Impat Simulation of Extreme Wind Generated issiles on Radioative Waste Storage Failities G. Barbella Sogin S.p.A. Via Torino 6 00184 Rome (Italy), barbella@sogin.it Abstrat: The strutural design of temporary storage failities for radioative waste generally requires the fulfillment of highly severe performane riteria if ompared to onventional buildings [1, 2, 4]. The present work fouses on the study of the behavior of a steel door subjet to the impat of extreme wind generated missiles, whih is one of the most demanding external events to aount for. Two different kinds of senario are onsidered: the first involves quasirigid small objets with limited mass and high translational veloity (hard impat); the latter involves massive objets with smaller veloity and larger ontat areas (soft impat). Contat is here modeled both by imposing the nonpenetration onstraint in average terms, and by approximating the distribution of ontat pressure over the impat area. oreover, the nonlinear material behavior modeling allows one to aount for the energy dissipation apabilities of the struture. In the ommon pratie the strutural response is often evaluated by using approximated methods, based on simple models, and/or on the expliit definition of ontat fores [2, 3]. However, these methods do not ompletely apture the evolution of the phenomenon and, in some ases, may lead to inorret design hoies. For these reasons, the evaluation of the mutual exhange of fores during the impat event should be determined by an expliit multi-body interation analysis. shielding panel supporting hinge stiffening ribs Keywords: ontat, impat analysis, strutural dynamis. 1. Introdution The simulation of impat phenomena is one of the most hallenging issues in the field of omputational mehanis applied to engineering, mostly due to the rapid improvement of highperformane omputing apabilities, as well as in the implementation of effiient numerial methods. In the present work, the strutural response of a steel door under impat loading is simulated. As above mentioned the effet of the ollision with two different objets is studied: a irular steel tube and an automobile, representative of hard and soft impat respetively. The door struture is omposed by a 10 mm thik steel panel with stiffening ribs and supported by a frame of UPN, HEA and IPE beams. All the parts are assembled with welded joints. A 3D view of the door struture is displayed in Figure 1, while Table 1 lists the main harateristis of the design missiles. supporting frame braket pin-hole Figure 1. Bak side 3D view of the single leaf of the steel door. The dimensions are 2.1 (width) 4.0 m (height). Table 1. ain harateristis of the missiles. automobile steel tube mass [kg] 1000.00 35.0 veloity [m/s] 12.25 24.5 kineti energy [kj] 75.00 10.5 linear momentum [kg m/s] 12250.00 857.5 The formulation of ontat adopted in the present work, desribed in Setion 2, is based on simple mapping tools diretly implemented and available in Comsol ultiphysis. Provided that
the shape and the extension of the interfae area do not hange during the impat event, or equivalently, that the variation of these harateristis is reasonably negligible, this approah guarantees redued omputational times if ompared to lassial ontat formulations, sine no searh algorithm is required. 2. odel formulation The numerial simulations are arried out by using Comsol ultiphysis Strutural ehanis and Nonlinear Strutural aterials modules. Geometrial data of the door are aquired with the CAD Import module. Both the impating objets (target and missile) are expliitly inluded in the finite element models. In partiular, solid, for 2D and 3D simulations, and shell elements, for 3D simulations only, are used. Contat is here modeled exploiting the general extrusion tool under the model definition node of the odel Builder window. Let genext1 and genext2 be the mapping operators ating on the faing ontat surfaes of the target and the missile respetively. Then, the expression Δ ( u) u u u genext2( u), (1) T defined on the target surfae, provides the differene between the values of the field variable u omputed on the target and on the missile. Analogously, the same quantity an be evaluated on the missile impat surfae as ΔT ( u) ut u genext1( u) u. (2) If u is the displaement omponent along the normal to the ontat surfae, eqs (1) and (2) provide the loal gap distane between the impating objets, denoted as d in the following. The ontat fore per unit area is then expressed as a funtion of the gap distane and its time derivative: k d d h( d ) h( d) F ( u, u ), (3) in whih k and are the elasti spring and the visous damper penalty onstants ating on the ontat interfae, while h(x) is the unit step funtion entered in x = 0. It an be easily verified that F is non-null when d > 0, whih defines the penetration ondition of the impating objets. Analogously, the visous term is non-null when the penetration value is inreasing. As an alternative, when the impat area is small if ompared to the harateristi strutural dimension of the target, as in the ase of the steel tube ollision, one an onsider