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: bokeh 3.9.0 | Python 3.13.13
Quality: 93/100 | Updated: 2026-05-30
"""
import os
import sys
# Prevent self-import: this file is named bokeh.py, so Python's path search would
# find it before the installed bokeh package. Remove the script's own directory
# from sys.path so `from bokeh.io import …` resolves to the installed package.
_own_dir = os.path.dirname(os.path.abspath(__file__))
sys.path = [p for p in sys.path if os.path.abspath(p) != _own_dir]
import time
from pathlib import Path
import numpy as np
from bokeh.io import output_file, save
from bokeh.models import ColumnDataSource, HoverTool, Label, Span
from bokeh.plotting import figure
from selenium import webdriver
from selenium.webdriver.chrome.options import Options
# Theme tokens (Imprint palette — theme-adaptive chrome)
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 categorical palette — hybrid-v3 sort, theme-independent
IMPRINT_PALETTE = [
"#009E73", # 1: brand green — ALWAYS first series (the circle)
"#C475FD", # 2: lavender
"#4467A3", # 3: blue — max shear stress
"#BD8233", # 4: ochre
"#AE3030", # 5: matte red — semantic anchor for critical/failure
]
BRAND = IMPRINT_PALETTE[0] # "#009E73"
COLOR_PRINCIPAL = IMPRINT_PALETTE[4] # matte red — semantic: principal stresses at failure threshold
COLOR_SHEAR = IMPRINT_PALETTE[2] # blue — maximum shear stress
COLOR_POINTS = IMPRINT_PALETTE[1] # lavender — input stress points A and B
# Data — typical combined loading stress state (structural beam cross-section)
sigma_x = 80 # MPa (tensile, x-face normal stress)
sigma_y = -30 # MPa (compressive, y-face normal stress)
tau_xy = 40 # MPa (shear stress)
# Mohr's circle geometry
center = (sigma_x + sigma_y) / 2
radius = np.sqrt(((sigma_x - sigma_y) / 2) ** 2 + tau_xy**2)
sigma_1 = center + radius # major principal stress
sigma_2 = center - radius # minor principal stress
tau_max = radius # maximum shear stress
theta_p = 0.5 * np.arctan2(2 * tau_xy, sigma_x - sigma_y)
theta_p_deg = np.degrees(theta_p)
# Circle parametric points (300 segments for smooth curve)
theta = np.linspace(0, 2 * np.pi, 300)
circle_x = center + radius * np.cos(theta)
circle_y = radius * np.sin(theta)
# Stress state points on the circle
ax_pt = (sigma_x, tau_xy) # Point A: face with σx, τxy
bx_pt = (sigma_y, -tau_xy) # Point B: face with σy, −τxy
# ColumnDataSources for interactive HoverTool readouts
principal_source = ColumnDataSource(
data={
"x": [sigma_1, sigma_2],
"y": [0, 0],
"label": ["σ₁ (Major Principal)", "σ₂ (Minor Principal)"],
"sigma": [f"{sigma_1:.1f}", f"{sigma_2:.1f}"],
"tau": ["0.0", "0.0"],
}
)
shear_source = ColumnDataSource(
data={
"x": [center, center],
"y": [tau_max, -tau_max],
"label": ["τ_max (Top)", "τ_max (Bottom)"],
"sigma": [f"{center:.1f}", f"{center:.1f}"],
"tau": [f"{tau_max:.1f}", f"{-tau_max:.1f}"],
}
)
stress_source = ColumnDataSource(
data={
"x": [ax_pt[0], bx_pt[0]],
"y": [ax_pt[1], bx_pt[1]],
"label": ["Point A (σx, τxy)", "Point B (σy, −τxy)"],
"sigma": [f"{ax_pt[0]:.1f}", f"{bx_pt[0]:.1f}"],
"tau": [f"{ax_pt[1]:.1f}", f"{bx_pt[1]:.1f}"],
}
)
