Beams, shear force and bending moments for QCE Engineering Unit 3

What this dot point is asking

QCAA wants you to look inside a loaded beam and quantify the two internal effects that bending produces: shear force and bending moment. You need to find their values at points along the beam, identify where the bending moment is greatest, and explain why that location is where the beam is most likely to fail. This extends the equilibrium and stress work to the everyday structural element of a floor joist, lintel or bridge girder.

The answer

Shear force and bending moment

When a beam bends, each internal cross-section carries two internal reactions:

• Shear force $V$: the net force acting transverse to the beam axis, tending to slide one part of the beam past the other.
• Bending moment $M$: the net turning effect about the section, tending to curve the beam.

To find them at a chosen point, make an imaginary cut there and consider the equilibrium of the part to one side. The shear force is the algebraic sum of the vertical forces on that part; the bending moment is the algebraic sum of their moments taken about the cut:

Sign conventions vary, so the key is consistency: sagging (concave up) bending is conventionally positive.

Where the beam is most stressed

Bending moment varies along the beam, and the section carrying the largest bending moment is the most highly stressed. The bending stress at the outer fibre of a beam is:

where $y$ is the distance from the neutral axis to the outer fibre and $I$ is the second moment of area of the cross-section. Because $\sigma$ is proportional to $M$, the point of maximum bending moment is the point of maximum bending stress, and therefore the most likely failure location. A deeper section (larger $I$) or a higher-strength material reduces the stress.

Standard results worth knowing

For a simply supported beam of span $L$:

• Central point load $W$: maximum bending moment $M_{max} = \dfrac{WL}{4}$ at midspan.
• Uniformly distributed load of total weight $W$: maximum bending moment $M_{max} = \dfrac{WL}{8}$ at midspan.

Deflection (how far the beam sags) depends on the load, the span cubed or to the fourth power, and on $EI$, the product of Young's modulus and second moment of area. A serviceable beam must satisfy both a strength limit (stress below the allowable) and a deflection limit.

Why this matters for civil structures

Every floor, roof and bridge deck is a beam problem. Engineers locate the maximum bending moment, compute the bending stress, and pick a section deep enough to keep that stress safe while also limiting deflection. This is why structural beams are deep rather than wide and why an I-section concentrates material at the top and bottom, far from the neutral axis where it does the most good.