Loading According to EN 1991-1-4 and Safety Against Overturning of Circular Cylinders

Technical Article

This article describes the determination of force coefficients by using a wind load and the calculation of a stability factor due to overturning.

  • Factor of safety against overturning < 1: The structural component is at risk of overturning.
  • Factor of safety against overturning = 1: The stability moment and the overturning moment are equal. The model is unstable and it cannot be ruled out that it is overturning.
  • Factor of safety against overturning > 1: The model is not at risk of overturning.

Example

As an example, there is a circular cylinder with a diameter of 2.5 m and a height of 6 m. It is located in the wind load zone 2 with the terrain category 3.

Figure 01 - Wind Load

Fundamental value of the basic velocity:

$${\mathrm v}_{\mathrm b0}\;=\;25.0\;\mathrm m/\mathrm s$$

Directional factor:

$${\mathrm c}_\mathrm{dir}\;=\;1$$

Season factor:

$${\mathrm c}_\mathrm{season}\;=\;1$$

Air density at the atmospheric pressure of 1,013 hPa and T = 10° C:

$$\mathrm\rho\;=\;1.25\;\mathrm{kg}/\mathrm m^3$$

Kinematic viscosity of the air:

$$\mathrm v\;=\;15\;\cdot\;10^{-6}$$

Basic velocity:

$${\mathrm v}_\mathrm b\;=\;{\mathrm c}_\mathrm{dir}\;\cdot\;{\mathrm c}_\mathrm{season}\;\cdot\;{\mathrm v}_{\mathrm b0}\;=\;25.0\;\mathrm m/\mathrm s$$

Basic velocity pressure:

$${\mathrm q}_\mathrm b\;=\;1/2\;\cdot\;\mathrm\rho\;\cdot\;{\mathrm v}_\mathrm b^2\;=\;0.391\;\mathrm{kN}/\mathrm m^2$$

Peak velocity pressure:

$${\mathrm q}_\mathrm p\;=\;1.5\;\cdot\;{\mathrm q}_\mathrm b\;=\;0.586\;\mathrm{kN}/\mathrm m^2$$

Peak velocity:

$${\mathrm v}_\mathrm{ze}\;=\;\sqrt{\frac{2\;\cdot\;{\mathrm q}_\mathrm p}{\mathrm\rho}}\;=\;30.619\;\mathrm m/\mathrm s$$

Equivalent surface roughness:

$$\mathrm k\;=\;0.2\;\mathrm{mm}\;(\mathrm{galvanized}\;\mathrm{steel})$$

Ratio of equivalent surface roughness and width:

$$\frac{\mathrm k}{\mathrm b}\;=\;8\;\cdot\;10^{-5}$$

Reynolds number:

$${\mathrm R}_\mathrm e\;=\;\frac{\mathrm b\;\cdot\;{\mathrm v}_\mathrm{ze}}{\mathrm v}\;=\;5.1\;\cdot\;10^6$$

Force coefficient of cylinders without free-end flow:

$${\mathrm c}_{\mathrm f0}\;=\;1.2\;\cdot\;\frac{0.18\;\cdot\;\log\;({\displaystyle\frac{1\;\cdot\;\mathrm k}{\mathrm b}})}{1\;+\;0.4\;\cdot\;\log\;({\displaystyle\frac{{\mathrm R}_\mathrm e}{10^6}})}\;=\;0.7666$$

Effective slenderness:

$$\mathrm\lambda\;=\;\frac{\mathrm l}{\mathrm b}\;=\;3.2$$

End-effect factor:

$${\mathrm\psi}_\mathrm\lambda\;=\;0.65$$

Structural factor:

$${\mathrm c}_\mathrm s{\mathrm c}_\mathrm d\;=\;1$$

Reference area:

$${\mathrm A}_\mathrm{ref}\;=\;\mathrm l\;\cdot\;\mathrm b\;=\;20\;\mathrm m^2$$

Force coefficient:

$${\mathrm c}_\mathrm f\;=\;{\mathrm c}_{\mathrm f0}\;\cdot\;{\mathrm\psi}_\mathrm\lambda\;=\;0.498$$

Wind force:

$${\mathrm F}_\mathrm w\;=\;{\mathrm c}_\mathrm s{\mathrm c}_\mathrm d\;\cdot\;{\mathrm c}_\mathrm f\;\cdot\;{\mathrm q}_\mathrm p\;\cdot\;{\mathrm A}_\mathrm{ref}\;=\;5.835\;\mathrm{kN}$$

Surface load due to the wind:

$${\mathrm F}_\mathrm w\;=\;\frac{{\mathrm F}_\mathrm w}{{\mathrm A}_\mathrm{ref}}\;=\;0.292\;\mathrm{kN}/\mathrm m^2$$
Stability Factor due to Overturning

Height of the circular cylinder:

$$\mathrm h\;=\;6\;\mathrm m$$

Distance between supports:

$$\mathrm a\;=\;1.35\;\mathrm m$$

Dead load:

$${\mathrm F}_\mathrm G\;=\;18.495\;\mathrm{kN}$$

Overturning moment:

$${\mathrm M}_\mathrm K\;=\;{\mathrm F}_\mathrm w\;\cdot\;\frac{\mathrm h}2\;=\;13.128\;\mathrm{kNm}$$

Stability moment:

$${\mathrm M}_\mathrm S\;=\;{\mathrm F}_\mathrm G\;\cdot\;\frac{\mathrm a}2\;=\;12.484\;\mathrm{kNm}$$

Factor of safety against overturning:

$$\mathrm\eta\;=\;\frac{{\mathrm M}_\mathrm S}{{\mathrm M}_\mathrm K}\;=\;0.951$$

If using RFEM for the calculation, you can recognize from the position of the resultants that they are within its extension behind the overturning edge of the circular cylinder. Thus, the model would be unstable if the supports were not additionally secured against pulling out.

Figure 02 - Resultant Position

Reference

[1]   Eurocode 1: Actions on structures - Part 1‑4: General actions - Wind actions; EN 1991‑1‑4:2005 + A1:2010 + AC:2010
[2]   National Annex - Nationally determined parameters - Eurocode 1: Actions on structures - Part 1‑4: General actions - Wind actions.

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