CN110765695B - Simulation calculation method for obtaining crack propagation path of concrete gravity dam based on high-order finite element method - Google Patents
Simulation calculation method for obtaining crack propagation path of concrete gravity dam based on high-order finite element method Download PDFInfo
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Abstract
The invention relates to a simulation calculation method for acquiring a crack propagation path of a concrete gravity dam based on a high-order finite element method, belongs to the field of hydraulic and hydroelectric engineering, and particularly relates to analysis calculation for the crack propagation path of the concrete gravity dam containing cracks. And calculating a stress field and a displacement field of the concrete gravity dam with higher precision by using a high-order finite element method, further obtaining a stress intensity factor with high precision and a corresponding stress intensity factor, and finally obtaining a relatively accurate crack propagation path of the concrete gravity dam. The calculation method can be directly applied to the actual engineering problem of the concrete gravity dam, the simulation calculation of the crack propagation path is carried out on the concrete gravity dam containing the initial microcrack, and the calculation result is relatively higher in precision compared with other conventional calculation methods.
Description
Technical Field
The invention relates to a simulation calculation method for acquiring a crack propagation path of a concrete gravity dam based on a high-order finite element method, belongs to the field of hydraulic and hydroelectric engineering, and particularly relates to analysis calculation for the crack propagation path of the concrete gravity dam containing cracks.
Background
In recent years, china builds a plurality of large and small hydropower stations and reservoirs for flood control, power generation and irrigation, and also receives great attention as a water retaining building which is very important for the hydropower stations and the reservoirs. Concrete gravity dams hold a great deal of weight in water retaining buildings due to applicability, reliability, stability and convenience of construction. Due to the weak tensile property of the concrete material and the large volume property of the concrete gravity dam, the concrete gravity dam inevitably generates micro cracks during construction. The initial microcracks pose a potential threat to the safety of concrete gravity dams, and as the operating time of concrete gravity dams increases, the initial microcracks may further propagate, and may ultimately cause very serious accidents. Therefore, in terms of crack problems, the primary problem of safety evaluation of concrete gravity dams is to judge under what conditions existing cracks can propagate, and to predict crack propagation paths under various possible working conditions and corresponding safety degrees, so as to take control protection measures before further crack propagation endangers the dam body. However, there is currently little research on crack propagation.
The discontinuity problem is mostly analyzed by adopting a finite element propagation method, and the finite element propagation method also has unique advantages in crack propagation. However, the extended finite element method is developed based on the traditional finite element method, and has the disadvantages of poor precision and slow convergence rate. The method for acquiring the crack propagation path of the concrete gravity dam based on the high-order finite element method utilizes the advantages of high precision, high convergence rate and good stability of the high-order finite element method, so that the relatively accurate crack propagation path of the concrete gravity dam is acquired.
The crack propagation problem is almost studied by using the propagation finite element method based on the conventional finite element method, so that the following disadvantages exist:
(1) The traditional finite element method lacks an effective error estimation mode for the crack propagation problem, and the control of the final calculation result precision is relatively dependent on the experience of a researcher. In order to improve the accuracy of the calculation result, the meshes need to be divided again for many times so as to improve the convergence of the finite element solution;
(2) The traditional finite element method is based on a low-order interpolation function, so that the defects exist when a high-gradient stress and a strain field near the tip of a crack are simulated, in order to obtain higher calculation precision, grids need to be encrypted for multiple times to ensure enough grid density, and the pretreatment and calculation cost is higher;
(3) Due to the singularity of the stress field near the crack tip, the error between the stress near the crack tip and the strain field calculated based on the traditional finite element method is large, and the precision of the stress intensity factor derived from the stress field is low; if a higher accuracy of the stress intensity factor is desired, the preprocessing and calculation costs will increase.
The invention patent with the patent application number of 201910063707X mainly utilizes a p-type finite element method in combination with a contour integration method to derive a composite stress intensity factor, and the invention selectively discloses a crack propagation path of the concrete gravity dam meeting the engineering precision requirement through cyclic calculation based on a high-order finite element method.
