DAMAGE ASSESSMENT OF STRUCTURES AGAINST BLAST LOAD BASED ON MODE APPROXIMATION METHOD

Transcription

1 DAMAGE ASSESSMENT OF STRUCTURES AGIANST BLAST LOAD BASED ON MODE APPROXIMATION METHOD HUANG XIN 11 DAMAGE ASSESSMENT OF STRUCTURES AGAINST BLAST LOAD BASED ON MODE APPROXIMATION METHOD HUANG XIN SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING 11

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3 DAMAGE ASSESSMENT OF STRUCTURES AGAINST BLAST LOAD BASED ON MODE APPROXIMATION METHOD HUANG XIN SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING NANYANG TECHNOLOGICAL UNIVERSITY 11

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5 Daage Assessent of Structures against Blast Load Based on Mode Approxiation Method Huang Xin School of Civil & Environental Engineering A thesis subitted to Nanyang Technological University in fulfillent of the requireent for the degree of Doctor of Philosophy 11

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7 ACKNOWLEDGEMENT Foreost, I would like to express y sincere gratitude to y supervisor Dr. Ma Guowei for the continuous support of y Ph.D study and research, for his patience, otivation, enthusias, and iense knowledge. His guidance helped e in all the tie of research and writing of this thesis. I could not have iagined having a better supervisor and entor for y Ph.D study. Besides y supervisor, I would like to thank professors who bring e to this research field and give e guidance. Dr. Lu Yong, Dr. Zhao Zhiye, and Dr. Yang Yaowen, thank you for your strictness during class and patience when I asked questions. I also would like to show y gratitude to Dr. Hao Hong, Dr. Li Qinging, and Dr. Wu Chengqing, I could not finish y research work without the ideas and help fro you. I a very grateful to Dr. Yu Jianxing and Dr. Tan Soon Keat, referees of y Ph.D application. I will never forget that you gave e this opportunity studying abroad. I a thankful to Dr. Gho Wie Min as well for the project experience he gave e. I believe such experience will be helpful in the future. Thanks are also due to Bao Huirong, He Lei, and other classates and staff fro the office I have stayed and the group we used to work together. I would like to thank technical staff in the P.E. lab, Construction lab, and CADD lab in NTU who support y research and work. The financial support fro Nanyang Technological University and DSTA in the for of research scholarship is gratefully acknowledged. Lastly, and ost iportantly, I wish to thank y parents, Huang Weiguo and Chang Xiaoyun. They always support e and love e. Special thanks are given to y wife, Han Linlin, for her love and everything she does for e. I

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9 ABSTRACT Daage assessent plays an iportant role in the evaluation of the stability and strength of structures, which is significant for both the existing structures and those under construction. An effective daage assessent ethod based on the deforation behavior of a structure is essential in order to apply protective easures when there exists potential blast load risks. For a reinforced concrete structural eleent, the analysis becoes ore coplicated because the reinforced concrete always defors in a nonlinear way, especially in the post-failure stage. Although it is straightforward to use the single-degree-of-freedo (SDOF) approach to derive the structural response, the SDOF odel usually oversiplifies the structural deforation due to the liitation of its atheatical for and ignores the influence of the shear deforation in bending failure. In the present study, the ode approxiation ethod (MAM) is adopted to generate pressure-ipulse (P-I) equations and diagras. According to this ethod, the shear and bending responses can be considered siultaneously, and the cobined failure odes for structural eleents are included. Daage assessent for underground structures is also studied. The soil-structure interactions (SSI) due to external or internal explosion are siplified as daping or stiffness effect and theoretical solutions are derived based on the MAM. To obtain a ore accurate theoretical solution and consider the coplexities of the aterial strength and the SSI effect, the MAM has been subsequently extended to use a generalized integration procedure on scenarios of both surface and underground structures. Therefore, the pulse-shape effect and the nonlinearities of aterial strength and the coplicated SSI effect could be considered. Nuerical siulation of a reinforced concrete (RC) wall is carried out to verify the analytical study and extend the analysis into three diensional probles. The blast loads used in the siulation are calibrated by TM5-13 and soe available experiental data. The siulation results are copared with the results obtained fro theoretical solution. III

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11 TABLE OF CONTENTS ACKNOWLEDGEMENT... I ABSTRACT... III TABLE OF CONTENTS... V LIST OF TABLES... IX LIST OF FIGURES... X LIST OF SYMBOLS... XIII CHAPTER 1 INTRODUCTION Background Research objectives Thesis organization Originality and contributions... 5 CHAPTER LITERATURE REVIEW P-I diagra ethod P-I diagras based on SDOF syste P-I diagras based on ode approxiation ethod P-I diagras for bea eleent P-I diagras for plate eleent Design paraeters of RC eleent against blast load Material properties of reinforced concrete Design of RC structural eleent Suary... 6 CHAPTER 3 DAMAGE ASSESSMENT FOR UNDERGROUND STRUCTURES AGAINST EXTERNAL BLAST LOAD Introduction Siplification of soil-structure interaction Failure criteria and structural failure assessent P-I diagras and discussions V

12 Boundaries between different failure odes Daping effect due to soil-structural interaction Verification of continuity Pressure and ipulse effects Further discussions Case study Conclusions CHAPTER 4 DAMAGE ASSESSMENT FOR UNDERGROUND STRUCTURES AGAINST INTERNAL BLAST LOAD Introduction Siplification of soil effect Failure criteria Failure odes and response analysis P-I diagras and discussions Differentiation of failure odes Soil-structure interaction effect Verification of continuity Case study Concluding rearks CHAPTER 5 PULSE SHAPE EFFECT ON STRUCTURAL DAMAGE INDUCED BY BLAST LOAD Introduction General assuptions and failure criteria Integration procedure Derivation of P-I equations Mode 1 (Shear failure ode) Mode (Bending failure ode) Mode 3 (Cobined failure ode) Mode 4 (Bending failure ode with a plastic zone) Mode 5 (Cobined failure ode with a plastic zone) Result discussions VI

13 5.6. Conclusions CHAPTER 6 NON-CONSTANT SOIL-STRUCTURE INTERACTION EFFECT ON UNDERGROUND STRUCTURE DAMAGE TO BLAST LOAD Introduction Assuptions and failure criteria Derivation of P-I equations Underground structure against external blast load Underground structure against internal blast load P-I diagras and result discussion Underground structure (external blast load) Underground structure (internal blast load) Concluding rearks CHAPTER 7 NONLINEAR STRUCTURAL DEFORMATION EFFECT ON THE BLAST INDUCED STRUCTURAL DAMAGE Introduction Phase division Eleent resistance and failure criteria Failure odes and P-I equations Mode 1 shear failure ode Mode bending failure ode Mode 3 cobined failure ode P-I diagras and discussions Differentiation of failure odes Effect of resistance-deforation relationship Paraetric study Applications in underground scenarios Conclusions CHAPTER 8 VERIFICATION OF MAM BASED P-I DIAGRAMS BY NUMERICAL SIMULATIONS Siulation of blast load Siulation of surface RC wall against blast load VII

14 8..1. Geoetry odel Material properties Erosion criterion Boundary conditions Results and discussion Siulation of underground RC wall against external blast load Siulation of underground RC wall against internal blast load Concluding rearks CHAPTER 9 CONCLUSIONS AND RECOMMENDATIONS Conclusion and discussions on the present work Daage assessent for underground structures against blast load Generalized integration procedure as extension of ode approxiation ethod Validation by nuerical siulation Recoendations on the future work REFERENCE LIST OF PUBLICATIONS... 5 APPENDIX... 7 VIII

15 LIST OF TABLES Table - 1 Three regions of the P-I diagra... 8 Table - Coparison of siplified equations of P-I diagra Table - 3 DIF for design of RC eleents... 4 Table 3-1 Different daage level under epirical bending and shear failure criteria Table 3 - Velocity profile Table 3-3 Continuity verification Table 3-4 Case study Table 4-1 Continuity verification Table 4 - Case study... 8 Table 5-1 Different daage level under epirical bending and shear failure criteria Table 7-1 Velocity profile Table 7 - Loads and paraeters Table 8-1 Different charge weights IX

16 LIST OF FIGURES Fig. - 1 Idealized P-I diagra... 9 Fig. - Siplified blast load types... 9 Fig. - 3 Difference of loading shapes... 1 Fig. - 4 Claped bea odel Fig. - 5 Velocity profile for claped bea odel Fig. - 6 Final central displaceent paraeter versus b/l Fig. - 7 Differences aong pulse shapes and shear effect Fig. - 8 Difference between SDOF odel and rigid-plastic bea odel... Fig. - 9 Difference between boundary conditions... Fig. - 1 Variations of the diensionless transverse shear force and the diensionless radial bending oent with ξ during the transverse shear sliding phase: C: fully claped circular pate; S: siply supported circular plate... 1 Fig. 3-1 Underground structure and bea odel... 3 Fig. 3 - Square yield surface... 3 Fig. 3-3 Distribution of failure odes Fig. 3-4 Typical failure odes for different value of υ Fig. 3-5 Shear and bending failures in different soils Fig. 3-6 Coparison with blast result of surface structure... 5 Fig. 3-7 Pressure and ipulse effects... 5 Fig. 3-8 Case study Fig. 4-1 Soil-structure interaction in internal blast load scenario Fig. 4 - Idealized resistance-deflection curve for large deflections... 6 Fig. 4-3 Typical failure odes Fig. 4-4 Failures in different soils Fig. 4-5 Coparison with surface structures against blast load Fig. 4-6 Case study X

17 Fig. 5-1 Flow chart for surface structural eleent against blast load Fig. 5 - Pulse shape effect on shear failure for surface structure against blast load (fixed daage level) Fig. 5-3 Pulse shape effect on bending failure for surface structure against blast load (fixed daage level) Fig. 5-4 Pulse shape effect on surface structure against blast (fixed ipulse)... 1 Fig. 5-5 Error in the final tie of structural response... 1 Fig. 5-6 Error in the final structural displaceent Fig. 6-1 Flow chart for underground structure against external blast load Fig. 6 - Flow chart for underground structure against internal blast load Fig. 6-3 Pulse shape difference of shear failure for underground structure against external blast load Fig. 6-4 Pulse shape difference of bending failure for underground structure against external blast load... 1 Fig. 6-5 Difference of constant and non-constant daping effect... 1 Fig. 6-6 Pulse shape difference of shear failure for underground structure against internal blast load Fig. 6-7 Pulse shape difference of shear failure for underground structure against internal blast load Fig. 6-8 Difference of constant and non-constant soil stiffness effect Fig. 7-1 R-D relationships for reinforced concrete structural eleent: (a) Rigid-plastic odel; (b) Elastic-plastic hardening odel; (c) elastic-plastic soften odel; (d) More realistic RC structural deforation odel Fig. 7 - Idealized resistance-deforation relationship for RC eleents Fig. 7-3 Siply supported bea odel Fig. 7-4 P-I diagras of failure odes Fig. 7-5 Noralized ending ties and final responses Fig. 7-6 Model difference in daage levels Fig. 7-7 Paraetric study of K and K Fig. 7-8 P-I diagras of underground RC structure against external blast load Fig. 7-9 P-I diagras of underground RC structure against internal blast load XI

18 Fig D wedge odel Fig. 8 - Pressure contour of explosion Fig. 8-3 Section view of RC wall Fig. 8-4 Geoetry odel of RC wall Fig. 8-5 Vector contour of blast load (with RC structure odel) Fig. 8-6 Boundary conditions of RC wall Fig. 8-7 Boundary conditions of air (half section at X-plane) Fig. 8-8 Velocity and displaceent for case RC1 (light bending daage) Fig. 8-9 Velocity and displaceent for case RC (oderate bending daage) Fig. 8-1 Velocity and displaceent for case RC3 (severe bending daage) Fig Result coparison for surface blast cases Fig. 8-1 Velocity and displaceent for case RC-C1 (light bending daage) Fig Velocity and displaceent for case RC-C (oderate bending daage) Fig Velocity and displaceent for case RC-C3 (severe bending daage) Fig Result coparison for underground external blast cases Fig Soil layer in RC-K cases Fig Velocity and displaceent for case RC-K1 (light bending daage) Fig Velocity and displaceent for case RC-K (oderate bending daage) Fig Velocity and displaceent for case RC-K3 (severe bending daage) Fig. 8 - Result coparison for underground internal blast cases Fig. A - 1 Transverse velocity profiles... 8 XII

19 LIST OF SYMBOLS t,, c iterative tie step constant in stress wave attenuation equation ratio of centerline deflection to half span average shear strain aterial paraeter support rotation half length of plastic zone, velocity of the oving plastic hinge, half length of plastic zone as a function of tie acoustic ipedance of soil, free-field and interface pressure respectively f i a b c dd, ' f diensionless strength ratio depth of equivalent rectangular stress block width of copression face P- wave velocity distance fro extree copression fiber to centroid of tension/copression reinforceent respectively coupling factor of explosive energy f ' dc dynaic ultiate copressive strength of concrete f ds dynaic design stress for reinforceent h, e depth of eleent ass per unit length, effective unit ass n p p p b attenuation coefficient function of blast pressure constant pressure in blast load duration strength ratio which produces balanced conditions at ultiate strength XIII

20 t 1 t d final tie of plastic hinges oving blast load duration t ean ean tie of the pulse t, t end tie of elastic bending and shear stage respectively e se t f, ts, t final tie of total and shear deforation, the tie when aterial y uv, x yy, begins to yield respectively structural velocity and soil velocity respectively abscissa on the eleent velocity and acceleration of unit ass respectively y, ye, y, y axiu displaceent, elastic displaceent, velocity and acceleration of bending deforation respectively ys, yse, ys, y s axiu displaceent, elastic displaceent, velocity and A, A ' C s s acceleration of shear deforation respectively area of tension and copression reinforceent within the width b respectively equivalent daping coefficient * * * D, M, Q noralized displaceent, noralized bending resistance, and I noralized shear resistance respectively ipulse per unit area * * I, P diensionless ipulse and pressure respectively K equivalent daping coefficient K1, K, K diensionless rates of slope in ulti-linear deforation odel 3 L MQ, respectively half length of eleent transverse bending oent and shear force respectively M, Q M y, Q y M, Q e e bending and shear strength respectively bending and shear resistance in ulti-linear R-D relationship axiu elastic bending and shear resistance respectively XIV

21 M u P, P, P, P e * R i SB, T T n V d ultiate unit resisting oent effective pressure, reflected pressure, peak pressure, external pressure respectively noralized general resistance unified fors for shear and bending failure criterion respectively general tie variable teporary variable of tie, n=1,,3,4,5,6,7 ultiate direct shear force XV

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23 CHAPTER 1 INTRODUCTION 1.1. Background Daage assessent plays an iportant role in the evaluation of the stability and strength of structures, which is significant for both the existing structures and those under construction. In an extree condition, such as a blast load scenario that is norally not considered in the original structural design of civilian structures, structural eleents ay experience daage in different degrees. An effective daage assessent ethod based on the deforation behavior of a structure is essential in order to apply protective easures when there are potential blast load risks. For a reinforced concrete (RC) structural eleent, the analysis becoes ore coplicated because the reinforced concrete always defors in a nonlinear way, especially in the post-failure stage. Early in the World War II, the pressure-ipulse (P-I) diagra ethod was first used to assess the daage of structural eleents and buildings against blast load. For a certain blast load, the P-I diagra indicates the safety status of the structural eleent. Researchers used to siplify a structural eleent by adopting an equivalent single-degree-of-freedo (SDOF) odel or a rigid-plastic bea odel to analyze the structural response against blast load, and ost of the structural blast designs are based on the SDOF approach. Based on the SDOF odel, the shear deforation effect was either neglected in the analysis or considered separately by another SDOF odel. For a ixed failure ode, the SDOF odel becoes invalid. Although it is straightforward to use the SDOF approach to derive the structural response, the SDOF odel has its liitations. It usually oversiplifies the structural deforation due to the liitation of its atheatical for and ignores the influence of the shear deforation in bending failure and vice versa, which ay yield unreasonable estiation of the structural failure. The localized transverse shear, which is known as a shear hinge, is an iportant feature of structural 1

24 deforation when a structural eleent is under blast load (Jones 1989, Jones 1997). Through experiental and analytical studies (Yu and Jones 1991, Krauthaer 1998), it has been realized that, when the span-to-height ratio of a structural eleent is relatively sall or when a detonation is at a close-in distance fro the structure, the shear failure likely occurs. If the duration of blast load is very short, the shear failure becoes doinant and cannot be ignored. Although both the shear and bending response of a structural eleent have been analyzed by an SDOF odel (Krauthaer et al. 1986), the two failure echaniss have to be considered separately due to the SDOF siplifications. Soeties shear failure is the ain cause of successive collapse of a structure. Bending failure often occurs near the iddle of a structural eleent which is siply supported or fixed at both ends, while shear failure occurs close to the supports where the shear force is always to be the axiu under a blast load. The classical ode approxiation ethod (MAM) for a rigid-plastic structural eleent has also been widely used for decades (Martin and Syonds 1966, Jones and Song 1986, Li and Jones 1999). The estiated deflections caused by the MAM in ost cases agree well with the final deflections observed in tests (Syonds and Chon 1979). Lellep and Torn (5) developed a ethod for the investigation of rigid-plastic beas subjected to ipulsive load. Theoretical solutions are derived for beas ade of perfectly plastic aterial obeying the square yield condition. According to this ethod, the shear and bending responses can be considered siultaneously, and the cobined failure odes for structural eleents are involved. Ma et al. (7) plotted the P-I diagras for bea structures based on the rigid-plastic structural odel. The MAM has the advantages in characterizing the cobined failure odes. The P-I diagras derived fro such odel agreed reasonably well with those fro an elastic-perfectly-plastic SDOF odel, especially for a severe daage case. In the classical MAM, the rigid-plastic odel is not effective for analysis of RC structures. It is because in practical structural engineering RC structures defor in a nonlinear way and the softening behavior significantly affects the blast resistance of a structural eleent. In ost of the previous analytical derivations, the structural

25 deforation was siplified as linear (rigid-plastic) or bi-linear (elastic, perfectly plastic or plastic hardening) odels. For ore accurate analysis of RC structures, the softening behavior cannot be neglected even in the light and oderate daage cases. Besides, nearly all of the existing research works were for surface structures. Seldo studies have been done on the daage analysis of underground structures, and the nonlinearity of the aterial and the non-constancy of soil-structure interaction (SSI) were rarely considered. With the developent of underground technology, a lot of underground structures are constructed or under construction, including tunnels, underground storage, underground protective structures, etc. Different fro design of surface structures, there is no coonly used standard or a reliable design reference available for underground structures. Many underground structures were designed and constructed based on code for surface structures. The propagation of blast wave in soil is different fro that in air, and the response of underground structures against blast loads with SSI. The present study ais to find out a convenient and unified way to carry out daage assessent of different structures. The MAM is further developed to generate P-I diagras for different surface and underground structures. 1.. Research objectives The priary objective of the present research is to extend the conventional MAM to the daage assessent of structural eleent in various blast loading and structural/aterial conditions. Starting fro the conventional MAM for inelastic structural response analysis, ore paraeters will be considered when deriving the P-I equations for surface or underground structures against blast load, for exaple, the SSI effect in structural response, the pulse shape effect on final displaceent of structural eleent, the nonlinearities of aterial strength, the effect of elastic deforation, and the effect of siplified SSI. The specific targets are outlined as follows: 3

26 to generate P-I diagras based on the MAM and rigid-plastic aterial odel for daage assessent of underground structures against blast load. In the external and internal blast load scenarios, the SSI will be siplified as daping and spring effect respectively. to work out a generalized integration ethod based on the MAM so that different pulse shapes of the blast load can be analyzed. to extend the generalized integration ethod to be applied to daage assessent of underground structures. Besides the pulse shape effect, ore siplified SSI will also be taken into account. to further extend the analytical solution to consider the effects of the elastic phase and nonlinearities of aterial strength based on the MAM siplifications. to verify the analytical solutions by using nuerical siulations Thesis organization This thesis is organized into 9 chapters. Chapter 1 introduces the background and the objectives of this thesis. Chapter reviews the P-I diagra ethod based on SDOF syste and ode approxiation ethod. The calculation of shear and bending strengths for structural eleent in TM5-13 is reviewed as well. Chapter 3 analyzes the underground structure against external blast load with consideration of SSI. It extends the P-I analysis based on the MAM fro surface structure to underground structure. Chapter 4 studies the underground structure against internal blast load with consideration of the SSI. Chapter 5 uses a sei-analytical integration ethod in derivation of P-I equations so that the pulse shape effect on P- I diagras based on the MAM for surface structure under blast load is studied. Chapter 6 extends the research work in Chapter 5 to underground structures and studies ore coplicated SSI effect. The pulse shaped effect is analyzed as well. Chapter 7 further extends the P-I diagra ethod based on the MAM and considers the nonlinearity of aterial strength and the effect of the elastic phase. Chapter 8 perfors nuerical siulation of an RC wall which verifies the analytical study in 4

27 earlier chapters, and also extends the analysis into three diensional probles. Chapter 9 suarizes the conclusions and gives recoendations for future study Originality and contributions All the research work done in this thesis is original. The MAM is used in the daage assessent for underground structures against blast load. The SSI is siplified as constant daping and spring coefficients for external and internal explosion scenarios respectively. When the general integration ethod is adopted in the procedure of solving the governing equations, it is feasible and easy to involve non-constant paraeters in P-I equations. Therefore, pulse shape effect, nonconstant SSI and piece-wise strengths of structural eleent are considered in daage assessents of both surface and underground structures against blast load. All results are shown in the for of P-I diagras and discussions are perfored. Nuerical siulations are attached to validate those P-I results. Findings fro the present research work would help engineers to optiize their design of structures against blast load. 5

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29 CHAPTER LITERATURE REVIEW Daage assessent is a critical issue in protective structure design against blast load. In each assessent, the structural response is the ost iportant factor aong others that designers and engineers care about. Based on the SDOF syste and MAM, the structural acceleration, velocity and displaceent of a structural eleent can be calculated. Consequently by plotting the P-I diagras, the structural daage can be assessed..1. P-I diagra ethod P-I diagra was first used to assess the daage extent of structural eleents and buildings in the World War II (Sith and Hetherington 1995). Different daage extents, such as coplete deolition, severe daage, relatively inor structural daage, were quantified by a series of curves. Stanford Research Institute contributed to the developent of P-I diagra fro 1959 to 1966 (Abrahason and Lindberg, 1976). They used P-I diagra to evaluate the effects of blast on aircraft structures in 195s and 196s. Kornhauser (1954) used velocity and acceleration as the coordinates, which are equivalent to the peak load and ipulse syste, to derive the curves for different structures. Coobs and Thornhill (196) identified that the peak load is very iportant for ipulsive loads. Researchers in Lovelace Foundation (Richond et al. 1966) studied the influences of the duration and the peak load on anial injuries. Mortion (1966) coputed the critical load curves for siple rigid-plastic syste. Abrahason and Lindberg (1976) brought up the closed-for forula for rigid-plastic syste. Sith and Hetherington (1994) analyzed the huan body response to blast loads with P-I diagra and categorized the daage with critical curves. Li and Meng (a) used the least-square ethod to constitute universal forula to fit the critical P-I diagra curves. 7

