Skew Arch - Notes

Notes

1. ^ The skew angle or angle of skew, θ is the angle between the centre line of the arch barrel and the perpendicular to the face of the arch. A regular arch is defined as having a zero angle of skew. The angle of obliquity, Ω is the complement of the angle of skew, though there is some confusion in a number of the 19th century texts where angle of skew and angle of obliquity tend to be used interchangeably.
2. ^ The skew span or span on the skew, S is the span of the arch measured parallel to its face. This is the actual span of the skew arch, for which it must be engineered, and it is always greater than the usable span.
3. ^ The square span or span on the square, s is the span of the arch measured perpendicular to the abutments. This is the usable span for the roadway beneath the arch (hence, it is also known as the clear span) and it is related to the skew span by the following formula: s = S cos θ .
4. ^ The rise of a skew arch is equal to the rise of a regular arch whose span is equal to the skew span of the skew bridge. A limiting case is the full-centred or semicircular skew arch, in which case the rise is equal to the radius of the arch, or half the skew span. For segmental, three-centred, and elliptical skew arches the rise is less than this limiting case.
5. ^ The term intrados is used because it is the mathematically correct term, referring to the curved surface of the inside of the arch barrel. The equivalent architectural term is soffit.
6. ^ Strictly speaking, the development of the face of a skew arch is not actually a straight line, but an S-shaped curve, the curvature of which becomes more pronounced with increasing angle of skew. Nicholson therefore added a straight line, called "the approximate line", between the ends of each face on the development drawing and then drew the courses perpendicular to it. The approximate line is tangential to the curve of the face only at the crown, with the difference increasing with distance away from that point.
7. ^ The 19th century texts use the word spiral to describe both lines and surfaces. The helix is a special case of the generic spiral and applies only to a line. It is used to describe the rifled appearance of the intrados of this particular class of skew arch: the courses follow helical paths between the imposts. The helicoid is a curved surface swept out by a radius moving in a helical path about an axial line. The bearing surfaces of a square threaded screw and its associated nut are helicoidal, and so are the bedding planes between adjacent courses of voussoirs in this class of skew arch.
8. ^ Equilibrated courses are ones built without residual shear stresses.
9. ^ This is the strict differential geometry definition of a cylinder, which includes both the right circular cylinder (the common cylinder with which everyone is familiar) and the right elliptic cylinder (also known as the cylindroid). If a helicoidal skew arch has a semicircular cross section, when taken on the square, perpendicular to the abutments, its barrel will have a shape based on the common cylinder (a semicylinder, in fact), and its cross section (taken on the skew, parallel with its faces) will be semi-elliptical. Segmental circular skew arches also have barrels based on the shape of the common cylinder, while those constructed with a semi-elliptical square section will have a semi-cylindroidal barrel and a flatter, wider semi-elliptical skew section. The extruded profile of a three-centred arch, strictly speaking, does not fall under this definition of a cylinder.