Operational Method for Continuous Beams

The standingpile has been applied quite extensive in our country, which is one of the three major structure forms of the quay. Take the internal force calculation of wharf track beam in Jiangsu Taizhou for example, this paper studied the bar system finite element method processing method of rigid support and elastic support. Using different bearing support mode to calculate the track beam internal force, the results show that the corresponding internal force of different models has big differences. The negative bending moment and shear force of point support model are large. The effect of support width and elastic support was considered in the continuous beam model of elastic distribution supporting and should be considered prior to other supporting. In the process of similar design, the bearing width can be increased. This is conducive to the uniform distribution of internal force, and make the structure more reasonable.

An analytical study is presented for the vibration characteristics of thin circular plates with rotational elastic edge support and resting on an elastic foundation. The Winkler model was used to describe this foundation. The exact analytical method is used to derive the frequency equation of the plate with elastic support and resting on this foundation. Parametric investigations of the vibration of circular plates with an restrained elastically edge against rotation and resting on an elastic foundation have been carried out with respect to various rotational stiffness parameters, foundation stiffness parameters, and boundary conditions. Twelve vibration modes are presented. The location of the stepped region with respect to rotational restraint changed very little and moved from the range of 0.014 26† [10.006 204 8‡]–1 134.227 [10.025 94] to 0.017 58 [14.001 133]–6 352.958 2 [14.658 356]. Here, † represents the rotational spring stiffness ratio, and ‡ represents the foundation stiffness ratio. Extensive data are tabulated so that pertinent conclusions can be arrived at on the influence of rotational edge restraint and the foundation parameters of a Winkler foundation on the natural frequencies of uniform thin isotropic circular plates.

Continuous Beams with Unilateral Elastic Supports

This paper aims to present the exact closed-form solutions for the free vibration of double-beam systems composed of two parallel beams connected by an arbitrary number of discrete elastic supports. The general solutions of the mode shapes of the double-beam system are derived employing the Laplace transform method from a perspective of the entire domain of beams without enforcement of any segmentation. A unified strategy applied to various boundary conditions is proposed to determine the independent constants involved in the general solutions, as well as the frequency equation. Numerical calculations are performed to verify the present solutions by comparing the results from the previous literature and finite element simulation, and to discuss the effects of support parameters (stiffness, location, and number) on the modal characteristics of the double-beam system in detail. Outcomes show that the support location plays a pivotal role in regulating the modal characteristics of the double-beam system; for each-order mode, there are one or more potential optimal positions to maximize the effect of the elastic support. The mode veering phenomenon is detected as the support parameters change. It is highlighted that, by introducing an amplitude similarity index, the proximity degree for the mode shapes of the two beams influenced by the support parameters can be evaluated quantitatively. The present analysis is greatly helpful to the optimal design, health monitoring, and vibration control of the double-beam system.

A discrete mechanical model for large-amplitude free vibrations of two-stepped functionally graded beams is developed in this study. The beam’s material properties vary through the thickness according to a power-law distribution between the metallic and ceramic phases. The continuous beam is replaced by an N-degree-of-freedom system of lumped masses, longitudinal, and torsional springs. Using Hamilton’s principle, the governing nonlinear algebraic equations are derived and solved through the single-mode approach (SMA) to obtain the nonlinear frequency-amplitude relationships. In addition, an Artificial Neural Network (ANN)-based surrogate model is proposed to provide fast and accurate predictions of the nonlinear-to-linear frequency ratio as a function of key parameters such as step ratio, step position, boundary conditions, and the power-law index. Trained on data generated by the discrete formulation, the surrogate attains excellent generalization with a drastic reduction in computation time. The combined discrete-ANN framework offers both physical interpretability and computational efficiency, making it suitable for rapid design and optimization of complex FGM beam structures.

The influence of unbalanced magnetic pull (UMP) and hydraulic seal force on the vibration of large rotor-bearing systems is studied. The UMP caused by rotor eccentricity imposes important effects on rotating machinery, especially for large generators such as water turbine generator sets, because these machines operate above their first critical speed in some instances and are supported by oil film bearings. A magnetic stiffness matrix for studying the effects of the UMP is proposed. The magnetic stiffness matrix can be generated by decomposing the expression of air gap magnetic field energy. Two vibration models are constructed using the Lagrange equation. The difference between the two models lies in the boundary support condition: one has rigid support and the other has elastic bearing support. The influence of the magnetic stiffness and elastic support on the critical speed of the rotor is studied using Lyapunov nonlinear vibration stability theory. The vibration amplitude of the rotor is calculated, taking the magnetic stiffness and horizontal centrifugal force into account. The unbalanced hydraulic seal force is produced because of the asymmetry of seal clearance. This imbalance is one of the factors that causes self-excited vibration in rotating machinery, and is as important as the UMP for large water turbine generator sets. The rotor-bearing system is supported by an oil film journal bearing, whose characteristic also impose considerable influence on vibration. On the basis of the above-mentioned conditions, a three-dimensional finite element model of the rotating system that includes the oil film journal bearing is constructed. The effect of the UMP and unbalanced hydraulic seal force is considered in the construction, and studied in relation to the magnetic parameters, seal parameters, journal bearing stiffness, and outer diameter of the rotating machine critical speed. Conclusions may benefit the dynamic design and optimized operation of large rotating machinery.

