A method for estimating the bending stiffness of a cracked beam with a small sample

Abstract

The present invention discloses a method for estimating the bending stiffness of a cracked beam with a small sample, which includes the following steps: applying a constant load force perpendicular to the surface of a cantilever plate, setting the load position point at the midpoint of the tip of the cantilever plate, and recording the maximum deformation position before the cantilever plate fails. At this time, the deformation amount is the maximum deformation amount; setting the cantilever plate model as a combination of a normal beam element and a cracked beam element, based on the maximum deformation amount, applying a preset crack to the cantilever plate model, and then applying a load to make the cantilever plate reach the maximum deformation position, and calculating the bending stiffness values of the normal beam and the cracked beam elements under different combination forms. The present invention expands the single-crack damage to multi-crack damage and considers the coupling factor between cracks. The set combination form of the normal beam and the cracked beam element is suitable for calculating the bending stiffness of a multi-number cracked beam element, has a wide application prospect, and at the same time improves the accuracy of the traditional cantilever plate crack damage estimation.

Description

Technical Field

[0001] The present invention relates to the technical field of equipment component maintenance in the field of mechanical fault diagnosis, and particularly relates to a method for estimating the bending stiffness of a cracked beam with a small sample. Background Art

[0002] With the complexity and integration of modern mechanical equipment, people have put forward higher and higher requirements for the safety and stability of equipment structures. The plate-beam structure is the basic structure of the equipment system, undertaking various engineering functions such as load bearing, driving, and conversion, and is widely used in aeroengines, water pumps, and automotive system structures. With the combined action of environmental corrosion loads, fatigue stress loads, and transient impact loads, the plate-beam structure is extremely prone to corrosion cracks, fatigue cracks, and impact fractures. In response to these defects, most of the existing crack diagnosis models are idealized models based on single cracks. For a single macroscopic crack (scale in millimeters), the idealized crack model is easy to model and has good model operation efficiency, but has the following defects: 1) The actual structural cracks are usually in the state of multiple cracks (including different lengths and positions). Therefore, it is difficult to meet the actual needs using a single macroscopic crack model, and the universality is poor, which brings certain challenges to the existing crack damage modeling and diagnosis. 2) The macroscopic single crack model ignores the mutual interaction between crack units, so the accuracy is limited. 3) The traditional macroscopic crack model has less research on the bending moment stiffness paradigm, especially in terms of the definition of bending moment stiffness under different crack scales and position distributions. Therefore, there is an urgent need for a comprehensive crack damage bending stiffness theoretical model that includes crack interaction, structural scale, and crack damage scale to provide theoretical and technical guidance for actual engineering damage.

[0003] Specifically, the current definition of the bending stiffness of the plate-beam structure mainly relies on the angular deflection of a normal plate-beam per unit length under a certain tensile load (a proportional relationship). When the plate-beam structure has crack damage, the bending stiffness of the structure itself will rapidly decrease and cause excessive angular deflection. Therefore, if the definition that ignores crack damage is used and the influence of cracks is ignored, it will cause obvious calculation errors in the traditional macroscopic single crack analysis model. Summary of the Invention

[0004] In order to solve the problems in the prior art, the present invention provides a method for estimating the bending stiffness of a cracked beam with a small sample, which improves the accuracy of the model and meets the requirements of the current beam stiffness calculation with cracks.

[0005] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A method for estimating the bending stiffness of a cracked beam with a small sample, comprising the following steps:

[0007] Fix one end of the cantilever plate model and apply a constant load force perpendicular to the surface of the cantilever plate at the other end. Set the load position point at the midpoint of the tip of the cantilever plate, and record the maximum deformation position before the cantilever plate fails. At this time, the deformation amount is the maximum deformation amount.

[0008] Set the cantilever plate model as a combination form of normal beam elements and cracked beam elements, including normal beam element - normal beam element combination, cracked beam element - cracked beam element combination, normal beam element - cracked beam element - normal beam element combination, and cracked beam element - normal beam element - cracked beam element combination; based on the maximum deformation amount, apply a pre - set crack to the cantilever plate model, and then apply a load to make the cantilever plate reach the maximum deformation position. Record the maximum crack deformation rotation angle at the maximum deformation position, and calculate the bending stiffness values of the normal beam and cracked beam elements in different combination forms according to the maximum crack deformation rotation angle.

[0009] A further improvement of the present invention is that the width of the cantilever plate is 1.5 times the length of the cantilever plate, and the crack depth is set to 0.3 times the length of the cantilever plate.

