• Journal of KSNVE
• /
• v.4 no.4
• /
• pp.451-457
• /
• 1994

A numerical method is presented to obtain natural frequencies and mode shapes of uniform elastic beams with static deflections due to dead loads. The differential equation governing the free vibration of beam taken into account the static deflection due to deal loads is derived and solved numerically. The hinged-hinged, clamped-clamped and clamped-hinged end constraints are applied in the numerical examples. As the numerical results, the lowest three nondimensional frequency parameters are reported as functions of nondimensional system parameters; the load parameters, and the slenderness rations. And some typical mode shapes of free vibrations are also presented in figures.

• Tunnel and Underground Space
• /
• v.33 no.3
• /
• pp.150-168
• /
• 2023

The anisotropic behavior of rocks is primarily attributed to the directional arrangement of rock-forming minerals and the distribution characteristics of microcracks. Notably, sedimentary and metamorphic rocks often exhibit distinct transverse isotropy in terms of their strength and deformation characteristics. Consequently, it is crucial to gain accurate insights into the deformation and failure characteristics of transversely isotropic rocks during rock mechanics design processes. The deformation of such rocks is described by five independent elastic constants, which are determined through laboratory testing. In this study, the characteristics of the directional variation of apparent elastic constants in transversely isotropic rocks were investigated using experimental data reported in the literature. To achieve this, the constitutive equation proposed by Mehrabadi & Cowin was introduced to calculate the apparent elastic constants more efficiently and systematically in a rotated Cartesian coordinate system. Four transversely isotropic rock types from the literature were selected, and the influence of changes in the orientation of the weak plane on the variations of the apparent elastic modulus, apparent shear modulus, and apparent Poisson's ratio was analyzed. Based on the investigation, a new constraint on the elastic constants has been proposed. If the proposed constraint is satisfied, the directional variation of the apparent elastic constants in transversely isotropic rocks aligns with intuitive predictions of their tendencies.

• Journal of Astronomy and Space Sciences
• /
• v.15 no.1
• /
• pp.209-220
• /
• 1998

The stability problem of steady motion of a rigid body with multi-elastic appendages and multi-liquid-filled cavities, in the presence of no external forces or torque, is considered in this paper. The flexible appendages are modeled as the clamped -free-free-free rectangular plates, or/and as the discrete mass- spring sub-system. The motion of liquid in every single ellipsoidal cavity is modeled as the uniform vortex motion with a finite number of degrees of freedom. Assuming that stationary holonomic constraints imposed on the body allow its rotation about a spatially fixed axis, the equation of motion for such a systematic configuration can be very complex. It consists of a set of ordinary differential equations for the motion of the rigid body, the uniform rotation of the contained liquids, the motion of discrete elastic parts, and a set of partial differential equations for the elastic appendages supplemented by appropriate initial and boundary conditions. In addition, for such a hybrid system, under suitable assumptions, their equations of motion have four types of first integrals, i.e., energy and area, Helmholtz' constancy of liquid - vortexes, and the constant of the Poisson equation of motion. Chetaev's effective method for constructing Liapunov functions in the form of a set of first integrals of the equations of the perturbed motion is employed to investigate the sufficient stability conditions of steady motions of the complete system in the sense of Liapunov, i.e., with respect to the variables determining the motion of the solid body and to some quantities which define integrally the motion of flexible appendages. These sufficient conditions take into account the vortexes of the contained liquids, the vibration of the flexible components, and coupling among the liquid-elasticity solid.

• KSCE Journal of Civil and Environmental Engineering Research
• /
• v.13 no.4
• /
• pp.251-258
• /
• 1993

The main purpose of this paper is to present both the natural frequencies and the buckling loads of beam-columns on Winkler-type foundations. The ordinary differential equations governing the free vibrations and the buckling loads of beam-columns on Winkler-type foundation are derived as nondimensional forms. The Runge-Kutta method and Determinant Search method are used to perform the integration of the differential equations and to determine the eigenvalues(natural frequencies and buckling loads), respectively. Hinged-hinged and damped-clamped end constraints are applied in numerical examples. The relation between frequency parameter and elastic foundation parameter is presented in figure. The effects of axial loads on the natural frequencies of beam-columns on elastic foundations are investigated and the relation between buckling load parameter and elastic foundation parameter is also analyzed. The relation between foundation rested ratio and frequency parameter, buckling load parameter are investigated. The beam-columns on non-homogeneous elastic foundation are analyzed and typical mode shapes are also presented.

• Journal of the Korea institute for structural maintenance and inspection
• /
• v.10 no.2
• /
• pp.123-133
• /
• 2006

The main objective of this study is to develop an optimization algorithm of framed structures with rigid and various semi-rigid connections using the multilevel dynamic programming and the sequential unconstrained minimization techniques (SUMT). The second-order elastic analysis is performed for steel framed structures. The second order elastic analysis is developed based on nonlinear beam-column theory considering the bowing effect. The following semi-rigid connections are considered; double web angle, top-seat angle and top-seat angle with web angle. We considered the three connection models, such as modified exponential, polynomial and three parameter model. The total weight of the structural steel is used as the objective function in the optimization process. The dimensions of steel cross section are selected as the design variables. The design constraints consist of strength requirements for axial, shear and flexural resistance and serviceability requirements.

