1. Introduction

1.1. Structural Analysis of Wings

Structural analysis refers to the understanding of how loads on physical structures and components affect their response and behaviour. This kind of analysis is done on structures that are required to withstand loads applied on them. It is an important part of design since the results of this analysis helps determine the safety factor of the structures; such that it is appropriate for its functions. This, in terms of the structures capability to resist deformations, support reactions, forces, and its basic stability itself is what is analysed without having to build a physical model.

During the design of any wing, it is crucial to keep a few aerodynamic and geometric constraints in mind. These affect the overall performance of the wing structurally too. Some of the vital constraints, among others, that must be given attention to are:

- Stall speed

- Boundary layer control

- Take-off and landing distance

- Stall angle

- Sweep angle

- Wing span

- Angle of attack

Also, the loads that an aircraft wing has to carry, support and withstand during its operational life cycle must be taken into consideration. Loads in general can be classified into three.

- Dead loads: these loads are those which are static or relatively constant over a relatively long period of time. For example, the weight of the wing is a dead load.

- Live loads: these are dynamic loads that are unstable and caused randomly with time. These may be caused due to a sudden impact, or due to vibrations.

- Cyclic loads: loads which are repetitive or vibrational that cause fatigue or failure are called cyclic loads.

The basic loads considered on a wing are:

- Lift and pressure distribution

- Weight of the fuel

- Weight of the wing itself

- Weight and position of the engine mounts

- The fuselage point load

- Wake turbulence vortices

Many a times, for the sake of simpler calculation and evaluation, some of the loads are assumed, approximated or neglected.

The most important one of them, which cannot be ignored, is probably the lift load which is generally considered as a straight load. When the lift does not coincide with the spars, twisting occurs. The lift is transferred to the spars through the ribs.

1.1.1. Basic Methods of Structural Analysis

There are basically two methods of structural analysis that are recognized and used. They are either analytical or numerical methods.

Analytical methods give closed form solutions and are mainly practical for linear elastic problems. Two main approaches of analytical method are the mechanics of materials approach and the elasticity theory approach.

Numerical methods can also be applied in two ways; either obtaining solutions to differential equations for displacements or stress estimations or using matrices for discrete-element approach. This discrete – element approach is also known as Finite Element Analysis (FEA) or Finite Element Method (FEM).

Since the analysis of wings requires non-linear computations, numerical methods are considered more appropriate for this field of application.

1.1.2. Relevant Research

Youhua Liu, of the Virginia Polytechnic Institute and State University, in his paper, ‘Efficient Methods for Structural Analysis of Built –Up Wings’ discusses a few techniques to evaluate the structural conduct of built – up wings.

One of the methods discussed, Equivalent Plate Analysis or simply EPA, uses an equivalent plate model, which was formed in order to investigate the static and free-vibration problems of built-up wing structures composed of skins, spars, and ribs. The model considers the transverse shear effects by considering the built-up wing as a plate by adhering to the Reissner-Mindlin theory (FSDT). Formulations are such

that there is no limitation on the wing thickness distribution. This EPA method is analysed by comparing the results that are attained using MSC/NASTRAN, for a set of examples including both static and dynamic problems. In general, the EPA’s main goal is to solve a wing problem by assuming that the wing behaves like a plate. This assumption is very reasonable as long as the wing has a small thickness-chord ratio.

The Equivalent Plate Analysis (EPA) can also be founded as a basis to create other efficient methods to incorporate tools for optimization while simultaneously processing an optimal design.

One of approaches discussed by Liu is to use the Artificial Neural Network (ANN), or simply called Neural Network (NN) in order to simulate the responses of wing structures. This can be applied in two ways; either directly or indirectly. The direct application makes use of the FEA or EPA method to generate results directly as the output. Alternatively, in the indirect application, the inner structure of the wing is joint to the skins to form an "equivalent" material. The constitutive matrix provides a relation between the stress vector and the strain vector. It is found that this EPA with indirect application of Neural Networks, which is also called an Equivalent Skin Analysis (ESA) of the wing structure, is a more efficient method than the EPA in addition to obtaining fairly good results.

Another methodology involves using the sensitivity techniques. Sensitivity techniques are frequently used in structural design practices for searching the optimal solutions near a baseline design. In this particular paper, Liu approximates the modal response of the general trapezoidal wing structures using shape sensitivities up to the second order.

