Approximate theories for solids with
rods, beams, membranes, plates and shells
10.5 Motion and Deformation of thin shells General theory
figure illustrates the problem to be solved.
The solid of interest is a shell with uniform thickness h.
The shell’s thickness is assumed to be much smaller than any relevant
in-plane dimension. The exterior
surface of the shell is subjected to a prescribed distribution of traction,
while the edge of the shell may either be supported so as to constrain its
motion, or may be subjected to prescribed forces. Our objective is to calculate the internal
forces in the shell, and to compute its deformed shape.
10.5.1 Coordinate systems and variables
characterizing deformation of shells
specify the position of a point on the mid-plane of the undeformed shell, we introduce a convenient curvilinear coordinate system
(examples include cylindrical or spherical
polar coordinates). Note that need not necessarily be distances along the
surface: for example, for a cylindrical shell, we would use the axial distance z and the angle as the coordinate system.
The position vector of a material particle on
the mid-section of the initial shell is denoted by
To characterize the orientation of an
arbitrary point in the undeformed shell, we introduce three basis vectors ,
Thus, are tangent to the coordinate lines in the undeformed shell, and is a unit vector perpendicular to the
mid-section of the shell. This basis is
called the covariant basis or natural basis for the coordinate
system. Note that the basis vectors are not unit vectors, and are not, in general,
Because are not orthogonal, it is convenient to
introduce a second set of basis vectors defined so that
where is the Kronecker delta symbol (the index i has been raised to match the indices
on the basis vectors), i.e. for i=j
and zero otherwise. This second triad of
vectors is called the contravariant basis or reciprocal
basis for the coordinate
contravariant basis vectors can be constructed by taking cross products of the
covariant basis vectors, as follows
position vector of an arbitrary point in the undeformed shell can be expressed
where is the perpendicular distance of the material
particle from the mid-section of the shell.
deformation, the mid-section of the shell is deformed to another smooth
surface. The point that lies at on the mid-section of the undeformed shell
moves to a new position after deformation.
To characterize the orientation of the
deformed shell, we introduce three basis vectors ,
with . Now, are tangent to the coordinate lines in the deformed shell, and is a unit vector perpendicular to the
mid-section of the deformed shell. We
can introduce a reciprocal basis in exactly the same way as for the undeformed
A few special vectors and tensors, such as the
angular velocity of the shell, and the internal stress couple in the shell are
most conveniently expressed in terms of vectors or . Special symbols will not be introduced for
these vectors; they will always be written out as a cross product.
10.5.2 Vectors and tensor components in
non-orthogonal bases: Covariant and Contravariant components
this section we introduce some additional notation that helps deal with the
complicated sets of basis vectors that characterize the deformation of a shell.
can be expressed as linear combinations of some subset of the twelve possible
basis vectors ,
or . For example, we can write an arbitrary vector
Here, the coefficients are called the contravariant components of a
and are called the covariant components of a.
Note that the contravariant components are coefficients of the covariant
basis vectors, and vice-versa.. The
reason for this confusing terminology is given below. Note also that the components do not in
general have the same units as the vector, because the basis vectors may have
The various components of a can be expressed as
To see the first result, take dot
products of with and recall that .
The `contravariant’ and `covariant’ terms assigned to and refer to the fact that they represent
projections of the vector a onto the
contravariant and covariant basis vectors, respectively. The raised and lowered indices for vector
components follow the same convention: raised indices indicate contravariant
components, while lowered indices represent covariant components.
can also be expressed as sums of nine dyadic products of various combinations
of basis vectors. For example, if S is a tensor we could write
or equivalent expressions in terms of and/or . Here are called contravariant and covariant
components of S, respectively, while
are called mixed tensor
components. The dot that appears before
the indices in the mixed tensors is introduced to identify whether the index
should be associated with the first or second basis vector in the dyadic
product (the dot appears before the index associated with the second basis
vector). For symmetric tensors, the dot
can be dropped.
The various components of S can be regarded as projections of the tensor onto the
contravariant or covariant basis vectors, as ,
with similar results for the mixed components.
