Static Deflection Research Articles

Mechanical Behavior of Honeybee Forewing with Flexible Resilin Joints and Stripes.

The flexibility of insect wings should be considered in the design of bionic micro flapping-wing aircraft. The honeybee is an ideal biomimetic object because its wings are small and possess a concise vein pattern. In this paper, we focus on resilin, an important flexible factor in honeybees' forewings. Both resilin joints and resilin stripes are considered in the finite element model, and their mechanical behaviors are studied comprehensively. Resilin was found to increase the static deflections in chordwise and spanwise directions by 1.4 times and 1.9 times, respectively. In modal analysis, natural frequencies of the first bending and first torsional modes were found to be decreased significantly-especially the latter, which was reduced from 500 Hz to 217 Hz-in terms of resilin joints and stripes, closely approaching flapping frequency. As a result, the rotational angle amplitude in dynamic responses is remarkable, with an amplification ratio of about six. It was also found that resilin joints and stripes together lead to well-cambered sections and improve the stress concentrations in dynamic deformation. As resilin is widespread in insect wings, the study could help our understanding of the flexible mechanism of wing structure and inspire the development of flexible airfoils.

Geometrically nonlinear static deflection of stiffened composite plates: A fifth-order equivalent model

Nonlinear reduced-order models (NLROM) have been developed to predict the geometrically nonlinear behaviour of structures with low computational cost. NLROMs built using non-intrusive methods are compatible with commercial FE software and have the potential for wide application in engineering. The challenge of non-intrusive NLROMs is to accurately reproduce the internal force of structures. In literature, the internal force has been assumed to be a cubic polynomial function of generalized coordinates. However, this assumption is effective only for structures with small rigid rotations but not for structures with significant rigid rotations such as cantilever structures. In this paper, we propose a fifth-order polynomial approximation of the internal force for such cases. The nonlinear stiffness tensors of the reduced-order model are built using explicit formulas, ensuring applicability to all structures. Our proposed reduced-order model accurately predicts the nonlinear static deflection of composite plates with complex geometries.

Sensing and control the static deflection of a cantilever beam by using piezoelectric patches

Abstract Different mechanical structures when comes under static and dynamic loadings then they respond. It is very important to sense it so that the dynamic behavior of the structure in terms of natural frequencies, its mode shapes, and viscous damping coefficients can be successfully predicted or measured. The accurate prediction of the dynamic response of the oscillating structure always plays an important role to design its physical parameters. However, in this paper, the simulated and experimental study is conducted to accurately sense the static deflection of the cantilever beam by using piezoelectric patches. The accurate prediction of the static deflection of the beam and its control through piezoelectric patches is the novelty of the present work. Mostly, the finite element model of the beam is developed to predict its dynamic behavior. The finite element modeling provides an approximate solution. In this regard, the finite element model updating technique such as the direct updating algorithm is used in this study to correct the simulated finite element model of the beam. The updated simulated finite element model is then used to predict the accurate value of the static deflection. It is observed that the maximum percentage error between the predicted and measured static deflection is 29.1%. The updated simulated finite element model is also used to design the controller. The direct output feedback controller is developed and it is found that the static deflection of the beam has been reduced by 46.65%.

Modeling Method for Static Large Deflection Problem of Curved Planar Beams in Compliant Mechanisms Based on a Novel Governing Equation

Abstract Slender beams serve as typical flexible components to transfer motion, force, and energy in compliant mechanisms. Therefore, the accurate and efficient kinetostatic modeling for the slender beams is highly needed in the synthesis and analysis of compliant mechanisms. Several impressive and great modeling methods have been completed by the pioneering researchers based on the governing equation of slender beams, which can solve the deflection of the beam while the beam-tip loads are known. However, parts of the beam-tip loads are unknown in some scenarios and the traditional governing equation becomes inefficient in these scenarios. The aim of this paper is to propose a novel modeling method for the compliant mechanism, which can solve the large deflection problem without iterative algorithm. In this research, a novel governing equation of slender beams has been established based on Castigliano’s principle and simplified by the differential geometry. In the proposed governing equation, the statically force equilibrium equations and geometric constraint equations between different slender beams can be established in the boundary conditions, therefore the model of compliant mechanisms can be solved directly even if parts of the beam’s tip loads and displacements are unknown. The proposed governing equation provides the deformed shapes and cross-sectional loads of the slender beams, which help the designer to check the mechanical interference and strength in the design process. The numerical examples are presented to demonstrate the feasibility of proposed method.

