Simple P I diagram for structural components based on ...
Transcript of Simple P I diagram for structural components based on ...
Advances in Concrete Construction, Vol. 10, No. 6 (2020) 509-514
DOI: https://doi.org/10.12989/acc.2020.10.6.509 509
Copyright ยฉ 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: [email protected] aPh.D. Candidate
E-mail: [email protected]
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 (๐ข๐, ๐ ๐)
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 ๐ข๐๐๐ฅ . When ๐ข๐ธ is too
small compared to ๐ข๐๐๐ฅ, ๐ข๐ธ 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
๐พ๐ฟ๐๐๏ฟฝฬ๏ฟฝ + ๐ ๐ข = ๐ (1)
where ๐ข is the deflection, ๐พ๐ฟ๐ is the load-mass factor for
perfectly plastic response, ๐ is the mass, and ๐ ๐ข is the
constant resistance. ๐ is the applied pressure defined as
๐ = ๐๐๐๐ฅ โ๐๐๐๐ฅ
๐ก๐๐ก when 0 โค ๐ก โค ๐ก๐
๐ = 0 when ๐ก๐ < ๐ก
(2)
where ๐๐๐๐ฅ is the peak pressure and ๐ก๐ is the pressure
duration.
When the initial deflection and velocity are set as zero,
the deflection histories are as follows
๐ข = โ1
6
๐๐๐๐ฅ
๐พ๐ฟ๐๐๐ก๐๐ก3 +
1
2
๐๐๐๐ฅโ๐ ๐ข
๐พ๐ฟ๐๐๐ก2 when 0 โค ๐ก โค ๐ก๐ (3)
๐ข = โ1
2
๐ ๐ข
๐พ๐ฟ๐๐๐ก2 +
1
2
๐๐๐๐ฅ๐ก๐
๐พ๐ฟ๐๐๐ก
โ1
6
๐๐๐๐ฅ๐ก๐2
๐พ๐ฟ๐๐
when ๐ก๐ < ๐ก (4)
The maximum deflection occurs when velocity is zero.
If deflection reaches the maximum before ๐ก๐, the time at
maximum deflection (๐ก๐๐๐ฅ) is as below
๐ก๐๐๐ฅ =2๐ก๐(๐๐๐๐ฅ โ ๐ ๐ข)
๐๐๐๐ฅ
(5)
It should be noted that ๐ก๐๐๐ฅ should be shorted than ๐ก๐.
Substituting Eq. (5) into Eq. (3), the maximum deflection
(๐ข๐๐๐ฅ) is obtained as follow
๐ข๐๐๐ฅ = ๐ข(๐ก๐๐๐ฅ) =
2
3
(1โ๐ ๐ข
๐๐๐๐ฅ)
3๐๐๐๐ฅ๐ก๐
2
๐พ๐ฟ๐๐
when ๐๐๐๐ฅ
๐ ๐ขโค 2 (6)
If deflection reaches the maximum after ๐ก๐, the time at
maximum deflection is
๐ก๐๐๐ฅ =๐๐๐๐ฅ๐ก๐
2๐ ๐ข (7)
where ๐ก๐๐๐ฅ should be longer than ๐ก๐. Substituting Eq. (7)
into Eq. (4), the maximum deflection is obtained as follows
๐ข๐๐๐ฅ =๐๐๐๐ฅ๐ก๐
2
๐พ๐ฟ๐๐(
๐๐๐๐ฅ
8๐ ๐ขโ
1
6) when
๐๐๐๐ฅ
๐ ๐ข> 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
๐ = ๐ก๐๐โ1 (๐ข๐๐๐ฅ
๐) (9)
where ๐ is the support rotation angle and ๐ 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 (๐ก๐ =2๐ผ2/๐๐๐๐ฅ), normalized pressure-impulse diagram equation
corresponding to a given damage level can be derived as
follows by Eqs. (6),(8) and (9)
๐ผ ฬ = โ3๏ฟฝฬ ๏ฟฝ๐ก๐๐๐
8(1โ1
๏ฟฝฬ ๏ฟฝ)
3 when 1 < ๏ฟฝฬ ๏ฟฝ < 2
Fig. 2 Searching algorithm for P-I diagram
๐ผ ฬ = โ2๐ก๐๐๐
(1โ4
3๏ฟฝฬ ๏ฟฝ) when ๏ฟฝฬ ๏ฟฝ โฅ 2 (10)
where ๐ผ ฬ and ๏ฟฝฬ ๏ฟฝ are the dimensionless impulse and peak
pressure, respectively, and are defined as:
๐ผ ฬ =๐ผ
โ๐พ๐ฟ๐๐๐ ๐ข๐
๏ฟฝฬ ๏ฟฝ =๐๐๐๐ฅ
๐ ๐ข
(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. PโI 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 ๐ผ 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 ๐ผ, 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 ๐ฟ๐ is as below
๐ฟ๐ = 1.3 ร (๏ฟฝฬ ๏ฟฝ๐ โ1 +1
tan(๐ผ)2) (12)
where ๏ฟฝฬ ๏ฟฝ๐ is solution of following equations.
๏ฟฝฬ ๏ฟฝ3 โ (3 +3 tan(๐) tan(๐ผ)2
8) ๏ฟฝฬ ๏ฟฝ2
+3๏ฟฝฬ ๏ฟฝ โ 1 = 0 when 1 < ๏ฟฝฬ ๏ฟฝ < 2
(13)
๏ฟฝฬ ๏ฟฝ2 โ4
3๏ฟฝฬ ๏ฟฝ โ 2 tan(๐) tan(๐ผ)2 = 0 when ๏ฟฝฬ ๏ฟฝ โฅ 2
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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
๐๐ธ๐๐ = ๐ ๐ข๐ข๐๐๐ฅ
๐๐ธ๐๐ = ๐ ๐ข๐ข๐๐๐ฅ โ1
2๐ ๐๐ข๐ โ
1
2(๐ ๐ข โ ๐ ๐)(๐ข๐ธ + ๐ข๐)
(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
((๐๐ธ๐๐ โ ๐๐ธ๐๐) ๐๐ธ๐๐โ ) 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 mmร1500
mmร150 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)
๐ผ (Pa*s) 3717.4 2400.4 1542.6 955.0
๐๐๐๐ฅ(KPa) 6645.0 2488.1 845.5 272
Normalized Pressure Characteristics (from Eq. 11)
๐ผ ฬ 0.87 0.56 0.36 0.22
๏ฟฝฬ ๏ฟฝ 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: ๐=360 kg/m2, ๐พ๐ฟ๐=0.66 and ๐ ๐ข=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.
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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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