Uncertainty Quantification of Linearized Stress in High-Pressure Spherical Air Storage Tanks Based on Non-Intrusive Polynomial Chaos Expansion

Abstract

The high-pressure spherical gas storage tank in a wind tunnel energy storage and gas supply system is a critical pressure-bearing component of the wind tunnel operation system. The linearized stress in its critical control region is a key parameter for structural safety assessment. Therefore, investigating and evaluating the linearized stress and its associated uncertainty in this region is essential for enhancing operational safety. In this study, a three-dimensional finite element model of the spherical tank was developed, and the critical control region was identified through stress linearization. The operating internal pressure, working temperature, and shell wall thickness were treated as random input variables. Based on the stress linearization results, the stability of the critical control location was assessed. For physically homogeneous intervals, a non-intrusive polynomial chaos expansion surrogate model was constructed, and a conditional uncertainty propagation model for the linearized stress was established. Compared with the Monte Carlo and GUM methods, the non-intrusive polynomial chaos expansion method achieves substantially higher computational efficiency while producing consistent evaluation results. The uncertainty analysis shows that the operating internal pressure is the dominant contributor to the uncertainty of the linearized stress, followed by the effective wall thickness of the spherical shell. In contrast, the working temperature has a minor effect, and the interactions among the input variables are weak.

1. Introduction

Wind tunnel testing platforms are indispensable infrastructure for high-speed aerodynamic research, with compressed air supply systems serving as the linchpin for generating high-pressure, stable, and continuous airflow in blowdown or energy-storage facilities [1,2,3]. In these systems, high-pressure spherical air storage tanks function as both energy reservoirs and pressure buffers, rendering them mission-critical pressure-bearing components that govern the reliability and operational efficiency of experiments. Spherical tanks are preferred in the petroleum, chemical, and energy sectors due to their superior structural efficiency and uniform stress distribution [4]. However, real-world engineering structures deviate from ideal, continuous shells. Geometric discontinuities—such as shell-to-support junctions, polar openings, and nozzle interfaces—induce complex redistributions of membrane and bending stresses, creating structural hotspots [5]. Stress concentration in these regions often far exceeds the average membrane stress, posing significant risks of local damage and crack initiation. Consequently, accurate identification of these critical zones and assessment of their linearized stresses are imperative to ensure the structural integrity of high-pressure spherical tanks.

Finite element (FE) methods are widely used for stress analysis of critical regions in spherical storage tanks. High-fidelity three-dimensional FE models can capture detailed stress distributions at support connections, reinforced openings, and local attachments, thereby supporting code-based strength assessment [6]. In pressure vessel design by analysis, however, raw FE stress fields are generally not used directly for structural evaluation. Instead, stresses are linearized along prescribed stress classification lines (SCLs) and decomposed into membrane, bending, and peak stress components [7]. The linearized membrane-plus-bending stress intensity, P_L + P_b, characterizes the structural stress level at geometric discontinuities and is therefore an important metric for assessing local structural integrity. Nevertheless, the extracted membrane and bending stress components depend strongly on the placement of SCLs, local reinforcement geometry, and boundary constraints [8]. For large spherical tanks with multiple discontinuities, deterministic FE analysis under a single load case is therefore insufficient to capture variations at critical locations and linearized stress responses resulting from changes in operating conditions and structural parameters.

In practical service, high-pressure spherical air storage tanks are subjected to a complex superposition of internal pressure, thermal loads, support constraints, manufacturing deviations, and environmental actions. For large-scale structures, external actions such as wind loads further compromise structural safety. Nevertheless, existing studies on spherical tanks predominantly employ deterministic finite element analysis, focusing primarily on local stress distributions, strength criteria, and code compliance under representative operating conditions. While these studies yield valuable insights for specific parameter sets, they fail to adequately quantify the statistical implications of operating-parameter fluctuations, material-property scatter, and fabrication tolerances on linearized stress assessment. As pressure-vessel safety assessment evolves from deterministic strength verification to reliability analysis and risk-informed decision-making, structural analysis methods that incorporate multiple sources of uncertainty have garnered increasing attention [9]. Consequently, uncertainty quantification must be integrated into conventional FE-based workflows to characterize the statistical behavior, risk levels, and dominant factors contributing to linearized stresses in structural hotspots.

The Monte Carlo (MC) method is routinely employed for random response statistics and structural reliability analysis [10]. However, its direct coupling with three-dimensional finite element (FE) models entails a prohibitive number of repeated evaluations, resulting in computational costs that make iterative assessments of complex structures impractical [11]. To mitigate this burden, surrogate modeling techniques have been extensively adopted to replace computationally intensive FE models. Among these, Polynomial Chaos Expansion (PCE) constructs an approximation of the mapping between random inputs and structural responses using orthogonal polynomial bases, enabling efficient uncertainty propagation with a limited number of high-fidelity samples [12]. In particular, Non-intrusive Polynomial Chaos Expansion (NI-PCE) builds the surrogate model exclusively from input–output samples without requiring modifications to the original FE solver. This feature renders it particularly compatible with commercial FE software for predicting the stochastic response of complex engineering structures [13]. Moreover, the PCE framework enables the analytical derivation of Sobol sensitivity indices from the expansion coefficients, thereby quantifying individual input contributions to the response variance [14]. A key limitation of current UQ frameworks for spherical tanks is the implicit assumption that critical stress locations remain stationary across the parameter space. In practice, these controlling regions migrate under varying thermal and pressure loads. Conventional single-surrogate approaches overlook this drift, yielding ambiguous response definitions in which the output maps to disparate structural locations or stress states under different input conditions. This violates physical consistency, compromising both the interpretability and reliability of the UQ results.

Recent advances have underscored the significance of nonlinear structural responses and multiphysics coupling under complex service conditions. Liu et al. [15] conducted a comprehensive review of the nonlinear dynamics of lightweight composite structures under thermo-electro-mechanical coupling, demonstrating that geometric nonlinearity, boundary effects, and multi-field interactions profoundly govern structural dynamics and failure modes. Addressing hydrogen energy infrastructure, Liu et al. [16] established a transient multiphysics model for a PEM hydrogen storage system, elucidating how renewable-energy fluctuations propagate through electrochemical–thermal–fluid couplings to compromise system pressure and safety. Collectively, these studies highlight that reliance on single-field or static-load analyses is inadequate for the holistic assessment of high-pressure gas storage and transport systems. Consequently, multiphysics-informed modeling and Uncertainty Quantification (UQ) have gained increasing traction in related fields, including fiber-optic sensing, composite structures, and high-dimensional systems [17,18,19,20,21,22].

