THIRD ORDER SHEAR DEFORMATION MODEL FOR LAMINATED SHELLS WITH FINITE ROTATIONS:FORMULATION AND CONSISTENT LINEARIZATION

The paper presents an approach for the formulation of general laminated shells based on a third order shear deformation theory. These shells undergo finite (unlimited in size) rotations and large overall motions but with small strains. A singularity-free parametrization of the rotation field is adopted. The constitutive equations, derived with respect to laminate curvilinear coordinates, are applicable to shell elements with an arbitrary number of orthotropic layers and where the material principal axes can vary from layer to layer. A careful consideration of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations leads to symmetric tangent stiffness matrices. The matrix formulation adopted here makes it possible to implement the present formulation within the framework of the finite element method as a straightforward task.

3-D Temperature Fields in Laminated Shells Subjected to Thermo-Loads

The temperature fields in the laminated shells were studied, including opencylindrical shells and cylindrical shells, according to the thermal theory. Analyticalsolution of the temperature in the shells with the known temperature on the surfaces waspresent. The thinning layer approach was introduced to simplify the three-dimensionalheat conduction equation. Firstly, the layered shell was divided into N thinner layers. Thegoverning equation was simplified by replacing the variable r by r0 in the center line ofevery thin layer. The general solutions of temperature satisfying the simplified threedimensionalgoverning equation in single-layered shell were deduced in the cylindricalcoordinate system. Then, the temperature and heat flux relationships between the surfacescould be found by transferring matrixes. According to the continuities of temperature andheat flux in the interface of the laminates, the temperature and heat flux relationships ofthe surfaces were derived. With the temperature condition on the surfaces, the unknowncoefficients in the general solution of temperature were obtained. Finally, the effects fromthe thinning layer approach were eliminated by analyzing different numbers of thin layers.The validity and accuracy of the proposal method were proved from the convergence andcomparison studies. And several numerical examples were studied to investigated thetemperature effects from surface temperatures, geometric size of the shells andcomposition of layers.

Snap-through behaviors and nonlinear vibrations of a bistable composite laminated cantilever shell:an experimental and numerical study

The snap-through behaviors and nonlinear vibrations are investigated for a bistable composite laminated cantilever shell subjected to transversal foundation excitation based on experimental and theoretical approaches.An improved experimental specimen is designed in order to satisfy the cantilever support boundary condition,which is composed of an asymmetric region and a symmetric region.The symmetric region of the experimental specimen is entirely clamped,which is rigidly connected to an electromagnetic shaker,while the asymmetric region remains free of constraint.Different motion paths are realized for the bistable cantilever shell by changing the input signal levels of the electromagnetic shaker,and the displacement responses of the shell are collected by the laser displacement sensors.The numerical simulation is conducted based on the established theoretical model of the bistable composite laminated cantilever shell,and an off-axis three-dimensional dynamic snap-through domain is obtained.The numerical solutions are in good agreement with the experimental results.The nonlinear stiffness characteristics,dynamic snap-through domain,and chaos and bifurcation behaviors of the shell are quantitatively analyzed.Due to the asymmetry of the boundary condition and the shell,the upper stable-state of the shell exhibits an obvious soft spring stiffness characteristic,and the lower stable-state shows a linear stiffness characteristic of the shell.

Theoretical and experimental investigation of the resonance responses and chaotic dynamics of a bistable laminated composite shell in the dynamic snap-through mode

The dynamic model of a bistable laminated composite shell simply supported by four corners is further developed to investigate the resonance responses and chaotic behaviors.The existence of the 1:1 resonance relationship between two order vibration modes of the system is verified.The resonance response of this class of bistable structures in the dynamic snap-through mode is investigated,and the four-dimensional(4D)nonlinear modulation equations are derived based on the 1:1 internal resonance relationship by means of the multiple scales method.The Hopf bifurcation and instability interval of the amplitude frequency and force amplitude curves are analyzed.The discussion focuses on investigating the effects of key parameters,e.g.,excitation amplitude,damping coefficient,and detuning parameters,on the resonance responses.The numerical simulations show that the foundation excitation and the degree of coupling between the vibration modes exert a substantial effect on the chaotic dynamics of the system.Furthermore,the significant motions under particular excitation conditions are visualized by bifurcation diagrams,time histories,phase portraits,three-dimensional(3D)phase portraits,and Poincare maps.Finally,the vibration experiment is carried out to study the amplitude frequency responses and bifurcation characteristics for the bistable laminated composite shell,yielding results that are qualitatively consistent with the theoretical results.

