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: altair 6.1.0 | Python 3.13.13
Quality: 91/100 | Updated: 2026-05-30
"""
import os
import sys
# altair.py shadows the altair package — remove this directory from sys.path first
_here = os.path.dirname(os.path.abspath(__file__))
sys.path[:] = [p for p in sys.path if os.path.realpath(p or os.getcwd()) != os.path.realpath(_here)]
import altair as alt
import numpy as np
import pandas as pd
from PIL import Image
# Theme tokens (Imprint style guide)
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
# Imprint palette positions used
BRAND = "#009E73" # pos 1 — stress state points + circle line
LAVENDER = "#C475FD" # pos 2 — max shear points
BLUE = "#4467A3" # pos 3 — principal stresses + 2θp arc
# Data — 2D stress state (MPa)
sigma_x = 80
sigma_y = -40
tau_xy = 30
center = (sigma_x + sigma_y) / 2
radius = np.sqrt(((sigma_x - sigma_y) / 2) ** 2 + tau_xy**2)
sigma_1 = center + radius
sigma_2 = center - radius
tau_max = radius
theta_p2 = np.degrees(np.arctan2(tau_xy, (sigma_x - sigma_y) / 2))
# Circle outline (parametric)
angles = np.linspace(0, 2 * np.pi, 361)
circle_df = pd.DataFrame(
{"sigma": center + radius * np.cos(angles), "tau": radius * np.sin(angles), "order": range(361)}
)
# Key stress points
stress_points = pd.DataFrame(
[
{
"sigma": sigma_x,
"tau": tau_xy,
"label": f"A ({sigma_x}, {tau_xy})",
"type": "Stress State",
"dx": 14,
"dy": -14,
},
{
"sigma": sigma_y,
"tau": -tau_xy,
"label": f"B ({sigma_y}, {-tau_xy})",
"type": "Stress State",
"dx": -18,
"dy": 14,
},
{"sigma": sigma_1, "tau": 0, "label": f"σ₁ = {sigma_1:.1f} MPa", "type": "Principal", "dx": 12, "dy": -16},
{"sigma": sigma_2, "tau": 0, "label": f"σ₂ = {sigma_2:.1f} MPa", "type": "Principal", "dx": -18, "dy": -16},
{
"sigma": center,
"tau": tau_max,
"label": f"τmax = {tau_max:.1f} MPa",
"type": "Max Shear",
"dx": 12,
"dy": -14,
},
{
"sigma": center,
"tau": -tau_max,
"label": f"−τmax = −{tau_max:.1f} MPa",
"type": "Max Shear",
"dx": 12,
"dy": 18,
},
]
)
# Diameter line A → B
diameter_df = pd.DataFrame({"sigma": [sigma_x, sigma_y], "tau": [tau_xy, -tau_xy]})
# 2θp angle arc
arc_r = radius * 0.25
arc_angles = np.linspace(0, np.radians(theta_p2), 50)
arc_df = pd.DataFrame(
{"sigma": center + arc_r * np.cos(arc_angles), "tau": arc_r * np.sin(arc_angles), "order": range(50)}
)
# Equal-aspect domains — square inner view (460×460) so circle renders as true circle
span = max(sigma_1 - sigma_2, 2 * tau_max) + 30
domain_sigma = [center - span / 2, center + span / 2]
domain_tau = [-span / 2, span / 2]
x_scale = alt.Scale(domain=domain_sigma)
y_scale = alt.Scale(domain=domain_tau)
# Reference lines through center
h_rule = (
alt.Chart(pd.DataFrame({"tau": [0]}))
.mark_rule(color=INK_SOFT, strokeWidth=1, opacity=0.5)
.encode(y=alt.Y("tau:Q", scale=y_scale))
)
v_rule = (
alt.Chart(pd.DataFrame({"sigma": [center]}))
.mark_rule(color=INK_SOFT, strokeWidth=1, opacity=0.4, strokeDash=[6, 4])
.encode(x=alt.X("sigma:Q", scale=x_scale))
)
# Circle
circle = (
alt.Chart(circle_df)
.mark_line(color=BRAND, strokeWidth=2.5)
.encode(
x=alt.X("sigma:Q", title="Normal Stress σ (MPa)", scale=x_scale),
y=alt.Y("tau:Q", title="Shear Stress τ (MPa)", scale=y_scale),
order="order:Q",
)
)
# Diameter line
diameter = (
alt.Chart(diameter_df)
.mark_line(color=BRAND, strokeWidth=1.5, strokeDash=[8, 5], opacity=0.5)
.encode(x="sigma:Q", y="tau:Q")
)
# 2θp arc — linked to principal plane rotation
arc = alt.Chart(arc_df).mark_line(color=BLUE, strokeWidth=2.5).encode(x="sigma:Q", y="tau:Q", order="order:Q")
# Angle label
angle_lbl_df = pd.DataFrame(
{
"sigma": [center + arc_r * 2.2 * np.cos(np.radians(theta_p2 / 2))],
