Here a typical I-section is taken as an example, other sections (like box-sections) are assessed in the same principal way. In Figure 5.5 an I-section having flanges of equal dimension but different strength grades is illustrated. The stress distribution for pure bending is shown. It is assumed that the area moment of inertia Iy,fic is determined according to Equation 5.2 using the timber compression flange as reference and that the tension flange has a much better strength grade than the compression flange. Note that this kind of asymmetry should be
Eq. 5.3
Figure 5.5: Stress distributions on an I-section (or box-section) used to determine its bending moment resistance My,Rd. Note that the “real” stress distributions indicated is valid if Ew < Ef,c < Ef,t.
In Figure 5.5 six dots indicate positions on the cross-section where the bending stress may become critically large in comparison to the strength of the relevant material. The absolute value of these stresses are determined using Navier’s equation for the fictitious cross-section, but also multiplying by the appropriate Ei ⁄ Er-ratio as in Equation 5.3. The stress for any of the dots is:
where zdot is the distance from the neutral axis to the dot and Er is the modulus of elasticity of the reference material (in this case the timber of the compression flange). It is now tempting to use Equation 3.6 in Section 3.1.2 to verify the resistance of the flanges. This is because the total stress affecting a timber flange can be split into two parts: one pure compressive or tensile stress at the centre of each flange and one pure bending stress, taken as the difference between the outermost edge stress and the centre stress. Finally, substituting these stresses into the appropriate interaction formula will verify the resistance. It has however turned out that such an approach is on the safe side compared to experimental results and also a bit cumbersome. The recommended approach is much simpler. Usually the flange depth is much smaller than the beam depth, in which case their resistance is almost entirely controlled by the compression or tensile strength of the flange material. Hence a reasonable design criterion would be to avoid stresses at the flange centres that are greater than the material strength at these points. But this approach is too optimistic if the flange depth becomes large in relation to the beam depth. This obstacle is overcome by realising that the bending strength is greater than the compression strength and much greater than the tensile strength for most timber materials. So the problem is solved by adding the additional requirement that the stress at the outermost fibres of the flanges must be smaller than the bending strength. For the dots in Figure 5.5 the following six verifications must be made.
Eq. 5.5 Compression flange, c Web panel,w Tension flange, t 1 c 2 3 t w My,Ed My,Ed f,c,max,d w,c,max,d z y z Fic titious s tr es s ”R eal ” s tr es s σf,c,d σw,t,max,d σf,t,max,d σf,t,d
where all σ-stresses are calculated using Equation 5.5; fc,0,d and fm,d are the compressive and bending strength of the timber in the compression flange, respectively; ft,0,d and fm,d are the tensile and bending strength of the timber in the tension flange, respectively; fc,w,d and ft,w,d are the compressive and tensile strength of the web, respectively.
Using fc,w,d and ft,w,d will slightly underestimate the resistance of the web panel/panels. The three main reasons for this are: 1) The strength values have been determined based on testing of fairly wide and uniformly stressed specimens. But in a beam just a small volume close to the edge is stressed to this level. 2) A lamination effect in the flange, where the timber will allow for some stress redistribution from the web into the flange before final web failure. 3) Web material of wood based sheeting has a tendency to break away the surface material if subjected to com- pression. This is more or less prevented for a surface that is glued to the timber. 4) The shear deformation of the web panel actually leads to a small increase of the axial stresses in the web, which slightly counteracts the favourable effects of remarks 1 to 3. By considering all these effects it should still be possible to increase these particular strength values by 10 percent, without hesitation. Such an increase is presently not promoted by the Eurocode.
Note that the verification criterion (Equation 5.6) does not cover the risk of lateral torsional buckling. The background to lateral torsional buckling of ordinary rectangular timber beams are treated in Section 3.2.3. Here some additional comments regarding thin webbed composite beams will be given. One very simple and easy to use method is to consider the compression flange to be an ordinary column that can buckle sideways without having to drag the rest of the cross-section with it. The column is cut out between two points at which the compression flange is sufficiently braced laterally. Pinned boundary conditions are assumed at these two points, that is the flange is treated as an Euler 2 column. Verification of the resistance is given by:
where kc is the reduction factor for flexural column buckling as described in Section 3.2.1. The bad thing with this model is that it grossly underestimates the bending resistance of the beam, especially for beams with large torsional stiffness G It,fic, like box-sections. Yet another problem is that Equation 5.7 is designed to work for one massive section of solid timber, but a flange usually has some web panels with different material properties attached to it. This is, however, a small and negligible problem in relation to neglecting the rest of the cross-section. A better approach to find the “true” resistance with regard to lateral torsional buckling is to rewrite Equation 5.7 as:
where My,LT,Rd is the moment resistance due to lateral torsional buckling, kcrit is the reduction factor covering up for lateral torsional buckling and My,Rd is the design resistance not
influenced by instability and taken from Equation 5.6. The reduction factor kcrit is obtained following the sequence: critical value → slenderness parameter → reduction factor, that is:
My,crit → λ → kcrit
The designation λ refer in the following sections to λcrit,rel unless otherwise stated.
