2.1. General Aircraft and Wing Data
The semi-tapered cantilever wing of the reference aircraft has a full span of 17.8 m and a reference area of 34.4 m
2 with an aspect ratio of 9.2. For the reference aircraft, the mass and stiffness properties of the aluminum wing box were determined based on the structural sizing and mass estimation methods developed by Elham [
3] and Torenbeek [
4], as described in detail in [
1].
Figure 2 shows a schematic of the wing platform, primary structure, and location of all other masses considered for each configuration. The heavy conventional turboprop engine with a mass of 576.2 kg is mounted on the inner wing at 30% of the half span and 8% of the local chord length relative to the leading edge. The fuel tank extends from the structural root at 9.3% to 92.2% of the half span. A retrofit scenario is assumed by removing the conventional fuel-powered propulsion system and integrating the distributed electric propulsion and battery modules into the wing structure. This alters the total mass and mass distribution of the wing, whereas the stiffness properties of the primary structure remain unchanged. The mass and distribution of the propulsion system and the battery modules for the electric configurations were also defined in the previous work [
1]. Each electric propulsion unit, comprising the propeller, electric engine, and switch, weighs 90.9 kg for the
DP4 and 61.3 kg for the
DP6 configuration. These concentrated masses are located near the leading edge at 30%, 65%, and 100% of the half span.
The mass breakdown of the wing and the eigenfrequencies of its first bending and torsional modes are provided in
Table 1. The powertrain mass includes the mass of the propulsion units and other powertrain components, such as electric cables. The mode shapes and eigenfrequencies are determined by modal analysis of the wing structure idealized as a beam stick model in the finite element software MSC Nastran (
https://hexagon.com/products/product-groups/computer-aided-engineering-software/msc-nastran accessed on 1 March 2025). The impact of the retrofitted powertrain on the modal properties of the wing is evident from the eigenfrequencies of the first bending and torsion mode of the wing structure. Despite a similar total wing mass for all three aircrafts, these eigenfrequencies are significantly lower for the wings with integrated distributed electric propulsion. More detailed information about the mass distribution and breakdown as well as the modal properties is given in [
1].
2.2. Load Cases and Load Spectrum
The load cases typically occurring in service are defined based on the detailed ground–air–ground (GAG) cycle as described by Grover [
5].
Figure 3 qualitatively shows the variation IN the nominal normal stresses in a lower skin panel of the wing structure over time for one GAG cycle. The following load cases are considered in this work for the fatigue analysis: 1-g cruise (relative to the ground static case), gusts (positive and negative vertical 1-cos gust), maneuvers (pull up and push over), and landing impacts (symmetric, no roll angle). Fatigue from cyclic loading is particularly relevant for the lower skin panels of the wing which during flight are almost permanently subjected to tensile stresses due to the upward bending of the wing as a result of the aerodynamic lift forces. The mostly constant positive mean stress (tension), superimposed by cyclic stresses induced by disturbances, is critical from a fatigue point of view. During taxiing, the mean stress in the lower skin panels is negative (compression) and no contribution to the accumulated damage is reported in the analysis of Grover [
5]. Therefore, taxiing loads are not considered in this fatigue analysis.
Statistical data recorded on a fleet of original Beechcraft 1900D aircrafts during operational service [
6] were used to define four reference missions for assessing the fatigue behavior of the wing for each aircraft configuration. The reference missions, as provided in
Table 2, were chosen to cover 90% of the flight missions and conditions in terms of distance, altitude, and speed. For each reference mission, only cruise conditions were assumed with constant altitude, speed, and aircraft mass, neglecting a reduction in fuel mass during the flight mission. The consideration of different cruise flight conditions ensures that overall, varying aerodynamic loads and aeroelastic behavior of the wing are accounted for.
