Structural stress analysis

• US 20050071091A1
• Filed: 11/16/2004
• Published: 03/31/2005
• Est. Priority Date: 11/17/2000
• Status: Active Grant

• Current Assignee: Jeong Kyun Hong, Jinmiao Zhang, Pingsha Dong
• Original Assignee: Jeong Kyun Hong, Jinmiao Zhang, Pingsha Dong
• Inventors: Dong, Pingsha, Hong, Jeong Kyun, Zhang, Jinmiao
• Granted Patent: US 7,089,124 B2
• Time in Patent Office: Days
• US Class Current: 702/42
• CPC Class Codes: B23K 31/02 relating to soldering or welding dip or wave soldering in the manufacture of printed circuits H05K3/34
• CPC Class Codes: G01L 5/00 Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes
• CPC Class Codes: G01M 5/0041 by determining deflection or stress
• CPC Class Codes: G01N 2203/0073 Fatigue
• CPC Class Codes: G01N 2203/0214 Calculations a priori without experimental data
• CPC Class Codes: G01N 2203/0216 Finite elements
• CPC Class Codes: G06F 30/23 using finite element methods [FEM] or finite difference methods [FDM]

Abstract

This need is met by the present invention wherein structural stress in a fatigue-prone region of a structure is determined by using the nodal forces and displacement values in the fatigue-prone region, or equilibrium-equivalent simple stress states consistent with elementary structural mechanics in the fatigue-prone region. The determination is substantially independent of mesh size and is particularly well-suited for applications where S-N curves are used in weld fatigue design and evaluation, where S represents nominal stress or stress range and N represents the number of cycles to failure. The present invention is directed to structural stress analysis through various combinations of modeling, calculation, and direct measurement schemes.

Claims

1. A method of analyzing structural stress σ

_s in a localized fatigue-prone region of a structure from a representation of the structure, said method comprising;

determining a through-thickness stress distribution σ

_x(y) along a selected cross section of said structure, wherein said localized region lies adjacent to said selected cross section, and wherein said stress distribution σ

_x(y) is determined from said representation of said structure;

determining a first component σ

_M of said structural structural stress σ

_s in said localized region by performing an operation having a result substantially equivalent to a result of the following first integration

${\mathrm{\sigma <br><br>}}_M=\frac{1}{t}<br><br>{\int <br><br>}_o^t<br><br>{\mathrm{\sigma <br><br>}}_X<br><br>\left(y\right)<br><br>d<br><br>y$

where σ

_x(y) represents said through-thickness stress distribution and t corresponds to the thickness of said structure in said selected cross section;

determining a second component σ

_B of said structural stress σ

_s in said localized region by performing an operation having a result substantially equivalent to a solution of the following equation for σ

_B

$\left(\frac{t^2}{2}\right)<br><br>{\mathrm{\sigma <br><br>}}_M+\left(\frac{t^2}{6}\right)<br><br>{\mathrm{\sigma <br><br>}}_B={\int <br><br>}_0^t<br><br>{\mathrm{\sigma <br><br>}}_x<br><br>\left(y\right)<br><br>y<br><br>d<br><br>y+\mathrm{\delta <br><br>}<br><br>{\int <br><br>}_0^t<br><br>t_{\mathrm{xy}}<br><br>\left(y\right)<br><br>d<br><br>y$

where y corresponds to a distance from y=0 to a material point of interest along said selected cross section, t corresponds to the thickness of said structure in said selected cross section, δ

is a value defined in said representation of said structure, σ

_x(y) represents said through-thickness stress distribution, and τ

_xy(y) represents a through-thickness shear stress distribution of said structure; and

calculating said structural stress σ

_s by combining said first component σ

_M of said structural stress and said second component σ

_B of said structural stress.

2. A method analyzing structural stress as claimed in claim 1 wherein said second component σ

_B of said structural stress σ

_s in said localized region is determined by defining the following relation where the shear stress distribution in the region is negligible;

$\mathrm{\delta <br><br>}<br><br>{\int <br><br>}_0^t<br><br>t_{\mathrm{xy}}<br><br>\left(y\right)<br><br>d<br><br>y=0$

3. A method of analyzing structural stress as claimed in claim 2 wherein said shear stress distribution τ

_xy(y) is determined from said structural representation.

4. A method of analyzing structural stress as claimed in claim 1 wherein said selected cross section of said structure is defined by a thickness t and includes a partial thickness fatigue crack extending a distance t_I from a surface of said localized region, and wherein said method comprises:

determining said first and second components σ

_M, σ

_B of said structural stress σ

_s in said localized region by performing respective operations having results substantially equivalent to respective results of four equations, where a first equation of said four equations defines a sub-component σ

_m′

of the structural stress σ

_s, a second equation of said four equations defines a force equilibrium condition, a third equation of said four equations defines a moment equilibrium condition, and a fourth equation of said four equations defines a stress continuity condition.

