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PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
A Modified High-Order Theory for Sandwich Beams under Contact Loading
F. Mortazavi and M. Sadighi
Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran
F. Mortazavi, M. Sadighi, "A Modified High-Order Theory for Sandwich Beams under Contact Loading", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 263, 2006. doi:10.4203/ccp.83.263
Keywords: sandwich beam, high-order theory, elasticity solution, contact loading, finite element analysis, soft core.
Due to their excessive ability in providing a high stiffness combined with a low weight, sandwich structures are widely used in various fields of application.
Using soft core materials such as plastic foams or nomex honeycombs in modern sandwich structures has caused them to show such complex behavior under relatively concentrated loads that may lead to unexpected failure of the structure if unnoticed. Thus, it has been an important research interest in the last decade to investigate the local behavior and develop the various methods of solution for light weight sandwich structures under static and dynamic loads. In many cases, the localized loading may involve contact loading, in which the size of the loaded area depends on the force, geometry of an external body termed the indentor, and the response of the sandwich structure. A very common example is in a standard laboratory test called the three-point bend test of a sandwich beam, in which the load is applied by means of a cylindrical loading pin.
Simple theories such as classical or first order shear deformation theory often treat contact loads as "point" loads. However it has been shown that these simpler theories are not capable of describing the surprisingly complex stress and strain fields adjacent to relatively concentrated loads. The non-planar deformed cross-section of the sandwich beam, observed experimentally, suggest the need for a model which takes into account the non-linear variations of the in-plane and vertical displacement field through the depth of the core. Frostig and Baruch  used variational principles to develop a high-order sandwich beam theory (HOSBT) which takes into account the transverse flexibility of the core. "High-Order" refers to the nonlinear patterns of transverse deflection and displacements through the depth of the core which the previous theories lack or ignore.
The problem of indentation of a beam of finite length by a rigid indentor has been of great practical and analytical interest since the 1980s, because the results deviate considerably from the Hertzian solution for a half-space as a result of bending effects. However, applications to sandwich beams has been studied recently by Petras and Sutcliffe  who utilized HOSBT formulations in conjunction with an indirect numerical method to obtain the contact stress distribution. In another similar work by Swanson and Kim , the elasticity solution has been used to calculate the stresses and strains of sandwich beams subjected to contact loading by a cylindrical indentor. However, the elasticity solution is restricted to only the simply supported boundary conditions which cannot anticipate the local effects at the supports in three point bending tests as well as other practical cases. In contrast, the high-order-based theories in which skins may be considered under different boundary conditions are sufficiently capable of dealing with various practical problems.
It is obvious that accurate prediction of the contact force-area relationship and consequently the resultant stresses is extremely dependent on the accuracy of the method used for calculating the displacements. Comparisons made here between HOSBT, elasticity and finite element analysis, represent poor agreements between the former and two latter solutions. Because of the presence of abrupt changes in the nature of the contact stress distribution, the assumption that cross sections of the faces remain normal to the centroid axis is no more applicable. Thus, it is necessary to make a few slight modifications to the high-order theory in order to eliminate the errors in contact analysis results which is the aim of the present work. This accomplishment has come into effect through the introduction of a novel modified form of the higher order theory termed hereafter MHOSBT. In this modification, Timoshenko's formulation has been utilized to express the constitutive and kinematic relations for skins as well as compatibility conditions between the skins and the core, instead of classical beam theory which is used in HOSBT. An analytical solution to the governing equations was found with the aid of trigonometric series for the case where the upper and lower skins are supposed to be simply supported. Following the approach used by Swanson and Kim  for contact modeling, the resultant stresses and displacements of sandwich beams in contact with a cylindrical indentor are obtained. The results then have been compared with the elasticity and finite element (FE) analysis. The results showed a very good agreement between the elasticity and FE analysis while the approximations that exist in HOSBT made the results significantly differ from the other two results. Also the comparisons proved that the modification made to the high-order theory, changed the results to agree well with the finite element and the elasticity analyses.
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