Stress concentration around an elliptical hole in a large rectangular plate subjected to linearly varying in-plane loading on two opposite edges

Journal Pre-proofs Stress concentration around an elliptical hole in a large rectangular plate subjected to linearly varying in-plane loading on two opposite edges Ashish Patel, Chaitanya K. Desai PII: DOI: Reference:

S0167-8442(19)30407-0 https://doi.org/10.1016/j.tafmec.2019.102432 TAFMEC 102432

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Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

24 July 2019 2 December 2019 2 December 2019

Please cite this article as: A. Patel, C.K. Desai, Stress concentration around an elliptical hole in a large rectangular plate subjected to linearly varying in-plane loading on two opposite edges, Theoretical and Applied Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.tafmec.2019.102432

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Stress concentration around an elliptical hole in a large rectangular plate subjected to linearly varying in-plane loading on two opposite edges Ashish Patela,∗, Chaitanya K. Desaib a

Assistant Professor, Department of Mechanical Engineering, Chhotubhai Gopalbhai Patel Institute of Technology, Uka Tarsadia University, Gopal-Vidyanagar, Maliba Campus, Surat (Gujarat) - 394350, India b Associate Professor, Department of Mechanical Engineering, C. K. Pithawala College of Engineering and Technology, Surat, Gujarat, 395007, India

Abstract Stress concentration in plates due to geometric irregularities such as holes and cracks is a crucial factor in design. The main objective of this paper is to derive the formula for tangential stress concentration factor around an elliptical hole in a large rectangular plate subjected to linearly varying inplane loading on two opposite edges. The problem is investigated by the two-dimensional theory of elasticity using Muskhelishvili’s complex variable method. The stress functions are evaluated using the conformal mapping method and Cauchy’s integral formula. The stress functions thus obtained are coded to compute the non-dimensional stress components in the plate. The stress concentration factors at the edge of the elliptical hole are compared for different aspect ratios under various in-plane loading conditions. The results indicate that the formula obtained for tangential stress matches to ∗

Corresponding author Email addresses: [email protected] (Ashish Patel), [email protected] (Chaitanya K. Desai)

Preprint submitted to Theoretical and Applied Fracture Mechanics

December 5, 2019

the Kang’s [1] formula when elliptical hole is reduced to a circular hole, by taking its aspect ratio equal to 1. Furthermore, the classical solutions proposed by Inglis and Kirsch are easily obtained as special cases of the problem presented in this work, which confirms the efficacy of the solution. A plane stress finite element model is prepared in ABAQUS and results are compared with present method for a particular case of linearly varying inplane loading. Keywords: Elliptical hole, Muskhelishvili’s complex variable method, Cauchy’s integral formula, stress concentration factor 1. Introduction Holes and openings are common in engineering structures for providing easy access to fasteners, minimizing weight, facilitating maintenance and many other service requirements. In practice different shape of holes are used for different applications. For example, manhole of any pressure vessel is either circular or elliptical while the window or door of an airplane is rectangular hole having some radius at corners. And, in some cases, holes are in the form of flaws in structures and this could cause serious problems of stress concentration. Particularly, when a structure is subjected to cyclic loads, cracks may initiate and grow in the vicinity of stress concentration. In order to ensure safety and reliability, it is necessary to predict the behavior of stress pattern around holes of different shapes under the application of various types of loading conditions. The first classic solution of stress field around a circular hole in an infinite plate subjected to uniaxial tension is presented by Kirsch [2] using real

2

variables. Since then, problems involving stress distribution around holes received a considerable amount of interest in literature. Inglis [3] worked out the stress distribution around the boundary of an elliptical hole in an infinite plate subjected to uniformly distributed axial load. Gao [4] proposed solution for infinite isotropic plate with elliptical hole. He introduced arbitrary biaxial loading condition to avoid superposition. By choosing appropriate biaxial loading factor, different types of loading conditions at infinity can be applied. Exhaustive literature on exact solutions of stress field around holes of different shapes in infinite isotropic and anisotropic plate for uniaxial, biaxial and shear edge loadings is presented by Savin [5] and Lekhnitski [6] using Muskhelishvili’s [7] complex variable method, which is a very powerful and elegant tool to solve two-dimensional problems of elasticity. Ukadgaonker and Awasare [8, 9, 10, 11] used Schwarz alternating technique to obtain the stress distribution around holes of various shapes (circular, elliptical, triangular, rectangular) with rounded corners using successive approximations. Simha and Mohapatra [12] obtained stress concentration around irregular holes using complex variable method. The stress field around polygonal holes in infinite and finite plate for different material properties and loading conditions is presented by Sharma [13, 14, 15], Chauhan and Sharma [16], Patel and Sharma [17]. The analytical solutions for voids and inclusions embedded in planes subjected to out-of-plane loading are also considered in literature [18, 19, 20]. Problems with similar settings are studied within strain gradient and couple stress elasticity [21, 22]. Functionally graded materials utilize the advantage of gradual spatial variation in elastic properties. The solutions of

