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Extra resources for Aircraft Structures 3E
The existence, therefore, of just three single-valued functions for displacement is an expression of the continuity or compatibility of displacement which we have presupposed. Since the six strains are defined in terms of three displacement functions then they must bear some relationship to each other and cannot have arbitrary values. These relationships are found as follows. 20) with respect to x and y gives a2y,yy- -a2 dv axay axay ax ~ a2 au +--axay ay or since the functions of u and v are continuous which may be written, using Eq.
15 Strain gauge rosette. the form of a rosette, as shown in Fig. 15. Suppose that and are the principal strains at the point, then if E,, &b and E, are the measured strains in the directions 8, (8 + a ) , (8 + a p) to we have, from the general direct strain relationship of Eq. 31) + E, = E ~ C O2S 8 + cIIsin2 8 (1S O ) since E, becomes E~ becomes qI and -yxy is zero since the x and y directions have become principal directions. Rewriting Eq. e. 53). 47). 56) 30 Basic elasticity Fig. 16 Experimental values of principal strain using Mohr's circle.
The determination of stresses in these regions requires, for some problems, separate calculation (see Chapter 11). Fig. 3 Stress distributions illustrating St. Venant‘s principle. -. 47) (for the case of plane stress) are used to determine the components of strain. 28). 11) in which a, b and c are constants representing movement of the body as a whole or rigid body displacements. Of these a and c represent pure translatory motions of the body while b is a small angular rotation of the body in the x y plane.