• Structural Engineering and Mechanics
• /
• v.67 no.5
• /
• pp.439-455
• /
• 2018

Within a probabilistic framework, this paper addresses the determination of the static structural response of beams and frames with partially restrained (semi-rigid) connections. The flexibility of the nodal connections is incorporated via an idealized linear-elastic behavior of the beam constraints through the use of rotational springs, which are here considered uncertain for taking into account the largely scattered results observed in experimental findings. The analysis is conducted via the Probabilistic Transformation Method, by modelling the spring stiffness terms (or equivalently, the fixity factors of the beam) as uniformly distributed random variables. The limit values of the Eurocode 3 fixity factors for steel semi-rigid connections are assumed. The exact probability density function of a few indicators of the structural response is derived and discussed in order to identify to what extent the uncertainty of the beam constraints affects the resulting beam response. Some design considerations arise which point out the paramount importance of probability-based approaches whenever a comprehensive experimental background regarding the stiffness of the beam connection is lacking, for example in steel frames with semi-rigid connections or in precast reinforced concrete framed structures. Indeed, it is demonstrated that resorting to deterministic approaches may lead to misleading (and in some cases non-conservative) outcomes from a design viewpoint.

• Steel and Composite Structures
• /
• v.22 no.5
• /
• pp.1163-1192
• /
• 2016

This paper presents the cost optimization of a composite I beam floor system, designed to be made from a reinforced concrete slab and steel I sections. The optimization was performed by the mixed-integer non-linear programming (MINLP) approach. For this purpose, a number of different optimization models were developed that enable different design possibilities such as welded or standard steel I sections, plastic or elastic cross-section resistances, and different positions of the neutral axes. An accurate economic objective function of the self-manufacturing costs was developed and subjected to design, resistance and deflection (in)equality constraints. Dimensioning constraints were defined in accordance with Eurocode 4. The Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm was applied together with a two-phase MINLP strategy. A numerical example of the optimization of a composite I beam floor system, as presented at the end of this paper, demonstrates the applicability of the proposed approach. The optimal result includes the minimal produced costs of the structure, the optimal concrete and steel strengths, and dimensions.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.27 no.7
• /
• pp.1159-1169
• /
• 2003

This paper investigates the effects of T-stress and plastic hardening mismatch on the interfacial crack-tip stress field via finite element analyses. Plane strain elastic-plastic crack-tip fields are modeled with both MBL formulation and a full SEC specimen under pure bending. Modified Prandtl slip line fields illustrate the effects of T-stress on crack-tip constraint in homogeneous material. Compressive T-stress substantially reduces the interfacial crack-tip constraint, but increases the J-contribution by lower hardening material, J$\_$L/. For bimaterials with two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and J$\_$L/. The fracture toughness for bimaterial joints would consequently be much lower than that of lower hardening homogeneous material. The implication of unbalanced J-integral in bimaterials is also discussed.

• Journal of Institute of Control, Robotics and Systems
• /
• v.13 no.4
• /
• pp.320-328
• /
• 2007

This paper presents a stiffness modeling of a low-DOF parallel robot, which takes into account of elastic deformations of joints and links, A low-DOF parallel robot is defined as a spatial parallel robot which has less than six degrees of freedom. Differently from serial chains in a full 6-DOF parallel robot, some of those in a low-DOF parallel robot may be subject to constraint forces as well as actuation forces. The reaction forces due to actuations and constraints in each serial chain can be determined by making use of the theory of reciprocal screws. It is shown that the stiffness of an F-DOF parallel robot can be modeled such that the moving platform is supported by 6 springs related to the reciprocal screws of actuations (F) and constraints (6-F). A general $6{\times}6$ stiffness matrix is derived, which is the sum of the stiffness matrices of actuations and constraints, The compliance of each spring can be precisely determined by modeling the compliance of joints and links in a serial chain as follows; a link is modeled as an Euler beam and the compliance matrix of rotational or prismatic joint is modeled as a $6{\times}6$ diagonal matrix, where one diagonal element about the rotation axis or along the sliding direction is infinite. By summing joint and link compliance matrices with respect to a reference frame and applying unit reciprocal screw to the resulting compliance matrix of a serial chain, the compliance of a spring is determined by the resulting infinitesimal displacement. In order to illustrate this methodology, the stiffness of a Tricept parallel robot has been analyzed. Finally, a numerical example of the optimal design to maximize stiffness in a specified box-shape workspace is presented.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.14 no.3
• /
• pp.371-379
• /
• 2001

This paper deals with the free vibrations of horizontally curved members resting on elastic foundations with continuity effect. Taking into account the effects of rotatory inertia and shear deformation, differential equations governing the free vibrations of such beams are derived, in which the Pasternak foundation model is considered as the elastic foundation with continuity effect. The differential equations are solved numerically to calculate natural frequencies and mode shapes. The experiments were performed in which the natural frequencies of such curved beams in laboratorial scale were measured and these results agree quite well with the present numerical studies. In numerical examples, the circular, parabolic, sinusoidal and elliptic curved members with the hinged-hinged, hinged-clamped and clamped end constraints are considered. The parametric studies are conducted and the lowest four frequency parameters are reported in tables and figures as the non-dimensional forms. Also the typical mode shapes are presented.