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01-07 

1 

 

 

 

Article 

Reliability assessment and accelerated life testing 

in a metalworking plant 
Ovundah King Wofuru-Nyenke* 

 Department of Mechanical Engineering, Rivers State University, Rivers State, Nigeria 

A R T I C L E   I N F O 
 

Article history: 
Received 09 January 2024  
Received in revised form 
15 February 2024 
Accepted 23 February 2024 
 
Keywords:  
Reliability assessment, Life testing, Metalworking, 
Exponential model, Weibull model 
 
*Corresponding author 
Email address: 
ovundah.wofuru-nyenke@ust.edu.ng  
 
 
DOI: 10.55670/fpll.futech.3.3.1 
 

A B S T R A C T 
 

Reliability assessments are useful for determining how well products, systems, 
and services maintain their quality over the course of time and through various 
conditions. In this paper, a reliability assessment of a metalworking plant was 
conducted, as well as accelerated life testing of the plant's spot-welded and 
riveted products. The overall layout of the plant was complex, requiring the use 
of equations to calculate the reliability of stations being connected in series and 
parallel in order to determine the overall reliability of the system. Furthermore, 
equations for mean life and failure rate were used in determining the estimate 
of mean life for the tested components, as well as the rate at which the 
components fail, respectively. The results indicated that the metalworking plant 
had a reliability of 0.81 or 81%. Moreover, the results indicated that the failure 
data of the spot-welded products follow the exponential model, with the failure 
rate of the products being constant throughout the period under investigation. 
The failure data of the riveted products follow the Weibull model, increasing 
throughout the period under investigation. This study presents a procedure for 
aiding production and maintenance managers in conducting reliability 
assessments of their production systems. 

 

1. Introduction 

Metalworking plants are responsible for processing 
metals by shaping and reshaping them to create useful items, 
components, assemblies, and other large-scale structures. 
Metalworking is responsible for the production of large 
structures like buildings, ships, and bridges, as well as more 
precise engine parts and delicate items such as jewelry. The 
process has evolved from the prehistoric times of shaping 
metals using simple hand tools to more modern and highly 
technical processes [1]. The modern processes include 
forming through bulk forming processes or sheet and tube 
forming processes; cutting through milling, turning, 
threading, grinding, or filing; and joining through riveting, 
brazing, soldering, or welding. All these processes need high 
reliability in order to produce high-quality products 
consistently and on time. Reliability is the probability that a 
product, system, or service adequately performs its required 
function for a specific period of time, operating in a particular 
environment without failure [2]. Unreliable products, 
systems, and services can be caused by several reasons, such 
as the complexity of the system due to the increasing 
integration of mechanical, electronic, and software parts into 
systems and products. The complexity of these systems 
makes it impossible for them to run without multiple flaws 
being present [3-5]. Defective products and systems can lead 
to direct costs, such as product recalls, warranty issues, and 

legal obligations, as well as indirect costs, such as market 
share loss and customer relationship damage [6,7]. Reliability 
assessments are key for maintaining system stability, 
improving quality, and reducing losses in health, 
manufacturing, agricultural, and service provision systems. 
These assessments help identify sources of failure and aid in 
failure prevention and control [8]. It can be seen that with 
customers increased demand for high-quality and reliable 
products delivered on time, reliability assessments are an 
indispensable tool for production companies to meet 
customer demand. These companies need to be faster, better, 
and more economical than their competitors in meeting 
customer demand in order to thrive. Reliability assessment 
can be achieved through extensive testing using several 
techniques such as failure mode and effect analysis, Petri nets, 
fuzzy logic neural networks, etc. [9-11]. In cases where wrong 
or oversimplified reliability models are used for assessment 
and making decisions, system performance can be damaged, 
thereby affecting safety and security [12]. Reliability 
assessments are useful wherever there are products, systems, 
or services that provide the value of a specific quality in order 
to determine how well these products, systems, or services 
perform their function [13,14]. Life testing refers to 
experimental tests designed to ascertain the life expectancy 
of structures by testing the structure at specific stress 
conditions similar to those of normal operation conditions. 

