19.3.1 Wing

The main function of the wing is to provide the lift needed to keep the aircraft airborne. Lift is a pressure force that acts over the surface of the wing, and if the resultant total force is sufficient the aircraft flies aloft. Structurally, the wing acts like a long beam, with the upward lift force being balanced, usually in the middle, by the fuselage and main payload. This is a self‐equilibrating system, as there is no ground reaction or supports in that sense. Structurally, therefore, the wing transmits these external forces through the material structure to achieve equilibrium (Figure 19.3).

Of course, depending on the role of the aircraft, and the resulting configuration, the wing can also be required to transmit forces from wing mounted engines, undercarriage, fuel, hydraulic systems and so on. Although this makes for a more complex external loading picture the wing itself still acts like a beam a transmits the forces through the material.

19.3.2 Empennage/Tail

The empennage generates control forces for manoeuvring the aircraft. The individual elements of the empennage can be considered as wings in terms of the forces they transmit and carry within themselves. However, they also therefore transmit such forces through the adjacent structures, for example the fuselage. These forces can be very significant and can cause the structural designers difficulty in ensuring the integrity of the airframe.

19.3.3 Fuselage

In the main category of civil aircraft, the primary function of the fuselage is to provide the space and location for payload and passengers. It must hold them and keep them safe and warm. Conceptually, the weight loading is similar to the wing. In this case the distribution is for the weight of the passengers, assumed to be evenly spread over the fuselage floor (Figure 19.4). Thus far, the fuselage is also like a beam, with a uniformly distributed load over its length, supported by the lift force of the wings, acting usually in the centre of the beam. In addition, the fuselage experiences bending moments from the kinetics of the lifting surfaces.

The fuselage, however, has an additional need where the aircraft flies in the stratosphere and that is to resist pressure. To keep passengers safe, aircraft are pressurised, normally to a minimum pressure equivalent to an altitude of 7000 ft. This additional pressure load provides many challenges for the structural designer, and has been the cause of major catastrophes such as the Comet disasters.

19.3.4 Undercarriage

While the aircraft is on the ground, the entire weight is transmitted through the undercarriage to the ground. In this case they act like stocky legs supporting the great weight of the aircraft. However, when the aircraft is landing, the shock loads of striking the ground are absorbed the undercarriage and subsequently transmitted through the wing or fuselage structure to which they are attached. Chapter 9 gives a fuller description of undercarriage design and the corresponding loading scenarios. Undercarriages are therefore not as beam‐like as the main airframe elements.

19.3.5 Torsion

In addition to normal load pattern as briefly described before, aircraft components get subjected to torsional load; in simple terms, they get twisted in manoeuvres, asymmetric aerodynamic loadings and so on.

Twisting along the axis of a body occurs as a result of applied torque that produces shear force on the cross‐sectional area normal to the axis of the body. Twist causes deformation and becomes significant along a long body, for example, fuselage, wing and so on, as shown in Figure 19.5.

19.4.1 Structure

A structure is simply the assemblage of material components that transmits and resists the applied loads to keep the aerodynamic shape, keep the passengers safe and comfortable and ensure that the aircraft can fulfil its function over its life.

19.4.2 Load

The load is the force exerted on a surface or body. In the case of the whole aircraft, these are normally categorised as air or ground loads. As the aircraft is a dynamic system it encounters many loading scenarios during its operational life, from dynamic loads arising from manoeuvres to impact loads at landing, and inertial loads from releasing stores. In the context of structures, the definitions of various kinds of loads given Chapter 17 is recast here.

19.4.3 Limit Load

The limit load for an aircraft is the maximum load it is expected encounter once in its operational life.

19.4.4 Proof Load

The proof load is the maximum load the aircraft must withstand without detrimental distortion. Under proof loading conditions, the airframe must not distort permanently more than the specified proof strain (0.1 or 0.2%). This is particularly critical for structures such as flap tracks where any permanent distortion (plastic deformation) would impair functionality. Proof load is defined as:

The proof factor is a factor of safety applied to allow for uncertainty in specifying the limit load. In civil aviation, as regulations are more conservative in defining loads, the proof factor is normally taken as 1, but in military aircraft it is usually taken as 1.125.

19.4.5 Ultimate Load

The ultimate load is the maximum load the aircraft can resist without failure. Ultimate load is defined as:

The ultimate factor is a factor of safety applied to account for uncertainties in a range of aspects, such as the variation of material properties, the degradation of material properties over time, uncertainty in analysis and key assumptions and uncertainty in flight operations (e.g. flying outside the allowable manoeuvre envelope). The ultimate factor is normally taken as 1.5 for all aircraft types.

19.4.6 Structural Stiffness

Structural stiffness is the measure of how much the structure deflects under a load. It is defined as the slope of the load deflection curve. Structural stiffness has several variants referring to the type of load and deflection, for example, bending stiffness or torsional stiffness. The structure itself may be made from elements of many materials. The relation between stiffness, K, load, P, and deflection, δ, is given here.

19.1

19.6.1 Stress

Section 19.3 mentioned that there are fundamentally two types forces (loads) to be considered in setting up stress equations. The two of loads are given here:

1. Normal force, pull or push, acting perpendicular to surface.
2. Shear force acting tangential to the surface.

To keep illustration simple, the small amount of dilation on deformation taking place under the normal force application, as shown in Figure 19.2, is not shown in Figure 19.7 (details can be found in the references given). The stress types are as follows.

The definition of stress is to normalise of load applied on unit area that indicates the intensity of a force on a material (Figure 19.7).

1. Normal stress, σ, is the total applied normal force, P, divided by the area, A, over which it acts.
   19.2
   It can be a tensile stress or compressive stress, depending on whether it is pull or push and cane be identified with a subscript.
2. Shear stress, τ, is the total tangential force applied, F, divided by the area, A, over which it acts.
   19.3

   The concept of shear stress on thin surface, for example, edges of aircraft metal/composite sheets can be expressed as shear stress per unit length termed as ‘shear flow’ (not dealt with here).

