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A CLOSED-FORM SOLUTION FOR STRESS CONCENTRATION AROUND A CIRCULAR HOLE IN A L

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A CLOSED-FORM SOLUTION FOR STRESS CONCENTRATION AROUND A CIRCULAR HOLE IN A L Powered By Docstoc
					 INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME
                              AND TECHNOLOGY (IJMET)

ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)                                                         IJMET
Volume 4, Issue 5, September - October (2013), pp. 37-48
© IAEME: www.iaeme.com/ijmet.asp
Journal Impact Factor (2013): 5.7731 (Calculated by GISI)                    ©IAEME
www.jifactor.com




A CLOSED-FORM SOLUTION FOR STRESS CONCENTRATION AROUND
    A CIRCULAR HOLE IN A LINEARLY VARYING STRESS FIELD

                              Raghavendra Nilugal1, Dr. M. S. Hebbal 2
        1
            Lecturer, Sri Venkateshwara College of Engineering, Bengaluru, Karnataka, INDIA.
                              2
                                Professor, Dept. of Mechanical Engineering,
                     Basaveshwar Engineering College, Bagalkot, Karnataka, INDIA.


ABSTRACT

        This paper presents a closed-form solution, based on ‘Theory of Elasticity’ to determine the
stress concentration around a circular hole in an infinite isotropic plate subjected to linearly varying
stress. Numerical solutions such as Finite Element Method, Finite Difference Method and Boundary
Element Method can be employed to solve this problem. But these requires are approximate
methods, and requires high skill, computers and computer programs to obtain the solution. Whereas
the method present here does not need computers and computer programs. The equation developed
in the present work can be used to determine the stress field around the circular hole. The results
obtained are compared with FEA results and are in close agreement.

Keywords: Closed-form solution, linearly varying stress field, stress concentration.

1. INTRODUCTION

         The study of distribution of stress field in elastic medium has been going on since 1910 in
different part of the world undertaken by different research teams. Kolosov G. V. is the oldest name
in this field who initiated the study of stress concentration around holes of arbitrary shape and size in
elastic medium, which is subjected to different stresses.
         Cracks in structures often initiate and propagate from the locations of stress or strain
concentration. The stress and strain concentration locations are the critical structural details to
determine the crack initiation, growth and life of engineering structures. Despite of careful detail-
design, practically many structures contain stress and strain concentration due to discontinuities such
as holes. Publication by Pilkey (1997), Young (Young and Budynas, 2002) and Pilkey (2005) are
widely known for data on stress concentration factor for a wide variety of possible specimen
configurations.


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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME

        Shivakumar (1992) studied three dimensional results for plates with straight shank holes
subjected to various types of loadings using a FROTRAN program. They found that the maximum
stress concentration lies at the mid-thickness of the isotropic plate and drops near the free surface.
The formulation of the compatibility conditions on the boundary of an elastic continuum was
presented by Patnaik (2001). Kotousov (2002) presented analytical solutions for (3-D) stress
distribution around typical stress concentrators in an isotropic plate with arbitrary thickness. They
showed that for wide range of thickness to radius ratios stress concentration factor for a circular hole
is slightly perturbed from the plane strain solution. Yang (2008) showed that the location of
maximum stress concentration factor occurs on the mid plane only in thin plates and it moves away
from the mid plane of plate by increasing the plate thickness. Singh (2007) and Xu (2008) published
the Kirsch’s problem for stress-concentration due to a circular hole in a stressed plate. Solution
strategies to elasticity problems to get Beltrami-Michell compatibility equations and Navier’s
equations are published by Sadd (2009) and Saada (2009). Batista (2011) investigated the stress
concentration around a hole in an infinite plate using modified Muskhelishvili complex variable
method. They showed that concave corners of polygonal holes are stress free while convex corners
gave the circumferential stress component and also the stress concentration depends on the radius of
curvature of a convex corner. Mohammadi (2011) presented a general form for the stress
concentration factor in case of biaxial tension and Frobenius series solution for pure shear case to
calculate the stress concentration factor around a hole in an infinite plate made up of functionally
graded material subjected to uniform biaxial tension and pure shear. Sharma (2012) investigated and
reported that by controlling the corner radius of the polygonal hole, the stress concentration can be
kept within some permissible limit. As the corner radii approaches zero, stress concentration factor
approaches infinity.
        However, these literatures do not include information on stress concentration around a
circular hole in a plate subjected to linearly varying stress field. Hence, this paper presents a closed-
form solution to the stress concentration around a circular hole in an infinite isotropic plate subjected
to linearly varying stress field. Numerical methods like Finite Element Method, Finite Difference
Method and Boundary Element Method can be employed to compute the stress concentration around
holes but the advantage of closed-form (analytical) solution is that it provides one step best possible
solution without including the infinite series, integrals or limits. And also it does not require the
computers and computer programs to obtain the solution.

