In a recent article, Hutchinson 1 employed the Hellinger-Reissner variational principle to construct a beam theory of Timoshenko type, together with a new expression for the inherent shear coefficient κ, as
$κ=−21+νAIy2 C4+ν1−IxIy$
(1)
where
$C4=−∫∫{νx2−y2f1+2ν x y f2+21+νf12+f22}dxd y$
(2a)
$f1=−121+ν ∂χ∂x+ν x22+2−ν2y2$
(2b)
$f2=−121+ν ∂χ∂y+2+νxy.$
(2c)

In the above, the notation is largely the same as that of Hutchinson except that Love’s notation (2) has been employed for the beam cross-sectional coordinates, by replacing y by x, and z by y, in 1; z is then the beam axial coordinate. The motivation of Hutchinson appears to be the construction of a theory in which the shear coefficient takes on the “best” value; for beams of circular and thin rectangular cross section, these are widely accepted to be $κ=61+ν2/7+12ν+4ν2,$ and $κ=51+ν/6+5ν,$ respectively. The evidence to suggest that these values are “best” comes from comparison with available “exact” elastodynamic analyses and, to a lesser degree, from experiment, and is discussed in 1,3,4.

Expression (1) derived by Hutchinson is exactly equivalent to one derived by the present author and Prof. Mark Levinson 3,4 some two decades ago, which is (Ref. 3, Eq. (20))
$κ=−41+ν2 Iy221+νA∫∫xχ+xy2dx dy+2ν1+νIyIy−Ix+νA∫∫x2−y22∂χ∂x+νx22+2−ν2y2+xy∂χ∂y+2+νxydx dy.$
(3)

It is remarkable that three quite different approaches should lead to the same expression for the coefficient. Hutchinson’s use of Hellinger-Reissner overcomes the compromises inevitable in a beam theory, allowing “best” choices for both stress and displacement fields, which may be incompatible. In 4, Stephen and Levinson adapted the procedure of Cowper 5, but argued that the stress distribution within a beam performing long wavelength flexural vibration would be approximated better by gravity force body loading (see Love 2, Chapter 16), rather than tip loading of a cantilevered beam, as assumed in 5. The former has shearing force varying linearly with axial coordinate, while for the latter shearing force is constant. In 3, the coefficient was obtained from the curvature correction during bending, again due to gravity loading (again see Chapter 16 of Love).

Demonstration of the equivalence of the two formulas is somewhat lengthy, and is based upon usage of Green’s formula
$∫∫∂g∂x−∂f∂ydx dy=∮f dx+g dy,$
(4)
and a knowledge of the normal derivative of the (harmonic) flexure function χ on the boundary of the cross section, that is
$dχdn=−νx22+2−ν2y2cosx,n−2+νxy cosy,n.$
(5)
The key step is recognition that the term within Hutchinson’s coefficient $C4$ involving the area integral of the sum of the squares of the terms $f1$ and $f2,$ in turn involves the area integral of the sum $∂χ/∂x2+∂χ/∂y2.$ Transformation of such terms within potential theory is well documented; see, for example, Sokolnikoff 6. For the present problem an outline of the procedure is as follows:construct
$∂∂x χ+xy2 ∂χ∂x=χ+xy2 ∂2χ∂x2+∂χ∂x+y2 ∂χ∂x$
(6a)
$∂∂y χ+xy2 ∂χ∂y=χ+xy2 ∂2χ∂y2+∂χ∂y+2xy ∂χ∂y$
(6b)
and add, noting that χ is harmonic, to give
$∂∂x χ+xy2 ∂χ∂x+∂∂y χ+xy2 ∂χ∂y=∂χ∂x2+∂χ∂y2+y2 ∂χ∂x+2xy ∂χ∂y.$
(7)
Integrate over the cross section, and transform the left-hand side (LHS) of the above using Green’s formula, to give
$LHS=∮χ+xy2 dχdn ds$
(8)
where direction cosines $cosx,n=dx/dn=dy/ds,$ and $cosy,n=dy/dn=−dx/ds$ have been employed.
Substitute for the normal derivative of χ according to (5), and convert back to an area integral to give
$∫∫∂χ∂x2+∂χ∂y2dx dy+∫∫y2 ∂χ∂x+2xy ∂χ∂ydx dy=−∫∫21+νxχ+xy2dx dy−∫∫2+νxy ∂χ∂y dx dy−∫∫νx22+2−ν2y2 ∂χ∂x dx dy−∫∫4+5ν2x2y2+2−ν2y4dx dy.$
(9)
Next, expand Hutchinson’s expression for coefficient $C4,$ and substitute the above, when one finds
$C4=∫∫xχ+xy2dx dy+∫∫ νx2−y241+ν ∂χ∂x+νx22+2−ν2y2dx dy+∫∫ νxy21+ν ∂χ∂y+2+νxydx dy.$
(10)
Lastly substitute the above into Eq. (1) to give expression (3). Not surprisingly, the values of the coefficient for the circular cross section, both solid, hollow and thin-walled, and for the elliptic cross section calculated in 1, are identical to those given in 3,4. Similarly, Hutchinson’s expression for the rectangular cross section reduces to the “best” value of $κ=51+ν/6+5ν$ as one approaches plane stress conditions.

A further very interesting feature of 1, Figs. 3 and 4, is the possibility of the shear coefficient taking a negative value for the combination of large width to depth ratio, and for large Poisson’s ratio. The effect of a negative coefficient would be to stiffen the structure, leading to a natural frequency higher than that predicted by Euler-Bernoulli theory. However, as one would not normally employ Timoshenko theory for a beam having a large width to depth ratio, this result may turn out to be of little importance. Nevertheless, the physical implication of a possible negative coefficient requires further consideration.

Finally, it is noted that while the above values for the coefficient may be widely accepted as the best, paradoxically Cowper’s values appear to be the more widely used; it is to be hoped that investigators will in future make greater use of these “best” values.

1.
Hutchinson
,
J. R.
,
2001
, “
Shear Coefficients for Timoshenko Beam Theory
,”
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,
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2.
Love, A. E. H., 1944, A Treatise on the Mathematical Theory of Elasticity, Dover, New York.
3.
Stephen
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N. G.
,
1980
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4.
Stephen
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N. G.
, and
Levinson
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M.
,
1979
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A Second Order Beam Theory
,”
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5.
Cowper
,
G. R.
,
1966
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The Shear Coefficient in Timoshenko Beam Theory
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6.
Sokolnikoff, I. S., 1956, Mathematical Theory of Elasticity, McGraw-Hill, New York.