01
Stress & Strain
Stress — Kya Hota Hai?
HIGH PRIORITY
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Jab kisi body pe external force lagta hai, toh internal resistance develop hoti hai. Yeh internal resistance per unit area hi Stress hai. Simple words mein — material ke andar kitna "khicha" hai.
Normal Stress (σ)
σ = P / A
P = Force (N), A = Cross-section area (m²)
Shear Stress (τ)
τ = V / A
V = Shear Force, A = Area parallel to force
Normal Strain (ε)
ε = δL / L
δL = change in length, L = original length. Dimensionless.
Shear Strain (γ)
γ = τ / G
G = Modulus of Rigidity. Angle of distortion (radians).
Thermal Stress
σ = E·α·ΔT
α = coefficient of thermal expansion, ΔT = temp change. Only when expansion is restricted.
Factor of Safety
FOS = σ_yield / σ_working
Always > 1. Static loading: 2–4. Dynamic: 6–12.
IIT Tip: Thermal stress ka formula galat likhna common mistake hai — ΔT ke saath E aur α dono multiply hote hain. Units: σ in Pa (N/m²) ya MPa (N/mm²). 1 MPa = 1 N/mm².
Memory Trick: "PAL" — P over A gives Length stress. σ = P/A, ε = δL/L. Dono mein numerator "change" hai, denominator "original" hai.
02
Elastic Constants
E, G, K, μ — Relationships
HIGH PRIORITY
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Yeh 4 elastic constants ek material ki stiffness aur deformation behavior define karte hain. GATE mein relationship formulas directly pooche jaate hain — har ek yaad hona chahiye.
Young's Modulus (E)
E = σ / ε
Tensile stress ÷ Tensile strain. Steel ≈ 200 GPa, Al ≈ 70 GPa.
Poisson's Ratio (μ or ν)
μ = −ε_lateral / ε_axial
Range: 0 to 0.5. Steel ≈ 0.3, Rubber ≈ 0.5, Cork ≈ 0.
Modulus of Rigidity (G)
G = E / [2(1+μ)]
Shear modulus. Steel ≈ 80 GPa.
Bulk Modulus (K)
K = E / [3(1−2μ)]
Volumetric stress ÷ Volumetric strain. When μ = 0.5, K = ∞ (incompressible).
E, G, K Relation
E = 9KG / (3K+G)
Master formula — connects all 3 moduli.
Lateral Strain
ε_lat = −μ · σ/E
Negative sign — lateral contraction when axially stretched.
Common Mistake: Bulk modulus formula mein (1−2μ) hai, Modulus of Rigidity mein (1+μ) hai. Dono yaad karo, confuse mat ho.
GATE Fact: μ kabhi 0 se kam ya 0.5 se zyada nahi ho sakta real materials ke liye. Agar problem mein μ = 0.5 diya hai, toh K → ∞ (incompressible material).
03
SFD & BMD
Shear Force & Bending Moment Diagrams
HIGH PRIORITY
▼
Beam analysis ka foundation. SFD aur BMD se pata chalta hai beam ke kis hisse pe maximum stress hai. GATE mein 2-3 questions har saal aate hain — sirf cases yaad karo.
Relationship — Key Rule
dV/dx = −w(x)
w = distributed load (N/m). SFD slope = −UDL intensity.
BM-SF Relationship
dM/dx = V
BMD slope = Shear Force value at that point.
Simply Supported + Central Load
M_max = WL/4
At centre. V = ±W/2 throughout.
Simply Supported + UDL
M_max = wL²/8
At centre. Parabolic BMD. V_max = wL/2 at supports.
Cantilever + Point Load (tip)
M_max = WL
At fixed end. V = W throughout length.
Cantilever + UDL
M_max = wL²/2
At fixed end. Parabolic BMD.
Golden Rule: Jahan V = 0, wahan M = maximum. Yeh ek line GATE ke 90% SFD-BMD problems solve kar deti hai.
Pattern: Point load → SFD mein jump, BMD mein kink (triangle shape). UDL → SFD linear slope, BMD parabolic. Concentrated moment → BMD mein jump.
04
Bending Stress
Flexure Formula & Section Modulus
HIGH PRIORITY
▼
Beam bending ke waqt neutral axis ke upar compression, neeche tension hota hai (ya ulta). Yeh distribution linear hoti hai — neutral axis pe zero, extreme fibres pe maximum. Yahi Flexure Formula keh ta hai.
