A stability determination method for simply supported and fixed supported beams

Abstract

This invention provides a method for determining the stability of simply supported and fixed beams. The method includes: establishing a system of equilibrium differential equations for an elastic beam and determining the simply supported boundary conditions; performing numerical integration based on the equilibrium differential equations and the simply supported boundary conditions to obtain the shapes and corresponding parameter information of all simply supported and fixed beams; measuring the actual arc length and actual support distance of the simply supported and fixed beam to be determined; determining the corresponding moment-rotation curve based on the actual arc length, actual support distance, and the shapes and corresponding parameter information of all simply supported and fixed beams obtained from numerical integration, and determining the critical value of the fixed-end rotation angle based on the moment-rotation angle curve; measuring the actual value of the fixed-end rotation angle of the simply supported and fixed beam to be determined, and determining the stability of the simply supported and fixed beam to be determined based on the relationship between the actual value of the fixed-end rotation angle and the critical value of the fixed-end rotation angle. This invention solves the technical problem of complex and computationally intensive stability determination processes for simply supported and fixed beams in the prior art.

Description

Technical Field

[0001] This invention relates to the field of buckling elastic beam stability analysis technology, and in particular to a method for determining the stability of a simply supported fixed beam. Background Technology

[0002] Shape mutations describe the process by which a structure abruptly changes from one shape to another stable shape. These mutations involve significant deformation before and after the change, and the rate of change is extremely rapid. In engineering, structural shape can significantly improve energy harvesting efficiency, increase the deployment ratio of deployable structures, and enable automated state recognition of switches. Simply supported beams are frequently used as fundamental components in these complex systems, and studying the shape mutation problem of simply supported beams helps in understanding the stability and shape mutation issues of complex systems.

[0003] Abrupt changes in the shape of a buckling elastic beam signify structural instability. Traditional methods determine stability using load-displacement curves and identification of the beam's natural frequencies. However, different control processes correspond to different load-displacement curves, and continuous shape changes during control correspond to continuous changes in the natural frequency. These two factors make traditional methods complex and computationally intensive when solving stability problems. Summary of the Invention

[0004] This invention provides a method for determining the stability of simply supported beams, which can solve the technical problems of complex stability determination process and large numerical calculation volume in the prior art.

[0005] According to one aspect of the present invention, a method for determining the stability of a simply supported beam is provided, the method comprising:

[0006] S1. Establish the system of equilibrium differential equations for the elastic beam and determine the simply supported boundary conditions;

[0007] S2, numerical integration is performed based on the equilibrium differential equations and simply supported boundary conditions to obtain the shape and corresponding parameter information of all simply supported fixed beams;

[0008] S3, measure the actual arc length of the simply supported and fixed beam to be judged and the actual support distance between the simply supported end and the fixed end;

[0009] S4. Based on the actual arc length, actual support distance, and the shapes and corresponding parameter information of all simply supported and fixed beams obtained in S2, determine the corresponding moment-rotation curve, and determine the critical value of the fixed end rotation angle based on the moment-rotation curve.

[0010] S5. Measure the actual value of the fixed-end rotation angle of the simply supported beam to be judged, and determine the stability of the simply supported beam to be judged based on the relationship between the actual value of the fixed-end rotation angle and the critical value of the fixed-end rotation angle.

[0011] Furthermore, the parameter information includes arc length, support distance, fixed support rotation angle, and bending moment.

[0012] Furthermore, based on the actual arc length, actual support distance, and the shapes and corresponding parameter information of all simply supported and fixed beams obtained in S2, the corresponding moment-rotation curves are determined, including:

[0013] Calculate the actual ratio of the actual arc length to the actual support distance;

[0014] Calculate the ratio of arc length to support distance for each simply supported beam shape obtained in S2, and select the simply supported beam shapes that correspond to the ratios that are equal to the actual ratios.

[0015] The corresponding moment-rotation curve is obtained by fitting the fixed end rotation angle and bending moment corresponding to the selected simply supported beam shape.

