A calculation method for the upper bound of plastic shakedown considering the strain hardening effect of thin plates

Abstract

The present invention proposes a method for calculating the upper bound shakedown load considering the strain hardening effect of thin plates, belonging to the field of physical technology, and comprising the following steps: ST1. Prepare calculation data to obtain the smooth generalized strain-displacement velocity relationship matrix corresponding to each triangular sub-domain of the thin plate; ST2. Calculate the generalized elastic stress field to obtain the smooth generalized elastic stress fields under the action of corner loads and constant loads respectively; ST3. At the initial iteration, assume that the entire thin plate is in a non-yield state and solve the linear equations; ST4. At the h-th (h≥1) iteration, according to the calculation results of the (h-1)-th iteration, integrate and solve the corresponding linear equations, and finally obtain the shakedown upper bound load multiplier at the h-th iteration, and determine whether to terminate the iteration; ST5. Post-process the calculation results. The present invention solves the problems of establishing and linearizing the minimization iteration format for shakedown upper bound analysis considering the strain hardening effect of thin plates, and realizes the accurate, efficient and stable solution of the shakedown upper bound load considering the strain hardening effect of thin plates.

Description

Technical Field

[0001] The present invention belongs to the technical field of plastic mechanics, and particularly relates to a method for calculating the upper bound of plastic shakedown load considering the strain strengthening effect of thin plates. Background Art

[0002] Thin plates are a structural form widely used in the fields of aerospace, aviation, nuclear energy, shipbuilding, etc. Accurately and reliably calculating the plastic ultimate bearing capacity of thin plate structures under cyclic loads is of great significance for the optimal design and safety assessment of plate structures. Shakedown analysis is a direct and effective method for studying plastic failure of structures under cyclic loads. It does not involve the load change history required for elastoplastic incremental analysis and has strong practicability and operability. Compared with conventional elastic analysis, the shakedown load obtained from shakedown analysis can be used as an important parameter reflecting the plastic failure of structures, and can better reflect the actual bearing potential and safety level of structures. It has been widely applied in the design codes and regulations of many engineering structures.

[0003] At present, the theoretical research and engineering applications of shakedown analysis have made great progress, but the research work on shakedown analysis of thin plate structures is relatively less. The theoretical and experimental research on shakedown analysis of thin plates is restricted by conditions such as structural geometry, boundaries, and applied loads. At present, only a very small amount of research has been reported. Numerical calculation can accurately solve the ultimate bearing capacity of relatively complex thin plates, so it has always been one of the research hotspots in this field. Many scholars have combined mature numerical analysis techniques with the mathematical programming theory of shakedown analysis, focusing on the efficient optimization algorithms and numerical solution procedures for shakedown analysis of thin plates, and have achieved some remarkable research results. Since many metals or alloys have obvious strain strengthening properties, and the strain strengthening effect of materials has a great influence on the shakedown of structures, the shakedown load obtained by ignoring the strain strengthening effect is on the conservative side. However, at present, none of these limited research results on shakedown analysis of thin plates consider the influence of the strain strengthening effect on the shakedown of thin plates. Therefore, it is very necessary to develop an accurate and efficient solution procedure using a numerical calculation method with good accuracy and high efficiency to study the influence of the strain strengthening effect of materials on the shakedown failure of thin plate structures. Summary of the Invention

[0004] The present invention provides a method for calculating the upper bound of plastic shakedown load considering the strain strengthening effect of thin plates, based on the C 1 Nodal Natural Element Method (C 1 -nodal-NEM), aiming to solve the establishment and linearization solution problems of the minimization format for shakedown upper bound analysis considering the strain strengthening effect of thin plates, and to achieve accurate, efficient, and stable solution of the upper bound of plastic shakedown load considering the strain strengthening effect of thin plates.

[0005] The object of the present invention is achieved by the following technical solutions:

[0006] A method for calculating the upper bound shakedown load considering the strain hardening effect of thin plates, comprising the following steps:

[0007] ST1. Prepare calculation data to obtain the smooth generalized strain-displacement velocity relationship matrix corresponding to each triangular sub-domain of the thin plate structure;

[0008] ST2. Calculate the generalized elastic stress field, and respectively obtain the smooth generalized elastic stress fields corresponding to each triangular sub-domain of the thin plate structure under the action of corner loads and constant loads;

[0009] ST3. At the initial iteration (h = 0), assume that the entire thin plate structure is in a non-yield state, solve the linear equations, and sequentially obtain the Lagrange multipliers, residual displacement increments, smooth generalized plastic strains corresponding to each triangular sub-domain under the action of each corner load, and the shakedown upper bound load multiplier considering the strain hardening effect of the thin plate;

[0010] ST4. At the h-th (h≥1) iteration, according to the calculation results of the (h - 1)-th iteration, integrate and solve the corresponding linear equations, and sequentially obtain the Lagrange multipliers, residual displacement increments, smooth generalized plastic strains corresponding to each triangular sub-domain under the action of each corner load, and the shakedown upper bound load multiplier considering the strain hardening effect of the thin plate, and judge whether to terminate the iterative calculation according to the iterative convergence condition;

[0011] ST5. Post-process the calculation results to obtain the plastic dissipation power of each node of the thin plate structure;

