On a geometrically exact curved/twisted beam theory under rigid cross-section assumption

 A geometrically exact curved/ twisted beam theory, that assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal frames of reference starting from a 3-D beam theory. The relevant engineering strain measures with an initial curvature correction term at any material point on the current beam cross-section, that are conjugate to the first Piola-Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The stress resultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimensional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are considered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the initial curvature correction term. Along with the beam theory, some basic concepts associated with finite rotations are also summarized in a manner that is easy to understand. Received: 17 June 2002 / Accepted: 21 January 2003 The work was partly sponsored by a grant (CDAAH04-95-1-0175) from the Army Research Office with Dr. Gary Anderson as the grant monitor. We would also like to thank Prof. Raymond Plaut of Dept. of Civil and Environmental Engineering at Virginia Polytechnic Institute and State University for his technical help.     

A new higher-order hybrid-mixed curved beam element

The purpose of this work is to show the successful use of nodeless degrees of freedom in developing a highly accurate, locking free hybrid-mixed C⁰ curved beam element. In the performance evaluation process of the present field-consistent higher-order element, the effect of field consistency and the role of higher-order interpolation on both displacement-type and hybrid-mixed-type elements are carefully examined. Several benchmark tests confirm the superior behaviour of the present element. © 1998 John Wiley & Sons, Ltd.     

Adhesive elements for stress analysis of bonded patch to curved thin-walled structures

 Bonded composite patching has been recognized as an efficient and economical method to extend the service life of cracked aluminum components. However, current analysis methods and empirical databases for designing composite bonded joint and patch repair are limited to flat plate and/or flat laminate geometries, and the effect of curvature on the performance and durability of composite bonded joints and repairs is not known. This paper presents a novel finite element formulation for developing adhesive elements for conducting quick stress analysis of bonded repairs to curved structures. Illustrative examples are presented to demonstrate the effect of curvature and the effect of patch location, i.e., internal and external patches, patch size and patch thickness on stresses in adhesive layer. Received: 24 April 2002 / Accepted: 10 October 2002     

A symmetric Galerkin BEM implementation for 3D elastostatic problems with an extension to curved elements

 Like the finite element method (FEM), the symmetric Galerkin boundary element method (SGBEM) can produce symmetric system matrices. While widely developed for two dimensional problems, the 3D-applications of the SGBEM are very rare. This paper deals with the regularization of the singular integrals in the case of 3D elastostatic problems. It is shown that the integration formulas can be extended to curved elements. In contrast to other techniques, the Kelvin fundamental solutions are used with no need to introduce the new kernel functions. The accuracy of the developed integration formulas is verified on a problem with known analytical solution. Received 6 November 2000     

Refined triangular discrete Mindlin flat shell elements

In this paper flat shell elements are formed by the assemblage of discrete Mindlin plate elements RDKTM and either the constant strain membrane element CST or the Allmans membrane element with drilling degrees of freedom LST. The element RDKTM is a robust Mindlin plate element, which can perform uniformly thick and thin plate bending analysis. It also passes the patch test for thin plate bending, and its convergence for very thin plates can be ensured theoretically. The singularity of the stiffness matrix and membrane locking are studied for the present elements. Numerical examples are presented to show that the present models indeed possess properties of simple formulations, high accuracy for thin and thick shells, and it is free from shear locking for thin plate/shell analysis.     

Free vibration analysis of arches using curved beam elements

The natural frequencies and mode shapes for the radial (in‐plane) bending vibrations of the uniform circular arches were investigated by means of the finite arch (curved beam) elements. Instead of the complicated explicit shape functions of the arch element given by the existing literature, the simple implicit shape functions associated with the tangential, radial (or normal) and rotational displacements of the arch element were derived and presented in matrix form. Based on the relationship between the nodal forces and the nodal displacements of a two‐node six‐degree‐of‐freedom arch element, the elemental stiffness matrix was derived, and based on the equation of kinetic energy and the implicit shape functions of an arch element the elemental consistent mass matrix with rotary inertia effect considered was obtained. Assembly of the foregoing elemental property matrices yields the overall stiffness and mass matrices of the complete curved beam. The standard techniques were used to determine the natural frequencies and mode shapes for the curved beam with various boundary conditions and subtended angles. In addition to the typical circular arches with constant curvatures, a hybrid beam constructed by using an arch segment connected with a straight beam segment at each of its two ends was also studied. For simplicity, a lumped mass model for the arch element was also presented. All numerical results were compared with the existing literature or those obtained from the finite element method based on the conventional straight beam element and good agreements were achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

A mixed finite element formulation for timoshenko beam on winkler foundation

 A mixed formulation for Timoshenko beam element on Winkler foundation has been derived by defining the total curvature in terms of the bending moment and its second order derivation. Displacement and moment have been chosen as primary variables, while slope and first derivation of moment have been chosen as secondary variables. The behaviour matrix for Timoshenko beam element has been obtained in mixed form by using weak formulation with equilibrium and compatibility equations. The presented formulation makes the analysis of beams free of shear locking. Received: 10 July 2002 / Accepted: 14 January 2003     

A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams

In the present work, a finite element approach is developed for the static analysis of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through-thickness stretching. The formulation is general in the sense that it can be used to compare the influence of different structural theories, through static and dynamic analyses of curved nanobeams. The governing equations derived here are solved introducing a 3-nodes beam element. The formulation is validated considering problems for which solutions are available. A comparative study is done here by different theories obtained through the formulation. The effects of various structural parameters such as thickness ratio, beam length, rise of the curved beam, loadings, boundary conditions, and nonlocal scale parameter are brought out on the static bending behaviors of curved nanobeams.     

