8
2 Literature Review
Fig. 2.1 Various hypotheses
for plates and shells
Since the thickness is very small compared to the in-plane dimensions, thin-walled
structures can be considered as 2-dimensional (2D) surfaces using proper throughthickness hypothesis, as shown in Fig. 2.1. The resulting elements are called plate
or shell elements and the resulting method is 2D FE method. The through-thickness
hypothesis is usually described as plate/shell theories in the literature, which defines
the displacement distribution law through the thickness. Compared to solid elements,
2D plate and shell elements have the features of relatively high accuracy and less
computational time. Such that plate and shell elements are frequently used in smart
structures. In the plate/shell hypothesis, it assumes that the thickness remain constant
during the structural deformation, by which the transverse normal strain is neglected.
In addition, if one of the in-plane dimensions reduces to in the order of the thickness,
the structures can be treated as a line using the Bernoulli or Timoshenko beam
hypothesis. The resulting elements are called 1D line elements.
2.1.1 Kirchhoff-Love Hypothesis
The simplest plate/shell hypothesis is the Kirchhoff-Love hypothesis, known as classical plate/shell theory (CLT). The Kirchhoff-Love hypothesis assumes that a vector
normal to the mid-surface in the undeformed configuration remains normal after
deformation. A large number of papers were developed FE models with 2D element
using the Kirchhoff-Love hypothesis. Tzou and Gadre [14], Lee [15] developed
numerical models for PVDF bonded multi-layered thin plates and shells based on
the Love’s equation. Applying the Kirchhoff-Love hypothesis, Kioua and Mirza [16]
constructed a linear finite element model for bending and twisting analysis of piezoelectric shallow shells. Many other studies have developed FE models for static and
dynamic analysis of piezoelectric smart structures, see e.g. Lam et al. [17], Saravanos [18], Liu et al. [19]. Additionally, the classical plate theory was implemented
into the analysis of vibration suppression using proportional feedback control [20]
and optimal control [21].
From the assumption of classical plate/shell theory, the resulting numerical models neglect transverse shear strains. Due to the neglect of shear strains, a certain
computational error may arise in the model. However, the error is negligible if the
thickness are small enough. Therefore, the classical plate/shell theory is only valid for
2 Literature Review
Fig. 2.1 Various hypotheses
for plates and shells
Since the thickness is very small compared to the in-plane dimensions, thin-walled
structures can be considered as 2-dimensional (2D) surfaces using proper throughthickness hypothesis, as shown in Fig. 2.1. The resulting elements are called plate
or shell elements and the resulting method is 2D FE method. The through-thickness
hypothesis is usually described as plate/shell theories in the literature, which defines
the displacement distribution law through the thickness. Compared to solid elements,
2D plate and shell elements have the features of relatively high accuracy and less
computational time. Such that plate and shell elements are frequently used in smart
structures. In the plate/shell hypothesis, it assumes that the thickness remain constant
during the structural deformation, by which the transverse normal strain is neglected.
In addition, if one of the in-plane dimensions reduces to in the order of the thickness,
the structures can be treated as a line using the Bernoulli or Timoshenko beam
hypothesis. The resulting elements are called 1D line elements.
2.1.1 Kirchhoff-Love Hypothesis
The simplest plate/shell hypothesis is the Kirchhoff-Love hypothesis, known as classical plate/shell theory (CLT). The Kirchhoff-Love hypothesis assumes that a vector
normal to the mid-surface in the undeformed configuration remains normal after
deformation. A large number of papers were developed FE models with 2D element
using the Kirchhoff-Love hypothesis. Tzou and Gadre [14], Lee [15] developed
numerical models for PVDF bonded multi-layered thin plates and shells based on
the Love’s equation. Applying the Kirchhoff-Love hypothesis, Kioua and Mirza [16]
constructed a linear finite element model for bending and twisting analysis of piezoelectric shallow shells. Many other studies have developed FE models for static and
dynamic analysis of piezoelectric smart structures, see e.g. Lam et al. [17], Saravanos [18], Liu et al. [19]. Additionally, the classical plate theory was implemented
into the analysis of vibration suppression using proportional feedback control [20]
and optimal control [21].
From the assumption of classical plate/shell theory, the resulting numerical models neglect transverse shear strains. Due to the neglect of shear strains, a certain
computational error may arise in the model. However, the error is negligible if the
thickness are small enough. Therefore, the classical plate/shell theory is only valid for