JP2011027484A - Method and program for calculating ratchet strain - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a rational method of calculating a ratchet strain relatively easily and precisely, the ratchet strain accumulated in a pipe when the pipe receiving internal pressure is subjected to repetitive operation. <P>SOLUTION: The method of calculating a ratchet strain includes: a first step of setting a calculation formula for calculating ratchet strain per repetition cycle in a straight pipe model for receiving axial repetition tensile and compression; a second step of obtaining three stress ranges comprising an axial film stress range, an axial bend stress range, and a circumferential bend stress range, and for calculating an equivalent stress range comprising the square root of sum of squares of the three stress ranges in target piping repeatedly subjected to tensile and compression and a bend; and a third step of calculating ratchet strain per repetition cycle of the target piping from the calculated equivalent stress range. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a ratchet strain calculation method for calculating ratchet strain accumulated in a pipe when a pipe receiving internal pressure is repeatedly acted on.

  In power plants and the like, the post-earthquake soundness assessment of piping that is most structurally affected by earthquakes is an important issue. In order to understand the behavior of piping systems during earthquakes, strength demonstration tests of actual piping systems have been conducted at home and abroad, and it has become clear that ratchet fatigue is the main damage mode of piping systems (for example, Non-patent documents 1 and 2).

When the pipe receiving the internal pressure is subjected to excessive repetitive displacement, deformation that swells in the circumferential direction (ratchet deformation) occurs, the axial fatigue strength decreases, and a through crack is likely to occur (see FIG. 14). This phenomenon is called ratchet fatigue. In order to evaluate the soundness of piping against ratchet fatigue, it is necessary to establish an evaluation method for ratchet deformation and ratchet fatigue strength.
Among these, ratchet fatigue strength is evaluated by a combination of repeated damage and ductile wear by conducting a uniaxial ratchet fatigue test of SUS304 steel and a repeated tensile torsion test of a pipe specimen subjected to internal pressure. There are examples (see Non-Patent Documents 3 and 4).
On the other hand, as a theoretical study on ratchet deformation, there is a study on a thermal ratchet, but a theoretical study on a mechanical ratchet has hardly been conducted. The theory of plastic deformation characteristics of the elbow part, which tends to be a particularly weak point in piping systems, has been elucidated (see Non-Patent Document 5), but the mechanical ratchet due to repetitive action has not yet been elucidated.

  In order to evaluate the ratchet deformation, it is necessary to calculate the ratchet strain. However, the ratchet strain is caused by plastic deformation, depends on the deformation of the entire piping system, and further has an influence of elastic follow-up (see Patent Document 1). Therefore, a rational calculation method for ratchet distortion has not been established yet. For this reason, the soundness evaluation of piping after an earthquake is a situation in which the design criteria (for example, ASME Section 3 or JEAG4601) based on elastic deformation are used for evaluation.

S. W. Tagart, Jr., Y. K. Tang, D.D. J. Guzy and S. Ranganath, “Piping dynamicreliability and code rule change recommendations”, Nuclear Engineering and Design, Vol.123, pp.373-385, 1990 K. Suzuki, H. Abe, Y. Sasaki, S. Kanno, T. Sato, H.C. Yokota and K. Suzuki, “Seismic Proving Test on the Ultimate Strength of Piping System (Component Tests), JSME 2000 Annual Conference, pp.833-834, 2000 Y. Asada, “FailureCriterion on Low-Cycle Fatigue with Excessive Progressive Deformation”, HPI ETD Committee, 98ETD1-6, 1998 J. Namaizawa, K. Ueno, A. Ishikawa, Y. Asada, “LifePrediction Technique for Ratcheting Fatigue”, PVP-Vol.266, Creep Fatigue, and Leak-Before-Break Assessment, 1993 A. Suzuki, “A Study of Simplified Inelastic Analysis Method for Piping Systems”, IHI Engineering Review, Vol.23, No.3, 1983

JP 2009-58224 A

However, it cannot be denied that the design standard based on elastic deformation results in a relatively large safety factor.
Furthermore, even if the ratchet distortion can be accurately calculated, a method that requires excessive calculation time and labor is not practical for application to the actual soundness evaluation of piping.

  Accordingly, an object of the present invention is to provide a rational ratchet strain calculation method capable of calculating the ratchet strain accumulated in the piping when the piping subjected to the internal pressure is repeatedly acted on relatively easily and accurately. .

  The present invention relates to a ratchet strain calculation method for calculating ratchet strain accumulated in a pipe when the pipe receiving the internal pressure is subjected to repeated action, in a straight pipe model that repeatedly receives tensile compression in the axial direction. In the first step of setting the calculation formula for calculating the ratchet strain per repeated cycle from the directional stress range and the circumferential stress due to internal pressure, and in the target pipe that repeatedly undergoes tensile compression and bending, the axial film stress range and the axis Three stress ranges consisting of a directional bending stress range and a circumferential bending stress range are obtained, and one stress range selected from the three stress ranges or a square root of the sum of squares of two or more stress ranges. A second step of calculating the equivalent stress range, and a calculation formula set in the first step by substituting the calculated equivalent stress range into the axial stress range. More, a third step of calculating the ratchet strain per repetitive cycle of the target pipe, a ratchet distortion calculating method comprising.