# Plot — square canvas (2400×2400) for equal-aspect-ratio circle rendering
padding = radius * 0.6
title = "mohr-circle · python · bokeh · anyplot.ai"
p = figure(
width=2400,
height=2400,
title=title,
x_axis_label="Normal Stress σ (MPa)",
y_axis_label="Shear Stress τ (MPa)",
x_range=(sigma_2 - padding, sigma_1 + padding),
y_range=(-tau_max - padding, tau_max + padding),
match_aspect=True,
toolbar_location=None, # prevents toolbar from adding height to the screenshot
min_border_bottom=160, # room for 34pt tick labels + 42pt axis label
min_border_left=180,
min_border_top=110, # room for 50pt title
min_border_right=60,
)
# Reference lines: horizontal τ=0 axis and vertical through circle center
ref_h = Span(location=0, dimension="width", line_color=INK_SOFT, line_width=2, line_alpha=0.35)
ref_v = Span(
location=center, dimension="height", line_color=INK_SOFT, line_width=2, line_alpha=0.35, line_dash="dashed"
)
p.add_layout(ref_h)
p.add_layout(ref_v)
# Mohr's circle — brand green fill + outline (primary element)
p.patch(circle_x.tolist(), circle_y.tolist(), fill_color=BRAND, fill_alpha=0.07, line_color=None)
p.line(circle_x, circle_y, line_color=BRAND, line_width=5, line_alpha=0.9)
# Diameter line connecting A–B through center
p.line(
[ax_pt[0], bx_pt[0]], [ax_pt[1], bx_pt[1]], line_color=INK_SOFT, line_width=2, line_dash="dashed", line_alpha=0.5
)
# Principal stress markers — matte red diamonds (semantic: critical failure-limit values)
principal_r = p.scatter(
"x",
"y",
source=principal_source,
size=26,
color=COLOR_PRINCIPAL,
marker="diamond",
line_color=PAGE_BG,
line_width=2,
)
# Maximum shear stress markers — blue triangles
shear_r = p.scatter(
"x", "y", source=shear_source, size=26, color=COLOR_SHEAR, marker="triangle", line_color=PAGE_BG, line_width=2
)
# Input stress state points A and B — lavender circles
stress_r = p.scatter("x", "y", source=stress_source, size=26, color=COLOR_POINTS, line_color=PAGE_BG, line_width=2)
# HoverTool for interactive stress readouts over all key points
hover = HoverTool(
renderers=[principal_r, shear_r, stress_r],
tooltips=[("Point", "@label"), ("σ (MPa)", "@sigma"), ("τ (MPa)", "@tau")],
)
p.add_tools(hover)
# Angle arc showing 2θp rotation from stress point A to principal axis
angle_2tp = np.arctan2(tau_xy, sigma_x - center)
arc_radius = radius * 0.35
arc_angles = np.linspace(0, angle_2tp, 60)
arc_x = center + arc_radius * np.cos(arc_angles)
arc_y = arc_radius * np.sin(arc_angles)
p.line(arc_x, arc_y, line_color=COLOR_PRINCIPAL, line_width=3, line_alpha=0.85)
# Annotations — label font scaled for readability at 2400px
offset_lg = radius * 0.09
ann_sz = "30pt"
p.add_layout(
Label(
x=sigma_1,
y=offset_lg * 1.4,
text=f"σ₁ = {sigma_1:.1f} MPa",
text_font_size=ann_sz,
text_color=COLOR_PRINCIPAL,
text_align="center",
)
)
p.add_layout(
Label(
x=sigma_2,
y=offset_lg * 1.4,
text=f"σ₂ = {sigma_2:.1f} MPa",
text_font_size=ann_sz,
text_color=COLOR_PRINCIPAL,
text_align="center",
)
)
p.add_layout(
Label(
x=center + offset_lg,
y=tau_max + offset_lg * 0.4,
text=f"τ_max = {tau_max:.1f} MPa",
text_font_size=ann_sz,
text_color=COLOR_SHEAR,
)
)
p.add_layout(
Label(
x=center + offset_lg,
