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This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

  • An independent structural reviewer must check the illustrated models, prediction feedback, geometry and full interactive input ranges.
  • Confirm sway and inflection illustrations for their particular stiffness and load cases; symmetry arguments assume a stable unique elastic solution.
  • Universal displacement-direction and fixed-base inflection claims must be limited to the illustrated elastic load cases.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

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Module 7 · Course notes

Reading a frame: sway, rotation and moment transfer

A frame is more than a beam on legs. Its joints are rigid, so a beam and the columns it sits on bend together — moment flows around the corner. Under sideways load the whole frame leans, or sways. Learning to see the sway, the joint rotations and the way moment transfers is what lets you judge a frame before analysing it.

The order of thinking is the same as for a beam, with one extra first move: find the sway. Then draw the joint rotations, then each member's curvature, then the moments.

Rigid joints share a rotation. Where a beam meets a column at a rigid corner, both members turn through the same angle. That is why a column bends when only the beam is loaded: the beam's end rotation is forced on the column top, and moment transfers around the corner. A column is almost never in pure axial force.

Bending moment

cantilever frame — a load on the beam bends the column
A vertical load on the cantilevering beam is carried back through the rigid corner into the column, which develops a bending moment all the way to its fixed base. The corner transfers moment; nothing here is axial-only.

Under horizontal load a frame sways. The columns lean and the beam translates sideways. How much it sways, and where the moment goes, depends on the bases:

  • Pinned bases carry no moment. The columns bend in single curvature and the frame sways a lot.
  • Fixed bases carry moment. The columns bend in double curvature (an inflection point partway up) and the frame sways much less.

Bending moment

fixed-base portal under horizontal load
Moment develops at the fixed bases and at the rigid knees; the columns bend in double curvature. A pinned-base version would show zero moment at the bases and would sway much more.

Worked example

Worked example — a pinned-base portal pushed sideways

A portal frame with pinned bases carries a horizontal load at the top of one column. Read the sway, then the curvature, then the moment.

  1. Step 1 — sway

    The pinned bases hold the column feet in place but let them rotate. Nothing resists the horizontal push except the bending of the frame, so it leans over — the whole frame sways in the direction of the load, and the beam translates sideways with it.

  2. Step 2 — joint rotations and member curvature

    The rigid knees rotate as the frame leans. Each column bends in single curvature between its pinned base (zero moment) and the knee. The beam bends too, because the knees force rotation on its ends. Watch the columns curve and the beam bow.

    Deflected shape

  3. Step 3 — bending moment

    The moment is zero at both pinned bases and grows to a maximum at the knees, where the columns and beam meet. It is drawn on the tension side of each member. The two knees carry opposite-sense moments — the frame's response to being pushed sideways.

    Bending moment

Every exercise below is one of these ideas on a different frame — comparing bases, columns bending under gravity, a symmetric frame that does not sway, a braced frame, and a cantilever frame. Find the sway first, then reason from the shape to the moment.

How to read these problems

The three-step method

  1. 1Points of certainty. Match the displacement and rotation that each support prescribes, including any settlement. Infer the displacement direction from the whole structure and load case; it need not be downward at every loaded point.
  2. 2Deflected shape and reaction directions. Sketch the compatible deflected shape. To find a reaction's direction, imagine removing that support: the direction that pushes the structure back to its place is the reaction's sense (it may be a hold-down).
  3. 3Bending moment and contraflexure. Draw the bending-moment diagram on the tension side and check it against the shape: hogging where the curve is convex-up, sagging where convex-down, zero at pins and at every contraflexure.

Rules that must always hold

  • 1.Moment is zero at a genuine moment release and at an unloaded simple end. A continuous member can carry moment over a translational support.
  • 2.A bending-moment diagram crosses the baseline exactly at a point of contraflexure.
  • 3.Under a distributed load the bending-moment diagram is curved; under point loads alone it is straight lines.
  • 4.At a fully fixed support the deflected shape leaves the support with zero rotation (tangent along the member).
  • 5.For an initially straight elastic beam without imposed curvature, zero curvature implies zero bending moment.
  • 6.The moment is drawn on the tension side: sagging below the member, hogging above it.

Now predict for yourself

Frame behaviour exercises

12 exercises — base fixity and sway, columns bending under gravity, a symmetric frame that does not sway, a braced frame and a cantilever frame. Predict, then reveal an explanation checked against a real elastic solve.

Start the exercises →

This lesson is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions.