2D stress analysis of flat parts

A flat part such as a bracket, a lug, a gusset or a link carries its loads through its outline. Where the outline narrows, turns or has a hole, the stress rises above the nominal value of the hand formulas, by the stress concentration factor. Charts give that factor for standard notches and holes only. For any other outline, it follows from a solution of the elasticity equations. The tool meshes the part, solves these equations with the finite element method and shows the stress over the whole part, with its peak, the deformed shape and the reaction at every support.

Plane stress

The part is thin compared with its outline, and its loads act in its plane, so the stress across the thickness is zero. The tool solves the plane-stress elasticity equations for the displacement, with Young's modulus E and Poisson's ratio ν of the material and the thickness t. The thickness carries the loads, so the same force on a part twice as thick gives half the stress. The von Mises stress σ_v = √(σ_x² − σ_xσ_y + σ_y² + 3τ_xy²) combines the stresses into the one value that ductile metals yield at.

Edge and hole numbers

Every edge of the outline gets a number, from E1 at the lowest left corner around the part, and every hole one number for its whole boundary, H1, H2 and so on from the left. The supports and loads refer to these numbers, and the drawing shows them, with each edge's length and each hole's size on hover. The Propagation column carries a support or a load on along the outline, with planex2d's selection propagation.

• Minimal keeps the edge alone.
• Coincident adds the edges joined to it end to end, which on a closed outline is all of it.
• Tangent adds the edges that continue it tangentially, such as the lines on either side of a fillet.
• Curvature adds the edges that continue its curvature as well.

Supports

A fixed edge or hole cannot move in any direction. Fixed in x holds only the x direction and lets the edge slide along y, and fixed in y the other way round, which models a symmetry line or a roller. A fixed hole stands for a rigid bolt or pin. The supports must hold the part against moving and turning, and the tool says so when they do not.

Loads

• A force acts on an edge, spread evenly along it, at an angle from the x axis.
• A pressure acts normal to an edge or a hole's wall and pushes into the part. A negative pressure pulls.
• A bolt load acts on a hole the way a pin presses on it. It pushes on the half of the hole that faces the force, with the pressure p = 2F/(πr) · cos φ along the wall, where φ is measured from the direction of the force.

Results

The results are the peak von Mises stress and where it is, the largest displacement, the safety factor against yield, which is the yield strength over the peak stress, and the reaction force at every support. The drawing colors the part by the von Mises stress, marks its peak with a cross and shows the deformed outline as a dashed line, enlarged so that it can be seen. The legend shows the mesh on request.

The list beside the Results heading colors the part by another result of the same solution instead. It offers the normal stresses σ_x and σ_y, the shear stress τ_xy, the principal stresses σ₁ and σ₂ and the displacement. The stresses among them have a sign, so their color scale is centered on zero, with positive values in red and negative ones in blue. For the normal and principal stresses, tension is red and compression blue. The results then also give the range of the chosen stress, and the drawing marks its largest and smallest values.

At a sharp inside corner and at the end of a fully supported edge, the stress is infinite in theory, so the value there grows as the mesh is refined. The tool says when the peak sits at such a point.

Simplify outline

Drawings often carry arcs and holes as chains of short lines. Their corners would show as false stress peaks and would give the outline many edge numbers. With Simplify outline on, planex2d fits arcs to such chains, within a ten-thousandth of the part's size, before the part is numbered and meshed. Turn it off to analyze the outline exactly as drawn. Where the fit misses an arc, the contour checker fits arcs with a tolerance and an edge length of your choice, closes gaps and removes tiny edges, and saves the cleaned drawing for the stress analysis.

Units

Lengths and displacements are in millimeters, forces in newtons, pressures and stresses in MPa and Young's modulus in GPa. A file is read in the unit it names and converted to millimeters.

1. Load

   The page loads Python and planex2d into this tab, and NGSolve with the first solve, about 23 MB the first time. Your browser keeps them in its cache, so later visits start faster.

2. Read

   planex2d reads the file, joins its closed contours into the part, simplifies the outline and numbers its edges and holes.

