Elasticity_ Barber

84 6 End effects
problems therefore decay more slowly and this is confirmed by the fact that
the real part of the first eigenvalue of the symmetric equation (6.21) is only
λ R b=2.1.
Figure 6.3: Leading term in self-equilibrated tractions for (a) symmetric
loading and (b) antisymmetric loading.
6.3 Other Saint-Venant problems
The general strategy used in §6.2 can be applied to other curvilinear coördinate
systems to correct the errors incurred by imposing the weak boundary conditions
on appropriate edges. The essential steps are:-
(i) Define a coördinate system (ξ, η) such that the boundaries on which the
strong conditions are applied are of the form, η = constant.
(ii) Find a class of separated-variable biharmonic functions containing a parameter
(λ in the above case).
(iii) Set up a system of four homogeneous equations for the coefficients of
each function, based on the four traction-free boundary conditions for the
corrective solution on the edges η = constant.
(iv) Find the eigenvalues of the parameter for which the system has a nontrivial
solution and the corresponding eigenfunctions, which are then used
as the terms in an eigenfunction expansion to define a general form for
the corrective field.
(v) Determine the coefficients in the eigenfunction expansion from the prescribed
inhomogeneous boundary conditions on the end ξ = constant.
6.4 Mathieu’s solution
The method described in §6.2 is particularly suitable for rectangular bodies
of relatively large aspect ratio, since the weak solution is then quite accurate
over a substantial part of the domain and the corrections at the two ends are
essentially independent of each other. In other cases, and particularly for the
square b = a, we lose these advantages and the inconvenience of solving the
eigenvalue equations (6.21, 6.22) tilts the balance in favour of an alternative