Curved Shear Panel Theory as an Enabler for Gradient-Based Wing Optimization ^†

Abstract

In the preliminary design of aircraft structures, efficient modelling techniques are essential to balance accuracy and computational cost. Shear Panel Theory (SPT) offers a simple yet effective idealisation of thin-walled, stiffened structures such as wings. It captures more structural detail—like ribs, sweep and taper—than traditional beam idealisation and would otherwise require detailed finite element analysis. However, compared to a finite element model, the degrees of freedom of the structure as well as the meshing effort are significantly reduced, as SPT idealisation uses a structural element approach. This improves mass estimation and structural response calculation and makes SPT particularly well-suited for optimisation tasks in early design phases. This work presents a methodology to derive structural properties of wing segments based on NACA airfoils using SPT. This offers adjustment of the wing’s geometry for use in aeroelastic analysis and enables fast evaluation of structural behaviour and gradient computation, supporting integration into multidisciplinary design optimisation frameworks. The proposed methodology advances the use of idealised structural models in aircraft design by bridging the gap between high-fidelity analysis and system-level aeroelastic simulations, supporting faster and more informed early design iterations.

1. Introduction

Aircraft design is increasingly influenced by multidisciplinary requirements, concerning structure, environmental impact [1], cost and manufacturing [2,3,4]. This necessitates close cooperation in co-development. In the field of wing design, this co-development exists in the form of aeroelasticity as an independent discipline, where the structural design and the aerodynamics are strongly coupled due to the interaction of elastic and dynamic loads as well as aerodynamic loads [5,6]. The interaction of the three sub-disciplines has many influencing factors in wing design and is referred to as aeroelastic tailoring. The literature describes various strategies for aeroelastic tailoring of wing structures, for example, through anisotropic and unbalanced laminate structures [7], sweep angles [8,9] or the choice of rib arrangement and orientation [8,9,10,11].

To manage complexity, a formalised and consistent hierarchical modelling approach can be adopted [12]. To process the available information and obtain the information required for the specific design stage, appropriate models must be selected. Particularly in the early design phases, a very large design space must be investigated, and the key influencing factors must be understood, which requires simple and efficient models [13]. In the context of aeroelasticity, so-called reduced-order models are used for this purpose [14,15]. Various strategies have been developed [14,15,16], including VABS [17], static Guyan reduction [18] and dynamic Craig Bampton reduction [19]. However, these models often use CAD-based Finite Element Method (FEM) models, which allow parameter studies to be carried out, but still make it difficult to efficiently investigate different configurations.

Before the establishment of FEM, the shear panel theory (SPT) was used for the structural assessment of wings [20]. This theory involves the discretisation of structures into shear panels and surrounding stiffeners. Since this theory is capable of calculating swept, tapered and curved wing structures and makes the force flows in the structure easy to comprehend, it was a common calculation method. In the past, several shear panel formulations were derived and implemented in FEM, which were general quadrilaterals, but all flat [21,22,23]. Bäß and Schröder [24] derived a three-dimensional formulation of a simply curved panel, which can be used to efficiently calculate wing structures with curved profiles. In this paper, the curved SPT-formulation by Bäß and Schröder [24] will be investigated with regard to its suitability for calculating the structural response of wings with curved NACA profiles for the concept and preliminary design phases. For this purpose, a NACA wing, based on [10], is examined with the SPT model and a FEM reference model, as well as coarser FEM models, and the structural mechanical properties that are of particular interest for the interaction in aeroelasticity, such as the elastic axis, are compared. Furthermore, the influence of wing spar positions and NACA numbers on the elastic axis is investigated to show that the SPT can be used as an efficient model for gradient-based optimisation due to its reduced number of degrees of freedom.

