## Introduction

Are you wondering how a web opening would influence the lateral-torsional buckling resistance of your beam? Check it precisely with a Consteel Superbeam based analysis

It is often required to let services pass through the web of beams. In such cases the common solution is to provide the required number of opening in the webplate. Such an opening can have a circular or rectangular shape, depending on the amount, size and shape of pipes or ventilation or cable trays.

Beams must be designed to have the required against lateral-torsional buckling. The design procedure defined in Eurocode 3 is based on the evaluation of the critical bending moment value which provides the slenderness value, needed to calculate the reduction factor used for the design verification.

There is no analytical formula provided in the code for beams with web openings. Would the neglection of such cutouts cause a miscalculated and unsafe estimation of the critical moment value?

The following demonstration will be made with a 6 meters long simple supported floor beam with a welded section.

Exposed to a linear load of 10 kN/m, the critical bending moment value of the solid web beam can be obtained by performing a Linear Buckling Analysis (LBA) with Consteel.

The obtained critical multiplier for the first buckling mode is 3.00 which means that the actually applied load intensity can be multiplied by 3.00 to reach the critical load level. The corresponding critical moment will have the value of Mcr = 3.0 * 47.18 = 141.54 kNm yielding a slenderness of 1.286 (Mpl,Rd = 234.20 kNm) and a lateral-torsional buckling resistance of 0.394 * 234.20 = 92.27 kNm. With this value the actual utilization ratio is at 51%.

How would this value change if a rectangular opening needs to be cut into the web of this beam?

## Analytical formula for critical bending moment

By looking to the analytical formula (ENV 1993-1-1 F.4) to calculate the critical moment of double symmetric sections loaded at eccentric load application point it becomes obvious that the section properties having effect on the moment value are Iz, Iw and It.

An opening in the web has no effect on the first two values and has very little effect on the last one. As it has been already shown in previous article, the presence of such an opening can have effect on the vertical deflection, but as long as the lateral stiffness of a beam is much lower than it’s strong axis stiffness, the vertical deflections can be neglected when the lateral-torsional buckling resistance is calculated. The usual linear buckling analysis (LBA) performed also by Consteel neglects the pre-buckling deformations.

Therefore one can expect that in general web openings can be disregarded when the critical moment value is calculated.

## Analysis with Consteel Superbeam

Beam finite elements cannot natively consider the presence of web openings. In order to obtain the precise analysis result, it is possible to use shell finite elements. The new Superbeam functionality comes as a solution in such cases. Instead of using beam finite elements, let’s use shell elements!

Opening can be positioned easily along the web, either as an individual opening or as a group of openings placed equidistantly. The opening can be rectangular, circular or even hexagonal. Circular openings can be completed with an additional circular ring stiffener.

The rectangular opening for this example can be easily defined with this tool. As there is no need to provide any additional opening on the remaining part of the beam, only the first part which includes the opening will be 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.

As the result of the new analysis using Consteel Superbeam with a mixed model from shell and beam elements, the obtained value of the first critical multiplier is 2.99 which is virtually identical to the value obtained with the beam with solid web.

This confirms the feeling that in general such a web opening may not significantly impact the lateral-torsional buckling resistance of a simple supported beam.

But this can change when for example the same beam would have fixed end conditions. In this case the region with the opening is close to the position of change of sign of bending moment, plus the weakened bottom T shape is subject to elevated compression combined with an unstiffened edge of the opening which may result in a distortion of the cross-section which causes a stiffness reduction and therefore lower critical load level.

In cases where such section distortion happens, another violation of the basic assumptions of Bernoulli-Navier Hypothesis used also by the 7DOF beam finite element happens.

In order to get the most precise analysis results, the use of shell elements is recommended at locations where the assumptions of a beam finite element are significantly violated. Consteel Superbeam provides to the designer a very efficient tool to analyse such critical parts locally with shell elements and continue to use the well established 7DOF beam elements elsewhere. This provides an optimum compromise between analysis result precision and size of finite elements model and solution time.

## Modeling of tapered elements

Stability calculation of tapered members is always a difficult problem despite its popularity in steel hall construction.

Generally in analysis software for the stability analysis a segmented but uniform beam element method is used where a member with I or H cross section and with variable web depth is divided into n segments and the depth of each segment is taken equal to the real depth measured at the middle of the segment. The lengths of the segments were taken equal, except at both ends where additional shorter segments are added in order the better approximate the real depth of the elements to be modeled. Such model captures correctly the in-plane displacements, but cannot consider accurately the additional torsion coming from the axial stresses due to warping in the flanges which are not parallel with the reference line in case of tapered elements.

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In everyday practice frames of pre-engineered metal buildings are often designed as 2D structures. Industrial buildings often have partial mezzanine floors, attached to one of the main columns, to suit the technology. Additionally, such buildings often have above the roof platforms for machineries.

When it comes to seismic design, as long as seismicity is not deemed to be a strongly controlling factor for final design, the mezzanines are just attached to the same type of frames as used at other non-seismic locations and are locally strengthened, if necessary. Only the horizontal component of the seismic effect is considered in most of the cases.

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It is known that the lateral torsional buckling resistance of a simple supported beam can be significantly increased if its compressed flange is connected adequately to a trapezoidal decking.

Eurocode allows to consider the compressed flange as fully restrained laterally, if the following condition is met (EN 1993-1-1 BB2.):

If the condition is not met, Eurocode doesn’t provide too much guidance how the contribution of a less rigid sheeting can be considered.

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