Elasticity solution of laminated beams with temperature-dependent material properties under a combination of uniform thermo-load and mechanical loads

An exact solution for simply-supported laminated beams with material properties variable with temperature under a combination of uniform thermo-load and mechanical loads was investigated,based on the two-dimensional(2-D)thermo-elasticity theory.Firstly,the beam was divided into a series of layers with uniform material properties along the interfaces of the beam.The uniform thermo-load acted on each layer was transformed into a combination of the normal surface forces acted at the two ends and the transverse thermo-load.Secondly,the state space method was employed to obtain the general solutions of displacements and stresses in an arbitrary layer.Thirdly,based on the interfacial continuity conditions between adjacent layers,the relations of displacement and stress components between the top and bottom layers of the beam were recursively derived by use of the transfer-matrix method.The unknowns in the solutions can be solved by the mechanical loads acted on the top and bottom surfaces.The convergence of the present solutions was checked.The comparative study of the present solutions with the Timoshenko’s solutions and the finite element(FE)solutions was carried out.The effects of material properties variable with temperature on the thermo-elastic behavior of laminated beams were discussed in detail.

Analytical solution of temperature in laminated beams subjected to general thermal boundary conditions

The temperature distribution in laminated beams underging thermal boundary conditions has been investigated.The thermal boundary conditions are general and include various combinations of prescribed heat fluxes and temperatures at the edges.An analytical solution of temperature for the laminated beam is present on the basis of the heat conduction theory in this paper.The proposed method is applicable to the beams with arbitrary thickness and layer numbers.Due to the complexity of the boundary conditions,the temperature field to be determined was considered from two sources.The first part was the temperature field from the complex temperature conditions at two edges of the laminated beam.The solution for the temperature of the first part was constructed to satisfy temperature boundary conditions at two edges.The second part was the temperature field from the upper and lower surface temperatures without taking account of the thermal conditions at two edges.In this part,the exact solution for the temperature was obtained based on the heat conduction theory.The convergence of the solution was examined by analyzing terms of Fourier series.The validity and feasibility of the proposed method was verified by comparing theoretical results with numerical results due to the equivalent single layer approach and the finite element method(FEM).The influences of surface temperatures,beam thicknesses,layer numbers and material properties with respects to the solution of the temperature field of the beam were investigated via a series of parametric studies.

Elasticity solution of laminated beams subjected to thermo-loads

According to the two-dimensional(2-D) thermo-elasticity theory, the exact elasticity solution of the simply supported laminated beams subjected to thermo-loads was studied. An analytical method was presented to obtain the temperature, displacement and stress fields in the beam. Firstly, the general solutions of temperature, displacements and stresses for a single-layered simply supported beam were obtained by solving the 2-D heat conduction equation and the 2-D elasticity equations, respectively. Then, based on the continuity of temperature, heat flux, displacements and stresses on the interface of two adjacent layers, the formulae of temperature, displacements and stresses between the lowest layer and the top layer of the beam were derived out in a recurrent manner. Finally, the unknown coefficients in the solutions were determined by the use of the upper surface and lower surface conditions of the beam. The distributions of temperature, displacement and stress in the beam were obtained by substituting these coefficients back to the recurrence formulae and the solutions. The excellent convergence of the present method has been demonstrated and the results obtained by the present method agree well with those from the finite element method. The effects of surface temperatures, thickness, layer number and material properties of the plate on the temperature distribution were discussed in detail. Numerical results reveal that the displacements and stresses monotonically increase with the increase of surface temperatures. In particular, the horizontal stresses are discontinuous at the interface.

The analysis of composite laminated beams using a 2D interpolating meshless technique

Laminated composite materials are widely implemented in several engineering constructions. For its relative light weight, these materials are suitable for aerospace, military, marine, and automotive structural applications. To obtain safe and economical structures, the modelling analysis accuracy is highly relevant. Since meshless methods in the recent years achieved a remarkable progress in computational mechanics, the present work uses one of the most flexible and stable interpolation meshless technique available in the literature—the Radial Point Interpolation Method(RPIM).Here, a 2 D approach is considered to numerically analyse composite laminated beams. Both the meshless formulation and the equilibrium equations ruling the studied physical phenomenon are presented with detail. Several benchmark beam examples are studied and the results are compared with exact solutions available in the literature and the results obtained from a commercial finite element software. The results show the efficiency and accuracy of the proposed numeric technique.

