## Abstract

Recent numerical evidence indicates that a parabolic funicular is not necessarily the optimal structural form to carry a uniform load between pinned supports. When the constituent material is capable of resisting equal limiting tensile and compressive stresses, a more efficient structure can be identified, comprising a central parabolic section and networks of truss bars emerging from the supports. In the current article, a precise geometry for this latter structure is identified, avoiding the inconsistencies that render the parabolic form non-optimal. Explicit analytical expressions for the geometry, stress and virtual-displacement fields within and above the structure are presented. Furthermore, a suitable displacement field below the structure is computed numerically and shown to satisfy the Michell–Hemp optimality criteria, hence formally establishing the global optimality of this new structural form.

## 1. Introduction

While studying one of the oldest and most fundamental problems in structural engineering, Darwich *et al.* (2010) recently obtained convincing numerical evidence to suggest the existence of a new structure that is more optimal than the commonly accepted benchmark.

The problem considered was this: what is the least-volume structure to carry a uniformly distributed vertical load between two-level fixed-pin supports, assuming that the structure is confined to occupy the half-plane lying above (or, alternatively, below) the line intersecting the supports? It is assumed that the self-weight of the structure is negligible compared with the applied load, and that the vertical points of application of the loads are not defined *a priori*, but rather the loads are allowed to migrate through the design space to enable their optimal points of application to be found.

Since the seventeenth century, it has commonly been assumed that the resulting structure will be a parabolic funicular, with the loads being applied directly to either an arch in compression or a suspended cable in tension. It can easily be proved that the least-volume solution in this case is obtained when the angle between the parabolic funicular and the vertical at the support is equal to 30^{°}, e.g. Rozvany & Prager (1979).

Typical textbook demonstrations of the requirement for the structure to be a parabolic funicular rely on the assumption that the structure must be free of bending stresses. This naturally follows on from an assumption that the material has zero tensile (or, alternatively, zero compressive) strength. Nevertheless, this does not answer the question of whether the parabolic funicular remains the optimal solution when the requirement of zero tensile strength is relaxed. Rozvany & Wang (1983) shed some light on this issue by proving the optimality of the parabolic funicular subject to certain additional assumptions specific to so-called ‘Prager structures’ (the optimal forms that result when all structural forces are required to be of the same sign, and when external loads have migrated to their optimal locations). This led Wang & Rozvany (1983) to prove that the parabolic funicular is optimal when the magnitude of the limiting tensile stress does not exceed one-third that of the limiting compressive stress. However (and perhaps surprisingly), until recently, what might be considered to be the ‘standard’ case, where limiting tensile and compressive stresses are equal, appears to have received little attention.

The numerical results presented by Darwich *et al.* (2010) provide an invaluable insight of the form of a lower volume structure for this standard case. Making use of large-scale layout optimization techniques developed by Gilbert & Tyas (2003), the researchers found that the optimum structure actually comprises truss structures (apparently of the form of Hencky-net fans) connecting the supports to the springing points of a central parabolic section. The purpose of the present study is to definitively prove the existence of such a structure, by establishing explicit analytical expressions for the geometry, stress and global virtual-displacement fields that satisfy the relevant optimality criteria.

## 2. Initial geometrical considerations and proposed layout

The theory of optimum structures is underpinned by the criteria formulated by Michell (1904) and Hemp (1973) (see also Rozvany 1996). According to these criteria (henceforth referred to as the Michell–Hemp criteria), in a least-volume pin-joined framework, all axial force-carrying members must be stressed to limiting values in either tension *σ*_{T} or compression *σ*_{C}. At the same time, when the structure is subject to a virtual deformation, the virtual strains along trajectories where no load-carrying members exist must remain within the following limits:
2.1where *ε* is the virtual strain, *ϵ* is a positive infinitesimal and *σ* = (*σ*_{C} + *σ*_{T})/2. The associated virtual deformation field must satisfy the kinematic restrictions imposed on the solution and be continuous, though not necessarily smooth, across the entire problem domain.

