CN115859733B - Crack T beam unit damage degree calculation method by Gaussian process regression - Google Patents
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Abstract
The invention discloses a crack T beam unit damage degree calculation method based on Gaussian process regression. The method comprises the following steps: setting proper number of measuring points for the crack T beam; calculating the relative height zeta of the crack according to the crack height and the section height of the T beam; calculating crack add-on spring rate parameter from ζCan be calculated according to a stress intensity factor manual; calculating a crack stress diffusion angle alpha (ζ), and calculating according to the equivalent crack unit line stiffness through a rectangular cross-section beam: calculating the moment of inertia of the undamaged T beam section; calculating the moment of inertia of the stress-diffusing portion beams Duan Jiemian; calculating the damage degree of the T beam unit by a stress diffusion angle method; and (5) calculating super parameters in the Gaussian process regression model by adopting a maximum likelihood estimation method, and calculating the damage degree of the T beam unit for correcting the crack height. The invention provides a theoretical calculation method for the damage degree of a crack T-beam unit, which provides theoretical basis for designing and calculating the actual damage degree of a T-beam when carrying out a damage degree quantitative test.
Description
Technical Field
The invention belongs to the field of structural health monitoring, relates to a method for calculating the theoretical damage degree of a beam structure, and particularly relates to a method for calculating the damage degree of a crack T beam unit by Gaussian process regression.
Background
In recent years, old bridges in China are more and more, and the problems are more and more remarkable. Among various bridge damage forms, cracks are a common damage form, and have great influence on the bearing capacity and the subsequent service life of the bridge. Although many studies have been made on the damage identification method of the beam structure based on the static index, the calculation method of the transverse crack theory damage is not so many. The distance between the measurement points is usually fixed during the damage recognition, and when the structure finds damage, it is likely that local damage is generated between the two measurement points, at this time, what is the equivalent damage degree between the two measurement points? The problem is a key problem of reasonably reading the damage quantitative index result, and the literature report for test verification is fresh at present because the damage degree quantitative difficulty is high.
T-beams are commonly used in various bridges due to their good load-bearing capacity and relatively simple construction processes. At present, a systematic and accurate theoretical method for calculating the T beam crack stress intensity factor does not exist, so that the theoretical damage degree of the transverse crack unit can not be directly calculated according to the T beam transverse crack stress intensity factor, and the relative error between the result of the crack rectangular beam unit damage degree calculation formula and the damage degree of the T beam crack unit is larger, and the method is not applicable.
Disclosure of Invention
Aiming at the problem of calculating the unit damage degree theoretical value of the T beam crack damage, the invention provides a crack T beam unit damage degree calculating method by Gaussian process regression.
The invention discloses a crack T beam unit damage degree calculation method by Gaussian process regression, which comprises the following steps:
(1) Setting proper number of measuring points for the crack T beam, wherein the beams Duan Ji between adjacent measuring points are a unit, and the length of each measuring point unit is delta l;
(2) According to the crack height h cr And the T beam section height h, calculating the crack relative height ζ, ζ=h cr /h；
(3) Calculating crack additional spring rate parameter according to crack relative height ζCan be calculated according to a stress intensity factor manual;
(4) Calculating a crack stress diffusion angle alpha (ζ), and calculating according to an equivalent crack unit line stiffness method through a rectangular cross-section beam, wherein the calculation formula is as follows:
wherein h is the section height of the rectangular beam, h cr For crack height, I 0 Is the moment of inertia of the cross section of the rectangular beam,b is the width of the rectangular beam section, N is the number of beams Duan Huafen on one side of the stress diffusion portion, I 0dm Moment of inertia corresponding to the rectangular beam section of the mth section of the stress diffusion section +.>h 0m Is the m-th beam Duan Jiemian height; h is a 0m ＝h-f(h cr )，f(h cr ) Calculating according to a specific stress diffusion mode as a stress diffusion function;
(5) Calculating the moment of inertia I of the undamaged T beam section:
area moment S of undamaged T beam cross section 0 ：
Wherein b 1 、b 2 Respectively the widths of webs and top plates of the T beam sections, h 1 、h 2 The heights of webs and top plates of the sections of the T beams are respectively;
t beam cross-sectional area A 0 ：
A 0 ＝b 1 h 1 +b 2 h 2 ；
Neutral axis coordinate y of undamaged section co ：
According to the parallel axis displacement theorem, the top plate and the web plate are divided into two parts, and the moment of inertia of the web plate is I 1 The moment of inertia of the top plate is I 2 The total moment of inertia of the T beam section is I:
I＝I 1 +I 2 ；
(6) The moment of inertia of the stress-diffusing portion beam Duan Jiemian was calculated:
the beam section on one side of the stress diffusion part has a length of l 2 By stress diffusion model, l 2 ＝h cr tan. Alpha. (ζ), equivalent to length l 2 N small sections of N are connected in series, N is the number of beams Duan Huafen on one side of the stress diffusion part, each small section is of a T-shaped section, the height of the midpoint position of the small section is taken as the height of the small section, and the height h of the stress-free area of the m-th small section crm ：
a) Web cracking
The web height of the m-th small section is h 1dm ：
Area moment S of cross section of the mth paragraph m ：
Cross-sectional area A of the mth section m ：
A m ＝b 1 h 1dm +b 2 h 2 ；
Neutral axis coordinate y of the mth paragraph cm ：
According to the parallel axis displacement theorem, the top plate and the web plate are divided into two parts, and the moment of inertia of the web plate 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 ：
I wm ＝I 1m +I 2m ；
b) Roof cracking