Experimental definition of lateral stiffness of a pneumatic wheel of the MTZ-82 “Belarus” tractor

BACKGROUND: Due to the excessive vertical and angular vibrations of suspensionless vehicles used in agriculture and road construction that occur during movement, the velocity of their motion is limited, which, in the conditions of a constant traffic flow, reduces the road capacity. Therefore, the search for ways to improve the vibration-isolating properties of wheeled suspensionless vehicles is a relevant issue, the solution of which affects not only on traffic safety and driving comfort, but also on average speed and fuel efficiency.
 AIMS: To determine the lateral stiffness of the pneumatic wheel from the rear axle of the MTZ-82 Belarus tractor at different tire pressures, basing on bench tests.
 METHODS: On the basis of the hydraulic pulsation test rig of the Automatic Units department of the VolgGTU, a special-purpose movable supporting and measuring device was developed and assembled, the feature of which is that 4 rollers with flanges are installed on top of the hydraulic actuator force sensor, a flat rectangular base plate is supported on these rollers and connected with a vertical frame of the rig through a force-measuring device by means of a screw mechanism. The tested wheel is mounted on the base plate, the wheels axle is fixed on a vertically-moving traverse, on top of which weights creating the necessary vertical force are fixed. The test procedure consisted in measuring the lateral force and lateral displacement of the base plate until the tire slip occurred at the following tire pressures: at the recommended pressure of 0.16 MPa and at reduced pressures of 0.12, 0.08 and 0.04 MPa.
 RESULTS: Based on the test results, the elastic characteristic curves of the lateral stiffness of the tested wheel were obtained. These characteristic curves have a regressive form, as their slope significantly decreases with a significant reduce in tire pressure. Thus, when the tire pressure is reduced from 0.16 to 0.12 MPa, there is practically no difference in the obtained values of lateral stiffness, which reaches 112.5 kN/m. After a 2-fold tire pressure reduction, the lateral stiffness decreases by 7%, and after a 4-fold reduction, by 28%. At the same time, the static tire deflection increases from 22 to 32 mm, which significantly increases the contact patch of the tire with the ground.
 CONCLUSIONS: By means of the tests carried out, it is established that tire pressure reduction by 2550% of the recommended value does not have a significant effect on the loss of lateral stiffness of the pneumatic wheel of the rear axle of the MTZ-82 Belarus tractor, which is important to use in order to increase traction and improve ride comfort of wheeled tractors.

Free wake panel method simulations of a highly flexible wing in flutter and gusts

This paper presents low speed fluid structure interaction simulations of a highly flexible wing at various flow conditions, including flutter and excitation from sinusoidal gusts. Such wings are becoming more relevant in recent years, due to their potential for improving aerodynamics and reducing weight, while their flutter characteristics are particularly challenging to address, as the modal properties of the wings change as deflections increase. Calculations are based on time domain coupling of a geometrically exact beam structural model and a 3D free wake panel method, modeling the outer surface of the wing, which allow for nonlinear effects in terms of geometrical deformations and the flow at low computational cost. Static and aeroelastic wing deflections are in line with experimental data of the Pazy wing, which is a benchmark for highly flexible wings from Technion. Two flutter mechanisms are predicted within 1 to 3 m/s of the experimental range. An analysis of the flutter modes is performed, showing that the second torsion mode plays a role in flutter, something that had not been published before. Limit cycle oscillations are achieved and are shown to compare well with reference data, with the frequency being within 1% of the experimental value. Finally, results of gust simulations of the Pazy wing are compared to data from experiments and corrections for the wind tunnel measurements are proposed, which should facilitate future validation efforts. This work serves as a contribution to the Pazy wing dataset and is a step towards mid-fidelity simulations for more complex configurations.

