61 Euler buckling of columns
The first significant contribution to the theory of the buckling of columns was made as early as 1744 by Euler. His classical approach is still valid, and likely to remain so, for slender columns possessing a variety of end restraints. Our initial discussion is therefore a presentation of the Euler theory for the small elastic deflection of perfect columns. However, we investigate first the nature of buckling and the difference between theory and practice. It is common experience that if an...
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
51 Pure bending of thin plates
The thin rectangular plate of Fig. 5.1 is subjected to pure bending moments of intensity Mx and My per unit length uniformly distributed along its edges. The former bending moment is applied along the edges parallel to the v axis, the latter along the edges parallel to the v axis. We shall assume that these bending moments are positive when they produce compression at the upper surface of the plate and tension at the lower. If we further assume that the displacement of the plate in a direction...
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...
References 1
1 Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, 2nd edition, McGraw-Hill Book Company, New York, 1961. 2 Gerard, G., Introduction to Structural Stability Theory, McGraw-Hill Book Company, New Yprk. 1962. 3 Murray, N. W., Introduction to the Theory of Thin-walled Structures, Oxford Engineering Science Series, Oxford, 1984. 4 Handbook of Aeronautics No. Structural Principles and Data, 4th edition. The Royal Aeronautical Society, 1952. 5 Bleich, F., Buckling Strength of Metal...
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...
Gl
P.3.3 Show that the warping function ip kxy, in which k is an unknown constant, may be used to solve the torsion problem for the elliptical section of Example 3.1. is the correct solution for a bar having a cross-section in the form of the equilateral triangle shown in Fig. P.3.4. Determine the shear stress distribution, the rate of twist and the warping of the cross-section. Find the position and magnitude of the maximum shear stress.
References Qcs
1 Zbrozek, J. K., Atmospheric gusts present state of the art and further research . Roy. Aero. Soc., Jan. 1965. 2 Cox. R. A., A comparative study of aircraft gust analysis procedures. J. Roy. Aero. Soc Oct. 1970. 3 BisplingholT. R. L Ashley, H. and Halfman, R. L Aeroelasticity. Addison-Wesley Publishing Co. Inc., Cambridge, Mass 1955. 4 Babister, A. W Aircraft Stability and Control, Pergamon Press, London. 1961. 5 Zbrozek, J. K Gust Alleviation Factor. R. and M. No. 2970. May 1953. 6 Handbook...
Pit
Vertically q23 sin lt j gt rd f gt Jo The shear flows in the remaining walls are constant and the solution proceeds as before. 11.3 Thin-walled rectangular section beam subjected to torsion In Example 9.7 we determined the warping distribution in a thin-walled rectangular section beam which was not subjected to structural constraint. This free warping distribution u0 was found to be linear around a cross-section and uniform along the length of the beam having values at the corners of The effect...
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...
Problems 1
P.3.1 Show that the stress function 4 gt k r2 a2 is applicable to the solution of a solid circular section bar of radius a. Determine the stress distribution in the bar in terms of the applied torque, the rate of twist and the warping of the cross-section. Is it possible to use this stress function in the solution for a circular bar of hollow section P.3.2 Deduce a suitable warping function for the circular section bar of P.3.1 and hence derive the expressions for stress distribution and rate...
Ac
Calculate the tail load necessary for equilibrium in the turn. The necessary data are given in the usual notation as follows Weight W 133 500 N dCL da 4.5 rad P.8.6 The aircraft for which the stalling speed Vs in level flight is 46.5 m s has a maximum allowable manoeuvre load factor ri of 4.0. In assessing gyroscopic effects on the engine mounting the following two cases are to be considered a pull-out at maximum permissible rate from a dive in symmetric flight, the angle of the flight path to...
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 BredtBatho 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....
Problems Bwg
P.8.1 The aircraft shown in Fig. 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...
Info Vdi
The shear strain at a point in a body is defined as the change in the angle between two mutually perpendicular lines at the point. Therefore, if the shear strain in the xz plane is then the angle between the displaced line elements O'A' and O'C' in Fig. 1.12 is 7r 2 - radians. Now cosA'O'C' cos tt 2 - jx, sin jxz and as 7iz is small then cos A'O'C' jx . From the trigonometrical relationships for a triangle cos A O C 2 0'A' 0'C' We have previously shown, in Eq. 1.17 , that But for small...