Addressing these limitations, this study focuses on a 3000 m³ high-pressure spherical air storage tank. First, the stationarity of critical stress locations is examined using finite-element-based stress linearization. Subsequently, for parameter regimes exhibiting stable hotspots, Non-intrusive Polynomial Chaos Expansion (NI-PCE) surrogates are constructed to perform uncertainty quantification and global sensitivity analysis of linearized stresses. A core innovation lies in integrating location-stability diagnostics directly into the UQ workflow to eliminate physically ambiguous response definitions that arise from hotspot migration. This strategy substantially enhances the physical consistency, reliability, and engineering interpretability of probabilistic assessments under complex loading conditions.

2. Finite Element Model and Linearized Stress Assessment Criteria for a Spherical Tank

2.1. Research Object and Structural Characteristics

The subject of this investigation is a 3000 m³ high-pressure spherical air storage tank integrated into a wind-tunnel energy storage facility’s air-supply system. Key structural constituents include the spherical shell, support columns, polar openings, and local reinforcement details. The design and fabrication strictly adhere to the provisions of GB/T 12337-2014 Steel Spherical Tanks [23], with the main design parameters detailed in

Table 1. Principal structural parameters of the spherical storage tank.

Parameter	Value	Parameter	Value
Nominal volume/m3	3000	Inner diameter of spherical shell/mm	18,000
Operating pressure/MPa	2.0	Shell material	Q370R
Design pressure/MPa	2.1	Number of support columns, n	10
Operating temperature/°C	−10 to 50	Outer diameter of support column/mm	628
Design temperature/°C	−10/60	Thickness of support column/mm	14
Stored medium	Dry air	Support column material	Q370R
Corrosion allowance/mm	1.5	Nominal shell thickness/mm	48

.

A three-dimensional solid finite element (FE) model of the tank was constructed to represent the full structural assembly, including the spherical shell, supporting columns, polar openings, and reinforcement structures (Figure 1). The geometry was generated in SolidWorks 2024, exported in STEP format, and imported into the ANSYS Workbench 2020 R1 environment for meshing, load application, and static structural analysis. Adhering to design-by-analysis requirements, local geometric details (e.g., weld toes and reinforcement profiles) were intentionally omitted to focus on primary membrane and bending stresses (P_L + P_b) induced by structural discontinuities. Local mesh refinement was implemented in critical regions—specifically, shell–column junctions and polar openings—to capture stress gradients accurately. A unified protocol for modeling, meshing, and post-processing was maintained across all candidate locations to ensure comparability of results.

2.2. Material Properties, Boundary Conditions, and Loading

The spherical shell is constructed from Q370R steel plate and modeled herein as a linearly elastic solid. Material properties, summarized in

Table 2. Material parameters for Q370R steel in FE analysis.

Parameter	Value
Elastic modulus, E/GPa	201
Poisson’s ratio, μ	0.30
Density, ρ/kg·m−3	7850
Coefficient of thermal expansion, α/°C−1	1.2 × 10−5
Reference temperature for thermal strain, Tref/°C	20

, were extracted from the equipment design documentation. In accordance with GB/T 150.2-2011 Pressure Vessels—Part 2: Materials [24], the coefficient of linear thermal expansion is assigned a value of α = 1.2 × 10⁻⁵ °C⁻¹, consistent with recommended values for carbon and low-alloy steels. Thermal strain calculations are referenced to a uniform ambient temperature of T_ref = 20 °C.

Boundary conditions include full constraints at the support column bases to replicate foundation fixation. Superimposed loads consist of internal pressure on the shell’s inner surface and gravity (g = 9.81 m/s²). A uniform thermal regime is applied, with strains calculated via Equation (1):

$${\epsilon }_{th}=\alpha (T-T_{ref})$$

(1)

Here, ϵ_th denotes the thermal strain, α is the coefficient of thermal expansion, T is the analysis temperature, and T_ref is the reference temperature for thermal strain calculation.

Under a uniform temperature rise, the spherical shell would ideally expand freely; however, the fixed constraints at the column bases restrict thermal expansion along the column axes. This kinematic restraint induces significant thermally induced bending stresses in the shell–column junction and adjacent regions. These stresses superimpose with the mechanical stresses caused by internal pressure and gravity, collectively governing the final stress state at the candidate locations.

2.3. Stress Linearization Method and Selection of Evaluation Metrics

In accordance with the pressure vessel design-by-analysis code, the stress tensors derived from the FE analysis are linearized along prescribed Stress Classification Lines (SCLs) and decomposed into membrane (P_L), bending (P_b), and nonlinear peak stress components. In this study, stress linearization was executed within the ANSYS Workbench 2020 R1 environment using its integrated path-integration tools. The procedure strictly conforms to the Design by Analysis for Pressure Vessels GB/T 4732-2024 [25], wherein stress components are integrated through the wall thickness along the defined SCLs.

The membrane stress P_L along the wall thickness direction is expressed as:

$$P_L={\displaystyle \frac{1}{t}}{\displaystyle {\int }_{-t/2}^{t/2}\sigma }(x)dx$$

(2)

The bending stress P_b is defined as:

$$P_{\mathrm{b}}={\displaystyle \frac{6}{t^2}}{\displaystyle {\int }_{-t/2}^{t/2}\sigma }(x)xdx$$

(3)

where t is the shell thickness, σ(x) is the through-thickness stress distribution, and x is the local coordinate through the thickness.

Owing to the complex geometry of the shell–column junction, the Stress Classification Line (SCL) was established as a through-thickness trajectory normal to the shell mid-surface, positioned centrally within the junction while excluding local geometric singularities such as weld toes. The SCL spans the entire wall thickness, extending from the inner to the outer surface. During post-processing, the finite-element stress tensor for each SCL was linearized to extract the membrane and bending stress tensors. Subsequently, the membrane stress intensity (P_L), bending stress intensity (P_b), and their combined intensity (P_L + P_b) were computed in strict accordance with the pressure vessel design-by-analysis code.