  According to this configuration, after setting the ratchet strain calculation formula with a simple straight pipe model in which only internal pressure and repeated axial tension and compression act, the complex stress state (stress range) of the target pipe that also undergoes repeated bending can be determined. By replacing with one equivalent value (equivalent stress range), the ratchet strain of the target pipe can be calculated with high accuracy using the same calculation formula.

Here, the axial stress range is an axial virtual elastic stress range Δσ ^E _XX which is a stress range acting when the straight pipe model is assumed to be an elastic body, and the ratchet strain Δε ^R _YY per the repeated cycle The calculation formula to calculate is the elastic shakedown limit axial direction, which is the limit stress range where elastic shakedown that shows elastic behavior occurs after the elastoplastic cycle, obtained from the circumferential stress σ _YY and the yield stress σ _y due to internal pressure Using the stress range Δσ ^S _XX , it is expressed as follows, and the three stress ranges are preferably obtained by elastic analysis.
Δσ ^S _XX = √ {σ _YY ² −4 (σ _YY ² −σ _y ² )} (1)
Δε ^R _YY = 3σ _YY × (Δσ ^E _XX / Δσ ^S _XX −1) / E (2)
here,
√ {}: Symbol for calculating the square root of the result in parentheses E: Young's modulus

  According to the calculation formula for ratchet strain, when the axial virtual elastic stress range is larger than the elastic shakedown limit axial stress range, the ratchet strain is generated according to the magnitude, and with less parameters. Ratchet distortion can be calculated easily. Furthermore, by using the axial virtual elastic stress range in the calculation formula, the ratchet phenomenon in the plastic region can be easily evaluated by elastic analysis.

  Further, the target pipe is a pipe having an elbow part, and the second step is an elastic follow-up obtained from a ratio between a deformation amount of the elbow part by inelastic analysis and a deformation amount of the elbow part by elastic analysis. Preferably, the method includes calculating a coefficient, and correcting the equivalent stress range using the calculated elastic follow-up coefficient.

  According to this configuration, by correcting the equivalent stress range with the elastic tracking coefficient for the target pipe having an elbow part, the ratchet strain can be calculated easily and appropriately reflecting the effect of deformation due to plasticization of the elbow part. it can.

  Here, in the correction, the corrected circumferential bending stress range is calculated by multiplying the circumferential bending stress range by the elastic tracking coefficient, and the axial film stress range, the axial bending stress range, and the corrected circumferential bending stress are calculated. Even if one of the stress ranges is selected to include at least the modified circumferential bending stress range, or a corrected equivalent stress range composed of the square root of the sum of squares of each of the two or more stress ranges is calculated. Good.

  Alternatively, in the correction, the corrected equivalent stress range may be calculated by multiplying the equivalent stress range by the elastic tracking coefficient.

  A more rational ratchet strain can be calculated by appropriately reflecting the shape characteristics of the elbow by multiplying the elastic bending coefficient by a circumferential bending stress range that is particularly easily amplified in the elbow portion. Further, the ratchet strain can be calculated on the safer side by correcting the entire equivalent stress range by multiplying by the elastic follow-up coefficient.

In addition, the elastic follow-up coefficient decreases with the progress of repeated cycles due to the effect of the overall hardening of the target pipe as a whole due to strain hardening, and the yield used to calculate the shake-down limit axial stress range Δσ ^S _XX The stress σ _y is preferably increased with the progress of the repeated cycle due to the influence of local hardening in which the elbow portion is locally hardened by strain hardening and strain accumulation.

  According to this configuration, it is possible to calculate a more rational ratchet strain by reflecting the change of the ratchet strain with the progress of the repeated cycle by considering the influence of the overall curing and the local curing.

  According to the ratchet strain calculation method according to the present invention, the ratchet strain accumulated in the pipe when the pipe receiving the internal pressure is repeatedly acted can be calculated relatively easily and with high accuracy.

It is a schematic diagram of a piping model concerning an embodiment of the present invention. It is explanatory drawing of the mechanism of ratchet distortion, and is a schematic diagram showing the relationship between axial direction stress and circumferential direction strain. It is explanatory drawing of the mechanism of ratchet distortion, and is a schematic diagram showing transition of the circumferential distortion for every repetition cycle. It is a flowchart showing the calculation method of the ratchet distortion concerning the 1st Embodiment of this invention. It is explanatory drawing of the calculation formula of the ratchet distortion concerning the 1st Embodiment of this invention, and is a schematic diagram showing the relationship between the circumferential stress stress and axial direction stress in a plastic region. It is a ratchet distortion diagram calculated | required by the calculation formula of the ratchet distortion concerning the 1st Embodiment of this invention. It is a flowchart showing the calculation method of the ratchet distortion concerning the 2nd Embodiment of this invention. It is a mesh figure showing an example of the model used by the three-dimensional FEM elasticity analysis of the elbow pipe concerning the 2nd Embodiment of this invention. It is a schematic diagram showing an example of the beam model used by the simple inelastic analysis of the elbow pipe concerning the 2nd Embodiment of this invention. It is a diagram showing the member characteristic used by the simple inelastic analysis of the elbow pipe concerning the 2nd Embodiment of this invention. It is a scatter diagram which compared the calculation result of ratchet distortion by Example 1 and comparative example 1. It is a diagram showing the relationship between the fulcrum reaction force N by the simple inelastic analysis of Example 3, and the curvature k in an elbow center cross section. It is a diagram showing transition of ratchet distortion accompanying the progress of the number of repeated cycles for Example 3, Example 4, and Comparative Example 3. It is a multi-axis ratchet fatigue test result for demonstrating ratchet fatigue.

