Harmonic response: load phase angles on a pinned beam (antiphase forces cancel a symmetric mode)

Harmonic analysisPASS · 0.32%

Two equal transverse forces at L/4 and 3L/4 of a pinned beam. In phase they excite only the symmetric modes; in antiphase (phase 180°) the modal force of every symmetric mode is exactly zero and only the antisymmetric modes respond; with a 90° shift the first modal force is F·sin(π/4)·(1 + i). The responses are checked against the Euler-Bernoulli modal series with complex modal forces, and the two cancellations against zero.

Tolerance:: 5%

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Problem statement

Geometry: Prismatic beam L = 1.5 m, square section 25 × 25 mm (48 beam elements), pinned at one end and on a roller at the other.
Material: Steel: E = 200 GPa, ν = 0.3, ρ = 7850 kg/m³ (lumped translational mass). Modal damping ζ = 0.02.
Boundary conditions: Two transverse forces of 100 N at x = L/4 and x = 3L/4; phases 0°/0°, 0°/180° and 0°/90°. 12 modes (six bending pairs).
Measured quantity: Mid-span and quarter-span amplitude at the first two natural frequencies against the series; |u_mid(f₁)| in antiphase divided by in phase (must be below 1e-6).

WebCAE Terminal

Paste these commands into the Terminal panel inside WebCAE. They use the same text DSL as the application; run one command per line:

# The card numbers come from cases/run_harmonic_verification.mjs (native modal + complex modal forces of loads with phase angles).
# This script reproduces the model and its native modal solution; on solid bodies the phase is set per load: "load surface number 2 force=100 component=y phase=180".
units mm
material E=2e11 nu=0.3 density=7850
beam length=1500 section=rect w=25 h=25 n=48 scheme=simply-supported P=1
solve modal nmodes=12
probe frequency mode=1 id="f1"

Results comparison

Reference value	f₁ = 25.4312 Hz, u_mid(f₁0/0) = 37.6313 mm, u_¼(f₂0/180) = 3.32619 mm, cancellation ratio = 0
Our result	f₁ = 25.4221 Hz, u_mid = 37.6391 mm, u_¼ = 3.32238 mm, cancellation ratio = 6.82e-13
Deviation	0.32%
Tolerance	5%
Status	PASS · 0.32%
Element family	Beam element
Mesh details	48 Euler–Bernoulli beam elements, pinned–roller; 12 modes (6 bending pairs); modal: webcae_solve (native, sidecar)

Updated: 2026-10-21 · WebCAE solver