CN109145405B - Steel structure fiber model damage evaluation method - Google Patents

Steel structure fiber model damage evaluation method Download PDF

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CN109145405B
CN109145405B CN201810861145.9A CN201810861145A CN109145405B CN 109145405 B CN109145405 B CN 109145405B CN 201810861145 A CN201810861145 A CN 201810861145A CN 109145405 B CN109145405 B CN 109145405B
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蒋瓅
瞿革
倪建公
李庆武
肖炳辉
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China Shipbuilding NDRI Engineering Co Ltd
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Abstract

The invention discloses a damage evaluation method of a steel structure fiber model, which is characterized in that elastic-plastic analysis of a general finite element of the steel structure fiber model is carried out to obtain an edge fiber plastic strain cloud picture, the damage and damage conditions of a component in the dynamic response process of the whole structure are visually reflected, the failure and damage of the component are controlled by the deformation limit and the material deformation limit of the component, and the safety of the whole structure is judged and evaluated according to the bending damage of the component and the plastic deformation of the part of the component except a plastic corner section and the condition of judging the tension and pressure damage of the component. Compared with the prior art, the method has the advantages of simple analysis process and high working efficiency, and particularly provides favorable basis and reference for the damage and the destruction of important components, so that the failure destruction of the components in the dynamic response process can meet the deformation limit of the FEMA components and the strain limit of materials.

Description

Steel structure fiber model damage evaluation method
Technical Field
The invention relates to the technical field of building structure safety design, in particular to a steel structure fiber model damage evaluation method based on the FEMA standard.
Background
When an engineering structure is designed, the safety of the whole structure under the action of earthquake and accidental load is usually judged through dynamic elastoplasticity analysis, and the failure and the damage of a component are usually considered. At present, the finite element models commonly used for the elasto-plastic analysis comprise a plastic hinge model and a fiber model, wherein the plastic hinge model is a relatively macroscopic model, gives the performance state of a component according to the force-deformation relationship and is commonly used in software such as Sap2000, midas/Gen and the like; the fiber model is a relatively microscopic model, divides a section into a plurality of fibers, describes the performance state of a component through the uniaxial stress-strain relation of materials, and is commonly used in ABAQUS, ANSYS/LS-DYNA, MSC.MARC, perform-3D and other software.
With the continuous improvement of the computer level and the continuous improvement of the requirement of the calculation precision, the application of the dynamic elastoplasticity analysis based on the fiber model in the actual engineering is gradually increased. As for evaluation criteria, the current national specifications do not clearly specify the failure damage of the component, but are classified into six grades of no damage (intact), slight damage, mild damage, moderate damage, more severe damage and severe damage according to the damage condition of the component. In order to better evaluate the performance level of the engineering structure, the member performance evaluation indexes in the FEMA specification are widely applied at the present stage and have high acceptance, and the indexes are shown in the following table 1:
TABLE 1 corresponding relationship table of plastic strain and each evaluation standard of steel
Degree of damage of high gauge Without damage Slight damage Mild damage Moderate damage Is relatively seriously damaged
FEMA Specification <B B～IO IO～LS LS～CP >CP
General purposePlastic strain index 1 epsilon p /ε y 0 0～1 1～3 3～6 >6
Common plastic strain index 2 epsilon p /ε y 0 0～2 2～4 4～6 >6
The standard divides the plastic stage of the component into a strain strengthening section (BC section) and a strength loss section (CDE section) based on the relation between the internal force of the component and macroscopic deformation, correspondingly divides the performance indexes of the component into three stages of immediate use (IO), life Safety (LS) and near Collapse (CP) within the plastic rotation range allowed by design (not exceeding the C point), and gives the plastic rotation angle and the deformation extreme value of different components corresponding to each performance level.
In the prior art, evaluation indexes are based on a macroscopic component, in order to combine the macroscopic evaluation indexes with a microscopic fiber beam unit model analysis result, when most of actual engineering is used for dynamic elastoplasticity analysis based on ABAQUS, perform-3D and SAusage, the ratio of steel plastic strain epsilon p to yield strain epsilon y is used as an index for judging the damage degree of the component by taking the FEMA specification as reference, and the limit plastic strain of the steel is set to be 2.5%. At present, the damage degree indexes of the components based on strain are not uniform, meanwhile, the damage form of the components cannot be sufficiently reflected through material deformation limit control, and the damage degree of the components is evaluated through the plastic strain of steel and still has great difference with the evaluation standard in the FEMA standard.
