Civil Engineering Concepts & Formulas: Learn, Revise & Practise

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Current coverage: mathematics, structural mechanics and analysis, concrete structures, geotechnical engineering, fluid mechanics and hydraulics, hydrology and irrigation, environmental engineering, and transportation. This is a growing collection based on the supplied formula sheet; it does not yet cover the entire Civil Engineering syllabus.

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Cross-sectionFormula

Parabolic camber

How does the drop from the crown vary with transverse distance?

$$y=\frac{2x^2}{\mathrm{nW}}$$

W: full carriageway width (m); x: horizontal distance from crown (m); y: downward drop (m); 1 in n: average crown-to-edge crossfall.

A parabolic surface is flat at the crown and becomes steeper towards the edge.

Check: This convention uses average crossfall. If 1/n denotes the local edge slope instead, the coefficient changes to 1/(nW).

Understand / derivation

1. Write y = a x² with zero slope at the crown.
2. At x = W/2, the specified average crossfall gives y = W/(2n).
3. Substitution gives a = 2/(nW). The local edge slope is 2/n.

Worked example

W = 7 m, n = 50, x = 3.5 m: y = 0.070 m = 70 mm.

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Sight distanceFormula

Stopping and intermediate sight distance

Can you separate reaction distance from braking distance?

$$\mathrm{SSD}=\mathrm{vt}+\frac{v^2}{2g(f\pm G)}$$$$\mathrm{ISD}=2\mathrm{SSD}$$

v: m/s; t: s; g: 9.81 m/s²; f: longitudinal friction; G: grade magnitude as a decimal. SSD and ISD: m.

A driver travels during perception and reaction, then needs distance to dissipate kinetic energy.

Check: A 3% grade is 0.03. Convert V in km/h using v = V/3.6. For downhill braking, f − G must stay positive.

Understand / derivation

1. Reaction distance = vt.
2. Work-energy gives braking distance v²/[2g(f ± G)] using the small-slope approximation.
3. Use +G uphill and −G downhill; on level ground set G = 0. ISD is twice SSD in this teaching convention.

Worked example

v = 20 m/s, t = 2.5 s, f = 0.35, G = 0: SSD = 108.25 m; ISD = 216.50 m.

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Sight distanceFormula

Overtaking sight distance

What are the three distance components and the overtaking time?

$$\mathrm{OSD}=d_1+d_2+d_3$$$$d_1=v_{b}t;d_2=2s+v_{b}T;d_3=\mathrm{vT}$$$$T=\sqrt{4s/a}$$

v_b: slower vehicle speed (m/s); v: opposing vehicle speed (m/s); t: reaction time (s); T: overtaking time (s); s: spacing (m); a: acceleration (m/s²).

The model adds the overtaker’s reaction travel, its passing travel, and the opposing vehicle’s travel.

Check: Constant acceleration and equal initial speeds are assumptions. Spacing and acceleration must follow the question’s model.

Understand / derivation

1. The overtaker initially follows at v_b; reaction travel is v_b t.
2. It gains 2s relative distance: aT²/2 = 2s, hence T = √(4s/a).
3. Passing travel is v_b T + 2s; opposing travel is vT.

Worked example

v_b = 15 m/s, v = 25 m/s, t = 2 s, s = 20 m, a = 1 m/s²: T = 8.94 s and OSD = 427.77 m.

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Horizontal curvesFormula

Superelevation and extra widening

What balances lateral acceleration, and why is the carriageway widened?

$$e+f=\frac{v^2}{\mathrm{gR}}$$$$W_e=\frac{{\mathrm{nl}}^2}{2R}+\frac{V}{9.5\sqrt{R}}$$

v: m/s; V: km/h; R and wheelbase l: m; n: lane count; e: superelevation decimal; f: lateral friction; W_e: m.

Banking and lateral friction share the cornering demand. Widening accounts for off-tracking and driver clearance.

Check: This equilibrium equation alone is not the complete IRC superelevation design procedure. Never use km/h in the v²/(gR) form.

Understand / derivation

1. For small banking angles, lateral equilibrium gives e + f ≈ v²/(gR).
2. Front/rear wheel geometry gives off-tracking approximately l²/(2R) per lane.
3. Add the empirical psychological allowance V/(9.5√R).

Worked example

v = 20 m/s, R = 250 m: e + f = 0.1631. With n = 2, l = 6 m and V = 72 km/h, W_e = 0.623 m.

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TransitionsFormula

Transition length: comfort

How is transition length related to allowable jerk?

$$L_s=\frac{v^3}{\mathrm{CR}}$$

L_s and R: m; v: m/s; C: permissible rate of change of lateral acceleration, m/s³.

A transition introduces lateral acceleration gradually.

Check: This is one criterion. Compare comfort, runoff and any specified empirical minimum; adopt the governing length.

Understand / derivation

1. Lateral acceleration changes from zero to v²/R.
2. Travel time through the transition is L_s/v.
3. Divide acceleration change by time: C = v³/(R L_s); rearrange.

Worked example

v = 20 m/s, R = 250 m, C = 0.5 m/s³: L_s = 64.00 m.

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TransitionsFormula

Transition length: superelevation runoff

What length develops the full edge rise about the inner edge?

$$L_s=e{N}_r(W+W_e)$$

N_r: denominator of relative gradient 1 in N_r; e: decimal; W and W_e: m; L_s: m.

The outer edge needs enough longitudinal distance to rise without an abrupt relative slope.

Check: This is the zero-crossfall-to-full-bank, inner-edge rotation model. Initial camber removal and centre-line rotation need their own geometry.

Understand / derivation

1. Total width B = W + W_e.
2. Final outer-to-inner edge rise E = eB.
3. For relative gradient E/L_s = 1/N_r, obtain L_s = N_r eB.

Worked example

e = 0.07, N_r = 150, W = 7.0 m, W_e = 0.6 m: L_s = 79.8 m.

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TransitionsFormula

Transition length: empirical criterion

Which empirical expression corresponds to each terrain category?

$$L_s=\frac{2.7V^2}{R}$$$$L_s=\frac{V^2}{R}$$

V: km/h; R and L_s: m. First equation: plain/rolling; second: steep/mountainous.

These are conventional IRC-based teaching expressions reproduced in the supplied formula sheet.

Check: Use the specified code edition or examination convention. These expressions are not claimed to be universal current design requirements.

Understand / derivation

1. These coefficients are empirical, not a force-balance derivation.
2. The equivalent m/s coefficients are approximately 35 and 12.96.
3. Compare this length with the comfort and runoff results.

Worked example

V = 72 km/h, R = 250 m: plain/rolling gives 55.99 m; steep/mountainous gives 20.74 m.

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TransitionsConcept

Spiral transition: the key property

What varies linearly along an ideal spiral transition?

$$1/r\left(x\right)=\frac{x}{{\mathrm{RL}}_s}$$

x: distance along the transition (m), 0 ≤ x ≤ L_s; r(x): local radius (m); R: final circular radius (m).

Curvature grows continuously from zero on the tangent to 1/R at the circular curve.

Check: The supplied sheet’s “spiral recommended” entry is a concept. This card adds its mathematical property.

Understand / derivation

1. At constant speed, constant jerk requires lateral acceleration to grow linearly with travel time.
2. Since distance is proportional to time and acceleration is v²/r, curvature must grow linearly with distance.
3. The endpoint condition is 1/r(L_s) = 1/R. This is the clothoid/spiral property.

Worked example

R = 250 m and L_s = 60 m: halfway along the spiral, r = 500 m.

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TransitionsFormula

Shift of a spiral

How far is the circular curve shifted to insert the transition?

$$p=\frac{{L_s}^2}{24R}$$

p: shift (m); L_s: transition length (m); R: circular-curve radius (m).

The circle moves inward to accommodate the transition.

Check: Shift is p here. The symbol S is reserved for sight distance to avoid confusion with your source sheet.

Understand / derivation

1. For small deflections, the spiral endpoint ordinate is approximately L_s²/(6R).
2. The corresponding circle offset contribution is approximately L_s²/(8R).
3. Their difference is L_s²/(24R).

Worked example

L_s = 60 m and R = 250 m: p = 0.60 m.

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Horizontal curvesFormula

Setback: sight distance within the curve

What is the clearance when the circular curve is at least as long as S?

$$m=R(1-\cos (\frac{S}{2R}\left)\right)$$

m, R, sight distance S and curve length L_c: m; L_c ≥ S. Trigonometric argument S/(2R): radians.

The sight line cuts across the inside of the curve; clearance is the middle ordinate.

Check: Use the vehicle-path radius and consistent clearance reference. A multi-lane centre-line setback needs the lane offset.

Understand / derivation

1. The arc subtends θ = S/R radians.
2. The radial projection of half the sight chord is R cos(θ/2).
3. Subtract this from R to obtain the setback.

Worked example

R = 250 m, S = 100 m, L_c = 150 m: m = 4.98 m.

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Horizontal curvesFormula

Setback: sight distance beyond the curve

How is the tangent extension included when L_c is shorter than S?

$$m=R(1-\cos (\theta /2\left)\right)+\frac{S-L_c}{2}\sin (\theta /2)$$$$\theta =\frac{L_c}{R}$$

m, R, S and L_c: m; L_c < S; θ: radians.

Part of the required sight distance lies on the adjoining tangents.

Check: θ uses L_c/R in this case, not S/R. This is the symmetric single-path teaching model.

Understand / derivation

1. The circular part contributes R[1 − cos(θ/2)].
2. Half the excess sight distance is (S − L_c)/2.
3. Project this tangent portion in the setback direction and add it.

Worked example

R = 250 m, S = 150 m, L_c = 100 m: m = 9.95 m.

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Horizontal curvesFormula

Curve resistance: projection model

How much tractive force is lost in the elementary projection model?

$$\mathrm{CR}=T(1-\cos \alpha )$$

T: tractive force (N or kN); α: steering/directional angle between the modelled force directions; CR: same force unit.

Only a component of the applied force acts along the required wheel direction.

Check: This is an elementary teaching model, not a complete tyre/vehicle resistance model. α is not automatically the road’s total intersection angle.

Understand / derivation

1. Applied tractive force is T.
2. Its useful projection is T cos α.
3. The difference is T − T cos α.

Worked example

T = 10 kN and α = 10°: CR = 0.1519 kN.

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Vertical curvesFormula

Grade compensation

Which percentage reduction is selected on a horizontal curve?

$$\mathrm{GC}=\min (\frac{30+R}{R},\frac{75}{R})\%$$

R: horizontal radius (m); GC: percentage points of gradient reduction.

A reduced longitudinal grade offsets additional resistance on a curve.

Check: Subtract percentage points: 6 − 0.75, not 6 × (1 − 0.0075). This empirical expression is not derived from the projection model alone.

Understand / derivation

1. Calculate both empirical percentage allowances.
2. Choose the smaller of the two.
3. Apply the question’s grade limits and code convention; compensation is not required for grades flatter than 4% in the cited teaching notes.

Worked example

R = 100 m: allowances are 1.30% and 0.75%; use 0.75 percentage points. A 6% grade would become 5.25%.

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Vertical curvesFormula

Summit curve for SSD: long curve

What length provides SSD when L is at least S?

$$L=\frac{{\mathrm{NS}}^2}{{\left(\sqrt{2h_1}+\sqrt{2h_2}\right)}^2}$$

N = |g_1 − g_2| with signed decimal grades; L, S, eye height h_1 and object height h_2: m. Require L ≥ S.

The crest blocks the sight line between the driver’s eye and the object.

Check: Use heights specified by the question. 1.2 m and 0.15 m are the supplied teaching convention, not an assertion about every code edition.

Understand / derivation

1. For a parabola, a = N/(2L).
2. Tangency geometry gives h_1 = aS_1² and h_2 = aS_2².
3. Use S = S_1 + S_2, then solve for L.

Worked example

N = 0.04, S = 120 m, h_1 = 1.2 m, h_2 = 0.15 m: L = 131.00 m, which satisfies L ≥ S.

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Vertical curvesFormula

Summit curve for SSD: short curve

What changes when the sight line extends beyond both curve ends?

$$L=2S-\frac{{\left(\sqrt{2h_1}+\sqrt{2h_2}\right)}^2}{N}$$

N: algebraic difference of signed decimal grades; L, S, h_1 and h_2: m. Require 0 < L < S.

The sight line includes tangent portions beyond the crest parabola.

Check: Do not accept a negative computed length as a construction length. Check the branch and any minimum length requirements.

Understand / derivation

1. Minimising the tangent sight geometry gives S = L/2 + (√h_1 + √h_2)²/N.
2. Multiply by two and rearrange to obtain L.
3. Check the result against the assumed L < S branch.

Worked example

N = 0.02, S = 120 m, h_1 = 1.2 m, h_2 = 0.15 m: L = 20.15 m, so the short-curve branch is consistent.

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Vertical curvesFormula

Summit curve for OSD: long curve

How does the crest formula simplify for equal sight heights?

$$L=\frac{{\mathrm{NS}}^2}{8H}$$

N: decimal grade difference; S: required overtaking sight distance (m); H: equal eye/object height (m); L: m. Require L ≥ S.

For overtaking, the sight line connects two equal-height vehicle reference points.

Check: Use the OSD specified or calculated for the question. Do not substitute SSD merely because the general formula looks similar.

Understand / derivation

1. Set h_1 = h_2 = H in the general long-crest formula.
2. The denominator becomes (2√(2H))² = 8H.
3. Substitute the required OSD for S.

Worked example

N = 0.06, S = 300 m, H = 1.2 m: L = 562.50 m, satisfying L ≥ S.

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Vertical curvesFormula

Summit curve for OSD: short curve

What is the short-crest formula for equal heights?

$$L=2S-\frac{8H}{N}$$

N: decimal grade difference; S, H and L: m; equal eye/object heights H. Require 0 < L < S.

Equal sight heights simplify the general short-crest equation.

Check: Grade difference is the magnitude of the difference of signed grades: +3% and −2% give N = 0.05.

Understand / derivation

1. Start with the short-crest SSD/OSD equation.
2. Set both heights equal to H; the height term becomes 8H.
3. Rearrange and verify the short-curve assumption.

Worked example

N = 0.02, S = 300 m, H = 1.2 m: L = 120.00 m, satisfying L < S.

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Vertical curvesFormula

Valley curve: headlight, long curve

How does headlight height and beam angle control the long sag curve?

$$L=\frac{{\mathrm{NS}}^2}{2h+2S\tan \alpha }$$

N: decimal grade difference; L, S and headlight height h: m; α: upward beam angle. Require L ≥ S.

At night, the headlight beam must reach the road far enough ahead.

Check: For α = 1°, use tan(π/180) in a radians-based calculator. Use h and α from the question/code convention.

Understand / derivation

1. Beam height at sight distance S is h + S tan α.
2. Equate this with the parabolic ordinate NS²/(2L).
3. Solve for L and check L ≥ S.

Worked example

N = 0.06, S = 120 m, h = 0.75 m, α = 1°: L = 151.87 m, satisfying L ≥ S.

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Vertical curvesFormula

Valley curve: headlight, short curve

What is the headlight criterion when L is shorter than S?

$$L=2S-\frac{2h+2S\tan \alpha }{N}$$

N: decimal grade difference; L, S and h: m; α: beam angle. Require 0 < L < S.

The headlight sight line extends beyond the sag curve.

Check: A sag design must satisfy both headlight visibility and the adopted comfort model, plus any specified minimum length.

Understand / derivation

1. Use the short-curve geometric relation N(2S − L) = 2h + 2S tan α.
2. Divide by N and solve for L.
3. Check the branch before comparing with the comfort criterion.

Worked example

N = 0.04, S = 120 m, h = 0.75 m, α = 1°: L = 97.77 m, satisfying L < S.

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Vertical curvesFormula

Valley curve: comfort criterion

What is the total length for the two-transition comfort model?

$$L=2\sqrt{\frac{{\mathrm{Nv}}^3}{C}}$$

L: total length (m); N: decimal grade difference; v: m/s; C: allowable jerk (m/s³).

This conventional model uses two equal transition portions. The square root is correct.

Check: Nv³/C has units m², so its square root has units m. Do not change it to a cube root or use the horizontal empirical C without instruction.

Understand / derivation

1. Let each transition portion have length l.
2. In this teaching model C = v³/(Rl) and R = l/N; hence l² = Nv³/C.
3. The total length is L = 2l.

Worked example

N = 0.04, v = 20 m/s, C = 0.6 m/s³: L = 46.19 m.

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TransitionsFormula

Permissible jerk for horizontal transitions

What expression and limits are used for C in this convention?

$$C=\frac{80}{75+V}$$$$0.5\le C\le 0.8$$

V: km/h; C: m/s³. Apply the lower and upper bounds after evaluating the expression.

The empirical allowable jerk reduces with speed, within specified limits.

Check: This card applies to horizontal transitions. The sag comfort example uses the separately specified C = 0.6 m/s³.

Understand / derivation

1. Compute the raw value 80/(75 + V).
2. If it is below 0.5 use 0.5; if above 0.8 use 0.8.
3. Use this C in the horizontal-transition comfort equation when this convention is specified.

Worked example

V = 72 km/h: raw C = 0.5442 m/s³, within bounds. At V = 100 km/h, raw C = 0.4571; adopt 0.5 m/s³.

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Highway materials and bituminous mixesFormula

Aggregate flakiness and elongation indices

Recall aggregate flakiness and elongation indices. State the definitions and conditions before you apply it.

$$FI=100\frac{M_{flaky}}{M_{tested}},EI=100\frac{M_{elongated}}{M_{tested}}$$

For the stated test fractions: flaky thickness <0.6 times mean sieve size; elongated length >1.8 times mean sieve size.

Understand / derivation

Separate particles using the relevant thickness/length gauge and compare their mass with the eligible test sample mass.

Worked example

A 1000 g eligible sample with 150 g flaky particles has FI=15%.

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Check: Use the prescribed eligible size fractions, mean sieve size and separate/combined-index procedure; these are mass-based indices.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 22, 23

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Highway materials and bituminous mixesFormula

Angularity number

Recall angularity number. State the definitions and conditions before you apply it.

$$AN=67-100\frac{W}{{\rho }_{w}V{G}_s}$$

W: aggregate mass in the calibrated cylinder; V: cylinder volume; G_s: aggregate specific gravity; use compatible units.

Understand / derivation

The measured solid-volume percentage is 100W/(ρ_wVG_s); angularity number measures its shortfall from a reference packing value of 67.

Worked example

If solid volume fraction=0.62, AN=67-62=5.

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Check: The source’s cylinder-water-weight form is equivalent when ρ_wV is used instead of volume alone.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 24

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Highway materials and bituminous mixesConcept

Bitumen penetration and Marshall flow units

Explain bitumen penetration and Marshall flow units. State the definitions and conditions before you apply it.

The conventional penetration test reports tenths of a millimetre under its specified load, temperature and duration; the source uses 100 g and 25°C. Traditional Marshall flow gauge divisions represent 0.25 mm.

Understand / derivation

A result is meaningful only with its test procedure: penetration measures consistency under a standard needle, while flow measures deformation at the stability test condition.

Worked example

Penetration 65 means 6.5 mm needle penetration under the specified test. Twelve 0.25 mm flow units mean 3 mm.

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Check: Include the prescribed penetration duration (commonly 5 s under the stated conventional test). Do not interpret these gauge values as direct material moduli.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 25, 30

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Highway materials and bituminous mixesFormula

Mixture specific gravity and bulk density

Recall mixture specific gravity and bulk density. State the definitions and conditions before you apply it.

$$G_{mix}=\frac{\sum W_i}{\sum W_i/G_i},G_{mb}=\frac{M}{{\rho }_{w}V}$$

Wᵢ: component masses; Gᵢ: compatible component specific gravities; G_mb: compacted bulk specific gravity.

Understand / derivation

Add component volumes Wᵢ/(ρ_wGᵢ) rather than averaging their densities arithmetically.

Worked example

Equal masses with G₁=2 and G₂=3 give combined specific gravity 2/(1/2+1/3)=2.4.

Original CivilEase learning example; not an exam PYQ.

Check: Maximum theoretical gravity G_mm, aggregate bulk G_sb and effective G_se have distinct definitions. The simple component formula assumes the chosen volume/absorption basis.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 26, 27

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Highway materials and bituminous mixesFormula

Bituminous-mixture volumetric properties

Recall bituminous-mixture volumetric properties. State the definitions and conditions before you apply it.

$$V_a=100(1-\frac{G_{mb}}{G_{mm}}),V_b=\frac{P_b{G}_{mb}}{G_b}$$$$VMA=100-\frac{G_{mb}{P}_s}{G_{sb}},VFB=100\frac{VMA-V_a}{VMA}$$

P_b,P_s: binder/aggregate mass percentages of total mix; G_b: binder gravity; G_sb: aggregate bulk gravity; V values percent by bulk mix volume.

Understand / derivation

Air voids compare compacted density with air-free maximum density. VMA is the bulk volume outside aggregate particles; binder-filled fraction is VMA minus air.

Worked example

G_mb=2.4 and G_mm=2.5 give V_a=4%. If VMA=16%, VFB=75%.

Original CivilEase learning example; not an exam PYQ.

Check: Binder absorption changes effective binder volume. Do not equate total binder V_b with effective binder in VMA without the compatible assumptions.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 28, 29

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Flexible and rigid pavement designFormula

Cumulative design traffic with growth

Recall cumulative design traffic with growth. State the definitions and conditions before you apply it.

$$N=365A\frac{(1+r{)}^n-1}{r}FD,A=P(1+r{)}^x$$

A: commercial vehicles/day at opening; P: count at survey; x: years to opening; n: design years; r: annual fractional growth; F: vehicle damage factor; D: lane factor.

Understand / derivation

Sum the geometric sequence A+A(1+r)+... over n years and multiply by 365, damage factor and lane fraction.

Worked example

A=1000, r=0.05, n=10 and FD=1 give N≈4.5909 million standard axles.

Original CivilEase learning example; not an exam PYQ.

Check: Corrected source denominator: growth factor is divided by r, not n. For r=0 use the limit n. Divide N by 10⁶ for msa.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 31

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Flexible and rigid pavement designConcept

Lane-distribution factors: source-sheet conventions

Explain lane-distribution factors: source-sheet conventions. State the definitions and conditions before you apply it.

The source lists undivided two-lane/four-lane factors 0.75/0.40, and divided-carriageway factors 0.75, 0.60, 0.45 for two, three and four lanes in a direction.

Understand / derivation

A design-lane factor converts commercial traffic on the specified carriageway/direction basis to the critical lane. Directional split and lane distribution are distinct operations.

Worked example

If directional traffic=2000 vehicles/day and the applicable design-lane fraction is 0.75, design-lane traffic=1500/day.

Original CivilEase learning example; not an exam PYQ.

Check: Retained as older study values, not universal current IRC factors. Confirm edition, number of lanes per direction and whether the traffic input is one-way or two-way before applying any factor.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 32, 33

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Flexible and rigid pavement designFormula

Fourth-power axle-load equivalency

Recall fourth-power axle-load equivalency. State the definitions and conditions before you apply it.

$$EALF\approx {\left(\frac{P}{80\mathrm{k}\mathrm{N}}\right)}^4$$

P: single-axle load in the source’s simplified 80 kN reference convention.

Understand / derivation

The empirical fourth-power approximation represents strongly nonlinear pavement damage with axle load.

Worked example

A 100 kN axle gives equivalency≈(100/80)⁴=2.4414 standard axles.

Original CivilEase learning example; not an exam PYQ.

Check: A fleet VDF is the weighted total equivalent standard axles per commercial vehicle, not this expression for every axle configuration. Tandem/tridem and prescribed design methods differ.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 34

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Flexible and rigid pavement designFormula

California bearing ratio

Recall california bearing ratio. State the definitions and conditions before you apply it.

$$CBR\%=100\frac{P_{test}}{P_{standard}}$$

Conventional source reference loads: 1370 kgf at 2.5 mm and 2055 kgf at 5 mm penetration; source reference stresses about 70 and 105 kgf/cm².

Understand / derivation

Compare measured penetration resistance with the standardized crushed-stone reference at the same penetration.

Worked example

At 2.5 mm, a 685 kgf test load gives CBR=50%.

Original CivilEase learning example; not an exam PYQ.

Check: The source omits the percent multiplier. Keep force units consistent and use the specified standard’s reference values and repeat-test rule when the 5 mm result exceeds the 2.5 mm result.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 35, 36

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Flexible and rigid pavement designFormula

ESWL by logarithmic interpolation

Recall eSWL by logarithmic interpolation. State the definitions and conditions before you apply it.

$$\log P_e=\log P+\frac{\log z-\log z_1}{\log z_2-\log z_1}\log 2$$

P: load on each dual wheel; P_e: equivalent single-wheel load; source idealized depth anchors z₁=d/2 and z₂=2s with compatible tyre-gap/spacing definitions.

Understand / derivation

Interpolate a straight line in log(load)-log(depth) space between separate-wheel and combined-load limiting behaviours.

Worked example

At the geometric-mean depth z=√(z₁z₂), log interpolation gives P_e=√2 P.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s first depth symbol is ambiguous. Determine clear spacing d and centre spacing s from the prescribed dual-wheel diagram; do not interchange them.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 37

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Flexible and rigid pavement designFormula

Radius of relative stiffness of a concrete slab

Recall radius of relative stiffness of a concrete slab. State the definitions and conditions before you apply it.

$$\ell ={\left[\frac{E{h}^3}{12k(1-{\nu }^2)}\right]}^{1/4}$$

E,ν: concrete elastic constants; h: slab thickness; k: subgrade reaction (force/length³).

Understand / derivation

Balance flexural plate stiffness Eh³/[12(1-ν²)] against Winkler support stiffness k to obtain a characteristic load-spreading length.

Worked example

Doubling thickness increases ℓ by 2^(3/4)≈1.682 with other properties fixed.

Original CivilEase learning example; not an exam PYQ.

Check: Use consistent units; ℓ is a length, not a dimensionless coefficient.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 38

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Flexible and rigid pavement designFormula

Subgrade reaction and plate-size effect

Recall subgrade reaction and plate-size effect. State the definitions and conditions before you apply it.

$$k=p/\delta ;k\approx \frac{E}{1.18a}(\text{rigid}),k\approx \frac{E}{1.5a}(\text{flexible});k_1{a}_1\approx k_2{a}_2$$

p: bearing pressure; δ: settlement; a: plate radius; the source’s 1.18/1.5 constants correspond approximately to elastic ν=0.5 assumptions.

Understand / derivation

For a plate on an elastic half-space, settlement scales with pa/E, so k scales with E/a rather than being an intrinsic soil constant.

Worked example

p=1 kgf/cm² at δ=0.125 cm gives k=8 kgf/cm³.

Original CivilEase learning example; not an exam PYQ.