30 It has been widely aditted that the P-I diagra has three regions: an ipulsive controlled region, a peak pressure controlled region, and a peak pressure and ipulse cobination controlled region (Abrahason and Lindberg 1976, Sith and Hetherington 1994, Li and Meng a). The division of the three regions is shown in Table -1. Table - 1 Three regions of the P-I diagra Abrahason, 1971 Sith, 1994 Ipulse controlled td.5t.4 t d Peak pressure and ipulse cobination.5t td 1.5T.4 t d 4 controlled Peak pressure controlled td 1.5T 4 t d where: T is the natural period, ω is the natural vibration frequency and t d is the blast load duration. The P-I diagra categorizes into part I, II, and III as illustrated in Fig. -1 (idealized P-I diagra in (Li and Meng a)). Although there is sall difference about the specific value for the dividing, they are siilar in principle because they are all based on the coparison with the natural period of structure or structural ebers. 8

31 Pressure Fig. - 1 Idealized P-I diagra There are three representative load types as shown in Fig. -, i.e., (1) rectangular load with step rise, constant value, and step down, () triangular load with step rise and linear depreciation, (3) exponential load with step rise and exponential depreciation. For the sae structural eleent, the rectangular load gives the lower bound of the threshold curve, while the exponential load results in the upper bound. For the triangular load, the corresponding curve is between the two boundary curves. The axiu difference between the lower bound and the upper bound could be up to 4% (Abrahason and Lindberg 1976). Li and Meng (b) also discussed the difference due to pulse shape effect (Fig. -3). Rectangular load Triangular load Exponential load Duration t dr t dt t de Fig. - Siplified blast load types 9

32 LA: rectangular load; LB: triangle load; LC: Exponential load. Fig. - 3 Difference of loading shapes The advantage of the P-I diagra is to siplify the judgent for the daage of structural ebers or structures. According to the P-I diagra, a certain load with the peak pressure and ipulse above the critical curve will result in daage of the structures, while the structure is safe if the peak pressure and ipulse cobination locates below the curve. The critical or threshold curve of the P-I diagra is plotted by the peak load and ipulse cobinations which produces the axiu displaceent that the structure can undertake. Though there is sall difference aong the shapes of the critical curves of the P-I diagra, the siilarity suggests a unifor approxiate expression for all load cases. There have been such endeavors and the results are suarized in Table -. 1

33 Table - Coparison of siplified equations of P-I diagra t I e = f P(t)dt ty ty t f, P e = I e tean P y P y eans static yield load; I e eans ideal ipulse, ty 6 5 (I e I ( I e I Py Pe Py Pe n 1 (i 1) n +.5 Pe 5Pe Denotation to sybols P : the agnitude of ideal load that produces the critical displaceent; I : the ideal ipulse that produces the critical displaceent. t ean = 1 I e (t t y )P(t)dt and tf are the ties when structural eleent yield and stops oving respectively, P(t) is the loading function of tie, Pe is the effective pressure, tean is the ean loading duration. x ax : the axiu displaceent; K: the stiffness; M: the ass. n 1, = β + β 1 d + β d d = x + y Where β, β 1, β, x, and y are coefficients got by curve-fitting., Applicable situations Rigid-plastic syste, linear spring-ass syste, all three kinds of ipulse Plastic structural ebers such as siply supported bea, siply supported circular plate and circular cylindrical shell under various ipulse Alost all the possible cases Elastic structures and dynaic loading such as descending ipulse load Forula ( P P 1) ( I I 1) = 1 ) (1 P y ) (1 4P y p = F = 1 Kx ax I x ax KM = 1 ) = 1 for ) = 1 for Abraha son (1971) Zhu (1986) Sith (1994) Li () 11

34 .. P-I diagras based on SDOF syste Aong the existing daage assessent ethods, the siplified Single-Degree-Of- Freedo (SDOF) syste has been widely used to analyze the structural response under blast load for a variety of structures or structural eleents. A structural eber can be siplified as a SDOF odel and described by the following equilibriu equation (Biggs, 1964): My Ky F f t (.1) where M is the equivalent luped ass, y is the structural acceleration, K is the equivalent structural stiffness, F f(t) is the equivalent applied force and y is the structural deforation. The solution of Eq. (.1) is: y F t y t y cos t sin t f sin t d M (.) where y is the initial displaceent, y is the initial velocity, and ω is the natural vibration circular frequency. Usually the initial conditions are: y t y y t y (.3) For siplification, the ipulsive load is norally assued to start at its axiu value and then descend to zero when t=t d, where t d is the duration of the blast load. The negative pressure phase is neglected. Most of the structural blast designs were based on the SDOF approach (TM5-13, 199). Although various extensions and applications of the SDOF approach have been carried out, in any cases it oversiplifies a structure or a structural eleent due to the liitation of its atheatical for. Based on the SDOF odel, the shear deforation effect was either neglected in the analysis (Li and Meng a; Capidelli and Viola 7; Fallah and Louca 7) or considered separately by 1

35 another SDOF odel (Krauthaer et al. 1986). For a ixed failure ode, the SDOF odel becoes invalid. The SDOF syste has its advantages in deriving analytically the structural response. In any cases of practical engineering, it has been used to give a preliinary assessent of structural daage induced by a blast load. However, due to its inherent liitation, the SDOF odel oversiplifies the structural eleents and neglects the influence of shear deforation, or it can only analyze the shear and bending response separately. In a blast event, structures at close-in distance ay experience localized transverse shear failure which defors as a shear hinge (Syonds 1968, Jones 1989, Jones 1997). Depending on different loading rates, intensities, and aterial properties, the localized shear deforation can be isotheral rupture (Menkes and Opat 1973) or adiabatic shear bending failure (Kalthoff and Winkler 1987, Kalthoff 199). The analytical solutions for the dynaic plastic shear/bending response of beas with general boundary conditions were studied (Li and Jones 1991) including both fullyclaped and siply-supported beas when subjected to blast loading. Based on experiental and analytical results (Yu and Jones 1991, Krauthaer 1998), it has been realized that, when the span-to-height ratio of a structural eleent is relatively sall or when a detonation is at close-in distance fro the structure, the shear failure probably occurs. If duration of blast load is sufficiently short, shear failure becoes doinant and cannot be ignored and soeties shear failure is the ain cause of successive collapse of a structure. Bending failure often occurs near the iddle of a structural eleent which is siply supported or fixed at both ends, while shear failure occurs close to the supports where the shear force is often to be the axiu under a blast load..3. P-I diagras based on ode approxiation ethod The proble of estiating peranent deforations of structures subjected to high intensity dynaic loading started to receive increasing attention in the early 195s. At that tie, above the coplexity of the proble of elastic structures subjected to transient loading, additional coplicating factors which are of iportance included the dissipation of energy in plastic work, elastic unloading fro plastic states, strength hardening, the dependence of yield stress upon rate of strain, geoetry 13

36 change, and other non-linear effects. Very few solutions of dynaic loading proble had recognized all these factors. More coonly, one or ore of the factors were assued to doinate the behavior of the structure, and all others are neglected or approxiated. The widely used approxiation was the replaceent of distributed ass of structure by one or ore point asses. Such approxiation had been used in conjunction with a variety of idealizations of the aterial behavior. In fairly siple structures, the actual distribution of ass, elastic stiffness and yield stress can be considered. Solutions for an elastic, perfect plastic aterial had been found by soe researchers, e.g. Bleich and Salvadori (1955) and Seiler et al. (1956), but were difficult if the load agnitudes were uch larger than those causing the yield stress to be reached. Martin and Syonds (1966) suggested the MAM to provide a rational ethod of constructing an SDOF approxiations for ipulsively loaded etal structures which are analyzed according to an eleentary rigid-plastic theory. The approxiation follows autoatically fro a chosen ode shape. Such ethod was based on the eleentary rigid-plastic theory which involved in two idealizations: 1) for the purpose of the dynaic loading proble under consideration, a ductile aterial is represented by a rigid perfectly plastic constitutive equation. All elastic effects are in consequence excluded. ) geoetry changes are assued to be sall, and the yield stress is assued to be independent of the rate of strain. There were two exaples to show the difference of solutions between SDOF approxiation and ode approxiation ethod. One exaple of the claped bea indicated that, the final central displaceent can be obtained as a function of b/l (width of the load along the bea over total length of the bea as shown in Fig. -4, and velocity for different phases can be referred to Fig. -5). The final central displaceent paraeter versus the ratio b/l in Fig. -6 shows the accuracy of MAM. The displaceents in the ode solution overestiate the actual displaceents in soe regions and underestiate the in others. Another exaple of the cantilever with an attached tip ass showed that, the final tip displaceent was a function of the paraeter γ, and hence the accuracy of the approxiation depended on γ. 14

37 Based on the analysis and discussion, Martin and Syonds (1966) tried to rationalize the setting up of an SDOF approxiation in eleentary rigid-plastic theory for ipulse loading. The ethod required a ode shape be chosen, thereafter the equivalent spring force, ass, and the initial ode velocity followed without further assuptions. A good approxiation solution was fro sall initial difference, i.e. sall copared to the initial energy in any other solutions. It could be concluded that the ode approxiation can be reasonably good. The work done by Martin and Syonds in 1966 showed that, by using the ode approxiation ethod, good result for bea eleent can be obtained in the analysis of ipulse load and the error of ode shape assuption is acceptable. Fig. - 4 Claped bea odel Fig. - 5 Velocity profile for claped bea odel 15

38 Fig. - 6 Final central displaceent paraeter versus b/l The classical ode approxiation ethod for rigid-plastic structural eleents analysis is already well known. Research work has been done on different constraints and load distributions (Martin and Syonds 1966; Syonds and Chon 1979; Jones and Song 1986; Liu and Jones 1988; Li and Jones 1999; Alves and Jones ; Jones and Jones ; Lellep and Torn 5; Li and Jones 5a). By coparison with the SDOF results, it shows that, although the rigid-plastic odel neglects the elastic deforation, the estiated deflections by the ode approxiation ethod in ost cases agree well with the final deflections observed in tests (Syonds and Chon 1979). The pressure-ipulse (P-I) diagras derived fro a rigid-plastic odel are very close to those fro elastic, perfectly plastic SDOF odel, especially when severe daage occurs to the structure eleent. Lellep and Torn (5) analyzed rigid-plastic bea daage subjected to an ipulsive load by assuing the perfectly plastic aterial obeying the square yield criterion. According to this ethod, shear and bending responses can be analyzed 16

39 together and thus cobined failure odes for beas can be considered. Ma et al. (7) further developed a P-I diagra ethod for daage assessent of surface structures subjected to blast load in consideration of both bending and shear daage. An explicit analytical solution for the P-I diagras were obtained which could be conveniently used for assessing structural daage P-I diagras for bea eleent The MAM was widely used for different eleent odels and aterial odels. Soe research works based on the MAM are suarized below: Nonaka (1977) analyzed the shear and bending response of a rigid-plastic bea eleent under blast-type loading. He used a siply supported unifor bea with liit oent M and liit shear Q to analyze the structural response under blasttype load. The blast-type load was siplified as pressure P(t) uniforly distributed over the whole span. Five failure odes were assued including rotation with single hinge, rotation with plastic zone, sliding, sliding and rotation with single hinge, and sliding and rotation with plastic zone. Such five failure odes indicated the shear failure, bending failure, and cobined failure odes. Results of the analysis showed that, the axiu shear agnitude was always attained at the bea ends where the bending oent was zero, while the bending oent was the axiu at the iddle portion of the bea where the shear force was zero. For a yield curve saller than the assued polygon, there ay be a possibility of reaching the yield liit under the M-Q interaction. It was iportant to state that any yield curves in the analysis of Nonaka was valid only when ( M/M o -1)( Q/Q -1)=. Soe of the iportant conclusions were generated such as that, the shear deforation reduced the bending part of the kinetic energy being absorbed in shearing, and for ipulsive loading with the ratio μ =p L/4M high enough, shearing always took place and this effect becae axiu aong blast-type loads with the sae total ipulse as shown in Fig

40 Fig. - 7 Differences aong pulse shapes and shear effect It was also concluded that, in the case of an ordinary load pulse with μ of the order greater than ten, the peranent deforation was well approxiated by treating the loading as ipulsive, which gave the largest plastic deforation for the sae total ipulse; when μ was of the order of ten or less, the finiteness of the load agnitude should be taken into account, and if μ was not uch larger than unity, the exact load-tie relation had to be considered. Jones and Song (1986) did the siilar analysis of shear and bending response of a rigid-plastic bea, but under partly distributed blast-type loading. The blast-type loading was assued as a pressure distributed along the bea syetrically about the bea center but in the region with half length of L 1. There were also five possibilities of otion which represented the shear failure ode, bending failure odes, and cobined failure odes but slight different fro work of Nonaka (1977) since the loading was different. The results showed that, dynaic response was governed by nine different patterns of initial otion, when the influence of transverse shear force was retained in the yield condition as well as bending oent. The initial velocity fields in the analysis were controlled by paraeters that related to shear strength, loading agnitude, and loading distribution. The influence of transverse shear on the axiu transverse displaceent were negligible when ν is bigger than 1.5 and the loading distribution η was in the range of.5 and 1, but becae ore iportant as η decreased. It was also confired that based on other researcher s work, rotational inertia effects turned out to be 18

41 uniportant for ost practical probles and ignoring such effect only cause less than 1% of the error in the axiu transverse displaceent. The pulse shape effect was also briefly discussed that the triangular and exponential pressure pulse loadings produced less transverse shear sliding and saller bending deforations than a rectangular pressure pulse loading did. Youngdahl (1997) used the MAM for strain-hardening cantilever beas where it was hypothesized that deflection shape satisfies soothness and continuity requireents. Arbitrary load distributions and pulse shapes were considered and the differential equations were solved nuerically. Li and Jones (1999) discussed the shear and adiabatic shear failures in an ipulsively loaded fully claped bea. The results showed that, with an increasing ipulsive pressure load there was a direct transition between a transverse shear failure ode and an adiabatic shear failure ode. Yu and Chen () did a further study of plastic shear failure of ipulsively loaded claped beas. The interaction between the shear force and bending oent, and the weakening of the sliding sections during the failing process were ephatically considered. Lellep and Torn (5) used the MAM on different boundary condition, one side claped and another side siply supported. The results differed fro those obtained fro beas with both ends siply supported or claped, and calculations showed that the shear sliding was ore essential for shorter beas. Ma et al. (7) used the MAM on P-I diagra ethod for cobined failure odes of rigid-plastic beas, thus the daage assessent of real structures can be carried out straightforwardly. In the paper of Ma et al. (7) the results were copared with SDOF approach to show the difference of those two odels in Fig. -8. The difference between boundary conditions (siply supported and fully claped) was discussed that, if the paraeter ν is replaced by ν, the P I equations for the fully claped bea are exactly the sae as that for the siply supported bea as shown in Fig. -9. The works done in Chapter 3 to 7 of this thesis are based on the basic idea of Ma et al. (7), and it is extended to underground structures and a generalized integration procedure is proposed. Derivation of P-I equations for surface structure against blast load (Ma et al. 7) is shown in Section Appendix. 19

42 Fig. - 8 Difference between SDOF odel and rigid-plastic bea odel Fig. - 9 Difference between boundary conditions

43 .3.. P-I diagras for plate eleent Early in 196s, Jones (1968) studied the finite deflections of a siply supported rigid-plastic annular plate under a rectangular pressure pulse. It was founded that the deforation predicted by his analysis was saller than those given by the bending only theory even for axiu deflections of the order of the plate thickness. Then Shen and Jones (1993) used the MAM to analyze the dynaic response and failure of fully claped circular plates under a rectangular pressure pulse. Good agreeent was obtained between the theoretical predictions for the peranent transverse deflections and the corresponding experiental results of Bodner and Syonds (1979), and Teeling-Sith and Nurick (1991) recorded on strain rate sensitive plates. Li and Jones published their results in 1994 of fully claped circular plates with transverse shear effects under blast loading. Siilar to the analysis of bea eleent, the five failure odes were studied and P-I curves were generated to discuss the effect of strength ratio ν. It was also realized that the relationships existed between the theoretical solutions for siply supported and fully claped circular plates, which were siilar to the observations ade for beas in Li and Jones (1994) as shown in Fig. -1. Fig. - 1 Variations of the diensionless transverse shear force and the diensionless radial bending oent with ξ during the transverse shear sliding phase: C: fully claped circular pate; S: siply supported circular plate 1

44 Chen and Li (3) presented a shear plugging odel to predict the ballistic liit and residual velocity of a etallic circular plate ipacted by a blunt rigid projectile. Besides shear effect, plate bending, ebrane stretching and local indentation/penetration were included in the odel. Siple forulae and good predictions were given. Chen et al. (5) extended their research works and the critical condition for the initiation of an adiabatic shear failure in the plate was deterined according to the axiu shear stress criterion. The plastic response of a circular cylindrical shell to dynaic loadings was studied by Lellep and Torn (4) that, siilar results to the bea odel (Lellep and Torn, 5) were obtained. It was interesting to note that, the solutions differed fro those corresponding to shells with siply supported or claped ends respectively. It was noted by Lellep and Torn (4) that in the case of tubes with identical supports at both ends the solution (except for the longitudinal bending oent) was not sensitive to the support conditions. That was siilar for beas and circular plates..4. Design paraeters of RC eleent against blast load The aterial properties of RC and design procedure of RC structural eleent against blast load are reviewed as follows. The calculated bending and shear capacities of structural eleent will be used in the analytical studies in following chapters Material properties of reinforced concrete Concrete is a ixture of ceent paste and aggregate, each of which has an essentially linear and brittle stress-strain relationship in copression. Brittle aterials tend to develop tensile fractures perpendicular to the direction of the largest tensile strain. Thus, when concrete is subjected to uniaxial copressive loading, cracks tend to develop parallel to the axiu copressive stress. Although concrete is ade up of essentially elastic, brittle aterials, its stress-strain curve is nonlinear and appears to be soewhat ductile (MacGregor and Wight, 5).

45 Generally, the concrete strength is taken to refer to the uniaxial copressive strength as easured by a copression test of a standard test cylinder, because this test is used to onitor the concrete strength for quality control of acceptance purposes. Factors affecting concrete copressive strength include water/ceent ratio, type of ceent, age of concrete, rate of loading, and so on. The tensile strength of concrete falls between 8 and 15 percent of the copressive strength, however the actual value is strongly affected by the type of test carried out to deterine the tensile strength, the type of aggregate, the copressive strength of the concrete, and the presence of a copressive stress transverse to the tensile stress (Raphael 1984, McNeely and Lash 1963). Because concrete is weak in tension, it is reinforced with steel bars or wires that resist the tensile stresses. The ost coon types of reinforceent for nonprestressed ebers are hot-rolled defored bars and wire fabric. When a structural eleent is subjected to blast load, it exhibits a higher strength than a siilar eleent subjected to a static loading. This increase in strength for both the concrete and reinforceent is attributed to the rapid rates of strain that occur in dynaically loaded ebers. The higher the strain rate, the higher the copressive strength of concrete and the higher the yield and ultiate strength of the reinforceent. Therefore in the design of protective RC structure, the Dynaic Increase Factor (DIF) should be considered for aterial strength. The design values of DIF presented in Table -3 vary not only for the design ranges and type of aterial but also with the state of stress (bending, diagonal tension, direct shear, bond, and copression) in the aterial (TM5-13, 199). 3

46 Table - 3 DIF for design of RC eleents Type of stress Far design range Close-in design range Reinforcing bars Concrete Reinforcing bars Concrete f dy /f y f du /f u f' dc /f' c f dy /f y f du /f u f' dc /f' c Bending Diagonal tension Direct shear Bond Copression Design of RC structural eleent The bending and shear capacities of RC structural eleent would follow the following calculation. Considering the possible structural daage involved in the present study, only ultiate oent capacity and direct shear capacity in TM5-13 are shown Ultiate oent capacity The ultiate unit resisting oent M u of a rectangular section of width b with tension reinforceent is given by: in which: / / Mu As fds b d a (.4) a A f /.85 bf ' (.5) s ds dc where A s is the area of tension reinforceent within the width b, f ds is the dynaic design stress for reinforceent, d is the distance fro extree copression fiber to the centroid of tension reinforceent, a is the depth of equivalent rectangular stress block, b is the width of copression face, f' dc is the dynaic ultiate copressive strength of concrete. The reinforceent ratio p is defined as: 4

47 p A / bd (.6) To prevent sudden copression failures, the reinforceent ratio p ust not exceed.75 of the ratio p b which produces balanced conditions at ultiate strength and is given by: s.85 ' / 87 / 87 pb K1 f dc fds fds (.7) where K 1 equals to.85 for f' dc up to 7.58 MPa and is reduced.5 for each MPa in excess of 7.58 MPa. For a rectangular section of width b with copression reinforceent, the ultiate unit resisting oent is: in which: ' / / ' / ' Mu As A s fds b d a A s fds b d d (.8) ' /.85 ' a A A f bf (.9) s s ds dc where A' s is the area of copression reinforceent within the width b, d' is the distance fro extree copression fiber to the centroid of copression reinforceent, a is the depth of equivalent rectangular stress block. The reinforceent ratio p' is: p' A' / bd ' (.1).4... Direct shear capacity Direct shear failure of a eber is characterized by the rapid propagation of a vertical crack through the depth of the eber. This crack is usually located at the supports where the axiu shear stresses occur. Failure of this type is possible even in ebers reinforced for diagonal tension. If the support rotation θ is greater than, or if a section with any support rotation is in net tension, then the ultiate direct shear capacity of the concrete V d is zero and diagonal bars are required to take all direct shear. s 5

48 If the design support rotation θ is less than or equal to, or if the section with any rotation θ is siply supported (total oent capacity of adjoining eleents at the support ust be significantly less than the oent capacity of the section being checked for direct shear), then the ultiate direct shear force V d that can be resisted by the concrete in a slab is: V.16 f ' bd (.11) d The above design procedure is based on the SDOF syste analysis. Diensions of concrete structural eleent and reinforceent can be calculated effectively, but it is insufficient to carry out daage assessent of RC structural eleent since only one failure ode (shear or bending) can be considered at one tie and such analysis cannot be used to the cobined failure ode. Therefore, the ode approxiation ethod is highly recoended to be used in daage assessent of RC structural eleent. The shear and bending capacity calculated fro above procedure will be used in the derivation of P-I equations in the following chapters. In the rigid-plastic aterial odel, M u and V d will be the constant bending and shear strength respectively. In the consideration of nonlinear aterial odel, M u and V d will be the axiu bending and shear strength respectively..5. Suary Fro the above review on P-I diagra ethod and design or RC eleent against blast load, it can be suarized as follows: dc The SDOF syste could give out P-I results quickly and P-I diagras based on such odel can be used as a preliinary assessent of structural daage induced by a blast load. However, the SDOF odel oversiplifies the structural eleents and cannot analyze shear and bending response at the sae tie. The MAM is suitable for analyzing the structural response against blast loading, especially for bea eleent. It can analyze cobined ode so that shear and bending failure can be studied siultaneously. Result of daage 6

49 assessent fro such ethod can be generated siply by reading points fro P-I diagras, and it agrees well with that fro SDOF syste. Paraeters in the design procedure of RC eleent can be used in P-I diagras based on MAM. The procedure can be further siplified by adopting results of the present research work, and can be extended to nonconstant paraeters easily. 7