Although the Michell–Hemp optimality criteria provide an important insight into the properties of optimum structures, the problem of actually finding such structures remains challenging, with only a handful of analytical solutions constructed over the course of the last century (e.g. Chan 1962, 1963; Hemp 1973; Rozvany & Gollub 1990; Rozvany *et al.* 1997; Graczykowski & Lewiński 2006*a*,*b*, 2007*a*,*b*, Lewiński & Rozvany 2007; 2008*a*,*b*). This prompted Rozvany & Prager (1979) to consider the simpler scenario in which one of the limiting stresses, i.e. *σ*_{T} or *σ*_{C}, was taken to be equal to zero, and the vertical positions of the applied loads were left unspecified, to be chosen while searching for the optimal structure. The ensuing analysis offered direct and straightforward formulation of a large number of explicit and practically relevant solutions, all of which were shown to be necessarily funicular in form. Even though the strains in so-constructed Prager structures are allowed to become unbounded, they remain bounded for most ‘well-behaved’ load distributions (Rozvany & Wang 1983), hence implying that these structures may also be interpreted as Michell structures for optimally placed loads and for sufficiently small values of one of the limiting stresses.

In particular, Rozvany & Wang (1983) showed that, for a uniformly distributed load *w* (per unit length), a symmetrical parabolic funicular with is the optimum structure provided that the allowable tensile stress *σ*_{T} = 0 (figure 1). The total volume of this structure is given by
2.2If *ϕ* is the angle measured from the horizontal line of support to the local normal to the funicular, then *ϕ* = −*π*/6 at the left-hand support point of the optimal parabolic arch. The principal strains in the funicular can be expressed as
2.3
where the subscripts I and II refer to the principal strains directed normal to and along the parabola, respectively. Wang & Rozvany (1983) analysed these expressions to show that the parabolic arch only satisfies the Michell–Hemp criteria if *σ*_{C} ≥ 3*σ*_{T}. Since we assume in this paper that the limiting tensile and compressive stresses are equal, i.e. *σ*_{C} = *σ*_{T} = *σ*, the conclusion is that the optimal parabolic arch must violate the Michell–Hemp optimality conditions (2.1) in our case.

An alternative way to interpret equation (2.3) is to state that when the limiting tensile and compressive stresses are of equal magnitude, a parabolic funicular satisfies the Michell conditions if and only if *ϕ* ≤ −*π*/4 or *ϕ* ≥ *π*/4. When *ϕ* = ±*π*/4, both principal strains in the parabolic rib are at their maximum allowable limits, and the only way to continue a Michell structure beyond these points must be to have regions of mutually orthogonal members, strained to their tensile and compressive limits. Supported by the results of the recent numerical study by Darwich *et al.* (2010), we shall therefore seek a new structure comprising a central parabolic section that ends when *ϕ* = ±*π*/4, and which is connected to the supports by Hencky-net-fan regions composed of mutually orthogonal members.

The left half of the proposed layout is shown schematically in figure 2. It is assumed that the structure lies in the region *y* ≥ 0 and is symmetrical about the centre line *x* = *L*. The fan emerges from a singular point at the support (0,0), and terminates at (*X*,*Y*), where *ϕ* = −*π*/4, whence a parabolic funicular extends to the top of the structure (*L*,*H*), such that *L* is the half-span and *H* is the height of the structure at mid-span. These assumptions, together with the requirement for the structure to be symmetric with respect to the centre line *x* = *L*, immediately yield the equation of the parabola,
2.4

The fan section of the structure comprises a continuum of mutually orthogonal members described by a curvilinear coordinate system (*α*,*β*), with the origin placed at the support and (0,0) being the coordinates of the top member at the support (figure 2). Coordinate *α* traverses across the fan, so that the lines of constant *α* follow the compression members, with e.g. line *α* = 0 following the top boundary of the fan; the lines of constant *β* traverse along the tension members, with *β* = *ϕ*_{1} at the top of the fan. The resulting fan may be characterized by
2.5
see also Hemp (1973, p. 79). The geometry of the fan has a number of similarities to the structure analysed by Hemp (1974). The main difference between the problem statements is that in Hemp’s study, all parts of the uniformly distributed load were applied along the *x*-axis, rather than being allowed to migrate vertically in order to produce the structure of the overall lowest volume for the given span. Hemp transmitted the applied loads to the lower boundary of a fan structure via vertical tension bars that, being loaded members of the optimal structure, were required to be strained to their limiting values, and which hence defined the trajectory of the principal strain in this region. Our use of ‘transmissible’ loads, which are free to migrate vertically, has led to a different expression for *ϕ* and to different equilibrium conditions along the lower boundary of the fan, which have necessitated some important differences in the analytical approach adopted. Hence, although we shall frequently refer to elements of the solution procedure presented by Hemp, key elements of our new methodology will become evident as we progress.

