Risk balancing in structural problems

Structural Safety Vol. 19, No. 1, pp. 67-77, 1997 ELSEVIER

PII: S0167-4730(96)00038-0

© 1997 Elsevier Science Ltd. All rights reserved Printed in The Netherlands 0167-4730/97 $17.00 + .00

Risk balancing in structural problems 1 D a v i d G. E l m s Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand

Abstract A balanced-risk design is one in which the risks of failure in different modes are balanced against one another in such a way that the total risk is minimised, for a fixed overall safety expenditure. The problem is formulated in general terms. It is then simplified, and expressed in terms of readily-understood quantities so that the process is as transparent as possible. Its simplicity makes it usable on a spreadsheet. The technique is primarily applicable at the rough early design stage of a structural project. Examples of application are given in the areas of preliminary bridge and fire design and the importance of sensitivity studies is demonstrated. © 1997 Elsevier Science Ltd. All rights reserved.

Keywords: Risk; Optimisation; Balance; Design; Structure; Bridge; Fire

1. Introduction In 1972, Wiggins wrote a paper introducing the concept of "balanced risk" [1]. The idea was intriguing, but its use at that stage was simple. Essentially, it was proposed that buildings should be designed such that the risks associated with them in earthquakes should be the same even though the occupancies differed. However, such a straightforward definition did not seem to apply to the more fundamental questions arising in structural design. Consider, for example, the question of building code loading provisions. One could assume that a balanced code would be one in which the code provisions for live load, wind and earthquake would in some way be balanced such that the code was no more severe for one type of loading than for another. Similarly, for a single structure, it could be assumed that a balanced-risk design would be one in which in some sense the risks arising from different failure modes and from different loading conditions would all be balanced in relation to each other, with, again, none being predominant or more severe than another. In neither case did Wiggins's simple formulation apply. The problems were real, though, and so a more precise formulation was sought. At first, the problem seemed vague and difficult to quantify. Eventually, though, it turned out that, as with many problems, the difficulty lay not so much in the solution as in the appropriate formulation of the problem. 1 Discussion is open until September 1997 (please submit your discussion paper to the editor, Ross B. Corotis). 67

D.G. Elms

68

The initial formulation was set down in 1979 [2], with a more readable description following later [3]. Although the technique found occasional use, it was limited by the numerical complexity required and the resulting practical difficulties. Now, however, the widespread use of spreadsheets has meant that it is a relatively easy matter to carry out the calculations. The present paper revives the earlier work [2], and while retaining the original basic formulation, expands and reformulates it to make the solution far more generally applicable. Indeed, in its present form the technique has become a useful risk management tool whose range of applicability goes far beyond that of the structural engineering problems for which it was originally intended. One final general point must be made before embarking on detail. It is, that the intent of the risk-balancing technique given here is that it should be used at the level of a preliminary strategic overview. It has been formulated consistently with this use in mind. It uses a number of simplifying assumptions, and these would not be compatible with use of the technique at a detailed design level. The technique is intended to provide an overview in the preliminary setting up of a design, or whatever problem is being addressed. It provides a foundation on which more detailed work can be based.

2. Formulation

2.1. General form Let us suppose that a fixed sum of money C is to be allocated to reducing risk or improving safety. The value of C can be varied later in the analysis to examine the sensitivity of the results to the initial assumption. However, at this point C is taken as a fixed quantity. We can assume that C can be distributed among m expenditure modes or fields. Thus if the amount spent in expenditure mode j is a j, then m

~-'~aj=C.

(1)

j=l We can also assume that we are trying to reduce the risk of failure of the structure or whatever it is we are considering, and that failure can occur in one of n failure modes. What is required is the values of the expenditures aj such that the expected cost of failure F is minimised, where F /

F

ci(a 1..... am)Pi(a I ..... am)

~

(2)

i=1

and

Ci

ci(a

i .....

Pi = Pi(al . . . . .

am) = COSt consequences of failure in mode am) = failure frequency (probability/year)

i in mode i.

