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ISSN : 1738-1894(Print)
ISSN : 2288-5471(Online)

Journal of Nuclear Fuel Cycle and Waste Technology Vol.20 No.2 pp.133-149
DOI : https://doi.org/10.7733/jnfcwt.2022.011

Design Optimization of an Impact Limiter Considering Material Uncertainties

Jongmin Lim*

, Woo-Seok Choi

Korea Atomic Energy Research Institute, 111, Daedeok-daero 989beon-gil, Yuseong-gu, Daejeon 34057, Republic of Korea

^* Corresponding Author.
Jongmin Lim, Korea Atomic Energy Research Institute, E-mail: jmlim86@kaeri.re.kr, Tel: +82-42-868-2564

Received October 27, 2021 ; Revised November 29, 2021 ; Approved February 9, 2022

Abstract

The design of a wooden impact limiter equipped to a transportation cask for radioactive materials was optimized. According to International Atomic Energy Agency Safety Standards, 9 m drop tests should be performed on the transportation cask to evaluate its structural integrity in a hypothetical accident condition. For impact resistance, the size of the impact limiter should be properly determined for the impact limiter to absorb the impact energy and reduce the impact force. Therefore, the design parameters of the impact limiter were optimized to obtain a feasible optimal design. The design feasibility criteria were investigated, and several objectives were defined to obtain various design solutions. Furthermore, a probabilistic approach was introduced considering the uncertainties included in an engineering system. The uncertainty of material properties was assumed to be a random variable, and the probabilistic feasibility, based on the stochastic approach, was evaluated using reliability. Monte Carlo simulation was used to calculate the reliability to ensure a proper safety margin under the influence of uncertainties. The proposed methodology can provide a useful approach for the preliminary design of the impact limiter prior to the detailed design stage.

Key Words : Transportation cask , Drop test , Impact limiter , Optimization , Material uncertainty

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1. Introduction

According to IAEA Safety Standards [1-2], 9 meters free drop tests are performed in order to evaluate structural integrity of a transportation package in a hypothetical accident condition. To reduce impact force delivered to package containments, an impacts limiter is usually equipped to the transportation package due to its absorbing performance. To maintain the structural integrity in the accident condition, it is important to determine proper dimensions of the impact limiter so that the impact limiter smoothly absorbs the impact energy. Impact resistance is usually commensurate with the size of the impact limiter, and the size of the impact limiter is proportional to production cost and handling efficiency. It indicates that there is a trade-off between the absorbing performance and the cost in the design of the impact limiter. In order to obtain a design solution seeking the minimized cost and satisfying the required performance, Choi and Seo [3] proposed the simple sizing optimization technique for the impact limiter. In addition to Choi and Seo’s approach, several improvements are developed in this paper: 1) crushed volume calculation according to all drop angles to find the drop angle causing maximum damage, 2) evaluation of specific absorbable energy according to a grain angle of a wood material, 3) definition of various objective functions for the design optimization, and 4) probabilistic design to consider influences of uncertainties in the impact limiter.

The optimization technique has been widely utilized in nuclear engineering [4-6]. Many optimization models of those studies are based on an implicit function including a computational analysis for the calculation of objective and constraint functions. Considering that one evaluation of the implicit function corresponds to one structural analysis in the optimization process, heavy computational costs might be required during the optimization design stage. Since the computational impact analysis of a metal cask [7-9] requires considerable computation time, it is difficult to apply the optimization technique with the implicit function for the design of the impact limiter. The proposed technique evaluates design feasibility using explicit functions, indicating that computational analysis is not included in this work. Therefore, design results can be readily and quickly obtained with negligible computational costs. Furthermore, the obtained design contains the proper conservatism evaluated using a stochastic approach under consideration of several uncertainties. For this reason, the design obtained from the proposed technique can be a good stepping-stone for a preliminary design process so that the whole time and cost for the design of the impact limiter are efficiently reduced.

This paper is organized as follows. In Section 2, a calculation of impact and absorbable energy of the impact limiter is addressed. Section 3 addresses the proposed optimization technique for the impact limiter and demonstrates results of the proposed design technique. In Section 4, stochastic uncertainties in designing the impact limiter are defined and the probabilistic design procedure is proposed with its results. Finally, Section 5 gives conclusions.

