Notes on
Stationarity and extremum
principles in mechanics
(with applications to optimal design)
Pauli Pedersen
Department of Mechanical Engineering, Solid Mechanics
Technical University of Denmark
Nils Koppels Allè, Building 404, DK-2800 Kgs.Lyngby, Denmark
email: [email protected]
WORKING PRINT
December 9, 2008
c
ii Pauli
Pedersen: Stationarity and extremum principles in mechanics
Stationarity and extremum
principles in mechanics
c
Copyright 2008
by Pauli Pedersen,
ISBN 06
Preface
The energy principles in mechanics play an important role, not only for analysis but also for design synthesis and optimization. However, when teaching
mechanics it is mostly found difficult to communicate a basic understanding
of these principles.
What is the reason for this situation that so many teachers agree with?
Should the reason be related to the students, to the teachers, or to the available textbooks? The present small book attempts to give an alternative nontraditionally presentation of the subject. The presentation in chapters 2 - 6 has
earlier been used in a course on elasticity, anisotropy and laminates.
A primary idea is to separate the mathematical derivation of an identity from the specific interpretations of this identity. Then also separate the
stationarity principles from the extremum principles, and finally balance the
physical interpretation of the non-physical variations, where also the aspect of
infinitesimal variations is important. Hopefully, this alternative presentation
will appeal to some readers.
The chapters 7 - 9 with direct relation to optimal design use to a large
extend the basic principles in mechanics, but also introduces new results from
design variations, i.e., the sensitivity analysis for design. These chapters are
influences by recently published papers, and here serve as examples to illustrate the simplicities that may results from using the basic principles in
mechanics.
Kgs. Lyngby, Winter 2008
Pauli Pedersen
iii
c
iv Pauli
Pedersen: Stationarity and extremum principles in mechanics
Contents
Preface
iii
Contents
v
1 Introduction
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Layout of contents . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1
2 Stationarity principles
in mechanics
2.1 The work equation, an identity . . . . . . . .
2.2 Symbols and definitions . . . . . . . . . . . .
2.3 Real stress field and real displacement field .
2.4 Real stress field and virtual displacement field
2.5 Virtual stress field and real displacement field
2.6 Summing up . . . . . . . . . . . . . . . . . .
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3 Extremum principles
in mechanics
3.1 Principle of minimum total potential energy
3.2 Principle of minimum total
complementary(stress) potential energy . .
3.3 Overview of principles and their relations .
3.4 Summing up . . . . . . . . . . . . . . . . .
4 Potential relations and derivatives
4.1 Equilibrium . . . . . . . . . . . .
4.2 Relations with power law elasticity
4.3 Derivatives of elastic potentials . .
4.4 Summing up . . . . . . . . . . . .
5 Energy densities
in matrix notation
5.1 Strain and stress energy densities
5.2 Energy densities in
1D non-linear elasticity . . . . .
5.3 Energy densities in
2D and 3D non-linear elasticity .
5.4 Summing up . . . . . . . . . . .
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v
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vi Pauli
Pedersen: Stationarity and extremum principles in mechanics
6 Elastic energy in beam models
6.1 Elastic energy in a straight beam . . . . . .
6.2 Results for simple (Bernoulli-Euler) beams
6.3 Beam solutions by stress(complementary)
principles . . . . . . . . . . . . . . . . . .
6.4 Summing up . . . . . . . . . . . . . . . . .
7 Some necessary conditions for optimality
7.1 Non-constrained problems . . . . . . . . .
7.2 Problems with a single constraint . . . . . .
7.3 Size optimization for stiffness and strength .
7.3.1 Size design with optimal stiffness .
7.3.2 Size design with optimal strength .
7.4 Shape optimization for stiffness and strength
7.4.1 Shape design with optimal stiffness
7.4.2 Shape design with optimal strength
7.5 Conditions with a
simple shape parametrization . . . . . . . .
7.5.1 Possible iterative procedure . . . .
7.6 Summing up . . . . . . . . . . . . . . . . .
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8 Analytical beam design
8.1 Optimality criterion for beam design . . . . . . .
8.1.1 Treated boundary conditions and loads .
8.1.2 Solutions in general . . . . . . . . . . .
8.2 Bernoulli-Euler cantilever beams . . . . . . . . .
8.2.1 Optimal compliances . . . . . . . . . . .
8.3 Timoshenko cantilever beams . . . . . . . . . . .
8.3.1 Design of beams with n = 1 cross sections
8.4 Examples of beam cross sections . . . . . . . . .
8.5 Summing up . . . . . . . . . . . . . . . . . . . .
9 The ultimate optimal material
9.1 The individual constitutive parameters
9.2 Sensitivity analysis . . . . . . . . . .
9.3 Final optimization . . . . . . . . . . .
9.4 Numerical aspects and comparison
with isotropic material . . . . . . . .
9.5 Summing up . . . . . . . . . . . . . .
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References
61
Index
63
Chapter 1
Introduction
1.1
Background
The authors background for the present notes are years of teaching energy principles in a course on elasticity, based on (Pedersen 1998a). As stated in the preface
it is mostly found difficult to communicate a basic understanding of these principles. However, when succeeding, the students has an important tool for many future
applications.
With a research background in optimal design, the unification that can be obtained using energy principles is found important, especially when the design sensitivity analysis is build on these principles. The obtained results are then valid for
1D, 2D and 3D problems, for anisotropy as well as for isotropy, often for non-linear
as well as non linear elasticity, and for analytically research as well as for numerical
research, see (Pedersen 1998b).
With the present notes, the notions in earlier notes and papers are unified. Limiting these notes to mostly analytically derivations without numerically results, the
total number of pages are few, but hopefully not too dry.
1.2
Layout of contents
In (Pedersen 1998a) the present chapters 2 and 3 are combined in a single chapter,
but it is found advantageously to separate the stationarity principles in chapter 2 with
focus on the principle of virtual work. This principle is of most importance and
is based on only a few assumptions, but with a rather abstract interpretation and
therefore not immediately easy to communicate.
Chapter 3 contains a graphical overview of the stationarity as well as the extremum principles. The extremum principles are mainly used as arguments for choosing approximations based on a stationarity principle.
The main assumption behind chapter 4 is a proportionality relation between complementary energy (stress energy) and strain energy. From this follows proportionality relations between total potentials, strain energy, stress energy, external potential, and compliance. The names of stress energy and complementary energy are
used synonymously, but in the notations only the superscript σ is applied, not the
superscript C as in (Pedersen 1998a). Focus is thus put on the fact that complementary principles have stress/force as the primary variables, while strain principles
has strain/displacement as the primary variable and superscript is applied for their
related quantities.
General simple first order design sensitivities is a result of derivatives on the
1
c
2 Pauli
Pedersen: Stationarity and extremum principles in mechanics
energy level. These results are obtained in fields of fixed strains or fixed stresses,
and these results do not involve approximations. Chapter 4 is earlier published as an
appendix in (Pedersen 2003).
To be specific about the proportionality relation between strain energy and stress
energy, chapter 5 presents a power law elasticity model, that is the most simple extension from linear elasticity. For this reversible model secant stiffnesses as well as tangent stiffnesses are derived for a three dimensional model, covering also anisotropic
materials.
Although being a one dimensional model, the beam is among the most important
structural elements. It is therefore natural to include as an example in chapter 6, the
use of energy principles in a linear elastic straight beam. These handbook formulas
are applied in chapter 8 for design optimization.
Before this design optimization a number of necessary conditions for design optimization are needed. The description in chapter 7 also focuses on the use of superelliptic description of shapes in two dimensional models, a description which has
often been successfully applied.
Chapter 8 contains results from recent research, see (Pedersen and Pedersen
2008), and demonstrates that optimal design based directly on an energy approach
has promising aspects. It has been possible to find analytically described optimal
beam designs, even for short beams where the Timoshenko beam theory must be
used.
The notes finish with chapter 9 on ultimate optimal material, related to compliance optimization,. i.e., again for an energy objective. The results in the reference
research (Bendsøe, Guedes, Haber, Pedersen and Taylor 1994) are extended to be
valid for non linear elastic materials as in (Pedersen 1998a). In active research on optimal design, this free material model is expected to play a major role. In the present
notes this is taken as a further example on the importance of energy approaches,
added design sensitivity analysis.
Chapter 2
Stationarity principles
in mechanics
Energy principles play a central role in mechanics, but surprisingly few books treat
the subject in a structured way. It is difficult to get an overview of the many different
principles, and important questions are not presented, especially in relation to the
necessary conditions for a certain principle.
The present chapter is written as an alternative to the classical presentations, as by
(Langhaar 1962) and (Washizu 1975). We shall show that all principles are specific
interpretations of the same identity, and necessary conditions will not be introduced
until absolutely needed. We shall refer only to a Cartesian 3-D coordinate system,
and traditional tensor notation for summation and differentiation is applied.
2.1
Goal of
the chapter
Tensor
notation
The work equation, an identity
Without physical interpretation of the quantities involved, we shall derive an important identity. The superscripts a and b are only part of the names (not powers) and
their use will be explained later. From
1 b
a b
a
b
b
a 1 b
b
σij ij = σij ij − (vi,j + vj,i ) + σij
(v + vj,i
)
(2.1)
2
2 i,j
No physical
interpretation
and
(2.2)
a b
a b
a
a
b
a b
σij
vj,i = σji
vi,j = (σji
− σij
)vi,j
+ σij
vi,j
follows
Z
Z
V
V
a b
σij
ij dV
−
Z
V
a
σij
bij
1 a
a
b
(σ − σij
)vi,j
dV −
2 ji
Z
1 b
b
− (vi,j + vj,i ) dV −
2
V
(2.3)
a b
σij
vi,j dV = 0
In (2.3) V is the volume of the domain of interest. Now let A be the surface that
bounds this domain and nj the outward normal at a point of the surface. Then, using
the theorem of divergence, the last part of (2.3) is rewritten to
Z
Z
Z Z
a b
a b
a
a b
a
σij
vi,j dV =
(σij
vi ),j − σij,j
vib dV =
σij
vi nj dA −
σij,j
vib dV
V
V
A
V
(2.4)
3
Using the
theorem of
divergence
c
4 Pauli
Pedersen: Stationarity and extremum principles in mechanics
The
identity
Once more, by adding and subtracting the same quantities, we obtain the identity from which the stationarity principles of mechanics can be read without further
calculations
Z
Z
1 b
b
a
a b
b
σij ij − (vi,j + vj,i ) dV −
σij ij dV −
2
V
Z
ZV
Z
1 a
a
a b
a
b
Tia vib dA+
(σji − σij )vi,j dV − (σij − Ti )vi dA −
2
A
ZV
Z A
a
a b
a b
(σij,j + pi )vi dV −
pi vi dV = 0
(2.5)
V
V
With the superscripts a and b we have indicated certain relations, and we shall
now assume that all quantities with index a are related and that all quantities with
index b are related. However, no relations exist between quantities with different
index, and the equations therefore still have no physical interpretations.
Let bij = bij (x) be a strain field derived from a displacement field vib = vib (x)
with small strain assumption (engineering strains, linear strains, Cauchy strains)
1 b
b
bij = (vi,j
+ vj,i
) (small strain assumption)
2
(2.6)
a = σ a (x) be a stress field with moment and force equilibrium with the
and let σij
ij
field of volume forces pai = pai (x) and surface traction’s Tia = Tia (x)
a
a
σij
= σji
a
σij,j
= −pai
(2.7)
a
σij
nj = Tia
The work
equation
With (2.6) and (2.7) the identity (2.5) reduces to
Z
Z
Z
a b
σij
ij dV =
Tia vib dA +
pai vib dV
V
A
(2.8)
V
which is often called the work equation, although it is merely an identity and does
not express physical work when the a and b fields are not related.
2.2
Real
fields
Virtual
fields
Symbols and definitions
For a better overview, the stationarity principles of mechanics are divided into four
groups covering the possible combinations of real fields (indexed by a superscript 0)
and virtual fields (without index).
The virtual fields are assumed to be sufficiently differentiable and admissible,
but otherwise arbitrary and non-physical. An admissible displacement field must be
kinematically admissible, i.e. it must satisfy the boundary conditions. An admissible
stress field must be statically admissible, i.e., it must satisfy force equilibrium.
A somewhat repeated definition of the quantities in the energy principles may be
useful before the individual principles are stated and proved, all by specific interpretations of the work equation (2.8).
Geometry variables:
V =
Volume of the continuum or structure
A=
Surface area that bounds the volume V
(2.9)
Stationarity principles 5
State variables at position x in the volume:
ij = ij (x) =
description of strain state at position x
σij = σij (x) =
description of stress state at position x
Ti = Ti (x) =
surface traction =
force (per unit area) in direction i at position x of A
pi = pi (x) =
force (per unit volume) in direction i at position x in V
(2.10)
Work and energy quantities in strains and displacements:
W = W (vi ) =
Z Z
A
vi
Ti (ṽi )dṽi dA +
Z Z
V
0
u = u (ij ) =
U =
Z
Z
Work of
external loads
vi
pi (ṽi )dṽi dV
0
Strain
energy density
ij
σij (˜
ij )d˜
ij
0
Total
strain energy
u dV
V
(2.11)
Π = U − W Complementary
work of
external loads
Complementary work and energy quantities in stresses and forces:
Z Z pi
Z Z Ti
σ
σ
vi (p̃i )dp̃i dV
vi (T̃i )dT̃i dA +
W = W (Ti , pi ) =
A
V
0
uσ = uσ (σij ) =
Uσ =
Z
Z
0
Stress
energy density
σij
ij (σ̃ij )dσ̃ij
0
Total
stress energy
uσ dV
V
(2.12)
Πσ = U σ − W σ
Φ = W (dead loads, i.e.,W σ = 0) =
Ti vi dA +
A
Z
pi vi dV
V
Compliance
from
elastic energy
Φ = U + Uσ = W + Wσ
Φ = −Uext
Total potential
complementary
energy
Compliance of
external loads
Compliance relations:
Z
Total potential
energy
(2.13)
Compliance
from
potential of
external loads
c
6 Pauli
Pedersen: Stationarity and extremum principles in mechanics
2.3
Real stress field and real displacement field
0
As a first example of use of the work equation (2.8) we insert the real stress field σ ij
0
0
0
in equilibrium with the real load field Ti , pi , and the real displacement field vi from
which the real strain field 0ij is derived. We get
Z
Z
Z
0 0
0 0
p0i vi0 dV
(2.14)
Ti vi dA +
σij ij dV =
V
A
V
For arbitrary constitutive relations we have
Z
Z 0
Z σ 0 0
ij
ij ij
0 0
σij (˜
ij )d˜
ij +
d(σ̃ij ˜ij ) =
σij ij =
0
0
0
σij
ij (σ̃ij )dσ̃ij
(2.15)
0
which, together with the definitions in (2.11) - (2.12), gives
0 0
σij
ij = u0 + uσ0
Z
Zero sum
of total
potentials
V
0 0
σij
ij dV = U 0 + U σ0
Analogously for the external loads we get
Z
Z
0 0
Ti vi dA +
p0i vi0 dV = W 0 + W σ0
A
(2.16)
(2.17)
V
and (2.14) can thus be written
U 0 + U σ0 − (W 0 + W σ0 ) = Π0 + Πσ0 = 0
(2.18)
i.e., the sum of the real total potentials is zero.
