# Optimum structure to carry a uniform load between pinned supports

## Authors

- First Online:

- Received:
- Revised:
- Accepted:

DOI: 10.1007/s00158-009-0467-0

- Cite this article as:
- Darwich, W., Gilbert, M. & Tyas, A. Struct Multidisc Optim (2010) 42: 33. doi:10.1007/s00158-009-0467-0

## 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 optimizationLayout optimizationParabolaArch## 1 Introduction

The study of the most efficient structure to carry a uniformly distributed load between two pinned supports has 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 \(\sqrt{3}L/2\), with an associated volume of \(4w L^2/\sqrt{3}\sigma_0\), where *w* is the intensity of the applied uniformly distributed loading and *σ*_{0} 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.

*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.

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-of-action. 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 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

*m*potential truss bars,

*n*nodes, and a single load case, may be stated as follows:

Where *V* is the total structural volume, \(\mathbf{c^{\text{T}}}=\{l_1/\sigma_1^+,\)\(-l_1/\sigma_1^-, l_2/\sigma_2^+, -l_2/\sigma_2^-, \ldots, -l_m/\sigma_m^- \}\), \(\mathbf{q^{\text{T}}}=\{q_1^+, -q_1^-,\)\(q_2^+, -q_2^-,\ldots, -q_m^- \}\), and **B** is a suitable (2*n* ×2*m*) equilibrium matrix. Also \(\mathbf{f^{\text{T}}} = \{f_1^x, f_1^y, f_2^x, f_2^y, \ldots, f_n^y\}\) and \(l_i, q_i^+, q_i^-\), \(\sigma_i^+, \sigma_i^-\) represent respectively the length and tensile and compressive forces and stresses in bar *i*. Finally, \(f_j^x, f_j^y\) 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 \(q_i^+, q_i^-\) in **q**.

**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 \(\tilde{\mathbf{f}}\) which are LP variables can be included, giving a revised equilibrium constraint \(\mathbf{Bq} - \tilde{\mathbf{f}} = 0\), 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 \(\hat{f}^y_{AB}\), the requisite constraint is simply:

*y*direction forces applied to nodes A and B respectively. Or, expressed in matrix-vector form:

Where **H** is a 2*p* ×2 *n* matrix, and where *p* is the number of node groups to which external loads are applied, \(\hat{\mathbf{f}}^{{\text{T}}}= \{\hat{f}^x_1, \hat{f}^y_1,\hat{f}^x_2,\hat{f}^y_2...,\hat{f}^x_p, \hat{f}^y_p\}\). The LP variables are now the bar forces in **q** and the nodal loads in \(\tilde{\mathbf{f}}\), where \(\tilde{\mathbf{f}}^{\text{T}}= \{\tilde{f}_1^x,\)\( \tilde{f}_1^y, \tilde{f}_2^x, \tilde{f}_2^y, ..., \tilde{f}_n^x, \tilde{f}_n^y\}\).

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).

*W*applied vertically, and uniformly distributed over a horizontal space between two pinned supports which is discretized using

*n*

_{x}equally spaced divisions between nodes. The uniformly distributed load may be discretized in various ways, e.g: (i) applying

*n*

_{x}− 1 point loads of magnitude

*W*/(

*n*

_{x}− 1) to nodes between the supports (henceforth referred to as Type-I load discretization), or (ii) applying

*n*

_{x}− 1 point loads of magnitude

*W*/

*n*

_{x}to nodes between the supports, with two ‘lost’ point loads of magnitude \(\frac{1}{2}W/n_x \) applied to the support nodes (Type-II load discretization). However it may be observed that these load discretizations will respectively over- and under-estimate 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:

*r*

_{I}= (

*n*

_{x}+ 1)/

*n*

_{x}; \(r_{II} = (n_x+1)(n_x-1)/n_x^2\), where

*r*

_{I}and

*r*

_{II}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

*r*

_{III}= 1. The three types of load discretization and the associated error values are summarized in Table 1.

Influence of load discretization type on applied bending moment

| Magnitude of load applied to nodes in the span | Ratio of areas under discretized and continuous BMDs ( | Sample error when |
---|---|---|---|

Type-I |
| ( | +1% |

Type-II |
| \((n_x+1)(n_x-1)/n_x^2\) | −0.01% |

Type-III |
| 1 | 0% |

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 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).

*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 (2

*L*) and the load intensity (*w*) were taken as unity. For this case the volume of the optimum single-rib parabola is \(1/\sqrt{3}=0.577350 \). (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, Δ*y*, was always taken as half the spacing between nodes in the*x*direction, Δ*x*(where here Δ*x*= 2*L*/*n*_{x}, and where*n*_{x}is the number of divisions between nodes in the*x*direction across the full span). - 3To 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,*n*_{x}, exceeded 200, a restricted design domain was used. The restricted domain was made sufficiently large to comfortably contain the*n*_{x}= 200 solution.^{2}Thus considering rectangles*R*_{1}(*x*_{1}= 0,*y*_{1}= 0:*x*_{2 }= 0.3,*y*_{2 }= 0.47),*R*_{2}(0.3,0:0.5,0.47), and circles:*C*_{1}(*x*_{centre}= 1.12,*y*_{centre}= − 0.58:*radius*= 1.27),*C*_{2}(0.9, − 0.46: 1),*C*_{3}(0.52, 0.1: 0.36),*C*_{4}(0.53, − 0.005: 0.42), the set of permissible nodes was taken as \(N \!=\! \{x:(x \!\in\! R_1\!\cap\! C_1 :x \!\ni\! C_2 )\) or \((x \!\in\! R_2\cap C_3 :x \ni C_4 )\};\) (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 Δ

*l*, where Δ*l*= \(\sqrt{\Delta x^2 + \Delta y^2}\)).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 30Δ

*l*.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).

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).

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 *n*_{x} > 200). Thus using *R*_{3}(0,0:0.235, 0.47), *R*_{4}(0.235,0:0.5,0.47), circles *C*_{5}( − 0.73, 0.75:1.06), *C*_{6}(1.04, − 0.50: 1.157), *C*_{7}(0.54, 0.07:0.4), *C*_{8}(0.51, 0.06: 0.37), the set of permissible nodes when *n*_{x} > 200 was taken as *N* = {*x*:(*x* ∈ *R*_{3} ∩ *C*_{5} ∩ *C*_{6} ) or \((x \!\in\! R_4\cap C_7 :x \ni C_8 )\}\) (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.

*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.

^{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,

*ε*

_{1}and

*ε*

_{2}respectively, will be:

*θ*is the angle between the parabolic rib and the vertical at any point on the rib.

Thus, the optimal Prager parabola with \(\theta\geq30\deg\) satisfies the Michell-Hemp conditions if (and only if) *σ*^{ }− ≥ 3*σ*^{ + }. 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 \({\sigma^-}\ngeq\ 3{\sigma^+}\), *θ* must be greater than \(30\deg\) 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 \({\sigma^-}\ngeq\ 3{\sigma^+}\) 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 net-type 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.

*θ*

_{1}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

*θ*

_{1}given in Table 2 were obtained by inspection from the computed optimal layouts, and clearly compare well with the analytical values derived from (8).)

## 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.

For 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 σ*_{0}’.

Note that if an over-restrictive domain is used the computed volume will tend to grossly overestimate the true optimal volume.

N.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.