 

 

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Open Access Journal 

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August 2024| Volume 03 | Issue 03 | Pages 01-07 

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OK. Wofuru-Nyenke /Future Technology                                                                                  August 2024| Volume 03 | Issue 03 | Pages 01-07 

2 

 

The process is aimed at measuring one or more reliability 
characteristics of experimental units under consideration. 
Most metalworking products available today are highly 
reliable with high mean times to failure due to substantial 
improvements in science and technology. Therefore, 
obtaining adequate lifetime distribution data and associated 
parameters in a timely manner using conventional life testing 
experiments is difficult. As a result of this, reliability analysts 
have resorted to accelerated life testing, where tested items 
are subjected to environmental conditions far more severe 
than normal operating conditions [15]. This causes the tested 
items to fail more quickly, drastically reducing the time 
required for the test and the number of items that need to be 
tested. Product failure can be due to careless planning, 
substandard raw materials, wear-out, or fatigue. Products put 
together by various mechanical means, such as welding and 
riveting, are prone to failure at joints, necessitating life testing 
of these products [16-18]. Many systems, such as 
manufacturing supply chains, service providers as well as 
agricultural systems, need reliability assessment in order to 
improve productivity and meet customer needs better [19, 
20]. Therefore, the aim of this research is to present a 
procedure for reliability assessment and product life testing, 
thereby aiding production and maintenance managers in 
implementing these concepts in their production systems. 
The following sections present the methods utilized as well as 
the results of the metalworking plant reliability assessment 
and accelerated life testing study. 

2. Methodology 

The reliability assessment was conducted in a 
metalworking plant that produces various sheet metal 
products. The metalworking plant consists of five sections: 
shearing sections, press shops, fabrication sections, paint 
shops, and assembly sections. At the shearing sections, mild 
steel sheets are cut into various sizes and shapes depending 
on the nature of the product to be manufactured. At the press 
shops, the processed materials from the shearing department 
undergo various changes in shape and dimensions through 
notching, bending, hole piercing, and embossment. At the 
fabrication sections, additional attachments are welded to 
processed materials from the press shop by spot welding. The 
materials from the press shop and fabrication department are 
moved to the paint shop, where they are hung on conveyors, 
spray painted, and dried in a special oven. In the assembly 
sections, all the individual parts of the products to be 
manufactured are fastened and assembled to create complete 
units by riveting. Therefore, the focus of this study is the 
calculation of the overall reliability of the entire 
metalworking plant, as well as accelerated life testing of spot 
welded and riveted components manufactured by the plant. 
This serves to gain an overall understanding of the reliability 
of the plant and develop expressions for predicting the failure 
rate and mean life of the manufactured components. 

2.1 Overall reliability of the metalworking plant 
The five sections of the metalworking plant consist of 

fourteen individual stations, which are connected in series 
and parallel. Therefore, the entire plant is a complex system, 
and the two basic equations for calculating the reliability of 
series and parallel systems can be combined to calculate the 
overall manufacturing plant reliability. The equation for 
calculating the reliability of a series system was obtained 
from Ebeling [21]: 

RS = ∏ Ri
n
i=1              (1) 

where RS is the reliability of the series system, and Ri is the 
reliability of the ith component. 
The equation for calculating the reliability of a parallel system 
was obtained from Ebeling [21]: 

RP = 1 − ∏ (1 − Ri)
n
i=1             (2) 

where RP is the reliability of the parallel system, and Ri is the 
reliability of the ith component. 
The procedure for evaluating the overall metalworking plant 
reliability is to replace the parallel sections having various 
individual reliabilities with an equivalent section having a 
single reliability, then evaluate the resulting series system, 
equivalent to the original system. 

2.2 Nonreplacement accelerated life testing 
The spot welded and riveted components were subjected 

to non-replacement accelerated life tests in order to develop 
the failure rate prediction expressions. The components were 
subjected to constant/static loading tests to assess how well 
the component would withstand a sustained load without 
failure. Failure, in this case, refers to the component breaking 
apart or separating at the joined spots. The two models 
investigated in the component life testing are the exponential 
model and the Weibull model. 

2.2.1 Exponential model 
The exponential model is suitable for describing the 

failure-time distribution of a component when the failure rate 
of the component is constant throughout the period under 
investigation. By the exponential model, the failure-time 
distribution of each component was obtained from Lawless 
[22] as: 

f(t) =  α ∙  e−αt              t > 0, where α >  0                (3) 

where α is the constant failure rate, and t is the observed 
failure times. 
If n components are put on test and life testing is discontinued 
after a fixed number of components have failed, r (𝑟 ≤  𝑛), 
and the observed failure times are t1 ≤ t2 ≤ ⋯ ≤  tr. The 

mean life of the component is 𝜇 =  
1

∝
. 

According to Lawless [22], the estimate of mean life can also 
be expressed as 

μ̂ =
Tr

r
              (4) 

where Tr is the accumulated life of the test until the rth failure 
occurs, and r is the number of failures. The failure rate is 

estimated by 1 μ̂⁄ . 