Once again, it is important to note that the
components do not have a convenient physical
interpretation. In general the components
do not even have the same units as the tensor itself, because the basis vectors
themselves have units.
various sets of components can be related by defining the components of the fundamental
tensor or metric tensor g as follows
can define in terms of and/or in the same way. With these definitions, we see that
the covariant and contravariant basis vectors are related by
metric tensor can be expressed as the sum of three dyadic products ,
with a similar expression for . From this expression we see that
g is therefore its own inverse it is a representation of the identity tensor.
10.5.3 Additional Deformation Measures
and Kinematic Relations
infinitesimal line element that lies in the mid-section of the undeformed
plate can be expressed in terms of infinitesimal changes in the coordinates as
length ds of can be computed as
This expression is known as the first fundamental form for
the surface. A similar expression can be
constructed for the deformed surface.
variation of the normal vectors and with position in the mid-plane of the shell
play a particularly important role in describing the shape and deformation of
the shell, because they characterize its curvature and bending. To quantify this variation, let and be the vectors normal to the surface at positions
and in the undeformed shell, with a similar
notation for the deformed shell. We introduce symmetric curvature tensors and that satisfy
The curvatures and are called surface tensors, because
they transform like tensors under changes of surface coordinates.
curvature components can be expressed in terms of their covariant,
contravariant or mixed components. This
can be used to deduce expressions such as
curvature components can be calculated from the position vector of the
mid-plane of the shell, using any of the following expressions
The mixed components follow as ,
where are the components of the metric tensor
defined in Sect 10.5.2.
magnitude of the curvature of a shell is quantified by principal curvatures - these are simply the principal values of .
and Gaussian curvature are also used.
also need to calculate the variation of the remaining basis vectors with
position in the surface. These are
quantified by Christoffel symbols of the second kind which satisfy
The Christoffel symbols are functions of position on
the surface, and can be related to the position vector of the mid-plane of the
shell and its curvature components as
Some relationships between the time
derivatives of these various kinematic quantities are also needed in subsequent
calculations. The rate of change in
shape of the shell can be characterized by the velocity of its middle surface . The velocity vector can be described as
components in any of the various bases: the representation is particularly useful.
The time derivatives of the basis vectors are a convenient way to characterize the rate
of change of bending of the shell.
These are related to the velocity of the shell’s mid-plane by
We will also need to calculate the time
derivative of the vector normal to the mid-plane of the shell . Since is a unit vector, its time derivative can be
quantified by an angular velocity vector ,
defined so that
The components of can be related to as
Recalling that ,
we see also that
Finally we may write this as
The components can also be regarded as the components of the
angular velocity vector in a basis in the sense that
The time derivative of the curvature tensor ,
is related to by
The components of the time derivative of curvature can
therefore be expressed in terms of as
It is important to note that are not equal to the time derivatives of the
We will also need to characterize the linear
and angular acceleration of the shell.
The linear acceleration can be quantified by the acceleration of the mid-plane
The angular acceleration of the shell can be
characterized by the angular acceleration of the normal to its mid-plane, . The angular acceleration can be related to
the acceleration of the mid-plane of the shell as follows
where we have used and noted that
to obtain the final result.
It is convenient to express in the form ,
where and the can be related to the velocity and
acceleration of the mid-plane of the shell as follows
These results show that
where we have used and noted
to obtain the second line.
HEALTH WARNING: The sign convention used to
characterize the curvature of a shell can be confusing. In the convention used here, a convex surface has positive curvature. For
example, a spherical shell with coordinate system chosen so that points radially out of the sphere would have
two equal positive principal curvatures. The mathematical analysis of curved
surfaces usually uses the opposite sign convention for curvature, and a few
texts on shell theory use curvature measures with the opposite sign to the one
10.5.4 Approximating the displacement
and velocity field
position vector of a material point in the shell before deformation can be
expressed as ,
where is the distance of the material particle from
the mid-section of the shell.
deformation of the material point that has coordinates in the undeformed shell moves to a new
position, which can be expressed as
This is a completely general expression.
