Plate buckling strength is an important aspect of offshore steel construction design. We will show how the problem of both general FEA analysis (strength evaluation, displacement, and deflection checks) and plate buckling check according to ABS or DNV rules is solved with the help of FEA and code checking tools. The most complicated task in performing a plate buckling check for a big structure, like a complete ship design, on a general Finite Element Analysis model is to define a big amount of plates and the dimensions of these plates to be verified.

For the precise FEA analysis, the model has to have a fine mesh with small enough finite elements to guarantee the correct results. But at the same time, each plate field must be treated as one separate structural member for plate buckling checks. With the help of SDC Verifier it is possible to break the boundaries of general FEA Analysis and enable the code checking directly in** Simcenter 3D**, **Femap**, **Ansys,** etc. by enabling the automatic recognition of structural items. The recognition of plates, stiffeners, and girders is based on mesh connectivity and can be performed on any structure which is built with 2D or, in some cases, even 3D elements. The structural members are defined automatically and mesh independently. This allows an engineer to have a model with fine mesh for precise results of the general finite element analysis and a list of structural members for code checking.

This article was originally posted in SNAMES 40th Annual Journal.

## Introduction

Plates are commonly used in the design of ships, offshore structures, aircraft, civil, and other engineering structures. Each plate should be verified as it influences the strength and stability of the whole construction. There are two main failure modes of a plated structural item that can lead to sudden damage: material failure and structural instability, which is called buckling. Most plated structures are capable of carrying tensile loading, but may be poor in resisting compressive forces. Usually, buckling effects take place suddenly and may lead to severe or even catastrophic structural failure.

That’s why it is very important to understand the buckling capacities of the plates to avoid a collapse of the complete structure. Buckling analysis of the structure with a general FEA solver may seem like a quick and easy solution, since it provides a buckling load factor – a ratio of the buckling load to the currently applied load. As a result, you get a value of the factor that will cause buckling failure in case of multiplication of the load value with it. But this analysis result would be only for a panel that will fail first, which does not guarantee that the rest of the structure is safe.

This is where it becomes necessary to verify according to the industry standards. A lot of these documents already contain verification procedures or recommended practices for the plate buckling analysis. Here are some of the codes that are commonly used in the industry:

- DNV RP-C201 Buckling Strength of Plated Structures;
- DNV CN30 Buckling Strength Analysis of Bars and Frames, and Spherical Shells;
- ABS Plate Buckling and Ultimate Strength Assessment for Offshore Structures;
- Eurocode 3 – design of steel structures – Part 1-5: Plated structural elements.

In the case of code checks according to the standards, an engineer is able to perform the calculation of the value of utilization factor for every plate as a result. It is possible to do the calculation for each load or combination to ensure that the whole structure is safe. This procedure is usually not very simple since a lot of factors, characteristics, and coefficients should be taken into account.

With a certain knowledge, it’s possible to do the check according to the standard by hand. But in this case, an engineer is able to verify one plate under one loading condition at a time. Though, a typical offshore structure or ship design consists of thousands or even millions of plates and hundreds of load combinations. This is when automation is a must. **When it comes to the code checking in CAE, there are two ways:**

- To run the general finite element analysis for the design, which is mandatory to understand the behavior of the structure and obtain the results of stresses, displacements, forces, other outputs. And then to perform the Standard verification of important details with scripts, spreadsheets, or hand calculation.
- To use the general FEA analysis for the design and dedicated software for the code checking.

Both of these methods have certain drawbacks. Spreadsheet or hand calculation analysis is time-consuming, and it’s easy to miss a failure, since not always the most stressed or longest plates are the most subjected to buckling. Using dedicated code checking software is more precise, but since it requires having each structural member defined, it is necessary to build another model for code checking. Discernibly, double modeling leads to an increase in the overall completion time of the project, since every update and modification has to be done twice.

## Automatic Recognition

Since checks are done on structural items and not on finite elements, the best solutions for both execution time and accuracy of the results would be to use the extension for general FEA software (Ansys, Femap, Simcenter 3D) that is capable of the automatic recognition of the structural members mesh independently. SDC Verifier is software that follows this methodology. **The best solution to avoid double work is to have the same environment for both General FEA and code checks.**

Stiffened Panel Finder — is a tool to automatically recognize sections, panels, plates, stiffeners, and girders, and dimensions of these structural members. The detection is based on mesh connectivity and can be performed on any structure which is built with 2D (plate or shell elements) for plate members and both 1D (beams) or 2D finite elements for stiffeners and girders.

Detection is made automatically and meshes independently. This brings to an engineer the opportunity to have a model with fine mesh for precise results of the general finite element analysis and use the same model for calculation of Eurocode, ABS, or DNV plate buckling checks. At the first stage, Sections are defined by the global or custom coordinates. All the elements that lay in one plane (of course, with a certain angle of deviation which could be defined by the user in settings) are defined as sections. This allows detection of, for example, Frames, Decks, and Longitudinal sections of the ship. Hull is also automatically recognized as a custom section.

The next step is to define the plates on these sections; plates are also recognized automatically with borders at sections intersection, stiffeners, girders, or any other members perpendicular to the sections. A user always has control over recognition to add/remove or split the members manually. But if the mesh is fine enough, there is no need for manual interaction with the recognized structural members. Recognition is completely mesh-independent, any plate of the studied FEA model can consist of hundreds or even thousands of finite elements for precise stress analysis, and it will still be defined as one structural member for plate buckling checks.

Automatic recognition of the plates defines the following parameters for the code check: length and widths of the plate, direction, amount of edges, material type, and thickness. The analysis is based on stresses in each finite element of the plate or on the plate average stress.

