Minimizing stress concentrations around an elliptical hole by concentrated forces acting on the uniaxially loaded plate

Accepted Manuscript

Minimizing stress concentrations around an elliptical hole by concentrated forces acting on the uniaxially loaded plate Aizhong Lu , Ning Zhang , Guisen Zeng PII: DOI: Reference:

S0894-9166(16)30291-9 10.1016/j.camss.2017.07.002 CAMSS 36

To appear in:

Acta Mechanica Solida Sinica

Received date: Revised date: Accepted date:

19 November 2016 30 June 2017 3 July 2017

Please cite this article as: Aizhong Lu , Ning Zhang , Guisen Zeng , Minimizing stress concentrations around an elliptical hole by concentrated forces acting on the uniaxially loaded plate, Acta Mechanica Solida Sinica (2017), doi: 10.1016/j.camss.2017.07.002

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ACCEPTED MANUSCRIPT Minimizing stress concentrations around an elliptical hole by concentrated forces acting on the uniaxially loaded plate Aizhong Lu, Ning Zhang*, Guisen Zeng A.Z. Lu, N. Zhang (*), G.S. Zeng Institute of Hydroelectric and Geotechnical Engineering, North China Electric Power University, Beijing 102206, China

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*Correspondence to: N. Zhang e-mail: [email protected] Tel: 86-010-61772392 Fax: 86-010-61772234

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Postal address: No.2 Beinong Road Huilongguan, Changping District, 102206, Beijing China

Contract/grant sponsor: Natural Science Foundation of China; contract/grant number: 11572126, 11172101. ABSTRACT When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy’s

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integral formula. And then, based on the superposition principle, the analytical solutions for stress around an elliptical hole in an infinite plate subjected to a uniform far-field stress and concentrated forces, are obtained.

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Tangential stress concentration will occur on the hole boundary when only far-field uniform loads are applied. When concentrated forces are applied in the reversed directions of the uniform loads, tangential

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stress concentration on the hole boundary can be released significantly. In order to minimize the tangential stress concentration, we need to determine the optimum positions and values of the concentrated forces.

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Three different optimization methods are applied to achieve this aim. The results show that the tangential stress can be released significantly when the optimized concentrated forces are applied.

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KEY WORDS: Elliptical hole; Closed-form solution for stress; Cauchy’s integral method; Minimum stress concentration; Optimization I. Introduction Stress concentration on the boundary of an elliptical hole is important in mechanical design. How to predict and reduce stress concentration is of great practical interest. There are many ways to reduce the stress concentration around a hole, e.g. (1) to thicken the zone around the hole[1,2]; (2) to reinforce the hole by using the reinforcements around the hole[3-5]; (3) to introduce smaller holes on either side of the original

ACCEPTED MANUSCRIPT hole [6,7]; (4) to use multi-material fabrication techniques[8-12]. The method discussed in this work is to apply the jointing element (see Fig. 1), which is a common part in mechanical engineering. Under uniaxial tension in the far-field (Q in Fig. 1) of an infinite plate, the tangential stress reaches its extreme values at points A and B, with =(1+2a/b)Q and B=-Q. We can see that the maximum coefficient of tangential stress concentration is 1+2a/b at point A, especially when b is smaller, the coefficient becomes greater. If we preset two jointing elements before applying tension Q, four symmetrical

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concentrated forces will be generated once the tension is applied. Positions of the concentrated forces can be described by W and H (see Fig. 1). In order to release the stress concentration at the hole boundary, we first need to solve the stress distribution around the elliptical hole.

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Q

y

z1

B

P

P

L

M

H

z3

W

H

W

A

x P

P

z2 W

W

Jointing element

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PT

ED

z4

H

H

o

Q

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Fig. 1. An infinite plate with an elliptical hole under uniform load in far-field and four symmetrical concentraed forces.

The exact solution of stress field for the plane problem, an elliptical hole in an elastic infinite plane, can be easily solved using the analytical methods only under certain forms of loads. For example, when the uniform pressure acts in the far-field, or the radial and/or tangential uniform surface forces act on the hole boundary, we can deal with these issues using the method of conformal transformation and the Cauchy’s integral formula proposed by Muskhelishvili[13-15]. The main process involves mapping the hole contour and its exterior (on the z-plane) into the unit circle and its exterior (on the ζ-plane) by the mapping function

ACCEPTED MANUSCRIPT z=ω(ζ), and deriving the two complex potentials υ(ζ) and ψ(ζ). For the case that concentrated force acts in the plane (Fig. 2), the complex potentials were obtained by using an auxiliary function and the analytical continuation theorem[16,17]. In this paper, the Cauchy’s integral formula is to be used to solve the ellipse problem. The Cauchy’s integral is a kind of direct method, The two complex potentials υ(ζ) and ψ(ζ) can be directly solved according to the stress boundary conditions of the hole boundary, and the stress field can then be derived from υ(ζ) and ψ(ζ). In this way, the closed-form

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solution for stress can be obtained. The method proposed in this paper is more general than the current method. This method is suitable for not only elliptical hole but also other shapes.

