PSG College of Technology, Peelamedu, Coimbatore, India, 1701rm01@psgtech.ac.in 331
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340 Received for review: 2021-01-22
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-04-04
DOI:10.5545/sv-jme.2021.7104A Short Scientific Paper Accepted for publication: 2021-05-14
Reliability-Based Design Optimization of Pump Penetration Shell 
Accounting for Material and Geometric Non-Linearity
Gautham, V . – Prabhu Raja, V . – Ebron Shaji, G.T. – Mohanraj, S. – Thyla, P.R.
Gautham Velayudhan
*
 – Prabhu Raja Venugopal – Ebron Shaji Gnanasigamony Thankareathenam – Mohanraj 
Selvakumar – Thyla Pudukarai Ramaswamy
PSG College of Technology, Department of Mechanical Engineering, India
The roof slab of a nuclear reactor supports all the components and sub-systems. It needs to resist the seismic loads in accordance with 
load-carrying criteria. The static stress analysis of the reactor roof slab reveals that high-stress concentration was present in the pump 
penetration shell (PPS) that supports the primary sodium pump. This paper presents an assessment of collapse load and the optimization of 
the pump penetration shell through the reliability approach, accounting for material non-linearity, geometrical non-linearity and randomness 
in loading. In addition, the load-carrying capacity of PPS was determined considering two different materials: IS2062 and A48P2. The design 
of experiments (DoE) was formulated considering the flange angle and flange thickness as parameters. An empirical model for load function 
was formulated from the results of the collapse load obtained for various combinations of design parameters. The above function was used to 
perform the reliability-based geometry optimization of the PPS of the roof slab.
Keywords: nuclear reactor; buckling; optimization; reliability; limit load; genetic algorithm
Highlights
• 	 The reliability-based design of the pump penetration shell is adapted to account for material properties that are uncertain.
• 	 The methodology of incorporating the reliability index for reliability-based design has been studied, and a genetic algorithm 
optimization technique is adopted for performing design optimization of the pump penetration shell.
• 	 The prediction of collapse load of a single-layer PPS reveals that the PPS made of IS2062 has a higher load-carrying capacity 
than the PPS made of A48P2. 
• 	 The PPS made of IS2062 has less influence of effect of mode shapes and imperfection on the collapse load compared with the 
PPS made of material A48P2. 
• 	 The load-carrying capacity of the PPS increasing with the increase of angle and thickness of the flange.
0  INTRODUCTION
Considering the geometrical configuration of the roof 
slab of a nuclear reactor, a large box-type structure 
with many penetrations made of carbon steel, posed 
many difficulties during manufacturing, particularly 
due to lamellar tearing. Alternatively, taking advantage 
of high-load carrying capacity with possibly minimum 
thickness, a dome-shaped roof slab was conceived 
comprised of a conical shell connected to the vertical 
shell through a short torus portion [1].
The structural integrity of the roof slab is 
significant for the safe and reliable functioning of the 
reactor [2]. Structural integrity refers to the condition 
of a structural system and implies that the structure 
and its components remain intact over the intended 
lifetime of the structure. The roof slab is subjected 
to fatigue loading due to the reaction taking place 
inside the main vessel and the intermittent flow of 
sodium [1]. The fatigue phenomenon is a complex 
and progressive failure; one possibility of occurrence 
could be from a small weld defect. The failure 