Especially for a linear elastic material the definitions in (2.11) - (2.12) give
Z
1
U 0 = U σ0 =
σ 0 0 dV
(2.19)
2 V ij ij
In relation to the nature of the external forces, the concept of dead load is important. For dead loads the forces are independent of the displacement of their point of
action, say a gravity load. For a dead load we get no complementary(stress) work,
W σ0 = 0, and the work is thus
Z
Z
0
0 0
W =
Ti vi dA +
p0i vi0 dV
A
assuming W
Clapeyron’s
theorem
V
σ0
= 0 (dead load)
(2.20)
For a system with both linear elasticity and dead loads (2.14) gives
1
U 0 = W 0
2
(2.21)
and thus
External
potential
and
compliance
1
Π0 := U 0 − W 0 = − W 0 = −U 0
(2.22)
2
which is often called Clapeyron’s theorem for linear elasticity. The ”missing” energy
W 0 /2 is assumed to be dissipated before the static equilibrium with which we are
concerned.
Note, that if we by definition take −W 0 as given by (2.20) to be the external
potential Uext , then the assumption of dead load is not necessary, but then again
external potential is hardly a physical quantity. The quantity W 0 as given by (2.20)
is also named the compliance, i.e., Φ as defined in (2.13).
Stationarity principles 7
2.4
Real stress field and virtual displacement field
Assume that vi is a kinematically admissible displacement field and that ij is the
strain field derived from vi . Furthermore, as before, σ 0 , Ti0 , p0i are the real stress,
surface traction and volume force fields. Then the work equation (2.8) reads
Z
Z
Z
0
0
σij ij dV =
Ti vi dA +
p0i vi dV
(2.23)
V
A
V
To distinguish the work by the external loads from the work of the reactions we
divide the surface area A into
(2.24)
A = A T + Av
where AT is the surface area without displacement control and Av is the surface area
with given kinematic conditions. Furthermore, we describe the virtual field v i by a
variation δvi relative to the real field vi0
vi = vi0 + δvi ,
ij = 0ij + δij
(2.25)
which with ij = (vi,j + vj,i )/2 gives
1
δij = (δvi,j + δvj,i )
2
(2.26)
Now, as vi is assumed to be kinematically admissible, we have δvi = 0 on the
surface Av (but not necessarily vi = 0) , and thus (2.23) with (2.14) reduces to
Z
Z
Z
0
Ti0 δvi dA +
σij
δij dV =
p0i δvi dV
(2.27)
V
AT
V
Virtual
work
principle
This is called the virtual work principle or the principle of virtual displacements.
Note that the virtual displacements and strains in (2.27) are infinitesimal and express
energy and work variations without assumptions of linearity. Note also that the virtual
work principle is a principle about a state, not a process.
Often (2.23) and even (2.8) are also called the virtual work principle, but in this
book we shall assume the virtual displacements and the virtual strains to be
infinitesimal.
Because stresses are fixed in the virtual work principle, a direct physical interpretation is not clear. However, it can be read as an energy balance which is valid for
any kinematically admissible disturbance of the displacement field.
Some specific cases of use of the virtual work principle lead us to specialized
principles. Let us choose the very specific virtual displacement field
δvi = ∆v corresponding to the single load Q
δvi = 0 corresponding to all other external loads
∆ij derived from this field
then (2.27) reduces to
Z
V
0
σij
∆ij dV = Q∆v or Q =
(2.28)
∆U ∂U =
∆v
∂v
(2.29)
which is the first theorem of Castigliano. It is useful in determining stiffnesses. Note,
that this theorem is valid independent of the specific constitutive behaviour (σ =
Castigliano’s
1st theorem
c
8 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Unit
displacement
theorem for
linear
elasticity
Stationary
total
potential
energy
σ()). The force Q should be interpreted as a generalized force; thus, if Q is an
external moment, then v is the corresponding rotation.
Now a much used theorem is obtained from (2.29) if we assume linear elasticity,
because then we can set the displacement to ∆v = 1 and ∆ ij = 1ij for the resulting
strains from this unit displacement field and get
Z
0 1
σij
ij dV
(2.30)
Q=
V
Returning to general non-linear elastic materials, we can interpret the virtual
work principle as stationary potential energy. A potential is a scalar from which
work can be derived. Let us assume that a material has a potential and the external
loads has as well, then (2.29) states
δU = δW or δΠ = 0
2.5
(2.31)
Virtual stress field and real displacement field
A virtual stress field σij is a statically admissible field, i.e., in equilibrium with the
given external loads. Now, inserting also the real displacement field v i0 , and derived
strain field 0ij in (2.27), we get
Z
Z
Z
Z
0
0 0
0
Ti vi dA +
Ti vi dA +
p0i vi0 dV
(2.32)
σij ij dV =
V
V
Av
AT
On the surface Av , the surface traction’s Ti are the unknown reactions. Taking
the virtual stress field as
0
σij = σij
+ δσij
(2.33)
0 is the real stress field and δσ is an infinitesimal virtual stress field satiswhere σij
ij
fying
δσij = δσji , δσij,j = 0
δσij nj = δTi where δTi = 0 on AT
Stress
virtual work
principle
(2.34)
then using (2.14) we get
Z
V
δσij 0ij dV
=
Z
Av
Ti vi0 dA
(2.35)
which expresses the principle of complementary(stress) virtual work, also called the
principle of virtual stresses.
Choosing a specific virtual field
δσij = ∆σij where ∆σij nj = ∆Q corresponding to displacement v
∆T = ∆σij nj = 0 for all other places with prescribed vi 6= 0
(2.36)
Stationarity principles 9
we get from (2.35)
Z
V
∆σij 0ij dV = ∆Qv or v =
∆U σ
∂U σ
=
∆Q
∂Q
(2.37)
which is the second theorem named after Castigliano. It is valuable in determining
flexibilities.
Also, a unit theorem is obtained in complementary(stress) energies, read directly
1
from (2.37) when linear elasticity is assumed, i.e., ∆Q = 1 and ∆σ ij = σij
v=
Z
V
1 0
σij
ij dV
(2.38)
Finally, the parallel to stationary potential energy is the principle of stationary
complementary(stress) potential energy
δU σ = δW σ or δΠσ = 0
2.6
(2.39)
Summing up
• The identity (2.5) is obtained by rather simple mathematics and has no physical
interpretation.
• The work equation (2.8) involve two independent fields. A stress/force field
in equilibrium, these fields are given super index a. A displacement field with
derived strain field, these fields are given super index b.
• Work as well as complementary work must in general be determined by integration and therefore depend on the force/displacement function.
• Strain energy density as well as stress energy density is also determined by
integration, applying the constitutive relation.
• The notion of compliance is important in optimal design formulations and is
therefore specifically defined.
• With real stress field and virtual displacement field we get the virtual work
principle, and from this a number of more specific principles. The virtual
work principle is a principle about a state, not a process.
• With virtual stress field and real displacement field we get the complementary
virtual work principle (stress virtual work principle), and from this a number
of more specific complementary principles.
• Virtual displacements and virtual stresses are in general infinitesimal.
Castigliano’s
2nd theorem
Unit load
theorem for
linear
elasticity
Stationary
total stress
potential
energy
c
10 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Chapter 3
Extremum principles
in mechanics
The stationarity principles of mechanics are based on very few assumptions. The
principles of virtual work hold for any constitutive model and for any type of load,
and for potential systems these virtual principles give stationary potential energies.
Many approximation methods (like the finite element method) are based on and
uniquely specified by these stationarity principles. However, this does not give us
sufficient reason to choose an approximate solution that satisfies the same stationarity as the unknown real solution. Energy principles that in addition to stationarity
give extremum can justify our choice. We choose the approximation for which the
energy is closest to the real unknown energy. Furthermore, consistent approximation
methods are a reasonable choice also for problems where an extremum cannot be
proved.
3.1
Motivation
for
extremum
Principle of minimum total potential energy
We shall firstly prove the principle of minimum total potential energy δ 2 Π > 0, and
for this we need assumptions concerning the constitutive model as well as for the load
behaviour. Let us start with a single load Q (force or moment) and the corresponding displacement v (translation or rotation). For this force Q as a function of the
corresponding displacement v, we will assume the following single load behaviour
∂v
∂Q
≥ 0,
> 0, Q(v = 0) = 0
∂v
∂Q
(3.1)
as illustrated in Figure 3.1a).
We note that Q(v) is a function but, as ∂Q/∂v = 0 is a possibility, v(Q) is not
strictly a function. Non-linearity and change of sign for curvature is possible. From
the definition of W and W σ in (2.11) - (2.12) follows
Z v
∂W Q(ṽ)dṽ ⇒
= Q = Q(v)
W =
∂v
0
Z Q
∂W σ
σ
W =
v(Q̃)dQ̃ ⇒
= v = v(Q)
(3.2)
∂Q
0
As by (2.17) we have for this case of a single force
W + W σ = Qv
(3.3)
11
Assumptions
Single force
behaviour
PSfrag replacements
c
12 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Q
Q
W
Wσ
b
b
(W )
W
a
v
a)
v
a
(W )
b)
va
W
v
W
v
= Qb
= Qa
v
vb
Figure 3.1: a): Illustration of a possible relation between force Q and corresponding displacement v. The work W and the complimentary work W σ are shown as areas. b): Work
W of the force Q as a function of the displacement v that correspond to the force.
Also the sign of the curvature of W = W (v) and of W σ = W σ (Q) is known
from (3.1) and (3.2)
∂2W σ
∂Q
∂v
∂2W ≥
0,
>0
=
=
2
2
∂v
∂v
∂Q
∂Q
Work
function
(3.4)
From this it follows that we get a work function W = W (v), as illustrated in Figure
3.1b).
From the tangents shown in Figure 3.1b) we read the inequalities
(W )a + Qa (v b − v a ) ≤ (W )b
(W )b − Qb (v b − v a ) ≤ (W )a
(3.5)
which together gives an inequality valid for v b > v a as well as for v a < v b , i.e.,
convexity
Qa (v b − v a ) ≤ (W )b − (W )a
(3.6)
where the equality only holds for ∂Q/∂v ≡ 0 in all the actual interval from v a to v b
and naturally for v a = v b .
Same arguments hold for the stress quantities and thus we also have
v a (Qb − Qa ) ≤ (W σ )b − (W σ )a
Inequality
for mixed
product
(3.7)
with equality only for Qb = Qa because ∂v/∂Q > 0 is assumed in (3.1). Inserting
Qa v a = (W )a + (W σ )a from (3.3) in (3.6) or (3.7) we get the inequality for the
mixed products
Qa v b ≤ (W )b + (W σ )a
(3.8)
By summation and/or integration we can extend the above results to a load system.
Extremum principles 13
For a uniaxial stress/strain in terms of pure normal (σ, ) or, alternatively, pure
shear (τ, γ), we assume a function very parallel to the load displacement function
(3.1),
∂σ
∂
> 0,
> 0, σ( = 0) = 0
∂
∂σ
(3.9)
i.e., well-defined functions for σ = σ() as well as for = (σ) because strict
inequalities hold in (3.9). From the assumptions (3.9) follows in direct analogy to the
load-work arguments which lead to (3.6)-(3.8)
Uniaxial
constitutive
model
σ a (b − a ) < ub − ua for b 6= a
a (σ b − σ a ) < uσb − uσa for σ b 6= σ a
σ a b < ub + uσa for a 6= b
(3.10)
with the definitions of energy densities in (2.11) - (2.12) and the previously discussed
relation u + uσ = σ as stated in (2.16).
For a multidimensional stress/strain state a direct generalization is not easy, and
the assumption is therefore often stated directly as convexity of the energy density in
the six-dimensional strain/stress spaces. With tensor symbols this is written
a b
σij
(ij − aij ) < ub − ua for bij 6= aij
b
a
b
a
aij (σij
− σij
) < uσb − uσa for σij
6= σij
a b
σij
ij < ub + uσa for a 6= b and
σij ij = u + uσ for a = b
(3.11)
We now have the necessary inequalities to prove the extremum principles, and
0 and virtual disagain we start from the work equation (2.8). With real stresses σ ij
placements, strains (vi − vi0 ), (ij − 0ij ) we get
Z
V
0
σij
(ij
−
0ij )dV
=
Z
A
Ti0 (vi
−
vi0 )dA
+
Z
V
p0i (vi − vi0 )dV
From (3.11) follows that the left-hand side satisfies
Z
0
σij
(ij − 0ij )dV < U − U 0 for ij 6= 0ij
General
stress/strain
state
(3.12)
(3.13)
V
and the right-hand side for dead loads (∂Ti0 /∂vi = 0, ∂p0i /∂vi = 0) gives
Z
Z
p0i (vi − vi0 )dV = W − W 0
Ti0 (vi − vi0 )dA +
A
(3.14)
V
Using (3.13) as well as (3.14) in (3.12) we get the result
U − U 0 > W − W 0 or Π > Π0 for ij 6= 0ij
(3.15)
i.e., the extremum principle for total potential energy. We note that the dead load
assumption (3.14) is a necessary condition if we do not decide by definition to term
the right-hand side of (3.12) as the negative external potential energy.
Minimum
potential
c
14 Pauli
Pedersen: Stationarity and extremum principles in mechanics
3.2
Principle of minimum total
complementary(stress) potential energy
We can directly establish the complementary(stress) principle because all the necessary inequalities were derived in the Section 3.1. In the work equation (2.8) we now
insert vi0 , 0ij and the virtual stresses σij and get
Z
V
0ij (σij
−
0
σij
)dV
=
Z
A
vi0 (Ti
−
Ti0 )dA
+
Z
V
vi0 (pi − p0i )dV
The left-hand side satisfies
Z
0
0
0ij (σij − σij
)dV < U σ − U σ0 for σij 6= σij
(3.16)
(3.17)
V
Minimum
complementary
(stress)
potential
The main part of the right-hand side is zero when σij is statically admissible
because pi − p0i = 0 and Ti − Ti0 can only be different from zero at the reactions. For
dead loads this part will be W σ − W σ0 , and with (3.17) in (3.16) we get
0
U σ − U σ0 > W σ − W σ0 or Πσ > Πσ0 for σij 6= σij
i.e., the extremum principle for total complementary(stress) potential energy.
