Beam Bending & Deflection
Simply supported, cantilever, fixed beams — shear, moment, deflection, stress check (IS 456 / Eurocode)
🔩Beam Parameters
⚡ Loading
🔲 Section
⚙️ Material
📊Results—
Section Moment of Inertia I
—
Section Modulus Z = I/y
—
Radius of Gyration r
—
Max Bending Moment Mmax
—
Max Shear Force Vmax
—
Short-term Deflection δ
—
Long-term Deflection δ_LT (creep)
—
Bending Stress σ = M/Z
—
—
Stiffness EI (kN·m²)
—
L/δ Short-term (limit L/250)
—
L/δ Long-term (limit L/250)
—
Select a beam type to see formulae
Column Buckling & Capacity
Euler critical load, slenderness ratio, axial capacity check — IS 456 / Eurocode 3
🏛️Column Parameters
📊Column Results
Critical Load Pcr (theoretical)
—
Design Resistance Pd = Pcr / γM0 (γM0 = 1.10)
—
Cross-Section Area A
—
Min. Inertia Imin
—
Min. Radius of Gyration rmin
—
Slenderness λ = KL/r
—
Non-dim Slenderness λ̄ (IS 800)
—
Buckling Reduction Factor χ
—
Classification
—
Axial Stress σ = N/A
—
—
Safety Factor (Pcr/N)
—
Critical Stress σcr
—
IS 800 Cl.7.1.2 Perry-Robertson: λ̄ = √(fy/σe) | φ = 0.5[1+α(λ̄−0.2)+λ̄²] | χ = 1/[φ+√(φ²−λ̄²)] ≤ 1.0
fcd = χ×fy/γM0 | Pd = fcd×A | γM0 = 1.10 (IS 800 Table 5)
Buckling curves: Rect/Weld → c (α=0.49) | Hollow → b (α=0.34) | Rolled I → a (α=0.21)
Slender check: σe = π²E/λ² | Johnson intermediate: σcr = fy[1−fyλ²/(4π²E)] | λc = π√(2E/fy)
fcd = χ×fy/γM0 | Pd = fcd×A | γM0 = 1.10 (IS 800 Table 5)
Buckling curves: Rect/Weld → c (α=0.49) | Hollow → b (α=0.34) | Rolled I → a (α=0.21)
Slender check: σe = π²E/λ² | Johnson intermediate: σcr = fy[1−fyλ²/(4π²E)] | λc = π√(2E/fy)
Isolated Footing Design
Square/rectangular pad footing — IS 456 punching shear (k·0.25√fck), eccentric bearing pressure, bending moment by column-face method
🔩Footing Parameters
🏛️ Column & Load
🔲 Footing Geometry
⚖️ Eccentricity (optional)
📊Footing Results
Max Net Bearing Pressure q_max
—
Required Area A_req
—
Provided Area A_prov
—
Min Pressure q_min
—
Eccentricity ratio e/B, e/L
—
Factored BM Mu — B-direction
—
Factored BM Mu — L-direction
—
Punching Perimeter b₀ (at d/2)
—
Applied Punching Shear τ_v
—
k factor (IS 456 Cl.31.6.3)
—
Allowable τ_c = k × τ_co
—
Punching Shear Check
—
Ast — B direction (mm²/m)
—
Ast — L direction (mm²/m)
—
One-way Shear — B dir (IS 456 Cl.34.2.4)
—
One-way Shear — L dir
—
IS 456 Punching Shear (Cl.31.6.3): τ_c = k × 0.25√fck | k = 0.5 + β_c ≤ 1.0, β_c = short/long column side
Eccentric bearing: q = P/(B×L) × (1 ± 6e_x/B ± 6e_y/L)
Mu = 1.5 × q_max × cantilever² / 2 (IS 456 Cl.34.2, column face) | Ast = Mu / (0.87×fy×0.9×d)
One-way shear (Cl.34.2.4): critical at d from column face | τ_c from IS 456 Table 19 (pt-based)
Min steel (Cl.26.5.2.1): 0.15% bD (Fe415) / 0.12% bD (Fe500) | governing value shown above
Eccentric bearing: q = P/(B×L) × (1 ± 6e_x/B ± 6e_y/L)
Mu = 1.5 × q_max × cantilever² / 2 (IS 456 Cl.34.2, column face) | Ast = Mu / (0.87×fy×0.9×d)
One-way shear (Cl.34.2.4): critical at d from column face | τ_c from IS 456 Table 19 (pt-based)
Min steel (Cl.26.5.2.1): 0.15% bD (Fe415) / 0.12% bD (Fe500) | governing value shown above
Concrete Mix Design
Mix proportioning — IS 10262 / ACI 211 — cement, water, fine & coarse aggregate per m³
🧱Mix Design Inputs
📊Mix Results — Per m³ & Per Batch
Target Mean Strength fm (IS 10262)
—
Cement Content
—
Water Content
—
W/C Ratio (actual)
—
Fine Aggregate
—
Coarse Aggregate
—
Mix Ratio (C : FA : CA : W)
—
Fresh Concrete Density
—
Cement Bags (50 kg) / m³
—
IS 10262 / ACI 211 | fm = fck + 1.65σ (σ = 4 MPa for M20-M25; 5 MPa for M30+)
Water content per IS 10262 Table 2 (varies with agg size & slump) | Absolute volume method
⚠ Trial mixes required. Aggregate moisture correction not applied — adjust for field conditions.
Water content per IS 10262 Table 2 (varies with agg size & slump) | Absolute volume method
⚠ Trial mixes required. Aggregate moisture correction not applied — adjust for field conditions.
Steel Section Properties
I/H-section, channel, angle, SHS, CHS — area, inertia, moduli, plastic modulus, moment capacity
⚙️Section Inputs
⚙️ Material
📊Section Properties
Cross-Section Area A
—
Ixx (major axis)
—
Iyy (minor axis)
—
Zxx (elastic mod.)
—
Zyy (elastic mod.)
—
rx — radius of gyration
—
ry — radius of gyration
—
Plastic Modulus Zpx
—
Self Weight
—
Moment Capacity Mc (IS 800)
—
I = ΣbD³/12 | Z = I/y | r = √(I/A)
Zpx: I-sect = BfTf(hw+Tf) + Tw×hw²/4 | SHS = (B³−Bi³)/4 | CHS = (D³−Di³)/6
Mc = fy × Zpx / γM0 (γM0 = 1.10, IS 800) | ⚠ LTB not checked — verify λLT per IS 800 Cl.8.2
Zpx: I-sect = BfTf(hw+Tf) + Tw×hw²/4 | SHS = (B³−Bi³)/4 | CHS = (D³−Di³)/6
Mc = fy × Zpx / γM0 (γM0 = 1.10, IS 800) | ⚠ LTB not checked — verify λLT per IS 800 Cl.8.2
Pipe Flow & Hydraulics
Two distinct modes — Pressurised pipe (Darcy-Weisbach / Swamee-Jain) and Gravity/open-channel (Manning). Select the correct mode for your scenario.
Flow Mode:
Darcy-Weisbach — for closed pipes under pressure (pumped mains, distribution)
💧Pressurised Pipe Parameters
📊Hydraulic Results
Flow Velocity v
—
Reynolds Number Re
—
Flow Regime
—
Darcy-Weisbach Friction Factor f
—
Head Loss hf (Darcy-Weisbach)
—
—
Hydraulic Gradient hf/L
—
Velocity Head v²/2g
—
Pipe Flow Area
—
Mass Flow Rate
—
Darcy-Weisbach (pressurised closed pipe): hf = f×(L/D)×(v²/2g)
Swamee-Jain (turbulent, log₁₀): f = 0.25 / [log₁₀(ε/3.7D + 5.74/Re⁰·⁹)]²
Laminar (Re < 2300): f = 64/Re | Re = ρvD/μ
✗ Do NOT use Darcy-Weisbach for gravity channels — use Manning mode instead.
Swamee-Jain (turbulent, log₁₀): f = 0.25 / [log₁₀(ε/3.7D + 5.74/Re⁰·⁹)]²
Laminar (Re < 2300): f = 64/Re | Re = ρvD/μ
✗ Do NOT use Darcy-Weisbach for gravity channels — use Manning mode instead.
