KR100340217B1 - Numerical Analysis Method For Analyzing Residual Stress of Welding - Google Patents

Abstract

The present invention relates to a numerical analysis method for analyzing the welding residual stress, which can analyze the thermal stress and the welding residual stress distribution, size, thermal deformation, etc. of various structures with high accuracy.

The numerical analysis method for analyzing the welding residual stress of the present invention includes a first step of inputting initial data for an analysis model; A second step of calculating a bandwidth for the analysis model; A third step of setting an integration point coordinate of the Gaussian integral and a load function; A fourth step of defining coordinates for all the elements of the analysis model and constructing a strain rate matrix using a shape function; A fifth step of setting an initial temperature for each element, and initializing stress, strain, and displacement; A sixth step of inputting a temperature history of a process of performing welding and a temperature history of cooling to replace the temperature with a thermal load; A seventh step of setting material and mechanical properties such as yield stress, strain hardening coefficient and thermal expansion coefficient of each element of the analytical model; An eighth step of initializing a stiffness matrix and a node displacement with respect to a node coordinate system of each element, and then constructing and building an equivalent node force vector with the stiffness matrix of the element; A ninth step of calculating the total strain, the stress, the plastic strain, the equivalent plastic strain, and the sintering date by constructing and calculating the equilibrium equation; A step 10 of judging the elasticity and the plasticity of the analytical model material on the basis of the Von-mices yielding function; And outputting a result of the analysis model according to elasticity and plasticity.

Description

{Numerical Analysis Method for Analyzing Residual Stress of Welding}

The present invention relates to a welding residual stress analysis numerical value capable of highly analyzing thermal stress and welding residual stress distribution, size and thermal deformation of various structures subjected to thermal and external loads such as welding, mold, boiler, turbine and petrochemical plant And an analysis method.

So far, the problem of welding residual stress has been solved by analytical method and empirical equation. Therefore, the physical constant due to the temperature dependence of material can not be properly introduced and the plasticity problem can not be considered due to difficulty of analysis. Also, the problem of the strain hardening rule considering the time increment was not applied. The problems of the conventional welding residual stress analysis will be described in detail as follows.

If we consider the stresses on the inherent stress and the inherent strain generated at this time, assuming a model in which both ends of a rod with a length of ℓ and a cross-sectional area of A are fixed to a rigid wall, In other words, it is changed by the temperature change. Here, g is the inherent strain of the rod. Stresses occur in the rod for mechanical strain ε _m . In this case, the deformation degree ε, the intrinsic strain degree g and the mechanical strain degree ε _m of the rod are the same as those of ε-g = ε _m . Through this relationship, it was possible to know the generation process of the inherent strain due to the local temperature change.

The residual plastic deformation and the residual stress that occur when the maximum temperature and the maximum temperature are reached in the both-end fixed bar are used as shown in Table 1 below.

Here, Te represents the upper limit temperature (critical temperature) at which plastic deformation occurs during the entire heat rise, Tt represents the upper limit temperature which causes the plastic deformation during the temperature lowering process, and α represents a constant linear expansion coefficient regardless of the temperature . Then, T _M = θ _M -θ ₀ where θ _M is the temperature at which the material loses its deformation resistance, ie, the mechanical melting temperature, θ ₀ is the initial temperature of the rod, and the temperature dependence of the yield stress σ _Y and the modulus of elasticity E Is expressed by the following Equation 1

However, the problem of the process of analyzing the residual plastic deformation and the residual stress is that the linear expansion coefficient (α) of a certain material is used regardless of the temperature, and the temperature variation such as the yield stress (σ _Y ) and the elastic modulus (E) And that the dependent property is simply a weak equation.

In the analysis of the residual stresses in the butt welds, we calculated the residual stresses in the butt welds by deriving the weakly acidic equation through the experiment within a very limited range of materials such as mild steel or aluminum alloy as shown in Table 2 below.

The problem of obtaining the residual welding residual stress using the weak acid equation derived from the simple analytical method can not be applied to the shape of the diversified welded part, and it is mainly applied to the butt welded part which is the simplest form. The accuracy of which is the biggest problem.