(3)

As the minimisation is subject to the constraint of Eq. (1), we can write the function to be minimised as

G=~ciPi-A[

j~ = la j - C

(4)

D.G. Elms

69

where A, the Langrange multiplier, is a constant. The condition for a minimum is that all m values of the partial derivative of G with respect to aj are 0; that is n

Op i

i=11 Oa] p i + i=1

~ oaj ] -

=

(5)

This, together with Eq. (1), represents the general form of the solution. Note that the assumption has been made that both the cost consequences of failure and the probabilities of failure in the different failure modes are all smooth and continuous functions of the expenditures a~. Clearly, there will be few cases in practice where this will be true. However, as explained earlier, it is the intent that the formulation should be used at a broad and strategic level of analysis. Thus it is intended that the use of the formulation should be balanced against the crudeness of the initial assumptions, together with that of the simplifying assumptions contained in the next section, and of the data which are expected to be used. Thus the whole analysis and its use follows the balancing requirements of the "principle of consistent crudeness" formulated elsewhere [4,5].

2.2. Functional assumptions Expenditure on safety will reduce either Pi or Ci, or both. There are three assumptions that can be made for the generic shape of a continuous progressively reducing function. They are: a. Exponential reduction, of the form

y=aexp[-g(x/k)].

(6)

b. Inverse power law reduction, of the form y = a ( 1 + x / k ) -q

(7)

c. Curtailed power law reduction, of the form

y=a(1-x/k)

q (OO)

(8)

where a, k and q are constants, and g ( ) is an arbitrary function. In what follows, somewhat arbitrarily we will assume an exponential reduction for the probability of failure, and an inverse power law reduction for failure consequence. Thus for the probability of failure, we assume . . . . .

where: pO=i base expenditure failure frequency (or probability), rij--limiting probability reduction (as a fraction), u o = standard reduction expenditure. Note that the constants rij and uij are formulated in this way so that they can be relatively easily understood, and their values easily estimated. The first quantity, r~j, implies that for any failure mode, expenditure mode j cannot reduce the probability of failure beyond a certain limiting amount, no matter how great the expenditure. The "standard reduction expenditure" controls the speed at which an expenditure will begin to reduce the probability of failure. The quantity u/j can be thought of as the expenditure in mode j that will reduce the probability of failure in mode i by 0.632 (or about _~) of the possible reduction in that mode.

70

D.G. Elms

Assuming that no positive expenditure increases the probability of failure, then we have the restrictions

(10)

0 ~ Vij _< 1

and m

Y'~ rij[1-exp(-aJuij)]

< 1.

(11)

j=l

If the expenditures aj are regarded as increases over some standard or base expenditure, then negative values of aj mean that the base expenditure on safety in that mode was too great and should be reduced. Clearly, if negative values occur, the formulation must also ensure that p; < 1.0. For the reduction in failure consequence, assume

1 - __ si:(1- (1-[-aj//vij)

ci(a I ..... am)=C °

(12)

j=l

where: c ° = zero expenditure failure cost, sij = limiting consequence reduction, vii = standard reduction expenditure, qij = shape constant. Assuming that no expenditure will increase the cost consequence of failure, then the coefficients have the restrictions

0 < sij <

1

(13)

and m

2., Sij[X-

(1 +

aj/vij ) --qij]] .,

"~

< 1.

(14)

j=l

Note that for simplicity an additive formulation has been used for both probability and consequence functions. An alternative would be to put them in multiplicative form.

2.3. Solution From Eqs. (9) and (12) it follows that the derivatives of the failure probability and cost consequences are

Opi Oaj

-p°rij --exp(-aJuij

)

uij

(15)

and

o ( aj ) -(q'J+ 1) Oc-----Li- -ci siJqiJ 1 + - Oaij

uij

(16)

uij

The m equations [Eq. (5)] could be written in terms of Eqs. (9), (12), (15) and (16). However, this will not be done as the resulting expression is both cumbersome and unnecessary. Rather, we will proceed to the next step, which is to linearise the expression in order to obtain an explicit formulation for the set of expenditures aj.

