Cell Wall Length Research Articles

Research on a Honeycomb Structure for Pyroshock Isolation at the Spacecraft–Rocket Interface

Honeycomb is a splendid kind of structure for aerospace engineering with the advantages of light weight, good shock absorption, and good structural stability. This article aims to provide methods to isolate Pyroshocks based on a honeycomb structure and guarantee the safety of such equipment against a high-frequency shock response. According to stress wave theory, an equation for stress wave transmittance of the honeycomb structure is derived considering the effect of cell wall length and thickness, where desirable honeycomb parameters are obtained. The complexity of the transfer path of the honeycomb structure is exploited to build the spacecraft–rocket interface, which could increase the impedance of the stress wave dramatically. Both finite element analysis and experiments are carried out to validate the shock isolation strategies. The influence of parameters, such as cell wall length and the thickness of the stainless-steel honeycomb, on the isolation performance is analyzed. It is revealed that the honeycomb structure has a significant effect on Pyroshock isolation performance when the wall length of the honeycomb cell is 8 mm and the thickness of the cell is 0.1 mm.

Dynamic response of the hybrid honeycomb sandwich panel with zero Poisson’s ratio under low-speed impact

The dynamic response of the hybrid honeycomb sandwich panel under low-speed impact is studied. According to Hamilton’s principle and first-order shear deformation theory, the motion equation of the hybrid honeycomb sandwich panel is deduced. Then the Navier method and Duhamel’s integral are used to solve the vibration displacement of panels. Besides, the mass-spring (MS) model is used to calculate the contact force between the honeycomb sandwich panel and the impactor. The results of this model are compared with those of Abaqus and published studies, which verifies the feasibility of the theoretical model in this paper. Based on the developed theoretical model, the influences of the unit cell angle, cell wall length, cell wall thickness, core layer height and impact speed on the dynamic response of sandwich panels have been studied. The results show that under the same low-speed impact, the maximum lateral displacement of hybrid sandwich panels was 11.65% smaller than that of conventional sandwich panels when honeycomb parameter θ = 60°, and was 15.45% when honeycomb parameter The energy absorption character of the hybrid honeycomb sandwich panel are better than those of the traditional hexagonal honeycomb sandwich panel and the concave hexagonal honeycomb sandwich panel.

Energy absorption characteristics of a super hexagonal honeycomb under out-of-plane crushing

Designing innovative honeycomb materials with improved energy absorption capacity is a long-term pursuit to provide adequate impact protection for crucial structural components. The edge-based design has been regarded as an effective strategy to enhance the mechanical properties and energy absorption of honeycomb materials. This paper proposes a novel super hexagonal honeycomb (SHH) structure to pursue better energy absorption capacities. Its geometry is derived from the outline of a hexagonal honeycomb. The crushing performance of the SHH is investigated based on the out-of-plane compression experiments, and an explicit finite element model is established to explore the crashworthiness characteristics of the structure. Furthermore, a theoretical model based on the simplified super folding element (SSFE) method is established to predict the mean crushing force during the crushing process. The comparison with the experimental and numerical results verifies its accuracy. Besides, the crushing mechanisms and collapse modes of the constitutive Y and V elements in the SHH are discussed. The parametric analysis of the crashworthiness of the SHH is conducted in terms of the cell-wall length ratio, sub-corner angle and hierarchy. The results reveal that the crashworthiness of SHH is much superior to traditional hexagonal honeycomb and presents 106%, 62%, 104% and 162% improvement in the mean crushing force, crushing force efficiency, specific energy absorption and volumetric energy absorption, respectively. This work can inspire future research on lightweight edge-based honeycombs with excellent energy absorption capacity.

Study on the Cell Magnification Equivalent Method in Out-of-Plane Compression Simulations of Aluminum Honeycomb

The large scale and long calculation times are unavoidable problems in modeling honeycomb structures with large sizes and dense cells. The cell magnification equivalent is the main method to solve those problems. This study finds that honeycomb structures with the same thickness-to-length ratios have the same mechanical properties and energy absorption characteristics. The improved equivalent finite element models of honeycomb structures with the same thickness-to-length ratios were established and validated by experiments. Based on the validated finite element model of the equivalent honeycomb structures, the out-of-plane compression behaviors of honeycomb structures were analyzed by LS-DYNA software. The results show that the performance of honeycomb structures is not equivalent before and after cell magnification. Thus, the cell magnification results were further subjected to CORA (correlation analysis) to determine the magnification time and prove the accuracy of the cell magnification time through drop-weight impact tests. In addition, a first-order decay exponential function (ExpDec1) for predicting cell magnification time was obtained by analyzing the relationship between the cell wall length and the cell magnification time.

