PE Exam Exam > PE Exam Notes > Mechanical Engineering for PE > Cheatsheet: Stress Analysis
Cheatsheet: Stress Analysis
1. Fundamental Stress Concepts
1.1 Normal Stress
| Type | Formula & Description |
|---|---|
| Axial Stress | σ = P/A, where P = axial force, A = cross-sectional area |
| Tensile Stress | Positive normal stress (pulling apart) |
| Compressive Stress | Negative normal stress (pushing together) |
| Bending Stress | σ = My/I, where M = bending moment, y = distance from neutral axis, I = moment of inertia |
1.2 Shear Stress
| Type | Formula & Description |
|---|---|
| Direct Shear | τ = V/A, where V = shear force, A = shear area |
| Torsional Shear | τ = Tr/J, where T = torque, r = radial distance, J = polar moment of inertia |
| Transverse Shear | τ = VQ/(Ib), where V = shear force, Q = first moment of area, I = moment of inertia, b = width |
1.3 Bearing Stress
- σ_b = P/A, where A = projected bearing area
- Occurs at contact surfaces (pins, bolts, rivets)
- Acts perpendicular to contact surface
2. Stress Transformation
2.1 Plane Stress Equations
| Parameter | Formula |
|---|---|
| Normal Stress (rotated) | σ_x' = (σ_x + σ_y)/2 + (σ_x - σ_y)/2·cos(2θ) + τ_xy·sin(2θ) |
| Shear Stress (rotated) | τ_x'y' = -(σ_x - σ_y)/2·sin(2θ) + τ_xy·cos(2θ) |
2.2 Principal Stresses
| Parameter | Formula |
|---|---|
| Principal Stresses | σ_1,2 = (σ_x + σ_y)/2 ± √[((σ_x - σ_y)/2)² + τ_xy²] |
| Principal Angle | tan(2θ_p) = 2τ_xy/(σ_x - σ_y) |
| Maximum Shear Stress | τ_max = √[((σ_x - σ_y)/2)² + τ_xy²] = (σ_1 - σ_2)/2 |
| Max Shear Angle | θ_s = θ_p ± 45° |
2.3 Mohr's Circle
- Center: C = (σ_x + σ_y)/2
- Radius: R = √[((σ_x - σ_y)/2)² + τ_xy²]
- Plot point (σ_x, τ_xy) on horizontal-vertical axes
- Rotation of 2θ on circle = rotation of θ in element
- Clockwise shear stress plotted upward (sign convention)
- Principal stresses at τ = 0 intercepts
2.4 3D Stress State
| Parameter | Formula |
|---|---|
| Principal Stresses | σ_1 ≥ σ_2 ≥ σ_3 (eigenvalues of stress tensor) |
| Absolute Max Shear | τ_abs_max = (σ_1 - σ_3)/2 |
| Hydrostatic Stress | σ_h = (σ_1 + σ_2 + σ_3)/3 = (σ_x + σ_y + σ_z)/3 |
| Octahedral Shear | τ_oct = (1/3)√[(σ_1-σ_2)² + (σ_2-σ_3)² + (σ_3-σ_1)²] |
3. Combined Loading
3.1 Axial + Bending
- σ = P/A ± My/I (use ± for opposite sides)
- Maximum stress at extreme fiber: y = ±c
- Neutral axis shifts when axial load present
- Check both tension and compression sides
3.2 Bending + Torsion
| Parameter | Formula |
|---|---|
| Normal Stress | σ_x = My/I, σ_y = 0 |
| Shear Stress | τ_xy = Tr/J |
| Principal Stresses | σ_1,2 = (σ_x/2) ± √[(σ_x/2)² + τ_xy²] |
| Max Shear Stress | τ_max = √[(σ_x/2)² + τ_xy²] |
3.3 Pressure Vessels
3.3.1 Thin-Walled Cylinders
| Stress Component | Formula |
|---|---|
| Hoop Stress (circumferential) | σ_h = pr/t, where p = internal pressure, r = radius, t = thickness |
| Longitudinal Stress | σ_l = pr/(2t) |
| Radial Stress | σ_r ≈ 0 (thin-wall assumption: r/t ≥ 10) |
3.3.2 Thin-Walled Spheres
- σ = pr/(2t) (equal in all directions)
- Biaxial stress state
3.3.3 Thick-Walled Cylinders (Lamé Equations)
| Stress Component | Formula |
|---|---|
| Radial Stress | σ_r = A - B/r², where A and B are constants from boundary conditions |
| Hoop Stress | σ_h = A + B/r² |
| Constants | A = (p_i·r_i² - p_o·r_o²)/(r_o² - r_i²), B = (p_i - p_o)·r_i²·r_o²/(r_o² - r_i²) |
- p_i = internal pressure, p_o = external pressure, r_i = inner radius, r_o = outer radius
