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- 1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 275 PRESSURE-IMPULSE DIAGRAM FOR DAMAGE ASSESSMENT OF STRUCTURAL ELEMENTS SUBJECTED TO BLAST LOADS: A STATE-OF-ART REVIEW C. Sharmila1,* , N. Anandavalli2 , N. Arunachalam1 , Amar Prakash2 1 Bannari Amman Institute of Technology, Sathyamangalam, India 2 CSIR- Structural Engineering Research Centre, Chennai, India ABSTRACT In recent years, explosion incidents due to terrorist attack and accidental explosion have increased worldwide. It has been recognized that assessment of structural elements and buildings under blast loading is very important for protective design. Pressure- impulse (P-I) diagram is an isodamage curve used in the preliminary design of protective structures to establish safe response limits for a given blast loading. With a maximum displacement or damage level defined, P-I curve indicates the combination of load and impulse that will cause the specified failure. This paper presents the state-of-art review on theoretical and numerical methods for developing P-I diagram for structural elements. P-I diagram can be generated from closed form solution for idealized single degree of freedom (SDOF) system of the structural element subjected to a specific loading. Energy balance method is the most widely used method for generating P-I diagram. These methods are simple, but suitable for specific load pulses and structures. For complex geometry or irregular pulse shapes, numerical method has to be resorted. Advantages and suitability of each of the methods are brought out from a critical review. Keywords: P-I Diagram, Blast Loads, Single Degree of Freedom System. 1.0 INTRODUCTION The need for protection of buildings against explosion becomes obligatory. The use of explosives in military and industrialisation of society imposes to manufacture, store and handle INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET) ISSN 0976 – 6308 (Print) ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME: www.iaeme.com/ijciet.asp Journal Impact Factor (2014): 7.9290 (Calculated by GISI) www.jifactor.com IJCIET ©IAEME
- 2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 276 explosives in productive ways. Thus, the assessment of buildings against blast loads and prediction of suitable structure to withstand blast loads plays vital role in construction field. Pressure-impulse (P-I) diagrams are commonly used to assess the structural elements under blast loading circumstances [1-6, 9]. It is also known as load impulse or force impulse diagram [1]. Impulse of the load, I is defined as the area enclosed by the load time curve and the time axis and its magnitudes are given by the following expression. I= (1) where Q(t) is the load time curve and td is the duration of load pulse. Many factors which influence the P-I curve are shape of the load pulse, load time curve, plasticity and damping properties of the structural element [1]. It can also be used to assess human response to blast loading and to establish damage criterion to specific organs of the human being [1]. The single degree of freedom (SDOF) model is the most widely used method to determine the response of structures under blast loading [9-12]. But, this method is restricted to specific loading conditions and geometry of the structure. In this paper, critical literature review is done to identify the suitable method to develop P-I diagram for structural elements under blast loading. 2.0 CHARACTERISTICS OF PRESSURE-IMPULSE DIAGRAM Pressure-impulse diagram can be obtained by transforming the response spectrum (Fig.1a) by changing the set of axes [1]. In Fig. 1, xmax is the maximum dynamic displacement, K is the spring stiffness, Po is the peak force, M is the lumped mass, td is the load pulse duration, and T is the natural period of the system. The threshold curve divides the pressure impulse diagram into two distinct regimes as shown in Fig.1b. The combination of pressure and impulse which falls on the left and below the curve will not induce damage, while those to the right and above the graph will produce damage in excess of the allowable limit [1]. Pressure-impulse diagram is characterised into three different regimes (Fig.1b) based on the ratio between the natural frequency of the structural element and the duration of the forcing or load function. They are impulse response regime, quasi static regime, dynamic response regime [1, 7, 12]. In an impulsive response regime, duration of the blast load is very short compared with the natural period of the structural element. The duration of the load is such that the load has finished acting before the element had time to respond [8]. When the duration of the blast load is much longer than the natural period of the structural element, the loading is termed as quasi-static [8]. In this case, the blast loading magnitude may be considered constant while the element reaches its maximum deformation. In between the impulsive and quasi-static regimes, there is a more complicated, time- dependent regime, commonly called the dynamic regime. In this regime, the load duration is similar to the natural period of vibration of the structural element, and the duration of the load is similar to the time taken for the element to respond significantly [8].
