Skew arch

A skew arch is a method of construction that enables an arch bridge to span an obstacle at some angle other than a right angle. This results in the faces of the arch not being perpendicular to its abutments and its plan view being a parallelogram, rather than the rectangle that is the plan view of a regular, or "square" arch.
In the case of a masonry skew arch, the construction requires precise stonecutting, as the cuts do not form right angles, but once the principles were fully understood in the early 19th century, it became considerably easier and cheaper to build a skew arch of brick.
The problem of building skew arch masonry bridges was addressed by a number of early civil engineers and mathematicians, including Giovanni Barbara, William Chapman, Benjamin Outram, Peter Nicholson, George Stephenson, Edward Sang, Charles Fox, George W. Buck and William Froude.

History

Benjamin Outram and Store Street Aqueduct

Skew bridges are not a recent invention, having been built on exceptional occasions since Roman times, but they were little understood and rarely used before the advent of the railway. An early example of the skew arch is the Arco Barbara in the Floriana Lines fortifications in Malta, which was designed by the Maltese architect and military engineer Giovanni Barbara in 1726. Another notable exception is an aqueduct, designed by British engineer Benjamin Outram, constructed in masonry and completed in 1798, which still carries the Ashton Canal at an angle of 45° over Store Street in Manchester. Outram's design is believed to be based on work done on the Kildare Canal in Ireland in 1787, in which William Chapman introduced the segmental oblique arch to the design of Finlay Bridge at Naas, employing an arch barrel based on a circular segment that is smaller than a semicircle and which was repeated by Thomas Storey in 1830 in the bridge carrying the Haggerleases branch of the Stockton and Darlington Railway over the River Gaunless near Cockfield, County Durham with a skew angle of 63° and a skew span of, resulting in a clear span of and a rise of.
The common method they all used was to clad the timber centring with planks, known as "laggings", laid parallel with the abutments and carefully planed and levelled to approximate closely the required curve of the intrados of the arch. The positions of the courses in the vicinity of the crown were first marked out at right angles to the faces using long wooden straight-edges, then the remaining courses were marked out in parallel. The masons then laid the stones, cutting them to shape as required.
Contemporary designs by rival engineers were less successful and for a time skew bridges were considered weak in comparison with the regular, or "square" arch bridge and so were avoided if at all possible,
the alternatives being to construct the road or canal with a double bend, so as to allow it to cross the obstacle at right angles, or to build a regular arch bridge with the extra width or span necessary to clear the obstacle "on the square".
An example of the latter type of construction is Denbigh Hall Bridge, built in 1837 to carry the London and Birmingham Railway across Watling Street at an acute angle of only 25°.
Now a Grade II listed structure, the bridge is still in use today, carrying the busy West Coast Main Line. It was constructed in the form of a long gallery, some long and wide, consisting of iron girders resting on walls built parallel with the road; the girders, and consequently the faces of the bridge, being perpendicular to the roadway and the railway line being laid out obliquely across the top, the need to build a highly skewed bridge of span was therefore avoided.
The eminent canal engineer James Brindley never succeeding in working out a solution to the problem of constructing a strong skew arch and as a consequence all his overbridges were built at right angles to the waterway, with double bends in the roadway, where necessary, and to this day many of them cause inconvenience to their users. However, it was the coming of the railway, with its need to cross existing obstacles, such as rivers, roads, canals and other railways, in as straight a line as possible, that rekindled the civil engineer's interest in the skew arch bridge.