We will assume that the uniformly distributed load is applied directly to the central parabolic section of the structure, and to the lower boundary of the fan. However, the form and location of these loaded boundaries are not known *a priori* and hence the load positions must be optimized along with the structural layout. Rozvany & Wang (1983) introduced the concept of allowing loads to translate from an arbitrary pre-defined application point to an optimum position on the ‘real’ structure along ‘virtual’ bars with zero volume and infinite tensile strength. This implies that *ε*_{y}≡0 everywhere along the line of action between the arbitrary initial application point of a given load and the actual point of application on the structure. This condition is automatically satisfied throughout the parabolic rib (Rozvany & Wang 1983). Because we assume that the lower boundary of the fan must serve as the optimal curve for application of the external load (i.e. the transmissible loads migrate from some arbitrary location below the structure to their application point on the lower boundary of the fan), we require *ε*_{y}≡0 along the bottom of the fan and everywhere beneath it. Additionally, we assume that the structural members of the fan carrying tension and compression forces form a Hencky net, i.e. they are mutually orthogonal, and strained to the limiting values ±*ϵ*, as they must if the fan is to be an optimal Michell structure. These conditions immediately furnish the directions of the principal strains, and lead to the requirement that the angle *ϕ* = −*π*/4, or *α* + *β* = *ϕ*_{1} (equation (2.5)), must remain constant along the bottom of the fan. The necessary property can be furnished by aligning the lower boundary with one of the diagonals of the Hencky net, e.g. Hill (1998, p. 147). This completes the specification of the layout of the optimum structure.

## 3. Internal forces

Given the proposed layout, it is necessary to find a distribution of internal forces that ensures all parts of the structure are in equilibrium under the action of the uniformly distributed vertical applied load. Within the fan, equilibrium is governed by differential equations of the following form:
3.1
where *T*_{1} and *T*_{2} are the internal forces per unit coordinate difference in the direction of *α* and *β*, respectively (Hemp 1973, p. 81). The boundary conditions for this system can be formulated by matching the internal forces at the top of the fan with forces in the parabola. Furthermore, consideration of mid-span moment equilibrium allows the horizontal support reaction to be established, and by extension also the horizontal component of thrust at any section in the parabola,
3.2
The axial thrust in the parabolic section is equilibrated by a concentrated force in the rib forming the upper boundary of the fan. Since the internal members of the fan impose only normal forces on this rib as it extends from the central parabolic section down towards the support, the compressive force in the rib must remain constant, with the normal forces merely rotating the line of action of the force in the rib. Thus, the boundary condition for the upper member of the fan is
3.3
Another boundary condition follows from the horizontal equilibrium of internal forces acting on the infinitesimal element PQ of the bottom (figure 3). This gives , which, with equation (2.5), yields
3.4
The boundary-value problem (3.1), (3.3) and (3.4) is similar to the boundary-value problem considered by Hemp (1974), and can similarly be solved by an application of the Laplace transform. After inverting the transform, the resulting solution can be written as
3.5
and
3.6
with
3.7
where *I*_{n}(*x*) is the modified Bessel function of the first kind, and use has been made of the standard recurrence relationships between the Bessel functions (e.g. Gradshteyn & Ryzhik 2007).

## 4. Geometry of the fan

The force field (3.5) and (3.6) is given in terms of, as yet unspecified, orthogonal curvilinear coordinates (*α*,*β*). The general theory for Michell structures also furnishes a description for the geometry of the fan (Hemp 1973, p. 80). If *A* and *B* are the Lamé parameters converting d*α* and d*β* into the corresponding lengths *A* d*α* and *B* d*β*, they must satisfy the following system of partial differential equations:
4.1
The first boundary condition for this system is necessary to account for the singular point at the bottom support of the fan; it is given by
4.2
The second boundary condition is formulated by consideration of vertical equilibrium at the lower boundary of the fan (figure 3). The infinitesimal boundary element PQ is subjected to external load *w* d*x*, with the length d*x* given by . Vertical equilibrium of PQ produces the equality . Since *ϕ* = −*π*/4 everywhere along the lower boundary, reference to equations (2.5), (3.2), (3.4) and (3.5) yields
4.3

