Method for calculating damage degree of crack t-beam unit by stress diffusion angle method

Abstract

This invention discloses a method for calculating the damage degree of a cracked T-beam element using the stress diffusion angle method. The steps are as follows: Set an appropriate number of measuring points for the cracked T-beam; calculate the relative crack height ζ based on the crack height and the T-beam cross-sectional height; calculate the additional spring stiffness parameter of the crack based on ζ, which can be calculated according to the stress intensity factor handbook; calculate the crack stress diffusion angle α(ζ), using a rectangular cross-section beam and the equivalent crack element linear stiffness method, the calculation formula for the linear diffusion mode is α(ζ) = 74.5 - 28.895ζ; calculate the moment of inertia of the undamaged T-beam cross-section; calculate the moment of inertia of the beam segment cross-section in the stress diffusion part; calculate the damage degree of the T-beam element using the stress diffusion angle method; calculate the damage degree of the T-beam element with the corrected crack height. The number of beam segments N on one side of the stress diffusion part should not be less than 100. This invention proposes a theoretical calculation method for the damage degree of a cracked T-beam element, providing a theoretical basis for the design and calculation of the actual damage degree of the T-beam when conducting quantitative damage degree tests.

Description

Technical Field

[0001] This invention belongs to the field of structural health monitoring and relates to a method for calculating the theoretical damage degree of beam structures, specifically a method for calculating the damage degree of cracked T-beam elements using the stress diffusion angle method. Background Technology

[0002] In recent years, the number of old bridges in my country has been increasing, and the problems they present have become increasingly prominent. Among various forms of bridge damage, cracking is a relatively common one, significantly impacting the load-bearing capacity and subsequent lifespan of bridges. Although numerous studies have been conducted on damage identification methods for beam structures based on static indices, theoretical damage calculation methods for transverse cracks are still limited. Furthermore, the spacing between measuring points is usually fixed during damage identification. When damage is detected in the structure, it is likely that localized damage occurs between two measuring points. In this case, what is the equivalent damage level between the two measuring points? This question is crucial for the reasonable interpretation of quantitative damage index results. Due to the significant difficulty in quantifying the degree of damage, there are currently few reports in the literature on experimental verification.

[0003] T-beams are commonly used in various bridges due to their good load-bearing capacity and relatively simple construction process. Currently, there is no systematic and accurate theoretical method for calculating the crack stress intensity factor of T-beams. Therefore, the theoretical damage degree of elements containing transverse cracks cannot be directly calculated based on the transverse crack stress intensity factor of T-beams. Furthermore, the results calculated using the formula for the damage degree of rectangular beam elements with cracks have a large relative error compared to the damage degree of T-beam crack elements, and are not applicable. Summary of the Invention

[0004] To address the problem of calculating the theoretical value of element damage in T-beam crack damage, this invention proposes a method for calculating the element damage of cracked T-beams using the stress diffusion angle method.

[0005] The method for calculating the damage degree of a cracked T-beam element using the stress diffusion angle method described in this invention comprises the following steps:

[0006] (1) Set an appropriate number of measuring points for the cracked T-beam. The beam segment between adjacent measuring points is recorded as a unit, and the length of the measuring point unit is δl.

[0007] (2) Based on the crack height h cr Given the T-beam section height h, calculate the relative crack height ζ, ζ = h cr / h;

[0008] (3) Calculate the additional spring stiffness parameter of the crack based on the relative height ζ of the crack. It can be calculated according to the stress intensity factor handbook;

[0009] (4) Calculate the crack stress diffusion angle α(ζ) using a rectangular cross-section beam, based on the method of equivalent crack element linear stiffness. The calculation formula is as follows:

[0010]

[0011] Where h is the height of the rectangular beam section, h cr Where I is the crack height, and I0 is the moment of inertia of the rectangular beam section. b is the width of the rectangular beam section, N is the number of beam segments divided on one side of the stress diffusion section, and I 0dm Let be the moment of inertia corresponding to the rectangular beam section of the m-th segment in the stress diffusion part. h 0m h is the height of the m-th beam segment section. 0m =hf(h cr ), f(h cr The stress diffusion function is calculated based on the specific stress diffusion mode.

[0012] (5) Calculate the moment of inertia I of the undamaged T-beam section:

[0013] The area moment S0 of the undamaged T-beam cross-section:

[0014]

[0015] Where b1 and b2 are the widths of the web and top plate of the T-beam section, respectively, and h1 and h2 are the heights of the web and top plate of the T-beam section, respectively;

[0016] The cross-sectional area of ​​the T-beam is A0:

[0017] A0 = b1h1 + b2h2;

[0018] Neutral axis coordinates of the undamaged section (y) co :

[0019]

[0020] According to the parallel axis shift theorem, the top plate and web are divided into two parts. The moment of inertia of the web is I1, and the moment of inertia of the top plate is I2. The total moment of inertia of the T-beam section is I:

[0021]

[0022]

[0023] I = I1 + I2;

[0024] (6) Calculate the moment of inertia of the beam segment in the stress diffusion part:

[0025] The beam segment on one side of the stress diffusion section has a length of l2. Using the stress diffusion model, l2 = h. crtanα(ζ) is equivalent to N small segments of length l² / N connected in series, where N is the number of beam segments on one side of the stress diffusion region. Each small segment has a T-shaped cross-section, and the height of the midpoint of the small segment is taken as the height of the small segment. The height h of the stress-free zone of the m-th small segment is... crm :

[0026]

[0027] a) Web cracking

[0028] The web height of the m-th segment is h. 1dm :

[0029]

[0030] The area moment S of the cross-section of the m-th segment m :

[0031]

[0032] The cross-sectional area A of the m-th segment m :

[0033] A m =b1h 1dm +b2h2;

[0034] The neutral axis coordinate y of the m-th segment cm :

