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: seaborn 0.13.2 | Python 3.13.13
Quality: 90/100 | Updated: 2026-05-30
"""
import os
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import seaborn as sns
from matplotlib.lines import Line2D
# Theme tokens
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 — canonical order, first series always #009E73
IMPRINT_PALETTE = ["#009E73", "#C475FD", "#4467A3", "#BD8233", "#AE3030", "#2ABCCD", "#954477", "#99B314"]
COLOR_CIRCLE = IMPRINT_PALETTE[0] # brand green — Mohr's circle boundary
COLOR_STATE = IMPRINT_PALETTE[1] # lavender — stress state points A, B
COLOR_SIGMA = IMPRINT_PALETTE[2] # blue — principal stresses σ₁, σ₂
COLOR_SHEAR = IMPRINT_PALETTE[3] # ochre — max shear τmax
COLOR_ANGLE = IMPRINT_PALETTE[4] # matte red — principal angle 2θp arc
sns.set_theme(
style="ticks",
rc={
"figure.facecolor": PAGE_BG,
"axes.facecolor": PAGE_BG,
"axes.edgecolor": INK_SOFT,
"axes.labelcolor": INK,
"text.color": INK,
"xtick.color": INK_SOFT,
"ytick.color": INK_SOFT,
"grid.color": INK,
"grid.alpha": 0.12,
"legend.facecolor": ELEVATED_BG,
"legend.edgecolor": INK_SOFT,
},
)
# Data — stress state for a shaft under combined torsion and bending
sigma_x = 70
sigma_y = -50
tau_xy = 35
# Mohr's circle parameters
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
two_theta_p = np.degrees(np.arctan2(tau_xy, (sigma_x - sigma_y) / 2))
# Circle coordinates
theta = np.linspace(0, 2 * np.pi, 360)
circle_sigma = center + radius * np.cos(theta)
circle_tau = radius * np.sin(theta)
# Key points DataFrame for seaborn semantic mapping
points_df = pd.DataFrame(
{
"sigma": [sigma_x, sigma_y, sigma_1, sigma_2, center, center],
"tau": [tau_xy, -tau_xy, 0, 0, tau_max, -tau_max],
"Element": [
"Stress State (A, B)",
"Stress State (A, B)",
"Principal Stress (σ₁, σ₂)",
"Principal Stress (σ₁, σ₂)",
"Max Shear (τmax)",
"Max Shear (τmax)",
],
}
)
element_palette = {
"Stress State (A, B)": COLOR_STATE,
"Principal Stress (σ₁, σ₂)": COLOR_SIGMA,
"Max Shear (τmax)": COLOR_SHEAR,
}
element_markers = {"Stress State (A, B)": "o", "Principal Stress (σ₁, σ₂)": "D", "Max Shear (τmax)": "s"}
# Plot — square canvas for equal-aspect Mohr's circle (2400×2400 px)
fig, ax = plt.subplots(figsize=(6, 6), dpi=400, facecolor=PAGE_BG)
ax.set_facecolor(PAGE_BG)
# Mohr's circle with subtle fill
ax.plot(circle_sigma, circle_tau, color=COLOR_CIRCLE, linewidth=2.5, zorder=3)
ax.fill(circle_sigma, circle_tau, color=COLOR_CIRCLE, alpha=0.05, zorder=1)
# Reference lines through center
ax.axhline(y=0, color=INK_SOFT, linewidth=1.0, zorder=2)
ax.axvline(x=center, color=INK_MUTED, linewidth=0.8, linestyle="--", alpha=0.6, zorder=2)
# Diameter line connecting A and B
ax.plot([sigma_x, sigma_y], [tau_xy, -tau_xy], color=COLOR_STATE, linewidth=1.5, linestyle="--", alpha=0.5, zorder=3)
# Key points via seaborn scatterplot with hue + style semantic mapping
sns.scatterplot(
data=points_df,
x="sigma",
y="tau",
hue="Element",
style="Element",
markers=element_markers,
palette=element_palette,
s=220,
edgecolor=PAGE_BG,
linewidth=1.5,
zorder=5,
ax=ax,
legend=False,
)
# Center point
ax.scatter([center], [0], s=100, color=INK_SOFT, edgecolors=PAGE_BG, linewidth=1.5, zorder=5)
# Annotations — graduated visual hierarchy matching the spec requirements
# Input stress state info box (tertiary — small, monospace)
info_text = f"σx = {sigma_x} MPa\nσy = {sigma_y} MPa\nτxy = {tau_xy} MPa"
ax.text(
0.98,
0.98,
info_text,
transform=ax.transAxes,
fontsize=7,
verticalalignment="top",
horizontalalignment="right",
family="monospace",
color=INK_SOFT,
bbox={"boxstyle": "round,pad=0.4", "facecolor": ELEVATED_BG, "edgecolor": INK_MUTED, "alpha": 0.9},
linespacing=1.5,
)
# Center label (tertiary)
ax.annotate(
f"C ({center:.0f}, 0)",
xy=(center, 0),