In this study, P_L + P_b was used as the primary strength-evaluation metric for candidate locations and as the response variable Y in the uncertainty propagation analysis. This parameter effectively characterizes the overall structural stress state under prescribed loading and is well-suited for cross-regional comparison and surrogate model development. As illustrated in Figure 2, linearization paths were strategically distributed across critical regions, including the shell–column junction and the upper and lower polar openings. All paths were consistently aligned along the local thickness direction to extract stress distributions and derive the corresponding linearized stress results.

2.4. Credibility Verification of the Finite Element Model

2.4.1. Mesh Independence Verification

To ensure that the linearized stress results are insensitive to mesh discretization, a mesh convergence study was executed before the uncertainty propagation analysis. Six distinct mesh densities were established, and the membrane-plus-bending stress intensity (P_L + P_b) was extracted at three candidate critical locations: the shell–column junction, the upper polar opening, and the lower polar opening. Using the finest mesh scheme (100 mm) as the baseline, the maximum relative deviation for each coarser mesh was computed against this baseline, expressed as:

$${\delta }_{\max }={\max }_j\left({\displaystyle \frac{\left|Y_{i,j}-Y_{100,j}\right|}{Y_{100,j}}}\times 100\%\right)$$

(4)

where Y_i,j is the stress intensity for the i-th mesh density at location j, and Y_100,j is the baseline value from the 100 mm mesh. Locations include the shell–column junction and the upper and lower polar openings.

As summarized in

Table 3. Finite Element Stress Linearization Mesh Independence Verification.

Mesh (mm)	Total Nodes	Total Elements	Shell–Column Junction (PL + Pb) (MPa)	Upper Polar Hole (PL + Pb) (MPa)	Lower Polar Hole (PL + Pb) (MPa)	Max Relative Difference vs. 100 mm (%)
300	888,596	418,278	201.39	179.05	200.41	11.93
250	927,407	423,776	206.06	179.21	193.75	11.85
200	996,130	433,515	207.38	192.09	198.13	5.51
150	1,146,222	454,836	210.14	201.62	200.06	2.22
100	1,568,739	514,972	211.12	203.3	204.61	-
80	1,994,837	575,746	211.17	204.21	204.98	0.45

, the P_L + P_b results at all three critical locations demonstrate a clear convergence trend with mesh refinement. Refining the mesh from 300 mm to 150 mm reduced the maximum relative deviation against the 100 mm baseline from 11.93% to 2.22%. Further refinement to 80 mm yielded a negligible deviation of only 0.45%; however, this came at the expense of a prohibitive increase in computational overhead, with node and element counts rising from 1.57 million to 1.99 million and from 0.51 million to 0.58 million, respectively. Consequently, further mesh refinement yields diminishing returns in stress accuracy while imposing a substantial computational penalty.

Figure 3 presents the mesh convergence profiles to substantiate the numerical stability visually. While Figure 3a tracks the decay of linearized stress with mesh refinement, Figure 3b benchmarks the results by computing the relative deviation from the 100 mm baseline.

Balancing accuracy against computational expense, the 100 mm mesh was selected as the baseline for subsequent FE calculations, stress linearization, and UQ analyses.

2.4.2. Benchmark Verification of Analytical Solutions for Membrane Stress

To evaluate the fidelity of the finite element model in capturing the global membrane stress response, a benchmark case with t = 46.5 mm, P = 2 MPa, and T = 20 °C is used to compare with the analytical solution. For a thin-walled spherical shell subjected to uniform internal pressure, the theoretical membrane stress is expressed as:

$${\sigma }_m={\displaystyle \frac{PR}{2t}}$$

(5)

Here, σ_m represents the theoretical membrane stress for a spherical shell with internal pressure, mid-surface radius R, and wall thickness t. With R_set to 9000 mm, Equation (5) gives σ_m =193.55 MPa.

Numerically, the membrane stress was sampled at locations distant from geometric discontinuities (e.g., openings and attachments) and constraint boundaries to isolate the global response. The relative deviation is computed via:

$$\epsilon ={\displaystyle \frac{|{\sigma }_{m,\mathrm{FE}}-{\sigma }_{m,\mathrm{theory}}|}{{\sigma }_{m,\mathrm{theory}}}}\times 100\%$$

(6)

where σ_m,FE is the numerical membrane stress extracted from the intact spherical shell (excluding discontinuities), σ_m,theory is the corresponding analytical baseline, and ε is the relative deviation. The comparative results are summarized in

Table 4. Comparison of analytical and finite element membrane stresses in the main spherical-shell region.

Comparison Location	Analytical Membrane Stress/MPa	FE Membrane Stress/MPa	Relative Error/%
Main spherical-shell region	193.55	202.31	4.53

.

Table 4 shows that the FE model yields a membrane stress of 202.31 MPa, which deviates by only 4.53% from the theoretical value (193.55 MPa). This, combined with the proven mesh convergence, validates the numerical stability of the model and its linearization routine. The model is thus confirmed as a reliable foundation for identifying candidate locations and conducting rigorous UQ analyses.

2.5. Candidate Locations and Identification of Control Locations

Candidate locations were defined at the shell–column junction and polar openings. Using 175 full-factorial design points, P_L + P_b values were derived for each site. As plotted in Figure 4, the stresses exhibit clear temperature-dependent trends, facilitating the identification of critical zones under operational variability.

As illustrated in Figure 4, the P_L + P_b stress at the shell–column junction exhibits a relatively mild temperature dependence. Within the T < 40 °C range, it persistently surpasses that of the polar openings, establishing the junction as the stable dominant control location under low-temperature conditions. Conversely, stresses at the polar openings show a pronounced increase as temperature rises, approaching or even exceeding the junction stress once T ≥ 40 °C. This signifies a transition of the critical control location at elevated temperatures. The phenomenon is attributed to thermally induced constraint stresses at the fixed column bases, compounded by localized stiffness discontinuities in the opening-reinforcement zones.

To formalize this observation, the control response Y_max is defined as the maximum P_L + P_b value among the three candidate locations. This metric constitutes the worst-case linearized stress level under the prevailing operating condition:

$$Y_{max}(T,P,t)=max(Y_{shell\text{–}col},Y_{upper},Y_{lower})$$

(7)

Here, T, P, and t represent the temperature, pressure, and thickness, respectively; Yshell-col, Yupper, and Ylower are the P_L + P_b values at the three candidate sites.