Y1

y

P1

X1

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z1

b

L

o

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M

a

x

Fig. 2. Concentrated force on the outer region of an ellipse.

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To minimize stress concentration around the hole when the plate is subjected to uniaxial tension, we

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combine the stress solution with the optimization technique, the optimum designs of positions and values of the concentrated forces under different constrained conditions are conducted.

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II. Complex potentials using the Cauchy’s integral formula

2.1. Stress boundary conditions A concentrated force P1 acts at one point of an infinite plane z1 and the components in the x- and y-axes are X1 and Y1, respectively (Fig. 3). When there is no hole in the plane, the two complex potentials υ(1)(z) and ψ(1)(z) are[14]

（1）( z )  

X1  iY1 ln( z  z1 ) 2 (1   )

(1)

ACCEPTED MANUSCRIPT  (1) ( z ) 

 ( X1  iY1 ) ( X  iY1 ) 1 ln( z  z1 )  z1 1 2 (1   ) 2 (1   ) ( z  z1 )

Y1

(2)

P1

y X1

z1 L Yn

L o

x

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L-

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Xn +

Fig. 3. Surface forces on the pre-excavated boundary induced by concentrated forces. Under concentrated force P1, the components of surface force at any point of the proposed drilled boundary L are (Xn, Yn). Drilling the hole can be considered as applying the opposite surface force (-Xn, -Yn)

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on the boundary. Let υ(2)(z) and ψ(2)(z) denote the two complex potentials as the ellipse boundary is subjected to (-Xn, -Yn), the functions υ1(z) and ψ1(z) under the action of concentrated force and surface force

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(-Xn, -Yn) can be obtained by superposition as

X1  iY1 ln( z  z1 )  (2) ( z ) 2 (1   )

(3)

 ( X1  iY1 ) ( X  iY1 ) 1 ln( z  z1 )  z1 1  (2) ( z ) 2 (1   ) 2 (1   ) ( z  z1 )

(4)

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1 ( z )  (1) ( z )  (2) ( z )  

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 1 ( z )   (1) ( z )  (2) ( z ) 

where υ(2)(z) and ψ(2)(z) can be derived by the stress boundary condition of the hole boundary, Eq. (5).



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Because

L

( X n )ds  0 and



L

(Yn )ds  0 , υ(2)(z) and ψ(2)(z) are single valued complex potentials.

1 (t )  t1(t )  1 (t )  0, t  L

(5)

Substituting Eqs. (3) and (4) into Eq. (5), we have

X 1  iY1 ln(t  z1 ) 2 (1   ) X  iY1 t  ( X 1  iY1 ) ( X  iY1 ) 1  1  ln(t  z1 )  z1 1 2 (1   ) t  z1 2 (1   ) 2 (1   ) (t  z1 )

 (t )  (2) (t )  (2) (t )  z(2)

(6)

ACCEPTED MANUSCRIPT 2.2. Solution of complex potentials To map the outer region of an ellipse in the z-plane into the outer region of a unit circle in the ζ plane, the mapping function is

 m z   ( )  R       where R 

(7)

ab a b and m  . 2 ab

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Substituting the mapping function Eq. (7) into (2) ( z ) and  (2) ( z ) gives the expressions of (2) ( ) and  (2) ( ) . Here, for the sake of simplification, the same function symbol (2) (or  (2) ) is used. (2) ( ) and  (2) ( ) are single valued complex potentials in the outer region of the unit circle. Substituting

(2)   

      

    (2)    (2)

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Eq. (7) into Eq. (6), we have the following equation for solving (2) ( ) and  (2) ( ) ,

X 1  iY1  R 2  z1  Rm  ln   2 (1   )   

 ( X 1  iY1 )  Rm 2  z1  R  X 1  iY1 m  2 ( X  iY1 )   ln   z1 1  2 2 2 (1   )   2 (1   ) Rm  z1  R  2 (1   ) Rm  z1  R

(2)   1     d  2 i

1

    d

      (2) 1        d  2 i

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1 2 i

1 2 i

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Using Cauchy’s integral operator

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where σ is the boundary value of ζ, i.e.,   ei .

