A numerical calculation method for the upper and lower limit loads of plastic shakedown for ideal elastic-plastic thin plates

Abstract

The present invention proposes a numerical calculation method for the upper and lower limit loads of the plastic stability of an ideal elastic-plastic thin plate, which belongs to the field of physical technology, including: ST1, preparing calculation data to form a generalized strain-displacement velocity relationship matrix; ST2, constructing a generalized elastic stress field to obtain the smooth generalized elastic stress of each corner load; ST3, performing initial iteration, assuming that the entire thin plate is in a non-yielding state, solving and calculating; ST4, in the hth (h≥1)th iteration, solving the linear equation according to the result of the h-1th iteration, obtaining the Lagrange multiplier, smooth generalized plastic strain and generalized stress, and the upper and lower limit load multiplier of the hth iteration, and judging whether the iteration is terminated; ST5, post-processing of the calculation results. Based on the established iterative format for the upper limit analysis of the ideal elastic-plastic thin plate, the present invention solves the construction of a high-precision smooth generalized stress field for the lower limit analysis and the iterative solution of the lower limit load, and realizes the simultaneous accurate and efficient calculation of the upper and lower limit loads of the ideal elastic-plastic thin plate.

Description

Technical Field

[0001] The invention belongs to the technical field of plastic mechanics, and in particular relates to a numerical calculation method for upper and lower limit loads of plastic stability of an ideal elastic-plastic thin plate. Background Art

[0002] Thin plates are an important structural form. Their geometric characteristics are that the dimensions in the thickness direction are much smaller than those in the length and width directions. They are widely used in aerospace, aviation, shipbuilding, civil engineering and other engineering fields. Studying the ultimate bearing capacity of thin plate structures under cyclic loads and revealing their safety margins are of great significance for ensuring their safety and reliability. Shakedown analysis is an important branch of plastic mechanics. It does not involve loading history. It is a simple and effective method to study the cumulative plastic failure and alternating plastic failure of structures under cyclically changing loads. The obtained shakedown load is an important parameter reflecting the structural safety margin in many engineering design specifications.

[0003] The shakedown analysis of thin plates is based on Koiter's upper limit theorem (dynamic shakedown theory) and Melan's lower limit theorem (static shakedown theory). The upper limit theorem is applicable to plastic pressure processing and geotechnical engineering, while the lower limit theorem is applicable to structural engineering design and safety assessment. At present, compared with the large amount of systematic and in-depth research work on plastic limit analysis of thin plates, there is relatively little research work on shakedown analysis of thin plates. In these limited research works, theoretical analysis, experimental research and numerical calculation are usually used to obtain the upper or lower limit load of the thin plate based on the upper or lower limit theorem of the shakedown analysis. However, there is no numerical calculation method for directly carrying out the lower limit analysis of the shakedown based on the upper limit analysis of the thin plate.

[0004] With the continuous emergence and application of a large number of novel and excellent numerical methods, it is an important research direction for thin plate shakedown analysis to combine them with the mature planning theory of shakedown analysis and develop numerical solution algorithms with high calculation accuracy, stable performance and wide application range. 1 Nodal Natural Element Method (C 1 -nodal-NEM) is in C 1 Natural Unit Method (C 1 -NEM) based on the development of a novel meshless method suitable for the analysis of thin plate structures. 1The shape function is constructed by natural neighbor interpolation, and the calculation format is simple. It only requires node connection information, does not involve complex matrix operations, and does not require any artificial parameters. The triangular subdomain stable compatible integral (TSCI) is used for numerical integration, which has high calculation accuracy, good numerical stability, and convenient post-processing. It is particularly suitable for constructing the high-precision smooth generalized plastic strain field required for the upper limit analysis of the ideal elastic-plastic thin plate structure, and is also suitable for constructing the high-precision smooth generalized stress field required for the lower limit analysis based on the upper limit iterative format.

[0005] In summary, there is no numerical calculation method in the prior art that can directly carry out lower limit analysis of shakedown based on upper limit analysis of thin plates, and this needs to be provided. Summary of the invention

[0006] The present invention provides a numerical calculation method for the upper and lower limit loads of plastic stability of an ideal elastic-plastic thin plate. 1 The linearized direct iterative solution format for the upper limit analysis of the shakedown of an ideal elastic-plastic thin plate established by the node natural element method aims to solve the problems of constructing a high-precision smooth generalized stress field and iteratively solving the lower limit load for the lower limit analysis of the shakedown of an ideal elastic-plastic thin plate, and to achieve simultaneous accurate and efficient calculation of the upper and lower limit loads of the shakedown of an ideal elastic-plastic thin plate.

[0007] The objective of the present invention is achieved through the following technical solutions:

[0008] A numerical calculation method for upper and lower limit loads of plastic shakedown of an ideal elastic-plastic thin plate, comprising:

[0009] ST1. Prepare calculation data using C 1 The nodal natural element method forms a smooth generalized strain-displacement velocity relationship matrix corresponding to each triangular subdomain;

[0010] ST2, construct the generalized elastic stress field, and obtain the smooth generalized elastic stress corresponding to each triangular subdomain when the load acts at each corner point;

[0011] ST3, perform initial iteration according to the established iterative calculation format (h=0);

[0012] Assuming that the entire thin plate structure is in a non-yielding state, solve the linear equations and calculate the Lagrange multipliers at the initial iteration, the smooth generalized plastic strain corresponding to each triangular subdomain under the action of each corner point load, the smooth generalized stress corresponding to each triangular subdomain, and the upper and lower limit load multipliers of the ideal elastic-plastic thin plate.

