, a "jumping" from the subcritical pre-buckling path to the supercritical post-buckling path.

This concept will not be explained further here, because in the author's opinion, it is of little importance for design practice. However, it should be made to the article by Herbert Schmidt [2] in the Steel Calendar 2012, which gives a good overview of difficulties in the design using the GMNIA concept.

#### Example of a buckling safety design using the MNA/LBA concept

**Enter the structural system**

We want to design the steel shell for buckling shown in Figure 01. Basically, this structure is a typical case where an engineer familiar with the design of steel shells would hardly consider an FE analysis. Since the main aim of this article is to explain the topic of the buckling safety analysis according to the MNA/LBA concept to the reader, the simplest possible example will be used.

An important issue in nonlinear calculations or in branching analyzes of shell structures is the element size, since unfavorably selected FE mesh settings can lead to falsified results. In the technical literature, there are also various rough formulas, whereby the most reasonable approach is a (small) convergence study.

**Calculation using RFEM**

After entering the model and load and defining the appropriate FE mesh settings, the calculation can be started with the help of RFEM. First, the material nonlinear analysis is performed. The aim of this analysis is the plastic reference resistance, that is, the load increasing factor at which the entire shell would fail plastically. Ideally, the RF-MAT NL add-on module is required for this, as it is only in this way that nonlinear material properties are available. Alternatively, a linear-elastic calculation can be performed and then the plastic reference resistance can be approximately calculated with formula (8.24) from [3] . Figure 02 shows the deformation state after reaching the plastic reference _{resistance} r _{Rpl} = 11.90.

Figure 02 - Material Nonlinear Calculation

Subsequently, the linear critical analysis is performed, the order of which was chosen arbitrarily. It is also possible to prefer this analysis first and then continue with the MNA. The aim of the linear strain analysis is also to obtain a load increase factor, but this time the one where the perfect shell would buckle. This requires the RF-STABILITY add-on module, which allows you to perform linear junction analyzes and geometrically nonlinear calculations. This does not mean any GMNIA calculations. Figure 03 shows the first _{mode shape of} the considered shell for the eigenvalue of r _{Rcr} = 7.70.

Figure 03 - Linear Bifurcation Analysis (1st Eigenvalue)

**Buckling safety analysis**

The buckling analysis is shown as a whole in the following. Special attention should be paid to the four independent buckling parameters, which can be determined for most practical construction cases according to Annex D in [3] .

Plastic reference resistance from the MNA:

r _{Rpl} = 11.9

Critical load factor from the LBA:

r _{Rcr} = 7.70

Related Slenderness:

$${\overline{\mathrm\lambda}}_\mathrm{ov}\;=\;\sqrt{\frac{{\mathrm r}_\mathrm{Rpl}}{{\mathrm r}_\mathrm{Rcr}}}\;=\;\sqrt{\frac{11.9}{7.70}}\;=\;1.243$$

Elastic imperfection factor:

$${\mathrm\alpha}_\mathrm{ov}\;\approx\;{\mathrm\alpha}_\mathrm x\;=\;\frac{0.62}{1\;+\;1.91\;\cdot\;\left({\displaystyle\frac{{\mathrm{Δw}}_\mathrm k}{\mathrm t}}\right)^{1.44}}\;=\;\frac{0.62}{1\;+\;1.91\;\cdot\;\left({\displaystyle0.98}\right)^{1.44}}\;=\;0.217$

Plastic zone factor:

β _{ov} = 0.60

Buckling curve exponent:

η _{ov} = 0.60

Full plastic limit slenderness:

$${\overline{\mathrm\lambda}}_{0,\mathrm{ov}}\;=\;0.20$$

Semi-plastic limit slenderness:

$${\overline{\mathrm\lambda}}_{\mathrm p,\mathrm{ov}}\;=\;\sqrt{\frac{{\mathrm\alpha}_\mathrm{ov}}{1\;-\;{\mathrm\beta}_\mathrm{ov}}}\;=\;\sqrt{\frac{0.217}{1\;-\;0.60}}\;=\;0.737$$

Buckling reduction factor:

$$\begin{array}{l}{\overline{\mathrm\lambda}}_\mathrm{ov}\;=\;1.243\;>\;{\overline{\mathrm\lambda}}_{\mathrm p,\mathrm{ov}}\;=\;0.737\\\rightarrow\;\mathrm{Pure}\;\mathrm{elastic}\;\mathrm{buckling}\;\mathrm{is}\;\mathrm{available}.\\{\mathrm\chi}_\mathrm{ov}\;=\;\frac{{\mathrm\alpha}_\mathrm{ov}}{\overline{\mathrm\lambda}_\mathrm{ov}^2}\;=\;\frac{0.217}{1.243^2}\;=\;0.140\end{array}$$

Buckling Design:

$$\begin{array}{l}{\mathrm r}_\mathrm d\;=\;\frac{{\mathrm\chi}_\mathrm{ov}\;\cdot\;{\mathrm r}_\mathrm{Rpl}}{{\mathrm\gamma}_{\mathrm M1}}\;=\;\frac{0.140\;\cdot\;11.9}{1.10}\;=\;1.515\;>\;1.0\\\rightarrow\;\mathrm{Design}\;\mathrm{is}\;\mathrm{fulfilled}.\end{array}$$

The main problem of the design is the classification of the results obtained by the program into one of the classic buckling cases. In this case, loading is very simple: It is almost pure meridian pressure buckling. Thus, the independent buckling parameters are calculated according to Annex D 1.2 in EN 1993-1-6 [3] .

The result of the buckling analysis according to the MNA/LBA concept is a load increasing factor. In the example considered here, it is 1.515. That is: The load on the shell could be increased by more than 50%.

If the analysis is performed with the stress-based concept, a load increase factor of 1.398 would result for the present case, which shows that for the classic buckling cases such as the meridia compression buckling considered here no special gains are obtained by the numerically supported buckling safety analysis according to the MNA/LBA Concept can be achieved. However, it should be mentioned again here that this behaves differently as soon as local load introductions or supports lead to stress concentrations.

#### Summary

Modern, powerful and at the same time user-friendly FEM programs such as RFEM make the work of a calculating engineer a lot easier when proving the sufficient buckling resistance of a shell. Due to the more consistent use of computing technology in the MNA/LBA concept, it is usually possible to achieve more realistic and thus more economical results.

It should also be mentioned that an FE analysis is not useful for every shell structure because good analytical methods are available for classic buckling cases that can lead to a low documentation effort and at the same time to similarly economical results. However, if you find cases in design practice that cannot be assigned to a classic buckling, an FE analysis according to the MNA/LBA concept by RFEM with the RF-STABILITY and RF-MAT NL add-on modules is a real alternative to the classic ones Procedure.

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