Apparent Bending Stiffness Research Articles

Damage detection by means of broadband local wavenumber estimation in plate-like structures

This short communication evinces the ability of the CFAT methodology [Q. Leclère, JSV, 2015] to detect and characterize thickness-loss defects on homogeneous flat panels. A contactless measurement of the velocity field of an aluminium plate is performed by scanning laser vibrometry, from which the equivalent rigidity (apparent bending stiffness) is locally estimated within a wide frequency band in the ultrasonic domain. The method allows imaging thickness-reduction defects on the hidden face of an aluminium plate up to several hundreds of kilohertz. The first part details the experimental protocol, followed by a second enhancing the identification of the wavenumbers of symmetric S_0 and antisymmetric A_0 modes. The comparison with theoretical Lamb waves dispersion curves reveals the precision of the method. Finally, the methodology validates the possibility to precisely estimate the local material thickness and as a consequence the location, size and depth of a reference defect in the shape of a flat bottom hole (assuming homogeneous material properties of the plate).

Investigating the Out-of-Plane Bending Stiffness Properties in Hybrid Species Diagonal-Cross-Laminated Timber Panels

Since the introduction of Cross-laminated Timber (CLT) in Austria in the early 1990s, the adoption of this 90°-crosswise-laminated product has seen exponential growth worldwide. Compared to traditional laminated timber products (e.g., glulam), CLT provides improved dimensional stability but with reduced out-of-plane bending stiffness. To improve the bending stiffness, while maintaining relative dimensional stability, a modified orientation of the inner layers in a diagonal direction can be used. This novel product is Diagonal-Cross-laminated Timber (DCLT), a composite timber product, consisting of inner layers which are rotated at different angle-ply orientations between 0 and 90 degrees to the outer layers. To properly model the out-of-plane bending behavior of the DCLT, analytical models and finite element analysis (FEA) were used, and the results were validated by four-point bending tests performed on DCLT panels with angle-ply orientations of 10°, 20°, 40°, 70°, and a conventional CLT 90° panel. The results indicate that DCLT panels with angle-ply cross layers have a structural advantage in out-of-plane bending over traditional CLT (90°) panels. The apparent bending stiffness from DCLT 90° to DCLT ± 10° has an increase of 33%, 24%, and 35%, respectively, regarding the assessed methods of experimental, theoretical, and FEM modeling. Using these panels would allow for increased spans or load-carrying capacity for a given panel span-to-depth ratio. The development of DCLT and its introduction to the industry not only could enable the use of lower-quality timber that would not otherwise satisfy structural requirements for CLT but also could help reduce the fabrication cost of CLT due to utilizing lower amounts of fiber.

Bending and stretching behavior of graphene structures using continuum models calibrated with modal analysis

Two-dimensional nanomaterials exhibit specific mechanical characteristics essential for commercialization and industrial applications, such as nano electro-mechanical systems. When analyzing a mechanical field, it is important to determine these materials' bending stiffness and stretching properties. The apparent bending stiffness and stretching of graphene structures at the macro-scale differ from theoretical predictions at the nano-scale. This discrepancy results from thermally generated dynamic waves in these atomic-based structures. Therefore, characterization methods based on atomistic static methods were unable to capture these effects. This study uses hybrid atomistic-continuum models to apply modal analysis to determine these key parameters. The proposed approach exploits the advantages of both atomistic and continuum models. Our approach is based on optimization techniques such as the simulated annealing algorithm. The unknown parameters of the continuum models, such as the bending stiffness and stretching, were acquired from molecular dynamics simulations and continuum mechanics resonance frequencies. The effects of diameter on the mechanical properties and value of different mode shapes of graphene nanostructures are studied using this method, which incorporates the effects of nano and macro-scale models. The presented approach provides insight into the mechanical vibrational characteristics of two-dimensional graphene structures at an atomic level.