  EMBODIMENT OF THE INVENTION The form for implementing this invention is demonstrated in detail, referring an accompanying drawing below. The dimensions, materials, and other specific numerical values shown in the embodiments are merely examples for facilitating understanding of the invention, and do not limit the present invention unless otherwise specified.

  The present invention is a ratchet strain calculation method for calculating ratchet strain accumulated in a pipe when the pipe receiving the internal pressure is repeatedly acted on. The piping referred to here includes not only the straight pipe of FIG. 1 (a) but also a deformed pipe such as an elbow pipe of FIG. 1 (b) as shown in the schematic diagram of the piping model of FIG. The repeated action includes not only axial tensile and compressive deformation as shown in FIG. 1A but also axial and circumferential bending deformation. In the following, the subscript XX indicates the axial direction of the pipe, and YY indicates the circumferential direction of the pipe cross section.

The mechanism of ratchet distortion will be briefly described with reference to the explanatory diagrams shown in FIGS. In FIG. 2, the vertical axis is the axial stress σ _XX (tensile is positive), the horizontal axis is the circumferential strain ε _YY (tensile is positive), and as shown in FIG. axial repeated displacement 2.delta. _a indicates the relationship in the case where the action. FIG. 3 shows the transition of the circumferential strain ε _YY for each repeated cycle. 2 (a) and 3 (a) show cases of elastic behavior, and FIGS. 2 (b) and 3 (b) show cases of inelastic behavior, respectively.
One repetitive cycle is composed of the following four steps.
1/4 cycle: tensile displacement δa ₁ acts
2/4 cycle: δa ₁ return displacement δa ₂ acts
3/4 cycle: compression displacement δa ₃ acts
4/4 cycle: Return displacement δa _{4 of} δa ₃ acts

First, the case of elastic behavior will be described. As shown in FIG. 2A, first, a circumferential strain ε ₀ is generated by the internal pressure P and moves to the initial point P ₀ . Thereafter, axial stress σ _a (= δ _a × E, E: Young's modulus) corresponding to the axial repetitive displacement δ _a acts in the order of the tension side and the compression side. Circumferential distortion occurs. Due to the elastic behavior, after the end of one cycle, it returns to the initial point P ₀ again and no residual strain occurs. The same applies to FIG.

Next, the case of inelastic behavior will be described. As shown in FIG. 2 (b), first, it moves to the initial point P ₀ , and thereafter, an axial tensile stress corresponding to the axial tensile displacement δ _a1 acts, and it behaves elastically up to P ₁ . At P ₁ , the axial tensile plastic starting point σ _b1 is reached, and plastic deformation is performed up to P ₂ , and tensile plastic strain ε ₁ is generated in the circumferential direction. Thereafter, the actuator moves to P ₃ by the return displacement δ _a2 . Furthermore, an axial compressive stress corresponding to the axial compressive displacement δ _a3 acts, and elastic behavior is performed up to P ₄ . At P ₄ , the axial compression plastic starting point σ _b2 is reached, and plastic deformation is performed up to P ₅ , thereby generating a compressive plastic strain ε _{2 in the} circumferential direction.

At this time, the axial tensile plastic starting point σ _b1 becomes larger than the axial compressive plastic starting point σ _b2 due to the influence of the internal pressure P acting to generate circumferential tensile stress (details will be described later). Therefore, the circumferential tensile plastic strain ε ₁ becomes larger than the circumferential compressive plastic strain ε ₂ , and residual strain is generated at the time when the repeated cycle is completed (P ₆ ). This residual strain is the ratchet strain ε ^R. The same applies to FIG.

[First Embodiment]
The first embodiment is mainly directed to a ratchet strain calculation method for a thin straight pipe. A calculation method for a deformed pipe such as an elbow pipe will be described in detail in the second embodiment.

  FIG. 4 is a flowchart showing a ratchet distortion calculation method according to this embodiment. This flow starts with a first step S10 for setting a calculation formula for calculating ratchet strain in a straight pipe model subjected to internal pressure and axial tension compression, and a first step of calculating an equivalent stress range for a target pipe subject to internal pressure, tension compression and bending. 2 step S20, and 3rd step S30 which calculates the ratchet distortion of object piping by a calculation formula using the calculated equivalent stress range. Hereinafter, the first step S10 to the third step S30 will be described in detail step by step.