19.6.2 Strain

Strain is the non‐dimensionalised deformation of a material under stress and the two types are as follows.

1. Normal strain, ε, is the ratio of the extension or contraction, x, (or stretch/shrink) of the material under load to its original length, L. It is denoted:
   19.4
2. Shear strain, γ, is the angular deformation under tangential force applied (details can be found in the references given).
   19.5
   where Δx is the transverse displacement, that is, the angle between the sides of the original shape and deformed shape (Figure 19.7).

In case of torque, the twist angle, θ, produces circumferential displacement ‘dS’ on account of the applied torque straining the sold circular by the angle, γ, over the length, L.

Then dS = θR = γL where, R = r_max

or

19.6

19.6.3 Hooke's Law

The fundamental property of stress–strain relationship within the linear range of application, is shown in Figure 19.6. The stress–strain relationship obeys Hooke's Law, which states that within the linear range the ratio of stress by strain (stress/strain) is constant for both the normal and tangential loads.

The two types of stresses and strains related to stiffness are briefly described next.

1. Under normal load, the material stretches or contracts depending on whether it is pull or push. It is a fundamental property of the material obeying Hooke's Law is called the Elastic Modulus (E) or the Young's Modulus, named in honour of Thomas Young. It is defined as the slope of the stress–strain curve (Figure 19.6).
   19.7
   
2. Under tangential load, the material deforms as shown in Figures 19.2 and 19.7. Its fundamental property obeying Hooke's Law is called the Modulus of Rigidity (G). It is defined as follows.
   
   

19.7.1 Stiffness

Material stiffness is the measure of how much the material deforms. It is measured from a material's fundamental property of stress–strain relationship within the linear range of application, as shown in Figure 19.6.

19.7.2 Yield Stress

Yield Stress is the stress level in the material that will cause permanent deformation. It is normally defined where the resulting plastic strain is 0.1 or 0.2%. This stress level is often called the 0.1% proof stress.

19.7.3 Fracture Toughness

Fracture Toughness is the stress intensity at which a crack in the material will grow rapidly until failure. It is a measure of the material's ability to resist cracks.

19.7.4 Ductility

Ductility is the ability of a material to deform plastically without fracturing. That is how much the material can be stretched after yielding.

19.7.5 Durability

Durability in the context of aircraft is the ability of the component to perform its function over its life. In aircraft design this is usually in reference to the number of flight hours or load cycles that the component can continue to function over. Cracks exist in materials naturally due to flaws in the material structure and these can grow slowly under loading. As the crack lengthens the local stress increases, eventually reaching the fracture toughness level for the material and resulting in complete failure.

19.8.1 Material Selection

Material properties are the main drivers to make the choice to select a suitable aircraft component under the nature of the applied load on it. Over the years of development, engineers moved from the natural material, for example, wood to a large variety of man‐made alloys and composite materials made from natural materials; each type has different properties to suit specific applications in a selective manner to fabricate aircraft components. Some general materials of interest are given in Table 19.2 (Young's Modulus, E, GigaPascals) and Table 19.3 (density, mega grams m⁻³), see Ref. [7]. Brief introductory considerations are given in this section on material selection in aircraft design. Readers are suggested to refer to Refs [4, 5, 7–10] for detailed study.

If the maximum stress level required for the design is around 500 MPa, then steel, aluminium alloy, titanium alloy and carbon‐fibre reinforced plastics (CFRP) are all possibilities. The strengths of these materials are compared in a bar chart as shown in Figure 19.8.

A quick study of the tables and figure show that, while steel is much stronger than aluminium, it is much heavier. The lightest steel is about 2.5 times denser than the heaviest aluminium alloy. This suggests that the same structure can be made much lighter if aluminium is used. In fact, based on average values the strength‐to‐weight ratio of these two materials is pretty comparable, meaning that other factors are really needed to help. There is of course a major caveat here in that there is a huge variation in properties for specific material compositions. A further complication in material properties is the inverse relationship between ultimate strength and toughness. As materials are made stronger, they have a tendency to be more brittle and have a relatively lower fracture toughness. Thus, if durability is a concern then the material chosen is may have a lower ultimate strength. Therefore, the aircraft designers need to know other material other material properties, for example, their strength, fracture toughness, costs and so on. So, any comparison like these must be made really on the exact material being considered. Many engineers make mistakes by assuming average properties when the material they have to hand is quite different. That said, the average values here are useful to make the point.

Because of these contradictory indicators a variety of metrics have been used to aid in material selection. The most common of these are the ratios of strength‐to‐weight (specific strength) and stiffness to weight (specific stiffness). Other useful measures are modulus to strength and toughness to strength. The best tools for aiding this are the graphs (Figure 19.9) developed by Ashby [8, 9] as part of a materials selection tool suite.

Since cost is not a material property, it is dealt with separately in the next section. For materials with similar properties, cost varies from manufacture to manufacturer. A good example is that of building construction steel with similar specifications, the unit price is considerably lower in the Far East as compared to the prices in the West.

Another important consideration for material selection is the manufacturability of the component. This is dealt with briefly in Section 19.8.1.

19.8.2 Ashby Scatter Plots

Professor Martin Ashby of Cambridge University systematically classified and grouped materials within the class and their associated properties of interest in a series of scatter plots. The applicable ones for this book are given in Figure 19.9. These are (i) Young's Modulus versus density, (ii) Young's Modulus versus strength, (iii) strength versus density and (iv) fracture toughness versus strength and so on, can be found in Refs [8, 9]. In general, engineers like to have material of high strength, high stiffness, high toughness and low weight.