[0035]

[0036] According to the parallel axis shift theorem, the top plate and web are divided into two parts, and the moment of inertia of the web is I. 1m The moment of inertia of the top plate is I. 2m The total moment of inertia of the T-beam section is I. wm :

[0037]

[0038]

[0039] I wm =I 1m +I 2m ;

[0040] b) Cracks in the roof slab

[0041] The height of the top plate of the m-th segment is h. 2dm :

[0042]

[0043] The area moment S of the cross-section of the m-th segment m:

[0044]

[0045] The cross-sectional area A of the m-th segment m :

[0046] A m =b1h1+b2h 2dm ;

[0047] The neutral axis coordinate y of the m-th segment cm :

[0048]

[0049] According to the parallel axis shift theorem, the top plate and web are divided into two parts, and the moment of inertia of the web is I. 1m The moment of inertia of the top plate is I. 2m The total moment of inertia of the T-beam section is I. fm :

[0050]

[0051]

[0052] I fm =I 1m +I 2m ;

[0053] (7) Calculation of damage degree of T-beam element using stress diffusion angle method:

[0054] a) Web cracking

[0055] The linear stiffness K of the undamaged T-beam element with a measuring point element length δl:

[0056]

[0057] Where E is the elastic modulus of the material, and I is the moment of inertia of the undamaged T-beam cross section;

[0058] The linear stiffness K of an undamaged T-beam segment of length l1 nd :

[0059]

[0060] Where l1 is the length of the measuring point unit minus the length of the stress diffusion part, l1=(δl-2l2) / 2;

[0061] The linear stiffness K of the m-th segment of length l² / N m :

[0062]

[0063] The linear stiffness K of a T-beam stress diffusion segment, divided into N segments of length l2, is obtained using the beam segment series connection method. xf :

[0064]

[0065] By using the method of connecting beam segments in series, two undamaged segments of length l1 and two stress-diffusing beam segments of length l2 are connected in series to obtain the linear stiffness K of the cracked T-beam element. d :

[0066]

[0067] The damage degree D of the web crack element of the T-beam is derived by comprehensive derivation. eT-wc for:

[0068]

[0069] D eT-wc It can calculate the damage degree of T-beam elements with web cracks relatively accurately;

[0070] b) Cracks in the roof slab

[0071] Similar to web cracking, the damage degree D of the crack element in the top plate of the T-beam can be derived. eT-fc for:

[0072]

[0073] D eT-fc The calculation accuracy is higher than D eT-wc The crack height h is low; further adjustments could be considered. cr Improve calculation accuracy;

[0074] (8) Correct the crack height h cr Calculation of damage degree of T-beam element:

[0075] a) Web cracking

[0076] Considering the web crack height h cr The revised formula for calculating the degree of unit damage is D. eT-wco :

[0077]

[0078]

[0079] Among them, H cr This is the corrected crack height;

[0080] b) Cracks in the roof slab

[0081] Considering the crack height h in the top platecr The revised formula for calculating the degree of unit damage is D. eT-fco :

[0082]

[0083]

[0084] Correct crack height h cr The formula for calculating the damage level of T-beam elements is more accurate.

[0085] Specifically, in step (3), the additional spring stiffness parameter for the crack... It can be calculated as follows:

[0086]

[0087] Where ζ is the relative height of the crack, and F(ζ) is the crack stress intensity factor coefficient.

[0088] Specifically, in step (4), the crack stress diffusion angle α(ζ) can be calculated according to the linear diffusion model:

[0089] α(ζ) = 74.5 - 28.895ζ;

[0090] Where ζ is the relative height of the crack, ζ = h cr / h, where h is the height of the T-beam section. cr The crack height is α(ζ), and the unit is degrees.

[0091] Specifically, in step (1), the length δl of the measuring point unit is not less than the cross-sectional height h, and the number of measuring points is not less than 4.

[0092] Specifically, in steps (4), (6), (7), and (8), the number of beam segments N on one side of the stress diffusion section is not less than 100.

[0093] This invention proposes a crack stress diffusion model with equivalent linear stiffness based on a method for calculating the damage degree of a single-sided transverse crack in a rectangular beam. The damage degree of crack elements in the web and top plate of a T-beam is calculated according to the stress diffusion angle. Furthermore, the accuracy of the calculation of the damage degree of cracked T-beam elements is improved by correcting the crack height. The applicability of the method is verified through numerical examples, providing a theoretical basis for T-beam structural damage identification experiments. Attached Figure Description

[0094] Figure 1 This is a schematic diagram illustrating the calculation of the damage degree of the cracked T-beam unit in this invention.

[0095] Figure 2 This is the cracked beam element model of the present invention.

[0096] Figure 3 This is the cracked additional spring beam unit model of the present invention.

[0097] Figure 4 This is the Type I cracked beam model of the present invention.

[0098] Figure 5 This invention relates to a crack stress propagation model (linear type).

[0099] Figure 6 This is the equivalent part of the linear stress diffusion mode of the present invention.

[0100] Figure 7 This is the fitted curve of the α(ζ) formula of this invention.

[0101] Figure 8 This is the standard T-beam cross section of this invention.

[0102] Figure 9 This invention relates to the transverse crack unit of the web of a T-beam.

[0103] Figure 10 This is the transverse crack section of the web of the T-beam of the present invention.

[0104] Figure 11 This invention relates to the transverse crack unit on the top plate of the T-beam.

[0105] Figure 12 This is the transverse crack section of the top plate of the T-beam of the present invention.

[0106] Figure 13 This is a detailed diagram of the stress diffusion model for cracks in the web of a T-beam according to the present invention.

[0107] Figure 14 This is the cross-section of the web of the T-beam of this invention.

[0108] Figure 15 This invention relates to the stress diffusion beam segment of a T-beam.

