Axial Moment Research Articles

Engineering method for determining the axial moment of inertia of a circular segment

Engineering method for determining the axial moment of inertia of a circular segment

On some methods of determining proper frequencies torsional oscillations in metal-working machines

Forced torsional vibrations that occur in machine tool drives are one of the pressing problems of modern machine tool industry. The reasons for these fluctuations may be imbalances of rotating masses, structural imperfections of bearing units, operating or technological process features (for metal-cutting machines – eccentric clamping of the workpiece, processing of non-round workpieces, etc.). In this work, using the example of a real metal-cutting machine, the task of dynamic analysis of the drive of the main movement of the machine with determination of the natural frequencies of torsional vibrations is realized. The real object is replaced by a dynamic model (DM). When constructing the DM, the inertial and rigidity characteristics are determined from the conditions: – the kinetic energy of the object is equivalent to the kinetic energy of the DM (from this condition the axial moments of inertia of the concentrated masses of the row model are obtained); – the potential energy of elastic deformations of the object is equivalent to the potential energy of deformation of the DM (from this condition the rigidity coefficients of the DM are obtained). The natural frequencies of the row DM were determined in three ways: 1) by solving the frequency equation (exact method) in combination with the method of reducing the number of masses of A.P. Cherevkov; 2) partial systems method (fundamental frequency only); 3) method of residuals (Tolle). The following conclusions were made: 1) the main (smaller) frequency can be found by any of the methods considered (since the results of the three methods coincided); 2) when simplifying the DM using the method of partial systems, the criterion for the possibility of further simplifications over the specified frequency range should be taken into account; the transition from a three-mass DM to a two-mass one if the criterion in this calculation was not met led to an error in determining the value of the fundamental frequency of about 10%; 3) dynamic calculations of machines using DM and methods for determining natural frequencies make it possible, based on known frequencies of disturbing forces, to identify possible near-resonant operating modes and avoid them in advance by making changes to the design of the machine at the design stage

Eccentric compressive behavior of T-shaped welded multi-partition composite members

Six T-shaped welded multi-partition composite members were tested under eccentric compression loading by considering the influence of eccentricity and eccentric angle. The test results showed that no weld tearing occurred when the specimen was damaged, indicating that the two vertical welds could ensure the overall working performance of the specimen. The specimens exhibited the smallest bearing capacity when failing along the asymmetric axis, which was the most dangerous situation. The parametric analysis is conducted by a verified finite element model considering the influence of the slenderness ratio, section height-to-width ratio, section depth-to-width ratio, concrete and steel grade, steel ratio, and eccentric angle. If concrete contributes a larger proportion to the bearing capacity, the interaction curve of axial force and bending moment tends to protrude. Conversely, the interaction curve of axial force and bending moment gradually shrinks as the slenderness ratio increases. Based on this observation, a calculation method of T-shaped welded multi-partition composite members under eccentric compressive loading along the asymmetric axis is proposed by considering the effect of the slenderness ratio. Formulas for bearing capacity under biaxial eccentric compression are proposed based on the results of the parametric analysis.

Local buckling of the bottom flange in steel-concrete composite beams with trapezoidal corrugated webs subjected to end restraint at high temperatures

Steel-concrete composite beams with trapezoidal corrugated webs (SCBCW) are prone to rapid failure at high temperatures due to local instability of the bottom flange. The failure mechanism of SCBCW, which is an integral part of the holistic structure and subject to axial and rotational restraints from adjacent members at elevated temperatures, differs from conventionally simply supported steel-concrete composite beams with straight webs. To investigate the evolution of local buckling in the compressed flange of SCBCW under high temperature conditions, a composite beam model with frame restraints was developed using ABAQUS finite element software. In this model, the composite beam is connected to restraint frame columns through high-strength bolts. The changes in axial stress, bending moment, and deflection of the compressed flange of SCBCW were analyzed as temperature increased. Results indicate that at elevated temperatures, additional bending moments occur in the flange plane due to axial restraints, section temperature gradients, section negative bending moments and accordion effect caused by corrugated web on compressed bottom flange of SCBCW. This leads to a gradual increase in axial compressive stress before local buckling occurs followed by a decrease due to internal force redistribution upon onset of buckling. Studies show that initial buckling of bottom flange occurs on one side of flat section of web where axial compressive stress is relatively high while restraint effect on web is low; Subsequent local buckling of the beam then appears on opposite side adjacent flat section of web after initial buckle occure on one side. After local buckling appears on both sides of web tension bands appear on both sides where bottom flange experiences localized bucking. The setting of longitudinal stiffeners (LS) can effectively improve the local stability of the bottom flange of the composite beam, while the setting of transverse stiffeners (TS) has no significant impact.

