Slabs and Grillages

A plate model bends out of plane: a slab (plate-bending triangles) or a grillage (beams carrying transverse load and torsion). It is one of the two model domains; pick Slab / Grillage (out-of-plane) in New model. The DOFs per node are w, θx, θy.

This page is the orientation for plate models. The everyday topics — supports, loads, analysis, results, design — are covered for both domains on their own pages, linked below.

What is plate-specific

Topic	Plate specifics	Where
Sections	Plate section = material + thickness t + element (MITC3 shear-deformable, the default; DKT thin-plate); grillage beam reuses the bar section and gains a St-Venant torsion constant J = β·b·h³ (with G = E/2(1+ν))	Modelling
Supports	restrain w, θx, θy: SIMPLE, CLAMPED, CLAMP-X/Y, SYM-X/Y (symmetry lines)	Modelling
Loads	point Fz, Mx, My; beam qz; area pressure pz; Winkler kz; through-thickness thermal gradient ΔT	Loads
Analysis	linear, non-linear, modal and spectrum — vertical modes; no P-Delta	Analysis
Results	deflection field w/θx/θy; slab forces mx, my, mxy, m1, m2, vx, vy; Wood–Armer (W-A) design moments; grillage M/V/T diagrams; reactions Rz, Mx, My	Results
Templates	Slabs (rectangular slab, rib-stiffened, flat slab on columns, on grade, circular sector) and Grillages (beam grid, bridge deck)	Templates

By script

The engine reaches all of this without the interface:

from xdfem2d import Structure2D

s = Structure2D(domain='plate')
s.add_material('C30/37', elastic_modulus=33e6, unit_weight=25.0, poisson=0.2)
s.add_plate_section('SLAB', 'C30/37', thickness=0.20)      # MITC3 (default; formulation='DKT' for thin plates)
s.add_quad_section('SLABQ', 'C30/37', thickness=0.20, formulation='MITC4')  # quad slab (DKT4 for thin)
s.add_section('BEAM', 'C30/37', b=0.3, h=0.5)              # grillage (gains J)
# ... nodes, then triangles (add_tri_element, 3 nodes) and/or
#     quads (add_quad_element, 4 ccw nodes), and/or bars ...
s.add_support('SS', w=True)
s.add_area_load('S1', 'LC', pz=-10.0)   # pz targets a triangle OR a quad id (or a surface object)
r = s.calculate()

Slab element: MITC3 vs DKT (and their quad twins)

A slab triangle uses one of two plate-bending elements, chosen per section:

• MITC3 (the default) is shear-deformable (Mindlin–Reissner): it accounts for transverse-shear deflection, so it is valid for thick slabs as well as thin ones and reports a genuine shear field vx, vy. It converges to the exact Mindlin solution and passes the constant-moment patch test.
• DKT is the thin-plate (Kirchhoff) element: no shear deformation, very accurate for thin slabs on coarse meshes.

Use the default (MITC3) unless the slab is very thin and the mesh is coarse, where DKT converges with fewer elements. Both share the same nodes, DOFs, results and Wood–Armer design moments, so switching is just the section's formulation.

A slab can also be meshed with quads on a regular grid, which avoid the diagonal bias a triangle split introduces near supports and point loads. The quad slab elements are the 4-node twins of the two triangles, chosen the same way on a quad section:

• MITC4 — the shear-locking-free Mindlin quad, the 4-node analogue of MITC3 (the recommended slab quad).
• DKT4 — the thin-plate quad (four DKT triangles condensed), the analogue of DKT.

They carry the same moments/shears, Wood–Armer design and result views as the triangles, so a mixed tri/quad slab (quads in the regular field, triangles at an irregular edge) is one continuous model. See Quadrilateral elements for how to create them and the "prefer quads on a regular grid" guidance.

Runnable examples live in examples/example-slab.py (thin slab with the DKT element, checked against the Kirchhoff Navier series), examples/example-slab-mitc3.py (thick slab with the MITC3 element, checked against the exact Mindlin series — the transverse-shear deflection the DKT misses), examples/example-grillage.py (crossed two-beam grillage, load split) and examples/example-grillage-cantilever.py (cantilever bar checked against the bending F·L³/3EI and the St-Venant torsion T·L/GJ).