Solid Mechanics (Strength of Materials) – Complete Study Notes

Comprehensive chapter-wise notes covering every aspect of Solid Mechanics — elastic constants, stress and strain, Mohr's circle, principal stresses, theories of failure, bending, shear, torsion, deflection of beams, pressure vessels, columns and shear centre. All formulae, IS code references, SVG diagrams, solved examples and exam-focused tables included.

GATE ESE / IES SSC JE State PSC RRB JE

Ch 1 · Properties of Materials Ch 2 · Stress-Strain & Elastic Constants Ch 3 · SF & BM Diagrams Ch 4 · Principal Stress & Failure Theories Ch 5 · Deflection of Beams Ch 6 · Bending & Shear Stresses Ch 7 · Thick & Thin Cylinders Ch 8 · Torsion & Springs Ch 9 · Columns & Retaining Walls Ch 10 · Shear Centre & MI ★ Quick Revision

1Properties of Materials►

1.1 Mechanical Properties

Property	Definition	Unit / Symbol	Exam Relevance
Elasticity	Ability to regain original shape on removal of load	—	Basis of Hooke's law
Plasticity	Ability to retain deformation after load removal	—	Plastic hinge concept in RCC
Ductility	Ability to undergo large plastic deformation before fracture	% elongation; % reduction in area	Mild steel ductile; cast iron brittle
Brittleness	Fracture with little or no plastic deformation	—	Concrete, cast iron, glass
Toughness	Energy absorbed per unit volume up to fracture (area under σ–ε curve)	J/m³ or N·m/m³	Charpy/Izod impact tests
Resilience	Energy stored per unit volume within elastic limit = σ_y²/(2E)	J/m³	Springs, proof resilience
Hardness	Resistance to surface indentation/scratch	BHN, Vickers HV, Rockwell	Wear resistance
Creep	Slow, time-dependent plastic deformation under sustained stress below yield	Creep rate ε̇	High-temp applications; PSC losses
Fatigue	Failure under repeated cyclic loading below UTS	Endurance limit σ_e	S-N curve; notch sensitivity
Malleability	Ability to be hammered into thin sheets	—	Gold > Silver > Copper > Iron

1.2 Stress–Strain Curve for Mild Steel

Fig. 1.1 — Stress–Strain Curve for Mild Steel: O→A elastic; A upper yield, B lower yield; B→C strain hardening; C→D necking; D fracture

1.3 Key Points on Stress–Strain Curve

Point / Zone	Description	Key Value (typical)
Proportional limit	Hooke's law holds exactly (σ ∝ ε)	Below yield point
Elastic limit	Max stress with full elastic recovery (slightly above proportional limit)	≈ Proportional limit for metals
Upper yield point	Sudden drop; dislocations break free from solute atoms (Lüder's bands)	~250 MPa (Fe415 rebar)
Lower yield point	Stable plateau of plastic flow	~240 MPa
Ultimate Tensile Stress (UTS)	Maximum engineering stress; necking begins	~415 MPa (Fe415)
Fracture point	Actual failure; true stress > engineering stress	Ductile: cup-cone fracture

1.4 Factor of Safety and Proof Stress

Factor of Safety (FOS) = Ultimate Stress / Permissible (Working) Stress

Proof Stress (0.2% offset): stress at which 0.2% permanent strain occurs
— Used for materials without a well-defined yield point (Al alloys, HYSD bars)

Resilience (Modulus) = σ_y² / (2E) [energy stored per unit volume at yield]
Toughness ≈ area under complete σ–ε curve ≈ σ_avg × ε_fracture

1.5 Material Comparison

Property	Mild Steel	HYSD Steel (Fe415)	Concrete (M25)	Cast Iron
E (GPa)	200–210	200	25 (≈5000√f_ck MPa)	100–170
σ_y (MPa)	250	415	25 (f_ck)	No yield
UTS (MPa)	400–500	485 min	—	140–350
Behaviour	Ductile, defined yield	Ductile, no clear yield	Brittle in tension	Brittle
Poisson's ratio ν	0.25–0.30	0.25–0.30	0.15–0.20	0.25

📝 GATE Tip: E for steel = 2×10⁵ MPa (200 GPa). E for concrete = 5000√f_ck MPa (IS 456). Poisson's ratio for steel ν = 0.3; for concrete ν = 0.15 to 0.2. These values appear in almost every numerical question.

2Simple Stress–Strain & Elastic Constants►

2.1 Stress and Strain Definitions

Normal Stress: σ = P / A (Pa = N/m²; MPa = N/mm²)
Shear Stress: τ = V / A (shear force / area)
Normal Strain: ε = δ / L (dimensionless; δ = deformation, L = original length)
Shear Strain: γ = τ / G (angular distortion in radians)
Volumetric Strain: e = ΔV/V = ε_x + ε_y + ε_z (for triaxial loading)

2.2 Elastic Constants

Constant	Symbol	Definition	Typical Range (steel)
Young's Modulus	E	σ / ε (axial stress / axial strain)	200–210 GPa
Shear Modulus (Rigidity)	G	τ / γ (shear stress / shear strain)	80–82 GPa
Bulk Modulus	K	Hydrostatic stress / volumetric strain = σ / e	160–170 GPa
Poisson's Ratio	ν	Lateral strain / longitudinal strain (magnitude)	0.25–0.30

2.3 Inter-Relations Between Elastic Constants

G = E / [2(1 + ν)]
K = E / [3(1 − 2ν)]
E = 9KG / (3K + G)
ν = (3K − 2G) / (2(3K + G))

