## C

Fig. 4.15 Analysis of a propped cantilever by the method of complementary energy. is an example of a singly redundant beam structure for which total complementary energy readily yields a solution. The total complementary energy of the system is, with the notation of Eq. (4.18) where Ac and AB are the deflections at C and B respectively. Usually, in problems of this type, AB is either zero for a rigid support, or a known amount (sometimes in terms of R ) for a sinking support. Hence, for a...

## 336

Fig. 9.49 Idealized fuselage section of Example 9.12. The section has Cy as an axis of symmetry and resists a bending moment Mx 100 kNm. Equation (9.6) therefore reduces to The origin of axes Cxy coincides with the position of the centroid of the direct stress carrying area which, in this case, is the centroid of the boom areas. Thus, taking moments of area about boom 9 (6 x 640 + 6 x 600 + 2 x 620 + 2 x 850) 640 x 1200 + 2 x 600 x 1140 and column (5) is completed using Eq. (i). The derivation...

## W

P.9.4 A beam, simply supported at each end, has a thin-walled cross-section shown in Fig. P.9.4. If a uniformly distributed loading of intensity w unit length acts on the beam in the plane of the lower, horizontal flange, calculate the maximum P.9.7 A thin-walled cantilever has a constant cross-section of uniform thickness with the dimensions shown in Fig. P.9.7. It is subjected to a system of point loads acting in the planes of the walls of the section in the directions shown. Calculate the...

## 961 Warping of the crosssection

We saw in Section 3.4 that a thin rectangular strip suffers warping across its thickness when subjected to torsion. In the same way a thin-walled open section beam will warp across its thickness. This warping, wu may be deduced by comparing Fig. 9.36(b) with Fig. 3.10 and using Eq. (3.32), thus In addition to warping across the thickness, the cross-section of the beam will warp in a similar manner to that of a closed section beam. From Fig. 9.16 Referring the tangential displacement vt to the...

## 69 Local instability

We distinguished in the introductory remarks to this chapter between primary and secondary (or local) instability. The latter form of buckling usually occurs in the flanges and webs of thin-walled columns having an effective slenderness ratio, lc r, < 20. For e r > 80 this type of column is susceptible to primary instability. In the intermediate range of lc r between 20 and 80, buckling occurs by a combination of both primary and secondary modes. Thin-walled columns are encountered in...

## 12

The section is symmetrical about Cy so that lxy 0 and since Sx 0 the shear flow distribution in the closed section 3456 is, from Eq. (9.35) Also the shear load is applied through the shear centre of the complete section, i.e. along the axis of symmetry, so that in the open portions 123 and 678 the shear flow distribution is, from Eq. (9.34) We note that the shear flow is zero at the points 1 and 8 and therefore the analysis may conveniently, though not necessarily, begin at either of these...

## Preface to Third Edition

The publication of a third edition and its accompanying solutions manual has allowed me to take a close look at the contents of the book and also to test the accuracy of the answers to the examples in the text and the problems set at the end of each chapter. I have reorganized the book into two parts as opposed, previously, to three. Part I, Elasticity, contains, as before, the first six chapters which are essentially the same except for the addition of two illustrative examples in Chapter 1...

## 65 Energy method for the calculation of buckling loads in columns

The fact that the total potential energy of an elastic body possesses a stationary value in an equilibrium state may be used to investigate the neutral equilibrium of a buckled Fig. 6.12 Shortening of a column due to buckling. Fig. 6.12 Shortening of a column due to buckling. column. In particular, the energy method is extremely useful when the deflected form of the buckled column is unknown and has to be 'guessed'. First, we shall consider the pin-ended column shown in its buckled position in...

## 33 The membrane analogy

Prandtl suggested an extremely useful analogy relating the torsion of an arbitrarily shaped bar to the deflected shape of a membrane. The latter is a thin sheet of material which relies for its resistance to transverse loads on internal in-plane or membrane forces. Suppose that a membrane has the same external shape as the cross-section of a torsion bar (Fig. 3.7(a)). It supports a transverse uniform pressure q and is restrained along its edges by a uniform tensile force jV unit length as shown...

