Topology 2/3 (limiting case between topologies 2 and 3)
This topology with \(h=d/\sqrt 2 \) is discussed first, because for this case we have a complete adjoint strain fields for all analytical solutions, some of which are shown in Fig. 2. The optimal solution for Topology 2/3 depends on the relative magnitude of (L − d) and h.
If (L − d) = 0, then we have the solution in Fig. 2a, which is the classical solution by Michell (1904). The strain fields for the quarter domain consist of two T-regions: one has constant principal directions, the other one consists of a circular fan. This solution is in fact valid even when h > h
1, because the T-regions shown in Fig. 2a can be extended even to the entire plane, satisfying the optimality condition of Michell in (2).
The layout in Fig. 2a will become part of all other topologies in Fig. 2b–d (shown partially on the left hand side of these diagrams). The right hand side of these layouts consists of concentrated horizontal bars along the domain boundaries. It is important to note that these optimal topologies are only valid if the domain has a limited height, i.e. \(h=d/\sqrt 2 \).
This is because in Fig. 2b, for example, kinematic continuity would not be satisfied if we moved upwards the present domain boundary (the strains in the two T-regions would cause an overlapping of the horizontal displacements). However, since the adjoint strain field in O-regions may be non-unique (see Section 4.2), there should be some other adjoint displacement field for the present problem of restricted height, which can be extended for greater height values.
In Fig. 2b and d the central part of the adjoint strain field consists of T-regions and Z-regions, see the definitions above. In Fig. 2c, however, there is also an R-region, with a horizontal principal strain of \(\bar{{\varepsilon }}_1 =-k\). The value of \(\bar{{\varepsilon }}_2 \) can be calculated from the condition that the strains must be zero in the direction of the region boundary with the Z-region. Then elementary calculations give \(\bar{{\varepsilon }}_2 =k\tan^2\alpha \). This implies \(\bar{{\varepsilon }}_2 =0\) and \(\bar{{\varepsilon }}_2 =k\), respectively, for Fig. 2b (with α = 0) and Fig. 2d (with α = π/4).
The solution in Fig. 2b was actually presented by the second author previously (Rozvany 2011).
Extending the region pattern in Fig. 2b to d, we can also construct optimal strain fields for any value of L − d > 2h.
For 0 < L − d < h the optimal topology is discussed in greater detail subsequently. For this case the optimal adjoint strain field is shown in Fig. 3a. Each quarter of the central, memberless O-region consists of a T-region and a V-region (see Section 1). The state of adjoint strains in these two regions, respectively, is represented by the Mohr-circles in Fig. 3b and c. It can be seen from Fig. 3b that the strain along the boundary of the T- and V-regions is
$$ \bar{{\varepsilon }}_{\rm B} =k\,\cos\left( {2\alpha } \right). $$
(4)
Moreover, one can infer from Fig. 3c that for the V-region, we have
$$ k\,\cos\left( {2\alpha } \right)=\left( {\bar{{\varepsilon }}_{1} /2} \right)\left( {1+\cos\left( {2\alpha } \right)} \right)\Rightarrow \bar{{\varepsilon }}_{1} =k\big({1-\tan^{2}\alpha }\big). $$
(5)
It can be seen that for α = 0 and α = 45° (5) gives the correct \(\bar{{\varepsilon }}_{1} \) values for the limiting cases in Fig. 2a and b. The above adjoint strain field satisfies all optimality criteria of Michell (1904) (see also (2) in this paper). With this addition, we have complete analytical solution for all aspect ratios of Type 2/3 topologies, and also for the O-region of all Type 3 topologies.
Topology 1
The topology of the optimal truss for h = ∞ (or h ≥ h
1, see Fig. 1b) can be constructed as follows. We can infer from numerical results that the optimal layout is that of a ‘Michell cantilever’Footnote 1 and a horizontal bar. The geometry of this structure is fully determined by three angles: θ
1, θ
2 and γ
2, shown in Fig. 4.
