Flexural Stiffness Distribution Research Articles

Determination of flexural stiffness of composite timber beams using DIC and wavelet transform of deflection lines

The article presents an experiment of four-point bending of timber I-beams, the deflections of which were measured using digital image correlation (DIC). The deflection functions were subjected to a wavelet transform operation, which used the second derivative of the Gaussian distribution to determine the curvature of the beams’ axis. On this basis, the distribution of flexural stiffness of the elements was estimated, showing the potential usefulness of the DIC technique in production control and laboratory testing of prefabricated timber elements. The limitations of the proposed research method related to the need to perform numerical integration operations on discrete experimental data subject to measurement errors are also presented

Damage Identification of Simply Supported Bridges Using Static Responses: Unified Framework and Feasibility Study

Simply supported bridges are the main bridge types in many transportation systems, and their damage can significantly reduce their load-carrying capacity. To detect possible damage, the time history and spatial distribution of the static responses of bridges (deflection, rotation, and strain influence lines/deformation curves) have recently been proposed as important indicators due to their good local damage detection ability and low testing cost. This paper attempts to establish connections between different static curve-based damage indicators through the various forms of Maxwell-Betti’s law. Damage indicators related to seven static curves are then rewritten as a unified framework. The framework states that all these static curves are directly related to the flexural stiffness distribution of the main girder for the simply supported bridge. Then, the theoretical formulations for the difference between the static curves of bridges in intact and damaged states are derived, and the response difference surfaces (RDSs) are plotted to visualize the sensitivity of different static curves to damage. Sensor placement suggestions for stiffness degradation evaluation are presented based on the damage sensitivity analysis at the end of this paper. The results of this study provide comprehensive theoretical support for static response-based damage identification of simply supported bridges.

Measurement of natural frequencies and mode shapes of transparent insect wings using common-path ESPI.

In this study, a common-path electronic speckle pattern interferometry system which upholds the natural property of transparency of insect's wings has been developed to measure the wings' natural frequencies and mode shapes for the first time. A novel base-exciting method was designed to enable the simultaneous application of sinusoidal and static forces to excite wings and introduce an additional phase. The moiré effect induced by the amplitude modulation was employed to accurately recognize the resonance state. Subsequently, the mode shapes were visualized by phase-shifting and real-time frame subtraction. Eight pairs of forewings from cicadas were investigated. The first three order natural frequencies of the wings are approximately 145 Hz, 272 Hz and 394 Hz, respectively, which are dispersed to prevent modal coupling. The cambered mode shapes exhibit a strongly spanwise-chordwise anisotropy flexural stiffness distribution, generally dominated by bending and twisting deformation. The details of the high-order mode shapes show that the tip exhibits distinct deformation, indicating more flexibility to cope with external impact load, and the nodal lines usually comply with the direction of the wing veins in higher modes, substantiating the fact that the veins play an important role as stiffeners of the membrane. The results are in excellent agreement with the dynamic performance of previous studies, which will potentially affect a broader community of optical measurement specialists and entomologists to enhance our understanding of time-averaged interferograms and insect flights.

Stiffness Estimation of Girder Bridges Using Influence Lines Identified from Vehicle-Induced Structural Responses

The influence line is an important static property in bridges. The shape and magnitude of the girder bridge influence line contain stiffness information for the bridge supports and beam. This research sought to directly identify the rotational stiffness of bridge supports and the flexural stiffness of beams from extracted influence lines, which provides a quantification method for bridge damage and performance degradation. The paper first reviews the influence line identification method from vehicle-induced bridge responses. Then, the complete theoretical formulations of the deflection, rotation, and strain influence line for a general girder bridge are derived. Based on these derivations, the rotational stiffness at two supports and the equivalent flexural stiffness value of the beam are identified. Next, a certain span of a continuous girder bridge is converted to an equivalent simply supported system, and the flexural stiffness distribution of the beam is estimated based on the bending moment-curvature relationship and Betti’s law. The paper then presents a three-span continuous girder bridge verification example. The stiffness estimation results under different test conditions combined with the bridge influence line identification algorithm are presented to verify the reliability of the proposed method. This stiffness estimation method can be applied to baseline-free bridge performance evaluations based on vehicle-induced bridge responses.