2. Materials and Methods

In this study, a NACA 5310 wing model was investigated, which is based on the wing model of [10] and is shown in Figure 1. The wing has a span of 630 mm and a chord length of 100 mm. Ten ribs are defined and evenly spaced at 63 mm intervals along the span. The entire structure is made of steel, with a Young’s modulus of $E=$210,000$\mathrm{MPa}$, a Poisson’s ratio of $\nu =0.3$, and density of $\rho =7.85\times {10}^{-9}\mathrm{t}/{\mathrm{mm}}^3$. The skin thickness of the wing is $1\mathrm{mm}$, and the ribs have a thickness of $0.5\mathrm{mm}$. At the four intersection points of the spars and outer profile, stringers with a parametrised cross-sectional area A are defined.

To assess the applicability of the curved SPT formulation for wing modelling, the SPT model is compared with a detailed FEM reference model and two additional FEM models with coarser meshes, which feature a reduced number of degrees of freedom to decrease computational cost. The following four models are investigated:

• A curved SPT model;
• A detailed FEM model serving as reference;
• A medium-mesh size FEM model;
• A coarse-mesh size FEM model.

The SPT model is discretised into structural elements based on the intersection lines of the skin, spars, and ribs. The nose, spar caps, and tail region are represented by curved SPT elements, whereas the spars and the ribs between the spars are modelled using rectangular SPT elements. The enclosed and surface areas required for the curved SPT elements are computed numerically from the NACA surface points.

Straight truss elements are used to model both the stringers and the frames around the ribs. The normal stiffness of the skin is smeared onto the adjacent truss elements. The FEM wing models are created in Abaqus using S4R shell elements and T3D2 truss elements.

Table 1. Definition of models and its mesh parameters.

Model	Approx. Global Element Size [mm]	Number of Elements	Number of Nodes	Degrees of Freedom
SPT	-	140	40	120
FEM	1.75	49,836	47,092	282,552
FEM mid	20	784	501	3006
FEM coarse	100	186	80	480

summarizes the mesh parameters, the resulting number of elements and nodes, and the total degrees of freedom of the respective models.

Two load cases are applied to the models. For the bending load case, a tip load of 1 N is applied in the z-direction to the four corner nodes of the spars. The average deflection of a cross-section is defined as

$$\begin{array}{c}\hfill u_z={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^4}{u}_{z,i},\end{array}$$

(1)

and the average rotation is defined as

$$\begin{array}{c}\hfill {\varphi }_x={\displaystyle \frac{1}{4}}{\displaystyle \sum _{i=1}^4}{\varphi }_{x,i}.\end{array}$$

(2)

The rotation ${\varphi }_{x,i}$ at the corner node i is computed from its change in orientation between the unloaded configuration (0) and the loaded configuration (1). Let $r_{i,0}$ and $r_{i,1}$ denote the 2D vectors (in the x-z plane) from the cross-section centre to corner node i in the unloaded and loaded states, respectively. Then, the signed angle between them is

$${\varphi }_{x,i}=atan2\left(-\left(r_{x,i,0}{r}_{z,i,1}-r_{z,i,0}{r}_{x,i,1}\right),r_{x,i,0}{r}_{x,i,1}+r_{z,i,0}{r}_{z,i,1}\right).$$

(3)

The bending stiffness is then given by

$$\begin{array}{c}\hfill k_b=Q_z/u_z.\end{array}$$

(4)

For the torsion load case, a torsional moment is applied at the tip by applying concentrated forces of 1 N in the negative z-direction to the front spar corner nodes and in the positive z-direction to the rear spar corner nodes. The torsional stiffness is defined as

$$\begin{array}{c}\hfill k_t=T_y/{\varphi }_y.\end{array}$$

(5)

The calculation of the elastic axis follows the approach presented in [25]. Transverse loads $Q_z$ and torsional loads $T_y$ are applied individually to each rib, and the resulting deflections and rotations are measured. The distance e of the shear centre from the line of action of the applied transverse force is then obtained from

$$\begin{array}{c}\hfill e=a{\displaystyle \frac{T_y}{Q_z}},\end{array}$$

(6)

with the scaling coefficient a given by

$$\begin{array}{c}\hfill a={\displaystyle \frac{{\varphi }_{y,Q}}{{\varphi }_{y,T}}},\end{array}$$

(7)

with ${\varphi }_{y,Q}$ denoting the torsional rotation due to a transverse load, and ${\varphi }_{y,T}$ indicating the torsional rotation due to a torsional moment.