The use of the EPA method was also made by Gern, Inman, and Kapania in their paper ‘Structural and Aeroelastic Modeling of General Planform Wings with Morphing Airfoils’ in the same way as the first paper. They applied the same model to analyse the roll performance of a flapless smart wing with morphing airfoils.

Dan Doherty talks about ‘Analytical Modelling of Aircraft Wing Loads’ by using MATLAB and a software called the Symbolic Math Toolbox. In this paper, he shows a method in which he derives analytical models of the loads and bending moments on the wing of a small passenger aircraft to determine whether the wing design meets the strength requirements.

The models are derived in the notebook interface in Symbolic Math Toolbox. Then the use of data management and analysis tools in MATLAB are used in order to simulate the models for different scenarios to verify that anticipated bending moments are within design limits. The analysis is done by evaluating the three primary loads that act on the aircraft wing: aerodynamic lift, load due to wing structure weight, and load due to the weight of the fuel contained in the wing.

Another research done by an AIAA member, Zweber, and others, titled ‘Structural and Manufacturing Analysis of a Wing Using the Adaptive Modelling Language’ examines the application of the Adaptive Modelling Language to the wing design process. Also known as AML, this software uses a unified part model paradigm. This implies that the part model of a wing can include all the data required for a panel method aerodynamic analysis, and structural analysis through an equivalent plate structural analysis and a finite element analysis. Since some of the information is needed by all three analyses, such as the span, chord lengths, sweep angles, using the part model concept simplifies the storage of the data and insures that all analyses are using the same values of the common information.

1.2. Finite Element Analysis

Finite element analysis (FEA) or Finite Element Method (FEM) is one of the most commonly used methods for numerical analyses of partial differential equation and is most often applied for structural analysis. It works by using mathematical tools of Eulers, or Runge – Kutta in order to approximate the partial differential equations to ordinary differential equations and then solve them. Other than structural analysis, FEA also helps solve elasticity problems. The method is especially advantageous in the civil, architectural and mechanical, and also aeronautical fields.

In the early 1940s, work done by Richard Courant is recognized as the beginning of the formulation of the finite element method. The concept behind the method is to select nodes or points within an interval or domain to create discretised elements. These elements were triangular and upon developments further into the years, lattice forms were also applied. These integrate in order to generate a web – like mesh system that solves each element. In the late 1950s, FEA was introduced for airframe and other structural analyses in the aeronautical field. From initiation of NASA, the NASTRAM software of FEA was developed. This used algorithms that used stiffness matrices.

For the sake of engineering structures of safety and durability, FEA simulates the physical system of a structure into a mathematical model by using matrix algebra. It is a computer aided program that helps engineers most in the designing phase of any object. Be it automobiles, aircrafts, or any other structure under stresses and lads, FEA based programs provide estimated visualizations of their responses. This is extremely helpful, since these approximations help optimize a design in order to increase safety, resistance, the life of the structure and in turn helps reduce damages and costs. This also helps engineering companies design in accordance to the client’s requirements.

When data is input in the software that defines the material, geometry and other physical properties such as density, etc., the results of built model undergoing loads

of any kind show the probable deformations and their approximate intensities. Also, the shape of deformation can be assumed.

Considering the geometry, on his paper on the FEA of wings and fuselage, Muhammad Ismaeel, P.E, comments that for the analysis of a wing, the structures that are basically considered are bars, beams, and plate elements along with shear flow panels.

1.2.1 Principle Procedure of the FEM

As David Roylance of the Massachusetts Institute of Technology points out in his paper, there are three critical steps of the FEA –

1. Pre – processing: the stage where the user generates the model of the structure to be analysed and defines the geometry. Also, node points are identified and the model is meshed. Loads and boundary value conditions are defined.

2. Analysis/Process: after the pre-processing stage where the initial data is fed, the software uses the inputs to form a matrix of linear or non-linear equations and solves them. The principle equation is

Kijuj = fi

where, uj represents displacements, fi the forces applied and Kij is defined as the stiffness matrix. This implies that for any node, there is the applied force f and its corresponding deflection u.

3. Post – processing: this stage displays the results of the structural analysis in a colour coded manner to indicate levels of stresses or deformations.

1.2.2. Advantages and disadvantages of the FEA

A few advantages of the Finite Element Method can be briefly listed as:

- It allows for an improved perception of the key design parameters.

- It eliminates the need for building a prototype and allows for testing activities to be done virtually to increase efficiency and for them to be conducted at minimal costs.