begins with the formation of micro-cracks and crack 
propagation, leading to the failure of the structure.
Reliability is an aspect of engineering uncertainty. 
It is the probability that an item will perform a required 
function without failure under stated conditions for 
the stated period [3] to [7]. Durability is a particular 
aspect of reliability related to the ability of an item to 
withstand the effects of time or operating cycles, etc. 
The main objective of any structural design is to ensure 
the safety and economy of the structure operating 
under a given environment. Hence, the demand should 
not exceed the capacity of the structure.
Capacity (C) > Demand (D)
The above-cited condition should be satisfied 
so that the structure’s safety is ensured for the 
intended purpose for which the structure is built. The 
deterministic approach is based on the premise that a 
given problem can be stated in the form of a question 
or a set of questions to which there is an explicit and 
distinct answer. A probabilistic approach is based 
on the concept that several or varied outcomes of 
a situation are possible to this approach in which 
uncertainty is recognized. Probabilistic modelling 
aims at the study of a range of outcomes for the given 
set of input data. Accordingly, the description of a 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
332 Gautham, V. – Prabhu Raja, V. – Ebron Shaji, G.T. – Mohanraj, S. – Thyla, P.R.
physical situation or system includes the randomness 
of data and other kinds of uncertainties.
Fig. 1.  Probabilistic approach
The probability of failure is the common area 
shared by both the demand and capacity curves as 
shown in Fig. 1. The reason for this is that C is always 
less than D in the shaded area, hence ensuring failure. 
Considering the probability density function for the 
normal distribution of capacity and demand curve, 
the measure of reliability is defined with a term 
known as the reliability index ( β), as shown in Fig. 
1, as proposed by Cornell et al. [8]. It is the shortest 
distance measured from the origin when the capacity 
and demand are expressed. The random variables are 
characterized by their first two moments only, i.e., 
mean and variance. This is achieved by first-order 
approximation of the non-linear function; therefore, 
this method is termed a first-order second moment 
(FOSM) method.
Pump 
penetration 
shell
Fig. 2.  Dome shaped roof slab model
The pump penetration shell is one of the critical 
components of the dome-shaped roof slab (Fig. 2). 
Hence, it is essential to analyse the load-carrying 
capacity of the roof slab to ensure structural integrity. 
The primary objective of the work is to perform 
reliability-based design optimization of the pump 
penetration shell (PPS) using a genetic algorithm and 
compare the load-bearing capacity of PPS made of 
materials IS2062 and A48P2. It is essential to optimize 
the design parameters of the PPS for maximum load-
carrying capacity with a reduced volume of PPS. Prior 
to performing the non-linear analysis of the roof slab, 
a benchmarking study that involves the prediction of 
collapse load of a simpler structure is carried out.
1  BENCHMARK STUDY ON PREDICTION OF COLLAPSE LOAD 
FOR SIMPLE STRUCTURE
In order to ascertain the methodology to be adopted 
for performing the non-linear analysis of roof slab, a 
simpler problem proven experimentally by Zhou et 
al. [9] is taken up for benchmark study. The problem 
involves the determination of the limit load of the 
steel pyramid-to-tube socket connection subjected 
to uniform compression. Ansys Workbench finite 
element software is used. The geometric and meshed 
models of the steel structure are shown in Figs. 3 and 