2. Impact and Absorbable Energy of Impact Limiter

In this section, impact and absorbable energy of the impact limiter are analytically calculated with several assumptions.

The transportation cask and impact limiter considered in this work are shown in Fig. 1. Two types of wood materials are used for the design of the impact limiter in this study: Red and Balsa woods. These woods are commonly used as an absorbing material. Red wood is relatively stiffer than Balsa wood. The absorbing performance of Red wood is higher than that of Balsa wood, but Red wood produces higher impact force delivered to the cask. In this study, Red wood with high stiffness is placed in the outer in order to absorb the impact force concentrated on the corner part of the impact limiter in a corner drop. Balsa wood is placed in the center in order to reduce the strong impact force caused by the large area of contact between the impact limiter and the target in an end-on drop.

Fig. 1

Dimensions of the impact limiter.

2.1 Impact and Absorbable Energy

Impact energy corresponding to a 9 meters free drop is equal to potential energy as

E_{Imp} = m_{T} g H

(1)

where E_Imp is the impact energy, m_T is the total mass of a transportation cask including the impact limiter, g is the gravitational acceleration, and H is the drop height (= 9 meters).

An absorbable energy of the impact limiter can be written as

E_{Abs} = e_{Red} (θ) V_{Red} (θ) + e_{Balsa} (90^{°} - θ) V_{Balsa} (θ)

(2)

In Eq. (2), θ is the drop angle defined in Fig. 2, E_Abs(θ) is the absorbable energy with respect to the drop angle, and e_Red and e_Balsa are the specific absorbable energy per unit volume of the Red and Balsa wood, respectively. V_Red and V_Balsa are the maximum crushed volume of parts of the Red and Balsa wood, respectively. The maximum crushed volume of each part will be dealt with in Section 2.2.

Fig. 2

Drop Angle θ.

Wood materials are used in the impact limiter for the absorbing behavior, and stress-strain curves of the used wood materials are shown in Fig. 3. As shown in Fig. 3, the mechanical properties of the wood materials strongly depend on loading directions.

Fig. 3

Stress-Strain curve of wood used in the impact limiter.

In order to obtain an off-axis s-s curve of the wood material, this study utilizes Hankinson’s formula given by [10]

σ_{δ} = \frac{σ_{V} σ_{H}}{σ_{V} s i n^{2} δ + σ_{H} c o s^{2} δ}

(3)

where σ_δ is the off-axis stress, σ_V and σ_H is the stress horizontal and vertical to the grain orientation. The off-axis angle δ is defined as an angle between the grain orientation of the wood and axis along the loading direction. The offaxis stress-strain relation of the wood material is obtained by applying Hankinson’s formula in Eq. (3) to the entire s-s curves. Specific absorbable energy per unit volume with respect to the off-axis angle δ is calculated by integrating the off-axis stress σ_δ(ε) over the strain ε. The specific absorbable energy of the Balsa and Red wood with respect to the off-axis angle δ is shown in Fig. 4.

Fig. 4

Specific absorbable energy according to grain angle.

2.2 Maximum Crushed Volume

Maximum crushed volume can be analytically calculated under several assumptions as below:

Assumption 1. The transportation cask translates vertically, indicating that the drop angle is constantly maintained during impact.
Assumption 2. Rebound of the transportation cask is not considered.
Assumption 3. The impact limiter can be crushed until the crush limitation plane shown in Fig. 5.

Fig. 5

Crush limitation plane (CLP).

Assumption 1, which enables to analytically calculate the crushable volume, means that the cask does not rotate about a contact point during the impact. Considering that a major factor of the cask rebound and an excessive deformation over the crush limitation plane is an insufficient capacity of the absorbable energy of the impact limiter, the assumption 2 and 3 naturally become valid with the proper design of the impact limiter which means that the absorbing performance of the impact limiter can endure the whole impact energy.

