Optimum structure to carry a uniform – Global Homework Experts

Struct Multidisc Optim (2010) 42:33–42
DOI 10.1007/s00158-009-0467-0
Optimum structure to carry a uniform load
between pinned supports
Wael Darwich · Matthew Gilbert · Andy Tyas
Received: 29 June 2009 / Revised: 16 October 2009 / Accepted: 26 November 2009 / Published online: 7 January 2010
c Springer-Verlag 2009
Abstract Since the time of Huygens in the 17th century
it has been believed that, if the weight of the structural
members themselves are negligible in comparison to the
applied load, the optimum structure to carry a uniformly distributed load between pinned supports will take the form of
a parabolic arch rib (or, equivalently, a suspended cable).
In this study, numerical layout optimization techniques are
used to demonstrate that when a standard material with
equal tension and compressive strength is involved, a simple
parabolic arch rib is not the true optimum structure. Instead,
a considerably more complex structural form, comprising
a central parabolic section and networks of truss bars in
the haunch regions, is found to possess a lower structural
Keywords Structural optimization · Layout optimization ·
Parabola · Arch
1 Introduction
The study of the most efficient structure to carry a uniformly distributed load between two pinned supports has
This research has been supported by Buro Happold Ltd (sponsorship
of Wael Darwich’s doctoral studies) and EPSRC (Advanced Research
Fellowship grant GR/S53329/01 held by Matthew Gilbert).
W. Darwich
· M. Gilbert (B) · A. Tyas
Department of Civil & Structural Engineering,
University of Sheffield, Mappin Street, Sheffield S1 3JD, UK
e-mail: [email protected]
W. Darwich
e-mail: [email protected]
A. Tyas
e-mail: [email protected]
been of interest to scientists and engineers since the 17th
century. Since that time it has generally been believed that,
when the self-weight of the structure itself is ignored, the
structure requiring the least volume of material comprises
a single parabolic arch rib (or equivalently a suspended
cable). When the supports are at the same level, and a distance 2
L apart, the central rise of this optimum parabola
can be shown to be
3L/2, with an associated volume of
wL2/3σ0, where w is the intensity of the applied uniformly distributed loading and σ0 is the limiting material
stress (Rozvany and Wang
1983). However, it appears that
the optimality of this form has never been formally proven
for the standard case of a material possessing equal limiting compressive and tensile stress. Instead published studies
have focused on demonstrating optimality for the more constrained problem involving material with only compressive
or tensile capacity, where the structure experiences zero
bending and shear.
The first scientist to have considered this type of problem
appears to have been Galileo, who hypothesized that a flexible cable of fixed weight per unit length spanning between
two pinned supports would take up the form of a parabola
1961). This hypothesis was later disproved by
Jungius in a work published in 1669, although Christiaan
Huygens appears to have come to the same conclusion in
1646 (Sylla
2003). The hyperbolic cosine curvature for this
problem was later identified by Johann Bernoulli, Gottfried
Wilhelm Leibniz, and Christiaan Huygens. This shape was
named the ‘catenary’ by Huygens. Significantly, Huygens
also demonstrated that, if the load was uniformly distributed
horizontally, then the cable would take up a parabolic
In the intervening years it has generally been assumed
that the parabolic shape will also be optimal even in the case
of the more general problem definition, where the form is

