Variable Flexural Stiffness Research Articles

Advancing the dynamic properties of a column with variable flexural stiffness in terms of structural mounting damping, external and internal damping

Advancing the dynamic properties of a column with variable flexural stiffness in terms of structural mounting damping, external and internal damping

Investigation of the moment redistribution for steel – concrete continuous composite beams

ABSTRACTThis study investigates moment redistribution phenomenon for pin ended double‐span steel‐concrete continuous composite beams and arise a question about moment redistribution for the continuous beams subjected to partially distributed or concentrated loading conditions. The overall objective of the study is to propose a methodology to fulfil the technical gaps adopted by Eurocode‐4, which considers the redistribution for only uniformly load condition. A robust iterative algorithm is developed to obtain moment‐curvature relations of the standard composite cross‐sections. Flexural stiffness‐curvature relations of the cross‐sections are predicted through the moment‐curvature relations. Flexural stiffness distribution along the composite beams is determined with curvature compatibility. Fourth order differential load‐deflection equation is evaluated with finite difference solution scheme for continuous beam conditions and variable flexural stiffness along the beam length. A non‐iterative incremental load‐deflection algorithm is developed to calculate moment distribution in continuous beams with variable flexural stiffness. A benchmark example is solved with Eurocode‐4 provisions to validate the developed algorithms. Moment redistribution from hogging (negative) zone to sagging (positive) zone of uniformly loaded double‐span continuous composite beam is illustrated considering the degree of flexural stiffness degradation in the composite medium. A question is arisen for moment redistribution in the continuous beams subjected to partially distributed or concentrated loading conditions.

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded (AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions. Furthermore, assuming polynomial distributions of Young’s modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.

Free vibration of axially functionally graded beams using the asymptotic development method

Axially functionally graded (AFG) beams, with variable coefficients in the governing equation, are a novel class of composites structures that have continuous variations in material properties from one component to another. In this paper, the asymptotic development method (ADM) is utilized to investigate the free vibration of uniform AFG beams with different boundary conditions. By decomposing the variable flexural stiffness and mass per unit length into reference invariant parts and variant parts, perturbation theory is introduced to obtain an approximate formula of the natural frequencies of the uniform AFG beams. The numerical results of the proposed method are confirmed by comparing the obtained results with those obtained via finite element analysis and the published literature results, the comparison reveals the proposed method yields an accurate estimate of the first three order natural frequencies of the AFG beam. Moreover, the influences of the gradient parameter and support conditions on the first three natural frequencies are discussed. The proposed analytical method is simple and efficient and can be used to conveniently analyze uniform AFG beams with arbitrary changes in the material properties along the axial direction.

Investigation of nonuniform rod elements stability by direct integration method

The calculation for the equilibrium stability of a rod system with variable stiffness by the direct integration method is proposed. The method is based on the exact solutions of the corresponding differential equations. Using this method, the rod stability problem with arbitrary continuous variable flexural stiffness is solved. The formulas for the parameters of the frame structure stability are expressed in the analytical form and the method of their numerical implementation is provided. The frame structures with binomial distribution of stiffness are considered.

Construction of a Solution of the Problem of Stability of a Bar with Arbitrary Continuous Parameters

We consider the problem of stability of a bar with arbitrary continuous variable flexural stiffness compressed by an arbitrary continuously applied variable axial longitudinal force. For the first time, we construct the exact solution of the corresponding differential equation of longitudinal bending. As a result, we obtain the formulas for displacements and internal forces in any cross section of the bar in the analytic form.

Ti–Ni Rods with Variable Stiffness for Spine Stabilization: Manufacture and Biomechanical Evaluation

A new concept of monolithic spinal rods with variable flexural stiffness is proposed to reduce the risk of adjacent segment degeneration and vertebral fracture, while providing adequate stability to the spine. The variability of mechanical properties is generated by locally annealing Ti–Ni shape memory alloy rods. Ten-minute Joule effect annealing allows the restoration of the superelasticity in the heated portion of the rod. Such processing also generates a mechanical property gradient between the heated and the unheated zones. A numerical model simulating the annealing temperature and the distributions of the mechanical properties was developed to optimize the Joule-heating strategy and to modulate the rod’s overall flexural stiffness. Subsequently, the rod model was included in a finite element model of a porcine lumbar spine to study the effect of the rod’s stiffness profiles on the spinal biomechanics.

