Curved girder bridge analysis

A Fourier Series Slope-Deflection Technique has been developed as a means of analyzing curved girder bridge systems, which may contain orthotropic decks. The technique incorporates both pure and warping torsional effects as well as bending effects. Girder deflections and internal forces are determined at any location along the respective girders. The resulting forces include bending moment, shear, pure torsion, warping torsion, and bimoment.

The entire slope-deflection analyses has been programmed for use on an IBM 7094 computer, FORTRAN IV language. The three programs for a (1) single span analysis, (2) two-span continuous structure, and (3) three-span continuous structure may be obtained from the authors.     

Cross-section optimal design of composite laminated thin-walled beams

This paper aims to perform optimal design of cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. The finite element method is used in discretizing the theory. For design sensitivity calculations, the cross-section is modelled throughout design elements. Geometrically, these elements may coincide with the laminates that constitute the cross-section. The developed formulation is based on the concept of adjoint structure. After a warping function is calculated for the cross-section, an adjoint problem may be formulated for each of the properties and a corresponding adjoint warping is determined. It can be applied in a unified way to open, closed or hybrid cross-sections. Design optimization is performed by nonlinear programming techniques. Laminate thickness and lamina orientations are considered as design variables.     

Nonuniform torsion of multi-material composite bars by the boundary element method

In this paper a boundary element method is developed for the nonuniform torsion of arbitrary constant cross-section multi-material composite bars. The materials have different elasticity and shear moduli and are firmly bonded together. The bar is subjected to an arbitrarily concentrated or distributed twisting moment while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, beside the Saint-Venant torsional shear stresses, the warping normal stresses are also computed. Two boundary value problems with respect to the variable along the beam angle of twist and to the warping function with respect to the shear center are formulated and solved employing a BEM approach. Both the warping and the torsion constants are computed by employing an effective Gaussian integration over domains of arbitrary shape. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The contribution of the normal stresses due to restrained warping is investigated.     

Solution of non-uniform torsion of bars by an integral equation method

In this paper, a boundary element method (BEM) is developed for the non-uniform torsion of simply or multiply connected cylindrical bars of arbitrary cross-section. The bar is subjected to an arbitrarily distributed twisting moment while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, besides the Saint-Venant torsional shear stresses, the warping normal stresses are also computed. Two boundary value problems with respect to the variable along the beam angle of twist and to the warping function are formulated and solved employing a BEM approach. Both the warping and the torsion constants are computed by employing an effective Gaussian integration over the domains of arbitrary shape. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The contribution of the normal stresses due to a restrained warping is investigated, by numerical examples, with great practical interest.     

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.     

Sectorial properties of straight thin-walled beams

The sectorial method of analysis for thin-walled beams subjected to torsional loading offers many advantages. In particular it enables warping restraint effects due to non-uniform torsion to be incorporated into a general beam theory covering all solid, thick-walled and thin-walled beams of non-deformable cross-section. The distribution of warping restraint stresses around the section is defined in a similar way as for bending by a system of sectorial co-ordinates and several additional geometrical terms. A wider understanding and acceptance of this useful method of analysis is only hindered by difficulties in calculating the various sectorial functions. Solutions are readily available for open sections and regular single cell closed sections. For more complex sections, particularly asymmetric or multicellular sections or those with tapering walls, pierced walls or bracing, these calculations are often tedious. A general computer program is described which analyses any cross-section with open or closed parts. Examples are given of its application to the analysis of several straight thin-walled beams.     

Warping shear stresses in nonuniform torsion of composite bars by BEM

In this paper a complete approximation of the nonuniform torsion problem of composite bars of arbitrary constant cross section using the boundary element method is developed. The composite bar consists of a matrix surrounding a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The bar is subjected to an arbitrarily concentrated or distributed twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, beside the Saint-Venant torsional shear stresses, the warping normal and shear stresses are also computed. Three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions are formulated and solved employing a pure BEM approach, that is only boundary discretization is used. Both the warping and the torsion constants together with the torsional shear stresses and the warping normal and shear stresses are computed. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The magnitude of the warping shear stresses due to restrained warping is investigated by numerical examples with great practical interest.     

