ATEAS V1(2):: American Transactions on Engineering & Applied Sciences

American Transactions
on Engineering & Applied Sciences
IN THIS ISSUE
A Detailed Analysis of Capillary
Viscometer
Fuzzy Logic Modeling Approach for
Risk Area Assessment for Hazardous
Materials Transportation
Computer Modeling of Internal
Pressure Autofrettage Process of a
Thick-Walled Cylinder with the
Bauschinger Effect
Types of Media for Seeds Germination
and Effect of BA on Mass Propagation
of Nepenthes mirabilis Druce
Numerical Analysis of Turbulent
Diffusion Combustion in Porous Media
Production of Hydrocarbons from
Palm Oil over NiMo Catalyst
Volume 1 No.2 (April 2012)
ISSN 2229-1652
eISSN 2229-1660
http://TuEngr.com/ATEAS

American Transactions on
Engineering & Applied Sciences
International Editorial Board
Editor-in-Chief
Zhong Hu, PhD
Associate Professor, South Dakota State University,
USA
Executive Editor
Boonsap Witchayangkoon, PhD
Associate Professor, Thammasat University,
THAILAND
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Editorial Research Board Members
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Professor Dr.Martin Tajmar (Dresden University of Technology, German )
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Professor Dr. Paolo Bassi ( Universita' di Bologna, Italy )
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Associate Prof. Dr. Burachat Chatveera (Thammasat University, Thailand)
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Assistant Prof. Dr. Jeremiah Neubert (University of North Dakota, USA)
Assistant Prof. Dr. Didem Ozevin (University of Illinois at Chicago, USA)
Assistant Prof. Dr. Deepak Gupta (Southeast Missouri State University, USA)
Assistant Prof. Dr. Xingmao (Samuel) Ma (Southern Illinois University Carbondale, USA)
Assistant Prof. Dr. Aree Taylor (Thammasat University, Thailand)
Assistant.Prof. Dr.Wuthichai Wongthatsanekorn (Thammasat University, Thailand )
Assistant Prof. Dr. Rasim Guldiken (University of South Florida, USA)
Assistant Prof. Dr. Jaruek Teerawong (Khon Kaen University, Thailand)
Assistant Prof. Dr. Luis A Montejo Valencia (University of Puerto Rico at Mayaguez)
Assistant Prof. Dr. Ying Deng (University of South Dakota, USA)
Assistant Prof. Dr. Apiwat Muttamara (Thammasat University, Thailand)
Assistant Prof. Dr. Yang Deng (Montclair State University USA)
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Dr. Monchai Pruekwilailert (Thammasat University, Thailand )
Dr. Piya Techateerawat (Thammasat University, Thailand )
Scientific and Technical Committee & Editorial Review Board on Engineering and Applied Sciences
Dr. Yong Li (Research Associate, University of Missouri-Kansas City, USA)
Dr. Ali H. Al-Jameel (University of Mosul, IRAQ)
Dr. MENG GUO (Research Scientist, University of Michigan, Ann Arbor)
Dr. Mohammad Hadi Dehghani Tafti (Tehran University of Medical Sciences)
2012 American Transactions on Engineering & Applied Sciences.

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ISSN 2229-1652 eISSN 2229-1660 http://tuengr.com/ATEAS
FEATURE PEER-REVIEWED ARTICLES for Vol.1 No.2 (April 2012)
A Detailed Analysis of Capillary Viscometer 107
Fuzzy Logic Modeling Approach for Risk Area
Assessment for Hazardous Materials Transportation
127
Computer Modeling of Internal Pressure Autofrettage
Process of a Thick-Walled Cylinder with the
Bauschinger Effect
143
Types of Media for Seeds Germination and Effect of BA
on Mass Propagation of Nepenthes mirabilis Druce
163
Numerical Analysis of Turbulent Diffusion Combustion
in Porous Media
173
Production of Hydrocarbons from Palm Oil over NiMo
Catalyst
183

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A Detailed Analysis of Capillary Viscometer
Prashanth Sridharan
a
, Abiodun Yakub
a
,
Charles Safarik
a
, and Rasim Guldiken
a*
a
Department of Mechanical Engineering, College of Engineering, University of South Florida, USA
A R T I C L E I N F O A B S T RA C T
Article history:
Received 13 December 2011
Received in revised form
19 January 2012
Accepted 19 January 2012
Available online
22 January 2012
Keywords:
Capillary
Viscometer
Viscosity
Surface Tension
The purpose of this paper is to understand how a capillary
viscometer is able to measure the viscosity of a fluid, which equals
time required to empty a given volume of liquid through an orifice. A
fluid analysis was done on a capillary viscometer in order to derive
equations to theoretically describe the viscometer. In addition,
physical experiments were undertaken in order to correlate empirical
data with theoretical models. Various fluids were tested and their
corresponding times were recorded. Time readings were taken at two
separate temperatures of 25o
C and 100o
C. The kinematic viscosity of
a fluid is measured in Saybolt Universal Seconds (SUS), which is
related to the kinematic viscosity of the tested fluid.
1. Introduction
The viscometer used consists of a cylindrical cup with a capillary tube at one end. The
cross-section of the viscometer is shown in Figure 1. It is assumed that the dimensions of the
capillary tube play a key role in the function of the viscometer. A fluid analysis was done to
determine how the dimensions of the viscometer affected its function.
2011 American Transactions on Engineering & Applied Sciences.2012 American Transactions on Engineering & Applied Sciences
*Corresponding author (R. Guldiken). Tel/Fax: 813-974-5628 E-mail address:
guldiken@usf.edu. 2012. American Transactions on Engineering & Applied Sciences.
Volume 1 No.2. ISSN 2229-1652 eISSN 2229-1660 Online Available at
http://TUENGR.COM/ATEAS/V01/107-126.pdf
107

Figure 1: Cross-section of Capillary Viscometer.
The following fluids were tested: water, honey, dish detergent (Ajax), mixtures of water and
detergent; Car oils: SAE 5W-30, SAE 10W-30, SAE 10W-40, SAE 50W; Gear oil: SAE 75W-90.
The reason detergent was used was to see possible relationships between surface tension and
viscosity, since dish detergent is commonly used as a surfactant to change surface tension in
various industries. Due to this, mixtures of water and dish detergent were tested to determine the
effect of surface tension on the viscometer. The concentration ratio of water to detergent was
varied 0% to 100%. The viscometer is tested according to regulations under the ASTM D88 and
D2161 Standards (ASTM, 1972). The D88 standard ensures careful controlled temperature,
causing negligible change in temperature during testing procedure. The time is in Saybolt
Universal Seconds, which dictates the time required for 60 mL of petroleum product to flow
through the calibrated orifice of a Saybolt Universal Viscometer (ASTM, 1972) . The viscometer
used is calibrated to this standard. The D2161 standard relates the relationship between the
kinematic viscosity units of Centistoke and Saybolt Universal Second (SUS) (ASTM, 1972).
Saybolt Universal Second is also referred to as a Saybolt Second Universal (SSU).
2. Mathematical Model
The Navier-Stokes and Continuity equations are used to develop a theoretical expression that
relates time taken for volume of fluid to empty to the dynamic viscosity of the fluid. We begin the
108 P. Sridharan, A. Yakub, C. Safarik, and R. Guldiken

analysis with the capillary tube itself in order to determine the velocity and volumetric flow rate,
after which ,we apply the results to the overall viscometer in order to determine viscosity in terms
of time (Hancock and Bush, 2002).
Table 1: Fluid Analysis Nomenclature.
Variable Definition
µ Dynamic viscosity
𝑣 Kinematic viscosity
ρ Density
r r-direction
z z-direction
θ theta-direction
𝑉θ Velocity in θ-direction
𝑉𝑧 Velocity in z-direction
𝑉𝑟 Velocity in r-direction
b Viscometer radius
h Fluid Column height
k Capillary Tube length
a Capillary Tube radius
g Gravitational constant
Q Volumetric flow rate
P Pressure
Figure 2: Capillary tube analysis coordinate system.
We wish to derive the velocity profile within the capillary tube. The following assumptions are
made for the derivation:
109

1. Fluid dynamic viscosity, µ, and density, ρ, remain constant.
2. Gravity occurs only in z-direction.
3. Pressure gradients occur only in z-direction.
4. The r and θ components of the velocity are equal to zero.
5. Flow is laminar and steady.
6. Temperature is constant.
7. Fluid is newtonian and incompressible.
The coordinate axis orientation of the analysis is shown in Figure 2.
The Navier-Stokes Equation in cylindrical coordinates for the z-direction is
𝜌 �
𝑑𝑉𝑧
𝑑𝑡
+ 𝑉𝑟
𝑑𝑉𝑧
𝑑𝑟
+
𝑉θ
𝑟
𝑑𝑉𝑧
𝑑θ
+ 𝑉𝑧
𝑑𝑉𝑧
𝑑𝑧
� =
−𝑑𝑝
𝑑𝑧
+ 𝜌𝑔 + 𝜇 �
1
𝑟
𝑑
𝑑𝑟
�𝑟
𝑑𝑉𝑧
𝑑𝑟
� +
1
𝑟2
𝑑2 𝑉 𝑍
𝑑𝜃2
+
𝑑2 𝑉 𝑍
𝑑𝑧2
� (1)
Applying assumption 7 to (2) the Incompressible Continuity Equation in cylindrical
coordinates, which is
1
𝑟
𝑑(𝑟𝑉𝑟)
𝑑𝑟
+
1
𝑟
𝑑𝑉θ
𝑑θ
+
𝑑𝑉𝑧
𝑑𝑧
= 0 (2)
Applying assumption 4 to (2) yields
𝑑𝑉𝑧
𝑑𝑧
= 0 (3)
Applying (3) along with Assumptions 1-5, and 7 to (1) simplifies it to
𝑟
𝜇
�
𝑑𝑝
𝑑𝑧
− 𝜌𝑔� =
𝑑
𝑑𝑟
�𝑟
𝑑𝑉𝑧
𝑑𝑟
� (4)
Integrating (4) twice to determine 𝑉𝑧(𝑟) yields
𝑉𝑧(𝑟) =
𝑟2
4𝜇
�
𝑑𝑝
𝑑𝑧
− 𝜌𝑔� + 𝐶1 𝑙𝑛(𝑟) + 𝐶2 (5)
C1 and C2 can be found by applying boundary conditions:
1.
𝑑𝑉𝑧
𝑑𝑟
�
𝑟=0
= 0 (Due to symmetry)
2. Vz(r =a) = 0 (Due to no slip)