The obtained boundary-value problem can again be solved by an application of the Laplace transform. Since the explicit transform of the boundary condition (4.3) is not practical, the inversion of the solution relies upon the use of the convolution theorem, which leads to the following expressions for the Lamé parameters:
4.4
and
4.5
The geometry of the structure in the Cartesian coordinate system can now be explicitly determined via the path integral
4.6
see Hemp (1973, p. 74). Generally, the computation of this double integral must be performed numerically; nevertheless, the integral can be evaluated explicitly to find the coordinates of the top of the fan in terms of the yet unknown fan angle *ϕ*_{1},
4.7
where
4.8
see appendix A for the details of this calculation. Function *G*(2*ϕ*_{1}) plays a prominent role in many expressions obtained in this paper. Since the parabolic rib must meet the top of the fan, point (*X*,*Y*) must satisfy equation (2.4), which, after substituting expressions (3.2) and (4.7), leads to the conclusion that
4.9
(N.B. In fact, the resulting equation presents the possibility of two different relationships between *G*(2*ϕ*_{1}) and *H*/*L*; however, the omitted relationship only leads to structures with non-unique contours of load application that do not satisfy the original problem requirements.) It can be shown by substituting equations (4.7), (3.2) and (4.9) into the equation of the parabola (2.4) that the line of thrust of the parabolic rib passes through the support points.

## 5. Virtual displacements and volume of the structure

The expressions for the virtual displacements within the parabolic funicular were previously derived by Rozvany & Wang (1983). Horizontal and vertical displacements within the parabola are given by
5.1
The first of these, together with the equation of the parabola (2.4) and the requirement that horizontal displacement must vanish at the top of the funicular to provide continuity with the opposite half of the structure, yields
5.2
If (*U*_{x},*V* _{y}) denote the Cartesian displacements at the top of the fan, relation (5.2) leads to the following boundary condition for *U*_{x}:
5.3
The vertical displacement equation in equation (5.1), together with equation (2.4), yields
5.4

Since translation is prohibited at the supports, the general expression in curvilinear coordinates for virtual displacements within the fan takes the form of
5.5
where *u* and *v* are the displacements along *Oα* and *Oβ*, and are expected to vanish at the supports, so that *u*_{0} = *v*_{0} = 0. Symbol *ω* denotes the rotation within the Hencky-net fan, given by the general formula *ω* = *ω*_{0}−2*ϵ*(*α*−*β*) (Hemp 1973, p. 75). Using equations (5.2) and (5.4), it is possible to determine the rotation within the parabolic rib, and consequently to enforce continuity of rotation at the top of the fan in order to establish the value of the constant *ω*_{0} = −*ϵ*(2*ϕ*_{1} + 1), so that
5.6
With the rotation within the fan known, double integral (5.5) provides an explicit expression for the displacements within the fan. It is again possible to integrate along the lower boundary to evaluate displacements at the top of the fan and to project them onto Cartesian axes, eventually yielding
5.7
and
5.8
see appendix B for more details. Relation (5.8) may now be used to recover vertical displacements within the funicular (5.4) in terms of *ϕ*_{1}. The angle *ϕ*_{1} itself is found by matching the horizontal displacement in the funicular (5.3) with the corresponding displacement at the top of the fan (5.7). The resulting equation,
5.9
has three solutions, but only *ϕ*_{1}≈0.32403 corresponds to a structure satisfying the original problem requirements. Given *ϕ*_{1}, one can establish the full geometry of the structure, a discretized version of which is presented in figure 4. It is worth noting that the height of the obtained structure *H*/*L*≈0.88944 is approximately 2.7 per cent greater than the height of the least-volume parabolic arch. The volume of the structure can also be obtained analytically, with the relevant derivations provided in appendix C. The resulting volume,
5.10
is approximately 0.3495 per cent lower than that of the least-volume parabolic funicular, as given by equation (2.2) (cf. the numerical result of Darwich *et al.* (2010), which is 0.3485 per cent lower than the least-volume parabolic arch).

## 6. Proof of optimality

Even though we have now established the precise geometry of a structure possessing a lower structural volume than that of the most efficient parabolic funicular, it has not yet been proved that the new structure is optimal for the problem in hand. The proof of optimality requires the construction of a continuous virtual-displacement field, compatible with the requirements of the Michell–Hemp criteria (2.1), covering the entire problem domain, which here comprises the half-plane lying above the straight line intersecting both supports.

