• Research in Mathematical Education
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• v.10 no.3 s.27
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• pp.169-187
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• 2006

Ubiquitous errors in algebra like $(x+y)^2=x^2+y^2$ are a constant reminder that most students' manipulation of algebraic symbols has become detached from structural principles. The U.S. mathematics education community (NCTM, 2000) has responded by shying away from algebra as a structural study, preferring instead to ground meaning in empirical domains of reference. A new analysis of such errors shows that students' detachment from structural meaning stems from an inadequate structural curriculum, not from the inherent difficulty of adopting an abstract perspective on expressions and equations. A structural curriculum is outlined that preserves the possibility of students' engaging fully with algebra as both an empirical and a structural study.

• Structural Engineering and Mechanics
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• v.52 no.2
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• pp.323-349
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• 2014

In this paper, linear buckling analysis of a curved sandwich beam with a flexible core is investigated. Derivation of equations for face sheets is accomplished via the classical theory of curved beam, whereas for the flexible core, the elasticity equations in polar coordinates are implemented. Employing the von-Karman type geometrical non-linearity in strain-displacement relations, nonlinear governing equations are resulted. Linear pre-buckling analysis is performed neglecting the rotation effects in pre-buckling state. Stability equations are concluded based on the adjacent equilibrium criterion. Considering the movable simply supported type of boundary conditions, suitable trigonometric solutions are adopted which satisfy the assumed edge conditions. The critical uniform load of the beam is obtained as a closed-form expression. Numerical results cover the effects of various parameters on the critical buckling load of the curved beam. It is shown that, face thickness, core thickness, core module, fiber angle of faces, stacking sequence of faces and openin angle of the beam all affect greatly on the buckling pressure of the beam and its buckled shape.

• Interaction and multiscale mechanics
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• v.1 no.1
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• pp.53-81
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• 2008

The theory of microlocal analysis of hyperbolic partial differential equations shows that the energy density associated to their high-frequency solutions satisfies transport equations, or radiative transfer equations for randomly heterogeneous materials with correlation lengths comparable to the (small) wavelength. The main limitation to the existing developments is the consideration of boundary or interface conditions for the energy and power flow densities. This paper deals with the high-frequency transport regime in coupled heterogeneous structures. An analytical model for the derivation of high-frequency power flow reflection/transmission coefficients at a beam or a plate junction is proposed. These results may be used in subsequent computations to solve numerically the transport equations for coupled systems, including interface conditions. Applications of this research concern the prediction of the transient response of slender structures impacted by acoustic or mechanical shocks.

• Structural Engineering and Mechanics
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• v.63 no.4
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• pp.481-495
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• 2017

The present article examines the static response of multilayered magneto-electro-elastic (MEE) beam in thermal environment through finite element (FE) methods. On the basis of the minimum total potential energy principle and the coupled constitutive equations of MEE material, the FE equilibrium equations of cantilever MEE beam is derived. Maxwell's equations are considered to establish the relation between electric field and electric potential; magnetic field and magnetic potential. A simple condensation approach is employed to solve the global FE equilibrium equations. Further, numerical evaluations are made to examine the influence of different in-plane and through-thickness temperature distributions on the multiphysics response of MEE beam. A parametric study is performed to evaluate the effect of stacking sequence and different temperature profiles on the direct and derived quantities of MEE beam. It is believed that the results presented in this article serve as a benchmark for accurate design and analysis of the MEE smart structures in thermal applications.

• Structural Engineering and Mechanics
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• v.19 no.1
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• pp.73-96
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• 2005

For the spatially coupled free vibration analysis of shear deformable thin-walled non-symmetric curved beam subjected to initial axial force, an exact dynamic element stiffness matrix of curved beam is evaluated. Firstly equations of motion and force-deformation relations are rigorously derived from the total potential energy for a curved beam element. Next a system of linear algebraic equations are constructed by introducing 14 displacement parameters and transforming the second order simultaneous differential equations into the first order simultaneous differential equations. And then explicit expressions for displacement parameters are numerically evaluated via eigensolutions and the exact $14{\times}14$ dynamic element stiffness matrix is determined using force-deformation relations. To demonstrate the accuracy and the reliability of this study, the spatially coupled natural frequencies of shear deformable thin-walled non-symmetric curved beams subjected to initial axial forces are evaluated and compared with analytical and FE solutions using isoparametric and Hermitian curved beam elements and results by ABAQUS's shell elements.