in Eq. (8), we have

 (2)   X  iY 1     d  2 1(1  1) 2 i

 Rm 2  z1  R  1 ln d         2 m  1 ( X  iY1 ) 1  1 d  z1 1 d  2 2 2 (1   ) 2 i  Rm  z1  R    Rm  z1  R   

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 R 2  z1  Rm  1  ( X 1  iY1 ) 1   ln  d     2 (1   ) 2 i    X 1  iY1 1 2 (1   ) 2 i



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

ζ in

1 2 i

1

    d

(8)

(9)

is any point external to γ. Integral is along the γ counterclockwise. From Eq. (9),

we have



(2)    a1 ln 1  

 m  1   b1 ln 1   1    1

 c1   1   1

(10)

ACCEPTED MANUSCRIPT X  iY1  ( X 1  iY1 ) where a1  1 , b1   , c1  2 (1   ) 2 (1   )





 1  m 2    2  m   1 1 1  1



2

1 1  m



a1

z1  z12  4mR 2 and  1  . 2R

ζ1 should take the value as  1  1 . The deriving process of Eq. (10) is given in Appendix A. Substituting Eq. (7) into 1 ( z ) and  1 ( z ) in Eqs. (3) and (4) gives the expressions of 1 ( z ) and

we have



    a1 ln    1   b1 ln 1  

Taking the conjugates of both sides of Eq. (8), we have

c1 1    1  1   1

   X  iY1  Rm 2  z1  R      (2)    1 (2) ln       2 (1   )   

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(2)   

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 1 ( z ) . Substituting Eq. (10) into Eq. (3) and taking the constants which have no effect on the stress as zero,

 ( X 1  iY1 )  R 2  z1  Rm  X 1  iY1 1  m 2 ( X  iY1 )   ln   z1 1  2 2 2 (1   )   2 (1   ) R  z1  Rm  2 (1   ) R  z1  Rm



 (2)    a1 ln 1 





 1  m   b1 ln 1   1    1

 1    a0     (2)    1  m    

(12)

(13)

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

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In the similar way, from Eq. (12), we have

(11)

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  2  m m    2  m3   1 1  1 1 a1 . where a0     1  m   12 

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The deriving process of Eq. (13) is given in Appendix B. Substituting Eq. (13) into Eq. (4) and taking the constants as zero, we have

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 1     a1 ln 1    1

 1  m 2   1 1    d  (2)   b1 ln    1   b0    1 0  1  m  2 m 

(14)

 2  m3  2 (  m) a1 . a1 and d0  1 where b0  1 12 m   12  1 ( 1  m) 2

   m  2 m  ( ) , we  (2) Because  ( )  a1 ln  R      z1   (2) ( ) and  ( )  a1   (    )   m      1 1   have

 1  m 2   m 2

    (2)

 1  m 2   m 2

 F  m  E   , where E and F are given in    1   m 1 

 ( )  a1 

ACCEPTED MANUSCRIPT Appendix B. Substituting this equation into Eq. (14) and taking the constants as zero, we have

 1  m  d1 1      a1 ln 1     b1 ln    1     1  2 m   1  where d1 





 1 1   1 1 m 1   1



2

1 1  m



2

  

 

(15)

a . 1

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III. Stresses as multiple concentrated forces are applied 3.1. Complex potentials

As multiple concentrated forces (Xj, Yj) (j=1, 2, …, n) are applied, the complex potentials can be derived from Eqs. (11) and (15) as n

n

j 1

2 (1   )



 j 1   j  j m j   j



,

2

 j  j m



a

j

bj  

 ( X j  iY j ) 2 (1   )

and  j 

2R





 1  m 2    2  m   j j  j  j cj   aj 2  j  j m

,

z j  z 2j  4mR 2

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dj 



X j  iY j

(16)

n  n  1  m 2  dj         b1 ln    j    2  j 1      m j  1 j 

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aj 

where

j 1

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 1      a1 ln 1    j 1 j  n

1  n cj   j  j 1 1   1

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  

     a j ln    j    b j ln 1 





(17)

,

. ζj should take the value as  j  1 .