[0109] Figure 16 This is the cross-section of the m-th segment of the web crack stress diffusion beam segment of the present invention.

[0110] Figure 17 This is a detailed diagram of the stress diffusion model of cracks in the top plate of the T-beam of this invention.

[0111] Figure 18 It is the cross-section of the m-th segment of the top plate crack stress diffusion beam segment of the present invention.

[0112] Figure 19 This is the T-beam model of the present invention.

[0113] Figure 20 This is the cross-section of the web crack in this invention.

[0114] Figure 21This is the web cracked beam model of the present invention.

[0115] Figure 22 This invention relates to the modeling of a T-beam with web cracks.

[0116] Figure 23 This is the cross-section of the crack in the top plate of the T-beam of the present invention.

[0117] Figure 24 This is the model of the cracked top plate of the T-beam of the present invention.

[0118] Figure 25 This invention relates to the modeling of a cracked beam on the top plate of a T-beam.

[0119] Figure 26 This is the crack stress propagation mode D in Embodiment 1 of the present invention. eT-wc Plotting the error against the sample.

[0120] Figure 27 This is the modified crack stress propagation mode D of Embodiment 1 of the present invention. eT-wco Plotting the error against the sample.

[0121] Figure 28 The damage degree D of the rectangular beam crack element in Embodiment 1 of the present invention. er-c Plotting the error against the sample.

[0122] Figure 29 The damage degree D of the rectangular beam crack element in Embodiment 1 of the present invention. er-ci Plotting the error against the sample.

[0123] Figure 30 This is the crack stress diffusion mode D in Embodiment 2 of the present invention. eT-fc Plotting the error against the sample.

[0124] Figure 31 This is the modified crack stress propagation mode D of Embodiment 2 of the present invention. eT-fco Plotting the error against the sample.

[0125] Figure 32 The damage degree D of the rectangular beam crack element in Embodiment 2 of the present invention. er-c Plotting the error against the sample.

[0126] Figure 33 The damage degree D of the rectangular beam crack element in Embodiment 2 of the present invention. er-ci Plotting the error against the sample. Detailed Implementation

[0127] The present invention will be further described below with reference to the accompanying drawings and embodiments. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements.

[0128] Figure 1This is a schematic diagram illustrating the calculation of damage degree of the cracked T-beam element in this invention. In the diagram, δl is the length of the measuring point element, and h, h cr b1 and b2 are the beam height and crack height of the T-beam section, respectively; b1 and b2 are the web and top plate widths of the T-beam section, respectively; h1 and h2 are the web and top plate heights of the T-beam section, respectively; EI and EI are the beam height and crack height of the T-beam section, respectively. d EI eq D represents the stiffness of the undamaged beam segment, the stiffness of the damaged beam segment, and the equivalent stiffness of the damaged beam segment, respectively. e denoted as the degree of damage to the element, and denoted as the quantity to be determined.

[0129] The method for calculating the damage degree of a cracked T-beam element using the stress diffusion angle method described in this invention is as follows:

[0130] I. Damage Degree D of Cracked Rectangular Beam Element e Calculation method

[0131] 1) Degree of damage to element linear stiffness

[0132] Based on the model of a cracked element equivalent to a cracked spring, the location of the crack is considered as a series of lengthless, massless, and stiff torsional springs. The cracked spring beam element model is as follows: Figure 2 , Figure 3 X in the figure n X n+1 Here, n and n+1 are the measurement point location numbers, δl is the measurement point unit length, and h is the measurement point location number. cr K represents the crack height. r Add spring stiffness to the crack, l x l is half the length of the undamaged portion of the measuring point unit. x =δl-l x .

[0133] The linear stiffness of the damage-free element is K:

[0134]

[0135] In the formula, E is the elastic modulus of the material, and I is the moment of inertia of the cross section.

[0136] By connecting the crack-added spring in series with the undamaged beam element, the equivalent linear stiffness of the element containing the crack damage is obtained as K. d :

[0137]

[0138] The formula for calculating the damage degree of cracked elements is D. e :

[0139]

[0140] 2) Additional spring for crack

[0141] The model diagram of the Type I cracked beam is shown below. Figure 4 In the figure, M is the bending moment at the beam end, L0 is the distance from the crack measuring point element to the left end of the beam, and L is the calculated span of the beam.

[0142] Chondros proposed a Type I crack calculation model, suggesting that when a beam structure develops a crack, an additional flexibility is generated at the crack location. This additional flexibility can be calculated based on the additional strain energy generated by the crack. According to Castigliano's theorem, the additional displacement θ generated by the crack in a structure under general loads is... * The load can be obtained by differentiating the additional strain energy generated by the structure:

[0143]

[0144] In the above formula: U F This is due to the additional strain energy generated by the cracks in the structure; M is the bending moment. Additional strain energy U F It can be obtained by integral calculation using J-integral:

[0145]

[0146] In the formula, b is the width of the beam section, h cr Let J be the crack height. The J integral can be calculated from the corresponding crack stress intensity factor. The specific formula for calculating the strain energy density J integral is as follows:

[0147]

[0148] In the formula, F(ζ) is a coefficient related to the stress intensity factor and the relative height of the crack, which can be selected from the stress intensity factor handbook according to the stress condition of the beam.

[0149] Additional angular displacement θ * for:

[0150]

[0151] Simplifying equation (7) yields:

[0152]

[0153] In the formula: ζ=h cr / h,

[0154] Differentiating equation (8) with respect to the bending moment M, we obtain the crack-additional spring flexibility c. * The expression is:

[0155]

[0156] Stiffness K of the spring with crackr The expression is:

[0157]

[0158] In the formula: ζ=h cr / h, It is the parameter calculation formula obtained by the transformation integral of the additional strain energy calculation formula; Φ(ζ) is the parameter calculation formula after the integration of the correction formula; And Φ(ζ) varies depending on the chosen F(ζ); I is the moment of inertia of the rectangular beam section.

