Hinged Boundary Conditions Research Articles

Stability analysis of nanobeams in hygrothermal environment based on a nonlocal strain gradient Timoshenko beam model under nonlinear thermal field

AbstractThis article is dedicated to analyzing the buckling behavior of nanobeam subjected to hygrothermal environments based on the principle of the Timoshenko beam theory. The hygroscopic environment has been considered as a linear stress field model, while the thermal environment is assumed to be a nonlinear stress field based on the Murnaghan model. The size-dependent effect of the nanobeam is captured by the nonlocal strain gradient theory (NSGT), and the governing equations of the proposed model have been derived by implementing a variational principle. The critical buckling loads have been calculated for the hinged–hinged boundary condition by incorporating the Navier approach and considering other elasticity theories such as classical elasticity theory, Eringen nonlocal elasticity theory, and strain gradient theory along with the NSGT. The present model is also validated with the pre-existing model in exceptional cases. Further, a parametric investigation has been performed to report the influence of various scaling parameters like hygroscopic environment, thermal environment, length-to-diameter ratio, small scale parameter, and length scale parameter on critical buckling loads by considering both the linear and nonlinear temperature distributions.

Hygro-Magnetic Vibration of the Single-Walled Carbon Nanotube with Nonlinear Temperature Distribution Based on a Modified Beam Theory and Nonlocal Strain Gradient Model

In this study, vibration analysis of single-walled carbon nanotube (SWCNT) has been carried out by using a refined beam theory, namely one variable shear deformation beam theory. This approach has one variable lesser than a contractual shear deformation theory such as first-order shear deformation theory (FSDT) and acts like classical beam approach but with considering shear deformations. The SWCNT has been placed in an axial or longitudinal magnetic field which is also exposed to both the hygroscopic as well as thermal environments. The thermal environment is considered as nonlinear thermal stress field based on the Murnaghan’s model whereas the hygroscopic environment is assumed as a linear stress field. The size effect of the SWCNT has been captured by both the nonlocal and gradient parameters by employing the Nonlocal Strain Gradient Theory (NSGT). Governing equation of motion of the proposed model has been developed by utilizing the extended Hamilton’s principle and the non-dimensional frequency parameters have been computed by incorporating the Navier’s approach for Hinged–Hinged (HH) boundary condition. The proposed model is validated with the existing model in special cases, by comparing the non-dimensional frequency parameters, displaying an excellent agreement. Further, a parametric study has been conducted to analyze the impact of nonlocal parameter, gradient parameter, thermal environment, hygroscopic environment, and magnetic field intensity on the non-dimensional frequency parameters. Also, results for some other theories like Classical Elasticity Theory (CET), Nonlocal Elasticity Theory (NET), and Strain Gradient Theory (SGT) have been presented along with the NSGT.

Long-time behavior for a class of extensible beams with nonlocal weak damping and critical nonlinearity

This paper is devoted to establishing the long-time behavior of extensible beam equation with the nonlocal weak damping on a bounded smooth domain of Rn with hinged (clamped) boundary condition. It proves the well-posedness by means of the monotone operator theory and the existence of a global attractor when the growth exponent of the nonlinearity f(u) is up to the critical case in natural energy space.

Decay Result for a Delay Viscoelastic Plate Equation

In this work, we consider a fourth-order plate equation that represents the downward displacement of suspension bridge in the presence of viscoelastic and delay damping. We establish an optimal and general decay estimates with partially hinged boundary conditions under suitable assumptions on the relaxation function. This result improves and generalizes earlier results in the literature such as Kirane and Said-Houari (Z Angew Math Phys 62:1065–1082, 2011) and Zhifeng (Z Angew Math Phys 66:727–745, 2015).

Failure conditions for stand-alone cold-frame greenhouse under snow loads

In South Korea, even snowfalls under the design snow loads are frequently known to cause significant deformations or failures of stand-alone cold-frame greenhouses. However, the possible causes of such events are not reflected in the structural design criteria currently used. To estimate the deformations related to structural failures, this study set up possible scenarios to be able to make the structural failures and examined the reasons through scaled-down experiments and nonlinear structural analyses. Hinge boundary conditions of supports and non-uniform snow loads occurred by the repetitive freeze–thaw cycles of the soils surrounding the supports without concrete foundations, and differences between internal and ambient temperatures, respectively, were considered. Two-dimensional scaled-down models of the greenhouses were constructed based on the similarity law. The experimental results were compared with those from nonlinear structural analyses of linear elastic–plastic models. For fixed boundary conditions at both ends and uniform snow loads, the relative errors of displacements between the experimentally obtained values and the results of the finite element analyses were within 6.10%. The results revealed that structural failures occurred in all cases with hinge boundary conditions for the supports at both ends for both uniform and non-uniform snow loads, but the greatest deformation behavior of the structure was observed with a hinge–hinge boundary condition and non-uniform snow loads where the horizontal displacement was similar to the vertical displacement. The temporary installations of diagonal braces in the greenhouses when heavy snowfalls are forecasted are therefore recommended to reduce the side sway.

