• Structural Engineering and Mechanics
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• 제65권1호
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• pp.9-17
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• 2018

This study concerns about calculating exact natural frequencies of frames using a single variable shear deformation theory (SVSDT) which considers the parabolic shear stress distribution across the cross section. Free vibration analyses are performed for multi-bay, multi-storey and multi-bay multi-storey type frame structures. Dynamic stiffness formulations are derived and used to obtain first five natural frequencies of frames. Different beam and column cross sections are considered to reveal their effects on free vibration analysis. The calculated natural frequencies are tabulated with the results obtained using Euler-Bernoulli Beam Theory (EBT) and Timoshenko Beam Theory (TBT). Moreover, the effects of inner and outer columns on natural frequencies are compared for multi-bay frames. Several mode shapes are plotted.

• Computers and Concrete
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• 제26권3호
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• pp.285-292
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• 2020

In engineering structures, to having the projected structure to serve all the engineering purposes, the theory to be used during the modeling stage is also of great importance. In the present work, an analytical solution of the free vibration of the beam composed of functionally graded materials (FGMs) is presented utilizing different beam theories. The comparison of supposed beam theory for free vibration of functionally graded (FG) beam is examined. For this aim, Euler-Bernoulli, Rayleigh, Shear, and Timoshenko beam theories are employed. The functionally graded material properties are assumed to vary continuously through the thickness direction of the beam with respect to the volume fraction of constituents. The governing equations of free vibration of FG beams are derived in the frameworks of four beam theories. Resulting equations are solved versus simply supported boundary conditions, analytically. To verify the results, comparisons are carried out with the available results. Parametrical studies are performed for discussing the effects of supposed beam theory, the variation of beam characteristics, and FGM properties on the free vibration of beams. In conclusion, it is found that the interaction between FGM properties and the supposed beam theory is of significance in terms of free vibration of the beams and that different beam theories need to be used depending on the characteristics of the beam in question.

• Smart Structures and Systems
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• 제13권1호
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• pp.55-80
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• 2014

The present work pays emphasis on investigating the effect of different types of debonding on the bending behaviour of active sandwich beam, consisting of both extension and shear actuators. An active sandwich beam finite element is formulated by using Timoshenko's beam theory, characterized by first order shear deformation for the core and Euler-Bernoulli's beam theory for the top and bottom faces. The problem of debondings of extension actuator and face are dealt with by employing four-region model for inner debonding and three-region model for the edge debonding respectively. Displacement based continuity conditions are enforced at the interfaces of different regions using penalty method. Firstly, piezoelectric actuation of healthy sandwich beam is assessed through deflection analysis. Then the effect of actuators' debondings with different boundary conditions on bending behavior is computationally evaluated and experimentally clamped-free case is validated. The results generated will be useful to address the damage tolerant design procedures for smart sandwich beam structures with structural control and health monitoring applications.

• Steel and Composite Structures
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• 제21권5호
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• pp.1045-1067
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• 2016

Concrete bridges with corrugated steel webs and prestressed by both internal and external tendons have emerged as one of the promising bridge forms. In view of the different behaviour of components and the large shear deformation of webs with negligible flexural stiffness, the assumption that plane sections remain plane may no longer be valid, and therefore the classical Euler-Bernoulli and Timoshenko beam models may not be applicable. In the design of this type of bridges, both the ultimate load and ductility should be examined, which requires the estimation of full-range behaviour. An analytical sandwich beam model and its corresponding beam finite element model for geometric and material nonlinear analysis are developed for this type of bridges considering the diaphragm effects. Different rotations are assigned to the flanges and corrugated steel webs to describe the displacements. The model accounts for the interaction between the axial and flexural deformations of the beam, and uses the actual stress-strain curves of materials considering their stress path-dependence. With a nonlinear kinematical theory, complete description of the nonlinear interaction between the external tendons and the beam is obtained. The numerical model proposed is verified by experiments.

• Structural Engineering and Mechanics
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• 제54권4호
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• pp.693-710
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• 2015

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler-Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

• Nuclear Engineering and Technology
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• 제52권12호
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• pp.2939-2948
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• 2020

This paper presents an analytic solution procedure of the rotationally restrained hinged-hinged beam subjected to transverse motions at supports based on EBT (Euler-Bernoulli beam theory). The EBT solutions are compared with the solutions based on TBT (Timoshenko beam theory) for a wide range of the rotational restraint parameter (kL/EI) of slender beams whose slenderness ratio is greater than 100. The comparison shows the followings. The internal loads such as bending moment and shearing force of an extremely thin beam obtained by EBT show a good agreement with those obtained by TBT. But the discrepancy between two solutions of internal loads tends to increase as the slenderness ratio decreases. A careful examination shows that the discrepancy of the internal loads originates from their dynamic components whereas their static components show a little difference between EBT and TBT. This result suggests that TBT should be employed even for slender beams to consider the rotational effect and the shear deformation effect on dynamic components of the internal loads. The influence of the parameter on boundary conditions is examined by manipulating the spring stiffness from zero to a sufficiently large value.