the averaged values, over the ontat area, of gap distane and gap veloity. The formula in eq (3) is used within the Comsol model to define the boundary loads applied to the target and the missile surfaes, onsidering for the gap evaluation expressions (1) and (2) respetively. Frition effets are negleted in this study. The penalty onstants k and have to be set in order to keep the penetration magnitude relatively small, and to avoid spurious high frequeny vibrations whih may arise due to loal modes, generally related to the harateristi elements dimension. The visous onstant is expressed in the following form: 2 k m, (4) where ξ is the non-dimensional damping fator and m is the effetive mass involved in the ontat proess. In this work, we onventionally set m equal to the total missile mass. Dynami equilibrium equations are solved via a step-by-step time integration algorithm based on seond order bakward differene formulas. This hoie is reommended for two main reasons: first, the evolution of ontat fores, whih are frequently subjet to large amplitude variations, needs extra robustness properties to be orretly aptured and to avoid stability issues; seondly, the intrinsi numerial damping whih haraterizes this family of time integration shemes, an be useful to redue the above mentioned spurious ontributions related to the large value of the penalty stiffness onstant. 3. odel validation Validation tests were performed on a simplified 2D axisymmetri elasti model of a irular plate hit by a ylindrial hollow tube, having the properties listed in Table 1. The diameter of the plate is 1 m and the thikness 10 mm, the external diameter of the tube is 76.2 mm (3 in) and a wall thikness of 6.8 mm is assumed. The lamped end ondition at the boundary of the plate is simulated by onentrating a large mass (say almost infinite) at
Figure 2. Geometry of the axisymmetri model of the irular plate (axes units are in mm). Figure 4. Axisymmetri model, omparison between the ontat fore and the variation of missile linear momentum. Figure 3. Partiular of the mesh of the axisymmetri model. the external border, as denoted in red in Figure 2. Contat fores are in this ase averaged over the impat area. A partiular of the mesh is shown in Figure 3. Finite elements with first order shape funtions are adopted in the model. The quality evaluation of the numerial results is arried out by monitoring the evolution of some key variables during the analyses, with the twofold objetive to hek the fulfillment of global energy and linear momentum balane priniples, and to set the relevant parameters and tolerane thresholds of the solution algorithm. Let onsider the linear momentum balane priniple applied to the missile body; one an write: F ( t) d dt ( x) a( x, t) dv d ( x) v( x, t) dv P dt ( t), (5) in whih ρ is the mass density, a and v are the aeleration and veloity, and P is the linear momentum of the missile. The time histories of omputed ontat fore and linear momentum derivative are ompared in Figure 4, showing a perfet math during the impat event, even though some small disrepanies an be observed afterwards. Figure 5. Axisymmetri model, omparison between missile and plate linear momentum. Let now onsider the whole system, inluding the missile and the plate: sine external fores are null, the global momentum must be onstant throughout the analysis, as it an be verified in Figure 5. The urves of kineti and strain energy of the system are displayed in Figure 6. The osillations of total energy during the impat are due to Figure 6. Axisymmetri model, time histories of kineti and strain energy of the system.
the aumulation of strain energy in the ontat spring layer, while a limited derease an be noted omparing the final and initial values of total energy. Suh a derease, whih is admissible in this ontext, is both due to the time integration sheme and to the ontat penalty damper at the interfae between the impating objets. Finally, it is worth notiing that the area under the omputed ontat fore urve is onsiderably larger than the value of initial linear momentum of the steel tube. This is justified sine the simulated impat is almost perfetly elasti, and the missile has a non-null rebound veloity after the ollision, so that the time integral of the ontat fore is equal to the differene between the linear momentum omputed after and before the ollision. 4. Hard impat The ase of hard impat on the door is simulated by onsidering a loal model of a single square frame of the shielding panel. This is justified sine the fundamental period of the door, approximately 0.031 s, is muh larger than the typial impat duration, meaning that the loal response of the steel panel and the global response of the door struture an be reasonably onsidered as deoupled. 