Advances in Concrete Construction, Vol. 10, No. 6 (2020) 509-514

DOI: https://doi.org/10.12989/acc.2020.10.6.509 509

Copyright © 2020 Techno-Press, Ltd. http://www.techno-press.org/?journal=acc&subpage=7 ISSN: 2287-5301 (Print), 2287-531X (Online)

1. Introduction

For preliminary blast resistance design, hydro code

based structural analysis is not feasible. Although, hydro

codes, such as AUTODYN (Toy and Sevim 2017; Sevim

and Toy 2020) and LS-Dyna (Hong et al. 2017), are capable

of estimating detailed failure mechanism, they require

severe calculation time and are sensitive to assumed

parameter values. Single Degree Of Freedom (SDOF)

model has been an alternative in the preliminary design

stage (US DoD 2008), where many calculations are needed.

The SDOF model consists of equivalent mass, damper and

resistance function based on assumed mode shaped. SDOF

can be applied to different types of structural types and

materials such as masonry walls (Edri and Yankelevsky

2018), reinforced concrete columns (Liu et al. 2019, Park et

al. 2014), beams (Nagata et al. 2018), slabs (Kee et al.

2019, Wang et al. 2013), steel columns (Al-Thairy 2016).

For more simplicity in the design stage, Pressure-

Impulse diagram (P-I) is generally accepted (Hou et al.

2018, Li and Meng 2002), which is an iso-damage curve

showing failure causing load as pressure and impulse. For

more convenient use, studies on normalization of P-I have

been made (Yu et al. 2018, Krauthammer et al. 2008, Li

and Meng 2002, Fallah and Louca 2007, Dragos and Wu

2013). Although, many assumptions were made for type of

loading shapes and resistance functions during

normalization, it has not been explained how these

assumptions affects the accuracy. In this study, closed form

solution of P-I diagram based on US DoD criteria (PDC

2008) is proposed and error caused by assumption in reality

Corresponding author, Associate Professor

E-mail: jip111@seoultech.ac.kr aPh.D. Candidate

E-mail: junghunkee@gmail.com

is evaluated. The proposed P-I curve is also validated

comparing to a field blast test results.

2. Solutions for P-I diagram

2.1 Closed form solution for dimensionless P-I diagram

Resistance function of SDOF system assuming no

tension or compression membrane response can be

expressed as Fig. 1, where the point of infection (𝑢𝑒, 𝑅𝑒)

only exists for indeterminate structural components. The

resistance function is determined by material properties and

structural geometry. The level of damage is determined by

dynamic maximum deflection 𝑢𝑚𝑎𝑥 . When 𝑢𝐸 is too

small compared to 𝑢𝑚𝑎𝑥, 𝑢𝐸 can be negligible. That is,

perfectly plastic resistance can be assumed. The validity of

this assumption will be explained in Section 3.

Fig. 1 bi-linear resistance curve

Simple P-I diagram for structural components based on support rotation angle criteria

Jung Hun Keea and Jong Yil Park

Department of Safety Engineering, Seoul National University of Science and Technology, Seoul 01811, Republic of Korea

(Received September 2, 2020, Revised October 16, 2020, Accepted October 21, 2020)

Abstract. In the preliminary design phase of explosion-proof structures, the use of P-I diagram is useful. Based on the fact that

the deformation criteria at failure or heavy damage is significantly larger than the yield deformation, a closed form solution of

normalized P-I diagram is proposed using the complete plastic resistance curve. When actual sizes and material properties of RC

structural component are considered, the complete plasticity assumption shows only a maximum error of 6% in terms of strain

energy, and a maximum difference of 9% of the amount of explosives in CWSD. Thru comparison with four field test results,

the same damage pattern was predicted in all four specimens.

Keywords: explosion; P-I diagram; closed form solution; support rotation angle

Jung Hun Kee and Jong Yil Park

When undamped perfectly plastic SDOF system is

subjected to a triangular load with zero rise time, dynamic

governing equation can be given as follows

𝐾𝐿𝑀𝑀�̈� + 𝑅𝑢 = 𝑃 (1)

where 𝑢 is the deflection, 𝐾𝐿𝑀 is the load-mass factor for

perfectly plastic response, 𝑀 is the mass, and 𝑅𝑢 is the

constant resistance. 𝑃 is the applied pressure defined as

𝑃 = 𝑃𝑚𝑎𝑥 −𝑃𝑚𝑎𝑥

𝑡𝑑𝑡 when 0 ≤ 𝑡 ≤ 𝑡𝑑

𝑃 = 0 when 𝑡𝑑 < 𝑡

(2)

where 𝑃𝑚𝑎𝑥 is the peak pressure and 𝑡𝑑 is the pressure

duration.

