**Introduction**

This verification example studies a simple fork supported beam member with welded section (flanges: 200-12 and 100-12; web: 400-8) subjected to bending about major axis. Constant bending moment due to concentrated end moments and triangular moment distribution from concentrated transverse force is examined for both orientations of the I-section. 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.

(more…)**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.

(more…)## 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.

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:

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.

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

**type. Watch the feature preview below and learn how to use the Coloring to make your model more perscpicuous.**

*colours from their cross-section***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

**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.

## 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

**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]).**

*Stiffener*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

b_{1} = width of the battens [mm]

t_{1} = thickness of the battens [mm]

b_{2} = width of the web stiffener [mm]

t_{2} = 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

b_{1} = width of channel web [mm]

t_{1} = thickness of channel web [mm]

b_{2} = width of channel flange [mm]

t_{2} = 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 |

*Table 1:**geometric parameters*

Name | Dimension | Value |
---|---|---|

Elastic modulus | [N/mm^{2}] | 200 |

Poisson ratio | [-] | 10 |

Yield strength | [N/mm^{2}] | 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.

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 |

*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.

#### 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

### First eigenshape – shell model

## 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

Comparison of chosen methods for estimation of critical lateral torsional buckling bending moment of web-tapered I-beams. In this article, the elastic critical bending moments of the web-tapered I-beams calculated by the analytical and numerical solutions developed last years by researchers involved in the topic were compared with own calculations carried out with available common tools. The main goal was to verify the accuracy and convergence of the results provided by different modern methods and different finite bar elements 1D with 7 degrees od freedom at the node (7DOF).

Click the button bellow to download and read the full article. (PL)

Download brochureD. Czepiżak, A. Machowiak: Comparison of chosen methods for estimation of critical lateral torsional buckling bending moment of web-tapered I-beams. Inżynieria i Budownictwo Nr 5–6/2021

Consteel 14 is a powerful analysis and design software for structural engineers. Watch our video how to get started with Consteel.

**Contents**

- Set analysis parameters
- Perform first and second order analysis
- Perform buckling analysis
- Analysis results in graphics and in tables
- Results: deformation, internal forces, reactions