Crashworthiness analysis of gradient fractal thin-walled structure

Inspired by the structure of bamboo, this study introduces a gradient fractal structure (GFS) design concept, which can be used in high-performance energy absorber. Crashworthiness performances of GFSs are investigated via numerical simulation. The 4-cell square thin-walled column with the cross-sectional dimensions of 36 mm × 36 mm reported in reference is used for the finite element model verification before the parametric analysis of GFS. Results show that the specific energy absorption (SEA) of 1st and 2nd order GFSs is 15% and 25% higher respectively compared to conventional square honeycombs under the same cell wall length and thickness. The gradient fractal distribution of materials not only increases the average folding number, but also lowers the force–displacement fluctuation and increases the crashworthiness performances of GFS. When the wall thickness increases, the SEA of GFS increases. However, large wall thickness will inevitably deteriorate the force–displacement fluctuation. Increasing the number of the outer components k also has similar tendency for crashworthiness performances. Simulations also suggest that reasonable wall thickness and height-side length ratio are two important factors for designing the GFSs with better energy absorption performances.

Influence of geometric parameters on free vibration behavior of an aluminum honeycomb core sandwich beam using experimentally validated finite element models

The hexagonal honeycomb core sandwich panels used in the satellite structure are subjected to severe vibration during launch. Therefore, the amounts of natural frequencies of these panels are of great importance for design engineers. Three-dimensional finite element modeling of the core considering all geometric parameters (i.e., a high-fidelity model) to achieve accurate results is not cost-effective. The honeycomb core is traditionally equivalent to a homogenized continuum core (i.e., a low-fidelity model) using simple analytical relations with ignoring the adhesive layer at the double cell-walls and radius of inclined cell-wall curvature. In this study, analytical formulations are first presented for the prediction of the equivalent elastic properties of a hexagonal aluminum honeycomb with considering all geometric parameters including adhesive layer thickness, cell-wall thickness, inclined cell-wall length, radius of inclined cell-wall curvature at the intersection, internal cell-wall angle, and honeycomb height. Then, two aluminum honeycomb core sandwich beams with free-free boundary conditions are modeled and analyzed in Abaqus finite element software, one with 3D high-fidelity core and the other with 3D low-fidelity core. In order to validate the results of the equivalent model, the modal analysis test was performed and the experimental natural frequencies were compared. The obtained results show a good agreement between the 3D low-fidelity and high-fidelity finite element models and experimental results. In addition, the influence of the above-mentioned geometric parameters has been investigated on the natural frequencies of a sandwich beam. [Formula: see text]

Design optimization of moderately thick hexagonal honeycomb sandwich plate with modified multi-objective particle swarm optimization by genetic algorithm (MOPSOGA)

Design optimization of moderately thick hexagonal honeycomb sandwich plate has been investigated via employing an improved multi-objective particle swarm optimization with genetic algorithm (MOPSOGA). Based on the first-order shear deformation theory (FSDT), governing equations of the plate are obtained. The equations are solved analytically. Total weight and maximum deflection of the plate under static gravity loads are considered to be objective functions of the problem. Core height, faces thickness, cell walls thickness, vertical and inclined cell wall length and the angle between inclined cell wall and horizontal line are set to be design variables of the problem. The geometrical and failure constrains are chosen to have desirable performance and stability of the sandwich plate. In the used multi-objective optimization technique, the optimum velocity parameter, inertia weight and acceleration coefficients for next iteration of the MOPSO are obtained by employing the genetic algorithm via minimizing generational distance between the sets of dominated and non-dominated particles in the previous iteration. Efficiency and accuracy of the proposed solution procedure are demonstrated and effects of different parameters on design optimization of the plate are studied. Also, TOPSIS multi-criteria decision-making method has been selected to report appreciate results from the Pareto-front curve of the MOPSOGA.

Are thick leaves, large mesophyll cells and small intercellular air spaces requisites for CAM?