Beam element resonance-based prediction and parametric analysis of sound transmission of laminated plates

Laminated plates are complex structures important to practical applications, so it is essential and valuable to develop efficient and accurate models for predicting their sound transmission loss. An efficient and unified approach based on the frequency-dependent bending stiffness of a beam element taken from a laminated plate is proposed to predict the sound transmission loss of the laminated plate. First, the finite element method is used to calculate the resonant frequencies of the multilayer beam under different boundary conditions, based on which the apparent bending stiffness of the corresponding multilayer plate is estimated. Next, with the aid of the so obtained apparent bending stiffness, the dynamic behavior of a multilayer composite plate is described by a simplified fourth-order governing equation. Finally, the sound transmission loss is conveniently calculated with the modal superposition method. In this approach, the influences can be conveniently and accurately simulated of the plate size, shear modulus of the core, loss factor and boundary conditions. Furthermore, the predicted sound transmission loss is validated through comparisons with the measured values of typical laminated plates. Finally, based on calculations with the proposed approach, the influence is revealed of the shear modulus on the coincidence region and sound transmission loss, and the relationship is discussed between the weighted sound reduction index and the shear modulus. These results suggest that to achieve good acoustic performance of a designed laminated plate, the coincidence region should be narrow and located at frequencies as high as possible.

A bending stiffness criterion for sandwich panels with high sound insulation and its realization through low specific modulus layers

The sound transmission loss of a sandwich panel is usually much lower than that given by the mass law for a homogeneous panel with the same weight in the mass-controlled frequency range. How to make the sound transmission loss of the sandwich panel close to the mass law and maintain a high static stiffness is a key acoustic issue in its design. In this paper, the critical apparent bending stiffness is defined as the apparent bending stiffness at which the sound transmission loss of a sandwich panel approaches that of the mass law. Then, a method is given to determine the critical apparent bending stiffness of sandwich panels, based on which a range of structural parameters can be obtained to guide the design of sandwich panels with high acoustic performance. Compared with the apparent bending stiffness of a typical sandwich panel, the critical apparent bending stiffness decays more rapidly with the increase of frequency, exhibiting a relatively high static stiffness and a relatively low dynamic stiffness. In order to make the apparent bending stiffness of a typical sandwich panel close to the critical apparent bending stiffness, the concept of introducing an elastic layer with low specific modulus in the middle of the core layer is proposed. Both theoretical and experimental results show that the introduced elastic layer with low specific modulus reduces the dynamic bending stiffness of the sandwich panel while a high static bending stiffness is maintained, thus significantly improving the broadband sound-insulation performance of the panel.

Effect of Layer Arrangement on Bending Strength of Cross-Laminated Timber (CLT) Manufactured from Poplar (Populus deltoides L.)

This study aimed to investigate the effect of layer arrangement on bending properties of CLT panels made from poplar (Populus deltoides L.). A total of 20 three-layer CLT panels with the same dimensions of 1300 × 360 × 48 mm3 (Length, Width, Thickness) were fabricated in five configurations: 0/30/0, 0/45/0, 0/90/0, 45/0/45, and 45/45/45. The apparent modulus of elasticity (MOEapp), modulus of rupture (MOR) and apparent bending stiffness (EIapp) values in major and minor axes of CLT panels were calculated using experimental bending testing. In the major axis, the highest values of MOR, MOEapp, and EIapp were obtained from the 0/30/0 arrangement, while the least values resulted from the arrangements of 90/60/90 and 90/45/90 in the minor axis. Besides, in all arrangements, the average of the experimental apparent bending stiffness values (EIapp,exp) of specimens was higher than that of the shear analogy apparent bending stiffness values (EIapp,shear). The bending and shear stress distribution values over the cross section of samples were also estimated using the finite element method. Moreover, the numerical apparent bending stiffness (EIapp,fem) values of samples were compared to experimental apparent bending stiffness (EIapp,exp) values. Based on experimental and finite element method results, in all groups of layer arrangements, the EIapp,fem values concurred well with the EIapp,exp values.

Experimental study on bending properties of cross-laminated timber-bamboo composites

Using bamboo and timber to manufacture cross-laminated timber-bamboo (CLTB) composites could utilize the advantages of two material to improve the mechanical properties and expand the raw material source of cross-laminated timber (CLT). To explore the feasibility of using engineered bamboo as CLT lamination, the rolling shear properties of two bamboo scrimbers and one bamboo plywood were evaluated. Two types of CLTB specimens, using bamboo scrimber as transverse layer or as both transverse layer and the outermost longitudinal layer, were developed in this study. The bending properties of CLTB and generic Spruce-pine-fir (SPF) CLT specimens were evaluated by dynamic tests and third-point bending tests, respectively. The shear analogy method was employed to calculate the apparent bending stiffness of CLT/CLTB specimen. The results indicated that the rolling shear modulus and strength of bamboo scrimber (group RB(F)-4.5) were 92.81% and 98.64% higher than those of bamboo plywood, respectively, and were 330.19% and 121.13% higher than those of SPF dimension lumber, respectively. The CLTB specimens, group BWBWB, had 23.69% and 60.50% higher apparent bending modulus and peak load than those of generic SPF CLT specimens. Most of the CLTB bending specimens had main failure mode of tensile failure in the bottom layer which replaced rolling shear failure in transverse bamboo scrimber layer. The apparent bending stiffness values of CLT/CLTB specimens obtained from dynamic test and shear analogy method were in good agreement with the static experimental values. The results presented in this paper can provide fundamental basis for supporting the potential engineering application of CLT panel products fabricated with bamboo scrimber.