D.G. Elms

7l

Equation (5) can be linearised using a first-order Taylor's series approximation about an assumed set of expenditures aj.o Let F 0o _- v a l u e of function of linearisation point a j,o f ,jo= derivative of function at linearisation point. Then Eq. (5) is approximated by

(Air q--Bijaj)

( j = 1 ..... m)

=/~

(17)

i=1

where

(18)

Aij = Fi°j- aj°fi'~ 8,j = L°j

(19)

and Fi°j and f/~ are given by (20)

F?j = a~a ,Sij + "lifiij

6ij

& -(1 +q,j)~'J Vij and where

Vij

-- ]lij Uij

(21)

o ( a,l-'+"

Olij = -- Ci siJqiJ I + Yij Uij ]

(22)

(23)

")lij = C°

-- ~ Sij(1 --(1

"-~

a~/vi;) -q')

(24)

j=l

aij = ~- pO ri j exp( - aj/u ij)"

(25)

uij From Eq. (17) the required solution is A- ~Aij

aj =

ni = 1 E Bij

(26)

i=1

where, from the constraint of Eq. (1),

Cq- ~ [ ~-~ A i j / ~ A=

j=l

i=1

i=1

nij l (27)

D.G. Elms

72

As an alternative solution, in some cases what is required is simply the relative values of a j, normalised so that one of them is given the value unity. If, say, the first is set to unity, that is, the new value a'1 = 1, we have a'i = a J a I

(28)

where aj is given by Eq. (26). An alternative constraint would be to specify the value of one of the values of aj. If, for example, the first expenditure is fixed at a~ = a~*, then we have, from Eq. (26) n

E(alBil +Ail-aij) aj =

i=,

n

(29)

E Bij i=1

The solutions are easily carried out on a spreadsheet, iterating on the assumed values of the expenditures. Uncertainty in the input data can be allowed for by using an add-on macro such as the widely-used program @Risk to the spreadsheet, then assigning distributions (normal, triangular, etc.) with appropriate parameters to the uncertain input. A Monte Carlo simulation is then carried out, and the outputs are expressed as means and standard deviations. A sensivity study can also be done at this point. Furthermore, the calculation could, and should, be repeated for different values of C to examine the effect of scale. This step is necessary because the relative magnitudes of the aj's can vary markedly with C. 2.4. Simplification A number of simplifications are possible for the above solution. One could be the case where a given expenditure affects either consequences, or failure probabilities, but not both. Another simplification could be that the expenditure modes are the same as the failure modes. This case was dealt with in the earlier paper [2]. Finally, we could consider the special case in which all expenditures only affect the probabilities of failure (or alternatively, only the consequences). Although all can be handled by the general formulation, we will expand on the last case mentioned as the solution is greatly simplified. Assume that expenditures affect failure probabilities alone. In such a case, Eq. (2) becomes

F = ~ ciPi(a l ..... am).

(30)

i=1

Equation (4) is unchanged but Eq. (5) becomes

n

Opi )

E Ci("~ i=1

k oaj

-- I~=0.

(31)

Using the assumption of Eq. (9), the partial derivatives of the failure probabilities are given by Eq.

D. G. Elms

73

(15). The results are still those given in Eqs. (26), (27) or (29), except that Eqs. (22) and (24) become Olij : 0

(32)

")/ij = Ci"

(33)

In such a case, Eqs. (20) and (21) simplify to Fi~ = Ci ~ij

(34)

fi°j = - ci 8 i / . i j .

(35)

In this form, the solution is identical to that presented earlier [2]. All examples that follow use the simplified form given in Eq. (30). This is because structural problems invariably are such that safety expenditure is aimed solely at failure frequency reduction. Situations where expenditure can be used to reduce failure consequences arise in, for example, rail transport safety or natural hazard response expenditure.

3. Examples 3.1. Risk-balanced bridge design Consider a multi-span concrete bridge crossing a shipping channel, and subjected to scour. The failure and expenditure modes are the same in this case, and are: (1) scour failure reduced by increasing the size and number of piles, (2) earthquake failure, reduced by increasing the size of shear connections and the amount of column transverse steel and (3) collision failure, reduced by increasing the span length. The safety investment C can be expressed as a percentage of the basic capital cost of the bridge, rather than in direct monetary terms. Let C = 20% in this case. The other costs will also be expressed as percentages. The major parameters are given in Table 1, while the limiting consequence reduction and standard expenditure coefficients are

r~ =

1.0

0

0

0

o.9

0

0

0

0.8

(36)

