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STATISTICAL ANALYSIS OF RANDOM EXPLOSION PARAMETERS AT VARIOUS
LEVELS OF DAMAGE TO STRUCTURES AND BUILDINGS
Tursunov Kadirjon Mukhammadjonovich
PhD in Technical Sciences, Professor, Head of the Department
Institute of Communications and Information Technologies, Ministry of Defense of the Republic of
Uzbekistan
Makhmudov Nemadulla Akhmatovich
Candidate of Technical Sciences, Professor of the Department
Academy of the Armed Forces of the Republic of Uzbekistan
Mamatkulov Alisher Azamatovich
Researcher Institute of Communications and Information Technologies,
Ministry of Defense of the Republic of Uzbekistan
Ummatkulov Farrukh Murodjonovich
Doctoral student of the Armed Forces Academy of the Republic of Uzbekistan
Abstract:
This article examines the issue of determining the resource capacity of structures and
buildings using statistical methods. Based on experimental data, mathematical modeling was used to
study the durability of objects and their condition under the influence of internal and external forces.
The assessment of the degree of damage to buildings and structures under the impact of explosive
substances was carried out using the Gaussian normal distribution. Statistical parameters such as the
coefficient of variation, arithmetic mean, variance, and standard deviation were determined. The
calculation results showed that the mathematical and statistical analysis of random explosion
processes corresponds to the experimental data.
Keywords:
statistical analysis, mathematical modeling, structural durability, Gaussian normal
distribution, coefficient of variation, arithmetic mean, variance, standard deviation, explosive
processes, building damage.
Using statistical methods, any experimental results can be mathematically modeled. The essence of
mathematical statistics lies in determining all parameters of the probability distribution functions of
random processes (phenomena). To find these parameters and their values, integral calculus and the
solution of differential equations are usually required. Such problems are addressed within the field
of mathematical analysis.
When determining the resource capacity of structures based on experimental data, the mathematical
expectation
�(�)
, the arithmetic mean
�
, and the behavior of structures and buildings under the
influence of external and internal forces (earthquakes, explosions, loads, pressure, etc.) are analyzed.
In particular, their oscillations or damages of varying degrees are considered, expressed through
deviations from the center of gravity
�(�)
variance and the standard
deviation ( ).
The resource is an indicator that determines how long a structure (building) can be operated and
how long it can continue functioning without interruption. The resource diagnosis is expressed
through the function
�
−��
. The condition of structures damaged as a result of a random explosion
has been proven to depend on the parameters of the explosive material. However, the degree of
correlation between the constituent components of explosives and their statistical parameters (
�(�)
,
х
,
�
6
z
coefficients of variation) is not always precisely determinable. The coefficient of variation
�
6
, expressed as a percentage, represents the ratio of the standard deviation to the arithmetic mean
and is determined by the formula:
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�
�
=
�(�)
�
.
If the mass of the explosive substance
�
�
increases and the distance between the charge and the
object
�
becomes relatively smaller (closer), then the pressure variation coefficient will be higher.
In all statistical distributions (Gaussian, Rayleigh, Student's t-distribution, Fisher-Snedecor,
Weibull-Gnedenko, etc.), three main parameters must be determined: the mathematical expectation
�(�) = �
�
, the variance of the random variable
�(�)
, and the standard deviation ( ). Among
them, the Gaussian normal distribution stands out due to its high accuracy in diagnosing the degree
of damage to structures and buildings, as well as in scientific research of their physical and
mechanical properties [1-3]. In a series of studies, experiments were conducted in which pressure
was recorded at various distances from the explosion epicenter. The impact of the explosion was
analyzed using a numerical model and subjected to statistical analysis [4].
For random TNT (trinitrotoluene) charges with a mass of
� = 10, . . 30
kg, the correspondence of
building damage points to the Gaussian normal distribution is scientifically analyzed. During the
calculations, the coefficient of variation was assumed to be
�
�
= 0,2
and was considered constant
for a series of random explosions.
The density (differential) function of the normal distribution can be expressed in the following form:
� � =
1
�(�) 2�
⋅ exp −
(�−�
�
)
2
2�
2
(�)
(1)
Here:
( ) — differential (density) function of the Gaussian distribution by mass;
�
�
� = ��
�
— standard deviation;
�
�
— unknown mathematical expectation,
= 2.71... — base of the natural logarithm.
