Elastic beam deformation analysis method and system based on fourth-order nonlinear differential equation

Abstract

This invention belongs to the field of mechanical analysis technology and provides a method and system for elastic beam deformation analysis based on fourth-order nonlinear differential equations. The method includes: obtaining the stress-strain relationship and lateral displacement function of the elastic beam; constructing a fourth-order nonlinear mechanical response function of the elastic beam based on the obtained stress-strain relationship and lateral displacement function; solving the constructed fourth-order nonlinear mechanical response function to identify the local buckling location and higher-order shear effects of the elastic beam; taking the minimum weight of the elastic beam as the objective function, considering the displacement and stress constraints of the elastic beam, performing sensitivity analysis of the elastic beam based on the obtained local buckling location and higher-order shear effects, adjusting the nonlinear coefficient of the elastic beam, and completing the elastic beam deformation analysis based on fourth-order nonlinear differential equations.

Description

Technical Field

[0001] This invention belongs to the field of mechanical analysis technology, specifically relating to a method and system for analyzing the deformation of elastic beams based on fourth-order nonlinear differential equations. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Currently, elastic beams are widely used in engineering. For example, in building structures, elastic beam theory is used in foundation design to bear and transfer the loads of the superstructure, improving the overall performance of the building and adapting to ground deformation and external forces such as earthquakes to enhance its seismic performance. In underground engineering, elastic beam theory can be used in the foundation design of underground structures such as tunnels and subways to ensure the stability and safety of underground structures. In transportation engineering, elastic foundation beam theory is widely used in main beams, approach bridges, and ramps to ensure the stability and safety of the overall bridge structure. In hydraulic engineering, elastic foundation beam theory can be applied to the foundation design of structures such as offshore platforms and offshore wind power to improve the structure's resistance to wind and waves. In the design of large-span buildings and cable-stayed bridges, elastic foundation beam theory helps to reveal the interaction mechanism between the foundation and the beam, predict the vibration response of the structure, thereby optimizing structural design and reducing the impact of natural disasters such as earthquakes.

[0004] Traditional elastic beam theory employs second-order partial differential equations, which cannot accurately describe the mechanical behavior under large deformations, high loads, or material nonlinearity. Furthermore, existing methods neglect higher-order deformations such as torsional effects and bending-torsional coupling, leading to safety hazards or excessive redundancy in structural design. While elastic beam theory can use second-order partial differential equations (such as the bending equation of a beam), it cannot be used to analyze and study elastic beams under large deformations, high loads, or material nonlinearity. Summary of the Invention

[0005] To address the aforementioned issues, this invention proposes a method and system for analyzing the deformation of elastic beams based on fourth-order nonlinear differential equations. Considering higher-order deformation effects and nonlinear mechanical effects, the method analyzes elastic beams based on fourth-order nonlinear differential equations, achieving a balance between lightweighting and safety in elastic beams, and providing a theoretical basis for complex engineering structural components.

[0006] According to some embodiments, the first aspect of the present invention provides a method for analyzing the deformation of an elastic beam based on a fourth-order nonlinear differential equation, employing the following technical solution:

[0007] A method for analyzing the deformation of an elastic beam based on a fourth-order nonlinear differential equation includes:

[0008] Obtain the stress-strain relationship and lateral displacement function of the elastic beam;

[0009] Based on the obtained stress-strain relationship and transverse displacement function, the fourth-order nonlinear mechanical response function of the elastic beam is constructed.

[0010] Solve the constructed fourth-order nonlinear mechanical response function to identify the local buckling location and higher-order shear effects of the elastic beam;

[0011] Taking the minimum weight of the elastic beam as the objective function, considering the displacement and stress constraints of the elastic beam, the sensitivity analysis of the elastic beam is carried out based on the obtained local buckling position and higher-order shear effect. The nonlinear coefficient of the elastic beam is adjusted to complete the deformation analysis of the elastic beam based on the fourth-order nonlinear differential equation.