Using also the earlier result (2.18) of Π0 + Πσ0 = 0 for only real fields we can
with (3.15) and (3.18) set up two-sided bounds on approximate solutions. For the real
solution we have (2.18), and by the sum of (3.15) and (3.18) we for an approximate
solution get
Π + Πσ > 0 or Π > −Πσ
Two-sided
bounds
(3.18)
(3.19)
Furthermore, substitution of Π0 = −Πσ0 in (3.15) and (3.18) then gives the
two-sided bounds
Π > Π0 > −Πσ
Πσ > Πσ0 > −Π
3.3
(3.20)
Overview of principles and their relations
Figure 5.1 illustrate the connections between the many different energy principles.
The indicated horizontal dash line shows the division between the stationarity and
the extremum principles. The indicated vertical dash line shows the division between
the strain principles and the complementary stress principles.
Extremum principles 15
b
b
a
σija = σji
)
+ vj,i
, bij = 12 (vi,j
a
σija nj = Tia , σij,j
= −pai
frag replacements
1st CASTIGLIANO
R
V
σija bij dV =
R
T a v b dA +
A i i
VIRTUAL WORK
PRINCIPLE
R
V
pai vib dV
2nd CASTIGLIANO
VIRTUAL
COMPLEMENTARY
(STRESS) PRINCIPLE
ASSUMPTIONS ON
LINEARITY
LOADS- AND CONSTITUTIVE
BEHAVIOUR
STRAIN
PRINCIPLES
STRESS
PRINCIPLES
UNIT - LOAD
UNIT - DISPLACEMENT
PRINCIPLE
LINEARITY
STATIONARITY
PRINCIPLES
PRINCIPLE
EXTREMUM
PRINCIPLES
MINIMUM OF TOTAL
ELASTIC POTENTIAL
MINIMUM OF TOTAL
COMPLEMENTARY
(STRESS) POTENTIAL
TWO-SIDED BOUNDS
MINIMUM OF INTERNAL
MINIMUM OF INTERNAL
ELASTIC
COMPLEMENTARY
STRAIN ENERGY
ELASTIC STRESS ENERGY
Figure 3.2: Overview of stationarity and extremum energy principles in mechanics.
c
16 Pauli
Pedersen: Stationarity and extremum principles in mechanics
3.4
Summing up
• Extremum principles serve as argument for choosing approximate solutions
that satisfy stationarity principles.
• A basic assumption for extremum principles is that a force(moment) is a strict
function of its corresponding translation(rotation).
• For the constitutive behaviour a basic assumption for extremum principles is
convexity of energy density in the six dimensional strain/stress space.
• The principle of minimum total potential energy and the principle of minimum total complementary energy together give two-sided bounds for the real
solution.
• The graphical overview in Figure 5.1 shows stationary as well as extremum
principles, strain principles as well as stress (complementary) principles.
Chapter 4
Potential relations and derivatives
This chapter shows some important potential relations and then the sensitivity analysis directly in energy terms, for linear as well as for non-linear power law elasticity.
These results are often used as a basis for other formulations. For detail on the results
in this chapter see (Pedersen 1998b) and (Masur 1970). Although not new, results
like (4.10) and (4.14) are not so well known, and not intuitively understandable,
even for the case of linear elasticity (p = 1). For optimal design these results give
rise to important simplifications.
4.1
Not well known
but important
Equilibrium
The general equation of energy equilibrium is
(4.1)
U + U σ + U ext = 0
with elastic strain energy U and elastic stress energy U σ (elastic complementary energy) from the corresponding densities u , uσ integrated over the structure/continuum
volume V
Z
Z
σ
uσ dV
(4.2)
u dV
and
U =
U =
Energy
equilibrium
Internal
potentials
V
V
and the external potential U ext is defined by
Z
Z
ext
U
:= −
Ti vi dA +
pi vi dV
A
External
potential
(4.3)
V
with surface traction’s Ti , volume forces pi , corresponding displacements vi , and
area A surrounding the volume V . The surface traction’s and the volume forces
are assumed to be given. The displacements vi (displacement field) with resulting
strains, stresses and energy densities are the solution for a given design, i.e. the
solution for a given static problem of elasticity.
Power law elastic materials resulting in the density relation uσ = pu everywhere
in the continuum or structure give in total U σ = pU and the equilibrium (4.1) is then
simplified to
(1 + p)U = −U ext
(4.4)
with 0 < p ≤ 1 being a material power law constant.
17
Assumed
power law
elasticity
c
18 Pauli
Pedersen: Stationarity and extremum principles in mechanics
4.2
Total
potentials
Relations with power law elasticity
Defining the total potential Π and the total complementary(stress) potential Πσ by
(4.5)
Π := U + U ext = −Πσ
using Π + Πσ = 0, give from (4.4) the relations
Π = −pU = −U σ =
Potential
relations
p
U ext = −Πσ
1+p
(4.6)
and by p > 0 and U > 0 get U σ > 0, Πσ > 0, Π < 0 and U ext < 0. From
this follows that design for a number of differently stated extremum problems are
equivalent and that their values at the extrema are related as
max Π = −min Πσ = −min pU = −min U σ = max
p
U ext
1+p
(4.7)
with p = 1 for the specific case of linear elasticity.
4.3
Derivatives of elastic potentials
The derivative of the total potential Π with respect to an arbitrary parameter, say a
design parameter h, is
dΠ /dh = (∂Π /∂h)fixed strains + (∂Π /∂) (d/dh) = (∂Π /∂h)fixed strains (4.8)
Design
independent
loads
because of stationary total potential ∂Π /∂ = 0 (virtual work principle) with respect
to kinematically admissible strain variations.
For design-independent external loads, (∂U ext /∂h)fixed strains = 0, the definition
(4.5) then gives
(∂Π /∂h)fixed strains = (∂U /∂h)fixed strains
(4.9)
and totally from (4.6), (4.8) and (4.9) get the result that is frequently used in design
optimization
Local
design
parameter
1
dU /dh = − (∂U /∂h)fixed strains
p
For a local design parameter he that only changes the design in the region e of
the structure/continuum this gives the possibility of a localized determination of the
sensitivity for the total elastic strain energy
1
dU /dhe = − (∂((ū )e Ve )/∂he )fixed strains
p
Remark
(4.10)
(4.11)
where (ū )e is the mean strain energy density in the region of he and where Ve is the
corresponding volume. Note that the only difference between linear (p = 1) and nonlinear material is the factor 1/p, and for a condition on stationarity dU /dhe = 0, p
has no influence.
Note, that the sensitivity is not physically localized, but still without approximation it is possible to determine the sensitivity localized.
Potential relations 19
For the complementary potentials even more simple results are available
dΠσ /dh = (∂Πσ /∂h)fixed stresses + (∂Πσ /∂σ) (dσ/dh) = (∂Πσ /∂h)fixed stresses
(4.12)
because of stationary total complementary potential ∂Πσ /∂σ = 0 (complementary
virtual work principle) with respect to statically admissible stress variations. From
U σ = Πσ then follows
dU σ /dh = (∂U σ /∂h)fixed stresses
(4.13)
which with the relation p(dU /dh) = (dU σ /h) and (4.13), (4.10) gives
(∂U σ /∂h)fixed stresses = − (∂U /∂h)fixed strains
(4.14)
valid also for p 6= 1. Formula (4.14) can also be found in the optimal design paper
by (Masur 1970).
4.4
Summing up
• With proportionality between strain energy and stress energy, simple proportional relations to external potential and to total potentials exist.
• From this follows that sensitivity analyses are simplified, and alternative optimization objectives can be chosen.
• First order sensitivities with fixed strain fields or fixed stress fields result from
the virtual work principle or from the virtual complementary(stress) work principle.
c
20 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Chapter 5
Energy densities
in matrix notation
For linear strain models {}, {δ} are the vectors of strain and variational strain, and
{σ}, {δσ} are the vectors of conjugated stress and conjugated variational stress. In
this chapter energy densities is described by the linear strain quantities, but a similar
description follows for non-linear strain notation.
5.1
Linear strain
notation
Strain and stress energy densities
With {}, {δ} being the vectors of strain and variational strain, and {σ}, {δσ} being
the vectors of conjugated stress and conjugated variational stress; then the variational
strain energy density δu is defined by
Z {}
T
{σ}T {d˜
}
(5.1)
δu = {σ} {δ}
⇒
u =
Strain energy
density
0
and thus having the same dimension as stress. The tilde ˜ indicate the difference
between integration variable {˜
} and final strain state {}, i.e., the stress function is
{σ} = {σ({˜
})}. The strain energy density is a function of strain, and from the strain
energy density function u stress is obtained by differentiation
{σ} =
du
= {σ({})} = [L̄]{}
{d}
and by its definition then the secant modulus [L̄].
The variational stress energy density δuσ is defined by
Z {σ}
σ
T
σ
{}T {dσ̃}
δu = {δσ} {}
⇒
u =
(5.2)
Stress energy
density
(5.3)
0
The stress energy density is a function of stress and from the stress energy density
function uσ , and the secant compliance relation [L̄]−1 is obtained by differentiation
{} =
duσ
= {({σ})} = [L̄]−1 {σ}
{dσ}
(5.4)
Together the definitions of variational strain energy density and variational stress
energy density gives
δ({σ}T {}) = δu + δuσ
(5.5)
21
c
22 Pauli
Pedersen: Stationarity and extremum principles in mechanics
and with integration from zero elastic energy density to final state follows
(5.6)
u + uσ = {σ}T {}
For a one dimensional case Figure 5.1 illustrates the definitions (5.1) and (5.3)
by areas, and the relation (5.6) is also directly recognized.
σ
uσ
PSfrag replacements
u
Figure 5.1: Illustration of strain energy density u (5.1) and of stress energy density uσ (5.3)
by areas.
5.2
Power law
elasticity
Energy densities in
1D non-linear elasticity
The analysis is restricted to power law non-linear analysis, because for this model
analytical explicit expressions for the constitutive matrices are obtained.
Although the general case of 3D anisotropic behaviour is described for this nonlinearity, at first a 1D model is treated. The model of power law non-linear elasticity
for a 1D model in terms of σ = σ() is
σ = Ẽp or more general σ = |
p−1
| E0 for ≥ 0
0
(5.7)
with the positive material parameters E0 , p and the limit of linearity 0 measured in
strain or in stress by σ0 . The material modulus for linear elasticity is E0 and thus
σ0 = E0 0 . Figure 5.2 shows the drastic influence of the power p in the possible
range 0 < p ≤ 1.
Energy densities in matrix notation 23
σ
E0
0.4
0.35
PSfrag replacements
=
p
p−1
0
p = 1.0
p = 0.9
p = 0.8
p = 0.6
p = 0.4
p = 0.2
p = 0.1
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.4
Figure 5.2: Illustration of the influence from power p in the applied power law non linear
elasticity. For strains smaller than the limiting strain 0 a linear behavior is assumed.
For convenience, here > 0 is assumed, and it follows that the secant modulus
Es and the tangent modulus Et are
σ
= Ẽp−1
dσ
= pẼp−1 = pEs
Et :=
d
Es :=
(5.8)
Figure 5.3 illustrates the definition of the Et modulus by the tangent at an actual (, σ)
value, and the Es modulus at the same point by the slope from the origin to the (, σ)
value.
σ
Et
PSfrag replacements
actual (, σ)
Es
Figure 5.3: Illustration of the secant modulus Es and the tangent modulus Et .
Secant and
tangent
modulus
c
24 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Inserting (5.7) in (5.1), the strain energy density for this material model is obtained
Z 1
˜p d˜
=
Ẽp+1
(5.9)
u = Ẽ
p
+
1
0
and from (5.6) with (5.21) get
u + uσ = σ = Ẽp+1
Proportional
relation
⇒
uσ =
p
Ẽp+1
p+1
(5.10)
The simple proportional relation
uσ = pu
(5.11)
between stress energy density and strain energy density often simplifies analysis (and
especially sensitivity analysis for optimal design) to a large extent and give rise to
a number of important general results. However, it should be kept in mind that the
power law non-linear elasticity is a restrictive model.
5.3
Effective
strain/stress
Energy densities in
2D and 3D non-linear elasticity
Extension to 2D and 3D models is not trivial, and the definitions of effective strain
and of effective stress must be chosen appropriately. In matrix notation the differential strain energy density du is similar to the variation in (5.1)
du = {σ}T {d}
(5.12)
In analogy with (5.7) the constitutive secant modulus is
p−1
p−1
e
e
{σ} =
E0 [α]{} ⇒ {σ} = [L̄]{} with [L̄] =
E0 [α]
0
0
(5.13)
Constitutive
secant
modulus
assuming linear elasticity for e ≤ 0 , and with the non-dimensional and constant
matrix [α] describing the relative moduli (isotropy as well as non-isotropy). The
reference strain is 0 and the corresponding reference modulus E0 . It follows from
(5.13) that at the reference strain 0 , the scalar secant modulus is independent of the
power p. The fact that the matrix [α] is constant means that the non-isotropic relations
are unchanged, only the stiffness magnitude changes through the factor p−1
e .
Inserting (5.13) in (5.12), with [α] being symmetric, give
p−1
e
{}T [α]{d}
(5.14)
du = E0
0
and the effective strain e must be defined so that {}T [α]{d} can be integrated.
From the energy related definition
2e := {}T [α]{}
(5.15)
follows by differentiation with [α] constant and symmetric
2e de = 2{}T [α]{d}
(5.16)
Energy densities in matrix notation 25
and thus inserted in (5.14) the one dimensional result
du =
E0
pe de
0p−1
= Ẽpe de
with
Ẽ =
E0
0p−1
(5.17)
which is integrated to obtain the relations proved for the 1D case, i.e.,
u =
1
Ẽp+1 and uσ = pu
p+1 e
(5.18)
The constitutive tangent modulus is obtained by differentiating (5.13) to get, with
the use of (5.16),
p−1
1−p
e
(5.19)
E0 [α] [I] − 2 {}{}T [α] {d} = [L]{d}
{dσ} =
0
e
or alternatively written for the constitutive tangent modulus
[L] =
e
0
p−1
E0
1−p
[α] − 2 {ζ}{ζ}T
e
with
{ζ} = [α]{}
(5.20)
showing the influence of the dyadic product {ζ}{ζ}T .
The model (5.13) with the definition (5.15) for non-linear elasticity may alternatively be derived from the strain energy potential u as determined in (5.21)
u =
Ẽ p+1
p+1 e
(5.21)
giving
{σ} =
Ẽ
du
de
=
(p + 1)pe
d{}
p+1
d{}
(5.22)
and from the definition of effective strain (5.15)
de
= −1
e [α]{}
d{}
(5.23)
{σ} = Ẽep−1 [α]{}
(5.24)
that inserted in (5.22) give
with Ẽ = E0 /0p−1 identical to the assumed model (5.13).