For the calculation of the maximum crushed volume with respect to the drop angle, two functions are defined in advance as follows

\begin{array}{l} f (r, p, q) = \int_{q}^{p} \int_{- \sqrt{r^{2} - x^{2}}}^{\sqrt{r^{2} - x^{2}}} (α x + β) d y d x \\ = {[- \frac{2 α {(r^{2} - x^{2})}^{3 / 2}}{3} + β (x \sqrt{r^{2} - x^{2}} + r^{2} t a n^{- 1} \frac{x}{\sqrt{r^{2} - x^{2}}})]}_{x = q}^{x = p} \end{array}

(4)

where α = tan θ, β = H₁− R₁tan θ

\begin{array}{l} g (r, h, p) = \int_{p}^{r} \int_{- \sqrt{r^{2} - x^{2}}}^{\sqrt{r^{2} - x^{2}}} (h) d y d x \\ = h {[x \sqrt{r^{2} - x^{2}} + r^{2} t a n^{- 1} \frac{x}{\sqrt{r^{2} - x^{2}}}]}_{x = p}^{x = r} \end{array}

(5)

Using Eq. (4) and (5), partial volumes of the cylinder shown in Figs. 6 and 7 can be readily calculated.

Fig. 6

Partial volume calculated from Eq. (4).

Fig. 7

Partial volume calculated from Eq. (5).

Formulas for the maximum crushed volume can be defined to six categories according to the values of a and b shown in Fig. 8. The values of a and b can be obtained as

a = - α R_{o} + β, b = α R_{o} + β

(6)

Fig. 8

Definition of a and b.

The each formula for the maximum crushed volume of the impact limiter is derived as below

1) a ≥ 0

$\begin{array}{l} V_{T} = \int_{- R_{o}}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (α x + β) d y d x = f (R_{o}, R_{o}, - R_{o}) \\ V_{B} = \int_{- R_{2}}^{R_{2}} \int_{- \sqrt{R_{2}^{2} - x^{2}}}^{\sqrt{R_{2}^{2} - x^{2}}} (α x + β) d y d x = f (R_{2}, R_{2}, - R_{2}) \\ V_{R} = V_{T} - V_{B} \end{array}$

where V_T is the total crushed volume, that is V_T = V_R + V_B, V_B is the volume of part of the Balsa wood, and V_R is the volume of part of the Red wood. A end-on drop is a special case of this category when a = H₁.

2) a < 0 & b < (H₁ + H₂)

$\begin{array}{l} V_{T} = \int_{C}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (α x + β) d y d x = f (R_{o}, R_{o}, c) \\ V_{B} = {\begin{array}{l} \int_{- R_{2}}^{R_{2}} \int_{- \sqrt{R_{2}^{2} - x^{2}}}^{\sqrt{R_{2}^{2} - x^{2}}} (α x + β) d y d x = f (R_{2}, R_{2}, - R_{2}) & , c \leq - R_{2} \\ \int_{c}^{R_{2}} \int_{- \sqrt{R_{2}^{2} - x^{2}}}^{\sqrt{R_{2}^{2} - x^{2}}} (α x + β) d y d x = f (R_{2}, R_{2}, C) & , - R_{2} < c \leq R_{2} \\ 0 & , c > R_{2} \end{array} \end{array}$

where $c = - \frac{β}{α}$ is shown in Fig. 9(a)

Fig. 9

Definition of values c, d, e, and k.

3) a < 0 & (H₁ + H₂) ≤ b < (H − 2H₂)

$\begin{array}{l} V_{T} = \int_{c}^{d} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (α x + β) d y d x + \int_{d}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (H_{1} + H_{2}) d y d x \\ = f (R_{o}, d, c) + g (R_{o}, R_{o}, d) \\ V_{B} = {\begin{array}{l} \int_{c}^{R_{2}} \int_{- \sqrt{R_{2}^{2} - x^{2}}}^{\sqrt{R_{2}^{2} - x^{2}}} (α x + β) d y d x = f (R_{2}, R_{2}, c) & , c \leq R_{2} \\ 0 & , c > R_{2} \end{array} \end{array}$

where $d = \frac{H_{1} + H_{2} - β}{α}$ is shown in Fig. 9(b)

4) a < 0 & (H − 2H₂) ≤ b < (H + 2H₁)

$\begin{matrix} V_{T} = \int_{c}^{d} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (α x + β) d y d x + \int_{d}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (H_{1} + H_{2}) d y d x \\ + \int_{e}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (α x + β) d y d x - \int_{e}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (H + 2 H_{2}) d y d x \\ = f (R_{o}, d, c) + g (R_{o}, H_{1} + H_{2}, R_{o}, d) + f (R_{o}, R_{o}, e) \\ - g (R_{o}, H - 2 H_{2}, e) \\ V_{B} = {\begin{array}{l} \int_{c}^{R_{2}} \int_{- \sqrt{R_{2}^{2} - x^{2}}}^{\sqrt{R_{2}^{2} - x^{2}}} (α x + β) d y d x = f (R_{2}, R_{2}, c) & , c \leq R_{2} \\ 0 & , c > R_{2} \end{array} \end{matrix}$

where $e = \frac{H - 2 H_{2} - β}{α}$ is shown in Fig. 9(c)