3. Solve

   NGSolve meshes and solves the part, again whenever a setting changes after the first solve.

4. Save

   The results are saved as a PDF report or a CSV table, made in the tab as well.

The finite element model

The mesh has elements of about a fortieth of the part's size, finer along arcs and holes, where they are a third of the radius, and it is curved to the exact outline. NGSolve solves the plane-stress equations on it with second-order elements and a sparse Cholesky factorization. The tests compare the solver with known answers. A bar in tension gives F/A exactly, a cantilever its tip deflection P L³/(3 E I) within 0.1 %, a plate with a hole its peak stress within 1 % of Heywood's formula, and the reactions balance every load.

Element order and accuracy estimate

The Solver group sets the element order, from 1 to 5, and whether the elements are curved to the outline. A higher order resolves the stress better on the same mesh and takes longer. Straight elements replace each arc with chords, whose corners at the mesh vertices raise the stress along holes and fillets.

Accuracy estimate, on by default, solves the part a second time on the same mesh with elements one order higher and shows the peak stress of that solution and how much it differs. The difference estimates the error of the peak stress at the chosen order, so a difference of a percent or less means that the mesh resolves the peak. A larger one calls for a higher order, unless the peak sits at a sharp inside corner or at the end of a supported edge, where it grows with every refinement and the tool says so. At order 2 a solve takes about 1.5 times as long with the estimate.

Supported files

Run it in Python

The same analysis takes a few lines with planex2d and NGSolve. The example is the plate with a hole sample, 400 × 100 mm and 5 mm thick, with a hole of 20 mm in its middle, fixed at its left edge and pulled with 20 kN at its right edge. planex2d draws the plate and names its edges, and NGSolve meshes it, finer along the hole, and solves it. See Meshing with netgen and the shoulder fillet example in the docs.

$ pip install "planex2d[occ]" ngsolve

import ngsolve as ngs
import planex as px

plate = (px.QuickSketch().Rectangle(400, 100).Face() - px.Circle((200, 50), 10).Face()).Face()
for e in plate.edges:
    e.name = {0: "left", 400: "right"}.get(e.midpoint.x, "free")
    if e.midpoint.y not in (0, 100) and e.name == "free":
        e.maxh = 10 / 3
mesh = ngs.Mesh(px.OCCGeometry(plate).GenerateMesh(maxh=10))
mesh.Curve(2)
fes = ngs.VectorH1(mesh, order=2, dirichlet="left")
u, v = fes.TnT()
E, nu, t = 210e3, 0.3, 5
mu, lam = E / 2 / (1 + nu), E * nu / (1 - nu**2)


def stress(w):
    e = ngs.Sym(ngs.Grad(w))
    return 2 * mu * e + lam * ngs.Trace(e) * ngs.Id(2)


a = ngs.BilinearForm(t * ngs.InnerProduct(stress(u), ngs.Sym(ngs.Grad(v))) * ngs.dx).Assemble()
f = ngs.LinearForm(20000 / 100 * v[0] * ngs.ds("right")).Assemble()
gf = ngs.GridFunction(fes)
gf.vec.data = a.mat.Inverse(fes.FreeDofs(), inverse="sparsecholesky") * f.vec
print("stress beside the hole", stress(gf)[0, 0](mesh(200, 39.99)), "MPa")

It prints a stress of about 127 MPa beside the hole, 2.54 times the net stress of 50 MPa in the section through the hole. Heywood's formula for this ratio of hole to width gives 2.51.

Privacy

Limits

• The tool solves linear elasticity in plane stress: small displacements, a linear material and loads in the part's plane.
• Contact, friction, plasticity, buckling, bending out of the plane and 3D parts are not included.
• A bolt load presses on its hole as a pin would, with an assumed pressure distribution. The bolt, its preload and the friction under its head are not modeled.
• The safety factor compares the peak stress with the yield strength only. It does not cover fatigue or the checks of a design code. The material values are typical minimum values. Use the ones certified for your material.
• The tool analyzes one connected part. A file must be planar, and its text, dimensions and hatches are not read.
• A part that fails or a result that looks wrong? Open an issue on GitHub and attach the file if you can.