Tolerance driven lightweight design and interface robustness of multi material aircraft horizontal tail structures

Abstract

This study addresses the challenge of balancing weight reduction with stiffness in aircraft horizontal tails by proposing a multi-material design strategy combining carbon fiber reinforced polymer (CFRP) spars, closed-cell foam cores, and aluminum alloy joints. A three-dimensional nonlinear finite element model was developed to quantitatively assess how manufacturing tolerances—specifically variations in adhesive layer thickness and foam core density—affect interfacial mechanical performance. The co-optimized structure achieved a single-wing mass of 17.8 kg, representing a 32% reduction compared to conventional all-metal designs, while limiting the maximum displacement to 188.8 mm. Sensitivity analysis revealed that a 0.2 mm decrease in adhesive thickness increased peak interfacial shear stress by 22%. Monte Carlo simulations identified adhesive thickness variability as the dominant factor, contributing 64% of the variance in overall displacement. Robustness optimization, incorporating ± 45° ply reinforcement and tolerance-aware design, reduced the standard deviation of displacement by 50% and increased the transverse shear modulus by 17.3%. Validation tests on scaled prototypes demonstrated that a gradient density compensation strategy reduced displacement variability by 41%. The calibrated finite element model showed strong agreement with experimental data, yielding a coefficient of determination (R²) of 0.96. This work establishes a process-structure-property framework to support the reliable design of multi-material aerospace structures, though the findings are based on a scaled prototype and specific material combinations, indicating a need for validation at full scale and with other material systems.

Introduction

Aircraft lightweight design is paramount for enhancing fuel efficiency, extending range, and reducing environmental impact¹. Among primary aerostructures, the horizontal tail is critical for longitudinal stability and control, making its structural efficiency a key design driver². Traditional horizontal tails, predominantly fabricated from aluminum alloys, offer proven manufacturability and damage tolerance³. However, their specific stiffness and strength are increasingly insufficient to meet the aggressive weight-saving targets of next-generation aircraft without compromising performance.

Composite materials, particularly carbon fiber reinforced polymers (CFRP), have emerged as the leading alternative due to their superior specific properties⁴. Landmark applications, such as the Boeing 787 and Airbus A350 wings, demonstrate weight savings exceeding 20%⁵. For secondary structures like horizontal tails, sandwich constructions with composite facesheets and lightweight cores (e.g., honeycomb or foam) are widely adopted for their high bending stiffness-to-weight ratio^6,7.

Despite these advances, the design of multi-material coupled structures—integrating dissimilar materials like CFRP, metallic joints, and polymeric foam cores—introduces significant challenges. The inherent stiffness mismatch at material interfaces can lead to severe stress concentrations, elevating the risk of adhesive bond failure or core crushing^8,9. Furthermore, the mechanical performance of such hybrid structures is highly sensitive to manufacturing process variations, which are often inevitable. For instance, the thickness of adhesive bonds and the density distribution of foam cores are subject to tolerances that can critically affect load transfer and global stiffness ^10,11. Current design practices frequently overlook the coupled effects of these manufacturing uncertainties, leading to a performance gap between as-designed and as-built structures and necessitating conservative safety factors that undermine weight-saving benefits¹².

Recent research has focused on optimizing multi-material structures for weight and stiffness ^13,14 and assessing their damage tolerance¹⁵. Numerical studies incorporating process-induced variations are also emerging¹⁶. However, a comprehensive methodology that integrates tolerance-driven design, quantitative uncertainty analysis of interfacial robustness, and experimental validation for aircraft horizontal tails remains lacking. Specifically, there is a need to: (1) quantify the sensitivity of key performance metrics (e.g., displacement, interfacial stress) to dominant manufacturing tolerances; (2) develop a robust optimization framework that minimizes performance variability; and (3) validate the predictions with tests that account for real-world process deviations.