Simultaneous evaluation of bending and shear stiffness of wood I-joists by transverse vibration tests

Prefabricated wood I-joists are widely used in lightweight wood constructions. Accurate and efficient measurements of their elastic properties are of great importance for predicting their mechanical behaviour during structural design. In this study, an alternative algorithm for simultaneous measurement of bending and shear stiffness of wood I-joists was developed for transverse vibration techniques based on Timoshenko beam theory. Sixty-five wood I-joists with thirteen configurations were tested by using the proposed modal testing method as well as traditional third-point bending tests. Based on the test results, the influences of section properties of wood I-joists on the shear rigidities were studied. It was found that the transverse vibration tests can be simply and efficiently implemented to determine the true and apparent bending stiffness of wood I-joists. In addition to the I-joist depth, the shear rigidities are also influenced by the web thickness and flange width. More importantly, shear deflection coefficients of tested wood I-joists are significantly larger than those theoretical values currently used by wood I-joist manufacturers.

Sodium ion transport across the endothelial glycocalyx layer under electric field conditions: A molecular dynamics study.

In the present research, the sodium ion transport across the endothelial glycocalyx layer (EGL) under an imposed electric field is investigated, for the first time, using a series of molecular dynamics simulations. The electric field is perpendicularly imposed on the EGL with varying strengths. The sodium ion molarity difference between the inner and outer layers of EGL, Δc, is used to quantify the sodium transport in the presence of the negatively charged glycocalyx sugar chains. Results suggest that a weak electric field increases Δc, regardless of whether the electric field is imposed perpendicularly inward or outward. By contrast, a strong electric field drives sodium ions to travel in the same orientation as the electric field. Scrutiny of the charge distribution of the glycocalyx sugar chains suggests that the electric field modifies the spatial layouts of glycocalyx atoms as it drives the transport of sodium ions. The modification in glycocalyx layouts further changes the inter-molecular interactions between glycocalyx sugar chains and sodium ions, thereby limiting the electric field control of ion transport. The sodium ions, in turn, alter the apparent bending stiffness of glycocalyx. Moreover, the negative charges of the glycocalyx sugar chains play an important role in maintaining structural stability of endothelial glycocalyx. Based on the findings, a hypothesis is proposed regarding the existence of a strength threshold of the electric field in controlling charged particles in the endothelium, which offers an alternative explanation for contrasting results in previous experimental observations.

Mechanical Properties of Glued‐Laminated Timber with Different Assembly Patterns

A glued‐laminated timber section with mixed‐grade laminae could make an efficient use of material strength and reduce the cost. A 4‐point bending test was conducted on a total of 18 specimens to investigate the mechanical properties of glued‐laminated timber. Uniform‐grade, asymmetrical mixed‐grade, and symmetrical mixed‐grade patterns were used to assemble the beam sections. The bending stiffness and reliability of the beams were assessed according to the experimental results. The influence of the assembly pattern on the bending behavior of glued‐laminated timber was investigated by finite element models. The results show that the assembly pattern of the section has little influence on the failure mode of glued‐laminated timber. Relative lower strength in compressive area of the section is beneficial for delaying the occurrence of the first crack on the glued‐laminated timber beam. An equation for apparent bending stiffness of glued‐laminated timber was proposed, whose results match well with the experimental results. The beam section assembled by the asymmetrical mixed‐grade pattern retains the higher level of safety compared to those assembled by uniform‐grade and symmetrical mixed‐grade patterns. The grade of the second bottom lamina in tensile has little influence on the performance of glued‐laminated timber, while lower grade laminae in compressive area of the section would cause a bending stiffness reduction at smaller deflection.