[0012] As a further technical limitation, a hyperelastic constitutive model is employed to obtain the stress-strain relationship of the elastic beam, i.e. ;in, and These represent the stress and strain of an elastic beam, respectively. These are the parameters of the hyperelastic constitutive model; For the nonlinear effect of the elastic beam; the obtained transverse displacement function is: u ( t That is, the elastic beam in position The dimensionless deformation at the location.

[0013] Furthermore, the fourth-order nonlinear mechanical response function of the constructed elastic beam is: Based on the constructed fourth-order nonlinear mechanical response function, the relationship between the deformation of the elastic beam, including the influence of displacement and derivatives of each order, and the load and material nonlinearity is described. The local buckling and shear deformation of the elastic beam are analyzed through the second-order and third-order differential equations of the transverse displacement function.

[0014] As a further technical limitation, the Galerkin method basis functions are used to characterize the vibration mode shape of the elastic beam. The obtained transverse displacement function is expanded, and the fourth-order nonlinear mechanical response function is transformed into a system of nonlinear algebraic equations. The fourth-order derivative terms are discretized by the finite difference method, and the system of nonlinear algebraic equations is solved iteratively.

[0015] As a further technical limitation, the curvature abrupt change point is identified by the second derivative in the solution of the fourth-order nonlinear mechanical response function, and the local buckling location of the elastic beam is identified based on the curvature abrupt change point; the influence of higher-order shear effects is quantified by combining the fourth-order model, and the shear deformation of the elastic beam is evaluated.

[0016] As a further technical limitation, the objective function adopted is to minimize the beam weight. ;in, For cross-sectional area, This represents the material density.

[0017] According to some embodiments, the second aspect of the present invention provides an elastic beam deformation analysis system based on a fourth-order nonlinear differential equation, employing the following technical solution:

[0018] A deformation analysis system for elastic beams based on fourth-order nonlinear differential equations includes:

[0019] The acquisition module is configured to acquire the stress-strain relationship and lateral displacement function of the elastic beam;

[0020] The construction module is configured to construct the fourth-order nonlinear mechanical response function of the elastic beam based on the obtained stress-strain relationship and lateral displacement function.

[0021] The identification module is configured to solve the constructed fourth-order nonlinear mechanical response function to identify the local buckling location and higher-order shear effects of the elastic beam.

[0022] The analysis module is configured to take the minimum weight of the elastic beam as the objective function, consider the displacement and stress constraints of the elastic beam, perform sensitivity analysis of the elastic beam based on the obtained local buckling position and higher-order shear effect, adjust the nonlinear coefficient of the elastic beam, and complete the deformation analysis of the elastic beam based on the fourth-order nonlinear differential equation.

[0023] According to some embodiments, a third aspect of the present invention provides a computer-readable storage medium, employing the following technical solution:

[0024] A computer-readable storage medium having a program stored thereon, which, when executed by a processor, implements the steps in the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in the first aspect of the present invention.

[0025] According to some embodiments, the fourth aspect of the present invention provides an electronic device, which adopts the following technical solution:

[0026] An electronic device includes a memory, a processor, and a program stored in the memory and running on the processor. When the processor executes the program, it implements the steps in the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in the first aspect of the present invention.

[0027] According to some embodiments, the fifth aspect of the present invention provides a computer program product, which adopts the following technical solution:

[0028] A computer program product includes software code, wherein the program in the software code performs the steps of the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in the first aspect of the present invention.

[0029] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0030] This invention considers higher-order deformation effects and nonlinear mechanical effects, and analyzes elastic beams based on fourth-order nonlinear differential equations to achieve a balance between lightweighting and safety of elastic beams, providing a theoretical basis for complex engineering structural components. Attached Figure Description

[0031] The accompanying drawings, which form part of this embodiment, are used to provide a further understanding of this embodiment. The illustrative embodiments and their descriptions are used to explain this embodiment and do not constitute an improper limitation of this embodiment.