3

stress concentration around various shaped holes in functionally graded (FG) plate is obtained by many researchers [23, 24, 25, 26, 27, 28]. Most research is focused on study of stress distribution around holes of various shapes in rectangular plates subjected to uniaxial, biaxial and shear loadings, but limited work is reported to consider the effect of linearly varying loading. Few problems on buckling of plates subjected to linearly varying in-plane loads are available in literature [29, 30, 31, 32]. Kang [1] presented the elastostatic solution of large rectangular plate with a central circular hole subjected to linearly varying in-plane stresses on two opposite edges. He introduced a loading factor α to avoid superposition. By choosing appropriate loading factor α, various types of loading condition can be applied. In the present work, an attempt is made to study the stress distribution around an elliptical hole in a large rectangular plate subjected to linearly varying loading on its two opposite edges. This study can be considered as an extension to Kang’s work, where the central circular hole is replaced with a central elliptical hole. A simple formula to calculate stress concentration factor (SCF) around elliptical hole is obtained for the case of linearly varying in-plane load. The formula derived uses three parameters, namely loading factor α, geometric ratio β, and the aspect ratio γ. The loading factor α facilitates the implementation of various in-plane loading conditions on the two opposite edges of the rectangular plate [1]. The geometric ratio β = a/h gives the relative size of elliptical hole to the height of the plate. However, the dimensions of elliptical hole are assumed small in comparison with the height h of the plate and thus the solution obtained is approximate and valid only for small values of geometric ratio. The aspect ratio γ = b/a is the

4

ratio of semi-axes of the elliptical hole. To the best of author’s knowledge, stress functions for a perforated rectangular plate with a central elliptical hole subjected to linearly varying in-plane loads are not reported in the literature. The paper is organized in the following manner. In the next section, the geometry of the plate, hypothesis and loading conditions are described briefly. In Sec. 3, complex variable method is used to determine the stress functions around the elliptical hole while a method to obtain stress components from these stress functions is described in Sec. 4. The variation in non-dimensional tangential stress around the edge of elliptical hole for different loading factors and aspect ratios is presented in Sec. 5. To confirm the validity of present method, comparison with FE results is made in Sec. 6. The stress concentration factors around the elliptical hole for various inplane loading conditions are obtained in Sec. 7. Salient conclusions from the work are enumerated in Sec. 8. 2. Problem definition A rectangular plate as shown in Fig. 1(a) of length L and height h has an elliptical hole of semi-axes a and b, located at its center [1]. The plate is assumed to be infinitely long (L → ∞). The dimensions of elliptical hole are small in comparison with the height h of the rectangular plate such that the state of stress at large distances from the hole is not disturbed by the presence of hole. With these assumptions, the solution obtained is approximate and valid only when the size of hole is small in comparison to the size of plate. The semi-axes of elliptical hole are parallel to sides of the rectangular plate. The material of the plate is assumed to be homogeneous, isotropic and linear 5

elastic. The origins of rectangular (x, y) and polar (r, θ) coordinate systems are located at the center of the plate. The x-axis of the rectangular coordinate system is along the axis of the elliptical hole. In the present work, the edge of the elliptical hole is assumed to traction free. The plate is subjected to linearly varying in-plane loading on two opposite edges, such that the intensity of compressive stress at y = −h/2 is σ0 and the intensity of tensile stress at y = h/2 is ασ0 (see Eq. (7)). Here α is a loading factor and by changing α, various cases of loading are obtained (see Fig. 1(b)). For example, α = −1 corresponds to the case of uniformly distributed compressive traction force. When α = 0, the traction force varies linearly from −σ0 at y = −h/2 to 0 at y = h/2. When α = 1, it is the case of pure in-plane bending moment. With other values of α in the range −1 < α < 1, it is a combination of bending and compression. 3. Determination of stress functions around the elliptical hole Plane elasticity problems can be solved using complex variable method. The method involves determination of two complex potential functions φ(z), ψ(z) of the complex variable z = x + iy. Particular forms of these potential functions exist for regions of different topology. The Kolosov-Muskhelishvili formulas for determining stress field in Cartesian co-ordinates are given by σx + σy = 2[φ0 (z) + φ0 (z)] = 4Re[φ0 (z)], 00

(1)