To make the paper self-consistent, it is useful to outline the so-called Lommel functions U
n
(·,·) that play the fundamental role in deriving coordinates and the displacement field in the Hencky net of mutually orthogonal members in region ABDC (see Fig. 4). Following the notation by Lewiński et al. (1994) we will use in the present paper two functions: G
n
(·,·) and F
n
(·,·), defined as
$$ G_n \left( {\alpha ,\beta } \right)=\left( {\frac{\alpha }{\beta }} \right)^{n/2}I_n \left( {2\sqrt {\alpha \beta } } \right) $$
(6)
and
$$ F_n \left( {\alpha ,\beta } \right)=\sum\limits_{m=0}^\infty {\left( {-1} \right)^mG_{2m+n} \left( {\alpha ,\beta } \right)} , $$
(7)
where I
n
(·) denotes the modified Bessel function of the first kind. Many important properties of functions G
n
and F
n
can be found in the paper mentioned before (Lewinski et al. 1994).
Now, the coordinates of the point D, connecting the horizontal bar with the Michell continuum, can be written as:
$$ \begin{array}{rll} x_{\rm D} &=&d\left[ \cos \left( \gamma_2 \right)\cos \left( \gamma_2 +\theta_2 -\theta_1 \right)\,F_0 \left( \theta_1 ,\theta_2 \right) \right. \\ &&{\kern8pt} +\,\sin \left( \theta_2 -\theta_1 \right)\;F_1 \left( \theta_1 ,\theta_2 \right) \\ &&{\kern8pt} -\left.\sin \left( \gamma_2 \right)\sin \left( \gamma_2 +\theta_2 -\theta_1 \right)\;F_2 \left( \theta_1 ,\theta_2 \right) \right], \\ y_{\rm D} &=&d\left[ \cos \left( \gamma_2 \right)\sin \left( \gamma_2 +\theta_2 -\theta_1 \right)\,F_0 \left( \theta_1 ,\theta_2 \right) \right. \\ &&{\kern8pt} +\,\cos \left( \theta_2 -\theta_1 \right)\;F_1 \left( \theta_1 ,\theta_2 \right) \\ &&{\kern8pt} +\left.\sin \left( \gamma_2 \right)\cos \left( \gamma_2 +\theta_2 -\theta_1 \right)\;F_2 \left( \theta_1 ,\theta_2 \right) \right]. \end{array} $$
(8)
Equations (8) are obtained by rearranging (2.9–11) of the paper by Sokół and Lewiński (2010). In the latter paper the displacement field of the domain RBDCNR (divided by appropriate sub-regions) was derived in detail. In particular, the adjoint displacements of points N and D (normalized with k = 1) are defined by:
$$ \begin{array}{rll} \emph{w}_x^{\rm N} &=&d\,\cos \left( {2\gamma_2 } \right), \\ \emph{w}_y^{\rm N} &=&d\,\left[ {-1+\sin \left( {2\gamma_2 } \right)} \right] \end{array} $$
(9)
and
$$ \begin{array}{rll} \emph{w}_x^{\rm D} &=&u^{\rm D}\cos \left( {\gamma_2 +\theta_2 -\theta _1 } \right)-\emph{v}^{\rm D}\sin \left( {\gamma_2 +\theta_2 -\theta_1 } \right), \\ \emph{w}_y^{\rm D} &=&u^{\rm D}\sin \left( {\gamma_2 +\theta_2 -\theta _1 } \right)+\emph{v}^{\rm D}\cos \left( {\gamma_2 +\theta_2 -\theta_1 } \right), \end{array} $$