In the first step S10, a straight pipe model that repeatedly undergoes tensile compression in the axial direction and the axial direction is prepared (S11). Next, a calculation formula for calculating the ratchet strain per repeated cycle is set from the axial stress range due to tension and compression and the circumferential stress σ _YY due to internal pressure (S12). For example, the calculation formula can be set as follows.

First, Mises' yield condition is given by the following equation (1).

Assuming that the plastic strain increment is proportional to the normal component of the yield surface, the plastic strain increment dε ^P _ij is given by the following equation (2).

Assuming that the internal pressure is constant and the circumferential stress and the radial stress do not change, dε ^P _ij is given by the following equation (3), and the axial strain increment dε _XX is given by equation (4). As a result, the axial stress increment dσ _XX is given by equation (5).

Substituting equation (5) into equation (3) and rearranging, finally, circumferential plastic strain increment dε ^P _YY
Is given by equation (6). The total strain increment dε _YY is given by equation (7).

Here, it is assumed that the material is an elastic perfect plastic body, and the stress state is considered when tensile and compression are repeated in the axial direction. If the radial stress is ignored, the axial stress σ _XX on the tension side and the compression side in the plastic deformation state is given by equation (8).

Next, consider the conditions under which ratchet deformation occurs. If the axial stress range is smaller than the difference between the axial stresses in tension and compression given by equation (8), a squeakdown that shows elastic behavior occurs after the elastoplastic cycle, and ratchet deformation does not occur. Therefore, the shake-down limit axial stress range Δσ ^S _XX is given by equation (9).

Here, the relationship between the equations (8) and (9) is shown in FIG. FIG. 5 shows values obtained by normalizing the circumferential stress stress σ _YY (positive tensile) on the vertical axis and the axial stress σ _XX (positive tensile) on the horizontal axis with the yield stress σ _y , respectively. The result of equation (8) is an ellipse tilted to the right (called Mises ellipse), and the result of equation (9) is expressed as the range of the horizontal axis when the vertical axis is fixed.
For example, the figure shows the range of σ _XX / σ _y (−0.65 to 1.15) when σ _YY / σ _y = 0.5. The value of the equation (8) corresponds to the above-described plastic starting point, and from this, when an internal pressure P that causes a circumferential tensile stress of 0.5 in the yield stress ratio is applied, the tensile-side plastic starting point σ _b1 ( It can be seen that the yield stress ratio 1.15) is larger than the compression side plastic starting point σ _b2 (yield stress ratio 0.65).

On the other hand, from the equation (6), the relationship between the axial virtual elastic stress range σ ^E _XX and the circumferential ratchet strain Δε ^R _YY per one cycle in tension and compression is finally expressed by using the Young's modulus E (10) Given in. Here, the virtual elastic stress range σ ^E _XX is a stress range acting when an elastic body is assumed. By using the axial virtual elastic stress range σ ^E _XX , the ratchet phenomenon in the plastic region can be easily evaluated by elastic analysis.

Further, when the stress and strain in the equation (10) are normalized by the yield stress and the yield strain, respectively, the equation (11) is obtained. The normalized value is marked with ~ on the upper side.

FIG. 6 shows a ratchet distortion diagram obtained from the equation (11). The vertical axis represents the axial virtual elastic stress range σ ^E _{XX due} to the internal pressure, and the horizontal axis represents the peripheral stress σ _YY due to the internal pressure, each normalized by the yield stress σ _y . Ε ^R in the figure is a ratchet strain Δε ^R _YY per cycle normalized by the yield strain. In the figure, the region indicated as Elastic is the elastic range, and the region indicated as Shakedown is the range where the elastic shakedown occurs even in the plastic region, and ratchet strain does not occur.
In the region where ratcheting occurs, ratchet distortion will occur. For example, when the horizontal axis is 0.5 (circumferential stress σ _YY due to internal pressure is 0.5 times the yield stress σ _y ), the vertical axis is 3.0 (the axial virtual elastic stress range σ ^E _XX is 3 times the yield stress σ _y ) For example, ε ^R = 1.0, and it can be seen that a ratchet strain ε ^R corresponding to the yield strain ε _y is generated.

The validity of the ratchet strain calculation formulas (10) and (11) has been verified by FEM elastic-plastic analysis. Therefore, according to this calculation formula, when the axial virtual elastic stress range Δσ ^E _XX is larger than the shakedown limit axial stress range Δσ ^S _XX, it is assumed that the ratchet strain ε ^R is generated according to the magnitude. Based on the above, the ratchet strain ε ^R can be calculated easily and with high accuracy with a small number of parameters.

  As described above, the ratchet strain calculation formula may be set not only by theoretical assumptions with some assumptions but also by multiple regression analysis of experimental results and analysis results.

Next, in the second step S20, first, a target pipe which is a calculation target of the ratchet strain ε ^R is specified, and its specification is confirmed (S21). Here, the specifications to be mainly confirmed are the tube thickness t, the inner diameter d, and the piping constraint conditions.

  Furthermore, a load displacement condition that acts on the target pipe is set (S22). In the first step S10, only internal pressure and repeated tensile compression in the axial direction are considered. However, in the second step S20, since actual piping is targeted, load displacement conditions are set in consideration of repeated bending in the axial direction and circumferential direction. To do.

