Loading time history for tornado-generated missiles

Nuclear Engineering and Design 51 (1979) 487-493 © North-Holland Publishing Company

487

LOADING TIME HISTORY FOR TORNADO-GENERATED MISSILES Anil K. KAR * 118-70 80th Road, Kew Gardens, New York, N Y 11415, USA

Received 29 June 1978

Nuclear power plant structures in the USA are designed for impact by tornado-generated missiles. The design load for flexure and shear can be obtained from the deceleration of the missile on impact. The paper gives a simple method to determine the deceleration of the most critical pipe missile. Results, obtained by the simple method, are compared with full-scale test results. The comparisons between the predicted and actual deceleration time histories show excellent correlation. 1. Introduction Nuclear power plant facilities and many other structures need protection against impact by tornado missiles. This protection is provided by designing the structures against the impactive effects of the missiles. The normal practice is to design for the prevention of local effects, e.g., perforation of the barrier and generation of scabbing of material from the backface, and to ensure that the barrier would not fail in the flexural or shear mode. Among others, the writer [1,2] has presented empirical formulas for determining the local effects of tornado-generated missiles. Predictions by the writer's method [1,2] lead to excellent correlations with fullscale test results for impact by postulated tornadogenerated and other missiles on reinforced concrete barriers. For the design for shear and flexure, it is necessary to determine the impactive load, imparted by the missile. Design loads, resulting from missile impacts, are governed by the absorption of kinetic energy from the missile by the target at its maximum permissible deflection. These loads can be limited by the yield, wrinkling, buckling, crushing or other local destructions of the missiles. The writer [3] has earlier presented a method for the determination of the impactive load. The method is based on the principle of colliding bodies. It has been assumed that the target is relatively hard and massive. The results, obtained by using the method, led to good correlation with test results for total support * Consulting Engineer.

reaction due to a 12-in (0.305 m) diameter, 13.73 ft (4.19 m) long pipe missile impacting on a 15-ft-square (4.58 m) slab, 18 in (0.458 m) thick at 202 ft/s (61.61

m/s). The process of determination of the load function during impact of a missile on a concrete barrier can be a complex one. This is specially so because of the limited stiffness of the structure and the local damage of the barrier, which is most often a reinforced concrete structure. Also, the reinforced concrete structure is not a homogeneous one. This could lead to further difficulty in developing an exact method for the determination of the load function. It is to be noted here that a missile impact at a particular point and in the most critical manner is more probabilistic than deterministic in nature. The response of a structure due to this impact is also not fully understood at this time [4]. All these make the development and use of a simple approximate load time history appropriate. This paper gives a method to determine the loading time history for the most critical tornado-generated pipe missiles. This is obtained from the deceleration of the missile. Calculated decelerations of the missiles upon impact are compared with test results. 2. Deceleration The elementary equation of motion of an object is F ( t ) = M£(t),

(1)

A.K. Kar /Loading time history for tornado-generated missiles

488

where Vo -- impact velocity of the missile; and

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WILLIAMSON

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TIME

Fig. 1. Different loading functions due to impact by tornadogenerated missiles. in which F(t) = load at time t at the interface of the missile and the barrier; M = mass of the missile; and 2(t) = deceleration of the missile at time t. Thus, if the deceleration of a missile on impact can be determined, the contact force on the barrier is easily obtained. Various loading functions have been proposed [3-8], see fig. 1. On the basis of elastic impact by a missile on a relatively rigid and massive barrier and empirical modifications, the writer [5] had proposed a linearly rising missile deceleration, which decays exponentially. Without such modifications, the writer [3] had proposed an exponentially decaying contact pressure. The first method led to good correlation with peak missile deceleration as obtained from tests [9]. However, the deceleration time history differed from the actual history. The second method led to good prediction of support reactions. The shape of the force-time history for an intermediate deformable pipe missile, as developed by McMahon et al. [6], resembles that of missile deceleration as observed in tests [9]. However, this force-time history is dependent on an assumed contact pressure at the interface of the missile and the barrier, and sufficient verification with test results have not been provided. Rotz [7] makes his recommendations by equating the initial impulse with an assumed crushing load, F, of the missile acting for time ti, obtained from

ti = MVo/F ,

(2)

(3)

in which P = collapse stress; and A = cross-sectional area of the missile. Williamson and ANy [8] consider the initial kinetic energy in developing the load function. It is assumed that the missile velocity decreases linearly during the time required by the missile to travel the depth of penetration. This load function also has no resemblance with what would have been obtained by using the actual decelerations as obtained in tests [9]. The writer [5,10] had recommended a method to obtain missile deceleration. Though it worked very well in the case of projectile penetration into earth media, and the derivations [5,10] for peak deceleration of the tornado missile yielded excellent results, recent results [9] for the most important pipe missile [11] indicate the need for improvement in the deceleration time history. This improvement is necessary because none of the existing methods leads to a correct load function. On the basis of recently available test results [9], modifications in the originally proposed time history [3,5] are made here. This has briefly been presented in ref. [4]. From the principle of colliding elastic bodies, the velocity V of the missile at time t can be obtained as

V = Vo exp(-PcACet/M),

(4)

in which V0 = normal impact velocity; Pc = density of of concrete; A = cross-sectional area of the missile; and Ce = speed of sound in concrete. By differentiating eq. (4) with respect to time, the deceleration of the missile I2 is obtained as

(I = Vo(-p cA Cc/M) exp(-peACct/M) .

