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Axisymmetric substitute structures for circular disks with noncentral holes
Pergamon
0045-7949(95)00437-8
Compurm & Slrurrures Vol 60. No 6, pp. 1047-1065, 1996 CopyrIght Q 1996 Elsevm Science Ltd Prmted I” Great Br~tam All righb reserved 0045-7949/96 $I 5 00 + 0 00
AXISYMMETRIC SUBSTITUTE STRUCTURES FOR CIRCULAR DISKS WITH NONCENTRAL HOLES M. Kiihl, G. Dhondt and J. Broede MTU Motoren- und Turbinen-Umon
Miinchen GmbH, Postfach 50 06 40, 80976 Munchen, Germany (Received 27 January 1995)
Abstract-For
economical finite element calculatrons of aero-engme dtsks it is advantageous to use an axisymmetric FE-model. However, dtsks often contain noncentral holes, for example bolt holes which disturb the axial symmetry. In the present article methods are described on how to find axisymmetric substitute structures for these local disturbances. The substitute structures have been constructed in such a way that the stresses they predict reproduce those in the original structure far away from the hole region. The stresses in the original structure were calculated by applying the formulas obtained by Muskhehshvili for a ring and for a hole in infinite space in an alternating way. An important asset of the substrtute structures is that they allow for the prediction of the stress peaks occurring at the hole surfaces. This has been achieved by the determination of appropriate stress concentration factors which must be applied to the respective model stresses. For the practically relevant cases, the developed substitute structures are independent of the specific load conditions and only depend on a single geometric parameter characterizing the relative distance between the holes. Copyright 0 1996 Elsevier Science Ltd
1. INTRODUCTION
Aero engine disks or similar structures often contain arrays of circular holes (e.g. bolt holes, air holes, oil holes) whose centers are situated on a circle around the disk center. This is illustrated by the sketch in Fig. 1 which can be considered as a part of the whole structure, cut along the circumference at r = r, and r = I,. These noncentral holes disturb the axial sym-
metry which, on the other hand, would be advantageous for economical finite element calculations. Therefore, in the present article the disk part with noncentral holes according to Fig. 1 is modeled by an axisymmetric structure. Figure 2 shows the model geometry which consists of three ring regions. The middle ring region (region 2) replaces the ring region of the original disk containing holes by an axisymmetric continuum as explained in detail in the following sections. The disk part shown in Fig. 1 respectively Fig. 2 is loaded by centrifugal forces due to rotation with angular frequency w, and by the radial stresses enI and CT,~ applied at the inner respectively, outer boundary. These radial stresses represent the internal stresses at the intersection lines at r = r, and r = r. (shear stresses are neglected). The model has to be determined in such a way that it is equivalent to the original disk in the sense of the following two requirements: (1) With increasing radial distance from the ring region 2 (Fig. 2) i.e. from the original hole region, the stresses of the model should tend to those in the original disk (Fig. 1). (2) The model should be capable of allowing for the calculation of concentrated peak stresses occur-
ring at the noncentral holes using the model stresses. For this purpose appropriate stress concentration factors have to be determined. To meet requirement (1) the condition is imposed that the hoop stresses rr,,, at r = r, and Q,,,~at r = r,, calculated by the model (Fig. 2) are equal to those in the original disk (mean value along the respective circumference). To this end, the considered mean hoop stresses at r = r, and r = r. for the disk with holes are calculated. The calculation can be done by the semi-analytical alternating technique described in Section 2. Other methods can be found in the literature (Refs [l-3]). In Section 3 three different axisymmetric models are introduced which yield the required hoop stresses. These three models are dependent on the geometry of the original disk as well as on the specific load conditions. The geometric dependency can be reduced to a single parameter containing the relative distance between the holes. With respect to the load dependency, it is advantageous to find a model which does not depend on the loads, at least for the practically relevant range of applications. Therefore, in Section 5 the three models are compared with respect to their dependency upon the loads. Finally, to meet requirement (2) stress concentration factors based on the stress results of the axisymmetric models are determined in Section 6.
2.
ITERATIVE CALCULATION OF STRESSES IN CIRCULAR DISKS WITH NONCENTRAL HOLES
In order to find the stresses in a circular disk with one concentric hole (also called a ring) and n noncen-
1047
1048
M. K6hl et al.
tral holes (cf. Fig. l), the so-called alternating technique will be applied. The basic idea of this method is to find a solution to the governing equations at stake satisfying only part of the boundary conditions. This should be an easier task than the original one. Next, another solution to the governing equations is superimposed making sure that the boundary conditions which were not yet satisfied are fulfilled. However, this latter solution usually destroys the fulfillment of the boundary conditions which were satisfied by the first solution. So the first solution must be applied again. This procedure is repeated until a convergent solution is obtained. For the ring with noncentral holes the solution of the ring without holes subject to centrifugal forces and inner and outer pressure will be taken as the starting solution. Analytical expressions for this starting solution are given by Timoshenko and Goodier [4]. The starting solution leads to stresses on the surface of the noncentral holes. Since these should be stress-free, the inverse stress must be applied. For the determination of the stress fields in the disk due to the application of the inverse stresses at just one hole, the solution of the hole problem in infinite space with arbitrary loading, given by Muskhelishvili [5], will be taken. Application of this solution to the first noncentral hole will free this hole from all stress. However, it will lead to additional stresses at all other noncentral holes and at the inner and outer boundary of the ring. This procedure is repeated for all noncentral holes. At this stage the boundary conditions at the inner and outer surfaces, which are not satisfied any more due to the noncentral hole stress fields, are restored by superimposing the solution of the circular ring with the inverse stress fields. The solution to this problem, which differs from the starting solution in that there is no centrifgual loading and the inner and outer loading is completely arbitrary, was again obtained by Muskhelishvili [5]. At this point the second iteration can start. This procedure is repeated until convergence for the stress fields is reached. 2.1. Solution to the problem of a hole in infinite space with arbitrary loading The center of the coordinate axes coincides with the center of the hole and an arbitrary point is denoted by the complex number z =x + iy. Let the loading on the hole be expressed in terms of a normal component u,~ positive towards the center of the hole, and a tangential component crD,positive clockwise. If the loading is written in a Fourier series as follows: a,, - IQ,”=
2
A, elk’,
k--m
+ @(z)l,
(2)
where @ and Y are two potentials and a bar denotes the complex conjugate. The potentials take the form ([5]): O(z)=
f
akzek,
(3)
bkz-k,
(4)
k=l
Y(z)=
g k=l
where A,R al=l+Kf a,=;i,R’, &=A,R”,
(6) n 23,
(7)
and b,=--
icA,R 1+lc’
b, = -A,RZ, b,,=(n -I)R2a,_2-
R”A_,,+=, n >3.
(9) (10)
R is the radius of the hole and 3-v K=I+V
(11)
for plain stress where v is Poisson’s coefficient. 2.2. Solution to the problem of a ring with arbitrary loading The center of the carthesian coordinate system coincides with the center of the ring. An arbitrary point is again denoted by the complex number z = x + iy. R, is the inside radius and R, is the outside radius of the ring. Let the loading on the inner and outer side be expressed in a Fourier series: CC err - iu,O= c A; elk0 --CD on the inside boundary
(12)
and
cc c,, - io,, = c A ; elka --21
(13)
on the outside boundary. Then the solution can again be written in terms of the potentials Q, and Y using eqn (2) where
(1)
then the stress field at z (an arbitrary point outside the hole) satisfies a,, + byv= W(z)
oYY- a,, + 2ioxY= 2[Z@‘(z) + Iv(z)],
Q(Z) =
f
akzk,
(14)
Y(z)=
f
bkzk,
(15)
1049
Axisymmetric substitute structures and Ab’R:-A;R:
“=
2(R;-
s-,
(16)
’
R;)
holes (pitch radius), rL the radius of the noncentral holes and NL the total number of noncentral holes. The holes are equidistantly spaced. In the example given here rO, rLM and rL were fixed and took the values r0 = 3
2A; R, (17)
“=m-(l++)(R;+R;)’
rLM-2 -
&R, -*-l-l+K?
rL = 0.1047
(1 + k)(R; - R:)E, - (RF*~+* - R;“+*)Bmk _k2)(R:_R;)*_(Ry+2_R:k+*)(R;*P+*_R;*k+*)’
‘k=(l
A~“RF~+* -A;Rc~+~,
Bk=
(20)
B_k=;i:k~:+2-Xk~f+2, b
-k)a,R’;+G_,R;k-AA;
=(l k
(21)
2
Rk-2 1
+l,
k=O,
+2 ,....
(22)
For large values of IkJ the above expressions can be simplified. For k > 0, R, > R, one gets: B,5. B_k-
-A;Rf-k, a:kR;+k, -k
ak
N
k-+oo,
(23)
k-a, ;i”k
(24) k
o. -+
R;+2kR;-Zk-x
3
(25)
and for k < 0, R2 > R, : Bkw A;R;-k,
k-+-co,
B _k~ -;i’_,Riik, ak*;jlkR;k,
k-r-co, k-*-co.
(26) (27) (28)
2.3. Example
k=+2 -
WI, [Ll, [L],
’ -+3,...,
(19)
whereas NL was varied, taking the values 10, 20, 30, 40 and 50. Figures 3-7 show the normal stress component in the circumferential direction along a noncentral hole for subsequent iterations. It is observed that an increasing number of holes requires an increasing amount of iterations in order to reach convergence. Furthermore, the hoop stress decreases at the inside (O’, 360”) and outside (180”) of the noncentral hole whereas an additional radial stress maximum is created in between (at about 90” and 270”) for increasing NL. 2.4. Comparison
with other techniques
First a comparison is made with the stress concentration factors obtained by a boundary integral method [l]. All the cases calculated in the article by Ang and Tan [1] were verified and are collected in Table 1 (NL = 30). A denotes the position along a noncentral hole closest to the disk center, B denotes the opposite point (Fig. 1). Two types of loading were considered: centrifugal loading (cases l-24) and external radial loading (cases 25-48). The stress concentration factors were calculated by dividing the local stress by a reference stress defined as the hoop stress at the inner boundary of the disk without noncentral holes for the centrifugal cases and the external radial loading for the other cases. The values obtained by
A rotating disk with NL holes was considered. Denoting the unit of length by [L], the unit of mass by [M] and the unit of time by [ T”j leaves the freedom to fix three independent quantities, e.g. o, p and r, (Fig. 1): o=
103
IT]-*,
p = 0.8 x IO-*
[Ml. [L]-3,
r, = 1
VI.
v was taken to be 0.3. rO, rLM, rL and NL are left as variables. r, denotes the inner radius, r, the outer radius, rtM the radius from the center of the disk to the circle connecting the centers of the noncentral
Fig. 1. Disk with noncentral
holes.
M. K6hl ef al.
10.50
is particularly effective if a lot of cases with different radii and numbers of holes are to be calculated. 3. SUBSTITUTE STRUCTURES FOR THE DISK
3.1. Axisymmetric (model 1 and 2)
Fig. 2. Model disk with axisymmetric substitute structure of the hole region.
