Efficiency of algorithms for shear stress amplitude
calculation in critical plane class fatigue criteria
A. Bernasconi
a,*, I.V. Papadopoulos
b aDipartimento di Meccanica, Politecnico di Milano, Via La Masa 34, I-20156 Milano, Italy
b
European Commission, Joint Research Centre, IPSC, I-21020 Ispra (VA), Italy Received 7 July 2004; received in revised form 25 November 2004; accepted 24 January 2005
Abstract
Fatigue criteria that belong to the critical plane class necessitate unambiguous definitions of the amplitude and mean value of the shear stress acting on a material plane. This is achieved through the construction of the minimum circle circumscribing the path described by the tip of the shear stress vector on each plane. By definition, the centre and the radius of this circle provide the mean shear stress and the shear stress amplitude, respectively. The search of the minimum enclosing circle is an optimisation problem for which efficient numerical solution schemes are required. Sev-eral algorithms exist for similar situations; however these are not necessarily related to the fatigue strength of metals. In this paper some algorithms are studied to assess their computational efficiency within the engineering framework of the application of fatigue criteria of the critical plane type.
2005 Elsevier B.V. All rights reserved.
Keywords: Fatigue; Critical plane; Shear stress amplitude; Smallest enclosing circle; Computational efficiency; Finite element method
1. Introduction
Amongst the numerous multiaxial fatigue theo-ries, the critical plane approach gave rise to several proposals for both stress-life and strain-life meth-odologies. In the high-cycle fatigue field, this
ap-proach consists of assuming as the measure of damage at a given point of a structure and along a given plane passing through this point, either the amplitude of the shear stress acting tangen-tially to this plane, or its combination (linear or non-linear) with the normal stress. Some of the most frequently cited theories based on this
ap-proach are due to Findley [1], McDiarmid [2]
and Dang Van et al. [3,4].
During cyclic loading, at a chosen point of a structure and a fixed material plane passing
0927-0256/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.01.005
*
Corresponding author. Tel.: +39 02 2399 8222; fax: +39 02 2399 8202.
E-mail address:andrea.bernasconi@polimi.it(A. Bernasco-ni).
through this point, the normal stress vector acting on this plane does not change direction, i.e. it re-mains perpendicular to the material plane. This is indifferent to whether the load is proportional or non-proportional. Therefore, to calculate the amplitude and mean value of the normal stress one only needs to consider its algebraic value, which is a real valued periodic function of time. Conversely, not only the length, but also the direc-tion of the shear stress vector acting tangentially to the fixed plane, may change with time. This poses a problem for correctly evaluating the shear stress amplitude and mean value. From the theoretical point of view, the radius of the Minimum Circum-scribed Circle (MCC), encompassing the plane curve described by the tip of the shear stress vector on each material plane, is at present the preferred definition of the shear stress amplitude. Actually, this proposal is applicable to any periodic stress path, because it is free from the conceptual and numerical drawbacks typical to other approaches as will be shown later in this paper.
The problem of finding the minimum circle cir-cumscribing a plane curve had originally been ad-dressed in its discrete form. Actually, the question of finding the smallest circle enclosing a given set of distinct points lying on a plane was formulated as early as 1857 by Sylvester[5], who also offered a solution using elementary geometrical methods in 1860 [6]. A geometrical solution was found
inde-pendently by Chrystal in 1885 [7]. The problem
reappeared in the sixties and early seventies in the field of Operations Research as a facilities loca-tion problem. For example reference, Toregas et al.[8]. In fatigue of materials, the concept of the MCC was first introduced by Dang Van et al.
[3,4], in the early eighties. Later on, also in the field
of fatigue, Papadopoulos [9] examined the
prob-lem in the n-dimensional Euclidean space present-ing specific applications in the 5-dimensional stress-deviator space. Recently, the MCC concept found renewed attention in robotics, e.g. to find the optimum layout of a robot arm supposed to pick up items lying at varying distances from it
[10].
From the short overview above, it is clear that finding the MCC is a common problem of many applications in various scientific fields and a
cer-tain number of algorithms have been created to solve it. The main attribute of each algorithm is the amount of running time required to find the solution and a secondary attribute is its ease of implementation. In the field of mechanics, contem-porary fatigue assessment of components and structures is performed by the finite element meth-od, often associated with the application of a fati-gue criterion of the critical plane type. This approach requires the examination of all the mate-rial planes passing through each point of a struc-ture, and thus generates a considerable amount of calculations. For engineers dealing with the fa-tigue assessment of large structures or components the speed of the algorithm used to determine the shear stress amplitude on each plane is of para-mount importance, and more efficient approaches are required.