To address these gaps, this study investigates the lightweight design and interface mechanical robustness of a multi-material horizontal tail structure comprising CFRP spars, closed-cell foam sandwich skins, and aluminum alloy joints. The specific objectives are:

1. To develop a high-fidelity 3D nonlinear finite element model capable of simulating the coupled mechanical response and the influence of adhesive and foam core manufacturing tolerances.

2. To perform a systematic sensitivity and Monte Carlo-based uncertainty quantification to identify the dominant tolerance parameters affecting structural performance.

3. To propose and implement a tolerance-driven robust optimization strategy to enhance performance stability against manufacturing variations.

4. To validate the numerical predictions through mechanical testing of scaled prototypes fabricated with controlled process deviations.

The remainder of this paper is organized as follows: “Methodology” details the methodology, including geometry, materials, finite element modeling, and the framework for tolerance analysis and robust optimization. “Results and discussion” presents the results and discussion on structural response, lightweight efficiency, tolerance sensitivity, optimization outcomes, and experimental validation. Finally, “ Conclusion” concludes the study and suggests future work.

Methodology

Geometric parameters and material configuration

The heterogeneous structure of the horizontal tail in this study is geometrically designed based on aerodynamic requirements and lightweight considerations (Fig. 1). It has a wingspan of 2988 mm, with a root chord length of 480 mm and a tip chord length of 470 mm, while the airfoil thickness tapers from 60 mm at the root to 30 mm at the tip along the span.

Fig. 1.

Geometry and dimensions of the horizontal tail finite element model. Key dimensions are annotated: wingspan (2988 mm), root chord (480 mm), tip chord (470 mm), root airfoil thickness (60 mm), and tip airfoil thickness (30 mm).

To balance weight and strength, a multi-material hybrid configuration is adopted: Spar: Optimized with T700-grade carbon fiber composites (unidirectional laminate stacking sequence optimized for load-bearing performance). Wing Surface: Features a foam sandwich structure, where the core consists of closed-cell polyurethane foam (density: 150 kg/m²) and the upper/lower skins are 1-mm-thick T700 composite laminates. Joints: Reinforced with aluminum alloy ribs (I-sectioned, 3 mm thick, 30 mm flange width) to enhance local stiffness. The surrounding skin in the joint area is thickened to 2 mm to mitigate stiffness discontinuities. The spatial arrangement of these materials within the assembly is depicted in Fig. 2, where distinct colors and patterns represent each material type, accompanied by a clear legend.

Fig. 2.

Material distribution within the horizontal tail structure. The legend identifies: Carbon Fiber Reinforced Polymer (CFRP) spar and skin (blue), closed-cell polyurethane foam core (orange), and aluminum alloy 7075-T6 joints/ribs (gray).

The mechanical properties of the constituent materials, as used in the finite element model, are summarized in Table 1. These values are based on manufacturer datasheets and standardized tests, as specified in the table.

Table 1.

Mechanical properties of materials.

Material	Elastic modulus (E)	Poisson‘s ratio (ν)	Density (ρ)	Strength	Source/standard
T700/Epoxy CFRP (unidirectional)	E1 = 135 GPa, E2 = E3 = 10 GPa	ν12 = 0.28, ν23 = 0.40	1700 kg/m3	Tensile: 2500 MPa (longitudinal)	Manufacturer data & ASTM D3039
Closed-cell polyurethane foam	Compressive modulus: 120 MPa	0.31	150 kg/m3	Compressive: 3.5 MPa	ASTM D1621
Aluminum alloy 7075-T6	71.7 GPa	0.33	2780 kg/m3	Yield: 503 MPa	MMPDS/ASTM E8/E9

Finite element modeling and nonlinear material constitutive laws

A high-fidelity three-dimensional nonlinear finite element model was developed in ABAQUS/Standard (version 2022) to simulate the mechanical response of the multi-material horizontal tail under static aerodynamic loading. The model explicitly accounts for geometric nonlinearities (large deformations) and key material nonlinearities, particularly in the foam core and adhesive interfaces.

Element types and mesh

The composite spar and skins were modeled using continuum shell elements (SC8R), which accurately capture bending and in-plane stresses while being computationally efficient for thin structures.