Applying Boundary Condition 1
𝐶1 = 0 (6)
Applying Boundary condition 2
C2 =
−a2
4μ
�
𝑑𝑝
𝑑𝑧
− 𝜌𝑔� (7)
Therefore (5) reduces to
𝑉𝑧(𝑟) =
r2
4μ
�
𝑑𝑝
𝑑𝑧
− 𝜌𝑔� +
−a2
4μ
�
𝑑𝑝
𝑑𝑧
− 𝜌𝑔� (8)
We wish to relate the volume of fluid emptied from the container in a given amount of time to
the viscosity of the fluid. Therefore, using (8), we must find an expression for the volumetric flow
rate (Q), which is
𝑄 = 2𝜋 ∫ 𝑟𝑉𝑧(𝑟)𝑑𝑟 =
𝑎
0
−πa4
8μ
�
𝑑𝑝
𝑑𝑧
− 𝜌𝑔� (9)
Solving for Q
𝑑𝑝
𝑑𝑧
= 𝜌𝑔 −
8𝜇𝑄
𝜋𝑎4 (10)
With (10) known, we can begin to extrapolate this information to the viscometer itself.
Figure 3 is used to accomplish this.
If, in terms of gauge pressure, pi is the inlet pressure to the capillary tube, and the outlet
pressure is zero. Then the pressure drop across the tube length, (10), can also be written as
𝑑𝑝
𝑑𝑧
=
𝑝 𝑖
𝑘
(11),
where k is defined in Figure 3. Setting (8) and (9) equal to each other and solving for pi yields
𝑝𝑖 =
8𝜇𝑄𝑘
𝜋𝑎4 − 𝜌𝑔𝑘 (12)
111

Figure 3: Capillary tube analysis coordinate system
Since the flow is gravity driven, it should be noted that pi is proportional only to the height of
the fluid column above it (From Figure 3). Therefore
𝑝𝑖 = 𝜌𝑔ℎ (13)
Setting (12) and (13) equal to each other and solving for h, the height of the fluid column,
ℎ =
8𝜇𝑄𝑘
𝜋𝜌𝑔𝑎4
− 𝑘 (14)
Note (14) can be rewritten in terms of kinematic viscosity rather than dynamic,
ℎ = 𝐶𝑣𝑄 − 𝑘 (15),
where 𝐶 =
8𝑘
𝜋𝑔𝑎4
and 𝑣 =
𝜇
𝜌
Rearranging for the kinematic viscosity yields
𝑣 =
ℎ + 𝑘
CQ
(16)
Q can also be expressed in terms of volume, from Figure 3, and time as

𝑄 =
𝜋𝑏2ℎ
𝑡
(17)
Substituting (17) into (16) yields
𝑣 =
ℎ + 𝑘
C𝜋𝑏2ℎ
𝑡 (18)
(16) may also be rewritten in terms of the viscometer dimensions as
𝑣 = ��
𝑔
8
� (
1
𝑏
)2
𝑎4
�
1
𝑘
+
1
ℎ
�� 𝑡 = 𝑀𝑡 (19),
where M is a constant. It is worth noting that (19) shows important insights into the sensitivity
of the function of the viscometer. The constant M is implicit and specific to each viscometer made,
which is dependent on the dimensions of the viscometer. Although M depends on the dimensions,
for proper calibration, the importance of each dimension must be known, such that each
dimension’s required tolerances can be assigned during manufacturing of the viscometer. (19) is
powerful in aiding with these insights.
It can be inferred from (19) that increasing or decreasing the capillary radius, a, exponentially
affects M since it is raised to the fourth power. Due to this, it can be seen that the capillary radius is
the most sensitive, and important, dimension of the viscometer in terms of its proper function. The
viscometer diameter, b, is the second most important dimension regarding the functioning of the
viscometer. It affects the function at an exponential rate, like capillary radius, but at a slower rate.
The lengths of the fluid column and capillary rank equally, but are last in line in dimensional
importance. Additionally, recalling from (15) about dynamic viscosity, (16) and (19) can be
rearranged as
𝜇 = 𝜌
ℎ + 𝑘
CQ
=
𝜌𝜋𝑔𝑎4(ℎ+𝑘)
8𝑘𝑄
(20)
𝜇 = 𝜌 ��
𝑔
8
� (
1
𝑏
)2
𝑎4
�
1
𝑘
+
1
ℎ
�� 𝑡 = 𝜌𝑀𝑡 (21),
113

where M is a constant. It is important to note that although, in terms of function of the
viscometer, the kinematic viscosity is not dependent on the density of the fluid, the relationship of
dynamic viscosity is density dependent. The kinematic viscosity mainly depends on the geometry
of the problem.
(a) (b)
Figure 4: Streamline Depictions of: (a) Bucket with hole in bottom and (b) viscometer; blue
indicates ~143 mm/s and yellow indicates ~36,800 mm/s velocity.
In addition to the above analysis, an elementary computational fluid simulation was done on
the tested viscometer. It is known that previous capillary viscometers existed, where the capillary
tube started at the bottom of the cup, not offset in height, k, as in Figure 3. It was assumed that fluid
flow accounted for this height offset. In other words, the reason for the height is assumed to be due
to the streamlines of the flow during use. In order to test this theory we must visualize streamlines
for different designs, and to get a general idea of how these streamlines change with the design.
Therefore, a CFD (Computational Fluid Dynamics) model was done. A model of the viscometer
was created using Solidworks. The FloExpress Simulation Module of Solidworks was used to run a
fluids simulation to predict streamlines of flow. The simulation input required specifying an inlet
and exit. The simulation required inlet conditions, while the outlet conditions were auto-set to be to
open air at STP. The inlet conditions that were input was a volumetric flow rate and inlet pressure,

which were 10 in3
/s, and 1 atm, respectively. The fluid was assumed be incompressible during the
simulation, which was done through an iterated Navier-Stokes equation reduction. The first case
considered was a tank with a hole at the bottom. The second case considered was the capillary
viscometer. The result of this analysis is shown in Figure 4, where the color of the line corresponds
to the speed of the flow; blue indicates lower speeds, whereas yellow indicate higher speeds.
Comparing the two pictures in Figure 4, it is easier to understand the reason for this height
offset. As can be seen in Figure 4a, the fluid that gets to the bottom of the tank undergoes
turbulence as it transitions into the capillary hole. In addition to this turbulence, slight rotation in
the flow can be seen as it enters the capillary. When looking at Figure 4b, lot of turbulence can also
be seen. The difference is that turbulence occurs due to vortices developing on the sides of the
capillary tube. These vortices occur in a way such that the turbulence does not affect the fluid
entering the capillary tube. Another observation is that there is minimum rotation in the flow.
In effect, the transition the flow undergoes going from the viscometer into the capillary is a lot
smoother when the capillary is offset in height. This allows Assumptions 4 and 5 of the Capillary
Tube Analysis to be more valid, causing the overall fluid analysis to have greater validity, causing
higher accuracy of (19). It can also be seen from Figure 4b that the result from (3) seems viable
since fully developed flow is depicted for most of the capillary tube. As an added case of support,
the Reynolds Number was calculated for the capillary tube, using water as the fluid, which was 95.
Note that this number is very low, so laminar flow is viable.
An interesting digress related to history is that Ford had a viscosity cup, Figure 5, which
attacked the previous problem in a different way (Wikipedia).
Figure 5: Ford Viscosity Cup (Wikipedia).
115

Notice the conical extrusion at the bottom of the cup. This conical profile allows the flow to
follow more of the streamline pattern as in Figure 4b. This ensures straight laminar entrance into
the capillary at the bottom. This method does have its disadvantages. Due to the profile of the
conical section, the fluid velocity accelerates as it gets near the outlet of the cup. This acceleration
may be more observable as angular rotation rather than laminar velocity. Although this does create
a stream tube as in 4b, there is a possibility of turbulence/angular rotation. The height offset as
shown in Figure 3 ensures a similar stream tube profile and also reduced chances of
turbulence/angular rotation.
The above analysis must then be applied for the viscometer used. Table 2 shows the measured
values for the dimensions of the viscometer being analyzed.
Table 2: Viscometer parameters.
a (mm) b (mm) h (mm) k (mm) g (mm/s)
1.19 19.05 95.25 9.525 9810
Substituting these values into (19), the approximate equation for the capillary viscometer is
𝑣 = 0.78254𝑡 (22)
The kinematic viscosity, from (21), is given in mm2
/s due to the units used from Table 2, and t
represents the SUS (Saybolt Universal Second). It is useful to note that 1 mm2
/s is referred to as 1
centiStoke (www.engineeringtoolbox.com).
Standard values of kinematic viscosity of water at 25o
C are known to be 1 centiStoke and 31
SUS (www.engineeringtoolbox.com). So why is it that, when plugging in 31 SUS for t in (22), the
corresponding kinematic viscosity is 24.3 centistokes? The reason for this answer takes a bit more
insight, where the ASTM D88 standard is referred. Remember that the original ASTM D88
standard for measuring viscosity uses 60 mL of liquid through a carefully calibrated orifice. What
is this calibration? This calibration is such that it takes a certain time, t, to empty 60 mL of a
standard, pre-agreed upon, liquid. Note that the time, t, and the liquid used are pre-agreed upon. For
example, the standard to be tested against is water. Through the ASTM D88 standard, it takes 31
seconds for 60 mL of water (at 25o
C) to empty from the viscometer.