Retaining Wall — Stability
Gravity / cantilever wall — sliding, overturning, bearing capacity checks (IS 456 / BS 8002)
🪨Wall & Soil Parameters
🏗️ Geometry
🌍 Soil Properties
📊Stability Results
Active Pressure Ka
—
Passive Pressure Kp
—
Horizontal Active Force Pa (kN/m run)
—
Total Vertical Load W (kN/m)
—
Overturning Moment Mo
—
Stabilising Moment Mr
—
FS Overturning (min 1.5)
—
FS Sliding (min 1.5)
—
Max Foundation Pressure q_max
—
Ka = tan²(45°−φ/2) | Pa = ½γH²Ka + qHKa | FSOvt = Mr/Mo
FSsl = μW/Pa (min 1.5) | e = B/2−(Mr−Mo)/W | q = W/B×(1±6e/B)
FSsl = μW/Pa (min 1.5) | e = B/2−(Mr−Mo)/W | q = W/B×(1±6e/B)
Earthwork Volume Calculator
Cut/fill by prismoidal & average end area, compaction, mass haul, truck loads
🌍Earthwork Inputs
🚛 Compaction & Haul
📊Volume & Haul Results
Bank Volume (in-situ)
—
Loose Volume (excavated)
—
Compacted Volume (fill)
—
Mass of Material
—
Truck Loads Required
—
Load Factor (Bank/Loose)
—
Cut / Fill Summary
—
Prismoidal Correction
—
% Volume Difference
—
Avg End Area: V = L/2×(A1+A2) | Prismoidal: V = L/6×(A1+4Am+A2)
Loose = Bank×(1+Swell/100) | Compacted = Bank×(1−Shrink/100)
Loose = Bank×(1+Swell/100) | Compacted = Bank×(1−Shrink/100)
Surveying & Road Geometry
Horizontal / vertical curves, coordinate geometry, sight distances — IRC / AASHTO
📏Survey Inputs
🔵 Horizontal Curve
📐 Vertical Curve
📌 Coordinate Geometry
📊Survey Results
Curve Length L = R × Δ (rad)
—
Tangent Length T
—
Mid-Ordinate M
—
External Distance E
—
Chord Length C
—
Degree of Curvature
—
Min Radius for Design Speed (IRC)
—
Vertical Curve Length
—
Grade Diff A = |G1−G2|
—
Rate of Change (%/m)
—
Bearing Point 1 → 2
—
Slope Distance
—
ΔX / ΔY Components
—
L = πRΔ/180 | T = R·tan(Δ/2) | M = R(1−cos(Δ/2))
Rmin = V²/127(e/100+f) | VCL(crest) = A×SSD²/658 | VCL(sag) = A×SSD/3.5+SSD
Rmin = V²/127(e/100+f) | VCL(crest) = A×SSD²/658 | VCL(sag) = A×SSD/3.5+SSD
Civil Engineering Study Guide
Comprehensive theory, derivations, design procedures, code references and worked examples — covering every calculator in this suite. Written for engineering students, graduate engineers and practitioners refreshing their fundamentals.
📐
Chapter 1 — Beam Bending, Shear & Deflection
Euler-Bernoulli · IS 456 · EC2
1.1 Euler-Bernoulli Beam Theory
The Euler-Bernoulli model makes two key assumptions: (a) plane sections remain plane after bending, and (b) the beam material obeys Hooke's law linearly. Together they give the fundamental curvature-moment relationship. The model is accurate when span-to-depth ratio L/d > 10. For deep beams (L/d < 5), significant shear deformation exists and Timoshenko beam theory must be used instead.
The governing differential equation of the elastic curve:
The governing differential equation of the elastic curve:
EI·(d²v/dx²) = M(x)
EI·(d³v/dx³) = V(x)
EI·(d⁴v/dx⁴) = −w(x) (distributed load)
Bending stress: σ = M·y / I
Shear stress (Jourawski): τ = V·Q / (I·b)
y = distance from neutral axis | Q = first moment of area above shear plane | b = width at shear plane
EI·(d³v/dx³) = V(x)
EI·(d⁴v/dx⁴) = −w(x) (distributed load)
Bending stress: σ = M·y / I
Shear stress (Jourawski): τ = V·Q / (I·b)
y = distance from neutral axis | Q = first moment of area above shear plane | b = width at shear plane
1.2 Section Properties — All Common Shapes
Rectangle b×d:
I = bd³/12 | Z = bd²/6 | r = d/√12
Solid circle D:
I = πD⁴/64 | Z = πD³/32 | r = D/4
Hollow rectangle (B×D outer, b×d inner):
I = (BD³ − bd³)/12
I-Section (doubly symmetric):
I = B·D³/12 − 2·(B−t_w)·(D−2t_f)³/12
Section modulus: Z = I/y_max
Radius of gyration: r = √(I/A)
I = bd³/12 | Z = bd²/6 | r = d/√12
Solid circle D:
I = πD⁴/64 | Z = πD³/32 | r = D/4
Hollow rectangle (B×D outer, b×d inner):
I = (BD³ − bd³)/12
I-Section (doubly symmetric):
I = B·D³/12 − 2·(B−t_w)·(D−2t_f)³/12
Section modulus: Z = I/y_max
Radius of gyration: r = √(I/A)
1.3 Standard Load Case Formulae
Simply supported, UDL w (kN/m):
M_max = wL²/8 (mid-span)
V_max = wL/2 (at supports)
δ_max = 5wL⁴/(384EI)
Simply supported, central point load P:
M_max = PL/4 | V_max = P/2
δ_max = PL³/(48EI)
Cantilever, UDL w:
M_max = wL²/2 (fixed root)
δ_max = wL⁴/(8EI) (free tip)
Cantilever, point load P at free end:
M_max = PL | δ_max = PL³/(3EI)
Fixed-fixed, UDL w:
M_end = −wL²/12 | M_mid = +wL²/24
δ_max = wL⁴/(384EI)
M_max = wL²/8 (mid-span)
V_max = wL/2 (at supports)
δ_max = 5wL⁴/(384EI)
Simply supported, central point load P:
M_max = PL/4 | V_max = P/2
δ_max = PL³/(48EI)
Cantilever, UDL w:
M_max = wL²/2 (fixed root)
δ_max = wL⁴/(8EI) (free tip)
Cantilever, point load P at free end:
M_max = PL | δ_max = PL³/(3EI)
Fixed-fixed, UDL w:
M_end = −wL²/12 | M_mid = +wL²/24
δ_max = wL⁴/(384EI)
1.4 Deflection Limits — IS 456 / IS 800
Deflection must satisfy two independent criteria under IS 456 Cl.23.2:
Short-term (elastic): calculated using instantaneous E. For steel, E = 200 GPa; for concrete, E_c = 5000√f_ck (MPa).
Long-term (creep + shrinkage): IS 456 applies a total deflection multiplier. EC2 uses effective modulus E_eff = E_c/(1 + φ), where φ is the creep coefficient (typically 1.5–3.0 depending on humidity and age at loading).
IS 456 limits:
— Total deflection ≤ span/250
— Post-construction ≤ span/350 or 20 mm
IS 800 steel beams:
— Live load only ≤ L/300
— Unfactored total ≤ L/250
Deflection governs design far more often than strength for spans > 8 m.
Short-term (elastic): calculated using instantaneous E. For steel, E = 200 GPa; for concrete, E_c = 5000√f_ck (MPa).
Long-term (creep + shrinkage): IS 456 applies a total deflection multiplier. EC2 uses effective modulus E_eff = E_c/(1 + φ), where φ is the creep coefficient (typically 1.5–3.0 depending on humidity and age at loading).
IS 456 limits:
— Total deflection ≤ span/250
— Post-construction ≤ span/350 or 20 mm
IS 800 steel beams:
— Live load only ≤ L/300
— Unfactored total ≤ L/250
Deflection governs design far more often than strength for spans > 8 m.
📐 Worked Example 1.1 — Simply Supported RC Beam
Problem: A simply supported beam, L = 6 m, carries UDL w = 25 kN/m (including self-weight). Section is 230 mm wide × 500 mm overall depth (d_eff = 460 mm), M25 concrete (E_c = 25 GPa), Fe415 steel.
Step 1 — Section properties (gross):
I_gross = bd³/12 = 230 × 500³ / 12 = 2,396 × 10⁶ mm⁴
Z = I/y = 2396 × 10⁶ / 250 = 9.58 × 10⁶ mm³
Step 2 — Internal forces (unfactored SLS):
M_max = wL²/8 = 25 × 6² / 8 = 112.5 kN·m
V_max = wL/2 = 25 × 6 / 2 = 75.0 kN
Step 3 — Bending stress (gross section, SLS):
σ_max = M/Z = 112.5 × 10⁶ / 9.58 × 10⁶ = 11.7 MPa
For concrete in bending: permissible = 8.5 MPa (M25 WSM, IS 456 Annex B) — overstressed on gross section; confirm by limit-state approach and cracked section analysis.
Step 4 — Short-term deflection:
δ = 5wL⁴/(384EI) = 5 × 25 × 6000⁴ / (384 × 25000 × 2396 × 10⁶) = 7.1 mm
Limit = L/250 = 24 mm → PASS ✓
Key lesson: A 230×500 beam at 25 kN/m over 6 m satisfies deflection on gross-section but needs limit-state reinforcement design using factored loads (1.5 × 25 = 37.5 kN/m → M_u = 168.75 kN·m). Check A_st requirement with M_u = 0.87·f_y·A_st·(d − 0.42·x_u).
Step 1 — Section properties (gross):
I_gross = bd³/12 = 230 × 500³ / 12 = 2,396 × 10⁶ mm⁴
Z = I/y = 2396 × 10⁶ / 250 = 9.58 × 10⁶ mm³
Step 2 — Internal forces (unfactored SLS):
M_max = wL²/8 = 25 × 6² / 8 = 112.5 kN·m
V_max = wL/2 = 25 × 6 / 2 = 75.0 kN
Step 3 — Bending stress (gross section, SLS):
σ_max = M/Z = 112.5 × 10⁶ / 9.58 × 10⁶ = 11.7 MPa
For concrete in bending: permissible = 8.5 MPa (M25 WSM, IS 456 Annex B) — overstressed on gross section; confirm by limit-state approach and cracked section analysis.