Limits of ν:
ν = 0 → K = E/3, G = E/2 (cork — no lateral strain)
ν = 0.5 → K → ∞ (incompressible material, rubber)
ν < 0 → auxetic material (rare, expands laterally under tension)
Theoretical range: −1 ≤ ν ≤ 0.5 (for isotropic elastic materials)

⭐ Key memory: G = E/[2(1+ν)] and K = E/[3(1−2ν)]. These two are the most tested inter-relations. For steel: E ≈ 2×10⁵ MPa, G ≈ 0.8×10⁵ MPa, ν ≈ 0.25 → G = 2×10⁵/[2×1.25] = 0.8×10⁵ ✓

2.4 Generalised Hooke's Law (3D)

ε_x = (1/E)[σ_x − ν(σ_y + σ_z)]
ε_y = (1/E)[σ_y − ν(σ_x + σ_z)]
ε_z = (1/E)[σ_z − ν(σ_x + σ_y)]

γ_xy = τ_xy / G; γ_yz = τ_yz / G; γ_zx = τ_zx / G

Volumetric strain: e = (1 − 2ν)(σ_x + σ_y + σ_z)/E = σ_hydro / K

2.5 Thermal Stress

Free expansion: δ_T = α · ΔT · L (no stress if unconstrained)

Thermal stress (fully constrained bar):
σ_T = E · α · ΔT (compressive if temperature rises)

Partially constrained (support settles by δ < δ_T):
σ = E · (α·ΔT·L − δ) / L

Fig. 2.1 — Fully constrained bar: thermal stress σ = EαΔT

Fig. 2.2 — Composite bar: equal deformation compatibility

2.6 Composite Bars and Statically Indeterminate Axial Problems

Compatibility (same deformation): δ_1 = δ_2
→ P_1·L/(A_1·E_1) = P_2·L/(A_2·E_2)
→ P_1/P_2 = A_1·E_1 / (A_2·E_2)

Equilibrium: P_external = P_1 + P_2 (if parallel)

Modular ratio: m = E_steel / E_concrete (used in RCC transformed section)
m = 280/(3σ_cbc) per IS 456 [WSM]

2.7 Plane Stress and Plane Strain

Condition	Plane Stress	Plane Strain
Definition	σ_z = τ_yz = τ_zx = 0; thin plates/discs	ε_z = γ_yz = γ_zx = 0; long dams, tunnels, retaining walls
σ_z	0 (given)	ν(σ_x + σ_y) — induced by constraint
ε_z	−ν(σ_x+σ_y)/E (non-zero)	0 (given by condition)
Effective E & ν	E, ν	E/(1−ν²), ν/(1−ν)
Examples	Thin plates, flanges under in-plane loads	Long dams, tunnels, wide beams

📝 GATE Tip: Plane stress → σ_z = 0 but ε_z ≠ 0. Plane strain → ε_z = 0 but σ_z ≠ 0. This is a classic source of confusion and is directly tested.

3Shear Force and Bending Moment Diagrams►

3.1 Sign Convention

ℹ️ Standard (Sagging Positive): Sagging BM is positive (+ve). Shear force: left portion has upward force → +ve SF.

Relationship between load, SF and BM:
dV/dx = −w(x) [rate of change of SF = negative of distributed load]
dM/dx = V(x) [rate of change of BM = SF]
d²M/dx² = −w(x)

Consequence: Where V = 0 → BM is maximum (or minimum)

3.2 Standard SFD/BMD Results

Beam Type	Loading	Max SF	Max BM	Location of Max BM
SS beam	Central point load P	P/2	PL/4	Midspan
SS beam	UDL w over full span	wL/2	wL²/8	Midspan
SS beam	Point load P at 'a' from A	Pb/L (at A), Pa/L (at B)	Pab/L	Under load
Cantilever	Point load P at free end	P	PL (hogging)	Fixed end
Cantilever	UDL w over full span	wL	wL²/2 (hogging)	Fixed end
Cantilever	UVL (0 at free, w at fixed)	wL/2	wL²/6	Fixed end
SS beam	UVL (0 at A, w at B)	wL/6 at A, wL/3 at B	wL²/(9√3) at x=L/√3	x = L/√3 from A

3.3 SFD/BMD Shapes for Standard Loads

Load type → SFD shape → BMD shape:
No load (span between loads) → Constant SF → Linear BM
Point load → Step (jump) in SF → Kink in BM
UDL → Linear SF → Parabolic (2nd degree) BM
UVL → Parabolic SF → Cubic (3rd degree) BM
Applied moment M₀ at a point → No change in SF → Step (jump) in BM

3.4 SFD and BMD — Simply Supported Beam with UDL

Fig. 3.1 — SS beam, UDL w: SFD linear (±wL/2), BMD parabolic (max = wL²/8 at midspan)

3.5 Point of Contraflexure

The point of contraflexure is where the bending moment changes sign (BM = 0, and the beam changes from sagging to hogging). It exists only in beams where the BMD crosses the zero axis.

