A testing method for a cantilever beam numerical technique based on improved constraints

Abstract

The present invention relates to the technical field of cantilever beam models, and particularly to a numerical technology test method for a cantilever beam based on improved constraints, including: establishing a cantilever beam model, where the cantilever beam has a constrained section and a free deformation section. The end face of the constrained section away from the free deformation section is the beam end section, the section in the middle of the constrained section and the free deformation section is the fixed end section, and the end face of the free deformation section away from the constrained section is the free end section; applying a point load to the free end section and applying a traction force to the constrained section. The traction force is applied to the upper and lower surfaces of the constrained section, and the direction of the traction force is opposite to the direction of the point load; performing a numerical technology test based on the cantilever beam model. Through the numerical technology test of the beam model of the present invention, the stress in the free deformation section of the beam is consistent with the distribution law and magnitude of the theoretical solution, and the stress singularity in the original model beam is eliminated.

Description

Technical Field

[0001] The present invention relates to the technical field of cantilever beam models, and in particular to a test method for cantilever beam numerical technology based on improved constraints. Background Art

[0002] Disclosing the information of this background art section is only intended to increase the understanding of the overall background of the present invention, and is not necessarily regarded as an admission or an implication in any form that this information constitutes the prior art already known to those of ordinary skill in the art.

[0003] A cantilever beam is a two-dimensional plane problem in elasticity mechanics. Timoshenko gave the solution to this problem, but the stress solutions at both ends of the cantilever beam are inaccurate. Many literatures use this cantilever beam point load test to quantitatively compare the displacements with the solutions of Timoshenko beams or finite element solutions (essentially still elastic mechanics solutions) to verify the accuracy and robustness of the proposed numerical technology. However, for the simulation of typical problems such as the axial stress and tangential stress symmetric about the neutral layer (the plane formed by the non-elongation or non-shortening changes in the beam under the action of external loads) in the cantilever beam, due to the singularity of the mathematical model, quantitative comparison cannot be carried out, and only qualitative analysis can be performed.

[0004] In addition, in the prior art, some cantilever beams under new constraint conditions are also attempted to be used to test the description degree of the numerical technology for displacements and stresses. For example, the original cross-section is rigidly constrained (restricting the translation and rotation in the x direction, y direction, and xoy plane), and is relaxed to restrict the translation in the x direction and the y direction at the neutral axis. However, it has also been confirmed that there is still a stress singularity phenomenon in the stress at this time, which undoubtedly greatly limits the use of this test case. Summary of the Invention

[0005] Aiming at the deficiencies of the prior art, the purpose of the embodiments of the present invention is to provide a test method for cantilever beam numerical technology based on improved constraints. Through the cantilever beam model with new boundary conditions, the needs of the above-mentioned numerical calculation test cases can be well met. Moreover, the model is a real physical model, rather than using the beam boundary conditions without real conditions as adopted in some literatures.

[0006] To achieve the above purpose, the embodiments of the present invention provide the following technical solutions:

[0007] A test method for cantilever beam numerical technology based on improved constraints, comprising:

[0008] A cantilever beam model is established. The cantilever beam has a constrained section and a free deformation section. The end face of the constrained section away from the free deformation section is the beam end section. The section in the middle of the constrained section and the free deformation section is the fixed end section. The end face of the free deformation section away from the constrained section is the free end section. A point load is applied to the free end section, and a traction force is applied to the constrained section. The traction force is applied to the upper and lower surfaces of the constrained section, and the direction of the traction force is opposite to the direction of the point load.

[0009] Numerical technical tests are carried out based on the cantilever beam model.

[0010] Preferably, the direction of the point load is vertically upward, and the direction of the traction force is vertically downward.

[0011] Preferably, the traction force adopts a uniform distribution form.

[0012] Preferably, with the center of the fixed end section as the origin, the axial direction as the x-axis, and the longitudinal direction as the y-axis, when the traction force adopts a uniform distribution form, the traction force makes the axial stress in the free deformation section of the beam show a distribution law linearly related to y, and at the same time makes the tangential stress show a parabolic distribution law related to y.

[0013] Preferably, the calculation method of the traction force value:

[0014] A deflection curve equation and a bending moment equation are established, and the deflection of any point on the entire beam is obtained from the continuity conditions and boundary conditions.

