Home · Mixed methods and locking

Solid elements versus beam elements

Same cantilever, side view. Top is the solid Q4 mesh (a 2-D continuum with thickness). Bottom is a 1-D beam — a line, not a brick. Numbers are MATLAB tip drops from the same script.

Why locking happens Gauss points

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Top — solid Q4 (2-D continuum)MATLAB |w| = —
Bottom — 1-D reduced TimoshenkoMATLAB |w| = —
—
σxx
—0—

Top = Q4 continuum. Bottom = 1-D beam axis. |w| from MATLAB.

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Agreement versus thickness

Same strip, same load, MATLAB tip values. Reduced Timoshenko tracks the solid Q4 from slender to thick. Full two-point shear locks when the beam is thin (panel C climbs toward 90%). Euler–Bernoulli has no shear, so it peels off when the beam is thick. Panel D overlays the solid mesh and the 1-D beam axis at four thicknesses (motion scaled the same way in each panel). The navy dashed line is the slider’s current h/L.

A · tip drop |w|

B · |w| / |w|Bernoulli

C · |difference from solid Q4|

solid Q4 reduced Timoshenko full-int. Timoshenko Euler–Bernoulli analytical Timoshenko current h/L

D · deformed shapes at four thicknesses — solid mesh and 1-D beam

solid Q4 reduced Timoshenko full-int. Timoshenko Euler–Bernoulli undeformed

What you are comparing

Top — solid elements. The strip is a 2-D continuum: Q4 quads along the length and through the thickness. No beam theory. This is the reference.

Bottom — 1-D beam elements. Euler–Bernoulli and Timoshenko are lines: deflection w(x) and rotation θ(x) only. The picture is that axis (nodes + a short tick for θ), not a solid of height h. Default is reduced Timoshenko (1-point shear). Tip values are MATLAB’s.

When they should match

  • Reduced Timoshenko (1-point shear) tracks the solid from thin to thick.
  • That is the beam you want if the job is to replace a solid mesh.

When they should not

  • Full-integration Timoshenko on a thin strip: the beam stays straight (shear locking).
  • Euler–Bernoulli on a thick strip: the beam is too stiff because it has no shear.
Thin truth ≈ P L³ / (3 EI)  ·  thick adds P L / (G As) of shear drop
Locking: two-point shear on a linear Timoshenko element  ·  Fix: one-point shear

MATLAB numbers (same script)

Cantilever tip drop from out_beam_solid_reduced/cantilever_vs_hL.dat. Larger number = more bending.

h/L|w|EB|w|red|w|full|w|Q4|w|an.TSnrmQ4

Finer mesh, still locked (thin strip 1/50)

Full integration at 32 elements is still only about half the true drop. Reduced is already fine at 4 elements.

n|w|EB|w|red|w|full|w|an.TS|w|Q4

3-D solid bricks agree too

A few 3-D hex meshes (B8) from the same script. They sit next to the 2-D solid, not next to the locked beam.

h/L|w|EB|w|red|w|Q4|w|B8ndof Q4ndof B8