4.1. odel desription The size of the plate onsidered in the analysis is 780 780 mm, with a thikness of 10 mm. Analogously to what done in Setion 3, the missile is modeled as an hollow tube having an external diameter of 3 in. Exploiting the symmetries of the problem, both in terms of geometry and loading, one height of the square plate is modeled only, as shown in Figure 7. The mesh is refined towards the impat area at the enter of the square, moreover, the disretization of the plate along the z axis is organized in layers, with elements of smaller thikness at the top and bottom faes. First order shape funtions finite elements are adopted. An elasti-plasti onstitutive law with Von ises yielding funtion and isotropi hardening is assumed for the plate, while the missile material is onsidered as linear elasti, with the same elasti properties. Table 2 lists the relevant parameters used. The ontat penalty parameters used are: Figure 7. Square plate model, boundary onditions and mesh of the model. k A i = 5.0 10 5 kn/m, ξ = 0, 1, (6) A i being the impat area. Table 2. aterial parameters adopted for the door steel panel. parameter symbol value Young modulus E 210000 Pa Poisson modulus ν 0.3 initial yielding stress σ y,0 410 Pa hardening modulus E iso 654 Pa 4.2. Results The resulting ontat fore and displaement at the enter of the plate are plotted in Figure 8, where the influene of the penalty damping is shown: if ξ = 1, high peak values of fore are observed in the very first time instants, while a smaller damping ratio is more onservative. The duration and the shape of the fore is similar to the urve obtained for the axisymmetri model of the irular plate. The maximum displaement, reahed at the instant t = 3 ms, is 40 mm, with 33 mm of permanent defletion. The development of plasti deformations is mostly loalized in proximity of the impat area, at the opposite side with respet to the ontat interfae, as it an be seen in Figure 9. The maximum effetive plasti strain is 14%, whih is ompatible with the harateristi failure parameters of the steel adopted, 20% being the minimum rupture elongation in a uniaxial test. In this ase, the resulting rebound veloity of the missile is about 6.1 m/s, meaning that during the impat the 94% of the kineti energy possessed by the steel tube (9.85 kj) is transferred to the plate.
Simplified formulas to evaluate the permanent defletion and the time instant orresponding to the peak displaement of the plate may be taken from [5, 6 ( 7.9.2)]. Under the hypothesis of a rigid-perfetly plasti material, the predition for a fully lamped square plate hit by a rigid mass, substituting geometrial and material data, is: 26 mm of permanent displaement and peak value reahed at ~ 2 ms. However, it has to be notied that the auray of this approximation dereases if the ratio between the masses of the missile and the plate is small (~ 0.73), as in the present ase. The apability of a struture to dissipate energy is often evaluated by the so alled pushover analysis, whih provides the harateristi fore-displaement urve, up to the failure of the system, determined by a maximum admissible dutility ratio. Figure 10 displays the fore-displaement urve obtained by applying an inreasing uniform load to the impat area of the plate. The unloading phase starts from the maximum displaement omputed in the dynami ontat simulation. Notie that, due to the boundary onstraints, the strutural response is influened by membrane ontributions from the very first steps of the test. The area inluded within the loading and unloading urves, whih is the energy dissipated in a quasi-steady yle, is about 7 kj. Therefore, in this ase an equivalent stati approah would have led to a larger demand of dutility. Figure 8. Square plate model, ontat fore and displaement time histories. Figure 9. Square plate model, ontour plot of effetive plasti strain at the instant t = 3 ms, deformed shape is not amplified. Figure 10. Pushover urve of the square plate. Fore is applied to the impat area, ontrol displaement is the maximum transverse defletion. 5. Soft impat In the ase of soft impat the analysis simulates the ollision with an automobile having a mass of 1000 kg and a veloity of 12.25 m/s. Experimental data are available [2] showing that the impat of a typial automobile on a rigid wall produes a reation fore time history whose shape an be analytially approximated as Q( t) a m v0 sin(20t), (7) defined within the interval (0 : 0.025 π) s, m and v 0 being the missile mass and initial veloity, while a is a oeffiient, expressed in s 1, suh that the integral of the urve desribed by eq (7) is equal to the linear momentum P = m v 0 of the automobile. One obtains a = 20 s 1, therefore, the peak value of Q(t) is 24.5 kn (mean value is 15.6 kn, orresponding to a mean deeleration of 15.9 g). This implies the assumption of a perfetly inelasti ollision, meaning that both bodies have the same motion after the impat. 5.1. issile alibration In the present work, the missile is modeled as a solid blok having the same global mass and