Progressive Collapse Behavior of a Precast Reinforced Concrete Frame System with Layered Beams

A possible way to improve the structural safety and robustness of precast building structures is to develop effective precast frame systems with layered beams, which combine prefabricated parts with cast-in situ ordinary concrete, high-performance concrete, fiber concrete, or FRP. The paper provides a new type of precast reinforced concrete frame system with layered beams for rapidly erected multi-story buildings resistant to accidental actions. Using a combination of the variational method and two-level design schemes, a simplified analytical model has been developed for structural analysis of the precast reinforced concrete frame system, both for serviceable and ultimate limit states as well as for accidental actions. The proposed model allows for determining shear deformations and the formation and opening of longitudinal cracks in the intermediate contact zone between precast and monolithic parts of reinforced concrete structural elements of the frame, as well as the formation and opening of normal cracks because of the action of axial tensile force or bending moment in these elements. The design model was validated by comparing the calculated and experimental data obtained from testing scaled models of the precast reinforced concrete frame system with layered beams. The paper investigates and thoroughly analyzes the factors affecting the stiffness and bearing capacity of the intermediate contact zone, discusses the criteria for the formation of shear cracks along the contact zone of precast and monolithic concrete, and examines the change in the stiffness and dissipative properties of layered elements at different stages of their static–dynamic loading. The robustness of the experimental models of the structural system was not ensured under the specified load, section dimensions, and reinforcement scheme. Following an accidental action, longitudinal cracks were observed in the contact joint between the monolithic and prefabricated parts in the layered beams. This occurred almost simultaneously with the opening of normal cracks in adjacent sections. A comprehensive analysis of the results indicated a satisfactory degree of agreement between the proposed semi-analytical model and the test data.

Shear buckling resistance and intermediate transverse stiffeners internal forces of full-scale steel plate girders – Experimental tests and numerical investigation

This paper presents the results from two full-scale transversally stiffened plate girder’s load tests in shear and bending. The plate girder’s internal stiffener forces and shear buckling resistance are estimated using the web and stiffener measured strains. These results show that the reported axial compression and bending moment of the intermediate transverse stiffeners for the tested plate girder are 70% of the FprEN 1993–1-5: 2022 [1] design force. The two full-scale plate girders fail due to buckling shear. Nonlinear FEM analyses based on the test girder geometry, measured initial geometric imperfections, and the material properties from the tensile tests, show excellent agreement with the test results, with less than 5% on the maximum applied forces and related displacements. The principal membrane stresses path that triggers the failure of the web plate are investigated based on numerical results and compared with the test measurements. Contrary to the assumption of a rotated stress field model, the principal compression stress obtained in the experimental test was higher than shear buckling stress in the vicinity of the intermediate transverse stiffener.

Comparison between the Dynamic Responses of Steel Buildings with Medium and Deep Columns under Several Seismic Intensities

Structural engineers often use deep columns in high seismic areas to reduce drifts, yet this somehow contradicts what is stated in some tests in the sense that even though deep columns may satisfy current seismic provisions, they can suffer premature twisting; this is an indication that a lot of research is needed in this area. Numerical and experimental studies have been conducted to estimate the response of steel buildings with medium and deep columns under the action of static and cyclic loading; however, studies accounting for the dynamic characteristics of buildings and strong motions are not common. In addition, responses in terms of local parameters have not been considered either. In this study, the nonlinear seismic responses of steel buildings with perimeter moment-resisting frames and medium (W14) columns are numerically calculated and compared to those of similar steel buildings with equivalent deep columns in terms of cost (W27 and larger). Low-, mid-, and high-rise steel building models with different dynamic characteristics, as well as several strong motions with different frequency contents, are considered. Results indicate that the drifts of the models with medium columns may be up to 140% greater than those of the models with deep columns. Significant reductions are also observed for top displacements, normalized interstory shears, and combined normalized axial loads and bending moments. Hence, the seismic demands of the buildings with deep columns may be much smaller than those of the buildings with medium columns and, therefore, the buildings with deep columns exhibit a superior behavior, which results in more economical designs. The reduction is greater for the case of low- and mid-rise buildings than for high-rise buildings. One of the reasons for this is that as medium columns are replaced by deep columns, the stiffness and the strength increase, which are lower in the tallest model.