5. A method of analyzing structural stress as claimed in claim 4 wherein said first equation is as follows:

${\mathrm{\sigma <br><br>}}_M^{\mathrm{\prime <br><br>}}=\frac{1}{t-t_1}<br><br>{\int <br><br>}_0^{t-t_1}<br><br>{\mathrm{\sigma <br><br>}}_x<br><br>\left(y\right)<br><br>d<br><br>y-\frac{1}{t-t_1}<br><br>{\int <br><br>}_0^{\mathrm{\delta <br><br>}}<br><br>{\mathrm{\tau <br><br>}}_{\mathrm{yx}}<br><br>\left(x\right)<br><br>d<br><br>x$

6. A method of analyzing structural stress as claimed in claim 4 wherein said force equilibrium condition is as follows:

σ

_Mt₁+σ

_m′

(t−

t₁)=σ

_mt

where

${\mathrm{\sigma <br><br>}}_m=\frac{1}{t_1}<br><br>{\int <br><br>}_0^{t_1}<br><br>{\mathrm{\sigma <br><br>}}_x<br><br>\left(y\right)<br><br>d<br><br>y+\frac{1}{t_1}<br><br>{\int <br><br>}_0^{\mathrm{\delta <br><br>}}<br><br>{\mathrm{\tau <br><br>}}_{\mathrm{yx}}<br><br>\left(x\right)<br><br>d<br><br>x$

7. A method of analyzing structural stress as claimed in claim 4 wherein σ

_m′ and

σ

_b′

comprise respective sub-components of said structural stress σ

_s and wherein said moment equilibrium condition is as follows;

$\left({\mathrm{\sigma <br><br>}}_M-{\mathrm{\sigma <br><br>}}_B\right)<br><br>t_1<br><br>\left(t-\frac{t_1}{2}\right)+{\mathrm{\sigma <br><br>}}_B<br><br>t_1<br><br>\left(t-\frac{t_1}{3}\right)+{\mathrm{\sigma <br><br>}}_m^{\mathrm{\prime <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\frac{{\left(t-t_1\right)}^2}{2}+{\mathrm{\sigma <br><br>}}_b^{\mathrm{\prime <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\frac{{\left(t-t_1\right)}^2}{6}={\mathrm{\sigma <br><br>}}_m\left(\frac{t^2}{2}\right)+{\mathrm{\sigma <br><br>}}_b\left(\frac{t^2}{6}\right)$

where

${\mathrm{\sigma <br><br>}}_m=\frac{1}{t_1}<br><br>{\int <br><br>}_0^{t_1}<br><br>{\mathrm{\sigma <br><br>}}_x<br><br>\left(y\right)<br><br>d<br><br>y$

(y)dy and σ

_B is determined by solving the following for σ

_B

$\left(\frac{t_1^2}{2}\right)<br><br>{\mathrm{\sigma <br><br>}}_m+\left(\frac{t_1^2}{6}\right)<br><br>{\mathrm{\sigma <br><br>}}_b={\int <br><br>}_0^{t_1}<br><br>{\mathrm{\sigma <br><br>}}_x<br><br>\left(y\right)<br><br>y<br><br>d<br><br>y+\mathrm{\delta <br><br>}<br><br>{\int <br><br>}_0^{t_1}<br><br>{\mathrm{\tau <br><br>}}_{\mathrm{xy}}<br><br>\left(y\right)<br><br>d<br><br>y+{\int <br><br>}_0^{\mathrm{\delta <br><br>}}<br><br>{\mathrm{\sigma <br><br>}}_y<br><br>\left(x\right)<br><br>x<br><br>d<br><br>x$

where δ

represents a value derived from a representation of said structure and τ

_xy(y) represents a through-thickness shear stress distribution of said structure, as determined from said representation of said structure.

8. A method of analyzing structural stress as claimed in claim 4 wherein σ

_m′ and

σ

_b′

comprise respective sub-components of said structural stress σ

_s and wherein said stress continuity condition at y=t−

t₁ is as follows;

σ

_M−

σ

_B=σ

_m′

+σ

_b′

9. A method of analyzing structural stress as claimed in claim 1 wherein said selected cross section of said structure is characterized by a thickness t and defines a non-monotonic through thickness stress distribution characterized by a minimum axial stress along a secondary axis lying in said localized region a distance t₂ below a surface of said structure, and wherein said method comprises:

determining said first and second components σ

_M, σ

_B of said structural stress σ

_s in said localized region by performing an operations having results substantially equivalent to a result of solving simultaneously first and second equations with two unknowns, σ

_M and σ

_B.