## Pr

The solution of Eqs (ix) involves the inversion of the matrix which may be carried out using any of the standard methods detailed in texts on matrix analysis. In this example Eqs (vii) and (viii) are clearly most easily solved directly however, the matrix approach illustrates the technique and serves as a useful introduction to the more detailed discussion in Chapter 12. A two-cell fuselage has circular frames with a rigidly attached straight member across the middle. The bending stiffness of...

## Info

The product terms in both integrals of Eq. (6.50) disappear on integration, leaving only integrated values of the squared terms. Thus Assigning a stationary value to the total potential energy of Eq. (6.51) with respect to each coefficient A in turn, then taking A as being typical, we have d(U + V) _ KEInAAn Tr2PCRn2An dA 2 3 21 We see that each term in Eq. (6.49) represents a particular deflected shape with a corresponding critical load. Hence the first term represents the deflection of the...

## 995 Alternative method for the calculation of shear flow distribution

Equation (9.73) may be rewritten in the form in which PT is the direct load in the rth boom. This form of the equation suggests an alternative approach to the determination of the effect of booms on the calculation of shear flow distributions in open and closed section beams. Let us suppose that the boom load varies linearly with z. This will be the case for a length of beam over which the shear force is constant. Equation (9.81) then becomes in which APT is the change in boom load over unit...

## 1036 Method of successive approximations torsion

It is clear from the torsion and shear loading of multicell wing sections that the greater the number of cells the greater the number of simultaneous equations requiring solution. Some modern aircraft have wings comprising a relatively large number of cells, for example, the Harrier wing shown in Fig. 7.8, so that the arithmetical labour involved becomes extremely tedious unless a computer is used an approximate but much more rapid method may therefore be preferable. The method of successive...

## 2

Which shows clearly that as Sa > S , N * oo. It has been found experimentally that N is inversely proportional to the mean stress as the latter varies in the region of 90 N mm2 while C is virtually constant. This suggests a method of determining a 'standard' endurance curve (corresponding to a mean stress level of 90 N mm2) from tests carried out on a few specimens at other mean stress levels. Suppose Sm is the mean stress level, not 90 N mm2, in tests carried out on a few specimens at an...

## 862 The graded gust

The 'graded' gust of Fig. 8.13(b) may be converted to an equivalent 'sharp-edged' gust by multiplying the maximum velocity in the gust by a gust alleviation factor, F. Thus Eq. (8.27) becomes Similar modifications are carried out on Eqs (8.25), (8.26), (8.28) and (8.32). The gust alleviation factor allows for some of the dynamic properties of the aircraft, including unsteady lift, and has been calculated taking into account the heaving motion (that is, the up and down motion with zero rate of...

## 1 1 1

So that for a member aligned with the x axis, joining nodes i and j subjected to nodal forces Fxi and Fxj, we have (FXA AE 1 -11 f FxJ L -I IJ hJ The solution proceeds in a similar manner to that given in the previous section for a spring or spring assembly. However, some modification is necessary since frameworks consist of members set at various angles to one another. Figure 12.3 shows a member of a framework inclined at an angle 6 to a set of arbitrary reference axes x, y. We shall refer...

## Bending shear and torsion of open and dosed thinwalled beams

In Chapter 7 we discussed the various types of structural component found in aircraft construction and the various loads they support. We saw that an aircraft is basically an assembly of stiffened shell structures ranging from the single cell closed section fuselage to multicellular wings and tail-surfaces each subjected to bending, shear, torsional and axial loads. Other, smaller portions of the structure consist of thin-walled channel, T-, Z-, 'top hat' or I-sections, which are used to...

