Probabilistic assessment of aircraft hazard for nuclear power plants

NUCLEAR ENGINEERING AND DESIGN 19 (1972) 333- 364

PaperJ4/l * ~ ~ . ~ .

N O R T H - H O L L A N D PUBLISIIING COMPANY FrS~ In~nwlie~d Con~encll Ill:

IIUlIIk'IW~. - . ~ - III!~I~ PROBABILISTIC ASSESSMENT OF AIRCRAFT HAZARD FOR NUCLEAR POWER PLANTS C.V. CHELAPATI

lwl~. ~21 ~

* a n d R.P. K E N N E D Y

tlolmes and Narver, Inc., Los Angeles, California, USA and I.B. W A L L

General Electric-APED, San Jose, California, USA Received 2 August 1971

As part of a general probabilistic safety analysis, the risk of structur',d damage to a nuclear power plant from aircraft crashes has been evaluated in a quantified manner. Frequency distributions of aircraft speed and weight and engine weight were constructed for small and large aircraft and for site locations adjacent to and remote from an airport. Based upon United States data an analysis of aircraft accidents is presented to establish the probability of an aircraft hitting a nuclear power plant. If an aircraft hits a building, either the whole building or just the local c o m p o n e n t may respond. It is shown that the response of the entire reactor building is negligible, and the damage to specific structural c o m p o n e n t s is of concern. C o m p o n e n t s of a reactor building may experience structural damage in several modes as a result of an aircraft crash, it is essential that all m o d e s of damage for any particular c o m p o n e n t are considered to establish the critical mode o f damage. Further, depending upon various parameters involved, the critical m o d e o f damage m a y vary for different c o m p o n e n t s . For the specific case of an aircraft striking a reactor building, three modes of damage could be delineated. The aircraft engine might perforate the structural c o m p o n e n t . This type of damage is classified as perforation mode of damage. The second m o d e is classified as collapse mode of damage, where a local collapse o f the structural system occurs. The third mode is classified as a cracking mode of damage, where the structural c o m p o n e n t ceases to function satisfactorily after the impact due to cracking. The 18-inch thick reinforced concrete sidewall o f a typical boiling water reactor plant located at the top floor of the reactor building is used as an example. The probability o f damage to this sidewall in the perforation and collapse modes is investigated. The results are compared to those obtained for the cracking mode of damage. Available empirical formulas for perforation of concrete are examined, and new formulas are proposed to cover the range o f parameters encountered in aircraft engine impact. Uncertainties in formulation are discussed, and the probability of damage by this mode is determined using Monte-Carlo methods. The conditional probability of local collapse of the wall panel is evaluated by using probabilistic approaches and yield line theory. The striking location (and thus the critical yield pattern), m o m e n t , and rotational capacities are all treated as r a n d o m variables. The probabilities of damage under the perforation and collapse modes are approximately o f the same order o f magnitude. Under the impact of an aircraft, the cracking mode o f damage is estimated using elastic analyses. Solutions are obtained using a finite-element idealization and considering the m a x i m u m reactive force due to an aircraft strike as a static load. For the 18-inch wall under consideration, it is predicted that the cracking mode o f damage occurs m u c h earlier than the other two modes of damage. It is shown that the impact load level predicted for cracking mode o f damage is very conservative. After a study o f all modes o f damage, it is concluded that the aircraft risk is usually acceptably low for the typical case studied here.

* Professor of Civil Engineering, California State College, Long Beach, California 90801, USA

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334

c.v. 67wlapati et aL. Prohabilistic assessment o f aircraft hazard for nuclear structures

I. Introduction Probabilistic approaches to the design, sitting, and safety analysis of nuclear power plants have been proposed by Farmer [I l, Wall [2], and Garrick [3]. Farmer and Wall established a limit-line which delineates between acceptable and unacceptable risks. A limitline, a complementary cumulative distribution function for fission product release magnitude, has been established for sites by statistical analysis of meteorology [2, 4]. To implement the method, all accident chains are systematically analyzed to determine their probability and associated fission product release magnitude: the combination is compared to the limit-line

I~.el. As an example of the method, the risk of damage a nuclear power plant from an aircraft crash is evaluated in a quantified manner. However, in the present paper only the concepts and probability of structural failure are discussed. The assessment of the risk due to fission product release has been discussed earlier by Wall and Augenstein [5] and Chelapati and Wall to

In the case of an aircraft striking a building, the region of impact is generally restricted to a local component. This impact energy is, in turn, transmitted to the whole structure. It has been shown that for a small aircraft strike, the response of the entire reactor building of a typical power plant is substantially less than that from an operating earthquake [7]. It can be shown in a similar manner that the response due to a large aircraft strike is approximately of the same magnitude as that from an operating basis earthquake. Further, it has been shown that the risk due to an operating basis earthquake is acceptably low; therefore, it can be concluded that the response of the total building is of no concern for an aircraft strike. Thus, this paper concentrates on the damage to a local component. Three modes of structural damage to a local component can be identified if an aircraft strikes the reactor building. The first type of damage is identified as the perforation mode where the aircraft engine perforates the structural component upon impact [6]. The second type of damage is caused by a collapse mode where the structural member yields considerably at all restraints so that it forms a mechanism. The load corresponding to the collapse mode is obtained by using yield line theory. In this mode of damage, the

structural component loses all of its integrity, and the [hlling debris from the aircraft or structure may enter the containment. The third type is identified as the cracking mode where the reinforced concrete structural member cracks inder impact. The loads corresponding to this criteria are obtained by making use of elastic analysis where the maximum moment reaches the ultimate moment of the section. Thus, the load obtained by using this criteria will not cause extensive deformations, and any equipment inside the structure is unharmed. In this mode of damage only one portion of the structure reaches the ultimate moment capacity, and thus the structure as a whole may still have considerable reserve capacity to counteract the load. Depending on the function of the structural barrier and tire equipment inside the structure, one might be interested in the perforation mode, collapse mode, of cracking mode of damage. The loads corresponding to the collapse and the cracking mode might be substantially different or very close, depending on the structural behavior of the component under loading. In this paper, all three modes of damage are discussed. The risk assessment approach to reactor safety is really an application of decision analysis to reactor design [8]. The assessment of the risk, probability, and consequences stemming from each possible initiatting event, e.g., earthquake shock, aircraft crash, pipe severance, tornado damage, provides perspective on the relative importance and contribution of each engineered safeguard in mitigating the overall risk. For decision-making within reactor design, delineation of significant and trivial risks only requires estimates of the probability and consequences within a factor of 10. Risk assessment is an aid, not the final arbiter in safety design. This paper contains: Analysis of aircraft accidents to establish the probability of an aircraft striking a nuclear power plant and the frequency distribution for aircraft: Discussion of various empirical perforation formulas and the development of a new empirical formula for the determination of perforation thickness; Determination of probability of perforation using the Monte-Carlo method; Determination of the probability of collapse mode of damage using yield line theory; Discussion of the cracking mode of damage using elastic theory and finite-element idealization.

335

C. E Chelapati et al., Probabilistic assessment o f aircraft hazard for nuclear structures

2. Probability of an aircraft strike 2.1. Aircraft accident data Aircraft accident data within the United States are published annually [9]. Ten and four-year periods up to and including 1968 were analyzed for air carrier operations and general aviation, respectively; the shorter period was used for general aviation since its rapid growth made the earlier years misleadingly low. The data are principally categorized into air carrier operations and general aviation, but the more useful breakdown for risk analysis is by aircraft weight with a division at 12500 pounds. Perusal of the briefs for air carrier accidents suggests that only accidents causing fatalities are significant with respect to an aircraft striking a building. Indeed, there is a strong correlation between accidents involving fatalities and those destroying the aircraft, and it is difficult to conceive of an aircraft striking a building without destruction and fatalities. The same conclusion seems reasonable for general aviation. For this reason, only fatal accidents were used to estimate the strike probabilities. Further statistical breakdowns [9] show that "collision with building" was the initial cause for only a small fraction of fatal accidents; therefore, the use of all fatal accidents is probably conservatively high with respect to aircraft risk. The aircraft accidents are also classified with respect to the proximity of the airport, i.e., those occurring within and beyond a 5-mile radius from the airport [9]. However, accidents on the airfield were excluded from the data. The summary of the aircraft accidents within the United States is shown in table 1.

Table 1 Annual aircraft accidents involving fatalities within the United States by operation, size, and location. Location of plant

Aircraft size Small • 12500 lbs Large > 12500 lbs

Beyond 5 miles Within 5 miles o f airport of airport Air General Air General carrier aviation Total carrier aviation Total 0 6.6

355.6 5.4

355.6 12.0

0 2.9

176.3 176.3 2.7 5.6

To estimate the probability of an aircraft hitting a nuclear power plant, the shadow areas of the reactor and turbine buildings and the switchyard were estimated. It was assumed that the aircraft has a strike angle of 10 degrees above the horizontal. For a typical boiling water reactor, the target areas are 0.093, 0.026, and 0.018 square miles for the reactor, turbine building, and switchyard, respectively. The number of operating airports within the United States was estimated to be 9200. From these estimates and table I, the probability of an aircraft striking some part of a nuclear power plant is summarized in table 2. Table 2 Probability of an aircraft striking a nuclear power plant per year. Location of plant Aircraft size

Small • 12500 Ibs Large > 12500 Ibs

Beyond 5 miles of airport

Within 5 miles of airport

1.4-5" 4.6-7

3.3-5 1.1-6

"1.4-5 denotes 1.4X 10-s. Three observations should be noted about the strike probabilities in table 2. First, they are average values throughout the United States and specific locations would be different depending upon their proximity to specific airports and traffic corridors. More importanly, only a fraction of the airports are probably capable of handling large aircraft. If this fraction is 10%, the average probability of a large aircraft striking a plant location within 5 miles of an airport would be increased to 1.1 X 10-5/year for those airports and approximately 4.6 × 10-6/year for the remainder. Second, the quoted strike probability near an airport is an average value over a 5-mile radius. A more detailed examination [10] shows that the strike probability varies approximately as 1/r 2, where r is the distance from the airport. That is, the strike probability for a plant located at 1/4 mile from an airport is about 60 times the average value and the probability at 5 miles is about equal to the quoted value for beyond 5 miles. Having noted these two refinements, they will not be considered further in this example. Third, the quoted strike probabilities are conservative estimates for the whole nuclear power plant including reactor and turbine buildings and switchyard. The probability of striking

336

C V. Chelapati et al., Probabilistic assessment of aircraft hazard for nuclear structures

a specific are a containing critical equipement, e.g., control-rod drive hydraulics, would be substantially lower; an approximate factor is estimated. The relative target areas of the reactor building, turbine building, and switchyard are 0.68, 0.19, and 0.13, respectively. Typically, a critical piece of equipment is located on one side of a building and would only be vulnerable to aircraft coming from a specific quadrant, a factor of 0.25. Further, a typical piece of equipement occupies only about 5 percent of a wall. Theretbre, the probability of striking a wall area opposite specific equipment is about i% of the strike probabilities of an aircraft striking the plant.

