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Extra resources for Aircraft Structures 3E
Substituting these values of u and Y in Eq. s. of this equation is a constant which means that F1( x )and F2(y ) must be constants, otherwise a variation of either x or y would destroy the equality. Denoting Fl ( x )by C and F2(y ) by D gives C+D=-- Pb2 8IG (viii) and so that and Py3 uPy3 h ( Y )=a-=+DY+H Therefore from Eqs (vii) u = - - - -Px2y 2EI uPy3 Py3 - Dy 6EI 6IG + + +H uPxy2 Px3 v = -+-+Cx+F 2EI 6EI The constants C , D, F and H are now determined from Eq. (viii) and the displacement boundary conditions imposed by the support.
The strain-displacement relationships are then used to form three equations involving the three displacements u, v and w. The boundary conditions for this method of solution must be specified as displacements. 20); the six unknown stresses follow from the equations expressing stress as functions of strain. It should be noted here that no use has been made of the compatibility equations. The fact that u, and UT are determined directly ensures that they are single-valued functions, thereby satisfying the requirement of compatibility.
9) we have E = -(2C +A ) The stress components are then a24 ox = -= Cx2 aY2 g - - 8= 4 A E'- ax2 #q5 x - + Dxy - (2C + A)y2 + Bxy + Cy2 B 2 5 = -axay - - - - 2- DY2 2cxy -2 The coefficients A , B, C and D are arbitrary and may be chosen to produce various loading conditions as in the previous examples. The obvious disadvantage of the inverse method is that we are determining problems to fit assumed solutions, whereas in structural analysis the reverse is the case. However, in some problems the shape of the body and the applied loading allow simplifying assumptions to be made, thereby enabling a solution to be obtained.
Aircraft Structures 3E by Megson