What would occur if the liquid being tested, whose viscosity is unknown, is very viscous, such
that the time required for it to empty is very large (on orders of tens of minutes to hours)? What
would occur if the liquid being tested is so inviscid, such that the time elapsed for emptying 60 mL
is on the orders of few seconds, not long enough to gather data? Both of the above scenarios, which
are encountered in larger frequency after the industrial revolution, would be hard to handle by one
size of viscometer. Both of the above scenarios can be solved by changing the height of the fluid
column, h, such that the time it took to empty that volume is kept in a second range that is workable
out in the field/application. But changing the fluid column height also changes the volume of the
fluid, causing it to differ from the traditional 60 mL, causing the test to not follow the ASTM
standard.
This is actually not a real problem. The reason why is because of the nature of the definition of
viscosity, tied in with ASTM. The unit known as viscosity is not a physical unit. It is a theoretical
unit which describes a physical presence, like gravity. This is the reason the ASTM D88 standard
was created. Since viscosities of fluids greatly differ and difficult to define, a theoretical zero-bar is
created, to which all other viscosities are related to. This zero-bar is through the ASTM D88
standard. Therefore the problem of changing the initial volume can be fixed if the time required for
that volume to empty can be analogous for the time it takes for 60 mL to empty. Due to this, the
relationship between units of kinematic viscosity (centistokes and SUS) is not related to the
geometry of the viscometer, which is displayed in Figure 6. It is also interesting to note that the
relationships in Figure 6 change depending on the temperature.
Figure 6: Centistoke to SUS conversion courtesy of "Standard Method for Conversion of
Kinematic Viscosity to Saybolt Universal Viscosity or to Saybolt Furol Viscosity" ASTM
Standard 2161 (ANSI M9101).
117

It is also interesting to note that, using the values from Table 2, the volume of the fluid column
in the viscometer tested is actually about 105 mL, not the standard 60 mL. The higher volume
allows for a larger range of viscosities to be tested, but only if the fluids fall within a specific
gravity range. But for this viscometer to work with 105 mL of initial volume and still relate to the
standard, the capillary radius is calibrated. The capillary radius is changed because it has the
highest effect on the function of the viscometer as discussed above. The capillary radius of the
viscometer is properly calibrated if it takes 31 seconds for 105 mL of water to empty through this
viscometer. Remembering that the time taken for discharge is the SUS (Saybolt Universal Second)
value for that fluid, the relationships from Figure 6 hold to convert between centistoke and SUS.
The key thing to note from above is that (19) and (21) contain a constant, M, inherent to the
geometry of the viscometer. It is important to note that the constant depends on the dimensions of
the viscometer. As long as the fluids being tested are within the specific gravity range of the
viscometer's calibration, the constant M, from (19), also equals the ratio of kinematic viscosity, in
centistokes, to kinematic viscosity, in SUS. This ratio is in accordance to the ASTM standard for
conversion between SSU and Centistokes (ASTM D2161). For every SSU value, there is a
corresponding Centistoke value at that temperature.
Therefore, for fluids with higher viscosities, the same capillary viscometer can be used, if the
dimensions are changed such that the stream tube of Figure 4b is valid, making the Navier-Stokes
Derivation valid, in addition to proper time calibration. If a fluid is very viscous, just changing the
fluid column height and the capillary radius to replicate the stream tube is not enough. In addition
to above, it must be ensured that the time it takes for the fluid to empty is in accordance with
analogous to the standard. If, by changing the height and capillary radius to account for the new
fluid, the viscometer's new volume composed of water emptying through the new capillary radius
does not equal 31 seconds (aka SUS), then all the times from the viscometer cannot be compared to
the standard. The balance between the two previous sentences is what ensures proper calibration of
the viscometer with assurance of proper functioning. It should be noted that the above is also only
for distinct densities or density ranges.
3. Study Details
The following fluids were tested: water, honey, dish detergent (Ajax), mixtures of water and
detergent; Car oils: SAE 5W-30, SAE 10W-30, SAE 10W-40, SAE 50W; Gear oil: SAE 75W-90.

These car/gear oils are engineered to have specific viscosities at two distinct temperatures during
operation. These temperatures are known as the cold start temperature and operation temperature.
The cold start temperature is the temperature at which the engine is turned on, which is also usually
when the oil is at its lowest temperature. The operation temperature is assumed to be 100 o
C
(Celsius). Ambient temperature of 25 o
C was chosen for simulating the cold start temperature due
to the fact that the temperature at which the engine is turned on can vary depending on altitude,
location, and other variables.
1. The following are procured: viscometer, stopwatch, fluids to be tested, containers, and a
gas flame.
2. The viscometer is cleaned thoroughly prior to each use.
3. A reservoir of liquid is heated over a gas flame, until the desired temperature is reached.
4. Once the fluid has become the desired temperature, the viscometer is submerged into the
fluid.
5. The viscometer is allowed to be submerged in the fluid to fill it up and held long enough to
ensure it, and the fluid, are at the desired temperature.
6. The viscometer is pulled out of the fluid; when the bottom face of the viscometer clears the
top of the liquid in the container, which allows the fluid to begin to fall out, the stopwatch is
started.
7. The liquid is allowed to empty from the viscometer.
8. The following times recorded from the stopwatch: start time, time at which flow change
from stream to drips occurs, and time when liquid flow stops.
Note: If needed, multiple trials can be done for same liquid to average the times by repeating
Steps 2-8. During the procedure, the temperature of the liquid was monitored to ensure temperature
was constant.
The viscometer acquired for the analysis was designed for fuel oils. Most of these fuel oils
have a specific gravity range between 0.8-1.0 (www.engineeringtoolbox.com). Therefore,
preliminary testing was done with automotive car and gear oils, due to the fact that most
automotive oils have specific gravities between 0.88-0.94 (www.engineeringtoolbox.com). The
119

results of the experimental procedure are summarized in Table 3 below.
Table 3: Summary of Experimental Results.
Time (s)
Oil Type 25 o
Celsius 100 o
Celsius
Stream to Drip Full Stop Stream to Drip Full Stop
C: 5W-30 268 406 - 70
C: 10W-30 289 469 - 70
C: 10W-40 383 581 - 84
C: 50W - 1395 - 97
G: 75W-90 318 775 - 81
In Table 3, the column “Stream to Drip” represents the time at which the fluid flow out of the
viscometer changed from a steady stream to drips. The "C" or "G" before the oil type dictates
whether it is crankcase or gear oil. If the fluid flow never changes from a stream to drip, no time is
recorded. Whether the fluid drips or streams depends on the viscosity and surface tension of the
fluid. The column “Full Stop” is the overall time it took the volume of fluid to empty from the
viscometer. It is important to note that the Full Stop time at 100 o
C corresponds to the kinematic
viscosity of the oil tested in Saybolt Universal Second. Car oils with an increasing number in
front of the W dictate an increase in viscosity. The physical representation of this can be inferred
from Table 1 since the Full Stop time increases as the number increases.
Table 4 shows the standard values (www.engineeringtoolbox.com), in SUS, of the oils tested
in Table 3. The values in Table 4 correspond to the Full Stop values at 100o
C in Table 3. As seen
from Table 4, it is worth noting that the experimentally found kinematic viscosity, in SUS, of
car/gear oils tested is within 5% of the standard value.
Table 4: Standard Values of Car/Gear Oils.
Oil Type Standard Value @ 100o
C (seconds)
C: 5W-30 70
C: 10W-30 70
C: 10W-40 85
C: 50W 110
G: 75W-90 74
The viscosities, at different temperatures, from Table 2 are in Figure 7.

Figure 7: Full Stop Time vs Temperature for Car/Gear oils.
Figure 8: ASTM Viscosity Chart.
0
500
1000
1500
25 100
Time(seconds)
Temperature (⁰C)
C: 5W-30
C: 10W-30
C: 10W-40
G: 75W-90
C: 50W
121

It is well known that detergent is used in industry as a type of surfactant to change the surface
tension of fluids. A set of trials were done were concentration of detergent to water was varied in
different mixtures of dish detergent (Ajax) and water. The time required to empty the viscometer
(SUS) was tabulated for these, which is shown in Table 5. The purpose of these trials is to discern
some type of relationship between surface tension and viscosity of Ajax and/or Water. It seems that
Table 5: SUS values of Mixtures of Ajax and Water.
Ratio of Ajax to Water (%) Time (seconds)
0 31
8.3 40
16.7 40
25 41
50 43
62.5 49
75 52
88 3651
100 3780
A graphical representation of Table 5 is shown in Figures 9 and 10. Figure 10 is a close up of
first seven rows of Table 5, whose correlation is not discernible from Figure 9.
Figure 9: Ajax Mixture % vs Time.
Figure 10: Close up of 0-75% range from Figure 9.

Recall, that the ASTM standard for determining viscosity requires constant temperature. The
reason is because viscosity is greatly affected by temperature; therefore to get an accurate viscosity
reading to hold for a specific temperature, isothermal conditions must be ensured. In order to
illustrate this, trials with honey were done, were the honey was allowed to cool during the run. The
temperatures tested were ambient conditions, 25o
C, and 49o
C, shown in Figure 11.
Figure 11: Honey SUS values at different temperatures.
4. Discussion
It can be seen from Figure 7 that all the car oils share a similar slope. This means that, in terms
of viscosity, their response corresponding to change in temperature is similar. It is also interesting
to note that the specific gravity range of the oils used is 0.88-0.94 (www.engineeringtoolbox.com).
Therefore, the observation of similar slopes may be tied into the fact that they share similar
densities, along with how their viscosity is affected by temperature.
In order to show the relationship of the measured data to applicable experience, Figure 8 was
created. Figure 8 shows a comparison of typical fuel oils ranging from lightest of oils, #1, to the
y = -13150x + 27250
0
2000
4000
6000
8000
10000
12000
14000
16000
25 49
Time(seconds)
Temperature (Celsius)
Honey
Linear (Honey)
123

heaviest, #6. It is important to note that the heaviest and most viscous fluid presented on the chart is
honey. In terms of specific gravities, the oils tested range between 0.88-0.94; water is
approximately 1; honey was measured to be approximately 2; pure Ajax was measured to be
approximately 0.8. Excluding honey, the specific gravity range for all liquids tested lie in the range
of 0.8-1 (within 20% of water). Although, from (21), the dynamic viscosity is density dependent,
Figure 6 shows wide range of viscosities corresponding to a small specific gravity range.
Making note of Figure 8, there is a narrow band of acceptable viscosity of 80 to 100 SUS
where a fuel oil must be heated in order to have clean combustion of the oil. Similarly the
accuracy of the fuel oil heating and circulation system has to maintain a fairly narrow range of
control. In other applications, such as paint spraying, coating etc. viscosity has to be accurately
controlled to prevent “orange peeling” or a wavy texture to the paint to enlarged droplet size.
Droplet size and fluid temperature are very dependent on one another.
As can be seen from Figures 9 and 10, the surfactant, dish detergent, does not affect the
viscosity of water as much as water affects the viscosity of it. It is interesting to note, that 100%
Ajax has an SUS value of approximately 3780. A decrease in concentration of just 15% decreases
the SUS value to 3619 (decrease of approximately 4.26% with respect to the value at 100% Ajax
concentration value). But a decrease of another 15% in Ajax concentration causes the SUS value to
decrease to 52 SUS., which is a decrease of approximately 98.6% of the 100% Ajax concentration
value.
Regarding Figure 11, the line named "Linear (Honey)" is a linear regression fitted to the
Honey data. The equation of this regression is shown on the Figure. We will use this equation to
extrapolate the viscosity of Honey (in SUS) at 100o
C. Plugging in 100 for x in the equation, the
corresponding y (time aka SUS) comes out to -1,287,750 seconds. Obviously, this is incorrect. The
reason for this error is due to the Honey trial at 49o
C. The time for the honey to empty is 950
seconds (15.8 minutes). During this time the honey decreased in temperature from 49o
C at the start

