Introduction

This verification example studies a simple fork supported beam member with welded section (flanges: 200-12; web: 400-8) subjected to bending about major axis. Constant bending moment due to concentrated end moments and triangular moment dsitribution from concentrated transverse force is examined. Critical moment and force of the member is calculated by hand and by the Consteel software using both 7 DOF beam finite element model and Superbeam function.

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Introduction of Consteel Superbeam

In general, Consteel uses 7 DOF beam elements for finite element analysis of steel structures which are adequate for most everyday design situations. It is also capable of using shell elements in order to get more precise results in cases where beam finite elements are not sufficient enough. With the new Superbeam function it is now possible to examine structural parts with the accuracy of shell elements but with the ease of using a beam element concerning definition, modification, model handling, etc. In practice, it means that 7DOF beams can be switched to shell elements (and back) at any stage of the design process.

Validation

The validation program aims to verify the full mechanical behavior of the Superbeam switched to and analyzed as shell elements within a structural model composed of 7DOF beam elements. The validation of the analysis of the shell finite elements was done before and it is clear that in the case of correctly set boundary conditions the results are the same as the beam model provided that the local web buckling effect is avoided because it can not be modeled with beam-theory. Therefore the accuracy of the mechanical behavior of the Superbeam basically depends on two major factors:

• 1. the automatic shell modeling and mesh of the Superbeam
  • When transforming a beam model in the structural analysis to shell model, several automatic transformations are done with the model objects (loads, supports, connected elements etc.) in order to yield a consistent mechanical model.

• 2. the mechanical consistency of the connections of Superbeam at the boundary to 7DOF nodes
  • To satisfy the mechanical consistency at the connecting nodes the Superbeam uses automatically set constraint elements at both ends. They ensure the compatibility of the complete displacement field (translations, rotations, and warping) with the adjacent 7DOF beam finite element node or with the 7DOF point support.

The validation studies prove that the beam analysis model is mechanically equivalent to the shell analysis model within the Superbeam by comparing the results of the two models. It is shown that

• in the case of models where the local plate-like specific behavior is not relevant (thick plates in the cross-secions) the results are the same
• in the case of models where the local plate-like specific behavior is relevant (thin plates in the cross-secions) the results can be different only because of this plate-like behavior (local buckling, cross-section distortion) while the isolated beam-like behavior is the same

Part 1

In this first part of the validation, we examined simply supported beams of welded I-sections with several different profile geometries. The full length of the beams was changed to Superbeam shell and so the consistency of results of both the shell elements and the constraints could be analyzed.

Structural models and analysis

In every case, the beam was first calculated with 7 DOF beam finite elements, after with Superbeam shell elements, and finally also as a full shell model with the same finite element sizes as the Superbeam shell. In full shell models, we applied rigid bodies along the edge of the web.

Linear buckling analysis was executed in order to compare the first buckling eigenvalues.

Our expectation was that the two kinds of shell models would produce very similar results which are by nature somewhat less favorable than the 7 DOF beam results, meaning that alfa critical values should be lower when using shell elements. To be able to compare the results related to global (lateral-torsional) buckling, the effect of local buckling of the web was to be avoided as much as possible so the examples were chosen accordingly.

Geometry

Loading

Two loading scenarios were considered on every section

Uniform moment:

Linear moment:

Result tables

First global buckling eigenvalue – α_crit

Consteel version CS15.1095

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Conclusion

As expected, the critical load parameters of beam models are always higher than the ones from the shell models. The difference between the results of the beam and shell models increases when the difference between the top and bottom flanges becomes more and more significant. It is because the straightness of the web starts to deviate (section distortion) which is already outside of beam theory, and so it is impossible to model with beam finite elements.

Despite the very different modeling techniques, the α_crit parameters of the two types of shell models are very similar – differences are within +/-3% which proves the applicability of the Suprerbeam as an alternative to the usual 7DOF beam finite elements for structural analysis. The use of Superbeam is recommended in cases where a more accurate analysis is desired by the designer.

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/mm², the lowest critical stress where local buckling occurs is 2.28*82.52=188.15 N/mm² 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/mm² 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.

The new versions of the EN 1993-1-1 (EC3-1-1) and the EN 1993-1-5 (EC3-1-5) standards have introduced the general method designing beam-column structures; see [1] and [2]. The design method requires 3D geometric model and finite element analysis. In a series of papers we present this general design approach. The parts of the series are the following:

• Part 1: 3D model based analysis using general beam-column FEM

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Papp F, Szalai J. New approaches in Eurocode 3 – efficient global structural design. Part 1: 3D model based analysis using general beam-column FEM. Terästiedote (Finnish Steel Bulletin), 5, 2010.