Numerical analysis of buried pipelines under seismic slope instability

Damage to buried pipes under seismic landslide actions has been reported in many post-earthquake reconnaissance. The landslide-pipe problem in the technical literature has been often investigated using simplified analytical methods. However, the analytical methods ignore the real mechanism of pipe response under natural dynamic slope instability. The dynamic slope instability is significantly influenced by its lateral boundary interface (LBI) characteristics. In this study, slope-pipe interaction (SPI) under seismic loading, focusing on the effect of LBI properties, is evaluated by continuum numerical simulation using the SANISAND constitutive model in FLAC3D. The results show that the geometry of the failure mass varies from 2D to 3D by increasing the stiffness at the slope boundaries (from smooth to hard) and the maximum pipe deformation decreases by around 40%. Moreover, the response components of maximum axial stress, bending moment, and shear stress of the pipe occur at the end sections of the buried pipe and near the boundaries of the landslide zone. However, the maximum pipe deflection occurs in the middle section of the pipe. The results of shear force-shear displacement curves demonstrate that the soil-pipe interaction stiffness is variable along the pipe length and can be estimated by a hyperbolic equation.

Optimization of Hanger Spacing of Steel Arch Bridges Using Dynamic Loads

Bridges are basic infrastructure that must be met to create regional connectivity in Indonesia. One type of bridge that is often used is a curved bridge, which has the advantages of high strength, attractiveness, aesthetics and economy. In order to accelerate the development of bridge infrastructure, an efficient innovation in curved bridge design is needed. The development of curved bridge structures to achieve efficient designs has received much attention in several decades. However, researchers have only focused on optimising the geometry variation of the arch height. Therefore, the aim of this research is to innovate the optimisation of the hanger spacing on the arch bridge structure. In order to obtain optimal results, a bridge model is carried out by varying the hanger spacing of the centre model with a hanger spacing ratio of (1.3 - 1.1 - 0.9 - 0.7), a flat model with a hanger spacing ratio of (1 - 1 - 1 - 1) and an edge model with a hanger spacing ratio of (0.7 - 0.9 - 1.1 - 1.3), so that from the three models, the effect of hanger location on three conditions is obtained. Each model is modelled in the SAP2000 software and given a bridge service load to obtain the internal forces and deflections that occur. The output of the internal force and deflection is then analysed to determine the effect of the location of the bridge service hanger. The serviceability of the bridge is also analysed by calculating the ratio between the weight of the bridge and the deflection that occurs. The results of the analysis show that the location of the hanger affects the performance of the arch bridge structure. The centre model bridge design produces the most efficient structural performance in resisting the compressive axial forces and moments that occur, and produces the least deflection. Meanwhile, the edge model will provide the most efficient structural performance in resisting tensile axial forces. By referring to the results of the bridge weight to deflection ratio analysis, it can be concluded that the centre model produces the most efficient structural design when compared to other curved bridge models.

Enhancing Structural Capacity Assessment with a Novel Failure Decision Function for Rectangular Reinforced Concrete Columns

This study introduces the Failure Decision Function, a novel approach for evaluating the structural capacity of rectangular reinforced concrete columns under axial forces and moments, both uniaxial and biaxial. The method simplifies existing practices, enhancing accuracy and integration into design software. The methodology hinges on deriving exact biaxial bending failure surfaces, utilizing integral expressions based on material properties and cross-sectional geometry. This direct integration process uncovers failure surface characteristics previously undocumented. Results confirm the utility of the Failure Decision Function through comparative analysis with established literature, showcasing its potential for simplifying and improving structural capacity assessments. The analytic procedure developed enables efficient computation of failure surfaces, streamlining the inclusion of these functions in structural engineering software in two key ways: (1) compiling a library of pre-calculated functions for quick capacity checks and (2) creating a dynamic application that generates these functions based on specific design parameters, allowing users to explore various load and moment scenarios. In conclusion, the Failure Decision Function represents a significant advancement in structural engineering design, offering an accurate and user-friendly method for assessing column performance under critical loading conditions.