Table 1
№
N
um
be
ro
f
ch
ar
ge
s(
�
)
Ch
ar
ge
m
as
s
�
(к
г)
Ra
nd
om
pr
ob
ab
ili
ty
�
�
�
=
4�
+
16
�
�
�
�
2
�(�)
�(�
2
)
�(�)
�(�)
�(�)
1. 1
20
0.05
1
1
3.
80
16
.1
0
21
.2
0
26
.5
6
5.
2
2. 2
24
0.10
2
4
3. 6
28
0.30
3
9
4. 3
32
0.15
4
16
5. 7
26
0.35
5
25
6. 1
40
0.05
6
36
If this statistical distribution is close to the mean value, the approximate equality
ℎ�(�)
holds. More
precise criteria corresponding to empirical and theoretical distribution laws will be proposed in the
future.
Table 2
�
1
2
3
4
5
6
�
0,05
0,10
0,30
0,15
0,35
0,05
�
2
1
4
9
16
25
36
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Substituting the variable into the formula
� = 4� + 6
, we write the statistical distribution for
�
and
�²
. Thus:
� � = 0,05 + 2 ∙ 0,10 + 3 ∙ 0,30 + 4 ∙ 0,15 + 5 ∙ 0,35 + 6 ∙ 0,05 = 3,80
� �
2
= 0,05 + 0,40 + 2,70 + 2,40 + 8,75 + 1,80 = 16,1
� � = 4 ∙ � � + 6 = 4 ∙ 3,8 + 6 = 21,2
� �
2
=
�
2
+12�+36
16
=
1
16
� �
2
−
3
4
� � +
9
8
= 16,1
� �
2
− 12� � + 36 = 16,1 ∙ 16
� �
2
= 257,6 + 12 ∙ 21,2 − 36 = 476
� � = � �
2
+ � �
2
= 476 − 449,44 = 26,56
� � = 5,15 ≈ 5,2
From this, it follows that:
� � =
1
� 2�
∙ exp −
(�−�(�))
2
2�
2
.
If
(�−31,2)
5,15
= �
, то
� � =
1
5,15 2�
∙ �
−
�2
2
= 0,19�
�
where
�
�
=
1
2�
∙ �
−
�2
2
The values of the function
�
�
– are provided in the second table. Using these values, we construct
the third table.
Table 3
№
�
�
�
�
� �
ℎ� �
1.
10
-2,15
0,039
0,0075
0,0303
2.
14
-1,38
0,154
0,029
0,117
3.
18
-0,62
0,331
0,063
0,252
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4.
22
0,15
0,396
0,075
0,301
5.
26
0,92
0,262
0,050
0,200
6.
30
1,69
0,096
0,038
0,1538
Conclusion
The degree of damage to buildings and structures was analyzed using mathematical and statistical
methods. A diagram was constructed to show the relationship between the mass of explosive
substances in the range of [10...30] kg and their quantity (frequency). The analysis demonstrated
that in the case of random explosions with a charge mass ranging from 10 to 30 kg and a number of
1 to 7, the points of destruction of buildings and structures follow the normal Gaussian distribution.
The results of the mathematical and statistical analysis confirmed the correspondence between the
experimental data and theoretical models.
References:
1. Mamatkulov A.A., Kodirov A.A., Makhmudov N.A., Kurbanbaev M.Sh., Tursunov K.M.
Mathematical modeling of strength levels and probability analysis of structural element damage in
buildings under random explosions. *Journal of Mechanics*, No. 3. Tashkent, 2024. pp. 114–119.
2. Belov N.N. et al. Calculation of reinforced concrete structures under explosive and impact loads.
*Northampton – Tomsk*, 2004. 465 p.
3. Belov N.N., Yugov N.T. et al. Calculation of the strength of steel-concrete columns under
explosive and impact loads. *Bulletin of TASU*, No. 2, 2007. pp. 132–138.
4. Mkrtichev O.V. Reliability of multi-element rod systems in engineering structures. Doctoral
dissertation in technical sciences. *Moscow State University of Civil Engineering*. Moscow, 2000.
324 p.
