Saint-Venant decay rates: A procedure for the prism of general cross-section

A procedure previously developed for the determination of decay rates for self-equilibrated loadings at one end of a pin-jointed framework consisting of repeated identical cells, wherein the decay factors are the eigenvalues of the single cell transfer matrix, is here further developed and applied to a prismatic continuum beam of general cross-section. A sectional length of beam is treated within ANSYS finite element code as a super element; nodes at both ends of the section are treated as master nodes and the stiffness matrix relating forces and displacements at these master nodes is constructed within ANSYS. Manipulation of this stiffness matrix within MATLAB gives the transfer matrix from which the eigenvalues and eigenvectors may be readily determined. Accuracy of the method is assessed by treating the plane strain strip, the plane strain sandwich strip, and the rod of circular cross-section, representing a selection of the examples for which exact analytical solutions are available, and is found to be very good in all cases.     

Exact dynamic stiffness matrix of non-symmetric thin-walled beams on elastic foundation using power series method

Exact dynamic element stiffness matrix for the flexural–torsional free vibration analysis of the shear deformable thin-walled beam with non-symmetric cross-section on two-types of elastic foundation is newly presented using power series method based on the technical computing program Mathematica. For this, the shear deformable beam on elastic foundation theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This beam includes the shear deformation effects due to the shear forces and the restrained warping torsion and due to the coupled effects between them, and rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. And then equations of motion and force–deformation relations are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components and the exact dynamic element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element solutions using the isoparametric beam elements. Particularly the influences of the coupled shear deformation on the vibrational behavior of non-symmetric beam on elastic foundation are investigated.     

Shear-Bending-Torsion Interaction in Structural Concrete Members: A Nonlinear Coupled Sectional Approach

Inclined cracks that take place in reinforced concrete elements due to tangential internal forces, such as shear and torsion, produce a non-isotropic response on the structure in the post-cracked regime and up to failure, also known as crack-induced-anisotropy. The result is that all six internal forces acting in a cross-section are generally coupled. A generalized beam formulation for the nonlinear coupled analysis of non-isotropic elements under six internal forces is presented. The theory is based on a cross-section analysis approach with both warping and distortion capabilities, which were proved necessary to correctly handle the problem with frame element analysis. In this paper, the non-linear mechanical aspects of cracked concrete structures under tangential forces are summarized. A state of the art review of beam formulations for the non-linear analysis of concrete structures is presented, and the approaches followed to account for the interaction of shear and torsion forces are discussed. After presenting the proposed formulation, its capabilities are shown by means of an application example of a cross section under coupled bending-shear and torsion, finally main conclusions are drawn.     

Analysis of beam-like lattice trusses

A simple procedure is presented for predicting the thermoelastic and free vibration responses of large repetitive beam-like trusses. The procedure is based on replacing the original lattice structure by an equivalent continuum beam model and obtaining closed-form (exact) solutions for the beam model. The equivalent beam model accounts for warping and shear deformation in the plane of the cross-section and is characterized by its thermoelastic strain and kinetic energies, from which the equations of motion and constitutive relations can be derived. The high accuracy of the results obtained by the proposed approach is demonstrated by means of numerical examples.     

Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained using two different nonlinear finite element formulations is examined. Both formulations can correctly describe arbitrary rigid body displacements and can be used in the large deformation analysis. The first formulation is based on the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation. The second formulation, the absolute nodal coordinate formulation (ANCF), relaxes this assumption and introduces modes that couple the deformation of the cross section and the axial and bending deformations. In the absolute nodal coordinate formulation, four different models are developed; a beam model based on a general continuum mechanics approach, a beam model based on an elastic line approach, a beam model based on an elastic line approach combined with the Hellinger–Reissner principle, and a plate model based on a general continuum mechanics approach. The use of the general continuum mechanics approach leads to a model that includes the ANCF coupled deformation modes. Because of these modes, the continuum mechanics model differs from the models based on the elastic line approach. In both the geometrically exact beam and the absolute nodal coordinate formulations, the centrifugal forces are formulated in terms of the element nodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotating beam is examined, and the results obtained using the two formulations are compared for different values of the beam angular velocity. The numerical comparative study presented in this investigation shows that when the effect of some ANCF coupled deformation modes is neglected, the eigenvalue solutions obtained using the geometrically exact beam and the absolute nodal coordinate formulations are in a good agreement. The results also show that as the effect of the centrifugal forces, which tend to increase the beam stiffness, increases, the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes less significant. It is shown in this paper that when the effect of the Poisson ration is neglected, the eigenvalue solution obtained using the absolute nodal coordinate formulation based on a general continuum mechanics approach is in a good agreement with the solution obtained using the geometrically exact beam model.     

