Abstract. 1. Introduction

Transcription

1 Design concept for reinforced concrete slab structures under soft impact loads A Miyamoto', M.W King* Department of Computer and Systems Engineering, Yamaguchi University, Tokwa-dai 2557, Ube 755, Japan ^Boral Concrete Sdn. Bhd., Taman Sri Tunas, Bay an Lepas, Penang, Malaysia Abstract The main objective of this paper is to propose a dynamic design procedure for concrete slab structures to soft impact loads. Firstly, the ultimate limit states of concrete structures are briefly introduced. And then, a design procedure for reinforced concrete(rc) slab structures under impact loads is discussed based on energy criterion and load criterion. Finally, a few case studies on design of RC slab structures are carried out as an example of the proposed dynamic design procedure. 1. Introduction Impact design methods of most structures are carried out by adopting an impact factor in the static design method. The dynamic forces during impact are converted into a static force of equal magnitude and treated in much the same way as other static loads. This static design method would not adequately describe an impact phenomenon and thus have only a limited amount of practical applicability. During impacts, an excitation of not only the first mode but also higher modes of vibration can be expected. Structures designed using the equivalent dynamic force would be able to withstand bending but not the shear, hence bringing about punching shear or concrete scabbing. These factors can only be totally considered if a dynamic design approach is adopted. In this paper, a design procedure for concrete slab structures under impact loads is discussed based on both energy criterion and load criterion. In the proposed dynamic design procedure, the indexes for impact resistance, such as impact load at failure, total energy, index of local deformation, volume displaced, etc.fl] are employed for determining structural modifications necessary in the design procedure. Furthermore, a case study of design of reinforced concrete guardrail(handrail) under vehicular collision is carried out as an example of the proposed dynamic design procedure. 2. Ultimate States of Concrete Structures The ultimate states of a concrete structure are necessary during the design

2 792 Structures Under Shock And Impact process. Since impact loading on concrete structures has a very low occurrence probability, the normal case would be to design the structure according to the ultimate limit states. A serviceability limit state would results in an uneconomical and conservative design. The ultimate states for different types of concrete structures are totally different. Some examples of ideal ultimate limit states for concrete structures in the field of civil and structural engineering are listed in Table 1. The ultimate limit state for rock sheds under impact loading can be set as structural failure, either in the bending or shear failure modes. But in the case of concrete handrails, the ultimate limit state would also be structural failure. But the resultant impact failure mode would be a very important factor. Another limit state for concrete handrails, especially in overhead expressways or Load characteristic Soft Hard Table 1: Ultimate states of concrete structures under impact loads Impactor Vehicle Ship Aircraft Rock Explosion Target / Structure Handrail / Barrier Building Bridge pier Bridge girder Bridge pier Offshore structure Marine structure Gravity platform Nuclear power plant Important structure Protective shelter Rock /Snow shed Protective shelter Ideal ultimate limit states Failure in structural element. Energy is absorbed by flexural deformation of structure. Punching shear and concrete scabbing should be prevented. Energy is absorbed by failure in structural element. Collapse of entire structural system should be prevented by allowing hinges to form at beam sections. Energy is absorbed by failure in structural element. But the stability of entire structure should be ensured, especially in the supporting forces. Furthermore, deformations that would cause movements to the upper structure should be checked. Failure in structural element. Large deformations would cause collapse of the girders from piers. Adequate flexibility should be allowed to prevent collapse from supporting piers. Energy is absorbed by failure in structural element. But the stability of entire structure should be ensured, especially in the supporting forces. Furthermore, deformations that would cause movements to the upper structure should be checked. At deep water levels, the piers should be designed to be rigid, i.e., collision energy should be absorbed by deformation of ship. Energy is absorbed by failure in structural element. But the stability of entire structural stability should be ensured. Single layered structure: Cracks should not be allowed for structures of extreme importance. In certain cases, cracking should be allowed but penetration and scabbing should be prevented at all cause. Double protection structure: Penetration and scabbing are allowable in the secondary structure. Cracks should not be aj lowed in the primary structure for extremely important structures. In certain cases, cracking should be allowed but penetration and scabbing should be prevented ai all cause in the primary structure. Failure in structural element. Single structure: Cracks should not be allowed for structures of extreme importance. In certain cases, cracking should be allowed but penetration and scabbing should be prevented at all cause. Double protection structure: Penetration and scabbing are allowable in the secondary structure. Cracks should not be allowed m the primary structure for extremely important structures. In certain cases, cracking should be allowed but penetration and scabbing should be prevented at all cause in the primary structure.