• Structural Engineering and Mechanics
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• v.39 no.3
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• pp.361-382
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• 2011

This paper presents the results of an investigation on shear strengthening of RC beams externally reinforced with CFRP composite. A total of six full-scale beams of four CFRP strengthened and two unstrengthened were tested in the absence of internal stirrups in the shear span. The strengthening configurations contained two styles: discrete uniformly spaced strips and customized wide strips over B-regions. The composite systems provided an increase in ultimate strength as compared to the unstrengthened beams. Among the three layouts that had the same area of CFRP, the highest contribution was provided by the customized layout that targeted the B-regions. A comparative study of the experimental results with published empirical equations was conducted in order to evaluate the assumed effective strains. The empirical equations were found to be unconservative. Nonlinear finite element (NLFE) models were developed for the beams. The models agreed with test results that targeting the B-region was more effective than distributing the same CFRP area in a discrete strip style over shear spans. Moreover, the numerical models predicted the contribution of different configurations better than the empirical equations.

• Structural Engineering and Mechanics
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• v.13 no.3
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• pp.329-343
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• 2002

In this research, the dynamic stability of an orthotropic elastic conical shell, with elasticity moduli and density varying in the thickness direction, subject to a uniform external pressure which is a power function of time, has been studied. After giving the fundamental relations, the dynamic stability and compatibility equations of a nonhomogeneous elastic orthotropic conical shell, subject to a uniform external pressure, have been derived. Applying Galerkin's method, these equations have been transformed to a pair of time dependent differential equations with variable coefficients. These differential equations are solved using the method given by Sachenkov and Baktieva (1978). Thus, general formulas have been obtained for the dynamic and static critical external pressures and the pertinent wave numbers, critical time, critical pressure impulse and dynamic factor. Finally, carrying out some computations, the effects of the nonhomogeneity, the loading speed, the variation of the semi-vertex angle and the power of time in the external pressure expression on the critical parameters have been studied.

• Structural Engineering and Mechanics
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• v.38 no.3
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• pp.283-300
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• 2011

A new conception of fundamental tasks in dynamics of the bridge-track-train systems (BTT), with the aim to evaluate moving load's models adequacy, has been developed. The 2D physical models of BTT systems, corresponding to the fundamental tasks, have been worked out taking into account one-way constraints between the moving unsprung masses and the track. A method for deriving the implicit equations of motion, governing vibrations of BTT systems' models, as well as algorithms for numerical integration of these equations, leading to the solutions of high accuracy and relatively short times of simulations, have been also developed. The derived equations and formulated algorithms constitute the basis for numerical simulation of vibrations of the considered systems.

• Structural Engineering and Mechanics
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• v.23 no.5
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• pp.525-542
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• 2006

On the basis of equilibrium equations for static electric and magnetic fields, two unknown functions related to electric and magnetic fields were firstly introduced to rewrite the governing equations, boundary conditions and initial conditions for mechanical field. Then by introducing a dependent variable and a special function satisfying the inhomogeneous mechanical boundary conditions, the governing equation for a new variable with homogeneous mechanical boundary conditions is obtained. By using the separation of variables technique as well as the electric and magnetic boundary conditions, the dynamic problem of a functionally graded magneto-electro-elastic hollow sphere under spherically symmetric deformation is transformed to two Volterra integral equations of the second kind about two unknown functions of time. Cubic Hermite polynomials are adopted to approximate the two undetermined functions at each time subinterval and the recursive formula for solving the integral equations is derived. Transient responses of displacements, stresses, electric and magnetic potentials are completely determined at the end. Numerical results are presented and discussed.

• Structural Engineering and Mechanics
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• v.68 no.6
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• pp.701-711
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• 2018

Herein, the thermo-magneto-elastic wave dispersion answers of functionally graded (FG) double-nanobeam systems (DNBSs) are surveyed implementing a nonlocal strain gradient theory (NSGT). The kinematic relations are derived employing the classical beam theory. Also, scale influences are covered precisely in the framework of NSGT. Moreover, Mori-Tanaka homogenization model is introduced in order to obtain the effective material properties of FG nanobeams. Meanwhile, effects of external forces such as thermal and Lorentz forces are included in this research. Also, based upon the Hamilton's principle, the Euler-Lagrange equations are developed; afterwards, these equations are incorporated with those of NSGT to reach the nonlocal governing equations of FG-DNBSs. Furthermore, according to an analytical approach, the governing equations are solved to obtain the wave frequencies and phase velocities of FG-DNBSs. At the end, some illustrations are rendered to clarify the influences of a wide range of involved parameters.