Bumps

Dents in the web in a welded double-T-beam under uniform axial pressure load. The red and blue lines are contour lines for moving the bridge in the opposite direction

In technical mechanics, bumps are :

• the evasion of plates , the load of which (essentially) represents a state of tension in the disk , out of their plane
• dodging shells whose load (essentially) a membrane voltage state represents
• the evasion of pipes in the direction of the surface normals under external overpressure .

The prerequisite for buckling is that compressive stresses exist in at least one direction in the plane of the plate or the shell surface .

Examples of plate buckling are buckling (becoming wavy) in the straps or webs of double-T or U-beams.

Mathematical acquisition

For the mathematical determination of the buckling, the equilibrium conditions must always be set up for the already buckled state of the component ( plate or shell ) ( second order theory , see below structural analysis ). The equations lead (if imperfections are neglected ) to an eigenvalue problem . The first eigenvalue then determines the smallest critical load at which the buckling can occur.

The eigenvalue problem is usually solved approximately by numerical methods , e.g. B. by means of the finite element method .

The first figure shows an example of a bulging figure, which in this case provides the lowest eigenvalue.

Buckling load

If the plate is held sideways, it is prevented from buckling . Instead, it takes on the double-curved shape of a bump , with the number of bumps depending on the aspect ratio. The buckling load is always greater than the buckling load. The real buckling load is always smaller than the ideal buckling load because of the inevitable imperfections . The yield point is decisive for compact cross-sections .

A plate that is not held on the side carries like a folding rod; in this case there is no buckling problem.

Buckling rod-like behavior

Buckling rod-like behavior on plate dents

In the case of panels with a large aspect ratio (first case in the figure), the stresses can be redistributed in the stiffened edges. Such plates do not have a behavior similar to that of a kink rod, instead they buckle.

In the case of panels with a low aspect ratio (second case in the figure) or heavily stiffened panels (third case in the figure), the bulge shape takes on a uniaxially curved shape and is more like a buckling bar than a plate. Such panels have almost no supercritical bearing reserves.

The buckling bar-like behavior is taken into account with a weighting factor that is calculated from the ideal critical buckling stress and the ideal critical buckling stress according to the standard : ${\displaystyle \xi }$ ${\displaystyle \sigma _{cr,p}}$ ${\displaystyle \sigma _{cr,c}}$

${\displaystyle \xi ={\frac {\sigma _{cr,p}}{\sigma _{cr,c}}}-1}$

The weighting factor decides whether there is pure bulging, pure buckling or a mixed form:

• if the buckling stress is very high, there is pure buckling:

${\displaystyle {\begin{aligned}\sigma _{cr,p}&\gg \sigma _{cr,c}\\\Rightarrow \xi &\gg 1\end{aligned}}}$

• if the buckling and buckling stresses are the same, pure buckling is present:

${\displaystyle {\begin{aligned}\sigma _{cr,p}&\approx \sigma _{cr,c}\\\Rightarrow \xi &\approx 0\end{aligned}}}$.

Proof of load-bearing capacity

The load-bearing capacity can be verified using two different models:

• Based on the model of the effective stresses, the maximum stress that can be absorbed is calculated and compared with the existing stress. In this model, the weakest part of the cross-section is decisive.
• According to the model of the effective widths, the effective widths are determined by the buckling, and the verification is carried out with the cross-section weakened in this way. This model brings higher load capacities because it includes the wearer as a whole.

The load-bearing capacity for unstiffened buckling areas can be verified with the following formula apparatus. The formulas come from the Eurocode  1993-1-5. According to DIN 18800-2 and DIN 18800-3, other symbols are used, but the same calculation is carried out in terms of content.

With (see below) as a reduction factor, the yield point can be reduced and the stress analysis based on the model of the effective stresses. ${\displaystyle \rho _{c}}$

Alternatively, the reduction factor can be used to calculate the effective web width and thus to carry out the cross-section analysis according to the effective width model.