[0013] ST4, perform the hth (h≥1) iteration;

[0014] According to the calculation results of the h-1th iteration, the corresponding linear equations are solved to obtain the Lagrange multiplier at the hth iteration, the smooth generalized plastic strain corresponding to each triangular subdomain under the action of each corner point load, the smooth generalized stress corresponding to each triangular subdomain, and the upper and lower limit load multipliers of the ideal elastic-plastic thin plate. The iteration is terminated according to the set error tolerance and convergence conditions.

[0015] ST5. Post-process the calculation results to obtain the plastic dissipated power and generalized Mises equivalent stress of each node of the thin plate structure, and draw a cloud diagram.

[0016] As a further optimization, the steps for preparing the calculation data include:

[0017] ST1.1. Prepare the discrete nodes, Delaunay triangles, displacement boundary nodes, loads, geometric parameters, material parameters, offset coefficient γ, and error tolerances vol1 and vol2 of the thin plate structure;

[0018] ST1.2, according to the information of the nodes, Delaunay triangles and displacement boundary nodes of the thin plate, use C 1 The stable and compatible integration scheme of the triangular subdomain of the nodal natural element method calculates each node x r (r = 1 ~ NP, NP is the total number of discrete nodes) Voronoi subdomain S r Each triangular subdomain RS is subdomain S r The total number of triangular subdomains) boundary Γ rs The smooth generalized strain-displacement velocity relationship matrix corresponding to the natural neighboring node I (I = 1 ~ np, np is the total number of natural neighboring nodes of each integration point) matrix:

[0019]

[0020] In the formula, x rs corresponds to the triangular subdomain S rs Virtual nodes;

[0021] A rs is a triangular subdomain S rs area;

[0022] n1 and n2 are the boundary Γ rs The tangent and normal vectors of ;

[0023] They are natural neighboring nodes IC 1 The first derivative of the natural adjacent shape function, superscript w, θ x ,θ yThey are the deflection of the thin plate, the rotation in the x direction, and the rotation in the y direction, respectively. The subscripts x and y indicate the first-order derivatives in the x direction and the y direction, respectively.

[0024] ST1.3, integrate each triangular subdomain S rs All integral points naturally adjacent to the corresponding nodes matrix( np is the total number of nodes after merging all natural adjacent nodes with the same integral points), forming each triangular subdomain S rs The corresponding smooth generalized strain-displacement velocity relationship matrix

[0025] As a further optimization, the steps of constructing the generalized elastic stress field specifically include:

[0026] ST2.1. Calculate and integrate the overall elastic stiffness matrix K of the thin plate structure e :

[0027]

[0028]

[0029] Where D b is the elastic relationship matrix;

[0030] is the bending stiffness of the thin plate;

[0031] E is Young's modulus;

[0032] h0 is the plate thickness;

[0033] v is Poisson’s ratio;

[0034] ST2.2, according to the range and number of groups of independent variable loads acting on the thin plate Determine corner loads and their total According to the Voronoi subdomain S r Area Equivalent uniform force in each corner point load to the equivalent load array of the node (k=1~l); directly apply the concentrated force in each corner point load to the node on which it acts; array the equivalent load of each node The overall elastic load array of the thin plate when the loads at each corner are integrated

[0035] ST2.3. Introduce displacement boundary conditions to modify the overall elastic stiffness matrix K of the thin plate e The overall elastic load array when the loads act on each corner point Solving the governing equations for thin plate elasticity problems Get the elastic displacement field of the thin plate when the load acts at each corner point

[0036] ST2.4. Calculate the S of each triangular subdomain when the load acts on each corner point according to the elastic constitutive relationship. rs The corresponding smooth generalized elastic stress

[0037]

[0038] As a further optimization, the initial iteration steps according to the established iterative calculation format specifically include:

[0039] ST3.1. Assume that the entire thin plate structure is in a non-yielding state. Calculate each triangle subdomain S at the initial iteration rs The corresponding intermediate variable

[0040]

[0041]

[0042]

[0043] Where M P =σ s h0 2 / 4 is the plastic limit bending moment;

[0044] σ s is the yield stress;

[0045] D is the constant matrix introduced to deal with the plastic incompressible condition of thin plates:

[0046]

[0047] ST3.2. Calculate and integrate the global stiffness matrix K of the ideal elastic-plastic thin plate at the initial iteration 0 and the overall variable load array F 0 :

[0048]

[0049]

[0050] ST3.3. Introduce displacement boundary conditions to modify the overall stiffness matrix K 0 and the overall variable load array F 0 , solve the linear equation K 0 (Δa1) 0 =F 0, and obtain the intermediate variable (Δa1) of the residual displacement increment at the initial iteration 0 ;

[0051] ST3.4. Solve the Lagrange multiplier λ at the initial iteration 0 :

[0052]

[0053] In the formula, the intermediate variable for:

[0054]

[0055] ST3.5, solve the initial iteration of the thin plate under the action of each corner point load in each triangular subdomain S rs The corresponding smooth generalized plastic strain

[0056]

[0057] ST3.6. Based on the calculated smooth generalized plastic strain, calculate the initial iteration S of each triangular subdomain. rs Corresponding variables

[0058]

[0059]

[0060] Where β1 is a positive number much less than 1 and is defined as:

[0061]

[0062] The shakedown upper limit load multiplier of the ideal elastic-plastic thin plate in the initial iteration is obtained as

[0063]

[0064] ST3.7. Calculate each triangle subdomain S during the initial iteration rs The corresponding smooth generalized stress and the load multiplier β rs,0 :

[0065]

[0066]

[0067] In the formula, the constant matrix A is:

[0068]

[0069] Take β rs,0 The minimum of the minimum value of the stable lower limit load multiplier of the ideal elastic-plastic thin plate at the initial iteration is obtained.