Disclosure of Invention
The invention aims to provide a steel structure fiber model damage evaluation method designed aiming at the defects of the prior art, which adopts the corresponding relation between microscopic material strain and FEMA macroscopic component performance indexes, intuitively reflects component damage and destruction conditions in the whole structure dynamic response process through finite element elasto-plastic analysis, realizes the control of component deformation limit and material deformation limit on component failure damage, ensures that the failure damage of the component in the numerical simulation in the dynamic response process meets both the FEMA component deformation limit and the material strain limit, and simultaneously, the damage degree of the residual structural components can be evaluated according to the component performance evaluation indexes of FEMA standard and can correspond to the evaluation indexes of conventional elasto-plastic analysis.
The purpose of the invention is realized by the following steps: a steel structure fiber model damage evaluation method comprises a beam calculated according to the following formulas (I) and (II) n Column (diagonal bracing) n ：
Beam
Figure BDA0001749709800000031
Column (diagonal bracing)
Figure BDA0001749709800000032
The method is characterized in that the steel structure fiber model is loaded and solved, then the damage condition of a component is judged, the safety of the structure is judged according to the dynamic response condition of the whole structure, and the specific judgment is carried out according to the following steps:
the method comprises the following steps: the plastic rotation angle of the component is determined according to the FEMA-356 standard (tables 5-6) to reach the values of the IO (immediately usable), LS (life safety) and CP (near collapse) three stages.
Step two: the ductility factor μ at the time of the three-stage leveling was calculated from the following formulas (XII) and (XIII) IO 、μ LS And mu CP And plastic strain epsilon of plastic corner section edge fiber IO 、ε LS And epsilon CP ：
Beam:
Figure BDA0001749709800000033
post, bracing:
Figure BDA0001749709800000034
step three: reading the result information after finite element analysis to obtain an edge fiber plastic strain cloud picture, and obtaining the edge fiber plastic strain epsilon according to the plastic corner section edge fiber plastic strain epsilon lp Determining which stage the plastic corner of the component reaches: if epsilon IO ≤ε lp ＜ε LS The plastic corner of the component reaches the IO stage; if epsilon LS ≤ε lp ＜ε CP The plastic corner of the component reaches the LS stage; if epsilon lp ≥ε CP The plastic corner of the component reaches the CP stage; whereby the degree of flexural damage of the member can be determined. The safety of the whole structure can be evaluated according to the plastic deformation of the part of the component except the plastic corner section and the condition of judging the pulling and pressing damage of the component.
The beam n Column (diagonal bracing) n The calculation adopts that the plastic corner deformation of all the beam, the column and the inclined strut member in the steel structure is set to be concentrated on the length l of the two ends of the member p Range, i.e. plastic corner section, and calculating the length l of the component and the plastic corner section l according to the component type, section parameters, material type, material ultimate plastic strain and FEMA component ultimate strength deformation p The ratio n is calculated by the following formulas (I) and (II) n Stud (diagonal bracing) n ：
Beam
Figure BDA0001749709800000041
Column (diagonal bracing)
Figure BDA0001749709800000042
In the formula: e is the modulus of elasticity; i is a section moment of inertia; epsilon u is the steel ultimate plastic strain (generally 2.5%;); mu.s a For plastic corners of members in strain-strengthened sections andratio of yield turns, mu a ＝θ C /θ y -1；θ C Is the angle of rotation when the component reaches the performance index ultimate strength C point of the FEMA component, theta y is the yield angle of the component, mu a The values can be referred to as the plastic turning angle a value when the component reaches the ultimate strength C in FEMA-356 (tables 5-6); gamma is the shape factor of the cross section; γ = W p /W n ，W p Is a plastic section modulus, W n Is the modulus of elasticity in section; f. of y Is the material yield strength; h is 1 The distance from the fiber at the edge of the cross section to the neutral axis, when the cross section is symmetrical with respect to the neutral axis, I = W n h 1 ＝W n h/2, i.e. beams n Column (diagonal bracing) n The method can be simplified into the following formulas (III) and (IV) for calculation, and then the formed fiber model is loaded and solved;
beam
Figure BDA0001749709800000043
Column (diagonal bracing)
Figure BDA0001749709800000044
The method for forming the steel structure fiber model by adopting the grid division is characterized in that in order to determine the distribution of fibers on the fiber section, the fiber model is formed by adopting an effective grid division method, and only one fiber beam unit is ensured in the plastic corner section of all the components.