Time-dependent global attractor for extensible Berger equation

In this paper, we investigate the asymptotic behavior of the nonautonomous Berger equationε(t)utt+Δ2u−(Q+∫Ω|∇u|2dx)Δu+g(ut)+φ(u)=f,t>τ, on a bounded smooth domain Ω⊂RN with hinged boundary condition, where ε(t) is a decreasing function vanishing at infinity. Under suitable assumptions, we establish an invariant time-dependent global attractor within the theory of process on time-dependent space.

Flow-induced vibration and stability analysis of carbon nanotubes based on the nonlocal strain gradient Timoshenko beam theory

A nonlocal strain gradient Timoshenko beam model is developed to study the vibration and instability analysis of the carbon nanotubes conveying nanoflow. The governing equations of motion and boundary conditions are derived by employing Hamilton’s principle, including the effects of moving fluid, material length scale and nonlocal parameters, Knudsen number and gravity force. The material length scale and nonlocal parameters are considered, in order to take into account the size effects. Also, to consider the small-size effects on the flow field, the Knudsen number is used as a discriminant parameter. The Galerkin approach is chosen to analyze the governing equations under clamped–clamped, clamped–hinged and hinged–hinged boundary conditions. It is found that the natural frequency and critical fluid velocity can be decreased by increasing the nonlocal parameter or decreasing the material length scale parameter. Furthermore, it is revealed that the critical flow velocity does not affected by two size-dependent parameters and various boundary conditions in the free molecular flow regime.

A nonlocal strain gradient theory for dynamic modeling of a rotary thermo piezo electrically actuated nano FG circular plate

Abstract In this paper, vibrational behavior of functionally graded piezoelectric material (FGPM) in a nano circular plate subjected to rotational and thermal loads is studied. Thermo-electro–elastic characteristics of FGPM nano circular plate are changed continuously along the thickness with the power-law model. This analysis is based on nonlocal strain gradient theory (NSGT), strain gradient theory (SGT) and nonlocal Eringen theory (NET) within the first order shear deformation theory (FSDT). Via Hamilton’s principle, the governing equations of FGPM nano circular plate obtained and solved using the differential quadrature method (DQM). It is seen that the vibrational behavior of FGPM nano circular plate is considerably affected by temperature rises, angular velocity, external voltage, FG power index, nonlocal parameter and material length scale for clamped and hinged boundary conditions.

Modeling and mechanical analysis of multiscale fiber-reinforced graphene composites: Nonlinear bending, thermal post-buckling and large amplitude vibration

In this paper, a mathematical model was developed to predict the effective material properties of graphene nanoplatelets/fiber/polymer multiscale composites (GFPMC). The large deflection, post-buckling and free nonlinear vibration of graphene nanoplatelets-reinforced multiscale composite beams were studied through a theoretical study. The governing equations of laminated nanocomposite beams were derived from the Euler–Bernoulli beam theory with von Kármán geometric nonlinearity. Halpin–Tsai equations and fiber micromechanics were used in hierarchy to predict the bulk material properties of the multiscale composite. Graphene nanoplatelets (GNPs) were assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. A semi-analytical approach was used to calculate the large static deflection and critical buckling temperature of multiscale multifunctional nanocomposite beams. A perturbation scheme was also employed to determine the nonlinear dynamic response and the nonlinear natural frequencies of the beams with clamped–clamped, and hinged–hinged boundary conditions. The effects of weight percentage of graphene nanoplatelets, volume fraction of fibers, and boundary conditions on the static deflection, thermal buckling and post-buckling and linear and nonlinear natural frequencies of the GFPMC beams were investigated in detail. The numerical results showed that the central deflection and natural frequency were significantly improved by a small percentage of GNPs. However, addition of GNPs led to a lower critical buckling temperature.

Magnetic field effect on free vibration of smart rotary functionally graded nano/microplates: A comparative study on modified couple stress theory and nonlocal elasticity theory

In this article, free vibration behavior of a rotating nano/microcircular plate constructed from functionally graded magneto-elastic material is simulated with the first-order shear deformation theory. For the sake of comparison, the nonlocal elasticity theory and the modified couple stress theory are employed to implement the small size effect in the natural frequencies behavior of the nano/microcircular plate. The governing equations of motion for functionally graded magneto-elastic material nano/microcircular plates are derived based on Hamilton’s principle; comparing the obtained results with those in the literature, they are in a good agreement. Finally, the governing equations are solved using the differential quadrature method. It is shown that the vibrational characteristics of functionally graded magneto-elastic material nano/microcircular plates are significantly affected by non-dimensional angular velocity, size dependency of the Eringen’s and the modified couple stress theories, and power law index for clamped and hinged boundary conditions. Results show that a critical point occurs by increasing the angular velocity and the effect of several parameters are changed after this point.