General analytical solution to bending of composite laminated beams with delaminations

Based on the first-order shear deformable beam theory, a refined model for composite beams containing a through-the-width delamination is presented, and the deformation at the delamination front is considered. Different from the ordinary delami- nated beam theory, each of the perfectly bonded portions of the new model is constructed as two separated beams along the interface without assuming a plane section at the de- lamination front. The governing equations of the delaminated portions and bonded ones are established, combined with continuity conditions of displacements and internal forces. Solutions of delaminated composite beams with different boundary conditions, delamina- tion locations and sizes axe shown in excellent agreement with the finite element results, showing efficiency and applicability of the present model.

Vibration suppression of magnetostrictive laminated beams resting on viscoelastic foundation

The vibration suppression analysis of a simply-supported laminated composite beam with magnetostrictive layers resting on visco-Pasternak’s foundation is presented.The constant gain distributed controller of the velocity feedback is utilized for the purpose of vibration damping.The formulation of displacement field is proposed according to Euler-Bernoulli’s classical beam theory(ECBT),Timoshenko’s first-order beam theory(TFBT),Reddy’s third-order shear deformation beam theory,and the simple sinusoidal shear deformation beam theory.Hamilton’s principle is utilized to give the equations of motion and then to describe the vibration of the current beam.Based on Navier’s approach,the solution of the dynamic system is obtained.The effects of the material properties,the modes,the thickness ratios,the lamination schemes,the magnitudes of the feedback coefficient,the position of magnetostrictive layers at the structure,and the foundation modules are extensively studied and discussed.

Force-displacement characteristics of simply supported beam laminated with shape memory alloys

As a preliminary step in the nonlinear design of shape memory alloy（SMA） composite structures,the force-displacement characteristics of the SMA layer are studied.The bilinear hysteretic model is adopted to describe the constitutive relationship of SMA material.Under the assumption that there is no point of SMA layer finishing martensitic phase transformation during the loading and unloading process,the generalized restoring force generated by SMA layer is deduced for the case that the simply supported beam vibrates in its first mode.The generalized force is expressed as piecewise-nonlinear hysteretic function of the beam transverse displacement.Furthermore the energy dissipated by SMA layer during one period is obtained by integration,then its dependencies are discussed on the vibration amplitude and the SMA＇s strain（Ms-Strain） value at the beginning of martensitic phase transformation.It is shown that SMA＇s energy dissipating capacity is proportional to the stiffness difference of bilinear model and nonlinearly dependent on Ms-Strain.The increasing rate of the dissipating capacity gradually reduces with the amplitude increasing.The condition corresponding to the maximum dissipating capacity is deduced for given value of the vibration amplitude.The obtained results are helpful for designing beams laminated with shape memory alloys.

Bending waves of a rectangular piezoelectric laminated beam

A simple nonlinear model is proposed in this paper to study the bending wave in a rectangular piezoelectric laminated beam of infinite length.Based on the constitutive relations for transversely isotropic piezoelectric materials and isotropic elastic materials,combined with some electric conditions,we derive the bending wave equation in a long rectangular piezoelectric laminated beam by using energy method.The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal beam direction.The shock-wave solution,solitary-wave solution and other exact solutions of the bending wave equation are obtained by the extended F-expansion method.And by using the reductive perturbation method we derive the nonlinear Schrodinger(NLS)equation,further more,the bright and dark solitons are obtained.For those soliton solutions,and some parameters derived by the process of solving soliton solutions,some conclusions are drawn by numerical analysis with some fixed conditions.

Exponential and polynomial decay for a laminated beam with Fourier's law of heat conduction and possible absence of structural damping

We study the well-posedness and decay properties of a one-dimensional thermoelastic laminated beam system either with or without structural damping,of which the heat conduction is given by Fourier's law effective in the rotation angle displacements.We show that the system is well-posed by using the Lumer-Philips theorem,and prove that the system is exponentially stable if and only if the wave speeds are equal,by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem.Further-more,we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal,by using the second-order energy method.When the speeds are not equal,whether the system without structural damping may has polynomial stability is left as an open problem.

