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Proc. Insrn Cio. Engrs, Part 2, 1981, 71, June, 597-600

8364 DISCUSSION

The estimation of the strength of masonry arches

J. HEYMAN

Dr C. Melbourne, Greater Manchester Council The Paper gives some interesting background to the MEXE/MOT method of assessment and raises some doubts in my mind. Does the Author know how the .provisional axle load given in BE 3/736 was derived from the Pippard analysis which assumed a single axle at the crown of a two-pinned arch, bearing in mind that the resulting allowable axle relates to a standard train of vehicles and not the single axle used in the analysis?

69. The plastic method of analysis presented bears many similarities with the method proposed by Pippard and Baker." In fact, for any particular arch the ultimate single knife-edge loading determined by the Pippard and Baker method compares fairly closely with values of P determined from Table 1. Although these methods, as presented in the literature, can be used to determine a permissible axle for construction and use vehicles they do not seem to have been extended to give guidance on the assessment of,masonry arches required to carry abnormal heavy loads.

70. At present 7500 abnormal load movements are notified each year to the Greater Manchester County Council, of which 250 are in excess of 100 tons. Many of the heavy load routes through the county pass over masonry bridges, posing the important problem of how best to assess the load carrying capacity of these bridges with respect to the abnormal heavy loads passing over them. Although the MEXE/MOT method gives some guidance, BE 3/736 specifically states that for spans greater than 60 ft and heavier vehicles the method should be used with caution.

71. 1 am investigating a method of assessment based on the Pippard and Baker method. Two fundamental criteria have to be established. First, the geometrical factor of safety, which I have taken to be 2. Second, the idealization of the axle loading for which Chettoe and Henderson' concluded that a disper- sal angle of 45" was reasonable if the fill over the arch was sound. (A dispersal angle of 30" appears reasonable if the fill is suspect.) This is contrary to BE 1/77,*' which indicates a dispersal angle of 30" for wheel loads on buried structures. Using these criteria in conjunction with the Pippard and Baker method and making allowance for the width of the arch, an estimate of the provisional axle load for a given axle configuration can be determined. The

Paper published: Proc. Instn Cio. Engrs, Part 2, 1980, 69, Dec., 921-937.

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engineer must then apply a condition factor in a similar manner to the BE 3/736 method to arrive at an allowable axle load.

72. Does the Author intend to extend his work to cover this problem?

M r J. W. Mann, M The Author has produced a commendably simple approximate solution to the problem, but how far is it applicable in practice? Most assessment work is concerned specifically with the passage of real vehicles over the bridge. How would the Author deal with multiple axle as opposed to single point loading, particularly on long spans? In the case of elliptical or semicircular arch rings, how would the hinge points near the springings be determined?

74. Most older arch structures needing assessment are not in good condi- tion. It therefore seems unwise to place too much confidence in the capacity of the joints in the arch ring to transmit tensile forces when. that capacity can be destroyed by a hairline crack. Such cracks occurring, say, as a result of settle- ment, would reduce the'effective thickness of the arch ring if tension is theor- etically permitted. Could the Author's method deal with a deformed arch?

75. As the Author rightly says ($4 61-62) the elastic method of analysis does not necessarily furnish an accurate reflection of the state of the arch when loaded. It is equally true to say that the calculated collapse load will not neces- sarily be that load which will actually cause the arch to collapse, owing to such factors as damage, cracking, distortion and abutment movement. In assessing a permissible load, a factor of safety must be applied to the calculated collapse load-which factor, as the Author points out($ 66), is open to question. The practical difficulty lies in determining a'factor which guarantees that there will be no damage to the arch ring under the maximum permitted load. I do not see how any such factor can be arrived at other than empirically. The only alterna- tive is testing to destruction.