When the initial deflection and velocity are set as zero,

the deflection histories are as follows

𝑢 = −1

6

𝑃𝑚𝑎𝑥

𝐾𝐿𝑀𝑀𝑡𝑑𝑡3 +

1

2

𝑃𝑚𝑎𝑥−𝑅𝑢

𝐾𝐿𝑀𝑀𝑡2 when 0 ≤ 𝑡 ≤ 𝑡𝑑 (3)

𝑢 = −1

2

𝑅𝑢

𝐾𝐿𝑀𝑀𝑡2 +

1

2

𝑃𝑚𝑎𝑥𝑡𝑑

𝐾𝐿𝑀𝑀𝑡

−1

6

𝑃𝑚𝑎𝑥𝑡𝑑2

𝐾𝐿𝑀𝑀

when 𝑡𝑑 < 𝑡 (4)

The maximum deflection occurs when velocity is zero.

If deflection reaches the maximum before 𝑡𝑑, the time at

maximum deflection (𝑡𝑚𝑎𝑥) is as below

𝑡𝑚𝑎𝑥 =2𝑡𝑑(𝑃𝑚𝑎𝑥 − 𝑅𝑢)

𝑃𝑚𝑎𝑥

(5)

It should be noted that 𝑡𝑚𝑎𝑥 should be shorted than 𝑡𝑑.

Substituting Eq. (5) into Eq. (3), the maximum deflection

(𝑢𝑚𝑎𝑥) is obtained as follow

𝑢𝑚𝑎𝑥 = 𝑢(𝑡𝑚𝑎𝑥) =

2

3

(1−𝑅𝑢

𝑃𝑚𝑎𝑥)

3𝑃𝑚𝑎𝑥𝑡𝑑

2

𝐾𝐿𝑀𝑀

when 𝑃𝑚𝑎𝑥

𝑅𝑢≤ 2 (6)

If deflection reaches the maximum after 𝑡𝑑, the time at

maximum deflection is

𝑡𝑚𝑎𝑥 =𝑃𝑚𝑎𝑥𝑡𝑑

2𝑅𝑢 (7)

where 𝑡𝑚𝑎𝑥 should be longer than 𝑡𝑑. Substituting Eq. (7)

into Eq. (4), the maximum deflection is obtained as follows

𝑢𝑚𝑎𝑥 =𝑃𝑚𝑎𝑥𝑡𝑑

2

𝐾𝐿𝑀𝑀(

𝑃𝑚𝑎𝑥

8𝑅𝑢−

1

6) when

𝑃𝑚𝑎𝑥

𝑅𝑢> 2 (8)

The degree of damage can be determined based on the

support rotation angle at maximum deflection for flexural

behavior. Then, support rotation angle is defined as below

𝜃 = 𝑡𝑎𝑛−1 (𝑢𝑚𝑎𝑥

𝑍) (9)

where 𝜃 is the support rotation angle and 𝑍 is the shortest

distance from a support to point of maximum deflection.

With dimensionless impulse and peak pressure and the

relationship of peak pressure, impulse, and duration (𝑡𝑑 =2𝐼2/𝑃𝑚𝑎𝑥), normalized pressure-impulse diagram equation

corresponding to a given damage level can be derived as

follows by Eqs. (6),(8) and (9)

𝐼 ̅ = √3�̅�𝑡𝑎𝑛𝜃

8(1−1

�̅�)

3 when 1 < �̅� < 2

Fig. 2 Searching algorithm for P-I diagram

𝐼 ̅ = √2𝑡𝑎𝑛𝜃

(1−4

3�̅�) when �̅� ≥ 2 (10)

where 𝐼 ̅ and �̅� are the dimensionless impulse and peak

pressure, respectively, and are defined as:

𝐼 ̅ =𝐼

√𝐾𝐿𝑀𝑀𝑅𝑢𝑍

�̅� =𝑃𝑚𝑎𝑥

𝑅𝑢

(11)

2.2 Numerical P-I diagram

The perfectly plastic resistance assumption should be

checked if it induces significant error. P-I diagrams based

on bilinear resistance curve are prepared to compare with

closed form solution (Eq. (11)). Since there no closed form

solution for bilinear one, numerical approach was

conducted. P–I diagrams can be generated with sufficient

numerically computed points, which represent pressure

characteristics causing a given damage level. Due to

massive computation loading from nonlinear resistance

functions of SDOF, a searching algorism is used as shown

in Fig. 2.

