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Basic calculations for Monier arch bridges
as carried out by Monash & Anderson.

This web page is devoted to the procedures used by Monash and Anderson, and their engineering assistants, to determine the profile for a Monier arch, and to calculate the resulting forces and stresses. It assumes that the reader has some basic knowledge of the mechanics of structures. It is restricted to the techniques used for M&A's early bridges, which were checked only for symmetrical uniformly distributed live load. The Upper Coliban Spillway Bridge is used as an example. Computations were sent to Sydney to be checked by W. J. Baltzer and F. M. Gummow. Baltzer had earlier used more complex procedures for the design and analysis of the Anderson Street (Morell) Bridge. After the collapse of the first King's Bridge at Bendigo, Monash obtained from him details of procedures for analysis for non-symmetrical and point loads, the most important 'point' loads being the axles of the steam rollers used in testing the bridges.

The process used for design was a sort of 'form-finding'. At this early stage in the development of reinforced concrete, M&A and their advisors were unaware of any method for taking into account the presence of the reinforcement in an arch cross-section subjected to combined axial load and bending moment. The grids of small diameter bars provided in the Monier system were therefore ignored in analysis, and the aim was to shape the curve of the arch to avoid tensile stresses under normal loading conditions. This was achieved by ensuring that the centreline of the profile coincided with the line of thrust due to the self weight of the arch, spandrel walls and filling. (Sometimes live load was included at this stage.) Checks were then made on varying live load conditions applied to the chosen form, to ensure that the thrust line did not deviate greatly from the centreline. Because the self weight of the bridge was enormous in comparison with the live load, this was rarely a problem in theory. (In practice it turned out that the arch curve as built often deviated considerably from the theoretical curve owing to deflection and subsidence of falsework, and this was a much more significant cause of bending stress.)

The left-hand side of the above drawing shows half of the longitudinal cross-section of a typical Monier arch bridge. The right hand side shows half of the side elevation. The arch profile is made up of three circular segments, as indicated by the radii. This is a simplified version of part of the working drawing for Ford's Ck Bridge, Mansfield. For a more complete extract click here.

The process of form-finding was iterative. All bridges were assumed symmetrical about the vertical centreline of the elevation, so that one half of the span could be treated as a 'free body' subjected to three forces: W, the total weight; R, the inclined reaction from the abutment; and H, the thrust in the crown exerted by the other half of the bridge.

diagram

Because of the symmetry, and in the absence of a point load at the crown, H was horizontal. Assuming that the desired form had already been found, both H and R would pass through the centreline of the arch thickness, while W passed through the centroid of the half-arch. The lines of action of three forces which are in equilibrium intersect. Thus R passed through the intersection point of W and H.

diagram

Hence a triangle of forces could be drawn. This gave the direction of R, while the magnitude of R and H could be determined by scaling from the known value of W. (H could also be obtained by taking moments about the abutment.)

diagram

This approach is evident in the drawing which J. S. Gregory produced for the Upper Coliban Spillway Bridge.
[600 × 636 pixels, 33KB] [1240 × 1576 pixels, 82KB]

In the actual calculation process, the spandrels and fill above the half-arch were conceived as broken into segments by taking vertical slices across the width of the bridge. For clarity, only four are shown in the figure below, but normally eight were taken. It was customary to work with a strip of arch adjacent to the edge and one foot wide. The weight of the live load, when included, was indicated on the drawings as a surcharge comprised of an equally heavy volume of fill. In the Coliban calculations it appears that when the weight of a segment was calculated the specific weights of reinforced concrete, mass concrete, and earth fill were simply taken as a uniform 1 cwt force (112 lbf) per cubic foot (17.6 kN/m3). When the arch was considered by itself (supporting its own weight during construction, or for an alternative scheme with timber superstructure) the specific weight of 'Monier' was taken as 150 lbf per cubic foot.

diagram

Taking moments about the springing point A:

H r = Sum (wi xi)

thus
H = ( Sum wi xi  ) / r

This approach is evident in the tabular calculations represented below. The same tables permitted the calculation of the total mass above the half-arch ( Sum wi ) and the position of its centroid so that the location of the force W could be established.

