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CivilComp Proceedings
ISSN 17593433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 196
NonOrthogonal Solutions for ThinWalled Members: Generalized Expressions for Stresses R.E. Erkmen and M. Mohareb
Department of Civil Engineering, University of Ottawa, Canada R.E. Erkmen, M. Mohareb, "NonOrthogonal Solutions for ThinWalled Members: Generalized Expressions for Stresses", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 196, 2006. doi:10.4203/ccp.83.196
Keywords: thinwalled members, open sections, Vlasov theory, normal stresses, shear stresses, interaction relations.
Summary
The Vlasov thinwalled beam theory [1] leads to four differential
equations of equilibrium. These equations relate the crosssection lateral, transverse,
and longitudinal displacement fields and the angle of twist of the crosssection. In
general, the four equilibrium conditions are coupled. Conventional solution
techniques attempt to eliminate the coupling between the differential equations of
equilibrium. This is realized by adopting an orthogonal system of coordinates. (i.e.,
adopting the centroidal principal axes, and adopting the line joining the section shear
centre and principal sectorial origin as a fixed radius). While this approach
successfully uncouples the equations of equilibrium, it generally suffers from a
number of drawbacks including: a) the necessity to perform coordinate
transformations, b) the difficulties in handling crosssection discontinuities where
members undergo stepwise crosssectional changes (e.g., members with holes,
flange copes, the presence of longitudinal stiffening plates, etc.), c) the boundary
conditions may remain or become uncoupled after transformation, and; d) the
adoption of an orthogonal system of coordinates may necessitate resolving the
internal and external forces along the orthogonal coordinate systems. As a result, the
use of the conventional solution technique has been limited to very simple
applications. A recent study [2,3] developed a general solution of
the Vlasov thinwalled beam theory under nonorthogonal coordinate systems in a
finite element context. The new solution scheme offers a number of modelling
advantages when compared to conventional solutions based on orthogonal
coordinates. The objective of the current paper is to complement the recent study [2,3]
by developing general expressions for normal and shearing
stresses under a nonorthogonal coordinate systems.
The paper starts by formulating the possible boundary conditions for thinwalled beams under a nonorthogonal coordinate system naturally arising from the principle of virtual work. Subsequently, the paper reviews established solution procedures for displacement field variables under (a) the direct solution of equilibrium conditions, and (b) finite element solutions. Consideration is given to solutions based on orthogonal and nonorthogonal coordinates. The discussion is extended to include stress computation schemes. Expressions for normal and shearing stresses are developed to relate the normal and shearing stresses to stressresultants. These are formulated under general nonorthogonal coordinate systems. The well known stress expressions under conventional orthogonal coordinate systems are recovered as a special case from the new expressions. The new normal and shearing stress expressions are combined using the vonMises yield criterion to develop a general interaction equation valid for general beams of open crosssection. The interaction equation is applicable in the elastic range of deformation and is quite general. It involves eight stress resultants: biaxial bending, biaxial shear, normal forces, the St. Venant twisting moment, warping twisting moments, and bimoments. References
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