The invention provides a novel computing technology for applying a high-order finite element method to the field of hydraulic and hydroelectric engineering based on a national science fund subsidy project (subsidy number: 51769011) so as to obtain a crack propagation path of a concrete gravity dam.
Disclosure of Invention
Aiming at the defects of the existing simulation calculation method for the crack propagation path of the concrete gravity dam, the invention provides a simulation calculation method for obtaining the crack propagation path of the concrete gravity dam based on a high-order finite element method. The invention is realized by the following technical scheme.
A simulation calculation method for obtaining a crack propagation path of a concrete gravity dam based on a high-order finite element method is characterized in that a stress field and a displacement field of the concrete gravity dam with higher precision are calculated by using the high-order finite element method, so that a high-precision stress intensity factor and a corresponding stress intensity factor are obtained, and finally, a relatively accurate crack propagation path of the concrete gravity dam is obtained, and the method comprises the following steps:
step 2, determining the position and the length of the initial crack, and creating the initial crack on a finite element model of the concrete gravity dam;
and automatically dividing grids for each circulation to analyze the crack propagation path of the concrete gravity dam, obtaining the stress intensity factor of the crack tip of each circulation step, solving the equivalent stress intensity factor, comparing the equivalent stress intensity factor with the fracture toughness of the concrete, taking the circulation step with the equivalent stress intensity factor being greater than the fracture toughness as an effective circulation, discarding the rest circulation steps, judging whether the difference between the equivalent stress intensity factor of the last step of the effective circulation step and the fracture toughness of the concrete is within an error allowable range, and if the difference is not within the error allowable range, returning to the step 3 to reduce the crack step length and increase the circulation times, and continuing to calculate.
The step 1 comprises the establishment of a concrete gravity dam geometric model, the setting of concrete material parameters, and the application of load and boundary conditions.
The step 4 specifically comprises the following steps:
step 4.1, presuming the order and the cycle number of the interpolation polynomial, wherein the order of the interpolation polynomial is 1-8;
4.2, calculating by a high-order finite element method for determining the order of an interpolation polynomial to obtain a displacement field and a stress strain field of the first circulation of the concrete gravity dam containing the cracks and obtain a crack expansion angle;
step 4.3, deriving the stress intensity factor K of each cycle by using a contour integral method I And K II ；
Step 4.4, calculating to obtain equivalent stress intensity factor of the first circulation
Step 4.5, completing the first step circulation of crack propagation through a crack propagation angle and a crack propagation step length;
4.6, starting the second-step circulation, repeating the steps 4.2 to 4.6, completing the second-step circulation of crack propagation, repeating the steps until all circulation times are completed, and sequentially comparing all calculated equivalent stress intensity factors of each step of circulation with the fracture toughness of the concrete;
if the equivalent stress intensity factors of all the circulation steps are larger than the concrete fracture toughness comparison, increasing the circulation times, returning to the step 4.1, if the equivalent stress intensity factors of some circulation steps are smaller than the concrete fracture toughness, finding out the equivalent stress intensity factor of the step closest to and larger than the concrete fracture toughness, obtaining the relative error between the equivalent stress intensity factor of the step and the concrete fracture toughness, judging whether the relative error meets the engineering precision requirement, if so, completing the crack expansion, if not, attempting to increase the insertion polynomial order to repeat the calculation, if not, greatly, reducing the crack expansion step length and increasing the circulation times, and if the product of the reduced crack expansion step length and the increased circulation times is larger than the product of the cycle step length of the step closest to and larger than the concrete fracture toughness of the equivalent stress intensity factor and the crack expansion step length before reduction, repeating the steps until the engineering precision requirement is met.