[0010] A further improvement of the present invention is that when fixing one end of the cantilever plate model, the fixing height is higher than the length of the cantilever plate, and the fixing direction is perpendicular to the surface direction of the cantilever plate.

[0011] A further improvement of the present invention is that the crack includes a tearing crack and a straight crack.

[0012] A further improvement of the present invention is that the setting spacing of the pre - set crack is 1.5 times the crack length.

[0013] A further improvement of the present invention is that the bending stiffness k of the normal beam element - normal beam element combination ii is calculated by the following formula:

[0014]

[0015] where f ii is the constant load force, and is the maximum deformation rotation angle.

[0016] A further improvement of the present invention is that the bending stiffness EI of the cracked beam element - normal beam element - cracked beam element mode combination 1 (δ) is calculated by the following formula:

[0017]

[0018] where E represents the elastic modulus, I 0 represents the cross - sectional moment of inertia of the normal beam element, I c is the cross - sectional moment of inertia of the cracked beam element, represents the crack coupling coefficient, δ represents the deformation rotation angle at the tip of the plate beam under tensile load, δ j1 represents the minimum deformation rotation angle of the left cracked beam element, δ j2 represents the maximum deformation rotation angle of the left cracked beam element, δ 1 represents the rotation angle of the left cracked beam element, p represents the power value; I ci represents the moment of inertia of the crack section of the left cracked beam, I cj represents the moment of inertia of the crack section of the right cracked beam, δ 2 represents the rotation angle of the right cracked beam element.

[0019] A further improvement of the present invention lies in that the flexural rigidity EI 2 of the normal beam element - cracked beam element - normal beam element mode combination is calculated by the following formula:

[0020]

[0021] where c 2 represents the maximum deformation rotation angle of the left normal beam element, c 1 represents the maximum deformation rotation angle of the right normal beam element, δ j represents the deformation of the middle cracked beam.

[0022] A further improvement of the present invention lies in that the flexural rigidity EI 3 of the cracked beam element - cracked beam element mode combination is calculated by the following formula:

[0023]

[0024] where χ 1 represents the maximum deformation rotation angle of the left cracked beam, χ 2 represents the maximum deformation rotation angle of the right cracked beam.

[0025] Compared with the prior art, the present invention has the following beneficial effects:

[0026] The method provided by the present invention overcomes the defect of traditional cantilever plates that only consider single cracks, expands the damage of single cracks to multi - crack damage and considers the coupling factors between cracks. The set combination form of normal beam and cracked beam elements is suitable for calculating the flexural rigidity of multiple cracked beam elements, has a wide application prospect, and at the same time improves the accuracy of crack damage estimation of traditional cantilever plates, which better meets the requirements of damage monitoring, fault diagnosis and metal maintenance of plate beam structures in modern industry. The calculation of bending moment rigidity considering crack damage has practical application significance. Especially under the premise of taking accuracy as the application goal, the calculation of flexural rigidity considering crack damage can significantly reduce the input of manpower and material resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 is the geometry of a normal beam under restraint and preloading;

[0028] Figure 2 is the deformation of a normal beam;

[0029] Figure 3 is the deformation of a single-crack beam;

[0030] Figure 4 is the deformation and position after combining normal beam elements and normal beam elements;

[0031] Figure 5 is the deformation and position after combining cracked beam elements and cracked beam elements;

[0032] Figure 6 is the deformation and position after combining a normal beam, a cracked beam, and normal beam elements;

[0033] Figure 7 is the deformation and position after combining a cracked beam, a normal beam, and cracked beam elements;

[0034] Figure 8 is the structural diagram of the finite element method simulation model;

[0035] Figure 9 is the deformation diagram in the y direction;

[0036] Figure 10 is the deformation diagram in the x direction. Specific implementation mode

[0037] The following further elaborates on the present invention with reference to the accompanying drawings, enabling those skilled in the art to implement it with reference to the description text.

[0038] Based on the bending stiffness of a traditional crack-free plate beam, the present invention proposes a comprehensive crack damage bending stiffness calculation method including crack interaction, structural scale, and crack damage scale.

[0039] A small-sample cracked beam bending stiffness estimation method of the present invention includes the following steps:

[0040] Step 1: Establish a solid cantilever plate model with specific dimensions according to industrial requirements, and collect the material parameters (including elastic modulus and Poisson's ratio, etc.) and size parameters (such as length, width, and height, etc.) of the cantilever plate.

[0041] The calculation formula for the shear factor of the solid cantilever plate is:

[0042]

[0043] where υ represents the Poisson's ratio of the cantilever plate material.