We now introduce a series of
approximations that are based on the assumptions that
- The shell is thin compared with its in-plane
- The principal radii of
curvature of the shell (both before and after deformation) are much larger
than the characteristic dimension of its cross section;
With this in
mind, we assume that can be approximated by a function of the form
where can be regarded as the first term in a Taylor expansion of with respect to . Note that
represents transverse shear deformation of the
shell, while quantifies the through-thickness stretching.
versions of plate theory exist, which use different approximations for the
shear deformation. Here, we will present
only simplest version, known as Kirchhoff shell theory, which is to assume that . This implies that material fibers that are
perpendicular to the mid-plane of the shell remain perpendicular to the
mid-plane of the deformed shell, and the shell does not change its thickness.
This reduces the displacement field to
The velocity field can be
while the acceleration is
where and denote the angular acceleration and angular
velocity of the unit vector normal to the mid-plane of the plate.
HEALTH WARNING: In addition to using this approximation to the
displacement and velocity field, Kirchhoff shell theory assumes that the
transverse stress vanishes in the shell. Strictly speaking, this is inconsistent with
the deformation. A more rigorous
approach would be to introduce a uniform transverse strain , which could be calculated as part of the
solution. However, this approach yields
results that are essentially indistinguishable from the plane-stress
10.5.5 Approximating the deformation gradient
The deformation gradient can
be approximated as
and are the metric tensors for shell before and
after deformation, and and are the curvature tensors for the mid-surface
of the shell before and after deformation, respectively. The three terms in the second formula for F can be interpreted as (i) the effects
of in-plane stretching of the shell; (ii) the effects of bending; and (iii) the
effects of a change in the shell’s thickness.
definition, the deformation gradient relates infinitesimal line elements in the
shell before and after deformation by . We wish to construct a tensor with these
- An infinitesimal line
element in the deformed shell can be expressed in terms of a small change
we have used
- This expression can be rearranged into the form
- An infinitesimal line element dx in the undeformed shell can also be related to as
can be re-written as
- Finally, note that
can substitute this result into (2) above to see that
the deformation gradient can be read off as the coefficient of dx.
- The approximate expression for F is obtained by assuming that for
a thin shell
To see this, multiply out ,
recall that and neglect the term of order .
Finally, substitute this approximation into the formula for F, multiply out the terms and neglect terms to obtain the approximation for F.
10.5.6 Other deformation measures.
is straightforward to calculate any other deformation of interest from the
deformation gradient. A few examples
that will be used in calculations to follow are listed below.
inverse of the deformation gradient can be approximated by
velocity gradient tensor ,
which relates the relative velocity of two material particles at positions and in the deformed shell as can be approximated by
are the covariant components of the time
derivative of the surface curvature tensor.
The Lagrange strain tensor
can be approximated by
where terms of order have been neglected, we have used and .
When we write constitutive
equations relating forces to deformations, it is convenient to introduce two
new strain measures defined as follows:
- The `mid-plane Lagrange strain tensor’ ,
and . The tensor quantifies length changes of
infinitesimal material elements in the mid-plane of the shell, in the sense
that the lengths and of a line element before and after deformation
are related by
- The `Curvature change tensor’ ,
which quantifies the additional stretch induced by bending and twisting
10.5.7 Representation of forces and moments in shells
figure shows a generic cross-section of the shell, in the deformed
configuration. To define measures of internal and external forces acting on the
shell, we define the following variables
A basis with vectors chosen following the scheme
described in 10.5.1. Vector and tensor
quantities will be quantified by their contravariant components in this basis
The body force acting on the rod ,
or in component form
tractions acting on the exterior surface of the shell . It is convenient to define separate variables
to characterize the tractions acting on the various parts of the shell, as
indicated in the sketch: the upper surface of the shell (denoted by ) is subjected to traction ;
the lower surface is subjected to ,
while the surface around the edge of the shell is subjected to
Cauchy stress within the shell .
External forces and
moments acting on
the shell are characterized by
- The external force per unit area acting on the
shell, . The force components can be calculated
from the tractions and body force acting on the shell as
- The external moment
per unit area acting on the shell. It is most convenient to express the
external moment as where the components can be calculated from the tractions and
body force as
The vector q
is work conjugate to the angular velocity of the normal to the mid-plane of the shell,
in the sense that is the rate of work done by the external
couple per unit area of the shell’s mid-plane.
