Shaft with Changing Moment Load,
Stiffness, and Radius
In the previous section on
circular bars and shafts, the bar cross section, internal moment, and stiffness
was assumed to be constant. This simplified the process of finding the angle of
twist. The total angle of twist for a shaft could be found by summing smaller segments
of the shaft, giving
If the segments are reduced to infinitesimal length, dx, then the summation
can be replaced with an integral, giving
where T(x) is the internal moment function, G(x) is a function describing the
changing stiffness, and J(x) is a function based on the cross section (changing
radius). The bar or shaft is still assumed to be circular.
It is important to note the applied moment, M(x) is not the same as the internal
moment, T(x). However, T(x) can be determined from applied moment and boundary
If T, G, and J are all simultaneously functions of x, the integral can be extremely
complex and a closed form solution may not be possible. In those cases, numerical
solution techniques are required.
In general, the internal moment is the most common parameter that will vary.
A good example is a drill where the moment builds up as the shaft goes deeper
into the material (oil drilling, screw, etc.).
axial structures, torsional shafts can have redundant boundary conditions
or supports. When the moment or torque in a shaft (or set of shafts) cannot be
determined with the basic summation of moments equation (ΣM
= 0), then some type of compatibility equations must be used.
Compatibility equations relate deflections or strains in a structure. Generally,
deflections are the easiest type of compatibility. For example, for two concentric
shafts, the angle of twist between them must be equal. On the other hand, two
shafts that are connected by gears or belts must have the arc length rotation
There is no single compatibility equation that will work for all indeterminate
torsion structures. This makes it difficult and requires the engineer to understand
the deflection process. However, two common compatibility relationships are given
in the two diagrams at the left.
The first type is two or more concentric shafts. Assuming the shafts are attached
(no slipping) the angle of twist must be the same for all shafts. For two shafts,
labeled as A and B in the diagram at the left, the angle of twist
needs to be equal.
θA = θB
Another typical indeterminate torsion structure is two separate
shafts that are connect through a set of gears at one end and fixed at the other.
Each shaft will take some of the applied torque (or moment) but the exact distribution
cannot be found with just static equilibrium equations.
Unlike the concentric shafts mentioned previously, the angle of twist between
the shafts will not be equal. However, they are related since the gears cannot
slip. The arc length change in both gears will be the same. Thus, the compatibility
equation will be,
θA rA = θB rB