Modeling of Semi-Rigid Composite Beam Made of Timber as Surface Model

Technical Article

There are several options to calculate a semi-rigid composite beam. They differ primarily in the type of modeling. Whereas the Gamma method ensures a simple modeling, additional efforts are required when using other methods (e.g. shear analogy) for the modeling which are, however, offset by the much more flexible application compared to the Gamma method.

Another option for modeling with a member model has been already shown in this article. In principle, it is possible to use this method also for surfaces. Since the modeling of, for example, coupling members is time-consuming, it is recommend to couple the surface directly to the member or the other surface. There are several options:

  • Coupling surface-surface with line hinge
  • Coupling member-surface with line release
  • Coupling surface-surface with contact solid

It is referred to the example from [1] Chapter E 8.6.2. The structural system and the cross-section dimensions are shown in Figure 01. The flexibility is considered with 133 N/mm² which results from a fastener spacing of 125 mm.

Figure 01 - Structural System and Cross-Section Dimension According to [1]

Coupling Surface-Surface with Line Hinge

The timber beam as well as the concrete will be modeled with surfaces here whereas the surface of the timber beam will be created perpendicular to the surface of the concrete. Since the transversal strain is considered for surface elements, an orthotropic material model has to be selected for timber beams. At the composite joint, the flexibility can then be implemented with a line hinge by defining the flexibility for the degree of freedom ux in the shape of a spring. For the example from [1], it amounts to 133 N/mm².

Figure 02 - Coupling Surface-Surface with Line Hinge

The advantage of this method is that the shear flow in the composite joint can be read off directly.  It is possible to create a cut in the vertical surface and to evaluate the basic internal force nxy. The results are identical with the shear analogy method from [1]. The decrease of the shear flow in the support area to 0.040 MN/m results from the singularity in this area and can be ignored.

Figure 03 - Evaluation of Shear Flow at the Level of the Composite Joint

One disadvantage might be that the stresses and superpositions have to be evaluated by using the surface internal forces. However, it is possible that a result member integrates the surface results of the timber surface here and a member design is thus ensured. To display the same centroidal distance of the single cross-sections, it would be necessary to arrange the concrete surface eccentrically by 35 mm. The influence in this case is very low, and therefore it is not considered here.

Coupling Member-Surface with Line Release

When using this method, the timber beam is modeled as member and connected eccentrically to the surface. Since the surface is normally not connected to the member, but continues (effect of continuity), no line hinge can be used here. In this case, it is necessary to use the line release. Thus, it is possible to release one of the components and to control the relation to each other by a line release type. In this case, the flexibility can be considered in the same way as for the line hinge by a line spring.

Figure 04 - Coupling Member-Surface with Line Release

The disadvantage of this method is that it is no more possible to read off the shear flow directly. In this case, a cut at the connection line of the surface to the member has to be created. This results in a shear flow to the left and right of this line. The result has to be added in this case. After the addition, the shear flow is identical to the previous method.

Figure 05 - Evaluation of Shear Flow at the Level of the Composite Joint

However, the advantage here is that the design is, for example, possible in the RF-TIMBER add-on module since the internal forces are proportionately available.

Coupling Surface-Surface with Contact Solid

Another option is to couple both surfaces with a contact solid. Both surfaces are modeled parallel to each other and a contact condition in the shape of a contact solid is established in between. The timber beam has to be also modeled  as orthotropic surface. The flexibility is implemented in this case by a surface spring. It is necessary here to extend the line spring 133 N/mm² by one plane by dividing the value with the width of the contact surface, thus 120 mm. The spring amounts to:

$\mathrm C\;=\;\frac{133\;\mathrm N/\mathrm{mm}²}{120\;\mathrm{mm}}\;=\;1.108\;\mathrm N/\mathrm{mm}³$

Figure 06 - Coupling Surface-Surface with Contact Solid

In this case again, the shear flow cannot be read off directly and has to be calculated from the shear stresses of the contact solid by multiplying the stresses again with 120 mm = 0.12 m. The result section can be exported to Excel and evaluated further there.

Figure 07 - Evaluation of Shear Stresses and Shear Flow at the Level of the Composite Joint

Since the surfaces are arranged geometrically, no eccentricity has to be defined. This modeling is certainly the most complex one and only makes sense if the composite components are surface elements (for example in the case of cross-laminated timber-concrete composite structures).

Reference

[1]   Blass, H. J.; Ehlbeck J.; Kreuzinger H.; Steck G.: Erläuterungen zu DIN 1052: Entwurf, Berechnung und Bemessung von Holzbauwerken, 2. Auflage. Karlsruhe: Bruderverlag, 2005

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