• Advances in nano research
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• v.7 no.6
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• pp.443-457
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• 2019

In this investigation, dynamic and bending behaviors of isolated protein microtubules are analyzed. Microtubules (MTs) can be considered as bio-composite structures that are elements of the cytoskeleton in eukaryotic cells and posses considerable roles in cellular activities. They have higher mechanical characteristics such as superior flexibility and stiffness. In the modeling purpose of microtubules according to a hollow beam element, a novel single variable sinusoidal beam model is proposed with the conjunction of modified strain gradient theory. The advantage of this model is found in its new displacement field involving only one unknown as the Euler-Bernoulli beam theory, which is even less than the Timoshenko beam theory. The equations of motion are constructed by considering Hamilton's principle. The obtained results are validated by comparing them with those given based on higher shear deformation beam theory containing a higher number of variables. A parametric investigation is established to examine the impacts of shear deformation, length scale coefficient, aspect ratio and shear modulus ratio on dynamic and bending behaviors of microtubules. It is remarked that when length scale coefficients are almost identical of the outer diameter of MTs, microstructure-dependent behavior becomes more important.

• Steel and Composite Structures
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• v.19 no.2
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• pp.369-384
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• 2015

This work presents a free vibration analysis of functionally graded metal-ceramic (FG) beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. For this purpose, a simple displacement field based on higher order shear deformation theory is implemented. The proposed theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The most interesting feature of this theory is that it accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factors. In addition, it has strong similarities with Euler-Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The rule of mixture is modified to describe and approximate material properties of the FG beams with porosity phases. By employing the Hamilton's principle, governing equations of motion for coupled axial-shear-flexural response are determined. The validity of the present theory is investigated by comparing some of the present results with those of the first-order and the other higher-order theories reported in the literature. Illustrative examples are given also to show the effects of varying gradients, porosity volume fraction, aspect ratios, and thickness to length ratios on the free vibration of the FG beams.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.31 no.6
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• pp.381-388
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• 2018

This paper presents a generalized finite element formulation for plastic hinge modeling based on lumped plasticity in the classical Euler-Bernoulli beam elements. In this approach, the plastic hinges are effectively modeled using proper enrichment functions describing weak discontinuities of the solution. The proposed methodology enables the insertion of plastic hinges at an arbitrary location without modifying the connectivity of elements. The formations of plastic hinges are instead achieved by hierarchically adding degrees of freedom to existing elements. Convergence analyses such as h- and p-extensions are performed to investigate the effectiveness of the proposed method. The analysis results indicate that the proposed generalized finite element method can achieve theoretical convergence rates for both cases where plastic hinges are located at nodes and within an element, thus demonstrating its accuracy.

• International journal of steel structures
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• v.18 no.4
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• pp.1440-1463
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• 2018

The Wagner coefficient is a key parameter used to describe the inelastic lateral-torsional buckling (LTB) behaviour of the I-beam, since even for a doubly-symmetric I-section with residual stress, it becomes a monosymmetric I-section due to the characteristics of the non-symmetrical distribution of plastic regions. However, so far no theoretical derivation on the energy equation and Wagner's coefficient have been presented due to the limitation of Vlasov's buckling theory. In order to simplify the nonlinear analysis and calculation, this paper presents a simplified mechanical model and an analytical solution for doubly-symmetric I-beams under pure bending, in which residual stresses and yielding are taken into account. According to the plate-beam theory proposed by the lead author, the energy equation for the inelastic LTB of an I-beam is derived in detail, using only the Euler-Bernoulli beam model and the Kirchhoff-plate model. In this derivation, the concept of the instantaneous shear centre is used and its position can be determined naturally by the condition that the coefficient of the cross-term in the strain energy should be zero; formulae for both the critical moment and the corresponding critical beam length are proposed based upon the analytical buckling equation. An analytical formula of the Wagner coefficient is obtained and the validity of Wagner hypothesis is reconfirmed. Finally, the accuracy of the analytical solution is verified by a FEM solution based upon a bi-modulus model of I-beams. It is found that the critical moments given by the analytical solution almost is identical to those given by Trahair's formulae, and hence the analytical solution can be used as a benchmark to verify the results obtained by other numerical algorithms for inelastic LTB behaviour.