0

σy − σx + 2iτxy = 2[¯ z φ (z) + ψ (z)], where φ(z) and ψ(z) are analytic functions of a complex variable also commonly referred to as the K-M potentials. The solution to particular problems 6

 0

 0 y

z r

b

h h

θ

oρ

x

a

--

0

--

L

0

(a)

 0

0.5 0

0

0.5 0

0

 0

 0

 0

 0

 0

  1

  0.5

 0

  0.5

 1

(b) Figure 1: (a) A rectangular plate with a central elliptical hole subjected to generalized in plane loading, and (b) examples of in-plane loading σx along the edge x = −L/2.

is then reduced to finding the appropriate potentials or stress functions that satisfy the boundary conditions. Since the rectangular plate contains an elliptical hole, which is a noncircular discontinuity, the method of conformal transformation is used. The region exterior to the elliptical hole in physical z-plane is mapped to a region 7

exterior to the unit circle in parametric ζ-plane as shown in Fig. 2. Any point z = x + iy = reiθ outside the elliptical hole in physical plane can be mapped to a corresponding point ζ = ξ + iη = ρeiϕ outside the unit circle (ρ = 1) in parametric plane by using the mapping function [7]:   m z = ω(ζ) = R ζ + , ζ

(2)

where R and m are determined by the semi-axes a and b as R=

y r b o

θ

a+b a−b , m= . 2 a+b

(3)

η

𝑧

ρ 𝑧 = 𝜔(ζ ) x

1

o

a

φ

ζ ξ

ζ-plane

z-plane

Figure 2: Complex mapping for infinite plane with elliptical hole.

The polar coordinates (ρ, ϕ) in parametric ζ-plane are elliptical coordinates in physical z-plane. Indeed the equation corresponding to the boundary of elliptical hole in physical plane is given by ρ = 1. By using transformation laws, the stresses in curvilinear coordinates can be written as [13]  0  φ (ζ) σρ + σϕ = 4Re 0 = 4Re[Φ(ζ)], ω (ζ) 2ζ 2 σϕ − σρ + 2iτρϕ = [ω(ζ)Φ0 (ζ) + ψ 0 (ζ)]. 2 0 ρ ω (ζ) 8

(4)

The boundary condition in the parametric plane is given by [13] f (t) = φ(t) +

ω(t) ω 0 (t)

φ0 (t) + ψ(t),

(5)

where t = eiϕ is the boundary value of ζ on the unit circle. The complex stress functions φ(ζ) and ψ(ζ) are obtained in two stages [9]. In the first stage, the rectangular plate is assumed to be homogeneous without any hole or discontinuity. The corresponding complex stress functions φ1 (ζ) and ψ1 (ζ) are obtained by mapping the physical z-plane on to the parametric ζ-plane. From these stress functions, the boundary condition at the imaginary elliptical boundary is determined. In the second stage, the two complex stress functions φ2 (ζ) and ψ2 (ζ) are obtained by application of negative of this boundary condition on the elliptical hole boundary. Addition of the stress functions obtained in both stages yields the actual stress functions near the elliptical hole as φ(ζ) = φ1 (ζ) + φ2 (ζ),

(6)

ψ(ζ) = ψ1 (ζ) + ψ2 (ζ). 3.1. First stage In the first stage, the rectangular plate is assumed to be homogeneous without any hole subjected to linearly varying loading. Therefore, the stress distribution throughout the plate can be written in terms of Kang’s [1] linearly varying loading conditions as follows: σx∞ =

σ0 (1 + α) σ0 (α − 1) y+ , h 2 (7)

σy∞ = 0, ∞ τxy = 0.

9

The corresponding stress functions φ1 (z) and ψ1 (z) are determined using Eq. (1) and Eq. (7) as σ0 z(α − 1) iσ0 z 2 (α + 1) − , 8 8h σ0 z(α − 1) iσ0 z 2 (α + 1) ψ1 (z) = − + . 4 8h φ1 (z) =

(8)

The parametric form of these stress functions can be obtained by substituting the mapping function z = ω(ζ) given by Eq. (2) in Eq. (8). Therefore,    2 2  R(ζ 2 + m) R (ζ + m)2 φ1 (ζ) = Λ1 − iΛ2 , ζ ζ2    2 2  R(ζ 2 + m) R (ζ + m)2 ψ1 (ζ) = −2Λ1 + iΛ2 , ζ ζ2

(9)

σ0 (α − 1) σ0 (α + 1) and Λ2 = . 8 8h Using the parametric form of stress functions given in Eq. (9) and the mapwhere Λ1 =

ping function given in Eq. (2), the boundary condition at the imaginary elliptical boundary in the homogeneous plate (or the boundary condition on the unit circle, ζ = t = eiϕ in the parametric plane) is obtained as f1 (t) = φ1 (t) +

