A beam model for large displacement analysis of flexibly connected thin-walled beam-type structures

This paper presents a beam formulation for large displacement analysis of beam-type structures with flexible connections. Within the framework of updated Lagrangian incremental formulation and the nonlinear displacement field of thin-walled cross-sections, which accounts for restrained warping and the second-order displacement terms due to large rotations, the equilibrium equations of a straight beam element are firstly developed. Due to the nonlinear displacement field, the geometric potential of semitangential moment is obtained for both the internal torsion and bending moments, respectively. Material nonlinearity is introduced for an elastic-perfectly plastic material through the plastic hinge formation at finite element ends. To account for the flexible connection behaviour, a special transformation procedure is developed. The numerical algorithm is implemented in a computer programme and its reliability is validated trough several test examples.     

Elastic biaxial bending and torsion of thin-walled members

This paper presents a detailed treatment of the non-linear elastic biaxial bending and torsion of thin-walled open section members. The treatment is valid for uniform members of linear elastic material, and is limited to small strains and rotations, and moderate deflections. Shear straining of the mid-surface of the member wall is neglected, and it is assumed that the member does not distort or buckle locally. The effects of initial deformations, loads, stresses, and strains are incorporated.

The treatment is based on non-linear strain-displacement relationships, and these are used to derive the non-linear equilibrium and tangent stiffness equations in forms which are suitable for computer solution by the finite element method.

Approximate linear and non-linear differential equilibrium equations are derived, as are the differential equilibrium equations and the energy equation for neutral equilibrium at bifurcation buckling, and these are then related to the classical equations developed by Timoshenko, Vlasov, and others.     

Lateral post-buckling analysis of thin-walled open section beams

Thin-walled beams with open sections are studied using a nonlinear model. This model is developed in the context of large displacements and small deformations, by accounting for bending-bending and bending-torsion couplings. The warping and shortening effects are considered in the torsion equilibrium equation. The governing coupled equilibrium equations obtained from Galerkin’s method are solved by a Newton–Raphson iterative process. It is established that the buckling loads are highly dependent on the pre-buckling deformations of the beam. The bifurcated branches are unstable and strongly influenced by shortening effects. Some comparisons are presented with the solutions commonly used in linear stability, like in the standard European steel code (Eurocode 3). The regular solutions appear to be very conservative, especially for I sections with large flanges.     

Numerical evaluation of the deformation behaviour of thick-walled hollow cylinders of shale

The stability of tunnels and boreholes in shales is a major problem. For example in the oil and gas industry, wellbore instability problems cost the industry many millions of dollars annually. In an attempt to minimise instability problems, detailed and careful analyses of the excavation process are often carried out at the planning stage. However, the accuracy of these analyses is highly dependent on the constitutive model adopted for the shale. One important feature of the constitutive behaviour is the dissipation of pore pressure within the shale. In this paper, two FLAC-based models are used to investigate the influence of induced pore water pressure and its dissipation on borehole deformation of a thick-walled hollow cylinder of synthetic shale. The two models are: a time-dependent model that incorporates coupled flow-mechanical interaction and a steady state time-independent analysis that only accounts for mechanical-induced pore pressure. In both models, a linear elastic-plastic constitutive model (Mohr–Coulomb) is used. Non-linear elastic-plastic and strain-softening constitutive models are also investigated in the coupled flow-mechanical analyses. The numerical predictions obtained using linear coupled Mohr–Coulomb, non-linear elastic-plastic and strain-softening constitutive models are assessed against experimental observations. The FLAC predictions use material parameters obtained from conventional laboratory tests.The investigation shows that there are large differences between predictions obtained from the coupled flow-mechanical analysis and the mechanical-induced pore pressure only simulation. The non-linear coupled Mohr–Coulomb numerical model is shown to be in good agreement with the results of the laboratory tests. The investigation also shows that the simple Mohr–Coulomb constitutive model can adequately predict the deformation in thick-walled hollow cylinders of shale. Further work needs to be done before the simple strain-softening constitutive model developed in this paper can be used confidently for shale. Otherwise, more sophisticated strain-softening constitutive models may have to be used.     

phimeca-soft

This paper provides a presentation of phimeca software for reliability analysis. The completion of this software has been achieved thanks to years of experience at the French Institute for Advanced Mechanics and phimeca engineering company is developing the commercial version. First, inherent hypotheses of first/second order and conditional simulation methods are briefly stated and their implementation discussed. Algorithms combining the physical model under concern – defined through an explicit formula, a finite element code computation or a user-defined function – and a stochastic model of the input are then presented to the reader. A special attention is given to the graphical user interface GUI in this paper so as to comply with the objectives of this special issue of “Structural Reliability Software” and the use of the GUI is illustrated through a very basic example. To conclude, a significant industrial application is presented demonstrating the capabilities of the software in the context of a large finite element model and a non-linear material behaviour.     

