arXiv:physics/0211004v1 [physics.ed-ph] 1 Nov 2002
Stress in Rotating Disks and Cylinders
Thomas B. Bahder
U. S. Army Research Laboratory
2800 Powder Mill Road
Adelphi, Maryland, USA 20783-1197
(Dated: February 2, 2008)
The solution of the classic problem of stress in a rotating elastic disk or cylinder, as solved in
standard texts on elasticity theory, has two features: dynamical equations are used that are valid
only in an inertial frame of reference, and quadratic terms are dropped in displacement gradient
in the definition of the strain. I show that, in an inertial frame of reference where the dynamical
equations are valid, it is incorrect to drop the quadratic terms because they are as large as the
linear terms that are kept. I provide an alternate formulation of the problem by transforming the
dynamical equations to a corotating frame of reference of the disk/cylinder, where dropping the
quadratic terms in displacement gradient is justified. The analysis shows that the classic textbook
derivation of stress and strain must be interpreted as being carried out in the corotating frame of
The problem of stresses in rotating disks and cylinders is important in practical applications to rotating machinery,
such as turbines and generators, and wherever large rotational speeds are used. The textbook problem of stresses
in elastic rotating disks and cylinders, using the assumption of plane strain or plane stress, is published in classic
texts, such as Love , Landau and Lifshitz , Nadai , Sechler , Timoshenko and Goodier , and Volterra and
Gaines . The standard approach presented in these texts has two characteristic features:
1. Newton’s second law of motion is applied in an inertial frame of reference to derive dynamical equations for the
continuum (see Eq. (1) below), and
2. quadratic terms in displacement gradient are dropped in the definition of the strain tensor (see Eq.(20) below).
In this paper, I show that, for a rotating elastic body, the second feature of the solution is inconsistent with the first:
dropping the quadratic terms in the displacement gradient is an unjustified approximation in an inertial frame of
reference. In what follows, I refer to the method that is employed in Ref. [1, 2, 3, 4, 5, 6] as ‘the standard method’,
and for brevity, I will refer to a cylinder as a generalization of both a disk and a cylinder.
The classic problem of stress in an elastic rotating cylinder is complex because the undeformed reference state
of the body is the non-rotating state. The deformed state is one of steady-state rotation. The analysis of the
problem must connect the non-rotating reference state to the rotating stressed/strained state. These two states are
typically connected by large angles of rotation. When large angles of rotation are present, the quadratic terms in the
displacement gradient cannot be dropped (in an inertial frame of reference) in the definition of the strain [7, 8, 9]. The
problem of stress analysis when large-angle rotations are present is well known and has been discussed by a number
of authors in general contexts, see for example [7, 8, 9]. However, large-angle rotations in the problem of a rotating
elastic cylinder have not been dealt with in a technically correct manner, because quadratic strain gradient terms are
incorrectly dropped in the ‘standard method’ [1, 2, 3, 4, 5, 6].
In this work, I formulate the elastic problem of a rotating cylinder in a frame of reference that is corotating with
the material. In this corotating frame, the quadratic terms in the displacement gradient can be dropped, and the
resulting differential equations are linear and can be solved.
In section II, I review the ‘standard method’ of solution used in Ref. [1, 2, 3, 4, 5, 6] and show that for a rotating
cylinder the displacement gradient in an inertial frame of reference is of order unity, and therefore quadratic terms
(in strain tensor definition) cannot be dropped when compared to the linear terms. Section III contains the bulk
of the analysis. I describe the corotating systems of coordinates and the transformation of the velocity field to the
corotating frame. I use the velocity transformation rules to transform the dynamical Eq. (1) from the inertial frame
to the corotating frame (see Eq. (61) or (62)), where extra terms arise known as the centrifugal acceleration and the
coriolis acceleration. In section IV, I write the explicit component equations for stress (in cylindrical coordinates)
for the rotating elastic cylinder in its corotating frame. To display the resulting solution concretely, I derive the
well-known formula for the stress in the rotating cylinder for the case of plane stress, as computed in the corotating
frame. Stress is an objective tensor, i.e., stress is independent of observer motion [10, 11], so the physical meaning
of stress in the corotating frame is the same as in the inertial frame. Therefore, the stress field components in the
corotating frame are equal to the stress field components in the inertial frame, see Eq. (36).