[0015] When the maximum deflection of the entire beam is equal to the maximum deflection of the free deformation section of the beam, the uniformly distributed load value T0 is obtained. Interpolation is carried out in the interval [0, T0] to obtain the shear stress distribution of the fixed end section of the beam. When it shows a parabolic distribution, the traction force value at this time is the required value.

[0016] Preferably, the traction force adopts a linear distribution form.

[0017] Preferably, when the traction force adopts a linear distribution form, for the maximum traction force T at the beam end section max and the minimum traction force T at the fixed end section min orthogonal tests of [0, 2T0] and [0, T0] are carried out. The same as the local load, when the shear stress distribution of the fixed end section is consistent with the theoretical parabolic distribution, the load value at this time is the required value.

[0018] Preferably, the traction force adopts a parabolic distribution form or a catenary distribution form.

[0019] Preferably, when the traction force adopts a parabolic distribution form or a catenary distribution form, the bending moment equation at this time is obtained, and the deflection of any point is obtained from the deflection curve equation, the continuity condition, and the boundary condition; the maximum load value T0 when the maximum deflection of the entire beam is equal to the maximum deflection of the free deformation section beam is obtained; for T min In the interval [0, T0], T max An orthogonal test is carried out on the beam in the interval [0, 2T0], and when the shear stress distribution at the fixed end section is consistent with the theoretical parabolic distribution, the load value at this time is the required value.

[0020] Preferably, the local area of the loading position is subjected to grid encryption or grid adaption.

[0021] One or more technical solutions provided in the embodiments of the present invention have at least the following technical effects or advantages:

[0022] 1. The present invention improves the cantilever beam model. Since the surface forces are tractioned on the upper and lower beam surfaces of the constrained section, the stress in the beam is redistributed. While ensuring that the displacement at the fixed end section of the cantilever beam model does not produce too large an error compared with the original model, the stress distribution law and magnitude in the free deformation section beam are consistent with the theoretical solution, thereby eliminating the stress singularity in the original model beam.

[0023] 2. The maximum displacement and the maximum axial stress solution of the improved beam model can converge during the grid independence test, and the stress distribution is correct. The error compared with the theoretical stress solution is controlled within 5%, fully meeting the needs of various numerical technology test cases. It not only realizes qualitative comparison but also can quantitatively compare the normal stress and shear stress in the beam with the traditional finite element method.

[0024] The advantages of the additional aspects of the present invention will be given in the following description, and some will become obvious from the following description, or will be understood through the practice of the present invention.

[0025] To make the above objects, features, and advantages of the present invention more obvious and understandable, the following preferred embodiments are specifically described below in conjunction with the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The specification drawings constituting a part of the present invention are used to provide a further understanding of the present invention. The schematic embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention.

[0027] Figure 1 is a schematic diagram of an existing cantilever beam model;

[0028] Figure 2 is a schematic diagram of the cantilever beam model provided by the embodiment of the present invention;

[0029] Figure 3 It is the calculation schematic diagram under the uniformly distributed load traction provided by the embodiments of the present invention;

[0030] Figure 4 It is the comparison diagram of the stress results between the cantilever beam model of the present invention and the traditional cantilever beam model.

[0031] In the figure:

[0032] 1. Beam end section; 2. Constraint section; 3. Fixed end section; 4. Free deformation section; 5. Free end section;

[0033] The distances or dimensions between each part are exaggerated for showing the positions of each part, and the schematic diagram is only for illustration. Detailed implementation manners

[0034] It should be noted that the following detailed descriptions are all exemplary and are intended to provide further explanations of the present invention. Unless otherwise specified, all technical and scientific terms used in the present invention have the same meanings as those commonly understood by those of ordinary skill in the technical field to which the present invention belongs.

[0035] It should be noted that the terms used herein are only for describing the specific implementation manners and are not intended to limit the exemplary embodiments according to the present invention. As used herein, unless the present invention clearly indicates otherwise, the singular forms are also intended to include the plural forms. In addition, it should also be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0036] Numerical technology, that is, numerical simulation technology and numerical simulation method, is a method of establishing a mathematical model that reflects the essence of engineering or physical problems, and transforming the original complex and difficult-to-directly-solve mathematical physics problems into directly-solvable mathematical calculations on a computer. Whether the numerical simulation technology can accurately reproduce the original problem depends on factors such as the accuracy of the mathematical model, the error of the numerical calculation method, and the rationality of the compiled program. Therefore, if the newly proposed numerical simulation technology is to be directly applied to practical problems, it is necessary to conduct verification tests on this numerical simulation technology to ensure the correctness of this simulation method.

