1.4 Geometric Properties of Sections

What the syllabus asks you to cover

Axes of Symmetry Center of Gravity: Built-up Plane Figures, Standard Steel Sections Moment of Inertia Radius of Gyration

Introduction to Geometric Properties

In structural engineering, the shape and distribution of material in a cross-section—not just the amount of material—determine its ability to resist bending, buckling, and torsion
These critical characteristics are quantified by the geometric properties of sections
These properties are fundamental to calculating stress, deflection, and stability in beams, columns, and other structural members
This unit explores the key geometric properties: axes of symmetry, the center of gravity, moment of inertia, and radius of gyration
This provides the mathematical tools to analyze built-up and standard structural shapes

1. Axes of Symmetry

Definition: An axis of symmetry is a line that divides a plane figure into two identical halves, each a mirror image of the other
The axis itself may be internal (within the figure) or external (outside the figure, like for a C-channel)
Types:
- Vertical Axis of Symmetry: Divides the figure into left and right mirror images
- Horizontal Axis of Symmetry: Divides the figure into top and bottom mirror images
Significance:
- For sections with at least one axis of symmetry, the centroid (Center of Gravity) always lies on that axis
- If a section has two perpendicular axes of symmetry, the centroid lies at their intersection
- This is true for rectangles, circles, I-sections, etc
- Symmetry simplifies the calculation of geometric properties
Examples:
- Rectangle, Circle, I-section: Two perpendicular axes of symmetry
- Angle (L-section), Channel (C-section), T-section: Only one axis of symmetry (or none for unequal legs)
- Z-section: Has no axis of symmetry (point symmetry only)

2. Center of Gravity (Centroid)

Definition: The Centroid (C.G.) or Center of Gravity (for a uniform density lamina) is the unique point where the entire area of the figure can be assumed to be concentrated for the purpose of calculating the first moment of area
It is the geometric center of the shape
Concept: It is the average location of all the points in the shape
For a uniform density material, the centroid coincides with the center of mass
Location Coordinates: For a composite area made up of $n$ simple shapes with known individual areas $A_i$ and centroids $(\bar{x}_i, \bar{y}_i)$
- $\bar{x} = \frac{\sum_{i=1}^{n} A_i \bar{x}*i}{\sum*{i=1}^{n} A_i}$
- $\bar{y} = \frac{\sum_{i=1}^{n} A_i \bar{y}*i}{\sum*{i=1}^{n} A_i}$
- Where $(\bar{x}, \bar{y})$ are the coordinates of the centroid of the composite area
Centroid of Built-up Plane Figures (Composite Sections):
- Method: The composite section is divided into simple geometric shapes (rectangles, triangles, circles) whose areas and centroid locations are known
- Procedure:
  - Divide the composite area into $n$ simple parts
  - Choose a convenient reference $x$ and $y$ axis
  - Calculate the area $A_i$ and the centroid coordinates $(\bar{x}_i, \bar{y}_i)$ of each part relative to the reference axes
  - Apply the composite centroid formula above
- Example: An I-section is divided into three rectangles (web and two flanges)
Centroid of Standard Steel Sections:
- The centroid locations for standard rolled steel sections (I-beams, channels, angles, tees) are tabulated in steel section handbooks (like IS 808 or AISC manuals)
- Key Point: For symmetrical sections (I-beam, channel about one axis), the centroid lies at the intersection of the axes of symmetry
- For unsymmetrical sections (angle, channel about both axes), the centroid must be looked up or calculated
- Example: For an unequal leg angle section (L 150 x 90 x 10), the centroid is not at the intersection of the center lines of the legs
- It is closer to the larger leg

3. Moment of Inertia (Second Moment of Area)

Definition: The Moment of Inertia ( $I$ ) of a plane area about an axis is a measure of its geometric stiffness or resistance to bending and deflection about that axis
It depends on both the shape of the area and the axis about which it is computed
Physical Significance:
- It quantifies how the area of a cross-section is distributed relative to a given axis
- A larger moment of inertia for a given area means the material is spread farther from the axis, resulting in greater bending stiffness
- In the flexure formula, bending stress $\sigma$ is inversely proportional to $I$
- $\sigma = \frac{My}{I}$
- Where $M$ is bending moment and $y$ is distance from neutral axis
Formulas:
- About Centroidal Axes: For a shape with area $A$
- Rectangle (width $b$ , depth $d$ )
  - $I_{xx} = \frac{bd^3}{12}$ (about horizontal centroidal axis)
  - $I_{yy} = \frac{db^3}{12}$ (about vertical centroidal axis)
- Circle (diameter $D$ )
  - $I_{xx} = I_{yy} = \frac{\pi D^4}{64}$
- Triangle (base $b$ , height $h$ )
  - $I_{base} = \frac{bh^3}{12}$ (about base)
Parallel Axis Theorem (Transfer Formula):
- Purpose: To calculate the moment of inertia about any axis parallel to a known centroidal axis
- Formula: $I = I_c + A d^2$
- Where:
  - $I$ = Moment of inertia about the new parallel axis
  - $I_c$ = Moment of inertia about the centroidal axis parallel to the new axis
  - $A$ = Area of the shape
  - $d$ = Perpendicular distance between the two parallel axes
- Application: This theorem is essential for finding the moment of inertia of composite/built-up sections
Moment of Inertia of Composite/Built-up Sections:
- Procedure:
  - Divide the composite section into simple parts
  - Find the centroid of the entire composite section (as described in Section 2)
  - For each part, calculate its moment of inertia about its own centroidal axis parallel to the desired composite axis
  - Use the Parallel Axis Theorem to transfer this moment of inertia to the centroidal axis of the composite section
  - $I_{part} = I_{c,part} + A_{part} d_{part}^2$
  - Where $d_{part}$ is the distance between the centroid of the part and the centroid of the composite section
  - Sum the contributions from all parts
  - $I_{total} = \sum (I_{c,part} + A_{part} d_{part}^2)$
Polar Moment of Inertia (J):
- Definition: Moment of inertia about an axis perpendicular to the plane (the z-axis)
- Formula: $J = I_{xx} + I_{yy}$
- Significance: Measures resistance to torsion (twisting)
- $\tau = \frac{T r}{J}$
- Where $T$ is torque