5) a < 0 & b > (H + 2H₁)

$\begin{matrix} V_{T} = \int_{c}^{d} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (α x + β) d y d x + \int_{d}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (H_{1} + H_{2}) d y d x \\ + \int_{e}^{k} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (α x + β) d y d x + \int_{k}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (H + 2 H_{1}) d y d x \\ - \int_{e}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} (H - 2 H_{2}) d y d x \\ = f (R_{o}, d, c) + g (R_{o}, H_{1} + H_{2}, R_{o}, d) + f (R_{o}, R_{o}, e) \\ + g (R_{o}, H + 2 H_{1}, k) - g (R_{o}, H - 2 H_{2}, e) \\ V_{B} = {\begin{array}{l} \int_{c}^{R_{2}} \int_{- \sqrt{R_{2}^{2} - x^{2}}}^{\sqrt{R_{2}^{2} - x^{2}}} (α x + β) d y d x = f (R_{2}, R_{2}, c) & , c \leq R_{2} \\ 0 & , c > R_{2} \end{array} \end{matrix}$

where $k = \frac{H + 2 H_{1} - β}{α}$ is shown in Fig. 9(d)

6) b = (H + 2H₁)

$V_{R} = \int_{R_{1}}^{R_{o}} \int_{- \sqrt{R_{o}^{2} - x^{2}}}^{\sqrt{R_{o}^{2} - x^{2}}} 2 (H_{1} + H_{2}) d y d x = g (R_{o}, 2 (H_{1} + H_{2}), R_{1}), V_{B} = 0$

Category 6) is a horizontal drop as shown in Fig. 9(e).

3. Design Optimization of Impact Limiter

A structural optimization consists of three components: an objective which the design seeks, constraints which the design satisfies, and design variables which the design decides. In this section, these components are formulated for an optimal design of the impact limiter.

3.1 Design Constraint

In order to obtain a feasible design solution, the design should satisfy constraints which can be evaluated through a structural analysis. A design criterion for this design is that the absorbable energy of the impact limiter should be greater than the impact energy in order to prevent the impact from being directly delivered to the cask without absorbing the energy. If the absorbable energy is less than the impact energy, the “locking-up” phenomenon occurs in a large area of the absorbing material, and the stiffness of the absorbing material dramatically increases. This can lead to a rapid increase in the impact force and deceleration. Furthermore, if the impact energy exceeds the limit of the absorbing capacity, the absorbing materials might be fractured. In this case, the cask can impact with the target directly, which means that the design of the impact limiter obviously fails. Because the absorbable energy has different values according to the drop angle, the design constraint can be formulated as

E_{Imp} \leq \underset{0 \leq θ \leq 90^{°}}{m i n} [E_{Abs} (θ)]

(7)

Equation (7) indicates that the design feasibility is evaluated at the drop angle with a maximum damage. Equation (7) can be re-written as a standard form for the design constraint as

G_{1} = \underset{0 \leq θ \leq 90^{°}}{m a x} [E_{Imp} - E_{Abs} (θ)] \leq 0

(8)

At every iteration during the optimization process, the drop angle with the maximum damage should be found to calculate the constraint function in Eq. (8), which means that a sub-problem is constructed within the optimization loop. After finding the drop angle with the maximum damage, the sensitivities of the constraint function are calculated using the finite difference method with the fixed drop angle.

In the end-on drop, the strong impact force is produced at the beginning of the impact due to the large area of the contact between the impact limiter and the target. The impact force produced at the beginning of the impact in the end-on drop can be assumed as

\begin{matrix} F_{Imp} = F_{Red} + F_{Balsa} \\ = σ_{Red}^{m a x} A_{Red} + σ_{Balsa}^{m a x} A_{Balsa} \end{matrix}

(9)

where F_Imp is the impact force delivered to the cask in the end-on drop. $σ_{Red}^{m a x}$ and $σ_{Balsa}^{m a x}$ is the maximum stress in stress-strain curves shown in Fig. 3, and A_Red and A_Balsa is the bottom area of the each wood part of the impact limiter.