Figure 5.4 shows the factor (e /0 )p−1 , that by the displacement iterations is
determined for each element. It follows from Figure 5.4 that domains close to e = 0
are most sensitive.
Constitutive
tangent
modulus
c
26 Pauli
Pedersen: Stationarity and extremum principles in mechanics
p−1
e
0
1
p = 0.9
0.8
PSfrag replacements
p = 0.8
0.6
0.4
p = 0.6
0.2
p = 0.4
p = 0.2
0
e
p = 0.1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 5.4: Illustration of the stiffness factor (e /0 )p−1 as function of the effective strain
e .
5.4
Summing up
• The notations u for strain energy density and uσ for stress energy density are
chosen as alternative to the use of complementary energy density in order to
focus on the relational dependence.
• The sum of these energy densities is denoted φ and only for linear elasticity is
u = uσ = 12 (u + uσ ) = 12 φ.
• However, for all non linear elasticity is φ = u + uσ .
• For power law non linear elasticity with power p is uσ = pu , and secant as
well as tangent constitutive matrices are analytical available.
• Note the misprint in the references (Pedersen 2005a), (Pedersen 2005b), and
(Pedersen 2006), where in formulas corresponding to 5.19 the denominator 2e
2 and should be η 2 , as here and in
for Green-Lagrange strains is written ηref
ef f
the early reference (Pedersen and Taylor 1993).
Chapter 6
Elastic energy in beam models
6.1
Elastic energy in a straight beam
In handbooks of strength of materials, like in (Sundstrøm 1998), we find the formula
for elastic energy (based on linear, isotropic elasticity) in a straight beam of length
b−a
U = U σ = U σN + U σT + U σM + U σMx
Z b 2
T2
M2
Mx2
N
+β
+
+
=
dx
2EA
2GA 2EI
2GK
a
Handbook
formulas
(6.1)
where the cross-sectional forces/moments are
N = normal force, T = transverse(shear) force
M = bending moment, Mx = torsional moment
The material parameters of the assumed isotropic linear elastic behaviour are
E = Young’s modulus, ν = Poisson’s ratio
E
G = shear modulus =
2(1 + ν)
and the cross-sectional constants are
A = area (tension/compression stiffness factor)
I = moment of inertia (bending stiffness factor)
K = torsional stiffness factor
β = factor from the shear stress distribution
Let us primarily prove the individual terms in (6.1) based on the definitions in
Chapter 2. With only a normal force N and stresses uniformly distributed over the
cross-section we get
N
σ
σ
σ2
2 E
, = , u = u σ =
=
=
A
E
2
2E
2
2
N
⇒ u = uσ =
2A2 E
σ=
(6.2)
27
Normal
force
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28 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Bending
moment
With uniformly distributed energy density, the energy per length is u σ A = N 2 /(2EA)
as stated in (6.1).
With only a bending moment M the stresses vary linearly through the height h
of the beam
Mz
−h
h
, for
≤z≤
I
2
2
2z2
M
⇒ u = uσ =
(6.3)
2I 2 E
R
and
the moment of inertia defined by I = A z 2 dA, the energy per length is
R with
σ
2
A u dA = M /(2EI) as stated in (6.1).
After the two cases of pure normal stress let us analyze the case of only a transverse force T . The distribution of shear stresses τ = τ (z) (τ = σ12 ) will depend on
the specific cross-sectional shape, and is here stated as
σ=
τ = τmax f (z) with f (z) = 0 for z = ±
Transverse
force
h
2
(6.4)
Then with engineering shear strain γ = 212 and µ as a cross-sectional constant,
determined by a function f (z), the analog to (6.2) is
τ
τγ
τ2
γ2G
µT
, τ = τmax f (z), γ = , u = uσ =
=
=
A
G
2
2G
2
2T 2
µ
⇒ u = uσ =
f 2 (z)
(6.5)
2A2 G
R
Integrating to energy per length A uσ dA, we see that the constant β in (6.1) must be
defined by
2
Z
µ
2
f (z)dA
(6.6)
β=
A
A
τmax =
Torsional
moment
For specific values of µ, β, see (Sundstrøm 1998) or an alternative handbook.
Finally the case of only a torsional moment, here restricted to circular crosssections for which the stress distribution with outer radius R is
τmax =
τmax r
Mx r
Mx R
, τ=
=
for rmin ≤ r ≤ R
K
R
K
(6.7)
In analog to (6.5) we then get
u = u σ =
Decoupled
energies
τ2
M 2 r2
= x2
2G
2K G
(6.8)
R
which with K = A r2 dA for circular cross-sections gives the energy per length as
stated in (6.1). The non-circular cross-sections is not covered here.
Energy is not linear in stresses as seen from (6.1), where N , T , M and M x are
often termed generalized stresses. We therefore need to prove that the simple addition
of the four energies is correct. For the normal stresses with both N and M we get
σ=
Mz
N 2 M 2 z 2 2N M z
N
+
⇒ σ2 = 2 +
+
A
I
A
I2
AI
(6.9)
Elastic energy in beam models 29
and the last
R term will not give rise to energy because by definition of the beam axis
we have A zdA = 0.
For the shear stresses with both T and Mx it is more simple to look at the work
of T and Mx instead of the elastic energy and then base the proof on the energy
principles of Chapter 2. The beam displacements from T give no rotation around the
beam length direction and therefore no work by Mx . Similar the beam cross-sectional
rotation from Mx give no transverse displacement and therefore no work by T . Thus
the decoupling’s in (6.1) are correct.
frag replacements6.2
Results for simple (Bernoulli-Euler) beams
M
MB
MA
x
Q
x
T
a)
b)
Figure 6.1: Slender beam examples. Case a) for cantilever with concentrated force at the
free end and case b) for a beam subjected to two end moments only.
In Figure 6.1 is shown slender beams and although N 6= 0 and T 6= 0 the elastic
energy from bending is often so dominating that we can simplify (6.1) to
Uσ =
Z
b
a
or expressed in displacements v from M =
U =
Z
b
a
EI
2
M2
dx
2EI
(6.10)
EId2 v/dx2
d2 v
dx2
2
Bending
energy only
by
dx
(6.11)
Let us in relation to the example in Figure 6.1a) discuss the error in neglecting
the term with T 2 in (6.1). With load Q we have T = −Q and M = Qx giving
U
σT
=
Z
L
0
Q2
Q2 L
β
dx = β
, U σM =
2GA
2GA
Z
L
0
Q2 x 2
Q2 L 3
dx =
2EI
6EI
(6.12)
Neglected
shear energy
From this follows
3βEI
6β(1 + ν)
U σT
=
=
σM
2
U
GAL
Γ2
p
with the slenderness ration Γ defined by Γ = L A/I
(6.13)
With β values of the order 1 and slenderness ratio mostly of the order 10-100 we see,
how dominating the bending energy is.
c
30 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Elementary
case
The important case of linearly varying moment shown in Figure 6.1b) give M (x) =
MA + (MB − MA )x/L and then from (6.10) with constant bending stiffness EI give
Z L
L
1
x 2
dx =
Uσ =
MA + (MB − MA )
(MA2 + MB2 + MA MB )
2EI 0
L
6EI
(6.14)
which could have given (6.12) directly for MA = QL and MB = 0.
6.3
Beam solutions by stress(complementary)
principles
PSfrag replacements
v0
x
x=0
Q
x=L
Figure 6.2: Slender cantilever beam problem with end force only.
The cantilever, slender beam problem is repeated in Figure 6.2 with displacement
v0 = v(x = 0) corresponding to the force Q. We shall first list v0 from solving the
differential equation (not shown)
QL3
3EI
How can we obtain this result with an energy principle?
v0 = v(x = 0) =
Four energy
solutions
(6.15)
• The stress(complementary) virtual work principle states δU σ = δW σ , which
with δW σ = v0 δQ and (6.12) gives (6.15) because v0 δQ = (2QL3 /(6EI))δQ
must hold independently of δQ.
• The 2nd Castigliano theorem states v0 = ∂U /∂Q and thus directly from
(6.12) v0 = 2QL3 /(6EI).
• The unit load theorem (2.38) for linear elasticity as here gives with σ 1 from
Q = 1 and 0 from Q:
Z LZ
Z LZ Qxz
1xz
1 0
v0 =
σ dAdx =
dAdx =
I
IE
0
A
0
A
Z L
Qx2
QL3
dx =
(6.16)
EI
3EI
0
σ
σ
σ
• The complementary
total potential
(6.12)
RL 2 2
R Q energy is Π = U − W , i.e., from
σ
Π = 0 Q x /(2EI)dx - 0 v(Q̃)dQ̃ and thus stationarity of Πσ with respect to variation of Q (∂Πσ /∂Q) gives the result when the following differentiation is applied
R
Q
d 0 v(Q̃)dQ̃
= v(Q) = v0
(6.17)
dQ
Minimum of Πσ (∂ 2 Πσ /∂Q2 > 0) gives L3 /(3EI) > 0, that is clearly satisfied.
Elastic energy in beam models 31
6.4
Summing up
• The elastic energy in a straight beam of linear, isotropic, homogeneous material is integrated along the beam axis, that contains the centers of cross sectional gravity. The formula 6.1 is taken from a handbook.
• Although energy is not linear in the cross sectional forces and moments, the
simple addition from each of these is proved.
• The coefficient β to the shear force component must be determined by integration for a specific cross section.
• As an example four different energy principles are shown to give the same
result.
c
32 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Chapter 7
Some necessary conditions for
optimality
In this chapter we primarily present two necessary conditions for optimal solutions
in general, i.e. not directly related to optimal design. The first condition is only valid
for non-constrained problems and the second condition is valid for problems with
only a single constraint in addition to the objective.
After the two first sections we then determine, expressed in physical terms, the
condition for solution of the most simple optimal design problems, for which sizes
(or field of size) optimize stiffness as well as strength. The optimal designs to these
two different problems are shown to be the same.
In close relation to the analysis for size optimization, we treat optimization of
shape, again to optimize stiffness as well as strength. The shape solution to these two
different problems is shown in many cases also to be the same, very much in parallel
to the result for size optimization.
A simple parametrization for shape optimization is finally described. The goal
of the present chapter is to obtain basic understanding of very simple optimal design
problems, without involving extended numerical calculations.
7.1
General
optimization
Design
optimization
Size and
shape
Non-constrained problems
The notion of Φ is often used for compliance as in (2.13), but in Sections 7.1 and 7.2
it is applied for a more general objective. A non-constrained optimization problem
may be defined as
Extremize Φ = Φ(he ) with variables he non-constrained
(7.1)
and a necessary condition for this is that the objective Φ is stationary with respect to
all the independent variables he
dΦ/dhe = 0 for all e
(7.2)
The optimization of material orientation is an important example of such a nonconstrained problem. Unfortunately, this problem is not simple to solve, because
many local optima exist. Each variable (orientational angle θ e in a domain e) has
several solutions to the stationarity condition (7.2).
33
Stationary
objective
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34 Pauli
Pedersen: Stationarity and extremum principles in mechanics
7.2
Problems with a single constraint
An optimization problem with only a single constraint may be defined as
Extremize Φ = Φ(he ) with variables he
constrained by g = g(he ) = 0
Converted
to nonconstrained
To obtain an optimality condition we convert this problem to a non-constrained problem, using a Lagrangian function L = Φ − λg to be made stationary for arbitrary
value of λ (the Lagrangian multiplier). It follows from (7.2) that a necessary optimality condition is
dL/dhe = 0 for all e
dΦ/dhe = λdg/dhe
Proportional
gradients
(7.3)
⇒
(7.4)
This general result we read as proportionality between the gradient of the objective and the gradient of the single constraint. The factor of proportionality, the Lagrangian multiplier λ, is determined by the constraint condition g(h e ) = 0, that
follows from dL/dλ = −g = 0.
The use of the optimality condition (7.4) for obtaining more general information
about optimal design is very important. It should be noted that behind this result is
the assumption that the constraint is active. Extensions to two and more constraints
are possible, but the uncertainty about the active constraints is then often a limiting
factor on the usefulness.
An alternative look at the problem is to postulate (7.4) and then see that dg = 0
implies dΦ = 0, i.e.
dΦ =
7.3
X ∂Φ
X ∂g
∆he = λ
∆he = λdg = 0
∂h
∂h
e
e
e
e
(7.5)
Size optimization for stiffness and strength
The theoretical results for size optimization are more developed than those for shape
optimization. Let us therefore start with some basis knowledge from size optimization, as it can be found in (Pedersen 1998b) for non-linear elasticity or in (Wasiutynski 1960) for linear elasticity.
7.3.1 Size design with optimal stiffness
Homogeneous
mass(volume)
dependence
If the objective is to minimize compliance (minimize elastic energy) for given total
mass then we have (for optimal stiffness design with homogeneous assumptions and
design independent loads): the ratio between sub-domain energy and sub-domain
mass should be the same in all the design sub-domains.
Let the P
design parameters
be he , then homogeneous mass relations are obtained
P
with M = e Me = e hm
M̄
e , where M is the total mass, Me is the mass in doe
main e, m is a given positive value, and M̄e is independent of P
the design parameters.
P
The homogeneous energy relations are obtained with U = e Ue = e hne Ūe ,
where U is the total strain energy, Ue is the strain energy in domain e, n is a given
positive value, and Ūe is explicitly independent of the design parameters.
Restricted to problems with constant mass density we get, in all design domains,
the same mean strain energy density. Furthermore, if the model has constant energy
Some necessary conditions for optimality 35
density within a design domain, then the result for the optimal design is uniform
strain energy density u∗ , i.e.
u∗e = ū for all free design domains
(7.6)
where lower and upper size constraints are not reached. The symbolism here is
a super-index ∗ related to the optimal design, and a overhead bar ¯ indicating a
constant value for each domain e (mean value).
Assume now that the necessary condition (7.6) give a global minimum solution,
then for any other design the total strain energy U is larger (or equal to)
X
X
X
X
X
Ve =
u∗e Ve (7.7)
Ve∗ = ū
U =
ue Ve ≥ U∗ =
u∗ Ve∗ = ū
e
e
e
e
Stiffest
design
e
where
is the optimal volume of the design domain e. For an P
alternativeP
design
with design volumes Ve we have the same total volume V V = e Ve = e Ve∗ .
From (7.7) we get
X
(ue − u∗e )Ve ≥ 0
(7.8)
Ve∗
e
7.3.2 Size design with optimal strength
With positive volumes Ve we read from (7.8), that at least one ue is not less than u∗e .
Thus if the strongest design is defined by minimum of maximum u e , then the stiffest
design characterized by the optimality condition (7.6) is also the strongest design.
We note that the strength may also be defined in relation to the von Mises stress
or an alternative effective stress, and these measures are not always proportional to
the energy density. For a detailed discussion of these aspects see (Pedersen 1998b).