34 W. Darwich et al.
Fig. 1 Optimal parabolic arch, (after Rozvany and Prager 1979)
unrestricted and both tensile and compressive stresses can
be resisted. The general presumption has been that, since
a parabolic arch can be considered to be a ‘perfect’ structure (as bending stresses are absent), this must also be the
most optimal solution to the more general problem definition. For example, Fuchs and Moses (
2000) calibrated
output from their numerical continuum optimization scheme
against the parabolic arch, obtaining good agreement (stating that ‘
the fit is almost perfect’) for the case where
both tensile and compressive stresses could be resisted.
However, there appears to be no
formal proof that the
parabolic arch is optimal in this case, something that distinguished researchers in the field have clearly been aware of.
Rozvany et al. (1982) have however formally proved that
when all cross-sections are uniformly stressed in compression (or tension) only, the optimum structure to carry a
uniform load must comprise a single layer of material taking
up a parabolic form, as shown on Fig.
Hemp (
1974), and Chan (1975) considered a related
problem where both tensile and compressive stresses can
be resisted and where a distributed load is applied along
the line connecting two level pinned supports, with the
supporting structure constrained to lie above this line, as
shown in Fig.
2. Both authors proposed structures comprising symmetrical ‘lobes’, each containing a network of
tension and compression elements lying below a principal
arch rib (Fig.
2). Hemp demonstrated that a form could
be found which was significantly lower in volume than the
simple parabola with vertical hangers, overturning previously held beliefs (e.g. Owen
1965). However, difficulties
were encountered obtaining a formal proof of optimality for this highly challenging problem, though Chan did
successfully obtain provably optimal solutions for specific
non-uniformly distributed loading patterns.
1For example, Rozvany and Prager (1979) were careful to preface
their work on optimal arch-grids as follows: ‘
It is either stipulated or
assumed without explicit proof that the optimal solution for a single
load condition consists of arches which are subject only to compression and no bending moments so that all cross-sections are stressed
uniformly to a given stress
Fig. 2 Hemp arch with hangers (after Hemp 1974)
McConnel (1974), a colleague of Hemp and Chan at
Oxford, addressed the same problem but instead used a two
stage numerical procedure to obtain near-optimal forms.
The first stage of his procedure involved the use of linear programming to identify the optimal layout of fully
stressed bars to carry the specified loading, following the
layout optimization procedure of Dorn et al. (
1964). This
technique involves discretization of a design domain using
nodes, which are then interconnected with potential truss
bars to create a fully connected ‘ground structure’ (or ‘structural universe’). The optimal subset of these is then sought
using optimization. The second stage of McConnel’s procedure involved adjusting the positions of the nodes (so-called
‘geometry optimization’), and was required partly because
the computational resources available at the time severely
limited the number of nodes that could be included in the
initial model.
In traditional numerical layout optimization formulations
(e.g. Dorn et al.
1964) applying a load at a given location
means that the optimal structure must always have one or
more structural elements connected to this location. Whilst
in some cases this is acceptable (e.g. consider a bridge with
a loaded horizontal deck that is required to be at a prescribed elevation), in other cases (e.g. consider a free-form
roof structure) it is potentially not acceptable as it violates
one of the goals of layout optimization, which is to identify the optimal form. To overcome this difficulty, loads can
instead be prescribed simply to act along specified lines-ofaction. Loads of this type were used by Rozvany and Prager
1979) to create so-called ‘Prager’ structures, and have more
recently been referred to as ‘transmissible’ loads (Fuchs and
Transmissible loads have previously been introduced into
topology optimization problem formulations by allowing
the point of action of the load to be transferred from some
arbitrary initial application point via ‘free’ bars aligned on
a given line of action onto the ‘real’ structure. For example,
Rozvany et al. (
1982), in seeking to identify optimal (compressive) funicular structures, allowed the load to transmit
through ‘virtual’ tension bars with infinite tensile strength
and which therefore had zero cross-sectional area. Considering a displacement-based structural optimization problem
formulation, such weightless bars experience zero virtual

Optimum structure between pinned supports 35
strain, and application of this concept is therefore the same
as constraining the virtual displacements parallel to the line
of action of the load to be constant everywhere along the line
of transmission (Fuchs and Moses
2000; Chiandussi et al.
2009). However, the present authors have not adopted the
aforementioned ‘free’ bar approach, also adopted by workers such as Yang et al. (
2005), instead choosing to use an
alternative transmissible loads formulation.
Whilst the present authors were attempting to validate
this alternative transmissible loads formulation, which was
being developed to extend the usefulness of layout optimization as a practical tool for use in industry, unexpected
results arose; these results were the main stimulus for the
study described in this paper. For this work the layout optimization formulation of Dorn et al. (
1964) was preferred
to the continuum formulation of Fuchs and Moses, as it
is more suitable for the types of ‘skeletal’ structures dealt
with by structural designers. Furthermore, this formulation
also appears capable of capturing high levels of topological
detail (e.g. see Gilbert and Tyas
2003), and can readily be
modified to treat transmissible loads (Gilbert et al.
In the next section the formulation is briefly described,
and then applied to the classical problem of identifying the
most optimal structure to carry a uniformly distributed load
between pinned supports.
2 Numerical formulation
2.1 Layout optimization with transmissible loads
The standard primal plastic layout optimization formulation
for a two-dimensional design domain, comprising
m potential truss bars, n nodes, and a single load case, may be stated
as follows:

order now
min V = cTq
subject to:
Bq = f

Where V is the total structural volume, cT = {l11+,

l11,l2 m}, qT = {q1+, q ,

2+, l22, . . . , lm1

q2+, q m}, and B is a suitable (2n × 2m) equi
librium matrix. Also fT = { f1x, f1y, f2x, f2y, . . . , fny} and
li,qi+,q i+, σirepresent respectively the length and