- The resultant force
per unit length acting on the external edge of the shell. The force per unit length can be
expressed as components as . The components are related to the
tractions acting on the external surface at the edge of the shell by
- The resultant moment
per unit length acting on the external edge of the shell. The moment per unit length can be
expressed as components as . The components are related to the
tractions acting on external surface at the edge of the shell
Internal forces and
moments within the
shell are characterized by three surface tensors ,
and , To visualize their physical significance,
suppose that the shell is cut through so as to expose an internal surface,
which lies perpendicular to the mid-plane of the shell. Let denote a unit vector normal to the internal
surface, and let ds denote an
infinitesimal line element that lies in the both the exposed surface and the
mid-plane of the shell. The exposed
surface is subjected to a distribution of traction, so that an small element of
area with dimensions is subjected to a resultant force and resultant moment . These forces and moments are related to ,
and as outlined below:
1. The in-plane stress resultant tensor quantifies internal forces that tend to
stretch and shear the shell in its own plane.
It is related to the internal tractions by ,
and its components can be calculated from the stress distribution in the shell
2. The transverse shearing stress tensor
quantifies internal forces that act to impose
the constraint that material fibers that are perpendicular to the mid-plane of
the shell before deformation remain perpendicular to the mid-plane after
deformation. Strictly speaking, in
Kirchoff shell theory it is a Lagrange multiplier, but can be regarded as quantifying
the transverse shear force . Its value cannot be computed from the
deformation of the shell, because the transverse shearing has been neglected:
instead, it must be determined by solving the equilibrium equations given in
the next section.
3. The internal moment tensor characterizes internal bending and twisting
moments in the shell. It is related to
the moment acting on internal through-thickness sections of the shell by . The components can be calculated from the internal stresses
in the shell as
The tensor is work conjugate to the gradient of the
angular velocity of the normal to the mid-plane of the shell ,
or alternatively, to the rate of change of curvature in the sense that
is the rate of
work done by M per unit area of the
mid-plane of the shell.
10.5.8 Equations of motion and boundary
We consider a shell with
thickness h and mass density .
The internal forces and moments must satisfy
operator represents the surface divergence, T, V
and M are the internal forces
defined in Sect 10.5.7; p and q are the external force and couple per
unit area acting on the shell, a is
the acceleration of the mid-plane and is the angular acceleration of the unit vector
normal to the mid-plane of the shell.
The two equations can be interpreted as linear and angular momentum
balance for an infinitesimal element of the shell. Note that:
1. If the system is in static equilibrium, the right hand
sides of all the equations of motion are zero.
2. In addition, in many dynamic problems, the right hand
sides of the angular momentum balance equations may be taken to be
approximately zero. For example, the
rotational inertia may be ignored when modeling the vibration of a shell. The rotational inertia terms can be important
if the shell is rotating rapidly: for example, they would influence the
out-of-plane vibration of a spinning disk.
equations of motion can also be expressed in terms of components of the various
force and moment tensors by substituting ,
and and recalling that
The result is
last equation shows that the stress resultant and moment tensors are not
symmetric. The asymmetry is small, and
is ignored in simplified versions of shell theory. However, there are a few special shell
geometries (a cylindrical shell subjected to torsional loading is one example)
where neglecting the asymmetry can lead to substantial errors.
boundary conditions for a shell
are complicated and confusing. To
understand them, it is helpful to visualize the possible types of motion that
can occur at the edge of a shell. The
edge of the shell is characterized by a curve C that lies in the mid-plane
of the shell, encircling in a counterclockwise sense. We let denote arc-length measured around C from some convenient origin, and use and denote unit vectors tangent and normal to C.