- This type of analysis is fast and fastens the designing phase which leads to more time invested in productivity and higher revenues.

- FEA computational programs are capable of handling extremely complex geometries and models.

- A wide range of engineering scenarios can be analysed by the programs. These include static structural, dynamic problems, thermal problems, fluid dynamics as well as electrostatic problems.

- The FEM is also capable of resolving indeterminate structures.

- Different kinds of complex loading can also be run through the program. For example, nodal loads, pressure, inertial or thermal forces, or even time or frequency dependant loads can be handled.

The disadvantages, although a few, are noteworthy. The following are identified by Hamdan, from Universiti Teknologi Malaysia, and by De Weck and Yong Kim from MIT, as the most important of them:

- The FEM does not obtain accurate results. It is based of mathematical formulations that compute just approximations.

- Errors in input data by the user affect the results by a significantly large margin. Also any inaccuracies by the user with relatively less experience with the software may often lead to incorrect results. For instance, weak choice of the type of elements or a poorly built model will cause inaccuracies.

- The effect of materials, geometry, and other such variable parameters on the stresses is not identified specifically. The relation between them cannot be solely examined.

- The finite element method itself consists of inherent errors due to round-off and accumulation of errors.

1.2.3. Relevant Works on FEM

A lot of literature review has been done based on the finite element method as can be seen from previous discussions. This is because the method can be widely applied to very specific kinds of problems in various situations. T. Bratanow and A. Ecer from University of Wisconsin discuss the different applications of finite element method in unsteady flow in their paper. They analysed the unsteady flow about a stationary as well as an oscillating NACA 0012 airfoil. They also applied the FEM to solve the Navier-Stokes formulation more accurately by defining the boundary layer more accurately.

Webster et Al proposed an adaptive finite element method for time variant and unsteady transonic compressible aerodynamics. The mesh that is generated is enriched by optimizing the element size ‘h’ and by controlling the shape of the elements as well.

J. T. Oden and L. C. Wellford Jr. analyse the flow of viscous fluids using the finite element method. In their work, they generalize formulations for 3 dimensional problems of heat conduction and flow of compressible and incompressible. Local approximations of parameters such as density, temperature, velocity are made to generate finite element analogues of the continuity, momentum and energy equations for compressible or incompressible fluids.

A.K. Noor and S.N. Atluri denoted the main reasons for future developments in Computational Structural Mechanics (CSM) (inclusive of FEA programs) as:

- The need for enhanced efficiency in production and engineering systems which are cost-effective.

- Sustenance of new and advanced high-tech industries such as the aerospace and aeronautics industry, transportation, petroleum, nuclear energy, and microelectronics.

1.3. Ansys

Ansys is an engineering-simulation-software that helps predict the structural responses of different products in the designing stage. It is one of the programs built on the basis of Finite Element Analysis.

The company, also called Ansys, is centred in Pennsylvania, US, and was founded in 1970; and is since one of the most successful simulation software developer.

1.3.1. Types of Structural Analysis

Ansys proves to be highly useful and capable in terms of the kinds of structural analyses it can run.

- Static Analysis – is used to determine displacements, stresses, etc. under static loading conditions. Ansys can solve either linear or nonlinear static analyses. Nonlinearities can include plasticity, stress stiffening, large deflection, large strain, hyper-elasticity, contact surfaces, and creep.

- Modal Analysis – is used to calculate the natural frequencies and mode shapes of a structure.

- Harmonic Analysis – is used to determine the response of a structure to harmonically time-varying loads.

- Transient Dynamic Analysis – is used to determine the response of a structure to arbitrarily time-varying loads. Non – linear problems can also be solved and analysed.

- Spectrum Analysis – is an extension of the modal analysis, used to calculate stresses and strains due to a response spectrum or a PSD input (random vibrations).

- Buckling Analysis - is used to calculate the buckling loads and determine the buckling mode shape. Both linear (eigenvalue) buckling and nonlinear buckling analyses are possible to carry out.

1.4. Different Subsonic Wing Configurations

Until date, a lot of different kinds of wings have been designed for the purpose of different characteristics. This section of the report gives an overlook at some of the most commonly known wings today.

Wings can be categorized mainly by their location, their planform shape and their sweep angle.

With reference to location, wings can be mainly classified as:

- Low – wing: the wing of the aircraft lies at the belly or at the lower part of the fuselage. This implies that the engines are not mounted on the wing and may be mounted at the rear of the aircraft.

