Shear centre

Shear centre

Shear centre of a section can be defined as a point about which the applied force is balanced by the set of shear forces obtained by summing the shear stresses over the section.

In unsymmetrical sections and in particular angle and channel sections, the summation of the Shear Stresses in each leg gives a set of Forces which must be in equilibrium with the applied Shearing Force.

(a) Consider the angle section which is bending about a principle axis and with a Shearing Force F at right angle to this axis. The sum of the Shear Stresses produces a force in the direction of each leg as shown above. It is clear that their resultant passes through the corner of the angle and unless F is applied through this point there will be a twisting of the angle as well as Bending. This point is known as The Shear Centre or Centre of Twist

(b) For a channel section with loading parallel to the Web, the total Shearing Force carried by the web must equal F and that in the flanges produces two equal and opposite horizontal forces. It can be seen that for equilibrium the applied load causing F must lie in a plane outside the channel as indicated. 

Note:
1. In case of a beam having two axes of symmetry, the shear centre coincides with the centroid.
2. In case of sections having one axis of symmetry, the shear centre does not coinside with the centroid but lies on the axis of symmetry.
3. When the load passes through the shear centre then there will be only bending in the cross section and no twisting.

Jet Engine Detail Design: The Compressor

In this post the design of jet engine compressors will be discussed leading to the definition of ballpark performance parameters. For smaller engines centrifugal (CF) compressors are used since they can handle smaller flow rates more effectively and are more compact than axial compressors. Axial compressors however have the advantage of a smaller frontal area for a given flow rate, can handle higher flow rates and generally have higher efficiencies than CF compressors. For larger turbines used on civil aircraft the most suitable compressor and turbine will be of the axial type. Early axial compressors were able to raise the pressure of the incoming area around 5-fold, while modern turbofan engines have pressure ratios in excess of 30:1.

Low pressure axial compressor scheme of the Olympus BOl.1 turbojet. (Photo Credit: Wikipedia)

Because the pressure rises in the direction of flow through the compressor there is an acute risk of the boundary layers separating on the compressor blades as they encounter this adverse pressure gradient. When this happens the performance of the compressor drops dramatically and compressor is said to stall. For this reason the compression is spread over a large number of compressor stages such that the smaller incremental increases in pressure across each stage allow engineers to obtain a large overall pressure ratio without incurring stall. A stage consists of a row of rotating blades called the rotor and a row of stationary blades called the stator. Each of these rows may consist of between 30-100 distinct blades and there may be up to 20 stages between the air inlet and compressor outlet. The role of the rotor blades is to accelerate the incoming air in order to increase the kinetic energy of the fluid. Across the stators the fluid is then decelerated and as a consequence the fluid pressure is increased. As the pressure and density increase across each stage the overall flow velocity is kept relatively constant by reducing the height of the blades from stage to stage. Thus the compressor tapers down from inlet to outlet.
In an attempt to reduce the number of compressor stages for a more compact engine, a designer’s goal is to maximise the pressure ratio across each stage. The stage pressure ratio R is given by the following expression,
Where  is the stage isentropic efficiency, T₀₁ is the total (stagnation) temperature, U the rotary speed of the compressor, C_a the axial speed of the fluid, c_p the coefficient of latent fusion at constant pressure, and b₁ and b₂ the angle of the rotor blade leading and trailing edge relative to the axial flow direction.

Diagram of an axial flow compressor. (Photo Credit: Wikipedia)