Also, from Lawless [22], the accumulated life to r failures for 
non-replacement tests is given by: 

 Tr = ∑ ti
r
i=1 + (n − r)tr           (5) 

To investigate if failure data follows the exponential model, a 
total time on test plot is made by plotting the total time on test 
until the ith failure, Ti, divided by the total time on test through 
the last (rth) observed failure, Tr, against i/r. If the plot follows 
a straight line along the 45-degree line, the failure data is 
exponential. However, if the plot is a curve above the 45-
degree line, the failure data follows an increasing hazard rate 
model, and the adequacy of the Weibull model can be checked. 

2.2.2 Weibull model 
When the failure rate of a component is not constant, 

but increasing and decreasing throughout a period under 
investigation, the Weibull model is more suitable for 
predicting the failure rate of such component. 



OK. Wofuru-Nyenke /Future Technology                                                                                  August 2024| Volume 03 | Issue 03 | Pages 01-07 

3 

 

According to Lawless [22], the Weibull distribution is given 
by: 

f(t) =  αβtβ−1e−∝tβ
           t > 0, where α > 0,   β > 0          (6) 

where f(t) is the probability density of the Weibull 
distribution at time t, α is the scale parameter and β is the 
shape parameter. 
From Johnson et al. [23], the mean of the Weibull distribution 
having the parameters α and β may be obtained by evaluating 
the integral: 

μ =  ∫ t ∙ αβtβ−1e−∝tβ
dt

∞

0
            (7) 

substituting 𝑢 =  𝛼𝑡𝛽 , we get 

μ = α
−1

β⁄
∫ u

1
β⁄

e−u∞

0
du            (8) 

The integral, ∫ 𝑢
1

𝛽⁄
𝑒−𝑢∞

0
𝑑𝑢, is the gamma function Γ (1 +  

1

𝛽
) 

evaluated at 1 + β−1, therefore the mean time to failure for the 
Weibull model is: 

μ =  α
−1

β⁄
Γ (1 + 

1

β
)           (9)  

From Johnson et al. [23], the equation for determining the 
shape parameter, β is: 

∑ 𝑡𝑖
𝛽

𝑙𝑛𝑡𝑖+(𝑛−𝑟)𝑡𝑟
𝛽

𝑙𝑛𝑡𝑟
𝑟
𝑖=1

∑ 𝑡
𝑖
𝛽𝑟

𝑖=1 +(𝑛−𝑟)𝑡𝑟
𝛽 − 

1

𝛽
−  

1

𝑟
∑ 𝑙𝑛𝑡𝑖

𝑟
𝑖=1 = 0       (10)  

where β is the shape parameter, r is the number of failures at 
which the test is terminated, n is the number of components 
being tested, tr is the time of the rth failure, ti is the time of 
the ith failure. 
Also from Johnson et al. [23], the equation for determining the 
scale parameter, α is: 

𝛼 =
1

1

𝑟
[∑ 𝑡𝑖

𝛽
+(𝑛−𝑟)𝑡𝑟

𝛽𝑟
𝑖=1 ]

          (11) 

where β is the shape parameter, r is the number of failures at 
which the test is terminated, n is the number of components 
put on the test, tr is the time of the rth failure, ti is the time of 
the ith failure. 
The α and β parameters are obtained by the maximum 
likelihood method. The method is implemented in Python 
programming language using the NumPy and SciPy libraries. 
From Johnson et. al [23], the Weibull failure-rate function is 
given by, 

z(t) =  αβtβ−1           (12) 

where 𝛼 is the scale parameter and 𝛽 is the shape parameter. 
According to Johnson et. al [23], the Weibull plot is a plot of 

ln ti versus Weibull score given by ln ln  
1

1−𝐹(𝑡𝑖)̂
, therefore: 

xi = ln ln 
1

1− F(ti)̂
                           (13)

     
and  

yi = ln ti           (14) 

From Johnson et. al [23], 𝐹(𝑡𝑖)̂ is given by  

𝐹(𝑡𝑖)̂ =
𝑖

𝑛+1
           (15) 

If the plotted points do not fall reasonably close to a straight 
line, the assumption that the underlying failure-time 
distribution is of the Weibull type is contradicted. 

3. Results and discussion 

This section presents the results of calculating the 
reliability of the entire metalworking plant as well as the 
mean life and failure rate values of the components in the 
non-replacement accelerated life testing study.  