The main technical limitations of the conventional analysis method are as follows. First, the physical constants (yield strength, elastic modulus, etc.) of materials that depend on temperature change were not introduced. Second, the problem of plasticity behavior beyond the elastic behavior limit could not be considered properly. Third, it was not possible to introduce the strain hardening rule according to the time increment. Due to these technical limitations, the conventional residual stress analysis method is limited to grasping the simple phenomenon of the welding residual stress, and the reliability of the result accuracy and prediction is insufficient.

Accordingly, an object of the present invention is to provide a numerical analysis method for analyzing the residual stress of welds, which can interpret the thermal stress and the weld residual stress distribution, size, thermal deformation, and the like of various structures with high accuracy by numerical methods.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a flow chart showing a stepwise numerical analysis method for analyzing welding residual stress according to an embodiment of the present invention; FIG.

FIG. 2 is a flow chart showing the step 26 of FIG. 1 step by step; FIG.

In order to achieve the above objects, a numerical analysis method for analyzing a welding residual stress according to the present invention includes a first step of inputting initial data for an analytical model; A second step of calculating a bandwidth for the analysis model; A third step of setting an integration point coordinate of the Gaussian integral and a load function; A fourth step of defining coordinates for all the elements of the analysis model and constructing a strain rate matrix using a shape function; A fifth step of setting an initial temperature for each element, and initializing stress, strain, and displacement; A sixth step of inputting a temperature history of a process of performing welding and a temperature history of cooling to replace the temperature with a thermal load; A seventh step of setting material and mechanical properties such as yield stress, strain hardening coefficient and thermal expansion coefficient of each element of the analytical model; An eighth step of initializing a stiffness matrix and a node displacement with respect to a node coordinate system of each element, and then constructing and building an equivalent node force vector with the stiffness matrix of the element; A ninth step of calculating the total strain, the stress, the plastic strain, the equivalent plastic strain, and the sintering date by constructing and calculating the equilibrium equation; A step 10 of judging the elasticity and the plasticity of the analytical model material on the basis of the Von-mices yielding function; And outputting a result of the analysis model according to elasticity and plasticity.

Other objects and advantages of the present invention will become apparent from the following description of preferred embodiments of the present invention with reference to the accompanying drawings.

Hereinafter, preferred embodiments of the present invention will be described in detail with reference to FIGS. 1 and 2. FIG.

The technical principles applied to the method for analyzing the residual welding stress according to the present invention are as follows. First, in the thermal stress problem, the unsteady heat conduction problem of the first continuum, that is, the temperature distribution that varies with time increment is obtained and used as the input data. Second, the physical properties of the constituent materials vary with temperature, so we obtain the relational expression between the joint force and the joint displacement considering these physical constants (elastic modulus, yield strength, etc.). Third, the isotropic material is considered for the most general analysis of weld residual stress. Fourth, the temperature dependence of the material constants is considered throughout the plasticity as well as the elasticity. Fifth, the relationship between strain and stress is expressed by total strain increment by introducing plastic flow theory. Sixth, the yielding condition of the Von-mises considering the linear isotropic hardening rule is used as the yield function in the burning zone.

In order to obtain the welding residual stress by a numerical method, a method of formulating various theories introduced in the present invention will be described as follows.

First, the stress and strain distributions are independent of the weld line length direction. Second, if a plane perpendicular to the longitudinal direction of the weld line exists before the deformation, it exists as a plane after deformation. These assumptions can be formulated by the following equations (2) and (3).

However, since the heat is isotropic, the shear strain due to heat does not occur. Therefore, Equation (3) can be expressed as Equation (4) below.

Third, assuming that the structure satisfies the equilibrium with the physical fitness and boundary condition, the equilibrium equation and the mechanical boundary condition are established. Therefore, the principle of virtual work can be formulated as the following equation (5).

Where {σ} is the stress vector, {ε} is the strain vector, {U} is the displacement vector, { } Is the water fitness vector per unit volume, { } Represents the surface force vector per unit volume. Accordingly, the strain-displacement relation of the thermal stress problem is expressed by the following equations (6) and (7).