[0032] Figure 1 This is a flowchart of the elastic beam deformation analysis method based on fourth-order nonlinear differential equations in Embodiment 1 of the present invention;

[0033] Figure 2 This is a structural block diagram of the elastic beam deformation analysis system based on fourth-order nonlinear differential equations in Embodiment 2 of the present invention. Detailed Implementation

[0034] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0035] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0036] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0037] In this invention, terms such as "upper," "lower," "left," "right," "front," "back," "vertical," "horizontal," "side," and "bottom" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. These terms are used only to facilitate the description of the structural relationships of the various components or elements of this invention and do not specifically refer to any component or element in this invention. They should not be construed as limiting the invention.

[0038] In this invention, terms such as "fixed connection," "connected," and "linked" should be interpreted broadly, indicating a fixed connection, an integral connection, or a detachable connection; a direct connection or an indirect connection through an intermediate medium. Those skilled in the art can determine the specific meaning of these terms in this invention based on the specific circumstances, and they should not be construed as limitations on the invention.

[0039] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0040] Example 1

[0041] Embodiment 1 of this invention introduces a method for analyzing the deformation of elastic beams based on fourth-order nonlinear differential equations.

[0042] like Figure 1 The method for analyzing the deformation of an elastic beam based on a fourth-order nonlinear differential equation, as shown, includes:

[0043] Obtain the stress-strain relationship and lateral displacement function of the elastic beam;

[0044] Based on the obtained stress-strain relationship and transverse displacement function, the fourth-order nonlinear mechanical response function of the elastic beam is constructed.

[0045] Solve the constructed fourth-order nonlinear mechanical response function to identify the local buckling location and higher-order shear effects of the elastic beam;

[0046] Taking the minimum weight of the elastic beam as the objective function, considering the displacement and stress constraints of the elastic beam, the sensitivity analysis of the elastic beam is carried out based on the obtained local buckling position and higher-order shear effect. The nonlinear coefficient of the elastic beam is adjusted to complete the deformation analysis of the elastic beam based on the fourth-order nonlinear differential equation.

[0047] This embodiment employs a hyperelastic constitutive model, and the stress-strain relationship is: where, and These represent the stress and strain of an elastic beam, respectively. These are the parameters of the hyperelastic constitutive model; For nonlinear effects on elastic beams (such as the hardening effect of rubber or biological tissue); the obtained transverse displacement function is: u ( t That is, the elastic beam in position The dimensionless deformation at the location.

[0048] This embodiment derives a fourth-order nonlinear boundary value problem (BVP) using the principle of virtual work, namely...

[0049] (P)

[0050] The existence of the positive solution, where the nonlinear term Is A continuous function, and . u ( t () represents the lateral displacement function of the beam, indicating the position of the beam. The deformation at the point (after dimensionless conversion). This represents the nonlinear mechanical response function, describing the relationship between beam deformation and load, material nonlinearity, including the influence of displacement and derivatives of various orders. and Analyze local buckling and shear deformation to guide engineering optimization.

[0051] Based on existing research on the existence of positive solutions to fourth-order differential equations, this paper uses the inequalities of eigenvalues ​​on Banach space and employs global bifurcation theory to prove the existence of positive solutions to this fourth-order nonlinear differential equation.

[0052] The following assumptions are always made in this problem:

[0053] (S1) Define auxiliary functions Assuming for middle Item and A linear transformation can be used to make... The function can be represented as ,in for ,Right now Established. Auxiliary functions that simplify nonlinear terms through linear transformations. f Separation displacement / slope ( α ) and curvature / shear force ( β )effect.)

[0054] (S2) yes A continuous function has a constant. satisfy, When hour, For everything Unanimously established, when hour, For everything The agreement is unanimously established. Linearization coefficients, representing small deformations ( ) and large deformation ( |) Weights of displacement and curvature.

[0055] (S3) For any ,have

[0056] (S4) has a constant. In order to satisfy When, the following function ( These are the lower bound control coefficients, ensuring the nonlinear term... f (linear dominance)

[0057] This embodiment uses global bifurcation theory and the Krein-Rutman theorem, combined with eigenvalue inequalities on Banach space, to prove the existence of the correct solution.

[0058] Constructing auxiliary functions α,β The nonlinear terms are simplified by linear transformation; the connectivity of the solution is analyzed using cone P and operator Q.

