Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 11, No. 3, 2024 

 

244 

Research	on	Ice	Load	Optimization	based	on	Monte	
Carlo	Algorithm	and	Finite	Element	Analysis	
Chang Liu 1, *, Mengfei Wang 2, Yixiang Zhou 1 

1 School of Computer and Information Technology, Beijing Jiaotong University, Beijing, China 
2 School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China 

* Corresponding author: Chang Liu (Email: 2783194591@qq.com) 

 

Abstract: This paper mainly focuses on the ice bearing capacity and ice vehicle layout optimization problems for in-depth 
research. By modeling the ice thickness and analyzing the ice stresses, the load-bearing capacity and safety margins of the ice 
were assessed. Firstly, the research considers the freezing time, environmental conditions, load factor, strength, temperature and 
salt influence coefficient, and obtains the exact value of the bearing capacity of the ice under different conditions and the safe 
distance of the ice truck. Secondly, Monte Carlo algorithm and finite element analysis are used to simulate the random 
distribution of the crowd and the influence of the ice truck on the ice surface stress, which provides an important reference for 
the safety management and planning of the ice surface activities. The research results show that under certain conditions, the ice 
surface can withstand a certain number of people, and the layout of the ice vehicle needs to consider the safe distance from the 
ice hole to ensure that the ice surface does not break and does not exceed the maximum allowable stress. These research results 
are of great significance for ensuring the safety of ice activities and improving the efficiency of ice utilization.  

Keywords: Finite Element Analysis; Monte Carlo Algorithm; Load-bearing Capacity. 
 

1. Introduction 
In winter, ice and snow activities become the first choice 

for leisure and entertainment for many citizens, and the 
natural ice rink attracts many tourists with its unique charm. 
However, the carrying capacity of the ice surface is the key 
factor to ensure the safety of these activities. With the change 
of climate and the diversification of people's activities, the 
research on the carrying capacity of ice surface becomes 
particularly important. 

In this paper, based on the perspective of spatial distance 
optimization, finite element analysis, Monte Carlo algorithm 
and genetic algorithm are used to deeply study the carrying 
capacity of ice rink. Firstly, we set up the ice thickness model 
and the maximum weight model to scientifically quantify the 
bearing capacity of the ice surface. On this basis, through the 
research of the ice bearing capacity model under unit area, the 
random distribution of the population on the ice surface and 
its influence on the stress on the ice surface are simulated. The 
research results of this paper will provide scientific basis for 
the security of ice and snow activities, and provide a new idea 
and method for the research of ice bearing capacity. 

2. Ice Rink Carrying Capacity Model  

2.1. Ice Thickness Model  
According to the freeze degree-day method, the thickness 

of ice is related to freezing time and environmental conditions. 
Use the following model to calculate the thickness of ice [1]: 

√                      (1) 

Where,	  is ice thickness and  is freezing degree-day, can 
be calculated by formula (2).   is a typical coefficient, 
usually related to ice sheet conditions, such as whether there 
is snow cover, whether there is wind blowing, etc. [2]. 

∑                     (2) 

Where,	  is the reference temperature, in this paper, the 
freezing point is chosen 0℃ as the reference temperature; 
, 1,⋯ , 		 is the average temperature of the 	  th 

measurement day. 
Figure 1 is a plot of ice surface thickness over time. 

2.2. Model of Maximum Load Bearing Weight 
It is assumed that the bearing capacity of natural ice can be 

regarded as the theory of thin plate on elastic foundation, 
whose thickness to width ratio is very small, floating on the 
water body, while the bearing capacity of ice and the thickness 
of ice  square relationship to deduce the maximum bearing 
capacity of ice , apply the following formula [3]: 

9.81                   (3) 

Which, 	   represents the load factor,  represents the 
strength and considers the ice crack coefficient ( 1.75 for 
no cracks, 2.0  for cracks) take temperature influence 
coefficient ,  represents the salt influence coefficient. 

The load bearing curve of the ice surface varies with the 
date, as shown in Figure 2. 

The yellow line in Figure 2 shows 1.7205	the load-
bearing capacity of the ice, the blue line shows 2.70 	the 
load-bearing capacity of the ice, and the green line shows the 
force of 5,000 people on the ice. During this period, the 
yellow and blue lines are above the green line, indicating that 
the ice can support 5,000 people during this time. 