[0037] For the newly proposed numerical technique for calculating the mechanical properties of solid materials, especially solids in a continuous medium, a method of testing stress and strain by loading a cantilever beam with a point load can be used to compare this numerical technique with existing numerical methods that have been widely recognized and mature in the industry, so as to verify its accuracy for the classic case of the cantilever beam point load test through this method. However, currently, only the displacement of the cantilever beam can be compared. Due to the shortcomings of the cantilever beam mathematical model, stress comparison cannot be carried out. Therefore, the present invention proposes a cantilever beam model based on improved constraints to more comprehensively and quantitatively test the numerical technique. As Figure 1 shown, in the traditional cantilever beam model, at the fixed-end section 3 of the cantilever beam (i.e., Figure 1 the beam section where x is 0 in

[0038] ), no movement or rotation is allowed, that is, the starting section of the cantilever beam is rigidly constrained, restricting the translation and rotation in the x direction, y direction, and xoy plane, that is, the translational displacements in the x and y directions are 0 and the rotational displacement in the xoy plane is 0. Figure 2 The present invention improves the traditional cantilever beam model and applies a traction force in the direction opposite to the loading force to the part where the upper and lower surfaces of the original beam are rigidly constrained (i.e.,

[0039] the part of the cantilever beam where x < 0 in

[0040] ). The traction force is a surface force. Since the model is to simulate the reaction force in the opposite direction on the constrained section of the beam in the actual situation, the traction force is applied to the upper and lower surfaces of the constrained section 2. The distribution form of the traction force is uniform distribution, linear distribution, but is not limited to these two distribution methods. For example, it can also be parabolic distribution and catenary distribution. Figure 3 shown (Uniform Traction refers to uniform load), establish a coordinate axis for the entire beam, establish the deflection curve equation and bending moment equation, and the deflection of any point on the entire beam can be obtained from the continuity conditions and boundary conditions:

[0041] Deflection curve equation:

[0042] EIω”=-M(x)

[0043] Bending moment equation:

[0044]

[0045] Continuity conditions:

[0046]

[0047] Boundary conditions:

[0048] ω1| x＝0 = 0 ω′1| x＝0 = 0

[0049] Where: E represents the elastic modulus of the beam, I represents the moment of inertia of the beam, M represents the bending moment in the beam, x represents the distance from the beam end section to any point within the beam length, T represents the magnitude of the uniformly distributed load, l0 represents the length of the constraint section 2, l represents the length of the entire beam, P represents the magnitude of the concentrated force, ω1 represents the deflection of the beam in the constraint section, ω2 represents the deflection of the beam in the free deformation section, ω1` represents the first derivative of the deflection of the beam in the constraint section, ω2` represents the first derivative of the deflection of the beam in the free deformation section, and ω`` represents the second derivative of the beam deflection.

[0050] Let the ratio of the length of the constraint section 2 to the length of the free deformation section 4 be a, and obtain the expression of T when the maximum deflection of the entire beam (free deformation section beam + constraint section beam) is equal to the maximum deflection of the free deformation section beam, and get:

[0051]

[0052] Where, T represents the magnitude of the uniformly distributed load, d represents the beam width, P represents the magnitude of the concentrated force, a represents the ratio of the constraint section length to the free deformation section length, and L represents the length of the free deformation section beam. The value of the uniformly distributed load T0 can be obtained from the formula. Furthermore, interpolation is performed within the interval [0, T0] to find the shear stress distribution of the beam fixed end section 3. When it shows a parabolic distribution, the traction value at this time is the required value.

[0053] For the case where the traction is a linear load, apply a vertically downward traction to the upper and lower surfaces of the beam section where the surface force traction is applied. When the traction distribution form is linear, for the maximum traction T Figure 2 at the beam end section 1 (i.e., max the leftmost section of the cantilever beam in min and the minimum traction T max at the fixed end section 3, perform orthogonal experiments in [0, 2T0] and [0, T0]. Similar to the case of applying a uniformly distributed load traction, when the shear stress distribution of the fixed end section 3 is consistent with the theoretical parabolic distribution, the T min at this time is the required value.