## 1

The two terms in Eq. (iii) may be evaluated separately, bearing in mind that only the beam ABC contributes to the first term while the complete structure contributes to the second. Evaluating the summation term by a tabular process we have Table 4.5. Summation of column (6) in Table 4.5 gives sr FiLi dFj__RL J_ 10 fr AiE dR 4E + A J The bending moment at any section of the beam between A and F is w 3n V3 , dM M -Pz - hence

## P

Taking into account only strains due to bending, calculate the distribution of bending moment around the frame in terms of the force P, the frame radius r and the angle 6. Ans. M Pr (0.160 0.080 cos0 0.1590 sin 0). P.4.17 The frame shown in Fig. P.4.17 consists of a semi-circular arc, centre B, radius a, of constant flexural rigidity EI jointed rigidly to a beam of constant flexural rigidity 2EI. The frame is subjected to an outward loading as shown arising from an internal pressure p0. Find...

## Index

Aerodynamic centre of a wing 221 Aerodynamic forces, see Loads on structural components Aeroelasticity 540-81 control effectiveness and reversal 546-51 aileron effectiveness 548, 551 aileron reversal speed 548, 551 flutter, see Flutter load distribution and divergence 541-6 divergence speed 543, 545, 546 swept wing divergence 545, 546 wing torsional divergence 541-5 structural vibration, see Structural vibration Aileron buzz, see Flutter Aileron effectiveness and reversal, see Aeroelasticity...

## 852 Correctly banked turn

From Eq. (8.19) we see that the load factor n in the turn is given by Also, dividing Eq. (8.17) by Eq. (8.18) Examination of Eq. (8.21) reveals that the tighter the turn the greater the angle of bank required to maintain horizontal flight. Furthermore, we see from Eq. (8.20) that an increase in bank angle results in an increased load factor. Aerodynamic theory shows that for a limiting value of n the minimum time taken to turn through a given angle at a given value of engine thrust occurs when...

## St l fy J

The curves represented by Eq. (8.71) may be divided into three regions. The first corresponds to a very slow crack growth rate (< 10s m cycle) where the curves approach a threshold value of stress intensity factor AA , corresponding to 4x10 m cycle, i.e. no crack growth. In the second region (10 8-106 m cycle) much of the crack life takes place and, for small ranges of AA , Eq. (8.71) may be represented by in which C and n depend on the material properties over small ranges of d dyV and AK, C...

## Mi

Equations (11.60) and (11.61) hold if the centre of twist coincides with the shear centre of the cross-section. To summarize, the centre of twist of a section of an open section beam carrying a pure torque is the shear centre of the section. We are now in a position to calculate rR. This may be done by evaluating Jc 4AR t ds in which 2Ar is given by Eq. (11.56). In general, the calculation may be lengthy unless the section has flat sides in which case a convenient analogy shortens the work...

## Ixl

Examination of Eq. (11.38) shows that q changes sign when cosh Az (sinh XL) XL, the solution of which gives a value of z less than L, i.e. Fig. 11.20 Cover of beam of Fig. 11.19. Fig. 11.20 Cover of beam of Fig. 11.19. Finally, the shear flow distributions are obtained from Eqs (11.39), thus _-0Pm_ PAX sinh X(L z) Oz 2 2B+A) cosh AL dPB2 _ -PAX sinh A(L-z) Again we see that each expression for direct stress, Eqs (11.47), (11.49) and (11.51). comprises a term which...

## 4

Which compares favourably with the result of Example 5.1. In this chapter we have dealt exclusively with small deflections of thin plates. For a plate subjected to large deflections the middle plane will be stretched due to bending so that Eq. (5.33) requires modification. The relevant theory is outside the scope of this book but may be found in a variety of references. Jaeger, J. C., Elementary Theory of Elastic Plates, Pergamon Press, New York, 1964. Timoshenko, S. P. and Woinowsky-Krieger,...