10. A method of analyzing structural stress as claimed in claim 9 wherein:

said structural stress σ

_s is calculated by combining said first component of σ

_M of said structural stress and said second component σ

_B of said structural stress with two additional structural stress parameters σ

_m′ and

σ

_b′ and

said additional structural stress parameters σ

_m′ and

σ

_b′

are related to said first and second components σ

_M, σ

_B as follows;

σ

_M−

σ

_B=σ

_m′

−

σ

_b′

11. A method of analyzing structural stress σ

_s in a localized fatigue-prone region of a structure from a three-dimensional finite element solid model of the structure, said method comprising;

identifying at least one local element for structural stress extraction, wherein said local element lies adjacent to said localized fatigue-prone region;

determining stress resultants f_x′

f_z′

, and m_y′

for said local element from said finite element solid model of said structure, wherein said stress resultants represent the sectional forces and moments for said local element; and

calculating structural stress σ

_s in said localized fatigue-prone region utilizing the following equation;

${\mathrm{\sigma <br><br>}}_s={\mathrm{\sigma <br><br>}}_M+{\mathrm{\sigma <br><br>}}_B=\frac{f_{x^{\mathrm{\prime <br><br>}}}}{t}+\frac{6<br><br>\left(m_{y^{\mathrm{\prime <br><br>}}}+\mathrm{\delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>f_{z^{\mathrm{\prime <br><br>}}}\right)}{t^2}$

where δ and

t represent dimensional values of said local element.

12. A technique for analyzing structural stress σ

_s in a localized fatigue-prone region of a structure defining a substantially monotonic through thickness stress distribution, said technique comprising;

measuring displacement in the vicinity of said fatigue-prone region by configuring at least one measuring device to

measure displacement along a first cross section of said structure offset a distance L from said fatigue-prone region, and

measure displacement along a second cross section of said structure offset a distance L-l from said fatigue prone region;

decomposing stress measurements at said strain gauges as follows

${\mathrm{\sigma <br><br>}}_b^B=\frac{1}{2}<br><br>\left({\mathrm{\sigma <br><br>}}_T^B-{\mathrm{\sigma <br><br>}}_B^B\right)$${\mathrm{\sigma <br><br>}}_b^C=\frac{1}{2}<br><br>\left({\mathrm{\sigma <br><br>}}_T^C-{\mathrm{\sigma <br><br>}}_B^C\right)$

where said superscript B corresponds to a measurement at said first cross section, said superscript C corresponds to a measurement at said second cross section, said subscript T corresponds to a measurement at a top surface of said structure, and said subscript B corresponds to a measurement at a bottom surface of said structure; and

approximating structural stress σ

_s in said fatigue-prone region as follows;

${\mathrm{\sigma <br><br>}}_s={\mathrm{\sigma <br><br>}}_T^C+\frac{L}{l}<br><br>\left({\mathrm{\sigma <br><br>}}_b^C-{\mathrm{\sigma <br><br>}}_b^B\right)$

where the distances L and l are measured in terms of fractions of thickness t.

13. A technique for analyzing structural stress σ

_s in a localized fatigue-prone region of a structure defining a non-monotonic through thickness stress distribution, said technique comprising;

measuring displacement in the vicinity of said fatigue-prone region by configuring at least one measuring device to

measure displacement along a first cross section of said structure offset a distance L from said fatigue-prone region,

measure displacement along a second cross section of said structure offset a distance L-l from said fatigue prone region, and

measure displacement along a third cross section of said structure, wherein said third cross section defines an approximately linear structural stress distribution.

decomposing stress measurements at said strain gauges as follows

σ

_b^B=(σ

_T^B−

σ

_m)
σ

_b^C=(σ

_T^C−

σ

_m)

where said superscript B corresponds to a measurement at said first cross section, said superscript C corresponds to a measurement at said second cross section, said subscript T corresponds to a measurement at a top surface of said structure, and σ

_m is a through thickness mean stress measurement taken at said third cross section; and

approximating structural stress σ

_s in said fatigue-prone region as follows

${\mathrm{\sigma <br><br>}}_s={\mathrm{\sigma <br><br>}}_T^C+\frac{L}{l}<br><br>\left({\mathrm{\sigma <br><br>}}_b^C-{\mathrm{\sigma <br><br>}}_b^B\right)$

where the distances L and l are measured in terms of fractions of thickness t.

