 # Stability Analysis of Two-Dimensional Structural Components on Example of Cross-Laminated Timber Wall 2

### Technical Article

The following article describes design using the equivalent member method according to  Section 6.3.2, performed on the example of cross-laminated timber wall susceptible to buckling described in Part 1 of this article series. The buckling analysis will be performed as a compressive stress analysis with reduced compressive strength. For this, the instability factor kc is determined, which depends primarily on the component slenderness and the support type.

The determination of the slenderness ratio requires the effective moment of inertia I, among others. This can be calculated from the bending stiffness (y-direction) of the surface (see Figure 3 in Part 1). Furthermore, the net area Anet is calculated, for which the components of the longitudinal layers in the y-direction are taken into account (see Figure 2). Since the lower percentile value of the critical buckling stress is to be determined, the fifth percentile value should be used for modulus of elasticity. For softwood timber, according to EN 338, this should be 2/3 of the mean value of the elastic modulus. The imperfection factor βc takes into account the precamber amplitude depending on the material. For members within the straightness limits, this factor is 0.2 (L/300) for solid timber, and 0.1 (L/500) for glued-laminated timber and laminated veneer lumber. Further calculations apply the imperfection factor of 0.2 for cross-laminated timber, according to  Annex K.6.3. Load duration is considered as "medium-term", resulting in a modification factor kmod of 0.8 for cross-laminated timber.

The instability factor which reduces the compressive strength is 0.37. As you can see in Figure 2, the resulting design value is 1.44 > 1.00. Thus, the stability design is not fulfilled.

To avoid the manual calculation, the equivalent member design can also be performed in the RF-TIMBER Pro add-on module. For this, the "result beam" member type is assigned to the corresponding member in the model (see Figure 3). This result beam has no additional stiffness and integrates the surface internal forces into the defined integration area. To be able to design this member in RF-TIMBER Pro, the corresponding cross-section and material has to be assigned to this member. In this case, the stiffness properties of the Stora Enso CLT 100 C5s deviate from the standard. Therefore, it is necessary to create a new user-defined material and to adjust the stiffness properties. In order to represent the moments of inertia for the design correctly, a cross-section with an effective width must be created. This can be reckoned back using the bending stiffness and the cross-section height (see Figure 3).

To obtain the same bending stiffness for the homogeneous member, we need the cross-section with a width of 92.56 mm and depth of 1,000 mm. Thus, the correct moment of inertia is determined when performing the buckling analysis. However, since the applied compression area Anet is too large in this case, it has to be reduced for the design. This reduction may be achieved by adjusting the effective length lef, for example. For this, the effective length lef,z,TIMBERPro is defined in Excel using the target value search, which results from the adjusted effective instability factor kc,z,ef (see Figure 4).

In this way, the adjusted effective length considers the cross-section area different from the effective cross-section in the buckling analysis. The design is the same as the manual calculation (see Figure 5).

If bending moments (due to the wind, for example) should be available in addition to the axial force, they can be considered in RF-TIMBER Pro with the same procedure, as the correct section modulus Sz is already considered for the bending stress. In the case of biaxial bending, the km factor can be additionally multiplied by the factor bef/bnet in the National Annex settings in order to determine the elastic section modulus Sy correctly.

#### Literature

  Eurocode 5: Design Of Timber Structures - Part 1-1: General - Common rules and rules for buildings; DIN EN 1995-1-1:2010-12  Eurocode 5: Design Of Timber Structures - Part 1-1: General - Common Rules And Rules For Buildings - National Specifications For The Implementation Of OENORM EN 1995-1-1, National Comments And National Supplements; ÖNORM B 1995-1-1:2015-06-15

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