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Reliability analysis of frame structures with brittle components
Structural Safety, 2 (1985) 281-290
281
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
RELIABILITY ANALYSIS OF FRAME STRUCTURES WITH BRITTLE COMPONENTS RICHARD M. BENNETT Department of Civil Engineering, University of Tennessee, Knoxville, TN 37996-2010 (U.S.A.) (Received August 3, 1984; accepted in revised form February 6, 1985)
Key words: Brittle failure, frames, probability theory, reliability, safety, structural systems
ABSTRACT
A review is provided of the recent developments in the reliability analysis of ductile (elastic-plastic) systems and of the stable configuration approach of structural system reliability analysis. These are combined to obtain a meth-
odology for evaluating the reliability of frames that have primarily ductile components but also a few brittle components. The methodology is applied to two frame structures.
INTRODUCTION
ductile (linear elastic-plastic) but that there are also a few connections that may fail in a brittle manner. The focus will be on handling the brittle behavior and it will be shown that the methodology developed can incorporate other than elastic-plastic behavior (e.g. nonlinear) in conjunction with the brittle components.
There have been many studies concerning the reliability of frame structures. However, most have invoked rather restrictive assumptions as to material behavior; in particular, m o s t studies have a s s u m e d perfectly elastic-plastic ductile behavior. The behavior of actual structures will in general be much more complex. Often there will be the possibility of brittle behavior; a member or connection may fracture and the load carrying capacity be reduced to zero. The proper evaluation of the reliability of the structure must take this into account. A methodology will be developed in this paper for analyzing frames with brittle components. It will be assumed that the behavior of the frame is 0167-4730/85/$03.30
REVIEW OF RELIABILITY ANALYSIS OF DUCTILE FRAMES Most of the work concerning the reliability analysis of ductile frames has been based on the formulations developed by Jorgenson and Goldberg [1] and Stevenson and Moses [2] in which each plastic mechanism is considered a
© 1985 Elsevier Science Publishers B.V.
282 potential failure mode and a performance function for each mode may be obtained through the principles of virtual work. For structures of any practical complexity, there will be many failure modes. Generally, however, only a limited number of these modes will be significant, that is, will have a major contribution to the unreliability of the structure. It is difficult to determine a priori which of the potential failure modes will be significant. Ang and Ma [3] developed a method to determine the significant modes based on nonlinear programming. Two different formulations of the optimization problem were suggested: one based on a general formulation of the failure modes and the other based on the linear combination of the independent elementary mechanisms. The first formulation leads to a constrained problem, where the constraints are defined by positive external virtual work and continuity of the structure. The second formulation leads to an unconstrained optimization problem. Murotsu et al. [4] and Grimmelt et al. [5] suggest a method based on the branch and bound optimization method. A significant mode is found by initially determining the hinge that is most likely to form first. Another hinge is selected on the basis of the maximum failure probability given the first hinge has formed. This process is repeated until enough hinges have formed to cause a plastic mechanism and thus a failure mode is identified, Other failure sequences are obtained by selecting other possible hinges at each stage, A hinge formation is not considered significant, and thus not considered any further, if it causes the probability of failure of the sequence to be less than a prescribed order of magnitude of the probability of failure of the most significant mode. Since the failure modes of a ductile frame are independent of the sequence of formation of hinges, the same failure mode may be generated through a number of sequences. Thus, the sequences
obtained must be reduced t o significant modes. Moses [6] and Gorman and Moses [7] developed a similar method in that significant sequences of hinge formation are found, but suggest a different approach for finding the sequences. All resistances are set equal to their respective mean values and the structure is incrementally loaded until a mechanism is formed and thus a significant mode is obtained. Other significant modes may be found by either preventing certain hinges from forming or examining the change in utilization coefficients as hinges form, where a utilization coefficient is defined as the increase in load in an element per unit increase in external load. Those elements which have a large increase in utilization with the failure of an element may be considered as potential candidates for failure.
REVIEW OF STABLE CONFIGURATION APPROACH The stable configuration approach can be most easily developed using graph theory [8]. A directed graph is constructed of all possible sequences of component failures that lead to the prescribed limit state. This will be called a failure graph. A branch represents the event a component fails and each node, except the terminal node, represents a stable configuration of the structure. Each path from the initial node to the terminal node represents a failure mode. A cut is defined as a set of branches containing one and only one branch from every path. If all the branches in a cut do not occur, that is, the components survive, the structure survives; there is no possible path from the initial node to the terminal node. Thus, if any cut does not occur the structure survives. Failure, therefore, is the event all cuts occur, where a cut is defined to occur if any branch in the cut occurs. The probability of failure is
283
obtained as:
151
Pv P(C, C2 C.) C, = B,] U ... U Bi. , . o .
(1)
S2
(2)
M2
where PF = probability of failure; C i = cut i occurs; B~s = branch j of cut i occurs. For a branch to occur, both the limit state for the pertinent component must be exceeded and the structure must be such that it will fail through that component. Thus, the probability of a branch occuring is
M1
(3)
M1
"3
r<
,Om
where X = vector of basic r a n d o m variables;
giy(X) = performance function for the pertinent component, defined such that g , j ( X ) < 0 is failure; F,j = event component 0 will fail before any other components related to the branches emanating from the beginning node of branch 0For a monotonically loaded structure, a structure in which the load on a surviving member never decreases following the failure of other members, the event ( X ~ F~j) is not necessary. In other words, the probability a branch occurs is: P(Bij)=P[(g,j(X))
Fig. 1.