Influence of geometric nonlinearity on static and dynamic response of flexible beam

The flexible beam theory is widely applied in engineering, due to that many engineering structures can be regarded as flexible beam. For the flexible beam under large rotation, researches have shown that consideration of geometric nonlinearity is necessary: beam deflection will be erroneously larger or diverge without consideration of geometric nonlinearity. The reason lies on the dynamic stiffening effect caused by geometric nonlinearity. The dynamic stiffening effect of flexible beam under large rotation is well studied already, however, most studies only focus on flexible beam with constant cross section and rarely considers flexible beam without rotation. In view of this, a flexible beam with variable and twisted cross section is adopted to analyze the influence of geometric nonlinearity, and the response of flexible beam without rotation or static response is also analyzed. The results of dynamic beam deflection with and without consideration of geometric nonlinearity is compared, the results of static beam deflection under different load is also compared. On the basis of result comparison, the influence of geometric nonlinearity on static and dynamic response of flexible beam is then explained.

ANSYS based Modal Analysis of a Fixed Beam of Uniform Cross Section with a Concentrated Mass at Mid Span

The report outlines ANSYS based modal and structural analysis of a fixed beam of dimension 20mm*20mm*1000mm that was constructed in SOLIDWORKS. Structural steel was selected as the material for analysis. The analysis was done in 3 phases, the first analysis was done without any crack and the other two analysis involved cracked beams with cracks created at different lengths. Suitable boundary and loading conditions were used to find the modes of frequencies. The resulting variation in static deflection, position of mode shapes representing transverse deflection and variation in position of equivalent stress corresponding to varying beam conditions with and without cracks are studied.

Vibration Analysis of Rotating Elements of A Motor

Abstract: With the industrial revolution, the range of applications for rotating machinery in daily life as well as for industrial use in manufacturing and processing, nuclear power plants, automobiles, oil & gas refineries, etc., has greatly expanded. And for all this rotating machinery, the problems brought on by bearing faults significantly contribute to machine failure. Machine Design courses usually teach students how to design a system consisting of a shaft and its bearings under rotating, bending, transverse, axial, and torsional loads. Most textbooks cover Rayleigh's and Holzer's methods, which do not cover other important dynamic effects in shaft and bearing system design. The diameter of the shaft is calculated and then the deflection of the shaft is evaluated by using static deflection formulas. However, this assumption can cause serious errors in shafts' deflection due to eccentric and nonaligned gears mounted on the shaft. The authors addressed this potential manufacturing defect issue in a Machine Design class as a term project, which required students to transfer and apply content knowledge from their dynamics and vibrations’ courses. This paper presents all aspects of this successful experience of implementing ABET strategies in the engineering classroom to maximize its reach and potential impact.

From thin plates to Ahmed bodies: linear and weakly nonlinear stability of rectangular prisms

We study the stability of laminar wakes past three-dimensional rectangular prisms. The width-to-height ratio is set to $W/H=1.2$ , while the length-to-height ratio $1/6< L/H<3$ covers a wide range of geometries from thin plates to elongated Ahmed bodies. First, global linear stability analysis yields a series of pitchfork and Hopf bifurcations: (i) at lower Reynolds numbers $Re$ , two stationary modes, $A$ and $B$ , become unstable, breaking the top/bottom and left/right planar symmetries, respectively; (ii) at larger $Re$ , two oscillatory modes become unstable and, again, each mode breaks one of the two symmetries. The critical $Re$ values of these four modes increase with $L/H$ , reproducing qualitatively the trend of stationary and oscillatory bifurcations in axisymmetric wakes (e.g. thin disk, sphere and bullet-shaped bodies). Next, a weakly nonlinear analysis based on the two stationary modes $A$ and $B$ yields coupled amplitude equations. For Ahmed bodies, as $Re$ increases, state $(A,0)$ appears first, followed by state $(0,B)$ . While there is a range of bistability of those two states, only $(0,B)$ remains stable at larger $Re$ , similar to the static wake deflection (across the larger base dimension) observed in the turbulent regime. The bifurcation sequence, including bistability and hysteresis, is validated with fully nonlinear direct numerical simulations, and is shown to be robust to variations in $W$ and $L$ in the range of common Ahmed bodies.

Biogenic polymer-based patches for congenital cardiac surgery: a feasibility study.