The pressure ratio across each stage can be maximised by increasing the rotary speed of the compressor U, the angle through which the fluid is turned across the rotor blades tan b₁ –tan b₂ and the axial speed of the fluid C_a through the compressor. However there is a limit on the extent of these three parameters.
1. The blade tip speed and therefore U is limited by stress considerations at the root. If the fan is assumed to be of constant cross-sectional area then the centrifugal stress at the root is given by,
Where U_t is the tip speed, is the density of the blade, and the ratio r_r/r_t is called the root-to-tip ratio of the blade. To prevent the blades from detaching from the hub and destroying the engine this root stress is not allowed to exceed a certain proof stress. It can be seen that the root stress is proportional to the square of the compressor rotational velocity and decreases as the blade length becomes shorter. Since the first compressor blades have the highest blade lengths they limit the maximum tip speed and therefore the efficiency of the compressor. It is therefore common to split the compressor into double or triple spool configurations such as a large fan, intermediate-pressure and high-pressure compressors that are rotating at three different speeds. In this manner the large diameter fan can rotate at lower speeds to satisfy the stress restrictions while the shorter blade high-pressure compressor may rotate at higher speeds.
However the rotational speed of the fan is typically constrained by more stringent stress considerations. In a turbofan engine the large diameter fan at the front of the engine acts as a single-stage compressor. In modern turbofan engines the fan divides the flow with most of the air going to the bypass duct to a propelling nozzle and only a small portion going into the core. The high root stresses caused by the long fan blades are often exacerbated by bird strikes. For mechanical reasons a lower limit of root-to-tip ratio of 0.35 is often employed. The flow impinging onto the fan is also at a very high Mach number since the cruising speed of civil aircraft is typically around M = 0.83. Supersonic flow inevitably terminates in a shock wave with a resulting increase in pressure and entropy over the compressor blades. Shock waves reduce the efficiency of the compressor blades since they disturb the flow over the profile that lead to boundary layer separation. Furthermore, these shock waves may cause unwanted vibrations of the fan blades that further reduce the efficiency of the compressor and increase noise. Therefore for reasons of efficiency, reducing noise and limiting the damage of bird strikes the tip speed of the fan is restricted, typically a relative Mach number of 1.6 is considered as the upper limit.

2. The axial speed C_a has to be maximised to optimise the pressure ratio and reduce the frontal area of the engine. Similar to the argument given above the axial speed is typically limited by compressibility effects of supersonic flow. As the pressure, static temperature and therefore the speed of sound increases from stage to stage, the compressibility effects are worst in the first stages. For the first stage the air enters axially such that by adding the orthogonal velocity vectors U and C_a we get V² = U² + C_a² where V is the speed relative to the blade. In modern engines V may be in the transonic region incurring quite large losses. In this respect twin-spool engines have the advantage that the lower-pressure compressor rotates at a lower speed, which reduces the compressibility problem.

3. The angle through which the fluid is turned across the rotor blades b is limited by the growth of the boundary layers. Compressor blades are aerofoils that function in the same manner as aeroplane wings. Therefore as the angle of attack or camber of aerofoil is increased to increase the rotation of the flow velocity vector, the adverse pressure gradient across the suction surface increases, until at some point the boundary layer separates. As the boundary layer separates the effective turning angle b is reduced such that the total pressure increase across the stage reduces.

The limits of U, C_a and b₁ – b₂ all place limits on the maximum pressure ratio that can be achieved in an axial compressor. Typical examples are U ≈ 350 m/s, C_a = 200 m/s, b₁ – b₂ < 45°.
Compressor blades are typically quite thin and are constructed from lightweight metallic alloys such as aluminium and titanium. The compressor blades feature an aerofoil section as shown in the Figure below. The centrifugal forces that act on the airflow are balanced by high-pressure air towards the tip of the blade. In order to obtain this higher pressure towards the tip the blade must be twisted from root to tip in order to change the angle of incidence on the flow, and therefore control the pressure variation over the blade.

Key References
Rolls-Royce (1996). The Jet Engine. Rolls Royce Technical Publications; 5th ed. edition

A Brief History of Aircraft Structures On August 19, 2012 · In Composite Materials, History, Novel Materials/Tailored Structures

A Brief History of Aircraft Structures

On August 19, 2012 · In Composite Materials, History, Novel Materials/Tailored Structures

