---
jupyter:
  description: 'Section properties with planex2d: the centroid, second moment of area, section modulus and radius of
    gyration of shapes from the textbook list, the parallel axis theorem on an I-beam, an IPE 200 and an unequal angle
    against the steel tables, and the principal axes by Mohr''s circle.'
---

# Section properties of steel profiles

How stiff is a beam's cross-section? In bending, a beam's stiffness is $EI$, the material's modulus times the second
moment of area of its section, $I = \int y^2\,dA$ about the axis through the section's centroid. Steel catalogs list $I$
for every rolled profile, with the elastic section modulus $W = I / y_{max}$, which gives the bending stress at the
outer fiber, and the radius of gyration $i = \sqrt{I/A}$, which enters the buckling of columns. All of them follow from
the profile's drawing alone. Here they are measured on the drawing and checked against the formulas and the catalogs.

`face.Measure()` returns them in one dictionary, from the face's exact lines and arcs:

| Key | Value |
|---|---|
| `"area"` | The area $A$ |
| `"center"` | The centroid |
| `"Ih"`, `"Iv"` | The second moments $I_h = \int v^2\,dA$ and $I_v = \int h^2\,dA$ about the horizontal and the vertical axis through the centroid |
| `"Ihv"` | The product of inertia $I_{hv} = \int h v\,dA$ |
| `"I1"`, `"I2"`, `"principal_angle"` | The principal moments, the largest and the smallest second moment about an axis through the centroid, and the angle of the axis of $I_1$ from $h$ in degrees |

![An IPE 200 and an unequal angle, each with its centroid and principal axes](../_static/gallery/section-properties.png){.result}

```python
import math

import matplotlib.pyplot as plt
import numpy as np
import planex as px
```

```python nbsphinx="hidden"
from docs_figures import FACE, INK, PLANEX, RED, Row
```

## Shapes from the list

The [list of second moments of area](https://en.wikipedia.org/wiki/List_of_second_moments_of_area) starts with the
simple shapes. The example measures four of them about their centroids (Figure 1):

- a rectangle of 40 × 60, with $I = bh^3/12$;
- a tube of 60 outside and 50 inside, with $I = \pi (D^4 - d^4)/64$;
- a half disc of radius 20, with $I = (\pi/8 - 8/(9\pi))\,r^4$ about its centroid, which lies $4r/(3\pi)$ above the
  diameter;
- a right triangle of 30 × 45, with $I = bh^3/36$.

```python
shapes = {
    "rectangle": (px.QuickSketch().RectangleC(40, 60).Face(), 40 * 60**3 / 12),
    "tube": (px.Circle((0, 0), 30).Face() - px.Circle((0, 0), 25).Face(), math.pi * (60**4 - 50**4) / 64),
    "half disc": (px.QuickSketch((-20, 0)).Line(40).Turn(90).Arc(20, 180).Face(),
                  (math.pi / 8 - 8 / (9 * math.pi)) * 20**4),
    "triangle": (px.QuickSketch().Line(30).Line((0, 45)).Close().Face(), 30 * 45**3 / 36),
}
for name, (face, exact) in shapes.items():
    print(name, "Ih", face.Measure()["Ih"], "exact", exact)
print("half disc centroid", shapes["half disc"][0].Measure()["center"][1], "exact", 4 * 20 / (3 * math.pi))
```

```python nbsphinx="hide-input"
Row(*[(face, name.capitalize(), {"marks": [face.Measure()["center"][:2]]}) for name, (face, _) in shapes.items()],
    caption="Four shapes from the list, their centroids marked")
```

## The parallel axis theorem

A built-up section adds the second moments of its parts, each moved to the common axis by the parallel axis theorem,
$I = \sum_i (I_i + A_i d_i^2)$, with $d_i$ the distance of the part's centroid from the axis. The example is an
IPE 200 without its root radii, a web of 5.6 × 200 and two flanges of 100 × 8.5, united into one face. The theorem
takes the web between the flanges, 183 high, and the two flanges at $d = 100 - 8.5/2$ from the axis:

```python
web = px.QuickSketch().RectangleC(5.6, 200).Face()
flanges = [px.QuickSketch((0, s * (100 - 8.5 / 2))).RectangleC(100, 8.5).Face() for s in (-1, 1)]
sharp = px.Unify([web, *flanges]).Face()
d = 100 - 8.5 / 2
print("Ih", sharp.Measure()["Ih"], "theorem", 5.6 * 183**3 / 12 + 2 * (100 * 8.5**3 / 12 + 100 * 8.5 * d**2))
```

## An IPE 200 against the table

A rolled IPE 200 has root radii of 12 between the web and the flanges. Filleted there, the section has the values of
the producers' tables, for example [Stad's](https://www.stad.fr/en/sections-products-ranges-ipe). The tables' $y$ axis
is the strong axis, horizontal here, so $I_y$ is `"Ih"` and $I_z$ is `"Iv"`. The section moduli divide by the distance
of the outer fiber, 100 from the strong axis and 50 from the weak one (Figure 2):

```python
ipe = sharp.Filleted([(s * 5.6 / 2, t * (100 - 8.5)) for s in (-1, 1) for t in (-1, 1)], 12)
m = ipe.Measure()
A = m["area"]
print("A", round(A / 100, 2), "cm², table 28.48")
print("Iy", round(m["Ih"] / 1e4, 1), "cm⁴, table 1943 | Wel,y", round(m["Ih"] / 100 / 1e3, 1), "cm³, table 194.3")
print("iy", round(math.sqrt(m["Ih"] / A) / 10, 2), "cm, table 8.26")
print("Iz", round(m["Iv"] / 1e4, 1), "cm⁴, table 142 | Wel,z", round(m["Iv"] / 50 / 1e3, 1), "cm³, table 28.5")
print("iz", round(math.sqrt(m["Iv"] / A) / 10, 2), "cm, table 2.24")
```

```python nbsphinx="hide-input"
Row((sharp, "Without the root radii"), (ipe, "IPE 200, the centroid marked", {"marks": [m["center"][:2]]}),
    caption="The IPE 200 from three rectangles, then filleted")
```