2. Definition of the shear stress amplitude
Let us consider a structure submitted to peri-odic loading. At a given point of the structure the applied stress tensor r varies periodically with time:
rðtÞ ¼ rðt þ T Þ ð1Þ
A material plane D passing through the point under consideration is identified by its unit normal vector n, seeFig. 1. The normal stress rN, which is
the projection of the stress vector r Æ n over n, is a vector acting perpendicularly to the plane D:
rNðtÞ ¼ n rðtÞ nð Þn ð2Þ
The amplitude rN,aand the mean value rN,mof the
normal stress can be easily found by considering its algebraic value rN, which is a real valued
peri-odic function of time:
rNðtÞ ¼ n rðtÞ n ð3Þ
Clearly, one has rN ;a¼ 1 2 maxt2T rNðtÞ mint2T rNðtÞ ð4Þ rN ;m¼ 1 2 maxt2T rNðtÞ þ mint2T rNðtÞ ð5Þ
The shear stress vector s, is the projection of r Æ n over the plane D,Fig. 1:
sðtÞ ¼ rðtÞ n ðn rðtÞ nÞn ð6Þ
Therefore, s acts tangentially to plane D, but not necessarily always in the same direction. For the property of periodic stress history the tip of the shear stress vector describes a closed curve W on D. The definitions of the shear stress amplitude and mean value are trivial problems only in the case of proportional loading. The stress tensor at a given point of a structure submitted to propor-tional periodic loading is written as
rðtÞ ¼ rafðtÞ þ rm; rm¼ kra ð7Þ
where rais a constant stress tensor, and f(t) is a
periodic function of time with values between [1, 1] and zero mean value, i.e. Tf(t) dt = 0 and
k is a scalar. Let us notice that if rmis not
propor-tional to ra, i.e. rm5kra, then the particular
loading given by Eq. (8) below is called affine
loading:
rðtÞ ¼ rafðtÞ þ rm; rm6¼ kra ð8Þ
For proportional loading, Eq.(7), the shear stress
vector derived from Eq. (6)becomes
sðtÞ ¼ ½f ðtÞ þ k½ra n ðn ra nÞn ð9Þ
The first factor in the right-hand side of the rela-tionship(9)above is a scalar function and the sec-ond factor a constant vector. Therefore, in this case the shear stress vector does not change direc-tion, and the curve W degenerates into a straight-line segment lying on a straight-line passing through the origin of zero shear stress. For a given plane D (i.e. for a given n) the problem is reduced to the evaluation of the amplitude and mean value of a real valued function of time given by
sðtÞ ¼ rk a n ðn ra nÞnk½f ðtÞ þ k ð10Þ
Clearly, one has
sa¼ rk a n ðn ra nÞnk ð11Þ
sm¼ k rk a n ðn ra nÞnk ð12Þ
In the more general case of non-proportional load-ing the shear stress amplitude has to be defined as a particular measure of the shear stress path W. In the past, various researchers have used three differ-ent definitions of the shear stress amplitude. They are briefly examined in the following sections. 2.1. The longest chord method
A first possible definition of the shear stress amplitude consists of the longest chord, or diame-ter, of the curve W, i.e. the longest straight line one can draw between two points belonging to the curve W. The mid point of the diameter of the curve W identifies the mean shear stress vector sm, the modulus of which provides the mean shear
stress value sm. The half-length of the diameter is
the shear stress amplitude sa, seeFig. 2(a).
Unfor-tunately, in some cases this solution is not unique. For instance, when the curve W is an acute-angled isosceles triangle, two distinct diameters exist, i.e. the two equal sides. This implies the existence of two different mean shear stress vectors and hence two possible mean shear stress values (see
Fig. 3).
Fig. 1. A material plane and the normal and tangential components of the stress tensor projection.
2.2. The longest projection method
This method consists of projecting the shear stress path W on every line of the plane D, emanat-ing from the origin O on D. The shear stress ampli-tude sais assumed to be equal to the half-length of
the longest of the W projections. The mid-point of this longest projection of W defines the tip of the mean shear stress vector, seeFig. 2(b). Though this definition recalls the role of the resolved shear stress in the nucleation of fatigue cracks, it is afflicted by drawbacks similar to those in the previous method. This can again be shown with
a shear stress path which is an acute-angled isosce-les triangle. The two longest projections exist along lines parallel to the two equal sides of the tri-angular path W, breaking down the sought after uniqueness of the solution.