[0016] Furthermore, determining the critical value of the rotation angle at the fixed support end based on the moment-rotation curve includes:

[0017] Find the saddle point bifurcation point on the moment-rotation curve;

[0018] The value of the fixed end rotation angle corresponding to the bifurcation point of the saddle point is taken as the critical value of the fixed end rotation angle.

[0019] Furthermore, determining the stability of the simply supported beam to be assessed based on the relationship between the actual value of the rotation angle at the fixed support end and the critical value of the rotation angle at the fixed support end includes:

[0020] Calculate the absolute value of the difference between the actual value of the fixed-end rotation angle and the critical value of the fixed-end rotation angle;

[0021] If the absolute value of the difference is less than a preset threshold, the simply supported beam to be judged is determined to be unstable; otherwise, the simply supported beam to be judged is determined to be stable.

[0022] Furthermore, the equilibrium differential equations are as follows:

[0023] ,

[0024] In the above formula, Indicates tangential force. Represents normal force, This represents the x-coordinate of any point on the arc. This represents the ordinate value of any point on the arc. express curvature at that point express The slope at that point.

[0025] Furthermore, the simply supported boundary conditions are:

[0026] ,

[0027] In the above formula, This represents the initial slope of the simply supported end.

[0028] Furthermore, the moment-rotation curve is based on the rotation angle at the fixed support end. Axis, with bending moment as The axis and the saddle point bifurcation point are the leftmost points on the moment-rotation curve.

[0029] This invention provides a method for determining the stability of simply supported beams. This method is based on numerical integration using established elastic beam equilibrium differential equations and determined simply supported boundary conditions. The resulting data, including the shapes and corresponding parameters of all simply supported beams, is stored in a database. When determining stability, data is selected from the database based on the actual arc length and actual support distance of the beam to be determined, yielding the corresponding moment-rotation curve. The critical value of the fixed-end rotation angle is then determined based on the moment-rotation curve. By comparing the actual value of the fixed-end rotation angle with the critical value, the stability of the simply supported beam can be determined. This method allows for a simple and rapid determination of the stability of simply supported beams, identifying their instability risk, and is convenient for engineering applications. Attached Figure Description

[0030] The accompanying drawings, which form part of this specification, are provided to further illustrate embodiments of the invention and, together with the textual description, explain the principles of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any creative effort.

[0031] Figure 1 A flowchart illustrating a method for determining the stability of a simply supported beam according to a specific embodiment of the present invention is shown.

[0032] Figure 2 A schematic diagram of the parameters of a simply supported beam according to a specific embodiment of the present invention is shown;

[0033] Figure 3 A schematic diagram of a simply supported beam obtained using a second-order buckling deformation shape according to a specific embodiment of the present invention is shown;

[0034] Figure 4 A schematic diagram showing the bending moment-rotation curve and the location of the saddle point bifurcation point provided according to a specific embodiment of the present invention is shown. Detailed Implementation

[0035] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[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 exemplary embodiments according to this application. 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] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps set forth in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.

[0038] like Figure 1As shown, a method for determining the stability of a simply supported and fixed beam is provided according to a specific embodiment of the present invention. The stability determination method includes: S1, establishing a system of equilibrium differential equations for an elastic beam and determining the simply supported boundary conditions; S2, performing numerical integration based on the system of equilibrium differential equations and the simply supported boundary conditions to obtain the shapes and corresponding parameter information of all simply supported and fixed beams; S3, measuring the actual arc length of the simply supported and fixed beam to be determined and the actual support distance between the simply supported end and the fixed end; S4, determining the corresponding moment-rotation curve based on the actual arc length, the actual support distance, and the shapes and corresponding parameter information of all simply supported and fixed beams obtained in S2, and determining the critical value of the fixed end rotation angle based on the moment-rotation curve; S5, measuring the actual value of the fixed end rotation angle of the simply supported and fixed beam to be determined, and determining the stability of the simply supported and fixed beam to be determined based on the relationship between the actual value of the fixed end rotation angle and the critical value of the fixed end rotation angle.

