Equivalent Flexural Rigidity Research Articles

Dynamics of a chain of permanent magnets

The dynamics of a straight chain of cylindrical neodymium magnets is considered. We showed that this system exhibits a behavior which is similar to that of a beam where the elastic rigidity acts like a restoring force. By using a Lagrangian approach, a linear model, accounting for different sets of boundary conditions, is derived. Specifically for the clamped-free case, eigenfrequencies and eigenmodes are determined. These theoretical results are compared to experiments. A good agreement is found, the discrepancies being attributed to the accumulation of defaults in the contact surface between the magnets. In the last part of the article, the equivalent beam behavior is sought by defining an equivalent flexural rigidity induced by magnetic dipoles interactions.

Equivalent Rigidity at the Minor Axis of Concrete Hollow Slab Based on Continuity Analysis

Basic assumptions are proposed for the continuity model of a tubular hollow slab combined with continuity analysis and calculation of a finite-element model; the continuity Equation of a tubular hollow slab at the minor axis supported at two ends of the hollow axle under a vertical even load is determined and solved. The overall equivalent flexural rigidity is then determined, which provides important conditions for the continuity analysis of tubular hollow floor supported along four sides.

Lateral Equivalent Flexural Rigidity for a Top-Hat Stiffened Composite Plate under Pure Bending

This paper is concerned with the analytical solution to lateral equivalent flexural rigidity for fiber-reinforced top-hat stiffened composite plate subjected to pure bending. The plate is divided into three parts consist of single-laminate, double-laminate and hat-laminate. After computing equivalent flexural rigidity of the three parts respectively, the overall rigidity is calculated based on the equality of relative rotation angle. An example has been adopted for analytical calculation and then been modeled by finite elements method. A comparison was made to verify the validity of analytical method.

Processing and Characterization of Continuous Fibre Tapes Co-Moulded with Long Fibre Reinforced Thermoplastics

An important advantage when designing with plastics is the ability to incorporate features such as ribs, grids and bosses in the part. Rib stiffened polymer matrix composites have been widely used in aerospace, automobile, and civil infrastructure applications due to their high impact and fatigue resistance, high strength and stiffness to weight ratio, and damage tolerance. However, for long fibre-reinforced polymer composites, the processing complexity increases for features including ribs, grids and bosses. An innovative method of replacing ribs is the use of pre-consolidated continuous fibre reinforced thermoplastic (CFRT) tapes that can be co-moulded with long fibre thermoplastics (LFT). This work focuses on processing and performance evaluation in terms of the static and dynamic properties of LFTs co-moulded with pre-consolidated CFRTs, referred to as endless long fibre thermoplastic (E-LFT). The E-LFT approach is an alternative to rib stiffened composites. The effect of the thickness and fibre type in tape reinforced LFT is compared to LFT (with and without ribs) of equivalent flexural rigidity for static flexure and low velocity impact (LVI) response. In all these conditions, E-LFT samples performed better that the LFTs with and without ribs. LFT samples with and without ribs exhibited a brittle failure, as opposed to the progressive failure exhibited by E-LFT.

Structural performance of light weight multicellular FRP composite bridge deck using finite element analysis

Fiber reinforced polymer (FRP) composite materials having advantages such as higher strength to weight than conventional engineering materials, non-corrosiveness and modularization, which should help engineers to obtain more efficient and cost effective structural materials and systems. Currently, FRP composites are becoming more popular in civil engineering applications. The objectives of this research are to study performance and behavior of light weight multi-cellular FRP composite bridge decks (both module and system levels) under various loading conditions through finite element modeling, and to validate analytical response of FRP composite bridge decks with data from laboratory evaluations. The relative deflection, equivalent flexural rigidity, failure load (mode) and load distribution factors (LDF) based on FE results have been compared with experimental data and discussed in detail. The finite element results showing good correlations with experimental data are presented in this work.