ω(t) ω 0 (t)

φ01 (t) + ψ1 (t)

   2 1 1 2 2 = 2Λ1 R(1 − m) t − − iΛ2 R (1 − m) t − . t t

(10)

3.2. Second stage In the second stage, the rectangular plate is assumed to be free from any far field loading. Only the negative of boundary condition f1 (t) (Eq. (10)) obtained in the first stage is applied on the elliptical boundary in physical plane and on the unit circle (ζ = t = eiϕ ) in parametric plane. Hence the

10

boundary condition in second stage is given by f2 (t) = −f1 (t) which gives φ2 (t) +

ω(t)

φ02 (t) + ψ2 (t) = −f1 (t),

(11)

ω(t) 0 φ (t) + ψ2 (t) = −f1 (t). ω 0 (t) 2

(12)

ω 0 (t)

with its conjugate form φ2 (t) +

1 dt and evaluating 2πi t − ζ the Cauchy’s contour integrals [9], the stress functions near the elliptical Multiplying both sides of Eqs. (11) and (12) by

boundary are found as 1 φ2 (ζ) = 2πi

I

f1 (t) dt t−ζ    2  R(m − 1) R (m − 1)2 = −2Λ1 + iΛ2 , ζ ζ2 I 1 f1 (t) ω(ζ)φ02 (ζ) ψ2 (ζ) = dt − 2πi t−ζ ω 0 (ζ)   2R(m − 1)(1 + m + mζ 2 − ζ 2 ) = Λ1 ζ(m − ζ 2 )  2  R (m − 1)2 (2 + m + 2mζ 2 − ζ 2 ) − iΛ2 . ζ 2 (m − ζ 2 )

(13)

The complex stress functions φ2 (ζ) and ψ2 (ζ) approach zero as ζ → ∞. This is expected as these stress functions are due to the presence of elliptical hole and the perturbation produced by elliptical hole should vanish far away from it. The actual complex stress functions φ(ζ) and ψ(ζ) are obtained by adding the stress functions obtained in first stage (Eq. (9)) and second stage (Eq. (13))

11

as follows: φ( ζ) = φ1 (ζ) + φ2 (ζ)    2  R(ζ 2 − m + 2) R (m − 1)2 − R2 (ζ 2 + m)2 = Λ1 + iΛ2 , ζ ζ2 ψ( ζ) = ψ1 (ζ) + ψ2 (ζ)   2R(m − 1)(1 + m + mζ 2 − ζ 2 ) 2R(ζ 2 + m) = Λ1 − ζ(m − ζ 2 ) ζ  2  2 2 2 2 2 R (m − 1) (2 + m + 2mζ − ζ ) R (ζ + m)2 − iΛ2 − . ζ 2 (m − ζ 2 ) ζ2

(14)

4. Stress components in rectangular plate The stress components in rectangular plate can be obtained by separating the real and imaginary parts given in Eq. (4). Thus the stress components turn out be " σρ = Re 2Φ(ζ) − " σϕ = Re 2Φ(ζ) + " τρϕ = Im

ζ2 ρ2 ω 0 (ζ)

ζ2 ρ2 ω 0 (ζ) ζ2 ρ2 ω 0 (ζ)

# (ω(ζ)Φ0 (ζ) + ψ 0 (ζ)) , # (ω(ζ)Φ0 (ζ) + ψ 0 (ζ)) ,

(15)

# (ω(ζ)Φ0 (ζ) + ψ 0 (ζ)) ,

where φ0 (ζ) , ω 0 (ζ) φ00 (ζ)ω 0 (ζ) − φ0 (ζ)ω 00 (ζ) Φ0 (ζ) = . (ω 0 (ζ))2 Φ(ζ) =

12

(16)

The derivatives of stress functions φ(ζ) and ψ(ζ) used in Eq. (15) and Eq. (16) can be obtained from Eq. (14) as follows    2 4  R(ζ 2 + m − 2) 2R (ζ − 2m + 1) 0 φ (ζ) = Λ1 − iΛ2 , ζ2 ζ3    2 4  2R(m − 2) 2R (ζ + 6m − 3) 00 φ (ζ) = −Λ1 − iΛ2 , ζ3 ζ4   2R(−ζ 6 + (m2 + m + 1)ζ 4 + (m3 − 2m2 + m − 3)ζ 2 + m) 0 ψ (ζ) = Λ1 ζ 2 (m − ζ 2 )2  2 8  2R (ζ − 2mζ 6 + (−2m3 + 5m2 − 4m + 1)ζ 4 + iΛ2 ζ 3 (m − ζ 2 )2  2  2R ((6m − 4)ζ 2 − 3m2 + 2m) + iΛ2 . ζ 3 (m − ζ 2 )2 (17) Similarly, the derivatives of mapping function ω(ζ) used in Eq. (15) and Eq. (16) can be obtained from Eq. (2) as follows   m 0 ω (ζ) = R 1 − 2 , ζ   2m ω 00 (ζ) = R . ζ3

