Stability Analysis of Two-Dimensional Structural Components on Example of Cross-Laminated Timber Wall 3
This article explains the alternative to the equivalent member method. It offers the option to determine internal forces of the wall susceptible to buckling according to the second-order analysis considering the imperfections and to subsequently perform the cross-section design for bending and compression.
In order to compare the results with the equivalent member method or to create the identical precondition, only the results of the wall section between the doors are considered. Since the load introduced to the respective wall section by door lintels is concentrated on the corner area of the door openings, there (locally) is also a larger axial force than in the middle of the wall section (see Figure 1).
Equivalent Member Method did not consider these local effects as they were calculated with a "blurred" axial force. In order to also consider this in the surface design (to get the same conditions), an average region is entered, which "distributes" the internal forces on the respective wall section (see Figure 2). The local stresses are considered in the design, of course, and will not be further explained in this article.
In order to consider pre-deformation without stress (imperfection) according to  Section 5.4.4(2), the RF-IMP add-on module generates a pre-deformed FE mesh from the buckling mode, which was determined in RF-STABILITY (see Figure 3 and 4). The value of 7.5 mm results from Equation 5.2 of .
To determine the internal forces according to the second-order analysis, it is necessary to activate the pre-deformed FE mesh in the Extra options of the respective load case or load combination (see Figure 5).
Thus, additional bending moments arise for the results in addition to the axial forces (see Figure 6), which must be considered in the design.
The subsequent calculation in RF-LAMINATE provides for the design ration of 94% for the wall section susceptible to buckling (see Figure 7). The design ratio resulting from the equivalent member method is 144 %. Due to the very low critical load factor, this difference should not be interpreted as linear at all.
The differences lead to a small, negligible part of the additional stiffness, which is caused by the door lintels when analyzing the surface model. However, the main difference between the calculation using Equivalent Member Method and the calculation according to the second-order analysis is caused by a different application of stiffnesses. While the equivalent member design uses the 5-percentile stiffness values, the design according to the second-order analysis applies the values for stiffness properties in accordance with , Section 2.2.2, or , Section NCI NA.9.3.3. However,  Section 8.5.1(2) and  provide that the individual structural components should be calculated with the 5-percentile stiffness values divided by the partial factor, and not with the values for stiffness properties. When calculating according to the second-order analysis, this affects the additional bending moment, which results from the pre-deformation. In addition to this, the limit design stress calculated according to the equivalent member method directly with kmod will be smaller while it barely changes when calculated according to the second-order analysis . Therefore, the stiffness should always be reduced additionally by the modification factor kmod in compliance with , Section E 8.5.1.
In order to analyze the various cases, Figure 8 shows what this actually means on a simplified structure. The load is reduced until the equivalent member method design is fulfilled (Case 4). For Case 1 to Case 3, the stability analysis was performed with internal forces on the pre-deformed model. In Case 1, the stiffness is considered with the design values. Case 2 is calculated with the 5-percentile stiffness values and Case 3 with the stiffness properties reduced by kmod. As confirmed in , the result with the best conformity is provided by the equivalent member method for Case 3.
If the reductions by kmod are not considered for the stiffness, the influence of moisture content and load duration on the stiffness properties, and thus on the determination of internal forces, is also not considered. Therefore, the design applying kmod of less than 1.0 can be incorrect. The modified stiffnesses can be considered for each load combination, for example, as shown in Figure 9.
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Figure 01 - Local Effects Relating to Load Introduction
Figure 02 - Left: Real Axial Force Distribution / Right: 'Blurred' Axial Force Distribution
Figure 03 - Generation of Pre-Deformation in RF-IMP
Figure 04 - Global Pre-Deformation of Wall
Figure 05 - Considering Pre-Deformation for Load Cases or Load Combinations
Figure 06 - Bending Moments Resulting from Calculation According to Second-Order Analysis
Figure 07 - Design Ratio of Wall Section Susceptible to Buckling
Figure 08 - Design Ratio Between Equivalent Member Method and Second-Order Analysis with Different Stiffnesses
Figure 09 - Stiffness Modification for Load Cases or Load Combinations
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