3 Structures Under Shock And Impact 193 in multi-story intersections (crossings), is the prevention of scabbing at the rear face, as concrete scabbing would likely cause a secondary disaster when falling onto a different traffic lane below. Concrete scabbing can be effectively reduced or even prevented by allowing a bending failure mode to occur. The formation of shear cones during shear failure would cause concrete scabbing to occur easily. Therefore, for the design of concrete handrails, the bending failure mode would be important. A different ultimate state would be for the design of nuclear power plants and its related facilities, where cracking in the concrete structure could cause leakage of radioactive materials into the atmosphere. In certain cases, where the structure is not highly radioactive, then prevention of perforation would be a more economical ultimate state. Structural integrity would be a major ultimate state when considering a structural system. For example, in the design of marine offshore structures or gravity platforms, or even the design of buildings, failure of structural elements would be the limit state, but the total structure should be intact after the impact. A loss in the bearing capacity of a structural element could cause the entire structure to collapse. The amount of permanent deformation is also the ultimate limit states in certain cases. For example, the permanent deformation in a bridge pier should be limited, or else it would cause damage or even collapse of the super-structure. The ultimate states for reinforced concrete handrail will be discussed a little more in detail here and followed by a case study for design in the following section. An ideal design procedure of concrete handrails for expressways is relatively difficult. An ideal handrail should be able to withstand the impact from a colliding vehicle. The handrail should not act as a solid barrier to stop the collision but more as a flexible wall that is capable of absorbing most of the impact collision energy. Therefore, it is necessary to design concrete handrails to fail under bending, as energy absorption is better during ductile type of failure. During the collision of a light vehicle or when the impact momentum is relatively small, the handrail should act as a rigid barrier and allow the deformation of the vehicle itself to absorb most of the impact energy. When the momentum of the collision is large, or when the collision speed is large, then the handrail should deform and absorb most of the impact energy, especially in the bending mode, and possibly let the vehicle not to have relatively large deformations. Therefore, the life or lives of occupants in the vehicle would not be highly endangered. In the case of collisions by large trucks or trailers, the structure should absorb the energy but not allow scabbing of concrete to occur at the rear face. 3. Design Concepts for Reinforced Concrete Structures The energy criterion would be the most efficient method of designing concrete structures under impact loads. The external energy from the impacting body (impactor) should be absorbed by the entire structure. The energy transformation process during impact loading of concrete structures is illustrated in Figure l[2j. The main part of the kinetic energy in an impacting body is transmitted to the concrete structure during impact collision. The energy transmitted to the concrete structure is then converted into the kinetic energy of the concrete structure and also the energy absorbed by the structure. The energy absorbed by the structure is converted into irrecoverable energy and also strain energy, which is recoverable. Formation of cracks and also fracture zones, friction, damping and etc. are the main source of the irrecoverable energy. When there is vibration in the concrete structure, then part of the strain energy is then re-transmitted as