(18)

For simplification, defining the geometric ratio β = a/h and aspect ratio γ = b/a. Therefore from Eq. (3), R = βh(1 + γ)/2 and m = (1 − γ)/(1 + γ). The non-dimensional stress components σρ /σ0 , σϕ /σ0 and τρϕ /σ0 in the plate can be easily computed by coding the following procedure. 1. Choose a point (ρ, ϕ) where the stress components are to be calculated. Calculate ζ = ρeiϕ . 2. To obtain non-dimensional stress components, set σ0 = 1 and h = 1. 3. Choose a suitable value of loading factor α (−1 ≤ α ≤ 1), geometric ratio β, and aspect ratio γ. Obtain R = βh(1 + γ)/2 and m = (1 − γ)/(1 + γ). 13

4. Evaluate the constants Λ1 = σ0 (α − 1)/8 and Λ2 = σ0 (α − 1)/8h. 5. Evaluate the mapping function ω(ζ) from Eq. (2) and its derivatives ω 0 (ζ), ω 00 (ζ) from Eq. (18). 6. Evaluate the stress functions φ(ζ), ψ(ζ) from Eq. (14) and their derivatives φ0 (ζ), φ00 (ζ), ψ 0 (ζ) from Eq. (17). Similarly, evaluate the functions Φ(ζ), Φ0 (ζ) from Eq. (16). 7. Evaluate the stress components from Eq. (15). 5. Tangential stress distribution around elliptical hole The stress components near the elliptical hole can be determined by substituting the stress functions φ(ζ) and ψ(ζ) from Eq. (14) into Eq. (4). At the edge of the elliptical hole, ζ = eiϕ and σρ = 0. Therefore from Eq. (4), the tangential stress around the elliptical hole is obtained as

 φ0 (ζ) σϕ = 4Re 0 ω (ζ) ζ=eiϕ  (1 − α)(m2 + 2 cos 2ϕ − 2m − 1) = σ0 2(m2 − 2m cos 2ϕ + 1)  R(1 + α)(m − 1)(sin 3ϕ − 2m sin ϕ − sin ϕ) + . h(m2 − 2m cos 2ϕ + 1) 

(19)

From Eq. (19), the non-dimensional tangential stress at the edge of elliptical hole becomes σϕ (1 − α)(m2 + 2 cos 2ϕ − 2m − 1) = σ0 2(m2 − 2m cos 2ϕ + 1) R(1 + α)(m − 1)(sin 3ϕ − 2m sin ϕ − sin ϕ) + . h(m2 − 2m cos 2ϕ + 1)

14

(20)

The variation of non-dimensional tangential stress σϕ /σ0 around the edge of elliptical hole with parametric angle ϕ/2π for aspect ratios γ = 0.25 (a weakly elliptical hole), γ = 1 (circular hole) and γ = 4 (a strongly elliptical hole) is shown in Fig. 3, Fig. 4 and Fig. 5 respectively. The above results are presented for the geometric ratio β = 0.01. From the figures, it can be concluded that the magnitude of non-dimensional tangential stress around the hole is maximum at ϕ = π/2 and/or ϕ = 3π/2. The magnitude of nondimensional tangential stress around the hole decreases with loading factor α and increases with aspect ratio γ. Also, as expected for the case of uniform compression (α = −1), the maximum magnitude of non-tangential stress is equal to 1 + 2b/a which is reflected in Fig. 3, Fig. 4 and Fig. 5 respectively. Kang [1] analyzed the exact stresses, strains and displacements in a rectangular plate with a central circular hole (a = b) under linear varying loading using Airy stress function method. Substituting a = b in Eq. (3) gives m = 0 and R = a = βh. For m = 0 and R = βh, the non-dimensional tangential stress given in Eq. (20) becomes σϕ 1 = (1 − α)(2 cos 2ϕ − 1) + β(1 + α)(sin ϕ − sin 3ϕ), σ0 2

(21)

which exactly matches to the result obtained by Kang [1]. Muskhelishvili presented the solution of an infinite plate weakened by an elliptical hole under uniform loading (α = −1) using complex variable method [7]. Substituting α = −1 into the non-dimensional tangential stress in Eq. (20) results in σϕ m2 + 2 cos 2ϕ − 2m − 1 = , σ0 m2 − 2m cos 2ϕ + 1