2.2. Frequeno, distributions for the characteristics o f striking aircraft The conditional probabilities of perforation or collapse of a structural component are functions of aircraft characteristics and speed. Frequency distributions for aircraft speed and weight, engine weight, and effective diameter are generated from the annual census of U.S. Civil Aircraft [ 11 ] published by the Federal Aviation Administration. Table 3 shows the number of eligible aircraft of December 31, 1968, by weight, piston versus turbine, and air carrier versus general aviation. For small aircraft, the piston engine category contributes over 90 percent of the class; this type was assumed to represent small aircraft. Table 3 Distribution of eligible United States aircraft. Aircraft type Air carrier General aviation Total

Small aircraft Large aircraft <~ 12500 lbs > 12500 Ibs ........................... Piston Turbine Piston Turbine 69

16

517

2302

117509

770

1225

1063

117578

786

1742

3365

The size, weight, and speed of an aircraft are, in general, a direct function of its horsepower. The 1967 census includes a breakdown of aircraft by horsepower and number and type of engines. From these statistics and data from catalogs, it is possible to construct a frequency distribution for aircraft speed and engine weight. A distinction should be made between site

locations, within and beyond 5 miles of an airport, with respect to probable aircraft speed. Within 5 miles of an airport, an aircraft is likely to be landing or taking off; a typical speed of 140% of stall was assumed if no other data were available. Away from an airport, an aircraft is most likely to be crusing at 75% power, but other possible speeds are 140% of stall (if in trouble) or maximum speed. For small aircraft, frequency distribution of 0.25, 0.5, and 0.25 were assumed for 140% of stall, 75% power and maximum speed, respectively. With these additional assumptions, four frequency distributions were generated for small and large aircraft striking within and beyond 5 miles of an airport. The data are presented in tables 4 and 5. The data are almost lognormally distributed, and this fact was used in subsequent analyses for the collapse mode of damage.

2.3. Determination o f a possible location for an aircraft strike In a nuclear power plant, a direct strike by an aircraft to the reactor building is of maximum concern. A typical boiling water reactor plant is taken as an example. From an examination of the plans of the reactor building, the longitudinal sidewall of the top floor is assessed to be the most vulnerable structurally. Fig. 1 shows schematically the reactor building being struck by an aircraft. Fig. 2 shows the dimensions and details of reinforcement of the sidewall, columns, and tloors. The wall panel is 24 feet 6 inches by 24 feet center to center. The thickness of the wall is 1 foot 6 inches with reinforcement of No. 9 bars at 9 inches center to center, running in both directions on both sides of the slab. Structurally, the wall is continuous over the columns and the floors, and is monolithic with the roof.

2.4. Ideal&ation o f airc'ra]? as a projectile In the frequency estimates in sections 2.1 and 2.2, both large and small aircraft, as well as single-engine and multi-engine aircraft, are included. To conduct the structural analysis in the latter part of this paper, the aircraft is assumed to be a nondeformable projectile. Further, it is assumed that the aircraft engine is the only part of the aircraft that offers enough intpact to the structure to cause damage. Other parts of the air-

337

C.V. Chelapati et al., Probabilistic assessment o f aircraft hazard for nuclear structures

craft, like wings and fuselage, offer less resistance and are assumed to break up upon impact. The above assumption is reasonable for single-engine small airplanes. For two engine small airplanes two points of impact have to be considered, but it is assumed that two engines are far enough apart that the solution obtained for one engine should be reasonable. However, for larger aircraft the fuselage also offers a great tesistance to crushing under impact. In this study, for the purposes of computation of perforation thickness, the engine weight is used only. It is believed that the thickness obtained using the characteristics of the engine gives more conservative results than that obtained by using the characteristics of the fuselage. For the collapse mode and cracking for the portion of the body weight associated with the engine.

Table 5 Frequency distribution for large aircraft projectiles. Frequency

Speed [mphl

Frequency

Speed [mph]

Effective diameter [inches]

Aircraft weight [lbsl

Within 5 miles of airport 0.24 0.38 0.25 0.04 0.06 0.03

67 70 84 98 105 105

230 340 560 560 560 800

33 36 43 43 43 46

1300 1800 2500 2900 5100 7000

E f f e c t i v e Aircraft diameter weight [inches] [lbsl

Within 5 miles of airport 0.17 0.14 o. 10 0.11 0.05 0.11 0.06 0.10 0.03 0.13

100 100 95 125 120 140 115 185 150 150

Table 4 l:requency distribution for small aircraft projectiles. Engine weight Jibs]

Engine weight [Ibs]

1470 2350 3000 1750 1500 450 3100 3100 3500 4200

55 53 54 33 35 22 44 44 38 53

21000 38000 85000 37000 70000 15000 70000 130000 150000 210000

Beyond 5 miles of airport .

0.17 0.14 0.10 O.ll 0.05 O.ll 0.06 O.10 0.03 0.13

175 280 340 300 375 510 565 550 600 610

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21000 38000 85000 37000 70000 15000 70000 130000 150000 210000

3. Perforation mode of damage Beyond 5 miles of airport 0.06 0.12 0.06

67 117 122

230

33

1300

0.10 0.18 0.10

70 125 150

340

36

1800

0.06 0.13 0.06

84 170 180

560

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98 200 220

560

43

2900

0.02 0.03 0.01

105 240 270

560

43

5100

0.01 0.015 0.005

105 240 280

800

46

7000

It has been assumed that the aircraft engine is a rigid body projectile and the reinforced concrete wall slab of the reactor building is the target. The analytical formulation of the problem of the aircraft impact and perforation to find the thickness of concrete so as to protect the equipment inside the structure is difficult and impractical. Thus, all the available formulae describing penetration pnenomena are empirical and are based on experimental data. As such, these empirical formulae are valid only within the ranges of the variables for which the experimental data are available. Many of the empirical formulae are based on the experiments conducted by striking bullets and bombs on reinforced concrete target slabs during Wold War II. The range of variables encountered in the problem of aircraft impact are in general different from the range

338

C. V. Chelapati et aL, Probabilistic assessment o f aircraft hazard for nuclear structures

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of variables for which the experiments were conducted• Since no other data is available, the available formulae are used in a number of studies without modifications, thus sometimes leading to very unreasonable results. Since the area of perforation mode of damage is relatively uncertain, an effort is made in this paper to review the most frequently used formulae and to come up with a perforation formula to be used in the case of perforation of reinforced concrete due to aircrafttype missiles.

3. 1. Description of penetration phenomenon Extensive work was done to develop terminal ballistics of concrete during World War II. A summary of the work done during that time was published 112]. While discussing the impact of projectiles on concrete, it is essential to distinguish between the terms, penetration and perforation and scabbing. Penetration depth is used for the depth to which a projectile enters a massive concrete target without passing through it. No evidence of bulging or rupture of the target on the backside can be observed. Thus, penetration depth is

C V. Chelapati et al., Probabilistic assessment of aircraft hazard for nuclear structures

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3.1.1. Historical development

perforation thickness is used specifically when the projectile just passes through the target completely.

Various empirical formulas for penetration and perforation thickness are given in appendix A. The Petty formula [eq. ( 1 - 1 ) of appendix A] was developed in 1910 and is the oldest of the available formulas. The formula gives the penetration depth, x, which is valid for targets with ticknesses over 3x. For slabs with thickness below 2x, perforation is implicitly assumed. For slabs with thicknesses between 3x and 2x, the depth of penetration is given by

That is, the exit velocity of the projectile after passing through the target is zero. The term scabbing thickness is used when the material on the backside of the target just begins to fall off. Thus, the tickness of target is implicitly associated with the perforation and scabbing thicknesses.

340

c v. Chelapati et aL, Probabilistic assessment of aircraft hazard for nuclear structures

x' [eq. ( I - 2 ) ] and is referred to as modified Petty formula [ 13]. Later, the BRL formula ( 2 - 1 ) was developed. In 1941, Beth [14] reviewed all the formulas that existed at that time and suggested an approximate theory of penetration and an empirical formula for penetration depth, x, of the nature x = K D Vad ~,

(1)

where K, a,/3

= empirical constants,

= the weight of projectile in pounds,

d

= the diameter of the projectile [inch],

V

= the velocity of the projectile [103 ft/sec.].

d = projectile diameter [inches],

= (x/2d) 2, for 0 ~< x/d <~ 2" G

(x/d) = (x/d)- 1, for x/d > 2.

For large diameter projectiles, experiments yielded the following relationship for perforation thickness (3)

e = perforation thickness [inches].

Based on the experimental results prior to 1943 from Ordnance Department, U.S. Army, and Ballistic Research Laboratory, the Army Corps of Engineers developed eq. ( 3 - 1 ) for the penetration depth, x. They also gave the relation between the penetration depth x and perforation thickness e and scabbing thickness s [eq. (3-2)]. Additional data was obtained during 1944 -1945, and the above relations were modified to those given by eq.(3-3). Beth [15] reviewed all the above information and suggested empirical relations similar to eq. (2), along with the ranges where this formula is applicable. Based on the conclusions reported by Beth and other available data. The National Defense Research Committee proposed an approximate theory of penetration for concrete leading to the empirical formula [121

x = penetration depth [inches],

and

where

W

where

o = striking velocity [ft/sec2].

e/d= 1.32+ 1.24(x/d) for3<~e/d<~ 18,

D =W/d 3= tke caliber density,

G (x/d) = (c/X/fc)NdO'2D (l-~-O v)l ~'

D = weight of projectile divided by d 3 [lb/in3],

(2)

It can be seen that this relation is the same as that suggested by the Army Corps of Engineers, but it is ,now valid only for values o f e / d between 3 and 18. It is obvious that when x = 0, e also must be equal to zero. For the projectiles and walls considered in this study, e/d is usually less than 3. Ref. [20] suggests parabolic extrapolation so that the curve will pass through the origin. The following expression was derived [16, 171: e/d = 3.19 (x/d) - 0.72 (x/d) 2, fore/d < 3. (4)

The probability of perforation is estimated as a function of thickness of reinforced concrete. The original experiments [12] on penetration found that the effect of reinforcing steel in resisting penetration was relatively small and did not justify increasing reinforcement beyond 1 percent. The average reinforcement of the wall exceeds 1 percent. It is, therefore,judged that the detailed design of reinforcement is not a significant variable with respect to perforation probabilities as a function of wall thickness. It should be emphasized finally that all the above empirical relations are based on the experiments conducted using high velocity, high caliber density, and smaller diameter projectiles. In the aircraft impact the problem is concerned with low velocity, low caliber density, and large diameter projectiles.