Structural Optimization Design of Flexible Anti-Collision Device for Bridge

Due to he complexity of the shipping collision process, the design of the protection device can not be accurately calculated by theoretical formula. Through the establishment of the anti-collision device in simplified model, this paper carries out the simulation analysis of the finite element values, and studies the key parameters that affects the performance of the anti-collision device: the geometric structure of inside and outside of steel, the combination of anti-collision circle (elastic element), the flexural rigidity of peripheral steel. And then we get the design parameters of the equivalent flexural rigidity of peripheral steel which is to ensure the that the peripheral steel won’t be partial collapse when there are some collision between some representative ships and it; and we also find that the appropriate increase in the flexural rigidity of around the vertex of the outer steel will improve the overall anti-collision ability of the anti-collision device.

Elastic Modulus of 304 Stainless Steel Coating by Cold Gas Dynamic Spraying

304 stainless steel coating was deposited on the IF steel substrate by cold gas dynamic spraying (CGDS), and the elastic modulus of the 304 stainless steel coating was studied. The elastic modulus of cold sprayed 304 stainless steel coating was measured using the three-point bend testing and the compound beam theory, and the other mechanic parameters (such as the equivalent flexural rigidity and the moment of inertia of area) of the coatings were also calculated using this compound beam theory. It is found that the calculated results using the above methods are accurate and reliable. The elastic modulus value of the cold sprayed 304 stainless steel coating is 1. 179 × 105 MPa, and it is slightly lower than the 304 stainless steel plate (about 2 × 105 MPa). It indicates that the elastic modulus of the cold sprayed coatings was quite different from the comparable bulk materials. The main reason is that the pores and other defects are existed in the coatings, and the elastic modulus of the coatings also depends on varies parameters such as the feed stock particle size, porosity, and processing parameters.

Equivalent flexural and torsional rigidity of hexagonal honeycomb

The flexural rigidity and torsional rigidity of an open honeycomb were evaluated by analyzing the bending and torsional deformation of cell plates with consideration of the constraint condition on the plate edges. The analysis of torsional rigidity showed that if the deformation of a honeycomb satisfies periodicity conditions, the ratio of M xy ∣ unit and M yx ∣ unit acting on each unit constituting the honeycomb becomes a specific value to satisfy the continuity condition of displacement. This specific value is obtained using an equation proposed in this study; the value depends only on the geometry of the honeycomb. As a consequence of the necessity that the ratio of M xy ∣ unit and M yx ∣ unit should become a specific value, if the ratio of torsional load moments M xy ∞ and M yx ∞ acting on the boundaries is not actually equal to the above-mentioned value, scattering appears near the boundary in the ratio of M xy ∣ unit and M yx ∣ unit , and the ratio approaches the above-mentioned specific value as the distance from the boundary increases. Therefore, even if the value ( M xy ∞ - M yx ∞ ) is kept the same, the scattering of deflection depends on the combination of torsional moments, and becomes smaller as the honeycomb model becomes larger compared with the size of a unit.

On the analytical approach for the bending/stretching of linearly elastic functionally graded rectangular plates with two opposite edges simply supported

In this article, a new analytical method for bending—stretching analysis of thick functionally graded (FG) rectangular plates is presented. Using this method, the governing equations of FG rectangular plates based on the first-order shear deformation or Mindlin plate theory are decoupled. Five coupled partial differential equations of the Mindlin FG plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function. It is analytically shown that by introducing an equivalent flexural rigidity, the equations of FG rectangular plate become similar to those of the homogeneous isotropic plate. Solving these equations, the solutions are obtained for the FG rectangular plate with two opposite edges simply supported. A comparison of the present results with available solutions from previous studies is made and a good agreement can be seen. Also, the numerical results for stress and deflection of the FG rectangular plate with various boundary conditions are obtained.

An analytical approach for stress analysis of functionally graded annular sector plates

In this paper, an exact analytical approach is used for bending analysis of functionally graded (FG) annular sector plates. The governing equilibrium equations are obtained based on the first order shear deformation plate theory. Introducing an analytical method, the coupled governing equilibrium equations are replaced by independent equations in term of transverse deflection and a new function. Using an equivalent flexural rigidity, the solutions of FG annular sector plates can be easily extracted from equation of homogeneous annular plates. Also, it is shown that the present method can provide accurate results. Finally, the effects of power of functionally graded material (FGM), plate thickness, inner to outer radius ratio and boundary conditions on the deflection and stresses of a functionally graded annular sector plate are studied.