of the trial to 32.2o
C. The honey is very viscous, requiring the time for a given volume to empty to
increase. If insulation is not present, temperature of the honey will change, especially if the time
required for trial increases. This decrease in temperature causes a change in viscosity, since
viscosity is temperature dependent. Due to this reason, the data from the 49o
C trial is very
erroneous, causing large errors in the linear regression. If the honey was well insulated, the linear
regression will be a better fit, giving a viable value when extrapolating to 100o
C.
5. Conclusion
It was found that of the main dimensions of the viscometer, the capillary radius and viscometer
radius greatly affected the performance of the viscometer compared to the fluid column height or
capillary tube height. The reason why the capillary tube was "inset" was found to help the transition
the fluid flow experiences going into the capillary tube. Creating the height offset minimized
turbulence and rotation of the flow entering the capillary tube. Constant temperature is essential to
the accuracy of the viscometer. Car/Gear oils were tested with the viscometer and all viscosity
results of the oils were with 5% of the real values.
Surface tension also was found to play a role in the viscometer. Once the surface tension force
is larger than the pressure force, in this case due to gravity, forcing the liquid down, the flow does
one of two things. If the flow was a steady stream out of the viscometer, it will either turn into
dripping flow and/or stop altogether. If the flow out of the viscometer was through drips to begin
with, the surface tension force then stops the flow of the fluid when it exceeds the pressure force.
6. References
American Society for Testing and Materials (ASTM), 1972. Library of Congress Catalog Card
Number: 70-180913.
125

Hancock, Matthew J. and Bush, John W. (2002). Fluid Pipes. Journal of Fluid Mechanics, vol.
466, 285-304.
Prashanth Sridharan is currently pursuing his Doctorate Degree in Mechanical Engineering at
University of South Florida, Tampa, Florida. Mr. Sridharan earned his Bachelor Degree in
Mechanical Engineering from University of Florida in 2010. Current research interests include
thermal science, alternative/renewable energy, and fluid dynamics.
Abiodun Yakub is an MS student in the Department of Mechanical Engineering at University of
South Florida, Tampa, Florida. He earned his B. Tech. from Ladoke Akintola University of
Technology (LAUTECH) Nigeria in 2004. Abiodun’s research interests include Dynamics and
Material science.
Charles Robert Safarik, received his Bachelor of Science Degree in Aerospace Engineering from The
Polytechnic Institute of New York, 1967, Masters in Mechanical Engineering, Pennsylvania State
University, 1978, and currently studying for a Ph.D. in Mechanical Engineering at the University of
South Florida. He was also a Registered Professional Engineer, Florida, from 1981- 2010, practicing in
Heat Transfer and Combustion Design and Development.
Dr. Rasim Guldiken is an Assistant Professor in the Department of Mechanical Engineering at
University of South Florida, Tampa, Florida. He earned his Ph.D. from Georgia Institute of
Technology in 2008. Dr. Guldiken’s research interests include Microfluidics and Bio-MEMS.
Peer Review: This article has been internationally peer-reviewed and accepted for publication
according to the guidelines given at the journal’s website.

Fuzzy Logic Modeling Approach for Risk Area
Assessment for Hazardous Materials
Transportation
Sanya Namee
a
, Boonsap Witchayangkoon
a*
, Ampol Karoonsoontawong
b
a
Department of Civil Engineering, Faculty of Engineering, Thammasat University, THAILAND
b
Department of Civil Engineering, Faculty of Engineering, King Mongkut’s University of Technology
Thonburi, THAILAND
Article history:
Received 01 December 2011
Received in revised form
20 January 2012
Accepted 26 January 2012
Available online
28 January 2012
Keywords:
Risk Area Assessment;
Hazardous Material;
Transportation;
Fuzzy Logic Modeling.
The assessment of area in risk of HazMat transportation is very
beneficial for the planning of the management of such area. We
prioritized the affected area using HazMat-Risk Area Index
(HazMatRAI) developed on the basis of Fuzzy Logic. The purpose of
such development is to reduce limits of the criteria used for the
assessment which we found exist when displaying data related to
Hazmat represented by iceberg. In this regard, we categorized type of
Membership Function according to Fuzzy set method in order to match
the existing criteria, both solid and abstract ones. The conditions of
Fuzzy Number and Characteristic are used respectively so that all risk
levels are covered. However, the displaying of HazMat-Risk Area
Index needs weighing of each criterion that is used for the assessment
which significance of each level varies. We used Saaty’s Analytic
Hierarchy Process (AHP) to establish weighing value obtained from
such assessment. Therefore it is beneficial for the preparation of area
with HazMatRAI value is high, hence proper preparation for the
management in case of critical situation.
2012 American Transactions on Engineering & Applied Sciences
*Corresponding author (B.Witchayangkoon). Tel/Fax: +66-2-5643001 Ext.3101.
E-mail address: wboon@engr.tu.ac.th. 2012. American Transactions on Engineering &
Applied Sciences. Volume 1 No.2 ISSN 2229-1652 eISSN 2229-1660. Online Available
at http://TUENGR.COM/ATEAS/V01/127-142.pdf
127

1. Introduction
Recently industrial sector has been growing rapidly. Industry involved with chemical
substances, nuclear, electrics, and petroleum are beneficial to the world, but at the same time they
come with complicated problems. These industries are generator where they need hazardous
material for the manufacturing process. Besides, some type of industry also produces hazardous
wastes. Major affects include the transportation of hazardous materials which occur everywhere in
pipe, rail, and road. It increases risk of people’s safety, life, property, and environment of the area
where transportation takes place. In the United States, we found that the transportation of
hazardous materials generates economic activities a great deal, for example, the transportation that
costs more than 2 billion dollars in the United States. Over all transportation increases to 20%
during 1997 – 2002 (USA Census Bureau, 2002) and transportation by truck is as high as 52.9%,
accidents on high way is 89%. For the accident, the serious ones are caused by the transportation of
hazardous material such as leaking or death, damage costs up to 31 billion dollars (about 80,000
dollars for 1 accident) (USA DOT, 2003). Despite our awareness that accident from hazardous
materials does not occur frequently (10-8
– 10-6
per vehicle per mile) (Erkut and Verter, 1995;
Zografos and Davis, 1989), the consequence is much to be concerned for every involved person or
everyone who is affected by the transportation of hazardous materials, involved people in the area,
government sector, transportation company, transportation vehicle, and people in risk. The
reduction of risk of transportation is the main purpose of every people involved in the
transportation of hazardous materials.
The National Fire Protection Agency (NFPA), 2008 has defined HazMat Risk that it is the
possibility and severity of sequence from the exposure to hazardous material. The result from this
definition is that the perception of hazardous material is always involved with leakage, and the
consequence of such leakage. Frequency of leakage depends on many factors e.g. possibility of
accident, possibility of leakage, and numbers of hazardous material transportation. Consequence
from the leakage depends on types of hazardous materials, amount of leakage, and duration from
the occurrence until the management. Hazardous material transportation can make people’s life
harmful, especially people who are living near transportation route. Besides, it also affects
environment. Although not frequent, if it occurs, it can lead to major disaster.
128 Sanya Namee, Boonsap Witchayangkoon, and Ampol Karoonsoontawong

Figure 1: The problem of hazardous material transportation is like an iceberg.
The inevitable truth in many countries is that the problem of hazardous material
transportation is like an iceberg. It is difficult to access the truth data about such transportation i.e.
pipe, rail, or road to see if it was operated with transparency. Avoidance and failure to comply to
the law, false information, ambiguous source of information, and the operation of officers that does
not cover all aspects, and the integration of involved units are all problems that have been hidden.
The preparation to handle the accident from hazardous material transportation plays an important
role in the safety of such transportation that results in the loss of life, property, and environment.
The major contributions of this paper are the guideline for the assessment of risk area from
hazardous materials using the theory of Fuzzy Set. The assessment is conducted under the
limitation of ambiguous factors in terms of both objective and subjective. Purpose of the
assessment is to obtain index for the identification of risk area from hazardous materials
2. Literature Review
In the past, problems of route management were handled by the development of model for
solving problems using single or multiple criteria. Purpose of single criteria model is to identify
one route or one network that minimizes risk (Glickman, 1983; Batta and Chiu, 1988; Karkazis and
Boffey, 1995). Multiple criteria model refers to route management on the basis of expense such as
travel time, expense of transportation, risk of accident, estimated numbers of affected people, risk
Assessable problem
• Evidence-based statistic data is available
• Specific responsible unit / organization
Problems difficult to assess
• Difficulty accessing data
• Ambiguity of data source
• Statistic data given is falsified
• Integration of responsible units
129

