A recent study by Rozvany and Sokól discussed an important topic in structural design: the allowance for support costs in the optimization process. This paper examines a frequently used kind of support —that of simple foundation with horizontal reaction by friction— that appears no covered for the Authors’ approach. A simple example is examined to illustrate the case and to apply the Authors’ method and the standard design method

Vázquez Espí, Mariano

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On the allowance for support costs in Prager-Rozvany’ optimal

layout theory

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Structural and Multidisciplinary Optimization manuscript No.
(will be inserted by the editor)
On the allowance for support costs in Prager-Rozvany’ optimal
layout theory
Mariano Va´zquez Esp´ı
Received: date / Accepted: date
Abstract A recent study by Rozvany and Soko´ l dis-
cussed an important topic in structural design: the al-
lowance for support costs in the optimization process.
This paper examines a frequently used kind of support
—that of simple foundation with horizontal reaction by
friction— that appears no covered for the Authors’ ap-
proach. A simple example is examined to illustrate the
case and to try of applying the Authors’ method; also
some solutions from standard standard design method
are included.
Keywords Support cost · Michell trusses · fixed
boundary problems · free loading problems
1 Introduction
The fixed boundary class of problems was exactly de-
fined by Cox (1965:116-117), as a different class of prob-
lems than those covered by Maxwell & Michell design
theory. In his remarkable book, Cox shown that al-
though the theories to tackle with these two classes have
different optimality criteria, both they lead to optimal
layouts on the basis of Hencky-Prandtl nets (1965:96).
The optimality criteria for the former class was after
formulated by Hemp (1973) in detailed form and by
many other authors.
The fixed boundary theory has a well-known draw-
back: “the reactions such as those at [fixed supports],
are in any case carried by some other bodies acting as
structures and the true picture of the economy achieved
should include the abutments.” (Owen, 1965:64). A con-
spicuous extension of this theory for the allowance of
M. Va´zquez Esp´ı E-mail: Mariano.Vazquez.Espi@UPM.es
Universidad Polite´cnica de Madrid. Research Group on Ar-
chitecture, Urban Planning and Sustainability. Avda. Juan
de Herrera, 4. 28040 Madrid, Espan˜a.
support cost has been derived by Rozvany and Soko´ l
(2012:§3–7) from the optimal layout theory of Prager
and Rozvany (1977), “which is based on ‘optimal plas-
tic design’” of Prager and Shield (1967). It could hope
that with this extension the shortcoming noted by Cox
and Owen could be filled.
2 Foundation with friction
The frequently used foundation with friction (see e.g.
Bow, 1873) is composed by a prismatic body with square
base A = a×a and height h. For an allowable soil stress
σS , A = |Y |/σS , being Y the vertical reaction to be ex-
erted to the structure by the foundation (see Fig. 1a).
If the static friction coefficient between foundation and
soil is µ, the foundation can bear too an horizontal re-
action X such that |X | ≤ −µY providing Y ≤ 0 ac-
cordingly with the sign convention of Fig. 1 of Rozvany
and Soko´ l (2012), i.e., the soil under the foundation will
be compressed under Y -action. For a given foundation
material, the total volume of the foundation can be ex-
pressed as λa3, being λ a given constant dependent of
foundation material and soil properties. Let us select
the foundation volume as the cost following Authors’,
then the function costs can be expressed as follows:
C(Y ) =