[0059] Global divergence theory:

[0060] (1) Let It is a Banach space, in which It is a cone. It has a non-empty interior and .

[0061] (2) yes A continuous positive operator on, and for ,have ,right and ,in It is a linear strongly positive compact operator. And satisfy when hour, right If local consistency holds, then there exists a set. An unbounded connected component and .like yes A linear weak function, and there exists , making ,but .

[0062] Global bifurcation theory allows for the analysis of the entire system, not just local characteristics. Using this theory, this embodiment can identify all possible solutions to the system and consider its stability and convergence globally.

[0063] The following embodiment defines a continuous space. (Function space, satisfying boundary conditions) (Function set) ;and Elements in the array satisfy the following condition: Existence ( : Curvature constraint coefficient (and u'' Related), making .

[0064] definition The norm in is ;but Since it is a Banach space, this embodiment can define a cone (a function containing non-negative displacement and non-positive curvature (downward convex deformation) on this normed space to ensure the physical rationality of the solution) If the cone is normal and has a non-empty interior, then the cone is normal and has a non-empty interior. .

[0065] set up It is a Banach space. It is a cone and satisfies set up It is a compact positive linear operator, and its spectral radius is... yes Positive eigenvalues ​​with positive characteristic functions.

[0066] When assumptions (S1), (S2), (S3), and (S4) are satisfied, and one of the following conditions is also satisfied:

[0067] (I) ,

[0068] (II) ,

[0069] Therefore, problem (P) has at least one correct solution. These are eigenvalues, associated with linear operators A and B or C and D, and determine the divergent behavior of the solution. rwei Parameters control the global existence conditions of solutions to differential equations.

[0070] Proof: Suppose there exists ( For higher-order nonlinear remainder terms, it represents f Nonlinear corrections for small and large deformations) enable

[0071] ;

[0072] And when Sometimes, (1)

[0073] Therefore, From this we can obtain Non-decreasing and (The upper bound function of the nonlinear remainder term).

[0074] Consider the following points of contention (2)

[0075] In the ordinary Disagreements arise, which is equivalent to ;

[0076] ( The Green function is used to represent the solution in integral form, corresponding to the response kernels for displacement and curvature, respectively. It is also necessary to define... Strongly continuous operators on X It can be known that .

[0077] Furthermore, this embodiment still needs to define

[0078]

[0079] (Nonlinear perturbation operator, characterizing higher-order nonlinear effects). For any satisfy

[0080] (curvature) and displacement (constraint coefficients).

[0081] From (1), we can know

[0082] ,

[0083] therefore exist Locally and Consistent. Q and F represent respectively Linear operators and nonlinear perturbation operators are used to construct bifurcation problems. ).

[0084] From (S3), we can obtain that if If it is a nontrivial solution to problem (2), then And there exists a set One is an unbounded connected component Make Obviously, any one of the forms... , If (I) and (II) are satisfied, the solutions to the divergent problem (2) are all solutions to the original problem (P).

[0085] It is now proven that in In the context of connected components Passing through the hyperplane That is, only proof is required. connect .make ( (The sequence of bifurcation parameters and the corresponding solution sequence, used to analyze the unboundedness of connected components) satisfies Obviously, for any Because when At that time, the divergence problem (2) has only trivial solutions, and The following is about The value of is discussed in two cases.

[0086] Scenario 1: At that time, proof is required. .

[0087] First, prove that there exists a constant. , making

[0088] (3)

[0089] but connect and .

[0090] Based on (3) and the above conditions, Consider the following questions .

[0091] make (The sequence of bifurcation parameters is used to analyze the unboundedness of connected components), because exist There is a bounded interval, therefore for exist ( (for its convergent subsequence) and .because It is non-subtractive, so for any , .and ,therefore, ;in, ( Representing the convergent subsequence, the solution sequence bounds of the divergence parameter sequence, thus, and therefore connect .

[0092] The following proves the existence of a constant. , making any According to the global divergence theory, it is only necessary to prove There is a linear weak function And exist , making From condition (S4), there exists a constant. , making .