7.4
Also best
strength
Shape optimization for stiffness and strength
In the following we use the same kind of reasoning to draw conclusions about shape
optimization, without involving a solution to the actual stress problem. Thus we
gain general knowledge, valuable for 3D and 2D-problems, for non-linear elastic as
well as for linear problems, for non-isotropic or isotropic problems, for any external,
design independent load. Also valid for non-homogeneous problems and independent
of the solution procedure.
In order to simplify the mathematics the design parametrization is chosen as illustrated in figure 7.1. An alternative parametrization with expansion in terms of shape
design functions is formulated in (Dems and Mroz 1978), a paper closely related to
this presentation.
We assume a homogeneous state for the strain energy density u e within the volume Ve related to the shape parameter he , say a constant stress finite element. Let us
now subject the shape to variation using only two parameters h i and hj . Furthermore,
let the total volume V of the structure (continuum) be fixed, then
∆V =
dVj
dV
dV
dVi
∆hi +
∆hj =
∆hi +
∆hj = 0
dhi
dhj
dhi
dhj
(7.9)
because we also assume the domain volumes to be depending only on one design
parameter and with a positive gradient (to be used later)
Ve = Ve (he ) and dVe /dhe > 0
General
knowledge
(7.10)
Localized
volume change
c
36 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Figure 7.1: Discretized design parametrization, showing two design domains i and j.
7.4.1 Shape design with optimal stiffness
In shape optimization for extremum elastic strain energy the increment of the objective corresponding to increments ∆hi , ∆hj is
∆U =
Design
independent
loads
Localized
energy
change
dU
dU
∆hi +
∆hj
dhi
dhj
which for power law non-linear elasticity σ = Ep can be written as
1 ∂U
∂U
∆U = −
∆hi +
∆hj
p ∂hi
∂hj
fixed strains
(7.11)
(7.12)
This is proved in (Pedersen 1998b) for design independent loads, and follows
from (4.10) in chapter 4. Therefore only the local energies U i = ui Vi and Uj =
uj Vj are involved and the variations in the strain energy densities need not be determined, because the constitutive relations are unchanged. We have
dVj
dVi
1
∆hi + uj
∆hj )
∆U = − (ui
p
dhi
dhj
(7.13)
and inserting (7.9) in (7.13) we obtain
1
dVi
∆U = − (ui − uj )
∆hi
p
dhi
(7.14)
A necessary condition for optimality ∆U = 0 with dVi /dhi > 0 is therefore
u i = u j .
With all design parameters, eq. (7.9) and (7.13) are written
X dVe
∆he
∆V =
dhe
e
1X
dVe
∆U = −
u e
∆he
(7.15)
p e
dhe
Constant
energy
density
and we conclude that a necessary condition for optimality ∆U = 0 with constraint
∆V = 0 is constant strain energy density ue . Thus for the stiffest design the energy
density along the shape(s) to be designed, here denoted u s , must be constant
us = ū
(7.16)
Some necessary conditions for optimality 37
7.4.2 Shape design with optimal strength
We now relate the stiffest design (minimum compliance) to the strongest design (minimum maximum strain energy density). Let us assume that the highest strain energy
density is at the shape to be designed. With index s referring to shape design domains
and index n referring to domains not subjected to design changes, this means that for
the stiffest design we assume
(7.17)
us = ū > un
A design domain that depends on design parameter is given index s (h s ) and a design
domain which is not subjected to design change is given index n (h n ). For the total
design domain we use index S and for the total domain not subjected to design, index
N . The total elastic strain energy U is obtained from
X
X
X
X
U = U S + U N =
U s +
U n =
u s V s +
un Vn ,i.e.,
s
U = ū
X
Vs +
s
X
n
s
n
(7.18)
u n V n
n
With unchanged domain N and for the stiffest design U > U∗ we obtain
X
X
X
X
u s V s +
u n V n >
ū Vs∗ +
u∗n Vn∗ ,i.e.,
s
n
X
s
(us − ū )Vs >
s
X
n
n
(u∗n
− un )Vn
(7.19)
as s ū Vs∗ = s ū Vs due to given total volume, and furthermore individual unchanged in the non-design domains Vn∗ = Vn .
The right hand side might be negative, so we can not directly draw conclusions
as from (7.8). However, in a complementary formulation with stress energies we can
prove that the right hand side is non-negative and then the proof holds.
The proof of increasing energy in the shape domain is as follows. We write the
total stress energy Uσ as the sum of stress energy in the shape domain UσS and stress
energy in the non-shape domain UσN and obtain
P
Basic
assumption
P
Uσ = U σ S + U σ N ⇒
dUσS
dUσN
dUσ
=
+
dh
dh
dh
From the principle of complementary virtual work follows
dUσ /dh = (∂Uσ /∂h)fixed stress field and we get
dUσ
∂UσS
∂UσN
=
+
dh
∂h
∂h fixed stress field
(7.20)
(7.21)
where the last term is zero when h has no direct influence on the non-shape domain.
Finally for the stiffest design we have dUσ /dh > 0 and from this we conclude
∂UσS
dUσ
1 dUS
=
=
>0
(7.22)
∂h fixed stress field
dh
p dh
Summarizing the theoretical results of this section; we have for the general threedimensional case with non-isotropic, power law non-linear elastic material in an nonhomogeneous structure, and for any design independent single load case that:
Detail of
proof
c
38 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Design for
stiffness
and strength
The minimum compliance shape design (stiffest shape design) has uniform energy
density along the designed shape, as far as the geometrical constraints make this
possible.
If we furthermore assume that the highest energy densities are found at the designed shape, then the stiffest design is also the strongest design, as defined by a
design which minimizes the maximum energy density.
Note that these results are obtained without calculating the stress/strain fields and
without specifying the constitutive behaviour. This behaviour need not be homogeneous and thus we can also include the multi-material case.
7.5
Good
experience
Conditions with a
simple shape parametrization
In the final conclusions in section 7.4.2, we have added the note ”as far as the geometrical constraints make this possible”. Also it was commented that normally the
shape parametrization implies such a geometrical constraint. In this section we use
a simple shape parametrization that makes a rather simple optimality condition possible. The limitations of using this simple parametrization can be evaluated by the
possibility to obtain almost uniform energy density distribution along the shape to be
designed. Many examples illustrate that the parametrization is in fact able to describe
optimal shapes in many cases.
Figure 7.2: A three parameter (α, β, η) description of an internal hole in a rectangular
domain, specified by A, B.
Figure 7.2 shows a single inclusion hole, where the shape of the boundary is modeled as a super-elliptic shape, described by only three non-dimensional parameters,
relative axes α, β and power η
x η y η
+
=1
αA
βB
Two or
only one
parameter
(7.23)
With known area of the hole we only have two parameters and if furthermore symmetry is enforced, say αA = βB, we only have one free parameter, which might
be the power η. Figure 7.3 shows the great flexibility even for this one parameter
description. This parametrization naturally has its limitation, but several examples
Some necessary conditions for optimality 39
Figure 7.3: Shapes giving equal area of the hole, with powers of the super-elliptic
shape being η = 0.75, 1.25. 1.75 and 3.00, respectively.
show its usefulness, and furthermore it can easily be extended to 3D-problems by
x η y η z η
+
=1
(7.24)
+
αA
βB
γC
In the 2D-model (7.23) the area of the hole is
Z αA
x η 1/η
4
βB 1 − (
) )
dx = 2αβABg(η)
αA
0
with the function g = g(η) defined by
η+1
2
1
Γ
/Γ
g(η) := Γ
η
η
η
(7.25)
(7.26)
where Γ is the Gamma-function. With the rectangular area being 4AB the relative
area of the hole φ (relative to the area 4AB) and the relative area of the solid (relative
density) ρ are
1
φ = αβg(η) = 1 − ρ
2
(7.27)
Relative
hole area
or density
An optimal design problem is formulated in order to extremize the elastic energy
U for constant relative area
Extremize U subject to φ(α, β, η) = φ̄
(7.28)
Within the possibilities of the three parameters α, β, η this also minimizes energy
concentration and returns constant energy density along the boundary of the hole,
as discussed in section 7.4.2. Using the result (7.12) from sensitivity analysis we
determine the differential of the elastic energy (p = 1 for linear elasticity)
1 ∂U
∂U
∂U
dU = −
dα +
dβ +
dη
(7.29)
p ∂α
∂β
∂η
fixed strains
and the differential of the constraint follows from (7.27) (using a formula manipulation program to differentiate the Gamma-functions)
dα dβ p(η)dη
+
+
dφ = φ
(7.30)
α
β
η2
Design
problem
c
40 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Available
functions
with the function p = p(η) defined by
2
1
η+1
p(η) := Ψ
−Ψ
−Ψ
η
η
η
(7.31)
where Ψ is the Psi-function. To illustrate that the functions g(η) and p(η) are wellbehaved functions we show in figure 7.4 these functions and there derivatives. These
functions are available in many libraries of computer routines.
Figure 7.4: Left g-function and right the p-function with their derivatives as a function
of the shape power η.
Optimality
condition
For model
by the FEM
The condition of dU = 0 when dφ = 0 is a necessary condition for optimality
and thus (as in general with only a single constraint) we from (7.29) and (7.30) get
the optimality condition by proportional gradients (7.4), i.e.
∂U
∂U
η 2 ∂U
(7.32)
α
=β
=
∂α
∂β
p(η) ∂η fixed strains
In a fixed strain field the energy densities u are constant and only the volumes of
domains (elements) connected to the hole boundary change. Thus in a finite element
formulation the optimality condition (7.32) is written
α
X
s
Strength or
stiffness
us
X ∂Vs
∂Vs
η 2 X ∂Vs
=β
us
=
us
∂α
∂β
p(η) s
∂η
s
(7.33)
where index s refers to an element connected to the hole boundary. The only information needed in addition to the results from analysis is ∂Vs /∂α, ∂Vs /∂β, ∂Vs /∂η,
i.e. only information from geometry. We note, P
in agreement with section 7.4.2, that
if us is constant along the hole boundary then s ∂Vs /∂α = ∂V /∂α = φ/α etc.,
and the optimality criterion (7.33) is satisfied by us φ = us φ = us φ. Thus a constant energy density along the boundary of the hole implies stationary total elastic
energy. However, we can have stationary energy without constant energy density, if
the possible designs are restricted. This is illustrated by the examples in chapter ??.
7.5.1 Possible iterative procedure
The problem is how to find a boundary shape that satisfies (7.32) or in finite element formulation (7.33). The heuristic approach of successive iterations could be to
estimate the Lagrange multiplier λ by the mean value
∂U
η 2 ∂U
1
∂U
λestimated =
+β
+
α
(7.34)
3
∂α
∂β
p(η) ∂η
Some necessary conditions for optimality 41
and then redefine α, β, η by
αnew = λ/(
∂U
η2
∂U
∂U
)old , βnew = λ/(
)old , (
)new = λ/(
)old
∂α
∂β
p(η)
∂η
(7.35)
with iterations on λ to satisfy the constraint of (7.28)
1
φnew = αnew βnew g(ηnew ) = φ̄
2
7.6
(7.36)
Estimated
multiplier
Summing up
In this chapter the important results to focus on are:
• For non-constrained problems the necessary optimality condition is stationarity of the objective with respect to all the independent variables.
• For problems with a single constraint the necessary optimality condition is
proportionality between the gradient of the objective and the gradient of the
single constraint.
• For size optimization of stiffness and strength the stiffest design is characterized by the optimality condition of uniform energy density and this design is
also the strongest design.
• For shape optimization of stiffness, the minimum compliance shape design
(stiffest shape design) has uniform energy density along the designed shape,
as far as the geometrical constraints make this possible.
• For shape optimization of strength, if we assume that the highest energy densities are found at the designed shape, then the strongest design, as defined
by a design which minimizes the maximum energy density, is also the stiffest
design.
• For shape optimization a simple super-elliptic description makes it possible to
design a wide spectrum of shapes, and the analytical treatment of this case is
almost as simple as the classic elliptic case.
• The super-elliptic description can be extended to include a skewness parameter, and then also describe triangular shapes, see (Pedersen 2004) for more
detail.
Collected
results
c
42 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Chapter 8
Analytical beam design
Beams are among the most important structural elements with classical results for
analysis as well as for design. Especially the statically determinate cases constitute a
basic knowledge in solid mechanics, treated in a one dimensional formulation. The
present chapter originates from a study published in (Pedersen and Pedersen 2008)
and describes the general aspects from an energy point of view, but only the most
simple examples of the statically determinate beams are included.
The analytical approach has two steps, first determining analytically the necessary optimality criterion as it is often found in the literature; for early reference on
compliance minimization see (Huang 1968) and (Masur 1970). Second analytical
step is to determine an explicit analytical solution to the optimality criterion and this
step is limited to statically determinate beams as the optimality criterion can then be
written independent of design. The problems of the present chapter are chosen so
that a full analytical approach is reasonable.
For long beams (Bernoulli-Euler beams) a number of optimal designs are analytically described as refereed in (Save, Prager and Sacchi 1985) and (Rozvany 1989),
with most focus on designs based on plastic collapse. These problems may be formulated to maximize stiffness or maximize eigenfrequency for a given amount of
material volume or mass, see the review (Olhoff and Taylor 1983). The solutions depend on the actual boundary conditions and load conditions in addition to the chosen
design type, i.e., the chosen relation between cross sectional area and cross sectional
moment of inertia. Bernoulli-Euler beam theory is applied in most treated cases , and
little analytical attention is given to short beams for which Timoshenko beam theory
is necessary.
In short beam structures the elastic energy from shear forces can not be neglected.
From a practical point-of-view more realistic designs are obtained when this additional energy is taken into account, but only a few optimal designs with analytical
description are then available. The goal of the present chapter is to present such
results from (Pedersen and Pedersen 2008).
Primarily, the statics of some statically determinate cases are presented with
one dimensional distribution of shear force T = T (x) and of bending moment
M = M (x). Material is assumed isotropic and when the beam has cross sectional
distribution of area A = A(x) and cross sectional moment of inertia I = I(x), then
the distribution of elastic energy per unit length is given. In beams the cross sectional
moment of inertia I = I(x) plays a major role and thus we need a relation between
area and moment of inertia. Three possibilities are treated in the present chapter
with I(x) proportional to A(x), to A2 (x), or to A3 (x). The cross sectional types
43
Statically
determinate
beams
Analytical
steps
Timoshenko
beam theory
c
44 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Cross sectional
size
Beam length
Use of optimal
design results
specifically treated in the present chapter are limited to those presented in Section
8.4. To illustrate the influence of beam length on the design, the common expression
I(x) = CAn (x) is further specified as C = γb2(2−n) with γ as a non-dimensional
constant in the range 0.08 ≤ γ ≤ 0.25 for the treated cases. Beam length L is
given relative to a cross sectional length parameter b, i.e., L = ηb, treating η as a
non-dimensional length parameter.