[0070]

[0071] As a further optimization, the hth (h≥1)th iteration step specifically includes:

[0072] ST4.1. Determine the variable based on the calculation results of the h-1th iteration The values ​​of each triangle subdomain S are calculated at the hth iteration. rs The corresponding intermediate variable

[0073]

[0074]

[0075]

[0076] ST4.2. Calculate and integrate the global stiffness matrix K of the ideal elastic-plastic thin plate at the hth iteration h and the overall variable load array F h :

[0077]

[0078]

[0079] ST4.3. Introduce displacement boundary conditions to modify the overall stiffness matrix K at the hth iteration h and the overall variable load array F h , solve the linear equation K h (Δa1) h =F h , and obtain the intermediate variable (Δa1) of the residual displacement increment at the hth iteration h ;

[0080] ST4.4. Lagrange multiplier λ at the hth iteration h :

[0081]

[0082] The intermediate variable in the formula for:

[0083]

[0084] ST4.5, in the hth iteration, each triangular subdomain S of the thin plate under the load at each corner point rsThe corresponding smooth generalized plastic strain

[0085]

[0086] ST4.6. Based on the calculated smooth generalized plastic strain, calculate the S of each triangular subdomain at the hth iteration. rs Corresponding variables

[0087]

[0088]

[0089] Then the shakedown upper limit load multiplier of the ideal elastic-plastic thin plate at the hth iteration is obtained:

[0090]

[0091] ST4.7. Calculate each triangle subdomain S at the hth iteration rs The corresponding smooth generalized stress and the load multiplier β rs,0 :

[0092]

[0093]

[0094] Take β rs,h The minimum of the minimum value of the lower limit load multiplier of the ideal elastic-plastic thin plate at the hth iteration is obtained.

[0095]

[0096] ST4.8. Determine whether the iteration is terminated based on the set error tolerance and the following convergence conditions:

[0097]

[0098] Where (Δa) h-1 =λ h-1 (Δa1) h-1 , is the residual displacement increment of the h-1th iteration;

[0099] (Δa) h =λ h (Δa1) h , is the residual displacement increment of the hth iteration.

[0100] As a further optimization, the post-processing steps of the calculation results specifically include:

[0101] ST5.1, Assume that the convergence condition is met when the hth iteration ends, calculate the triangle subdomains S when the iteration ends rs The corresponding smooth plastic dissipated power D h (x rs ), and then according to each triangular subdomain S rs With Voronoi subdomain S r The area relationship of each node is used to calculate the plastic dissipation power D h (x r ):

[0102]

[0103]

[0104] ST5.2, according to each triangular subdomain S obtained when the iteration is terminated rs The corresponding smooth generalized stress vector Calculate the corresponding smooth generalized Mises equivalent stress

[0105]

[0106] In the formula, and are the smooth generalized stress components in the x, y and xy directions obtained at the hth iteration, respectively;

[0107] Then, according to the area relationship, the generalized Mises equivalent stress of each node is calculated

[0108]

[0109]

[0110] ST5.3, according to the plastic dissipation power D of each discrete node of the thin plate structure h (x r ) and generalized Mises equivalent stress The plastic dissipated power cloud diagram and the generalized Mises equivalent stress cloud diagram were drawn using drawing software.

[0111] The beneficial technical effects achieved by the present invention are:

[0112] 1. The present invention can simultaneously calculate the upper and lower limit loads of the ideal elastic-plastic thin plate structure, can effectively deal with the nonlinearity of the objective function in the upper limit analysis, and overcome the strong nonlinearity of the inequality constraint in the lower limit analysis. It has the advantages of strong versatility, simple solution, and easy program implementation, and provides an accurate and efficient numerical calculation method for the plastic shakedown analysis of thin plates.

[0113] 2. C used in the present invention 1 Nodal Natural Element Method (C 1 -nodal-NEM), with C 1 The advantages of natural neighbor interpolation and stable compatible integral of triangular subdomains have high calculation accuracy, good numerical stability and convenient post-processing. They are very suitable for the construction of high-precision smooth generalized plastic strain field and smooth generalized stress field, iterative calculation and post-processing of numerical results in thin plate shakedown analysis.