Nomenclature

        Kt             Stress Concentration Factor
       σ max           Maximum Stress in N/mm2
       σ nom           Nominal Stress in N/mm2
       a               Hole Radius in mm
        r              Radius at any Point in the Plate in mm
       θ               Angle in degree
       h               Height of the Plate in mm
       σt              Stress at the Top Edge of the Plate in N/mm2
       σb              Stress at the Bottom Edge of the Plate in N/mm2
       σx, σ y         Normal Stress in Cartesian Co-ordinates in N/mm2
       τ xy            Shear Stress in Cartesian Co-ordinates in N/mm2
       φ               Stress Function


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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
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         σ r , σθ       Normal Stress in Polar Co-ordinates in N/mm2
         τ rθ           Shear Stress in Polar Co-ordinates in N/mm2
         b              Radius very much larger than Hole Radius in mm
         C1 to C16      Constants
         σ0, σu         Uniform Stress in N/mm2

2. STRESS CONCENTRATION AROUND A CIRCULAR HOLE

       The localization of maximum stress due to irregularities or abrupt change of cross section is
known as the stress concentration. Stress concentration is measured by stress concentration factor,
which is theoretically defined as the ratio between the maximum stresses in the neighborhood of a
geometric discontinuity to nominal applied stress. Mathematically it is representing as,

       σ max
Kt =                                                                                    (1)
       σ nom

2.1.     Closed-form Solution
        Redistribution of stress will take place if a small hole is made inside the elastic body
subjected to the stress distribution. However, such a phenomenon of stress concentration is of a
localized character, i.e., the effect of the hole is negligible at distances which are large in comparison
with the dimension of the hole.
        Consider a problem of stress concentration in a rectangular plate with a small circular hole of
radius and subjected to variable tensile stress of intensity at one end to at another end varying
linearly in the x direction as shown in Fig. 1. A small hole made in the middle of the plate will cause
stress redistribution in the plate. Since the hole is small compared with the plate width, the change in
the stress distribution will be localized in the neighborhood of the hole. Hence, the stresses acting far
away from the hole, i.e., at a distance several times the hole diameter, will remain unchanged as in
the plate with no hole. In other words, according to Saint-Venant’s principle, the change in stresses is
negligible at large distances of radius b compared with the radius of the hole a.




                Fig. 1: Plate with a circular hole subjected to linearly variable stress field




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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME

Considering plane stress with no body forces, the stresses with no hole are:

                                                   
(σ x )1 = 
                σt −σb  h    
                        + y  +σb               
                 h      2                                                       (2)
(σ ) = (τ )            =0                          
   y 1          xy 1                               

We have
     ∂ 2φ  σ − σ b   h  
σ x = 21 =  t       + y  +σb                                                     (3)
     ∂y       h  2      

Integrating Equation (3) twice with respect to y. We get,
      σ −σb  3  σt +σb  2
φ1 =  t      y +          y                                                      (4)
      6h          4 

We have, y = r sin θ in polar co-ordinates. Hence Equation (4) becomes,
      σ −σb  3                        σt +σb  2
φ1 =  t      r ( 3sin θ − sin 3θ ) +          r (1 − cos 2θ )                    (5)
      24h                             8 

