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System reliability of a rotating shaft of an aircraft tailplane
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Cornputt-rs & Slruchue~ Vol. 37,No. 5. pp. 833-836.1990 Printedin GreatBritain.
SYSTEM RELIABILITY OF A ROTATING AN AIRCRAFT TAILPLANE
SHAFT OF
YUANSHENG FENG and KAN NI Department of Aircraft Engineering, Northwestern Polytechnical University, Xian, Shaanxi Province, The People’s Republic of China (Received 15 November 1989)
Abstract-The rotating shaft of an aircraft tailplane can he considered as a structural system with several significant failure modes. Each significant failure mode is constituted by one critical cross-section of the shaft. T’he method of enumerating critical cross-sections is explained. All the significant failure modes are considered as series and correlative for computing the system reliability of the shaft.
1.INTRODUCTION
Substituting
eqn (2) into eqn (1) one obtains
A complex large scale structural system can be considered as a series system with several significant failure modes. The failure probability of single failure
(3)
mode can be computed by level 2 methods [l] and the correlation between each of the two failure modes is considered in [2]; therefore the structural system reliability can be obtained. For the practical structure, such as a rotating shaft of an aircraft tailplane, which is a continuous beam with two hinged supports and a outer cantilever beam segment, how to enumerate the significant failure modes of the continuous beam is a problem requiring solution. In this paper, this problem has been successfully solved and the system reliability of the beam are obtained with high accuracy. 2.
In the strength computation of aircraft structure, the design safety factorf, and reserve stress coefficient 9 are adopted. According to the definitions of these parameters, there is the following relation f, =_fk.
For a continuous beam, fd is the same along the axis; therefore, the value of /Ii is proportional to the value of ?,li. Because this beam is a statically determinate structure, the failure of each cross-section leads to the failure of the whole beam; therefore, all the crosssections are series in reliability analysis. The 20-30 segments along the shaft are separated by equal spaces. The values offCi are computed at the middle cross-section of each corresponding segment, and using thisfCi approximately indicates thef, of the ith segment. The minimum of these fCi is chosen. It can be indicated by
THE BASIC CONCEPT OF ENUMERATING SIGNIFICANT FAILURE MODES FOR A CONTINUOUS BEAM
For a continuous beam, in general, the material of the whole beam is the same; therefore, the coefficients of variations of strengths of different cross-sections are approximately the same. And the coefficients of variations of load effect of different cross-sections are also approximately the same by reason of that the external loads of the shaft are the same. Then, the reliability index pi can be defined as
where R, and S, are strength and load effect of i cross-sections, respectively, and V, and coefficients of variation of strength and load respectively. The symbol computational safety factor,& defined as f, = R/S.
min(.Li) =L
i=l-N
(2)
(5)
where N is the total number of segments, and subscript sl corresponds to the most serious significant failure modes. Adopting the following symbol
means V, are effect, can be
(4)
L==f,il!Ll~
('3
because the failure probability of ith cross-section is proportional to p,, i.e. proportional to& ifx, >f;,, these corresponding segments can be neglected. Assume that altogether P(P, , P2, . . . , P,) segments 833
Y.
834
FENG
and K. Nr
are omitted, then there are N - P segments which need to be discussed further. In general, the values of scl are chosen from 1.2 to 1.5 (the greater SC,,, the less x,). If the mode correlation coefficient p between two different cross-sections j and k is larger than pf, (for example, per = 0.9), these two cross-sections can be considered as only one serious section by reason of that if p approaches 1, the effect of the secondary failure mode can be neglected. Based on the physical concept and computational results, the mode correlation coefficient approaches I if the distance between two adjacent cross-sections is more or less equal. Therefore, first, M = N - P significant crosssections (M, , M2, . . . , M,,,) are chosen by their smaller values of fci; second, the boundaries of the adjacent segments are obtained by the condition that at these boundaries p = pc,, and of course the cross-sections located at the inner of this seg ment are p > pc,. It is obvious that instead of the whole adjacent segment only one cross-section needs to be considered. All these are shown in Fig. 1, where B,, and & mean left and right boundaries of ith adjacent segment, respectively+ In practical terms, during computing the boundaries of the adjacent segment, if this boundary beyond the original bound of natural separation segment, and the nearest segment has been omitted by smaller value of xi, there is no need to further compute extensively. If the boundary of the adjacent segment is less than the original one, there is a reserve region shown in Fig. 1 as a dotted region. In the reserve regions the next significant failure modes and their adjacent segments are then obtained by similar concept and computation, this is shown in Fig. 2, where Ri means ith reserve region. Successive computations proceed until there is no further reserve region. After all the M, significant cross-sections are obtained, the method of narrow reliability bounds can be used to compute the failure probability of the whole shaft.
