Stability Analysis of Two-Dimensional Structural Components on Example of Cross-Laminated Timber Wall 3
As an alternative to replacement bar method in this paper, the possibility will be explained to determine the internal forces of the risk of bending wall 2nd order theory taking into account imperfections and then perform a measurement of the cross section for bending and pressure.
In order to compare the results with the equivalent member method or to create identical conditions, only the results of the wall section between the doors are considered. Since the load that is introduced by the lintels into the wall section to be considered is concentrated on the corner area of the door openings, there (local) also results in a greater axial force than in the center of the wall section (see Figure 1).
In the equivalent member method, these local effects are not taken into account because a "smeared" axial force is expected. To take this into account in the surface design as well (to create identical conditions), a smoothing area is inserted that "smears" the internal forces over the wall section to be considered (see Figure 2). Of course, the local stresses have to be considered in the design, but this is not discussed further here.
In order to consider the stress-free pre-deformation (imperfection) according to , Chapter 5.4.4 (2), a pre-deformed FE mesh is generated from the eigenmode, which was determined in RF-STABILITY, using the RF-IMP add-on module (see Figure 3 and 4). The gauge results from  Equation 5.2 to 7.5 mm.
To determine the internal forces with second-order theory and imperfection, the pre-deformed FE mesh must be activated in the additional options of the load case or load combination (see Figure 5).
Thus, for the results, there are additional bending moments in addition to the normal forces (see Figure 6), which have to be considered in the design.
The subsequent design in RF-LAMINATE provides a utilization ratio of 94% for the wall section susceptible to buckling (see Figure 7). The utilization with the equivalent member method amounts to 144%. Due to the very small critical load factor, this difference cannot be interpreted as linear.
The differences are caused to a small, negligible part by the additional stiffness that results from the door lintels when analyzing the surface model. However, the main difference between the calculation with the equivalent member method and the calculation with the second-order theory is due to the differently applied stiffnesses. Whereas in the equivalent member design, the slenderness is calculated with the 5% quantile values of the stiffnesses, in the design with second-order theory, the design values of the stiffnesses are calculated according to  Chapter 2.2.2 or  Chapter NCI NA.9.3.3 . In , Chapter 8.5.1 (2) and  , it is pointed out, however, that when calculating single components, the 5% quantile value of the stiffness parameters divided by the partial safety factor is to be calculated and not with the design values. This affects the additional bending moment resulting from the pre-deformation in the calculation according to the second-order theory. In addition, the limit design stress according to the equivalent member method is directly smaller with k mod , while the limit design stress hardly changes according to the second-order analysis  . Therefore, strictly speaking, the stiffness should be additionally reduced by the modification factor k mod according to  Chapter E 8.5.1.
To analyze the different cases, Figure 8 shows what this means in concrete terms in a simplified system. The load is reduced to such an extent that the design is performed using the equivalent member method (Case 4). For cases 1 to 3, the stability analysis with internal forces was performed on the pre-deformed model. In case 1, the stiffness is taken into account with the design values. Case 2 calculates with the 5% quantile value of the stiffness parameters and Case 3 additionally with the stiffness parameters reduced by k mod . As also confirmed in  , the best match with the equivalent member method results for case 3.
If the reductions due to k mod for the stiffness are not observed, the influence of the timber moisture and the load duration on the stiffness parameters and thus on the determination of the internal forces is not taken into account. Thus, the design can be on the uncertain side with k mod less than 1.0. The adjusted stiffnesses can be taken into account for each load combination, for example as shown in Figure 9.
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Figure 02 - 2 - Left: Real Axial Force Distribution / Right: "Blurred" Axial Force Distribution
Figure 08 - Design Ratio Between Equivalent Member Method and Second-Order Analysis with Different Stiffnesses
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