Ang and Tan are denoted by the subscript ,,*. Please note that the expression K,J(& - R,) in Ref. [I] should read (Rb - R,)/(R, - R,). The deviation from the present semi-analytical technique lies in between 0% and about 4%. In most cases the agreement is very good. Furthermore, the hoop stresses u+, obtained with the present method at the noncentral holes were compared for selected geometries and centrifugal loading with the ones obtained using the finite element method (Table 2, w = 103[qe2, p = 0.8 x 10e8[M] . [Llm3, r, = l[L]], rLM= O.l047[L], NL = 30). Again the deviations are relatively small (0.1 to about 3%). Since remeshing constitutes a high cost in finite element calculations the present alternating technique
,-
s w 0.07
z! Y v) a
0.06
: _c 0.05
0.04
0.03
0.02
0.01
stresses
Let the hole region in the original structure (Fig. 1) be replaced by the ring region 2 in Fig. 2. Since the original structure in this region is interrupted by the holes, the formation of hoop stresses is disturbed compared to those in a disk without holes. The forces are transmitted predominantly by radial stresses, This fact suggests the substitution of the non-axisymmetric ring region with holes by an axisymmetric continuum (Fig. 8a) which is capable of taking up radial stresses only (uniaxial stress state). Since the loading of the disk by radial stresses u,,, at the inner radius r,, 0,0x at the outer radius r, and by centrifugal forces due to rotation w is known, the additional requirement of given inner and outer hoop stresses (7,11resp. oro3 uniquely determines the stress distribution and the displacements within the inner ring region 1 and the outer ring region 3 (the shear stresses at Y = r, and r = r, are assumed negligible). Consequently, at the boundaries of the middle ring region 2 the radial stresses o,,, and oaj as well as the displacements u,~ and a,) are known (see eqns (A3) and (A6)-(A9) in Appendix A). In order to fulfil the equilibrium of radial forces as well as the contmulty of radial displacements, the
.................. i............... .................
iterotio” 2
n
0 0.06
z 0-J
without hoop
0.1
I= -i- 0.09 s
models
iteration
3
........ :. .......
1051
Axisymmetric substitute structures
,.
0.1
I
E
7
. +...................
1.....................
0.09
A
iterotion 2 ... ....-
?
5
0.08
cc m v)
0.07
0 % a
0.06
:
_c 0.05
0.04
0.03
0.02
0.01
I ,,,
0 40
80
120
160
2w
240
280
320
2.30
520
I 360
Fig. 4. Disk with 20 holes.
-
0’
“;
z
7
0.09
?
s
O.OE
e5 :
0.07
f -w v) a
0.06
El L 0.05
0.04
0.03
0.02
0.01
0 40
80
120
160
200
240
360 0
Fig. 5. Disk with 30 holes. CAS 60,6-P.
co>
M. Kahl et al.
1052 _ 0.1 ‘: E 0.09 7 ? f 0.08 z In m 0.07 F u cn 0.
0 0 1
0.06
0.05
0.04
0.03
0.02
0.01
0 40
80
120
160
Zoo
240
280
320
360
Fig. 6. Disk with 40 holes.
iteration
2 ........
............ (................ .................... ...........
............ ......................
ot~~‘~““““~~‘l”“~““‘~“” 40 SO
I20
160
200
Fig. 7. Disk with 50 holes.
240
Axisymmetric Table
1. Comparison
substitute
structures
factors
with literature
of stress concentration
1053 results by Ang and Tan [l]
KB Centrifugal load 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075
1.659 1.553 1.459 1.389 1.326 1.227 1.141 1.063 1.173 1.086 1.009 0.937 1.504 1.407 1.323 1.263 1.227 1.144 1.070 1.002 1.109 I.036 0.969 0.903
0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8
2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 2.0 2.0 2.0 2.0 30 3.0 3.0 3.0 4.0 4.0 4.0 4.0
1.615 1.509 1.419 1.342 1.296 1.186 1.114 1.039 1.152 1.068 0.993 0.922 1.508 1.411 1.327 1.254 1.227 1.149 1.064 0.999 1.092 1.020 0.952 0.888 External
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075
2.0 2.0 2.0 2.0 3.0 3.0 3.0 30 4.0 4.0 4.0 4.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0
4.667 4.439 4.253 4.107 3.188 3.036 2.919 2.826 2.709 2.598 2.516 2.455 4.364 4.165 4.007 3.883 2.988 2.873 2.786 2.717 2.581 2.499 2.439 2.394
0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8
4.542 4.321 4.142 3.997 3.124 2.951 2.864 2.779 2.668 2.562 2.484 2.425 4.357 4.156 3.997 3.862 2.995 2.877 2.768 2.705 2.546 2.466 2.408 2.363
following transitional conditions between the middle ring region and the inner ring region at r = r,? respectively outer ring region at r = r,, must be met: Force equilibrium:
F,, = Fo,; Fo2= F,,. Continuity
2.7 2.9 29 3.4 2.3 3.5 2.4 2.2 1.8 1.7 1.7 1.7 -0.2 -0.2 -0.3 0.7 -0.1 -0.4 0.6 0.3 1.5 1.5 1.8 1.7
1.545 1.452 1.371 1.328 1.271 1.180 1.098 1.024 1.141 1.059 0.984 0.912 1.367 1.285 1.219 1.218 1.162 1.086 1.016 0.954 1.071 1.001 0.935 0.867
1.502 1.412 1.332 1.277 1.242 1.134 1.072 0.999 1 121 1.040 0.968 0.897 1.366 1.286 1.220 1.194 1.161 1.091 1.011 0.949 1.053 0.983 0.918 0.854
2.8 2.9 2.9 4.0 23 4.1 2.4 2.5 1.8 1.8 1.6 1.7 0.1 0 -0.1 2.0 0.1 -0.5 0.5 0.5 1.7 1.8 1.9 1.0
4.423 4.239 4.091 3.984 3.104 2.973 2.870 2.788 2.668 2.569 2.495 2.439 4.100 3.956 3.845 3.744 2.907 2.814 2.740 2.681 2.543 2.473 2.420 2.381
4.309 4.135 3.991 4.149 3.042 2.883 2.820 2.743 2.628 2.534 2.465 2.412 4.095 3.952 3.841 3.734 2.902 2.824 2.730 2.670 2.510 2.441 2.390 2.351
2.6 2.5 2.5 -4.0 2.0 3.1 1.8 1.7 1.5 1.4 1.2 1.1 0.1 0.1 0.1 0.3 0.2 -0.4 0.4 0.4 1.3 1.3 1.2 1.3
radial load 2.8 2.7 2.7 2.8 2.1 2.9 1.9 1.7 1.5 1.4 1.3 1.2 0.2 0.2 0.3 0.5 -0.2 -0.1 0.6 0.4 1.4 1.3 1.3 1.3
The forces in eqn (29) are the resultants of the radial stresses a, at the respective radial coordinate r, integrated over the intersection area,
A (r) = 2mb*(r),
(31)
of displacements: 42
=
%I;
&2
=
43
(30)
where b*(r) in ring regions 1 and 3 is equal to the constant axia1 width b of the original disk with holes
1054
M. Kiihl et af. (Fig. l), while a fictitious width in the middle ring region 2 (Fig. 8a) is now defined by
which leads to the radially constant area A(r) = b . d = constant,
(33)
all over region 2. The parameter d can be considered as a fictitious thickness. This formulation allows us to consider the middle ring region 2 mathematically equivalent to a bar with constant rectangular cross section b ’ d (see analysis in Appendix B). Furthermore, it reveals the correspondence between the middle ring continuum 2 which is not allowed to transmit hoop stresses and its frequently applied finite element realization by means of plane stress elements (in the radial-axial-plane) having the thickness d (Fig. 8b). From the above, the force equilibrium (eqn (29)) can be rewritten in terms of the radial stresses and the thickness d as
fJn2
=-_D
2nrt2 d
WI1
2nr2,
cm2
=-un3.
(34)
d
The inner and outer radial stresses rrrn2,Q,,~and radial displacements u,* and u,~ in the middle ring region 2 are related by eqns (B2) and (B4) in Appendix B. Combining those with eqns (30) and (34), the following two equations are obtained: 271 7
1
(r12urol
-
(35)
r23(3,3)=2P202(r13-r:2),
2nr12h3- r12) %3 -
41
=
E2d P2W2 +
000
Pit-ii
% d
PPP 000
ddd
2E2 -[
“” 3 r23 -
rL(r23 - r12) --j---
62
1.
(36)
These equations contain the quantities oral, Q,~ and u,r and u,~from the inner and outer ring region which are known as already stated above, and the parameters p2, E2, d and 1 from the middle ring region (note that, according to Fig. 2, 1 is contained in r,2 = rLM - lr, and r23= rLM+ ;Ir,; i.e. ,I determines the radial extension of the middle ring region). Finally, to complete the model, the four parameters p2, E,, d and 2 must be determined. Two methods are introduced in the following sub-sections: in the first method, the two material parameters p2 and E2 are calculated from eqns (35) and (36) after setting d = 2xr,, and 1 = 1. In the second approach, p2 and E2 are set equal to the values p respectively E of the inner-outer ring region, and the geometric parameters d and I are determined from eqns (35) and (36).
Axisymmetric substitute structures
a) Mathematical
b) Equivalent Finite Element Model
Substitute Model
1055
fixed by setting them equal to p respectively E of the inner-outer ring region (i.e. region 1 and 3 according to Fig. 2 which have the same p- and E-values as the original disk with holes). Elimination of the thickness d from eqns (35) and (36) yields a nonlinear equation for the determination of 1 which is contained in rj2 and r23:
+
6,3r23(2r12
3
2
- 3r,2r2,
+ r :3 )] = 0.
(39)
Numerical calculations suggest that in the range of relevant load parameters there exists for every load parameter combination a unique solution 1 in the interval 0 < 1 < 1. After solving 1 from eqn (39), d follows explicitly from eqn (36): 2nr12(r23-
d= E&,3
-
P2W2 U,I) - 2
r12brd
ri2(r2,
43 - 42 - ri2) - ~ 3
I.
Fig. 8. Comparison of the mathematical substitute model with its eqmvalent finite element model.
(40)
Model 1: model adaptation by two material parameters. The geometric parameters d and 1 are iixed by setting d = 2nrLM, i.e. equal to the mean circum-
Again, attention should be paid to the case for which the denominator in eqn (40) tends to zero which implies that d tends to infinity. This method is also applicable for o = 0 (nonrotating disks or similar nonrotating structures).
ference of the middle ring region 2, and 1 = 1, which means that the radial extension of the middle ring region is equal to the diameter of the noncentral holes of the original disk. Assuming w # 0, the parameter pZ follows explicitly from eqn (35) as
2(r,2a,l - r230,3) P2 = rLM(ri3
Substitution of E2:
-
G2b2
(37)
’
of eqn (37) into eqn (36) yields the value
For u,~-u,, tending to zero E2 tends to infinity. In case of a nonrotating disk, i.e. w = 0, eqn (35) cannot be fulfilled by arbitrary Q,, and G,,~values. In this case only an optimized value of E2 can be provided approximating both of the boundary conditions. Since the parameter p2 affects only the centrifugal forces, it has no influence if w = 0. Model 2: model adaptation using two geometric parameters. The material parameters p2 and E, are
3.2. Axisymmetric model with hoop stresses (model 3) The choice of a continuum without hoop stresses for the middle ring region in Section 3.1 was based on the consideration that-due to the holes-the hoop stresses in the original structure are interrupted in that region. This is true if the relative distance between the holes is rather small. In case of only a few holes around the circumference, large parts of the middle ring region are far away from the neighboring holes and are consequently able to take up hoop
stresses, so that the approach of Section 3.1 might not be optimal. This particularly applies to the first method introduced in Section 3.1, while the second method may be able to balance this effect by means of the parameter I, which reduces the radial extension of the region without hoop stresses as its value decreases. In the current section, the middle ring region 2 for fixed 1 = 1 (Fig. 2) is modelled (model 3) by a standard axisymmetric continuum (just as the inner
M. K6hl et al.
1056
Model
1
Model
Model
2
3
Model with d-reduction only
Model with ignored holes
Fig. 9. Illustration of the different models. and outer ring region) which is able to take up radial stresses as well as hoop stresses (biaxial stress states). The model simulates the loss of stiffness and mass due to the holes by a reduction of the parameters E2 and pz. The boundary conditions at the interface between the middle ring region and the inner respectively outer ring region can be formulated in terms of the radial stresses, since for all three ring regions the same width b is assumed, and in terms of the displacements: ori, = c,,1; 42
=
%I;
~ro2 = cri3r 42 = 24i3.
