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Structural Optimization 12, 244-250 (~) Springer-Verlag 1996
Some
shortcomings
in Michell's
truss
theory
G.I.N.R.ozvany
FB 10, Essen University, Postfach 103764, D-45117 Essen, Germany
Abstract
Michell (1904) derived his well-known optimality
criteria for trusses in the context of unequal permissible stresses in
tension and compression. For the same design constraints, Hemp
(1973) and the author (e.g. 1989) have obtained optimality conditions which are different from those of Michell, and also lead,
in general, to different, and lighter trusses. The reasons for this
contradiction are examined and it is found that for unequal permissible stresses Michell's optimality conditions are only valid for
a highly restricted class of support conditions.
compression members a constant strain value (say Ige ] = k),
such that the sign of the strain (:e) and that of the member
force (re) is the same (sgn :e = sgn re). In all other directions the absolute value of the strain must not exceed the
above value (I:] <_ k).
For any statically determinate truss with stress constraints, the volume can be calculated from the "primal" formula
v =
1
Michell's (1904) milestone contribution to truss layout optimization is undoubtedly both revolutionary and ingenious:
first, it introduced all essential elements of what we now call
layout theory, continuum-type optimality criteria, structural
universe and adjoint strain fields; second, he achieved all this
around the turn of the century, when almost nothing was
known about the essential techniques of structural optimization. For this reason, Michell had to rely on greatly imaginative ideas for obtaining general conditions of optimality.
The other leading researcher of this field is Hemp, who
has produced the only systematic, extensive and rigorous
treatment of least-weight trusses for stress constraints (Hemp
1973). Hemp has derived somewhat different optimality criteria from those of Michell. The same modified criteria have
been obtained by the author by making use of the PraterShield (1967) theory of optimal plastic design or the KuhnTucker conditions.
It is certainly not the author's aim to belittle Michell's immense effort; the more so, since the latter is not in a position
to answer these comments.
The main purpose of this paper is, therefore, to provide
a constructive explanation of the apparent discrepancy between Hemp's and Miehell's criteria and to delimit the range
of validity of the latter. This study was prompted by the author's examination of the early literature for a forthcoming
book on topology optimization.
2 A c o m p a r i s o n o f o p t i m a l i t y criteria by Michell
and Hemp
We consider trusses for which the permissible stresses ~O, a+
for compression and tension are different. This means that
for any truss element e (e = 1 , . . . , E), the stresses are constrained by the inequalities
:+.
+-
fe>O
Introduction
-o-~" < o-e <
feLe/
IeLe/,'
,
fe<o
where L e are the member lengths and compressive forces are
given a negative sign. Since Michell trusses can always be
made statically determinate (Sved 1954), the volume equation (2) is valid for both plastic design and elastic design.
For solutions based oi: Michell's criteria, the volume can also
be calculated from the "dual formula", which consists of the
sum of the products of the external loads PJ (j = 1 , . . . , J)
and the corresponding "comparison" displacements gJ, multiplied by ( 1 / 2 ) ( 1 / c ~+ + 1 / ~ o ) if we take k = 1. Examples
of these volume equations are given by Michell (1904, pp.
594-596).
In the optimality criteria of Hemp (1973) or the author
(e.g. Rozvany 1989; Rozvany and Gollub 1990), the "adjoint"
strains along tensile members must be g = 1/~0+, along compressive members g = - l / n o , and along other elements of
the space any convex combination of these two.
The dual formula for the optimal volume (V) is then simply the sum of products, of loads PJ and the corresponding
adjoint displacements K3.
The above optimality criteria and dual formulae are compared in Table 1.
Table I. A comparison of optimality ciiteria and dual volume
equations
fe>0
fe < 0
Michell (1904)
N=k
~ : -k
fe=o
-k<-g<k
dual
(for k = 1)
Hemp (1973)
:=
:=
vo, umo
3
o
< : _<
~. PJ~J
J
IV
(i)
Considering this class of problems, Michell's (1904) "test
deformation" must have in the direction of both tensile and
As expected, the two sets of criteria may lead to different
truss volumes, both of which cannot be optimal.
245
3
Illustrative example
Consider a vertical point load at a distance "a" from an infinitely long vertical support (Fig. la). Let the permissible
stress in tension and compression, respectively, be ~0+ = 1
and ~r~- = 1. The "comparison" displacements (~, ~) based
on Michell's criteria (with k = 1) are in the x and y directions
=_ 0,
~ = 2z.
