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

Abstract

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.

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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
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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=R2bσr
kln r
R+
Nk
X
n=1 Bk,n R
r2n
+Ck,n R
r2n−2cos 2nϕσ0.(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