## 873 Fatigue strength of components

In Section 8.2 we discussed the effect of stress level on the number of cycles to failure of a material such as mild steel. As the stress level is decreased the number of cycles to failure increases, resulting in a fatigue endurance curve (the S-N curve) of the type shown in Fig. 8.2. Such a curve corresponds to the average value of N at each stress amplitude since there will be a wide range of values of N for the given stress even under carefully controlled conditions the ratio of maximum N to...

## 1270

12 56 840, 23 783, 34 1083, 3g 27 68, 75 106, I6 202 We now 'cut' the top skin panels in each cell and calculate the 'open section' shear flows using Eq. (9.75) which, since the wing section is idealized, singly symmetrical (as far as the direct stress carrying area is concerned) and is subjected to a vertical shear load only, reduces to where, from Example 10.6, Ixx 809 x 106mm4. Thus, from Eq. (i) 86.8 x 103 , 4 Since qb 0 at each 'cut', then qb 0 for the skin panels 12,23 and 34. The...

## 874 Prediction of aircraft fatigue life

We have seen that an aircraft suffers fatigue damage during all phases of the ground-air-ground cycle. The various contributions to this damage may be calculated separately and hence the safe life of the aircraft in terms of the number of flights calculated. In the ground-air-ground cycle the maximum vertical acceleration during take-off is .2g for a take-off from a runway or 1.5g for a take-off from grass. It is assumed that these accelerations occur at zero lift and therefore produce...

## 56 Energy method for the bending of thin plates

Two types of solution are obtainable for thin plate bending problems by the application of the principle of the stationary value of the total potential energy of the plate and its external loading. The first, in which the form of the deflected shape of the plate is known, produces an exact solution the second, the Rayleigh Ritz method, assumes an approximate deflected shape in the form of a series having a finite number of terms chosen to satisfy the boundary conditions of the problem and also...

## YJ

We may define aileron effectiveness at speeds below the reversal speed in terms of the lift ALR produced by an aileron deflection on a rigid wing. Thus Hence, substituting in Eq. (13.19) for AL from Eq. (13.16) and ALR from Eq. (13.20), we have Equation (13.21) may be expressed in terms of the wing divergence speed Vd and aileron reversal speed VT, using Eqs (13.3) and (13.18) respectively hence aileron effectiveness -- 2* (13.22) We see that when Fj Vr, which occurs when dCL d (dCyi dQ e, then...

## 95 Torsion of closed section beams

A closed section beam subjected to a pure torque T as shown in Fig. 9.27 does not. in the absence of an axial constraint, develop a direct stress system. It follows that the equilibrium conditions of Eqs (9.22) and (9.23) reduce to dq ds 0 and dq dz 0 respectively. These relationships may only be satisfied simultaneously by a constant value of q. We deduce, therefore, that the application of a pure torque to a closed section beam results in the development of a constant shear flow in the beam...

## 42 Total potential energy

In the spring-mass system shown in its unstrained position in Fig. 4.3(a) we normally define the potential energy of the mass as the product of its weight, Mg, and its height, h, above some arbitrarily fixed datum. In other words it possesses energy by virtue of its position. After deflection to an equilibrium state (Fig. 4.3(b)), the mass has lost an amount of potential energy equal to Mgy. Thus we may associate deflection with a loss of potential energy. Alternatively, we may argue that the...

## E

Where E is a constant known as the modulus of elasticity or Young's modulus. Equation (1.40) is an expression of Hooke's Law. Further, ex is accompanied by in which v is a constant termed Poisson's Ratio. For a body subjected to direct stresses < rv, ar and a. the direct strains are, from Eqs (1.40), (1.41) and the principle of superposition (see Chapter 4, Section 4.9) v ay + (Tz) u(ax + crz) az u tTx + (7,,) Suppose now that, at some arbitrary point in a material, there are principal...