- 3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 277 Fig 1a: Typical response spectrum [1] Fig 1b: Typical P-I diagram [1] 3.0 METHODS TO DEVELOP P-I DIAGRAM The various method used to develop P-I diagram are • Energy balance method • Analytical method • Numerical method 3.1 Energy balance method Krauthammer et al., [14] used energy balance method to obtain Pressure-Impulse diagram. This is the commonly used method to determine P-I diagram. This is based on the principle of conservation of mechanical energy. It is assumed that due to inertia effects, the initial total energy imparted to the system is in the form of kinetic energy only. Thus, by equating this energy to the total strain energy stored in the system, expression for impulsive asymptote is obtained. To obtain the expression for quasi static loading regime, the load is assumed to be constant, before the deformation is achieved. Thus, by equating the work done by load to the total strain energy gained by the system, the expression for the quasi-static asymptote is obtained. Mathematically it can be expressed as, K.E = S.E impulsive asymptote (2) W.E = S.E quasi-static asymptote (3) where K.E is the kinetic energy of the system at time zero, S.E is the strain energy of the system at maximum displacement, and W.E is the work done by the load to displace the system from rest to the maximum displacement. For perfectly elastic systems the expressions are [14] K.E = (4) IMPULSIVE REGIME DYNAMIC REGIME QUASI-STATIC REGIME INITIAL TANGENT IMPULSIVE REGIME DYNAMIC REGIME QUASI-STATIC REGIME
- 4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 278 W.E = Po xmax (5) S.E = ½ (Kx2 max) (6) where I= impulse M= lumped mass xmax = maximum dynamic displacement K = spring stiffness Po = peak force Substituting above equations in (2) and (3) the dimensionless impulsive and quasi-static asymptotes can be obtained. This method is applicable only to the impulsive and quasi-static regimes and it is limited to simple structural systems, resistance models, and load functions. For complex geometries, numerical methods have to be adopted. 3.2 Analytical model Single degree of freedom system is commonly used to derive Pressure-Impulse diagram. The structural elements are converted in to an equivalent spring mass system known as single degree of freedom system (SDOF) as shown in Fig. 2 [15, 16]. The equilibrium equation which represents the single degree of freedom system is as follows, Mÿ + Cẏ + Ky = F(t) (7) where M = equivalent lumped mass C = damping coefficient K = stiffness of the system F(t) = resistance function. Fig 2: Structure idealised as spring mass system
- 5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 279 Li and Meng [13] considered that the overall structural behaviour is dominated by elastic response and hence, equation (7) can be reduced to Mÿ + Ky = F(t) (8) The maximum deflection of the structure can be determined through maximum loading intensity and loading impulse for a given loading shape as [13], = g (p,i) (9) where p = , (10) i = = p (11) if yc corresponds to certain structural damage level, = 1 , thus g ( p, i) = 1 (12) Through dimensional analysis the equation to obtain Pressure-impulse diagram can be obtained. They are as follows, p = (13) and impulse values can be obtained through equation (11). To eliminate pulse loading shape effects on Pressure-impulse diagram in elastic perfectly plastic response Li and Meng [18] introduced a new procedure. A general descending pulse load is described by Li and Meng [13]: f(τ) = (1-λ ) exp (-γ ) for 0 ≤ τ ≤ (14) f(τ) = 0 for τ > (15) α and loading pulse shape are the parameters which influence the P-I diagram defined by equation(12). The influence of α on the P-I diagram can be separated, i.e g = 1 (16) where h1 (α) = 0.2993+1.6065α-0.9448α2 (17a) h2 (α) = 0.0204+2.0150α-1.0216α2 (17b) The loading shape effect on P-I diagram can be eliminated by using the geometric method proposed by Li and Meng [13].
