Yeoshua Frostig

Buckling of sandwich panels with a flexible core—high-order theory

Abstract The buckling behavior of sandwich panels with a core that is flexible in the out-of-plane direction, also denoted as “soft” core including high-order effects, is presented. The buckling analysis consists of the formulation of the linear and the nonlinear governing equations along with the boundary conditions. The sandwich panel construction is general and consists of two skin-panels, metallic or composite laminated symmetric that may be unidentical and a flexible isotropic or orthotropic core made of foam or a low strength honeycomb. The analysis uses a high-order theory formulation, which permits nonlinear distortions of the cross-section plane of the core as well as changes in its height. The analysis determines the bifurcation loads along with the associated mode shapes, local or overall buckling modes, as well as deformations, internal resultants and stresses at skin-core interface layers due to imperfections. Numerical results using closed form solutions for simply-supported panels, with identical and non-identical skin-panels subjected to compressive inplane loads as well as imperfection analysis results, are presented. The results reveal that, under some structural configurations, the local buckling mode is a critical, rather than a global one as a result of the out-of-plane flexibility of the core.

Behavior of delaminated sandwich beam with transversely flexible core — high order theory

Abstract The bending behavior of a general sandwich beam, delaminated (debonded) at one of the skin-core interfaces, with transversely flexible core, based on variational principles is analytically investigated. The beam construction consists of upper and lower, metallic or composite laminated symmetric skins, and a soft core of a foam or low-strength honeycomb type. The delamination considered is a crack (debond) in which the crack faces may be in contact vertically, but can slip horizontally with respect to one another. The elastic analysis consists of a two-dimensional formulation for the core, in longitudinal and transverse directions, combined with a beam theory formulation for the skins. The effects of the vertical flexibility of the core, in the undelaminated and the delaminated regions, with and without contact, on the behavior are considered. The use of a high-order theory yields a non-linear displacement field in the core, in the undelaminated region, and determines the shear and the peeling (normal) stresses at the skin-core interfaces in the delaminated and the fully bonded regions, as well as at the crack tips. Any type of loading, distributed, localized or concentrated, located either at the upper or the lower skin, or at both, as well as any type of boundary and continuity conditions differing from one skin to the other and to the core at the same section, are allowed. The effect of the delamination length and location on the overall behavior and on the peeling stresses at the skin-core interfaces, are studied.

Experiments and analytical comparison of RC beams strengthened with CFRP composites

This paper deals with strengthening, upgrading, and rehabilitation of existing reinforced concrete structures using externally bonded composite materials. Five strengthened, retrofitted, or rehabilitated reinforced concrete beams are experimentally and analytically investigated. Emphasis in placed on the stress concentration that arises near the edge of the fiber reinforced plastic strip, the failure modes triggered by these edge effects, and the means for the prevention of such modes of failure. Three beams are tested with various edge configurations that include wrapping the edge region with vertical composite straps and special forms of the adhesive layer at its edge. The last two beams are preloaded up to failure before strengthening and the ability to rehabilitate members that endured progressive or even total damage is examined. The results reveal a significant improvement in the serviceability and strength of the tested beams and demonstrate that the method is suitable for the rehabilitation of severely damaged structural members. They also reveal the efficiency of the various edge designs and their ability to control the characteristic brittle failure modes. The analytical results are obtained through the Closed-Form High-Order model and are in good agreement with the experiment ones. The analytical and experimental results are also used for a preliminary quantitative evaluation of a fracture mechanics based failure criterion for the strengthened beam.