[0012] As a further optimization, the step of preparing calculation data specifically includes:

[0013] ST1.1. Prepare the discrete nodes, Delaunay triangles, displacement boundary nodes, loads (constant loads, variable loads), geometric parameters, material parameters (Young's modulus E, Poisson's ratio v, yield stress Y, hardening parameter ), offset coefficient γ, error tolerances vol1 and vol2 of the thin plate structure;

[0014] ST1.2. According to the information of the nodes, Delaunay triangles and displacement boundary nodes of the thin plate, use the C 1 node natural element method to sequentially determine the vertices and their counterclockwise arrangement order of the Voronoi sub-domains S r (r = 1 to NP, NP is the total number of discrete nodes) around each node x r , the coordinates of the vertices of the Voronoi structure after a small offset, and each triangular sub-domain S rs ( RS is the sub-domain S r The area A of the total number of triangular sub-domains of rs and the tangential vector n of each side 1 and the normal vector n 2 and the natural neighboring nodes I (I = 1~np, where np is the total number of natural neighboring nodes of each integration point determined by the empty circle criterion) of each integration point on each side and their C 1 The first derivative of the natural neighboring shape function x rs is the virtual node corresponding to the triangular sub-domain S rs with superscripts w, θ x and θ y being the deflection of the thin plate, the curl in the x direction, and the curl in the y direction respectively, and subscripts x and y represent the first derivative with respect to the x direction and the y direction respectively. Then calculate the smooth generalized strain-displacement velocity relationship matrix corresponding to the natural neighboring nodes of each integration point

[0015]

[0016] In the formula, Γ rs is the boundary of the sub-domain S rs ;

[0017] ST1.3. "Match the right seats" to integrate all the rs matrices corresponding to the natural neighboring nodes of all integration points ( is the total number of nodes after merging the same natural neighboring nodes of all integration points), and obtain the smooth generalized strain-displacement velocity relationship matrix corresponding to each triangular sub-domain S rs

[0018]

[0019] As a further optimization, the specific steps for calculating the generalized elastic stress field include:

[0020] ST2.1. According to the smooth generalized strain-displacement velocity relationship matrix corresponding to each triangular sub-domain and the elastic constitutive relationship, calculate and integrate the overall elastic stiffness matrix K of the thin plate e :

[0021]

[0022] In the formula, D b is the elastic relationship matrix;

[0023] is the bending stiffness of the thin plate; ​

[0024] h 0 is the plate thickness;

[0025] ST2.2. Determine the corner load and its total number according to the range and number of groups of each group of independently variable loads acting on the thin plate For the uniformly distributed force in each corner load, according to the area of the Voronoi sub-domain S equivalent it to the equivalent load array acting on the node r area For the concentrated force in each corner load, directly apply it to the node where it acts. From the equivalent load arrays of each node "Match the right seat" to integrate the overall elastic load array of the thin plate when each corner load acts

[0026]

[0027] ST2.3. Introduce displacement boundary conditions to modify the overall elastic stiffness matrix K e and the overall elastic load array when each corner load acts Solve the linear control equation of the elastic problem of the thin plate to obtain the elastic displacement field of the thin plate when each corner load acts

[0028] rs ST2.4. According to the elastic constitutive relation and the elastic displacement field, calculate the smooth generalized elastic stress corresponding to each triangular sub-domain S when each corner load acts

[0029]

[0030] ST2.5. Imitating steps ST2.2 to 2.4, calculate the smooth generalized elastic stress corresponding to each triangular sub-domain S when the constant load acts rs

[0031] As a further optimization, at the initial iteration, assume that the entire thin plate structure is in a non-yield state. The specific steps for solving the linear equations include:

[0032] ST3.1. Assume that the entire thin plate structure is in a non-yield state, take and calculate the intermediate variables corresponding to each triangular sub-domain S at the initial iteration in turn

[0033] rs

[0034] ​​​​Wherein, M P = Yh 0 2 / 4 is the plastic limit moment;

[0035]

[0036] Q is a positive definite symmetric constant matrix, and Q -1 = 2D / 3, where D is a constant matrix introduced to handle the plastic incompressibility condition of the thin plate:

[0037]

[0038] ST3.2. Calculate and integrate the overall constant load array F 0 , the overall stiffness matrix K 0 considering the strain hardening effect of the thin plate during the initial iteration 1 0 and the overall variable load array F

[0039]

[0040] ST3.3. Introduce displacement boundary conditions to modify the overall constant load array F 0 , the overall stiffness matrix K 0 considering the strain hardening effect of the thin plate during the initial iteration 1 0 , and the overall variable load array F 0 (Δa 0 ) 0 = F 0 and K 0 (Δa 1 ) 0 = F 1 0 , and obtain the intermediate variables (Δa 0 ) 0 and (Δa 1 ) 0 of the residual displacement increment during the initial iteration;

[0041] ST3.4. Solve the Lagrange multiplier λ during the initial iteration 0 :

[0042]

[0043] Wherein, the intermediate variables and are respectively:

[0044]

[0045] ST3.5. Solve the residual displacement increment (Δa) during the initial iteration​​​0 and the smooth generalized plastic strains corresponding to each triangular sub-domain S under the action of each corner load rs