Effect of flexural stiffness distribution of a ski on the ski–snow contact pressure in a carved turn

The design of the flexural stiffness distribution of a ski is one of the important factors in its development. On the other hand, the ski–snow contact pressure during a turn is thought to be an important factor in a skier’s feeling. However, the relationship between the flexural stiffness distribution and ski–snow contact pressure has not been made clear. The purpose of this study was to understand how flexural stiffness distribution affects the contacting pressure, applying load, and ski–snow contacting condition through measurement on these items. The measurement system consisted of a ski deflection sensor beam with bending sensors, pressure sensors installed near the ski edge, and four points of load cells connected between the binding plate and the ski. Measurements were performed on the different flexural stiffness distributions of Skis A and B on hard snow and soft snow surfaces. The peak of the flexural stiffness of Ski A was on the boot center, and Ski B was stiffer with a peak under the heel of the boot. An increase in the flexural stiffness under the heel of the boot for Ski B caused less deflection in the rear part of the ski, and the applied force from the binding plate shifted forward. By fitting the ski deflection on the snow groove, so the rear section of the ski after the pressure peak fitted on the sliding surface of the snow groove, the compression zone increased and the sliding zone was reduced in Ski B. The pressure peak in Ski A was under the boot center, whereas, for Ski B it was behind the boot center. These results show that changes in flexural stiffness distribution alter ski deflection and ski–snow contact pressure.

Geometry of the nuclear envelope determines its flexural stiffness.

During closed mitosis in fission yeast, growing microtubules push onto the nuclear envelope to deform it, which results in fission into two daughter nuclei. The resistance of the envelope to bending, quantified by the flexural stiffness, helps determine the microtubule-dependent nuclear shape transformations. Computational models of envelope mechanics have assumed values of the flexural stiffness of the envelope based on simple scaling arguments. The validity of these estimates is in doubt, however, owing to the complex structure of the nuclear envelope. Here, we performed computational analysis of the bending of the nuclear envelope under applied force using a model that accounts for envelope geometry. Our calculations show that the effective bending modulus of the nuclear envelope is an order of magnitude larger than a single membrane and approximately five times greater than the nuclear lamina. This large bending modulus is in part due to the 45 nm separation between the two membranes, which supports larger bending moments in the structure. Further, the effective bending modulus is highly sensitive to the geometry of the nuclear envelope, ranging from twofold to an order magnitude larger than the corresponding single membrane. These results suggest that spatial variations in geometry and mechanical environment of the envelope may cause a spatial distribution of flexural stiffness in the same nucleus. Overall, our calculations support the possibility that the nuclear envelope may balance significant mechanical stresses in yeast and in cells from higher organisms.

Continuously Variable Stiffness Mechanism Using Nonuniform Patterns on Coaxial Tubes for Continuum Microsurgical Robot

Variable stiffness enables the safe and effective operation of the minimally invasive surgical instruments. In this article, we propose a continuously variable stiffness mechanism of the scalable tubular structure. The mechanism consists of multiple coaxial nitinol tubes, and each tube has an anisotropic distribution of flexural stiffness created by nonuniform through-hole patterning. The stiffness of the mechanism is varied by relative rotation and translation among the tubes, resulting in flexural stiffness difference up to 7.2 times in the direction of load. Its flexural stiffnesses along principal axes are independently controlled by the suggested counterrotation algorithm. The stiffness change is validated through analytical modeling, FEM simulation, and the experiments. Thanks to its physically embodied intelligence, the mechanism has a simple scalable structure and the response time is immediate. We applied this mechanism to control the stiffness of the steerable needle. Varying the stiffness grants the additional degree of freedom to control the needle's trajectory, which can expand the workspace of the steerable needle.