Frame structure.
moment capacities of the columns are perfectly correlated and are statistically independent of the moment capacity of the beam. The moment-carrying component of the base plates may fail in a brittle manner; the capacities of the base plates are assumed to be statistically independent of each other and of the member resistances. All random variables are taken to be normally distributed with statistics as shown in Table 1. The columns are specified to be W14 × 68 sections and the beam to be a W24 × 62 section.
Construction of failure graph The possible plastic mechanisms for this structure are shown in Fig. 2 and the failure graph is shown in Fig. 3. Shown on the failure graph are the cuts that would be necessary for
TABLE 1 Statistics of random variables of frame (Fig. 1)
ture may fial.
Variable
Mean
C.O.V.
M1
36 k N - m
0.15
PROCEDURE FOR ANALYSIS OF FRAMES
M2
0.15
M3 M4
48 k N - m 40 k N - m 40 k N - m
Consider the one-story one-bay frame shown in Fig. 1. It is assumed that the plastic
$1
20kN
0.10
S2
10 k N
0.20 0.20 0.30
284
/ / / i 1
i
~
]
3
/
~
~
l
~
-
]
4
/"----~
I ~--'5
6 Fig. 2. Plastic mechanismsof frame structure (Fig. 1). analyzing the structure using the stable configuration approach. The failure graph was obtained by considering the failures that could occur initially. Either a plastic mechanism could form, causing the structure to collapse, or a base plate could fail, not necessarily causing collapse of the structure. After a base plate fails the support is treated as a pinned support. Subsequently, either a mechanism could form or the other base plate fail. Obviously, only three plastic hinges would now be needed for a sway mechanism. After both base plates fail the structure will collapse when two hinges form in the structure. The failure graph shown is not quite complete as it is also possible for a base plate to fail after some plastic hinges have formed, but not enough have formed to cause a mechanism. For example, a hinge may form in the upper right corner and then the right base plate fail. Although it is theoretically unconservative to not include these possibilities, the unconservatism will be counteracted by the
assumption of monotonic loading. An investigation of these additional branches revealed that many of the branches were insignificant (that is, had a low probability) a n d / o r were highly correlated with the event the base plate fails before the formation of any plastic hinge. Thus, neglecting these events will introduce negligible error. The failure graph concept, and thus the stable configuration approach, may be applied to structural behavior other than elastic-plastic. If nonlinear behavior is prescribed, the branches corresponding to plastic mechanisms may be replaced with a branch (or branches) representing collapse without brittle failures and the probabilities of the branches evaluated using the methodology developed by Kam et al. [9]. Wide ranges of material behavior may therefore be accommodated with the proposed procedure.
Probability of branches The probability of a branch occurring may be obtained assuming monotonic loading (eqn. 4) and using advanced first-order methods [10] whereby the generally nonlinear failure surface is approximated by a linear failure surface in independent standard normal space. Probability of c u t s The probability of a cut involves the determination of the probability of a union of events. The correlation coefficient between two events may be obtained as
Pij
p~j = aTaj (5) where = correlation coefficient between events i and j; etj = vector of direction cosines of linearized failure surface. Either close second-order bounds [11,12] or an approximate method (e.g. the PNET method [3]) may be used to evaluate the probability of the union. If only bounds are obtained, the geometric average of the two closest
285
bounds may be used as a suitable point estimate. Often a cut will contain m a n y branches representing plastic collapse mechanisms. One
Mechamsm Mechanism Mechan,sm Mechamsm Mechanism Mechanism
• Ic b
of the methods discussed previously may be used to identify the significant mechanisms a priori, thus making the calculations tractable.
r ~
\
\
a
M3 Fails ~
\
t
I ] /
/
~
/
1 2 3 4 5
Mechanism 6
/
M4 Fails
/ / ~
/
/
i, ~//
~ ~
Mechanism Mechanism Mechanism Mechanism Mechanism
/ ~
\
dI
1 2 3 4 5 6
Mech. 1 Mech. 2 Mech. 3
~/~ ~
a /
~
"
c
I'~_. ~ i
~h.
~/
4
\
'~ M3 Fails
M4 Fails
\ Mechanism 1 Mechanism 2 Mechanism Mechanism Mechanism Mechanism
Ib Fig. 3. Failure graph of frame structure (Fig. 1).
3 4 5 6
Mech. 6
j~/ I
I
if--II 11 I
j
'l
1
I
dj
286
Probability of failure
Even so, the upper b o u n d will often reduce to
The probability of failure of the structure is the probability of the intersection of the cuts. Second-order bounds for the intersection of events may be used to obtain the system failure probability. These bounds are based on the pairwise union of events and are as follows: >/max/~
i
_ i=2 J<' <~P ( C , ) + ~
i=2
i--1
I min{i - 2 + P(C~)
- Y'~ P(C, U Cj),0 j= 1
(6) (7)
}
where F - - CIC 2 • C,,. The sharpest lower b o u n d may be obtained using an algorithm similar to Hunter [12]. This is as follows: (1) Determine P*F = ZP(Cs)" (2) Find the m i n i m u m of ( P ( C ] U C,)}, i > 1. Without loss of generality, assume this occurs at i = 2. Set P*F = P~ - P(C] U C2). (3) Find the m i n i m u m of {P(C] u C,), P ( C 2 u C~)}, i > 2. Again without loss of generality, assume the m i n i m u m is P(C] u C3). Set P~ = P ~ - P(C] U C3). (4) Continue the process, i.e., at step k subtract the m i n i m u m of { P(Cj U C~)}, j ~< k, i > k, where it is assumed that the m i n i m u m occurs at i = k + 1. If all the cuts are positively dependent, that is, P(C~[~)>/(Cs), the upper bound may be improved to PF ~< P(C1 ) +
{ i-i ) min P ( C ~ ) - I - I P(Ci u cj),0 i= 2 j= 1 (8)
(9)
If all combinations of the union of three or more cuts are equivalent to the union of just two cuts, the probability of the intersection can be expanded in an analogue to Boole's formulas, as follows:
[0,]
P(C,)
i=]
PF
PF=min{P(C,)+P(Cj)-P(CiuCj)}
" + E E E P ( C,, U Ci,- U C6) -...