Check: State the test settlement level and plate geometry. Size correction, soil layering and stress dependence limit extrapolation.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 39, 40, 41, 42

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Flexible and rigid pavement designFormula

Tyre rigidity factor

Recall tyre rigidity factor. State the definitions and conditions before you apply it.

$$RF=\frac{p_{contact}}{p_{inflation}}$$

Both pressures in the same unit; RF: source tyre-rigidity factor.

Understand / derivation

Tyre carcass stiffness can make average contact pressure differ from inflation pressure; the ratio describes the assumed contact model.

Worked example

Contact=0.75 MPa and inflation=0.70 MPa give RF≈1.071.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s 7 kgf/cm² crossover is a historical empirical observation, not a universal modern tyre law. Actual footprints and pressures should be measured/modelled appropriately.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 43

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Flexible and rigid pavement designFormula

Thermal expansion and joint movement

Recall thermal expansion and joint movement. State the definitions and conditions before you apply it.

$$\Delta L=\alpha L\Delta T$$

α: thermal expansion coefficient per °C; L: contributing slab length; ΔT: temperature rise.

Understand / derivation

Uniform thermal strain αΔT times original length gives free expansion; joint movement depends on which slab lengths feed the joint.

Worked example

α=10⁻⁵/°C, L=30 m and ΔT=40°C give expansion=12 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Joint width also depends on installation temperature, multiple-slab movement, sealant and restraint.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 44

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Flexible and rigid pavement designFormula

Slab friction balance and tie-bar steel area

Recall slab friction balance and tie-bar steel area. State the definitions and conditions before you apply it.

$$\frac{L_c}{2}{\gamma }_{c}hf={\sigma }_{c}h,L_c=\frac{2{\sigma }_c}{{\gamma }_{c}f}$$$$A_s=\frac{bh{\gamma }_{c}f}{{\sigma }_s}$$

γ_c: concrete unit weight; f: friction coefficient; L_c: contraction-joint spacing; b: slab width held by longitudinal joint per unit length; h: thickness.

Understand / derivation

Integrate base friction over the tributary slab area. Equate it to concrete tensile resistance for joint spacing or tie-steel tension for steel area.

Worked example

b=3.5 m, h=0.25 m, γ_c=24 kN/m³, f=1.5 and σ_s=140000 kN/m² give A_s=225 mm² per metre.

Original CivilEase learning example; not an exam PYQ.

Check: Use force-based unit weight and the correct tributary width. Tie bars hold longitudinal joints; they do not replace dowels for transverse load transfer.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 45, 46

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Flexible and rigid pavement designFormula

Tie-bar anchorage length

Recall tie-bar anchorage length. State the definitions and conditions before you apply it.

$$\frac{L}{2}\pi \varphi {\tau }_{bd}=\frac{\pi {\varphi }^2}{4}{\sigma }_s,L=\frac{\varphi {\sigma }_s}{2{\tau }_{bd}}$$

L: total ideal tie length; φ: bar diameter; τ_bd: permissible bond; σ_s: steel stress.

Understand / derivation

The bar force must be developed by bond over half the total length on each side of the joint.

Worked example

φ=12 mm, σ_s=140 MPa and τ_bd=1.4 MPa give ideal L=600 mm before allowances.

Original CivilEase learning example; not an exam PYQ.

Check: Add the required joint, placement and loss-of-bond allowances from the specified design procedure.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 47

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Traffic-flow relationshipsFormula

Poisson traffic arrivals and exponential headways

Recall poisson traffic arrivals and exponential headways. State the definitions and conditions before you apply it.

$$P(N=x)=e^{-\lambda t}\frac{(\lambda t{)}^x}{x!},P(H\ge t)=e^{-\lambda t}$$

λ: arrivals per unit time; t: interval/headway in the same time unit.

Understand / derivation

A homogeneous Poisson process has independent increments; the probability of no arrival over t equals the survival probability of its exponential headway.

Worked example

λ=0.1 vehicle/s gives probability of a gap at least 10 s equal to e⁻¹≈0.3679.

Original CivilEase learning example; not an exam PYQ.

Check: Platooning, signals and interactions often violate independent stationary arrivals.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 48, 49

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Traffic-flow relationshipsFormula

Greenshields model and the flow-density identity

Recall greenshields model and the flow-density identity. State the definitions and conditions before you apply it.

$$v=v_f(1-k/k_j),q=kv,k=1000/s(s\mathrm{i}\mathrm{n}\mathrm{m})$$

v_f: free-flow speed; k_j: jam density; k: vehicles/km; v: km/h; s: mean front-to-front spacing in m.

Understand / derivation

Vehicle conservation gives q=kv. A linear speed-density assumption produces the parabolic flow-density curve and capacity at k_j/2.

Worked example

v_f=80 km/h and k_j=150 veh/km give q_max=v_fk_j/4=3000 veh/h.

Original CivilEase learning example; not an exam PYQ.

Check: Spacing includes vehicle length; use space-mean speed in the fundamental identity.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 50, 51

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Traffic studiesFormula

Peak-hour factor versus annual design-hour factor

Recall peak-hour factor versus annual design-hour factor. State the definitions and conditions before you apply it.

$$PHF=\frac{V_{hour}}{4V_{15,max}},K_{30}=\frac{V_{30}}{AADT}$$

V₁₅,max: highest 15-minute count within the hour; V₃₀: 30th highest hourly volume of the year.

Understand / derivation

PHF measures within-hour peaking. K₃₀ compares an annual design-hour volume with average daily traffic; they measure different time-scale features.

Worked example

V_hour=1000 and V₁₅,max=300 give PHF≈0.8333. V₃₀=1000 and AADT=10000 give K₃₀=0.10.

Original CivilEase learning example; not an exam PYQ.

Check: Corrected the sheet: V₃₀/AADT is a K-factor, not another PHF formula.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 52

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Traffic studiesFormula

Time-mean and space-mean speeds

Recall time-mean and space-mean speeds. State the definitions and conditions before you apply it.

$$v_t=\frac{1}{n}\sum _i{v}_i,v_s=\frac{n}{{\sum }_{i}1/v_i}=\frac{nL}{{\sum }_i{t}_i}$$

Equal-weight speed observations; tᵢ: traversal time over the same length L.

Understand / derivation

At a point, average observed speeds arithmetically. For a common road length, divide total vehicle-distance by total vehicle-time to obtain the harmonic mean.

Worked example

For 40 and 60 km/h, v_t=50 km/h and v_s=48 km/h.

Original CivilEase learning example; not an exam PYQ.

Check: Corrected the source’s inverted/dimensionally incomplete travel-time expression. Equal-length travel times and consistent sampling weights are essential.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 53, 54

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Traffic studiesFormula

Moving-observer estimate of flow and travel time

Recall moving-observer estimate of flow and travel time. State the definitions and conditions before you apply it.

$$q=\frac{n_a+n_y}{t_a+t_w},\overline{t}=t_w-\frac{n_y}{q}$$

n_a: opposite-direction encounters; n_y: vehicles overtaking the test car minus vehicles it overtakes while travelling with the stream; t_a,t_w: against/with-stream trip times.

Understand / derivation

Over the combined observer trips, encountered vehicles plus the net overtake adjustment estimate flow; the net overtake correction adjusts test-car time to stream mean time.

Worked example

n_a=100, n_y=20, t_a=5 min and t_w=5 min give q=12 veh/min and mean time≈3.333 min.

Original CivilEase learning example; not an exam PYQ.

Check: Sign conventions for n_y vary; define it before applying the time correction.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 55

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Traffic studiesConcept

Percentile speeds and annual design-hour conventions

Explain percentile speeds and annual design-hour conventions. State the definitions and conditions before you apply it.

The 85th-percentile speed is a traditional operating-speed input to speed management; the source cites the 98th-percentile speed as a geometric-design convention and the 30th highest hourly volume as a design-hour convention.

Understand / derivation

A percentile summarizes the observed distribution; a selected design percentile or design hour balances service and exceedance frequency.

Worked example

An 85th-percentile speed of 60 km/h means about 85% of the observed sample travels at or below 60 km/h.

Original CivilEase learning example; not an exam PYQ.

Check: These are historical study conventions, not rules that alone establish a safe speed limit or design speed. Modern selection also considers road function, terrain, risk, vulnerable users and the specified design standard.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 56, 57, 58

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Signs, signals, intersections and interchangesFormula

Webster's optimum signal-cycle estimate

Recall webster's optimum signal-cycle estimate. State the definitions and conditions before you apply it.

$$C_0=\frac{1.5L+5}{1-Y}$$

L: total lost time per cycle in s; Y: sum of critical flow/saturation-flow ratios for the phases; C₀ in s.

Understand / derivation

The empirical delay-minimizing cycle balances lost-time penalty against queueing delay as the intersection approaches saturation.

Worked example

L=12 s and Y=0.75 give C₀=92 s.

Original CivilEase learning example; not an exam PYQ.

Check: Requires Y<1 and suitable undersaturated conditions; pedestrian timing, coordination and minimum greens impose additional constraints.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 59

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Highway capacity and level of serviceFormula

Legacy rotary weaving-capacity estimate

Recall legacy rotary weaving-capacity estimate. State the definitions and conditions before you apply it.

$$Q=\frac{280w(1+e/w)(1-p/3)}{1+w/l}$$

Q: PCU/h; w: weaving width m; e: average entry width m; l: weaving length m; p: fraction of weaving traffic.

Understand / derivation

The older empirical rotary equation adjusts a width-based capacity for entry widening, weaving share and relative section length.

Worked example

w=10 m, e=6 m, p=0.5 and l=50 m give Q≈3111 PCU/h.

Original CivilEase learning example; not an exam PYQ.

Check: Use only within the equation’s geometry/traffic calibration and specified older method; it is not a universal modern roundabout-capacity model.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 60

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Airport runway and taxiway designFormula

Elementary runway-length corrections

Recall elementary runway-length corrections. State the definitions and conditions before you apply it.

$$T_r=T_a+\frac{T_m-T_a}{3},T_s=15-0.0065h$$$$L_e=L_0(1+0.07h/300),L_T=L_e[1+0.01(T_r-T_s)],L_g=L_T(1+0.20G_e)$$

h: elevation m; temperatures °C; G_e: effective runway gradient in percent as a numeric percent (1 for 1%); T_a,T_m: specified hottest-month means.

Understand / derivation

Apply the conventional elevation correction, then the temperature allowance above standard atmosphere, then the stated gradient correction.

Worked example

L₀=1000 m and h=300 m give L_e=1070 m. If T_r-T_s=10°C, L_T=1177 m before gradient.

Original CivilEase learning example; not an exam PYQ.

Check: These are elementary study rules; use only positive temperature correction where prescribed and observe the method’s limits. Actual runway design uses aircraft performance, operational requirements and applicable aerodrome standards.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 61, 62, 63, 64, 65

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Airport runway and taxiway designFormula

Taxiway exit turning radius from lateral friction

Recall taxiway exit turning radius from lateral friction. State the definitions and conditions before you apply it.

$$R\approx \frac{V^2}{125f}$$

V: km/h; R: m; f: assumed lateral-friction factor under the source approximation.

Understand / derivation

Lateral acceleration v²/R is limited by the allowed friction acceleration fg; converting km/h produces a factor near 127, with 125 used in the stated empirical form.

Worked example

V=50 km/h and f=0.13 give R≈153.85 m under the 125-factor convention.

Original CivilEase learning example; not an exam PYQ.

Check: Aircraft-specific steering, geometry and operational limits also govern; do not use an assumed road-friction value without justification.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 66

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Airport runway and taxiway designFormula

Taxiway geometric clearance and historical radius conventions

Recall taxiway geometric clearance and historical radius conventions. State the definitions and conditions before you apply it.

$$R=\frac{0.388W^2}{T/2-S},S=6+\frac{t_g}{2}$$

Source convention: W aircraft wheelbase m; T taxiway width m; t_g main-gear tread m; S clearance-plus-half-tread parameter m.

Understand / derivation

A turning aircraft’s main-gear path cuts inside its nose-gear path; the legacy equation relates allowable inward cut to wheelbase and available paved width.

Worked example

W=15 m, T=23 m and t_g=8 m give S=10 m and R≈58.2 m.

Original CivilEase learning example; not an exam PYQ.

Check: Requires T/2>S and the matching geometry. The sheet’s 120 m subsonic/180 m supersonic radii are historical generic examples, not aircraft-independent design minima.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 67, 68

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Railway track: speed and cantFormula

Railway curve degree and grade compensation

Recall railway curve degree and grade compensation. State the definitions and conditions before you apply it.

$$D_{deg}\approx \frac{1720}{R_m};GC=0.04D_{deg}\%(BG),0.03D_{deg}\%(MG),0.02D_{deg}\%(NG)$$

R in m; D_deg: source’s approximately 30 m chord/arc railway curve-degree convention.

Understand / derivation

The curve-degree relation approximates angle=arc/radius. Grade compensation reduces the ruling grade to allow for extra curvature resistance.

Worked example

R=430 m gives D≈4° and the source broad-gauge compensation=0.16 percentage points.

Original CivilEase learning example; not an exam PYQ.

Check: The chord/arc definition and railway/gauge standard matter; never mix with road-curve degree or percent-as-fraction.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 69, 70

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Railway track: speed and cantFormula

Versine of a circular curve

Recall versine of a circular curve. State the definitions and conditions before you apply it.

$$v=R-\sqrt{R^2-c^2/4}\approx \frac{c^2}{8R}$$

c: chord length; v: mid-ordinate/versine; R: radius.

Understand / derivation

Apply Pythagoras to the half chord. Expanding √(1-x) gives the small-chord approximation.

Worked example

c=20 m and R=500 m give approximate versine=0.1 m.

Original CivilEase learning example; not an exam PYQ.

Check: The approximation needs c/R small; use consistent length units.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 71

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Railway track: speed and cantFormula

Railway equilibrium cant and cant deficiency

Recall railway equilibrium cant and cant deficiency. State the definitions and conditions before you apply it.

$$e_{eq}=\frac{G{V}^2}{127R},e_{def}=e_{eq}-e_{provided}$$

V km/h; R m; effective gauge G and cant e in the same length unit.

Understand / derivation

Balancing lateral centrifugal acceleration with the gravity component on the inclined track gives e/G≈v²/(gR).

Worked example

G=1750 mm, V=80 km/h and R=800 m give e_eq≈110.24 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Corrected the sheet’s wording: theoretical/equilibrium cant equals provided cant plus cant deficiency, not equilibrium cant plus deficiency. Use the specified effective gauge and cant limits.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 72, 73

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Railway track: speed and cantFormula

Empirical gauge widening and flange lap

Recall empirical gauge widening and flange lap. State the definitions and conditions before you apply it.

$$d_{cm}\approx \frac{13(B+L_f{)}^2}{R},L_f=0.02\sqrt{h_f^2+D_w{h}_f}$$

B,L_f,R in m in the widening rule; h_f,D_w in cm in the source’s flange-lap formula, producing L_f in m.

Understand / derivation

A rigid wheelbase negotiates a curved track with inward geometric offset; an effective wheelbase including flange lap enters the approximate widening relation.

Worked example

If effective B+L_f=4 m and R=400 m, d≈0.52 cm.

Original CivilEase learning example; not an exam PYQ.

Check: Empirical older relation; confirm the source geometry, flange dimensions and prescribed railway limits. Mixing cm and mm in L_f creates a large error.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 74

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Railway track: speed and cantFormula

Railway transition length and cubic parabola

Recall railway transition length and cubic parabola. State the definitions and conditions before you apply it.

$$L_s=\frac{v^3}{C_{j}R}\approx 3.28\frac{v^3}{R},y\approx \frac{x^3}{6L_{s}R}$$

v in m/s; R,L_s,x,y in m; 3.28 corresponds to C_j≈0.305 m/s³; x measured from the transition tangent point.

Understand / derivation

A linearly increasing curvature gives lateral-acceleration rate C_j=v³/(RL_s). Integrating the small-angle curvature relation gives the cubic parabola.

Worked example

v=20 m/s and R=400 m give L_s≈65.6 m with the 3.28 convention.

Original CivilEase learning example; not an exam PYQ.

Check: Cant-rate/runoff and track-specific criteria may govern instead. The source’s usual cubic-parabola choice is a railway study convention; do not apply v in km/h to the 3.28 coefficient.

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Supplied formula sheet · page 8 · Transportation Engineering · entries 75, 76, 77

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Linear algebraFormula

Consistency of a linear system

Recall consistency of a linear system. State the definitions and conditions before you apply it.

$$rank(A)\ne rank([A|b])\Rightarrow \text{no solution}$$$$r=r_a=n\Rightarrow \text{unique};r=r_a<n\Rightarrow \text{infinitely many}$$

A: coefficient matrix; [A|b]: augmented matrix; n: unknowns; r and r_a: their ranks.

Understand / derivation

Row reduction preserves the solution set. A contradictory row makes a system inconsistent; otherwise each free variable adds a degree of freedom.

Worked example

For x+y=2 and 2x+2y=4, both ranks are 1<2: infinitely many solutions. For the second right side 5, the augmented rank becomes 2: no solution.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 1 · Mathematics · entries 1

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Linear algebraFormula

Homogeneous systems and nullity

Recall homogeneous systems and nullity. State the definitions and conditions before you apply it.

$$Ax=0,\dim \ker A=n-rank(A)$$

n: number of unknowns; rank: number of pivots.

Understand / derivation

The zero solution always exists. Nonzero solutions exist exactly when at least one free variable remains; their independent directions equal the nullity.

Worked example

A 3-column matrix of rank 2 has one independent null vector and infinitely many scalar multiples.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet confuses rank with nullity: the number of independent solutions is n-r, not r.

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Supplied formula sheet · page 1 · Mathematics · entries 2

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Calculus and vector calculusFormula

Taylor and geometric series

Recall taylor and geometric series. State the definitions and conditions before you apply it.

$$f(x+h)=f(x)+hf\prime (x)+\frac{h^2}{2!}f\prime \prime (x)+\cdots$$$$\frac{1}{1-x}=\sum _{n=0}^{\infty }{x}^n,\frac{1}{1+x}=\sum _{n=0}^{\infty }(-x{)}^n$$

h: increment; n: nonnegative integer.

Understand / derivation

Taylor terms come from matching successive derivatives at the expansion point. The geometric sum follows by multiplying the partial sum by 1-x and taking its limit.

Worked example

At x=0.1, 1/(1-x)=1+0.1+0.01+...=1.1111...

Original CivilEase learning example; not an exam PYQ.

Check: The geometric series requires |x|<1. A Taylor expansion needs appropriate smoothness and remainder/convergence conditions.

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Supplied formula sheet · page 1 · Mathematics · entries 3, 18, 19

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Calculus and vector calculusFormula

Classifying a stationary point

Recall classifying a stationary point. State the definitions and conditions before you apply it.

$$D=f_{xx}{f}_{yy}-f_{xy}^2;D>0,f_{xx}>0:min;D>0,f_{xx}<0:max;D<0:\text{saddle}$$

Evaluate derivatives at a stationary point where f_x=f_y=0.

Understand / derivation

The Hessian quadratic form controls the leading change near a stationary point. Positive definiteness gives a minimum and an indefinite form gives a saddle.

Worked example

For f=x²-y² at the origin, D=2(-2)-0=-4: a saddle.

Original CivilEase learning example; not an exam PYQ.

Check: D=0 is inconclusive; first find a stationary point.

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Supplied formula sheet · page 1 · Mathematics · entries 4

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Calculus and vector calculusFormula

Green's theorem

Recall green's theorem. State the definitions and conditions before you apply it.

$${\oint }_C(Pdx+Qdy)={\iint }_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$$

C: positively oriented simple closed boundary of plane region D.

Understand / derivation

The local circulation contributions inside the region cancel in pairs, leaving circulation along the outer boundary.

Worked example

With P=-y/2 and Q=x/2 around a unit circle, the integral equals its area π.

Original CivilEase learning example; not an exam PYQ.

Check: Reverse the boundary orientation and the line integral changes sign.

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Supplied formula sheet · page 1 · Mathematics · entries 5

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Calculus and vector calculusFormula

Stokes' theorem

Recall stokes' theorem. State the definitions and conditions before you apply it.

$${\oint }_C\mathbf{F}·d\mathbf{r}={\iint }_S(\nabla \times \mathbf{F})·\mathbf{n}dS$$

C: boundary of oriented surface S; n: compatible unit normal.

Understand / derivation

Surface curl accumulates local circulation. Interior edges cancel, leaving the circulation around the boundary.

Worked example

For F=(-y/2,x/2,0) over a unit disk with upward normal, both sides equal π.

Original CivilEase learning example; not an exam PYQ.

Check: Use the right-hand-rule pairing of boundary direction and surface normal.

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Supplied formula sheet · page 1 · Mathematics · entries 6

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Calculus and vector calculusFormula

Divergence theorem

Recall divergence theorem. State the definitions and conditions before you apply it.

$${∯}_S\mathbf{F}·\mathbf{n}dS={\iiint }_V\nabla ·\mathbf{F}dV$$

S: closed boundary of volume V; n: outward unit normal.

Understand / derivation

The divergence measures local outward flux per unit volume; summing local fluxes cancels internal interfaces.

Worked example

For F=(x,y,z) in the unit cube, divergence=3 and outward flux=3.

Original CivilEase learning example; not an exam PYQ.

Check: The surface must be closed; use outward normals.

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Supplied formula sheet · page 1 · Mathematics · entries 7

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Ordinary differential equationsFormula

First-order linear differential equation

Recall first-order linear differential equation. State the definitions and conditions before you apply it.

$$y\prime +P(x)y=Q(x),I=e^{\int Pdx},Iy=\int IQdx+C$$

I: integrating factor; C: integration constant.

Understand / derivation

Multiplying by I turns the left side into the product derivative (Iy)′ because I′=PI.

Worked example

For y′+2y=4, y=2+Ce⁻²ˣ. If y(0)=1, C=-1.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 1 · Mathematics · entries 8

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Ordinary differential equationsFormula

Integrating factors for differential forms

Recall integrating factors for differential forms. State the definitions and conditions before you apply it.

$$Mdx+Ndy=0;M_y=N_x\text{for exactness}$$$$I(x)=e^{\int (M_y-N_x)/Ndx},I(y)=e^{\int (N_x-M_y)/Mdy}$$$$I=\frac{1}{xM+yN}(\text{homogeneous});I=\frac{1}{xM-yN}(M=yf(xy),N=xg(xy))$$

M,N: coefficients of dx,dy; subscripts denote partial derivatives.

Understand / derivation

Choose an integrating factor that makes ∂(IM)/∂y=∂(IN)/∂x. The x-only or y-only factors follow by separating that condition.

Worked example

For M=2y, N=x, (M_y-N_x)/N=1/x, so I=x. Then d(x²y)=0 and x²y=C.

Original CivilEase learning example; not an exam PYQ.

Check: Each factor has its own structural condition. Divisors must be nonzero; verify exactness after multiplication.

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Supplied formula sheet · page 1 · Mathematics · entries 9, 10, 11, 12

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Ordinary differential equationsFormula

Potential function for an exact ODE

Recall potential function for an exact ODE. State the definitions and conditions before you apply it.

$$F_x=M,F_y=N,F(x,y)=\int Mdx+g(y)=C$$

M_y=N_x is the exactness condition.

Understand / derivation

Integrate M with y held constant, then differentiate with respect to y and choose g′(y) to match N.

Worked example

For 2x dx+2y dy=0, F=x²+y² and the solution is x²+y²=C.

Original CivilEase learning example; not an exam PYQ.

Check: Do not integrate M and N independently and add both complete integrals; that double counts shared terms.

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Supplied formula sheet · page 1 · Mathematics · entries 13

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Ordinary differential equationsFormula

Particular integrals of constant-coefficient ODEs

Recall particular integrals of constant-coefficient ODEs. State the definitions and conditions before you apply it.

$$F(D)y=e^{ax}:y_p=\frac{e^{ax}}{F(a)}(F(a)\ne 0)$$$$F(D)y=e^{ax}V(x):y_p=e^{ax}\frac{1}{F(D+a)}V(x)$$$$F(D^2)y=\sin (ax+b):y_p=\frac{\sin (ax+b)}{F(-a^2)}(F(-a^2)\ne 0)$$

D=d/dx; F: polynomial differential operator.

Understand / derivation

D(e^{ax}u)=e^{ax}(D+a)u gives the shift rule. For sine/cosine forcing, use the real/imaginary parts of a complex exponential.

Worked example

For y′-2y=e³ˣ, F(3)=1, so y_p=e³ˣ. For y′-2y=e²ˣ, resonance gives y_p=xe²ˣ.

Original CivilEase learning example; not an exam PYQ.

Check: If F(a)=0, division by F(a) is invalid. Multiply the trial by x for a simple resonance, or x to the root multiplicity.

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Supplied formula sheet · page 1 · Mathematics · entries 14, 15, 16

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Ordinary differential equationsFormula

Polynomial forcing in a constant-coefficient ODE

Recall polynomial forcing in a constant-coefficient ODE. State the definitions and conditions before you apply it.

$$F(D)y=P_m(x),y_p=F(D{)}^{-1}{P}_m(x)$$

P_m: polynomial forcing of degree m; D=d/dx; F: polynomial differential operator.

Understand / derivation

Use an undetermined polynomial trial. If zero is a root of F of multiplicity s, multiply the trial polynomial by x^s; substitute and equate coefficients.

Worked example

For y′-y=x, try y_p=ax+b. Substitution gives -a=1 and a-b=0, hence y_p=-x-1.

Original CivilEase learning example; not an exam PYQ.

Check: The inverse-operator notation in the sheet is symbolic: derivatives act on a function, not ordinary numbers. Resonance changes the trial.

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Supplied formula sheet · page 1 · Mathematics · entries 17

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Ordinary differential equationsFormula

Euler-Cauchy equations and the logarithmic substitution

Recall euler-Cauchy equations and the logarithmic substitution. State the definitions and conditions before you apply it.

$$x=e^t,x{y}_x=D_{t}y,x^2{y}_{xx}=D_t(D_t-1)y,x^3{y}_{xxx}=D_t(D_t-1)(D_t-2)y$$

D_t=d/dt; work on an interval with x>0 (or use t=ln|x|).

Understand / derivation

The chain rule changes powers of x times derivatives into constant-coefficient operators. Solve the transformed ODE, including its complementary solution.

Worked example

For x²y″+xy′-y=0, the transformed equation is y_tt-y=0, so y=C₁x+C₂/x.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s inverse-operator shorthand is interpreted through this substitution, not as ordinary algebra on derivatives.

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Supplied formula sheet · page 1 · Mathematics · entries 20

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Calculus and vector calculusFormula

Differentiating an integral with moving limits

Recall differentiating an integral with moving limits. State the definitions and conditions before you apply it.

$$\frac{d}{dx}{\int }_{a(x)}^{b(x)}h(t,x)dt={\int }_a^b{h}_x(t,x)dt+h(b(x),x)b\prime (x)-h(a(x),x)a\prime (x)$$

a,b: differentiable limits; h: integrand with suitable continuous partial derivative.