3. Results and Discussion

The bending and torsional stiffness of the wing for the respective models are shown in Figure 2 as a function of the stringers’ cross-sectional area. Figure 2a shows that the SPT model closely matches the FEM reference model across the entire range. This indicates that the curved shear panel formulation captures the global bending behaviour with high accuracy, while being extremely low in computational cost. In contrast, the mid and coarse FEM models show deviations for smaller stiffener areas, which suggests that these mesh levels are not sufficient to capture the bending behaviour.

A similar trend is observed for the torsional stiffness in Figure 2b. While the SPT model remains in good agreement with the reference, the mid and coarse FEM models exhibit significant deviations throughout the entire range of stiffener areas. This reinforces that coarse shell models tend to underestimate torsional stiffness, primarily due to their inability to correctly model the enclosed areas of the cells. All models, however, correctly reproduce the qualitative influence of warping restriction.

Figure 3a presents the bending–torsion coupling induced by the offset between the line of action of the transverse load and the shear centre. The SPT model reproduces this coupling well, while the mid and coarse FEM models show deviations of up to 25%. Figure 3b shows the calculated position of the elastic axis along the span. Again, the SPT model exhibits a close match with the FEM reference, whereas the mid and coarse FEM models show noticeable discrepancies.

Besides stiffness and the elastic axis, the eigenmodes and eigenfrequencies are also of interest in wing design. Figure 4 illustrates the first three bending and torsional eigenfrequencies for the investigated models. Overall, the models agree well with the reference. The coarse model shows the largest deviation—about 10% in the second eigenmode—while the SPT model deviates by less than 5% in the third bending mode.

In aeroelastic tailoring, shifting the aeroelastic axis is of major interest, as it allows designers to control bending–torsion coupling. The SPT model with a stringer cross-sectional area of $1000{\mathrm{mm}}^2$ is used to demonstrate how the elastic axis can be manipulated through structural design decisions. Figure 5 shows the resulting shear centre shift at the wing tip for variations in the front (a) and rear (b) spar positions. The sensitivity of the shear centre location is greater for variations of the rear spar.

Figure 6 illustrates the influence of NACA airfoil parameters on the elastic axis position along the span. The largest changes occur due to variations in the airfoil thickness, see Figure 6a. In contrast, variations in camber and maximum camber position, Figure 6b,c, have only a minor effect on the elastic axis position.

The system size could further be reduced by applying the concept of flexibility influence coefficients [5] to the SPT model. For this purpose, uniform shear loads and torsional loads are applied to each cross-section, analogous to the calculation of the elastic axis, and the average deflection and rotation are measured. Since the z-displacement, x-rotation and y-rotation are of primary interest in wing design, the system reduces to one node per cross section with three degrees of freedom.

Overall, the combined results confirm that the curved SPT reproduces the bending stiffness, torsional stiffness, bending–torsion coupling, elastic-axis location, and eigenfrequencies of the reference FEM model with high accuracy. Compared to mid and coarse FEM meshes, the SPT model delivers significantly improved predictive capability at a small fraction of computational cost. This makes SPT a powerful tool for preliminary wing-structure design and aeroelastic tailoring studies.

4. Conclusions

This work demonstrates the applicability of the curved SPT as a structural model for aeroelastic wing design in early design stages. The curved SPT model is able to accurately model multi-cell wings with curved surfaces. The computational cost compared to FEM models with a reduced number of elements is extremely low, while providing more accurate results in the calculation of bending and torsional stiffness, elastic axis position and dynamic properties. The results further show that structural design decisions, such as spar placement or modifications to NACA airfoil parameters, can be used to tailor the aeroelastic response by shifting the elastic axis. Due to the low computational cost and accuracy, the curved SPT is well-suited for gradient-based optimisation of aeroelastic wing structures, enabling efficient exploration of design variations in the preliminary design phase.