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If only uniaxial tension Q is applied, the corresponding two complex potentials are

 ( ) 

RQ  2 m   4   

RQ   2 (1  m  m2 )  1  ( )   2   (m   2 ) 

(18)

(19)

3.2. Stress solution The stress components can be solved according to the following two equations

      4 Re[ ( )  ( )] 

4 Re[ ( )] R

(20)

ACCEPTED MANUSCRIPT       2i  

                           2      [   ] 

2 2

1

2

(21)

where  ( ) and  ( ) should be the superpositions of Eqs. (16) and (18), Eqs. (17) and (19), respectively. σρ, σθ and τρθ are the stress components in orthogonal curvilinear coordinates in the z plane. On the hole boundary L, ρ=1, σρ and σθ are the normal and tangential stresses along the boundary, respectively. When there is no load on the hole boundary, the normal stress σρ=0, and the tangential stress is given as

n

aj

j 1

  j

where      

n

bj

j 1

 ( j  1)



n

 j 1

c j 1 (1   1 )

2



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 ( )  2   R(1  m /  )  

   4 Re[ ( )   ( )]  4 Re 

(22)

RQ  2  m  1 2  . 4   

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3.3. Comparison with the numerical results using ANSYS software

The parameters are taken as follows: Poisson’s ratio μ=0.2; dimensions of the ellipse a=2 and b=1; and tension in the far-field Q=2. Considering four concentrated forces applied at z1, z2, z3 and z4 (Fig. 1), respectively. The values of the four concentrated forces are

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X1  iY1  3.0  1.0i , X 2  iY2  3.0  1.0i , X 3  iY3  3.0  1.0i and X 4  iY4  3.0  1.0i .

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Two positions are considered, which are W=1.0, H=1.0 and W=2.1, H=0.6. The former is closer to the hole boundary. All parameters are in international units, i.e., tension Q in Pa, dimensions a, b, W and H in m,

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and concentrated force (Xj, Yj) in N.

To achieve results with high accuracy, a numerical model with large dimension of 100x100 is used to

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simulate the infinite plane. The planar 4-node isoparametric element is used. Totally, 170202 elements and 171128 nodes are generated. 360 nodes are on the hole boundary, and the interval between adjacent nodes is

AC

in the range of 0.015-0.035. Considering symmetry, 1/4 of the ellipse and the neighboring region are given in Fig. 4. The constraints are set on the symmetrical axis, i.e., the horizontal constraints on the y-axis and the vertical constraints on the x-axis (see Fig. 4).

ACCEPTED MANUSCRIPT

y

o

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x

Fig. 4. Mesh and constraints for numerical simulation.

The stresses on the hole boundary and the x-axis calculated using the analytical and numerical methods are illustrated in Fig. 5. When no loads act on the hole boundary, there is only tangential stress σθ. Due to

12

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symmetry, only σx and σy will appear on the x-axis.

10 8 6

10

20

30

40

50

60



70

80

2.0

-4 -6

(c)

9

(x)

0.0 0

2

4

6

8 x

10

12

14

16

W=1.0, H=1.0, Analytical solution W=1.0, H=1.0, Numerical solution W=2.1, H=0.6, Analytical solution W=2.1, H=0.6, Numerical solution

CE

8

0.5

PT

-8

1.0

90

ED

0

M



2 0

7

AC

6 y

W=1.0, H=1.0, Analytical solution W=1.0, H=1.0, Numerical solution W=2.1, H=0.6, Analytical solution W=2.1, H=0.6, Numerical solution

(b)

1.5

4

-2

2.5

W=1.0, H=1.0, Analytical solution W=1.0, H=1.0, Numerical solution W=2.1, H=0.6, Analytical solution W=2.1, H=0.6, Numerical solution

(a)

5 4 3 2 1

0

2

4

6

8

10

12

14

16

x

Fig. 5. Comparisons between the analytical and numerical results. (a) σθ on the hole boundary; (b) σx on the x-axis; (c) σy on the x-axis.

ACCEPTED MANUSCRIPT It can be seen from Fig. 5 that the analytical and numerical results agree with each other very well. Only as the concentrated forces are close to the hole boundary, e.g. when W=1.0 and H=1.0, there exist slight differences between the analytical and numerical results. IV. Optimization criterion and methods Under the action of Q and P, the tangential stress on the hole boundary is not uniform. The greatest

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tangential stress concentration will occur at certain points. As mentioned earlier, we need to reduce the greatest stress concentration. So, the optimization criterion is to minimize the absolute value of the greatest tangential stress concentration[18] (Lu et al., 2014).