λ
(
−Y
σS
) 3
2
if Y ≤ 0
∞ if Y > 0
(1)
C(X) =
{
0 if Y ≤ 0 and |X | ≤ −µY
∞ if Y > 0 or |X | > −µY
(2)
2 Mariano Va´zquez Esp´ı
The infinity values mean that the corresponding value
of the reaction can not be borne by the foundation.
Accordingly with Authors’ notation, we have:
∀X,Y : R(X,Y ) ∈
{
λ
(
−Y
σS
) 3
2
,∞
}
(3)
3 The Authors’ method
For the allowance of support cost, the Authors propose
to add to the optimality criterion on ‘adjoint’ strains,
new conditions on ‘adjoint’ displacements at supports
points. These displacements are given by generalised
gradients of reaction cost functions, i.e., ∂R/∂X and
∂R/∂Y—replaced by subgradients when appropriated.
From (3), it is clear that R is a “homogeneous”
function of order 3
2
(Rozvany, 1976:40). But from its
“constituents”, (1) and (2), it is clear also that the
cost function in this case has not a well defined gradi-
ent in the X,Y -space, neither fulfils the assumptions of
the Prager-Shield optimality criterion (1976:40–48).We
have of course well-defined derivatives for some of the
constituents, but R(X,Y ) = constant does not define a
closed, continuous surface, neither R(X,Y ) ≤ constant
does a convex domain—in ‘plastic’ jargon, perhaps it
can be said that we have no “flow rule” in this case.
Hence, the case can be tackled neither with Authors’
Eqs. (3) and (4) nor with the rules for the special cases
‘with non-separable variables’ or ‘with slope disconti-
nuities’ considered by the Authors. Therefore, it is not
covered by the Authors’ approach in the Writer’s view.
Of course, this fact does not point out any mistake
in Authors’, only a key shortcoming in the underlying
theories—of equal nature that the drawback noted by
Cox and Owen.
Against this conclusion, it can be argued that I have
not taken into account the self-weight of the foundation
itself and that if I would have done so the function R
would have been continuous in half of its domain at
least. I do not follow this approach for two main reasons:
(i) I want to show a simple example without technical
complexity; and (ii) to take into account the self-weight
of the foundation would require to make the same with
the structure itself, coming us very far of the realm of
the Authors’ paper.
Perhaps the Authors can enlightened the Writer with
some other rule that the latter is not aware of.
4 Standard design method
With the standard design method (see e.g. Cox, 1965;
Owen, 1965: or Cervera Bravo (2008) for a contempo-
rary writting), each value of X = −νY with |ν| ≤ µ
X
Y ≤ 0
A
−X
Y ≤ 0
B
P
C
L
L/2
D
E G
F
(a) The solution domain is the half plane over AB line.
The given load is P . The foundations are under A y B.
(b) Solution for µ = 0:
V = 1.435PL
(c) Solution for µ = 0.25:
V = 1.310PL
(d) Solution for µ = 0.50:
V = 1.190PL
(e) Solution for µ = 0.75:
V = 1.084PL
With µ = 1, minV = PL for the layout ACB.
Fig. 1 An illustrative example (Cox, 1965:127).
and Y ≤ 0 leads to a given Maxwell’s problem without
any kinematic support conditions (this is the Michell’s
approach: the “free loading” defined by Cox). All these
problems form a family —dependent on ν— that covers
all possible solutions for the given design problem. The
designer’s target is to find the optimal solution within
this family.
For the sake of brevity, let us consider an illustrative
example, see Fig. 1a. The vertical reactions are stati-
cally determined as |Y | = P/2. The horizontal ones are
restricted to maximum friction force, i.e., to |ν| ≤ µ for
the given friction coefficient µ.
By elementary calculations, the minimal internal
force in E with direction AB will be a traction P/2
without horizontal reactions, hence the designer would
wish X = P/2 that corresponds to µ = 1 —and to the
optimal solution ACB. Although this value is not at-
tainable with normal soil conditions, this fact leads to
the conclusion that we must select |ν| = µ when µ < 1,
i.e., to design accounting for all the friction force at our
disposal as X is free-cost anyway. To fix |ν| is the same
to select one problem from the family.
On the allowance for support costs in Prager-Rozvany’ optimal layout theory 3
Once we have one Maxwell’s problem, we can search
for the optimal layout that minimise the Michell’s func-
tional, the “quantity” of structure V (1904:Eq. (3)). We
can compute as an alternative the volume of the struc-
ture —to remain within Authors’ realm—with the well-
known formula (Cox, 1965:87, Eq. (121); Owen, 1965:53,
Eq. (18); Barnett, 1966:20, Eq. (5)):
V =
1
2
{(
1
P
+
1
Q
)
· V +