"tau": [arc_r * 2.2 * np.sin(np.radians(theta_p2 / 2))],
}
)
angle_lbl = (
alt.Chart(angle_lbl_df)
.mark_text(text=f"2θp = {theta_p2:.1f}°", fontSize=11, fontWeight="bold", color=BLUE)
.encode(x="sigma:Q", y="tau:Q")
)
# Interactive hover selection (enhances HTML output)
highlight = alt.selection_point(fields=["type"], on="pointerover")
# Stress points with Imprint palette colors and hover interaction
points = (
alt.Chart(stress_points)
.mark_point(filled=True, strokeWidth=2, stroke=PAGE_BG)
.encode(
x="sigma:Q",
y="tau:Q",
color=alt.Color(
"type:N",
scale=alt.Scale(domain=["Stress State", "Principal", "Max Shear"], range=[BRAND, BLUE, LAVENDER]),
legend=alt.Legend(
title=None,
orient="bottom-right",
direction="vertical",
symbolSize=180,
symbolStrokeWidth=0,
labelFontSize=10,
padding=8,
offset=8,
cornerRadius=4,
),
),
size=alt.condition(highlight, alt.value(420), alt.value(300)),
opacity=alt.condition(highlight, alt.value(1.0), alt.value(0.85)),
tooltip=[
alt.Tooltip("label:N", title="Point"),
alt.Tooltip("sigma:Q", title="σ (MPa)", format=".1f"),
alt.Tooltip("tau:Q", title="τ (MPa)", format=".1f"),
alt.Tooltip("type:N", title="Category"),
],
)
.add_params(highlight)
)
# Center point
center_pt_df = pd.DataFrame({"sigma": [center], "tau": [0]})
center_pt = (
alt.Chart(center_pt_df)
.mark_point(size=180, filled=True, color=INK_MUTED, strokeWidth=2, stroke=PAGE_BG)
.encode(x="sigma:Q", y="tau:Q")
)
center_lbl = (
alt.Chart(center_pt_df)
.mark_text(text=f"C ({center:.0f}, 0)", fontSize=10, color=INK_SOFT, dy=18)
.encode(x="sigma:Q", y="tau:Q")
)
# Annotation labels — data-space offsets (px_to_data = span per inner-view pixel)
px_to_data = span / 460
stress_points["lbl_sigma"] = stress_points["sigma"] + stress_points["dx"] * px_to_data
stress_points["lbl_tau"] = stress_points["tau"] - stress_points["dy"] * px_to_data
labels = (
alt.Chart(stress_points)
.mark_text(fontSize=11, fontWeight="bold", color=INK)
.encode(x="lbl_sigma:Q", y="lbl_tau:Q", text="label:N")
)
# Title and subtitle
title_text = "mohr-circle · python · altair · anyplot.ai"
subtitle_text = f"2D Stress Transformation — σx={sigma_x}, σy={sigma_y}, τxy={tau_xy} MPa"
# Compose chart — square inner view (460×460) preserves circular aspect ratio
chart = (
alt.layer(h_rule, v_rule, circle, diameter, arc, points, center_pt, labels, center_lbl, angle_lbl)
.properties(
width=460,
height=460,
background=PAGE_BG,
title=alt.Title(
title_text,
fontSize=16,
fontWeight="bold",
color=INK,
subtitle=subtitle_text,
subtitleFontSize=11,
subtitleColor=INK_SOFT,
),
)
.configure_view(fill=PAGE_BG, strokeWidth=0, continuousWidth=460, continuousHeight=460)
.configure_axis(
labelFontSize=10,
titleFontSize=12,
titleColor=INK,
labelColor=INK_SOFT,
tickColor=INK_SOFT,
grid=True,
gridOpacity=0.12,
gridColor=INK,
domain=False,
)
.configure_legend(
fillColor=ELEVATED_BG,
strokeColor=INK_SOFT,
labelColor=INK_SOFT,
titleColor=INK,
labelFontSize=10,
titleFontSize=10,
)
.configure_title(color=INK)
)
# Save — target 2400×2400; pad with PAGE_BG if vl-convert lands short
TW, TH = 2400, 2400
chart.save(f"plot-{THEME}.png", scale_factor=4.0)
_img = Image.open(f"plot-{THEME}.png").convert("RGB")
_w, _h = _img.size
if _w > TW or _h > TH:
raise SystemExit(
f"altair vl-convert produced {_w}×{_h}, exceeds target {TW}×{TH}. "
f"Shrink chart .properties(width=, height=) values and re-render."
)
if _w < TW or _h < TH:
_canvas = Image.new("RGB", (TW, TH), PAGE_BG)
_canvas.paste(_img, ((TW - _w) // 2, (TH - _h) // 2))
_canvas.save(f"plot-{THEME}.png")
chart.save(f"plot-{THEME}.html")
Part of Mohr's Circle for Stress Analysis on anyplot.ai.