In the end-on drop, the deceleration for rigid motion of the cask is assumed as

G^{Dec} = \frac{F_{Imp}}{m_{Total}}

(10)

where G^Dec is the deceleration for rigid motion of the cask in the end-on drop, and m_Total is the total mass of the cask including the impact limiter.

The deceleration of the cask in Eq. (10) in the end-on drop can be limited by the constraint as

G_{2} = G^{Dec} - G^{Target} \leq 0

(11)

where G^Target is the target deceleration of the cask in the end-on drop, which is the condition for the design to satisfy. The target deceleration is defined by the user, and the structural analysis for the detailed model of the transportation cask might be required to define the target deceleration before determining the dimensions of the impact limiter. For example, G^Target can be determined as the maximum inertial force in the vertical direction that the confinement structure or the internal contents such as spent nuclear fuels can withstand in order to maintain the structural integrity.

The constraint G₂ limits the excessive increase of the bottom area of the impact limiter, thereby preventing the high impact force in the end-on drop.

3.2 Formulation of Design Optimization of Impact Limiter

Among the dimensions shown in Fig. 1, W, H₁ and H₂ are selected as design variables. According to the design variables, the total volume of the impact limiter and the constraint function are calculated during the optimization process. A standard form for the design optimization can be formulated as

Find d = [W, H₁, H₂]

minimize f (d)

subject to $G_{1} = \underset{0 \leq θ \leq 90^{°}}{m a x} [E_{Imp} (d) - E_{Abs} (d, θ)] \leq 0$

\begin{matrix} G_{2} = G^{max} - G^{Target} \leq 0 \\ 0 < W \leq W^{Up}, 0 < H_{1} \leq H_{1}^{Up}, 0 < H_{2} \leq H_{2}^{Up} \end{matrix}

(12)

In Eq. (12), d is the vector of the design variables, f (d) is the objective function, and W^Up, H₁^Up and H₂^Up are the upper limits of the design variables, which means that the design variables cannot exceed the upper limits.

According to a goal of the design, various objective functions can be defined as following;

1) Minimization of a specific dimension for the efficient handling and transportation

$minimize f_{1} (d) = ω_{1} W + ω_{2} H_{1} + ω_{3} H_{2}$

(13)

where ω_i is a weighting factor for a designer to control priority of i^th variable.

2) Minimization of design change from the initial design

$\begin{matrix} minimize f_{2} (d) = ω_{1} {(W^{Initial} - W)}^{2} + ω_{2} {(H_{1}^{Initial} - H_{1})}^{2} \\ + ω_{3} {(H_{2}^{Initial} - H_{2})}^{2} \end{matrix}$

(14)

where W^Initial, $H_{1}^{Initial}$ and, $H_{2}^{Initial}$ are initial dimensions of the impact limiter.

3) Minimization of the amount of materials used to produce the impact limiter

$minimize f_{3} (d) = V_{IL}$

(15)

where V_IL is the volume of the impact limiter.

4) Maximization of the dissipated energy within limits to the amount of the used materials

$maximize f_{4} (d) = \underset{0 \leq θ \leq 90^{°}}{m i n} [E_{Abs} (d, θ)]$

(16)

For the objective 4), the below constraint is additionally considered for the limits of the amount of the used materials.

subject to V_{T} - V_{Allowable} \leq 0

(17)

where V_Allowable is the limit of the amount of the used materials.

3.3 Design Feasibility of Initial Design

Table 1 shows the initial design of the impact limiter. In order to evaluate the design feasibility based on Eq. (7), the absorbable energy was calculated as shown in Fig. 10.

Table 1

Design specification of an initial design

Fig. 10

Design feasibility of the initial design.

As illustrated in Fig. 10, the impact energy exceeds the absorbable energy at the drop angle from about 40° to 85°, which indicates that the impact energy exceeds the limit of the absorbing capacity. In this case, the structural integrity of the transportation cask cannot be guaranteed since the impact force is directly delivered to the cask body. This infeasible design can be improved using the design optimization which will be addressed in the following section.

