• Structural Engineering and Mechanics
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• v.51 no.5
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• pp.773-792
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• 2014

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

• Journal of Mechanical Science and Technology
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• v.19 no.11
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• pp.2016-2024
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• 2005

In this work, a mixed beam approach that combines both the stiffness and the flexibility methods has been performed to analyze the coupled composite blades with closed, two-cell cross-sections. The Reissner's semi-complementary energy functional is used to derive the beam force-displacement relations. Only the membrane part of the shell wall is taken into account to make the analysis simple and also to deliver a clear picture of the mixed method. All the cross section stiffness coefficients as well as the distribution of shear across the section are evaluated in a closed-form through the beam formulation. The theory is validated against experimental test data, detailed finite element analysis results, and other analytical results for coupled composite blades with a two-cell airfoil section. Despite the simple kinematic model adopted in the theory, an accuracy comparable to that of two-dimensional finite element analysis has been obtained for cases considered in this study.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2004.10a
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• pp.886-889
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• 2004

A simple macro triangle with drilling d.o.f.'s is proposed for plane stress problems based on IET(Individual element test) and finite element template. Three-node triangular element has geometrical advantages in preprocessing but suffers from bad performance comparing to other shapes of elements -especially quadrilateral. Main purpose of this study is to construct a high-performance linear triangular element with limited supplementary d.o.f.'s. A triangle is divided by three sub-triangles with drilling d.o.f.'s. The sub-triangle stiffness come from IET passing force-lumping matrix, so this assures the consistency of the element. The macro element strategy takes care of the element‘s stability and accuracy like higher-order stiffness in the F.E. template. The resulting element fits on the uses of conventional three-node. Benchmark examples show proposed element in closed form stiffness from CAS (Computer algebra system) gives the improved results without more computational efforts than others.

• Structural Engineering and Mechanics
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• v.4 no.3
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• pp.277-301
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• 1996

A finite element model of a beam element with flexible connections is used to investigate the effect of the randomness in the stiffness values on the modal properties of the structural system. The linear behavior of the connections is described by a set of random fixity factors. The element mass and stiffness matrices are function of these random parameters. The associated eigenvalue problem leads to eigenvalues and eigenvectors which are also random variables. A second order perturbation technique is used for the solution of this random eigenproblem. Closed form expressions for the 1st and 2nd order derivatives of the element matrices with respect to the fixity factors are presented. The mean and the variance of the eigenvalues and vibration modes are obtained in terms of these derivatives. Two numerical examples are presented and the results are validated with those obtained by a Monte-Carlo simulation. It is found that an almost linear statistical relation exists between the eigenproperties and the stiffness of the connections.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2000.10a
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• pp.369-376
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• 2000

For the general loading condition and boundary condition, it is very difficult to obtain closed-form solutions for buckling loads and natural frequencies of thin-walled structures because its behaviour is very complex due to the coupling effect of bending and torsional behaviour. Consequently most of previous finite element formulations introduced approximate displacement fields using shape functions as Hermitian polynomials, isoparametric interpoation function, and so on. The purpose of this study is to calculate the exact displacement field of a thin-walled straight beam element with the non-symmetric cross section and present a consistent derivation of the exact dynamic stiffness matrix. An exact dynamic element stiffness matrix is established from Vlasov's coupled differential equations for a uniform beam element of non-symmetric thin-walled cross section. This numerical technique is accomplished via a generalized linear eigenvalue problem by introducing 14 displacement parameters and a system of linear algebraic equations with complex matrices. The natural frequencies are evaluated for the non-symmetric thin-walled straight beam structure, and the results are compared with available solutions in order to verify validity and accuracy of the proposed procedures.

• Computers and Concrete
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• v.18 no.6
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• pp.1097-1112
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• 2016

Fire loading causes a critical collapse of RC (Reinforced Concrete) Structures since the embedded steels inside are relative week against high elevated temperature. Several numerical frameworks for fire resistance have been proposed, however they have limitations such as unstable convergence and long calculation period. In the work, 2-D nonlinear FE technique is proposed using Galerkin method for RC structures under fire loading. Closed-form element stiffness with a triangular element is adopted and verified with fire test on three RC slabs with different fire loading conditions. Several simulations are also performed considering fire loading conditions, water contents, and cover depth. The proposed numerical technique can handle time-dependent fire loading, convection, radiation, and material properties. The proposed technique can be improved through early-aged concrete behavior like moisture transport which varies with external temperature.