II. STANDARD SOLUTION METHOD
In the ‘standard method’ [1, 2, 3, 4, 5, 6], the stress analysis of elastic rotating cylinders starts with the dynamical
equations, which, in generalized curvilinear coordinates are given by [2, 10, 11]
;j+ ρfk= ρak
where σkjare the contravariant components of the stress tensor, fkis the vector body force, and akis the acceleration
vector. In Eq. (1), repeated indices are summed and the semicolon indicates covariant differentiation with respect to
the coordinates. Expressed in terms of the velocity field in spatial coordinates, the acceleration is given by [10, 11]
where vais the velocity field, and the semicolon indicates covariant differentiation with respect to the coordinates,
time dependent. Equation (1) is derived by applying Newton’s second law of motion to an element of the medium.
Newton’s second law is valid only in an inertial frame of reference, and consequently the validity of Eq. (1) is limited
to inertial frames of reference.
In the ‘standard method’ of solution, Eq. (1) is applied by invoking an “effective body force”, of magnitude equal to
the centrifugal force in the rotating frame. In the inertial frame, there is actually no effective force (such as Coriolis
or cetrifugal force). For the case of a body rotating about its principle axis, a more careful determination of the terms
fk− akin Eq. (1) comes from setting the body force to zero (or setting equal to some applied force) and computing
the material acceleration akfor a given body motion. For a rigid body, or a uniform density elastic cylinder that is
rotating about its axis of symmetry at a constant angular velocity ωo, the Cartesian velocity field components are:
v1= −ωoy, v2= ωox, and v3= 0, where superscripts 1,2,3 indicate components on the Cartesian basis vectors
associated with the x,y,z-axes (in the inertial frame). Corresponding to this velocity field, the cylindrical components
of the acceleration field are given by
;jis called the convective term. In Eq. (1), the stress σkj, acceleration ak, and body force fk, are generally
¯ ak=∂¯ vk
+ ¯ vb¯ vk
where I have chosen the z-axis as the symmetry axis and the bar over the components indicates that they are in the
inertial frame of reference in cylindrical coordinates. For the case where there are no body forces, with the acceleration
in Eq. (3), Eq. (1) in cylindrical coordinates leads to the three equations
,1+ ¯ σ12
,2+ ¯ σ13
− r¯ σ22= −ρrω2
,1+ ¯ σ22
,2+ ¯ σ23
r¯ σ12= 0
,1+ ¯ σ23
,2+ ¯ σ33
where the superscripts 1,2,3 enumerate tensor components on the r,φ,z coordinate basis vectors respectively, in
cylindrical coordinates and the commas indicate partial differentiation with respect to these coordinates.
For steady rotation at a uniform angular velocity ωo, and assuming the absence of elastic waves, there is rotational
symmetry about the z-axis so the stress components do not depend on azimuthal angle φ. Therefore, all derivatives
with respect to φ are zero, leading to the equations:
,1+ ¯ σ13
− r¯ σ22= −ρrω2
,1+ ¯ σ23
r¯ σ12= 0
,1+ ¯ σ33
I introduce physical components of stress, σrr, σφφ,σzz, σrφ, σrz, and σφz, with units of force per unit area and
which are related to the tensor components ¯ σ11, ¯ σ22, ¯ σ33, ¯ σ12, ¯ σ13, and ¯ σ23, by [10, 11]
¯ σrr= ¯ σ11, (10)
Substitution of this solution into Eq. (86)and (87) I obtain
˜ σrr= C1+ C2
r′ 2−3 + ν
r′ 2−1 + 3ν
˜ σφφ= C1− C2
The stresses at r′= 0 must remain finite, so I take C2= 0. Applying the boundary condition on the long peripheral
surface, Eq. (79) leads to C1= (3 + ν)ρω2
ob2/8 and the stresses
3 + ν
?(3 + ν)b2− (1 + 3ν)r′ 2?