The following adopted
equation
[a] - a, = 0,
can be (7)
where [a] is the allowable stress of material, a, is the equivalent stress, and both these variables are random ones. The load effects of the shaft cross-section are bending moment, torque and shearing force. Because the shearing stress caused by the torque is much greater than by the shearing force, the effect of the shearing force can be neglected. The normal stress caused by the bending moment and the shearing stress by the torque of ith crosssection are
63) (9) respectively, where P_,, is the applied resultant force of tailplane, Z, and dei are effective length and distance, rf is the radius of ith cross-section, Kp is the plastic corrective coefficient, Z& and Z,*are axis and polar moments of inertia of ith cross-section, and C,, and Czi are corresponding constants. Von Misis’es energy criterion is used to compute effective equivalent stress, uei, as api = J_.
(10)
Substituting eqns (8) and (9) into (lo), one obtains a, = Pspp$5-z
= p*i%iJ C*,
(11)
where C, is also a constant. Substituting eqn (11) into (7), one obtains [a] - PBPPC, = 0,
(12)
where [a] and Pwp are random variables. Let us take the coefficients of variation of allowable stress and applied external load as Vs., = 0.07 and VP= 0.14, respectively. The reliability index /Ii is
3. CONCRETE COMPUTATION
The shaft of a tailplane of certain aircraft is a round tube continuous beam. Its outer radii are the same and its inner radii are different along the axis; therefore, the axis and polar moments of inertia of cross-sections along the axis of the shaft are different.
82‘
safety margin
(13) where W indicates allowable.
&,
%,
B3,
Fig. 1. Division of shaft.
b.,
mean and subscript
‘a’ means
Rotating shaft of an aircraft tailplane
835
M7
Me I
‘6 I
I , I
I
[I
/’
I /
/
Rl
4
4
Fig. 2. Reserve regions of shaft.
Define the symbol f, as the computational factor of ith cross-section, i.e.
Then,
(20)
(14)
f,i = LJC.0 I VW = ~.J,J(G Up,app ). Substituting
safety
Let symbol K be
eqn (14) into (13), one obtains
K = C,/C,.
(21)
(15)
fli=&&&
Then, i.e.
(22) (15a) Now the computational procedure for correlation coefficient is explained as follows. There are two safety margin equations with standard normal variables Zi as follows a,Z,+a,Z*+a,=O 61z, + b,Z* + b, = 0. The correlation coefficient above two equations is
(16)
corresponding
to the
when K-1, p+l. Using this formula we can determine the farthest boundaries round the centrical cross-section of a certain segment under the condition p = 0.9 at these locations. If two cross-sections are located at different adjacent segments, assume that the allowable [uk] and [a,] can be considered as two different variables because their space distance is long enough (comparing with actual case, this may lead to somewhat conservative effect). Therefore, ck Pkl
,Ea,bi P=
,_-
J
_
*
(17)
The normal variables [a] and P_,r can be transferred to the standard normal variables in terms of the following linear transfers
-- [al-
z,
_
ucv )
_ 2-
P.PP - UP,SPP(18) u
P.aPP
’
where u is the standard deviation. Assume that in all the adjacent segments, because the two different cross-sections are quite close, the variable [a] can be considered as the same. Substituting eqn (18) into (12), one obtains -
GUP,~PPZ~
+
+
c,
‘&pp
G~,&,,NJ~,
(23) +
C:&,,)
It is obvious that in general pk, is less than pij. After the concrete computation, there are six significant cross-sections altogether in the shaft. The values of fczi are f,, = 1.997, fcs2= 2.117, LJ = 2.521, Lti = 2.593, fess = 2.746, Lfi = 2.874.
z
CT0.0
u~.uzl
=
(U,,a
-
C,
Up,ppp) = 0 (19a)
and
U~JI - +'p,nppZ2 + Ww - c,Up.app) = 0. WW
Here the
f,,,,
is taken as 1.5. Then, one obtains
/I, = 5.039,
/!I2= 5.479,
/Y,= 6.752,
/I4 = 6.949,
& = 7.342,
/Is = 7.646.