Currently used patch materials in congenital cardiac surgery do not grow, renew, or remodel. Patch calcification occurs more rapidly in pediatric patients eventually leading to reoperations. Bacterial cellulose (BC) as a biogenic polymer offers high tensile strength, biocompatibility, and hemocompatibility. Thus, we further investigated the biomechanical properties of BC for use as patch material. The BC-producing bacteria Acetobacter xylinum were cultured in different environments to investigate optimal culturing conditions. For mechanical characterization, an established method of inflation for biaxial testing was used. The applied static pressure and deflection height of the BC patch were measured. Furthermore, a displacement and strain distribution analysis was performed and compared to a standard xenograft pericardial patch. The examination of the culturing conditions revealed that the BC became homogenous and stable when cultivated at 29°C, 60% oxygen concentration, and culturing medium exchange every third day for a total culturing period of 12 days. The estimated elastic modulus of the BC patches ranged from 200 to 530 MPa compared to 230 MPa for the pericardial patch. The strain distributions, calculated from preloaded (2 mmHg) to 80 mmHg inflation, show BC patch strains ranging between 0.6% and 4%, which was comparable to the pericardial patch. However, the pressure at rupture and peak deflection height varied greatly, ranging from 67 to around 200 mmHg and 0.96 to 5.28 mm, respectively. The same patch thickness does not automatically result in the same material properties indicating that the manufacturing conditions have a significant impact on durability. BC patches can achieve comparable results to pericardial patches in terms of strain behavior as well as in the maximum applied pressure that can be withstood without rupture. Bacterial cellulose patches could be a promising material worth further research.

Geometrically nonlinear effects in wing aeroelastic dynamics at large deflections

This paper investigates geometrically nonlinear effects due to nonlinear kinematics and follower aerodynamics in the aeroelastic dynamics of a very flexible wing. The test case is the Pazy wing, a benchmark wing model for geometrically nonlinear aeroelastic studies involving large deflections in low-speed flow. The work builds on a low-order model of the wing composed of a geometrically nonlinear beam coupled with potential flow thin airfoil theory. Nonfollower aerodynamics underestimates static deflections by up to 10%, whereas linear kinematics overestimates them by up to 50%. At low static deflections, the wing hump mode flutter mechanism is driven by three-dimensional unsteady aerodynamic effects. As static deflections increase, the flutter onset and offset speeds are driven by curvature effects that can only be captured by nonlinear kinematics, which reduce the natural frequencies associated with torsion and in-plane bending motions. Follower aerodynamics shrinks the hump mode instability region and shifts it to 1–4% lower flow speeds with no change in the flutter frequency, the qualitative trends, and the amplitude of limit-cycle oscillations. Overall, nonlinear kinematics is the primary geometrical nonlinearity influencing the wing aeroelastic behavior: the modal and stability scenarios with and without follower aerodynamics collapse onto the same curves when presented as functions of the tip deflection. However, geometrical nonlinearities alone miss the subcritical nature of the hump mode flutter mechanism, which is attributed to aerodynamic stall and its interaction with large deflections. This work expands the growing literature on the aeroelastic dynamics of very flexible wings and will ease future studies considering aerodynamic nonlinear effects.

Static, dynamic and buckling responses of random functionally graded beams reinforced by graphene platelets

This work proposes a multiscale numerical method that can be used to study the static, dynamic and buckling behavior of functionally graded (FG) beams reinforced with graphene platelets (GPLs), whose random gradation is achieved through tailoring the internal porosity and GPLs reinforcement, either along the transverse or the longitudinal direction. The multiscale numerical method consists of microscale homogenization and structural analysis: micromechanics homogenization methods are introduced to predict the elastic moduli of nanocomposite layers with finite thickness along the transverse direction; Explicitly-expressed composite finite elements are then developed through the proposed variational principle and Hamilton principle. In the meantime, the corresponding governing equations are derived for FG beams with Timoshenko theory. On this basis, the static bending deflection, fundamental frequency and critical buckling load of the FG beams are obtained through the proposed FG finite elements. The effectiveness and accuracy of the proposed method are verified against the numerical results in the existing literature and experimental results in 3D printing with good agreement. Based on the credence of the present method, the effects of boundary conditions, gradient distributions of GPLs and pores, microscopic parameters of GPLs and structural geometric dimension on the static and dynamic performance of the FG beams are systematically analyzed, where the bi-directional gradient distribution pattern corresponding to the GPLs along the thickness direction and the pore along the axial direction demonstrates the best improvement of both bending and dynamic performance.