of some special population group, and property damage (Zografos and Davis, 1989; McCoord and
leu, 1995; List and Turnquist, 1994). Route management and scheduling help us find out the
problems. In this regard, we need to identify travel time and the point that mitigation team has to
stop before reaching the scene (Cox, 1984; Cox and Turnquist, 1986; Nozick et al., 1997).
Research by Lassarre (1993) and lepofsky et al (1993) has explained the Decision Support System:
DSS covering the analysis of danger from transportation and accident management, identifying
following topics a) risk assessment on the basis of accident possibility, leakage, consequence, and
risk b) identify optimum route between two points on the basis of multiple criteria such as duration,
possibility of accident, and population in risk c) identification of the outcome from hazardous
material and the assessment of evacuation and the identification of existing road usage d) traffic
management on the affected scene.
Weigkricht and Fedra (1995) and Brainard et al (1996) introduced management of hazardous
material transportation route indicating the route between two points by using multiple criteria and
weighing. Coutinho – rodrigues et al (1997) introduced DSS for routing and positioning of rescue
team. Feature of DSS is the geographical display of the unaffected route for problem solving and
decision making. The system integrates various techniques for solving various problems. When
making consideration, users might create his/her own way of problem solving by changing weight
of expense under the decision or setting the lowest point to the highest point of expense. Frank et al
(2000) developed DSS to choose the route between origins to destinations, each point matched.
Criteria used for route selection includes population who are in risk and travel time. Erkut e al
(2007) discussed about the routing of hazardous material transportation that it is a very important
decision to reduce risk. To be specific, risk of hazardous material transportation can be
dramatically reduced if it is well planned i.e. selecting the route with least possibility of accident,
control consequence, and try to find the way to rescue without obstacles. Zografos and
Androutsopoulos (2008) studied supportive system for making decision about hazardous material
transportation and how to respond emergency situation, and scope of risk management includes
logistics for hazardous material and the decision to respond emergency situation. The developed
system can be applicable to a) the preparation of route selection for hazardous material
transportation with lowest expense and risk b) identification of rescue team that can access the
scene with minimum travel time and safety before service arrives c) finding out the route for rescue

team d) identification of the best evacuation plan. The developed system is used for the
management of hazardous material being transported in road network under the area of Thriasion
Pedion at Attica, Greece.
Research related to the study of criteria used for risk assessment includes Saccomanno and
Chan (1985). It introduced the model that let us see the consequence of accident towards
population. In face, this model needs two criteria which are minimum risk and minimum
possibility of accident. The third criteria is the economic aspect of problems such as expense of
truck. Zografos and Davis (1989) developed a method for making decision with multiple
objectives. The 4 objectives that were considered include I) population in risk II) property damage
III) expense of truck operation and IV) risk of expansion by establishing capacity of network
connecting point.
Leonelli et al (2000) developed optimum route using mathematical program for route
calculation. Result of the calculation is the selection of route that only reduces expense. Frand et al
(2000) developed spatial decision support system (SDSS) for selecting the route for hazardous
material transportation. GIS environment model has been developed to create route image, while a
mathematical program has also been set to evaluate the use of such route. The purpose of this
model is to reduce travel time between origin and destination. However the actual goal is to
emphasis on the limitation, travel time, possibility of accident on such route, involved population,
and risk of population, all of the mentioned help establish the limitation of this model. Risk of
population has been established by the possibility of accident, multiplied by number of population
in that area.
Most of the studies emphasis on the analysis of transportation route to find out the route with
minimum risk, and the importance has been given to road network with highest chance of accident.
In this study, we assess the risk of area that might be affected from hazardous material
transportation including piping system, railing system, and road network. The result from
assessment can identify level of risk of each area so that each area is able to get prepared for the
prevention of accident in an appropriate manner.
131

3. Fuzzy Set Theory
Recently there has been an attempt to establish model and develop mathematical process for
solving problems of the system that is quite complicated including statistics, formula, or equation
that most fits to specific problem. Most engineering solution analyzes data in two ways that is
subjective and objective. General problem of engineering task is the necessity to manage uncertain
data i.e. uncertainty of numbers from the measurement or experiment, and the certainty of the
denotation. Fuzzy set theory is a new field of mathematical originated to handle subjective data. It
is accepted that it is a theory that can handle such problem properly.
The analysis for making decision regarding the area in risk of hazardous material
transportation for the management of disastrous situation under the certainty and limitation to data
access needs the analysis and decision making with multiple criteria. The main challenges of this
study are the consideration of criteria that might make the transportation harmful, either through
piping or railing system, road network, area categorization on the basis of Boolean Logic, and
evaluation limitation. Therefore we need to use Fuzzy Logic to solve problems that are still
ambiguous or unidentified. Besides, the process used for making decision can be implemented in
both quantitative and qualitative criteria, and some criteria are very outstanding.
The first person who introduced Fuzzy Set theory is Lofti A Zadeh, a professor of Computer of
California University, Berkley. He introduced his article regarding “Fuzzy Sets” (Zadeh L.A.,
1965). Zadeh defined fuzzy sets as sets whose elements have degrees of membership. Considered
sets are viewed in a function called Membership Function. Each member of the set is represented
by Membership Value which ranges between 0 – 1. When considering the Ordinary Sets, we found
that degree of membership of each set is represented by either value between 0 and 1, which means
no membership value at all, or complete value of membership respectively. Generally we found
that sometimes we cannot be so sure that something is qualified enough to be a member of that set
or not. We can see that fuzzy set theory if more flexible as partial membership is allowed in the set,
which is represented by degree, or the acceptance of change from being a non-member (0) until
being a complete member (1). Fuzzy Set theory (Zadeh L.A., 1965) leads to the idea of fuzzy
mathematics in various fields, especially in Electronic Engineering and Control that still uses the
fundamental of fuzzy set theory (Zadeh L.A., 1973). I hereby would like to mention fundamental
idea of fuzzy set, as mentioned by Zadeh, that fuzzy set can explain mathematics as follow:

According to the definition of fuzzy set that needs function of membership as a method to
establish qualification, fuzzy set A can by represented by member x, and membership degree of
such value as follow:
𝐴 = {(𝑥, 𝜇 𝐴(𝑥))|𝑥 ∈ 𝑈} (1)
Given that U has degree of membership for A, following symbols are used:
𝐴 = ∫ 𝜇 𝐴(𝑥)𝑈
𝑥� (2)
Fuzzy set A in Relative Universe (U) is set from characteristic by membership function
µA : U  [0 , 1] i.e µA (x) is value of each member x in U which identifies grade of
membership of x in fuzzy set A. In this regard, fuzzy set is considered classical set or crisp set.
This Membership function is called characteristic function. For classical set, there are only 2
value which are 0 and 1 i.e. 0 and 1 represents non-membership, and membership in the set
respectively. The example of Figure 2 represents characteristic of Boolean set and fuzzy set. Here
we use “fuzzy set” to explain, which means the set defined in function (1) where A and B represent
any fuzzy set and U represents Relative Universe (U). We found that fuzzy set is different from
classical set because fuzzy set has no specific scope. Concept of fuzzy set facilitates the
establishment of framework that goes along with ordinary framework, but it is even more ordinary.
Fuzzy framework lets us have natural way to handle problems of uncertainty, which is involved
with the uncertainty of how to categorize membership, rather than random method.
4. The Risk Assessment Criteria
The risk assessment of area with the consideration of piping system, railing system, and road is
a complicated process. Basically we need to consider many aspects including location, route
significance and geographical characteristics. Researches in the past used various tools for
assessment, which can be categorized as follow: safety, minimum travel time, minimum
transportation time, population in risk, environmental quality, and geographical characteristics as
shown in Table 1. When considered these factors, we have two topics that reflect the risk of area:
a) risk caused by various criteria used for the assessment and b) risk as a result from route
133

significance. In accordance to the assessment of risk are, we divided risk scale into 5 subsets as
follow
R = {R1, R2, R3, R4, R5} (3)
= {most risk, much risk, risk, less risk, least risk}
4.1 Membership Function Deviation
To successfully use fuzzy set, it depends on appropriateness of membership function either
quantitative assessment or qualitative assessment, which can be used for the identification of
membership function. When considered the complication and ambiguous source of information,
we can use 2 types of membership function
Table 1: Assessment Criteria for the Area in Risk of Hazardous Material Transportation
Main-Criteria Sub-Criteria
Membership
Function
Weight
Type of
transportation in
the area
Distance to transportation system
if transported by road
Function I 0.045
0.062
if transported by rail
Function I 0.013
if transported by pipe
Function I 0.004
Significance of
being a route for
HazMat
transportation
Transportation system to
manufacturer / pier / industrial
area is available in the area
Function II 0.027
0.040Number of gas station available in
transportation system
Function II 0.009
Transportation system available in
the area that reduces distance /
duration of transportation
Function II 0.004
Risk condition of
road in the area
Road characteristics that are risks
of accident
Function II 0.027
0.131Number of accidents occurred in
the past
Function II 0.020
Number of Hazmat transportation
trucks
Function II 0.084
Danger if
accident occurs
in case of explosion / fire
Function I 0.283
0.314
in case of leakage
Function I 0.031
Benefits of the
area
Characteristics of urban Function II 0.237
0.453Population density Function II 0.173
Distance to town center Function I 0.043

4.1.1 Membership Function I of Fuzzy Number Model
The criteria for risk assessment of the area in risk as indicated in the manual of emergency
response 2008 indicated different dangerous area in case of hazardous material leakage depending
on the severity of each hazardous material such as hazardous liquid (ammonia), flammable liquid
gas (LPG), and flammable liquid (fuel). All of these are hazardous materials used for model
development. According to the manual, it suggested that the area be restricted 100 – 200 meters
from the scene. In case of fire, evacuate the area in the radius of 1.6 kilometers (DOT, 2008). The
recommended distance is used for setting up impact area.
Table 2: Sample of membership function for distance to transportation system in case of
explosion/fire.
Risk Scale Membership Function Thresholds
Most risk
U(x) = 1
U(x) = (400-x)/200
U(x) = 0
x ≤ 200 m
200 m < x ≤ 400 m
x > 400 m
Much risk
U(x) = 0
U(x) = (x-200)/200
U(x) = (600-x)/200
U(x) = 0
x ≤ 200 m
200 m < x ≤ 400 m
400 m < x ≤ 600 m
x > 600 m
risk
U(x) = 0
U(x) = (x-400)/200
U(x) = (800-x)/200
U(x) = 0
x ≤ 400 m
400 m < x ≤ 600 m
600 m < x ≤ 800 m
x > 800 m
Less risk
U(x) = 0
U(x) = (x-600)/200
U(x) = (1000-x)/200
U(x) = 0
x ≤ 600 m
600 m < x ≤ 800 m
800 m < x ≤ 1000 m
x > 1000 m
Least risk
U(x) = 0
U(x) = (x-800)/200
U(x) = 1
x > 800 m
800 m < x ≤ 1000 m
x > 1000 m
Criteria for the assessment of risk area from hazardous material transportation in terms of distance had
been used to set Membership Function in this article. For example, Membership Function for distance from
the scene is the function of Fuzzy Number, as shown in Figure 3 and Table 2.
135