         Hence the stresses in the problem of plate with no hole subjected to a uniaxial varying tensile
stress field is
          1 ∂φ1 1 ∂ 2φ1
(σ r )1 =       +
          r ∂r r 2 ∂θ 2
              σ −σb                          σt +σb            σt +σb 
∴ (σ r )1 =  t       r ( sin 3θ + sin θ ) +          cos 2θ +                  (6a)
              4h                             4                 4 

         ∂ 2φ1
(σ θ )1 =
         ∂r 2
              σ −σb                           σt +σb                 σt +σb 
∴ (σ θ )1 =  t       r ( 3sin θ − sin 3θ ) −               cos 2θ +             (6b)
              4h                              4                      4 

                ∂  1 ∂φ1 
(τ rθ )1 = −             
                ∂r  r ∂θ 
                σ − σ b                              σt +σb           
∴ (τ rθ )1 = −  t            r ( cos θ − cos 3θ ) +            sin 2θ           (6c)
                4 h                                  4                

        Now suppose that a hole of radius a in the centre at the origin ‘o’ is drilled through the plate.
A new system of stresses σ r , σ θ and τ rθ will be produced inside the plate. These stresses must
satisfy the following boundary conditions:

When r = a , σ r = τ rθ = 0                                                           (7a)

When r = b = ∞ , σ r = (σ r )1 , σ θ = (σ θ )1 (Ragab (1998)) and τ rθ = (τ rθ )1     (7b)


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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME

These stresses can be derived from the biharmonic equation.

        ∂ 2 1 ∂ 1 ∂ 2   ∂ 2φ 1 ∂φ 1 ∂ 2φ 
∇ 4φ =  2 +     +                 +    +           =0                                   (8)
        ∂r  r ∂r r 2 ∂θ 2   ∂r 2 r ∂r r 2 ∂θ 2 
                                                

Assume this unknown stress function φ to be of the same form as the stress function given by
Equation (5), thus

φ ( r , θ ) = f1 ( r ) sin θ + f 2 ( r ) sin 3θ + f3 ( r ) + f 4 ( r ) cos 2θ            (9)

         Substituting Equation (9) in Equation (8), we get,

 d 2 1 d 1   d 2 f1 1 df1 1 f1         d 2 1 d 9   d 2 f 2 1 df 2 9 f 2 
 2+      − 2 2 +         − 2  sin θ +  2 +     −          +      − 2  sin 3θ
 dr  r dr r   dr    r dr r             dr  r dr r 2   dr 2 r dr    r 
   d 2 1 d   d 2 f 3 1 df 3   d 2 1 d          4   d 2 f 1 df 4 4 f 4 
+ 2 +           2 +             + 2 +        − 2   24 +        − 2  cos 2θ = 0
   dr     r dr   dr        r dr   dr     r dr r   dr      r dr   r 
Or
 d 2 1 d 1   d 2 f1 1 df1 f1                               
 2+          − 2 2 +               − 2=0                   
 dr     r dr r   dr           r dr r                       
d  2
         1 d 9   d f 2 1 df 2 9 f 2 
                         2                                     
 2    +      − 2 2 +               − 2 =0                  
 dr     r dr r   dr           r dr   r                     
                                                                                      (10)
 d 2 1 d   d 2 f 3 1 df 3                                  
 2+                 +          =0                          
 dr     r dr   dr 2 r dr 
                                                               
 d2 1 d         4   d 2 f 4 1 df 4 4 f 4                   
 2+          − 2 2 +               − 2 =0                  
 dr     r dr r   dr           r dr   r                     

           The general solution of the above equations are,
                             C                            
f1 ( r ) = C1r + C2 r ln r + 3 + C4 r 3                   
                              r
                                                          
            C5            C7                              
f2 ( r ) =            3
                + C6 r + 3 + C8 r  5
                                                          
             r             r                                                           (11)
f 3 ( r ) = C9 + C10 ln r + C11r + C12 r ln r
                                 2      2                 
                                                          