C = constant, fc = 28-day compressive strength of concrete

[psi], N = nose shape factor for projectile,

3.1.2. Comparison o f experimental data with empirical formulas The NDRC report gave detailed experimental data on high velocity bullets striking 9-inch and 22-inch

C. V. Chelapati et aL, Probabilistic assessment of hazard for nuclear structures

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slabs. Fig. 3 compares the penetration depth,x, using the four formulas with the experimental data. As expected, the Corps of Engineers' formula and the HNNDRC formula agree very well with the experimental data. The Petry and BRL formulas show a different trend and yield larger values for x at higher velocities. At 200 mph all the formulas gave similar values. Fig. 4 shows the perforation thickness, e, using the four different formulas. For the Petry formula a value of 2x is plotted as the perforation thickness since it is below this value that the slabs will be perforated. Again, it can be seen that the HN-NDRC and Corps of Engineers' formula give good correlation with the

experimental values. The BRL and Petry formulas do not follow the trend. They give higher values at larger velocities and lower values at lower velocities. Table 6 shows the penetration thickness for four different projectiles ranging from a 1.70 pound, 1.46inch diameter bullet to a 1000-pound, 12-inch diameter inert bomb. The NDRC report gave figures showing actual craters formed when these projectiles strike the target slabs of thicknesses varying from 9 inches to 75 inches. However, all these projectiles hit with a 20-degree obliquity. The normal depth of penetration Xp, in the case of oblique impact is related to the penetration depth, x, by the relation x = Xp/COS ¢, where ¢

342

C: V. Chelapati et al., Probabilistic assessment o f hazard for nuclear structures

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Fig. 4. Comparison of perforation thicknesses,e, using various formulas with experimental data. is the angle of obliquity. Observation of table 6 indicates the HN-NDRC formula gives good correlation with experimental results. The other formulas give higher values of penetration. In the first three cases the depth of penetration, Xp, is less than one-third of the thickness of target slab, and thus the penetration formulas are applicable as such. However, for the last case with a 1000-pound bomb, the value Of Xp is about half the thickness of slab; and thus the penetration formulas are not applicable as shown. This conclusion is indicated by the larger experimental value of 48

inches compared to 37.8 inches given by the HN-NDRC formula. The HN-NDRC formula predicts a perforation of this target at 800 mph. It can be stated, based on the results shown here, that the HN-NDRC formula gives the best estimate of the experimental values. This is due to the fact that the HN-NDRC formula is based on extensive experimental studies performed during 1941 to 1946 while the Petry, BRL, and Corps of Engineers' formulas are based on experimental results available prior to the NDRC studies.

343

C. V. Chelapatiet al., Probabilisticassesment of hazard for nuclear structures Table 6 Comparison of penetration depth, x, for four different projectiles using various penetration formulas. Penetration depth X [inches] Projectile Case weight [lbs] 1.70

Projectile C a l i b e r diameter d density D [inchesl [lb/in 3]

Experimental values [inches]

Corps Velocity of V engi- HN[mphl Petry BRL neers NDRC

20° Oblique Normal Thickness penetration penetration of slab Xp X = X p / 0 . 7 5 [inches]

1.46

0.546

682

6.5

5.3

4.0

3.5

12.55

2.95

0.489

629

10.5

9.6

7.5

6.6

5.0

6.7

19.0

85.5

6.10

0.377

678

18.0 19.7 15.6 13.9

10.0

13.3

30.0

12.00

0.579

661

52.9 65.5 48.2 37.8

36.0

48.0

75.0

1000

3.1.3. Aplication of empirical formulas to the case of impact of large aircraft The four empirical formulas discussed earlier are applied to two cases of impact of typical large jet aircraft engines; one weighing 4000 pounds with a 36inch diameter, and the other weighing 6000 pounds with a 60-inch diameter. The perforation thicknesses are plotted against the velocity of the aircraft. The ultimate strength of concrete is assumed to be 3200 psi. The results are shown in fig. 5 and 6. For both cases, the Petry and BRL formulas give very low values of perforation thickness for low velocities. For the 6000 pound engine at 100 mph, the Petry and BRL formulas give values of 1.4 and 2.30 inches, respectively. When one visualizes that this projectile is equivalent to a small truck, one can very well judge intuitively that a 1.4-inch concrete barrier will shatter under impact and will in no way stop the truck after impact. Even more dramatic evidence can be obtained by using a value of 50 mph (approximate cruising speed on a freeway). The Perry formula gives 0.4 inch and the BRL formula gives 1 inch for perforation thickness for 50 mph. The unrealistic values obtained by the Petry and BRL formulas are due to application of these formulas to the ranges of variables for which they were not designed. Fig. 5 and 6 also show the results of the application of the Corps of Engineers' formula as orginally given

2.7

3.9

9.0

for comparative purposes. It can be seen that this formula gives very large values. However, it should be remembered that the Corps of Engineers formula is valid only for values ofe/d between 3 and 8. The NDRC report pointed out this limitation and suggested a parabolic lift for values ofe/d below 3 so that it passes through origin. Holmes & Narver, Inc., in their studies conducted by Kennedy [ 16] and Chelapati [17] incorporated the NDRC recommendations, and the resulting plot is designated as HN-NDRC in fig. 5 and 6. Chelapati [17], in addition, gave formulas for the upper and lower bounds of penetration depth, perforation, and scabbing thicknesses.

3.1.4. Propagation of errors in empirical formula for perforation thickness One of the auxiliary benefits of a probabilistic approach to safety is the ranking of paramaters with respect to their contribution to the overall uncertainty. From this information, the value of refining a specific parameter (reducing its uncertainty) by additional experiments, etc., may be assessed. The relative contribution of each parameter in eq. (2) and (4) was estimated by the propagation of errors [19]; the dependence of the parameters d, D, and o was properly incorporated. For the case of small aircraft and plants located within 5 miles of an airport, the uncertainty in the projectile description (i.e., size, weight, and

344

C. I/. Chelapati et al., Probabilistic assesment o f hazard for nuclear structures 140

I

I

I

I

WEIGHT OF PROJECTILE

I

4000 Ih.

OlkMETER OF PROJECTILE

36 in.

ULTIMATE COMPRESSIVESTRENGTH OF CONCRETE 3200 p . s . i . SHAPE FACTOR . . . . . . . . . . . 0.85 CALIBER DENSITY . . . . . . . . . . . . . 0.086 1"~-3 Lb.

120

I00

c:

v

80

w o-

40-

.~:"":::'":""

! 20:

....

O0

Jtz~'~" "1

1O0

I

/

200

I

300

400

VELOCITY. V,

I

500

61

(m.p.h,)

I:ig. 5. Comparison of perforation thickness, e, using various formulas for a large projectile.

speed) contributes about 90% to the overall uncertainty. The other variables (concrete strength, nose shape, and the empirical relationships themselves) contribute 2.4, 2.7, and 4.4%, respectively. For the other aircraft sizes and plant locations, the projectile uncertainty is even more dominant. Therefore, it is evident that further experiments to refine the penetration formula would not significantly reduce the overall uncertainty and would not be justified economically.

3.2.6bnditional probability of perforation The conditional probability of perforation of a given thickness of reinforced concrete was estimated by a simple Monte-Carlo technique. Frequency distri-

butions were assigned to each major variable in eq. (2, 3), and (4); from a random sampling, a perforation thickness was calculated. From 10000 trials, the complementary cumulative distribution function for perforation thickness was generated. Field measurements of the compressive strength of t concretefc,indicated that the data can be assumed to be normally distributed [7] with a mean of 4500 psi and 3o limits of 3000 and 6000 psi. The nose shape factor, N, ranges from 0.72 for fiat-nosed to 1.14 for very sharp projectiles. For this study, N was assumed normally distributed with a mean value of 0.85 and 30 limits of 0.7 and 1.0. The variables d, D, and o describe the projectile diameter, density, and velocity. By consideration of the

345

C. V. Chelapati et al., Probabilistic assessment o f hazard for nuclear structures

180

140

I

I

I

I

I

WEIGHTOF PROJECTILE 8000 lb. DIAMETEROF PROJECTILE 60 in. ULTIMATE COMPRESSIVESTRENGTHOF CONCRETE 3200 p.s.i. SHAPE FACTOR 0.85 CALIBERDENSITY __ 0.028---Ib" In.

120 --

•

cDRPSOF ENgllLEERS

,.... . . . . . . . . . . . .

*

•

•

•

•

•

•

•

~

J

"'"

c

100 .

°

80

60

40

~,~. ....

1 ~.~" OM 0

IOO

200

300

400

500

600

VELOCITY, V, {m.p.h.)

Fig. 6. Comparison of perforation thickness, e, using various formulas for a large projectile. empirical formulations, it was concluded that the aircraft engines were the most significant projectiles by virtue of their relatively high weight per unit area. The variables d, D, and o are dependent and are all described by the frequency distributions for small and large aircraft; in addition, the velocity distribution was a function of the plant location, within or beyond 5 miles of an airport. In addition to the frequency distribution on missties and the uncertainty in concrete strength, etc., there is uncertainty in the empirical relationship of eq.

