2D Linear Elasticity Simulator — Plane Stress vs Plane Strain
A tool that compares the two assumptions used to reduce a 3D elasticity problem to 2D: plane stress and plane strain. Enter the in-plane stresses σx, σy and τxy and Hooke's law gives the strain components, the out-of-plane component and the von Mises stress in real time, showing how the result changes between a thin plate and a thick body.
Parameters
Young's modulus E
GPa
Elastic modulus of the material. About 200 GPa for steel
Poisson's ratio ν
Degree of lateral contraction. About 0.3 for steel
Stress σx
MPa
Normal stress in x (positive = tension)
Stress σy
MPa
Normal stress in y (positive = tension)
Shear stress τxy
MPa
In-plane shear stress
Analysis mode
Switches how the out-of-plane direction is treated
Results
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Normal strain εx (×10⁻³)
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Normal strain εy (×10⁻³)
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Shear strain γxy (×10⁻³)
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Out-of-plane component
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von Mises stress (MPa)
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Shear modulus G (GPa)
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Stressed material element — deformation animation
A square material element under σx, σy and τxy. The dashed outline is the original shape; the filled shape is the exaggerated deformed shape (colour shows the von Mises stress level).
Instead of the out-of-plane strain being free, it is constrained to zero and an out-of-plane stress σz appears. The shear modulus is common to both modes: G=E/(2(1+ν)).
What is 2D Linear Elasticity?
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In mechanics of materials I keep seeing "plane stress" and "plane strain". They sound alike and I can't tell them apart. What's the difference?
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Both are assumptions that let you solve a problem that is really 3D on a flat 2D sheet. The key is how you treat the depth (out-of-plane, z) direction. Plane stress says "the z-direction stress is zero"; plane strain says "the z-direction strain is zero". Roughly: use plane stress for thin things and plane strain for thick things.
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Why does thin versus thick decide which one to use?
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When you pull a thin plate in its plane, the depth is so small that it can shrink freely in z. So the z-direction stress is essentially zero — that's plane stress. Now take a dam cross-section that is metres deep. The slice you are looking at has identical material packed in front of and behind it, so it cannot stretch or shrink in z because the neighbouring material holds it back. So the z-direction strain is zero — that's plane strain.
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I see. So switching the "Analysis mode" on the top right changes the result even with the same σx and σy because of that.
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Exactly. In plane stress σz=0, but the plate freely contracts in depth through the Poisson effect, so εz takes the non-zero value -ν(σx+σy)/E. Plane strain is the opposite: εz is forced to 0, and as the reaction an out-of-plane stress σz=ν(σx+σy) appears. Whether that σz exists or not changes even the von Mises stress. Switch the mode and compare the "Out-of-plane component" card and the von Mises number.
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The von Mises stress is the one used for yield prediction, right? Does its formula change in 2D?
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The form changes. In plane stress σz=0, so it becomes the tidy expression √(σx²-σxσy+σy²+3τxy²). In plane strain σz is not zero, so you use the general triaxial form √(0.5((σx-σy)²+(σy-σz)²+(σz-σx)²)+3τxy²). In practice you decide "this part is thin, so plane stress is fine" and solve a 2D model — but if that judgement is wrong, you misread the stress. That is why it pays to compute both first and get a feel for them.
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So before starting a 2D FEM analysis it's good to check the ballpark with this tool.
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Right. FEM provides separate "plane-stress elements" and "plane-strain elements", and choosing the wrong one means solving a physically different problem. If you already know, from this tool, the strain and von Mises for a point given σx, σy and τxy, then when the FEM result is off by an order of magnitude you immediately think "ah, I picked the wrong mode". The Mohr's circle below is also a classic way to read off the in-plane principal stresses and the maximum shear at a glance.
Frequently Asked Questions
Both are assumptions that reduce a 3D elasticity problem to 2D. Plane stress assumes the out-of-plane stress is zero (σz=0) and applies to bodies, such as thin plates, that are free to deform in the thickness direction. The out-of-plane strain εz = -ν(σx+σy)/E is not zero. Plane strain assumes the out-of-plane strain is zero (εz=0) and applies to bodies, such as dams or long extrusions, whose thickness deformation is constrained. An out-of-plane stress σz = ν(σx+σy) then appears. For the same in-plane stresses, the strains and the von Mises stress differ because the out-of-plane component is treated differently.
For plane stress: εx=(σx-νσy)/E, εy=(σy-νσx)/E and γxy=τxy/G, where G is the shear modulus G=E/(2(1+ν)). For plane strain the contribution of the out-of-plane stress σz is included, giving εx=((1-ν²)σx-ν(1+ν)σy)/E and εy=((1-ν²)σy-ν(1+ν)σx)/E. The shear strain γxy=τxy/G is the same in both modes. This tool switches the formulas based on the chosen mode and shows the three strain components together with the out-of-plane component.
The von Mises equivalent stress collapses a multi-axial stress state into a single scalar used for yield prediction. For plane stress (σz=0) it takes the compact form σ_vm=√(σx²-σxσy+σy²+3τxy²). For plane strain σz=ν(σx+σy) is not zero, so the general form σ_vm=√(0.5((σx-σy)²+(σy-σz)²+(σz-σx)²)+3τxy²) is used. For the same in-plane stresses the plane-strain value differs by the contribution of σz, so the likelihood of yielding also changes.