Inter-well internal resonance analysis of rectangular asymmetric cross-ply bistable composite laminated cantilever shell under transverse foundation excitation

The chaotic dynamic snap-through and complex nonlinear vibrations are investigated in a rectangular asymmetric cross-ply bistable composite laminated cantilever shell,in cases of 1:2 inter-well internal resonance and primary resonance.The transverse foundation excitation is applied to the fixed end of the structure,and the other end is in a free state.The first-order approximate multiple scales method is employed to perform the perturbation analysis on the dimensionless two-degree-of-freedom ordinary differential motion control equation.The four-dimensional averaged equations are derived in both polar and rectangular coordinate forms.Deriving from the obtained frequency-amplitude and force-amplitude response curves,a detailed analysis is conducted to examine the impacts of excitation amplitude,damping coefficient,and tuning parameter on the nonlinear internal resonance characteristics of the system.The nonlinear softening characteristic is exhibited in the upper stable-state,while the lower stable-state demonstrates the softening and linearity characteristics.Numerical simulation is carried out using the fourth-order Runge-Kutta method,and a series of nonlinear response curves are plotted.Increasing the excitation amplitude further elucidates the global bifurcation and chaotic dynamic snap-through characteristics of the bistable cantilever shell.

BUCKLING AND POSTBUCKLING OF LAMINATED THIN CYLINDRICAL SHELLS UNDER HYGROTHERMAL ENVIRONMENTS

The influence of hygrothermal effects on the buckling and postbuckling of composite laminated cylindrical shells subjected to axial compression is investigated using a micro-to-macro-mechanical analytical model. The material properties of the composite are affected hy the variation of temperature and moisture, and are hosed on a micromechanical model of a laminate. The governing equations are based on the classical laminated shell theory, and including hygrothermal effects. The nonlinear prebuckling deformations and initial geometric imperfections of the shell were both taken into account. A boundary layer theory of shell buckling was extended to the case of laminated cylindrical shells under hygrothermal environments, and a singular peturbation technique was employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical shells under different sets of environmental conditions. The influences played by temperature rise, the degree of moisture concentration, fiber volume fraction, shell geometric parameter, total number of plies, stacking sequences and initial geometric imperfections are studied.

A unified prediction solution for vibro-acoustic analysis of composite laminated elliptical shells immersed in air

A semi-analytical method to conduct vibro-acoustic analysis of a composite laminated elliptical shell immersed in air is proposed.A variational method and multi-segment technique are used to formulate the dynamic model.The sound radiation of the exterior fluid field is calculated by a spectral Kirchhoff–Helmholtz integral formulation.The variables containing displacements and sound pressure are expanded by the combination of Fourier series and Chebyshev orthogonal polynomials.The collocation points are introduced to construct an algebraic system of acoustic integral equations,where these points are distributed on the roots of Chebyshev polynomials,and the non-uniqueness solution of system is eliminated by a combined Helmholtz integral.Numerical examples for sound radiation problems of composite laminated elliptical shells are presented and individual contributions of the circumferential modes to the acoustical results of composite laminated elliptical shells are also given.The effects of geometric and material parameters on sound radiation of composite laminated elliptical shells are also investigated.

Wave transmission through laminated composite double-walled cylindrical shell lined with porous materials

A study on free harmonic wave propagation in a double-walled cylindrical shell, whose walls sandwich a layer of porous materials, is presented within the framework of the classic theory for laminated composite shells. One of the most effective components of the wave propagation through the porous core is estimated with the aid of a flat panel with the same geometrical properties. By considering the effective wave component, the porous layer is modeled as a fluid with equivalent properties. Thus, the model is simplified as a double-walled cylindrical shell trapping the fluid media. Finally, the transmission loss （TL） of the structure is estimated in a broadband frequency, and then the results are compared.

