Formula Sheet: Stress Analysis

Normal Stress

Axial Stress

Direct axial stress due to axial force:

\[\sigma = \frac{P}{A}\]

• σ = normal stress (psi or Pa)
• P = axial force (lb or N)
• A = cross-sectional area (in² or m²)
• Positive for tension, negative for compression

Bending Stress

Flexural stress in beams:

\[\sigma = \frac{My}{I} = \frac{M}{S}\]

• σ = bending stress at distance y from neutral axis (psi or Pa)
• M = bending moment (lb·in or N·m)
• y = distance from neutral axis to point of interest (in or m)
• I = second moment of area (in⁴ or m⁴)
• S = section modulus = I/c (in³ or m³)
• c = distance from neutral axis to extreme fiber (in or m)

Maximum bending stress occurs at extreme fiber:

\[\sigma_{max} = \frac{Mc}{I}\]

Bearing Stress

Bearing stress at contact surfaces:

\[\sigma_b = \frac{P}{A_b}\]

• σ_b = bearing stress (psi or Pa)
• P = applied load (lb or N)
• A_b = bearing area (in² or m²)

Shear Stress

Direct Shear Stress

Direct shear in pins, bolts, and connections:

\[\tau = \frac{V}{A}\]

• τ = shear stress (psi or Pa)
• V = shear force (lb or N)
• A = cross-sectional area resisting shear (in² or m²)

Single shear: Load passes through one cross-section

Double shear: Load passes through two cross-sections, effective area = 2A

Transverse Shear Stress in Beams

Shear stress distribution in beams:

\[\tau = \frac{VQ}{Ib}\]

• τ = shear stress at location of interest (psi or Pa)
• V = transverse shear force (lb or N)
• Q = first moment of area above (or below) the point = A'ȳ' (in³ or m³)
• I = second moment of area of entire cross-section (in⁴ or m⁴)
• b = width of cross-section at the point of interest (in or m)
• A' = area above (or below) the point where stress is calculated
• ȳ' = distance from neutral axis to centroid of A'

Maximum shear stress in rectangular beams:

\[\tau_{max} = \frac{3V}{2A} = 1.5\tau_{avg}\]

• Occurs at neutral axis
• A = total cross-sectional area

Maximum shear stress in circular solid beams:

\[\tau_{max} = \frac{4V}{3A}\]

• Occurs at neutral axis

Torsional Shear Stress

Shear stress due to torsion in circular shafts:

\[\tau = \frac{T\rho}{J} = \frac{Tr}{J}\]

• τ = torsional shear stress (psi or Pa)
• T = applied torque (lb·in or N·m)
• ρ or r = radial distance from center to point of interest (in or m)
• J = polar moment of inertia (in⁴ or m⁴)

Maximum torsional shear stress (at outer surface):

\[\tau_{max} = \frac{Tc}{J} = \frac{T}{Z_p}\]

• c = outer radius (in or m)
• Z_p = polar section modulus = J/c (in³ or m³)

Polar moment of inertia for solid circular shaft:

\[J = \frac{\pi d^4}{32} = \frac{\pi c^4}{2}\]

• d = diameter (in or m)
• c = radius (in or m)

Polar moment of inertia for hollow circular shaft:

\[J = \frac{\pi (d_o^4 - d_i^4)}{32} = \frac{\pi (c_o^4 - c_i^4)}{2}\]

• d_o = outer diameter, d_i = inner diameter
• c_o = outer radius, c_i = inner radius

Power-torque relationship:

\[T = \frac{P}{\omega} = \frac{63,025 \times HP}{n}\]

• P = power (W or HP)
• ω = angular velocity (rad/s)
• HP = horsepower
• n = rotational speed (rpm)
• T = torque (lb·in when using HP formula)

Combined Stresses

Combined Axial and Bending

Superposition principle for combined normal stresses:

\[\sigma = \frac{P}{A} \pm \frac{Mc}{I}\]

• Use + for tension side, - for compression side
• Check both extreme fibers

Principal Stresses

Principal stresses for 2D (plane) stress state:

\[\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}\]

• σ₁ = maximum principal stress (psi or Pa)
• σ₂ = minimum principal stress (psi or Pa)
• σ_x = normal stress in x-direction (psi or Pa)
• σ_y = normal stress in y-direction (psi or Pa)
• τ_xy = shear stress (psi or Pa)

Principal angle (orientation of principal planes):

\[\tan(2\theta_p) = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}\]

• θ_p = angle from x-axis to principal plane (degrees or radians)
• Two solutions 90° apart