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.

Effect of flexural stiffness distribution of a fin on propulsion performance

Experimental investigations on biomimetic mechanical fins have been carried out by researchers to study and mimic the behaviour of fish. Flexible fins are proven to be better than rigid fins in terms of thrust force generation and efficiency when the degree of flexibility is chosen appropriately. It is observed that fishes can modulate their fin stiffness by muscle coactivation. Also, fins of different variety of fishes are developed with different stiffnesses through natural evolution to cope up with different living conditions. Inspired by the natural design of fish fins with varying flexural stiffness along the chord, in this work, a search-based numerical optimization study is conducted to investigate on the optimal flexural stiffness distribution yielding maximum thrust force. For this, a dynamic model of a flexible fin is developed through multi-body dynamics approach by considering the flexible fin as multiple rigid segments connected with torsion springs. Blade element method is utilized to compute the hydrodynamic forces acting on the fin segments. Based on the design optimizations several flat plate fins of trapezoidal geometry are developed with varied distribution of stiffness. Experiments are conducted with the fabricated fins to validate the optimization results for maximized propulsion performance.

Artificial flapping wings mimicking flexural stiffness distribution of hummingbird feathers

Artificial flapping wings mimicking flexural stiffness distribution of hummingbird feathers

The relationship between pectoral fin ray stiffness and swimming behavior in Labridae: insights into design, performance and ecology.

The functional capabilities of flexible, propulsive appendages are directly influenced by their mechanical properties. The fins of fishes have undergone extraordinary evolutionary diversification in structure and function, which raises questions of how fin mechanics relate to swimming behavior. In the fish family Labridae, pectoral fin swimming behavior ranges from rowing to flapping. Rowers are more maneuverable than flappers, but flappers generate greater thrust at high speeds and achieve greater mechanical efficiency at all speeds. Interspecific differences in hydrodynamic capability are largely dependent on fin kinematics and deformation, and are expected to correlate with fin stiffness. Here we examine fin ray stiffness in two closely related species that employ divergent swimming behaviors, the flapping Gomphosus varius and the rowing Halichoeres bivittatus To determine the spatial distribution of flexural stiffness across the fin, we performed three-point bending tests at the center of the proximal, middle and distal regions of four equally spaced fin rays. Pectoral fin ray flexural stiffness ranged from 0.0001 to 1.5109 µNm2, and the proximal regions of G. varius fin rays were nearly an order of magnitude stiffer than those of H. bivittatus In both species, fin ray flexural stiffness decreased exponentially along the proximodistal span of fin rays, and flexural stiffness decreased along the fin chord from the leading to the trailing edge. Furthermore, the proportion of fin area occupied by fin rays was significantly greater in G. varius than in H. bivittatus, suggesting that the proportion of fin ray to fin area contributes to differences in fin mechanics.

Design of shape-adaptive wind turbine blades using Differential Stiffness Bend–Twist coupling

The design of horizontal axis wind turbines (HAWT) has faced limitations in size and uncontrollable wind behaviour. Shape-adaptive wind turbine blades are proposed as one of the solutions to these design constraints, as they can improve the performance of the blades under varying operating conditions without increasing the size of the blades. A novel Differential Stiffness Bend–Twist coupling (DSBT)11Differential Stiffness Bend–Twist. concept is proposed in this paper to control the deformation behaviour of the blades. The DSBT concept achieves bend–twist coupling in the global structure while allowing the main deformation of the structural sub-components to remain in pure bending. By eliminating the need to design bend–twist coupled parts, this simplifies the material design and potentially enhances the manufacturability. An analytical conceptual model was developed to study the fundamental behaviour of DSBT structures under bending loads. The feasibility analysis demonstrated that the DSBT concept could achieve bend–twist coupling by altering the flexural stiffness distribution of stiffeners attached to the skin of the structure. A preliminary design scheme for the DSBT blades was also developed with the material design of stiffeners identified as an area to be optimised. The genetic algorithm (GA) was coupled with the Finite Element Method to search for the optimal flexure stiffness values. GA was reused to evaluate the required composite layup based on the stiffness requirements. Thus the design satisfies the requirements for strength and stiffness to enable DSBT.