Figure 2 Sample of Membership Function: Fuzzy Number
4.1.2 Membership Function II of Character
For Membership Function II of characteristics just like in Figure 3, generally it has
mathematical formula as follow
0 when x = Vi
U(x) = i = 1, 2, 3, …, m (4)
1 when x ≠ Vi
Characteristic Membership Function is seen as special type of fuzzy set. Actually normal
set can be used just like this. Or we can say that when U(x) has only point 0 and 1, fuzzy set will
automatically become non fuzzy set. In this research, characteristic function is used for the
assessment of risk area such as the area with transportation to manufacturer / pier / industrial area
in the area, and amount of hazardous material being transported. However they do not indicate
that there is a clear frame or it is difficult to check. Characteristic function will be used for the
cases that these data is not available, and it is difficult to establish characteristic function from the
assessment according to Membership Function I of Fuzzy Number. Therefore, the membership
function value has only 0 or 1. Regarding danger, it can be categorized into 5 levels as usual.
The estimation of involved amount of each criteria that uses Membership Function II for the
assessment makes us know that it can occur in 2 types which are i) amount and risk level with
direct variation and ii) amount and risk level with reverse variation, as shown in the Figure 3.
{

Figure 3: Sample of Characteristic Membership Function.
4.2 Weighting
The assessment of risk area uses Saaty’s Analytic Hierarchy Process (AHP) to set weight of
each criteria related to the risk area. AHP is a mathematics method used for setting priority of each
criteria for making decision. The process consists of 3 parts which are identification and ordering,
assessment and comparison of elements in order, and integration using solution algorithm of
comparison result of every step. Scale for the comparison of priority (Huizingh and Virolijk, 1994)
consists of 9 levels of qualitative value: Equally Preferred, Equally to Moderately, Moderately
Preferred, Moderately to Strongly, Strongly Preferred, Strongly to Very Strongly, Very Strongly
Preferred, Very Strongly to Extremely, Extremely Preferred. Quantitative value had been set from
1 to 9 respectively. Calculation result from AHP is shown in Table 1.
137

5. Risk Assessment Model for Areas in Risk of Hazardous Materials
Transportation Developed from Fuzzy Sets
We can see that there are 14 criteria for the assessment, as shown in Table 1. Each criteria is
different from each other and can be described as criteria set as follow:
M = {M1, M2,…. Mi, Mn}
Where Mi; i = 1, 2, 3, … n represents membership value of each risk area according to the
criteria used for assessment.
As mentioned in 4.2, each criteria has different significance which can be represented in form
of sets as follow:
W = {W1,W2,…. Wi, Wn}
Where Wi; i = 1, 2, 3 … n represents weight of criteria used in the assessment and size of
matrix is n x 1
To divide sets for decision making for the assessment of area R, it can be done as follow:
R = {R1, R2, ..., Rj, Rm}
Whereas Rj; j = 1, 2, .., m represents decision value of each level. Value of each risk set
consists of 5 levels including 0.9, 0.7, 0.5, 0.3, and 0.1 ranging from most risk to least risk and
matrix size is 1 x m
The area to be assessed has criteria data at i-th, which can be displayed in fuzzy matrix of M as
follow:
M11 M12 . . . M1m
M21 M22 . . . M2m
Mij = . . . . . .
. . . Mij . .
. . . . . .
Mn1 Mn2 . . . Mnm
(Matrix 1)

Matrix displaying Mij shows membership value of the area to be assessed where i is in risk
level j
Matrix 1 with Mij is level of membership of area to be assessed of criteria i. It is a significant
model of how fuzzy is represented by data used for the assessment. Mij can be calculated using
membership value that is related to risk level. When combined with set of weight, the assessment to
find index value for the categorization of area in risk of hazardous material transportation will be
using model that uses set of R and M before going to weighing of each criteria with W.
The calculation for HazMat-Risk Area Index: HazMatRAI needs the relation of Mij through
weighing using Wi on the basis of the significance of each criteria, just like Saaty’s Analytic
Hierarchy Process (AHP) as follow:
HazMatRAI = � 𝑊𝑖 � Mij
M
j=1
Rj
N
i=1
(5)
This Fuzzy Number model was developed due to the limitation of Boolean logic. Boolean
logic uses simple scope to identify risk level of an area e.g. most risk, much risk, risk, less risk, or
least risk. Area that has distance from transportation system less than 200 meters is considered
most risk, 200 – 600 meters is much risk, 600 – 800 meters is risk, 800 – 1,000 meters is less risk,
and more than 1,000 meters is least risk. When there are two areas which have distance from
transportation system 395 meters and 405 meters respectively, if fire occurs, these two areas are
assessed R1 (most risk) and R2 (much risk) although these two areas are close to each other. We can
avoid this limitation by using membership function of Fuzzy Number. With this method, the two
areas will be assessed by calculating membership function in order to obtain changes of risk in the
area. It can be clearly seen when using membership function i.e. the assessment of 395-meter area
will be ((R1|0.025, R2|0.975, R3|0, R4|0, R5|0) and the 405-meter area will be (R1|0, R2|0.975,
R3|0.025, R4|0, R5|0) instead of being assessed as two completely different areas. However, these
two areas are considered much risk as they are in the scope of µ R2 = 0.975. This method also tell us
that the 395-meter area tends to “have most risk” (R1|0.025) and it will be never be categorized as
“much risk” (R3|0.025), while the 405-meter area tends to become the area with only “risk”
139

(R3|0.025) as well. We can clearly see changes of risk level when using membership function of
Fuzzy Number.
The calculation of HazMat-Risk Area Index (HazMatRAI) as mentioned above is the evaluation
of every criterion for weighing. It is reliable enough to be used for the assessment of area in risk of
hazardous material transportation, and it accommodates area diversity under the limitation of data
access. Such index can be used to identify risk level by making comparison of the calculated values
as HazMatRAI that uses comparison of related value ranging from biggest one to smallest one.
6. Conclusion
Planning for the management of disaster caused by hazardous material transportation needs to
pay much attention to transportation system. This study has established criterions for the
assessment of area in risk and it covers all land transportation, with most emphasis on road. We
found that transportation by road has more risk of accident than other systems. However facts
about areas in risk of hazardous material transportation are rare and difficult to access. that’s why
the analysis cannot be done clearly. Using Fuzzy Set for the assessment of both objective and
subjective criteria is another way to develop model in order to obtain value that can be used in the
comparison of risk in the area. Literature reviews and relevant researches tell us that criterions used
for the assessment always emphasis on transportation by car and route network. Implementation of
study result has much effect towards the management of disaster for the local authority, including
the planning for establishment of HazMat team.
Result obtained from Fuzzy Set model is HazMat-Risk Area Index (HazMatRAI) which is used
to identify value of such area. Besides it can be used for comparison of risk level ranging from
biggest one to smallest one.
The next step of model development is to find the value of HazMat-Risk Area Index. In this
regard, many things can be done such as establishing weighing value of each criteria using various
expertise to establish such weighing value. Besides, the establishment of membership level of each
objective criteria can use Geographic Information System (GIS) to help categorize in order to
display geographical data more clearly. However, the idea of this study is to support decision
making for the assessment under ambiguous context in an appropriate manner.

7. References
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approaches to managing spent nuclear fuel transport. Ph.D. Dissertation, Cornell
University, Ithaca, New York.
Cox, E.G. and Turnquist. M.A., 1986. Scheduling truck shipments of hazardous materials in the
presence of curfews. Transportation Research Record 1063.
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Devlin, Edward S., 2007. Crisis Management Planning and Execution. New York: Taylor &
Francis Group.
Ghada, M.H., 2004 Risk Base Decision Support Model for Planning Emergency Response for
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area. Accident Analysis Prevention 15.
Hazarika, S., 1987. Bhopal: The lessons of a tragedy. Penguin Book. New Delhi.
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materials. European Journal of operational Research (86/2).
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List, G. and Turnquist, M., 1994. Estimating truck travel pattern in urban areas. Transportation
Research Record 1430.
McCord, M.R. and Leu. A.Y.C., 1995. Sensitivity of optimal hazmat routes to limited preference
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of Hazardous Material.
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141

National Fire Protection Association, 2001. Code and Standard- Massachusetts United States.
Available from :(http://www.nfpa.org) Access November 2008
Sikich, Geary W., 1996. “All Hazards” Crisis Management Planning. Highland: Pennwell Books.
Smith, D., 2005. What ‘s in a name? The nature of crisis and disaster-a search for Signature
qualities. Working Paper. University of Liverpool Science Enterprise Centre. Liverpool.
Zadeh, L.A., 1965. Fuzzy Sets. Information and control, Vol.8
Zografos, K.G. and Androutsopoulos. K.N., 2008. A decision support system for integrated
hazardous materials routing and emergency response decisions. Transportation Research
Part C 16.
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hazardous materials. ASCE Transportation Engineering Journal (115/6).
S. Namee is currently a PhD candidate in Department of Civil Engineering at Thammasat University.
He has been working at the Department of Disaster Prevention and Mitigation, Ministry of Interior,
THAILAND. His research interests encompass hazardous material transport.
Dr. B. Witchayangkoon is an Associate Professor of Department of Civil Engineering at Thammasat
University. He received his B.Eng. from King Mongkut’s University of Technology Thonburi with
Honors in 1991. He continued his PhD study at University of Maine, USA, where he obtained his PhD
in Spatial Information Science & Engineering. Dr. Witchayangkoon current interests involve
applications of emerging technologies to engineering.
Dr. A. Karoonsoontawong is an Assistant Professor of Department of Civil Engineering at King
Mongkut’s University of Technology Thonburi. He received his B.Eng. from Chulalongkorn
University with Honors in 1997. He received his M.S. and Ph.D. in Transportation Engineering in
2002 and 2006, respectively, from The University of Texas at Austin, USA. Dr. Ampol is interested in
transportation network modeling, logistical distribution network optimization, and applied
operations research.
Peer Review: This article has been internationally peer-reviewed and accepted for publication
according to the guidelines given at the journal’s website.

