In considering the design of masonry arch bridges it is important to remember that there are over 40 000 examples in the UK alone. Most have already exceeded the Department of Transport's design life of 120 years and therefore must be considered as an archive of good practice and proportion. Traditionally, the span to rise ratio varies from 2:1 (semi-circular) to 10:1 but is usually in the range of 3: 1 to 6:1 with the 'ideal' taken as 4:1. Clearly, the shape of the intrados is of great importance as this dictates of the complexity of the temporary support or centring. Nowadays, this usually takes the form of a proprietary metal system of supports upon which timber transverse joints are secured and a plywood facing attached. Care must be taken to limit the centring deflections during construction. It is good practice to provide a method statement which requires the barrel to be constructed at an even rate from each springing. The centring should not be removed until the spandrel walls (if the bridge is to be a filled spandrel bridge) have been taken to the string course and the wing walls extended sufficiently to resist the horizontal thrust and also contribute to the stability of the abutment. It is important to take a holistic approach and consider the interaction between the several elements of the bridge. It should not be forgotten that small settlements or rotations of the piers and abutments offer 'releases' to the structure and thus can contribute to the formation of a mechanism.
The initial sizing of the structure can be undertaken using any of a number of empirical equations.
Rankine advocated that the barrel thickness, d, for a segmental arch of radius R should be:
Trautwine suggested (for span = L):
d = 1.0[0.061 + 0.138P(R + 0.5L)] (first class cut stone)
(second class cut stone) d = 1.13[0.061 + 0.138p(R + 0.5L)] (brickwork)
Rennie and Stephenson related arch barrel thickness to span and radius, respectively.
Historically, the addition of haunches to the arch ring at the abutments has been considered good practice (and in keeping with the usual assumption for two-hinged arch analysis
Figure 41 Empirical rule for abutment sizes t = 0.06C + 0.03r = 0.6
Figure 41 Empirical rule for abutment sizes that the second moment of area of the barrel varies as the secant of the tangent to the centreline).
The suggested ratio of springing to crown thickness varied from 1.2 to 2. In stonework and concrete this is relatively easy to achieve but in brickwork the haunching is usually concrete and there is a reliance upon the bond between it and the brickwork.
Abutment sizes for a 'gravity' solution (as opposed to a reinforced concrete or masonry solution) have been suggested by Baker (1909) where the abutment thickness, t, in metres is given by:
where L is the span (metres) and H is the height from the top of the foundation to the springing line (metres).
Both the haunching rules and Trautwine's rule for abutment sizes are shown in Figure 41.
In the case of multi-span bridges, the thickness or piers vary. Rankine (1904) suggests a range from 1/10 to 1/4 of the span, the latter offering sufficient support to cater for the removal of one of the spans. Historically, the most common thickness for intermediate piers is from 1/6 to 1/7 of the span. These suggested thicknesses make no reference to the height of the pier which is an important parameter when considering any out of balance forces between adjacent spans that have to be accommodated by the pier. Of equal importance, then, is the slenderness ratio of the pier. Recent laboratory tests have demonstrated that even with a slenderness (height/thickness) ratio of 3.4, the failure mechanism involves at least two spans. It is therefore suggested that a slenderness ratio of less than two is required to ensure the independent behaviour of each span; otherwise any analysis must consider the whole structure. In any case, the stability of the piers must be verified.
Modern limit state codes require both serviceability and ultimate limit states to be considered. In the case of the masonry arch, serviceability criteria are difficult to quantify. Generally, serviceability criteria limit deflection for aesthetic or functional reasons and cracking for durability considerations. For masonry arches the stress levels are generally very low and hence deformations are small. Additionally, any cracking which might occur in response to de-centring and 'bedding-in' of the structure is usually only an indication that it is working and should cause no concern. On the other hand, it is important that the structure is designed to avoid proliferation of micro-cracking under repeated loading (fatigue); particularly if this could lead to ring separation in multi-ring brickwork arch barrels. It has been observed in field and laboratory tests that the first hinge forms at approximately half the ultimate load carrying capacity of each bridge. Using this observation and in the absence of any more reliable criteria, it is suggested that for HA type loading a partial load factor 7FL of 3.4 should be used for the critical axle (knife-edge load) and a 7FL of 1.9 for the other live loads. Whilst the HB type abnormal loading a 7FL of 2.0 may be used. These loadings should be applied in conjunction with a partial load factor for all the dead load and superimposed dead load of 1.2 when they are adversely affecting the structure and 1.0 when they are resisting hinge formation. Initial design procedure:
1 Choose span and rise.
2 Select materials to be used.
3 Determine trial section using a selection of empirical equations.
4 Ignoring horizontal soil pressures, calculate the required arch barrel thickness using a simple 'block' mechanism for the ultimate load condition. This can be assumed to comprise of appropriately factored HA loading.
5 Check the compressive stress based on 0.1 (arch thickness) or 100 mm whichever is the greater.
Compressive stress 4 Au f where Au is a coefficient of 0.35 for concrete grades 15 and 20, 0.4 for concrete grades 25 and above, and 0.44 for masonry; f is the characteristic cube strength of concrete, fcu, or the compressive strength of masonry fK as appropriate (allowance for 7M has been made).
6 Check that the radial shear at the crown is less than 0.4 (horizontal reactions).
7 In the case of multi-ring brickwork arches check that the horizontal mid-depth shear stress at the crown is less than 0.15 N/mm2.
8 Check abutment stability and stress levels.
9 Check foundation stability and stress levels.
Consider an 8 m span 2m-rise segmental brickwork arch bridge with a total available construction depth of 0.9 m at the crown. Following the initial design procedure:
2 Empirical equations:
Rankine = 0.19PR = 0.19^5 = 0.42m Heintzerling = 0.4 + 0.028R = 0.54m Trautwine = 1.33(0.061 + 0.138p(R + 0.5L))
It is therefore likely that the arch ring thickness will be about 0.5 m - it follows that a construction depth of 0.9 m is probably enough.
A simple 'block' mechanism analysis using a 1 m 'slice' of the bridge can be undertaken to determine the barrel thickness. The minimum lane width of 2.5 m is used with a 7FL of 3.4 for the KEL and a 7FL of 1.9 for the UDL. The KEL = (3.43120)/2.5 = 163.2kN/m width
The hinges are assumed to occur at A, B, C and D. It is also assumed that the soil offers no horizontal support and that 7FL for the self-weight on the loaded side of the arch is 1.2 while on the unloaded side of the arch it is assumed to be unity, as shown in Figure 42.
3 By taking moments about C, B and A for the free bodies to the right of each of the hinges three independent equations can be written and solved or the unknowns VD, H and t:
Making allowance for the potential loss of mortar depth to the intrados and rounding to the nearest half brick -assume a barrel thickness of 450 mm.
4 At the crown the compressive force = H = 447 kN so the characteristic compressive strength required (447 x 103)/(0.44 x 100 x 103) = 10.2N/mm2
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