[Muskelisvili 80th Anniversary Volume, pp. 403408, Moscow 1972]
1
Introduction
The considerations which follow resulted from endeavours to understand, appreciate and reconcile different statements in the literature on the subject of the centre of shear and of the centre of twist, and on the conditions under which these centres would or would not coincide. The outcome of our considerations was an approach which to us seemed quite different from what had been done previously, and rather more appropriate to the question, as we hope to make evident in what follows.
Specifically, our first object is to show that coincidence of the centres of twist and of shear may be taken to be no more than a natural consequence of a reasonable formulation of the problems of torsion and of flexure in the theory of beams.
Our second object is to show that an explicit, approximate determination of the location of these centres may be based on the Saint-Venant solutions of the problems of torsion and flexure (which by themselves are known to leave these centres arbitrary), in conjunction with a direct-methods-of-the-calculus-of-variations-type use of the principle of minimum complementary energy.
2
A Formulation of the Problems of Torsion and Flexure
We consider a linear elastic prismatical body, with boundaries defined, in an ( x, y, z) co-ordinate system, by means of a cylindrical surface f(x, y) = 0 and two parallel planes z = 0 and z = L.
We designate displacements by ux, uy, uz and stresses by Vx,Vy,Vz,Wxy,Wxz,Wyz and we assume that the usual three-dimensional homogeneous equations of linear elasticity hold. We further assume that the cylindrical boundary portion of the body is free of tractions and that the plane boundary portion z = 0 is fixed.
In regard to the plane boundary portion z = L we assume the absence of normal tractions while at the same time its tangential displacement components correspond to a plane rigid body translation and rotation, i.e., we stipulate
where U, V, and : are given constants.
It is evident that we may, in conjunction with the above set of prescribed boundary conditions, consider a set of overall tractions consisting of transverse
*With W. T. Tsai.
forces P and Q and of a torque T defined by
In view of the linearity of the problem, P, Q and T will come out to be linear combinations of U, V and : and it may be concluded that, inversely, U, V and : are related to P, Q and T in the form
It may further be concluded that Eqs. (3) retain their form for such generalizations of the stated problem as are obtained upon replacing the given conditions of end fixity at z = 0 by homogeneous linear relations of the form gi(ux, uy, uz,Vz,Wxz,Wyz) = 0 for i = 1, 2, 3, and by replacing the condition Vz = 0 for z = L by any homogeneous linear relation g(uz,Vz) = 0 for z = L.
3
The Centre of Twist and the Centre of Shear
The possibility of defining points in the cross-sections of a prismatical beam which may be designated as centre of twist and as centre of shear, respectively, depends on the rationality of the concept of the cross-sections of the beam rigidly translating and rotating in their own planes, at least approximately. We propose here to sharpen these definitions by confining them to sections which are prescribed to translate and rotate rigidly in their own planes, i.e., to the end cross-section of the prismatical beam for which the boundary conditions (1) and its mentioned generalizations are stipulated*.
Definition of Centre of Twist
With end cross-section displacements prescribed in the form ux = U y:, uy = V + x: we define the co-ordinates xT, yT of the centre of twist at those values of x and y for which ux = uy = 0, while at the same time P = Q = 0, i.e.,
Introduction of Eq. (3) into Eq. (4) expresses yT and xT in terms of influence coefficients, as follows:
*Given that the end cross-section translates and rotates rigidly, interior cross-sections may or may not also translate and rotate rigidly, exactly or approximately. To the extent that they do, by virtue of geometrical and material properties of the beam, one may ask for the dependence of the location of the centres of twist and of shear on distance from the end section of the beam, as has been done sometime earlier, in 1955, by the first-named author, in regard to the centre of twist.
Definition of Centre of Shear
We define co-ordinates xs, ys of the centre of shear as the co-ordinates of the point of intersection of the lines of action of the end forces P and Q for the case that (1) there is no rotation of the end section, and (2) the total torque about the point x = y = 0 (and any other point in the cross-section) is due to the forces P and Q.
Setting in accordance with the above definition, in Eqs. (3)
we obtain from the third equation in (3) the relation
Since (7) must hold for arbitrary ratios P/Q we have as expressions for ys and xs
Conditions for Coincidence of Centre of Twist and Centre of Shear
Inspection of Eqs. (8), (5) and (3) reveals that a sufficient condition for the coincidence of the centre of twist and the centre of shear is that the influence coefficient matrix C in Eqs. (3) be symmetric.
Since this matrix will be symmetric provided a strain energy function exists for the beam problem, coincidence of the two centres is established in the foregoing for most cases of practical interest.
Conversely, we may expect that the centre of twist and the centre of shear, as defined in the present account, will in general not coincide with each other for beams with material and/or support condition properties of such nature that a strain energy function does not exist for them.
4
The Principle of Minimum Complementary Energy for the Problems of Torsion and Flexure
We know that among all states of stress and displacement which satisfy the differential equations of equilibrium and the given surface traction conditions for f(x, y) = 0 and z = L, the state which also satisfies the stress-displacement differential equations and the given displacement boundary conditions for z = 0 and z = L is determined by a variational equation G3 = 0 where
Here W is the complementary energy density of the material of the beam, and Eq. (9) may be appropriately generalized for more general boundary conditions for z = 0 and z = L, of the kind noted in the paragraph following Eqs. (3).
In accordance with our earlier discussion it is our object to use the equation G3 = 0, on the basis of suitable approximative assumptions for the state of stress, in order to obtain approximate values of the integrals , in terms of the parameters U, V and : in (9).
To utilize (9) in a practical sense it suggests itself that we limit ourselves to cases for which part of the approximate assumptions for the state of stress consists in stipulating, as in Saint-Venant's theory of torsion and flexure, that
We further assume that the material of the beam is homogeneous and transversely isotropic so that, with (10),
where E and G are independent of x, y, z.