Development of Laser Spot Welded Stainless Steel Double Skin Panels for Civil Transports

Manufacturing technology for stainless steel double skin panels using laser spot welding was developed and some static strength properties were carried out. And this panel has been examined for the feasibility of practical utilization. The results obtained are summarized as follows;(1) Manufacturing technology was established on which there are invisible welding marks and the corrosion resistance and appearance are equivalent to original metal surface.(2) Calculate method of the moment of inertia of area of this panel is established.(3) This panel is approximately 75% lighter than the bulk material in equivalent flexural rigidity.(4) Any laser spot welded site is not broken in region of below maximum load as well as localized buckling in mechanical tests.(5) In application of this panel to vessels and vehicles, simulation model of stiffness and strength of this structures life prediction of this panel based on fatigue strength analysis for panel reliability prediction technology is required.

비구속형 점탄성 제진층을 갖는 보의 제진층 길이 최적화

Length of an unconstrained viscoelastic damping layer on beams is determined to maximizeloss factor using a numerical search method. The fractional derivative model can describe damping characteristics of viscoelastic damping materials accurately, and is used to represent nonlinearity of complex modulus with frequencies and temperatures. Equivalent flexural rigidity of the unconstrained beam is obtained using Ross, Ungar, Kelvin[RUK] equation. The loss factors of partially covered unconstrained beam are calculated by a modal strain energy method. Optimal lengths of the unconstrained viscoelastic damping layer of beams are identified with ambient temperatures and thickness ratios of beam and damping layer by using a finite-difference-based steepest descent method.

Identification of Coefficients for Continuous System's Equation of Motion Using One-Dimensional Vibration Experiments.

Equations of motion for one-dimensional continuous system are studied to identify unknown coefficients of that equation. Several new formulae are presented to calculate unknown coefficients. Experiments are achieved to verify accuracy of the present method. Good agreement suggests that the present method is reliable to identify unknown coefficients of the equation of motion. Following are applications of the present method. (1) Sound absorbing materials are tested. Complex sound propagation speed and complex effective density of the material are identified. (2) Wave dissipative offshore structure is also tested. Amplitude-dependent complex wave propagation speed in that structure are identified. (3) Perforated panels are tested to determine pressure loss coefficient under alternating flow condition. (4) Three-layered vibration damping steel beams are tested to determine complex equivalent flexural rigidity of the beam. Loss factor of the beam are measured in successive frequencies. (5) Cables with tension are tested. Both tension and flexural rigidity of the cable are identified using least square method from several resonance frequencies.

Parameters Back-Calculation for Concrete Pavement with Two Slab Layers

This paper presents a new procedure for back-calculating the properties of a concrete pavement with two slab layers using deflection measurements from nondestructive deflection testing. The two slab layers are first represented by an equivalent single slab layer based on the principle of equivalent flexural rigidity. In the back-calculation analysis, the pavement deflection response under load is evaluated using an analytical model for a slab supported on an elastic foundation. Employing a closed-form back-calculation algorithm, it is shown that the back-calculated equivalent radius of relative stiffness and subgrade modulus are unique and independent of the characteristics of the two slab layers. It follows that the equivalent flexural rigidity for the two slab layers is also unique. The next step employs a trial-and-error algorithm to back-derive the moduli of the two slab layers by matching the equivalent flexural rigidity. An upper bound and a lower bound to the modulus can be established for the slab layer with the higher modulus. Similarly, an upper bound to the modulus can be found for the slab layer with the lower modulus. Four numerical examples are presented to demonstrate the use of the proposed back-calculation method.

