What is the formula for thermal stress in a fully constrained member?

For a fully constrained homogeneous member with a temperature change ΔT, the thermal stress is σ = −EαΔT, where E is Young's modulus and α is the coefficient of thermal expansion (CTE). The negative sign indicates that a temperature rise (ΔT > 0) produces compressive stress. For example, steel (E=200 GPa, α=12×10⁻⁶/°C) with ΔT=100°C gives σ = −200×10⁹ × 12×10⁻⁶ × 100 = −240 MPa (compression).

How is bimetal beam warpage calculated using Timoshenko theory?

Timoshenko's (1925) bimetal strip theory gives the curvature κ = 6(α₂−α₁)ΔT(1+m)² / {h[3(1+m)² + (1+mn)(m² + 1/(mn))]} where m = t₁/t₂ (thickness ratio), n = E₁/E₂ (modulus ratio), and h = t₁+t₂ (total thickness). The tip deflection of a bimetal beam of length L is δ = κL²/8. The curvature is proportional to the CTE difference (α₂−α₁) and temperature change ΔT.

How can solder joint thermal fatigue in electronics be prevented?

CTE mismatch between a PCB (FR4, α≈17 ppm/°C) and ceramic components (α≈10 ppm/°C) generates cyclic shear strain in solder joints during thermal cycling. Each cycle accumulates inelastic strain, leading to fatigue cracking. Mitigation strategies include: (1) selecting low-modulus compliant solder alloys; (2) underfill encapsulation to redistribute stress; (3) minimising component size (interface stress scales with chip length); (4) using low-CTE board materials such as metal-core PCBs.

When should analytical thermal stress formulas be used versus FEA?

Analytical solutions such as the Timoshenko bimetal formula are valid for homogeneous, uniform-temperature, linear-elastic conditions and are ideal for rapid parametric studies and FEA result validation during early design. FEA becomes necessary for: complex 3D geometry, non-uniform temperature fields, nonlinear material behaviour (plasticity, creep), contact interactions, and multi-layer stacks beyond two layers. This tool's analytical solution applies to two-layer beams with aspect ratio >> 1 (width >> thickness).

Thermal Stress & Bimetal Warpage Calculator — Free Online Calculator

Thermal Stress & Bimetal Warpage Calculator

Interactively compute constrained thermal expansion, bimetal bending, and bilayer interface stress based on Timoshenko theory. Useful for solder joint stress analysis and bimetal thermometer design.

$\sigma = \dfrac{E\alpha\Delta T}{1-\nu}$, $\quad \delta = \dfrac{\kappa L^2}{2}$, $\quad \kappa = \dfrac{6(\alpha_1-\alpha_2)\Delta T(t_1+t_2)}{t_1^2 \cdot f(m,n)}$

Analysis Type & Parameters

Analysis Type

E — Elastic Modulus200 GPa

α — CTE12.0 ppm/K

ν — Poisson's Ratio0.30

ΔT — Temperature Change100 °C

L — Member Length100 mm

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Tip Deflection δ

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Radius of Curvature R

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MPa

Interface Stress σ_int

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μm

Free Expansion ΔL

Deformed Shape (Beam Cross-Section Schematic)

Through-Thickness Stress σ(z)

Tip Deflection δ vs ΔT

Thermal Stress & Bimetal Warpage Calculator

Theoretical Background (Timoshenko, 1925)