Nonlinear Slippage of Tensile Armor Layers of Unbonded Flexible Riser Subjected to Irregular Loads

The unbonded flexible riser has been increasingly applied in the ocean engineering industry to transport oil and gas resources from the seabed to offshore platforms. The slippage of helical layers, especially the tensile armor layers of unbonded flexible risers, contribute to the nonlinear hysteresis phenomenon, which is a research hotspot and difficulty. In this paper, on the basis of a typical eight-layer unbonded flexible riser model, the nonlinear slippage of a tensile armor layer and the corresponding nonlinear behavior of an unbonded flexible riser subjected to irregular external loads are studied by numerical modeling with detailed cross-sectional properties of the helical layers, and are verified through a theoretical method considering the coupled effect of the external loads on the unbonded flexible riser. Firstly, the balance equation of each layer considering the effect of external loads is established based on functional principles, and the overall theoretical model of the unbonded flexible riser is set up in consideration of the contact between adjacent layers. Secondly, the numerical modeling of each separate layer within the unbonded flexible riser, including the actual geometry of the carcass and pressure armor layer, is established, and solid elements are applied to all the interlayers, thus simulating the nonlinear contact and friction between and within interlayers. Finally, after verification through test data, the behavior of the unbonded flexible riser under the cyclic axial force, torsion, bending moment, and irregular external and internal pressure is studied. The results show that the tensile armor layer can slip under irregular loads. Additionally, some suggestions related to the analysis of unbonded flexible risers under irregular loads are drawn in the end.

A recursive trapezoid-based algorithm designed to compute the strength- and stiffness-related geometric properties of beams with polygonal cross-sections

A new recursive trapezoid-based algorithm is proposed for calculating the strength- and stiffness-related geometric characteristics of polygonal beam sections. By taking as input the coordinate vertices of the polygon, the approach may effectively be applied to determine the cross-section area and the position of its centroid along with the axial area moments of inertia and product of inertia relative to the cross-sections’ centroidal axes. As the method combines the theory of planar moments with the well-known trapezoid numerical method, it provides a straightforward tabular procedure that can be easily implemented using any spreadsheet software. In addition, aiming to determine the principal area moments of inertia, a computation matrix formalism based on the theory of orthogonal diagonalization of symmetric matrices is put forward as an alternative to the classical method of vanishing the first derivative and solving for the principal directions. Some benchmark examples are worked out to prove the relevance and accuracy of the proposed approach for several beam sections ranging from the most common triangular and quadrilateral ones to very complex open and closed polygonal profiles. The solution presented here appears to be virtually foolproof compared to the conventional one, as it does not necessitate the decomposition of complex polygonal cross-section area into elementary surfaces of different shapes, then use the parallel axes theorem, and so forth; rather, it requires only insertion of the coordinates of the vertices of the polygon, turn by turn in the clockwise order, relative to an arbitrary cartesian coordinate system, and no further steps.

Analysis of lumbar spine loading during walking in patients with chronic low back pain and healthy controls: An OpenSim-Based study.

Low back pain (LBP) is one of the most prevalent and disabling disease worldwide. However, the specific biomechanical changes due to LBP are still controversial. The purpose of this study was to estimate the lumbar and lower limb kinematics, lumbar moments and loads, muscle forces and activation during walking in healthy adults and LBP. A total of 18 healthy controls and 19 patients with chronic LBP were tested for walking at a comfortable speed. The kinematic and dynamic data of the subjects were collected by 3D motion capture system and force plates respectively, and then the motion simulation was performed by OpenSim. The OpenSim musculoskeletal model was used to calculate lumbar, hip, knee and ankle joint angle variations, lumbar moments and loads, muscle forces and activation of eight major lumbar muscles. In our results, significant lower lumbar axial rotation angle, lumbar flexion/extension and axial rotation moments, as well as the muscle forces of the four muscles and muscle activation of two muscles were found in patients with LBP than those of the healthy controls (p < 0.05). This study may help providing theoretical support for the evaluation and rehabilitation treatment intervention of patients with LBP.