2, . . . , qi, σtensile and compressive forces and stresses in bar i. Finally,
f jx, f jy are x and y load components applied to node j. This
formulation can be solved using linear programming (LP),
with the problem variables being the bar forces
qi+,qiin q.
B y B A
f A
ˆ y
f AB
Fig. 3 Applying a load to a node group (comprising nodes A and B)
In the standard formulation given in (1) the nodal loads
f are fixed quantities, making up the right hand side of
the equilibrium constraint (
1b). However, consider a variation on the basic formulation in which further loads can be
included which are allocated to groups of nodes rather than
to individual nodes (Gilbert et al.
2005); also see Fig. 3,
which shows a load shared by two nodes A and B in the
ground structure. In this case, additional nodal loads
˜ f which
are LP variables can be included, giving a revised equilibrium constraint
Bq – ˜ f = 0, assuming for simplicity
that there are now only transmissible rather than fixed loads
involved. Additional constraints also need to be introduced
to ensure that the total load applied to a given group can
be specified. e.g. for a node group AB comprising nodes A
and B, which is subject to a total load in the
y direction of
fˆAB y , the requisite constraint is simply:
f˜Ay + f˜By = fˆAB y (2)
f˜Ay and f˜By are the y direction forces applied to
nodes A and B respectively. Or, expressed in matrix-vector
HAB˜ fAB = ˆ fAB = 1 0 1 0 0 1 0 1
fˆAB x
fˆAB y
Where in this case
fˆAB x = 0. It follows that the LP formulation for a problem involving transmissible loads can be
written in full as:
V = cTq (4a)
36 W. Darwich et al.
subject to:

