Strength of Materials — Axial and Torsional Loading
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
Stress-Strain Fundamentals
- Normal stress (σ) = P/A; Strain (ε) = δ/L
- Hooke’s Law: σ = E·ε (E = modulus of elasticity, ~200 GPa for steel)
- Poisson’s ratio (ν) = −(lateral strain)/(axial strain), range 0–0.5
- Thermal strain: ε_th = α·ΔT; Thermal stress: σ_th = E·α·ΔT
Torsion of Circular Shafts
- Torsional shear stress: τ = T·r/J
- Polar moment of inertia: J = πd⁴/32 (solid); J = π(D⁴−d⁴)/32 (hollow)
- Torque-power relation: P = 2πNT/60 (N in RPM, T in N·m, P in watts)
- Angle of twist: θ = T·L/(J·G)
Key Formulas Table
| Parameter | Formula | Units |
|---|---|---|
| Normal Stress | σ = P/A | MPa |
| Strain | ε = δ/L | dimensionless |
| Hooke’s Law | σ = E·ε | MPa |
| Poisson’s Ratio | ν = −ε_lat/ε_ax | dimensionless |
| Thermal Stress | σ = E·α·ΔT | MPa |
| Torsional Stress | τ = T·r/J | MPa |
| Angle of Twist | θ = T·L/(J·G) | rad |
⚡ GATE Tip: Combined loading questions appear almost every year. Always resolve into axial, torsion, and bending components first.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Stress-Strain and Hooke’s Law
When an axial load P is applied to a bar of uniform cross-section, the resulting normal stress is σ = P/A. The deformation δ follows from ε = δ/L, giving δ = PL/(AE). This is the foundational equation for axially loaded members.
Hooke’s Law in its standard form σ = E·ε connects stress and strain through the modulus of elasticity (Young’s modulus). For structural steel, E ≈ 200 GPa; for aluminum, E ≈ 70 GPa. The relationship is linear-elastic — valid only up to the proportional limit.
Poisson’s ratio ν describes how a material contracts laterally when stretched axially. Most metals have ν ≈ 0.3. For incompressible materials (rubber, soil under confinement), ν → 0.5. GATE commonly tests that ν cannot exceed 0.5.
Thermal Stress
When temperature changes and expansion/contraction is restrained, thermal stress develops:
σ_th = E·α·ΔT
where α is the coefficient of thermal expansion. If the bar is free to expand, no stress develops — only restrained thermal loading causes stress. This is a classic GATE trap: students forget that “free thermal expansion” means zero stress, not zero strain.
Torsion of Circular Shafts
Circular shafts under torque T develop torsional shear stress varying linearly from zero at the center to maximum at the outer surface:
τ_max = T·r/J = T·d/(2J)
where J is the polar moment of inertia.
Solid circular shaft: J = πd⁴/32 Hollow circular shaft: J = π(D⁴ − d⁴)/32
The angle of twist over length L is:
θ = T·L/(J·G) (in radians)
where G is the shear modulus, related to E and ν: G = E/[2(1+ν)].
⚡ GATE Watch: Questions often ask for minimum diameter given permissible shear stress and transmitted torque. Set τ_max = τ_permissible and solve for d.
Power Transmission
P = T·ω = 2πNT/60
A shaft rotating at N RPM transmitting torque T (N·m) carries power P in watts. Often combined with torsional stress equations.
Combined Loading
Members can experience simultaneous axial load, torsion, and bending. The principal stresses are found using:
σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ_xy²]
The maximum shear stress theory for ductile materials:
τ_max = √[((σ_x − σ_y)/2)² + τ_xy²]
⚡ Common GATE Mistake: For combined axial + bending, direct stress σ = P/A ± My/I. Don’t forget the sign convention — tensile stress is positive.
Example Problem
A steel bar (E = 200 GPa, ν = 0.3) with cross-section 20 mm × 40 mm carries an axial tensile load of 100 kN. Find σ, ε, and lateral strain.
Solution: σ = P/A = 100×10³/(20×40×10⁻⁶) = 125 MPa ε = σ/E = 125×10⁶/(200×10⁹) = 6.25×10⁻⁴ Lateral strain = −ν·ε = −0.3 × 6.25×10⁻⁴ = −1.875×10⁻⁴
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Advanced Concepts in Axial and Torsional Loading
Strain Energy Under Axial Loading
The strain energy stored in a bar under axial load is:
U = ∫₀^δ P dδ = P²L/(2AE) = (σ²/2E)·AL
For a prismatic bar, this simplifies to U = σ²AL/(2E). The strain energy density (per unit volume) is u = σ²/(2E).
Thermal Stress — Restrained vs Free Expansion
Three thermal stress scenarios appear in GATE:
- Both ends fixed: σ = E·α·ΔT (compressive if heated)
- One end fixed, one end free: stress = 0, deformation = α·ΔT·L
- Pre-stress + thermal: Superpose thermal stress on existing stress state
Torsion of Non-Circular Sections
Only circular shafts carry pure torsion without warping. For non-circular sections (rectangular, elliptical), torsion produces warping and the simple τ = Tr/J formula does not apply. GATE focuses on circular sections, but rectangular thin sections use τ_max = T/(k·a·b²) where a ≥ b.
Combined Torsion and Bending
When a shaft carries both bending moment M and torque T:
- Find bending stress: σ_b = My/I
- Find torsional stress: τ_t = Tr/J
- Compute principal stress or equivalent moment
Equivalent torque concept: Replace combined M and T with an equivalent torque T_eq:
T_eq = √(M² + T²)
Equivalent bending moment:
M_eq = √(M² + T²)
⚡ Exam Weightage: Axial + torsion typically carries 6–8 marks per paper (1 question). Combined loading is more frequent than isolated topics.
Stress Concentration
In members with abrupt section changes (holes, keyways, shoulders), local stress exceeds nominal stress. The stress concentration factor K_t multiplies nominal stress: σ_max = K_t × σ_nominal. For fillet transitions, K_t depends on the ratio of radii.
Example — Torsional Power Transmission
A hollow steel shaft (D = 80 mm, d = 60 mm) transmits 200 kW at 300 RPM. If τ_permissible = 50 MPa, verify adequacy.
Solution: P = 200×10³ W; N = 300 RPM T = P×60/(2πN) = 200×10³×60/(2π×300) = 6366 N·m J = π(D⁴−d⁴)/32 = π(80⁴−60⁴)/32 = π(4.096×10⁸−1.296×10⁸)/32 = 8.796×10⁶ mm⁴ = 8.796×10⁻⁶ m⁴ τ_max = T·r/J = 6366×0.04/8.796×10⁻⁶ = 28.9 MPa < 50 MPa ✓ Adequate
GATE Previous Year Pattern
| Year | Topic | Marks |
|---|---|---|
| 2022 | Thermal stress + axial | 2 |
| 2021 | Torsion + angle of twist | 2 |
| 2020 | Combined loading, principal stress | 5 |
| 2019 | Power transmission + shaft design | 2 |
Content adapted based on your selected roadmap duration. Switch tiers using the selector above.