Step 4 — Short-term deflection:
δ = 5wL⁴/(384EI) = 5 × 25 × 6000⁴ / (384 × 25000 × 2396 × 10⁶) = 7.1 mm
Limit = L/250 = 24 mm → PASS ✓
Key lesson: A 230×500 beam at 25 kN/m over 6 m satisfies deflection on gross-section but needs limit-state reinforcement design using factored loads (1.5 × 25 = 37.5 kN/m → M_u = 168.75 kN·m). Check A_st requirement with M_u = 0.87·f_y·A_st·(d − 0.42·x_u).
🏛️
Chapter 2 — Column Buckling & Axial Capacity
Euler · Perry-Robertson · IS 800 · IS 456
2.1 Euler Buckling — Derivation
Euler (1744) solved the differential equation EI·d²v/dx² + P·v = 0 for a pin-ended strut. The smallest load that produces a deflected (buckled) equilibrium shape is the critical load. Real columns have initial geometric imperfections, residual stresses from rolling/welding, and load eccentricities — so the actual failure load is always less than Euler's theoretical value.
Euler (theoretical, pin-pin):
P_cr = π²EI / L²
σ_cr = π²E / λ²
λ = KL/r (slenderness ratio)
λ_c = π√(2E/f_y) (transition slenderness)
IS 800 Perry-Robertson (Cl.7.1.2):
λ̄ = √(f_y/σ_e) = KL/(rπ√(E/f_y))
φ = 0.5[1 + α(λ̄ − 0.2) + λ̄²]
χ = 1/[φ + √(φ² − λ̄²)] ≤ 1.0
f_cd = χ·f_y/γ_M0 | P_d = f_cd · A
P_cr = π²EI / L²
σ_cr = π²E / λ²
λ = KL/r (slenderness ratio)
λ_c = π√(2E/f_y) (transition slenderness)
IS 800 Perry-Robertson (Cl.7.1.2):
λ̄ = √(f_y/σ_e) = KL/(rπ√(E/f_y))
φ = 0.5[1 + α(λ̄ − 0.2) + λ̄²]
χ = 1/[φ + √(φ² − λ̄²)] ≤ 1.0
f_cd = χ·f_y/γ_M0 | P_d = f_cd · A
2.2 Effective Length Factors K
K modifies the actual length to give the equivalent pin-ended length. A column fixed at both ends has exactly half the effective length — and therefore four times the buckling resistance — of a pin-ended column of the same length.
| End Conditions | K | P_cr vs Pin-Pin |
|---|---|---|
| Fixed – Fixed | 0.50 | 4.0 × |
| Fixed – Pin | 0.70 | 2.04 × |
| Pin – Pin (base) | 1.00 | 1.0 × |
| Fixed – Free (flagpole) | 2.00 | 0.25 × |
In practice, fixity is partial. IS 800 Table 11 gives effective length ratios for columns in various frame types (sway / non-sway). Never assume K = 0.5 unless full fixity is verified by the connection design.
2.3 Buckling Curves — IS 800 / EC3
Residual stresses from manufacturing vary by section type, so IS 800 / EC3 assign an imperfection factor α to each shape:
Curve a (α = 0.21): Hot-rolled I-sections (strong axis), CHS hot-finished
Curve b (α = 0.34): Hot-rolled I (weak axis), welded I (strong axis), RHS/SHS cold-formed
Curve c (α = 0.49): Welded I (weak axis), rolled channels, L-angles, solid rectangles
Curve d (α = 0.76): Welded sections, thick plates t > 100 mm
Higher α → more imperfection-sensitive → more severe strength reduction. This is the physical reason why slender thin-walled welded sections carry less load than stocky hot-rolled ones at the same slenderness.
Curve a (α = 0.21): Hot-rolled I-sections (strong axis), CHS hot-finished
Curve b (α = 0.34): Hot-rolled I (weak axis), welded I (strong axis), RHS/SHS cold-formed
Curve c (α = 0.49): Welded I (weak axis), rolled channels, L-angles, solid rectangles
Curve d (α = 0.76): Welded sections, thick plates t > 100 mm
Higher α → more imperfection-sensitive → more severe strength reduction. This is the physical reason why slender thin-walled welded sections carry less load than stocky hot-rolled ones at the same slenderness.
2.4 IS 456 Short & Slender Columns (RC)
IS 456 Cl.25.1.2 — Short column criterion:
Column is short if l_ex/D ≤ 12 AND l_ey/b ≤ 12
where l_e = effective length, D = depth, b = width.
Short column capacity (IS 456 Cl.39.3):
P_u = 0.4·f_ck·A_c + 0.67·f_y·A_sc
Slender column additional moment:
M_add = P_u·e_a; e_a = (l_e²)/(2000b) for uniaxial
Design total moment M_total = M_primary + M_add and verify using P-M interaction diagram (IS 456 Annex G or SP-16).
Minimum eccentricity (IS 456 Cl.25.4):
e_min = max(L/500 + D/30 , 20 mm)
Column is short if l_ex/D ≤ 12 AND l_ey/b ≤ 12
where l_e = effective length, D = depth, b = width.
Short column capacity (IS 456 Cl.39.3):
P_u = 0.4·f_ck·A_c + 0.67·f_y·A_sc
Slender column additional moment:
M_add = P_u·e_a; e_a = (l_e²)/(2000b) for uniaxial
Design total moment M_total = M_primary + M_add and verify using P-M interaction diagram (IS 456 Annex G or SP-16).
Minimum eccentricity (IS 456 Cl.25.4):
e_min = max(L/500 + D/30 , 20 mm)
📐 Worked Example 2.1 — Steel UC Column (IS 800)
Problem: UC 200×200×60 kg/m (A = 7,600 mm², r_min = 52.0 mm), L_actual = 4.0 m, both ends pin (K = 1.0), Fe345 (f_y = 345 MPa, E = 200 GPa). Applied N = 900 kN. Check adequacy.
Step 1 — Slenderness: λ = KL/r = 1.0 × 4000 / 52 = 76.9 (IS 800 max for compression members = 180 ✓)
Step 2 — Euler stress: σ_e = π²E/λ² = π² × 200,000 / 76.9² = 334.0 MPa
Step 3 — Non-dimensional slenderness: λ̄ = √(345/334.0) = 1.016
Step 4 — Buckling curve c for welded UC (α = 0.49):
φ = 0.5[1 + 0.49(1.016 − 0.2) + 1.016²] = 0.5[1 + 0.400 + 1.032] = 1.216
χ = 1/[1.216 + √(1.216² − 1.016²)] = 1/[1.216 + √(0.445)] = 1/1.883 = 0.531
Step 5 — Design resistance: f_cd = 0.531 × 345 / 1.10 = 166.5 MPa
P_d = 166.5 × 7,600 / 1000 = 1,265 kN > 900 kN → PASS ✓ (UCR = 0.71)
Step 1 — Slenderness: λ = KL/r = 1.0 × 4000 / 52 = 76.9 (IS 800 max for compression members = 180 ✓)
Step 2 — Euler stress: σ_e = π²E/λ² = π² × 200,000 / 76.9² = 334.0 MPa
Step 3 — Non-dimensional slenderness: λ̄ = √(345/334.0) = 1.016
Step 4 — Buckling curve c for welded UC (α = 0.49):
φ = 0.5[1 + 0.49(1.016 − 0.2) + 1.016²] = 0.5[1 + 0.400 + 1.032] = 1.216
χ = 1/[1.216 + √(1.216² − 1.016²)] = 1/[1.216 + √(0.445)] = 1/1.883 = 0.531
Step 5 — Design resistance: f_cd = 0.531 × 345 / 1.10 = 166.5 MPa
P_d = 166.5 × 7,600 / 1000 = 1,265 kN > 900 kN → PASS ✓ (UCR = 0.71)
🔩
Chapter 3 — Isolated Footing Design
IS 456 Cl.31–34 · ACI 318
3.1 Bearing Pressure & Sizing
The footing plan area must keep net soil pressure below the SBC (Safe Bearing Capacity) determined by field plate-load test (IS 1888) or analytically from soil parameters (IS 6403).