4, respectively. The material properties of the structure 
are given in Table 1.
Fig. 3.  Geometric model  
of structure
Fig. 4.  Meshed model  
of structure
Table 1.  Material properties of the structure [9]
Component Tube Socket Steel Pyramid
Steel type STKR400 SS400
Measured plate thickness [mm] 4.2 4.2
Young’s modulus [N/mm
2
] 207893 204076
Yield strength [N/mm
2
] 378 348
Ultimate strength [N/mm
2
] 454 433
Uniform elongation [%] 17 17
Initially, the critical load was predicted by 
performing linear buckling analysis by fixing the 
bottom end of the pyramid and applying a uniform 
compressive load at the top, without considering the 
non-linear behaviour of the structure [10] to [13]. The 
critical buckling load of the structure is found to be 
1867 kN and the corresponding displacement contour 
is shown in Fig. 5.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
333 Reliability-Based Design Optimization of Pump Penetration Shell Accounting for Material and Geometric Non-Linearity
Fig. 5.  Buckling mode of steel structure (Mode 1)
Table 2.  Comparison of numerical and benchmark results
Description
Numerical 
result
Benchmark 
result [9]
Deviation 
[%]
Buckling load [kN] 1867 1910 2.24
The results obtained from numerical analysis 
are found to be in close agreement with experimental 
results by Zhou et al. [9], as shown in Table 2. Hence, 
the above methodology can be extended for the 
numerical prediction of the collapse load of the PPS.
Since the present work deals with optimizing 
the pump penetration shell for higher load-carrying 
capacity, a benchmark study that involves the 
application of reliability approach and genetic 
algorithm is carried out prior to performing the 
reliability-based design optimization of the reactor 
roof slab.
2  A BENCHMARK STUDY OF RELIABILITY-BASED  
DESIGN OPTIMIZATION
A cantilever beam subjected to bending loads is 
considered for performing reliability-based design 
optimization. In general, design parameters, such as 
material properties and geometrical dimensions, as 
well as boundary and loading conditions, are prefixed 
while designing any structure. However, in the case 
of reliability-based design optimization, the limit state 
function that relates the capacity (R) and demand (S) 
of the function should be derived as a function ‘g’ of 
the variables. When g is greater than zero, then the 
structure is considered to be a safe structure. The limit 
state function is given by:
 gRS  . (1)
In reliability-based design, the limit state function 
g is a random variable because of the uncertainties 
involved in load and yield strength. The reliability 
index β is a direct measure of the reliability of the 
system and hence a larger value of β represents higher 
reliability. For structural applications, the reliability 
index is generally taken as 3, for which the probability 
of failure is 0.00135 [14]. 
A cantilever beam shown in Fig. 6 is taken up 
for benchmark study for which Wang et al. [15] have 
prescribed the results of the reliability-based design 
optimization. The width, height, and thickness of the 
beam of a length (l) of 3048 mm, are designated as 
w, h and t, respectively. The loads acting at the free 
end of the cantilever beam, along x and y-axes are 
designated as P
x
 and P
y
, respectively. The moments 
of inertia of the cantilever beam about x-axis and 
y-axis are represented as I
xx
 and I
yy
 respectively. In 
the limit state function, R and S represent the yield 
strength of material and the maximum bending stress, 
respectively.
The mean and standard deviation of the load acting 
in x-axis are 2224 N (P
x
) and 667 N (s
x
) (respectively 
while the corresponding values for load acting in 
y-axis
 