[0054] For other forms of traction distribution: ① The bending moment equation at this time can be obtained, and the deflection at any point can be obtained from the deflection curve equation, continuity conditions, and boundary conditions; ② Obtain the maximum load value T0 when the maximum deflection of the entire beam (free deformation section beam + constraint section beam) is equal to the maximum deflection of the free deformation section beam; ③ For T min in the interval [0, T0], Tmax An orthogonal experiment was carried out on the beam within the interval [0, 2T0]. When the shear stress distribution at the fixed-end section 3 is consistent with the theoretical parabolic distribution, the T at this time max and T min is what is required.

[0055] The magnitude and position of the point load are the same as those of the original model, but grid encryption or grid adaptation technology can be performed on the local area of the loading position to eliminate the stress concentration problem similar to the actual situation. Grid encryption means: dividing the grids near the action point of the concentrated force into finer and denser ones, so as to obtain more accurate results near the action point of the concentrated force; grid adaptation means: dynamically changing the grid structure within the calculation area according to the actual needs of the calculation and the characteristics of the problem. In the calculation area where the physical quantity changes violently (such as the action point of the concentrated force), fine grids with a smaller spatial scale are used for calculation; in the area where the physical quantity changes slowly (such as far from the action point of the concentrated force and the constraint), coarse grids with a larger spatial scale are used for calculation, so as to improve the calculation efficiency.

[0056] The present invention improves the cantilever beam model. Due to the surface force traction on the upper and lower beam surfaces of the constraint section, the stress in the beam is redistributed. While ensuring that the displacement at the fixed-end section 3 of the cantilever beam model does not produce too large an error compared with the original model, the stress distribution law and magnitude in the free deformation section of the beam are consistent with the theoretical solution, thereby eliminating the stress singularity in the beam of the original model.

[0057] The maximum displacement and the solution of the maximum axial stress of the improved beam model can converge during the grid independence test (when the finite element calculation results gradually tend to be stable as the grid division of the structure becomes finer and finer, the grid density is no longer a factor affecting the accuracy of the finite element calculation results), and the stress distribution is correct, and the error from the theoretical stress solution is controlled within 5%. It fully meets the needs of various numerical technology test cases (that is, various structural numerical calculation methods, such as the discrete element method, the boundary element method, the smoothed particle hydrodynamics method, etc.), and can quantitatively compare the normal stress and shear stress in the beam with the traditional finite element method.

[0058] The following further illustrates the specific implementation manners of the present invention in conjunction with specific embodiments:

[0059] In different numerical technologies (such as the finite element method, the discrete element method, the smoothed particle hydrodynamics method, etc.), first establish the geometric model of the cantilever beam: the length of the free deformation section 4 of the beam is 0.201 m, the length of the constraint section 2 is 0.1 times the length of the free deformation section 4 of the beam, that is, the length of the constraint section 2 is 0.0201 m, the beam height is 0.006196 m, the elastic modulus is 7E10 Pa, and the Poisson's ratio is 0.33.

[0060] Then, displacement constraints in the x and y directions are imposed on the fixed-end section 3 of the beam; a point load with a vertical upward direction and a magnitude of 5000 N is applied to the free-end section 5.

[0061] A traction force in the vertical downward direction is applied to the upper and lower surfaces of the beam section where the surface force traction is applied. When the traction force distribution form is a uniform load, substitute into the following formula:

[0062]

[0063] Among them, T represents the magnitude of the uniform load, d represents the beam width (2D in this example, take 1), P represents the magnitude of the concentrated force, a represents the ratio of the length of the constrained section to the length of the free deformation section, L represents the length of the free deformation section of the beam. When the maximum deflection of the entire beam (free deformation section of the beam + constrained section of the beam) is equal to the maximum deflection of the free deformation section of the beam, the traction force T0 at this time is 1.0E7 Pa. Interpolate the traction force within [0, T0], and find the shear stress distribution of the fixed-end section 3 of the beam when using different numerical techniques. When it shows a parabolic distribution, the traction force value at this time is the required value. In this example, the traction force is 0.01T0 to 0.1T0. At this time, the errors between the axial stress and displacement of the beam and the theoretical solution are 2.97% and 28.87%.