xytext=(center - 6, -radius * 0.26),
fontsize=7,
color=INK_SOFT,
ha="right",
arrowprops={"arrowstyle": "->", "color": INK_SOFT, "lw": 1.0},
zorder=6,
)
# Stress point A (secondary)
ax.annotate(
f"A ({sigma_x}, {tau_xy})",
xy=(sigma_x, tau_xy),
xytext=(sigma_x + 10, tau_xy + 18),
fontsize=8,
fontweight="bold",
color=COLOR_STATE,
ha="left",
arrowprops={"arrowstyle": "->", "color": COLOR_STATE, "lw": 1.3},
zorder=6,
)
# Stress point B (secondary)
ax.annotate(
f"B ({sigma_y}, {-tau_xy})",
xy=(sigma_y, -tau_xy),
xytext=(sigma_y - 12, -tau_xy - 18),
fontsize=8,
fontweight="bold",
color=COLOR_STATE,
ha="right",
arrowprops={"arrowstyle": "->", "color": COLOR_STATE, "lw": 1.3},
zorder=6,
)
# Principal stresses (primary — boxed, bold, prominent arrows)
ax.annotate(
f"σ₁ = {sigma_1:.1f} MPa",
xy=(sigma_1, 0),
xytext=(center + radius * 0.42, -radius * 0.46),
fontsize=8,
fontweight="bold",
color=COLOR_SIGMA,
ha="center",
arrowprops={"arrowstyle": "-|>", "color": COLOR_SIGMA, "lw": 1.5, "mutation_scale": 12},
bbox={"boxstyle": "round,pad=0.25", "facecolor": ELEVATED_BG, "edgecolor": COLOR_SIGMA, "alpha": 0.92},
zorder=6,
)
ax.annotate(
f"σ₂ = {sigma_2:.1f} MPa",
xy=(sigma_2, 0),
xytext=(center - radius * 0.42, -radius * 0.46),
fontsize=8,
fontweight="bold",
color=COLOR_SIGMA,
ha="center",
arrowprops={"arrowstyle": "-|>", "color": COLOR_SIGMA, "lw": 1.5, "mutation_scale": 12},
bbox={"boxstyle": "round,pad=0.25", "facecolor": ELEVATED_BG, "edgecolor": COLOR_SIGMA, "alpha": 0.92},
zorder=6,
)
# Max shear stress (secondary)
ax.annotate(
f"τmax = {tau_max:.1f} MPa",
xy=(center, tau_max),
xytext=(center + radius * 0.52, tau_max + 6),
fontsize=8,
fontweight="bold",
color=COLOR_SHEAR,
arrowprops={"arrowstyle": "->", "color": COLOR_SHEAR, "lw": 1.3},
zorder=6,
)
# 2θp angle arc (from positive x-axis to the stress state point A direction)
arc_angles = np.linspace(0, np.radians(two_theta_p), 50)
arc_r = radius * 0.3
arc_x = center + arc_r * np.cos(arc_angles)
arc_y = arc_r * np.sin(arc_angles)
ax.plot(arc_x, arc_y, color=COLOR_ANGLE, linewidth=2.0, zorder=4)
mid_angle = np.radians(two_theta_p / 2)
ax.text(
center + arc_r * 1.5 * np.cos(mid_angle),
arc_r * 1.5 * np.sin(mid_angle),
f"2θp = {two_theta_p:.1f}°",
fontsize=7,
fontweight="bold",
color=COLOR_ANGLE,
ha="left",
va="bottom",
)
# Legend
legend_handles = [
Line2D([0], [0], color=COLOR_CIRCLE, linewidth=2.5, label="Mohr's Circle"),
Line2D(
[0],
[0],
marker="o",
color="w",
markerfacecolor=COLOR_STATE,
markersize=8,
markeredgecolor=PAGE_BG,
markeredgewidth=1.2,
label="Stress State (A, B)",
),
Line2D(
[0],
[0],
marker="D",
color="w",
markerfacecolor=COLOR_SIGMA,
markersize=8,
markeredgecolor=PAGE_BG,
markeredgewidth=1.2,
label="Principal Stress (σ₁, σ₂)",
),
Line2D(
[0],
[0],
marker="s",
color="w",
markerfacecolor=COLOR_SHEAR,
markersize=8,
markeredgecolor=PAGE_BG,
markeredgewidth=1.2,
label="Max Shear (τmax)",
),
Line2D([0], [0], color=COLOR_ANGLE, linewidth=2.0, label="Principal Angle (2θp)"),
]
ax.legend(handles=legend_handles, fontsize=7, loc="lower left", framealpha=0.9, edgecolor=INK_SOFT)
# Style
title = "mohr-circle · python · seaborn · anyplot.ai"
ax.set_xlabel("Normal Stress σ (MPa)", fontsize=10, color=INK)
ax.set_ylabel("Shear Stress τ (MPa)", fontsize=10, color=INK)
ax.set_title(title, fontsize=12, fontweight="medium", color=INK, pad=12)
ax.tick_params(axis="both", labelsize=8, colors=INK_SOFT)
ax.set_aspect("equal")
# Grid — both axes for this engineering coordinate diagram
ax.yaxis.grid(True, alpha=0.12, linewidth=0.7, color=INK)
ax.xaxis.grid(True, alpha=0.12, linewidth=0.7, color=INK)
sns.despine(ax=ax)
# Save — no bbox_inches to preserve exact 2400×2400 canvas
plt.savefig(f"plot-{THEME}.png", dpi=400, facecolor=PAGE_BG)
plt.close()
Part of Mohr's Circle for Stress Analysis on anyplot.ai.