[0114] 3. The upper and lower limit loads of the thin plate obtained by the present invention, the plastic dissipated power cloud map that can truly reflect the failure mode of the thin plate, and the generalized Mises equivalent stress cloud map can provide important basis and reference for the optimal design and safety assessment of the thin plate structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] Figure 1 It is an implementation step diagram of the present invention;

[0116] Figure 2 It is a schematic diagram of the triangular subdomain division of the present invention;

[0117] Figure 3 is a schematic diagram of a simply supported square thin plate subjected to a uniformly distributed force q for shakedown upper limit analysis in a specific embodiment of the present invention;

[0118] Figure 4 yes Figure 3 Schematic diagram of the distribution of 441 discrete nodes of the simply supported square thin plate;

[0119] Figure 5 yes Figure 3 Schematic diagram of the distribution of 800 Delaunay triangles in a simply supported square thin plate;

[0120] Figure 6 yes Figure 3 Iterative convergence curve of the upper and lower limit load multipliers of the simply supported square thin plate;

[0121] Figure 7 yes Figure 3 Plastic dissipation power cloud diagram of the simply supported square thin plate in the shakedown limit state under working condition 1 (10 6 N·m);

[0122] Figure 8 yes Figure 3 Generalized Mises stress distribution cloud diagram of the simply supported square thin plate in the shakedown limit state under working condition 1 (MPa);

[0123] Fig. 9 yes Figure 3 Plastic dissipation power cloud diagram of the simply supported square thin plate in the shakedown limit state under working condition 2 (106 N·m);

[0124] Fig.10 yes Figure 3 Generalized Mises stress distribution cloud diagram (MPa) of the simply supported square thin plate in the steady state limit state under working condition 2. DETAILED DESCRIPTION

[0125] The technical solution of the present invention is further described in detail below in conjunction with the accompanying drawings and specific implementation methods. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative work are within the scope of protection claimed by the present invention.

[0126] The present invention is based on the excellent performance of C 1 The node natural element method constructs a high-precision smooth generalized plastic strain increment and a linearized direct iterative solution format for the upper limit analysis of the ideal elastic-plastic thin plate structure. On this basis, a high-precision smooth generalized stress field and an iterative calculation format for the lower limit analysis are constructed accordingly, providing a numerical calculation method that can accurately and efficiently calculate the upper and lower limit loads of the ideal elastic-plastic thin plate at the same time.

[0127] A specific embodiment of a numerical calculation method for the upper and lower limit loads of the plastic shakedown of an ideal elastic-plastic thin plate is provided. The method establishes a minimization mathematical programming format for the upper limit analysis of the shakedown of an ideal elastic-plastic thin plate based on the Koiter theorem of shakedown analysis. 1 Nodal Natural Element Method (C 1 -nodal-NEM) has the advantages of high calculation accuracy, good numerical stability and convenient post-processing. 1 The smooth generalized plastic strain increment of thin plates is constructed by natural neighbor interpolation and the triangle subdomain stable compatible integration scheme (TSCI), and a C-based 1 -nodal-NEM is a linearized direct iterative format for the shakedown upper limit analysis of discretized thin plates. In each iteration, the corresponding linear equations are corrected and solved according to the calculation results of the previous iteration, and then the high-precision smooth generalized plastic strain field and shakedown upper limit load multiplier of the thin plate are obtained.

[0128] On the basis of the linearized direct iterative format of the upper limit analysis of thin plate shakedown, a smooth generalized stress field satisfying the equilibrium equations and boundary conditions was constructed. Then, according to the Melan theorem of shakedown analysis, the strong nonlinear constraints were overcome and the iterative calculation format of the lower limit load multiplier of the thin plate shakedown was determined.

[0129] The numerical calculation method of this specific embodiment can simultaneously and accurately and efficiently solve the upper and lower limit loads of the ideal elastic-plastic thin plate shakedown analysis, and obtain the plastic dissipated power cloud map and the generalized Mises equivalent stress cloud map that can reflect the plastic failure mode of the structure, such as Figure 1 As shown, the specific steps include:

[0130] ST1. Prepare calculation data

[0131] ST1.1. Prepare the discrete nodes, Delaunay triangles, displacement boundary nodes, loads (base value of variable loads, number of corner loads l), geometric parameters, material parameters (Young's modulus E, Poisson's ratio v, yield stress σ s ), offset coefficient γ, error tolerance vol1 and vol2.

[0132] ST1.2, according to the information of the nodes, Delaunay triangles and displacement boundary nodes of the thin plate, use C 1 Nodal Natural Element Method (C 1 -nodal-NEM) triangular subdomain stable and compatible integration scheme, calculating each node x r (like Figure 2 As shown, r = 1 ~ NP, NP is the total number of discrete nodes) Voronoi subdomain S r Each triangular subdomain S rs ( RS is subdomain S r The total number of triangular subdomains) boundary Γ rs The smooth generalized strain-displacement velocity relationship matrix corresponding to the natural neighboring node I (I = 1 ~ np, np is the total number of natural neighboring nodes of each integration point) matrix:

[0133]

[0134] In formula (1), x rs corresponds to the triangular subdomain S rs A virtual node rs is a triangular subdomain S rs The area of ​​​​the boundary Γ is n1 and n2 respectively. rs The tangent and normal vectors of They are natural neighboring nodes IC 1 The first derivative of the natural adjacent shape function, superscript w, θ x ,θ y They are the deflection of the thin plate, the rotation in the x direction, and the rotation in the y direction. The subscripts x and y represent the first-order derivatives in the x direction and y direction respectively.

[0135] ST1.3. Integrate each triangular subdomain S according to the “matching” method rs All integral points naturally adjacent to the corresponding nodes matrix( The total number of nodes after merging all natural adjacent nodes with the same integral points) is formed in each triangular subdomain S rs The corresponding smooth generalized strain-displacement velocity relationship matrix

[0136] ST2. Constructing generalized elastic stress field

[0137] ST2.1. Calculate and “match” the overall elastic stiffness matrix K of the integrated thin plate structure e :

[0138]

[0139]

[0140] In formula (3), D b is the elastic relationship matrix, is the bending stiffness of the thin plate, and h0 is the plate thickness.

