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Introduction

As it is important to have a clear overview of the structural model, the visualization of the analysis results is also essential when it comes to effective design process. From Consteel 15 we use an advanced method for deformation representation which makes it smooth and realistic.

Description

Civil engineering software in general use the traditional beam-type deformation representation where the section is shown on the deformation of the reference line. There are some consequences of this representation mode that can be disturbing for the users. The best example is an eccentric support, where the deformed shape is visualized as if the supported point would’ve moved. The reference line indeed moved but the supported point not – the representation can not show that.

Traditional deformation representation at eccentric support

With Consteel’s advanced deformation representation not only the position of the reference line points are calculated and the section is only shown automatically, but the positions of all the decorated points of the section are calculated during a post-process and so it is possible to represent the real deformations. As a consequence it is also visible that the supported points stay in position.

Consteel 15 advanced deformation representation at eccentric support

Coloring has also improved with this representation. With the traditional way, the section was colored based on only the deformations of the reference line, so the same color applied for the whole cross-section. With the advanced technique, colors are assigned to the decorated points, and so coloring can change within a cross-section.

It is important to know that the analysis results are the same as before, only the representation of the deformations are now more realistic.

Other issues connecting to visualization which were often raised by our customers are also fixed with this advanced representation method, e.g. at frame corners the connecting flanges are moving together as in reality. Warping at member ends can also be well inspected.

To see examples of what Consteel’s advanced deformation representation can give you, check out our feature preview video:

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 χLT.

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 Mcr = 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 Mcr as follows

This results Mcr = 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.

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Perfect the understanding of your structure with advanced buckling sensitivity results illustrated on proper mode shape and colored internal force diagrams.

Civil engineering software in general use the traditional beam-type deformation representation where the section is shown on the deformation of the reference line. In Consteel 15 we use an advanced method for deformation representation which makes it smooth and realistic. The analysis results are the same, but with the improved visualisation the real 3D behavior of the structure can be better seen.

Have you ever tried to visualize the stress distribution of a cross-section from the colored representation? To make it easier for you, we are now introducing Stress diagrams. Watch the video below to learn how to use this feature.

Good model and result visualization leads to better understanding and correct interpretation of any data model compared to texts or tables. With the help of Coloring by section feature, you will be able to switch to a new model view where the members get colours from their cross-section type. Watch the feature preview below and learn how to use the Coloring to make your model more perscpicuous.

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:

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

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

Steel grade S235

Loading

Two loading scenarios were considered on every section

Uniform moment:

Uniform moment

Linear moment:

Linear moment

Result tables

First global buckling eigenvalue – αcrit

Consteel version CS15.1095

Unifrom member – uniform moment
Uniform member – linear moment
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Uniform member with unequal flanges – uniform moment
Uniform member with unequal flanges – linear moment
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Tapered member – uniform moment
Tapered member – linear moment
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Haunched member – uniform moment
Haunched member – linear moment
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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 αcrit 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 σcr 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 λp 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 fy = 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 σcr = 7.72*82.52=637.0 N/mm2 resulting a slenderness of 0.606 with fy = 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.

Theoretical background

According to the beam-column theory, two types of torsional effects exist.

Saint-Venant torsional component

Some closed thin-walled cross-sections produce only uniform St. Venant torsion if subjected to torsion. For these, only shear stress τoccurs.

Fig. 1: rotated section [1.]

The non-uniform torsional component

Open cross-sections might produce also normal stresses as a result of torsion.[1.]

Fig. 2: effect of the warping in a thin-walled open section [1.]

Warping causes in-plane bending moments in the flanges. From the bending moment arise both shear and normal stresses as it can be seen in Fig. 2 above.

Discrete warping restraint

The load-bearing capacity of a thin-walled open section against lateral-torsional buckling can be increased by improving the section’s warping stiffness. This can be done by adding additional stiffeners to the section at the right locations, which will reduce the relative rotation between the flanges due to the torsional stiffness of this stiffener. In Consteel, such stiffener can be added to a Superbeam using the special Stiffener tool. Consteel will automatically create a warping support in the position of the stiffener, the stiffness of which is calculated using the formulas below. Of course, warping support can also be defined manually by specifying the correct stiffness value, calculated with the same formulas (see literature [3]).