50 8

51 CHAPTER 3 DAMAGE ASSESSMENT FOR UNDERGROUND STRUCTURES AGAINST EXTERNAL BLAST LOAD 3.1. Introduction The present study extended the daage assessent ethod, P-I diagras based on MAM for surface structures by Ma et al. (7) to underground structures by considering soil-structure interactions (SSI) which was proposed by Weidlinger and Hinan (1988). Coparison between the present study and the results of surface structures has been carried out and the siplified SSI effect has been discussed. To verify the continuity of the results for different failure odes, verifications are carried out on soe special conditions. Effect of the blast load pressure and duration on the daage of an underground structural eleent is discussed. A case study is also given to deonstrate the applicability of the proposed ethod. 3.. Siplification of soil-structure interaction Considering a reinforced concrete box-type structure which is buried in soil, the explosion occurs at a certain distance fro the structure as scheatically shown in Fig. 3-1(a). Taking out one unit strip fro the structural wall, the structure can be siplified into a siply supported bea odel which is subjected to external blast load as shown in Fig. 3-1(b). The present analysis assues a rigid-plastic deforation to the bea odel and the interaction between soil and structure is considered. The bending strength of the bea odel is denoted as M, and the transverse shear strength is Q, the ass per unit length of the bea odel is. A blast pressure P i is uniforly distributed over the span of bea. The SSI is siplified as a daping effect as shown in Fig. 3-1(b), which will be derived in detail below. Half of the bea is analyzed because of syetry. For siplicity, the square yield criterion presented in Fig. 3- is used for the bending and shear failure 9

52 analysis in the present study. The detailed discussion on the accuracy of the square yield criterion has been given by de Oliveira and Jones (1978). (a) Box-type underground structure (b) A unit strip bea odel under external blast load Fig. 3-1 Underground structure and bea odel Fig. 3 - Square yield surface The rotational inertia effect on the bea dynaic failure has been investigated by other researchers (de Oliveira and Jones 1979, Jones and de Oliveira 1979, de Oliveira 198). It has been verified that the rotational inertia was less iportant for ost practical cases. Considering this effect would ake the atheatical derivations uch ore coplex while the reductions in axiu transverse 3

53 displaceents was less than 1% even in extree cases as exained by de Oliveira and Jones (1979). The continuity requireents at a discontinuous interface have also been discussed by Syonds (1968) and Jones (1989). In the analysis of underground structures, SSI ay have significant effects on the structural response. Siplification of SSI was discussed by any researchers (Wolf 1985, Krauthaer et al. 1986, Weidlinger and Hinan 1988). In the present study, the ethod by Weidlinger and Hinan (1988) is adopted, and it is briefly introduced below. The siplest forulation of the free-field stress wave generated by conventional weapons is the sei-epirical forula fro TM (1986). It gives the peak aplitude of the stress wave at the arrival tie as: R P f c 13 W n (3.1) where P is the peak pressure, β=.47 for P in MPa, ρc in MPa-s/, W in N, R in, f is the coupling factor of the explosive, ρc is the acoustic ipedance of soil, R is the distance easured fro the center of an explosion to the structure, W is the equivalent TNT weight, and n is an attenuation coefficient. The value of n differs fro different soil types which were discussed by Sith and Hetherington (1994). For exaple, n=3. for loose dry sand, n=.75 for dense sand, and n=1.5 for saturated sandy clay. The decay of the stress wave in tie is given by: a P R, t P e tt (3.) where t a =R/c is the arrival tie of stress wave and c is the P-wave velocity. When neglecting the curvature of the wave front, the particle velocity in the free field is approxiated by the linear plane wave relation: v R, t P R, t (3.3) c 31

54 Further siplifying the free-field stress by neglecting difference of the distances fro the center of explosion to different part of a structure eleent, the free field pressure at the interface between the soil and structure can be expressed as: P R, t (3.4) f The loading experienced by the structural eleent can be divided into two parts, i.e., the free-field pressure at the interface, and f i, the interface pressure due to difference between the structural velocity The interface pressure is defined as: Thus the total interaction load is: ut, and the soil velocity vt. i c v u (3.5) i f i f P c v u (3.6) By considering Eqs. (3.3) and (3.4) and eliinating the velocity ter v(t) in Eq. (3.6), Eq. (3.6) transfors into: In the current analysis, Eq. (3.7) is transfored to: P c u (3.7) i f P p ( c) u p Cy i (3.8) where p is the constant pressure in blast load duration acting on bea eleent, C is the daping coefficient, and y is the structural velocity. Therefore the SSI is siplified as a daping coefficient. Because the soil at interface can only transit copression, the daping coefficient will equal to zero when f cuin Eq. (3.7). Under the case of a real blast load, the SSI becoes zero before the blast duration t d, since the blast pressure decrease with tie. While in the present study, the pressure on structural eleent p is assued as a constant within the blast duration, therefore C= only at t=t d. 3

55 The siplification of SSI is based on the elastic wave propagation theory which liits the application of current results within oderate and far-range underground explosion cases, and the close-in range cases with low charge weight. The reflected pressure is obtained fro TM5-13 (199) directly, and the SSI is considered only when structural eleent starts to response. The actual load on eleent in a real case should be further studied considering non-elastic wave propagation theories and a coplicated soil aterial odel, and it is not in the scope of current analysis Failure criteria and structural failure assessent According to the previous results done by other researchers (Yu and Jones 1991, Krauthaer 1998) for siply supported or claped beas, the ratio of centerlinedeflection to half-span is used as the criterion of the bending failure, since the largest ductile plastic deforation usually appears at the id-span due to bending effects. The axiu transverse displaceent due to bending, and the direct shear failure at the eleent supports can be expressed as follow: y L ys h (3.9) where y is the axiu transverse displaceent due to bending, L is the half length of the bea odel, β is the ratio of centerline deflection to the half span, y s is the axiu transverse displaceent due to direct shear, γ υ is the average shear strain, δ is the aterial paraeter obtained fro experiental results, h is the depth of the structural eleent. Norally δ varies in for different aterials. In the present study, δ is defined as.8 which is the sae as that used by Ma et al. (7) for surface structures. Table 3-1 shows different daage level under epirical bending and shear failure criteria (Krauthaer 1998). The criteria are applied to reinforced concrete structure eleent. 33

56 Table 3-1 Different daage level under epirical bending and shear failure criteria Light Moderate Severe Type of failure Criteria Daage Daage Daage (%) (%) (%) Shear Average shear strain 1 3 Bending Ratio of centerline deflection to half span For convenience in perforing analytical failure analysis of underground structures, a rectangular pulse load with equivalent pressure and ipulse as given in the seiepirical Eq. (3.1) is used to represent blast load, where the pulse shape effect is neglected. The rectangular pulse load can be expressed by a side-on blast load p, and blast load duration t d. The pulse shape effect has been discussed by Youngdahl (197, 1971), Li and Meng (b), and Li and Jones (5b) which can be eliinated by using the Youngdahl s (1971) correlation paraeter ethod. It has been validated that the pulse shape effect to the P-I diagras is inor especially in the quasi-static load region and the ipulsive load region. The daping effect of SSI is assued to exist only when t<t d. This assuption is discussed in the following section. The governing equation of the eleent is expressed as: Q Pi y x (3.1) When substitute Eq. (3.8) into Eq. (3.1), Eq. (3.1) can be rewritten as: Q p y Cy x (3.11) where Q is the transverse shear force, x is the abscissa on the eleent, is the ass per unit length of the eleent, y is the acceleration of structural eleent. 34

57 Siilar to the analysis for surface structures, there are totally five possible transverse velocity profiles which include one shear failure ode, two bending failure odes, and two cobined failure odes, as illustrated in Fig A diensionless strength ratio is introduced as below: QL M (3.1) where Q and M are respectively the shear and bending strength of the eleent. Table 3- shows the velocity profile of different phases under different odes. Mode 1 contains shear failure only. Mode is for siple bending failure which has a plastic hinge at the center of the eleent. Mode 3 can be considered as a cobination of ode 1 and ode. Mode 4 is the coplex bending failure ode which has a plastic zone at the iddle of the eleent. And ode 5 is the cobination of ode 1 and ode 4. Differentiation of the five failure odes is the sae as that for surface structures. However, the SSI has been incorporated in the derivations. The x-axis starts at the id of the bea and only half of the bea is considered due to syetry of the structure and load as seen in Fig. 3-1(b). 35

58 S: Shear failure; B: Bending failure; C: Cobined failure Fig. 3-3 Distribution of failure odes Table 3 - Velocity profile Mode 1 Mode Mode 3 Mode 4 Mode 5 Phase 1 Phase Phase 3 N.A. N.A. Phase 4 N.A. N.A. N.A. N.A. 36

59 M Mode 1. 1 and p L, shear failure ode. The shear failure ode provides that direct shear failure occurs at the two supports where carries the axiu shear force. Bending deforation will not occur in this case. The shear failure occurs only when the paraeter υ is saller than one which indicates a very low shear-to-bending strength ratio. There are totally two phases in this ode including a loading phase (phase 1) and a post-loading phase (phase ) which are assued to end at t d and t f respectively. In phase 1, the governing equation is: Q p y Cy x s s (3.13) The boundary and initial conditions are:, Q x Q x L Q (3.14) y t, y t (3.15) s By integrating Eq. (3.13) with respect to x and substituting Eq. (3.14), Q can be solved. Then integrates the equation of Q with respect to tie twice, the expressions of structural displaceent, velocity, and acceleration can be obtained. At the end of phase 1 when t=t d, the axiu displaceent eleent y t are solved. s d In phase, the governing equation becoes: s y t and the velocity of the s d Q ys x (3.16) By using Eq. (3.14) as the boundary condition, y t and s d y t as the initial conditions for phase otion, Eq. (3.16) can be solved under the sae procedure when solve Eq. (3.13) in phase 1. When the structural velocity equals to zero, the otion terination tie t f of phase is deterined as: s d 37

60 t f Ct d 1e pl Q td (3.17) QC The final transverse displaceent due to shear is derived as: Mode Q t f td ys t f ys td t f td ys td (3.18) L M M, or 1.5 and L L and p 4 3 M 6M p, bending failure ode. L L There are also two phases in this ode including a loading phase (phase 1) and a post-loading phase (phase ) which end at t d and t f respectively, as given in Table 3-. In phase 1, the governing equation is: Q x x p y 1 Cy 1 x L L (3.19) The boundary and initial conditions are: Q x, Q x L Q, M x M, M x L (3.) y t, y t (3.1) By integrating Eq. (3.19) with respect to tie, at the end of phase 1 when t=t d, the axiu displaceent deforation d y t are solved. In phase, the governing equation changes to: d y t and the velocity of the eleent due to bending Q x y 1 x L (3.) 38

61 By using Eq. (3.) as the boundary condition and y t and conditions, the terination tie t of phase can be solved as: f d y t as the initial d t f Ct d 1e pl M td (3.3) MC The final bending displaceent is then derived as: 3M t f td y t f y td t f td y td (3.4) L When t=t f, the velocity of the bea otion becoes zero. Mode and p 4 3 M, cobined failure ode. L Mode 3 is the cobination of ode 1 and ode. Both shear failure and bending failure occur to the eleent. The shear failure occurs at the two supports while the bending failure induces plastic hinge at the id-span of the eleent. Deforation of the eleent includes one loading phase (phase 1) and two post-loading phases (phase and phase 3) which are assued to end at t d, t s, and t f respectively. In phase 1, the governing equation is given by: Q x x p ys y ys 1 Cys C y ys 1 x L L (3.5) with the sae boundary and initial conditions given in Eqs. (3.), (3.15) and (3.1). Siilar to ode 1 and ode, at the end of phase 1 when t=t d, the axiu displaceent and velocity of the eleent due to shear and bending deforation can be deterined. In phase, the governing equation is: Q x ys y ys 1 x L (3.6) 39

62 The boundary conditions are still the sae as the first phase, while the final velocities and displaceents in phase 1 are used as the initial condition of phase otion. Solving Eq. (3.6), at the end of phase when t=t s, the displaceent due to shear stops, while the bending displaceent reains to the next phase otion. The ending tie and axiu displaceent of shear is derived as below, and the axiu displaceent and velocity due to bending otion can then be solved. t s 6t M 4t Q L L y t Q L 3M d d s d (3.7) M Q Lt t 3 s d s s s d s d s d y t y t t t y t (3.8) L where t s is deterined when the shear velocity becoes zero when the shear deforation stops. In phase 3, only the bending failure induced otion reains, and the governing equation is the sae as Eq. (3.). Siilarly, the otion terination tie t f and the final displaceent y (t f ) are deterined as: t f y ts L (3.9) 3M 3M t f ts y t f y ts t f ts y ts (3.3) L Mode and 6M 8M L 3L p, bending failure ode. The bending failure with a plateau deforation at the central portion of the eleent ay occur when the blast load is sufficiently intensive. Different fro ode, two plastic hinges are generated offset fro the id-span of the eleent. Three phases including one loading phase (phase 1) and two post-loading phases (phase and phase 3) which end at t d, t 1, and t f respectively are considered. In phase 1, the governing equation is expressed as: 4

63 Q L x L x p y Cy x L L (3.31) where ξ is the distance of the plastic hinge fro the id-span. The boundary conditions are as follows: Q x, Q x L Q, M x M, M x L (3.3) And the initial conditions are the sae as Eq. (3.1). Thus Eqs. (3.3) and (3.1) are used to deterine the integral constants when Eq. (3.31) is integrated. At the end of the loading period, when t=t d, the plastic hinge location indicated by ξ is derived as: 6 pm L (3.33) p And the displaceent and velocity of the bea odel due to bending at the end of phase 1 are then obtained. In phase, the blast load has been released, and the velocity profile is the sae as that in phase 1. However, the two plastic hinges start to ove toward the id-span of the eleent. At the end of phase, the two plastic hinges eet at the id-span and phase 3 otion then starts. The governing equation of the phase otion is: Q L x y x L (3.34) where ξ is the half length of plastic zone which is a function of tie. By considering the boundary condition given as: Q x, Q x L Q, M x M, M x L (3.35) The final displaceent and velocity of the first phase as the initial conditions, Eq. (3.34) of phase otion can be solved. The terination tie t 1 of phase is given by: 41

64 Ctd 1 e pl Ctd t1 e 1 t 6CM C d (3.36) In phase 3, the governing equation is the sae as Eq. (3.) and it can be solved in a siilar way as it has been done for ode and ode 3. At the end of phase 3, when t=t f, the otion stops and the final tie t f is deterined as: t f Ct d p 1 e L t1 (3.37) 3M The final bending displaceent y at the id-span when t= t f is given by: 3M t f t1 y t f y t1 t f t1 y t1 (3.38) L Mode and p 8M 3L, cobined failure ode. Mode 5 is the ost coplicated ode which cobined ode 1 and ode 4. There are altogether four phases including one loading phase (phase 1) and three postloading phases (phase, phase 3, and phase 4). They are assued to end at t d, t s, t 1, and t f respectively. In phase 1, both shear and bending deforation occur. The governing equation is: Q L x L x p ys y ys Cys C y ys (3.39) x L L The boundary conditions are the sae as Eq. (3.3), and the initial conditions of ode 1 and ode 4 are also cobined as: y t, y t, y t, y t (3.4) s s 4

65 At the end of phase 1 when t=t d, the plastic hinge location ξ is shown in Eq. (3.41) below, and the final displaceent and velocity of phase 1 due to shear and bending deforation can then be derived. 3M L (3.41) Q In phase, the velocity profile is the sae as that in phase 1, while the shear deforation tends to stop. The governing equation becoes: Q ys y ys L x x L (3.4) Based on the sae boundary condition given in Eq. (3.3) and the final displaceent and velocity of phase 1, Eq. (3.4) can then be solved. At the end of phase when t=t s, the shear deforation stops. The terination tie and the final shear displaceent of phase are given by: t s Ct d 1 e 3pM Q CtdQ (3.43) CQ Q ts td s s s d s d s d 3M y t y t t t y t (3.44) In phase 3, the bending deforation reains the sae velocity profile as that in phase of ode 4, and the two plastic hinges start to ove toward the id-span of the eleent. The governing equation is the sae as Eq. (3.34). Siilarly, at the end of phase 3, when t=t 1, the two plastic hinges coincides at the id-span, and: Ct d p 1 e 6M t1 L ts p 6CM (3.45) Siilarly, in phase 4 of ode 5, the governing equation is the sae as Eq. (3.). At the end of phase 4 when t=t f the otion stops and: 43

66 t f Ct d pl 1 e t1 (3.46) 3CM The final bending displaceent is: 3M t f t1 y t f y t1 t f t1 y t1 (3.47) L 3.4. P-I diagras and discussions In each of the five odes discussed previously, the final shear displaceent y s for the direct shear failure and the final bending displaceent y for bending failure, or both for cobined failure can be derived. Given a certain failure criterion in ters of the axiu shear and bending displaceents, P-I diagras for different odes can then be obtained. Although the five failure odes and the analysis of the structural response are very siilar to that done for the surface structures (Ma et al. 7), the present study focuses on the daage assessent for underground structures against external blast load. Define diensionless variables P* and I* of the pressure and ipulse of a blast load as: * pl P M * Ctd P I * (3.48) Fro the equations for final displaceents induced by shear and bending failure, the P-I equations can be represented in unified fors as follows: * * S P, I h * * B P, I L (3.49) where the shear and bending failure criteria given respectively in Eq. (3.9) are used. 44

67 S(P*,I*) and B(P*,I*) are iplicit expressions with respect to the noralized pressure and ipulse for shear and bending failure ode respectively. S(P*,I*) equals to the right part of Eqs. (3.18), (3.8), or (3.44) as the shear failure P-I equation, and B(P*,I*) equals to the right part of Eqs. (3.4), (3.3), (3.38), or (3.47) to represent the bending failure P-I equation Boundaries between different failure odes By deterining relevant paraeters, the P-I diagras can be plotted for all five failure odes. Differentiation between different odes given in Fig. 3-3 is deterined by the requireent of the otion initiation. For exaple, for ode 1, the acceleration induced by the shear force at the supports should be larger than zero, and the axiu bending oent should be saller than the bending strength of the eleent. Therefore p M L and 1 are derived, respectively. For ode, the initial acceleration due to bending should be larger than zero, while the axiu shear force should be less than the shear strength. So and p M M, or 1.5 and L L M 6M p are L L required. Siilarly, all the boundaries between different failure odes can be deterined. Three typical P-I diagras of shear and bending failures are plotted in Fig ) When υ=.8, which is in the range of υ<1, only shear failure exists as shown in Fig. 3-4(a) (ode 1). Region A and region B indicate shear failure and no failure respectively. ) As shown in Fig. 3-4(b) when υ=1., which satisfies and p M 4 3, the eleent fails with ode 3, which is a cobination of L ode 1 and ode. Two P-I diagras, for shear failure and bending failure respectively, are plotted based on the failure criteria given in Eqs. (3.9). There are four regions in this case which indicate four different failure types of the eleent. In region A, which corresponds to larger pressure and ipulse, cobined failure occurs since both the axiu shear displaceent and the 45

68 axiu bending displaceent exceed the failure threshold given by the shear and failure criteria. In region D, the pair of pressure and ipulse locates above the shear failure diagra but below the bending failure diagra, which indicates only shear failure occurred to the eleent. Siilarly, in region B, the pair of pressure and ipulse exceeds the bending failure diagra while it is below the shear failure diagra, which defines bending failure for the eleent. In region C, the eleent reains safe due to the pressure and ipulse pair is below both diagras. 3) When υ=1.8, which is in the range of υ>1.5, ode 4 and ode 5 ay occur. Fig. 3-4(c) shows the P-I diagras of ode 5 which is siilar to ode with 4 different regions. It is seen that shear failure occurs when the pressure reaches a certain level. The P-I diagra for shear failure is zooed up in Fig. 3-4(d). (a) Typical failure ode 1. (ode 1) 46

69 (b) Typical failure ode (ode 3) See Fig. 3-4 (d) (c) Typical failure ode 1.5 (ode 5) 47

70 (d) Typical failure ode 1.5 (ode 5 shear failure) Fig. 3-4 Typical failure odes for different value of υ Daping effect due to soil-structural interaction Different soil has different acoustic ipendence, and the daping effect due to SSI is investigated as below. Considering three different cases with dry sand and backfills (ρc=.497 MPa s/), dense sands (ρc=.995 MPa s/), and saturated sandy clay (ρc=.941 MPa s/) respectively, the P-I diagras for shear and bending failure odes are plotted and copared in Fig It shows that the saturated sandy clay gives higher P-I diagras for both shear and bending failure, which indicates that the acoustic ipendence has significant effect on the structural failure. 48

71 (a) Shear failure in different soils (b) Bending failure in different soils Fig. 3-5 Shear and bending failures in different soils Coparison is also done to the daage assessent results for surface structures when the daping coefficient is set sufficient sall (for exaple, ρc=1 Pa s/). The coparison is done for all the five failure odes, and here ode 3 is chosen as an exaple, as shown in Fig It can be seen that both shear failure and bending 49

72 failure diagras atch well with those for surface structures. It verifies that the present P-I diagras are valid and the P-I diagra ethod suggested by Ma et al. (7) has been successfully extended to daage assessent for underground structures when the decoupling of SSI is considered. It should be noted that a zero daping coefficient is not allowed in the P-I equations in the present study, however, the present results can be applied to daage assessent for surface structures by assigning a sufficient sall daping coefficient in the equations. Fig. 3-6 Coparison with blast result of surface structure Verification of continuity Since different failure odes give different final shear and/or bending displaceent, the solutions at the boundaries between different odes in Fig. 3-3 should be continuous. It eans that two odes should give the sae P-I diagra at their sharing boundary line. To verify the continuity of the equations, three checking points as shown in Fig. 3-3 have been exained. CP1 ( p shared by ode and ode 3, CP ( p 6.M L ode 3, ode 4 and ode 5, and CP3 ( p and M L 3.6M L and 1. ) is ) is shared by ode, and 1.8 ) is shared by 5

73 ode 4 and ode 5. The verification results are given in Table 3-3 which indicate that the results of ode, 3, 4, and 5 atch very well at the three checking points. Table 3-3 Continuity verification Mode P * I * P * I * P * I * N.A. N.A. 3 N.A. N.A N.A. N.A. 5 N.A. N.A Pressure and ipulse effects When the echanical paraeters of the eleent are given, the pressure and ipulse will have different effects on the axiu displaceent due to shear or bending deforation. Fig. 3-7 takes ode 3 as an exaple and the surrounding soil is dry sand and backfills. It shows that the tendency of the shear and bending displaceents vary with the noralized pressure and ipulse. The axiu displaceent due to shear failure is uch ore sensitive to the pressure than the axiu bending displaceent as shown in Fig. 3-7(a). However, the influence of the noralized ipulse on both of the axiu shear and bending displaceents is very siilar. The axiu shear and bending displaceents increase alost linearly with the increase of the noralized ipulse as shown in Fig. 3-7(b). 51

74 (a) Maxiu displaceent versus noralized pressure (b) Maxiu displaceent versus noralized ipulse Fig. 3-7 Pressure and ipulse effects Further discussions Researchers soeties use a claped bea odel for underground structure analysis. The difference of the P-I diagras between a siple supported bea and a fixed bea has been discussed for surface structures by Ma et al. (7). Daage 5