The corresponding
single failure probabilities
Pfi are
P,, = 0.2342 x 10-6,
Pfl= 0.2197 x 10-7,
Pfl = 0.7722 x lo-“,
Pf4 = 0.195 x lo-“,
P,s = 0.112 x IO- I2 )
P,,=O.113
x IO- 13 ,
Y. FENG and K. NI
836
Based on these probability data, there are only two significant failure modes needed, and&, can be taken as 1.2 by reason of that fl,, > 5. Therefore, one obtains f,] = 1.997,
fcsl = 2.117,
/I, = 5.039,
/?I2= 5.497,
P,, = 0.2342 x 10-16, Pn = 0.2197 x lo-‘. Because there are only two significant failure modes, the accurate solution of the system probability P,$ can be obtained as
where P,,2 is the two-order joint probability. on concrete computation, one obtains PI, = 0.663 x lo-
P,,=O.2561
IO
Based
,
x 10-6.
Here
(PfiP/,)/P,, = 9.35%. It means if only one most serious failure mode is considered, the actual system failure probability should be increased by 9.35% over the single failure probability. It is emphatically noted that the following computational method for beam system reliability is mistaken, This method adopts the following expression pr = 1 P,i Ali/ c Al,
Jr,=
‘P,Xax/l, j0
(24a)
where 1is the total length of the beam. In these cases, Pfiindicates the mean failure probability, but this probability is not the actual system failure probability. Because the beam is a statically determinate structure, the failure of any cross-section can lead to the failure of system, and the mean failure probability, in general, is far less than the system failure probability. The conclusion is obvious that this kind of method for computing mean failure probability is unacceptable. 4. CONCLUSIONS
PfS= p,, + p,z - p/12,
plz = 0.4542,
or
(24)
1. Because the original design method of the shaft is not based on reliability optimum design, the distribution of single failure probability along the axis is not so reasonable. In general, to decrease the structural weight and ensure high reliability, it is necessary to have a lot of significant failure modes, and their /Ii are quite close and their values are quite high. But it may be difficult for manufacture. 2. For a statically determinate structural system, only computing one single failure probability of one most serious failure mode or computing the mean failure probability of all the failure modes are both unacceptable. 3. Using the concept of correlativity and magnitude of f,i or /Ii to enumerate the significant crosssection, is an efficient and highly accurate method for a shaft or other continuous structures. REFERENCES
and M. J. Baker, Sfructural Reliability Theory and its Applications. Springer, New
1. P. Thoft-Christensen
York (1982). 2. 0. Ditlevsen, Narrow reliability bounds for structural systems. J. struct. Mech. 7(4), 453-472 (1979).
Cornputt-rs & Slruchue~ Vol. 37,No. 5. pp. 833-836.1990 Printedin GreatBritain.
SYSTEM RELIABILITY OF A ROTATING AN AIRCRAFT TAILPLANE
SHAFT OF
YUANSHENG FENG and KAN NI Department of Aircraft Engineering, Northwestern Polytechnical University, Xian, Shaanxi Province, The People’s Republic of China (Received 15 November 1989)
Abstract-The rotating shaft of an aircraft tailplane can he considered as a structural system with several significant failure modes. Each significant failure mode is constituted by one critical cross-section of the shaft. T’he method of enumerating critical cross-sections is explained. All the significant failure modes are considered as series and correlative for computing the system reliability of the shaft.
1.INTRODUCTION
Substituting
eqn (2) into eqn (1) one obtains
A complex large scale structural system can be considered as a series system with several significant failure modes. The failure probability of single failure
(3)
mode can be computed by level 2 methods [l] and the correlation between each of the two failure modes is considered in [2]; therefore the structural system reliability can be obtained. For the practical structure, such as a rotating shaft of an aircraft tailplane, which is a continuous beam with two hinged supports and a outer cantilever beam segment, how to enumerate the significant failure modes of the continuous beam is a problem requiring solution. In this paper, this problem has been successfully solved and the system reliability of the beam are obtained with high accuracy. 2.
In the strength computation of aircraft structure, the design safety factorf, and reserve stress coefficient 9 are adopted. According to the definitions of these parameters, there is the following relation f, =_fk.
For a continuous beam, fd is the same along the axis; therefore, the value of /Ii is proportional to the value of ?,li. Because this beam is a statically determinate structure, the failure of each cross-section leads to the failure of the whole beam; therefore, all the crosssections are series in reliability analysis. The 20-30 segments along the shaft are separated by equal spaces. The values offCi are computed at the middle cross-section of each corresponding segment, and using thisfCi approximately indicates thef, of the ith segment. The minimum of these fCi is chosen. It can be indicated by
THE BASIC CONCEPT OF ENUMERATING SIGNIFICANT FAILURE MODES FOR A CONTINUOUS BEAM
For a continuous beam, in general, the material of the whole beam is the same; therefore, the coefficients of variations of strengths of different cross-sections are approximately the same. And the coefficients of variations of load effect of different cross-sections are also approximately the same by reason of that the external loads of the shaft are the same. Then, the reliability index pi can be defined as
where R, and S, are strength and load effect of i cross-sections, respectively, and V, and coefficients of variation of strength and load respectively. The symbol computational safety factor,& defined as f, = R/S.