The specific steps of calculating the displacement field and the stress strain field of each cycle in the step 4.2 are as follows:
step 4.2.1, automatically dividing grids according to a concrete gravity dam finite element model containing initial cracks, combining the specified interpolation polynomial order, and solving a structural displacement array a
According to the equation: ka = F (1)
Wherein: k is a total integral rigidity matrix, K = ∑ Sigma e G T K e G，(K e Is a matrix of cell stiffness);
a is a structure displacement array;
f is a structural node load array, and F = ∑ Σ e G T f e (e represents a unit);
g represents a conversion matrix of the degree of freedom of the structure node and the degree of freedom of the unit node, and T represents transposition;
then according to formula K of the unit rigidity matrix e ＝∫ Ω B T DBdΩ (2)
And the unit equivalent node load array
Wherein: nodal force within a cell
External nodal force
In the above formula (2) to formula (5), Ω is represented as the inside of the cell; b = LN I B denotes a strain matrix, L denotes a differential operator, N I Representing a matrix of shape functions, I representing the index of the cell, Γ t Represents the outer boundary of the cell; d represents the stress matrix, b represents the physical strength,
representing load boundary conditions, wherein the quantities are calculated according to the concrete material properties, the loads and the boundary conditions determined in the step 1;
substituting the above formula (2) to the above formula (5) into the above formula (1), and solving the linear equation set (1) to obtain a structural displacement array a; then according to the formula a I = Ga further yielding a cell-shifted array a I (ii) a Finally according to the formula u = N I a I ，ε＝Lu，σ＝Dε＝DBa I And calculating the displacement field u, the strain field epsilon and the stress field sigma.
Wherein the shape function matrix N I The calculation adopts a high-order interpolation polynomial in a high-order finite element method, a typical two-dimensional high-order finite element method shape function takes a Legendre orthogonal polynomial as a base, a two-dimensional quadrilateral unit (shown in figure 1) is taken as an example to explain the high-order finite element method, and a point P is arranged under a coordinate system xi O eta 1 (-1,-1)、P 2 (-1,1)、P 3 (1, 1) and P 4 (-1, 1) each being quadrilateralFour vertices of a cell, Γ 1 (η＝-1)、Γ 2 (ξ＝1)、Γ 3 (η = 1) and Γ 4 (ξ = -1) are four sides of the quadrilateral unit respectively, and the step of constructing the shape function is as follows:
(1) when p =1, the node mode shape function is the same as the conventional lagrangian type interpolation function, and the four-node quadrilateral unit is expanded as follows:
(2) when p is greater than or equal to 2, the edge mode shape function N i (k) Is corresponding to the edge gamma k Associated shape function, and edge
Γ 1 The associated shape function of = { (ξ, -1), -1 ≦ ξ ≦ 1} is defined as follows:
wherein:
P n (t) is a Legendre polynomial with an order n of 0 or more
Similarly, define the edge Γ k (2 ≦ k ≦ 4) the associated shape function is as follows:
(3) when p ≧ 4, the internal mode shape function:
the method adopts a high-order finite element method, the order of the interpolation polynomial can be gradually increased from p =1, when the order of the interpolation polynomial is improved, the stiffness matrix of the previous low order can be continuously used, only the calculation of the high-order part needs to be increased, the repeated calculation of the stiffness matrix of the low-order part is avoided, the method has better usability, the calculation of the pretreatment is reduced, and the calculation cost is saved.
Step 4.2.2, according to the obtained stress field and displacement field, deriving a stress intensity factor by a contour integration method, further obtaining a crack expansion angle and combining the determined crack expansion step length to complete the crack expansion; meanwhile, solving a corresponding stress intensity factor to judge whether the crack propagation in the step is an effective step;
the invention utilizes the contour integral method to derive the stress intensity factor. The contour integral method is a method based on Betti's mutual equivalence theorem, and the stress intensity factor of the concrete gravity dam crack is derived by adopting the obtained displacement and stress strain on the integral path, and the method is super-convergent when the stress intensity factor is derived. The two-dimensional elastic crack field Ω, as shown in FIG. 2, defines a local Cartesian coordinate system (x) at the crack front 1 ,x 2 ) Remember u (x) 1 ,x 2 ) As a local coordinate system (x) 1 ,x 2 ) Position in lower elastic crack field omegaAnd (5) moving the field.
