ML19275B421

Elastic & Viscoplastic Impact Bending Response Analysis of Nuclear Shipping Cask Structures

ML19275B421
Person / Time
Issue date:	11/12/1979
From:	Rei-Mei Shieh NRC OFFICE OF NUCLEAR MATERIAL SAFETY & SAFEGUARDS (NMSS)
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NUDOCS 7911140210
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{{#Wiki_filter:. D M jD D ' 3 3 [d LAJU edalL ~ ~ - ELASTIC AND VISCOPLASTIC IMPACT BENDINO RESPONSE ANALYSIS Or* NUCLEAR SHIPPING CASK STRUCTURES By R. C. Shich Transportation Branch U. S. Nucicar Regulatory Commission Washington, D.C. 20555 ABSTRACT Within the f ramework uf lum,ed mass / upgraded the side impsets. In order to protect a cask struc-beam theory in which rotatory alastic chear ture and its contents from excessive impact damages, deformation ef f ects are considcted, the title analy-a large production cask is usually equipped with end sis is made for the case of large realistic lead-and/or intermediate impact limiters. To demonstrate shieded, cylindrical stainless steel shipping casks that a cask is adequately designed to withstand the equippsd with end impact limiters. A computerized side or other bending impact loadings,for over years, study is lia. -.-.,gd fuc. ic t w onse t. bu.. ar.alyzed r.:: an elementat; analysis of elastic and elastic-viscoplastic beams element */pe structure subjected to quasi-static and fra,e! and subsequently used in the cask icpact loading provided that no shell buckling can occur. response analysfs study. Because the length / diameter ratio of a typical large irradiated spent fuel shipping cask is usually small Three types of impact limiter reaction force (2 N 4), it is essential to include shear deformation Tulsen ere considered and three simplified analysis and rotatory inertia effects in the analysis, i.e., techniques (i.e. quasi-static, dynamic amplification to use Timoshenko beam theory in the analysis assuming ~ factor and elementary beam analysis techniques) used that beam bending type analysis is adequate for the in shipping cask design are evaluated. In particular, cask design problem. It should be noted that Tim-ef fects of shear deformation and rotatory inertia on oshenko beam theory itself represents a simplification elastic impact responses and strain rate sensitivity to a more sophisticated shell or three-dimensional effects on inelastic dynamic cask response behavior clastici. theory. are studied. Appropriate guidelines are formulated for (1) general use of these techniques in impact design analysis and (2) treating rtcain rate censi-tivity effects on material st eength propertics in conjunction with use of elat sic, limit and clastic-A cylindrical shipping cask structure is assumed plastic design analysis methods. to be equipped with cnd impact limiters so that the cask can be modeled as an assemblage of bean elements 1. INTRODUCTION in studying the gross transverse cask bending Impact responses. Federal Code of Regulatinns, Title 10 Part 71 requires that a radioactive material shipping cask be The main purposes of this paper besides dcv lop-designed to withstand, among other:,, 30-foot drop ment and introduction of appropriate clastic and impact loadings onto an essentially unyiciding surf ace clastic /viscoplastic impact bending response analysis in a location and orientation that will cause the techniques for beam elec+nt type structures are maximum damage to the cask structure and/or its (1) to study clastic and viscoplastic cask impact contents. One uf such critical drop orientation is bending response behavior, (ii) to show certain 3