## The principal axes of an angle

An unequal angle has no axis of symmetry, so its strong axis is tilted. The example is an L 150 × 100 × 10, the long
leg upright, with a root radius of 12 inside the corner and toe radii of 6 at the ends of the legs. Drawn with the pen
and filleted at its three corners, it has the values of the section table, here as
[RoyMech lists them](https://roymech.org/Useful_Tables/Sections/U_AngleA_dim_prop.html). They are the second moments
and the product of inertia about the centroid, and the principal moments $I_1$ and $I_2$, which the table calls $I_u$
and $I_v$, with $\tan\alpha$ of the axis of $I_1$ (Figure 3):

```python
angle = px.QuickSketch().Line(100).Turn(90).Line(10).Turn(90).Line(90).Turn(-90).Line(140).Turn(90).Line(10).Close()
profile = angle.Face().Filleted((10, 10), 12).Filleted([(100, 10), (10, 150)], 6)
m = profile.Measure()
print("A", round(m["area"] / 100, 2), "cm², table 24.2")
print("centroid", [round(c / 10, 2) for c in m["center"][:2]], "cm, table 2.34 and 4.81")
print("Ih", round(m["Ih"] / 1e4, 1), "| Iv", round(m["Iv"] / 1e4, 1), "| Ihv", round(m["Ihv"] / 1e4, 1),
      "cm⁴, table 553, 198, -192")
print("I1", round(m["I1"] / 1e4, 1), "| I2", round(m["I2"] / 1e4, 1), "cm⁴, table 637, 114")
print("tan α", round(math.tan(math.radians(m["principal_angle"])), 3), "table 0.438")
```

```python nbsphinx="hide-input"
center, turn = m["center"][:2], math.radians(m["principal_angle"])
axes = [px.Line((center[0] - 90 * math.cos(a), center[1] - 90 * math.sin(a)),
                (center[0] + 90 * math.cos(a), center[1] + 90 * math.sin(a))) for a in (turn, turn + math.pi / 2)]
Row((px.Compound([profile, *axes]), "The principal axes through the centroid", {"marks": [center]}),
    caption="The unequal angle L 150 × 100 × 10")
```

The principal values follow from $I_h$, $I_v$ and $I_{hv}$ by Mohr's circle. It is centered at $(I_h + I_v)/2$ with the
radius $\sqrt{((I_h - I_v)/2)^2 + I_{hv}^2}$, and the principal axis turns from $h$ by $\alpha$ with
$\tan 2\alpha = -2 I_{hv} / (I_h - I_v)$:

```python
c, R = (m["Ih"] + m["Iv"]) / 2, math.hypot((m["Ih"] - m["Iv"]) / 2, m["Ihv"])
print("I1", round((c + R) / 1e4, 1), "| I2", round((c - R) / 1e4, 1), "cm⁴")
print("α", math.degrees(math.atan2(-2 * m["Ihv"], m["Ih"] - m["Iv"])) / 2, "measured", m["principal_angle"])
```

The circle carries the two points $(I_h, I_{hv})$ and $(I_v, -I_{hv})$ in blue, and the principal moments, red, lie
where it crosses the axis:

```python nbsphinx="hide-input"
t = np.linspace(0, 2 * math.pi, 200)
fig, ax = plt.subplots(figsize=(4.6, 3.4))
ax.plot((c + R * np.cos(t)) / 1e4, R * np.sin(t) / 1e4, color=INK, lw=1.6)
ax.plot([m["Ih"] / 1e4, m["Iv"] / 1e4], [m["Ihv"] / 1e4, -m["Ihv"] / 1e4], "o-", color=PLANEX)
ax.plot([(c - R) / 1e4, (c + R) / 1e4], [0, 0], "x", color=RED, markersize=9, mew=2)
ax.set_aspect("equal")
ax.set_xlabel("second moment in cm⁴", color=INK)
ax.set_ylabel("product of inertia in cm⁴", color=INK)
plt.show()
```

```python nbsphinx="hidden"
import sys
sys.path.insert(0, "..")
from gallery import save_figure
shift = (110, -95)
profiles = ((ipe, ipe.Measure()["center"][:2], 0.0, 120),
            (profile.Translated(shift), np.add(m["center"][:2], shift), math.radians(m["principal_angle"]), 95))
fig, ax = plt.subplots(figsize=(6, 4))
for face, center, turn, half in profiles:
    face.Plot(ax=ax, face_col="#c3cbd3", line_col=INK, arc_col=INK, spline_col=INK, legend=False)
    for a in (turn, turn + math.pi / 2):
        ax.plot([center[0] - half * math.cos(a), center[0] + half * math.cos(a)],
                [center[1] - half * math.sin(a), center[1] + half * math.sin(a)], color=RED, lw=1.3, dashes=(6, 3),
                zorder=5)
    ax.plot(*center, "x", color=RED, ms=8, mew=2, zorder=6)
save_figure(fig, "section-properties", bare=True)
```