The analysis is performed by reflecting the set load displacement condition, and the stress range (difference between the maximum stress and the minimum stress in one cycle) acting on the pipe is calculated for each type of stress (S23). The following three stress ranges are calculated. In addition, an elastic analysis is used for calculation of the stress range here. This is because the ratchet strain ε ^R can be evaluated using the result of the elastic analysis in the calculation formula of the first step, as described above, in order to simplify the structure calculation.
ΔX: Axial membrane stress range ΔX _b : Axial bending stress range ΔY _b : Circumferential bending stress range

Among these, the axial film stress range ΔX corresponds to the stress range due to axial tension and compression targeted in the calculation formula of the first step. That is, when only the axial film stress range ΔX acts on the target pipe, the ratchet strain ε ^R can be calculated by directly applying the calculation formula set in the first step. However, when the axial bending stress range [Delta] X _b and circumferential bending stress range [Delta] Y _b is applied, since the stress state acting on the pipe is different, it is impossible to apply this calculation formula directly.

Therefore, the three stress ranges are replaced with one equivalent stress range ΔX _eq (S24). Here, the equivalent stress range ΔX _eq is given by equation (12). The reason why the square root of the sum of squares is used is that it is generally used when obtaining a vector sum of stresses.

Next, in the third step S30, the equivalent stress range ΔX _eq is substituted into the virtual elastic stress range σ ^E _XX of the equation (10), and the ratchet strain Δε ^R is calculated using the equation (S31).
In addition, not only the equivalent stress range ΔX _eq is calculated using the three stress ranges as shown in the equation (12), but one or more stress ranges are selected and the equivalent stress range is selected using the selected stress ranges. ΔX _eq may be calculated. This is because when some stress ranges are excellent, there is no great difference in calculation accuracy even if only the excellent stress range is used.

  Finally, the end of the repeated cycle is determined (S32). If the cycle continues, the process returns to S22, the load displacement condition is updated, and the other second step S20 and third step S30 are performed. Do. In S31, an increment of ratchet distortion in each cycle and a cumulative value up to that time are calculated. If it is determined that the cycle is complete, the flow is complete.

  As described above, according to the present embodiment, after setting the ratchet strain calculation formula with a simple straight pipe model in which only the internal pressure and repeated axial tension and compression act, the complex stress of the target pipe that also acts repeatedly bending By replacing the state (stress range) with one equivalent value (equivalent stress range), the ratchet strain of the target pipe can be calculated relatively easily and accurately using the same calculation formula.

[Second Embodiment]
The second embodiment is mainly directed to a method for calculating the ratchet strain ε ^R for a deformed pipe such as an elbow pipe. Here, the elbow pipe is typically described as an example, but it goes without saying that the method of the present embodiment can also be applied to other deformed pipes such as a T-shaped pipe.

  FIG. 7 shows a ratchet distortion calculation flow according to the second embodiment. In the first embodiment shown in FIG. 4, the first step S10 and the third step S30 are common, but the second step S20 is partially different. Therefore, the description of the first step S10 is omitted, and the second step S20 and subsequent steps will be described.

  In 2nd step S20 of this embodiment, the object elbow pipe which is the calculation object of ratchet distortion is specified, and the specification is checked (S21). Here, the specifications to be confirmed include not only the tube thickness t and the inner diameter d, but also the elbow bending radius R. Further, a load displacement condition acting on the target elbow pipe is set (S22).

Next, the above-described three stress ranges (axial film stress range ΔX, axial bending stress range ΔX _b , circumferential bending stress range ΔY _b ) are calculated by elastic analysis, reflecting the set load displacement conditions. (S23). It should be noted that the elbow part behaves differently from the normal beam theory, such as when the bending in the axial direction acts, the circumferential bending is dominant at the elbow center. Therefore, it is preferable to perform a three-dimensional FEM elastic analysis using a model as shown in FIG. 8 and calculate the stress range reflecting the stress concentration in the elbow part.

  Next, a simple inelastic analysis is performed reflecting the set load displacement condition, and an elastic follow-up coefficient q obtained from the ratio of the amount of deformation of the elbow part by the inelastic analysis and the amount of deformation of the elbow part by the elastic analysis. Is calculated (S25).

  The simple inelastic analysis can be performed using a beam model as shown in FIG. As for the member characteristics (bending rigidity EI) of this model, different characteristics are set bilinearly in the straight pipe portion and the elbow portion as shown in FIG. In this embodiment, the member characteristic by the elastic perfect plastic body shown to Fig.10 (a) is made into object. Among these, the member characteristics of the straight pipe can be set using beam theory.

On the other hand, the member characteristics of the elbow part are set by the following method. As the initial bending stiffness (EI ₀ ′) of the elbow portion, a value obtained by reducing the initial bending stiffness (EI ₀ ) of the straight pipe portion using a pipe coefficient λ described later is used. For this reduction, a known theory may be used, and details are omitted here (see Non-Patent Document 5 etc.).

Next, the plastic starting point of the elbow is obtained. Here, the procedure will be described for the elbow portion where the in-plane bending moment M _i and the internal pressure P act.