76. The geometrical factor of safety propounded by the Author in reference 1 is a guide, but only a guide, to the fitness and economy of design of the arch in relation to the loads it carries. For assessment purposes it does not say with what factor of safety a real load can be carried within permitted limits. Using a geometrical factor of safety of 3 instead of 2 would ensure that there could be no miscalculation due to the fact that some parts of the arch ring might not be able to bear tensile forces. Would the Author modify his calculated load still further by empirical factors reflecting the actual condition of the arch, e.g. as specified in BE 3/73?6

77. A simple method was developed in the early 1970s on British Railways (Western Region) for the assessment of arched bridges. It was evolved because

, existing methods of analysis were found to be either inadequate or inordinately protracted for three particular structures. These were: a very flat arch with span :rise ratio of 11 : 1 which fell outside the scope of BE 3/73; a pointed arch the shape of which required special treatment; an arch with asymmetrical dead loading due to a sloping road surface, which was required to carry an extremely heavy abnormal load.

78. The WR method can be applied to any shape of arch and any position and type of loading. It proceeds along the same line of approach as the Paper but uses Eddy's theorem to construct the equations from which the profile of the line of thrust is determined. The method can be used either to predict the

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8 3 6 4

consequences of a particular loading or to arrive at a maximum permissible loading within set limits and criteria.

79. I would agree with the Author's selection of the quarter point as being critical for point loading. In fact, the successive trial of several points in that region which he used in analysing the Teston Bridge" is also invoked in the WR method.

80. The advantages of the WR method, apart from simplicity and speed, are that it makes no assumptions about the shape or thickness of the arch ring but deals directly with the actual profile; the use of Eddy's theorem means that a complete thrust line can be derived and its acceptability verified by drawing. This method has been successfully used since 1972 for arched bridges which for various reasons could not be assessed by either the M O T or BRB standard methods.

P r o f e s s o r H e y m a n The Paper was concerned with the derivation of a quick method for determining the maximum value of axle load that could be permitted to cross a masonry bridge. Thus the results were intended to be directly comparable with those given by the Pippard analysis,' which in turn formed the basis of BE 3/73.6

82. Dr Melbourne remarks on the problem of a standard train of ,vehicles, and on the problem of the abnormal load, and Mr Mann also refers to the passage of real vehicles with multiple axles. The solution of these problems requires the drawing of funicular polygons for the various loading cases to be considered, as was mentioned in 9 40.

83. Thus a quick assessment by the method of the Paper must be followed if necessary by an accurate analysis. For bridges of small span the single axle will in general be the most critical loading condition; a multiple-axle vehicle of reasonable length will produce a more balanced loading than the single axle at quarter span. This was certainly the case in the extended analysis that was undertaken of Teston bridge," which admittedly had a main span of only 7.2 m.

84. It is indeed likely that up to a certain limit (perhaps the 60 ft specified by BE 3/73) the multiple-axle loading will not be critical, and the quick method will give a satisfactory analysis. However, there is no difficulty in drawing the funicu- lar polygon for any given pattern of loading, and such a procedure is valid for arches of large span and of unusual shape.

85. I agree with Mr Mann that the final basis for the selection of the value of the geometrical factor of safety must be empirical. I believe, and so apparently does Mr Melbourne, that a value of 2 is adequate. It is, however, wrong to assume that a value of 3 would ensure that tension is not developed in the arch ring.

86. It is certainly possible to design an arch ring so that, on the drawing board, the line of thrust is contained within the middle third. In practice, how- ever, the imperfections of the construction will inevitably compel the thrust line to depart from the middle third. I t is at this point that the designer can invoke the powerful safe theorem of plasticity (M 29-23). Whatever the practical imper- fections, the arch will retain, without diminution, its calculated margin of safety.

87. Thus the fact that an arch may be cracked is not relevant to the calcula- tion of its safety. Deformations must not, of course, be so large that the overall geometry is seriously upset, but this is commonplace in any structural theory.

88. In general, the assumptions made in the Paper are conservative. For

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example, loads were taken to be transmitted without dispersal through the fill. It is evident that any dispersal which occurs, whether 30" or 45", will be beneficial. In fact, the matter is not critical for arches with relatively small cover.

89. The method of the Paper was not intended to replace the traditional methods of analysis. It is an approximate method and for complex loading systems a final check must be made using a graphical method.

References 21. PIPPARD A. J. S. and BAKER J. F. The analysis of engineering structures, 3rd edn.

22. MINISTRY OF TRANSPORT. Standard highway loadings. Ministry of Transport, London, Arnold, London, 1957.

1977, Tech. Memo. (Bridges) BE 1/77.

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