(5)

Because of the limited rigidity of the barrier, the effective deceleration l,;'eff is assumed to be given by (6)

V e f f = [,T~2" ,

in which xit = 0 . 6 3 ( k b a r r i e r / k m i s s i l e ) O .

16 ~

1.0.

(7)

The stiffness of the missile, kmissile, is given by kmissil e = AE/L ,

(8)

where L is the length of the missile. In eq. (7), kbarrie r is the stiffness of the barrier acting as a spring. In deter-

A.K. Kar /Loading time history for tornado-generated missiles m i n i n g kbarrier, the uncracked properties of the barrier are considered. E is the modulus of elasticity of steel. It was originally [3,5] prpposed that the magnitude of the deceleration starts from zero at t = 0, linearly increases and peaks at time 7", given by

A I000

i //\\ '/ \ \

800

~,= t~(depth of penetration into barrie D VoI.SO

.

(9)

On the basis of new test results [9], it is here recommended that ~'= 0.0464/depth of penetration in feet] ~0.3

489

!

600

(10 )

in which Vo is expressed in f/s. Eq. (10) can be easily modified when the parametric values are in the CGS units. The depth of penetration can be calculated as recommended by the writer [1,2]. After reaching the peak at t = 7",the missile decelerates linearly and stops at time t = 2~r. A triangular deceleration function, ~(t~ with equal rise and decay times is thus obtained. The load time history is obtained from eq. (1). The deceleration time history, recommended here, may not be applicable for cases where the missile and barrier characteristics are significantly different, e.g., a slender rod impacting on a thick concrete wall, in which case there may develop significant lateral vibrations of the rod. This could lead to significant departures from the smooth rise and decay curves, as obtained above. Also, it should be mentioned here that there is no one-to-one relationship between the load function and depth of penetration. The preceding recommendations have been made simply as a convenience in determining the time history for the problem under consideration.

3. Comparisons with test results Deceleration time histories, as recommended in the preceding section, are compared with test results [9]. The test deceleration pulses of the impacting missiles were obtained from streak camera films taken at 3000 frames/s. Missile displacement versus time was digitized from the films and differentiated twice to yield the deceleration curves shown in thin lines in figs. 2 to 12. The recommended, as above, deceleration curves are shown as triangular pulses in these figures.

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Fig. 2. Missile deceleration; prediction and results from test no. 3 [9] (see table 1).

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Fig. 3. Missile deceleration; prediction and results from test no. 4 [9] (see table 1).

A.K. Kar / Loading time history for tornado-generated missiles

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Fig. 4. Missile deceleration; prediction and results from test no. 5 [9] (see table 1).

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Fig. 6. Missile deceleration; prediction and results from test no. 9 [9] (see table 1).

Figs. 2, 3 and 5 to 12 represent the deceleration of the 12-in (0.305 m) Schedule 40 pipe impacting on concrete barriers, 12 in (0.305 m) to 24 in (0.61 m)

thick. The test data, obtained from ref. [9], are given in table 1. Fig. 4 shows the deceleration of a 3-in (76.2 ram) 600

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Fig. 5. Missile deceleration; prediction and results from test no. 8 [9] (see table 1).

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Fig. 7. Missile deceleration; prediction and results from test no. 11 [9] (see table 1).

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Fig. 8. Missile deceleration; prediction and results from test no. 14 [91 (see table 1).

Fig. 10. Missile deceleration; prediction and results from test no. 16 [9] (see table 1).

schedule 40 pipe impacting on a 12-in (0.305 m) thick reinforced concrete barrier. In obtaining the duration of deceleration, the depths of penetration, used in eq. (10), are as given in ref. [9].

For a given design problem, this depth of penetration can be determined by using the simple formulas proposed by the writer [1,2]. For the proposed deceleration in fig. 12, the calculated depth of penetration

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Fig. 11. Missile deceleration; prediction and results from test no. 17 [9] (see table 1).

492

A.K. Kar /Loading time history for tornado-generated missiles In preparing figs. 2 to 12, the stiffness of the barrier has been obtained from

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in which Ec = modulus of elasticity o f concrete; I = moment of inertia of the concrete section per unit width; and b = unsupported span of the test slabs. The minor increase of the modulus during dynamic loading has not been considered. Also, in determining the moment of inertia of the slab, the full concrete depth has been used. The presence of reinforcement has been neglected. Should any modification in the quantities, b, E and 1 de desired, it can be easily done by making a corresponding change in eq. (7). However, it is believed that for the complex problem, these changes are unwarranted For different boundary conditions, eq. (11) should be modified accordingly. The deviation of the peak decelerations are found to be maximum in fig. 12 (+24.8%) and fig. 8 (-18.7%). As explained earlier, the test data for fig. 12 showed significant scatter.