Biomechanical assessment of the stabilization capacity of monolithic spinal rods with different flexural stiffness and anchoring arrangement

BackgroundSpinal disorders can be treated by several means including fusion surgery. Rigid posterior instrumentations are used to obtain the stability needed for fusion. However, the abrupt stiffness variation between the stabilized and intact segments leads to proximal junctional kyphosis. The concept of spinal rods with variable flexural stiffness is proposed to create a more gradual transition at the end of the instrumentation. MethodBiomechanical tests were conducted on porcine spine segments (L1–L6) to assess the stabilization capacity of spinal rods with different flexural stiffness. Dual-rod fusion constructs containing three kinds of rods (Ti, Ti–Ni superelastic, and Ti–Ni half stiff-half superelastic) were implanted using two anchor arrangements: pedicle screws at all levels or pedicle screws at all levels except for upper instrumented vertebra in which case pedicle screws were replaced with transverse process hooks. Specimens were loaded in forward flexion, extension, and lateral bending before and after implantation of the fusion constructs. The effects of different rods on specimen stiffness, vertebra mobility, intradiscal pressures, and anchor forces were evaluated. FindingThe differences in rod properties had a moderate impact on the biomechanics of the instrumented spine when only pedicle screws were used. However, this effect was amplified when transverse process hooks were used as proximal anchors. InterpretationCombining transverse hooks and softer (Ti–Ni superelastic and Ti–Ni half stiff-half superelastic) rods provided more motion at the upper instrumented level and applied less force on the anchors, potentially improving the load sharing capacity of the instrumentation.

In-vitro assessment of the stabilization capacity of monolithic spinal rods with variable flexural stiffness: Methodology and examples.

The concept of a monolithic Ti-Ni spinal rod with variable flexural stiffness is proposed to reduce the risks associated with spinal fusion. The variable stiffness is conferred to the rod using the Joule-heating local annealing technique. To assess the stabilization capacity of such a spinal rod, in vitro experiments on porcine spine models are carried out. This paper describes the methodology followed to evaluate the effect of Ti-Ni rods compared to conventional titanium rods. Validation of the methodology and examples of results obtained are also presented.

Implementation of a 3D porcine lumbar finite element model for the simulation of monolithic spinal rods with variable flexural stiffness.

Monolithic superelastic-elastoplastic spinal rods (MSER) are promising candidates to provide (i) dynamic stabilisation in spinal segments prone to mechanical stress concentration and adjacent segment disease and (ii) to provide fusion-ready stabilization in spinal segments at risk of implant failure. However, the stiffness distributions along the rod's longitudinal axis that best meet clinical requirements remain unknown. The present study is part of a mixed numerical experimental research project and aims at the implementation of a 3D finite element model of the porcine lumbar spine to study the role of MSER material properties and stiffness distributions on the intradiscal pressure distribution in the adjacent segment. In this paper, preliminary intradiscal pressure predictions obtained at one functional spinal unit are presented. Due to a lack of porcine material property data, these predictions were obtained on the basis of uncalibrated human vertebral disc data which were taken from the literature. The results indicate that human annulus and nucleus data predict experimental porcine in vivo and in vitro data reasonably well for the compressive forces of varying magnitudes.

