Here a transformed cross-section merely means a fictitious cross-section where the width of all parts except one has been changed such that the same modulus of elasticity can be assumed for the entire cross-section. The width is measured parallel to the relevant axis of bending. If a cross-section is made of two or more materials having different modulus of elasticity a transformed cross-section will simplify equations and actual design calculations. Using a transformed cross-section is by no means necessary from a theoretical point of view, but it brings practical simplifications. Figure 5.4 shows the transformation from a real crosssection into a fictitious cross-section.
For a cross-section made up of 3 pieces as shown in Figure 5.4 the fictitious widths are determined as:
if part 2 is taken as reference. A proper value of the modulus of elasticity must be used for each material making up the cross-section. For wood based materials this value will in the long run be affected by creep, as discussed in Section 2.4.7. The influence of creep is two folded for beams with a composite cross-section. First, the deflection and deformation of the beam increase with time. Second, the internal distribution of stresses between members will change with time. The outcome is that members that are more prone to creep will be unloaded, while members less sensitive to creep will carry more of the load with increasing time. Yet another problem is that two different moduli are given in material property tables for timber materials, for example in Eurocode 5. One value corresponds to a lower percentile value (usually 5 %) and one corresponds to the mean value of the modulus. Best practice is here to always use the mean values of the modulus for each material whenever a fictitious cross-section is needed. The simple reason for this is that we must try to predict the most likely distribution of stresses
Eq. 5.1 Figure 5.4: Going for a fictitious or transformed cross-section. a) The real physical cross-section,
b) fictitious cross-section drawn to scale, if part 2 is taken as reference with E1 = 2, E2 and E3 = E2/2. E1 E2 a) b) b1 b1,fic b3,fic b2 b2 E3 b3 E2 E2 E2 GC GC y z zfic 2 2 1 1 3 3
this reasoning applies to finding the best stress distribution affecting a composite cross-section and may not be entirely correct for instability problems, which is discussed in connection to Equation 5.4 below.
Eurocode 5 uses the concepts of instantaneous and final conditions (“inst” and “fin”) to handle the creep problem. In a more general case we have n parts where i = 1, 2, …, n; and if one of the i-values is associated with the reference material, r, the fictitious width of the ith part is:
Here ULS and SLS refer to the ultimate limit state and the serviceability limit state,
respectively. ki,def is the creep factor of the ith member as if subjected to permanent loading and
specified in Eurocode 5 for different materials. ψ2 is the reduction factor for quasipermanent loading and should be determined for the load causing the largest stress. If that is a permanent action, a value of 1 should be used. When and how to use the different μi-values in Equation 5.2 is explained as:
μi,ULS,inst = μi,SLS,inst is used in the ULS or SLS under instantaneous conditions, that is as if
all design loads are applied during a period shorter than a year or so. This value should, in the ULS, be used to check the resistance of those members that are most prone to creep.
μi,ULS,fin is used in the ULS under final conditions, that is towards the end of a design life of
some 50 years or so. This value should, in the ULS, be used to check the resistance of those members that are less sensitive to creep. The technique takes creep into account in an approximate way. Especially large variable loads act for short periods of time, under which not much creep will take place.
μi,SLS,fin is used in the SLS under final conditions to check deflections and deformations after
a long time like 50 years. This value will give an estimation of creep deformations towards the safe side, as only the creep factor for permanent load kdef is used. In practical design calculations the centre of gravity (GC), second area moment of inertia and bending stiffness are calculated for the fictitious cross-section, and in case of bending about the y-axis usually denoted as zfic, Iy,fic and Er Iy,fic. Note that zfic is measured from an arbitrarily chosen point of reference and that GC is the neutral axis in case of pure bending, that is no
Observe that when stresses are calculated for the fictitious section, such that they are assumed
to be uniformly distributed along the width bi,fic for a given z-coordinate, they are also fictitious and must be smeared out over the real width using the appropriate value of μi from Equation 5.2 in the expression:
For global stability problems such as column buckling and lateral torsional buckling of composite beams, the recommendation to always use the mean value of the elastic and shear moduli will be in conflict to the definition of slenderness parameters (see Sections 3.2.1 and 3.2.2), which are generally defined as:
The conflict is that the slenderness parameter should use the most probable ratio, which is achieved if both the numerator and denominator are based on either characteristic or mean values of the involved material parameters. Only characteristic material properties are
available for the numerator (the resistance is based on fm,k, fc,k, ft,k, etc.), while both characteristic and mean values are available for the denominator (the critical load depends on E, G, etc. and not on any strength parameters). It is therefore recommended that E- and G-values used in stiffness expressions are taken as (lower) characteristic values, while the fictitious section is still based on mean values of the moduli. Typical stiffness values involved in finding critical loads are Er Iy,fic, Er Iz,fic, Er Iw,fic and Gr It,fic for y-axis bending, z-axis bending, warping and torsion, respectively.
The difference is usually smaller than 5 percent in the final design resistance of a typical com- posite cross section, if comparing results obtained using either characteristic or mean values of the elastic moduli involved in determining the dimensions of the fictitious cross-section. If global instability is a part of the analysis the difference usually stays within 10 percent, if mixing characteristic and mean values when determining the slenderness parameter according to Equation 5.4. Hence, we conclude that a principal mistake regarding the used moduli will have a small impact on the final result, if compared to the large scatter of material data and the magnitude of partial coefficients used. The same also holds if mixing up the μi-values given in Equation 5.2, which in general will have a small influence on the final result. One can really ask if all the extra work associated with these different μi-values are worth the effort in practical design situations.