4 194 Structures Under Shock And Impact kinetic energy back to the concrete structure, as indicated in the figure by broken lines. Besides the energy criterion, most design specifications for structures under impact loading specify the impact load-time function as the design impact load. Figure 2 shows the idealized design impact load for impact of Phantom fighter aircraft into concrete nuclear reactors at a speed of 215km/sec[3]. This design impact load is used for design of certain concrete nuclear reactors in Germany. In such a case, an impact load criterion must be applied for design of the structure. Therefore, a dynamic design method, based on both the energy and load criteria, is proposed here. Figure 3 shows the proposed design method for concrete structures. Energy not transmitted into concrete structure Energy transmitted into concrete structure We Absc rbed ene rgy v \A ^ fa^^ *** > f ^" ^ Irrecov erable ene rgy ^^ V /i ^ Kinetic energy Wk A Strain energy Ws Figure 1: Energy transformation process of impact collision 12, TIME IN SECONDS Figure 2: Aircraft impact loading function

5 Structures Under Shock And Impact 195 [Outline of type of structure (1) ] is design impact load specified? (2) Yes No I Preliminary study & survey of possible impact collisions (3) y ' I Determination of design impact conditions (4) Results of design for static loads (8) I Selection of type of structure (7) ]Setting of structural dimensions (9) \^' Dynamic structural analysis (10)] No Is energy cntenon satisfied? (11) Is failure condition exceeded? (12) Yes Figure 3: Flow of dynamic design method for structures under impact load Bold lines in the flow chart indicate the safety provision according to energy criterion while the broken lines indicate safety provisions according to load criterion. The flow of the proposed design method can be explained as follows, where the numbers indicated correspond to the sequence of numbering in the flow: (l)outline of type of structure - The type of structure to be designed is selected, i.e., concrete handrail, concrete pier, etc.

6 796 Structures Under Shock And Impact (2)Is design impact load specified? - If the design impact load is specified for the particular case, then further considerations regarding the impact load is not necessary. If the design impact load is not specified, then considerations of the type and characteristics of the impact load function must be predicted based on a numerical or empirical procedure. (3)Preliminary study and survey of possible impact collisions - The type of impacting body that could possibly impact on the structure and also the possible collision speeds and also collision angles have to be surveyed. (4)Determination of design impact conditions - Based on the results obtained from the survey in Step (3), the design impact conditions (type of impacting body, collision angle, collision speed, etc.) are selected based on statistical considerations. (5)Is safety factor required? - It is considered that the safety factor can be calculated by applying the energy criterion. In the case of load criterion, only a straight-forward procedure is possible, where the structure is checked for failure under the specified impact load condition. (6)Calculation of design impact load - The impact design load is predicted based on numerical models. In this dissertation, application of the multi-mass model is proposed for predicting the design impact load function. (T)Selection of type of structure - Further details of the concrete structure is selected. By considering the magnitude of the specified design impact load or specified design impact collision, a structure that can effectively withstand the impact collision is selected, i.e., reinforced concrte structure or prestressed concrete structure, simple supports or fixed supports, etc. (S)Results of design for static loads - At present, most structures are not designed specifically for impact loading conditions. The structure is usually designed to withstand the necessary design loads, such as dead load, live load, earthquake load, etc. Once the dimensions of the structure have been determined based on the specified static design codes to withstand the normal loading conditions, the impact loading condition is then applied to the structure. (9)Setting of structural dimensions - If design of the structure to resist normal loading conditions in Step (8) has been performed, then the structural dimensions are fixed. But if no preliminary design under normal static loading conditions have been performed, then the structural dimensions are selected. (lo)dynamic structural analysis - Dynamic structural analysis is then performed based on the specified design impact load or design impact condition. In the context of this dissertation, the dynamic nonlinear finite element analysis proposed in earlier paper[6] is applied. The "linked" procedure!?] is applied for the safety provision by energy criterion while the normal finite element procedure is applied for the safety provision by load criterion. (1 l)is energy criterion satisfied? - Safety provision based on the energy criterion is applied. Details are given in the following section. (12)Is failure condition exceeded? - Safety provision based on the load criterion is applied. Details are given in the following section. 3.1 Safety Provision according to Energy Criterion The safety factor for design of concrete structures under impact loads can be determined by applying the energy criterion. In the present limit state design for concrete structures under static loads, the safety provision is determined by the ratio of design ultimate setional forces and sectional forces during loading by specified loads. When designing concrete structures subjected to impact loads,

