• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.9
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• pp.1779-1786
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• 2002

When the topology optimization is applied to the design of compliant mechanism, unexpected displacements of input and output port are generated since the displacement control is not included in the formulation. To devise a more precise mechanism, displacement constraint is formulated using the mutual potential energy concept and added to multi-objective function defined with flexibility and stiffness of a structure. The optimization problem is resolved by using Finite Element Method(FEM) and Sequential Linear Programming(SLP). Design examples of compliant mechanism with displacement constraint are presented to validate the proposed design method.

• Structural Engineering and Mechanics
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• v.81 no.3
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• pp.335-348
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• 2022

It is well known that after finding the displacement in the structural mechanics, strain and stress can be obtained in the straight-forward process. The main purpose of this paper is to unify the displacement functions for solving the solid body. By performing mathematical operations, three sets of these key relationships are found in this paper. All of them are written in the Cartesian Coordinates and in terms of a simple function. Both analytical and numerical approaches are utilized to validate the correctness of the presented formulations. Since all required conditions for the bodies with self-equilibrated loadings are satisfied accurately, the authors' relations can solve these kinds of problems. This fact is studied in-depth by solving some numerical examples. It is found that a very simple function can be used for each formulation instead of ten different and complex displacement potentials defined by previous studies.

• Computers and Concrete
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• v.4 no.2
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• pp.135-156
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• 2007

Three 3D nonlinear finite-element models are developed to study the behavior of concrete beams and plates with and without external reinforcement by fibre-reinforced plastic (FRP). All three models are formulated based upon the 3D theory of elasticity. The stress model is modified from the element developed by Ramtekkar, et al. (2002) to incorporate material nonlinearity in the formulation. Both transverse stress and displacement components are used as nodal degrees-of-freedom to ensure the continuity of both stress and displacement components between the elements. The displacement model uses only displacement components as nodal degrees-of-freedom. The transition model has both stress and displacement components as nodal degrees-of-freedom on one surface, and only displacement components as nodal degrees-of-freedom on the opposite surface. The transition model serves as a connector between the stress and the displacement models. The developed models are validated by comparing the results of the analyses with an existing experimental result. Parametric studies of the effects of the externally reinforced FRP on the load capacity of reinforced concrete (RC) beams and concrete plates are performed to demonstrate the practicality and the efficiency of the proposed models.

• Structural Engineering and Mechanics
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• v.67 no.4
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• pp.317-326
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• 2018

The iterative decomposition coupling formulation of the precise integration finite element method (FEM) and the time domain boundary element method (TD-BEM) is presented for elstodynamic problems. In the formulation, the FEM node and the BEM node are not required to be coincident on the common interface between FEM and BEM sub-domains, therefore, the FEM and BEM are independently discretized. The force and displacement converting matrices are used to transfer data between FEM and BEM nodes on the common interface between the FEM and BEM sub-domains, to renew the nodal variables in the process of the iterations for the un-coincident FEM node and BEM node. The iterative coupling formulation for elastodynamics in current paper is of high modeling accuracy, due to the semi-analytical solution incorporated in the precise integration finite element method. The decomposition coupling formulation for elastodynamics is verified by examples of a cantilever bar under a Heaviside-type force and a harmonic load.

• Structural Engineering and Mechanics
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• v.36 no.5
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• pp.545-560
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• 2010

Two or more distinct materials are combined into a single functionally graded material (FGM) where the microstructural composition and properties change gradually. Thermal post-buckling behavior of uniform slender FGM beams is investigated independently using the classical Rayleigh-Ritz (RR) formulation and the versatile Finite Element Analysis (FEA) formulation developed in this paper. The von-Karman strain-displacement relations are used to account for moderately large deflections of FGM beams. Bending-extension coupling arising due to heterogeneity of material through the thickness is included. Simply supported and clamped beams with axially immovable ends are considered in the present study. Post-buckling load versus deflection curves and buckled mode shapes obtained from both the RR and FEA formulations for different volume fraction exponents show an excellent agreement with the available literature results for simply supported ends. Response of the FGM beam with clamped ends is studied for the first time and the results from both the RR and FEA formulations show a very good agreement. Though the response of the FGM beam could have been studied more accurately by FEA formulation alone, the authors aim to apply the RR formulation is to find an approximate closed form post-buckling solutions for the FGM beams. Further, the use of the RR formulation clearly demonstrates the effect of bending-extension coupling on the post-buckling response of the FGM beams.

