16702x

001625

Dipl.-Ing. (FH) Gerhard Rehm

2020-02-28

Lateral-Torsional Buckling in Timber Construction | Theory

Slender bending beams that have a large h/w ratio and are 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 Image 01). This is called flexural-torsional buckling or lateral buckling. Similar to 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.

1 Lateral Buckling of Single-Span Beam

Symbols used:

L	Beam length
E	Modulus of elasticity
G	Shear modulus
I_z	Second moment of area about the weak axis
I_T	Torsion moment of inertia
I_ω	Warping resistance
a_z	Distance of the load application from the shear center
e	Distance of the member elastic foundation from the 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 the 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.

M_{crit} = \frac{π}{L} \cdot \sqrt{{EI}_{z} \cdot {GI}_{T} \cdot (1 + \frac{{EI}_{ω}}{{GI}_{T}} \cdot \frac{π^{2}}{L^{2}})}

Critical Moment

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; thus, the part in the parentheses is omitted.

M_{crit} = \frac{π}{L} \cdot \sqrt{{EI}_{z} \cdot {GI}_{T}}

Critical Moment Without Warping Stiffness

Since there are many more cases in structural analysis than those 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.

l_{ef} = \frac{L}{a_{1} \cdot [1 - a_{2} \cdot \frac{a_{z}}{L} \cdot \sqrt{\frac{{EI}_{z}}{{GI}_{T}}}]}

Effective Length

a_z is the distance of the load application from the shear center.

2 Descriptions on 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 a₁ and a₂ are shown in Image 03.

3 Lateral Buckling Factors

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% quantile values of the stiffnesses. Thus, the following results for timber structures:

M_{crit} = \frac{π \cdot \sqrt{E_{0.05} \cdot I_{z} \cdot G_{0.05} \cdot I_{T}}}{l_{ef}}

Critical Moment for Timber Structures

The critical bending stress results in:

σ_{crit} = \frac{π \cdot \sqrt{E_{0.05} \cdot I_{z} \cdot G_{0.05} \cdot I_{T}}}{l_{ef} \cdot W_{y}}

Critical Bending Stress

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].

l_{ef} = \frac{L}{a_{1} \cdot [1 - a_{2} \cdot \frac{a_{z}}{L} \cdot \sqrt{\frac{{EI}_{z}}{{GI}_{T}}}]} \cdot \frac{1}{α \cdot β}

Effective Length Considering Elastic Rotational Spring, Restraint, or Member Elastic Foundation

where

α = \sqrt{\frac{1}{1 \frac{3.5 \cdot {GI}_{T}}{K_{G} \cdot L}}}

β = \sqrt{(1 \frac{K_{y} \cdot L^{4}}{{EI}_{z} \cdot π^{4}}) \cdot (1 \frac{(K_{θ} e^{2} \cdot K_{y}) \cdot L^{2}}{{GI}_{T} \cdot π^{2}})} + \frac{e \cdot K_{y} \cdot L^{3}}{\sqrt{{EI}_{z} \cdot {GI}_{T}} \cdot π^{3}}

Calculation of α and β

4 Descriptions on 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.

Finding the solution for such eigenvalue issues in a skillful way will be described with different examples in the next article.

Knowledge Base

KB 001625 | Lateral-Torsional Buckling in Timber Construction | Theory

Author

Mr. Rehm is responsible for developing products for timber structures, and he provides technical support for customers.

Links

Structural Analysis and Design Software for Timber Structures

References

Timoshenko, S., & Gere, J. M. (1961). Theory of Elastic Stability. (2nd ed.). New York: McGraw-Hill.
DIN. (2013). National Annex – Eurocode 5: Design of Timber Structures – Part 1-1: General – Common Rules and Rules for Buildings, DIN EN 1995-1-1/NA:2013-08. Berlin: Beuth Verlag GmbH.

;

Lateral-Torsional Buckling in Timber Construction | Theory

Analytical Determination of Mcrit

Analytical Determination of M_crit