The loading solution of the steel structure fiber model is to ensure that the failure damage of the member in the numerical simulation in the dynamic response process meets the deformation limit and the material strain limit of the FEMA member, and meanwhile, the damage degree of the residual structural member can be evaluated according to the member performance evaluation index of the FEMA standard.
The ratio of the plastic corner to the yield corner mu a The plastic rotation angle a value of the component in the FEMA-356 standard (tables 5-6) when the component reaches the ultimate strength C is shown as mu a ＝θ C /θ y -1；θ C The corner is the corner when the component reaches the performance index ultimate strength C point of the FEMA component; theta y Is the yield angle of the component.
The describedBeam n The derivation of the calculation formula is as follows:
(1) For a fiber model in finite element software (ABAQUS, ANSYS/LS-DYNA), when the plastic strain of a certain fiber on a fiber beam unit reaches failure strain, the fiber can be withdrawn from work; when all the fibers are removed from operation, the fiber beam unit is killed, since the maximum plastic strain of the fiber beam unit generally occurs in the edge fibers, the fiber beam unit is removed from operation firstly, and then other fibers are failed in the process of internal force redistribution until the unit is killed. According to the failure characteristic, the component reaches the ultimate strength (corresponding to the C point in the FEMA component performance index) when the edge fiber is withdrawn from the work, and therefore the corresponding relation between the macroscopic deformation of the component and the microscopic strain of the fiber is deduced.
Since the numerical simulation satisfies the assumption of a flat section, the strain on the section of the member is proportional to the distance from the neutral axis, and the relationship between the curvature phi and the fiber strain epsilon at the edge of the member can be deduced according to the mathematical meaning of the curvature of the member as shown in the following formula (V):
φ≈tanφ＝ε/h 1 ； (V)
wherein phi is the curvature; ε is the strain of the cross-sectional edge fiber.
(2) The plastic corner deformation of the setting member is concentrated at both ends l of the member p And in the length range, namely plastic corner sections are arranged at two ends of the component, the curvature phi in the length range is kept constant, and the relation of the fiber strain epsilon of the edge with the component corner theta according to the formula (V) is represented by the following formula (VI):
θ＝φl p ≈εl p /h 1 。 (VI)
(3) According to the formula (5-1) in the FEMA-356 standard, the yield angle theta of the flexural member y Calculated according to the following formula (VII):
Figure BDA0001749709800000051
the rotation angle theta of the component reaching the performance index ultimate strength C point of the FEMA component c Calculated according to the following formula (VIII):
Figure BDA0001749709800000061
(4) Assuming that the component angle reaches theta C When the plastic strain of the edge fiber of the plastic corner section reaches the failure strain epsilon u While setting the strain at which the edge fiber yields to ε e ＝f y Based on the formulas (VI) and (VIII), the ultimate strength of a member represented by the formula (IX) shown below was determined by theta C And plastic strain epsilon of fiber at edge of plastic corner section u The relation is as follows:
Figure BDA0001749709800000062
thereby pushing out the beam of the formula (I) n And (4) calculating a formula.
The post (bracing) n The derivation of the calculation formula is as follows:
for bending and stretch-bending members such as columns and inclined struts, the influence of axial compression ratio is also considered, and the yield angle theta is calculated according to the formula (5-2) in FEMA-356 y Calculated according to the following formula (X):
Figure BDA0001749709800000063
wherein, P is axial force; p ye Is the yield axial force of the section, P ye ＝A fy . The FEMA-356 standard indicates when the axial pressure P is less than the compressive strength P CL When the steel member is 50 percent, the failure of the steel member is controlled by deformation, and the steel member is expressed as ductile stress performance; on the contrary, the force is controlled by force, which shows brittle stress performance, and the failure of the component is mainly destabilized and destroyed. In the range of deformation control, when P/P is found by the FEMA-356 standard (tables 5-6) CL <0.2 time mu a =4, when 0.2<P/P CL <0.5 time mu a =1, the member deformation index greatly differs depending on the shaft pressure P. The factor taking into account the stability of the member is P CL <P ye While taking into account the shaftThe adverse effect of the pressure P on the deformation limit of the component and the uncertainty of the axial force P in the dynamic response process are uniformly taken as P =0.2P ye Performing a calculation of theta y ＝W pf y l /(7.5E I ) (ii) a The relation between the curvature and the strain of the fiber at the edge of the plastic corner section is phi = (in epsilon-epsilon)/h 1 Where, in ε is the central axis strain, ε is =0.2P ye /(EA)＝f y and/5E. Thereby, a member rotation angle θ represented by the following formula (XI) is obtained C Plastic strain epsilon of plastic corner section edge fiber u The relation of (1):
Figure BDA0001749709800000071
and then push out (II) type column (diagonal bracing) n And (4) calculating a formula.