Bending Equation (Flexure Formula)
M/I = σ/y = E/R
M = BM, I = MI, y = dist from NA, R = radius of curvature.
Bending Stress
σ = M·y / I
Maximum at extreme fibre where y = y_max.
Section Modulus (Z)
Z = I / y_max
σ_max = M/Z. Higher Z = stronger section for same M.
Rectangular Section — I
I = bd³/12
b = width, d = depth. Z = bd²/6.
Circular Section — I
I = πd⁴/64
Z = πd³/32. Hollow: I = π(D⁴−d⁴)/64.
Radius of Curvature
R = EI / M
Larger EI = stiffer beam = larger R = less curvature.
Vaibhav's Tip: I-section zyada efficient kyun hai? Kyunki zyada material extreme fibre ke paas (high y) rakhta hai — maximum Z milta hai same weight mein. Yahi real beams mein I-section use hone ki wajah hai.
05
Shear Stress in Beams
Shear Stress Distribution
MEDIUM PRIORITY
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Bending stress linear hoti hai, but shear stress parabolic hoti hai. Neutral axis pe shear stress maximum hota hai (bending stress = 0 wahan), extreme fibres pe zero. Yeh concept confuse karta hai beginners ko.
General Shear Stress Formula
τ = VQ / Ib
V = SF, Q = first moment of area above the point, b = width at that level.
Rectangular Section — τ_max
τ_max = 1.5 × V/A
At neutral axis. 1.5 times average shear stress.
Circular Section — τ_max
τ_max = 4V / 3A
= 4/3 × average shear stress. At neutral axis.
τ_max / τ_avg Ratios
Rect: 1.5 | Circ: 1.33
Diamond: 1.5 at NA (same as rect, but at 45° orientation check separately).
Yaad karo: "1.5 for Rectangle, 4/3 for Circle." GATE mein directly puchha jaata hai — "rectangular section mein max shear stress avg ka kitna guna?"
06
Torsion
Torsion of Circular Shafts
HIGH PRIORITY
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Shaft pe twisting moment (torque) lagane se shear stress develop hoti hai. Yeh Flexure formula ki tarah hi hai — bas M ki jagah T, I ki jagah J, aur σ ki jagah τ. Yeh analogy yaad raho, sab easy ho jaata hai.
Torsion Equation
T/J = τ/r = Gθ/L
T = Torque, J = Polar MI, r = radius, θ = angle of twist.
Shear Stress due to Torsion
τ_max = T·R / J
R = outer radius. Max at surface, zero at centre.
Polar MI — Solid Shaft
J = πd⁴ / 32
Note: J = 2I (for circular section). J = πR⁴/2.
Polar MI — Hollow Shaft
J = π(D⁴−d⁴) / 32
D = outer dia, d = inner dia.
Angle of Twist
θ = TL / GJ
θ in radians. GJ = torsional rigidity.
Power Transmitted
P = 2πNT / 60
P in Watts, N in RPM, T in N·m.
Hollow vs Solid Shaft: Same weight mein hollow shaft zyada torque carry kar sakta hai. Why? Kyunki material outer radius ke paas hota hai jahan τ max hoti hai. Real shafts (drive shafts, axles) hollow kyun hote hain — yahi reason hai.
Common Mistake: J = πd⁴/32 (polar), I = πd⁴/64 (planar). Dono mein difference sirf 32 vs 64 hai. Torsion mein J use karo, Bending mein I. J = 2I for circular section.
07
Beam Deflection
Deflection Formulas — Standard Cases
HIGH PRIORITY
▼
GATE mein deflection formulas directly diye jaate hain ya directly pooche jaate hain. Yeh standard cases ratta maaro — derivation nahi chahiye, values yaad karo.
| Beam Type | Load | Max Deflection (δ) | Location |
|---|---|---|---|
| Cantilever | Point load W at tip | WL³ / 3EI | At free end |
| Cantilever | UDL w over full length | wL⁴ / 8EI | At free end |
| Simply Supported | Central point load W | WL³ / 48EI | At centre |
| Simply Supported | UDL w over full length | 5wL⁴ / 384EI | At centre |
| Fixed Both Ends | Central point load W | WL³ / 192EI | At centre |
Shortcut: Cantilever point load = WL³/3EI. Simply supported central = WL³/48EI. Ratio = 48/3 = 16. Simply supported is 16 times stiffer than cantilever. Interview mein poochha jaata hai yeh!