3.4 Design Optimization of Impact Limiter

Based on the formulation in Eq. (12), the design optimization of the impact limiter is performed. For the optimization algorithm, the sequential quadratic programming (SQP) method [11] is utilized in this study. SQP is one of popular iterative methods for the nonlinear constrained optimization. In SQP, the design is sequentially updated using gradients of the objective and constraint functions until stopping criteria are satisfied. Results of the design optimization of the impact limiter are summarized in Fig. 11 and Table 2.

Fig. 11

Optimal design of impact limiter.

Table 2

Optimal design of impact limiter

In Table 2, while the initial design does not satisfy the design criterion, all of the optimal designs were obtained as a feasible design. In design 1, the design variable W was minimized since the highest weighting factor is given to the design variable W in Eq. (13). Because design 2 seeks the minimized change of the design, the amount of the design change was minimized. The objectives of the design 3 and design 4 are the minimization of the impact limiter volume and the maximization of the absorbable energy, respectively. Even though the dimensions are considerably changed, design 3 and design 4 are properly improved in terms of their objectives. In design 3, the feasible design was achieved using the least amount of the absorbing materials. In design 4, the absorbable energy was maximized within the allowable volume of the impact limiter limited to 1.40 m³. In design 2 and 3, the absorbable energy is almost equal to the impact energy, indicating no margin of safety for the design criteria of Eq. (7).

For the initial and optimum design, the computational analysis is performed using Abaqus/Explicit [12]. For stability of the computational analysis, an element deletion option is used, indicating that an element is eroded when an equivalent plastic strain of the element reaches to limitation. The limit value of the equivalent plastic strain is set to 0.62 in this work. The cask drops at the height of 9 meters, and the drop angle is 45° in this case. Finite element modeling can include the steel sheet as well as the wood material. Furthermore, the dynamic behavior of the cask by the 9 meters drop test is simulated without the assumptions stated in Section 2.2. On the other hand, the evaluation using the proposed method does not include the effect of the steel sheet, and several assumptions to derive the explicit function are introduced in this work. These points lead to a more conservative evaluation of safety in the proposed design methodology.

Results of the computational analysis are illustrated in Fig. 12. For the initial design, the elements that no longer absorb the impact energy are eroded, and the cask body directly impacts with the target due to the absence of elements with the absorbable energy. It indicates that the absorbable energy is not sufficient to absorb the whole impact energy, so the considerable impact force is delivered to the main body. As illustrated in Fig. 13(a), the reaction force dramatically increases when the right front of the main body reaches to the ground. This insufficient performance of the impact limiter could lead to serious deterioration of the integrity of the cask. It is shown in Fig. 12(b) that the total absorbable energy at this drop angle is enough to afford the total impact energy. The reaction force shown in Fig. 13(b) also indicates that the impact limiter smoothly absorbs the impact energy.

Fig. 12

Results of FEA for the initial and optimal design.

Fig. 13

Reaction force for the initial and optimal design.

4. Probabilistic Design of Impact Limiter

It is known that a wood material has high material uncertainties due to various factors such as moisture content, knotholes, bulk density, and so on [13]. Since these uncertainties in an engineering system could cause unintended degradation of system performance, safety margins should be properly determined in order to consider influences of these uncertainties. In a conventional design method, a safety factor has been used to provide the safety margin, but the safety factor is empirically determined in most cases. If experience about successful designs is insufficient, the design might undergo many trials and errors in the engineering design. For this reason, a probabilistic approach is applied to the design of the impact limiter in this work. The uncertainties included in the engineering system are assumed to be random variables in this work. Design feasibility based on statistical theory is evaluated by calculating a probability of failure.

4.1 Definition of Uncertainties Included in Design of Impact Limiter

In a statistics-based design, factors causing the uncertainties are assumed as random variables with the probability distribution and coefficient of variation. The random variables considered in this study are summarized in Table 3.