[0141] ST2.2. Determine the corner load and its total number based on the range and number of groups v of each group of independent variable loads acting on the thin plate. According to the Voronoi subdomain S r Area Equivalent uniform force in each corner point load to the equivalent load array of the node Directly apply the concentrated force in each corner point load to the node on which it acts. Array the equivalent load of each node The overall elastic load array of the thin plate when the loads at each corner are integrated ST2.3. Introduce displacement boundary conditions to modify the overall elastic stiffness matrix K of the thin plate e The overall elastic load array when the loads act on each corner point Solving the governing equations for thin plate elasticity problems Get the elastic displacement field of the thin plate when the load acts at each corner point

[0142] ST2.4. Calculate the S of each triangular subdomain when the load acts on each corner point according to the elastic constitutive relationship. rs The corresponding smooth generalized elastic stress

[0143]

[0144] ST3, initial iteration (h = 0)

[0145] ST3.1. Assume that the entire thin plate structure is in a non-yielding state. Calculate each triangle subdomain S at the initial iteration rs The corresponding intermediate variable

[0146]

[0147]

[0148]

[0149] In formula (5), M P =σ s h0 2 / 4 is the plastic limit bending moment, and D is the constant matrix introduced to deal with the plastic incompressible condition of thin plates:

[0150]

[0151] ST3.2. Calculate and integrate the global stiffness matrix K of the ideal elastic-plastic thin plate at the initial iteration 0 and the overall variable load array F 0 :

[0152]

[0153]

[0154] ST3.3. Introduce displacement boundary conditions to modify the overall stiffness matrix K 0 and the overall variable load array F 0 , solve the linear equation K 0 (Δa1) 0 =F 0 , and obtain the intermediate variable (Δa1) of the residual displacement increment at the initial iteration 0 .

[0155] ST3.4. Solve the Lagrange multiplier λ at the initial iteration 0 :

[0156]

[0157] The intermediate variable in formula (11) for:

[0158]

[0159] ST3.5, solve the initial iteration of the thin plate under the action of each corner point load in each triangular subdomain S rs The corresponding smooth generalized plastic strain

[0160]

[0161] ST3.6. Based on the calculated smooth generalized plastic strain, calculate the initial iteration S of each triangular subdomain. rs Corresponding variables

[0162]

[0163]

[0164] β1 in formula (15) is a positive number much smaller than 1 and is defined as:

[0165]

[0166] The shakedown upper limit load multiplier of the ideal elastic-plastic thin plate in the initial iteration is obtained as

[0167]

[0168] ST3.7. Calculate each triangle subdomain S during the initial iteration rs The corresponding smooth generalized stress and the load multiplier β rs,0 :

[0169]

[0170]

[0171] The constant matrix A in formula (19) is:

[0172]

[0173] Take β rs,0 The minimum of the minimum value of the stable lower limit load multiplier of the ideal elastic-plastic thin plate at the initial iteration is obtained.

[0174]

[0175] ST4, hth (h≥1) iteration

[0176] ST4.1. Determine the variable based on the calculation results of the h-1th iteration The values ​​of each triangle subdomain S are calculated at the hth iteration. rs The corresponding intermediate variable

[0177]

[0178]

[0179]

[0180] ST4.2. Calculate and integrate the global stiffness matrix K of the ideal elastic-plastic thin plate at the hth iteration h and the overall variable load array F h :

[0181]

[0182]

[0183] ST4.3. Introduce displacement boundary conditions to modify the overall stiffness matrix K at the hth iteration h and the overall variable load array F h , solve the linear equation K h (Δa1) h =F h , and obtain the intermediate variable (Δa1) of the residual displacement increment at the hth iteration h .

[0184] ST4.4. Solve the Lagrange multiplier λ at the hth iteration h :

[0185]

[0186] The intermediate variable in formula (27) for:

[0187]

[0188] ST4.5. Solve the h-th iteration for each triangular subdomain S under the action of the load at each corner point of the thin plate. rs The corresponding smooth generalized plastic strain

[0189]

[0190] ST4.6. Based on the calculated smooth generalized plastic strain, calculate the S of each triangular subdomain at the hth iteration. rs Corresponding variables

[0191]

[0192]

[0193] Then the shakedown upper limit load multiplier of the ideal elastic-plastic thin plate at the hth iteration is obtained:

[0194]

[0195] ST4.7. Calculate each triangle subdomain S at the hth iteration rs The corresponding smooth generalized stress and the load multiplier β rs,0 :

[0196]

[0197]

[0198] Take β rs,h The minimum of , we can get the lower limit load multiplier s_ of the ideal elastic-plastic thin plate at the hth iteration. h :

[0199]

[0200] ST4.8. Determine whether the iteration is terminated based on the set error tolerance and the following convergence conditions:

[0201]

[0202] In formula (36), (Δa) h-1 =λ h-1 (Δa1) h-1 and (Δa) h =λ h (Δa1) h are the residual displacement increments of the h-1th and hth iterations respectively.