- Maximum stress at inner surface
4. Stress Concentration
4.1 Stress Concentration Factor
| Parameter | Description |
|---|---|
| K_t (theoretical) | σ_max = K_t·σ_nom, where σ_nom = nominal stress without discontinuity |
| K_f (fatigue) | K_f = 1 + q(K_t - 1), where q = notch sensitivity (0 ≤ q ≤ 1) |
4.2 Common Geometries
- Circular hole in plate (tension): K_t = 3.0
- Semicircular notch in plate (tension): K_t = 2.6 to 3.0
- Fillet radius: K_t depends on r/d ratio (smaller radius = higher K_t)
- Shaft with shoulder: K_t = 1.3 to 2.5 (depends on geometry)
4.3 Application Rules
- Use K_t for static brittle materials
- Use K_f for fatigue loading
- Ductile materials under static load: stress concentration less critical (yielding redistributes stress)
- Avoid sharp corners: use fillets with adequate radius
5. Failure Theories
5.1 Ductile Material Theories
| Theory | Criterion |
|---|---|
| Maximum Shear Stress (Tresca) | τ_max = (σ_1 - σ_3)/2 ≤ S_y/2, where S_y = yield strength |
| Distortion Energy (von Mises) | σ_e = √[((σ_1-σ_2)² + (σ_2-σ_3)² + (σ_3-σ_1)²)/2] ≤ S_y |
| Von Mises (plane stress) | σ_e = √(σ_x² - σ_x·σ_y + σ_y² + 3τ_xy²) ≤ S_y |
| Von Mises (3D) | σ_e = √[(σ_x-σ_y)² + (σ_y-σ_z)² + (σ_z-σ_x)² + 6(τ_xy² + τ_yz² + τ_zx²)]/√2 ≤ S_y |
- Von Mises more accurate and conservative for ductile materials
- Both predict yielding, not fracture
5.2 Brittle Material Theories
| Theory | Criterion |
|---|---|
| Maximum Normal Stress (Rankine) | σ_1 ≤ S_ut (tension) or σ_3 ≥ -S_uc (compression) |
| Mohr-Coulomb | |σ_1|/S_ut - |σ_3|/S_uc ≤ 1 |
| Modified Mohr | Uses Mohr circles with different tension/compression strengths |
- S_ut = ultimate tensile strength, S_uc = ultimate compressive strength
- Brittle materials: S_uc > S_ut
5.3 Fatigue Failure
| Parameter | Formula/Description |
|---|---|
| Endurance Limit (steel) | S_e' = 0.5·S_ut for S_ut ≤ 200 ksi (1400 MPa) |
| Modified Endurance | S_e = k_a·k_b·k_c·k_d·k_e·k_f·S_e' |
| k_a (surface) | k_a = a·S_ut^b (ground: a=1.58, b=-0.085; machined: a=4.51, b=-0.265) |
| k_b (size) | k_b = (d/0.3)^(-0.107) for 0.11 ≤ d ≤ 2 in; k_b = 0.91·d^(-0.157) for d > 2 in |
| k_c (loading) | Bending: 1.0; Axial: 0.85; Torsion: 0.59 |
| k_d (temperature) | Room temp: 1.0; elevated temp: <> |
| k_e (reliability) | 50%: 1.0; 90%: 0.897; 95%: 0.868; 99%: 0.814; 99.9%: 0.753 |
| k_f (miscellaneous) | Corrosion, residual stress, etc. |
5.3.1 Stress-Life Method
| Parameter | Formula |
|---|---|
| S-N Curve | S_f = a·N^b, where N = cycles to failure |
| Fatigue Strength | S_f = [(0.9·S_ut)²/S_e]·N^(1/3·log(0.9·S_ut/S_e)) |
| Safety Factor | n = S_f/σ_a (fully reversed) |
5.3.2 Mean and Alternating Stress
| Parameter | Formula |
|---|---|
| Mean Stress | σ_m = (σ_max + σ_min)/2 |
| Alternating Stress | σ_a = (σ_max - σ_min)/2 |
| Stress Ratio | R = σ_min/σ_max |
| Goodman Line | σ_a/S_e + σ_m/S_ut = 1/n |
| Gerber Parabola | σ_a/S_e + (σ_m/S_ut)² = 1/n |
| Soderberg Line | σ_a/S_e + σ_m/S_y = 1/n (conservative) |
| Modified Goodman | n = 1/(σ_a/S_e + σ_m/S_ut) |
6. Strain and Deformation
6.1 Strain Definitions
| Type | Formula |
|---|---|
| Normal Strain | ε = δ/L = ΔL/L_0 |
| Shear Strain | γ = tan(θ) ≈ θ (radians) for small angles |
| Engineering Strain | ε_eng = (L - L_0)/L_0 |
| True Strain | ε_true = ln(L/L_0) = ln(1 + ε_eng) |
6.2 Hooke's Law
| Relationship | Formula |
|---|---|
| Uniaxial | σ = E·ε, where E = Young's modulus |