[0044] To ensure that the cantilever plate can generate sufficient deformation deflection, the width of the cantilever plate can be set to 1.5 times the length of the cantilever plate, and the crack depth can be set to 0.3 times the length of the cantilever plate.

[0045] Step 2: After completing the parameter statistics in Step 1, fix one end of the cantilever plate (the fixed height is higher than the length of the cantilever plate, the fixed direction is perpendicular to the plane of the cantilever plate, and the fixing method is not limited to welding and bolt connection, etc.), and apply a constant load force perpendicular to the plane of the cantilever plate at the other end relative to the fixed end. The load position point is set at the midpoint of the tip of the cantilever plate. Ensure that the cantilever plate continuously undergoes stable deformation, and record the maximum deformation position before the cantilever plate fails (not failing means that the cantilever plate does not exhibit irreversible deformation). The deformation at this time is the maximum deformation.

[0046] Step 3: Set the solid cantilever plate model as a combination of normal beam elements and cracked beam elements. By setting various combination types of beam elements such as normal beam element - normal beam element combination ("positive - positive"), cracked beam element - cracked beam element mode combination ("crack - crack"), normal beam element - cracked beam element - normal beam element mode combination ("positive - crack - positive"), cracked beam element - normal beam element - cracked beam element mode combination ("crack - positive - crack"), etc., form a sample of damage types containing various damage crack distribution types on the cantilever plate, and calculate the bending stiffness values of the normal beam and cracked beam elements within different combination sample types.

[0047] In the present invention, only tearing - type and straight - crack types are considered for cracks, and other crack forms such as inclined cracks, opening cracks, and slip cracks are not considered.

[0048] Specifically, after completing the deformation test in Step 2, based on the maximum deformation in Step 2, then apply a pre - set crack on the cantilever plate (the setting spacing of the crack can be 1.5 times the crack length). Then apply a load until the cantilever plate reaches the maximum deformation position, and record the maximum crack deformation angle and the crack position at the maximum deformation position at this time.

[0049] Thereafter, according to the finite element method theory, decompose the cantilever plate under a constant tensile load force into n segments (the length l of each segment i is equal to the average value of the sum of the widths of each crack and the crack spacing), define any two adjacent cracks (numbered j1 and j2), and assume the distance between the two cracks is lc 1 and lc 2 . At this time, the solid cantilever plate model with cracks can be regarded as an overall model composed of multiple beam elements elastically connected.

[0050] For the connection between normal beam elements, from the relative curvature values between two adjacent beam elements, the bending stiffness k of the "positive - positive" mode combination connection between two adjacent normal beam elements can be knownii It can be expressed as:

[0051]

[0052] Wherein, f ii is the constant load force, is the maximum deformation rotation angle.

[0053] The combined bending stiffness of the crack beam element - normal beam element - crack beam element mode, that is, the combined bending stiffness EI of the "crack - normal - crack" mode 1 (δ):

[0054]

[0055] Wherein, E represents the elastic modulus, and I 0 represents the cross - sectional moment of inertia of the normal beam element, and I c is the cross - sectional moment of inertia of the crack beam element, represents the crack coupling coefficient, δ represents the deformation rotation angle at the tip of the plate beam under the tensile load, and δ j1 represents the minimum deformation rotation angle of the left - hand crack beam element, and δ j2 represents the maximum deformation rotation angle of the left - hand crack beam element, and δ 1 represents the rotation angle of the left - hand crack beam element, p represents the power value; and I ci represents the moment of inertia of the crack section of the left - hand crack beam, and I cj represents the moment of inertia of the crack section of the right - hand crack beam, and δ 2 represents the rotation angle of the right - hand crack beam element.

[0056] The combined bending stiffness of the normal beam element - crack beam element - normal beam element mode, that is, the combined bending stiffness EI of the "normal - crack - normal" mode 2 is:

[0057]

[0058] In the formula, c 2 represents the maximum deformation rotation angle of the left - hand normal beam element, and c 1 represents the maximum deformation rotation angle of the right - hand normal beam element, and δ j represents the deformation of the middle crack beam.

[0059] The combined bending stiffness of the crack beam element - crack beam element mode, that is, the combined bending stiffness EI of the "crack - crack" mode 3 is:

[0060]

[0061] In the formula, χ 1 represents the maximum deformation rotation angle of the left - hand crack beam, and χ 2It represents the maximum deformation rotation angle of the crack beam on the right side.