3.1 Calculating the reliability of the metalworking plant 
The five sections of the metalworking plant consist of 

fourteen individual stations, which are connected in series 
and parallel. The shearing section consists of two shearing 
stations connected in parallel; the press shop section consists 
of three stations connected in parallel; the fabrication section 
consists of two stations connected in parallel; the paint shop 
section consists of four stations connected in parallel; and the 
assembly section consists of three stations connected in 
parallel. Therefore, the entire plant is a complex system, and 
the two basic equations for calculating the reliability of series 
and parallel systems were combined to calculate the overall 
manufacturing plant reliability. For resolving the parallel 
stations into equivalent single stations, the equation was 
used, while for resolving the series stations into equivalent 
single stations, the equation was used. Figure 1 shows the 
metalworking plant layout and the various reliabilities of the 
individual sections. 
The two individual parallel shearing stations can be replaced 
by a single shearing section having a reliability of  

Rshearing =  1 – [(1 –  0.8)(1 –  0.85)] = 0.97        

The three individual parallel press shop stations can be 
replaced by a single press shop section having a reliability of  

Rpress shop = 1 − [(1 − 0.65)2(1 − 0.7)] = 0.96        

The two individual parallel fabrication stations can be 
replaced by a single fabrication section having a reliability of  

Rfabrication = 1 − [(1 − 0.75)(1 − 0.7)] = 0.93  

The four (4) individual parallel paint shop stations can be 
replaced by a single paint shop section having a reliability of  

Rpaint shop = 1 − [(1 − 0.60)2(1 − 0.65)(1 − 0.7)] = 0.98  

The three individual parallel assembly stations can be 
replaced by a single assembly section having a reliability of  

Rassembly = 1 − [(1 − 0.75)(1 − 0.55)(1 − 0.60)] = 0.96  

Finally, the resulting series system, equivalent to the original 
system, has a reliability of  

Rplant = Rshearing  ×  Rpress shop  ×  Rfabrication ×

Rpaint shop × Rassembly  

= 0.97 × 0.96 × 0.93 × 0.98 × 0.96 = 0.81  

From the foregoing, the overall reliability of the metalworking 
plant can be improved by replacing certain stations with low 
reliabilities by several similar stations connected in parallel. 
This is because if the manufacturing plant consists of a 
number (n) of similar independent stations connected in 
parallel, the high-reliability parallel stations will make up for 
the low-reliability parallel stations, thereby increasing the 
reliability of the section. Also, the section will fail to function 
only if all n stations fail.  

 

 

 



OK. Wofuru-Nyenke /Future Technology                                                                                  August 2024| Volume 03 | Issue 03 | Pages 01-07 

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Figure 1. Metalworking plant reliability 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 1. Data from spot-welded components life test 

i 
Failure times, ti 

(hours) 

Accumulated life, Ti 

(hours) 
𝑇𝑖

𝑇𝑟
⁄  𝑖

𝑟⁄  𝐹(𝑡𝑖)̂ 𝑦𝑖  𝑥𝑖 

1 186 11160 0.185474 0.1 0.02 5.23 -4.10 

2 231 13815 0.229599 0.2 0.03 5.44 -3.40 

3 501 29475 0.489862 0.3 0.05 6.22 -2.99 

4 541 31755 0.527755 0.4 0.07 6.29 -2.69 

5 626 36515 0.606864 0.5 0.08 6.44 -2.46 

6 701 40640 0.67542 0.6 0.09 6.55 -2.27 

7 771 44420 0.738242 0.7 0.11 6.65 -2.10 

8 961 54490 0.905601 0.8 0.13 6.87 -1.96 

9 1,031 58130 0.966096 0.9 0.15 6.94 -1.83 

10 1,071 60170 1 1 0.16 6.98 -1.72 

 



OK. Wofuru-Nyenke /Future Technology                                                                                  August 2024| Volume 03 | Issue 03 | Pages 01-07 

5 

 

3.2 Calculating mean life and failure rate of 
manufactured components 
During the non-replacement accelerated life test, nw = 60 

units of components joined by spot welding and nr = 60 units 
of components fastened by riveting were tested, and the test 
was truncated after r = 10 items failed from each of the spot-
welded and riveted components. Table 1 shows the results 
from the spot-welded components life test.  
While Table 2 shows the results from the riveted components 
life test. Figure 2 is an exponential model plot of Scaled Time 
on Test (Ti/Tr) versus i/r for the spot-welded components. 
From Figure 2, the plot seems to follow a straight line along 
the 45-degree line, therefore the failure data is exponential. 
From equation (4) the mean life of the spot-welded 
components is calculated as  