7 Structures Under Shock And Impact 197 the structure should be designed to act as a solid rigid body under impact loads with a high occurrence frequency, but for impact loads that are not likely to occur, the structure should deform and absorb part of the kinetic energy. The energy transmitted into a concrete structure during soft impact collisions can be assumed as, Energy transmitted into concrete structure ( E } = (Kinetic energy in impacting body before impact collision) - (Energy dispersed in the impacting body by plastic deformation) - (Kinetic energy in impacting body at time of separation from concrete structure) - (Energy lost : sound, friction, etc.) (1) If the concrete structure is capable of absorbing all of the energy transmitted during the impact collision, then the structure would be intact and structural failure would not occur. But if the structure is not capable of absorbing all of the energy transmitted during impact collision, then structural failure is likely to occur. Consequently, structural modifications, such as changes to dimensions of the structure, amount of reinforcement, material parameters, etc. have to be carried out. The safety provision for the energy criterion can be performed by calculating the ultimate absorption energy ( E\ If the ultimate absorption energy is larger than the energy transmitted into structure ( E), then failure will not occur in the particular structure. It is considered that the ultimate absorption energy of a structure is dependent of the loading rate of the load function [4, 5]. Therefore, the loading rate for the design impact condition has to be evaluated and the corresponding ultimate absorption energy must be calculated. Figure 4 shows the definitions of calculations of energy absorbed by structure and ultimate absorption energy. If the structure does not fail under the specified design impact loads, then the energy absorbed ( E J is taken as the area enclosed by the impact force function and the midspan deflection axis, at maximum deflection. The amount of energy absorbable by a structure is related to the deformations, therefore the maximum deflection is selected. The amount of impulse transmitted during the impact collision at maximum deflection is referred to as / in the context of this dissertation. For the similar loading rate, i.e., v =v, dynamic analysis is performed until failure in the concrete structure, as indicated in the figure. The final amount of energy absorbed is called the ultimate absorption energy, E, while the corresponding amount of impulse at ultimate failure is indicated by /. Since the impact failure mode is similar for the same loading rate, the final failure conditions for both analyses would be relatively similar. The safety factor for the concrete structure could then be expressed as, Efu I Ef x / Safety factor for impact loading, y = -jf- - -f^ -^ (2) ' ~ L A * " If the safety factor is larger than unity, then the structure would not fa il. Concepts of failure energy in relation with loading rate is illustrated in Figure 5. Under a low loading rate, the bending failure mode is dominant but as the loading rate is increased, then the punching shear failure mode would be more prominent. Therefore the envelope for failure energy of concrete structures

8 198 Structures Under Shock Ami Impact INPUT OUTPUT Deflection (6) Deflection (<J) Ie : Impulse at maximum deflection Iu : Impulse at ultimate failure Efe : Energy absorbed by structure at maximum deflection Efu : Ultimate absorption energy Figure 4: Calculation of energy absorbed by structure and ultimate absorption energy Failure energy (Ef) A EKPS') VLl VL2 VL4 Loading rate (vq : Failure envelope for impact toads - min { EKB), EKPS)} --> EKB') :Structural modifications EKPS) --> EKPS') : Structural mod (f(cations Figure 5: Concept of failure energy under impact load