The height of the top plate of the m th small section is h 2dm ：
Transverse of the mth paragraphArea moment S of cross section m ：
Cross-sectional area A of the mth section m ：
A m ＝b 1 h 1 +b 2 h 2dm ；
Neutral axis coordinate y of the mth paragraph cm ：
According to the parallel axis displacement theorem, the top plate and the web plate are divided into two parts, and the moment of inertia of the web plate 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 ：
I fm ＝I 1m +I 2m ；
(7) Calculating the damage degree of the T beam unit by a stress diffusion angle method:
a) Web cracking
Undamaged T beam cell line stiffness K for measurement point cell length δl:
wherein E is the elastic modulus of the material, and I is the moment of inertia of the undamaged T beam section;
length of l 1 Is not damaged T-beam Duan Xian stiffness K nd ：
Wherein l 1 Is the length of the measuring point unit minus half the length of the stress diffusion part, l 1 ＝(δl-2l 2 )/2；
Length of l 2 Mth small section line stiffness of/N K m ：
Length of l 2 The T beam stress diffusion section divided into N sections is used for obtaining the line rigidity K by using a beam section series connection method xf ：
By means of beam sections connected in series, two beams of length l 1 Is provided and has a length l 2 The stress diffusion beam sections of the beam are connected in series to obtain the linear rigidity K of the crack-containing T-beam unit d ：
Comprehensively deducing damage degree D of crack unit of T beam web eT-wc The method comprises the following steps:
D eT-wc the damage degree of the web crack T beam unit can be accurately calculated;
b) Roof cracking
Similar to web cracking, the damage degree D of the crack unit of the T-beam top plate can be deduced eT-fc The method comprises the following steps:
D eT-fc is calculated with accuracy higher than D eT-wc Low, further correction of crack height h is contemplated cr The calculation accuracy is improved;
(8) Regression correction of crack height h in Gaussian process cr The damage degree of the T beam unit is calculated:
a) Gaussian process regression
According to the characteristics of the T-shaped section, taking the ith training sample for web crack
For roof cracking, the ith training sample is taken
Wherein x is i For the ith training sample, x i k For training sample x i K=1, 2, 3;
m training samples are taken, and corresponding outputs, namely correction coefficient column vectors, are as follows:
μ＝[μ 1 μ 2 …μ i …μ M ] T ；
wherein M is the number of training samples, μ is the correction coefficient column vector corresponding to the training samples, μ i I.e. with training sample x, as the i-th element in μ i A corresponding output;
the covariance matrix is:
wherein C is covariance matrix, C ij Omega is the element of the ith row and jth column in covariance matrix C 0 、ω 1 、ω 2 、ω 3 Is a super parameter;
wherein C is * For sample x to be analyzed * Covariance row vector of c) j Is C * The j-th element, x * Is a sample to be analyzed; can obtain the sample x to be analyzed * The corresponding correction coefficients are:
μ * ＝C * C -1 μ；
wherein mu * For sample x to be analyzed * Corresponding correction coefficient C -1 An inverse matrix of C;
b) Web cracking considers web crack height h cr The calculation formula of the modified unit damage degree is D eT-wco ：
H cr ＝μ*h cr ；
Wherein H is cr The crack height after correction;
c) Roof cracking
Considering the roof crack height h cr The calculation formula of the modified unit damage degree is D eT-fco ：
H cr ＝μ*h cr ；
Correcting crack height h cr The accuracy of the T beam unit damage degree calculation formula is higher.
Specifically, in step (3), the crack adds a spring rate parameterThe method can be calculated as follows:
wherein ζ is the relative height of the crack, and F (ζ) is the crack stress intensity factor coefficient.
Specifically, in the step (4), the crack stress diffusion angle α (ζ) may be specifically calculated according to a linear diffusion mode:
α(ζ)＝74.5-28.895ζ；
wherein ζ is the relative height of the crack, ζ=h cr And h is the section height of the T beam, h cr The units of α (ζ) are degrees for the crack height.
Specifically, in the step (8), the super-parameters are calculated by using a maximum likelihood estimation method MLE or a markov chain monte carlo method MCMC.
Specifically, in the step (1), the length δl of the measuring point unit is not less than the section height h, and the number of measuring points is not less than 4.
Specifically, in the steps (4), (6), (7) and (8), the number N of the beams Duan Huafen on the stress diffusing portion side is not less than 100.
The invention provides a crack stress diffusion model with equivalent linear stiffness based on a rectangular beam unilateral transverse crack damage degree calculation method, calculates the damage degree of crack units of a T beam web plate and a top plate according to the stress diffusion angle of the crack stress diffusion model, further improves the calculation precision of the crack T beam unit damage degree by a Gaussian process regression correction method, verifies the applicability of the method by a calculation example, and provides a theoretical basis for a T beam structure damage identification test.
Drawings
FIG. 1 is a schematic view of the calculation of the damage degree of a crack T beam unit according to the present invention.
FIG. 2 is a crack beam unit model of the present invention.
FIG. 3 is a crack attachment spring beam unit model of the present invention.
FIG. 4 is a model of a type I crack beam according to the present invention.
FIG. 5 is a crack stress propagation model (straight line) of the present invention.
Fig. 6 is an equivalent portion of a linear stress diffusion mode of the present invention.
Fig. 7 is a graph of the α (ζ) formula fit of the present invention.
Fig. 8 is a standard T-beam cross-section of the present invention.
Fig. 9 is a T-beam web transverse crack element of the present invention.
Fig. 10 is a transverse crack cross-section of a T-beam web of the present invention.
Fig. 11 is a T-beam roof transverse crack element of the present invention.
Fig. 12 is a cross section of a transverse crack in a T-beam roof of the present invention.
Fig. 13 is a detailed view of a T-beam web crack stress propagation model of the present invention.
Fig. 14 is a T-beam web section of the present invention.
Fig. 15 is a T-beam stress diffusion beam section of the present invention.
Figure 16 is a Liang Duandi m section cross section of a web crack stress propagation according to the invention.
Fig. 17 is a detailed view of a T-beam roof crack stress propagation model of the present invention.
Fig. 18 is a top plate crack stress propagation Liang Duandi m small section cross section of the present invention.
