Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

138
TORSION
If the strain is small, tan γ ≈ γ ; therefore,
∆
γ = ρ φ
∆x
As the length ∆x of the shaft segment decreases to zero, the shear strain becomes
d
γ = ρ φ
(6.1)
dx
The quantity dφ/ dx is the angle of twist per unit length. Note that Equation (6.1) is linear
with respect to the radial coordinate ρ; therefore, the shear strain at the shaft centerline
(i.e., ρ = 0) is zero, while the largest shear strain occurs for the largest value of
ρ (i.e., ρ = c), which occurs on the outermost surface of the shaft.
γ max
c d φ
= (6.2)
dx
Equations (6.1) and (6.2) can be combined to express the shear strain at any radial coordinate
ρ in terms of the maximum shear strain:
ρ
γ ρ = γ
c
max
(6.3)
Note that these equations are valid for elastic or inelastic action and for homogeneous or
heterogeneous materials, provided that the strains are not too large (i.e., tan γ ≈ γ ). Problems
and examples in this book will be assumed to satisfy that requirement.
6.3 Torsional Shear Stress
ρ
τ ρ
τ
max
If the assumption is now made that Hooke’s law applies, then the shear strain γ can be related
to the shear stress τ by the relationship τ = Gγ [Equation (3.5)], where G is the shear modulus
(also called the modulus of rigidity). This assumption is valid if the shear stresses remain
below the proportional limit for the shaft material. Using Hooke’s law, we can express Equation
(6.3) in terms of τ to give the relationship between the shear stress τ ρ at any radial coordinate
ρ and the maximum shear stress τ max , which occurs on the outermost surface of the shaft
τmax
τ ρ
c
(i.e., ρ = c): 2 ρ
τ = τ
(6.4)
ρ
c max
FIGURE 6.5 Linear variation
of shear stress intensity as a
function of radial coordinate ρ.
τ ρ
τ ρ
τ ρ
τ ρ
ρ
As with shear strain, shear stress in a circular shaft increases linearly in intensity as the radial
distance ρ from the centerline of the shaft increases. The maximum shear stress intensity occurs
on the outermost surface of the shaft. The variation in the magnitude (intensity) of the
shear stress is illustrated in Figure 6.5. Furthermore, shear stress never acts solely on a
single surface. Shear stress on a cross-sectional surface is always accompanied by shear
stress of equal magnitude acting on a longitudinal surface, as depicted in Figure 6.6.
The relationship between the torque T transmitted by a shaft and the shear stress τ ρ
developed internally in the shaft must be developed. Consider a very small portion dA of a
cross-sectional surface (Figure 6.7). In response to torque T, shear stresses τ ρ are developed
on the surface of the cross section on area dA, which is located at a radial distance of ρ from
FIGURE 6.6 Shear stresses
act on both cross-sectional and
longitudinal planes.
2
In keeping with the notation presented in Section 1.5, the shear stress τ ρ should actually be designated τ x θ to
indicate that it acts on the x face in the direction of increasing θ. However, for the elementary theory of torsion of
circular sections discussed in this book, the shear stress on any transverse plane always acts perpendicular to the
radial direction at any point. Consequently, the formal double-subscript notation for shear stress is not needed for
accuracy and can be omitted here.