Mohr's Circle is a graphical method used to determine principal stresses, maximum shear stress, and stress transformations from a given 2D stress state. A circle is drawn on a normal stress (σ) vs. shear stress (τ) plane, with the center at ((σx + σy) / 2, 0) and a radius derived from the stress components. It is an essential tool in mechanical and civil engineering for visualizing how stress components change under coordinate rotation.
# anyplot.ai
# mohr-circle: Mohr's Circle for Stress Analysis
# Library: makie 0.22.10 | Julia 1.11.9
# Quality: 90/100 | Created: 2026-05-30
using CairoMakie
using Colors
using Random
Random.seed!(42)
# Theme tokens — Imprint palette, theme-adaptive chrome
const THEME = get(ENV, "ANYPLOT_THEME", "light")
const PAGE_BG = THEME == "light" ? colorant"#FAF8F1" : colorant"#1A1A17"
const INK = THEME == "light" ? colorant"#1A1A17" : colorant"#F0EFE8"
const INK_SOFT = THEME == "light" ? colorant"#4A4A44" : colorant"#B8B7B0"
const INK_MUTED = THEME == "light" ? colorant"#6B6A63" : colorant"#A8A79F"
const IMPRINT_PALETTE = [
colorant"#009E73", # 1 — brand green (Imprint palette — always first series)
colorant"#C475FD", # 2 — lavender
colorant"#4467A3", # 3 — blue
colorant"#BD8233", # 4 — ochre
colorant"#AE3030", # 5 — matte red (semantic anchor for critical values)
colorant"#2ABCCD", # 6 — cyan
colorant"#954477", # 7 — rose
colorant"#99B314", # 8 — lime
]
# Stress state: steel plate under combined biaxial loading and shear (MPa)
# Pythagorean triple (3-4-5) gives clean radius = 50 MPa
const sigma_x = 80.0
const sigma_y = 20.0
const tau_xy = 40.0
# Mohr's circle geometry
const center_c = (sigma_x + sigma_y) / 2.0
const radius = sqrt(((sigma_x - sigma_y) / 2.0)^2 + tau_xy^2)
const sigma1 = center_c + radius
const sigma2 = center_c - radius
const tau_max = radius
const two_theta_p = atan(tau_xy, sigma_x - center_c)
# Circle (parametric, 360 points)
const n_pts = 360
const angles = LinRange(0.0, 2π, n_pts)
const circle_x = center_c .+ radius .* cos.(angles)
const circle_y = radius .* sin.(angles)
# Title
const title_str = "mohr-circle · julia · makie · anyplot.ai"
const n_title = length(title_str)
const title_sz = round(Int, 20 * (n_title > 67 ? 67.0 / n_title : 1.0))
# Figure — square canvas (2400×2400 via px_per_unit=2) for true circle aspect ratio
fig = Figure(
size = (1200, 1200),
fontsize = 14,
backgroundcolor = PAGE_BG,
)
ax = Axis(
fig[1, 1];
title = title_str,
titlesize = title_sz,
titlecolor = INK,
xlabel = "Normal Stress σ (MPa)",
ylabel = "Shear Stress τ (MPa)",
xlabelsize = 14,
ylabelsize = 14,
xlabelcolor = INK,
ylabelcolor = INK,
xticklabelsize = 12,
yticklabelsize = 12,
xticklabelcolor = INK_SOFT,
yticklabelcolor = INK_SOFT,
xtickcolor = INK_SOFT,
ytickcolor = INK_SOFT,
backgroundcolor = PAGE_BG,
topspinevisible = false,
rightspinevisible = false,
leftspinecolor = INK_SOFT,
bottomspinecolor = INK_SOFT,
xgridcolor = RGBAf(INK.r, INK.g, INK.b, 0.10),
ygridcolor = RGBAf(INK.r, INK.g, INK.b, 0.10),
xminorgridvisible = false,
yminorgridvisible = false,
aspect = DataAspect(),
)
# Axis limits — equal ranges so DataAspect() fills the square canvas
const pad = 30.0
xlims!(ax, sigma2 - pad, sigma1 + pad)
ylims!(ax, -(tau_max + pad), tau_max + pad)
# Reference line: σ-axis (τ = 0)
hlines!(ax, [0.0]; color = (INK_SOFT, 0.55), linewidth = 1.2)
# Reference line: vertical through center C
vlines!(ax, [center_c]; color = (INK_SOFT, 0.4), linewidth = 1.0, linestyle = :dash)
# Mohr's circle — primary element, Imprint palette position 1 (green)
lines!(ax, circle_x, circle_y; color = IMPRINT_PALETTE[1], linewidth = 2.8)