For propped cantilever (fixed at A, roller at B), UDL w:
MA = −wL²/8 (hogging), RA = 3wL/8, RB = 5wL/8
BM = 0 (at support B) and at x = 3L/4 from fixed end A → point of contraflexure

3.6 Relationship Between Load, SF, and BM — Key Rules

The SFD between two point loads is horizontal (constant)
Under a UDL, SFD is linear, BMD is parabolic
At a free end → V = 0 and M = 0 (unless a point load/moment is applied)
At a roller/pin → M = 0 (but V may be non-zero)
At a fixed support → V and M are both non-zero reactions
A couple (moment M₀) applied at a point → SFD unchanged, BMD jumps by M₀

4Principal Stress–Strain & Theories of Failure►

4.1 Stress Transformation (2D)

On an inclined plane at angle θ to the x-face:

σ_θ = (σ_x + σ_y)/2 + (σ_x − σ_y)/2 · cos2θ + τ_xy · sin2θ
τ_θ = −(σ_x − σ_y)/2 · sin2θ + τ_xy · cos2θ

Principal stresses (τ = 0):
σ₁, σ₂ = (σ_x + σ_y)/2 ± √[(σ_x − σ_y)²/4 + τ_xy²]

Max shear stress: τ_max = √[(σ_x − σ_y)²/4 + τ_xy²] = (σ₁ − σ₂)/2

Angle of principal planes: tan(2θ_p) = 2τ_xy / (σ_x − σ_y)
Principal planes are at 90° to each other; max shear planes at 45° to principal planes

4.2 Mohr's Circle of Stress

Fig. 4.1 — Mohr's Circle: Centre = ((σ_x+σ_y)/2, 0), Radius R = τ_max = (σ₁−σ₂)/2; σ₁, σ₂ at ends of horizontal diameter

4.3 Mohr's Circle Construction — Step by Step

Draw σ-axis (horizontal) and τ-axis (vertical)
Plot point X = (σ_x, +τ_xy) and point Y = (σ_y, −τ_xy)
Centre C = midpoint of XY = ((σ_x+σ_y)/2, 0)
Radius R = |CX| = √[(σ_x−σ_y)²/4 + τ_xy²]
Principal stresses: σ₁ = σ_avg + R, σ₂ = σ_avg − R (on σ-axis)
Max shear stress: τ_max = R (at top/bottom of circle)
Angle 2θ_p on circle = angle from X to horizontal (right); → θ_p on actual element

4.4 Mohr's Circle of Strain

Replace σ with ε and τ with γ/2 in all formulae:
Centre: ε_avg = (ε_x + ε_y)/2
Radius: R_ε = √[(ε_x − ε_y)²/4 + (γ_xy/2)²]
Principal strains: ε₁, ε₂ = ε_avg ± R_ε
Max shear strain: γ_max = 2R_ε = ε₁ − ε₂

Strain rosette (0°–45°–90° rosette):
ε_x = ε₀°, ε_y = ε₉₀°, γ_xy = 2ε₄₅° − ε₀° − ε₉₀°

4.5 Theories of Failure

Failure theories predict when a material transitions from safe to unsafe under multiaxial stress, by comparison with the uniaxial yield stress σ_y (or UTS σ_u).

Theory	Failure Criterion	Safe for	Best Suited for
Maximum Principal Stress (Rankine)	σ₁ ≤ σ_y	σ₁ < σ_y	Brittle materials (cast iron, concrete)
Maximum Shear Stress (Tresca / Guest)	τ_max ≤ σ_y/2 → (σ₁−σ₂) ≤ σ_y	(σ₁−σ₂) < σ_y	Ductile metals (conservative)
Max Principal Strain (St. Venant)	ε₁ ≤ σ_y/E → σ₁−ν(σ₂+σ₃) ≤ σ_y	σ₁−ν(σ₂+σ₃) < σ_y	Brittle (less common)
Max Strain Energy (Beltrami-Haigh)	σ₁²+σ₂²+σ₃²−2ν(σ₁σ₂+σ₂σ₃+σ₃σ₁) ≤ σ_y²	—	Obsolete; not used practically
Distortion Energy (von Mises)	σ_e = √[½((σ₁−σ₂)²+(σ₂−σ₃)²+(σ₃−σ₁)²)] ≤ σ_y	σ_e < σ_y	Ductile metals (most accurate)
Max Strain Energy of Distortion (Hencky)	Same as von Mises	—	Same as von Mises

4.6 Failure Criteria for Biaxial Stress (σ₃ = 0)

Rankine (Max Principal Stress): σ₁ = σ_y
Tresca (Max Shear Stress): σ₁ − σ₂ = σ_y (if σ₁ and σ₂ have opposite signs)
σ₁ = σ_y (if same sign — Tresca = Rankine in same-sign quadrants)
von Mises (Distortion Energy): σ₁² − σ₁σ₂ + σ₂² = σ_y² (ellipse in σ₁−σ₂ space)

For pure shear (σ₁ = −σ₂ = τ):
Tresca: τ_y = σ_y/2
von Mises: τ_y = σ_y/√3 ≈ 0.577 σ_y
von Mises is more accurate (agrees with experiments); Tresca is conservative (safe side).

⭐ Key comparison: For pure shear, Tresca gives τ_y = 0.5σ_y and von Mises gives τ_y = 0.577σ_y. von Mises predicts ~15% more capacity — important for GATE numericals. Use Tresca for conservative design; von Mises for accurate prediction of ductile failure.

4.7 Solved Example — Mohr's Circle

Given: σ_x = 80 MPa, σ_y = 20 MPa, τ_xy = 30 MPa
σ_avg = (80+20)/2 = 50 MPa
R = √[(80−20)²/4 + 30²] = √[900 + 900] = √1800 = 42.43 MPa
σ₁ = 50 + 42.43 = 92.43 MPa
σ₂ = 50 − 42.43 = 7.57 MPa
τ_max = 42.43 MPa
2θ_p = arctan(2×30/(80−20)) = arctan(1) = 45° → θ_p = 22.5°
Check von Mises: σ_e = √(92.43² − 92.43×7.57 + 7.57²) = √(8543 − 699 + 57) ≈ 89 MPa

5Deflection of Beams►

5.1 Elastic Curve Equation

Euler-Bernoulli beam equation:
EI · d²y/dx² = M(x) [approx for small deflections]
EI · d³y/dx³ = V(x) = dM/dx
EI · d⁴y/dx⁴ = −w(x)