Concentric load:
A_req = (P + W_sw) / q_a (m²)
q_net = P / (B × L) (kN/m²)
Biaxial eccentricity (IS 456 Cl.34.2.3):
q = P/(B·L) × (1 ± 6e_x/B ± 6e_y/L)
No-tension condition: e_x ≤ B/6 , e_y ≤ L/6
Typical SBC (IS 1893):
Rock ≥ 330 kN/m² | Gravel 100–300 | Stiff clay 100–200 | Soft clay 50–100
A_req = (P + W_sw) / q_a (m²)
q_net = P / (B × L) (kN/m²)
Biaxial eccentricity (IS 456 Cl.34.2.3):
q = P/(B·L) × (1 ± 6e_x/B ± 6e_y/L)
No-tension condition: e_x ≤ B/6 , e_y ≤ L/6
Typical SBC (IS 1893):
Rock ≥ 330 kN/m² | Gravel 100–300 | Stiff clay 100–200 | Soft clay 50–100
3.2 Punching Shear — IS 456 Cl.31.6
The column tends to punch through the footing slab on a critical perimeter located at d/2 from the column face on all four sides. The punching shear stress must not exceed the allowable:
Critical perimeter: b₀ = 2(b_c + d_c + 2×d/2) + 2(d_c + d/2 + d/2)
= 2(b_c + d) + 2(d_c + d) = 2(b_c + d_c + 2d)
Allowable punching stress:
τ_c = k × 0.25√f_ck (IS 456 Cl.31.6.3)
k = 0.5 + β_c ≤ 1.0 where β_c = shorter / longer column face dimension
Critical perimeter: b₀ = 2(b_c + d_c + 2×d/2) + 2(d_c + d/2 + d/2)
= 2(b_c + d) + 2(d_c + d) = 2(b_c + d_c + 2d)
Allowable punching stress:
τ_c = k × 0.25√f_ck (IS 456 Cl.31.6.3)
k = 0.5 + β_c ≤ 1.0 where β_c = shorter / longer column face dimension
Applied shear: τ_v = V_u_punch / (b₀ × d)
M25 concrete, k=1.0: τ_c = 1.0 × 0.25 × √25 = 1.25 MPa
M30 concrete, k=1.0: τ_c = 1.37 MPa
M25 concrete, k=1.0: τ_c = 1.0 × 0.25 × √25 = 1.25 MPa
M30 concrete, k=1.0: τ_c = 1.37 MPa
3.3 Flexural Steel Design
The footing cantilevers beyond the column face. The critical section for bending is at the face of the column. Factored moment (IS 456 Cl.34.2.3.2):
M_u = 1.5 × q_max × a² / 2
where a = cantilever length = (B/2 − b_c/2) in B-direction
Steel area (limit-state):
A_st = M_u / (0.87 × f_y × 0.9 × d)
(0.9 is lever-arm approximation; iterate for accuracy)
Minimum steel (IS 456 Cl.26.5.2.1):
Fe415: A_st_min = 0.15% bD
Fe500: A_st_min = 0.12% bD
Development length check:
L_d = 47φ (Fe415, M25) — bars must extend L_d beyond column face
M_u = 1.5 × q_max × a² / 2
where a = cantilever length = (B/2 − b_c/2) in B-direction
Steel area (limit-state):
A_st = M_u / (0.87 × f_y × 0.9 × d)
(0.9 is lever-arm approximation; iterate for accuracy)
Minimum steel (IS 456 Cl.26.5.2.1):
Fe415: A_st_min = 0.15% bD
Fe500: A_st_min = 0.12% bD
Development length check:
L_d = 47φ (Fe415, M25) — bars must extend L_d beyond column face
3.4 One-Way Shear & 7-Step Design Procedure
One-way (beam) shear is checked at distance d from the column face per IS 456 Cl.34.2.4. If τ_v > τ_c (from IS 456 Table 19 for given p_t and f_ck), increase effective depth — avoid shear reinforcement in footings.
Complete 7-step design procedure:
1. Get soil SBC; subtract overburden γ_soil × D_f to get net q_a
2. Find A_req = (P + 10% SW) / q_net; choose B, L
3. Assume d = 0.10B as first estimate for 1000–2000 kN loads
4. Compute factored q and check punching shear at d/2
5. Check one-way shear at d from face; adjust d if needed
6. Design flexural steel both directions; check A_st_min
7. Verify eccentricity ratio e/B and e/L ≤ 1/6 (no tension)
Complete 7-step design procedure:
1. Get soil SBC; subtract overburden γ_soil × D_f to get net q_a
2. Find A_req = (P + 10% SW) / q_net; choose B, L
3. Assume d = 0.10B as first estimate for 1000–2000 kN loads
4. Compute factored q and check punching shear at d/2
5. Check one-way shear at d from face; adjust d if needed
6. Design flexural steel both directions; check A_st_min
7. Verify eccentricity ratio e/B and e/L ≤ 1/6 (no tension)
🧱
Chapter 4 — Concrete Mix Design
IS 10262:2019 · ACI 211 · IS 456 Table 3
4.1 Abrams Law & w/c Ratio
Duff Abrams (1918) established empirically that for a given set of materials, concrete compressive strength is inversely proportional to the water-cement ratio. This remains the single most important principle in mix design. A lower w/c also means denser paste, lower permeability and longer durability.
The penalty for adding water is severe: every 10 kg/m³ extra water (to gain workability) reduces 28-day strength by approximately 3–5 MPa. Superplasticisers allow workability to be increased without adding water.
The penalty for adding water is severe: every 10 kg/m³ extra water (to gain workability) reduces 28-day strength by approximately 3–5 MPa. Superplasticisers allow workability to be increased without adding water.
Target mean strength (IS 10262:2019):
f_cm = f_ck + k·s
k = 1.65 (5% defectives) | s = std deviation (MPa)
s = 4 MPa (M20–M25) | 5 MPa (M30–M50)
Max w/c ratios (IS 456 Table 5):
M25: 0.50 | M30: 0.45 | M40: 0.40 | M45: 0.35
Min cement content (IS 456):
M20: 300 kg/m³ | M25: 300 | M30: 320 | M35: 340
f_cm = f_ck + k·s
k = 1.65 (5% defectives) | s = std deviation (MPa)
s = 4 MPa (M20–M25) | 5 MPa (M30–M50)
Max w/c ratios (IS 456 Table 5):
M25: 0.50 | M30: 0.45 | M40: 0.40 | M45: 0.35
Min cement content (IS 456):
M20: 300 kg/m³ | M25: 300 | M30: 320 | M35: 340
4.2 IS 10262:2019 Mix Design Procedure
Step 1: Target mean strength f_cm = f_ck + 1.65s
Step 2: w/c ratio from IS 10262 Table or ACI 211 Table 6.3.4. Use minimum of (strength requirement, durability requirement).
Step 3: Water content from IS 10262 Table 2 — based on max aggregate size and workability (slump). Apply corrections for admixtures (superplasticiser reduces water by 20–30%).
Step 4: Cement content C = W/(w/c). Check C ≥ C_min from IS 456 Table 5.
Step 5: Volume of aggregates from absolute volume method:
1 m³ = C/ρ_c + W/ρ_w + FA/ρ_fa + CA/ρ_ca + Air
Step 6: FA/CA split from IS 10262 Table 3 (fineness modulus of FA, max CA size).
Step 7: Trial mix — adjust for actual aggregate moisture, absorption, and target slump. Confirm with cube tests at 7 and 28 days (IS 516).
Step 2: w/c ratio from IS 10262 Table or ACI 211 Table 6.3.4. Use minimum of (strength requirement, durability requirement).
Step 3: Water content from IS 10262 Table 2 — based on max aggregate size and workability (slump). Apply corrections for admixtures (superplasticiser reduces water by 20–30%).
Step 4: Cement content C = W/(w/c). Check C ≥ C_min from IS 456 Table 5.
Step 5: Volume of aggregates from absolute volume method:
1 m³ = C/ρ_c + W/ρ_w + FA/ρ_fa + CA/ρ_ca + Air
Step 6: FA/CA split from IS 10262 Table 3 (fineness modulus of FA, max CA size).
Step 7: Trial mix — adjust for actual aggregate moisture, absorption, and target slump. Confirm with cube tests at 7 and 28 days (IS 516).
4.3 Workability — Slump vs Application
| Slump (mm) | Application |
|---|---|
| 0 – 25 | Mass gravity dams, roads |
| 25 – 75 | Shallow foundations, lightly reinforced |
| 75 – 125 | Standard RC columns, beams, slabs |
| 125 – 175 | Heavily reinforced sections |
| > 175 | Pumped concrete, SCC |
Each 25 mm increase in slump from water addition reduces strength by ~3 MPa. Use superplasticiser to gain workability without strength loss.
4.4 Exposure Conditions — IS 456 Table 3
Mild: Interior protected. Min M20, max w/c 0.55, cover 20 mm
Moderate: Exterior sheltered, buried in non-aggressive soil. M25, w/c 0.50, cover 30 mm
Severe: Coastal zone, alternate wet/dry, chlorides. M30, w/c 0.45, cover 45 mm
Very Severe: Seawater spray, de-icing salts. M35, w/c 0.40, cover 50 mm
Extreme: Tidal zone, acid attack. M40, w/c 0.35, cover 75 mm
Supplementary cementitious materials (fly ash, GGBS, silica fume) can improve durability while reducing cement content — IS 10262:2019 Annex A allows up to 35% fly ash replacement by mass.
Moderate: Exterior sheltered, buried in non-aggressive soil. M25, w/c 0.50, cover 30 mm
Severe: Coastal zone, alternate wet/dry, chlorides. M30, w/c 0.45, cover 45 mm
Very Severe: Seawater spray, de-icing salts. M35, w/c 0.40, cover 50 mm
Extreme: Tidal zone, acid attack. M40, w/c 0.35, cover 75 mm
Supplementary cementitious materials (fly ash, GGBS, silica fume) can improve durability while reducing cement content — IS 10262:2019 Annex A allows up to 35% fly ash replacement by mass.