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MILLISECONDS

Fig. 12. Missile deceleration; prediction and results from test no. 18 [91 (see table 1). [1,2], instead o f the test [9] penetration depth, has been used for the heavy solid line. This has been done because t h e t e s t result for test no. 18 [9] showed large "test scatter". The time history, using the test penetration depth, is represented by the discontinuous line.

4. Design The load time history, F(t), is obtained from eq. (I). In a design, an equivalent static load is determined

Table 1 Summary of Tornado Impact Test Data (after ref. [9]) Test no.

Missile pipe

Weight (lbs)

Velocity (fps)

Panel thickness (in)

Unsupported span of square slab (ft)

28-day concrete cylinder strength (psi)

Penetration (in)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

3 4 5 8 9 11 14 15 16 17 18

12" 12" 3" 12" 12" 12" 12" 12" 12" 12" 12"

743 743 78 743 743 743 743 743 743 743 743

202 198 212 202 143 98 92 152 92 157 213

18 18 12 24 18 12 12 18 12 18 18

15 15 15 15 15 15 15 15 15 15 15

3350 3560 3340 3795 4325 3595 3545 4205 3350 4070 4420

7.0 6.8 4.6 6.8 5.0 4.5 3.9 5.3 3.5 4.1 6.8 a)

a) Value calculated by method in refs. [1,2]. Note: 1 in = 2.54 cm; 1 ft = 0.305 m; 1 fps = 0.305 m/s; 1 psi = 6.9 kN/m 2.

A.K. Kar / Loading time history for tornado-generated missiles corresponding to F(t) and the dynamic properties of the barrier. An observation of the test results for the deceleration time history indicates that it would be theoretically intractable to reproduce the curves at the two ends. It is thus recommended that the effects of the two end zones of the actual time history be considered by increasing the equivalent static force by 10 to 15%. Fig. 13 shows that the equfvalent static load does not increase monotonically with decrease in the natural period, T, of the barrier. Thus, to cover computational uncertainties, it is here recommended that this period be varied by +15%, and three equivalent static loads determined corresponding to the three structural periods. The largest load is increased by 10 to 14%, as described in the preceding paragraph, and used in design. Though fig. 13 represents elastic response of the barrier, the elastic-plastic response function shows similar non-monotonic trends. For the level of elastoplastic design see ref. [4]. After the loading time history has been developed, and modifications have been made in the equivalent static load, as outlined above, the barrier is designed for the modified load. The dynamic localized nature of the loading calls for special attention to be given to the structural response. Ref. [4] considers the phenomenon of dynamic localized loading, and the 0~ o ¢j u.

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Fig. 13. Maximum response of one-degree elastic systems (undamped) subjected to isosceles triangular load pulses.

493

recommendations therein differ very significantly from the present design practices.

5. Summary and conclusions Nuclear power plant structures in many areas are designed for impact loads from tornado-generated missiles. Different mathods exist for the determination of the impactive load imparted by a missile. Recent tests have shown the inadequacy of these methods. The load time history can be obtained from the deceleration time history of the missile. The paper provides a simple method to determine this time history for pipe missiles. Results, obtained by this method, are compared with full-scale test results. The comparisons between the predicted and actual deceleration time histories show excellent correlation.

References [1] A.K. Kar, J. Struct. Div. Am. Soc. Civil Eng. 104 (1978) ST1. [2] A.K. Kar, J. Struct. Div. Am. Soc. Civil Eng. 104 (1978) ST5. [3] A.K. Kar, Nucl. Eng. Des., 47 (1978) 107. [4] A.K. Kar, Design for Tornado-Generated Missiles and Aircraft, American Society of Civil Engineres Spring Convention and Exh~it, Pittsburgh, 24-28 April, 1978. [5] A.K. Kar, Trans. 4th Int. Conf. Structural Mechanics in Reactor Technology, eds. Th.A. Jaeger and B.A. Boley (North-Holland, Amsterdam, 1977) paper J 10/3. [6] P.M. McMahon, B.L. Meyers and K.P. Buchert, Trans. 4th Int. Conf. Structural Mechanics in Reactor Technology, eds. Th. A. Jaeger and B.A. Boley (North-Holland, Amsterdam, 1977) paper J 10/2. [7] J.V. Rotz, Evaluation of Tornado Missile Impact Effects on Structures, Symposium on Tornadoes - Assessment of Knowledge and Implications for Man, Texas Technical University, Lubbock, Texas, June 1976. [8] R.A. Williamson and R.R. Alvy, Impact Effects of Fragments Striking Srtuctural Elements (Holmes and Narver, Anaheim, CA, 1973). [9] A.E. Stephenson, Final Report, EPRI NP-440, Palo Alto, CA (1977). [10] A.K. Kar, Trans. 4th Int. Conf. Structural Mechanics in Reactor Technology, eds. Th. A. Jaeger and B.A. Boley (North-Holland, Amsterdam, 1977) paper J 9/8. [ 11 ] Missiles Generated by Natural Phenomena, US Nuclear Regulatory Commission, Standard Review Plan, Sec. 3.5.1.4, (1975).