(3)
Then the principal strains become -fl,II = -4-1 at 4-45 ~ to the
vertical. For a proof, see e.g. the book by Rozvany (1989, pp.
324-326), which discusses a problem with equal permissible
stresses. In the vertical direction, we have : = 0, which
satisfies the kinematic boundary conditions. It follows then
from Michell's (1904) criteria that the two bar layout in Fig.
lb should be optimal. In this paper, compressive bars are
indicated in broken line. For the layout in Fig. lb, we have
the "primal" and "dual" volumes
V= ~
_ _v ~ ] -
+ x/2
(1/3) = 4
V = (2)(1)1 [1 + (1T3)] = 4,
(4)
4
A critical e x a m i n a t i o n o f Miehell's p r o o f
Michell's (1904) derivation of his optimality criteria consists
of two parts. The first part, which relies on a theorem by
Maxwell (1872), is only necessary for unequal permissible
stresses in tension and compression. If the latter are the
same then the second part by itself is a complete proof.
4.1 The first part of Michell's proof (proof of equivalence
of weight minimization for unequal and for equal permissible
stresses in tension and compression)
In the first part of his proof, Michell correctly formulates the
primal volume equation (2) herein. In addition, he quotes a
theorem of Maxwell (1872), which states
EfeL e = c,
which are in agreement.
(s)
e
\
where C is the same constant for any statically admissible
truss layout considering a given set of external loads. This
implies that if we find a solution minimizing a given linear
combination of C and V, then the same solution will minimize
V:
min(aV+be)~minV
(fora>0, b>0).
(9)
Since the LHS of (8) can be expressed as ~-~ef eLe =
~ f c > 0 f i Le + ~ f ~ < 0 f ene, by choosing a = 2a+cr O and
b = or0+- ~O, Michell obtained
L~=2
v'g/2
1
:i
= 2x/3(1) = 2v/-3 = 3.4641,
(7)
which are also in agreement, but significantly lower than the
results from Michell's criteria. Both solutions are definitely
feasible. Does this mean that Michell's theory is not generally
valid? To answer this question we must carefully examine
Michell's proof.
~
~A
La-=2/V~\ A
( a V + b C ) = 0r0+ + ~ O ) (Ekfe>0 f i L e - fe<0E f e l t )
c~
If elL e ,
=
(10)
e
~ X
Y
(a)
= 0
P = 2x
(b)
~ = -2x
P = 2v~x
(c)
where e -- (~0+ % n o ) is a given constant. Then (9) and (10)
imply that instead of minimizing V in (2), we can minimize
the function in (10).
In the special case of q0+ -- ~o -- ~0, we have the problem
Fig. 1. (a) A simple layout problem with the stress constraint
(-1/3) < a~ < 1; (b) solution based on Michell's criteria; (c) the
author's solution
minV=--c~01 e ~ ] f e [ / e ,
Using now Hemp's optimality criteria, we have the "adjoint" displacements (e.g. Rozvany and Gollub 1990)
which is a scaled version of (10) with c = 1/~ 0. This first
part of Michell's proof comes therefore to the surprising conclusion that instead of optimizing for two different permissi-
:
=
:
=
(5)
giving the adjoint principal strains
:i = i = i / % + ,
ji = -3 =
(6)
at 30 ~ and 60 ~ to the vertical. In the vertical direction, we
have again : = 0. The corresponding optimal truss is shown
in Fig. lc, in which compressive members are indicated by
broken line.
The "primM" and "dual" volumes then become
-
2 l+vf~2(i/3---) -
(11)
ble stresses in tension and compression, we can optimize the
truss as if we had the same permissible stress for both. Is this
possible in view of the example in Section 3?
4.2 The second part of Michell's proof (a sufficient condition of optimality for equal permissible stresses in tension and
compression)
We present here a slightly modified version of Michell's proof
by minimizing the function in (11) instead of that in the I%ttS
of (10). The aim of this modification is to show directly that
Michell's theory is valid for equal permissible stresses.
246
The layout which is to be shown of minimum weight is
denoted by Michell by "m" and some arbitrary layout by "a".
The external loads for both are the same: p3 (j = 1 .... , J).
Miehell then uses a virtual strain field ("test deformation",
which we denote by ~) with the property
_< 1/(r0,
(12)
where we have or0+ = aO = a0 as in (11). This virtual strain
field produces the virtual displacements ~J at the loads PJ.