⚙️
Chapter 5 — Steel Section Properties & Classification
IS 808 · IS 800:2007 · EN 1993-1-1
5.1 Fundamental Section Parameters
Every steel design starts with five geometric parameters. They govern different aspects of structural behaviour:
A — Area [mm²]: axial capacity, self-weight
I_x, I_y — Moment of inertia [mm⁴]: flexural stiffness EI, deflection
Z_el — Elastic section modulus [mm³]: first yield moment M_el = f_y × Z_el
Z_pl — Plastic section modulus [mm³]: full plastic moment M_p = f_y × Z_pl
r_x, r_y — Radius of gyration [mm]: governs slenderness and buckling
Shape factor f = Z_pl / Z_el: typical values 1.12 (I-section), 1.50 (rectangle), 1.27 (circle)
A — Area [mm²]: axial capacity, self-weight
I_x, I_y — Moment of inertia [mm⁴]: flexural stiffness EI, deflection
Z_el — Elastic section modulus [mm³]: first yield moment M_el = f_y × Z_el
Z_pl — Plastic section modulus [mm³]: full plastic moment M_p = f_y × Z_pl
r_x, r_y — Radius of gyration [mm]: governs slenderness and buckling
Shape factor f = Z_pl / Z_el: typical values 1.12 (I-section), 1.50 (rectangle), 1.27 (circle)
5.2 Section Classification (IS 800 / EC3)
Local plate buckling limits the moment capacity if the compression elements (flange, web) are too thin relative to their width. IS 800 Table 2 / EC3 Table 5.2 define four classes based on width-to-thickness ratios, corrected for yield strength: ε = √(250/f_y)
Class 1 (Plastic): Forms plastic hinge with sufficient rotation capacity for plastic analysis. Flange: c/t_f ≤ 8.4ε. Web: d/t_w ≤ 84ε.
Class 2 (Compact): Reaches M_p but limited rotation. Flange: ≤ 9.4ε. Web: ≤ 105ε.
Class 3 (Semi-compact): Outer fibre reaches f_y but local buckling prevents full plasticity. Use M_el only.
Class 4 (Slender): Local buckling before yield. Requires effective section analysis (reduced area/modulus).
Class 1 (Plastic): Forms plastic hinge with sufficient rotation capacity for plastic analysis. Flange: c/t_f ≤ 8.4ε. Web: d/t_w ≤ 84ε.
Class 2 (Compact): Reaches M_p but limited rotation. Flange: ≤ 9.4ε. Web: ≤ 105ε.
Class 3 (Semi-compact): Outer fibre reaches f_y but local buckling prevents full plasticity. Use M_el only.
Class 4 (Slender): Local buckling before yield. Requires effective section analysis (reduced area/modulus).
5.3 Plastic vs Elastic Moment — IS 800
Elastic (Class 3 or 4):
M_el = f_y × Z_el / γ_M0
Plastic (Class 1 or 2):
M_p = f_y × Z_pl / γ_M0
γ_M0 = 1.10 (IS 800 Table 5)
Example — ISMB 300 (approx values):
Z_el ≈ 573 cm³ | Z_pl ≈ 651 cm³
Shape factor = 651/573 = 1.14
M_el = 250×573,000/1.10 = 130.2 kN·m
M_p = 250×651,000/1.10 = 147.9 kN·m
Gain from plastic design = +13.6%
M_el = f_y × Z_el / γ_M0
Plastic (Class 1 or 2):
M_p = f_y × Z_pl / γ_M0
γ_M0 = 1.10 (IS 800 Table 5)
Example — ISMB 300 (approx values):
Z_el ≈ 573 cm³ | Z_pl ≈ 651 cm³
Shape factor = 651/573 = 1.14
M_el = 250×573,000/1.10 = 130.2 kN·m
M_p = 250×651,000/1.10 = 147.9 kN·m
Gain from plastic design = +13.6%
5.4 Lateral-Torsional Buckling (LTB)
An unrestrained beam can fail by the compression flange buckling sideways before reaching M_p. IS 800 Cl.8.2 defines a LTB reduction factor χ_LT analogous to column buckling, governed by the non-dimensional slenderness λ_LT.
Key parameters:
L_LT — unrestrained compression flange length
r_y — weak-axis radius of gyration
I_w — warping constant [mm⁶]
J — St Venant torsion constant [mm⁴]
Design rule: If λ_LT ≤ 0.4 → χ_LT = 1.0 (no reduction).
Prevention: Lateral restraints to compression flange at ≤ L_y intervals; use rigid purlins, bracing, or composite action.
Key parameters:
L_LT — unrestrained compression flange length
r_y — weak-axis radius of gyration
I_w — warping constant [mm⁶]
J — St Venant torsion constant [mm⁴]
Design rule: If λ_LT ≤ 0.4 → χ_LT = 1.0 (no reduction).
Prevention: Lateral restraints to compression flange at ≤ L_y intervals; use rigid purlins, bracing, or composite action.
💧
Chapter 6 — Pipe Flow & Open Channel Hydraulics
Darcy-Weisbach · Manning · IS 1742 · IS 4111
6.1 Darcy-Weisbach — Pressurised Closed Pipes
The most theoretically rigorous head-loss equation, valid for all flow regimes and pipe sizes. The friction factor f is found from the Moody diagram or directly from the Colebrook-White / Swamee-Jain equations.
Darcy-Weisbach:
h_f = f × (L/D) × (v²/2g)
Re = ρvD/μ = vD/ν
Laminar (Re < 2000): f = 64/Re
Turbulent — Swamee-Jain (explicit, ±1%):
f = 0.25/[log₁₀(ε/3.7D + 5.74/Re⁰·⁹)]²
Roughness ε: New steel 0.046 mm
Concrete 0.3–3 mm | PVC 0.0015 mm
Cast iron 0.26 mm | Galv. steel 0.15 mm
h_f = f × (L/D) × (v²/2g)
Re = ρvD/μ = vD/ν
Laminar (Re < 2000): f = 64/Re
Turbulent — Swamee-Jain (explicit, ±1%):
f = 0.25/[log₁₀(ε/3.7D + 5.74/Re⁰·⁹)]²
Roughness ε: New steel 0.046 mm
Concrete 0.3–3 mm | PVC 0.0015 mm
Cast iron 0.26 mm | Galv. steel 0.15 mm
6.2 Manning's Equation — Gravity Open Channel
Manning's equation is the workhorse of drainage and sewer design. It assumes uniform, steady, full-bore gravity flow where the hydraulic gradient equals the channel bed slope S.
Manning:
Q = (1/n) · A · R^(2/3) · S^(1/2)
v = (1/n) · R^(2/3) · S^(1/2)
R = A/P (hydraulic radius)
Circular (full flow, D):
A = πD²/4 | P = πD | R = D/4
Rectangular (B wide, y deep):
A = By | P = B+2y | R = By/(B+2y)
Trapezoidal (B base, side slope z:1):
A = (B+zy)y | P = B+2y√(1+z²)
n values: PVC 0.009 | Conc. pipe 0.011
Lined channel 0.013 | Earth clean 0.025
Q = (1/n) · A · R^(2/3) · S^(1/2)
v = (1/n) · R^(2/3) · S^(1/2)
R = A/P (hydraulic radius)
Circular (full flow, D):
A = πD²/4 | P = πD | R = D/4
Rectangular (B wide, y deep):
A = By | P = B+2y | R = By/(B+2y)
Trapezoidal (B base, side slope z:1):
A = (B+zy)y | P = B+2y√(1+z²)
n values: PVC 0.009 | Conc. pipe 0.011
Lined channel 0.013 | Earth clean 0.025
6.3 Flow Regimes, Froude Number & Design Velocities
Reynolds number classification:
Re < 2000: Laminar (f = 64/Re)
2000–4000: Transition — avoid in design
Re > 4000: Turbulent (use Moody / Swamee-Jain)
Froude number (open channel):
Fr = v / √(g·y_h) where y_h = hydraulic depth
Fr < 1: Subcritical — normal for sewers and channels
Fr = 1: Critical — maximum discharge per unit energy
Fr > 1: Supercritical — prone to hydraulic jumps
Design velocity limits (IS 1742, IS 4111):
v_min ≥ 0.6 m/s (self-cleaning in sewers)
v_max ≤ 3.0 m/s (erosion, concrete lining)
Water mains economic range: 0.9–1.8 m/s
Re < 2000: Laminar (f = 64/Re)
2000–4000: Transition — avoid in design
Re > 4000: Turbulent (use Moody / Swamee-Jain)
Froude number (open channel):
Fr = v / √(g·y_h) where y_h = hydraulic depth
Fr < 1: Subcritical — normal for sewers and channels
Fr = 1: Critical — maximum discharge per unit energy
Fr > 1: Supercritical — prone to hydraulic jumps
Design velocity limits (IS 1742, IS 4111):
v_min ≥ 0.6 m/s (self-cleaning in sewers)
v_max ≤ 3.0 m/s (erosion, concrete lining)
Water mains economic range: 0.9–1.8 m/s
6.4 Minor Losses & Pipeline Economics
Total head loss = friction loss + minor losses:
h_minor = K · v²/2g
Loss coefficients K (typical):
Sharp entry: K = 0.50
Exit to reservoir: K = 1.00
Gate valve (full open): K = 0.10
Globe valve: K = 6–10
90° elbow (short radius): K = 0.9
Tee (flow through run): K = 0.2
Pipe sizing economics:
Increasing pipe diameter by one nominal size reduces head loss by approximately 60% (since h_f ∝ D⁻⁵). Optimal pipe size minimises (capital cost + pumping energy cost) over the design life, typically 30–50 years.
h_minor = K · v²/2g
Loss coefficients K (typical):
Sharp entry: K = 0.50
Exit to reservoir: K = 1.00
Gate valve (full open): K = 0.10
Globe valve: K = 6–10
90° elbow (short radius): K = 0.9
Tee (flow through run): K = 0.2
Pipe sizing economics:
Increasing pipe diameter by one nominal size reduces head loss by approximately 60% (since h_f ∝ D⁻⁵). Optimal pipe size minimises (capital cost + pumping energy cost) over the design life, typically 30–50 years.