Boundary conditions:
Fixed end: y = 0, dy/dx = 0
Simply supported: y = 0, M = 0 (d²y/dx² = 0)
Free end: M = 0, V = 0 (unless load applied)
Internal hinge: M = 0 at hinge; y and dy/dx continuous

5.2 Standard Deflection Formulae

Beam	Loading	Max Deflection δ_max	Location
SS beam	Central point load P	PL³/(48EI)	Midspan
SS beam	UDL w, full span	5wL⁴/(384EI)	Midspan
SS beam	Load P at 'a' from A (a<b)	Pa(3L²−4a²)b/(48EIL) at midspan approx	Exact: x=√(L²−b²)/3 from A if a>b
Cantilever	Point load P at free end	PL³/(3EI)	Free end
Cantilever	UDL w, full span	wL⁴/(8EI)	Free end
Cantilever	UDL w over half-span from fixed end	7wL⁴/(384EI)	Free end
Propped cantilever	UDL w	wL⁴/(185EI)	~0.42L from fixed end
Fixed-fixed beam	Central P	PL³/(192EI)	Midspan
Fixed-fixed beam	UDL w	wL⁴/(384EI)	Midspan

5.3 Methods for Computing Deflections

Double Integration Method

EI · y'' = M(x) → integrate twice → apply BCs to find constants C₁, C₂
Slope: θ = dy/dx | Deflection: y (positive downward convention or per problem)

Macaulay's Method (Singularity Functions)

Use <x−a>ⁿ brackets — integrated as (x−a)ⁿ⁺¹/(n+1) only when (x−a) > 0
Very efficient for multiple point loads / partial UDLs

Example: SS beam, load P at x = a:
EIy'' = RA·x − P<x−a>¹
EIy' = RA·x²/2 − P<x−a>²/2 + C₁
EIy = RA·x³/6 − P<x−a>³/6 + C₁x + C₂

Moment Area Method (Mohr's Theorems)

Theorem I: Change in slope between two points A and B
θ_AB = (1/EI) × Area of M/EI diagram between A and B

Theorem II: Tangential deviation of B from tangent at A
t_BA = (1/EI) × (Area of M/EI diagram between A and B) × x̄_B
where x̄_B = distance of centroid of M/EI diagram from B

Best used for: cantilevers, overhanging beams, non-prismatic beams

Conjugate Beam Method

In the conjugate beam: load = M/EI diagram of real beam
Shear in conjugate beam = slope in real beam
Moment in conjugate beam = deflection in real beam

Boundary conditions for conjugate beam:
Real free end → Conjugate fixed end (non-zero slope and deflection → non-zero V and M)
Real fixed end → Conjugate free end (zero slope and deflection → zero V and M)
Real pin/roller → Conjugate pin/roller (same)

5.4 Virtual Work / Unit Load Method

Deflection at point C in direction of unit load:
δ_C = ∫(M · m̄ / EI) dx

M = real bending moment (under actual loads)
m̄ = virtual bending moment (due to unit load at C in desired direction)

Slope: θ_C = ∫(M · m̄_θ / EI) dx (m̄_θ due to unit couple at C)

📝 GATE Tip: For a SS beam with UDL, max deflection = 5wL⁴/(384EI). For cantilever with tip load: PL³/(3EI). Ratio of these two: (PL³/3EI) / (5wL⁴/384EI) — frequently asked to compare. Memorise the "5 in 384" for UDL SS beam.

6Bending & Shear Stresses in Beams►

6.1 Flexure Formula (Bending Stress)

Euler-Bernoulli Bending Formula:
σ/y = M/I = E/R

σ = M·y/I (bending stress at distance y from NA)
σ_max = M·y_max/I = M/Z

Where:
M = bending moment at section
I = second moment of area (moment of inertia) about NA
y = distance from neutral axis (NA)
Z = I/y_max = section modulus (m³ or mm³)
R = radius of curvature of neutral axis
E = Young's modulus

⭐ Key: Bending stress is zero at NA and maximum at the extreme fibres. Sagging BM → tension at bottom, compression at top. Hogging BM → opposite.

6.2 Section Modulus and Moment of Resistance

Section	I (about NA)	y_max	Z = I/y
Rectangle (b × d)	bd³/12	d/2	bd²/6
Solid Circle (dia D)	πD⁴/64	D/2	πD³/32
Hollow Circle (D, d)	π(D⁴−d⁴)/64	D/2	π(D⁴−d⁴)/(32D)
Triangle (b, h)	bh³/36	2h/3 (from apex)	bh²/24
I-Section	BD³/12 − 2·[(B−b)d³/12]	D/2	I/(D/2)

6.3 Shear Stress in Beams

Shear stress at distance y from NA:
τ = V·Q / (I·b)

Where:
V = shear force at section
Q = A'·ȳ' = first moment of area of section above (or below) y about NA
I = moment of inertia of full section about NA
b = width of section at distance y

Maximum shear stress for standard sections:
Rectangle: τ_max = 1.5 × V/A (at NA) [parabolic distribution]
Solid Circle: τ_max = 4V/(3A) = 4/3 × V/A (at NA)
I-section: τ_max ≈ V/A_web (approximately uniform in web)
Thin-walled hollow circle: τ_max = 2V/A

Fig. 6.1 — Rectangular section: parabolic shear stress, τ_max = 1.5V/A at NA

Fig. 6.2 — I-section: shear concentrated in web; τ_max ≈ V/A_web at NA

6.4 Shear Flow in Thin-Walled Sections

Shear flow: q = V·Q / I (force per unit length along the wall)

In open thin-walled sections (channels, angles):
q varies along the wall; zero at free edges
Shear centre is the point through which V must pass for no twisting