                             C15                          
f 4 ( r ) = C13 r + C14 r + 2 + C16
                 2       4
                                                          
                             r                            

Hence the function φ ( r , θ ) becomes.
                          C3                   C            C7          
φ =  C1r + C2 r ln r +       + C4 r 3  sin θ +  5 + C6 r 3 + 3 + C8 r 5  sin 3θ
                          r                     r           r           
                                                                                        (12)
                                                                  C         
    + ( C9 + C10 ln r + C11r + C12 r ln r ) +  C13 r 2 + C14 r 4 + 15 + C16  cos 2θ
                                2         2
                                                                      2
                                                                   r        

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Using Equation (12) and the stress-Airy Stress Function relationships

     1 ∂φ 1 ∂ 2φ
σr =     +                                                                               (13a)
     r ∂r r 2 ∂θ 2

    ∂ 2φ
σθ = 2                                                                                   (13b)
    ∂r

           ∂  1 ∂φ 
τ rθ = −                                                                               (13c)
           ∂r  r ∂θ 

We get,
       C 2C                      10C          12C            
σ r =  2 − 3 3 + 2C4 r  sin θ −  3 5 + 6C6 r + 5 7 + 4C8 r 3  sin 3θ
       r   r                     r             r             
                                                                                         (14a)
             C                                        6C    4C 
           +  10 + 2C11 + C12 (1 + 2 ln r )  −  2C13 + 415 + 216  cos 2θ
                2
             r                                         r     r 

      C        2C                    2C             12C              
σ θ =  2 + 3 3 + 6C4 r  sin θ +  3 5 + 6C6 r + 5 7 + 20C8 r 3  sin 3θ
       r   r                     r             r              
                                                                                         (14b)
              C                                                      6C 
           +  − 10 + 2C11 + C12 ( 3 + 2 ln r )  +  2C13 + 12C14 r 2 + 415  cos 2θ
                 2
              r                                                       r 

           C     2C                     6C              12C              
τ rθ = −  2 − 3 3 + 2C4 r  cos θ −  − 3 5 + 6C6 r − 5 7 + 12C8 r 3  cos 3θ
          r   r                     r               r              
                                                                                         (14c)
                                6C    2C 
            +  2C13 + 6C14 r 2 − 415 − 216  sin 2θ
                                 r     r 

Where C’s are constants.
Since for r → ∞ , σ r , σ θ and τ rθ must be finite.

∴ C8 = C12 = C14 = 0                                                                      (15)

The remaining constants are determined from boundary conditions given by Equations (7).
Here single terms are took separately from the Equations (14) and solved for the constants for ease of
evaluation.
Consider the terms having sin θ and cos θ as coefficient from Equations (14) and apply boundary
conditions from Equations (7). We get,

 C2 2C3      
 2 − 4 + 2C4  = 0                                                                     (16a)
 a   a       

   C2 2C                C2 2C        
−  2 − 43 + 2C4  = 0 ⇒  2 − 43 + 2C4  = 0                                           (16b)
  a    a               a    a        

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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
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 C2 2C3                  σt −σb             C2 2C3        σt −σb 
 2 − 4 + 2C4  r sin θ =          r sin θ ⇒  2 − 4 + 2C4  =        (16c)
b    b                   4h                b    b         4h 

 C2 2C3                  σt −σb                  C 2 2 C3          σ t − σ b  (16d)
 2 + 4 + 6C4  r sin θ =             3 r sin θ ⇒  2 + 4 + 6C4  = 3            
b    b                   4h                     b     b            4h 

   C2 2C                      σ −σb                C2 2C3        σ t − σ b  (16e)
−  2 − 43 + 2C4  r cos θ = −  t         r cos θ ⇒  2 − 4 + 2C4  =          
   b   b                      4h                   b   b         4h 

          a    
By taking  = 0  and solving Equations (16) we get,
           b   

                   σ −σb          σt −σb 
C2 = 0 , C3 = a 4  t      , C4 =                                                  (17)
                   8h             8h 