- Mid – wing: the wing is fixed at the centre of the fuselage and. This configuration allows for sufficient ground clearance for the engines and is common for commercial aircrafts.

- High – wing: these wings are mounted at the top of the fuselage. These kinds of wings allow for ground clearance but access to interior of the plane is difficult. Also, the engines mounted on the wings tend to cause higher levels of noise in the plane. For these reasons, high wings are seen mainly in military aircrafts.

Figure 1: Wing Positions (Sky High)

With respect to planform shapes of the wings, there are several configurations.

- Elliptical wing: the wing shapes out in an elliptical manner span wise and is curved at the end. These wings account for an optimal lift distribution and seem to be the most efficient wings in terms of their aerodynamic performance. However, they are comparatively more challenging to build.

Figure 2: Elliptical Wing

- Constant chord wing: this wing has the frame of constant airfoils along the span of the whole wing. This means that the leading edge and trailing edge are parallel to each other. This wing is also known as rectangular wing.These wings, although simple to build, have poor aerodynamic properties like lower lift and higher drag.

Figure 3: Rectangular Wing

- Tapered wing: the chord of the wing narrows down from the root of the wing through to its tip, the leading edge and trailing edge coming closer together at the tip with straight edges as the wing tips. This type of wing is probably the most common wing in this day. It is a moderately simple wing to build and has good aerodynamic performance as well.

Figure 4: Tapered Wing

- Reverse – tapered wing: the win widens from the root to the tip of the wing. This is a very poor design of a wing structurally, since it amounts to instability in terms of the weight.

Figure 5: Reverse - Tapered Wing

- Trapezoidal wing: the wing is a tapered wing with a small wing span and hence has a small aspect ratio.

Figure 6: Trapezoidal Wing

- Delta wing: A delta wing is one where the trailing edge is straight, the leading edge sweeping way back in order to form a triangular shaped wing.

The structural efficiency of the wing is beneficial. However, the wing has a low wing loading capability. There are many variations of a delta wing. The figure illustrates the simple or tailless delta wing.

Figure 7: Delta Wing

The sweep angle of the wing is a major factor in its performance. Wings may be distinguished by the different sweep characteristics.

- Straight wing: is similar to the tapered wing with the trailing edge and leading edge swept towards each other.This wing is considered to be most efficient structurally and is the most common form of sweep.

Figure 8: Straight Wing

- Swept wing: the leading edge and trailing edge are both swept backwards with different angles to meet with straight edges.Swept wings are commonly used for aircrafts that fly at upper-subsonic ranges and are used widely for commercial aircrafts. This is because they are aerodynamically stable.

Figure 9: Swept Wing

- Forward swept wing: the leading and tailing edges are swept forward but are structurally not as stable and are aeroelastically, very prone to flutter.

Figure 10: Forward Swept Wing

- Variable sweep wing: the wing is capable of changing its sweep angle for different requirements. These wings are used mainly for military and combat aircrafts.

Figure 11: Variable Sweep Wing

1.5. Aeroelasticity

One of the most famous and fitting definitions of Aeroelasticity was given by Arthur Collar in the year 1947 as:

"The study of the mutual interaction that takes place within the triangle of the inertial, elastic, and aerodynamic forces acting on structural members exposed to an airstream, and the influence of this study on design."

This essentially means that aeroelasticity is the science that explores the effect of the inertial, elastic or aerodynamic forces on any structure that is subjected to an airstream and their impact on the design of the structure.

Since aircraft structures are not completely rigid bodies, deformations could result in variations in the aerodynamic forces. These variations or additions of loads could further increase the structural deformations. This phenomenon may occur as a loop

wherein two situations can be of result. One possibility is that the interactions may decrease in their magnitude and reach a state of equilibrium. Another chance is that, resonance occurs and the interactions increase, which may be disastrous.

Hodges and Pierce of Georgia Institute of Technology pointed out that aeroelasticity may be categorized into three:

- Structural Dynamics: study of the interaction of elasticity and dynamics.

- Static Aeroelasticity: study of the interaction between aerodynamic and elasticity on an elastic structure.

- Dynamic Aeroelasticity: examines the interaction between all; inertial, elastic, and aerodynamic forces.The most important topic pertaining to wings is the phenomenon of flutter, which is a form of dynamic instability.