The advantages of Ashby plots are that an overview of material properties can be visualised in few graphs to make a quick comparison of the relative merits to make a decision. These graphs are useful in giving potential material choices and can provide alternatives that were perhaps not originally at the forefront of the designer's mind.

19.8.3 Material Cost Considerations

Of the manufacturing considerations in material choice, the cost of making the structure from the chosen material is critical. Again, an exotic material may provide excellent physical properties but may require new specialist equipment to machine or fabricate. It may be further complicated by being available only from a single supplier or small number of suppliers. A sole supplier can be in a powerful position and significantly affect the final cost of the aircraft.

Another factor, even with more common materials, is the effect of the world markets. Aluminium, for example, is traded in global stock markets and the price can vary widely over time. This is also true for some of the minor additives to make aluminium alloys. Again, it is crucial to know the exact grade of material to have confidence in cost, otherwise the designer must be aware of the risk.

Fundamentally, most companies have estimates for the cost of material for a given weight and use these at design stage. The company will have corporate knowledge of the additional costs of manufacture using materials with which it is familiar. It is where the designer strays into uncharted waters that risk becomes an issue. This was certainly the case for many manufacturers in building the latest generation of composite aircraft where in many cases a completely new factory was built for the manufacturing programme. These are huge risks and while at the design stage this seems far away, some consideration is always helpful, as the risk can then be taken with some confidence.

Compromise on material properties is required on account of economic factors to remain competitive in the market without compromising aircraft safety but possibly on performance, mainly arising from substitution with a heavier material. An Ashby scatter plot of material cost is given in Figure 19.10.

A more exacting cost of some relevant material, in terms of US$/kg, is given in Figure 19.11 in bar chart form to facilitate relative cost comparison. The comparison allows compromises to be made, if required. It is to note that the material cost varies according to global markets. Therefore, accuracy of the two figures cannot be guaranteed.

19.8.4 Manufacture

While the detailed manufacturing programme is not yet on the horizon for the designer, at this point it is important that some heed is paid to it here. When selecting a material it is very tempting to refer to the kind of diagrams shown here or to use other look up tables available from many accessible resources to garner a range of possible materials. It is a common sight for students, and even experienced engineers when under pressure, to make a decision because a material with the appropriate properties has been identified. However, the many physical characteristics of materials also affect the manufacturing processes. Some, such as titanium and CFRP are very difficult to machine, some are very difficult to bond and some materials when together react and corrode. There are an additional and simple questions for any company: Is the material readily available and is the supply is in sufficient quantity for the project to completion? Are there appropriate manufacturing tools and systems in place to handle the material? These apparently mundane considerations can cause the designer to rethink and perhaps reshape the design so that a more accessible and cost‐effective material can be used.

An addition to the older example of Boeing767 given in Figure 19.1, is an example of a current example of the Airbus380 in Figure 19.12. It shows component‐wise material selection.

19.8.5 Integrated Decisions

The preceding sections have indicated four key considerations in material choice. There will always be others in a full design programme, but these four reain dominant. The last and most important consideration is that these aspects are integrated. The decision on the material choice should be based on a consideration of all four aspects: material properties, shape/configuration, manufacture and cost. It is here that the trade‐offs begin and choices are exercised such as accepting some weight gain for a cost saving, accepting higher cost for a reduction in weight or shaping the material to keep the stresses low so that a common and well understood material can be used.

19.8.6 Design Exercise

All the information given in Section 19.8 prove useful for the design exercise given here. As an exercise, consider a cantilever beam from above. By using any available resources for material properties and costs, try to enumerate a number of cross‐sectional shapes that will keep the cost of each design of the beam within a 10% margin.

As a further exercise, try to design the beam to be half the original weight, but still within 10% cost of the original. If that is too easy, then try half again and so on. Be imaginative with the cross‐sections. You can, for example, use hollow cross‐sections.

19.9.1 Bending

For example, in Figure 19.13 a simple cantilever beam with a rectangular cross‐section under bending is shown with one end fixed and one end free. It is a simplistic representation of a wing, the free end is the tip and fixed end attached to fuselage. The maximum stress in the cantilever beam like wing spar is at the root joining the fuselage. The beam transverse cross‐section has a width of ‘b’ and depth of ‘2d’ and the neutral axis passes through the centroid at middle of the rectangular beam cross‐section; ‘y’ is the distance from the neutral axis.

In the Figure 19.13, without any axial stress, the cantilever beam of length, L, is in pure bending under the vertical load, P, at the free end. This gives the maximum bending moment, M, at the fixed end expressed as.

19.8

The shear force is constant along its length. The distribution of shear force and bending moment along the cantilever beam is shown in Figure 19.13.

The loading condition will deflect the beam in to a shallow uniform curvature causing tension above the neutral axis and an equal amount of compression below it. The opposite forces above and below the neutral axis act as resistive forces from pure bending.

The tension makes the upper half stretch and vice versa at the bottom half on account of compression. Therefore, the cross‐section has stress distributed in linear variation from ‘0’ at the neutral axis to maximum value at the top and bottom level. The total force, F, equates to zero, top half and bottom have equal and opposite magnitudes. This expressed mathematically as follows.

Let at any elemental area dA of the beam transverse cross‐section at a distance ‘y’ from the neutral axis experiences stress, σ_x = σ_{x_max} × (y/d) with linear variation.

Then the elemental force, dF, on an elemental area dA of the beam transverse cross‐section is as follows.

where σ_{x_max} and ‘d’ have constant values.

Integrating over the cross‐section area,

Elemental moment, dM on account of elemental force is

Integrating moment,

19.9

The term ∫y²dA) is the area moment of inertia, I, of about the neutral axis.