μ̂ =
60,170

10
= 6,017 hours  

and the failure rate of the spot-welded components is: 

Failure rate =  
1

6,017
= 0.00017 failures per hour  

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

This is equivalent to 0.17 failure per thousand hours. 
Figure 3 is an exponential model plot of Scaled Time on Test 
(Ti/Tr) versus i/r for the riveted components. From Figure 3, 
the plot is a curve above the 45-degree line, therefore, the 
failure data follows an increasing hazard rate model, and the 
adequacy of the Weibull model needs to be checked. Figure 4 
is a Weibull model plot of the riveted components data. From 
Figure 4, the majority of the plotted points fall reasonably 
close to a straight line, therefore the assumption that the 
underlying failure-time distribution is of the Weibull type 
cannot be contradicted.  Therefore, the maximum likelihood 
estimators are computed using Python and the values are α = 
405.86 and β = 2.72. Hence, from equation (9) the mean time 
to failure for the riveted components is: 

μ = (405.86)−1
2.72⁄ Γ (1 +

1

2.72
) = 0.098 hours  

Also, from equation (12), the failure rate function is given by 

Z(t) = (405.86)(2.72)t2.72−1 = 1103.94t1.72  

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Table 2. Data from riveted components life test 

i 
Failure times, ti 

(hours) 

Accumulated life, Ti 

(hours) 
𝑇𝑖

𝑇𝑟
⁄  𝑖

𝑟⁄  𝐹(𝑡𝑖)̂ 𝑦𝑖  𝑥𝑖 

1 
171 10260 0.304957793 0.1 0.02 5.14 -4.10 

2 
205 12266 0.364582095 0.2 0.03 5.32 -3.40 

3 
216 12904 0.383545357 0.3 0.05 5.38 -2.99 

4 
251 14899 0.442842706 0.4 0.07 5.53 -2.69 

5 
326 19099 0.56767923 0.5 0.08 5.79 -2.46 

6 
381 22124 0.65759125 0.6 0.10 5.94 -2.27 

7 
391 22664 0.67364166 0.7 0.11 5.97 -2.10 

8 
491 27964 0.831173463 0.8 0.13 6.20 -1.96 

9 
561 31604 0.939365117 0.9 0.15 6.33 -1.83 

10 
601 33644 1 1 0.16 6.40 -1.72 

 



OK. Wofuru-Nyenke /Future Technology                                                                                  August 2024| Volume 03 | Issue 03 | Pages 01-07 

6 

 

 
Figure 2. Exponential model plot for spot-welded components 

 
 

 
Figure 3. Exponential model plot for riveted components 

 

 
Figure 4. Weibull model plot for riveted components 

 
 

4. Conclusion 

Regular reliability assessments on manufacturing plants 
are very crucial for improving the overall productivity of the 
plants. Hence, production managers are usually faced with the 
responsibility of determining the probability that products, 
systems and services will carry out their functions adequately 
for a specific period of time without failure. Therefore, the aim 
of this research was to present a procedure for reliability 
assessment and product life testing, thereby aiding 
production and maintenance managers in implementing 

these concepts in their production systems. This study 
assessed the reliability of a complex metalworking plant 
having stations connected in series and parallel. Though, the 
reliability of the plant was calculated to be 0.81, the overall 
reliability of the metalworking plant can be improved by 
replacing certain stations with low reliabilities with several 
similar stations connected in parallel. This is because if the 
manufacturing plant consists of a number (n) of similar 
independent stations connected in parallel, the high 
reliability parallel stations will make up for the low reliability 
parallel stations, thereby increasing the reliability of the 
section. Also, the section will fail to function only if all n 
stations fail. Accelerated life testing of spot-welded and 
riveted components was conducted, and it was shown that the 
spot-welded components failure data followed the 
exponential model. However, the failure data of the riveted 
components showed an increasing hazard rate, thereby 
following the Weibull model. This study presents a procedure 
for aiding production and maintenance managers in 
conducting reliability assessments of their production 
systems. 

Ethical issue 
The author is aware of and complies with best practices in 
publication ethics, specifically with regard to authorship 
(avoidance of guest authorship), dual submission, 
manipulation of figures, competing interests, and compliance 
with policies on research ethics. The author adheres to 
publication requirements that the submitted work is original 
and has not been published elsewhere. 

Data availability statement 
Datasets analyzed during the current study are available and 

can be given following a reasonable request from the 

corresponding author. 

Conflict of interest 

The author declares no potential conflict of interest. 

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