The false skew arch

The strength of a regular arch comes from the fact that the mass of the structure and its superincumbent load cause lines of force that are carried by the stones into the ground and the abutments without producing any tendency for the stones to slide with respect to one another. This is due to the fact that the courses of stone are laid parallel to the abutments, which in a regular arch causes them also to lie perpendicular to its faces. For only slightly oblique bridges, where the skew angle is less than approximately 15° it is possible to use the same construction method, laying the stones in courses parallel to the abutments. The result is known as a "false" skew arch and analysis of the forces within it shows that in each corner where the face forms an acute angle with an abutment there are resultant forces that are not perpendicular to the planes of the stone courses whose tendency is to push the stones out of the face, the only resistance to this being provided by friction and the adhesion of the mortar between the stones.
An example of such a false skew arch is the Colorado Street Bridge in Saint Paul, Minnesota.
Before starting work on Store Street Aqueduct, Outram built a number of false skew arches, one of them with a skew angle as great as 19°, as accommodation bridges across the Huddersfield Narrow Canal. The fact that these inherently weak structures are still standing today is attributed to their light loading.

A more rigorous approach

When considering the balance of forces within a regular arch, in which all courses of masonry that make up the barrel are parallel with its abutments and perpendicular to its faces, it is convenient to consider it as a two-dimensional object by taking a vertical section through the body of the arch and parallel with its faces, thereby ignoring any variation in loading along the length of its barrel. In an oblique or skew arch the axis of the barrel is deliberately not perpendicular to the faces, the deviation from perpendicularity being known as the skew angle or the "obliquity" of the arch.
For this reason a skew arch needs to be thought of as a three-dimensional object and by considering the direction of the lines of force within the barrel the optimum orientation for the courses of stonework that make the barrel can be decided.

The helicoidal skew arch

A characteristic of the regular arch is that the courses of stones run parallel to the abutments and perpendicular to the faces. In an oblique arch these two conditions cannot both be met because the faces and the abutments are deliberately not perpendicular. Since skew angles greater than about 15° are required for many applications, mathematicians and engineers such as Chapman abandoned the idea of laying the courses of stones parallel to the abutments and considered the alternative of laying the courses perpendicular to the faces of the arch, and accepting the fact that they would then no longer run parallel to the abutments. Though Outram's Store Street Aqueduct was constructed with this principle in mind, it was done so empirically, with the masons cutting each voussoir stone as it was required, and it was not until 1828 that details of the technique were published in a form that was useful to other engineers and stonemasons.

Peter Nicholson's helicoidal method in stone

In his book A Popular and Practical Treatise on Masonry and Stone-cutting, Scottish architect, mathematician, cabinet-maker and engineer Peter Nicholson first set out in clear and understandable terms a workable method for determining the shape and position of the stones required for the construction of a strong skew arch that enabled them to be prepared in advance of the actual construction process.
Nicholson approached the problem by constructing a development of the intrados of the arch from the plan and elevation drawings, effectively unrolling and flattening the surface, then drawing the courses perpendicular to the faces, adding the header joints perpendicular to the courses, then finally rolling up the development diagram by projecting the detail of the intrados back onto the plan and elevation drawings, a technique also used by others who would later offer alternative solutions to the problem. This method resulted in the courses of stone voussoirs making up the barrel of the skew arch following parallel helical paths between the abutments, giving the view along the barrel an attractive rifled appearance. Although these courses meet the arch faces at right angles at the crown of the arch, the nearer they are to the springing line the greater their deviation from perpendicularity. Thus Nicholson's method is not the perfect solution, but it is a workable one that has one great advantage over more purist alternatives, namely that since the helical courses run parallel to each other, all the voussoir stones can be cut to the same pattern, the only exceptions being the ring stones, or quoins, where the barrel meets the faces of the arch, each of which is unique but has an identical copy in the other face.
Nicholson never pretended to have invented the skew arch but in his later work The Guide to Railway Masonry, containing a Complete Treatise on the Oblique Arch, he does claim to have invented the method for producing the templates that enabled the accurate cutting of the voussoir stones used in all skew bridges built between the years 1828 and 1836, citing testimonials from the builders of major works, such as the Croft Viaduct at Croft-on-Tees near Darlington. However, by 1836 a young engineer called Charles Fox had improved on Nicholson's helicoidal method and other writers were proposing alternative approaches to the problem.