In the table reproduced below, the effective half-span is taken as 20.08 feet and is split into eight vertical segments each of width K = 20.08 / 8 = 2.51 feet. The centre of gravity of each segment lies at the centroid of its area as seen in elevation. This is assumed to be midway between its vertical edges. The distances from the springing point to each centroid are expressed throughout in terms of K. For a one-foot wide slice in the direction of the span, the volume of each segment is one foot multiplied by its area as seen in elevation, i.e. 1 × K × (average depth). The average depths have been scaled from the drawing as 15.4, 11.8, etc. As the unit weight of all materials is taken as 1 cwt per cubic foot, the weight of a segment is simply 1 × K × (av. depth) × 1 = K × (av. depth). In column 2 the weight W of the one-foot-wide slice of the half-span is summed as 62.62 K = 157.17 cwt. Its first moment about the abutment (Column 3) is 184.29 K2 = 1161 foot-cwt. Hence the centroid lies 1161/157 or about 7.38 feet from the abutment. With these facts it is now possible to obtain the magnitudes of H and R and the direction of R.

Final set of calculations for Coliban Spillway Bridge "accepted design" with masonry spandrels.
by J.S.Gregory, 21 August 1901 (edited for this website.)
Dead load plus half live load. Span = 39'-4", Rise = 13'.
K = 20.08 / 8 = 2.51

Column 1Column 2Column 3Column 4Column 5
Lever arm
from springing.
Weight of segment.Moment of weight about
springing.
Segments
grouped in twos.
Segments
grouped in fours.
K × 1/2
K × 3/2
15.40 K
11.80 K
27.207.70 K2
17.70 K2
25.4025.40 K2 / 27.20 K= 2.3473.95 K2 / 43.70 K = 4.24
K × 5/2
K × 7/2
9.20 K
7.30 K
16.5023.00 K2
25.55 K2
48.5548.55 K2 / 16.50 K = 7.38
K × 9/2
K × 11/2
5.86 K
4.86 K
10.7226.37 K2
26.73 K2
 53.1053.10 K2 / 10.72 K = 12.43  110.34 K2 / 18.92 K = 14.63
K × 13/2
K × 15/2
4.25 K
3.95 K
8.2027.62 K2
29.62 K2
57.2457.24 K2 / 8.20 K = 17.52
 62.62 K 184.29 K2   

Distance of centre of gravity of whole from abutment point = 184.29 K2 / 62.62 K = 7.38'
Horizontal thrust = 184.29 × 6.3 / 13.25 = 87.6 cwt.
Vertical Reaction = 157.17 cwt.

In the calculation for horizontal thrust 184.29 × 6.3 / 13.25, the 6.3 is K2 and the 13.25 is 13'-3", the rise from the abutment "hinge" to the centreline of the arch at the crown i.e. to the level of the horizontal thrust in the crown. The vertical reaction at the abutment must equal the total weight of the segments, 157.17.

To trace the full pressure curve within the arch the vertical slices are grouped first into four groups of two (Column 4). The positions of the centres of gravity is determined for each group. In Column 5 two groups of four segments are taken. This process can be traced through the system of symbols at the top of the drawing, consisting of small concentric circles:
Four small circles indicate the position of the total load W. Part way down its line of action, the intersecting lines of H and R can be seen.
Three small circles indicate the weight of the two groups of four segments. The points where their lines of action cut H and R are joined by a construction line.
This process is repeated until the level of the individual segment is reached, resulting in the thrust line, shown dashed.

If the thrust curve differed significantly from the initially-assumed profile of the arch, the arch shape would be adjusted to fit the pressure curve, and the calculations repeated using revised segment weights. Generally, only two iterations were needed to achieve satisfactory agreement.

Footnote

A promise was made in early dossiers, to publish a monograph on technical aspects of the Monier arch bridges designed and built by Monash & Anderson. This promise has not been fulfilled. There is some technical discussion of arch failure in the dossier on King's Bridge, Bendigo. It is unlikely that any more comment on technical aspects will appear on this website.