# Buckling

## Overview

Buckling is the sudden change or deformation of a structural component under a load. In the simplest terms buckling refers to the instability and failure of a structure as it compresses under a load and the sideways deflection that occurs before the strength limit of the material is reached.

From a mathematical approach, buckling is often referred to as a bifurcation problem because there can be more than one solution to the same load levels. At the bifurcation point (also expressed as the point of instability or buckling point) various load carrying capacities branch out. Finding the most stable of these bifurcation paths is the aim of buckling analysis.

This explanation will focus on global buckling and member buckling; outlining key modelling considerations and analysis methods.

### Global vs local buckling

Buckling issues can be global: affecting the entire structure (entire truss, entire shell, entire building) or local: confined to a single area (e.g., a plate in a connection node), a single member (e.g., a column, beam, etc.).

The failure of a single part or member can endanger or limit the capacity of the whole structure and, in certain cases, activate global effects. For some structures, such as thin wall shell structures, grid shells, etc., local, and global behaviour are closely related. For this reason, global and local bucking are generally to be considered related. These two aspects can be studied separately under certain circumstances or assumptions in which the problem can be simplified.

Often, local buckling checks of a single member (e.g., column, beam, etc.) is part of the typical steel member design. This can be studied and checked using standard methods and formulations as a post-processing process, based on the initial geometry and internal forces defined in a linear static analysis.

In GSA, the following analysis types can be used to evaluate buckling problems: linear buckling analysis (modal buckling analysis), nonlinear buckling analysis (nonlinear static analysis), and member buckling checks within the steel designer. The following sections provides insights into these analysis approaches and an overview of each type.

## Key modelling considerations that influence buckling analyses

There are several aspects to consider when evaluating the buckling resistance of a structure or member. The main ones are listed below. Each factor can be considered in different ways, depending on the level of approximation in the calculation of these parameters, requiring different types of analysis with different levels of complexity.

### Support conditions

The support condition of a member or the whole structure influences its buckling lengths and indirectly its buckling resistance. In particular, the rotational capacity at the end nodes, and therefore the rotational stiffness of the connection, is one of the key aspects that can dominantly influence the result. For these reasons, and because no connection is infinitely stiff, a correct estimation of the connection stiffness is a key aspect of the analyses. In practice, safe assumptions can be made (e.g., ignoring the bending resistance of the connection).

### Loads

The buckling resistance of a structure, or part of it (independent from the method used to evaluate it) depends on both the mechanical properties and geometry of the structure itself and the applied load. Therefore, buckling must be explicitly evaluated for each load configuration and under the exact boundary conditions. In other words, the buckling resistance or buckling factor of a structure or member is not an inherent property of the structure but can vary with changing loads.

### Imperfections in the structure

Imperfections of the member and entire structure (such as initial curvature of the profiles, initial deformation of the entire structure, etc.) in different cases strongly influence the structural behaviour.

For a single member, the influence of the imperfection can be estimated by a standardised method (which varies from country to country) or by direct modelling of the imperfection and considering analysis based on more complex methods.

For a structure in its globality, one must consider global imperfections not just member imperfections. For example, unevenness in a spherical shell or tank, or the initial inclination of a frame system are typical examples of global imperfections.

As it is impossible to define all this in a formal way, in practice it is common to consider some ideal initial deformations of the structure and scale them with some arbitrarily defined values. In this way, some worst-case scenarios are defined and used instead of the perfect geometry as a basis for calculation, to ensure adequate margin for error are conservatively retained.

### Second order effects (geometric nonlinearity)

Generally, the effects of the deformed geometry (second order effects) should be considered if they increase the action effects significantly, or significantly modify the structural behaviour. In the case of buckling analyses, the consideration of second order effects is possible using a geometric nonlinear analysis. This is strongly suggested when a structure is subjected to large deflections or rotations, snap-through, load stiffening, or relevant changes in the stiffness matrix between the initial and final states.

### Material

Since buckling usually occurs before the material reaches its strength limit, structures prone to buckling are generally designed considering linear elastic materials. In some special cases, an elasto-plastic or fully plastic low material may be required.