(2, 3) and (4). This uncertainty was handled by introducing uncertainty into the empirical constants and the exponent for v in eq. (2). From experiments [15, 20] with 37 and 155 millimeter projectiles, values of the constant Cwere found to range from 146 to 192 with an average of 176; its value was assumed normally distributed with a mean of 180 and 3o limits of 140 and 220. it was also apparent that uncertainty existed in the empirical exponent of 1.8 in eq. (2). From the scatter of the experimental data, maximum and minimum values of 1.9 and 1.7 were estimated; and these

346

6: V. Chelapati et aL, Probabilistic assessment of hazard for nuclear structures

values were assumed to be 3o limits of a normal distribution. No experimental data was available from which to judge the uncertainty in the exponent of d; however, the sensitivity of perforation thickness to this variable is small so the exponent was assumed constant at 0.2. The scatter in the experimental points leading to eq. (3) is about +- 10%. Two additional equations, bounding the experimental data, were assumed to represent the 99% prediction interval. These equations were extrapolated to zero in the same manner as eq. (4). The additional uncertainty was introduced into e/d by assuming a normal distribution about the mean regression line with 3o limits of -+ 10%. In the analysis it was conservatively assumed that the missile struck the wall perpendicularly. The penetration depth is sharply reduced by oblique impact. For example, impact at 30 degrees from the perpendicular would reduce the penetration depth by approximately 42 percent for projectiles with velocities in the range 1,000 to 2,000 fps [12]. As obliquity is increased, ricochet may occur. By using a computer program, 10000 trials were simulated and the conditional complementary cumulative distribution function for perforation thickness of reinforced concrete was estimated. The summary of results for the four distributions are shown in Tables 7 and 8. It can be seen from Table 7 that for a small aircraft the mean perforation thickness is 6.3 inches when the plant is located within 5 miles of the airport. The standard deviation is 1.8 inches. For a large aircraft the mean perforation thickness is 54.9 inches when the plant is located outside the 5-mile limit from the airport. The probabilities of a strike, the conditional probabilities of perforation given a strike, and the absolute probabilities of perforation are summarized in ~able 8 as a function of plant location and thickness of reinforced concrete. It can be seen that the absolute probability of perforation of an 18-inch, thick reinforced concrete wall is 6 × 10 -7 for small aircraft strike and 10 .6 for a large aircraft impact. Values are also given for other thicknesses. For a 6-foot thick wall the absolute probability of perforation is zero for small aircraft and 1.5 × 10.7 for large aircraft. The perforation probabilities stated in table 8 are associated with the missile just passing through the wall, i.e., exit velocity is zero. However, it should be noted that even a slight reduction of wall thickness, say 10%,

results in a residual velocity of almost 40% of the striking velocity [ 17, 18]. The strike probabilities are average values for the whole plant, and all absolute probabilities should be multiplied by approximately 0.01 to obtain the probability of perforating a wall opposite specific equipment. Furthermore, the probabilities should be modified for specific airports and for plants located within 2 miles of an airport or on the flight path. Table 7 Summary of perforation thickness as function of aircraft type and plant location. Perforation thickness [inches] Plant location Aircraft Standard 95th 99th from airport type Mean deviation percentile percentile Small

6.3

1.8

10

12

Large

20.9

7.7

36

38

Small

10.2

4.4

19

24

Large

54.9 26.4

100

108

~<5 miles

> 5 miles

4. Collapse mode of damage This section is concerned with estimating the conditional probability of collapse for the typical reinforced concrete wall panel described in section 2.3. The purpose is twofold, consisting first of predicting the conditional probability of wall panel failure and secondly of illustrating the use of simple probablilistic techniques to account for the uncertainties associated with the prediction. An energy balance technique is used to determine the factor of safety against a flexural failure of a single wall panel. The factor of safety is defined as the ratio of the strain energy capacity of the individual wall panel to the kinetic energy absorbed by the wall panel as a result of aircraft impact. Probability density functions are estimated for the strain energy capacity and the absorbed kinetic energy from which a probability density function is determined for the factor of safety. The probability of flexural failut~ of a wall panel is equivalent to the probability of the factor of safety being less than unity.

347

C. V. Chelapati et al., Probabilistic assessment o f hazard for nuclear structures

Table 8 Probabilities of perforation as a function of plant location and concrete thickness. Probability (b) of perforation (conditional; absolute) .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Thickness of reinforced concrete Plant location from airport 5 miles

> 5 miles

Aircraft type

Probability of strike/year

1 foot

Small

3.3-5 (a)

0.003; 1-7

0

Large

1.1-6

0.96; 1-6

0.52;5.7-7

Total

3.4--5

1-6

6-7

3-7

0

Small

1.4-5

0.28;3.9-6

0.06;8.4-7

0.01 ;1.4-7

0

Large

4.6-7

1.0;4.6-7

1.0;4-7

0.84;3.9-7

0.32;1.5-7

Total

1.4-5

1-6

5-7

1.5-7

.

4-6

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1.5 feet

.

.

.

.

.

.

.

2 feet

6 feet

0

0

0.28; 3.1- 7

0

(a) 3.3-5 denotes 3.3 X 10"5. (b) Multiply all absolute values by 0.01 for damage probabilities for specific equipment.

The strain energy capacity for the individual wall is subdivided into the elastic strain energy occurring prior to yielding and the nonelastic strain energy occurring along lines of yielding on the slab. The elastic strain energy is considered negligible and is ignored. The yield line pattern is determined for any aircraft engine impact location on the slab. The moment and rotational capacities are evaluated for the slab. Based upon assuming a rigid-plastic moment-curvature diagram for the behavior of the wall panel along the yield lines, the nonelastic strain energy capacity per unit length of yield line is the product of the moment capacity per unit length times the rotational capacity. The total nonelastic strain energy capacity is then equal to the product of the nonelastic strain energy per unit length times the total length of the critical yield lines. The engine striking location and thus the critical yield pattern and the moment and rotational capacities are all treated as random variables, and the median and standard deviation values are evaluated for each. Median values and standard deviations for the impact kinetic energy applied by an aircraft engine to a wall panel are evaluated for four broad classes of impacting aircraft consisting of small one-engine and twoengine private aircraft traveling at landing and takeoff velocities (close to an airport); and large multiengine

commercial aircraft both close to and distant from an airport. Next, the kinetic energy absorbed by the wall panel is evaluated as a function of the impact kinetic energy of the impacting aircraft, the degree of plastic versus elastic impact, and the ratio of the impacting weight to the effective wall weight. Median values and standard deviations are determined for each significant parameter affecting the absorbed kinetic energy. 4.1. Distribution o f variables

While evaluating the perforation mode of damage, a normal distribution was assumed for the variables and the distribution for the perforation mode of damage was generated using Monte-Carlo procedures. Thus, the procedures, of necessity, are practical only when large digital computers are used and when the cost per one set of calculations is small so that large numbers of solutions can be run without considerable expense. A different approach is used in this case to generate the distribution for the collapse mode without the use of computers [21]. A lognormal distribution is used for all the variables. In the calculations several variables are in the form of products or quotients. if the variables are assumed to have been distributed Iognormally, it is a simple matter to derive the distribution for functions of variables involving products or quotients. Also, for small coefficients of variations,

348

C. v. Chelapati et al., Probabilistic assessment of hazard ]or nuclear structures

the difference between lognormal and normal distribution is not very significant except in the case of very low probabilities. The lognormal distribution is used often in civil engineering applications [22]. A brief description of the lognormal distribution is given in appendix B. The two parameters required to describe completely the properties of a lognormally distributed variable, x, are the median X and the standard derivation 6 x of the corresponding norreally distributed variable .~ where ~ is equal to log x; 6 x is called a lognormal standarcJ deviation. In addition to the values of X and °x, it is very informative to have the 90% confidence interval values for the variable x, which can be representative of a lower and upper limit on the variable. For all the variables used here, the above four parameters (mean, logarithmic standard deviation, and lower and upper condifidence limits) are given with the following notation: x(median, log standard deviation), [90% confidence interval], ( X , ~x), [x(O.05), x(0.950)].

(5)

4.2. Ultimate m o m e n t capacity, m u, o f the wall For the wall slab under consideration, it can be shown that it is underreinforced, indicating that the reinforcing steel would yield prior to the crushing of concrete. This condition is beneficial since it increases the ductility of the wall. It can also be shown that the compression steel is ineffective at ultimate load. The ultimate moment capacity per unit length is thus given by m u = As./~, de,

(6)

where A s is the area of steel,)'y is the yield stress of steel and d e is the effective lever arm for the slabs as shown in fig. 7 and is given by d e = d - ½klC,

(7)

and

Af c = ~E~b.

(8)

4500 psi with a standard deviation Of 500 psi 17]. Since aircraft impact is of very short duration, the rate of straining could be expected to increase the compressive strength by 10 to 20%. For this study the following parameters are used to describe the ultimate dynamic compressive strength of concrete: fc' (5200, 0.12), [4270, 6340] psi.

The minimum yield stress for reinforcing steel is specified as 60000 psi. From experimental data of similar steels the mean stress is found to be 15% more than the specified minimum. Further. an increase in yield stress between 0 and 8% is expected because of the in)pact nature of the loading. The nlinimum values of k I and k 2 are defined by ACI building Code (1963) conservatively at 0.79 and 0~85 [23], and it can be assumed that the mean values of tl~ese coefficients are approximately 5% greater. The area of steel, As, is taken as constant and is equal to 1.33 square inch/foot width. The parameters used for/i,, kl, and k 2 are given below: Ji' (72000, 0.08), 163100, 822001 psi,

(10)

k I (0.85, 0.03), 0.81,0.891.

(1 I)

k 2 (0.9, 0.03), [0.86, 0.95].

(12)

Using the values above, the parameters for c and k i t are obtained as c (2.01,0.15), [1.57, 2.57], klC (1.71,0.15), [1.33, 2.20].

(13) (14~

Thus, the standard deviation for klC is approximately 0.25 inches. The tensile steel depth, d, can be reasonably assumed to be lognormally distributed with a me dian value of 15.56 inches and a standard deviation of 0.25 inches for workmanship errors. For both the variables, d and k i t , the coefficients of variation are very sma'.l and thus, the difference between the assumptions of lognormal and normal distribution is not appreciable. Thus, the median of d e can beapproximated by median o f d e = 1 5 . 5 6 - ½ × 1.71 = 14.7 inches. and

Field data for concrete with a minimum specified ultimate compression strength of 3750 psi produced concrete with a mean compressive strength,f~., of

(9)

standard deviation Ode

(15) 0.28.

C V. Chelapati et al., Probabilistic assessment o f hazard f o r nuclear structures

349

: 2,44"

15.56"f

(a)

WALL PANEL PROPERTIESPER UNIT WIDTH v

Cc=T =klk2f c . ~

~,: :~!,:~::.-, ~

.L~

NEUTRAL " I S : ' ~

}I~:NEUTRAL IX ' i 'i;;i/,'~:, ~:.".'.:: ,: -<:,':~'~',.t,-.~ ~ : : f.,:~),tzt,.,~

,..% . . ,

,1• t.. ,,a..

Ld

;.;;.,.S,)~.~,~;:~;3,;~...,~

ks c .f, ,t~........... " t~'.:', : '~:i .:~ '~;~ • ";..~.:,,..' d =d.f.~ "-~

,,~ ','-';~:; :.:..':t,,::,;" 'i ~.: ." , a;..,,~

-

.:'.'...?:

...

."

~-{~:';:'?'~; .' :..'.T.':'.. -'"~:-:'.';':

./;.;"~:.';".~,L'.'i:"~:.~ ,;i:;.'.t

L .

5

T =A

Te =

(b)

ACTUAL STRESSCONOITION IN SLAB AT ULTIMATE MOIENT

(c)

Fig. 7. C r o s s s e c t i o n

Thus, the coefficient of variation and logarithmic standard deviation of d e are given by

Ode ~ Vale = 0.28/14.7 = 0.02.