Equation 19.9 takes the final form M = (σ_{x_max}/d) × I.or

19.10

The tension makes the upper half to stretch and vice versa at the bottom half on account of compression. Consider the strain due elongation to be ε_x along its axis. The loading condition will deflect the beam in to shallow curvature of radius R. It can be shown [1–3] that for small deflection the following relation holds.

19.11

The distortion of the transverse cross‐section on account of tension and compression is negligibly small and is ignored here. Then applying the elastic property 19.4 in terms of Young's Modulus, E, the following can be written.

Using Eq. 19.11, at any distance y from the neutral axis, this becomes as follows:

19.12

where y = σ_xd/σ_{x_max}.

From Eq. 19.6, Eq. 19.12 can be written as follows

19.13

this is known as elastic flexure formula

Noting that the rectangular shaped area moment of inertia of about the neutral axis is as follows.

Then for rectangular cross‐section cantilever beam, Eq. 19.12 becomes:

19.14

The maximum stress is at y = d/2, that makes

19.15

The equation indicates that the stress in the beam is inversely proportional to the depth of the beam. The longer the ‘d’ (beam depth) is, the lesser is the maximum stress, σ_{x_max}. If space is available, then for the beam cross‐section area, a deeper beam is preferred. The example given next demonstrates this.

In Figure 19.14, if b₁ is set to be 0.8b, then for the same cross‐sectional area, d₁ becomes 1.25d with a net reduction in stress of 20%. In a real design, if such changes are possible then lighter or cheaper materials may be used, or the section is reduced to reduce weight. It is possible to make multiple gains sometimes, by reducing the area and using a lighter material at the same time.

This now provides more freedom in material selection. If the stress is very high, then being able to reduce it may bring other materials into play that can offer benefits in terms of weight or cost or both. For example, reducing the maximum stress from 600 MPa to 300 MPa brings aluminium alloy into competition with titanium, carbon fibre and steel.

Although this example is simple, it is very relevant for the selection of material for an aircraft fuselage or wing. In most design cases a single wing is idealised as a cantilever beam that, although it has a more complicated cross‐section, still behaves very much like a beam. It is possible therefore to get very quick estimates of the order of magnitude of stresses in a wing to inform the initial selection of materials.

19.9.2 Torsion

Following on with the idealisation of the wing as a beam it becomes clear that torsional loads arise naturally since the elastic centre and the centre of lift are not coincident (Figure 19.15). Torsion is particularly important because twist of the wing directly influences the angle incidence and hence the lift. So, in addition to the stresses the angle of twist is a key response that needs to be known.

When torque is applied to a circular bar, in this case a solid cylindrical shaft, the following assumptions are taken to derive the torque equations.

1. No distortion takes place on the cross‐section perpendicular to the cylinder's axis.
2. Shear strain varies linear from 0 at the centre to γ_max at the peripheral surface.
3. Valid only in the linear range of stress–strain relation obeying Hooke's law, that is, (τ/γ) = constant = G.

This implies that shear stress,

19.16

linearly varying from 0 at the centre to τ_max at the peripheral surface.

Therefore, force, F, on the cross‐sectional area, A, is given by

19.17

where, dA is the thin concentric cross‐sectional strip at radius, r.

Torque,

Hence,

19.18

where

19.19

the polar moment of inertia.

For a given tube, τ_max and D = 2R are constants.

On substituting, the torque Eq. 19.18 can be written as

19.20

Normally, applied torque, T, can be measured and is known, thus τ_max can computed.

19.21

For a solid circular cylinder,

19.22

Airframe structures are largely thin‐walled tubes and hence this basic theory is modified to produce expressions and relationships that are more appropriate for the structural designer. In the case of hollow circular sections:

For thick tubes,

19.23

where B is the internal radius of the tube.

For thin tubes of a thickness, t, the stress variation within the thin cross‐section section is small and can be approximated by taking the average value at R_{ave_thin tube}. This gives

19.24

As aerofoil shapes are complex this theory has been extended significantly as a general theory of thin‐walled tubes, as shown next. Although the treatment is extensive, there are two key equations of use to the wing designer, which can be used to find the skin thickness and the wing twist. Before developing these equations, there are a couple of useful points to note. Firstly, in simple bending, all forces act through the neutral axis, also known as the elastic axis. In symmetric sections, this causes bending about the two main beam axes, but in unsymmetrical sections it also causes twist. The twist is around a point in the section called the shear centre, which is defined as the point in the section through which a shear force will cause no twist. The distance between the shear centre and the elastic centre is a critical design parameter for the structure and should be minimised if possible. Since the centre of lift is typically not through the shear centre of the wing section it means that wings always have a torsional load to deal with. A fuller treatment of these concepts is beyond the scope of this book, but these are useful key points to bear in mind in the structural design of the wing.

The neutral axis passes through the centroid of the area and the equations here are developed assuming that the torque acts along this axis. The underlying assumption in this is that the torsional load is independent from the bending load. This highly idealised assumption is conservative but perfectly suited to this conceptual stage of design to obtain initial estimates of stress and skin thickness.

Figure 19.16 shows a non‐circular uniform cross‐section tube with a constant thickness, t. It is subjected to torsion, T, about a longitudinal axis passing through its shear centre. The torque, T, produces shear stress, τ, at the tube cross‐section, as shown on the elemental circumference length, ds that associates with its complementary stress along its longitudinal direction as shown in the figure.

Then the elemental force dF on the elemental cross‐section of length ds can be written as follows.

19.25

This is often written in terms of the shear per unit thickness, which is called the shear flow, q, written as:

19.26

Force equilibrium gives that force per unit length along the circumference is constant. That means, for the torque loading the shear flow, q, is also constant along the circumference. This is true for non‐uniform thickness along the circumference, that is, if thickness, t, reduces then stress, τ, increases and vice versa.

Then torque, T, can be obtained by integrating the whole circumference.