Maximum Shear Stress

Maximum in-plane shear stress:

\[\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}\]

Alternatively:

\[\tau_{max} = \frac{\sigma_1 - \sigma_2}{2}\]

Angle to maximum shear stress plane:

\[\theta_s = \theta_p \pm 45°\]

Stress Transformation Equations

Normal stress on inclined plane:

\[\sigma_{\theta} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos(2\theta) + \tau_{xy}\sin(2\theta)\]

Shear stress on inclined plane:

\[\tau_{\theta} = -\frac{\sigma_x - \sigma_y}{2}\sin(2\theta) + \tau_{xy}\cos(2\theta)\]

• θ = angle measured counterclockwise from x-axis (degrees or radians)

Mohr's Circle

Construction Parameters

Center of Mohr's circle:

\[C = \frac{\sigma_x + \sigma_y}{2}\]

Radius of Mohr's circle:

\[R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}\]

Key Points on Mohr's Circle

• Principal stresses: σ₁ = C + R, σ₂ = C - R
• Maximum shear stress: τ_max = R
• Convention: Normal stress on horizontal axis, shear stress on vertical axis
• Sign convention: Tensile stress positive, compressive stress negative; shear causing clockwise rotation positive

Three-Dimensional Stress State

Principal Stresses in 3D

For three-dimensional state with σ_x, σ_y, σ_z, τ_xy, τ_yz, τ_xz:

Absolute maximum shear stress:

\[\tau_{abs,max} = \frac{\sigma_{max} - \sigma_{min}}{2}\]

• σ_max = largest principal stress among σ₁, σ₂, σ₃
• σ_min = smallest principal stress among σ₁, σ₂, σ₃

Strain

Normal Strain

Engineering normal strain:

\[\epsilon = \frac{\Delta L}{L_0} = \frac{L - L_0}{L_0}\]

• ε = normal strain (dimensionless or in/in, mm/mm)
• ΔL = change in length
• L₀ = original length
• L = final length

Shear Strain

Engineering shear strain:

\[\gamma = \frac{\Delta x}{h} = \tan(\phi) \approx \phi\]

• γ = shear strain (radians, dimensionless)
• Δx = horizontal displacement
• h = height
• φ = angle of deformation (radians, for small angles)

Axial Deformation

Elongation or contraction under axial load:

\[\delta = \frac{PL}{AE}\]

• δ = deformation/displacement (in or m)
• P = axial force (lb or N)
• L = original length (in or m)
• A = cross-sectional area (in² or m²)
• E = modulus of elasticity (psi or Pa)

For non-uniform members:

\[\delta = \sum_{i=1}^{n} \frac{P_i L_i}{A_i E_i}\]

Torsional Deformation

Angle of twist in circular shafts:

\[\phi = \frac{TL}{GJ}\]

• φ = angle of twist (radians)
• T = applied torque (lb·in or N·m)
• L = length of shaft (in or m)
• G = shear modulus of elasticity (psi or Pa)
• J = polar moment of inertia (in⁴ or m⁴)

Stress-Strain Relationships

Hooke's Law

For normal stress and strain:

\[\sigma = E\epsilon\]

• E = modulus of elasticity (Young's modulus) (psi or Pa)
• Valid only in elastic region

For shear stress and strain:

\[\tau = G\gamma\]

• G = shear modulus (modulus of rigidity) (psi or Pa)

Poisson's Ratio

Lateral strain relationship:

\[\nu = -\frac{\epsilon_{lateral}}{\epsilon_{axial}} = -\frac{\epsilon_y}{\epsilon_x} = -\frac{\epsilon_z}{\epsilon_x}\]

• ν = Poisson's ratio (dimensionless)
• Typical range: 0.25 to 0.35 for metals
• Negative sign indicates lateral contraction with axial extension

Generalized Hooke's Law

Strain in x-direction for 3D stress state:

\[\epsilon_x = \frac{1}{E}[\sigma_x - \nu(\sigma_y + \sigma_z)]\]

Strain in y-direction:

\[\epsilon_y = \frac{1}{E}[\sigma_y - \nu(\sigma_x + \sigma_z)]\]

Strain in z-direction:

\[\epsilon_z = \frac{1}{E}[\sigma_z - \nu(\sigma_x + \sigma_y)]\]

Shear strains:

\[\gamma_{xy} = \frac{\tau_{xy}}{G}, \quad \gamma_{yz} = \frac{\tau_{yz}}{G}, \quad \gamma_{xz} = \frac{\tau_{xz}}{G}\]