Understand / derivation

Separate the change of the integrand from the gain at the upper limit and loss at the lower limit.

Worked example

For I(x)=∫₀ˣ t²dt, I′(x)=x². For ∫ₓ²ˣ t dt, the rule gives 4x-x=3x.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 1 · Mathematics · entries 21

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Partial differential equations and Fourier seriesFormula

Fourier series on a 2π-periodic interval

Recall fourier series on a 2π-periodic interval. State the definitions and conditions before you apply it.

$$f(x)~\frac{a_0}{2}+\sum _{n=1}^{\infty }[a_n\cos nx+b_n\sin nx]$$$$a_n=\frac{1}{\pi }{\int }_{\alpha }^{\alpha +2\pi }f(x)\cos nxdx,b_n=\frac{1}{\pi }{\int }_{\alpha }^{\alpha +2\pi }f(x)\sin nxdx$$

For a₀ set cos(0x)=1; the integral spans one complete period.

Understand / derivation

Multiply the expansion by a basis sine/cosine and integrate. Orthogonality removes all other frequency terms.

Worked example

A constant f=3 has a₀=6 and all aₙ,bₙ=0 for n≥1.

Original CivilEase learning example; not an exam PYQ.

Check: At a jump, the usual convergence value is the average of the one-sided limits.

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Supplied formula sheet · page 1 · Mathematics · entries 37, 38

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Partial differential equations and Fourier seriesFormula

Fourier series for period 2L

Recall fourier series for period 2L. State the definitions and conditions before you apply it.

$$f(x)~\frac{a_0}{2}+\sum _{n=1}^{\infty }[a_n\cos \frac{n\pi x}{L}+b_n\sin \frac{n\pi x}{L}]$$$$a_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\cos \frac{n\pi x}{L}dx,b_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\sin \frac{n\pi x}{L}dx$$

L: half-period; any interval of length 2L is valid with the same coordinate convention.

Understand / derivation

Rescale the independent variable to convert the 2L period into 2π, then apply orthogonality.

Worked example

For f=2 on a period of length 4, L=2 and a₀=4.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 1 · Mathematics · entries 39, 40

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Partial differential equations and Fourier seriesFormula

Half-range sine and cosine expansions

Recall half-range sine and cosine expansions. State the definitions and conditions before you apply it.

$$b_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \frac{n\pi x}{L}dx;a_n=\frac{2}{L}{\int }_0^{L}f(x)\cos \frac{n\pi x}{L}dx$$

Sine series: odd extension; cosine series: even extension; a₀ uses the cosine formula with n=0.

Understand / derivation

Extend the function to [-L,L]. Odd functions have only sine terms; even functions have only cosine terms.

Worked example

For f(x)=1 on (0,π), the sine coefficients are 4/(nπ) for odd n and 0 for even n.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 1 · Mathematics · entries 41, 42

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Probability and statisticsFormula

Mean and variance of a random variable

Recall mean and variance of a random variable. State the definitions and conditions before you apply it.

$$\mu =\sum _{x}xp(x)\text{or}\int xf(x)dx;{\sigma }^2=E[(X-\mu {)}^2]=E[X^2]-{\mu }^2$$

p: probability mass; f: probability density; probabilities sum or integrate to 1.

Understand / derivation

Expand (X-μ)² and use E[X]=μ to obtain the second-moment form.

Worked example

For a fair die, μ=3.5 and variance=35/12≈2.9167.

Original CivilEase learning example; not an exam PYQ.

Check: For grouped data, use normalized frequencies. Population variance is distinct from the unbiased sample estimator.

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Supplied formula sheet · page 1 · Mathematics · entries 43, 44

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Probability and statisticsFormula

Poisson distribution

Recall poisson distribution. State the definitions and conditions before you apply it.

$$P(X=k)=e^{-\lambda }\frac{{\lambda }^k}{k!},E[X]=Var(X)=\lambda$$

λ: expected event count in the chosen interval; k=0,1,2,...

Understand / derivation

Independent rare events with a constant rate lead to the Poisson count model. Its mean and variance both equal λ.

Worked example

For λ=2, P(X=0)=e⁻²≈0.1353.

Original CivilEase learning example; not an exam PYQ.

Check: The symbol μ in the sheet denotes the mean count, not the standard deviation.

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Supplied formula sheet · page 1 · Mathematics · entries 45

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Probability and statisticsFormula

Normal probability density

Recall normal probability density. State the definitions and conditions before you apply it.

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp [-\frac{(x-\mu {)}^2}{2{\sigma }^2}],Z=\frac{X-\mu }{\sigma }$$

μ: mean; σ>0: standard deviation; x spans the real line.

Understand / derivation

Standardization shifts and rescales a normal variable to a standard normal variable. Probabilities are areas under the density.

Worked example

If μ=100 and σ=15, X=115 corresponds to Z=1.

Original CivilEase learning example; not an exam PYQ.

Check: A density value is not a point probability; continuous P(X=x)=0.

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Supplied formula sheet · page 1 · Mathematics · entries 46

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Probability and statisticsFormula

Least-squares straight-line regression

Recall least-squares straight-line regression. State the definitions and conditions before you apply it.

$$y=a+bx,\sum y=na+b\sum x,\sum xy=a\sum x+b\sum x^2$$

n: observations; a: intercept; b: slope.

Understand / derivation

Set the derivatives of Σ(yᵢ-a-bxᵢ)² with respect to a,b to zero to obtain the normal equations.

Worked example

The points (0,1),(1,3),(2,5) give a=1,b=2 and zero residuals.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 1 · Mathematics · entries 47

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Numerical methodsFormula

Lagrange interpolation

Recall lagrange interpolation. State the definitions and conditions before you apply it.

$$p_n(x)=\sum _{i=0}^n{y}_i\prod _{j\ne i}\frac{x-x_j}{x_i-x_j}$$

Distinct xᵢ: interpolation nodes; yᵢ: data values.

Understand / derivation

Each basis product is 1 at its own node and 0 at every other node; the weighted sum reproduces all supplied ordinates.

Worked example

For (0,1),(2,5), p₁(1)=1(2-1)/2+5(1/2)=3.

Original CivilEase learning example; not an exam PYQ.

Check: Interpolation through data does not justify distant extrapolation.

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Supplied formula sheet · page 1 · Mathematics · entries 49

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Numerical methodsFormula

Finite-difference differentiation

Recall finite-difference differentiation. State the definitions and conditions before you apply it.

$$f\prime (x_0)=\frac{1}{h}(\Delta y_0-\frac{{\Delta }^2{y}_0}{2}+\frac{{\Delta }^3{y}_0}{3}-\cdots )$$$$f\prime (x_n)=\frac{1}{h}(\nabla y_n+\frac{{\nabla }^2{y}_n}{2}+\frac{{\nabla }^3{y}_n}{3}+\cdots )$$

h: equal node spacing; Δ: forward difference; ∇: backward difference.

Understand / derivation

Differentiate the Newton forward/backward interpolation polynomial at the end node.

Worked example

For y=x² at x=0,1,2: Δy₀=1 and Δ²y₀=2. The derivative at 0 is (1-2/2)/1=0.

Original CivilEase learning example; not an exam PYQ.

Check: Use equally spaced data and the correct endpoint convention.

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Supplied formula sheet · page 1 · Mathematics · entries 50, 51

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Numerical methodsFormula

Newton-Raphson root iteration

Recall newton-Raphson root iteration. State the definitions and conditions before you apply it.

$$x_{n+1}=x_n-\frac{f(x_n)}{f\prime (x_n)}$$

xₙ: current estimate; f′ must be nonzero at the iteration.

Understand / derivation

The tangent line at xₙ crosses the x-axis at the next iterate.

Worked example

For f=x²-2 and x₀=1, x₁=1.5 and x₂≈1.4167.

Original CivilEase learning example; not an exam PYQ.

Check: Convergence depends on the starting point and function; check both residual and step size.

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Supplied formula sheet · page 1 · Mathematics · entries 52

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Numerical methodsFormula

Composite trapezoidal integration and error

Recall composite trapezoidal integration and error. State the definitions and conditions before you apply it.

$$T=h[\frac{y_0+y_n}{2}+{\sum }_{i=1}^{n-1}{y}_i],I-T=-\frac{b-a}{12}{h}^{2}f\prime \prime (\xi )$$

h=(b-a)/n; yᵢ=f(a+ih); ξ lies in the interval under the usual smoothness assumptions.

Understand / derivation

Integrate straight-line segments between adjacent samples. Their local cubic errors accumulate to a second-order global error.

Worked example

For ∫₀¹x²dx with h=0.5, T=0.375 and exact I=1/3; I-T=-1/24.

Original CivilEase learning example; not an exam PYQ.

Check: The error sign here is exact integral minus approximation.

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Supplied formula sheet · page 1 · Mathematics · entries 53, 55

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Numerical methodsFormula

Composite Simpson's one-third rule and error

Recall composite Simpson's one-third rule and error. State the definitions and conditions before you apply it.

$$S=\frac{h}{3}[y_0+y_n+4{\sum }_{i\mathrm{o}\mathrm{d}\mathrm{d}}{y}_i+2{\sum }_{i\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},0<i<n}{y}_i]$$$$I-S=-\frac{b-a}{180}{h}^4{f}^{(4)}(\xi )$$

h=(b-a)/n; n must be even.

Understand / derivation

Integrate a quadratic interpolant on every pair of subintervals; cancellation makes the rule exact for cubic polynomials.

Worked example

For ∫₀¹x²dx with n=2, S=(0.5/3)[0+1+4(0.25)]=1/3.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 1 · Mathematics · entries 54, 56

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Numerical methodsFormula

Euler's explicit ODE step

Recall euler's explicit ODE step. State the definitions and conditions before you apply it.

$$y_{k+1}=y_k+hf(t_k,y_k),y\prime =f(t,y)$$

h: time step; (tₖ,yₖ): current approximate solution.

Understand / derivation

A first-order Taylor expansion replaces the derivative by the ODE right-hand side at the start of the step.

Worked example

For y′=y, y(0)=1 and h=0.1, y₁=1.1; exact e⁰·¹≈1.10517.

Original CivilEase learning example; not an exam PYQ.

Check: Local error is O(h²); accumulated global error is O(h) under regularity assumptions.

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Supplied formula sheet · page 1 · Mathematics · entries 57

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Stress, strain and material behaviourFormula

Poisson's ratio

Recall poisson's ratio. State the definitions and conditions before you apply it.

$$\nu =-\frac{{ϵ}_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}}}{{ϵ}_{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}}$$

Strains are signed and dimensionless; ν is dimensionless.

Understand / derivation

A tensile axial strain usually accompanies a negative lateral strain. The minus sign makes the conventional Poisson ratio positive for such materials.

Worked example

Axial strain=0.001 and lateral strain=-0.0003 give ν=0.30.

Original CivilEase learning example; not an exam PYQ.

Check: The unsigned ratio in the source sheet omits the conventional minus sign.

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Supplied formula sheet · page 2 · Strength of material · entries 1

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Stress, strain and material behaviourFormula

Three-dimensional isotropic Hooke law

Recall three-dimensional isotropic Hooke law. State the definitions and conditions before you apply it.

$${ϵ}_x=\frac{{\sigma }_x-\nu ({\sigma }_y+{\sigma }_z)}{E};{ϵ}_y=\frac{{\sigma }_y-\nu ({\sigma }_x+{\sigma }_z)}{E};{ϵ}_z=\frac{{\sigma }_z-\nu ({\sigma }_x+{\sigma }_y)}{E}$$

E: Young modulus; σ: normal stresses in consistent units; ν: Poisson ratio.

Understand / derivation

Superpose the direct axial strain caused by each stress and the two lateral Poisson contractions.

Worked example

For uniaxial σₓ=200 MPa, E=200000 MPa and ν=0.3: εₓ=0.001 and εᵧ=ε_z=-0.0003.

Original CivilEase learning example; not an exam PYQ.

Check: Assumes small-strain linear isotropic elasticity.

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Supplied formula sheet · page 2 · Strength of material · entries 2, 3, 4

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Stress, strain and material behaviourFormula

Relationships between elastic constants

Recall relationships between elastic constants. State the definitions and conditions before you apply it.

$$G=\frac{E}{2(1+\nu )},K=\frac{E}{3(1-2\nu )}$$

G: shear modulus; K: bulk modulus; all moduli have stress units.

Understand / derivation

Apply isotropic Hooke law to pure shear and equal triaxial stress to relate the independent elastic constants.

Worked example

E=210 GPa, ν=0.3 give G≈80.77 GPa and K=175 GPa.

Original CivilEase learning example; not an exam PYQ.

Check: The incompressible limit ν→0.5 makes K large; these are isotropic relations.

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Supplied formula sheet · page 2 · Strength of material · entries 5, 6

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Bending and shear in beamsFormula

Simple elastic bending

Recall simple elastic bending. State the definitions and conditions before you apply it.

$$\frac{M}{I}=\frac{\sigma }{y}=\frac{E}{R_c}$$

M: bending moment; I: second moment of area; y: distance from neutral axis; R_c: curvature radius.

Understand / derivation

Plane sections remaining plane give strain ε=y/R_c; Hooke law gives stress, and integrating σy over the section gives M=EI/R_c.

Worked example

M=10 kN·m, I=8×10⁻⁶ m⁴ and y=0.1 m give σ=125 MPa.

Original CivilEase learning example; not an exam PYQ.

Check: Elastic homogeneous beam, small curvature and the appropriate neutral axis are assumed.

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Supplied formula sheet · page 2 · Strength of material · entries 7

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Bending and shear in beamsFormula

Transverse shear stress in a beam

Recall transverse shear stress in a beam. State the definitions and conditions before you apply it.

$$\tau =\frac{VQ}{Ib}$$

V: shear force; Q: first moment of area above/below the level; b: local width; I: section second moment.

Understand / derivation

Longitudinal equilibrium between adjacent beam sections relates the change in bending force to the shear flow VQ/I.

Worked example

For a rectangular section of area 0.02 m² under V=20 kN, τ_max=1.5V/A=1.5 MPa.

Original CivilEase learning example; not an exam PYQ.

Check: Use local width, not necessarily the overall flange width.

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Supplied formula sheet · page 2 · Strength of material · entries 8

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Uniform torsionFormula

Elastic torsion of a circular shaft

Recall elastic torsion of a circular shaft. State the definitions and conditions before you apply it.

$$\frac{T}{J}=\frac{\tau }{r}=\frac{G\theta }{L}$$

T: torque; J: polar second moment; θ: twist in radians; L: shaft length.

Understand / derivation

For a circular shaft, shear strain grows linearly with radius. Integrating torque from τr over the cross-section gives T=GJθ/L.

Worked example

A solid 50 mm shaft under 1 kN·m has τ_max=16T/(πd³)≈40.74 MPa.

Original CivilEase learning example; not an exam PYQ.

Check: The circular-section formula does not directly apply to arbitrary noncircular shafts.

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Supplied formula sheet · page 2 · Strength of material · entries 9

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Uniform torsionFormula

Equivalent bending moment and torque

Recall equivalent bending moment and torque. State the definitions and conditions before you apply it.

$$M_e=\frac{M+\sqrt{M^2+T^2}}{2},T_e=\sqrt{M^2+T^2}$$

M,T: bending moment and torque on a circular shaft; M_e,T_e: equivalent measures.

Understand / derivation

At the surface, combine σ=32M/(πd³) and τ=16T/(πd³). The maximum principal stress gives M_e; maximum shear gives T_e.

Worked example

M=3 and T=4 kN·m give T_e=5 and M_e=4 kN·m.

Original CivilEase learning example; not an exam PYQ.

Check: These correspond to different failure criteria; they are not interchangeable design capacities.

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Supplied formula sheet · page 2 · Strength of material · entries 10, 11

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Deflections, work and energy methodsFormula

Laminated semi-elliptic leaf spring

Recall laminated semi-elliptic leaf spring. State the definitions and conditions before you apply it.

$$\sigma =\frac{3WL}{2nb{t}^2},\delta =\frac{3W{L}^3}{8Enb{t}^3}=\frac{\sigma L^2}{4Et}$$

W: central load; L: full span; n: effective leaves; b,t: leaf width/thickness.

Understand / derivation

Idealize the leaves as a constant-strength triangular assembly in bending; substitution of the bending stress gives the second deflection form.

Worked example

If σ=400 MPa, L=1000 mm, E=200000 MPa and t=10 mm, δ=50 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Full-span and half-span conventions differ. This is an idealized spring model, not a fatigue or contact design.

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Supplied formula sheet · page 2 · Strength of material · entries 12

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Uniform torsionFormula

Close-coiled helical spring deflection

Recall close-coiled helical spring deflection. State the definitions and conditions before you apply it.

$$\delta =\frac{64W{R}^{3}n}{G{d}^4}=\frac{8W{D}_m^{3}n}{G{d}^4}$$

W: axial load; R: mean coil radius; D_m=2R; d: wire diameter; n: active turns.

Understand / derivation

The wire primarily twists under torque WR. Castigliano differentiation of torsional strain energy gives the axial extension.

Worked example

W=100 N, D_m=50 mm, n=10, G=80000 MPa and d=5 mm give δ=20 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Small helix angle and elastic close-coiled behaviour; stress concentration needs separate checks.

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Supplied formula sheet · page 2 · Strength of material · entries 13

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Stress transformation and Mohr’s circleFormula

Plane-stress transformation and principal stresses

Recall plane-stress transformation and principal stresses. State the definitions and conditions before you apply it.

$${\sigma }_n=\frac{{\sigma }_x+{\sigma }_y}{2}+\frac{{\sigma }_x-{\sigma }_y}{2}\cos 2\theta +{\tau }_{xy}\sin 2\theta$$$${\tau }_n=-\frac{{\sigma }_x-{\sigma }_y}{2}\sin 2\theta +{\tau }_{xy}\cos 2\theta$$$${\sigma }_{1,2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$$

θ: angle of plane normal from x; stresses use a consistent sign convention.

Understand / derivation

Rotate the stress tensor or use force equilibrium on an inclined wedge. At principal planes shear vanishes; the Mohr-circle radius gives the principal difference.

Worked example

For σₓ=100 MPa, σᵧ=0 and τ=0: principal stresses are 100 and 0 MPa; at 45°, σ_n=50 MPa and τ_n=-50 MPa.

Original CivilEase learning example; not an exam PYQ.

Check: The absolute 3D maximum shear also considers the out-of-plane principal stress.

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Supplied formula sheet · page 2 · Strength of material · entries 14, 15, 16, 17

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Column bucklingFormula

Euler buckling with effective length

Recall euler buckling with effective length. State the definitions and conditions before you apply it.

$$P_{cr}=\frac{{\pi }^{2}EI}{L_e^2},L_e=KL$$

Ideal K: pin-pin 1; fixed-free 2; fixed-fixed 0.5; fixed-pin approximately 0.699.

Understand / derivation

Solving EI y″+Py=0 with the end restraints gives the first nontrivial buckling mode.

Worked example

If π²EI/L²=100 kN, ideal fixed-fixed capacity is 400 kN and fixed-free capacity is 25 kN.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet uses 2π²EI/L² for fixed-pin as an approximation; π²/(0.699L)² is more precise. Slender elastic columns only.

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Supplied formula sheet · page 2 · Strength of material · entries 18, 19

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Column bucklingFormula

Rankine-Gordon column model

Recall rankine-Gordon column model. State the definitions and conditions before you apply it.

$$P_R=\frac{{\sigma }_{c}A}{1+a(L_e/k{)}^2},a=\frac{{\sigma }_c}{{\pi }^{2}E}$$

σ_c: crushing stress; k=√(I/A): radius of gyration; L_e: effective length.

Understand / derivation

The reciprocal combination 1/P_R=1/P_crushing+1/P_Euler recovers crushing for a short column and Euler behaviour for a slender one.

Worked example

If σ_cA=1000 kN and a(L_e/k)²=1, P_R=500 kN.

Original CivilEase learning example; not an exam PYQ.

Check: An empirical interpolation; modern code resistance curves require their own prescribed factors.

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Supplied formula sheet · page 2 · Strength of material · entries 20

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Statically indeterminate structural analysisFormula

Fixed-end moments: point load, UDL and triangular load

Recall fixed-end moments: point load, UDL and triangular load. State the definitions and conditions before you apply it.

$${\overline{M}}_{AB}=-\frac{Pa{b}^2}{L^2},{\overline{M}}_{BA}=\frac{P{a}^{2}b}{L^2}$$$$\mathrm{U}\mathrm{D}\mathrm{L}:(-w{L}^2/12,+w{L}^2/12)$$$$\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}(0\to w):(-w{L}^2/30,+w{L}^2/20)$$

a: load distance from A; b=L-a; w: load per length, or triangular peak at B. Clockwise member-end moment convention.

Understand / derivation

Restrain both end rotations and impose compatibility on the elastic beam. Fixed-end moments are the load terms in slope-deflection equations.

Worked example

A 12 kN/m UDL over 6 m gives member-end moments -36 and +36 kN·m.

Original CivilEase learning example; not an exam PYQ.

Check: End signs depend on the convention. The triangular peak must be at the stated end.

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Supplied formula sheet · page 2 · Structural analysis · entries 1, 2, 3

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Statically indeterminate structural analysisFormula

Fixed-end moments from an applied couple

Recall fixed-end moments from an applied couple. State the definitions and conditions before you apply it.

$${\overline{M}}_{AB}=\frac{M_{0}b(3a-L)}{L^2},{\overline{M}}_{BA}=\frac{M_{0}a(3b-L)}{L^2}$$

M₀: clockwise applied couple at distance a from A, b=L-a; same member-end sign convention as the sheet.

Understand / derivation

Obtain the couple case as the limiting difference of two opposite nearby point loads in the point-load fixed-end moment expressions.

Worked example

For a=b=L/2, both member-end moments are M₀/4 in this convention.

Original CivilEase learning example; not an exam PYQ.

Check: Always draw the external couple and check the end-moment sign convention before substitution.

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Supplied formula sheet · page 2 · Structural analysis · entries 4

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Deflections, work and energy methodsFormula

Strain energy in axial, bending, torsion and shear deformation

Recall strain energy in axial, bending, torsion and shear deformation. State the definitions and conditions before you apply it.

$$U=\int \frac{N^2}{2EA}dx+\int \frac{M^2}{2EI}dx+\int \frac{T^2}{2GJ}dx+\int \frac{V^2}{2\kappa GA}dx$$

N,M,T,V: internal actions; κ: shear correction factor (5/6 for a rectangle in the common energy model).

Understand / derivation

For a linear elastic action, energy equals half action times displacement; integrate the energy density along each member.

Worked example

A uniform axial bar with P=10 kN, L=2 m, E=200 GPa, A=100 mm² stores P²L/(2EA)=5 J.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s 1.2 rectangular shear multiplier equals 1/κ. Use the correct energy model for the section.

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Supplied formula sheet · page 2 · Structural analysis · entries 5, 6

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Arches and cablesFormula

Horizontal thrust in a two-hinged arch

Recall horizontal thrust in a two-hinged arch. State the definitions and conditions before you apply it.

$$H=\frac{\int M_{0}y/(EI)ds-{\Delta }_{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}}{\int y^2/(EI)ds}$$

M₀: simply supported beam moment; y: arch ordinate; Δ_imposed: compatible support/thermal movement term with a defined sign.

Understand / derivation

Write M=M₀-Hy and impose zero net horizontal displacement by differentiating bending strain energy with respect to H.

Worked example

With no imposed movement, numerator=20 and denominator=4 in consistent units give H=5 force units.

Original CivilEase learning example; not an exam PYQ.

Check: Include thermal, axial and support-spread effects only under an explicit sign/kinematic convention; the sheet’s compact thermal term is not universal.

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Supplied formula sheet · page 2 · Structural analysis · entries 7

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Arches and cablesFormula

Parabolic cable under uniform horizontal load

Recall parabolic cable under uniform horizontal load. State the definitions and conditions before you apply it.

$$H=\frac{w{L}^2}{8f},s\approx L+\frac{8f^2}{3L}$$

w: load per horizontal length; L: level-support span; f: sag; s: cable length.

Understand / derivation

Equilibrium gives a parabolic profile. Integrating √(1+y′²) and using the small-slope expansion gives the length approximation.

Worked example

w=10 kN/m, L=40 m, f=5 m give H=400 kN and approximate s=41.67 m.

Original CivilEase learning example; not an exam PYQ.

Check: The cable is parabolic for load uniform per horizontal length; self-weight per cable length gives a catenary.

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Supplied formula sheet · page 2 · Structural analysis · entries 8

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Statically indeterminate structural analysisFormula

Slope-deflection equations

Recall slope-deflection equations. State the definitions and conditions before you apply it.

$$M_{AB}={\overline{M}}_{AB}+\frac{2EI}{L}(2{\theta }_A+{\theta }_B-3\Delta /L)$$$$M_{BA}={\overline{M}}_{BA}+\frac{2EI}{L}(2{\theta }_B+{\theta }_A-3\Delta /L)$$

θ_A,θ_B: end rotations; Δ: relative joint displacement consistent with clockwise chord rotation Δ/L.

Understand / derivation

Decompose member deformation into its two end rotations and chord rotation, then superpose the corresponding end moments.

Worked example

For Δ=0 and θ_B=0, the additional M_AB is 4EIθ_A/L and M_BA is 2EIθ_A/L.

Original CivilEase learning example; not an exam PYQ.

Check: Prismatic elastic member; joint equilibrium and support conditions complete the solution.

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Supplied formula sheet · page 2 · Structural analysis · entries 9, 10

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Statically indeterminate structural analysisFormula

Clapeyron's three-moment theorem

Recall clapeyron's three-moment theorem. State the definitions and conditions before you apply it.

$$\frac{M_A{L}_1}{I_1}+2M_B(\frac{L_1}{I_1}+\frac{L_2}{I_2})+\frac{M_C{L}_2}{I_2}=-6(\frac{A_1{\overline{x}}_1}{I_1{L}_1}+\frac{A_2{\overline{x}}_2}{I_2{L}_2})$$

Aᵢ: area of simply supported bending-moment diagram; x̄₁ from A and x̄₂ from C; equal E and level supports.

Understand / derivation

Equate the rotation at the common support calculated from the adjacent elastic-curve areas. Equal I reduces to the sheet’s compact form.

Worked example

For two equal spans L under equal UDL w and simple outer ends, M_B=-wL²/8.

Original CivilEase learning example; not an exam PYQ.

Check: Unequal support levels or variable stiffness require the corresponding settlement/stiffness terms.

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Supplied formula sheet · page 2 · Structural analysis · entries 11

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Deflections, work and energy methodsFormula

Unit-load deflection of a pin-jointed truss

Recall unit-load deflection of a pin-jointed truss. State the definitions and conditions before you apply it.

$$\delta =\sum _i\frac{N_i{n}_i{L}_i}{A_i{E}_i}$$

Nᵢ: actual axial force; nᵢ: force under a unit load in the desired displacement direction.

