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Received: 10 July 2020 Accepted: 15 November 2020

DOI: 10.1002/pamm.202000155

Closed-form stress solution of open holes in a ﬁnite domain

Minh Nguyen-Hoang1,∗and Wilfried Becker1

1Institute of Structural Mechanics, Technical University of Darmstadt, Franziska-Braun-Straße 7, 64287 Darmstadt,

Germany

Open holes are essential design features in many technical structures. For a lightweight optimal design precise stress analysis

tools are vital. Focus of this contribution is the stress ﬁeld determination of an open hole in a rectangular plate of ﬁnite width

by analytical means, particularly the Airy stress function. This is beneﬁcial in terms of computational effort. To obtain results

of high precision, the stress boundary conditions at the straight edges as well as along the hole boundary must be fulﬁlled.

This is reached by supplementing the ﬁeld modelling the corresponding inﬁnite dimensions problem – the Kirsch solution –

with auxiliary functions.

© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH

1 Introduction

Open holes are commonly used design features and cause stress concentrations. Providing accurate and efﬁcient assessment

tools is crucial to obtain a both safe and lightweight optimal design. For reasons of fast computation and implementation,

analytical methods to determine the stress ﬁeld are focussed. A convenient means to model the two-dimensional stress state of

isotropic plates is given by the Airy stress function method. To determine the ﬁnite domain problem’s stress ﬁeld, the inﬁnite

open-hole solution by Kirsch [2] is taken and supplemented by auxiliary functions enabling to model stress-free boundaries.

2 Determination of the stress ﬁeld

Let us assume a plane stress state, an isotropic plate material and vanishing body forces. Then, the governing equations

for equilibrium, compatibility and Hooke’s law can be expressed by the Airy stress function Ffulﬁlling the differential

equation [4]

∆∆F = 0 with ∆=∂2F

∂x2+∂2F

∂y2.(1)

The corresponding stress components can be derived using

σx=∂2F

∂y2, σy=∂2F

∂x2, τxy =−∂2F

∂x∂ y , σr=1

r

∂F

∂r +1

r2

∂2F

∂ϕ2, σϕ=∂2F

∂r2, τrϕ =−∂

∂r 1

r

∂F

∂ϕ .(2)

The Airy stress function Fneeds to be chosen such that the boundary conditions

lim

y→±∞ σy(x, y) = σ0, σr(R, ϕ) = τrϕ (R, ϕ)=0, σx(±w/2, y) = τxy (±w/2, y )=0 (3)

of the open-hole problem shown in Fig. 1 are fulﬁlled. In doing so, the stress function Finf describing the corresponding

inﬁnite domain problem [2] shall be taken. Then, three types of auxiliary functions are iteratively supplemented to fulﬁl all

boundary conditions in Eq. (3) simultaneously. The ﬁrst type of auxiliary stress function is based on a periodic arrangement

of the inﬁnite domain stress ﬁeld enabling to cancel shear stresses along the edges x=±w/2(Fig. 2). The second type is

xϕ

y

σ0

σ0

r

∅d

w

Fig. 1: Problem setting. Therein, wrepresents the width, d=

2Rthe hole diameter and σ0the external load.

τ+

xy

τ−

xy

τ+

xy

τ−

xy τ+

xy

τ−

xy

... ...

Finf

1

Finf

3Finf

2

Fig. 2: Periodic arrangement of the inﬁnite dimensions problem Finf .

∗Corresponding author: e-mail nguyen- hoang@fsm.tu-darmstadt.de, phone +49 6151 1626146

This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution

in any medium, provided the original work is properly cited, the use is non-commercial and no modiﬁcations or adaptations are made.

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2 of 2 Section 4: Structural mechanics

dedicated to cancel nonzero stresses σx(±w/2, y)based on Fourier series expansion of arising deviations. The corresponding

stress function of the iteration khas the general form

FSE⊥

k/σ0=aky2+

Nk

X

n=1

φk,n(x)Ak ,n cos αk,n ywith

φk,n(x) = Cφk,n

1cosh αk,n x+Cφk,n

2sinh αk,n x+Cφk,n

3xcosh αk,n x+Cφk,n

4xsinh αk,n x.

(4)

The free coefﬁcients can be calculated by equating the stress components of the stress function using Eq. (2) with that of the

deviations’ Fourier series expansion. The last type of the auxiliary functions enables to cure violated hole boundary conditions,

which may arise due to the other two. In doing so, the stress function FHBC

kexpands the violations in a Fourier series and for

the present doubly-symmetric open-hole problem reads

FHBC

k=R2bσr

kln r

R+

Nk

X

n=1 Bk,n R

r2n

+Ck,n R

r2n−2cos 2nϕσ0.(5)

However, each of the correction functions either addresses the straight edges or the hole boundary and may interfere with the

other. Fulﬁlled stress-free conditions along all are then achieved by applying the correction functions iteratively. For plates

bounded by w/d ≥3, three iterations revealed to yield sufﬁciently accurate results. The present calculus is detailed in [1].

3 Results and discussion

The characteristic stresses for the plate geometries w/d ={3,10}are validated against Finite Element results of ABAQU S.

The latter ratio is chosen such that ﬁnite width effects have decayed and an inﬁnite plate model is applicable. In Fig. 3,

the present calculus’ stresses reveal excellent agreement. In the net section stress decay σy(ξ , 0) relevant for crack initiation

assessment, for the plate w/d = 3 with the most pronounced ﬁnite width effects in the study, errors are only of the order

1 %. The well-known approximation by Tan [3], however, leads to errors up to 36 %. Note that failure analysis conducted

by nonlocal criteria such as the Theory of Critical Distances or Finite Fracture Mechanics requires the stress evaluation in the

hole vicinity. Let us roughly estimate it to be within ξ≤0.2, then Tan’s maximum error therein is 8.3 %.

0π

2π3π

22π

0

2

ϕ[−]−→

σϕ(R, ϕ)/σ0[−]−→

0 0.2 0.4 0.6 0.8 1

1

2

3

Airy

Tan

o: FE

ξ

ϕ

σ0

σ0

∅d

w

w/d ={3,10}

ξ[−]−→

σy(ξ, 0)/σ0[−]−→

Fig. 3: Circumferential and net section stresses.

4 Conclusion

In the present work, the stress ﬁeld of the isotropic, open-hole problem with ﬁnite width is treated using the Airy stress

function. In doing so, the corresponding inﬁnite domain problem is supplemented by auxiliary functions based on a novel

periodic arrangement technique and Fourier series expansion. This enables the continuous modelling of stress-free edges

revealing excellent agreement to results of Finite Element calculation with errors of only 1 %.

Acknowledgements Open access funding enabled and organized by Projekt DEAL.

References

[1] M. Nguyen-Hoang and W. Becker, In: Analysis of Shells, Plates, and Beams (Editors: H. Altenbach et al., Springer, Cham, 2020)

[2] G. Kirsch, Zeitschrift des Vereins deutscher Ingenieure 42, 797–807 (1898)

[3] S. C. Tan, Journal of Composite Materials 22 (11), 1080-1097 (1988)

[4] S. Timoshenko and J. N.Goodier, Theory of Elasticity (McGraw-Hill Book Company, Inc., New York, 1970)

© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH www.gamm-proceedings.com