[0064] A traction force in the vertical downward direction is applied to the upper and lower surfaces of the beam section where the surface force traction is applied. When the traction force distribution form is a linear distribution, substitute into the following formula:

[0065]

[0066] Among them, T represents the magnitude of the uniform load, d represents the beam width, P represents the magnitude of the concentrated force, a represents the ratio of the length of the constrained section to the length of the free deformation section, L represents the length of the free deformation section of the beam, and the maximum traction force obtained. The maximum traction force T at the beam end section 1 max and the minimum traction force T at the fixed-end section 3 min Conduct orthogonal experiments in [0, 2T0] and [0, T0], and find the shear stress distribution of the fixed-end section 3 of the beam when using different numerical techniques. When it shows a parabolic distribution and is basically consistent with the theoretical shear stress magnitude, determine the optimal T max and T min . In this case, when T max and T min are respectively in the intervals [0.1T0, 2T0] and [0, 0.01T0], the shear stress distribution of the beam is correct at this time. In particular, when T max and T min are 2T0 and 0.01T0, the shear stress of the beam is almost consistent with the theoretical solution, and the errors between the axial stress and displacement of the beam and the theoretical solution are 3.06% and 9.47%.

[0067] The technical advantages and specific implementation methods of the present invention will be described below in combination with the simulation of the cantilever beam problem using the discrete element numerical technique.

[0068] First, a geometric model is established. The length of the free deformation section 4 of the beam is 0.201 m, and the length of the constraint section 2 is 0.1 times the length of the free deformation section 4 of the beam, that is, the length of the constraint section 2 is 0.0201 m. The beam height is 0.006196 m, the elastic modulus is 7E10 Pa, the Poisson's ratio is 0.33, and the density is 2800 Kg / m 3 Using the Fish language in the PFC software, particles are generated in the manner of face-centered cubic closest packing, and a parallel bond contact model is specified between particles to generate Bond bonds. Macroscopic parameters corresponding to the elastic modulus and Poisson's ratio, such as normal stiffness and stiffness ratio, are assigned to the particles and Bond bonds.

[0069] In the traditional model, a vertical upward force of 5000 N is applied to the particle in the lower right of the free end, and a fixed constraint is applied to the particles at the beam end section, that is, U x = U y = U xy = 0 (horizontal displacement, vertical displacement, rotation angle are 0).

[0070] In the model of the present invention, a vertical upward force of 5000 N is applied to the particle in the lower right of the free end, and a fixed constraint is applied to the particles at the beam end section, that is, U x = U y = U xy = 0 (horizontal displacement, vertical displacement, rotation angle are 0);

[0071] Linear traction forces are applied to the particles on the upper and lower surfaces of the constraint section beam. If the number of particles on the upper and lower surfaces of the constraint section is n, then traction forces with a magnitude of are applied to the particles on the upper and lower surfaces of the constraint section, where R is the radius of the particle; T max , T min are the traction forces acting on the beam end section and the fixed end section. Here, 2T0 and 0.01T0 are taken, where T0 is 1.0E7 Pa; a represents the ratio of the length of the constraint section to the length of the free deformation section. Here, 0.1 is taken; d represents the beam width (2D in this example, 1 is taken); x is as shown in Figure 3 , which is the distance from the beam end section to any point within the beam length. Here, it refers to the horizontal distance from the nth particle on the upper and lower surfaces of the beam to the origin.

[0072] Two models are used to test the displacement and stress of the cantilever beam under different particle radii, that is, a particle size independence test similar to the mesh independence test in finite element is carried out. The stress results are as shown in Figure 4As shown, as the particle radius decreases, the displacements of the cantilever beam under the traditional model and the model of the present invention both tend to converge; the axial stress of the cantilever beam using the model of the present invention gradually converges, and the relative errors of the maximum axial stresses of the particles and the Bond bonds gradually tend to converge, while the traditional model shows a phenomenon of gradually increasing particle stress, that is, stress divergence.