📐 Worked Example 6.1 — Water Main Sizing
Problem: Design a steel water main to carry Q = 0.05 m³/s over L = 500 m with available head h = 10 m. Pipe roughness ε = 0.046 mm, ν = 1.0 × 10⁻⁶ m²/s (water at 20°C).
Trial D = 150 mm (0.15 m):
v = Q/A = 0.05/(π × 0.15²/4) = 2.83 m/s (too high for mains — try D = 200 mm)
D = 200 mm: v = 0.05/(π × 0.04/4) = 1.59 m/s ✓ (within 0.9–1.8 economic range)
Re = 1.59 × 0.20 / 1.0×10⁻⁶ = 318,000 (turbulent)
Friction factor (Swamee-Jain):
f = 0.25/[log₁₀(0.046/(3.7×200) + 5.74/318000⁰·⁹)]² = 0.0172
Head loss: h_f = 0.0172 × (500/0.20) × (1.59²/19.62) = 5.6 m < 10 m available → PASS ✓ (headroom 4.4 m for fittings and growth)
Answer: Specify DN200 welded steel pipe. Check pressure class for operating pressure.
Trial D = 150 mm (0.15 m):
v = Q/A = 0.05/(π × 0.15²/4) = 2.83 m/s (too high for mains — try D = 200 mm)
D = 200 mm: v = 0.05/(π × 0.04/4) = 1.59 m/s ✓ (within 0.9–1.8 economic range)
Re = 1.59 × 0.20 / 1.0×10⁻⁶ = 318,000 (turbulent)
Friction factor (Swamee-Jain):
f = 0.25/[log₁₀(0.046/(3.7×200) + 5.74/318000⁰·⁹)]² = 0.0172
Head loss: h_f = 0.0172 × (500/0.20) × (1.59²/19.62) = 5.6 m < 10 m available → PASS ✓ (headroom 4.4 m for fittings and growth)
Answer: Specify DN200 welded steel pipe. Check pressure class for operating pressure.
🪨
Chapter 7 — Retaining Wall Stability
Rankine · Coulomb · BS 8002 · IS 456
7.1 Rankine Earth Pressure Theory
Rankine (1857) assumed a smooth (frictionless) vertical wall and planar failure surface in the retained soil. His solution is a lower bound on passive resistance and an upper bound on active pressure — therefore conservative for design. For granular soils (c = 0) with horizontal backfill:
K_a = tan²(45° − φ/2) = (1−sinφ)/(1+sinφ)
K_p = tan²(45° + φ/2) = (1+sinφ)/(1−sinφ)
Active pressure at depth z:
p_a(z) = γ·z·K_a − 2c·√K_a
Total active thrust, cohesionless soil:
P_a = ½γH²K_a + q·H·K_a
(acts at H/3 from base for triangular component,
H/2 for uniform surcharge component q)
Typical φ values: Dense gravel 40° | Loose sand 28–32° | Stiff clay 20–25°
K_p = tan²(45° + φ/2) = (1+sinφ)/(1−sinφ)
Active pressure at depth z:
p_a(z) = γ·z·K_a − 2c·√K_a
Total active thrust, cohesionless soil:
P_a = ½γH²K_a + q·H·K_a
(acts at H/3 from base for triangular component,
H/2 for uniform surcharge component q)
Typical φ values: Dense gravel 40° | Loose sand 28–32° | Stiff clay 20–25°
7.2 Three Stability Checks
1. Overturning (IS 456 / BS 8002 FS ≥ 1.5–2.0):
FS_OT = ΣM_restoring / ΣM_overturning
M_r = W_stem·x₁ + W_base·x₂ + W_soil·x₃
M_o = P_a_tri · H/3 + P_a_surcharge · H/2
2. Sliding (FS ≥ 1.5):
FS_sl = (μ · ΣV + P_p) / P_a
μ = tan(δ) ≈ tan(2φ/3) (base on soil)
3. Bearing (≤ q_allowable):
e = B/2 − (M_r − M_o)/ΣV
q_max = ΣV/B · (1 + 6e/B)
q_min = ΣV/B · (1 − 6e/B) ≥ 0 (no tension)
Condition: e ≤ B/6 for no tension at base
FS_OT = ΣM_restoring / ΣM_overturning
M_r = W_stem·x₁ + W_base·x₂ + W_soil·x₃
M_o = P_a_tri · H/3 + P_a_surcharge · H/2
2. Sliding (FS ≥ 1.5):
FS_sl = (μ · ΣV + P_p) / P_a
μ = tan(δ) ≈ tan(2φ/3) (base on soil)
3. Bearing (≤ q_allowable):
e = B/2 − (M_r − M_o)/ΣV
q_max = ΣV/B · (1 + 6e/B)
q_min = ΣV/B · (1 − 6e/B) ≥ 0 (no tension)
Condition: e ≤ B/6 for no tension at base
7.3 Drainage — The Most Critical Detail
The majority of retaining wall failures are caused by inadequate drainage rather than structural inadequacy. Water build-up behind the wall creates hydrostatic pressure of 9.81 kN/m³ per metre depth — far exceeding the soil active pressure increment.
For a 4 m wall with water to the top:
P_water = ½ × 9.81 × 4² = 78.5 kN/m — approximately equal to the entire active soil force for dense sand. Doubling the total lateral force collapses the FS.
Required drainage (BS 8002 / IS 456 SP:24):
— Weepholes ≥ 75 mm dia at ≤ 1.0 m centres in lowest 300 mm
— Granular filter layer (gravel) behind wall stem
— Perforated back drain pipe to daylight
— Geotextile filter to prevent fine migration
Never design assuming drainage unless it is permanently maintained. For critical walls, assume saturated conditions.
For a 4 m wall with water to the top:
P_water = ½ × 9.81 × 4² = 78.5 kN/m — approximately equal to the entire active soil force for dense sand. Doubling the total lateral force collapses the FS.
Required drainage (BS 8002 / IS 456 SP:24):
— Weepholes ≥ 75 mm dia at ≤ 1.0 m centres in lowest 300 mm
— Granular filter layer (gravel) behind wall stem
— Perforated back drain pipe to daylight
— Geotextile filter to prevent fine migration
Never design assuming drainage unless it is permanently maintained. For critical walls, assume saturated conditions.
7.4 Stem Design & Detailing
The wall stem acts as a vertical cantilever fixed at the base. Factored moment at the base of stem of height h_s (from ULS earth pressure):
M_u_stem = 1.5 × [γ·K_a·h_s³/6 + q·K_a·h_s²/2]
Design horizontal reinforcement (tension in back face):
A_st = M_u / (0.87 × f_y × 0.9 × d_stem)
Minimum horizontal steel (IS 456 Cl.26.5.2):
0.12% bD (Fe500) for temperature and shrinkage
Typical detailing for H = 3–5 m walls:
— 16 mm dia. bars @ 150 mm c/c (back face, horizontal)
— Starter bars from base: full lap ≥ L_d = 47φ
— 10 mm bars @ 200 mm c/c (distribution, front face)
— Cover: 50 mm (in soil), 40 mm (protected face)
M_u_stem = 1.5 × [γ·K_a·h_s³/6 + q·K_a·h_s²/2]
Design horizontal reinforcement (tension in back face):
A_st = M_u / (0.87 × f_y × 0.9 × d_stem)
Minimum horizontal steel (IS 456 Cl.26.5.2):
0.12% bD (Fe500) for temperature and shrinkage
Typical detailing for H = 3–5 m walls:
— 16 mm dia. bars @ 150 mm c/c (back face, horizontal)
— Starter bars from base: full lap ≥ L_d = 47φ
— 10 mm bars @ 200 mm c/c (distribution, front face)
— Cover: 50 mm (in soil), 40 mm (protected face)
🌍
Chapter 8 — Earthwork Volume Calculation
Prismoidal Formula · IS 2720 · MORT&H 5th Ed.
8.1 Volume Methods & Their Accuracy
Cross-section areas are computed at regular chainages (typically every 20 m for roads) from levelling surveys or digital terrain models. Volume between two sections is then calculated by integration.
Average End-Area (Prismatoid):
V = L/2 × (A₁ + A₂)
Simple, always overestimates volume
Prismoidal Formula (exact for prismoids):
V = L/6 × (A₁ + 4A_m + A₂)
A_m = area of mid-section (not average of A₁, A₂)
Prismoidal correction to AEA:
C_p = (L/12) × (b₁−b₂)(h₁−h₂) for simple trapezia
Prismoidal formula is 3–5% more accurate. Use when A₁/A₂ > 1.5 or for complex cross-sections.