In closed thin-walled sections (box, tube):
q = V·Q/I + q₀ (q₀ = statically indeterminate shear flow)

6.5 Composite Beams (Transformed Section)

Transform all materials to equivalent area of one material (usually concrete or steel):
A_transformed = A_original × (E_material / E_reference)
Modular ratio m = E₁/E₂

Neutral axis: ΣA_t · ȳ = 0 (centroid of transformed section)
Bending stress in material 1: σ₁ = M·y/(I_t)
Bending stress in material 2: σ₂ = m · M·y/(I_t)

6.6 Flitched Beams

A timber beam reinforced with steel plates on two faces. Both materials undergo same curvature (1/R), same strain at same y from NA:

Equal curvature: σ_steel/E_s = σ_timber/E_t at same y
∴ σ_steel = m × σ_timber (m = E_s/E_t ≈ 20 for steel-timber)
Moment of resistance: M = M_steel + M_timber

📝 GATE/ESE Tip: The relationship τ = VQ/(Ib) is the Jourawski formula. For an I-beam, the web carries ~80–90% of the shear force even though it has a much smaller area than the flanges. This is why web buckling (shear buckling) is critical in plate girders.

7Thick & Thin Cylinders and Spheres►

7.1 Thin Cylinders (t/D < 1/20 or t/r < 1/10)

Internal pressure p, mean radius r, thickness t:

Hoop (circumferential) stress: σ_H = pd/(2t) = pr/t
Longitudinal stress: σ_L = pd/(4t) = pr/(2t) = σ_H/2

Volumetric strain of cylinder:
e_v = (pd/2tE)[5/2 − 2ν] = (pd/2tE)[(5−4ν)/2]

Hoop strain: ε_H = (σ_H − ν·σ_L)/E = (pd/2tE)(1 − ν/2)
Longitudinal strain: ε_L = (σ_L − ν·σ_H)/E = (pd/2tE)(1/2 − ν)
Volumetric strain: e_v = 2ε_H + ε_L = (pd/2tE)(5/2 − 2ν)

Fig. 7.1 — Thin cylinder: Hoop stress σ_H = pd/(2t), Longitudinal stress σ_L = pd/(4t); σ_H = 2σ_L

7.2 Thin Spherical Shell

Both hoop and longitudinal stresses are equal:
σ = pd/(4t) = pr/(2t)

Volumetric strain: e_v = 3ε = (3pd/4tE)(1 − ν)

Comparison: Sphere is more efficient — same stress in all directions; needs ~half the steel of a cylinder for same p and d.

7.3 Thick Cylinders — Lamé's Equations

When t/d ≥ 1/20, stress variation through thickness is significant. Lamé's equations apply:

Radial stress: σ_r = A − B/r²
Hoop stress: σ_H = A + B/r²

For internal pressure p_i only (p_o = 0), r_i = inner radius, r_o = outer radius:
A = p_i · r_i² / (r_o² − r_i²)
B = p_i · r_i² · r_o² / (r_o² − r_i²)

Hoop stress (max at inner radius):
σ_H (max) = p_i · (r_o² + r_i²) / (r_o² − r_i²)

Radial stress: σ_r = −p_i at r = r_i (compressive), σ_r = 0 at r = r_o

Boundary Conditions

At r = r_i: σ_r = −p_i (negative = compressive, pointing inward)
At r = r_o: σ_r = 0 (free outer surface, assuming no external pressure)

7.4 Thick Cylinder — Stress Distribution

Fig. 7.2 — Thick cylinder (Lamé): Hoop stress σ_H is maximum at inner radius and decreases hyperbolically; Radial stress σ_r = −p_i at inner surface, 0 at outer

7.5 Compound Cylinders and Shrink-Fit

A compound cylinder (outer cylinder shrunk onto inner) introduces residual compressive hoop stress at the inner surface, allowing higher working pressure.

Junction pressure p_j (from interference δ = r_j × (e_outer_inner − e_inner_outer)):
δ = p_j · r_j/E × [(r_o² + r_j²)/(r_o² − r_j²) + ν_outer + (r_j² + r_i²)/(r_j² − r_i²) − ν_inner]

Net hoop stress = prestress (compressive) + operating pressure stress

📝 GATE Tip: For thin cylinder: Hoop stress = 2 × Longitudinal stress. This is the most frequent fact. For thick cylinder at inner surface: σ_H = p_i(r_o²+r_i²)/(r_o²−r_i²) — maximum — and σ_r = −p_i (compressive).

8Torsion in Shafts & Springs►

8.1 Torsion Formula

T/J = τ/r = Gφ/L

Where:
T = applied torque (N·mm or N·m)
J = polar moment of inertia of cross-section
τ = shear stress at radius r from axis
G = modulus of rigidity (shear modulus)
φ = angle of twist (radians)
L = length of shaft

Maximum shear stress (at outer surface, r = R):
τ_max = T·R/J = T/Z_p

Angle of twist: φ = TL/(GJ)

8.2 Polar Moment of Inertia J

Section	J	Z_p (Polar Section Modulus)
Solid circular shaft (dia D)	πD⁴/32	πD³/16
Hollow circular shaft (D, d)	π(D⁴−d⁴)/32	π(D⁴−d⁴)/(16D)
Thin-walled tube (mean radius r, thickness t)	2πr³t	2πr²t
Rectangle (non-circular, approx)	Uses St. Venant correction; J ≠ I_p exactly	—

Fig. 8.1 — Circular shaft under torsion: shear stress zero at axis, maximum at outer surface; angle of twist φ = TL/GJ