In mathematics, this is a minimization problem using optimization techniques, for which the corresponding objective function can be given as

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F  max    [0, /2]

The mathematical model can be expressed as

min F ( X )

s.t. g j ( X )  0, j  1, 2, ..., M ,

(23)

(24)

M

X  (W , H , P)T

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where gj(X)>0 is the constrained condition. For the issue discussed in this work, the constrained conditions are

(25)

PT

W  0, H  0, P  0, W 2 / a 2  H 2 / b2  1

The optimization methods used in this paper includes the unconstrained nonlinear simplex method, the

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constrained mixed penalty function method and the constrained differential evolution method. The differential evolution can ensure that the results are globally optimal solution, but it cannot treat certain

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constrained conditions, such as W2/a2+H2/b2>1. Sometimes the solutions are not in the feasible region, for which, we have to use the mixed penalty function. V. Optimization on a circular hole In this section, we take a=b=2, Q=1 and μ=0.2. a=b means the hole is a circle, one special case of the elliptical hole. 5.1. Optimizations on W, H and P The optimizing process should be conducted under the constrained condition of Eq. (25), as well as

ACCEPTED MANUSCRIPT under Ws-W>0, Hs-H>0 and Ps-P>0, where Ws, Hs and Ps are the upper limits of W, H and P, respectively. For the sake of analyzing, we take the same upper limit for all three parameters. The optimized results are listed in Table 1 under different constraint intervals, i.e., [0, 10000], [0, 1000], [0, 100], [0, 10], [0, 5] and [0, 2.5]. To ensure the correctness of the results, the mixed penalty function method and the constrained differential evolution method are used for each case. Only when the results by the two methods are the same, can they be considered as the correct optimization results. All the corresponding stresses of max in all

Table 1 Optimization results of W, H and P. W

max

W:H:P

5.17  10-7

1:1.4142:2.7768

999.98

4.57  10-5

1:1.4142:2.7768

99.99

4.57  10-3

1:1.4137:2.7764

9.99

0.2717

1:2.1931:2.6811

5.00

0.3661

1:2.5707:2.9555

2.50

1.1744

1:1.5523:1.5523

H

3562.83

5038.60

[0, 1000]

360.12

509.28

[0, 100]

36.01

50.91

[0, 10]

3.73

8.02

[0, 5]

1.69

4.35

[0, 2.5]

1.61

2.50

9893.76

M

[0, 10000]

P

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Intervals

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examples of this paper are tension, so we directly use σθmax to denote the maximum absolute value.

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Under the same constraint interval for each parameter, P always approaches to or equals the corresponding upper limit. A larger interval results in greater optimized values of W, H and P but smaller

PT

tangential stress concentration. This law applies to the intervals beyond those listed in Table 1. When the interval is relatively large, the values of W, H and P are almost in the same ratio, i.e., 1:1.4142:2.7768;

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which however, is not the case when the interval is small. When the interval is relatively large, the maximum tangential stress is very small, and almost

AC

approaches to zero. It is remarkably different from the case where no concentrated forces are applied. When no concentrated forces are applied, the maximum tangential stress is 3Q, occurring at point A. When W and H are small, or if the concentrated forces are applied near the hole boundary, the maximum tangential stress at point A can still be decreased significantly. Similar to the last three intervals, the optimized σθmax is decreased by 90.94%, 87.80% and 60.85%, respectively. The tangential stress distributions on the hole boundary for the three cases are illustrated in Fig. 6, in which θ=0° and 90° denote points A and B, respectively.

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W=3.73, H=8.02, P=9.99,max=0.27 W=1.69, H=4.35, P=5.00,max=0.37 W=1.61, H=2.50, P=2.50,max=1.17 No concentrated forces,max=3.00

2

C4



C3 1

C1

C2

-1 0

10

20

30

40

A

50

60

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0

70

80

90

B



Fig. 6. Tangential stress distributions on the hole boundary.

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Fig. 6 shows that, when W=3.73, H=8.02 and P=9.99, the tangential stress on the hole boundary is relatively uniform and the maximum value σθmax=0.2717, happening at point C1 where θ=55°. When W=1.69, H=4.35 and P=5.00, the tangential stress is still relatively uniform with σθmax=0.3661 at point C2 (θ=68°). When the concentrated force gets closer to the hole boundary, W=1.61, H=2.50 and P=2.50, the tangential

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stress is not uniform anymore, with σθmax=1.1744 happening at points C3 (θ=0°) and C4 (θ=54°) simultaneously. But the compressive stress happens at the vertex, B, with σθ=-1.0511, which is almost the

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same as the one without any concentrated forces applied. 5.2. Optimizations on W and H

PT

From the previous discussion, we know that in order to minimize tangential stress concentration, we

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need a relatively great concentrated force P, which may be difficult to achieve in practice. Thus, we just search for the optimized position of concentrated force (W, H) under a given value of P. For this situation,

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the optimization results can be obtained using the unconstrained nonlinear simplex method and verified by the other two optimization methods. The optimization results for different values of P are listed in Table 2. Fig. 7 illustrates the tangential stress distributions on the hole boundary when P=0.1, 0.4, 0.8 and 1.5. Table 2 Optimization results of W and H. P

