A New Co-rotational Procedures for Plane Beam Element of Large Rotation and Large Deflection (2)nn.docx

A New Co-rotational Procedure for Nonlinear Analysis of Cables of Moderately Large Deflection with High TensionKim Changwan1, Cai Songbai1,2 (1. Department of Mechanical Design and Production Engineering, School of Engineering, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul 143-701, Republic of Korea，Tel: 82-2-4503543, Fax: 82-2-4475886, Email: goodantkonkuk.ac.kr2. College of Resources and Environmental Science, Hunan Normal University, No 36, Lushan South Road, Changsha, Hunan, 410081, P. R. China, Tel: 86-731-8872535, Fax: 86-731-8872535, Email: songbai_)Abstract：Although many cable or cable-like beam elements of large deflection have been developed, much of this early work was surprisingly sophisticated and worth re-visiting. Starting from geometric and physical relations of the cable or cable-like beam element, this work has successfully applied co-rotational procedure to derive non-linear tangent stiffness matrix of cable or cable-like beam element in large but small-strained rotational displacement. Interestingly, this tangent stiffness matrix is asymmetric, and a non-linear finite element iteration procedure is also established by the Newton-Raphson technique and the corresponding FORTRAN program NPFSAP has been worked out. Numerical results of high accuracy for beams, square and circular frames are thus obtained. Computations indicate that formulations for the nonlinear element are of validation and that the nonlinear solution procedures of convergence. The method and its program NPFSAP could be used for engineering designers to accurately analyze beams and frames. Key words：co-rotational procedure; finite element method; asymmetric tangent stiffness matrix; plane frame; high accuracy1Introduction1. IntroductionCables of high tension have widely been used in many fields as civil, mechanical and aeronautical engineering to transmit forces and carry payloads over great distances. Despite the great interest surrounding these problems, most of current finite elements methods can model such phenomena only partially. As a matter of fact, the high tension cables under loading could cause excessive tension stresses in the cable leading to catastrophic failures, such as premature cable breakage (Phillips, 1949). Many studies have been carried out for the analysis of high tension cable problems using the classic cable theory and the finite element method. However, these methods often computed the nonlinear behavior of cables such ultimate bearing capacity through a linearised analysis, which may give inaccurate results in certain cases as the classic cable theory becomes inaccurate when the tension strain is very high in any part of the cable. The limitations of the existing approaches create a need for alternative approaches to treat the high tension cable problems. Many works have been carried out to alleviate the high tension cable problems by higher order terms and bending stiffness of cable. Among them, only the approach based on the beam theory is a natural extension of the classic cable theory and has a sound theoretical foundation and physical meaning. The modeling of the high tension cables with a realistic and robust description of cable dynamics inevitably leads to a complex mathematical problem and consequently requires numerical solution techniques, such as, finite difference (Koh et al., 1999), and finite element (Zhu et al., 2001; Rizzo, 1991). Among all the numerical methods, the finite element method is probably the most appealing technique. The main advantage of the finite element method over other methods is its capability to handle complex geometries with multiple cable branches or different cable properties along the cable length in an algorithmic fashion. There are many types of beam elements available in the literatures for modeling cable systems (Schrefler and Odorizzi, 1983). Detailed reviews of beam element formulations can be found in the works of Raveendranath et al. (1999, 2000, 2001) and Bucalem and Bathe (1995). It has been noticed that the relatively high accuracy approaches is generally coupled with much more complicated mathematical formulations and cumbersome numerical computations. Naturally, finite beam elements are very common, the design of failure-free installation procedures for submarine pipelines 1, dynamic analysis of marine pipes during operation 2, flight simulation of flexible aircraft 3, and stability analysis of frame structures 48, are examples of such applications. a huge amount of research has been carried out within this topic and many kinds of such elements which include both geometrical and material non-linearities have been presented. However, two problems remain. The first one concerns a balance of computational accuracy, convergence and efficiency. The second problem is related to material nonlinearity in the sense that most of elements are too crude to model correctly such phenomena. As an example, it will be shown in this work that elements which neglect hardening or use yield criteria expressed in function of stress resultants cannot be used. In