8.3 Power Transmission

Power: P = T · ω = T × 2πN/60

Where: P in Watts, T in N·m, N = rpm, ω = rad/s

T = P × 60 / (2πN) = 9550 × P(kW) / N(rpm) [in N·m]

8.4 Shafts in Series and Parallel

Series shafts (same torque T, different φ):
φ_total = φ₁ + φ₂ = TL₁/(G₁J₁) + TL₂/(G₂J₂)

Parallel shafts (same φ, shared T):
T = T₁ + T₂
φ₁ = φ₂ → T₁L₁/(G₁J₁) = T₂L₂/(G₂J₂)
→ T₁/T₂ = G₁J₁L₂/(G₂J₂L₁)

8.5 Combined Bending and Torsion

For a circular shaft under BM = M and Torque = T simultaneously:

Equivalent bending moment (using max normal stress):
M_e = (1/2)[M + √(M² + T²)] [Rankine/Tresca criterion]

Equivalent torque (using max shear stress):
T_e = √(M² + T²) [both normal and shear stresses combined]

Design: σ_max = M_e × 32 / (πD³) ≤ σ_permissible
or: τ_max = T_e × 16 / (πD³) ≤ τ_permissible

8.6 Close-Coiled Helical Springs

Shear stress in wire: τ = 8WD/(πd³) × K_w
Wahl's stress correction factor: K_w = (4C−1)/(4C−4) + 0.615/C
Spring index: C = D/d (D = mean coil dia, d = wire dia)

Axial deflection: δ = 8WD³n/(Gd⁴) = WL/(GJ) [where n = number of active coils]

Stiffness: k = W/δ = Gd⁴/(8D³n)

Strain energy stored: U = W²D³n/(Gd⁴) = W·δ/2 = τ²/(4G) × Volume_wire

8.7 Springs in Series and Parallel

Series springs (same load, deflections add):
1/k_eq = 1/k₁ + 1/k₂ + ... → δ_total = δ₁ + δ₂

Parallel springs (same deflection, loads add):
k_eq = k₁ + k₂ + ... → W_total = W₁ + W₂

8.8 Open-Coiled and Closely Coiled Springs

Type	Helix Angle α	Loading	Stress type
Close-coiled	α ≈ 0 (small)	Axial load W	Pure torsion in wire
Open-coiled	α significant	Axial load W	Torsion + bending in wire
Close-coiled	α ≈ 0	Applied torque M	Pure bending in wire

📝 GATE Tip: For close-coiled springs under axial load, wire is in pure torsion. Key formula: δ = 8WD³n/(Gd⁴). Stiffness k ∝ Gd⁴/(D³n). Doubling D reduces k by 8×; doubling d increases k by 16×.

9Theory of Columns & Retaining Walls►

9.1 Classification of Columns

Type	Slenderness Ratio λ = L_eff/r	Failure Mode
Short column	λ < 40 (steel); λ < 12 (concrete)	Material crushing/yielding
Long / slender column	λ > 120	Elastic buckling (Euler)
Intermediate column	40 ≤ λ ≤ 120	Inelastic buckling (Johnson/Rankine)

9.2 Euler's Buckling Load

Euler's critical (buckling) load:
P_cr = π²EI / L_eff²

Effective length L_eff for various end conditions:
Both ends pinned: L_eff = L
One fixed, one free: L_eff = 2L
Both ends fixed: L_eff = L/2
One fixed, one pinned: L_eff = L/√2 ≈ 0.707L

Critical stress: σ_cr = P_cr/A = π²E/(λ²) where λ = L_eff/r
r = radius of gyration = √(I/A)

Euler's formula valid only when σ_cr < proportional limit (σ_p)

Fig. 9.1 — Column end conditions and effective lengths: Pin-Pin L_eff=L; Fixed-Free L_eff=2L; Fixed-Fixed L_eff=L/2; Fixed-Pin L_eff=0.707L

9.3 Rankine-Gordon Formula (Intermediate Columns)

1/P_cr = 1/P_crushing + 1/P_Euler
P_cr = σ_c · A / [1 + a(L_eff/r)²]

Where:
σ_c = crushing stress (yield or compressive strength)
a = Rankine's constant = σ_c / (π²E) [dimensionless]

For steel: a = 1/7500 (approx)
For cast iron: a = 1/1600
For timber: a = 1/3000

9.4 Eccentrically Loaded Columns (Secant Formula)

For column with eccentric load P at eccentricity e:
σ_max = P/A [1 + (e·c/k²)·sec(L_eff/(2k)·√(P/AE))]

Where: k = radius of gyration, c = distance from centroid to extreme fibre

Core of section (no tension): load must be within kern
Rectangular section: kern is a rhombus with diagonals b/3 and d/3
Circular section: kern is a circle of diameter D/4

9.5 Retaining Walls — Lateral Earth Pressure

Rankine's Theory

Active earth pressure coefficient: K_a = (1−sinφ)/(1+sinφ) = tan²(45° − φ/2)
Passive earth pressure coefficient: K_p = (1+sinφ)/(1−sinφ) = tan²(45° + φ/2)
K_a × K_p = 1

Active pressure at depth z: p_a = K_a·γ·z − 2c·√K_a (c-φ soil)
For cohesionless soil (c=0): p_a = K_a·γ·z

Total active force: P_a = ½·K_a·γ·H² (per unit length, acting at H/3 from base)
Total passive force: P_p = ½·K_p·γ·H² (acting at H/3 from base)

Coulomb's Theory (Wedge Theory)

K_a (Coulomb) = sin²(α+φ) / [sin²α · (1 + √(sin(φ+δ)·sin(φ−β)/(sin(α−δ)·sin(α+β))))²]