In the stress-strain relationship of the equations (6) and (7), each element is constructed as shown in the following equations (8) to (10).

Where {α} T = ε ^t is the thermal strain, α is the instantaneous coefficient of linear expansion, and T is the temperature.

Fourth, the former strain ε {} is a sum of the elastic deformation ratio ε ^{e} and {heat deformation rate ε ^t} can be expressed as Equation (11).

Fifth, the stress and strain are calculated from the Hooke's law as shown in the following equation (12).

Where [D ^e ] is an elastic stress-strain matrix

If the effect of the stress increment considering the temperature dependency of the physical properties of the material is [C] dT, the stress-strain relationship is expressed by the following equation (13).

First, the relation between the stress-strain rate in the elasticity region is as follows.

Here, the elastic deformation ratio ε ^e {} is a function of the stress {σ} and temperature T.

The increment of the elastic strain rate is configured as shown in the following Equation (16), and the total strain rate increment is constructed as shown in Equation (17).

The equations (14) to (17) summarize the equations (14) to (17) to obtain a constitutive equation for the stress increment in the elastic region as shown in the following Equation (18).

Here, E ₁ is the modulus of elasticity after temperature change ΔT, {C} to be.

Second, the relationship between the stress-strain in the plastic zone and the yield stress (σ _Y ) As a function of ^{_{σ Y (T, W P)}} , and the yield function F is configured as shown in Equation 19 from the stress {σ} and yield stress (σ _Y).

In other words, the material surrenders when dF = df-df_0 = 0 is satisfied.

Therefore, when the material is in a load state in the burning zone, the condition of dF = 0 must be satisfied as shown in the following equation (20).

here, A change in equivalent stress, The work hardening, Is the temperature increment.

Assuming that the yielding function is a plastic potential, the plastic strain is given by the following equation (21).

here, Is the shear stress, and lambda is the positive scalar quantity. Also, Equation 20 follows the Von-mises yield condition.

The total strain increment {dε} is the sum of the tan-plasticity and the thermal strain rate as shown in the following equation (22).

Thus, a constitutive equation for the stress increment in the burnout zone is obtained as shown in the following equation (23).

Third, considering the relationship between the joint force and the joint displacement, the element equivalent joint force increment {dF} is expressed by the following equation (24) according to the principle of the virtual work,

Since the action of external force is not considered in the thermal stress analysis by welding, the equilibrium condition equation for each element is expressed by the following equation (25).

here, Is an equivalent nodal force due to heat, Is the stiffness matrix of the element.

Therefore, the numerical analysis method for analyzing the welding residual stress according to the present invention is constructed based on the above-described equations.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a flow chart showing a stepwise numerical analysis method for analyzing welding residual stress according to an embodiment of the present invention. FIG.

In step 2, various data about the analytical model are input. In detail, the number of elements, the number of points, the coordinates of each node, and the boundary condition are input to the analysis model, and the temperature distribution obtained from the program for 2-dimensional thermal conductivity analysis and the element having the heat source and the element having the heat effect (HAZ) . In step 4, the bandwidth for the analytical model is calculated, and in step 6, the integral point coordinates of the Gaussian integration and the weight function are set. Next, in steps 8 and 10, coordinates for all elements of the analytical model are defined and a strain matrix, i.e. a Jacobian matrix [B], is constructed using a shape function. In step 12, the initial temperature is set. In this case, the room temperature is set at the nodal point in each element of the analysis model. In step 14, stress, strain, and displacement are initialized.

Then, in step 16, since the heat distribution history is heat load, the temperature history of the process of performing the welding and the temperature history of the cooling are input. In this case, the average temperature of each node of the analysis model element is calculated as the element temperature. In step 18, the element temperature is replaced by the thermal load. In step 20, the material and mechanical properties for the analytical model are set. In this case, yield stress, strain hardening coefficient and thermal expansion rate of the element are defined according to the temperature change. In the next step 22, the stiffness matrix and the node displacement of the joint coordinate system are initialized. In step 24, the stiffness matrix of the local coordinate system element And an equivalent nodal force vector . Then, in step 26, the stiffness matrix [K] of the element is superimposed and the equivalent joint force vector {dL} is superimposed to make the local coordinate global.