1.5.1. Flutter

Flutter refers to the vibration of flexible structures when the natural frequencies of the structure mingle with the aerodynamic forces. That is, aerodynamic forces increase vibrations and this hence increases the aerodynamics loads and so on. This is known as a positive feedback and this is what causes flutter.

Flutter is a catastrophic phenomenon that occurs in strong airstreams. It is a dynamic instability in occurs in elastic bodies. It can lead to structural failure and hence is an issue that requires much attention and consideration.

Flutter may take place at any elastic part of the aircraft. The component that is most affected by flutter is the wing of the aircraft. Although on a smaller scale, other airfoil-shaped parts such as the tail, rudders, elevators, and stabilizers also experience flutter.

Flutter is a highly undesirable occurrence in aircrafts since it can cause instabilities that can be fatal. Hence, it is extremely studied in order to understand it and come up with solutions so as to fix it.

C. Herbert et al. discussed a few possibilities of fixes. Flutter can be analysed computationally and considerations during the design stage can help make changes before the aircraft is built, tested and flown.

The maximum air speed is a design parameter that could help prevent flutter. At the critical air speed, the oscillation becomes steady and this can be designed to be low to provide a margin of safety. Another approach is to vary the distribution of mass so that it moves the centre of gravity nearer to the centre of twist. This helps to uncouple the torsion and the bending to reduce flutter. Yet another simplistic

measure that can be taken is to strengthen the structure by increasing the stiffness-to-mass ratios within itself so as to have an increase in its natural frequencies. When the flutter is lowest and is at the least harmful level, it is perceived as a buzz. This is a safe and manageable degree of vibration.

1.6. Tuning of the Wing’s Natural Frequencies

The aircraft wing is susceptible to several vibrational excitations. The list is rather large but the main factors appear below.

- Wind Gusts

- Engine Noise

- Engine Shaft Vibration

- Engine Vibration at the pylon

- Fuselage Vibration

- Turbulence

Each of these phenomena is a vibrational load the wing must suffer from during flight. These loads can cause vibrations on the wing structure. In a typical scenario, the input vibrational load is eventually damped out and the system dissipates the energy.

However, if the frequency of that applied vibrational load is close to the modal/natural frequency of the structure, the damping is at its minimum and the system stores the vibrational energy. Even a small load can cause large amplitude vibrations and this is very likely to result in a resonance disaster.

Hence, the first step to tackle this problem is to carry out the modal analysis for the wing. Once this is done the designer can check the expected frequency of the vibrational loads versus the calculated modal frequencies.

Theoretically, as long as the modal and input frequencies are not equal, the said structure can be labelled safe as far as a possible resonance disaster is concerned. However, to add a factor of safety, the designer may need to adjust the design in a manner that puts the modal frequencies in a range that is away from the excitation frequencies by a predetermined percentage.

2. Different types of Analyses

This section intends to discuss the main analyses performed and describe their procedures briefly.

2.1. Static Structural

Static structural analysis can be used to simulate the response of a particular structure to non-time varying loads. Also, one may apply forces that are time-varying, but which can be approximated as time independent such as seismic loads etc.

The following is a typical analysis to determine the deformation map for three boxes that are bonded to each other.

To create the geometry in Ansys, the following steps must be followed:

1. Create a new project/analysis 2. Double click on geometry. This opens the design modeller.3. Click Create > Primitive > Box.4. Hitting generate displays the box. 5. In the same way, another beam/box can be created such that they meet each

other at a face.

This is shown below.

Figure 12: Example - Creating Geometry

Once this geometry has been drawn, the fixed supports, interfaces, the forces must be specified. This is shown below.

Figure 13: Example - Forces Applied

The mesh used is displayed below:

Figure 14: Example - Mesh

The geometry is then meshed as shown above. A solution for the total deformation produces the following map.

Figure 15: Example - Solution/Results

2.2. Modal Analysis

Modal Analysis is conducted in order to view the response of structured when under vibrational excitation. The dynamic behaviour of these structures with a vibrational input is studied and the shapes of deformations are attained.

This section illustrates the modal analysis performed on a solid wing structure based on the Boeing Dreamliner - 787 model.

In order to conduct the analysis on Ansys, the geometry is first imported on to the Workbench.

-To accomplish this import:

1. Create the geometry in ProEngineer and save as .igs file2. Import the geometry in Ansys using the import option under file.3. Next drag and drop the Modal Analysis module into the workspace.4. Drag the “geometry” from the Geometry module into the “ model “ in the

Modal Analysis module.5. Double click on Model to open the Modal – Mechanical interface.