(16)

Using the above results in eq. (6), the parameters for m u are given as rnu (I 17000, 0.082), [102000, 134000] ft lb/ft. (17)

through

AS~IEO STRESS CONDITION IN SLAB AT ULTIMATE MOMENT wall panel.

zation is assumed with an ultimate moment of 117 foot kips/foot. The ultimate curvature, $u, is given by ~u/C where ~u is the ultimate strain capacity of concrete. Using the data for beams given by Mattock [241, file ultimate strain ~u can be assumed to have a median value and logarithmic standard deviation as given by median ~u = 0.003 + 0.75/Z, (18)

4.3. Strain energy capacity of wall E s' The collapse analysis is based upon the premise that the collapse mode of damage occurs when the strain energy that can be absorbed by an admissible collapse mechanism is less than the kinetic energy imparted to the wall by striking aircraft. Fig. 8 shows the moment-curvature diagram for yielding zone of the wall panel. A rigid plastic ideali-

o~ u = 1 5 / 2 ,

where Z is the effective shear span as given by the distance from maximum moment to zero moment measured in inches. It is difficult to estimate the effective value of Z for the wall, and yet this value significantly influences ~u. It is expected that the effective value, Z, ranges between 70 and 100 inches, and therefore ~u ranges between 0.005 and 0.019. This

350

C 1/. Chelapati ct al., Probahilistic assessment o] hazard ]or nuclear structures

180

I

]

1t50:

i

1

1

I

-- ULTIMATEHOHENT, .,...,..~,.,..~/ liOT CONSIOERINGSTI~AIN HAROENIHG,mu~II7 ft.-kips.

ULTIMATEMOMENT, CONSIDERIHOSTRAIN HAROENING~.140 f!..klps"

ft.

ft.

140

120

,

=,

ii

m..~17 !t.:k,~ o

~100

'-:

ft.

"-~

\

~YIELD MOMENT~112 ft~-~l--O~-' ft.

~ - EQUIVALENTRIGID PLASTIC MOMENT-CURVATUREOIAGRAH

80

1 60

40

~ULTIMATE CURVATURE . =0.005 ifl. "1

\~YiELD CURVATURE ,,y: 0.0002 In. "z

20

Oi 0

1

0,001

0.002

0.003

0.004 0.005 CURVATURE,., (in. -1)

0.006

I 0.007

0.008

Fig. 8. Moment-curvaturediagram for yielding zone of wall panel. range on ~u can be obtained by assuming a median value of 0.01 with a logarithmic standard deviation of 0.4 which leads to ~u (0.01,0.4), [0.005, 0.019].

(19)

using eq. (13) and (18), the parameters for ultimate curvature, $u, are given by ~bu (0.005, 0.43), [0.0025, 0.0102] inches"1 (20) Alternate calculations made for $u using the results given by Gaston [25] yielded similar results. The ultimate rotational capacity in the vicinity of yield line is given by ru = flu Dy,

(21 )

where Dy is the effective width of yield zone. The strain energy per unit length of yield line with idealized rigid plastic properties (fig. 8) is given by Es' = mu ru = mu ~u Dy,

(22)

The parameters for the effective width, Dy, are also

estimated using the results given by Mattock and are shown below: Dy (14.7, 0.2), [10.6, 20.4] inches.

(23)

Using the data given above, the following parameters are obtained for r u and E~: r u (0.074.0.47), [0.034, 0.161] radians,

(24)

E~ (8.66, 0.48), 63.92, 19.10] foot kips/ft.

(25)

Note that the energy capacity, E~, has a very wide spread ranging from a 5% probability that it is less than 4 foot kips per foot to a 5% probability that is greater than 19 foot kips per foot. Further, note that the variability of the strain energy capacity is completely governed by the variability of the rotational capacity. This dominance is a direct result of the fact that there is inexact knowledge concerning the rotational capacity of concrete yield lines while the ultimate moment capacity is reasonable well defined.

351

C. V. Chelapati et al., Probabilistic assessment o f hazard for nuclear structures

4.4. K i n e t i c energy, E A, a b s o r b e d b y wall slab The total kinetic energy, Ec, of the impacting aircraft is given by

1.0

O.g

E c-- Wc V: z / 2 g ,

where lec is the total loaded aircraft weight, V is aircraft velocity at impact, and g is the acceleration due to gravity which is 32.2 ft/sec 2. Most of this impacting energy is, however, not absorbed by the wall panel being impacted. First of all, after the impact the velocity of the aircraft is not zero, especially if one consideres the impact to be nearly elastic, in which case the aircraft rebounds off of the wall with a velocity. Thus, after impact the aircraft retains kinetic energy. Secondly, if the impact is nearly plastic, considerable kinetic ernergy is lost during the impact itself. Lastly, energy is used to deform portions of the impacting aircraft. Only those portions of the aircraft which are essentially rigid (primaril~¢ the engines) will transfer significant energy to the wall. For the small aircraft which has a rather flexible airframe, it is expected that the most of kinetic ernergy not associated with the engines and will be used to crumple the airframe during impact. Thus, the effective impacting kinetic energy per engine is given by

0.!

0.7

.(,,) .,c

0.8

,? i o

0.5

i-

OO O fO ~ °

0.4

G o~,SB° "

.,S •

0.2 / 0.1

lew: ;:0Kw where K w is a factor close to unity, as discussed in section 4.6, and must be treated as a random variable and le T is the total weight of a single panel. The energy absorbed by the wall upon impact can be approximated by (28)

|---!

j,o**.~_ PLASTIC: /

""°

tS

,e#

0.1

0.2

0.3

(27)

where We.; is the weight of the engine, and K I is a factor to account for the effective weight of other portions of the aircraft that might be associated with the engine and is called the effective rigid body factor. This is a random variable and its properties must be subjectively estimated. I4/I is called the effective rigid body weight of engine. The energy absorbed depends upon the effective rigid body weight of the wall and the impacting aircraft as well as the nature of impact. The effective rigid body weight of the wall is defined by

E A = F A E I,

s~ osB~s BwS

0.3

0 E I = WI V2/ 2g = K i WE2gV2 - K1 /z'E'

[A = FAEI 1

I

(26)

0.4 0.5 0.8 0.7 WEI~4TRATIO, /_~_v/

0.8

0.9

l-'ig. 9. Absorption factor F A as a function of the weight ratio (Wi/Ww) for rigid body impact.

where F A is the absorption factor. Fig. 9 shows the relationship between F A and W l / W w . For low ranges of W I / W w below 0.4, the factor F A can be assumed to be linear and is given by F A = K A (WI/ Ww),

(30)

where K A is a variable which ranges from 0.7 to 1.000 for plastic impact and from 2 to 4 for elastic impact. Using the above relations, the kinetic energy absorbed by the wall is given by EA

(31) Kw

WT

.

,

where/:A is the normalized kinetic energy which is independent of the characteristics of the wall. If the strain energy capacity of the wall panel for the for-

1.0

352

('. 1'. ('helapati et aL. Probat~ilistic a.~sessment tffhazard .tbr nuclear structures

mation of a collapse mechanism is k'~, then the factor of safety can be defined as Factor of safety (F.S.) = 1:']1: A = K~J:s/l:" ~ .

(32)

The conditional probability of a collapse mode of damage is defined as tile probability that the factor of safety is less than one. 4.5. N o r m a l i z e d absorbed k i n e t i c energy, 1: A Tile parameters for L"A are determined by using eq. 131 ) and finding the distributions for K I, K A, and IVl-l:'l.I. IVT is the weight of the wall panel and can be considered constant and equal to 1 14 kips. 4.5.1. D istribution o f variables K I and K A The factors K I and K A are dependent on the size of an impacting aircraft. The analysis developed here is primarily applicable to small aircraft where the engine is the only relatively rigid element and the impact area is also relatively small (2 to 3 feet in diameter) compared to the dimensions of the wall panel. In the case of large aircraft, the fuselage itself offers a resistance under impact with a large impact area ( I 0 to 15 feet in diameter) compared to the dimension of the wall. Also, it is very probable that the wall will be hit by the nose of the aircraft rather than by the engines. Thus, the assumption of rigid body engine impact is not quite valid. Further, the supporting system for the wall may also fail in addition to the wall itself. l lowever, in order to obtain an estimate of the large aircraft impact problem, it is felt desirable to extend the analysis to the large aircraft. Some of the uncertainties arising are taken into account by increasing the variability' on the coefficients K 1 and K A. The following parameters for K 1 and K A are judged representative for the present study: Small aircraft: K l (1.4.0.2), [1.01, 1.95], (33) KA(1.2,0.1),

[1.02, 1.42[.

Large aircraft: h'~ (3.2, 0.5), 11.40, 7.301, h'a(0.8, 0.3), 10.49, 1.31 I.

4.5.2. Distribution o f variabh' Wl: E l. The frequency distrioulions for the four divisions of mlpacting aircraft were given m tables 4 and 5. Observation of the data shows that there is strong correlalion between variables It'l! and t:l:. : and thus the product variable I¢1:I1:'i: is treated as a random variable rather than treating Wt.- and t:l: as separate independent random variables. Figs. 10 and I 1 show the plots of cumulative normal probabilities versus the natural logarithm of normalized Wt: t:'1.I . The plots are drawn on cumulative normal probability paper. On such a paper the logarithm of lognormally distributed data will plot as a straight line with a median value occurring at the 50~ cumulative probability and with a slope equal to the log standard deviation. The data shows that a straight line fits quite reasonably, and thus the assumption that the variable is lognormally distributed is quite good. Based on tile plots of fig. 10 and I 1, tile following parameters are obtained for variable IVI.:k.l.: . Small aircraft: Within 5 miles WI.k'F. (27.3. 1.02), [5.1, 146.9], Beyond 5 milesWI-I:'F (71.9, 1.22), 19.6,538.2]:

I,arge aircraft : Within 5 miles Wl.El: ( 2 0 8 0 , 1 . 5 1 ) . [ 1 7 0 , 2 5 1 3 0 ] . Beyond 5 milesWl.:t:r

(18600, 1.80), [950, 362550].

Study of this infornmtion shows that there is a great variability for the factor WEE E. As an example, for small aircraft with the plant located beyond 5 miles from the airport, the median for WEI:"E is 71.9 which corresponds to a single-engine aircraft with an engine weight of 350 pounds and traveling at a velocity of 133 miles per hour. I lowever, m tile same classification, there are larger aircraft with engine weight of 800 pounds traveling at 280 miles per hour and smaller aircraft with engine weight of 230 pounds traveling at 67 miles per hour, thus giving a great spread for this classification. Using the distributions for K a, K 1, and WEEt.: . the parameters for the distribution ofl:'~ are calculated. Table 9 indicates typical aircraft engine weight and speed representing the median/_: A and the parameters for 1: A for all tile four classifications.

