Konrad Basler

Solid Cross Sections

This first part deals with solid prismatic shafts with cross-sectional area F which transmit a given torsionai moment T. The rotation of the shaft and the shear stresses are to be determined. The exact solution of this problem for noncircular sections was first provided by Saint -Venant.1

Skew Supported Members with Saint-Venant Torsion

The two usual assumptions are made. The members are slender, e. g. the spans exceed by far the largest extension of the cross sections, and Saint-Venant torsion alone provides a suitable basis for analysis. The torsionai property of a member is thus described completely by its torsionai rigidity G K which is again given in Fig. 4.1 for a rectangular and a monocellular, hollow cross section [v. Eqs. (1.9) and (2.5)].

Schief gelagerte Stäbe bei Saint-Venantscher Torsion

Es werden die beiden ublichen Voraussetzungen gemacht: erstens, das die Stabstatik Gultigkeit hat, d. h. die Spannweiten gegenuber den Querschnittsabmessungen gros sind, und zweitens, das der Wolbanteil zur Aufnahme der Torsion vernachlassigbar klein ist. In Rechnung wird also nur die Saint-Venantsche Torsionssteifigkeit G K gesetzt, was fur Plattenbrucken und einzellige Hohlquerschnitte in Abb. 4.1 definiert ist [s. Gl. (1.9) und (2.5)].

Torsionsmomente und Stabverdrehung bei Saint-Venantscher Torsion

In den beiden vorangegangenen Kapiteln 1 und 2 ist das Saint-Venantsche Torsionsmoment fur einen Querschnitt als gegeben betrachtet worden. Die Bestimmung des Torsionsmomentes T fur jede Stabstelle ist ein Problem der Stabstatik, dem verschiedene Kapitel dieses Buches gewidmet sind. In diesem und dem nachsten Kapitel wird die Berechnung der Schnittgrose T fur reine Saint-Venantsche Torsion (T s ) behandelt, im Kapitel 7 und 8 fur reine Wolbtorsion (T ω), und im Kapitel 9 fur gemischte Torsion, gebildet aus Saint-Venantscher und Wolbtorsion zusammen, (T = T s + T ω). Unter welchen Voraussetzungen Saint-Venantsche Torsion fur sich betrachtet werden darf, ist im Kapitel 10 angedeutet. Im allgemeinen wird dies bei allen gedrungenen Voll- und Hohlquerschnitten der Fall sein

Closed, Thin-Walled Cross Sections

To be consistent with the heading of the first chapter, “Solid Cross Sections”, the second chapter should have been titled “Hollow Cross Sections”. Since in most practical applications the walls are thin compared with the dimensions of the cross section, this chapter will deal with this special case with only few exceptions. A criterion as to what may be considered to be a thin-walled cross section is given in Section 2.2b.

Core Point Method

The three-element equations which were derived for the analysis of a folded plate are very similar to the three-moment equations for a continuous beam. There are therefore as many possible methods of solution as there are methods to solve the three-moment equations. There exists, for instance, a stress distribution method in complete analogy to the moment distribution method1.

Warping Torsion in Skew Supported Members

Skew supported members in which Saint-Venant torsion is predominant were treated in Chapter 4. The same structural systems will now be analyzed under the assumption that the Saint-Venant torsionai resistance may be neglected and that only warping resistance need be considered. Intermediate systems for which the two types of torsion should be treated jointly will be discussed in Chapter 10.

Relations Between the Folded-Plate Theory and the Theory of Slender Members

It is the purpose of this chapter to examine the differences, similarities and possible equivalence of the folded-plate theory and the theory of slender members.

Torsional Moments and Member Rotation for Saint-Venant Torsion

In the previous two chapters, the torsionai moment T acting on a cross section was assumed to be known. The determination of this torsionai moment T along the length of a member, however, is a problem of structural analysis to which various chapters of this book will be devoted. This and the next chapter will be concerned with the evaluation of the torsionai moment T s for pure Saint-Venant torsion while Chapters 7 and 8 will deal with pure warping torsion T ω. Chapter 9, finally, will consider a combination of Saint-Venant and warping torsion, T = T s + T ω. Chapter 10 summarizes the relative degree by which either Saint-Venant torsion or warping torsion dominate behavior, giving consideration to shape of cross-section, type of load, and slenderness of the member.

General Analysis of Folded Plates

In order to arrive at a general solution for this new type of structure, the individual plates are first considered separately whereupon the conditions for the compatibility at the hinges are formulated.