Direct evaluation of reaction forces and moments using isoparametric elements

The use of isoparametric finite elements for plates, shells, solids and other structures is by now widespread. The quadratic family of elements, in general, gives the optimum results. Very good displacements are obtained and these are used to evaluate stresses (or stress resultants). When it comes to reactions at the boundaries, the generalized nodal forces are often treated as useless since it is usually not obvious how to relate them to the distributed boundary forces and moments. They usually show sharp variation and even reversal of sign between neighboring nodes. In this paper it is shown how to derive the distributed reaction forces and moments directly from the corresponding generalized forces. Several numerical examples are given to show the excellent agreement with exact and other known solutions.     

A systematic method for efficient computation of full and subsets Zernike moments

A new method is proposed for fast and accurate computation of Zernike moments. This method presents a novel formula for computing exact Zernike moments by using exact complex moments where the exact values of complex moments are computed by mathematical integration of the monomials over digital image pixels. The proposed method is applicable to compute the full set of Zernike moments as well as the subsets of individual order, repetition and an individual moment. A comparison with other conventional methods is performed. The results show the superiority of the proposed method.     

HIGH-PERFORMANCE COMPUTING FOR ROBOT DYNAMIC PARAMETERS LEARNING

The robot control problem deals with the computation of the actuating forces/torques in order to provide the desired motion of the end effector. Knowledge of the exact dynamic characteristics of robot manipulators is one of the most significant factors that affects the design of control systems, because the controller performance is directly dependent on the accuracy of the dynamic model. The behavior of dynamic models is normally complicated because of the geometrically varying arm configuration, uncertain loads handled by the end effector, and nonlinear effects resulting from the high interactions among joints. These problems are further complicated by the discrepancy in measuring the arm dynamic parameters (e.g., first moments, radius of gyration of each link). Unless these factors are considered while developing the manipulator dynamic model, the performance of the controller is not expected to meet the given requirements. Thus, the development of an efficient and computationally viable method for estimating the dynamic parameters of a robot arm in situ is essential. In this paper the problem of estimating these parameters in real-time situations is addressed. First, an efficient dynamic model, which is linear in the dynamic parameters, is derived based on the Lagrange-Euler representation. By using this model, the dynamic parameters can easily be isolated and regrouped. Second, to enable real-time applications, the algorithm for computing the dynamic model and the estimation of parameters is distributed     

Numerical study of a steel sub-frame in fire

Steel-framed buildings are generally designed with “simple” shear-resisting connections, and lateral forces are resisted by vertical bracing and shear walls. When a beam is considered then the effects of the longitudinal restraints by the adjacent structure and the rotational restraint by the connections has to be taken into account. Because of structural interaction, the beam behaviour at elevated temperature is rather complex.This paper presents a numerical parametric study of a structural system consisting of an exposed steel beam restrained between a pair of fire protected steel columns. The structural sub-frame is modelled using 3D shell elements, thereby taking into account the effect of the local failure modes, and the realistic behaviour of the sub-frame exposed to natural fire. The numerical model accounts for the initial geometrical imperfections, nonlinear temperature gradient over the cross-section, geometrical and material nonlinearity and temperature dependent material properties.Results obtained using a general Finite Element software – LUSAS and a fire dedicated software – SAFIR, are compared. The influence of following variables: beam span/depth ratio, lateral restraint, gradient temperature within the cross-section and mechanical load level is presented in the paper. The failure modes, the development of the internal forces and displacements throughout the analysis are considered to exemplify the effects of the variables considered.     

Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns

An improved numerical method to exactly evaluate 14 × 14 dynamic and static element stiffness matrices is proposed for the spatial free vibration and stability analysis of nonsymmetric thin-walled straight beams subjected to eccentrically axial loads. Firstly equations of motion and force-deformation relations are rigorously derived from the total potential energy for a uniform beam element with nonsymmetric thin-walled cross-section. Next a system of linear algebraic equations with nonsymmetric matrices is constructed by introducing 14 displacement parameters and transforming the higher order simultaneous differential equation into the first order simultaneous equation. And then explicit expressions for displacement parameters are exactly evaluated by solving a generalized eigenproblem with complex eigenvalues. Finally exact element stiffness matrices are determined using force-deformation relations. Particularly straightforward application of the present method may not give the exact static stiffness because of existence of multiple zero eigenvalues in case of static buckling problems. Accordingly, a modified numerical method to resolve this difficulty is developed for two cases depending on the initial state of stress resultants. In order to demonstrate the validity and the accuracy of this method, the natural frequencies and buckling loads of nonsymmetric thin-walled beam-columns having bending-torsional deformation modes are evaluated and compared with analytical and F.E. solutions or results analyzed by ABAQUS’s shell element.     

A torsion bending element for thin-walled beams with open and closed cross sections

A finite element is formulated for the torsion problems of thin-walled beams. The element is based on Benscoter's beam theory, which is valid for open and also closed cross-sections. The non-polynomial interpolation presented in this paper allows the exact static solution to be obtained with only one element. Numerical results are presented for three thin-walled cantilever beams, one with a channel cross-section and the two others with rectangular cross-sections. The influence of the transverse shear strain is investigated and the different models of torsion are compared. For one example, the results obtained with one-dimensional torsion elements are compared with those obtained using shell elements.     

Uniform strength for large deflections of cantilever beams under end point load

Analysis is conducted for slender beams with a varying cross-section under large non-linear elastic deformation. A thickness variation function is derived to achieve optimal - constant maximum bending stress distribution along the beam for inclined end load of arbitrary direction. Closed form solutions are derived for the large deflections that correspond to the various loading conditions. The analysis is repeated for a beam with optimally varying width (for arbitrary end force) and the width variation function is also determined.     

Study of the ligament tension and cross-section deformation using nonlinear finite element/multibody system algorithms

In this investigation, the effects of the knee-joint movements on the ligament tension and cross-section deformation are examined using large displacement nonlinear finite element/multibody system formulations. Two knee-joint models that employ different constitutive equations and significantly different deformation kinematics are developed and implemented to analyze the ligament dynamics in a computational solution procedure that integrates large displacement finite element and multibody system algorithms. The first model employs a lower fidelity large displacement cable element that does not capture the cross-section deformations and allows for using only nonlinear classical beam theory with a linear Hookean material law instead of a general continuum mechanics approach. In the second model, a higher fidelity large displacement beam model that captures more coupled deformation modes including Poisson modes as well the cross-section deformation is used. This higher fidelity model also allows for a straight forward implementation of general nonlinear constitutive models, such as Neo Hookean material laws, based on a general continuum mechanics approach. Cauchy stress tensor and Nanson’s formula are used to obtain an accurate expression for the ligament tension forces, which as shown in this investigation depend on the ligament cross section deformation. The two models are implemented in a general multibody system algorithm that allows introducing general constraint and force functions. The finite element/multibody system computational algorithm used in this investigation is based on an optimum sparse matrix structure and ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The results obtained in this investigation show that models that ignore coupled deformation modes including some Poisson modes and the cross-section deformations can lead to inaccurate prediction of the ligament forces. These simpler models, as demonstrated in this investigation, can be used to obtain only simplified expressions for the ligament tensions. A three-dimensional knee-joint model that consists of five bodies including two flexible bodies that represent the medial collateral ligament (MCL) and lateral collateral ligament (LCL) is used in the numerical comparative study presented in this paper. The large displacement procedure presented in this investigation can be applied to other types of Ligaments, Muscles, and Soft Tissues (LMST) in biomechanics applications.     