Exact modes for post-buckling characteristics of nonlocal nanobeams in a longitudinal magnetic field

• Exact mode solution is utilized to analyze postbuckling dynamics of the nanobeam in a longitudinal magnetic field. • The introduced magnetic field can increase the critical buckling load to enhance the stability of the nanobeam. • The nonlocal impact results in a decrease in the critical buckling load. • The nonlocal effect can speed up the activation of the internal resonance between the first two modes. • A significant difference is obtained for the first frequency when considering classical and exact mode solutions. An exact mode solution that investigates the prebuckling and postbuckling characteristics of nonlocal nanobeams with fixed–fixed, hinged–hinged, and fixed–hinged boundary conditions in a longitudinal magnetic field is determined. The geometric nonlinearity arising from mid-plane stretching is considered to obtain the nonlinear governing equation of motion by virtue of Hamilton's principle. The influences of the nonlocal and magnetic parameters on the prebuckling and postbuckling dynamics of nanobeams with various boundary conditions are evaluated, indicating that the critical buckling force can be decreased with the increase of the nonlocal parameter while can be increased with increasing the magnetic parameter. It is demonstrated that the first natural frequency of the nanobeam with fixed–fixed and fixed–hinged conditions in postbuckling configuration is increased from zero to a constant value for higher values of the nonlocal parameter with increasing the axial force. The second natural frequency of the buckled nanobeam is always decreased with an increase of the nonlocal parameter. The results show that the internal resonance between the first and second modes of the postbuckling nanobeams can be quickly and easily activated by increasing the nonlocal parameters, especially for fixed–fixed and hinged–hinged boundary conditions. In addition, the results obtained by exact mode solution are compared those obtained by classical mode solution. It is found that the classical mode is valid only for nonlocal nanobeams with the hinged–hinged boundary conditions.

Asymptotic behaviour of a suspension bridge problem

In this paper, we consider a fourth-order viscoelastic plate equation with infinite memory, and with partially hinged boundary conditions. We investigate the asymptotic behaviour of solutions. This present paper improves earlier results in the literature and allow an extended range of relaxation functions.

An analytical investigation on the wrinkling of aluminium alloys during stamping using macro-scale structural tooling surfaces

Structural surface texturing is believed to be a promising approach to modify tribological and thermal performances of tooling for sheet-stamping processes. However, a fundamental study on the surface-texturing design and resulting material deformation is currently lacking. In this paper, an advanced analytical buckling model specifically for the utilisation of textured tools at macro-scale, comprising dislocation-driven material model, isotropic yield criteria, bifurcation theory and Donnell-Mushtari-Vlasov (DMV) shell structure theory, was established. The developed analytical buckling model was validated by cylindrical deep-drawing experiments. Further finite element (FE) simulations with the implementation of material model via user-defined subroutine were also used to validate the bucking model for large surface texture designs. Effects of theoretical assumptions, such as yield criterion, boundary condition and test-piece geometry, on the accuracy of model prediction for wrinkling were investigated. It was found that the von Mises yield criterion and hinged boundary condition exhibited more accurate predictions. In addition, the DMV shell theory made this model most representative for large structural texturing designs. Furthermore, the implementation of induced shear strain component has an important effect on precisely predicting the wrinkling occurrence. The advanced analytical models developed in this study combine various classical mechanics, structure stability and material modelling together, which provides a useful tool for tooling engineers to analyse structural designs.

Thermal effect on dynamics of thin and thick composite laminated microbeams by modified couple stress theory for different boundary conditions

In this article, thermal effect on free vibration behavior of composite laminated microbeams based on the modified couple stress theory is presented. The proposed anisotropic model is developed by using a variational formulation. The governing equations and boundary conditions are obtained based on a modified couple stress theory and using the principle of minimum potential energy and considering different beam theories, i.e., Euler–Bernoulli, Timoshenko and Reddy beam theories. Unlike the classical beam theories, this model contains a material length scale parameter and can capture the size effect. Free vibration of a simply supported beam is solved by utilizing Fourier series. In addition, the fundamental frequency is achieved by using the generalized differential quadrature method for four types of cross-ply laminations with clamped–clamped, clamped–hinged and hinged–hinged boundary conditions for different beam theories. For investigating different parameters including temperature changes, material length scale parameter, beam thickness, some numerical results on different cross-ply laminated beams are presented. The fundamental frequency of different thin and thick beam theories is investigated by increasing slenderness ratio and thermal loads. The results prove that the modified couple stress theory increases the natural frequency under the thermal effects for free vibration of composite laminated microbeams.

