Structures and loads
A structure is a thing that keeps its shape and carries weight, such as a building, a bridge or a shelf. A load is anything that pushes on it.
- Dead load: the weight of the structure itself (beams, slabs, walls). It never goes away.
- Live load: things that come and go, such as people, furniture and vehicles.
- Wind load: sideways push of moving air.
- Earthquake load: shaking of the ground makes the building sway.
- Snow load: weight of snow on the roof in cold places.
A point load acts at one spot (a column on a beam) and is measured in kN. A distributed load is spread along the length (a wall or a slab) and is measured in kN per metre. A distributed load of w kN/m over a length L is the same as a single force w × L at the middle of that length.
The load path: roof → beam → column → footing → ground. Every part must pass the load on safely.
Equilibrium of forces
A structure at rest is in equilibrium: it does not move and does not turn. For forces in one plane, three simple rules hold.
- ΣFx = 0: total of forces left equals total of forces right.
- ΣFy = 0: total of forces up equals total of forces down.
- ΣM = 0: turning effects clockwise equal turning effects anticlockwise about any point.
A moment is force × distance from the turning point (unit kN·m). It is why a long spanner opens a tight nut more easily.
Beam with load W at distance a from the left support A and span L: take moments about A: RB × L = W × a, so RB = W·a/L. Then RA = W − RB.
Supports and reactions
A support holds the structure. Its push back on the structure is called a reaction. The type of support decides which movements it stops.
- Roller support: stops up-down movement only. 1 reaction (vertical).
- Pin (hinge) support: stops up-down and left-right, but allows turning. 2 reactions (vertical and horizontal).
- Fixed support: stops movement and turning, like a post cast in concrete. 3 reactions (vertical, horizontal, moment).
For a beam with only vertical loads, the horizontal reaction at the pin is zero. For a uniformly distributed load w over span L, each of two end supports carries wL/2.
Stability and determinacy
A structure is stable if the supports stop every possible movement. It is unstable if some movement is free. Example: a beam on two rollers can slide sideways, so it is unstable.
Count the unknown reactions r. For a flat structure we have 3 equations (ΣFx, ΣFy, ΣM).
- r = 3 and well placed: statically determinate. Equations are enough to find all reactions (pin + roller).
- r > 3: statically indeterminate. Extra reactions, so more than the three equations is needed (pin + pin gives r = 4, degree 1).
- r < 3: unstable.
Indeterminate structures are not bad: they are often stronger, but they need more advanced methods. Remember also that three reactions meeting at one point or all parallel can still leave a structure unstable.
Try it: the stick and bag test
Try it. Hang a bag (about 2 kg) on the middle of a broom held by two friends at the ends. Both feel the same weight. Slide the bag towards one friend. Predict who feels more, then check. Now use the 3D above: set load 6 kN, place 2 m, and read RA and RB. They add to 6 kN.
Key formulas and definitions
- ΣFx = 0, ΣFy = 0, ΣM = 0
- Moment = force × perpendicular distance (kN·m)
- Point load W at distance a from A, span L: RB = W·a/L, RA = W − RB
- Uniform load w on span L: total = w·L, each support = wL/2
- Determinacy (plane): r = 3 determinate, r > 3 indeterminate, r < 3 unstable
Worked examples
1. A beam of span 6 m has a 12 kN load 2 m from the left support A. Find RA and RB.
Moments about A: RB × 6 = 12 × 2 = 24, so RB = 4 kN. RA = 12 − 4 = 8 kN. Check: 8 + 4 = 12 kN.
2. A slab load of 5 kN/m acts over a beam of span 8 m. Find the reaction at each support.
Total load = 5 × 8 = 40 kN, acting in the middle. Each support carries 40 ÷ 2 = 20 kN.
3. A beam has one pin and one roller support. Is it determinate? What if both ends are pins?
Pin gives 2, roller gives 1: r = 3, so it is statically determinate. Pin + pin gives r = 4, more than 3, so it is statically indeterminate to the first degree.
Common mistakes
- Forgetting the self weight (dead load) of the beam itself.
- Taking moments but using the sloping distance instead of the perpendicular distance.
- Mixing units: kN with N, or m with mm, in one equation.
- Thinking more supports are always better: extra supports make the structure indeterminate and need more work to solve.