2D Finite Element Method - Plane Stress & von Mises Explained

What the finite element method does in 2D

The finite element method (FEM) turns a structural problem that has no simple formula into a large but routine system of linear equations. The region - a frame, a truss, or a continuous plate - is divided into many small elements joined at nodes. For each element the relationship between the forces and displacements at its nodes is written as an element stiffness matrix. These are assembled into one global stiffness matrix $\mathbf{K}$, and the equilibrium of the whole structure is expressed as a single matrix equation:

where $\mathbf{u}$ collects the unknown nodal displacements and $\mathbf{F}$ the applied nodal forces. After imposing the supports (known-zero displacements), the reduced system is solved for $\mathbf{u}$. Everything else - reactions, member forces, and the stress field - is recovered from those displacements. This tool covers two families of 2D element: plane-stress triangles for the stress field of a plate, and frame and truss line elements for member forces.

Plane stress - the stress field of a thin plate

A flat plate that is thin and loaded only in its own plane is in a state of plane stress: the stress through the thickness is taken as zero, and the material is free to contract out of plane. The stress at a point is then described by three components - two normal stresses and one shear stress:

For an isotropic linear-elastic material with Young’s modulus $E$ and Poisson’s ratio $\nu$, stress and strain are linked by the plane-stress constitutive (elasticity) matrix $\mathbf{D}$:

Plane stress suits thin panels, gusset and base plates, brackets, deep beams and plates with holes. For a thick body restrained in the third direction, plane strain is more appropriate (it uses a slightly different $\mathbf{D}$).

The constant-strain triangle (CST) element

This tool meshes the plate into three-node triangles. With linear displacement interpolation across a triangle, the strain - and hence the stress - is constant within each element, which is why it is called a constant-strain triangle. Each node has two degrees of freedom, $u$ and $v$, so an element has six.

The element strains are obtained from the nodal displacements $\mathbf{u}_e$ through the strain–displacement matrix $\mathbf{B}$ (constant, built from the node coordinates and the triangle area $A$), and the element stiffness follows from the virtual-work integral, which for a constant $\mathbf{B}$ and thickness $t$ reduces to a simple product:

After the global solve, the stress in each triangle is recovered directly:

Because CST stresses are piecewise-constant, the tool averages the surrounding element stresses at each node (area-weighted) to produce a smooth, continuous contour. CSTs are robust and easy to verify but relatively stiff, so refining the mesh improves accuracy - particularly near corners and holes where the stress changes rapidly.

Von Mises and principal stresses

To judge how heavily the material is worked, the three stress components are combined into the von Mises stress - a single scalar that maps onto yielding of a ductile material. When it reaches the yield strength, yielding begins, which is why it is the natural quantity to plot as a contour:

The principal stresses - the extreme normal stresses, on the planes where shear vanishes - are also reported:

Stress concentration around a hole

A hole or notch interrupts the flow of stress and forces it to divert, raising the local stress well above the average. For a small circular hole in a wide plate under uniaxial tension, classical elasticity gives a peak stress about three times the far-field value - a stress concentration factor $K_t \approx 3$. The plate-with-hole preset reproduces this: the von Mises contour peaks at the sides of the hole, and refining the mesh sharpens the peak.

Frame and truss elements (the member-force side)

The same $\mathbf{K}\mathbf{u}=\mathbf{F}$ machinery solves skeletal structures. A 2D frame element has three degrees of freedom per node - $u$, $v$ and a rotation $\theta$ - and combines axial stiffness with Euler–Bernoulli bending. Its local stiffness matrix, for axial rigidity $EA$, flexural rigidity $EI$ and length $L$, is the standard 6×6:

Each element is rotated from local to global axes by a transformation $\mathbf{k}_{\text{global}} = \mathbf{T}^{\mathsf T}\mathbf{k}_{\text{local}}\mathbf{T}$ before assembly. A truss bar is the special case with the bending terms released, leaving only axial stiffness $EA/L$- pin-jointed members that carry axial force alone. Distributed loads are applied as equivalent nodal (fixed-end) forces, and the member’s internal axial force, shear and bending moment are recovered from the solved end displacements.

Quantities and units used by this tool

Accuracy, verification and limits

The solver has been checked against problems with known answers. A uniform-tension patch test reproduces a perfectly uniform stress field, confirming the element and assembly are correct. A slender cantilever plate matches Euler–Bernoulli beam theory for tip deflection to within about one percent at a moderate mesh, and the bending stress converges toward the theoretical $My/I$ value as the mesh is refined. The frame solver reproduces the textbook reactions and moments for simply supported, cantilever, propped-cantilever, portal-frame and truss problems.

As with any linear FEM, keep the assumptions in mind:

• Linear elastic - no yielding or plasticity; the stress can exceed the yield strength in the model, which simply tells you yielding would occur.
• First order (small displacements) - no large-deflection or P-delta effects, and no buckling.
• Mesh dependence - coarse meshes underestimate peak stresses; refine near holes, notches and re-entrant corners and check that the result has stabilised.
• Singularities - stress at a sharp re-entrant corner or a point load grows without bound as the mesh refines; treat such peaks with judgement.

Used within these limits, 2D FEM is an excellent way to see where a member is most stressed and how a structure deforms - a complement to the closed-form checks in the rest of the toolkit.