353

C. V. Chelapati et al.. Probabilistic assessment of hazard ]or nuclear stn~ctures 6 . 0 =

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Fig. 10. Cumulative normal probability plot of natural logarithm of WEEE for small aircraft. Table 9. Representative aircraft characteristics and parameters for/::A" Aircraft size

Plant within 5 miles of airport

Plant beyond 5 miles of airport

Small aircraft Representative median Representative median median velocity 82 mph velocity 133 mph corresponds to Piper Cherokee EA (5.6, 1.06),11,32] EA(14.8, 1.26), [2,118] with engine weight of 350 lbs .

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4.6. Distribution ]br K w b,'~ The yield line method is used [ 2 6 - 2 9 ] to determine the collapse load capacity of the slab. This method is used because of the simplicity o f analysis and the experimental evidence that the yield line analysis predicts the collapse load within reasonable limits. For the purpose of the application of yield line method, the wall slab is considered 23 feet square with fixed boundaries. Although this assumption is quite reasonable for the top and b o t t o m edges o f the wall, it is not very good to represent the continuity over columns. However, for the sake of keeping the complexity of yield line pattern to a reasonable level, this assumption was made. Tile yield lines are curved in general, but these are approximated by straight lines and the assumed yield line pattern for the collapse mode of damage is shown in fig. 12. For small aircraft impact the impacting engine load is applied over a small area ot" the wall panel. This load is assumed to be a single concentrated point load on the wall panel. Results obtained for small aircraft

354

C] V. Chelapati et al., Probabilistie assessment o f hazard ]or nuclear structures 4.0

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engine impact are estimated to be accurate to within I0 to 15%. However, this assumption is not very reasonable for larger aircraft. Thus, it is mentioned to use the results for large aircraft problems cautiously by understanding the limitations and assumptions. As mentioned earlier, the wall is reinforced both ways and on both faces with equal reinforcement. So the slab can be assumed to be isotropic with yield moment m u. It can be shown [21] that the total strain energy capacity of the wall at collapse is equal to 8

t:"s = 2 l:'~ ~ l= 21:'~1, i=1

(35)

where 1:'~ is the nonelastic stra'n energy capacity per unit length as defined in section 4.3 and ~. is the total length of the negative yield lines as shown in fig. 12. The effective weight Ww of the wall is defined as that which must be concentrated at the point of impact on an equivalent weightless slab so that it will have the same kinetic energy as the actual slab when the

point of impact is subjected to unit velocity. The value of Ww for the yield pattern shown can be derived [21 ] to be 8 Ww

I4LF ~ ~ihi ' 12 LhL v i= 1

(36)

The variables L h, L v, ~i, and h i are defined in fig. 12. Kwis then given by 10 W w / W T as defined by eq. (28). For any specific location of impact, the specific yield pattern is obtained so that it gives the minimunr collapse load. Since the two variables/:s and K w are not independent, the distribution is needed for tile variable K w E s. Johansen [26] gave an analytical solution for a square plate with a concentrated load at the center using a curved yield pattern which is more accurate than an approximate linear segmental yield line pattern. On the square plate with concentrated load m the center, the value of K w E s for the segmentally linear yield line is 1177 greater than the corresponding load obtained with curved yield lines. It is assumed that this error is approximately the same

C. V. Chelapati et al.. Probabilistic assessment of hazard for nuclear structures

355

Lh

al

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Fig. 12. Approximate yield pattern for concentrated load on rectangular slab with restrained edges. for other locations of impact and the 11% correction is incorporated into eq. (38). When the impact is very close to the boundary, the assumed yield pattern is no longer valid; however, the same yield pattern is used as it is felt that refinements in this area are not warranted at the present time. Because of symmetry, only ~ of the slab is used for calculations. Table 10 shows the load capacity ratio Fp, strain energy capacity ratio FE, and the effective weight ratio F w for eight locations of the ~-segment of the slab. The ratios Fp, F E, and F w are parameters normalized with respect to the corresponding values obtained for a square slab with a concentrated load at the center using curved yield pattern and are given by 8 t=

where L is the side length of a square slab. It can be shown that K w E s = 7.42 L E~, [ F E Fw).

The distribution for F E F w is shown in fig. 13. It can be observed that only the innermost 5% of the slab has values of/''/:. F w greater than 1.09, while the outermost 15% of the slab has values o f F E F w less than 0.64. These two limits can be satisfied by assuming a lognormal distribution for F E F w with the following parameters: / ' ~ . F w (0.78, 0.2), [0.56, 1.09].

(39)

Using the above parameters for F E F w and the properties of E~ as given by eq. (25), the properties for the distribution o f K ~ E s are given by K w E s (1150, 0.53), [480, 2760].

l

(38)

(40)

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(37)

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i= 1

4.7. Factor o f safety The factor of safety is defined by eq. (32) as K w Es/ E A. Table 11 summarizes the results of probability of collapse for the four different cases under study. It can

~ V. Chelapati et al., Probabilistic assessment o f hazard[or nuclear structures

356

be seen that the conditional probability of collapse is practically zero for small aircraft and is more than 50% for larger aircraft. Note, however, that these values apply to one wall panel whose collapse does not necessarily imply that the reactor or critical equipment is damaged.

Table 10. Evaluation of load capacity ratio, strain energy capacity ratio, and effective weight ratio for typical locations of concentrated load on square slab with restrained edges.

allL b i l l

Xll L x2/ L x3l L x4/ L Fp

FE

Fw

0.5 0.7 0.9 1.0 0.7 0.9 0.8 0.85

0.293 0.363 0.450 0.50 0.410 0.478 0.469 0.498

1.06 1.04 1.00 0.90 1.03 0.99 1.00 0.98

1.05 1.03 0.90 0.63 0.99 0.86 0.88 0.80

0.5 0.5 0.5 0.5 0.7 0.7 0.8 0.85

0.293 0.252 0.284 0.50 0.352 0.426 0.434 0.495

0.293 0.252 0.284 0.50 0.176 0.155 0.117 0.088

0.293 0.363 0.450 0.50 0.352 0.465 0.434 0.495

1.2

1.06 1.14 1.75 " 1.22 1.80 1.50 1.77

4.8. Additional comments Table 11 not only presents median factor of safety information for local flexural failure of a wall panel but also the 90% confidence interval on this factor of safety. One should note the very wide spread between the lower and upper bounds on this interval. In all cases the upper bound on the factor of safety is at least 50 times greater than the lower bound. This, of course, suggest considerable

1

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1.0

357

C. V. Chelapati et al., Probabilistic assessment o f hazard for nuclear structures

Table 11. Median factor of safety and conditional probability for collapse mode of damage of individual wall panel due to aircraft crash. Small aircraft Within 5 miles Beyond 5 miles of airport of airport

Large aircraft Within5 miles Beyond5 miles of airport of airport

Median factor of safety

205

78

0.77

0.086

Log standard deviation for factor of safety

1.18

1.37

1.78

2.06

90% confidence interval for factor of safety

29 -1440

8.1 -750

0.04 -14.5

0.003 -2.6

Conditional probability of collapse mode of damage of wall panel [%]

0

0

56

86

uncertainty concerning the factor of safety against a local fiexural mode of wall failure. Most of the uncertainty stems from the large range in the effective aircraft kinetic energy applied to the wall at impact and is not due to uncertainties concerning the structural behavior of the wall. In the face of such great uncertainties concerning the applied load, it must be coneluded that an accurate and costly analysis of the structural behavior of the wall is unwarranted and the most simple structural analyses and investigations of material strengths are probably preferable. Basically, the analytical techniques and material investigations employed in this study are overly precise unless the applied loading is more accurately defined. The overwhelmingly most significant factor influencing the applied kinetic energy due to aircraft impact and, thus, influencing the standard deviation on the factor of safety is the large variability in the type of aircraft which might impact the wall. Specific aircraft were not considered in this study. Rather, aircraft were categorized into four broad classes consisting of small aircraft close to and far from an airport, plus large aircraft both close to and far from the airport. Each of the four classes cover a wide range of aircraft which have a large spread on aircraft weight, engine weight, and aircraft velocities. Thus, the kinetic energy absorbed by the individual wall panel E~, as a result of one of the classes of aircraft impact has a logarithmic standard deviation much larger than the logarithmic standard deviation for any other parameter. This condition results in the logarithmic standard deviation for the overall factor of safety being not much greater

than that for the absorbed kinetic energy E~. The only way to significantly reduce the spread on the 90% confidence interval for the factor of safety is to specify a much larger number of classes for the impacting aircraft so that each class has a much smaller range on aircraft weight, engine weight, and aircraft velocities. Unless this refinement is made, no other refinement appears to be warranted for the analyses conducted in this study. The second most significant influence upon the spread in the confidence interval on the factor of safety is the uncertainty concerning the strain energy capacity per unit length of yield line E s. This strain energy capacity per unit length has a logarithmic standard deviation of 0.48 which results in a ratio of approximately 5 between the upper bound and lowex bound on the 90% confidence interval for E s. The strain energy capacity per unit length is the product of the moment capacity times the rotational capacity of the wall panel. The logarithmic standard deviation is roughly six times as great for the rotational capacity as for the moment capacity. Thus, the uncertainty concerning the strain energy capacity per unit length is governed completely by the uncertainty concerning the rotational capacity of the slab. Reduction in the degree of uncertainty concerning the rational capacity of the yield lines would require increased experimentally derived knowledge concerning the behavior of concrete slabs beyond the yield point rather than increased analytical work. The third most significant parameter influencing the degree of uncertainty in the factor of safety was the .

F

.

