OPTIMIZATION CONTROL OF COMPOSITE LAMINATES FOR MAXIMUM THERMAL BUCKLING AND MINIMUM VIBRATIONAL RESPONSE

The problem of maximizing the thermal buckling and minimizing the vibrational response of composite laminates is solved using optimal design and active control procedures. The problem is formulated based on a first-order shear deformation laminate theory with various cases of boundary conditions. The design objective is to maximize thermal buckling using ply thickness and the fiber orientation angle as design variables. The active control objective is to minimize the laminate vibrational response with the minimum possible expenditure of control energy. The vibrational response is expressed in terms of the total elastic energy of the laminate and a penalty functional of closed-loop control force. Liapunov-Bellman theory is used to obtain solutions for controlled deflections and optimal control force. Comparative examples are given for angle-ply antisymmetric laminates subjected to a uniform temperature distribution. A general representation for the design variables is presented such that the ply thickness is a function of the number of layers. Some of the obtained numerical results are compared with their counterparts in the literature.     

THERMAL BUCKLING OF THICK ANTISYMMETRIC ANGLE-PLY LAMINATES

The buckling behavior of moderately thick antisymmetric angle-ply laminates that are simply supported and subject to a uniform temperature rise is analyzed. Transverse shear deformation is accounted for by employing the thermoelastic version of the Reissner-Mindlin theory. Results for the classical thin-plate theory are obtained as a special case. Numerical results are presented for fiber-reinforced laminates and show the effects of ply orientation, number of layers, plate thickness, and aspect ratio on the critical buckling temperature. Finally, an optimization procedure is proposed for the design of laminates having maximum resistance to thermal buckling.     

INFLUENCE OF DAMAGE ON THE THERMAL RESPONSE OF GRAPHITE-EPOXY LAMINATES

A “constrained ” displacement finite element approach for studying the influence of transverse cracks and delaminations on the thermal response of laminated composites is presented. Typical results are given in the form of percent retention curves for the coefficient of thermal expansion as a function of crack density. Cross-ply and quasi-isotropic T300/5208 graphite-epoxy laminates are considered. It is shown that transverse cracks can have a significant influence on the coefficient of thermal expansion, but delaminations located symmetrically about the laminate mid-plane have no influence on thermal expansion.     

THERMAL BUCKLING ANALYSIS OF ISOTROPIC AND COMPOSITE PLATES WITH A HOLE

Abstract

We analyze thermal buckling of both circular isotropic plates and square antisymmetric angle-ply laminates with a hole in the middle, and subject to a uniform temperature rise, by either closed form solution for the former or finite-element method for the latter. Thin-plate theory is used to analyze the isotropic plates. However, a high-order displacement theory including high-order terms along the transverse direction taking into account transverse normal strain is used in the case of laminates. Results for the isotropic plates indicate that in contrast to the reduction in mechanical buckling loads due to the hole, the thermal buckling temperature actually rises as the size of the hole increases, which indicates that the effect on stress reduction exceeds that on stiffness decrease. Results are more complicated for laminated plates, due to anisotropy.     

GEOMETRICALLY NONLINEAR ANALYSIS OF TIMOSHENKO BEAMS UNDER THERMOMECHANICAL LOADINGS

Considering the axial extension and the transversal shear deformation, geometrically nonlinear governing equations for static deformations of Timoshenko beams subjected to thermal as well as mechanical loadings are formulated. As an example, on the basis of the governing equations, thermal postbuckling response of an immovably pinned-fixed Timoshenko beam subjected to a static transversely nonuniform temperature rise is numerically analyzed by using a shooting method. Characteristic curves showing the relationships between the beam deformation and temperature rise are presented. The thermal postbuckled configurations and the equilibrium paths of the beam are presented. In particular, the effects of shear deformation on the buckling response are quantitatively investigated. The numerical results show, as we know, that shear deformation effects become significant with decrease of the slenderness and with increase of the shear flexibility.     

A procedure for the approximated response history analysis of linear thermoelastic structures

This article presents a procedure for the approximated response history analysis of linear thermoelastic structures subjected to pure thermal loading based on eigenvalue analysis. The underlying assumption is that the mechanical system response does not affect heat transfer. Typically, the mechanical response of a structure subjected to pure thermal loading is such that inertia forces can be neglected but there are instances, however, where this is not the case. An approach to make this determination is also presented for a generic pair of retained mechanical and thermal modal coordinates and then extended to the multiple degree-of-freedom case.     