min(.Li) =L
i=l-N
(2)
(5)
where N is the total number of segments, and subscript sl corresponds to the most serious significant failure modes. Adopting the following symbol
means V, are effect, can be
(4)
L==f,il!Ll~
('3
because the failure probability of ith cross-section is proportional to p,, i.e. proportional to& ifx, >f;,, these corresponding segments can be neglected. Assume that altogether P(P, , P2, . . . , P,) segments 833
Y.
834
FENG
and K. Nr
are omitted, then there are N - P segments which need to be discussed further. In general, the values of scl are chosen from 1.2 to 1.5 (the greater SC,,, the less x,). If the mode correlation coefficient p between two different cross-sections j and k is larger than pf, (for example, per = 0.9), these two cross-sections can be considered as only one serious section by reason of that if p approaches 1, the effect of the secondary failure mode can be neglected. Based on the physical concept and computational results, the mode correlation coefficient approaches I if the distance between two adjacent cross-sections is more or less equal. Therefore, first, M = N - P significant crosssections (M, , M2, . . . , M,,,) are chosen by their smaller values of fci; second, the boundaries of the adjacent segments are obtained by the condition that at these boundaries p = pc,, and of course the cross-sections located at the inner of this seg ment are p > pc,. It is obvious that instead of the whole adjacent segment only one cross-section needs to be considered. All these are shown in Fig. 1, where B,, and & mean left and right boundaries of ith adjacent segment, respectively+ In practical terms, during computing the boundaries of the adjacent segment, if this boundary beyond the original bound of natural separation segment, and the nearest segment has been omitted by smaller value of xi, there is no need to further compute extensively. If the boundary of the adjacent segment is less than the original one, there is a reserve region shown in Fig. 1 as a dotted region. In the reserve regions the next significant failure modes and their adjacent segments are then obtained by similar concept and computation, this is shown in Fig. 2, where Ri means ith reserve region. Successive computations proceed until there is no further reserve region. After all the M, significant cross-sections are obtained, the method of narrow reliability bounds can be used to compute the failure probability of the whole shaft.
The following adopted
equation
[a] - a, = 0,
can be (7)
where [a] is the allowable stress of material, a, is the equivalent stress, and both these variables are random ones. The load effects of the shaft cross-section are bending moment, torque and shearing force. Because the shearing stress caused by the torque is much greater than by the shearing force, the effect of the shearing force can be neglected. The normal stress caused by the bending moment and the shearing stress by the torque of ith crosssection are
63) (9) respectively, where P_,, is the applied resultant force of tailplane, Z, and dei are effective length and distance, rf is the radius of ith cross-section, Kp is the plastic corrective coefficient, Z& and Z,*are axis and polar moments of inertia of ith cross-section, and C,, and Czi are corresponding constants. Von Misis’es energy criterion is used to compute effective equivalent stress, uei, as api = J_.
(10)
Substituting eqns (8) and (9) into (lo), one obtains a, = Pspp$5-z
= p*i%iJ C*,
(11)
where C, is also a constant. Substituting eqn (11) into (7), one obtains [a] - PBPPC, = 0,
(12)
where [a] and Pwp are random variables. Let us take the coefficients of variation of allowable stress and applied external load as Vs., = 0.07 and VP= 0.14, respectively. The reliability index /Ii is
3. CONCRETE COMPUTATION
The shaft of a tailplane of certain aircraft is a round tube continuous beam. Its outer radii are the same and its inner radii are different along the axis; therefore, the axis and polar moments of inertia of cross-sections along the axis of the shaft are different.
82‘
safety margin
(13) where W indicates allowable.
&,
%,
B3,
Fig. 1. Division of shaft.
b.,
mean and subscript
‘a’ means
Rotating shaft of an aircraft tailplane
835
M7
Me I
‘6 I
I , I
I
[I
/’
I /
/
Rl
4
4
Fig. 2. Reserve regions of shaft.
Define the symbol f, as the computational factor of ith cross-section, i.e.
Then,
(20)
(14)
f,i = LJC.0 I VW = ~.J,J(G Up,app ). Substituting
safety
Let symbol K be
eqn (14) into (13), one obtains
K = C,/C,.
(21)
(15)
fli=&&&
Then, i.e.