15

(22)

0.5

,=1 0

<' =<0

, = 0:5 ,=0

-0.5

, = !0:5 -1

, = !1 -1.5 0

0.25

0.5

0.75

1

'=2: Figure 3: The non-dimensional tangential stress distribution σϕ /σ0 around elliptical hole for γ = 0.25 and β = 0.01.

which is same as the result presented by Muskhelishvili for the case of uniaxial compression [7]. 6. Comparison with FE results The finite element method is proved to be an effective method for solving complex structural problems. The ABAQUS software is used here to verify the correctness of the present solution. The parameters of the model for comparison are: a = 1 mm, b = 3 mm, L = 100 mm, h = 100 mm, σ0 = 1 16

1 0.5 0

,=1

<' =<0

-0.5

, = 0:5

-1 -1.5

,=0

-2

, = !0:5

-2.5

, = !1

-3 0

0.25

0.5

0.75

1

'=2: Figure 4: The non-dimensional tangential stress distribution σϕ /σ0 around elliptical hole for γ = 1 and β = 0.01.

7 MPa and α = −0.5. A linearly varying in-plane load, σx∞ = (0.005y − 0.75) MPa (see Eq. (7)) is applied on the opposite edges of the rectangular plate (x = −50 mm and x = 50 mm). In the FEM analysis, a eight-node plane stress element (CP8SR) is employed with local mesh refinements near the elliptical hole. The plate is divided into sufficient number of elements (altogether 21236 finite elements) such that the approximate solution can be considered as a converged one. The finite element model and the mesh near 17

1 0 -1

,=1 -2

<' =<0

-3

, = 0:5

-4

,=0

-5 -6

, = !0:5

-7 -8

, = !1

-9 0

0.25

0.5

0.75

1

'=2: Figure 5: The non-dimensional tangential stress distribution σϕ /σ0 around elliptical hole for γ = 4 and β = 0.01.

the elliptical hole are shown in Fig. 6. The tangential stress σϕ around the hole boundary is compared in Fig. 7. The absolute percentage error between the FEM solution and the present solution at ϕ = 90◦ and ϕ = 270◦ is about 2.5694% and 2.5911%, respectively. It can be seen that the stress results of present method are in good agreement with the stress results of the ABAQUS finite element method when the size of plate is large compared to the size of hole.

18

The FEM analysis is repeated for a smaller plate with parameters: a = 1 mm, b = 3 mm, L = 25 mm, h = 25 mm, σ0 = 1 MPa and α = −0.5. A linearly varying in-plane load, σx∞ = (0.002y − 0.75) MPa (see Eq. (7)) is applied on the opposite edges of the rectangular plate (x = −12.5 mm and x = 12.5 mm). A eight-node plane stress element (CP8SR) is employed and the plate is divided into 21800 elements. The tangential stress σϕ around the hole boundary is compared in Fig. 8. The absolute percentage error between the FEM solution and the present solution at ϕ = 90◦ and ϕ = 270◦ is about 16.1443% and 2.9044%, respectively. It can be seen that the difference between the stress results of present method and the ABAQUS finite element method is greater when the size of plate is comparable to the size of hole. This is expected since the present solution is valid only when the size of hole is very small compared to the size of plate. The results are summarized in Table 1.

Figure 6: Finite element model and the mesh near the elliptical hole.

19

Figure 7: Contrast of tangential stress σϕ around hole boundary between the present method and the ABAQUS finite element method for a = 1 mm, b = 3 mm, L = 100 mm, h = 100 mm, σ0 = 1 MPa and α = −0.5.

Figure 8: Contrast of tangential stress σϕ around hole boundary between the present method and the ABAQUS finite element method for a = 1 mm, b = 3 mm, L = 25 mm, h = 25 mm, σ0 = 1 MPa and α = −0.5.

20

Table 1: Contrast between FEM solution and present solution at ϕ = 90◦ and ϕ = 270◦ for σ0 = 1 MPa, α = −0.5, a = 1 mm and b = 3 mm.