fact, concerning beam elements, attempts to correctly model plastic behavior problems and compute material post-bifurcation paths are not very common in literature. To develop efficient non-linear beam elements which are accurate enough in order to model elastic and elasto-plastic nonlinear behavior problems is desirable therefore become a subject of considerable interest among researchers. Usually, the geometric non-linearity in non-linear finite element models for thin beams is treated by including the rotation-related quadratic terms in the strain-displacement equations (i.e. the von Karman non-linear strains; see References 24, 25). However, the beam finite element in large rotation problems suffers from one inherent drawback: it is restricted to small rotations between two successive load increments during the deformation process. The corotational approach overcomes the aforementioned drawback and provides a non-linear framework in which standard linear beam finite elements can be utilized locally. In a nonlinear elastoplastic analysis of high tension cable under extremely loading condition, a number of models have been used to simulate the constitutive relationship of the material. The most common models used are: the elastic perfectly plastic model, the piecewise linear model, the Ramberg-Osgood model and the isotropic and kinematic hardening models. Among these, the bilinear hardening model has been used widely. However, a bilinear model introduces a sharp transition from the elastic to the plastic regimes. Further, it cannot be used to simulate the Bauschinger effect accurately since a simplifying assumption has to be made such as that the total elastic range of the material remains constant. Trilinear hardening models have been suggested , but this destroys the simplicity of the bilinear model without addressing the sharp transition from one regime to another. To provide a smooth transition between regions of elasticity and plasticity, the Ramberg-Osgood model has been used. Mondkar and Powell6, Noor and Peters7 and Zhu et al. s have employed this model to analyse the dynamic responses of truss-like structures. Uzgider9 has applied the inverse of the Ramberg-Osgood model to simulate the moment-rotation and axial force-axial deformation relationships. In this paper, 4 kinds kinematic hardening material model is used in conjunction 5 large strain measures for high tension of cable. The resulting finite element formulations are implemented in an appropriate computer program and validated by theoretical analyses and experimental investigations. Several examples are presented to demonstrate the applicability of the method.2 Corotational formulation for local moderate deflection and large strain of tensionAs shown on Fig 1 is a plane beam element. It is assummed that coordinates of node A in global system is and for node B, then the projection length of beam element along -axis and -axis are, respectively and . Furthermore,the initial length of beam element is assumed as and is with oblique angle to x-axis,and the global displacements of node A and node B of beam element are represented by and , respectively. So the new coordinates of node A and node B of deformed element are and . Assuming that the deformation of beam element in local coordinate system is large strain in tension and small strain for deflection and that the oblique angle with respect to x-axis and the length of deformed element are and ,then the streching effect of beam element in local coordinate system can be calculate byABlylxABQiQjNiMiNjMjxyXYO0Fig 1 Beam element 图1 大转动小变形梁元 （1） （2） （3） （4） （5）The rotation angle of node A and node B are denoted by and , which are （6） （7）And the axial strain of the beam element can be calculated by the strain measures as engineering strain measure, (8) Greens strain measure, (9) Log strain measure, (10) Almansis strain measure, (11)Hyperbolic strain measure, (12) The tensile stresses of the element for engineering strain measure, Green strain measure, log strain measure and Almansis measure are, respectively assumed as , and. Thus, for the geometrical nonlinear analysis, a linear strain-stress relationship is assumed for all 5 strain measures. i.e., (13) Where is the steel elastic modulus. Assuming that the beams deflection in the local coordinate system is moderately large, calculated by elasticity as (14) (15)For local material nonlinearity only (16) (17) (18) (19) (20) (21) (22)The increment of axial strain in the x-direction can, using a above large strain measure, be expressed as (23)Where, for engineering strain (24) For Greens strain (25)For log strain (26)For Almansis strain (27)For Hyperbolic strain (28)Assuming plane sections remain plane, the displacement due to element bending in the x-direction, at distance from the centroid is given by and the strain is then given by ,the strain increment due to element bending in the x-direction, at distance from the centroid is given by (29)Letting (30)Combining (18) and (23), gives the increment of total strain at distance from the centroid of beam element in the x-direction as (31)Where (32)For local material nonlinearity only (33) Then element local nodal force can be com