Points for the P-I diagram are searched by changing the

angle 𝛼 at regular intervals. Due to the shape

characteristics of the P-I diagram, a relatively small number

of points are generated in the impact and the quasi-static

region, and a large number of points are generated in the

dynamic region where the variation is severe.

At a given angle 𝛼, the initial loading point is set at 1.3

times the distance from the origin to the proposed P-I

diagram (Eq. (13)). The initial distance 𝐿𝑖 is as below

𝐿𝑖 = 1.3 × (�̅�𝑠√1 +1

tan(𝛼)2) (12)

where �̅�𝑠 is solution of following equations.

�̅�3 − (3 +3 tan(𝜃) tan(𝛼)2

8) �̅�2

+3�̅� − 1 = 0 when 1 < �̅� < 2

(13)

�̅�2 −4

3�̅� − 2 tan(𝜃) tan(𝛼)2 = 0 when �̅� ≥ 2

510

Simple P-I diagram for structural components based on support rotation angle criteria

Table 1 Practical ranges of two-way reinforced concrete

slab parameters

Parameters Ranges

Geometry

Boundary Condition Four Sides Simply Supported,

Four Sides Fixed Supported,

Long Span Length (mm) 4,000~6,000

Short Span Length (mm) 2,000~4,000

Thickness (mm) 150~250

Material Properties

Concrete Compressive Strength (MPa) 18~60

Reinforcement Ratio 0.001~0.005

Yield Strength of Reinforcement (MPa) 300~500

At each point, the maximum deflection is numerically

calculated. The loading is reduced from the initial point

toward the origin with a constant step until damage does not

occur. Then, the load is increased at smaller steps until

damage occurs. This process is repeated until the error is

less than a certain level.

3. Application of perfectly plastic closed form solution

3.1 Error from perfectly-plastic assumption

Strain energies up to the maximum deflection for

perfectly plastic resistance and elastic-perfectly plastic

resistance is as follows

Table 2 Practical ranges of owe-way reinforced concrete

slab parameters

Parameters Ranges

Geometry

Boundary Condition Fixed-Fixed, Simple-Simple

Span Length (mm) 2,000~4,000

Thickness (mm) 150~250

Material Properties

Concrete Compressive Strength (MPa) 18~60

Reinforcement Ratio 0.001~0.005

Yield Strength of Reinforcement (MPa) 300~500

𝑆𝐸𝑝𝑝 = 𝑅𝑢𝑢𝑚𝑎𝑥

𝑆𝐸𝑒𝑝 = 𝑅𝑢𝑢𝑚𝑎𝑥 −1

2𝑅𝑒𝑢𝑒 −

1

2(𝑅𝑢 − 𝑅𝑒)(𝑢𝐸 + 𝑢𝑒)

(22)

As the energy difference is smaller, the reliability of the

closed form P-I diagram increases. The energy differences

of the structural components with practical ranges are

analyzed, of which characteristics are shown in Table 1 and

2. The range of each variable of existing structures is set

based on Ellefsen and Fordyce (2012), who collected data

from 41 countries.

Maximum and minimum energy difference ratios

((𝑆𝐸𝑝𝑝 − 𝑆𝐸𝑒𝑝) 𝑆𝐸𝑝𝑝⁄ ) are given in Table 3 corresponding

to each damage level of 5o and 10o support rotation criteria,

where maximum difference is 6.0% at all simply supported

two-way RC slab.

Fig. 3 shows closed form P-I diagrams from perfectly

plastic resistance and numerical P-I one from elastic-

perfectly plastic resistance when the energy difference is

(a) Two-way slab (all simply supported) (b) Two-way slab(all fixed supported)

(c) One-way slab (simply supported) (d) One-way slab (fixed supported)

Fig. 3 Closed form and numerical P-I in the case of maximum energy difference

511

Jung Hun Kee and Jong Yil Park

Table 3 Energy difference of numerical and closed form P-I

curves

Structural components Energy Difference (%) from Eq. (22)

5o rotation angle 10o rotation angle

Fixed-fixed, one-way RC

slabs 0.1~2.5 0.0~1.2

Simple-simple, one-way

RC slabs 0.1~5.2 0.1~2.5

All fixed, two-way RC

slabs 0.2~4.8 0.1~2.3

All simple, two-way RC

slabs 0.2~6.0 0.1~2.9

Fig. 4 Charge weight-standoff distance diagram

maximum.

