## Introduction

During the lifetime of a steel structure changes often happen. These changes usually result an increase of loads acting on some of its elements which therefore may need to be strengthened.

Strengthening is usually done by welding additional steel plates to the existing members. In the case of I sections, usually, the flanges are reinforced to increase the bending moment capacity or the web is stiffened to avoid local buckling or crippling at support regions.

This paper will focus on the increase of bending moment capacity.

## Lateral-torsional buckling resistance

The usual practice is to either increase the compression flange thickness by adding additional plates to it, or by widening it with the help of angles, as can be seen in the pictures below.

Although these can be very efficient ways to increase the bending moment capacity of a beam, welding on site is a complex process and might require the temporary removal of structural or non-structural elements connected to the flange of the beam. Welding especially “above the head” is difficult, the quality of weld seam needs to be properly checked.

Bending moment capacity of a beam might be limited by lateral-torsional buckling. If the section is not sufficiently restrained laterally against torsion, its actual load-bearing capacity will be lower than the value which depends purely on its section resistance.

In such cases, if the LTB behaviour could be directly improved, there would be no need to strengthen its cross-section along its full length. Here comes the **Superbeam** as a possible help.

Additional lateral restraining elements are often difficult to be added, therefore this is often not an option.

If we look at what LTB resistance of an I section depends on, we can see, that if we don’t want to change its cross section along its full length, it depends on the value of the reduction factor responsible to consider lateral-torsional buckling χ_{LT}.

This reduction factor is calculated from the slenderness value of the beam, which needs to be improved (reduced) to result a lower, more favourable reduction factor.

Without changing the cross section, the only way to do this is by improving the critical moment value. Increasing this value can be made not only by changing the cross-section but also by changing the boundary conditions.

The value of parameters ‘k’ and ‘kw’ depend on the boundary conditions, where ‘k’ means a factor which depends on how the section is fixed against weak axis bending at its ends and ‘kw’ means a factor which depends on how the section is fixed against warping. Warping is the phenomenon when the upper and lower flange of an I section twist in opposite directions.

To change the end conditions is typically difficult, but a certain limitation of the twist of flanges relative to each other ie. preventing or limiting warping might be possible. Limitation of this twist can be obtained by connecting the flanges by an additional element which has non-zero torsional stiffness. This torsional stiffness will prevent the counter-rotation of the flanges and therefore the warping and allowing to consider a ‘kw’ value different than 1.0 in this formula.

Consteel supports several such strengthening profiles and can determine the torsional stiffness to be considered in preventing or limiting warping.

## Analysis with Consteel Superbeam

Let’s take the following case. We have a simple supported 5 m long beam loaded by a uniform load of 20 kN/m acting at the top flange, on top of its self weight, without any intermediate lateral support. Its section is a welded I profile, made of S235, 10 mm thick plates, flange width of 200 mm and total section depth of 320 mm.

As we can expect, in the case of such a large unbraced length, the bending moment resistance would be strongly limited by lateral-torsional buckling, and therefore we can expect that strengthening by the proposed method is viable.

The critical moment of this beam is obtained in Consteel using linear buckling analysis with 7DOF beam elements option of the Superbeam, which has found the critical multiplier of 2.88.

This results M_{cr} = 2.88*64.18=184,84 kNm and a slenderness λ of 1,036 and reduction factor of 0,519.

The final bending moment resistance is 103 kNm.

Let’s further assume that this resistance needs to be increased by 30% due to new requirements. Let’s see whether a successful strengthening without modifications of the cross-section would be possible.

Let’s insert small vertical hot-rolled UPE 200 profiles at both side the web, connecting the flanges close to the extremities of the beam, without touching the components of end connection (potential stiffeners, bolts, etc) where welding might be difficult.

The addition of these U profiles will be converted automatically into an elastic warping stiffness with the value of 1003,24 kNm2/(rad/m)) which will result an elevated M_{cr} as follows

This results M_{cr} = 6.91*64.18=443,48 kNm and a slenderness λ of 0,669 and reduction factor of 0,744.

The final bending moment resistance is 147,5 kNm. This is an increase of 43% which is perfectly enough in our case.

The correctness of the analysis can be directly verified using the alternative, shell element based analysis mode of the Superbeam.

The result is almost identical (6.79 vs 6.91, difference less than 2%) to the value obtained with the 7DOF beam element based analysis, which confirms its correctness.

## Conclusion

Consteel Superbeam gives interesting new opportunities for the designer, which includes also a cost-efficient strengthening option of existing structures.

More details about the background of the calculations you can find in our article about Discrete warping restraint.

Download model file## 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.

**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.

Download model fileThe 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 M_{cr} = 3.0 * 47.18 = 141.54 kNm yielding a slenderness of 1.286 (M_{pl,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 I_{z}, I_{w} and I_{t}.

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.

Download model fileThis 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.

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 openings in the web plate. Such an opening can have a circular or rectangular shape, depending on the amount, size and shape of pipes or ventilation or cable trays.

If the structural engineer has the freedom to position these openings along the beam, where to place them? What would be its effect on the deflection of the beam?

The effect of such openings on the deflection is more important when the length of the opening along the beam is increased. As circular openings are made with equal length and depth, they are usually less critical than rectangular openings.

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 deflection at mid-span of the solid web beam is 4.6 mm.

Download model fileLet’s assume that a 250 mm deep rectangular opening with a length of 400 mm needs to be provided on the web, at a distance of 300 mm from the left support.

**Traditional analysis with beam finite elements**

Consteel 7DOF beam finite elements are very powerful, but cannot consider natively such opening. The usual approach is to build a Vierendeel-type of model, by using additional beam elements with a T shape section „above” and „below” the opening. These additional beam elements are defined eccentrically to the reference line of the solid-web beam.

Eccentricities can be easily defined in Consteel using both smart and traditional link elements.

The deflection with this refined model will be equal to 4.8 mm.

Download model file**Analysis with Consteel Superbeam**

In order to find a more 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 of 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 precise deflection of 5.2 mm can be obtained.

Download model fileThis deflection is higher than the value obtained with the solution using the Vierendeel-type of structure using beam finite elements.

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.

In our series we have shown in Part 1 and Part 2 how the spring stiffness „K” is determined for and edge and for an intermediate stiffener. In this part we will show how to proceed further.

The goal is to determine how efficiently can this stiffener support the connected compressed plate. To consider local buckling of the compressed plate the effective widths will be calculated. These widths can be either calculated for a plate supported at both ends or for a plate supported at one end only, using Table 4.1 and Table 4.2, respectively.

gateDistortional buckling – Further secret formulas of EN 1993-1-3. Check our article for the first secret formula if you hadn’t read it yet.

In our series we continue with the second „secret” formula of EN 1993-1-3.

This formula (5.11) is used when the ability of an intermediate stiffener to stabilize a compressed web plate is studied.

gate## Part 2 – Imperfection factors

The Eurocode EN 1993-1-1 offers basically two methods for the buckling verification of members:

(1) based on buckling reduction factors (buckling curves) and

(2) based on equivalent geometrical imperfections.

gate## 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.

EN 1993-1-3 contains 3 „secret” formulas. The first two are used to determine the effective cross section due to distortional buckling when edge or intermediate stiffeners are involved. The third is used to calculate the distortion of the whole cross section when analyzed with a connected sheeting.

gateIn 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.

gate