Aircraft have changed enormously over the last century from the early Wright Flyer flown at Kittyhawk to the supersonic SR-71 Blackbird flown today. Of course the developments in aeronautical engineering can be broken down into separate divisions that have developed at different rates: a) the aerodynamics, b) power plant engineering, c) control, radios and navigation aids, d) airframe engineering (e.g. hydraulic/electrical systems, interior fittings etc.), and finally e) the structural design. For example, power plants have developed in two large steps separated by a series of sudden burst of ingenuity. In order to facilitate the first successful flight the Wright Brothers had to find a light yet powerful engine system. The next stride was the ingenious invention of the jet engine prior and during WWII by Sir Frank Whittle and Hans von Ohain. In between, the power output of piston engines “increased almost 200 times from 12 bhp to over 2000 bhp in just 40 years, with only a ten times increase in mass (3) “. As will be outlined in this article, the design of aerospace structures on the other hand has only made one fundamental stride forward, but this change was sufficient to change the complete design principle of modern aircraft. Today however, the strict environmental legislation and advent of the composite era may induce further leaps in structural design.

Fig. 1. A schematic drawing of the Wright Flyer (1)

Fig. 2. The modern supersonic SR-71 Blackbird (2)

1) Wire Braced Structures
If we look at the early design of aircraft such as the Wright Flyer in Figure 1 there can really be no misunderstanding of the construction style. The entire aircraft, including most notably the wings, forward and rear structures were all constructed from rectangular frames that were prevented from shearing (forming a parallelogram) or collapsing by diagonally stretched wire. There were two major innovative thoughts behind this design philosophy. Firstly, the idea that two parallel wings would facilitate a lighter yet stronger structure than a single wing, and secondly, that these two wings could be supported with two light wires rather than with a single, thicker wooden member. The structural advantage of the biplane construction is that the two wings, vertical struts and wires form a deep light beam, which is more resistant to bending and twisting than a single wing. Much like a composite sandwich beam it can be treated as two stiff outer skins for high bending rigidity connected by a lightweight “core” to provide resistance to shear and torsion.

Fig. 3. Cutaway drawing of the 1917 Sopwith Camel (3)

Fig. 4. Cutaway drawing of the 1935 Hawker Hurricane (3)

The biplane construction with wire bracing was the most notable feature of aircraft construction for much of the following years and paired nicely with lightweight materials such as bamboo and spruce (Figure 3). Wood is a composite of cellulose fibres embedded in a matrix of lignin and the early aeronautical engineers knew to take advantage of its high specific strength and stiffness. Strangely enough, after the era of metals we are now returning back to the composite roots of aircraft, albeit in a more advanced fashion. The biplane era lasted until the 1930s at which point metal was taking over as the prime aerospace material. Initially the design philosophy was not adapted to take full advantage of thin sheet metal manufacturing techniques such that wooden spars and struts were just replaced by thinner metal tubing. Consequently there remained a striking similarity in construction between a 1917 (Figure 3) and a 1931 (Figure 4) fighter. Even though some thin metal sheets were being used these components generally did not carry much load such that the main fuselage structure featured 4 horizontal longerons supported by vertical struts and wire bracing. This so called “Warren Girder” design can also be seen in some of earliest monoplane wing constructions such as the 1935 Hawker Hurricane. Aeronautical engineers were initially “unsure how to combine the new metal construction with a traditional fabric covering (3)” used on earlier aircraft. The onset of WWII meant that some safe and conservative design decisions were made to facilitate monoplane wings and the “Warren Girder” principle was directly copied to the internal framework of monoplane wings (Figure 5). These early designs were far from optimised and perfectly characterise the transition period between wire-frame structures and the semi-monocoque structures we use today.

Fig. 5. The Hawker Hurricane wing construction (3).

2) Semi-Monocoque Structures
The internal cross-bracing was initially acceptable for the early single or double seater aircraft, but would obviously not provide enough room for larger passenger aircrafts. To overcome this, inspiration was taken from the long tradition and expertise in boat building which had already been applied to construct the fuselages of early wooden flying boats. The highest standards of yacht construction at the time featured “bent wooden frames and double or triple skins…with a clear varnished finish…and presented a much more open and usable fuselage interior (3)”. The well-established boat building techniques were thus passed on to aircraft construction to produce newer aircraft with very smooth, aerodynamic profiles.

