Element formulations

What each element type actually is. If a result surprises you, the explanation is usually here.

Bar

The simplest element: two nodes, axial force only.

With no rotational degrees of freedom, a bar cannot form a moment-release mechanism. A genuine truss chord should therefore be a bar, not a beam released at both ends.

Beam

Two nodes, six degrees of freedom each: axial, two shear, torsion and two bending.

The cubic interpolation is why a beam needs no meshing in a linear static analysis. A cubic is the exact deflected shape of a prismatic member loaded only at its ends ($EI{w}^{\prime \prime \prime \prime }=0$). Under a member load it is not, but the nodal displacements still are, because the load enters through its equivalent nodal forces. The forces between the nodes are then recovered from the end forces and the member load by statics, not read off the cubic. Subdividing such a beam changes where results are reported, not what they are.

That holds for the elastic stiffness matrix only. The geometric stiffness matrix behind stability analysis and the consistent mass matrix behind modal analysis use the same cubic functions, but for those problems the cubic is an approximation, so both converge with subdivision instead of arriving at the answer. A pin-ended column modelled as a single element buckles at

roughly 22% high. Two elements per member bring it inside 1%, four inside 0.1%.

Mode shapes have the same limit: a member with no internal node cannot represent a mode that has one, so higher modes need at least as many elements per member as the shape has half-waves. Beam subdivision therefore has a global target and a per-member override like any other mesh setting, and modal analysis enforces a minimum on top of both.

Shear deformation

Slender members follow Euler–Bernoulli behaviour: plane sections remain normal to the axis and shear deformation is neglected. For stocky members (span-to-depth below about 10) shear deformation is not negligible, and Timoshenko behaviour, which includes it, gives noticeably larger deflections. The section's shear areas control this.

What a beam cannot represent

• Cross-section distortion or local buckling. Local buckling of a Class 4 section is handled by the steel design check through its effective section; distortion is not handled at all.
• Warping restraint, unless explicitly modelled.
• Non-prismatic behaviour within a single element; a tapered member needs subdivision.

Shell

One element covers every surface in the model: slabs, rafts, shear walls, cores, deep beams and shells. It carries out-of-plane bending and in-plane membrane action together, so a wall meeting a slab is one continuous mesh, not two formulations joined by a constraint.

It is a four-node flat shell built from four parts, each the one the literature settled on for its job:

Why the mixed interpolation matters

A displacement-based Reissner–Mindlin element locks as the thickness drops: the transverse shear strains it derives from the displacement field cannot go to zero everywhere, so a thin plate comes out far too stiff. MITC4 samples the transverse shear strains at the element edges and interpolates them separately, which removes the constraint that causes locking.

In practice:

• One element for any thickness. A thin floor slab and a thick raft use the same formulation. You do not select an element from the span-to-depth ratio.
• No stabilisation to tune. Reduced integration is a different fix for the same problem. It introduces spurious zero-energy modes that need hourglass control, with a factor somebody has to choose. This element needs neither: a single element has six zero-energy modes, the six rigid-body motions, and no others.
• Distorted meshes degrade gracefully. The element passes the patch test in membrane and in bending, so skewed elements around an opening or a re-entrant corner lose accuracy gradually.

Walls and deep beams: the membrane

The plain bilinear membrane that MITC4 is usually paired with locks in in-plane bending. Bending a four-node element in its own plane drags a parasitic shear strain along with it, so a wall, a deep beam or a box-girder web modelled with shells comes out far too stiff unless it is meshed many elements deep: a cantilever wall one element deep reaches about 68 % of its true tip deflection.

The membrane here is enriched with four enhanced strain modes, condensed out inside each element, which absorb that parasitic shear. The same wall reaches 99 % with one element through its depth. The enhanced modes reproduce every uniform strain state exactly, so the patch test still holds, and they enter the stresses and resultants you read as well as the stiffness.

Drilling degrees of freedom

The element carries a drilling degree of freedom ${\theta }_z$, the in-plane rotation about the surface normal, so it has all six degrees of freedom at every node.

