• Structural Engineering and Mechanics
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• v.65 no.6
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• pp.731-739
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• 2018

This paper presents an optimal pattern for distributing stiffness along a framed tube structure through an analytic equation, which may be used during the preliminary design stage. Most studies in this field are computationally intensive and time consuming, while a hand-calculation method, as presented here, is a more suitable tool for sensitivity analyses and parametric studies. Approach in development of the analytic model is to minimize the mean compliance (external work) for a given volume of material. A variational statement of the problem is made, and a specified deformation-profile is obtained as the necessary condition for a minimum; enforcing this condition, stiffness is then computed. Due to some near-zero values for stiffness, the problem is modified by considering a lower bound constraint. To deal with this constraint, the design domain is assumed to be divided into two zones of constant stiffness and constant curvature; and the problem is restated in terms of these concepts. It will be shown that this methodology allows for easy computation of stiffness through an analytic and dimensionless equation, valid in any system of units. To show practicality of the proposed method, a tubed-system structure with uniform stiffness distribution is redesigned using the proposed model. Comparative analyses of the results reveal that in addition to simplicity of the proposed method, it provides a rather high degree of accuracy for real-world problems.

• Structural Engineering and Mechanics
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• v.51 no.3
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• pp.377-402
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• 2014

In this study, optimal distribution of springs which supports a cantilever beam is investigated to minimize two objective functions defined. The optimal size and location of the springs are ascertained to minimize the tip deflection of the cantilever beam. Afterwards, the optimization problem of springs is set up to minimize the tip absolute acceleration of the beam. The Fourier Transform is applied on the equation of motion and the response of the structure is defined in terms of transfer functions. By using any structural mode, the proposed method is applied to find optimal stiffness and location of springs which supports a cantilever beam. The stiffness coefficients of springs are chosen as the design variables. There is an active constraint on the sum of the stiffness coefficients and there are passive constraints on the upper and lower bounds of the stiffness coefficients. Optimality criteria are derived by using the Lagrange Multipliers. Gradient information required for solution of the optimization problem is analytically derived. Optimal designs obtained are compared with the uniform design in terms of frequency responses and time response. Numerical results show that the proposed method is considerably effective to determine optimal stiffness coefficients and locations of the springs.

• 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 the Korean Society for Precision Engineering
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• v.16 no.3 s.96
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• pp.103-108
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• 1999

A problem in the grinding with a small diametric wheel is the decrease of wheel speed. In order to resolve this problem, an accelerating unit which increases the wheel speed is recommended. In this paper, the accelerating effect of an accelerating unit has been investigated through the side-cut grinding experiments performed with a vitrified bonded CBN wheel in a machining center(MC). The static stiffness, normal force, and machining error were measured in the experiments. As the accelerating unit is attached on the column of machining center, the static stiffness of tool system is largely decreased. But as the wheel speed increased by the accelerating unit, this problem is overcome and machining efficiency is improved. The lesser the quill stiffness was, the higher the accelerating effect became.

• International Journal of Automotive Technology
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• v.8 no.3
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• pp.337-342
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• 2007

A multi-objective optimization technique was implemented to obtain optimal topologies of the inner reinforcement for a vehicle's hood simultaneously considering the static stiffness of bending and torsion and natural frequency. In addition, a smoothing scheme was used to suppress the checkerboard patterns in the ESO method. Two models with different curvature were chosen in order to investigate the effect of curvature on the static stiffness and natural frequency of the inner reinforcement. A scale factor was employed to properly reflect the effect of each objective function. From several combinations of weighting factors, a Pareto-optimal topology solution was obtained. As the weighting factor for the elastic strain efficiency went from 1 to 0, the optimal topologies transmitted from the optimal topology of a static stiffness problem to that of a natural frequency problem. It was also found that the higher curvature model had a larger static stiffness and natural frequency than the lower curvature model. From the results, it is concluded that the ESO method with a smoothing scheme was effectively applied to topology optimization of the inner reinforcement of a vehicle's hood.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.17 no.1 s.118
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• pp.80-87
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• 2007

In this paper, a method for identifying the location and extent of a damage in a structure using residual forces was presented. Element stiffness matrix reduction parameters in a finite element model were used to describe the damaged structure mathematically. The element stiffness matrix reduction parameters were determined by minimizing a global error derived from dynamic residual vectors, which were obtained by introducing a simulated experimental data into the eigenvalue problem. Genetic algorithm was used to get the solution set of element stiffness reduction parameters. The proposed scheme was verified using Euler-Bernoulli beam. The results were presented in the form of tables and charts.

• Structural Engineering and Mechanics
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• v.38 no.2
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• pp.195-210
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• 2011

Bernoulli-Euler beam theory is used to develop an exact dynamic stiffness matrix for the flexural-torsional coupled motion of a three-dimensional, axially loaded, thin-walled beam of doubly asymmetric cross-section. This is achieved through solution of the differential equations governing the motion of the beam including warping stiffness. The uniform distribution of mass in the member is also accounted for exactly, thus necessitating the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick-Williams algorithm. Finally, examples are given to confirm the accuracy of the theory presented, together with an assessment of the effects of axial load and loading eccentricity.

• Computational Structural Engineering : An International Journal
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• v.2 no.1
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• pp.33-42
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• 2002

A computationally efficient stiffness design method for building structures is proposed in which dynamic soil-structure interaction based on the wave-propagation theory is taken into account. A sway-rocking shear building model with appropriate ground impedances derived from the cone models due to Meek and Wolf (1994) is used as a simplified design model. Two representative models, i.e. a structure on a homogeneous half-space ground and a structure on a soil layer on rigid rock, are considered. Super-structure stiffness satisfying a desired stiffness performance condition are determined via an inverse problem formulation for a prescribed ground-surface response spectrum. It is shown through a simple yet reasonably accurate model that the ground conditions, e.g. homogeneous half-space or soil layer on rigid rock (frequency-dependence of impedance functions), ground properties (shear wave velocity), depth of surface ground, have extensive influence on the super-structure design.

• Structural Engineering and Mechanics
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• v.39 no.4
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• pp.541-558
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• 2011

This paper presents an analysis of stochastic eigenvalue problem of plane bar structures. Particular attention is paid to the effect of spatial variations of the flexural properties of the structure on the first four eigenvalues. The problem of spatial variations of the structure properties and their effect on the first four eigenvalues is analyzed in detail. The stochastic eigenvalue problem was solved independently by stochastic finite element method (stochastic FEM) and Monte Carlo techniques. It was revealed that the spatial variations of the structural parameters along the structure may substantially affect the eigenvalues with quite wide gap between the two extreme cases of zero- and full-correlation. This is particularly evident for the multi-segment structures for which technology may dictate natural bounds of zero- and full-correlation cases.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1996.04a
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• pp.182-189
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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 developed and examined. 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.