[0159] 3) Formula for calculating the damage degree of transverse crack elements

[0160] For a transverse crack in a beam structure element, based on the crack stress intensity factor and the crack-added spring stiffness, combined with the crack series spring theory and the linear stiffness damage degree calculation method, the damage degree calculation formula for a rectangular beam element with a transverse crack is derived. From equations (1) to (3) and equation (10), the damage degree D of the rectangular beam element with a transverse crack is derived. e :

[0161]

[0162] In the above formula: E is the elastic modulus, I is the moment of inertia, ζ is the relative height of the crack, and ζ = h cr / h, where h is the beam height. cr Let δ be the crack height and δl be the element length at the measuring point. The element damage degree of a transverse crack on one side of a rectangular beam is denoted as D. er-c (D e The damage level is indicated by "Damage", where "r" represents a rectangular beam and "c" represents a crack.

[0163] Additional spring stiffness parameters for cracks Calculation method:

[0164] From the stress intensity factor handbook, the coefficient F(ζ) of the stress intensity factor for a single-sided crack in pure bending and The formula is as follows:

[0165] F(ζ) = 1.122 - 1.40ζ + 7.33ζ 2 -13.08ζ 3 +14.0ζ 4 (12)

[0166]

[0167] Equation (12) is used in a range and with an accuracy of ζ<0.6. The calculation error of the stress intensity factor is within 0.2%. The length of the measuring point unit δl is not less than the section height 2h. The case analysis shows that when δl is not less than the section height h, the effect is also good.

[0168] II. Crack Damage Equivalent Stress Diffusion Model

[0169] Due to the influence of cracks, a stress-free zone exists at and around the crack location. This leads to a significant discrepancy between the damage calculated directly based on the series stiffness of the beam segments for slotted or inclined damage and the actual damage. Taking this as a starting point, the stress diffusion effect of the crack can be considered as diffusion along a certain angle, called the stress diffusion angle of the crack. The stiffness of the portion outside the stress diffusion zone is neglected, thus causing damage. The stiffness of the stress diffusion portion is then calculated segment by segment, and then incorporated into the damage element for damage degree calculation, thereby obtaining the crack damage of other beam sections. Based on the common rectangular beam transverse crack series spring model, the location of the crack is treated as an added lengthless, massless, stiff spring for structural damage calculation.

[0170] The transverse crack stress propagation model is an equivalent model of beam structure damage calculation that removes the stress-free zone caused by the crack, replacing the crack-added spring model. It assumes the stress propagation mode is linear. Figure 5 In the figure, the gray area represents the assumed stress-free region, h cr α is the crack height, α is the stress diffusion angle on one side of the crack, δl is the length of the measuring point element, and according to the stress intensity factor handbook, δl = 2h; l1 is half the length of the undamaged part of the measuring point element, l2 is half the length of the stress diffusion region, 2l2 = δl - 2l1; h cr The formula relating l2 and α is: tanα=l2 / h cr The coordinate system in the figure has the crack tip as the origin, the beam length direction as the x-axis, and the beam height direction as the y-axis.

[0171] The stress diffusion angle of a linear crack is calculated using the equivalent crack element stiffness method, based on the crack element stiffness K calculated using a rectangular beam series spring model. d The crack element linear stiffness K is equivalent to the crack stress diffusion angle calculated. dSDA Therefore, the stress diffusion angle α can be calculated. The theoretical derivation is as follows:

[0172] The stress intensity factor, as a criterion for whether a crack will continue to propagate, is related to the local stress state. However, the degree of crack damage is a characterization of the regional impact of the crack on the beam. Without considering whether the crack will continue to propagate, the degree of element damage caused by the crack is independent of the stress state. The stiffness K of a crack-connected spring element...d Equations (1), (2), (10), and (13) are used for calculation. The linear stiffness K of the undamaged element is calculated using equation (1), and the linear stiffness K of the undamaged part on one side of the cracked element is... nd :

[0173]

[0174] The stress diffusion portion on one side can be considered as a beam segment of length l2 composed of N small segments of equal length but unequal height connected in series. In actual calculations, N = 100 indicates that the calculation has converged. Figure 6 .

[0175] Stress diffusion length l1 and h cr The relationship between the length of the lossless segment l2 and the length of the lossless segment is: l2 = h cr ·tanα, δl=2l1+2l2, the linear stiffness K of the m-th segment of the stress diffusion part on one side of the crack damage element. xm :

[0176]

[0177] The stiffness K of this part is calculated using the beam segment series stiffness calculation method. x :

[0178]

[0179] By connecting the four parts in series, the equivalent linear stiffness K of the element containing the crack stress diffusion angle is obtained. dSDA :

[0180]

[0181] The method of using the equivalent crack element linear stiffness, i.e., K d =K dSDA The crack stress diffusion angle α of the equivalent damage to the rectangular beam can be obtained, and its calculation formula is as follows:

[0182]

[0183] Where h is the height of the rectangular beam section, h cr Where I is the crack height, and I0 is the moment of inertia of the rectangular beam section. b is the width of the rectangular beam section, N is the number of beam segments in the stress diffusion section, and I 0dm Let m be the moment of inertia corresponding to the rectangular beam section of segment m. h 0m The height of the m-th beam segment section; ζ is the relative height of the crack, ζ = h cr / h, The spring stiffness parameter for the crack is added and calculated according to formula (13).