Understand / derivation

Virtual work equates the unit external load times displacement to the sum of actual member extension times virtual member force.

Worked example

One bar with N=10 kN, unit-load force n=1, L=2 m and EA=20 MN gives δ=1 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Use signed member forces consistently; joint rigidity and bending are neglected.

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Supplied formula sheet · page 2 · Structural analysis · entries 12

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Statically indeterminate structural analysisFormula

Redundant force by consistent deformation

Recall redundant force by consistent deformation. State the definitions and conditions before you apply it.

$$X=-\frac{\sum N_i{n}_i{L}_i/(A_i{E}_i)+\sum n_i{\alpha }_i\Delta T_i{L}_i-{\delta }_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{e}\mathrm{d}}}{\sum n_i^2{L}_i/(A_i{E}_i)}$$

X: redundant; nᵢ: unit-redundant member forces; prescribed displacement uses the same positive direction.

Understand / derivation

Release the redundant, compute load/thermal displacement, then add the displacement caused by X until compatibility is satisfied.

Worked example

If free displacement is 2 mm and flexibility is 0.5 mm/kN, the restraining redundant is -4 kN.

Original CivilEase learning example; not an exam PYQ.

Check: The minus sign follows the chosen positive redundant direction. The source’s Q sign requires this convention.

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Supplied formula sheet · page 2 · Structural analysis · entries 13

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Stiffness matrix methodConcept

Stiffness and flexibility coefficients

Explain stiffness and flexibility coefficients. State the definitions and conditions before you apply it.

$$F_i=\sum _j{k}_{ij}{d}_j,d_i=\sum _j{f}_{ij}{F}_j$$

kᵢⱼ: force at restrained coordinate i for unit displacement j; fᵢⱼ: displacement i from unit force j.

Understand / derivation

Columns of each matrix are obtained by applying a unit input at one coordinate. After restraints remove rigid-body modes, the matrices are inverses.

Worked example

A single degree of freedom with k=20 kN/mm has f=0.05 mm/kN.

Original CivilEase learning example; not an exam PYQ.

Check: Units differ: stiffness is force/displacement; flexibility is its reciprocal.

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Supplied formula sheet · page 2 · Structural analysis · entries 14

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Stiffness matrix methodFormula

Translational stiffness of a beam end

Recall translational stiffness of a beam end. State the definitions and conditions before you apply it.

$$k_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}-\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}=\frac{12EI}{L^3},f=\frac{L^3}{12EI};k_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}-\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{d}}=\frac{3EI}{L^3},f=\frac{L^3}{3EI}$$

The translated end is restrained against rotation in the first case; the other end is fixed. In the second, the translated end is a moment-free pin.

Understand / derivation

Solve the elastic beam with unit relative transverse translation under the stated rotational restraints.

Worked example

For EI/L³=10 kN/mm, the stiffnesses are 120 and 30 kN/mm.

Original CivilEase learning example; not an exam PYQ.

Check: These stiffnesses describe relative joint translation, not the unrelated end-rotation stiffness 4EI/L or 3EI/L.

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Supplied formula sheet · page 2 · Structural analysis · entries 15, 16

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Concrete design concepts and material behaviourFormula

Target mean concrete strength

Recall target mean concrete strength. State the definitions and conditions before you apply it.

$$f_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}=f_{ck}+1.65s$$

f_ck: characteristic cube strength; s: established strength standard deviation, both MPa.

Understand / derivation

For an approximately normal strength distribution, placing the lower 5th percentile at f_ck puts the mean 1.645 standard deviations above it.

Worked example

M25 with s=4 MPa gives target mean≈31.6 MPa.

Original CivilEase learning example; not an exam PYQ.

Check: A mix-design statistical target is distinct from factored structural design strength.

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Supplied formula sheet · page 2 · RCC · entries 1

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Concrete design concepts and material behaviourFormula

Concrete modulus and flexural tensile strength

Recall concrete modulus and flexural tensile strength. State the definitions and conditions before you apply it.

$$E_c=5000\sqrt{f_{ck}},f_{cr}=0.7\sqrt{f_{ck}}$$

f_ck, E_c and f_cr in N/mm² (MPa).

Understand / derivation

These are empirical normal-weight concrete study relations, not elastic identities. The square-root dependence represents observed strength-stiffness trends.

Worked example

For f_ck=25 MPa, E_c=25000 MPa and f_cr=3.5 MPa.

Original CivilEase learning example; not an exam PYQ.

Check: Use the code/material relation required by the question; aggregates and age influence actual stiffness.

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Supplied formula sheet · page 2 · RCC · entries 2

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Concrete design concepts and material behaviourFormula

Tension-steel strain at flexural failure

Recall tension-steel strain at flexural failure. State the definitions and conditions before you apply it.

$${ϵ}_{st}\ge 0.002+\frac{0.87f_y}{E_s}$$

f_y,E_s in consistent stress units; strain dimensionless.

Understand / derivation

The limit ensures a ductile tension-controlled section: steel strain includes an elastic part plus a minimum inelastic component.

Worked example

For f_y=415 MPa and E_s=200000 MPa, the threshold is about 0.003805.

Original CivilEase learning example; not an exam PYQ.

Check: This is a code-based limit-state convention; it is not the general steel stress-strain curve.

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Supplied formula sheet · page 2 · RCC · entries 3

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Concrete design concepts and material behaviourFormula

Steel modulus and WSM modular ratio

Recall steel modulus and WSM modular ratio. State the definitions and conditions before you apply it.

$$E_s\approx 2\times {10}^5\mathrm{M}\mathrm{P}\mathrm{a},m=\frac{280}{3{\sigma }_{cbc}}$$

σ_cbc: permissible concrete flexural compression in MPa; m: working-stress modular ratio.

Understand / derivation

The WSM modular ratio transforms steel area to an equivalent concrete area and includes a prescribed long-term allowance.

Worked example

For σ_cbc=7 MPa, m=13.33.

Original CivilEase learning example; not an exam PYQ.

Check: Do not substitute f_ck for σ_cbc or confuse this WSM ratio with E_s/E_c in an instantaneous elastic model.

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Supplied formula sheet · page 2 · RCC · entries 4

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Reinforced concrete beamsFormula

Effective flange width of T- and L-beams

Recall effective flange width of T- and L-beams. State the definitions and conditions before you apply it.

$$b_f=\frac{L_0}{6}+b_w+6D_f(T);b_f=\frac{L_0}{12}+b_w+3D_f(L)$$$$b_f=\frac{L_0}{L_0/b+4}+b_w(\text{isolated T});b_f=\frac{0.5L_0}{L_0/b+4}+b_w(\text{isolated L})$$

L₀: distance between zero-moment points; b_w: web width; D_f: slab/flange thickness; b: available flange outstand parameter under the stated rule.

Understand / derivation

Shear lag prevents the entire physical slab width from acting uniformly. The empirical effective width must not exceed the actual available width.

Worked example

For L₀=6000 mm, b_w=300 mm, D_f=100 mm, the general T-beam expression gives 1900 mm before the physical-width cap.

Original CivilEase learning example; not an exam PYQ.

Check: These are source-sheet IS-style study rules. Match isolated-beam definitions and physical caps to the specified code.

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Supplied formula sheet · page 2 · RCC · entries 5, 6, 7, 8

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Reinforced concrete beamsFormula

Neutral-axis depth and limiting depth

Recall neutral-axis depth and limiting depth. State the definitions and conditions before you apply it.

$$\frac{x_u}{d}=\frac{0.87f_y{A}_{st}}{0.36f_{ck}bd};\frac{x_{u,max}}{d}=\frac{0.0035}{0.0055+0.87f_y/E_s}$$

b,d: rectangular breadth/effective depth; A_st: tension steel area.

Understand / derivation

Equate concrete compression 0.36f_ck b x_u to steel tension 0.87f_yA_st; strain compatibility sets the limiting axis.

Worked example

For Fe415 and E_s=200000 MPa, x_u,max/d≈0.480.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s alternative relation using p_t,lim is equivalent only when p_t=100A_st/(bd).

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Supplied formula sheet · page 2 · RCC · entries 9, 12

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Reinforced concrete beamsFormula

Moment resistance of a singly reinforced rectangular section

Recall moment resistance of a singly reinforced rectangular section. State the definitions and conditions before you apply it.

$$M_u=0.87f_y{A}_{st}(d-0.42x_u)=0.36f_{ck}b{x}_u(d-0.42x_u)$$$$M_u=0.87f_y{A}_{st}d(1-\frac{f_y{A}_{st}}{f_{ck}bd})$$

Use N,mm,MPa for moment in N·mm; the last expression uses the rounded 0.42×0.87/0.36≈1.015 coefficient as 1.

Understand / derivation

Moment resistance equals the compression/tension resultant multiplied by the internal lever arm. Substitution of x_u gives the steel-area form.

Worked example

For concrete compression 500 kN and lever arm 400 mm, the resisting moment is 200 kN·m.

Original CivilEase learning example; not an exam PYQ.

Check: The first two expressions are preferred. Apply only within the valid neutral-axis/ductility limit, not to over-reinforced failure.

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Supplied formula sheet · page 2 · RCC · entries 10, 11, 13

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Reinforced concrete beamsFormula

Compression-resultant depth in a flanged beam

Recall compression-resultant depth in a flanged beam. State the definitions and conditions before you apply it.

$$y_f=0.15x_u+0.65D_f\le D_f$$

y_f: effective flange compression depth in the relevant IS-style flanged stress-block case.

Understand / derivation

The equivalent flange contribution must reflect whether the compression block covers the whole flange or only part of it.

Worked example

x_u=200 mm and D_f=100 mm give y_f=95 mm before the D_f cap.

Original CivilEase learning example; not an exam PYQ.

Check: This is a case-specific expression, not the neutral-axis location. Use the correct flange-thickness/neutral-axis branch.

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Supplied formula sheet · page 2 · RCC · entries 14

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Reinforced concrete beamsFormula

Doubly reinforced beam decomposition

Recall doubly reinforced beam decomposition. State the definitions and conditions before you apply it.

$$0.87f_y{A}_{st2}=(f_{sc}-f_{cc})A_{sc}$$$$M_u=M_{u,lim}+0.87f_y{A}_{st2}(d-d\prime )$$

A_st2: extra tension steel beyond the limiting singly reinforced section; A_sc: compression steel; f_sc,f_cc: steel and displaced-concrete stresses at compression-steel level.

Understand / derivation

Split resistance into a limiting singly reinforced concrete block and an added steel couple. Subtract concrete displaced by compression steel to avoid double counting.

Worked example

An extra tension resultant of 100 kN at a 350 mm lever arm adds 35 kN·m.

Original CivilEase learning example; not an exam PYQ.

Check: Compression steel need not reach 0.87f_y; obtain its stress from compatible strain.

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Supplied formula sheet · page 2 · RCC · entries 15, 16

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Reinforced concrete beamsFormula

Minimum beam tension reinforcement

Recall minimum beam tension reinforcement. State the definitions and conditions before you apply it.

$$\frac{A_{st,min}}{bd}=\frac{0.85}{f_y}$$

b,d in mm; f_y in N/mm²; A_st in mm².

Understand / derivation

A minimum steel ratio supplies post-cracking tensile resistance and prevents a brittle loss of load capacity at first cracking.

Worked example

For b=300 mm, d=500 mm and Fe415, A_st,min≈307 mm².

Original CivilEase learning example; not an exam PYQ.

Check: This beam minimum does not replace slab or column minimum reinforcement rules.

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Supplied formula sheet · page 2 · RCC · entries 17

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Concrete slabsFormula

Minimum slab reinforcement

Recall minimum slab reinforcement. State the definitions and conditions before you apply it.

$$A_{s,min}=0.0012A_g(\text{HYSD});A_{s,min}=0.0015A_g(\text{mild steel})$$

A_g: gross concrete cross-sectional area of the considered slab strip.

Understand / derivation

Distributed reinforcement controls cracking and provides minimum tensile resistance; the percentage is based on gross area.

Worked example

For a 1000 mm strip 150 mm thick, 0.12% gives 180 mm² per metre.

Original CivilEase learning example; not an exam PYQ.

Check: The familiar source-sheet Fe415/Fe250 labels represent material categories under its code convention.

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Supplied formula sheet · page 2 · RCC · entries 18

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Reinforced concrete beamsFormula

Maximum longitudinal reinforcement: source-sheet rule

Recall maximum longitudinal reinforcement: source-sheet rule. State the definitions and conditions before you apply it.

$$A_s\le 0.04A_g$$

A_g: gross concrete area; the source applies the 4% rule to beam tension/compression reinforcement.

Understand / derivation

An upper steel limit protects concrete placement, bond and ductile behaviour. Actual design also needs spacing and neutral-axis checks.

Worked example

A 300×500 mm section has 4% of gross area equal to 6000 mm².

Original CivilEase learning example; not an exam PYQ.

Check: Do not use this beam limit as a universal limit for every member; column and lap-zone requirements differ.

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Supplied formula sheet · page 2 · RCC · entries 19

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Reinforced concrete beamsFormula

Balanced neutral-axis depth in WSM

Recall balanced neutral-axis depth in WSM. State the definitions and conditions before you apply it.

$$k=\frac{1}{1+{\sigma }_{st}/(m{\sigma }_{cbc})}$$

k=x/d; σ_st: permissible steel tension; σ_cbc: permissible concrete compression; m: WSM modular ratio.

Understand / derivation

Linear strain compatibility gives σ_st/(mσ_cbc)=(1-k)/k; rearrange for k.

Worked example

If σ_st/(mσ_cbc)=2, k=1/3.

Original CivilEase learning example; not an exam PYQ.

Check: Use working stresses; factored strengths from limit-state design do not belong in this relation.

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Supplied formula sheet · page 2 · RCC · entries 20

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Concrete slabsFormula

Span-to-effective-depth study limits

Recall span-to-effective-depth study limits. State the definitions and conditions before you apply it.

$$(L/d{)}_{basic}=7(\text{cantilever}),20(\text{simply supported}),26(\text{continuous})$$

L: effective span; d: effective depth. For the source-sheet continuous-slab shortcut, L/d≈35 for Fe250 and 40 for Fe415 under its restricted loading/span conditions.

Understand / derivation

Deflection depends strongly on span/depth and is adjusted for steel, stress and loading. Base ratios are screening values, not complete checks.

Worked example

A simply supported 4 m span with unmodified ratio 20 suggests d≈200 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Apply relevant modification factors and code conditions. The 10/L span correction applies to eligible simply supported/continuous beams over 10 m; cantilevers need explicit calculation. The slab shortcut must not be used without its conditions.

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Supplied formula sheet · page 2 · RCC · entries 21, 29

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Bond and development lengthFormula

Development length

Recall development length. State the definitions and conditions before you apply it.

$$L_d=\frac{\varphi {\sigma }_s}{4{\tau }_{bd}}$$

φ: bar diameter; σ_s: design steel stress; τ_bd: design bond stress in consistent units.

Understand / derivation

Equate axial bar force (πφ²/4)σ_s to bond resistance πφL_dτ_bd.

Worked example

φ=16 mm, σ_s=360 MPa and τ_bd=1.6 MPa give L_d=900 mm.

Original CivilEase learning example; not an exam PYQ.

Check: The source uses bond increases of 60% for eligible deformed bars and 25% for bars in compression; apply only as prescribed by the code/question.

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Supplied formula sheet · page 2 · RCC · entries 22

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Bond and development lengthFormula

Anchorage at simple supports and points of inflection

Recall anchorage at simple supports and points of inflection. State the definitions and conditions before you apply it.

$$L_d\le \frac{M_1}{V}+L_0$$

M₁: moment resistance of section with reinforcement anchored at that location; V: design shear; L₀: permitted anchorage beyond the support/inflection point.

Understand / derivation

As moment rises away from a low-moment location, the bar force must be developed quickly enough through bond. M₁/V provides an equivalent available length.

Worked example

If M₁/V=0.6 m and L₀=0.2 m, the basic permitted development length is 0.8 m.

Original CivilEase learning example; not an exam PYQ.

Check: A source-sheet 1.3 enhancement applies only under the specified confining compression condition. This is not a substitute for all bar-curtailment rules.

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Supplied formula sheet · page 2 · RCC · entries 23

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Shear and torsion in concrete membersFormula

Concrete-beam shear demand and stirrup resistance

Recall concrete-beam shear demand and stirrup resistance. State the definitions and conditions before you apply it.

$${\tau }_v=\frac{V_u}{bd},V_{u,net}=V_u+\frac{M_u}{d}\tan \beta$$$$V_{us}=\frac{0.87f_y{A}_{sv}d}{s_v}$$

β: signed taper angle where applicable; A_sv: stirrup leg area across the shear plane; s_v: longitudinal spacing.

Understand / derivation

Vertical stirrups crossing an idealized diagonal crack form tensile ties; their total force over a depth d gives the shear resistance.

Worked example

Fe415, A_sv=100 mm², d=450 mm and s_v=200 mm give V_us≈81.2 kN.

Original CivilEase learning example; not an exam PYQ.

Check: Taper contribution can add or subtract depending on geometry; choose the sign from equilibrium. Shear capacity also needs concrete and maximum-stress checks.

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Supplied formula sheet · page 2 · RCC · entries 24, 25

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Shear and torsion in concrete membersFormula

Minimum and maximum vertical-stirrup spacing

Recall minimum and maximum vertical-stirrup spacing. State the definitions and conditions before you apply it.

$$\frac{A_{sv}}{b{s}_v}\ge \frac{0.4}{0.87f_y},s_v\le min(0.75d,300\mathrm{m}\mathrm{m})$$

f_y in N/mm²; b,d,s_v in mm.

Understand / derivation

Minimum distributed ties provide crack control and robustness even when calculated shear steel is small; maximum spacing limits unbridged diagonal cracks.

Worked example

For d=300 mm, the maximum vertical stirrup spacing is 225 mm before any stricter demand limit.

Original CivilEase learning example; not an exam PYQ.

Check: Use the code cap on the steel grade permitted in shear calculations.

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Supplied formula sheet · page 2 · RCC · entries 26, 27

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Shear and torsion in concrete membersFormula

Equivalent actions for combined torsion and shear

Recall equivalent actions for combined torsion and shear. State the definitions and conditions before you apply it.

$$V_e=V_u+\frac{1.6T_u}{b},M_e=M_u+\frac{T_u}{1.7}(1+\frac{D}{b})$$

T_u: torsional moment; b,D: breadth and overall depth; V_e,M_e: equivalent design actions.

Understand / derivation

The code-equivalent method represents torsional shear and longitudinal tension through augmented shear and bending demands.

Worked example

With T_u=10 kN·m and b=0.25 m, the added equivalent shear is 64 kN.

Original CivilEase learning example; not an exam PYQ.

Check: Use overall depth D in the torsional moment expression, not the effective depth d shown ambiguously in the sheet.

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Supplied formula sheet · page 2 · RCC · entries 28

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Concrete slabsFormula

Punching-shear strength study expression

Recall punching-shear strength study expression. State the definitions and conditions before you apply it.

$${\tau }_{c,p}=k_{s}0.25\sqrt{f_{ck}},k_s=min(1,0.5+{\beta }_c)$$

β_c: short/long column dimension ratio; strengths in N/mm².

Understand / derivation

Two-way shear around a column differs from one-way beam shear. Aspect ratio modifies the nominal punching resistance.

Worked example

For a square column and M25, k_s=1 and τ_c,p=1.25 MPa.

Original CivilEase learning example; not an exam PYQ.

Check: The critical perimeter and moment-transfer effects must be checked separately.

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Supplied formula sheet · page 2 · RCC · entries 30

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Concrete columnsFormula

Minimum column eccentricity

Recall minimum column eccentricity. State the definitions and conditions before you apply it.

$$e_{x,min}=max(\frac{L}{500}+\frac{D}{30},20\mathrm{m}\mathrm{m}),e_{y,min}=max(\frac{L}{500}+\frac{b}{30},20\mathrm{m}\mathrm{m})$$

L: unsupported length; D,b: lateral dimensions in the corresponding bending planes, all mm.

Understand / derivation

Perfectly concentric loading is unrealistic; a prescribed minimum eccentricity allows for unavoidable imperfections.

Worked example

L=3000 mm and D=400 mm give 6+13.33=19.33 mm, so adopt 20 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Match the relevant dimension to the bending plane; slenderness still requires a separate assessment.

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Supplied formula sheet · page 2 · RCC · entries 31, 32

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Concrete columnsFormula

Axial strength of a short tied column

Recall axial strength of a short tied column. State the definitions and conditions before you apply it.

$$P_u=0.4f_{ck}{A}_c+0.67f_y{A}_{sc}$$

A_c: concrete area excluding steel; A_sc: longitudinal steel area.

Understand / derivation

The resistance is the sum of prescribed concrete and steel contributions under the code’s small-eccentricity conditions.

Worked example

For M25, Fe415, A_c=156000 mm² and A_sc=4000 mm², P_u≈2672.2 kN.

Original CivilEase learning example; not an exam PYQ.

Check: Only for eligible short columns with the specified eccentricity limit; general columns require interaction analysis.

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Supplied formula sheet · page 2 · RCC · entries 33

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Concrete columnsFormula

Helical reinforcement requirement

Recall helical reinforcement requirement. State the definitions and conditions before you apply it.

$$\frac{V_h}{V_c}\ge 0.36(\frac{A_g}{A_{core}}-1)\frac{f_{ck}}{f_y}$$

V_h/V_c: volume ratio of helix steel to core concrete; A_core: confined core area.

Understand / derivation

Closely spaced helix steel confines the concrete core and compensates for lost cover contribution under the prescribed model.

Worked example

If A_g/A_core=1.25 and f_ck/f_y=25/415, the minimum volume ratio is about 0.00542.

Original CivilEase learning example; not an exam PYQ.

Check: Use the code definition of core diameter. The sheet’s shorthand for helix mean diameter must account for cover and bar diameter consistently.

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Supplied formula sheet · page 2 · RCC · entries 34

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Concrete columnsFormula

Column slenderness reduction: legacy study relation

Recall column slenderness reduction: legacy study relation. State the definitions and conditions before you apply it.

$$C_r=1.25-\frac{L_e}{48b}$$

L_e/b: effective-length to lateral-dimension ratio under the source’s older working-stress reduction rule.

Understand / derivation

The empirical reduction represents decreasing allowable axial resistance as slenderness grows.

Worked example

For L_e/b=20, C_r≈0.833.

Original CivilEase learning example; not an exam PYQ.

Check: Legacy WSM relation retained for interpretation of older questions; use current specified slender-column/second-order provisions for design.

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Supplied formula sheet · page 2 · RCC · entries 35

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Concrete footingsFormula

Local concrete bearing at a footing interface

Recall local concrete bearing at a footing interface. State the definitions and conditions before you apply it.

$${\sigma }_{br}=0.45f_{ck}\sqrt{\frac{A_1}{A_2}}\le 0.9f_{ck}$$

A₂: loaded area; A₁: supporting area geometrically similar and concentric under the code definition.

Understand / derivation

Confinement from the surrounding supporting concrete can increase local bearing resistance, but the increase is capped.

Worked example

For A₁/A₂=4 and f_ck=25 MPa, σ_br=22.5 MPa.

Original CivilEase learning example; not an exam PYQ.

Check: The footprint geometry must satisfy the defined supporting-area construction; it is not simply the whole footing area.

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Supplied formula sheet · page 2 · RCC · entries 36

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Soil classification, phase relationships and index propertiesFormula

Flow index and toughness index

Recall flow index and toughness index. State the definitions and conditions before you apply it.

$$I_f=\frac{w_1-w_2}{{\log }_{10}(N_2/N_1)},I_t=\frac{I_p}{I_f}$$

w: water content in percent; N: liquid-limit cup blows; I_p: plasticity index.

Understand / derivation

On the semilog flow curve, I_f is the water-content drop per tenfold increase in blows. Toughness normalizes the plastic range by this slope.

Worked example

w₁=40%, w₂=30%, N₁=10 and N₂=100 give I_f=10; if I_p=20%, I_t=2.

Original CivilEase learning example; not an exam PYQ.

Check: Use matched water contents/blow counts and percentage-point differences.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 1, 2

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Soil classification, phase relationships and index propertiesFormula

Gradation coefficients

Recall gradation coefficients. State the definitions and conditions before you apply it.

$$C_u=\frac{D_{60}}{D_{10}},C_c=\frac{D_{30}^2}{D_{10}{D}_{60}}$$

D_p: particle diameter at p percent finer, using the same length unit.

Understand / derivation

C_u measures the size spread while C_c checks the middle of the grading curve against the end sizes.

Worked example

D₁₀=0.1 mm, D₃₀=0.3 mm and D₆₀=0.9 mm give C_u=9 and C_c=1.

Original CivilEase learning example; not an exam PYQ.

Check: Well-graded classification also depends on soil type and fines; these two numbers alone do not classify every soil.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 3, 4

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Soil classification, phase relationships and index propertiesFormula

Plasticity, liquidity, consistency and activity

Recall plasticity, liquidity, consistency and activity. State the definitions and conditions before you apply it.

$$I_p=w_L-w_P,I_s=w_P-w_S,I_L=\frac{w-w_P}{I_p},I_c=\frac{w_L-w}{I_p}=1-I_L$$$$I_p=0.73(w_L-20)(A\text{-line}),A=\frac{I_p}{\%\text{ clay-sized fraction}}$$

w_L,w_P,w_S: liquid, plastic and shrinkage limits; w: natural water content, all percent.

Understand / derivation

The indices compare the current water content to the plastic range; the A-line is an empirical classification boundary, not a material identity.

Worked example

w_L=50%, w_P=20%, w=35% give I_p=30%, I_L=0.5 and I_c=0.5. With 20% clay fraction, activity=1.5.

Original CivilEase learning example; not an exam PYQ.

Check: Use the clay fraction as a percentage when I_p is in percentage points; do not substitute a fraction without conversion.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 5, 6, 7, 8, 9, 10

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Soil classification, phase relationships and index propertiesFormula

Relative density of granular soil

Recall relative density of granular soil. State the definitions and conditions before you apply it.

$$D_r=\frac{e_{max}-e}{e_{max}-e_{min}}=\frac{1/{\gamma }_{d,min}-1/{\gamma }_d}{1/{\gamma }_{d,min}-1/{\gamma }_{d,max}}$$

e: void ratio; γ_d: dry unit weight; limiting values are loose/dense states.

Understand / derivation

Using γ_d=G_sγ_w/(1+e) eliminates the common grain-density factor and produces the reciprocal-unit-weight form.

Worked example

e_max=0.9, e_min=0.5 and e=0.6 give D_r=0.75=75%.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 11

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Permeability, seepage and flow netsFormula

Refraction of seepage across an interface

Recall refraction of seepage across an interface. State the definitions and conditions before you apply it.

$$\frac{k_1}{k_2}=\frac{\tan {\alpha }_1}{\tan {\alpha }_2}$$

α: flow-line angle measured from the interface normal; k: hydraulic conductivity.