9 Structures Under Shock And Impact 199 under different loading rates may be assumed as, (3) where, E(B) E/PS) : Failure energy for bending failure mode, : Failure energy for punching shear failure mode. During the design of concrete structures according to the energy criteria in Figure 5, the following conditions would apply: l)design criteria ( E^ v^) - No failure in structure. Dominant deformation mode is bending. 2)Design criteria (E^ v^) ~ Failure in structure by bending. Structural modifications are necessary to improve the failure energy for bending failure modetoe^b'). 3)Design criteria ( E^ v ^) - No failure in structure. Dominant deformation mode is shear. If shear failure should be prevented at all risk, structural modifications should be performed by improving the failure energy for punching shear failure modetoeips'). 4)Design criteria (E^ v^) - Failure in structure by punching shear. Structural modifications are necessary. If the failure mode is not important, then improvements to level E(PS') would be adequate. Or else improvements to level E(B') would be necessary. The flow of safety provision according to energy criterion is shown in Figure 6. The flow can be explained as follows: (13)Safety provision according to energy criterion - Start of safety consideration. (14)Setting of limit states - The limit states for design of the structure is determined. (15)Is safety factor > 1.0? - The safety factor is considered based on Eq.(2). (16)Is failure mode important? - If the resultant impact failure mode is not a limit state specified, then the design procedure will end at this stage, (17)Determination of impact failure mode - The failure mode is determined based on results of the dynamic analysis. ( 18)Possibility of punching shear failure? - This corresponds to E(PS) > E(B) in Figure 5. If punching shear failure is not likely, then the design ends at this stage. (19)StructuraI modifications based on index of impact resistance - Modifications such as amount of steel reinforcement, structural dimensions, material characteristics, etc. are performed based on evaluations of impact resistance propertiesfll]. 3.2 Safety Provision according to Load Criterion The safety provision according to load criterion is relatively simple, in comparison to the energy criterion. The safety factor is not considered in this case. The flow of safety provision according to load criterion is shown in Figure 7. The flow is as follows: (20)Safety provision according to load criterion - Start of safety consideration. The design impact load is applied when specified. (21)Setting of limit states - The limit states for design of the structure is determined, (22) A re the limit states exceeded? - A check on whether the limit states have been exceeded is performed based on dynamic analysis results.

10 200 Structures Under Shock And Impact [Safety provision according to energy criterion (13) Structural modifications based on index of impact resistance (19) [Setting of limit states (14) Is safety factor > 1.07(15) Safety factor I. Is impact failure mode Important? (16) Determination of Impact failure mode (17)] Possibility of punching shear failure? (18) Figure 6: Flow of safety provision according to energy criterion (23)Structural modifications based on index of impact resistance - Modifications such as amount of steel reinforcement, structural dimensions, material characteristics, etc. are performed based on evaluations of impact resistance properties. 4. Case Study of Design of Reinforced Concrte Handrail As a case study of the dynamic design procedure proposed, the design of reinforced concrete handrail (guardrail) under vehicular impact is discussed. The numbers in brackets indicated below are in relation to the numbers in the flow of the design procedure indicated in Figures 3, 6, and 7. The case study is as follows: (l)outline of type of structure - Concrete handrail. (2)ls design impact load specified? - No. (3)Preliminary study & survey of possible impact collisions - Vehicular collisions are considered.

11 Structures Under Shock And Impact 201 (4)Determination of design impact collision - The design impact collision is selected as in Figure 8. It simulates a medium sized car colliding at a relatively low speed. For convenience, the angle of collision is set at 90 degrees (perpendicular to handrail). (5)1 s safety factor required? - Yes. (T)Selection of type of structure - Reinforced concrete, fixed cantilever at bottom of handrail. (S)Results of design for static loads - A normal concrete handrail is selected. (9)Setting of structural dimensions - The dimensions and model according to the layered finite element procedure are shown in Figures 9(a) and 9(b), respectively. (lo)dynamic structural analysis - The linked procedure in Reference 7 is applied. The main results are given in Table 2, Figures 10, 11 and 12. The results for this case is referred to as "Case 1" in the Table and Figures. (ll)is energy criterion satisfied? - Failure by concrete crushing at rear face is indicated in the analysis (refer Table 2). Proceed to Step (9). (9')Setting of structural dimensions - The amount of steel reinforcement in the Structural modifications based on index of impact resistance (23) Safety provision according to load criterion [Design impact load] (20) Setting of limit states (21) Figure 7: Flow of safety provision according to load criterion v=7.5m/sec (27km/hr) RC handrail VVO.etf W,=0.3tf - Reinforcement ki=^=kg=2tf/cm Collision angle = 90 Figure 8 Setting of design impact condition of RC handrail