[0184] Calculate the fitting formula for α according to the pure bending stress intensity factor calculation formula provided in the stress intensity factor manual. According to the stress intensity factor manual, equation (13) is applicable to the damage situation where ζ = h cr / h ≤ 0.6. Use the data of ζ = h cr / h ≤ 0.6 to fit the calculation formula for α. The fitting curve graph is as Figure 7 , where R 2 is the goodness of fit, and its value range is (0, 1). The closer R 2 is to 1, the better the fitting effect. The calculation formula for the relative relationship between the crack stress diffusion angle α and ζ is as follows:

[0185] α(ζ) = 74.5 - 28.895ζ (19)

[0186] II. Damage degree of crack T-beam element by stress diffusion angle method

[0187] 1) Simplified diagram of T-beam crack damage

[0188] The theoretical derivation takes the simplified model of a standard T-beam as an example, and the schematic diagram is as Figure 8 . The single transverse crack of the T-beam is divided into two types: the transverse crack of the top plate and the transverse crack of the web. The schematic diagram is as Figures 9-12 . In the figure, X n , X n+1 is the measurement point position number, h is the total beam height, h1 is the web height, h2 is the top plate height, b1 is the web width, b2 is the top plate width, h cr is the crack depth, and δl is the length of the measurement point unit; the shaded area in the figure is the part with crack damage. Only consider the web crack with a height less than the web height, that is, h cr <h1; only consider the top plate crack with a height less than the top plate height, that is, h cr <h2.

[0189] 2) Transverse crack of the web

[0190] The derivation of the calculation method for the damage degree of the web crack (web crack of T-beam) unit takes a web crack as an example. The model schematic diagram is Figure 13 . Only consider the web crack with a height less than the web height, that is, h cr <h1. The cross-section of the web of an undamaged T-beam is as Figure 14 . Figure 13 In it, X n , X n+1 is the measurement point position number, h is the total beam height, h1 is the web height, h2 is the top plate height, h crWhere α is the crack height, α is the stress diffusion angle of the crack on one side, l1 is half the length of the undamaged part of the measuring point element, l2 is half the length of the stress diffusion region, and 2l2=δl-2l1; h cr The formula relating l2 and α is: tanα=l2 / h cr .

[0191] Figure 14 In the diagram, b1 is the width of the web, b2 is the width of the top plate, and y c This represents the distance in the y-direction from the centroid of the cross section to the origin of the coordinate system, and is also the position of the neutral axis. Figure 14 Based on the coordinate system shown, the area moment S0 of the undamaged T-beam cross-section is:

[0192]

[0193] The cross-sectional area of ​​the T-beam is A0:

[0194] A0=b1h1+b2h2 (21)

[0195] Undamaged neutral axis coordinates y co :

[0196]

[0197] According to the parallel axis shift theorem, the top plate and web are divided into two parts. The moment of inertia of the web is I1, the moment of inertia of the top plate is I2, and the total moment of inertia of the T-beam section is I.

[0198]

[0199]

[0200] I = I1 + I2 (25)

[0201] Linear stiffness K of an undamaged T-beam element of length δl:

[0202]

[0203] The linear stiffness K of an undamaged T-beam segment of length l1 nd :

[0204]

[0205] Next, the linear stiffness K of the stress propagation segment of the T-beam web crack is calculated. x The diffusion angle α is calculated using equation (19). According to the stress diffusion model, the stress diffusion portion on one side can be equivalent to N small segments of length l² / N connected in series, as shown in the schematic diagram. Figure 15 . Figure 15Each segment is a T-beam section, and the height of the segment is taken as the web height at the middle position of the segment. A schematic diagram of the m-th segment is shown below. Figure 16 With the bottom of the effective cross-section as the coordinate system, the height h of the stress-free zone of the m-th segment. crm :

[0206]

[0207] The web height of the m-th segment is h. 1dm :

[0208]

[0209] The area moment S of the cross-section of the m-th segment m :

[0210]

[0211] The cross-sectional area A of the m-th segment m :

[0212] A m =b1h 1dm +b2h2 (31)

[0213] The neutral axis coordinate y of the m-th segment cm :

[0214]

[0215] According to the parallel axis shift theorem, the top plate and web are divided into two parts, and the moment of inertia of the web is I. 1m The moment of inertia of the top plate is I. 2m The total moment of inertia of the T-beam section is I. wm .

[0216]

[0217]

[0218] I wm =I 1m +I 2m (35)

[0219] The linear stiffness K of the m-th segment of length l² / N m :

[0220]

[0221] The linear stiffness K of a T-beam stress diffusion segment, divided into N segments of length l2, is obtained using the beam segment series connection method. xf :

[0222]

[0223] Using the method of connecting beam segments in series, two undamaged segments with a length of l1 and two stress-diffusion beam segments with a length of l2 are connected in series. According to Equation (27) and Equation (37), the linear stiffness K of the cracked T-beam element is derived. d :

[0224]

[0225] Based on the above formula derivation, the damage degree D of the transverse crack element in the T-beam web eT-wc is:

[0226]

[0227] 3) Transverse crack in the top plate

[0228] The calculation method for the damage degree of the T-beam top plate crack (flange crack of T-beam) element is derived taking a top plate crack as an example. The schematic diagram of the model is Figure 17 . Only consider the top plate crack with a height less than the top plate height, that is, h cr <h2. The cross-sectional schematic diagram of the T-beam top plate crack is as Figure 18 . Figure 17 In, X n , X n+1 is the measurement point position number, h is the total beam height, h1 is the web height, h2 is the top plate height, h cr is the crack height, α is the stress diffusion angle on one side of the crack, l1 is the half length of the undamaged part of the measurement point unit, l2 is the half length of the stress diffusion area, 2l2 = δl - 2l1; h cr , l2 and α have the following formula relationship: tanα = l2 / h cr .

[0229] In the case of damage to the T-beam top plate crack, the linear stiffness K of the undamaged unit and the linear stiffness K of the undamaged beam segment of the crack damage unit nd are calculated in the same way as for the web crack, using Equation (26) and Equation (27).