Bq – ˜ f = 0
˜ f = ˆ f
0 or ˜ fk 0} k = 1….p

Where H is a 2p ×2n matrix, and where p is the number
of node groups to which external loads are applied,
ˆ fT =
fˆ1x, fˆ1y, fˆ2x, fˆ2y…, fˆpx, fˆpy}. The LP variables are now the
bar forces in
q and the nodal loads in ˜ f, where ˜ fT = { f˜1x,
f˜1y, f˜2x, f˜2y, …, f˜nx, f˜ny}.
Using this formulation it should be noted that a given
load applied to a node group may be found to be shared
between several nodes in the optimum solution (i.e. there
is no constraint that stipulates that the loads must be
applied along a contiguous surface). However, constraint
4e) ensures that all loads in a given node group are of the
same sign.
2.2 A surprising result
As was previously mentioned, when Fuchs and Moses
2000) wished to validate their transmissible loads formulation they applied this to the uniform load between pinned
supports problem, with equal limiting tensile and compressive stresses. Though initially unaware of this work, the
present authors followed the same course, initially finding that when increasing the number of nodes used in the
layout optimization, the solution ‘
appeared to converge
towards the theoretical optimum
[parabolic arch] solution
(Gilbert et al.
2005). However, it was also observed that
in the solution a number of ‘
secondary members’ radiated
out from the supports. At the time it was suspected that
the deviation from the parabolic form was probably due
to nodal discretization error. Further numerical optimizations were therefore subsequently conducted by Darwich
et al. (
2007), who used an increased number of nodes in
the design domain to try to verify this latter conjecture. This
led to two unexpected findings: firstly, the ‘secondary members’ identified previously did not disappear with increasing
numbers of nodes, and secondly the volumes of the corresponding synthesized structures
reduced to below that of the
parabolic arch
These findings were clearly surprising, and it was
decided that more in-depth investigations should be undertaken to verify their correctness. These will now be
2.3 Numerical confirmation
The transmissible loads layout optimization formulation
described previously was used, and all optimization problems were set up using a purpose written C
++ software
application designed to interface with a robust 3rd-party
linear programming (LP) solver (Mosek version 5.0, build
105). The software was either run on a PC with 2Gb of
memory and running Microsoft Windows XP, or, when large
numbers of nodes were used, on an AMD Opteron-based
Sun workstation with 16Gb memory and running Scientific
2.3.1 Influence of discretization of the load
As a uniformly distributed vertical load must in reality be
applied as a series of point loads, applied to groups of nodes
located at fixed
x-positions, it is useful to first examine the
influence of load discretization on the solutions obtained.
(Note that here discretization of the load is considered separately from discretization of the topology of the structure
due to nodal discretization—which is obviously also important, and which will normally ensure that the computed
volume is an over-estimate of the true optimum volume).
Consider the case of a uniformly distributed external
load of total magnitude
W applied vertically, and uniformly
distributed over a horizontal space between two pinned supports which is discretized using
nx equally spaced divisions
between nodes. The uniformly distributed load may be discretized in various ways, e.g: (i) applying
nx 1 point
loads of magnitude
W/(nx 1) to nodes between the supports (henceforth referred to as Type-I load discretization),
or (ii) applying
nx 1 point loads of magnitude W/nx
to nodes between the supports, with two ‘lost’ point loads
of magnitude
1 2 W/nx applied to the support nodes (TypeII load discretization). However it may be observed that
these load discretizations will respectively over- and underestimate the magnitude of the bending moment produced
by a continuous uniform load. One way of quantifying the
error involved is to compute the ratio of the areas under
the discretized and continuous bending moment diagrams.
This ratio can be computed as follows:
rI = (nx + 1)/nx;
rI I = (nx + 1)(nx 1)/n2x, where rI and rI I are the ratios
for load discretization Type-I and Type-II respectively. This
leads naturally to a third load discretization type, Type-III,
for which the load magnitude is scaled to ensure the ratio
rI I I = 1. The three types of load discretization and the