19.27

where r is mean radius at ds.

The integral, ∮rds gives twice the cross‐sectional area at its mean radius r.

Therefore,

19.28

this is known as the Bredt–Batho formula.

The Bredt–Batho formula is directly useful for estimating the skin thickness needed for a given material choice. The shear stress at any point of the thin tube cross‐section (can be non‐uniform thickness) can be computed by.

19.29

The angle of twist, θ, arising from the applied torque can be obtained by the following expression (not derived).

19.30

where G is the modulus of rigidity.

References [4–6] may be consulted for its derivation as well for non‐circular cross‐section.

19.9.3 Buckling

Buckling is an unstable behaviour, a deformation that can take place on account of axial/in‐plane compressive load applied to slender column, thin plate, shell structures and so on. Characteristically, buckling shows large lateral deformation compared to axial deformation. This occurs if there is local misaligned (eccentric) loading and/or imperfections in the material.

Within the elastic limits of the material, such deformation is tolerated as deformation recovers to normality as soon as load is withdrawn. In flight, many aircraft exhibit skin buckling on wing/fuselage surfaces under flight load, loosely described as ‘tin canning’ or ‘wrinkling’. Buckling is not desirable, but if well designed, buckling accommodates flight load variation. Stiffeners/ribs are used to keep buckling deformation within acceptable limits. Modern high performance aircraft have integrally milled skins with low level of buckling.

Only the metal structure buckling characteristics are discussed briefly here. Non‐metals also exhibit bucking. Readers may refer to [1, 12] for detailed study. A brief overview is summarised here.

• Structural components under axial/in‐plane compression loads are susceptible to buckling – unstable out‐of‐plane deflection when the applied compression stress reaches a certain level, known as critical buckling stress.
• Depending on the structural geometry and material properties, the compressive stress at which a structure will buckle can be well below that of the material compressive yield stress.
• When defining the allowable compressive design stress, buckling stress must also be considered alongside the material yield stress.
• Buckling on aircraft structures usually takes the form of ‘global’ buckling (buckling of large or full structural components usually resulting in structural failure) or ‘local’ buckling (buckling of a small sub‐section of a larger component that may not result in structural failure).

The various buckling behaviours applicable to aircraft structures can be considered in two parts: (i) beam/column buckling and (ii) plate buckling, as discussed next at an introductory level. Figure 19.17 shows the basic examples of the two types of buckling. Buckling deformation can be of different patterns.

19.9.3.1 Column/Beam Buckling

Figure 19.17a shows a basic example of buckling of a slender column; a beam like structure supported at one or both ends, such as the stringers on stiffened panels, are susceptible to column buckling. A long, thin column will generally fail due to flexural or Euler elastic buckling and at relatively low buckling loads. The buckling stress of these columns can be predicted by the elastic Euler equation 13, 14].

19.31

where

σ_Col = Critical column buckling stress.

E = Material Young's Modulus.

L_e = Column effective length.

I = Column second moment of inertia.

A = Column cross‐sectional area.

The column buckling stress is dependent on the particular conditions at each end, which can be free, or range from simply supported through to fixed joint (clamped). This is accounted for in the above equation by the use of an ‘effective’ length parameter, L_e. Figure 19.17a indicates how the column effective length varies with column end boundary conditions.

The boundary conditions on the real structure will vary with joint idealisation and manufacture and assembly methods. More rigid end boundary conditions lead to increased critical buckling stresses. Therefore, for the conservative design of columns constrained at both ends, it should be assumed that the ends are simply supported and that the effective length is equal to the actual column length.

Beams/columns that are shorter and sturdier are more resistant to buckling, but their buckling behaviour can be more complex. In these cases the column failure mechanisms usually involve the interaction of global buckling with localised plastic yielding. Further details can be found in the Secant [14] and Euler–Johnson [1] analysis methods for local‐flexural interaction.

It should be noted that buckling of a beam/column typically constitutes global failure of that structural component and the predicted critical column buckling stress should therefore correspond to an ultimate load condition.

19.9.3.2 Plate Buckling

Flat sheets, supported on three or more sides, can also buckle under in‐plane compression loads. On an aircraft, these flat sheets usually comprise the wing and fuselage skin components, as well as internal spar or rib webs.

Buckling of these flat sheets between the supporting structures, such as ribs, frames or stringers, is typically considered a ‘local’ instability. Buckling of the plate sections may not lead to failure of the structure providing the supporting structure remains stable after the local plate sections have buckled. Such structures are described as ‘post‐buckling’ – the total structure remains stable and safe even when local buckling of sub‐sections occurs.

The critical buckling stress of a flat uniformly thick plate under uniaxial in‐plane compression loading can be found with the following equation;

19.32

where

σ_Pl = critical plate buckling stress.

E = material Young's Modulus.

ν = material Poisson's ratio.

t = thickness of the plate section.

b = width of the compression loaded edges.

K = buckling coefficient.

Similar to column buckling, the critical plate buckling stress is dependent on the support conditions on each side of the plate. The more rigid the support constraint, the more resistant to buckling the plate. Figure 19.17b indicates how the plate buckling varies with the end boundary conditions.

The various levels of constraint/support on each side of the plate is accounted for by the buckling coefficient, K. Various external sources describe the definition of these buckling coefficients in great detail ([12–14]), however, Figure 19.17b describes the three main edge support scenarios and subsequent buckling coefficients which should be used for conservative structural design.

As noted before, a larger stable structure can accommodate local plate buckling of its secondary elements. These ‘post‐buckled’ structures can offer considerable weight savings, however, the local reduction in stiffness and deformation of the flat plate geometry must be taken into consideration. For example, buckling of the external skin on a commercial aircraft skin can have significant impact on aerodynamic performance. Subsequently, local buckling of the wing skin on a commercial aircraft would not typically be allowed below the limit load. However, on the external fuselage skin where the aerodynamic surface is less influential, local plate buckling could be allowed to occur as low as 50% of the limit load.