(22) (15a) Now the computational procedure for correlation coefficient is explained as follows. There are two safety margin equations with standard normal variables Zi as follows a,Z,+a,Z*+a,=O 61z, + b,Z* + b, = 0. The correlation coefficient above two equations is
(16)
corresponding
to the
when K-1, p+l. Using this formula we can determine the farthest boundaries round the centrical cross-section of a certain segment under the condition p = 0.9 at these locations. If two cross-sections are located at different adjacent segments, assume that the allowable [uk] and [a,] can be considered as two different variables because their space distance is long enough (comparing with actual case, this may lead to somewhat conservative effect). Therefore, ck Pkl
,Ea,bi P=
,_-
J
_
*
(17)
The normal variables [a] and P_,r can be transferred to the standard normal variables in terms of the following linear transfers
-- [al-
z,
_
ucv )
_ 2-
P.PP - UP,SPP(18) u
P.aPP
’
where u is the standard deviation. Assume that in all the adjacent segments, because the two different cross-sections are quite close, the variable [a] can be considered as the same. Substituting eqn (18) into (12), one obtains -
GUP,~PPZ~
+
+
c,
‘&pp
G~,&,,NJ~,
(23) +
C:&,,)
It is obvious that in general pk, is less than pij. After the concrete computation, there are six significant cross-sections altogether in the shaft. The values of fczi are f,, = 1.997, fcs2= 2.117, LJ = 2.521, Lti = 2.593, fess = 2.746, Lfi = 2.874.
z
CT0.0
u~.uzl
=
(U,,a
-
C,
Up,ppp) = 0 (19a)
and
U~JI - +'p,nppZ2 + Ww - c,Up.app) = 0. WW
Here the
f,,,,
is taken as 1.5. Then, one obtains
/I, = 5.039,
/!I2= 5.479,
/Y,= 6.752,
/I4 = 6.949,
& = 7.342,
/Is = 7.646.
The corresponding
single failure probabilities
Pfi are
P,, = 0.2342 x 10-6,
Pfl= 0.2197 x 10-7,
Pfl = 0.7722 x lo-“,
Pf4 = 0.195 x lo-“,
P,s = 0.112 x IO- I2 )
P,,=O.113
x IO- 13 ,
Y. FENG and K. NI
836
Based on these probability data, there are only two significant failure modes needed, and&, can be taken as 1.2 by reason of that fl,, > 5. Therefore, one obtains f,] = 1.997,
fcsl = 2.117,
/I, = 5.039,
/?I2= 5.497,
P,, = 0.2342 x 10-16, Pn = 0.2197 x lo-‘. Because there are only two significant failure modes, the accurate solution of the system probability P,$ can be obtained as
where P,,2 is the two-order joint probability. on concrete computation, one obtains PI, = 0.663 x lo-
P,,=O.2561
IO
Based
,
x 10-6.
Here
(PfiP/,)/P,, = 9.35%. It means if only one most serious failure mode is considered, the actual system failure probability should be increased by 9.35% over the single failure probability. It is emphatically noted that the following computational method for beam system reliability is mistaken, This method adopts the following expression pr = 1 P,i Ali/ c Al,
Jr,=
‘P,Xax/l, j0
(24a)
where 1is the total length of the beam. In these cases, Pfiindicates the mean failure probability, but this probability is not the actual system failure probability. Because the beam is a statically determinate structure, the failure of any cross-section can lead to the failure of system, and the mean failure probability, in general, is far less than the system failure probability. The conclusion is obvious that this kind of method for computing mean failure probability is unacceptable. 4. CONCLUSIONS
PfS= p,, + p,z - p/12,
plz = 0.4542,
or
(24)
1. Because the original design method of the shaft is not based on reliability optimum design, the distribution of single failure probability along the axis is not so reasonable. In general, to decrease the structural weight and ensure high reliability, it is necessary to have a lot of significant failure modes, and their /Ii are quite close and their values are quite high. But it may be difficult for manufacture. 2. For a statically determinate structural system, only computing one single failure probability of one most serious failure mode or computing the mean failure probability of all the failure modes are both unacceptable. 3. Using the concept of correlativity and magnitude of f,i or /Ii to enumerate the significant crosssection, is an efficient and highly accurate method for a shaft or other continuous structures. REFERENCES
and M. J. Baker, Sfructural Reliability Theory and its Applications. Springer, New
1. P. Thoft-Christensen
York (1982). 2. 0. Ditlevsen, Narrow reliability bounds for structural systems. J. struct. Mech. 7(4), 453-472 (1979).