associated error values are summarized in Table
Now whereas in Darwich et al. (
2007) Type-I load discretization was used, principally to ensure there was no
danger of under-estimating the load effect and hence potentially also the structural volume, it is clear from Table
1 that
Optimum structure between pinned supports 37
Table 1 Influence of load
discretization type on applied
bending moment
Magnitude of load applied Ratio of areas under discretized Sample error when
nx = 100
to nodes in the span and continuous BMDs (
r) [100 × (1 r)]

W/(nx 1)
(nx + 1)/nx
(nx + 1)(nx 1)/n2x
Type-III W nx/(nx + 1)(nx 1)

the error involved is much larger than when Type-II load
discretization is used. However in the latter case there is
a slight danger of under-estimating the structural volume.
Consequently for this study load discretization Type-III was
2.3.2 Validation of numerical model: ‘compression only’
As mentioned previously, Rozvany et al. (1982) proved that
when the constituent material is capable of carrying compression only, the form of the optimum structure
must be a
parabolic arch
. Hence a study was undertaken to validate
that the numerical model described earlier was capable of
correctly replicating such a form (if the answer to this question was ‘yes’ then confidence in results from the model
for the case of a standard material, with equal limiting
compressive and tensile stresses, would be increased).
Although ideally the limiting tensile stress
σ + would be
set to zero for all potential truss bars in the design space,
in numerical layout optimization models it is usually convenient to arrange nodes on a rectilinear grid, with the
consequence that it is impossible to precisely replicate the
parabolic form. This meant that a small non-zero tensile
stress had to be specified to ensure a stable structure could
be identified (100
σ + = σ = 1 for all potential bars in the
design space). It was hoped that as the number of nodes was
increased, a closer and closer approximation of the analytical parabolic solution would result. Further details of the
model are as follows:
1. The span
(2L) and the load intensity (w) were taken as
unity. For this case the volume of the optimum singlerib parabola is 1
/3 = 0.577350. (Note in practice
only half the domain needed to be modelled due to symmetry; consequently the volumes presented are twice
those actually computed).
2. The nodes were laid out on a rectilinear grid but to facilitate accurate modelling in the flat crown region of the
arch, the spacing between nodes in the
y direction, y,
was always taken as half the spacing between nodes in
x direction, x (where here x = 2L/nx, and
nx is the number of divisions between nodes in
x direction across the full span).
3. To reduce the number of potential truss bars to be considered in the optimization, and hence the required computational resources, various measures were taken:
– Since the optimum structure only occupies a small
proportion of the initial rectangular design domain,
when the number of divisions between nodes in
x direction, nx, exceeded 200, a restricted
design domain was used. The restricted domain was
made sufficiently large to comfortably contain the
nx = 200 solution.2 Thus considering rectangles
R1(x1 =0, y1 =0 : x2 =0.3, y2 =0.47), R2(0.3, 0 :
0.5, 0.47), and circles: C1(xcentre =1.12, ycentre =
0.58 : radius = 1.27), C2(0.9,0.46 : 1),
C3(0.52, 0.1 : 0.36), C4(0.53, 0.005 : 0.42), the
set of permissible nodes was taken as
N ={x : (x
R1 C1 : x C2) or (x R2 C3 : x C4)};
(considering only the half span actually modelled).
– The adaptive ‘member adding’ procedure proposed
by Gilbert and Tyas (
2003) was used to limit the
size of linear programming problem to be solved
(the starting set of bars interconnected only neighbouring nodes, i.e. contained bars of maximum
l, where l = x2 + y2).
– Since the optimal form was expected to primarily comprise of relatively short bars, the maximum length of potential bars added at subsequent
iterations was restricted to 30
– Finally, as bars which overlap others serve no useful purpose in this problem, these were filtered out
using the well known ‘greatest common divisor’
algorithm (Knuth
Output from the model when up to 1000 divisions
between nodes across the full span were used are provided
in Fig.
4a. To give an indication of the size of the numerical
problems, when 1000 nodal divisions were used (500 in the
2Note that if an over-restrictive domain is used the computed volume
will tend to grossly overestimate the true optimal volume.