The following types of stiffeners can be used:

The general formula which can be used to determine the stiffness of the discrete warping restraint is the following:

where,

Rω = the stiffness of the discrete warping restraint

G = shear modulus

GIt = the Saint-Venan torsional constant

h = height of the stiffener

Effect of the different stiffener types

Web stiffener

where

b = width of the web stiffener [mm]

t = thickness of the web stiffener [mm]

h = height of the web stiffener [mm]

Fig. 3: web stiffener

T – stiffener

where

b1 = width of the battens [mm]

t1 = thickness of the battens [mm]

b2 = width of the web stiffener [mm]

t2 = thickness of the web stiffener [mm]

h = height of the web stiffener [mm]

Fig. 4: T–stiffener

L-stiffener

where

b = width of the L-section [mm]

t = thickness of the L-section [mm]

h = height of the L-section [mm]

Fig. 5: L–stiffener

Channel stiffener

where

b1 = width of channel web [mm]

t1 = thickness of channel web [mm]

b2 = width of channel flange [mm]

t2 = thickness of channel flange [mm]

h = height of the web stiffener [mm]

Fig. 6: Channel stiffener

Numerical example

The following example will show the increase of the lateral-torsional buckling resistance of a simple supported structural beam strengthened with a box stiffeners. The effect of such additional plates can be clearly visible when shell finite elements are used.

Shell model

Fig. 7 shows a simple fork supported structural member with welded cross-section modeled with shell finite elements and subjected to a uniform load along the member length acting at the level of the top flange.

Table 1. and Table 2. contain the geometric parameters and material properties of the double symmetric I section. The total length of the beam member is 5000 mm, the eccentricity of the line load is 150 mm in direction z.

Fig. 7: simple supported, double symmetric structural member modeled by shell elements
NameDimensionValue
Width of the top Flange[mm]200
Thickness of the top Flange [mm] 10
Web height [mm] 300
Web thickness [mm] 10
Width of the bottom Flange [mm] 200
Thickness of the bottom Flange [mm] 10

Table 1: geometric parameters
NameDimensionValue
Elastic modulus[N/mm2]200
Poisson ratio[-] 10
Yield strength [N/mm2] 300

Table 2: material properties

Box stiffener

The box stiffeners are located near the supports as can be seen in Fig. 8. Table 3. contains the geometric parameters of the box stiffeners.

Fig. 8: the structural shell member with added box stiffeners
NameDimensionValue
Width of the web stiffener[mm]100
Thickness of the battens [mm] 100
Total width of the box stiffener [mm] 200
Height of the plates [mm] 300
Thickness of the plates [mm] 10

Table 3: geometric parameters of the box stiffeners

7DOF beam model

The same effect in a model using 7DOF beam finite elements can be obtained when discrete warping spring supports are defined at the location of the box stiffeners.

Fig. 9: beam member supported with fork supports and loaded with eccentric uniform load

Discrete warping stiffness calculated by hand

The warping spring stiffness corresponding to the box stiffeners can be calculated as shown in the previous chapter [2.1.4]. The formula is the following:

Results

Lateral torsional buckling shapes can be seen in the following pictures (Fig. 10 and Fig 11.). The obtained elastic critical buckling factor with the 7DOF beam model and with the shell model with and without the box stiffeners are compared in Table 4. The results show good agreement. The lateral-torsional buckling resistances of the original and strengthened structural members are calculated using EN 1993-1-1 and shown in Table 5.

First eigenshape – beam model

Fig. 10: first eigenshape of the 7DOF beam model

First eigenshape – shell model

Fig. 11: first eigenshape of the shell model
ResultModelDimensionValue
Eigenvalue7DOF beam model[-]6.85
Eigenvalue Shell model [-] 6.76
Table 4: elastic critical load factors
ResultModelDimensionValue
Lateral torsional buckling resistance Mb,RdOriginal beam[kNm]101.07
Lateral torsional buckling resistance Mb,Rd Strengthened beam [kNm] 146.15
Table 5: lateral-torsional buckling resistances

Bibliography

[1.] Web page: https://structural-analyser.com/domains/SteelDesign/Torsion/

[2.] Web page: http://www.isa.fh-trier.de/home/Projekte/Nagolnij/index_63.html

[3.] Univ.-Prof. em. Dr.-Ing. Rolf Kindmann, Stahlbau, Teil 2: Stabilität und Theorie II. Ordnung, April 2008

[4.] Pascal Händler, The Bearing Behaviour of Warping Springs in Torsionally Loaded I – Beams, April 2016

The results of analyzes of frames with a span of 12, 15 and 18 m have been presented. Their minimum mass was assumed as the optimization criterion. The finite element method was used in the calculations. The results of calculations in the form of: structure mass, bar resistance coefficient and checking the SLS condition were presented in the tables.

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A. Wojnar, G. Wiatrowic: Analysis of steel comsumption of portal frames made of hot-rolled and cold-formed sections. Inżynieria i Budownictwo Nr 3/2021