An efficient thick beam theory and finite element model with zig-zag sublaminate approximations

Averill, Ronald C.; Yip, Y. C.

doi:10.2514/6.1995-1211

Cited by 14 publications

(16 citation statements)

References 5 publications

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“…Since TOT has the same number of displacement variables as the present theory and does not require an arbitrary shear correction factor, results are also compared with the coupled TOT [16]. Simply [25,31] are considered: The frequencies ω n and the modal entities are non-dimensionalised as follows with S = a/ h:…”

Section: Results and Assessmentmentioning

confidence: 99%

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

Kapuria

Achary

2005

Arch Appl Mech

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A recently developed coupled third-order zigzag theory for the statics of piezoelectric hybrid crossply plates is extended to dynamics. The theory combines a third-order zigzag approximation for the in-plane displacements and a sub-layerwise linear approximation for the electric potential, considering all components of the electric field. The nonuniform variation of the transverse displacement due to the piezoelectric field is accounted for. The conditions for the absence of shear traction at the top and bottom surfaces and continuity of transverse shear stresses in the presence of electromechanical loading are satisfied exactly, thereby reducing the number of displacement variables to five, which is the same as in a first-or third-order equivalent single-layer theory. The governing equations of motion are derived from the extended Hamilton's principle. The theory is assessed by comparing the Navier solutions for the free and forced harmonic vibration response of simply supported plates with the exact three-dimensional piezoelasticity solutions. Comparisons for hybrid test, composite and sandwich plates establish that the present theory is quite accurate for the dynamic response of moderately thick plates.

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Section: Results and Assessmentmentioning

confidence: 99%

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

Kapuria

Achary

2005

Arch Appl Mech

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“…The stacking order is mentioned from the bottom. The five-ply substrate of beam (a) is a good test case, [21]. It has plys of thickness 0:09h=0:225h=0:135h=0:18h=0:27h of materials 1=2=3=1=3 which have highly inhomogeneous stiffness in tension and shear.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…The reference plane z ¼ 0 either passes through or is the bottom surface of the k 0 -th layer. For a beam with small width, the usual assumptions for mathematical simplification of 1D model made by other researchers, [8,15,21], which are retained in the present theory, are: assume plane state of stress ðr y ¼ s yz ¼ s xy ¼ 0Þ, neglect transverse normal stress ðr z ' 0Þ and assume the axial and transverse displacements u; w and electric potential / to be independent of y ð) electric field component E y ¼ À/ ;y ¼ 0Þ. The strain-displacement and electric field-potential relations for directions x; z are where a subscript comma denotes differentiation.…”

Section: Formulation Of Layerwise Statical Theorymentioning

confidence: 99%

An efficient coupled layerwise theory for static analysis of piezoelectric sandwich beams

Kapuria

Dumir

Ahmed

2003

Archive of Applied Mechanics (Ingenieur Archiv)

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An efficient one-dimensional model is developed for the statics of piezoelectric sandwich beams. Third-order zigzag approximation is used for axial displacement, and the potential is approximated as piecewise linear. The displacement field is expressed in terms of three primary displacement variables and the electric potential variables by satisfying the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces. The deflection field accounts for the piezoelectric transverse normal strain. The governing equations are derived using a variational principle. The present results agree very well with the exact solution for thin and thick highly inhomogeneous simply supported hybrid sandwich beams. The developed theory can accurately model open and closed circuit boundary conditions. IntroductionComposite laminates and sandwich structures with some piezoelectric layers, acting as sensors and actuators to achieve desired control, form part of a new generation of adaptive structures. Sandwich beams have high ratio of flexural stiffness to weight resulting in lower deflection, higher buckling load and higher natural frequencies compared to beams of other constructions. Piezoelectric layers are incorporated in sandwich beams for active control. Sandwich structures offer advantage of placement of the electrodes for the piezoelectric layers. A review of threedimensional (3D) continuum-based approaches, 2D theories for plates and shells and 1D theories for beams, along with their comparative study for plates under static loading, has been presented in [1]. Analytical 3D solutions are available only for some specific shapes and boundary conditions, [2, 3]. The 3D finite element (FE) analysis, [4], results in large problem size which may become computationally costly for practical dynamics and control problems. Hence, efficient accurate electromechanical coupled 2D plate and 1D beam models are required without too much loss of accuracy compared to 3D models. Early works used elastic beam models, [5][6][7], with effective forces and moments due to induced strain of actuators. A discrete layer theory with layerwise approximation of displacements was developed for elastic laminated beams with induced actuation strain in [8]. Classical laminate theory (CLT), [9], firstorder shear deformation theory (FSDT), [10], and the refined third-order theory, [11,12] have been applied without electromechanical coupling to hybrid beams and plates. Coupled CLT, FSDT,[13][14][15], and third-order, [16,17], solutions for hybrid beams and plates including the charge equation of electrostatics and electromechanical coupling have been reported. In [18], coupled discrete layer theory (DLT) was presented using layerwise approximation for displacement and potential, which yields accurate results for thin and thick beams. However, it is expensive for practical problems since the number of displacement unknowns depends on the