Three load case-specific parameters (incremental maneuver load factor
for maneuvers, derived gust velocity
for gusts, and vertical landing impact velocity
for landing impacts) as well as their statistical frequency of occurrence per flight and mission were determined and varied based on the referenced statistical data for actual operational flight and ground loads for the Beechcraft 1900D fleet [
6]. To keep the computational effort acceptable, the extensive statistical data were discretized into bins. For example,
Figure 4 illustrates the discretization of the incremental maneuver load factor and the derived gust velocity at the flight level between 14,500 and 19,500 ft (for mission 3 at a cruise altitude of 17,000 ft), as well as the vertical landing impact velocity. The data points represent the total occurrence of the respective parameter value. For each bin, the mean value of the bin edges was used as the representative value for the load case. In addition to the four 1-g cases, this discretization results in a total number of ten different gust and maneuver load cases for each reference mission, and seven different landing impact load cases. The total number of occurrences of each load was defined as the sum of all data points inside the respective bin.
2.3. Modeling Approaches and Calculation Methods
The primary structure is an aluminum wing box consisting of skin panels (lower and upper) and spar webs (front and rear). It is idealized as a beam stick model along its elastic axis and discretized by 26 beam elements per half wing. The mass distribution is modeled using the lumped mass matrix for the wing structure, fuel, and batteries, considering additional concentrated masses for the propulsion units and other system components. The structural damping of the wing is modeled with a modal damping of 2% for all modes.
The analysis of the flight load cases is performed by finite element analysis with MSC Nastran. The aerodynamic loads are computed based on the Doublet Lattice method. The coupling of the aerodynamic and structural model shown in
Figure 5a is realized via finite beam splines. A fixed boundary condition is applied at the structural wing root. Assuming quasi-static behavior for maneuvers, static aeroelastic analyses (SOL 144) for the 1-g cruise and maneuver load cases are conducted. For the gust load cases, a dynamic aeroelastic analysis (SOL 146) of vertical 1-cos gusts defined according to CS 23.333 [
7] is performed. The total response is obtained by superposition with the solution of the 1-g cruise load case.
The landing impact load case is analyzed based on a dynamic response analysis implemented as a Python algorithm. First, the aircraft is modeled as a multibody system, see
Figure 5b, with rigid bodies connected to each other via spring, damper, and coupling elements. An analytical model according to Milwitzky and Cook [
8] is used to idealize the oleo-pneumatic landing gear to compute the time-dependent impact forces. The pitch angle of the aircraft is set to 3 ° as the average value in operational service according to the statistical data [
6]. Only symmetric landing load cases are considered in this study.
As a second step, the calculated landing impact loads are applied to the structural model of the wing depicted in
Figure 5c. The transient loads are applied at the fuselage center node. The dynamic response of the wing is obtained by modal analysis in combination with the phase plane method. The dynamic loads acting on the wing are calculated by implementation of the Williams method [
9]. The aerodynamic loads are assumed to be constant during the landing impact phase and equal to the landing weight of the aircraft. Again, the total solution is obtained by the superposition of the static and dynamic solution.
The fatigue analysis is conducted for the skin panels of the wing box. The alloy is assumed to be AL2024-T3 for the lower skin panels and AL7075-T6 for the upper panels. The rainflow counting method is used to discretize and determine load cycles as a combination of, firstly, value pairs of mean stress and stress amplitude, and secondly, their frequency of occurrences over the assumed service life. Considering elastic behavior and a safe life approach, the damage for each load cycle is calculated by the Palmgren–Miner rule yielding the damage parameter
D. The cumulative fatigue damage is obtained by summation of the damage for all load cycles and indicates failure for
. The S-N curves for the aluminum alloys were taken from the handbook MIL-HDBK-5J [
10]. The allowable stresses consider a notch factor
of 2.0 to approximate the effect of stress raisers, such as holes and rivets, which are paramount for the fatigue analysis but cannot be captured by the beam model. To adjust the S-N curves for varying mean stresses, the Walker equation [
11] is used and calibrated by the material data given in the referenced handbook.