Large-deflection and postbuckling of beam-columns with non-linear semi-rigid connections including shear and axial effects

The non-linear large deflection-small strain analysis and post-buckling behavior of an out-of-plumb Timoshenko beam-column of symmetrical cross section subjected to end loads (forces and moments) with non-linear bending connections at both ends, and its top end partially restrained against transverse and longitudinal translations are developed in a classical manner. A set of non-linear equations based on the “modified shear equation” that includes the effects of (1) shear deformation and the shear component of the applied axial forces; and (2) shortening of the beam-column due to both axial forces and “bowing” are presented. The proposed method and corresponding equations can be used in the large deflection-small strain analysis of Timoshenko beam-columns with non-linear bending connections, as well as lateral and longitudinal non-linear restraints at the top end. This paper is an extension of previous work presented by the senior author on the large deflection and post-buckling behavior of Timoshenko beam-columns with linear elastic semi-rigid connections and linear elastic lateral bracing. Three comprehensive examples are included that show the effectiveness of the proposed method and corresponding equations. Results obtained in the three examples are verified against analytical solutions available in the technical literature and against results from models using the FEM program ABAQUS.     

Large-deflection and post-buckling behavior of slender beam-columns with non-linear end-restraints

The large-deflection analysis and post-buckling behavior of laterally braced or unbraced slender beam-columns of symmetrical cross section subjected to end loads (forces and moments) with both ends partially restrained against rotation, including the effects of out-of-plumbness, are developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions utilized herein are those presented previously by Aristizabal-Ochoa [1]. The proposed method can be used in the large-deflection analysis and post-buckling behavior of elastic slender beam-columns with rigid, semi-rigid, and simple flexural connections at both ends including linear and non-linear inelastic connections like those that suffer from flexural degradation (such as flexural cracking and elasto-plastic connections) or flexural stiffening. Only bending strains are considered in the proposed analysis. Results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for laterally braced or unbraced slender beam-columns under bending caused by end loads. The large-deflection analysis and post-buckling behavior of slender beam-columns with both supports partially restrained against rotation and with sway inhibited or uninhibited are complex problems requiring the simultaneous solution of two coupled non-linear equations with elliptical integrals whose unknowns are the limits of the integrals. The validity of the proposed method and equations are verified against solutions available in the technical literature. Three comprehensive examples are included that show the effects of linear and non-linear connections at both ends on the large-deflection analysis and post-buckling behavior of slender beam-columns.     

A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation

The aim of this paper is to develop a new method of analyzing the non-linear deflection behavior of an infinite beam on a non-linear elastic foundation. Non-linear beam problems have traditionally been dealt with by semi-analytical approaches that involve small perturbations or by numerical methods, such as the non-linear finite element method. In this paper, in contrast, a transformed non-linear integral equation that governs non-linear beam deflection behavior is formulated to develop a new method for non-linear solutions. The proposed method requires an iteration to solve non-linear problems, but is fairly simple and straightforward to apply. It also converges quickly, whereas traditional non-linear solution procedures are generally quite complex in application. Mathematical analysis of the proposed method is performed. In addition, illustrative examples are presented to demonstrate the validity of the method developed in the present study.     

Non-linear forced periodic oscillations of laminates with curved fibres by the shooting method

Large-amplitude forced vibration, before damage onset, of variable stiffness composite laminated plates with curvilinear fibres are studied. The fibre paths considered change linearly in relation to one Cartesian coordinate. The plates are rectangular and with clamped edges. The displacement field is modelled by a third order shear deformation theory and the equations of motion, in the time domain, are obtained using a p-version finite element method. The in-plane inertia is neglected, still taking into consideration the in-plane displacements, and the model is statically condensed. The condensed model is transformed to modal coordinates in order to have a reduced model with a smaller number of degrees-of-freedom. A shooting method using fifth-order Runge–Kutta method, as well as adaptive stepsize control, is used to find periodic solutions of the equations of motion. Frequency-response curves of composite laminates with different curvilinear fibre angles and various thicknesses are plotted and compared. Tsai–Wu criterion is employed in order to predict the damage onset. When it is detected that damaged started, the continuation method is interrupted and no further points of the response curve are computed. The reason behind this interruption is that the model does not include the effects of damage. Examples of bifurcations are presented and studied in detail, using projections of trajectories in a phase plane and Fourier spectra. The time histories and frequency spectra of steady-state stresses are plotted for VSCL plates with different fibre angles. The steady-state stresses are also displayed for bifurcated branches of the solutions.     