The requirements of the Michell–Hemp criteria can result in several distinct types of regions in the virtual-displacement field, e.g. as set out by Rozvany *et al.* (1995; table 1). *T*-type regions are the regions most commonly associated with Michell structures; these feature systems of mutually orthogonal members everywhere strained to maximum allowable limits. The fans in the new structure are examples of *T*-type regions. It is also possible to have regions in which only one strain is everywhere at its maximum allowable limit, the field in this case becoming a single set of non-intersecting curves. Depending on whether the maximum strain is tensile or compressive, these regions are denoted *R*^{T} or *R*^{C}. The parabolic rib in our structure resides inside an *R*^{C}-type region. The *O*-type regions, in which neither principal strain is everywhere at the maximum allowable limit given in equation (2.1), are also possible. Clearly, regions of this kind cannot include load-carrying members since, by definition, members in such regions will not be fully strained. Moreover, the resulting strain fields can generally be expected to be non-unique, and we will take advantage of this non-uniqueness when constructing a feasible virtual-displacement field below the optimal structure.

The displacement field within our structure has been fully specified in the preceding sections. Thus, we only need to construct a virtual-displacement field covering the rest of the upper half-plane. It comprises two *O*-type regions, together with three *T*-type regions and one *R*^{C}-type region, as shown in figure 5. In this figure, load-carrying members are shown as bold lines, trajectories of limiting compressive virtual strains are shown as light solid lines and those of limiting tensile virtual strain by dashed lines. The task is now to identify suitable displacement fields within each of the identified regions, while also ensuring continuity of the resulting global virtual-displacement field.

### (a) The field above the structure

The area above the optimal structure delimited by the upper boundary of the fan (*α* = 0) and two outward normals at *β* = 0 and *β* = *ϕ*_{1} is denoted as the region *T*_{II}. The *T*-type field within *T*_{II} can be constructed by following the procedure used by Hemp (1974). Specifically, the compression contours following the upper boundary of the fan are projected along the straight lines in tension, normal to the fan. If the definition of *β* is kept the same as within the fan, and *α* is modified to represent the normal distance from a point to the upper boundary of the fan (*α* < 0 within *T*_{II}), then the Hencky net is characterized by
6.1
see equations (2.5), (4.5) and (5.6). The corresponding displacement field is found using equation (5.5), and may be written as
6.2
It is readily verified that field (6.2) matches with the field at the top of the fan when *α* = 0; see equation (5.5) used together with equations (2.5), (4.4), (4.5) and (5.6).

The field within the region *T*_{I} (figure 5) may be filled by a fan of straight lines and circular arcs (cf. appendix 4 in Hemp 1974). If variable *α* within *T*_{I} is chosen to represent the distance from a point to the origin, then both definitions of *α* match at the boundary between *T*_{I} and *T*_{II}. Variable *β* can then be redefined to measure the polar angle from the top boundary of *T*_{I}, so that *β* = 0 at the boundary between *T*_{I} and *T*_{II}, and *β* < 0 below it. The appropriate geometry may be defined using
6.3
The resulting displacement field
6.4
matches with the field within *T*_{II} along the line *β* = 0 (equation (6.2)).

The field within the region *T*_{III} is filled by a system of orthogonal straight lines, with a local origin at the top of the fan (figure 5). Coordinate *α* is chosen to follow the lines of maximum tensile strain (*α* < 0 above the structure), which matches the definition of *α* within the region *T*_{II}. Coordinate *β* within *T*_{III} measures the normal distance from the boundary between *T*_{II} and *T*_{III} to a point of interest. This field is given by
6.5
The corresponding displacements must be matched to the displacements at the top of the fan. The application of formula (5.5), with *u*_{0} = *U* and *v*_{0} = *V* , yields
6.6
see also appendix B. The resulting displacements are correct at the top of the fan by construction. It is also easy to verify that the field (6.6)_{β = 0} also matches with the displacement field within *T*_{II}, see (6.2)_{β = ϕ1}. Interestingly, the displacements along the vertical line *x* = *X*, given within *T*_{III} by *α* + *β* = 0, remain constant. Consequently, the displacement field within the parabolic rib (5.2) and (5.4) can be translated along vertical lines to fill in the strip *X* ≤ *x* ≤ 2*L*−*X*, *y* ≥ *Y* . The resulting region *R*^{C}_{I} only contains a single parabolic member and continuously extends the field within *T*_{III} to cover the rest of the space above the structure.