First, a reference stress σ _R (reference stress σ _RMi due to in-plane bending moment M _i , reference stress σ _RP due to internal pressure P) of the elbow portion is obtained from equation (13), and equivalent reference stress is obtained from equation (14) (non- (See Patent Document 5). Next, the equivalent reference stress is substituted into the Mises stress in the equation (1) to determine the yield, and the time when the yield is reached is set as the plastic start point.

An inelastic analysis in which a forced displacement is applied to the beam model shown in FIG. 9 is performed using the set member characteristics, and the curvature k ^P of the elbow portion is obtained. Further, an elastic analysis is performed in which the member characteristics of the straight pipe portion and the elbow portion are respectively set to initial stiffness, and the curvature k ^E of the elbow portion is similarly obtained. Using the result, the elastic follow-up coefficient q is calculated by the equation (15).
(Equation 15)
q = k ^P / k ^E ------------------------------ Equation (15)

  As shown in FIG. 10A, since the plastic starting point of the elbow part is lower than that of the straight pipe part, it can be seen that the elbow part becomes plastic before the straight pipe part. After plasticization, the bending rigidity of the elbow part decreases, so that deformation concentrates only on the elbow part. In order to reflect this influence, the stress range calculated | required by process S23 is correct | amended using the calculated elastic tracking coefficient q.

As this correction method, the circumferential bending stress range [Delta] Y _b elastic coefficient of following q modifications circumferential bending by multiplying the stress in the range _{ΔY b '(= ΔY b ×} q) calculates (S26), modified by (16) There is a method of calculating the equivalent stress range ΔX _eq ′ (S24). According to this correction, a more rational ratchet distortion can be achieved by appropriately reflecting the shape characteristics of the elbow by multiplying the elastic bending coefficient by the circumferential bending stress range that is particularly easily amplified in the elbow. It can be calculated.

It should be noted that not only the corrected equivalent stress range ΔX _eq ′ is calculated using the three stress ranges as in equation (16), but also one or more stress ranges so as to include at least the corrected circumferential bending stress range ΔY _b ′. And the corrected equivalent stress range ΔX _eq ′ may be calculated using the selected stress range.

As another correction method, (12) there is a method of calculating the elastic coefficient of following modified by multiplying the q equivalent stress range _{ΔX eq '(= ΔX eq ×} q) in the equivalent stress range [Delta] X _eq determined by the formula (shown ) By correcting the entire equivalent stress range by multiplying by the elastic follow-up coefficient q, the ratchet strain can be calculated more safely.

Using the calculated corrected equivalent stress range ΔX _eq ′, ratchet strain is calculated in the third step S30 (S31). Since the subsequent processing is the same as that of the first embodiment, description thereof is omitted.

  As described above, according to the second embodiment, by correcting the equivalent stress range with the elastic follow-up coefficient q for the target pipe having the elbow part, the influence of deformation due to plasticization of the elbow part can be simplified and simplified. The ratchet distortion can be calculated appropriately.

[Third Embodiment]
Since the second embodiment is intended for an elastic perfect plastic body, the ratchet strain Δε ^R per repeated cycle does not change, and the ratchet strain ε ^R increases in proportion to the repeated cycle. However, it is known that the ratchet strain ε ^R actually converges to a constant value as the cyclic cycle progresses due to the effect of strain hardening (Nuclear Safety Infrastructure Organization, “Nuclear Facility Seismic Reliability”). (Refer to "Practical test piping system ultimate strength seismic verification test" (Heisei 10 to 15)).

  Therefore, in the third embodiment, in addition to the second embodiment, the effect of strain hardening is improved. Here, the influence of the overall curing in which the target pipe is cured as a whole by strain hardening and the effect of the local curing in which the elbow part is locally cured by strain hardening and strain accumulation are considered.

First, a method that takes into account the influence of overall curing will be described. Here, a member characteristic of the elbow part reflecting the effect of strain hardening is set, a simple inelastic analysis is performed using the set member characteristic, and an elastic follow-up coefficient q is calculated for each repeated cycle (S25).
A method of simple inelastic analysis that reflects the effect of strain hardening on the elbow where the in-plane bending moment M _i and the internal pressure P act is as follows.

First, as in the second embodiment, the equivalent reference stress is obtained from the result of the reference stress σ _R of the elbow part of the expression (13) by the expression (14), and the equivalent reference stress is calculated as the Mises stress of the expression (1). Substitution is performed to determine the yield, and the point in time when the yield is reached is set as the plastic start point.

After plasticization, the reference plastic strain increment Δε ^P _RMi is obtained by the following equation (17). Further, using the reference plastic strain increment Δε ^P _RMi , the plastic curvature increment Δk ^P _Mi is obtained from the equation (18).

The tangential bending stiffness in the plastic region of the elbow portion is set from the relationship between the moment increment ΔM acting on the elbow portion and the obtained plastic curvature increment Δk ^P _Mi. As a result, as shown in FIG. 10B, the bending rigidity of the elbow portion is increased even after plasticization. Therefore, using the member characteristics shown in FIG. 10B in consideration of the effect of overall hardening, the curvature k ^P of the elbow portion is smaller than that using the member characteristics shown in FIG. The coefficient q is also reduced. As a result, the elastic follow-up coefficient q, which is an additional coefficient of the stress range, decreases with the progress of the repeated cycle, and the ratchet strain ε ^R also decreases.