2.3. The minimum circumscribed circle method Based on the uniqueness of the Minimum Cir-cumscribed Circle (MCC) encompassing a closed curve in a plane, this method defines the shear stress amplitude as the radius of the MCC and the mean shear stress as the vector pointing form
Fig. 2. Three possible definitions of the shear stress amplitude: (a) the longest chord; (b) the longest projection; (c) the minimum circumscribed circle.
the origin O to the centre of the MCC, see Fig.
3(c). The mean shear stress vector then has the
property of minimising the maximum distance of the shear stress tip from the centre of the MCC: sm¼ min
s0 maxt2T sðtÞ s
0
k k ð13Þ
Therefore, the shear stress amplitude can be evalu-ated as
sa¼ max
t2T ksðtÞ smk ð14Þ
This method is the only one, amongst the three existing, to allow for a unique mean shear stress value (and amplitude) for any possible shear stress path and thus for any periodic non-proportional or proportional loading[11].
3. Numerical methods for finding the minimum circumscribed circle
In practical applications of fatigue criteria, the analytical expression of the stress paths is gen-erally unknown. Instead, a discrete stress time his-tory is available, as might be obtained by a sequence of load cases in finite element calcula-tions. The curve W is then approximated by a poly-gon whose vertices form a set of n points. The problem of finding the MCC of W is then equiva-lent to the computational geometry problem of finding the smallest circle enclosing all these points, for which some algorithms exists.
3.1. Points combination algorithms
The minimum circumscribed circle to a plane polygon with n vertices is either one of the circles with a diameter equal to a line segment joining any two vertices of the polygon or one of the cir-cumcircles of all the triangles formed from every three vertices of the polygon. Accordingly, the group of methods examined in this section takes its name from the finite algorithm. This method examines all pairs of points (i.e. chords), and all triple sets of points (i.e. triangles), and chooses the smallest circle determined by them that still encloses the set of n points (i.e. the vertices of the polygon assumed to accurately approximate the curve W).
The solution is exact (apart from numerical pre-cision and cut-off errors), but the computational time can be high because it is necessary to build all possible circles and verify they contain all the other points. Let us assume that each one of the following calculations is an elementary operation (i.e. it counts one operation):
• drawing a circle of given diameter; • drawing the circumcircle of a triangle; • calculating the distance between two points; • comparing two lengths (distances).
Then the detailed complexity analysis of this algorithm is straightforward. The number of cir-cles based on pairs of points is the number of
combinations of n points by 2, denoted as Cn
2.
For each of these circles, the distance from the centre of the circle to the remaining n 2 points has to be calculated in order to verify the circle encloses these points, making the number of operations equal toðn 2ÞCn
2. The number of
cir-cles based on triple sets of points is equal to the combinations of n points by 3, i.e. Cn
3. Following
the above method, and after checking the circle encloses all remaining points, we arrive at ðn 3ÞCn
3 operations. Thus, we have a total of
operations equal to nop¼ ðn 2ÞCn2þ ðn 3ÞC n 3 ¼ ðn 2Þ n! 2!ðn 2Þ!þ ðn 3Þ n! 3!ðn 3Þ! ð15Þ
Fig. 3. Example of a load path having more than one mean shear stress vector, as defined by the longest chord method.
This further simplifies to nop¼ 1 2nðn 1Þðn 2Þ þ 1 6nðn 1Þðn 2Þðn 3Þ ð16Þ The above result clarifies that the number of oper-ations of this obvious Points Combination Algo-rithm (PCA) is a fourth order polynomial of the number of input points n. Translated into the Big-O notation this reads ‘‘the algorithm runs in O(n4) time’’. For an increased n to better approxi-mate the shear stress path W, the number of oper-ations can be too high, thus practically preventing the application of the method in the assessment of large structures.
Clearly, it is not necessary to search through all possible circles and thus the method can be modi-fied in order to reduce the number of calculations
required. Weber et al. [12] proposed one such
modification in order to reduce the calculation duration of the critical plane approach in fatigue. Their proposal is mainly a variant of the algorithm introduced by Elzinga and Hearn[13]. As usual, it is assumed that the curve W is approximated by a polygon of n vertices denoted as Pi, i = 1 to n. The
algorithm consists of the following steps (see
Fig. 4):
1. Consider all the line segments defined by each pair of n vertices (i.e. chords) and choose the
longest one. Denote as P1 and P2 the points
defining this longest chord and build the circle S0 of centre K0 with it as the diameter. The
operations done so far are Cn2 to calculate all
the chords, plus Cn2 1 comparisons to find
the longest one, and one more operation to draw the circle with a diameter equal to the lon-gest chord, hence a total of 2Cn2 operations.