In optimal design formulation for minimum compliance with point wise design
variables, a necessary optimality criterion is uniform elastic energy density. In beam
design with the cross sectional area A = A(x) as design parameter the optimality
criterion must be stated for this two dimensional design domain. The gradient of
the elastic energy per unit length with respect to area change, must as a necessary
condition be the same for all areas, i.e., at all positions x. For the simplest cases, i.e.,
all Bernoulli-Euler beams and Timoshenko beams with I(x) linearly depending on
A(x) this means that the mean value of elastic energy density should be the same for
all areas. However, this later statement of the optimality criterion is not valid for all
Timoshenko beams.
Four cases of statically determinate beams are analyzed in (Pedersen and Pedersen 2008): Bernoulli-Euler cantilevers, Bernoulli-Euler simply supported, Timoshenko cantilevers, and Timoshenko simply supported. For each of these four beam
models three load cases are applied, and in addition to the cross sectional parameter
n = 1, 2, 3 in An (x), this adds up to 36 individual cases. As examples in the present
chapter are only presented the cantilever beams.
In two and three dimensional design of beam like structures, this knowledge on
A∗ (x) can be used to compare with numerically obtained results or as initial designs.
The presented values of compliance decrease also give possibility for comparing with
alternatively obtained results.
8.1
Optimality criterion for beam design
The optimization problem is stated as
Minimize compliance Φ
for a given volume V =
Compliance
objective
Z
L
A(x)dx
(8.1)
0
where the cross sectional area A(x) is integrated along the beam axis x from 0 to
length L. The length is specified relative to a cross sectional reference length b as
L = ηb with the influence of the non-dimensional parameter η, for short beams
showed in the range 1 ≤ η ≤ 5. The area function A(x) is the design function to be
optimized, then denoted A∗ (x).
The compliance Φ is the work of external dead loads and may be evaluated as the
sum of internal elastic energies (stress energy + strain energy) Φ = U σ + U . Using
beam theory under the assumption of zero normal force and zero torsional moment
(with linear elasticity Φ = 2Uσ = 2U ) the compliance is
Φ=
Z
L
0
φ(x)dx with φ(x) = β
M 2 (x)
T 2 (x)
+
GA(x)
EI(x)
(8.2)
where φ(x) is twice the stress elastic energy per unit length, T (x) the shear force,
M (x) the bending moment, I(x) the cross sectional moment of inertia, G the shear
Analytical beam designs 45
modulus, and E Young’s modulus. For isotropic material we have G determined by
Poisson’s ratio ν as E = 2(1 + ν)G. The factor from the cross sectional distribution
of shear stresses is β, here approximated to be unchanged along the beam, and being
in the range 1 < β < 2.
In Section 8.4 different models of type I(x) = CAn (x) are shown, with values of
the power n = 1, 2, 3 and correspondingly different dimensions of the quantity C that
does not depend on the actual design variable. Introducing the model C = γb 2(2−n)
give
I(x) = γb2(2−n) An (x) for n = 1, 2, or 3
(8.3)
Model for
moment of
inertia
where b is a cross sectional reference length, assumed constant for the optimization.
Section 8.4 shows examples where for n = 1 the size b is the width w or the height
1
≤ γ ≤ 41 . For n = 2 the size b disappear in (8.3) and as reference value
h with 12
√
p
1
b a kind of mean value V /L may be chosen, for this case 4π
≤ γ ≤ 183 . Finally
1
for n = 3 the size b is again the width w and γ = 12
. Totally the non-dimensional
parameter γ is in the range 0.08 ≤ γ ≤ 0.25 for the cases shown in Section 8.4.
Twice the stress elastic energy per unit length, (8.2) and (8.3) is
2
T (x) M 2 (x)b2n
1
α
+ 4 n
φ(x) =
γE
A(x)
b A (x)
with α = 2γβ(1 + ν)
(8.4)
For practical cases the defined non-dimensional parameter α is of the order 1. With
a non-dimensional parenthesis and a design independent factor, then (8.4) is written
Q2
(T (x)/Q)2 (M (x)/(Qb))2
+
φ(x) =
α
γEb2
A(x)/b2
(A(x)/b2 )n
with α = 2γβ(1 + ν)
(8.5)
Compliance
per unit
length
where Q is the total external force. The shear force distribution T (x) and the moment
distribution M (x) are independent of design for statically determinate cases.
The optimality condition for an objective with a single active constraint is given
by proportional gradients, as stated in (7.4). For the compliance problem (8.1) with
A(x) as design parameter and λ̄ as a positive constant this give
Z L
Z L
dΦ
dφ(x)
dV
1dx or
=
dx = −λ̄
= −λ̄
dA(x)
dA(x)
0 dA(x)
0
Z L
dφ(x)
+ λ̄ dx = 0
(8.6)
dA(x)
0
to hold for any variation of A(x). For the statically determinate cases only A(x) in
(8.4) is varying and differentiation gives
dφ(x)
1
T 2 (x)
M 2 (x)b2n
=
−α 2
− n 4 n+1
(8.7)
dA(x)
γE
A (x)
b A
(x)
and the necessary condition (8.6) to hold at all x may be expressed by a new nondimensional positive constant λ, defined by
λ=α
(M (x)/(Qb))2
(T (x)/Q)2
+
n
(A∗ (x)/b2 )2
(A∗ (x)/b2 )n+1
(8.8)
Optimality
criterion
c
46 Pauli
Pedersen: Stationarity and extremum principles in mechanics
where A∗ (x) is the optimal area distribution. The optimality criterion (8.8) is valid in
general for the studied cases, and the optimal area function can be determined from
(8.8). For Bernoulli-Euler beams (α = 0) and for Timoshenko beams with n = 1 the
constant λ is proportional to the mean value of elastic energy density, as seen from
(8.5), i.e., φ(x)/A(x).
8.1.1 Treated boundary conditions and loads
Figure 8.1 shows the elementary cases that are all treated in (Pedersen and Pedersen
2008), all being statically determinate. The cantilever cases 1), 2) and 3) give most
simple force/moment distributions (8.9) and case 4) is in reality identical to case 1)
when half the length is designed. The other simply supported cases 5) and 6) have
additional force/moment components as stated in (8.10).
PSfrag replacements
x
x T (x)
T (x)
M (x)
M (x)
L/2
Q
1)
Q
4)
Q
2)
5)
Q
3)
L
Q
Q
6)
L
Figure 8.1: Boundary conditions and loads for 6 beam cases.
Cantilever
static
distributions
For the cantilever cases with 0 ≤ x ≤ L = ηb or 0 ≤ x̃ ≤ η, with x̃ = x/b
1)
T (x) = Q ⇒
M (x) = −Qx ⇒
2)
3)
Q
x⇒
L
Q
M (x) = − x2 ⇒
2L
Q
T (x) = 2 x2 ⇒
L
Q
M (x) = − 2 x3 ⇒
3L
T (x) =
T (x)
=1
Q
M (x)
= −x̃
Qb
T (x)
x̃
=
Q
η
M (x)
x̃2
=−
Qb
2η
T (x)
x̃2
= 2
Q
η
x̃3
M (x)
=− 2
Qb
3η
(8.9)
Analytical beam designs 47
For the simply supported cases with 0 ≤ x ≤ L = ηb or 0 ≤ x̃ ≤ η, omitting case 4
5)
6)
Q Q
+ x⇒
2
L
Q 2
Q
x ⇒
M (x) = x −
2
2L
Q
Q
T (x) = − + 2 x2 ⇒
3
L
Q
Q 3
M (x) = x −
x ⇒
3
3L2
T (x) = −
T (x)
1 x̃
=− +
Q
2 η
M (x)
x̃ x̃2
= −
Qb
2 2η
T (x)
1 x̃2
=− + 2
Q
3 η
M (x)
x̃
x̃3
= − 2
Qb
3 3η
Simply
supported
static
distributions
(8.10)
8.1.2 Solutions in general
For the Bernoulli-Euler beams the shear force term in the optimality criterion (8.8)
is omitted and the solution is (proportional to)
s
M (x) 2
n+1
∗
(
)
(8.11)
[A (x)]B ∝
Qb
The influence of the shear force in Timoshenko beams complicates the solution
for n = 2 and 3, but for n = 1 the simplicity remains
s
M (x) 2
T (x) 2
∗
α(
[A (x)]T,n=1 ∝
) +(
)
(8.12)
Q
Qb
although then depending on the value of α (of the order 1).
For n = 2 and 3 the optimality criterion (8.8) is rewritten to polynomial form
λ(
(A∗ (x) n+1
T (x) 2 (A∗ (x) n−1
M (x) 2
) (
) =0
)
−
α(
)
− n(
2
2
b
Q
b
Qb
A∗ (x)
b2
=
T,n=3
v
s
u
u α T (x)
α T (x) 2 2 3 M (x) 2
t
2
+ (
(
) +
(
)
)
2λ Q
2λ Q
λ Qb
Timoshenko
design
formula
for n = 1
(8.13)
with solutions that depend on the constant λ, and iterations are necessary. Solutions
are obtained in the inverse sense, that a specified λ directly gives a corresponding
volume.
For n = 3 a second order polynomial in the squared (A∗ (x)/b2 ) with the solution
BernoulliEuler
design
formula
Timoshenko
design
formula
for n = 3
(8.14)
For n = 2 a third order polynomial in (A∗ (x)/b2 ), also with a simplified solution
because the component of second order is zero
∗
A (x)
=
b2
T,n=2
v
s
u
u
3
−α T (x) 2 3
1
M
(x)
1 M (x) 2 2
t
2
+
(
) +
(
) ) +
(
)
λ Qb
λ Qb
3λ Q
v
s
u
u
3
−α T (x) 2 3
M
(x)
1
1 M (x) 2 2
t
2
(8.15)
(
) −
(
) ) +
(
)
λ Qb
λ Qb
3λ Q
Timoshenko
design
formula
for n = 2
c
48 Pauli
Pedersen: Stationarity and extremum principles in mechanics
In the following sections, specific solutions for cantilever beams are presented
with focus on the optimal area function A∗ (x), on the integration to satisfy the volume constraint and on the values of decrease in compliance relative to the compliance
for a uniform beam.
8.2
Bernoulli-Euler cantilever beams
For the case of Bernoulli-Euler cantilever beams the optimal solution is not depending
on the length of the beam, and the elastic energy per unit length (8.4) simplifies to
x2m
M 2 (x)
= C̃ n
EI(x)
A (x)
Q2
with C̃ = 2
m γEL2(m−1) b2(2−n)
φ(x) =
(8.16)
using the cross sectional modeling (8.3) for n = 1, 2 and 3, and the load cases 1), 2),
3) correspond to m = 1, 2, 3, respectively. The optimality criterion then state that
φ(x)/A(x) should be constant and simplifies to
2m
(8.17)
A∗ (x) = Kx n+1
The constant K is determined by the volume constraint (8.1)
V =
Z
L
A∗ (x)dx = K
0
Z
L
2m
x n+1 dx
0
2m+n+1
n+1
=K
L n+1
2m + n + 1
2m + n + 1
V
giving K =
2m+n+1
n+1
L n+1
BernoulliEuler
cantilever
designs
(8.18)
The optimal designs are illustrated in Figure 8.2, corresponding to combinations
of m = 1, 2, 3, and n = 1, 2, 3, with the non-dimensional length coordinate 0 ≤
x/L ≤ 1 in order to focus on the form rather than satisfying a common volume
constraint.
2m
x n+1
1
2m
n+1
0.8
=
1
2
2
3
1
4
3
3
2
2 3
0.6
PSfrag replacements 0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
L
Figure 8.2: For Bernoulli-Euler cantilever beams, illustration of optimal cross sectional area
distributions.
Analytical beam designs 49
8.2.1 Optimal compliances
For a uniform beam (cylindrical) the size of the area is A(x) = V /L and the total
energy from (8.16) with C̃ inserted is
L
Z
L n L 2m
φ(x)dx = C̃( )
Φ=
x dx
V
0
0
1
b2(n−2) Ln+3 Q2
= 2
m (2m + 1)
γEV n
Z
(8.19)
For the optimal designs in (8.17) the area distributions give
∗
Φ =
C̃
Z
L
0
Z
L
∗
φ (x)dx = C̃
0
x2m
Z
C̃
2mn dx =
Kn
K n x n+1
L
x2m
dx =
(A∗ (x))n
Z
L
0
2m
x n+1 dx =
0
C̃V
K n+1
(8.20)
R L 2m
using (8.18) for the integral 0 x n+1 dx. Inserting the obtained value for K (also in
(8.18)) and C̃ from (8.16) finally give
∗
Φ =
Z
L
0
φ∗ (x)dx =
b2(n−2) Ln+3 Q2
(n + 1)n+1
m2 (2m + n + 1)n+1
γEV n
(8.21)
The ratio of the compliances (8.21) and (8.19) is
Φ∗
(2m + 1)(n + 1)n+1
=
Φ
(2m + n + 1)n+1
(8.22)
with resulting values for the different combinations of load case power m = 1, 2, 3
and cross section parameterization power n = 1, 2, 3 in Table 8.1. As seen from this
table we can for unchanged volume have considerable decrease in the compliance by
using the optimal forms in Figure 8.2. The notion ”form” is here preferred, because
the present problems can not be characterized as shape design problems (the beam
axis is unchanged).
Ratios Φ∗ /Φ
n=1
n=2
n=3
m=1 m=2
0.750 0.556
0.648 0.394
0.593 0.313
m=3
0.438
0.259
0.179
Table 8.1: For Bernoulli-Euler cantilever beams, values for the ratios Φ∗ /Φ for different
combinations of load case m and design parameterization power n.
8.3
Timoshenko cantilever beams
Short beams are beams where the energy from shear forces must be taken into account (Timoshenko beams). Thus the influence of the T (x) distributions in (8.9) and
(8.10) must be taken into account for obtaining optimal designs. Note that the singular designs, which for Bernoulli-Euler beams follows from A ∗ (x) = 0 implied by
Optimally
obtained
compliance
c
50 Pauli
Pedersen: Stationarity and extremum principles in mechanics
M (x) = 0, are then treated more properly if T (x) 6= 0 at the same position. For
more extended discussion on singularities of optimal beams, see (Olhoff and Niordson 1979).