When the stress of the component is in the force control range (P is more than or equal to 0.5P) ye ) When the component fails, the failure damage of the component is controlled by the deformation limit of the material and the stability of the component, and the set material failure effect is changed into epsilon u The method can be satisfied by solving a basic equation of large deformation dynamics, and can be realized by finite element software.
Compared with the prior art, the method has the advantages that the elastic-plastic analysis of the micro material strain and FEMA macroscopic component performance general finite element is realized, the damage and destruction conditions of the component in the whole structure dynamic response process are intuitively reflected, the control of the component failure destruction by the component deformation limit and the material deformation limit is realized, the failure destruction of the component in the dynamic response process meets both the FEMA component deformation limit and the material strain limit, meanwhile, the damage degree of the residual structural component can be evaluated according to the component performance evaluation index of the FEMA standard, the analysis process is simple, the working efficiency is high, and particularly, favorable basis and reference are provided for the damage and destruction of important components.
Drawings
FIG. 1 is a flow chart of the operation of the present invention;
FIG. 2 is a schematic view of a fiber model of a component;
FIG. 3 is a schematic view of an overall construction model of embodiment 1;
FIG. 4 is a schematic representation of FEMA component performance metrics;
fig. 5 is a cloud plot of plastic strain for example 1.
Detailed Description
Referring to the attached figure 1, the method of the invention comprises the following steps:
the first step is as follows: firstly, setting the length of two ends of all beams, columns and diagonal bracing members in a steel structure as l p And determining lengths l and l of the component according to the component type, section parameters, material type, material ultimate plastic strain and FEMA component ultimate strength deformation p The ratio n of (A) to (B);
the second step is that: counting n values of all components, determining a value range of n, verifying the n value with small influence on stress, deformation and failure of the components in the value range through finite element analysis, and if the influence is large, adjusting the mesh division of a finite element model;
the third step: effective meshing is adopted to form a finite element model to define a fiber section, only one fiber unit is ensured in the plastic corner section of all the components, and then loading solving is carried out.
The fourth step: and judging the damage condition of the member based on the member performance index in the FEMA specification, and judging the safety of the structure according to the dynamic response condition of the whole structure.
Example 1
The invention relates to a method for carrying out safety judgment on an overall structure through an edge fiber plastic strain cloud picture obtained by a steel structure fiber model, which comprises the following specific steps:
a. calculating the ratio n of the length of the member to the length of the shaped corner segment
Referring to FIG. 2, for all members of beams, columns and braces in a steel structure, a length l is set such that plastic corner deformation is concentrated at both ends of the member p The lengths l and l of the component are determined according to the following formulas (I) and (II) according to the component type, section parameters, material type, material ultimate plastic strain and FEMA component ultimate strength deformation p Beam of the ratio n Column, column n And diagonal bracing n And (3) value calculation:
beam
Figure BDA0001749709800000081
Column (diagonal bracing)
Figure BDA0001749709800000082
Wherein E is the modulus of elasticity; i is a section moment of inertia; epsilon u The material limit plastic strain of the steel is 2.5 percent of the current engineering common numerical value; mu.s a The ratio of the plastic angle 2 to the yield angle of the component in the strain-strengthened section, μ a ＝θ C /θ y -1；θ C Is the angle of rotation, theta, at which the component reaches the ultimate strength C point of the FEMA component y Is the yield angle of the member, mu a The values can be referred to the plastic corner a values when the component reaches the ultimate strength C in the FEMA-356 standard (tables 5-6); γ is a shape coefficient of a cross section, γ = W p /W n ，W p Is a plastic section modulus, W n Is the modulus of elasticity in section; f. of y Is the material yield strength; h is 1 The distance of the cross-sectional edge fiber 8 from the neutral axis. When the cross section is symmetrical with respect to the neutral axis, then I = W n h 1 ＝W n The h/2,n value can be simplified to the following formula (III) and (IV):
beam
Figure BDA0001749709800000091
Column (diagonal bracing)
Figure BDA0001749709800000092
Referring to the attached figure 3, when the high-rise steel structure 1 is subjected to the progressive collapse resistance analysis of the bottom corner post 3, firstly, all beams are calculated according to the formulas (I) and (II) n Column, column n And diagonal bracing n The value is obtained.