2. Check if the circle S0 contains all other n 2
points Pi, i = 3 to n; if it does, this is the
MCC, otherwise from now on we know that the researched MCC has at least three points in common with the curve W (approximated by the n points Pi). The operations performed
in this step are (n 2) calculations of distances from the centre K0of S0to the points Pi, i = 3 to
n, plus (n 3) comparisons to find the longest of the distances K0Pi, i = 3 to n. One more
com-parison of the longest of the K0Pi distances
against the radius of S0 is needed to check if
S0covers all remaining points. Hence the
num-ber of operations performed in this step is
(n 2) + (n 3) + 1 = 2(n 2). If the MCC
has been found at the end of this step then the MCC is located after precisely 2Cn
2þ 2ðn 2Þ ¼
nðn 1Þ þ 2ðn 2Þ operations. Clearly, this is the least number of operations (best case) one can hope to perform to find the solution apply-ing this algorithm. If the MCC is not found at step 2 then proceed to step 3.
3. Pick the point that is farthest from the centre of S0, already located in the previous step, label it
P3and build the circumcircle S1of the triangle
P1P2P3the centre of which is labelled K1. Only
one operation (i.e. drawing the circle S1) is
per-formed in step 3.
4. Check if the circle S1 contains all the other
n 3 points, that is calculate the n 3 dis-tances K1Pi, i = 4 to n, find the longest one in
(n 4) comparisons, call it K1P4, and check if
K1P4 is smaller than the radius of the circle
S1. If it is, S1is the MCC, if not, pick the point
P4that is farthest from the centre of S1and with
points P1, P2, P3and P4go to the next step. All
these steps require (n 3) + (n 4) + 1 =
2(n 3) operations.
Fig. 4. Construction of the MCC by the modified Point Combination Algorithm.
5. The four points P1, P2, P3and P4, taken three
at a time, define four circles, the one of which determined by P1, P2 and P3is already
exam-ined i.e. circle S1. Amongst the remaining three
circles, one, denoted as S2, is the minimum
cir-cumscribed circle to the four points P1, P2, P3,
P4. Therefore, three circumcircles are drawn
(i.e. three operations) and for each circumcircle one check is done to see if the remaining point is covered (three more distance calculations). It is not excluded that all three circles cover the four points, in which case we need two more com-parisons to find the smallest one. At most, eight operations are performed is this step.
6. If the circle S2 contains the remaining n 3
points, it is the MCC. As in step 4 2(n 3)
operations are needed to perform this task. If
S2 does not cover the remaining points, keep
the three points that defined S2, add the point
lying farthest from the centre of the circle S2
and with these four points go back to step 5.
Clearly, the total number of operations of the above implementation of this algorithm depends on how many times would be needed to repeat steps 5 and 6 in finding the MCC. This is not a straightforward calculation. Let us assume that in step 5 above the smallest circle covering P1,
P2, P3, P4(i.e. S2) is the circumcircle of the triangle
P2P3P4. If after the check in step 6, it turns out
that S2 is not the MCC, then we disregard point
P1and we add the point lying farthest from the
cir-cle S2, labelled P5, to the three points P2, P3, P4.
With points P2, P3, P4 and P5 we repeat step 5,
which leads to the smallest covering circle of these four points. But this does not ensure that the dis-regarded point P1is covered. Clearly, this situation
arises any time that steps 5 and 6 are repeated. Therefore, one may wonder if this algorithm con-verges to the solution or if the possibility exists for the algorithm to be entrapped in an endless repetition scheme. However, it can be demon-strated that the circles built at each repetition of step 5 are monotonically increasing in radius. Then, since there are a finite number of three-point circles, the process is finite. The demonstration that the circles built at each repetition of step 5
are monotonically increasing in radius is very sim-ilar to the one provided in the paper by Elzinga and Hearn [13]. It is estimated that (n 3) repeti-tions of steps 5 and 6 are sufficient to reach the solution. Therefore the total number of operations
for these two steps amounts to (8 + 2(n 3))
(n 3). Summing the number of operations for
each step of this implementation of the Weber et al. algorithm, one has a total number of opera-tions equal to
nop¼ 3½nðn 1Þ 3 ð17Þ
Although it is possible for different implementa-tions of this algorithm to lead to a different total number of operations than shown above, it is clear that the running time of this algorithm is O(n2)
[14].