(41)
E2
2&
K3 + VI43 + (1 - vH21rI243 - I(1 - VP:3 +
(3 + v)ri2Ir23hi~
(45)
tends to zero then the solutions E2 and p2 of eqns (43) and (44) tend to infinity.
(42)
Hence ring region 2 can be considered as a disk with centrifugal loading p2w2, outer pressure cti3, and mner pressure u,~, uniquely defining Ui2and u,,~[eqns (A3)-(A5)]. Substituting these into eqn (41) yields a system of two linear equations for the determination of p2 and E2: - 4JI + V~,ol = n r12 r23 -r12
Just as model 1 in Section 3.1, this method can only provide optimally approximated, not exact, values for E2 in the case where the disk does not rotate (i.e. w = 0), since in this case eqns (43) and (44) can generally not be fulfilled simultaneously for given a,, and ori values. If the expression
43 + 4 ori - 2 Qrol rz3 - r12
(43) Gi3 + ri2 H2 E2 - Ui3 + vBe3 = -2 o,oI + 2 Ori r23 - r12 r23 - 112 r23
WI
4. EXAMPLE CALCULATIONS OF THE STRESSES INNER AND OUTER RING REGION
IN THE
Each of the three models introduced in Section 3 yields identical stresses in the ring regions r, < r < rLM - r, and rLM + rL < r < r, , since the given quantities w and c,,, , oti, for the inner respectively cW3,oro3for the outer ring region uniquely determine the stresses in the respective region. Additionallyfor the purpose of comparison-stresses were calculated by two other approaches: (I) a model where the holes are ignored (i.e. disk without holes); (II) a frequently applied method where the loss of volume due to the holes is taken into account by a proportional reduction of thickness d in the ring region 2 (cf. model 2, but with constant 1 = 1). Figure 9 compares the five different models. Example calculations were performed for a rotating
1057
Axisymmetric substitute structures IO
10
h
B --c-
9
9
, I7
10
1”’
9
I
\
6.8
8
I .I
I
I
I
8
E I
7
7
I
7
I
1
I
I
6
6
\
5
6
I....----...-.---..
I
I
’
I
.!
-d
I
,
, .\_
i_
4
‘\
.A
j_
3
\.
\
.
I
2 1
_
.._.__
-d-reduction
II 1
3
r/c
b)
------‘exact
only
I1
I
3
2 r/r,
c)
Fig. 10. Comparison of hoop stresses (20 holes; o,, = u,~ = 0).
disk with inner radius ri, outer radius r0 = 3ri, radius of the centre of the noncentral holes r,, = 2ri and hole radius r, = 0.1047ri. Three cases are considered:
in Figs 12 and 13 and for the third case in Figs 14 and 15. In addition to the exact stresses calculated according to Section 2, each figure shows three different approaches: (a) the stresses modelled according to this report (models 1, 2 and 3); (b) the stresses of the abovementioned model (I); (c) the stresses of the abovementioned model (II). The curves (a) show a very good agreement between the exact and the modelled values up to a small distance from the hole region-in contrast to the curves (b) and (c)--indicating the good quality of the present approach.
(1) a disk without radial loads (i.e. cti, = u,~~= 0) containing 20 noncentral holes; (2) a disk without radial loads containing 40 holes; (3) a disk containing 55 holes with the radial loads a,, = c,~, = l.25rfpw2. The exact as well as the modelled hoop and radial stresses along a radial line (ri < r d r,) passing through the center of a noncentral hole are plotted for the first case in Figs 10 and 11, for the second case
,_j
21----
I” .._.&.Ji.
, .* _
i
T
!
I
“.,j.....,.........._ I’ i !
.
. ,.
’, .:....i _____.___._ ,
, ..i..,I I .i..,
1 2
: i
..;.
,
..i.
I + I I 4 I I , I \ I
..j. ’
;;g
0.4
e5 ..._.I --.... o)= I
..;.
t
I , 0.2
t-/r,
i
. . . ..I
..i.
I 1
\ 3
+....
1
2 b)
3
0”“”
2
1
r/ri
Fig. 11. Comparison of radial stresses (20 holes; uti, = Q,,~= 0).
c)
3 r/r,
1058
M. K6hl ef al.
$
-
8
-
!.
i I !..:.*
Ni_ 7
I
\
i ..................! .:.
i I I I .,I .:.I I I I I I .;. I
I I
I I .:.I
ti-
.I
6
I I I I II 1.. .:. I I
5
I
Y
I
.
c
4
3
.). ..I .____,........_.
.:.
?\
‘.\\ \\ .._... ,.’ .‘._
x. \
______. exact
______. exlJct
-
-
modelled
t
holes ignored
Fig. 12. Comparison of hoop stresses (40 holes; u,,, = cm, = 0).
5. COMPARISON OF THE THREE MODELS WITH RESPECT TO PARAMETER DEPENDENCY UPON THE LOADS
The three methods of replacing the ring region containing holes by axisymmetric continua introduced in the previous sections require the determination of parameters which depend upon the geometry and upon the applied loads. Numerical calculations for different geometries showed that for the relevant geometric configurations as well as for the relevant range of the loads the geometric influence is well represented by a single
A
“3 4 Nc V
1.8
1.8
1.6
1.6
parameter q which is defined by the angle CLintroduced in Fig. 16 as the ratio of 2a NL to 2n, i.e. the proportion of the angle taken by a number of NL holes to the angle of a whole circle: q:=!!La H
=N,arcsinrL. x
(46) rLM
This parameter varies between the values q = 0 (no holes) and q = 1 (touching holes). For the calculations performed, the same disk radii as for the examples in Section 4 were chosen. Thus, the radial
\ b’
1.4
‘\ _.;._‘r._..........
1.2
! ,
0.6
0.8
.I
0.6
.:
:
I’
L a>
1
3
2
1
r/c
b)
r/r;
Fig. 13. Comparison of radial stresses (40 holes; cti, = u,~ = 0).
2 c)
3
t-/r,
1059
Axisymmetric substitute structures
8
8
6
5
--__--.
exact
-
holes
-r----.
exact
4
3i
t
ignored
3
2
1
b)
3
2
1
r/r;
c>
r/r;
Fig. 14. Comparison of hoop stresses (55 holes; unl = CT,~#O).
respectively 3.2. (model 2 and 3) show only little parameter dependency within the region of practically relevant load values. In order to demonstrate these results, the parameters of the respective methods were collected from the calculation of a rotating disk without radial stresses at the inner and outer circular boundaries (i.e. o#O and CT~]=CT~~=0). The normalized parameters, EJE and p2/p respectively d/(2nrtM) and A, are plotted for model 1 in Fig. 17, for model 2 in Fig. 18, and for model 3 in Fig. 19. Note that Fig. 17 shows extremely high EJE-values for small q which could be interpreted by saying that the model tries to
distance between the center of the holes and the inner respectively outer boundary (i.e. r,, - ri and r, - rLM) is about 10 times the hole radius. Since an axisymmetric model is looked for which does not depend upon the applied loads, the three methods were compared with respect to the load dependency of their parameters. The method of Section 3.1 (model 1) showed the greatest influence of the loads, particularly for small q-values. This result is reasonable since the method uses a continuum without hoop stresses which is not adequate in case of only a few noncentral holes as already assumed in Section 3. The other two methods of Section 3.1.
\ b’
2.4
2.8
2.8
2.4
2.4
2
2
1.6
1.6
1.6
1.2
.. . . . .. .. .
1.2
..L.i.....35 I : , Ii,
0.8
0.8
0.8
_
:
.... ... .. ...y+._
, I ;
,;, 0.4
0.4
Ii,
2
1
a>
3 r/r,
3
2
1
bj
,+
..___
u@
2;
m:
- .... ... ... ....+.. Ii,
.
s
Ii,
I
’ : :
0 1
r/ri
Fig. 15. Comparison of radial stresses (55 holes; btil = u,,,~+ 0).
2
4
r/ri
1060
j:.j._ ::.__ .A
M. Kohl er al.
09 0.8
j
i
:
07
.; :
06
:
0.5
.._:_
/ ;.
.i_
; ._
:
: _
:
,.
:
/
:
/
:__
:
j
._.!
:
:
0.4 0.3
a=arcsin(r,/r,,)
9
0.2 0.1 0
0
I 0 1
I
I
0.3
0.4
I
0.5
I
I
I
0.6
0.7
0.8
II 0.9 D
Fig. 16. Definition of angle a and of the ID-, OD- and RAD-positions. balance the lack of required stiffness in circumferential direction by an increase of radial stiffness. This confirms the above-mentioned inadequacy of model 1 for small q. Next, model-calculations using the parameters of Figs 17-l 9 were performed for load cases including radial stresses a,, and/or u,~. The hoop stresses c,,, at the inner and Q,,~ at the outer boundary obtained by this approach are in general different from the required values calculated from the original disk containing noncentral holes. Therefore, the dependency of the three models upon different load conditions can be demonstrated in terms of the relative deviations,
I
0 2
Fig. 18. Parameters for model 2.
For the rotating disk (i.e. w # 0), these deviations in percent as functions of q are presented in Fig. 20 for a,, = 1,25rfpw* and a,, = 0, in Fig. 21 for uri, = 0 and oro3= 1,25rfpo*, and in Fig. 22 for ori, = a ro3 = 1,25r:pw*. Each figure contains five deviation curves. Apart from the three curves for each method, there is-just as in Figs 10-15 (cf. illustration in Fig. 9)-one that shows the deviation for the case that the noncentral holes are simply neglected, i.e. for the disk without noncentral holes, and another one which corresponds to a model where the loss of volume due to the holes was taken into account by a proportional reduction of the thickness d. A comparison of the curves makes clear that using any of the three methods introduced in this report yields better results than either neglecting the holes or
1 0.9 0.8 28
0.7
2.4
06 0.5
16
0.4
12
0.3
0.8
0.2
0.4
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
I 0.6
I 0.7
II 0.9 77
rl
Fig. 17. Parameters for model 1.
I 0.8
Fig. 19. Parameters for model 3.
1061
Axisymmetric substitute structures
TX C
20 16
-
;
12
-
.:
a
-
4
-
0 -4
-
-a -12
I., ._...
,
<
,. :
2
-<
0
c
-4
:
-
‘. .’ I I
0
.: :
4
-
-‘I3 -20
i i
.
0.1
...
...- ‘. 0
.
.. .'0 0
I 3
0 2 0
I 0.4
I
_a
model 1 model 2 model 3
-12
A 0 d-reduct,on holes ignored only I 1 1 1
0 5 0.6
0 7 0.8
1
09
.
:
:
.
,
.
:
,_ :
-16
1
I..
,
,
-201
” 0 01
0.2
0.3
’
”
0.4
-i
.
:
I.
, ’
0 5 0.6
”
0 7 0.8
’
09
‘I 20
20 ,6
d I1 : %d:l 2
16
:’ 0 model. 3 +_...;.A holes Ignored o d-reduction only . --. .- . _:
:
1.2 a
1 rl
_
0 model 1 0 model 2 -. 0 model 3 ,_ A holes ignored ; 0 d-reductloo , .i. I ,
:
.._____
.
12-,
..:.
:-’
4
only
0 -4 -a -12 -16 -20
0
..:.
i
I 0.1
I 0.2
,__ , I 0.3
I 0.4
.e
.__,
:..
,.
:
I
I
I
I
0.5
0.6
0.7
0.8
.E
h
1 0.9
1 17
Fig. 20. Deviation of hoop stresses for w # 0, on, # 0, a,0, = 0.
-
c
.-
:‘
12
-
a
b”
4
z
:
: ,
.:
,
:
i--
0
“:‘ .; ‘_____ :. :
I
I I 0 2 0 3 0.4
.,._ ,
I
I
I
0.5
0.6
0.7
I 1. 0 a 0.9
1 rt
Fig. 22. Deviation of hoop stresses for w # 0, un, = 0,) Z 0.
i 0 model 1
0 model 2 I 0 model 3 A holes ignored 0 d-reduction only i i i.. ., i... ,_
,.