On stress concentration in the bending of sandwich beams with transversely flexible core

Abstract Stress concentrations in sandwich beams with a ‘soft’ core subjected to bending loading are investigated and the level of stresses is analytically determined. The cases discussed include stress concentration effects (i) in the vicinity of concentrated loads and supporting zones; (ii) at the edges of debonding regions-edge and inner delamination types at the interface layer between the skins and the core; and in the vicinity of vertical cut-off connections; and (iii) at the location of diaphragms that are bonded and unbonded with the adjacent core and embedded in it. The results are determined with the aid of a variational rigorous, analytical, systematic elastic high-order theory that uses closed form solutions. This theory is applicable to any type of sandwich construction with and without discrete diaphragms (i) to any type of loading, concentrated or distributed; and (ii) to any type of boundary conditions including cases in which at the same section the conditions at the upper skin are different from those at the lower skin. The stress concentration effects are presented in terms of deflections, internal forces and normal stresses (peeling) in the interface layers between the skin and the core. A parametric study is conducted and investigates the level of stresses as a result of (i) the vertical modulus of elasticity of the core; (ii) the delamination length; (iii) the reinforcing diaphragm in a typical connection; and (iv) the presence of diaphragms, either bonded or unbonded with the adjacent core, at specific locations.

Analysis of Sandwich Beams With a Compliant Core and With In-Plane Rigidity—Extended High-Order Sandwich Panel Theory Versus Elasticity

A new one-dimensional high-order theory for orthotropic elastic sandwich beams is formulated. This new theory is an extension of the high-order sandwich panel theory (HSAPT) and includes the in-plane rigidity of the core. In this theory, in which the compressibility of the soft core in the transverse direction is also considered, the displacement field of the core has the same functional structure as in the high-order sandwich panel theory. Hence, the transverse displacement in the core is of second order in the transverse coordinate and the in-plane displacements are of third order in the transverse coordinate. The novelty of this theory is that it allows for three generalized coordinates in the core (the axial and transverse displacements at the centroid of the core and the rotation at the centroid of the core) instead of just one (midpoint transverse displacement) commonly adopted in other available theories. It is proven, by comparison to the elasticity solution, that this approach results in superior accuracy, especially for the cases of stiffer cores, for which cases the other available sandwich computational models cannot predict correctly the stress fields involved. Thus, this theory, referred to as the “extended high-order sandwich panel theory” (EHSAPT), can be used with any combinations of core and face sheets and not only the very “soft” cores that the other theories demand. The theory is derived so that all core=face sheet displacement continuity conditions are fulfilled. The governing equations as well as the boundary conditions are derived via a variational principle. The solution procedure is outlined and numerical results for the simply supported case of transverse distributed loading are produced for several typical sandwich configurations. These results are compared with the corresponding ones from the elasticity solution. Furthermore, the results using the classical sandwich model without shear, the first-order shear, and the earlier HSAPT are also presented for completeness. The comparison among these numerical results shows that the solution from the current theory is very close to that of the elasticity in terms of both the displacements and stress or strains, especially the shear stress distributions in the core for a wide range of cores. Finally, it should be noted that the theory is formulated for sandwich panels with a generally asymmetric geometric layout. [DOI: 10.1115/1.4005550]

CLASSICAL AND HIGH ORDER COMPUTATIONAL MODELS IN THE ANALYSIS OF MODERN SANDWICH PANELS

Abstract The classical and the high-order computational models of unidirectional sandwich panels with incompressible and compressible cores are presented. The significant theoretical and practical differences are discussed and elaborated through some numerical examples of typical sandwich panels. The classical models considered for the incompressible panel consists of two variants of the well-known splitted rigidity approach. The first one, due to Allen and Plantema and many others, assumes that the plane section of the shear substructure takes a specific ‘zigzag’ pattern with no in-plane deformation in the face sheets and a vertical one when the flexural rigidity of the faces is ignored. The second model, due to Frostig, assumes that the plane section of the core in the shear substructure remains vertical and the face sheets are subjected to in-plane deformation as well as flexural ones. They are compared with the accurate incompressible model, denoted as ordinary sandwich panel theory (OSPT) and with the high-order sandwich panel theory (HSAPT) based on a variational approach. In case of a sandwich panel with a compressible core the elastic foundation models based are compared with the high-order one. The governing equations and the appropriate boundary conditions of the classical models have been rederived to clarify the ambiguity involved in the definition of the boundary conditions of the various computational models. The cases of simply supported panel, cantilevered and a two-span panel are used to demonstrate numerically the differences in the overall response of the panel as well as in the near vicinity of the localized loads and supports.