W

H

A

B

max

0.01

2.0240

0.2471

2.9491

-1.0000

2.9491

0.02

2.0450

0.3147

2.9215

-1.0000

2.9215

ACCEPTED MANUSCRIPT 2.0552

0.5936

2.7804

-0.9990

2.7804

0.2

2.0715

0.7775

2.6561

-0.9965

2.6561

0.4

2.0530

1.0438

2.4585

-0.9896

2.4585

0.6

2.0223

1.2490

2.2925

-0.9819

2.2925

0.8

1.9953

1.4215

2.1440

-0.9756

2.1440

1.0

1.9664

1.5759

2.0073

-0.9727

2.0073

1.5

1.8755

1.9170

1.7004

-0.9887

1.7004

2.0

1.7495

2.2198

1.4266

2.5

1.6305

2.5010

1.1742

3.0

1.4626

2.7703

0.9377

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0.1

-1.0329

1.4266

-1.0506

1.1742

-0.9253

0.9377

No concentrated forces , max=3.00

3.5

P=0.4, W=2.0530, H=1.0438, max=2.46

C1

3.0

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P=0.1, W=2.0552, H=0.5936,max=2.78

C2

2.5

P=1.5, W=1.8755, H=1.9170, max=1.70

C4

2.0 1.5 1.0

M



P=0.8, W=1.9953, H=1.4215, max=2.14

C3

0.5

-0.5 -1.0 -1.5

10

PT

0

ED

0.0

A

20

30

40

50

60

70



80

90

B

CE

Fig. 7. Tangential stress distributions on the hole boundary under different values of P. The results indicate that σθmax decreases with increasing P. For each case, σθmax can always be found at

AC

two positions: one is always at point A and the other gets farther from A with increasing P. However, the tangential stress at point B is almost unaffected by P, with the value being around -1, almost the same as the one without any concentrated forces applied. We can say that the concentrated force has a great effect on the tangential stress concentration at point A but almost no effect at point B. It can also be seen from Table 2 that the optimized H increases with increasing P, but the optimized W does not change in a monotone way. When P is small, the optimized W is always close to the radius of the circle and increases with P. When P is large, W is smaller than the radius and decreases with increasing P.

ACCEPTED MANUSCRIPT 5.3. Optimizations on W and H under the constrained condition W≥a From Section 5.2, when the concentrated force P≥0.8, the optimized W
Optimization results under W≥a. W

H

A

B

max

0.8

2.0000

1.4208

2.1441

-0.9756

2.1441

1.0

2.0000

1.5709

2.0082

-0.9727

2.0082

1.5

2.0000

1.9110

1.7101

-0.9870

1.7101

2.0

2.0000

2.2271

1.4541

-1.0220

1.4541

2.5

2.0000

2.5336

1.2276

-1.0390

1.2276

3.0

2.0000

2.8390

1.0240

-0.9973

1.0240

M

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P

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Table 3

The data indicate that, when concentrated force P is greater than 0.8, the optimized W under the

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constrained condition W≥a always equals to radius a. And the corresponding maximum tangential stress concentration is slightly greater than the results listed in Table 2. The optimized H also increases with P and

PT

is just slightly greater than that in Table 2.

To illustrate the effect of position on stress concentration, Fig. 8 presents the tangential stress

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distributions on the hole boundary for different values of H. When H is relatively small, the tangential stress near the point is affected significantly by the concentrated force. For example, when H=0.2, the tangential

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stress σθ=-17.25 at θ=5°, but sharply increases to 18.99 at θ=6.5°. The maximum tangential stress σθmax is always found at only one point, except for the case when H=1.5709, σθmax is found at two points, point A and C, where θ=35° (see Fig. 8).

ACCEPTED MANUSCRIPT ,

10

P=1.0, W=2.0, H=0.2, max=18.99 P=1.0, W=2.0, H=0.5, max=5.09

8

P=1.0, W=2.0, H=1.0, max=3.05

6

P=1.0, W=2.0, H=1.5709, max=2.01 P=1.0, W=2.0, H=2.5, max=2.28

4

c



2 0 -2 -4

-8

,

-10 0

10

20

30

40

50 

A

60

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-6

70

80

90

B

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Fig. 8. Tangential stresses on the hole boundary for different values of H. VI. Optimization on an elliptical hole

For the elliptical hole, we conduct the optimization on W and H under a given P and the constrained situation W≥a. In this section, we take a=2, P=1, Q=1 and μ=0.2. The mixed penalty function and the

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differential evolution are used in this case. The optimization results for different values of b are listed in Table 4.

