LateralTorsional Buckling in Timber Construction  Examples 1
Technical Article
The article titled LateralTorsional Buckling in Timber Construction  Theory explains the theoretical background for the analytical determination of the critical bending moment M_{crit} or the critical bending stress σ_{crit} for the lateral buckling of a bending beam. This article uses examples to verify the analytical solution with the result from the eigenvalue analysis.
L 

beam length 
b 

beam width 
h 

beam height 
E 

modulus of elasticity 
G 

shear modulus 
Iz 

second moment of area around minor axis 
IT 

torsional constant 
az 

distance of load application from shear center 
SingleSpan Beam with Lateral and Torsional Restraint and Without Intermediate Support
L 

18  m  
b 

160  mm  
h 

1,400  mm  
a_{z} 

700  mm  
I_{z} 

477,866,667  mm^{4}  
I_{T} 

1,773,842,967  mm^{4}  
E_{0.05} 

10,400  N/mm²  
G_{0.05} 

540  N/mm²  
For the singlespan beam with lateral and torsional restraint without intermediate supports (see Figure 01), the equivalent member length results in the case of a load application at the top to:
${\mathrm{l}}_{\mathrm{ef}}=\frac{\mathrm{L}}{{\mathrm{a}}_{1}\xb7\left[1{\mathrm{a}}_{2}\xb7{\displaystyle \frac{{\mathrm{a}}_{\mathrm{z}}}{\mathrm{L}}}\xb7\sqrt{{\displaystyle \frac{{\mathrm{EI}}_{\mathrm{z}}}{{\mathrm{GI}}_{\mathrm{T}}}}}\right]}=18.26\mathrm{m}$
The factors a1 and a2 can be seen in Figure 02 according to the moment distribution.
The critical bending moment can then be calculated as follows:
${\mathrm{M}}_{\mathrm{crit}}=\frac{\mathrm{\pi}\xb7\sqrt{{\mathrm{E}}_{0.05}\xb7{\mathrm{I}}_{\mathrm{z}}\xb7{\mathrm{G}}_{0.05}\xb7{\mathrm{I}}_{\mathrm{T}}}}{{\mathrm{l}}_{\mathrm{ef}}}=375.42\mathrm{kNm}$
In this example, we do not increase the product of the 5% quantiles of the stiffness values due to the homogenization of beams made of glued laminated timber.
For more complex systems, it may be advantageous to determine the critical loads, moments, or stresses using the eigenvalue solver. Use the RF/FELTB addon module, which is based on the finite element method, to calculate the stability loads of sets of members. An elastic material behavior is assumed for a geometric nonlinear behavior. The critical load factor is important for timber construction. This indicates the factor by which the load can be multiplied before the system becomes unstable.
For this example, the beam is loaded with a unit load of 1 kN/m. For this, the bending moment results in:
$\mathrm{M}=\frac{\mathrm{q}\xb7{\mathrm{L}}^{2}}{8}=\frac{1.00\xb718.{00}^{2}}{8}=40.50\mathrm{kNm}$
Since the lower quantile value of the critical moment is to be determined, the 5% quantiles must be used for the stiffness values E and G. To do this, you have to create a userdefined material that is only used in the addon module. For this material, the stiffness parameters E and G have to be replaced.
Then, define the lateral and torsional restraints. It is important to ensure that the degree of freedom φ_{Z} also needs to be solved.
You have to set the load eccentrically so that it acts on top of the beam.
In the details, it is still necessary to deactivate the reduction of the stiffness by the partial safety factor γ_{M} (see Figure 07). Alternatively, you can set the partial safety factor to 1.0 directly in the userdefined material.
The calculation results in a critical load factor of 9.3333 (see Figure 08). If the load is multiplied by this factor, the upper flange will deflect and the system becomes unstable.
The following applies for the critical moment:
${\mathrm{M}}_{\mathrm{crit}}=9.3333\xb740.50\mathrm{kNm}=378.00\mathrm{kNm}$
This corresponds very well with the result of the analytical solution.
SingleSpan Beam with Lateral and Torsional Restraint and Intermediate Support
The beam is now supported as rigidly fixed laterally in the third points by a stiffening structure.
Since the moment distribution in the middle area is almost constant, a constant moment distribution is assumed for the lateral buckling length coeffcient. Thus, the value of a_{1} is 1.0 and a_{2} is 0. The effective length with L = 6.0 m results in
${\mathrm{l}}_{\mathrm{ef}}=\frac{\mathrm{L}}{{\mathrm{a}}_{1}\xb7\left[1{\mathrm{a}}_{2}\xb7{\displaystyle \frac{{\mathrm{a}}_{\mathrm{z}}}{\mathrm{L}}}\xb7\sqrt{{\displaystyle \frac{{\mathrm{EI}}_{\mathrm{z}}}{{\mathrm{GI}}_{\mathrm{T}}}}}\right]}=6,00\mathrm{m}$
and the critical moment in
${\mathrm{M}}_{\mathrm{crit}}=\frac{\mathrm{\pi}\xb7\sqrt{{\mathrm{E}}_{0.05}\xb7{\mathrm{I}}_{\mathrm{z}}\xb7{\mathrm{G}}_{0.05}\xb7{\mathrm{I}}_{\mathrm{T}}}}{{\mathrm{l}}_{\mathrm{ef}}}=1,142.41\mathrm{kNm}$
The eigenvalue solver results in a critical load factor of 26.1735, taking into account the intermediate supports at the shear center (see Figure 10).
The following applies for the critical moment:
${\mathrm{M}}_{\mathrm{crit}}=26.1735\xb740.50\mathrm{kNm}=1,060.03\mathrm{kNm}$
If the intermediate support acts on the upper side (see Figure 11), the critical load factor becomes larger (32.5325) because this position has a more favorable effect on the lateral buckling behavior of the beam.
${\mathrm{M}}_{\mathrm{crit}}=32.5325\xb740.50\mathrm{kNm}=1,317.57\mathrm{kNm}$
The analytical approximation is also relatively good for this case.
Alternative Analysis on a Surface Model
You can also use RFEM and the RFSTABILITY addon module to calculate the critical load factors. For this, it is necessary that you model the beam as an orthotropic surface. The results from RFSTABILITY correspond very well with the member calculation from RF/FELTB. The first mode shape and the corresponding critical load factor are shown in Figure 12.
System  M_{crit} Analytical  M_{crit} RF/FELTB  M_{crit} RFSTABILITY 

without intermediate support  375.42 kNm  378.00 kNm  378.55 kNm 
with intermediate support in shear center  1,142.41 kNm  1,060.03 kNm  1,085.81 kNm 
with intermediate support on top flange    1,317.57 kNm  1,455.98 kNm 
For most cases, it is probably sufficient to determine the critical bending moment M_{crit} or the critical bending stress σ_{crit} using the analytical equations from the literature. For special cases, two options for implementation using Dlubal programs were shown. While the RF/FELTB addon module is used to perform the calculation using members, the RFSTABILITY addon module allows you to perform even more complex stability designs. One example is a lateral and torsional restraint that is not arranged over the entire beam height. This can be analyzed easily with a surface model.
Author
Dipl.Ing. (FH) Gerhard Rehm
Product Engineering & Customer Support
Mr. Rehm is responsible for the development of products for timber structures, and provides technical support for customers.
Keywords
Lateral buckling Lateraltorsional buckling Eigenvalue
Downloads
Links
 LateralTorsional Buckling in Timber Structures: Theory
 Timber Structural Analysis and Design Software
 Stability Analysis Software
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