Vibro-acoustic properties of sandwich structures

The vibro-acoustic properties of three-layered sandwich elements depend on the bending stiffness of the structure. It is found that the apparent bending stiffness of a sandwich beam depends on frequency, its length and boundary conditions. A beam with free ends appears stiffer than the same beam clamped at both ends. The apparent bending stiffness of a sandwich beam with clamped ends decreases as the length of the beam is reduced. The opposite is the case for a beam with free ends. A method for estimating the material parameters of a sandwich beam is described. The frequency and space averages of the point mobility of a finite sandwich beam is found to be equal to the point mobility of an infinite sandwich beam of the same type. Coupling loss factors between sandwich beam elements for some simple types of junctions are given.

Assessment of the apparent bending stiffness and damping of multilayer plates; modelling and experiment

In the context of aeronautics, automotive and construction applications, the design of light multilayer plates with optimized vibroacoustical damping and isolation performances remains a major industrial challenge and a hot topic of research. This paper focuses on the vibrational behavior of three-layered sandwich composite plates in a broad-band frequency range. Several aspects are studied through measurement techniques and analytical modelling of a steel/polymer/steel plate sandwich system. A contactless measurement of the velocity field of plates using a scanning laser vibrometer is performed, from which the equivalent single layer complex rigidity (apparent bending stiffness and apparent damping) in the mid/high frequency ranges is estimated. The results are combined with low/mid frequency estimations obtained with a high-resolution modal analysis method so that the frequency dependent equivalent Young's modulus and equivalent loss factor of the composite plate are identified for the whole [40 Hz-20 kHz] frequency band. The results are in very good agreement with an equivalent single layer analytical modelling based on wave propagation analysis (model of Guyader). The comparison with this model allows identifying the frequency dependent complex modulus of the polymer core layer through inverse resolution. Dynamical mechanical analysis measurements are also performed on the polymer layer alone and compared with the values obtained through the inverse method. Again, a good agreement between these two estimations over the broad-band frequency range demonstrates the validity of the approach.

EXPERIMENTAL STUDY OF THE BENDING PERFORMANCE OF HOLLOW GLULAM BEAMS

Hollow glulam beam has some advantages that the traditional solid glulam beam does not have, such as the convenience for wiring construction and comparably light weight. Four-point bending tests of three solid glulam beams and 15 hollow glulam beams with various sizes of rectangular holes produced from small-diameter larch timber were conducted to investigate the influence of the hollow ratio and wall thickness on bending stiffness and load capacity. The midspan deflection, cross-section strain, and ultimate load were obtained from the tests, and the detailed failure modes and apparent MOE for all specimens are reported. Hollow glulam beams with the hollow ratio ranged from 25% to 40%, and the wall thickness greater than 20 mm after the assumption of plane section under bending moment. The apparent bending stiffness and ductility of hollow glulam beam were less than those of solid glulam beam, and the apparent MOE is 0.86 times the elastic modulus value calculated by theory of elasticity. In addition, a calculation formula for the ultimate bending moment is proposed.

Flexural elasticity of woodpile lattice beams

Abstract Flexure of slender structures, composed of filaments in a woodpile arrangement, is theoretically studied. Expressions for the apparent bending stiffness are derived. The model is validated experimentally using three-point bending. Computer simulations show that bending is accompanied by lattice shear for increased porosity. A shear-inclusive micromechanical model for two different stacking arrangements is developed. This is further refined by including shear deformation within the filaments, which is supported by numerical simulations for relatively short filament overhang. The apparent flexure and shear of lattice beams are attributed primarily to stretch and flexure of the filaments, respectively. Asymptotic formulas for the effective bending stiffness in terms of the relative density are presented. The apparent bending stiffness scales linearly with the volume fraction, whereas the apparent shear stiffness scales with the cube of the volume fraction. Exclusion of lattice shear leads to errors for by over an order of magnitude in the extreme, unlike void-free solid beams where shear is a higher order effect. Excellent quantitative agreement with numerical results is obtained for a wide range of porosity and external aspect ratio, when lattice shear is included in the model.