Large deflection of elastic composite circular springs under uniaxial compression

The equivalent flexural rigidity approach for large deflection analysis has been applied to orthotropic midplane symmetric laminated circular rings under uniaxial compression. The analytical approach incorporates the exact expression of the moment-curvature relationship for the bending of symmetric laminated beams. Non-linear spring behaviour is obtained and verified by experimental testings. The circumferential strain distributions around the rings are compared with that obtained by a small deflection strain energy approach and found to provide a more accurate prediction, especially at high loads where the deflection is large. Finite element calculations performed with some of the ring models show good correlation with both analytical and experimental results and the distribution of circumferential strains at increased external load is illustrated.

An Analysis of the Vibration of Fabrics Considering the Bending Hysteresis.

The dynamic bending response of a woven fabric was theoretically estimated from its static bending property including hysteresis. The hysteresis loop obtained by pure bending test was simulated by the Rate madel, based on which the equivalent flexural rigidity and damping coefficient were estimated. The response of the model strip of fabric with these parameters to forced oscillation was analyzed by a nonlinear FEM. The results showed that the calculated results agreed well with the experimental ones in value as well as in dependence on the amplitude of forced vibration.

A biomechanical analog of curve progression and orthotic stabilization in idiopathic scoliosis

A biomechanical analog of curve progression and orthotic stabilization in idiopathic scoliosis has been developed using the classical theory of curved beam-columns. The interaction of the spinal musculature and other supporting structures is incorporated in the model using an equivalent flexural rigidity. The stability of a given scoliotic curve relative to a normal spine is described in terms of the so-called critical load ratio ( P c P e ). This dimensionless quantity appears in the exact solution of the governing differential equation and boundary conditions. It is defined as the ratio of the load bearing capacity of a scoliotic spine ( P c ) to that of a normal spine where the load bearing capacity of a normal spine is defined as Euler's buckling load ( P e ). The computation of P c P e is based upon a maximum allowable moment criterion. This model is used to study the effect of the degree of initial curvature and curve pattern in the frontal plane on the stability of untreated idiopathic scoliosis. Although restricted to two-dimensions, the model appears to demonstrate the synergistic effects of end support, transverse loading, and curve correction on improvement in relative stability of an orthotically supported scoliotic curve. The results of this study are in qualitative agreement with clinical findings that are based on long-term studies of natural history of idiopathic scoliosis and of patients undergoing orthotic management for scoliosis.

A Method for the Estimation of Collapse Loads of Corner Girders and Struts

The equivalent thickness of deflected panels under compression and bending moment are estimated by the energy method. The effective flexural rigidity of columns with rather small slenderness ratio are estimated by using Johnson's and Euler's formula. Using these equivalent thickness and effective flexural rigidity, the collapse load of corner girders and struts are calculated by elastic finite element method. The collapse load and collapse mechanism are compared with experimental results. This method will be usefull at the initial design stage for the simplicity.

Ultimate Compressive Strength of Stiffened Plates (2nd report)

The present paper investigates the ultimate compressive strength of a simply-supported wide panel reinforced longitudinally by multiple stiffeners. It was clarified in the previous paper that in the case of one stiffener, the ultimate behaviors depend upon the equivalent flexural rigidity γe of the stiffener and they are classified in the three modes by the specific values of γe : γ1 and γ2. The same tendencies are observied in the case of multiply stiffened panels ; weak stiffeners (γe < γ1) can not prevent panels from overall deformation, while strong stiffeners (γe>γ2) are effective to be panel breakers and only local deformations are dominent even at the ultimate strength. If γ1<γe<γ2, overall deformation interacts with local buckling mode. It is also shown that the ultimate strength largely depends upon the in-plane constraint conditions. The simple approximation formulas are proposed for estimation of the ultimate strength.

Determination of the elastic and damping properties of the human femur in bending

The results of an experimental investigation of the deflection of the human femur subjected to a bending moment are presented. The existence of two principal bending planes is established. The equivalent flexural rigidity of the dry bone is calculated for both static and dynamic loading. It is found that the human femur has a nonlinear elastic characteristic. The stiffness and damping properties obtained in static and dynamic testing are compared.