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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)
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IJCIET
©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].

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

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

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].

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)

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.

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.

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.
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