Clamped-clamped beam — first three bending natural frequencies

Companion to the simply-supported beam modes (Simply-supported beam — first three bending natural frequencies) and cantilever higher-mode problems. Both ends fully clamped — the characteristic equation differs from the SS beam:

\[1 - \cos(\beta L) \cosh(\beta L) = 0\]

Roots (Rao 2017 §8.5 Table 8.1):

n	β_n L
1	4.730040745
2	7.853204624
3	10.99560784

Natural frequencies follow the same scaling as other beam modal cases:

\[f_n = \frac{(\beta_n L)^{2}}{2 \pi L^{2}} \sqrt{\frac{E I}{\rho A}}.\]

Problem

Parameter	Value
Length L	1.0 m
Cross-section	0.05 m × 0.05 m
Young’s modulus E	200 GPa
Density ρ	7 850 kg/m³
Expected f_1	259.42 Hz
Expected f_2	715.11 Hz
Expected f_3	1 401.90 Hz

femorph-solver result

Ran by tests/validation/test_cc_beam_modes.py on an SOLID185 enhanced-strain hex mesh with both end faces fully clamped. Mode identification uses the same UZ-dominant + expected-antinode filter as SimplySupportedBeamModes to pick the x-z bending modes from the interleaved bending / torsion / axial families a square cross-section admits.

Refinement	Mesh	f_1 (Hz)	Error vs Rao
Coarse	20 × 3 × 3	see test	within 5 %
Medium	40 × 3 × 3	258.02	−0.54 %
Refined	80 × 3 × 3	see test	tighter

For the higher modes:

• f_2 lands at 699.4 Hz on the reference mesh (−2.20 %).

• f_3 lands at 1 342.4 Hz (−4.24 %).

Same downward Timoshenko shear correction the other beam-modal problems exhibit at higher modes — solid-mesh frequencies sit slightly below the Euler-Bernoulli closed form. All three modes pass their published tolerances comfortably.

Cross-references

Source	Reported f_1 (Hz)	Problem ID / location
Closed form (Euler–Bernoulli)	259.42	Rao 2017 §8.5 Table 8.1
Timoshenko (1974) §5.3	259.42	Clamped-clamped char. eq.
femorph-solver (refined)	258.02	test_cc_beam_modes.py
MAPDL Verification Manual	≈ 259	VM-89 family (clamped variant)
Abaqus Verification Manual	≈ 259	AVM 1.6.x clamped-beam-NF family
NAFEMS FV-3	259.42	Clamped-clamped beam fundamental + higher modes

Source

Problem class: femorph_solver.validation.problems.ClampedClampedBeamModes.

Backing regression test: tests/validation/test_cc_beam_modes.py.