## Introduction

Beam with welded I sections are often executed with slender webs. This is mainly due to the recognition that the main contributors to bending stiffness of a beam are the flanges. The web plate’s main role is to safely keep these flanges away from each other and carry the shear stresses which might be present. Significant weight saving can be achieved with the use of slender webs, but there are some aspects to take care about.

When slender web plates are exposed to longitudinal, uniform normal stresses, above a certain stress level its distribution will no longer remain uniform. A compressed region of a plate distant from its lateral supports may buckle in a direction perpendicular to the acting external normal stresses, causing a subsequent transfer of stresses from the affected region to other neighbouring regions remaining in their unbuckled position.

This buckling remains limited to a part of the plate keeping other parts intact and therefore is called as local buckling. Local buckling usually does not result an immediate collapse of the structure, due to possibility of the stresses to redistribute and often even a substantial amount of further load increases are possible.

The tendency of a compressed plate to suffer local buckling is characterized by its slenderness value defined by the following formula

where σ_{cr} is the critical stress level above of which the stress redistribution and local buckling starts to appear. A higher critical stress will result in lower slenderness value which indicates that the plate can carry higher compressive stresses without the initiation of local buckling.

## Analysis of cross-sections with beam finite elements

The well-known beam finite elements used by usual structural design software do not “see” the internal composition of the cross-section. During structural analysis the sections are represented by certain integrated cross-sections properties assuming the validity of several assumptions including the Bernoulli-Navier Hypothesis and the non-deformability of the cross-section. A local buckling of any of its internal plates would violate these assumptions making hard to create the equivalent cross-sections properties.

In the modern design practice followed by Eurocode the phenomenon of local buckling is handled by the use of effective section properties. Regions subject to possible local buckling of compressed plates of a cross-sections are “eliminated” and the section properties are calculated based on the remaining parts of the cross-sections.

Design verifications use these effective cross-section properties to calculate the resistance of cross-sections exposed compressive forces. When required by Eurocode, the effect of appearance of local buckling can also be reflected in a structural analysis using beam finite elements with the use of effective cross-section properties, instead of the original gross section properties. This is mainly required to prove serviceability criteria.

## Analysis of cross-sections with Consteel Superbeam

The Consteel Superbeam function makes possible to confirm directly the presence of local buckling using the same beam element based model, but using a mixed beam and shell finite element modelling and analysis approach. Using the Superbeam tool, complete structural members or parts of them can be alternatively modelled with shell elements and the rest can still be modelled with beam finite elements. Using this technique, the total degrees of freedom of the model can be kept as low as possible. When using Superbeam, the designer has the choice whether to use beam or shell finite elements, as appropriate.

Contrary to beam finite elements, modelling with shell finite elements doesn’t have the previously mentioned limitations. This approach can fully consider the shape and location of the cross-section’s internal components instead of the use of an integrated overall section property. When a linear buckling analysis (LBA) is performed, the critical stress multipliers corresponding to the actual stress distribution can be obtained. Additionally to the load multipliers, the corresponding buckling shapes are also available, giving direct indication on the location, shape and appearance of local buckling within the compressed parts of the cross-section.

The use of effective cross-section concept is very convenient but there might be cases when more insight view is desired. The following example gives an idea where the Superbeam function can be helpful.

## Demonstrative example

Let’s consider a 12 m long simple supported welded beam with the following parameters

The beam is laterally restrained at third points at the level of its upper flange. The beam is loaded with its self-weight plus a uniformly distributed load of 10 kN/m acting at the level of upper flange.

When the beam is analysed with 7DOF beam finite elements, one can obtain the critical load multiplier of 5.2 of the global buckling mode, which is lateral-torsional buckling (LTB) in this case.

The beam finite element cannot give any visible indication about possible local buckling in compressed plates of the cross-sections.

As the maximum bending moment occurs in the middle third of this beam, it seems enough to analyse this part mode deeply with the Superbeam function. An LBA with the mixed beam and shell model gives comparable critical multiplier of 5.22 with some numeric perturbances in the part modelled with shell elements.