[0062] By combining the bending stiffness of the entire plate - beam structure according to the above - mentioned several modes, the bending stiffness of the cracked plate - beam in the case of multiple crack modes can be obtained.

[0063] The bending stiffness and the structural stiffness matrix mentioned above are not the same concept. The structural stiffness k can be obtained by performing an integral operation of the product of the bending stiffness and the shape function N(.), and the expression is:

[0064]

[0065] Example 1

[0066] A small - sample crack stiffness estimation method of the present invention includes the following steps:

[0067] The initial constraint in the present invention refers to fixing either one end of the cantilever plate or fixing both ends simultaneously.

[0068] Fix one end of the overall cantilever plate (the present invention takes the cantilever plate as an example for illustration). In the global coordinate system, as Figure 1 shown, the right - hand direction along the normal beam length is the x b positive direction, the upward direction of the height is the y b positive direction, and the direction perpendicular to the paper surface of the width is the z b positive direction. Statistically, under the action of the initial constraint and the initial tensile force F, the length l l of the normal cantilever plate (since the cracked beam and the non - cracked beam are the same at the initial moment, they are unified into a normal beam model), the width b of the cantilever plate, and the height h of the cantilever plate are measured. Set the material parameters of the cantilever plate (including elastic modulus, Poisson's ratio, etc.), and calculate the associated property parameters such as the cross - section moment of inertia I and the shear factor of the cantilever plate. The calculation formula for the shear factor κ of the solid cantilever plate beam is:

[0069]

[0070] where υ represents the Poisson's ratio of the material, and the calculation formula for the non - cracked cross - section moment of inertia I 0 of the normal cantilever plate is:

[0071] I 0 = bh 3 / 12

[0072] In the formula, b is the width of the normal beam, and h is the height of the normal beam.

[0073] The calculation formula for the cross - section moment of inertia I c of the cracked cantilever plate is:

[0074]

[0075] In the formula, A c is the converted effective crack cross-sectional area, and h c is the crack depth in the height direction of the beam.

[0076] Refer to Figure 2 , fix one end of the cantilever plate (the fixing height is higher than the plate length, and the fixing direction is perpendicular to the plate surface direction), and apply a constant load force perpendicular to the plate surface at the other end relative to the fixed end. The load position point is set at the midpoint of the plate tip. Ensure that the cantilever plate continuously undergoes stable deformation, and record the maximum deformation position before the plate beam fails (not failing means that the cantilever plate does not exhibit irreversible deformation after the tensile force disappears). In the global coordinate system, statistically calculate the comprehensive bending angle θ 1 of the normal beam, the stress f 1 of the left cross-section of the normal beam, and the stress f' 1 of the right cross-section of the normal beam, and connect and constrain the left and right cross-sections x b , y b , z b of the normal beam, as well as the x b rotation direction, y b rotation direction, and z b rotation direction degrees of freedom (respectively represented as the x b direction degree of freedom direction p 1 of the left cross-section of the normal beam, the x b direction degree of freedom direction p' 1 of the right cross-section of the normal beam, the y b direction degree of freedom direction p 2 of the left cross-section of the normal beam, the y b direction degree of freedom direction p' 2 of the right cross-section of the normal beam, the z b direction degree of freedom direction p 3 of the left cross-section of the normal beam, the z b direction degree of freedom direction p' 3 of the right cross-section of the normal beam, the y b direction rotation degree of freedom direction p 4 of the left cross-section of the normal beam, the y b direction rotation degree of freedom direction p' 4 of the right cross-section of the normal beam, the z b direction steering degree of freedom direction p 5 of the left cross-section of the normal beam, the z b direction steering degree of freedom direction p' 5 of the right cross-section of the normal beam, the x b direction rotation degree of freedom direction p 6 of the left cross-section of the normal beam, the x b direction rotation degree of freedom direction p' 6 ) for connection and constraint. The setting conditions for the constraint relationship are:

[0077] 1) For the element with the lateral extrusion deformation of the beam less than 0.3 times the angular displacement, the lateral displacement is approximately ignored, and the degrees of freedom in the lateral direction of the normal beam element are constrained at this time.

[0078] 2) For the beam element at the fixed end of the beam, all degrees of freedom of the fixed end section are discarded, and the other end is trimmed according to the standard in 1).

[0079] 3) For the constraint setting in the central region of the beam, for the element with the deformation amount of degrees of freedom in each direction greater than 0.3 times the angular displacement, all degrees of freedom constraints in all directions are retained. According to the setting conditions of the constraint relationship, the degrees of freedom of the cross-sections at both ends of the beam are restricted.

