# Lateral-torsional buckling in timber construction: Theory

### Technical Article

Slender bending beams with a large h/b ratio and loaded parallel to the minor axis tend to have stability problems. This is due to the deflection of the compression chord.

The beam undergoes a lateral displacement with simultaneous rotation (see Figure 01). This is called flexural-torsional buckling or tilting. Similar to the flexural buckling, in which a member buckles suddenly when the Euler buckling load is reached, the compression chord deviates from a critical tipping load during flexural-torsional buckling. This results in a critical bending moment M_{crit} , which results in a critical bending_{stress} stress σ_{crit} .

Figure 01 - Tilting a single-span beam

Symbols used:

L ... Beam length

E ... is the modulus of elasticity,

G ... is the shear modulus,

I_{z} ... Moment of inertia about the minor axis

I_{T} ... Torsion moment of inertia

I_{ω} ... Warping resistance_{z} ... Distance of load application from shear center

e ... Distance of member elastic foundation from shear center

K_{G} ... Elastic torsion spring on support in Nmm

K_{Θ} ... Elastic rotational restraint in N

K_{y} ... Elastic member elastic foundation in N/mm²

#### Analytical determination of M_{crit}

To determine the bending moment under which a beam becomes unstable, analytical solutions are available to the engineer in the literature, but these are limited in their application. In [1] , the following equation is derived for a single-span beam with double-hinged single-span beams 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 warping-free cross-sections (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.

${\mathrm l}_{\mathrm{ef}}\;=\;\frac{\mathrm L}{{\mathrm a}_1\;\cdot\;\left[1\;-\;{\mathrm a}_2\;\cdot\;{\displaystyle\frac{{\mathrm a}_{\mathrm z}}{\mathrm L}\;}\;\cdot\sqrt{\displaystyle\frac{{\mathrm{EI}}_{\mathrm z}}{{\mathrm{GI}}_{\mathrm T}}}\right]}$

Where 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, a_{z} with negative signs must be considered. The coefficients a_{1} and a_{2} are shown in Figure 03.

The different systems are to be understood as follows:

- Single-span beam with double-hinged fork supports
- Restrained beam
- Cantilever arm with fork support at the free end
- Beam fixed on both sides
- Single-span beam with one-sided restraint
- Two-span beam
- Fork-supported continuous beam - infield
- Fork-supported continuous beam - outer span

In the standards, the tilt analysis is proposed according to the equivalent member method. The critical moment has to be calculated with the 5% -quantil values of the stiffnesses. Thus, the following results for timber construction:

${\mathrm M}_{\mathrm{crit}}\;=\;\frac{\mathrm\pi\;\cdot\;\sqrt{{\mathrm E}_{0,05}\;\cdot\;{\mathrm I}_{\mathrm z}\;\cdot\;{\mathrm G}_{0,05}\;\cdot\;{\mathrm I}_{\mathrm T}}}{{\mathrm l}_{\mathrm{ef}}}$

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 torsion spring (for example resulting from the flexibility of the fork support) 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 torsional spring K_{G} on the support is considered as infinitely stiff, α = 1. The elastic rotational restraint K_{Θ} is usually not taken into account in timber construction, 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 tilting behavior of a beam. However, it should be noted that the previous equation is limited in its application. Strictly speaking, this is only valid if there is a deflection in a large sinusoidal arc. If the member foundation is too stiff, 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 problems in a skilful way is described in different examples in the next article.

#### Keywords

Tilt Lateral-torsional buckling Eigenvalue

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