[0046] (Δa) 0 =(Δa 0 ) 0 +λ 0 (Δa 1 ) 0

[0047]

[0048] ST3.6. Calculate the variables corresponding to each triangular sub-domain S at the initial iteration in turn according to the obtained residual displacement increment and smooth generalized plastic strain rs and

[0049]

[0050]

[0051] where β 1 and β 2 are both positive numbers much smaller than 1 and are respectively set as:

[0052]

[0053] Furthermore, the shakedown upper bound load multiplier s considering the strain hardening effect of the thin plate at the initial iteration is obtained 0 :

[0054]

[0055] As a further optimization, at the h (h≥1) -th iteration, according to the calculation results of the (h - 1) -th iteration, the steps of integrating and solving the corresponding linear equations specifically include:

[0056] ST4.1. Determine the values of the variables and according to the calculation results of the (h - 1) -th iteration, and calculate the intermediate variables corresponding to each triangular sub-domain S at the h -th iteration in turn rs

[0057]

[0058] ST4.2. Calculate and integrate the overall stiffness matrix K h considering the strain hardening effect of the thin plate at the h -th iteration 1 ​​​h :

[0059]

[0060] ST4.3. Introduce displacement boundary conditions to modify the overall constant load array F 0 , the overall stiffness matrix K at the h-th iteration h and the overall variable load array F 1 h , and solve the linear equations K h (Δa 0 ) h = F 0 and K h (Δa 1 ) h = F 1 h , to obtain the intermediate variables (Δa 0 ) h and (Δa 1 ) h ;

[0061] ST4.4. Solve the Lagrange multiplier λ at the h-th iteration h :

[0062]

[0063] where the intermediate variables and are respectively:

[0064]

[0065]

[0066] ST4.5. Solve the residual displacement increment (Δa) at the h-th iteration h and the smooth generalized plastic strain rs corresponding to each triangular subdomain S

[0067] (Δa) h = (Δa 0 ) h + λ h (Δa 1 ) h

[0068]

[0069] ST4.6. According to the obtained residual displacement increment and smooth generalized plastic strain, calculate each triangular subdomain S at the h-th iteration in turnrs The corresponding variable and μ h+1 (x rs )、

[0070]

[0071]

[0072] Thus, the shakedown upper bound load multiplier s considering the thin plate strain hardening effect at the h-th iteration is calculated h :

[0073]

[0074] ST4.7. According to the set error tolerances vol1, vol2 and the following convergence conditions, determine whether to terminate the iteration:

[0075] ||(Δa) h -(Δa) h-1 || / ||(Δa) h-1 ||≤vol1, |s h -s h-1 | / s h-1 ≤vol2.

[0076] As a further optimization, the post-processing steps of the calculation results specifically include:

[0077] ST5.1. Assume that the convergence conditions are satisfied when the h-th iteration terminates, and calculate the smooth plastic dissipation power D rs corresponding to each triangular subdomain S h (x rs ):

[0078]

[0079] ST5.2. According to the area relationship between each triangular subdomain S rs and the Voronoi subdomain S r , obtain the plastic dissipation power D h (x r ) of each node of the thin plate structure:

[0080]

[0081] The beneficial technical effects achieved by the present invention are:

[0082] 1. The calculation method developed in the present invention has the advantages of relatively simple format, strong versatility, easy program implementation, high calculation accuracy and efficiency, and good numerical stability. It can accurately obtain the upper limit load of the thin plate structure under different strengthening models, making up for the shortcomings that the current thin plate structure shakedown analysis is relatively small and the strain strengthening effect has not been considered. It is a further enrichment and development of the plastic shakedown analysis of thin plate structures.

[0083] 2. C used in the present invention 1 Nodal Natural Element Method (C 1 -nodal-NEM) has both C 1 The advantages of natural neighbor interpolation and stable and compatible integrals in triangular subdomains only require the calculation of C 1 The first-order derivative of the shape function is naturally approached without calculating its second-order derivative, which is very beneficial to construct a high-precision smooth generalized strain field for the shakedown upper limit analysis of thin plates, and the post-processing of the numerical calculation results is convenient.

[0084] 3. The stable loads of different material models of thin plates obtained by the present invention and the plastic dissipated power cloud diagram that can truly reflect the stable failure mode of the structure can not only provide a reference for the engineering design and safety assessment of thin plate structures, but also provide a reference for further in-depth research on the influence of strain hardening effect on the stable failure of thin plate structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] Figure 1 It is a flowchart of the implementation process of the present invention;

[0086] Figure 2 It is a schematic diagram of the triangular subdomain division of the present invention;

[0087] Figure 3 Schematic diagram of a circular thin plate clamped by a uniformly distributed force q for shakedown upper limit analysis in a specific embodiment of the present invention;

[0088] Figure 4 yes Figure 3 Schematic diagram of 801 discrete nodes of the clamped circular thin plate;

[0089] Figure 5 yes Figure 3 Schematic diagram of the distribution of 1536 Delaunay triangles in the circular plate with supports;

[0090] Figure 6 yes Figure 3 Iterative convergence curve of the shakedown upper limit load multiplier of the clamped circular thin plate;

[0091] Figure 7 yes Figure 3 The circular plate with support in the material model Working condition 0≤q≤q maxPlastic dissipation power contour at (10 6 N·m);

[0092] Figure 8 is Figure 4 for the clamped circular thin plate in the material model under the condition 0 ≤ q ≤ q max Plastic dissipation power contour at (10 6 N·m);

[0093] Figure 9 is Figure 5 for the clamped circular thin plate in the material model and under the condition -q max ≤ q ≤ q max Plastic dissipation power contour at (10 6 N·m). Detailed implementation manners

[0094] The technical solution of the present invention will be further described in detail below in conjunction with the accompanying drawings and specific implementation manners. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative efforts fall within the scope of the present invention to be protected.