Representing the neighborhood Ω surrounding the crack front in a local polar coordinate system (r, θ) s Of the displacement field of (1). When sub-field Ω s Already close enough to the crack tip, the subdomain Ω assumes a completely in-plane strain state s The internal displacement field can be approximated as follows:
in the formula, K I And K II Corresponding to type I and type II stress intensity factors, respectively, G is the shear modulus, kappa is the Kolosov constant,
t for (u) (r, θ) to represent the traction force vector derived from the displacement field. Stress intensity factor K I And K II Can pass through the contour gamma 2 ，Γ 3 And gamma 4 The line integral over (c) calculation yields:
in the formula:
the derived functions of the type i and type ii stress intensity factors are shown separately.
And with
Respectively represent by
And with
A derived tractive effort vector.
The results calculated by equations (6) and (7) are accurate, but the solution of u and the corresponding traction force T (u) Unknown, the present invention uses a high order finite element method to obtain an approximate numerical solution. Contour line gamma 2 ，Γ 3 And gamma 4 Independent of the mesh for solving an approximate numerical solution with a high-order finite element method, and the contour Γ 2 The selection of which is more flexible and does not have to be close to the front end of the crack. Since the accuracy of the numerical solution result in the innermost layer unit at the crack tip is low and the error of the stress intensity factor obtained here is large, the shroud line Γ is selected 2 In general, the cell is selected to encompass the innermost layer near the tip of the fissure. The contour of the contour integration method in said step 4.2.2 is far away and contains the split tip, and since the outer elements of the split tip are affected singularly, the contour should also contain the innermost elements outside the split tip.
The concrete gravity dam finite element model in the 4.2.1 is used for automatically dividing the grids without dividing the grids in advance or setting the sizes of the grids, and the automatic grid division in each step is mainly used for taking cracks which finish expansion in the previous step as unit boundaries, so that the crack behavior of the concrete gravity dam is more conveniently described.
The invention has the beneficial effects that:
(1) The method can reduce the calculation cost, accelerate the convergence speed and improve the calculation precision and the calculation efficiency.
(2) The calculation method can be directly applied to the actual engineering problem of the concrete gravity dam, the simulation calculation of the crack propagation path is carried out on the concrete gravity dam containing the initial microcrack, and the calculation result is relatively higher in precision compared with other conventional calculation methods.
Drawings
FIG. 1 is a schematic diagram of a standard quadrilateral female cell to which the present invention relates;
FIG. 2 is a schematic diagram of a crack tip neighborhood under a global coordinate and local coordinate system according to the present invention;
FIG. 3 is a geometric model of a concrete gravity dam containing incipient cracks according to example 1 of the present invention;
FIG. 4 is a cloud of displacement field calculated by a finite element model of a concrete gravity dam in embodiment 1 of the present invention;
FIG. 5 is a cloud of stress fields calculated by a finite element model of a concrete gravity dam in example 1 of the present invention;
FIG. 6 is a schematic view of the crack propagation path of the concrete gravity dam in example 1 of the present invention;
FIG. 7 is a deformation diagram of a concrete gravity dam according to example 1 of the present invention;
fig. 8 is a cloud chart of local stress at the crack tip of the concrete gravity dam after the expansion is completed in the embodiment 1 of the invention.
Detailed Description
The invention is further described with reference to the following drawings and detailed description.
Example 1
A geometric model of a concrete gravity dam containing initial cracks is shown in 3, wherein the height H =100m of the dam, the width A =6m of the dam top, the width B =20 of the dam bottom, the upstream dam slope is vertical, and the gradient of the downstream dam slope is 1.7; the dam body is made of concrete materials, the elastic modulus E =22GPa, the Poisson ratio v =0.250 and the density p =2700kg/m are considered 3 (ii) a Fracture toughness K of concrete c ＝575314Pa·m 1 / 2 . The upstream face of the dam body bears hydrostatic pressure, the worst load condition is considered, and the water level is assumed to be flush with the elevation of the dam crest. Assuming that a crack exists at the lower position of the upstream face of the dam body, and the crack length is L =10.00m, for convenience of description and calculation, the coordinates of the end points at the two ends of the crack are (0.00, 13.5) and (10.00, 13.5) in consideration of establishing a coordinate system xoy with the heel of the upstream face intersecting with the bottom face of the dam as an origin of coordinates.