Fig. 19 is a T-beam model of the present invention.
FIG. 20 is a web crack cross-section of the present invention.
Figure 21 is a web crack Liang Moxing of the present invention.
Fig. 22 is a T-beam web crack Liang Jianmo of the present invention.
FIG. 23 is a cross section of a crack in a T-beam roof of the present invention.
Fig. 24 is a T-beam roof crack Liang Moxing of the present invention.
Fig. 25 is a T-beam roof crack Liang Jianmo of the present invention.
FIG. 26 shows a crack stress propagation pattern D according to an embodiment of the present invention eT-wc And a sample error map.
FIG. 27 shows a modified crack stress propagation pattern D according to an embodiment of the invention eT-wco And a sample error map.
FIG. 28 is a graph showing the damage degree D of a crack unit of a rectangular beam according to an embodiment of the present invention er-c And a sample error map.
FIG. 29 is a graph showing the damage degree D of a crack cell of a rectangular beam according to an embodiment of the present invention er-ci And a sample error map.
FIG. 30 shows a second crack stress propagation pattern D according to an embodiment of the present invention eT-fc And a sample error map.
FIG. 31 is a graph showing a crack stress propagation pattern D after correction according to the second embodiment of the present invention eT-fco And a sample error map.
FIG. 32 is a graph showing the degree of damage D of a crack unit of a second rectangular beam according to an embodiment of the present invention er-c And a sample error map.
FIG. 33 is a graph showing the degree of damage D of a crack unit of a second rectangular beam according to an embodiment of the present invention er-ci And a sample error map.
Detailed Description
The present invention is further described below with reference to the drawings and examples, wherein like reference numerals in the various drawings refer to the same or similar elements unless otherwise specified.
FIG. 1 is a schematic diagram showing the calculation of damage degree of a crack T beam unit according to the present invention, wherein δl is the length of a measuring point unit, h cr Beam height and crack height, b, of the T beam section respectively 1 、b 2 Respectively the widths of webs and top plates of the T beam sections, h 1 、h 2 The heights of the web plate and the top plate of the T beam section are respectively EI and EI d 、EI eq Respectively, an undamaged beam Duan Gangdu, a damaged beam section rigidity and an equivalent rigidity of the damaged beam section, D e The unit damage degree is the quantity to be calculated.
The invention relates to a crack T beam unit damage degree calculation method by Gaussian process regression, which comprises the following specific contents:
1. degree of damage D of crack rectangular beam unit e Calculation method
1) Degree of Unit line stiffness damage
Based on the fact that the crack-containing cell is equivalent to a crack-containing additional spring model, the point where a crack is to be generated is regarded as a series of torsion springs with no length and no mass and rigidity. The crack-added spring beam unit model is shown in fig. 2 and 3, in which X n 、X n+1 The position numbers of the measuring points are n, n+1, delta l, h cr For crack height, K r Adding spring rate to crack, l x The length of the undamaged part of the measuring point unit is half, l x ＝δl-l x 。
The line stiffness of the atraumatic unit is K:
wherein E is the elastic modulus of the material, and I is the section moment of inertia.
The crack additional spring is connected in series and enters the nondestructive beam unit to obtain the equivalent linear stiffness K containing the crack damage unit d ：
The damage degree calculation formula of the crack-containing unit is D e ：
2) Crack-added spring
The model diagram of the I-type crack beam is shown in figure 4, wherein M is the bending moment of the beam end and L 0 The distance between the crack measuring point unit and the left end of the beam is calculated and the L is the calculated span of the beam.
Chondros proposes that with a type i crack calculation model, when a crack is generated in the beam structure, additional compliance is generated at the crack, and the additional compliance can be calculated according to additional strain energy generated by the crack. According to Castigliano theorem, the additional displacement theta of the structure due to cracks under normal load * The load can be derived from the additional strain energy generated by the structure:
in the above formula: u (U) F Because of the additional strain energy of the structure due to the crack, M is the bending moment. Additional strain energy U F The integral calculation can be carried out by J integral to obtain:
wherein b is the width of the cross section of the beam, h cr Is the crack height. The J integral can be calculated by the corresponding crack stress intensity factor, and the specific calculation formula of the strain energy density J integral is as follows:
wherein F (ζ) is a coefficient of the stress intensity factor related to the relative height of the crack, and can be selected from a stress intensity factor manual according to the stress condition of the beam.
Additional angular displacement theta * The method comprises the following steps:
the simplified formula (7) is obtained:
wherein: ζ=h cr /h，
Deriving the bending moment M from the step (8) to obtain the crack additional spring flexibility c * The expression of (2) is:
stiffness K of crack-added spring r The expression of (2) is:
wherein: ζ=h cr /h，Is a parameter calculation formula obtained from the transformation integral of the additional strain energy calculation formula; Φ (ζ) is a parameter calculation formula after correction formula integration; />And Φ (ζ) is different depending on the selected F (ζ); i is the cross-sectional moment of inertia of the rectangular beam.
3) Formula for calculating damage degree of transverse crack unit
And (3) adding spring stiffness to the transverse crack of a certain unit of the beam structure according to the crack stress intensity factor, and deducing a damage degree calculation formula of the rectangular beam unit containing the transverse crack by combining a crack series spring theory and a line stiffness damage degree calculation method. The damage degree D of the rectangular beam unit containing the transverse crack is jointly deduced from the formulas (1) - (3) and the formula (10) e ：
In the above formula: e is the modulus of elasticity, and the modulus of elasticity,i is the moment of inertia, ζ is the relative height of the crack, ζ=h cr And h is the height of the beam, h cr And delta l is the length of the measuring point unit. The unit damage degree of the unilateral transverse crack of the rectangular beam is marked as D er-c (D e The Damage degree Damage is shown, r is a Rectangular beam, and c is a Crack.