## Global buckling analysis methods

Global buckling analyses usually refer to analyses of the entire structure. In general, each individual structure can be subject to a global buckling problem and requires global analysis. Some structural systems (e.g., shells and grid shells) are more sensitive to this problem and require more advanced methods. For others (e.g., plane frames in buildings), simpler methods are sufficient.

For this reason, different methods are available. In general, it is advisable to start with a simpler method and progress to a more complex one only if the simple method is not sufficient. The following is a list of the more commonly used methods.

A further summary of analysis options for linear and nonlinear buckling analysis types can be found at the end of this section.

### Linear buckling analysis (modal buckling analysis)

Linear buckling analysis (referred to as **Modal buckling analysis** in GSA) allows you to perform linear elastic buckling with or without consideration of global imperfections in the structure. The imperfections can be considered in the initial geometry as an imposed displacement of all finite element nodes without generating initial stresses in the structure. The shape and vector of the imposed displacement should be defined by the engineer based on a specific consideration (e.g., the shape of the first buckling mode by a specific maximum norm vector).

A modal buckling analysis looks at the potential for buckling on a structure with a given load by finding its eigenvalues. Unlike a modal dynamic analysis, which characterises the structure, a buckling analysis characterises the structure for a given load. When a column is subjected to an increasing compressive axial load its ability to carry transverse load is reduced until it can no longer carry any transverse load, and the *Euler load* is reached. For a given axial load the ratio of the Euler load to the actual load is the load factor for that mode. If the column is supported at its midpoint, then the first mode will be suppressed and higher modes with corresponding higher loads factors are possible. The modal buckling analysis extends this concept from a simple column to the whole structure, to determine global modes of buckling. Sufficient modes should be selected for the buckling analysis to ensure that all the modes of concern are identified.

The theoretical buckling strength of a structure is estimated as a multiplication factor of the applied loads, based on the initial geometry of the structure (considered linear elastic) and for a defined load configuration. This kind of analysis can be defined as a modal buckling analysis in GSA.

Note:See the theory entry on eigenvalue buckling analysis for more information including modelling implications and what the results mean.

Multiplying the applied loads by the multiplication factor gives the bucking strength, namely the maximum load level at the critical bifurcation point (boundary between stable and unstable state of the structure). Each buckling load (or buckling factor) is associated with a buckling mode shape or instability shape. This represents the shape of the structure under bucking.

A linear elastic buckling analysis defines an upper limit to the strength of the structure and should only be considered as a theoretical buckling strength. In reality, due to imperfections and non-linear behaviour, this theoretical buckling strength cannot be achieved in a real structure, and collapse will occur at much lower loads. In addition, due to imperfections and non-linear behaviour, a structure may be much more prone to buckling under some specific buckling modes than others (e.g., because the imperfection configuration is close to the buckling shape). For this reason, not only the first buckling factor and shape are relevant, but also other major buckling modes (e.g., second, third) should be considered. Code and standards provide a simplified method which limit the bucking strength to account for imperfections etc. and allow it to continue to use the linear elastic buckling method. Alternatively, the nonlinear method can be used.

Key modelling consideration | How to account for each consideration with modal buckling analysis in GSA |
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Support conditions | Support degrees of freedom can be defined as fixed, free or by a linear spring; however, nonlinear springs can not be used. Note that if a symmetry boundary condition is used, asymmetrical buckling modes may not be captured |

Loads | Applied loads should be the load case under consideration. The distribution of the load matters but the magnitude is not as important. |

Initial imperfections | Initial imperfections can be included and can be defined in the advanced settings of this analysis task. |

Geometric nonlinearity | It is not possible to account for geometric nonlinearity with modal buckling analysis. |

Material nonlinearity | It is not possible to account for material nonlinearity with modal buckling analysis. |

### Nonlinear buckling analysis (nonlinear static analysis)

With nonlinear static analysis, it is possible to perform nonlinear bucking analysis. This method considers geometric nonlinearities (second order effects: P-Δ and P-δ) and defines the buckling factor at the final geometric and load configuration.