V = L/2 × (A₁ + A₂)
Simple, always overestimates volume
Prismoidal Formula (exact for prismoids):
V = L/6 × (A₁ + 4A_m + A₂)
A_m = area of mid-section (not average of A₁, A₂)
Prismoidal correction to AEA:
C_p = (L/12) × (b₁−b₂)(h₁−h₂) for simple trapezia
Prismoidal formula is 3–5% more accurate. Use when A₁/A₂ > 1.5 or for complex cross-sections.
8.2 Bank, Loose & Compacted States
The same mass of soil occupies different volumes in three states. Failing to apply conversion factors is one of the most common and costly errors in earthwork contracting.
Bank (in-situ, undisturbed): Reference state. Volumes from cross-sections are bank volume.
Loose (excavated, in truck):
V_loose = V_bank × (1 + swell/100)
Compacted (fill, after rolling):
V_compact = V_bank × (1 − shrinkage/100)
Load factor: LF = V_bank/V_loose
Used to size truck fleet: a 10 m³ truck carries (10 × LF) bank m³
Bank (in-situ, undisturbed): Reference state. Volumes from cross-sections are bank volume.
Loose (excavated, in truck):
V_loose = V_bank × (1 + swell/100)
Compacted (fill, after rolling):
V_compact = V_bank × (1 − shrinkage/100)
Load factor: LF = V_bank/V_loose
Used to size truck fleet: a 10 m³ truck carries (10 × LF) bank m³
Typical values (IS 3764 guidance):
Loose sand: swell 12%, shrink 7%, LF 0.89
Dense clay: swell 25%, shrink 13%, LF 0.80
Weathered rock: swell 35%, shrink 3%, LF 0.74
Loose sand: swell 12%, shrink 7%, LF 0.89
Dense clay: swell 25%, shrink 13%, LF 0.80
Weathered rock: swell 35%, shrink 3%, LF 0.74
8.3 Proctor Compaction & Field Control
The Standard Proctor test (IS 2720 Part 7) determines the Maximum Dry Density (MDD) and Optimum Moisture Content (OMC) — the water content at which maximum compaction is achieved for a given compactive effort.
Modified Proctor (IS 2720 Part 8): 4.5× greater compactive energy, used for heavy compaction (airport runways, major highways).
Field compaction specifications (MORT&H 5th Ed.):
— Subgrade top 500 mm: ≥ 97% of MDD
— Embankment fill: ≥ 95% of MDD
— Layer thickness (loose): 200–250 mm
— Field test: Core cutter (IS 2720 Pt 29) or sand replacement (IS 2720 Pt 28)
Typical MDD / OMC values:
Gravel: 2.0–2.2 t/m³, OMC 7–12%
Sandy clay: 1.75–1.95 t/m³, OMC 10–16%
Heavy clay: 1.5–1.7 t/m³, OMC 18–25%
Modified Proctor (IS 2720 Part 8): 4.5× greater compactive energy, used for heavy compaction (airport runways, major highways).
Field compaction specifications (MORT&H 5th Ed.):
— Subgrade top 500 mm: ≥ 97% of MDD
— Embankment fill: ≥ 95% of MDD
— Layer thickness (loose): 200–250 mm
— Field test: Core cutter (IS 2720 Pt 29) or sand replacement (IS 2720 Pt 28)
Typical MDD / OMC values:
Gravel: 2.0–2.2 t/m³, OMC 7–12%
Sandy clay: 1.75–1.95 t/m³, OMC 10–16%
Heavy clay: 1.5–1.7 t/m³, OMC 18–25%
8.4 Mass Haul & Borrow/Waste
The mass haul curve (Brückner curve) is a plot of cumulative earthwork volume (positive for cut, negative for fill) along the road alignment. It allows the engineer to:
— Find balanced sections where cut exactly fills adjacent fill
— Determine economic haul distance (below which it is cheaper to haul than borrow)
— Identify where borrow pits or spoil tips are needed
Free haul distance: distance within which haulage cost is included in unit rates (typically 50–500 m in Indian contracts).
Economic haul distance: distance beyond which importing borrow is cheaper than hauling excavated material.
MORT&H Cl.301 and 305 govern the specifications for roadway embankment and subgrade preparation including CBR testing requirements for pavement thickness design.
— Find balanced sections where cut exactly fills adjacent fill
— Determine economic haul distance (below which it is cheaper to haul than borrow)
— Identify where borrow pits or spoil tips are needed
Free haul distance: distance within which haulage cost is included in unit rates (typically 50–500 m in Indian contracts).
Economic haul distance: distance beyond which importing borrow is cheaper than hauling excavated material.
MORT&H Cl.301 and 305 govern the specifications for roadway embankment and subgrade preparation including CBR testing requirements for pavement thickness design.
📏
Chapter 9 — Surveying & Road Geometry Design
IRC SP:23 · IRC 66 · IS 4447 · AASHTO
9.1 Horizontal Curve Elements — Full Derivation
A simple circular curve of radius R joins two tangents that meet at the Point of Intersection (PI) with deflection angle Δ (total angle turned through). Key elements derived from circle geometry:
L = π·R·Δ/180 (arc length, m)
T = R·tan(Δ/2) (tangent length — PC to PI)
M = R·(1 − cos(Δ/2)) (mid-ordinate)
E = R·(sec(Δ/2) − 1) (external distance)
C = 2R·sin(Δ/2) (long chord — PC to PT)
D_c = 5730/R (degree of curvature, arc def.)
PC = Point of Curvature (start of curve)
PT = Point of Tangency (end of curve)
Δ in degrees throughout
T = R·tan(Δ/2) (tangent length — PC to PI)
M = R·(1 − cos(Δ/2)) (mid-ordinate)
E = R·(sec(Δ/2) − 1) (external distance)
C = 2R·sin(Δ/2) (long chord — PC to PT)
D_c = 5730/R (degree of curvature, arc def.)
PC = Point of Curvature (start of curve)
PT = Point of Tangency (end of curve)
Δ in degrees throughout
9.2 Minimum Radius — IRC Design Speed
The minimum radius balances centrifugal acceleration against friction and superelevation:
R_min = V² / [127(e/100 + f)]
R_min = V² / [127(e/100 + f)]
| Speed (km/h) | R_min (m) | f coeff |
|---|---|---|
| 120 km/h (NH/Exp.) | 510 | 0.12 |
| 100 km/h | 360 | 0.14 |
| 80 km/h | 230 | 0.15 |
| 60 km/h | 130 | 0.17 |
| 40 km/h | 60 | 0.20 |
e = 7% max superelevation (IRC SP:23). For mountain roads e = 10% permitted.
9.3 Vertical Curves — Crest & Sag
Vertical curves join two grades G1 (%) and G2 (%) with a parabola. Length is governed by sight distance requirements.
A = |G1 − G2| (grade difference, %)
Crest curve — SSD governs (SSD < L):
L = A·SSD² / 658 (h₁=1.2 m, h₂=0.15 m, IRC)
Crest curve — SSD > L (short curve):
L = 2·SSD − 658/A
Sag curve — headlight criterion:
L = A·SSD / (3.5 + SSD/L) (iterate)
≈ A·SSD / 3.5 + SSD (first approximation)
Rate of change comfort limit:
Sag: A/L ≤ 0.6 %/m | Crest: A/L ≤ 0.3 %/m
Crest curve — SSD governs (SSD < L):
L = A·SSD² / 658 (h₁=1.2 m, h₂=0.15 m, IRC)
Crest curve — SSD > L (short curve):
L = 2·SSD − 658/A
Sag curve — headlight criterion:
L = A·SSD / (3.5 + SSD/L) (iterate)
≈ A·SSD / 3.5 + SSD (first approximation)
Rate of change comfort limit:
Sag: A/L ≤ 0.6 %/m | Crest: A/L ≤ 0.3 %/m
9.4 Stopping Sight Distance (SSD) — IRC 66
SSD is the minimum distance a driver needs to perceive a hazard, react, and stop before collision. IRC 66 uses reaction time t = 2.5 s and longitudinal friction f_lon.
SSD formula:
SSD = 0.278·V·t + V²/(254·f_lon)
where V in km/h, t = 2.5 s
IRC 66 SSD values:
V=120 km/h: SSD = 250 m, f = 0.35
V=100 km/h: SSD = 180 m, f = 0.36
V= 80 km/h: SSD = 120 m, f = 0.38
V= 60 km/h: SSD = 80 m, f = 0.40
V= 40 km/h: SSD = 45 m, f = 0.40
Intermediate Sight Distance (ISD): 2 × SSD — used for overtaking sight analysis. Overtaking SSD (OSD) ≈ 8 × SSD.
SSD formula:
SSD = 0.278·V·t + V²/(254·f_lon)
where V in km/h, t = 2.5 s
IRC 66 SSD values:
V=120 km/h: SSD = 250 m, f = 0.35
V=100 km/h: SSD = 180 m, f = 0.36
V= 80 km/h: SSD = 120 m, f = 0.38
V= 60 km/h: SSD = 80 m, f = 0.40
V= 40 km/h: SSD = 45 m, f = 0.40
Intermediate Sight Distance (ISD): 2 × SSD — used for overtaking sight analysis. Overtaking SSD (OSD) ≈ 8 × SSD.