veloity of the referene automobile. The dimensions of the blok, in meters, are 1 1.6 4 (H L P), H L being the impat area. The progressive damage of the automobile strutures and the resulting internal dissipation of kineti energy is globally reprodued by onsidering an elasti-plasti onstitutive law with Von ises yielding funtion and isotropi hardening. The mehanial harateristis of the material and the mass distribution are tuned, under the hypothesis of infinitely rigid target, in order to obtain a reation fore time history whih approximately reprodues the urve Q(t). The ontat penalty parameters used in the simulations are: k A i = 5.0 10 5 kn/m, ξ = 0. (8) The resulting mean values of peak penetration obtained in the tests do not exeed some tenth of millimeter, whih is ompatible with the infinitely rigid target onstraint. Several numerial tests were made to investigate the sensitivity of the model to the variation of eah parameter. Table 3 lists the optimal ombination of parameters identified by the tests for the material onstitutive law; while the mass is distributed as follows: 97.5% of the mass is onentrated on the fae opposite to the impat side, while the remaining 2.5% is uniformly distributed on the volume of the blok. Table 3. aterial parameters adopted for the automobile solid model. parameter symbol value Young modulus E 100000 kpa Poisson modulus ν 0.3 initial yielding stress σ y,0 4.9 kpa hardening modulus E iso 500 kpa As an example, Figures 11 and 12 show the influene of the mass distribution and the hardening modulus value on the shape and the duration of the ontat fore time history, in omparison with the target urve. As noted in Setion 3, the area under the urves of the omputed ontat fore is still larger than the value of initial linear momentum of the automobile, even though to a lesser extent. In partiular, in this ase the rebound veloity is muh smaller than the initial value. Suh a ondition is very lose to, although moderately more onservative than the above stated hypothesis of perfetly inelasti ollision at the basis of the analyti approximation in eq (7). Figure 11. issile model alibration, influene of mass distribution. The values in perent are referred to the amount of mass onentrated on the fae opposite to the impat side. Figure 12. issile model alibration, influene of isotropi hardening modulus, defined as E iso = k iso E. 5.2. odel desription As desribed in Setion 1, the door struture is basially made of a supporting frame and a shielding panel. In the FE model, shell elements Figure 13. Door model, 3D view of the mesh inluding the missile.
are used for the latter, while all the remaining parts are disretized with solid elements. The material properties of the missile are desribed above (see Setion 5.1). As for the door struture, plasti behavior is assumed for the parts modeled with solid elements only, with σ y,0 = 235 Pa, and E iso = 235 Pa, whilst the shielding panel is onsidered as linear elasti. This is justified beause of the large value of initial yielding stress of the panel steel, meaning that no plasti deformations are expeted for the automobile impat. Contat fores are omputed as a funtion of the loal gap value, to aount for the nonuniform distribution of pressure due to the rigidity of the different regions of the impat area. 5.3. Results Figure 14 (top) shows the omparison between the omputed ontat fore and the analyti approximation of eq (7). The dynami interation between the impating objets has redued the peak value of the fore, inreasing its duration. The maximum penetration observed over the ontat interfae is smaller than ~ 0.5 mm. The transverse displaements of the door are reported in Figure 14 (bottom): the maximum values obtained by using the analyti input fore and through an expliit impat simulation are similar. oreover, reations at the supports are reported in Figure 15: the use of eq (7) as input Figure 15. Door model, support reations time histories, (solid lines) impat simulation, (dashed lines) input fore eq (7). fore provides slightly larger peak values, while the results of the expliit impat analysis are haraterized by a wider frequeny ontent, meaning that this latter allows one to obtain a more aurate desription of the phenomenon. 6. Conlusions A proedure for the dynami simulation of impat phenomena, within the Comsol ultiphysis framework, is presented and tested. The mutual exhange of ontat fores between the impating objets is expliitly modeled. Validation tests on a simple ase are performed showing that linear momentum and energy balane priniples are fulfilled. oreover, a real-life example of a steel door under extreme wind generated missiles is analyzed and the results are shown, giving evidene that Comsol ultiphysis is a useful tool also for the dynami simulation of impat problems. 7. Referenes Figure 14. Door model, global ontat fore and maximum transverse displaement time histories. 1. ACI 349-06. Code Requirements for Nulear Safety-Related Conrete Strutures and Commentary, Amerian Conrete Institute (2007) 2. BC TOP 9A Rev 02, Design of Strutures for issile Impat, Behtel Power Corporation (1974) 3. Biggs J., Introdution to Strutural Dynamis, Graw Hill (1964) 4. DOE STD 3014-2006, Aident Analysis for Airraft Crash into Hazardous Failities, U.S. Department of Energy (2006) 5. Jones N, On the ass Impat Loading of Dutile Plates, Defene Siene Journal, 53(1), 15 24 (2003) 6. Jones N, Strutural Impat, 2 nd edition, Cambridge University Press (2012)