In addition to the global buckling mode, the Superbeam based model can also provide local buckling modes of the middle third part of the beam. The first buckling mode with a critical multiplier of 2.28 shows clearly the expected local buckling in the upper compressed part of the web.

A transverse section of the relevant buckling mode shows clearly that the buckling mode shape has a maximum ordinate around the middle of the upper half of the web plate.

A local buckling shape of this kind does not necessarily mean automatically that the member has such a slender web where the design calculation should be performed with effective section properties. Effective properties shall be used, if the reduction factor ρ for internal compression element (the web) defined in EN 1993-1-5 with the formula (4.2) yields a value less than 1.0. This is the expected case when the plate slenderness λ_{p} has a value higher than 0.673.

As the maximum normal stress in the web plate obtained with a linear elastic analysis is 82.52 N/mm^{2}, the lowest critical stress where local buckling occurs is 2.28*82.52=188.15 N/mm^{2} resulting a slenderness of 1.12 with f_{y} = 235 MPa and a ρ reduction factor of 0.811.

As this number is lower than 1.0, it confirms the presence of such a slender compressed web, which will be unable to carry the elastic stress distribution calculated on the gross cross-section and therefore as a response to the appearance of local buckling, reduced cross section properties must be used for design verifications.

Just to remember, the effective cross-section calculated with beam finite elements clearly shows the corresponding eliminated part of the web plate.

## How could be avoided such a section reduction?

For example, an additional stiff enough longitudinal stiffener could be positioned close to the level of the maximum amplitude of the first buckling shape with the expectation that it will increase substantially the critical load factor corresponding to the buckling shape responsible for the reduction. Or even completely eliminate such a buckling shape. To make it efficient, additional vertical stiffeners are also recommended at the ends of the horizontal stiffener.

The standard procedure of Eurocode for the determination of effective cross-sections cannot consider the presence of such longitudinal stiffeners. Longitudinal stiffeners – among other features – can easily be placed using the auxiliary tool of Superbeam function, together with vertical stiffeners at both ends of the middle third of the beam.

When a 10 mm thick longitudinal stiffener is welded to left and right from the web at the critical level – close to the maximum ordinate of the corresponding buckling mode shape – a new buckling mode shape is obtained, with a higher critical multiplier of 7.72. This is almost 3.5 times higher than the value without the longitudinal stiffener.

By repeating the previous calculation to obtain the necessary slenderness value of the web plate, one gets σ_{cr }= 7.72*82.52=637.0 N/mm^{2} resulting a slenderness of 0.606 with f_{y} = 235 MPa which falls already under the limit of 0.673 therefore no reduction is necessary to consider possible local buckling.

Of course, it is important to remember, that additionally it must be also confirmed, that the stiffener has high enough stiffness to allow to consider it as an efficient lateral line support for the web plate.

EN 1993-1-5 Chapter 4.5 needs to be followed to confirm this.

With the use of Superbeam analysis tool, the designer gets the chance to receive in-depth information about the analyzed structure, making possbible to find the most ideal solution to handle buckling related problems.

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In this paper a numerical study is presented which examines a steel frame with two different finite element programs. Stability failure is more frequent in a lot of cases than strength failure hence it is important to focus on these failure modes: global, in-plane-, out-of-plane -, lateral-torsional- and local buckling. Three models were used with different elements such as shell elements and 7 DOF beam elements. 7 DOF beam elements were used in the first model, shell elements were used in the other two. The first of the shell models gave too much local buckling shapes therefore it was improved with local constraints and that is the third model where global buckling shapes can be examined. There are three different procedures to calculate the resistance: (i) the general method, (ii) the method of the reduction factors, and (iii) the simulation. The analysis results of the different programs and design methods were compared to each other and to the manual calculation based on the Eurocode 3 standards.

Download articleTóth A. , Joó A.: Comparative study of steel frame modelling levels and Eurocode based design methods, “Model Validation and Simulation” – Graduate Courses for Structural Engineering Applications, Bauhaus Summer School 2015