b*d e.. a,. p p shortconinps i merr.1 sfilified cask impact mNnent e ss load carried (sh)/(kg) bending ter.ponse analysin techniques and (iii) treatment of r aterial te 128,087/58,og7 strain rate ef fects on material strength properties "jf provide guidance f or currect ~ The study ir. done within the in cask design. framework of an upgraded beam theory in which effects Outer Shell 24,004/11,158 of rotatory inertia and clastic shear def erma - tions are included in the.malysis. For gings (2) 42,314/19,190 The basic acsu:.; tions and theoretical forn.ula-Impact tion of elastic and elastic /viseplastic beam element limiters (2) 5,000/1.134 type (frame) structures are given and the developed TOTAL 200,000/90,720 coaputer program described in section 2 and Appendix A. The computerized study is subsequently applied to stud, C.e elastic and inelastic responses of large 24odel II Cask (Sincie Cornosite Shell Type) - The lead-shicided spent fuel shipping steel casks sub. cask nc,dcl is shown in Fig. 3. The total weight of jected to side impact loading in Sections 3 and 4, the cask is 170 kips (77,270 kg) of which the weight respectively. The dynamic yielding behavior of the of two impact limiters is 3 kips (2,268 kg). The cask and a cantilever beam under drop impact loaoing cross sect' ion of the composite shell is identical to is Liso studied in Section 4. that of Model I Cask. 2. BASIC ASSUMPTI9NS AND COMPUTER PROGRAM As a first approximation, the cask shell assembly will be modeled as an assemb} age of beam elements The computerized study used throughout the carrying finite number of lumped mass points as shown present study has the following capabilities and in Figs. 2 and 3. Only one-half of the structural limitations: model is shown due 'to symmetry. Fach mass point is subjected to the downward velocity of 528 inch /aec a. The structure or strt.ctural system can be closely (30 MPH = 48.27 Km/hr) and mass gravity force. The modeled as an assemblage of finite beam elements masses are lu:nped at joints (nedal points) by as-that all lie in the plane of two-dimensional signing one-half of the total mass that is carried by motions. Out of plane motion is not permitted. the adjacent beam elements to the joint in question. Rotatory inertia at each joint is calculated by b. The structural material is elastic or rate using the formula sensitive elastic-plastic and follo.rs linear n m strainhardening law /rurely kinematic hardening I =I y[ (r 2+r 2)

• 1 y i g

1 2 rules under static loading condition and visco-i=1 i i 12 plastic constitutive power law similar to that wherem is the 1-th inass point, n is the number of of Malvern [2] (see Fig. 1 and Appendix A) under beamelkmet.'= surroundingthejoinfinquestion. dynamic loading condition, r33 and r2 are, respectively the inner and outer w circular non-seede d Ge M ra c. Inelantic deformations are predominantly flex-cylindrial beam element, and Eg the 1-th beam ele-3 ment length. or the present.tudy, the contents d. Lumped mass / upgraded beam theory in which elastic c ntent weight is significant compared with the total shear deformation and rotatory inertia effects I* E

• are considered (i.e., Timoshenko beam theory for the dynamic response analysis is usually required the elastic case) Is used in the theoretical 15]. The shear correction factors K used in cal-fo th' culating the shear flexibility coefficients e.

Linear and rotational displacements may be arbi-p = 12 EI/KC*At (see Eq. 14 in Appendix A) trarily large. n the Cwpn formla [5], an ase A brief formulation of the structural (frame) g, 6J1+v)(1+(ri/re)2)2 analysis probica is given in Appendix A. An associa-ted computer program, RATE 1 has been developed and (7+6v)[1+(rg/r2)a ]' +4(5+3v)(r3/r2)# verified numerically and experimentally (see Fig. 9)- in which v is the Poisson ratio, C* the shear modulus, 3. SIDE IMPACT ELASTIC RESPONSE ANALYSIS OF LEAD-E the Young modulus, I the second moment of inertia. A = cross-sectional area and i the beam element SHIELDED STAINLESS STEEL CASK length. The following material constants are used in Consider a lead-shielded cylindrical stainless computing K and y steel cask drop impacted horizontally onto a flat 6 rigid surface from a height of 30 feet (9.144M). Two Steci: E - 29 x 10 psi = 200 CPa cask models are considered in this study: 6

• =

2 x 10 psi = 77.22 CPa Model I Cask (Two Concentric ';hcIl Type) - The cask model is shown in Fig. 2 which represent, a v = 0.3 realistic cask design. The total weight of the cask 6 is W = 200 kips (90.72 Mg). The distribution of the Lead: E = 2.3 x 10 psi = 15.86 CPa weight (mass load) is as follows: 6

