Torsion Design of Glulam Beams

Technical Article

Long-span glued-laminated beams are usually supported by a reinforced concrete column with torsional restraints.

Figure 01 - Beam with Torsional Restraints Subjected to Uniform Load (Source: [3])

On these supports, torsion moments occur which have to be designed according to [2], Section 6.1.9:

$$\frac{{\mathrm\tau}_{\mathrm{tor},\mathrm d}}{{\mathrm k}_\mathrm{shape}\;\cdot\;{\mathrm f}_{\mathrm v,\mathrm d}}\;+\;(\frac{{\mathrm\tau}_{\mathrm y,\mathrm d}}{{\mathrm f}_{\mathrm v,\mathrm d}})²\;+\;(\frac{{\mathrm\tau}_{\mathrm z,\mathrm d}}{{\mathrm f}_{\mathrm v,\mathrm d}})²$$

The superposition of internal forces from shear force and torsion should prevent cracks on the rigid support.

Figure 02 - Cracks in Glulam Beam (Source: [4])

The torsional moment on the end supports is caused by beam deflection in the case of a sine‑shaped load (cf. Figure 03).

Figure 03 - Beam Deflection

According to [1], a value of l/400 should be set for the precamber. This is based on the minimum requirement of stiffening the secondary supporting system. More information can be found in [3], for example.

However, the current structural member analysis methods cannot detect torsion on supports. In addition, many calculation programs do not allow for consideration of the cross‑section warping. Since the calculation is often performed in 2D structural frame analysis programs, the limiting criterion is provided in [2], Section NCI to 9.2.5.3 (Expression 2):

$${\mathrm\lambda}_\mathrm{ef}\;=\;{\mathrm l}_\mathrm{ef}\;\cdot\;\frac{\mathrm h}{\mathrm b²}\;\leq\;225$$

If the slenderness ratio of the beam is below this value, the torsional stress components can be neglected.

Calculation in RX-TIMBER Glued-Laminated Timber

The following example clarifies this relation.

System

Span  =  25 m
Material  =  GL24c
Cross-Section  =  12 cm / 242 cm (without apex wedge)

Figure 04 - Geometry of Beam

The beam is subjected to a uniformly distributed load of 13.5 kN/m. The dead load is neglected.

The governing design is the torsional stress analysis specified in Expression 1. In this case, lef is the same as the span length of 2.46 m. Spacing of supports for lateral‑torsional buckling can only be applied if the horizontal stiffening of the secondary supporting system is < l/ 500 or l/1,000. This is not applied here.

$$\begin{array}{l}{\mathrm\lambda}_\mathrm{ef}\;=\;{\mathrm l}_\mathrm{ef}\;\cdot\;\frac{\mathrm h}{\mathrm b²}\;=\;2,460\;\mathrm{cm}\;\cdot\;\frac{240\;\mathrm{cm}}{(12\;\mathrm{cm})²}\;=\;4,100\;>\;225\\\frac{{\mathrm\tau}_{\mathrm{tor},\mathrm d}}{{\mathrm k}_\mathrm{shape}\;\cdot\;{\mathrm f}_{\mathrm v,\mathrm d}}\;+\;\left(\frac{{\mathrm\tau}_{\mathrm z,\mathrm d}}{{\mathrm f}_{\mathrm v,\mathrm d}}\right)^2\;=\;\frac{0.11\;\mathrm{kN}/\mathrm{cm}²}{1.3\;\cdot\;0.16\;\mathrm{kN}/\mathrm{cm}²}\;+\;\left(\frac{0.12\;\mathrm{kN}/\mathrm{cm}²}{0.16\;\mathrm{kN}/\mathrm{cm}²}\right)^2\;=\;1.1\end{array}$$