The physical stress components in Eq. (92) and (93) are in the corotating frame S′. However, due to the transformation
between the corotating frame and the inertial frame in Eq. (36), and the coordinate transformation in Eq. (35), the
corotating frame components in Eq. (92) and (93) are equal to the inertial frame components of stress. Using the
expressions in the rotating frame, such as Eq. (75)—(77), expressions for plane strain and other boundary conditions
can be derived for rotating cylinders, disks and annular rings, see Ref. [1, 2, 3, 4, 5, 6].
The classic problem of stress in rotating disks or cylinders is important in applications to turbines, generators, and
whenever large rotational speeds exist. The textbook problem of stress in perfectly elastic disks or cylinders is solved
in standard texts [1, 2, 3, 4, 5, 6]. The ‘standard method’ of solution begins with Eq. (1) and drops terms that are
quadratic in strain gradient in the definition of the strain, see Eq. (20). Equation (1) is valid only in an inertial frame
of reference, since it is derived from Newton’s second law of motion, which itself is only valid in an inertial reference
In this work, I have shown that dropping the terms quadratic in the displacement gradient (in Eq. (20)) is incorrect
in the inertial frame in which Eq. (1) is applied in the ‘standard method’ of solution [1, 2, 3, 4, 5, 6]. I provide an
alternative formulation of the rotating elastic cylinder problem in a frame of reference that is corotating with the
cylinder. In this corotating frame, I derive the dynamical equation for the stress (see Eq. (61) or (62)) and I show that
terms quadratic in the displacement gradient can be dropped because they are small (for moderate angular speed of
rotation). This analysis in the corotating frame shows that the ‘standard method’ of solution [1, 2, 3, 4, 5, 6] should
be interpreted as being carried out in the corotating frame of reference of the cylinder.
Furthermore, when stresses are computed in rotating disks or cylinders composed of materials that have more
complex constitutive equations, such as elastic-plastic or viscoelastic behavior, one must carefully justify dropping
the quadratic terms in displacement gradients. If dropping these terms cannot be justified, then the problem can be
analyzed in a rotating frame, using the derived Eq. (61) or (62). Another practical application of the stress Eq. (62)
in the rotating frame is to study elastic waves in bodies during rotation, where coriolis effects may play a role.
The author thanks Dr. W. C. McCorkle, U. S. Army Aviation and Missile Command, for suggesting this problem
and providing numerous discussions. The author thanks Howard Brandt for discussions and pointing out Ref. .
APPENDIX A: CONVENTIONS
I specify tensor components on coordinate (non-holonomic) basis vectors using numerical indices, 1,2,3, such as
¯ σ12. For physical components, which have the dimensions associated with that quantity, I use lettered indices, such
as ¯ σrφ. In addition, I must distinguish between four coordinate systems: Cartesian and cylindrical coordinates in the
inertial frame S and Cartesian and cylindrical in the corotating frame S′. I use zk= (x,y,z) and xk= (r,φ,z) for
Cartesian and cylindrical coordinates in inertial frame S, respectively. In corotating frame S′, I use z′k= (x′,y′,z′)
and x′ k= (r′,φ′,z′) for Cartesian and cylindrical coordinates, respectively. For distinguishing components in these
four coordinate systems, I use an additional mark as follows: absence of mark and a bar, for Cartesian and cylindrical
TABLE I: Coordinates
Inertial Frame S Rotating Frame S′
TABLE II: Tensor Components
Inertial Frame S
Rotating Frame S′
a σ′ab(z′k) = σ′
˜ σab(x′k), ˜ eik, ˜ ωb
Cartesian σab(zk) = σab, eik, ωb
Cylindrical¯ σab(xk), ¯ eik, ¯ ωb
ik, ω′ b
components in inertial frame S, respectively. For components in the corotating frame S′, I use a prime and a tilde,
for Cartesian and cylindrical components, respectively. See Table I and II.
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;k, so that the full strain tensor eik= 0.