Understand / derivation

Continuity of normal discharge and continuity of tangential head gradient combine with Darcy law to give the tangent ratio.

Worked example

If k₁/k₂=2 and α₂=20°, α₁=atan(2tan20°)≈36.1°.

Original CivilEase learning example; not an exam PYQ.

Check: Angles measured from the interface itself give a different reciprocal convention.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 12

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Permeability, seepage and flow netsFormula

Pore geometry and intrinsic permeability

Recall pore geometry and intrinsic permeability. State the definitions and conditions before you apply it.

$$k=C\frac{{\gamma }_w}{\mu }\frac{e^3}{1+e}{d}^2,K=\frac{k\mu }{{\gamma }_w}$$

k: hydraulic conductivity (m/s); K: intrinsic permeability (m²); μ: dynamic viscosity; d: representative grain size; C: empirical geometry factor.

Understand / derivation

Darcy conductivity separates into a pore-geometry property K and the fluid ratio γ_w/μ. The grain-size expression is an idealized correlation.

Worked example

For k=10⁻⁴ m/s, μ=0.001 Pa·s and γ_w=9810 N/m³, K≈1.02×10⁻¹¹ m².

Original CivilEase learning example; not an exam PYQ.

Check: C is not universal; grain distribution, fabric and tortuosity matter.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 13, 14

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Permeability, seepage and flow netsFormula

Steady radial well flow

Recall steady radial well flow. State the definitions and conditions before you apply it.

$$Q=\frac{\pi k(h_2^2-h_1^2)}{\ln (r_2/r_1)}(\text{unconfined});Q=\frac{2\pi kb(h_2-h_1)}{\ln (r_2/r_1)}(\text{confined})$$

Q: pumping discharge; r: radial distance; h: head above the impermeable base for unconfined flow; b: confined saturated thickness.

Understand / derivation

Apply Darcy law to a cylindrical flow surface and integrate dr/r between the observation wells.

Worked example

For confined b=10 m, k=10⁻⁴ m/s, head difference=2 m and r₂/r₁=10, Q≈0.00546 m³/s.

Original CivilEase learning example; not an exam PYQ.

Check: Steady, homogeneous isotropic aquifer and the Dupuit approximation for the unconfined expression.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 15, 16

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Permeability, seepage and flow netsFormula

Equivalent permeability of layered soils

Recall equivalent permeability of layered soils. State the definitions and conditions before you apply it.

$$k_h=\frac{\sum k_i{H}_i}{\sum H_i},k_v=\frac{\sum H_i}{\sum H_i/k_i},k_e=\sqrt{k_h{k}_v}$$

Hᵢ: layer thickness; h,v refer to flow parallel/perpendicular to horizontal layers.

Understand / derivation

Parallel layers share the same gradient and their flows add; layers in series share discharge and their head losses add.

Worked example

Two equal layers with k=1 and 9 units give k_h=5, k_v=1.8 and transformed-flow k_e=3.

Original CivilEase learning example; not an exam PYQ.

Check: The geometric mean is used in a transformed anisotropic flow-net analysis, not as a replacement for both directional conductivities.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 17, 18, 19

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Permeability, seepage and flow netsFormula

Laboratory permeability tests

Recall laboratory permeability tests. State the definitions and conditions before you apply it.

$$k=\frac{aL}{At}\ln \frac{h_1}{h_2}(\text{falling head});k=\frac{VL}{Ath}(\text{constant head})$$

a: standpipe area; A,L: specimen area/length; t: elapsed time; V: collected volume; h: head difference.

Understand / derivation

For constant head use Darcy discharge V/t=kAh/L. Falling-head integration uses a dh/dt=-kAh/L.

Worked example

V=100 cm³, L=10 cm, A=50 cm², t=100 s and h=20 cm give k=0.01 cm/s.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s q in the constant-head numerator denotes collected volume, not an already time-normalized discharge.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 20, 21

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Permeability, seepage and flow netsFormula

Flow-net seepage discharge

Recall flow-net seepage discharge. State the definitions and conditions before you apply it.

$$q=k_{e}H\frac{N_f}{N_d}$$

q: discharge per unit width for a two-dimensional net; H: total head loss; N_f: flow channels; N_d: potential drops.

Understand / derivation

Each approximately curvilinear-square cell transmits the same discharge under one equal head drop. Sum over flow channels.

Worked example

k=10⁻⁵ m/s, H=10 m and N_f/N_d=3/10 give q=3×10⁻⁵ m²/s.

Original CivilEase learning example; not an exam PYQ.

Check: For anisotropy draw/use the transformed net and its equivalent conductivity.

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Supplied formula sheet · page 3 · Geotechnical Engineering · entries 22

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Stress distribution in soilsFormula

Boussinesq vertical stress from a surface point load

Recall boussinesq vertical stress from a surface point load. State the definitions and conditions before you apply it.

$$\Delta {\sigma }_z=\frac{3Q}{2\pi z^2}{[1+(r/z{)}^2]}^{-5/2}$$

Q: vertical point load; z: depth; r: horizontal radial distance.

Understand / derivation

The elastic half-space point-load solution spreads load radially while satisfying vertical force equilibrium.

Worked example

Directly below Q=100 kN at z=2 m, Δσ_z≈11.94 kPa.

Original CivilEase learning example; not an exam PYQ.

Check: Homogeneous isotropic linear-elastic half-space; not a direct layered-soil settlement prediction.

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Stress distribution in soilsFormula

Westergaard point-load stress

Recall westergaard point-load stress. State the definitions and conditions before you apply it.

$$\Delta {\sigma }_z=\frac{Q}{2\pi z^2}\frac{c}{[c^2+(r/z{)}^2{]}^{3/2}},c=\sqrt{\frac{1-2\nu }{2-2\nu }}$$

ν: Poisson ratio in the constrained-layer idealization; Q,r,z as for a point load.

Understand / derivation

The model introduces closely spaced nonyielding horizontal reinforcement/layers that suppress lateral deformation; it differs from an isotropic half-space.

Worked example

For ν=0, c²=1/2 and directly below the load, Δσ_z=Q/(πz²).

Original CivilEase learning example; not an exam PYQ.

Check: Use the supplied model and ν; do not mix this coefficient with the Boussinesq influence factor.

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Stress distribution in soilsFormula

Vertical stress beneath an infinite line load

Recall vertical stress beneath an infinite line load. State the definitions and conditions before you apply it.

$$\Delta {\sigma }_z=\frac{2q_l}{\pi z}{[1+(x/z{)}^2]}^{-2}$$

q_l: vertical load per unit length; x: perpendicular horizontal offset; z: depth.

Understand / derivation

Integrate the point-load half-space solution along an infinitely long line.

Worked example

For q_l=100 kN/m and z=2 m directly below, Δσ_z≈31.83 kPa.

Original CivilEase learning example; not an exam PYQ.

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Stress distribution in soilsFormula

Stress beneath a uniformly loaded strip

Recall stress beneath a uniformly loaded strip. State the definitions and conditions before you apply it.

$$\Delta {\sigma }_z=\frac{q}{\pi }[({\beta }_1-{\beta }_2)+\sin ({\beta }_1-{\beta }_2)\cos ({\beta }_1+{\beta }_2)]$$$$\text{centre: }\Delta {\sigma }_z=\frac{q}{\pi }(2\theta +\sin 2\theta ),\theta ={\tan }^{-1}(B/2z)$$

q: surface pressure; β₁,β₂: signed edge angles from the vertical, in radians; B: total strip width.

Understand / derivation

Integrate the line-load stress across the loaded width; the edge angles express the geometry compactly.

Worked example

For B=2z, θ=π/4 and centre stress=q(π/2+1)/π≈0.8183q.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet uses half-width b in θ=atan(b/z); avoid a factor-of-two width error.

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Stress distribution in soilsFormula

Stress under the centre of a circular loaded area

Recall stress under the centre of a circular loaded area. State the definitions and conditions before you apply it.

$$\Delta {\sigma }_z=q[1-\frac{1}{(1+(R/z{)}^2{)}^{3/2}}]=q(1-{\cos }^3\theta )$$

R: loaded radius; z: depth; θ=atan(R/z).

Understand / derivation

Integrate the Boussinesq point-load solution over concentric rings.

Worked example

At z=R, Δσ_z=q[1-1/(2√2)]≈0.6464q.

Original CivilEase learning example; not an exam PYQ.

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Shear strength and stress pathsFormula

Mohr-Coulomb shear failure

Recall mohr-Coulomb shear failure. State the definitions and conditions before you apply it.

$${\tau }_f=c+{\sigma }_n\tan \varphi ,\sin \varphi =\frac{({\sigma }_1-{\sigma }_3)/2}{c\cot \varphi +({\sigma }_1+{\sigma }_3)/2}$$$${\sigma }_1={\sigma }_3{\tan }^2({45}^{\circ }+\varphi /2)+2c\tan ({45}^{\circ }+\varphi /2)$$

c: cohesion intercept; φ: friction angle; σ₁,σ₃: major/minor principal stresses.

Understand / derivation

At failure the Mohr circle is tangent to the shear-strength line. Circle radius divided by the distance to the strength-line intercept gives sinφ.

Worked example

For cohesionless soil with σ₁=300 and σ₃=100 kPa, sinφ=0.5 and φ=30°.

Original CivilEase learning example; not an exam PYQ.

Check: Use effective stresses with c′,φ′ for drained/effective-strength analysis; the cohesionless tanφ=τ/σ relation does not apply to c≠0.

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Subsurface investigation and field testsFormula

Vane shear strength

Recall vane shear strength. State the definitions and conditions before you apply it.

$$T=\pi c_u{D}^2(\frac{H}{2}+\frac{D}{6})(\text{both ends});T=\pi c_u{D}^2(\frac{H}{2}+\frac{D}{12})(\text{one end})$$

T: measured torque; D,H: vane diameter/height; c_u: undrained shear strength.

Understand / derivation

Integrate torque over the cylindrical side and one or two circular end surfaces assuming uniform mobilized undrained strength.

Worked example

For H=2D, the both-end factor is 7D/6 inside the bracket; measured T then gives c_u=6T/(7πD³).

Original CivilEase learning example; not an exam PYQ.

Check: Correct rod friction and choose the actual participating end surfaces.

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Foundation settlementFormula

Immediate elastic settlement

Recall immediate elastic settlement. State the definitions and conditions before you apply it.

$$S_i=\frac{qB(1-{\nu }^2)}{E}{I}_s$$

q: net foundation pressure; B: characteristic width; E,ν: soil elastic parameters; I_s: compatible influence factor.

Understand / derivation

Integrate the elastic displacement field over the loaded foundation area and represent shape, depth and rigidity through the factor.

Worked example

q=100 kPa, B=2 m, E=20000 kPa, ν=0.3 and I_s=1 give S_i=9.1 mm.

Original CivilEase learning example; not an exam PYQ.

Check: I_s must match the chosen width, footing rigidity and settlement point.

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Foundation settlementFormula

Footing settlement estimated from a plate-load test

Recall footing settlement estimated from a plate-load test. State the definitions and conditions before you apply it.

$$S_f=S_p{\left[\frac{B_f(B_p+0.3)}{B_p(B_f+0.3)}\right]}^2$$

B_f,B_p in metres; S_f,S_p: footing and plate settlement at corresponding pressure.

Understand / derivation

The empirical sand correction accounts for the influence of foundation size on the measured plate response.

Worked example

For B_p=0.3 m, B_f=1 m and S_p=5 mm, S_f≈11.83 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Sand-specific empirical relation; do not apply the same size law blindly to clay or extrapolate beyond the test conditions.

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Shear strength and stress pathsFormula

Skempton pore-pressure parameters

Recall skempton pore-pressure parameters. State the definitions and conditions before you apply it.

$$\Delta u=B[\Delta {\sigma }_3+A(\Delta {\sigma }_1-\Delta {\sigma }_3)]$$

B: response to isotropic total stress; A: response to deviator stress increment; u: pore pressure.

Understand / derivation

Separate a triaxial stress increment into an isotropic change and a deviatoric change, then superpose the pore-pressure responses.

Worked example

B=1, A=0.5, Δσ₃=20 and Δσ₁=100 kPa give Δu=60 kPa.

Original CivilEase learning example; not an exam PYQ.

Check: A depends on stress path and strain; B approaches 1 for fully saturated conditions but is not always exactly 1.

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Shear strength and stress pathsFormula

Failure line in p-q stress coordinates

Recall failure line in p-q stress coordinates. State the definitions and conditions before you apply it.

$$p=\frac{{\sigma }_1+{\sigma }_3}{2},q=\frac{{\sigma }_1-{\sigma }_3}{2},q=p\sin \varphi +c\cos \varphi$$$$m=\sin \varphi ,\varphi ={\sin }^{-1}m,c=k/\cos \varphi$$

q=mp+k; use total or effective stresses consistently.

Understand / derivation

The Mohr-circle radius is q and the centre is p. Tangency to the strength line gives q=c cosφ+p sinφ.

Worked example

For m=0.5 and k=10 kPa, φ=30° and c≈11.55 kPa.

Original CivilEase learning example; not an exam PYQ.

Check: Corrected source error: φ is arcsin(m), not arctan(m), for these p-q definitions. Other invariant coordinates have different slopes.

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Consolidation and time rate of settlementFormula

Compression index and empirical correlations

Recall compression index and empirical correlations. State the definitions and conditions before you apply it.

$$C_c=\frac{e_1-e_2}{{\log }_{10}({\sigma }_2\prime /{\sigma }_1\prime )},C_c\approx 0.009(w_L-10),C_{c,rem}\approx 0.007(w_L-10)$$

C_c: virgin compression slope in e-logσ′ space; w_L in percent.

Understand / derivation

A positive slope magnitude measures void-ratio loss per decade of effective-stress increase. The liquid-limit correlations are empirical estimates.

Worked example

If e drops 0.2 while effective stress rises from 100 to 1000 kPa, C_c=0.2.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet mislabels the 0.007 relation as a universal overconsolidated-soil rule; it is a remoulded-clay correlation, not the recompression index C_r.

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Consolidation and time rate of settlementFormula

Volume compressibility and consolidation coefficient

Recall volume compressibility and consolidation coefficient. State the definitions and conditions before you apply it.

$$\frac{\Delta H}{H_0}=\frac{\Delta e}{1+e_0},m_v=\frac{\Delta e}{(1+e_0)\Delta \sigma \prime },c_v=\frac{k}{m_v{\gamma }_w}$$

Compression losses ΔH,Δe taken positive; m_v has inverse stress units; c_v has length²/time units.

Understand / derivation

With solid volume unchanged, a void-ratio decrease shortens the soil layer. Darcy drainage plus compressibility gives the consolidation diffusion coefficient.

Worked example

k=10⁻⁹ m/s, m_v=10⁻⁴ m²/kN and γ_w=9.81 kN/m³ give c_v≈1.02×10⁻⁶ m²/s.

Original CivilEase learning example; not an exam PYQ.

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Consolidation and time rate of settlementFormula

Time factor and average degree of consolidation

Recall time factor and average degree of consolidation. State the definitions and conditions before you apply it.

$$T_v=\frac{c_{v}t}{H_{dr}^2};T_v\approx \frac{\pi }{4}{U}^2(U\le 0.6);T_v\approx -0.933{\log }_{10}(1-U)-0.085(U>0.6)$$

U: average consolidation as a fraction; H_dr: longest drainage path.

Understand / derivation

The one-dimensional diffusion solution is approximated by separate early-time and late-time expressions; solve t=T_vH_dr²/c_v.

Worked example

At U=0.5, T_v≈π/16=0.1963. A double-drained 4 m layer has H_dr=2 m.

Original CivilEase learning example; not an exam PYQ.

Check: Do not put 50 in place of 0.50 or use the full layer thickness for double drainage.

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Consolidation and time rate of settlementFormula

Primary consolidation settlement

Recall primary consolidation settlement. State the definitions and conditions before you apply it.

$$S=\frac{C_c{H}_0}{1+e_0}{\log }_{10}\frac{{\sigma }_f\prime }{{\sigma }_0\prime }(\mathrm{N}\mathrm{C})$$$$S=\frac{H_0}{1+e_0}[C_r{\log }_{10}\frac{{\sigma }_p\prime }{{\sigma }_0\prime }+C_c{\log }_{10}\frac{{\sigma }_f\prime }{{\sigma }_p\prime }]({\sigma }_f\prime >{\sigma }_p\prime )$$

σ′₀: initial effective stress; σ′_p: preconsolidation stress; σ′_f: final stress; C_r: recompression slope.

Understand / derivation

Convert e-logσ′ compression to layer strain. If loading crosses preconsolidation pressure, split the path into recompression and virgin compression.

Worked example

For NC H=4 m, e₀=1, C_c=0.2 and σ′ doubling, settlement≈0.1204 m.

Original CivilEase learning example; not an exam PYQ.

Check: If final stress stays below σ′_p, use C_r for the entire increment; do not apply the split formula with an invalid stress order.

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Subsurface investigation and field testsFormula

Sampler area ratio

Recall sampler area ratio. State the definitions and conditions before you apply it.

$$A_r=\frac{D_o^2-D_i^2}{D_i^2}\times 100\%$$

D_o,D_i: outer and inner diameters of the sampling tube cutting edge.

Understand / derivation

The ratio compares displaced tube-wall area to collected sample area, providing one indication of sampling disturbance.

Worked example

D_o=80 mm and D_i=75 mm give A_r≈13.78%.

Original CivilEase learning example; not an exam PYQ.

Check: An area ratio is not sufficient by itself to establish an undisturbed sample.

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Slope stabilityFormula

Taylor stability number

Recall taylor stability number. State the definitions and conditions before you apply it.

$$S_n=\frac{c}{F\gamma H},F=\frac{c}{S_n\gamma H}$$

c: cohesion; γ: unit weight; H: slope height; F: factor of safety using the compatible chart definition.

Understand / derivation

A dimensionless stability number packages slope geometry and failure-mechanism information for chart-based analysis.

Worked example

c=20 kPa, γ=20 kN/m³, H=5 m and S_n=0.1 give F=2.

Original CivilEase learning example; not an exam PYQ.

Check: Select the correct slope angle, friction angle, depth and pore-pressure chart.

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Shallow foundations and bearing capacityFormula

Terzaghi bearing-capacity expressions

Recall terzaghi bearing-capacity expressions. State the definitions and conditions before you apply it.

$$q_{ult}=c{N}_c+q_0{N}_q+\frac{1}{2}\gamma B{N}_{\gamma }(\text{strip})$$$$\text{square: }1.3c{N}_c+q_0{N}_q+0.4\gamma B{N}_{\gamma }$$$$\text{circular: }1.3c{N}_c+q_0{N}_q+0.3\gamma B{N}_{\gamma }$$$$\text{rectangular: }(1+0.3B/L)c{N}_c+q_0{N}_q+\frac{1}{2}\gamma B{N}_{\gamma }(1-0.2B/L)$$

q₀: effective overburden at base; B: footing width/diameter; L≥B; γ: appropriate effective unit weight.

Understand / derivation

Superpose cohesion, surcharge and self-weight terms under the assumed general-shear mechanism; empirical shape factors modify finite footings.

Worked example

For a strip with φ=0, c=50 kPa, N_c=5.7 and q₀=20 kPa, q_ult=305 kPa when N_q=1 and N_γ=0.

Original CivilEase learning example; not an exam PYQ.

Check: General-shear, level-base/load assumptions; water table, eccentricity, local shear and safety factors need separate treatment.

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Shallow foundations and bearing capacityFormula

Meyerhof factors and effective footing area

Recall meyerhof factors and effective footing area. State the definitions and conditions before you apply it.

$$q_{ult}=c{N}_c{s}_c{d}_c{i}_c+q_0{N}_q{s}_q{d}_q{i}_q+\frac{1}{2}\gamma B\prime N_{\gamma }{s}_{\gamma }{d}_{\gamma }{i}_{\gamma }$$$$B\prime =B-2e_B,L\prime =L-2e_L$$

s,d,i: shape, depth and inclination factors from one consistent method; e: eccentricities.

Understand / derivation

Represent load eccentricity through an effective loaded area and apply a compatible set of modifiers to each bearing-capacity term.

Worked example

B=2 m and e_B=0.2 m give B′=1.6 m.

Original CivilEase learning example; not an exam PYQ.

Check: Do not mix factors from different authors or use a nonpositive effective dimension.

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Shallow foundations and bearing capacityFormula

Skempton undrained bearing capacity

Recall skempton undrained bearing capacity. State the definitions and conditions before you apply it.

$$q_{net,ult}=c_u{N}_c,N_c\approx 5(1+0.2D_f/B)(1+0.2B/L)$$

Undrained φ_u=0 clay; B≤L; source-sheet approximation limits D_f/B to 2.5.

Understand / derivation

With φ_u=0 the net ultimate capacity is governed by undrained cohesion and a shape/depth factor.

Worked example

At D_f/B=1 and B/L=1, N_c≈7.2, so c_u=40 kPa gives q_net,ult≈288 kPa.

Original CivilEase learning example; not an exam PYQ.

Check: Use the specified Skempton charts/factors where required; this compact approximation is not a universal exact factor.

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Deep foundations and pile groupsFormula

Engineering News dynamic pile formula

Recall engineering News dynamic pile formula. State the definitions and conditions before you apply it.

$$Q=\frac{{\eta }_{h}Wh}{s+C}$$

W: hammer weight as force; h: drop; η_h: hammer efficiency; s: permanent set/blow; C: empirical allowance.

Understand / derivation

Equate an effective impact energy η_hWh to penetration work Q(s+C).

Worked example

η_hWh=10 kJ and s+C=0.01 m give Q=1000 kN in the stated convention.

Original CivilEase learning example; not an exam PYQ.

Check: Legacy empirical relation: allowable versus ultimate resistance and safety factor depend on the formulation. The sheet quotes C=2.54 cm for drop and 0.254 cm for steam hammers; do not combine with metres.

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Deep foundations and pile groupsFormula

Hiley dynamic resistance

Recall hiley dynamic resistance. State the definitions and conditions before you apply it.

$$Q=\frac{{\eta }_h{\eta }_{b}Wh}{s+\frac{1}{2}(C_1+C_2+C_3)},{\eta }_b=\frac{W+e^{2}P}{W+P}(W\ge eP)$$$${\eta }_b=\frac{W+e^{2}P}{W+P}-{\left(\frac{W-eP}{W+P}\right)}^2(W<eP)$$

P: pile/helmet weight; e: restitution; C₁,C₂,C₃: compatible elastic compression allowances.

Understand / derivation

Impact energy is reduced by hammer and impact efficiencies; resistance does work over permanent set plus half the elastic compression.

Worked example

Effective impact energy 8 kJ, set=4 mm and total compression=8 mm give Q=1000 kN.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s mixed-unit compression constants are replaced by explicitly defined lengths. The second branch accounts for rebound when W<eP; use the specified complete model. A driving estimate is not a verified static design resistance.

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Deep foundations and pile groupsFormula

Danish dynamic pile formula

Recall danish dynamic pile formula. State the definitions and conditions before you apply it.

$$Q=\frac{{\eta }_{h}Wh}{s+S_0/2},S_0=\sqrt{\frac{2{\eta }_{h}WhL}{AE}}$$

L,A,E: pile length, area and modulus; s,S₀ in consistent length units.

Understand / derivation

The formula estimates the pile’s elastic compression directly from the input energy and axial stiffness.

Worked example

η_hWh=10 kJ and L/(AE)=10⁻⁹ m/N give S₀≈4.47 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Legacy empirical model; use compatible energy and force units and the required safety interpretation.

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Deep foundations and pile groupsFormula

Static axial pile capacity in clay and sand

Recall static axial pile capacity in clay and sand. State the definitions and conditions before you apply it.

$$Q_u=Q_b+Q_s;Q_u=A_b{c}_u{N}_c+\alpha c_u{A}_s(\text{clay})$$$$Q_u=A_b{\sigma }_b\prime N_q+\int K{\sigma }_v\prime \tan \delta d{A}_s(\text{sand})$$$$Q_s=\lambda ({\overline{\sigma }}_v\prime +2c_u)A_s(\lambda \text{-method})$$

A_b,A_s: base/shaft areas; α,λ,K,δ: method-specific empirical parameters.

Understand / derivation

Add base end bearing and shaft resistance, integrating along depth where effective stress changes.

Worked example

A_b=0.1 m², c_u=50 kPa and N_c=9 give base resistance=45 kN before adding shaft resistance.

Original CivilEase learning example; not an exam PYQ.

Check: A fixed universal critical depth of 15D is not assumed: use the model/question’s limiting effective-stress rule and pile installation conditions.

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Deep foundations and pile groupsFormula

Pile-group capacity: individual and block modes

Recall pile-group capacity: individual and block modes. State the definitions and conditions before you apply it.

$$Q_{group}\le min(\eta N{Q}_{single},Q_{block}),Q_{block}=A_{bg}{c}_u{N}_c+A_{sg}{c}_u$$

N: piles; η: group efficiency if used; A_bg,A_sg: group-block base and perimeter shaft areas in clay.

Understand / derivation

A group can fail as separate piles or as a common soil block; the weaker mechanism controls.

Worked example

Nine 200 kN piles with η=0.8 give an individual-pile mode of 1440 kN; a 1200 kN block estimate controls.

Original CivilEase learning example; not an exam PYQ.

Check: Settlement, negative skin friction and interaction need separate checks.

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Earth pressure and sheet pilesFormula

Rankine earth pressure and pore water

Recall rankine earth pressure and pore water. State the definitions and conditions before you apply it.

$$K_a=\frac{1-\sin \varphi \prime }{1+\sin \varphi \prime },K_p=1/K_a$$$${\sigma }_{ha}=K_a{\sigma }_v\prime -2c\prime \sqrt{K_a}+u;{\sigma }_{hp}=K_p{\sigma }_v\prime +2c\prime \sqrt{K_p}+u$$

σ_v′: effective vertical stress; u: pore pressure; K values for smooth vertical wall and level backfill.

Understand / derivation

Tangency of the active/passive Mohr circle to the shear envelope produces the stress ratios; add water pressure separately.

Worked example

For φ′=30°, K_a=1/3 and K_p=3. With c′=0, σ_v′=90 kPa and u=20 kPa, active total pressure=50 kPa.

Original CivilEase learning example; not an exam PYQ.

Check: Do not multiply pore pressure by K_a or allow tensile soil-water pressure without considering cracking.

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Earth pressure and sheet pilesFormula

Tension-crack depth and unsupported cohesive cut

Recall tension-crack depth and unsupported cohesive cut. State the definitions and conditions before you apply it.

$$z_t=\frac{2c}{\gamma \sqrt{K_a}},H_{unsupported}\approx \frac{4c}{\gamma \sqrt{K_a}}=2z_t$$

Dry homogeneous c-φ soil; z_t: depth where the ideal active pressure changes from tension to compression.

Understand / derivation

Set K_aγz-2c√K_a=0 for the crack depth. Requiring zero net resultant for the full unsupported cut gives twice that depth in the idealized model.