α = inclination of back of wall to horizontal
β = inclination of backfill surface to horizontal
δ = wall friction angle (0 to 2φ/3)
φ = angle of internal friction of soil

Rankine = Coulomb with α=90°, β=0°, δ=0° (smooth vertical wall, horizontal backfill)

9.6 Stability of Retaining Walls

Stability Check	Requirement	FOS
Overturning	FOS = M_restoring / M_overturning	≥ 1.5 (normal), ≥ 2.0 (earthquake)
Sliding	FOS = μ·(ΣV) / P_a (+ passive resistance)	≥ 1.5
Bearing failure	q_max ≤ q_allowable	≥ 2–3
Eccentricity check	e = B/2 − a ≤ B/6 (for no tension)	—

📝 GATE Tip: L_eff memorisation: "Pin-Pin: Full | Fixed-Free: Double | Fixed-Fixed: Half | Fixed-Pin: Root-Half (0.707L)". Euler's formula is valid only for long columns; Rankine for intermediate. The effective length is the key input to both.

10Shear Centre, Moment of Inertia & Principal Axes►

10.1 Moment of Inertia (Second Moment of Area)

I_x = ∫y² dA (about x-axis / NA)
I_y = ∫x² dA (about y-axis)
I_xy = ∫xy dA (product of inertia)
J = I_x + I_y (polar moment of inertia = I_p)

10.2 Standard MI Values

Shape	I_xx (centroidal)	I_yy (centroidal)	J
Rectangle b×d	bd³/12	db³/12	bd(b²+d²)/12
Solid circle D	πD⁴/64	πD⁴/64	πD⁴/32
Triangle (base b, h)	bh³/36	bh³/48 (about median)	—
Semicircle (R)	(π/8 − 8/9π)R⁴ ≈ 0.11R⁴	πR⁴/8	—
Thin circular ring (R, t)	πR³t	πR³t	2πR³t

10.3 Parallel Axis Theorem

I_x = I_x̄ + A·d²

where I_x̄ = MI about centroidal axis parallel to x
A = area of section
d = distance between the two parallel axes

NEVER add d² alone — must be A·d². A common exam error!

10.4 Perpendicular Axis Theorem (2D)

For a planar lamina:
I_z = I_x + I_y = J (polar MI)

This relates out-of-plane MI to in-plane MIs.

10.5 Principal Moments of Inertia and Principal Axes

Just as principal stresses are the extreme values of normal stress at a point, principal moments of inertia are the extreme (maximum and minimum) values of MI achieved by rotating the axes. The product of inertia I_xy = 0 about principal axes.

I_max, I_min = (I_x + I_y)/2 ± √[(I_x − I_y)²/4 + I_xy²]

Angle of principal axes:
tan(2θ_p) = −2I_xy / (I_x − I_y)

Check: I_max + I_min = I_x + I_y (sum invariant)
I_max × I_min − I_xy² = I_x·I_y − I_xy² (product invariant)

⭐ Notice the analogy: principal MI formulae are identical to principal stress formulae with σ replaced by I and τ replaced by I_xy. Mohr's circle construction works for both!

10.6 Mohr's Circle of MI

Construct exactly like Mohr's stress circle:

Plot point X = (I_x, +I_xy) and Y = (I_y, −I_xy)
Centre = ((I_x+I_y)/2, 0)
Radius R = √[(I_x−I_y)²/4 + I_xy²]
I_max = centre + R, I_min = centre − R

10.7 Shear Centre (Flexural Centre)

The shear centre (SC) is the point in the cross-section through which the transverse shear force must pass to produce pure bending (no twisting). If load passes through SC → bending only. If load does not pass through SC → combined bending + torsion.

Shear centre location found by equating external moment of shear force about SC
to sum of moment of shear flows in all elements about SC:
e = ΣQ_flange × h / V (for open thin-walled sections)

Fig. 10.1 — Channel section: shear centre SC lies outside the section, to the left of the web

Fig. 10.2 — Shear centre: at corner for angle, at centroid for I-section & circle

10.8 Shear Centre Locations — Summary

Section	Shear Centre Location	Notes
Solid circular / hollow circular	At geometric centre	Symmetric section
I-section / H-section	At centroid (intersection of axes of symmetry)	Doubly symmetric
Channel (C-section)	Outside the web, on the axis of symmetry	e = 3b²/(6b + h·t_w/t_f)
Angle section (equal legs)	At the corner of the angle	No shear in either leg passing through corner
T-section	At intersection of web and flange	On axis of symmetry
Z-section	At centroid	Point-symmetric section
Semi-circular thin ring	At 4R/π ≈ 1.27R from centre (outside)	On axis of symmetry

10.9 Unsymmetric Bending

When the bending moment is not in a plane of symmetry, or when I_xy ≠ 0 (oblique bending), the neutral axis is not perpendicular to the applied moment. The beam deflects in the direction perpendicular to the NA, which is not the direction of loading.

Bending stress at point (x, y):
σ = (M_x·I_y − M_y·I_xy)·y/D + (M_y·I_x − M_x·I_xy)·x/D
D = I_x·I_y − I_xy²

For oblique loading at angle α to principal x-axis:
σ = M·cosα·y/I_x − M·sinα·x/I_y (if I_xy = 0, i.e., x-y are principal axes)

Neutral axis angle β: tan β = (I_x/I_y)·tan α
NA is NOT perpendicular to load unless I_x = I_y (circular) or loading in plane of symmetry

⚠️ Key ESE concept: For unsymmetric sections (angle, Z-section), the deflection is NOT in the direction of the applied load. The beam deflects in the direction of the principal axis closest to the load. Designers must account for this to prevent unexpected lateral deflection.