• = 0.84 x 10 psi = 5.792 CPr v = 0.43 3P0 D'3'Y f Rk 2030 002 o

A. Cr m.ri mn Study of Dypylf and ouv.1-St atic innact n.ately 1.4 and 1.35, respectively, for the Tir.oshenko go.ses of Cm.I n Suh V ted to kretanru11r IPt r.udel c.me. The correspondh.g t actor for the cr;uiva-1.imi t e r Forc e l'ul acs lent single degree of frecion r.ystem with the funda-mental cask oscillation period of approximately 9.1 (a) Model II Cask Lade - Consider the case in seconds is found to be approximately 1.33 [9,12]. which the impact limited reaction force Again the an.plification f actors f or the continuous pulse is of rectangular type with force system are seen to be larger than that of the equiva-intensity F = 100 W (i.e. 100 g-load) and lent one degree system. duration time of 0.0138 sec. 1he clastic bending response M (t) at mid span section It appears that the main reason for existence of A (section A) is shown in Fig. 3 as a function discrepancies between the bending response results of of time for the 10 and 20 (5 and 10 for the Timoshenko and elemer.tary becm models is due to one-half structure) element cases. De difference in fundamchtal oscillation periods (T = results for these two cases are seen t 9.3 ms and 7.1 ms, respectively). This can be seen dif fer only slightly; the maximum moment from the dynamic amplifica.,lon factor spectrum (for difference is only 1.52. Hence, in what one-degree elastic system) shown in Fig. 5. For a follows unless otherwise noted, the 10-rise tice/ oscillation period ratio (tr/T) between 0 element model (see Fig. 3) is used for the to 1, the factor decreases with decrease in T for a Model II cask response study and 20 given rise time (tr/T=0.71 and 0.93 for the Timoshenko elements (see Fig. 2) is used for the and elementary beam models, respectively). It should Model I cask response study. he noted that the fundamental period of a beam-element structure based on Ti::.oshenko beam theory is ne peak moment is f ound to be M = 3.2M. W. = always larger than that based on elementary beam A static yield moment = 268,000 kip - in = 3.088 Cg-mm) theory. In addition to load pulse shape and risa occuring at about 5.3 ms after initial imiact. The time, certain structural boundary conditions are also peak dynamic anplification factor (i.e. peak dynamic / expected to cause appreciable difference in bending static moment ratio with the static moment obtained stress results between those based on Timoshenko and by applying 101 times of static cask weight statically elementary beam theories due to redistribution of to the simply supported cask bec.m) for M, is found stresses. to be =2 as opposed to 2 for the single Begree of freedom system [9]. 4. DYNAMIC YIELDING AND INELASTIC RESTONSE OF CASK (b) Model II Cask Case - The dynamic bending moment results at locations A and B (based In the preceding section, the cask was assumed on the Timoshenko beam model) are shown to behave elast-ically even though it was stressed in Fig. 5 for the case of a rectangular into inelastic range of deformation. Although the impact limiter pulse of 37.55 W with dura-elastic analysis results do not represent actual cask tion time t* = 0.0354 sec. response behavior, they are useful in cask design in conjunction with a set of elastic design criteria Also shown in Fig. 4 are the static moment [13, 14]. In this section, dynamic yielding behavior results at locations A and B (obtained statically by and inciastic response analysis of the casks and amplifying the static weights at various mass points other structural components are presented. with a factor of 37.55). The peak dynamic loading (amplification) factors at locations A, B, C and D In general, the analysis of "short" beam struc- , 7.37 and ...a requircs to cuncider shear deformation (and m A+1 h. <..r ra 2..I,, o ~ -. 4.~ -,. r*? " 7 for a single also rotatory inertia) effects on both elastic and degree of freedom system subjected to a rcetangular inelastic deformation components. However, the force pulse of sufficient long duration (> half of computerized study described in Section 2, which esc 111ation period of the system). Hence, use of the neglects inelastic shear deformation effects, should latter in cask impact analysis may lead to noncon-also be applicable to the particular cask in: pact servative results -in this case. Treblem at hand (i.e., sid e n impacts of cask with end impact limiters) in which inelastic deformations B. Effects of Shear Deformation and Rotatory are mostly concentrated in the mid-cask section Inertia on Cask Impact Response Under Constant regions where shear forces and thus, shear deformations Limiter Force Pulse with Finite Rise Time are either zero or small. The second type of impact limiter reaction force A. Inclastic Dynamic Response Analysis of Model I pulse F(t) chosen for studying effects of shear -Cask (l'in 2) deformation and rotatory ine-tia on the Model I ccsk dynamic response behavior is a constant force pulse To perform inelastic analysis, besides those of of 80 W with the rise time of 6.5 ms. The bending elastic material constants, it is also necesscry to moment responses at locations A and B for both have information on static yield stresses o, strain-Timoshenko and elementary beam element cask models hardeningparametersnandstrainratesensStivity are shown in Fig 5. IIere, the response patterns parameters D* = (o /D)"and n for both 304 stain 1 css g between the results of Timoshenko and elcmentary beam steel and chemical 1 cad (cf. Appendix A). For stain-models are seen to be drastically different; the less steel, o = 30 ksi (206.85 MPa). D* = 100 s9, o peak bcnding noments (MA and M ) at locations A and n = 10 and n = 0.014 (c* < 5%) are used in the fol-3 B for the former case are found to be respectively lowing cask impact response analysis. The values of 37Z and 27% larger than those for the latter (ele-last three material constants are those used in [6]. sientary beam model) case. The dynamic anplification factors (in which the static analysis is based on 80 Because contribution of Icad shell layer on the v/ loading) for HA and MB are found to be approxi-overall composite beam bending strength is small and