Elements of the shell that lie on C
have four independent degrees of freedom, as
- The material element can have an arbitrary
velocity, with three components
- The material element can rotate about the tangent
To visualize this motion, imagine that the shell is supported around C by a hinge.
motion of the edge of the shell can be prescribed by constraining one or more
of these degrees of freedom. Alternatively,
the edge of the shell can be subjected to one or more of four generalized forces,
which are work-conjugate to these degrees of freedom. The generalized forces can be expressed in
terms of the forces and couples acting on the edge of the shell as
- is work-conjugate to the in-plane
displacement of the shell
- is work-conjugate to the out-of-plane
displacement of the edge of the shell
- is work-conjugate to the rotation of the
shell about its edge.
four boundary conditions can be expressed in terms of these forces as follows:
Derivation: Measures of internal force, the equilibrium
equations, and the boundary conditions emerge naturally from the principle of
virtual work. The principle of virtual
work states that, for any deformable solid that is in static equilibrium, the
Cauchy stress distribution must satisfy
all virtual velocity fields and compatible velocity gradients . The virtual velocity field in the shell must
have the same general form as the actual velocity, as outlined in Section
10.2.4. The virtual velocity can
therefore be characterized by the virtual velocity of the mid-plane of the
shell . It is convenient to introduce the time
derivative of the normal vector to the plate’s mid-plane as an additional kinematic variable, which
must of course be compatible with . We will show the following:
virtual work principle can be expressed in terms of the generalized deformation
measures and forces defined in the preceding sections as
virtual work equation is satisfied for all and compatible ,
then the internal forces and moments must satisfy the following equilibrium
as well as the boundary
conditions listed above.
The last equilibrium equation does not emerge from the virtual work
principle. Instead, this equation is a consequence of the symmety of the Cauchy
stress tensor ,
as shown below. It is automatically satisfied if the components and are calculated by integrating the stresses
through the thickness of the shell.
However, for some statically
determinate shell problems it is possible to avoid evaluating these integrals
explicitly, in which case the equilibrium equation is useful.
the virtual work equation in terms of generalized force measures is a
straightforward but lengthy algebraic exercise.
- When applying the
virtual work principle, we will need to integrate over the volume of the
shell. It is convenient to write
the volume integral as separate integrals over the mid-plane of the shell
and through its thickness, as follows
where the term of order has been neglected. Substituting this result into the volume
the area integral is understood to be taken over the mid-plane of the shell.
- Similarly, the integrals over the outer surface
of the shell can be separated into integrals taken over the upper and
lower surfaces of the shell ( and ), together with an integral over the
surface at the edge of the shell ,
Following the procedure in step (1), the integrals
over and can be expressed in terms of integrals taken
over the mid-plane of the shell as
integral over can be reduced to a line integral taken around
the curve(s) bounding the edge of the shell, as
The procedure to derive this result is very similar to
the steps required to simplify the volume integral and is left as an exercise.
- Next, consider the integrand
To reduce this to a scalar combination of the
components of the various tensors and vectors, substitute ,
together with the kinematic formulas:
- Substituting this
result into the first integral of the virtual work principle, reducing the
volume integral to an integral over the mid-plane of the shell, and using
the definitions of and gives
- Similar manipulations can be used to reduce the
remaining terms in the virtual work principle to
the equations in (4) and (5) into the virtual work equation gives the first
Next, we show that the
equilibrium equations and boundary conditions follow from the virtual work
- The virtual work
equation must first be augmented by a Lagrange multiplier to enforce
compatibility between the velocity field and the time derivative of the vector
normal to the shell’s mid plane . To this end, we regard the unit vector as an independent degree of freedom, and
introduce a vector valued Lagrange multiplier which must satisfy
for all admissible variations ,
and . The second integral can simply be added to
the virtual work equation to ensure compatibility of and
- The augmented virtual
work equation now reads
- The remaining steps
are routine, but fiddly. The
partial derivatives of and must be removed by integrating by
parts. This is accomplished by
applying the surface divergence theorem, which states that if is a differentiable vector field on an
area A of the surface, and C is the curve bounding A, then
is the outward normal to C. To see how to use this
integrand can be re-written as
the surface divergence theorem to the first term on the right hand side of this
equation shows that
substituting and and remembering to differentiate the basis
vectors gives the component form
- Applying the procedure
outlined in step (3) to similar terms, the virtual work equation can be
This equation must be satisfied for all and ,
which immediately gives the equilibrium equations.
- Some further algebra
is required to derive the boundary conditions. It is tempting to conclude
that coefficients of and in the boundary terms must all vanish,
but this is not the case, because and are related by compatibility equations.