Judge by the ratio of the body's thickness (depth) to its in-plane dimensions. If the thickness is much smaller than the in-plane size and the body can stretch freely out of plane, use plane stress: gear teeth, thin sheet-metal parts and panels carrying in-plane loads. If the thickness is large and out-of-plane deformation is constrained by surrounding material, use plane strain: a dam cross-section, a tunnel, a long extrusion or the mid-section of a pipe. For intermediate thicknesses, the practical approach is to compute both and take the more conservative result.
Real-World Applications
Thin-plate and sheet-metal design (plane stress): Thin metal plates under in-plane loads, brackets, gear-tooth profiles and linkage plates are treated as plane stress, because the plate thickness is small compared with the in-plane dimensions and the through-thickness stress can be neglected. Stress concentration around a hole and the stress distribution near a notch — the most common cases in machine design — fall in this category, and FEM also uses plane-stress elements as the standard choice.
Civil and geotechnical structures (plane strain): A concrete dam cross-section, the ground around a tunnel, retaining walls and long embankments — structures whose cross-section continues uniformly along the depth — are analysed as plane strain. Identical material continues in front of and behind the slice being studied, so out-of-plane deformation is constrained. Missing the resulting out-of-plane stress σz would underestimate the stress state.
Long machine components (plane strain): The mid-section of a long pipe, extruded profiles, the middle of a rotating shaft and the cross-section of a thick-walled cylinder are also typical plane-strain cases. Away from the ends, where end effects do not reach, the cross-section deforms uniformly along the length so εz≈0 holds. In the stress analysis of thick-walled pressure vessels, the mid-section is first approximated with plane strain as the starting point.
Mode-selection check for CAE modelling: When you build a 2D model in FEM, the first decision is "plane-stress element or plane-strain element". Being able to compare the strains and von Mises of both modes instantly for a single stress state, as this tool does, lets you confirm the modelling choice intuitively. Use it for a hand-calculation estimate before the detailed analysis, and as a sanity check: if the FEM result is off by an order of magnitude, suspect a wrong element type.
Common Misconceptions and Pitfalls
The biggest misconception is thinking that in plane stress the out-of-plane strain is also zero, and in plane strain the out-of-plane stress is also zero. The truth is the opposite. In plane stress σz=0, but the out-of-plane strain εz=-ν(σx+σy)/E is not zero (the plate contracts in depth). In plane strain εz=0, but the out-of-plane stress σz=ν(σx+σy) is not zero (a constraint reaction appears). Mix up which quantity — stress or strain — the mode sets to zero, and you completely misread the meaning of the result. The "Out-of-plane component" card in this tool switches between displaying εz and σz depending on which is meaningful for the chosen mode.
Next, assuming that the same in-plane stresses give the same von Mises stress. Even with identical σx, σy and τxy, the von Mises stress does not match between plane stress and plane strain. In plane strain a third stress component σz=ν(σx+σy) is added, so the value is computed with the general triaxial formula and departs from the plane-stress value. The difference is especially noticeable when σx and σy are both large and of the same sign, making σz large. For yield prediction, always use the von Mises of the correct mode.
Finally, do not forget the limits of linear elasticity itself. This tool handles only linear elasticity based on Hooke's law. Once the stress exceeds the material's yield stress (roughly 200-400 MPa for steel) the material deforms plastically and stress and strain are no longer proportional. In the region where the von Mises stress exceeds yield, the strain computed here comes out smaller than the real value. Large deformations beyond a few percent strain, and problems involving buckling or contact, are also outside linear elasticity. Always keep in mind that linear elasticity is valid only under the assumption of "small deformation, before yield".
How to Use
Enter Young's modulus E (GPa) and Poisson's ratio ν for your material—typical values: aluminum ν=0.33, steel ν=0.30, concrete ν=0.20.
Input in-plane stresses σx and σy (MPa) representing your load case on a 2D element.
Toggle between Plane Stress (σz=0, free out-of-plane strain) and Plane Strain (εz=0, constrained thickness) to observe how each assumption changes εx, εy, γxy, and von Mises stress.
Worked Example
Steel plate under biaxial loading: E=210 GPa, ν=0.30, σx=150 MPa, σy=50 MPa. Plane Stress mode yields εx≈0.686×10⁻³, εy≈−0.071×10⁻³. Switching to Plane Strain increases εx to 0.643×10⁻³ while locking εz=0, raising von Mises from 139 MPa to 161 MPa due to additional out-of-plane reaction stress σz=60 MPa.
Practical Notes
Use Plane Stress for thin shells, membranes, and surface-mounted sensors where out-of-plane constraint is absent.
Use Plane Strain for thick dams, foundation slabs, and long cylinders where thickness far exceeds in-plane dimensions and axial deformation is negligible.
Shear modulus G=E/[2(1+ν)] auto-computes; mismatched ν values between modes highlight artificial stiffening in constrained conditions.