Next, a method for considering the influence of local curing will be described. The ratchet strain ε ^R accumulates in the elbow portion as the cycle progresses, but the yield stress σ _y increases in the next cycle when strain hardening is taken into consideration. As a result, the shake-down limit axial stress range Δσ ^S _XX obtained by equation (9) also increases, and the ratchet strain Δε ^R per repeated cycle obtained by equation (10) decreases.

As described above, according to the third embodiment, by considering the influence of global hardening and local hardening, it is further rationalized by reflecting that the ratchet strain increment Δε ^R changes with the progress of repeated cycles. Ratchet distortion ε ^R can be calculated.

In the above description, the ratchet distortion calculation method from the first embodiment to the third embodiment has been described in detail. However, the same technical idea is applied to a computer program, and the computer is processed. It may be done. It is possible to calculate the ratchet distortion ε ^R more easily and accurately by performing the above-described arithmetic processing for calculating the ratchet distortion in a calculation area such as a CPU of a computer. Note that the use of a computer is not necessarily required for the ratchet distortion calculation method according to these embodiments, and the computer may be partially used or the use thereof may be reduced as much as possible.

EXAMPLES Next, although an Example demonstrates this invention in detail, referring drawings, this invention is not limited only to a following example.
[Example 1, Comparative Example 1]
In Example 1, straight tube ratchet distortion was calculated using the method of the first embodiment. Moreover, in the comparative example 1, the ratchet distortion of the same straight pipe was computed by the three-dimensional FEM elastic-plastic analysis. A total of 22 cases with different stress ranges were studied.

  FIG. 11 compares ratchet strains according to Example 1 and Comparative Example 1. As a result, in most cases, the result of Example 1 is slightly higher than the result of Comparative Example 1, and the method of the first embodiment is a ratchet distortion evaluation method that has a good approximation on the safe side in a straight pipe. I can say that.

  In some cases, the result of Example 1 is lower than the result of Comparative Example 1, and the result on the unsafe side is recognized. This is the case when all three stress ranges are 16 times the yield stress. It is a result. Since these are not considered to have such a large value at the same time, there is no problem as a ratchet distortion evaluation method.

[Example 2, Comparative Example 2]
In Example 2, the ratchet distortion of the 90-degree elbow tube having the following specifications was calculated using the method of the second embodiment. In Comparative Example 2, ratchet strain of the same elbow pipe was calculated by three-dimensional FEM elasto-plastic analysis.
Elbow pipe specifications)
Structure dimensions Pipe radius r 104mm, thickness t 8.2mm
Elbow bending radius R 305mm
Load condition Internal pressure P 21. 67MPa
Repeated displacement D ± 57.5mm
Material properties Young's modulus E 195Gpa
Yield stress σ _y 550MPa (Elastoplastic)

In Example 2, the result of calculating the stress range (normalized by the yield stress σ _y ) by elastic analysis is as follows. It can be seen that the circumferential bending stress range ΔYb is more than twice that of the others.
Normalized axial film stress range: 0.98
Normalized axial bending stress range: 2.45
Normalized circumferential bending stress range: 5.74

As a result of simple inelastic analysis, the elastic follow-up coefficient was found to be 4.02, and the normalized corrected equivalent stress range was 23.23 in terms of yield stress σ _y ratio from equation (16). The circumferential stress due to the internal pressure was 0.5 in terms of yield stress σ _y ratio. As a result, the ratchet strain was 17.7 in terms of yield strain ε _y ratio, and was calculated to be about 5.0%.

  On the other hand, in Comparative Example 2, the ratchet strain was calculated to be 4.8%. Therefore, it can be said that the method of the second embodiment is a ratchet strain evaluation method that is close to the safe side at the elbow portion of the elastic perfect plastic body.

[Example 3, Example 4, Comparative Example 3]
In Example 3, the ratchet strain of the same elbow pipe as Example 2 was calculated using the method of the third embodiment. In Example 4, ratchet strain was calculated for the case where the influence of local hardening was ignored (that is, only the influence of overall hardening was considered) as a reference. In Comparative Example 3, the ratchet strain of the same elbow pipe was calculated by three-dimensional FEM elasto-plastic analysis.
The material was linearly isotropically cured, and the strain hardening coefficient H was 5 Gpa.

  FIG. 12 shows the relationship between the fulcrum reaction force N by the simple inelastic analysis of Example 3 and the curvature k at the elbow center cross section. The curvature of the elbow decreases with repeated cycles due to the overall curing. Using the elastic follow-up coefficient for each cycle obtained by this analysis, the corrected equivalent stress range for each cycle was calculated according to equation (16) to determine the ratchet strain. However, in the reference stress method, tension / compression was not distinguished, and it was considered that one cycle of pipe tension / compression was represented by a 2-cycle analysis.