Since errors from the resistance shape assumption is

difficult to be explained intuitively in P-I diagrams, TNT

Charge Weight-Standoff Distance(CWSD) diagram can be

introduced. Kingery-Bulmash equation (Kingery and

Bulmash 1984) is applied for the pressure and impulse

values at a given TNT and standoff distance. Fig. 4 shows

CWSD diagrams from perfectly plastic resistance and

elastic-perfectly plastic one causing 5o support rotation.

Structural component is the simply supported one-way RC

slab with parameters causing 5.2% energy difference. The

range of TNT amounts is from 100 kg to 5,900 kg.

Maximum difference of standoff distance at a given TNT

weight is 9.4% at 100kg surface detonation.

Considering the geometry and material range of the real

structural components, Eq. (10) with perfectly plastic

resistance assumption yields applicable P-I diagram when

the support rotation angle criterion is greater than 5o.

3.2 Comparison with field test results

To validate the proposed closed form P-I diagram, the

damage patterns observed in the real size test (Lee et al.

2017) are compared. In the test, four 2050 mm×1500

mm×150 mm fixed-fixed one-way RC slabs are located to

have a standoff distance of 5 m, 7 m, 10 m, and 15 m

Table 4 Component damage level (PDC 2008)

Damage

Level

Support rotation

angle Description of Component Damage

Blowout Over 10 o

Component is overwhelmed by the

blast load causing debris with

significant velocities

Hazardous

Failure 5 o ~ 10 o

Component has failed, and debris

velocities range from insignificant to

very significant

Heavy

Damage 2 o ~ 5 o

Component has not failed, but it has

significant permanent deflections

causing it to be unrepairable

Moderate

Damage

Elastic

deflection ~ 2 o

Components has some permanent

deflection. It is generally repairable, if

necessary, although replacement may

be more economical and aesthetic

Table 5 Loading characteristics

Specimen A B C D

Applied Pressure Characteristics

(calculated from Kingery-Bulmash)

𝐼 (Pa*s) 3717.4 2400.4 1542.6 955.0

𝑃𝑚𝑎𝑥(KPa) 6645.0 2488.1 845.5 272

Normalized Pressure Characteristics (from Eq. 11)

𝐼 ̅ 0.87 0.56 0.36 0.22

�̅� 88.8 33.2 11.3 3.6

Fig. 5 Loading of test data in proposed P-I diagram

around 100 kg TNT, of which SDOF characteristics are as

follows: 𝑀=360 kg/m2, 𝐾𝐿𝑀=0.66 and 𝑅𝑢=74.87 kPa. The

damage criteria from PDC (PDC-TR-06-08 2008) is

adopted as shown in Table 4.

Since the strain energy difference from the perfectly

plastic assumption in one-way RC slab is 1.9% for 2o of

support rotation angle criteria, 0.7% for 5o, 0.4% for 10o, 2o

of support rotation angle criteria also can be used for the

tested slab. Fig. 5 shows the closed form P-I diagrams and

combination of the dimensionless loading of the test.

Applied pressure characteristics and normalized ones are

given in Table 3. Table 4 summarizes anticipated damages

from Eq. (10) and observed field test results, showing that

application of proposed P-I is feasible.

512

Simple P-I diagram for structural components based on support rotation angle criteria

4. Conclusions

In the range of sizes and material properties of widely

used RC structural members, the difference of yielding

deformation and failure deformation is significantly large. It

was shown that the strain energy difference between the

elasto-plastic resistance and perfectly plastic one is less than

6% at failure deformation.

Based on this observation, the closed solution of the P-I

diagram was proposed from the perfectly plastic resistance

curve and support rotation angle criteria. The P-I curve was

normalized to apply to various materials and damage

criteria. In addition, it is applicable to steel structural types.

In the case of steel structural components, the difference

between yield and failure deformation is generally larger

than that of RC components, so the proposed P-I equation

will be applicable. For verification, the proposed P-I curve

equation was compared with the real scaled experimental

data of four one-way RC slabs, and successfully predicted

the damage pattern in all cases.

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Table 6 Validation of closed form solution

Specimen A B C D

Observed

Damage

Blowout Hazardous Failure Heavy Damage Moderate Damage

Damage

from Eq. (10) Blowout Hazardous Failure Heavy Damage Moderate Damage

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Jung Hun Kee and Jong Yil Park

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JK

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