Worked example

For φ=0, c=20 kPa and γ=20 kN/m³, z_t=2 m and ideal limiting cut height=4 m.

Original CivilEase learning example; not an exam PYQ.

Check: Not a safe excavation depth: water-filled cracks, surcharge, fissures and safety requirements change the problem.

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Supplied formula sheet · page 4 · Geotechnical Engineering · entries 68

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Earth pressure and sheet pilesFormula

Coulomb active and passive coefficients

Recall coulomb active and passive coefficients. State the definitions and conditions before you apply it.

$$K_a=\frac{{\sin }^2(\beta +\varphi )}{{\sin }^2\beta \sin (\beta -\delta ){[1+\sqrt{\frac{\sin (\varphi +\delta )\sin (\varphi -i)}{\sin (\beta -\delta )\sin (\beta +i)}}]}^2}$$$$K_p=\frac{{\sin }^2(\beta -\varphi )}{{\sin }^2\beta \sin (\beta +\delta ){[1-\sqrt{\frac{\sin (\varphi +\delta )\sin (\varphi +i)}{\sin (\beta +\delta )\sin (\beta +i)}}]}^2}$$

β: wall-back angle from horizontal; i: backfill slope; δ: wall friction; φ: soil friction; use the exact compatible geometry/sign convention.

Understand / derivation

Force equilibrium of a trial sliding wedge gives a thrust depending on slip-plane angle; extremizing the thrust yields the coefficient.

Worked example

For β=90°, δ=i=0 and φ=30°, K_a=1/3 and K_p=3, recovering Rankine.

Original CivilEase learning example; not an exam PYQ.

Check: Coulomb passive estimates can be unreliable for large wall friction. Check admissible geometry and denominator before use.

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Supplied formula sheet · page 4 · Geotechnical Engineering · entries 69, 70

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Earth pressure and sheet pilesFormula

Rankine coefficients for inclined backfill

Recall rankine coefficients for inclined backfill. State the definitions and conditions before you apply it.

$$K_a=\cos i\frac{\cos i-\sqrt{{\cos }^{2}i-{\cos }^2\varphi }}{\cos i+\sqrt{{\cos }^{2}i-{\cos }^2\varphi }},K_p=\cos i\frac{\cos i+\sqrt{{\cos }^{2}i-{\cos }^2\varphi }}{\cos i-\sqrt{{\cos }^{2}i-{\cos }^2\varphi }}$$$$P_{a,p}=\frac{1}{2}\gamma H^2{K}_{a,p}$$

i: surface slope; H: wall height; cohesionless soil; thrust is parallel to the ground surface in the Rankine idealization.

Understand / derivation

The inclined stress field changes both the stress ratio and the thrust direction. This definition absorbs cos(i) into K.

Worked example

At i=0 and φ=30°, the coefficients become 1/3 and 3.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet leaves cos(i) outside K and multiplies the thrust by it. Either definition works, but never apply cos(i) twice.

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Supplied formula sheet · page 4 · Geotechnical Engineering · entries 71, 72

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Subsurface investigation and field testsFormula

SPT dilatancy correction

Recall sPT dilatancy correction. State the definitions and conditions before you apply it.

$$N_d=15+\frac{1}{2}(N_c-15)(N_c>15)$$

N_c: corrected SPT count before dilatancy adjustment; N_d: dilatancy-adjusted count.

Understand / derivation

Rapid shearing in saturated fine sand/silt can create negative pore pressures and an artificially high penetration resistance.

Worked example

N_c=25 gives N_d=20; if N_c≤15, no such reduction is made.

Original CivilEase learning example; not an exam PYQ.

Check: Apply only to relevant saturated fine sand/silt conditions after the required overburden/energy corrections.

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Supplied formula sheet · page 4 · Geotechnical Engineering · entries 73

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Effective stress and quicksandFormula

Critical hydraulic gradient

Recall critical hydraulic gradient. State the definitions and conditions before you apply it.

$$i_{cr}=\frac{G_s-1}{1+e}=\frac{\gamma \prime }{{\gamma }_w}$$

G_s: specific gravity of solids; e: void ratio; γ′: submerged unit weight.

Understand / derivation

Upward seepage force iγ_w per unit volume cancels submerged soil weight at the critical condition, reducing effective stress to zero.

Worked example

G_s=2.65 and e=0.65 give i_cr=1.

Original CivilEase learning example; not an exam PYQ.

Check: Boiling/piping safety depends on actual exit gradients and soil stability; average gradient alone can be misleading.

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Supplied formula sheet · page 4 · Geotechnical Engineering · entries 74

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Fluid properties and fluid staticsFormula

Newton's law of viscosity

Recall newton's law of viscosity. State the definitions and conditions before you apply it.

$$\tau =\mu \frac{du}{dy}$$

μ: dynamic viscosity (Pa·s); du/dy: velocity gradient (s⁻¹).

Understand / derivation

For a Newtonian fluid, shear stress is proportional to the local rate of shear strain.

Worked example

μ=0.001 Pa·s and du/dy=1000 s⁻¹ give τ=1 Pa.

Original CivilEase learning example; not an exam PYQ.

Check: Not every fluid is Newtonian; distinguish dynamic viscosity μ from kinematic viscosity ν=μ/ρ.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 1

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Continuity, momentum and energy equationsFormula

Continuity equation

Recall continuity equation. State the definitions and conditions before you apply it.

$$\frac{\partial \rho }{\partial t}+\nabla ·(\rho \mathbf{v})=0,\frac{D\rho }{Dt}+\rho \nabla ·\mathbf{v}=0$$

ρ: density; v: velocity; D/Dt: material derivative.

Understand / derivation

Mass conservation for a differential control volume gives the conservative form. Product expansion gives the material form.

Worked example

For incompressible flow, ρ is constant and ∂u/∂x+∂v/∂y+∂w/∂z=0.

Original CivilEase learning example; not an exam PYQ.

Check: Corrected the sheet’s malformed dot-product term: the material form needs velocity divergence.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 2, 3

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Continuity, momentum and energy equationsFormula

Linear momentum balance

Recall linear momentum balance. State the definitions and conditions before you apply it.

$$\sum \mathbf{F}=\frac{d}{dt}{\int }_{CV}\rho \mathbf{v}dV+\sum _{out}\dot{m}\mathbf{v}-\sum _{in}\dot{m}\mathbf{v}$$

F: net external force on the fluid; velocities are vectors in an inertial frame.

Understand / derivation

Newton’s second law includes momentum stored in the volume and momentum carried through its boundary.

Worked example

For steady water flow Q=0.1 m³/s with x-velocity rising from 2 to 5 m/s, net fluid force is ρQΔv=300 N.

Original CivilEase learning example; not an exam PYQ.

Check: Pressure, weight and support reactions all belong in the external-force balance.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 4

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Continuity, momentum and energy equationsFormula

Jet propulsion: velocity-frame bookkeeping

Recall jet propulsion: velocity-frame bookkeeping. State the definitions and conditions before you apply it.

$$F=\dot{m}(V_{out}-V_{in})\text{along a chosen axis}$$

V_in,V_out: inertial velocities; ṁ: actual mass rate crossing the moving-device boundary.

Understand / derivation

Use relative velocity to calculate flux through a moving opening but inertial velocity to calculate momentum change. This avoids mixing frames in jet-propulsion formulas.

Worked example

If ṁ=10 kg/s and the device produces a 20 m/s momentum change, the thrust magnitude is 200 N.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s compact (v+u) expressions are frame-dependent; define inlet, outlet and body velocities before using a special-case thrust formula.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 5

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Continuity, momentum and energy equationsFormula

Bernoulli's energy equation

Recall bernoulli's energy equation. State the definitions and conditions before you apply it.

$$\frac{p}{\rho g}+\frac{V^2}{2g}+z=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$

Each term is head (m); p: pressure; V: speed; z: elevation.

Understand / derivation

For steady inviscid incompressible flow along a streamline, pressure work, kinetic energy and potential energy exchange without loss.

Worked example

At equal elevation with negligible upstream velocity, a 5 m pressure-head drop gives V=√(2g×5)≈9.90 m/s.

Original CivilEase learning example; not an exam PYQ.

Check: Include pumps, turbines, loss terms and kinetic-energy coefficients when the ideal assumptions do not apply.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 6

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Fluid properties and fluid staticsFormula

Hydrostatic pressure gradients

Recall hydrostatic pressure gradients. State the definitions and conditions before you apply it.

$$\frac{\partial p}{\partial x}=0,\frac{\partial p}{\partial y}=0,\frac{\partial p}{\partial z}=-\rho g$$

z: upward vertical coordinate in a static fluid.

Understand / derivation

Static force equilibrium leaves pressure balancing gravity; integrate vertically to p₂-p₁=ρg(z₁-z₂).

Worked example

At 2 m below a water surface, gauge pressure≈19.62 kPa.

Original CivilEase learning example; not an exam PYQ.

Check: For variable density integrate ρ(z)g; sign changes if the vertical coordinate points downward.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 7

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Laminar and turbulent flowFormula

Laminar flow in a circular pipe

Recall laminar flow in a circular pipe. State the definitions and conditions before you apply it.

$$\tau (r)=-\frac{r}{2}\frac{dp}{dx}=\frac{\rho g{h}_f}{2L}r$$$$u(r)=-\frac{1}{4\mu }\frac{dp}{dx}(R^2-r^2),V=\frac{u_{max}}{2}$$$$\Delta p=\frac{32\mu VL}{D^2}=\frac{128\mu QL}{\pi D^4},f_D=\frac{64}{Re}$$

R=D/2; f_D: Darcy friction factor; Re=ρVD/μ.

Understand / derivation

Balance pressure force and shear on a cylindrical fluid core, use τ=μdu/dr and integrate with no slip.

Worked example

For Re=1000, Darcy f_D=0.064. The corresponding Fanning factor is 0.016.

Original CivilEase learning example; not an exam PYQ.

Check: Steady fully developed laminar Newtonian flow. Do not use f_D=64/Re for turbulent pipe flow.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 8, 9, 10, 11

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Laminar and turbulent flowFormula

One-seventh-power turbulent velocity profile

Recall one-seventh-power turbulent velocity profile. State the definitions and conditions before you apply it.

$$\frac{u}{u_{max}}={(1-\frac{r}{R})}^{1/7}$$

r: distance from pipe centre; R: pipe radius.

Understand / derivation

This empirical profile is flatter than the laminar parabola, representing enhanced momentum mixing by turbulence.

Worked example

At r/R=0.5, u/u_max=0.5^(1/7)≈0.9057.

Original CivilEase learning example; not an exam PYQ.

Check: Approximation for suitable smooth-pipe turbulent conditions; it does not resolve the near-wall viscous region.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 12

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Flow in pipes and pipe networksFormula

Darcy-Weisbach loss and equivalent pipes

Recall darcy-Weisbach loss and equivalent pipes. State the definitions and conditions before you apply it.

$$h_f=f_D\frac{L}{D}\frac{V^2}{2g},\frac{f_e{L}_e}{D_e^5}=\sum _i\frac{f_i{L}_i}{D_i^5}$$

The equivalent-pipe relation assumes the same steady discharge through pipes in series.

Understand / derivation

Use V=4Q/(πD²) in the Darcy equation: for fixed Q, loss is proportional to fL/D⁵. Series head losses add.

Worked example

With equal f, doubling D at fixed Q reduces the loss to 1/32 for the same length.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s L/D⁵ equality assumes equal friction factors; fittings and local losses need separate terms.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 13, 14

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Flow in pipes and pipe networksFormula

Siphon summit pressure and total head

Recall siphon summit pressure and total head. State the definitions and conditions before you apply it.

$$\frac{p_a-p_c}{\gamma }=z_c-z_A+\frac{V^2}{2g}+h_{L,A\to c}$$

A: upstream free surface; c: siphon summit; p_c: summit absolute pressure; γ=ρg.

Understand / derivation

Apply the energy equation from the reservoir to the summit. Elevation, acceleration and entrance/pipe losses reduce summit pressure.

Worked example

If the summit is 3 m above the reservoir and velocity plus upstream losses total 1 m, its pressure is 4 m water head below atmospheric.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s 0.5V²/(2g) and 1.5V²/(2g) terms assume particular entrance/exit-loss coefficients. Check vapor pressure and losses explicitly.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 15

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Boundary layer, lift and dragFormula

Boundary-layer integral thicknesses

Recall boundary-layer integral thicknesses. State the definitions and conditions before you apply it.

$${\delta }^{*}={\int }_0^{\infty }(1-\frac{u}{U})dy;\theta ={\int }_0^{\infty }\frac{u}{U}(1-\frac{u}{U})dy$$$${\delta }_E={\int }_0^{\infty }\frac{u}{U}[1-{\left(\frac{u}{U}\right)}^2]dy$$

U: external velocity; δ*: displacement thickness; θ: momentum thickness; δ_E: energy thickness.

Understand / derivation

Compare actual mass, momentum and energy transport with an ideal uniform stream to express each deficit as an equivalent thickness.

Worked example

For a linear profile u/U=y/δ on 0≤y≤δ, δ*=δ/2, θ=δ/6 and δ_E=δ/4.

Original CivilEase learning example; not an exam PYQ.

Check: Thickness definitions differ; do not substitute one for another in integral balances.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 16, 17, 18

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Boundary layer, lift and dragFormula

Blasius laminar flat-plate relations

Recall blasius laminar flat-plate relations. State the definitions and conditions before you apply it.

$$\frac{\delta }{x}\approx \frac{5}{\sqrt{R{e}_x}},\frac{{\delta }^{*}}{x}=\frac{1.7208}{\sqrt{R{e}_x}},\frac{\theta }{x}=\frac{0.664}{\sqrt{R{e}_x}}$$$$C_{f,x}=\frac{0.664}{\sqrt{R{e}_x}},{\overline{C}}_f=\frac{1.328}{\sqrt{R{e}_L}},D={\overline{C}}_f\frac{\rho U^2}{2}A$$

Re_x=Ux/ν; A: wetted plate area for the face(s) considered.

Understand / derivation

The similarity solution for a zero-pressure-gradient laminar boundary layer produces the local relations; integrate local wall shear for mean drag.

Worked example

At Re_L=10⁵, mean skin-friction coefficient≈0.00420.

Original CivilEase learning example; not an exam PYQ.

Check: Smooth flat plate, incompressible laminar attached flow; transition and pressure gradients change the coefficients.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 19, 20, 21, 22, 23

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Boundary layer, lift and dragFormula

Viscous-sublayer estimate

Recall viscous-sublayer estimate. State the definitions and conditions before you apply it.

$${\delta }_v\approx 11.6\frac{\nu }{u_{*}},u_{*}=\sqrt{{\tau }_w/\rho }$$

u*: friction velocity; ν: kinematic viscosity.

Understand / derivation

The wall coordinate y⁺=yu*/ν gives an approximate edge of the linear near-wall velocity region.

Worked example

ν=10⁻⁶ m²/s and u*=0.05 m/s give δ_v≈0.232 mm.

Original CivilEase learning example; not an exam PYQ.

Check: The numerical boundary is conventional; use the stated friction-factor definition if replacing u* by mean velocity.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 24

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Fluid properties and fluid staticsFormula

Hydrostatic force and centre of pressure

Recall hydrostatic force and centre of pressure. State the definitions and conditions before you apply it.

$$F=\rho gA\overline{h},h_{cp}=\overline{h}+\frac{I_G{\sin }^2\theta }{A\overline{h}}$$

h̄: vertical centroid depth; I_G: centroidal area moment about an in-plane axis parallel to the free surface; θ: plane inclination from horizontal.

Understand / derivation

Pressure grows linearly with vertical depth. Integrate force and its moment over the submerged plane.

Worked example

A vertical 2×1 m plate spanning depths 1–3 m has F≈39.24 kN and h_cp≈2.167 m.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s alternate centre-pressure coordinate is along the plane; do not mix it with vertical depth.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 25

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Fluid properties and fluid staticsFormula

Metacentric height

Recall metacentric height. State the definitions and conditions before you apply it.

$$GM=BM-BG,BM=\frac{I_{wp}}{\nabla }$$

I_wp: waterplane second moment about heel axis; ∇: displaced volume; BG: signed vertical separation of buoyancy centre and mass centre.

Understand / derivation

A small heel shifts the buoyancy line; geometric buoyancy analysis gives BM=I_wp/∇. Positive GM gives initial restoring stability.

Worked example

BM=2 m and BG=0.8 m give GM=1.2 m.

Original CivilEase learning example; not an exam PYQ.

Check: This is small-angle initial stability, not the full finite-angle righting-arm curve.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 26

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Flow measurement and weirsFormula

Triangular sharp-crested weir

Recall triangular sharp-crested weir. State the definitions and conditions before you apply it.

$$Q=\frac{8}{15}{C}_d\sqrt{2g}\tan (\theta /2)H^{5/2}$$

θ: notch angle; H: head above apex; C_d: calibrated discharge coefficient.

Understand / derivation

Integrate strip discharge C_d b(y)√(2gy) over the triangular opening.

Worked example

For a 90° notch, C_d=0.6 and H=0.2 m, Q≈0.02536 m³/s.

Original CivilEase learning example; not an exam PYQ.

Check: Free, ventilated nappe and applicable approach-velocity/head-measurement conditions are required.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 27

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Flow measurement and weirsFormula

Horizontal jet trajectory and outlet speed

Recall horizontal jet trajectory and outlet speed. State the definitions and conditions before you apply it.

$$V=\sqrt{\frac{g{x}^2}{2y}}$$

x: horizontal distance; y: vertical downward drop from the outlet.

Understand / derivation

For a horizontal jet x=Vt and y=gt²/2. Eliminate t to recover the outlet speed.

Worked example

x=2 m and y=0.5 m give V≈6.26 m/s.

Original CivilEase learning example; not an exam PYQ.

Check: Neglect air drag and take the outlet direction as horizontal.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 28

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Flow measurement and weirsFormula

Venturimeter discharge and manometer head

Recall venturimeter discharge and manometer head. State the definitions and conditions before you apply it.

$$Q=C_d\frac{A_1{A}_2}{\sqrt{A_1^2-A_2^2}}\sqrt{2gh},h=x(\frac{{\rho }_m}{\rho }-1)$$

A₁,A₂: inlet/throat areas; h: piezometric head difference in flowing-fluid units; x: heavy-fluid differential-manometer reading.

Understand / derivation

Combine continuity with Bernoulli, then multiply the ideal discharge by the coefficient C_d.

Worked example

For water with mercury manometer x=0.1 m and relative density 13.6, equivalent head h=1.26 m.

Original CivilEase learning example; not an exam PYQ.

Check: The simple manometer conversion assumes the corresponding geometry and a heavier immiscible manometer fluid.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 29

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Potential flow and fluid kinematicsFormula

Velocity potential and stream function: one convention

Recall velocity potential and stream function: one convention. State the definitions and conditions before you apply it.

$$u=-{\varphi }_x=-{\psi }_y,v=-{\varphi }_y={\psi }_x,Q_{1\to 2}={\psi }_1-{\psi }_2$$

u,v: Cartesian velocity components; discharge is per unit width between streamlines.

Understand / derivation

The stream-function derivatives satisfy incompressible continuity identically; a velocity potential makes the flow irrotational where smooth.

Worked example

With ψ=-2y, u=2 and v=0. Between y=0 and 3 m, discharge per width=6 m²/s.

Original CivilEase learning example; not an exam PYQ.

Check: Some texts use opposite signs for φ or ψ. Keep the derivative and discharge conventions together.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 30, 31, 32

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Dimensional analysis and hydraulic similitudeFormula

Reynolds and Froude model similarity

Recall reynolds and Froude model similarity. State the definitions and conditions before you apply it.

$$Re=\frac{\rho VL}{\mu },Fr=\frac{V}{\sqrt{gL}};\frac{{\rho }_r{V}_r{L}_r}{{\mu }_r}=1,\frac{V_r}{\sqrt{g_r{L}_r}}=1$$

Subscript r: model/prototype ratio; L: characteristic length.

Understand / derivation

Match the ratio of inertia to the dominant viscous or gravity force to preserve the corresponding flow behaviour.

Worked example

Under Froude similarity at equal gravity, a 1:25 length model has velocity ratio 1:5.

Original CivilEase learning example; not an exam PYQ.

Check: Matching one dimensionless group does not ensure all forces are simultaneously similar.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 33, 34

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Potential flow and fluid kinematicsFormula

Material acceleration of a fluid particle

Recall material acceleration of a fluid particle. State the definitions and conditions before you apply it.

$$\mathbf{a}=\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}·\nabla )\mathbf{v}$$

Local term: change at a fixed point; convective term: change along the particle path.

Understand / derivation

Apply the multivariable chain rule to v(x(t),y(t),z(t),t). Each component contains its local derivative plus spatial derivatives multiplied by the three velocity components.

Worked example

For steady 1D u=2x with x in metres, aₓ=u du/dx=4x m/s².

Original CivilEase learning example; not an exam PYQ.

Check: A steady flow can have nonzero acceleration due to spatial variation.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 35, 36, 37

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Potential flow and fluid kinematicsFormula

Vorticity, angular velocity and circulation

Recall vorticity, angular velocity and circulation. State the definitions and conditions before you apply it.

$$\mathit{\zeta }=\nabla \times \mathbf{v},\mathit{\omega }=\frac{1}{2}\mathit{\zeta },\Gamma =\oint \mathbf{v}·d\mathbf{r}=\iint \mathit{\zeta }·\mathbf{n}dA$$

ζ: vorticity; ω: local rigid-body angular velocity; Γ: circulation.

Understand / derivation

The antisymmetric part of the velocity gradient describes local rotation. Stokes theorem connects circulation to vorticity flux.

Worked example

For rigid rotation v=(-Ωy,Ωx), ζ_z=2Ω and Γ=2ΩA.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s circulation=vorticity×area applies only to uniform normal vorticity.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 38, 39, 40

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Continuity, momentum and energy equationsFormula

Jet impact on a stationary plate

Recall jet impact on a stationary plate. State the definitions and conditions before you apply it.

$$F_n=\rho Q{V}_n,F=\rho a{V}^2(\text{normal plate})$$$$Q_1=\frac{Q}{2}(1+\cos \theta ),Q_2=\frac{Q}{2}(1-\cos \theta )$$

θ: angle between incident jet and plate; Q₁,Q₂: ideal opposite streams along a smooth plate.

Understand / derivation

Normal outlet velocity vanishes at a smooth plate. Continuity and tangential momentum give the split stream discharges.

Worked example

For θ=90°, the normal plate splits the ideal jet equally: Q₁=Q₂=Q/2.

Original CivilEase learning example; not an exam PYQ.

Check: These are stationary, smooth, ideal-jet relations; moving plates require relative mass flux.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 41, 42

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Dimensional analysis and hydraulic similitudeFormula

Pump and turbine specific speed

Recall pump and turbine specific speed. State the definitions and conditions before you apply it.

$$N_{s,p}=\frac{N\sqrt{Q}}{H^{3/4}},N_{s,t}=\frac{N\sqrt{P}}{H^{5/4}}$$

N: rotational speed; Q: discharge; H: head; P: turbine power under the prescribed unit convention.

Understand / derivation

Combine similarity/affinity laws to describe geometrically similar machines operating at comparable conditions.

Worked example

For N=1000 rpm, Q=0.1 m³/s and H=10 m, pump N_s≈56.23 in that conventional unit system.

Original CivilEase learning example; not an exam PYQ.

Check: Conventional specific speed is unit dependent. State rpm, flow and power units; it is not automatically dimensionless.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 43

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Channel hydraulicsFormula

Critical open-channel flow

Recall critical open-channel flow. State the definitions and conditions before you apply it.

$$\frac{Q^{2}T}{g{A}^3}=1,Q_c=\sqrt{\frac{g{A}^3}{T}};y_c^3=\frac{2y_1^2{y}_2^2}{y_1+y_2},E=\frac{y_1^2+y_1{y}_2+y_2^2}{y_1+y_2}$$

A,T: flow area and top width; y₁,y₂: alternate depths at equal specific energy in a rectangular channel.

Understand / derivation

Set dE/dy=0 at fixed discharge for the critical condition. Equating specific energies at alternate depths gives the rectangular identities.

Worked example

For a rectangular channel at y_c=1 m, critical specific energy is 1.5 m.

Original CivilEase learning example; not an exam PYQ.

Check: Alternate depths share specific energy; conjugate/sequent depths of a hydraulic jump share momentum instead.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 44, 45, 46

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Channel hydraulicsFormula

Gradually varied-flow equation

Recall gradually varied-flow equation. State the definitions and conditions before you apply it.

$$\frac{dy}{dx}=\frac{S_0-S_f}{1-F{r}^2},F{r}^2=\frac{Q^{2}T}{g{A}^3}$$

S₀: bed slope; S_f: friction/energy slope; x: distance downstream.

Understand / derivation

Differentiate the energy equation along a prismatic channel and separate the geometry-dependent kinetic-energy term.

Worked example

S₀=0.001, S_f=0.0005 and Fr=0.5 give dy/dx≈0.000667.

Original CivilEase learning example; not an exam PYQ.

Check: The gradually varied model is singular near critical flow and does not describe the rapidly varied jump roller.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 47

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Channel hydraulicsFormula

Hydraulic jump in a rectangular channel

Recall hydraulic jump in a rectangular channel. State the definitions and conditions before you apply it.

$$M=\frac{Q^2}{gA}+A\overline{y},M_1=M_2$$$$\frac{y_2}{y_1}=\frac{\sqrt{1+8F{r}_1^2}-1}{2},\Delta E=\frac{(y_2-y_1{)}^3}{4y_1{y}_2}$$

M: momentum function/specific force; ȳ: centroid depth below free surface; y₂: sequent depth.

Understand / derivation

Neglect external friction over the short jump, equate hydrostatic momentum functions and use continuity. Energy is dissipated, not conserved.

Worked example

For Fr₁=2 and y₁=1 m, y₂≈2.3723 m and ΔE≈0.2724 m.

Original CivilEase learning example; not an exam PYQ.

Check: Corrected the source’s radical: it is 1+8Fr₁². Use these depth and energy expressions only for the stated rectangular idealization.

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Supplied formula sheet · page 5 · Fluid mechanics · entries 48, 49, 50

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Hydrologic cycle, precipitation and infiltrationConcept

Cyclones and anticyclones

Explain cyclones and anticyclones. State the definitions and conditions before you apply it.

Northern-hemisphere circulation: cyclonic flow is typically counterclockwise around low pressure; anticyclonic flow is clockwise around high pressure.

Understand / derivation

The pressure-gradient force and Coriolis deflection determine the large-scale rotational sense, with surface friction modifying the ideal circulation.

Worked example

A low-pressure tropical cyclone north of the equator rotates counterclockwise; the sense reverses in the Southern Hemisphere.

Original CivilEase learning example; not an exam PYQ.

Check: The source describes Northern Hemisphere systems. This is not a universal clockwise/anticlockwise rule at all latitudes.

