On the optimality of Hemp’s arch with vertical hangers

Abstract

In 1974 W. S. Hemp constructed a prototype structure to carry a uniformly distributed load between two pinned supports. Although Hemp’s structure had a significantly lower volume than a parabolic arch with vertical hangers, it was shown to fail the Michell optimality criteria, and therefore to be non-optimal. In this paper we demonstrate that if limiting compressive and tensile stresses are unequal then Hemp’s structure is optimal for the half-plane provided the ratio of limiting tensile to compressive stresses falls below a certain threshold. An analytical proof is presented and the finding is confirmed by results from large scale numerical layout optimization simulations.

Appendix

Various tests can be devised to confirm the correctness of the analytical derivations presented in this paper. One such test involves computing the integral of the horizontal strains along the line joining the supports, at the bottom of the optimum structure, and checking that this equals zero; details of this test follow.

Using the notation of Section 3.2, and referring to (18) and (27), the sought for integral can be expressed as the sum of two terms

$$ \begin{array}{rll} && \int_0^l \varepsilon'_x|_{y=0} \textrm{d} x\\ &=&\int_0^{\phi_1} \varepsilon'_2|_{\alpha'=0} \frac{\partial x}{\partial\beta'} \,\textrm{d}\beta'\varepsilon'_2|_{\alpha'=0}B(\phi_1-\beta',\beta') \,\textrm{d}\beta'\\ &=&2\ell\sigma\left(\frac{1}{\sigma_C}+\frac{1}{\sigma_T}\right) \int_0^{\phi_1}\int_0^{\beta'} A(\phi_1-\zeta,\zeta) \;\textrm{d}\zeta\,\textrm{d} \beta'\\ &&\quad -\frac{\ell\sigma}{\sigma_C}\int_0^{\phi_1} B(\phi_1-\beta',\beta')\textrm{d}\beta'\,. \end{array} $$

(40)

Reference to (4) immediately indicates that

$$ \int_0^{\phi_1} B(\phi_1-\beta',\beta')\textrm{d}\beta'=x(0,\phi_1)=l\,. $$

(41)

The remaining double (in fact, triple, see (2)) integral on the right hand side of (40) can be expressed in terms of the already known integral H ₀(ϕ ₁), by changing the order of integration:

$$ \begin{array}{rll} && \int_0^{\phi_1}\int_0^{\beta'} A(\phi_1-\zeta,\zeta) \,\textrm{d}\zeta\,\textrm{d}\beta'\\ &=&\int_0^{\phi_1} \left( \int_{\zeta}^{\phi_1}\textrm{d}\beta' \right) A(\phi_ 1-\zeta,\zeta) \,\textrm{d}\zeta\\ &=&\int_0^{\phi_1} (\phi_1-\zeta) A(\phi_1-\zeta,\zeta) \,\textrm{d}\zeta=\frac{l^2}{2h} H_0(\phi_1)\,, \end{array} $$

(42)

see also (14). Equations (41) and (42), when considered in conjunction with identity (15), lead to an immediate conclusion that

$$ \int_0^l \varepsilon'_x|_{\alpha'=0} \textrm{d} x =2\ell\sigma\left(\frac{1}{\sigma_C}+\frac{1}{\sigma_T}\right)\frac{l^2}{2h} H_0(\phi_1) -\frac{\ell\sigma}{\sigma_C} l=0\,, $$

(43)

which is as should be expected.