The aluminum alloy ribs and the closed-cell polyurethane foam core were modeled using 8-node linear brick elements with reduced integration (C3D8R) to avoid shear locking. A convergence study was performed, and a global element size of 10 mm was selected, with local refinement to 3 mm in the joint and spar-skin interface regions. The final mesh comprised approximately 850,000 elements.

Material constitutive models and interface modeling

CFRP Laminate: The T700/Epoxy unidirectional plies were modeled as a linear elastic orthotropic material. A layered composite section definition was used to assign the stacking sequence ([0°/90°/±45°] s) and individual ply orientations to the shell elements. Aluminum Alloy 7075-T6: Modeled as an isotropic, linear elastic-perfectly plastic material, with plasticity defined using the von Mises yield criterion and the yield strength listed in Table 1.

Closed-cell Polyurethane Foam: To capture its compressive nonlinearity and potential crushing, the foam was modeled using the crushable foam plasticity model with volumetric hardening available in ABAQUS. The input parameters (yield stress ratio, hydrostatic yield stress vs. volumetric strain) were calibrated against uniaxial compression test data per ASTM D1621. Adhesive Interface: The bonding between the spar, skin, and ribs was initially simulated using cohesive surface interaction with a traction-separation law (initially linear elastic, followed by a quadratic nominal stress criterion for damage initiation and a power-law criterion for evolution). This allows for the simulation of interfacial debonding. For the initial sensitivity studies where perfect bonding was assumed, “Tie” constraints were used as a simplification. The transition between these modeling approaches and their implications. The stress–strain relationship for a unidirectional CFRP ply in its material coordinate system (1,2,3) is given by:

where is the stress vector, is the strain vector, and C is the stiffness matrix. For a transversely isotropic ply (with 2–3 plane as isotropic),

with , , , , C₆₆ = G₁₂. E₁, E₂ are longitudinal and transverse moduli, ν_ij are Poisson’s ratios, and G₁₂ is the in-plane shear modulus. The closed-cell foam plasticity is governed by a volumetric hardening law. The yield surface F in the p-q stress space (p: hydrostatic pressure, q: Mises equivalent stress) is defined as:

where α defines the shape of the yield ellipse, p₀ is the initial hydrostatic tensile strength, and B is the hardening function dependent on the plastic volumetric strain .

Boundary conditions, loads, and validation approach

The root section of the horizontal tail, representing the attachment to the fuselage, was assigned an encastré (fully fixed) boundary condition.

The aerodynamic load was simplified as a quasi-static, uniformly distributed pressure of 2.5 kPa acting on the lower wing surface, resulting in a total lift-equivalent force of approximately 1400 N. This simplification is justified for a preliminary static strength and stiffness assessment. To validate the model’s predictive capability, a separate local sub-model of the critical joint region was created. The boundary conditions for this sub-model were imported from the global model analysis. This sub-model, featuring a refined mesh and the cohesive interface definition, was used for detailed interfacial stress analysis and later compared with strain gauge data from the prototype tests.

Manufacturing methods for constituent materials and prototypes

The scaled prototypes for experimental validation were manufactured to closely reflect the multi-material design and to intentionally introduce controlled process variations.

CFRP Components (Spar and Skins): The T700 carbon fiber/epoxy prepreg (Hexcel^®) was laid up on mold tools according to the designed ply book. The laminates were cured in an autoclave following the manufacturer’s recommended cycle (125 °C, 6 bar pressure for 120 min). Post-curing, the parts were ultrasonically scanned (C-scan) to ensure quality and absence of major voids or delaminations. Foam Core: The closed-cell polyurethane foam cores were machined from commercial panels (density 150 kg/m² ± 7.5 kg/m²) using a CNC milling machine to achieve the tapered geometry. Density variation within and between panels was characterized by cutting small coupons from different locations and measuring mass and volume, confirming a normal distribution with the stated tolerance.

