Semi-numerical analysis of rectangular plates in bending

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

Chapter 8 - Basic Equations and Solution Procedure

The solution of elastostatic problems and the principles of minimum potential energy, minimum complementary work, under fuzzy boundary conditions

The aim of this research is to analyze buckling and bending behavior of a sandwich Reddy beam with porous core and composite face sheets reinforced by boron nitride nanotubes (BNNTs) and shape memory alloy (SMA) wires resting on Vlasov\'s foundation. To this end, first, displacement field\'s equations are written based on the higher-order shear deformation theory (HSDT). And also, to model the SMA wire properties, constitutive equation of Brinson is used. Then, by utilizing the principle of minimum potential energy, the governing equations are derived and also, Navier\'s analytical solution is applied to solve the governing equations of the sandwich beam. The effect of some important parameters such as SMA temperature, the volume fraction of SMA, the coefficient of porosity, different patterns of BNNTs and porous distributions on the behavior of buckling and bending of the sandwich beam are investigated. The obtained results show that when SMA wires are in martensite phase, the maximum deflection of the sandwich beam decreases and the critical buckling load increases significantly. Furthermore, the porosity coefficient plays an important role in the maximum deflection and the critical buckling load. It is concluded that increasing porosity coefficient, regardless of porous distribution, leads to an increase in the critical buckling load and a decrease in the maximum deflection of the sandwich beam.

Nonlinear electromechanical bending of bi-modular piezoelectric laminated beams

Bending and free vibration analyses of ribbed plates with a hole based on the FSDT meshless method

Static bending of a twisted beam subjected to distributed transverse loads is established by using the Timoshenko beam theory and applying the principle of minimum potential energy. The closed-form solution for the static deflection of cantilever twisted beam is then derived. The effects of the twist angle, thickness ratio and shear deformation on the static bending characteristics of twist beams are investigated and discussed.

Solving contact forces with the competition between potential energy and entropy in elastic mechanics

This paper investigates the thermal mechanical bending response of symmetric functionally graded material (FGM) plates. This article proposes a thermodynamic analysis model of both the FGM plate and FGM sandwich plate, and the model only involves four control equations and four unknown variables. The control equation is based on the refined shear deformation theory and the principle of minimum potential energy. The Navier method is used to solve the control equation. According to the method, numerical examples are provided for the thermo-mechanical bending of the symmetric FGM plate and FGM sandwich plate under a simply supported boundary condition, and the accuracy of the model is verified. Finally, parameter analysis is conducted to investigate the effects of the volume fraction index, side-to-thickness ratio, thermal load, and changes in core thickness on the thermal mechanical bending behavior of the symmetric FGM plate and FGM sandwich plate in detail. It was found that the deflection of the FGM plate is greater than that of the FGM sandwich plate, while the normal stress of the FGM plate is smaller than that of the FGM sandwich plate. Moreover, the FGM plate and FGM sandwich plate are sensitive to nonlinear temperature changes.

Finite element interpolations of arbitrary polynomial degree which are exactly and minimally C1 continuous across interelement boundaries are studied. A perspective view of the techniques proposed by other authors is presented and some new possibilities are considered. The problem is discussed with reference to the plate and shell finite element approximations, based on the principle of minimum potential energy. The approach also permits identification of sufficient conditions for guaranteeing full rank of stiffness matrices based on modified principles of potential energy. The results of this analysis can be used to investigate the linear dependencies among nodal variables which also occur in the formulation of other fourth‐order problems.

Bending and vibration of CFRP-faced rectangular sandwich plates

According to the nonlocal strain gradient and modified couple stress theories, a unified size-dependent plate model is proposed for the vibration analysis of rectangular magneto-electro-thermo-elastic (METE) nanoplates to analyze the rates of its surface energy layers. The METE nanoplate is composed of BiTiO3–CoFe2O4 materilas. The developed model can simultaneously catch nonlocal, strain gradient, and modified couple stress models, in addition to surface energy effects. Using the refined plate theory, we have developed the non-classical governing equations based on the principle of minimum potential energy. The refined plate theory is employed for the first time to get the governing equations through the modified couple stress theory. Analytical solutions are obtained for the natural frequency of rectangular METE nanoplates under the influence of various external in-plane loadings. Verification of the proposed model is done based on the comparison of the degenerated results with those previously accessible in the literature. It is demonstrated that the nonlocal scale parameter exerts an opposite impact on the degree of surface layers, as compared with those of the strain gradient and modified couple stress scale parameters. So, by enhancing the nonlocal scale parameter, it would be possible to increase the effects of surface energy layers on the natural frequencies ratio of METE nanoplate, while the influence of the surface energy layers could be reduced with the enhancement of the strain gradient and modified couple stress scale parameters.

Analytical solutions and in-situ measurements on the internal forces of segmental lining produced in the assembling process

The shear lag effect in high-rise framed tube structures is a significant issue when considering temperature variations. This effect can result in uneven axial forces in the columns, where the axial force in the middle part of the structure is smaller than in the peripheral parts. Such a phenomenon can lead to material waste or structural unsafety if the structure is not properly designed. This study proposes a two-parameter displacement function that accounts for the difference in amplitude between the two sides of a flange framed tube. A governing equation is established to describe the shear lag effect under temperature variations. The principle of minimum potential energy is used to analyze the shear lag effect in framed tube structures subjected to thermal loading. To analyze the shear lag effect, the framed tube is idealized as an equivalent continuum solid web tube structure. An analytical solution is derived based on the proposed governing equation. Furthermore, several examples are provided to illustrate the validity of the proposed theory. The study reveals that temperature variations significantly affect the shear lag effect in framed tube structures, leading to non-uniform axial force distributions that depend on the magnitude and distribution of the thermal load. The proposed analytical model, which is validated through finite element analysis, offers a simplified yet accurate method for predicting temperature-induced stresses and deformations.

A strip element method is presented to determine bending solutions of orthotropic plates. In this method, the plate is discretized into strip elements in one direction. By using the principle of minimum potential energy to a strip element, a set of fourth-order ordinary differential equations with respect to the other direction are derived. By assembling the corresponding equations of adjacent strip elements, differential equations for the whole plate are obtained. These equations can be solved analytically by using the Fourier transform method. The deflection distribution is determined by using the boundary conditions. Numerical examples for an orthotropic plate are presented and it is shown that the strip element method is an efficient and accurate numerical method for the plate bending analysis.