[0203] ST5. Post-processing of calculation results

[0204] ST5.1, Assume that the convergence condition is met when the hth iteration ends, calculate the triangle subdomains S when the iteration ends rs The corresponding smooth plastic dissipated power D h (x rs ), and then according to each triangular subdomain S rs With Voronoi subdomain S r The area relationship of each node is used to calculate the plastic dissipation power D h (x r ):

[0205]

[0206]

[0207] ST5.2, according to each triangular subdomain S obtained when the iteration is terminated rsThe corresponding smooth generalized stress vector and are the smooth generalized stress components in the x, y and xy directions obtained in the hth iteration respectively, and the corresponding smooth generalized Mises equivalent stress is calculated Then, according to the area relationship, the generalized Mises equivalent stress of each node is calculated

[0208]

[0209] ST5.3, according to the plastic dissipation power D of each discrete node of the thin plate structure h (x r ) and generalized Mises equivalent stress Relevant drawing software is used to draw the plastic dissipated power cloud diagram and the generalized Mises equivalent stress cloud diagram which can intuitively reflect the plastic failure mode of the thin plate.

[0210] The following is a further explanation of the above specific embodiment by taking a set of specific data as an example:

[0211] like Figure 3 As shown, the uniformly distributed variable load (0≤q≤q max ) is taken as an example to analyze the upper and lower limits of the shakedown of a simply supported (w = 0) square thin plate under two working conditions (working condition 1: simply supported on four sides; working condition 2: simply supported on three sides and free on one side). The following steps are taken:

[0212] Step 1: Prepare the discrete nodes of the thin plate (such as Figure 4 As shown, the total number is NP = 441), Delaunay triangles (such as Figure 5 As shown, the total number is 800), displacement boundary nodes ( Figure 4 The nodes on the edge of the square, the number of boundary nodes for Condition 1 and Condition 2 are 80 and 61 respectively), the reference value of the variable load q = 1.0N, the number of corner loads l = 2, the plate thickness h0 = 0.01m, the side length 2a = 1.0m, the Young's modulus E = 210GPa, the Poisson's ratio v = 0.3, the yield stress σ s =200MPa, deviation coefficient γ = 1.0×10 -5 、Error tolerance vol1=vol2=1.0×10 -4 And other information.

[0213] Step 2: Use C 1 The nodal natural element method calculates the smooth generalized strain-displacement-velocity relationship matrix corresponding to the natural adjacent nodes of each integration point, and forms the smooth generalized strain-displacement-velocity relationship matrix corresponding to each triangular subdomain.

[0214] Step 3: Integrate the overall elastic stiffness matrix of the thin plate structure and the overall elastic load matrix when the loads act at each corner point.

[0215] Step 4: Introduce displacement boundary conditions to modify the overall elastic stiffness matrix of the thin plate and the overall elastic load matrix when each corner point load acts, solve the control equation, and obtain the elastic displacement field of the thin plate when each corner point load acts and the smooth generalized elastic stress corresponding to each triangular subdomain at this time.

[0216] Step 5: Calculate the relevant intermediate variables corresponding to each triangular subdomain at the initial iteration, and calculate and integrate the overall stiffness matrix and the overall variable load array of the ideal elastic-plastic thin plate.

[0217] Step 6: Introduce displacement boundary conditions to modify the overall stiffness matrix and the overall variable load array, solve the linear equations, and obtain the intermediate variables of the residual displacement increment at the initial iteration, the Lagrange multiplier, the smooth generalized plastic strain corresponding to each triangular subdomain under the action of each corner point load, the smooth generalized stress corresponding to each triangular subdomain, and the upper and lower limit load multipliers of the ideal elastic-plastic thin plate.

[0218] Step 7: According to the calculation results of the h-1th iteration, calculate the relevant intermediate variables corresponding to each triangular subdomain at the hth iteration, and integrate the overall stiffness matrix and the overall variable load array of the ideal elastic-plastic thin plate at the hth iteration.

[0219] Step 8: Introduce displacement boundary conditions to modify the overall stiffness matrix and the overall variable load matrix at the h-th iteration, solve the linear equations, and obtain the intermediate variables of the residual displacement increment at the h-th iteration, the Lagrange multiplier, the smooth generalized plastic strain corresponding to each triangular subdomain under the action of each corner point load, the smooth generalized stress corresponding to each triangular subdomain, and the upper and lower limit load multipliers of the ideal elastic-plastic thin plate.

[0220] Step 9: Determine whether the iteration is terminated based on the set error tolerance and convergence conditions.

[0221] Step 10: Calculate the smooth plastic dissipated power and smooth generalized Mises equivalent stress of each triangular subdomain when the iteration is terminated, and then calculate the plastic dissipated power and generalized Mises equivalent stress of each node according to the area relationship between each triangular subdomain and the Voronoi subdomain, and draw the corresponding cloud map using relevant drawing software.

[0222] Table 1 shows the C-based 1 Nodal Natural Element Method (C 1-nodal-NEM, using the triangular subdomain stable compatible integral) to obtain the numerical solution and calculation time, and also gives the lower limit finite element numerical solution based on the temperature parameter method [1], based on C 1 Natural Unit Method (C 1 -NEM) and ACM (both using traditional background grid integration) numerical solutions and computational time.