Adhesive Bonding and Assembly: The surfaces to be bonded were mechanically abraded and cleaned. A two-part epoxy paste adhesive (Henkel Loctite^® EA 9396) was selected for its aerospace certification and controlled flow properties. The adhesive thickness was controlled using calibrated glass bead spacers mixed into the adhesive or fine wire meshes placed between parts, creating nominal thicknesses of 0.8 mm, 1.0 mm, and 1.2 mm for different specimen sets. The assembly was clamped and cured at room temperature for 72 h. Aluminum Joints: The 7075-T6 aluminum ribs were machined via 5-axis CNC milling. Before bonding, the faying surfaces underwent a phosphoric acid anodize (PAA) treatment per ASTM D3933 to enhance adhesion.

Tolerance analysis and robust optimization framework

Monte Carlo simulation for uncertainty quantification

The probabilistic analysis followed this procedure:

1. Parameter selection and distributions: Two key manufacturing variables were identified as stochastic: adhesive bondline thickness (t_adj) and foam core density (ρ_foam). Based on process capability data, t_adj was modeled as a normal distribution: N (µ = 1.0 mm, σ = 0.067 mm) to represent a ± 0.2 mm tolerance range (± 3σ). ρ_foam was modeled as N (µ = 150 kg/m³, σ = 2.5 kg/m³) for a ± 5% tolerance.

2. Sampling: A Latin Hypercube Sampling (LHS) scheme (Fig. 3) was used to generate 200 statistically independent sample sets of (t_adj, ρ_foam). LHS ensures better stratification over the parameter space compared to simple random sampling.

3. Simulation procedure: For each of the 200 parameter sets, a deterministic finite element analysis was run using a Python script to automate model parameter updates, job submission, and result extraction in ABAQUS.

4. Statistical evaluation: The outputs of interest—maximum tip displacement (U_max) and maximum compressive strain in the foam core (ε_foam−max)—were collected. Their empirical distributions were analyzed to calculate means, standard deviations, and 95% confidence intervals. Sobol’ indices were computed via post-processing the LHS results to perform global sensitivity analysis (GSA), quantifying the fractional contribution of each input parameter’s variance to the output variance.

Fig. 3.

Compression strain contour plot.

Tolerance-driven robust optimization formulation

The goal was to find a design that minimizes the performance sensitivity to the identified tolerances. A single-loop robust optimization approach was employed: Objective: Minimize the combined weighted sum of the mean (µ) and standard deviation (σ) of the tip displacement.

Minimize:

where w₁ and w₂ are weighting factors set to 0.7 and 0.3, respectively, prioritizing performance stability (σ). Design Variables: Skin thickness increment (∆t_skin), adhesive nominal thickness (t_adj−nom), and ply orientation of the outer layers. The tolerance-driven design is formulated as a single-loop robust optimization:

where d: Design variable vector (skin thickness increment, nominal adhesive thickness, ply angles). f(d): Objective function minimizing weighted sum of mean (µ) and standard deviation (σ) of max displacement. w₁, w₂: Weighting factors (0.7, 0.3). g₁, g₂, g₃: Constraints on mean mass increase, mean peak adhesive stress, and ply angle manufacturability.The agreement between calibrated FE model predictions (P_i) and experimental data (E_i) is quantified by the coefficient of determination:

Parameter explanation: : Coefficient of determination. , : i-th experimental and predicted value (e.g., displacement at a given load). : Mean of experimental data. n: Number of data points. =0.96 indicates excellent predictive capability.

Constraints: (1) Mean mass increase ≤ 1.5 kg; (2) Mean peak stress in adhesive ≤ 80% of its allowable strength; (3) All ply angles within manufacturable set [0°, ± 45°, 90°].

Method: The optimization was performed using an Adaptive Response Surface Method (ARSM) integrated with the Monte Carlo simulation. A Kriging surrogate model was iteratively built and refined to approximate the relationship between design variables and the statistical moments (µ, σ) of the responses, reducing the computational cost of numerous FE calls.

Results and discussion

Structural performance under design load

The static structural response of the multi-material horizontal tail under the equivalent aerodynamic design load is presented first to establish a baseline.