-a
!
__!_. :
_‘
,
__,
:
,/ :
,
;I$:1
I
1 2
._( -...i
1. . : .. ... : ..:..: . . ... i- ._. : .._; . .: :_. ..,
-12 -16 -20
z-201
-0
Q model c .-
-20
-0
I 0.1
J
0 -4
6 .+ 0 ,> aI D
:
G
-12 - _.(... ; _,fj ___:. _.,
,2
-.
I
..: ..:
0
’
0.1
’
0.2
’
0.3
’
0.4
’
0.5
.: : I
.:
0.6
’
0 7 0.8
’
0.9
I 0.6
I 0.7
I 0.8
, 0.9
’
, ’
1
i : ;:zl:l$ i A holes
ignored @ d-reduction only a
*)
&
4
%
0 -4
s , -0
0
I 0.1
I 0.2
I 0.3
I 0.4
I 0.5
,
;
I I I 0 6 0 7 0.8
I 0.9
I ‘I
Fig. 21. Deviation of hoop stresses for o #O, CT,,= 0, 04 f 0.
-a
.-
-12
.?
-16
5 -0
-20
-
0
.
:
I 0.1
I 0.2
.
j 0.3
I 0.4
I 0.5
‘I Fig. 23. Deviation of hoop stresses for w = 0. o,, =O, c,, # 0.
1062
M. Kohl er al.
28 27 26 25 24 23 22 21 2 19 18 17 16
0 Fig. 24. Stress concentration factor K,,
STRESS CONCENTRATION
02
03
04
05
06
07
08
These concentrated stresses should be derived from the model, which was achieved by introducing stress concentration factors. The exact concentrated stresses around the noncentral holes were calculated by the method in Section 2. The highest values are located at three points: at the inner point r = rLM - rL (called the ID-position), at the outer point r = rLM + rL (the OD-position) and, for great q-values, at the point r = r, (the RAD-position) defined by the tangent through the disk centre (Fig. 16). In Figs 10, 12 and 14 the stress peaks at the ID- and OD-position are well illustrated. In order to reproduce the stress peaks at these three points by using the model stresses, the exact stresses are written as a linear combination of the local model hoop stresses and model radial stresses. For the IDand OD-position this yields:
FACTORS
For real applications it is essential to determine the concentrated stresses at the surfaces of the holes.
06
06-
'
t
01 0
’ 01
’
0 2
’
0 3
’
0.4
’
0.5
’
0.6
’
0 7
[
Fig. 26. Stress concentration factor K,,.
reducing only the thickness d. It is also obvious that the above-mentioned decreasing validity of model 1 for decreasing q-values leads to greater deviations. Models 2 and 3 yield small deviations; model 2 seems to be a little better. Remarkable results were revealed by an additional calculation of the deviations for the nonrotating structure (i.e. w = 0) with an outer radial load (T,~~# 0 and on3 = 0, plotted in Fig. 23. Although the model parameters were determined for the rotating disk (i.e. w # 0) without inner or outer radial loads, the deviations for the models 2 and 3 are still small. Remember that it was not possible to find an exact solution for model 3 in the nonrotating load case. These results indicate that the applicability of the models 2 and 3 reaches far beyond the specific load case for which their parameters are calculated. 6.
01
’
’
0.8
0.9 T
Fig. 25. Stress concentration factor K,,.
Fig. 27. Stress concentration
factor
K,,.
IO63
Axisymmetric substitute structures
o ID-Pa o ID-Pa
: a,-deviotfon
0 OD-Pm
0
-4
III 01
0
02
III 04
03
RAD-Pos
05
: a,-dewotlon : a.-dewotlon
1 07
06
q
11 08
: a,-deviation
OD-Pos
a,-dewotion
0 RAD-Pos
I 1
09
0
I
I
I
I
I
I
01
02
03
04
05
06
a,-deviation
1 07
I
I
08
09
77
for
w # 0,
Fig.
of the regions has to be taken outside of region 2 (note that cr,is not continuous at the boundary), and for the RAD-position:
Calculations were performed for a number of different pairs of load combinations. The resulting concentration factors showed a certain scatter, particularly for K,, and K,, if instead of the above selected mean values local model stresses in region 2 were used. The choice of the mean values, however, reduced the scatter. Based on these results, unique concentration factor curves were selected. They are plotted in Figs 24-27. Their quality is assessed by the accuracy of the modelled stress peaks at the ID-, OD- and RADposition using model 3 together with the parameters according to Fig. 19. This is illustrated for some examples in Figs 28-32. In these figures the deviation
where o~modc’) at the boundary
O(mcW uwCt)
=
,
Where
_ K,, gjmdel),
L
1y
(49)
Ir 1-q
apmW
and
qjmoW
are mean values of their respective values at the ID- and OD-position outside of region 2. Note that the modelled radial stresses are scaled by 1 - n in order to consider the stress increase due to the reduction of the circumferential cross section in the hole region, and thus limiting the magnitude of the concentration factor. The determination of the stress concentration factors K,,, K,,, K,,and K,,requires for each q-value two calculations of the disk subjected to different loads, which yields two linearly independent equations for eqn (48) as well as for eqn (49). &Or-
30. Deviation
of peak stresses un, = 0, or03# 0.
Fig. 28. Deviation of peak stresses for o # 0, CT,,= 0, u,al = 0.
(50) between the modelled and exact peak stresses in percent is plotted as a function of the geometric
I ID-Pos
: o,-devlotmn
1
OD-Pos
o,-devlatlon
0
RAD-Pos
: o,-devlotmn
0 OD-Pos
I
ID-Pas
: a,-devhon a,-deviation
I
.
0
RAD-Pos
I
I 06
u.-deviabn
:
-4
I 0
01
I 02
I 03
I 04
I 05
I
I
I
I
0.6
0.7
08
09
1
77 Fig. 29. Deviation of peak stresses for w #O, CT,,# 0, 6,,, = 0.
-4
l0
i 01
02
I 03
i 0.4
05
I 07
I 08
I 09
1 r
Fig. 31. Deviation of peak stresses for w # 0. CT,,,= o,,,~# 0.
1064 6\”
M. Kohl et al. 10
: I
~
: ; Y 0
,
:
0
:
ID-Pas.:
REFERENCES
;
1. M. E. Ang and C. L. Tan, Stress concentrations at holes in thin rotating disks. J. Strain Anal. 23,223-225 (1988). 2. W. A. Green, G. T. J. Hooper and R. Hetherington, Stress distribution in rotating disks with noncentral holes. Aeronaut. Q. 15, 107-121 (1964). 3. H. G. Edmunds, Stress concentrations at holes m rotating disks. Engr 618620 (1954). 4. S. P. Timoshenko and J. N. Goodier, Theory of Elasficiry, 3rd edn, International Student Edition. McGrawHill, New York (1970). 5. N. I. Muskhelishvili, Some Eusrc Problems of the Mathematical Theory of Elasticity. Noordhoff, GroningenHolland (1953).
o,-deviation
0 OD-PCS.:
a,-devmtion
0
: o,-devmtion
RAD-Pos
4
E 2 “0 c 0 .+0 .z
0
-2
APPENDIX A: STRESSES AND DISPLACEMENTS IN THIN RING (WITHOUT NONCENTRAL HOLES)
u
-4
-
0
0.1
02
0.3
04
0.5
0.6
07
0.8
09
A
1 77
Fig. 32. Deviation of peak stresses for w = 0, o,,, = 0, 0,) # 0.
parameter r~ according to eqn (46). The load parameters in Figs 29-32 are identical to those in Figs 20-23; Fig. 28 shows the additional load case w # 0 and CT,,,= a,,,, = 0. The large deviation values of up to +26% for small 1 values at the RAD-positionsparticularly in Fig. 28-are insignificant since for small r~the stresses at the RAD-positions are much smaller than at the ID- and OD-positions. Significant stress peaks at the RAD-positions emerge only for higher q values. Apart from this, the deviations in Figs 28-3 1 are smaller than 3%, predominantly with a positive sign which means a conservative approach with respect to subsequent life prediction. The deviations for the nonrotating disk (Fig. 32) do not exceed 4%.
The stresses and deflections in a ring rotating with circular frequency o and subjected to inner and outer radial load o, respectively u,,, are under plane stress conditions [4]:
u,(r) =
rfr:(un
-urn)
ri-rf
2 2 1 ,-+ w, - u,r,
r2
ra - rf
(Al) u,(r) = -
2 2 umro - unrl
rfra(u, -cm) 1 .-+ r2 ri -r: 3+v +Tpo2
i-z- rf
64.3
(
u(r) = i [u,(r) - vu,(r)].
According to eqn (A2) the hoop stresses u, at the inner and (I,~ at the outer boundary, r = r, respectively r = rO. are:
(A4) ura = 7.
r* + r2
2r2
&“~+~u~
CONCLUSIONS
+Tpm(ri+$r:).
Three different axisymmetric substitute models for the circular disk with noncentral holes have been introduced. These models fulfil the two necessary requirements for a model to qualify, i.e. they reproduce the stresses of the original structure at large distances from the hole region, and the stress peaks occurring at the hole surfaces can be calculated using appropriate stress concentration factors. Furthermore, each model is determined by two parameters which for most practical purposes depend geometrically on the relative distance between the holes only. However, an additionally favoured model feature, the independence from specific load conditions, was not adequately met by model 1. Models 2 and 3 on the other hand are both of good quality with respect to load-independency. For finite element calculations the choice of model 3 would have the advantage that the corresponding axisymmetric ring element is of the same type as the one used in the bulk of the structure.
(A5)
Instead of known inner and outer radial stresses u, respectively u,~, for the application in Section 3.1 and 3.2 either u,, and un, are given (ring region 1 according to Fig. 2), or, u,~ and, ura3 (ring region 3) are known. Therefore, explictt expresstons for u,~, and e,O1as functions of a,, , u,,, in ring region 1, and for u,~ and u,,~as functions of cm3, u,,~ in ring region 3 are required. From eqns (A4) and (A5) one obtains:
-~pwi(r:z+~r:)].
=rl3
=
--
ri - r& =,03 2rL
[
ri + 4, - 2 cm3 r.
-
r23
(A6)
>I.
(A7)
1065
Axisymmetric substitute structures APPENDIX Rr RADIAL STRESSES AND
DISPLACEMENTS IN THE MIDDLE RING REGION ACCORDING TO SECTION 3.1
As explained in Section 3.1, the middle ring region (region 2 in Fig. 2) can be considered mathematically equivalent to a bar with constant rectangular cross section A = 6 d. According to Fig. Bl, the equilibrium of forces requires the equality between the force difference F,, - For and the centrifugal forces integrated over the interval r,>Q r < ro2:
r23
r01 F,z - Far =
s r=r,*
Ap,02r
dr = f b . dp2w2(r$, - rf,).
@I)
Applying the relations F,2= b d un2 and Fo2 = b du,, , the difference of inner and outer radial stresses can be expressed by 1 (B2) O”Z- ~K,z= 2 p202(r& - r:,). ui2 The radial stress u,(r) at an arbitrary location r, r,2 < r Q r,, , follows from eqn (B2) by replacing u,~*by u,(r) and r2, by r: Fig. Bl. Bar substitute structure of the middle ring region.
u,(r) = ufi2- ip2c02(r2 - rf,).