[0080] The bending stiffness includes the bending stiffness of a normal crack-free cantilever plate, the bending stiffness with a single crack, and the bending stiffness with multiple cracks. The derivation processes of their respective plate-beam bending stiffnesses can be expressed as follows.

[0081] (1) Derivation process of calculating the bending stiffness of a normal plate-beam:

[0082] According to the deformation theory of the plate, the bending stiffness K of the normal plate-beam is derived n , and there is:

[0083] M L = FL

[0084]

[0085] Among them, F is the load force at the tip of the cantilever plate, and the acting direction is perpendicular to the parallel direction of the cross-section. L is the length of the cantilever plate, M L is the bending moment of the plate-beam, is the deformation rotation angle at the tip of the plate-beam under the tensile load.

[0086] 2) Derivation process of the bending stiffness with a single crack:

[0087] See Figure 3 , in the global coordinate system, the comprehensive bending angle θ of the cracked beam is statistically analyzed 2 , the crack depth h c , the stress f on the left cross-section of the cracked beam 2 and the stress f' on the right cross-section of the cracked beam 2 , the crack position l in the curvature direction under the global coordinate c , the length l of the cracked beam l , the height h of the crack element c . Set the degrees of freedom relationship of the crack element to be consistent with the degrees of freedom constraint relationship of the crack-free element,

[0088] The setting conditions of the constraint relationship are:

[0089] 1) For the elements where the lateral extrusion deformation of the beam is less than 0.3 times the angular displacement, the lateral displacement is approximately ignored, and at this time, the degrees of freedom in the lateral direction of the normal beam element are constrained.

[0090] 2) For the beam elements located at the fixed end of the beam, all the degrees of freedom of the fixed-end section are discarded, and the other end is trimmed according to the standard in Constraint 1).

[0091] 3) For the elements in the central region where the deformation of the degrees of freedom in all directions is greater than 0.3 times the angular displacement, all the degrees of freedom in all directions are retained.

[0092] According to Castigliano's principle, considering the action of the vertical tensile load, a "tear-type" crack is generated in the middle section of the cantilever plate. The additional displacement u caused by a single crack under the load i There is:

[0093] Additional displacement u i :

[0094] In the formula, U c is the strain energy released by the crack, and P i is the crack stress.

[0095] The strain energy U released by the crack c :

[0096] In the formula, J is the strain energy density function, and h c is the crack depth.

[0097] The strain energy density function J:

[0098] In the formula, μ is the Poisson's ratio, i is the number of segments of the beam, and E b is the Young's modulus of the i-th segment of the beam, and K Ii is the strain energy density of the beam corresponding to the i-th segment.

[0099] The additional strain energy K of the mode-I crack I5 :

[0100]

[0101] In the formula, γ is the ratio of the crack depth to the beam height, b is the width of the cantilever plate, and h is the height of the cantilever plate.

[0102] At this time, the simplified expression of the crack section stiffness kc can be expressed as:

[0103]

[0104] According to Kirchhoff plate theory, by integrating the crack section stiffness \(k_c\) into the stiffness of the cantilever plate structure, the equivalent bending stiffness \(K\) of the cantilever plate with a single crack can be obtained. nc It is:

[0105]

[0106] K nc =K n +k_c*f(y)

[0107] where \(y\) 0 is the starting coordinate of the crack in the \(y\) b direction, \(y\) c is the coordinate of the crack tip point in the \(y\) b direction, \(K\) n is the stiffness of the normal cantilever plate, and \(f(y)\) is the function of the crack opening change when the cantilever plate reaches the limit position under the load force.

[0108] (3) Derivation process of the bending stiffness of multiple cracks considering various connection methods:

[0109] According to the finite element method theory, the cantilever plate under a constant tensile load is decomposed into \(n\) segments (the length of each segment is equal to the average value of the sum of the widths of each crack and the crack spacing). Define any two adjacent cracks (numbered \(j\) 1 and \(j\) 2 ). Assume the distance between the two segments of cracks is \(l_c\) 1 and \(l_c\) 2 . At this time, the beam model with cracks can be regarded as an overall model composed of multiple beam models connected elastically. For the connection between normal beam elements, from the relative curvature values between two adjacent beam elements, it can be known that between two adjacent normal beam elements, that is, the connection bending stiffness \(K\) in the "positive - positive" mode ii . Similarly, for the connection bending stiffness \(k\) ij between the cracked beam element and the normal beam element, based on the curvature obtained on the crack stiffness element and the stress values at the positions of the two end points on both sides of the element, the expression of the connection bending stiffness between the normal beam element and the cracked beam element can be obtained, as follows:

