ENG1011 Chap.11 Shear Force and Bending Moment Diagrams
Shear Force and Bending Moment Diagrams
Shear Force and Bending Moment Diagrams is Week 8, which looks inside a beam. Cutting a beam at any point exposes two internal actions: a shear force across the section and a bending moment about it. Both are found the same way as reactions, by keeping one side of the cut and applying equilibrium, with a sign convention that ties each sign to its effect on the beam rather than to an arrow's direction.
Positive shear acts upward on the left face of a segment, and a positive, sagging moment bends the segment so that its top shortens. A shear force diagram plots shear along the beam and a bending moment diagram plots moment. Two relations make them quick to draw: the shear diagram falls at the rate of the downward load, and the slope of the moment diagram equals the shear.
The moment therefore peaks where the shear passes through zero, and the change in moment between two points equals the area of the shear diagram between them.
Point loads cause steps in shear, and applied couples cause steps in moment.
Week 8 is tested with diagram recognition and with numbers read from diagrams: choosing the correct shear and moment pair for a loading, then finding a shear value, two moment values, the maximum moment and the point where the moment changes sign.
The repeater practice test sets all of these on one beam with a uniform load, a couple and an overhanging point load.
What this chapter covers
- 01
Internal shear force and bending moment
- 02
Cutting a beam and keeping one side
- 03
The sign convention for shear and moment
- 04
General equations for shear and moment in x
- 05
Load, shear and moment relations
- 06
Shapes under uniform, triangular and point loads
- 07
Steps caused by applied couples
- 08
Finding the maximum moment where shear is zero
Worked example · free
Shear and moment at the wall of a uniformly loaded cantilever
- 1Cut just right of the wall and keep the free part, which carries 2 times 3 = 6 kN of load centred 1.5 m from the cut.
- 1Vertical balance on the free part: V minus 6 = 0 with V drawn positive, so V = 6 kN.
- 1Moments about the cut: the load turns the free part clockwise with 6 times 1.5 = 9 kN m, so M = minus 9 kN m, hogging.
Key terms
- Shear force
- The internal force acting across a beam's cross-section, found from vertical equilibrium of one side of a cut.
- Bending moment
- The internal moment acting about a beam's cross-section, found from moment equilibrium of one side of a cut.
- Shear force diagram
- A plot of internal shear force against position along a beam.
- Sagging moment
- A bending moment that makes a beam curve with its top shortened and its bottom stretched.
- Hogging moment
- A bending moment that makes a beam curve with its top stretched, as over a cantilever support.
Shear Force and Bending Moment Diagrams FAQ
Where is the maximum bending moment on a beam?
At a point where the shear force passes through zero, or at a point where the shear jumps across zero because of a point load. Find that position from the shear diagram first, then calculate the moment there by cutting the beam.
Does it matter which side of the cut I keep?
No, both sides give the same shear and moment if the signs follow the convention. Keep the side with fewer loads and no unknown reactions, which is often the free end of a cantilever or a short overhang.
How does an applied couple show up on the diagrams?
It leaves the shear diagram unchanged but makes the moment diagram jump by the size of the couple at the point where it acts. Missing that jump is a common error, because nothing on the shear diagram warns you about it.
What shape does the moment diagram take under a uniform load?
A parabola. The shear under a uniform load is a sloping straight line, and since the moment's slope equals the shear, the moment curve changes slope steadily and peaks where the shear line crosses zero.
Exam move
Draw the beam, shear diagram and moment diagram stacked with key points aligned, and check every segment against the shape rules. Practise writing the general equations for shear and moment in x for one segment at a time, and always confirm that the moment returns to zero at pinned and roller ends. Draw every diagram you practise with key values labelled, and confirm the moment at each support from both sides of the beam.
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