The methods outlined here model the plate and column buckling assuming that the column or plate behaviour is ideal. In reality such perfect conditions are rare and more often than not the column or plate structure has built in imperfections or is loaded out‐of‐plane caused by general misalignment of the loads. Any geometric imperfection or loading eccentricity will cause the structure to buckle at a critical stress lower than that predicted using these methods.

19.13.1 Composite Materials, Fibre and Fabric

The fibres are naturally unidirectional with strength along the length. Normally, fibres are woven into fabric, which provides strength in many possible directions. The most common ones are woven orthogonally. The fibres are woven into fabric in such a way that fibres cross each other or in groups, with several possibilities for distributing the fibre properties in different directions as woven. The basis of weaving is to lay vertical (warp) strands and horizontal (weft or fill) stands alternately crossing each other over and under. The woven fabric is then laid in layers sandwiched with resin in between as a bonding compound. In each chosen orientation of fibre pattern to form a composite laminate, the ply number is the number of fabric layers laid.

Some of the most well‐known weave patterns are plain, twill, satin, harness weave, basket weave, unidirectional weave and bidirectional weave. In aerospace usage, the plain, twill and satin weave types are dominant, possibly the twin is the most commonly used. These are briefly described in the following and also shown in Figure 19.19. Other types are not dealt with here.

19.13.2 Types of Synthetic Composite Material Fibre

There are many types of synthetic composite fibres, for example, fibreglass (E‐ or S‐glass), polyester, carbon, aramid (trade name: Kevlar, Nomex, Conex etc.), boron, graphite and so on, with a wide range of physical properties and cost. Selection of material is based on the suitability for the application at low cost. Currently, carbon fibre is dominantly in use.

19.13.3 Matrix Bond for Composite Material Fibre

Although the composite fibres in themselves are strong, and when woven into fabric sheets they provide strength in different directions, they need to be brought together to really become a super strong material. To achieve this, a matrix material is used to bond fibres and sheets together to make a solid structural unit (Figure 19.20). The matrix material is often a resin of sorts, which when cured forms a solid connection between the fibres and helps them all act as a single unit rather than a loose collection of strands or sheets. The matrix is the support and bonding material that acts as adhesive to bind fabric as reinforcement to transfer and distribute load throughout the structure.

The matrix material is therefore a critical part of the composite material. There are three main types matrix as follows.

1. Polymer
2. Metal
3. Ceramic

Composites most often use a polymer matrix, as these resins are more accessible and cost effective. The liquid resin can be more easily poured over the composite fabric and then cured so it sets hard. This type of hard resin is called a thermoset. There is a number of types of resin such as acrylic, polyester or polyurethane, but the most common in aircraft manufacture is epoxy resin. In complex structures the resin is drawn through the woven fabric lay‐up and cured under pressure in an autoclave.

It is of course also possible to have metal and ceramic matrix materials, but these are far less common and remain a very specialist area of development.

19.13.4 Strength Characteristics of Composite Materials

Since the fibre is the dominant load carrying element within a composite lay‐up it is very amenable to designing a material structure that is tailored to specific loading. For example, if most of the end load on a structure acts at 45° then aligning the fibres that way can ensure the material is most effective. This tailoring of the material is one of the most powerful aspects of composites. The challenge, however, is for the analyst to understand fully the many complex loading scenarios and stress distributions in the material to effectively design the material. The governing equations for composite materials are complex and this is no easy task.

The analysis of a composite material to determine failure stresses is an even more complex procedure and even now theories are still being developed that can characterise the failure mechanisms accurately and reliably. Although the fibre is very strong, the resin matrix is far less so and can crack or fail under a much lower load. If such a failure occurs then layers of the composite may separate and no longer act as a unit. This is delamination. There are many other modes of failure, such as fibre pull out, matrix cracking and so on.

As this is such a complicated analysis, most designers use simple approximations to allow some estimation of strength appropriate to this stage in design. Typically, the structure may be treated as a metallic like structure and the stresses compared on a similar basis. This gives an estimate of the potential strength of the structure and hence also its weight. Specialist stress analysts can work through the details at a later stage.

19.13.5 Honeycomb Structural Panels

The preceding sections noted the key materials and general considerations for their use an application in aircraft. When this is brought together with further consideration of the structural configurations and how both geometry (i.e. shape) and material are combined to produce a strong, stiff panel, other ‘types’ of construction become very useful. One common form arises from the need to increase stiffness and bending strength of panels without having to add much weight. By separating the outer skin layers and filling the interior with a strong shear resistant configuration, a structural strength and stiffness increases significantly. The interior layer is made from hexagonal shells and for that reason these are called ‘honeycomb panels’.

Honeycomb panels are extensively used in aircraft industry, mostly of metal fabrication but they can also be made of non‐metals, for example, Nomex/Kevlar core with Kevlar/carbon‐fibre surface skins layers and so on. Honeycomb panels may be seen as composite structure comprising of top and bottom sheets, sandwiched in between a honeycomb core panel glued with adhesive layers (Figure 19.21). It is named honeycomb as it resembles with columnar hexagonal bee‐hive honeycomb. Strength improves with thicker core and skins. A high quality strong adhesive layer is essential, otherwise delamination can occur. Honeycomb panels are orthotropic requiring orientation to suit load application, best applied normally.

These are generally purpose‐built panels to suit applications and are supplied by manufactures in plate form coated with PVDF (polymerising vinylidenedifluoride) film to protect from scratches, corrosions and so on, during transportation and in storage. If required, custom‐made honeycomb panels are designed and manufactured to suit the structural requirements. With appropriate manufacturing process stress‐free curved honeycomb are manufactured. Figure 19.21 shows use of a curved honeycomb panel as the wing leading edge.