[0039] In this invention, the parameter information includes arc length, support distance, fixed-end rotation angle, and bending moment. The critical value of the fixed-end rotation angle is the critical point corresponding to the abrupt change in shape of the simply supported beam. Therefore, determining the relationship between the actual value of the fixed-end rotation angle and the critical value of the fixed-end rotation angle can determine the stability of the simply supported beam. Specifically, the closer the actual value of the fixed-end rotation angle is to the critical value of the fixed-end rotation angle, the lower the stability of the simply supported beam, and correspondingly, the higher the risk of instability.

[0040] This configuration provides a method for determining the stability of simply supported beams. This method is based on numerical integration using established elastic beam equilibrium differential equations and defined simply supported boundary conditions. The resulting data, including the shapes and corresponding parameters of all simply supported beams, is stored in a database. When determining stability, data is filtered from the database based on the actual arc length and actual support distance of the beam to be determined, yielding the corresponding moment-rotation curve. The critical value of the fixed-end rotation angle is then determined based on this curve. The stability of the simply supported beam is determined by comparing the actual fixed-end rotation angle with the critical value. This method allows for a simple and rapid determination of the stability of simply supported beams, identifying their instability risk, and is convenient for engineering applications. Compared with existing technologies, this invention solves the problems of complex stability determination processes and large numerical computation volumes in existing technologies.

[0041] Further, in this embodiment of the invention, determining the corresponding moment-rotation curve based on the actual arc length, actual support distance, and all simply supported beam shapes and corresponding parameter information obtained in S2 includes: calculating the actual ratio of the actual arc length to the actual support distance; calculating the ratio of the arc length to the support distance corresponding to each simply supported beam shape obtained in S2, and selecting the simply supported beam shape corresponding to the ratio equal to the actual ratio; and fitting the fixed end rotation angle and bending moment corresponding to the selected simply supported beam shape to obtain the corresponding moment-rotation curve.

[0042] Furthermore, in this embodiment of the invention, determining the critical value of the fixed-end rotation angle based on the moment-rotation curve includes: finding the saddle point bifurcation point on the moment-rotation curve; and using the value of the fixed-end rotation angle corresponding to the saddle point bifurcation point as the critical value of the fixed-end rotation angle. Wherein, the aforementioned moment-rotation curve uses the fixed-end rotation angle as... Axis, with bending moment as The axis and the saddle point bifurcation point are the leftmost points on the moment-rotation curve.

[0043] Further, as a specific embodiment of the present invention, determining the stability of a simply supported beam to be assessed based on the relationship between the actual value of the fixed-end rotation angle and the critical value of the fixed-end rotation angle includes: calculating the absolute value of the difference between the actual value of the fixed-end rotation angle and the critical value of the fixed-end rotation angle; determining whether the absolute value of the difference is less than a preset threshold; if so, the simply supported beam to be assessed is determined to be unstable; otherwise, the simply supported beam to be assessed is determined to be stable. The preset threshold is determined based on the actual situation. To more accurately determine the stability of a simply supported beam, in another embodiment of the present invention, multiple numerical intervals are set, each representing a corresponding stability level. The stability level of the simply supported beam is determined by determining which numerical interval the absolute value of the difference belongs to, thereby taking corresponding measures.

[0044] To facilitate a clearer understanding of the stability determination method for simply supported beams provided by this invention, the following will be combined with... Figure 2 , Figure 3 and Figure 4 The above processes are explained in detail using practical application examples. Those skilled in the art will understand that this example is only for the purpose of making a clearer understanding of the stability determination method for simply supported beams provided by the present invention, and does not impose any technical limitations on it.

[0045] like Figure 2 As shown, an elastic beam, also known as a simply supported beam, includes one simply supported end. and fixed branch end Simple branch end and fixed branch end The distance between the supports is Simple branch end and fixed branch end The arc length between them is the beam length. The fixed support end rotation angle is The bending moment at the fixed support end is The equilibrium differential equations established for it are as follows:

[0046] ,

[0047] In the above formula, Indicates tangential force. Represents normal force, This represents the x-coordinate of any point on the arc. This represents the ordinate value of any point on the arc. express curvature at that point express The slope at which point is located, and any point on the arc mentioned here refers to any point on the elastic beam.