3.2. The incremental algorithm
The construction of the MCC is also required
by the criterion of Dang Van [3], who proposed
an incremental procedure [4]. The corresponding
algorithm was inspired from methods employed in the incremental theory of plasticity. An explicit formulation of this Incremental Algorithm (IA) is provided below. As usual, the curve W is approxi-mated by a polygon with n vertices numbered from 1 to n. Vertex number 1 is chosen arbitrarily on the curve W. The other vertices are numbered consec-utively along W. Each vertex is described by a shear stress vector denoted as s(i), i = 1 to n. The steps to follow are (see Fig. 5):
1. Guess an initial value for the mean shear stress vector denoted as sð1Þ
m ; build a circle with centre
at the tip of sð1Þ
m and radius R1. Actually, R1is
chosen equal to zero.
2. Calculate the distance D1 between the current
mean shear stress vector sð1Þ
m and the first vertex
s(1), D1¼ ksð1Þ sð1Þm k.
3. Increment the circle radius by j(D1 R1), 0 <
j< 1. The new radius is R2= R1+ j(D1 R1).
4. Displace the centre of the circle towards a new position, denoted as sð2Þ
m , along the direction of
the vector ðsð2Þ sð1Þ
m Þ. The precise position of
the new centre is calculated according to the rule, sð2Þ
5. The current state is a circle of radius R2with its
centre at the tip of sð2Þ
m . Calculate the distance
D2between the current mean shear stress vector
sð2Þm and the second vertex s (2)
, D2¼ ksð2Þ sð1Þm k.
6. Build a new circle with radius R3= R2+ j
(D2 R2) and centred at the tip of the vector
sð3Þ
m ¼ sð2Þm þ ð1 jÞð1 R2=D2Þðsð3Þ sð2Þm Þ.
7. After j steps as above, the current state is a cir-cle centred on the tip of sðjÞ
m with radius Rj.
Upon calculating the distance Dj¼ ksðjÞ
sðjÞ
mk, one determines the subsequent circle with
radius Rj+1= Rj+ j(Dj Rj) centred at the tip
of the next mean shear stress vector
sðjþ1Þ
m ¼ sðjÞm þ ð1 jÞð1 Rj=DjÞðsðjþ1Þ sðjÞmÞ.
8. The iterations stop when the increment of both the radius and the centre of the circles fall below fixed tolerances, i.e. Rj+1 Rj6er and sjþ1m
sj
mk 6 es.
The number of iterations depends on both the ini-tial guess and the value of the constant j; the accu-racy is dependent on the tolerance, it is higher for lower values of j and for a higher number of points n. Unfortunately, a decrease of j, and/or and in-crease of n, causes longer running times. In the numerical experiments presented in this paper and for the tolerances chosen, this algorithm ran in lin-ear time, O(n). Weber et al.[12]studied the
conver-gence of the incremental algorithm. They found that convergence could not be ensured if the radius in-crease (Rj+1 Rj) at each step is less than the
dis-placementksðjþ1Þ
m sðjÞmk of the centre of the circle.
3.3. Optimisation algorithms
As described in a previous paper[15], the prob-lem stated in Eq.(13)belongs to the class of opti-misation problems known as minimax, for the solution of which some algorithms are available in commercial mathematical software packages or programming libraries. Examples include the function fminimax contained in the Optimisation Package of Matlab[16], the Sequential Quadratic Programming routine developed by Zhou and Tits
[17], or the Solvopt algorithm developed by
Kuntsevich and Kappel[18].
Without entering into the details of the structure of these algorithms, the Matlab fminimax routine, which performs a Sequential Quadratic Program-ming, Quasi-Newton, line search was selected. As for the Incremental Algorithm, the efficiency of optimisation routines depends on the initial guess and the tolerances set for convergence.
3.4. Randomised algorithms
Randomised algorithms are powerful calcula-tion tools for many problems. A simple and effi-cient algorithm that solves the problem of finding the smallest disc enclosing a set of n points in a plane can be found in Ref.[10]and is summarised as follows:
1. The algorithm derives its name from its initial random permutation P1, P2, . . . , Pnof n points,
the first two points of which, P1 and P2, are
used to build the initial circle D1 having its
diameter defined by P1and P2, while the others
are added one by one (seeFig. 6).
2. When adding the point Pi, with i = 3 to n, let us
call Di the current circle. If the point Pi being
added lies inside the current circle, i.e. Pi2 Di
(point P3 in Fig. 6), the next point is added
and the current circle remains unchanged, i.e. Di= Di1; if the point Pi being added lies
out-side the circle, i.e. Pi62 Di(point P4in Fig. 6), Fig. 5. Construction of the MCC by the Incremental
the circle containing this and all the other points that have already been added, will have Pi on
the boundary. This lemma can be easily proven, as reported in Ref.[10]. In this case the subset of i points is passed to a subroutine performing the operations described in the next step. The circle found by this subroutine will be the cur-rent circle Di.