The non-linear dynamic response of the Euler-Bernoulli beam with an arbitrary number of switching cracks

In this study the non-linear dynamic response of the Euler-Bernoulli beam in presence of multiple concentrated switching cracks (i.e. cracks that are either fully open or fully closed) is addressed. The overall behaviour of such a beam is non-linear due to the opening and closing of the cracks during the dynamic response; however, it can be regarded as a sequence of linear phases each of them characterised by different number and positions of the cracks in open state. In the paper the non-linear response of the beam with switching cracks is evaluated by determining the exact modal properties of the beam in each linear phase and evaluating the corresponding time history linear response through modal superposition analysis. Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control has been enforced with the aim of establishing the minimum number of linear modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented in order to illustrate the efficiency of the proposed approach for the evaluation of the non-linear dynamic response of beams with multiple switching cracks. In particular, the behaviour under different boundary conditions both for harmonic loading and free vibrations has been investigated.     

变截面高层筒体结构的有限元线法整体稳定和二阶分析

本文用有限元线法对变截面的高层简体结构进行空间整体稳定和二阶分析。先把实际框筒结构分段连续化为正交各向异性折板结构；用有限元线法，通过有限条元半离散化，取结线上位移为基本未知函数，由势能驻值原理建立稳定和二阶分析的常微分方程组；由常微分方程求解器直接求解。     

Development of the large increment method for elastic perfectly plastic analysis of plane frame structures under monotonic loading

The displacement-based finite element method dominates current practice for material nonlinear analysis of structures. However, there are several characteristics that may limit the effectiveness of this approach. In particular, for elastoplastic analysis, the displacement method relies upon a step-by-step incremental approach stemming from flow theory and also requires significant mesh refinement to resolve behavior in plastic zones. This leads to computational inefficiencies that, in turn, encourage the reconsideration of force-based approaches for elastoplastic problems.One of these force algorithms that has been recently developed is the large increment method. The main advantage of the flexibility-based large increment method (LIM) over the displacement method is that it separates the global equilibrium and compatibility equations from the local constitutive relations. Consequently, LIM can reach the solution in one large increment or in a few large steps, thus, avoiding the development of cumulative errors. This paper discusses the extension of the large increment methodology for the nonlinear analysis of plane frame structures controlled by an elastic, perfectly plastic material model. The discussion focuses on the power of LIM to handle these nonlinear problems, especially when plastic hinges form in the frame and ultimately as the structure approaches the collapse stage. Illustrative planar frame examples are presented and the results are compared with those obtained from a standard displacement method.     

Transient analysis of Cosserat rod with inextensibility and unshearability constraints using the least-squares finite element model

In this study, time-dependent fully discretized least-squares finite element model is developed for the transient response of Cosserat rod having inextensibility and unshearability constraints to simulate a surgical thread in space. Starting from the kinematics of the rod for large deformation, the linear and angular momentum equations along with constraint conditions for the sake of completeness are derived. Then, the α-family of time derivarive approximation is used to reduce the governing equations of motion to obtain a semi-discretized system of equations, which are then fully discretized using the least-squares approach to obtain the non-linear finite element equations. Newton׳s method is utilized to solve the non-linear finite element equations. Dynamic response due to impulse force and time-dependent follower force at the free end of the rod is presented as numerical examples.