Figure 16: Example - Geometry

The material used for the wing, as agreed before is Aluminum and must be specified as such in the materials section under engineering data.

-Meshing

The geometry must now be meshed. To do this:

1. Click on Mesh in the Outline tree.2. Right Click and choose sizing.3. Using the edge selection tool to select the leading and trailing edge of the

wing.4. In the details window (geometry) click apply. 5. For the sizing criteria use “number of divisions” for type and set it to 50.6. Select “Hard” for behaviour.7. Now create new sizing criteria.8. Repeat steps 3, 4, 5 and 6, but this time choose the upper and lower edges of

the root and tip airfoils.9. Finally click on generate mesh

This generates the meshed structure as shown.

Figure 17: Example - Mesh

-Fixing Supports

Finally we must now create a fixed support at the wing root, to simulate the existence of the aircraft itself to do this the following steps must be completed.

1. Right – click Modal in the Outline view.2. Go to insert > Fixed Support3. Next using the face selection tool, click on the root of the wing.4. In the Details Window, in geometry, click apply.5. This creates the fixed support we need at the root.

-Solution

To solve for the modal frequencies we must tell Ansys to solve for the first 6 mode shapes of the structure.

1. Right click on solution in the Outline view.2. Go to insert > Deformation > Total Deformation.3. In the details view for “Total Deformation “, ensure that mode is 1.4. Change the name for Total Deformation to Mode Shape 1.5. Repeats Steps 3, 4, 5 for a total of 6 times. Each time this is done, the mode

number must be changed to reflect the respective mode that is being solved for.

6. Stop at mode 6.7. Finally click on the Solve button to begin the solution process.

Once this is done, the Mode Shapes can be viewed for each mode. These are displayed below.

Figure 18: Mode Shape 1

Figure 19: Mode Shape 2

Figure 20: Mode Shape 3

Figure 21: Mode Shape 4

Figure 22: Mode Shape 5

Figure 23: Mode Shape 6

The modal frequencies are also displayed in tabular form as shown:

Figure 24: Modal Frequencies

Conclusion:

The modal analysis suggests that any excitation or vibration that is supplied to the said structure at the given frequency will create a large deformation in the structure.

Sources of these vibrations can be from:

- Turbulence

- Engine Noise

- Engine Shaft Vibrations

- Vibrations from the fuselage

If it is found that any known sources of the above vibrations are at the same frequencies as the calculated modal frequencies, adjustments need to be made at the vibration source or in the design of the structure, to ensure that the two frequencies are not equal.

3. Evolution of the 3D Model

The most important fact considered was the absence of an apt reference of a wing model of a commercial aircraft with its internal structure that was complex enough to be close to being realistic and at the same time simple enough to be within the capacity of the system that based Ansys.

Accordingly, as the learning process furthered, the models evolved from being simple to complex, to being compatible with the current system on which Ansys runs.

To begin with the learning of Ansys, a very simplified model of a Boeing 787-8 wing was designed such that it had no internal structure, i.e. a hollow wing. The dimensions of the wing were obtained from the official site belonging to the Boeing.

The geometric dimensions are:

Wing chord at tip 5.55 feetWing chord at planform break 21.06 feetWing chord at root (gross) 38.94 feetWing chord at c/line (gross) 45.00 feetWing chord at c/line (notional trapezoidal) 30.83 feet

Planform break location 0.320 (fraction of exposed semi-span)Thickness break location 0.333 (fraction of exposed semi-span)

Figure 25: Hollow wing, Boeing 787-8.

Figure 26: Another view of the hollow wing.

Now that the methodology of how to conduct analyses in Ansys was realized, there was a move to make the wing model more inclined towards replicating a realistic one. This move comprised of numerous iterations due to the lack of a proper source that could avail proper aircraft wing dimensions with the respective internal structures.

Therefore, initially, the ribs of the wing were designed based on an electronic reference (vBulletin Solutions, Inc., 2008). These ribs were covered by the aircraft wing skin of 5mm thickness.

Figure 27: Aircraft wing ribs.

Designing the skin of the wing in a 3D-software proved to be one of the greatest challenges, as it was supposed to firmly cover the ribs as well as not to exceed the number of nodes to mesh. This was quite a task as a number of errors were encountered such as the skin not properly touching all the ribs (i.e. presence of gaps or penetration into the skin), number of faces on the skin due to the employment of a number of polylines to obtain a perfect curvature to imitate the airfoil, etc.