📐 Worked Example 9.1 — Crest Vertical Curve Design
Problem: National Highway design speed 80 km/h. Grade G1 = +3.5%, G2 = −2.0%. Check and design the vertical curve.
Step 1 — Stopping Sight Distance (IRC 66):
SSD = 0.278 × 80 × 2.5 + 80²/(254 × 0.38) = 55.6 + 66.5 = 122.1 m → use 120 m
Step 2 — Grade difference: A = |3.5 − (−2.0)| = 5.5%
Step 3 — Vertical curve length, assume SSD < L:
L = A × SSD² / 658 = 5.5 × 120² / 658 = 120.4 m
Check: SSD = 120 ≈ L = 120.4 → near equal — both formulae give same result here ✓
Step 4 — Rate of change check: A/L = 5.5/120 = 0.046 %/m < 0.30 %/m → comfortable ✓
Answer: Provide L = 125 m parabolic crest curve centred on the PI. Chainage of highest point: x = L·G1/(G1−G2) = 125 × 3.5/5.5 = 79.5 m from PC.
Step 1 — Stopping Sight Distance (IRC 66):
SSD = 0.278 × 80 × 2.5 + 80²/(254 × 0.38) = 55.6 + 66.5 = 122.1 m → use 120 m
Step 2 — Grade difference: A = |3.5 − (−2.0)| = 5.5%
Step 3 — Vertical curve length, assume SSD < L:
L = A × SSD² / 658 = 5.5 × 120² / 658 = 120.4 m
Check: SSD = 120 ≈ L = 120.4 → near equal — both formulae give same result here ✓
Step 4 — Rate of change check: A/L = 5.5/120 = 0.046 %/m < 0.30 %/m → comfortable ✓
Answer: Provide L = 125 m parabolic crest curve centred on the PI. Chainage of highest point: x = L·G1/(G1−G2) = 125 × 3.5/5.5 = 79.5 m from PC.
📋
Standards Reference, Common Design Errors & Glossary
⛔ 12 Common Civil Engineering Design Errors
1. Forgetting long-term creep in RC deflection — always check both short-term and long-term
2. Using gross (uncracked) section for SLS deflection in cracked RC beams
3. Assuming K = 0.5 (fixed-fixed) without verifying connection fixity
4. No drainage behind retaining walls — hydrostatic pressure is the #1 wall failure cause
5. Neglecting eccentricity in footings under combined axial + moment (P + M) loads
6. Confusing bank, loose and compacted volumes when pricing earthwork contracts
7. Using Manning's equation for pressurised pipes — it is invalid for closed-pipe pressure flow
8. Using Darcy-Weisbach for open channels — use Manning or Chezy instead
9. Designing to R_min without verifying superelevation is constructable on the terrain
10. Not applying γ_M0 = 1.10 partial factor to IS 800 steel resistance values
11. Adding water to gain concrete slump — each 10 kg/m³ extra water ≈ −3 to −5 MPa strength
12. Designing RC slender columns without adding IS 456 Cl.39.7 additional moment M_add
2. Using gross (uncracked) section for SLS deflection in cracked RC beams
3. Assuming K = 0.5 (fixed-fixed) without verifying connection fixity
4. No drainage behind retaining walls — hydrostatic pressure is the #1 wall failure cause
5. Neglecting eccentricity in footings under combined axial + moment (P + M) loads
6. Confusing bank, loose and compacted volumes when pricing earthwork contracts
7. Using Manning's equation for pressurised pipes — it is invalid for closed-pipe pressure flow
8. Using Darcy-Weisbach for open channels — use Manning or Chezy instead
9. Designing to R_min without verifying superelevation is constructable on the terrain
10. Not applying γ_M0 = 1.10 partial factor to IS 800 steel resistance values
11. Adding water to gain concrete slump — each 10 kg/m³ extra water ≈ −3 to −5 MPa strength
12. Designing RC slender columns without adding IS 456 Cl.39.7 additional moment M_add
Glossary of Key Symbols & Abbreviations
E — Young's modulus (GPa)
I — Second moment of area (mm⁴)
Z_el — Elastic section modulus (mm³)
Z_pl — Plastic section modulus (mm³)
r — Radius of gyration (mm)
λ — Slenderness ratio KL/r
λ̄ — Non-dimensional slenderness (IS 800)
χ — Buckling reduction factor
γ_M0 — Partial safety factor = 1.10 (IS 800)
f_y — Steel yield strength (MPa)
f_ck — Concrete char. strength (MPa)
w/c — Water-cement ratio (by mass)
K_a — Active earth pressure coefficient
K_p — Passive earth pressure coefficient
R — Hydraulic radius A/P (m)
Fr — Froude number v/√(g·y)
Re — Reynolds number ρvD/μ
SBC — Safe Bearing Capacity (kN/m²)
SSD — Stopping Sight Distance (m)
FS — Factor of Safety
MDD — Maximum Dry Density (t/m³)
OMC — Optimum Moisture Content (%)
I — Second moment of area (mm⁴)
Z_el — Elastic section modulus (mm³)
Z_pl — Plastic section modulus (mm³)
r — Radius of gyration (mm)
λ — Slenderness ratio KL/r
λ̄ — Non-dimensional slenderness (IS 800)
χ — Buckling reduction factor
γ_M0 — Partial safety factor = 1.10 (IS 800)
f_y — Steel yield strength (MPa)
f_ck — Concrete char. strength (MPa)
w/c — Water-cement ratio (by mass)
K_a — Active earth pressure coefficient
K_p — Passive earth pressure coefficient
R — Hydraulic radius A/P (m)
Fr — Froude number v/√(g·y)
Re — Reynolds number ρvD/μ
SBC — Safe Bearing Capacity (kN/m²)
SSD — Stopping Sight Distance (m)
FS — Factor of Safety
MDD — Maximum Dry Density (t/m³)
OMC — Optimum Moisture Content (%)
⚠ Calculator Scope & Engineering Disclaimer
All calculators in MULTICALCI.COM are first-pass sizing and checking tools. They implement simplified analytical models that are appropriate for preliminary design, learning and quick verification — they are not a substitute for rigorous analysis, geotechnical investigation, specialist software or licensed engineering sign-off. Specific limitations by module:
Beam/Column: Linear elastic analysis only. Does not cover seismic / wind dynamic effects, second-order P-Δ moments, beam-column interaction (combined axial + bending), connection design, or fatigue. Use STAAD.Pro, ETABS, SAP2000, or Robot for final design.
Footing: Isolated pad footings only. Combined footings, strip footings, raft foundations and pile foundations require dedicated analysis. Settlement (consolidation, Terzaghi) is not computed — this is critical for soft clay sites.
Concrete Mix: Simplified volume-method proportioning. Actual site mix requires trial batches, slump tests and compressive strength tests at 7 and 28 days per IS 516. Aggregate moisture corrections must be applied in the field.
Pipe Hydraulics: Uniform, steady flow assumed. Transient/water-hammer analysis (pipe surge, valve closure) requires AFT Impulse, HAMMER, or EPANET. Wave speed and surge pressure are not computed.
Retaining Wall: Rankine active pressure for horizontal backfill, cohesionless soil. Inclined backfill, cohesive soils, seismic loading (IS 1893 Part 1 Annex), surcharge on slopes and global slope stability require Coulomb / log-spiral / slip circle (SLOPE/W) analysis.
Earthwork: Simplified cross-section area inputs. Real earthwork requires survey data imported into specialist software (Civil 3D, 12d, TOPCON tools) with terrain modelling.
All results are indicative — verify with certified engineering documents and a licenced civil or structural engineer before any construction or procurement decision.
Beam/Column: Linear elastic analysis only. Does not cover seismic / wind dynamic effects, second-order P-Δ moments, beam-column interaction (combined axial + bending), connection design, or fatigue. Use STAAD.Pro, ETABS, SAP2000, or Robot for final design.
Footing: Isolated pad footings only. Combined footings, strip footings, raft foundations and pile foundations require dedicated analysis. Settlement (consolidation, Terzaghi) is not computed — this is critical for soft clay sites.
Concrete Mix: Simplified volume-method proportioning. Actual site mix requires trial batches, slump tests and compressive strength tests at 7 and 28 days per IS 516. Aggregate moisture corrections must be applied in the field.
Pipe Hydraulics: Uniform, steady flow assumed. Transient/water-hammer analysis (pipe surge, valve closure) requires AFT Impulse, HAMMER, or EPANET. Wave speed and surge pressure are not computed.
Retaining Wall: Rankine active pressure for horizontal backfill, cohesionless soil. Inclined backfill, cohesive soils, seismic loading (IS 1893 Part 1 Annex), surcharge on slopes and global slope stability require Coulomb / log-spiral / slip circle (SLOPE/W) analysis.
Earthwork: Simplified cross-section area inputs. Real earthwork requires survey data imported into specialist software (Civil 3D, 12d, TOPCON tools) with terrain modelling.
All results are indicative — verify with certified engineering documents and a licenced civil or structural engineer before any construction or procurement decision.