Lateral buckling of thin-walled beam-column elements under combined axial and bending loads

Based on a non-linear stability model, analytical solutions are derived for simply supported beam-column elements with bi-symmetric I sections under combined bending and axial forces. An unique compact closed-form is used for some representative load cases needed in design. It includes first-order bending distribution, load height level, pre-buckling deflection effects and presence of axial loads. The proposed solutions are validated by recourse to non-linear FEM software where shell elements are used in mesh process. The agreement of the proposed solutions with bifurcations observed on non-linear equilibrium paths is good. It is proved that classical linear stability solutions underestimate the real resistance of such element in lateral buckling stability especially for I section with large flanges. Numerical study of incidence of axial forces on lateral buckling resistance of redundant beams is carried out. When axial displacements of a beam are prevented important tension axial forces are generated in the beam. This results in important reduction of displacements and for some sections, the beam behaviour becomes non-linear without any bifurcation.     

Non-linear model for stability of thin-walled composite beams with shear deformation

A geometrically non-linear theory for thin-walled composite beams is developed for both open and closed cross-sections and taking into account shear flexibility (bending and warping shear). This non-linear formulation is used for analyzing the static stability of beams made of composite materials subjected to concentrated end moments, concentrated forces, or uniformly distributed loads. Composite is assumed to be made of symmetric balanced laminates or especially orthotropic laminates. In order to solve the non-linear differential system, Ritz's method is first applied. Then, the resulting algebraic equilibrium equations are solved by means of an incremental Newton–Rapshon method. This paper investigates numerically the flexural–torsional and lateral buckling and post-buckling behavior of simply supported beams, pointing out the influence of shear–deformation for different laminate stacking sequence and the pre-buckling deflections effect on buckling loads. The numerical results show that the classical predictions of lateral buckling are conservative when the pre-buckling displacements are not negligible, and a non-linear buckling analysis may be required for reliable solutions.     

Flexural–torsional post-buckling analysis of thin-walled elements with open sections

The post-buckling analysis of thin-walled elements under compression is investigated. A nonlinear model is developed by using nonlinear relationships between curvatures and bending moments. Warping and shortening effects are considered in the torsion equilibrium equation. Based on Galerkin's method, a nonlinear algebraic system is obtained for simply supported boundary conditions. The three resulting equations in bending and torsion are highly coupled and the Newton–Raphson algorithm with displacement control is adopted for the solution. The post-buckling equilibrium curves are obtained for various sections shapes, such as bisymmetric and monosymmetric sections. The importance of the shortening effect is outlined.     

Numerical simulation model of vibration responses of rectangular plates embedded with piezoelectric actuators

A numerical simulation model for random large amplitude vibration control of composite plate using piezoelectric material is presented. The H_∞ control design is employed to suppress the large amplitude vibrations of composites plates under random loading. The numerical simulation model is developed and based on the finite element method. The finite element governing equation includes fully coupled structural and electrical nodal degrees of freedom, and consider the von Karman large amplitude vibration. The modal reduction method using the structural modes is adopted to reduce the finite element equations into a set of modal equations with fewer degrees of freedom. The modal equations are then employed for controller design and time domain simulation. In the simulations without control, the value of the linear mode to the nonlinear deflection is quantified; and the minimum number of linear modes needed for accurate model is obtained. In the simulations with control, it is shown that the truncated modes, which are neglected in the control design, deteriorate the controller performance. Generally, the vibration reduction level is not monotonically increasing with the size of the piezoelectric actuator. The optimal piezoelectric actuator size depends on the excitation level. For higher excitation level, optimal actuator size is larger. The H_∞ controller based on the linear finite element formulation gives better vibration reduction for small amplitude vibration, but it still gives reasonable performance for large amplitude vibration provided that the piezoelectric actuator is big and powerful enough.     

The geometric non-linear analysis of thin-walled structures by finite strips

The geometric non-linear analysis of prismatic thin-walled structures is presented. The theory is based on the moderately large displacement assumption, giving non-linear strain-displacement relations, but linear curvature-displacement relations. The corresponding non-linear equilibrium equations are produced by the principle of stationary potential energy using the finite strip discretisation. The equilibrium equations are solved using incremental and incremental-iterative numerical methods. Thus for the simple case of the square plate in edge compression, the self-correcting incremental method gives satisfactory results. For more complex examples of loading and structural type, a variable load incremental-iterative method is adopted. It is shown that the finite strip method used in conjunction with this method can be applied in particular to problems containing load maxima.     