3. Permute the subset of points P1, P2, . . . , Pi1
again (in the example ofFig. 6, this new permu-tation is P3, P1, P2), and add them one by one,
keeping point Pi, denoted from now on as Q
(i.e. Pi Q) on the boundary. The current circle
is now the one having its diameter defined by the first point of the new permutation (P3inFig. 6)
and Q. If while adding point Pj, with j = 2 to
(i 1), it is found it lies inside the current circle Dj (point P1of Fig. 6), then the current circle
remains unchanged, i.e. Dj= Dj1; otherwise, if
point Pj, in this case denoted as Q2(i.e. Pj Q2),
falls outside (point P2ofFig. 6), the smallest
cir-cle containing the j 1 points that have already
been added will have both Q and Q2 on its
boundary. Once found, this circle will become the current circle Dj. To find it, as in the previous
step, the subset of j points and point Q are passed to a second subroutine which will find the smallest circle containing the j 1 points and having Q and Q2on its boundary. The
oper-ations performed by this second subroutine are described in the next step.
4. First build the circle having its diameter defined by points Q and Q2. Then add the third subset
of remaining points P1,P2, . . . , Pj1, one by
one, keeping Q and Q2 on the boundary of
the circle. Therefore if the point being added, let us call it Pk, with k = 1 to (j 1), lies inside
the current circle, this remains unchanged, i.e Dk= Dk1. Otherwise, if point Pkfalls outside,
the current circle is the one built over Q and Q2and point Pk. It is clear that the circle
con-taining all j 1 points and having Q and Q2
on its boundary, is one of the following: • The circle having its diameter defined by Q
and Q2.
• The circle passing through Q and Q2, and a
third point, which is the last found outside the current circle.
When step 4 is completed, the procedure returns to step 3. The circle found at step 4 now becomes the current circle for step 3. Point Pj+1is added
follow-ing the order of the permutation P1, P2, . . . , Pi1.
When all points P1, P2, . . . , Pi1 have been
pro-cessed, with the subroutine of step 4 being called whenever necessary, the procedure returns to step 2. The circle found at the end of step 3, which
is the smallest circle containing points P1,
P2, . . . , Pi1and having point Q on its boundary,
becomes the current circle for step 2. Now one adds the remaining points of the first random per-mutation P1, P2, . . . , Pn, processing point Pi+1.
Whenever necessary, the subroutine of step 3 is called. The algorithm stops when the last point of the initial random permutation, i.e Pn, has been
processed.
The current circle found at the end of the proce-dure is the MCC. As in the case of PCA and, with the same meaning of the expression, the solution is exact. The running time obviously depends on the random permutation, but it can be proven[10]that the expected running time is O(n).
4. Examples
The methods presented in the preceding sec-tions were implemented in Matlab 6.5 script files and run on a Pentium IV, 1.8 GHz processor,
Fig. 6. Construction of the MCC by the Randomised Algorithm.
512 MB RAM, personal computer. Three different load paths were chosen as benchmark applications for the aforementioned methods. Each of them was built with an increasing number of points to test the dependence of both the running times and the number of operations required on the number of points.
The value of the constant j for the incremental method was set to 0.05, and the initial guess for both this and the optimisation method was chosen as the mean value over time of the suand sv
com-ponents. The tolerance was set to a value that al-lowed the same accuracy in the determination of the radius sa, and of the modulusksmk, as found
with the Points Combination Algorithm using six significant figures.
4.1. Shear stress components of different frequencies A first test load path, proposed in Ref.[15], has been employed to compare the efficiency of the numerical methods in finding the centre and the radius of the MCC. It is given by
su¼ 100 sinðxtÞ; sv¼ 100 sinð2xt p=4Þ ð18Þ
This stress path has the property of having four distinct points of contact with the enclosing circle and is represented in Fig. 7. All the algorithms led to the correct solution with the same accuracy. The values of the radius of the MCC, i.e. the shear
stress amplitude sa, and the modulus of the centre
vector, i.e. the modulus ksmk of the mean shear
stress vector, as obtained using all methods are
reported in Table 1 for an increasing number of
input points. It is noticed that su,m= 0 because of
the symmetry of the load path along the svaxis.