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Table 4 Optimization results of an elliptical hole. W

H

A

PT

b

B

max1

max2

(max2-max1)/max2

(*)

2.0000

0.1368

5.4165

-1.0080

5.4165

41.0

86.79%

0.5

2.0000

0.3787

3.6370

-0.9969

3.6370

9.0

59.59%

1.0

2.0000

0.7347

2.8468

-0.9429

2.8468

5.0

43.06%

2.0105

1.3340

2.4429

-0.9057

2.4429

8/3

33.38%

2.0000

1.5709

2.0082

-0.9727

2.0082

3

33.06%

2.5

2.0712

2.0035

1.9995

-0.8984

1.9995

2.6

24.60%

3.0

2.1044

2.4615

1.8582

-0.9051

1.8582

7/3

20.36%

5.0

2.0423

4.4851

1.5188

-0.9298

1.5188

1.8

15.62%

2.0

AC

1.5

CE

0.1

Note: σθmax2 (*) denotes the maximum tangential stress on the hole boundary when no concentrated forces

ACCEPTED MANUSCRIPT are applied. Because the length of the horizontal semi-axis is given, i.e., a=2, the vertical semi-axis b in Table 5 increases from 0.1 to 5.0, meaning the hole shape turns from a flat ellipse in the horizontal direction to a circle and then to a flat ellipse in the vertical direction. As can be seen from the optimization results, the optimized value of W is almost unaffected by b, keeping the value being or approaching to a; but the optimized H increases with increasing b and is smaller than b, except when b takes very small values such as

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0.1 (see Table 4). Thus, in order to release stress concentration, the concentrated force should be applied near the corner of the circumscribed rectangle.

Combining the data in Table 4 with tangential stress distributions (Fig. 9), it can be seen that the maximum tangential stress decreases with increasing b. Similar to the circle case, the maximum value can

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also be found at two points including point A, and the tangential stress at point B is slightly affected by b, with the value roughly equaling -1, which is almost the same as the one without any concentrated forces applied. The position of concentrated force affects the tangential stress at point A more significantly than that at point B, which is consistent with the circle case. 4.0

b=1.0, W=2.0000, H=0.7347, max=2.85

M

3.5

C2

3.0

C3

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2.5 2.0 

b=0.5, W=2.0000, H=0.3787, max=3.64

C1

1.5

b=2.0, W=2.0000, H=1.5709, max=2.01 b=2.5, W=2.0712, H=2.0035, max=2.00

C4 C5

PT

1.0

b=1.5, W=2.0105, H=1.3340, max=2.44

0.5

CE

0.0

-0.5

AC

-1.0

0

10

20

30

40

A

50 

60

70

80

90

B

Fig. 9. Tangential stress on the boundary of the ellipse for different b. VII. Conclusions

The jointing elements applied near the hole can supply concentrated forces. The optimized positions and values of concentrated forces can significantly release the tangential stress concentration on the hole boundary when the plate is subjected to uniaxial loads. Cauchy’s integral formula is used to solve the complex potentials. The stress distribution is verified by

ACCEPTED MANUSCRIPT numerical analyses using ANSYS software. Based on the analytical solutions for stress, optimizations on the position and value of concentrated force are conducted. In determining complex potentials, the method proposed in this paper is more general than most other methods. This method is applicable to not only an elliptical hole but also holes of other shapes. The tangential stress concentration can be released effectively by applying a concentrated force at the optimum position. The maximum tangential stress concentration is generally found at two points, one of

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which is always located at the two vertexes in the direction of the concentrated force, where the tangential stress is affected most significantly. However, the tangential stress on the other two vertexes in the direction perpendicular to the concentrated force is scarcely affected by the concentrated force. Funding

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This work was supported by the National Natural Science Foundation of China [grant numbers 11172101, 11572126]. Appendix A

M

The terms in Eq. (9),

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    R 2  z1  Rm  m  ln     ln   R 1   ln 1    ln 1      1    1   Rm 2  z1  R   m   1  ln    ln  R 1  ln 1    ln 1    1       1 



PT





CE

m  2 1  A B       2 Rm  z1  R Rm    1  1    1 m 

Rm  z1  R

AC

2



 B  1  m  , C  m  A m   m     m  m 1  m 1

where

 1  C D    Rm    1  1    1 m 

2

2

1

2

1

1

2

2

1

1

and D 

1 2

2

1  m

.

  m  1 1  Because  1  1 , the terms ln 1  and (2)   in Eq. (9) are the  , ln 1  ,   1    1    1  1 boundary values of the complex potentials in the outer region of the unit circle. Taking the integral constants

ACCEPTED MANUSCRIPT  m as zero, the Cauchy’s integrals of them are  ln 1    1

  1  ,  ln 1     1

 1 and (2)   , ,    1  1 

    m     1     and  (2)   respectively. The terms ln 1   , ln 1  , , (2) ,     m    m        1  1 1  1  are the boundary values of the complex potentials in the inner region of the unit circle, the integrals of which are all zero.