It is commonly accepted that the leaf of a crassulacean acid metabolism (CAM) plant is thick, with large mesophyll cells and vacuoles that can accommodate the malic acid produced during the night. The link between mesophyll characteristics and CAM mode, whether obligate or C3/CAM, was evaluated. Published values of the carbon isotopic ratio (δ 13C) as an indicator of CAM, leaf thickness, leaf micrographs and other evidence of CAM operation were used to correlate cell density, cell area, the proportion of intercellular space in the mesophyll (IAS) and the length of cell wall facing the intercellular air spaces (Lmes/A) with CAM mode. Based on 81 species and relatively unrelated families (15) belonging to nine orders, neither leaf thickness nor mesophyll traits helped explain the degree of CAM expression. A strong correlation was found between leaf thickness and δ 13C in some species of Crassulaceae and between leaf thickness and nocturnal acid accumulation in a few obligate CAM species of Bromeliaceae but, when all 81 species were pooled together, no significant changes with δ 13C were observed in cell density, cell area, IAS or Lmes/A. An influence of phylogeny on leaf anatomy was evidenced in a few cases but this precluded generalization for widely separate taxa containing CAM species. The possible relationships between leaf anatomy and CAM mode should be interpreted cautiously.

Dynamic indentation of auxetic and non-auxetic honeycombs under large deformation

It is generally acknowledged that the indentation resistance or hardness of auxetic materials is higher than that of their conventional counterparts under elastic deformation. However, this property of the auxetic material may not always be superior to that of the non-auxetic materials when the deformation is relatively large with plasticity considered. In this study, we come up with an index to quantitatively depict the indentation resistance of the hexagonal honeycombs under large deformation. The indentation resistance of both the auxetic and non-auxetic hexagonal honeycombs is compared and discussed. Results show that in the premise of honeycombs possessing the same relative density, the indentation resistance of auxetic hexagonal honeycombs is not always higher than that of the non-auxetic honeycombs. This phenomenon is verified by the numerical simulations. Further analysis shows that there is a critical value of the absolute value of Poisson’s ratio, which is determined by the cell-wall length ratio, to estimate the higher indentation resistance between the auxetic and non-auxetic hexagonal honeycombs. The influence of indentation velocity is also analyzed based on numerical simulations. This present work is supposed to shed light on the design and evaluation of the indentation resistance for both auxetic and conventional honeycombs.

Dynamic crushing response of auxetic honeycombs under large deformation: Theoretical analysis and numerical simulation

In order to comprehensively understand the dynamic response of auxetic honeycombs, theoretical analysis are conducted to predict the NPR effect and the crushing stress of the re-entrant hexagonal honeycomb. The honeycomb’s crushing stress is a function of the cell’s geometric parameters, crushing velocity and the mechanical property of the cell-wall material. Results show that the crushing stress enhances with the increasing crushing velocity. A dynamic sensitivity index is employed to quantitatively evaluate this enhancement. It is shown that small cell-wall angle, low relative density or high cell-wall length ratio of the honeycomb attribute high velocity-sensitivity to the crushing stress. The Poisson’s ratio of the re-entrant honeycomb is also expressed as a function of the cell’s geometric parameters. It is revealed that the NPR effect enhances with the increasing cell-wall angle and the decreasing cell-wall length ratio. All the theoretical predictions are verified by numerical simulations. Besides, an interesting phenomenon is noticed that the crushing velocity has significant influence on the honeycomb’s NPR effect at the early stage of crushing. However, this influence almost vanishes when the overall strain is larger than about 0.2. This present work is supposed to shed light on the design of the auxetic honeycomb.

In-plane energy absorption evaluation of 3D printed polymeric honeycombs

ABSTRACTHoneycomb structures have been widely used for impact protection and energy absorption applications. However, studies on such structures have been mainly limited to metallic honeycombs with little research done on polymeric honeycomb structures. The purpose of this paper is to study the in-plane static compressive crushing behaviour and energy absorption capacity of 3D printed polymeric honeycomb structures of different unit cell thicknesses. Uniaxial quasi-static compression tests were performed on hexagonal honeycomb structures of varying wall thicknesses printed in Nylon 12 by fused deposition modelling 3D printing. Numerical simulations using ABAQUS/Explicit finite element analysis software were performed to determine the compressive behaviour of the same honeycomb structures. Gibson and Ashby’s analytical model was also used to determine the theoretical plateau stresses of the honeycombs. The experimental compressive behaviour, plateau stress and energy absorption capacity of the honeycombs were compared with numerical and analytical values. It was found that experimental results obtained are in good agreement with computational numerical solutions and with the theoretical model for all cell wall thickness to cell wall length ratios. It was also observed that the plastic deformation of hexagonal honeycombs are different in its two main in-plane directions and 3D printing can be applied to generate honeycomb structures of varying geometry and energy absorption capacity with predictable performance.