Figure 4 shows the displacement contour, with the maximum displacement of 188.8 mm occurring at the wingtip. This is 5.6% below the predefined aeroelastic design limit of 200 mm, thereby defining the stated safety margin. The displacement field exhibits a smooth gradient from root to tip, correlating with the spanwise reduction in bending stiffness due to the tapered geometry. Figure 4 reveals the critical role of the joint region.

Fig. 4.

Displacement contour plot of the baseline multi-material horizontal tail under design load.

A significant strain concentration is observed at the interface between the aluminum rib and the composite skin, with a peak tensile strain of 4210 µε. This value exceeds the allowable strain for the adjacent composite laminate (3200 µε) by approximately 1.3 times, highlighting the stress intensification caused by the abrupt stiffness transition. Conversely, the compressive strain in the foam core and skins remains well within limits, with a maximum of − 3461 µε. The shear stress distribution (Fig. 5) shows a maximum of 5.5 MPa in the spar, which is only 6.9% of the laminate’s shear strength, indicating the adhesive joints are not the critical failure path under pure static bending.

Fig. 5.

Shear stress (S12) contour (in MPa) in the composite spar.

Lightweight efficiency and comparative analysis

The mass breakdown of the optimized single-wing structure was calculated via the FE model’s volume integration.

The total mass is 17.8 kg, representing a 32% reduction compared to a reference all-aluminum 7075-T6 design of equivalent outer mold line and stiffness performance, which was estimated to weigh 26.2 kg. This significant saving is attributed to the high specific stiffness of the CFRP spar (contributing 37.1% of the mass but carrying the primary bending load) and the efficient use of the lightweight foam core (12.4% of mass). To contextualize this result, a comparative analysis with literature data is provided in Table 2. While direct comparison is challenging due to differences in scale, loading, and exact configuration, the achieved weight saving aligns with the upper range of values reported for composite-based tail structures^17,19, and notably surpasses that of simpler foam-sandwich-only designs²⁰. This underscores the benefit of the targeted multi-material, stiffness-graded strategy.

Table 2.

Comparative analysis of lightweight performance against literature studies on tail empennage structures.

Study (source)	Primary materials	Key finding/weight saving	Context
This work	CFRP, foam, Al alloy	32% vs. equiv. all-metal design	Full horizontal tail, static stiffness-driven
Herrera et al.	CFRP sandwich	15–25% (optimized design)	Horizontal tailplane, multi-objective optimization
Xin et al.	All-foam sandwich composite	~ 20% (implied)	UAV horizontal tail, strength validation
Banhart et al.	Metal foam sandwich	~ 30% (potential)	Empennage panels, conceptual study

Influence of manufacturing tolerances and sensitivity analysis

The impact of discrete tolerance changes was first examined (Figs. 6 and 7).

Fig. 6.

Impact of manufacturing tolerances on performance metrics.

Fig. 7.

Comparison of performance before and after robust optimization.

Reducing the adhesive thickness from the nominal 1.0 mm to 0.8 mm (a − 0.2 mm shift) caused the peak transverse shear stress at the spar-skin interface to increase from 5.31 MPa to 6.48 MPa (a 22% increase). Concurrently, a 5% reduction in foam core density (from 150 to 142.5 kg/m³) led to a 3.4% increase in wingtip displacement (188.8 mm to 195.3 mm) and an 18% increase in core compressive strain. The coupled effects are summarized in Table 3, demonstrating a non-linear interaction where combined variations cause greater performance degradation than the sum of individual effects.

Table 3.

Influence of coupled manufacturing parameter variations on key performance metrics.