Elastic Constants Relationship

Relationship between E, G, and ν:

\[G = \frac{E}{2(1 + \nu)}\]

Bulk modulus:

\[K = \frac{E}{3(1 - 2\nu)}\]

• K = bulk modulus (psi or Pa)

Strain Energy

Axial Loading

Strain energy stored in axially loaded member:

\[U = \frac{P^2L}{2AE} = \frac{\sigma^2 V}{2E}\]

• U = strain energy (lb·in or J)
• V = volume = AL

Strain energy density:

\[u = \frac{U}{V} = \frac{\sigma^2}{2E} = \frac{\sigma\epsilon}{2}\]

• u = strain energy per unit volume (psi or Pa)

Torsional Loading

Strain energy in torsion:

\[U = \frac{T^2L}{2GJ}\]

Bending

Strain energy in bending:

\[U = \int_0^L \frac{M^2}{2EI} dx\]

Modulus of Resilience

Modulus of resilience (energy absorbed up to yield point):

\[u_r = \frac{\sigma_y^2}{2E}\]

• u_r = resilience (psi or Pa)
• σ_y = yield strength

Modulus of Toughness

Modulus of toughness (total energy absorbed up to fracture):

\[u_t = \int_0^{\epsilon_f} \sigma \, d\epsilon\]

• u_t = toughness (psi or Pa)
• ε_f = strain at fracture
• Approximated by area under stress-strain curve

Failure Theories

Maximum Normal Stress Theory (Rankine)

Failure criterion:

\[\sigma_1 \geq \sigma_{yield} \quad \text{or} \quad \sigma_3 \leq -\sigma_{yield}\]

• Used primarily for brittle materials
• Conservative for ductile materials

Maximum Shear Stress Theory (Tresca)

Failure criterion:

\[\tau_{max} = \frac{\sigma_1 - \sigma_3}{2} \geq \frac{\sigma_{yield}}{2}\]

Or equivalently:

\[\sigma_1 - \sigma_3 \geq \sigma_{yield}\]

• Commonly used for ductile materials
• Conservative and simple

Maximum Distortion Energy Theory (von Mises)

Failure criterion for 3D stress state:

\[\sigma_{von \ Mises} = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}} \geq \sigma_{yield}\]

For plane stress (σ₃ = 0):

\[\sigma_{von \ Mises} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2} \geq \sigma_{yield}\]

For biaxial stress with shear:

\[\sigma_{von \ Mises} = \sqrt{\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2} \geq \sigma_{yield}\]

• Most accurate for ductile materials
• Preferred in modern design

Maximum Normal Strain Theory (Saint-Venant)

Failure criterion:

\[\epsilon_1 \geq \epsilon_{yield} = \frac{\sigma_{yield}}{E}\]

Factor of Safety

Factor of safety based on yield:

\[FS = \frac{\sigma_{yield}}{\sigma_{applied}}\]

Factor of safety based on ultimate strength:

\[FS = \frac{\sigma_{ultimate}}{\sigma_{applied}}\]

• Typical values: FS = 1.5 to 4 depending on application and uncertainty

Stress Concentrations

Stress Concentration Factor

Maximum stress with stress concentration:

\[\sigma_{max} = K_t \sigma_{nominal}\]

• K_t = theoretical stress concentration factor (dimensionless)
• σ_nominal = stress calculated using net cross-section

For notched members in tension:

\[\sigma_{nominal} = \frac{P}{A_{net}} = \frac{P}{(w - d)t}\]

• w = width of member
• d = hole or notch diameter
• t = thickness

Common Stress Concentration Factors

• Circular hole in infinite plate: K_t ≈ 3.0
• Semicircular notch: K_t depends on notch radius and geometry
• Shoulder fillet: K_t depends on radius ratio and diameter ratio
• Values obtained from charts or tables in NCEES handbook

Fatigue Stress Concentration Factor

Fatigue notch factor:

\[K_f = 1 + q(K_t - 1)\]

• K_f = fatigue stress concentration factor
• q = notch sensitivity factor (0 ≤ q ≤ 1)
• q = 0 for no sensitivity, q = 1 for full sensitivity

Thermal Stress

Thermal Strain

Free thermal strain:

\[\epsilon_T = \alpha \Delta T\]

• ε_T = thermal strain (dimensionless)
• α = coefficient of thermal expansion (1/°F or 1/°C)
• ΔT = temperature change (°F or °C)

Free thermal deformation:

\[\delta_T = \alpha (\Delta T) L\]

Thermal Stress in Constrained Members

Thermal stress when deformation is fully restrained:

\[\sigma_T = E\alpha \Delta T\]

