Page 361 - Aircraft Stuctures for Engineering Student

342 Open and closed, thin-walled beams

9.10 Deflection of open and closed sectio

Bending, shear and torsional deflections of thin-walled beams are readily obtained by
application of the unit load method described in Section 4.8.
The displacement in a given direction due to torsion is given directly by the last of
Eqs (4.27), thus

(9.83)

where J, the torsion constant, depends on the type of beam under consideration. For
an open section beam J is given by either of Eqs (9.59) whereas in the case of a closed
section beam J = 4A2/($ds/t) (Eq. (9.52)) for a constant shear modulus.
Expressions for the bending and shear displacements of unsymmetrical thin-walled
beams may also be determined by the unit load method. They are complex for the
general case and are most easily derived from first principles by considering the
complementary energy of the elastic body in terms of stresses and strains rather
than loads and displacements. In Chapter 4 we observed that the theorem of the prin-
ciple of the stationary value of the total complementary energy of an elastic system is
equivalent to the application of the principle of virtual work where virtual forces act
through real displacements. We may therefore specify that in our expression for total
complementary energy the displacements are the actual displacements produced by
the applied loads while the virtual force system is the unit load.
Considering deflections due to bending, we see, from Eq. (4.12), that the increment
in total complementary energy due to the application of a virtual unit load is

where o~,~ is the direct bending stress at any point in the beam cross-section corre-
sponding to the unit load and E,,~ is the strain at the point produced by the actual
loading system. Further, AM is the actual displacement due to bending at the point
of application and in the direction of the unit load. Since the system is in equilibrium
under the action of the unit load the above expression must equal zero (see Eq. (4.12)).
Hence

(9.84)

From Eq. (9.6) and the third of Eqs (1.42)

where the suffixes 1 and 0 refer to the unit and actual loading systems and x, J' are the
coordinates of any point in the cross-section referred to a centroidal system of axes.
Substituting for o~,~ and E,,~ in Eq. (9.84) and remembering that jA .Y* dA = I,,.,