Eq. 5.7
In determining λ the bending moment resistance My,Rk is needed, which is the characteristic resistance without taking lateral instability into account. This value is easily obtained using Equation 5.6 with characteristic values rather than design values. Next step is to find the critical bending moment My,cr for the actual support and loading conditions at hand. Note that this is a theoretical value obtained for a beam without any imperfections whatsoever. The basic idea is that the critical value captures the influence of geometry, boundary conditions and how the load is entering into the beam. Normally, My,crit is calculated based on some elementary load case from the literature. Cross-sectional properties needed are typically E Iz,fic, E Iw,fic and
G It,fic for z-axis bending, warping and torsion, respectively. They should be determined as
discussed in Section 5.1.1.1, that is Iz,fic, Iw,fic and It,fic are calculated based on mean values of elastic and shear moduli, while E and G in front of them should reflect the characteristic value. It is not straightforward to find My,crit, but not unrealistically problematic. Guidance is given in many handbooks such as StBK-K2 (1973). Next, the slenderness parameter for lateral torsional buckling is obtained as:
The value of My,crit should be taken as close as possible to the design section xdim where the beam is subjected to a combination of stiff axis bending, weak axis bending and torsion. Sometimes it is obvious where xdim is located, sometimes one has to make a qualified estimation. It is of course always on the safe side to choose xdim such that My,crit has its maximum, but that may not always be the same position as where the beam section will fail.
Finally, kcrit can be read from a suitable design curve of an instability chart. Presently, only one design curve exists for lateral torsional buckling of timber beams, and that curve is actually valid for sections of rectangular structural timber and glulam. No specific curve exists for more complicated sections. In Figure 5.6 the one and only curve for lateral stability (as given in Section 3.2.3.3) is brought together with the two curves used for ordinary column buckling of structural timber and glulam (as given in Section 3.2.1). A fourth extremely important curve 1 ⁄ λ2 is also depicted. This curve exactly represents the theoretical critical
load of a beam or column without any imperfections, that is critical bending moment in case of a beam and critical axial load in case of column buckling. You may easily verify this by rewriting 1 ⁄ λ2 using Equation 5.9 and then substituting into Equation 5.8, which results in
the critical bending moment. It works the same for flexural column buckling, except that the bending moment is replaced by axial force. In either case it is important to realise that the critical curve 1 ⁄ λ2 represents an absolute upper limit of the resistance, no postbuckling
resistance is possible.
The whole idea behind the chart of Figure 5.6 is that the influence of geometry, support and loading arrangements enters via λ, while the buckling curves only corrects the resistance with respect to all relevant imperfections, where the most important ones for wood based materials are non-linear material and bow imperfections. The critical load is determined under the assump tion of a linear elastic material that can sustain infinitely large stress. In reality
kcrit or kc can never exceed unity. We see that the two kc-curves give a slightly greater reduction
than the kcrit-curve. It can also be shown that a curve for lateral torsional buckling must give smaller reductions than for flexural column buckling. Since no specific curve exist for lateral torsional buckling of composite I- and box-sections it is here proposed that the present kcrit- curve can be used also for composite beams, especially since most composite beams have smaller bow imperfections than timber beams with rectangular cross-sections. And if one for some reason feel uncertain about the applicability of the kcrit-curve it should be safe to use the upper kc-curve for flexural buckling of glulam.
The procedure to find the resistance with regard to lateral torsional buckling is summarized as follows: The characteristic short term properties should be used when calculating the slender- ness according to Equation 5.9, since the slenderness only reflects geometry, boundary
conditions and load arrangement. But after determining the reduction factor kcrit as a function of λ the reduction should be applied in Equation 5.8, where My,Rd should be calculated based on design values modified for duration of load with kmod according to section 2.4.4, also see Section 3.2 in Volume 2. It is recommended to determine the slenderness parameter λ under instantaneous (short term) conditions only, since a determination under final conditions (long term) would have a small and insignificant impact on the ultimate resistance. But note that in finding My,Rd itself both “inst” and “fin” conditions may have to be considered.