S

358

C v. CTtelapati e t al.. Probabilist& asses srnen t o f hazard f o r nuclear structures

lack of detailed knowledge concerning the impact phenomenon itself. Uncertainties existed concerning the impact energy absorption coefficient K A which is related to the coefficient of restitution of the impact (i.e., degree of plastic versus elastic impact) and the ratio of the effective weight of the impacting aircraft versus the effective wall panel weight. Further uncertainties excisted concerning the effective impact weight coefficient K I for the impacting aircraft. This coefficient is required to define the effective rigid body impacting kinetic energy applied to an individual wall panel as a result of aircraft impact. Both coefficients K I and K A were more uncertain for the larger commercial aircraft impact than for the small private aircraft impact. Considerable investigation of any available data concerning past aircraft impacts plus experimental derived data will be required to significantly reduce the uncertainties concerning K A and K 1. The last parameter to have any noticeable influence on the uncertainty of the factor of safety was the yieldline failure pattern which is a function of the location of engine impact on the wall panel. The influence of the yield-line pattern on both the strain energy capacity and on effective slab weight are expressed by the product F E F w , which has a logarithmic standard deviation of 0.2. This low standard deviation results in the location of impact causing a ratio of less than two between the upper and lower bounds on the 90% confidence interval for the factor of safety. Thus, refinements in estimating the location of engine impact or usage of a more exact segmentally linear yield-line pattern to represent the actual curved pattern are not warranted. The moment capacity had a logarithmic standard deviation of only 0.082. As such, it had no influence on the uncertainty in the factor of safety and could have been. treated as a constant rather than a random variable. The slab is definitely underreinforced and for such a slab it is interesting to note that the logarithmic standard deviation for the moment capacity is exclusively governed by the uncertainty in the yield stress for the tensile reinforcements. One of the major advantages to performing a probabilistic failure analysis rather than a deterministic failure analysis is that the relative significance of the uncertainty associated with each of the variables considered in the analysis can be assessed. This assessment enables one to determine where to concentrate one's efforts in an attempt to decrease the uncertainty concern-

ing failure. Furthermore, a probabilistic analysis enables one to assess the accuracy associated with a predicted factor of safety against failure. If a deterministic analysis had been made of a local flexural failure of the wall panel when subjected to a particular class of aircraft impact, one would have obtained only a single number for the factor of safety. Because of the possible wide spread in the factor of safety associated with this problem, it would be misleading to only estimate a single factor of safety rather than a spread on the factor of safety. 5. Cracking mode of damage 5. 1. A n a l y t i c a l p r o c e d u r e

The conditional probability for a perforation mode of damage for a small aircraft impact on a plant beyond 5 miles of airport is 0.06 for an 18-inch wall and the probability is essentially zero for the collapse mode of damage. Thus, for small aircraft impact the probabilities of wall perforation or wall colapse are very low. In section 1, it was mentioned that the cracking mode of damage is defined when the maximum moment at any section based on elastic analysis reaches the ultimate moment at that section. Beyond that level, extensive plastic deformations occur, eventually collapsing the wall, as defined by collapse mode. In this section, deterministic results will be predicted for the cracking mode of damage for the specific case of a small aircraft striking a plant which is beyond 5 miles from the airport. The median aircraft for this class can be represented by an aircraft with an engine weight of 350 pounds and traveling at 133 mph. The coefficients presented in the collapse mode calculations are used while calculating the absorbed energy, and the equivalent weight of wall for the determination of the cracking mode. However, in this case it is assumed that this energy is stored elastically. The equivalent properties of the wall are found using a finite-element idealization. Because of the versatitily and adaptability, it is felt desirable to model the structure as accurately as possible. The effects of neighboring panels are included by using a total of six panels. As shown in fig. 14, the model consists of 185 nodal points, 324 triangular elements, and 40 beam elements. After a careful review of current literature, it was decided to adopt a compatible triangular finit element, known in the literature as the HCT (Hsieh, Clough, Tother element. This element has properties so that the d~placements and slopes normal to the bound:lries ~re compa-

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C 1/. Chelapatiet al., Probabilisticassessmentof hazardfor nuclearstructures

360

tible along the edges of the discrete triangular elements that make up the model [30]. This element has been used in several studies, and the approximations involved, including the convergence to analytical solutions, have been studied [31 - 3 4 ] . Further, because of their triangular nature, these elements can be adopted easily for various configurations. Keller [31 ] studied several problems and his computer program, RIPL II, has a feature to include the effect on stiffeners. This feature is valuable, since the colunms can be considered as stiffeners to the walls. The results given here are obtained using a modified version of RIPL II. In this study the details of the analysis are not discussed and the final results are used directly.

5.2. Numerical calculations Weight of small aircraft engine WE = 350 lb, Impacting velocity Impact factor

V = 133 mph, K I = 1.4,

Effective impacting energy E I = KI(W E V2/2g) = 289.5 kip ft, Absorption factor Effective wall weight factor Absorbed kinetic energy by wall, EA, using eq. (31)

which corresponds to an equivalent maximum static force of /°max = Ke ~max = 576 kips. Under this load the finite element analysis gave results for bending moments Mxx, Myy, and twisting moments of Mxy and other pertinent information. The maximum moment occurred under the load at the center and is given by the average of the moments in eight elements around the center as

Mxx = 73.7 f t / k i p s / f t , Mvy = 93.9 f t / k i p s / f t , Mxy = 5.6 ft/ kips/ ft. 5.3. Factor of safety

K w = 1.31, = 11.4 kip ft.

Modulus of elasticity of concrete E c = 3.5 X 106 psi v -- 0 . 1 5

Moment of inertia of uncracked section lg = 5832 in4/ft Moment of inertia of cracked section

6ma x = o.475 inches,

K A = 1.2,

For the finite-element analysis of the sidewall the following properties are used.

Poisson's ratio

Assuming the wall as an equivalent spring storing impacting energy as strain energy at maximum deflection, the following relation can be written: 2 ½K e 6 =E A. max Thus, the maximum deflection is given by

I T = 1853 m4/ft

Average moment of intemia ½(lg +IT) = 3842.5 in4/ft Using the above data, the static finite element analysis was done for a concentrated load at the center of the top middle panel. The equivalent stiffness of the slab is given by Ke = Concentrated load at center = 14550 kips/ft deflection at center

The twisting moment is very small and can be neglected. The values of bending moments Mxx and Myy shown are close to the ultimate moment capacity m u of 117 kips/ft. Defining the factor of safety for cracking behavior as the ratio between mu/Mma x, the factor of safety is approximately equal to 1.25. However, the factor of safety, as defined previously, was about 78 for the collapse mode. The above analysis indicates that there is a large increase in load capacity associated with the collapse mode or perforation mode of damage as opposed to the cracking mode. It also points out that results based on an elastic behavior are unduly conservative so far as wall failure is concerned. It is true that in the above analysis, many assumptions were made about the phenomenon of impacting aircraft, but these areas can be improved by gathering more information and experimental evidence. 6. Summary This paper provides an example of a risk assesment approach to nuclear power plant safety. From the avail-

361

C V. Chelapti et al., Probabilisti¢ assessment o f hazard for nuclear structures

able accident data, the probability of aircraft strike is estimated for four different classifications. The aircraft is considered larger or smaller depending whether its weight is more or less than 12000 lbs. The location of the plant is classified according to whether it is located within or beyond 5.mile radius from the airport. The conditional probability of an aircraft striking a nuclear power plant is of the order of 1.4 × 10-5 to 4.6 × 10.7 per year, depending upon the classification. Three modes of damage to an 18-inch thick outside reinforced concrete wall of a typical boiling water reactor plant which could be hit by an aircraft are delineated as perforation, collapse, and cracking modes of damage. The available perforation formulas are discussed and it is shown that some of these formulas yield very unconservative values for perforation thickness. New empirical formulas for perforation are proposed based on available data for the range of variables associated with aircraft impact. A Monte-Carlo approach is used to compute the conditional probability of perforation. It is shown for small aircraft that the probability of perforation is zero or 0.06 depending upon whether the plant is located within or beyond a 5-mile limit from the airport. Similar figures for large aircraft are 0.52 and 1.00. The collapse mode of damage is estimated using log normal distribution for the variables and the yield line theory. For the collapse mode of damage the corre-

sponding figures are 0, 0, 0.56, and 0.86. Thus, it is concluded that the conditional probability is negligible for small aircraft and above 50% for a large aircraft. The factor of safety for the cracking mode of damage is estimated using the elastic finite element analysis. For the case of small aircraft and for a plant located beyond five miles of the airport, the factor of safety is around 1.25, whereas for the collapse mode it is about 78. This large spread in factor of safety between these two modes indicates that it is very unconservafive to design the nuclear reactor building for purely elastic deformations under aircraft impact, or viewing differently, there is a great reserve capacity for the wall against collapse. Table 12 shows the absolute and relative probabilities of perforation and collapse mode of damages for the four classifications and they range from 4.0 × 10 -7 to 13.0 × 10-7. These factors must be reduced by another 100 so that the strike against a critical location can be estimated. Thus, from the information presented here, it can be concluded that aircraft risk is usually acceptably low for the typical case studied here.

Table 12 Probability of damage to an 18-inch wall due to an impact of aircraft.

Plant locatior~ . Aircraft from airport ~cj type

Probability of strike/year

,g 5 miles

3. 3 - 5 (a)

> 5 miles

Small

Perforation mode of damage

Collapse mode of damage

Conditional probability

Conditional probability

0

0.52

Absolute probability 0

5.7-7

0

Large

I. 1-6

Total

3.4-5 (b)

Small

1.4-5

0.06

8.4-7

0

Large

4. 6-7

1.00

4.6-7

O.88

Total

1 . 4 - 5 (b)

0.56

5.7-7 (b)

13.0-7 (b)

(a) 3. 3 - 5 denotes 3.3 X 10"5.

(b) Multiply these values by 0.01 for damage probabilities for specific e~uigment. (c) Actual strike probability near an airport varies approximately as 1/rz, where r is the distance from the airport.

Absolute probability 0 6.2-7 6.2-7 (b) 0 4. O-7 4. 0 - 7 (b)

C.V. Chelapati et al., Probabilistic assessment of hazard lbr nuclear structures

362

Appendix A. Penetration formulas for concrete

A. 3. Army Corps o f l:'ngmeers' Jormula

A. 1. Petry formula

The Army Corps of Engineers' formula 1or reinforced concrete of infinite thickness is given by

The following penetration formulas are compiled from existing literature for convenience. The Petry Formula was first reported in 1910 and is given below: IAI - A 4 I : x=KoAplOgl0(l+v2/215000),

(I 1 )

where x = depth of penetration into an infinite thickness, [ft], Ko= a coefficient depending on the nature ot" the concrete, and it is 0.00799 for massive concrete, 0.00426 for normal reinforced concrete, such as would be used in building construction, and 0.00284 for specially reinforced concrete, Ap= weight of missile per foot 2 of projected area [pounds per foot2], v = velocity of the missile [ft/sec]. The formula given [A4] by the bureau of Yards and Docks, U.S. Department of the Navy, in 1950 to calculate penetration depth x ' into a reinforced concrete slab whose thickness is between Zv and 3x is: ,

x =x { l+e

-4

lIT~D)-21

},

(1-2)

where x ' = actual depth of penetration [ft], and 7 = thickness of the slab [ft]. It is mentioned that for slabs whose thickness is less than 2x, perforation of target occurs. Ref. [AI] indicates that this formula yields the least conservative thickness for penetration and is not recommended for the calct, lations.