Behaviour and strength of hollow flange channel sections under torsion and bending

Hollow flange channel section is a cold-formed high-strength and thin-walled steel section with a unique shape including two rectangular hollow flanges and a slender web. Due to its mono-symmetric characteristics, it will also be subjected to torsion when subjected to transverse loads in practical applications. Past research on steel beams subject to torsion has concentrated on open sections while very few steel design standards give suitable design rules for torsion design. Since the hollow flange channel section is different from conventional open sections, its torsional behaviour remains unknown to researchers. Therefore the elastic behaviour of hollow flange channel sections subject to uniform and non-uniform torsion, and combined torsion and bending was investigated using the solutions of appropriate differential equilibrium equations. The section torsion shear flow, warping normal stress distribution, and section constants including torsion constant and warping constant were obtained. The results were compared with those from finite element analyses that verified the accuracy of analytical solutions. Parametric studies were undertaken for simply supported beams subject to a uniformly distributed torque and a uniformly distributed transverse load applied away from the shear centre. This paper presents the details of this research into the elastic behaviour and strength of hollow flange channel sections subject to torsion and bending and the results.     

Linear and non-linear stability analyses of thin-walled beams with monosymmetric I sections

The paper investigates beam lateral buckling stability according to linear and non-linear models. First, the classical linear stability solutions are derived from the stability equation in the case of monosymmetric cross-sections. Bending distribution, load height parameter and Wagner's coefficient effects are taken into account. In the second step, they are extended to non-linear stability by considering pre-buckling deformation and improved solutions are then obtained. Based on a finite element model developed for large torsion of thin-walled beams with open sections, the stability of beams under gradient moments (M₀, ψM₀, ?1≤ψ≤1) is particularly investigated. It is then concluded that beam lateral buckling resistance depends not only on pre-buckling deformation but also on section shape and load distribution. For bisymmetric I beam, closed form solutions are possible and pre-buckling deformations have an incidence. In the case of beams with monosymmetric I and Tee sections, effects of pre-buckling deflections are important only when the largest flange is in compression under M₀ and positive gradient moment. Analytical solutions are possible. For negative gradient moments all available solutions fail and numerical solutions are more powerful. Effect of gradient moments on stability of redundant beams is investigated at the end. Under such boundary conditions, important axial forces are present due to non-linear beam deformation. These forces, omitted in literature, have an incidence on stability. The element is then concerned with beam-column behaviour rather than beam stability.     

Non-linear analysis of unbolted base plates by the FEM and experimental validation

Unbolted base plates are often used in civil engineering as structural connections in storage racks construction. The aim of this work is to describe the development of a numerical model to simulate accurately the connection between columns and foundation in metallic structures, which constitute any frame in automated storage systems. In this way, the bending stiffness of the column can be modeled in the analysis of these structures, in order to approach the real behavior in service, and these values can be included in linear beams of structural analysis programs, such as ESCAL3D [del Coz Diaz JJ, Ordieres Mere B, Siare Dominguez FJ, Bello García A, Felgueros Fernández D. J Constr Res 1998; 46:273–5]. In this study, a non-linear structural behavior of the model occurs due to the changing status of the contact surfaces and point-to-point contacts, the geometric non-linearities of the model and the material non-linearities, such as plasticity and surface friction. The finite element method is a general technique for numerical solution of differential and integral equations in science and engineering. Thus the finite element approach has been carried out in two phases. Firstly, a pre-buckling analysis has been accomplished and secondly, the above-mentioned non-linear analysis has been performed, updating the geometry of the finite element model to the deformed configuration for the first mode buckling. A total of four load cases were analyzed, with different compressive load and imposed lateral displacement. In order to validate the results some experimental models were tested to compare with the numerical model, so that a good agreement and better correlations were obtained between both.     

The computational post buckling analysis of fuselage stiffened panels loaded in compression

Fuselage panels are commonly fabricated as skin–stringer constructions, which are permitted to locally buckle under normal flight loads. The current analysis methodologies used to determine the post buckling response behaviour of stiffened panels relies on applying simplifying assumptions with semi-empirical/empirical data. Using the finite element method and employing non-linear material and geometric analysis procedures, it is possible to model the post buckling behaviour of stiffened panels without having to place the same emphases on simplifying assumptions or empirical data. Investigation of element, mesh, idealisation, imperfection and solution procedure selection has been undertaken, with results validated against mechanical tests. The research undertaken has demonstrated that using a commercial implicit code, the finite element method can be used successfully to model the post buckling behaviour of flat riveted panels. The work has generated a series of guidelines for the non-linear computational analysis of flat riveted panels subjected to uniform axial compression.     