Hybrid‐mixed curved beam elements with increased degrees of freedom for static and vibration analyses

In this paper, hybrid‐mixed elements for static and vibration analyses of curved beams are presented. The proposed elements based on the Hellinger–Reissner variational principle employ the consistent stress parameters corresponding to the displacement fields with additional internal nodeless degrees of freedom in order to enhance the numerical performance. Elimination of the stress parameters by the stationary condition and condensation of internal nodeless degrees of freedom by Guyan reduction are carried out in the element formulation. This study shows how much the order of internal nodeless displacement functions and the type of mass matrix affect the numerical performance of hybrid‐mixed curved beam elements in static and dynamic analyses. Various numerical examples confirm that the proposed elements with increased internal nodeless degrees of freedom generate superior accuracy in the prediction of bending behaviours and high vibration modes. Copyright © 2006 John Wiley & Sons, Ltd.     

A mixed finite element method for beam and frame problems

 In this work we consider solutions for the Euler-Bernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation. A finite element solution method is presented from a three-field variational form based on an extension of the Hu–Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects – identical to the behavior of flexibility based elements. The advantages of the approach are illustrated with a few numerical examples. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

A numerical study of the similarity of fully developed laminar flows in orthogonally rotating rectangular ducts and stationary curved rectangular ducts of arbitrary aspect ratio

 The present study showed that a quantitative analogy of fully developed laminar flow in orthogonally rotating rectangular ducts and stationary curved rectangular ducts of arbitrary aspect ratio could be established. In order to clarify the similarity of the two flows, the dimensionless parameters K _LR=Re/(Ro)^1/2 and the Rossby number, Ro=w _m /Ωd _h , in a rotating straight duct were used as a set corresponding to the Dean number, K _LC=Re/λ^1/2, and curvature ratio, λ=R/d _h , in a stationary curved duct. Under the condition that the value of the Rossby number and the curvature ratio was large enough, the flow field satisfied the `asymptotic invariance property'; there were strong quantitative similarities between the two flows such as in the friction factors, flow patterns, and maximum axial velocity magnitudes for the same values of K _LR and K _LC. Based on these similarities, it is possible to predict the flow characteristics in rotating ducts by considering the flow in stationary curved ducts, and vice versa. Received: 10 September 2001 / Accepted: 13 May 2002     

Planar four node piezoelectric elements with drilling degrees of freedom

Several new planar four node piezoelectric elements with drilling degrees of freedom are presented. We begin by deriving two families of variational formulations accounting for piezoelectricity and in‐plane rotations. The first family retains the skew‐symmetric part of the stress tensor, while in the second, the skew part of stress is eliminated from the functional. The finite elements derived from two of the variational formulations derived in this paper are investigated. The first element is based on an ‘irreducible’ form, while the other is based on a fully mixed functional, with both stress and electric flux density assumed. Our new elements are shown to be accurate and robust in comparison with a number of existing elements, for several benchmark test problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Solution of plane elasticity problems for orthotropic bodies with cracks by the displacement discontinuity method

 The force problem of fracture mechanics for orthotropic body is solved by boundary element method. The equations of plane strain state are used. The key point of this paper is obtaining the influence functions for finite segment with constant displacement discontinuities in the main axes of orthotropy. They have been deduced by means of two-dimensional Fourier transform and theory of generalized functions. The straight crack under the action of uniform tension in the infinity is considered by using obtained influence functions. The calculations were carried out for isotropic and two orthotropic materials. Received 6 November 2000     

Experimental validation of a computationally efficient beam element for combined finite–discrete element modelling of structures in distress

 The combined finite/discrete element method is adopted to model the pre-failure and post failure transient dynamics of reinforced concrete structures. For this purpose a novel beam element is introduced in order to increase CPU and RAM efficiency. In this paper the accuracy and reliability of this element is assessed when used in dynamic loading conditions. Experiments, which have already been undertaken at the Swiss Federal Institute of Technology, are used for comparison and validation. The results indicate that the element introduced is capable of accurately modelling inertia and contact effects in pre and post failure dynamics, up to collapse. Received: 1 August 2002 / Accepted: 9 January 2002 A debt of gratitude is owed to Prof. Bachman and Mrs. N. Ammann for their kind and sincerely appreciated assistance in the provision of the reports from the Swiss Federal Institute of Technology. The assistance provided by Dr. W. Ammann and Dr. S. Heubbe-Walker in the translation of the reports is also gratefully acknowledged.     

