• Steel and Composite Structures
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• v.33 no.3
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• pp.403-432
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• 2019

Axially functionally graded (AFG) beams are a new class of composite structures that have continuous variations in material and/or geometrical parameters along the axial direction. In this study, the exact analytical solutions for the free vibration of AFG and uniform beams with general elastic supports are obtained by using Euler-Bernoulli beam theory. The elastic supports are modeled with linear rotational and lateral translational springs. Moreover, the material and/or geometrical properties of the AFG beams are assumed to vary continuously and together along the length of the beam according to the power-law forms. Accordingly, the accuracy, efficiency and capability of the proposed formulations are demonstrated by comparing the responses of the numerical examples with the available solutions. In the following, the effects of the elastic end restraints and AFG parameters, namely, gradient index and gradient coefficient, on the values of the first three natural frequencies of the AFG and uniform beams are investigated comprehensively. The analytical solutions are presented in tabular and graphical forms and can be used as the benchmark solutions. Furthermore, the results presented herein can be utilized for design of inhomogeneous beams with various supporting conditions.

• Journal of KSNVE
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• v.8 no.2
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• pp.297-305
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• 1998

The finite element method is used for the study of the eigenvalue problems of partially fixed end beams with intermediate elastic supports. The elastic critical loads and natural frquencies of the beams are investigated by changing the numbers of elastic supports and their stiffness, and also by changing Kinney's fixity factor, $f_a$. The relationship between two eigenvalues is established by calculating the corresponding values of $(w/w_n)^2$ through changing $(P/P_{cr})$ values. The results of this study are as follows : (1) The elastic critical loads and natural frequencies of beams increase with increases in Kinney's fixity factor, $f_a$ and with the increased numbers of intermediate elastic supports. (2) The relationship between elastic critical loads and the natural frequencies of partially fixed end beams with intermediated elastic supports is $P/P_{cr} + (w/w_n)^2/ = 1$ without regard to Kinney's fixity factor, the stiffness of elastic supports, or the number of elastic supports.

• Structural Engineering and Mechanics
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• v.57 no.6
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• pp.969-988
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• 2016

Vibration analysis of the beams on elastic foundation has gained the great interest of many researchers. In the literature, there are many studies that focus on the free vibration analysis of the beams on one or two parameter elastic foundations. On the other hand, there are no sufficient studies especially focus on the comparison of dynamic response including the bending moment and shear force of the beams resting on Winkler and two parameter foundations. In this study, dynamic response of the axially loaded Timoshenko beams resting on modified Vlasov type elastic soil was investigated by using the separation of variables method. Governing equations were obtained by assuming that the material had linear elastic behaviour and mass of the beam was distributed along its length. Numerical analysis were provided and presented in figures to find out the differences between the modified Vlasov model and conventional Winkler type foundation. Furthermore, the effect of shear deformation of elastic soil on the dynamic response of the beam was investigated.

• Structural Engineering and Mechanics
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• v.15 no.6
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• pp.705-716
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• 2003

Gradient elastic flexural beams are dynamically analysed by analytic means. The governing equation of flexural beam motion is obtained by combining the Bernoulli-Euler beam theory and the simple gradient elasticity theory due to Aifantis. All possible boundary conditions (classical and non-classical or gradient type) are obtained with the aid of a variational statement. A wave propagation analysis reveals the existence of wave dispersion in gradient elastic beams. Free vibrations of gradient elastic beams are analysed and natural frequencies and modal shapes are obtained. Forced vibrations of these beams are also analysed with the aid of the Laplace transform with respect to time and their response to loads with any time variation is obtained. Numerical examples are presented for both free and forced vibrations of a simply supported and a cantilever beam, respectively, in order to assess the gradient effect on the natural frequencies, modal shapes and beam response.

• Structural Engineering and Mechanics
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• v.81 no.6
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• pp.751-767
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• 2022

In the hogging bending moment area, continuous composite beams are subjected to the ultimate limit state of lateral-torsional buckling (LTB), which depends on web stiffness as well as concrete slab and shear connection stiffnesses. The design of the LTB and the determination of the elastic critical moment are produced approximately, using the European Standard EN 1994-1-1:2004, for continuous composite steel beams, but is applicable only for those with a plane web steel profile. Also, and from the previous researches, the elastic critical moment of the continuous composite beams with corrugated sinusoidal web steel profiles was determined. In this paper, a finite element analysis (FEA) model was developed using the ANSYS 16 software, to determine the elastic critical moments of continuous composite steel beams with various corrugated web profiles, such as trapezoidal, zigzag, and rectangular profiles, which were evaluated against numerical data of the sinusoidal one from the literature. Ultimately, the failure load of a composite steel beam with various web profiles was predicted by studying 46 models, based on FEA modeling, and a procedure for predicting the elastic critical moment of composite beams with various web steel profiles was proposed. When compared to sinusoidal web profiles, the trapezoidal, zigzag, and rectangular web profiles required an average increase in load capacity and stiffness of 7%, 17.5%, and 28%, respectively, according to the finite element analysis. Also, the rectangular web steel profile has a greater stiffness and load capacity. In contrast, the sinusoidal web has lower values for these characteristics.

