Design of Tapered Single-Span Beam According to Eurocode 3

Technical Article

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In the following, a single-span beam subjected to bending and compression will be designed by means of the RF-/STEEL EC3 add-on module according to EN 1993-1-1. Since the beam is designed as a tapered cross-section and does not represent a uniform structural component, the design must be performed either according to the General Method according to clause 6.3.4 of EN 1993-1-1 or the second-order analysis. Both options will be analyzed and compared, whereby an additional design by partial internal forces is available for the calculation according to the second-order analysis. From this, the design is divided into three steps:

  1. Design according to clause 6.3.4 of EN 1993-1-1 (General Method)
  2. Design According to Second-Order Analysis, Elastic (Warping Torsion Analysis)
  3. Design According to Second-Order Analysis, Plastic (Warping Torsion Analysis and Partial Internal Forces Method)

1 - System and Loading

The welded I-section, steel grade S235, has the following dimensions in mm:
Height = 500/300
Width = 200
Web thickness = 14
Flange thickness = 14
Weld thickness = 4

Figure 01 - 1 - System and Loading

Design according to General Method 6.3.4 EN 1993-1-1

The beam design is performed in RF-/STEEL EC3 as a set of members. Since sets of members in RF-/STEEL EC3 are designed according to the General Method by default, no further settings are necessary. In window "1.7 Nodal Supports" and the associated window, the boundary conditions of the set of members can be checked very well. The orientation of the local coordinate system is also of interest for the check. To do this, activate the local axis system by clicking the button of the same name under the partial window. As is clear from the boundary conditions of the nodal supports, the degrees of freedom that characterize the failure from the frame plane are included in the design of the General Method. In this example, the nodal supports are defined as fork supports. The preset supports already correspond to this support type and thus the calculation can be started directly.

Figure 02 - 2 - Entering Nodal Supports

The design according to the General Method is fulfilled and specified with 0.97. The critical factor α cr, op is 1,647.

Figure 03 - 3 - Result Table

The failure mode can be checked in a separate window, which can be opened by clicking the Mode Shapes… button to the right of the maximum design.

Figure 04 - 4 - Mode Shape

Figure 05 - 5 - Graphical Display of Results

Design according to second-order analysis with RF-/STEEL Warping Torsion

In order to compare the results of the design according to the General Method with the design according to the second-order analysis, the design case is first duplicated via "File" -> "Copy Case". The new design case can now be adjusted for the inputs for the design according to the second-order analysis. The design according to the second-order analysis taking into account the warping is carried out as a equivalent stress design and can be activated via "Details" -> "Warping Torsion".

This design method is only available for sets of members. Analogous to the first design case, the nodal supports have to be checked and adjusted. As can be seen in the input table of nodal supports, the add-on module extension RF-/STEEL Warping Torsion not only considers four degrees of freedom, but seven. In this example, it is important that the end of the set of members is supported in X-direction, otherwise the axial force will not be introduced into the structural component.

Figure 06 - Entering Nodal Supports

For the following design, it is not only the input of nodal supports that is important, but also the specification of an imperfection. This can be found, for example, in the National Annex to EN 1993-1-1. Table NA.2 provides the corresponding information for this example: For a welded I-section with h/b> 2 and an elastic cross-section ratio, e 0/l = 1/300 is to be applied. This value may have to be doubled if the related slenderness ratio is in a range from 0.7 to 1.3. In the first design case, the related slenderness ratio can be read by the value λ transversely, op, according to the General Method. In this example, the value 1/300 is applied for the precamber. Subsequently, the design can be carried out.

Figure 07 - Definition of Imperfection

The design is fulfilled and is specified with 0.90. The critical branch value is 1.651.

Figure 08 - Result Table

Figure 09 - Graphical Display of Results

Design according to the second-order analysis with RF-/STEEL Warping Torsion and RF-/STEEL Plasticity

To achieve a more economical design, the RF-/STEEL Plasticity extension is available for RF-/STEEL EC3. This allows you to design the internal forces according to the second-order analysis from the warping torsion analysis for the stability analysis of a set of members, using the partial internal force method according to Kindmann, or for general cross-sections using the simplex method.

After copying the second design case, you can activate the plastic design with "Details" -> "Plasticity". By copying the second design case, the correct nodal supports have already been transferred, but the imperfection must be checked and adjusted. Table NA.2 shows a value of 1/200 for the plastic design of the welded I-section with h/b> 2.

Figure 10 - Definition of Imperfection

The design can now be performed and is fulfilled.

Figure 11 - Result Table

Figure 12 - Graphical Display of Results

Summary

RF-/STEEL EC3 provides basically two design methods for tapered components. In addition to the integrated General Method according to clause 6.3.4, it is also possible to perform a second-order analysis including warping by using the RF-/STEEL Warping Torsion add-on module. However, the warping torsion analysis can also be used for other cross-sections and load states.

To enable an even more economical design, it is possible to perform a plastic design according to the partial internal force method or the simplex method, using the RF-/STEEL Plasticity add-on module, based on the warping analysis.

Reference

[1]   DIN EN 1993-1-1: 2010-12 with NA: 2015-08

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Design of steel members according to Eurocode 3

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