Loads are applied incrementally and at each increment the new stiffness of the structure and load configuration is calculated. The process continues until a small change in load level causes a large change in displacement. This is considered as the transition point stable and instable state of the structure. This method can be used with or without consideration of geometric initial imperfections and generally allows for non-linear materials.

In GSA, it is possible to define nonlinear materials for 1D members and elements. To do a **nonlinear buckling analysis in GSA**, you can run a **nonlinear-static analysis** and incrementally increase the load. A **modal buckling analysis** or **component buckling analysis** can be used to help estimate the maximum axial force to apply.

Key modelling considerations | How to account for each consideration with nonlinear static analysis in GSA |
---|---|

Support conditions | User can choose to model any type of support condition that best represents their structure including nonlinear springs. |

Loads | For a nonlinear static analysis, loads are applied incrementally up to the specified load. It is possible to terminate the analysis prematurely if it is not converging (likely due to buckling). |

Initial imperfections | Initial imperfections can be modeled directly by creating a new model from deformed geometry of another analysis case. Or by using applied displacement loads. |

Geometric nonlinearity | Geometric nonlinearity will be considered in the analysis if beam and shell element slenderness effect is checked in the analysis task definition . |

Material nonlinearity | It is possible to use nonlinear materials for 1D members and elements with a nonlinear static analysis. |

#### Imperfections in the structure

It is possible and can be useful to specify an imperfect initial geometry. The results of an eigenvalue buckling can be scaled to give an appropriate imperfection. Using the ** Model > Model tools > Create new model from deformed geometry** menu command allows the user to select an analysis case and specify either a scale factor or maximum imperfection to apply to the displacements to be used to update the model geometry. Initial imperfections in nonlinear analyses can also be created using Applied displacement loads.

#### Material nonlinearity

As explained in the previous part, since buckling usually occurs before the material reaches its strength limit, structures prone to buckling are generally designed considering linear elastic materials. In some special cases, an elasto-plastic or fully plastic low material may be required. In terms of global buckling, even if it is not very common to consider an elasto-plastic material, it is important to be sure that in no part of the structure is the yield limit exceeded. Indeed, if there is softening of any part of the structure due to plasticisation, this could be relevant in defining the buckling resistance of the structure itself and/or the correct load path.

Material nonlinearlity can be included in a nonlinear static analysis. A material nonlinear analysis (MNA) can be used to find the plastic limit load, i.e. the load that leads to plastic collapse of the structure. This analysis is usually developed using small displacement theory as a basic assumption, neglecting second order effects and imperfections, but taking into account the elastic-plastic behaviour of the material or some part of it. In most cases, it is possible to define where plastic hinges will form based on other preliminary analyses of the structure. This type of analysis is much more closely related to capacity analysis than to buckling analysis.

#### Geometric nonlinearity combined with imperfections and/or material nonlinearity

##### Geometrically nonlinear elastic analysis with imperfections (GNIA)

In this type of analysis, the geometrically non-linear nature of the structure with imperfections is considered. This type of analysis is useful for evaluating the performance of structures that are essentially compressed sensitive to the effects of imperfections, but with stress values in the material that do not significantly exceed the yield value.

##### Geometrically and materially non-linear analysis (GMNA)

The combination of the previous analyses with an elasto-plastic material leads to this type of analysis, which provides the geometrically non-linear plastic collapse multiplier that determines the instability of the structure assumed to be free of defects.

##### Geometrically and materially non-linear analysis with imperfections (GMNIA)

This analysis is the most complete and provides and is therefore the best approximation to the real structure, allowing both the stiffness reductions due to geometry changes and the elasto-plastic material behaviour to be captured; but it is also, of course, the most complex.

### Summary of global buckling analyses

Below is a summary of various analysis options by combining the two analysis types with the various options to account for imperfections, large deformations and material nonlinearity.

Some of the more complex methods are rarely used in practice. In any case, if used, they should be backed up with simpler methods and defined and analysed with sound engineering knowledge and experience.