Parameter change	Max shear stress (MPa)	Core comp. strain	Tip disp. (mm)	Skin bending stress increase
Adhesive: 1.0 → 0.8 mm	+ 22% (5.31 → 6.48)	–	+ 0.7% (→190.1)	+ 15.3%
Foam density: -5%	–	+ 18% (0.032 → 0.0379)	+ 3.4% (→195.3)	+ 23.7%
Combined effect	+ 32% (→7.02)	+ 28% (→0.041)	+ 6.8% (→201.6)	+ 37.2%

The Monte Carlo simulation, following the procedure, provided a statistical description of performance variability (Figs. 8 and 9). The maximum displacement (U_max) followed a near-normal distribution with a mean (µ) of 189.1 mm and a standard deviation (σ) of 6.2 mm. The Sobol’ sensitivity indices were calculated to quantify the contribution of each input parameter’s variance to the output variance. This confirms that adhesive thickness variability is the dominant source of uncertainty, accounting for 64% of the variance in displacement.

Fig. 8.

Monte Carlo simulation: displacement distribution.

Fig. 9.

Global sensitivity analysis (sobol indices).

The foam density, while significant for local strain, has a lesser effect on global stiffness. The non-negligible interaction term (12%) justifies the use of a coupled analysis over a one-factor-at-a-time approach. The input distributions for this analysis were based on measured process capability data from our prototype manufacturing, not merely assumed.

Robust optimization results and performance stabilization

Applying the tolerance-driven robust optimization framework yielded a design that is less sensitive to manufacturing variations²¹.

The key changes included: tightening the adhesive thickness tolerance to ± 0.15 mm, introducing a [± 45°] ply sequence in the skin near joints, and a slight increase in local skin thickness. The results, summarized in Table 4, show a marked improvement in robustness. The standard deviation (σ) of the maximum displacement was reduced by 50% (from 6.2 mm to 3.1 mm), indicating much tighter performance distribution. The probability of exceeding the allowable core strain dropped by 75.9%. These gains came with a modest mass penalty of 1.3 kg (+ 7.3% over the baseline 17.8 kg), preserving a net 24% weight saving over the all-metal design. The transverse shear modulus, critical for joint integrity, increased by 17.3%.

Table 4.

Comparison of performance variability before and after robust optimization.

Performance metric	Baseline design	Robust optimized design	Relative change
Mean max. displacement, µ (mm)	189.1	187.5	− 0.8%
Std. dev. of disp., σ (mm)	6.2	3.1	− 50%
Strain exceedance probability	8.7%	2.1%	− 75.9%
Mass (kg)	17.8	19.1	+ 7.3%
Transverse shear modulus (GPa)	4.2	4.93	+ 17.3%

Experimental validation and model calibration

Test setup and data acquisition

Scaled prototype tests were conducted to validate the numerical predictions (Fig. 10). The compressive strain was measured using 350-Ω resistance strain gauges (Tokyo Sokki Kenkyujo Co., Ltd., type FLA-2-350) mounted at critical locations, including rosettes at joint interfaces. The gauges were oriented to measure longitudinal, transverse, and shear strains. Calibration was performed against a certified strain simulator prior to testing. Displacement was tracked using five laser displacement sensors (Keyence LK-G500, accuracy ± 0.01 mm) positioned along the span. The load was applied via a servo-hydraulic actuator (MTS 322) with a load cell, following a quasi-static displacement-controlled protocol.

Fig. 10.

Experimental validation of process deviations in heterogeneous tail structure.

Comparison of numerical and experimental results

A direct comparison for the specimen with nominal 1.2 mm adhesive thickness is presented in Table 5. The agreement in joint strain is excellent (2.3% error). The discrepancy in tip displacement (7.4 mm or 3.8%) was systematically investigated. Post-test CT scanning revealed a gradient in foam core density across the wing span, varying by up to ± 6.8% from the nominal value, rather than the assumed uniform distribution. This spatial variability was the primary error source.

Table 5.

Comparison of key experimental and simulation results for the nominal specimen.

