Design of Tapered Single-Span Beam According to Eurocode 3
The following article describes the design of a single‑span beam subjected to bending and compression, which is performed according to EN 1993‑1‑1 in the RF‑/STEEL EC3 add‑on module. Since the beam is modelled with a tapered cross‑section, thus not a uniform structural component, the design must be performed either according to General Method in compliance with Chap. 6.3.4 of EN 1993‑1‑1, or according to the second‑order analysis.
Both options will be explained and compared, and for the calculation according to the second‑order analysis, there is an additional design format using Partial Internal Forces Method (PIFM) available. Therefore, the design is divided into three steps:
- Design according to Chap. 6.3.4 of EN 1993‑1‑1 (General Method)
- Design according to the second‑order analysis, elastic (warping torsion analysis)
- Design according to the second‑order analysis, plastic (warping torsion analysis and Partial Internal Forces Method)
System and Loading
A welded I-section of the S235 steel grade has the following dimensions in [mm]:
Design According to General Method 6.3.4 EN 1993‑1‑1
The beam design is performed as a set of member design in RF‑/STEEL EC3. Since sets of members are designed in RF‑/STEEL EC3 according to General Method by default, no further settings are required. In Window ‘1.7 Nodal Support’ and the corresponding partial view, you can easily check the boundary conditions of the set of members.
You can also check the orientation of the local coordinate system there. The local axis system can be activated by clicking the corresponding button under the partial view graphic. As it is clear in the boundary conditions of nodal supports, there are degrees of freedom in the design according to General Method, which characterize the frame plane failure. The nodal supports are to be defined as lateral and torsional restraints in this example. The preset supports already correspond to this support type so the calculation can be started directly.
The design according to General Method is fulfilled and gives a result of 0.97. The critical buckling value αcr,op is 1.647.
You can check the failure mode in a separate partial view window, which can be opened by clicking the [Mode Shapes] button on the right of the maximum design ratio.
Design According to Second-Order Analysis with RF-/STEEL Warping Torsion
In order to compare the results of the design according to General Method with the design according to the second‑order analysis, the design case is duplicated by clicking ‘File’ → ‘Copy Case’. Now, the new design case can be adjusted for the design according to the second‑order analysis. The design according to the second‑order analysis, taking into account the warping, is performed as equivalent stress design and can be selected in ‘Details’ → ‘Warping Torsion’.
This design method is available for sets of members only. As in the first design case, the nodal supports have to be checked and adjusted. As you can see in the window for entering nodal supports, the module extension RF‑/STEEL Warping Torsion does not only consider four degrees of freedom but seven of them. In our example, it is important to provide member ends in the X‑direction with free supports, otherwise the axial force is not applied to the component.
For the following design, it is not only important to enter the nodal supports, but also to specify the imperfection in particular. You can find this in the National Annex to EN 1993‑1‑1, for example. Table NA.2 provides the relevant data for our example: e0 / 1 = 1 / 300 applies for a welded I‑section with h/b > 2. This value has to be doubled if the slenderness ratio is in a range from 0.7 to 1.3. The slenderness ratio can be set by the value λcr,op in the first design case according to General Method. In our example, the value of 1/300 is set for the precamber. Finally, the design can be carried out.
The design is fulfilled and gives a result of 0.90. The critical buckling value is 1.651.
Design According to Second‑Order Analysis with RF‑/STEEL Warping Torsion and RF-/STEEL Plasticity
For more efficient design, there is the RF‑/STEEL Plasticity extension of the RF‑/STEEL EC3 add‑on module available. It allows you to analyse internal forces according to the second‑order analysis from the warping torsion analysis with Partial Internal Forces Method according to Kindmann for stability analysis of a set of members, or with Simplex Method for general cross‑sections.
After copying the second design case, the plastic design can be activated in ‘Details’ → ‘Plasticity’. By copying the second design case, the correct nodal supports have already been adopted. However, it is necessary to check and adjust the imperfection. Table NA.2 indicates a value of 1/200 for the plastic design of welded I‑section with h/b > 2.
Now, the design can be performed and is fulfilled.
For tapered structural components, there are two design methods available in RF‑/STEEL EC3. In addition to the integrated General Method according to Chap. 6.3.4 of EN 1993‑1‑1, you can perform the design according to the second‑order analysis, including consideration of warping in the RF‑/STEEL Warping Torsion module extension. Moreover, the warping torsion analysis can also apply to other cross‑sections and load situations.
For more efficient design, you can perform plastic design according to Partial Internal Forces Method (PIFM) or according to Simplex Method in the RF‑/STEEL Plasticity module extension in addition to the warping torsion analysis.
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Figure 01 - System and Loading
Figure 02 - Entering Nodal Supports
Figure 03 - Result Table
Figure 04 - Mode Shape
Figure 05 - Graphical Display of Results
Figure 06 - Entering Nodal Supports
Figure 07 - Definition of Imperfection
Figure 08 - Result Table
Figure 09 - Graphical Display of Results
Figure 10 - Definition of Imperfection
Figure 11 - Result Table
Figure 12 - Graphical Display of Results