The non-dimensional moment distribution for the cases 1), 2) and 3), corresponding to m = 1, 2, 3, is
M (x)
x̃m
=
for m = 1, 2, or 3
Qb
mη m−1
(8.23)
and the non-dimensional shear force distribution for these cases is
T (x)
x̃m−1
= m−1 for m = 1, 2, or 3
Q
η
(8.24)
Inserting specifically these distributions, twice the stress elastic energy per unit length
(8.5) is
!
x̃2(m−1) /η 2(m−1) x̃2m /(m2 η 2(m−1) )
Q2
α
+
(8.25)
φ(x) =
γEb2
A(x)/b2
(A(x)/b2 )n
8.3.1 Design of beams with n = 1 cross sections
Beams with n = 1 cross sections are for the cases in Section 8.4 related to thin walled
beams or to pure width design. For these cases the general solution is (8.12). Inserting
the distributions (8.24) and (8.23) the solution, keeping m as parameter, is
A∗ (x) ∝
q
αx̃2(m−1) + x̃2m /m2
(8.26)
valid for the load cases 1), 2), and 3).
For a single load at the free end, load case 1) with m = 1, we get
A∗ (x) = KT 1
p
α + x̃2
(8.27)
The constant KT 1 is determined by the volume constraint, i.e. by integration of
(8.27)
Z ηp
α + x̃2 dx̃ with x̃ = x/b ⇒
V = KT 1 b
0
KT 1
p
4V p
/ 2η α + η 2 − α ln α + 2α ln (η + α + η 2 )
=
b
(8.28)
Figure 8.3 shows the optimal area distributions in (8.27) for a number of chosen
values of α = 0.4, 0.6, 0.8, 1, 1.5, 2, 3, and the beam length is chosen as L = 4b, i.e.,
η = 4. As expected the effect of taking the shear force into account (the parameter
α 6= 0) is mainly noted at the tip end of the cantilever beam, where the optimal area
√
is KT 1 α. Further along the beam, a linear change of design area is seen, like for
the Bernoulli-Euler beams with 2m/(n + 1) = 1, as seen in Figure 8.2.
For a uniform load, load case 2) with m = 2, the solution is
A∗ (x) = KT 2 x̃
p
4α + x̃2
(8.29)
Analytical beam designs 51
A∗ /KT 1
5
4
L = 4b (η = 4)
3
2
PSfrag replacements
1
α = 3, 2, 1.5, 1 0.8, 0.6, 0.4
0
0
0.5
1
1.5
2
2.5
3
3.5
x/b
4
Figure 8.3: The optimal design distributions for cross sectional areas, corresponding to load
case 1) are illustrated for a Timoshenko cantilever beam which account for the shear force.
The cross sectional moment of inertia is assumed proportional to the cross sectional area.
Note, that KT 1 is depending on α as given in (8.28).
and the volume constraint gives
KT 2
Z η p
V
2
x̃ 4α + x̃ dx̃
= /
b
0
3V / (4α + η 2 )3/2 − 8α3/2
=
b
(8.30)
For a triangular load distribution, load case 3) with m = 3, the solution is
p
A∗ (x) = KT 3 x̃2 9α + x̃2
(8.31)
and the volume constraint gives
Z η p
V
x̃2 9α + x̃2 dx̃)
KT 3 = /(
b
0
p
p
8V
/(η(9α + 2η 2 ) 9α + η 2 + 81α2 ln (3 α)−
=
b
p
81α2 ln (η + 9α + η 2 ))
(8.32)
The optimal area distributions in (8.29) and (8.31) are presented in Figure 8.4.
These designs are very similar to the optimal designs for Bernoulli-Euler beams with
2m/(n + 1) = 2 and 3 in Figure 8.2, but here the influence of the parameter α value
is shown.
Optimal compliances
To determine how much is obtained by the designs (8.27), (8.29) and (8.31) relative
to the compliance for a uniform design, the three energies per length are written
Timoshenko
cantilever
designs
c
52 Pauli
Pedersen: Stationarity and extremum principles in mechanics
A∗ /KT 3 for upper bunch of curves
A∗ /KT 2 for lower bunch of curves
100
80
L = 4b (η = 4)
60
PSfrag replacements
α = 3, 2, 1.5, 1, 0.8, 0.6, 0.4
40
20
0
0
0.5
1
1.5
2
2.5
3
3.5
4
x/b
Figure 8.4: The optimal design distributions for cross sectional areas, are illustrated for a
Timoshenko cantilever beam which account for the shear force. The lower bunch of curves
correspond to load case 2) and the upper bunch of curves correspond to load case 3). The
cross sectional moment of inertia is assumed proportional to the cross sectional area. Note,
that KT 2 and KT 3 are depending on α as given in (8.30) and (8.32).
together, i.e., (8.25) for n = 1
φ(x) =
η 2(1−m) Q2 2(m−1)
αx̃
+ x̃2m /m2
γEA(x)
(8.33)
For the design of uniform area A(x) = V /(ηb) the integrated compliance is
Z
η 3−2m b2 Q2 η 2(m−1)
Φ=
x̃
(α + x̃2 /m2 )dx̃
γEV
0
Q2 b2 η 2 (αm2 (2m + 1) + η 2 (2m − 1))
= 2
(8.34)
m γE
V (4m2 − 1)
For the optimal designs (8.27), (8.29) and (8.31) the integrated compliance is for
n = 1 determined directly by the constants KT m , as presented in (8.28), (8.30) and
(8.32).
Z ηb
V
Q2
∗
(8.35)
Φ =
φ∗ (x)dx = 2
m γE KT2 m
0
The resulting ratios Φ∗ /Φ are then only a function of the non-dimensional parameters
α, η and m.
Φ∗
V2
4m2 − 1
= 2 2
2
2
Φ
b KT m η (αm (2m + 1) + η 2 (2m − 1))
(8.36)
Table 8.2 list, for a number of combinations of values α = 1 and 2, η = 1, 3 and 5, the
obtained relative decrease in compliance, corresponding to the load cases 1), 2), and
3) (m = 1, 2 and 3).
Analytical beam designs 53
α, η
m=1
m=2
m=3
1, 1
0.988
0.733
0.549
1, 3
0.888
0.662
0.515
1, 5
0.828
0.616
0.486
2, 1
0.996
0.741
0.552
2, 3
0.931
0.692
0.531
2, 5
0.867
0.646
0.506
Table 8.2: Values for the ratios Φ∗ /Φ for different combinations of shear parameter α and
load case m. The cross sectional moment of inertia is assumed proportional to the cross
sectional area (n = 1).
In relation to the obtained decrease in compliance, we note that the shear parameter α do not have a significant influence, neither do the length parameter, i.e.,
the value of the non-dimensional parameter η. Table 8.2 shows that not so much is
obtained for shorter beams and for higher values of α, all as expected and as it is
directly seen from (8.36). The load distribution has most influence on this relative
decrease of compliance. To compare with the resulting decrease for Bernoulli-Euler
beams , the case of α, η = 1, 5 is chosen, and for the three load cases (m = 1, 2, 3) Table 8.2 shows for the Timoshenko beam model that the compliances by optimization
decreased to 82.8%, 61.6%, and 48.6%, respectively. With the Bernoulli-Euler beam
model the corresponding compliances by optimization decreased to 75%, 55.6%, and
43.8%, according to Table 8.1.
w
Sfrag replacements
I(x) =
w
w = 2r
1
A2 (x)
12
I(x) =
w
1
A2 (x)
4π
I(x) =
h
√
3 2
A (x)
18
I(x) =
I(x) =
I(x) '
w2
A(x)
6
I(x) '
h2
A(x)
12
1
A3 (x)
12w2
w2
A(x)
8
I(x) '
Optimally
obtained
compliance
w2
A(x)
12
I(x) '
h2
A(x)
4
Figure 8.5: Moment of inertia expressed by area for different cross sections. Solid square
with side length w and below thin walled. Solid circular with radius r = w/2 and below
thin walled. Solid equal sided triangular with side length w and below thin walled. Finally
rectangular with width w and height h and below a pure flange approximation. For the thin
walled cross sections the wall thickness is applied as design parameter.
c
54 Pauli
Pedersen: Stationarity and extremum principles in mechanics
8.4
Examples of beam cross sections
Figure 8.5 gives for a number of beam cross sections, the moment of inertia expressed
in terms of the cross sectional area.
For thin walled beams the assumed uniform thickness t is also assumed relatively
small and thus approximated directly by the areas. For the square 4wt(x) = A(x),
for the circular πwt(x) = A(x), for the triangular 3wt(x) = A(x) and for the ”flange
or sandwich” beam 2wt(x) = A(x).
For the circular and the equal sided triangular cross sections I(x) is relative to
any direction through the center of gravity. For the rectangular cross sections I(x) is
relative to the symmetry lines (also through the center of gravity).
8.5
Summing up
• A large number of analytically (mostly explicit) optimal area distributions are
derived for compliance minimization with a given structural volume. This is
possible only for statically determinate cases, where the shear force distribution T (x) and the moment distribution M (x) are independent of design. The
graphical displays of these designs give the background for basic understanding of the influence from boundary conditions, from load distribution, and
from the cross sectional modeling. Dealing with both long beams (BernoulliEuler model) and with short beams (Timoshenko beam model), the influence
from shear force is made clear. All these different models are designed based
on an energy approach to directly obtain optimality criteria analytically.
• The direct energy approach has enabled a unified analysis for the specific
cases, i.e., for three problems of cantilever beams and three problems of simple supported beams, each with three cross sectional types, i.e., 18 cases for
Bernoulli-Euler beams and 18 cases for Timoshenko beams.
• The effect of optimization relative to uniform cross sectional design (cylindrical beam) also depends on these aspects, with optimal compliance values often
half the original or even less, but also almost unchanged for specific cases.
• Taking the area distribution as design variable, then a numerical two or three
dimensional model of a beam like structure may be compared. The obtained
optimal volume distribution may be compared with A∗ (x)dx from the present
study or A∗ (x) may be used in an initial design for a similar or more complicated problem. Many cases are studied and the main parameters are related to
the integer values m = 1, 2, 3 and n = 1, 2, 3. The load distribution determines
m and the chosen cross sectional type is modeled with the power n.
• For long (Bernoulli-Euler)
√ cantilever beams the optimal designs (8.17) are
n+1
given by the function
x̃2m with Figure 8.2 showing these different forms.
Values for the obtained ratio of decreased compliance, relative to a uniform
beam (cylindrical beam), are listed in Table 8.1.
• For long (Bernoulli-Euler) simply supported beams the optimal beam designs
also depend on the length, expressed by the ratio of length over the chosen
cross sectional length parameter η = L/b. For these results see (Pedersen
and Pedersen 2008), that also show results for Timoshenko simply supported
beams.
Analytical beam designs 55
• For short (Timoshenko) cantilever beams the optimal designs, furthermore,
are depending on the parameter α that describes the relative influence from
the shear force. This parameter α = 2γβ(1 + ν) is depending on Poisson’s
ratio ν for the assumed isotropic material, on the shear stress distribution by β,
and on a factor from cross sectional type γ. A practical range of 0.5 < α < 2
is included in the chapter. For the ”thin walled” modeling (or pure width design) of n = 1 the optimal design functions (8.27), (8.29) and (8.31) are still
simple. Figure 8.3 shows optimal area functions for a load concentrated at the
free end and Figure 8.4 shows results for distributed loads. Details on constraint scaling are presented and the obtained ratio of decreased compliances
for chosen values of α, η and m are listed in Table 8.2.
• The cross sectional modeling cases of n = 2 and 3 give rise to a change from
direct scaling in order to satisfy the volume constraint for the two solutions
(8.14) and (8.15) that depend implicitly on the non-dimensional Lagrangian
multiplier λ. This constant is determined by simple bisection.
c
56 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Chapter 9
The ultimate optimal material
9.1
The individual constitutive parameters
In ultimate optimal material design, also named free material design, we represent
the material properties in the most general form possible for an elastic continuum,
namely the unrestricted set of components in positive semi-definite constitutive matrices.
For a given material (given constitutive relations), we normally measure cost by
the amount of material, say by thickness or density. With the free material we need
a measure of the ”amount of a matrix”, and cost is then measured on the basis of
invariants of these matrices.
With reference to the paper by (Bendsøe et al. 1994), we extend the results obtained in that paper to be valid also for power law non-linear elasticity, as done in
(Pedersen 1998b). If we choose as cost constraint the Frobenius norm (length of a
matrix) of the constitutive matrix, then the analytical proof of the optimal constitutive
matrix, even for 3D-problems, is rather direct.
9.2
Free
material
Frobenius
norm
Sensitivity analysis
With localized sensitivity analysis as shown in chapter 4, and also given specifically
by (4.10), we have
du
1 ∂u
1 ∂u
(9.1)
=−
= −Ve
dh
p ∂h fixed strains
p ∂he fixed strains
where Ve is the volume of the domain of the localized design variable h e , (here a
component of the constitutive matrix). Thus minimum total strain energy u implies
maximum strain energy density u in a fixed strain field. (In domains of non-constant
strain energy density, the notion of mean value ū should be used). The strain energy density depends homogeneously on the squared effective strain e , see (5.15) in
chapter 5. The problem formulation can therefore be stated as
Maximize 2e := {}T [α]{} subject to Frobenius([α]) = F([α]) = 1
(9.2)
where the matrix [α] describes the non-dimensional part of the constitutive matrix in
the secant formulation (??).
In the invariant formulation for [α] we can choose the coordinate system of principal strains
{}T = {{1 2 3 } {0 0 0}}
(9.3)
57
Localized
determined
sensitivities
Design
problem
c
58 Pauli
Pedersen: Stationarity and extremum principles in mechanics
Principal
strains only
Direct
conclusions
and obtain


α1111 α1122 α1133  1 
2
2e = {}T [α]{} = {1 2 3 }  α1122 α2222 α2233 


α1133 α2233 α3333
3

(9.4)
Now, the Frobenius norm of a matrix is defined as the square root of the sum
of the squares of all the elements of the matrix (equal to the squared length of the
contracted vector). It thus follows directly that for optimality, the matrix elements
not involved in (9.4) must be zero. This means directly that also for the non-linear,
power law materials we have:
• the optimal material is orthotropic
• principal directions of material, strain and stress are aligned
• there is no shear stiffness
This result for linear elastic material is proved in (Bendsøe et al. 1994), based
also on a constraint on the trace of the constitutive matrix. Here, the extension to
non-linear elastic material follows directly from the localized sensitivity result (9.1).
For simplicity of proof we have chosen the Frobenius norm as the constraint.