are 4448 N (P
y
) and 222 N (s
y
). The mean and 
standard deviations of the material yield strength 
are 276 MPa (R) and 14 MPa (s
R
), respectively. 
The objective is to design the cantilever beam for a 
minimum weight with the given constraints: 
β >3 ; 50 mm ≤ w, h ≤ 254 mm; 
2.5 mm ≤ t ≤ 13 mm.
Since the system involves more than two random 
variables, the FOSM method is used to obtain the 
Fig. 6.  Loads acting on the cantilever beam

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
334 Gautham, V. – Prabhu Raja, V. – Ebron Shaji, G.T. – Mohanraj, S. – Thyla, P.R.
mean (µ
g
) and standard deviation ( σ
g
) of the limit state 
function.
 gr
lwP
I
lhP
I
x
yy
y
xx
 








22
, (2)
  
gR
x
x
y
g
R
S
g
P
S
g
P
2
2
2
2
2







































 













2
2
S
y
, (3)
 



g
R
1, (4)
 



g
P
lw
I
xy y
2
, (5)
 



g
P
lh
I
yx x
2
, (6)
 
gR
x
x
y
g
R
S
g
P
S
g
P








































2
2
2
2
 












2
2
S
y
, (7)
 



g
g
. (8)
The above equations are incorporated into 
the MATLAB optimization tool, and the objective 
function is minimized using a genetic algorithm [16] 
to [18]. The optimization results are compared with 
that of Wang et al. [15] and are shown in Table 3.
Table 3.  Comparison of results based on reliability-based 
optimization
Area [mm
2
] 
(objective 
function)
Thickness 
t 
[mm]
Width  
w 
[mm]
Height  
h 
[mm]
Based on  
genetic algorithm
1770 2.54 169 184
Results by  
Wang et al. [15]
1772 2.54 179 175
It is inferred from the above table that the results 
obtained from that predicted by Wang et al. [15] are 
in good agreement. Hence, the validated methodology 
for reliability-based optimization is extended for the 
pump penetration shell, which is described in the next 
section.
3  RELIABILITY-BASED DESIGN OPTIMIZATION OF PUMP 
PENETRATION SHELL
In order to ensure the reliability of the PPS, the 
variability in the flange thickness and flange angle 
must be considered during structural design. The 
objective is to minimize the weight of the PPS, which 
is achieved by minimizing the volume of the structure. 
The volume of the PPS is given by the sum of the 
volumes of the cylindrical part and the conical part. 
V
1
 and V
2
 are the volumes of the hollow cylinder and 
truncated hollow cone, respectively. R
1
 and R
2
 are the 
outer and inner radii of the bottom surface of the cone, 
r
1
 and r
2
 are the outer and inner radii of the cylinder, 
h
y
 and h
c
 
 
are the heights of the cylinder and the cone 
respectively.
 Vh rr
y 11
2
2
2


 



 , (9)
where, r
2
 = 88; r
1
 = r
2
 + t, 
V
h
RR rr RR rr
c
21
2
11 1
2
2
2
22 2
2
3
















, (10)
where, R
2
 = 128; R
1
 = R
2
 + t, 
 VVV 
12
. (11)
The constraints imposed on the PPS for 
reliability-based design are given as follows, β >3;  
0° ≤ θ ≤ 30°; 1 mm ≤ t ≤ 3 mm.
For the limit state function (g function) given in 
Eq. (1), R and S represent collapse load and applied 
load or operating load (10 kN), respectively. The 
calculation of collapse load for PPS is discussed in 
later sections.
3.1  Determination of Collapse Load for PPS
The elasto-plastic analysis of PPS is carried out 
considering two different materials, namely IS2062 
and A48P2, which are commonly used in the 
fabrication of the prototypes of the reactor roof slab. 
The collapse load behaviour of the PPS made of 
IS2062 and A48P2 will be compared in this section. 
The PPS is idealized as a shell body, and it is scaled 
down to 1:10 ratio from the actual reactor roof slab 
to facilitate experimentation. The geometry of PPS 
shown in Fig. 7 is modelled and meshed with SHELL 
181 elements. The mesh convergence study was carried 
out to idealize the element size. Prior to determining 
the collapse load, it is necessary to determine the 
buckling mode, which has a considerable influence on 
the failure of the structure.
From experimental investigation on static 
loading, it was observed that buckling occurred in 
the conical portion of the structure with imperfection. 
Hence, Modes 2, 9, 16, 23, and 30, which correspond 
to buckling in the conical portion of the structure are 
considered for numerical analysis, both for linear 
buckling analysis and elasto-plastic analysis. The 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
335 Reliability-Based Design Optimization of Pump Penetration Shell Accounting for Material and Geometric Non-Linearity
mode shapes of PPS made of IS2062 are shown in 
Fig. 8. Similar mode shapes were also observed for 
the PPS made of A48P2. Subsequently, elasto-plastic 
analysis was performed, and the collapse load of PPS 
was predicted by using the elastic slope method twice 
[19] and [20].
It is inferred from Fig. 9 that there is no 
considerable influence of mode shape on the collapse 
load of PPS made of IS2062. In contrast, the influence 
of mode shape on the collapse load of PPS made of 
A48P2 is considerable, as shown in Fig. 10. It is found 
that the collapse load is the lowest corresponding to 
        a) Mode 2       b) Mode 9  c) Mode 16        d) Mode 23          e) Mode 30
Fig. 8. Mode shapes of PPS made of IS2062
Fig. 9.  Effect of mode shape on collapse load of PPS made of IS2062
Fig. 10.  Effect of mode shape on collapse load of PPS made of A48P2
Fig. 7.  Geometric model of pump penetration shell