+(-1)"+IP
J=' C i
]
(10)
where the repeated summation signs ~ . . . mean the summation over all integers i], i 2 ..... /,.subject to 1 <~ij<~n, i a < i : < ... < i,.. Substitution would then yield an expression for the probability of failure in terms of just the unions of two cuts. Since the cuts are themselves composed of the union of events and the associative property holds for unions, the probability of the union of two cuts will be the probability of the union of all branches in the two cuts. A reasonable estimate of the probability of failure can usually be obtained by considering only a few cuts; generally only a few cuts besides the cut representing initial system damage, that is, the cut containing all branches emanating from the initial node, need to be included. The significant cuts, those with small probabilities, can be found either through engineering j u d g e m e n t - - i d e n t i f y i n g the configurations in which the structure is likely to carry the l o a d - - o r t h r o u g h the reliability analysis. If a branch has a high probability compared with other branches in the cut, it is wise to select a cut containing the stable configuration obtained when that branch occurs. Neglecting cuts will be conservative. As more cuts are included in the analysis, the calculated probability of failure will approach the true probability of failure from above.
287
Returning to the example frame, the probability of cut a - a is determined to be P[Ca_a] = 0.1324. This is the probability of damage, where damage is defined as the failure of the moment-carrying capacity of a base plate, and may be used as an upper bound to the probability of failure of the system. The reliability indices, fl, for each of the branches emanating from the initial node are shown in Table 2. It is observed that the reliability index against failure of the right base plate (M4) is much lower than the other reliability indices. Thus, the right base plate will probably fail first and the stable configuration with it failed is considered, that is, cut b - b is included in the formulation. The probability of cut b - b is P [ C b _ b ] = 0.3221 and P[C1-a (") C b - b ] = 0.3611. The estimated probability of failure of the system is: pv = P[Ca_a ] + P[Cb_b ] - P [ Ca_ a (.J Cb_ b ] = 0.0934 (11) Including other cuts did not change the estimated probability of failure. Thus, the system failure probability can be obtained in this case using only two cuts. To examine the effect of non-monotonic loading and not including the full failure graph, 200 Monte Carlo trials were performed. The reason for the relatively small number of simulations was the complicated analysis required when taking into account
TABLE 2 Reliability indices of branches emanating from initial node of failure graph of Fig. 2 Branch fl Mechanism Mechanism Mechanism Mechanism Mechanism Mechanism M 3 fails M 4 fails
1 2 3 4 5 6
2.212 3.228 3.302 3.018 1.817 2.259 2.615 1.241
the non-monotonic loading, i.e., using eqn. (3) instead of eqn. (4) to determine the probability of a branch occuring. This should be sufficient to examine the effect or error of monotonic loading. Among the 200 trials, 16 failures were observed using the monotonic loading formulation, giving a probability of failure of Pe = 0.080. Taking into account the non-monotonic loading and assuming that the load S~ is applied followed by the load $2, resulted in 13 failures or a probability of failure of PF = 0.065. Alternatively, assuming $2 is applied followed by the load S~, resulted in the same 13 failures plus an additional failure not obrained previously for a failure probability of PF = 0.070. The additional failed structure was not obtained assuming monotonic loading. The failed structure corresponded to the right base plate failing; the two loads then provided counteracting moments at the left base. The full load S~ provided enough of a counteracting moment to prevent failure. However, the vertical load that caused failure at the right base plate did not provide enough of a counteracting moment to prevent failure of the left base plate. After failure of the left base plate, the structure collapsed. The possibility of a counteracting moment not being present was not taken into account in the monotonic loading formulation and therefore this failure was missed. However, the monotonic loading formulation still gave conservative results. In this example, radically different loading paths apparently had little effect on the reliability of the structure. The assumption of monotonic loading caused the probability of failure to be overestimated by about 20%.
AN UNSYMMETRICAL TWO-STORY TWO-BAY FRAME
Consider the unsymmetrical two-story twobay frame shown in Fig. 4. Similar to the
288
~F2
P "1
u4 u3
] ~
"3
2P
~F1 "2
}
I
1
~F3 "s
~~
~0m
~'-"8"'
"1
I'X-M6 10m "
Mr ,
11=8.5×107mm4
14=8.5×107mm 4
12=18.7xlO7rnm 4
15=15.6x107mm 4
13=6.5x107mm 4 Fig. 4. Unsymmetrical two-story two-bat frame.
simple framed structure considered above, it is assumed that failure will occur through the formation of plastic hinge mechanisms and the moment-carrying capacity of the base plates may fail in a brittle manner. The statistics of the random variables are summarized in Table 3. Sections labeled with the same symbol in Fig. 4 are assumed to be perfectly correlated. All random variables are assumed tO be statistically independent normal variates Three cuts are used with the stable config-
uration approach, as shown in the failure graph in Fig. 5. The first cut, a-a, corresponds to initial damage. From the analysis of this cut it was determined that the probability of failure of the base plates at the center and °n the right ( f l = 2"84 f°r M7 and fl = 2.57 for Ms) was higher than the probability of failure of the left base plate (fl = 3.66 for M 6 ) . Therefore, two more cuts were included (cuts b - b and c-c) related to the configurations with the center base plate failed and the right base plate failed, respectively. Based on the results of the previous exampie and to simplify calculations, the possibility of a failure of a base plate after the formation of a plastic hinge somewhere in the structure was not considered. The probabilities for each of the cuts and the union of any two cuts are given in Table 4. These were obtained by first determining the significant plastic mechanisms (see [3]). For example, out of at least 111 possible mechanisms with no base plate failed, only 8 are significant. Using the bounds of eqns. 6 and 9, the probability of failure is 0.00673 ~
PF=P[Ca-a] + P [ f b - b ] "1"P [Cc_c]
TABLE 3 Statistics of random variables of two-story two-bay
-P[Ca_
a LI C b _ b ] -
frame (Fig. 4)
-P[Cb_
b 1,3 Cc_c]
Variable
Mean
C.O.V.