• Compression if temperature increases (for restrained expansion)
• Tension if temperature decreases (for restrained contraction)

Combined Mechanical and Thermal Loading

Total strain:

\[\epsilon_{total} = \epsilon_{mechanical} + \epsilon_{thermal} = \frac{\sigma}{E} + \alpha \Delta T\]

Total deformation:

\[\delta_{total} = \frac{PL}{AE} + \alpha (\Delta T) L\]

Pressure Vessels

Thin-Walled Cylindrical Pressure Vessels

Hoop stress (circumferential stress):

\[\sigma_1 = \sigma_{hoop} = \frac{pr}{t}\]

• σ₁ or σ_hoop = hoop stress (psi or Pa)
• p = internal pressure (psi or Pa)
• r = inner radius (in or m)
• t = wall thickness (in or m)
• Valid for r/t ≥ 10

Longitudinal stress (axial stress):

\[\sigma_2 = \sigma_{long} = \frac{pr}{2t}\]

• Also called axial stress
• Acts parallel to vessel axis

Radial stress:

\[\sigma_3 = \sigma_{radial} \approx 0\]

• Negligible in thin-walled vessels
• σ₃ = -p at inner surface for thick-walled analysis

Thin-Walled Spherical Pressure Vessels

Stress in spherical vessel:

\[\sigma = \frac{pr}{2t}\]

• Same in all directions on the surface
• Equal to longitudinal stress in cylinder

Thick-Walled Cylinders (Lamé Equations)

Radial stress at radius r:

\[\sigma_r = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} - \frac{r_i^2 r_o^2 (p_i - p_o)}{r^2(r_o^2 - r_i^2)}\]

Tangential (hoop) stress at radius r:

\[\sigma_{\theta} = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} + \frac{r_i^2 r_o^2 (p_i - p_o)}{r^2(r_o^2 - r_i^2)}\]

• p_i = internal pressure
• p_o = external pressure
• r_i = inner radius
• r_o = outer radius
• r = radius at point of interest

For internal pressure only (p_o = 0):

\[\sigma_r = \frac{p_i r_i^2}{r_o^2 - r_i^2}\left(1 - \frac{r_o^2}{r^2}\right)\] \[\sigma_{\theta} = \frac{p_i r_i^2}{r_o^2 - r_i^2}\left(1 + \frac{r_o^2}{r^2}\right)\]

Contact Stress (Hertzian Contact)

Contact Between Cylinders

Maximum contact stress for two cylinders in contact:

\[\sigma_{max} = 0.798\sqrt{\frac{P(1/r_1 + 1/r_2)}{L(1 - \nu_1^2)/E_1 + (1 - \nu_2^2)/E_2}}\]

• P = normal force (lb or N)
• r₁, r₂ = radii of cylinders (in or m)
• L = length of contact (in or m)
• E₁, E₂ = elastic moduli
• ν₁, ν₂ = Poisson's ratios

Contact Between Spheres

Maximum contact stress for two spheres in contact:

\[\sigma_{max} = 0.918\sqrt[3]{\frac{P(1/r_1 + 1/r_2)^2}{[(1 - \nu_1^2)/E_1 + (1 - \nu_2^2)/E_2]^2}}\]

Curved Beams

Winkler-Bach Formula

Stress in curved beam:

\[\sigma = \frac{My}{Ae(r_n - y)}\]

• M = bending moment (positive for increasing curvature)
• y = distance from neutral axis to point of interest
• A = cross-sectional area
• e = distance from centroidal axis to neutral axis = r_c - r_n
• r_c = radius to centroid
• r_n = radius to neutral axis

Neutral axis location:

\[r_n = \frac{A}{\int \frac{dA}{r}}\]

• Integration over cross-section
• Values available in tables for common shapes

Residual Stress

Elastic-Plastic Loading

Residual stress after yielding and unloading:

\[\sigma_{residual} = \sigma_{applied} - \sigma_{elastic \ unload}\]

• Develops when material is loaded beyond yield and then unloaded
• Unloading follows elastic path

Important Sign Conventions and Notes

• Tensile stress: positive
• Compressive stress: negative
• Tensile strain: positive (elongation)
• Compressive strain: negative (contraction)
• Shear stress sign: varies by convention; typically positive if causing clockwise rotation on element face
• Principal planes: planes of zero shear stress
• Maximum shear planes: oriented 45° from principal planes
• Plane stress: σ₃ = 0 (thin members)
• Plane strain: ε₃ = 0 (thick members, restrained deformation)