The panel cores can be other shapes than hexagonal, for example, corrugated cores or even foam filler but the hexagonal cross‐section is most efficient. Small aircraft use a wire‐cut foam filler as these are easy and cheap fabricate, yet support loading adequately.

Listed here are some of the advantages and disadvantages of using honeycomb panels.

Advantages:

Honeycomb panels offer the highest strength‐to‐weight ratio for the materials used. They have excellent rigidity at minimal weight and offer high stiffness.

Honeycomb panels act as insulators for noise vibration propagation: they are used as nacelle acoustic liners.

They are light weight, facilitating easier fabrication processes save on labour cost. This also reduces the part‐count and maintenance costs are relatively low.

Disadvantages:

They are not suitable for concentrated high loads.

Manufacturing quality issues may result into delamination, not easily visible.

If not sealed properly, honeycomb panels are prone to water ingress that is not easily detectable and can cause severe corrosion.

Storage of water filling the hollow cells in the exposed panels can occur and weight gain occurs in a hidden manner. If this happens to the elevator, then flying may prove unsafe.

Applications:

Honeycomb panels suit normal distributed loads. High concentrated load applications should be avoided. The following are some examples of typical applications in aircraft industries.

19.14.1 Skin

As noted before, in the early days of aviation, most of the structure was an open framework, the wing being the only major element with a surface, being the main lifting surface. As knowledge of aerodynamics, and particularly drag, improved it was recognised that a skin covering the frame enhanced performance significantly. The skin had no structural function, its presence there was to form an impermeable aerodynamic surface.

As structural design methods and philosophies developed, it was recognised that a non‐structural skin was a dead weight and its presence could be used more effectively. By using a stressed skin construction, the skin could help improve structural stiffness and strength. But the continuing growth in the size of aircraft to carry many passengers rather than a single adventurous aviator and, subsequently, the advent of stratospheric flight made the need for a structural skin essential, with the loads in a pressurised fuselage being significantly more testing than in early aircraft. This led to the dominance of the semi‐monocoque form of airframe that is so prevalent today. In this, the skin both performs its aerodynamic function and carries pressure and some bending loads.

In current designs, functionally, the skin transmits aerodynamic forces to the ribs and stringers in the wing, and similarly to the frames and stiffeners in the fuselage. Additionally, it resists shear from both torsional and bending loads.

19.14.2 Fuselage

Chapter 5 deals with geometric considerations of fuselage design. A typical structural layout of load bearing fuselage components is shown in Figure 19.22. They constitute of circumferential ring‐like frames placed to normal and longerons running along the fuselage frame offering structural integrity to withstand imposed load. The skeletal frame is wrapped with a cover, in the figure this is an aluminium alloy skin that also supports tensional load. Figure 19.22b depicts a C‐series fuselage frame in a production line showing frames and longerons, as well the floorboard that supports the passenger cabin and the below‐floorboard container space (also see Figure 7.16b).

In most modern commercial aircraft the fuselage carries passengers in the stratosphere and it must therefore be pressurised. Of course, for smaller, lighter aircraft, pressurisation is not necessary because the fuselage there mostly has a different purpose, which is to hold the empennage in the right place. In the case of pressurised aircraft, the fuselage is typically circular as this is the most efficient from a stress perspective and is also relatively easy to manufacture. However, in practical terms, a floor is needed to attach seats to and, since this creates an underfloor space, this is used to carry cargo. Consequently, the fuselage is sometimes made slightly elliptical in shape to allow the opportunity to maximise the combination of both passengers and cargo.

Since the fuselage is a pressure vessel it must be sealed. This is done through bulkheads fore and aft (Figure 19.23). The bulkheads are heavy structures. From a manufacturing perspective, it is easier to make them flat stiffened plates, but this is not the most efficient structurally, so they are sometimes made with a slight curvature. This makes them more expensive, but more efficient. Normally, the whole fuselage is pressurised but some aircraft have used a pressurised floor, which is difficult to design and can add weight. Pressure bulkheads cause additional design challenges with the need for cables, ducts and so on to pass through, all of which must be sealed.

19.14.3 Wings

The wing of any aircraft is the critical system to be concerned about. It provides the lift needed for the aircraft to remain aloft and as such has very stringent geometric constraints both in its cross‐section and deflected shape under load, both of which are critical to performance. Under the skin of the wing, the structure is conceptually similar to the fuselage. The main structural element in the wing is the spar. The spar carries the bending loads that are in effect reacting the forces to keep the whole aircraft flying (Figure 19.24). The wing also has stiffeners, more commonly called stringers, in the wing that help the spars with the bending load and help to stabilise the skin to ensure it maintains its aerodynamic shape and does not buckle. It also has ribs periodically spaced along its length that

maintain the aerofoil shape and help to stabilise the skin and the stringers.

Although the primary function of the wing is to provide lift, this is also where control surfaces can be most effectively placed and where fuel can be stored. In many aircraft, the engines and undercarriage are also hung from the wing, adding both complexity and weight to the structure. At this point in the design though, the key thing to consider is the influence of the control surfaces and the need to store fuel. Both these factors directly influence the strength and stiffness of the wing. In particular, the location of front and rear spars is directly connected to these decisions.

Figure 19.25 shows this. If the location of the rear spar is dictated by the chord size of the main flap (c_f), from which it will be attached, then it can be seen that this choice directly impacts the available volume of fuel, if it is stored in the wing centre, and also the stiffness off the wing in that this is now defining the size of the torsion box. Both the torsional stiffness, J, and the shear stress in the skin, τ, are affected directly by the area of the torsion box.