A. 2. BRL fi~rmula The Ballistic Research Laboratory's formula for penetration into concrete is given by x = 6 Wd 0"2 ( o )4/3

where x = penetration depth of missile for reinforced concrete of infinite thickness [inch[, tg = weight of missile [pound], d = diameter of missile [inch], v = velocity of missile [ft/sec]. The perforation thickness e is given by 1.3x.

x = Scl / 2 \ d 2 ]

]0~

+0.5d ,

13-1)

where s c is strength of concrete in psi; W is the weight of the projectile in p o u n d s ; d is the diameter of the projectile in inches: and v is the velocity of the projectile in feet per second. The pertoration and scabbing thicknesses respectively are given by e = 1.32d + 1.24x, s = 2.12d + 1.36x. (3--2) Eq. (3--2) has been based on experimental results obtained prior to 1943 [A5, A6]. The coefficients were revised in 1944 to the following equations based on additional data [A7[ relating to 0.5 caliber bullet penetration tests: e = 1.23d + 1.07x. .~' = 2 . 2 8 d + 1.13x. (3--3)

A. 4. RcR'rences [A 1 ] R.C. Gwaltney, Missile Generation and protection in Light-Water-Cooled Power Reactor Plants, ORNL-NSl('22, Oak Ridge National Laboratory. Oak Ridge, rennesse September 1968. [A2] l'.J. Samuely and (.W. llamann, Civil Protection 1The Architectural Press, london, England). 1939. [A31 C.R. Russell, Reactor Safeguards lThe MacMillan Company, New York, N.Y.L 1962. [A4] A. Amirikian, Design of Protective Structures, RcportNP-3726, Bureau of Yards and Docks, Department ol the Nay,,, August 195{I. [A5] HTect of Impact and kxplosion, National Defense Research Committee. Summary Technical Report on 1TM ivision 2. Volume 1. Washington, 1).('.. 1946. [A6I R.A. Beth and J.(;. Stipc, Jr., Penetration and I'xplosion l'ests on (_'onlretc Slabs, Report I : DATA : ('I'PAB Interim Report No. 21L January 1943. IA7] J.G. Stipe, Jr., M.F. l)e Reus, J.T. Pittcngcr, and ICJ. llansen Ballistic Tests on Concrete Slabs. ('1"1) Interim Report No. 28. June 1944.

Appendix B. Lognormal distribution A random variablex is said to be lognorm',dly distributed if its natural logarithm,~=logx, is normallydistributed It has been observed that the phenomenon which is the result of a multiplicative mechanism fits the lognormal

C. V. Chelapati et al.. Probabilistic assessment of hazard for nuclear structures

363

Table B 1. Notations for lognormal and normal distributions. Variable

Median

Lognormal distribution

x

X

Normal distribution

x

distribution b e t t e r than a normal distribution. The lognormal distribution is skewed w i t h zero probability for negative range o f variable and thus describes the distribution o f certain variables better than the normal distribution. Table B I describes the notations for the parameters o f lognormal distribution and the corresponding normal distribution. The relations b e t w e e n the parameters o f a lognorreally distributed variable x and the parameters of the corresponding normally distributed variable 2'are given below: Variable x = exp,, Median X = exp.~, Mean )7 = exp (X + ½~,2), Variance o2 exp ( 2 . ~ + 20xz) - exp ( 2 X + 6x~), X Standard deviation

% = ~ x2 Coefficient o f variation

v x =Ox/J?

=~Txp(o~)2-i.

For variables with logarithmic standard d e v i a t i o n , o x, less than 0.6 the coefficient o f variation V x can be taken to be a p p r o x i m a t e l y equal tOOx, within an error of less than 9%. If a r a n d o m variable is defined by d = a r bS/c t, where a, b and c are lognormally distributed variables and r, s, and t are k n o w n e x p o n e n t s , then it can be shown than d is also a lognormally distributed variable with the parameters given by median, D = A r Bs/ct, where A, B, and C are median values o f a, b, and c and by logarithmic variance o f d given by o2 =r2O~a + S 2 6 b + t 2 0 2

Mean

Standard deviation

Coefficient of variation

ox

I"x

0x

"~X

where ~la, ~'b, and ffc are standard deviations of a, b, and c respectively.

References [ 1 ] F.R. Farmer, Siting Criteria - - A new Approach, Paper SM-89/34, IAEA Symposium, Containment and Siting, Vienna, April 1967. [2l I.B. Wall, Probabilistic Assessment of Risk for Reactor Design and Siting, Trans. Am. Nucl. Sot., 12 ~1969) 169. [31 B.J. Garrick, Unified Systems Safety Analysis for Nuclear Power Plants, Ph.D. I)issertation, University of California, Los Angeles, June 1968. [4] J.R. Beattie, Review of Hazards and Some Thoughts on Safety and Siting, BNES Symposium, Safety and siting, London, March 1969. 151 I.B. Wall and R.C. Augenstein, Probabilistic Assessment or" Aircraft flazard for Nuclear Power Plants -I, Trans. Am. Nucl. Sot., 13, 1 (1970). [6] C.V. Chelapati and I.B. Wall, Probabilistic Assessment of Aircraft ttazard for Nuclear Power Plants - II, Trans. Am. Nucl. Soc., 13, 1 (1970). [7] C.V. Chelapati and I.B. Wall, Probabilistic Assesment of Seismic Risk for Nuclear Power Plants, irans. Am. Nucl. Sot., 12, 2 (1969). [8] I.B. Wall, Lecture Notes for UCLA Short ('ourse on Nuclear Reactor Safety, September 8 - 12, 1969. [91 Annual Review of U.S. Air Carrier and General Aviation Accidents, National Transportation Safety Board, Department of Transportation. [ 10] Shorehanl Nuclear Power Station. Docket No. 50-322, Amendment No. 3 to License Application. l.ong Island Lighting Company, l'ebrt,ary 5. 1969. [11 ] Census of U.S. Civil Aircraft, Office of Management Systems, I ederal Aviation Administration, l)eparnnent of Transportation. [12] Effect ~t" hnpact :rod Explosion, National Defense Research ('ommittee, Summary Technical Report on Division 2, Volume 1, Washington, I).('., 1946. [ 13] A. Amirikian. Design of Protective Structures. Report NP-3726. NAVI)OCKS P-51, I:;ureau of Yards and Docks, Department of Navy, August 1950. [ 14] R.A. Beth, Penetration of Projectiles in Concrete, I'I'AB Int~.rim Report No. 3, November 1941.

364

C, I ". Ctwlapati et al.. Probabilistic assessment o l h a z a r d ] o r tmch,ar structures

'1151 R.A. Beth, l mal Report on Concrete Penetration. ()SR1)-6459, NDR(" Report A-388. November 1946. [16] R.P. Kennedy, |'Tfects of an Aircraft ('rash l n t ~ . a ( ' o n crete Reactor I:~uilding, tlohnes & Narver, Inc., Los Angeles, ('alifornia, July 1966. 117[ ('.V. ('helapati, Probability of Perforation of a Reactor Building Due to an Aircraft Crash, 11N-212, Ih)lmes & Narver, Inc., l,os Angeles, California, January 19711. 1181 L.\\.Saffian~,nd R.M. Rindner, Safety l)esign('riteria for I'ixplosives Manufacturing and Storage Facilities, 1 ethnical Memorandum 1300, Amlnunitiorl Engineering l)irectorate, Picatinny Arsenal, l)over, Nev. Jersey,, November 1963. [191 (LJ. Ilahn, and S.S. Shapiro, Statistical Models m lmgineermg I John Wiley & Sons, New York1, 1967. [201 R.A. Beth, ('oncrete Penetration, OSR1)-4856, N I ) R ( Report :',-319, "vlarch 1945. [21 I R.t'. Kennedy. and ('.V. Chelapa!i, Conditional t'robability of a Local I'lexural Wall l.ailure for a Reactor Building as a P,esuh of Aircraft Impact, 11N-70-8082-5, tlolmes & Narver, Inc., Los Angeles, California, June 1970. 1221 Jack R. Benjamin, and ('. Allin (ornell, Probability, Statistics and Decisions for Civil Engineers [McGraw Hill Book (Oml'~any, New York), 1970. [23] A('l Standard Building ('ride Requirements for Reinforced ('oncrete, American Concrete Institute, l)elroit, Michigan. June 1963. [241 A.II. Mattock. Rotational Capacity of ttinging Regions m Reinforced ( o n c r e t e Beams, Proceedings of tile International S vnlposium on I:[exural Mechanics of Reinforced Concrete, American Society of Civil Engineers, 1965. [251 J.R. Gaston, ('.1'. Siess, and N.M. Newmark, An Investigation of the l, oad-Deformation Characteristics of Reinh)rced ('oncrete Beams Lip to the Point of Failure. Rept~rt No. 40, University of Illinois Structural Research Series, I)ecember 1952.

26] K.~,~,.Johansen, Yield-line lheory, (ement anti (.'oncrete Association, l.ondon, lmgland. 1962. tOrginal in Danish. 1943). 271 1.i. llognestad, Yield-I ine l h c o r y for the L Rimate Flexural Strength of Reintorced Concrete Slabs, .4(1 Journal ( '~larch 1953k 281 I'.M. Ierguscm, Reinforced Concrete t. undamcntal~, v, ith I mphasis on Lqtimate Strength (Johll Wiley & Sons, New York), 1958. 29] N.M. Newmark, and J.l). Ilaltjwanger, Air I,,rce Design Manual - Principles and Practices for Design of llardened Structures, AFSW('-TDt~,-62-138, Air t'orce Special \ \ e a p o n s ('enter, Kirtland Air I.orce Base, New Mexico, l)ecember 1962. 30[ R.V.'. Clough, and J.l.. T o t h e r , l.inite I.ilemen! Stiffness Matrices for the Analysis of Plate Bending, Proceedings ol ('onference on Matrix Methods in Structural Mechanics. Air Force Institute of Technology. Wright-Patterson Air Force Base, Ohio, October 1965. 31 ] J.L. Keller, Tile Analysis of Irregularly Shaped and Stitfened Plate Structures, Sandia I,aboratories. l.ivcrmore, ('alifornia, June 1969. [32] I.IASI~ 1. O/EAC, l"lastic Analysis for Structural I ngineermg, User's }.lallual, Control Data ('orporati~m, l o s Angeles [)ahl ('enter, Los Angeles, ('alifornia. August 1968 [33] C.V. ('hclapati et al., Probabilistic Analysis of Nuclear Reactor ( ' o n t a i m n e n t Structures, Part 2: I'inite t.~lement Analysis of l)rywell Due It) l,osx- (ff -('oolant Accident. IIN-203, l loh'nes & Narver, Inc.. l,os Angeles, ('alifornia, June 1969. 1341 R.II. Gallagher. Analysis of Plate and Shell Structures. S y m p o s i u m on Application of l. inite Flement Methods m (ivil Engineering, Vanderbih University, Nashville. Fennessee, Nm, cmber 1969.