• Steel and Composite Structures
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• v.46 no.5
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• pp.709-718
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• 2023

The absence of an important portion of the web plate in steel beams with multiple circular perforations, cellular beams, causes the web plate to undergo distortions prior to and during lateral torsional buckling (LTB). The conventional LTB equations in the codes and literature underestimate the buckling moments of cellular beams due to web distortions. The present study is an attempt to develop analytical methods for estimating the elastic buckling moments of cellular beams. The proposed methods rely on the reductions in the torsional and warping rigidities of the beams due to web distortions and the reductions in the weak-axis bending and torsional rigidities due to the presence of web openings. To test the accuracy of the analytical estimates from proposed solutions, a total of 114 finite element analyses were conducted for six different standard IPEO sections and varying unbraced lengths within the elastic limits. These analyses clearly indicated that the LTB solutions in the AISC 360-16 and AS4100:2020 codes overestimate the buckling loads of cellular beams within elastic limits, particularly at shorter span lengths. The LDB solutions in the literature and the Eurocode 3 LTB solution, on the other hand, provided conservative buckling moment estimates along the entire range of elastic buckling.

• Structural Engineering and Mechanics
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• v.52 no.5
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• pp.1033-1049
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• 2014

The free vibration of functionally graded material (FGM) beams on an elastic foundation and spring supports is investigated. Young's modulus, mass density and width of the beam are assumed to vary in thickness and axial directions respectively following the exponential law. The spring supports are also taken into account at both ends of the beam. An analytical formulation is suggested to obtain eigen solutions of the FGM beams. Numerical analyses, based on finite element method by using a beam finite element developed in this study, are performed in order to show the legitimacy of the analytical solutions. Some results for the natural frequencies of the FGM beams are given considering the effect of various structural parameters. It is also shown that the spring supports show the greatest effect on the natural frequencies of FGM beams.

• Computers and Concrete
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• v.32 no.6
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• pp.543-551
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• 2023

In this article, thermal post-buckling and primary resonance of the porous functionally graded material (FGM) beams in thermal environment considering the geometric imperfection are studied, the material properties of FGM beams are assumed to vary along the thickness of the beam, meanwhile, the porosity volume fraction, geometric imperfection, temperature, and the elastic foundation are considered, using the Euler-Lagrange equation, the nonlinear vibration equations are derived, after the dimensionless processing, the dimensionless equations of motion can be obtained. Then, the two-step perturbation method is applied to solve the vibration problems, the resonance and thermal post-buckling response relations are obtained. Finally, the functionally graded index, the porosity volume fraction, temperature, geometric imperfection, and the elastic foundation on the resonance behaviors of the FGM beams are presented. It can be found that these parameters can influence the thermal post-buckling and primary resonance problems.

• Structural Engineering and Mechanics
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• v.18 no.3
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• pp.303-314
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• 2004

An artificial neural network (ANN) application is presented for flexural and axial vibration analysis of elastic beams with various support conditions. The first three natural frequencies of beams are obtained using multi layer neural network based back-propagation error learning algorithm. The natural frequencies of beams are calculated for six different boundary conditions via direct solution of governing differential equations of beams and Rayleigh's approximate method. The training of the network has been made using these data only flexural vibration case. The trained neural network, however, had been tested for cantilever beam (C-F), and both end free (F-F) in case the axial vibration, and clamped-clamped (C-C), and Guided-Pinned (G-P) support condition in case the flexural vibrations which were not included in the training set. The results found by using artificial neural network are sufficiently close to the theoretical results. It has been demonstrated that the artificial neural network approach applied in this study is highly successful for the purposes of free vibration analysis of elastic beams.

• Structural Engineering and Mechanics
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• v.64 no.1
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• pp.23-31
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• 2017

In the study, dynamic behavior of laminated orthotropic beams on elastic foundation is investigated. Consistent model presented here combines the finite element solution of the system with SAP2000 software and the calculation of soil parameters with MATLAB software using Modified Vlasov Model type elastic foundation. For this purpose, a computing tool is coded in MATLAB which employs Open Application Programming Interface (OAPI) feature of SAP2000 to provide two-way data flow during execution. Firstly, an example is taken from the literature to demonstrate the accuracy of the consistent FEM-Vlasov Model. Subsequently, the effects of boundary conditions, subsoil depth, elasticity modulus of subsoil, slenderness ratio, velocity of moving load and lamination scheme on the behavior of laminated orthotropic beams on elastic foundation are investigated on a new numerical example. It can be concluded that it is really convenient to use OAPI feature of SAP2000 to model this complex behavior of laminated orthotropic beams on elastic foundation under moving load.