Define shear force and bending moment

Shear force (SF)

The shear force at a section of a beam is the algebraic sum of all the vertical forces (loads and reactions) acting on either side of the section. It is the force tending to slide one part of the beam relative to the other along the section, and it is the same, in magnitude, whether taken from the left or the right part. Unit: N or kN.

Sign convention: shear force is positive when the part to the left of the section is pushed upward relative to the right part (i.e. the resultant of the vertical forces on the left of the section acts upward); this produces a clockwise shearing effect on the element.

Bending moment (BM)

The bending moment at a section is the algebraic sum of the moments, about the centroid of that section, of all the forces (loads and reactions) acting on either side of the section. It is the moment that tends to bend the beam and is resisted by the internal stresses. Unit: N·m or kN·m.

Sign convention: bending moment is positive (sagging) when it bends the beam concave upward, causing compression in the top fibres and tension in the bottom fibres; it is negative (hogging) when the beam bends concave downward, with tension on top.

 Positive (sagging)         Negative (hogging)
   ⌣  tension at bottom       ⌢  tension at top

Example: for a simply supported beam of span $L$ with a central load $P$, the shear force is $+P/2$ on the left half and $-P/2$ on the right half, and the bending moment at mid-span is $+PL/4$.