[0095] A specific embodiment of a method for calculating the upper bound of plastic shakedown load considering the strain hardening effect of thin plates. According to Koiter's theorem of shakedown analysis and the two-surface yield criterion, a mathematical programming format for upper bound analysis of plastic shakedown considering the strain hardening effect of thin plates is established, and a C 1 nodal natural element method (C 1 -nodal-NEM) with excellent performance such as high calculation accuracy and efficiency, good numerical stability, and convenient post-processing is used to construct the smooth generalized plastic strain increment of the thin plate, and the nonlinear minimization programming problem containing time integration and equality constraints is discretized. The theory is used to process the generalized plastic strain increment corresponding to the load at each corner point in the load domain to eliminate the time integration in the programming format; a set of direct iteration formats are established to linearize and solve the nonlinear problem. In each iterative solution, the objective function and constraint conditions are modified accordingly according to the calculation results of the previous time, realizing the linearization of the nonlinear objective function, and transforming each iterative calculation into the solution of a set of linear equations. Finally, the smooth generalized plastic strain field, the upper bound of shakedown load, and the plastic dissipation power considering the strain hardening effect of thin plates are obtained.

[0096] As Figure 1 shown, the method for calculating the upper bound of plastic shakedown load considering the strain hardening effect of thin plates in this specific embodiment includes the following steps:

[0097] ST1. Preparation of calculation data

[0098] ST1.1. Preparation of discrete nodes, Delaunay triangles, displacement boundary nodes, loads (constant load, variable load), geometric parameters, material parameters (Young's modulus E, Poisson's ratio v, yield stress Y, hardening parameter ), offset coefficient γ, error tolerances vol1 and vol2.

[0099] ST1.2. According to the information of the nodes, Delaunay triangles and displacement boundary nodes of the thin plate, using the C 1 node natural element method to sequentially determine the vertices of the Voronoi subdomain S r (as Figure 2 shown, r = 1 to NP, where NP is the total number of discrete nodes) and their counterclockwise arrangement order, the coordinates of the vertices of the Voronoi structure after a small offset, the area A r of each triangular subdomain S rs ( RS is the total number of triangular subdomains of subdomain S r ), the tangential vector n rs and normal vector n 1 of each edge, the natural neighboring nodes I (I = 1 to np, where np is the total number of natural neighboring nodes of each integration point determined by the empty circle criterion) of each integration point on each edge and their C 2 first-order derivatives of the natural neighboring shape functions 1 . x rs is the virtual node corresponding to the triangular subdomain S rs , and the superscripts w, θ x , θ y are the deflection, the curl in the x direction, and the curl in the y direction of the thin plate respectively, and the subscripts x and y represent the first-order derivatives with respect to the x direction and y direction respectively. Furthermore, calculate the smooth generalized strain-displacement velocity relationship matrix corresponding to the natural neighboring nodes of each integration point

[0100]

[0101] In Equation (1), Γ rs is the boundary of subdomain S rs .

[0102] ST1.3. "Match and integrate" all the matrices corresponding to the natural neighboring nodes of all the integration points of each triangular subdomain S rs ( ) (which is the total number of nodes after merging all natural neighboring nodes with the same integral points), to obtain each triangular subdomain S rs The corresponding smooth generalized strain-displacement velocity relationship matrix

[0103]

[0104] ST2. Calculate the generalized elastic stress field

[0105] ST2.1. According to the smooth generalized strain-displacement velocity relationship matrix corresponding to each triangular subdomain and the elastic constitutive relationship, calculate and integrate the overall elastic stiffness matrix K of the thin plate e :

[0106]

[0107] In Equation (4), D b is the elastic relationship matrix, is the bending stiffness of the thin plate, h 0 is the plate thickness

[0108] ST2.2. According to the range and number of groups of each set of independently varying loads acting on the thin plate Determine the corner point loads and their total number For the uniformly distributed force in each corner point load, according to the area of the Voronoi subdomain S r of equivalent it to the equivalent load column array acting on the nodes For the concentrated force in each corner point load, directly apply it to the node where it acts. From the equivalent load column arrays of each node "Match the right seat" to integrate the overall elastic load column array of the thin plate when each corner point load acts

[0109]

[0110] ST2.3. Introduce displacement boundary conditions to modify the overall elastic stiffness matrix K of the thin plate e and the overall elastic load column array when each corner point load acts Solve the linear control equation of the thin plate elastic problem to obtain the elastic displacement field of the thin plate when each corner point load acts

[0111] ST2.4. According to the elastic constitutive relationship and the elastic displacement field, calculate the smooth generalized elastic stress corresponding to each triangular subdomain S when each corner point load acts rs