[0048] Furthermore, the simply supported boundary conditions are determined as follows:

[0049] ,

[0050] In the above formula, This represents the initial slope of the simply supported end.

[0051] The fourth-order Jung-Kutta method can be used to quickly solve for the second-order buckling deformation shape given an initial slope. Numerical integration of this second-order buckling deformation shape yields the shape of a simply supported beam. During numerical integration, different arc lengths (i.e., step lengths) correspond to different simply supported beam shapes, such as... Figure 3 As shown, arrive There are two simply supported beams between them, one of which is a simply supported beam with the shape of... For simple branches, For fixed supports, simply supported supports The initial slope is , fixed end The turning angle is , and The distance between the supports is , and The arc length between them is By changing the initial angle of the second-order buckling deformation shape, the shapes and corresponding parameter information of all simply supported and fixed beams can be obtained. These parameters include arc length, support distance, fixed-end rotation angle, and bending moment. The parameter information of simply supported and fixed beams can be stored using the scaling principle, thus avoiding repetitive numerical integration processes.

[0052] Measure the actual arc length of the simply supported beam to be judged. Distance from actual support Calculate the ratio of the actual arc length to the actual support distance, and denot it as... Then based on the ratio From the previously stored information on simply supported beam shapes and their corresponding parameters, select suitable simply supported beam shapes and their corresponding parameters, and fit them to form corresponding moment-rotation curves. Specifically, from the stored information on simply supported beam shapes and their corresponding parameters, select shapes where the ratio of arc length to support distance on each second-order buckling deformation shape is equal to... For a simply supported beam, by changing the shape of the second-order buckling deformation, it is possible to obtain all beams that satisfy the condition (the ratio of arc length to support distance is equal to...). For simply supported beams, the bending moment of each simply supported beam is selected. and fixed support end angle Curve fitting can yield the complete moment-rotation curve. Furthermore, it can also be obtained by first... ( (Representing the bending stiffness coefficient of a simply supported beam) Calculate the curvature at the fixed ends of each simply supported beam. Then, using the fixed support end corner for Axis, to fix the curvature of the support ends for Axis fitting yields the moment-rotation curve. This fitting process can identify multiple solutions near the saddle point and bifurcation point, avoiding the omission of solutions. A typical moment-rotation curve is shown below. Figure 4 As shown.

[0053] Figure 4 In the middle, with the fixed support end rotation angle The decrease, Gradually and continuously increase to the maximum value, which is the bending moment. Continue to increase to the maximum value; when the fixed support end rotation angle As it continues to decrease, Gradually decrease to the saddle point bifurcation point, which is the bending moment. Gradually decrease to the saddle point and bifurcation point (The leftmost point on the diagram); fixed support end rotation angle Beyond the saddle point bifurcation, no corresponding simply supported beam exists, resulting in a sudden shape change. Therefore, the saddle point bifurcation... The corresponding critical value of the fixed end rotation angle is used as the critical point at which the shape of a simply supported fixed beam undergoes abrupt change.

[0054] In engineering applications, the arc length and distance between supports of a simply supported fixed beam can be quickly determined by measurement. This allows us to use the aforementioned method to determine the corresponding moment-rotation curve and the location of its saddle point bifurcation. Based on the location of the saddle point bifurcation, the critical value of the fixed-end rotation angle can be determined. Measurement can determine the fixed-end rotation angle of a simply supported fixed beam in engineering. The closer the measured fixed-end rotation angle value is to the critical value of the fixed-end rotation angle corresponding to the saddle point bifurcation, the higher the risk of instability and the lower the stability of the simply supported fixed beam.

[0055] In summary, this invention provides a method for determining the stability of simply supported beams. This method is based on numerical integration using established elastic beam equilibrium differential equations and determined simply supported boundary conditions. This yields the shapes and corresponding parameter information of all simply supported beams, which are stored in a database. When determining stability, data is filtered from the database based on the actual arc length and actual support distance of the simply supported beam to be determined, resulting in the corresponding moment-rotation curve. The critical value of the fixed-end rotation angle is then determined based on the moment-rotation curve. The stability of the simply supported beam is determined by comparing the actual value of the fixed-end rotation angle with the critical value. This method allows for a simple and rapid determination of the stability of simply supported beams, identifying their instability risk, and is convenient for engineering applications. Compared with existing technologies, the technical solution of this invention solves the technical problems of complex stability determination processes and large numerical calculations in existing technologies for simply supported beams.