[0223] Table 1 Shaker load multipliers and calculation time for simply supported square thin plates

[0224]

[0225] From Table 1 we can see that:

[0226] 1. The numerical solution of the upper and lower limits of the ideal elastic-plastic square thin plate structure obtained by this specific embodiment is consistent with the numerical solution based on C 1 The upper and lower limit numerical solutions obtained by -NEM and ACM are almost equal, and these numerical solutions are in good agreement with the lower limit numerical solution obtained based on the temperature parameter method [1]. This shows that the solution method described in this specific embodiment can accurately and effectively calculate the upper and lower limit loads of the ideal elastic-plastic square thin plate structure at the same time.

[0227] 2. The calculation time of using the present invention is similar to that of using C 1 - The computation time of NEM and ACM is comparable.

[0228] Figure 6 The iterative convergence curve for obtaining the upper and lower limit load multipliers in Table 1 is given. It can be seen that the solution method described in this specific embodiment, C 1 The convergence curves calculated by -NEM and ACM are almost the same, and only 19 and 16 iterations are needed to efficiently obtain the upper and lower limit load multipliers of simply supported square thin plates that meet the calculation accuracy under two working conditions.

[0229] Figures 7-10 The plastic dissipated power distribution cloud diagram and the generalized Mises equivalent stress cloud diagram of the shakedown limit state of the simply supported square thin plate obtained by post-processing in this specific embodiment are given. These cloud diagrams are consistent, symmetrical and reasonable, and can truly and intuitively reflect the plastic shakedown failure mode of the simply supported square thin plate.

[0230] In summary, the present invention is based on C with excellent performance 1 The node natural element method, based on the established linearized direct iterative solution format for the upper limit analysis of ideal elastic-plastic thin plates, solves the problems of constructing a high-precision smooth generalized stress field for the lower limit analysis and iterative solution of the lower limit load. It can simultaneously and accurately and efficiently calculate the upper and lower limit loads of ideal elastic-plastic thin plates, and has the technical effects of strong versatility, simple solution, easy program implementation, and convenient post-processing.

Claims

1. A numerical calculation method for the upper and lower limit loads of plastic stability of an ideal elastic-plastic thin plate, characterized in that: include; ST1. Prepare calculation data using C 1 The nodal natural element method forms a smooth generalized strain-displacement velocity relationship matrix corresponding to each triangular subdomain; ST2, construct the generalized elastic stress field, and obtain the smooth generalized elastic stress corresponding to each triangular subdomain when the load acts at each corner point; ST3. Perform initial iteration according to the established iterative calculation format, and use h to represent the number of iterations. At this time, h=0; Assuming that the entire thin plate structure is in a non-yielding state, solve the linear equations and calculate the Lagrange multipliers at the initial iteration, the smooth generalized plastic strain corresponding to each triangular subdomain under the action of each corner point load, the smooth generalized stress corresponding to each triangular subdomain, and the upper and lower limit load multipliers of the ideal elastic-plastic thin plate. ST4, perform the hth iteration, at this time the number of iterations h≥1; According to the calculation results of the h-1th iteration, the corresponding linear equations are solved to obtain the Lagrange multiplier at the hth iteration, the smooth generalized plastic strain corresponding to each triangular subdomain under the action of each corner point load, the smooth generalized stress corresponding to each triangular subdomain, and the upper and lower limit load multipliers of the ideal elastic-plastic thin plate. The iteration is terminated according to the set error tolerance and convergence conditions. ST5. Post-process the calculation results to obtain the plastic dissipated power and generalized Mises equivalent stress of each node of the thin plate structure, and draw a cloud diagram.

2. The numerical calculation method according to claim 1, characterized in that: The step of preparing the calculation data specifically includes: ST1.

1. Prepare the discrete nodes, Delaunay triangles, displacement boundary nodes, loads, geometric parameters, material parameters, offset coefficient γ, and error tolerances vol1 and vol2 of the thin plate structure; ST1.2, according to the information of the nodes, Delaunay triangles and displacement boundary nodes of the thin plate, use C 1 The stable and compatible integration scheme of the triangular subdomain of the nodal natural element method calculates each node x r In the Voronoi subdomain S r Each triangular subdomain S rs The boundary of Γ rs The smooth generalized strain-displacement velocity relationship matrix corresponding to the natural neighboring node I of each integral point on matrix: In the formula, node x r The subscript r = 1 ~ NP, NP is the total number of discrete nodes; RS is subdomain S r The total number of triangular subdomains; I = 1 ~ np, np is the total number of natural neighboring nodes of each integral point; x rs corresponds to the triangular subdomain S rs Virtual nodes; A rs is a triangular subdomain S rs area; n1 and n2 are the boundary Γ rs The tangent and normal vectors of ; They are natural neighboring nodes IC 1 The first derivative of the natural adjacent shape function, superscript w, θ x ,θ y They are the deflection of the thin plate, the rotation in the x direction, and the rotation in the y direction, respectively. The subscripts x and y indicate the first-order derivatives in the x direction and the y direction, respectively. ST1.3, integrate each triangular subdomain S rs All integral points naturally adjacent to the corresponding nodes matrix, Subscript in To merge all the nodes with the same natural neighboring nodes, each triangular subdomain S rs The corresponding smooth generalized strain-displacement velocity relationship matrix 3. The numerical calculation method according to claim 2, characterized in that: The step of constructing a generalized elastic stress field specifically includes: ST2.