[0230] Next, calculate the linear stiffness K of the stress diffusion beam segment of the T-beam top plate crack x . According to the stress diffusion model, the beam segment on one side of the stress diffusion part is equivalent to N small segments with a length of l2 / N connected in series. The schematic diagram is as Figure 15 .

[0231] Figure 15 In each small segment is the T-beam cross-section, and the beam height at the middle position of the small segment is taken as the height of the small segment. The schematic diagram of the m-th small segment is as Figure 18 , taking the bottom of the effective cross-section as the coordinate system, the stress-free zone height h of the m-th small segmentcrm :

[0232]

[0233] The height of the top plate of the m-th segment is h. 2dm :

[0234]

[0235] The area moment S of the cross-section of the m-th segment m :

[0236]

[0237] The cross-sectional area A of the m-th segment m :

[0238] A m =b1h1+b2h 2dm (43)

[0239] The neutral axis coordinate y of the m-th segment cm :

[0240]

[0241] According to the parallel axis shift theorem, the top plate and web are divided into two parts, and the moment of inertia of the web is I. 1m The moment of inertia of the top plate is I. 2m The total moment of inertia of the T-beam section is I. fm .

[0242]

[0243]

[0244] I fm =I 1m +I 2m (47)

[0245] The linear stiffness K of the m-th segment of length l² / N m :

[0246]

[0247] The linear stiffness K of a T-beam stress diffusion segment, divided into N segments of length l2, is obtained using the beam segment series connection method. xd :

[0248]

[0249] By using the method of connecting beam segments in series, two undamaged segments of length l1 and two stress-diffusing beam segments of length l2 are connected in series to obtain the linear stiffness K of the cracked T-beam element. d :

[0250]

[0251] Based on the above formula derivation, the damage degree D of the crack element in the top plate of the T-beam is... eT-fc for:

[0252]

[0253] III. Damage Degree of Cracked T-Beam Element with Modified Crack Height

[0254] 1) Correction of the formula for calculating the damage degree of transverse crack elements in the web

[0255] For the crack height h in equation (39) cr The damage assessment calculation was revised and optimized. First, multiple finite element models of cracked T-beams were created using Ansys to obtain multiple sets of damage assessment data. The T-beam models are shown below. Figure 19 As shown. The span is 500mm, with each element divided into 50mm sections, for a total of 10 elements and 11 nodes (the numbers in the top row of the model are the element numbers, and the numbers in the bottom row are the node numbers). The material's elastic modulus is 2.06 × 10⁻⁶. 5 MPa, density 7.9 g / cm³ 3 Poisson's ratio is 0.25; measuring point length δl = 50. The basic sample diagram of the damage model is as follows: Figure 20 , Figure 21 Ansys was used for modeling, and Solid186 was selected for calculation. The web crack modeling diagram is shown below. Figure 22 .

[0256] Model actual damage level D e0 The deflection curvature damage identification theory (54) is used for quantitative analysis. The deflection curvature damage identification theory for beam structures is a damage identification method based on the difference in deflection curvature before and after damage at each node of the beam structure, enabling damage localization and quantification. The deflection curvatures of node n before and after damage are as follows:

[0257]

[0258]

[0259] In the formula w n w represents the deflection at node n. n "" represents the curvature of node n, and the subscripts 'u' and 'd' represent the undamaged state and the damaged state, respectively.

[0260] The quantitative formula for the degree of unit damage is:

[0261]

[0262] The damage level data quantified by equation (54) is used as the theoretical damage level D. e0 Write an optimization function to calculate D using formula (39). eT-wc The correction was performed to obtain the corrected crack height H. cr Substituting into the original formula, the corrected form for the crack height is as follows:

[0263]

[0264] The correction function for web crack damage in T-beams is as follows: The corrected formula for calculating web crack damage in T-beams is:

[0265]

[0266] In the formula, the subscript o denotes optimization, stress diffusion angle α, and moment of inertia I of the damaged section. wm still using h cr , ζ=h cr / h, h=h1+h2 are used for calculation. D e0 With D eT-wco The minimum sum of squared errors f is used as the criterion for evaluating the quality of the fitting correction method. f is calculated by equation (57).

[0267]

[0268] The sample data are shown in Tables 1 to 4. The second group of sample data is basically the same as the first group. In fact, in the subsequent correction function... It was not used in the optimized fitting, but a case error analysis was performed on the second set of data.

[0269] Taking three-factor correction as an example, three-factor correction refers to... The correction function is derived based on three optimization factors. The two-factor correction function is obtained through optimal fitting based on the two-factor correction results. The calculation formulas are divided into product type and exponential type, and the function is obtained by optimization fitting. See Tables 5 and 6.

[0270] Table 1 Damage Degree of Web Crack Element in T-Beam (D) e0 Sample data 1 (h1=26, h2=4, b1=4)

[0271]

[0272] Table 2 Damage Degree of Web Crack Element in T-Beam (D) e0Sample data 2 (h1=22, h2=8, b1=4)

[0273]

[0274] Table 3 Damage Degree of Web Crack Units in T-Beams (D) e0 Sample data 3 (h1=18, h2=12, b1=4)

[0275]

[0276] Table 4 Sample data of damage degree of web crack element in T-beam (h1=14, h2=16, b1=4)

[0277]

[0278] Table 5. Three-Factor Correction Function for Web Crack Damage Height of T-Beams (Product type)

[0279]

[0280]

[0281] Table 6. Three-Factor Correction Function for Web Crack Damage Height of T-Beams (Exponential type)

[0282]

[0283]

[0284] From the two tables above, the minimum value of f for the final correction function is 0.0074, and its correction parameters are shown in Table 7.

[0285] Table 7 Optimal Correction Function for Three Factors of Web Crack Damage Height in T-Beams

[0286]

[0287] Conclusion: The exponential optimization effect was not improved. The single-factor correction function formula 1 or the three-factor correction function product formula 12 was adopted as the final correction result. The final correction function calculation formula is shown in Table 8.