[0056] For ease of description, spatial relative terms such as "above," "on top of," "on the upper surface of," "above," etc., are used herein to describe the spatial positional relationship of a device or feature as shown in the figures to other devices or features. It should be understood that spatial relative terms are intended to encompass different orientations in use or operation beyond the orientation of the device as described in the figures. For example, if the device in the figures were inverted, a device described as "above" or "on top of" other devices or structures would subsequently be positioned as "below" or "under" other devices or structures. Thus, the exemplary term "above" can include both "above" and "below." The device may also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatial relative descriptions used herein will be interpreted accordingly.

[0057] Furthermore, it should be noted that the use of terms such as "first" and "second" to define components is merely for the purpose of distinguishing the corresponding components. Unless otherwise stated, the above terms have no special meaning and therefore should not be construed as limiting the scope of protection of this invention.

[0058] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method of determining the stability of a simply supported fixed beam, characterized by, The stability determination method includes: S1. Establish the system of equilibrium differential equations for the elastic beam and determine the simply supported boundary conditions; S2, Numerical integration is performed based on the equilibrium differential equations and the simply supported boundary conditions to obtain the shapes and corresponding parameter information of all simply supported fixed beams; S3, measure the actual arc length of the simply supported and fixed beam to be judged and the actual support distance between the simply supported end and the fixed end; S4. Determine the corresponding moment-rotation curve based on the actual arc length, the actual support distance, and the shapes and corresponding parameter information of all simply supported and fixed beams obtained in S2, and determine the critical value of the fixed end rotation angle based on the moment-rotation curve. S5, measure the actual value of the fixed end rotation angle of the simply supported beam to be judged, and determine the stability of the simply supported beam to be judged based on the relationship between the actual value of the fixed end rotation angle and the critical value of the fixed end rotation angle. The parameter information includes arc length, support distance, fixed end rotation angle, and bending moment. Determining the corresponding bending moment-rotation curve based on the actual arc length, the actual support distance, and all simply supported beam shapes and their corresponding parameter information obtained in S2 includes: calculating the actual ratio of the actual arc length to the actual support distance; calculating the ratio of the arc length to the support distance for each simply supported beam shape obtained in S2, and selecting the simply supported beam shapes with ratios equal to the actual ratios; and fitting the fixed end rotation angle and bending moment corresponding to the selected simply supported beam shapes to obtain the corresponding bending moment-rotation curve. Determining the critical value of the fixed-end rotation angle based on the bending moment-rotation curve includes: finding the saddle point bifurcation point on the bending moment-rotation curve; taking the value of the fixed-end rotation angle corresponding to the saddle point bifurcation point as the critical value of the fixed-end rotation angle; the bending moment-rotation curve uses the fixed-end rotation angle as... Axis, with bending moment as The saddle point bifurcation point is the leftmost point on the bending moment-rotation curve.

2. The stability determination method according to claim 1, characterized in that, Determining the stability of the simply supported beam to be judged based on the relationship between the actual value of the fixed-end rotation angle and the critical value of the fixed-end rotation angle includes: Calculate the absolute value of the difference between the actual value of the fixed support end rotation angle and the critical value of the fixed support end rotation angle; Determine whether the absolute value of the difference is less than a preset threshold. If it is, determine that the simply supported beam to be determined is unstable. If it is not, determine that the simply supported beam to be determined is stable.

3. The stability determination method according to claim 2, characterized in that, The equilibrium differential equations are as follows: ， In the above formula, Indicates tangential force. Represents normal force, This represents the x-coordinate of any point on the arc. This represents the ordinate value of any point on the arc. express curvature at that point express The slope at that point.

4. The stability determination method according to claim 3, characterized in that, The simply supported boundary conditions are: ， In the above formula, This represents the initial slope of the simply supported end.

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