Response spectrum of a cantilever beam: modal masses and base shear (Euler-Bernoulli)
Response spectrum analysisPASS · 0.48%Effective modal mass fractions of the first three bending modes of a slender cantilever, the base shear and the tip displacement combined by SRSS over the five bending modes of a piecewise-linear design spectrum, checked against the closed-form Euler-Bernoulli modal participation (roots of cos·cosh = -1, integrals of the mode shapes).Effective modal mass fractions of the first three bending modes of a slender cantilever, the base shear and the tip displacement combined by SRSS over the five bending modes of a piecewise-linear design spectrum, checked against the closed-form Euler-Bernoulli… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Prismatic cantilever, L = 2 m, rectangular section 50 × 100 mm (beam elements, 80 per span).
- Material
- Steel: E = 200 GPa, ν = 0.3, ρ = 7850 kg/m³ (lumped translational mass).
- Boundary conditions
- Fully fixed at the root. Base excitation along the weak bending direction (dir=z), SRSS mode combination, missing-mass correction off, 8 modes.
- Measured quantity
- Effective mass of modes 1, 3, 5 (the z-bending family) as a share of the total beam mass, the base shear Σ(M_eff·Sa)² and the tip displacement Σ(Γ·Sa/ω²·φ(L))² against the analytic values; the design spectrum is the piecewise-linear table 4.0 → 2.5 → 1.0 → 0.3 m/s².
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:
units mm
material E=2e11 nu=0.3 density=7850
beam length=2000 section=rect w=50 h=100 n=80 scheme=cantilever P=1
solve spectrum code=custom points="0.001,4;0.02,4;0.1,2.5;0.3,1;2,0.3" abscissa=period unit=m/s2 dir=z combine=srss missingmass=off nmodes=8
probe spectrum_effective_mass mode=1 dir=z id="Meff1"
probe spectrum_effective_mass mode=3 dir=z id="Meff2"
probe spectrum_effective_mass mode=5 dir=z id="Meff3"
probe spectrum_reference_mass dir=z id="Mref"
probe spectrum_cumulative_mass dir=z id="cum"
probe spectrum_base_shear dir=z id="V"
probe spectrum_displacement id="u"Results comparison
| Reference value | M_eff/M = 61.3118.836.47 %V_z = 137.641 N|u| = 0.968358 mm |
| Our result | M_eff/M = 61.3218.876.50 %V_z = 137.707 N|u| = 0.968775 mm |
| Deviation | 0.48% |
| Tolerance | 5% |
| Status | PASS · 0.48% |
| Element family | Beam element |
| Mesh details | 80 Euler–Bernoulli beam elements (STRUCT_ELEM.BEAM), lumped translational mass, 486; modal: webcae_solve (native), 8 modes |
Updated: 2026-10-21 · WebCAE solver