## Pl 3ei

The fictitious load method of the framework example may be employed in the solution of beam deflection problems where we require deflections at positions on the beam other than concentrated load points. Suppose that we are to find the tip deflection AT of the cantilever of the previous example in which the concentrated load has been replaced by a uniformly distributed load of intensity w per unit

## 67 Inelastic buckling of plates

For plates having small values of b t the critical stress may exceed the elastic limit of the material of the plate. In such a situation, Eq. (6.57) is no longer applicable since, as we saw in the case of columns, E becomes dependent on stress as does Poisson's ratio v. These effects are usually included in a plasticity correction factor r so that Eq. (6.57) becomes where E and v are elastic values of Young's modulus and Poisson's ratio. In the linearly elastic region t 1, which means that Eq....

## 03 200000

(83 0.3 x 65) 3.175 x 104 (65 - 0.3 x 83) 2.005 x 10-4 (83 + 65) -2.220 x 10-4 In this case, since there are no shear stresses on the given planes, ax and < rv are principal stresses so that ex and ey are the principal strains and are in the directions of < jx and < jy. It follows from Eq. (1.15) that the maximum shear stress (in the plane of the stresses) is acting on planes at 45 to the principal planes. Further, using Eq. (1.45), the maximum shear strain is so that 7max 1.17 x 10 4 on...

## Zjl

We note in Eq. (iv) that the shear flow is not zero when s2 0 but equal to the value obtained by inserting si A 2 in Eq. (iii), i.e. q2 QA2Sy h. Integration of Eq. (iv) yields This distribution is symmetrical about Cx with a maximum value at s2 h 2(y 0) and the shear flow is positive at all points in the web. The shear flow distribution in the upper flange may be deduced from antisymmetry so that the complete distribution is of the form shown in Fig. 9.20.

## 74 Fabrication of structural components

The introduction of all-metal, stressed skin aircraft resulted in methods and types of fabrication which remain in use to the present day. However, improvements in engine performance and advances in aerodynamics have led to higher maximum lift, higher speeds and therefore to higher wing loadings so that improved techniques of fabrication are necessary, particularly in the construction of wings. The increase in wing loading from about 350 N nr for 1917 18 aircraft to around 4800 N m2 for Fig....

## 72 Loads on structural components

The structure of an aircraft is required to support two distinct classes of load the lirst, termed ground loads, includes all loads encountered by the aircraft during movement or transportation on the ground such as taxiing and landing loads, towing and hoisting loads while the second, air loads, comprises loads imposed on the structure during flight by manoeuvres and gusts. In addition, aircraft designed for a particular role encounter loads peculiar to their sphere of operation. Carrier born...

## 612 Flexuraltorsional buckling of thinwalled columns

It is recommended that the reading of this section be delayed until after Section 11.5 has been studied. In some instances thin-walled columns of open cross-section do not buckle in bending as predicted by the Euler theory but twist without bending, or bend and twist simultaneously, producing flexural-torsional buckling. The solution of this type of problem relies on the theory presented in Section 11.5 for the torsion of open section beams subjected to warping (axial) restraint. Initially,...

## 71 Materials of aircraft construction

Several factors influence the selection of the structural material for an aircraft, but amongst these strength allied to lightness is probably the most important. Other properties having varying, though sometimes critical significance are stiffness, toughness, resistance to corrosion, fatigue and the effects of environmental heating, ease of fabrication, availability and consistency of supply and, not least important, cost. The main groups of materials used in aircraft construction have been...

## 92 General stress strain and displacement relationships for open and single cell closed section thinwalled beams

We shall establish in this section the equations of equilibrium and expressions for strain which are necessary for the analysis of open section beams supporting shear loads and closed section beams carrying shear and torsional loads. The analysis of open section beams subjected to torsion requires a different approach and is discussed separately in Section 9.6. The relationships are established from first principles for the particular case of thin-walled sections in preference to the adaption...