• Steel and Composite Structures
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• 제31권5호
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• pp.489-502
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• 2019

This article presents a unified mathematical model to investigate free and forced vibration responses of perforated thin and thick beams. Analytical models of the equivalent geometrical and material characteristics for regularly squared perforated beam are developed. Because of the shear deformation regime increasing in perforated structures, the investigation of dynamical behaviors of these structures becomes more complicated and effects of rotary inertia and shear deformation should be considered. So, both Euler-Bernoulli and Timoshenko beam theories are proposed for thin and short (thick) beams, respectively. Mathematical closed forms for the eigenvalues and the corresponding eigenvectors as well as the forced vibration time response are derived. The validity of the developed analytical procedure is verified by comparing the obtained results with both analytical and numerical analyses and good agreement is detected. Numerical studies are presented to illustrate effects of beam slenderness ratio, filling ratio, as well as the number of holes on the dynamic behavior of perforated beams. The obtained results and concluding remarks are helpful in mechanical design and industrial applications of large devices and small systems (MEMS) based on perforated structure.

• Coupled systems mechanics
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• 제6권3호
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• pp.251-272
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• 2017

In this paper, the thermo-mechanical vibration characteristics of functionally graded (FG) nanobeams subjected to three types of thermal loading including uniform, linear and non-linear temperature change are investigated in the framework of third-order shear deformation beam theory which captures both the microstructural and shear deformation effects without the need for any shear correction factors. Material properties of FG nanobeam are assumed to be temperature-dependent and vary gradually along the thickness according to the power-law form. Hence, applying a third-order shear deformation beam theory (TSDBT) with more rigorous kinetics of displacements to anticipate the behaviors of FG nanobeams is more appropriate than using other theories. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. The obtained results are compared with those predicted by the nonlocal Euler-Bernoulli beam theory and nonlocal Timoshenko beam theory and it is revealed that the proposed modeling can accurately predict the vibration responses of FG nanobeams. The obtained results are presented for the thermo-mechanical vibration analysis of the FG nanobeams such as the effects of material graduation, nonlocal parameter, mode number, slenderness ratio and thermal loading in detail. The present study is associated to aerospace, mechanical and nuclear engineering structures which are under thermal loads.

• Structural Engineering and Mechanics
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• 제2권3호
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• pp.285-294
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• 1994

Based on a fast and accurate method for the stationary random seismic response analysis for discretized structures(Lin 1992, Lin et al. 1992), a Ritz method for dealing with such responses of continuous systems in developed. This method is studied quantitatively, using cantilever shear beams for simplicity and clarity. The process can be naturally extended to deal with various boundary conditions as well as non-uniform Bernoulli-Euler beams, or even Timoshenko beams. Algorithms for both proportionally and non-proportionally damped responses are described. For all of such damping cases, it is not necessary to solve for the natural vibrations of the beams. The solution procedure is very simple, and equally efficient for a white or a non-white ground excitation spectrum. Two examples are given where various power spectral density functions, variances, covariances and second spectral moments of displacement, internal force response, and their derivatives are calculated and analyses. Some Ritz solutions are compared with "exact" CQC solutions.

• Wind and Structures
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• 제24권1호
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• pp.43-57
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• 2017

The use of nanotechnology materials and applications in the construction industry should be considered for enhancing material properties. However, the nonlinear buckling of an embedded straight concrete columns reinforced with silicon dioxide ($SiO_2$) nanoparticles is investigated in the present study. The column is simulated mathematically with Euler-Bernoulli and Timoshenko beam models. Agglomeration effects and the characteristics of the equivalent composite are determined using Mori-Tanaka approach. The foundation around the column is simulated with spring and shear layer. The governing equations are derived using energy method and Hamilton's principal. Differential quadrature method (DQM) is used in order to obtain the buckling load of structure. The influences of volume percent of $SiO_2$ nanoparticles, geometrical parameters and agglomeration on the buckling of column are investigated. Numerical results indicate that considering agglomeration effects leads to decrease in buckling load of structure.