The girder ZL with the cross section of H680 multiplied by 400 multiplied by 32 and the material strength of Q345 1 For example, according to the type of the component and the symmetry of the section, the method adopts the formula (III)And (4) calculating. Calculating the section parameters to obtain gamma =1.16; finding E = 2.06X 105N/mm according to the type of material 2 、f y =345MPa; material limit plastic strain epsilon u ＝2.5％。
Referring to FIG. 4, referring to the FEMA-356 standard (tables 5-6), the plastic rotation angle a at which the component reaches the ultimate strength C is determined to be 4 θ y, i.e., μ a Is 4.
Substituting the parameters into formula (III) to calculate the girder ZL 1 N =16.5, i.e. the length of the plastic corner section 2 is 1/16.5 of the length of the component.
The steel column Z with the section of 9633800X 500X 28 and the material strength of Q345 1 For example, the calculation is performed by using the formula (IV) according to the type of the component and the symmetry of the cross section. Calculating the section parameters to obtain gamma =1.22; finding E = 2.06X 105N/mm according to the type of material 2 、f y =345MPa; material limit plastic strain epsilon u Taking out 2.5 percent.
Referring to FIG. 4, referring to FEMA-356 standard (tables 5-6), the plastic rotation angle a at which the member reaches ultimate strength C is determined to be 4 θ y I.e. mu a Is 4.
Substituting the parameters into formula (IV), and calculating to obtain the steel column Z 1 N =19.3, i.e. the length of the plastic corner section 2 is 1/19.3 of the length of the component.
b. Forming fiber model for loading solution
Referring to fig. 2, the distribution of fibers 11 on a fiber section 10 is determined, a fiber model 13 is formed on a high-rise steel structure 1 by adopting an effective grid 12 dividing method, only one fiber beam unit 14 is ensured in a plastic corner section 2 of all components, and then loading solving is carried out.
c. Evaluating the damage of a component and the safety of the structure
Reading the result information after finite element analysis to obtain an edge fiber plastic strain cloud chart, and obtaining the edge fiber plastic strain epsilon according to the plastic corner section edge fiber plastic strain epsilon lp Determining which stage the plastic corner of the component reaches: if epsilon IO ≤ε lp ＜ε LS The plastic corner of the component reaches the IO stage; if epsilon LS ≤ε lp ＜ε CP Then the component is plastically deformedThe corner reaches LS stage; if epsilon lp ≥ε CP The plastic corner of the component reaches the CP stage; whereby flexural damage of the member can be determined. And evaluating the whole structure according to the plastic deformation of the part of the component except the plastic corner section and judging the condition of the pulling and pressing damage of the component, and judging the safety of the structure according to the dynamic response condition of the whole structure.