Non-contact experimental assessment of apparent dynamic stiffness of constrained-layer damping sandwich plates in a broad frequency range using a Nd:YAG pump laser and a laser Doppler vibrometer

This paper concerns the assessment of the dynamic stiffness of plates using non-contact excitation and non-contact measurements. In particular, a constrained-layer damping sandwich plate is considered, of which the apparent dynamic stiffness as function of frequency is determined. The experimental results compare very well with an analytical model that computes the frequency dependent apparent bending stiffness of the constrained-layer damping sandwich plate. A Nd:YAG laser is used for excitation of the plate and a laser Doppler vibrometer is used to measure its dynamic response along a line on the plate, thus reducing the measurement effort. Using advanced data acquisition and data processing techniques, a signal-to-noise ratio of up to 100dB was obtained, yielding estimates of the flexural wavenumber and dynamic stiffness of the plate up to a frequency of 50kHz. It is shown that the Prony method and the wavenumber fit method yield a much improved wavenumber resolution as compared to the spatial Fourier transform method. An additional advantage of the wavenumber fit approach is that it allows the accuracy of the fit to be determined. The accuracy was estimated at 1rad/m (best relative wavenumber resolution 2‰).

System transverse in-plane shear stiffness of pultruded GFRP bridge decks

The static transverse behavior of two pultruded GFRP deck systems with trapezoidal (DS) and triangular (AS) cell cross-sectional geometry was experimentally investigated in order to study their transverse in-plane shear stiffness. Symmetric three-point bending experiments up to failure were performed on 200-mm-wide beams. Their stiffness, strength and failure modes were compared. Different load transfer mechanisms were found in the DS (frame-dominated) and AS (truss-governed) systems depending on the cell geometry. The DS beams exhibited a lower apparent bending stiffness (24–30 times less) and degree of composite action between the flanges (14–17 times less) than the AS beams. These dissimilarities were attributed to the lower transverse in-plane shear stiffness provided by the trapezoidal core than by the triangular core. The low bound values for both system in-plane shear moduli were estimated from the experimental deflection results. The system in-plane shear modulus of the DS beams represented approximately 2–3% of that of the AS beams.

Prediction of Sound Transmission Loss for Finite Sandwich Panels Based on a Test Procedure on Beam Elements

An approach on the prediction of sound transmission loss for a finite sandwich panel with honeycomb core is described in the paper. The sandwich panel is treated as orthotropic and the apparent bending stiffness in two principal directions is estimated by means of simple tests on beam elements cut from the sandwich panel. Utilizing orthotropic panel theory, together with the obtained bending stiffness in two directions, the sound transmission loss of simply-supported sandwich panel is predicted by the modal expansion method. Simulation results indicated that dimension, orthotropy, and loss factor may play important roles on sound transmission loss of sandwich panel. The predicted transmission loss is compared with measured data and the agreement is reasonable. This approach may provide an efficient tool to predict the sound transmission loss of finite sandwich panels.

PREDICTION AND MEASUREMENT OF SOME DYNAMIC PROPERTIES OF SANDWICH STRUCTURES WITH HONEYCOMB AND FOAM CORES

Some dynamical properties of sandwich beams and plates are discussed. The types of elements investigated are three-layered structures with lightweight honeycomb or foam cores with thin laminates bonded to each side of the core. A six order differential equation governing the apparent bending of sandwich beams is derived using Hamilton's principle. Bending, shear and rotation are considered. Boundary conditions for free, clamped and simply supported beams are formulated. The apparent bending stiffness of sandwich beams is found to depend on the frequency and the boundary conditions for the structure. Simple measurements on sandwich beams are used to determine the bending stiffness of the entire structure and at the same time the bending stiffness of the laminates as well as the shear stiffness of the core. A method for the prediction of eigenfrequencies and modes of vibration are presented. Eigenfrequencies for rectangular and orthotropic sandwich plates are calculated using the Rayleigh–Ritz technique assuming frequency dependent material parameters. Predicted and measured results are compared.

Wave propagation in and sound transmission through sandwich plates

Some dynamical and acoustical properties of sandwich plates are investigated. The types of sandwich elements discussed are three-layered plates with a thick lightweight core, with thin and comparatively stiff laminates bonded to each side of the core. In the model derived it is assumed that the laminates and core are isotropic. The laminates are treated as thin plates, whereas the deflection in the core is described by means of the general field equations. This means that bending shear and rotation, as well as longitudinal deflection, are considered in the core. Wavenumbers, loss factors and apparent bending stiffness for symmetric and asymmetric plates are derived. In addition, the sound transmission loss for sandwich plates is discussed. Measured and predicted results are compared. It is found that bending stiffness and loss factor not only depend on material parameters and plate geometries but also on frequency. The core thickness is very critical for the sound transmission loss of a sandwich plate. Sandwich plates are frequently used in the shipbuilding industry for light and fast passenger vessels. The effects of a fluid load on a sandwich plate is therefore also included.