M1 M2 M3 3'/4 M5 M6 M7 Ms F1 F2 F3 P
10 kN-m 19 kN-m 9 kN-m 11 kN-m 15 kN-m 14 kN-m 14 kN-m 14 kNom 7.6 kN 4.0 kN 5.2 kN 1.4 kN
0.15 0.15 0.15 0.15 0.15 0.20 0.20 0.20 0.15 0.25 0.25 0.25
P [ C a _ a U Cc_c]
+ P[Ca_at,..) Cb_b t..) Cc_c]
(13)
TABLE 4 Probabilities of cuts Ca-a Cb-b Cc-c Ca_a 0.01334 0.03401 * 0.03892 Cb- b 0.02940 0.05527 Co_c 0.03692 * off-diagonal terms represent probabilities of respective unions
289 •
M6 Fails a~
cI b
I
"
I
°
.
I ~ I <~
--
o
Plastic Collapse
M7 Fails M8 Fails
I \
==
Plastic Collapse
~
Plastic Collapse
- - - -
I
Fails
IV
y
\,~
: ~
I \I bI
;
.
.
.
-0 a,,s
C
Collapse
Plastic Collapse .
.
.
=
.
.
.
:
•
Ic
Fig. 5. Failure graph of structure (Fig. 4).
Examine the term (Ca_a U Cb_ b U Cc_c). Cut b - b contains all the branches in cut a - a except the branch labeled " M 7 fails" emanating from the initial node of the failure graph, However, this branch is in cut c-c. Thus, all the branches in cut a - a are in either cut b - b or cut c - c and (Ca_a U Cb_ b U Cc_c) = (Cb_ b U Co_c). Substituting into eqn. (13) PF = P[fa_a]-~- P [ f b _ b ] "~-P[Cc_c] - P [ C , , _ a U Cb_ b] -- P [ C a _ a U Co_c] (14)
The probability of failure is therefore PF = 0.00673, which was the lower bound obtained previously, The above estimate appears reasonable for two reasons. First, the inclusion of several other cuts did not have any noticeable effect on the previously computed result. Secondly, the probability of plastic collapse ignoring the possibility of base plate failure is PF = 0.00612. It is likely that a plastic hinge will
form in a column before the corresponding base plate fractures; therefore, it is reasonable that there would only be a slight increase in the probability of failure when including the possibility of base plate failure.
CONCLUSIONS The stable configuration approach provides a viable means for evaluating the reliability of frame structures with brittle components. The approach is general in nature and can incorporate research results concerning ductile frames and nonlinear structures. Monotonic loading may be assumed in the analysis and generally only a few cuts will be necessary to obtain an accurate estimate of the probability of failure. These cuts are easily identified either through the analysis or engineering judgement.
290
REFERENCES 1 J.L. Jorgenson and J.E. Goldberg, Probability of plastic collapse failure, J. Struct. Div., ASCE, 95 (ST8) (1969) 1743-1761. 2 J. Stevenson and F. Moses, Reliability analysis of frame structures. J. Struct. Div., ASCE, 96 (STll) (1970) 2409-2427. 3 A.H-S. Ang and H-F. Ma, On the reliability of structural systems. In: T. Moan and M. Shinozuka (Eds.), Structural Safety and Reliability. Proc. 3rd Int. Conf. on Struct. Safety and Reliability, Trondheim, Norway, 1981, Elsevier, Amsterdam, 1981, pp. 295-314. 4 Y. Murotsu, H. Okada, K. Taguchi, M. Grimmelt and M. Yonezawa, Automatic generation of stochasticaily dominant failure modes of frame structures. Struct. Safety, 2(1) (1984) 17-25. 5 M.J. Grimmelt, G.I. Scheller and Y. Murotsu, On the evaluation of collapse probabilities. In: W.F. Chen and A.D.M. Lewis (Eds.), Recent Advances in Engineering Mechanics and Their Impact on Civil
6 7
8
9
10
11
12
Engineering Practice, Proc. 4th ASCE-EMD Speciality Conf., Purdue University, 1983, pp. 859-862. F. Moses, System reliability developments in structural engineering, Struct. Safety, 1(1)(1982)3-13. M.R. Gorman and F. Moses, Reliability of Structural Systems. Report No. 79-2, Case Western Reserve University, 1979. R.M. Bennett and A.H-S. Ang, Investigation of Methods for Structural System Reliability. Structural Research Series No. 510, University of Illinois, September, 1983. T-Y. Kam, R.B. Corotis and E.C. Rossow, Reliability of nonlinear framed structures. J. Struct. Eng., ASCE, 109(7) (1983) 1585-1601. R. Rackwitz and F. Fiessler, Structural reliability under combined random load sequences. Comp. Struct., 9 (5) (1978) 489-494. E.G. Kounias, Bounds for a probability of a union, with applications. Ann. Math. Stat., 39 (6) (1968) 2154-2158. D. Hunter, An upper bound for the probability of a union. Appl. Probability, 13 (3) (1976) 597-603.