[0093] for ,make ,but It is a linear weak function, and by the Krein-Rutman theorem, it exists. ( : Positive characteristic function, corresponding to the fundamental frequency mode shape or instability mode of the beam), take ,but and In fact, According to the global divergence theory, .

[0094] Scenario 2: If at this time there exists satisfy ,

[0095] and Then, in the same case 1, it can be proved that...

[0096]

[0097] Therefore, it is proven. connect According to the global bifurcation theory, there exists at least one fourth-order nonlinear boundary value problem (P).

[0098] This embodiment applies the solution to optimize beam structure design, for example, by predicting local buckling or shear deformation to improve seismic performance. Fourth-order nonlinear problems play a crucial role in solving local deformation problems. For example, in local buckling, beams may buckle locally under large loads, especially when the beam is long or has an irregular cross-section. Fourth-order nonlinear problems can account for this local buckling phenomenon, providing more accurate analysis. In shear deformation, traditional elastic beam theory typically assumes that shear deformation is negligible, but in some cases, such as thinner beams or under high loads, shear deformation can become significant. Fourth-order nonlinear problems can address the impact of this shear deformation on the overall beam behavior.

[0099] This embodiment considers higher-order deformation effects and nonlinear mechanical effects, and analyzes the elastic beam based on fourth-order nonlinear differential equations to achieve a balance between lightweighting and safety of the elastic beam, providing a theoretical basis for complex engineering structural components.

[0100] Example 2

[0101] Embodiment 2 of the present invention introduces an elastic beam deformation analysis system based on a fourth-order nonlinear differential equation.

[0102] like Figure 2The system shown is an elastic beam deformation analysis system based on a fourth-order nonlinear differential equation, comprising:

[0103] The acquisition module is configured to acquire the stress-strain relationship and lateral displacement function of the elastic beam;

[0104] The construction module is configured to construct the fourth-order nonlinear mechanical response function of the elastic beam based on the obtained stress-strain relationship and lateral displacement function.

[0105] The identification module is configured to solve the constructed fourth-order nonlinear mechanical response function to identify the local buckling location and higher-order shear effects of the elastic beam.

[0106] The analysis module is configured to take the minimum weight of the elastic beam as the objective function, consider the displacement and stress constraints of the elastic beam, perform sensitivity analysis of the elastic beam based on the obtained local buckling position and higher-order shear effect, adjust the nonlinear coefficient of the elastic beam, and complete the deformation analysis of the elastic beam based on the fourth-order nonlinear differential equation.

[0107] The detailed steps are the same as those provided in Example 1 for the elastic beam deformation analysis method based on fourth-order nonlinear differential equations, and will not be repeated here.

[0108] Example 3

[0109] Embodiment 3 of the present invention provides a computer-readable storage medium.

[0110] A computer-readable storage medium having a program stored thereon, which, when executed by a processor, implements the steps in the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in Embodiment 1 of the present invention.

[0111] The detailed steps are the same as those provided in Example 1 for the elastic beam deformation analysis method based on fourth-order nonlinear differential equations, and will not be repeated here.

[0112] Example 4

[0113] Embodiment 4 of the present invention provides an electronic device.

[0114] An electronic device includes a memory, a processor, and a program stored in the memory and running on the processor. When the processor executes the program, it implements the steps in the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in Embodiment 1 of the present invention.

[0115] The detailed steps are the same as those provided in Example 1 for the elastic beam deformation analysis method based on fourth-order nonlinear differential equations, and will not be repeated here.

[0116] Example 5

[0117] Embodiment 5 of the present invention provides a computer program product.

[0118] A computer program product includes software code, wherein the program in the software code performs the steps of the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in Embodiment 1 of the present invention.

[0119] The detailed steps are the same as those provided in Example 1 for the elastic beam deformation analysis method based on fourth-order nonlinear differential equations, and will not be repeated here.

[0120] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code. The solutions in the embodiments of the present invention can be implemented using various computer languages, such as the object-oriented programming language Java and the interpreted scripting language JavaScript.