δ → m or mm
E → N/m² or MPa
I → m⁴ or mm⁴
W → N
L → m or mm
08
Columns & Buckling
Euler's Buckling — End Conditions
MEDIUM PRIORITY
▼
Lambi aur patli column pe compressive load lagate hain toh woh suddenly side mein mur jaati hai — yahi buckling hai. Euler ka formula critical load deta hai. End conditions se effective length badalta hai — yeh GATE ka favourite topic hai.
Euler's Critical Load
P_cr = π²EI / L_e²
L_e = effective length. Buckling tab hoga jab P ≥ P_cr.
Both Ends Pinned
L_e = L
Standard case. P_cr = π²EI/L².
Both Ends Fixed
L_e = L/2
Strongest case. P_cr = 4π²EI/L² (4× pinned-pinned).
One Fixed, One Free
L_e = 2L
Weakest case. P_cr = π²EI/4L² (1/4× pinned-pinned).
One Fixed, One Pinned
L_e = 0.7L
P_cr = 2π²EI/L² (approx 2× pinned-pinned).
Slenderness Ratio
λ = L_e / k
k = √(I/A) = radius of gyration. Long column: λ > 120.
GATE Pattern: "Column A ka critical load kya hai agar end conditions change ho?" — Directly effective length formula se karo. Both fixed = 4× stronger. One fixed free = 4× weaker. Yeh ratio yaad karo.
09
Strain Energy
Resilience, Toughness & Energy Methods
MEDIUM PRIORITY
▼
Jab material deform hota hai, energy store hoti hai usmein — yahi strain energy hai. Ye concept springs, impact loading, aur Castigliano's theorem mein use hoti hai.
Strain Energy (Axial)
U = σ²V / 2E
= P²L / 2AE. V = volume, A = area, L = length.
Modulus of Resilience
u_r = σ_y² / 2E
Energy per unit volume upto elastic limit. Springs chahiye? High resilience material lao.
Strain Energy in Bending
U = ∫M²dx / 2EI
Integrate over full beam length.
Strain Energy in Torsion
U = T²L / 2GJ
Compare with axial: P²L/2AE. Same structure!
Castigliano's Theorem
δ = ∂U / ∂P
Deflection = partial derivative of strain energy w.r.t. that load.
Modulus of Toughness
u_t = area under σ-ε curve
Total energy per unit volume upto fracture. Toughness ≠ Hardness.
Resilience vs Toughness: Resilience = energy stored elastically (springs need this). Toughness = total energy before fracture (impact tools need this). High hardness ≠ High toughness — yeh GATE mein puchha jaata hai.
⚡
GATE Quick Reference Sheet
SOM — ONE PAGE REVISION
// Print karo, wall pe lagao, exam se 1 din pehle dekho
Normal Stress
σ = P/A
Normal Strain
ε = δL/L
Young's Modulus
E = σ/ε
Shear Modulus
G = E/2(1+μ)
Bulk Modulus
K = E/3(1-2μ)
E, G, K Link
E = 9KG/(3K+G)
Thermal Stress
σ = EαΔT
Flexure Formula
M/I = σ/y = E/R
Section Modulus
Z = I/y_max
Rect MI (bending)
I = bd³/12
Circ MI (bending)
I = πd⁴/64
Polar MI (torsion)
J = πd⁴/32
Torsion Equation
T/J = τ/r = Gθ/L
Angle of Twist
θ = TL/GJ
Power & Torque
P = 2πNT/60
τ_max (Rectangle)
τ = 1.5 × V/A
τ_max (Circle)
τ = 4V/3A
SS + Central Load
δ = WL³/48EI
Cantilever + Load
δ = WL³/3EI
SS + UDL
δ = 5wL⁴/384EI
Euler Critical Load
P_cr = π²EI/Le²
Slenderness Ratio
λ = Le/k
Strain Energy
U = σ²V/2E
Resilience
u_r = σ_y²/2E
STANDARD VALUES — YAAD KARO
// GATE mein directly diye nahi jaate, yaad hone chahiye
| Material | E (GPa) | G (GPa) | μ | α (×10⁻⁶/°C) |
|---|---|---|---|---|
| Mild Steel | 200–210 | 80 | 0.28–0.30 | 12 |
| Aluminium | 70 | 26 | 0.33 | 23 |
| Copper | 120 | 45 | 0.34 | 17 |
| Cast Iron | 100–170 | 40–70 | 0.26 | 10–12 |
| Rubber | ~0.001–0.01 | — | ~0.5 | — |