The core of this research is finding the momentless shape of a two-pin arch with the use of static equilibrium, highlighting the excellence of its application, which has not been attempted hitherto. This work has been preceded by the assessment of known shapes of arches that advances understanding the behaviour of two-pin arches as a function of chosen arch form. A comprehensive assessment of known arch shapes and a comparison of their structural performance are carried out numerically, which has not hitherto been done. The degree of credibility of approximate analysis methods of two-pin arches consisting of masonry design (Curtin et al. 2006) and virtual work based on bending action only (Megson, 2005), is also evaluated for the first time.
In this chapter, the principal findings of the author’s PhD work in association with the effect of arch form on its structural action and the optimal form of two-pin rib arches are presented. The general behaviour of a two-pin arch of constant cross-section is presented in Section 6.1. Section 6.2 scopes a comparison of two alternative approximate methods of arch analysis with a numerical method using GSA software and the second theorem of Castigliano. A concise review of analytical finding of the form of a momentless two-pin arch and the application of a momentless arch are given in Section 6.3. Section 6.4 is used to present conclusions from the new contribution to knowledge and understanding. Finally, various research areas for future work are presented in Section 6.5.
6.1. Discussion of the general behaviour of the two-pin arch
The behaviour of two-pin arches of known shapes was studied comprehensively and compared to each other in Chapter 3. Although the behaviour of a circular arch and comparison of the buckling load of known arch shapes have been investigated
widely, the inclusive performance of arch structures based on their structural form has received little attention. The collapse of Gerrards Cross Tunnel (NCE 2005) in 2005 was an example of the importance of choosing the best arch shape, depending on load cases during execution and operation
Common arch shapes including catenary, parabolic and circular forms with constant cross-sections of concrete were statically analysed numerically using GSA finite element software for different ratios of UDL:SW, and for L:h ratios between 2–10 in
Section 3.5. According to the Timoshenko stability equation (Timoshenko and Gere 1961), the arches were stable for the considered applied loading. The arches were then modelled using 80 straight elements following a sensitivity analysis for the variation of bending moments. The criterion of developing minimum combined stress due to bending and thrust was used to find the optimal common arch shape and best range of L:h ratio. The three arch forms were evaluated using four different load cases. The maximum sagging bending moment with parabolic and catenary arches reduced when L:h increased from 2 to 5 and for higher ratios the maximum bending moments increased. The minimum sagging bending moment of circular arch was observed at L:h = 8. It was seen that increasing the L:h ratio to 300 led the maximum bending moment to grow and be close to that of the equivalent horizontal beam. The maximum sagging and hogging bending moments for the catenary and circular arches occurred at approximately the same location which are respectively at arch crown and at both sides, different for the parabolic arch. The sagging bending moments of parabolic and catenary arches are greater than the hogging ones for any L:h ratio. However, in the case of the circular arch, the maximum hogging bending moment was higher than maximum sagging bending moment when L:h < 6, showing the dominance of tensile forces when the loading was a general combination of uniformly distributed load (UDL) and self-weight (SW). The absolute value of maximum bending moment of the parabolic and catenary arches ratio between 4–5; this ratio was between 7–8 for
the circular arch. The displacement and shear forces for the arches gave distributions over the arch shape that are similar in shape to that of the bending moment. Since the deformation of the parabolic arch with UDL+SW was different to the catenary and circular arches, the parabolic arch is expected to fail with the different mode. The deformed shape of each arch was similar for any load condition, excluding the patch loading condition. The geometry of different arch forms became almost identical for L:h ratios abovef 7. However, it was shown that thereremains differences in the structural response, even at the L:h ratio of 10. One important finding from the author’s work is that small changes in shape may noticeably affect the structural results. The presence of tensile stress in the arch was inevitable when the load case had the patch load over half of the arch length. In general, the maximum stress for the parabolic and catenary arches had its minimum when 2 <
L:h < 4; it was between 4–6 for the circular arch. This range of L:h ratios was referred to as the optimal ratios. To generalise this result, the arch made of a hollow steel cross-section that had the same second moment of area as the concrete solid cross-section was analysed with load case UDL+SW. Although the magnitude of the combined stresses from the steel arch was different to that of the concrete one, the overall behaviour was the same.
The relation between the maximum thrust of the arches and the L:h ratios was almost linear, with the maximum value of thrust at the supports. The difference in the horizontal reaction forces for the three arch shapes was indistinguishable when ratio L:h exceeded 5. There were noticeable differences for the bending moments and combined stresses. This indicates the importance of calculating the horizontal reaction force. Since the horizontal reaction force has an increasing linear relation with increasing L:h ratio, a lower L:h ratio is preferable to design for this action. It has been shown that the arch mass is not largely affected by shape, and that it decreases with increasing L:h ratio. The best range of L:h ratios assuming mass