Synchrotron-based X-ray fluorescence microscopy enables multiscale spatial visualization of ions involved in fungal lignocellulose deconstruction

The role of ions in the fungal decay process of lignocellulose biomaterials, and more broadly fungal metabolism, has implications for diverse research disciplines ranging from plant pathology and forest ecology, to carbon sequestration. Despite the importance of ions in fungal decay mechanisms, the spatial distribution and quantification of ions in lignocellulosic cell walls and fungal hyphae during decay is not known. Here we employ synchrotron-based X-ray fluorescence microscopy (XFM) to map and quantify physiologically relevant ions, such as K, Ca, Mn, Fe, and Zn, in wood being decayed by the model brown rot fungus Serpula lacrymans. Two-dimensional XFM maps were obtained to study the ion spatial distributions from mm to submicron length scales in wood, fungal hyphae with the dried extracellular matrix (ECM) from the fungus, and Ca oxalate crystals. Three-dimensional ion volume reconstructions were also acquired of wood cell walls and hyphae with ECM. Results show that the fungus actively transports some ions, such as Fe, into the wood and controls the distribution of ions at both the bulk wood and cell wall length scales. These measurements provide new insights into the movement of ions during decay and illustrate how synchrotron-based XFM is uniquely suited study these ions.

Image-based correlation between the meso-scale structure and deformation of closed-cell foam

In the correlation between structural parameters and compressive behaviour of cellular materials, previous studies have mostly focused on averaged structural parameters and bulk material properties for different samples. This study focuses on the meso-scale correlation between structure and deformation in a 2D foam sample generated from a computed tomography slice of Alporas™ foam, for which quasi-static compression was simulated using 2D image-based finite element modelling. First, a comprehensive meso-scale structural characterisation of the 2D foam was carried out to determine the size, aspect ratio, orientation and anisotropy of individual cells, as well as the length, straightness, inclination and thickness of individual cell walls. Measurements were then conducted to obtain the axial distributions of local structural parameters averaged laterally to compression axis. Second, the meso-scale deformation was characterised by cell-wall strain, cell area ratio, digital image correlation strain and local compressive engineering strain. According to the results, the through-width sub-regions over an axial length between the average (lower bound) and the maximum (upper bound) of cell size should be used to characterise the meso-scale heterogeneity of the cell structure and deformation. It was found that the first crush band forms in a sub-region where the ratio of cell-wall thickness to cell-wall length is a minimum, in which the collapse deformation is dominated by the plastic bending and buckling of cell walls. Other morphological parameters have secondary effect on the initiation of crush band in the 2D foam. The finding of this study suggests that the measurement of local structural properties is crucial for the identification of the “weakest” region which determines the initiation of collapse and hence the corresponding collapse load of a heterogeneous cellular material.

Orthotropic elastic constants of 2D cellular structures with variously arranged square cells: The effect of filleted wall junctions

Cellular materials are characterized by a complex interconnected structure of struts or plates and shells, which make up the cells edges and faces. Their structure can be advantageously engineered in order to tailor their properties according to the specific application. This aspect makes them particularly attractive for the manufacturing of bone prosthetics, where the elastic modulus of the implant should match that of the bone in order to avoid loosening due to the stress shielding phenomenon. In this regard, the ability to design a component with the desired mechanical response is crucial. For this reason, the present paper evaluates the stiffness of 2D cellular structures with variously arranged square cells. Specifically, two spatial arrangements are considered: the former one is a regular square cell honeycomb, while in the latter the square cells are staggered by a prescribed offset of half of the cell wall length. An analytical model based on classical beam theory is proposed to identify the effect of stretching and bending actions on the stiffness of a single cell by applying the periodic boundary conditions. The theoretical beam model is fitted on the results from a 2D Finite Elements model based on plane elements via an extensive parametric analysis. In this way, semi-analytical formulas are proposed to calculate the stiffness in large domains of the geometric parameters: wall thickness to edge length ratio in the interval [0.04,0.20] and fillet radius to edge length ratio in the interval [0,0.15].