[0112] ​

[0113] ST2.5. Follow the steps ST2.2 to 2.4 to calculate the smooth generalized elastic stress of each triangular sub-domain S under the action of a constant load. rs The corresponding smooth generalized elastic stress

[0114] ST3. Initial iteration (h = 0)

[0115] ST3.1. Assume that the entire thin plate structure is in a non-yield state, and take and to calculate the intermediate variables corresponding to each triangular sub-domain S at the initial iteration in turn. rs The corresponding intermediate variables

[0116]

[0117] In equations (7) and (10), M P = Yh 0 2 / 4 is the plastic limit moment; Q is a positive definite symmetric constant matrix, and Q -1 = 2D / 3, where D is a constant matrix introduced to handle the plastic incompressibility condition of the thin plate:

[0118]

[0119] ST3.2. Calculate and integrate the overall constant load array F 0 , the overall stiffness matrix K considering the strain hardening effect of the thin plate at the initial iteration 0 and the overall variable load array F 1 0 :

[0120]

[0121] ST3.3. Introduce the displacement boundary conditions to modify the overall constant load array F 0 , the overall stiffness matrix K considering the strain hardening effect of the thin plate at the initial iteration 0 and the overall variable load array F 1 0 , and solve the linear equations K 0 (Δa 0 ) 0 = F 0 and K 0 (Δa 1 ) 0 = F 1 0 , to obtain the intermediate variable of the residual displacement increment at the initial iteration (Δa 0 ) 0and (Δa 1 ) 0 。

[0122] ST3.4, Solve for the Lagrange multiplier λ at the initial iteration 0 :

[0123]

[0124] The intermediate variables in Equation (15) and are respectively:[[]]

[0125]

[0126] ST3.5, Solve for the residual displacement increment (Δa) at the initial iteration 0 and the smooth generalized plastic strain corresponding to each triangular subdomain S rs under the action of each corner load

[0127]

[0128] ST3.6, According to the obtained residual displacement increment and smooth generalized plastic strain, sequentially calculate the variables rs corresponding to each triangular subdomain S and μ 1 (xrs),

[0129]

[0130] β in Equation (21) 1 and β in Equation (23) 2 are both positive numbers much less than 1, and are respectively set as:[[]]

[0131]

[0132] Furthermore, obtain the shakedown upper bound load multiplier s considering the strain hardening effect of the thin plate at the initial iteration 0 :

[0133]

[0134] ST4. The h (h≥1) - th iteration

[0135] ST4.1, Determine the values of the variables and according to the calculation results of the (h - 1) - th iteration, and sequentially calculate the intermediate variables rs corresponding to each triangular subdomain S at the h - th iteration

[0136]

[0137] ST4.2, Calculate and integrate the global stiffness matrix K considering the thin plate strain hardening effect at the h-th iteration h and the global variable load array F 1 h :

[0138]

[0139] ST4.3, Introduce displacement boundary conditions to modify the global constant load array F 0 , the global stiffness matrix K at the h-th iteration h and the global variable load array F 1 h , and solve the linear equations K h (Δa 0 ) h = F 0 and K h (Δa 1 ) h = F 1 h , to obtain the intermediate variables of the residual displacement increment at the h-th iteration (Δa 0 ) h and (Δa 1 ) h .

[0140] ST4.4, Solve the Lagrange multiplier λ at the h-th iteration h :

[0141]

[0142] The intermediate variables in Equation (33) and are respectively:

[0143]

[0144] ST4.5, Solve the residual displacement increment (Δa) at the h-th iteration h and the smooth generalized plastic strain corresponding to each triangular subdomain S rs under the action of corner point loads

[0145] (Δa) h = (Δa 0 ) h + λ h (Δa 1 ) h (36)

[0146]

[0147] ST4.6. Calculate the variables corresponding to each triangular sub - domain \(S\) at the \(h\) - th iteration in sequence according to the obtained residual displacement increment and smooth generalized plastic strain. rs corresponding variables and \(\mu\) h+1 (x rs )、

[0148]

[0149] Thus, calculate the shakedown upper - bound load multiplier \(s\) considering the strain - hardening effect of the thin plate at the \(h\) - th iteration: h :

[0150]

[0151] ST4.7. According to the set error tolerances \(vol1\), \(vol2\) and the following convergence conditions, determine whether to terminate the iteration:

[0152] \(\left\|\left(\Delta a\right)\right.\) h \(-\left(\Delta a\right)\) h-1 \(\left\|\right. / \left\|\left(\Delta a\right)\right.\) h-1 \(\left\|\right.\leq vol1,\left|s\right.\) h \(-s\) h-1 \(\left| / \right.s\) h-1 \(\leq vol2(43)\)

[0153] ST5. Post - processing of calculation results

[0154] ST5.1. Assume that the convergence condition is satisfied when the \(h\) - th iteration terminates, and calculate the smooth plastic dissipation power \(D\) corresponding to each triangular sub - domain \(S\) at the termination of the iteration: rs corresponding smooth plastic dissipation power \(D\) h (x rs ):

[0155]

[0156] ST5.2. According to the area relationship between each triangular sub - domain \(S\) rs and the Voronoi sub - domain \(S\) r , obtain the plastic dissipation power \(D\) of each node of the thin - plate structure: h (x r ):