Table 5. Statistical proportion of candidate locations serving as the controlling location.

Temperature Range	Shell-Column Connection	Upper Polar Opening	Lower Polar Opening
All samples	120/175 (68.57%)	45/175 (25.71%)	10/175 (5.71%)
T < 40 °C	105/105 (100%)	0 (0%)	0 (0%)
T ≥ 40 °C	15/70 (21.43%)	45/70 (64.29%)	10/70 (14.29%)

statistically delineates the dominance of each location as the control point (Ymax) over varying temperature intervals.

As evidenced by Table 5, the shell–column junction serves as the unequivocal control location across all 105 samples within the low-temperature regime (T < 40 °C), confirming its stability within the sampled parametric domain. Consequently, the uncertainty quantification and sensitivity analyses presented in Chapters 3 and 4 exclusively target this junction within said regime. This focused scope ensures that the response variable Y (i.e., P_L + P_b at the junction) maintains an unambiguous and physically consistent interpretation for surrogate modeling. The behavior in the high-temperature regime (T ≥ 40 °C), characterized by control-location migration, will be addressed separately.

2.6. Random Input Variables and Their Probability Distributions

To quantify the effects of operational fluctuations and manufacturing deviations, the internal pressure, uniform operating temperature, and shell thickness were selected as random input variables. The probability distributions adopted herein are engineering-oriented probabilistic models derived from equipment design parameters, the design temperature envelope, corrosion allowances, and specified parameter tolerances. It is important to note that these distributions are not fitted to long-term operational monitoring data but are conservatively estimated from design specifications to ensure robustness. The statistical characteristics of these inputs are summarized in

Table 6. Probability distributions and statistical parameters of the random input variables.

Random Variable	Probability Distribution	Parameters	Unit
T	Truncated normal	$\mu =25,\sigma =10,bounds:[-10,60]$	°C
P	Normal	$\mu =2.0,\sigma =0.14$	MPa
t	Uniform	$bounds$: [45.0–48.0]	mm

.

The shell thickness t refers to the effective load-bearing thickness, derived by deducting the corrosion allowance (1.5 mm) from the nominal thickness (48 mm). It is modeled as a uniform distribution bounded between [45.0, 48.0] mm, which corresponds to a standard deviation of approximately 0.866 mm.

Due to the lack of empirical thickness statistics, the effective wall thickness is modeled as a uniform distribution over the specified range, thereby avoiding extraneous prior assumptions. These probability distributions constitute the foundational input premises for constructing the NI-PCE surrogate model and performing uncertainty propagation analyses. Consequently, the results must be interpreted as conditional statistical responses contingent upon these input probabilistic models.

Discretized sampling levels were established as follows: internal pressure P at [1.8, 1.9, 2.0, 2.1, 2.2] MPa; temperature T at [−10, 0, 20, 40, 60] °C; and wall thickness t at [45.0, 45.5, 46.0, 46.5, 47.0, 47.5, 48.0] mm. This discrete set spans the principal parametric envelope of this study. It serves to calibrate the mapping relationship between the input variables and the linearized stress intensity at candidate evaluation locations.

3. Non-Intrusive Polynomial Chaos Expansion Surrogate Model and Uncertainty Propagation

3.1. Response Quantity and Input Variables

As established in Section 2.5 and summarized in Table 5, a regime shift in the maximum-stress control point occurs when the temperature exceeds 40 °C. Constructing a single surrogate model over the entire parameter domain would conflate response characteristics associated with distinct physical locations. This heterogeneity would severely compromise the model’s physical interpretability and substantially degrade predictive fidelity, as evidenced by the poor global model performance (R² = 0.5241) reported in

Table 7. Performance comparison of NI-PCE models.

Model Type	Temperature Interval	Order p	Basis Num. M	${\mathit{R}}_{\mathit{C}\mathit{V}}^{\mathbf{2}}$	$\mathit{R}\mathit{M}\mathit{S}{\mathit{E}}_{\mathit{C}\mathit{V}}$	$\mathit{M}\mathit{A}\mathit{P}{\mathit{E}}_{\mathit{C}\mathit{V}}$
Global Model	−10 °C ≤ T ≤ 60 °C	4	35	0.5241	14.6925	5.14
Low-Temp Seg.	T < 40 °C	4	35	0.9774	2.33	0.97
High-Temp Seg.	T ≥ 40 °C	3	20	0.3814	22.4	10.22

.

To circumvent this limitation, a regime-based piecewise modeling strategy is adopted. Specifically, a non-intrusive polynomial chaos expansion (NI-PCE) model is constructed exclusively for the low-temperature regime (T < 40 °C), where the critical location remains unambiguously anchored at the shell–column junction. Consequently, the membrane-plus-bending stress (P_L + P_b) at this junction is defined as the sole response variable Y for uncertainty propagation. The mapping relationship between Y and the input variables is formulated as:

$$Y=f(P,T,t),T<40°\mathrm{C}$$

(8)

The uncertainty propagation results presented hereafter pertain to the statistical characteristics conditional upon the low-temperature regime (T < 40 °C). Specifically, the mean, standard deviation, probability intervals, and exceedance probabilities reflect the conditional statistics of the shell–column junction response within this confined thermal domain. Conversely, for the high-temperature regime (T ≥ 40 °C), the migration of the control location across disparate structural components precludes the definition of a single, consistent response variable for uncertainty propagation. Instead, the characteristics of this transient control behavior are subsequently analyzed using discrete finite-element sample results.

3.2. Construction of the NI-PCE Surrogate Model

A Non-Intrusive Polynomial Chaos Expansion (NI-PCE) surrogate model is established for the linearized stress response of the membrane-plus-bending at the shell-column junction in the low-temperature segment. The core of this method approximates the response as a sum of multivariate orthogonal polynomial functions of standard normal random variables ξ [25]:

$$\widehat{Y}(\xi )={\displaystyle \sum _{j=0}^{M-1}{c}_j}{\mathsf{\Psi }}_j(\xi )$$

(9)

where $\widehat{Y}$ is the predicted value, c_j are the expansion coefficients, ${\mathsf{\Psi }}_j(\xi )$ are the multivariate Hermite orthogonal polynomial basis functions, and M is the total number of basis functions.