(10)
where
$$ \begin{array}{rll} u^{\rm D}\!/d&=&2\theta_1 \sin \left( {\gamma_2 } \right)G_0 \left( {\theta _1 ,\theta_2 } \right) \\ &&+\,\cos \left( {\gamma_2 } \right)\left[ {G_0 \left( {\theta_1 ,\theta_2 } \right)+2\theta_2 G_1 \left( {\theta_1 ,\theta_2 } \right)} \right] \\ &&+\left[ {\cos \left( {\gamma_2 } \right)-\sin \left( {\gamma_2 } \right)} \right]\left[ {F_1 \left( {\theta_1 ,\theta_2 } \right)-F_2 \left( {\theta_1 ,\theta_2 } \right)} \right], \\ \emph{v}^{\rm D}/d&=&-2\theta_2 \sin \left( {\gamma_2 } \right)G_1 \left( {\theta _1 ,\theta_2 } \right) \\ &&\!-\left( {1+2\theta_2 } \right)\cos \left( {\gamma_2 } \right)G_0 \left( {\theta_1 ,\theta_2 } \right) \\ &&\!+\!\left[ {\cos \left( {\gamma_2 } \right)-\sin \left( {\gamma_2 } \right)} \right]\!\left[ {F_1 \left( {\theta_1 ,\theta_2 } \right)+F_2 \left( {\theta_1 ,\theta_2 } \right)} \right]. \end{array} $$
(11)
The (8)–(11) are general and valid for any angles: θ
1, θ
2 and γ
2. They can be simplified considerably for a specific problem. For the problem under consideration, one can show that the angles θ
1, θ
2 and γ
2 must satisfy
$$ \gamma_2 =\pi /4\quad{\rm and}\quad\theta_1 =\theta_2 +\gamma_2 . $$
(12)
The first relation under (12) comes directly from the zero value of the horizontal displacement at point N: \(\emph{w}_x^{\rm N} =0\) and (9)1. The second constraint can be derived from the fact that the upper chord BD should connect smoothly to horizontal bar DS at the point D, see (5.1) by Sokół and Lewiński (2010) and the detailed explanation given there. Now, by (12) the coordinates of point D can be expressed as
$$ \begin{array}{rll} x_{\rm D} &=&\frac{d}{\sqrt 2 }\left[ {F_0 \left( {\pi /4+\theta_2 ,\theta _2 } \right)+F_1 \left( {\pi /4+\theta_2 ,\theta_2 } \right)} \right]\,, \\ y_{\rm D} &=&\frac{d}{\sqrt 2 }\,\left[ {F_1 \left( {\pi /4+\theta_2 ,\theta_2 } \right)+F_2 \left( {\pi /4+\theta_2 ,\theta_2 } \right)} \right]\,, \end{array} $$
(13)
while the displacements of the point D can be written as
$$ \begin{array}{rll} \emph{w}_x^{\rm D} &=&\frac{d}{\sqrt 2 }\left[ {\frac{\pi }{2}\;G_0 \left( {\pi /4+\theta_2 ,\theta_2 } \right)+g\left( {\theta_2 } \right)} \right], \\ \emph{w}_y^{\rm D} &=&\frac{-d}{\sqrt 2 }g\left( {\theta_2 } \right), \end{array} $$
(14)
where the auxiliary function
$$ g\left( \alpha \right)=\left( {1+2\alpha } \right)G_0 \left( {\pi /4+\alpha ,\alpha } \right)+2\alpha \,G_1 \left( {\pi /4+\alpha ,\alpha } \right) $$
(15)
is introduced to shorten the notation.