For both layouts "m" and "a" the external virtual work is
)"~j PJsJ. Assume now that for the layout "m" along all bars
in tension (re > 0) the virtual strain is ~ = 1/~ 0 and along
those in compression (re < 0) it is ~ = -1/o" 0. This means
that the internal work for the layout "m" is ~ e lf,~lL~/o'O,
and then by the principle of virtual work we have
E Pj~j = E If~lLel~176
j
(13)
For the arbitrary layout "a" the principle of virtual work
and (12) imply
E Pj~j = E f~'~eLea < E ]feli~eiLe -< E [felLealcro"
j
e
~
e
(14)
Then by (13) and (14), we have
If
~
e
,lLm/o'o
<_
(15)
I f ~ ". l L , U o o ,
e
which shows that the layout satisfying Michell's optimality
criteria has indeed the smallest volume. Since we have not
shown that such a layout exists, this proof establishes a sufficient condition of optimality.
5
plane, for example, a pin and a roller would allow such
a displacement field, showing the identity of the statical
and kinematical conditions above.
Another kinematic reason for the severe restriction on
Michell's theory is the fact that for the same optimal layout
both the optimality criteria of general validity and Michell's
different criteria must produce a kinematically admissible displacement field. This is only possible with a very high degree
of kinematic freedom, i.e. very few supports. In the test example in Section 3 (Fig. 1), there exists an infinite number
of possible reactions along the vertical support and hence
Michell's criteria are not valid for that problem. However, in
the case of equal permissible stresses (~0+ = nO) , Michell's
criteria are valid for any support condition.
It will be seen from the next section that the modified optimality criteria introduced by Hemp and used by the author
are not subject to such restrictions.
6
P r o o f s of o p t i m a l i t y criteria o f g e n e r a l validity
Three proofs of the corrected optimality criteria for ~0+ # ~O
are given in this paper. The first one is based on the PraterShield (1967) criterion of optimal plastic design, which is a
necessary and sufficient condition. The second proof establishes these criteria in a finite dimensional design space and
is based on the Kuhn-Tucker conditions. The third proof is
tIemp's (1973) corrected version of Michell's proof for the
amended criteria as a sufficient condition.
6.1 Proof based on the Prater-Shield (1967) theory of optimal plastic design
In this theory, the total cost ~ is expressed as the integral of
a convex and piecewise smooth specific cost function r
E v a l u a t i o n o f Miehell's p r o o f
The second part of the proof corresponds to equal permissible
stresses in tension and compression (~r0+ = r
It is valid for
any number of rigid supports, because at these kinematically
admissible virtual displacements must be zero and hence they
do not influence the value of the external virtual work.
Michell's optimality criteria can only be extended to his
theory with different permissible stresses (%+ # a'~) by
virtue of the first part of the proof, which relies on Maxwell's
(1872) theorem.
The limits of validity of the latter can be explained as
follows.
9 From a statical point of view, Maxwell's theorem is restricted to a given set of external forces. This means that
the first part of Michell's proof is only valid for problems in
which the reactions do not depend on the layout. In other
words, the number of supports must be sufficiently small
to make all possible layouts externally statically determinate, i.e. all reaction components must be uniquely determined by the equilibrium conditions. For a plane truss,
for example, the supports may consist of a pin (two reaction components) and a roller (one component) and then
the three reaction components are uniquely given by the
three equilibrium equations.
9 From a kinematical point of view, Maxwell's theorem is
restricted to support conditions which allow a virtual displacement field consisting of uniform dilatation. In the
l
k
= L r
dO,
(16)
where f denotes the stress resultants for a cross-section, subject to equi]ibrium. In any smooth region, the adjoint (in
Prager's terminology "associated") strain components are
given by
~i = Or
(17)
and at slope discontinuities of r any convex combination of
the adjoint strain values for the adjacent smooth regions can
be taken.
The above theory is based on the minimum complementary energy theorem for the adjoint structure which, in turn,
is a necessary and sufficient condition. In our problem we
have
(forf>O)
r
~+,
(forf<O)
r
o ,
(t8)
and hence the Prager-Shield condition gives
1
(forfe >0)
-~e = g+'
(forfe<o)
~e__
(for
= o)
1
1 <
O.0
,
< !