This step 26 will be described in detail. After the displacement-strain matrix [B] of the element is formed in step 26a as shown in Fig. 2, the elasticity and firing of the material are judged in step 26b. Here, when the material is determined to be elastic, the elasticity [C ^e ] of the material is calculated in step 24c, and then the elastic stress-strain matrix [D ^e ] is constructed in step 24d. On the other hand, the process proceeds from the step 26b to step 24e, if the material is determined by firing small Toughness [C ^P] one after the firing stresses in step 24f calculate - constitutes the strain matrix [D ^P]. Then, in step 24g, the stiffness matrix of the element And in step 24h, the equivalent nodal point force .

Then, in step 28 of FIG. 1, the equilibrium equation is constructed and calculated to calculate the displacement. Then, in step 30, the plastic elements are checked, and the lower limit determination (?) And the strain rate increments d?, ^D ?, ^D ? ^P , , d? ^P are calculated. In step 32, the total strain ε, the stress σ, the plastic strain ε ^P , the plastic strain , And the sintering day? ^P. Then, in step 34, it is judged whether or not the analytical model material is elastic or fired. In this case, whether or not the material is elastic or fired can be determined by the Poisson- ). If HI is greater than or equal to 0.95, it is determined to be fired. If HI is less than 0.95, it is determined to be elastic. In step 36, the results for the analytical model, i.e., temperature, displacement, stress, plastic strain and equivalent stress, are output. In the case of multi-layer-multi-pass, the result of each pass is superimposed at step 38 in step 38, and final result data is output at step 40 and the process is terminated. The resultant value outputted in the step 40 can be easily grasped by using general graphic software (Grapher, Tecplot, etc.).

As described above, according to the numerical analysis method for analyzing the welding residual stress according to the present invention, accurate values can be derived more than conventional simple analysis solutions, and time and cost can be saved. Accordingly, it is very advantageous in designing and manufacturing various structures by using the numerical analysis method for analyzing the residual stress of welding according to the present invention. In addition, according to the numerical analysis method for analyzing the welding residual stress according to the present invention, it is possible to prevent fatal defects that might occur in various structures by controlling the deformation and defects that occur during welding in consideration of thermal conduction phenomenon and over- It is possible to prevent material loss. Furthermore, according to the numerical analysis method for analyzing the welding residual stress according to the present invention, it is possible to calculate the optimal welding condition, to calculate the economical and efficient welding condition, and to data of the welding condition so that the non- Thereby reducing the number of process steps. In addition, the numerical analysis method for analyzing the welding residual stress according to the present invention is compatible with other graphic software, and the result can be easily grasped.

It will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit or scope of the invention. Therefore, the technical scope of the present invention should not be limited to the contents described in the detailed description of the specification, but should be defined by the claims.

Claims (2)

A first step of inputting initial data for an analysis model; A second step of calculating a bandwidth for the analysis model; A third step of setting an integration point coordinate of the Gaussian integral and a load function; A fourth step of defining coordinates for all the elements of the analysis model and constructing a strain rate matrix using a shape function; A fifth step of setting an initial temperature for each of the elements and initializing stress, strain, and displacement; A sixth step of inputting a temperature history of a process of performing welding and a temperature history of cooling to replace the temperature with a thermal load; A seventh step of setting material and mechanical properties such as yield stress, strain hardening coefficient and thermal expansion coefficient of each element of the analytical model; An eighth step of initializing a stiffness matrix and a node displacement with respect to a node coordinate system of each element and then constructing an equivalent node force vector with the stiffness matrix of the element, A ninth step of calculating the total strain, the stress, the plastic strain, the equivalent plastic strain, and the sintering date by constructing and calculating the equilibrium equation; A step 10 of judging the elasticity and plasticity of the analytical model material on the basis of a Von-mices yielding function, And outputting a result of the analytical model according to the elasticity and the plasticity. The method according to claim 1, Further comprising the step of superimposing the results for each pass when the welding for the analytical model is multi-layered and multi-pass, and outputting a final result.