The interaction of plane frames with elastic foundation having normal and shear moduli of subgrade reactions

The interaction of plane frames with an elastic foundation, of the Winkler type, having normal and shear moduli of subgrade reactions was studied.

An exact stiffness matrix for a beam element on an elastic foundation having only a normal modulus of subgrade reaction was modified to include the shear modulus of subgrade reaction of the foundation as well as the axial force in the beam. A computer program was written and used in two case studies. In the first case study the convergence of the modified element was tested: in the second it was employed to study the effect of the foundation normal and shear moduli of subgrade reactions on the bending moments in a selected plane frame. The results indicated that bending moments might be considerably affected according to the type of frame and loading.     

Mathematical observations in structural dynamics

In dynamic structural analysis, the basic relations between forces and displacements for a beam element subjected to axial, torsional or flexural vibration are obtained either by solving the appropriate equation of motion or by using an approximate method. The exact equation leads to the dynamic stiffness matrix while the approximate method results in the superposition of elastic and inertial forces represented respectively by the stiffness and mass matrices.The common procedure in finding the natural frequencies is to set the determinant of the dynamic stiffness matrix for the system equal to zero. The approximate method leads to an eigenvalue type problem while the exact method results in a transcendental equation of trigonometric and hyperbolic functions. The natural frequencies in a region of interest are found by a systematic search in the determinntal function.The purpose of this paper is to show that the search technique cannot be applied for certain values of the argument at which the determinantal function is not defined. It is proved that the natural frequencies of any isolated member in the system are critical values for the determinantal function. A practical method is given to obviate the difficulty in order to find the natural frequencies from the determinant, including the critical values at which the dynamic stiffness matrix is not defined. Also, as part of this investigation, the mathematical relation is established between the dynamic stiffness matrix derived by the approximate finite element method and the results obtained from the exact Bernoulli-Euler equation for flexural vibration or the wave equation for axial or torsional vibration.     

An explicit expression for the tangent-stiffness of a finitely deformed 3-D beam and its use in the analysis of space frames

Simplified procedures for finite-deformation analyses of space frames, using one beam element to model each member of the frame, are presented. Each element can undergo three-dimensional. arbitrarily large, rigid motions as well as moderately large non-rigid rotations. Each element can withstand three moments and three forces. The nonlinear bending-stretching coupling in each element is accounted for. By obtaining exact solutions to the appropriate governing differential equations, an explicit expression for the tangent-stiffness matrix of each element, valid at any stage during a wide range of finite deformations, is derived. An arc length method is used to incrementally compute the large deformation behavior of space frames. Several examples which illustrate the efficiency and simplicity of the developed procedures are presented. While the finitely deformed frame is assumed to remain elastic in the present paper, a plastic hinge method, wherein a hinge is assumed to form at an arbitrary location in the element, is presented in a companion paper.     

Stiffness method of thin-walled beams with closed cross-section

The aim of the present study is to present a general stiffness method capable of analyzing three-dimensional thin-walled straight beams with closed cross-sections. The method based on the assumptions introduced by Benscoter is suited for automatic computation on computers. Starting from the principle of virtual displacement, an exact stiffness matrix and vector of fixed-end reactions for the analysis of thin-walled beam with an arbitrary closed cross-section are derived. The method is illustrated by example.     

Analysis of a beam column on elastic foundation

The analysis of beams on elastic Winkler foundation is very common in engineering. In many applications, transverse as well as axial forces exist. An exact analytical solution of a finite element beam column resting on a Winkler foundation is performed from which the exact stiffness terms are determined. The stiffness matrix is incorporated into a common beam program. Nodes are required only at points of discontinuity in stiffness, loading, or supports. Comparisons are made with case results appearing in literature.