Crack-added spring rate parameterIs calculated by the following steps:
coefficient F (ζ) and coefficient F (ζ) of stress intensity factor of single-sided crackThe formula is as follows:
F(ζ)＝1.122-1.40ζ+7.33ζ 2 -13.08ζ 3 +14.0ζ 4 (12)
the using range and precision of the formula (12) are zeta <0.6, the calculation error of the stress intensity factor is within 0.2%, the length delta l of the measuring point unit is not smaller than the section height of 2h, and the calculation analysis shows that the effect is good when delta l is not smaller than the section height of h.
2. Crack damage equivalent stress diffusion model
Because of the influence of the crack, there is no stress area at the position and around the crack, the relative error between the damage degree calculated directly according to the rigidity of the beam Duan Chuanlian and the actual damage degree is great, and the crack can be taken as a starting point, the stress diffusion effect generated by the crack is regarded as diffusing along a certain angle, the angle is called a crack stress diffusion angle (Stress Diffusion Angle of Crack), the part except the stress diffusion is not counted for rigidity so as to generate damage, then the rigidity of the stress diffusion part is calculated in sections, and then the damage degree calculation is performed by connecting the sections into a damage unit in series, so that the crack damage of other section beams can be obtained. According to a common rectangular beam transverse crack series spring model, the position where the crack is generated is regarded as a spring with rigidity and without length mass, so that structural damage calculation is performed.
The transverse crack stress propagation model is a beam structure damage calculation model which is equivalent to a crack-added spring model and is used for removing a stress-free area of the structure generated by cracks. The stress diffusion mode is assumed to be linear (fig. 5). In the figure, the gray areas are assumed unstressed areas, h cr Taking the height of the crack, alpha is the stress diffusion angle at one side of the crack, δl is the length of a measuring point unit, and δl=2h according to a stress intensity factor manual; l (L) 1 For the half length of the undamaged part of the measuring point unit, l 2 Is half length of stress diffusion region 2l 2 ＝δl-2l 1 ；h cr 、l 2 The formula relationship with α exists as: tan α=l 2 /h cr The method comprises the steps of carrying out a first treatment on the surface of the In the figure, the coordinate system takes the peak of the crack tip as an origin, the length direction of the beam as an x axis and the height direction of the beam as a y axis.
The calculation of the linear crack stress diffusion angle adopts an equivalent crack element line stiffness method, and the crack element line stiffness K is calculated according to a rectangular beam series spring model d Crack element line stiffness K equivalent to crack stress spread angle calculation dSDA Thereby back-calculating the stress relief angle α. The theoretical derivation is as follows:
the stress intensity factor is used as a criterion for whether the crack continues to develop or not and is related to the local stress state. However, the degree of crack damage is an indication of the regional effect of the crack on the beam, and the degree of unit damage caused by the crack is independent of the stress state regardless of whether the crack continues to develop. Crack series spring unit stiffness K d The calculation is performed using the formulas (1), (2), (10) and (13). The undamaged unit line rigidity K is calculated by the method (1), and the undamaged part line rigidity K at one side of the crack unit nd ：
One side stress diffusion portion is regarded as N equal-length unequal-heightThe length of the small sections formed by series connection is l 2 In actual calculation, n=100, i.e. the calculation convergence has been reached, as in fig. 6.
Stress diffusion length l 1 And h cr Length of lossless segment l 2 The relation of (2) is: l (L) 2 ＝h cr ·tanα，δl＝2l 1 +2l 2 Line stiffness K of mth small section of stress diffusion part at one side of crack damage unit xm ：
The rigidity K of the part is calculated by using the rigidity calculation method of the beam Duan Chuanlian x ：
The four parts are connected in series to obtain the equivalent linear rigidity K of the crack stress diffusion angle part unit dSDA ：
Method for applying equivalent crack element line stiffness, i.e. K d ＝K dSDA The crack stress diffusion angle alpha of the equivalent damage of the rectangular beam can be obtained, and the calculation formula is as follows:
wherein h is the section height of the rectangular beam, h cr For crack height, I 0 Is the moment of inertia of the cross section of the rectangular beam,b is the width of the rectangular beam section, N is the number of beams Duan Huafen of the stress diffusion portion, I 0dm Moment of inertia corresponding to the rectangular beam section of the mth section,/->h 0m Is the m-th beam Duan Jiemian height; />ζ is the relative height of the crack, ζ=h cr /h，/>A spring rate parameter is added to the crack, calculated according to equation (13).
