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Stress Calculator – Tensile, Shear, Bending & Thermal Stress Calculator with Formulas

Free comprehensive engineering stress calculator for tensile stress, shear stress, bending stress, thermal stress, bearing stress, hoop stress, and principal stresses. Includes formulas, material properties, unit conversions, and step-by-step calculations for mechanical and structural engineering.
Stress Calculator

Stress Calculator - Comprehensive Engineering Stress Analysis Tool

A stress calculator is a fundamental engineering tool that computes various types of mechanical stress including tensile stress, shear stress, bending stress, thermal stress, and bearing stress based on applied forces, material properties, and geometric parameters. Understanding stress analysis is critical in mechanical engineering, civil engineering, materials science, structural design, and aerospace applications where predicting material behavior under load determines safety, reliability, and performance. Whether calculating normal stress in tension members, bending stress in beams, hoop stress in pressure vessels, or maximum shear stress for failure analysis, accurate stress calculations prevent structural failures and optimize material usage.

Stress represents the internal force per unit area within a material and is mathematically defined as \(\sigma = \frac{F}{A}\) for normal stress, where F is the applied force and A is the cross-sectional area. Different loading conditions produce different stress types: tensile stress occurs when forces pull a material apart, compressive stress when forces push together, shear stress when forces act parallel to surfaces, and bending stress when moments cause curvature. The relationship between stress and strain (deformation) follows Hooke's Law in the elastic region: \(\sigma = E \varepsilon\), where E is Young's modulus (elastic modulus) and ε is strain. Materials have characteristic strength values including yield strength (where permanent deformation begins), ultimate tensile strength (maximum stress before failure), and allowable stress (design stress with safety factors).

Our comprehensive stress calculator offers eight specialized calculation modes to address diverse engineering scenarios. You can calculate tensile and compressive stress from axial loads, determine shear stress in bolts and pins, analyze bending stress in beams using moment and section modulus, compute thermal stress from temperature changes and coefficient of expansion, calculate bearing stress at contact surfaces, determine hoop stress and longitudinal stress in pressure vessels and pipes, find principal stresses and maximum shear stress using Mohr's circle analysis, and calculate Young's modulus from stress-strain data. Each mode provides detailed step-by-step solutions showing formulas, unit conversions, and intermediate calculations, making this tool invaluable for mechanical engineers, structural engineers, students, designers, and anyone performing strength analysis and material selection in engineering projects.

Interactive Stress Calculator

Calculate Tensile or Compressive Stress

Calculate Shear Stress

Calculate Bending Stress in Beams

Calculate Thermal Stress

Calculate Bearing Stress

Calculate Hoop Stress (Pressure Vessel)

Calculate Principal Stresses

Calculate Young's Modulus (Elastic Modulus)

Essential Stress Formulas

Normal Stress (Tensile/Compressive)

Normal stress occurs when force acts perpendicular to the cross-sectional area.

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

Where σ is normal stress (Pa or psi), F is applied force (N or lbf), and A is cross-sectional area (m² or in²). Positive values indicate tension, negative indicate compression.

Shear Stress

Shear stress acts parallel to the surface and causes one layer to slide over another.

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

Where τ (tau) is shear stress, V is shear force, and A is the area resisting shear. Common in bolted connections, pins, and structural joints.

Bending Stress in Beams

Bending stress varies linearly across the beam cross-section, maximum at the outermost fibers.

\[\sigma_b = \frac{M \cdot c}{I} = \frac{M}{S}\]

Where M is bending moment, c is distance from neutral axis to extreme fiber, I is moment of inertia, and S is section modulus (S = I/c).

Thermal Stress

Thermal stress develops when temperature changes cause expansion or contraction that is constrained.

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

Where E is Young's modulus, α (alpha) is coefficient of thermal expansion (1/°C or 1/°F), and ΔT is temperature change.

Bearing Stress

Bearing stress occurs at contact surfaces between components, like bolts in holes or supports under beams.

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

Where F is the bearing force and A_b is the projected bearing area (diameter × thickness for cylindrical bearings).

Hoop Stress (Circumferential Stress)

Hoop stress acts circumferentially in pressure vessels and pipes, resisting internal pressure.

\[\sigma_h = \frac{P \cdot r}{t}\]

Where P is internal pressure, r is inner radius, and t is wall thickness. Longitudinal stress is σ_l = Pr/(2t).

Principal Stresses

Principal stresses are maximum and minimum normal stresses at a point, acting on planes with zero shear stress.

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

Where σ₁ is maximum principal stress, σ₂ is minimum principal stress, and τ_xy is shear stress.

Maximum Shear Stress

Maximum shear stress occurs at 45° to principal stress directions.

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

This is half the difference between principal stresses and determines yielding in ductile materials (Tresca criterion).

Stress-Strain Relationship (Hooke's Law)

In the elastic region, stress is proportional to strain with Young's modulus as the constant.

\[\sigma = E \cdot \varepsilon \quad \text{where} \quad \varepsilon = \frac{\Delta L}{L_0}\]

Where ε is strain (dimensionless), ΔL is change in length, and L₀ is original length. E is Young's modulus (elastic modulus).

Ultimate and Yield Strength

Material strength properties define failure criteria and design limits.

\[\sigma_y = \text{Yield Strength} \quad \sigma_u = \text{Ultimate Tensile Strength}\] \[\sigma_{allow} = \frac{\sigma_y}{SF} \quad \text{or} \quad \sigma_{allow} = \frac{\sigma_u}{SF}\]

Where SF is safety factor (typically 1.5-4). Yield strength marks onset of plastic deformation; ultimate strength is maximum stress before fracture.

Material Properties Reference Table

MaterialYoung's Modulus (GPa)Yield Strength (MPa)Ultimate Strength (MPa)Thermal Expansion (10⁻⁶/°C)
Structural Steel (A36)200250400-55011.7
Aluminum 6061-T66927631023.6
Stainless Steel 30419321550517.3
Titanium Ti-6Al-4V1148809508.6
Copper (annealed)1177022016.5
Concrete (typical)30-30-50 (compression)10-14
Wood (Douglas Fir)13-50-803-5
Glass (soda-lime)69-50 (brittle)9

Stress Units Conversion Table

From/ToPascal (Pa)Megapascal (MPa)PSIksi
1 Pascal (Pa)11 × 10⁻⁶1.450 × 10⁻⁴1.450 × 10⁻⁷
1 Megapascal (MPa)1 × 10⁶1145.040.145
1 PSI6894.766.895 × 10⁻³10.001
1 ksi6.895 × 10⁶6.89510001

Key Takeaways

  • Normal stress σ = F/A - force per unit area, positive for tension, negative for compression
  • Shear stress τ = V/A - acts parallel to surface, causes sliding between layers
  • Bending stress σ_b = Mc/I - maximum at outer fibers, zero at neutral axis
  • Thermal stress σ_T = EαΔT - develops when constrained thermal expansion/contraction occurs
  • Hoop stress σ_h = Pr/t - circumferential stress in pressure vessels, pipes, and cylinders
  • Principal stresses are maximum/minimum normal stresses at 45° to shear stress planes
  • Maximum shear stress τ_max = (σ₁ - σ₂)/2 - determines yielding in ductile materials
  • Hooke's Law σ = Eε - linear stress-strain relationship in elastic region
  • Yield strength marks permanent deformation; ultimate strength is maximum before fracture
  • Safety factors (1.5-4) applied to allowable stress ensure design safety margins
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