12 202 Structures Under Shock And Impact longitudinal direction of front face (tension side) is increased from 15cm pitch to 10cm pitch because failure by concrete in compression occurred at the rear face. Details of the cross sectional areas are indicated in Table 3. The analysis is now referred to as Case 2. (lo')dynamic structural analysis - The results are indicated in Table 2, Figures 10, Hand 13 as Case 2... (11 ')Is energy criterion satisfied? - Failure by concrete crushing at front lace is indicated in the analysis (refer Table 2). Proceed to Step (9). (9")Setting of structural dimensions - The amount of steel reinforcement in the longitudinal direction of rear face (compression side) is increased from 15cm pitch to 10cm pitch because of concrete crushing in the front face. Details of the cross sectional areas are indicated in Table 3. The analysis is now referred to as Case (Detail of handrail) r- 10 X (unit : mm) x." : _.,_^-- (a) Details of RC handrail Y A C.L. Concrete layer A i c C ; : / j CX i i k Ar- ; o T C% : XX, II / 0 ^ ; O N O ; % j O X T \ ' _ ' ^/ '; 's@'l 00=800 4(a) 150=600 ^3(0)200=600 4 in) \ ^-D 2000 Reinforceme ntlcayer C.L (Plan) (Unit: mm) (Section D-D) (b) Layered finite element meshes for RC handrail(l/2 portion) Figure 9: Dimensions and model of RC handrail

13 Structures Under Shock And Impact 203 (10")Dynamic structural analysis - The results are indicated in Table 2, Figures 10, 11, 14 and 15 as Case 3. (1 OIs energy criterion satisfied? - No failure in structure. Proceed to Step (13). (13)Safety provision according to energy criterion - Proceed to Step (14). (14)Setting of limit states - Structural failure by bending mode to prevent concrete scabbing. (15)Is safety factor > Safety factor is calculated. The ultimate absorption energy is calculated by the finite element method and the results are indicated in Table 2 as " Ultimate". The safety factor is calculated based on results in Table 2 as; Table 2: Main results of dynamic analysis for design of RC handrail Case Ultimate Loading rate (tf/msec) Load at failure (tf) (30.86)* Deflection at failure (mm) (0.50)** 1.41 Cracking load (tf) Concrete plasticity load (tf)*** Reinforcement yielding load (tf) Case Ultimate Index of local deformation (x lo^/cm^) Impulse (kgf-sec) Total energy (kgf-cm) **** 5540 Failure mode***** PS PS (B)****** B * Maximum impact load ** Maximum deflection *** Plasticity in compression ****Total energy at maximum deflection ***** B: Bending, PS: Punching shear ***** Main deformation mode Failure condition Concrete crushing at rear face Concrete crushing at front face Concrete crushing at front face (Center of handrail) (Side end of handrail) Figure 10: Distribution of transverse deflection at failure(case 1, 2, 3)

14 204 Structures Under Shock And Impact (Top of handrail) (Bottom of handrail) * 1 ' ' ' Deflection (mm) Figure 11: Distribution of longitudinal deflection at failure(case 1, 2, 3) 30- S 20 o OJ a Deflection (mm) Figure 12: Impact force - central deflection relation for Case 1