Computer Modeling of Internal Pressure
Autofrettage Process of a Thick-Walled Cylinder
with the Bauschinger Effect
Zhong Hu
a*
, and Sudhir Puttagunta
a
a
Department of Mechanical Engineering, South Dakota State University, USA
Article history:
Received January 13, 2012
Accepted January 27, 2012
Available online
January, 28 2012
Keywords:
Thick-walled cylinder;
Internal Pressurize;
Autofrettage;
Bauschinger effect;
Finite Element Analysis
In this paper, the internally pressure overloading autofrettage
process of a thick-walled cylinder has been numerically investigated.
The corresponding axi-symmetric and plane-stress finite element
model has been employed. The elasto-plastic material model with
nonlinear strain-hardening and kinematic hardening (the Bauschinger
effect) was adopted. The residual stresses in the thick-walled cylinder
induced by internal autofrettage pressure have been investigated and
optimized. The optimum autofrettage pressure and the maximum
reduction percentage of the von Mises stress in the autofrettaged
thick-walled cylinder under the elastic-limit working pressure have
been found, the differences of stress and strain distribution between
adopting the Bauschinger-effect and the non-Bauschinger-effect have
been compared.
1. Introduction
Thick-walled cylinders subjected to high internal pressure and/or elevated temperature are
widely used in the nuclear and chemical industries involving pressures as high as 1380 MPa and
2012 American Transactions on Engineering & Applied Sciences
*Corresponding author (Z.Hu). Tel/Fax: +1-605-688-4817. E-mail address:
Zhong.Hu@sdstate.edu. 2012. American Transactions on Engineering & Applied
Sciences. Volume 1 No.2 ISSN 2229-1652 eISSN 2229-1660. Online Available at
143

temperatures of up to 300 °C, (Ford et al. 1981) especially for military applications involving
transient peak internal pressures as high as 350 MPa and temperature of up to 1500 °C inside the
gun barrel in a ballistic event. (Bundy et al. 1996) In the absence of residual stresses, cracks usually
form at the bore where the hoop stress developed by the working pressure is highest. (Daniels
1942; Zapfec 1942; Bush 1988; Masu and Graggs 1992) To prevent such failure and to increase the
pressure-carrying capacity, a common practice is autofrettage treatment of the cylinder prior to use.
Autofrettage is used to introduce advantageous favorable compressive residual hoop stress inside
wall of a cylinder and result in an increase in the fatigue lifetime of the component. There are
basically three types of autofrettage. These are carried out by hydraulic pressurization, by
mechanically pushing an oversized mandrel, or by the pressure of powder gas, (Davidson and
Kendall 1970; Malik and Khushnood 2003) in which hydraulic and powder gas pressurization are
based on the same principal and strengthening mechanism. In general, vessels under high pressure
require a strict analysis for an optimum design for reliable and secure operational performance.
Prediction of residual stress field and optimization of the autofrettage processes’ parameters are
some of the key issues in this context, which normally involve a careful evaluation of the related
modeling, simulation and experimental details. (Davidson et al. 1963; Chu and Vasilakis 1973;
Shannon 1974; Tan and Lee 1983; Gao 1992; Avitzur 1994; Kandil 1996; Lazzarin and Livieri
1997; Zhu and Yang 1998; Venter et al. 2000; Gao 2003; Iremonger and Kalsi 2003; Kihiu et al.
2003; Parker et al. 2003; Perry and Aboudi 2003; Zhao et al. 2003; Perl and Perry 2006; Bihamta et
al. 2007; Hojjati and Hassani 2007; Korsunsky 2007; Gibson 2008; Perry and Perl 2008; Ayob et
al. 2009; Darijani et al. 2009) Efforts are continually made in the regarding aspects.
Overloading pressure autofrettage process involves the application of high pressure to the
inner surface of a cylinder, until the desired extent of plastic deformation is achieved. Analytical
solution of pressure autofrettage of a constant cross-section cylinder, subject to some end
conditions, is possible through the use of simplifying assumptions, such as choice of yield criteria
and material compressibility and, critically, material stress-strain behavior. On the other hand,
autofrettage causes large plastic strains around the inner surface of a cylinder, which noticeably
causes the early onset of non-linearity when remove the autofrettage pressure in the unloading
process – a kinematic hardening phenomenon termed the Bauschinger effect. This non-linearity
typically causes significant deviation from those material models that are often assumed. The effect
is most pronounced around the inner surface, and in turn has a significant effect on the residual
144 Zhong Hu, and Sudhir Puttagunta

stresses developed when the autofrettage load is removed, especially as it can cause reverse
yielding to occur when it otherwise would not be expected. Research has been done on this issue
with theoretical analysis mostly based on bilinear kinematic hardening (linear elastic and linear
hardening) material model which is a good approximation for small strain. (Lazzarin and Livieri
1997; Venter et al. 2000; Kihiu et al. 2003; Parker et al. 2003; Perry and Aboudi 2003; Huang 2005;
Perl and Perry 2006; Korsunsky 2007). However, the practical material model is of nonlinear
kinematic hardening and with equivalent strain up to 0.5~1% in the autofrettage process, which
gives complexity to theoretical analysis using nonlinear kinematic hardening material model. In
this paper, the internally pressure overloading autofrettage process will be numerically
investigated. An axi-symmetric and plane stress (for open-ended cylinder) finite element model
will be presented. The elasto-plastic nonlinear material constitutive relationship will be adopted,
incorporating a nonlinear kinematic hardening (the Bauschinger effect) for which no analytical
solution exists. The effects of the autofrettage pressure on the residual stresses in a thick-walled
cylinder will be evaluated. The percentage of stress reduction by autofrettage treatment will be
calculated based on von Mises yield criterion. The optimum autofrettage pressure will be found.
The differences of stress and strain distribution between adopting the Bauschinger-effect and the
non-Bauschinger-effect will be compared.
2. Mathematical Model
In this work, the thick-walled cylinder is made of stainless steel AISI 304. An elasto-plastic
governing equations for material behavior with a homogeneous and isotropic hardening model is
used. The true stress – true strain behavior of the strain hardening material follows the Hooke’s law
in the elastic region, and, for comparison purpose, the power-law hardening in the plastic region,
(Hojjati and Hassani 2007)
𝜎𝜎 = �
𝐸𝐸𝐸𝐸 𝜖𝜖 < 𝜖𝜖𝑦𝑦
𝐾𝐾𝜖𝜖 𝑛𝑛
𝜖𝜖 ≥ 𝜖𝜖𝑦𝑦
(1),
where σ and 𝜖𝜖 are true stress and true strain, respectively. 𝜖𝜖𝑦𝑦 is the strain at the yield point. E is
the modulus of elasticity. K is a material constant equal to 𝐾𝐾 = 𝜎𝜎𝑦𝑦 𝜖𝜖𝑦𝑦
−𝑛𝑛
= 𝜎𝜎𝑦𝑦
1−𝑛𝑛
𝐸𝐸 𝑛𝑛
, and n is the
strain-hardening exponent of the material (0 ≤ n < 1). σy is the yield stress. Table 1 lists the basic
145

parameters and material data and derived material parameters to be used in the FEA modeling.
(Peckner and Bernstein 1977; Harvey 1982; Boyer et al. 1985; ASM International Handbook
Committee 1990). Figure 1 shows the elasto-plastic stress-strain relationship with kinematic
hardening (the Bauschinger effect). Commercially available software ANSYS has been used for
finite element modeling of the autofrettaged thick-walled cylinder. (Swanson Analysis System Inc.
2011) The finite element model is shown in Figure 2. The element PLANE183 with the capacity of
elastic and plastic material nonlinearity and non-linear kinematic hardening (the
Bauschinger-effect) has been adopted, which is an eight-node plane-stress 2-D element with higher
accuracy quadratic shape function. In order to get reasonable accuracy, more elements are used
near inner surface and outer surface of the cylinder, see Figure 2.
Table 1: Model dimensions and material properties.
Material of the Cylinder AISI304
Modulus of Elasticity E 196.0 GPa
Poisson's Ratio υ 0.29
Yield Strength 𝜎𝜎𝑦𝑦 152.0 MPa*
Strain at Yield Point 𝐸𝐸𝑦𝑦 7.755×10-4
Strain-Hardening Exponent n 0.2510
Material Constant K 917.4 MPa
Inner Radius a 60 mm
Outer Radius b 90 mm
Maximum Working Pressure pi 47.22MPa
* corresponding to 215MPa at 0.2% offset and ultimate tensile strength of 505 MPa.
Figure 1: The elasto-plastic stress – strain model with kinematic hardening (the Bauschinger
effect).

Figure 2: Finite element model of a plane-stress thick-walled cylinder.
3. Modeling Results and Discussions
Consider a thick-walled cylinder having inner radius a and outer radius b and subjected to the
internal pressure pi as shown in Figure 2. The material will obey the Hooke’s law when it is within
the elastic region. This allow us to use the Lame’s equations for calculating the hoop stress, 𝜎𝜎𝜃𝜃, and
radial stress, 𝜎𝜎𝑟𝑟, along the thickness of the cylinder, when the ends of the cylinder are open and
unconstrained so that the cylinder is in a condition of plane stress. (Ugural 2008)
𝜎𝜎𝜃𝜃 =
𝑎𝑎2 𝑝𝑝𝑖𝑖
𝑏𝑏2−𝑎𝑎2 �1 + �
𝑏𝑏2
𝑟𝑟2�� (2),
𝜎𝜎𝑟𝑟 =
𝑏𝑏2−𝑎𝑎2
�1 − �
𝑏𝑏2
𝑟𝑟2
�� (3),
𝜎𝜎𝑧𝑧 = 0 (4),
Therefore, the von Mises (equivalent) stress is
𝜎𝜎𝑖𝑖 = �𝜎𝜎𝜃𝜃
2
+ 𝜎𝜎𝑟𝑟
2
− 𝜎𝜎𝜃𝜃 𝜎𝜎𝑟𝑟�
1
2
=
�1 + 3 �
𝑏𝑏
𝑟𝑟
�
4
�
1
2
(5),
The radial displacement is
147