9.3
Positive
definite
Final optimization
The further analysis relates only to the sub-matrix in (9.4). To fulfill the condition of
being positive definite, we have as necessary conditions
α1111 > 0, α2222 > 0, α3333 > 0
2
2
2
α1111 α2222 > α1122
, α1111 α3333 > α1133
, α2222 α3333 > α2233
(9.5)
The problem formulation (9.1) can now be written as
Maximize 2e = α1111 21 + α2222 22 + α3333 23 +
2α1122 1 2 + 2α1133 1 3 + 2α2233 2 3
(9.6)
New design
problem
constrained by (9.5) and by given Frobenius norm F
Optimality
condition
The general necessary condition for optimality is proportional gradients (see
(7.4) in chapter 7), i.e. for this specific case
2
2
2
2
2
2
F 2 − 1 = α1111
+ α2222
+ α3333
+ 2α1122
+ 2α1133
+ 2α2233
−1=0
d(2e )/dαiijj = λd(F 2 )/dαiijj
(9.7)
(9.8)
with the same λ for all αiijj . This gives the result
21
Optimal
modulus
matrix
22
23
1 2
1 3
2 3
=
=
(9.9)
α1111
α2222
α3333
α1122
α1133
α2233
and we can finally write the resulting constitutive matrix in the directions of principal
strains/stresses (evaluating λ to satisfy (9.7):
 2

1 1 2 1 3 0 0 0
 1 2 22 2 3 0 0 0 


 1 3 2 3 23 0 0 0 
1


[α]optimal =
(9.10)

0
0
0
0
0
0
(1 + 2 + 3 )2 


 0
0
0
0 0 0 
0
0
0
0 0 0
=
=
=
Ultimate optimal material 59
9.4
Numerical aspects and comparison
with isotropic material
The result (9.10) is valid also for power law, non-linear elastic materials. We note that
the matrix in (9.10) has only one non-zero eigenvalue and that the material therefore
only has stiffness in relation to the specified strain condition. For the ultimate optimal
material, the effective strain e , the strain energy density u , and the Frobenius norm
F are
2e = 21 + 22 + 23
1
u = E
(2 + 22 + 23 )(p+1)/2
p+1 1
F = 1
(9.11)
We can obtain the same effective strain and strain energy density with an isotropic,
zero Poisson’s ratio material [α] = [I], but then the corresponding Frobenius norm is
F = 6, i.e. the material cost is six times greater. As shown in (Bendsøe et al. 1994),
the zero Poisson’s ratio material may be valuable in numerical calculation, because
of the degeneracy of the ultimate optimal material.
9.5
Summing up
In this chapter the important results to focus on are:
• The ultimate optimal material is very degenerate and is only stable in relation
to the specific strain state for which it is designed.
• The obtained solution is also valid for power law non-linear elastic materials,
and simple arguments lead to the obtained analytical solution.
• The direct comparison with isotropic, zero Poisson’s ratio material is most
interesting, and can be used for obtaining numerical solutions to specific problems.
Collected
results
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60 Pauli
Pedersen: Stationarity and extremum principles in mechanics
References
Bendsøe, M. P., Guedes, J. M., Haber, R. B., Pedersen, P. and Taylor, J. E. (1994), ‘An analytical model to predict optimal material properties in the context of optimal structural
design’, J. Applied Mechanics 61, 930–937.
Dems, K. and Mroz, Z. (1978), ‘Multiparameter structural shape optimization by finite element method’, Int. J. Numer. Meth. Engng. 13, 247–263.
Huang, N. C. (1968), ‘Optimal design of elastic structures for maximum stiffness’, Int. J.
Solids Structures 4, 689–700.
Langhaar, H. L. (1962), Energy methods in applied mechanics, John Wiley and Sons, Inc.
Masur, E. F. (1970), ‘Optimum stiffness and strength of elastic structures’, J. Eng. Mech.
Div., ASCE EM5, 621–649.
Olhoff, N. and Niordson, F. I. (1979), ‘Some problems concerning singularities of optimal
beams and columns’, Z A M M 59, T16–T26.
Olhoff, N. and Taylor, J. E. (1983), ‘On structural optimization’, Journal of Applied Mechanics 50, 1139–1151. 58 references.
Pedersen, N. L. (2004), ‘Optimization of holes in plates for control of eigenfrequencies’,
Struct. Multidisc. Optim. 28(1), 1–10.
Pedersen, P. (1998a), Elasticity - Anisotropy - Laminates with Matrix Formulation, Finite
Element and Index to Matrices, Solid Mechanics, DTU, Kgs. Lyngby, Denmark. 320
pages - also available at http://www.fam.web.mek.dtu.dk/html/pp.html.
Pedersen, P. (1998b), ‘Some general optimal design results using anisotropic power law nonlinear elasticity’, Structural Optimization 15, 73–80.
Pedersen, P. (2003), ‘A note on design of fiber-nets for maximum stiffness’, J. of Elasticity
73, 127–145.
Pedersen, P. (2005a), ‘Analytical stiffness matrices with Green-Lagrange strain measure’, Int.
J. Numer. Meth. Engng. 62, 334–352.
Pedersen, P. (2005b), ‘Axisymmetric analytical stiffness matrices with Green-Lagrange
strains’, Computational Mechanics 35, 227–235.
Pedersen, P. (2006), ‘Analytical stiffness matrices for tetrahedral elements’, Computer Methods in Applied Mechanics and Engineering 196, 261–278.
Pedersen, P. and Pedersen, N. L. (2008), ‘Analytical optimal designs for long and short statically determinate beam structures’, Struct. Multidisc. Optim. pp. 1–15. on line.
Pedersen, P. and Taylor, J. E. (1993), Optimal design based on power-law non-linear elasticity, in P. Pedersen, ed., ‘Optimal Design with Advanced Materials,’, Elsevier, pp. 51–66.
Rozvany, G. N. I. (1989), Structural Design via Optimality Criteria, Kluwer, Dordrecht, The
Netherlands.
Save, M., Prager, W. and Sacchi, G. (1985), Structural optimization, optimality criteria,
Vol. 1, Plenum Press.
61
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62 Pauli
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Sundstrøm, B., ed. (1998), Handbok och Formelsamling i Hållfasthetslaera (in Swedish),
KTH, Stockholm. 398 pages.
Washizu, K. (1975), Variational methods in elasticity and plasticity, Pergamon Press, Ltd.
Wasiutynski, Z. (1960), ‘On the congruency of the forming according to the minimum potential energy with that according to equal strength’, Bull. de l’Academie Polonaise des
Sciences, Serie des Sciences Techniques 8(6), 259–268.
Index
admissible
displacement field, 4
stress field, 4
Also best strength, 35
Analytical steps, 43
Assumed power law elasticity, 17
Assumptions, 11
Available functions, 40
Basic assumption, 37
beam cross sections
rectangular, 53
solid circular, 53
solid equal sided triangular, 53
solid square, 53
thin walled, 53
Beam length, 44
Beam solutions by stress(complementary) principles, 30
Bending energy only, 29
Bending moment, 28
Bernoulli-Euler cantilever beams, 48
Bernoulli-Euler cantilever designs, 48
Bernoulli-Euler design formula, 47
cantilever, slender beam, 30
Castigliano’s 1st theorem, 7
Castigliano’s 2nd theorem, 9
Cauchy strains, 4
Clapeyron’s theorem, 6
Collected results, 41, 59
complementary energy density, 26
Complementary work of external loads, 5
complementary(stress) virtual work, 8
compliance decrease, 44
Compliance from elastic energy, 5
Compliance from potential of external loads, 5
compliance minimization, 43
Compliance objective, 44
Compliance of external loads, 5
Compliance per unit length, 45
Conditions with a simple shape parametrization, 38
Constant energy density, 36
constitutive
individual parameter, 57
matrix
non-dimensional, 57
positive semi-definite matrix, 57
Constitutive secant modulus, 24
Constitutive tangent modulus, 25
Converted to non-constrained, 34
corresponding displacement, 11
cross sectional reference length, 45
Cross sectional size, 44
cross-sectional constants, 27
cross-sectional forces/moments, 27
curvature, 11
dead load, 6, 13
Decoupled energies, 28
Derivatives of elastic potentials, 18
Design for stiffness and strength, 38
Design independent loads, 18, 36
Design optimization, 33
Design problem, 39, 57
Detail of proof, 37
differential strain energy density, 24
Direct conclusions, 58
direction
principal strain/stress, 58
displacement control, 7
displacement field, 4
distribution of shear stresses, 45
domain
non-shape, 37
shape, 37
effective
strain, 24, 57
stress, 24
Effective strain/stress, 24
Elastic energy in a straight beam, 27
elastic strain energy, 17
elastic stress energy, 17
elasticity
power law non-linear, 36, 57
Elementary case, 30
Energy densities in 1D non-linear elasticity, 22
Energy densities in 2D and 3D non-linear elasticity, 24
Energy equilibrium, 17
engineering strains, 4
Equilibrium, 17
equilibrium simplified, 17
Estimated multiplier, 41
Examples of beam cross sections, 54
explicit analytical solution, 43
External potential, 17
External potential and compliance, 6
extrema relations, 18
extremum principle
for total complementary(stress)potential energy, 14
for total potential energy, 13
Final optimization, 58
finite element, 40
fixed stress field, 37
For model by the FEM, 40
force equilibrium, 4
Four energy solutions, 30
Free material, 57
Frobenius norm, 57
gamma-function, 39
General knowledge, 35
General optimization, 33
General stress/strain state, 13
63
c
64 Pauli
Pedersen: Stationarity and extremum principles in mechanics
geometrical constraint, 38
given kinematic conditions, 7
Goal of the chapter, 3
Good experience, 38
gradient of the elastic energy, 44
Frobenius, 58
Normal force, 27
Not well known but important, 17
Numerical aspects and comparison with isotropic material, 59
Handbook formulas, 27
heuristic approach
successive iteration, 40
homogeneous
energy relation, 34
mass relation, 34
Homogeneous mass(volume) dependence, 34
optimal design formulation, 44
optimal designs, 48
optimal material
aligned, 58
degenerate, 59
no shear stiffness, 58
non-zero eigenvalue, 59
orthotropic, 58
Optimal modulus matrix, 58
Optimality condition, 40, 58
optimality condition, 34
Optimality criterion, 45
optimality criterion, 46
Optimality criterion for beam design, 44
Optimally obtained compliance, 49, 53
optimize
material orientation, 33
stiffness, 33
strength, 33
outward normal, 3
Overview of principles and their relations, 14
identity, 4
Inequality for mixed product, 12
influence of beam length, 44
Internal potentials, 17
invariant
matrix, 57
matrix length, 57
kinematically admissible, 4
Lagrange
function, 34
multiplier, 34, 40
linear elastic material, 6
linear elasticity and dead loads, 6
Linear strain notation, 21
linear strains, 4
Local design parameter, 18
Localized determined sensitivities, 57
Localized energy change, 36
Localized volume change, 35
material design
free material, 57
ultimate design, 57
material parameters, 27
matrix invariant, 57
maximize
eigenfrequency, 43
stiffness, 43
maximum strain energy density, 57
minimize compliance, 44
minimum
compliance, 37
maximum strain energy density, 37
total strain energy, 57
Minimum complementary (stress) potential, 14
Minimum potential, 13
mixed products, 12
Model for moment of inertia, 45
moment distribution, 45
moment equilibrium, 4
Motivation for extremum, 11
multidimensional stress/strain state, 13
necessary condition, 45
a single constraint, 34
non-constrained, 33
positive definite, 58
proportional gradient, 58
proportionality, 34
necessary optimality criterion, 43
Neglected shear energy, 29
New design problem, 58
No physical interpretation, 3
Non-constrained problems, 33
norm
parametrization
boundary shape, 38
super-ellipse, 38
Poisson’s ratio, 45
zero, 59
Positive definite, 58
Potential relations, 18
Power law elasticity, 22
Principal strains only, 58
principle
of minimum total potential energy, 11
overview, 14
Principle of minimum total complementary(stress) potential energy, 14
Principle of minimum total potential energy, 11
principle of virtual displacements, 7
principle of virtual stresses, 8
Problems with a single constraint, 34
Proportional gradients, 34
Proportional relation, 24
psi-function, 40
ratio of the compliances, 49
real
displacement field, 6
load field, 6
strain field, 6
stress field, 6
Real fields, 4
Real stress field and real displacement field, 6
Real stress field and virtual displacement field, 7
Relations with power law elasticity, 18
relative decrease in compliance, 52
Relative hole area or density, 39
Remark, 18
Results for simple (Bernoulli-Euler) beams, 29
Secant and tangent modulus, 23
secant formulation, 57
Sensitivity analysis, 57
sensitivity analysis
localized, 57
shape optimization
References 65
stiffness, 36
Shape optimization for stiffness and strength, 35
shear force distribution, 45
shear modulus, 45
shear stresses, 28
Simply supported static distributions, 47
Single force behaviour, 11
single load at the free end, 50
singularities of optimal beams, 50
six-dimensional strain/stress spaces, 13
Size and shape, 33
size optimization
stiffness, 34
strength, 35
Size optimization for stiffness and strength, 34
specialized principles, 7
statically admissible, 4
Statically determinate beams, 43
statically determinate cases, 45
Stationary objective, 33
stationary total complementary potential, 19
stationary total potential, 18
Stationary total potential energy, 8
Stationary total stress potential energy, 9
Stiffest design, 35
stiffest design
shape, 36
size, 35
strain
effective, 57
principal, 57
Strain and stress energy densities, 21
strain by differentiation, 21
Strain energy density, 5, 21
strain field, 4
strength
von Mises, 35
Strength or stiffness, 40
stress by differentiation, 21
stress energy
complementary formulation, 37
Stress energy density, 5, 21
stress field, 4
Stress virtual work principle, 8
strongest design
shape, 38
size, 35
successive iteration
heuristic approach, 40
Summing up
chapter 8, 54
chapter 6, 31
chapter 5, 26
chapter 3, 16
chapter 7, 41
chapter 4, 19
chapter 2, 9
chapter 9, 59
super-elliptic shape, 38
surface area, 7
surface traction’s, 4, 17
Symbols and definitions, 4
tangent modulus, 23
Tensor notation, 3
The identity, 4
The individual constitutive parameters, 57
The work equation, 4
The work equation, an identity, 3
theorem
stiffest shape design, 37
strongest shape design, 38
theorem of divergence, 3
Timoshenko beam theory, 43
Timoshenko cantilever beams, 49
Timoshenko cantilever designs, 51
Timoshenko design formula for n = 1, 47
Timoshenko design formula for n = 2, 47
Torsional moment, 28
total external force, 45
Total potential complementary energy, 5
Total potential energy, 5
Total potentials, 18
Total strain energy, 5
Total stress energy, 5
Transverse force, 28
triangular load distribution, 51
Two or only one parameter, 38
Two-sided bounds, 14
Uniaxial constitutive model, 13
uniform load, 50
uniform strain energy density, 35
unit displacement field, 8
Unit displacement theorem for linear elasticity, 8
Unit load theorem for linear elasticity, 9
Use of optimal design results, 44
Using the theorem of divergence, 3
variational strain, 21
variational strain energy density, 21
variational stress energy density, 21
Virtual fields, 4
Virtual stress field and real displacement field, 8
Virtual work principle, 7
volume forces, 4
work equation, 4
Work function, 12
Work of external loads, 5
Young’s modulus, 45
Zero sum of total potentials, 6