Note that both the horizontal and vertical displacements of the point N defined in (9) are equal to zero for γ
2 = π/4. The reason for this strange result comes from the fact that the displacements (9) and (10) are derived by means of a rigid rotation around point R. The magnitude of this rotation is not known in advance but can be determined from the boundary conditions. This procedure was described in detail by Sokół and Lewiński (2011a, see (23) and (24) there). The displacements of points N and D adjusted in this way (without normalization) can be written as
$$ \begin{array}{rll} \overline{{\emph{w}}}_x^{\rm N} &=&k\emph{w}_x^{\rm N} \\ \overline{{\emph{w}}}_y^{\rm N} &=&k\left( {\emph{w}_y^{\rm N} +\psi \,d} \right) \end{array} $$
(16)
and
$$ \begin{array}{rll} \overline{{\emph{w}}}_x^{\rm D} &=&k\left( {\emph{w}_x^{\rm D} -\psi y_{\rm D} } \right) \\ \hfill \overline{{\emph{w}}}_y^{\rm D} &=&k\left( {\emph{w}_y^{\rm D} +\psi \,x_{\rm D} } \right) \end{array} $$
(17)
where ψ denotes the angle of the rigid rotation. It can be determined from the zero value of the horizontal displacement of point S, lying at the vertical axis of symmetry (see Fig. 4). We know that for fully stressed truss the virtual elongation of the bar DS is equal to its length (L − x
D), thus
$$ \overline{{\emph{w}}}_x^{\rm S} =\overline{{\emph{w}}}_x^{\rm D} +k\left( {L-x_{\rm D} } \right)=0. $$
(18)
From (18) and (17)1 one can easily deduce that
$$ \psi =\frac{L-x_{\rm D} +\emph{w}_x^{\rm D} }{y_{\rm D} }. $$
(19)
Now, the adjusted (properly rotated) displacement \(\overline{{\emph{w}}}_y^{\rm N} \) can be written as
$$ \overline{{\emph{w}}}_y^{\rm N} =\frac{k\,d}{y_{\rm D} }\left( {L-x_{\rm D} +\emph{w}_x^{\rm D} } \right) $$
(20)
where x
D, y
D and \(\emph{w}_x^{\rm D} \) are given in (13) and (14).
The forces for the layout in Fig. 4 are given by: Q
y
= P/2, F
x
= Q
y
d/y
D and F
y
= 0. The horizontal reaction Q
x
at point N can be obtained from Chan’s formula (see (2.47) in the paper by Sokół and Lewiński (2010)), while the reactions at point R can easily be obtained from the equilibrium equations. These reaction forces, however, do not generate any virtual work in (3), because the corresponding virtual displacements are zero (points R and N in Fig. 4).
The volume of one quarter of the full structure can be calculated by adding the volume of the Michell’s continuum and the volume of the horizontal bar DS (see Sokół and Lewiński 2010). However, it is more elegant and convenient to use the dual method (see (3))
$$ V=Q_y \overline{{\emph{w}}}_y^N =\frac{Pd}{2y_{\rm D} \sigma_p }\left( {L-x_{\rm D} +\emph{w}_x^{\rm D} } \right). $$
(21)
The volume in (21) is a function of one variable V = V(θ
2) and its minimum can be determined from the necessary condition V′(θ
2) = 0. This leads to the transcendental equation
$$ \xi \,q\left( {\theta_2 } \right)=\sqrt 2 , $$
(22)
where ξ = d/L and function q(θ
2) is defined by
$$ \begin{array}{rll} q\left( {\theta_2 } \right)&=&\left( {\pi /2+4\theta_2 } \right)\left[ {F_1 \left( {\pi /4+\theta_2 ,\theta_2 } \right)} \right. \\ &&-\left. F_0 \left( {\pi /4+\theta_2 ,\theta_2 } \right) \right]+2F_1 \left( {\pi /4+\theta_2 ,\theta_2 } \right) \\ &&+\,2\theta_2 \left[ {G_0 \left( {\pi /4+\theta_2 ,\theta_2 } \right)-G_1 \left( {\pi /4+\theta_2 ,\theta_2 } \right)} \right].\\ \end{array} $$
(23)
Equation (22) uniquely defines the optimal angle θ
2 because the function q(θ
2) is monotonic (it is a decreasing function for θ
2 ≥ 0 which starts from \(q\left( 0 \right)=\sqrt 2 \) and then asymptotically approaches 0 for θ
2 → ∞). The lower limit of ξ for which the solution of (22) is valid for the rectangular domain shown in Fig. 1a is equal to ξ = 0.182027. This corresponds to θ
2 = π/4 which means that the upper chord RBDS starts vertically from the support. For lower values of ξ the external fans extend beyond vertical lines drawn above the supports and the solution is formally infeasible. However, if we allow the horizontal expansion of our rectangle domain outside the supports, we can obtain the feasible solution for ξ < 0.182027. In this case the lowest limit of ξ for which the solution of (22) makes physical sense is equal to ξ = 0.0477491. This corresponds to θ
2 = π/2 and θ
1 = 3π/4. For lower values of ξ the internal circular fan goes outside the symmetry line connecting two supports, and that is obviously infeasible. Thus we can conclude that for ξ < 0.0477491 the optimal solution is not known.