281
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
RELIABILITY ANALYSIS OF FRAME STRUCTURES WITH BRITTLE COMPONENTS RICHARD M. BENNETT Department of Civil Engineering, University of Tennessee, Knoxville, TN 37996-2010 (U.S.A.) (Received August 3, 1984; accepted in revised form February 6, 1985)
Key words: Brittle failure, frames, probability theory, reliability, safety, structural systems
ABSTRACT
A review is provided of the recent developments in the reliability analysis of ductile (elastic-plastic) systems and of the stable configuration approach of structural system reliability analysis. These are combined to obtain a meth-
odology for evaluating the reliability of frames that have primarily ductile components but also a few brittle components. The methodology is applied to two frame structures.
INTRODUCTION
ductile (linear elastic-plastic) but that there are also a few connections that may fail in a brittle manner. The focus will be on handling the brittle behavior and it will be shown that the methodology developed can incorporate other than elastic-plastic behavior (e.g. nonlinear) in conjunction with the brittle components.
There have been many studies concerning the reliability of frame structures. However, most have invoked rather restrictive assumptions as to material behavior; in particular, m o s t studies have a s s u m e d perfectly elastic-plastic ductile behavior. The behavior of actual structures will in general be much more complex. Often there will be the possibility of brittle behavior; a member or connection may fracture and the load carrying capacity be reduced to zero. The proper evaluation of the reliability of the structure must take this into account. A methodology will be developed in this paper for analyzing frames with brittle components. It will be assumed that the behavior of the frame is 0167-4730/85/$03.30
REVIEW OF RELIABILITY ANALYSIS OF DUCTILE FRAMES Most of the work concerning the reliability analysis of ductile frames has been based on the formulations developed by Jorgenson and Goldberg [1] and Stevenson and Moses [2] in which each plastic mechanism is considered a
© 1985 Elsevier Science Publishers B.V.
282 potential failure mode and a performance function for each mode may be obtained through the principles of virtual work. For structures of any practical complexity, there will be many failure modes. Generally, however, only a limited number of these modes will be significant, that is, will have a major contribution to the unreliability of the structure. It is difficult to determine a priori which of the potential failure modes will be significant. Ang and Ma [3] developed a method to determine the significant modes based on nonlinear programming. Two different formulations of the optimization problem were suggested: one based on a general formulation of the failure modes and the other based on the linear combination of the independent elementary mechanisms. The first formulation leads to a constrained problem, where the constraints are defined by positive external virtual work and continuity of the structure. The second formulation leads to an unconstrained optimization problem. Murotsu et al. [4] and Grimmelt et al. [5] suggest a method based on the branch and bound optimization method. A significant mode is found by initially determining the hinge that is most likely to form first. Another hinge is selected on the basis of the maximum failure probability given the first hinge has formed. This process is repeated until enough hinges have formed to cause a plastic mechanism and thus a failure mode is identified, Other failure sequences are obtained by selecting other possible hinges at each stage, A hinge formation is not considered significant, and thus not considered any further, if it causes the probability of failure of the sequence to be less than a prescribed order of magnitude of the probability of failure of the most significant mode. Since the failure modes of a ductile frame are independent of the sequence of formation of hinges, the same failure mode may be generated through a number of sequences. Thus, the sequences
obtained must be reduced t o significant modes. Moses [6] and Gorman and Moses [7] developed a similar method in that significant sequences of hinge formation are found, but suggest a different approach for finding the sequences. All resistances are set equal to their respective mean values and the structure is incrementally loaded until a mechanism is formed and thus a significant mode is obtained. Other significant modes may be found by either preventing certain hinges from forming or examining the change in utilization coefficients as hinges form, where a utilization coefficient is defined as the increase in load in an element per unit increase in external load. Those elements which have a large increase in utilization with the failure of an element may be considered as potential candidates for failure.
REVIEW OF STABLE CONFIGURATION APPROACH The stable configuration approach can be most easily developed using graph theory [8]. A directed graph is constructed of all possible sequences of component failures that lead to the prescribed limit state. This will be called a failure graph. A branch represents the event a component fails and each node, except the terminal node, represents a stable configuration of the structure. Each path from the initial node to the terminal node represents a failure mode. A cut is defined as a set of branches containing one and only one branch from every path. If all the branches in a cut do not occur, that is, the components survive, the structure survives; there is no possible path from the initial node to the terminal node. Thus, if any cut does not occur the structure survives. Failure, therefore, is the event all cuts occur, where a cut is defined to occur if any branch in the cut occurs. The probability of failure is
283
obtained as:
151
Pv P(C, C2 C.) C, = B,] U ... U Bi. , . o .
(1)
S2
(2)
M2
where PF = probability of failure; C i = cut i occurs; B~s = branch j of cut i occurs. For a branch to occur, both the limit state for the pertinent component must be exceeded and the structure must be such that it will fail through that component. Thus, the probability of a branch occuring is
M1
(3)
M1
"3
r<
,Om
where X = vector of basic r a n d o m variables;
giy(X) = performance function for the pertinent component, defined such that g , j ( X ) < 0 is failure; F,j = event component 0 will fail before any other components related to the branches emanating from the beginning node of branch 0For a monotonically loaded structure, a structure in which the load on a surviving member never decreases following the failure of other members, the event ( X ~ F~j) is not necessary. In other words, the probability a branch occurs is: P(Bij)=P[(g,j(X))
Fig. 1.