5%

uij =

-

3%

(37) 8%

Note that because of the correspondence of the failure and expenditure modes, in this case the

D.G.Elms

74 Table 1 Parameter

Mode

aj o

pO Co i

1

2

3

5% 0.01 30%

5% 0.10 120%

10% 0.10 60%

coefficients are expressed as diagonal matrices. Note also that the values of u/j corresponding to the zero values of rij are not given values. They are indeterminate. The initial expenditure estimates show that it is assumed likely that increasing the span length will be more expensive than expenditure in the other two modes. The 30% failure cost for scour rests on the assumption that the problem, if it exists, will be found in time before a catastrophic failure occurs. On the other hand, earthquake failure, should it occur, is likely to have severe consequences, both physical and social, so that a cost figure of 120% is justified. The results of a ship collision could also be reasonably severe, so a cost of 60% is assigned. A substantial reduction in earthquake failure probability would occur with only a small level of expenditure in mode 2, so that u22 is taken to be 3%: greater expenditures would be required in the other two modes to achieve an equivalent performance improvement. The limiting failure reduction r H in mode 1 is taken to be 1.0 as it is assumed that given sufficient expenditure the bridge could be made completely safe from scour. Only a 90% safety level could be achieved for earthquake risk because of the high degree of uncertainty in earthquake behaviour and also because of the finite probability of human error. The limiting failure reduction for mode 3 is set at 0.8 as it is judged that complete protection from collision could not be achieved. Performing the calculation, the results are A = 0.058, a 1 = - 3 . 1 % scour, a 2 7.7% earthquake, a 3 = 15.3% collision. The negative result for expenditure in the first mode simply means that the standard situation (design) occurring before any expenditure on safety in the first mode (scour) represented too great an expenditure on scour prevention. =

3.2. Risk-balancedfire design Consider the design of the fire prevention measures in a building. The total expenditure to be allocated to fire protection is assumed to be $ 3 0 / m 2. The three failure modes to be considered are: (1) smoke spread blocks exits, with consequent loss of life, (2) structural collapse takes place after exit of occupants and (3) excessive fire, smoke and water damage occurs. Three expenditure modes are considered. They are: (1) degree of compartmentalisation, (2) degree of fuel load control and (3) insulation of structural members. Values for the relevant parameters and coefficients are as follows:

Pi°=~

ci=~

$/m2

a ) ° = ~ $1m2

(38)

D. G. Elms

0.5

~j =

%=

75

0.2

0

0

0

1.0

0.7

0.1

0

30

20

-

-

-

a.0

20

80

-

(39)

(40)

Note that the costs c i relate to a total building cost of $ 3 0 0 / m 2. Loss of life would cause the failure cost to greatly exceed this figure which is why the first failure mode is, at $1500, many times greater than the original building cost. Note also that the initial assumed modal expenditures do not sum to $ 3 0 m 2. It is not necessary to make such an initial assumption. The results stabilise after three iterations to: A = 0.0164, a I = $ 3 0 . 3 / m 2, a 2 = - $ 1 . 2 / m 2, a 3 = $ 0 . 9 / m 2. Clearly, by far the greatest expenditure should be in compartmentalisation. The existing expenditure on fuel load control is already greater than necessary. Note that negative values are allowable if the expenditures aj are interpreted as expenditure above an initial base level.

3.3. Sensitivity The risk-balancing technique set out in this paper is intended to be used only as a rough preliminary guide for risk-relevant investment distribution. Nevertheless, a single calculation can be misleading, and in practical terms it is important to investigate the sensitivity of the results to changes in the input parameters. This is readily done if the calculations are set out on a spreadsheet. It is particularly easy to look at marginal sensitivities by using a proprietary spreadsheet macro such as @Risk, as mentioned earlier. However, it will often be appropriate to investigate the variation of the results following large changes to the inputs. As an illustration, consider the results of varying the total safety-related expenditure C in the risk-balanced fire design example given above. The initial value of C was taken as $ 3 0 / m 2, but it might in practice be within a far wider range. The variation in modal expenditures for a range of values of C is given in Table 2 and plotted in Fig. 1. In both the table and the figure, the results for the modal expenditures are given in percentage rather than absolute terms, to enable comparisons to