Figure 28: Wing skin (2D-wireframe view)

Figure 29: Wing skin (Realistic view)

Figure 30: Wing skin with ribs. (2D-wireframe view)

Since the system is subjected to low memory processor, the presence of holes and trusses rendered the generation of millions of nodes and eventually ran out of memory while meshing the model to perform analyses. Hence, the model was modified such that none of the ribs contained any truss members of holes within them.

Figure 31: Simplified Ribs.

Proceeding with the analyses, it was realized that the skin of the wing model had a number of faces that resulted in the steep decrement in analyses speed. In order to encounter this issue, the wing skin was smoothened as much as possible to facilitate smooth analysis. Also, the ribs were placed appropriately to be attached to the skin.

Figure 32: Smoothened skin with modified ribs.

Later, to raise the complexity of the model to the next level, the front and rear spars were added to obtain more realistic results. Both, rectangular plates and cylindrical rods had been used as front and rear spars to perform analyses.

Figure 33: Rectangular spars along with ribs and skin.

Figure 34: Trial of circular rods as spars.

Figure 35: X-Ray view of the wing model.

In order to keep the complexity of the model within the capacity of the system in which Ansys runs, the model was kept void of stringers and longerons.

The dimensions of the evolved model are:

Wing chord at tip: 4.73 m

Wing chord at root: 11.5941 m

Wing span: 26.544 m

Figure 36: Evolved Model (Realistic view).

Figure 37: Evolved Model (3D-wireframe view).

Lastly, a simplified model of a Boeing 787-8 wing was designed such that it had airfoil plates instead of ribs, a cylindrical front spar, a cylindrical rear spar, and two longerons along the upper surface of the wing and two longerons along the lower surface of the wing. This model served to reduce the complications arose during the analyses due to system memory issues.

Figure 38: Boeing 787-8 wing model with simplified internal structure.

Figure 39: Boeing 787-8 wing model with simplified structure (3D-wireframe view).

Thus, by the end of Project I, learning the process of designing a realistic model in 3D was thoroughly realized and accomplished. And this leads to one of the scopes for Project II which is to design a thorough realistic model; alongside the presence of a system with improved memory and processor to support the analysis. Also, the trials and errors undergone while designing the model resulted in the better realizations of minute details that need to be considered to attain appropriate results.

4. Attempts and Results Obtained

During the course of the semester, a number of obstacles were faced and many more issues were dealt with. The purpose of this section is to give an overview of the activities, procedure, and the work done during Project 1.

4.1. Trial 1

Initially, in order to gain experience with the software, the wing used was a simplistic wing that neglected the internal structure. The pressure distribution was applied over the wing in order to perform the analysis.

4.1.1. Pressure Distribution over the Wing

Pressure Distribution Approximation:

The Coefficient of Pressure was obtained which enabled us to extrapolate the pressure distribution in the chord-wise direction and different spans.

Assuming an altitude of 25000 feet and a cruising speed of 576.12 feet/s, which is about 80% of the cruising speed, we were able generate a complete set of chord-wise pressure distributions at different spans.

This pressure distribution was approximated by using the pressure coefficient data. Below are the tabulated graphs for the coefficient of pressure distribution that were used to estimate the pressure distribution.

The following graphs show us the cp distributions at different spans locations.

0 0.2 0.4 0.6 0.8 1 1.2

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 6.1 % Span

CpCp

x/c chord wise

Figure 40: Cp distribution at 6.1% span.

0 0.2 0.4 0.6 0.8 1 1.2

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 15.6 % Span

CpCp

Figure 41: Cp distribution at 15.6% span.

0 0.2 0.4 0.6 0.8 1 1.2

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 20.36 % Span

CpCp

Figure 42: Cp distribution at 20.36% span

0 0.2 0.4 0.6 0.8 1 1.2

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 25.12 % Span

CpCp

Figure 43: Cp distribution at 25.12% span.

0 0.2 0.4 0.6 0.8 1 1.2

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @29.88% Span

CpCp

Figure 44: Cp distribution at 29.88% span.

0 0.2 0.4 0.6 0.8 1 1.2

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 34.65 % Span

CpCp

Figure 45: Cp distribution at 34.65% span.

0 0.2 0.4 0.6 0.8 1 1.2

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 40.87 % Span

CpCp

Figure 46: Cp distribution at 40.87% span.