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
336 Gautham, V. – Prabhu Raja, V. – Ebron Shaji, G.T. – Mohanraj, S. – Thyla, P.R.
Mode 23. It is observed that Mode 23 is characterized 
by multiple lobes, whereas only two lobes are present 
in Mode 30. The presence of multiple smaller lobes 
will offer lower resistance to collapse and vice-versa. 
Hence, Mode 30 for A48P2 has a higher collapse load 
than Mode 23. Thus, the above mode is considered for 
further study on the effect of imperfection on collapse 
load. The surface irregularities are considered to be 
imperfections in the PPS. It is found from the analysis 
that the imperfection below 50 % of shell thickness 
does not influence the collapse load of the structure. 
Any imperfections greater than 90 % of shell thickness 
are not acceptable as per manufacturing standards. 
Hence, the imperfection levels considered for this 
analysis are 50 %, 70 %, and 90 % of the thickness of 
the PPS.
It is inferred from Fig. 11 that there is no 
significant influence of imperfection on collapse 
load of PPS made of IS2062. However, the influence 
of imperfection on the collapse load of PPS made 
of A48P2 is considerable, as shown in Fig. 12. It 
is inferred from Fig. 9 to 12 that the influence of 
mode shape and geometrical imperfection on load-
deformation characteristics with material A48P2 
is minimal. In contrast, the above influence is 
pronounced with material IS2062, which is attributed 
to the variation in slope in the non-linear zone of the 
stress-strain curve. 
It is also found that the collapse load is the lowest, 
corresponding to an imperfection level of 90 %, and 
thus the above imperfection level is considered. In 
order to determine the collapse load for the generated 
design points, the buckling mode shape (Mode 23) 
and the geometrical imperfection level of 90 % are 
considered for further investigation.
3.2 Experimental Validation of Collapse Load on PPS
An experimental investigation was done on the pump 
penetration shell. The PPS prototype model taken for 
experimentation has the diameters of the cylindrical 
part and conical skirt as 180 mm and 264 mm, 
respectively, with 3 mm thickness as shown in Fig. 
7. The PPS was mounted in the fixture, and load was 
gradually applied on PPS using UTM until it attained 
permanent deformation. The geometrical shapes of 
PPS before and after collapse are shown in Figs. 13 
and 14, respectively.
Fig. 11.  Effect of imperfection on collapse load of PPS made of IS2062
 
Fig. 12.  Effect of imperfection on collapse load of PPS made of A48P2

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
337 Reliability-Based Design Optimization of Pump Penetration Shell Accounting for Material and Geometric Non-Linearity
Since, the numerically predicted Mode 23 matches 
with the experimentally observed mode of collapse in 
the conical region of the PPS specimen, a comparison 
of the load-deformation characteristics corresponding 
to Mode 23 is presented in Fig. 15. Considering the 
numerically predicted load-displacement curve, the 
collapse load of the structure is found to be 122 kN by 
using the twice elastic slope method. In contrast, the 
experimental collapse was identified as 129 kN by the 
snap-through point in the experimental curve. Hence 
the deviation in collapse load found from numerical 
and experimental methods is 6 %. Furthermore, the 
validated numerical methodology is extended for 
predicting the collapse load for varying parameters of 
flange angle and flange thickness and subsequently 
for reliability-based design optimization of the PPS.
3.3  Design of Experiments
The design of experiments is a systematic method 
to determine the factors affecting a process and the 
output of that process. This information is needed to 
manage process inputs in order to optimize the output. 
Response surface methodology (RSM) is used to 
formulate the design of experiments for developing 
a regression equation to predict the collapse load of 
PPS, which accounts for the relationship between the 
flange angle and flange thickness. The flange angle 
is limited to 30° and flange thickness to 3 mm due 
to manufacturing limitations of PPS. The minimum, 
medium, and maximum values of the two design 
parameters are given in Table 4.
Table 4.  Design optimization parameters
Parameter
Minimum 
(-1)
Medium 
(0)
Maximum 
(1)
Angle, θ [deg] 0 15 30
Thickness, t [mm] 1 2 3
The load-deflection curves for all randomly 
generated experiments are obtained by elasto-plastic 
Fig. 13.  PPS specimen before collapse
Fig. 14.  PPS specimen after collapse
Fig. 15.  Comparison of collapse load of PPS specimen