Application of homotopy perturbation method and inverse prediction of thermal parameters for an annular fin subjected to thermal load

In this work, application of the homotopy perturbation method (HPM) and an inverse solution for estimating unknown thermal parameters such as the variable thermal conductivity parameter (β), the thermogeometric parameter (K), and the nondimensional coefficient of thermal expansion (χ) in an annular fin subjected to thermal stresses is presented. Initially, to obtain the nondimensional temperature distribution from the heat equation, the forward method is employed using an approximate analytical solution based on HPM. Thereafter, a closed form solution for the temperature-dependent thermal stresses is obtained using the classical theory of thermoelasticity coupled with HPM solution containing the temperature distribution. Next, for satisfying a particular stress criterion which makes relevance in selecting appropriate configurations for selecting the finned system, unknown thermal parameters are obtained using an inverse approach based on the Nelder–Mead simplex search minimization technique. The objective function is taken as the sum of square of the residuals between the measured stress field and an initially guessed value which is updated iteratively. It is found that more than one type of temperature distribution may yield a given stress distribution, thereby giving rise to different fin efficiencies. The agreement between the actual and the predicted results was found to be satisfactory.     

THERMAL BUCKLING AND POSTBUCKLING OF SYMMETRICALLY LAMINATED COMPOSITE PLATES

The thermal buckling and postbuckling response of symmetrically laminated composite plates are discussed. Using variational methods in conjunction with a Ray-leigh-Ritz formulation, thermal buckling and postbuckling are investigated for two laminates, a ( ±45/0/90) _s and a ( ± 45/0₂ ) _s, under two different simple support conditions, fixed and sliding. These laminates are subjected to the condition of a uniform temperature change. The effects of the principal material axes not being aligned with the edges of the plate, referred to here as material axis skewing, are also investigated. Although differences between buckling temperatures for the two support conditions were small, support conditions can have a large influence on thermal postbuckling response. In general, plates with fixed simple supports defied more than plates with sliding simple supports. In addition, support conditions can influence modal interaction. Skewing of the material axis decreases the buckling temperatures of both laminates and, like fixed support conditions, causes increased postbuckling deflections. Skewing also influences modal interaction.     

NONLINEAR RESPONSE OF FLAT AND CURVED PANELS SUBJECTED TO THERMOMECHANICAL LOADS

The results of an analytical study of the nonlinear response of flat and curved panels subjected to pre-existing, nondestabilizing lateral pressure and thermal loads and to mechanical edge loads are presented. The mechanical loads include uniaxial compression loads and combinations of uniaxial compression and transverse tension or compression loads that are increased monotonically into the postbuckling response range of the panels. The structural model used to analyze the panels is based on a higher order shell theory that includes transverse-shear flexibility, initial geometric imperfections, and von Karman_type geometric nonlinearities. The edges of a panel are modeled as simply supported edges with the displacement normal to the edge face either unrestrained or fully restrained. Results are presented for transversely isotropic single-layer panels and three-layer sandwich panels that illustrate how the temperature field, initial imperfections, lateral pressure loads, and mechanical edge loads interact to change the character of the nonlinear panel response. Some response curves are presented that have classic unstable, asymmetric bifurcation behavior and intense snap-through instabilities. Other results show that, for some cases, these interactions can reduce the intensity of snap-through instabilities and even eliminate this form of instability altogether for certain ranges of loading and structural parameters. In addition, results are presented that show how transverse-shear flexibility affects the interactions of the temperature field, the initial imperfections, the lateral pressure loads, and, thus, the character of the nonlinear panel response. One important finding of the present study is that linear bifurcation buckling analyses may not indicate adequately the onset of significant nonlinear deformations of a geometrically perfect, shallow curved panel for certain combined mechanical loading conditions. This finding may affect current preliminary design practice.     

Investigation of Aerothermoelastic Behaviors of Functionally Graded Panels in Supersonic Flows

Considering the potentials of Functionally Graded Panels (FGPs) in aerospace field, it is necessary to study the aerothermoelastic behaviors of FGPs in supersonic flows. In this study, Piston Theory Aerodynamics (PTA) and Eckert reference enthalpy method are used to model aerodynamic force and heating, respectively. The 2-D heat conduction equation is solved and the impact of elevated temperature on the mechanical properties of FGPs is considered to build an aerothermoelastic two-way coupling model of FGPs, and Finite Element Method (FEM) is used to approach the solution. As the results, it is found that there exist three different regions in the bifurcation diagram, namely, thermal buckling region, critical region and flutter region. Due to the inhomogeneous distribution of thermal expansion coefficient, the panel buckles up first and then buckles down via vibration, as thermal buckling happens. Also, irregular vibrations are observed in the critical region of bifurcation diagram. In the flutter region, the dynamic behavior of FGPs is discontinuous and very sensitive to initial conditions. With the impact of aerothermoelastic two-way coupling, different FGPs behaviors lead to the differences in temperature distribution. In particular, the final buckling position and vibration center move to lower positions, and lower temperature region near leading edge is left in the FGPs, because of thermal moment. Also, regular vibrations, rather than irregular vibrations, are easy to extract more principal and regular POD (Proper Orthogonal Decomposition) modes. The results presented could be applied to the analysis and design of Functionally Graded Panels in supersonic flows.     