Table 3

Random variables in design of impact limiter

4.2 Calculation of Failure Probability Using MCS

Reliability can be defined as a probability to satisfy the design constraint in Eq. (8) under the influences of the uncertainties as

R = Pr [G = E_{Imp} - E_{Abs} (θ^{*}) \geq 0] = \int_{G \geq 0} f_{x} (x) d x

(18)

where R is defined as the reliability, θ* is the drop angle maximizing the damage to the impact limiter such that $\underset{0 \leq θ \leq 90^{°}}{m a x} [E_{Imp} - E_{Abs} (θ)]$ , and f_x(x) is a joint distribution function of the random variables. The probability of failure is then defined as

P_{f} = 1 - R = Pr [G = E_{Imp} - E_{Abs} (θ^{*}) < 0] = \int_{G< 0} f_{x} (x) d x

(19)

where P_f is the probability of failure, which means the probability of violating the design constraint in Eq. (8). In order to evaluate the probability of failure, Monte Carlo simulation (MCS) [14] is utilized in this study. In MCS, sufficient samples are generated, and an indicator function in Eq. (20) is then evaluated for the generated samples.

I_{Ω f} (x) = {\begin{array}{l} 1, & x \in Ω_{f} \\ 0, & otherwise \end{array}

(20)

where Ω_f ≡ {x : G(x) > 0}

When the indicator function in Eq. (20) is evaluated, it is quite inefficient to find the drop angle θ* for all samples. Therefore, the drop angle maximizing the damage is found at mean values of random variables, and the performance function G(x) in Eq. (8) is then evaluated using the drop angle θ* already found about the mean values. Using Eq. (20), Eq. (19) can be rewritten as

P_{f} = \int_{R^{NRV}} I_{Ω f} (x) f_{x} (x) d x = E [I_{Ω f} (x)] = \sum_{i = 1}^{N} \frac{I_{Ω f} (x_{i})}{N}

(21)

where N is the number of samples. For accurate estimation of the failure probability, 10⁷ samples are used in this work.

In the probabilistic design, the probabilistic constraint is defined instead of the original constraint in Eq. (8) as

P_{f} = Pr [G = E_{Imp} - E_{Abs} (θ^{*}) < 0] \leq P_{f}^{Target}

(22)

where $P_{f}^{Target}$ is the target probability of failure determined by a designer.

4.3 Sensitivity Analysis of Failure Probability

A gradient-based iterative optimization method such as SQP requires sensitivities of objective and constraint functions during the optimization process. In MCS, the sensitivity of the failure probability with respect to the mean values can be obtained [15] as

\begin{matrix} \frac{\partial P_{f}}{\partial μ_{j}} = \int_{R^{NRV}} I_{Ω_{f}} (x) \frac{\partial f_{x} (x)}{\partial μ_{j}} d x = \int_{R^{NRV}} I_{Ω_{f}} (x) \frac{\partial ln f_{x} (x)}{\partial μ_{j}} f_{x} (x) d x \\ = E [I_{Ω_{f}} (x) \frac{\partial ln f_{x} (x)}{\partial μ_{j}}] = \sum_{i = 1}^{N} [\frac{I_{Ω_{f}} (x_{i})}{N} \frac{\partial ln f_{x} (x_{i})}{\partial μ_{j}}] \end{matrix}

(23)

Since the probability density function is given, the sensitivity of the failure probability can be readily calculated during the calculation of the failure probability by MCS. Through Eq. (23), the sensitivity of the failure probability with respect to mean values is as follows

\frac{\partial P_{f}}{\partial μ} = {[\frac{\partial P_{f}}{\partial e_{Red}} \frac{\partial P_{f}}{\partial e_{Balsa}} \frac{\partial P_{f}}{\partial V_{Red}} \frac{\partial P_{f}}{\partial V_{Balsa}} \frac{\partial P_{f}}{\partial m_{T}}]}^{T}

(24)

Using a chain rule, the sensitivities with respect to the design variable d_j are calculated as

\begin{matrix} \frac{\partial P_{f}}{\partial d_{j}} = \frac{\partial P_{f}}{\partial e_{Red}} \frac{\partial e_{Red}}{\partial d_{j}} + \frac{\partial P_{f}}{\partial e_{Balsa}} \frac{\partial e_{Balsa}}{\partial d_{j}} + \frac{\partial P_{f}}{\partial V_{Red}} \frac{\partial V_{Red}}{\partial d_{j}} \\ + \frac{\partial P_{f}}{\partial V_{Balsa}} \frac{\partial V_{Balsa}}{\partial d_{j}} + \frac{\partial P_{f}}{\partial m_{T}} \frac{\partial m_{T}}{\partial d_{j}} \end{matrix}

(25)

4.4 Probabilistic Design of Impact Limiter

The optimization formulation for the probabilistic design of the impact limiter can be written as

Find d = [W, H₁, H₂]

minimize f (d)

subject to $P_{f} = Pr [\underset{0 \leq θ \leq 90^{°}}{m a x} [E_{Imp} - E_{Abs} (θ)] > 0] \leq P_{f}^{Target}$

0 < W \leq W^{Up}, 0 < H_{1} \leq H_{1} \leq H_{1}^{Up}, 0 < H_{2} \leq H_{2}^{Up}

(26)

Results of the probabilistic design according to target reliability are summarized in Fig. 14 and Table 4. The objective of the probabilistic design is selected to be the objective 2 in Eq. (14).