38 W. Darwich et al.
half span), a total of 41,783 nodes and 530,712,246 potential bars were present, and the problem required 7 h 20 min
CPU time and 6Gb of memory to solve.
It is clear from Fig.
4a that the numerical model is
capable of closely approximating both the form and volume of the optimal parabola. The volume computed when
1000 nodal divisions were used was 0.577391, which is
just 0.0071% greater than that corresponding to the optimal
parabola. Extrapolation techniques were then used to estimate the volume when an infinite number of nodal divisions
were present (see
Appendix for details). An extrapolated

volume of V= 0.577350 was calculated, which is

precisely the same as that of the optimal parabola (to 6
significant figures).
See Fig. 5 for
magnified view
nx Volume %diff divs, Nodal nx Volume %diff
2 0.666666 15.47% 320 0.577526 0.0304%
20 0.583459 1.058% 340 0.577513 0.0282%
40 0.580207 0.4948% 360 0.577504 0.0266%
60 0.578755 0.2433% 380 0.577494 0.0249%
80 0.578328 0.1693% 400 0.577484 0.0232%
100 0.578150 0.1385% 420 0.577475 0.0216%
120 0.577988 0.1105% 440 0.577469 0.0206%
140 0.577851 0.0867% 460 0.577463 0.0195%
160 0.577765 0.0718% 480 0.577457 0.0185%
180 0.577718 0.0637% 500 0.577452 0.0176%
200 0.577684 0.0578% 600 0.577432 0.0142%
220 0.577642 0.0505% 700 0.577415 0.0112%
240 0.577612 0.0453% 800 0.577405 0.0095%
260 0.577585 0.0407% 900 0.577397 0.0081%
280 0.577559 0.0362% 1000 0.577391 0.0071%
300 0.577540 0.0329%
* 0.577350 0.0000%
* Extrapolated volume (see Appendix for details)
nx Volume %diff divs, Nodal nx Volume %diff
2 0.666666 15.47% 320 0.575512 -0.3184%
20 0.583459 1.058% 340 0.575499 -0.3206%
40 0.578446 0.1898% 360 0.575486 -0.3229%
60 0.577022 -0.0569% 380 0.575476 -0.3246%
80 0.576462 -0.1539% 400 0.575466 -0.3264%
100 0.576200 -0.1992% 420 0.575457 -0.3279%
120 0.576006 -0.2328% 440 0.575451 -0.3290%
140 0.575867 -0.2569% 460 0.575445 -0.3300%
160 0.575782 -0.2716% 480 0.575438 -0.3312%
180 0.575724 -0.2817% 500 0.575434 -0.3319%
200 0.575672 -0.2907% 600 0.575414 -0.3354%
220 0.575632 -0.2976% 700 0.575397 -0.3383%
240 0.575600 -0.3032% 800 0.575387 -0.3400%
260 0.575571 -0.3082% 900 0.575379 -0.3414%
280 0.575547 -0.3123% 1000 0.575373 -0.3425%
300 0.575526 -0.3160%
* 0.575338 -0.3485%
* Extrapolated volume (see Appendix for details)
Part of σ = 100σ + structure
from (a) shown for comparison
Fig. 4 Numerical results: a σ = 100σ +; b σ = σ + (layouts shown were obtained using 1000 nodal divisions; the %diff is relative to the
volume of the optimum single-rib parabolic arch, 1
Optimum structure between pinned supports 39
Fig. 5 Magnified view of part
of optimum
σ = σ + structure
shown in Fig.
4b (1000 nodal
As the results appeared to indicate that the numerical procedure was working well, this was then applied to the more
standard problem in which equal tensile and compressive
stresses could be resisted.
2.3.3 Numerical results when tension and compression
can be resisted
The procedure described in the preceding section was
repeated in order to model the
σ = σ + case. Apart from
changing the limiting tensile stress (now taking
σ + = σ ),
the only change was to use a different restricted design
domain to reflect the different form of the emerging solution (when
nx > 200). Thus using R3(0, 0 : 0.235,
0.47), R4(0.235, 0 : 0.5, 0.47), circles C5(0.73, 0.75 :
1.06), C6(1.04, 0.50 : 1.157), C7(0.54, 0.07 : 0.4),
C8(0.51, 0.06 : 0.37), the set of permissible nodes when
nx > 200 was taken as N = {x : (x R3 C5 C6) or
(x R4 C7 : x C8)} (considering only the half span
actually modelled).
Output from the model is given in Fig.
4b. When 1000
nodal divisions were used (500 in the half span), a total of
45,869 nodes and 639,579,837 potential bars were present,
and the problem required 33h 8mins CPU time and 8.2Gb
of memory to solve.
The difference in the form of the optimal structure shown
in Fig.
4b, compared with the optimal parabolic structure shown in Fig. 4a, is clearly apparent. Figure 5 shows
part of the ‘microstructure’ lying below the main compression rib, showing the level of detail that can be captured
using the numerical layout optimization procedure. This
‘microstructure’ comprises near-orthogonal bars apparently
arranged in the form of a Hencky net, qualitatively similar to
those identified in the problems examined by Hemp (
and by Chan (
1975). The volume computed when 1000
nodal divisions were used was 0.575373, which is 0.3425%
lower than that corresponding to the optimal parabola. The
extrapolated volume was computed to be 0.575338, which
is 0.3485% lower than the volume of the optimal parabola.
3 Discussion
It has been demonstrated numerically that the parabolic
arch rib is no longer optimal when the constituent material possesses equal compressive and tensile strengths. The
existence of a structure that is more optimal than the Prager
parabola presented by Rozvany and Wang (
1983) is, at first
sight, surprising. However, the fact that the Prager parabola
cannot be the most optimal structure when allowable tension
and compression stresses are equal is in fact implicit in the
results presented in that work, as shall now be demonstrated.
The Michell-Hemp conditions for optimality of a framework carrying a given load require that everywhere in the
design space, the virtual strains are within the following

σ + (5)
The optimal Prager parabola is required to be a funicular, comprising only compressive elements (or alternatively, only tensile elements for the inverted topology; the
compression-only case will be assumed here). To ensure
the absence of tensile elements in their structure, Rozvany
and Wang set the allowable tensile stress to zero, so that (
could be rewritten as:

ε ≤ ∞ (6)
In the transmissible load formulation presented by
Rozvany and Wang, the uniformly distributed vertical load

40 W. Darwich et al.
Table 2 The influence of the σ +ratio on the form of the optimal structure to carry a uniform load (200 nodal divisions; the %diff is relative
to the half span volume of the optimum single-rib parabolic arch
= 1/3)
= 3 = 2 = 1 2
= 3 = 12
Volume 0.577684