75 assessent for underground structures by siplifying the structure to be siply supported gives a conservative analysis and it will lead to a safer design of the underground structures. By using siilar derivations as given in the present study, the results of a fixed bea odel can also be obtained correspondingly. When the underground structures are in a iddle or far range fro the explosion, the blast wave acting on the structures is very close to a plane wave. In a close-in range, the load acting on the bea odel can be siplified as a cycloidally, exponentially, sine-shapely, or linearly descending pressure fro the bea center to the two ends. Again, using a uniforly distributed load with a consistent peak pressure will give ore conservative analysis results which ensure a safer design of underground structures. Furtherore, the pulse shape effect produces error less than 1% (Oliveira and Jones 1979). In the present study, the daping effect is assued to exist in phase 1 of all the five odes, but it disappears in the following phases. This is reasonable because as soon as the blast load is released, the interaction between the surrounding soil and the structure becoes inor. Such assuption also leads to conservative predictions of bea shear and bending failures. In a real case of underground explosion, especially when an explosion occurs near a structure, the stress wave is very unlikely to reain as elastic. While based on assuptions of the present study, only elastic wave propagation can be considered as shown in Eqs. (3.5) to (3.8). This leads to that, results of the present study are ore accurate in cases of oderate and far explosions, but the pressure in close-in explosion should be checked carefully before plotting P-I diagras Case study To verify the applicability of the developed P-I diagras, a case study is carried out for a box-shape underground structure. R is the distance fro the explosion center to the side wall as shown in Fig. 3-1(a). Daage assessent for the side wall is carried out and one unit strip of the side wall is considered for siplicity. In a typical underground structure, the side wall is ade up of laced reinforced concrete with stirrups to increase its shear and bending strength. The transverse 53

76 reinforceent ratio is between.4 and.6 percent. Details of such blast design can be referred to the TM5-13 anual, and are oitted here. The soil layer is dry sand and relevant paraeters are given below. Half span: L = Thickness of the side wall: h =. Unit length ass of the side wall: = 5 kg Shear strength of the side wall: Q = kn Bending strength of the side wall: M = 17.7 kn Soil acoustic ipedance: ρc =.497 MPa s/ Charge weight: W = 15 kg Once the distance R is deterined, by using the above paraeters, the constant pressure p and the blast duration t d can be approxiated by using the seiepirical Eqs. (3.1) and (3.). Subsequently, the noralized pressure and ipulse are calculated. In the present case study, we took three different standoff distances, i.e., R=6.4, 7.3 and 1, respectively (see Table 3-4). Table 3-4 Case study Scaled Distance Duration Point Distance (R) (t d ) Z Noralized Ipulse Z s Z s Z s I * Pressure (p ).46 MPa MPa.667 MPa Noralized Pressure P *

77 The strength ratio υ is calculated as 1.374, and the failure criteria of Eq. (3.9) in which β =.5% and γ υ =1% are used. Judging fro Fig. 3-3, the failure ode is ode 3, which is a cobination of shear and bending failures. As shown in Fig. 3-8, point Z1 is in region A, point Z is in region B, and point Z3 is in region C. That eans, when the scaled distance is 1.83, the structure will endure both shear and bending failure; when the scaled distance is 1.465, the structure will endure bending failure only; when the scaled distance is 1.978, the structure is safe. Fig. 3-8 Case study 3.6. Conclusions The present analysis is based on a rigid-plastic eleent odel and the MAM to assess daage of underground structures against external blast load. The developed P-I diagras consider siplified SSI and they can be applied conveniently to daage assessent of the underground structures in the for of different failure odes. For a given strength ratio of υ, different regions for shear, bending, and cobined failure odes according to the P-I diagras can be deterined. Coparison has been done to the results of surface structures which verifies the validity of the developed P-I diagras. 55

78 The effect of soil cannot be ignored in estiating the blast load applied to structure. With the increase of acoustic ipendence, interaction between structure and surrounding soil becoes ore significant. The axiu shear displaceent is ore sensitive to the pressure applied than the axiu bending displaceent, while the effect of ipulse to the axiu shear and bending displaceents is siilar. The case study shows that, the proposed ethod can be effectively applied to daage assessent of underground structures against blast load. 56

79 CHAPTER 4 DAMAGE ASSESSMENT FOR UNDERGROUND STRUCTURES AGAINST INTERNAL BLAST LOAD 4.1. Introduction Most existing works focused on daage assessent of surface structures. Very few involved in failure of underground structures. In fact, the coplexity of soil property akes it difficult to perfor structure response analysis when the SSI is considered. Fro the result of previous chapter, it is understood that the surrounding soil interacts with the underground structure and such effect cannot be ignored. When an underground structure is under internal blast load, the soil acts as an elastic support which absorbs part of the blast energy during an internal blast event occurred to an underground structure. In the present study, the SSI is siplified as an equivalent spring coefficient. The rigid plastic odel with the ode approxiation ethod is adopted. P-I diagras are subsequently developed for structural eleent by considering the SSI. The results of the present study can also be applied to the scenario when an explosion occurs near a retaining wall, over a foundation slab, or baseent, etc. Coparison between the present study and daage assessent for surface structures is carried out to evaluate the SSI effect. P-I diagras of different daage levels are also plotted. To verify the continuity of different failure odes, verifications are done by checking a few sharing points of different failure odes. A case study for an underground RC structure has also been conducted to show the applicability of the proposed daage assessent ethod. 4.. Siplification of soil effect During the physical process when a blast wave interacts with underground structure, as shown in Fig. 4-1, the soil ediu is considered as a acroscopic hoogeneous aterial which can be siplified to an elastic foundation to support the structure. 57

80 Therefore the SSI effect is siplified to a spring support which distributes over the structural eleent. Such siplification ignores the change of soil stiffness during response of structural eleent, and would under-estiate structural daage especially in a close-in explosion. But it is still suitable for cases of oderate and far-range explosions, and cases of close-in explosion with low charge weight. Previous research works showed that, soil stiffness is frequency-dependent, but those were based on seisic wave and cyclic loading. Soe of the conclusions also showed that: R-waves are predoinant when away fro vibration center (Athanasopoulos et al. ); When bulk odulus of soil is high, error in static and dynaic SSI is low (ülker-kaustell et al. 1); Soil stiffness degrades in soft clay due to cyclic loading (Li and Assiaki, 1); SSI is significant in stiff soil (Steheyer and Rizos, 8), etc. In the present study, only the first half circle is adopted to calculate the axiu structural response, therefore the degradation and dynaic effects of SSI are not so evident and are ignored. For the sae reason, any kind of soil daping is ignored, although there are viscous daping, hysteretic daping and radiation daping of soil in a real underground explosion case. The SSI effect exists in the deforation phase till the otion of eleent vanishes. Change of soil density during deforation is not considered; therefore the average stiffness of soil is adopted according to the analytical and laboratorial results (Sawangsuriya et al. 1974, Dutta et al. 4). Such siplification ais on getting result efficiently in the analysis of the structure response by considering the SSI effect. 58

81 Fig. 4-1 Soil-structure interaction in internal blast load scenario 4.3. Failure criteria A typical resistance-deflection curve for laterally restrained eleents is shown in Fig. 4- (TM5-13, 199). The initial portion of the curve is priarily due to the flexural action. The ultiate flexural resistance is aintained until degrees of support rotation is produced. At this support rotation, the concrete begins to crush and the eleent loses flexural capacity. If adequate single leg stirrups were provided, the flexural action would be extended to 4 degrees. However, due to the presence of the continuous reinforceent and adequate lateral restraint, a tensile ebrane action is developed. The resistance due to this action increases with increasing deflection up to incipient failure at approxiately 1 degrees support rotation. In order to siplify the analysis, the resistance is assued to be due to plastic action throughout the entire range of behavior. To approxiate the energy absorbed under the actual resistance-deflection curve, the deflection of the idealized curve is liited to 8 degrees support rotation. Design for this axiu deflection would produce incipient failure conditions. Existing studies (Yu and Jones 1991, Krauthaer 1998) also suggested use the ratio of centerline-deflection to halfspan as the criterion of bending failure, since the largest ductile plastic deforation usually appeared at the id-span due to bending effects. 59

82 Fig. 4 - Idealized resistance-deflection curve for large deflections The axiu bending and shear deforations of the eleent are defined the sae as Eqs. (3.9). For a reinforced concrete structure eber, the dynaic bending strength M varies for different cross section designs. According to TM5-13, for type I design, there is no crushing or spalling in concrete; only cracking appears on the tension side and the M can be expressed as: M Af s b ds a d (4.1) in which: Af s ds a (4.).85 bf ' where A s is the area of tension reinforceent within the width, f ds is the dynaic design stress for reinforceent, b is the width of copression face, d is the distance fro extree copression fiber to centroid of tension reinforceent, a is the depth of equivalent rectangular stress block, f dc is the dynaic ultiate copressive strength of concrete. In the cross section type II, cracking appears on the tension side of eber while crushing appears on the copression side. The M can be shown as: dc 6

83 M A f d b s ds c (4.3) where A s shows the area of tension or copression reinforceent within the width b, d c is the distance between the centroids of the copression and the tension reinforceent. In the cross section type III, disengageent of concrete appears on both tension and copression sides of the eber. The bending strength M is expressed the sae as in Eq. (4.3). The dynaic shear strength of a rectangular section reinforced concrete structure eber Q can be expressed as: Q.16 f ' dc bd (4.4) The sae daage level under epirical bending and shear failure criteria fro results of Yu and Jones (1991) and Krauthaer (1998) are used as those in Table 3-1. There are three levels of daage, including light daage, oderate daage, and severe daage. For shear design, average shear strain γ v indicates different daage levels, while the support rotation β is for bending Failure odes and response analysis The internal blast load is siplified as a rectangular pulse load with agnitude of p and duration of t d, while the pulse shape effect is eliinated. The pulse shape effect has been discussed by Youngdahl (197, 1971), Li and Meng (b), and Li and Jones (5b) that the loading shape effects on the P-I diagras of a rigid, perfectly plastic SDOF odel can be eliinated by using the Yougdahl s correlation paraeter ethod. For convenience of P-I equations derivation while not losing the generality of the solution, the rectangular pulse load with equivalent pressure and ipulse is adopted in the present study. The pulse shape effect on the P I diagras of a rigid, perfectly plastic SDOF odel can be eliinated by using the Youngdahl s (1971) correlation paraeter ethod as follows. 61

84 t f I Ptdt ty 1 t f tean t ty P t dt I (4.5) ty I Pe tean where I is the ipulse, t y is the tie when aterial begins to yield, t f is the end tie of total deforation, P(t) is the external pressure, t ean is the ean tie, P e is the effective pressure. As discussed in section 4., the governing equation is Eq. (3.1). Since the spring effect exists in all the phases, the governing equation is expressed as: Q p y Ky x (4.6) where Q is the transverse shear force, x is the abscissa on the eleent, is the ass per unit length, y is the acceleration of unit ass, K is the equivalent spring coefficient to represent the SSI effect, and y is the displaceent of unit ass. The sae as the analysis of surface structure, there are totally five possible transverse velocity profiles including one pure shear failure ode, two pure bending failure odes, and two cobined failure odes as shown in Fig A diensionless strength ratio ν is introduced the sae as that in Eq. (3.1). The velocity profile of different phases under different odes is the sae as Table 3-. Mode 1 contains shear failure only. Mode is for siple bending failure which has a plastic hinge at the center of the eleent. Mode 3 can be considered as a cobination of ode 1 and ode. Mode 4 is the coplex bending failure ode which has a plastic zone at the iddle of the eleent. And ode 5 is the cobination of ode 1 and ode 4. Differentiation of the five failure odes is the sae as that for surface structures. Derivation of the displaceent tie history for the five failure odes is suarized as follows. The x-axis starts at the id of the bea and only half of a strip bea is considered due to syetry as seen in Fig

85 M Mode 1. 1 and p L, shear failure ode. This shear failure ode provides direct shear failure occurs at the two supports where carries the axiu shear force, while bending failure does not occur. In this ode, the diensionless paraeter ν is less than one which indicates a very low shear-to-bending strength ratio. There are totally two phases in this ode including a loading phase (phase 1) and a post-loading phase (phase ) which end at t d and t f respectively. In phase 1, the governing equation is: Q p y Ky x s s (4.7) where y s and respectively. y s are the acceleration and displaceent due to shear force The boundary and initial conditions are: where ys is the velocity due to shear force., Q x Q x L Q (4.8) y t, y t (4.9) s s Integrating Eq. (4.7) with respect to tie, at the end of phase 1 when t=t d, the axiu displaceent and velocity of the eleent are shown as follows: y s t y d s t d pl Q Kt d cos 1 KL p L Q Kt K L d sin (4.1) (4.11) In phase, the governing equation transfers to: 63

86 Q y s Ky x (4.1) Using Eq. (4.8) as the boundary condition and Eqs. (4.1)-(4.11) as the initial conditions to solve Eq. (4.1), at the end of phase, the terination tie t f will be: t f Kt d sin pl Q arctan t Ktd Kt d pl cos Q cos pl d (4.13) The eleent s final transverse displaceent due to shear is: where 1 f d y t T y t T K sin 1 cos 1 d y t KL Q Q KL KL s s f s d (4.14) T K t t. Mode and 4 3 M L p M L, or 1.5 and M L p 6M L, bending failure ode. A bending failure ode will occur when the above conditions are satisfied. A plastic hinge is generated at the id-span of the eleent. There are totally two phases in this ode including a loading phase (phase 1) and a post-loading phase (phase ) which end at t d and t f respectively. In phase 1, the governing equation is: Q x x p y 1 Ky 1 x L L (4.15) where respectively. y and y are the acceleration and displaceent due to bending force The boundary and the initial conditions are: Q x, Q x L Q, M x M, M x L (4.16) 64

87 where y is the velocity due to bending force. y t, y t (4.17) Integrating Eq. (4.15), at the end of phase 1 when t=t d, the axiu displaceent and velocity of the eleent due to bending are shown respectively as follows: y y t d 3 pl M Kt d 1 cos KL t d 3 pl M Kt d sin KL (4.18) (4.19) In phase, the governing equation changes to: Q y 1 x Ky 1 x x L L (4.) Using Eq. (4.16) as the boundary conditions and Eqs. (4.18)-(4.19) as the initial conditions, the final tie t f will be: t f K L y t d arctan t K KL ytd 3M d (4.1) The final bending displaceent of the eleent is solved as: y Mode t f d sin T1 y cos 1 3 t T KL y t M d 3M (4.) K KL KL and p 4 3 M, cobined failure ode. L Mode 3 is the cobination of ode 1 and ode. Both shear failure and bending failure occur to the eleent. The shear failure occurs at the two supports, while the bending failure induces plastic hinge at the id-span of the eleent. There are three phases including one loading phase (phase 1) and two post-loading phases (phase and phase 3) which end at t d, t s, and t f respectively. 65

88 In phase 1, the governing equation is: Q x x p ys y ys 1 Kys K y ys 1 x L L (4.3) with the sae boundary and initial conditions given in Eqs. (4.16), (4.9), and (4.17). Siilar to ode 1 and ode, after integrating Eq. (4.3), at the end of phase 1 when t=t d, the axiu displaceent and velocity of the eleent due to shear and bending are given respectively as follows: In phase, the governing equation is: y y s y y t s t d d t t Kt d 1 cos pl 4Q L 6M (4.4) KL d Kt d sin pl 4Q L 6M (4.5) K L Kt d 1 cos pl Q L 6M (4.6) KL d Kt d sin pl Q L 6M (4.7) K L Q ys y ys 1 x Kys K y ys 1 x x L L (4.8) The boundary conditions are still the sae as in the first phase, while the velocities and displaceents given in Eqs. (4.4)-(4.7) are used as the initial conditions. Solving Eq. (4.8), at the end of phase when t=t s, the displaceent due to shear stops first, while the bending displaceent reains to the next phase. The ending tie t s and the axiu displaceent and velocity due to shear and bending are shown respectively as follows: 66

89 t s K L ys t d arctan t K ystd KL 4QL 6M d (4.9) y s t s K L y t sin T KL y t 4Q L 6M cos T 4Q L 6M s d s d KL (4.3) y t s K L ys td sin T KL ys td Q L 6M cos T Q L (4.31) KL y where t s s d sin 6 sin K L y td T KL y td QL M T (4.3) KL T K t t. In phase 3, only the bending failure induced otion reains, and the governing equation is the sae as Eq. (4.). Siilarly, the otion terination tie t f and the final displaceent y (t f ) are deterined as: t f K L y t s arctan t K KL tts 3M s (4.33) KL y ts 3M cos T3 3M y t f sin T3 y ts (4.34) K KL KL T K t t. where 3 f s Mode and 6M L p 8M 3L, bending failure ode. In this ode, the bending failure with a plateau deforation at the central portion of the eleent occurs when the blast load is sufficiently intensive. Different fro ode, two plastic hinges are generated offset fro the id-span of the eleent. There are totally three phases including one loading phase (phase 1) and two postloading phases (phase and phase 3) which end at t d, t 1, and t f respectively. In phase 1, the governing equation is expressed as: 67

90 Q L x L x p y Ky x L L (4.35) where ξ is the distance of the plastic hinge fro the id-span. The boundary conditions are as follows: Q x, Q x L Q, M x M, M x L (4.36) And the initial conditions are the sae as Eqs. (4.9)and (4.17). Thus Eqs. (4.36), (4.9), and (4.17) are used to deterine the integral constants when Eq. (4.35) is integrated. At the end of the loading period, when t=t d, the plastic hinge location which is indicated by ξ and the ending displaceent and velocity of phase 1 are derived respectively as follows: 6 pm L (4.37) p y y t d t 1 cos Kt d p (4.38) K d sin Kt d p (4.39) K In phase, the blast load has been released, and the velocity profile is the sae as that given in phase 1. However, the two plastic hinges ove toward the id-span of the eleent. At the end of phase, the two plastic hinges eet at the id-span and phase 3 otion starts then. The governing equation of phase is: Q y L x L x Ky x L L (4.4) where ξ is the distance between the plastic hinge and the id-span of the eleent. 68

91 The boundary conditions are: Q x, Q x L Q, M x M, M x L (4.41) Eqs. (4.38) and (4.39) are used as the initial condition to solve Eq. (4.4). At the end of phase when t=t 1, the two plastic hinges stops oving. The ending tie of phase and the axiu displaceent and velocity due to bending are shown respectively as follows: where A 4Q L 6M t ln p B pc 6BM A ln C BL A 1 td (4.4), B y t K L, C 6M Q L s d 1 d 1 d d. y t y t t t y t (4.43) y t 1 Kt d p sin (4.44) K In phase 3, the governing equation is the sae as Eq. (4.), and it can be solved in a siilar way as it was done for ode and ode 3. At the end of phase 3, when t=t f, the otion stops and the final tie t f is deterined as: t f pl sin Ktd arctan t K KL y t1 3M 1 (4.45) The final bending displaceent y (t f ) is: KL y t1 3M 3M y t f y t1 sin T4 cos T4 (4.46) K KL KL T K t t. where 4 f 1 69

92 Mode 5, 1.5 and p 8M 3L, cobined failure ode. Mode 5 is the ost coplicated ode as ode 1 and ode 4 are cobined. There are four phases including one loading phase (phase 1) and three post-loading phases (phase, phase 3, and phase 4) which end at t d, t s, t 1, and t f respectively. In phase 1, both shear and bending deforation occur. The governing equation is: Q L x L x p ys y ys Kys K y ys (4.47) x L L The boundary conditions are the sae as those in Eq. (4.16), and initial conditions are the sae as Eqs. (4.9) and (4.17). After integrating Eq. (4.47), at the end of phase 1 when t=t d, the plastic hinge location, the axiu displaceent and velocity of this phase are shown respectively as follows: 3M L (4.48) Q y s t y d s t Kt d 1 cos 3pM Q (4.49) 3KM d y Kt d 3pM Q sin (4.5) 3 K M y t d t 1 cos Kt d p (4.51) K d sin Kt d p (4.5) K In phase, the velocity profile is the sae as that of the previous phase, while the shear deforation tends to vanish. The governing equation is: 7

93 Q s s L x y y y Kys K y ys L x x L L (4.53) Based on the sae boundary conditions given in Eq. (4.15), and the initial conditions of Eqs. (4.49)-(4.5), Eq. (4.53) can be integrated with respect to tie. At the end of phase when t=t s, the shear deforation stops. The displaceent and velocity at the ending tie of phase due to shear and bending are shown respectively as follows: t s ystd K L arctan K 4 Q L 6 M y t K L s d (4.54) T A B A cos ys ts ys td sin T K K L K L T A B A cos ys ts ys td sin T K K L K L (4.55) (4.56) y t y t (4.57) s d In phase 3, the bending deforation reains the sae velocity profile as that in phase of ode 4, and the two plastic hinges start to ove toward the id-span of the eleent. The governing equation is the sae as given in Eq. (4.4). Siilarly, at the end of phase 3, when t=t 1, the two plastic hinges coincides at the id-span, and the ending displaceent and velocity of phase 3 are deduced as follows: t ln p E pf 6EM D ln F EL D 1 td (4.58) where D p sin Ktd K, E p Ktd s s cos 1, F 3M. y t y t t t y t (4.59) y t y t (4.6) 1 s 71

94 Siilarly, in phase 4 of ode 5, the governing equation is the sae as Eq. (4.). At the end of phase 4 when t=t f, the otion stops. t f pl sin Ktd arctan t K KL y t1 3M 1 (4.61) The final bending displaceent is: KL y t1 3M 3M y t f yn t1 sin T4 cos T4 (4.6) K KL KL 4.5. P-I diagras and discussions In each of the five odes that discussed in section 4.4, the final shear displaceent y s for direct shear failure and the final bending displaceent y for bending failure, or both for cobined failure can be derived. Based on the failure criteria discussed in section 4.3, P-I diagras for different odes can then be derived. Define diensionless variables P* and I* of the pressure and ipulse of a blast load as: P pl (4.63) M * I * Kt P (4.64) L * d Fro the equations for final displaceents induced by shear and bending failure, the P-I diagras can be represented in unified fors as follows: S P I h y (4.65) * *, s B P I L y (4.66) * *, where y s is the axiu displaceent due to shear which equals to y s (t f ) in a shear failure ode, and y s (t s ) in a cobined failure ode; y is the axiu displaceent due to bending which equals to y (t f ); S(P*,I*) and B(P*,I*) are 7

95 iplicit expressions with respect to the noralized pressure and ipulse for shear and bending according to failure criteria respectively. S(P*,I*) equals to the right part of Eqs. (4.14), (4.3), or (4.55) which gave the axiu shear deforation, and B(P*,I*) equals to the right part of Eqs. (4.), (4.34), (4.46), or (4.6) which represent the axiu bending deforation. Based on Eqs. (4.65) and (4.66), the P-I diagras corresponding to shear and bending failures can be drawn Differentiation of failure odes By deterining relative paraeters, the P-I diagras can be plotted for all five failure odes. Differentiation of the failure odes given in Fig. 3-3 is deterined by the requireent of the otion initiation. For exaple, for ode 1, the acceleration induced by the shear force at the supports should be larger than zero, and the axiu bending oent should be saller than the bending strength of the eleent. Therefore p M L and 1are derived, respectively. For ode, the initial acceleration due to bending should be larger than zero, while the axiu shear force should be less than the shear strength. Therefore and 4 3 M L p M L, or 1.5 and M L p 6M L are required. For ode 3, by cobining the two requireents that accelerations due to both shear and bending are larger than zero, the equations of boundaries are derived as p M 4 3 L. The ode 4 has the sae requireents as and ode and the boundaries are 1.5 and p 8M 3L. On the other hand, the ode 5 has the sae requireents as ode 3, and the boundaries are 1.5 and 6M L p 8M 3L. Three typical P-I diagras of shear and bending failures are plotted in Fig ) When.8 and p M L, only shear failure exists as shown in Fig. 4-3(a) (ode 1). Region A and region B indicate shear failure and no failure respectively. 73