Calculating a fitting alpha according to a calculation formula of a pure bending stress intensity factor provided by a stress intensity factor manual, wherein the formula (13) is applicable to ζ=h according to a description of the stress intensity factor manual cr In the case of damage with/h.ltoreq.0.6, ζ=h is used cr The calculation formula of data fitting alpha of/h less than or equal to 0.6 is shown in figure 7, wherein the fitting curve is R 2 For the goodness of fit, the value range is (0, 1), R 2 The closer to 1, the better the fitting effect. The calculation formula of the relative relation between the crack stress diffusion angle alpha and zeta is as follows:
α(ζ)＝74.5-28.895ζ (19)
2. stress diffusion angle method crack T beam unit damage degree
1) T beam crack damage sketch
The theoretical derivation is exemplified by a standard T-beam simplified model, schematically shown in fig. 8.T Liang Shan transverse cracks are divided into two types, namely a top plate transverse crack and a web transverse crack, and schematic diagrams are shown in figures 9-12. In the figure, X n ，X n+1 Numbering the positions of the measuring points, wherein h is the height of the total beam, and h 1 For the web height, h 2 Is the height of the top plate, b 1 For web width b 2 For the width of the top plate, h cr The depth of the crack is shown as delta l, and the length of the measuring point unit is shown as delta l; the hatched area in the figure is the part of the crack damage. Considering only web cracks of a height less than the web height, i.e. h cr ＜h 1 The method comprises the steps of carrying out a first treatment on the surface of the Considering only roof cracks of height less than the height of the roof, i.e. h cr ＜h 2 。
2) Transverse crack of web
The calculation method of damage degree of a T beam web crack (web crack of T-beam) unit is derived by taking a web crack as an example, and a model schematic diagram is shown in FIG. 13. Considering only web cracks of a height less than the web height, i.e. h cr ＜h 1 . The undamaged T-beam web cross section is shown in fig. 14. In FIG. 13, X n ，X n+1 Numbering the positions of the measuring points, wherein h is the height of the total beam, and h 1 For the web height, h 2 Is the height of the top plate, h cr For the crack height, α is the crack stress spread angle on one side, l 1 For the half length of the undamaged part of the measuring point unit, l 2 Is half length of stress diffusion region 2l 2 ＝δl-2l 1 ；h cr 、l 2 The formula relationship with α exists as: tan α=l 2 /h cr 。
In FIG. 14, b 1 For web width b 2 For roof width, y c The distance of the cross-section centroid from the origin of coordinates in the y-direction is also the position of the neutral axis. Based on the coordinate system shown in FIG. 14, the area moment S of the undamaged T-beam cross-section 0 ：
T beam cross-sectional area A 0 ：
A 0 ＝b 1 h 1 +b 2 h 2 (21)
Undamaged neutral axis coordinate y co ：
According to the parallel axis displacement theorem, the top plate and the web plate are divided into two parts, and the moment of inertia of the web plate is I 1 The moment of inertia of the top plate is I 2 The total moment of inertia of the T beam section is I.
I＝I 1 +I 2 (25)
Undamaged T beam cell line stiffness K of length δl:
length of l 1 Is not damaged T-beam Duan Xian stiffness K nd ：
/>
Next, calculating the line stiffness K of the T-beam web crack stress diffusion beam section x . The diffusion angle α is calculated by the formula (19), and the stress diffusion model shows that the one-side stress diffusion portion is equivalent to be l in length 2 N small segments of/N are connected in series, and the schematic diagram is shown in FIG. 15. In fig. 15, each small section is a T-beam section, and the height of the web plate at the middle of the small section is the height of the small section. The schematic diagram of the mth small section is shown in FIG. 16, in which the bottom of the effective section is taken as a coordinate system, and the height h of the stress-free region of the mth small section crm ：
The web height of the m-th small section is h 1dm ：
Area moment S of cross section of the mth paragraph m ：
Cross-sectional area A of the mth section m ：
A m ＝b 1 h 1dm +b 2 h 2 (31)
Neutral axis coordinate y of the mth paragraph cm ：
According to the parallel axis displacement theorem, the top plate and the web plate are divided into two parts, and the moment of inertia of the web plate 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 。
I wm ＝I 1m +I 2m (35)
Length of l 2 Mth small section line stiffness of/N K m ：
Length of l 2 The T beam stress diffusion section divided into N sections is used for obtaining the line rigidity K by using a beam section series connection method xf ：
By means of beam sections connected in series, two beams of length l 1 Is provided and has a length l 2 The stress diffusion beam sections of (a) are connected in series, and the linear rigidity K of the crack-containing T beam unit is deduced according to the formulas (27) and (37) d ：
/>
Deriving by combining the formulas, and obtaining the damage degree D of the transverse crack unit of the T-beam web eT-wc The method comprises the following steps:
3) Transverse crack in roof
The calculation method of the damage degree of the T-beam roof crack (flange crack of T-beam) unit is deduced by a roof crack example, and a model schematic diagram is shown in fig. 17. Considering only roof cracks of height less than the height of the roof, i.e. h cr ＜h 2 . The schematic of the crack cross section of the T-beam top plate is shown in figure 18. In FIG. 17, X n ，X n+1 Numbering the positions of the measuring points, wherein h is the height of the total beam, and h 1 For the web height, h 2 Is the height of the top plate, h cr For the crack height, α is the crack stress spread angle on one side, l 1 For the half length of the undamaged part of the measuring point unit, l 2 Is half length of stress diffusion region 2l 2 ＝δl-2l 1 ；h cr 、l 2 The formula relationship with α exists as: tan α=l 2 /h cr 。
Under the condition of crack damage of a T-beam top plate, the stiffness K of a nondestructive unit line and the stiffness K of a crack damage unit nondestructive beam Duan Xian nd As with the calculation of web cracks, the calculation is performed by the formulas (26) and (27).
Next, calculating the line stiffness K of the T-beam roof crack stress diffusion beam section x . As can be seen from the stress diffusion model, the stress diffusion portion side beam Duan Dengxiao is made to be l in length 2 N small segments of/N are connected in series, and the schematic diagram is shown in FIG. 15.
In fig. 15, each small section is a T-beam section, and the beam height at the middle of the small section is the height of the small section. The schematic diagram of the mth small section is shown in FIG. 18, in which the bottom of the effective section is taken as the coordinate system, and the height h of the stress-free region of the mth small section crm ：
The height of the top plate of the m th small section is h 2dm ：
Area moment S of cross section of the mth paragraph m ：
Cross-sectional area A of the mth section m ：
A m ＝b 1 h 1 +b 2 h 2dm (43)
Neutral axis coordinate y of the mth paragraph cm ：
According to the parallel axis displacement theorem, the top plate and the web plate are divided into two parts, and the moment of inertia of the web plate 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 。
I fm ＝I 1m +I 2m (47)
Length of l 2 Mth small section line stiffness of/N K m ：
Length of l 2 The T beam stress diffusion section divided into N sections is used for obtaining the line rigidity K by using a beam section series connection method xd ：
By means of beam sections connected in series, two beams of length l 1 Is provided and has a length l 2 The stress diffusion beam sections of the crack T-beam units are connected in series to obtain the line rigidity K of the crack T-beam units d ：
Deriving by combining the formulas, and damaging degree D of crack unit of T-beam top plate eT-fc The method comprises the following steps:
3. gaussian process regression correction crack height crack T beam unit damage degree
1) Gaussian process regression
According to the characteristics of the T-shaped section, taking the ith training sample for web crack
For roof cracking, the ith training sample is taken
Wherein x is i For the (i) th training sample,for training sample x i K=1, 2, 3.