Fig. 14

Results of probabilistic design.

Table 4

Results of probabilistic design

In Table 4, the safety margin is defined as

Safety Margin = \frac{E_{Abs} (θ^{*})}{E_{Imp}}

(27)

As shown in Table 4, the desired level of reliability can be achieved using the probabilistic design. The higher reliability is required, the more conservative design is obtained. It is noted that the reliability defined in this study means the probability of satisfying the constraint in Eq. (8). It does not mean that the design violating the constraint cannot pass the regulatory that is 9 m drop test here. Furthermore, the analytical model in this study already includes several factors leading to the conservative design: 1) from the assumptions for the analytical calculation in section 2.2, and 2) from not considering the steel sheet in the impact limiter. Therefore, the design satisfying the above 99% reliability might be too conservative to reflect the actual uncertainties included in the engineering system. For this reason, the optimal design satisfying the 95% reliability is proposed in this study.

Fig. 15 illustrates computational impact analysis of the deterministic and probabilistic optimal design under the material uncertainty. The 3 σ worst case scenario is assumed in order to consider the material uncertainty, which means that degraded absorbable energy ê_Red is used in the impact analysis such that

Pr [X_{e_{Red}} \leq {\hat{e}}_{Red}] = Φ (- 3)

(28)

Fig. 15

Results of FEA for the deterministic and probabilistic optimal design.

where $X_{e_{Red}}$ is the random variable for the absorbable energy, and Φ(·) is the cumulative distribution function (CDF) of a standard normal variable.

In contrast to the results of the computational analysis without consideration of the material uncertainty, the deterministic optimal design cannot endure the impact under the influences of the material uncertainty. Therefore, the significant impact force is delivered to the main body of the cask as shown in Figs. 15(a) and 16(a). Since the safety margin is considered during the stochastic design optimization process, the impact limiter of the probabilistic optimal design smoothly absorbs the impact energy without dramatic increase of the impact force. Even though the probabilistic optimal design is obtained with the safety margin corresponding to 95% reliability level (1.645 σ), this design solution can afford up to 3 σ worst case due to several assumptions in the proposed design methodology which makes the safety evaluation more conservative.

Fig. 16

Reaction force of deterministic and probabilistic optimal design.

5. Conclusions

The probabilistic design methodology for an impact limiter was proposed in this work. Formulations to calculate the absorbable energy of the impact limiter were derived under several assumptions, which enables to evaluate design feasibility using an explicit function without a computational analysis. Several objective functions were also defined to seek various design goals. The design solution obtained by the proposed design optimization was verified using the computational impact analysis. For the initial design, the impact limiter was not able to absorb the whole impact energy, so the impact was considerably delivered to the main body of the cask. On the other hand, it was shown that the optimal design afforded the whole impact energy, because the design solution was obtained under the design criterion. Furthermore, the probabilistic design optimization was performed to ensure the proper safety margin under the uncertainties. It was shown that the probabilistic design solution held the design feasibility even in the worst case scenario. The proposed design methodology is based on explicit functions, which means the proposed method requires negligible computational costs. Repeated computational analyses during the design optimization can cause the excessive computational burden. Thus, the design obtained from the proposed technique can be a good stepping stone for the detailed design process including the computational analysis so that the whole time and cost for the design of the impact limiter is efficiently reduced. In the future work, the efficient design methodology including the computational impact analysis will be developed for the impact limiter.

Acknowledgements

This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry and Energy (MOTIE) of the Republic of Korea (No. 20201710200010 and No. 20211710200020). This work was partially supported by the Institute for Korea Spent Nuclear Fuel (iKSNF) and National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT, MSIT) (No. 2021M2E1A1085229).

Figures

Tables

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