96 ) As shown in Fig. 4-3(b) when 1. p M 4 3 L, the eleent and fails in the ode 3, which is the cobination of ode 1 and ode. Two P-I diagras, for shear failure and bending failure respectively, are plotted based on Eqs. (4.65)-(4.66). There are four regions in this case which indicate four different failure types of the eleent. In region A, which corresponds to larger pressure and ipulse, cobined failure occurs since both the axiu shear displaceent and the axiu bending displaceent exceed the failure threshold given by the shear and bending failure criteria. In region D, the pair of pressure and ipulse locates above the shear failure diagra but below the bending failure diagra, which indicates only shear failure occurred to the eleent. Siilarly, in region B, the pair of pressure and ipulse exceeds the bending failure diagra while it is below the shear failure diagra, which defines bending failure for the eleent. In region C, the eleent reains safe due to the pressure and ipulse pair is below both diagras. 3) When 1.8 and p 6M L, ode 4 and ode 5 ay occur. Fig. 4-3(c) shows the P-I diagras of ode 5 which is siilar to ode with 4 different regions. (a) Typical failure ode 1. (Mode 1) 74

97 (b) Typical failure ode (Mode 3) (c) Typical failure ode 1.5 (Mode 5) Fig. 4-3 Typical failure odes 75

98 4.5.. Soil-structure interaction effect The boundary conditions of different failure odes are exactly the sae as those of surface structures. This is because that, whether the spring effect acts on the bea or not, the initial conditions for all the failure odes reain the sae. Different soil has different density and copressibility, therefore the spring effect varies in all kinds of soil types. Considering a coparison with plastic beads-sand ixture (K=.39 MN/), ediu sands (K=3.83 MN/), and crushed rock (K=4.83 MN/) (Sawangsuriya et al. 1974), P-I diagras for shear and bending failure are plotted and copared in Fig It shows that, these three types of soil are characterized as low stiffness, interediate stiffness, and high stiffness. The low stiffness aterial is a ixture of 5% by volue of sand and 5% by volue of plastic beads. The interediate stiffness aterial is a ediu of uniforly-graded quartz sand. The high stiffness aterial is 19- crushed lie rock. It can be seen fro Fig. 4-4 that, the crushed rock has the highest stiffness and has significant effect on the structural failure. (a) Shear failure in different soils 76

99 (b) Bending failure in different soils Fig. 4-4 Failures in different soils Coparison is also done to the daage assessent results for surface structures when the spring coefficient is set sufficient sall (for exaple, K=1 N/) as shown in Fig It can be seen that both the shear failure and bending failure diagras atch well with those for surface structures. It verifies the present P-I diagras and the P-I diagra ethod suggested by Ma et al. (7) has been successfully extended to daage assessent for underground structures when the decoupling of SSI is considered. Furtherore, the present results can be applied to daage assessent for surface structures by assigning a sufficient sall spring coefficient in the equations. 77

100 Fig. 4-5 Coparison with surface structures against blast load Verification of continuity Since different failure odes give different final shear and/or bending displaceent, the solutions at the boundaries of different odes in Fig. 3-3 should be continuous. It eans that two odes should give the sae P-I diagra at their sharing boundary line. To verify the continuity of the equations, three checking points as shown in Fig. 3-3 are exained. CP1 ( p 3.6M L and 1. ) is shared by ode and ode 3, CP ( p 6.M L and 1.5 ) is shared by ode, ode 3, ode 4 and ode 5, and CP3 ( p 8.64M L and 1.8 ) is shared by ode 4 and ode 5. CP1 and CP3 are chosen randoly fro the boundary lines. The verification is shown in Table 4-1 which indicates the diagras of Mode, 3, 4, and 5 atch very well at the check points. 78

101 Table 4-1 Continuity verification Mode P* I * P* I * P* I * N.A. N.A N.A. N.A. 4 N.A. N.A N.A. N.A Case study To verify the applicability of the developed P-I diagras, a case study is carried out by considering a box-shape underground structure. An explosion occurred inside the underground structure as scheatically shown in Fig Daage assessent for the side wall is carried out and one unit strip of side wall is considered for siplicity. The soil layer is ediu sands and relative paraeters are given below. Half eleent length: L= Eleent height: h=. Unit ass of eleent: =5 kg/ Shear resistance of eleent: Q =5.8 kn Bending resistance of eleent: M =153.3 kn Soil spring coefficient: K=3.83 MN/ Charge weight: W=15 kg Once the distance R is confired, using the above paraeters, the constant pressure p and blast duration t d can be approxiated by using relative equations or read fro charts in TM5-13. Subsequently, the noralized pressure and ipulse are 79

102 calculated. In the present case study, three different distances R=1.7, 14. and 15.6, are considered respectively (see Table 4-). Table 4 - Case study Point Scaled Noralized Distance Duration Distance Ipulse (R) (t d ) (Z) (I * ) Z s /kg 1/ Z s.45 /kg 1/ Z s.76 /kg 1/ Pressure (p ) 1.86 MPa.66 MPa.76 MPa Noralized Pressure (P * ) ν is calculated as 1.374, the failure criteria of Eqs. (4.65) and (4.66) in which 3.49% and 1% are used for light daage criteria. Judging fro Fig. 3-3, the failure ode is Mode 5, which is the ost coplex cobination of shear and bending failures. As shown in Fig. 4-6, point Z1 is in region A, point Z is in region B, and point Z3 is in region C. That eans, when the scaled distance is 1.674, the structure will endure both light shear and light bending daage; when the scaled distance is.45, the structure will endure light bending daage only; when the scaled distance is.76, the structure is safe. 8

103 Fig. 4-6 Case study 4.7. Concluding rearks The present study derives P-I diagras for underground structure subjected to internal blast load. The SSI is considered and a rigid-plastic bea odel is applied in order to consider the cobined effect of both shear and bending failure. Verification shows that the present analysis is a successful extension of daage assessent of surface structures to underground structures. Results show that with increase of the soil stiffness, the SSI and its effect to the structural daage becoes very significant. An internal detonation generates very coplicated blast load to the underground structure walls, ceiling and floor, which is affected by any factors, such as venting, geoetry of the underground structure, charge weight and the way of charge placeent, etc. In the present study, the distance fro the centroid of the charge to the bea eleent changes along the bea. For exaple, when the charge is placed at the floor of the underground structure, the axiu distance will be 4L R, where R is the perpendicular distance fro the centroid of the charge to the bea eleent. Therefore the scaled distance is sensitive to the ratio of L/R. To ensure the 81

104 difference of the peak pressure along the bea within 1%, L/R should be less than.9. Under the above considerations, the present study assued that the blast load is uniforly acted to the structural eleent which gives a conservative assessent of structural daage and leads to a safer design for underground protective structures. And the localized daage due to unevenly distributed blast load is not in the scope of the present study. The structural eleent in analysis is siplified to a siply supported bea odel which ignores the rigidity at the supports. This results in conservative assessent of structural daage and gives a safe design of the structure although adjustent of the P-I diagras can be done for a fixed bea as it has been done for surface structures (Ma et al. 7). The rigid-plastic odel ignores the elastic deforation stage which ay cause discrepancies at the inor daage case, however it can well represent the structural deforation behavior when the structure undergoes ediate and large deforations. In the present analysis, a rectangular pulse shape is adopted. The pulse shape effect to the P-I diagras is inor especially in the two extree ipulsive and quasistatic cases. Besides, the rate dependence of the soil stiffness is also not considered in the analysis which is ainly due to that experiental data on the soil stiffness rate dependence is not available. Discussion of the P-I diagras shows that, the effect of soil cannot be ignored in estiating the dynaic response of underground structures against internal blast load. With the increase of soil stiffness, interaction between structure and surrounding soil becoes ore significant, and ore blast energy is absorbed by the soil. Coparison between the present result and the result of Ma et al. (7) shows that, result of present study can be easily applied to surface structure by setting a sall enough spring coefficient K. The case study shows that, daage assessent can be carried out by using the P-I diagra based on the ode approxiation ethod. 8

105 CHAPTER 5 PULSE SHAPE EFFECT ON STRUCTURAL DAMAGE INDUCED BY BLAST LOAD 5.1. Introduction In the early research work of Youngdahl s (1971), the pulse shape effect was eliinated by using a correlation paraeter ethod as shown in Eq. (4.5). Li and Meng (b), and Li and Jones (5b) also gave out soe useful results. It has been claied that the pulse shape effect to the P-I diagras is inor especially in the quasi-static load region and the ipulsive load region. However, such conclusions were drawn based on the analysis of the SDOF syste. Since the analysis using the MAM always siplifies the blast load to be a rectangular pulse, the pulse shape effect based on the MAM is still not clear, and the error caused by such difference is unknown. In the daage assessent using P-I diagras, the MAM used by Ma et al. (7) can be extended by using a generalized integration procedure so that rectangular, triangular, exponential or any other pulse-shaped pressure can be analyzed. Therefore the pulse shape effect can be considered in the analysis. In another aspect, the blast loads used by Li and Meng (b) were based on the sae load agnitude and duration. Their result contains the difference in pulse shapes and the difference in ipulse of each load type. In the present study, the blast load is siplified in three types with the sae ipulse and axiu pressure agnitude. Details of applying a generalized integration procedure to analyze the pulse shape effect for surface structure against blast load will be given. Both shear and bending failure odes will be adopted to show the difference. 83

106 5.. General assuptions and failure criteria To siplify the calculation, the shear and bending resistances of the bea eleent are assued to be constants. That is to say the rigid-plastic aterial odel is adopted. This assuption leads to the lower bound solution of the proble since the independent axiu bending and shear strength criteria used in analysis circuscribe an interactive shear-bending yield surface. The axiu displaceent of the bea eleent is again adopted as the criterion when generating the P-I diagras. To ensure the MAM is applicable in each failure ode, sall displaceent assuption should be used as Martin and Syonds (1966) did, and the yield stress is assued to be independent of the rate of strain. The failure odes include shear failure ode, bending failure ode, cobined failure ode, bending failure ode with a plastic zone, and cobined failure ode with a plastic zone, which are the sae as those in Ma et al. (7) as shown in Fig The failure criteria of the structural eleent under bending and shear are the sae as used in Chapter 3 and Integration procedure In the present study, a siply supported bea eleent under uniforly distributed blast pressure is adopted as the odel to illustrate the surface structure under blast load. The derivation procedure of P-I equations reains the sae as previous chapters, therefore the governing equations, boundary conditions, initial conditions, and results of structural response are the sae in each phase of each ode as those in Ma et al.'s work (7). Since a general shape of the pulse loading is assued and to siplify the derivation of P-I equations, a generalized integration procedure for different cases have to be adopted. In the work of Ma et al. (7) and Chapters 3 and 4, the blast load is a function of tie in the governing equations which couples with other paraeters such as structural velocity or displaceent. It becoes difficult and even ipossible to solve the governing equation without decoupling this "redundant" tie-dependent 84

107 paraeter. By using the integration procedure, the differential equations can be solved easily without losing accuracy. Therefore, the general integration procedure is adopted. In the cases of surface structure under blast load, the pressure changes with tie. The aterial properties (shear strength Q and bending strength M ) and the blast pressure p should be obtained first, so that the failure ode can be deterined. Starting fro the first phase (loading phase), the blast duration t d is divided into plenty of tie steps. In each tie step, the pressure will be calculated according to the blast load shape equation and the structural responses obtained will be used as the initial conditions for the next tie step till the end of current phase. In the postloading phase(s), the siilar procedure will be used and the final structural displaceent can be calculated. The flowchart in Fig. 5-1 shows the generalized integration procedure. 85

108 Calculate ν and p to deterine which failure ode to be used Input aterial properties and data of bea odel Set y y t, p pt and calculate y, Solve y, y, p, y at the end of this tie step, and set t t t No t=t d? Yes Enter post-load phase and set yy, as the initial conditions Solve yyy,,, and set t t t No End of phase? Yes Output final displaceent Fig. 5-1 Flow chart for surface structural eleent against blast load 86

109 5.4. Derivation of P-I equations Before carrying out the nuerical calculation, the blast pressure p and the strength ratio ν (.5 QL / M) should be given, and consequently the failure ode can be decided. Three different load cases are used in the present study, i.e., 1) Rectangular load, p p when t td p when t td (5.1) ) Triangular load, p p 1 / t td when t t p when t t d d (5.) 3) Exponential load, 1 / p when t t.75 tt / p p t t d d e when t td d (5.3) Then the aterial properties (shear resistance Q, bending resistance M ), geoetrical properties of the bea (unit ass, half length L) should be inputted to the integration procedure. For coparison reason, the present study uses the three pulse shapes only. More general pulse shapes with an arbitrary pressure-tie history can also be applied. In this section, the iterative tie step is set as t t / n, where n is a pre-defined constant which satisfies the precision requireent of the integration Mode 1 (Shear failure ode) For ode 1 failure, the governing equations in phase 1 and phase are the sae as given in Eq. (5.4) and (5.5) respectively. The only difference is that the pressuretie history of the blast load becoes arbitrary and it can be represented by p=p(t) in the above equations. d 87

110 Q p y x s (5.4) Q ys x (5.5) In phase 1, the boundary conditions and the initial conditions are the sae as in Eq. (3.14) and (3.15). Then Eq. (5.4) is integrated with respect to tie t. In every integration step, the value of p needs be updated according to the load function. The equations for integration are derived as: y s t i i p t L Q (5.6) L ys ti 1 ys ti ys ti t (5.7) ys ti 1 ys ti ys ti t (5.8) After each tie step, set t i 1 t i t. At the end of this phase when t td, the axiu velocity and displaceent of the bea are y t and s d y t respectively. In phase, the boundary conditions reain the sae as in the previous phase, while the initial conditions change to:, y t t y t y t t y t (5.9) s d s d s d s d The integration equations for structural axiu velocity and displaceent reain the sae as Eqs. (5.7) and (5.8), while the axiu shear acceleration is as follow: The integration stops when ys ( t tf y s t i Q (5.1) L ), and the final displaceent is s f s d y t Mode (Bending failure ode) In phase 1, the pulse pressure acts on the bea and the governing equation for each tie step is: 88

111 Q x p y 1 x L (5.11) The boundary conditions and the initial conditions are the sae as Eq. (3.) and (3.1). In every integration step, the value of p needs to be updated according to the load function. The equations for integration are derived as: y t i 3p ti L M (5.1) L y ti 1 y ti y ti t (5.13) y ti 1 y ti y ti t (5.14) After each tie step, set t i 1 t i t. At the end of this phase when t td, the axiu velocity and displaceent of the bea are y t and y t respectively. Phase is the only post-loading phase and the governing equation is: d d Q x y 1 x L (5.15) The boundary conditions reain the sae as Eq. (3.), while the initial conditions change to:, y t t y t y t t y t (5.16) d d d d The integration equations for structural axiu velocity and displaceent reain the sae as Eqs. (5.13) and (5.14), while the axiu bending acceleration is as follow: y 3M (5.17) L 89

112 The integration stops when y ( t tf y t. ), and the final displaceent is f Mode 3 (Cobined failure ode) During phase 1, the governing equation is: Q x p ys y ys 1 x L (5.18) The initial conditions are as Eqs. (3.15) and (3.1), and the boundary conditions are the sae as Eq. (3.). The equations for the shear and bending responses integration are the sae as in Eqs. (5.7), (5.8), (5.13), and (5.14), while the axiu shear and bending acceleration are as follows: y y s t i t i i 6 4 p t M Q (5.19) L i 6 p t M Q (5.) L After each tie step, set t i 1 t i t. At the end of this phase when t td, the axiu velocity and displaceent of the bea for shear and bending are y t, y t and s d d y t respectively. d y t, In phase, shear deforation will terinate at the end, and the bending deforation will continue. The governing equation is: s d Q x ys y ys 1 x L (5.1) The boundary conditions reain the sae as Eq. (3.), while the initial conditions change to Eqs. (5.9) and (5.16) for shear and bending respectively. The integration equations for structural axiu velocity and displaceent reain the sae as Eqs. (5.7), (5.8), (5.13), and (5.14), while the axiu shear and bending acceleration are as follows: 9

113 y y s 6M 4Q L (5.) L t i Q L 6M (5.3) L t The integration of this phase stops when ys ( t ts displaceent is s s i ), and the final shear y t. At the end of this phase, the axiu bending velocity and displaceent are y t and s y t. s In the following phase 3, the integration procedure is siilar to phase of ode. Only the difference is that the initial conditions changes to:, y t t y t y t t y t (5.4) s s s s The integration of this phase stops when y ( t tf displaceent is f y t. ), and the final bending Mode 4 (Bending failure ode with a plastic zone) Mode 4 is the bending failure with a plateau deforation at the central portion of the eleent which ay occur when the blast load is sufficiently intensive. Different fro ode, two plastic hinges are generated offset fro the id-span of the eleent. In the range of x L, the governing equation is: Q L p y x x L (5.5) where ξ is the distance of the plastic hinge fro the id-span and L 6 M / p. The initial conditions are the sae as those in Eq. (3.1) and the boundary conditions are as those in Eq. (3.3). 91

114 The equations for the bending responses integration are the sae as in Eqs. (5.13) and (5.14), while the axiu bending acceleration is: y t i pt i (5.6) After each tie step, set t i 1 t i t. At the end of this phase when t td, the axiu velocity and displaceent of the bea are y t and y t respectively. In phase which is the first post-loading phase the two plastic hinges ove toward the id-span of the bea and eet at the id-span. The governing equation is: d d Q y x Lx L (5.7) The initial conditions are as Eq. (5.16) and the boundary conditions are as Eq. (3.35). The integration equations for structural axiu velocity and displaceent reain the sae as Eqs. (5.13) and (5.14), while the axiu bending acceleration equals to zero. The integration of this phase stops when t ( t t1 ), and the bending displaceent at the end of this phase is y t. 1 In phase 3, i.e., the second post-loading phase, bending deforation will terinate at the end. The iterative procedure is siilar to phase of ode and the only difference is that, the initial condition changes to:, y t t y t y t t y t (5.8) The integration of this phase stops when y ( t tf displaceent is f y t. ), and the final bending 9

115 Mode 5 (Cobined failure ode with a plastic zone) Mode 5 is the ost coplicated ode which cobined ode 1 and ode 4. The governing equation for each tie step in the range of x L for phase 1 is: Q p ys y ys L x x L (5.9) The initial conditions are as Eqs. (3.14) and (3.), and the boundary conditions are as Eq. (3.3). The equations for the shear and bending responses integration are the sae as in Eqs. (5.7), (5.8), (5.13), and (5.14), while the axiu shear and bending acceleration are as follows: y y s t i t i L p L 4Q L 6M L p L Q L 6M (5.3) (5.31) After each tie step, set t i 1 t i t. At the end of this phase when t td, the axiu velocity and displaceent of the bea for shear and bending are y t, y t and s d d y t respectively. d y t, In phase, which is the first post-loading phase, the shear deforation will terinate at the end, and the bending deforation will continue. The governing equation is: s d Q ys y ys L x x L (5.3) The boundary conditions reain the sae as Eq. (3.3), while the initial conditions change to Eqs. (5.9) and (5.16) for shear and bending respectively. 93

116 The integration equations for structural axiu velocity and displaceent reain the sae as Eqs. (5.7), (5.8), (5.13), and (5.14), while the axiu shear and bending acceleration are as follows: y s t i Q L (5.33) The integration of this phase stops when ys ( t ts displaceent is s s y t (5.34) i ), and the final shear y t. At the end of this phase, the axiu bending velocity and displaceent are y t and s y t. s The integration for phase 3 is siilar to phase of ode 4. The integration of this phase stops when t phase is y t. 1 ( t t1 ), and the bending displaceent at the end of this In phase 4, bending deforation will terinate at the end. The integration procedure is siilar to phase 3 of ode 4, and it stops when y ( t tf the final bending displaceent is f y t. ), and 5.5. Result discussions Based on the integration ethod, the final displaceents induced by shear or bending can be derived nuerically, and the P-I diagras can be represented in a siilar way. It should be entioned that the present integration analysis uses a constant load in each tie step in spite of the pulse shape of the blast load. Analytical results can thus be derived for each tie step. The integration ethod adopted here is sei-analytical which has the advantage to consider arbitrary pulse shapes and thus the effect of the pulse shape to the P-I diagras can be analyzed. Daages to the structure are categorized into three levels as shown in Table 5-1. It lists different daage levels of an RC bea eleent according to epirical shear failure criteria of Yu and Jones (1991) and Krauthaer (1998) which are for the 94

117 structures without protective design. The bending daage levels are derived fro the daage of RC ebers in TM5-13 (199). There are three levels of daage: light daage, oderate daage, and severe daage. For shear design, average shear strain γ ν indicates different daage levels, while support rotation β is applied for bending. By using the failure criteria and daage level in table 5-1, the P-I equations can be represented in unified fors as Eq. (3.49). The diensionless variables P* and I* of the pressure and ipulse of a blast load are defined as: P p L p L Q M (5.35) * I p t p t Q 4M L * d d (5.36) Table 5-1 Different daage level under epirical bending and shear failure criteria Failure Criteria Light Daage Moderate Daage Severe Daage Shear Average shear strain 1% % 3% Bending Ratio of centerline deflection to half span 6.993% 14.54% 1.56% The pulse shape effect on P I diagras has been early discussed by Youngdahl s (1971) for a rigid, perfectly plastic SDOF odel. The correlation paraeter ethod is shown in Eq. (4.5). Such ethod was widely used, but it is not verified for the application in MAM. Based on the sei-analytical results, the pulse shape effect can be presented. The three kind of load shapes are shown in Fig. -. The pressure agnitude reains the sae aong each type of loads but the blast duration differs. The t dr, t dt, and t de, in Fig. - are blast durations for rectangular load, triangular load, and exponential load respectively. In noral procedure of plotting P-I 95

118 diagras, the blast pressure and duration are input and the structural displaceents are output. But in Fig. 5- and 5-3, the blast pressure and the structural displaceents are input and the blast durations are calculated reversely. In Fig. 5-4, the blast durations are calculated according to Eq. (4.5). In the following figures, T load eans the triangular load, E load denotes the exponential load, and R load denotes the rectangular load. When a daage level of the structural eleent is deterined, the P-I diagras for the pulse shape effect of surface structure under blast load at different shear and bending daage levels are shown in Fig. 5- and 5-3. It indicates that to reach the sae daage level, higher pressure and ipulse are required for the exponential pulse load and the result fro the rectangular load can be considered as the upper bound solution which is relatively conservative. The pulse effect in shear failure is inor in all regions of P-I diagras, but it is significant in bending failure, especially in the dynaic region and soe part of the ipulse region. (a) Light daage 96

119 (b) Moderate daage (c) Severe daage Fig. 5 - Pulse shape effect on shear failure for surface structure against blast load (fixed daage level) 97