[0073] This case first illustrates the effectiveness of the model proposed by the present invention; secondly, the stress results obtained by the discrete element numerical method, whether in terms of the distribution law or the numerical value, have very small errors compared with the analytical solution (or the finite element solution), and the numerical technology test method of the cantilever beam with improved constraints proposed by the present invention also proves that the discrete element numerical method using the parallel bond contact model can accurately simulate classical solid mechanics problems.

[0074] Although the specific implementation manners of the present invention have been described above in conjunction with the accompanying drawings, it is not a limitation on the protection scope of the present invention. Those skilled in the art should understand that based on the technical solutions of the present invention, various modifications or deformations that can be made by those skilled in the art without creative efforts are still within the protection scope of the present invention.

[0075] Finally, it should be noted that if there is no conflict, the embodiments of the present invention and the various features in the embodiments can be combined with each other, and all are within the protection scope of the present invention. In addition, all or part of the steps in the above method can be executed in a computer system such as a set of computer-executable instructions, and although the steps are listed in the order of 1, 2, 3..., in some cases, the steps shown or described can be executed in a different order than here.

Claims

1. A testing method for a cantilever beam numerical technique based on improved constraints, characterized in that Including: Establish a cantilever beam model. The cantilever beam has a constrained section and a freely deformable section. The end face of the constrained section away from the freely deformable section is the beam end section. The section in the middle of the constrained section and the freely deformable section is the fixed end section. The end face of the freely deformable section away from the constrained section is the free end section. Apply a point load to the free end section and apply a traction force to the constrained section. The traction force is applied to the upper and lower surfaces of the constrained section, and the direction of the traction force is opposite to the direction of the point load. Conduct a numerical technology test based on the cantilever beam model. The traction force adopts a linear distribution form, and the maximum traction force T at the beam end section max and the minimum traction force T at the fixed end section min are subjected to orthogonal tests in the ranges of [0, 2T0] and [0, T0]. Similar to the local load, when the shear stress distribution at the fixed end section is consistent with the theoretical parabolic distribution, the load value at this time is the required value.

2. The method for testing the numerical technology of a cantilever beam based on improved constraints according to claim 1, characterized in that The direction of the point load is vertically upward, and the direction of the traction force is vertically downward.

3. The numerical technology test method for a cantilever beam based on improved constraints according to claim 1, characterized in that, The traction force adopts a uniform distribution form.

4. The numerical technology test method for a cantilever beam based on improved constraints according to claim 3, characterized in that, Taking the center of the fixed end section as the origin, the axial direction as the x-axis, and the longitudinal direction as the y-axis, when the traction force adopts a uniform distribution form, the axial stress in the freely deformable section of the beam presents a distribution law linearly related to y, and at the same time, the tangential stress presents a parabolic distribution law related to y.

5. The numerical technology test method for a cantilever beam based on improved constraints according to claim 4, characterized in that, Calculation method of the traction force value: Establish a deflection curve equation and a bending moment equation, and obtain the deflection of any point on the entire beam by the continuity condition and the boundary condition. When the maximum deflection of the entire beam is equal to the maximum deflection of the freely deformable section of the beam, obtain the uniformly distributed load value T0, conduct interpolation in the interval [0, T0], and obtain the shear stress distribution of the fixed end section of the beam. When it presents a parabolic distribution, the traction force value at this time is the required value.

6. The method for testing a cantilever beam numerical technique based on improved constraints as claimed in claim 1, wherein, The traction force adopts a parabolic distribution form or a catenary distribution form.

7. The method for testing a cantilever beam numerical technique based on an improved constraint according to claim 6, wherein When the traction force adopts a parabolic distribution form or a catenary distribution form, the bending moment equation at this time is obtained, and the deflection at any point is obtained from the deflection curve equation, continuity conditions, and boundary conditions; the maximum load value T0 when the maximum deflection of the entire beam is equal to the maximum deflection of the freely deformed section of the beam is obtained; for T min In the interval [0, T0], T max An orthogonal test is carried out on the beam in the interval [0, 2T0], and when the shear stress distribution at the fixed end section is consistent with the theoretical parabolic distribution, the load value at this time is the required value.

8. The method for testing the numerical technology of a cantilever beam based on improved constraints according to claim 1, wherein Perform grid encryption or grid adaption on the local area of the loading position.

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