in order to reasonably sinplify the une of RATE 2 shown in Tig. 83 is replotted in Fig. 8b as a function U /I is the rotational deformation code, the lead material constants, deduced from the of t, where GA 3 e the cantilever tip uass deficction. experimental data of [7] are taker. as a

• 370 psi at section A and U3 (2.57 MPa), n = 0.014, D* = 0.145 s I, on = 10 The dynanic yield moment, on the other hand, is 1/n

=Mo + DI* K approximately governed by My (c* 1 5%) With n-values equal for three components of the It can be shown [8] that M at section A for the =U composite shell, the D1* value in Eq. (1) of Appendix cantilever Npact problem is related to Og B A for the composite shell is an algebraic sum of the and C = 1 - lM hf l by o A individual shell component. + DI* [(n+1)(n+2)0 1/n (M ^ > M ) A yA(t) = M M To account for lead yiciding at low yield stress, (n+2-([*A the bending stiffacss of the elastic portion of static bilinear moment-curvature relationship is This equation is plotted in Fig. 8b as curve 2. taken to be 0.912 that (EI) of the fully elastic Intersection point of curves 1 and 2 in Fig. 8b composite beam cross-section (see Fig. 6); the in-defines the time t and dynamic yield moment value y at which the cross-section A vields M*yg = M g(tdynamically.7)ecausePfyA(t) decreases gra elastic portion starts at M = M, where M.is the static yield moment of fully yielded composite beam B cross-section. g = (M ) max ? M, the lM!"M to MyA " M at M 4 o B at A o o Consider now the case in which Model I cask is dynamic yielding mu'st occur af ter hfAl exceeds M and o drop impacted horizontally onto a flat unyielding beare it attains its relative maximum (or minimum) surface from a height of 30 feet. The impact value at which curvature rate vanishes. 1 hat is, in limiters are assumed to be of stainless steel simple an inertia loading system, once a cross-section (or fin type whose force pulse characteristics are material point) is bended (or stressed) beyond its idealized as the ones shown in the upper figure of static yield moment ( or stress) value, the section Fig. 7 (cf. [10]). The corresponding elastic and (or material point) will eventually ycild dynamically elastic-viscoplastic bcnding moment and relative no matter how large is the dynamic yield stress. displacement responses at location A are shown in the lower figure of Fig. 7. The maximum elastic moment C. Guide 11ner for Treating Strain Rate Sensitivity M is f und to be 2.5 McA(MoA = static yield moment Effects on Material Strength Properties A at location A) and elastic dynamic amplification factor (corresponding to the static loading of 180 W) (a) Design Based on Elastic Analysis Methods - From the above studies on material yielding is found to be 1.14. behavior it is apparent that if a design In the inelastic case, it is found that due to criterion is based on material yield strength material strain rate sensitivity properties, dynamic (i.e., a material point is not allowed to yield moment and subsequent flow moment are approxi-yield), then the yield strength to be used mately 60% higher than those (M = 1 % 1.1 ffoA) f r comparison with stresses should be the correspondingtothestrainrathinsensitivecase. st.stic yield strength. No credit should be taken for increase in yield strength due to The downsard peak relative displacements uA ~ "O at location A with respect to cask end displacements for strain rate effects. The same conclusion the inelastic case is seen to be 25% to 30% higher also applies to the other material strength than those of the corresponding elastic case indi-properties (such as ultimate strength). cating occurrence of appreciable inelastic deformation (b) Design Based on Dynamic Limit (collapse in the cask. Load) Analysis Methods - I-this case, In order yielding of a cross-section does not mean B. Evnamic Yic1 ding of a Cantilever P, cam to better understand dynamic yielding behavior, let's collapse of the structure unless the latter is statically determinate. Therefore, it consider the impact problem of a mass-less, rate-sensitive clastic, perfectly-plastic cantilever beam is obvious that the dynamic yield moment (or stress) can be used at various dynamically carrying a tip mass W = 1595 lb and subjected to yicided (plastic hinge) locations except as shown in various initial impact velocities Vo Fig. 8. The beam cross-section is a hollow rectan-for the last plastic hinge location at formation of which a structural collapse gular one with width, depth and wall thickness of 6 x 4 x 3/16 in (147 x 98 x 4.76 mm) and mild steel mechanism being initiated. The yield 3 ksi (206.85 Gra), moment or stress value to be used in this material properties E = 30 x 10 o = 30 ksi (206.85 MPA), n = 0, n = 5 and D* = last hinge location again should be the a o /D)" - 40.4 s~I. The elastic and elastic-visco-static yield value, o plastic bending responses at location A of the beam under various impact velocities are shown in Fig 8a. (c) Design Based on Elastic 41astic Analysis Methods - In this case, it is obvious that In each case, once the beam cross-section moment full credit of strain rate effects on exceeds its static yield value M, the cross-section (or moment) can be taken into o flow stress yields eventually although dynamic yield moment is account. larger than the static one, To explain this phenomenon more c1carly, Ict's CONCLUDING E121 ARKS consider the case of drop impact a V = 2 mph The title analysis has been made under the regulatory (3.22 Km/hr). The clastic solution 30-foot hard surface drop impact loading and within 2 the context of an upgraded lumped mass / bean element O /L PfA A B rtructural theory in which both rotatory inertia and = clastic shear deformation offrets are included. The impact limiter force pulse characterintics we-c 2030 00^