- 6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 280 A dimension less parameters has been introduced by Fallah and Louca [19] to derive unique differential equation for all elastic-plastic- hardening and softening SDOF models which are subjected to detonation blast loading. The equations to derive impulsive and quasi-static asymptotes are, For quasi-static asymptote, (18) For impulsive asymptote, (19) where I= total impulse, Fm= maximum force on the system, K = elastic stiffness, M = lumped mass of the SDOF system, ym = maximum structural deflection. α, Ψ, θ are dimensionless parameters, defined as α = (yel /yc ), Ψ2 = (Kβ/ K) , θ= +1 , -1 for elastic plastic hardening and elastic plastic softening respectively. Wei et al., [20] used two loosely coupled SDOF system to model the flexural and shear failure of one-way reinforced concrete slab under blast load. Analytical formulae has been generated as, (P-Po) (I-Io)n = 0.33( 1.5 (20) where n = the failure mode factor ( i.e. 0.6 for flexure mode, 0.5 for shear mode), Po = the pressure asymptote for damage degree (kPa), Io = the impulsive asymptote for damage degree ( kPa-ms). 3.3 Numerical method Structural response to blast loads are obtained using numerical analysis [21, 22, 23], Damage criterion is defined. A series of numerical simulation is carried out for various combinations of pressure and duration. Blast loads corresponding to the RC column damage will be plotted in the Pressure-Impulse space together with the damage level. Curve- fitting is adopted to generate Pressure-impulse curves. A numerical model of RC column has been created by Shi et al., [24]. A new damage criterion for RC column is defined based on the residual axial load carrying capacity and bond slip effect between concrete and steel is take in to account. Parametric studies are carried out to determine the influence of column dimension, concrete strength, longitudinal and transverse reinforcement on Pressure-impulse diagram. Based on numerical results analytical formula has been derived to predict pressure and impulse asymptote for various degrees of damage as 0.2, 0.5, and 0.8. They are Po (0.2) = 1000 [0.007 exp( )+0.069( )+0.034 exp( ) – 0.835 ln( ) + ( )1.804 + 0.067 ln ( ) – 0.168], (21) Io (0.2) = 1000 [0.053 exp( )+0.107( )+0.021 exp( ) + ( )-0.207 +1.203 exp( ) - 0.943 ln ( ) – 2.686], (22)
- 7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 281 Po (0.5) = 1000 [0.143 ln( )+0.320 ln( )+0.063 exp( ) + ( )-1.390 +2.639 ( ) +0.318 ln ( ) – 2.271], (23) Io (0.5) = 1000 [0.837( )+0.036 ( )+0.235exp( ) + ( )-0.274 +2.271exp ( ) -0.998 ln ( ) – 5.286], (24) Po (0.8) = 1000 [0.062 ln( )+0.238 ( )+0.291ln( ) -1.676 ln( )+2.439 ln ( ) +0.210 ln ( ) + 1.563], (25) Io (0.8) = 1000 [3.448 ( )-0.254 ( )+1.200 ( ) -0.521+ 6.993 ( ) -2.759 ln ( )-2.035]. (26) where Po(D) = Pressure asymptote, Io(D) = impulsive asymptote, ρs = transverse reinforcement ratio, ρ = longitudinal reinforcement ratio, f΄c = concrete strength, H = column height, h = column depth, b = column width. 4.0 P-I DIAGRAM USING SDOF SYSTEM A reinforced concrete (RC) slab is taken up for numerical study. Dimension of the slab is 15 m x 3.5 m, while its thickness is 450 mm. Slab is provided with 0.39% main reinforcement. Grade of concrete is M25, while high strength steel is used for reinforcement. Slab is subjected to air blast loading due to an explosive charge (TNT) of weight 30000 kg. RC slab is converted into an equivalent SDOF system with effective mass of 740 kg and equivalent stiffness of 717094 N/m using IS 4991(1968). Location of explosive charge is varied to obtain the pressure-impulse diagram. Response of perfectly elastic SDOF system for triangular load pulses is calculated analytically for various scaled distances given in Table 1. Normalised pressure and impulse are obtained using the following equation [1], P̅ = (27) I̅ = (28) where, = peak over pressure K= stiffness of slab M= mass = maximum displacement I= impulse = ( )/2. Pressure impulse curve for the RC slab is shown in Fig.3.