[0157]

[0158] The following takes a set of specific data as an example to further illustrate the above - mentioned specific embodiments:

[0159] Such as Figure 3As shown, for the simply supported circular thin plate considering the strain hardening effect under two working conditions (two working conditions considering variable uniform force: 0 ≤ q ≤ q max , -q max ≤ q ≤ q max , without considering the constant load), and two material models (ideal elastoplastic finite kinematic hardening ), the shakedown upper bound analysis is taken as an example, and the implementation method is as follows:

[0160] Step 1: Prepare the discrete nodes of the thin plate (as Figure 4 shown, the total number is NP = 801), Delaunay triangles (as Figure 5 shown, the total number is 1536), displacement boundary nodes ( Figure 4 the nodes on the middle arc, the total number is 64), the reference value q = 1.0 N of the variable load, the number l of corner loads, the plate thickness h 0 = 0.01 m, the radius R = 1.0 m, the Young's modulus E = 210 GPa, the Poisson's ratio v = 0.3, the yield stress Y = 200 MPa, the hardening parameters and the offset coefficient γ = 1.0×10 -5 , the error tolerances vol1 = vol2 = 1.0×10 -4 and other information.

[0161] Step 2: Calculate and form the smooth generalized strain-displacement velocity relationship matrix corresponding to the natural neighboring nodes of each integration point, and "seat" and integrate the smooth generalized strain-displacement velocity relationship matrices corresponding to all triangular subdomains.

[0162] Step 3: Calculate and integrate the overall elastic stiffness matrix of the thin plate and the overall elastic load column matrix when each corner load acts.

[0163] Step 4: Introduce the displacement boundary conditions to modify the overall elastic stiffness matrix of the thin plate and the overall elastic load column matrix when each corner load acts, and solve to obtain the elastic displacement field of the thin plate when each corner load acts and the corresponding smooth generalized elastic stress fields of each triangular subdomain at this time.

[0164] Step 5: Calculate the relevant intermediate variables corresponding to each triangular subdomain at the initial iteration in turn, and calculate and integrate the overall stiffness matrix and the overall variable load column matrix considering the strain hardening effect of the thin plate at the initial iteration.

[0165] Step 6: Introduce displacement boundary conditions to modify the global stiffness matrix and the global variable load array considering the thin plate strain hardening effect at the initial iteration, solve the corresponding linear equations, and successively obtain the intermediate variables of the residual displacement increment, Lagrange multipliers, residual displacement increment, smooth generalized plastic strains corresponding to each triangular subdomain under the action of each corner load, and the shakedown upper bound load multiplier considering the thin plate strain hardening effect.

[0166] Step 7: Calculate the relevant intermediate variables corresponding to each triangular subdomain at the h-th iteration according to the calculation results of the (h - 1)-th iteration, and calculate and integrate the global stiffness matrix and the global variable load array considering the thin plate strain hardening effect at the h-th iteration.

[0167] Step 8: Introduce displacement boundary conditions to modify the global stiffness matrix and the global variable load array at the h-th iteration, solve the corresponding linear equations, and successively obtain the intermediate variables of the residual displacement increment, Lagrange multipliers, residual displacement increment, smooth generalized plastic strains corresponding to each triangular subdomain under the action of each corner load, and the shakedown upper bound load multiplier considering the thin plate strain hardening effect.

[0168] Step 9: Determine whether to terminate the iteration according to the set error tolerance and convergence conditions.

[0169] Step 10: Calculate the plastic dissipation power corresponding to each triangular subdomain at the termination of the iteration, and obtain the plastic dissipation power of each node of the thin plate structure according to the area relationship between each triangular subdomain and the Voronoi subdomain.

[0170] Table 1 presents the shakedown upper bound load multiplier and the calculation time obtained by using this specific embodiment (based on the C 1 nodal natural element method, C 1 -nodal-NEM, using stable compatible node integration for triangular subdomains) to solve, and also presents the theoretical solution [1] based on the unified strength theory, the numerical solution [2] obtained by using the C 1 natural element method (C 1 -NEM, using traditional background grid integration) and the calculation time.

[0171] Table 1 Shakedown upper bound load multiplier and calculation time of clamped circular thin plates

[0172]

[0173] As can be seen from Table 1:

[0174] 1. In the ideal elastoplastic model when, the shakedown upper bound numerical solutions obtained by using this specific embodiment are respectively in good agreement with the theoretical solution [1] based on the unified strength theory and the C1 The numerical solution of -NEM is in good agreement, indicating that this specific embodiment has good computational accuracy.

[0175] 2. In the finite kinematic hardening model, for the clamped circular thin plate under -q max ≤q≤q max working condition, it is alternating plastic failure. The shakedown load multipliers obtained by using this specific embodiment and C 1 -NEM are the same as those in the perfectly elastoplastic case respectively; under the working condition of 0≤q≤q max it is cumulative plastic failure. The shakedown load multipliers obtained by using this specific embodiment and C 1 -NEM are increased by 22.371% and 19.861% respectively compared with those in the perfectly elastoplastic case, which verifies the conclusion that neglecting the strengthening effect of materials will make the obtained shakedown load conservative.

