Hip Shift and Hip Drop
Example Calculations

The Hip Shift/Drop calculation determines the placement of the Hip rafter shoulders to permit the jack rafters and purlins to intercept or "plane" without backing the Hip.

Plumb Section extracted from Hip Roof
Hip Rafter Shift Ratio in terms of Rise/Run
Hip Shift Ratios in terms of Rise/Run
The length labelled Run is equal for both Plan Angles

Width of Hip Rafter to the Major Span side of the Hip Rafter
= Major Plan Angle Rise
Width of Hip Rafter to the Minor Span side of the Hip Rafter
= Minor Plan Angle Rise
Overall Width of Hip Rafter
= Major Plan Angle Rise + Minor Plan Angle Rise

The proportion of the Hip Rafter to the Major Span side of the Hip Rafter
= Major Plan Angle Rise ÷ (Major Plan Angle Rise + Minor Plan Angle Rise)
The proportion of the Hip Rafter to the Minor Span side of the Hip Rafter
= Minor Plan Angle Rise ÷ (Major Plan Angle Rise + Minor Plan Angle Rise)

Equal 6/12 Pitches, Plan Angle between Eaves = 135°
The Hip rafters are symmetrical about their centerlines and need not be shifted. The Plumb Backing Angle may be determined using the following formulas:

Plumb Backing Angle Formulas
Plumb Backing Angle Geometry and Formula
Solving the Hip Drop
Plumb Backing Angle
= arctan (tan 24.79409° ÷ tan 67.50000°) = 10.83214°
Hip Rafter Width = 1.5"
Hip Drop = ½ Hip Rafter Width × tan Plumb Backing Angle
Hip Drop = .75" × tan 10.83214° = .14351" ( approx. 5/32" )

Since the terms of the second formula within the parentheses equal
tan Plumb Backing Angle the value of the angle need not be solved to find the Hip Drop.
Plan Angle Rise = Common Run = 12
Plan Angle Run = 4.97056
Hip Rise = Common Rise = 6
Hip Run = 12.98871
Hip Drop = ½ Hip Rafter Width × tan Plumb Backing Angle
Hip Drop = .75" × [(6 × 4.97056) ÷ (12.98871 × 12)]
= .14351" ( approx. 5/32" )

Trigonometric scaling of Hip Rafter Shift Ratio
Trigonometric Solution of Hip Shift Ratios
A Unit Dimension is assigned to the Run

Width of Hip Rafter to the Major Span side of the Hip Rafter
= tan Major Plan Angle
Width of Hip Rafter to the Minor Span side of the Hip Rafter
= tan Minor Plan Angle
Overall Width of Hip Rafter
= tan Major Plan Angle + tan Minor Plan Angle

The proportion of the Hip Rafter to the Major Span side of the Hip Rafter
= tan Major Plan Angle ÷ (tan Major Plan Angle + tan Minor Plan Angle)
The proportion of the Hip Rafter to the Minor Span side of the Hip Rafter
= tan Minor Plan Angle ÷ (tan Major Plan Angle + tan Minor Plan Angle)

Major Pitch = 6/12, Minor Pitch = 10/12, Plan Angle between Eaves = 90°
Determining the Hip Shift/Drop Ratios
6/12 Plan Angle = 59.03624°
10/12 Plan Angle = 30.96376°
6/12 Shift Ratio
= tan 59.03624° ÷ (tan 59.03624° + tan 30.96376° ) = .73529
10/12 Shift Ratio
= tan 30.96376° ÷ (tan 59.03624° + tan 30.96376° ) = .26471

Hip Shift Ratio = (.73529 – .26471) ÷ 2 = .23529
or
Hip Shift Ratio = .50000 – .26471 = .23529
or
Hip Shift Ratio = .73529 – .50000 = .23529
The Hip rafter is shifted toward the 6/12 side of the roof

6/12 Pumb Backing Angle = 14.42642°
10/12 Pumb Backing Angle = 35.54874°
Hip Drop Ratio = .26471 × tan 35.54874° = .18915
or
Hip Drop Ratio = .73529 × tan 14.42642° = .18915

Adjusting the Ratios to the Hip Rafter Width
Hip Rafter Width = 1.5"
6/12 Side Dimension = 1.5" × .73529 = 1.10294" ( approx. 1 3/32" )
10/12 Side Dimension = 1.5" × .26471 = .39707" ( approx. 13/32" )
Hip Shift = 1.5" × .23529 = .35294" ( approx. 11/32" )
... aligning the theoretical ridge line of the Hip rafter over the Hip run

Hip Drop = 1.5" × .18915 = .28373" ( approx. 9/32" )
... aligning the shoulders of the Hip rafter to permit the jack rafters/purlins to plane in at the same height

Major Pitch = 8/12, Minor Pitch = 10/12, Plan Angle between Eaves = 120°
Determining the Hip Shift/Drop Ratios
8/12 Plan Angle = 70.89339°
10/12 Plan Angle = 49.10661°
8/12 Shift Ratio
= tan 70.89339° ÷ (tan 70.89339° + tan 49.10661° ) = .71429
10/12 Shift Ratio
= tan 49.10661° ÷ (tan 70.89339° + tan 49.10661° ) = .28571

Hip Shift Ratio = (.71429 – .28571) ÷ 2 = .21429
or
Hip Shift Ratio = .50000 – .28571 = .21429
or
Hip Shift Ratio = .71429 – .50000 = .21429
The Hip rafter is shifted toward the 8/12 side of the roof

8/12 Pumb Backing Angle = 12.30999°
10/12 Pumb Backing Angle = 28.61444°
Hip Drop Ratio = .28571 × tan 28.61444° = .15587
or
Hip Drop Ratio = .71429 × tan 12.30999° = .15587

Adjusting the Ratios to the Hip Rafter Width
Hip Rafter Width = 1.5"
8/12 Side Dimension = 1.5" × .71429 = 1.07143" ( approx. 1 1/16" )
10/12 Side Dimension = 1.5" × .28571 = .42857" ( approx. 7/16" )
Hip Shift = 1.5" × .21429 = .32143" ( approx. 5/16" )
... aligning the theoretical ridge line of the Hip rafter over the Hip run

Hip Drop = 1.5" × .15587 = .23380" ( approx. 7/32" )
... aligning the shoulders of the Hip rafter to permit the jack rafters/purlins to plane in at the same height

Joe Bartok
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