        Similarly by considering the terms sin 3θ and cos 3θ , 1 and cos 2θ and sin 2θ as
coefficients from Equations (14) and applying boundary conditions from Equations (7) we get,

          σ −σb            σt −σb            6  σt −σb 
C5 = a 4  t      , C6 = −          , C7 = − a                                   (18)
          8h               24h                  12h 


             σ +σb           σt +σb 
C10 = − a 2  t      , C11 =                                                       (19)
             4               8 


         σ +σb             4  σt +σb           2  σt +σb 
C13 = −  t      , C15 = − a           , C16 = a                                 (20)
         8                    8                   4 

Respectively.
Substitute Equations (17), (18), (19) and (20) in Equations (14) and by solving we get the stress
distribution around circular hole subjected to the linearly variable stress field in polar co-ordinate
system.

          σt −σb  a                 σ t − σ b   4a 5a 
                           4                                  6     4
    σr =          1 − 4  r sin θ +             1 + 6 − 4  r sin 3θ
          4h   r                    4h               r      r 
∴                                                                                    (21a)
              σ + σ b   a   σ t + σ b   3a 4a 
                              2                           4      2
           + t         1− 2  +             1 + 4 − 2  cos 2θ
              4  r   4                           r       r 

      σt −σb       a4              σ t − σ b   4a a 
                                                            6    4
σθ =           3 + r 4  r sin θ −  4h  1 + r 6 − r 4  r sin 3θ
      4h                                                    
                                                                                     (21b)
          σ + σ b   a   σ t + σ b   3a 
                             2                          4
       + t         1 + 2  −               1 + r 4  cos 2θ
          4  r   4                                 


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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
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         σt −σb  a                  σ t − σ b   4a 3a 
                            4                                6  4
τ rθ = −          1 − 4  r cos θ +              1 − 6 + 4  r cos 3θ
         4h   r                     4h              r   r 
                                                                                   (21c)
             σ + σ b   3a 2a 
                                4     2
          − t          1 − 4 + 2  sin 2θ
             4             r     r 

2.2.     Localized Stress Concentration Factor
        From the present work, nominal stresses cannot be obtained that are used to find the stress
concentration factor given by the Equation (1). So a new term can be used to calculate the stress
concentration factor and it is called as localized stress concentration factor. Localized SCF is defined
as the ratio of stress at a point with hole to the stress at the same point without hole. Mathematically
it can be expressed as:
                                              Stress With Hole at a Point
 Localized Stress Concentration Factor =                                              (22)
                                             Stress Without Hole at a Point

3.      VERIFICATION OF THE PROPOSED CLOSED-FORM SOLUTION

       Conditions of the stress concentration around circular hole subjected to uniform stress are
applied to the new Equations (21) for verification of the present work. Uniform stress case is the
well-known method. Nominal condition for uniform stress is σ t = σ b .
       Stress concentration equations around a circular hole subjected to uniform stress given by
Kirsch’s problem are

      σ      a2   σ          3a 4 4a 2     
σ r =  0  1 − 2  +  0      1+ 4 − 2         cos 2θ                         (23a)
       2     r   2            r    r       

       σ0   a2   σ 0    3a 4 
σ θ =    1 + 2  −    1 + 4  cos 2θ                                         (23b)
       2    r   2        r 

         σ       3a 4 2a 2 
τ rθ = −  0   1 − 4 + 2  sin 2θ                                                (23c)
          2       r    r 

Apply the condition σ t = σ b = σ u in the Equations (24) we get,

      σ      a2   σ       3a 4 4a 2         
σ r =  u  1 − 2  +  u  1 + 4 − 2             cos 2θ                        (24a)
       2     r   2         r    r           

     σ      a2   σ             3a 4 
σθ =  u  1 + 2  −  u        1 + 4  cos 2θ                                  (24b)
      2     r   2               r 

         σ       3a 4 2a 2 
τ rθ = −  u   1 − 4 + 2  sin 2θ                                                (24c)
          2       r    r 


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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME

       By comparing Equations (23) and (24), it can be said that the present method gives well
agreement with the Kirsch’s problem. Hence present method can be used to determine the stress
concentration around a circular hole subjected to linearly varying stress.