1. Calculate and integrate the overall elastic stiffness matrix K of the thin plate structure e : Where D b is the elastic relationship matrix; is the bending stiffness of the thin plate; E is Young's modulus; h0 is the plate thickness; v is Poisson’s ratio; ST2.2, according to the range and number of groups of independent variable loads acting on the thin plate Determine corner loads and their total According to the Voronoi subdomain S r Area Equivalent uniform force in each corner point load to the equivalent load array of the node The subscript k in the formula is 1 to 1; the concentrated force in each corner point load is directly applied to the node on which it acts; the equivalent load array of each node The overall elastic load array of the thin plate when the loads at each corner are integrated ST2.

3. Introduce displacement boundary conditions to modify the overall elastic stiffness matrix K of the thin plate e The overall elastic load array when the loads act on each corner point Solving the governing equations for thin plate elasticity problems Get the elastic displacement field of the thin plate when the load acts at each corner point ST2.

4. Calculate the S of each triangular subdomain when the load acts on each corner point according to the elastic constitutive relationship. rs The corresponding smooth generalized elastic stress 4. The numerical calculation method according to claim 3, characterized in that: The initial iteration step according to the established iterative calculation format specifically includes: ST3.

1. Assume that the entire thin plate structure is in a non-yielding state and take the variable Calculate each triangle subdomain S at the initial iteration rs The corresponding intermediate variable Where M P =σ s h0 2 / 4 is the plastic limit bending moment; σ s is the yield stress; D is the constant matrix introduced to deal with the plastic incompressible condition of thin plates: ST3.

2. Calculate and integrate the global stiffness matrix K of the ideal elastic-plastic thin plate at the initial iteration 0 and the overall variable load array F 0 : ST3.

3. Introduce displacement boundary conditions to modify the overall stiffness matrix K 0 and the overall variable load array F 0 , solve the linear equation K 0 (Δa1) 0 =F 0 , and obtain the intermediate variable (Δa1) of the residual displacement increment at the initial iteration 0 ; ST3.

4. Solve the Lagrange multiplier λ at the initial iteration 0 : In the formula, the intermediate variable for: ST3.5, solve the initial iteration of the thin plate under the action of each corner point load in each triangular subdomain S rs The corresponding smooth generalized plastic strain ST3.

6. Based on the calculated smooth generalized plastic strain, calculate the initial iteration S of each triangular subdomain. rs Corresponding variables Where β1 is a positive number much less than 1 and is defined as: The shakedown upper limit load multiplier of the ideal elastic-plastic thin plate in the initial iteration is obtained as ST3.

7. Calculate each triangle subdomain S during the initial iteration rs The corresponding smooth generalized stress and the load multiplier β rs,0 : In the formula, the constant matrix A is: Take β rs,0 The minimum of the minimum value of the stable lower limit load multiplier of the ideal elastic-plastic thin plate at the initial iteration is obtained. :

5. The numerical calculation method according to claim 4, characterized in that: The step ST4 specifically includes: ST4.

1. Determine the variable based on the calculation results of the h-1th iteration The values ​​of each triangle subdomain S are calculated at the hth iteration. rs The corresponding intermediate variable ST4.

2. Calculate and integrate the global stiffness matrix K of the ideal elastic-plastic thin plate at the hth iteration h and the overall variable load array F h : ST4.

3. Introduce displacement boundary conditions to modify the overall stiffness matrix K at the hth iteration h and the overall variable load array F h , solve the linear equation K h (Δa1) h =F h , and obtain the intermediate variable (Δa1) of the residual displacement increment at the hth iteration h ; ST4.

4. Lagrange multiplier λ at the hth iteration h : The intermediate variable in the formula for: ST4.5, in the hth iteration, each triangular subdomain S of the thin plate under the load at each corner point rs The corresponding smooth generalized plastic strain ST4.

6. Based on the calculated smooth generalized plastic strain, calculate the S of each triangular subdomain at the hth iteration. rs Corresponding variables Then the shakedown upper limit load multiplier of the ideal elastic-plastic thin plate at the hth iteration is obtained: ST4.

7. Calculate each triangle subdomain S at the hth iteration rs The corresponding smooth generalized stress and the load multiplier β rs,0 : Take β rs,h The minimum of the minimum value of the lower limit load multiplier of the ideal elastic-plastic thin plate at the hth iteration is obtained. ST4.

8. Determine whether the iteration is terminated based on the set error tolerance and the following convergence conditions: Where (Δa) h-1 =λ h-1 (Δa1) h-1 , is the residual displacement increment of the h-1th iteration; (Δa) h =λ h (Δa1) h , is the residual displacement increment of the hth iteration.

6. The numerical calculation method according to claim 5, characterized in that: The calculation result post-processing step specifically includes: ST5.1, Assume that the convergence condition is met when the hth iteration ends, calculate the triangle subdomains S when the iteration ends rs The corresponding smooth plastic dissipated power D h (x rs ), and then according to each triangular subdomain S rs With Voronoi subdomain S r The area relationship of each node is used to calculate the plastic dissipation power D h (x r ): ST5.2, according to each triangular subdomain S obtained when the iteration is terminated rs The corresponding smooth generalized stress vector Calculate the corresponding smooth generalized Mises equivalent stress In the formula, and are the smoothed generalized stress components in the x, y and xy directions obtained at the hth iteration, respectively; Then, according to the area relationship, the generalized Mises equivalent stress of each node is calculated ST5.3, according to the plastic dissipation power D of each discrete node of the thin plate structure h (x r ) and generalized Mises equivalent stress The plastic dissipated power cloud diagram and the generalized Mises equivalent stress cloud diagram were drawn using drawing software.

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