step 2, determining the position (0, 13.5) and the length L =10m of the initial crack, and creating the initial crack ((0, 13.5) - (10, 13.5)) on a finite element model of the concrete gravity dam;
calculating displacement field and stress strain field of each cycle, obtaining a displacement field cloud chart of the cycle of 715 with interpolation polynomial order p =5 as shown in fig. 4, and obtaining a stress field cloud chart through calculation as shown in fig. 6
The grids are automatically divided in each circulation to analyze the crack propagation path of the concrete gravity dam, the stress intensity factor of the crack tip in each circulation step is obtained, and the equivalent stress intensity factor is obtained (the equivalent stress intensity factor is considered as
Comparing the equivalent stress intensity factor with the fracture toughness of the concrete, taking the cycle step in which the equivalent stress intensity factor is greater than the fracture toughness as an effective cycle, and leaving the rest of the cycle steps, judging whether the difference between the equivalent stress intensity factor of the last step of the effective cycle step and the fracture toughness of the concrete is within an error allowable range, if not, judging whether the difference is within the error allowable rangeAnd (4) returning to the steps (3) and (4) to reduce the cracking step length and increase the cycle number within the error allowable range, and continuing to calculate.
Through the above steps, the coordinate table of the crack tip point on the crack propagation path is calculated as follows (due to space, this example only shows the important 702-715 cycles out of 725 cycles, and 716-725 cycles are invalid cycles):
TABLE 1 coordinates of crack cusp points on crack propagation path and table for judging whether cracks propagate
In this example, in cycle step 716, K Ⅰ ＝0.56178，K Ⅱ =0.066490, the equivalent stress intensity factor K is calculated e ＝0.56570<K c =0.575314, wherein K c In order to obtain the fracture toughness of concrete (which is a parameter for describing the fracture performance of concrete itself, the concrete is actually a material between brittle and plastic, and the fracture toughness of the concrete is correspondingly reduced along with the increase of the strength of the concrete, namely, the fracture toughness of the high-strength concrete is more brittle than that of the common concrete), the 716 th cycle step and the following cycle steps are ineffective cycle steps and are omitted. The 715 th and previous loop steps are effective loop steps, and the last step in the effective loop steps is the error (K) of the 715 th step e -K c )/K c And =1.484%, which meets the engineering precision requirement, so the calculation is stopped. The calculation is the obtained crack propagation path of the concrete gravity dam, the crack propagation path of the concrete gravity dam is shown in a graph in fig. 6, the deformation graph of the concrete gravity dam is shown in a graph in fig. 7, and the result is consistent with engineering experience, namely, the crack gradually extends to the bottom of the dam body of the concrete gravity dam. The x-direction and y-direction local stress field cloud pictures of the crack tip of the concrete gravity dam after the expansion is finished are shown in fig. 8, and the result of fig. 8 is identical with the fracture mechanics conclusion, namely the stress concentration phenomenon at the crack tip is obvious and the stress at the crack tip is increased sharply.
The concrete gravity dam model applied in the calculation is simple in structure, the load and the constraint are simplified more, the required calculation cost is lower, the calculation efficiency and the calculation accuracy are higher, and the convergence speed is higher as can be seen from the table. When the model of the concrete gravity dam is closer to the actual engineering model, the model structure is correspondingly more complicated, the order of the interpolation polynomial can be increased to obtain higher precision and faster convergence rate, and finally the crack propagation path of the concrete gravity dam meeting the engineering precision requirement is obtained.
As can be seen from the above table, when the crack propagation path of the concrete gravity dam is analyzed and calculated, the pretreatment is relatively less, the convergence rate is relatively high, and the numerical solution of the crack propagation path of the concrete gravity dam with relatively high calculation accuracy is obtained through relatively low calculation cost.
While the present invention has been described in detail with reference to the embodiments, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.