Nondestructive damage detection in Euler-Bernoulli beams using nodal curvatures-Part I: Theory and numerical verification

SUMMARY This paper deals with the problem of nondestructively detecting, locating, and quantifying damage in beam-type structures. To achieve this objective, structural responses collected prior and subsequent to damage are related to the changes in physical properties of the structure. The proposed methodology is based on the moment–curvature relations of the Euler–Bernoulli beam theory and the assumption that internal stress resultants are invariant before and after damage. Damage is expressed in terms of local decreases in the flexural stiffness of structural members. These decreases are shown to cause singularities in the curvature profile of the beam. Utilizing fundamental equations of solid mechanics, we relate discontinuities in the flexural stiffness distribution to the pre-damage and post-damage nodal curvatures. The resulting system of linear equations can be solved to obtain specific element damage indices. The performance of the proposed methodology is evaluated using numerically generated experiments. It is shown that the location, the extent, and the severity of damage in beams may be successfully identified with the proposed technique provided that the structure conforms to the deformations dictated by the Euler–Bernoulli beam theory. The practicality of the methodology under field conditions is demonstrated in the accompanying paper Nondestructive damage detection in Euler-Bernoulli beams using nodal curvatures-Part II: Field measurements. Copyright © 2013 John Wiley & Sons, Ltd.

Maxima and minima of moments in hyperstatic spans

When a distributed load is applied to a hyperstatic (statically indeterminate) beam span of two distinct and given section flexural stiffnessses, maxima or minima of the critical moments within the span arise when a physically meaningful coincidence exists between the moment diagram and the flexural stiffness diagram for the span. In this paper, an insightful geometric argument is employed to identify that physically meaningful phenomenon for a span with both ends encastered. Symbolic analysis is then used to derive the coincidence for spans with rotational springs of any stiffness between zero (free rotation) and infinity (encastered) at one end or both ends. The vital role of implicit differentiation in enabling serendipitous discovery of the result is highlighted. Examples are used to show how the phenomenon and associated theory may be used to obviate iterative computations, thereby providing rapid and useful insight into the process of optimising fibre-reinforced polymer/steel plate layouts which create two section stiffnesses when connected to timber or steel beams. For reinforced concrete or steel–concrete composite spans it is shown that the phenomenon automatically occurs, and the effects on steel-to-concrete shear transfer of uncertainties in the flexural stiffness distribution are discussed again by recourse to the underlying theory.

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.

Direct method for analysis of flexible cantilever beam subjected to two follower forces

The static analysis of the flexible non-uniform cantilever beams under a tip-concentrated and intermediate follower forces is considered. The angles of inclination of the concentrated forces with respect to the deformed axis of the beam remain unchanged during deformation. The governing non-linear boundary-value problem is reduced to an initial-value problem by change of variables. The resulting problem can be solved without iterations. It is shown that there are no critical loads in the Euler sense (divergence) for any flexural–stiffness distribution and angles of inclination of the follower forces. In particular, if the follower forces are tangential, the rectilinear shape of the non-uniform cantilever beam is the only possible equilibrium configuration. In this paper some equilibrium configurations of the uniform cantilever under normal or tangential follower forces are presented using direct method.