A phase-field model for the brittle fracture of Euler–Bernoulli beams coupling stretching and bending

Damage gradient models seek to simulate fracture mechanics by modulating the material stiffness. Within this framework, a singular scalar field representing damage is commonly utilized to globally decrease elastic energy. However, when considering structural models like beams and plates, this approach often fails to adequately capture important aspects due to the interaction between stretching and bending contributions. In this work, we propose a model for planar Euler–Bernoulli beams that incorporates the following features: firstly, we utilize two phase-field damage parameters to describe material damaging, specifically addressing the ‘erosion’ occurring above and below the original, undamaged beam; secondly, we assume a simple linear dependence of the axial stress field on the thickness coordinate, along with linear dependence on the axial force and bending moment. By appropriately identifying the constitutive response, our model effectively considers the coupling between stretching and bending induced during through-the-thickness damage, demonstrating good agreement with 2D observations.

Shear performance of tapered box girders with corrugated steel webs in extradosed cable-stayed bridges under cable forces

The integration of box girders with corrugated steel webs (CSWs) and an extradosed cable-stayed design gives rise to an innovative and distinctive bridge design, recognized as the extradosed cable-stayed bridge with CSWs. The synergistic integration of the technical benefits facilitates expanding the maximum span and enhancing the adaptability of this structural configuration. Subjected to cable forces, the main bridge superstructure featuring a box girder with CSWs experiences vertical shear, substantial axial force, and bending moments. Shear buckling on the CSWs emerges as a prominent issue influencing design considerations. In response to heightened internal forces near support, the main girder is typically configured with variable height. The Resal effect in the tapered girder leads to additional shear stresses from axial forces and bending moments, consequently modifying the effective shear stress on the CSWs. Therefore, traditional basic calculation assumptions that solely account for the vertical component of cable forces in shear stress calculations are frequently inaccurate. This study presents a modified theoretical method for calculating shear stress in tapered box girders with CSWs subjected to cable forces by considering the Resal effect. The proposed modified method incorporates not only the vertical shear but also considers bending moments and axial forces. The finite element simulation and experimental results showcase its remarkable accuracy, concurrently validating the presence of the Resal effect in tapered box girders with CSWs under cable forces. Additionally, this study quantitatively analyzes the shear force redistribution phenomenon induced by vertical shear, axial force, and bending moments considering the Resal effect. This reveals the shear performance of tapered box girders with CSWs under cable forces. The study is expected to provide a strategy and theoretical guideline for rapidly and efficiently obtaining the shear stresses of tapered box girders with CSWs in extradosed cable-stayed bridges under cable forces.

Eccentric compression behaviour of normal and ultra-high performance concrete-filled stainless steel tube stub columns

To investigate the mechanical properties of normal concrete-filled stainless steel tubes (CFSST) and ultra-high performance concrete-filled stainless steel tubes (UFSST) under eccentric compression, 6 axial compression specimens and 18 eccentric compression specimens were designed. The effect of different concrete strengths, eccentricity and diameter-to-thickness ratios (D/t) on the damage modes, load-displacement curves, load-strain curves, and lateral deflection distribution curves of the specimens were investigated. The test results showed that stainless steel tubes exhibited different degrees of buckling after failure. The load-displacement curve of CFSST presents a slight upward trend after the elastic-plastic phase, whereas that of the UFSST enters a slowly decreasing phase. And the increase in eccentricity leads to a decrease in the load bearing capacity, ductility, and initial stiffness of the specimen. As the D/t decreases, the load carrying capacity of the specimen increases. Additionally, parametric analysis was conducted on CFSST and UFSST specimens, including the eccentricity, concrete strength, stainless steel yield strength, and D/t. Compared with the design codes of EC 4, AISC 360, and GB/T 51446, the ultimate axial force-bending moment (Nu-Mu) curve was inaccurate. Thus, a prediction model for the Nu-Mu curve of CFSST and UFSST stub columns was proposed, and the predicted results of the proposed model were in good agreement with the experimental results.

