Boley–Barber problem: thermally induced vibrations of a cantilever beam
Coupled transient thermoelasticityPASS · 0.45%A slender cantilever beam subjected to sudden surface heat flux q₀ on its top face undergoes coupled bending oscillations around the evolving quasi-static thermal curvature κ_T(t) because the thermal diffusion time τ_d = h²/a is comparable to the fundamental flexural period T₁ (Boley parameter B = T₁/τ_d = 1). Peak dynamic tip deflection is verified against the closed-form Euler–Bernoulli modal series.A slender cantilever beam subjected to sudden surface heat flux q₀ on its top face undergoes coupled bending oscillations around the evolving quasi-static thermal curvature κ_T(t) because the thermal diffusion time τ_d = h²/a is comparable to the fundamental f… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Cantilever beam L = 40 mm, cross-section 1 × 1 mm; quadratic tetrahedra (tet10), element size 0.25 mm (13 179 elements).
- Material
- Steel: E = 200 GPa, ν = 0.3, ρ = 7800 kg/m³, α = 1.2 × 10⁻⁵ K⁻¹, c_p = 460 J/(kg·K), conductivity tuned to B = T₁/τ_d = 1 (T₁ = 1.9560 ms).
- Boundary conditions
- Root face (surface number 1, x = 0) clamped; top face (surface number 6, z = 1 mm) subjected to sudden heat flux q₀ = 1 MW/m²; other faces insulated. Dynamic Newmark (β = 0.25, γ = 0.5), Δt = T₁/400 = 4.89 μs, t_end = 3·T₁.
- Measured quantity
- Peak dynamic transverse tip deflection |w_tip|_max compared against the 12-mode Euler–Bernoulli analytical series (4.6297 μm, dynamic amplification factor DAF = 1.783).
WebCAE Terminal
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units mm
material E=200000000000 nu=0.3 density=7800 kappa=1834.33 specific_heat=460 alpha=0.000012
primitive box dx=40 dy=1 dz=1
mesh tet size=0.25 order=2
fix surface number 1
thermal surface number 6 flux=1000000
solve thermoelastic transient mode=dynamic dt=0.00000489006 tend=0.00586807 t0=0 theta=0.5 probes="40,0.5,0.5" frames=2 outputs=1200 stress=off
probe transient_peak dir=z id="peak"Results comparison
| Reference value | |w_tip|_max = 4.6297 µm (DAF = 1.783) |
| Our result | |w_tip|_max = 4.6504 µm (DAF = 1.791) |
| Deviation | 0.45% |
| Tolerance | 5% |
| Status | PASS · 0.45% |
| Mesh details | tet10, size 0.25 mm (13 179 tets); 1200 steps of Δt = 4.89 µs, heat balance residual 4.14e-15 |
Updated: 2026-10-27 · version not recorded