Effect of residual stresses on the buckling behaviour of glass beams

1. Effect of residual stresses on the buckling behaviour of glass beams Jan Belis, Rudy Van Impe, Andrea Piras Introduction Lateral torsional buckling is a stability problem that causes the beam to deform out of its own plane and which evokes complicated stress situations due to bending and torsion in the element. In literature, different influence factors of the buckling load –also called critical load– are already discussed. The generally rather slender rectangular cross-section of glass beams combined with the higher design stresses of prestressed glasses enhances the risk of buckling, since the load at which fracture occurs will usually be lower than the buckling load. This is, however not the only effect of the prestressing treatment of the glass on its buckling behaviour. Normal forces acting in longitudinal direction on the beam due to residual stresses will also directly influence the value of the critical load. This is examined by means of a numerical buckling model of toughened glass. From classical structural analysis theory it is known that a normal compressive force along the longitudinal axis of a structural member will reduce its effective torsional stiffness. The opposite will be true when a normal tensile force is applied. This effect is sometimes referred to in literature as the effect of Wagner. The idea elaborated in the present contribution is that the important residual stresses in thermally “improved” glass types will cause normal stresses along the longitudinal axis of the glass element, which can influence the structural behaviour, especially when torsion is involved. This is truly the case for lateral torsional buckling of toughened glass beams. Static analysis No thermal calculations are performed in order to simulate the real toughening process. A stress pattern is artificially introduced in the model by means of a static calculation step in the analysis. In order to do so, the model has been divided throughout the thickness by parallel planes. Each resulting “slice” is allocated an initial constant stress level, fixed at the extremes and determined as the numerical mean of the desired parabolic stresses elsewhere. The simplified stress distribution is illustrated below. The stresses of the stepwise function can easily be imported in the finite elements software, where a static analysis can redistribute the stresses. In this way, a stress distribution very similar to the original parabola can be created. In addition, the resulting stress patterns near the edges are qualitatively also very similar to those found in literature. A good quantitative comparison is difficult due to the lack of reference material, but for the purpose of this study the corresponding deviations of the buckling load are of minor importance. Execution of a parametric study allowed to determine an acceptable element mesh. FACADES, FLOORS, ROOFS, bUCKLING RESIDUAL STRESSES Buckling analysis All described simulations are eigenvalue calculations; the obtained critical load is the elastic buckling load and the corresponding deformed geometry gives only formal deformation information according to the different eigenmodes. Comparative numerical simulations have been executed for beams with annealed glass as well as toughened glass, where the value of the maximum compressive stress at the surfaces is varied from 84 N/mm² to 160 N/mm². The division of the beam in multiple long “slices”, necessary for the residual stress simulation, results in a very fine mesh. For this reason even a rather simple elastic buckling analysis is rather time consuming. Compared to the buckling behaviour of the basic annealed float glass beam, the general effect of the thermally induced stresses is unfavourable. For both prestress levels the critical load is reduced: a drop of 4.08 % is noticed if maximum compressive residual stresses are equal to 84 N/mm², while the maximum stress level of 160 N/mm² corresponds to a decrease of the buckling load of 7.73 %. REFERENCES • LITERATURE • J. Belis, R. Van Impe, B. De Meester, G. Lagae, K.B. Katnam: Stability Approach of the Dimensioning of Glass Beams, Proceedings of ISAAG International Symposium on the Application of Architectural Glass - Engineering and architectural design of glass – München, 2004 • A. Luible: Stabilität von Tragelementen aus Glas, Dissertation EPFL thèse 3014, Lausanne, 2004 • D. Vandepitte: Berekening van Constructies, Vol. II, E. Story-Scientia, Gent, Antwerpen, Brussel, Leuven, 1980 • W. Laufs: Ein Bemessungskonzept zur Festigkeit thermisch vorgespannter Gläser, dissertation, Schriftenreihe Stahlbau – RWTH Aachen Heft 45, Shaker Verlag, Aachen, 2000 • A.S. Redner, G.K. Bhat: Advanced method of measuring surface stress using digital image analysis based readout, Proceedings of Glass Processing Days, Tampere, 1999, p. 671-673 LABORATORIUM VOOR MODELONDERZOEK Faculteit Ingenieurswetenschappen Vakgroep Bouwkundige Constructies Universiteit Gent ir.-arch. Jan Belis tel: +32 09 264 54 78 fax: +32 09 264 58 38 Jan.Belis@UGent.be LABORATORY FOR RESEARCH ON STRUCTURAL MODELS Faculty of Engineering Sciences Department of Structural Engineering Ghent University Technologiepark-Zwijnaarde 904 B-9052 Zwijnaarde Belgium http://www.LMO.UGent.be CONTACT