Table 2 Sensitivity of relative expenditures to variations in total safety expenditure C ( $ / m 2)

a 1 (%)

a 2 (%)

a 3 (%)

Expected risk

20 30 40 50 60

124.2 100.9 88.7 81.4 76.5

- 28.6 - 3.9 8.5 16 21

3.4 3.0 2.7 2.6 2.5

1.644 1.461 1.314 1.195 1.098

76

D.G. Elms 140

I

120 100 8O

--al

6O

......... a2 ..... a3

K 4o ~= 20

--7--ZSZ----~--v.-~Z2L--~--T-TT"7~ZZT-----Z---~---......... "30 -20

40

50

. ...-""

-40

C ($/sq.m) Fig. 1. Variation in relative modal expenditures.

be made more readily. It can be seen that the low original value of - 3.9% for a2, the expenditure on fuel load control, is misleading. Quite by chance, the curve for a 2 happens to cross zero very close to the initially chosen C value of $ 3 0 / m 2, but it has in fact a relatively steep slope. If the same calculations are carried out while, however, varying the standard reduction expenditure coefficients Uij in proportion to C (both are in terms of $/m2), then, unlike Fig. 1, there is virtually no variation in the optimal modal relative expenditures. The variation comes about due to proportionate differences between C and the expenditure coefficients. In practice, though, it would be expected that the standard reduction expenditure coefficients would be independent of C: the uij's are fundamental to the situation and are not design parameters, whereas the choice of C is very much a question of the choice of the designer. Table 2 also shows values of expected risk, calculated according to Eq. (2). It can be seen that a threefold increase in safety expenditure reduces the expected risk by about a third in this particular problem. Such an analysis is sometimes useful to the designer in trying to determine the most appropriate total expenditure on risk. In many cases expenditure is largely determined by code considerations and further analyses of this type would not be justified. However, the analysis is relevant and useful when dealing with larger questions, such as the development and appropriate levels of the codes themselves.

4. Conclusion

The risk-balancing technique described above is widely applicable. The examples given in this paper have referred to structural problems. However, there are many other engineering situations which could benefit from such an analysis. Examples where it has either been used or considered for application are: rail transport risk management, code formulation [6], local authority response to natural hazards [7], civil defence expenditure profiles, and fire service risk management.

D.G. Elms

77

Finally, to repeat an earlier comment, it must be understood that the technique is intended as a broad-brush and reasonably easily used tool for strategic planning. It has been developed with this in mind, and it should not be applied to detailed design problems. Software has been developed using the methods of this paper as an add-on macro for use with Microsoft Excel.

References [1] Wiggins, J. H. The balanced risk concept: new approach to earthquake building code. Civil Engineering, ASCE, 1972, August, 55-59. [2] Elms, D. G. Risk balancing for code formulation and design. In Proc. 3rd Int. Conf. on the Application of Statistics and Reliability in Soil and Structural Engineering, Sydney, Australia, 1979, pp. 701-713. [3] Elms, D. G. Risk balancing approaches. In Optimisation and Artificial Intelligence in Civil Engineering, ed. B. H. V. Topping, vol. 1. Kluwer Academic Publishers, Dordrecht, 1992, pp. 61-70. [4] Elms, D. G. The principle of consistent crudeness. In Proc. Workshop on Civil Engineering Applications of Fuzzy Sets, Purdue University, IN, 1985, pp. 35-44. [5] Elms, D. G. Consistent crudeness in system construction. In Optimiation and Artificial Intelligence in CiL'il Engineering, ed. B. H. V. Topping, vol. 1. Kluwer Academic Publishers, Dordrecht, 1992, pp. 71-85. [6] Elms, D. G. Rational derivation of risk factors. In Proc. Seventh Australasian Conf. on the Mechanics of Structures and Materials, Perth, 1980, pp. 149-153. [7] Elms, D. G., Berrill, J. B. and Darwin, D. J. Appropriate distribution of resources for optimum risk reduction. In Large Earthquakes in New Zealand: Anticipation, Precaution, Reconstruction, Misc. series No. 5, Royal Society of New Zealand, Wellington, 1981, pp. 69-75.