15 Structures Under Shock And Impact Deflection (mm) Figure 13: Impact force - central deflection relation lor Case 2 Table 3: Structural modifications of RC handrail Tension steel (2nd layer in Figure 9) Longitudinal Transverse direction direction \ A,(cm2) A,(cm2) Case 1 Case 2 Case (13@D13) (20@D13) (20@D13) (80D13) (8@D13) (8@D13) Compression steel (7th layer in Figure 9) Longitudinal Transverse direction direction A,(cm2) A, (cm 2) (13OD13) (13@D13) (20OD13) (8@D13) (8GD13) (8@D13) Safety factor, y. = Efu ^Ultitimate E3 X I Ultimate 5540 x x224.0 = 2.99(> 1.0) Proceed to Step (16). (16)ls impact failure mode important? - Yes. (17)Determination of impact failure mode - Bending failure mode is indicated in

16 206 Structures Under Shock And Impact the results in Table 2. Proceed to Step (18). (IS)Possibility of punching shear failure - No. The design of the concrete handrail ends at this stage with the dimensions used in "Case 3" being selected, with a safety factor of c 0.8- L &0.6-L o o i=1075mm \ _h=775mm.. ^~ -^ h=675mrri" h: height from bottom of handrail (Center of handrail) (Side end of handrail) Figure 14: Distribution of transverse deflection at failure(case 3) CT3 Q Deflection (mm) Figure 15: Impact force - central deflection relation for Case 3 0.6

17 5. Conclusions Structures Under Shock Ami Impact 207 In this paper, a dynamic design procedure for concrete structures subjected to impact loads is proposed, based on the energy criterion and the load criterion. And also, a case study for designing a reinforced concrete handrail under impact collision from a light vehicle is illustrated. The main results of this paper can be summarized as follows; (l)limit states for different types of structures under different impact loading conditions are necessary. Designing a structure according to the ultimate limit states would produce a more rational design result. When designing concrete structures under impact loading, the allowable failure modes should also be specified. (2)A dynamic design procedure that is capable of considering the impact failure modes and also the safety factor is proposed based on an energy criterion. Furthermore, to allow application of specified design impact loads in design, a load criterion is also proposed. For safety provision by the energy criterion, the safety factor is taken as a ratio between the ultimate absorption energy and the energy absorbed by the structure under the specified design impact conditions. The design procedure would allow a rational and effective design of reinforced concrete structures to be performed. (3)A case study of design of reinforced concrete handrail for vehicle impact conditions is performed to point out the applicability of the proposed design procedure. The results indicate that the proposed design procedure can be practically applied on full scale structures. References 1. Miyamoto, A., Ishibashi, T. & Mito, M. Non-Linear Dynamic Analysis and Design Concepts for RC Beams under Impulsive Loads, JSCE Journal of Structural Engineering, 1994, 40A, Koyanagi, W., Rokugo, T. & Horiguchi, H. Failure Condition of Steel Fiber Reinforced Concrete Beam Element under Repeated Impact Loading, Transactions of the Japan Cement Association, 1984, 38, Kamil, H., Krutzik, N., Kost, G. & Sharpe, R. An Overview of Major Aspects of the Aircraft Impact Problem, Nuclear Engineering and Design, 1978, 46, King, M. W., Miyamoto, A. & Nishimura, A. Failure Criteria and Analysis of Failure Modes for Concrete Slabs Under Impulsive Loads, Memoirs of the Graduate School of Science and Technology, Kobe University, 1991, 9-A, Miyamoto, A., King, M. W. & Fujii, M. Analysis of Failure Modes for Reinforced Concrete Slabs Under Impulsive Loads, Journal of the American Concrete Institute, 1991, 88, 5, Miyamoto, A. & King, M. W. Concrete Structures under Soft Impact Loads, Chapter 5, Shock and Impact on Structures, eds Brebbia, C. A. & Sanchez- Galves, V., pp , Computational Mechanics Publications, Southampton, UK & Boston, USA, Miyamoto, A., King, M. W. & Fujii, M. Integrated Analytical Procedure for Concrete Slabs under Impact Loads, ASCE Journal of Structural Engineering, 1994, 120,6,