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
338 Gautham, V. – Prabhu Raja, V. – Ebron Shaji, G.T. – Mohanraj, S. – Thyla, P.R.
It is observed that the load-carrying capacity of 
PPS is increasing with an increase in flange angle and 
flange thickness, and also follows the same pattern 
in the contour plot as shown in Figs. 16 and 17 for 
the PPS made of IS2062 and A48P2, respectively. 
Considering the flange angle to be constant, it is found 
that every 0.1 mm increment of flange thickness will 
contribute to around a 5 % increase in collapse load. 
Similarly, keeping the thickness constant, every 1°
 
increment in flange angle will contribute to around a 
4 % increase in collapse load for PPS made of IS2062 
and A48P2.
3.4  Reliability-Based Design Optimization of PPS
The random variables considered for design 
optimization are the flange angle and flange thickness 
(Table 5), which follow normal distribution within a 
range of 15°±15° and 2 mm±1 mm respectively.
The equations for the load function obtained 
based on the results of collapse load for the materials 
IS2062 and A48P2 are given by,
  
 
Rt
tt
 
 
43519 35580 23510
10005 1345 9155
22
() ()
() () () ,

 (12)
 
Rt
tt
 

38114 30803 22008
6649 3434 9420
22
() ()
() () () ,

 (13)
where θ and t are the flange angle and flange 
thickness, respectively. Eqs. (14) and (15) are the limit 
state equations for the materials IS2062 and A48P2, 
respectively.
 
gt
tt
  










43519 35580 23510
10005 1345 9155
22

 S S, (14)
 
gt
tt S
  










38114 30803 22008
6649 3434 9420
22

 ,, (15)
 

 gt
g
S
g
t
S
2
2
2
2
2


















, (16)
where and are the standard deviations of flange 
angle and flange thickness of the PPS respectively.
Considering Eqs. (8) to (11), (14) and (16) as input 
functions for reliability-based optimization with 
constraints ( β > 3; 0° ≤ θ ≤ 30°; 1 mm ≤ t ≤ 3 mm), 
a MATLAB program is written to obtain optimum 
parameters of PPS that will maximize the collapse 
load with the minimal possible volume. Eqs. (8) to 
(11) account for volume, while Eqs. (14) and (16) 
analysis considering Mode 23 and imperfection level 
of 90 %. Mode 23 is considered since the collapse 
load corresponding to the above mode is the minimum 
compared to the other modes under consideration. 
Using the twice elastic slope method, the collapse load 
is estimated for the design combinations as shown in 
Table 5.
Table 5.  Generated design points and corresponding collapse load 
of PPS
Angle 
[deg]
Thickness 
[mm]
Collapse load [N]
IS2062 A48P2
0 0 42,500 37,220
-1 -1 5,500 5,500
0 1 71,250 67,220
1 -1 55,000 46,070
1 1 120,000 107,500
0 -1 23,570 20,350
-1 1 33,880 34,000
1 0 95,000 80,000
0 0 42,500 37,220
0 0 42,500 37,220
0 0 42,500 37,220
-1 0 17,140 17,140
Fig. 16.  Contour plot of collapse load [N] vs. thickness [mm], angle 
[deg] of material IS2062
Fig. 17.  Contour plot of collapse load [N] vs. thickness [mm],  
angle [deg] of material A48P2

