Beam Sign Conventions: Stop Losing Marks

More correct answers get thrown away over signs than over maths. Here is what the conventions actually mean and how to keep them straight.

Bending moment: sagging positive

The near-universal structural convention: sagging is positive. A beam that sags (smiles) has positive moment; a beam that hogs (frowns) over a support has negative moment.

• Simply supported under gravity: positive everywhere.
• Continuous over a support under gravity: negative over the support.

If your continuous-beam support moment comes out positive under downward load, something is wrong — the beam cannot sag where it is held up.

Shear: the left-face rule

Positive shear rotates the element clockwise: on the left face of a cut, upward is positive. Equivalently, walking left to right, the shear diagram drops by the magnitude of each downward load.

The practical consequence: for a simply supported beam under downward load, shear starts positive at the left reaction and ends negative at the right. If your diagram starts negative, you either took the wrong reaction or the wrong face.

The trap: different books, different shear

Some US textbooks and some software flip the shear convention. The moment diagram is unambiguous (sagging positive is nearly universal), but always check which shear convention a reference uses before copying numbers. The relationship dM/dx = V holds in the sagging-positive/left-face-upward system; in the flipped system it is dM/dx = −V.

Reactions

Reactions are vectors — report them with direction. A support “reaction” that comes out negative means the support would have to pull down; if it is a roller that cannot do that, the beam lifts off and your model is wrong (or the loading case is uplift, which needs hold-downs).

In this calculator

• Moments: sagging positive. Continuous-beam support moments are negative under gravity.
• Shear: left-face-upward positive, so dM/dx = V exactly as the diagrams show.
• Deflection: downward is negative (consistent with y-up coordinates).
• Applied loads: enter magnitudes as positive downward; the calculator handles signs internally.