14. A computer-readable medium encoded with a computer program for analyzing structural stress σ

_s in a localized fatigue-prone region of a structure, said program being operative to;

determine a through-thickness stress distribution σ

_x(y) along a selected cross section of said structure, wherein said localized region lies adjacent to said selected cross section;

determine a membrane component σ

_M and a bending component σ

_B of said structural stress σ

_s in said localized region from said through thickness stress distribution σ

_x(Y), wherein said membrane and bending components comprise simple structural stress distributions that are equilibrium-equivalent to said through thickness stress distribution σ

_x(y); and

calculate said structural stress σ

_s by combining said first component σ

_M of said structural stress and said second component σ

_B of said structural stress.

15. A computer-readable medium encoded with a computer program for analyzing structural stress σ

_s in a localized fatigue-prone region of a structure, said program being operative to;

identify at least one local element for structural stress extraction, wherein said local element lies adjacent to said localized fatigue-prone region;

determine stress resultants f_x′

, f_z′

, and m_y′

for said local element from said finite element solid model of said structure, wherein said stress resultants represent the sectional forces and moments for said local element; and

calculate structural stress σ

_s in said localized fatigue-prone region utilizing the following equation

${\mathrm{\sigma <br><br>}}_s={\mathrm{\sigma <br><br>}}_M+{\mathrm{\sigma <br><br>}}_B=\frac{f_{x^{\mathrm{\prime <br><br>}}}}{t}+\frac{6<br><br>\left(m_{y^{\mathrm{\prime <br><br>}}}+\mathrm{\delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>f_{z^{\mathrm{\prime <br><br>}}}\right)}{t^2}$

where δ and

t represent dimensional values of said local element.

16. A system for analyzing structural stress σ

_s in a localized fatigue-prone region of a structure, said system including a controller programmed to;

determine a through-thickness stress distribution σ

_x(y) along a selected cross section of said structure, wherein said localized region lies adjacent to said selected cross section;

determine a membrane component σ

_M and a bending component σ

_B of said structural stress σ

_s in said localized region from said through-thickness stress distribution σ

_x(y), wherein said membrane and bending components comprise simple structural stress distributions that are equilibrium-equivalent to said through thickness stress distribution σ

_x(y); and

calculate said structural stress σ

_s by combining said first component σ

_M of said structural stress and said second component σ

_B of said structural stress.

17. A system for analyzing structural stress or in a localized fatigue-prone region of a structure, said system including a controller programmed to:

identify at least one local element for structural stress extraction, wherein said local element lies adjacent to said localized fatigue-prone region;

determine stress resultants f_x′

, f_z′

, and m_y′

for said local element from said finite element solid model of said structure, wherein said stress resultants represent the sectional forces and moments for said local element; and

calculate structural stress σ

_s in said localized fatigue-prone region utilizing the following equation

${\mathrm{\sigma <br><br>}}_s={\mathrm{\sigma <br><br>}}_M+{\mathrm{\sigma <br><br>}}_B=\frac{f_{x^{\mathrm{\prime <br><br>}}}}{t}+\frac{6<br><br>\left(m_{y^{\mathrm{\prime <br><br>}}}+\mathrm{\delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>f_{z^{\mathrm{\prime <br><br>}}}\right)}{t^2}$

where δ and

t represent dimensional values of said local element.

18. A system for analyzing structural stress σ

_s in a localized fatigue-prone region of a structure, said system comprising;

at least one measuring device configured to measure displacement in the vicinity of said fatigue-prone region by

measuring displacement along a first cross section of said structure offset a distance L from said fatigue-prone region, and

measuring displacement along a second cross section of said structure offset a distance L-l from said fatigue prone region; and

a controller programmed to

decompose stress measurements at said strain gauges as follows

${\mathrm{\sigma <br><br>}}_b^B=\frac{1}{2}<br><br>\left({\mathrm{\sigma <br><br>}}_T^B-{\mathrm{\sigma <br><br>}}_B^B\right)$${\mathrm{\sigma <br><br>}}_b^C=\frac{1}{2}<br><br>\left({\mathrm{\sigma <br><br>}}_T^C-{\mathrm{\sigma <br><br>}}_B^C\right)$

where said superscript B corresponds to a measurement at said first cross section, said superscript C corresponds to a measurement at said second cross section, said subscript T corresponds to a measurement at a top surface of said structure, and said subscript B corresponds to a measurement at a bottom surface of said structure, and