%diff Approx. θ1
30° 50° 55°
Analytic. 1 30° 50.77° 54.74°
σ σ
– + σ σ – + σ σ – + σ σ – + σ σ – +
is transmitted to the ‘real’ structure by means of ‘virtual’
vertical bars that have infinite tensile strength.
3 There is
therefore zero virtual strain in these vertical bars and compatibility requires that along the parabolic arch rib, the
vertical virtual strain is also zero. The implication of this,
when considered together with (
5), is that the principal
strains along and perpendicular to the parabolic rib,
ε1 and
ε2 respectively, will be:
ε1 = –
ε2 =
cot2 θ (7b)
θ is the angle between the parabolic rib and the
vertical at any point on the rib.
Using this reasoning, Rozvany and Wang demonstrated
that the Prager parabola is a special case of a Michell-Hemp
structure, with particular limits on the relative tensile and
compressive stresses. In fact their requirement that the tensile stress must be zero is overly onerous; it can be deduced
from (
5) and (7) that the principal strain given in (7b)
satisfies the Michell-Hemp strain conditions provided:
θ tan1 σ +
3N.B. These ‘virtual’ bars act only to transmit the load to the most optimal point of application; they cannot form integral parts of the ‘real’
Thus, the optimal Prager parabola with θ 30o satisfies
the Michell-Hemp conditions if (and only if)
σ 3σ +.
The possibility of the existence of some other, more optimal,
topology is therefore raised, e.g. for the case when
σ =σ +.
Clearly any section in the actual optimal structure that is
a parabolic arch rib directly carrying the applied load must
satisfy (
8). In other words, if σ 3σ +, θ must be greater
than 30
o everywhere in the parabola. However, as Rozvany
and Wang’s results demonstrate, a parabolic arch rib which
has this geometry and spans the full distance between the
supports must be less efficient than the Prager parabola.
Therefore, whilst the optimal structure for
σ 3σ + may
include a parabolic section satisfying (
8), this section cannot extend over the whole span. In fact such a parabolic
section must be confined to the central section of the span
where the slope of the parabola is sufficiently low, with
other sections of structure that satisfy (
5) being required in
the vicinity of the supports. A suitable form may comprise
a central parabolic arch rib, with orthogonal Hencky nettype sections adjacent to each support. This appears to be
the form of structure presented in Figs.
4b and 5 for the case
of equal tensile and compressive strengths.
In general (
8) also indicates that the parabolic region
should become more restricted in extent around the centre
of the span as the ratio of compressive to tensile strengths
decreases, emerging from the Hencky net-type section at an
θ1 when (8) becomes an equality. Sample numerical
results for different
σ + ratios are presented in Table 2,
which clearly confirm this. (These results were obtained
by using the procedure outlined previously, but for sake
of simplicity using 200 nodal divisions and an unrestricted
rectangular permissible design domain. The approximate
values of
θ1 given in Table 2 were obtained by inspection
from the computed optimal layouts, and clearly compare
well with the analytical values derived from (
Optimum structure between pinned supports 41
Fig. 6 Computed volume vs.
nodal refinement (for
σ + = σ = 1 and
σ + = σ = 1 cases)
0 200 400 600 800 1000
4 Conclusions
Although it has been widely believed since the time of
Christiaan Huygens in the 17th century that a single
parabolic arch rib (or suspended cable) is the most optimal form to carry a uniformly distributed load between
pinned supports, here large-scale numerical layout optimization techniques have been used to demonstrate that,
when the constituent material possesses equal tensile and
compressive strength, this is not the case. Instead a considerably more complex structural form, comprising a central
parabolic section and networks of truss bars in the haunch
regions, is found to possess a lower structural volume.
Appendix: Extrapolation of numerically computed data
to estimate volume for infinite nodal density
In a recent review article Rozvany (2009) observes that
‘Most of the authors in numerical topology optimization
simply compare their solutions visually with the exact optimal truss topology and are satisfied with a vague resemblance’. He goes on to suggest that a rigorous verification
approach should comprise the following steps:
1. For a given set of constraints, derive numerically the
optimal topology for various numbers of elements.
2. Calculate the structural volume (weight) for each
3. Extrapolate the volume (weight) value for infinite number of elements.
4. Compare this extrapolated value with that calculated
analytically for the exact benchmarks.
These steps have been applied to the numerical data presented in Fig.
6.4 These data appear to follow a relation of
the form:
Vnx = V+ knx α (9)
Vnx is the numerically computed optimal volume for
nx equally spaced nodal divisions across the span, Vis the
volume when
nx → ∞, and k and α are positive constants.
Using (
9), a non-linear least-squares approach was used
to find best-fit values for
V, k and α. The resulting trend
lines and values for
Vare shown in Fig. 6.
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