0 0.2 0.4 0.6 0.8 1 1.2

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 53.66 % Span

CpCp

Figure 47: Cp distribution at 53.66% span.

0 0.2 0.4 0.6 0.8 1 1.2

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 58.46 % Span

CpCp

Figure 48: Cp distribution at 58.46% span.

0 0.2 0.4 0.6 0.8 1 1.2

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 63.22 % Span

CpCp

Figure 49: Cp distribution at 63.22% span.

0 0.2 0.4 0.6 0.8 1 1.2

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 67.89 % Span

CpCp

Figure 50: Cp distribution at 67.89% span.

0 0.2 0.4 0.6 0.8 1 1.2

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 72.47% Span

CpCp

Figure 51: Cp distribution at 72.47% span.

0 0.2 0.4 0.6 0.8 1 1.2

-2.5

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 77.5 % Span

CpCp

Figure 52: Cp distribution at 77.5% span.

0 0.2 0.4 0.6 0.8 1 1.2

-2

-1.5

-1

-0.5

0

0.5

Cp Distribution @ 87.03% Span

CpCp

Figure 53: Cp distribution at 87.03% span.

0 0.2 0.4 0.6 0.8 1 1.2

-1.5

-1

-0.5

0

0.5

1

Cp Distribution @ 96.5 % Span

CpCp

Figure 54: Cp distribution at 96.5% span.

0 0.2 0.4 0.6 0.8 1 1.2

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Cp Distribution @ 98.93 % Span

CpCp

Figure 55: Cp distribution at 98.93% span.

The above data was represented as a fourth order polynomial by using the curve-fitting tool in Matlab.

The polynomials that were obtained for the span-wise and chord-wise direction are as represented below.

Chord Wise Pressure Distribution:

Linear model Poly4:

f(x) = p1*x^4 + p2*x^3 + p3*x^2 + p4*x + p5

Coefficients of polynomial:

p1 = 6.337e+04 p2 = -1.503e+05 p3 = 1.274e+05 p4 = -4.531e+04 p5 = 5930

Span Wise Pressure Distribution:

g(x) = p1*x^4 + p2*x^3 + p3*x^2 + p4*x + p5

Coefficients (with 95% confidence bounds):

p1 = 0.01352 p2 = -0.5317 p3 = 10.4 p4 = -40.22 p5 = -5000

4.1.2. The Static Structural Analysis

The static structural analysis when performed on the hollow wing had the following results.

Figure 56: Deformation

Figure 57: Stress Distribution

Figure 58: Strain Distribution

4.1.3. Modal Analysis

The figures pertaining to the modal analysis are as follows:

Figure 59: Geometry of the Wing

Figure 60: Meshed Wing

Figure 61: Mode Shape 1

Figure 62: Mode Shape 2

Figure 63: Mode Shape 3

Figure 64: Mode Shape 4

Figure 65: Mode Shape 5

Figure 66: Mode Shape 6

4.2. Trial 2

The model was made more complex by adding the internal structure. Ribs were added in order to support loads acting on the wing.

However, during the analysis, the meshing phase posed certain issues. Although, the meshing was completed, the number of nodes produced due to meshing of the structure, was too large; in terms of the functions the computer hardware had to handle. The configuration of the computer was inadequate for the magnitude of the analysis that needed to be done. Due to this, the computer would hang and stop responding and therefore the analysis could not proceed.

Figure 67: Mesh of the Internal Ribs

Figure 68: Mesh of the Outer Skin

4.3. Trial 3

In order to work with the current configuration of the computer, the design model was made less complex. This simplified model consisted of fewer faces and less intricate parts. This considerably reduced the surfaces that needed to be meshed and solved.

This model was successfully meshed and solved under the provided input loads.

However, stress concentrations were found at the root of the wing as well as the wing tips.

4.4. Trial 4

Upon redesigning the model, and adding thick plates at the tip and root of the wing, the stress concentrations were effectively eliminated.

During the Static Structural Analysis, the appropriate expected results with regard to the deflections, stresses and strains, were attained. Conversely, when the modal analysis was conducted, the software showed an error suggesting the structure had

a problem. There were local oscillations at very low frequencies which suggested that the structure was wobbly and probably not fastened appropriately.

Constraints in time prohibited further work to be done using the model.

4.5. Trial 5

The structure was remodelled to make an even simpler design. The number of faces and bodies was reduced to the significant minimum that is required for sustaining the applied forces.