Nonlinear analysis of anisotropic rods using curvature transformation and including warping

A model for analyzing the nonlinear behavior of Isotropic rods was recently presented, based on principal curvature transformations, use of generalized coordinates, and adopting Euler-Bernoulli and inextensionality assumptions. It was shown to be sufficiently accurate for many design purposes, when the resulting deflection shapes can be reasonably expected to be well approximated by a superposition of free vibration modes. This is, apparently, the case for a large number of nonlinear beam bending-torsion dynamics problems including, for example, helicopter rotor blades. The present paper extends that model so as to include anisotropic materials and a general accounting for warping of cross-sections, different from most approaches in the literature. Rigid movement of cross-sections, such as associated with pure extension and shear deformations due to transverse shear or torsion, are considered as special cases of warping, but rigid rotations which remain normal to the deformed neutral axis, such as occur in Euler-Bernoulli beam bending, are not. This leads to a general and efficient model, easy to apply in many useful cases. Examples of this model's utility are presented in the form of studies of the tension of a pretwisted rod and the bending of a beam with sandwich construction. Good agreement is obtained with exact analytic results, and modal convergence behavior is shown to be quite different for global deflection behavior, on the one hand, and local deflections of the cross-sections, on the other hand, near boundaries with warping constraint, such as blade roots.     

Analysis of distortional stress in curved I girder bridges

A simplified means of determining more accurately the stress distribution in a curved girder system considering web distortion is presented. The distortional response of a single span curved plate girder bridge is determined utilizing a finite difference procedure. The analysis finds the maximum bending, warping and distortional stresses along each girder.

A UNIVAC 1108 computer was used in the development of the simplified equations which relate the bending and warping stresses to the induced distortional stress. The program written in FORTRAN V, calculates the stresses developed in a four plate girder system having any number of diaphragms along its span.     

Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections. I: Stability formulation and closed-form solutions

For spatial stability analysis of shear deformable thin-walled curved beams with non-symmetric cross-sections, an improved analytical formulation is proposed. Firstly the displacement field is introduced considering the second order terms of semi-tangential rotations. Next an elastic strain energy is derived by using transformation equations of displacement parameters and stress resultants and considering shear deformation effects due to shear forces and restrained warping torsion. And then the potential energy due to initial stress resultants is consistently derived with accurate calculation of Wagner effect. In addition, closed-form solutions for in-plane and lateral-torsional buckling loads of curved beams subjected to uniform compression and pure bending are newly derived. In the companion paper, FE procedures are developed by using curved and straight beam elements with arbitrary thin-walled sections. In numerical examples, to illustrate accuracy and validity of this study, closed-form solutions for in-plane and out-of-plane buckling loads are presented and compared with those obtained from analytical solutions by other researchers.     

Interactive simulation of one-dimensional flexible parts

In this paper, we present a system for simulating one dimensional flexible parts such as cables or hose. The modelling of bending and torsion follows the Cosserat model. For this purpose we use a generalized spring-mass system and describe its configuration by a carefully chosen set of coordinates. Gravity and contact forces as well as the forces responsible for length conservation are expressed in Cartesian coordinates. But bending and torsion effects can be dealt with more effectively by using quaternions to represent the orientation of the segments joining two neighbouring mass points. This augmented system allows an easy formulation of all interactions with the best appropriate coordinate type and yields a strongly banded Hessian matrix. An energy minimizing process accounts for a solution exempt from the oscillations that are typical of spring-mass systems. The whole system is numerically stable and can be solved at interactive frame rates. It is integrated in a virtual reality software for use in applications such as cable routing and assembly simulation.     

Numerical method for analysing open thin-walled structures under interaction of bending and torsion

A computer program is developed to analyse concrete beams of open thin-walled sections, at different stages of loading from zero load to failure. The program is divided into two parts; the first part deals with the beam from zero load to cracking. Of course, loading is combined bending, shear and torsion (warping torsion and St Venant's torsion). In this part of the program, Vlassov's theory has been used. The cracking load is defined as that load which causes principal tensile stresses equal to the tensile strength of concrete. The second part of the program deals with all post-cracking stages of loading from cracking point to failure. An iteration procedure is used until full convergence occurs at a particular cross-section. The geometrical properties are calculated; these include the contribution of steel in the cross-section and that of concrete in the compressive zones. The mathematical model is given. The computer results are compared with earlier experimental results, and the two sets of results show reasonable agreement. The program is written in FORTRAN.     

曲线梁桥在车辆载荷下的动力响应研究

近年来随着我国公路交通的快速发展,曲线梁桥被大量工程采用.车辆过桥时的曲线桥振动涉及车辆三向载荷和桥梁弯扭耦合振动,有必要对其进行精确建模并深入研究.本文以某五跨连续曲线箱梁桥为背景,采用ABAQUS有限元软件建立该桥梁的模型,并利用Adams建立了三轴重型汽车整车模型,计算得到了不同工况下的轮胎力.通过编写基于Fortran语言的DLOAD和UTRACLOAD荷载子程序,将六个轮胎载荷作用于曲线桥上,分别进行了不同曲率半径、车速、载重和路面不平顺等条件下的曲线梁桥振动响应计算.研究表明:随着曲率半径的增加,桥梁跨中位移呈现略微下降的趋势;车辆行驶速度对跨中的横向位移及支座的支反力有较大影响;路面不平度等级越高,桥梁竖向挠度及弯矩的冲击效应越大.     