[0110] See Figure 4 , in the global coordinate system, the combination of the normal beam element and the normal beam element is in the "positive - positive" mode, where the coordinate of the left normal beam element in the global coordinate system is \(n\) 1 , the coordinate of the right normal beam element in the global coordinate system is \(n\) 2 , the bending deformation amount (i.e., angle) of the left normal beam element is \(\Delta n\) 1 , and the deformation amount (i.e., angle) of the right normal beam element is \(\Delta n\) 2At this time, the beam model can be regarded as an integral model composed of multiple beam models elastically connected. For the connection between normal beam elements, from the relative curvature values between two adjacent beam elements, the combined bending stiffness k of the combination of two adjacent normal beam elements in the "positive - positive" mode can be known ii It can be expressed as:

[0111]

[0112] where f ii is the vector stress in the one - way direction of the normal beam segment, l i is the decomposed beam segment length, is the bending angular deflection of the unit beam segment deformation.

[0113] See Figure 5 , in the global coordinate system, the combination of the cracked beam element and the cracked beam element is simply called the "crack - crack" mode. Let the coordinate of the left cracked beam element be c 1 , and the coordinate of the right cracked beam element be set as c 2 , the distances from the crack to the left and right boundaries of the cracked beam element are l c , let the bending deformation of the left cracked beam element be Δl 1 , and the bending deformation of the right cracked beam element be set as Δl 2 , the crack - induced deformation δ 1 of the left beam element caused by the crack opening, and the crack - induced deformation of the right beam element is δ 2 , the distance between the central position points of the crack amounts is Δn. Based on the single - crack bending stiffness, the expression of the combined bending stiffness EI c-c (δ) is:

[0114]

[0115] χ 1 = l c + 0.5*(δ 1 + Δl 1 )

[0116] χ 2 = χ 1 + Δn + 0.5*(δ 2 + Δl 2 ) + l c

[0117] where E represents the elastic modulus, I 0 represents the section moment of inertia of the normal beam element, I c is the section moment of inertia of the cracked beam element. Combining the connection bending stiffness k ii between the aforementioned normal beam elements and the equivalent bending stiffness K of the cantilever plate with a single cracknc , is the crack coupling coefficient, and the following conversion relationship exists:

[0118] k ii = EI 0

[0119]

[0120] See Figure 6 , in the global coordinate system, the combination of connecting normal beam elements, cracked beam elements, and normal beam elements in sequence is simply referred to as the "positive - crack - positive" mode. In this mode, the geometric center positions of the left - hand and right - hand normal beam elements are defined as nn 1 and nn 2 (the corresponding distance is L nn1 and L nn2 ), based on the length l Figure 3 of the crack relative to the starting point of the element shown in c , the bending deformation (i.e., angular displacement) of the left - hand normal beam element, the deformation (i.e., angular displacement) of the middle cracked beam element, and the deformation (angular displacement) of the right - hand normal beam element are set as Δn i , Δn j2 and Δn j , and the induced deformation (additional angular deflection) of the middle cracked beam element is set as δ j2 , and the bending stiffness of the two - side normal beams can be obtained as:

[0121]

[0122]

[0123] where F is the constant load force. In addition, the bending stiffness of the central crack element is:

[0124]

[0125] Combining the additional crack deflections induced by the two - side normal beam elements in the central crack element, and further the influence of the additional bending stiffness caused (using EI ci to represent the bending stiffness of the crack element after the combination of the left - hand normal beam element and the crack element respectively, and EI cj to represent the bending stiffness of the crack element after the combination of the right - hand normal beam element and the crack element respectively), the expression of the combined bending stiffness EI n-c-n (δ) in the "positive - crack - positive" mode is:

[0126]

[0127] where E represents the elastic modulus, and I 0 represents the cross - sectional moment of inertia of the normal beam element, Ic is the moment of inertia of the cross-section of the cracked beam element, c 1 and c 2 The size is:

[0128] c 1 = Δn i + 0.5*(Δn j2 + δ j2 )

[0129] c 2 = lc + Δn i + Δn j2 + δ j2 + Δn j ;