Plate Geometry

ϕ

h = L = 100 mm

h = L = 25 mm

σϕ (FEM)

σϕ (Present)

Absolute Error

90◦

−5.0600 MPa

−5.1900 MPa

2.5694%

270◦

−5.1759 MPa

−5.3100 MPa

2.5911%

90◦

−5.5452 MPa

−4.6500 MPa

16.1443%

270◦

−6.0250 MPa

−5.8500 MPa

2.9044%

7. Stress concentration factor around elliptical hole and some numerical results The maximum non-dimensional tangential stress (σϕ )max /σ0 around elliptical hole is called stress concentration factor (SCF). The points A, B, C, D on elliptical hole in physical z-plane correspond to the points A0 , B 0 , C 0 , D0 on unit circle in parametric ζ-plane as shown in Fig. 9. The non-dimensional tangential stresses at points A, B, C, D on elliptical hole are obtained by substituting ϕ = 0, ϕ = π/2, ϕ = π, ϕ = 3π/2 respectively in Eq. (20) as follows 

 σϕ 1 = (1 − α), σ0 2  A σϕ 1 = βγ(1 + α)(1 + γ) − (1 − α)(1 + 2γ), σ0 2  B σϕ 1 = (1 − α), σ0 2  C σϕ 1 = −βγ(1 + α)(1 + γ) − (1 − α)(1 + 2γ). σ0 D 2

(23)

Stress concentration factors for various conditions of loading are discussed below. 21

y

η

B

B’

𝑧 = 𝜔(ζ )

b C

o

A

x

1

C’

D

o

A’

ξ

D’ a

ζ-plane

z-plane

Figure 9: The points A, B, C, D on elliptical hole in physical z-plane and the corresponding points A0 , B 0 , C 0 , D0 on unit circle in parametric ζ-plane.

7.1. Uniform compression When the plate is subjected to uniform compression, the value of loading factor α = −1. Therefore by substituting α = −1 in Eq. (23), the nondimensional tangential stresses at the points A, B, C, D are obtained as   σϕ = 1, σ0 A   σϕ = −(1 + 2γ), σ0 B   (24) σϕ = 1, σ0 C   σϕ = −(1 + 2γ). σ0 D For γ > 0, the maximum magnitude of tangential stress on the elliptical hole occurs at the points B and D. Therefore, the stress concentration factor for the case of uniform compression is obtained as   b SCF = −(1 + 2γ) = − 1 + 2 , a

(25)

which is the well-known solution proposed by Inglis [3]. Substituting γ = 1 in Eq. (25) gives SCF = −3, which corresponds to the classic case of a circular 22

hole [2]. The negative sign in Eq. (25) indicates that the opposite edges of the plate are subjected to compression. Also, SCF is independent of the geometric ratio β. 7.2. Combined bending and compression When the plate is subjected to combined bending and compression, the value of loading factor lies between -1 and 1 (−1 < α < 1). For γ > 0, the maximum magnitude of tangential stress around the elliptical hole occurs at the point D and from Eq. (23), the corresponding stress concentration factor can be written as 1 SCF = −βγ(1 + α)(1 + γ) − (1 − α)(1 + 2γ). 2

(26)

It can observed that the stress concentration factor in the present case of combined bending and compression, depends upon geometric ratio β. The stress concentration factor for the particular case of triangular loading can be obtained by substituting α = 0 in Eq. (26) as 1 SCF = −βγ(1 + γ) − (1 + 2γ). 2

(27)

7.3. Pure bending When the plate is subjected to pure in-plane bending, the value of loading factor α = 1. Therefore by substituting α = 1 in Eq. (23), the nondimensional tangential stresses at the points A, B, C and D are obtained

23

as 

 σϕ = 0, σ0 A   σϕ = 2βγ(1 + γ), σ0 B   σϕ = 0, σ0 C   σϕ = −2βγ(1 + γ). σ0 D

(28)

For γ > 0, the maximum magnitude of tangential stress around the elliptical hole occurs at points B and D. In this case, the tangential stress at point B is tensile (positive) and the tangential stress at point D is compressive (negative). Thus, the stress concentration factor for the case of pure in-plane bending is obtained as SCF = ±2βγ(1 + γ).

(29)

It can be observed that the stress concentration factor increases linearly with β = a/h. Substituting γ = 1 in Eq. (29) gives SCF = ±4β = ±4a/h, which corresponds to the case of a circular hole [33]. The variation of stress concentration factor (SCF) with aspect ratio γ for β = 0.01 is shown in Fig. 10. For a given loading factor α, the magnitude of SCF increases with aspect ratio γ and for a given value of aspect ratio γ, the magnitude of SCF decreases with loading factor α. The variation of SCF with geometric ratio β for γ = 1 is shown in Fig. 11. For uniform compression (α = −1), the SCF is independent of geometric ratio β. However, for loading conditions other than uniform compression (−1 < α ≤ 1), the magnitude of SCF increases with geometric ratio β. Also, the SCF is more affected by 24

geometric ratio β when the value of loading factor α is greater. Table 2 to Table 4 show variation in stress concentration factors for different values of α, β and γ.

SCF

0 -2

,=1

-4

, = 0:5

-6

,=0

-8

, = !0:5

-10 , = !1

-12 0

1

2

3

4

5

. Figure 10: Variation in stress concentration factor (SCF) with aspect ratio γ for β = 0.01.