Two-equation method for heat transfer efficiency in metal honeycombs: An analytical solution

An analytical model based on the two-equation method is proposed in this study to evaluate the heat transfer performance in sandwiched metal honeycomb heat exchangers under forced convection conditions. The local thermal non-equilibrium between a cooling fluid and solid honeycombs is considered. The validity and the accuracy of the analytical results are verified by numerical simulation. Compared with the convectional corrugated wall, effective medium, and transfer matrix models, the present analytical predictions are closer to the finite element simulation results in a wide range of relative density. According to the analytical solutions, the effects of cell wall length, relative thickness of the heat exchanger, fluid-to-solid thermal conductivity ratio on flow characteristics, and heat transfer performance are subsequently examined to obtain the optimum design for compact heat exchangers that require high heat transfer performance. Results show the mutual influence between cell wall length and relative thickness. The use of a metal honeycomb in compact heat exchangers can significantly enhance heat transfer with a low pressure drop.

RETRACTED: The Nonlinear Compressive Response and Deformation of an Auxetic Cellular Structure under In-Plane Loading

At the request of the Author, the following article Zhang, W, Hou, W, Hu, Ping and Ma, Z (2014) The Nonlinear Compressive Response and Deformation of an Auxetic Cellular Structure under In-Plane Loading Advances in Mechanical Engineering published 17 November 2014. doi: 10.1155/2014/214681has been retracted due to errors discovered by the authors. On Page 3, the definition of component force in Equation 4 is incorrect. (2) On Page 4, the definition of component force in Equation 11 is incorrect. Moreover this equation should not have parameterM(length of cell wall), because a mistake was made in the process of calculation. Because of the above reasons, the conclusion obtained from the mechanical model is incorrect and should instead state that the Elastic Buckling and Plastic Collapse are both yield modes of this structure (3) On Page 5, the FEA model used in this article is not appropriate, because the deformation of single cell is not homogeneous, which means that the geometrical non-linear effect on single cell model is greater. So in the actual whole structure we may not obtain the results that were described in Page 6 Paragraph 2. (4) The data in figures 8 (page 6) and 9 (page 7) is incorrect and the values of effective Young’s modulus and plateau stress are much larger than reasonable values. The definition of effective stress is wrong in the FEA model, which means the effective stress should be calculated by the total width of cell instead of length of horizontal cell wall. For example, in Figure 8, the plateau stress can reach 140Mpa, this is not reasonable because there are many holes in this cellular structure, and its mechanical properties should be much lower than material properties of cell wall. The reasonable plateau stress should be around 2Mpa. The authors takes responsibility for these errors and regret the publication of invalid results. The nonlinear compressive response and deformation of an auxetic cellular structure that has periodic negative Poisson's ratio (NPR) cell were studied through theoretical and numerical method. The in-plane deformation and process of absorbing energy of this NPR cellular structure were discussed. The relative density of NPR cell was determined by geometric parameters and a theoretical approach to the strength of NPR cell was carried out to analyze the failure process under in-plane compression. Thus, to prove the effective of unit cell mechanical model, the size effect of this NPR cellular structure was discussed. The simulations of two different NPR cellular structures under in-plane compression were generated and their effective properties, strain-stress curves, and absorbed energy were analyzed. It can be concluded that the NPR cell that has lower relative density may have better capability of absorbing energy with enhanced Young's modulus. Finally the parametric analyses related to relative density were presented with NPR unit cell model and a set of quasistatic compressive experiments was carried out, which can be used to reveal the nonlinear properties of NPR cellular structure and to prove the effectiveness of theoretical method.

Dynamic crushing behavior and energy absorption of honeycombs with density gradient

This paper presents an analytical study of the in-plane dynamic crushing and energy absorption of hexagonal honeycombs with density gradients under different impact loading. Explicit dynamic finite element method simulations are carried out by using ANSYS/LS-DYNA. Firstly, under the assumption that the cell wall length is the same, a density-graded honeycomb mode is established by the variation of the cell wall thicknesses along the crushing direction. The effects of density gradient and impact velocity on the crushing deformation modes, plateau stresses and energy absorption characteristics of the specimens are explored in detail. Numerical results show that except for the impact velocity, the dynamic crushing performance and energy absorption abilities of honeycombs also rely on the density/strength gradients. The weakest layer is suggested to be placed at the impact end or the output end, and the strongest layer at the intermediate stage to achieve higher energy absorbing efficiency. According to the one-dimensional shock wave theory, the simple empirical formulae for graded honeycombs to predict the plateau stress are given under high-impact velocities. These results will provide some useful guides in the multi-objective optimization dynamic design and shock energy absorbing control of sandwich structures.