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“…They are a drawback since they prevent the formulation of C 0 computationally efficient finite elements. The C 1 continuity requirement of conventional ZZ models that does not exhibit a piecewise variation of the transverse displacement has been circumvented by Averill [33] and Averill and Yip [34] using interdependent interpolation (see, Tessler and Dong [51]) and penalty function concepts (see, Reddy [53]). In this paper, the following representation is introduced to convert first-order derivatives into new variables:…”

Section: Displacement and Stress Fieldsmentioning

confidence: 99%

Layerwise mixed element with sublaminates approximation and 3D zig‐zag field, for analysis of local effects in laminated and sandwich composites

Icardi

2006

Numerical Meth Engineering

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SUMMARYThe analysis of damaged sandwich composites and thick laminates remains a still challenging task, owing to their intricate distributions of displacements and stresses across the thickness. The arduous task is capturing these distributions with an affordable computational effort. The use of sublaminate zig-zag models appears promising since they allow to group several physical layers into a sublaminate without violating the continuity of interlaminar stresses. To contribute to research in this field, a mixed, sublaminate element is developed in which displacements and stresses are approximated by two independent zig-zag models. Like for any zig-zag model, the interfacial contact conditions are fulfilled a priori suitably choosing the continuity functions and constants. In addition to the contact conditions on interlaminar shears, here those on the transverse normal stress and stress gradient are also fulfilled, as required by the elasticity theory. This uncustomary feature implies the presence of unwise terms which need C 2 continuous shape functions. To overcome this drawback, C 1 and C 2 terms are eliminated from the displacement field, as allowed by the mixed approach, then the interlaminar stresses obtained by the C 0 -made displacement model are cast into a separate zig-zag representation which restores their continuity at the interfaces. As nodal d.o.f. the three displacements and the three interlaminar stresses at the upper and lower faces are assumed, in order to fulfil the contact conditions assembling the sublaminates. Standard, C 0 , serendipity interpolation polynomials are used to represent both internal displacements and stress fields. As a consequence of this choice, the intra-element stress equilibria are fulfilled in an approximate integral form. This makes easier the development of the element, but does not compromise its accuracy and convergence rate, as shown by former applications. A postprocessing procedure is developed which improves the estimation of interlaminar stresses and helps to reduce the number of sublaminates. The result is that the analysis can be carried out using a single computational layer across the thickness and a reasonably fine in-plane discretization, even with distinctly different material properties of layers and in presence of damage. The overall computational time is reduced to an half of that required by a conventional sublaminate model, like present one deprived of zig-zag terms and postprocessing procedure.

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An efficient thick beam theory and finite element model with zig-zag sublaminate approximations

Cited by 14 publications

References 5 publications

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

An efficient coupled layerwise theory for static analysis of piezoelectric sandwich beams

Layerwise mixed element with sublaminates approximation and 3D zig‐zag field, for analysis of local effects in laminated and sandwich composites

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