Stress analysis of historical roof trusses poses specific problems which are otherwise not common in structural engineering. As opposed to the analysis of modern constructions, the idealization ofthe structure is not straightforward, in particular because the stiffnesses of the joints and the support conditions are not well defined. On the other hand, these modeling decisions have a decisive influence on the results, as we shall demonstrate by a little example.
In the following, we present some preliminary results from the analysis of a principal girder of the Baumburg roof under symmetric loading. Very similar analyses have also been conducted by Lewandoski & Levin (2003) for historic American scissor trusses. Although these 19th century American trusses are only superficially akin to our baroque trusses, but differ conceptually in many details, our results compare well to those already reported by Lewandoski & Levin (2003).
In the finite element analyses presented, we have considered the "liegender Stuhl" girders as frames with rigid corners. The rafters are in contact with the tilted columns of the "liegender Stuhl". All traditional timber joints have been considered as perfectly hinged,
Figure 14. Baumburg roof. Normal forces obtained for a model based on fixed supports. Thick lines: compressive forces. Thin lines: tensile forces.
quasi pin-jointed connections. By contrast, all connections employing more than one iron bolt have been modeled as rigid because the stiffness of such a connection is much higher than that of a wooden pin and a pair of iron bolts permits transfer of moments.
Firstly, we study the effect of the support conditions. Allowing the supports to slide freely, we obtain what one would probably expect, namely, tensile stresses in the scissor braces (Fig. 13). The computation demonstrates that the king post performs a good job as a tie-rod between what is essentially the upper and lower chords of a triangulated truss, constituted by the rafters and the scissor braces. In this model, the upper portions of the braces act as struts supporting the rafters, but only with minor effect. As a natural consequence of the overall structural response, the lower collar beam is also in tension, which contradicts expectation and will not be permitted by the joinery. However, neglecting the effect of the collar beam will not change the picture much.
By contrast, if we modify the model such that the inner supports of the frame are considered fixed, the stress in the cross-braces reverses sign (Fig. 14); the scissor braces then turn into some kind of an arch supporting the lower collar beam.
This model may be criticized on the grounds that the support will never be perfectly fixed in reality, and it is true that the scissor braces regain their function as tie-beams immediately once one forces even a very small outward displacement on the inner supports.
However, the role of the scissor braces is also strongly dependent on the king-post. If we assume that the king-post is imperfectly fixed to either the ridge of the roof or to the collar beam, the scissor braces are once again transformed into an arch and get compressive stresses (Fig. 15).
In summary, the loads of a scissor-braced truss may either be carried by a simple triangulated, statically determinate truss constituted by the scissor braces and the rafters, with the essential contribution of the
king-post, or, alternatively, by some kind of a polygonal arch. The transition between these two distinct and essentially incompatible load-carrying structures depends strongly on local modeling assumptions and stiffness ratios. Somewhat surprisingly, the "liegender Stuhl" frame plays only a secondary role in the overall stress distribution.
The true situation is much more difficult to assess than could be demonstrated in this greatly simplified example. Roofs which do not have a tie-beam are much more sensitive to such modeling assumptions than are roofs with a closed rafter-tie-beam triangle.
Based on experiences similar to the one presented, the authors are currently working on more detailed computational modeling of typical baroque roof trusses which lack tie-beams, including the application of interval arithmetic and fuzzy set theory.
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