To construct the Hermite polynomial chaos basis functions, each physical input variable is mapped into the independent standard normal space via an isoprobabilistic transformation. For any input variable Xi, the corresponding standard normal variable is expressed as:

$${\xi }_i={\mathsf{\Phi }}^{-1}(F_{X_i}(x_i))$$

(10)

where $F_{X_i}$ is the cumulative distribution function (CDF) of $X_i$, and Φ − 1 is the inverse CDF of the standard normal distribution. The operating internal pressure is standardized directly according to its normal distribution; the operating temperature is transformed using the CDF of the truncated normal distribution; and the effective shell thickness is transformed using the CDF of the uniform distribution. After this transformation, the surrogate model is built in the standard normal variable space $\xi =[{\xi }_P,{\xi }_T,{\xi }_t]$. Since P, T, and t are assumed independent, the transformed standard normal variables are also independent.

For three-dimensional input variables, candidate models are constructed using a total-order truncation strategy:

$$M={\displaystyle \frac{(d+p)!}{d!p!}}$$

(11)

where d = 3 is the input dimension, and p is the polynomial order. To determine the appropriate order for the low-temperature segment surrogate model, second-order, third-order, and fourth-order NI-PCE candidate models are constructed. Their prediction accuracy is evaluated via the 5-fold cross-validation described in Section 3.3. The model with the highest prediction accuracy and best generalization performance is ultimately selected for uncertainty propagation and sensitivity analysis of the linearized stress at the shell-column junction in the low-temperature segment.

3.3. Coefficient Identification and Cross-Validation

After determining the polynomial basis functions and the standard normal variables corresponding to the sample points, the coefficient vector c is solved using the least squares method:

$$\widehat{c}=\arg {\min }_c{‖Y-\Psi c‖}_2^2$$

(12)

where Y and $\widehat{Y}$ are the vectors of finite element responses and basis function matrices for the low-temperature segment, respectively.

To evaluate the generalization capability of models across varying polynomial orders, a 5-fold cross-validation (CV) procedure was implemented. The low-temperature dataset D_Low was partitioned into five subsets. An iterative training-and-validation cycle was executed, wherein the model was trained on four subsets and validated on the remaining hold-out set, repeated five times to ensure complete coverage of the sample space. The model’s predictive accuracy is benchmarked using the coefficient of determination ($R_{CV}^2$), root mean square error ($RMS{E}_{CV}$), and mean absolute percentage error ($MAP{E}_{CV}$):

$$R_{CV}^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^N{\left(Y^{(i)}-{\widehat{Y}}_{CV}^{(i)}\right)}^2}}{{\displaystyle \sum _{i=1}^N{\left(Y^{(i)}-\overline{Y}\right)}^2}}}$$

(13)

$$RMS{E}_{CV}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left(Y^{(i)}-{\widehat{Y}}_{CV}^{(i)}\right)}^2}}$$

(14)

$$MAP{E}_{CV}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left|\frac{Y^{(i)}-{\widehat{Y}}_{CV}^{(i)}}{Y^{(i)}}\right|}\times 100\%$$

(15)

where N = 105 is the number of low-temperature samples and $\overline{Y}$ is the response mean.

The final model order is determined based on $R_{CV}^2$, $RMS{E}_{CV}$, and $MAP{E}_{CV}$ as comprehensive evaluation metrics. Once the order is fixed, all 105 low-temperature samples are used to re-estimate the model coefficients, yielding the final NI-PCE surrogate model for subsequent uncertainty propagation and sensitivity analysis.

3.4. Uncertainty Propagation Method Based on NI-PCE

According to the probability distributions of input variables given in Section 2.6, Nm = 2 × 10⁵ random samples within the T < 40 °C condition are generated. These input samples are mapped to the standard normal space via the transformation described in Section 3.2 and substituted into the low-temperature NI-PCE surrogate model to obtain predicted samples of the membrane-plus-bending linearized stress at the shell-column junction ${\{{\widehat{Y}}^{(k)}\}}_{k=1}^{N_m}$.

Based on these predicted samples, the mean, standard deviation, 95% probability interval, and exceedance probability of the linearized stress at the shell-column junction in the low-temperature segment are calculated. The response mean μ_Y and standard deviation σ_Y are computed as follows:

$${\mu }_Y={\displaystyle \frac{1}{N_m}}{\displaystyle \sum _{k=1}^{N_m}{\widehat{Y}}^{(k)}}$$

(16)

$${\sigma }_Y=\sqrt{{\displaystyle \frac{1}{N_m-1}}{\displaystyle \sum _{k=1}^{N_m}{\left({\widehat{Y}}^{(k)}-{\mu }_Y\right)}^2}}$$

(17)

The 95% probability interval for the response was expressed in quantile form:

$$I_{95\%}=\left[Q_{2.5\%}(\widehat{Y}),Q_{97.5\%}(\widehat{Y})\right]$$

(18)

Based on the design stress intensity $S_m = 212 \mathrm{MPa}$, the evaluation limit for membrane plus bending stress is set as $Y_{lim}=1.5S_m$. The probability of stress exceeding this limit is calculated as:

$${\widehat{P}}_f={\displaystyle \frac{1}{N_m}}{\displaystyle \sum _{k=1}^{N_m}I}\left({\widehat{Y}}^{(k)}>Y_{\lim }\right)$$

(19)

where $I(\cdot )$ is the indicator function.

3.5. Comparison with GUM and Sobol Global Sensitivity Analysis

To systematically compare the discrepancies between local linear uncertainty propagation and global surrogate-based propagation, the Guide to the Expression of Uncertainty in Measurement (GUM) framework is adopted as the reference benchmark [26]. Furthermore, variance-based global sensitivity analysis is conducted using Sobol indices to quantify the fractional contribution of each input variable to the response variance within the low-temperature regime [27].

The GUM method is based on the local sensitivity coefficients near the input mean point and utilizes the law of propagation of uncertainty to calculate the combined standard uncertainty:

$$u_c^2(Y)={\left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }P}}\right)}^2{u}^2(P)+{\left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }T}}\right)}^2{u}^2(T)+{\left({\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }t}}\right)}^2{u}^2(t)$$

(20)

It provides a 95% expanded uncertainty interval corresponding to a coverage factor k = 2, serving as a reference for local linear propagation results to be compared with the NI-PCE-MC results.