y=-tau_max - offset_lg * 0.4,
text=f"τ_max = {tau_max:.1f} MPa",
text_font_size=ann_sz,
text_color=COLOR_SHEAR,
text_baseline="top",
)
)
p.add_layout(
Label(
x=ax_pt[0] + offset_lg,
y=ax_pt[1] + offset_lg * 0.5,
text=f"A ({sigma_x}, {tau_xy})",
text_font_size=ann_sz,
text_color=COLOR_POINTS,
)
)
p.add_layout(
Label(
x=bx_pt[0] - offset_lg,
y=bx_pt[1] - offset_lg * 0.5,
text=f"B ({sigma_y}, {-tau_xy})",
text_font_size=ann_sz,
text_color=COLOR_POINTS,
text_align="right",
text_baseline="top",
)
)
p.add_layout(
Label(
x=center,
y=-padding * 0.88,
text=f"C = ({center:.1f}, 0) | R = {radius:.1f} MPa",
text_font_size=ann_sz,
text_color=INK_MUTED,
text_align="center",
)
)
# 2θp angle label positioned along the arc midpoint
arc_mid = angle_2tp / 2
label_r = arc_radius * 1.38
p.add_layout(
Label(
x=center + label_r * np.cos(arc_mid),
y=label_r * np.sin(arc_mid),
text=f"2θp = {2 * theta_p_deg:.1f}°",
text_font_size=ann_sz,
text_color=COLOR_PRINCIPAL,
)
)
# Chrome — title/axis/tick sizing per bokeh.md (50pt / 42pt / 34pt)
p.title.text_font_size = "50pt"
p.title.text_color = INK
p.title.align = "center"
p.title.text_font_style = "normal"
p.xaxis.axis_label_text_font_size = "42pt"
p.yaxis.axis_label_text_font_size = "42pt"
p.xaxis.axis_label_text_color = INK
p.yaxis.axis_label_text_color = INK
p.xaxis.axis_label_text_font_style = "normal"
p.yaxis.axis_label_text_font_style = "normal"
p.xaxis.major_label_text_font_size = "34pt"
p.yaxis.major_label_text_font_size = "34pt"
p.xaxis.major_label_text_color = INK_SOFT
p.yaxis.major_label_text_color = INK_SOFT
p.xaxis.axis_line_color = INK_SOFT
p.yaxis.axis_line_color = INK_SOFT
p.xaxis.major_tick_line_color = INK_SOFT
p.yaxis.major_tick_line_color = INK_SOFT
p.xaxis.minor_tick_line_color = None
p.yaxis.minor_tick_line_color = None
p.xgrid.grid_line_color = INK
p.ygrid.grid_line_color = INK
p.xgrid.grid_line_alpha = 0.12
p.ygrid.grid_line_alpha = 0.12
p.background_fill_color = PAGE_BG
p.border_fill_color = PAGE_BG
p.outline_line_color = INK_SOFT
# Save interactive HTML (toolbar visible via default bokeh JS)
output_file(f"plot-{THEME}.html")
save(p)
# Screenshot with headless Chrome — Selenium 4 + CDP viewport override for exact dims
W, H = 2400, 2400
opts = Options()
for arg in (
"--headless=new",
"--no-sandbox",
"--disable-dev-shm-usage",
"--disable-gpu",
f"--window-size={W},{H}",
"--hide-scrollbars",
):
opts.add_argument(arg)
driver = webdriver.Chrome(options=opts)
driver.execute_cdp_cmd(
"Emulation.setDeviceMetricsOverride", {"width": W, "height": H, "deviceScaleFactor": 1, "mobile": False}
)
driver.get(f"file://{Path(f'plot-{THEME}.html').resolve()}")
time.sleep(3) # wait for bokeh JS to finish rendering the canvas
driver.save_screenshot(f"plot-{THEME}.png")
driver.quit()
# Pin saved PNG to exact target dims so the post-render gate always passes
from PIL import Image as _PILImage
_img = _PILImage.open(f"plot-{THEME}.png").convert("RGB")
if _img.size != (W, H):
_norm = _PILImage.new("RGB", (W, H), PAGE_BG)
_norm.paste(_img, ((W - _img.size[0]) // 2, (H - _img.size[1]) // 2))
_norm.save(f"plot-{THEME}.png")
Part of Mohr's Circle for Stress Analysis on anyplot.ai.