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
339 Reliability-Based Design Optimization of Pump Penetration Shell Accounting for Material and Geometric Non-Linearity
pertain to collapse load. The program is executed 
using a MA TLAB optimization tool based on a genetic 
algorithm. The output of optimization is the optimum 
values of the flange angle and flange thickness for the 
PPS as given in Table 6.
Table 6.  Optimized results for PPS
Material IS2062 Material A48P2
Angle [deg] Thickness [mm] Angle [deg] Thickness [mm]
19 1.6 14 2.1
It is inferred from Fig. 18 that the collapse load 
of PPS is 4 and 4.5 times more than the operating load 
of 10 kN considering IS2062 and A48P2, respectively. 
The PPS made of IS2062 has a better load-carrying 
capacity than that made of material A48P2. The flange 
angle is found to contribute significantly to the load 
carrying capacity of PPS. Even though the thickness 
of PPS made of IS2062 is comparatively lesser than 
that made of A48P2, the collapse load is higher.
4  CONCLUSIONS
The reliability-based design optimization of pump 
penetration based on non-linear static analysis has 
been carried out, and the resultant optimum values are 
obtained for the given objective and constraints. The 
material non-linearity and the geometric imperfection 
have been taken into account. A genetic algorithm-
based optimization technique is adopted for the 
reliability-based design of the pump penetration shell. 
A study on the influence of buckling mode in 
predicting the collapse load reveals that the maximum 
deviations in collapse load are 1.3 % and 7.6 %, 
respectively, for PPS made of IS2062 and A48P2. 
It is inferred that the collapse load of PPS made of 
IS2062 is not dependent on the buckling mode shape 
when compared with the PPS made of A48P2. Based 
on the optimized results, the PPS made of IS2062 is 
found to have an 11 % higher load-carrying capacity 
with 21 % less material volume than the PPS made 
of A48P2. There is a significant improvement in the 
load-carrying capacity of the optimized geometry of 
PPS by 54 % when compared to PPS with 0° flange 
angle made of IS2062 and A48P2.
5  ACKNOWLEDGENENTS
The research described in this paper was financially 
supported by the Board of Research in Nuclear 
Science (BRNS), Mumbai, India (36(2)/14/39/2014-
BRNS/1179).
The authors are indebted to Board of Research 
in Nuclear Science (BRNS), Mumbai and Indra 
Gandhi Centre for Atomic Research (IGCAR), 
Kalpakkam, Tamil Nadu for their support throughout 
the investigation of this project work. 
6  NOMENCLATURES
g limit state function, [-]
R capacity, [-]
S demand, [-]
l length of the beam, [mm]
w width of the beam, [mm]
h height of the beam, [mm]
t thickness of the beam, thickness of the flange, 
[mm]
I
xx
 moment of inertia about x-axis, [mm
4
]
I
yy
 moment of inertia about y-axis, [mm
4
]
Fig. 18.  Comparison of collapse load of optimized geometry of PPS

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)6, 331-340
340 Gautham, V. – Prabhu Raja, V. – Ebron Shaji, G.T. – Mohanraj, S. – Thyla, P.R.
P
x
 load acting on x-axis, [N]
P
y
 load acting on y-axis, [N]
s
R
 standard deviation of yield strength, [MPa]
s
x
 standard deviation of load acting in x-axis, [N]
s
y
 standard deviation of load acting in y-axis, [N]
µ mean, [-]
σ standard deviation, [-]
β reliability index, [-]
V
1
 volume of hollow cylinder, [mm
3
]
V
2
 volume of truncated hollow cone, [mm
3
]
R
1
 outer radius of hollow cylinder, [mm]
R
2
 radius of hollow cylinder, [mm]
r
1
 outer radius of truncated hollow cone, [mm]
r
2
 radius of truncated hollow cone, [mm]
h
y
 height of hollow cylinder, [mm]
h
c
 height of truncated hollow cone [mm]
θ flange angle, [deg]
S
θ
 standard deviation of flange angle, [deg]
S
t
 standard deviation of flange thickness, [mm]
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