ANALYSIS OF LAMINATED PIEZOELECTRIC CYLINDRICAL SHELLS

A new method is developed for three-dimensional stress analysis of laminated piezoelectric cylindrical shell with simple support. The shell can be subjected to various applied loadings, including distributed body force, inner and outer surface traction and potential. Each layer of the shell can be piezoelectric or elastic/dielectric, with perfect bonding assumed between each interface. The governing equations are solved by the state-space technique. Numerical results are presented to show the sensing and actuating effects of three-layered piezoelectric cylindrical shell.

STACKING SEQUENCE OPTIMIZA-TION OF LAMINATED COMPOSITE CYLINDER SHELL FOR MAXIMAL BUCKLING LOAD

A new optimization method for the optimization of stacking of composite glass fiber laminates is developed. The fiber orientation and angle of the layers of the cylindrical shells are sought considering the buckling load. The proposed optimization algorithm applies both finite element analysis and the mode-pursuing sampling （MPS）method. The algorithms suggest the optimal stacking sequence for achieving the maximal buckling load. The procedure is implemented by integrating ANSYS and MATLAB. The stacking sequence designing for the symmetric angle-ply three-layered and five-layered composite cylinder shells is presented to illustrate the optimization process, respectively. Compared with the genetic algorithms, the proposed optimization method is much faster and efficient for composite staking sequence plan.

A 3-Node Co-Rotational Triangular Finite Element for Non-Smooth,Folded and Multi-Shell Laminated Composite Structures

Based on the first-order shear deformation theory,a 3-node co-rotational triangular finite element formulation is developed for large deformation modeling of non-smooth,folded and multi-shell laminated composite structures.The two smaller components of the mid-surface normal vector of shell at a node are defined as nodal rotational variables in the co-rotational local coordinate system.In the global coordinate system,two smaller components of one vector,together with the smallest or second smallest component of another vector,of an orthogonal triad at a node on a non-smooth intersection of plates and/or shells are defined as rotational variables,whereas the two smaller components of the mid-surface normal vector at a node on the smooth part of the plate or shell(away from non-smooth intersections)are defined as rotational variables.All these vectorial rotational variables can be updated in an additive manner during an incremental solution procedure,and thus improve the computational efficiency in the nonlinear solution of these composite shell structures.Due to the commutativity of all nodal variables in calculating of the second derivatives of the local nodal variables with respect to global nodal variables,and the second derivatives of the strain energy functional with respect to local nodal variables,symmetric tangent stiffness matrices in local and global coordinate systems are obtained.To overcome shear locking,the assumed transverse shear strains obtained from the line-integration approach are employed.The reliability and computational accuracy of the present 3-node triangular shell finite element are verified through modeling two patch tests,several smooth and non-smooth laminated composite shells undergoing large displacements and large rotations.

A state space solution for the bending problem of thick laminated piezoelectric open cylindrical shells

Based on the theories of three-dimensional elasticity and piezoelectricity, and by assuming appropriate boundary functions, we established a state equation of piezoelectric cylindrical shells. By using the transfer matrix method, we presented an analytical solution that satisfies all the arbitrary boundary conditions at boundary edges, as well as on upper and bottom surfaces. Our solution takes into account all the independent elastic and piezoelectric constants for a piezoelectric orthotropy, and satisfies continuity conditions between plies of the laminates. The principle of the present method and corresponding results can be widely used in many engineering fields and be applied to assess the effectiveness of various approximate and numerical models.

Transient impact responses of laminated composite cylindrical shells

The generalized ray method(GRM) has been successfully used to study the transient elastic wave transmitting in the beams,planar trusses,space frames and infinite layered media.In this letter,the GRM is extended to investigate the early short time transient responses of laminated composite cylindrical shells under impact load.By using the Laplace transformation and referring to the boundary conditions,the ray groups transmitting in the finite laminated cylindrical shells under the shock load are obtained and the transient response related to each ray group can be derived via FFT algorithm.From the numerical results,it is shown that the early short time transient accelerations of the laminated composite cylindrical shell under impact loads are very large.But the short time transient shear strain and displacement are very small.

