# Design of Tapered Column According to EN 1993-1-1

### Technical Article

The following structure is covered as Example IV.10 in [1] “Comment on Eurocode 3.” For a support with linearly varying cross‑section, the sufficient ultimate limit state design (cross‑section check and stability analysis) is to be performed. Due to the unequal structural component, it is necessary to perform the stability analysis (from the main support direction) using the method according to Section 6.3.4, or alternatively, according to the second‑order analysis.

#### System

On the tension side of the cross‑section, there is a continuous support in the Y‑direction (lateral rotation axis).

#### Loads

Design loads:

#### Cross-Section Classification

With the existing design loads, the cross‑section does not reach the ultimate limit state curve. Therefore, the internal forces must be increased up to the ultimate state. For this, there are two options:

- Linearly increase all internal forces until the ultimate state is reached (see Figure 02 left, the second option (default) in Details)
- Increase only M
_{Ed}to reach the ultimate state (see Figure 02 right, the first option in Details)

Figure 02 - Classification of Cross‑Sections

Both options and methods lead to very different results, from a maximum elastic design in the top third to a completely possible plastic design ration of the cross‑section over the entire column height.

In the present stability failure, no increment of the axial force arises, but only an increment of the moments due to deformations and the second‑order analysis. Therefore, the second option is selected.

#### Minimum Amplifier α_{ult,k}

In this case, the cross‑section design ration is determined by using the linear plastic interaction (see [2] Eq. 6.2). This must be activated in Details, since RF‑/STEEL EC3 performs the design for the cross‑sections of Class 1 or 2 according to Eq. 6.31 or 6.41 of [2] by default.

In compliance with Section 6.3.4. (2), it may be necessary to calculate the minimum load amplifier α_{ult,k} to reach the characteristic resistance in the main plane with all effects of imperfections and the second‑order analysis.

The check, as far as deformations affect the internal forces, is determined according to Equation 5.1 in [2]:

$${\mathrm\alpha}_{\mathrm{cr}\;=\;}\frac{{\mathrm N}_\mathrm{cr}}{{\mathrm N}_\mathrm{Ed}}\;>\;10$$In this case, α_{cr} should be determined by RF‑/STEEL EC3 and RF‑/STEEL Warping Torsion. The best way is to generate a separate module case and define intermediate lateral restraints for the set of members in order to enforce the first mode shape with “buckling in the major axis direction.”

The cross‑section design ratio and thus the minimum load amplifier α_{ult,k} can be calculated with the internal forces according to the linear static analysis. The following ratios and factors then arise along the member length.

Figure 04 - Minimum Amplifiers and Design Ratios

#### Slenderness of Structural Component and Reduction Factor χ_{op}

Determination of the reduction factor χ_{op} requires the slenderness ratio λ_{op} to take into account flexural buckling or lateral-torsional buckling. This is calculated according to Equation 6.64 of [2]:

where

During the design according to 6.3.4, the RF‑/STEEL EC3 solver determines the minimum load amplifier to reach the elastic critical load of the structural component with regards to lateral or lateral torsional buckling. Properties of the underlying structural system are specified in Window 1.4 and 1.7 as follows.

Figure 05 - Properties of Structural System

Based on the reference literature, elastic warping restraints were waived, although they would be justified due to the base plate and also the present restraint on the column head. The calculation result is:

Thus, it is possible to determine the slenderness of the structural component according to [1], Section 6.3.4:

$${\mathrm\lambda}_\mathrm{op}\;=\;\sqrt{\frac{2.097}{3.23}}\;=\;0.805$$The buckling curve can be selected in compliance with National Annex (NDP to 6.3.4 (1)) according to Table NA.4:

In the case of combined effects, the following minimum load amplifier should be used:

#### Component Design

The actual design is now performed according to [2], 6.3.4 (2), Equation 6.63:

$$\frac{{\mathrm\chi}_\mathrm{op}\;\cdot\;{\mathrm\alpha}_{\mathrm{ult},\mathrm k}}{{\mathrm\gamma}_{\mathrm M,1}}\;>\;1.0$$Adjustment of the equation in terms of the design ratio:

$$\frac{{\mathrm\gamma}_{\mathrm M,1}}{{\mathrm\chi}_\mathrm{op}\;\cdot\;{\mathrm\alpha}_{\mathrm{ult},\mathrm k}}\;<\;1.0$$ $$\frac{1.1}{0.659\;\cdot\;2.097}\;\;=\;0.80\;<\;1.0$$#### Reference

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