In many cases, the initial sizes from the concept design show that these choices are all easily accommodated for. However, it is very easy to get to a point where a thin wing, needing a larger flap, has an extended range to cover, and suddenly these calculations and corresponding decisions become very important. This simple check is often omitted by students, but it can cause a rethink in the overall concept of the aircraft.

19.14.4 Spars

The primary structural element of the wing, and therefore arguably for the whole aircraft, is the main spar (Figure 19.26). The main spar is essentially a structural beam that carries the wing bending loads. If the spar can be located and designed correctly much of the rest of the design falls easily into place. The spar can be a single entity but wings often have two, one front and one rear, so that together with the skin they form a closed box, which is very strong in torsion. Much of the structural strength of a wing is due to the torsion box. The spar itself is usually a single machined piece, sometime including lightening holes to reduce weight. In the past it was often built up with many rivets and was a heavy component. The spar is idealised as a beam and designed very much as any typically cantilever beam is designed.

Because the wing both bends and twists and, to counter this, the torsion box arrangement (i.e. a box beam) is used, there are usually two spars, one front and one rear. The front one usually takes the greater portion of the bending load and a ratio of 2:3 front to 1:3 rear is a common assumption at this point in the design.

For structural analysis, the spar is usually idealised as a beam with booms and a shear web. The assumption being that the spar caps take all the end load from bending and that the web takes all the shear load. This idealisation is shown in Figure 19.27.

This very simple idealisation is a key concept and allows for a quick estimation of the proposed sizes. It allows for a sanity check on the proposed dimensions of the wing and the material choice. A simple example is shown in Section 19.15 later.

19.14.5 Ribs and Stringers

The main function of the ribs in the wing is to maintain aerodynamic shape of the aerofoil, critical for performance. The ribs act in concert with the skin to distribute the air loads through the structure. They also perform an important stabilising function for the stringers by providing end support for sections and increasing the buckling strength of the structure. Figure 19.28 shows typical wing spars, ribs, stringers and skin.

19.14.6 Fuselage Structural Considerations

Although a circular cross‐section is the most desirable from stressing (minimise weight) and manufacture (minimise cost) points of view, the market requirements for the below‐cabin floor‐space arrangement could result in a cross‐section elongated to an oval or elliptical shape. The Boeing 747 with a narrower upper deck width is a unique oval shape in the partial length that it extends. This partial length of the upper deck helps cross‐sectional area distribution (see Section 5.10.2) and area ruling. Figure 19.29 shows that wing‐box structure passes below the floorboard (near mid‐wing aircraft). It shows underfloor space to accommodate LD3 containers snugly fitted into a circular cross‐section fuselage either side of wing box.

A bulkheads are load bearing frames placed transverse to the hull, in this case these are wall‐like frames placed transverse to the fuselage shell. Aircraft cruising above 15 000 ft altitude need to be pressurised for passenger safety. All large commercial aircraft have a front pressure bulkhead at the nose cone and an aft pressure bulkheads (or rear pressure bulkhead at the tail cone of the aircraft – Figure 19.23) to seal the fuselage shell from the flight deck in front to the entire usable length of passenger cabin in a leak‐proof manner to keep the interior pressurised. Sealing of electrical cables and hydraulic pipes passing through is done as required.

19.14.7 Assembly and Wing Box

Although the preceding sections cover the main elements of the airframe it is worth mentioning two other key elements. The first is the wing carry through box that is prevalent in modern jet aircraft. As noted previously, the torsion box, that is, the section between the front and rear spars, is the key load carrying structure in the aircraft. To maintain structural integrity, this box normally runs through the fuselage and maintains its structural form. The idea being to allow the spar to run interrupted from wing tip to wing tip as any joint will reduce structural strength. There are exceptions to this, but they are all known to be very difficult design challenges. Interestingly, in larger aircraft, the design of the carry through itself is a major design challenge due to the loading intensity at this point.

Figure 4.7 shows typical choices for wing position in relation to fuselage. Figure 19.30 shows wing‐box structure passes below the floorboard (near mid‐wing aircraft).

Where the wing meets the fuselage is also a challenging structural for the fuselage. In this case the main fuselage frames here must react the wing shear and torsion loads. Typically, the frames will be very heavy and attached to the wing structure with pins to ensure that only direct axial loads are transferred into the frame itself.

19.14.8 Empennage

Structurally, H‐ and V‐tails are like small wings with smaller planform areas and aspect ratios carrying considerably less load, just what is required for stability and control. They are not. practically required to contribute to aircraft lift, hence there is no requirement for high lift devices. However, the control load has to be sustained by a long and strong fuselage.

The wing joint to the fuselage must ensure the fuselage shell keeps the passenger cabin space to offer maximum passenger space for comfort and movement, hence where and how it is attached requires careful consideration as discussed in Section 4.4.2. This is not the case with the empennage mounted at the tail cone where its narrow volume only houses the control linkages and accessories. Empennage structural layout does not interfere with positioning of control linkages and other devices. While there is similarity with wing structures, the empennage structural design is simpler.

The V‐tail is attached to the fuselage at the aircraft symmetric plane; typically, its spars are attached to the near circular fuselage frames. It looks like a banjo and is loosely known as banjo‐frame (Figure 19.31a). The rudder is kept free by hinging along the fin trailing end.

Structural considerations of the H‐tail attachment to the fuselage are dictated by its position with respect to the fuselage, arising from aerodynamic, stability and control considerations. Aerodynamic loads generated by the empennage require a strong and stiff fuselage structure so that fuselage flexure does not adversely affect empennage control effectiveness. The attachment joint H‐tail requires special considerations because part or the whole of the H‐tail can be movable (Figure 19.31b). The H‐tail attachment could be at the fuselage (low tail) or at the V‐tail (T‐tail/mid‐tail).

A multi‐spar empennage arrangement is shown in (Figure 19.31c).