Swept and curved wings: a numerical approach based on generalized lifting-line theory

 Whenever the lifting-line is used for curved and swept wings with high aspect ratios, available software shows weaknesses. Actually, when considering sweep and curvature, most of these programs use the normal component of the incident flow in an empirical extension of Prandtl's model, which is theoretically founded only in the case of straight unswept wings. Recent theory based on the matched asymptotic expansions technique shows that, in addition to this 2D-type correction, extra terms have to be considered in order to express the three-dimensional induced velocity, when computing the spanwise variation of the circulation. These terms require finite parts in Hadamard's sense integrals computation. On these theoretical foundations, the computational approach presented in this paper improves on the empirical approach for swept, curved lifting-lines, including all the necessary corrections. The validity of the approach is examined in a simple application compared with available analytical and numerical results. Considering these results, it can be said that the model described here offers real improvements over the usual empirical numerical computations for arbitrarily shaped lifting-lines. Received: 26 February 2001 / Accepted: 14 June 2002     

A new model for the analysis of composite steel – concrete slab and beam structures with deformable connection

 In this paper a solution to the bending problem of reinforced concrete slabs stiffened by steel beams with deformable connection including creep and shrinkage effect is presented. The adopted model takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beam, due to combined response of the system. The analysis consists in isolating the beams from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface producing lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beams are related with the interface slip through the shear connector stiffness. Any distribution of connectors along the interface and any linear or non-linear load–slip relationship or partial shear connection for the shear connectors can be handled. The creep and shrinkage effect relative with the time of the casting and the time of the loading of the plate is taken into account. The solution of the arising plate and beam problems, which are nonlinearly coupled, is achieved using the analog equation method (AEM). The adopted model compared with those ignoring the inplane forces and deformations, approaches more reliable the actual response of the plate–beams system. Moreover, it permits the evaluation of the shear forces at the interfaces, the knowledge of which is very important in the design of composite steel–concrete structures. Received: 21 June 2002 / Accepted: 5 March 2003     

A three-dimensional FE analysis of large deformations for impact loadings using tetrahedral elements

 A three-dimensional dynamic program for the anaysis of large deformations in contact-penetration problems is developed using the finite element Lagrangian method with explicit time integration. By incorporating a tetrahedral element, which allows a single-point integration without a special hourglass control scheme, this program can be more effective to the present problem. The position code algorithm is used to search contact surface. Eroding surfaces are also considered. The defense node algorithm was slightly modified for the calculation of contact forces. A study of obliquity effects on metallic plate perforation and ricochet processes in thin plates impacted by a sphere was conducted. It is well simulated that on separation of two parts of the sphere, the portion still within the crater tends to perforate, while the portion in contact with the plate surface ricochets. This deformation pattern is observed in experiments, especially at high obliquities. A long rod that impacts an oblique steel plate at high impact velocity was also simulated in order to study the dynamics of the rod caused by the three dimensional asymmetric contact. The agreement between simulated and experimental results is quite good. Fracture phenomena occuring at high obliquity deserves further investigations. Received: 20 February 2002 / Accepted: 20 September 2002     

Body-force-driven multiplicity and stability of combined free and forced convection in rotating curved ducts: Coriolis force

 A numerical study is made on the fully developed bifurcation structure and stability of forced convection in a rotating curved duct of square cross-section. Solution structure is determined as variation of a parameter that indicates the effect of rotation (Coriolis-force-driven multiplicity). Three solutions for the flows in a stationary curved duct obtained in the work of Yang and Wang [1] are used as initial solutions of continuation calculations to unfold the solution branches. Twenty-one solution branches are found comparing with five obtained by Selmi and Nandakumar [2]. Dynamic responses of the multiple solutions to finite random disturbances are examined by the direct transient computation. Results show that characteristics of physically realizable fully developed flows changes significantly with variation of effect of rotation. Fourteen sub-ranges are identified according to characteristics of physically realizable solutions. As rotation effect changes, possible physically realizable fully-developed flows can be stable steady 2-cell state, stable multi-cell state, temporal periodic oscillation between symmetric/asymmetric 2-cell/4-cell flows, temporal oscillation with intermittency, temporal chaotic oscillation and temporal oscillation with pseudo intermittency. Among these possible physically realizable fully developed flows, stable multi-cell state and stable steady 2-cell state exist as dual stable. And oscillation with pseudo intermittency is a new phenomenon. In addition to the temporal oscillation with intermittency, sudden shift from stationary stable solution to temporal chaotic oscillation is identified to be another way of onset of chaos. Received: 20 December 2001 / Accepted: 5 August 2002 The financial support from the Research Grants Council of Hong Kong (RGC, Project No: HKU7086/00 E), the CRCG and the Outstanding Young Researcher Award of the University of Hong Kong to LW is gratefully acknowledged.