- - 0 .0+
(19)
247
6.2 Proof of extended optimality criteria as a necessary condillon
This proof is based on the Kuhn-Tucker conditions and proof
of convexity. Similar proofs have been used by the author
since the early seventies (Charett and Rozvany 1972; Rozvany 1976) for various extensions of the Prager-Shield (1967)
theory. We can state our problem as
minZAeLe
,
(20)
f_~e _ A e _< 0,
(21)
subject to
pJuJ. Then for the layout "m" we have by the principle
3
of virtual work
+
j
(22)
A e < O,
:
0,
(231
n/
o =
AmL m = l/m,
(27)
since f e = A e o . + and f e = _ A e o . o , respectively, for f e >
O and f e < O.
For the arbitrary layout "a" the work equation and its
consequences are
j
=
e
<_ ~
-
,L
fe<o
=
fe
S
-
fe>o
AX
oLY
re<0
f ~ L [ / % = ~ ' A e L e = 'ca, (28)
f~L~/z + - ~
fe>o
-
fe>O
fe>o
e
due to the inequality (26). Then (27) and (28) imply
j
where A e is the cross-sectional area of the member e and
(~e, ~ j ) represent a kinematically admissible virtual displacement field. By the virtual displacement theorem, (23) ensures equilibrium of ( f e , p j ) . Using the Lagrange multipliers A+, )~- and a = 1 for the constraints (21)-(23), we have
the Kuhn-Tucker conditions for variations of f e and At:
L e - A+ - )~- = 0,
-geLe -~+
(24)
~ro '
implying (19).
The cross-sectional areas A e are convex (although nonsmooth) functions of fe:
(forfe>0)
A e - ~r~_
f e,~j
f e_ . (25)
r0
The sum of such functions in (20) is also convex in f t . If
we incorporate (25) into (20), the only remaining constraints
are those of equilibrium, which are linear. Hence the constrained problem remains convex.
This means that (19) represents a necessary and sufficient condition, provided that the solution exists. For a truss
with a finite number of members, existence can readily be
shown (e.g. Hemp 1973). Proof of existence requires a more
advanced mathematical treatment for an infinite number of
truss members but this aspect is rather theoretical from an
engineering point of view.
The internal and external forces need to be only statically admissible in this proof, and may include any number
of reactions.
(forfe<0)
A e-
6.3 Proof of extended optimality criteria as a sufficient condition (after Hemp 1973)
Hemp's proof is an amended version of the second part of
Michell's (1904) proof. It does not rely on Maxwell's (1872)
theorem. Again, we use a virtual displacement field, but with
the property
(-1/~rO) _< ~ < (1/c~+).
(26)
The layout "m" has the property that along tensile bars
= ( 1 / a + ) and along compressive bars ~ = ( - 1 / ~ O ) .
The external virtual work for both layouts "m" and "a" is
Vm < Va.
(29)
7 Implications of the modified optimality criteria
for known optimal solutions
It was shown in Section 3 that the correct optimality criteria for the problem considered there result in a significant
reduction in the truss weight. As explained in Section 5, the
correctness of a solution based on Michell's criteria depends
on the number of unknown reaction components. Looking
at examples in Michell's (1904) paper, his very first example, involving a circular support and a point load (Fig. 2a
herein), seems to be incorrect for a + r crO. This means that
his equation for the optimal volume (in our notation)
V = Pa
log
+
,
(30)
would need to be revised. In (30) "a" is the distance of the
vertical load P from the centre of the circular support and
r 0 is the radius of the latter. For (a - r0) << r 0 this problem
reduces to the one in Section 3 (Fig. 1) for which Michell's
criteria have been demonstrated to give the wrong solution.
In the considered problem the reactions are distributed
along the circular support and depend on the layout. Hence
Michell's optimality criteria for ~r+ # c~O are not valid. In
actual fact, if we take again c~+ = 1, ~rO --- 1/3, then we
must have e I = 1, r = - 3 along the tensile and compressive bars. If these were at • ~ to the supporting line (as in
Michell's Fig. 1), then we would violate the kinematic boundary conditions. As in Section 3, the bars must enclose 300
and 60 ~ with the circular support (and with any circle having
the same centre) for optimality. This agrees with the correct
solution in Fig. lc which is a limiting case for the present
problem for (a - r 0 ) / r 0 --+ 0.
On the other hand, the second example of Michell (1904)
has only two vertical reactions (Fig. 2b) and hence Maxwell's
(1872) theorem holds for that problem, ensuring the true
optimality of this solution. Indeed, using e I = 1, e l I =
- 3 would not upset kinematic admissibility of this solution,
because the two supports allow enough kinematic freedom
248
"~..-'~J'~
~
7x, /
",, I
xx--
a
s//SS ~ s
number of reaction components is four, which is statically
indeterminate. In the case of a line support, if we have a
vertical load close to that support with a0+ # ~ro, then the
solution is definitely of the type in Fig. lc. The logical extension of this solution for say ~0+ = 1, c~O = 1/3 is shown
in Fig. 3b.