Substitition of eqn (A6) into eqn (A5) and of eqn (A7) into eqn (A4) gives:
The displacement-increment Au = ud - u,~ is obtained by integrating the strain r(r) = u,(r)/Ez:
(B3)
‘02 u,(r)dr =’ E2
XP
r:,-rf----
:;:
1
(A8l
s ,=,a rf2)
-iZp2d(r2-
dr 1
r’, + r&
cn3 = 2r:,
r: - rz3
a,03--u&f8 2rL
2
r23 - rt2 =---ucr,,+p2w
3fV
2E2
352
rl(ra + r&) - 2r$ 2rk
1. (A9)
4 - - ri2 . 3
1
(B4)
0045-7949(95)00437-8
Compurm & Slrurrures Vol 60. No 6, pp. 1047-1065, 1996 CopyrIght Q 1996 Elsevm Science Ltd Prmted I” Great Br~tam All righb reserved 0045-7949/96 $I 5 00 + 0 00
AXISYMMETRIC SUBSTITUTE STRUCTURES FOR CIRCULAR DISKS WITH NONCENTRAL HOLES M. Kiihl, G. Dhondt and J. Broede MTU Motoren- und Turbinen-Umon
Miinchen GmbH, Postfach 50 06 40, 80976 Munchen, Germany (Received 27 January 1995)
Abstract-For
economical finite element calculatrons of aero-engme dtsks it is advantageous to use an axisymmetric FE-model. However, dtsks often contain noncentral holes, for example bolt holes which disturb the axial symmetry. In the present article methods are described on how to find axisymmetric substitute structures for these local disturbances. The substitute structures have been constructed in such a way that the stresses they predict reproduce those in the original structure far away from the hole region. The stresses in the original structure were calculated by applying the formulas obtained by Muskhehshvili for a ring and for a hole in infinite space in an alternating way. An important asset of the substrtute structures is that they allow for the prediction of the stress peaks occurring at the hole surfaces. This has been achieved by the determination of appropriate stress concentration factors which must be applied to the respective model stresses. For the practically relevant cases, the developed substitute structures are independent of the specific load conditions and only depend on a single geometric parameter characterizing the relative distance between the holes. Copyright 0 1996 Elsevier Science Ltd
1. INTRODUCTION
Aero engine disks or similar structures often contain arrays of circular holes (e.g. bolt holes, air holes, oil holes) whose centers are situated on a circle around the disk center. This is illustrated by the sketch in Fig. 1 which can be considered as a part of the whole structure, cut along the circumference at r = r, and r = I,. These noncentral holes disturb the axial sym-
metry which, on the other hand, would be advantageous for economical finite element calculations. Therefore, in the present article the disk part with noncentral holes according to Fig. 1 is modeled by an axisymmetric structure. Figure 2 shows the model geometry which consists of three ring regions. The middle ring region (region 2) replaces the ring region of the original disk containing holes by an axisymmetric continuum as explained in detail in the following sections. The disk part shown in Fig. 1 respectively Fig. 2 is loaded by centrifugal forces due to rotation with angular frequency w, and by the radial stresses enI and CT,~ applied at the inner respectively, outer boundary. These radial stresses represent the internal stresses at the intersection lines at r = r, and r = r. (shear stresses are neglected). The model has to be determined in such a way that it is equivalent to the original disk in the sense of the following two requirements: (1) With increasing radial distance from the ring region 2 (Fig. 2) i.e. from the original hole region, the stresses of the model should tend to those in the original disk (Fig. 1). (2) The model should be capable of allowing for the calculation of concentrated peak stresses occur-
ring at the noncentral holes using the model stresses. For this purpose appropriate stress concentration factors have to be determined. To meet requirement (1) the condition is imposed that the hoop stresses rr,,, at r = r, and Q,,,~at r = r,, calculated by the model (Fig. 2) are equal to those in the original disk (mean value along the respective circumference). To this end, the considered mean hoop stresses at r = r, and r = r. for the disk with holes are calculated. The calculation can be done by the semi-analytical alternating technique described in Section 2. Other methods can be found in the literature (Refs [l-3]). In Section 3 three different axisymmetric models are introduced which yield the required hoop stresses. These three models are dependent on the geometry of the original disk as well as on the specific load conditions. The geometric dependency can be reduced to a single parameter containing the relative distance between the holes. With respect to the load dependency, it is advantageous to find a model which does not depend on the loads, at least for the practically relevant range of applications. Therefore, in Section 5 the three models are compared with respect to their dependency upon the loads. Finally, to meet requirement (2) stress concentration factors based on the stress results of the axisymmetric models are determined in Section 6.
2.
ITERATIVE CALCULATION OF STRESSES IN CIRCULAR DISKS WITH NONCENTRAL HOLES
In order to find the stresses in a circular disk with one concentric hole (also called a ring) and n noncen-
1047
1048
M. K6hl et al.
tral holes (cf. Fig. l), the so-called alternating technique will be applied. The basic idea of this method is to find a solution to the governing equations at stake satisfying only part of the boundary conditions. This should be an easier task than the original one. Next, another solution to the governing equations is superimposed making sure that the boundary conditions which were not yet satisfied are fulfilled. However, this latter solution usually destroys the fulfillment of the boundary conditions which were satisfied by the first solution. So the first solution must be applied again. This procedure is repeated until a convergent solution is obtained. For the ring with noncentral holes the solution of the ring without holes subject to centrifugal forces and inner and outer pressure will be taken as the starting solution. Analytical expressions for this starting solution are given by Timoshenko and Goodier [4]. The starting solution leads to stresses on the surface of the noncentral holes. Since these should be stress-free, the inverse stress must be applied. For the determination of the stress fields in the disk due to the application of the inverse stresses at just one hole, the solution of the hole problem in infinite space with arbitrary loading, given by Muskhelishvili [5], will be taken. Application of this solution to the first noncentral hole will free this hole from all stress. However, it will lead to additional stresses at all other noncentral holes and at the inner and outer boundary of the ring. This procedure is repeated for all noncentral holes. At this stage the boundary conditions at the inner and outer surfaces, which are not satisfied any more due to the noncentral hole stress fields, are restored by superimposing the solution of the circular ring with the inverse stress fields. The solution to this problem, which differs from the starting solution in that there is no centrifgual loading and the inner and outer loading is completely arbitrary, was again obtained by Muskhelishvili [5]. At this point the second iteration can start. This procedure is repeated until convergence for the stress fields is reached. 2.1. Solution to the problem of a hole in infinite space with arbitrary loading The center of the coordinate axes coincides with the center of the hole and an arbitrary point is denoted by the complex number z =x + iy. Let the loading on the hole be expressed in terms of a normal component u,~ positive towards the center of the hole, and a tangential component crD,positive clockwise. If the loading is written in a Fourier series as follows: a,, - IQ,”=
2
A, elk’,
k--m
+ @(z)l,
(2)
where @ and Y are two potentials and a bar denotes the complex conjugate. The potentials take the form ([5]): O(z)=
f
akzek,
(3)
bkz-k,
(4)
k=l
Y(z)=
g k=l
where A,R al=l+Kf a,=;i,R’, &=A,R”,
(6) n 23,
(7)
and b,=--
icA,R 1+lc’
b, = -A,RZ, b,,=(n -I)R2a,_2-
R”A_,,+=, n >3.
(9) (10)
R is the radius of the hole and 3-v K=I+V
(11)
for plain stress where v is Poisson’s coefficient. 2.2. Solution to the problem of a ring with arbitrary loading The center of the carthesian coordinate system coincides with the center of the ring. An arbitrary point is again denoted by the complex number z = x + iy. R, is the inside radius and R, is the outside radius of the ring. Let the loading on the inner and outer side be expressed in a Fourier series: CC err - iu,O= c A; elk0 --CD on the inside boundary
(12)
and
cc c,, - io,, = c A ; elka --21
(13)
on the outside boundary. Then the solution can again be written in terms of the potentials Q, and Y using eqn (2) where
(1)
then the stress field at z (an arbitrary point outside the hole) satisfies a,, + byv= W(z)
oYY- a,, + 2ioxY= 2[Z@‘(z) + Iv(z)],
Q(Z) =
f
akzk,
(14)
Y(z)=
f
bkzk,
(15)
1049
Axisymmetric substitute structures and Ab’R:-A;R:
“=
2(R;-
s-,
(16)
’
R;)
holes (pitch radius), rL the radius of the noncentral holes and NL the total number of noncentral holes. The holes are equidistantly spaced. In the example given here rO, rLM and rL were fixed and took the values r0 = 3
2A; R, (17)
“=m-(l++)(R;+R;)’
rLM-2 -
&R, -*-l-l+K?
rL = 0.1047
(1 + k)(R; - R:)E, - (RF*~+* - R;“+*)Bmk _k2)(R:_R;)*_(Ry+2_R:k+*)(R;*P+*_R;*k+*)’
‘k=(l
A~“RF~+* -A;Rc~+~,
Bk=
(20)
B_k=;i:k~:+2-Xk~f+2, b
-k)a,R’;+G_,R;k-AA;
=(l k
(21)
2
Rk-2 1
+l,
k=O,
+2 ,....
(22)
For large values of IkJ the above expressions can be simplified. For k > 0, R, > R, one gets: B,5. B_k-
-A;Rf-k, a:kR;+k, -k
ak
N
k-+oo,
(23)
k-a, ;i”k
(24) k
o. -+
R;+2kR;-Zk-x
3
(25)
and for k < 0, R2 > R, : Bkw A;R;-k,
k-+-co,
B _k~ -;i’_,Riik, ak*;jlkR;k,
k-r-co, k-*-co.
(26) (27) (28)
2.3. Example
k=+2 -
WI, [Ll, [L],
’ -+3,...,
(19)
whereas NL was varied, taking the values 10, 20, 30, 40 and 50. Figures 3-7 show the normal stress component in the circumferential direction along a noncentral hole for subsequent iterations. It is observed that an increasing number of holes requires an increasing amount of iterations in order to reach convergence. Furthermore, the hoop stress decreases at the inside (O’, 360”) and outside (180”) of the noncentral hole whereas an additional radial stress maximum is created in between (at about 90” and 270”) for increasing NL. 2.4. Comparison
with other techniques
First a comparison is made with the stress concentration factors obtained by a boundary integral method [l]. All the cases calculated in the article by Ang and Tan [1] were verified and are collected in Table 1 (NL = 30). A denotes the position along a noncentral hole closest to the disk center, B denotes the opposite point (Fig. 1). Two types of loading were considered: centrifugal loading (cases l-24) and external radial loading (cases 25-48). The stress concentration factors were calculated by dividing the local stress by a reference stress defined as the hoop stress at the inner boundary of the disk without noncentral holes for the centrifugal cases and the external radial loading for the other cases. The values obtained by
A rotating disk with NL holes was considered. Denoting the unit of length by [L], the unit of mass by [M] and the unit of time by [ T”j leaves the freedom to fix three independent quantities, e.g. o, p and r, (Fig. 1): o=
103
IT]-*,
p = 0.8 x IO-*
[Ml. [L]-3,
r, = 1
VI.
v was taken to be 0.3. rO, rLM, rL and NL are left as variables. r, denotes the inner radius, r, the outer radius, rtM the radius from the center of the disk to the circle connecting the centers of the noncentral
Fig. 1. Disk with noncentral
holes.
M. K6hl ef al.
10.50
is particularly effective if a lot of cases with different radii and numbers of holes are to be calculated. 3. SUBSTITUTE STRUCTURES FOR THE DISK
3.1. Axisymmetric (model 1 and 2)
Fig. 2. Model disk with axisymmetric substitute structure of the hole region.