Axial compressive stability design of longitudinal asymmetrical shuttle-shaped columns

Longitudinal asymmetrical shuttle-shaped column (LA-SSC) has found its wide application when subjected to combined axial compression and horizontal point load in practical engineering. This paper focuses on elastic and elasto-plastic buckling behaviors of the LA-SSC only under axial compression and the resulting outcome could provide an indispensable basis for further establishing buckling strength prediction of LA-SSC under axial compression and horizontal point load. At first, the elastic buckling behavior of tapered columns (TC) and longitudinal symmetrical shuttle-shaped columns (LS-SSC) are derived in theory and verified by finite element numerical results, and corresponding elastic buckling modes are discussed. Subsequently the elastic buckling load prediction of LA-SSCs is established first and the corresponding normalized slender ratio, as a key design parameter, is generalized. Finally, the stability capacity of LA-SSCs is involved in theoretical analysis and finite element (FE) numerical calculation under axial pressure. Through investigating the axial load-lateral displacement curves and axial stress distributions the buckled LA-SSCs, the effects of different parameters (such as overall length, plate thickness, and steel strength) on the axial compression stability capacity of LS-SSCs are explored. As a result, the complete design formulas for axial compression stability capacity of LA-SSCs are presented first and compared with current major design codes in various countries. It is observed that the proposed strength design formulas are in good correspondence with the FE numerical results. The proposed formulas could provide references and guidance for the axial compression buckling strength of LA-SSCs under axial compression and they further supply fundamentals for establishing the strength design recommendation of LA-SSCs under axial compression and bending moments.

Failure Mechanism and Structural Safety Assessment of the Primary Support Structure of Soft Rock Tunnel: A Case Study

Based on a soft rock tunnel in a mountainous area of northwest Yunnan Province, a refined numerical model of different support forms considering structural interaction is established to comprehensively evaluate the structural failure mechanisms and mechanical responses of different support forms. Research shows that compared to steel rib structures, the axial force and bending moment of sprayed concrete structures is larger under different combinations of support structures. Compared with a single-layer support structure, the stress distribution at each position of the double-layer I-shaped steel structure is more uniform, and the sprayed concrete structure only experiences compression damage at the corner of the side wall. As the strength of sprayed concrete increases, the stress distribution of sprayed concrete at the arch waist becomes more uniform. The stress concentration state of sprayed concrete at the corner of the tunnel wall and the plastic yield state of the steel rib structure have also been improved. The higher the concrete strength, the lower the stress ratio on the steel rib’s inner and outer sides.

Experimental and FE analysis of the behavior of steel column base connections under tension and bending moment

Most design procedures represent the steel base connection by assuming either a perfectly hinged or a fully rigid connection. However, these kinds of connections are, in fact, semi-rigid or flexible and their actual behavior is complicated. Most of the previous studies investigated these base connections experimentally under moments and axial compression only, which is completely different in the case of axial tension and moment. Therefore, experimental and numerical investigations with variable base plate thicknesses and diameters and arrangements of the anchor bolts were carried out to verify the behavior of this kind of base connection under axial tension and moment. Full-scale steel columns with hinged base plate connections were tested under the effect of two simultaneous actions, axial tension force and bending moment. Moreover, Finite Element (FE) models were developed and verified against the experimental results to conduct a parametric study. The investigated parameters were the base plate thickness and anchor bolt diameter and arrangement. The base plate connections under the effect of axial tension with moment instead of axial compression caused reductions in the rotational stiffness up to 99%. In addition, using only two bolts in this type of connection concentrated stresses on the base plate that led to big deformations. The small base plate thickness deformed easily under the effect of tension force which induced big initial gaps between the steel base plate and the Base beam and subsequently decreased the moment resistance and rotational stiffness of the base plate connection. The big elongation of the anchor bolts with small diameters increased the gap between the steel plate and the Base beam.

Elastic-plastic response for the foam-filled sandwich circular tube under internal blast loading

The dynamic response and the blast resistance of an aluminum foam-filled sandwich circular tube under internal blast loading was investigated theoretically. The deformation of the structure is decoupled into three phases corresponding to blast loading interacting with the internal tube, foam core crushing and external tube deformation. During the process, the theoretical model includes both the circumferential membrane forces and the axial moment components and considers an elastic-plastic strengthening model for internal/external tubes. The proposed model is tested against the experimental results for the mid-point deflection of the external/internal tubes. The influence of the physical and geometrical parameters of the structure on their deformation is investigated. It is shown that the dimensionless mid-point deflection of the external tube decreases as the tangent modulus of the internal/external tube increases, while the tangent modulus of the internal tube has a larger effect on the mid-point deflection of the external tube. The minimum mass of a sandwich circular tube can be found when the deformation of the external tube is specified. The research results can provide theoretical basis for lightweight structure design.