Energy absorption of sandwiched honeycombs with facesheets under in-plane crushing

AbstractThe in-plane crushing and energy absorption of sandwiched honeycomb cores with facesheets are examined through finite element simulations. Assuming no debonding between the facesheet and honeycomb core (which would be the case if manufacturing techniques such as brazing are used to produce very strong bonds between the facesheeet and the core), intracellular buckling mode for thin facesheets, and wrinkling mode for thick facesheets are observed. In the dimpling mode, deformation is governed by the core, honeycomb vertical cell walls do not deform, and the inclined wall deformation does not vary through the cell depth. In the wrinkling mode, deformation is governed by the facesheet, the vertical cell walls deform significantly, and the inclined cell wall deformation varies through the cell depth. Increasing cell angle increased Specific Energy Absorption (SEA) for honeycombs with thin facesheets. Decreasing vertical cell wall length increased SEA for honeycombs with thick facesheets. Increasing wall thickness and decreasing core depth increased SEA for honeycombs with thin and thick facesheets. With geometric changes, SEA increased ~3 times over the baseline configurations. For a given keel beam dimensions, using fewer rows of larger cells reduces the effective non-dimensional core-depth, thereby increasing the effect of the facesheet and the SEA significantly. The SEA of sandwiched honeycombs with facesheets in in-plane crushing appears to be competitive with, or better than, SEA honeycombs in out-of-plane crushing.

Application of Small-Angle X-Ray Scattering to Predict Microfibril Angle in <i>Acacia mangium</i> Wood

Partially crystalline cellulose microfibrils are wound helically around the longitudinal axis of the wood cell. A method is presented for the measurement, using small-angle X-ray scattering (SAXS), of the microfibril angle, (MFA) and the associated standard deviation for the cellulose microfibrils in the S2 layer of the cell walls of Acacia mangium wood. The length and orientation of the microfibrils of the cell walls in the irradiated volume of the thin samples are measured using SAXS and scanning electron microscope, (SEM). The undetermined parameters in the analysis are the MFA, (M) and the standard deviation (σФ) of the intensity distribution arising from the wandering of the fibril orientation about the mean value. Nine separate pairs of values are determined for nine different values of the angle of the incidence of the X-ray beam relative to the normal to the radial direction in the sample. The results show good agreement. The curve distribution of scattered intensity for the real cell wall structure is compared with that calculated with that assembly of rectangular cells with the same ratio of transverse to radial cell wall length. It is demonstrated that for β = 45°, the peaks in the curve intensity distribution for the real and the rectangular cells coincide. If this peak position is Ф45, Then the MFA can be determined from the relation M = tan-1 (tan Ф45 / cos 45°), which is precise for rectangular cells.

Estimation of the Cellulose Microfibril Angle in Acacia mangium Wood Using Small-Angle X-Ray Scattering

The term microfibril angle, MFA in wood science refers to the angle between the direction of the helicalwindings of cellulose microfibrils in the secondary cell wall, S2 layer of fibres and the long axis of the cell wall.A method is presented for the measurement, using small-angle X-ray scattering (SAXS), of the microfibril angle,(MFA) and the associated standard deviation for the cellulose microfibrils in the S2 layer of the cell walls ofAcacia mangium wood. The length and orientation of the microfibrils of the cell walls in the irradiated volume ofthe thin samples are measured using SAXS and scanning electron microscope, (SEM). The undeterminedparameters in the analysis are the MFA, (M) and the standard deviation (σ?) of the intensity distribution arisingfrom the wandering of the fibril orientation about the mean value. Nine separate pairs of values are determinedfor nine different values of the angle of the incidence of the X-ray beam relative to the normal to the radialdirection in the sample. The results show good agreement. The curve distribution of scattered intensity for thereal cell wall structure is compared with that calculated with that assembly of rectangular cells with the sameratio of transverse to radial cell wall length. It is demonstrated that for β = 45° (where β is the angle between theplane face of the wood samples and the radial direction) the peaks in the curve intensity distribution for the realand the rectangular cells coincide. If this peak position is ?45, Then the MFA can be determined from the relationM = tan-1 (tan ?45 / cos 45°), which is precise for rectangular cells.