★Quick Revision – All Formulae & Mnemonics►

Elastic Constants

G = E / [2(1+ν)] | K = E / [3(1−2ν)] | E = 9KG/(3K+G)
Steel: E = 2×10⁵ MPa, G = 0.8×10⁵ MPa, ν = 0.25–0.30
Concrete: E = 5000√f_ck MPa, ν = 0.15–0.20
Limits: −1 ≤ ν ≤ 0.5; ν = 0.5 → incompressible (rubber)

Stress Transformation & Mohr's Circle

σ₁,σ₂ = (σ_x+σ_y)/2 ± √[(σ_x−σ_y)²/4 + τ²_xy]
τ_max = (σ₁−σ₂)/2 = Radius of Mohr's circle
tan(2θ_p) = 2τ_xy/(σ_x−σ_y) [angle of principal planes]
Max shear planes at 45° to principal planes always.

Theories of Failure (Biaxial, σ₃=0)

Rankine: σ₁ = σ_y
Tresca: σ₁−σ₂ = σ_y (opp. signs); τ_yield = σ_y/2
von Mises: σ₁²−σ₁σ₂+σ₂² = σ_y²; τ_yield = σ_y/√3 ≈ 0.577σ_y
von Mises > Tresca (less conservative) → use for ductile metals

Bending and Flexure

Deflection of Beams

EIy'' = M(x) → integrate twice with BCs
SS + central P: δ = PL³/48EI
SS + UDL w: δ = 5wL⁴/384EI ← MEMORISE "5 in 384"
Cantilever + P: δ = PL³/3EI
Cantilever + UDL: δ = wL⁴/8EI
Fixed-fixed + central P: δ = PL³/192EI (=PL³/48EI ÷ 4)

Thin Cylinders & Spheres

Cylinder: σ_H = pd/2t (hoop); σ_L = pd/4t = σ_H/2 (longitudinal)
Sphere: σ = pd/4t (equal in all directions)
Thick cylinder (Lamé): σ_H = A + B/r²; σ_r = A − B/r²
Max hoop (inner radius): σ_H = p_i(r_o²+r_i²)/(r_o²−r_i²)

Torsion and Springs

T/J = τ/r = Gφ/L | φ = TL/GJ | τ_max = TD/2J = 16T/πD³
J: solid = πD⁴/32; hollow = π(D⁴−d⁴)/32
Spring: τ = 8WD/πd³ × K_w | δ = 8WD³n/Gd⁴ | k = Gd⁴/8D³n
Series: 1/k_eq = Σ(1/k_i); Parallel: k_eq = Σk_i

Columns

Euler: P_cr = π²EI/L_eff² | σ_cr = π²E/λ² (λ = L_eff/r)
End conditions: Pin-Pin: L_eff = L; Fixed-Free: 2L; Fixed-Fixed: L/2; Fixed-Pin: L/√2
Rankine: P_cr = σ_c·A / (1 + a·λ²); a = σ_c/(π²E)

Earth Pressure

K_a = tan²(45−φ/2) = (1−sinφ)/(1+sinφ)
K_p = tan²(45+φ/2) = (1+sinφ)/(1−sinφ) | K_a·K_p = 1
P_a = ½·K_a·γ·H² (at H/3 from base); P_p = ½·K_p·γ·H²

Shear Centre & MI

I_max,min = (I_x+I_y)/2 ± √[(I_x−I_y)²/4 + I_xy²]
Parallel axis: I = I_centroid + A·d² (MUST include A)
Shear centre: where load must pass for no twist
SC at corner → angle; SC at centroid → I, circle, Z; SC outside → channel

Key Mnemonics

Elastic constants: "G gives shape change (Shear); K keeps volume (Kompressibility); E for Everything axial"

Failure theories (increasing allowance): Rankine < Tresca < von Mises
"Rankine is Rigid (brittle); Tresca is Tough (ductile, safe); von Mises is the Most accurate"

Thin cylinder stresses: "Hoop is Half-done on top: pd/2t | Longitudinal is half of Hoop: pd/4t"

Effective lengths of columns:
"Fixed-Free = 2 (worst); Pin-Pin = 1; Fixed-Pin = √½; Fixed-Fixed = ½ (best)"
Or: 2L — L — 0.7L — 0.5L (decreasing → increasing strength)

Spring analogy: "Series springs are WEAK (less k); Parallel springs are STRONG (more k)"
"Springs in Series → like resistors in parallel for k"

Bending formula: σ/y = M/I = E/R → "σ over y = M over I = E over R → SIMON is very Emotional"

Shear centre: "When load passes through SC → Section bends cleanly; off-centre → it twists meanly"

Exam-Angle Comparison

Topic	GATE Focus	ESE Focus	SSC JE Focus
Elastic constants	Numerical: find G or K given E, ν	Derivation + numerical; 3D Hooke's law	Definitions; standard values
Mohr's circle	Principal stresses + angle; τ_max	Strain rosette; Mohr's circle of strain	Identify σ₁, σ₂; τ_max
Failure theories	Tresca vs von Mises; pure shear comparison	All 5 theories; plot on σ₁-σ₂ plane	Name and criterion only
Deflection	All 5 methods; propped cantilever	Virtual work; non-prismatic beams	Standard formulae only
Torsion	Solid/hollow shaft; power transmission	Combined bending+torsion; equivalent M and T	φ = TL/GJ formula
Columns	Euler load; effective length; Rankine	Eccentric load; secant formula; design	Types; Euler formula
Cylinders	Thin: σ_H = pd/2t; vol strain; sphere	Thick: Lamé; compound cylinder; shrink-fit	σ_H and σ_L only
Shear centre	Channel section e formula; angle at corner	Unsymmetric bending; principal axes of MI	Definition only