Branching behavior in the nonlinear response of sandwich panels with a transversely flexible core

Abstract A branching behavior of sandwich panels with a transversely flexible (“soft”) core subjected to longitudinal external forces is investigated using a geometrically nonlinear analysis. The study is based on a closed form high-order theory that allows for a general analysis without resort to the classical mode decoupling approach. The governing equations and the associated boundary conditions are presented, and the appropriate boundary conditions resulting from using edge beams are derived. An efficient path-following algorithm based on the quasi-Newton global framework has been developed. It provides a powerful numerical tool for determining the branching behavior which consists of a sequence of equilibrium states of the sandwich panel as a function of the external loading factor. Application of the general numerical analysis to the “soft” core sandwich panels reveals that they possess a complicated branching behavior with limit points and secondary bifurcations. It is shown that the wrinkling of the face sheets does not necessarily identify the buckling of the panel as a whole and in many cases it is a result of the nonlinear response. The localized buckling modes are found in some cases to be the critical ones rather than the usual sinusoidal buckling patterns. It is further shown that variations in the geometry, boundary conditions and mechanical properties of the panel constituents can lead to a qualitative shift in its nonlinear response from an imperfection-sensitive, “shell-wise” response, to an imperfection-nonsensitive, “plate-wise” one.

Boundary condition effects in free vibrations of higher-order soft sandwich beams

Natural motions of sandwich beams with a transversely flexible- (soft-) core are analyzed based on a higher-order theory formulation. The theory does not resort to the use of presumed displacement patterns and permits imposition of the different support conditions at the same boundary section. Finite differences are used to approximate the governing equations, and the deflated iterative Arnoldi algorithm is applied to solve the algebraic eigenvalue problem. Free vibration predictions of the higher-order theory are shown to be in good agreement with experiments reported in the literature. The face sheet deflections of the sandwich beams with nonidentical support conditions at the same boundary are different in the close vicinity of that boundary. The interaction between the face sheets and the core plays a crucial role in the vibration response of sandwich beams with a soft core. The parametric study shows that the qualitative sequence of the antisymmetric (global) and symmetric (local) vibration modes varies with face sheet thickness and that there are sandwich beam layouts for which some higher vibration modes arise from the interaction of the basic modes.

Free Vibrations of Curved Sandwich Beams with a Transversely Flexible Core

A high-order model for free vibrations of singly curved sandwich beams is presented. The model takes into account the transverse flexibility of the sandwich core while the faces of the sandwich are treated as thin beams. Linear equations of motion as well as the natural boundary conditions are derived. Verification of the model is performed for straight beams via a number of asymptotic cases. For the straight and curved sandwich beams, it is shown that there exist four eigenmodes. Their nature is clarified, and simple theoretical estimations for the eigenfrequencies are obtained. A numerical analysis of free vibration of the simply supported beams is carried out. Effects of design parameters of the sandwich constituents on the eigenmodes and their appropriate frequencies are investigated. The developed model may find its use in the context of various applications of curved sandwich members in the high-performance vehicles.

Stresses and failure patterns in the bending of sandwich beams with transversely flexible cores and laminated composite skins

The stresses and failure maps in a sandwich beam that consists of a transversely flexible compressible core between two laminated composite skins, are presented. The stresses and the failure maps are determined using a general, systematic rigorous, and high-order analysis that is based on variational principles, and includes the flexibility effects of the core on the global and local bending behavior of the beam. The analysis uses closed form solutions for any type of skin construction, symmetric or unsymmetric laminated composite layups, any type of core, compressible or incompressible, any type of loading, concentrated or distributed, and any types of boundary and continuity conditions that may differ from one skin to the other, even in the same section. Failure patterns are determined with the aid of the analytical description of the longitudinal stresses in the skins and the principal stresses through the thickness of the core. The stresses in the core and the skins, along with an appropriate failure criteria, for a specified three point bending beam, are demonstrated in the form of principal stresses, failure and failure load maps, that indicate possible failure patterns and locations.