A multi-layered variable stiffness device based on smart form closure actuators

This contribution describes the properties and limitations of multi-layered mechanical devices with variable flexural stiffness. Such structures are supposed to be components of new smart, self-sensing and self-controlling composite materials for lightweight constructions. To enable a proper stiffness control, reliable actuators with high actuation capabilities based on smart materials are used. Those actuators are either driven by electroactive polymers (EAPs) or shape memory alloy (SMA) wires. They control the area moment of inertia of the multi-layered bending structures. To change the area moment of inertia and, hence, the flexural stiffness of an multi-layered beam within a wide range, it is necessary to stack as many layers as possible over each other. The fundamental function of this approach is demonstrated with a three-layer stack consisting of three independent layers and four form closure actuators driven by SMAs. This experimental set-up was able to change its bending stiffness k by a factor of 14.6, with a minimum and maximum stiffness of kmin = 0.11 N mm−1 and kmax = 1.73 N mm−1, respectively. The usage of four independently controllable actuators yields nine independent flexural-stiffness states of the beam. Both analytical and numerical calculations have shown good agreement with the measured stiffness values.

Flexural vibrations of a rod of variable stiffness and distributed mass

A general approach to constructing mathematical models of the transverse vibrations of structures for different dynamic forces is considered. The variable flexural stiffness and the distributed mass are taken into account for the case when their ratio is expressed by a quadratic dependence on the dimensionless radius of inertia of the cross section. The free flexural vibrations and dynamic loads in an elastic rod, modelled by the structure of a lattice tower, are considered as an example. The effect of the model parameters on the value of the first natural frequency is investigated taking into account the presence of an inertial load. Expressions are obtained for the dynamic coefficients when the load is suddenly applied.

Monolithic superelastic rods with variable flexural stiffness for spinal fusion: Modeling of the processing–properties relationship

The concept of a monolithic Ti–Ni spinal rod with variable flexural stiffness is proposed to reduce the risks associated with spinal fusion. The variable stiffness is conferred to the rod using the Joule-heating local annealing technique. The annealing temperature and the mechanical properties’ distributions resulted from this thermal treatment are numerically modeled and experimentally measured. To illustrate the possible applications of such a modeling approach, two case studies are presented: (a) optimization of the Joule-heating strategy to reduce annealing time, and (b) modulation of the rod's overall flexural stiffness using partial annealing. A numerical model of a human spine coupled with the model of the variable flexural stiffness spinal rod developed in this work can ultimately be used to maximize the stabilization capability of spinal instrumentation, while simultaneously decreasing the risks associated with spinal fusion.

Monolithic superelastic rods with variable flexural stiffness for spinal fusion: simplified finite element analysis of an instrumented spine segment.

Rigid instrumentations have been widely used for spinal fusion but they come with complications, such as adjacent disc degeneration. Dynamic instrumentations have been tested but their efficiency (stabilization capability) and reliability (mechanical integrity of the implant) have yet to be proven. A monolithic Ti-Ni spinal rod with variable flexural stiffness is proposed to reduce the risks associated with spinal fusion while maintaining adequate stabilization. This publication presents a simplified numerical model capable of evaluating the eventual benefits of a Ti-Ni spinal rod with variable flexural stiffness. A simplified instrumented spine segment model composed of six vertebrae and five discs has been developed. Two types of spinal rods were evaluated: Classic Ti instrumentation and Ti-Ni rods with variable stiffness. Both instrumentations were tested using two anchor configurations: pedicle screws only or a screws-cable combination. The all-screws configuration does not allow much motion with either classic Ti or variable Ti-Ni rods. The combination of a Ti rod with screws-cable anchoring allows more motion and, therefore, lower adjacent disk pressure, but puts extremely high stresses on the rod and anchors. The combination of the variable Ti-Ni rod and screws-cable anchoring leads to a significant decrease in adjacent disk pressure, without increasing stresses and pullout forces in the spinal instrumentation.

Manufacturing of monolithic superelastic rods with variable properties for spinal correction: Feasibility study

A new concept of monolithic spinal rod with variable flexural stiffness is proposed to reduce the risk of adjacent segment degeneration and fracture associated with rigid spinal fixation techniques while providing adequate stability to the spine. The concept is based on the use of Ti–Ni shape memory alloy rods subjected to different processing schedules implying local annealing, cold work, or a combination of both. A feasibility study of the concurrent technological routes is performed by comparing their potential to locally control material microstructure and properties.

