The previous article broke down the three pillars behind every linear elastic analysis: equilibrium, kinematics and the constitutive law. All three hold exactly the same way in a full 3D solid as in a flat sheet —the physics doesn't care. What changes, a lot, is the cost of solving it. In simulation engineering there's a persistent temptation to model everything in 3D "because reality is 3D": it's a common mistake, and an expensive one. A full solid where it isn't needed doesn't just blow up solve time —it also multiplies the odds of numerical and meshing errors that a simpler model would never have let through. Tools like Creo Simulate or Creo Ansys Simulation build in three special stress states —plane stress, plane strain and axisymmetry— that let you exploit a part's symmetry and geometric proportions to solve on a surface what would otherwise demand a full volume.
Plane stress: thinness as a hypothesis
Plane stress applies when one of the three dimensions —the thickness— is significantly smaller than the other two, and the loads act exclusively within the part's main plane. Under that assumption, the normal and shear stresses in the thickness direction are taken as zero across the whole domain:
There's a common conceptual trap here: since there's no stress in z, it's tempting to assume there's no strain in z either. That's false. Through the Poisson effect, the material contracts or expands through the thickness even though σz is zero —strain εz exists, is non-zero, and depends directly on the in-plane stresses. What drops out of the system is a stress, not a movement.
It's the natural state of thin sheet metal, cladding panels, thin cylindrical gears, or something as everyday as a spanner, as long as the load stays within its own plane and there's no appreciable stress gradient through the thickness.
In Creo, setting a model to «2D Plane Stress» isn't just a label: the solver distributes loads per unit area or volume so that a two-dimensional surface reproduces the same behaviour the real sheet would show across its full thickness —and, by dropping the z-related rows and columns from the constitutive matrix [D], the problem shrinks by the same proportion.
Plane strain: the domain of the massive
Plane strain solves the opposite problem: bodies that are extremely long in one direction, with a constant cross-section and a load that doesn't vary along that length. Here what vanishes isn't a stress but a movement —the material is assumed to be so confined by the adjacent mass that strain in the longitudinal direction is zero:
And here's the exact mirror of the earlier trap: unlike plane stress, plane strain does have a non-zero normal stress σz. The material "wants" to strain longitudinally through the Poisson effect, but since it's prevented from doing so by the material surrounding it in that direction, that tendency turns into internal stress instead of displacement. Each state zeroes one quantity and frees the other —never both at once.
It's the natural simplification for concrete dams, tunnels, retaining walls, rolling mill rollers, or thick-walled cylinders such as the body of a massive hydraulic actuator. Modelling one metre of a ten-kilometre tunnel is enough to understand the structural behaviour of the entire structure —the sections on either side act as a constraint, not as part of the model.
One nuance worth keeping in mind: in a plane strain model, a load applied on a curve or edge of the 2D section physically represents an area load on the real surface that curve stands in for. Mixing up that equivalence —treating a line load as if it were a point load, or the reverse— is the most common way to introduce an order-of-magnitude error the software will never flag.
Axisymmetry: the power of radial symmetry
Of the three, axisymmetry is the most elegant simplification because it doesn't require the part to be thin or long: it requires it to revolve. It applies to any body generated by revolving a flat section around a central axis, as long as geometry, loads and constraints are all symmetric about that axis.
The software switches reference frame: instead of (x, y, z) it works in cylindrical coordinates (r, θ, z). Symmetry forces the tangential displacement to zero —v = 0— and means nothing, not geometry, not load, not result, depends on the angle θ. That's what lets the problem be solved on a single radial-axial plane instead of a full volume.
The computational advantage isn't marginal: it turns a massive volumetric problem into an area analysis on the r-z plane. That makes it possible to use extremely fine meshes in critical regions —the root of a thread, a fillet radius— that would be unworkable at the same density in a full 3D model.
It fits parts that look nothing alike: pressure vessels —cylindrical tanks with hemispherical heads, spherical tanks—, rotors, turbine shafts, flywheels, bushings, and bolted connections when the thread's helix angle is ignored.
In Creo, an axisymmetric model reduces the entire volume to a single flat face on the XY plane, with the Y axis as the axis of symmetry. The software mentally reconstructs the solid by sweeping that face 360° around the axis —without meshing a single element outside that plane.
How to choose between the three
The decision doesn't come down to preference —it comes down to three questions about the part's geometry and load path:
- Plane stress. Is the thickness much smaller than the other two dimensions, with all the load acting inside that plane? The key variable is thickness, treated as a simple parameter.
- Plane strain. Is the length much greater than the cross-section, with loads perpendicular to the long axis and constant along it? The key variable is a unit thickness for the slice that stands in for the whole part.
- Axisymmetry. Is the part a solid of revolution, with geometry, loads and constraints symmetric about an axis? The key variables become radius and angle, not the three Cartesian coordinates.
Overusing these simplifications is a common source of error. The most frequent case: modelling a thin plate as plane stress when the real loads make it bend out of its own plane. In that scenario the assumption doesn't hold —what's needed is a shell element, which does capture bending and transverse shear, not a simplification built for purely membrane loading.
Why (and when) it's worth reducing 3D to 2D
- Less memory, less solve time. A 2D model needs a fraction of the RAM and CPU time of its 3D equivalent —headroom that can go straight into far more ambitious optimisation or sensitivity studies.
- Finer meshes where they matter. With far fewer elements overall, you can afford a much higher mesh density at stress concentrations or fillet radii without solve time exploding.
- Less surface area for error. Fewer degrees of freedom, fewer boundary conditions to define, fewer chances to slip in a meshing or loading mistake that goes unnoticed.
What doesn't change with more compute power
In a future with practically unlimited compute thanks to the cloud, the reasonable question isn't whether 2D simplification will still be possible, but whether it'll still be necessary. And it's worth being precise here: the limit was never just raw compute power. A full 3D model still demands more of the engineer's time to build, mesh, debug and interpret, and it produces orders of magnitude more results to review. Reducing a problem from 3D to 2D isn't approximating the physics: it's recognising the physics the geometry was already handing you, and letting the software stop ignoring it. As long as the engineer's time stays the most expensive resource in the process —more so than any CPU core in the cloud— identifying the dominant physics and choosing the stress state that best represents it will stay part of the judgement call, not just the calculation.