8. Conclusions In the present work, the tangential stress distribution around an elliptical hole in a large rectangular plate subjected to linearly varying loading is presented using complex variable method. The stress functions obtained are 25

0 -0.5

,=1

SCF

-1 , = 0:5

-1.5 ,=0

-2 -2.5

, = !0:5

, = !1

-3 0

0.02

0.04

0.06

0.08

0.1

Figure 11: Variation in stress concentration factor (SCF) with geometric ratio β for γ = 1. Table 2: Stress concentration factors around elliptical hole for γ = 0.25.

α = −1

α = −0.5

0.01 −1.5000

−1.1266

−0.7531 −0.3797 ±0.0063

0.02 −1.5000

−1.1281

−0.7562 −0.3844 ±0.0125

0.03 −1.5000

−1.1297

−0.7594 −0.3891 ±0.0187

0.04 −1.5000

−1.1312

−0.7625 −0.3937 ±0.0250

0.05 −1.5000

−1.1328

−0.7656 −0.3984 ±0.0313

β

α=0

26

α = 0.5

α=1

Table 3: Stress concentration factors around elliptical hole for γ = 1.

α = −1

α = −0.5

0.01 −3.0000

−2.2600

−1.5200 −0.7800 ±0.0400

0.02 −3.0000

−2.2700

−1.5400 −0.8100 ±0.0800

0.03 −3.0000

−2.2800

−1.5600 −0.8400 ±0.1200

0.04 −3.0000

−2.2900

−1.5800 −0.8700 ±0.1600

0.05 −3.0000

−2.3000

−1.6000 −0.9000 ±0.2000

β

α=0

α = 0.5

α=1

Table 4: Stress concentration factors around elliptical hole for γ = 4.

α = −1

α = −0.5

0.01 −9.0000

−6.8500

−4.7000 −2.5500 ±0.4000

0.02 −9.0000

−6.9500

−4.9000 −2.8500 ±0.8000

0.03 −9.0000

−7.0500

−5.1000 −3.1500 ±1.2000

0.04 −9.0000

−7.1500

−5.3000 −3.4500 ±1.6000

0.05 −9.0000

−7.2500

−5.5000 −3.7500 ±2.0000

β

α=0

α = 0.5

α=1

used to compute the non-dimensional stress components in the plate. The variation of non-dimensional tangential stress around elliptical hole is plotted for different cases of linearly varying in-plane loading and aspect ratio. A simple analytical formula to calculate the tangential stress concentration factor SCF around the elliptical hole is derived. It is observed that the magnitude of stress concentration depends upon hole size (β = a/h), aspect ratio (γ = b/a), and the loading condition. The loading factor α facilitates the implementation of various in-plane loading conditions. The following points are concluded: 1. For a given value of loading factor α and hole size β, the magnitude of 27

stress concentration factor SCF increases with increase in aspect ratio γ. 2. For uniform compression (α = −1), the stress concentration factor is independent of hole size β. For loading conditions other than uniform compression (−1 < α ≤ 1), the magnitude of stress concentration factor increases with increase in hole size β. 3. For a given value of hole size β and aspect ratio γ, the magnitude of stress concentration factor decreases with increase in α. 4. It should be pointed out that the solution obtained is approximate and it is valid only when the length of rectangular plate is large (L → ∞) and the size of elliptical hole is very small in comparison to the height h of the plate. The results for uniform compression can be easily generated by considering the loading factor α = −1 [7]. Furthermore, the tangential stress distribution obtained for the case of circular hole (γ = 1) matches exactly with the earlier result obtained by Kang [1]. References [1] J.-H. Kang, Exact solutions of stresses, strains, and displacements of a perforated rectangular plate by a central circular hole subjected to linearly varying in-plane normal stresses on two opposite edges, International Journal of Mechanical Sciences 84 (2014) 18–24. doi:https://doi.org/10.1016/j.ijmecsci.2014.03.023. [2] G. Kirsch, Die theorie der elastizitat und die bedurfnisse der festigkeitslehre, Zeitschrift des Vereines Deutscher Ingenieure 42 (1898) 797–807. 28

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Declaration of interests    √The authors declare that they have no known competing financial interestsor personal relationships  that could have appeared to influence the work reported in this paper.    ☐The authors declare the following financial interests/personal relationships which may be considered  as potential competing interests:      

   

Stress distribution in a large rectangular plate with a central elliptical hole.  The plate is subjected to linearly varying in‐plane loading.  Muskhelishvili's complex variable method is used.   SCF formula is proposed as a function of hole size, aspect ratio and loading condtion.