Finite Element Analysis of Thermal Buckling of Rectangular Laminated Composite Plates with Circular Cut-Out

In this work, thermal buckling analysis of symmetric and anti-symmetric laminated composite plates with a cut-out is presented. The plate is assumed to be subjected to a uniform temperature rise for different boundary conditions. The thermal buckling analysis is performed using the code developed in MATLAB software. The stiffness matrices and thermal force vector are derived according to first-order shear deformation theory (FSDT). To have more control on the mesh pattern around the cut-out, convenient meshes are manually constructed using a mesh generation algorithm in which mesh density around the hole can be controlled. The results of FEM code is compared with ABAQUS's solutions and with those available in the literature. After that, the effects of cut-out size, boundary conditions, plate aspect ratio, and stacking sequence on critical thermal buckling temperature are investigated for symmetric and anti-symmetric plates. Also, plates with cut-out located at positions other than the center of plate are investigated and useful conclusions are derived from the numerical results.     

Multi-Objective Optimization of Turbulent Tube Flows Over Diamond-Shaped Turbulators

In this paper, experimentally derived correlations of heat transfer and pressure drop are used in a Pareto-based multi-objective optimization approach to find the best possible combinations of heat transfer and pressure drop in a tube fitted with diamond-shaped turbulators in tandem arrangements. The design variables are two geometrical parameters of diamond-shaped turbulators, namely, cone angle and tail length ratio, Reynolds number, and Prandtl number. The objectives are maximizing the nondimensional heat transfer coefficient and minimizing the nondimensional pressure drop in a round tube fitted with diamond-shaped turbulators. It is shown that some interesting and important relationships as useful optimal design principles involved in the thermal performance of round tube fitted with diamond-shaped turbulators can be discovered by the Paret- based multi-objective optimization approach.     

Variable kinematic shell elements for composite laminates accounting for hygrothermal effects

This article presents advanced shell models for the steady-state hygrothermal analysis of composite laminates. The Carrera Unified Formulation is used to derive refined models that include both layer-wise and equivalent single layer models. The governing equations are derived from the principle of virtual displacement taking into account thermal and hygroscopic effects. The geometrical relations for the exact cylindrical geometry are here considered. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. The mixed interpolation of tensorial component method is applied to a nine-node shell element to contrast the membrane and shear locking phenomena. Simply supported cross-ply cylindrical shells with antisymmetrical lamination subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness/curvature ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are presented. Variable kinematics are compared regarding both accuracy and computational costs. The results show that all the kinematics can approximate the transverse shear stress distribution through the thickness with satisfactory accuracy when su?cient expansion terms are adopted. In some cases, miscellaneous expansions can lead to significant reductions in computational costs. The results presented here can be used as benchmark solutions for future works.     

Thermal Buckling of Piezolaminated Plates Subjected to Different Loading Conditions

A thermal buckling analysis is presented for simply supported rectangular laminated composite plates that are covered with top and bottom piezoelectric actuators, and subjected to the combined action of thermal load and constant applied actuator voltage. The thermomechanical properties of composite and piezoelectric materials are assumed to be linear functions of the temperature. The formulations of the equations are based on the higher-order laminated plate theory of Reddy and using the Sanders nonlinear kinematic relations. The closed-form solutions for the buckling temperature are obtained through the Galerkin procedure and solving the resultant eigenvalue problem, which are convenient to be used in engineering design applications. Numerical examples are presented to verify the proposed method. The effects of the plate geometry, fiber orientation in composite layers, lay-up configuration, different utilized piezoelectric materials, temperature dependency of material properties, thermal conductivity, and energy generation on the buckling load are investigated.     

Buckling of functionally graded plates under thermal,axial, and shear in-plane loading using complex finite strip formulation

A complex finite strip method was used to study the buckling of functionally graded plates (FGPs) under thermal and mechanical (longitudinal, transverse, and shear in-plane) loading. The mechanical characteristics of FGPs were assumed to vary through the thickness, according to power law distribution. The nonlinear temperature distribution in the direction of the plate thickness was assumed according to thermal conduction steady state conditions. In complex finite strip method, the polynomial Hermitian functions were assumed in the transverse direction and the complex exponential functions were used in the longitudinal direction to evaluate the standard and geometric stiffness matrices that have the ability of calculating the critical shear stress in contrast to trigonometric shape functions. The solution was obtained by the minimization of the total potential energy and solving the corresponding eigenvalue problem. In addition, numerical results for FGPs with different boundary conditions were presented and compared with those available in the literature and the interaction curves of mechanical and thermal buckling capacity of FGPs were obtained.     