Neglect of shear defornation and rot a ory incrtia assumed known and characteristic duration time period o was assued much longer t han the cask fundamental effects in beam type dynamic bcnding revonse oscillation period, which is usually the case. of a*short' cask may 1 cad to nonconservative results. Theoretical fornlation of the analytical For a given cask under 30-foot hard surface predictive techniques for both elastic ar.d elastic / o viscoplastic beam element type structures, together drop impact loading, the magnitude of dynamic with an associated computer program were presented yield stress (moment) at a material point first. Based on the computerized study, a comparison (cross-section) is a function of impact limiter study of quasi-static and dynamic clastic impact characteristics. In general, if the elastic responses of composite cask shell structures was made stress (uoment)/ static yield stress (moment) and effects of both rotatory interia and shear ratio (>1) is larger, the larger will be the deformation on elastic cask dynamic responses were dynamic yield stress (coment). studicd. In addition, yielding bchavior and strain rate sensitivity effects on inelast,1c response behav-Guidelines for using dynamic material strength for of composite (stainicss steel / lead /otainless properties in conjunction with elastic, limit and steel) cask shell and carbon steel bean structures elastic-plastic analysis design criteria such as were studied. Specifically, from these studies, it the ones given in ASME Boiler and Pressure Vessel was concluded or reaffirmed thats Cod,e, Section III, were given in Section 4C. Numerical results of the present study are presented in figures in Appendix B. Use of the quasi-static method of impact analysis o needs to exercise caution because it may lead to nonconservative results ciuc to dynamic amplifica-tion effects. To find appropriate dynamic amplification factors, a multi-degrees of freedom /O system model is, in general, required. go "g - LM l! ML REFERENCES 1. Shich, R. C., "Large Displacement Matrix Analysis 8. Shieh, R. C., " Strain Rate Sensitivity Effects of Elastic /Viscoplastic Plane Frame Structures," on Crash Response and Dynamic Yield homent Journal of Engineering for Industry, Vol. 79, Formulas for Crash Prediction of Automobile Series B, No. 14, pp. 1238-1244, Nov. 1975. Bumpers," presented at the AIAA/ASME/SAE Six-teenth Structures. Structural Dynsmic and 2.