[0130] See Figure 7 , in the global coordinate system, the sequential combination of the cracked beam element, the normal beam element, and the cracked beam element is simply referred to as the "crack-normal-crack" mode. In this mode, the coordinates of the left beam element in the global coordinate system are designated as nn' 1 , and the coordinates of the right beam element in the global coordinate system are designated as nn' 2 , the normal bending deformation of the middle beam element is designated as Δn j , the normal bending deformation of the left cracked beam element is Δn' i1 , the crack-induced deformation of the left beam element is δ i1 ; the normal bending deformation of the right cracked beam element is Δn' j2 , and the crack-induced deformation of the right beam element is δ j2 . The length of the crack of the left crack element relative to the starting point of the left crack element is set as l c1 , and the length of the crack of the right crack element relative to the end point of the right crack element is set as l c2 . According to the initial crack starting position, the expression for the combined bending stiffness of the "crack-normal-crack" mode is:

[0131]

[0132] Among them, E represents the elastic modulus, I 0 represents the moment of inertia of the cross-section of the normal beam element, I c is the moment of inertia of the cracked cross-section of the cracked cantilever plate, c j1 and c j2 The sizes can be obtained by conversion according to the following relationship:

[0133] c j1 = lc 1 + 0.5*(Δn' j1 + δ j1 );

[0134] c j2 = δj1 +Δn j +0.5*(Δn' j2 +δ j2 )+lc 2 ;

[0135] According to Figures 1 to 7 the bending beam stiffness calculation methods for cracked beams, normal beams and composite beams, for a beam with multiple cracks, by using the above-mentioned multi-mode call accumulation, the calculation of the bending stiffness of the overall cracked beam can be realized.

[0136] Based on the calculation results of the bending stiffness of the overall cracked beam, the effectiveness and accuracy of the cracked stiffness calculation method are verified below by numerical simulation methods.

[0137] To verify and illustrate the technical effects adopted in this method, this embodiment will select the traditional technical method and compare it with the implementation effect of using this method, and compare the test results by means of scientific demonstration to verify the real effects and effectiveness of this method.

[0138] The traditional beam stiffness calculation methods mainly use the finite element method and the simple beam method for calculation. Among them, the finite element method is widely used in industry. However, for large-scale structures, the calculation period is too long and the analysis accuracy is strictly limited by the quality of the mesh, resulting in large calculation errors. For the simple beam method, most traditional beam methods focus on the calculation of the bending stiffness without cracks, and there is little research on the calculation of the bending stiffness of cracked beams with different crack arrangements. Therefore, this method proposes to decompose the beam segment by segment to realize the combination of cracked beams and normal beams. On the basis of absorbing the advantages of high calculation efficiency of the beam method, it provides the bending calculation of beams with different crack distribution forms. The following is the test comparison process of using the traditional method and the method of the present invention.

[0139] 1) Establish a plate beam model with a single crack as Figure 8 shown, and set the model size parameters, specifically including the parameters in the following table:

[0140] Length of the plate Width of the plate Height of the plate Crack location Magnitude and direction of the load 20m 600m 2m 7m -y direction, 0.1 N / m Bending deformation angle Young's modulus Shear modulus Crack depth Poisson's ratio 85.9976° 210000 MPa 81000 Mpa 0.8m 0.3

[0141] 2) Calculate the bending stiffness by the traditional finite element method

[0142] The bending deformation of the cracked bending beam under the load in the -y direction is calculated by using the traditional finite element method. The results are shown in the figure. The maximum deformation in the y direction is 0.03713 m, and the deformation in the x direction is 0.0031 m.

[0143] According to Figure 9 and Figure 10 the simulation values in, combined with the classical beam theory, at this time the bending stiffness in the y direction is defined as the ratio of the bending constant force f and the deformation δ, and the calculation formula is:

[0144]

[0145] According to the above formula, the flexural rigidity can be calculated as 2.693 N / m.

[0146] 3) The theoretical calculation formula for the flexural rigidity of the traditional beam method is:

[0147]

[0148] Combined with the foregoing parameters, the theoretically calculated flexural rigidity is 2.6 N / m, and the error compared with the simulation result shown in step 2) is 3.4%.

[0149] 4) Applying the calculation method of the present invention, the beam model belongs to the cracked beam model for description. The equivalent flexural rigidity K of the cantilever plate with a single crack nc is:

[0150]

[0151] where y0 is the coordinate of the crack starting point in the y b direction, yc is the coordinate of the crack tip point in the y b direction, K n is the stiffness of the normal cantilever plate, f(y) is the crack opening function when the cantilever plate reaches the limit position under the load force, and the crack section stiffness kc = 1.03. According to the model parameter values noted in Table 1, the flexural rigidity value of the cracked beam can be calculated as 2.69002 N / m, and the error compared with the simulation value in step 2) is ≤0.00002%, with good accuracy.