• Advances in concrete construction
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• v.11 no.1
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• pp.81-88
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• 2021

Rice Husk Ash (RHA) geopolymer paste activated by sodium aluminate were characterized by X-ray diffractogram (XRD), scanning electron microscope (SEM), energy dispersion X-Ray analysis (EDAX)and fourier transform infrared spectroscopy (FTIR). Five series of RHA geopolymer specimens were prepared by varying the Si/Al ratio as 1.5, 2.0, 2.5, 3.0 and 3.5. The paper focuses on the correlation of microstructure with hardened state parameters like bulk density, apparent porosity, sorptivity, water absorption and compressive strength. XRD analysis peaks indicates quartz, cristobalite and gibbsite for raw RHA and new peaks corresponding to Zeolite A in geopolymer specimens. In general, SEM micrographs show interconnected pores and loosely packed geopolymer matrix except for specimens made with Si/Al of 2.0 which exhibited comparatively better matrix. Incorporation of Al from sodium aluminate were confirmed with the stretching and bending vibration of Si-O-Si and O-Si-O observations from the FTIR analysis of geopolymer specimen. The dense microstructure of SA2.0 correlate into better performance in terms of 28 days maximum compressive strength of 16.96 MPa and minimum for porosity, absorption and sorptivity among the specimens. However, due to the higher water demand to make the paste workable, the value of porosity, absorption and sorptivity were reportedly higher as compared with other geopolymer systems. Correlation regression equations were proposed to validate the interrelation between physical parameters and mechanical strength. RHA geopolymer shows comparatively lower compressive strength as compared to Fly ash geopolymer.

• Advances in nano research
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• v.10 no.1
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• pp.15-24
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• 2021

Microtubules (MTs) are the main part of the cytoskeleton in living eukaryotic cells. In this article, a mechanical model of MT buckling, considering the modified strain gradient theory, is analytically examined. The MT is assumed as a cylindrical beam and a new single variable trigonometric beam theory is developed in conjunction with a modified strain gradient model. The main benefit of the present formulation is shown in its new kinematic where we found only one unknown as the Euler-Bernoulli beam model, which is even less than the Timoshenko beam model. The governing equations are deduced by considering virtual work principle. The effectiveness of the present method is checked by comparing the obtained results with those reported by other higher shear deformation beam theory involving a higher number of unknowns. It is shown that microstructure-dependent response is more important when material length scale parameters are closer to the outer diameter of MTs. Also, it can be confirmed that influences of shear deformation become more considerable for smaller shear modulus and aspect ratios.

• Structural Engineering and Mechanics
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• v.86 no.3
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• pp.291-309
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• 2023

This paper addresses the finite element modeling of functionally graded porous (FGP) beams for free vibration and buckling behaviour cases. The formulated finite element is based on simple and efficient higher order shear deformation theory. The key feature of this formulation is that it deals with Euler-Bernoulli beam theory with only three unknowns without requiring any shear correction factor. In fact, the presented two-noded beam element has three degrees of freedom per node, and the discrete model guarantees the interelement continuity by using both C⁰ and C¹ continuities for the displacement field and its first derivative shape functions, respectively. The weak form of the governing equations is obtained from the Hamilton principle of FGP beams to generate the elementary stiffness, geometric, and mass matrices. By deploying the isoparametric coordinate system, the derived elementary matrices are computed using the Gauss quadrature rule. To overcome the shear-locking phenomenon, the reduced integration technique is used for the shear strain energy. Furthermore, the effect of porosity distribution patterns on the free vibration and buckling behaviours of porous functionally graded beams in various parameters is investigated. The obtained results extend and improve those predicted previously by alternative existing theories, in which significant parameters such as material distribution, geometrical configuration, boundary conditions, and porosity distributions are considered and discussed in detailed numerical comparisons. Determining the impacts of these parameters on natural frequencies and critical buckling loads play an essential role in the manufacturing process of such materials and their related mechanical modeling in aerospace, nuclear, civil, and other structures.