Coefficient of each leveling IO /L S /C P Plastic strain epsilon with plastic corner section edge fiber IO /L S /C P The relationship (XII) and (XIII) is as follows:
beam:
Figure BDA0001749709800000101
column (diagonal):
Figure BDA0001749709800000102
referring to fig. 2, a girder ZL in a high-rise steel structure 1 1 Mu of a 4, looking up the FEMA-356 standard (tables 5-6), the ductility coefficients corresponding to the IO, LS, and CP levels are 0.25, 2, and 3, respectively, and the plastic strain of the cross-section edge fiber 8 of the corresponding plastic corner section 2 is calculated according to the formula (XII) as ε IO ＝0.50％、ε LS =1.43% and ∈ CP =1.97%; steel column Z 1 Mu of a 4, looking up the FEMA-356 standard (tables 5-6), the ductility factors corresponding to IO, LS, CP levels are 0.25, 2, and 3, respectively, and the plastic strain of the fiber 8 at the cross-section edge of the corresponding plastic corner section 2 is calculated according to the equation (XIII) IO ＝0.52％、ε LS =1.45% and ε CP ＝1.97％。
Referring to fig. 2 and 5, according to the plastic deformation cloud chart of the whole structure and the local components, as can be seen from the strain condition of the section edge fiber 8 of the plastic corner section 2, the plastic corners of most components do not reach the IO level, and the components with large plastic deformation are mainly concentrated on alternative paths: the maximum plastic corner of the beam reaches the IO level and is represented as flexural damage; the maximum plastic strain of the inclined strut is 0.67 percent, occurs in a non-plastic corner section and is expressed as compression damage; the column did not undergo significant plastic deformation, from which it was found that the structural members of the overall structure were subjected to bending and less compressive damage.
Referring to the attached figure 3, in the overall structure dynamic response process of the high-rise steel structure 1 after the bottom corner columns 3 are dismantled, the maximum plastic strain of all components is 1.09 percent and does not reach 2.5 percent, namely no component fails due to the fact that plastic deformation reaches the limit; most components are not obviously damaged, and the fact that the integral structure has certain continuous collapse resistance under the dismantling working condition is shown.
The above examples are only for further illustration of the present invention and are not intended to limit the present invention, and all equivalent implementations of the present invention should be included within the scope of the claims of the present invention.

Claims (1)

1. A damage evaluation method for a steel structure fiber model comprises a beam calculated according to the following formulas (I) and (II) n Column (diagonal bracing) n ：
Figure FDA0004130666960000011
Figure FDA0004130666960000012
Wherein E is the modulus of elasticity; i is a section moment of inertia; epsilon u Is the steel ultimate plastic strain; f. of y Is the material yield strength; mu.s a The ratio of the plastic corner to the yield corner of the component in the strain-strengthened section; gamma is the shape factor of the cross section; wn is the elastic section modulus; h is 1 Is the distance from the cross-sectional edge fiber to the neutral axis;
the method is characterized in that the steel structure fiber model is loaded and solved, then the damage condition of a component is judged, the safety of the structure is judged according to the dynamic response condition of the whole structure, and the specific judgment is carried out according to the following steps:
the method comprises the following steps: determining the numerical values of the plastic corner of the component reaching the IO stage, the LS stage and the CP stage according to the FEMA-356 standard;
step two: determining ductility factors mu IO, mu LS and mu CP corresponding to the three-stage level by referring to FEMA-356 standard, and calculating ductility factor mu corresponding to the three-stage level according to the following formulas (XII) and (XIII) IO 、μ LS And mu CP And plastic strain epsilon of plastic corner section edge fiber IO 、ε LS And epsilon CP ：
Beam:
Figure FDA0004130666960000013
column, bracing:
Figure FDA0004130666960000014
step three: reading the result information after finite element analysis to obtain an edge fiber plastic strain cloud chart, and obtaining the edge fiber plastic strain epsilon according to the plastic corner section edge fiber plastic strain epsilon lp Determining which stage the plastic corner of the component reaches: if epsilon IO ≤ε lp ＜ε LS The plastic corner of the component reaches the IO stage; if epsilon LS ≤ε lp ＜ε CP The plastic corner of the component reaches the LS stage; if epsilon lp ≥ε CP The plastic corner of the component reaches the CP stage; therefore, the bending damage degree of the component can be judged, and the safety of the whole structure can be evaluated according to the plastic deformation of the component except the plastic corner section and the condition of judging the pulling and pressing damage of the component; the grid is divided into a steel structure fiber model according to the following steps:
the first step is as follows: firstly, setting plastic corner sections with length lp at two ends of all beams, columns and diagonal bracing members in a steel structure, and determining the ratio n of the length l of each member to the length lp according to the type, section parameters, material type, material limit plastic strain and FEMA member limit strength deformation of the members; the second step: counting n values of all components, determining a value range of n, verifying the n value with small influence on stress, deformation and failure of the components in the value range through finite element analysis, and if the influence is large, adjusting the mesh division of a finite element model; the third step: effective meshing is adopted to form a finite element model to define a fiber section, only one fiber unit is ensured in the plastic corner section of all the components, and then loading solving is carried out.
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