Finite Element Analysis of a Curved Elastic Arch under Traction

Model a curved elastic structure in plane stress and use shear strain to color the deformed configuration across load levels

This example studies a thin quarter-arch modeled in plane stress, fixed along its horizontal base and subjected to a vertical traction applied on its vertical edge. The deformation is computed for a sequence of increasing load values, and the resulting shear strain is used to consistently color the deformed geometry across all load levels.

Specify inner and outer radii to represent a thin curved structure, restricting the angular span to a right angle and discretizing the region so that it can be used in a finite element method analysis:

In[1]:=

{r1,r2}={90,100};Ω=DiscretizeRegion[Annulus[{0,0},{r1,r2},{0,π/2}]]

Out[1]=

Load the finite element method utilities so that region-to-mesh conversion is available for later sampling and post-processing steps:

In[2]:=

Needs["NDSolve`FEM`"];

Declare the unknown displacement components 𝑢(𝑥, 𝑦) and 𝑣(𝑥, 𝑦), which describe horizontal and vertical motion of each material point. Additionally, assume linear elasticity, where stresses are proportional to strains, and select a plane stress formulation, appropriate for thin structures where stresses perpendicular to the plane are negligible:

In[3]:=

vars={{u[x,y],v[x,y]},{x,y}};pars="YoungModulus"3010^6,"PoissonRatio"0.3,"Thickness"0.1,"SolidMechanicsModelForm""PlaneStress";

Define the largest vertical traction that will be applied on the boundary:

In[4]:=

maxLoad=10000;

Apply a boundary traction on the edge 𝑥 = 0 and fix the edge 𝑦 = 0, then solve for the displacement field. Interpret the traction vector

{0,-maxLoad}

as a load with no horizontal component and a downward vertical component. In a plane-stress setting, this traction represents force per unit length of boundary and per unit thickness:

In[5]:=

displacement=NDSolveValue[{SolidMechanicsPDEComponent[vars,pars]SolidBoundaryLoadValue[x0,vars,pars,"Pressure"{0,-maxLoad}],SolidFixedCondition[y0,vars,pars]},{u[x,y],v[x,y]},{x,y}∈Ω]

Out[5]=

InterpolatingFunction

Domain: {{0.,100.},{0.,100.}}

Output: scalar

[x,y],InterpolatingFunction

Domain: {{0.,100.},{0.,100.}}

Output: scalar

[x,y]

The shear strain

𝑥𝑦

measures angular distortion and is defined as

𝑥𝑦

(

∂

∂

∂

∂

), which captures shape change without pure stretching.

Compute the small-strain tensor derived from displacement gradients and select its shear component:

In[6]:=

strain=SolidMechanicsStrain[vars,pars,displacement];shearStrain=strain〚1,2〛

Out[6]=

InterpolatingFunction

Domain: {{0.,100.},{0.,100.}}

Output: scalar

[x,y]

Convert the computational region into a finite element mesh and evaluate the shear strain at all mesh nodes. Use the resulting discrete strain values to determine the minimum and maximum shear strain over the domain, which provides a stable and reproducible range for defining a fixed color scale in all subsequent visualizations:

In[7]:=

pts=MeshCoordinates[Ω];vals=(shearStrain/.Thread[{x,y}#])&/@pts;vals=Select[vals,NumericQ];{minStrain,maxStrain}=MinMax[vals]

Out[7]=

{-0.0184575,0.0195177}

Map shear strain values to colors using a fixed scale based on the previously computed global strain range:

In[8]:=

colorfunction=ColorData["Rainbow"][Rescale[#5,{minStrain,maxStrain}]]&;

Create a single color legend corresponding to the global shear-strain range so that all visualizations can be interpreted using the same reference scale:

In[9]:=

legend=BarLegend[{ColorData["Rainbow"][Rescale[#1,{minStrain,maxStrain}]]&,{minStrain,maxStrain}},LegendLabel"Shear Strain"];

Solve the elasticity problem for a sequence of increasing traction values, evaluate the corresponding shear strain field, and visualize the deformed geometry colored by shear strain using the fixed color scale:

In[10]:=

tb=Table[displacement=NDSolveValue[{SolidMechanicsPDEComponent[vars,pars]SolidBoundaryLoadValue[x0,vars,pars,"Pressure"{0,-load}],SolidFixedCondition[y0,vars,pars]},{u[x,y],v[x,y]},{x,y}∈Ω];strain=SolidMechanicsStrain[vars,pars,displacement];shearStrain=strain〚1,2〛;Legended[VectorDisplacementPlot[{displacement,shearStrain},{x,y}∈Ω,ColorFunctioncolorfunction,ColorFunctionScalingFalse,PlotRange{{-30,110},{0,110}},VectorSizesFull,PlotLabelRow[{"Stress= ",load}]],legend],{load,0,maxLoad,maxLoad/10}];

Display the sequence of deformed configurations and step through the applied load levels to observe how shear strain evolves as the traction increases:

In[11]:=

SlideShowVideo[Rasterize[#,RasterSize1000]&/@tb]

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Finite Element Analysis of a Curved Elastic Arch under Traction

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Finite Element Analysis of a Curved Elastic Arch under Traction

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