Optimal arch shape of long-span open-spandrel arch bridges under vertical permanent loads

An arch with an optimal shape can resist loads more efficiently with higher safety. Long-span open-spandrel arch bridges need to have an optimal arch shape. This paper presents a study of obtaining the optimal arch shape of long-span open-spandrel arch bridges under vertical permanent loads. The mathematical model of the optimal arch shape was established firstly for open-spandrel arch bridges. Then the solution procedure of the mathe-matical model was determined. Finally, the mathematical model was applied to two arch bridge cases. The re-sults show that the optimal shape of open-spandrel arch bridges is the linear combination of the thrust lines under both the arch self-weight and the point loads (the deck weight plus the spandrel column or hanger weight). When subjected to the arch self-weight, the arch with variable cross-section has a thrust line equation that is the total of the hyperbolic cosine function, the hyperbolic sine function, and the quadratic polynomial. As for the point loads, the corresponding thrust line is a polyline. The two long-span arch bridge cases demonstrate that the proposed mathematical model is highly accurate and efficient for practical application. The bending moments, bending stress to axial stress ratios, eccentricities, and the square sum of eccentricities generated in the optimal arch shape are significantly less than those generated in conventional arch shapes, such as parabola, catenary, polygon, and spline. The results highlight the advantages of the optimal arch shape presented in this paper over conventional arch shapes, mostly of the parabola and the catenary. The point loads are assumed to be uniformly distributed along the arch span and the arch self-weight is assumed to be uniformly distributed along the arch length in conventional approaches, which may cause significant inaccuracies in forming an arch shape. The mathematical model proposed in this paper avoids the inaccuracies caused by the above-described assumptions, and greatly enhances the calculation accuracy. As a result, the mathematical method proposed in this paper has a wide range of applications in bridge engineering.