Lateral-Torsional Buckling in Timber Construction: Theory
Technical Article
Slender bending beams with a large h/w ratio and loaded parallel to the minor axis tend to have stability issues. This is due to the deflection of the compression chord.
The beam has a lateral displacement with simultaneous rotation (see Figure 01). This is called flexural-torsional buckling or lateral buckling. Similar to the flexural buckling, where a member buckles suddenly when the Euler load is reached, the compression chord shifts from a critical lateral buckling load during flexural-torsional buckling. This results in a critical bending moment M_{crit}, which results in a critical stress for lateral buckling σ_{crit}.
Figure 01 - Lateral Buckling of a Single-Span Beam
Symbols used:
L ... beam length
E ... modulus of elasticity
G ... shear modulus
I_{z} ... torsional constant about the minor axis
I_{T} ... torsional constant
I_{ω} ... warping resistance
a_{z} ... distance of load application from shear center
e ... distance of member elastic foundation from shear center
K_{G} ... elastic rotational spring on support in Nmm
K_{Θ} ... elastic rotational restraint in N
K_{y} ... elastic member foundation in N/mm²
Analytical Determination of M_{crit}
To determine the bending moment where a beam becomes unstable, the engineer can use analytical solutions from literature, but these are limited in their application. In [1], the following equation is derived for a single-span beam with lateral and torsional restraints and hinged at both sides with a constant bending moment and load application at the shear center.
${\mathrm M}_{\mathrm{crit}}\;=\;\frac{\mathrm\pi}{\mathrm L}\;\cdot\;\sqrt{{\mathrm{EI}}_{\mathrm z}\;\cdot\;{\mathrm{GI}}_{\mathrm T}\;\cdot\;\left(1\;+\;\frac{{\mathrm{EI}}_{\mathrm\omega}}{{\mathrm{GI}}_{\mathrm T}}\;\cdot\;\frac{\mathrm\pi^2}{\mathrm L^2}\right)}$
In the case of cross-sections not free to warp (for example, a narrow rectangular cross-section in timber construction), the warping stiffness can be set to zero and thus the part in the parenthesis is omitted.
${\mathrm M}_{\mathrm{crit}}\;=\;\frac{\mathrm\pi}{\mathrm L}\;\cdot\;\sqrt{{\mathrm{EI}}_{\mathrm z}\;\cdot\;{\mathrm{GI}}_{\mathrm T}}$
Since there are many more cases in structural analysis than the ones mentioned above, correction factors have been introduced in order to take into account, for example, deviating moment distributions, support situations and a different load application. For this, the length of the beam is modified with the factors and results in an effective length l_{ef}. This is described in [2], among others, as follows.
${\mathrm l}_{\mathrm{ef}}\;=\;\frac{\mathrm L}{{\mathrm a}_1\;\cdot\;\left[1\;-\;{\mathrm a}_2\;\cdot\;{\displaystyle\frac{{\mathrm a}_{\mathrm z}}{\mathrm L}\;}\;\cdot\sqrt{\displaystyle\frac{{\mathrm{EI}}_{\mathrm z}}{{\mathrm{GI}}_{\mathrm T}}}\right]}$
a_{z} is the distance of the load application from the shear center.
Figure 02 - Descriptions on the Rectangular Cross-Section
If the load acts on the bottom side of the beam, you have to consider a_{z} with negative signs. The coefficients a_{1} and a_{2} are shown in Figure 03.