𝑢𝑢 =
𝑎𝑎2 𝑝𝑝𝑖𝑖 𝑟𝑟
𝐸𝐸(𝑏𝑏2−𝑎𝑎2)
�(1 − 𝜈𝜈) + (1 + 𝜈𝜈)
𝑏𝑏2
𝑟𝑟2
� (6),
Obviously, the maximum von Mises stress is at r = a. Assuming von Mises yield criterion
applied, i.e., 𝜎𝜎𝑖𝑖 ≤ 𝜎𝜎𝑦𝑦, so by substituting the data from Table 1, the maximum applied working
pressure (the maximum internal pressure without causing yielding) is
𝑝𝑝𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚
= 𝜎𝜎𝑦𝑦
𝑎𝑎2
�1 + 3 �
𝑏𝑏
𝑎𝑎
�
4
�
−
1
2
= 47.2 (MPa) (7),
Figure 3 shows the analytical and modeling results of stress components and radial
displacement along the thickness of the cylinder subjected to the maximum working pressure (pw
max = 47.2 MPa). The modeling results are well agreed with the analytical results from Lame’s
equations, indicating the reliability of the model employed in the numerical analysis. Figure 3 also
shows that the maximum von Mises stress and hoop stress located at the inner surface of the
cylinder, and the hoop stress is the major stress component causing yield failure.
Figure 3: Analytical and modeling results of stresses and radial displacement along the thickness
of the cylinder subjected to the maximum working pressure (pw max = 47.2 MPa).
3.4E-5
3.5E-5
3.6E-5
3.7E-5
3.8E-5
3.9E-5
4.0E-5
4.1E-5
4.2E-5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 20 40 60 80 100
StressRatioσ/σy
(r-a)/(b-a)×100 (%)
σθ/σy by Eq.(2)
σθ/σy by Modeling
σr/σy by Eq.(3)
σr/σy by Modeling
σi /σy by Eq.(5)
σi/σy by Modeling
u by Eq.(6)
Displacementu(m)

Figure 4: The relationship of elastic-plastic interface ρ and the autofrettage pressure pa.
As the internal pressure increasing, yielding (plastic deformation) starts from cylinder’s inner
surface and gradually extends towards cylinder’s outer surface. This process is basically an
elastic-plastic deformation process. The relation between internal autofrettage pressure, pa, and the
radius of the elastic-plastic boundary (interface) in plane stress condition, ρ, is determined by (Gao
1992; Gao 2003)
⎩
⎪⎪
⎨
⎪⎪
⎧𝑝𝑝𝑎𝑎
𝜎𝜎𝑦𝑦
=
2
√3
�
cos�𝜑𝜑 𝜌𝜌+𝜑𝜑 𝑛𝑛�
cos(𝜑𝜑𝑎𝑎+𝜑𝜑 𝑛𝑛)
�
�3𝑛𝑛2+𝑛𝑛�
�3𝑛𝑛2+1�
𝑒𝑒𝑒𝑒𝑒𝑒 �
√3𝑛𝑛(𝑛𝑛−1)
3𝑛𝑛2+1
�𝜑𝜑𝑎𝑎 − 𝜑𝜑𝜌𝜌�� cos 𝜑𝜑𝑎𝑎 ,
𝜌𝜌 = 𝑎𝑎�
sin(𝜑𝜑𝑎𝑎+𝜋𝜋/6)
sin�𝜑𝜑 𝜌𝜌+𝜋𝜋/6�
�
cos�𝜑𝜑 𝜌𝜌+𝜑𝜑 𝑛𝑛�
cos(𝜑𝜑𝑎𝑎+𝜑𝜑 𝑛𝑛)
�
2𝑛𝑛
�3𝑛𝑛2+1�
𝑒𝑒𝑒𝑒𝑒𝑒 �
√3
2
1−𝑛𝑛2
3𝑛𝑛2+1
�𝜑𝜑𝜌𝜌 − 𝜑𝜑𝑎𝑎�� ,
𝜑𝜑𝜌𝜌 = cos−1
�
√3
2
�𝑏𝑏2−𝜌𝜌2�
�3𝑏𝑏4+𝜌𝜌4
� ,
(8),
Where
𝜑𝜑𝑛𝑛 = cos−1
�
√3𝑛𝑛
√3𝑛𝑛2+1
� (9),
During autofrettage process, the elastic-limit pressure pe (pressure at which yielding
commences at inner surface) is obviously
𝑝𝑝𝑒𝑒
𝑝𝑝 𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚
= 1 and 𝜌𝜌 = 𝑎𝑎 in the given case, and the
plastic-limit pressure py (pressure at which plasticity has spread throughout the cylinder) is
1
1.1
1.2
1.3
1.4
1.5
1.6
0 20 40 60 80 100
AutofrettagePressureRatio
pa/pwmax
Elasto-plastic interface (ρ-a)/(b-a)*100 (%)
pa/pw max by Eq.(8) from Refs. (Gao
1992 and 2003)
pa/pw max by Modeling
149

𝑝𝑝𝑦𝑦
= 1.543 and 𝜌𝜌 = 𝑏𝑏 in the given case. Figure 4 shows the relation of elastic-plastic interface
ρ and the autofrettage pressure pa obtained by modeling and analytical approaches, they are well
agreed.
Figures 5-7 show the autofrettage stress distributions for the thick-walled cylinder subjected to
different internal autofrettage pressure range of
𝑝𝑝𝑎𝑎
= 1~1.6 by modeling. Hoop stress and
von Mises stress in Figures 5 and 7 clearly indicate that by increasing applied autofrettage pressure,
elastic-plastic interface (the turning point of the curve corresponding to the position ρ in Figure 4)
moves towards the outer surface of the cylinder and eventually reaches the outer surface, while
radial stress in Figure 6 indicate that the radial stress in compression, with highest compressive
stress in the inner surface and zero in the outer surface, increasing as the autofrettage pressure
increasing.
Figure 5: Autofrettage hoop stress distribution under different autofrettage pressure.
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0 20 40 60 80 100
AutofrettageHoopStressRatioσa
θ/σy
(r-a)/(b-a)*100 (%)
by Modeling under pw max
by Modeling under 1.1 pw max

Figure 6: Autofrettage radial stress distribution under different autofrettage pressure.
Figure 7: Autofrettage von Mises stress distribution under different autofrettage pressure.
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0 20 40 60 80 100
AutofrettageRadialStressRatioσa
r/σy
(r-a)/(b-a)*100 (%)
by Modeling by pw max
0.4
0.6
0.8
1.0
1.2
1.4
0 20 40 60 80 100
AutofrettagevonMisesStressRatioσa
i/σy
(r-a)/(b-a)*100 (%)
by Modeling under pw max
151

When the internal autofrettage pressure is removed, the elastic deformation is trying to resume
its original shape while the plastic deformation is resisting this process so that residual stresses
within the cylinder have been induced, and the so-called pressure autofrettage process has been
accomplished. This unloading process basically is treated elastically. However, reverse yielding is
possible if earlier onset happens due to the Bauschinger effect. Figures 8-10 show the residual
stress distribution after removed autofrettage pressure. From Figure 8, it shows a compressive
residual hoop stress near the inner surface generated which is favorable to the thick-walled cylinder
when it is under internal working pressure (partially cancelling the tensile hoop stress induced by
the working pressure), and a tensile residual hoop stress near the outer surface of the cylinder.
Figure 9 shows a relative smaller but compressive residual radial stress left inside of the thick-wall,
satisfying the boundary conditions of zero radial stress on inner surface and outer surface of the
cylinder. Figure 10 shows the residual von Mises stress with higher values basically near the inner
surface and outer surface of the cylinder. The turning points of the curves in Figures 8 and 10 are
almost the same as in Figures 5 and 7 indicates that the earlier onset of the reverse yielding is very
less, not changing the elastic-plastic interface very much.
Figure 8: Residual hoop stress distribution after removed the autofrettage pressure.
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0 20 40 60 80 100
ResidualHoopStressRatioσr
θ/σy
(r-a)/(b-a)*100 (%)
by Modeling with autofrettage of 1.1 pw max

Figure 9: Residual radial stress distribution after removed the autofrettage pressure.
Figure 10: Residual von Mises stress distribution after removed the autofrettage pressure.
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0 20 40 60 80 100
ResidualRadialStressRatioσr
r/σy
(r-a)/(b-a)*100 (%)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 20 40 60 80 100
ResidualvonMisesStressRatioσr
i/σy
(r-a)/(b-a)*100 (%)
153

When an autofrettaged thick-walled cylinder is applied to an internal working pressure, this
reloading process is treated elastically, since a plastic deformation is not desired in the application
of the cylinders based on the static yield design criterion. Figures 11-13 show the stress
distributions of an autofrettaged thick-walled cylinder after reloading by an elastic-limit pressure
(i.e., the maximum working pressure 𝑝𝑝𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚 = 47.2 MPa). It shows more uniform distribution of
the stress components and von Mises stress throughout the thickness of the cylinder due to the
autofrettage treatment, which makes maximum von Mises stress in the cylinder less than that
without autofrettage treatment. However, radial stress has not been changed very much, see Figure
12. Figure 14 shows the relationship of the final maximum von Mises stress versus autofrettage
pressure for a thick-walled cylinder under the maximum working pressure (pw max = 47.2 MPa). It
clearly shows the optimal autofrettage pressure is about 1.5 times the maximum working pressure,
and the maximum von Mises stress reduction with this autofrettage pressure is about 28%.
Figure 11: Final hoop stress distribution of an autofrettaged thick-walled cylinder under the
maximum working pressure (pw max = 47.2 MPa).
0.4
0.5
0.6
0.7
0.8
0.9
0 20 40 60 80 100
FinalHoopStressRatioσf
θ/σy
(r-a)/(b-a)×100 (%)
by Eq.(2) without autofrettage
by Modeling without autofrettage

Figure 12: Final radial stress distribution of an autofrettaged thick-walled cylinder under the
Figure 13: Final von Mises stress distribution of an autofrettaged thick-walled cylinder under the
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0 20 40 60 80 100FinalRadialStressRatioσf
r/σy
(r-a)/(b-a)×100 (%)
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100
FinalvonMisesStressRatioσf
i/σy
(r-a)/(b-a)×100 (%)
by Modeling with autofrettage of 1.1 pw
max
by Modeling with autofrettage of 1.2 pw
max
155

ATEAS V1(2):: American Transactions on Engineering & Applied Sciences

ATEAS V1(2):: American Transactions on Engineering & Applied Sciences

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ATEAS V1(2):: American Transactions on Engineering & Applied Sciences