Ang and Tan are denoted by the subscript ,,*. Please note that the expression K,J(& - R,) in Ref. [I] should read (Rb - R,)/(R, - R,). The deviation from the present semi-analytical technique lies in between 0% and about 4%. In most cases the agreement is very good. Furthermore, the hoop stresses u+, obtained with the present method at the noncentral holes were compared for selected geometries and centrifugal loading with the ones obtained using the finite element method (Table 2, w = 103[qe2, p = 0.8 x 10e8[M] . [Llm3, r, = l[L]], rLM= O.l047[L], NL = 30). Again the deviations are relatively small (0.1 to about 3%). Since remeshing constitutes a high cost in finite element calculations the present alternating technique
,-
s w 0.07
z! Y v) a
0.06
: _c 0.05
0.04
0.03
0.02
0.01
stresses
Let the hole region in the original structure (Fig. 1) be replaced by the ring region 2 in Fig. 2. Since the original structure in this region is interrupted by the holes, the formation of hoop stresses is disturbed compared to those in a disk without holes. The forces are transmitted predominantly by radial stresses, This fact suggests the substitution of the non-axisymmetric ring region with holes by an axisymmetric continuum (Fig. 8a) which is capable of taking up radial stresses only (uniaxial stress state). Since the loading of the disk by radial stresses u,,, at the inner radius r,, 0,0x at the outer radius r, and by centrifugal forces due to rotation w is known, the additional requirement of given inner and outer hoop stresses (7,11resp. oro3 uniquely determines the stress distribution and the displacements within the inner ring region 1 and the outer ring region 3 (the shear stresses at Y = r, and r = r, are assumed negligible). Consequently, at the boundaries of the middle ring region 2 the radial stresses o,,, and oaj as well as the displacements u,~ and a,) are known (see eqns (A3) and (A6)-(A9) in Appendix A). In order to fulfil the equilibrium of radial forces as well as the contmulty of radial displacements, the
.................. i............... .................
iterotio” 2
n
0 0.06
z 0-J
without hoop
0.1
I= -i- 0.09 s
models
iteration
3
........ :. .......
1051
Axisymmetric substitute structures
,.
0.1
I
E
7
. +...................
1.....................
0.09
A
iterotion 2 ... ....-
?
5
0.08
cc m v)
0.07
0 % a
0.06
:
_c 0.05
0.04
0.03
0.02
0.01
I ,,,
0 40
80
120
160
2w
240
280
320
2.30
520
I 360
Fig. 4. Disk with 20 holes.
-
0’
“;
z
7
0.09
?
s
O.OE
e5 :
0.07
f -w v) a
0.06
El L 0.05
0.04
0.03
0.02
0.01
0 40
80
120
160
200
240
360 0
Fig. 5. Disk with 30 holes. CAS 60,6-P.
co>
M. Kahl et al.
1052 _ 0.1 ‘: E 0.09 7 ? f 0.08 z In m 0.07 F u cn 0.
0 0 1
0.06
0.05
0.04
0.03
0.02
0.01
0 40
80
120
160
Zoo
240
280
320
360
Fig. 6. Disk with 40 holes.
iteration
2 ........
............ (................ .................... ...........
............ ......................
ot~~‘~““““~~‘l”“~““‘~“” 40 SO
I20
160
200
Fig. 7. Disk with 50 holes.
240
Axisymmetric Table
1. Comparison
substitute
structures
factors
with literature
of stress concentration
1053 results by Ang and Tan [l]
KB Centrifugal load 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075
1.659 1.553 1.459 1.389 1.326 1.227 1.141 1.063 1.173 1.086 1.009 0.937 1.504 1.407 1.323 1.263 1.227 1.144 1.070 1.002 1.109 I.036 0.969 0.903
0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8
2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 2.0 2.0 2.0 2.0 30 3.0 3.0 3.0 4.0 4.0 4.0 4.0
1.615 1.509 1.419 1.342 1.296 1.186 1.114 1.039 1.152 1.068 0.993 0.922 1.508 1.411 1.327 1.254 1.227 1.149 1.064 0.999 1.092 1.020 0.952 0.888 External
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075
2.0 2.0 2.0 2.0 3.0 3.0 3.0 30 4.0 4.0 4.0 4.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0
4.667 4.439 4.253 4.107 3.188 3.036 2.919 2.826 2.709 2.598 2.516 2.455 4.364 4.165 4.007 3.883 2.988 2.873 2.786 2.717 2.581 2.499 2.439 2.394
0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8
4.542 4.321 4.142 3.997 3.124 2.951 2.864 2.779 2.668 2.562 2.484 2.425 4.357 4.156 3.997 3.862 2.995 2.877 2.768 2.705 2.546 2.466 2.408 2.363
following transitional conditions between the middle ring region and the inner ring region at r = r,? respectively outer ring region at r = r,, must be met: Force equilibrium:
F,, = Fo,; Fo2= F,,. Continuity
2.7 2.9 29 3.4 2.3 3.5 2.4 2.2 1.8 1.7 1.7 1.7 -0.2 -0.2 -0.3 0.7 -0.1 -0.4 0.6 0.3 1.5 1.5 1.8 1.7
1.545 1.452 1.371 1.328 1.271 1.180 1.098 1.024 1.141 1.059 0.984 0.912 1.367 1.285 1.219 1.218 1.162 1.086 1.016 0.954 1.071 1.001 0.935 0.867
1.502 1.412 1.332 1.277 1.242 1.134 1.072 0.999 1 121 1.040 0.968 0.897 1.366 1.286 1.220 1.194 1.161 1.091 1.011 0.949 1.053 0.983 0.918 0.854
2.8 2.9 2.9 4.0 23 4.1 2.4 2.5 1.8 1.8 1.6 1.7 0.1 0 -0.1 2.0 0.1 -0.5 0.5 0.5 1.7 1.8 1.9 1.0
4.423 4.239 4.091 3.984 3.104 2.973 2.870 2.788 2.668 2.569 2.495 2.439 4.100 3.956 3.845 3.744 2.907 2.814 2.740 2.681 2.543 2.473 2.420 2.381
4.309 4.135 3.991 4.149 3.042 2.883 2.820 2.743 2.628 2.534 2.465 2.412 4.095 3.952 3.841 3.734 2.902 2.824 2.730 2.670 2.510 2.441 2.390 2.351
2.6 2.5 2.5 -4.0 2.0 3.1 1.8 1.7 1.5 1.4 1.2 1.1 0.1 0.1 0.1 0.3 0.2 -0.4 0.4 0.4 1.3 1.3 1.2 1.3
radial load 2.8 2.7 2.7 2.8 2.1 2.9 1.9 1.7 1.5 1.4 1.3 1.2 0.2 0.2 0.3 0.5 -0.2 -0.1 0.6 0.4 1.4 1.3 1.3 1.3
The forces in eqn (29) are the resultants of the radial stresses a, at the respective radial coordinate r, integrated over the intersection area,
A (r) = 2mb*(r),
(31)
of displacements: 42
=
%I;
&2
=
43
(30)
where b*(r) in ring regions 1 and 3 is equal to the constant axia1 width b of the original disk with holes
1054
M. Kiihl et af. (Fig. l), while a fictitious width in the middle ring region 2 (Fig. 8a) is now defined by
which leads to the radially constant area A(r) = b . d = constant,
(33)
all over region 2. The parameter d can be considered as a fictitious thickness. This formulation allows us to consider the middle ring region 2 mathematically equivalent to a bar with constant rectangular cross section b ’ d (see analysis in Appendix B). Furthermore, it reveals the correspondence between the middle ring continuum 2 which is not allowed to transmit hoop stresses and its frequently applied finite element realization by means of plane stress elements (in the radial-axial-plane) having the thickness d (Fig. 8b). From the above, the force equilibrium (eqn (29)) can be rewritten in terms of the radial stresses and the thickness d as
fJn2
=-_D
2nrt2 d
WI1
2nr2,
cm2
=-un3.
(34)
d
The inner and outer radial stresses rrrn2,Q,,~and radial displacements u,* and u,~ in the middle ring region 2 are related by eqns (B2) and (B4) in Appendix B. Combining those with eqns (30) and (34), the following two equations are obtained: 271 7
1
(r12urol
-
(35)
r23(3,3)=2P202(r13-r:2),
2nr12h3- r12) %3 -
41
=
E2d P2W2 +
000
Pit-ii
% d
PPP 000
ddd
2E2 -[
“” 3 r23 -
rL(r23 - r12) --j---
62
1.
(36)
These equations contain the quantities oral, Q,~ and u,r and u,~from the inner and outer ring region which are known as already stated above, and the parameters p2, E2, d and 1 from the middle ring region (note that, according to Fig. 2, 1 is contained in r,2 = rLM - lr, and r23= rLM+ ;Ir,; i.e. ,I determines the radial extension of the middle ring region). Finally, to complete the model, the four parameters p2, E,, d and 2 must be determined. Two methods are introduced in the following sub-sections: in the first method, the two material parameters p2 and E2 are calculated from eqns (35) and (36) after setting d = 2xr,, and 1 = 1. In the second approach, p2 and E2 are set equal to the values p respectively E of the inner-outer ring region, and the geometric parameters d and I are determined from eqns (35) and (36).
Axisymmetric substitute structures
a) Mathematical
b) Equivalent Finite Element Model
Substitute Model
1055
fixed by setting them equal to p respectively E of the inner-outer ring region (i.e. region 1 and 3 according to Fig. 2 which have the same p- and E-values as the original disk with holes). Elimination of the thickness d from eqns (35) and (36) yields a nonlinear equation for the determination of 1 which is contained in rj2 and r23:
+
6,3r23(2r12
3
2
- 3r,2r2,
+ r :3 )] = 0.
(39)
Numerical calculations suggest that in the range of relevant load parameters there exists for every load parameter combination a unique solution 1 in the interval 0 < 1 < 1. After solving 1 from eqn (39), d follows explicitly from eqn (36): 2nr12(r23-
d= E&,3
-
P2W2 U,I) - 2
r12brd
ri2(r2,
43 - 42 - ri2) - ~ 3
I.
Fig. 8. Comparison of the mathematical substitute model with its eqmvalent finite element model.
(40)
Model 1: model adaptation by two material parameters. The geometric parameters d and 1 are iixed by setting d = 2nrLM, i.e. equal to the mean circum-
Again, attention should be paid to the case for which the denominator in eqn (40) tends to zero which implies that d tends to infinity. This method is also applicable for o = 0 (nonrotating disks or similar nonrotating structures).
ference of the middle ring region 2, and 1 = 1, which means that the radial extension of the middle ring region is equal to the diameter of the noncentral holes of the original disk. Assuming w # 0, the parameter pZ follows explicitly from eqn (35) as
2(r,2a,l - r230,3) P2 = rLM(ri3
Substitution of E2:
-
G2b2
(37)
’
of eqn (37) into eqn (36) yields the value
For u,~-u,, tending to zero E2 tends to infinity. In case of a nonrotating disk, i.e. w = 0, eqn (35) cannot be fulfilled by arbitrary Q,, and G,,~values. In this case only an optimized value of E2 can be provided approximating both of the boundary conditions. Since the parameter p2 affects only the centrifugal forces, it has no influence if w = 0. Model 2: model adaptation using two geometric parameters. The material parameters p2 and E, are
3.2. Axisymmetric model with hoop stresses (model 3) The choice of a continuum without hoop stresses for the middle ring region in Section 3.1 was based on the consideration that-due to the holes-the hoop stresses in the original structure are interrupted in that region. This is true if the relative distance between the holes is rather small. In case of only a few holes around the circumference, large parts of the middle ring region are far away from the neighboring holes and are consequently able to take up hoop
stresses, so that the approach of Section 3.1 might not be optimal. This particularly applies to the first method introduced in Section 3.1, while the second method may be able to balance this effect by means of the parameter I, which reduces the radial extension of the region without hoop stresses as its value decreases. In the current section, the middle ring region 2 for fixed 1 = 1 (Fig. 2) is modelled (model 3) by a standard axisymmetric continuum (just as the inner
M. K6hl et al.
1056
Model
1
Model
Model
2
3
Model with d-reduction only
Model with ignored holes
Fig. 9. Illustration of the different models. and outer ring region) which is able to take up radial stresses as well as hoop stresses (biaxial stress states). The model simulates the loss of stiffness and mass due to the holes by a reduction of the parameters E2 and pz. The boundary conditions at the interface between the middle ring region and the inner respectively outer ring region can be formulated in terms of the radial stresses, since for all three ring regions the same width b is assumed, and in terms of the displacements: ori, = c,,1; 42
=
%I;
~ro2 = cri3r 42 = 24i3.
(41)
E2
2&
K3 + VI43 + (1 - vH21rI243 - I(1 - VP:3 +
(3 + v)ri2Ir23hi~
(45)
tends to zero then the solutions E2 and p2 of eqns (43) and (44) tend to infinity.