Under the assumption of independent input variables, the contribution of each input variable to the response variance is quantified based on the variance decomposition principle. The first-order Sobol index Si and the total-effect Sobol index STi are defined as:

$$S_i={\displaystyle \frac{{\mathrm{Var}}_{X_i}\left[\mathrm{E}(Y|X_i)\right]}{\mathrm{Var}(Y)}}$$

(21)

$$S_{Ti}=1-{\displaystyle \frac{{\mathrm{Var}}_{X_{~i}}\left[\mathrm{E}(Y|X_{~i})\right]}{\mathrm{Var}(Y)}}$$

(22)

where S_i represents the independent contribution of variable X_i, and S_Ti denotes the total contribution of variable X_i, including its interactions with other variables.

4. Results and Discussion

4.1. Validation and Accuracy Assessment of the NI-PCE Model

To validate the segmented modeling strategy, both global (−10 ≤ T ≤ 60 °C) and temperature-segmented NI-PCE models were constructed. A 5-fold cross-validation was employed to evaluate their prediction accuracy, with the results summarized in Table 5.

As summarized in Table 7, the fourth-order total-degree Hermite polynomial chaos expansion (PCE) model demonstrates exceptional accuracy in full-sample cross-validation (R² = 0.9893, RMSE = 1.45 MPa, MAPE = 0.58%), confirming its ability to replicate the underlying physics of the low-temperature dataset faithfully. However, while robust, full-sample cross-validation remains inherently susceptible to an optimistic bias, as the model may over-adapt to the specific characteristics of the available sample distribution. To rigorously assess generalization capability on unseen data, an independent test set was introduced. Specifically, the 105 finite element samples within the low-temperature regime were randomly partitioned into a training set (84 samples) and a test set (21 samples) at a 4:1 ratio. The training set facilitated polynomial order selection and coefficient calibration, whereas the independent test set was strictly sequestered from the training process and reserved exclusively for final performance evaluation. The outcomes of this validation are presented in

Table 8. Independent test-set validation results of the surrogate model in the low-temperature regime.

Evaluation Case	R2	RMSE/MPa	MAPE/%
5-fold cross-validation on the training set	0.9032	0.9302	0.58
Independent test set	0.9695	2.7576	1.16

.

As summarized in Table 8, the fourth-order NI-PCE model yields a coefficient of determination (R²) of 0.9695 and a root mean square error (RMSE) of 2.76 MPa on the strictly sequestered independent test set. While the RMSE shows a modest increase relative to the cross-validation results on the training set, it remains well below 3 MPa, with the mean absolute percentage error (MAPE) remaining at a negligibly low level of approximately 1.2%. Crucially, the R² value remains exceptionally high at 0.97, confirming that the model retains robust predictive fidelity for unseen data. This marginal attenuation in predictive precision is characteristic of expected statistical fluctuations in out-of-sample evaluation and does not indicate overfitting. Overall, independent validation demonstrates that the fourth-order model possesses excellent generalization capability without evident overfitting. Consequently, it is fully justified to employ this model for uncertainty propagation analysis of the linearized stress at the shell–column junction within the low-temperature regime. The high predictive veracity of this model also furnishes a reliable computational foundation for the subsequent global sensitivity analysis.

Figure 5 illustrates the predictive performance of the surrogate model for the low-temperature regime on the independent test set. In Figure 5a, the predicted values are tightly distributed around the y = x line, showing no obvious systematic bias. Figure 5b presents the residual distribution, which is approximately normal and concentrated near zero. These results provide visual confirmation that the fourth-order NI-PCE model reliably captures the mapping relationship between the operating internal pressure, uniform temperature, and effective shell thickness and the linearized stress at the shell–column junction.

4.2. Uncertainty Propagation Results for the Low-Temperature Segment

By integrating the validated high-precision NI-PCE surrogate model with the Monte Carlo method (NI-PCE-MC), Nm = 2 × 10⁵ samples were generated based on the input variable probability distributions, strictly constrained to the T < 40 °C condition. This yielded the probabilistic statistical characteristics of the linearized stress at the shell-column junction for the low-temperature segment, as presented in

Table 9. Uncertainty propagation results of linearized stress in the low-temperature segment.

Statistical Metric	Low-Temperature Segment (T < 40 °C)
Mean μY/MPa	187.27
Standard Deviation σY/MPa	9.4
95% Probability Interval/MPa	[170.39, 204.19]
Exceedance Probability Pf/%	≤0.001

.

Since the analysis object in this section is restricted to the low-temperature control zone, the statistics in Table 9 reflect the distribution characteristics of the membrane plus bending (P_L + P_b) stress at the shell-column junction within this specific temperature interval. The results indicate that, under the considered fluctuations of operating internal pressure, working temperature, and effective shell thickness, the mean P_L + P_b at the shell-column junction is 187.3 MPa, with a standard deviation of 9.4 MPa. The 95% probability interval is [170.37, 204.23] MPa, which remains below the evaluation limit of Ylim = 1.5 Sm = 318 MPa. Under the input variable distributions, low-temperature conditions, and finite element modeling assumptions defined in this study, no exceedance samples were observed in the surrogate model Monte Carlo simulations, suggesting a low risk of P_L + P_b exceedance for the shell-column junction in the low-temperature segment.

Figure 6 illustrates the probability density distribution of the linearized stress at the shell-column junction in the low-temperature segment, derived from the aforementioned large-scale Monte Carlo sampling. The distribution exhibits a concentrated, symmetric, unimodal profile, indicating that the linearized stress response in this region is relatively centralized within the low-temperature interval and generally moderately sensitive to fluctuations in the input parameters. The concentrated region of the distribution corresponds precisely to the 95% probability interval of [170.37, 204.23] MPa and lies entirely to the left of the assessment threshold (Ylim = 318 MPa).