- 8. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 282 Table 1: Scaled distance and peak overpressure No. Scaled distance, z (m) Peak overpressure (N/m2 ) No. Scaled distance, z (m) Peak overpressure (N/m2 ) 1. 0.6437 2468478.40 10. 1.9310 222758.70 2. 0.8046 1590410.00 11. 1.9953 206348.10 3. 0.9011 1245990.00 12. 2.0597 191558.30 4. 0.9494 1114300.00 13. 2.1884 166435.90 5. 0.9655 1076819.00 14. 2.2528 155495.50 6. 1.1264 768867.00 15. 2.5746 114570.30 7. 1.2873 569609.90 16. 2.8965 87928.40 8. 1.4482 437008.20 17. 3.2183 69795.70 9. 1.6091 341381.00 18. 3.5401 57031.90 Fig 3: P-I Diagram for RC slab using SDOF system 5.0 DISCUSSION In SDOF approach, there is a probability for underestimation and overestimation of blast resistant capacity of the structural elements at various loading regimes due to material idealization and negligence of strain rate effect [24]. Thus, SDOF approach is suitable for simple problems and for specific shape of load pulses. Energy balance method is simple, but applicable only to impulsive and quasi-static regimes. In case of complex geometries and varying shape of pulse loads numerical methods produces accurate results, and it has been verified by shi et al., [24] with experimental test results.
- 9. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 283 6.0 CONCLUSION In this paper, three major methods to develop Pressure-impulse diagram for various damage criterions are discussed. Pressure impulse diagram is found to be greatly influenced by shape of pulse load, geometry of the structure and material properties. SDOF system is the most commonly used method for simple geometrical structures, however for complex geometries and for irregular shape of pulse loads, it becomes unsuitable. Despite numerical methods are complex, it provides better results compared to SDOF system. Hence, numerical method is suggested for developing P-I diagrams for new structural elements. ACKNOWLEDGEMENTS The authors wish to thank Dr. J. Rajasankar for his valuable suggestions. The authors wish to acknowledge the support rendered by staff of Shock and Vibration Group. Paper is being published with kind permission of Director, CSIR-Structural Engineering Research Centre, Chennai. REFERENCES 1. Krauthammer, T., “Modern protective structures”, CRS Press, 2008. 2. Ma, G.W., Shia, H.J., Shub, D.W., “P–I diagram method for combined failure modes of rigid- plastic beams”, International Journal of Impact Engineering, Vol. 34, pp. 1081-1094, 2007. 3. Dragos, J., Wu, C., Haskett, M., Oehlers, D., “Derivation of Normalized Pressure Impulse Curves for Flexural Ultra High Performance Concrete Slabs”, ASCE Journal of Structural Engineering, Vol. 139, No. 6, June 2013. 4. Smith, P., Hetherington J., “Blast and ballistic loading of structures”, Great Britain, London: Butterworth-Heinemann Ltd, pp.166-169, 1994. 5. Lan, S.R., Crawford, J.C., “Evaluation of the blast resistance of metal deck proofs”, proceeding of the fifth Asia-Pacific conference on shock & impact loads on structures, Changsha, Hunan, China, pp. 3–12, 2003. 6. Mutalib, A.A., and Hao, H., “Development of P-I diagrams for FRP strengthened RC columns”, International Journal of Impact Engineering, Vol. 38, pp.290-304, 2011. 7. Marchand, M.A., and Alfawakhiri, F., "Blast and Progressive Collapse", AISC, Inc., 2004. 8. Yandzio, E., and Gough, M., “Protection of Buildings against Explosions”, The Steel Construction Institute Silwood Park. 9. Dragos, J., Wu, C., “A new general approach to derive normalised pressure impulse curves”, Vol.62, pp.1-12, 2013. 10. Morison, C.M., "Dynamic response of walls and slabs by single-degree-of-freedom analysis— a critical review and revision", International Journal of Impact Engineering, Vol. 32, pp.1214–1247, 2006. 11. Huang, X., Ma, G.W., Li, J.C., “Damage Assessment of Reinforced Concrete Structural Elements Subjected to Blast Load”, International Journal of Protective Structures,Vol.1, No.1, 2010. 12. Shi Y, Li ZX, Hao H, “Bond slip modelling and its effects on numerical analysis of blast- induced responses of RC columns”, Structural Engineering & Mechanics, Vol.32, No.2, pp.251-267, 2009. 13. Li, Q.M., and Meng, H., “Pressure Impulse diagram for blast loads based on Dimensional analysis and SDOF model”, Journal of Engineering mechanics, Vol. 128, No.1, Jan 2002.