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Supplied formula sheet · page 4 · Hydrology · entries 1, 2

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Hydrologic cycle, precipitation and infiltrationFormula

Rain-gauge network adequacy

Recall rain-gauge network adequacy. State the definitions and conditions before you apply it.

$$N={\left(\frac{C_v}{ϵ}\right)}^2,C_v=100\frac{s_P}{\overline{P}},s_P=\sqrt{\frac{\sum (P_i-\overline{P}{)}^2}{m-1}}$$

C_v and allowable percentage error ε use the same percent scale; m: existing gauges.

Understand / derivation

Approximate standard error of a spatial mean scales as s/√N; solving the relative-error target gives N.

Worked example

C_v=20% and ε=5% give N=16 gauges in the idealized estimate.

Original CivilEase learning example; not an exam PYQ.

Check: Round up and assess spatial independence/representativeness; closely correlated gauges weaken this approximation.

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Supplied formula sheet · page 4 · Hydrology · entries 3

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Hydrologic cycle, precipitation and infiltrationFormula

Normal-ratio estimate of missing rainfall

Recall normal-ratio estimate of missing rainfall. State the definitions and conditions before you apply it.

$$P_x=\frac{N_x}{m}\sum _{i=1}^m\frac{P_i}{N_i}$$

N: normal annual rainfall; Pᵢ: concurrent observed rainfall at neighbouring gauges.

Understand / derivation

Normalize each station’s observed rain by its own long-term normal, average those ratios, and rescale to the missing station.

Worked example

N_x=1000 mm, two neighbour ratios 0.8 and 1.0 give P_x=900 mm.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 4 · Hydrology · entries 4

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Hydrologic cycle, precipitation and infiltrationFormula

Areal rainfall: Thiessen and isohyetal methods

Recall areal rainfall: Thiessen and isohyetal methods. State the definitions and conditions before you apply it.

$$\overline{P}=\frac{\sum A_i{P}_i}{A}(\text{Thiessen});\overline{P}=\frac{\sum a_i(P_i+P_{i+1})/2}{A}(\text{isohyetal})$$

Aᵢ: station influence polygon; aᵢ: area between successive isohyets; A: total watershed area.

Understand / derivation

Integrate rainfall over area by dividing the basin into zones assigned a representative rainfall, then divide by total area.

Worked example

Equal areas receiving 20 and 40 mm give mean rainfall=30 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Isohyetal estimates assume the average of adjacent contour values represents the intervening strip.

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Supplied formula sheet · page 4 · Hydrology · entries 5, 6

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Flood estimation and routingFormula

Rainfall exceedance, return period and dependable rainfall

Recall rainfall exceedance, return period and dependable rainfall. State the definitions and conditions before you apply it.

$$P=\frac{1}{T},P_{\ge 1}=1-(1-P{)}^n,P(r)=(\genfrac{}{}{0ex}{}{n}{r})P^r(1-P{)}^{n-r};P_{rank}=\frac{m}{N+1}$$

T: return period in years; n: independent years; m: descending rainfall rank among N observations.

Understand / derivation

Use Bernoulli trials for annual exceedances. The Weibull plotting position estimates the exceedance probability of a ranked observation.

Worked example

A 75%-dependable rainfall has exceedance probability 0.75 and nominal return period 1/0.75≈1.33 years.

Original CivilEase learning example; not an exam PYQ.

Check: Return period is an average interval, not a schedule; independence and stationarity are assumptions.

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Supplied formula sheet · page 4 · Hydrology · entries 7, 8

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Hydrologic cycle, precipitation and infiltrationConcept

Evaporation control and evapotranspiration measurement

Explain evaporation control and evapotranspiration measurement. State the definitions and conditions before you apply it.

Cetyl-alcohol monomolecular films can reduce open-water evaporation in controlled applications. A lysimeter measures components of the soil-water balance used to estimate evapotranspiration.

Understand / derivation

A film impedes transfer at the water surface; a lysimeter accounts for water added, stored and drained around a soil/plant sample.

Worked example

If a lysimeter receives 20 mm, drains 5 mm and gains 3 mm in storage, ET=20-5-3=12 mm when other terms are negligible.

Original CivilEase learning example; not an exam PYQ.

Check: Chemical use requires site-specific environmental approval; a study fact is not a recommendation to dose a reservoir.

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Supplied formula sheet · page 4 · Hydrology · entries 9, 10

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Hydrologic cycle, precipitation and infiltrationFormula

Horton's infiltration-capacity equation

Recall horton's infiltration-capacity equation. State the definitions and conditions before you apply it.

$$f(t)=f_c+(f_0-f_c)e^{-kt}$$

f₀: initial capacity; f_c: final capacity; k: decay constant; t: elapsed time in a compatible unit.

Understand / derivation

The assumed rate of decline df/dt=-k(f-f_c) integrates to an exponential approach to f_c.

Worked example

f₀=30, f_c=10 mm/h, k=1 h⁻¹ and t=1 h give f≈17.36 mm/h.

Original CivilEase learning example; not an exam PYQ.

Check: Actual infiltration cannot exceed rainfall/water supply; intermittent wetting may need recovery modelling.

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Supplied formula sheet · page 4 · Hydrology · entries 11

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Runoff and hydrograph analysisFormula

φ-index and W-index

Recall φ-index and W-index. State the definitions and conditions before you apply it.

$$\varphi :\sum max(0,i_j-\varphi )\Delta t=R;W=\frac{P-R-I_a}{t_e}$$

P: storm rain; R: direct-runoff depth; I_a: initial abstraction; t_e: applicable rainfall duration.

Understand / derivation

The φ-index is the constant loss rate whose excess rainfall volume equals direct runoff. W separates initial abstraction before averaging continuing losses.

Worked example

For constant 20 mm/h rain over 2 h and 20 mm runoff, φ=10 mm/h if the same rate applies throughout.

Original CivilEase learning example; not an exam PYQ.

Check: Choose the contributing rainfall periods consistently; φ is not generally total rain divided by duration.

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Supplied formula sheet · page 4 · Hydrology · entries 12, 13

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Runoff and hydrograph analysisFormula

Stream velocity from point measurements

Recall stream velocity from point measurements. State the definitions and conditions before you apply it.

$$\overline{v}\approx v_{0.6};\overline{v}\approx \frac{v_{0.2}+v_{0.8}}{2}$$

Subscripts: fractions of flow depth measured below the water surface.

Understand / derivation

The measured vertical velocity profile is approximated by a representative 0.6-depth value or the average of 0.2 and 0.8-depth readings.

Worked example

Readings 1.2 and 0.8 m/s give vertical mean≈1.0 m/s.

Original CivilEase learning example; not an exam PYQ.

Check: Discharge also requires lateral subsection areas and appropriate spatial averaging.

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Supplied formula sheet · page 4 · Hydrology · entries 14

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Runoff and hydrograph analysisFormula

Stream discharge by tracer dilution

Recall stream discharge by tracer dilution. State the definitions and conditions before you apply it.

$$Q_t{C}_t+Q{C}_0=(Q+Q_t)C_2,Q=Q_t\frac{C_t-C_2}{C_2-C_0}$$

Q_t: injected flow; C_t: tracer concentration; Q: original stream flow; C₀,C₂: background and fully mixed concentrations.

Understand / derivation

Conservation of tracer mass gives the mixing balance; solve for the unknown stream discharge.

Worked example

Q_t=0.01 m³/s, C_t=100, C₀=0 and C₂=1 in consistent concentration units give Q=0.99 m³/s.

Original CivilEase learning example; not an exam PYQ.

Check: Requires conservative tracer, complete mixing and suitable steady conditions.

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Supplied formula sheet · page 4 · Hydrology · entries 15

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Runoff and hydrograph analysisFormula

Unit-hydrograph equilibrium discharge

Recall unit-hydrograph equilibrium discharge. State the definitions and conditions before you apply it.

$$Q_e=2.778\frac{A}{D}$$

A: basin area in km²; D: duration in hours; 1 cm uniform rainfall excess; Q_e in m³/s.

Understand / derivation

A depth of 0.01 m over A×10⁶ m² gives 10000A m³; divide by 3600D seconds.

Worked example

For A=100 km² and D=2 h, Q_e=138.9 m³/s.

Original CivilEase learning example; not an exam PYQ.

Check: The coefficient is tied to a 1 cm unit depth and these units.

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Supplied formula sheet · page 4 · Hydrology · entries 16

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Flood estimation and routingFormula

Gumbel reduced variate and design flood

Recall gumbel reduced variate and design flood. State the definitions and conditions before you apply it.

$$y_T=-\ln [-\ln (1-1/T)],x_T=\overline{x}+K_{T}s,K_T=\frac{y_T-{\overline{y}}_n}{s_{y,n}}$$

T>1: return period; x̄,s: sample flood mean and standard deviation; ȳ_n,s_y,n: reduced-variate constants for the chosen sample-size method.

Understand / derivation

Map the nonexceedance probability 1-1/T through the Gumbel distribution and transform to the fitted flood scale.

Worked example

T=10 gives y_T≈2.2504. If x̄=100, s=20 and K_T=1.3, x_T=126 discharge units.

Original CivilEase learning example; not an exam PYQ.

Check: Use a consistent fitting method and sample-size constants; a return-level estimate has sampling uncertainty.

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Supplied formula sheet · page 4 · Hydrology · entries 17

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Flood estimation and routingFormula

Design-life exceedance risk

Recall design-life exceedance risk. State the definitions and conditions before you apply it.

$$R=1-{(1-\frac{1}{T})}^n$$

n: service-life years; T: annual-exceedance return period.

Understand / derivation

The probability of no exceedance in n independent years is (1-1/T)ⁿ; subtract it from one.

Worked example

For T=100 years and n=30 years, R≈26.0%.

Original CivilEase learning example; not an exam PYQ.

Check: Annual stationarity and independence are assumptions; climate/nonstationarity can alter the risk.

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Supplied formula sheet · page 4 · Hydrology · entries 18

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Flood estimation and routingFormula

Muskingum routing coefficients

Recall muskingum routing coefficients. State the definitions and conditions before you apply it.

$$O_2=C_0{I}_2+C_1{I}_1+C_2{O}_1$$$$C_0=\frac{-Kx+\Delta t/2}{K(1-x)+\Delta t/2},C_1=\frac{Kx+\Delta t/2}{K(1-x)+\Delta t/2},C_2=\frac{K(1-x)-\Delta t/2}{K(1-x)+\Delta t/2}$$

K: routing storage time; x: weighting factor; I,O: inflow/outflow; subscripts denote time steps.

Understand / derivation

Combine continuity over the step with S=K[xI+(1-x)O] and solve for the new outflow. C₀+C₁+C₂=1.

Worked example

K=2 h, x=0.2 and Δt=1 h give C₀≈0.04762, C₁≈0.42857, C₂≈0.52381.

Original CivilEase learning example; not an exam PYQ.

Check: Nonnegative coefficients require 2Kx≤Δt≤2K(1-x). K and time step must share units.

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Supplied formula sheet · page 4 · Hydrology · entries 19

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Groundwater hydrology and well hydraulicsFormula

Specific yield, retention and porosity

Recall specific yield, retention and porosity. State the definitions and conditions before you apply it.

$$n=S_y+S_r$$

n: porosity; S_y: gravity-drainable water fraction; S_r: retained-water fraction, all based on bulk volume.

Understand / derivation

Split the pore water of an initially saturated volume into the portion that drains under gravity and the portion retained.

Worked example

n=0.35 and S_r=0.10 give S_y=0.25.

Original CivilEase learning example; not an exam PYQ.

Check: This is the hydrologic definition of specific retention, not the degree of saturation symbol sometimes also written S_r.

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Supplied formula sheet · page 4 · Hydrology · entries 20

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Groundwater hydrology and well hydraulicsFormula

Well recuperation test

Recall well recuperation test. State the definitions and conditions before you apply it.

$$C_w=\frac{A_w}{t}\ln \frac{s_1}{s_2},Q=C_{w}s$$

A_w: well cross-sectional area; s₁,s₂: drawdowns at start/end of recovery interval t; C_w: specific capacity under the linear inflow model.

Understand / derivation

During recovery, A_w ds/dt=-C_ws; integrate to relate the logarithmic decay of drawdown to inflow capacity.

Worked example

A_w=10 m², s₁/s₂=2 and t=1 h give C_w≈6.93 m²/h; at 2 m drawdown, Q≈13.86 m³/h.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s permeability notation is ambiguous: conversion to soil hydraulic conductivity additionally requires an effective seepage area/length model.

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Supplied formula sheet · page 4 · Hydrology · entries 21

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Crop water requirements and irrigation methodsFormula

Time and maximum area for surface irrigation

Recall time and maximum area for surface irrigation. State the definitions and conditions before you apply it.

$$t=\frac{y}{f}\ln \frac{Q}{Q-fA},A_{max}=\frac{Q}{f}$$

Q: constant inflow; f: constant infiltration rate as length/time; A: wetted area; y: uniform surface water depth in the ideal border-advance model.

Understand / derivation

Water balance is Qdt=y dA+fA dt. Separate and integrate from dry area zero to A.

Worked example

Q=0.02 m³/s and f=10⁻⁵ m/s give limiting area 2000 m²; finite time is needed for any A below the limit.

Original CivilEase learning example; not an exam PYQ.

Check: The model assumes constant infiltration and a uniform advancing depth; t diverges as A approaches Q/f.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 1, 2

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Crop water requirements and irrigation methodsFormula

Sodium adsorption ratio

Recall sodium adsorption ratio. State the definitions and conditions before you apply it.

$$SAR=\frac{[N{a}^{+}]}{\sqrt{([C{a}^{2+}]+[M{g}^{2+}])/2}}$$

Ion concentrations expressed consistently in milliequivalents per litre.

Understand / derivation

The ratio compares sodium with the stabilizing divalent ions and indicates sodium-related soil-structure risk in irrigation water.

Worked example

Na=6, Ca=4 and Mg=4 meq/L give SAR=3.

Original CivilEase learning example; not an exam PYQ.

Check: Do not substitute mg/L directly; convert each ion using its equivalent weight.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 3

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Crop water requirements and irrigation methodsFormula

Duty, delta and base period

Recall duty, delta and base period. State the definitions and conditions before you apply it.

$$\Delta =8.64\frac{B}{D},A=QD$$

Δ: seasonal water depth in m; B: base period in days; D: duty in hectares per cumec; Q: m³/s; A: hectares.

Understand / derivation

One cumec supplies 86400B m³ over B days. Divide by D×10000 m² to obtain depth.

Worked example

B=120 days and D=1000 ha/cumec give Δ=1.0368 m; Q=2 m³/s irrigates 2000 ha.

Original CivilEase learning example; not an exam PYQ.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 4, 5

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Crop water requirements and irrigation methodsFormula

Irrigation conveyance, application, storage and distribution efficiencies

Recall irrigation conveyance, application, storage and distribution efficiencies. State the definitions and conditions before you apply it.

$${\eta }_c=\frac{V_{delivered}}{V_{supplied}},{\eta }_a=\frac{V_{rootstored}}{V_{delivered}},{\eta }_s=\frac{V_{rootstored}}{V_{rootneeded}},{\eta }_d=1-\frac{\overline{d}}{\overline{D}}$$

D̄: mean infiltrated depth; d̄: mean absolute deviation of depth; multiply fractions by 100 for percent.

Understand / derivation

Each efficiency compares water achieved at one stage to the relevant input or need. Distribution efficiency measures nonuniformity separately.

Worked example

If 100 units enter a canal, 80 reach the field and 60 remain in the root zone, η_c=80%, η_a=75% and combined efficiency=60%.

Original CivilEase learning example; not an exam PYQ.

Check: Do not confuse application efficiency with storage efficiency; their denominators differ.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 6, 7, 8, 9

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Crop water requirements and irrigation methodsFormula

Net crop irrigation requirement and field losses

Recall net crop irrigation requirement and field losses. State the definitions and conditions before you apply it.

$$CIR=C_u-R_e,NIR=CIR+P_{deep}+W_{special}$$

C_u: crop consumptive use; R_e: effective rain; P_deep: required/unavoidable percolation in the stated accounting; W_special: other specified needs.

Understand / derivation

Subtract effective rainfall from consumptive demand, then add the nonconsumptive water requirements defined by the problem.

Worked example

C_u=500 mm, effective rain=150 mm and specified percolation=50 mm give requirement=400 mm.

Original CivilEase learning example; not an exam PYQ.

Check: Terminology varies: define whether NIR includes special leaching/percolation needs; gross supply also accounts for efficiency.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 10, 11

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Crop water requirements and irrigation methodsFormula

Blaney-Criddle consumptive-use estimate

Recall blaney-Criddle consumptive-use estimate. State the definitions and conditions before you apply it.

$$C_u=K\frac{p}{40}(1.8T+32)$$

Source-sheet monthly form: T in °C; p: monthly percentage of annual daylight hours; K: calibrated crop coefficient; C_u in cm.

Understand / derivation

The empirical temperature-daylight factor is scaled by a crop coefficient. Modern variants use different constants and units.

Worked example

T=20°C, p=8 and K=0.8 give C_u=10.88 cm in this convention.

Original CivilEase learning example; not an exam PYQ.

Check: Do not mix this older °F-based conversion with a modern FAO form using mm/day and a different p definition.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 12

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Crop water requirements and irrigation methodsFormula

Root-zone water depth from a moisture deficit

Recall root-zone water depth from a moisture deficit. State the definitions and conditions before you apply it.

$$d_w=\frac{{\gamma }_d}{{\gamma }_w}{d}_r(w_{FC}-w_i)$$

d_r: root-zone depth; γ_d: dry soil unit weight; w_FC,w_i: gravimetric water contents as fractions.

Understand / derivation

Convert the gravimetric moisture deficit to water volume using dry-soil mass per unit bulk volume, then integrate through root depth.

Worked example

γ_d/γ_w=1.5, d_r=0.6 m and moisture deficit=0.10 give d_w=0.09 m=90 mm.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s OMC label is replaced by initial moisture w_i: optimum compaction moisture is not the usual irrigation target.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 13

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Canals and cross-drainage structuresFormula

Tractive shear stress in a uniform channel

Recall tractive shear stress in a uniform channel. State the definitions and conditions before you apply it.

$${\tau }_0={\gamma }_{w}RS$$

R: hydraulic radius; S: energy/bed slope for uniform flow.

Understand / derivation

Balance the downstream water-weight component against boundary shear over a reach.

Worked example

R=1 m and S=0.001 give mean boundary shear≈9.81 Pa.

Original CivilEase learning example; not an exam PYQ.

Check: Mean boundary shear is not the maximum local bed/bank stress in every section.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 14

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Canals and cross-drainage structuresFormula

Kennedy's critical-velocity relation

Recall kennedy's critical-velocity relation. State the definitions and conditions before you apply it.

$$V_0=0.55m{D}^{0.64},n_{Manning}\approx \frac{d_{mm}^{1/6}}{26}$$

V₀ in m/s; D: flow depth in m; m: Kennedy critical-velocity ratio; the roughness correlation uses grain size in mm.

Understand / derivation

Kennedy’s empirical critical velocity attempts to balance silt-carrying action and deposition. Grain-based roughness is a separate empirical estimate.

Worked example

For D=2 m and m=1, V₀≈0.857 m/s. d=1 mm gives n≈0.0385 in the source roughness correlation.

Original CivilEase learning example; not an exam PYQ.

Check: The roughness estimate is a coarse empirical rule and not Manning n for every sediment bed. Kennedy m is not Manning n.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 15, 16

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Canals and cross-drainage structuresFormula

Kutter's resistance formula

Recall kutter's resistance formula. State the definitions and conditions before you apply it.

$$V=C\sqrt{RS},C=\frac{23+0.00155/S+1/n}{1+(23+0.00155/S)n/\sqrt{R}}$$

R in m; S dimensionless; n: Kutter roughness parameter in this metric convention.

Understand / derivation

The empirical Chezy coefficient varies with roughness, slope and hydraulic radius; substitute it into the uniform-flow velocity law.

Worked example

If the computed C=50 and RS=0.001 m, V≈1.581 m/s.

Original CivilEase learning example; not an exam PYQ.

Check: The numerical constants and roughness parameter are unit-system dependent; do not mix them with a different Chezy/Kutter convention.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 17

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Canals and cross-drainage structuresFormula

Lacey's regime relations

Recall lacey's regime relations. State the definitions and conditions before you apply it.

$$V={\left(\frac{Q{f}^2}{140}\right)}^{1/6},R=\frac{2.5V^2}{f},P=4.75\sqrt{Q},S=\frac{f^{5/3}}{3340Q^{1/6}}$$

Metric regime equations: Q m³/s, V m/s, R,P m; f: Lacey silt factor.

Understand / derivation

The empirical equations describe a stable alluvial channel transporting the sediment characteristic of f; they must be used as one consistent regime set.

Worked example

For Q=100 m³/s, wetted perimeter P=47.5 m.

Original CivilEase learning example; not an exam PYQ.

Check: Regime assumptions exclude arbitrary rigid-lined sections and transient sediment conditions.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 18, 19, 20, 21

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Canals and cross-drainage structuresFormula

Local scour from unit discharge

Recall local scour from unit discharge. State the definitions and conditions before you apply it.

$$D_s=1.35{\left(\frac{q^2}{f}\right)}^{1/3}$$

D_s: scour depth below the reference high-water surface in m; q: discharge per metre width in m²/s; f: silt factor.

Understand / derivation

This empirical Lacey-based expression connects scour depth to unit-width hydraulic loading and sediment size.

Worked example

q=10 m²/s and f=1 give D_s≈6.27 m.

Original CivilEase learning example; not an exam PYQ.

Check: Reference elevation and location multipliers matter; this depth is not automatically measured below the bed.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 22

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Canals and cross-drainage structuresFormula

Lacey velocity written using radius and slope

Recall lacey velocity written using radius and slope. State the definitions and conditions before you apply it.

$$V=10.8R^{2/3}{S}^{1/3}$$

R in m; S dimensionless; V m/s in the source’s regime convention.

Understand / derivation

Eliminate the silt factor between compatible Lacey regime equations to express velocity through radius and slope.

Worked example

R=1 m and S=0.001 give V=1.08 m/s.

Original CivilEase learning example; not an exam PYQ.

Check: This is not the Manning equation, whose slope exponent is 1/2.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 23

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Canals and cross-drainage structuresFormula

Rounded-bed lined-channel section geometry

Recall rounded-bed lined-channel section geometry. State the definitions and conditions before you apply it.

$$A=y^2(\theta +\cot \theta ),P=2y(\theta +\cot \theta )$$$$A=By+y^2(\theta +\cot \theta ),P=B+2y(\theta +\cot \theta )$$

First pair: source’s curved-bed triangular family; second adds a horizontal bottom width B. θ in radians; y is the compatible depth/radius parameter.

Understand / derivation

The areas and wetted lengths combine circular-sector and straight-side contributions. The section sketch and tangent geometry are essential.

Worked example

For the formula’s θ=π/4, y=1 m and B=0, A≈1.7854 m² and P≈3.5708 m.

Original CivilEase learning example; not an exam PYQ.

Check: These are not ordinary straight-sided triangular/trapezoidal formulas. Without the rounded-bed geometry use A=zy² or By+zy² and the corresponding straight wetted perimeter.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 24, 25

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Canals and cross-drainage structuresFormula

Scour allowance and launching-apron length

Recall scour allowance and launching-apron length. State the definitions and conditions before you apply it.

$$R_s=0.47(Q/f{)}^{1/3},D=x{R}_s-y_{HFL},L_{apron}\approx \sqrt{5}D$$

R_s: normal scour depth below HFL; x: location multiplier; y_HFL: existing flow depth above bed; D: additional scour below bed.

Understand / derivation

Subtract existing water depth from predicted total scour depth. A launched apron on an assumed 2H:1V slope has sloping length √(D²+(2D)²).

Worked example

If xR_s=8 m and water depth=5 m, D=3 m and apron sloping length≈6.71 m.

Original CivilEase learning example; not an exam PYQ.

Check: Launch geometry, thickness, launching behaviour and protection design require the specified standards; negative D is not a scour allowance.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 26, 27

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Dams, spillways and hydraulic structuresFormula

Floor thickness against uplift

Recall floor thickness against uplift. State the definitions and conditions before you apply it.

$$t=\frac{h\prime }{G},t=\frac{h}{G-1}$$

G: floor material specific gravity; h′: uplift pressure head measured above floor bottom; h: residual head above floor top under the corresponding submergence model.

Understand / derivation

Equate resisting floor weight to uplift. If the uplift head includes the thickness itself, h′=h+t and Gt=h+t.

Worked example

For G=2.4 and residual top-referenced head h=2 m, t=2/(2.4-1)≈1.43 m before safety allowance.

Original CivilEase learning example; not an exam PYQ.

Check: The two expressions use different head reference conventions; include overlying water, drainage and safety factors explicitly.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 28, 29

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Dams, spillways and hydraulic structuresFormula

Khosla pile-interference correction

Recall khosla pile-interference correction. State the definitions and conditions before you apply it.

$$C\%=19\sqrt{\frac{D}{b}}\frac{d+D}{b}$$

b: horizontal distance between neighbouring piles; D: neighbouring pile depth below the reference floor level; d: depth of the pile being corrected.

Understand / derivation

Nearby cutoffs alter the ideal elementary pressure distribution; the empirical correction modifies the percentage potential.

Worked example

D=4 m, d=4 m and b=20 m give a correction magnitude≈3.40 percentage points.

Original CivilEase learning example; not an exam PYQ.

Check: This C is a pressure correction, not a creep coefficient. Add/subtract according to the upstream/downstream interference rule for the exact profile.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 30

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Dams, spillways and hydraulic structuresFormula

Elementary Khosla exit gradient and floor-end potentials

Recall elementary Khosla exit gradient and floor-end potentials. State the definitions and conditions before you apply it.

$$\alpha =b/d,\lambda =\frac{1+\sqrt{1+{\alpha }^2}}{2},G_E=\frac{H}{\pi d\sqrt{\lambda }}$$$${\Phi }_D=\frac{1}{\pi }{\cos }^{-1}\frac{\lambda -2}{\lambda },{\Phi }_E=\frac{1}{\pi }{\cos }^{-1}\frac{\lambda -1}{\lambda }$$

H: total head; b: floor length; d: cutoff depth; Φ: fractional potential in the elementary end-pile profile.

Understand / derivation

Potential-flow solution around the idealized end cutoff provides both residual potentials and exit gradient; multiply Φ by 100 for percent.

Worked example

For b/d=4, λ≈2.5616; with H=d=1 m, G_E≈0.1989.

Original CivilEase learning example; not an exam PYQ.