## 410 The reciprocal theorem

A , ,, gt , al2P2 au,P, Fig. 4.24 Linearly elastic body subjected to loads P,. P2. P3, P Fig. 4.24 Linearly elastic body subjected to loads P,. P2. P3, P at the points of application of the complete system of loads are then A, anPi anP2 4- ai3P3 ainP A2 a2iPi a22P2 a23P3 a2nPn A3 31A a32P2 a33P3 a3nP

## 1041 Fuselage frames

We have noted that fuselage frames transfer loads to the fuselage shell and provide column support for the longitudinal stringers. The frames generally take the form Fig. 10.46 Support of load having a component normal to a web. Fig. 10.46 Support of load having a component normal to a web. of open rings so that the interior of the fuselage is not obstructed. They are connected continuously around their peripheries to the fuselage shell and are not necessarily circular in form but will usually...

## Elementary aeroelasticity

Aircraft structures, being extremely flexible, are prone to distortion under load. When these loads are caused by aerodynamic forces, which themselves depend on the geometry of the structure and the orientation of the various structural components to the surrounding airflow, then structural distortion results in changes in aerodynamic load, leading to further distortion and so on. The interaction of aerodynamic and elastic forces is known as aeroelasticity. Two distinct types of aeroelastic...

## 610 Instability of stiffened panels

It is clear from Eq. 6.58 that plates having large values of b 1 buckle at low values of critical stress. An effective method of reducing this parameter is to introduce stifleners along the length of the plate thereby dividing a wide sheet into a number of smaller and more stable plates. Alternatively, the sheet may be divided into a series of wide short columns by stifleners attached across its width. In the former type of structure the longitudinal stifleners carry part of the compressive...

## 83 Aircraft inertia loads

The maximum loads on the components of an aircraft's structure generally occur when the aircraft is undergoing some form of acceleration or deceleration, such as in landings, take-offs and manoeuvres within the flight and gust envelopes. Thus, before a structural component can be designed, the inertia loads corresponding to these accelerations and decelerations must be calculated. For these purposes we shall suppose that an aircraft is a rigid body and represent it by a rigid mass. in. as shown...

## Stress analysis of aircraft components

In Chapter 9 we established the basic theory for the analysis of open and closed section thin-walled beams subjected to bending, shear and torsional loads. In addition, methods of idealizing stringer stiffened sections into sections more amenable to analysis were presented. We now extend the analysis to actual aircraft components including tapered beams, fuselages, wings, frames and ribs also included are the effects of cut-outs in wings and fuselages. Finally, an introduction is given to the...

## 951 Displacements associated with the Bredt Batho shear flow

The relationship between q and shear strain 7 established in Eq. 9.39 , namely is valid for the pure torsion case where q is constant. Differentiating this expression with respect to z we have In the absence of direct stresses the longitudinal strain dw dz e. is zero so that d20 d 2u p cos sin ip 0 dz dzz dz For Eq. 9.51 to hold for all points around the section wall, in other words for all values of ifr It follows that 6 Az B, u Cz D, v Ez F, where A, B, C, D, E and F are unknown constants....

## Bending Moment Diagram Fuselage Structure

P.8.1 a weighs 135 kN and has landed such that at the instant of impact the ground reaction on each main undercarriage wheel is 200 kN and its vertical velocity is 3.5 m s. If each undercarriage wheel weighs 2.25 kN and is attached to an oleo strut, as shown in Fig. P.8.1 h , calculate the axial load and bending moment in the strut the strut may be assumed to be vertical. Determine also the shortening of the strut when the vertical velocity of the aircraft is...

## Energy methods of structural analysis

In Chapter 2 we have seen that the elasticity method of structural analysis embodies the determination of stresses and or displacements by employing equations of equilibrium and compatibility in conjunction with the relevant force-displacement or stress-strain relationships. A powerful alternative but equally fundamental approach is the use of energy methods. These, while providing exact solutions for many structural problems, find their greatest use in the rapid approximate solution of...