## Verification procedure from the user point of view

Despite the fact that material properties, forces, stresses are defined in the FEA program and plate dimensions and types are automatically recognized, some parameters still should be defined by the user. For example, DNV RP-C201 Plate/Stiffener Buckling (2010) requires user input for a characteristic called Resulting Material Factor. During the analysis procedure, buckling resistance will be divided by this factor. By default, this factor is 1.15, but an engineer may change this value taking into account the type of structure or consequences of failure.

It is also possible to define a thickness factor that allows to increase/decrease all plate thicknesses quickly without re-solving the model. For example, a thickness factor of 1.2 means a thickness increase of 20%, which leads to a stress decrease.

An important decision has to be made about what stresses to use. It is possible to use plate average stress; this will result in one buckling factor result on each plate. A more conservative approach is to use stresses of every element for the analysis; in this case, the maximum buckling factor from all the elements of a plate would be presented as a resulting buckling factor of the whole plate.

If the option to use plate average stress is turned off, then there are two options to define elemental stress: average element stress or minimum element midplane stress (which is maximum compressive stress).

Basically, the parameters and decisions described above are the only engineer’s input in case of automatic code checking. The rest of the calculation is done by a code checking program: standard outputs of the FEA solver and parameters of the model are used as variables for formulas to define plate buckling factor as a result. **The benefit of SDC Verifier as a code checking tool is also that all the formulas are open and refer to the standards, so it is easy to follow the calculation procedure, possible to find the source of the problem quickly, and even modify the existing formulas if customization of the checks is necessary**. It is also possible to see the intermediate results values.

Software completely follows the verification procedure of the selected standard. At the first step of the code check, plate length, width, and thickness are retrieved from the recognition, and compressive Stresses Sx, Sy, and Sxy are calculated in plate direction. Then the Slenderness and Buckling resistance for both X and Y directions are checked. Every formula is open and has a description, names of the intermediate variables. For example, the slenderness formula (used to calculate the buckling resistance in the X direction of the plate) from the DNV check is represented below:

Different types of variables are highlighted with different colors, and description refers to the formula from the standard. In the final step, Buckling factors are calculated for X, Y, and XY (Shear) direction, as well as Maximum overall directional and combined Buckling Factors.

## Results of the Automated Plate Buckling Checks

### Result tables

As a result of this automated verification procedure, the user will get a Buckling factor for every plate of the whole structure in minutes, rather than days spent with spreadsheets or hand calculations. Moreover, the calculation could be done for multiple load combinations and envelope groups of loads. This means that the results of the analysis, which are typically presented in detailed buckling factor tables for every section/plate, are automatically prepared for each loading condition.

A wide variety of tables are available, and results can be presented over any load or selection. The extreme table type shows the maximum value for the complete selection, and Expand table type presents the value for every item of this selection to be quite extensive.

In addition to the buckling factor, the following parameters results can also be listed in the table:

- Plate Width;
- Plate Thickness;
- Sx in plate direction;
- Sy in plate direction;
- Sxy in plate directions;
- Equivalent Stress.

The interface of the tables allows presenting not only the final results but also the calculation details – all the formulas with intermediate resulting values of the parameters used for the calculation. This provides an engineer with an extra instrument to control the calculation and leaves much less room for an error.

### Result plots

The graphical interface of the FEA programs is used to visualize the buckling factor or any other output (including the recognition details) values for any user-defined selection.

This provides a user with full control of the view, including the positioning of the model, plotting style, legend settings. Views are stored and can be used to present the results of general FEA analysis, as well as code checking results, for any load or selection.

### Automatic reporting

Since the calculation core allows to get the results for individual loads and load combinations, and SDC Verifier has an interface to present the results with tables and plots, it’s easy to prepare an automated template-based structure for the report generation. Typically, the report contains the following parts:

- Model Setup – information about materials, properties, loads and boundary conditions, the basis of calculation, formulas used for the analysis (automatically added because of the open interface of the code-checking software);
- Results – automatically sorted by load, or by selection results of finite element analysis and verification according to standards;
- Summary – a short explanation of the main results and comparison with the allowable values.

Automation of the reporting process helps to save time on the repetitive documentation routine. It also reduces the deadline pressure: since the report structure is set, there is no need to create a new report in case of modifications or design changes. The engineer has only to update the model, run the calculation, and regenerate the report.

## Conclusions

The code checking approach described in this article brings to marine designers and naval architects the understanding of alternatives for the usual code checking workflow and describes the ways to save time on routine and repetitive tasks by automating the verification for a complete model in a single CAE environment.

In addition to the time-saving benefits, usage of code checking extensions for the General FEA programs allows to:

- Check the quality of modeling with the help of recognition tools.
- Understand the behavior of a studied structure, by analyzing all possible loading conditions and defining the governing ones.
- Analyze the critical parameters for the checks.
- Quickly improve the design. By using the thickness factors and modifying the plate dimensions. Or with the help of powerful editors in the general FEA tools and instant update of the simulation data code-checking extension.
- Compare different design approaches of loading conditions in one user-friendly CAE environment.

## References

- Timoshenko, S. P. and Gere, J. Theory of Elastic Stability, 2nd edition, McGraw-Hill, 1961
- Recommended Practice. Det Norske Veritas. DNV-RP-C201 Buckling Strength of Plated Structures. October 2010.
- Webinar: Plate buckling checks. SDC Verifier for ANSYS
- Webinar SDC Verifier for Simcenter. Introduction and Plate Buckling Demo