120 (a) Light daage (b) Moderate daage 98

121 (c) Severe daage Fig. 5-3 Pulse shape effect on bending failure for surface structure against blast load (fixed daage level) Taking oderate shear and bending daage level as an exaple, when using results of Youngdahl (1971) to eliinate the pulse shape effect, the relationship aong P-I diagras under the sae ipulse and pressure agnitude for different load shapes is shown in Fig In this exaple, although the load durations t dr <t dt <t de, the final displaceent of structural eleent under the exponential load is the sallest, eanwhile that under the rectangular load is the largest. Such result coincides with that of Jones and Song (1986). It also indicates that the shear failure is sensitive to the pulse shape in the dynaic region of P-I diagras and the bending failure is sensitive in both the dynaic and ipulse regions. Moreover, the error caused by pulse shape effect in shear failure is relatively sall, and the error in bending failure cannot be ignored. Generally speaking, the shear failure is not sensitive to the pulse shape effect as the bending failure does. It is reasonable since: a) shear displaceent stops before structural eleent reaches its final bending displaceent, and is uch saller than the bending displaceent; b) structural eleent has less tie to response on shear displaceent than bending, therefore shear displaceent is ainly decided by agnitude of blast load, while bending displaceent is controlled by load agnitude and pulse shape as well. 99

122 (a) Moderate shear daage (b) Moderate bending daage Fig. 5-4 Pulse shape effect on surface structure against blast (fixed ipulse) Fig. 5-5 and 5-6 show the relative errors of final tie and displaceent for the shear and bending failures at different daage levels. At the light daage level, the errors of t s, t f, y s, and y f are 1%, 9%, 1% and 11% respectively. At the oderate daage level, the errors change to 15%, %, 14%, and %. The errors at the severe 1

123 daage level are the sallest which are 5%, 5%, 5%, and 6% respectively. The largest errors exist in oderate daage level, and the error at light or severe daage level is relatively sall. The reason of such phenoenon exists in the pulse shape effect on P-I diagras as explained previously. At light and severe daage levels it is closer to the quasi-static and ipulse region of P-I diagras respectively, and the pressure and ipulse resulted in the oderate level are ore likely to fall in the dynaic region. Such phenoenon is only valid for structural eleent under blast load. (a) Final tie of shear displaceent 11

124 (b) Final tie of bending failure Fig. 5-5 Error in the final tie of structural response (a) Final displaceent of shear failure 1

125 (b) Final displaceent of bending failure Fig. 5-6 Error in the final structural displaceent 5.6. Conclusions In the analysis of surface structure, the error caused by the pulse shape difference is very liited in shear displaceent, while the effect is distinct in bending displaceent. The pulse effect in shear failure is inor in all regions of P-I diagras, but it is significant in the dynaic region and the ipulse region in bending failure. The reason of such phenoenon could be explained by the properties of cobined failure odes. In the two cobined failure odes (odes 3 and 5), the shear displaceent always stops before the bending displaceent, and is saller than the final bending displaceent. The pulse shape effect on shear displaceent is not as significant as that on bending displaceent. Results fro the present study verify that, under the sae ipulse, the final displaceent of structural eleent under exponential load is the sallest eanwhile that under rectangular load is the largest. The largest errors exist in oderate daage level, and the error in light or severe daage level is relatively sall. In 13

126 another aspect, errors in quasi-static and ipulse regions in P-I diagras are inor, but the error in dynaic region is significant in all daage levels. Youngdahl (1971) stated that the pulse shape effect to the P-I diagras of bending displaceent is inor especially in the quasi-static load region and the ipulsive load region. But the result fro the present study shows that P-I diagras ay also be sensitive to pulse shape effect in the ipulsive region for both shear and bending displaceents. That is to say the correlation paraeter ethod ay not be applied to P-I diagras based on the MAM. Generally speaking, results under triangular load and rectangular load overestiate the final displaceents of both shear and bending, and the pulse shape effect cannot be ignored. In the following chapter, the generalized integration ethod will be applied to the P-I equations for underground structure against blast external and internal blast load. Further extension will be carried out. 14

127 CHAPTER 6 NON-CONSTANT SOIL-STRUCTURE INTERACTION EFFECT ON UNDERGROUND STRUCTURE DAMAGE TO BLAST LOAD 6.1. Introduction In Chapters 3 and 4, the P-I diagras based on the MAM are extended fro surface structures to underground structures. The SSI is siplified to be a daping or stiffness coefficient for underground structure against external or internal blast load respectively. Research result shows that, soil is a coplex aterial which cannot be easily siplified as a hoogenous and linear aterial. Therefore, the assuption of constant SSI coefficient ay lead to unavoidable error when calculating the axiu structural deforation, and such error will affect the accuracy of the evaluated structural daage. This chapter extends the MAM introduced in Chapter 3 and 4. Non-constant SSI is considered for underground structures subjected to external or internal explosions. Based on a siilar integration procedure as introduced in Chapter 5, the P-I diagras of the underground structures are derived sei-analytically and the nonconstant SSI effect is therefore included. Both scenarios of underground structures against external and internal blast loads are adopted and the results are copared to those given in Chapter 3 and 4, and the pulse shape effect will also be discussed as done in Chapter Assuptions and failure criteria The assuptions and failure criteria used in the present study reain the sae as those in chapters 3 to 5. The rigid-plastic aterial odel is again adopted which leads to the lower bound solution of the proble since the independent axiu bending and shear strength criteria used in analysis circuscribe an interactive shear-bending yield surface. 15

128 The axiu deforation occurred in the bea eleent is again adopted as the failure criterion. As the ajor advantage of the MAM, the failure odes still include shear failure, bending failure, cobined failure, bending failure with a plastic zone, and cobined failure with a plastic zone, which are the sae as those in Ma et al. (7) as shown in Fig These analytical solutions kineatically adissible only Derivation of P-I equations In the present study, a siply supported bea under uniforly distributed blast pressure is adopted. The derivation of P-I equations reains the sae as previous chapters, therefore the governing equations, boundary conditions, initial conditions, and results of structural response are the sae in each phase of each ode, except that the daping coefficient and the soil stiffness varies over the response tie. The integration procedures for different cases are discussed as follows. In the cases of underground structure under external blast load, the pressure and daping coefficient changes with tie. Again, the aterial properties (shear strength Q and bending strength M ) and the axiu blast pressure p should be obtained first, so that the corresponding failure ode fro the five possible odes can be deterined. In the loading phase, the whole duration is divided into sufficiently sall tie steps. In each tie step, the pressure will be calculated according to the blast load shape equation, the daping coefficient C will be calculated based on the structural response tie, and the structural responses obtained will be used as the initial conditions for the next tie step till the end of this phase. In the post-loading phase(s), the siilar procedure of integration will be used and the final structural displaceent can be calculated. In the post-loading phase(s), the daping effect is ignored, and the reason was discussed in section The flowchart in Fig. 6-1 shows the general procedure of such generalized integration procedure. 16

129 In the cases of underground structure under internal blast load, the pressure and soil stiffness effect changes with tie. Again, the failure ode can be deterined by the aterial properties (shear strength Q and bending strength M ) and the axiu blast pressure p. In the loading phase, the whole duration is divided into sufficient sall tie steps. In each tie step, the pressure p and soil stiffness K will be updated according to the structural displaceent, and the structural responses will be used as the initial conditions for the next tie step till the end of this phase. In the post-loading phase(s), the siilar procedure of integration will be used and the final structural displaceent can be calculated. In the post-loading phase(s), the stiffness effect exists throughout the phases since the bea odel always contacts with the soil. The flowchart in Fig. 6- shows the flow of such generalized integration procedure. It should be entioned that the integration procedures give sei-analytical solutions of underground structural response to arbitrary loading and non-constant soil property. The current analysis is applicable to a ore general case for the assessent of structural daage against blast loading. With nuerical integrations being involved, P-I curves with respect to different case can be conveniently generated and the MAM becoes ore effective in structural daage assessent. Again in each tie step, since a constant load, a constant daping coefficient or a soil stiffness are considered, an analytical solution is obtained siilar to Chapter 3 and 4. The result of the current step becoes the initial condition of the next step, and thus the explicit integration solves the inelastic dynaic response of the structure eleent efficiently. 17

130 Calculate ν and p to deterine which failure ode to be used Input aterial properties and data of bea odel Set y y t, p pt, C C t,and calculate y Solve y, y, p, y, C at the end of this tie step, and set t t t No t=t d? Yes Enter post-load phase and set yy, as the initial conditions Solve yyy,,, and set t t t No End of phase? Yes Output final displaceent Fig. 6-1 Flow chart for underground structure against external blast load 18

131 Calculate ν and p to deterine which failure ode to be used Input aterial properties and data of bea odel Set y y t, p pt, K K y,and calculate y Solve y, y, p, y, K at the end of this tie step, and set t t t No t=t d? Yes Enter post-load phase and set y, y, K as the initial conditions Solve y, y, y, K, and set t t t No End of phase? Yes Output final displaceent Fig. 6 - Flow chart for underground structure against internal blast load 19

132 Underground structure against external blast load The procedure and ost of the equations for generalized integration procedure of underground structure under external blast load are the sae as those for surface structure under blast load in Chapter 5. Only the differences are stated below. The SSI is siplified as a daping coefficient C which is a function of tie. The governing equation cannot be solved by iteration ethod used in previous chapters since it couples with y. Therefore, C is treated as a constant in every tie step when using the general integration ethod and the tie step is saller enough to ake sure the error caused by such dealing ethod of C does not affect result of structural displaceent so uch. Mode 1 (Shear failure ode) Phase 1: Due to the non-constant daping coefficient effect, the governing equation becoes: Q p ys Cy x s (6.1) where C(t) is the daping effect, which is a function of tie. And the axiu shear acceleration for integration is: y s t i Ct i i e p t L Q (6.) L Phase : All equations are the sae as those for surface structure since the daping coefficient vanishes in the post-loading phase. Mode (Bending failure ode) Phase 1: The governing equation changes to: Q x x p y 1 Cy 1 x L L (6.3) The axiu bending acceleration for integration is: 11

133 y t i Ct i i 3e p t L M (6.4) L Phase : All equations are the sae as those for surface structure. Mode 3 (Cobined failure ode) Phase 1: The governing equation is: Q p ys y ys 1 x Cys C y ys 1 x x L L (6.5) For integration, the axiu shear and bending acceleration are as follows: y y s t i t i Ct i i 6 4 e p t L M Q L (6.6) L Ct i i 6 e p t L M Q L (6.7) L Phase : All equations are the sae as those for surface structure. Phase 3: All equations are the sae as those for surface structure. Mode 4 (Bending failure ode with a plastic zone) Phase 1: The governing equation changes to: Q L x L x p y Cy x L L (6.8) The axiu bending acceleration for integration are as follows: y t i Ct i e pti (6.9) Phase : All equations are the sae as those for surface structure. Phase 3: All equations are the sae as those for surface structure. 111

134 Mode 5 (Cobined failure ode with a plastic zone) Phase 1: The governing equation is: Q L x L x p y y y Cy C y y x L L s s s s (6.1) The axiu shear and bending acceleration for integration are as follows: Ct i e ys ti p ti L 4Q L 6M L Ct i e y ti p ti L Q L 6M L (6.11) (6.1) Phase : All equations are the sae as those for surface structure. Phase 3: All equations are the sae as those for surface structure. Phase 4: All equations are the sae as those for surface structure Underground structure against internal blast load The procedure and ost of the equations for generalized integration procedure of underground structure under internal blast load are the sae as those for surface structure under blast load. The soil stiffness K is a function of the structural displaceent but it is treated as a constant in every tie step of integration. The governing equations cannot be solved by the iteration ethod used in previous chapters since it couples with structural displaceent. Therefore, K is treated as a constant in every tie step when using the general integration ethod and the tie step is saller enough to ake sure the error caused by such dealing ethod of K does not affect result of structural displaceent so uch. Only the differences are stated below. Mode 1 (Shear failure ode) Phase 1: The governing equation changes to: 11

135 Q p ys Ky x s (6.13) The axiu shear acceleration for integration is: y s t Phase : The governing equation is: i Kt i sin p ti L Q (6.14) K L Q y s Ky x s (6.15) The axiu shear acceleration in this phase for integration changes to: y s t i Mode (Bending failure ode) Phase 1: The governing equation is: Kt i Kt i sin K ys td cos y s td LK Q (6.16) L Q x x p y 1 Ky 1 x L L (6.17) The axiu bending acceleration for integration changes to: y t Phase : The governing equation is: i Kt i 3cos pt i L M (6.18) L Q y 1 x Ky 1 x x L L (6.19) Here the axiu bending acceleration for integration changes to: 113

136 y t i Kt i Kt i cos y td KL 3M sin K y td (6.) L Mode 3 (Cobined failure ode) Phase 1: The governing equation is: Q p ys y ys 1 x Kys K y ys 1 x x L L (6.1) The axiu shear and bending acceleration changes to: y y s t i t Phase : The governing equation is: i Kt i cos pt i L 4QL 6M (6.) L Kt i cos pt i L QL 6M (6.3) L Q ys y ys 1 x Kys K y ys 1 x x L L (6.4) The axiu shear and bending acceleration changes to: y y s t i t i 3 Kt i Kt i cos K 6M 4QL ys td KL sin K L ys td (6.5) KL KL 3 Kt i Kt i cos K QL 6M y td KL sin K L y td (6.6) KL KL Phase 3: The governing equation is the sae as Eq. (6.19). The axiu bending acceleration is as Eq. (6.). 114

137 Mode 4 (Bending failure ode with a plastic zone) Phase 1: The governing equation changes to: Q L x L x p y Ky x L L (6.7) The axiu bending acceleration for integration is: y t i Kt i pti L M 3cos L (6.8) Phase : The governing equation is: Q L x L x y Ky x L L (6.9) The axiu bending acceleration for integration is zero. Phase 3: The governing equation is the sae as Eq. (6.19), and the axiu bending acceleration for integration is: y t i Kt i Kt i sin K yt1 cos yt1 KL 3M (6.3) L Mode 5 (Cobined failure with a plastic zone) Phase 1: The governing equation changes to: Q L x L x p y y y Ky K y y x L L s s s s (6.31) The axiu shear and bending acceleration are: y s t i Kt i cos pti L 4Q L 6M L (6.3) 115

138 y t i Kt i cos pti L Q L 6M L (6.33) Phase : The governing equation is: Q L x L x y s y y s Ky s K y y s x L L (6.34) The axiu shear and bending acceleration change to: y s t i 3 Kt i Kt i sin K ys td M cos K Q 3ys ti KM (6.35) KM 3KM y t Phase 3: The governing equation is: i Kt i Kt i sin K y td cos Ky td (6.36) Q L x L x y Ky x L L (6.37) The axiu bending acceleration equals to zero. Phase 4: The governing equation reains the sae as Eq. (6.19), and the axiu bending acceleration is: y t i Kt i Kt i cos yt1 KL 3M sin K yt1 (6.38) L 6.4. P-I diagras and result discussion The noralized pressure (P * ) and noralized ipulse (I * ) used for P-I diagras in the following reain the sae as defined in Eq. (3.48). 116

139 Underground structure (external blast load) The SSI in underground structure against external blast load scenario is siplified as a daping coefficient. Such assuption is based on the physical basis of SSI in the external explosion cases. The daping effect will definitely becoe saller during the duration of blast load since the contact area between soil and structure changes fro full contact to non-contact when the blast wave is reflected on the surface RC wall. The soil layer will copletely be separated fro RC wall when the velocity of soil layer is saller than that of structure according to Eq. (3.5); and the tie ay be at the end of loading or even earlier, which depends on soil properties and intense of load. The daping coefficient, which represents the siplification of soil-structure interaction, can be expressed as Eq. (6.39). The daping factor in the present study is assued attenuating with tie and vanishing when the blast load stops. Such siplification is based on the fact that the SSI starts when the blast load acting on the structure and attenuates to zero when the blast load stops. The change of SSI agnitude ay be different in real case, while it is not in the scope of current study. C 1 C t t t d (6.39) where C is the axiu daping effect during blast load. Fig. 6-3 and 6-4 show the effect of the non-constant daping for underground structure under external blast load. Like the surface structure under blast load, the pulse shape effect on bending deforation at different daage levels is ore significant than that on shear deforation. The P-I diagras of shear and bending deforations are sensitive to the pulse shape effect in the ipulse and dynaic regions. Coparing with Fig. 5-3, it indicates that the pulse shape effect on the ipulse and dynaic region in P-I diagras of shear deforation cannot be ignored for underground structures under external blast load. The only reason should be the siplified SSI used in the derivation of P-I equations. The pulse shape effect on bending deforation is not as significant as that on surface structures but still cannot be neglected. 117

140 (a) Light daage (b) Moderate daage 118

141 (c) Severe daage Fig. 6-3 Pulse shape difference of shear failure for underground structure against external blast load (a) Light daage 119

142 (b) Moderate daage (c) Severe daage Fig. 6-4 Pulse shape difference of bending failure for underground structure against external blast load The difference of constant and non-constant daping effect for shear and bending failure ode is shown in Fig The oderate daage level is adopted as an exaple. It can be seen fro Fig. 6-5 a) that, the non-constancy of daping effect 1

143 on shear deforation is sensitive in the ipulsive and dynaic regions of P-I diagras, and these two P-I curves intersect in the quasi-static region. It is because that, under the sae pressure and ipulse, the non-constancy of daping affects the final structural displaceent especially when the blast pressure is low. Fig. 6-5 b) shows that the pulse shape effect on bending deforation is siilar to that on shear deforation, but no intersection of P-I curves. Generally, Fig. 6-5 shows that the P-I diagras are sensitive to the non-constancy of SSI in the dynaic and ipulsive region. The reason lies in the fact that the effect of daping on the final displaceent of structural eleent is based on the structural velocity. In the quasi-static region of P-I diagras, the structural velocity is relatively low therefore the daping effect is liited. But in the dynaic and ipulsive region, the effect of daping increase with the axiu structural velocity. The constant daping effect will absorb ore energy than the nonconstant daping effect so that under the sae pressure and ipulse the structural displaceents for shear and bending with non-constant daping coefficient are saller than that with a constant daping coefficient. It can be seen as well fro Fig. 11 that, the result of non-constant daping effect is close to the result of without considering the soil daping effect. This phenoenon soehow verifies that neglecting daping effect in blast effect analysis is reasonable, but the accuracy of result in dynaic region of P-I diagra should be further discussed. 11

144 a) Shear failure b) Bending failure Fig. 6-5 Difference of constant and non-constant daping effect 1

145 6.4.. Underground structure (internal blast load) The soil-structure interaction is siplified as the soil stiffness effect, and the K is expressed as a piecewise function as below: K y y K Kc K y y yc Kc y y c c (6.4) where K is the soil stiffness before copression, K c is the soil stiffness after full copression, y c is the displaceent when the soil is assued as fully copressed. The siplification of SSI in underground structure against internal blast load scenario is to adopt a piece-wise stiffness coefficient which is a function of structural displaceent. Such assuption is based on the physical properties of soil before and after copression, which presents the hardening of soil aterial under copression in acro physical scope. Before copression, the soil is assued as elastic base with stiffness K, and its stiffness increases to K c when the soil is fully copressed and the change of stiffness is assued as linear. Fig. 6-6 and 6-7 show the non-constant soil stiffness effect with pulse shape difference of underground structure under internal blast load. Different fro the pulse shape effect on underground structure against external blast load, the effect in present study is significant in the quasi-static region in P-I diagras for shear deforation as well. It is because the soil stiffness effect exists throughout the process of structural deforation. In the quasi-static region and under the sae ipulse, the effective pressure of rectangular load is higher than that of triangular or exponential load, therefore the displaceent of structural eleent under rectangular load is the largest. Such fact exists in the bending deforation, but the pulse shape effect is not so significant in the quasi-static region because the bending deforation is norally uch larger than the shear deforation. 13

146 (a) Light daage (b) Moderate daage 14

147 (c) Severe daage Fig. 6-6 Pulse shape difference of shear failure for underground structure against internal blast load (a) Light daage 15

148 (b) Moderate daage (c) Severe daage Fig. 6-7 Pulse shape difference of shear failure for underground structure against internal blast load The difference of constant and non-constant soil stiffness effect on shear and bending failure ode is shown in Fig The non-constancy of SSI affects the P-I diagras in the sae way as in Fig It once ore indicates that the P-I 16

149 diagras are sensitive to the non-constancy of SSI in the dynaic and ipulsive regions. Again, the constant soil stiffness will absorb ore energy than the nonconstant soil stiffness so that under the sae pressure and ipulse the structural displaceents for shear and bending failure with non-constant soil stiffness are saller than that with constant one. This explains the reason why the result of nonconstant soil stiffness is close to that without considering SSI. Results in Figs. 6-5 and 6-8 show that, the structural displaceent given by constant SSI used in previous study leads to a risky design of underground RC protective structure. The non-constant SSI is ore reasonable and highly recoended. a) Shear failure 17

150 b) Bending failure Fig. 6-8 Difference of constant and non-constant soil stiffness effect 6.5. Concluding rearks The P-I diagra result is affected by different factors such as pulse shapes, daage levels, non-constancy of daping or spring effect, and so on. In the analysis of underground structure against external blast load, the pulse shape effect and the non-constancy of daping effect are considered. Fro the P-I diagras, it can be concluded that the pulse shape effect still exists in such analysis. The result in shear failure shows that, the error between triangular load and exponential load is relative sall, and the over-estiation fro rectangular load cannot be ignored. In the analysis of bending failure, results fro triangular load and exponential load are closer. Both triangular load and rectangular load overestiate the structural response. The P-I diagras of shear and bending deforations are sensitive to the pulse shape effect in the ipulse and dynaic regions. The pulse shape effect on bending deforation is not as significant as that on surface structures but still cannot be neglected. The P-I diagras are sensitive to the non-constancy of SSI in the dynaic and ipulse region. That eans P-I 18

151 diagras for underground structures against external blast load is ore sensitive to non-constancy of SSI when the blast duration is short. The analysis of underground structure against internal blast load shows that, the result of triangular load is closer to the exponential load. Again, both triangular and rectangular loads overestiate the final displaceents due to shear and bending. The result also shows that the P-I diagras for underground structures against internal blast load are ore sensitive to non-constancy of SSI when the structural displaceent is under control of ipulse and dynaic load. Siilar conclusions can be drawn as to those for underground structure under external blast load. Technically, the generalized integration procedure can be easily applied to a ore coplicated calculation, for exaple a cobined failure with the consideration of ore accurate blast load tie history, nonlinear aterial properties, pulse shape effect, and non-constant soil-structure interaction, by cobining the results fro Chapter 3 into the present chapter. Such idea is realized in the following chapter. 19

152 13

153 CHAPTER 7 NONLINEAR STRUCTURAL DEFORMATION EFFECT ON THE BLAST INDUCED STRUCTURAL DAMAGE 7.1. Introduction An effective daage assessent ethod based on realistic deforation behavior of a structure is essential in order to apply protective easures when there exist potential blast load risks. For an RC structural eleent, the analysis becoes ore coplicated because the RC always defors in a nonlinear way, especially in the post-failure stage. This chapter extends the work done in Chapters 3 to 6 to derive the P-I diagras for an RC eleent by using a ulti-linear resistance-deforation odel. The three failure odes, i.e. the pure shear failure ode, the pure bending failure ode, and the cobined shear and bending failure ode, which are denoted as odes 1,, and 3, respectively in the previous chapters are considered in the present analysis. A ulti-linear resistance-deforation odel including a coplex softening regie which is ore appropriate to represent the deforation behavior of RC structures is eployed, and the non-constant SSI is also considered for underground structure scenarios. The effect of the elastic stage is deonstrated. The siplification of the ulti-linear deforation behavior is cobined with the MAM. The results fro the present analysis are copared to show the effect of the softening behavior of RC structural eleent on different failure odes. The present ethod can be applied to various RC structures with different elastic and post-elastic behaviors. 7.. Phase division A siply supported bea eleent is analyzed and both the elastic and post-elastic stages are considered. The difference of the boundary conditions was discussed by Ma et al. (7). The siply supported bea eleent is again adopted in the 131