Related bulging slimness:

${\displaystyle {\overline {\lambda }}_{p}={\frac {b}{t\cdot 28{,}43\cdot \varepsilon \cdot {\sqrt {k_{\sigma }}}}}}$

With

• the sheet thickness t
• ${\displaystyle \varepsilon ={\sqrt {\frac {235}{f_{yk}}}}}$
  • the yield point ${\displaystyle f_{yk}}$
• the buckling value (calculated according to DIN EN 1993-1-5 or from curve tables according to Klöppel / Scheer / Möller). ${\displaystyle k_{\sigma }}$

Reduction factor for dents:

${\displaystyle \rho _{p}={\frac {{\overline {\lambda }}_{p}-0{,}055\cdot (3+\psi )}{{\overline {\lambda }}_{p}^{2}}}}$

with the edge stress ratio . ${\displaystyle \psi }$

Ideal buckling stress:

${\displaystyle \sigma _{cr,c}={\frac {\pi ^{2}\cdot E\cdot t^{2}}{10,92\cdot a^{2}}}}$

with the E-module  E.

Related kink slenderness:

${\displaystyle {\overline {\lambda }}_{ki}={\sqrt {\frac {f_{yk}}{\sigma _{cr,c}}}}}$

Reduction factor for buckling:

${\displaystyle \chi _{c}={\frac {1}{k+{\sqrt {k^{2}-{\overline {\lambda }}_{ki}^{2}}}}}}$

With

• ${\displaystyle k=0{,}5\cdot (1+\alpha \cdot ({\overline {\lambda }}_{ki}-0{,}2)+{\overline {\lambda }}_{ki}^{2})}$
  • ${\displaystyle \alpha }$ depends on the carrier.

Weighting factor: s. O.

Interaction between bumps and kinks:

${\displaystyle \rho _{c}=(\rho _{p}-\chi _{c})\cdot \xi \cdot (2-\xi )+\chi _{c}}$

Calculation of the load-bearing capacity for pipes

In the case of pipes, a distinction is made between two types of buckling stress .

Dents due to the circular pressure on axial cylinders

The jacket of the pipe is deformed into a chessboard-like pattern. Assuming a metallic material that has a Poisson's ratio of 0.3 (e.g. steel), the problem is simplified. In theory the strength is much lower than in practice, which is determined as follows:

• for seamless tubes :

${\displaystyle \sigma _{Beulung}=0{,}5\cdot E{\frac {t}{d}}}$

• for welded pipes :

${\displaystyle \sigma _{Beulung}=0{,}3\cdot E{\frac {t}{d}}}$

each with

• the modulus of elasticity ${\displaystyle E}$
• the wall thickness ${\displaystyle t}$
• the diameter . ${\displaystyle d}$

Dents due to external overpressure or internal negative pressure

The general formula is described in AD Merkblatt 2000 - B6. Assuming a metallic material that has a Poisson's ratio of 0.3, the problem is greatly simplified and in theory there is a significantly higher external overpressure than in practice, which is described as follows, provided that : ${\displaystyle l/d>3}$

${\displaystyle p_{Beulung}=2{,}2\cdot E\left({\frac {t}{d}}\right)^{3}}$

If the relationship 0.2 << 5 is maintained, the lower load-bearing capacity determined in tests applies again with: ${\displaystyle l/r}$

${\displaystyle p_{Beulung}=0{,}65\cdot E\cdot {\frac {r}{l}}\cdot \left({\frac {t}{r}}\right)^{\frac {5}{2}}}$

in which

${\displaystyle p_{Beulung}}$is the allowable pressure, the modulus of elasticity, the wall thickness, the diameter, the mean radius of the cylinder jacket and the length of the cylinder. ${\displaystyle E}$ ${\displaystyle t}$ ${\displaystyle d}$ ${\displaystyle r}$ ${\displaystyle l}$

In order to obtain the permissible values, the reduction due to the selected safety concept must also be taken into account.

Web links

Wikibooks: Plate dents  - learning and teaching materials

literature

• Kurt Klöppel , Joachim Scheer , KH Möller: Buckling values ​​of stiffened rectangular plates. Verlag W. Ernst & Sohn, 1960, 1968 (Part 2), Reprint 2001, ISBN 3-433-02828-1
• DIN 18800-2 11-90 Steel structures Stability cases, buckling of bars and structures
• DIN 18800-3 11-90 Steel structures, stability cases, plate dents
• DIN EN 1993 Eurocode 3 Dimensioning and construction of steel structures Part 1-5 2006: February 2007 Plate-shaped components

See also

• Bulging stiffness

Individual evidence

1. ↑ ^{a b} Günther Holzmann, Heinz Meyer, Georg Schumpich: Technical mechanics: strength theory; Page 307
2. ^ Günther Holzmann, Heinz Meyer, Georg Schumpich: Technical Mechanics: Strength of Materials; Page 306
3. ↑ Anton Schweizer: Project planning aid ( Memento of the original dated November 7, 2011 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.