(
1
P
−
1
Q
)
· C
}
(4)
where —following Michell’s notation— P,Q are the al-
lowable stresses in tension and compression, respec-
tively; and C is the “static constant” of Owen (1965)—
see Michell (1904:Eq. (1)): C = − 1
2
PL(1 + µ) in this
example. Notice that in doing so we do not account any
cost for vertical and horizontal reactions because these
reactions are invariable—their cost are constant and fi-
nite for all the solutions in the search space. Of course,
once a solution is selected as the appropriated one, the
designer adds to the cost of the structure the cost of
the foundations.
It should be noted too that the optimal layout will
be independent of P,Q for any given µ—but, never-
theless, volume varies as indicated by (4). This fact is
important and its proof is simple. Let us consider the
variation of V within the feasible solution space:
δV =
1
2
(
1
P
+
1
Q
)
· δV (5)
as δC = 0. Hence ∀P,Q : δV = 0 ⇔ δV = 0 and any
optimal layout for V will be optimal too for V for any
P,Q couple.
The non-optimal solutions in Fig. 1 were obtained
by a simulated annealing code (Va´zquez Esp´ı, 1995). All
they suggest the existence of Michell solutions that fulfil
his second theorem. These solutions seem to correspond
to a fan CDE (a T-region following Authors’), and a
region EDB, see Fig. 1a. The latter will be composed
of a T-region (EDFG) and a R-region (GFB). With
µ = 0, GFB area leads to zero; on the other hand CDE
and EDFG areas will be zero with µ = 1. With this
hypothesis, the internal force in the arch CDFB will be
constant and equal to NCDFB = P
√
1 + µ2/2 and the
angle of the arch in B with the vertical direction will
be exactly arctanµ. The angle of the fan can then be
obtained by elementary equilibrium calculations, and
the frontier GF would be obtained by the minimum
condition on V . The dashed-lines in Fig. 1 represents
the first part of these calculus.
It should be stressed that when a given optimal de-
sign —completely defined— will be analysed with any
standard, suitable code, the Michell virtual displace-
ment field will not be obtained, because the difference
between this field and the actual field of the designed
structure: in the latter the horizontal displacement of
foundations is non-linear in respect to its contact with
the ground—including a perfect rigid one—, the yield
tension strain can be different than that of the com-
pression —e.g., because difference in Young’s Modulus
yet when allowable stresses have equal value—, or the
self-weight of both structure and foundations have to be
accounted, etc. This fact is a consequence of the static
approach of Maxwell and Michell (“free loading” with-
out any kinematic support conditions): the displace-
ment fields used in the theory are virtual ones ever,
and that of the Michell’s theorems is only required to
be of bounded absolute strain and continuous.
This point is a key one to formulate any design the-
ory, and it is worth of further analysis. The design prob-
lem with an unique load conditions is defined by given
useful loads and some planes for support the structure
(e.g., the soil plane for a standard building). Accord-
ingly with Hemp (1958:1), “The theory [of structural
design] ought to be in a position to tackle the design
problem directly, that is, to begin with the given forces
and to produce by calculation the best structure that
will safely carry them”. The first pass in the designing
process is to look for some appropriate shapes. As at
this very time the designer has not selected any struc-
tural material, the theory must be formulated without
any constitutive equations. Moreover, the designers nei-
ther has any structure to be analysed, hence it makes
no sense to use structural analysis models as rolled or
pinned supports. The outstanding merit of Maxwell in
1870 was to formulate his design theory only with equi-
librium equations so it could be useful for the designer.
And the equal remarkable improvement of Michell was
to use the compatibility equations only to derive his
optimality criterion on optimal solutions for Maxwell’s
problems, being respectful with Maxwell’s fundamental
axioms.
5 Conclusion
The case of simple foundations with friction has solu-
tion within the standard design theory of Maxwell &
Michell—the “free loading” of Cox—, but it is not ob-
vious that it can be tackled with the method proposed
by the Authors.
The Writer will be grateful to the Authors for fur-
ther comments and criticism.
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