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Research Papers

Shear in Structural Stability: On the Engesser–Haringx Discord

[+] Author and Article Information

Delft University of Technology, Private: Klinkenbergerweg 74, 6711 ML Ede, The Netherlandsj.blaauwendraad@tudelft.nl

J. Appl. Mech 77(3), 031005 (Feb 04, 2010) (8 pages) doi:10.1115/1.3197142 History: Received December 22, 2008; Revised February 23, 2009; Published February 04, 2010; Online February 04, 2010

Abstract

Since Haringx introduced his stability hypothesis for the buckling prediction of helical springs over 60 years ago, discussion is on whether or not the older hypothesis of Engesser should be replaced in structural engineering for stability studies of shear-weak members. The accuracy and applicability of both theories for structures has been subject of study in the past by others, but quantitative information about the accuracy for structural members is not provided. This is the main subject of this paper. The second goal is to explain the experimental evidence that the critical buckling load of a sandwich beam-column surpasses the shear buckling load $GAs$, which is commonly not expected on basis of the Engesser hypothesis. The key difference between the two theories regards the relationship, which is adopted in the deformed state between the shear force in the beam and the compressive load. It is shown for a wide range of the ratio of shear and flexural rigidity to which extent the two theories agree and/or conflict with each other. The Haringx theory predicts critical buckling loads which are exceeding the value $GAs$, which is not possible in the Engesser approach. That sandwich columns have critical buckling loads larger than $GAs$ does, however, not imply the preference of the Haringx hypothesis. This is illustrated by the introduction of the thought experiment of a compressed cable along the central axis of a beam-column in deriving governing differential equations and finding a solution for three different cases of increasing complexity: (i) a compressed member of either flexural or shear deformation, (ii) a compressed member of both flexural and shear deformations, and (iii) a compressed sandwich column. It appears that the Engesser hypothesis leads to a critical buckling load larger than $GAs$ for layered cross section shapes and predicts the sandwich behavior very satisfactory, whereas the Haringx hypothesis then seriously overestimates the critical buckling load. The fact that the latter hypothesis is perfectly confirmed for helical springs (and elastomeric bearings) has no meaning for shear-weak members in structural engineering. Then, the Haringx hypothesis should be avoided. It is strongly recommended to investigate the stability of the structural members on the basis of the Engesser hypothesis.

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Figures

Figure 1

Results of theories and tests for helical springs (1)

Figure 2

Hypotheses of Engesser and Haringx on normal force direction, resulting in different ratio of shear force V and compressive load P

Figure 3

Simply-supported member; force and moment in cross section

Figure 4

Ratios Pcr/Pb and Pcr/Ps as function of Ps/Pb and Pb/Ps; plots for standard and modified theories

Figure 5

Vierendeel girder; example of “member” in pure shear

Figure 6

A beam-column can be considered as a combination of a beam and a strut chain. Beam and strut chain are drawn in a deflected state.

Figure 7

Equilibrating forces on infinitesimal small compressed cable part; drawn at positive second derivative of the deflection w

Figure 8

Scheme of relevant quantities and their relationships in a member with both flexural and shear deformations

Figure 9

Definition of curvature and shear strain; sign convention for moment and shear force

Figure 10

Geometry of sandwich

Figure 11

Definition of the slip deformation and section forces in a sandwich; sign convention

Figure 12

Scheme of relevant quantities and their relationships in a sandwich member

Figure 13

Comparison of Eq. 37 (Engesser hypothesis) with test results of Attard and Hunt (18)

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