approximate structural stress σ

_s in said fatigue-prone region as follows

${\mathrm{\sigma <br><br>}}_s={\mathrm{\sigma <br><br>}}_T^C+\frac{L}{l}<br><br>\left({\mathrm{\sigma <br><br>}}_b^C-{\mathrm{\sigma <br><br>}}_b^B\right)$

where the distances L and l are measured in terms of fractions of thickness t

19. A system for analyzing structural stress σ

_s in a localized fatigue-prone region of a structure, said system comprising;

at least one measuring device configured to measure displacement in the vicinity of said fatigue-prone region by

measuring displacement along a first cross section of said structure offset a distance L from said fatigue-prone region,

measuring displacement along a second cross section of said structure offset a distance L-l from said fatigue prone region; and

measuring displacement along a third cross section of said structure, wherein said third cross section defines an approximately linear structural stress distribution; and

a controller programmed to

decompose stress measurements at said strain gauges as follows

σ

_b^B=(σ

_T^B−

σ

_m)
σ

_b^C=(σ

_T<sup>C−

σ</sup>

_m)

where said superscript B corresponds to a measurement at said first cross section, said superscript C corresponds to a measurement at said second cross section, said subscript T corresponds to a measurement at a top surface of said structure, and σ

_m is a through thickness mean stress measurement taken at said third cross section; and

approximate structural stress a, in said fatigue-prone region as follows

${\mathrm{\sigma <br><br>}}_s={\mathrm{\sigma <br><br>}}_T^C+\frac{L}{l}<br><br>\left({\mathrm{\sigma <br><br>}}_b^C-{\mathrm{\sigma <br><br>}}_b^B\right)$

where the distances L and l are measured in terms of fractions of thickness t.

20. A method of analyzing structural stress in a localized fatigue-prone region of a structure, said method comprising:

determining a through-thickness stress distribution along a selected cross section of said structure, wherein said localized region lies adjacent to said selected cross section and said through-thickness stress distribution is determined from a finite element model of said structure;

determining a membrane component and a bending component of said structural stress in said localized region from said through thickness stress distribution, wherein said membrane and bending components comprise simple structural stress distributions that are equilibrium-equivalent to said through thickness stress distribution; and

calculating said structural stress by combining said first component of said structural stress and said second component of said structural stress, wherein said membrane component and said bending component are determined such that said structural stress calculation is substantially independent of the size, shape and distribution of mesh elements defining said finite element model of said structure.

21. A method of analyzing structural stress as claimed in claim 20 wherein said membrane component and said bending component are determined such that said structural stress is substantially independent of mesh size over a mesh size range extending from about 0.16t and 0.1t, along respective x and y axes, to about 2t and t along respective x and y axes, where t represents the thickness of said structure.

22. A method of analyzing structural stress as claimed in claim 20 wherein said membrane component and said bending component are determined such that said structural stress is substantially independent of mesh size over a mesh size range extending from about 0.008t and 0.02t, along respective x and y axes, to about 0.4t and 0.5t along respective x and y axes.

23. A method of analyzing structural stress as claimed in claim 20 wherein said membrane component and said bending component are determined such that said structural stress is substantially independent of mesh size over a mesh size range extending up to about 2t and t along respective x and y axes.

24. A method of analyzing structural stress a, in a localized fatigue-prone region of a structure, said method comprising:

determining a through-thickness stress distribution σ

_x(y) along a selected cross section of said structure, wherein said localized region lies adjacent to said selected cross section;

determining a membrane component σ

_M and a bending component σ

_B of said structural stress σ

_s in said localized region from said through thickness stress distribution σ

_x(y), wherein said membrane and bending components comprise simple structural stress distributions that are equilibrium-equivalent to said through thickness stress distribution σ

_x(y); and

calculating said structural stress σ

_s by combining said first component σ

_M of said structural stress and said second component σ

_B of said structural stress.

25. A method of analyzing structural stress as claimed in claim 24 wherein said membrane component σ

_M and said bending component σ

_B are determined as a function of the following variables;

a distance variable y corresponding to a distance from y=0 to a material point of interest along said selected cross section; and

a thickness variable t corresponding to a thickness of said structure in said selected cross section.

26. A method of analyzing structural stress as claimed in claim 25 wherein said membrane component σ

_M and said bending component σ

_B are determined as a function of the following variables;

a distance variable y corresponding to a distance from y=0 to a material point of interest along said selected cross section;

a thickness variable t corresponding to a thickness of said structure in said selected cross section;

a finite element value δ

representing a value defined in a finite element representation of said structure; and

a distribution T_xy(y) representing a through-thickness shear stress distribution of said structure derived from a finite element model of said structure.