  FIG. 13 shows the transition of ratchet distortion with the increase in the number of repeated cycles for Example 3, Example 4, and Comparative Example 3. In both Example 3 and Example 4, it can be seen that the increase in ratchet distortion for each repeated cycle is reduced. Further, Example 3 (considering global curing and local curing) is closer to Comparative Example 3 than Example 4 (considering only overall curing). Therefore, it can be said that the method of the third embodiment is a ratchet strain evaluation method that is close to the safe side in the elbow portion that is strain-hardened.

ε ^R _YY circumferential ratchet strain Δε ^R _YY circumferential ratchet strain per repeated cycle Δσ ^E _XX axial virtual elastic stress range Δσ ^S _XX elastic shakedown critical axial stress range σ _YY circumferential stress σ _y yield stress E Young's modulus ΔX Axial membrane stress range ΔX _b Axial bending stress range ΔY _b Circumferential bending stress range ΔX _eq Equivalent stress range ΔX _eq 'Modified equivalent stress range

Claims (7)

A ratchet strain calculation method for calculating ratchet strain accumulated in a pipe when a pipe subjected to internal pressure is repeatedly acted on,
In a straight pipe model subjected to repeated tensile compression in the axial direction, a first step of setting a calculation formula for calculating ratchet strain per repeated cycle from an axial stress range due to tensile compression and a circumferential stress due to internal pressure;
In a target pipe that repeatedly undergoes tensile compression and bending, three stress ranges including an axial film stress range, an axial bending stress range, and a circumferential bending stress range are obtained, and one of the three stress ranges is selected. A second step of calculating a stress range or an equivalent stress range consisting of a square root of the sum of squares of each of the two or more stress ranges;
A third step of substituting the calculated equivalent stress range into the axial stress range and calculating a ratchet strain per repetitive cycle of the target pipe according to the calculation formula set in the first step. Calculation method. The axial stress range is an axial virtual elastic stress range Δσ ^E _XX that is a stress range that acts when the straight pipe model is assumed to be an elastic body,
The calculation formula for calculating the ratchet strain Δε ^R _YY per repetitive cycle is the limit of occurrence of elastic shakedown that shows elastic behavior after the elastoplastic cycle, which is obtained from the circumferential stress σ _YY and yield stress σ _y due to internal pressure. Using the elastic shakedown limit axial stress range Δσ ^S _XX , which is the stress range, it is expressed as
The ratchet strain calculation method according to claim 1, wherein the three stress ranges are obtained by elastic analysis.
Δσ ^S _XX = √ {σ _YY ² −4 (σ _YY ² −σ _y ² )} (1)
Δε ^R _YY = 3σ _YY × (Δσ ^E _XX / Δσ ^S _XX −1) / E (2)
here,
√ {}: Symbol for calculating the square root of the result in parentheses E: Young's modulus The target pipe is a pipe having an elbow part,
The second step includes
Calculating the elastic follow-up coefficient determined from the ratio of the amount of deformation of the elbow part by inelastic analysis and the amount of deformation of the elbow part by elastic analysis;
The ratchet strain calculation method according to claim 2, comprising correcting the equivalent stress range using the calculated elastic follow-up coefficient. In the correction,
Multiplying the circumferential bending stress range by the elastic tracking coefficient to calculate a corrected circumferential bending stress range,
One stress range selected so as to include at least the modified circumferential bending stress range from the axial film stress range, the axial bending stress range, and the modified circumferential bending stress range, or each of two or more 4. The ratchet strain calculation method according to claim 3, wherein a corrected equivalent stress range including a square root of a sum of squares of the stress range is calculated. In the correction,
The ratchet strain calculation method according to claim 3, wherein a corrected equivalent stress range is calculated by multiplying the equivalent stress range by the elastic tracking coefficient. The elastic follow-up coefficient decreases with the progress of the repeated cycle due to the influence of the overall curing in which the target pipe is cured as a whole by strain hardening,
The yield stress σ _y used to calculate the shakedown limit axial stress range Δσ ^S _XX increases with the progress of repeated cycles due to the effect of local hardening in which the elbow part hardens locally due to strain hardening and strain accumulation. The ratchet distortion calculation method according to claim 3, wherein: In a computer, a program for calculating ratchet strain that accumulates in a pipe when the pipe receiving the internal pressure is repeatedly acted on,
On the computer,
In the storage area of the computer, in a straight pipe model subjected to repeated tensile compression in the axial direction, a calculation formula for calculating ratchet strain per repeated cycle is stored from the axial stress range due to tensile compression and the circumferential stress due to internal pressure. A first step to:
In a target pipe that repeatedly undergoes tensile compression and bending, three stress ranges including an axial film stress range, an axial bending stress range, and a circumferential bending stress range are obtained, and one of the three stress ranges is selected. A second step of calculating a stress range or an equivalent stress range consisting of a square root of the sum of squares of each of the two or more stress ranges;
A ratchet that executes a third step of substituting the calculated equivalent stress range into the axial stress range and calculating a ratchet strain per repeated cycle of the target pipe according to the calculation formula stored in the first step. A program for calculating distortion.

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