[0152] Therefore, combined with the finite element simulation value in step 1), it can be seen by comparison that the accuracy of the cracked flexural rigidity calculation method in the present invention is better than that of the traditional beam flexural rigidity method. For large mechanical plate beam structures, it saves the calculation cost of the traditional finite element method and improves the calculation accuracy, and is approximately consistent with the simulation result.

[0153] The method provided by the present invention overcomes the defect of only considering a single crack in the traditional cantilever plate, expands the single crack damage to multi-crack damage and considers the coupling factor between cracks. The set combination form of the normal beam and cracked beam elements is suitable for the calculation of the flexural rigidity of multi-number cracked beam elements, has a wide range of application prospects, and also improves the accuracy of the traditional cantilever plate crack damage estimation, and is more in line with the requirements of damage monitoring, fault diagnosis and metal maintenance of plate beam structures in modern industry.

Claims

1. A method for estimating the bending stiffness of a cracked beam with a small sample size, characterized in that, it includes the following steps: Fix one end of the cantilever plate model, apply a constant load perpendicular to the plane of the cantilever plate at the other end, set the load position point at the midpoint of the tip of the cantilever plate, and record the maximum deformation position before the cantilever plate fails. At this time, the deformation amount is the maximum deformation amount; Set the cantilever plate model as a combination form of normal beam elements and cracked beam elements, including normal beam element-normal beam element combination, cracked beam element-cracked beam element combination, normal beam element-cracked beam element-normal beam element combination, and cracked beam element-normal beam element-cracked beam element combination; based on the maximum deformation amount, apply a pre-set crack to the cantilever plate model, and then apply a load to make the cantilever plate reach the maximum deformation position, and record the maximum deformation rotation angle of the crack at the maximum deformation position. According to the maximum deformation rotation angle of the crack, calculate the bending stiffness values of the normal beam and the cracked beam elements in different combination forms; Normal beam element - Bending stiffness k of the combination of normal beam elements ii It is calculated by the following formula: Among them, f ii is a constant load force, is the maximum deformation rotation angle; Flexural stiffness EI of the combination mode of cracked beam element - normal beam element - cracked beam element 1 (δ) is calculated by the following formula: Among them, E represents the elastic modulus, and I 0 represents the cross-sectional moment of inertia of a normal beam element, and I c is the cross-sectional moment of inertia of the cracked beam element, θ represents the crack coupling coefficient, δ represents the deformation rotation angle at the tip of the plate beam under tensile load, and δ j1 represents the minimum deformation rotation angle of the left cracked beam element, and δ j2 represents the maximum deformation rotation angle of the left cracked beam element, and δ 1 represents the rotation angle of the left cracked beam element, p represents the power value; and I ci represents the moment of inertia of the cracked section of the left cracked beam, and I cj represents the moment of inertia of the cracked section of the right cracked beam, and δ 2 represents the rotation angle of the right cracked beam element; Flexural stiffness EI of the combination mode of normal beam element - cracked beam element - normal beam element 2 (δ) is calculated by the following formula: where, c 2 represents the maximum deformation rotation angle of the left normal beam element, c 1 represents the maximum deformation rotation angle of the right normal beam element, and δ j represents the deformation of the middle cracked beam; Cracked beam element - flexural stiffness EI of the cracked beam element mode combination 3 (δ) is calculated by the following formula: EI 3 (δ) = 2 * E(I 0 - θI c ), χ 1 ≤ δ ≤ χ 2 where χ 1 represents the maximum deformation rotation angle of the left cracked beam, and χ 2 represents the maximum deformation rotation angle of the right cracked beam.

2. The method for estimating the bending stiffness of a cracked beam with a small sample size according to claim 1, characterized in that, the width of the cantilever plate is 1.5 times the length of the cantilever plate, and the crack depth is set to 0.3 times the length of the cantilever plate.

3. The method for estimating the bending stiffness of a cracked beam with a small sample size according to claim 1, characterized in that, when fixing one end of the cantilever plate model, the fixing height is higher than the length of the cantilever plate, and the fixing direction is perpendicular to the plane direction of the cantilever plate.

4. The method for estimating the bending stiffness of a cracked beam with a small sample size according to claim 1, characterized in that, the crack includes a tearing crack and a straight crack.

5. The method for estimating the bending stiffness of a cracked beam with a small sample size according to claim 1, characterized in that, the setting spacing of the pre-set crack is 1.5 times the crack length.

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