ANALYSIS OF TILT‐BUCKLING OF EULER COLUMNS WITH VARYING FLEXURAL STIFFNESS USING HOMOTOPY PERTURBATION METHOD

In this paper, the Homotopy Perturbation Method (HPM), is introduced for elastic stability analysis of tilt‐buckled columns with variable flexural stiffness. Buckling loads and corresponding mode shapes are determined considering different types of variations in flexural stiffness of columns. The proposed approach is an efficient technique for the elastic stability analysis of specified problems.

Aerodynamic effects of flexibility in flapping wings

Recent work on the aerodynamics of flapping flight reveals fundamental differences in the mechanisms of aerodynamic force generation between fixed and flapping wings. When fixed wings translate at high angles of attack, they periodically generate and shed leading and trailing edge vortices as reflected in their fluctuating aerodynamic force traces and associated flow visualization. In contrast, wings flapping at high angles of attack generate stable leading edge vorticity, which persists throughout the duration of the stroke and enhances mean aerodynamic forces. Here, we show that aerodynamic forces can be controlled by altering the trailing edge flexibility of a flapping wing. We used a dynamically scaled mechanical model of flapping flight (Re approximately 2000) to measure the aerodynamic forces on flapping wings of variable flexural stiffness (EI). For low to medium angles of attack, as flexibility of the wing increases, its ability to generate aerodynamic forces decreases monotonically but its lift-to-drag ratios remain approximately constant. The instantaneous force traces reveal no major differences in the underlying modes of force generation for flexible and rigid wings, but the magnitude of force, the angle of net force vector and centre of pressure all vary systematically with wing flexibility. Even a rudimentary framework of wing veins is sufficient to restore the ability of flexible wings to generate forces at near-rigid values. Thus, the magnitude of force generation can be controlled by modulating the trailing edge flexibility and thereby controlling the magnitude of the leading edge vorticity. To characterize this, we have generated a detailed database of aerodynamic forces as a function of several variables including material properties, kinematics, aerodynamic forces and centre of pressure, which can also be used to help validate computational models of aeroelastic flapping wings. These experiments will also be useful for wing design for small robotic insects and, to a limited extent, in understanding the aerodynamics of flapping insect wings.

Exact Solutions for the Generalized Euler’s Problem

This paper is concerned with buckling analysis of a nonuniform column with classical∕nonclassical boundary conditions and subjected to a concentrated axial force and distributed variable axial loading, namely, the generalized Euler’s problem. Exact solutions are derived for the buckling problem of nonuniform columns with variable flexural stiffness and under distributed variable axial loading expressed in terms of polynomial functions. Then, more complicated buckling problems are considered such as that the distribution of flexural stiffness of a nonuniform column is an arbitrary function, and the distribution of axial loading acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for such problems is reduced to Bessel equations and other solvable equations for seven cases by means of functional transformations. A class of exact solutions for the generalized Euler’s problem involved a nonuniform column subjected to an axial concentrated force and axially distributed variable loading is obtained herein for the first time in literature.

Determination of Critical Buckling Loads for Euler Columns of Variable Flexural Stiffness with a Continuous Elastic Restraint Using Homotopy Perturbation Method

Article Determination of Critical Buckling Loads for Euler Columns of Variable Flexural Stiffness with a Continuous Elastic Restraint Using Homotopy Perturbation Method was published on February 1, 2009 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 10, issue 2).

The problem of the bending of a beam lying on an elastic base

The plane contact problem of the transmission of a normal force of specified strength onto an elastic anisotropic, wedge-shaped plate by an elastic beam of variable flexural stiffness is considered. The beam is coupled to one of the edges of the plate and its other edge is stress-free. The solution of the problem is obtained in closed form by reducing it to a Karleman boundary-value problem with shear for a strip. A conclusion is reached concerning the nature of the discontinuity of the normal contact stress at the vertex of the wedge.