This matters where a beam frames into a wall. A membrane element without a drilling DOF has nothing to resist the in-plane moment a beam brings to the node. The connection then behaves as a pin whether you intended one or not, and the moment disappears into a redistribution you did not ask for. Here it transfers: a beam landing on a core wall is continuous with it unless you release it deliberately.

The drilling stiffness is not a spring to the ground. Following Hughes and Brezzi, the rotation ${\theta }_z$ is tied to the rotation the membrane itself undergoes,

by a penalty $\frac{1}{2}\gamma t\int ({\theta }_z-\omega {)}^{2}dA$ with $\gamma =0.1G$. The penalty vanishes for every rigid motion, so it never resists a rigid rotation, and being a fraction of the shear modulus it does not depend on the units the model is in. That is what lets a curved shell work: where facets meet at an angle, one element's drilling rotation is its neighbour's bending rotation, and a spring there stiffens the shell several times over. A pinched hemisphere modelled with such a spring can come out at a third of the published deflection, or at the right one, depending on the units the model is in; with this one it reaches the published value whatever the units.

Warped elements

A four-node element on a curved or twisted surface is rarely flat: its corners sit off their mean plane. The element is formulated on that mean plane, and each node's motion is carried onto it as a rigid offset, so a rigid rotation does not strain a warped element. Without this correction a twisted shell locks: the 90° twisted-beam benchmark reaches 0.29 of its reference when thick and almost nothing when thin. With it, both come within 1.5 %, and refining the mesh converges instead of drifting.

What a shell cannot represent

• Curved geometry is approximated by flat facets, so curvature accuracy depends on mesh density in a way a flat slab's does not. The warping correction keeps a warped element from locking, but it does not make a facet curved: watch warping on curved surfaces.
• In-plane bending on strongly tapered elements. The enhanced membrane is exact on rectangles and close on parallelograms, but every four-node membrane that passes the patch test stiffens on a trapezoid (MacNeal, 1987). A wall meshed one element deep in steeply leaning elements reaches about 80 % of its tip deflection, against 99 % on rectangles. The mesher keeps elements close to square for this reason.
• Through-thickness stress variation beyond the assumed linear distribution.
• Local plate buckling of the surface itself. No design check covers a plate either; a buckling analysis on the shell model is the closest tool.

Springs and rigid arms

• Springs add a stiffness between a node and ground. Any of the six directions can be free, rigid or flexible, in global axes or in a skew frame you define (an inclined bearing, a support on a raking face).
• Line supports are the same idea distributed along a line: a stiffness per unit length along a whole member or along one edge of a slab. Rigid directions hold every analysis node on the line; flexible ones are lumped onto those nodes by tributary length, so the total is the line stiffness times the length however finely the line is meshed. This is the Winkler bed for a grade beam, a strip footing or a wall under a slab.
• All of these are linear in both directions: they resist tension exactly as they resist compression, which matters under uplift. Compression-only and tension-only supports are not available.
• Rigid arms constrain degrees of freedom rather than adding stiffness, so they do not degrade conditioning the way a very stiff element does.

Integration and result recovery

Element matrices are integrated numerically at Gauss points. Stresses and internal forces are most accurate at the integration points and are carried to the corners for display. Which way that is done is your choice, and it travels with the analysis setup: two recovery methods are two post-processings of the same solve, so they belong to the setup, not the model.

Mass

For modal analysis, mass is derived from material density and section or thickness geometry, distributed either consistently with the element's interpolation or lumped onto the diagonal. Consistent mass bounds the true frequencies from above; lumped mass usually lands below them, though that is not guaranteed. Consistent is the default, being the more accurate of the two for the same number of degrees of freedom.

Non-structural mass must be added deliberately: as a point mass or rotary inertia at a node, along a member, or by explicitly converting a load case through a mass group. It is never inferred: a force and the mass that would produce it are independent statements, and guessing one from the other is a guess about gravity.