4.3. Comparison with GUM Method for Uncertainty Propagation

To underscore the necessity of global uncertainty quantification, the results of the NI-PCE-MC framework are benchmarked against the Guide to the Expression of Uncertainty in Measurement (GUM) method, which relies on local linear approximation. As outlined in Section 3.5, the GUM approach propagates uncertainty via a first-order Taylor expansion, using local sensitivity coefficients derived from the surrogate model at the nominal operating point (P = 2.0 MPa, T = 25 °C, t = 46.5 mm). The quantitative comparison between these two methodologies is presented in

Table 10. Comparison of uncertainty propagation results obtained by GUM and NI-PCE-MC.

Method	Mean Stress/MPa	Standard Uncertainty/MPa
GUM at the sample center	183.92	6.69
NI-PCE-MC with (2 × 105) simulations	187.27	9.41

.

Table 10 reveals a critical insight: while both methods predict similar mean stresses, the GUM method drastically underpredicts the standard uncertainty compared to NI-PCE-MC. This discrepancy arises because GUM relies on local gradients at a single point, failing to capture the nonlinear response surface away from the mean. By performing global sampling, NI-PCE-MC yields a wider, more realistic confidence interval. This comprehensive characterization is crucial, as it rectifies the GUM method’s conservative bias, ensuring that the full spectrum of operational risks is accounted for in the low-temperature regime.

4.4. Global Sensitivity Analysis Based on Sobol Indices

To identify the primary sources of uncertainty in the low-temperature segment stress, the Sobol indices for each input variable were analytically calculated based on the NI-PCE model. The results are summarized in

Table 11. Sobol Sensitivity Indices for the Low-Temperature Regime.

Input Variable	First-Order Index Si	Total-Effect Index STi
P	0.969	0.968
T	0.009	0.009
t	0.025	0.024

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As summarized in Table 11 and Figure 7, the operating internal pressure is the predominant contributor to the uncertainty in the linearized stress at the shell–column junction in the low-temperature regime, with a first-order Sobol index of 0.969. This dominance is mechanistically grounded in the linear proportionality between the primary global membrane stress and internal pressure (Equation (5)), whereby fluctuations in P account for the vast majority of the variance in the response. The effective shell thickness exerts a secondary yet discernible influence, with a first-order Sobol index of 0.025, reflecting the modulating effect of thickness variations on the local bending stress (P_b) via alterations in the cross-sectional bending stiffness. In contrast, the uniform operating temperature contributes negligibly (first-order Sobol index of 0.009). This indicates that, within the low-temperature range, the constraint-induced thermal bending stresses have not fully manifested and remain subordinate to the mechanically driven stress from internal pressure. The close correspondence between the total-effect and first-order Sobol indices indicates that inter-variable interactions are negligible, with the output variance being predominantly governed by the independent main effects. Although temperature exerts only a marginal influence on local stress fluctuations in the low-temperature regime, it remains a pivotal physical parameter that can trigger regime transitions across the full operational envelope. As established in Section 2.5, when the temperature exceeds 40 °C, the coupling between accumulated thermal strain and fixed column constraints induces a shift in the critical evaluation location. This transition shifts the stress-governance mechanism from a membrane-dominated state to a more complex, bending-dominated paradigm.

The results conclusively demonstrate that, within the low-temperature regime, the uncertainty of the linearized stress at the shell–column junction is predominantly governed by fluctuations in the operating internal pressure. However, as the temperature transitions into the high-temperature regime, a migration of the structural control location is triggered. Consequently, elucidating the evolution patterns of this control location across varying thermal intervals is essential for a comprehensive structural assessment.

4.5. Analysis of Controlling Location Shift in the High-Temperature Interval

As discussed in Section 2.5, the structural control location shifts when entering the high-temperature interval (T ≥ 40 °C). To further investigate this phenomenon, the number of samples for which each candidate location served as the control location was statistically analyzed across different temperature intervals, with the results presented in Figure 8.

As shown in Figure 8, among all 175 samples, the shell-column junction, upper polar opening, and lower polar opening served as the control location in 120, 45, and 10 samples, respectively. Within the low-temperature interval (T < 40 °C), the shell-column junction was the control location in all 105 samples, demonstrating stability in this range. However, in the high-temperature interval (T ≥ 40 °C), the upper polar opening became the control location in 45 samples (64.3% of high-temperature samples). In comparison, the shell-column and lower polar opening accounted for 15 and 10 samples, respectively. This indicates a clear shift in the control location from the shell-column junction to the opening zones in high-temperature scenarios.

To quantify this phenomenon, we define the transition probability for the high-temperature regime as the proportion of samples where the control location shifts away from the shell–column junction:

$$P_{\mathrm{shift}}={\displaystyle \frac{N_{\mathrm{up}}+N_{\mathrm{low}}}{N_{\mathrm{HT}}}}={\displaystyle \frac{45+10}{70}}=78.6\%$$

(23)

The results indicate that within the high-temperature regime (T ≥ 40 °C), the probability of a critical-location transition peaks at a substantial 78.6%. This aligns precisely with the marked degradation in surrogate model performance observed in Table 7 ($R_{CV}^2$ = 0.3814), wherein a single-location response metric proves incapable of capturing the inherent non-stationary dynamics associated with the migrating control location.

To further compare potential extreme linearized stress levels across the full parameter space,

Table 12. Comparison of baseline conditions and sample space extremes for different candidate locations.

Candidate Location	Sample Space Extreme PL + Pb (MPa)	Corresponding Extreme Condition (T, P, t)
Shell-column connection	237.79	(60, 2.2, 48)
Upper polar opening	246.66	(60, 2.2, 46)
Lower polar opening	234.39	(60, 2.2, 45.5)

lists the sample-space extremes of P_L + P_b for each candidate location.

Table 12 demonstrates that although the shell–column junction exhibits elevated stress under the reference operating condition, the upper polar opening attains the absolute maximum stress across the full sample space, reaching 246.66 MPa. This value surpasses the peak stresses observed at the shell–column junction (237.79 MPa) and the lower polar opening (234.39 MPa). Consequently, the controlling location identified under the reference condition does not invariably coincide with the most critical location over the entire parametric domain.