• Proceedings of the KSR Conference
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• 2000.11a
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• pp.358-365
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• 2000

For the general case of loading conditions and boundary conditions, it is very difficult to obtain closed form solutions for buckling loads and natural frequencies of thin-walled structures because its behaviour is very complex due to the coupling effect of bending and torsional behaviour. In consequence, most of previous finite element formulations are introduce approximate displacement fields to use shape functions as Hermitian polynomials, and so on. The Purpose of this study is to presents a consistent derivation of exact dynamic stiffness matrices of thin-walled straight beams, to be used ill tile free vibration analysis, in which almost types of boundary conditions are exist An exact dynamic element stiffness matrix is established from governing equations for a uniform beam element of nonsymmetric thin-walled cross section. This numerical technique is accomplished via a generalized linear eigenvalue problem by introducing 14 displacement parameters and a system of linear algebraic equations with complex matrices. The natural frequency is evaluated for the thin-walled straight beam structure, and the results are compared with analytic solutions in order to verify the accuracy of this study.

• Earthquakes and Structures
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• v.26 no.2
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• pp.163-174
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• 2024

Dynamic equilibrium equations for finite element analysis were derived for the free field one-dimensional shear wave propagation through the horizontally layered soil deposits with the elastic half-space. We expressed Rayleigh's viscous damping consisting of mass and stiffness proportional terms. We considered two cases where damping matrices are defined in the total and relative displacement fields. Two forms of equilibrium equations are presented; one in terms of total motions and the other in terms of relative motions. To evaluate the performance of new equilibrium equations, we conducted two sets of site response analyses and directly compared them with the exact closed-form frequency domain solution. Results show that the base shear force as earthquake load represents the simpler form of equilibrium equation to be used for the finite element method. Conventional finite element procedure using base acceleration as earthquake load predicts exact solution reasonably well even in soil deposits with unrealistically high damping.

• Steel and Composite Structures
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• v.7 no.3
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• pp.219-240
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• 2007

An analytical-numerical procedure has been presented in this paper to take into account the nonlinear effects of concrete cracking and time-dependent effects of creep and shrinkage in the concrete portion of the continuous composite beams under service load. The procedure is analytical at the element level and numerical at the structural level. The cracked span length beam element consisting of uncracked zone in middle and cracked zones near the ends has been proposed to reduce the computational effort. The progressive nature of cracking of concrete has been taken into account by division of the time into a number of time intervals. Closed form expressions for stiffness matrix, load vector, crack lengths and mid-span deflection of the beam element have been presented in order to reduce the computational effort and bookkeeping. The procedure has been validated by comparison with the experimental and analytical results reported elsewhere and with FEM. The procedure can be readily extended for the analysis of composite building frames where saving in computational effort would be very considerable.

• Structural Engineering and Mechanics
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• v.60 no.1
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• pp.111-130
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• 2016

Two new elements with six degrees of freedom are proposed by applying the equilibrium conditions and strain-displacement equations. The first element is formulated for the infinite ratio of beam radius to thickness. In the second one, theory of the thick beam is used. Advantage of these elements is that by utilizing only one element, the exact solution will be obtained. Due to incorporating equilibrium conditions in the presented formulations, both proposed elements gave the precise internal forces. By solving some numerical tests, the high performance of the recommended formulations and also, interaction effects of the bending and axial forces will be demonstrated. While the second element has less error than the first one in thick regimes, the first element can be used for all regimes due to simplicity and good convergence. Based on static responses, it can be deduced that the first element is efficient for all the range of structural characteristics. The free vibration analysis will be performed using the first element. The results of static and dynamic tests show no deficiency, such as, shear and membrane locking and excessive stiff structural behavior.