Check: Use the matching elementary profile and point labels. Apply interference, floor-thickness and slope corrections where required.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 31, 32

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Dams, spillways and hydraulic structuresFormula

Intermediate-pile Khosla potential parameters

Recall intermediate-pile Khosla potential parameters. State the definitions and conditions before you apply it.

$${\lambda }_1=\frac{\sqrt{1+{\alpha }_1^2}+\sqrt{1+{\alpha }_2^2}}{2},{\lambda }_2=\frac{\sqrt{1+{\alpha }_1^2}-\sqrt{1+{\alpha }_2^2}}{2}$$$${\Phi }_C=\frac{1}{\pi }{\cos }^{-1}\frac{2{\lambda }_2+1}{2{\lambda }_1},{\Phi }_D=\frac{1}{\pi }{\cos }^{-1}\frac{{\lambda }_2}{{\lambda }_1},{\Phi }_E=\frac{1}{\pi }{\cos }^{-1}\frac{2{\lambda }_2-1}{2{\lambda }_1}$$

α₁=b₁/d and α₂=b₂/d for the matching elementary intermediate-pile geometry.

Understand / derivation

The distances to the two floor ends determine a potential distribution on either side and at the tip of an intermediate cutoff.

Worked example

For equal b₁,b₂, λ₂=0 and the tip potential Φ_D=0.5.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s diagrams are absent: identify C,D,E from the prescribed Khosla elementary profile before applying these expressions.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 33

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Canals and cross-drainage structuresConcept

Canal outlet classification

Explain canal outlet classification. State the definitions and conditions before you apply it.

Nonmodular: discharge depends on both upstream and downstream water levels. Semimodular/flexible: depends on upstream head while insensitive to downstream level over the working range. Modular: ideally independent of both over a specified range.

Understand / derivation

Classify outlets by which hydraulic heads govern their discharge, not by the name alone.

Worked example

A freely discharging open flume is semimodular within its modular range; a drowned pipe behaves as a nonmodular outlet.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet lists masonry sluices/wooden shoots as nonmodular, pipe/open/venturi flumes as semimodular and Gibb/Kennedy/Foote devices as modular; actual submergence can change operation.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 34, 35, 36

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Canals and cross-drainage structuresFormula

Outlet flexibility, setting and sensitivity

Recall outlet flexibility, setting and sensitivity. State the definitions and conditions before you apply it.

$$F=\frac{dq/q}{dQ/Q}=\frac{m}{n}\frac{y}{H},S=\frac{dq/q}{dy/y}=m\frac{y}{H}=nF,\text{setting}=H/y$$

q∝H^m: outlet flow; Q∝y^n: canal flow; H: working outlet head; y: canal depth.

Understand / derivation

Logarithmically differentiate the two discharge laws. For a fixed crest change in head equals change in canal depth, giving the depth/head factor.

Worked example

m=0.5, n=1.5 and y/H=3 give F=1 and S=1.5: proportional behaviour.

Original CivilEase learning example; not an exam PYQ.

Check: F=1 is proportional; F>1 hyperproportional; F<1 subproportional. The sheet’s comparison m/n versus H/y is equivalent.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 37, 38, 39, 40, 41, 42

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Canals and cross-drainage structuresConcept

Cross-drainage structures

Explain cross-drainage structures. State the definitions and conditions before you apply it.

Aqueduct: canal above drain with free-surface drain flow. Siphon aqueduct: canal above drain with the drain barrel under pressure. Superpassage: drain above canal with free-surface canal flow. Canal siphon: drain above canal with canal flow under pressure.

Understand / derivation

Use two decisions: which watercourse is above, and whether the lower crossing flows freely or under pressure.

Worked example

A drain passing beneath a canal in a pressure barrel is a siphon aqueduct.

Original CivilEase learning example; not an exam PYQ.

Check: Do not interpret “siphon” as requiring a vacuum-driven inverted-U device here; it denotes pressure flow through the crossing barrel.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 43, 44, 45, 46

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Dams, spillways and hydraulic structuresFormula

Principal stress at a sloping dam face

Recall principal stress at a sloping dam face. State the definitions and conditions before you apply it.

$${\sigma }_1={\sigma }_v{\sec }^2\alpha -p_w{\tan }^2\alpha$$

σ_v: vertical normal stress at the face; p_w: water pressure normal to the face; α: face angle from vertical.

Understand / derivation

Use stress transformation at a boundary where water applies normal pressure and no tangential traction; combine with the vertical stress component.

Worked example

For a dry face p_w=0 and α=30°, σ₁=(4/3)σ_v.

Original CivilEase learning example; not an exam PYQ.

Check: The relevant face geometry and principal direction must match the stress transformation; compression signs must be consistent.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 47

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Dams, spillways and hydraulic structuresFormula

Elementary triangular gravity-dam section

Recall elementary triangular gravity-dam section. State the definitions and conditions before you apply it.

$$B=\frac{H}{\sqrt{G-C}},{\sigma }_{toe}={\gamma }_{w}H(G-C+1),H_{max}=\frac{f_{allow}}{{\gamma }_w(G-C+1)}$$

G: dam specific gravity; C: uplift coefficient in the elementary model; H,B: height/base width.

Understand / derivation

The no-tension/middle-third condition for the elementary triangular profile balances dam weight, hydrostatic force and the assumed uplift.

Worked example

G=2.4, C=1 and H=10 m give B≈8.45 m and toe stress≈235.4 kPa.

Original CivilEase learning example; not an exam PYQ.

Check: Elementary study model only: seismic loading, tailwater, galleries, sliding and actual cross-section require full stability checks.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 48, 49

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Dams, spillways and hydraulic structuresFormula

Ogee spillway downstream profile

Recall ogee spillway downstream profile. State the definitions and conditions before you apply it.

$$x^{1.85}=2H_d^{0.85}y$$

x: horizontal distance downstream of crest; y: vertical distance downward; H_d: design head.

Understand / derivation

The empirical standard profile approximates the lower nappe of a sharp-crested weir at the design head to maintain near-atmospheric pressure.

Worked example

At H_d=1 m and x=1 m, y=0.5 m.

Original CivilEase learning example; not an exam PYQ.

Check: This is the standard vertical-upstream-face profile convention; approach geometry and off-design head change pressure/performance.

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Supplied formula sheet · page 6 · Irrigation Engineering · entries 50

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Water demand and distributionFormula

Population forecasting methods

Recall population forecasting methods. State the definitions and conditions before you apply it.

$$P_n=P_0+n\overline{x}(\text{arithmetic});P_n=P_0(1+r{)}^n(\text{geometric})$$$$P_n=P_0+n\overline{x}+\frac{n(n+1)}{2}\overline{y}(\text{incremental})$$

x̄: average increase per time period; r: fractional compound growth rate; ȳ: average change in the increment.

Understand / derivation

A constant absolute increase gives a line; a constant percentage increase compounds; a constant incremental change sums the arithmetic sequence of increases.

Worked example

P₀=10000 and x̄=500 per decade give P₂=11000 by arithmetic increase. At r=5%, geometric P₂=11025.

Original CivilEase learning example; not an exam PYQ.

Check: If r is reported in percent, divide by 100. Geometric-mean growth uses the product of (1+rᵢ), not generally the product of raw percentage rates.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 1, 2, 3

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Water and wastewater qualityFormula

Carbonate and noncarbonate hardness

Recall carbonate and noncarbonate hardness. State the definitions and conditions before you apply it.

$$CH=min(TH,Alk),NCH=TH-CH$$

All three concentrations expressed as mg/L as CaCO₃.

Understand / derivation

Carbonate hardness is limited both by available hardness cations and the alkalinity that can balance them; remaining hardness is noncarbonate.

Worked example

TH=250 and alkalinity=180 mg/L as CaCO₃ give CH=180 and NCH=70.

Original CivilEase learning example; not an exam PYQ.

Check: Do not compare raw ionic mg/L values with CaCO₃-equivalent alkalinity.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 4, 5

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Water and wastewater qualityFormula

CaCO₃ equivalents of hardness and alkalinity

Recall caCO₃ equivalents of hardness and alkalinity. State the definitions and conditions before you apply it.

$$C_{asCaC{O}_3}=C_i\frac{50}{E{W}_i}$$$$TH\approx 2.5[C{a}^{2+}]+4.167[M{g}^{2+}];Alk=\frac{50}{30}[C{O}_3^{2-}]+\frac{50}{61}[HC{O}_3^{-}]+\frac{50}{17}[O{H}^{-}]$$

Cᵢ in mg/L; EW: molar mass/charge in g/equivalent; dilute-water alkalinity expression neglects H⁺/other weak-acid terms.

Understand / derivation

One equivalent of CaCO₃ weighs 50 g. Convert each ionic equivalent to the same reference mass before adding.

Worked example

40 mg/L Ca²⁺ contributes 100 mg/L as CaCO₃; 24 mg/L Mg²⁺ contributes about 100.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s Al³⁺ term uses 50/9; include additional hardness cations only when appropriate. Alkalinity is not a synonym for pH.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 6, 7

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Water and wastewater qualityConcept

Total Kjeldahl nitrogen

Explain total Kjeldahl nitrogen. State the definitions and conditions before you apply it.

$$TKN=\text{organic nitrogen}+\text{ammoniacal nitrogen}$$

Concentrations are reported as nitrogen mass, usually mg N/L.

Understand / derivation

The Kjeldahl digestion measures organic nitrogen and ammonia/ammonium, not generally nitrate and nitrite.

Worked example

Organic N=3 and ammonia N=5 mg N/L give TKN=8 mg N/L.

Original CivilEase learning example; not an exam PYQ.

Check: Total nitrogen also includes oxidized nitrogen not included in TKN.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 8

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Water and wastewater qualityConcept

Drinking-water study values and microbiological safety

Explain drinking-water study values and microbiological safety. State the definitions and conditions before you apply it.

Selected IS 10500:2012 base acceptable values: colour 5 Hazen; pH 6.5–8.5; turbidity 1 NTU; TDS 500 mg/L; chloride 250 mg/L; sulphate 200 mg/L; nitrate 45 mg/L; fluoride 1 mg/L; total hardness and alkalinity 200 mg/L as CaCO₃; Ca 75, Mg 30 and Zn 5 mg/L. E. coli/thermotolerant coliforms should not be detectable in a 100 mL sample under the base drinking-water requirement.

Understand / derivation

Separate physical/chemical acceptability from microbiological safety. “Acceptable” and “permissible in the absence of an alternative source” are different columns in the standard.

Worked example

A fluoride result of 1.2 mg/L exceeds the base acceptable value 1.0 even though the base alternative-source permissible value is 1.5 mg/L.

Original CivilEase learning example; not an exam PYQ.

Check: These are dated base-standard study values, not a current compliance certificate. The sheet’s iron 0.3 mg/L and free residual chlorine 0.2 mg/L are retained as historical/exam conventions only; amendments, sampling location, contact time and emergency chlorination conditions must be checked in the specified standard.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 9, 10, 11

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Water and wastewater qualityFormula

MPN dilution series and Thomas estimate

Recall mPN dilution series and Thomas estimate. State the definitions and conditions before you apply it.

$$MPN/100\mathrm{m}\mathrm{L}\approx \frac{100P}{\sqrt{N_v{T}_v}}$$

P: total positive tubes; N_v: sample volume in negative tubes; T_v: total sample volume in all tubes, in mL.

Understand / derivation

Multiple-tube fermentation estimates a probable organism density from positive/negative responses at known dilutions. Thomas gives a compact approximation.

Worked example

P=5, N_v=10 mL and T_v=100 mL give estimate≈15.81 per 100 mL.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s 10, 1 and 0.1 mL series is one test setup; use its specified tube counts and an appropriate MPN table/confidence interval.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 12, 13

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Water treatment and physical-chemical processesFormula

Stokes settling and ideal sedimentation removal

Recall stokes settling and ideal sedimentation removal. State the definitions and conditions before you apply it.

$$v_s=\frac{g({\rho }_p-\rho )d^2}{18\mu }=\frac{g(G-1)d^2}{18\nu };\eta =min(1,v_s/v_o)$$

v_o=Q/A: surface overflow rate; d: particle diameter; dilute discrete particles.

Understand / derivation

Balance particle submerged weight with Stokes drag for terminal velocity. In an ideal horizontal-flow tank, the fraction captured for v_s<v_o scales as v_s/v_o.

Worked example

v_s=0.5 mm/s and v_o=1 mm/s give ideal removal=50%.

Original CivilEase learning example; not an exam PYQ.

Check: Stokes law requires low particle Reynolds number; flocculent settling needs a different model.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 14, 15

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Water treatment and physical-chemical processesConcept

Coagulation and chlorine disinfectant form

Explain coagulation and chlorine disinfectant form. State the definitions and conditions before you apply it.

Alum, ferric/ferrous salts and related coagulants destabilize suspended particles. Alum dosage and effective pH depend on raw-water alkalinity and jar testing. Hypochlorous acid (HOCl) is generally a more effective free-chlorine disinfectant than hypochlorite ion (OCl⁻).

Understand / derivation

Coagulation relies on charge neutralization and/or precipitate enmeshment. Chlorine speciation shifts with pH, changing disinfection effectiveness.

Worked example

At a lower pH within a suitable treatment range, a larger fraction of free chlorine is HOCl; compare performance using actual contact time and dose.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s alum pH 6.5–8.3 is a study range, not a universal optimum. “Most destructive disinfectant” is replaced with the specific HOCl-versus-OCl⁻ comparison.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 16, 23, 24

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Water treatment and physical-chemical processesFormula

Alum reaction and stoichiometric products

Recall alum reaction and stoichiometric products. State the definitions and conditions before you apply it.

$$A{l}_2(S{O}_4{)}_3·18H_{2}O+3Ca(HC{O}_3{)}_2\to 3CaS{O}_4+2Al(OH{)}_3+6C{O}_2+18H_{2}O$$$$\frac{\text{alkalinity consumed}}{\text{alum}}=\frac{300}{666};\frac{Al(OH{)}_3}{\text{alum}}=\frac{156}{666};\frac{C{O}_2}{\text{alum}}=\frac{264}{666}$$

Alum basis: the 18-water hydrate with approximate molar mass 666 g/mol. Alkalinity is as CaCO₃; product ratios are actual product mass.

Understand / derivation

Balance the reaction, multiply moles by molar mass, and convert six equivalents of alkalinity to 300 g CaCO₃ equivalent.

Worked example

A 10 mg/L alum dose theoretically consumes about 4.50 mg/L alkalinity as CaCO₃ and produces 2.34 mg/L Al(OH)₃.

Original CivilEase learning example; not an exam PYQ.

Check: Corrected source units: 408/666 is CaSO₄ mass per alum mass, corresponding to 300/666 as CaCO₃ hardness, not 408/666 as CaCO₃. Sludge and CO₂ mass are not “as CaCO₃”. Actual sludge includes other solids.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 17, 18, 19, 20, 21

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Water treatment and physical-chemical processesFormula

Velocity gradient for rapid mixing/flocculation

Recall velocity gradient for rapid mixing/flocculation. State the definitions and conditions before you apply it.

$$G=\sqrt{\frac{P}{\mu V}}$$

G: mean velocity gradient (s⁻¹); P: dissipated power (W); μ: dynamic viscosity; V: mixed water volume.

Understand / derivation

Equate mechanical power dissipation to viscous dissipation μVG².

Worked example

P=100 W, μ=0.001 Pa·s and V=10 m³ give G=100 s⁻¹.

Original CivilEase learning example; not an exam PYQ.

Check: Motor input power is not necessarily all dissipated in the water. Contact duration is a separate design parameter.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 22

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Water treatment and physical-chemical processesFormula

Lime-soda softening stoichiometric demand

Recall lime-soda softening stoichiometric demand. State the definitions and conditions before you apply it.

$$CaO\mathrm{d}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d}=\frac{56}{100}(CH+M{g}_{asCaC{O}_3})+\frac{56}{44}C{O}_2$$$$N{a}_{2}C{O}_3\mathrm{d}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d}=\frac{106}{100}NCH$$

All demands are mg/L of pure chemical; CH,NCH and Mg contribution as CaCO₃; free CO₂ in actual mg/L.

Understand / derivation

Lime removes carbonate hardness and provides additional hydroxide for magnesium precipitation; soda supplies carbonate for noncarbonate calcium hardness.

Worked example

CH=100, Mg contribution=50 and CO₂=0 mg/L give CaO=84 mg/L. NCH=50 gives Na₂CO₃=53 mg/L.

Original CivilEase learning example; not an exam PYQ.

Check: Mg contribution is the total magnesium hardness, not raw Mg mg/L. Add purity, excess dose and other acid/alkalinity demands explicitly.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 25, 26

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Biological treatment, sludge and reuseFormula

First-order BOD exertion and temperature dependence

Recall first-order BOD exertion and temperature dependence. State the definitions and conditions before you apply it.

$$BO{D}_t=L_0(1-{10}^{-k_{b}t}),k_{b,T}=k_{b,20}{\theta }^{T-20},\theta \approx 1.047$$

L₀: ultimate carbonaceous BOD; k_b: base-10 rate constant per day; t: days; T: °C.

Understand / derivation

If remaining biodegradable demand decays exponentially, exerted demand equals initial minus remaining. Temperature changes the empirical rate constant.

Worked example

For k_b=0.1 d⁻¹ and t=5 d, exerted fraction=1-10⁻⁰·⁵≈0.6838.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s 0.794 and 0.630 daily retention factors correspond approximately to k_b=0.1 and 0.2, not universal 20°C/37°C constants. Natural-exponential k_e=2.303k_b.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 27, 28, 29

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Biological treatment, sludge and reuseFormula

NRC trickling-filter efficiency and recirculation factor

Recall nRC trickling-filter efficiency and recirculation factor. State the definitions and conditions before you apply it.

$$E\%=\frac{100}{1+0.44\sqrt{W/(VF)}},F=\frac{1+R}{(1+0.1R{)}^2}$$

W: organic loading kg/day; V: filter volume m³; R: recycle/fresh-flow ratio; F=1 with no recycle.

Understand / derivation

The empirical NRC relation connects removal efficiency to loading per effective media volume; recycle changes the apparent repeated-contact factor.

Worked example

W/V=0.25 kg/(m³·day), F=1 give E≈81.97%.

Original CivilEase learning example; not an exam PYQ.

Check: Equivalent coefficient 0.0044 uses loading in kg/(hectare-metre·day), 10000 times the SI volumetric loading. Do not mix the two unit bases.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 30, 31

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Biological treatment, sludge and reuseFormula

Aeration HRT, volumetric loading and F/M

Recall aeration HRT, volumetric loading and F/M. State the definitions and conditions before you apply it.

$$HRT=\frac{V}{Q},L_v=\frac{Q{S}_0}{V},\frac{F}{M}=\frac{Q{S}_0}{VX}$$

Q: wastewater flow; V: aeration volume; S₀: applied BOD concentration; X: biomass concentration with a defined MLSS/MLVSS basis.

Understand / derivation

Divide stored water volume by throughput for residence time, applied substrate mass rate by volume for loading, and substrate rate by biomass inventory for F/M.

Worked example

Q=1000 m³/day, V=500 m³, S₀=0.2 kg/m³ and X=3 kg/m³ give HRT=0.5 day, L_v=0.4 kg/m³/day and F/M≈0.133/day.

Original CivilEase learning example; not an exam PYQ.

Check: Define whether food is influent BOD or removed BOD and whether biomass uses MLSS or MLVSS.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 32, 33, 34

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Biological treatment, sludge and reuseFormula

Mean cell residence time

Recall mean cell residence time. State the definitions and conditions before you apply it.

$${\theta }_c=\frac{VX}{Q_w{X}_r+Q_e{X}_e}$$

Q_w: sludge wasting flow; X_r: wasted/return-sludge concentration under the selected wasting point; Q_e,X_e: effluent flow/solids.

Understand / derivation

Sludge age is the system biomass inventory divided by its net solids loss rate.

Worked example

Inventory=1500 kg and daily losses=100 kg/day give θ_c=15 days.

Original CivilEase learning example; not an exam PYQ.

Check: The wasting location changes the correct solids concentration; include all relevant biomass inventories/losses.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 35

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Biological treatment, sludge and reuseFormula

Sludge volume index and return-sludge ratio

Recall sludge volume index and return-sludge ratio. State the definitions and conditions before you apply it.

$$SVI=\frac{V_{30}}{X}(X\mathrm{i}\mathrm{n}\mathrm{g}/\mathrm{L});\frac{Q_r}{Q}=\frac{X}{X_r-X},X_r\approx \frac{{10}^6}{SVI}(\mathrm{m}\mathrm{g}/\mathrm{L})$$

V₃₀: settled sludge volume in mL/L after 30 min; SVI in mL/g.

Understand / derivation

A clarifier solids balance neglecting effluent solids gives the return-flow ratio. The SVI approximation assumes the settled volume represents the return-sludge concentration.

Worked example

SVI=100 mL/g suggests X_r≈10000 mg/L; at X=3000 mg/L, Q_r/Q≈0.4286.

Original CivilEase learning example; not an exam PYQ.

Check: The X_r-from-SVI approximation is idealized and should not replace measured underflow concentration.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 36, 37

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Unit processes, reaction kinetics and reactorsFormula

Mixing fresh and recycled streams

Recall mixing fresh and recycled streams. State the definitions and conditions before you apply it.

$$C_{mix}=\frac{Q_s{C}_s+Q_r{C}_r}{Q_s+Q_r}$$

Q: volume flow; C: concentration on the same mass basis.

Understand / derivation

Conservation of constituent mass in a zero-storage mixing junction gives a flow-weighted mean.

Worked example

Equal 1 m³/s flows at 10 and 30 mg/L mix to 20 mg/L.

Original CivilEase learning example; not an exam PYQ.

Check: Use concentrations, not stream volumes, in the numerator factors; the source’s V symbols are ambiguous.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 38

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Biological treatment, sludge and reuseFormula

Streeter-Phelps oxygen deficit and critical time

Recall streeter-Phelps oxygen deficit and critical time. State the definitions and conditions before you apply it.

$$D(t)=\frac{k_d{L}_0}{k_r-k_d}(e^{-k_{d}t}-e^{-k_{r}t})+D_i{e}^{-k_{r}t}$$$$t_c=\frac{\ln \{(k_r/k_d)[1-D_i(k_r-k_d)/(k_d{L}_0)]\}}{k_r-k_d},k_r{D}_c=k_d{L}_0{e}^{-k_d{t}_c}$$

k_d,k_r: natural-exponential deoxygenation/reaeration rates; D_i: initial DO deficit; L₀: ultimate BOD; D_c=D(t_c).

Understand / derivation

Solve dD/dt=k_dL₀e⁻ᵏᵈᵗ-k_rD. At an interior maximum set dD/dt=0, giving the critical condition.

Worked example

For D_i=0, L₀=10 mg/L, k_d=0.1/day and k_r=0.2/day, t_c=ln2/0.1≈6.93 days and D_c=2.5 mg/L.

Original CivilEase learning example; not an exam PYQ.

Check: The source’s standalone critical-deficit power identity is corrected through the differential equation. The logarithm argument must be positive and t_c must lie in the reach; k_r=k_d needs the limiting solution. Do not mix base-10 and natural rate constants.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 39, 40, 41

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Air pollution and combustionConcept

Primary and secondary air pollutants

Explain primary and secondary air pollutants. State the definitions and conditions before you apply it.

Primary pollutants are emitted directly, such as CO, SO₂, many NOₓ species, hydrocarbons and primary particulate matter. Secondary pollutants form through atmospheric reactions, including ozone, PAN and secondary particulate matter.

Understand / derivation

Classify by formation pathway. A pollutant family such as particulate matter can have both primary and secondary components.

Worked example

Ozone formed from photochemical reactions involving NOₓ and volatile organic compounds is secondary.

Original CivilEase learning example; not an exam PYQ.

Check: Smog and aerosols are mixtures/categories, not single chemical species; the same category can contain both primary and secondary components.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 42, 43

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Air pollution and combustionFormula

Atmospheric lapse rate and stability

Recall atmospheric lapse rate and stability. State the definitions and conditions before you apply it.

$$ELR<{\Gamma }_d:\text{dry stable};ELR>{\Gamma }_d:\text{dry unstable}$$

ELR=-dT_environment/dz; Γ_d: dry adiabatic lapse rate; both positive for temperature decreasing upward.

Understand / derivation

Compare a displaced unsaturated parcel cooling at Γ_d with its environment. If it becomes colder/denser aloft, buoyancy restores it.

Worked example

ELR=6°C/km compared with dry Γ_d≈9.8°C/km indicates dry static stability.

Original CivilEase learning example; not an exam PYQ.

Check: Saturated parcels use the moist adiabatic lapse rate; equality is neutral and inversions are strongly stable in this simplified comparison.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 44, 45

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Noise pollution (older syllabus topics)Formula

Sound level, energy addition, averaging and distance

Recall sound level, energy addition, averaging and distance. State the definitions and conditions before you apply it.

$$L_p=20{\log }_{10}(p_{rms}/p_0),p_0=20\mu \mathrm{P}\mathrm{a}$$$$L_{sum}=10{\log }_{10}\sum _i{10}^{L_i/10},L_{eq}=10{\log }_{10}\frac{{\sum }_i{t}_i{10}^{L_i/10}}{{\sum }_i{t}_i}$$$$L_2=L_1-20{\log }_{10}(r_2/r_1)$$

L in dB; independent sound sources for the energy sum; point source in a free field for the distance rule.

Understand / derivation

Mean-square pressure/energy is proportional to 10^(L/10). Add energies before taking the logarithm; spherical spreading gives pressure inversely proportional to distance.

Worked example

Two independent 60 dB sources produce about 63.01 dB. Doubling distance from an ideal point source reduces level by about 6.02 dB.

Original CivilEase learning example; not an exam PYQ.

Check: The sheet’s pressure-amplitude “average” is not the usual energy-equivalent sound level and is replaced by L_eq. Noise is retained as supplied broader Civil Engineering content, although outside the uploaded syllabus’s explicit Environmental section.

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Supplied formula sheet · page 7 · Environmental Engineering · entries 46, 47, 48, 49, 50

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One place to keep learning

This collection will grow across the Civil Engineering syllabus. Closely related entries are grouped into learning cards; original card IDs preserve saved revision marks. Only subjects and topics with published cards appear in the filters. The current modules span the subjects in the supplied formula sheet. Steel design, surveying, construction management and many other syllabus topics will be expanded in future releases.

Learn, then practise: reveal an explanation or formula, open a derivation or example, and try related questions inside the card. Direct matches and broader topic practice are labelled separately. Each question identifies its source exam; GATE questions are available now, with other exams planned.

Revision marks are saved in this browser on this device.

Sources: the supplied eight-page Civil All Formulas sheet, reviewed and rewritten with explicit notation, unit conventions and corrections; the supplied 2027 Civil Engineering syllabus for the mathematics scope; NPTEL teaching notes and the base IS 10500:2012 water-quality reference where indicated. The original geometry cards retain their NPTEL links. Examples and explanations are written for this collection. Primary references: IIT Bombay/NPTEL traffic-flow relations; BIS IS 10500:2012 base reference. Older empirical/code-based relations are labelled for study use; follow the edition and conditions specified by a question or project.