• Steel and Composite Structures
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• v.44 no.5
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• pp.721-740
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• 2022

In this paper, an accurate kinematic model has been developed to study the mechanical response of functionally graded (FG) sandwich beams, mainly covering the bending, buckling and free vibration problems. The studied structure with homogeneous hardcore and softcore is considered to be simply supported in the edges. The present model uses a new refined shear deformation beam theory (RSDBT) in which the displacement field is improved over the other existing high-order shear deformation beam theories (HSDBTs). The present model provides good accuracy and considers a nonlinear transverse shear deformation shape function, since it is constructed with only two unknown variables as the Euler-Bernoulli beam theory but complies with the shear stress-free boundary conditions on the upper and lower surfaces of the beam without employing shear correction factors. The sandwich beams are composed of two FG skins and a homogeneous core wherein the material properties of the skins are assumed to vary gradually and continuously in the thickness direction according to the power-law distribution of volume fraction of the constituents. The governing equations are drawn by implementing Hamilton's principle and solved by means of the Navier's technique. Numerical computations in the non-dimensional terms of transverse displacement, stresses, critical buckling load and natural frequencies obtained by using the proposed model are compared with those predicted by other beam theories to confirm the performance of the proposed theory and to verify the accuracy of the kinematic model.

• Advances in nano research
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• v.15 no.3
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• pp.215-224
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• 2023

Nonlinearity plays an important role in control systems and the application of design. For this reason, in addition to linear vibrations, nonlinear vibrations of the stepped nanobeam are also discussed in this manuscript. This study investigated the vibrations of stepped nanobeams according to Eringen's nonlocal elasticity theory. Eringen's nonlocal elasticity theory was used to capture the nanoscale effect. The nanoscale stepped Euler Bernoulli beam is considered. The equations of motion representing the motion of the beam are found by Hamilton's principle. The equations were subjected to nondimensionalization to make them independent of the dimensions and physical structure of the material. The equations of motion were found using the multi-time scale method, which is one of the approximate solution methods, perturbation methods. The first section of the series obtained from the perturbation solution represents a linear problem. The linear problem's natural frequencies are found for the simple-simple boundary condition. The second-order part of the perturbation solution is the nonlinear terms and is used as corrections to the linear problem. The system's amplitude and phase modulation equations are found in the results part of the problem. Nonlinear frequency-amplitude, and external frequency-amplitude relationships are discussed. The location of the step, the radius ratios of the steps, and the changes of the small-scale parameter of the theory were investigated and their effects on nonlinear vibrations under simple-simple boundary conditions were observed by making comparisons. The results are presented via tables and graphs. The current beam model can assist in designing and fabricating integrated such as nano-sensors and nano-actuators.

• Steel and Composite Structures
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• v.49 no.5
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• pp.487-502
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• 2023

In the realm of nanotechnology, the nonlocal strain gradient theory takes center stage as it scrutinizes the behavior of spinning cantilever nanobeams and nanotubes, pivotal components supporting various mechanical movements in sport structures. The dynamics of these structures have sparked debates within the scientific community, with some contending that nonlocal cantilever models fail to predict dynamic softening, while others propose that they can indeed exhibit stiffness softening characteristics. To address these disparities, this paper investigates the dynamic response of a nonlocal cantilever cylindrical beam under the influence of external discontinuous dynamic loads. The study employs four distinct models: the Euler-Bernoulli beam model, Timoshenko beam model, higher-order beam model, and a novel higher-order tube model. These models account for the effects of functionally graded materials (FGMs) in the radial tube direction, giving rise to nanotubes with varying properties. The Hamilton principle is employed to formulate the governing differential equations and precise boundary conditions. These equations are subsequently solved using the generalized differential quadrature element technique (GDQEM). This research not only advances our understanding of the dynamic behavior of nanotubes but also reveals the intriguing phenomena of both hardening and softening in the nonlocal parameter within cantilever nanostructures. Moreover, the findings hold promise for practical applications, including drug delivery, where the controlled vibrations of nanotubes can enhance the precision and efficiency of medication transport within the human body. By exploring the multifaceted characteristics of nanotubes, this study not only contributes to the design and manufacturing of rotating nanostructures but also offers insights into their potential role in revolutionizing drug delivery systems.