### (b) The field beneath the structure

We have been unable to construct an analytically tractable combination of *T*-type and *R*-type fields to cover the space below the structure. Nevertheless, since this space does not contain any load-bearing members, the problem can be greatly simplified by considering ‘suboptimal’ displacement fields, as e.g. discussed by Rozvany & Gollub (1990), Rozvany *et al.* (1997) or Melchers (2005). The transmissibility of the applied loads can be ensured by following Rozvany & Wang (1983) who introduce rigid weightless bars to connect the points of actual load application to the bottom of the fan. This assumption results in vertical strains vanishing everywhere below the structure, i.e. *ε*_{y}≡0.

The field within the region *O*_{I} is constructed by additionally constraining the horizontal strains so that *ε*_{x}≡0. One consequence of these assumptions is that the horizontal displacement at a point within *O*_{I} is equal to the horizontal displacement at the point of the bottom boundary of the fan with the same ordinate. Also, the vertical displacement at a point within *O*_{I} is equal to the vertical displacement at the point of the bottom boundary of the fan with the same abscissa. Importantly, the magnitudes of the associated principal strains are then equal to the magnitude of the shearing strain. By invoking expressions (B1) and (B2), one can compute
6.7
where *β*′ and *β*′′ are the points on the bottom of the fan with the same horizontal and vertical displacement as at the current point, so that *ϕ*_{1} ≥ *β*′′ ≥ *β*′ ≥ 0. The value of the corresponding shearing strain,
6.8
is clearly within the allowable limits: −*ϵ* ≤ *ε*_{xy} ≤ −*ϵ* + 2*ϵϕ*_{1}. Therefore, the field within the region *O*_{I} satisfies the requirements of the Michell–Hemp criteria.

The transmissibility of loads below the parabolic rib implies that the vertical displacements within the region *O*_{II} remain constant along verticals and can be found explicitly from equation (5.4). The horizontal displacements within the region *O*_{II} can be chosen by scaling the horizontal displacements in the parabolic rib (5.2) to ensure continuity with the field within *O*_{I}, i.e.
6.9
where *X* ≤ *x* ≤ *L*, 0 ≤ *y* ≤ *Y* ; *U*_{x} is given by equation (5.7) and *u*_{x}(*X*,*y*) = *u*_{x}(*β*)|_{β = β(y)} is computed using equations (B1) and (B2). After several straightforward algebraic manipulations, one can obtain explicit expressions for the Cartesian strains within *O*_{II} in the form
6.10
and
6.11
where, as before, *β*′′(*y*) denotes the point at the bottom of the fan with the same ordinate as the current point. Although relatively simple, expressions (6.10) and (6.11) cannot be used to perform direct analytical analysis of the principal strains. Therefore, we computed principal-strain values numerically to six significant figures using a fine mesh, and found them to remain within the allowable limits across the whole of region *O*_{II}. The resulting principal-strain values are plotted in figure 6. In view of the simple analytical structure of equations (6.10) and (6.11), the presented numerical evidence would appear to be sufficient to allow us to conclude that the displacement field described fully satisfies the requirements of the Michell–Hemp criteria (2.1) across the whole of region *O*_{II}. Thus, we have now completed the construction of a continuous displacement field that covers the entire half-plane, and that fully satisfies the requirements of the Michell–Hemp criteria. This therefore completes the formal proof of global optimality of the structure.

## 7. Discussion

In the preceding sections, the steps required to establish the optimality of a hybrid structure that combines elements of a ‘Prager’ structure (the transmissible loads giving rise to a funicular across part of the span) and elements of a conventional ‘Michell’ structure (involving regions of orthogonally arranged truss bars) have been described. It appears to be the first time that the formal optimality of a structure that combines both these elements has been established in the literature, and also the first time a non-funicular optimal structure has been identified when transmissible loads are involved.

The approach used in our analysis essentially involves three stages. Firstly, the region over which a Prager structural form is optimal, and where virtual-strain requirements are satisfied, is found. Secondly, the boundaries beyond which a Prager structure would violate the virtual-strain limits are identified. As a final step, conditions at these boundaries are used to help define the forms of the richer Michell structures that must lie beyond these boundaries. It is likely that this approach is applicable to a wider range of problems.

