There exists several interpretations for the term “torsion-free bending” on which the definition of shear center (§ 1.2) is based.

Consider a uniform cantilever beam (Fig. A 1.1) of length /, built-in at the right end (z = /), and loaded at the left end by a force 8 which acts in the negative jy-axis direction. The origin is taken at the centroid of



the cross section at the free end. The horizontal axis x and the vertical axis у are the principal axes of the cross section.

First Analytical Definition. The stresses and deflection of this beam can be found by Saint-Venant’s theory of bending. Let u, v represent the components of displacement in the x, у axes’ directions, respectively. It is well known that the “rotation” ш of an element in the cross section of the beam is expressed by the equation


The rate of change of w in the axial direction is

where exz and eyz are the shearing-strain components. By Hooke’s law, one obtains

^ = J_ дтхг

dz 2G l 3x dy

where rxz and ryz are shearing-stress components.

It turns out that the right-hand side of Eq. 2 can be expressed very simply in terms of the stress function used in Saint-Venant’s theory. The result indicates that the “local twist” 3a>/3z at different points in the cross section has different values. It is impossible to have this zero for all elements of the cross section.

It seems natural to define the torsion.-free bending by the condition that the average value of the local twist over the whole section vanishes.* Hence, the first analytic definition of a torsion-free bending is


By using Eq. 3, the problem of “torsion-free” bending can be solved in a classical way. The shear center is then determined from the fact that the resultant of the shearing stresses in the section and the load S must be equal and opposite and have the same moment arm about the z axis. The distance І of the resultant from the у axis is therefore the abscissa of the shear center. Equating the moment about the z axis of the shearing stresses rxy and ryz with that of S, one obtains

f = £ J J(z V – yrj dx dy (4)

In this formulation, the position of the shear center depends on the Poisson’s ratio y.

Second Analytical Definition. Trefftz proposes-}- another definition of torsion-free bending on the basis of energy considerations. If a beam is twisted by a couple M at the free end, so that that end rotates through an angle a, the elastic strain energy stored in the beam is equal to the work done by the couple during deformation

A1 = IMat (5)

* This is the view taken by J. N. Goodier, J. Aeronaut. Sci. 11, 272-280 (1944). R. D. Specht, in a note to J. Applied Mech. 10, A-235-236 (1943), attributed this definition to A. C. Stevenson {Phil. Trans. Roy. Soc. London, A. 237, 161-229 (1938- 1939).) This definition agrees with the works of Timoshenko.

t E. Trefftz, Z. angew. Math. u. Mech. 15, 220-225 (1935). For a different point of view, which leads to results agreeing with Trefftz’s definition, see P. Cicala, Atti. R. Acc. Sci. Torino 70, 356-371 (1935), and A. Weinstein, Quart. Applied Math, 5, 97-99 (1947).

On the other hand, if a single force S is applied at the free end where the deflection is d, then the elastic strain energy is

A2 = iSd (6)

In combined action of the couple M and the force 8, the elastic strain energy stored in the beam is, in general, different from the sum Ax + A2. Let us apply the torsional moment first, so that it does the work Av Then apply the bending load 8, keeping M fixed. The load S does the work A2, while the moment M must do additional work corresponding to the angle of rotation of the section induced by the action of 8. Trefftz defines the torsion-free bending by the condition that this energy of “interaction” be zero. Alternately, if a shear acts through the “shear center,” so that the beam is in “pure” bending, and then a torque is added, the elastic energy in the beam is simply the sum of elastic energies due to the torsion and the “pure” bending alone. The order of applica­tion of S and M is evidently immaterial to this definition.

Trefftz derives an expression for the coordinates of the shear center with the aid of the “warping” function in St.-Venant’s torsion theory.

The shear center so determined is independent of the Poisson’s ratio ц and the average value of the local twist over the cross section, given by the integral in Eq. 3, does not always vanish.

When the section is symmetrical about the у axis, the two definitions yield the same location of the shear center. It can also be shown that, for a single-cell closed thin-walled section, the two definitions agree, whereas, for a multicell thin-walled section, the two definitions in general disagree.

Example. For a semicircular cylinder the abscissa of the shear center is

(a) By the first definition,[42]

8 3 + 4fi * ~ Ш 1 + p, °

(b) By the second definition, f

For a Poisson’s ratio /л — 0.3; the locations of the shear center P are shown in Fig. A 1.2.

Thin-Walled Sections—Third Definition. For a single-cell closed thin – walled tube, it is shown in § 1.2 that the shear flow q at a point s is

S f*

4 = 4a + 7 У* ds (7)

і Js0

where q0 is the value of q at s0, and s is the distance measured along the wall. In order to find the shear flow, it is necessary to determine q0. In aeronautical literature it is customary to define a “pure” bending of the tube as one in which the value of q0 is so chosen that the strain energy

Fig. A 1.2. Locations of shear center according to
different definitions.

stored in the tube is a minimum. Now, the elastic strain energy per unit length of the tube, due to the shearing strain, is

where the integral is taken over the entire section, pure bending,

But гq/Эq0 = 1 according to Eq. 7; hence, the condition for pure bending is

The last result can be generalized to multicelled tubes. For an и-celled tube it is necessary to determine и integration constants of the nature of

qQ above. It can be shown that an equation like 10 holds for every possible closed circuit drawn along the walls. Since there are n inde­pendent circuits for an и-celled tube, the и integration constants can be uniquely determined.

Having determined the shear flow, the abscissa of the shear center is obtained from Eq. 18, § 1.2.