Malvern, L., "The Propagation of Longitudinal Material Conference, May 27-29, 1975, Denver, Waves of Plastic Deformation in a Bar of Mate-Colo.,iATAA Paper No. 75-791.

rial Exhibiting a Strain-Rate Effect," Journal of Applied Mechanics, Vol. 18, 1951, 9. Biggs, J. M., Introduction to Structural Dynamics, pp. 203-208. McGraw-II111 Inc., N.Y., 1964. 3. Cowper, C., and Symonds, P., " Strain liardening 10. Sagartz, M. " Dynamic Tests of Metallic Impact and Strain Rate Effects in the Impact Loading Limiters," Sandia Laboratories Report No. of Cantilever Beams," Technical Report No. 28, SAN 77-1939, January 1978. Brown University, Providence, R.I., Sept. 1957 11. Shieh, R. C., "Thermoclastic Vibration and 4. Manjoine, M., " Influence of Rate of Stiain and Damping for Circular Timoshenko Beams," TRANS. Temperature on Yield Stresses of Milo Steel," ASME, Journal of Applicd Mechanics, Vol. 42, Journal of Applied Mechanics, Vol. 11, TRANS. Series E, No. 2, 1975, pp. 405-410. ASME, Vol. 66, 1944, p. A-211. 12. Kornhauser, M., Structrual Effects of Impact, 5. Cowper, G. R., "The Shear Coefficient in Spartan Book, Inc., Baltimore, 1964 Timoshenko Beam Theory," TPANS. ASME, Journal of Applied Mechanics, Vol. 33, Series E, No. 2,' 13. ASME Boiler and Pressere vessel Code Section TTI, 1966, pp. 335-340.

1977, 6.

Forrestal, M. and Sc'gartz, M., " Elastic-14. U.S. Nuclear P.cgulatory Commission Regulatory Plastic Response of 304 Stain 1 css Steel Beams Guide 7.6, "Desige Criteria for the Structural to Impulse Loads," Sandia Laboratories Report Analysis of Shipping Cask Containment," March SAND 77-2101, January 1978. 1978. 7. Mok, C. and Duffy, J., "The Dynamic Stress-15. Pickel, T. W., Jr. " Evaluation of Nuclear Strain Relation of Metals as Determined From System Requirements for Acconmodating Seismic Impact Tests with a liard Ball," International Ef fects," Nuclear Engineering and Design, 20, Journal of Mechanical Science, Vol. 7, 1972, pp.-323-337. pp. 355-371, 1964. 20 H 005

ICFEliCNCES (Cont'd) 16. Mindlin, R. D., " Dynamic of Package Crushioning," 17. Shich, R. C., " transient Im> e t Kesponse Analysis bell System Tech. Journal, Vol. 24, Nos. 3-4, of Rate Sensitive. Elastic-Plastic Beams and 1954, pp. 353-461. Frames," to be published. APPENDIX A ^* M" Consider a mass-less beam element of length 1, cross-liere p = a /S, (i = M, N), g - EI, ~ g g N sectional arca A and second moment of inertia 1. In teams of stress resultants (bending moment M ard (DI*)", and S = (DA)", and (gi for o ( (g i1 N axial f orce N) and strain resultants (curvnture K (i =.1, 2) are the i-the end inelastic zone spreading and centroidal axial strain c), the constitutive lengths. equation corresponding to that given in Fig. I can be written as In the presence of strainhardening (n Y o), the force-deformation equations are approximately given by Eq. (4) with ($ ) given by (cf. Eqs. 1) K = "El + (O),; M=M + Eln(K El) P DI* o (1) (i)=[S]-1(;n) ((;n), (;n1,jn2, ; } (50 .n g p c = g + (N-N) ; h=N + EAn(c g ) ~ (b") (5b) N o 7 ($)=[a]~ (S) + [6] in which the effect of bending moment / axial force interaction on inelastic deformation has been that is, neglected. In Eqs. (1), M - M = 0 if lMltlblandHz {g) + (,l] {S ) = [G]{e) (n O) (6) i H if lNl<jNl;M and N the static yield moment and where [a] is a (3x3) diagonal matrix whose diagonal axial force res cctive y, and n and D = o /D* " are g S - S' the material strain rate sensitivity constants, n the elements are EI, El, EA, and S = t g strainhardening parameter, the dots indicate material time derivatives, and (S = static parts of S ),1.a., in view of Eos. (1), s g (5)+[H](b")= C) (*) (n(0) t y(n+1)/n dA (2) 1*= ,(S) = (S) -(s'),(S* ) = n[a]{x) + (1-n) (S,) 'A For a hollow circular cross-section of outer and inner radii r and r, wherc (S,) is the static yield stress resultant 2 g vector. The parameters C for the case of entire I 2n6 (3n+1)/n (3n+1)/ T[(2n+1)/2nl (3) beam elemeat being in inelastic state now are given ~ ~5i1 I'2 - 'l T[(3n+1)/2n] I 3 by: where T(a) is a gan:ma function of its argument a. (g = S MSy+S) (i=1, 2) (7) g 2 Within the context of elementary beam theory, the and for the.other cases (partially elastic and beam element end force-deformation ({S)-{e)) rela. ion-partially inelastic) ships for the clastic / perfectly visioplastic case determined from (n-o) have been obtained in [1) as