A computationally efficient approach to swimming monofin optimization

Monofins provide swimmers with an efficient alternative to the standard pair of fins. For example, all short and long distance human swimming records have been established using monofins. Current monofin design is mostly empirical, so the objectives of this work are to analyze monofin propulsion through coupled fluid–structure simulation and to optimize its flexural stiffness distribution. The optimization process maximizes the propulsive power provided by the monofin with a constraint on the total expended power. To be able to carry out the optimization of the coupled fluid–structure system, which is numerically costly to evaluate, the following simplifications are proposed: (1) a 2-D unsteady, inviscid, and incompressible fluid flow is considered; (2) the swimmer is composed of linear articulated segments, whose kinematics is imposed and identified from experimental data; and (3) the monofin is represented by rigid bars linked by torsional springs. For various allowable swimmer powers, optimal 2-D stiffness distributions are obtained using the Globalized and Bounded Nelder–Mead algorithm. Finally, an identification procedure is described to translate the optimal 2-D stiffness distributions into 3-D thickness profiles for a given monofin planform shape.

Stability of Nonuniform Cracked Bars Under Arbitrarily Distributed Axial Loading

This study is concerned with the stability analysis of nonuniform bars with arbitrary number of cracks and with arbitrary distribution of flexural stiffness or arbitrarily distributed axial loading. The homogeneous solutions of the governing differential equation for buckling of a nonuniform uncracked bar are derived for several important cases. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks. Then a new approach that combines the exact buckling solution of a nonuniform uncracked bar, the model of massless rotational spring, and the transfer matrix method is presented for the title problem. The main advantage of the proposed method is that the eigenvalue equation for buckling of a nonuniform bar with an arbitrary number of cracks, arbitrary distribution of flexural stiffness, or arbitrarily distributed axial loading can be conveniently determined from a second-order determinant. The decrease in the determinant order as compared with other methods leads to significant savings in the computational effort. A numerical example is given to illustrate the reliability of the proposed approach through comparisons with numerical solutions and to study the effect of cracks on the stability of a nonuniform bar. Nomenclature ai = depth of the ith crack Ci =fl exibility of the ith rotational spring Ci1, Ci2 = constants of integration 1F2(a1; a2; a3; a4) =h ypergeometric series f () =fl exibility function hi = height of cross section at xi Jν(x) = first Bessel function of the vth order Ki(x) =fl exural stiffness at x in the ith

Classes of exact solutions for buckling of multi-step non-uniform columns with an arbitrary number of cracks subjected to concentrated and distributed axial loads

The governing differential equation for buckling of a multi-step non-uniform column subjected to combined concentrated and distributed axial loads, each step of which has an arbitrary number of cracks with or without spring supports, is expressed in the form of bending moment. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the column. In this paper, the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of functional transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for six important cases. The exact buckling solutions of one-step non-uniform columns are thus found. Then a new approach that combines the exact buckling solution of a one-step column and the transfer matrix method is presented to establish the eigenvalue equation for buckling of a multi-step non-uniform column with spring supports. The main advantage of the proposed method is that the eigenvalue equation for buckling of a non-uniform column with an arbitrary number of cracks, any kind of two end supports and various spring supports at intermediate points can be conveniently determined from a second order determinant. Due to the decrease in the determinant’s order as compared with previously developed procedures the computational time required by the present method can be reduced significantly. A numerical example is given to examine the accuracy of the proposed method and to investigate the effect of cracks on buckling of a multi-step non-uniform column.

Exact solutions for buckling of non-uniform columns under axial concentrated and distributed loading

In this paper, the buckling problem of non-uniform columns subjected to axial concentrated and distributed loading is studied. The expression for describing the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a non-uniform column with arbitrary distribution of flexural stiffness or axial forces is reduced to a second-order differential equation without the first-order derivative by means of functional transformations. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 12 cases, several of which are important in engineering practice. The exact solutions that represent a class of exact functional solutions for the buckling problem of non-uniform columns subjected to axial concentrated and distributed loading are obtained. In order to illustrate the proposed method, a numerical example is given in the last part of this paper.