The boundary terms must be expressed in terms of four independent degrees
of freedom. To this end, recall
so the integral around the boundary can be re-written as
- The vector can be expressed in terms of components
parallel and perpendicular to the boundary of the plate C, as
and represent unit vectors tangent and normal to C, and is an independent degree of freedom that
represents the rotation of the shell about . Finally, we integrate by parts to see that
terms associated with the ends of C
vanish because C is a closed curve).
- Substituting the result of (6) back into (5)
condition must be satisfied for all ,
which gives the boundary conditions.
Finally, we must derive the
last equilibrium equation . Using the definitions of ,
and noting that ,
it is straightforward to show that
Adding these two equations
gives the last equilibrium equation.
10.5.9 Constitutive equations relating
forces to deformation measures in elastic shells
internal forces in a shell are related to its deformation by the stress-strain
law for the material. Here, we give
force-deformation equations for an isotropic elastic shell which experiences
small shape changes (but possibly large rotations).
Shape changes are
characterized using the following deformation measures, defined in Sections
- The in-plane
components of the metric tensors for the mid-plane of the shell before and
after deformation are denoted
- The `mid-plane Lagrange strain tensor’ ,
- The `Curvature change tensor’ ,
which quantifies the bending and twisting the shell.
forces are characterized using the stress resultant tensor components and internal moment components defined in Section 10.5.7.
shell is assumed to have a uniform thickness h, and is assumed to be made from an isotropic, linear elastic
solid, with Young’s modulus E and Poisson’s ratio . We assume for simplicity that the shell is
homogeneous, and neglect thermal expansion (the effects of thermal expansion
are included an example problem solved in Section 10.7.4)
is convenient to introduce a plane stress elasticity tensor with components
relations can then be expressed as
For all but a few very rare
shell geometries these expressions may be approximated by
- We have assumed that
the material in the shell experiences small distortions, but arbitrary
rotations. Material behavior can
therefore be modeled using the generalized Hooke’s law described in
Section 3.3, which relates the Material stress to the Lagrange strain
using the isotropic linear elastic constitutive equations.
- We assume that the
shell is in a state of plane stress, so that the material stress tensor
has the form
- The Lagrange
strain tensor is given in Section
- The material stress is
related to the Lagrange strain by the plane stress version of the linear
elastic constitutive equations, which can be expressed as
where is the in-plane component of the metric
tensor associated with the undeformed shell (this replaces the identity tensor
in the Cartesian version of the constitutive equations)
- The Cauchy stress is
related to the material stress by . Substituting the formulas for from (4) and approximating F as ,
we find after some algebra that
- The components of the
stress-resultant tensor are given by
Substituting the formula for stress components into this
expression and integrating through the thickness of the shell gives
Here, terms of order have been neglected.
- The components of the internal
moment tensor are
Substituting the formula for stress components into this
expression and integrating through the thickness of the shell gives
where terms of order and higher have been neglected.
10.5.10 Strain energy and kinetic energy of an elastic
It is useful to express the strain energy and kinetic energy
of a deformed shell in terms of the motion and deformation of its
mid-plane. To this end:
an isotropic, linear elastic shell, with Young’s modulus E,
Poission’s ratio ,
mass density and thickness h. Denote the
contravariant components of the tensor of elastic constants by
- Let and denote the covariant components of the
mid-plane Lagrange strain tensor, defined in the preceding section,
- Let and denote the linear and angular velocity of
the mid-plane of the shell.
The total strain energy of the shell can be
all but a very few special shell geometries this result may be approximated by
kinetic energy can be calculated using the formula
second term in the expression for the kinetic energy represents the rotational
energy. In many practical problems, such as vibration of a shell, the
rotational energy can be neglected.
Derivation: The strain energy density in the shell is given by ,
where is the material stress tensor and E is the Lagrange strain tensor,
defined in the preceding section. The
stress can be expressed in terms of the strain using the constitutive equation,
while the strain can be expressed in terms of and using step (3) in Section 10.5.8. Integrating over the volume of material in
the shell, and evaluating the integral through the shell’s thickness explicitly
gives the result stated. The kinetic
energy is calculated using the formula for the velocity field in 10.2.8.