Frame structure.
moment capacities of the columns are perfectly correlated and are statistically independent of the moment capacity of the beam. The moment-carrying component of the base plates may fail in a brittle manner; the capacities of the base plates are assumed to be statistically independent of each other and of the member resistances. All random variables are taken to be normally distributed with statistics as shown in Table 1. The columns are specified to be W14 × 68 sections and the beam to be a W24 × 62 section.
Construction of failure graph The possible plastic mechanisms for this structure are shown in Fig. 2 and the failure graph is shown in Fig. 3. Shown on the failure graph are the cuts that would be necessary for
TABLE 1 Statistics of random variables of frame (Fig. 1)
ture may fial.
Variable
Mean
C.O.V.
M1
36 k N - m
0.15
PROCEDURE FOR ANALYSIS OF FRAMES
M2
0.15
M3 M4
48 k N - m 40 k N - m 40 k N - m
Consider the one-story one-bay frame shown in Fig. 1. It is assumed that the plastic
$1
20kN
0.10
S2
10 k N
0.20 0.20 0.30
284
/ / / i 1
i
~
]
3
/
~
~
l
~
-
]
4
/"----~
I ~--'5
6 Fig. 2. Plastic mechanismsof frame structure (Fig. 1). analyzing the structure using the stable configuration approach. The failure graph was obtained by considering the failures that could occur initially. Either a plastic mechanism could form, causing the structure to collapse, or a base plate could fail, not necessarily causing collapse of the structure. After a base plate fails the support is treated as a pinned support. Subsequently, either a mechanism could form or the other base plate fail. Obviously, only three plastic hinges would now be needed for a sway mechanism. After both base plates fail the structure will collapse when two hinges form in the structure. The failure graph shown is not quite complete as it is also possible for a base plate to fail after some plastic hinges have formed, but not enough have formed to cause a mechanism. For example, a hinge may form in the upper right corner and then the right base plate fail. Although it is theoretically unconservative to not include these possibilities, the unconservatism will be counteracted by the
assumption of monotonic loading. An investigation of these additional branches revealed that many of the branches were insignificant (that is, had a low probability) a n d / o r were highly correlated with the event the base plate fails before the formation of any plastic hinge. Thus, neglecting these events will introduce negligible error. The failure graph concept, and thus the stable configuration approach, may be applied to structural behavior other than elastic-plastic. If nonlinear behavior is prescribed, the branches corresponding to plastic mechanisms may be replaced with a branch (or branches) representing collapse without brittle failures and the probabilities of the branches evaluated using the methodology developed by Kam et al. [9]. Wide ranges of material behavior may therefore be accommodated with the proposed procedure.
Probability of branches The probability of a branch occurring may be obtained assuming monotonic loading (eqn. 4) and using advanced first-order methods [10] whereby the generally nonlinear failure surface is approximated by a linear failure surface in independent standard normal space. Probability of c u t s The probability of a cut involves the determination of the probability of a union of events. The correlation coefficient between two events may be obtained as
Pij
p~j = aTaj (5) where = correlation coefficient between events i and j; etj = vector of direction cosines of linearized failure surface. Either close second-order bounds [11,12] or an approximate method (e.g. the PNET method [3]) may be used to evaluate the probability of the union. If only bounds are obtained, the geometric average of the two closest
285
bounds may be used as a suitable point estimate. Often a cut will contain m a n y branches representing plastic collapse mechanisms. One
Mechamsm Mechanism Mechan,sm Mechamsm Mechanism Mechanism
• Ic b
of the methods discussed previously may be used to identify the significant mechanisms a priori, thus making the calculations tractable.
r ~
\
\
a
M3 Fails ~
\
t
I ] /
/
~
/
1 2 3 4 5
Mechanism 6
/
M4 Fails
/ / ~
/
/
i, ~//
~ ~
Mechanism Mechanism Mechanism Mechanism Mechanism
/ ~
\
dI
1 2 3 4 5 6
Mech. 1 Mech. 2 Mech. 3
~/~ ~
a /
~
"
c
I'~_. ~ i
~h.
~/
4
\
'~ M3 Fails
M4 Fails
\ Mechanism 1 Mechanism 2 Mechanism Mechanism Mechanism Mechanism
Ib Fig. 3. Failure graph of frame structure (Fig. 1).
3 4 5 6
Mech. 6
j~/ I
I
if--II 11 I
j
'l
1
I
dj
286
Probability of failure
Even so, the upper b o u n d will often reduce to
The probability of failure of the structure is the probability of the intersection of the cuts. Second-order bounds for the intersection of events may be used to obtain the system failure probability. These bounds are based on the pairwise union of events and are as follows: >/max/~
i
_ i=2 J<' <~P ( C , ) + ~
i=2
i--1
I min{i - 2 + P(C~)
- Y'~ P(C, U Cj),0 j= 1
(6) (7)
}
where F - - CIC 2 • C,,. The sharpest lower b o u n d may be obtained using an algorithm similar to Hunter [12]. This is as follows: (1) Determine P*F = ZP(Cs)" (2) Find the m i n i m u m of ( P ( C ] U C,)}, i > 1. Without loss of generality, assume this occurs at i = 2. Set P*F = P~ - P(C] U C2). (3) Find the m i n i m u m of {P(C] u C,), P ( C 2 u C~)}, i > 2. Again without loss of generality, assume the m i n i m u m is P(C] u C3). Set P~ = P ~ - P(C] U C3). (4) Continue the process, i.e., at step k subtract the m i n i m u m of { P(Cj U C~)}, j ~< k, i > k, where it is assumed that the m i n i m u m occurs at i = k + 1. If all the cuts are positively dependent, that is, P(C~[~)>/(Cs), the upper bound may be improved to PF ~< P(C1 ) +
{ i-i ) min P ( C ~ ) - I - I P(Ci u cj),0 i= 2 j= 1 (8)
(9)
If all combinations of the union of three or more cuts are equivalent to the union of just two cuts, the probability of the intersection can be expanded in an analogue to Boole's formulas, as follows:
[0,]
P(C,)
i=]
PF
PF=min{P(C,)+P(Cj)-P(CiuCj)}
" + E E E P ( C,, U Ci,- U C6) -...