| Shear | τ = G·γ, where G = shear modulus |
| Elastic Moduli Relation | G = E/[2(1 + ν)] |
| Poisson's Ratio | ν = -ε_lateral/ε_axial (0 ≤ ν ≤ 0.5) |
6.3 Generalized Hooke's Law
| Strain Component | Formula |
|---|---|
| ε_x | ε_x = [σ_x - ν(σ_y + σ_z)]/E |
| ε_y | ε_y = [σ_y - ν(σ_x + σ_z)]/E |
| ε_z | ε_z = [σ_z - ν(σ_x + σ_y)]/E |
| γ_xy | γ_xy = τ_xy/G |
| γ_yz | γ_yz = τ_yz/G |
| γ_zx | γ_zx = τ_zx/G |
6.4 Strain Transformation
- Principal strains: ε_1,2 = (ε_x + ε_y)/2 ± √[((ε_x - ε_y)/2)² + (γ_xy/2)²]
- Principal angle: tan(2θ_p) = γ_xy/(ε_x - ε_y)
- Max shear strain: γ_max = √[(ε_x - ε_y)² + γ_xy²] = ε_1 - ε_2
- Mohr's circle for strain (same construction as stress)
6.5 Strain Energy
| Type | Formula |
|---|---|
| Strain Energy Density | u = σ²/(2E) = σ·ε/2 |
| Total Strain Energy | U = ∫u·dV = ∫(σ²/(2E))·dV |
| Axial Loading | U = P²L/(2AE) |
| Torsion | U = T²L/(2GJ) |
| Bending | U = ∫(M²/(2EI))·dx |
| Modulus of Resilience | u_r = S_y²/(2E) (energy to yield) |
| Modulus of Toughness | u_t = ∫σ·dε (area under stress-strain curve to fracture) |
7. Special Topics
7.1 Thermal Stress
| Parameter | Formula |
|---|---|
| Thermal Strain (free) | ε_T = α·ΔT, where α = coefficient of thermal expansion |
| Thermal Stress (constrained) | σ_T = -E·α·ΔT (compression if heated, tension if cooled) |
| Partially Constrained | σ = E(ε - α·ΔT) |
7.2 Contact Stress (Hertz)
| Configuration | Formula |
|---|---|
| Two Cylinders | σ_max = √[P·E*/(π·L·R*)], where E* = E/(1-ν²), R* = R_1·R_2/(R_1+R_2) |
| Sphere on Flat | σ_max = 0.918·∛(P·E*²/R²) |
| Contact Width (cylinders) | 2b = 2√[2P·R*/(π·L·E*)] |
7.3 Residual Stress
- Self-equilibrating stresses with no external load
- Sources: welding, heat treatment, cold working, machining
- Compressive residual stress beneficial for fatigue resistance
- Shot peening induces beneficial compressive residual stress
- Must sum to zero: ∫σ_res·dA = 0
7.4 Creep
| Stage | Description |
|---|---|
| Primary Creep | Decreasing strain rate; strain hardening dominates |
| Secondary Creep | Constant strain rate; balance of hardening and recovery |
| Tertiary Creep | Increasing strain rate; necking and damage accumulation |
- Time-dependent deformation under constant stress at elevated temperature
- Power law: ε̇ = A·σ^n·exp(-Q/RT)
- Design for secondary creep rate or rupture time
8. Design Considerations
8.1 Safety Factor
| Method | Formula |
|---|---|
| Based on Yield | n = S_y/σ_max or n = S_y/σ_e (von Mises) |
| Based on Ultimate | n = S_ut/σ_max |
| Based on Fatigue | n = S_e/σ_a (fully reversed) or use Goodman/Gerber |
- Recommended n = 1.5 to 4 depending on uncertainty and consequences
- Higher for uncertain loads, brittle materials, consequences of failure
8.2 Allowable Stress
- σ_allow = S_y/n (ductile materials)
- σ_allow = S_ut/n (brittle materials)
- τ_allow = S_sy/n, where S_sy = 0.5·S_y (ductile shear)
- Design requirement: σ_max ≤ σ_allow
8.3 Stress Analysis Process
- Identify loading: type, magnitude, direction, variability
- Determine stress state: normal, shear, combined
- Find maximum stress location and magnitude
- Apply stress concentration factors if applicable
- Calculate principal stresses and maximum shear stress
- Apply appropriate failure theory for material type
- Calculate safety factor and compare to allowable
- Check all critical locations and loading conditions
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