Hip Rafter Edge Bevel and Backing Angle

In complex roof framing there are 7 main roof framing angles.

1. Hip Rafter Backing Angle
2. Valley Sleeper Bevel Angle
3. Plain Titl (DP)
4. Main Plan Angle
5. Adjacent Plan Angle 
6. Main Profile Rafter Slope Angle
7. Adjacent Profile Rafter Slope Angle

In this article I will discuss the hip rafter backing angle and the valley sleeper bevel angle. These two angles determine the miter angle and and saw blade bevel angles for jack rafters and purlin rafters that are plumb to the earth and for jack rafters and purlin rafters that are skewed from the eave line. A thorough knowledge of these two angles is critical in complex roof framing using geometry or trigonometry.

The hip rafter edge bevel was avoided in the 12th century on some roof structures by rotating the hip rafter into the roof surface plane. However, by avoiding the hip rafter edge bevel, the carpenters-builders had to have a knowledge of the Valley Sleeper Bevel Angle. They didn't call it the Valley Sleeper Bevel Angle, but it's the same angle we use today for valley sleepers, layover valleys or hip rafters rotated into the roof surface.

The hip rafter edge bevel (Hip Rafter Backing Angle) is determined by the dihedral angle triangle. The dihedral angle is formed by the intersection of two adjacent planes. In hip roof framing the dihedral angle is formed by the intersection of the roof surface planes at the hip or valley rafter. The square edge of the hip rafter at this intersection must be edge beveled to align the edge of the hip or valley rafter with the adjoining roof surface plane. The hip rafter could also be lowered to align the edge of the hip with the adjoining roof surface plane ( hip drop).

There are several ways of geometrically drawing out the hip rafter geometry for determining the hip rafter edge bevel. Each geometric method will give the correct hip rafter backing angle. However, some methods are not that accurate for use with an bevel gauge or transferring the hip rafter edge bevel from the geometric lines of a drawing unless the drawing is a full scale geometric drawing of the roof.

Drawing showing the dihedral angle formed by two intersecting planes.

3D drawings of the roof framing kernels showing the different ways of drawing out the dihedral angle for the hip rafter edge bevel angle.

Geometric drawings of simple folded out roof framing kernels showing the different ways of drawing out the dihedral angle for the hip rafter edge bevel angle.

In this drawing of an irregular hip roof, folded out roof framing kernel. the hip rafter backing triangle is drawn perpendicular to the hip rafter run line at the peak of the roof framing kernel. This is the most accurate way to draw out the hip rafter backing angles. You can also use this same roof framing kernel for drawing the hip rafter rotated into the roof surface to determine the Valley Sleeper Bevel Angle.

Warning

This method of drawing out the hip rafter backing angle

only works on equal pitched roofs with a 90° eave angle.

However, this geometric method should be common knowledge for use in the field for simple hip roof framing.

Here's a classic drawing showing the hip rafter backing angles developed geometrically. This classic drawing of the hip rafter backing angles was more of an exercise in roof framing geometry for architects and road scholars of the 18th & 19th century. This drawing method only works when drawn full scale.

Most carpenters in the centuries pass would have drawn out the hip rafter in profile using the hip rafter shift geometry to determine the intersection of the Hip Rafter Shift Line & Level Line that determines the Hip Rafter Backing Depth Line. They didn't need the actual hip rafter backing angle. They would mark out the hip rafter backing depth lines on the sides of the  hip rafter and hand hew the hip rafter edge bevel.

There are three methods of backing out the hip rafter.

1. Standard Equal Pitch Roof, with Hip Rafter Material Centered on Hip Rafter Run Line with Equal Hip Rafter Backing Depths
2. Unequal Pitched Roofs, with Hip Rafter Material Centered on Hip Rafter run Line with Unequal Hip Rafter Backing Depths.
3. Unequal Pitched Roofs, with Hip Rafter Material Shifted on Hip Rafter Run Line for Equal Hip Rafter Backing Depths.

Standard Equal Pitch Roof, with Hip Rafter Material Centered on Hip Rafter Run Line with Equal Hip Rafter Backing Depths Examples.

Rectangular  Roof Framing
Equal Pitch Roof,
with Hip Rafter Material Centered on Hip Rafter Run Line
with Equal Hip Rafter Backing Depths

Pentagon Roof Framing
Equal Pitch Roof,
with Hip Rafter Material Centered on Hip Rafter Run Line
with Equal Hip Rafter Backing Depths

Hexagon Roof Framing
Equal Pitch Roof,
with Hip Rafter Material Centered on Hip Rafter Run Line
with Equal Hip Rafter Backing Depths

Octagon Roof Framing
Equal Pitch Roof,
with Hip Rafter Material Centered on Hip Rafter Run Line
with Equal Hip Rafter Backing Depths

For most modern irregular hip roof framing you will use the hip rafter shift geometry for equal height heel heights on each side of the hip rafter that's the same as the HAP of the common rafters. However, there can be situations where the hip rafter material is centered on the corner of the plate line for timber framing joints on irregular hip roofs or where  the fascia lines of an irregular hip roof make it more desirable to keep the hip rafter material centered on the corner of the fascia.

Drawing with unequal pitched roof with the hip rafter material centered on the corner of the eave line. This results in different backing depths on each side of the hip rafter. This would also require different height plate lines or different heights for the common rafter HAP. (HAP = Height Above Plate)

Unequal Pitched Roofs, with Hip Rafter Material Centered on Hip Rafter run Line with Unequal Hip Rafter Backing Depths Example.

Unequal Pitched Roof, with Hip Rafter Material Shifted on Hip Rafter Run Line for Equal Hip Rafter Backing Depths Example

Hip Rafter Shift Geometry