+(-1)"+IP
J=' C i
]
(10)
where the repeated summation signs ~ . . . mean the summation over all integers i], i 2 ..... /,.subject to 1 <~ij<~n, i a < i : < ... < i,.. Substitution would then yield an expression for the probability of failure in terms of just the unions of two cuts. Since the cuts are themselves composed of the union of events and the associative property holds for unions, the probability of the union of two cuts will be the probability of the union of all branches in the two cuts. A reasonable estimate of the probability of failure can usually be obtained by considering only a few cuts; generally only a few cuts besides the cut representing initial system damage, that is, the cut containing all branches emanating from the initial node, need to be included. The significant cuts, those with small probabilities, can be found either through engineering j u d g e m e n t - - i d e n t i f y i n g the configurations in which the structure is likely to carry the l o a d - - o r t h r o u g h the reliability analysis. If a branch has a high probability compared with other branches in the cut, it is wise to select a cut containing the stable configuration obtained when that branch occurs. Neglecting cuts will be conservative. As more cuts are included in the analysis, the calculated probability of failure will approach the true probability of failure from above.
287
Returning to the example frame, the probability of cut a - a is determined to be P[Ca_a] = 0.1324. This is the probability of damage, where damage is defined as the failure of the moment-carrying capacity of a base plate, and may be used as an upper bound to the probability of failure of the system. The reliability indices, fl, for each of the branches emanating from the initial node are shown in Table 2. It is observed that the reliability index against failure of the right base plate (M4) is much lower than the other reliability indices. Thus, the right base plate will probably fail first and the stable configuration with it failed is considered, that is, cut b - b is included in the formulation. The probability of cut b - b is P [ C b _ b ] = 0.3221 and P[C1-a (") C b - b ] = 0.3611. The estimated probability of failure of the system is: pv = P[Ca_a ] + P[Cb_b ] - P [ Ca_ a (.J Cb_ b ] = 0.0934 (11) Including other cuts did not change the estimated probability of failure. Thus, the system failure probability can be obtained in this case using only two cuts. To examine the effect of non-monotonic loading and not including the full failure graph, 200 Monte Carlo trials were performed. The reason for the relatively small number of simulations was the complicated analysis required when taking into account
TABLE 2 Reliability indices of branches emanating from initial node of failure graph of Fig. 2 Branch fl Mechanism Mechanism Mechanism Mechanism Mechanism Mechanism M 3 fails M 4 fails
1 2 3 4 5 6
2.212 3.228 3.302 3.018 1.817 2.259 2.615 1.241
the non-monotonic loading, i.e., using eqn. (3) instead of eqn. (4) to determine the probability of a branch occuring. This should be sufficient to examine the effect or error of monotonic loading. Among the 200 trials, 16 failures were observed using the monotonic loading formulation, giving a probability of failure of Pe = 0.080. Taking into account the non-monotonic loading and assuming that the load S~ is applied followed by the load $2, resulted in 13 failures or a probability of failure of PF = 0.065. Alternatively, assuming $2 is applied followed by the load S~, resulted in the same 13 failures plus an additional failure not obrained previously for a failure probability of PF = 0.070. The additional failed structure was not obtained assuming monotonic loading. The failed structure corresponded to the right base plate failing; the two loads then provided counteracting moments at the left base. The full load S~ provided enough of a counteracting moment to prevent failure. However, the vertical load that caused failure at the right base plate did not provide enough of a counteracting moment to prevent failure of the left base plate. After failure of the left base plate, the structure collapsed. The possibility of a counteracting moment not being present was not taken into account in the monotonic loading formulation and therefore this failure was missed. However, the monotonic loading formulation still gave conservative results. In this example, radically different loading paths apparently had little effect on the reliability of the structure. The assumption of monotonic loading caused the probability of failure to be overestimated by about 20%.
AN UNSYMMETRICAL TWO-STORY TWO-BAY FRAME
Consider the unsymmetrical two-story twobay frame shown in Fig. 4. Similar to the
288
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P "1
u4 u3
] ~
"3
2P
~F1 "2
}
I
1
~F3 "s
~~
~0m
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"1
I'X-M6 10m "
Mr ,
11=8.5×107mm4
14=8.5×107mm 4
12=18.7xlO7rnm 4
15=15.6x107mm 4
13=6.5x107mm 4 Fig. 4. Unsymmetrical two-story two-bat frame.
simple framed structure considered above, it is assumed that failure will occur through the formation of plastic hinge mechanisms and the moment-carrying capacity of the base plates may fail in a brittle manner. The statistics of the random variables are summarized in Table 3. Sections labeled with the same symbol in Fig. 4 are assumed to be perfectly correlated. All random variables are assumed tO be statistically independent normal variates Three cuts are used with the stable config-
uration approach, as shown in the failure graph in Fig. 5. The first cut, a-a, corresponds to initial damage. From the analysis of this cut it was determined that the probability of failure of the base plates at the center and °n the right ( f l = 2"84 f°r M7 and fl = 2.57 for Ms) was higher than the probability of failure of the left base plate (fl = 3.66 for M 6 ) . Therefore, two more cuts were included (cuts b - b and c-c) related to the configurations with the center base plate failed and the right base plate failed, respectively. Based on the results of the previous exampie and to simplify calculations, the possibility of a failure of a base plate after the formation of a plastic hinge somewhere in the structure was not considered. The probabilities for each of the cuts and the union of any two cuts are given in Table 4. These were obtained by first determining the significant plastic mechanisms (see [3]). For example, out of at least 111 possible mechanisms with no base plate failed, only 8 are significant. Using the bounds of eqns. 6 and 9, the probability of failure is 0.00673 ~
PF=P[Ca-a] + P [ f b - b ] "1"P [Cc_c]
TABLE 3 Statistics of random variables of two-story two-bay
-P[Ca_
a LI C b _ b ] -
frame (Fig. 4)
-P[Cb_
b 1,3 Cc_c]
Variable
Mean
C.O.V.