4.     NUMERICAL EXAMPLE

        Consider an example to solve the Equations (21) to observe the behavior of stresses around
a hole subjected to the linearly varying stress field and to compare the results obtained by the
equations with ANSYS by using following parameters:
Plate size =
Height, h =
Hole radius,
Normal Stress,
Normal Stress,
Young’s Modulus,
Poisson’s Ratio,
        Same values are taken for the model generation in ANSYS. PLANE82 element is used for
meshing.
        The parameters prescribed above are substituted in the Equations (21) for the analysis. Half-
symmetry of the model shown in Fig. 1 is considered for the evaluation.
At the inner side of the hole there is no applied stress is acting, hence    is zero at       (Equation
(7a)) and it is minimum along           ° and           ° as there is no circumferential stress is acting
on the plate. Radial stress approaches to the applied stress at the edge of the plate as       increases
as shown in Fig. 2.



                                        70

                                        60
            Radial Stress, σr (N/mm2)




                                        50                                                                      θ=-90

                                        40                                                                      θ=-60
                                                                                                                θ=-30
                                        30
                                                                                                                θ=0
                                        20
                                                                                                                θ=30
                                        10
                                                                                                                θ=60
                                        0                                                                       θ=90
                                             0         1      2       3          4     5       6       7

                                                                          r/a


                                                 Fig. 2: Radial Stress ( ) vs.       at different Angles ( ).




                                                                                45
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME

                                                              250



                         Circumferential Stress, σθ (N/mm2)
                                                              200

                                                              150                                                                                     θ=-90
                                                                                                                                                      θ=-60
                                                              100
                                                                                                                                                      θ=-30
                                                               50                                                                                     θ=0
                                                                                                                                                      θ=30
                                                                   0
                                                                                                                                                      θ=60
                                                               -50                                                                                    θ=90

                                                              -100
                                                                       0           1      2         3             4          5       6        7
                                                                                                         r/a

                                                                   Fig. 3: Circumferential Stress (             ) vs.       at different Angles ( )

        Circumferential stress ( ) for          ° and           ° at the edge of the hole is maximum
because force flow lines tend to modify their path around hole and it concentrate around edge of the
hole perpendicular to the load direction, as the radius increases stress decreases and at the end of the
plate it reaches the applied stress as shown in the Fig. 3. Shear stress ( ) is the resultant of the
radial stress ( ) and the circumferential stress ( ), hence it has the nature of both the normal
stresses as shown in the Fig. 4.


                                                              50
                                                              40
           Shear Stress, τrθ (N/mm2)




                                                              30
                                                                                                                                                        θ=-90
                                                              20
                                                                                                                                                        θ=-60
                                                              10
                                                                                                                                                        θ=-30
                                                              0
                                                                                                                                                        θ=0
                                                  -10
                                                  -20                                                                                                   θ=30

                                                  -30                                                                                                   θ=60
                                                  -40                                                                                                   θ=90
                                                  -50
                                                                   0           1         2          3             4          5        6           7
                                                                                                         r/a

                                                                           Fig. 4: Shear Stress (       ) vs.           at different Angles ( )




                                                                                                           46
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME


                                              3.5
                                               3
                  Concentration Factor        2.5
                    Localized Stress

                                                                                                                             θ=-90°
                                               2                                                                             θ=-60°
                                              1.5                                                                            θ=-30°
                                               1                                                                             θ=30°
                                              0.5                                                                            θ=60°

                                               0                                                                             θ=90°

                                             -0.5
                                                    0           1     2          3   r/a 4         5        6         7


                                                        Fig. 5: Localized SCF vs.         at different Angles ( )

        Stress concentration factor is maximum at the hole edge (              ) because the stress flow
lines tend to modify their paths around the discontinuities (hole), this increases the stress around hole
and stress concentration factor. As the radius increases, stress reduces; this decreases the stress
concentration factor. At the plate edge stress is nearly equal to the applied stress hence the stress
concentration factor is around one as shown in Fig. 5.
        Fig. 6 shows the comparison of present work with the ANSYS of normal stress ( ) versus
angle at radius          . Comparison shows that the results are well agreed. Here only one
comparison is presented, but the comparison of remaining stresses for different radii is also give the
close results.