Internal Forces and Stresses

$$\begin{array}{l}{\mathrm T}_{\mathrm M,\mathrm d}\;=\;\frac{{\mathrm M}_{\max,\mathrm d}}{80}\;=\;\frac{102,665\;\mathrm{kNcm}}{80}\;=\;12.8\;\mathrm{kNm}\\{\mathrm W}_\mathrm t\;=\;11,520\;\mathrm{cm}³\\{\mathrm\tau}_{\mathrm{tor},\mathrm d}\;=\;\frac{1,280\;\mathrm{kNcm}}{11,520\;\mathrm{cm}³}\;=\;0.11\;\mathrm{kN}/\mathrm{cm}²\\{\mathrm\tau}_\mathrm d\;=\;1.5\;\cdot\;\frac{{\mathrm V}_\mathrm d}{{\mathrm k}_\mathrm{cr}\;\cdot\;\mathrm b\;\cdot\;\mathrm h}\;=\;0.12\;\mathrm{kN}/\mathrm{cm}²\end{array}$$

Calculation Considering Warping Torsion

RF-/FE-LTB allows you to apply the eccentric compression force to the beam. Thus, the uniform load of 13.5 kN/m can be applied eccentrically to the beam.

Figure 05 - Eccentric Load Application in RF‑/FE‑LTB

As shown in Figure 05, the load eccentricity is set to 6 cm. Furthermore, lateral deformation of 6.15 cm is applied in accordance with [2] (NA.5).

$$\mathrm e\;=\;\frac{\mathrm l}{400}\;\cdot\;{\mathrm k}_\mathrm l\;=\;\frac{2,460\;\mathrm{cm}}{400}\;=\;6.15\;\mathrm{cm}$$

Based on the Bernoulli bending theory, RF‑/FE‑LTB can determine the critical load Fki and thus the ideal elastic critical moment Mki and the torsional buckling load Nki,phi.

The calculation is based on the second‑order torsional buckling theory. The cross‑section warping (7th degree of freedom) is also taken into account.

In order to consider the corresponding roof covering or stiffening due to the secondary supporting system, a rotational spring about the local x‑axis of the member is defined. The program converts this spring to the shear center M.

Figure 06 - Continuous Springs (from RF-/FE-LTB)

The rotational spring is only applied to obtain the deformation shown in Figure 02. A translational spring on the upper flange of the structure would be closer to reality. However, the required imperfection shape cannot be created due to the curvature of the beam. The imperfection shape would then fail in the middle as shown in Figure 07. In this way, the torsional moments would be reduced significantly.

Figure 07 - Failure Mode

With the rotational restraint of 500 kNm/m, the torsional moment of 9.8 kNm arises at the supports.

Figure 08 - Torsional Moments

Using this torsional moment, the design of [1] can be performed again in RX‑TIMBER Glued‑Laminated Beam. For this, the determined torsional moment is defined in RX‑TIMBER Glued‑Laminated Beam.

Figure 09 - Torsional Moments in RX‑TIMBER Glued‑Laminated Beam

$$\frac{0.085\;\mathrm{kN}/\mathrm{cm}²}{1.3\;\cdot\;0.16\;\mathrm{kN}/\mathrm{cm}²}\;+\;\left(\frac{0.12\;\mathrm{kN}/\mathrm{cm}²}{0.16\;\mathrm{kN}/\mathrm{cm}²}\right)^2\;=\;0.97\;<\;1$$

Summary

By considering warping stiffness of a cross‑section, you can design the structure in a significantly more effective way.

The difference from the general approach of Section 9.2.5 in [2] is even more serious when replacing a virtual rotational restraint by a translational spring stiffness of 915 N/mm for longitudinal deformation of a conventional nail in a coupling member, for example.

Reference

[1]   Eurocode 5: Design of timber structures - Part 1‑1: General - Common rules and rules for buildings; EN 1995‑1‑1:2010‑12
[2]   National Annex - Nationally determined parameters - Eurocode 5: Design of timber structures - Part 1‑1: General - Common rules and rules for buildings; DIN EN 1995‑1‑1/NA:2013‑08
[3]   Blass, H., Ehlbeck, J., Kreuzinger, H., & Steck, G. (2005). Erläuterungen zu DIN 1052:2004‑08 (2nd ed.). Cologne: Bruderverlag.
[4]   Winter, S. (2008). Bad Reichenhall und die Folgen (1st ed.). Munich: TU München.

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