• Structural Engineering and Mechanics
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• v.51 no.4
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• pp.685-705
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• 2014

This paper presents a new formulation for forward scalar wave simulations in semi-infinite media. Perfectly-Matched-Layers (PMLs) are used as a wave absorbing boundary layer to surround a finite computational domain truncated from the semi-infinite domain. In this work, a hybrid formulation was developed for the simulation of scalar wave motion in two-dimensional PML-truncated domains. In this formulation, displacements and stresses are considered as unknowns in the PML domain, while only displacements are considered to be unknowns in the interior domain. This formulation reduces computational cost compared to fully-mixed formulations. To obtain governing wave equations in the PML region, complex coordinate stretching transformation was introduced to equilibrium, constitutive, and compatibility equations in the frequency domain. Then, equations were converted back to the time-domain using the inverse Fourier transform. The resulting equations are mixed (contain both displacements and stresses), and are coupled with the displacement-only equation in the regular domain. The Newmark method was used for the time integration of the semi-discrete equations.

• Computational Structural Engineering
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• v.9 no.3
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• pp.169-177
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• 1996

The focus of this study is on the problem of the design of structure of undetermined topology. This problem has been regarded as being the most challenging of structural optimization problems, because of the difficulty of allowing topology to change. Conventional approaches break down when element sizes approach to zero, due to stiffness matrix singularity. In this study, a novel nonlinear programming formulation of the topology problem is presented. Its main feature is the ability to account for topology variation through zero element sizes. Stiffness matrix singularity is avoided by embedding the equilibrium equations as equality constraints in the optimization problem. Although the formulation is general, two dimensional plane elasticity examples are presented. The design problem is to find minimum weight of a plane structure of fixed geometry but variable topology, subject to constraints on stress and displacement. Variables are thicknesses of finite elements, and are permitted to assume zero sizes. The examples demonstrate that the formulation is effective for finding at least a locally minimal weight.

• Geophysics and Geophysical Exploration
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• v.6 no.2
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• pp.77-86
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• 2003

We designed a new time-domain, finite-difference, elastic wave modeling technique, based on a displacement formulation. which yields nearly correct solutions to Lamb's problem. Unlike the conventional, displacement-based, finite-difference method using a node-based grid set (where both displacements and material properties such as density and Lame constants are assigned to nodal points), in our new finite-difference method, we use a cell-based grid set (where displacements are still defined at nodal points but material properties within cells). In the case of using the cell-based grid set, stress-free conditions at the free surface are naturally described by the changes in the material properties without any additional free-surface boundary condition. Through numerical tests, we confirmed that the new second-order finite differences formulated in the cell-based grid let generate numerical solutions compatible with analytic solutions unlike the old second-order finite-differences formulated in the node-based grid set.

• Structural Engineering and Mechanics
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• v.11 no.2
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• pp.171-184
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• 2001

A direct discrete formulation suitable for the nonlinear analysis of masonry structures is presented. The numerical approach requires a pair of dual meshes, one for describing displacement fields, one for imposing equilibrium. Forces and displacements are directly used (instead of having to resort to a model derived from a set of differential equations). Associated and nonassociated flow laws are dealt with within a complementarity framework. The main features of the method and of the relevant computer code are discussed. Numerical examples are presented, showing that the numerical approach is able to describe plastic strains, damage effects and crack patterns in masonry structures.

• International Journal of Aeronautical and Space Sciences
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• v.13 no.2
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• pp.199-209
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• 2012

An advanced aeroelastic formulation for flutter analyses is presented in this paper. Refined 1D structural models were coupled with the doublet lattice method, and the g-method was used for flutter analyses. Structural models were developed in the framework of the Carrera Unified Formulation (CUF). Higher-order 1D structural models were obtained by using Taylor-like expansions of the cross-section displacement field of the structure. The order (N) of the expansion was considered as a free parameter since it can be arbitrarily chosen as an input of the analysis. Convergence studies on the order of the structural model can be straightforwardly conducted in order to establish the proper 1D structural model for a given problem. Flutter analyses were conducted on several wing configurations and the results were compared to those from literature. Results show the enhanced capabilities of CUF 1D in dealing with the flutter analysis of typical wing structures with high accuracy and low computational costs.