154 present analysis since such boundary condition will give out larger deforation for both shear and bending failure and lead to a safer design. In traditional analysis, the resistance-deforation (R-D) relationship of a structural eleent was usually expressed as an elastic-plastic odel, including elastic, perfectly plastic and elastic-hardening odels. To siplify the analysis, the elastic portion are norally ignored and thus a rigid-plastic as shown in Fig. 7-1 (a) or an elastic-plastic hardening odel as given in Fig. 7-1 (b) are often used. Such siplification is valid only for relatively ductile structures which are able to sustain blast and/or ipact loads to a certain degree. For RC coluns and beas, the structures always behave with an apparent softening portion and the R-D relationship is siilar to that in Fig. 7-1 (c) or (d). To account for such post-elastic behavior, in the present study, a ulti-linear odel shown in Fig. 7-, is used in the dynaic response analysis for shear, bending, and cobined failure of RC bea structures. R* in Fig. 7- denotes the noralized general resistance of an RC eleent which can be replaced by noralized shear resistance Q* or noralized bending resistance M* respectively. D* is the corresponding noralized deforation. The MAM and P-I diagras in Ma et al. (7) included five failure odes; while in the present study only the first three odes were adopted. It is because the ode 4 and ode 5 are based on the oving plastic hinges phenoenon which requires relative low shear resistance, low depth-to-span ratio, and high intensity loading acting on the eleent. In consideration that the RC eleent is relatively brittle and the softening behavior in the post-failure stages is obvious, those two odes with plastic zones are not considered. The pulse load with a pressure p is suggested to be lower than 6M /L, and the diensionless strength ratio υ for ode 3 is suggested to be in the range fro 1. to 1.5. Although the failure odes 4 and 5 in Ma et al. (7) do exist, for exaple, bea eleent ade of steel aterial under blast load in a certain distance, they are not in the scope of the present study for RC eleents. 13

155 (a) (b) (c) (d) Fig. 7-1 R-D relationships for reinforced concrete structural eleent: (a) Rigidplastic odel; (b) Elastic-plastic hardening odel; (c) elastic-plastic soften odel; (d) More realistic RC structural deforation odel Fig. 7 - Idealized resistance-deforation relationship for RC eleents 133

156 The structural response can be divided into several stages based on the ulti-linear R-D relationship. The eleent defors linearly in the elastic stage. A few postelastic stages subsequently follow in the for of shear, bending, or a cobined failure ode. In the shear failure ode (ode 1), there are two basic phases which are the loading phase (phase 1) and the post loading phase (phase ). Phase 1 is in the period that the blast loads acts on the bea eleent and t d denotes its ending tie. During this phase, the acceleration is positive; the velocity and displaceent of the bea keep increasing. In phase, the structural eleent is free of the blast load and the bea eleent reaches the axiu deforation at the end of this phase. Within this phase, the velocity decreases onotonically, while the displaceent continues to increase. The phase ends when the velocity equals zero, and the displaceent reaches its axiu value. The final response tie is denoted as t f. The above two phases are differentiated according to the action of the blast load; while the structural response can also be divided into an elastic stage (ends at t se ) and a few post-elastic stages according to the deforation of the structural eleent. There are three possibilities when considering different response stages. If the elastic stage ends in phase 1 (<t se <t d ), the entire process is categorized into a loading-elastic phase (phase 1a, <t t se ), a loading-post-elastic phase (phase 1b, t se <t t d ), and a post-loading-post-elastic phase (phase, t d <t t f ), where t is the tie variable of the structural response. If the elastic stage ends in phase, the entire process includes a loading-elastic phase (phase 1, <t<t d ), a post-loading-elastic phase (phase a, t d <t t se ), and a post-loading-post-elastic phase (phase b, t se <t t f ). If t se =t d, as a special case, the process degrades to two phases, i.e., a loading-elastic phase (phase 1) and a post-loading-post-elastic phase (phase ). Siilar to the shear failure ode, in the bending failure ode (ode ), there are also three possibilities of cobination, i.e., 1) phase 1a => phase 1b => phase ; ) phase 1 => phase a => phaseb; and 3) phase 1 => phase. The end tie of the elastic stage for the bending ode is denoted as t e. In the cobined shear and bending failure ode (ode 3), the division of the response phases becoes ore coplex. Except the loading phase, i.e., phase 1, 134

157 there are two post-loading phases with respect to the terinations of shear and bending deforations, respectively. The first post-loading phase (phase ) ends when the shear deforation terinates, and the second post-loading phase (phase 3) then starts and ends when the bending deforation stops. During phase 1, the acceleration of the bea odel is positive, the velocity and displaceent increase onotonically. In phase, the accelerations of both shear and bending forces becoe negative so that the corresponding velocities start to decrease, while the displaceents keep increasing. When the velocity induced by the shear deforation vanishes, the shear displaceent stops, and the ending tie is denoted as t s. On the other hand, the displaceent due to bending still increases. In phase 3, the velocity of bending deforation reduces to zero and the bea odel reaches its final displaceent. By cobining the three phases with the elastic and post-elastic deforation stages, there are any possibilities. For brief, only two possibilities are explained as exaples in the following context. If the elastic stages of both shear and bending end in phase 1 (<t se <t e <t d ) for an exaple, the entire response process can be put into a loading-elastic phase (phase 1a, <t t se ), a loading-bending-elastic phase (phase 1b, t se <t t e ), a loading-postelastic phase (phase 1c, t e <t t d ), a shear-terinating-post-elastic phase (phase, t d <t t s ), and a bending-terinating-post-elastic phase (phase 3, t s <t t f ). In this exaple, phase 1 is divided into phase 1a, phase 1b, and phase 1c. In phase 1a, both shear and bending deforations are in the elastic stages. In phase 1b, the shear deforation enters the post-elastic stage while the bending deforation is still elastic. In phase 1c, both shear and bending deforations are in the post-elastic stages. Phase is the first post-loading phase in which the shear deforation will stop at the end, and the bending deforation continues. Phase 3 is the second postloading phase in which the bending deforation terinates at the end. If the shear-elastic stage ends in phase 1 but the bending-elastic stage ends in phase for another exaple, the entire process can be categorized into a loading-elastic phase (phase 1a, <t<t se ), a loading-bending-elastic phase (phase 1b, t se <t t d ), a post-loading-bending-elastic phase (phase a, t d <t t e ), a shear-terinating-postelastic phase (phase b, t e <t t s ), and a bending-terinating-post-elastic phase 135

158 (phase 3, t s <t t f ). Phase 1a, phase 1b and phase 3 are the sae as in the previous exaple, while phase is divided into phase a and phase b. Phase a indicates the post-loading phase in which shear deforation is in the post-elastic stage, while the bending deforation is still in the elastic stage. Phase b is a post-loading phase in which both the shear and bending deforations are in post-elastic stages. Any other possible cobination can be derived in analogy to one of the aforeentioned cases. It should be entioned that a rectangular pulse load is considered in the present analysis. In the elastic stage, the syste is siplified into an equivalent SDOF odel to calculate its acceleration, velocity and displaceent, while in the post-elastic stages, a bilinear plastic softening odel for both the shear and bending deforation as shown in Fig. 7- is adopted Eleent resistance and failure criteria In ost of the reported research works in cobining the shear and bending deforation effect, to siplify the derivation and calculation, the R-D relationship was assued as either rigid-plastic (e.g. Ma et al. 7) or bi-linear (e.g. Fallah and Louca 7). Such siplification generally overestiates the resistance of an RC structural eleent. In the present study, the shear and bending deforations are analyzed based on the assuption of independent interactions between shear and bending resistance, which eans the shear-bending yield surface is rectangular which is the sae as that in Ma et al. (7). The assuption used in the present study leads to the lower bound solution of the proble since the independent axiu bending and shear strength criteria used in the analysis circuscribe an interactive shear-bending yield surface. When elastic deforation is considered, it shall still be true since with the sae ipact energy, larger deforation will be resulted if a convex interactive shear-bending yield surface is adopted. When elastic deforation is included, the concept of bound solution is still valid since the independent axiu bending and shear strength criteria used in the analysis circuscribe an interactive shear-bending yield surface. The stress-strain relationship of the steel reinforceent is rearkably different fro the concrete which behaves very differently in copression and tension conditions 136

159 (Nawy 3, McCorac 6). When a structural eleent is ade of RC, its shear and bending resistances are very coplicated. The design code TM5-13 (199) gives out a guide on the perissible deforation of RC bea and slab in the design procedure. The eleent behaves linearly in the elastic phase and nonlinearly in the post-elastic phase. The support rotation is adopted as the definition of daage level. For civilian structures or structures without protective design, such daage definition is also suitable if certain odification is applied accordingly. There are various criteria for assessing daage of structures. Soe of the are based on strain energy definition at either the aterial level or the structure level. Soe others cobined the structural deforation and strain energy quantities for assessing structural daage. For exaple, the daage at the aterial level is better for daage assessent if the failure of reinforced concrete structure/eleent is spalling and penetration. Such definition of daage is uch ore accurate than deforation. However, for the sake of convenience, the axiu deforation occurred in a structure (bea, colun and slab) has been widely used for assessing structural daage in practice. Therefore, the deforation in structure level is adopted as the daage criterion in the present study. In the present study, the failure criteria of shear and bending reain the sae as those in the previous chapters. The axiu bending deforation at the center of the bea eleent and the axiu shear deforation at the eleent supports can be expressed as Eqs. (3.49) Failure odes and P-I equations The pulse shape difference on P-I diagra has been discussed by Youngdahl (197, 1971), and Li and Meng (b) that it can be eliinated by using the Yougdahl s correlation paraeter ethod. For the convenience of the P-I equations derivation while not losing the generality of the solution, the rectangular pulse load with a agnitude of p and a duration of t d is again adopted in the present study. The possible transverse velocity profiles which indicate the shear failure ode, the bending failure ode, and the cobined failure ode are shown in Fig. 3-3 (Ma et al. 7). The diensionless strength ratio is also used, i.e.: 137

160 QL e (7.1) M where Q e and M e are the axiu elastic shear and bending resistance respectively; L is the half length of the bea eleent. e The Q e and M e in Eq. (7.1) are assued to be independent fro each other and obey the rectangular yield surface. The general governing equation is expressed as: Q p y ( loading) x Q y ( post loading ) x (7.) where Q is the transverse shear force; x is the abscissa on the eleent; p is the constant pressure of the blast load; is the ass per unit length; y is the acceleration of a unit ass. The x-axis starts fro the id-span of the bea and only half of the bea is considered due to syetry as shown in Fig Fig. 7-3 Siply supported bea odel Table 7-1 shows the phase velocity profiles for different odes. Both the elastic and the post-elastic stages follow these profiles. Mode 1 contains the shear failure only. 138

161 Mode indicates the bending failure which has a plastic hinge at the center of the eleent. Mode 3 can be considered as the cobination of ode 1 and ode. Table 7-1 Velocity profile Mode 1 Mode Mode 3 Phase 1 Phase Phase 3 N.A. N.A. Details of derivation of the displaceent tie history for the three odes are given in the following context Mode 1 shear failure ode When 1 and p Me / L, the shear failure ode will occur. The shear failure ode deonstrates that shear failure at the two supports which carry the axiu shear force, while the bending failure does not occur. In this ode, the diensionless paraeter υ is less than 1, which indicates a very low shear-tobending resistance ratio. The following context studies a case which includes two possible post-elastic stages. Derivation of the equations for other cases can be referred to this exaple. There are totally three phases in this ode including loading-elastic phase (phase 1a, <t t se ), loading-post-elastic phase (phase 1b, t se <t t d ), and post-loading-postelastic phase (phase, t d <t t f ) which end at t se, t d and t f respectively. In phase 1a, the SDOF syste is adopted to present the elastic behavior of the RC eleent. Once the eleent enters the post-elastic phases (phase 1b and phase), the ode approxiation ethod used by Ma et al. (7) is adopted. 139

162 Phase 1a is in the elastic stage with the blast loading action. The axiu acceleration, velocity, and displaceent are calculated by the SDOF odel as follows: ys p, y p t s, y p t s (7.3) where y, y, and y are the acceleration, velocity, and displaceent variables respectively; the suffix s indicates the quantities by the shear force only; t is the tie variable. The distribution patterns of these quantities can be deduced referring to the velocity profile. At the end of this phase, the axiu velocity and displaceent are denoted as y t, and s se y t respectively. s se Phase 1b is in the post-elastic stage within the blast duration. The governing equation is: Q p x y s (7.4) The boundary conditions and initial conditions are:, s t, y y t Q x Q x L Q y y T y t T s s se s s se (7.5) where Q (y) is the shear resistance of the bea eleent that deduced fro the shear-resistance-displaceent curve; T is the general tie variable which equals to t-t se in this phase. t Setting the iterative tie step as t d t n se, where n is a pre-defined constant which satisfies the precision requireent of the integration. Then Eq. (7.4) is integrated with respect to tie. In every integration step, the value of Q (y) needs be updated due to the change of displaceent. The equations for integration are derived as: 14

163 pl Q ys ti t y t y t L s i1 s i (7.6) y s ti pl Q y t t 1 ys( ti ) t ys( t ) (7.7) L s i i At the end of phase 1b when t=t d, the axiu velocity and displaceent of the eleent are y t and s d y t respectively. s d Phase is a post-loading-post-elastic phase, and the governing equation is: Q ys x (7.8) The boundary conditions reain the sae as for the previous phase, while the initial conditions change to: where T is equal to t-t d. The equations for integration are:, y T y t y T y t (7.9) s s d s s d Q ys t s i 1 i t y t ys ti (7.1) L ti Q ys t ys ti1 ys ti t ys ti (7.11) L The iterative tie step t is preliinarily estiated the sae as that for the previous phase. Siilarly, the final axiu displaceent of the eleent can be yielded as s f y t where t f is the final tie when the velocity of eleent reaches zero Mode bending failure ode When 1 and M / L p M 4 3 / L, in the bending failure ode, a e e plastic hinge fors at the id-span of the eleent where the axiu bending oent occurs. 141

164 Siilar to ode 1, the case with two post-elastic deforation phases is taken as an exaple. The derivation and integration procedures for ode are siilar to those for ode 1. Equations for other possible cases can refer to this exaple. There are totally three phases in this ode including a loading-elastic phase (phase 1a, <t t e ), a loading-post-elastic phase (phase 1b, t e <t t d ), and a post-loadingpost-elastic phase (phase, t d <t t f ) which ends at t e, t d and t f respectively. The sae as ode 1, the SDOF syste and the ode approxiation ethod are adopted to represent the elastic and post-elastic stages. Phase 1a is the elastic phase with the blast loading acting on the eleent. The axiu acceleration, velocity, and displaceent can be derived as: p p t p t y y y (7.1), e e e where e is the effective ass used for the elastic phase in the bending failure ode; the suffix indicates the relevant quantities are incurred by the bending oent. At the end of this phase, the axiu velocity and displaceent are expressed as y t and e y t, respectively. e Phase 1b is the post-elastic phase within the blast loading duration. The governing equation is: Q x p y 1 x L (7.13) The boundary conditions and initial conditions are:,, Q x M x M y t y T y t y T y t e e (7.14) where M (y) is the bending resistance of the bea eleent deduced fro the bending-resistance-displaceent curve; T is the response tie. 14

165 td te Setting the iterative tie step to t, Eq. (7.13) is then integrated with n respect to tie. In every integration step, the value of M (y) should be updated since the displaceent of eleent changes. The equations for integration can be derived as: t 3pL 6M y i t y ti1 y ti (7.15) L t 3pL 6M y i t y ti1 y ti t y ti (7.16) 4L At the end of phase 1b when t=t d, the axiu velocity and displaceent of eleent are y t and d y t respectively. d Phase is the unique post-loading-post-elastic phase, and the governing equation is written as: Q x y 1 x L (7.17) The boundary conditions reain the sae as for the previous phase, while the initial conditions change to: where T is equal to t-t d. The equations for integration are:, y T y t y T y t (7.18) d d 3M y ti t y ti1 y ti (7.19) L 3M y ti t ys ti1 y ti t y ti (7.) 4L where the tie step t is set the sae as that for the last phase. 143

166 The axiu displaceent of the eleent is denoted as f y t when the velocity of eleent reaches zero Mode 3 cobined failure ode Mode 3 is the cobination of ode 1 and ode. Its sub-phases are uch ore than those for ode 1 or ode. The shear failure occurs at the two supports, while the bending failure induces a plastic hinge at the id-span of the eleent. One of the possible cases with the ost post-elastic deforation phases is taken as an exaple. There are totally five sub-phases in this case including a loading-elastic phase (phase 1a, <t t se ), a loading-bending-elastic phase (phase 1b, t se <t t e ), a loading-post-elastic phase (phase 1c, t e <t t d ), a shear-terinating-post-elastic phase (phase, t d <t t s ), and a bending-terinating-post-elastic phase (phase 3, t s <t t f ). Equations for other cases can refer to the derivation for this exaple. Again, the SDOF syste and the ode approxiation ethod are adopted for the elastic and post-elastic stages respectively. Phase 1a is the first phase in which the blast load acts on the eleent and both shear and bending responses reain elastic. The axiu acceleration, velocity, and displaceent due to shear and bending are the sae as in Eqs. (7.3) and (7.1). At the end of this phase, the axiu velocity and displaceent are given as y t, y t, y t and s se s se se y t. se Phase 1b is the loading-bending-elastic phase within the blast loading duration when the shear deforation falls into the post-elastic stage while the bending response still reains elastic. The governing equation is the sae as Eq. (7.4), and the boundary and initial conditions are the sae as Eq. (7.5). The equations for integration in this phase are as follows: s ti i p t 4Q y t 6M y t t ys ti1 y s ti (7.1) L L 144

167 y s s ti 3 i Q y t M y t t ( )(7.) L L p t i1 y ( ) s ti t ys ti t te t where the tie step t equals to n se. By the sae integration ethod as for ode 1, at the end of phase 1b, the axiu velocity and displaceent are given as y t, y t, y t and s e s e e y t. Phase 1c is the post-elastic phase for both shear and bending responses till the end of the blast duration. The governing equation for the cobined deforations is given as: e Q x p ys y1 x L (7.3) The boundary conditions and initial conditions are:, s t Q x Q x L Q y M x M y t, M x L,, ys T ys te ys T ys te y T y t y T y t e e (7.4) (7.5) where T equals to t-t e. The equations for the shear response integration are the sae as Eqs. (7.1) and (7.), while those for the bending response are given as follows: y s ti i p t Q y t 6M y t t y ti1 y ti (7.6) L L s ti 3 i Q y t M y t t y( ti ) t y( ti ) (7.7) L L p t i1 t td t where the tie step t equals to n e. 145

168 At the end of this phase, the response quantities are y t, y t, d y t. s d s d y t and Phase is the post-loading-post-elastic phase, at the end of which, the shear response terinates. The governing equation for shear and bending is: d Q x ys y1 x L (7.8) The boundary conditions are the sae as in Eq. (7.4), and the initial conditions are updated as:,, ys T ys td ys T ys td y T y t y T y t d d (7.9) where T=t-t d. The equations of integration for the shear and bending response are as follows: s ti i 4Q y t 6M y t t ysti 1 y sti L L Q ys ti t 3M ytit ysti1 y ( ) ( ) s ti t ys ti L L (7.3) s ti i Q y t 6M y t t y ti1 y ti L L Q ysti t 3M yti t y L L ti1 y ( ) ti t y ti ( ) (7.31) td where the tie step t is set to be t n e. At the end of this phase when t=t s, the velocity due to the shear force equals to zero, and the response quantities are y t, y t and s s s y t. s 146

169 Phase 3 is the last phase in which the total response of structural eleent will terinate. The governing equation is the sae as in Eq. (7.17)and the boundary conditions are the sae as in Eq. (7.4). The initial conditions are as follow:, y T y t y T y t (7.3) s s where T equals to t-t s. The equations of integration for the shear and bending response are as follows: 3M y ti t i1 i y t y t L (7.33) y i 3M y t t ( ( ti) (7.34) L ti 1 y ) ti t y td t where the tie step t is set as n e. At the end of this phase when t=t f, the velocity due to the bending oent equals to zero, and the final responses is f y t. In all the above odes, the end tie of the elastic phases (t se and t e ) can be calculated as: t t 7.5. P-I diagras and discussions se e y y s y y t s se t t se t e e (7.35) Define diensionless pressure P* and ipulse I* for the blast load as: * pl pl P Q M (7.36) p p 147

170 I p t p t Q 4M L * d d p p (7.37) Fro the equations for the final displaceents induced by the shear and bending failure, the P-I diagras can be represented in unified fors as: S P I h y (7.38) * *, s B P I L y (7.39) * *, where y s is the axiu displaceent due to shear that equals to y s (t f ) in the shear failure ode or y s (t s ) in the cobined failure ode; y is the axiu displaceent due to bending which equals to y (t f ); S(P*,I*) and B(P*,I*) are iplicit expressions with respect to the noralized pressure and ipulse for shear and bending deforation odes according to the failure criteria Differentiation of failure odes Differentiation of failure odes as given in Fig. 3-3 is deterined in view of the otion initiation, i.e., the cobination of p, t d, and ν. For ode 1, the acceleration induced by the shear force at the supports should be larger than zero at the beginning of post-loading phase, and the axiu bending oent should be saller than the bending strength of the eleent. Therefore, p M / L and 1 should be satisfied. For ode, the initial acceleration due to bending should be positive at the beginning of post-loading phase, while the axiu shear force should be less than the shear strength. Therefore, 1 and M / L p M 4 3 / L are required. For ode 3, cobine the requireents of ode 1 and, 1 p M 4 3 / L becoe the and boundary lines. The definition of the failure odes is siilar to that by Ma et al. (7). The shear or bending failure are always and ust be initiated within the loading phase since the differentiation of failure odes is decided by the initiation conditions of post-elastic structural response. In another word, if the blast load does 148