Consistent with the statistical findings in Table 5, the upper polar opening not only constitutes the predominant control location in the high-temperature regime but also corresponds to the global maximum observed across all samples. This confirms that, under the current experimental design and boundary-condition assumptions, the upper polar opening is the primary structural hotspot for evaluation in the high-temperature regime. The underlying physical mechanism aligns with the analyses in Section 2.2 and Section 2.5: at elevated temperatures, restrained thermal expansion due to the fixed column bases induces substantial constraint thermal stresses; concurrently, the local stiffness discontinuity in the reinforced region of the upper polar opening amplifies its susceptibility to bending stress, resulting in stress levels that overtake those at the shell–column junction.

From a structural reliability perspective, the observed migration of the critical location indicates that a single-point assessment, tied to a fixed response location, risks missing the globally most critical condition across the full parameter space, particularly in the high-temperature regime. To preclude underestimation of risk due to this location shift, the subsequent envelope-based reliability analysis uses the global maximum linearized stress, denoted as Y_max = max (Yshell-col, Yupper, Ylower), as the unified safety metric. Once the allowable stress limit (Y_lim) is stipulated by relevant design codes (e.g., GB/T 4732 or ASME Section VIII), the limit-state function can be defined as:

$$g(X)=Y_{lim}-Y_{max}(X)$$

(24)

where X represents the vector of uncertain input variables. Failure occurs when g(X) ≤ 0, which allows calculation of the failure probability (P_f) and the reliability index (β). A comprehensive reliability assessment would further require the characterization of material strength uncertainty, a clear definition of the failure mode, and the specification of a target reliability index. While these aspects fall outside the scope of this study, the uncertainty propagation framework developed herein provides the essential probabilistic foundation for such engineering assessments.

4.6. Influence of Support Form on Multi-Location Stress Fields

To investigate the influence of boundary conditions on structural response, this section compares the linearized stress P_L + P_b under fixed and sliding support configurations. The sliding support model was configured to reflect practical engineering: one column served as a guide column with full constraints, while the remaining nine columns had their in-plane displacement degrees of freedom (Ux,Uy) released to simulate the sliding effect within elongated anchor bolt holes. Analyses were conducted for a shell thickness of t = 46.5 mm across the full operational load spectrum, as shown in

Table 13. Comparison of multi-location stresses under different support forms (t = 46.5 mm).

T (°C)	P (MPa)	Support Type	Shell-Column (MPa)	Up.Polar (MPa)	Low.Polar (MPa)	Diff. Ratio
40	1.8	Fixed	188.90	183.33	177.66	0.02
		Sliding	188.93	183.33	177.66
40	2.2	Fixed	230.18	219.79	212.83	0.02
		Sliding	230.22	219.79	212.84
60	1.8	Fixed	192.67	205.41	198.65	0.02
		Sliding	192.70	205.41	198.67
60	2.2	Fixed	233.95	241.86	233.74	1.39
		Sliding	237.21	241.86	233.76

.

Data analysis shows that the effect of the support configuration exhibits a distinct dependence on temperature and load (Figure 9). Within the mid-to-low temperature range (T ≤ 40 °C), the variation induced by sliding supports is negligible, with a maximum difference ratio of only 0.02%. According to thin-shell theory, the structural response under these working conditions is predominantly governed by internal pressure-induced membrane stress (theoretical range: 174.2~191.4 MPa). Owing to the relatively small temperature differential (ΔT ≤ 20 °C), the contribution of additional bending stress arising from restrained thermal expansion is minimal, rendering variations in support stiffness inconsequential to the overall stress level. At the elevated temperature of 60 °C, the effect of support flexibility becomes apparent but exhibits significant load sensitivity. Under the low-pressure condition (P = 1.8 MPa), the stress fluctuation caused by sliding supports remains below 0.02%, indicating that the boundary restraint effect induced by thermal expansion is not yet activated. However, under the high-pressure condition (P = 2.2 MPa), sliding supports increase the stress at the shell-column junction by approximately 1.4% (i.e., localized failure of the stress relief effect). Nevertheless, the control location remains consistently anchored at the upper polar nozzle, and the stress levels in the polar regions exhibit low sensitivity to changes in support form. This indicates that support flexibility merely induces weak local perturbations proximal to the columns under specific loading conditions and does not alter the control location migration pattern induced by thermo-mechanical coupling. Consequently, the partitioned uncertainty quantification framework proposed herein is robust to variations in support configuration.

5. Conclusions

This study establishes a conditional uncertainty quantification (UQ) framework for high-pressure spherical air storage tanks in wind tunnel energy supply systems, integrating finite-element (FE) stress linearization with Non-Intrusive Polynomial Chaos Expansion (NI-PCE). The framework resolves the non-stationary migration of the critical control location induced by thermo-mechanical coupling, thereby circumventing the physical heterogeneity risks inherent in conventional global modeling approaches. FE sample analysis reveals a pronounced dependence of the membrane-plus-bending stress control location on the thermal regime: the shell–column junction remains the stable control point in the low-temperature regime, whereas the control location progressively shifts toward the nozzle region as temperature increases. This mechanism underscores that prior identification of operational sensitivity is a prerequisite for UQ, invalidating the assumption of a single, invariant control location across the full temperature domain. Leveraging this stability analysis, a high-fidelity NI-PCE surrogate model was constructed for the low-temperature regime, effectively replacing costly FE computations to enable efficient probabilistic propagation. UQ results indicate that stress responses remain within safety limits under the specified operational fluctuations. Global sensitivity analysis further identifies operating pressure as the dominant source of uncertainty, with shell thickness playing a secondary role. While temperature contributes marginally in the low-temperature regime, it is the pivotal physical parameter triggering the regime transition in structural behavior at elevated temperatures.

For industrial practice, specific monitoring priorities are dictated: the shell-column junction is the stable control point for low-temperature operations (T < 40 °C), while the upper polar opening becomes the primary hotspot in high-temperature scenarios (T ≥ 40 °C), accounting for 78.6% of control cases. Crucially, uncertainty assessments must integrate this location-migration mechanism to avoid underestimating extreme stresses (e.g., 246.66 MPa), thereby ensuring full-envelope operational safety.

Overall, this work establishes an “Identify–Partition–Quantify” analytical paradigm and refines the physical prerequisites for UQ. Although the migration of the control location precluded direct calculation of a full-domain failure probability, the proposed framework provides a rigorous probabilistic foundation for envelope-based reliability assessments that incorporate location-competition effects. This methodology is readily applicable to the safety evaluation of pressure equipment in the petrochemical and energy storage industries.