- 10. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 275-284 © IAEME 284 14. Krauthammer, T., Astarlioglu, S., Blasko, J., Soh, T.B., Ng, P.H., “Pressure impulse diagram for behaviour assessment of structural components”, International Journal of Impact Engineering, Vol.38, pp. 771-783, 2008. 15. Biggs, J.M., “Introduction to Structural Dynamics “, McGraw-Hill, New York, 1964. 16. Syed, Z.I., Mendis, P., Lam, N.T.K, Ngo, T, “Concrete damage assessment for blast load using pressure-impulse diagrams”, Earthquake Engineering in Australia, Canberra 24-26, November 2006. 17. Fallah, A.S., Nwankwo, E., Louca, L.A., “Pressure-Impulse diagram for blast loaded continuous beams based on dimensional analysis”, ASME Journal of Applied Mechanics, Vol. 80, Sep 2013. 18. Li, Q.M., and Meng, H., “Pulse loading shape effects on P-I diagram of an elastic- plastic, SDOF structural model”, International Journal of Mechanical Sciences, Vol. 44, 2002. 19. Fallah, A.S., and Louca, L.A., “Pressure–impulse diagrams for elastic-plastic hardening and softening single-degree-of-freedom models subjected to blast loading”, International Journal of Impact Engineering, Vol.34, pp. 823–842,2007. 20. Wei, W., Duo, Z., Fang-yun, LU., Fu-jing, T., Song-chuan, W.,“ Pressure-impulse diagram with multiple failure modes of one-way reinforced concrete slab under blast loading using SDOF method”, Journal of Central South University, Vol. 20, pp. 510-519, 2013. 21. Huang, X., Li, J.C., Ma, G.W., “Damage analysis of RC column/beam subject to blast load. In: Proceeding of the Eighth International Conference on Shock & Impact Loads on Structures, Adelaide”, Australia, 2009. 22. Shi, Y., Hao, H., Li, Z.X., “Numerical simulation of blast wave interaction with structures columns”, Shock Waves, Vol. 17, pp.113-133, 2007. 23. Chan, S., Fawaz, Z., Behdinan, K., Amid, R., “Ballistic limit prediction using numerical model with progressive damage capability”, Composite Structures, Vol.77, pp.466-474, 2007. 24. Shi, Y., Hao, H., Li, Z.X., “Numerical derivation of Pressure-impulse diagrams for prediction of RC column damage to blast loads”, International Journal of Impact Engineering”, Vol. 35, pp. 1213-1227, 2008. 25. IS: 4991-1968, “Criteria for blast resistant design of structures for explosions above ground”. 26. Mohammed S. Al-Ansari, “Building Response to Blast and Earthquake Loading”, International Journal of Civil Engineering & Technology (IJCIET), Volume 3, Issue 2, 2012, pp. 327 - 346, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316. 27. Prof.M.R.Wakchaure and S.T.Borole, “Comparison of the Lateral Deflection at Midpoint of Long & Short Side Column Under Blast Loading”, International Journal of Civil Engineering & Technology (IJCIET), Volume 4, Issue 4, 2013, pp. 106 - 112, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316. 28. Shaikh Zahoor Khalid and S.B. Shinde, “Seismic Response of FRP Strengthened RC Frame”, International Journal of Civil Engineering & Technology (IJCIET), Volume 3, Issue 2, 2012, pp. 305 - 321, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316. 29. Dr. T.Ch.Madhavi, Shanmukha Kavya .V, Siddhartha Das, Sri Prashanth .V and Vetrivel .V, “Composite Action of Ferrocement Slabs under Static and Cyclic Loading”, International Journal of Civil Engineering & Technology (IJCIET), Volume 4, Issue 3, 2013, pp. 57 - 62, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.

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