[0176] 3. As Figure 6 shown, overall (except for individual cases, such as the material model under the working condition of 0≤q≤q max ), the number of iteration steps for calculation using this specific embodiment is usually less, and its iterative calculation time is only about 70% of the calculation time spent using C 1 -NEM. This shows that the C 1 -nodal-NEM based on the stable compatible nodal integration of triangular subdomains used in this specific embodiment usually has higher iterative calculation efficiency compared with the C 1 -NEM using the traditional background grid integration.

[0177] Figure 6 The iterative convergence curve diagram for obtaining the shakedown upper limit load multiplier of the clamped circular thin plate in Table 1 is given. It can be seen that this specific embodiment can monotonically and stably converge to the required computational accuracy after 11 - 78 iterations.

[0178] Figures 7 - 9 The plastic dissipation power contour diagram of the shakedown limit state of the clamped circular thin plate obtained by post - processing using this specific embodiment is shown, which truly and intuitively reflects the plastic shakedown failure mode of the clamped circular thin plate.

[0179] In summary, the present invention adopts the C 1 nodal natural element method with excellent performance, solves the problems of establishing and linearly solving the minimization iterative format for the shakedown upper limit analysis considering the strain strengthening effect of thin plates, can accurately, efficiently and stably solve the shakedown upper limit load considering the strain strengthening effect of thin plates, and has the technical effects of relatively simple format, strong versatility, easy program implementation, high computational accuracy and efficiency, good numerical stability, and convenient post - processing.

Claims

1. A method for calculating the upper bound shakedown load considering the strain hardening effect of thin plates, characterized in that, it includes the following steps: ST1. Prepare calculation data to obtain the smooth generalized strain-displacement velocity relationship matrix corresponding to each triangular subdomain of the thin plate structure, specifically including: ST1.

1. Prepare the discrete nodes, Delaunay triangles, displacement boundary nodes, loads, geometric parameters, material parameters, offset coefficient γ, error tolerances vol1 and vol2 of the thin plate structure; The loads include: constant loads, variable loads; The material parameters include: Young's modulus E, Poisson's ratio v, yield stress Y, and hardening parameter ST1.

2. According to the information of the nodes of the thin plate, Delaunay triangles, and displacement boundary nodes, the natural node element method is used to sequentially determine the vertices of the Voronoi subdomain S 1 surrounding each node x r and their counterclockwise arrangement order, the coordinates of the vertices of the Voronoi structure after a small offset, the area A r of each triangular subdomain S rs , the tangential vector n rs of each edge and the normal vector n 1 , the natural neighboring nodes I of each integration point on each edge and their C 2 first-order derivatives of the natural neighboring shape functions 1 ​ r = 1 to NP, where NP is the total number of discrete nodes; RS is the total number of triangular sub-domains of sub-domain S r ; I = 1 to np, where np is the total number of natural neighboring nodes of each integration point determined by the empty circle criterion; x rs is the virtual node corresponding to the triangular subdomain S rs , and the superscripts w, θ x , θ y are respectively the deflection of the thin plate, the curl in the x direction, and the curl in the y direction. The subscripts x and y respectively denote the first-order derivatives with respect to the x direction and the y direction, and then calculate the smooth generalized strain-displacement velocity relationship matrix corresponding to the natural neighboring nodes of each integration point where Γ rs is the boundary of the subdomain S rs ; ST1.

3. Integrate each triangular sub-domain S by "matching the right seats" rs corresponding to the natural neighboring nodes of all integration points matrix is the total number of nodes after merging the same natural neighboring nodes of all integration points, obtaining each triangular sub-domain S rs corresponding smooth generalized strain-displacement velocity relationship matrix ST2. Calculate the generalized elastic stress field, and respectively obtain the smooth generalized elastic stress fields corresponding to each triangular subdomain of the thin plate structure under the action of corner loads and constant loads, specifically including: ST2.

1. Calculate and integrate the global elastic stiffness matrix K of the thin plate according to the smooth generalized strain-displacement velocity relationship matrix and elastic constitutive relationship corresponding to each triangular subdomain e : where D b is the elastic relationship matrix; is the flexural rigidity of the thin plate; h 0 is the plate thickness; ST2.2 Determine the corner load and its total number according to the range and number of groups of each set of independently varying loads acting on the thin plate Determine the corner load and its total number For the uniformly distributed force in each corner load, according to the area of the Voronoi subdomain S r Area Equivalent it to the equivalent load array acting on the node For the concentrated force in each corner load, directly apply it to the node where it acts; from the equivalent load arrays of each node "Match the right seat" to integrate the overall elastic load array of the thin plate when each corner load acts ST2.

3. Introduce displacement boundary conditions to modify the overall elastic stiffness matrix K of the thin plate e and the overall elastic load column array under the action of each corner load Solve the linear control equation for the elastic problem of the thin plate Obtain the elastic displacement field of the thin plate under the action of each corner load ST2.