I
(a)
\
~%'~
T
tension bars
compression bars
(b)
Fig. 2. Examples for differing permissible stresses by Michell: (a)
invalid, nonoptimal solution; (b) valid, optimal solution
(a)
(b)
~
Fig. 4. Another solution by Michell which is invalid for differing
permissible stresses
Michell's (1904) space-truss example (Fig. 4) would also
need to be modified in accordance with the amended optimality criteria: instead of having tensile and compressive
members of the same length at an equal angle, the compressive members would become shorter and the tensile members
longer for a O < %+.
8 Nonselfadjointness and singularity of the probl e m s c o n s i d e r e d in M i c h e l l ' s t h e o r y
As explained earlier, the correct real and adjoint strains (e, ~)
for the considered problem are as follows.
Real strains
( f o r f e > 0) ~e = Cr+o/E,
( f o r f e < 0) c ~ : - % / E ,
(31)
where E is Young's modulus.
Adjoint strains
(forf ~>0)
~e=l/- +,
(for/"<0)
(fory e = o)
(c)
Fig. 3. Optimal cantilever truss for equal permissible stresses (a)
and unequal permissible stresses (b)
to admit both the generally valid optimal adjoint strain field
and the one with restricted validity.
Another often quoted solution is shown in Figs. 3a and
b. This is often called the "Michell-cantilever" in numerical literature although it should be called the "Chan-ChanLewinski et al. cantilever" because various parts (Fig. 3b) of
it were derived by A.S.L. Chan (1960), H.S.Y. Chan (1963)
and Lewinski et al. (1994). It could also be called the "Hempcantilever", because the latter supervised the two Chans'
work and also reviewed their results in detail in his book
(Hemp 1973). In this problem, we have an infinite number
of potential reaction forces along the support A C (Fig. 3a).
Even if the supports are restricted to pins at A and C, the
- 1 1 % < ~ < 1/~ + 9
ge=_l/%,
(32)
In the case of ~r0+ = ~O' the real and adjoint strain fields
are proportional and then the problem is termed selfadjoint.
This has many advantages, and has been treated extensively
even in the recent literature (e.g. Bendseet al. 1994). However, it can be seen from (31) and (32) that for c~0+ r c~O
the two fields are nonproportional and hence the problem
becomes nonselfadjoint, which is still convex (see Section 6).
However, the problem can be solved satisfactorily both analytically and numerically by using the optimality criteria in
(22).
It can also be seen from (31) that no real strains are specified for f e = 0 (elements of space without a member, i.e.
along a vanishing member). This is quite correct because
once a member disappears (with f e = O, A e = 0), the stress
condition need not be observed. In fact, it was shown by Rozvany and Birker (1994) that for the optimal solution in Fig.
lc the real strain in certain directions is outside the range
249
- a o / E < c < a+o/E.
(33)
This means that the solution is singular in the sense that
certain stress conditions would need to be suddenly removed
during the computational procedure. This difficulty was originally pointed out by Sved and Ginos (1968) and discussed extensively by others (e.g. Kirsch 1990; Cheng and Jiang 1992;
Rozvany and Birker 1994). In the case of one load condition, which was considered by Michell (1904), the solution is
known to be statically determinate (Sved 1954). This means
that it is sufficient to consider equilibrium conditions only
and hence the problem of singularity can be avoided both in
analytical and in numerical (in this case linear programming
or optimality criteria) formulations.
Michell trusses, which was based in part on known properties of Hencky-Prandtl nets for slip lines in two-dimensional
perfectly plastic flow. His general equations for unequal permissible stresses also confirm the validity of solutions that
differ from those based on Michell's original theory for such
stresses, some given in Michell's (1904) paper.
For example, Hemp's (1973) equation (4.25) states the
kinematic boundary conditions for the adjoint strain field
along '% fixed line of support", which in our notation can
be stated as
= + 1 / ~ ,
Adc~ = - t - ~
9 R e l e v a n t a s p e c t s of H e m p ' s
Michell's truss t h e o r y
(1973) review of
The development of the correcled Michell criteria can be
traced through Hemp's (1973) book and his earlier research
reports (Hemp 1958, 1968), which reveal the following.
9 In his earliest research report on this topic, Hemp (1958)
discusses in detail Michell's proof but does not mention
that Michell's criteria are restricted to a narrow class of
support conditions.