M training samples are taken, and corresponding outputs, namely correction coefficient column vectors, are as follows:
μ＝[μ 1 μ 2 … μ i … μ M ] T (54)
wherein M is the number of training samples, μ is the correction coefficient column vector corresponding to the training samples, μ i I.e. with training sample x, as the i-th element in μ i And a corresponding output.
The covariance matrix is:
/>
wherein C is covariance matrix, C ij Omega is the element of the ith row and jth column in covariance matrix C 0 、ω 1 、ω 2 、ω 3 For the super-parameters, the super-parameters are calculated by using a maximum likelihood estimation method (Maximum Likelihood Estimate, MLE) or a markov chain monte carlo method (Markov Chain Monte Carlo, MCMC).
C * ＝[c 1 c 2 … c j … c M ] (57)
Wherein C is * For sample x to be analyzed * Covariance row vector of c) j Is C * The j-th element, x * Is the sample to be analyzed.
Can obtain the sample x to be analyzed * The corresponding correction coefficients are:
μ * ＝C * C -1 μ (60)
wherein mu * For sample x to be analyzed * Corresponding correction coefficient C -1 Is the inverse of C.
2) Correction of web transverse crack unit damage degree calculation formula
Height h of crack in formula (39) cr And correcting and optimizing the damage degree calculation effect. First, ansys is used for establishing a plurality of crack T-beam finite element models, so that a plurality of groups of damage degree data are obtained, and the T-beam models are shown in figure 19. The span is 500mm,50mm divides a unit, 10 units in total and 11 nodes (the upper row of numbers in the figure are unit numbers and the lower row of numbers are node numbers). The elastic modulus of the material is 2.06 multiplied by 10 5 MPa, density of 7.9g/cm 3 The method comprises the steps of carrying out a first treatment on the surface of the Poisson's ratio is 0.25; station length δl=50. The basic sample of the damage model is shown in fig. 20 and 21. Ansys modeling is adopted, solid186 model calculation is adopted, and a web crack modeling diagram is shown in FIG. 22.
Degree of actual damage of model D e0 And quantifying by adopting a deflection curvature damage identification theoretical formula (63). The beam structure deflection curvature damage identification theory is a damage identification method for realizing damage positioning and damage quantification based on deflection curvature difference before and after each node of the beam structure is damaged. The deflection curvatures before and after the damage of the n-number node on the structure are respectively as follows:
w in n Represents the deflection of the n-number node, w n The n-node curvature is represented, and the subscripts 'u' and'd' respectively represent an undamaged state and a damaged state.
The quantitative formula of the unit damage degree is as follows:
the damage degree data quantified by the formula (63) is used as a theoretical damage degree D e0 D calculated for equation (39) using Gaussian process regression eT-wc Correcting to obtain a corrected crack height H cr Substituting the original formula to calculate, the correction form of the crack height is as follows:
H cr ＝μ*h cr (64)
μ is a correction coefficient of the crack height of the T-beam web, and the corrected crack damage calculation formula of the T-beam web is as follows:
in the formula, the subscript o represents optimization, stress diffusion angle alpha and damage section moment of inertia I wm Still adopt h cr ，ζ＝h cr /h，h＝h 1 +h 2 And (5) performing calculation.
Sample data are shown in tables 1 to 4, the sample data in the group 2 are basically the same as those in the group 1, the actual Gaussian process is not used as a test sample, and the sample error analysis is carried out on the data in the group 2.
TABLE 1 degree of damage D to beam web crack units e0 Sample data 1 (h 1 ＝26，h 2 ＝4，b 1 ＝4)
Table 2T degree of beam web crack element damage D e0 Sample data 2 (h 1 ＝22，h 2 ＝8，b 1 ＝4)
Table 3T degree of beam web crack element damage D e0 Sample data 3 (h 1 ＝18，h 2 ＝12，b 1 ＝4)
Table 4T web crack cell damage degree De sample data 4 (h 1 ＝14，h 2 ＝16，b 1 ＝4)
The value of the super parameter obtained by adopting the maximum likelihood estimation is as follows:
further, the coefficient of the covariance matrix C can be obtained as:
the sample x to be analyzed can be obtained by the method (60) * Corresponding correction coefficient mu * The modified cell damage degree can be obtained.
2) Formula correction for calculating damage degree of transverse crack unit of top plate
Crack height h in formula (51) using Gaussian process regression cr Parameter correction is carried out, and damage degree D is optimized eT-fc And calculating the effect. First, ansys is used for establishing a plurality of crack T beam finite element models, the basic parameters of the models are the same as those of the upper section, and the basic sample diagrams of the damage models are shown in fig. 23 and 24. Ansys modeling is adopted, solid186 model calculation is adopted, and a modeling diagram is shown in FIG. 25.
And D calculated from formula (51) eT-fc Parameter correction is carried out to obtain the bestPost-chemical crack height H cr Substituting the original formula to calculate, the correction function form of the crack height is as follows:
H cr ＝μ*h cr (68)
the corrected T-beam top plate crack damage calculation formula is as follows:
in the formula (69), the stress diffusion angle alpha and the damage section moment of inertia I fm Still adopt h cr ，ζ＝h cr /h，h＝h 1 +h 2 And (5) performing calculation.
Sample data are shown in tables 5 to 7.
