# Liquid Hydrogen Tank Problem

The 5-dimensional liquid hydrogen tank problem is a reliability analysis benchmark problem (Bichon et al., 2011). The problem consists in quantifying the failure probability of a liquid hydrogen fuel tank on a space launch vehicle. The structure of the tank is subjected to stresses caused by ullage pressure, head pressure, axial forces due to acceleration, and bending and shear stresses caused by the weight of the fuel.

## Description

Three different failure modes exist for the tank:

• Von Mises stress failure (P_{VM})
• Isotropic strength failure (P_{IS})
• Honeycomb buckling failure (P_{HB})

The limit state function of the overall system is then defined as:

g(\mathbf{x}) = \min{(P_{VM}, P_{IS}, P_{HB})},

where:

P_{VM} = \frac{8.4 \cdot 10^4 \cdot t_{\text{plate}}}{\sqrt{N_x^2 + N_y^2 - N_x N_y + 3 N_{xy}^2}} - 1,
P_{IS}=\frac{8.4 \cdot 10^4 \cdot t_{\text{plate}}}{|N_y|} - 1,
\begin{split} P_{HB} = & 0.847 + 0.96 x_1 + 0.986 x_2 - 0.216 x_3 \\ & + 0.077 x_1^2 + 0.11 x_2^2 + 0.007 x_3^2 \\ & + 0.378 x_1 x_2 - 0.106 x_1 x_3 - 0.11 x_2 x_3, \end{split}

and:

• x_1 = 4 (t_{\text{plate}} - 0.075)
• x_2 = 20 (t_h - 0.1)
• x_3 = -6 \cdot 10^3 (\frac{1}{N_{xy}} + 0.003)

while the input variables of the performance function are \mathbf{x} = \{t_{\text{plate}}, t_{h}, N_x, N_y, N_{xy}\}.

The failure event is defined as g(\mathbf{x}) \leq 0 and the failure probability P_f = \mathbb{P}[g(\mathbf{x}) \leq 0].

## Inputs

The five input variables are modeled as independent Gaussian random variables.

No Variable Distribution Parameters Description
1 t_{\text{plate}} Gaussian \mu_{t_\text{plate}} = 0.07433
\sigma_{t_\text{plate}} = 0.005
Plate thickness
2 t_{h} Gaussian \mu_{t_h} = 0.1
\sigma_{t_h} = 0.01
Honeycomb thickness
3 N_x Gaussian \mu_{N_x} = 13
\sigma_{N_x} = 60
4 N_y Gaussian \mu_{N_y} = 4751
\sigma_{N_y} = 48
5 N_{xy} Gaussian \mu_{N_{xy}} = -648
\sigma_{N_{xy}} = 11

## Reference values

Some reference values for the failure probability P_f from the literature are shown in the table below. Note that the values from Bichon et al. (2011) are average values over 20 replications.

Method N \hat{P_f} \text{CoV}[\hat{P_f}] Source
MCS (LHS) 10^4 7.0 \times 10^{-4} - Bichon et al. (2011)
MCS (LHS) 10^5 6.92 \times 10^{-4} - Bichon et al. (2011)
MCS (LHS) 10^6 6.97 \times 10^{-4} - Bichon et al. (2011)
MCS 10^6 7.1 \times 10^{-4} 3.75\% UQLab v1.2.1

## Resources

The vectorized implementation of the limit state function for the liquid hydrogen tank problem in MATLAB as well as the script file with the model and probabilistic inputs definitions for the function in UQLAB can be downloaded below:

uq_liquidHydrogenTank.zip (3.0 KB)

The contents of the file are:

Filename Description
uq_liquidHydrogenTank.m vectorized implementation of the limit state function of the liquid hydrogen tank problem in MATLAB
uq_Example_liquidHydrogenTank.m definitions for the model and probabilistic inputs in UQLab
LICENSE license for the function (BSD 3-Clause)

## References

• B. J. Bichon, J. M. McFarland, and S. Mahadevan, “Efficient surrogate models for reliability analysis of systems with multiple failure modes,” Reliability Engineering & System Safety, vol. 96, no. 10, pp. 1386-1395, 2011. DOI:10.1016/j.ress.2011.05.008

In the variables input, may I know why only 5 variables are allowed?

How many additional variables do you want to vary? Which ones?
For sure you can vary additional variables in the function for your own application, but the problem is presented here (following the paper) as a benchmark problem, so the problem (including the input specification) has to be specific and well defined. The reference values, in turn, correspond to this specification.

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