Frames: thrust and frame action

This is a static copy of the chapter for search engines. The interactive version has animated figures, check questions, and buttons that load the example into the calculator.

Close a bent bar down onto two supports and you get a frame. What makes a frame more than the sum of its members is that the corners are rigid: the beam cannot rotate at its end without dragging the column with it. That coupling is called frame action, and it is why a portal frame sags far less than a simply supported beam of the same span.

Whether a frame is a frame at all is decided by the supports. Two pins let the corners carry moment and the base push outwards (the thrust H); a roller removes the thrust and the beam behaves almost like a simply supported one.

The thrust

For a portal frame with two pinned bases, a uniform load q on the beam, span l, column height h and constant EI, the horizontal reaction is

H = q·l²4·h·(2k + 3), k = I_beam / lI_col / h

and the moment at each corner is simply M_corner = H·h, which then reduces the mid-span moment to q·l²/8 − H·h. FrameLab reproduces this formula to within a fraction of a per cent — you can check it yourself with the example below.

Stiffness ratio changes the answer

Note what the k in that formula means: the relative stiffness of beam and column decides how much moment goes where. Make the columns very stiff (k → 0) and H → q·l²/12h, so the corners take a lot. Make them very flexible (k → ∞) and H → 0, so the beam ends up carrying the full simple-beam moment. This is the single most important intuition about indeterminate structures: load flows towards stiffness.

Horizontal loads

A wind load on one column is where frames really earn their keep. With pinned bases the frame sways and the moment diagram is antisymmetric; with fixed bases the sway drops dramatically and the base moments jump up. Try both.

Pitched (gable) frames

Tilt the beam into two rafters and the frame becomes a gable frame. Two things change: the rafters now carry a large axial compression (they are half-way to being an arch), and the load per metre along the rafter is smaller than the load per metre of plan, by a factor cos of the pitch — remember that when you convert a snow load into a q value.

💡 In FrameLab, q is always per metre of the member itself. For a snow load of s_k per square metre of plan on a rafter at angle α with spacing b, use q = s_k · b · cos α.

Examples

Portal frame under wind on one column, and a gable frame under a snow-like load on both rafters.

Example — self-weight only

A four-segment polygonal arch on two pins, carrying nothing but its own weight. Look at how small the moments are compared with the axial compression: this is what "arch action" means.