Metric	Experimental value	Simulation (baseline model)	Error	Primary identified cause
Peak joint strain (µε)	3270	3195	+ 2.3%	Minor adhesive porosity
Tip displacement (mm)	196.2	188.8	+ 3.8%	Spatial foam density gradient
Load at initial nonlinearity (kN)	1.18	1.25	-5.6%	Adhesive plasticity initiation

Model calibration and enhanced predictive capability

To bridge this gap, the FE model was enhanced by incorporating the measured density gradient field (via kernel density estimation of CT data) instead of a single uniform value^22–26. Furthermore, a cohesive zone model was activated for the adhesive interfaces in the joint sub-model. After this calibration, the displacement prediction error reduced to 2.0% (192.4 mm vs. 196.2 mm). The coefficient of determination (R²) between the calibrated model predictions and experimental load-displacement curves for all test specimens improved to 0.96 (Fig. 11), confirming high predictive fidelity. This process of identifying error sources and model updating directly strengthens the validation claim.

Fig. 11.

Load-displacement curves for different adhesive thicknesses.

Repeatability and reliability considerations

Three identical specimens were fabricated and tested for the nominal adhesive thickness case to assess repeatability^27–32. The coefficient of variation (COV) for the peak displacement was 1.8%, and for the peak joint strain was 2.5%, indicating good manufacturing and test repeatability. While the current study focuses on static performance variability, the link between this statistical variability and reliability metrics like probability of failure is crucial for design^32–37. Based on the obtained displacement distribution (µ = 187.5 mm, σ = 3.1 mm after optimization) and assuming a normal distribution (Fig. 12), the probability of exceeding the 200 mm limit is less than 5 × 10⁻⁵, demonstrating high structural reliability under the considered manufacturing uncertainties.

Fig. 12.

Influence of manufacturing tolerances on structural response.

Conclusion

This study presented a comprehensive investigation into the lightweight design and interfacial robustness of a multi-material aircraft horizontal tail structure. A synergistic design incorporating CFRP spars, foam-cored sandwich skins, and aluminum alloy joints achieved a 32% weight reduction while maintaining structural stiffness under design loads.

1. The core contribution of this work lies in the development and validation of a tolerance-aware robust design methodology. By integrating high-fidelity finite element modeling with a Monte Carlo-based uncertainty quantification framework, we systematically quantified the impact of key manufacturing tolerances—specifically adhesive bondline thickness and foam core density. The global sensitivity analysis conclusively identified adhesive thickness variability as the dominant factor, responsible for approximately 64% of the variance in global tip displacement. This finding directs critical attention to process control in bonding operations.

2. Building on this insight, a tolerance-driven robust optimization was implemented. The optimized design demonstrably enhanced performance stability, reducing the standard deviation of the maximum displacement by 50% and the probability of exceeding critical strain limits by over 75%, albeit with a modest 7.3% mass increase. This trade-off underscores the value of designing for robustness against inevitable process variations.

3. Experimental validation using prototypes fabricated with controlled deviations confirmed the model’s high predictive capability, particularly for localized joint strains. Discrepancies in global displacement were primarily attributed to spatial gradients in foam core density—a factor often overlooked in deterministic design. Calibrating the model with measured density fields and incorporating a cohesive zone model for the adhesive improved the agreement significantly (R² = 0.96), closing the gap between simulation and reality.

Limitations and Future Work: While this study focused on static stiffness and strain, future work should extend the framework to fatigue and damage tolerance analysis under cyclic loads, which are critical for certifying primary aircraft structures. Furthermore, the probabilistic analysis could be expanded to include a wider range of uncertainties, such as ply angle misalignment and environmental degradation of adhesive properties. Finally, exploring automated fiber placement or additive manufacturing of integrated structures could provide alternative pathways to mitigate the interfacial challenges identified herein.

Author contributions

This manuscript is original, unpublished, and not under review elsewhere. All authors contributed to: 1. M.L.: FEM development, optimization. 2. B.W.: Experimental validation, data curation. 3. C.L.: Conceptualization, supervision. No conflicts of interest exist.

Data availability

The datasets generated and analysed during the current study are available from the corresponding author on reasonable request. Project support The 2025 Open Fund of the IOT Technology Application Transportation Industry R&D Center [Grant Number: (202502)].    

Declarations

Competing interests

The authors declare no competing interests.