## Optimum structure to carry a uniform – Global Homework Experts

Struct Multidisc Optim (2010) 42:33–42

DOI 10.1007/s00158-009-0467-0

RESEARCH PAPER

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

volume.

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 2L apart, the central rise of this optimum parabola

can be shown to be √3L/2, with an associated volume of

4wL2/√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

(Lockwood 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

profile.

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.1

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. 1.

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 σ0’.

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

Moses 2000).

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. 2005).

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:

min V = cTq subject to: Bq = f q ≥ 0 |
(1a) |

(1b) (1c) |

Where V is the total structural volume, cT = {l1/σ1+,

–l1/σ1–,l2/σ | m–}, qT = {q1+, –q | , |

2+, –l2/σ2–, . . . , –lm/σ1–

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+, σi– represent 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+,qi– in q.

B y B A

f

y

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

in 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

magnitude fˆAB y , the requisite constraint is simply:

f˜Ay + f˜By = fˆAB y (2)

Where f˜Ay and f˜By are the y direction forces applied to

nodes A and B respectively. Or, expressed in matrix-vector

form:

HAB˜ fAB = ˆ fAB = 1 0 1 0 0 1 0 1

⎡⎢⎢⎢⎣

f˜Ax

f˜Ay

f˜Bx

f˜By

⎤⎥⎥⎥⎦

=

fˆAB x

fˆAB y

(3)

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:

min V = cTq (4a)

36 W. Darwich et al.

subject to:

Bq – ˜ f = 0 H˜ f = ˆ f q ≥ 0 ˜fk ≥ 0 or ˜ fk ≤ 0} k = 1….p |
(4b) (4c) (4d) (4e) |

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

described.

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

Linux.

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 1.

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)]

Type-I Type-II |
W/(nx – 1) W/nx |
(nx + 1)/nx (nx + 1)(nx – 1)/n2x 1 |
+1% –0.01% 0% |

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

used.

2.3.2 Validation of numerical model: ‘compression only’

arch

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 (100σ + = 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

the x direction, x (where here x = 2L/nx, and

where nx is the number of divisions between nodes in

the 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

the 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

length 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 30l.

– 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 1997).

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

(a)

(b)

Nodal

divs, 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)

Nodal

divs, 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/√3)

Optimum structure between pinned supports 39

Fig. 5 Magnified view of part

of optimum σ – = σ + structure

shown in Fig. 4b (1000 nodal

divisions)

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 (1974)

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

limits:

–

1

σ –

≤ ε ≤

1

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 (5)

could be rewritten as:

–

1

σ –

≤ 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 +/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

Layout

Volume 0.577684

0.577656 0.0530% 35° 35.26° |
0.575672 -0.2907% 45° 45° |

%diff Approx. θ1

θ

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 = –

1

σ –

(7a)

ε2 =

1

σ –

cot2 θ (7b)

where 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 ≥ tan–1 σ +

σ –

(8)

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’

structure.

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 30o 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

angle θ1 when (8) becomes an equality. Sample numerical

results for different σ –/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 (8).)

Optimum structure between pinned supports 41

Fig. 6 Computed volume vs.

nodal refinement (for

100σ + = σ – = 1 and

σ + = σ – = 1 cases)

0.575

0.576

0.577

0.578

0.579

0.580

0 200 400 600 800 1000

n

x

V

nx

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

solution.

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∞ + kn–x α (9)

where Vnx is the numerically computed optimal volume for

nx equally spaced nodal divisions across the span, V∞ is 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 V∞ are shown in Fig. 6.

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