Table 5T extent of roof beam cracking unit damage D e0 Sample data 1 (h 1 ＝26，h 2 ＝4，b 1 ＝4)
Table 6T extent of roof cracking unit damage D e0 Sample data 1 (h 1 ＝22，h 2 ＝8，b 1 ＝4)
TABLE 7 degree of roof beam roof crack unit damage D e0 Sample data 1 (h 1 ＝18，h 2 ＝12，b 1 ＝4)
The value of the super parameter obtained by adopting the maximum likelihood estimation is as follows:
further, the coefficient of the covariance matrix C can be obtained as:
the sample x to be analyzed can be obtained by the method (60) * Corresponding correction coefficient mu * The modified cell damage degree can be obtained.
To the extent D of unit damage e The higher calculation precision is obtained, delta l/h is more than or equal to 1, the number of measuring points is not less than 4, and equidistant measuring points can be adopted generally.
The number N of beams Duan Fenduan on the side of the stress diffusing portion is not less than 100.
Embodiment one: t beam web crack calculation example
The damage degree analysis of the T beam model is performed on the models shown in tables 1 to 4 as shown in fig. 19, and since the beam section on the side of the stress diffusion portion needs to be divided into N sections for analysis, n=200 is taken in the example, and the working conditions are many, and the calculation is performed by adopting the programming method. Obtaining unmodified D eT-wc 、D eT-wco D of rectangular beam er-c And using section moment of inertia damage (I) cr /I) 1/3 Conversion crack relative height ζ=h cr /h is substituted into D calculated by (11) er-ci Error maps with the degree of sample damage are shown in fig. 26 to 29.
As can be seen from comparison of the four graphs, the D is calculated directly by adopting the crack stress diffusion method eT-wc The average value of the relative error between the model sample and the damage degree is below 6%, and the calculation effect is better than that of a rectangular beam formula; d calculated by a modified damage degree calculation formula of crack stress diffusion unit eT-wco The error of the sample damage degree is less than 0.5% for training samples, and the error of most samples is less than 1% for the second group (sample group 2) of test samples, the maximum error is less than 2.5%, and the calculation effect after correction is good; d obtained by adopting rectangular beam crack unit damage degree calculation formula er-c And D er-ci The relative error with the sample is large, and the calculation effect is poor.
Embodiment two: t-beam roof crack calculation example
T beam model As shown in FIG. 19, damage degree analysis was performed on the models of tables 5 to 7, unmodified D eT-fc 、D eT-fco D of rectangular beam er-c And using section moment of inertia damage (I) cr /I) 1/3 Conversion crack relative height ζ=h cr /h is substituted into D calculated by (11) er-ci Error maps with the degree of sample damage are shown in fig. 30 to 33.
As can be seen from the comparison of the figures, D is calculated directly by the crack stress propagation method eT-fc The relative error between the model sample damage degree and the model sample damage degree is large, the calculation effect is poor, and the formula is necessary to be corrected; d calculated by a modified crack stress diffusion mode T beam roof crack unit damage degree calculation formula eT-fco The error with the damage degree of the sample is less than 1.5%, and the effect is good; d obtained by adopting rectangular beam crack unit damage degree calculation formula er-c And D er-ci The relative error with the sample is large and the calculation effect is poor.
Embodiment III: t beam web and roof cracking calculation
The first and second embodiments mainly analyze the effect of the training sample, and to further verify the effectiveness of the method, another model is selected, the T-beam model is shown in fig. 19, the section parameters are not completely identical to the sample models, and other parameters are identical. The basic parameter is (h 1 ＝20，h 2 ＝10，b 1 ＝4，b 2 ＝36)。
1) The damage conditions of the web are shown in Table 8:
table 8T beam 5 Unit web crack damage Condition
The damage degree vs. specific table 9,D eT-wc The error of (C) is less than 3%, the effect is good, D eT-wco The error of the steel is less than 1.6%, the effect is better, and the steel is not greatly different in general, so that the steel can be directly used for crack damage of the T beam webD eT-wc 。
Table 9T comparison of crack damage level of web
2) The damage conditions of the top plate are shown in table 10:
table 10T beam 5 unit roof crack damage condition
The degree of damage is shown in Table 11, D eT-fco The relative errors are smaller than 1%, and the formula calculation effect is good. And D is eT-fc The error is smaller when the damage is small (such as when the damage degree is 10%), and the error is also increased and larger after the damage degree is increased. Therefore, for the crack of the T beam top plate, the accuracy can be ensured by adopting a corrected formula to calculate the theoretical damage degree.
Table 11T roof beam crack damage extent comparison
The above description is only of 3 embodiments of the present invention, and all equivalent changes and modifications according to the claims of the present invention are included in the scope of the present invention.