The running times are compared by plotting them as a function of the number of input points as seen inFig. 8. This verifies their runtime depen-dence on n. The dependepen-dence is linear, i.e. O(n), for the Incremental Algorithm, the fminimax routine, and the Randomised Algorithm. It can also be seen that the relationship is quadratic, i.e. O(n2), as
pre-dicted by Eq. (17), for the Points Combination
Algorithm of Weber et al.[12]. The running times of the other PCA, the one that examines all pairs and triple sets of points, are reported on a broken axis due to its values being considerably higher than those achieved with the other methods.
Fur-Fig. 7. The first example load path, made of 30 points, and its MCC.
Table 1
Values of saandksmk as obtained for the first load path
n sa ksmk
30 0.124737E+03 0.180570E+02
60 0.124879E+03 0.178393E+02
120 0.124983E+03 0.176776E+02
240 0.124983E+03 0.176776E+02
Fig. 8. The running times of the four algorithms as a function of the number of points of the first example load path (mean values for the Randomised Algorithm are over 1000 repetitions).
thermore, due to its inefficiency this method is not considered in the following examples. For the case of the Randomised Algorithm, the possibility of having large differences in running times depending on the random permutation of points required that the routine was called 1000 times and the mean value of the running times reported. As
shown in Fig. 9, the number of function
evalua-tions of the Randomised Algorithm has a mean va-lue increasing linearly with the number of input points; however, the scatter band can be wide en-ough to make maximum values as high as three times the mean value.
Due to the symmetry of the load path, the search of the MCC could have been solely per-formed along the svaxis. However, the load path
is still appropriate for testing the convergence of the different algorithms in the presence of more than three contact points between it and the MCC. In order to test the algorithms in the more general case where a MCC centre must be exam-ined along both the suand svaxis, asymmetric load
paths have been employed.
4.2. Fully non-symmetric load paths
The first load path was proposed by Weber et al.
[12]as a benchmark curve to test their method. It is
made of 10 points. The corresponding su and sv
coordinates are reported in Table 2. To increase the number of points, cubic spline interpolation was used, with periodic end conditions. This al-lowed for increasing the number of points without
Fig. 9. The mean value of the number of iterations over 1000 samples, as required by the Randomised Algorithm, as a function of the number of points and the scatter band.
Table 2
Load path proposed by Weber et al.[12]
Points 1 2 3 4 5 6 7 8 9 10
su 14 11 7 5 6 7 11 15 21 22
sv 15 81/4 11 + 5(3)1/2 15 6 11 5(3)1/2 7/4 9 10 11
Fig. 10. The second example load path and its MCC with a varying number of base points: n = 10 (a) and n = 40 (b).
altering the asymmetry of the stress path. InFig. 10
the original load path is drawn, together with the one obtained by interpolation made of 40 points. The value of the radius of the MCC, i.e. sa, and
the coordinates of the centre of the MCC are re-ported inTable 3. The comparison of the running times reported in Fig. 11highlights the change in the dependence of the running time of the fminimax routine on the number of points n. Notice that it is not linear as in the first example. This is probably due to the line search performed along both the suand svaxis. Furthermore, the best performance
is once again obtained by adopting the Random-ised Algorithm, particularly for high values of n. Also as before, the running times of the Random-ised Algorithm reported in the graph are the mean values calculated over 1000 runs.
The second non-symmetric load path, see Fig.
12, is defined by the following relations:
su¼ 100 sinðxtÞ þ 50 cosð2xtÞ þ 50;
sv¼ 100 sinð2xt p=3Þ þ 50 sinðxtÞ:
ð19Þ The value of the radius of the MCC, i.e. saand the
coordinates of the centre of the MCC are reported in Table 4; the running times are reported in
Table 3
Values of sa, su,mand sv,mobtained for the second load path
n sa su,m sv,m
10 0.100000E+02 0.120000E+02 0.110000E+02 20 0.103454E+02 0.124503E+02 0.108670E+02 40 0.103565E+02 0.124402E+02 0.108827E+02 80 0.103827E+02 0.125021E+02 0.108830E+02 160 0.103851E+02 0.124996E+02 0.108800E+02
Fig. 11. The running times of the four algorithms as a function of the number of points of the second example load path (mean values for the Randomised Algorithm are over 1000 repetitions).
Fig. 12. The third example load path and its MCC.
Table 4
Values of sa, su,mand sv,mobtained for the third load path
n sa su,m sv,m
30 0.139370E+03 0.393432E+02 0.238853E+02 60 0.139847E+03 0.392287E+02 0.234603E+02 120 0.140058E+03 0.393420E+02 0.236898E+02 240 0.140074E+03 0.393240E+02 0.236808E+02
Fig. 13. The running times of the four algorithms as a function of the number of points of the third example load path (mean values for the Randomised Algorithm are over 1000 repetitions).