Fig. 6. Semi monocoque fuselage construction of an early wooden flying boat (4)

The major advantage of this type of construction is that the outer skin of the fuselage and wing no longer just define the shape and aerodynamic profile of the aircraft, but become an active load-carrying member of the structure as well. Thus, the structure becomes “multifunctional” and more efficient, unlike the braced fuselage which would be just as strong without the fabric covering the girders. As a consequence the whole structure is generally at a uniform and lower stress level, reducing stress concentrations and giving better fatigue life. Finally, as the majority of the material is located at the outer surface of the structure the second and polar moments of area, and therefore the bending and torsional rigidities are much increased. On the other hand, the thin-skinned construction means that compression and shear buckling become the most likely forms of failure. In order to increase the critical buckling loads the skins are stiffened by stringers and broken up into smaller sections by spars and ribs.

Fig. 7. Components of a semi monocoque wing (5)

Because the external skin is now a working part of the structure this type of construction became to be known as stressed skin or semi-monocoque, where monocoque means  ”shell in one piece” and “semi” is an english addition to describe the discrete discontinuities of internal stiffeners. The adoption of the semi-monocoque construction and a change from wood to metal naturally coincided since sheet metal production allowed a variety of thin skins to be easily manufactured quite cheaply, with better surface finish and superior material properties. Furthermore, metal construction was conducive to riveting which would overcome the adhesive problems of early wooden semi-monocoque aircraft such as the deHavilland Mosquito.

Fig. 8. Cutaway Drawing of the recently released A400M aircraft (7).

Figure 8 shows the typical construction of a modern aircraft. There have been numerous different structural arrangements over the past number of years but all generally feature some sort of vertical stiffener (ribs in the wings and rings in the fuselage) and longitudinal stiffener (called stringers). Over the years the main driver has been towards a) a reduction in the number of rivets by reverting to bonded assembly or ideally manufacturing separate components as a single piece and b) understanding the effects and growth of cracks under static and fatigue loading by building structures that can easily be inspected or have multiple redundancies (load paths). The design and manufacturing methods of semi-monocoque aircraft are now so automated that the development of a new aluminium, medium sized airliner “could be regarded as a routine exercise (1)”. However, the continuing legislative pressure to reduce weight and fuel consumption provides enough incentive for further development.
3) Sandwich Structures and Composite Materials
One of the major disadvantages of thin-skinned structures is their lack of rigidity under compressive loading which gives them a tendency to buckle. A sheet of paper nicely illustrates this point, since it is quite strong in tension but will provide no support under compression. One way of improving the rigidity of thin panels is by increasing the bending stiffness with the aid of external stiffeners, which at the same time break the structure up into smaller sections. The critical buckling load is a function of the square of the width of the plate over which the load is applied. Therefore skins can be made 4 times stronger in buckling by just cutting the width in half. As a wing bends upwards the main compressive loads act on the top skin along the length of the wing and therefore a large number of stringers are visible across the width.

Fig. 8. Buckling analysis of a stiffened wing panel. The stiffeners break the buckling mode shapes into smaller wavelengths that require higher energy to form compared to a single wave (7)

Another technique to provide more rigidity is sandwich construction. This generally features a very lightweight core, such as a honeycomb lattice or a foam, sandwiched between two thin yet stiff outer panels. Here the role of the sandwich core is to carry any shear loads and separate the two skins as far as possible. The second moment of area is a function of the cube of the depth and therefore the bending rigidity is greatly increased with this technique. Ideally, in this manner it would be possible to design an entire fuselage without any internal rings or stringers and the Beech Starship is an excellent example of a successful application. However, there are problems of forming honeycomb cores onto doubly curved shells since the material is susceptible to strong anticlastic curvature, forming a saddle shape when bent in one direction. Furthermore, there are problems with condensation and water ingress into the honeycomb cells and the ability to guarantee a good bond surface between the core and the outer skins. There is the possibility to use foam cores instead, but these tend to be heavier with lower mechanical properties. Perhaps the current trend is away from sandwich construction (10).