Bending of Bi-directional inhomogeneous nanoplates using microstructure-dependent higher-order shear deformation theory

In contrast of experimental analysis in the mechanical behavior of the complicated structures, the progress of theoretical research has many difficulties, especially with regard to the modelling of structures. The main objective of this research is to analyze analytically the static problem of a new model of functionally graded materials, namely the “bi-coated FGM” plate. The materials are graded continuously through 2-directional by using a power law function. Two types of coated FG plates are investigated, Hardcore and Softcore FG plates. Based on the generalized field of displacement, a Quasi-3D higher-order shear deformation plate theory is proposed in this work by reducing the number of variables from six to five variables. The equilibrium equations are performed based on the principle of virtual work and solved using the Galerkin method to cover different boundary conditions. The nonlocal strain gradient theory is employed to capture the microstructure-dependent effect. The model is validated compared to recent relevant research interests. Numerical results are performed to present the impact of hardcore and softcore distributions, gradation indices, and boundary conditions on static bending deflection and stresses of bi-coated FGM plate.

Mathematical Model of Flexible Link Dynamics in Marine Tethered Systems Considering Torsion and its Influence on Tension Force

Abstract The rigidity in bending of a flexible link (is an important characteristic that should be considered during regular service conditions. The tension and bending with torsion of wire ropes are also significant factors. This study proposed a method to calculate the vectors of the generalised forces of bending of flexible links. One of the causes of torsional stresses in the power plant of underwater tethered systems is the interaction with ship equipment, such as spiral winding on the winch drum, friction on the flanges of the pulleys or winch drums, and bends on various blocks and rolls that cause torsion. The source of torsional stresses in the FL may also be related to manufacturing, storage, transportation, and its placement on the ship’s winch drums. Torsion can lead to a decrease in the tensile strength due to load redistribution between power elements, or even a violation of their structure. In some cases, torsion significantly affects the movement of the underwater tethered system as a whole. The development of a mathematical model to describe the marine tethered systems dynamics, taking into account the effect of torsion, is important and relevant. The mathematical model of the marine tethered systems dynamics was improved and solved by accounting for the generalised forces of the torsion rigidity of the flexible link, using an algorithm and computer program. The influence of the bending and torsional rigidity of the FL on its deflection and tensile strength were considered based on the example of two problems. The developed program’s working window image shows the simulated parameters and the initial position of the flexible link. The results show that torsion has almost no effect on the shape of the a flexible link’s deflection in the X0Z plane, but leads to a deviation from the X0Z plane when calculating the static deflection of the flexible link. When the carrier vessel is stationary and the submersible vehicle has no restrictions on movement and has positive buoyancy, torsion leads to a three-dimensional change in the shape of the flexible link both in the X0Z plane and in the X0Y plane. The tension force of the flexible link along its length is distributed unevenly, and the torsion of the flexible link can lead to significant changes in its shape, the trajectory of towed objects, and the forces acting on the elements of the marine tethered systems

Study Membrane Solarelasticity Using a Wave Model and a Corpuscular Model of Light

The difference between solarelastic interaction and aeroelastic interaction is illustrated from the perspective of external forces. Membrane solarelastic responses of the solar cell and solar sail are studied through a wave model and a corpuscular model of light, respectively, where the light intensity and phase are considered in the wave model to calculate the solar radiation pressure but the phase of light is neglected in the corpuscular model. The effects of the membrane optical properties, the thickness, and the size on the solarelastic flutter instability are investigated. The solar radiation pressure is divided into a part depending on the sail deformation and a part independent of sail deformation to investigate their respective influences. The results show that the former terms result in membrane flutter and the latter term results in membrane static deflection. A comparison is conducted between the wave model and the corpuscular model on the flutter boundaries and membrane responses. The membrane reflectivity is coupled with membrane stiffness by the membrane thickness in the wave model, but it is uncoupled in the corpuscular model. Therefore, the wave model has an advantage over the corpuscular model when evaluating the thickness effect of membrane reflectivity.