2.3. Model of Ice Bearing Capacity Per Unit 
Area 

2.3.1. Model Proposed 
Consider the load-bearing capacity of the local ice surface, 

i.e. the maximum number of people that can be 
accommodated per unit area of the ice surface. After the water 
surface is frozen, the ice sheet floats on the water surface, and 
the ice sheet is an elastic, homogeneous and isotropic material 



 

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plate, which acts on the elastic basis [4]. Under the action of 
load, the ice has deflection deformation and internal tensile 
and compressive stress. However, because the ice is a brittle 
material, under the action of allowable load P. The ice strain 

is small, which conforms to Hooke's Law. Moreover, since the 
compressive strength of the ice is obviously greater than the 
tensile strength, the ice failure is determined by the ultimate 
tensile stress at the bottom of the layer. Application formula: 

 
Figure 1. Curve of ice thickness with time 

 

 
Figure 2. Curve of ice surface load bearing with date 

 

0.275 1 log	         (4) 

Where, the Young's modulus represents 	 Pa ,  the 
foundation modulus , g,  represents the area of the 
load area, when 1.724  , 	 √1.6 0.675  . 
when ⩾ 1.724 , . 

The stress value of the distance from the center of the load 
area  where the uniformly distributed load is applied on the 
infinite ice sheet is: 

0.249
/
exp	

.
            (5) 

Where, 	   represents the characteristic length and the 

calculation method is	
/

. 

2.3.2. Algorithm Proposed 
To realize the random distribution of the crowd, we will 

adopt Monte Carlo algorithm. The algorithm generates many 
random points in a given area through random sampling to 
simulate the distribution of the crowd. The steps of the 
algorithm are shown in Table 1: 

 

Table 1. Steps of Monte Carlo algorithm 
Procedure Description 

Step 1: Define 
the variables 

Determine the number of simulations and the 
number of people required. 

Step 2: Random 
sampling 

Coordinate points are randomly selected on 
the unit area. 

Step 3: 
Simulation 

process 

For each random point, run the simulation 
process in the unit area, using the hook stock 
calculate the distance from that point to the 

center.

Step 4: Output 
the result 

The output result is a list of distances from 
the center for each person in the simulated 

population distribution. 

2.3.3. Model Solving 
Formula (4) represents the maximum bearing capacity of 

the ice surface when tourists gather together , and formula 
(5) represents the bending stress of the ice surface when a 
tourist stands x	 meters away from the center of the area. 
When 	tourists in a unit area are randomly distributed, the 
bending stress of the ice surface  at that moment is compared 
with the maximum bearing capacity of the ice surface  to 
determine how many people can be accommodated in the unit 



 

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area. Compared with the actual maximum number of people 
that can be accommodated in the unit area, judge whether the 
ice will break if part of 5,000 people gathers together, as 
shown in Figure 3: 

 
Figure 3. Relationship curve between thickness and maximum 

number of people per unit area 
 

It can be seen from Figure 3 that the ice can withstand the 
crowd gathering per unit area during the Open Day. 

3. Ice Surface Stress Optimization 
Model based on Finite Element 
Analysis 

The layout of the skaters in the rink is optimized to 
minimize the proximity of the single and 10-person skaters to 
the ice hole, while ensuring that the ice does not break and 
does not exceed the maximum allowable stress of the ice. 

3.1. Model Building 
Stress distribution often becomes complicated when ice 

caves are present, especially around the edges and near the 
holes. A detailed description of the specific stress distribution 
around the hole has not been found in the available literature. 
Finite element analysis is an effective tool for dealing with 
such complex geometry. This method can accurately simulate 
the stress distribution around the hole by dividing the problem 
area into multiple small units and applying the appropriate 
approximation function to each unit. 

(1) Optimization objectives: 
Optimize the closest distance between the single ice truck 

and the hole, and optimize the closest distance between the 
10-person team and the hole. 