M1 M2 M3 3'/4 M5 M6 M7 Ms F1 F2 F3 P
10 kN-m 19 kN-m 9 kN-m 11 kN-m 15 kN-m 14 kN-m 14 kN-m 14 kNom 7.6 kN 4.0 kN 5.2 kN 1.4 kN
0.15 0.15 0.15 0.15 0.15 0.20 0.20 0.20 0.15 0.25 0.25 0.25
P [ C a _ a U Cc_c]
+ P[Ca_at,..) Cb_b t..) Cc_c]
(13)
TABLE 4 Probabilities of cuts Ca-a Cb-b Cc-c Ca_a 0.01334 0.03401 * 0.03892 Cb- b 0.02940 0.05527 Co_c 0.03692 * off-diagonal terms represent probabilities of respective unions
289 •
M6 Fails a~
cI b
I
"
I
°
.
I ~ I <~
--
o
Plastic Collapse
M7 Fails M8 Fails
I \
==
Plastic Collapse
~
Plastic Collapse
- - - -
I
Fails
IV
y
\,~
: ~
I \I bI
;
.
.
.
-0 a,,s
C
Collapse
Plastic Collapse .
.
.
=
.
.
.
:
•
Ic
Fig. 5. Failure graph of structure (Fig. 4).
Examine the term (Ca_a U Cb_ b U Cc_c). Cut b - b contains all the branches in cut a - a except the branch labeled " M 7 fails" emanating from the initial node of the failure graph, However, this branch is in cut c-c. Thus, all the branches in cut a - a are in either cut b - b or cut c - c and (Ca_a U Cb_ b U Cc_c) = (Cb_ b U Co_c). Substituting into eqn. (13) PF = P[fa_a]-~- P [ f b _ b ] "~-P[Cc_c] - P [ C , , _ a U Cb_ b] -- P [ C a _ a U Co_c] (14)
The probability of failure is therefore PF = 0.00673, which was the lower bound obtained previously, The above estimate appears reasonable for two reasons. First, the inclusion of several other cuts did not have any noticeable effect on the previously computed result. Secondly, the probability of plastic collapse ignoring the possibility of base plate failure is PF = 0.00612. It is likely that a plastic hinge will
form in a column before the corresponding base plate fractures; therefore, it is reasonable that there would only be a slight increase in the probability of failure when including the possibility of base plate failure.
CONCLUSIONS The stable configuration approach provides a viable means for evaluating the reliability of frame structures with brittle components. The approach is general in nature and can incorporate research results concerning ductile frames and nonlinear structures. Monotonic loading may be assumed in the analysis and generally only a few cuts will be necessary to obtain an accurate estimate of the probability of failure. These cuts are easily identified either through the analysis or engineering judgement.
290
REFERENCES 1 J.L. Jorgenson and J.E. Goldberg, Probability of plastic collapse failure, J. Struct. Div., ASCE, 95 (ST8) (1969) 1743-1761. 2 J. Stevenson and F. Moses, Reliability analysis of frame structures. J. Struct. Div., ASCE, 96 (STll) (1970) 2409-2427. 3 A.H-S. Ang and H-F. Ma, On the reliability of structural systems. In: T. Moan and M. Shinozuka (Eds.), Structural Safety and Reliability. Proc. 3rd Int. Conf. on Struct. Safety and Reliability, Trondheim, Norway, 1981, Elsevier, Amsterdam, 1981, pp. 295-314. 4 Y. Murotsu, H. Okada, K. Taguchi, M. Grimmelt and M. Yonezawa, Automatic generation of stochasticaily dominant failure modes of frame structures. Struct. Safety, 2(1) (1984) 17-25. 5 M.J. Grimmelt, G.I. Scheller and Y. Murotsu, On the evaluation of collapse probabilities. In: W.F. Chen and A.D.M. Lewis (Eds.), Recent Advances in Engineering Mechanics and Their Impact on Civil
6 7
8
9
10
11
12
Engineering Practice, Proc. 4th ASCE-EMD Speciality Conf., Purdue University, 1983, pp. 859-862. F. Moses, System reliability developments in structural engineering, Struct. Safety, 1(1)(1982)3-13. M.R. Gorman and F. Moses, Reliability of Structural Systems. Report No. 79-2, Case Western Reserve University, 1979. R.M. Bennett and A.H-S. Ang, Investigation of Methods for Structural System Reliability. Structural Research Series No. 510, University of Illinois, September, 1983. T-Y. Kam, R.B. Corotis and E.C. Rossow, Reliability of nonlinear framed structures. J. Struct. Eng., ASCE, 109(7) (1983) 1585-1601. R. Rackwitz and F. Fiessler, Structural reliability under combined random load sequences. Comp. Struct., 9 (5) (1978) 489-494. E.G. Kounias, Bounds for a probability of a union, with applications. Ann. Math. Stat., 39 (6) (1968) 2154-2158. D. Hunter, An upper bound for the probability of a union. Appl. Probability, 13 (3) (1976) 597-603.