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.

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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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 τ_t occurs.

The non-uniform torsional component

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

• Web stiffeners
• T – stiffener
• L – stiffener
• Box stiffener
• Channel –stiffener

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

where,

R_ω = the stiffness of the discrete warping restraint

G = shear modulus

GI_t = 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]

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]

L-stiffener

where

b = width of the L-section [mm]

t = thickness of the L-section [mm]

h = height of the L-section [mm]

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]

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.

Name	Dimension	Value
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

Name	Dimension	Value
Elastic modulus	[N/mm2]	200
Poisson ratio	[-]	10
Yield strength	[N/mm2]	300

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.

Name	Dimension	Value
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

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.

Discrete warping stiffness calculated by hand

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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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