9 In a later research report, Hemp (1968) develops his theory only for trusses with equal permissible stresses in tension and compression and thereby avoids the issue discussed in this paper.
9 In Hemp's (1973) book, the corrected optimality criteria
for different tensile and compressive permissible stresses
[see (19) herein] are derived. However, Hemp very generously attributes his own improved optimality criteria to
Michell ("Michell's sufficient conditions"), although they
are substantially different from, and of more general validity than those of Michell (1904).
9 For all his examples that are not valid for different permissible stresses, correctly only one permissible stress (a)
is mentioned in IIemp's volume equations.
9 For all his examples that are valid for different permissible stresses, the volume equations correctly contain both
( a c ) and (aT) , which are the two permissible stresses in
tIemp's notation.
9 No example is given by Hemp for unequal permissible
stresses, where the solution for those stresses would be
different from the solution for equal permissible stresses
(like our Fig. lc). These would demonstrate that Michel's
criteria are not always valid.
It appears, therefore, that Hemp (1973) had a deep understanding of the limits of validity of Michell's original criteria,
but in his wisdom and modesty, decided to correct Michell's
conditions without taking credit for this important contribution. He also avoided all results which would have indirectly
inferred that Michell's criteria are not quite correct for unequal permissible stresses. With this generous gesture, Hemp
clearly intended to keep Michell's well-deserved historical image untarnished. The main aim of this paper is to put the
historical record right in recognizing Hemp's contribution and
to prevent misapplications of Michell's criteria in the future.
Hemp (1973) also developed a very elegant unified theory in curvilinear coordinates for deriving the geometry of
(34)
Bdfl,
(35)
where w is the rotation along the supporting line and (35)
"gives the angle between ~- or ~-lines and the tangent to the
line of support" (see Fig. 5 herein).
line
support
/
~
O~
Fig. 5. Optimal member directions along a supporting line (Hemp
1973)
For a0+ -= 1, a O = 1/3 and A -- B, for example (34) and
(35) give
w = x/~,
03=30 ~ ,
(36)
0~=-60 ~
(37)
where the meaning of the angles 0a and 0/~ is explained in
Fig. 5. This confirms the solution in Fig. lc and the triangular region in Fig. 3c (for which both the (~- and ~-lines are
straight with A = B) and also the nonoptimality of Michell's
solution in Fig. 2a [for which we also have A = B (Hemp
1973, p. 88)1.
It is also interesting to observe that Prager (e.g. 1974)
only dealt with trusses having equal permissible stresses
(a0+ =
10
Conclusions
The following conclusions can be drawn from this paper.
(1) For different permissible stresses in tension and compression, which was the case treated by Michell (1904), his
optimality criteria are valid only for a highly restricted
class of support conditions. This means that the number
of potential supports are to be so few that all possible
reaction components can be uniquely determined from
equilibrium conditions.
250
(2) For plane trusses, for example, this means that the maximum number of potential reaction components is three
which, for example, is the case when the potential supports are restricted to one pin and one roller.
(3) For unequal permissible stresses Hemp (1973) derived corrected optimality criteria which were also obtained independently by the author.
(4) It was demonstrated on a simple example (vertical supporting line with a vertical load) that for the considered
class of problems tIemp's (1973) optimality criteria may
yield a different (and much lighter) structure than that
of Michell.
(5) Referring to Michell's (1904) examples, already the very
first of these does not seem to be correct for different
permissible stresses, although Michell's volume formula
specifically refers to this case. This implies that Michell
was not aware of the limits of validity of his theory.
(6) For equal permissible stresses in tension and compression,
Michell's (1904) criteria have a general validity.
(7) In recognition of deriving the correct optimality criteria,
least-weight trusses for a0+ r ~O shall be termed Hempstructures.
(8) A special class of Hemp-structures with or0+ = 0 was
treated extensively from the late seventies by Prager,
Rozvany and associates (e.g. Rozvany and Prager 1979).
forces. Cranfield College of Aeronautics Report No. 167
Acknowledgements
Rozvany, G.I.N. 1989: Structural design via optimality criteria.
Dordrecht: Kluwer
The author is indebted to the Deutsche Forschungsgemeinschaft
(Project Ro 744/6) and to the NATO Scientific and Environmental Affairs Division for financial support; to Anne Fischer (text
processing) and Elke Becker (drafting) for their help in preparing
the manuscript; and to a referee for some excellent suggestions.
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