Although the problem considered here may appear highly idealized, involving as it does only a single load case and also Hencky-net-fan regions that would be prohibitively expensive to fabricate using traditional means, it is worth noting that the present study has produced a precise volume that is valuable when assessing the comparative efficiency of real structures. Furthermore, as Rozvany (1998) and Lewiński & Rozvany (2007, 2008*a*,*b*) have noted, the development of computational layout optimization approaches over recent decades (e.g. Sigmund 2000) has led to a need for non-trivial analytical benchmark problems for the purpose of assessing the accuracy of approximate numerical solutions. In this context, the fact that the optimal structure presented here has a volume only fractionally less than that of the optimal parabolic funicular is perhaps of less importance than the fact that the optimal structure has been shown to be markedly different in form from that of a parabolic funicular.

Finally, it should be pointed out that the structural form identified has been formally proved to be globally optimal only when the load is applied to the underside of the structure. This means that the existence of a yet more optimal structure, for which this requirement is not imposed, cannot be discounted. However, the very close proximity of the solution obtained here to the numerical solution of Darwich *et al.* (2010), which was obtained without the need for this requirement to be imposed, would appear to render it highly improbable that in practice a more optimal structure exists.

## 8. Conclusions

This paper has addressed the problem of how to most efficiently carry a uniformly distributed load between pinned supports using a material possessing equal tensile and compressive strength. The geometry of a structure with a total volume 0.3495 per cent lower than that of an equivalent parabolic funicular has been derived. The structure comprises a central parabolic section and Hencky-net-fan regions emerging from the supports, confirming previous numerical findings (Darwich *et al.* 2010). Additionally, the relevant stress and virtual-displacement fields have been checked in a rigorous manner to formally establish the global optimality of this newly derived structural form.

## Acknowledgements

A.V.P. gratefully acknowledges that his work on this paper was supported by the ‘BRIEF’ Research Award from Brunel University.

## Appendix A. Coordinates of the top of the fan

The integral (4.6) can be evaluated along the lower boundary of the fan, which leads to the following expression for the coordinates of the top:
A1
Since *I*_{−1}(*x*) = *I*_{1}(*x*), the infinite sums within the definitions of the Lamé parameters (4.4) and (4.5) reduce to identical expressions at the lower boundary, which leads to the conclusion that
A2
where equation (3.7) is used. Using the indefinite integrals,
A3
found in Gradshteyn & Ryzhik (2007), a new function may be introduced,
A4
The direct application of equations (A1), (A2) and (A4) leads then to an explicit definition of *X* in terms of *G*(2*ϕ*_{1}), see equations (4.7) and (4.8). It is worth remarking that for 0 < *ϕ*_{1} < 1, an alternative, faster converging, expression for *G*(2*ϕ*_{1}) may be obtained by using the recurrence relationships for Bessel functions, yielding
A5

The evaluation of the second integral on the right-hand side of equation (A2) requires the change of the integration order, so that
A6
which, after referring to equations (A1) and (A2), gives the expression for *Y* in equation (4.7).

## Appendix B. Displacements at the top of the fan

The path integral (5.5) can be written more explicitly when the path is chosen to follow the lower boundary of the fan, resulting in the following expressions for the curvilinear displacements:
B1
and
B2
The displacements at the top of the fan (*U*,*V*) can then be found by selecting *β* = *ϕ*_{1} in equations (B1) and (B2). The rotation is known, see equation (5.6), hence the use of integrals (A2) and (A6), computed in appendix A, immediately leads to significant simplifications
B3
and
B4
The similarities within definitions (4.4) and (4.5) enable one to establish an analogue of the relation (A2) in the form
B5
which in view of the fact that leads to the expression for the vertical displacement (5.8). The remaining integral can also be tackled using integration by parts and changing the integration order, which, after simplification, leads to
B6
Since , expression (5.7), which defines the horizontal displacement at the top of the fan, can be obtained as the direct consequence of (B3)–(B6).

## Appendix C. The volume of the optimum structure

The volume of the resulting structure may be calculated from the virtual work of the distributed load divided by *ϵσ* (Hemp 1973, p. 72). It is convenient to write the resulting integral as the following sum:
C1
with the integral term reflecting the contribution of the fan and the remaining terms representing the contribution of the parabolic rib, see equations (4.6) and (5.4). Once again, the remaining integral can be evaluated explicitly using equations (4.4), (4.5) and (B1), (B2). The application of integration by parts and several changes of integration order yield
C2
The substitution of this result back into equation (C1), followed by the use of equations (3.2), (4.9), (5.8) and (5.9), immediately leads to expression (5.10).

- Received July 19, 2010.
- Accepted September 17, 2010.

- This journal is © 2011 The Royal Society