#### Linear buckling analyses

Options | Imperfections | Large deformations | Material nonlinearity |
---|---|---|---|

Linear buckling analysis | ✗ | ✗ | ✗ |

Linear buckling analysis with initial imperfections | ✓ | ✗ | ✗ |

#### Nonlinear buckling analyses

## Member buckling in steel design checks

As different characteristics such as slenderness, profile class, imperfections, buckling curves etc. influence the buckling resistance of a single profile, standards and codes have defined appropriate formulations to define the buckling resistance of a member subjected to compression, bending, torsion or a combination of these. This is a post-processing analysis that can be performed directly in GSA using the **Steel member design** module or using separate tools.

### Types of member buckling

#### Flexural

Also known as column buckling, this is the simpler type of buckling. It applies to any member under compressive load, whether it is open-walled or box section, symmetrical or asymmetrical about one or both axes. The buckling load can be derived from *Euler's Colum* buckling formula.

#### Torsional

This type of buckling occurs in open-walled sections such as cruciform sections.

#### Flexural torsional

Flexural torsional buckling occurs when sections that are not symmetrical about their two transverse axes are used as compression members. Examples of such sections are channel sections and T-sections. Flexural torsional buckling starts in the same way as simple flexural buckling.

#### Lateral torsional

Lateral torsional buckling occurs when an unrestrained beam (beams where the compression flange is free to move laterally and rotate) deforms away from its longitudinal axis, causing both lateral displacement and twisting.

### How to take member buckling into account

The buckling of members should be considered under two aspects: the design of the member itself, and the softening effect that the buckling of the individual member, or of several members at the same time, could produce within the structure, thus influencing the global buckling or even inducing a global buckling collapse.

The design of the member itself can be done in GSA using the **Steel member design** module.

The reduction in stiffness of the whole structure due to buckling of a single member could be generated (in some cases) using a non-linear buckling analysis considering the second order effects P-Δ and P-δ or using more advanced methods such as the *Dallard method*.

#### Steel member design

The member design considers the imperfection of the individual member (e.g., initial curvature), which modifies the buckling curves and thus reduces the theoretical Euler buckling factor. Other aspects, such as second-order effects, are considered with amplification factors. Material nonlinearity is usually neglected.

An important parameter of these analyses is the definition of the appropriate buckling length. This can be defined based on knowledge of the stiffness of the connection elements and the boundary conditions.

Key modelling considerations | How to account for each consideration with steel design in GSA |
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Support conditions | Done through definition of member end and intermediate supports or through a buckling length override. |

Loads | Based on ultimate limit state load combinations. |

Initial imperfections | Initial imperfections within a member is accounted for since code equations reduce the theoretrical euler buckling factor. |

Geometric nonlinearity | Any second order effects within the member (P-delta effects) are included in code equations through the reduced Euler buckling factor. If the global model does not include second order effects, it can be accounted for in member design with moment amplification factors. If the global model does include second order effects, no additional action needed. |

Material nonlinearity | Material yielding considered in design equations. |

#### Linear buckling analysis

As well as the global analysis, a linear buckling analysis could be used for the member analyses. This has the same advantages and disadvantages as described in the global buckling section. Doing that for a single member could be useful only in the case of complex geometry or boundary conditions (e.g., a curved element with multiple supporting points).

It is important to note that a linear elastic buckling analysis defines an upper limit to the strength of the member and should only be considered as a theoretical buckling strength. Due to imperfections and non-linear behaviour, this theoretical buckling strength cannot be achieved in a real structure and will collapse at much lower loads. For this reason, a linear buckling analysis for a single member is sufficient to assess the sensitivity of the member to buckling. Normally further detailed analysis is required and only for very large buckling factors (normally greater than 10) could the member buckling analyses be considered sufficient.

#### Nonlinear buckling analysis

The application of nonlinear buckling analysis is the same for a global structure or a single member, only the scale of the object changes. This method could be used in the case of complex profiles and/or complex boundary conditions. With correct application of imperfections and analysis of a sufficient number of load cases, this option has proven to be very effective particularly in design cases.