# Diameter line from A to B (construction line)
lines!(ax, [sigma_x, sigma_y], [tau_xy, -tau_xy];
color = (INK_MUTED, 0.55), linewidth = 1.4, linestyle = :dash)
# Arc showing the principal-plane angle 2θp (measured at center C)
const arc_r = radius * 0.30
const arc_angs = LinRange(0.0, two_theta_p, 50)
const arc_x = center_c .+ arc_r .* cos.(arc_angs)
const arc_y = arc_r .* sin.(arc_angs)
lines!(ax, arc_x, arc_y; color = IMPRINT_PALETTE[4], linewidth = 2.2)
# Center point C
scatter!(ax, [center_c], [0.0]; color = INK, markersize = 9, strokewidth = 0)
# Point A: (σx, τxy) — x-face of the stress element
scatter!(ax, [sigma_x], [tau_xy];
color = IMPRINT_PALETTE[2], markersize = 17,
strokewidth = 1.5, strokecolor = PAGE_BG)
# Point B: (σy, −τxy) — y-face of the stress element
scatter!(ax, [sigma_y], [-tau_xy];
color = IMPRINT_PALETTE[3], markersize = 17,
strokewidth = 1.5, strokecolor = PAGE_BG)
# Principal stress points σ1, σ2 (circle intersects σ-axis)
scatter!(ax, [sigma1, sigma2], [0.0, 0.0];
color = IMPRINT_PALETTE[5], markersize = 17, marker = :diamond,
strokewidth = 1.5, strokecolor = PAGE_BG)
# τ_max points (top and bottom of circle)
scatter!(ax, [center_c, center_c], [tau_max, -tau_max];
color = IMPRINT_PALETTE[4], markersize = 17,
strokewidth = 1.5, strokecolor = PAGE_BG)
# Annotations
const nudge = 3.0
const voff = radius * 0.07
text!(ax, sigma1 + nudge, -voff;
text = "σ₁ = $(round(Int, sigma1)) MPa",
color = INK,
fontsize = 12,
align = (:left, :top))
text!(ax, sigma2 - nudge, -voff;
text = "σ₂ = $(round(Int, sigma2)) MPa",
color = INK,
fontsize = 12,
align = (:right, :top))
text!(ax, center_c + nudge, tau_max + voff;
text = "τmax = $(round(Int, tau_max)) MPa",
color = INK,
fontsize = 12,
align = (:left, :bottom))
text!(ax, center_c - nudge, -(tau_max + voff);
text = "−τmax",
color = INK,
fontsize = 12,
align = (:right, :top))
text!(ax, sigma_x + nudge, tau_xy;
text = "A ($(round(Int, sigma_x)), $(round(Int, tau_xy))) MPa",
color = IMPRINT_PALETTE[2],
fontsize = 11,
align = (:left, :center))
text!(ax, sigma_y - nudge, -tau_xy;
text = "B ($(round(Int, sigma_y)), $(round(Int, -tau_xy))) MPa",
color = IMPRINT_PALETTE[3],
fontsize = 11,
align = (:right, :center))
text!(ax, center_c, voff;
text = "C",
color = INK,
fontsize = 13,
align = (:center, :bottom))
# 2θp arc label
const mid_arc = two_theta_p / 2.0
text!(ax, center_c + arc_r * 1.45 * cos(mid_arc), arc_r * 1.45 * sin(mid_arc);
text = "2θₚ ≈ $(round(rad2deg(two_theta_p), digits=1))°",
color = IMPRINT_PALETTE[4],
fontsize = 11,
align = (:left, :center))
save("plot-$(THEME).png", fig; px_per_unit = 2)
Runnable source as JSON, for any HTTP client: https://api.anyplot.ai/specs/mohr-circle/makie/code. Any spec id and library id listed in llms-full.txt fit the same URL shape; every URL below is complete and callable.
{
"spec_id": "mohr-circle",
"language": "julia",
"library": "makie",
"page": "https://anyplot.ai/mohr-circle/julia/makie",
"hub": "https://anyplot.ai/mohr-circle",
"code_json": "https://api.anyplot.ai/specs/mohr-circle/makie/code",
"spec_json": "https://api.anyplot.ai/specs/mohr-circle",
"render_light_png": "https://storage.googleapis.com/anyplot-images/plots/mohr-circle/julia/makie/plot-light.png",
"render_dark_png": "https://storage.googleapis.com/anyplot-images/plots/mohr-circle/julia/makie/plot-dark.png",
"quality_score": 90.0,
"license": "MIT",
"guide": "https://anyplot.ai/llms.txt"
}Part of Mohr's Circle for Stress Analysis on anyplot.ai.