A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue

We introduce here a nonlocal operator as a natural generalization to the biharmonic operator that appears in plate theory. This operator is built in the nonlocal calculus framework defined by Du et al. and its connected with the recent theory of peridynamics. For the steady state equation coupled with different boundary conditions we show existence and uniqueness of solutions, as well as regularity of solutions. The boundary conditions considered are nonlocal counterparts of the classical clamped and hinged boundary conditions. For each system we show convergence of the nonlocal solutions to their local equivalents using compactness arguments developed by Bourgain, Brezis and Mironescu.

The global attractor for a suspension bridge with memory and partially hinged boundary conditions

Recently, Ferrero and Gazzola, [Disc. Cont. Dyn. Syst. 35: 5879–5908 (2015)], suggested and investigated a rectangular plate model describing the statics and dynamics of a suspension bridge. The plate is assumed to be hinged on its vertical edges and free on its remaining horizontal edges. This reliable model aims to describe more accurately the motion of suspension bridges compared to all previous known models. In the present paper, we consider a plate equation in the presence of memory and subject to the above‐mentioned boundary conditions. We give a rigorous well‐posedness result and establish the existence of a global attractor.

Thermal postbuckling analysis of anisotropic laminated beams with tubular cross-section based on higher-order theory

Buckling of composite riser assumed beam structure is one of numerous engineering challenges in deep water pipeline design. Thermal postbuckling analysis of shear deformable anisotropic laminated composite beams with tubular cross-section subjected to uniform, linear and non-linear temperature distribution through the thickness resting on a two-parameter elastic foundation is presented. The material of each layer for the composite beam with tubular section is assumed to be linearly elastic and fiber-reinforced. The governing equations are introduced by using high-order shear deformation beam model with a von Kármán-type of kinematic nonlinearity. Composite beams with clamped–clamped, clamped–hinged, and hinged–hinged boundary conditions are considered. A numerical solution for nonlinear partial-integral differential form in terms of the transverse deflection by using Galerkin׳s method is employed to determine the buckling temperatures and postbuckling equilibrium paths of anisotropic laminated beams with different types of temperature distribution through the thickness. The numerical illustration concern the thermal postbuckling response of laminated beams with different types of boundary conditions, ply arrangements (lay-ups), geometric and physical properties. The results reveal that the geometric and physical properties, temperature dependent properties, initial geometry imperfection, boundary conditions and elastic foundation have a significant effect on thermal postbuckling behavior of anisotropic laminated composite tubular beams.

Non-linear buckling of elliptical curved beams

In this paper, a study on non-linear buckling and postbuckling behavior of elliptical curved beams is presented. An isogeometric analysis framework with the use of an arc-length iteration technique is developed based on the shear deformable theory and accounts for the geometric non-linearity in von Kármán sense. The proposed method is capable of exactly modeling the geometry of elliptical curved beams by means of NURBS (non-uniform rational B-splines) basis functions. The solutions of the present model are validated by comparing with experimental and numerical results of the literature. Numerical examples are carried out to investigate the effects of different geometric parameters on the buckling and postbuckling of elliptical curved beams with hinged–hinged, clamped–clamped and clamped–hinged boundary conditions.

Post-buckling and Large Amplitude Free Vibration Analysis of Composite Beams: Simple Intuitive Formulation

Post-buckling and large amplitude free vibration analysis of composite beams with axially immovable ends is investigated in the present study using a simple intuitive formulation. Geometric nonlinearity of Von-Karman type is considered in the analysis which accounts for mid-plane stretching action of the beam. Intuitive formulation uses only two parameters: the critical bifurcation point and the axial stretching force developed due to membrane stretching action of the beam. Hinged–hinged, clamped–clamped and clamped–hinged boundary conditions are considered. Numerical accuracy of the proposed analytical closed-form solutions obtained from the intuitive formulation are compared to available finite element solutions for symmetric and asymmetric layup schemes of laminated composite beam which indicates the confidence gained on the present formulation.

Analytic evaluation of fire performance according to boundary conditions and lengths for steel columns built with SM 520 and SM 570 at high temperatures

The fire performance of H-section columns made of high strength structural steels can be sustained by the coating of fire protective material. The thickness of fire protective material evaluated from fire test can be used any kind of grades, lengths, and boundary conditions of H-section columns. While, the H-section column designed and constructed in a building can have various grades of structural steels, lengths, and boundary conditions. To suggest a new guideline of fire protection material for H-section column made of high strength structural steels, fire engineering design was adopted. The facts derived from this study showed that the use of fire protection material derived from load fire tests with SS 400, 3500mm, and hinge to hinge boundary condition into H-section column made of SM 520 and SM 570 with a longer column length and other boundary condition is determined to be safe.