(2) Constraints: 
First, consider the situation of an individual ice truck. The 

goal is to maximize the distance  between the vehicle 
and the ice hole while sliding around the area with the hole, 
ensuring that it does not cause the ice to break, so: 

only ice_hole               (6) 

To move on to the next stage, we need to consider the 
situation of a convoy of ten vehicles coasting around the area 
with a hole. The total length of this convoy is , where,  
is the number of vehicles in the convoy,	  is the length of a 
single vehicle. Since the mass of the convoy is evenly 
distributed, and the motion of the entire convoy is regarded as 
the motion of the center of mass. Our goal is to ensure that the 
entire convoy does not slide into the ice hole, so the distance 
between the center of mass and the center of the ice hole more  

should be at least: 

more /2 ice_hole          (7) 

At the same time, ensure that the entire team maximizes its 
distance from the ice hole when sliding around the area with 
a hole, so that the critical distance between the team and the 
ice hole is expressed , so there are: 

more                       (8) 

Due to the existence of the load of the ice hole, the stress 
distribution on the ice surface must be within an acceptable 
range. The stress of the ice sheet when a single ice vehicle is 
sliding to a certain position	 only   and the stress of the ice 
sheet when the whole team is sliding to a certain more  
position should meet: 

only 	 max                 (9) 

more 	               (10) 

Which,  represents the maximum bearing capacity of 
the ice to a single ice vehicle, ' represents the maximum 
bearing capacity of the ice to the ice vehicle fleet. 

The large data prediction model for the user's electricity 
consumption is implemented in the Clementine software.  

3.2. Model Solving 
After finite element analysis, we can obtain the maximum 

stress of the ice surface under the pressure exerted by a single 
ice skater or a team of 10, as shown in Figure 4. According to 
the literature, the maximum allowable stress of ice is 0.98912 
MPa [5]. Currently, however, it is still not possible to 
determine the optimal distance for a single skater or a team of 
10. Therefore, we will use genetic optimization algorithm to 
optimize the distance between the single ice skater or the team 
of 10 people to the ice hole.  

 

 
Figure 4. Ice surface stress distribution at 1 m from the ice hole by 

an ice truck 

3.3. Result Analysis 
Through matlab solution, we obtained the safety range of 

the single ice truck and the 10-person team, as shown in 
Figure 5 and Figure 6. 

By setting the minimum moving radius of the ice vehicle 
as 0.1 meters and the maximum moving radius as 10 meters, 
we get that when the distance between the ice vehicle and the 
center of the hole is greater than or equal to 3.550705 meters, 



 

247 

the single ice vehicle is in a safe state. It is shown in Figure 5. 
Considering the length and constraints of the convoy, and if 
the total length of the 10-person convoy is 7 meters, the 
minimum moving radius of the convoy is 3.6 meters and the 

maximum moving radius is 10 meters. We can get that when 
the distance between the convoy and the center of the hole is 
greater than or equal to 8.737758 meters, the convoy is in a 
safe state. It is shown in Figure 6. 

 

 

Figure 5. Simulation results of single ice vehicle Figure 6. Simulation results of a 10-person team

3.4. Model Checking 
After obtaining the safe range, use Solidworks software to 

build a solid model to simulate the distribution of people on 
the ice rink. The model was then meshed with the help of its 

built-in Simulation plug-in. Through the analysis of the grid 
model, the maximum stress value in the ice rink is obtained, 
and the rationality of the results is tested. The results obtained 
are shown in Figure 7 and Figure 8 below. 

 

 

Figure 7. Diagram of static stress analysis for one person Figure 8. Static stress analysis diagram of vehicle fleet

According to the above analysis results, it is found that the 
maximum stress value of the ice rink remains below the 
allowable stress value within the minimum distance range 
obtained. 0.98912MPa . This conclusion not only validates 
the reasonableness of the safety range set by us, but also 
further proves the structural stability and safety of the ice rink 
under this condition. 

4. Conclusion 
This paper makes an in-depth study on the bearing capacity 

of ice surface, and reveals the spatio-temporal variation and 
influencing factors of the bearing capacity of ice surface by 
constructing the ice thickness model and the maximum 
bearing weight model. At the same time, a detailed ice surface 
model is established by finite element analysis (FEA), which 
can accurately simulate the ice surface pressure generated by 
the ice vehicle moving on the ice surface and calculate the 
maximum stress value in the area around the ice cave. With 
the distance between the ice truck and the ice cave as the 
objective function, the spatial distance of the ice truck is 
optimized by using genetic algorithm. Finally, it is concluded 
that when the distance between the single ice truck and the 
center of the hole is greater than or equal to 3.550705 meters, 

and the distance between the 10-person team and the center 
of the hole is greater than or equal to 8.737758 meters, it is in 
a safe state. It provides theoretical support for ice rink 
management and safety evaluation. 

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