Stability endangerment

A rod that is subjected to compression can be at risk of buckling

There is a risk of stability in the case of slender components that are loaded with a load relevant to buckling (or buckling relevant) and must be verified according to the theory of the second order . The best-known example of a threat to stability is buckling in which a rod gives way to the side under an axial compressive force. When buckling, it follows from a deflection (e.g. an imperfection ) that there is a lever arm between the rod axis and the line of action of the normal force, which in the second order theory leads to a bending moment . This bending moment leads to a one-sided load on the cross-section, which further increases the deflection or deflection. Mathematically, this leads to a branching load for a perfectly straight member:

${\displaystyle N_{\mathrm {cr} }={\frac {\pi ^{2}\,EI}{s^{2}}}}$

• ${\displaystyle EI}$: Bending stiffness
• ${\displaystyle \pi \,}$: Circle number
• ${\displaystyle s\,}$: Buckling length

With this critical load, the rod is mathematically in an indifferent equilibrium, where the deflection is arbitrary. Structures must be dimensioned so that they have a low probability of failure, and one should also ensure that failure is announced and human lives can be saved in good time. Since buckling under a pure normal compressive force of a straight bar represents a sudden loss of equilibrium and deformations are not announced in advance, the probability that a component will fail due to stability failure must be minimized. The Eurocode regulates differently for different materials whether there is a risk to stability .

steel construction

According to current standards, there is a risk of stability if the compressive force acting on a rod reaches or exceeds ten percent of the ideal buckling compressive force, the following applies:

${\displaystyle N<0,\qquad N\geq {\frac {P_{Ki}}{10}}}$

If a member is dangerous to stability, it must be calculated according to the second order member theory. Furthermore, geometric substitute imperfections (so-called "pre-deformations") must be invoiced.

Concrete construction

According to Eurocode 2 , the standard for the dimensioning and construction of reinforced concrete and prestressed concrete structures , the effects according to the second order theory can be neglected if the slenderness is below a limit value that can be found in the National Annex.

The recommended value for the limit slenderness is:

${\displaystyle \lambda ={\frac {20\cdot A\cdot B}{\sqrt {n}}}}$

With

• ${\displaystyle A={\frac {1}{1+0{,}2\varphi _{ef}}}}$ (simplified, A = 0.7 can be assumed)
  • ${\displaystyle \varphi _{ef}}$... effective creep coefficient
• ${\displaystyle B={\sqrt {1+2\omega }}}$ (simplified, B = 1.1 can be assumed)
  • ${\displaystyle \omega ={\frac {A_{y}\cdot f_{yd}}{A_{c}\cdot f_{cd}}}}$ ... mechanical reinforcement level
• ${\displaystyle C=1,7-r_{m}}$ (simplified, C = 0.7 can be assumed)
  • ${\displaystyle r_{m}={\frac {M_{1}}{M_{2}}}}$
    • ${\displaystyle M_{1}}$and the end moments according to the first order theory with ${\displaystyle M_{2}}$ ${\displaystyle |M_{2}|>M_{1}}$

Timber construction

In Eurocode 5 , flexural buckling of compression rods should be assumed to:

${\displaystyle \lambda _{rel,z}={\frac {\lambda }{\pi }}\cdot {\sqrt {\frac {f_{c{,}0{,}k}}{E_{0{,}05}}}}}$

${\displaystyle \beta _{c}=0{,}2}$ for solid wood

${\displaystyle k_{z}=0{,}5\cdot (1+\beta _{c}\cdot (\lambda _{rel,z}-0{,}3)+\lambda _{rel,z}^{2})}$

${\displaystyle k_{c,z}={\frac {1}{k_{z}+{\sqrt {k_{z}^{2}-\lambda _{rel,z}^{2}}}}}}$

${\displaystyle {\frac {\sigma _{c,0,d}}{k_{c,z}\cdot f_{c,0,d}}}+k_{m}{\frac {\sigma _{m,y,d}}{f_{m,y,d}}}+{\frac {\sigma _{m,z,d}}{f_{m,z,d}}}\leq 1}$

credentials

1. ↑ ^{a b} Bernhard Pichler, Josef Eberhardsteiner : Structural Analysis VO - LVA-Nr . 202.065 . Ed .: E202 Institute for Mechanics of Materials and Structures - Faculty of Civil Engineering, TU Vienna - 1040 Vienna, Karlsplatz 13/202. SS 2017 edition. TU Verlag, Vienna 2017, ISBN 978-3-903024-41-0 , 24.1.1 Definition of the stability risk, p. 445 (516 pages, tuverlag.at ). Structural Analysis VO - LVA-Nr. 202.065 ( Memento of the original from March 13, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
2. ↑ H. Bruckner, R. Gelhaus, Alfons Goris, Dieter Herbeck, Frank Höfler, Hans-Georg Kempfert, Elmar Kuhlmann, Eberhard Lattermann, Erwin Memmert, Klaus Peters, Frank Preser, Helmut Rubin, Torsten Schoch, Rüdiger Wormuth: Entry stability threat in Beuth building dictionary . Ed .: Klaus-Jürgen Schneider, Rüdiger Wormuth. April 2016 (420 pages).
3. ↑ Jan Höffgen: Fundamentals of steel construction formula collection. May 2, 2014, accessed March 25, 2018 . 
4. ↑ ^{a b c d e f g h i j k l} CEN European Committee for Standardization: EN 1992-1-1: 2015-03-01 Eurocode 2: Design and construction of reinforced and prestressed concrete structures - Part 1-1: General design rules and rules for building construction; EN 1992-1-1: 2004 / A1: 2014 . 2014, 5.8.3.1 (1), p. 69 .
5. ↑ If the end moments and tension produce on the same side, it is to be assumed positive (i.e. ), otherwise it is to be assumed as negative. ${\displaystyle M_{1}}$ ${\displaystyle M_{2}}$ ${\displaystyle r_{m}}$ ${\displaystyle C\leq 1{,}7}$ ${\displaystyle r_{m}}$
6. ↑ ^{a b c d e} CEN European Committee for Standardization: EN 1995-1-1: 2015-03-01 Eurocode 5: Dimensioning and construction of wooden structures - Part 1-1: General - General rules and regulations for building construction; EN 1995-1-1: 2004 + AC: 2006 + A1: 2008 . December 2010, 6.3.2 Flexural buckling of compression rods, p. 45-46 .