Claims (6)
1. A method for calculating damage degree of crack T beam units by Gaussian process regression is characterized by comprising the following steps:
(1) The crack T beam is provided with proper number of measuring points, the beam Duan Ji between adjacent measuring points is a unit, and the length of the measuring point unit is；
(2) According to the crack heightAnd the beam section height h, calculating the crack relative height +.>，/>；
(3) According to the relative height of the crackCalculating crack additional spring stiffness parameter +.>According to a stress intensity factor manual;
(4) Calculating crack stress spread angleThe method is characterized in that the method comprises the following steps of calculating according to the equivalent crack cell line stiffness through a rectangular section beam, wherein the calculation formula is as follows:
；
wherein h is the height of the cross section of the beam, h cr As the height of the crack is set to be equal to the height of the crack,is rectangular beam section moment of inertia->B is the width of the rectangular beam section, N is the number of beams Duan Huafen on the side of the stress diffusion portion, +.>Moment of inertia corresponding to the rectangular beam section of the mth section of the stress diffusion section +.>，h 0m Is the m-th beam Duan Jiemian height; />，/>Calculating according to a specific stress diffusion mode as a stress diffusion function;
(5) Calculating the moment of inertia I of the undamaged T beam section:
area moment S of undamaged T beam cross section 0 ：
；
Wherein b 1 、b 2 Respectively the widths of webs and top plates of the T beam sections, h 1 、h 2 The heights of webs and top plates of the sections of the T beams are respectively;
t beam cross-sectional area A 0 ：
；
Neutral axis coordinate y of undamaged section co ：
；
According to the parallel axis displacement theorem, the top plate and the web plate are divided into two parts, and the moment of inertia of the web plate is I 1 The moment of inertia of the top plate is I 2 The total moment of inertia of the undamaged T beam section is I:
；
；
；
(6) The moment of inertia of the stress-diffusing portion beam Duan Jiemian was calculated:
the beam section on one side of the stress diffusion part has a length of l 2 By means of the stress diffusion model,equivalent it as longN is the number of beams Duan Huafen on one side of the stress diffusion part, each small section is of T-shaped section, the height of the midpoint position of the small section is taken as the height of the small section, and the height h of the stress-free area of the m-th small section crm ：
；
a) Web cracking
The web height of the m-th small section is h 1dm ：
；
Area moment S of cross section of the mth paragraph m ：
；
Cross-sectional area A of the mth section m ：
；
Neutral axis coordinate y of the mth paragraph cm ：
；
According to the parallel axis displacement theorem, the top plate and the web plate are divided into two parts, and the moment of inertia of the web plate is I 1m The moment of inertia of the top plate is I 2m The total moment of inertia of the cross section of the web cracking T-beam is I wm ：
；
；
；
b) Roof cracking
The height of the top plate of the m th small section is h 2dm ：
；
Area moment S of cross section of the mth paragraph m ：
；
Cross-sectional area A of the mth section m ：
；
Neutral axis coordinate y of the mth paragraph cm ：
；
According to the parallel axis displacement theorem, the top plate and the web plate are connectedIs divided into two parts, 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 section of the top plate cracking T-beam is I fm ：
；
；
；
(7) Calculating the damage degree of the T beam unit by a stress diffusion angle method:
a) Web cracking
Length of measuring point unitThe undamaged T beam cell line stiffness K:
；
wherein E is the elastic modulus of the material, and I is the moment of inertia of the undamaged T beam section;
long asIs not damaged T-beam Duan Xian stiffness K nd ：
；
Wherein,is the measurement point unit length minus half the stress diffusion section length, +.>；
Length ofThe mth minor segment line stiffness K m ：
；
Length ofThe T beam stress diffusion section divided into N sections is used for obtaining the line rigidity K by using a beam section series connection method xf ：
；
By using a method of connecting beam sections in series, two lengths areIs a non-invasive section and two lengths +.>The stress diffusion beam sections of (a) are connected in series to obtain the linear rigidity of the crack-containing T beam unit>：
；
Comprehensively deducing damage degree of crack unit of T beam webThe method comprises the following steps:
；
the damage degree of the web crack T beam unit can be accurately calculated;
b) Roof cracking
Similar to web cracking, the damage degree of crack units of the T-beam top plate can be deducedThe method comprises the following steps:
；
is less sensitive to the calculation accuracy of (2)>Low, further correction of crack height h is contemplated cr The calculation accuracy is improved;
(8) Regression correction of crack height h in Gaussian process cr The damage degree of the T beam unit is calculated:
a) Gaussian process regression
According to the characteristics of the T-shaped section, taking the ith training sample for web crack
；
For roof cracking, the ith training sample is taken
；
Wherein,for the ith trainingSample (S)>For training sample->K=1, 2, 3;
m training samples are taken, and corresponding outputs, namely correction coefficient column vectors, are as follows:
；
where M is the number of training samples,correction coefficient series vector corresponding to training sample, < ->Is->I.e. with training sample +.>A corresponding output;
the covariance matrix is:
；
；
wherein,is covariance matrix>Is covariance matrix->Elements of row i and column j->、/>、/>、/>Is a super parameter;
；
；
；
wherein,for the sample to be analyzed->Covariance row vector,/>Is->The j-th element of (a)>Is a sample to be analyzed;
can obtain the sample to be analyzedThe corresponding correction coefficients are:
；
wherein,for the sample to be analyzed->Corresponding correction factor,/>Is->An inverse matrix of (a);
b) Web cracking
Considering the web crack height h cr The corrected unit damage degree calculation formula is as follows：
；
；
Wherein,the crack height after correction;
c) Roof cracking
Consider roof crackingHeight h of the pattern cr The corrected unit damage degree calculation formula is as follows：
；
；
Correcting crack height h cr The accuracy of the T beam unit damage degree calculation formula is higher.
2. The method for calculating the damage degree of the crack T beam unit by Gaussian process regression according to claim 1, wherein the method comprises the following steps of: in step (3), the crack adds a spring rate parameterThe method can be calculated as follows:
；
；
wherein,for the relative height of the crack +.>Is a crack stress intensity factor coefficient.
3. The method for calculating the damage degree of the crack T beam unit by Gaussian process regression according to claim 1, wherein the method comprises the following steps of: in the step (4), crack stress diffusion angleThe method can be specifically calculated according to a linear diffusion mode:
；
wherein,for the relative height of the crack +.>H is the height of the beam section, +.>For crack height->In degrees.
4. The method for calculating the damage degree of the crack T beam unit by Gaussian process regression according to claim 1, wherein the method comprises the following steps of: in the step (8), the super-parameters are calculated by adopting a maximum likelihood estimation method MLE or a Markov chain Monte Carlo method MCMC.
5. The method for calculating the damage degree of the crack T beam unit by Gaussian process regression according to claim 1, wherein the method comprises the following steps of: in step (1), the length of the measuring point unitThe height h of the cross section of the beam is not smaller than that of the cross section of the beam, and the number of measuring points is not smaller than 4.
6. The method for calculating the damage degree of the crack T beam unit by Gaussian process regression according to claim 1, wherein the method comprises the following steps of: in the steps (4), (6), (7) and (8), the number N of the beams Duan Huafen on the stress diffusing portion side is not less than 100.
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