• "s(*i' (4)
• 1' 1"( 1 2 1 (S)+[H][S]{$p)=[G](e) where z - (71 and z = (1-(2)L
• y y

ere the bending (i=1,2) Here, the components S, eg g and axial (i=3) components cf the vectors (S) and (e) E 'E * "#E respectively (with clockwise and tensile components regarded positive) and (X ) the corresponding plastic parts of the end strain resultant vector (X), the [p](u)+[ [B] (S) = (P(t)) (9) p ~ matrices [C] and [11) are given by j=l '[G ) (o) ~ [H ) (o)" g, h '2 where subscript "j" stands for j-th beam element, 1 M . I 2, [p] is the diagonal mass matrix, [B({u)))) the g g [11] = j j-th beam element compatibility matrix associated [C] = y C 12 - 3 C /(n+2)], (y [1 - 3 (y/(n+2)]- with the joint displacements: 293 1 1 _ g [1 - 3 (7 (n+2)], ty [2 - 3 (y/(n+2)], () = [B] (*) (10)

• ail

( / 1 0 Equa tions (5)-(10), together with the appropriate =0 initial conditions, c,n be used to 1.olve (u), (S)g if Sy+S2 " "M 0 1 if Si+S3 and (x}g simultaneously. 2030 006

A general purpose ce:npister prorrum, RATE 1, has For a mass-ler.s (n - 0) cnd lateral force free been developed using the f ourth order Runge-Eutta (q = o) beam clonent, Eq s. (10, topthcr with the numerical integration techniques. Verifications end conditions can be sulved to yield of the computer program and analytical predictive techniques have been rade by comparing the RATE 1 S{ I}. [ce]{el} (13) results with the existing solution results (for linear power law case n=1) a'id experimental frame 2

• 2 drop impact response results with excellent analytical / experimental correlation results(See fi9 9) where The$e will be reported in detail elsewhere [17].

~4+u 2-p' Extension to Timoshenko Eeam Elernent Case 1+y 1+y '[C*] = E 2-9 4+p (p = 2) (14) 1 Let O(x,t) and v(x,t) be the cross-section rotational 1+y 1+p, gt and transverse displacement at coordinate x and time __t. The governing equations of motion for a uniform For a simply. supported Timoshenko beam, subjected Timoshenko beam element can be written as (5, 11] to initial transverse velocity v, it is readily shown that the natural frequencies of vibraticas 8(O' + v") - p A v + q O (11) are a0" - B(8 + v') - pIO = 0 ,e + [a -4p A.,In? (nif / t) ]! 1/2 2 ( ~L "1,2 2pApl where a = EI, B = KC*A (K = shear correction f actor, G* = shear nodulus) p = mass density, where q = lateral force per unit length of beam, and primes and dots indicate derivatives of the a = (pas + p]B) (nII/i)2 + pAB quantity with respect to coordinate x and time t, respectively; the other notations are defined as before. The bending momemt M(x,t) and shear force V(x,0) are related to 0 and v by V = B(v' + 0), M = a0' (12) APPENDIX B DTD D 3f C W

3

,,(51RI55) e' , (Star 55)

• A f>h "o 1 'E t

4*

• th s, I

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