4. Calculate the smooth generalized elastic stress corresponding to each triangular sub-domain S when each corner point load acts according to the elastic constitutive relationship and the elastic displacement field. rs The corresponding smooth generalized elastic stress ST2.5, calculate the smooth generalized elastic stress corresponding to each triangular subdomain S under the action of a constant load rs corresponding smooth generalized elastic stress ST3. At the initial iteration, h = 0, assume that the entire thin plate structure is in a non-yield state, solve the linear equations, and sequentially obtain the Lagrange multipliers, residual displacement increments, smooth generalized plastic strains corresponding to each triangular subdomain under the action of each corner load, and the shakedown upper bound load multiplier considering the strain hardening effect of the thin plate; The steps for solving the linear equations specifically include: ST3.

1. Assume that the entire thin plate structure is in a non-yield state, and take and to calculate the intermediate variables rs corresponding to each triangular subdomain S where M P = Yh 0 2 / 4 is the plastic limit moment; Q is a positive definite symmetric constant matrix, and Q -1 = 2D / 3, where D is a constant matrix introduced for dealing with the plastic incompressibility condition of thin plates: ST3.2, Calculate and integrate the global constant load array F 0 , the global stiffness matrix K considering the thin plate strain hardening effect at the initial iteration 0 and the global variable load array F 1 0 : ST3.

3. Introduce displacement boundary conditions to modify the overall constant load array F 0 and the overall stiffness matrix K 0 considering the strain hardening effect of the thin plate at the initial iteration 1 0 , and solve the linear equations K 0 (Δa 0 ) 0 = F 0 and K 0 (Δa 1 ) 0 = F 1 0 respectively, to obtain the intermediate variables (Δa 0 ) 0 and (Δa 1 ) 0 of the residual displacement increment at the initial iteration; ST3.4, Solve for the Lagrange multiplier λ at the initial iteration 0 : In the formula, the intermediate variables and are respectively: ST3.5, Solve for the residual displacement increment (Δa) at the initial iteration 0 and the smoothed generalized plastic strain corresponding to each triangular subdomain S rs under the action of each corner point load (△a) 0 =(△a 0 ) 0 +λ 0 (△a 1 ) 0 ST3.

6. Calculate the variables corresponding to each triangular subdomain S at the initial iteration in sequence according to the obtained residual displacement increment and smooth generalized plastic strain rs corresponding variables and μ 1 (x rs ), where β 1 and β 2 are both positive numbers much less than 1 and are respectively set as: Furthermore, the shakedown upper bound load multiplier s considering the thin plate strain hardening effect at the initial iteration is obtained 0 : ST4. At the hth (h ≥ 1) iteration, according to the calculation results of the (h - 1)th iteration, integrate and solve the corresponding linear equations, and sequentially obtain the Lagrange multipliers, residual displacement increments, smooth generalized plastic strains corresponding to each triangular subdomain under the action of each corner load, and the shakedown upper bound load multiplier considering the strain hardening effect of the thin plate, and judge whether to terminate the iterative calculation according to the iterative convergence condition; ST5. Post-process the calculation results to obtain the plastic dissipation power of each node of the thin plate structure.

2. The method for calculating the upper bound shakedown load considering the strain hardening effect of thin plates according to claim 1, characterized in that, the steps of integrating and solving the corresponding linear equations at the hth (h ≥ 1) iteration according to the calculation results of the (h - 1)th iteration specifically include: ST4.

1. Determine the values of variables and according to the calculation results of the (h - 1)-th iteration, and calculate the intermediate variables rs corresponding to each triangular sub-domain S ST4.

2. Calculate and integrate the global stiffness matrix K considering the thin plate strain hardening effect at the h-th iteration h and the global variable load array F 1 h : ST4.

3. Introduce displacement boundary conditions to modify the global constant load array F 0 , the global stiffness matrix K at the h-th iteration h and the global variable load array F 1 h , and solve the linear equations K h (Δa 0 ) h = F 0 and K h (Δa 1 ) h = F 1 h , to obtain the intermediate variables (Δa 0 ) h and (Δa 1 ) h ; ST4.4, Solve for the Lagrange multiplier λ at the h-th iteration h : In the formula, the intermediate variables and are respectively: ST4.5, Solve for the residual displacement increment (Δa) at the h-th iteration h and the smoothed generalized plastic strains corresponding to each triangular sub-domain S rs under the action of each corner load (△a) h =(△a 0 ) h +λ h (△a 1 ) h ST4.

6. Calculate the variables corresponding to each triangular subdomain \(S\) at the \(h\)-th iteration in sequence according to the obtained residual displacement increment and smooth generalized plastic strain rs corresponding variables and \(\mu\) h+1 (\(x\) rs ) Thus, the shakedown upper bound load multiplier s considering the thin plate strain hardening effect at the h-th iteration is calculated h : ST4.

7. Judge whether to terminate the iteration according to the set error tolerances vol1, vol2 and the following convergence conditions:

3. The method for calculating the upper bound shakedown load considering the strain hardening effect of thin plates according to claim 2, characterized in that, the steps of post-processing the calculation results specifically include: ST5.

1. Assume that the convergence condition is satisfied at the end of the h-th iteration, and calculate the smooth plastic dissipation power D corresponding to each triangular subdomain S at the end of the terminated iteration rs corresponding to h (x rs ): ST5.

2. Obtain the plastic dissipation power D rs of each node in the thin plate structure according to the area relationship between each triangular subdomain S r and the Voronoi subdomain S h (x r ):

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