Fig. 13. Again it is evident that the Randomised Algorithm is the best performer, even when the ex-treme values of the running times scatter band are considered.
5. Concluding remarks
Let us first notice that the concept of the mini-mum circumscribed circle is a quick and unequiv-ocal method of determining the shear stress amplitude for a number of loading conditions frequently used in multiaxial fatigue testing. The
affine loading Eq. (8) is such an example. The
shear stress acting on the plane D is given by sðtÞ ¼ rð afðtÞ þ rmÞ n
n r½ ð afðtÞ þ rmÞ nn ð20Þ
This simplifies to
sðtÞ ¼ f ðtÞ r½ a n ðn ra nÞn
þ r½ m n ðn rm nÞn ð21Þ
Examining Eq. (21), one can understand that the
path described by the tip of the shear stress vector is a line segment in the direction (raÆ n (n Æ
raÆ n)n), and centred on the point (rmÆ n (n Æ
rmÆ n)n). Therefore, the MCC to the shear stress
path is centred at this point and has a radius equal tokraÆ n (n Æ raÆ n)nk. So, for any affine loading,
the concept of the MCC readily allows the finding of the mean shear stress and amplitude:
sm¼ rk m n ðn rm nÞnk ð22Þ
sa¼ rk a n ðn ra nÞnk ð23Þ
For the case of proportional loading the MCC ap-proach leads to the same solution as that presented in Section 2. It is known that proportional loading is a particular case of affine loading in which rm= kra. Introducing this in Eq. (22) we recover
Eq.(12), which provides the mean shear stress act-ing on plane D in the case of proportional loadact-ing. Another class of multiaxial fatigue loading frequently used in fatigue testing is out-of-phase sinusoidal loading where stress components are given by
rijðtÞ ¼ rij;asinðxt uijÞ þ rij;m ð24Þ
where uijis the phase lag between the rij
compo-nent and a reference stress compocompo-nent. For
in-stance if rxx is the reference stress, then
obviously uxx= 0. Furthermore, uxy would be
the phase difference between rxy and rxx, and so
on. Because the loading path in the stress space has a centre of symmetry, which is given by the mean stress state rij,m, the mean shear stress on a
material plane is derived only from rij,m:
si;m¼ rij;mnj nð krkl;mnlÞni ð25Þ
Then the problem of finding the amplitude of the shear stress on this same plane, instead of being
a minimax problem solvable by Eq. (13), reduces
to a problem of a simple maximisation over time of the shear stress derived from the time dependent parts of the stress components. This result can be generalised also for non-sinusoidal loadings, pro-vided that the load path in the stress space pos-sesses a centre of symmetry.
Turning our attention to the performance of the algorithms examined we notice that the
Ran-domised Algorithm [10]and the Points
Combina-tion Algorithm by Weber et al. [12] can be
considered the best performing methods for load paths made of up to 40 points. For a higher number of points, the Randomised Algorithm and the Incremental Algorithm are preferred due to their running time linearity. Of these two, the Randomised Algorithm is free from the uncertainties typical to incremental methods which in some situations may have poor conver-gence or even fail. These considerations proved the Randomised Algorithm to be the most effi-cient of the methods studied in this paper; more-over, randomised algorithms are simpler and shorter than deterministic ones. This aspect com-pensates for the uncertainty in running times, as it is believed that the probability of a more com-plex deterministic algorithm failing is higher than the probability that a randomised algorithm fails to produce the correct answer in time. Finally, as the number of analysis points on a structure increases, the more the total running time will depend on the mean value of the running times of the algorithm.
Other approaches exist, thanks to the already mentioned vast choice of problems requiring a known MCC for their solution. The linear-time
algorithm of Megiddo [19] and/or the class of
genetic algorithms [20] are sometimes proposed;
however, due to its simplicity, the authors believe that the Randomised Algorithm should be pre-ferred in the engineering framework of the applica-tion of critical plane fatigue criteria.
Finally, the methods presented herein refer only to the class of critical plane criteria, but it is worth mentioning that fatigue criteria based on the stress
deviator [9] require the solution of the similar
problem of finding the smallest enclosing hyper-sphere of a given set of points in the 5-dimensional Euclidean space. Some of the existing randomised algorithms for plane problems can be adequately adapted for these higher dimension problems
[21], thus making them a powerful and versatile
computational tool. References
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