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.     

A thin-walled box beam finite element for curved bridge analysis

Practical design of single and multispan curved bridges requires an analysis procedure which is easy and economical to use, and provides a physical insight into structural response under general loading conditions. In the work presented, the thin-walled beam theory has been directly combined with the finite element technique to provide a new thin-walled box beam element. The beam element includes three extra degrees-of-freedom over the normal six degrees-of-freedom beam formulation, to take into account the warping and distortional effects as well as shear. The beam may be curved in space and variable cross-sections may be included. The performance of the box beam element has been compared favourably against results obtained from full 3D shell element analysis, differential equation solutions and experimental results.     

Analytical behavior of rectangular electrostatic torsion actuators with nonlinear spring bending

In this paper, we study the pull-in effect for rectangular electrostatic torsion actuators by using analytical calculations that include the higher order effects of nonlinear spring bending. The calculation approach speeds the design of such systems. The method is found to be suitable for actuators with single long beam springs where the ratio of the resonant frequencies for the torsion and bending modes is up to at least 3.5, in the region where bending dominates torsion. After fitting the theory in this paper to Coventor simulation results with three nonphysical coefficients, the fractional differences between Coventor simulation and analytical calculation results are smaller than 6%. The method is also suitable for at least one class of folded spring designs, with greatly decreased bending mode displacement. The main results are also verified by comparing them with published experimental results.     

Vibrations of grids by the finite element method

The finite element method is applied to the free vibration analysis of grids with arbitrary configuration. Grid bars are of solid or thin-walled doubly symmetric cross-section. Stiffness and consistent mass matrices for flexural behavior include the effects of shear deformation and rotary inertia in bending. The torsional behavior of solid sections is approximated by a linear displacement field, and of thin-walled sections, by a cubic. Rotary inertia in torsion is included in both cases and warping inertia, in the latter.

The computer program performs the free vibration analysis starting from the element stiffness and consistent mass matrices. A numerical solution of a thin-walled beam and a parametric solution of an orthogonal and a skew grid with solid and thin-walled bars are presented.     

Nonlinear,three-dimensional beam theory for dynamic analysis

For beams undergoing large motions but small strains, the displacement field can be decomposed into an arbitrarily large rigid-section motion and a warping field. When applying beam theory to dynamic problems, it is customary to assume that all inertial effects associated with warping are negligible. This paper examines this assumption in details. It is shown that inertial forces affect the beam’s dynamic response in two manners: (1) warping motion induces inertial forces directly, and (2) secondary warping arises that alters the beam’s constitutive laws. Numerical examples demonstrate the range of validity of the proposed approach for beams made of both homogeneous isotropic and heterogeneous anisotropic materials. For low-frequency warping, it is shown that inertial forces associated with warping and secondary warping resulting from inertial forces are negligible. To examine the dynamic behavior of beams over a wider range of frequencies, their dispersion curves, natural vibration frequencies, and mode shapes are evaluated using both one- and three-dimensional models; good correlation is observed between the two models. Applications of the proposed beam theory to multibody problems are also presented; here again, good correlation is observed between the prediction of beam models and of full three-dimensional analysis.     

Torsional vibrations of composite bars of variable cross-section by BEM

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of doubly symmetric composite bars of arbitrary variable cross-section. The composite bar consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The beam is subjected to an arbitrarily distributed dynamic twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. A distributed mass model system is employed which leads to the formulation of three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions. These problems are solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced torsional vibrations are considered and numerical examples are presented to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The discrepancy in the analysis of a thin-walled cross-section composite beam employing the BEM after calculating the torsion and warping constants adopting the thin tube theory demonstrates the importance of the proposed procedure even in thin-walled beams, since it approximates better the torsion and warping constants and takes also into account the warping of the walls of the cross-section.     

Design of a multilink-articulated wheeled pipeline inspection robot using only passive elastic joints

This paper presents a multilink-articulated robot with omni and hemispherical wheels (AIRo-2.1) for inspecting and exploring pipelines. To quickly adapt to winding pipes, holonomic rolling movement without moving forward and backward is useful. However, this requires the rolling actuators to replace the driving actuators at the expense of the driving force. Furthermore, so far the number of driving wheels and torsion springs, magnitude of driving forces, stiffness and natural angle of the spring that are required to adapt to various pipelines have not been clarified. In this paper, we investigate the possibility of high maneuverability of multilink-articulated robots in winding pipes with as few driving actuators as possible and only elastic joints (torsion springs) for body bending. We further validate its effectiveness by experimental verification.