Global Solution and Exponential Stability for a Laminated Beam with Fourier Thermal Law

This paper focuses on the long-time dynamics of a thermoelastic laminated beam modeled from the well-established Timoshenko theory.From mathematical point of view,the study system consists of three hyperbolic motion equations coupled with the parabolic equation governed by Fouriers law of heat conduction and,in consequence,does not belong to one of the classical categories of PDE.We have proved the well-posedness and exponential stability of the system.The well-posedness is given by Hille-Yosida theorem.For the exponential decay we applied the energy method by introducing a Lyapunov functional.

ANALYSIS OF ELASTO-PLASTIC POSTBUCKLING AND ENERGY RELEASE RATE FOR DELAMINATED FIBER METAL LAMINATED BEAMS IN THERMAL ENVIRONMENT

The elasto-plastic postbuckling of fiber metal laminated beams with delamination and the energy release rate along the delamination front are discussed in this paper. Considering geometrical nonlinearity, thermal environment and geometrical initial imperfection, the incremental nonlinear equilibrium equations of delaminated fiber metal laminated beams are established, which are solved using the differential quadrature method and iterative method. Based on these, according to the J-integral theory, the elasto-plastic energy release rate is studied. The effects of some important parameters on the elasto-plastic postbuckling behavior and energy release rate of the aramid reinforced aluminum laminated beams are discussed in details.

Experimental Vibration Analysis of a Cracked Laminated Glass Beam with Bolted Joint

In this study, in order to determine the very early load transfer behavior in the bolted joint connection, experimental dynamic analysis of different laminated glass beams including two surface cracks is considered. For this purpose, both three different plastic interlayers （i.e., three types of polyvinyl butyral--PVB） and three different glass-lamina thicknesses are taken into account. Effects of the plastic interlayer, thickness of the glass-lamina, number of surface cracks and their locations on the vibration characteristics/structural performances are examined experimentally. Vibration tests are performed to present free vibration characteristics of the laminated glass beams under clamped-free boundary conditions. Experimental dynamic analysis consists of six parts： （I） vibration analysis with no-crack and no-hole with a bolted joint; （I1） vibration analysis with a surface crack and no-hole with a bolted joint; （III） vibration analysis with two surface cracks and no-hole with a bolted joint; （IV） vibration analysis with no-crack and a hole with a bolted joint; （V） vibration analysis with a surface crack and a hole with a bolted joint; （VI） vibration analysis with two surface cracks and a hole with a bolted joint. For these experimental steps, an impact hammer with a force transducer is used to excite the uncracked or cracked composite beams through the selected points. After the excitation, the responses are obtained by an accelerometer. The vibration measurements are completed using a microprocessor-based data acquisition system and nCode GlyphWorks software. Results are given in tabular and graphical forms.

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Free vibration of statically thermal postbuckled functionally graded material （FGM） beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electro- mechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Mechanical Analysis of Sisal Fibers to Use as a Reinforced Material in Wood Beams

Natural fibers have recently raised attention for presenting adequate mechanical characteristics for reinforcement of structural elements. The use of both natural fibers, in especial Sisal fibers, in wood laminated beams and also wood from reforestation, is in accordance with the current economic interest and ecological appeal. Specifically, the strengthening of wood laminated beams with Sisal fibers is more effective for structures that require an increase in their structural capacity without a significant increase in height of the cross section. Furthermore, it is recommended that this type of reinforcement is used in wood structural elements where the elastic modulus is at least equal to the Sisal fibers. The composition of Sisal fibers is basically of cellulose, lignin and hemicelluloses. In particular, the amount of cellulose and the angle that the micro-fibers with the axis of the fiber characterize the failure strength and the modulus of elasticity. The average mechanical characteristics of the Sisal fiber are： tensile strength 347 to 378 （MPa） and elastic modulus 15.2 （GPa） whereas these properties are lower for strips of Sisal fibers. In this context, this paper deals with the analysis and the viability of the use of Sisal fibers in wood structures as a reinforced material.

A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.