[0121] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0122] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0123] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0124] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0125] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

[0126] The above description is merely a preferred embodiment of this practice and is not intended to limit the scope of this practice. Various modifications and variations can be made to this practice by those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of this practice should be included within the protection scope of this practice.

Claims

1. A method for analyzing the deformation of an elastic beam based on a fourth-order nonlinear differential equation, characterized in that, include: Obtain the stress-strain relationship and lateral displacement function of the elastic beam; Based on the obtained stress-strain relationship and transverse displacement function, the fourth-order nonlinear mechanical response function of the elastic beam is constructed. Solve the constructed fourth-order nonlinear mechanical response function to identify the local buckling location and higher-order shear effects of the elastic beam; Taking the minimum weight of the elastic beam as the objective function, considering the displacement and stress constraints of the elastic beam, the sensitivity analysis of the elastic beam is carried out based on the obtained local buckling position and higher-order shear effect. The nonlinear coefficient of the elastic beam is adjusted to complete the deformation analysis of the elastic beam based on the fourth-order nonlinear differential equation. A hyperelastic constitutive model is used to obtain the stress-strain relationship of the elastic beam, i.e. ;in, and These represent the stress and strain of an elastic beam, respectively. These are the parameters of the hyperelastic constitutive model; For the nonlinear effect of the elastic beam; the obtained transverse displacement function is: u ( t That is, the elastic beam in position The dimensionless deformation at the point; The fourth-order nonlinear mechanical response function of the constructed elastic beam is: Based on the constructed fourth-order nonlinear mechanical response function, the relationship between the deformation of the elastic beam, including the influence of displacement and derivatives of each order, and the load and material nonlinearity is described. The local buckling and shear deformation of the elastic beam are analyzed through the second-order and third-order differential equations of the transverse displacement function. The objective function used is to minimize the beam weight. ;in, For cross-sectional area, This represents the material density.

2. The method for analyzing the deformation of an elastic beam based on a fourth-order nonlinear differential equation as described in claim 1, characterized in that, The Galerkin method basis functions are used to characterize the vibration mode shape of the elastic beam. The obtained transverse displacement function is expanded, and the fourth-order nonlinear mechanical response function is transformed into a system of nonlinear algebraic equations. The fourth-order derivative terms are discretized by the finite difference method, and the system of nonlinear algebraic equations is solved iteratively.

3. The method for analyzing the deformation of an elastic beam based on a fourth-order nonlinear differential equation as described in claim 1, characterized in that, Curvature abrupt change points are identified by the second derivative in the solution of the fourth-order nonlinear mechanical response function. The local buckling location of the elastic beam is then identified based on the curvature abrupt change points. The influence of higher-order shear effects is quantified by combining the fourth-order model, and the shear deformation of the elastic beam is evaluated.

4. A deformation analysis system for an elastic beam based on a fourth-order nonlinear differential equation, employing the method described in claim 1, characterized in that, include: The acquisition module is configured to acquire the stress-strain relationship and lateral displacement function of the elastic beam; The construction module is configured to construct the fourth-order nonlinear mechanical response function of the elastic beam based on the obtained stress-strain relationship and lateral displacement function. The identification module is configured to solve the constructed fourth-order nonlinear mechanical response function to identify the local buckling location and higher-order shear effects of the elastic beam. The analysis module is configured to take the minimum weight of the elastic beam as the objective function, consider the displacement and stress constraints of the elastic beam, perform sensitivity analysis of the elastic beam based on the obtained local buckling position and higher-order shear effect, adjust the nonlinear coefficient of the elastic beam, and complete the deformation analysis of the elastic beam based on the fourth-order nonlinear differential equation.

5. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in any one of claims 1-3.

6. An electronic device comprising a memory, a processor, and a computer program stored in the memory and running on the processor, characterized in that, When the processor executes the program, it implements the steps of the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in any one of claims 1-3.

7. A computer program product, comprising software code, characterized in that, The program in the software code executes the steps of the elastic beam deformation analysis method based on fourth-order nonlinear differential equations as described in any one of claims 1-3.

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