(42)
Hence ring region 2 can be considered as a disk with centrifugal loading p2w2, outer pressure cti3, and mner pressure u,~, uniquely defining Ui2and u,,~[eqns (A3)-(A5)]. Substituting these into eqn (41) yields a system of two linear equations for the determination of p2 and E2: - 4JI + V~,ol = n r12 r23 -r12
Just as model 1 in Section 3.1, this method can only provide optimally approximated, not exact, values for E2 in the case where the disk does not rotate (i.e. w = 0), since in this case eqns (43) and (44) can generally not be fulfilled simultaneously for given a,, and ori values. If the expression
43 + 4 ori - 2 Qrol rz3 - r12
(43) Gi3 + ri2 H2 E2 - Ui3 + vBe3 = -2 o,oI + 2 Ori r23 - r12 r23 - 112 r23
WI
4. EXAMPLE CALCULATIONS OF THE STRESSES INNER AND OUTER RING REGION
IN THE
Each of the three models introduced in Section 3 yields identical stresses in the ring regions r, < r < rLM - r, and rLM + rL < r < r, , since the given quantities w and c,,, , oti, for the inner respectively cW3,oro3for the outer ring region uniquely determine the stresses in the respective region. Additionallyfor the purpose of comparison-stresses were calculated by two other approaches: (I) a model where the holes are ignored (i.e. disk without holes); (II) a frequently applied method where the loss of volume due to the holes is taken into account by a proportional reduction of thickness d in the ring region 2 (cf. model 2, but with constant 1 = 1). Figure 9 compares the five different models. Example calculations were performed for a rotating
1057
Axisymmetric substitute structures IO
10
h
B --c-
9
9
, I7
10
1”’
9
I
\
6.8
8
I .I
I
I
I
8
E I
7
7
I
7
I
1
I
I
6
6
\
5
6
I....----...-.---..
I
I
’
I
.!
-d
I
,
, .\_
i_
4
‘\
.A
j_
3
\.
\
.
I
2 1
_
.._.__
-d-reduction
II 1
3
r/c
b)
------‘exact
only
I1
I
3
2 r/r,
c)
Fig. 10. Comparison of hoop stresses (20 holes; o,, = u,~ = 0).
disk with inner radius ri, outer radius r0 = 3ri, radius of the centre of the noncentral holes r,, = 2ri and hole radius r, = 0.1047ri. Three cases are considered:
in Figs 12 and 13 and for the third case in Figs 14 and 15. In addition to the exact stresses calculated according to Section 2, each figure shows three different approaches: (a) the stresses modelled according to this report (models 1, 2 and 3); (b) the stresses of the abovementioned model (I); (c) the stresses of the abovementioned model (II). The curves (a) show a very good agreement between the exact and the modelled values up to a small distance from the hole region-in contrast to the curves (b) and (c)--indicating the good quality of the present approach.
(1) a disk without radial loads (i.e. cti, = u,~~= 0) containing 20 noncentral holes; (2) a disk without radial loads containing 40 holes; (3) a disk containing 55 holes with the radial loads a,, = c,~, = l.25rfpw2. The exact as well as the modelled hoop and radial stresses along a radial line (ri < r d r,) passing through the center of a noncentral hole are plotted for the first case in Figs 10 and 11, for the second case
,_j
21----
I” .._.&.Ji.
, .* _
i
T
!
I
“.,j.....,.........._ I’ i !
.
. ,.
’, .:....i _____.___._ ,
, ..i..,I I .i..,
1 2
: i
..;.
,
..i.
I + I I 4 I I , I \ I
..j. ’
;;g
0.4
e5 ..._.I --.... o)= I
..;.
t
I , 0.2
t-/r,
i
. . . ..I
..i.
I 1
\ 3
+....
1
2 b)
3
0”“”
2
1
r/ri
Fig. 11. Comparison of radial stresses (20 holes; uti, = Q,,~= 0).
c)
3 r/r,
1058
M. K6hl ef al.
$
-
8
-
!.
i I !..:.*
Ni_ 7
I
\
i ..................! .:.
i I I I .,I .:.I I I I I I .;. I
I I
I I .:.I
ti-
.I
6
I I I I II 1.. .:. I I
5
I
Y
I
.
c
4
3
.). ..I .____,........_.
.:.
?\
‘.\\ \\ .._... ,.’ .‘._
x. \
______. exact
______. exlJct
-
-
modelled
t
holes ignored
Fig. 12. Comparison of hoop stresses (40 holes; u,,, = cm, = 0).
5. COMPARISON OF THE THREE MODELS WITH RESPECT TO PARAMETER DEPENDENCY UPON THE LOADS
The three methods of replacing the ring region containing holes by axisymmetric continua introduced in the previous sections require the determination of parameters which depend upon the geometry and upon the applied loads. Numerical calculations for different geometries showed that for the relevant geometric configurations as well as for the relevant range of the loads the geometric influence is well represented by a single
A
“3 4 Nc V
1.8
1.8
1.6
1.6
parameter q which is defined by the angle CLintroduced in Fig. 16 as the ratio of 2a NL to 2n, i.e. the proportion of the angle taken by a number of NL holes to the angle of a whole circle: q:=!!La H
=N,arcsinrL. x
(46) rLM
This parameter varies between the values q = 0 (no holes) and q = 1 (touching holes). For the calculations performed, the same disk radii as for the examples in Section 4 were chosen. Thus, the radial
\ b’
1.4
‘\ _.;._‘r._..........
1.2
! ,
0.6
0.8
.I
0.6
.:
:
I’
L a>
1
3
2
1
r/c
b)
r/r;
Fig. 13. Comparison of radial stresses (40 holes; cti, = u,~ = 0).
2 c)
3
t-/r,
1059
Axisymmetric substitute structures
8
8
6
5
--__--.
exact
-
holes
-r----.
exact
4
3i
t
ignored
3
2
1
b)
3
2
1
r/r;
c>
r/r;
Fig. 14. Comparison of hoop stresses (55 holes; unl = CT,~#O).
respectively 3.2. (model 2 and 3) show only little parameter dependency within the region of practically relevant load values. In order to demonstrate these results, the parameters of the respective methods were collected from the calculation of a rotating disk without radial stresses at the inner and outer circular boundaries (i.e. o#O and CT~]=CT~~=0). The normalized parameters, EJE and p2/p respectively d/(2nrtM) and A, are plotted for model 1 in Fig. 17, for model 2 in Fig. 18, and for model 3 in Fig. 19. Note that Fig. 17 shows extremely high EJE-values for small q which could be interpreted by saying that the model tries to
distance between the center of the holes and the inner respectively outer boundary (i.e. r,, - ri and r, - rLM) is about 10 times the hole radius. Since an axisymmetric model is looked for which does not depend upon the applied loads, the three methods were compared with respect to the load dependency of their parameters. The method of Section 3.1 (model 1) showed the greatest influence of the loads, particularly for small q-values. This result is reasonable since the method uses a continuum without hoop stresses which is not adequate in case of only a few noncentral holes as already assumed in Section 3. The other two methods of Section 3.1.
\ b’
2.4
2.8
2.8
2.4
2.4
2
2
1.6
1.6
1.6
1.2
.. . . . .. .. .
1.2
..L.i.....35 I : , Ii,
0.8
0.8
0.8
_
:
.... ... .. ...y+._
, I ;
,;, 0.4
0.4
Ii,
2
1
a>
3 r/r,
3
2
1
bj
,+
..___
u@
2;
m:
- .... ... ... ....+.. Ii,
.
s
Ii,
I
’ : :
0 1
r/ri
Fig. 15. Comparison of radial stresses (55 holes; btil = u,,,~+ 0).
2
4
r/ri
1060
j:.j._ ::.__ .A
M. Kohl er al.
09 0.8
j
i
:
07
.; :
06
:
0.5
.._:_
/ ;.
.i_
; ._
:
: _
:
,.
:
/
:
/
:__
:
j
._.!
:
:
0.4 0.3
a=arcsin(r,/r,,)
9
0.2 0.1 0
0
I 0 1
I
I
0.3
0.4
I
0.5
I
I
I
0.6
0.7
0.8
II 0.9 D
Fig. 16. Definition of angle a and of the ID-, OD- and RAD-positions. balance the lack of required stiffness in circumferential direction by an increase of radial stiffness. This confirms the above-mentioned inadequacy of model 1 for small q. Next, model-calculations using the parameters of Figs 17-l 9 were performed for load cases including radial stresses a,, and/or u,~. The hoop stresses c,,, at the inner and Q,,~ at the outer boundary obtained by this approach are in general different from the required values calculated from the original disk containing noncentral holes. Therefore, the dependency of the three models upon different load conditions can be demonstrated in terms of the relative deviations,
I
0 2
Fig. 18. Parameters for model 2.
For the rotating disk (i.e. w # 0), these deviations in percent as functions of q are presented in Fig. 20 for a,, = 1,25rfpw* and a,, = 0, in Fig. 21 for uri, = 0 and oro3= 1,25rfpo*, and in Fig. 22 for ori, = a ro3 = 1,25r:pw*. Each figure contains five deviation curves. Apart from the three curves for each method, there is-just as in Figs 10-15 (cf. illustration in Fig. 9)-one that shows the deviation for the case that the noncentral holes are simply neglected, i.e. for the disk without noncentral holes, and another one which corresponds to a model where the loss of volume due to the holes was taken into account by a proportional reduction of the thickness d. A comparison of the curves makes clear that using any of the three methods introduced in this report yields better results than either neglecting the holes or
1 0.9 0.8 28
0.7
2.4
06 0.5
16
0.4
12
0.3
0.8
0.2
0.4
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
I 0.6
I 0.7
II 0.9 77
rl
Fig. 17. Parameters for model 1.
I 0.8
Fig. 19. Parameters for model 3.
1061
Axisymmetric substitute structures
TX C
20 16
-
;
12
-
.:
a
-
4
-
0 -4
-
-a -12
I., ._...
,
<
,. :
2
-<
0
c
-4
:
-
‘. .’ I I
0
.: :
4
-
-‘I3 -20
i i
.
0.1
...
...- ‘. 0
.
.. .'0 0
I 3
0 2 0
I 0.4
I
_a
model 1 model 2 model 3
-12
A 0 d-reduct,on holes ignored only I 1 1 1
0 5 0.6
0 7 0.8
1
09
.
:
:
.
,
.
:
,_ :
-16
1
I..
,
,
-201
” 0 01
0.2
0.3
’
”
0.4
-i
.
:
I.
, ’
0 5 0.6
”
0 7 0.8
’
09
‘I 20
20 ,6
d I1 : %d:l 2
16
:’ 0 model. 3 +_...;.A holes Ignored o d-reduction only . --. .- . _:
:
1.2 a
1 rl
_
0 model 1 0 model 2 -. 0 model 3 ,_ A holes ignored ; 0 d-reductloo , .i. I ,
:
.._____
.
12-,
..:.
:-’
4
only
0 -4 -a -12 -16 -20
0
..:.
i
I 0.1
I 0.2
,__ , I 0.3
I 0.4
.e
.__,
:..
,.
:
I
I
I
I
0.5
0.6
0.7
0.8
.E
h
1 0.9
1 17
Fig. 20. Deviation of hoop stresses for w # 0, on, # 0, a,0, = 0.
-
c
.-
:‘
12
-
a
b”
4
z
:
: ,
.:
,
:
i--
0
“:‘ .; ‘_____ :. :
I
I I 0 2 0 3 0.4
.,._ ,
I
I
I
0.5
0.6
0.7
I 1. 0 a 0.9
1 rt
Fig. 22. Deviation of hoop stresses for w # 0, un, = 0,) Z 0.
i 0 model 1
0 model 2 I 0 model 3 A holes ignored 0 d-reduction only i i i.. ., i... ,_
,.
-a
!
