Horizontal thrust of a gable frame

Loads on plan

The sample frame spans 9.8 m with a rise of 2.8 m to the ridge and a roof pitch of 30 degrees. It carries a strip of roof 4.35 m wide: a dead load of 1.2 kPa measured on the slope and a live load of 0.7 kPa on plan. The dead load is converted to plan area by dividing by the cosine of the pitch, then both become a line load on the frame. Results from the engine:

Three thrusts

• The shortcut H = V a / h, with a the horizontal run of one rafter (half the span) and h the rise. It treats the whole reaction as acting at the ridge and ignores the load on the rafter.
• Pinned ridge: moments about the ridge of one half give H = (V a - w a^2 / 2) / h, which is the same as w L^2 / (8 h).
• Rigid ridge, pinned feet and the same stiffness along both rafters: H = 5 w a^2 / (8 h).

The shortcut gives exactly twice the pinned value, for any span and any uniform load, because it leaves out the moment of the load on the rafter. The rigid ridge gives five quarters of the pinned value, here a ratio of 1.25: the ridge moment it carries pushes the feet further apart.

Which one to use

The ridge as built decides. A joint with little moment capacity behaves close to a pin; a welded or haunched ridge is closer to the rigid case. When a frame model gives the rigid value but the joint is really a pin, release the moment at the ridge node in the model and the thrust drops to the pinned value. The member self weight is not in the loads above, so a model that includes it will show a little more, about two percent for this frame.

What this is, and is not

The sheet source

The sample, with its decision diagram:

# Gable frame: thrust at the supports

A symmetric pitched roof frame under a uniform load. The vertical reactions are simple, but the horizontal
thrust depends on what the ridge is: a pin carries no moment, a continuous (rigid) ridge carries some. Check
which one the real joint is before trusting a thrust.

```mermaid
flowchart TD
  A[Loads on plan: w] --> B[Vertical reaction V = w L / 2]
  B --> C{Ridge joint?}
  C -- pin --> D[Moments about the ridge: H = V a - w a^2 / 2, divided by h]
  C -- rigid --> E[Two-pinned frame: H = 5 w a^2 / 8 h]
  D --> F[Compare with the model]
  E --> F
```

## Inputs
L: 9.8 m | span between the supports
h: 2.8 m | rise from the supports to the ridge
trib: 4.35 m | tributary width of the frame
g_s: 1.2 kPa | dead load per area of roof slope
q: 0.7 kPa | live load per plan area
pitch: 30 deg | roof pitch

## Load on plan
g: g_s / cos(pitch) | dead load converted to plan area
w: (g + q) * trib -> kN/m | line load on the frame, per metre of span
a: L / 2 | horizontal run of one rafter

## Vertical reactions
V: w * L / 2 -> kN | each support

## Horizontal thrust
H_hand: V * a / h -> kN | V a / h: treats the whole reaction as acting at the ridge and ignores the load on the rafter
H_pin: (V * a - w * a^2 / 2) / h -> kN | pinned ridge: moments about the ridge of one half
H_pin_check: w * L^2 / (8 * h) -> kN | the same thing in one line
H_rigid: 5 * w * a^2 / (8 * h) -> kN | rigid ridge, pinned feet, constant stiffness along the rafters

## What the rigid ridge does
M_ridge: H_rigid * h - V * a + w * a^2 / 2 -> kN*m | hogging moment at the ridge (zero for a pin)
ratio: H_rigid / H_pin | the thrust is larger when the ridge resists moment

The rigid case assumes the same stiffness along both rafters and ignores axial and shear deformation. Member
self weight is not in the load above, so a model that includes it will show about two per cent more.
If a pin is what is really built, release the moment at the ridge node in the model and the thrust drops to the pinned value.