Simply-supported beam — first three bending natural frequencies

A prismatic slender beam of length \(L\), cross-section \(w \times h\), simply supported at both ends, vibrates transversely with mode shapes \(w_n(x) = \sin(n \pi x / L)\) and natural frequencies

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

For \(L / h = 20\) the first few bending modes dominate the low end of the spectrum; torsional + axial modes appear well above them.

Problem

Parameter	Value
Length L	1.0 m
Cross-section	0.05 m × 0.05 m (square)
Young’s modulus E	200 GPa
Poisson’s ratio ν	0.30
Density ρ	7 850 kg/m³
Expected f_1	114.44 Hz
Expected f_2	457.76 Hz (4 × f₁)
Expected f_3	1 030.0 Hz (9 × f₁)

Analytical reference

Euler–Bernoulli separation-of-variables solution for a simply-supported prismatic beam (Rao, Mechanical Vibrations 6th ed., §8.5; Meirovitch, Fundamentals of Vibrations §7.4).

femorph-solver result

Ran by tests/validation/test_ss_beam_modes.py on an SOLID185 enhanced-strain hex mesh with the same knife-edge support convention as the static SS-beam problem. Mode identification uses a post-processing filter: pick the first finite-frequency mode that is (a) dominated by UZ motion (≥ 70 % of kinetic energy) and (b) has the expected number of antinodes along the top-face centerline. This isolates the pure x-z-plane bending modes from the y-bending, torsional, and axial families the square cross-section admits at interleaved frequencies.

Refinement	Mesh	f_1 (Hz)	f_2 (Hz)	f_3 (Hz)
Coarse	20 × 3 × 3	113.62 (−0.7 %)	438.60 (−4.2 %)	1 072.41 (+4.1 %)
Medium	40 × 3 × 3	113.33 (−1.0 %)	433.35 (−5.3 %)	1 042.17 (+1.2 %)
Refined	80 × 3 × 3	113.23 (−1.1 %)	431.12 (−5.8 %)	1 031.49 (+0.2 %)

The persistent ~5 % shift on f_2 is the combined rotary- inertia + shear-deformation correction that Timoshenko beam theory captures but the pure Euler–Bernoulli reference omits. At L / h = 20 the correction is small but not negligible for the second bending mode.

Cross-references

Source	Reported f_1 (Hz)	Problem ID / location
Closed form (Euler–Bernoulli)	114.44	Rao 2017 §8.5
Meirovitch (2010) §7.4	114.44	SS beam transverse vibration
femorph-solver (refined)	113.23	test_ss_beam_modes.py
MAPDL Verification Manual	≈ 114.4	VM-89 Natural frequencies of a SS beam
Abaqus Verification Manual	≈ 114.4	AVM 1.6.x SS beam natural freq family

Source

Problem class: femorph_solver.validation.problems.SimplySupportedBeamModes.

Backing regression test: tests/validation/test_ss_beam_modes.py — three-mesh convergence sweep asserting each of f_1, f_2, f_3 passes its published tolerance and that the three frequencies are strictly ordered.