The three pillars of linear elastic analysis

The previous article showed how a real object gets carved into nodes and elements. But before the software can run the seven calculation steps we'll cover in the next article of this series, it has to lean on something much older than FEA itself: the theory of linear elasticity. Every solver, however polished its interface, is really solving three sets of equations at once —force equilibrium, kinematic compatibility of displacements, and the material's constitutive law— and none of the three is optional. Understanding those three pillars is what separates someone who interprets a stress map from someone who just looks at it.

A system of fifteen equations

For a solid in three dimensions, the complete solution has fifteen unknowns: three displacements (u, v, w), six strains —three stretching, three angular distortion— and six stresses, split the same way. Solving for them takes exactly fifteen equations, split into three groups:

3 equilibrium + 6 kinematic + 6 constitutive = 15

No single group is enough on its own, and no FEA result is valid if any one of the three fails silently while the other two look fine.

Pillar 1 — Equilibrium

Picture an infinitesimal cube inside the part. Normal stresses (σ) and shear stresses (τ) act on its faces; body forces such as gravity or inertia (f) act on its volume. For that cube to neither accelerate nor tear apart, the variation in stresses between opposite faces has to be exactly balanced by those body forces. In the x direction, that's written as:

∂σx/∂x + ∂τxy/∂y + ∂τxz/∂z + fx = 0

The other two directions repeat the same structure. Just three equations —but they hold at every one of the infinite points in the solid.

Fig. 1 — Equilibrium of the differential element: stresses on opposite faces almost cancel out; the small remaining difference is exactly balanced by the body forces f. The three pairs pulse in turn as a reminder that their sum, at every point, is always zero.

The software carries this principle over to the global scale in the stiffness equation, {F} = [K]{u}: the sum of external nodal forces has to equal the internal reactions generated by the structure. But there's fine print worth remembering —equilibrium is satisfied exactly only at the nodes. Inside each element it's an approximation, only as good as the mesh.

Equilibrium isn't a property of the result. It's a condition the software enforces exactly at the nodes —and only approximates everywhere else.

Pillar 2 — Kinematics

The second pillar has nothing to do with forces: it's pure geometry. Strain-displacement equations translate how the points of the body move (u, v, w) into how the material deforms (ε). There are six strain components at every point —three stretching, three angular distortion— and under the small-strain assumption they're obtained by differentiating the displacements:

εx = ∂u/∂x · γxy = ∂u/∂y + ∂v/∂x

Fig. 2 — The element breathes between its original shape (dashed outline) and its deformed shape: the stretch of the side gives εx, the change in corner angle gives γxy. Displacements u, v are solved for at the nodes first; strain is derived afterwards, point by point.

This is the pillar the software runs first: it solves for the nodal displacements —the primary unknown in {K}{u} = {F}— and only afterwards derives the strains inside each element from them. If the part deforms so much that its shape changes appreciably, this linear relationship stops holding and a geometrically non-linear analysis is needed.

Pillar 3 — The material

The third pillar bridges the other two: it converts strain into stress. For an elastic, linear, homogeneous and isotropic material, that relationship is the generalised Hooke's law, and it depends on just two constants —Young's modulus E and Poisson's ratio ν— grouped in the constitutive matrix [D]:

{σ} = [D] · {ε}

Fig. 3 — The [D] matrix translates strain into stress through E and ν. The ν terms couple εx with σy and vice versa; the bottom-right term, tied to the shear modulus G = E / [2(1+ν)], is independent of the rest. Below, the three simplifications commonly used depending on the part's geometry.

In practice, [D] is almost never used in its full 3D form. Depending on the part's geometry, it's simplified to:

Plane stress. Thin plates loaded in their own plane —σz is assumed to be zero.
Plane strain. Very long bodies, such as dams or tunnels —εz is assumed to be zero.
Axisymmetric state. Geometry and loads with rotational symmetry —a 3D problem collapses to an analysis in a single plane.

Choosing the wrong simplification doesn't throw a software error: it produces a numerically correct, physically false result.

How FEA ties them together

Solving this fifteen-equation system by hand is unworkable for any geometry beyond a prism or a cylinder. FEA doesn't solve it directly —it works around it: it discretises the body into finite elements and approximates the displacements inside each one with shape functions, simple polynomials defined at the nodes. When you hit "Solve", the software chains the three pillars into a single process: it uses kinematics to go from displacements to strains, uses the material law to go from strains to stresses, and uses the principle of virtual work —a form of equilibrium expressed in energy terms— to integrate both steps into the stiffness matrix of each element.

Validity depends on all three at once

A mistake in the boundary conditions breaks equilibrium. A mesh that fails to capture the strain gradient breaks kinematics. A badly entered elastic property breaks the material law. Any one of the three, on its own, invalidates the entire analysis —no matter how colourful the stress maps the software produces afterwards.

END