Civil structures and forces
A structure is something built to carry loads: a bridge, a building, a tower, a dam. Loads are the forces on it. Dead load is the weight of the structure itself. Live load is people, cars, furniture. Environmental loads are wind, snow, earthquake and water.
A force has a size, a direction and a point where it acts. Forces inside a member can be tension (pulling, the member gets longer), compression (pushing, shorter), shear (sliding) or bending (curving).
The structure rests on supports. A roller pushes only up. A pin pushes up and sideways. A fixed support also stops turning. The push of a support is called a reaction.
Calculating statically determinate structures
A structure is statically determinate when the three balance rules alone give every unknown reaction. The rules: ΣFx = 0, ΣFy = 0, ΣM = 0 (forces and turning effects cancel).
Simply supported beam of span L with load P at distance a from A (b = L − a): take moments about B, then RA = P b / L and RB = P a / L. Check: RA + RB = P.
The bending moment is the turning effect inside the beam. Under the point load it is M = P a b / L. For an even (uniform) load w per metre it is Mmax = w L² / 8 at midspan. For a cantilever (fixed at one end) with a tip load: M = P L at the fixed end. A truss, made of triangles, is solved joint by joint: each bar is only in tension or compression.
Strength of materials and member design
Stress is force per area: σ = F / A. Unit: pascal (Pa); 1 MPa = 1 N/mm². Strain is the change in length divided by the original length: ε = ΔL / L (no unit). For steel, stress and strain stay in proportion in the elastic range: σ = E ε, where E (Young's modulus) is about 200 GPa. So a bar stretches by ΔL = F L / (A E).
In bending, the top of the beam is squeezed and the bottom is stretched. The stress is biggest at the outer edge: σ = M / Z, where Z (section modulus) depends on the cross-section shape. A deep section has a big Z.
Member design means: find the load, find the moment, then choose a section so that the stress stays below the allowable stress (the strength divided by a safety factor, such as 1.5).
Try it
At home: put a ruler across two books and press in the middle. Press near one end. Where does it bend most? Place a coin load and watch.
In the 3D: predict first. If the load moves to the middle (a = 3 m), what will RA be? Then use the slider and check.
Key formulas and definitions
- ΣFx = 0, ΣFy = 0, ΣM = 0
- R_A = P b / L, R_B = P a / L
- M = P a b / L (point load); M = w L² / 8 (uniform load); M = P L (cantilever)
- σ = F / A; ε = ΔL / L; σ = E ε
- ΔL = F L / (A E)
- σ = M / Z (bending stress)
Worked examples
1. A 6 m beam carries a 30 kN load 2 m from support A. Find both reactions.
a = 2, b = 4. R_A = 30 × 4 / 6 = 20 kN. R_B = 30 × 2 / 6 = 10 kN. Check: 20 + 10 = 30.
2. Find the bending moment under that load.
M = P a b / L = 30 × 2 × 4 / 6 = 40 kNm. (Also M = R_A × 2 = 20 × 2 = 40.)
3. A beam of span 8 m carries a uniform load of 10 kN/m. Find the maximum moment.
M = w L² / 8 = 10 × 64 / 8 = 80 kNm at midspan.
4. A rod of area 500 mm² pulls with 50 kN. Find the stress.
σ = 50 000 N / 500 mm² = 100 N/mm² = 100 MPa.
5. A steel bar (E = 200 GPa) 2 m long, area 500 mm², carries 100 kN tension. Find the stretch.
ΔL = F L / (A E) = 100 000 × 2000 / (500 × 200 000) = 2 mm.
6. A beam has M = 40 kNm and section modulus Z = 800 cm³. Find the bending stress.
M = 40 × 10⁶ N·mm. Z = 800 × 10³ mm³. σ = 40 × 10⁶ / 800 × 10³ = 50 MPa.
Common mistakes
- Forgetting to include the beam's own weight (dead load).
- Mixing units: kN with N, or m with mm. Convert first. 1 MPa = 1 N/mm².
- Taking moments about different points and mixing the signs. Pick one point and keep one sign rule.
- Thinking the support nearer the load takes less. It takes more.