Claims (2)
1. A simulation calculation method for obtaining a crack propagation path of a concrete gravity dam based on a high-order finite element method is characterized by comprising the following steps: the method comprises the following steps of calculating a stress field and a displacement field of the concrete gravity dam with high precision by using a high-order finite element method, further obtaining a high-precision stress intensity factor and a corresponding stress intensity factor, and finally obtaining a relatively accurate crack propagation path of the concrete gravity dam, wherein the method comprises the following steps:
step 1, determining the geometric dimension of a concrete gravity dam engineering model, and establishing a finite element model of the concrete gravity dam;
step 2, determining the position and the length of the initial crack, and creating the initial crack on a finite element model of the concrete gravity dam;
step 3, setting an initial cracking direction, an initial cracking position and a cracking step length;
step 4, setting the order of interpolation polynomial and the cycle number;
each circulation automatically divides the grids to analyze the crack propagation path of the concrete gravity dam, obtains the stress intensity factor of the crack tip of each circulation step, calculates the equivalent stress intensity factor, compares the equivalent stress intensity factor with the fracture toughness of the concrete, the circulation step with the equivalent stress intensity factor larger than the fracture toughness is an effective circulation, the other circulation steps are left, judges whether the difference between the equivalent stress intensity factor of the last step of the effective circulation step and the fracture toughness of the concrete is within an error allowable range, if not, returns to the step 3 to reduce the crack step length and increase the circulation times, and continues to calculate;
the step 4 specifically comprises the following steps:
step 4.1, presuming the order of an interpolation polynomial and the cycle number, wherein the order of the interpolation polynomial is 1-8;
4.2, calculating to obtain a displacement field and a stress strain field of the first circulation of the concrete gravity dam containing the cracks by a high-order finite element method with determined interpolation polynomial order, and obtaining a crack expansion angle;
step 4.3, deriving the stress intensity factor K of each cycle by using a contour integral method I And K II ；
Step 4.4, calculating to obtain the equivalent stress intensity factor of the first circulation
Stress intensity factor K I And K II Can pass through the contour gamma 2 ，Γ 3 And gamma 4 The line integral over (c) calculation yields:
step 4.5, completing the first step circulation of crack propagation through a crack propagation angle and a crack propagation step length;
4.6, starting the second-step circulation, repeating the steps 4.2 to 4.6, completing the second-step circulation of crack propagation, repeating the steps until all circulation times are completed, and sequentially comparing all calculated equivalent stress intensity factors of each step of circulation with the fracture toughness of the concrete;
if the equivalent stress intensity factors of all the circulation steps are greater than the concrete fracture toughness comparison, increasing the circulation times, returning to the step 4.1, if the equivalent stress intensity factors of some circulation steps are less than the concrete fracture toughness, finding out the equivalent stress intensity factor of the step which is closest to and greater than the concrete fracture toughness, obtaining the relative error between the equivalent stress intensity factor of the step and the concrete fracture toughness, judging whether the relative error meets the engineering precision requirement, if so, completing the crack propagation, if not, trying to increase the interpolation polynomial order to repeat the calculation, if not, and if the relative error is greater than the engineering precision requirement, reducing the crack propagation step length and increasing the circulation times, and if the product of the reduced crack propagation step length and the increased circulation times is greater than the product of the circulation step length of the step which is closest to and greater than the concrete fracture toughness and the crack propagation step length before the reduction, repeating the steps until the engineering precision requirement is met.
2. The method for obtaining the simulation calculation of the crack propagation path of the concrete gravity dam based on the high-order finite element method according to claim 1, is characterized in that: the step 1 comprises the establishment of a concrete gravity dam geometric model, the setting of concrete material parameters, and the application of load and boundary conditions.
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| CN112347666B (en) * | 2020-09-15 | 2022-09-06 | 西北工业大学 | Numerical simulation method for interaction of high-temperature alloy cracks in tensile stress state |
| CN116038107B (en) * | 2022-09-30 | 2024-05-28 | 扬州市职业大学(扬州开放大学) | Device for processing recycled concrete and control method thereof |
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