Lagged strain of laminates in RC beams strengthened with fiber-reinforced polymer

Based on the theory of concrete structure, a new expression was derived for lagged strain of fiber-reinforced polymer （FLIP） laminates in reinforced concrete （RC） beams strengthened with FRP. The influence of different preloaded states and nonlinear stress-strain relationship of compressed concrete were both taken into account in this approach. Then a simplified expression was given by ignoring tensile resistance of concrete. Comparison of analytical predictions with experimental results indicates satisfactory accuracy of the procedures. The errors are less than 8% and 10% respectively when the tensile resistance of concrete is or not considered. While the maximum error of existing procedures is up to 60%.

PIEZOELECTRIC PROPERTIES OF SINGLE-STRAND DNA MOLECULAR BRUSH BIOLAYERS

The paper is devoted to investigations on nanomechanical behaviors of biochips in label-free biodetections. The chip consists of Si-layer, Ti-layer, Au-layer and single-strand DNA （ssDNA） molecular brush biolayer immobilized by self-assembly technology of thiol group. Unlike previous viewpoints, such as force-bending, entropy-bending and curvature electricity effect, etc., the piezoelectric effect of the biopolymer brush layer is viewed as the main factor that induces nanomechanical bending of biochips, and a classical macroscopic piezoelectric constitutive relation is used to describe the piezoelectric effect. A new laminated cantilever beam model with a piezoelectric biolayer in continuum mechanics, the linearized Poisson-Boltzmann equation in statistical mechanics and the scaling method in polyelectrolyte brush theory are combined to es- tablish a relationship between the nanomechanical deflection of DNA chips and the factors such as nanoscopic structural features of ssDNA molecules, buffer salt concentration, macroscopic mechanical/piezoelectric parameters of DNA chips etc. Curve fitting of experimental data shows that the sign of the piezoelectric constant of the biolayer may control the deflection direction of DNA chips during the packaging process.

Haar wavelet discretization method for free vibration study of laminated composite beam under generalized boundary conditions

Investigation on vibration of laminated composite beam(LCB)is an important issue owing to its wide use as fundamental component.In the present work,we study the free vibration of arbitrarily LCB with generalized elastic boundary condition(BC)by using Haar wavelet discretization method(HWDM).Timoshenko beam theory is utilized to model the free vibration of LCB.The LCB is first split into several segments,and then the displacement for each segment is obtained from the Haar wavelet series and their integral.Hamilton’s principle is applied to construct governing equations and the artificial spring boundary technique is adopted to obtain the general elastic boundary and continuity conditions at two ends of LCB.The vibration characteristics of beam with different fiber orientations and lamina numbers is investigated and its displacements are compared with those in previous works.Numerical results are shown graphically and demonstrate the validation of our method.

A semi-analytical method for forced vibration analysis of cracked laminated composite beam with general boundary condition

In this paper,a semi-analytical method for the forced vibration analysis of cracked laminated composite beam(CLCB)is investigated.One computational model is formulated by Timoshenko beam theory and its dynamic solution is solved using the Jacobi-Ritz method.The boundary conditions(BCs)at both ends of the CLCB are generalized by the application of artificial elastic springs,the CLCB is separated into two elements along the crack,the flexibility coefficient of fracture theory is used to model the essential continuous condition of the connective interface.All the allowable displacement functions used to analyze dynamic characteristics of CLCB are expressed by classical Jacobi orthogonal polynomials in a more general form.The accuracy of the proposed method is verified through the compare with results of the finite element method(software ABAQUS is used in this paper).On this basis,the parametric study for dynamic analysis characteristics of CLCB is performed to provide reference datum for engineers.

Research and optimization on laminated steel tube column-concrete beam joints with outer stiffening ring

In practical design,the joints with outer stiffening ring were used to connect concrete beams and laminated steel tube columns.The seismic behavior of joints with outer stiffening ring with various types was studied based on experiments.The bearing capacity,rigidity,ductility,energy dissipation capacity,deformation property and strain distribution of the joints with outer stiffening ring with various types were comprehensively evaluated based on the test results of three specimens under quasistatic cyclic loading and finite element analysis.The test results showed that the failure mode,hysteretic behavior,bearing capacity and rigidity degradation of the joints with outer stiffening ring with various types were nearly identical.Furthermore,the strain distribution of the outer stiffening ring of the three joints was nearly the same.The detailing recommendation for the outer strengthening rings was proposed for the concrete beam-laminated steel tube column joints with outer stiffening ring,in order to ensure the good seismic capacity of the joints.