171 not satisfy the requireents of specified failure ode, the bea eleent will not response according to velocity profile of that ode. Under such situation, yielding will never occur after the loading phase in the present study. Such liitation can be eliinated when the differentiation ethod of failure odes is updated in future works Effect of resistance-deforation relationship In the present study, the size of RC bea eleent in Barros and Dias (6) is adopted ( ), and the paraeters K1, K, and K3 for bending in the ulti-linear R-D relationship are generated by curve fitting of Barros and Dias (6) results. Meanwhile, the above three K values for shear in the R-D relationship are generated fro the results of Lee et al. (8). In the present study, K1, K, and K3 for shear is calculated as 1.75, -1.47, and -.1, while the values for bending are 1.53, -.99, and M p is 1.53kN in all the following calculations and coparisons. Denoting Q* and M* as the noralized shear and bending resistances respectively and D* for noralized displaceent: M M y M * Q Q( ys ) / Q * ( ) / (7.4) D* y / yse shear D* y / ye bending (7.41) Since the noralized shear and bending resistances Q* and M* are plotted with respect to noralized displaceent D*, the K values are diensionless. Fig. 7-4 shows the P-I diagras for the rigid-plastic odel, the elastic-rigid-plastic odel, and ulti-linear odel suggested in the present analysis, with respect to the three failure odes. It is seen that the differences caused by the R-D relationship cannot be ignored. Fig. 7-5 shows the coparison of the results fro the rigidplastic, bi-linear, and ulti-linear R-D relationships. As listed in Table 7-, three loading scenarios, i.e., blast loads at far, iddle, and close standoffs (load cases 1,, and 3, respectively), are assued for each failure ode which confor to the initial conditions applied in the derivation for the three different odes. 149

172 (a) Failure ode 1 (b) Failure ode 15

173 (c) Failure ode 3 Fig. 7-4 P-I diagras of failure odes (a) Coparison of shear failure 151

174 (b) Coparison of bending failure Fig. 7-5 Noralized ending ties and final responses Table 7 - Loads and paraeters Load case Pressure p (MPa) Duration t d (s) Mode 1 Mode Mode 3 Mode 1 Mode Mode Based on the coparisons shown in Fig. 7-5, which take the results of the ultilinear odel as a reference, the difference caused by the R-D relationship and loading scenarios can be stated as follows. The ending tie and final displaceent due to shear force are sensitive to the R-D relationship in all loading scenarios. The axiu difference in ode 1 can be 7.% and 44.9% respectively. In ode 3, the above values change to 38.3% and 59.6%. The coparison also indicates that, the differences in both ending tie and final displaceent decrease with the increase of the shear-to-bending strength ratio υ. Such differences decrease as well 15

175 fro the far range scenario to the close range scenario with the sae failure ode. The ending tie and final displaceent due to the bending deforation have siilar tendencies as those for the shear failure ode, but the axiu differences with regard to different R-D relationships are higher, especially for the final displaceent. Generally, the difference between the results fro the rigid-plastic and the bi-linear odels is influenced by the elastic deforation, while the difference between the results fro the bi-linear and ulti-linear odels is due to the softening behavior in the post-elastic phase. It can be stated that the present study deonstrates the rearkable effect of elastic deforation in daage analysis for RC structural eleents, which, thus, cannot be ignored. Adopting the rigid-plastic and the ulti-linear R-D odels respectively in the analysis of the failure ode 3, Fig. 7-6 shows the difference of the results for different daage levels. Generally, the effect is saller in the light daage level, while it increases when the daage level increases. This tendency is true for both the shear and bending failure odes, while the bending failure ode is ore sensitive to this effect. Such phenoenon can be explained by the effect of the postelastic stages in the failure ode which has also been illustrated in Fig (a) Coparison of shear failure 153

176 (b) Coparison of bending failure Fig. 7-6 Model difference in daage levels Paraetric study Since the bending and shear strength can be derived fro experiental results, the relationship of the resistance versus displaceent for both shear and bending failure can be in different profiles. The R-D curve as shown in Fig. 7- is applicable to different RC structural eleents. According to the results reported by Polak (1997), Hadi (5), Lee and Al-Mahaidi (8), Rizzo and Lorenzis (9), etc., the deforation property of reinforced concrete can be siplified as ulti-linear. The R-D relationship includes an ascending part, a descending part, and a relatively stable residual part with the slopes denoted as K 1, K, and K 3 respectively. K 1 is set equal to the elastic stiffness of the eleent and the resistance reaches the axiu at Q 1 * or M 1 *. K is of a negative value and this part ends at Q * or M *. The value of K 3 is relatively sall, which can be either negative or positive. Therefore, the resistance does not change a lot in the third part. The values of Q 1 *, M 1 *, Q *, and M * are based on the paraeters in rigid-plastic odel in previous coparison, and the K values for the shear and bending R-D relationships reain the sae as those in the ulti-linear odel. 154

177 For failure ode 3 as an exaple, Fig. 7-7 shows the effect of K and K 3. C1 indicates the ulti-linear odel in the previous calculation, and the results are considered as references which are noralized as one. C and C3 stand for changing of K 3 and K respectively. In this paraetric study, the values of K 3 and K for bending cases are deterined by two kinds of curve fittings of the experient result of Barros and Dias (6), while K 3 and K for shear cases are generated fro the results of Lee et al. (8) accordingly. And the increase rates based on values in C1 are taken as 73.41% and 1.43% for K 3 and K respectively. The ending tie and final displaceents in Fig. 7-7 are all noralized quantities. Fig. 7-7 Paraetric study of K and K3 It can be seen that, the responses due to bending failure are ore sensitive to the paraeters K than those due to shear failure. The differences for the ending tie and final displaceent due to the shear failure are only about.7% and.88% respectively, while those for the bending failure can be up to 4.1% and 1.79% respectively. The difference caused by varying K 3 is relatively sall for both the ending tie and the final displaceent due to both shear and bending deforation, 155

178 while the varying K affects the result evidently. The effect of varying K 3 and K are roughly siilar for all the three loading scenarios Applications in underground scenarios Using the generalized integration ethod, even ore coplicated paraeters can be considered in the calculation of underground structures under blast load. Taking failure ode 1 and ode as exaples, Fig. 7-8 and 7-9 show the P-I diagras of underground RC structure against external and internal blast respectively. These P-I diagras are based on the oderate daage level for both shear and bending, and the properties of different soils are the sae as those in Chapter 6. The axiu daping coefficients (C ) in non-constant SSI as expressed in Eq. (6.39) for dry sand and backfills, dense sands, and saturated sandy clay are C =.497 MPa s/, C =.995 MPa s/, and C =.941 MPa s/ respectively. The axiu spring coefficients (K ) in non-constant SSI as shown in Eq. (6.4) for plastic beads-sand ixture, ediu sands, and crushed rock are K =.39 MN/, K =3.83 MN/, and K =4.83 MN/ respectively. (a) Shear failure 156

179 (b) Bending failure Fig. 7-8 P-I diagras of underground RC structure against external blast load (a) Shear failure 157

180 (b) Bending failure Fig. 7-9 P-I diagras of underground RC structure against internal blast load 7.6. Conclusions The present study extends the MAM by adopting the generalized integration procedure and P-I diagras were plotted for shear, bending, and cobined failure odes by considering coplex paraeters. Besides the difference in failure odes, the paraeters also include effect of elastic phase, nonlinearity of structural deforation, pulse shape difference, and effect of non-constant SSI for underground scenarios. A ulti-linear R-D relationship is adopted in the present study which is suitable to assess the daage of different RC structural eleents with different designs. Fro the present study, the effect of different R-D relationships is rearkable. The effect of the elastic phase cannot be ignored for far, iddle, and close range loading scenarios. Generally in both shear and bending failures, the effect is ore proinent when the daage level increases, and bending failure is ore sensitive to this effect. Such phenoena can be explained by the effect of post-elastic or softening phases in the failure ode. 158

181 It was proved that using different deforation odes to satisfy the kineatic conditions of the bea has negligible effect on the axiu final deforation (Sith and Hetherington, 1994). But the result in the present study shows that, the elastic deforation has certain influence on the shear and bending plastic deforations, especially in the light daage level. In another word, the effect of the elastic stage increases with the decreasing of the plastic displaceent. The paraetric study shows that the responses due to the bending failure are ore sensitive to the post-elastic behavior than those due to the shear failure. The value of K 3 affects the ending tie and the final displaceent for shear and bending failure, and variation of K affects the results ore proinent. The effects of the variations of K 3 and K are roughly siilar for all the three loading scenarios. Therefore, it can be stated K is the ost critical paraeter in the R-D relationship to affect the siulation results rearkably. In the above calculations and analysis, the end tie of elastic bending stage and elastic shear stage were calculated first, so that the differentiation of phases for each failure ode can be carried out. In the present study, the t se is shorter than t e occasionally that does not ean the shear-elastic stage always stops earlier than the bending-elastic stage. The calculations of the ulti-linear deforation odel adopted in the present study are based on the SDOF syste and the ode approxiation ethod for elastic and post-elastic stages respectively. The calculations are carried out by a general integration procedure, that eans the ethod proposed in the present study can be easily applied to any kind of deforation odels without losing its accuracy. 159

182 16

183 CHAPTER 8 VERIFICATION OF MAM BASED P-I DIAGRAMS BY NUMERICAL SIMULATIONS In the previous chapters, the P-I diagra ethod based on the MAM is successfully extended fro surface structures to underground structures against external and internal blast load. Since the general integration procedure is adopted, coplicated paraeters can be considered in the derivation of P-I equations which include the ore accurate blast load tie history, the nonlinear structural deforational behavior, the pulse shape effect, the effect of the elastic phase, the non-constant SSI, and so on. In this chapter, nuerical siulations are carried out to validate the results in the previous chapters. The blast load is siulated in a -D odel and reapped to a 3- D odel of an RC wall. Three scenarios are siulated including surface structure against blast load, underground structure against external and internal blast load respectively. All the scenarios are siulated by the coercial code naed ANSYS-AUTODYN. Results fro the nuerical siulations are copared with that fro the analytical solutions in the for of P-I diagras. One strip fro the center of the RC wall is taken as the bea eleent used in theoretical analysis. The MAM is adopted to generate P-I diagras coparing with the results fro nuerical siulations. For shear resistance: Q.16 f ' dc b d (8.1) where f' dc is the dynaic strength of concrete, b is the width of copression face, d is the distance fro the extree copression fiber to the centroid of the tension reinforceent. 161

184 In the present study, f' dc equals to 35 MPa, b is.35 for one strip of the wall for analysis, and d=.15. Therefore the shear resistance of wall strip is KN. For bending resistance: M As fds / b d a / (8.) where A s is the area of tension reinforceent within the width b, f ds is the dynaic design stress for reinforceent, a is the depth of equivalent rectangular stress block a=a s f ds /(.85 b f' dc ). In the present study, A s =.74, f ds =414 MPa, so the bending resistance of the wall strip is kn. Accordingly, the diensionless strength ratio defined in Eq. 3.1 is ν= Siulation of blast load The blast load will refer to different scaled distances. Several cobinations of stand-off distance and equivalent TNT weight are listed in Table 8-1. Table 8-1 Different charge weights Distance Charge weight (case nuber) kg (RC1) 43 kg (RC) 69 kg (RC3) It is a coplex and tough task to study the dynaic response of an RC wall subjected to an explosion because the initiation of blast and the interaction between the wall and the blast wave are involved. In addition, coputational cost is very high due to the siulation of the blast wave propagation. In order to reduce coputational tie and ensure the accuracy of results, the generation of blast loading in air is based on reapping technique provided by ANSYS-AUTODYN, which is an effective ethod to calculate the initiation of detonation and the blast wave propagation in air. 16

185 In the nuerical siulation, the detonation is odeled using a wedge which is a coon ethod used in ANSYS-AUTODYN. The inforation obtained fro the wedge calculation is then written into a data file and subsequently reapped to a larger 3-D Euler air grid. The air grid loaded with blast pressure inforation is coupled to interact with the Lagrangian structural eleent, e.g. the RC wall odel in the present analysis. The first task of analysis is the siulation of explosion itself fro the detonation instant and the propagation of the blast wave in air. The solution tie is set to be very short referring to the arrival tie when the blast wave ipact on the RC wall. Just before the blast wave reaches the RC wall, the blast wave propagation siulation is stopped and the reapping file is generated based on the -D odel. The second task is the input of previous reapping file and analysis of the ipact effect and interaction with the wall of the blast wave. In this way, the results of -D analysis can be later apped in the 3-D odel representing the wall and the surrounded air volue drastically reducing the coputational cost of nuerical analysis. The procedure of -D siulation is described as follows. The -D axial odel is selected, and the basic units are, g, and s for length, ass, and tie respectively. The air and TNT aterial odels in ANSYS-AUTODYN s aterial library are adopted. The equation of state (EOS) for air is the ideal gas type which refers to Rogers and Mayhew (1994), while the EOS for TNT aterial is the JWL type which refers to Lee et al. (1973). A wedge, as shown in Fig. 8-1, is built as the air part with the axiu radius the sae as the distance fro center of charge to the surface of the RC wall in different cases. The air part is divided as 1 for one cell in eshing and the initial energy for air is J/kg. Later, the TNT aterial is filled at the center of the wedge as a circle, for exaple, with the radius of which represents 11 kg of TNT since the shape of charge is assued as a half-sphere. The nuber of cells across the radius is 1 with grade zoning of.1 as the increent (dx). The single detonation point is located at the center of charge. 163

186 Fig D wedge odel After the part is built up, the solver of Euler -D ulti-aterial is adopted to carry out the calculation. Fig. 8- shows one exaple of pressure contour at the end of calculation. Fig. 8 - Pressure contour of explosion 164

187 After the -D calculation is finished, the result is written in a data file which can be reapped to the 3-D siulation. Three scenarios will be shown in the following sections including surface structures against blast load, and underground structures against external and internal blast load respectively. 8.. Siulation of surface RC wall against blast load The nuerical siulation of a surface RC wall odel against blast load is carried out by the coercial code ANSYS-AUTODYN. Details of the siulation of the surface wall against blast load are as follows Geoetry odel The section view of design drawing for the RC wall is shown in Fig Fig. 8-3 Section view of RC wall The Cartesian coordinate is defined as that in Fig The X-axis is fro the botto center of the RC wall along with the long edge; the Y-axis is fro the botto center of RC wall pointing to the top of RC wall; and the Z-axis is fro the botto center of RC wall pointing to the center of charge. 165

188 a) Full odel b) Rebars Fig. 8-4 Geoetry odel of RC wall 166

189 The 3-D odel, which included the RC structure and the surrounding air, and blast load are syetrical to the YZ-plane in the Cartesian coordinate. To save the coputational tie, half odels for both RC wall and air are built and the full odels can be plotted by shown the irror according to X-plane. The geoetrical odel of RC wall as shown in Fig. 8-4 is the solid Largrange odel, and the rebars are built up as bea odel grouped to the RC nodes. In the present analysis, the total size of the RC wall odel is 35 (W) 44 (H) 115 (T). The radius and span of the rebar are 1 and 35 respectively. The solver for surrounding air is the Eular 3-D ulti-aterial. The width and height of air are 4 and 3 respectively which fully covered the RC wall odel. The length of the air odel is 14 fro the center of charge, and the RC wall odel is placed at 3 fro another end of the air odel to the front surface of the RC wall. The esh size of air and structure is defined as.1 in three axial directions. After the wall part and air part are built up, the data file fro -D calculation can be filled. Both the air and TNT aterials are reapped into the 3-D odel, and the reapping is syetrical to X-plane as well. Fig. 8-5 shows the vector contour of the blast load with the RC wall odel. Fig. 8-5 Vector contour of blast load (with RC structure odel) 167

190 In the solution control, the wrap-up criterion of energy friction is set to.5, and the starting tie of calculation is set the sae as the ending tie of -D calculation so that the blast wave would keep on propagating in the 3-D odel Material properties The aterial properties of air, concrete, and rebar in the present study is odified fro the original aterials naed air, conc-35_mpa, and steel_434 in the ANSYS-AUTODYN aterial library respectively. The air aterial is the sae as that used in D reapping calculation with the EOS of ideal gas. The concrete aterial is odified based on the original conc-35_mpa aterial which refers to Riedel (). The P_alpha EOS is adopted; the strength and failure odels are the RHT odels; and the erosion criterion is based on the geoetric strain. The steel odel is odified fro the original steel_434 odel fro Johnson and Cook (1985). The linear EOS equation and Johnson Cook strength odel are adopted for the steel aterial. The failure criterion is based on the principal strain and the erosion criterion is again based on the geoetric strain. In the present analysis, the dynaic ultiate copressive strength of concrete is 35 MPa and the dynaic design stress for reinforceent is 41 MPa which are calculated according to the dynaic increase factor given in TM5-13 (199). The coputational tie for each case is 1 s, which is long enough to evaluate the final structural displaceent after vibration. The nubers of eleents for RC block, steel rebars, and air aterial are 51, 18, and 4 respectively. The eleent types for RC, steel, and air are 3D solid eleent with RHT concrete odel, D link eleent with Johnson Cook strength odel, and 3D solid eleent with ideal gas EOS Erosion criterion In the generalized integration procedure, the efficiency of calculation is affected by the tie step and the quality of eleent which are decided by the shortest eleent 168

191 edge and the deforation of eleent in the present tie step. The erosion is a technique adopted in nuerical siulations to erode greatly distorted eleents so that the probles associated with the esh distortions caused by gross otions of a Lagrangian grid can be overcoe. It should also be noticed that, the eroded eleents will result in loss of internal energy, strength and ass, and they ay affect the overall results. Therefore, the erosion criterion should be decided with discretion. In general, the erosion criterion has to be selected according to loading conditions and objective in the work. In the present analysis, the RC structure is under high loading density (the blast load with certain distances). Under such condition, a large nuber of eleents ight reach the erosion criteria and would be eroded during coputational process. Norally the instantaneous geoetric strain is set to be 15% as the erosion criterion in penetration siulations, and the erosion liit varies between 1% and %. In the present analysis, 1% geoetric strain is set as the erosion criterion for both the concrete and steel aterials. Such consideration is based on the general assuption in nuerical siulation and the daage definition in the present study Boundary conditions The RC wall odel in the present analysis is siply supported on the long edges in the three axial directions as shown in Fig

192 Fig. 8-6 Boundary conditions of RC wall The flow-out boundary is added on the surrounding air but one surface is left unconstrained which indicates the ground to reflect blast wave. Fig. 8-7 shows half of the boundaries. In soe cases (e.g., buried wall under external and internal blast load), the flow-out boundary is applied to all surfaces of the air odel. 17

193 Z-VELOCITY (/s) Fig. 8-7 Boundary conditions of air (half section at X-plane) Results and discussion The tie histories of velocities and displaceents of the RC wall before 1 s after explosion under surface blast load are shown as Fig. 8-8 to 8-1 (as case RC1, RC, and RC3). The velocities and displaceents vibrate stably after the blast load is reflected by the RC wall. 171

194 Z-VELOCITY (/s) Z-DISPLACEMENT () a) Tie history of velocity b) Tie history of displaceent Fig. 8-8 Velocity and displaceent for case RC1 (light bending daage) a) Tie history of velocity 17

195 Z-VELOCITY (/s) Z-DISPLACEMENT () b) Tie history of displaceent Fig. 8-9 Velocity and displaceent for case RC (oderate bending daage) a) Tie history of velocity 173

196 Z-DISPLACEMENT () b) Tie history of displaceent Fig. 8-1 Velocity and displaceent for case RC3 (severe bending daage) Results fro the nuerical siulation are copared with theoretical solutions. The P-I diagras for nuerical siulation are generated by curve fitting of result data. Fig shows the two results of surface RC structure under blast load. The charge (TNT) weights used in the nuerical siulation are 17 kg, 43 kg, and 69 kg which present scaled distances of 1.978, 1.465, and 1.84 respectively. Such three loading cases are preliinarily designed that, the bending daage levels of the RC wall are light, oderate, and severe respectively. The shear deforations of RC wall are shown accordingly. Based on the ratio of shear to bending strength of the RC wall, failure ode 3, the cobined failure ode with one plastic hinge at the center and direct shear failure near the supports is adopted in the coparison. 174

197 a) Light bending daage b) Moderate bending daage 175

198 c) Severe bending daage Fig Result coparison for surface blast cases The P-I diagras in Fig show that: The nuerical siulations norally over-estiate the bending daage of surface RC structure under blast load. In the iddle range fro the center of explosion (oderate scaled distance, Fig b)), the bending daage generated fro the theoretical solution and the nuerical siulation are very close. The error in close-in range case (Fig c)) is acceptable, especially for the bending daage. The P-I diagras with respect to severe shear daage have soe discrepacies in the ipulsive load region. The error of the bending daage in far range case (Fig a)) becoes considerable especially in the ipulsive and dynaic regions of P-I diagra. However, the discrepacies with respect to oderate shear daage becoes inor based on Fig a). The errors of bending daage between the theoretical solutions and nuerical siulations are 9.16%, 1.37%, and 4.58% for light, oderate, and severe bending daage level respectively. 176

199 Z-VELOCITY (/s) The shear daage in the above three cases falls in oderate or severe level. The shear daage is ore severely presented than the bending daage. Such phenoena eans that the RC wall is very vulnerable to shear daage and strengthening of the shear resistance of the RC wall by using stirrups is essential Siulation of underground RC wall against external blast load Three load cases (RC-C1, RC-C, and RC-C3) for the scenario of underground RC wall against external blast load are siulated. The geoetric odel, aterial properties, erosion criterion, and boundary conditions reain the sae as those in section 8.. The SSI is siplified as a daping ratio and is applied on all the eleents of the RC part. According to general experience, the daping ratio is set to be 6% for siulating the SSI caused by dry sand and backfills used in Chapter 3. Although the blast wave propergation in soil is uch different fro that in air, for convenience of the siulation and coparison, the blast pressure and ipulse on the RC wall reain the sae as that in surface RC wall siulation. The difference of the blast wave propagation in different aterial is not in the scope of current study. The results of the RC wall response are shown in Fig. 8-1 to a) Tie history of velocity 177

200 Z-VELOCITY (/s) Z-DISPLACEMENT () b) Tie history of displaceent Fig. 8-1 Velocity and displaceent for case RC-C1 (light bending daage) a) Tie history of velocity 178

201 Z-VELOCITY (/s) Z-DISPLACEMENT () b) Tie history of displaceent Fig Velocity and displaceent for case RC-C (oderate bending daage) a) Tie history of velocity 179

202 Z-DISPLACEMENT () b) Tie history of displaceent Fig Velocity and displaceent for case RC-C3 (severe bending daage) Coparison between the P-I diagras of the underground RC wall against external blast load at different daage level is shown in Fig a) Light bending daage 18

203 b) Moderate bending daage c) Severe bending daage Fig Result coparison for underground external blast cases The P-I diagras for underground RC structure under external blast load show that: 181

204 Because of the SSI (the daping effect for underground RC structure against external blast load), the bending daage is reduced in all daage levels. Such attenuation effect is uch rearkable in the ipulsive and dynaic regions of P-I diagras. The attenuation percentages are 9.79%, 6.6%, and.58% for light, oderate, and severe bending daage respectively. The shear daage level is higher than bending daage level for different scaled distances. By coparing with the results of surface RC structure, the decreent of shear daage caused by the daping effect is 3.6%, 1.85%, and 3.19% in light, oderate, and severe bending daage cases respectively Siulation of underground RC wall against internal blast load The soil aterial odel used in nuerical siulation is a siplified elastic-plastic odel whose Young's odulus is defined according to Eq. (6.4). The interface of structure and soil is odeled as two contact faces, and the interaction is treated by the built-in solver of ANSYS-AUTODYN. Three load cases (RC-K1, RC-K, and RC-K3) of underground RC wall under internal blast load are siulated. The SSI is siulated as a block of soil attached to the rear face of RC wall as shown in Fig The blast wave on the RC wall reain the sae as that in surface RC wall siulation since the blast wave propagates in air before hitting on the RC wall. The result is shown in Fig to

205 Z-VELOCITY (/s) Fig Soil layer in RC-K cases a) Tie history of velocity 183