Keywords: sensitivity, laminated beam, piezoelectric, sensors-actuators, layerwise theory.

This paper presents the sensitivity analysis of laminated beams integrated with piezoelectric sensors and, or actuators, that are called "laminated smart structures". These novel structures have found many practical applications in aerospace and automotive industries. They are receiving growing attention in research community. However, due to the multidisciplinary nature of the problem and large number of parameters involved, the tremendous potential of laminated smart structures still remains in many aspects unexplored.

Sensitivity analysis can be used in the preliminary design and to reassess the mathematical models through providing the trends of important changes in parameters. In addition, the sensitivity analysis is the predominated contributor to the cost and time required in any gradient based optimization technique [1]. The core of any design procedure perhaps is the mathematical model used in the analysis.

Most of the existing works on laminated smart structures are based on equivalent single-layer (ESL) theories [2] which assume continuous displacements through the thickness leading to continuous transverse strain through the thickness. However, the stiffnesses of the adjacent layers are not equal. Thus, this assumption requires having discontinuity in transverse stress through the thickness which is contrary to the equilibrium of the inter-laminar forces. In general, the ESL theories provide acceptable results for relatively thin laminates. However, laminated smart structures are highly inhomogeneous due to application of different materials and specific geometry and require a robust theory with capability to incorporate the electrical and mechanical coupling effects and material inhomogenity through the thickness.

In this work, a layerwise displacement theory [3] is used as the mathematical model to develop a systematic sensitivity analysis for laminated beams with embedded and/bonded sensors and actuators. The layerwise formulation has been developed based on the piecewise displacement variation through the laminate thickness and has the capability to address the local through-thickness effect, such as the evolution of complicated stress-strain fields in smart composite structure and interfacial phenomena between the embedded micro-devises and passive composite plies in a smart laminate.

Sensitivity analysis is performed by development of the analytical gradients of the design constraint functions based on the layerwise displacement theory. In the static problems, the design constraints are the layer stress and nodal displacements. However, in dynamic problems, typically frequency constraints are concerned with. Sensitivity analysis techniques have been applied to investigate the influences of the variation of design variables including the layer thickness and location of piezoelectric patches on selected performance functions. Illustrated examples are presented in order to demonstrate some practical application of the developed sensitivity analysis and analytical gradients. It has been shown that the use of analytical gradients leads to faster convergence and higher accuracy in design sensitivity analysis compared to the numerical gradients based on the finite difference method.

1

Adelman, H.M. and Haftka, R.T., "Sensitivity Analysis of Discrete Structural Systems", AIAA Journal, 24(5), 823-829, 1986. doi:10.2514/3.48671

2

Reddy, J.N., "On Laminated Composite Plates with Integrated Sensors and Actuators", Engineering Structures, 21, 568-593, 1999. doi:10.1016/S0141-0296(97)00212-5

3

Robbins, D.H. and Reddy, J.N., "Analysis of Piezoelectrically Actuated Beams using a Layerwise Displacement Theory", Composite and Structure, 41(2), 265-279, 1991. doi:10.1016/0045-7949(91)90430-T

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