[0288] Table 8 Correction function for web crack damage height of T-beams (Exponential type)

[0289]

[0290] Calculate the correction function using the three-factor product formula. The final revised formula is D. eT-wco :

[0291]

[0292]

[0293] 2) Correction of the formula for calculating the damage degree of transverse crack elements in the top plate

[0294] For the crack height h in equation (51) cr Perform parameter correction to optimize the damage level D eT-fc Calculation results. First, multiple cracked T-beam finite element models were created using Ansys. The basic parameters of the models were the same as in the previous section. Basic damage model diagrams are shown below. Figure 23 , Figure 24 Ansys was used for modeling, and Solid186 was selected for calculation. The modeling diagram is shown below. Figure 25 .

[0295] Write a function to correct the fit, and calculate D from equation (51). eT-fc After parameter fitting and correction, the optimized crack height H is obtained. cr Substituting into the original formula, the corrected function for the crack height takes the following form:

[0296]

[0297] Ensure that when the cross-sectional dimensions of the T-beam are b1 = b2 When b2 approaches ∞ Take, take Substituting into equation (60), we get:

[0298]

[0299]

[0300] The revised formula for calculating crack damage in the top plate of the T-beam is as follows:

[0301]

[0302] In equation (63), the stress diffusion angle α and the moment of inertia of the damaged section I fm still using h cr , ζ=h cr The damage degree is calculated using the formula / h, where h = h1 + h2. The corrected formula calculates D as the damage level. eT-fco D e0 With D eT-fco The minimum sum of squared errors, f, is used to evaluate the merits of the parameter correction method. f is calculated using equation (4.43):

[0303]

[0304] The sample data are shown in Tables 9 to 11.

[0305] Table 9. Damage Degree of Crack Units in the Top Plate of T Beam (D) e0 Sample data 1 (h1=26, h2=4, b1=4)

[0306]

[0307] Table 10T Beam Top Plate Crack Element Damage Degree D e0 Sample data 1 (h1=22, h2=8, b1=4)

[0308]

[0309] Table 11 Damage Degree of Crack Units in T Beam Top Plate (D) e0 Sample data 1 (h1=18, h2=12, b1=4)

[0310]

[0311] Based on equation (61), construct the calculation of optimization factors. Correction function obtained by fitting the minimum sum of squared errors The calculation formula is shown in Table 12. In the table below, n1, n2, n3, and n4 are the optimization coefficients of the modified formula, and h cr Let h1 be the crack height, h2 be the web height, h1 be the top plate height, b1 be the web width, and b2 be the top plate width, where h = h1 + h2. This is based on dimensional optimization factors, such as size factors. Damage factors wait.

[0312] Table 12 Correction Function for Crack Damage Height on Top Plate of T Beam

[0313]

[0314]

[0315] Based on the optimization parameter calculation results in the table above, the correction formulas with the smallest f are numbered 14, 22, and 34. Final correction function. The calculation formulas and coefficients are shown in Table 13.

[0316] Table 13 Correction Formula for Crack Damage Height on Top Plate of T Beam coefficient

[0317]

[0318] Based on the principles of simplicity and minimal error, the revised formula D was ultimately selected. eT-fco :

[0319]

[0320]

[0321] To make the degree of unit damage D e To achieve high calculation accuracy, δl / h≥1 must be satisfied, and the number of measuring points must be no less than 4. Generally, equally spaced measuring points can be used.

[0322] The number of beam segments N on one side of the stress diffusion section shall not be less than 100.

[0323] Example 1: Calculation of web cracks in T-beams

[0324] T-beam model as follows Figure 19 Damage analysis was performed on the models in Tables 1 to 4. Since the beam segment on one side of the stress diffusion area needs to be divided into N segments for analysis, N = 200 was chosen in the example. Furthermore, due to the numerous working conditions, a programming method was used for calculation. The uncorrected D was obtained. eT-wc D eT-wco D of the rectangular beam er-c and the use of cross-sectional moment of inertia damage (I cr / I) 1 / 3 Convert the relative height of the crack ζ = h cr / h Substituted into equation (11) to calculate D er-ci Error plot with respect to sample damage level, such as Figures 26-29 .

[0325] A comparison of the four figures shows that D calculated directly using the crack stress diffusion method... eT-wc The relative error between the model sample damage degree and the actual damage degree is less than 6%, and the calculation effect is better than that of the rectangular beam formula; the crack stress diffusion element damage degree calculation formula after three-factor correction calculates D eT-wco The error between the damage degree and the sample damage degree is less than 2%, and the calculation effect after correction is very good; while the D obtained by using the damage degree calculation formula of rectangular beam crack element is much better. er-c and D er-ci The relative error with the sample is large, resulting in poor calculation performance.

[0326] Example 2: Case Study of Cracks in the Top Slab of a T-Beam

[0327] T-beam model as follows Figure 19 Damage degree analysis was performed on the models in Tables 9 to 11. The uncorrected D... eT-fc D eT-fco D of the rectangular beam er-c and the use of cross-sectional moment of inertia damage (I cr / I) 1 / 3 Calculate the relative height of the crack ζ = h cr / h Substituted into equation (11) to calculate Der-ci Error plot with respect to sample damage level, such as Figures 30-33 .

[0328] As can be seen from the comparison of the figures, D calculated directly using the crack stress propagation method... eT-fc The relative error between the formula and the damage level of the model sample is large, resulting in poor calculation performance. Therefore, it is necessary to revise the formula. The revised formula for calculating the damage level of the crack element in the T-beam top plate under the crack stress propagation mode yields a higher D. eT-fco The error between the measured damage level and the sample damage level is mostly less than 5%, which is a significant improvement over the calculation results before correction; while the D obtained by using the rectangular beam crack element damage level calculation formula is much better. er-c and D er-ci The relative error with the sample is large, resulting in poor computational performance.