                                   250

                                   200
       Normal Stress, σx (N/mm2)




                                   150
                                                                                                                      Present Work
                                   100                                                                                for r=25

                                    50                                                                                FEA for r=25

                                         0

                                   -50
                                             -180        -120       -60          0         60       120         180
                                                                          Angle, θ (Degrees)

                                         Fig. 6: Comparison of Normal Stress ( ) vs. Angle ( ) with ANSYS




                                                                                     47
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME

5.     CONCLUSIONS

        New equations for stress distribution around circular hole in an isotropic infinite plate in a
linearly varying stress field are formulated using closed-form solution and it is extended from the
Kirsch’s problem for ‘Stress-concentration due to a circular hole in a stressed plate’. Newly found
equations (21) show that it satisfies the boundary conditions from Fig. 2 to Fig. 4. The maximum
stress concentration is found at the edge of the hole at an angle of 90° from the load direction.
Localized stress concentration factor is calculated and it is observed that the localized SCF is
maximum at the hole and it decreases as radius increases. At the end of the plate localized SCF is
equal to 1 that means stresses are converged. The results obtained from new equations are compared
with the ANSYS and it is in closed agreement.

REFERENCES
[1]    Pilkey W. D., Peterson’s stress concentration factors Second edition, John Wily and Sons,
       United States, America, 1997.
[2]    Young W. C., Budynas R. G., Roark’s formulas for stress and strain Seventh edition,
       McGraw-Hill, United States, America, 2002.
[3]    Pilkey W. D., Formulas for stress, strain, and structural matrices Second edition, John Wily
       and Sons, Hoboken, New Jersey, 2005.
[4]    Shivakumar K. N., Newman J. C., Jr., Stress concentrations for straight-shank and
       countersunk holes in plates subjected to tension, bending, and pin loading, NASA TP-3192,
       1992.
[5]    Patnaik S. N., Hopkins D. A., Stress formulation in three-dimensional elasticity, NASA TP-
       210515, 2001.
[6]    Kotousov, Wang C. H., Three-dimensional stress constraint in an elastic plate with a notch,
       International Journal of Solids and Structures, 39, 4311-4326, 2002.
[7]    Yang Z., Kim C. B., Cho C., Beom H. G., The concentration of stress and strain in finite
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       Structures,       45, 713-731, 2008.
[8]    Singh S., Theory of elasticity Fourth edition, Khanna Publishers, Delhi, India, 2007.
[9]    Xu Z., Applied Elasticity New Age International Publishers, New Delhi, India, 2008.
[10]   Sadd M. H., Elasticity theory, applications, and numerics Elsevier, United States, America,
       2009.
[11]   Saada A. S., Elasticity theory and applications Second edition, J. Ross Publishers, U. S. A.,
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[12]   Batista M., On the stress concentration around a hole in an infinite plate subject to a uniform
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[13]   Mohammadi M., John R. Dryden, Liying Jiang, Stress concentration around a hole in a
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[14]   Sharma D. S., Stress distribution around polygonal holes, International Journal of Mechanical
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[15]   Ragab A. R., Bayoumi S. E., Engineering solid mechanics-fundamentals and applications
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[16]   Akash.D.A, Anand.A, G.V.Gnanendra Reddy and Sudev.L.J, “Determination of Stress
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       Technology (IJMET), Volume 4, Issue 2, 2013, pp. 373 - 382, ISSN Print: 0976 – 6340,
       ISSN Online: 0976 – 6359.
                                                  48

				
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