Figure 03 - Lateral Buckling Coefficients
The different systems have to be understood as follows:
- Single-span beam with lateral and torsional restraints and hinged at both sides
- Restrained beam
- Cantilever with lateral and torsional restraint at the free end
- Beam fixed on both sides
- Single-span beam with restraint on one side
- Two-span beam
- Continuous beam with lateral and torsional restraint - inner span
- Continuous beam with lateral and torsional restraint - outer span
The standards suggest the lateral buckling design according to the equivalent member method. You have to calculate the critical moment with the 5% quantil values of the stiffnesses. Thus, the following results for timber structures:
${\mathrm M}_{\mathrm{crit}}\;=\;\frac{\mathrm\pi\;\cdot\;\sqrt{{\mathrm E}_{0.05}\;\cdot\;{\mathrm I}_{\mathrm z}\;\cdot\;{\mathrm G}_{0.05}\;\cdot\;{\mathrm I}_{\mathrm T}}}{{\mathrm l}_{\mathrm{ef}}}$
The critical bending stress results in:
${\mathrm\sigma}_{\mathrm{crit}}\;=\;\frac{\mathrm\pi\;\cdot\;\sqrt{{\mathrm E}_{0.05}\;\cdot\;{\mathrm I}_{\mathrm z}\;\cdot\;{\mathrm G}_{0.05}\;\cdot\;{\mathrm I}_{\mathrm T}}}{{\mathrm l}_{\mathrm{ef}}\;\cdot\;{\mathrm W}_{\mathrm y}}$
If you want to consider an elastic rotational spring (for example resulting from the flexibility of the lateral and torsional restraint) on the support, an elastic rotational restraint (for example from trapezoidal sheeting) or an elastic member elastic foundation (for example from bracings), you can extend the previous equation as follows [2].
${\mathrm l}_{\mathrm{ef}}\;=\;\frac{\mathrm L}{{\mathrm a}_1\;\cdot\;\left[1\;-\;{\mathrm a}_2\;\cdot\;{\displaystyle\frac{{\mathrm a}_{\mathrm z}}{\mathrm L}}\;\cdot\;\sqrt{\displaystyle\frac{{\mathrm{EI}}_{\mathrm z}}{{\mathrm{GI}}_{\mathrm T}}}\right]}\;\cdot\;\frac1{\mathrm\alpha\;\cdot\;\mathrm\beta}$
Where
$\mathrm\alpha\;=\;\sqrt{\frac1{1\;+\;{\displaystyle\frac{3.5\;\cdot\;{\mathrm{GI}}_{\mathrm T}}{{\mathrm K}_{\mathrm G}\;\cdot\;\mathrm L}}}}\\\mathrm\beta\;=\;\sqrt{\left(1\;+\;\frac{{\mathrm K}_{\mathrm y}\;\cdot\;\mathrm L^4}{{\mathrm{EI}}_{\mathrm z}\;\cdot\;\mathrm\pi^4}\right)\;\cdot\;\left(1\;+\;\frac{\left({\mathrm K}_{\mathrm\theta}\;+\;\mathrm e^2\;\cdot\;{\mathrm K}_{\mathrm y}\right)\;\cdot\;\mathrm L^2}{{\mathrm{GI}}_{\mathrm T}\;\cdot\;\mathrm\pi^2}\right)}\;+\;\frac{\mathrm e\;\cdot\;{\mathrm K}_{\mathrm y}\;\cdot\;\mathrm L^3}{\sqrt{{\mathrm{EI}}_{\mathrm z}\;\cdot\;{\mathrm{GI}}_{\mathrm T}}\;\cdot\;\mathrm\pi^3}$
Figure 04 - Descriptions on the Rectangular Cross-Section with Elastic Springs
If the rotational spring K_{G} on the support is considered as infinitely rigid, α = 1. The elastic rotational restraint K_{Θ} is usually not taken into account in timber structures, as no studies are available here. Thus, the parameter K_{Θ} is included in the equation with the value 0. The elastic member elastic foundation K_{y}, resulting from a bracing or a shear panel, has a favorable effect on the lateral buckling behavior of a beam. However, please note that the previous equation has a limited application. Strictly speaking, it is only valid if there is a deflection in a large sinusoidal arc. If the member foundation is too rigid, this is no longer the case because the mode shape has several arcs along the beam. There is currently no definition of when the extended formula with α and β becomes invalid.
How to solve such eigenvalue issues in a skilful way will be described with different examples in the next article.
Keywords
Lateral buckling Lateral-torsional buckling Eigenvalue
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