[0329] The above descriptions are merely two embodiments of the present invention. All equivalent changes and modifications made within the scope of the claims of the present invention are within the scope of the present invention.

Claims

1. A method for calculating the damage degree of a cracked T-beam element using the stress diffusion angle method, characterized in that... Includes the following steps: (1) Set an appropriate number of measuring points for the cracked T-beam. The beam segment between adjacent measuring points is recorded as one unit, and the length of the measuring point unit is... ; (2) Based on the crack height Calculate the relative height of the crack based on the T-beam section height h. , ; (3) Based on the relative height of the crack Calculate the additional spring stiffness parameters for cracks Calculated according to the stress intensity factor handbook; (4) Calculate the crack stress diffusion angle The calculation formula is as follows: For a rectangular cross-section beam, the stiffness is calculated using the equivalent crack element linear stiffness method. ； in, The moment of inertia of the rectangular beam section, b is the width of the rectangular beam section, and N is the number of beam segments divided on one side of the stress diffusion region. Let be the moment of inertia corresponding to the rectangular beam section of the m-th segment in the stress diffusion part. h 0m The height of the m-th beam segment section; , This is the stress diffusion function, calculated based on the specific stress diffusion mode; (5) Calculate the moment of inertia I of the undamaged T-beam section: The area moment S0 of the undamaged T-beam cross-section: ； Where b1 and b2 are the widths of the web and top plate of the T-beam section, respectively, and h1 and h2 are the heights of the web and top plate of the T-beam section, respectively; The cross-sectional area of ​​the T-beam is A0: ； Neutral axis coordinates of the undamaged section (y) co : ； According to the parallel axis shift theorem, the top plate and web are divided into two parts. The moment of inertia of the web is I1, and the moment of inertia of the top plate is I2. The total moment of inertia of the T-beam section is I: ； ； ； (6) Calculate the moment of inertia of the beam segment in the stress diffusion part: The beam segment on one side of the stress diffusion section has a length of l2. Using the stress diffusion model, Equivalent to long It is composed of N small segments connected in series, each segment having a T-shaped cross-section. The height of the segment is taken as the height of the midpoint of the segment, and the height h of the stress-free zone of the m-th segment is... crm : ； a) Web cracking The web height of the m-th segment is h. 1dm : ； The area moment S of the cross-section of the m-th segment m : ； The cross-sectional area A of the m-th segment m : ； The neutral axis coordinate y of the m-th segment cm : ； According to the parallel axis shift theorem, the top plate and web are divided into two parts, and the moment of inertia of the web is I. 1m The moment of inertia of the top plate is I. 2m The total moment of inertia of the T-beam section is I. wm : ； ； ； b) Cracks in the roof slab The height of the top plate of the m-th segment is h. 2dm : ； The area moment S of the cross-section of the m-th segment m : ； The cross-sectional area A of the m-th segment m : ； The neutral axis coordinate y of the m-th segment cm : ； According to the parallel axis shift theorem, the top plate and web are divided into two parts, and the moment of inertia of the web is I. 1m The moment of inertia of the top plate is I. 2m The total moment of inertia of the T-beam section is I. fm : ； ； ； (7) Calculation of damage degree of T-beam element using stress diffusion angle method: a) Web cracking Length of measuring point unit Linear stiffness K of the undamaged T-beam element: ； Where E is the elastic modulus of the material, and I is the moment of inertia of the undamaged T-beam cross section; Length The linear stiffness K of the undamaged T-beam segment nd : ； in, The length of the measuring point element minus half the length of the stress diffusion portion. ; Length is The linear stiffness K of the m-th segment m : ； Length is The stress diffusion segment of the T-beam is divided into N segments, and its linear stiffness K is obtained by using the beam segment series method. xf : ； By using the method of connecting beam segments, two beams of length are connected. The undamaged segment and two segments of length are By connecting the stress-diffusion beam segments in series, the linear stiffness of the cracked T-beam element can be obtained. : ； The damage degree of the web crack element of the T-beam is derived by comprehensive deduction. for: ； It can calculate the damage degree of T-beam elements with web cracks; b) Cracks in the roof slab Similar to web cracking, the damage degree of the crack element in the top plate of the T-beam can be derived. for: ； The calculation accuracy is relatively high Low, further adjust crack height h cr Improve calculation accuracy; (8) Correct crack height h cr Calculation of damage degree of T-beam element: a) Web cracking Considering the web crack height h cr The revised formula for calculating the degree of unit damage is as follows: : ； ； in, This is the corrected crack height; b) Cracks in the roof slab Considering the crack height h in the top plate cr The revised formula for calculating the degree of unit damage is as follows: : ； ； Correct crack height h cr The formula for calculating the damage level of T-beam elements is more accurate.

2. The method for calculating the damage degree of a cracked T-beam element using the stress diffusion angle method according to claim 1, characterized in that: In step (3), the additional spring stiffness parameter of the crack Calculate using the following method: ； ； in, This represents the crack stress intensity factor coefficient.

3. The method for calculating the damage degree of a cracked T-beam element using the stress diffusion angle method according to claim 1, characterized in that: In step (4), the crack stress propagation angle Specifically, the calculation is based on the linear diffusion model: ； in, The unit is degrees.

4. The method for calculating the damage degree of a cracked T-beam element using the stress diffusion angle method according to claim 1, characterized in that: In step (1), the length of the measuring point unit The cross-sectional height must be no less than h, and the number of measuring points must be no less than 4.

5. The method for calculating the damage degree of a cracked T-beam element using the stress diffusion angle method according to claim 1, characterized in that: In steps (4), (6), (7), and (8), the number of beam segments N on one side of the stress diffusion section shall not be less than 100.

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