Delamination growth for composite laminated cylindrical shells under external pressure

The delamination growth may occur in delaminated cylindrical shells＇under external pressure. This will lead to failure of structure. By using the variational principle of moving boundary and considering the contact effect between delamination regions, in this work, the delamination growth was investigated for cylindrical shells under the action of external pressure. At the same time, according to the Griffith criterion, the formulas of energy release rate along the delamination front were obtained. In the numerical calculation, the delamination growth of axisymmetrical laminated cylindrical shells was analyzed, and the effects of delamination sizes and depths, the geometrical parameters, the material properties, and the laminate stacking sequences on delamination growth were discussed.

Analytical model of sound transmission through laminated composite cylindrical shells considering transverse shear deformation

Composite structures are often used in the aerospace industry due to the advantages offered by a high strength to weight ratio. Sound transmission through an infinite laminated composite cylindrical shell is studied in the context of the transmission of airborne sound into the aircraft interior. The shell is immersed in an external fluid medium and contains internal fluid. Airflow in the external fluid medium moves with a constant velocity. An exact solution is obtained by simultaneously solving the first-order shear deformation theory （FSDT） of a laminated composite shell and the acoustic wave equations. Transmission losses （TL） obtained from numerical solutions are compared with those of other authors. The effects of structural properties and flight conditions on TL are studied for a range of values, especially, the Mach number, stack sequences, and the angle of warp. Additionally, comparisons of the transmission losses are made between the classical thin shell theory （CST） and FSDT for laminated composite and isotropic cylindrical shells.

DYNAMIC RESPONSE OF LAMINATED ORTHOTROPIC SPHERICAL SHELLSINCLUDING TRANSVERSE SHEAR DEFORMATION AND ROTATORY INERTIA

In this paper.the equations of motion of axisymmetrically laminated cylindrical orthotropic spherical shells are derived.Theeffects of transverse shear deformation and rotatory inertia are considered.On this basis,the dynamic response of spherical shells under axisymmetric dynamic load is calculated using the finite difference method The effects of material parameters.structural parameters and transverse shear dgformation are discussed.

Thermal buckling of axisymmetrically laminated cylindrically orthotropic shallow spherical shells including transverse shear

The nonlinear thermal buckling of symmetrically laminated cylindrically orthotropic shallow spherical shell under temperature field and uniform pressure including transverse shear is studied. Also the analytic formulas for determining the critical buckling loads under different temperature fields are obtained by using the modified iteration method. The effect of transverse shear deformation and different temperature fields on critical buckling load is discussed.

THE EXACT SOLUTION FOR THICK LAMINATEDCANTILEVER CYLINDBICAL SHELL

Discarding any ussumption regarding displacement or strers models, the stateequation for orthotropy is established in a cylindrical system. The exact solution ispresented for the statics of thick closed laminated cantilever cylindrical shells. Everyequation of elasticity can be satisfied and all the elastic constants are taken intoaccount. Arbitrary precision of a desired order can be obtained.

ITERATIVE METHOD FOR LARGE DEFLECTION NONLINEAR PROBLEM OF LAMINATED COMPOSITE SHALLOW SHELLS AND PLATES

Based on the results by Wang,in this paper, the iterative method is presented for the study of large deflection nonlinear problem of laminated composite shallow shells and plates. The rectangular laminated composite shallow shells have been analyzed. The results have been compared with the small deflection linear analytical solution and finite element nonlinear solution. The results proved that the solution coincide with small deflection linear analytical solution in the condition of the low loads and finite element nonlinear solution in the condition of the high loads.

EXACT SOLUTION FOR LAMINATED CONTINUOUSOPEN CYLINDRICAL SHELLS

Discarding any assumption about displacement models and stress distribution andintroducing δ-function into the present study, we established the state equation for thecontinuous orthotropic open cylindrical shells. An thentical exocl solution is presentedfor the statics of thin. moderately thick and thick laminated continuous openrylindrical shells. Numerical results are obtained and compared with those calculatedusing SAP5.