A model considering the longitudinal track–bridge interaction in ballasted railway bridges subjected to high-speed trains

AbstractIn this paper, a dynamic interaction model of the coupled system of railway bridge, foundation, subsoil, ballast, track, and high-speed train is presented, with special emphasis on the longitudinal interaction between the track and the bridge structure, taking into account the flexibility of the ballast. After a description of the model of this interaction system, the equations of motion are given separately for each subsystem. The discretization of the bedded rails is performed by two different approaches. In the first approach, the deflection of the rails is expanded into the eigenfunctions of a finitely long bedded beam representing the rails. In the second, simplified approach, the track response is represented by a superposition of the static deflection of the infinitely long bedded beam due to a concentrated load. The coupling of the bridge structure with the track is achieved by a component mode synthesis technique, which in the first approach leads to a representation of the equations of motion in state-space. A discrete substructure technique is used to couple this subsystem with the train model. The two presented strategies are verified by comparison with results of a finite element model of this interaction system. Several application examples reveal the influence of the horizontal track–bridge interaction and other modeling parameters on the dynamic bridge response.

Exploring nonlinearity to enhance the sensitivity and bandwidth performance of a novel tapered beam micro-gyroscope

In this paper, nonlinearity is used to enhance the sensitivity and bandwidth performance of tapered beam micro-gyroscopes. Considering the electrostatic force nonlinearity and fringing field effects, the governing equations of system are derived by Euler angular transformation and Hamilton principle. The influences of parameters such as the shape factors and DC voltage on static deflection, pull-in voltage and natural frequency of tapered beam micro-gyroscope are analyzed by the Adomian decomposition method (ADM). Galerkin discretization is performed on the governing equations of the tapered beam micro-gyroscope. Frequency-response equation of the system is deduced and solved by applying the method of multi-scale (MMS). The convergence analysis of the frequency response curve is performed, and then verified by the Runge-Kutta method. Finally, the performance of the tapered beam micro-gyroscope is analyzed in terms of bandwidth, sensitivity, and calibration curve nonlinearity under the different shape factors. The maximum amplitude of response in the driving and sensing directions increases significantly with the change of shape factors of the tapered beam in the dynamics analysis. Compared with the straight beam micro-gyroscope, the tapered beam micro-gyroscope has the higher amplitude and pronounced nonlinear characteristics. Due to the electrostatic force nonlinearity, the amplitude of motion in the driving direction has a high response over a wide frequency band, while also effectively increasing the detection bandwidth. In addition, the nonlinear micro-gyroscope has less loss of sensitivity performance during damping fluctuations. Therefore, in view of the contradiction between bandwidth and sensitivity of the traditional micro-gyroscope, this paper proposes a solution from the perspective of structural design and nonlinear characteristics of the tapered beam micro-gyroscope.

Nonlinear static/dynamic behaviors of composite curved panels with piezoelectric patches in supersonic airflow

The nonlinear aeroelastic static deflection and dynamic response of the composite curved panel with rectilinear- and curvilinear-fiber lay-up configurations under supersonic flow are investigated. The geometrical nonlinear effect of the composite curved panel is depicted by the Von-Karman strain and the aerodynamic load is incorporated through the nonlinear third-order piston theory. The Newton-Raphson method is employed to determine the static deflection of the composite curved panel and the four-order Runge-Kutta method is for the dynamic response, respectively. After examining the accuracy of the established mechanical model through numerical comparisons with existing literatures, the parametric studies focusing on panel curvature, high-order piston aerodynamic force, fiber angle orientation, temperature rise, locations and applied voltages of the piezoelectric patch on nonlinear static deformation and dynamic response are investigated. The results are deliberated in detail, and it is observed that the panel curvature and fiber angle orientation significantly affect the aeroelastic static deflection and dynamic response of the curved panel under thermal environment. Moreover, high-order aerodynamic force enlarges the static deflection and dynamic response of the curved panels with a large curvature whereas the impact of applied voltages of the piezoelectric patch will be weaker.

Analytical Calculation of Static Deflection of Biperiodic Stepped Euler–Bernoulli Beam

In this paper, we investigate the lateral deflection of a simply supported periodic stepped beam under uniform load by using an analytical method. This study considers each element of the biperiodic stepped beam as a Euler–Bernoulli beam. By using the local coordinates alongside with the boundary and continuity conditions, the different coefficients for each element caused by the jump of the bending rigidity are calculated. The continuous deflection problem of the multi-stepped repetitive beam is formulated as a linear first-order difference equation with second member. With these coefficients, the deflection at mid-span of the biperiodic beam is analytically found in exact form. This deflection is satisfactory compared to the results of a finite element model based on beam discretization techniques using Hermitian cubic shape functions. The normalized deflection at mid span converges non-monotonically towards the homogenization beam model based on equivalent homogenized stiffness.