__!_. :
_‘
,
__,
:
,/ :
,
;I$:1
I
1 2
._( -...i
1. . : .. ... : ..:..: . . ... i- ._. : .._; . .: :_. ..,
-12 -16 -20
z-201
-0
Q model c .-
-20
-0
I 0.1
J
0 -4
6 .+ 0 ,> aI D
:
G
-12 - _.(... ; _,fj ___:. _.,
,2
-.
I
..: ..:
0
’
0.1
’
0.2
’
0.3
’
0.4
’
0.5
.: : I
.:
0.6
’
0 7 0.8
’
0.9
I 0.6
I 0.7
I 0.8
, 0.9
’
, ’
1
i : ;:zl:l$ i A holes
ignored @ d-reduction only a
*)
&
4
%
0 -4
s , -0
0
I 0.1
I 0.2
I 0.3
I 0.4
I 0.5
,
;
I I I 0 6 0 7 0.8
I 0.9
I ‘I
Fig. 21. Deviation of hoop stresses for o #O, CT,,= 0, 04 f 0.
-a
.-
-12
.?
-16
5 -0
-20
-
0
.
:
I 0.1
I 0.2
.
j 0.3
I 0.4
I 0.5
‘I Fig. 23. Deviation of hoop stresses for w = 0. o,, =O, c,, # 0.
1062
M. Kohl er al.
28 27 26 25 24 23 22 21 2 19 18 17 16
0 Fig. 24. Stress concentration factor K,,
STRESS CONCENTRATION
02
03
04
05
06
07
08
These concentrated stresses should be derived from the model, which was achieved by introducing stress concentration factors. The exact concentrated stresses around the noncentral holes were calculated by the method in Section 2. The highest values are located at three points: at the inner point r = rLM - rL (called the ID-position), at the outer point r = rLM + rL (the OD-position) and, for great q-values, at the point r = r, (the RAD-position) defined by the tangent through the disk centre (Fig. 16). In Figs 10, 12 and 14 the stress peaks at the ID- and OD-position are well illustrated. In order to reproduce the stress peaks at these three points by using the model stresses, the exact stresses are written as a linear combination of the local model hoop stresses and model radial stresses. For the IDand OD-position this yields:
FACTORS
For real applications it is essential to determine the concentrated stresses at the surfaces of the holes.
06
06-
'
t
01 0
’ 01
’
0 2
’
0 3
’
0.4
’
0.5
’
0.6
’
0 7
[
Fig. 26. Stress concentration factor K,,.
reducing only the thickness d. It is also obvious that the above-mentioned decreasing validity of model 1 for decreasing q-values leads to greater deviations. Models 2 and 3 yield small deviations; model 2 seems to be a little better. Remarkable results were revealed by an additional calculation of the deviations for the nonrotating structure (i.e. w = 0) with an outer radial load (T,~~# 0 and on3 = 0, plotted in Fig. 23. Although the model parameters were determined for the rotating disk (i.e. w # 0) without inner or outer radial loads, the deviations for the models 2 and 3 are still small. Remember that it was not possible to find an exact solution for model 3 in the nonrotating load case. These results indicate that the applicability of the models 2 and 3 reaches far beyond the specific load case for which their parameters are calculated. 6.
01
’
’
0.8
0.9 T
Fig. 25. Stress concentration factor K,,.
Fig. 27. Stress concentration
factor
K,,.
IO63
Axisymmetric substitute structures
o ID-Pa o ID-Pa
: a,-deviotfon
0 OD-Pm
0
-4
III 01
0
02
III 04
03
RAD-Pos
05
: a,-dewotlon : a.-dewotlon
1 07
06
q
11 08
: a,-deviation
OD-Pos
a,-dewotion
0 RAD-Pos
I 1
09
0
I
I
I
I
I
I
01
02
03
04
05
06
a,-deviation
1 07
I
I
08
09
77
for
w # 0,
Fig.
of the regions has to be taken outside of region 2 (note that cr,is not continuous at the boundary), and for the RAD-position:
Calculations were performed for a number of different pairs of load combinations. The resulting concentration factors showed a certain scatter, particularly for K,, and K,, if instead of the above selected mean values local model stresses in region 2 were used. The choice of the mean values, however, reduced the scatter. Based on these results, unique concentration factor curves were selected. They are plotted in Figs 24-27. Their quality is assessed by the accuracy of the modelled stress peaks at the ID-, OD- and RADposition using model 3 together with the parameters according to Fig. 19. This is illustrated for some examples in Figs 28-32. In these figures the deviation
where o~modc’) at the boundary
O(mcW uwCt)
=
,
Where
_ K,, gjmdel),
L
1y
(49)
Ir 1-q
apmW
and
qjmoW
are mean values of their respective values at the ID- and OD-position outside of region 2. Note that the modelled radial stresses are scaled by 1 - n in order to consider the stress increase due to the reduction of the circumferential cross section in the hole region, and thus limiting the magnitude of the concentration factor. The determination of the stress concentration factors K,,, K,,, K,,and K,,requires for each q-value two calculations of the disk subjected to different loads, which yields two linearly independent equations for eqn (48) as well as for eqn (49). &Or-
30. Deviation
of peak stresses un, = 0, or03# 0.
Fig. 28. Deviation of peak stresses for o # 0, CT,,= 0, u,al = 0.
(50) between the modelled and exact peak stresses in percent is plotted as a function of the geometric
I ID-Pos
: o,-devlotmn
1
OD-Pos
o,-devlatlon
0
RAD-Pos
: o,-devlotmn
0 OD-Pos
I
ID-Pas
: a,-devhon a,-deviation
I
.
0
RAD-Pos
I
I 06
u.-deviabn
:
-4
I 0
01
I 02
I 03
I 04
I 05
I
I
I
I
0.6
0.7
08
09
1
77 Fig. 29. Deviation of peak stresses for w #O, CT,,# 0, 6,,, = 0.
-4
l0
i 01
02
I 03
i 0.4
05
I 07
I 08
I 09
1 r
Fig. 31. Deviation of peak stresses for w # 0. CT,,,= o,,,~# 0.
1064 6\”
M. Kohl et al. 10
: I
~
: ; Y 0
,
:
0
:
ID-Pas.:
REFERENCES
;
1. M. E. Ang and C. L. Tan, Stress concentrations at holes in thin rotating disks. J. Strain Anal. 23,223-225 (1988). 2. W. A. Green, G. T. J. Hooper and R. Hetherington, Stress distribution in rotating disks with noncentral holes. Aeronaut. Q. 15, 107-121 (1964). 3. H. G. Edmunds, Stress concentrations at holes m rotating disks. Engr 618620 (1954). 4. S. P. Timoshenko and J. N. Goodier, Theory of Elasficiry, 3rd edn, International Student Edition. McGrawHill, New York (1970). 5. N. I. Muskhelishvili, Some Eusrc Problems of the Mathematical Theory of Elasticity. Noordhoff, GroningenHolland (1953).
o,-deviation
0 OD-PCS.:
a,-devmtion
0
: o,-devmtion
RAD-Pos
4
E 2 “0 c 0 .+0 .z
0
-2
APPENDIX A: STRESSES AND DISPLACEMENTS IN THIN RING (WITHOUT NONCENTRAL HOLES)
u
-4
-
0
0.1
02
0.3
04
0.5
0.6
07
0.8
09
A
1 77
Fig. 32. Deviation of peak stresses for w = 0, o,,, = 0, 0,) # 0.
parameter r~ according to eqn (46). The load parameters in Figs 29-32 are identical to those in Figs 20-23; Fig. 28 shows the additional load case w # 0 and CT,,,= a,,,, = 0. The large deviation values of up to +26% for small 1 values at the RAD-positionsparticularly in Fig. 28-are insignificant since for small r~the stresses at the RAD-positions are much smaller than at the ID- and OD-positions. Significant stress peaks at the RAD-positions emerge only for higher q values. Apart from this, the deviations in Figs 28-3 1 are smaller than 3%, predominantly with a positive sign which means a conservative approach with respect to subsequent life prediction. The deviations for the nonrotating disk (Fig. 32) do not exceed 4%.
The stresses and deflections in a ring rotating with circular frequency o and subjected to inner and outer radial load o, respectively u,,, are under plane stress conditions [4]:
u,(r) =
rfr:(un
-urn)
ri-rf
2 2 1 ,-+ w, - u,r,
r2
ra - rf
(Al) u,(r) = -
2 2 umro - unrl
rfra(u, -cm) 1 .-+ r2 ri -r: 3+v +Tpo2
i-z- rf
64.3
(
u(r) = i [u,(r) - vu,(r)].
According to eqn (A2) the hoop stresses u, at the inner and (I,~ at the outer boundary, r = r, respectively r = rO. are:
(A4) ura = 7.
r* + r2
2r2
&“~+~u~
CONCLUSIONS
+Tpm(ri+$r:).
Three different axisymmetric substitute models for the circular disk with noncentral holes have been introduced. These models fulfil the two necessary requirements for a model to qualify, i.e. they reproduce the stresses of the original structure at large distances from the hole region, and the stress peaks occurring at the hole surfaces can be calculated using appropriate stress concentration factors. Furthermore, each model is determined by two parameters which for most practical purposes depend geometrically on the relative distance between the holes only. However, an additionally favoured model feature, the independence from specific load conditions, was not adequately met by model 1. Models 2 and 3 on the other hand are both of good quality with respect to load-independency. For finite element calculations the choice of model 3 would have the advantage that the corresponding axisymmetric ring element is of the same type as the one used in the bulk of the structure.
(A5)
Instead of known inner and outer radial stresses u, respectively u,~, for the application in Section 3.1 and 3.2 either u,, and un, are given (ring region 1 according to Fig. 2), or, u,~ and, ura3 (ring region 3) are known. Therefore, explictt expresstons for u,~, and e,O1as functions of a,, , u,,, in ring region 1, and for u,~ and u,,~as functions of cm3, u,,~ in ring region 3 are required. From eqns (A4) and (A5) one obtains:
-~pwi(r:z+~r:)].
=rl3
=
--
ri - r& =,03 2rL
[
ri + 4, - 2 cm3 r.
-
r23
(A6)
>I.
(A7)
1065
Axisymmetric substitute structures APPENDIX Rr RADIAL STRESSES AND
DISPLACEMENTS IN THE MIDDLE RING REGION ACCORDING TO SECTION 3.1
As explained in Section 3.1, the middle ring region (region 2 in Fig. 2) can be considered mathematically equivalent to a bar with constant rectangular cross section A = 6 d. According to Fig. Bl, the equilibrium of forces requires the equality between the force difference F,, - For and the centrifugal forces integrated over the interval r,>Q r < ro2:
r23
r01 F,z - Far =
s r=r,*
Ap,02r
dr = f b . dp2w2(r$, - rf,).
@I)
Applying the relations F,2= b d un2 and Fo2 = b du,, , the difference of inner and outer radial stresses can be expressed by 1 (B2) O”Z- ~K,z= 2 p202(r& - r:,). ui2 The radial stress u,(r) at an arbitrary location r, r,2 < r Q r,, , follows from eqn (B2) by replacing u,~*by u,(r) and r2, by r: Fig. Bl. Bar substitute structure of the middle ring region.
u,(r) = ufi2- ip2c02(r2 - rf,).
Substitition of eqn (A6) into eqn (A5) and of eqn (A7) into eqn (A4) gives:
The displacement-increment Au = ud - u,~ is obtained by integrating the strain r(r) = u,(r)/Ez:
(B3)
‘02 u,(r)dr =’ E2
XP
r:,-rf----
:;:
1
(A8l
s ,=,a rf2)
-iZp2d(r2-
dr 1
r’, + r&
cn3 = 2r:,
r: - rz3
a,03--u&f8 2rL
2
r23 - rt2 =---ucr,,+p2w
3fV
2E2
352
rl(ra + r&) - 2r$ 2rk
1. (A9)
4 - - ri2 . 3
1
(B4)