Fig. 9. A carbon fibre composite/honeycomb sandwich panel (9)

Fig. 10. The Beech Starship whose fuselage was design using sandwich construction with minimal internal bulkheads and ribs (8)

One of the major applications of honeycomb structures has been in combination with composite materials. Stiff carbon composite panels are the ideal candidate for the outer skins and the whole assembly can be co-cured together in an autoclave without having to perform any secondary bonding operations. Furthermore, the incredible specific strength and stiffness of carbon composites makes this combination an  ultra lightweight yet resilient structure for aerospace applications. Indeed, we are now at the start of the “black” carbon age in commercial aircraft design. Apart from their excellent specific strength and stiffness properties composites exhibit the ability to tailor optimum mechanical properties by orientating the majority of plies in the direction of the load and allowing for less material waste during manufacture.  As a result, the first generation of commercial aircraft that contain large proportions of composite parts, such as the Boeing 787 Dreamliner and Airbus A350 XWB, are planned to enter service throughout the next years.

Fig. 11. Considerable delamination leading to catastrophic failure (11)

Considerable effort has been made to mature composite technology in order to reduce manufacturing costs, guarantee reliably high quality laminates, understand the highly complex failure criteria and built hierarchical, multifunctional or self-healing structures.  One of the major shortcomings is that the structural advantages of fibre-reinforced plastics must be viewed with respect to applications where the primary loads are aligned with the fibre direction. However, if a composite plate is subjected to significant out-of-plane stresses subsurface delaminations may develop between layers due to the weak through-thickness cohesive strength of the composite. These intralaminar delaminations are a significant problem as they are difficult to detect by visual inspection and may reduce the compressive strength of the laminate by up to 60%.
4) Novel Designs
With environmental legislation becoming ever so strict it is adamant that new concepts for lightweight and fuel efficient aircraft are found swiftly. Although the pressure on developing advanced composite materials is high it must be remembered that 100 years of innovation were required to reach the stage that large metal semi-monocoque structures could be manufactured in the 1940s and another 30 years to fully understand all failure criteria. Thus we may still require significant research and development before all current issues with composite materials are resolved. Apart from carbon fibre and other composites other researchers have been looking into completely redefining the shape of aircraft. Researchers at MIT have been developing the blended wing concept and NASA are exploring the technology of morphing or shape-changing aircraft, taking inspiration directly from nature.

Fig. 12. Illustration of the MIT Silent Aircraft concept (12).

Fig. 13. NASA morphing wing aircraft (13)

Whatever the final solution might look like the next 5o years in aerospace engineering will be incredibly innovative, ground-breaking and an exciting industry to be part of!



References
(3) Cutler, John (1992). Understanding Aircraft Structures. 2nd Edition. Blackwell Scientific Publications, Oxford.
(10) Potter, Kevin (1996). An Introduction to Composite Products: Design, Development and Manufacture. Springer, 5th Ed. Chapman & Hall, London.

Images
(1) http://www.pbs.org/wgbh/nova/wright/images/flye-lotech.gif
(2) http://thexodirectory.com/wp-content/uploads/2011/05/Air-to-air-overhead-front-view-of-an-SR-71A-460×361.jpg
(4) http://imgc.artprintimages.com/images/art-print/j-r-eyerman-workmen-building-flying-boat-that-was-designed-by-millionaire-howard-r-hughes_i-G-37-3793-OAAIF00Z.jpg
(5) http://www.nomenclaturo.com/wp-content/uploads/Airplane-Wing-Part-Diagram-Terminology.png
(6) http://pds13.egloos.com/pds/200906/24/60/a0118060_4a4194709ef22.jpg
(7) http://www.dnv.com/binaries/PULS-buckling_tcm4-284864.JPG
(8) http://www.bobscherer.com/Images/Pages/Starship/Starship%20page/NC-6%20Over%20Foggy%20Hills.jpg
(9)http://upload.wikimedia.org/wikipedia/commons/3/3d/Steinbichler_Shearography_Honeycomb_with_CFRP_Top_Layer_Artificial_failures_that_simulate_layer-core_delaminations_Material.jpg
(11) http://en.wikipedia.org/wiki/File:Delamination-CFRP.jpg
(12) http://silentaircraft.org/
(13) http://www.espaciolutacoot.com.mx/images/postcard/large/nave1.jpg