1. Centroid and Moment of Inertia
The centroid is the point where the total area/volume of a body is assumed concentrated. Moment of inertia measures a section’s resistance to bending. For a hollow circular section: I = (π/64)(D⁴ – d⁴). The centroid of a semicircle of radius r from its base is 4r/3π.
2. Stress, Strain, and Elastic Constants
- Volumetric strain: eᵥ = (P/AE)(1 – 2/m), where 1/m is Poisson’s ratio
- Relation between E, K (bulk modulus), μ (Poisson’s ratio): E = 3K(1 – 2μ)
- If modulus of rigidity = bulk modulus, Poisson’s ratio = 1/4
- Principal planes are where the resultant stress is purely normal stress (no shear)
3. Torsion of Shafts
Torsion formula: τ = 16T/(πd³). Polar modulus = ratio of polar moment of inertia to distance from center (radius). The ratio of average to maximum shear stress in a solid circular section is 2/3.
4. Columns and Buckling
Euler’s crippling load depends on end conditions. For one end fixed, one end hinged: Pcr = π²EI/(2l²) (using effective length = l/√2).
5. Beams — Shear Force, Bending Moment, Fixed & Continuous Beams
- Shear force = algebraic sum of transverse forces on one side of a section
- A three-hinged arch is statically determinate
- Fixed beam with UDL: max hogging BM = wl²/12
- The three-moment equation applies only when there’s no discontinuity (like hinges) within the span
6. Soil Classification and Index Properties
- Silt particles (IS: 1498) are smaller than 75 microns
- China clay = Kaolinite mineral
- Black cotton soil’s swelling nature is due to montmorillonite
7. Permeability, Seepage, and Consolidation
- Seepage quantity ∝ coefficient of permeability × head loss
- Uniformity coefficient = D₆₀/D₁₀ (from the given data ≈ 0.06/0.0005 = 120… check values)
- Compression index = decrease in void ratio per increase in effective stress (log scale)
- Degree of consolidation = ratio of settlement at any time to final settlement
8. Shear Strength of Soil and Bearing Capacity
- For saturated clay shear strength testing: unconfined compression test or triaxial test
- Relative density formula relates natural, max, and min void ratios
- Bearing capacity relations link safe, net, and ultimate values using factor of safety
9. Open Channel Hydraulics
- Most economical rectangular channel: hydraulic mean depth = half the depth of flow
- Normal depth corresponds to uniform flow
- Most economical section = maximum discharge for given area, or minimum wetted perimeter
- Hydraulic radius = area of flow / wetted perimeter
10. Surveying — Curves, Contours, and Bearings
- Relation between curve radius and degree of curve: R = 1718.9/D (for chord definition) or R = 1146/D
- Contour lines join points of equal elevation
- True bearing = Magnetic bearing ± declination (West declination is subtracted… actually added when West: True = Magnetic + Declination for West)
- Transition curves gradually introduce superelevation/curvature
- Whole circle bearing ranges from 0° to 360°
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