Thermal buckling analysis of laminated composite beams using hyperbolic refined shear deformation theory

In this article, thermal buckling of laminated composite beams, based on hyperbolic refined shear deformation theory, presented for the first time, is formulated using the principle of minimum total potential energy. Navier’s analytical solution is derived to analytically solve the differential equations and the thermal critical buckling is presented in closed-form solution. The effects of temperature distribution, length to thickness ratio, modulus ratio, and thermal expansion coefficient ratio on thermal buckling of isotropic, orthotropic and laminated composite beams are investigated. The accuracy of the numerical model is verified by comparison with the available results in the literature.     

Hygrothermal analysis of multilayered composite plates by variable kinematic finite elements

Advanced plate models with variable kinematics for steady state hygrothermal analysis of composite laminates are proposed. The refined models discussed include both layer-wise (LW) and equivalent single layer (ESL) models, and the Carrera Unified Formulation (CUF) is used. The mixed interpolation of tensorial component (MITC) method is applied to a nine-node element to contrast the shear locking phenomena. The governing equations are derived from the principle of virtual displacement (PVD) taking into account elastic mechanical, thermal and hygroscopic effects. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. Cross-ply plates with symmetrical lamination and simply supported edges subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are compared. Variable kinematics with a variety of thickness functions are compared regarding both accuracy and computational costs. The results show that all the kinematics proposed can approximate the transverse shear stress distribution through the thickness with satisfactory accuracy when su?cient expansion terms are adopted. In some cases, miscellaneous expansions can lead to significant reductions in computational costs. The results presented here can be used as benchmark solutions for future works.     

Transient Evolution of Inter Vessel Gap Pressure Due to Relative Thermal Expansion Between Two Vessels

In a typical liquid metal cooled fast breeder reactor (LMFBR), a cylindrical sodium filled main vessel, which carries the internals such as reactor core, pumps, intermediate heat exchangers etc. is surrounded by another vessel called safety vessel. The inter vessel gap is filled with nitrogen. During a thermal transient in the pool sodium, because of the relative delay involved in the thermal diffusion between MV and SV, they are subjected to relative thermal expansion or contraction between them. This in turn results in pressurisation and depressurisation of inter vessel gap nitrogen respectively. In order to obtain the external pressurization for the buckling design of MV, transient thermal models for obtaining the evolutions of MV, SV and inter gap nitrogen temperatures and hence their relative thermal expansion and inter vessel gap pressure have been developed. This paper gives the details of the mathematical model, assumptions made in the calculation and the results of the analysis.     

Two-dimensional thermal shock problem of generalized thermoelastic multilayered structures considering interfacial conditions

A two-dimensional model of the generalized thermoelasticity with one relaxation time is established. The resulting nondimensional coupled equations together with the Laplace and Fourier transform techniques are applied to a specific problem of multilayered structures considering thermal resistance subjected to thermal shock and traction-free surface. The solutions in the transformed domain are obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transform. Numerical results are represented graphically to estimate the effects of the thermal resistance and thermal conductivities on the temperature, displacement, and stress distributions.     

The thermal and mechanical behaviour of structural steel piping systems

The temperature, the deformation and the stress field in thermo-mechanical problems play a very important role in engineering applications. This paper presents a finite element algorithm developed to perform the thermal and mechanical analysis of structural steel piping systems subjected to elevated temperatures. The new pipe element with 22 degrees of freedom has a displacement field that results from the superposition of a beam displacement, with the displacement field associated with the section distortion. Having determined the temperature field, the consequent thermal displacement produced in the piping systems due to the thermal variation can be calculated. The temperature rise produces thermal expansion and a consequent increase of pipe length in the structural elements. For small values of the ratio of the pipe thickness to mean radius, the thermal behaviour can be calculated with adequate precision using a one-dimensional mesh approach, with thermal boundary conditions of an axisymmetric type across the pipe section. With this condition, several case studies of piping systems subjected to elevated temperatures and mechanical loads are presented and compared with corresponding results from commercial finite element codes. The main advantage of this formulation is associated with reduced time for mesh generation with a low number of elements and nodes. Considerable computational effort may be saved with the use of this finite pipe element.