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Substituting these integral results into Eq. (9) and taking the integral constants as zero, we can then obtain Eq. (10). Appendix B The two terms in Eq. (12)

where E 

  1 G H     , , 2  R  z1  Rm R    m  1    1 

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1  m 2 1 E F  m    2 R  z1  Rm R    m  1    1

 1 (1  m2 )  12 m  12  m3 F   H  G  , , and . m   12  12  m m   12  1  m   12 

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  1    m  1 1      in Because  1  1 , the terms ln 1  ,  (2)   and ,  , ln 1      (2)   1    m  1   1  Eq. (12) are the boundary values of the complex potentials in the outer region of the unit circle. The

    1    m    , respectively. The terms ln 1   , ln 1  (2)     1  1  

CE



PT

  m  1 1  Cauchy’s integrals of these terms are  ln 1  ,  (2)   and ,   ,  ln 1    m 1   1    1   1  , and (2)   are ,       1 1 

AC

the boundary values of the complex potentials in the inner region of the unit circle, the Cauchy’s integral of which are all zero.

Substituting these integral results into Eq. (12) and ignoring the constants, we can get Eq. (13). References [1] Rosen, T.T., Magne, K.N., Andreas, E., Design procedure for reducing the stress concentration around circular holes in laminated composites. Composites, 1995, 26, 815-828. [2] Vovk, L.M., Kozhevnikov. V.F., Semenov-Ezhov, I.E., Stress concentration around holes in plates and

ACCEPTED MANUSCRIPT lugs with symmetric reinforcements in uniaxial tension. Russ. Eng. Res., 2001, 21(11), 27-31. [3] Engels, S., Becker, W., Optimization of hole reinforcements by doublers. Struct. Multidisc Optim., 2000, 20, 57-66. [4] Giare, G.S., Shabahang, R., The Reduction of stress concentration around the hole in an isotropic plate using composite materials. Eng. Fract. Mech., 1989, 32(5), 757-166. [5] Savin, G.N., Stress concentration around holes. Pergamon Press, New York, 1961.

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[6] Erickson, P.E., Riley, W.F., Minimizing stress concentrations around circular hole in uniaxially loaded plates. Exp. Mech., 1978, 18(3), 97-100.

[7] Jindal, U.C., Reduction of stress concentration around a hole in a uniaxially loaded plate. J. Strain Anal. Eng., 1983, 18(2), 135-141.

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[8] Huang, J., Venkataraman, S., Rapoff, R.J., Haftka, R.T., Optimization of axisymmetric elastic modulus distributions around a hole for increased strength. J. Struct. Multidisc Optim., 2003, 25(4), 225-36. [9] Muc, A., Ulatowska, A., Local fibre reinforcement of holes in composite multilayered plates. Compos. Struct., 2012, 94, 1413-1419.

M

[10] Sburlati, R., Stress concentration factor due to a functionally graded ring around a hole in an isotropic plate. Int. J. Solids Struct., 2013, 50(22–23), 3649–3658.

ED

[11] Sburlati, R., Atashipour, S.R., Atashipour, S.A., Reduction of the stress concentration factor in a homogeneous panel with hole by using a functionally graded layer. Composite, Part B, 2014 61,

PT

99-109.

[12] Yang Q.Q., Gao C.F., Reduction of the stress concentration around an elliptic hole by using a

CE

functionally graded layer. Acta Mech., 2016, 227, 2427–2437. [13] Gao, X.L., A general solution of an infinite elastic plate with an elliptic hole under biaxial loading. Int.

AC

J. Pres. Ves. & Piping, 1996, 67, 95-104. [14] Muskhelishvili, N.I., Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen, 1963. [15] Timoshenko, S.P., Goodier, J.N., Theory of Elasticity, 3nd edn. McGraw-Hill, New York, 1970. [16] Chao, C.K., Chen, F.M., Revisit of an elliptic hole problem by using complex variable method. J. Chin. Inst. Eng., 2016, 39(1), 121-130. [17] Chen,Y.Z., A new analysis method of an infinite plate containing elliptical hole and applied by

ACCEPTED MANUSCRIPT concentrated forces and a couple. Shanghai J. Mech., 1994, 15(4), 64-66. [18] Lu, A.Z., Chen, H.Y., Qin, Y., Zhang, N., Shape optimization of the support section of a tunnel at great

AC

CE

PT

ED

M

AN US

CR IP T

depths. Comput. Geotech., 2014, 61,190-197.