Shear behaviour of continuous fibre-reinforced lightweight aggregate concrete beams

In the regulatory documents regulating the design of beam structures, insufficient attention is paid to checking the overall stability of thin-walled beams. Recommendations for testing the stability of single pin-ended beams and rigidly sealed cantilevers are given. There are no recommendations for calculating the stability of continuous multi-span beams. Insufficient attention to the general stability of thin-walled beams can be explained by the fact that the load on the beam structures is transmitted, as a rule, through other beams, decking, etc., that is, additional connections are imposed on the beam, preventing the rotation and lateral displacement of the loaded sections, which positively affects its stability. The specifics of the working conditions of crane beams, namely, the lack of additional connections and the level of load application, negatively affect their stability. The lack of recommendations for checking the overall stability of continuous crane beams leads to an unjustified assignment of geometric cross-section parameters during design, and, as a result, to serious accidents. In this paper, the solution of the equations of stability of a flat bending shape is given by the finite difference method. The derivatives of the deflection and the angle of torsion of the beam are replaced by the central finite differences of the second order. The boundary conditions are written for the extreme and intermediate supports. The use of the finite difference method in solving the stability equations allows obtaining a result for continuous multi-span crane beams, taking into account the level of load application and the influence of neighboring spans on the stability of the most loaded span.

The development of the electronic digital computer and the application of matrix algebra has made it possible for the structural engineer to analyze complex or highly redundant structures which previously had been prohibitive to analyze by the hand method of calculation. Papers recently written have utilized either the “flexibility” or “stiffness” matrix approach. It is the purpose of this paper to point out the common origin of these methods in energy and to discuss procedures for calculating the total energy of a structure. The concept of the “total” structure will be considered. Applications include analysis of a truss with elastic supports and a continuous beam.

Thermoelastic vibrations in a thin elliptic annulus plate with elastic supports

This paper proposes a modified matched interface and boundary (MMIB) method to analyze the free vibration of beams with various interfaces caused by steps, intermediate rigid and elastic supports, intermediate concentrated masses and spring-mass systems, etc. A new strategy is developed to determine the parameters in the iterative computation of MMIB. The MMIB procedures are established to deal with boundary conditions and various interface conditions, which overcomes the shortcoming of the traditional MIB. A number of examples are utilized to illustrate the performance of MMIB method. Numerical results indicate that the MMIB method is a highly accurate and convergent approach for solving interface problems.

Conventional design approaches for cold-formed steel (CFS) members were mainly based on elastic theory, including the effective width method and the direct strength method, where the influence of moment redistribution on the failure load was neglected. A Pseudo-Plastic Design Method (PPDM) has been presented for analysing multi-span CFS structures in order to enhance the economical of resistance prediction. This method is analogical to conventional plastic design theory as it introduces a pseudo-plastic moment resistance to accommodate the benefit of the redistribution of moments. In the present study, finite element modelling approach was introduced based on ANSYS package, and comparisons were made between numerical and PPDM results, in particular, on load-carrying capacity, pseudo-plastic hinges formation, moment redistribution and failure behaviour. Extensive sensitivity studies have been carried out in validation of PPDM application for full-scale continuous beams under various span number, span length, steel yield strength and cross section properties. The study shows that the design employing PPDM can lead to a better economic benefit than the aforementioned methods for CFS sigma continuous beams.

Axisymmetric vibration of point-supported circular plates

A simple, fast numerical method for the solution of a wide variety of electrochemical diffusion problems

The general theoretical development given in this paper considers the material to be orthotropic. Single Frobenius series solutions and double series solutions are obtained for orthotropic continuous plates. These solutions reduce to exact known solutions in the case of isotropic plates. For the purpose of numerical calculations, two cases of isotropic continuous plates—one over an intermediate rigid ring support and the other over an intermediate elastic ring support—are considered. For the case of a free plate over a rigid intermediate support, buckling loads increase as the support moves towards the center. In the case of an elastically supported fixed plate, the support elasticity considerably influences the buckling values, when the coefficient of support elasticity is more than 100. For lower modes a value of K 1 equal to 10,000 gives an almost fixed support. When the support is at distance of 0.355 times the radius of the plate the buckling resistance is found to be a maximum.