Torsion warping transmission at thin-walled frame joints: Kinematics, modelling and structural response

This paper reports on the use of simple kinematic models to simulate the torsion warping restraint and transmission at thin-walled frame joints in the context of beam finite element structural analysis. After reviewing the main concepts involved in the torsional behaviour of thin-walled members, the paper addresses the development of kinematic models aimed at simulating the torsion warping transmission at frame joints connecting two or more non-aligned plain channel (U-section) or I-section members. Finally, numerical results are presented and discussed, in order to illustrate the application and show the capabilities of the above kinematic models, which make it possible to use beam finite element models accounting for the joint torsion warping behaviour. For validation purposes, the beam finite element results obtained are compared with values yielded by rigorous shell finite element analyses.     

Simulation of a fully coupled thermo–hydro–mechanical system in freezing and thawing rock

This paper describes the application of a numerical model of thermal–mechanical–fluid flow coupling system to the simulation of laboratory freezing and thawing experiments on rocks. A theoretical formulation that accommodates a linear stress–strain constitutive relationship is presented and a two-dimensional (plane stress) numerical modeling is performed based on the finite element method applied to thermo–poro–elasticity. As the primary objective of this research is to simulate the freezing and thawing process, the developed code takes into account the phase change of pore water during freezing. It is found from the numerical simulation that a relatively good prediction can be made of temperature transfer and deformation behavior within the elastic deformation limit. In some cases, however, large deformation results from the freezing processes, leading to material failure. Future research should therefore take into account irreversible and plastic effects.     

开口截面钢-混凝土组合梁弯扭性能非线性分析

在钢筋混凝土变角软化桁架模型的基础上,提出了适于分析开口截面钢-混凝土组合梁弯扭性能的三维桁架模型。在弯扭作用下,组合梁截面各单元分别处于一维应力状态(体系1)和二维应力状态(体系2),体系1用来抵抗由弯矩和扭矩引起的截面纵向应力,体系2用来抵抗由扭矩引起的截面剪应力,两者通过截面的纵向应变协调和内力平衡条件联系起来。分析充分满足平衡条件、变形协调条件和材料本构方程。通过对部分试件的计算验证,结果表明该模型不仅可以用于预测组合梁的极限强度,而且为混凝土翼板开裂后组合梁全过程分析,提供了有效途径。     

Torsional buckling behaviour of stiffened cylinders under combined loading

In this study cylindrical boundary conditions for finite element analysis are formulated that allow torsional displacement and buckling of a sector of a cylinder of half axial height, and of a circumferential arc angle that will divide into 360°. Finite element tests are carried out on un-stiffened elastic cylinders to verify the method of analysis against classical elastic torsional buckling theory.Elastic–plastic limit point finite element tests are carried out on ring and stringer stiffened and stringer stiffened cylinders to investigate the effects of stiffeners on post-buckling behaviour in torsion.A stringer stiffened cylinder is subjected to many combinations of axial force and surface pressure in the elastic range of response and then tested to failure in torsion to investigate the effects of axial and surface pressure loads on the resistance to plastic collapse in torsion.     

Distortion (and Torsion) of rectangular thin-walled beams

A study of rectangular cross-section thin-walled beams under torsional, distortional and bimomental loads is presented. The assumed displacement field describes both torsional and distorsional behaviour, the shearing strain in the walls being taken into account. By a variational principle the equilibrium and boundary equations are derived and a physical interpretation of the natural conditions is given.A closed form solution is given with reference to the matrix form of the governing differential system together with a discussion of a sixth-order ‘uncoupled’ differential equation generalizing the BEF analogy.The present formulation shows that, in general, torsion and distortion are pair of coupled problems and a study of distortion without considering torsion would therefore not be legitimate.The comparison of the results predicted by the theory is in good agreement with experimental evidence.     

Exact solutions for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment

The exact solutions for torsional analysis of thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment are presented for the first time. For this, a general thin-walled composite beam theory with arbitrary lamination is developed by introducing Vlasov's assumption and the equilibrium equations and the force–deformation relations are derived from the energy principle. Applying the displacement state vector consisting of 14 displacement parameters and the nodal displacements at both ends of the beam, the displacement functions are derived exactly. Then, the exact stiffness matrix for torsional analysis is determined using the force–deformation relations. As a special case, the closed-form solutions for symmetrically laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the closed-form solutions and the finite element results using the Hermitian beam elements and ABAQUS's shell elements.