Examples of optimal layouts of Topology type 1 are shown in Fig. 5. For increasing ξ the angle θ
2 decreases and at the same time the height and the volume of the structure increase. Obviously, the final solution for ξ = 1 is identical with the well known Michell’s solution (1904) (with θ
2 = 0, θ
1 = π/4, and 4V = PL/σ
p
(2 + π)). All solutions have been confirmed by numerical calculations. The exact layouts are given on the left side and their numerical equivalents on the right.
The exact and numerically calculated volumes are compared in Section 3.
Topology 2
In this case y
D
= h and then by (13)2 we have
$$ h=\frac{d}{\sqrt 2 }\left[ {F_1 \left( {\pi /4+\theta_2 ,\theta_2 } \right)+F_2 \left( {\pi /4+\theta_2 ,\theta_2 } \right)} \right]. $$
(24)
Obviously this case is simpler than the previous one (compare (22), (23) and (24)). For the limiting case \(h=d/\sqrt 2 \) one can easily obtain θ
2 = 0 and then the total volume of the optimal truss becomes
$$ 4V=\frac{P{\kern 1pt}L}{\sigma_p }\left[ {2\sqrt 2 \left( {1-\xi } \right)+\left( {2+\pi } \right)\xi } \right]. $$
(25)
The above formula gives the correct volume for special cases. For ξ = 1 we obtain 4V = PL/σ
p
(2 + π), which is the solution by Michell (1904). For \(\xi =\sqrt 2 /\big( {1+\sqrt 2}{\kern1pt}\big)\) we obtain \(4V=PL/\sigma_p \left( {4+\pi } \right)\big( {2-\sqrt 2}{\kern1pt}\big)\), which can also be readily derived from the second author’s solution (Rozvany 2011, here Fig. 2b).
Three examples of Topology 2 with \(\xi =\sqrt 2 /( {1+\sqrt 2 } )\) and three different permissible heights are shown in Fig. 6. The topologies in Fig. 6e and f are actually of type 2/3 (see Section 2.1, Fig. 2).
Topology 3
The exact solutions for Topology 3 are not known at present, but the optimal layouts can be predicted on the basis of numerical solutions (e.g. those in Fig. 7). Here the value ξ = d/L = 0.5 was assigned to the length ratio. The layout in Fig. 7a was calculated for \(h=d/\sqrt 2 =L\,\sqrt 2 /4\approx 0.353553L\). It is the limiting case between topology types 2 and 3, hence this layout is similar to the one presented in Fig. 6e and f (see also Fig. 2 in Section 2.1). By progressively decreasing the height h of the structural domain for the numerical solutions, we find that additional T-regions develop in the optimal topology, similarly to a long cantilever (Lewiński et al. 1994), but following a different geometry (see Fig. 8).
In Fig. 8 the upper horizontal bar is connected to the chord of a circular fan having an angular width of 45° and a curved three-sided region, which is similar to that derived by Chan (for details see the paper by Lewiński et al. (1994)). The upper border of this region can be a straight line if the height is sufficiently small (Fig. 8b) or it starts with a straight section and smoothly passes into a curved section (Fig. 8a). It is to be noted that contrary to Topologies type 1 and 2 the external (upper) chord has not a constant cross section in the straight segment. The three-sided domain above the circular fan is connected with a region with straight members in one direction. This is the reason why the rest of the regions with Hencky nets are different from those derived for the long cantilever problem by Lewiński et al. (1994). Nevertheless, there are also some similarities in forming subsequent new regions if the permissible height is decreased. On the basis of many additional numerical tests we have established that switching from the first sub-type (Fig. 8a) to the second one (Fig. 8b) occurs at a height value in the range of h ∈ (0.24d, 0.25d).
The adjoint strain field in empty regions of Topology 3 can be filled the same way as described in Section 2.1 (Fig. 2).