4. Radius of Gyration

Definition: The Radius of Gyration ( $k$ ) is a distance that represents how far from the axis the entire area of the section could be concentrated and still have the same moment of inertia
Mathematical Definition: It is the root-mean-square distance of the area elements from the axis
- $k = \sqrt{\frac{I}{A}}$
- Where $I$ is the moment of inertia about the axis, and $A$ is the total area
Interpretation:
- A larger radius of gyration indicates that the area is distributed farther from the axis, leading to greater bending stiffness and buckling resistance
- It is a measure of the effectiveness of the cross-sectional shape in resisting buckling
Application in Column Design:
- The slenderness ratio of a column, which determines its buckling load (Euler's formula), is defined as
- $\lambda = \frac{L_e}{k_{min}}$
- Where $L_e$ is the effective length of the column, and $k_{min}$ is the minimum radius of gyration (about the axis with the smallest $k$ )
- Key Insight: For a given area, a shape with a higher minimum $k$ will have a lower slenderness ratio and thus a higher buckling load
Radius of Gyration for Common Sections:
- Rectangle (about centroidal axis parallel to depth)
  - $k_{xx} = \frac{d}{\sqrt{12}}$
- Circle
  - $k = \frac{D}{4}$

The geometric properties of sections—centroid, moment of inertia, and radius of gyration—are not merely abstract calculations
They are the fundamental link between the physical shape of a structural member and its engineering performance
The ability to compute these properties for complex, built-up sections using the principles of composite areas and the parallel axis theorem is an essential skill for designing efficient and safe beams, columns, and other structural elements

Pause and recall

Without looking back, explain the main idea, recall a key definition or formula, and describe when it applies. Include assumptions and units where relevant. Revisit any step you could not explain.

1. Axes of Symmetry

Definition: An axis of symmetry is a line that divides a plane figure into two identical halves, each a mirror image of the other

The axis itself may be internal (within the figure) or external (outside the figure, like for a C-channel)

Types:

Vertical Axis of Symmetry: Divides the figure into left and right mirror images
Horizontal Axis of Symmetry: Divides the figure into top and bottom mirror images

Significance:

For sections with at least one axis of symmetry, the centroid (Center of Gravity) always lies on that axis
If a section has two perpendicular axes of symmetry, the centroid lies at their intersection
This is true for rectangles, circles, I-sections, etc
Symmetry simplifies the calculation of geometric properties

Examples:

Rectangle, Circle, I-section: Two perpendicular axes of symmetry
Angle (L-section), Channel (C-section), T-section: Only one axis of symmetry (or none for unequal legs)
Z-section: Has no axis of symmetry (point symmetry only)

2. Center of Gravity (Centroid)

Definition: The Centroid (C.G.) or Center of Gravity (for a uniform density lamina) is the unique point where the entire area of the figure can be assumed to be concentrated for the purpose of calculating the first moment of area

It is the geometric center of the shape

Concept: It is the average location of all the points in the shape

For a uniform density material, the centroid coincides with the center of mass

Location Coordinates: For a composite area made up of

n

simple shapes with known individual areas

A_i

and centroids

(\bar{x}_i, \bar{y}_i)

$\bar{x} = \frac{\sum_{i=1}^{n} A_i \bar{x}*i}{\sum*{i=1}^{n} A_i}$
$\bar{y} = \frac{\sum_{i=1}^{n} A_i \bar{y}*i}{\sum*{i=1}^{n} A_i}$
Where $(\bar{x}, \bar{y})$ are the coordinates of the centroid of the composite area

Centroid of Built-up Plane Figures (Composite Sections):

Method: The composite section is divided into simple geometric shapes (rectangles, triangles, circles) whose areas and centroid locations are known
Procedure:
- Divide the composite area into $n$ simple parts
- Choose a convenient reference $x$ and $y$ axis
- Calculate the area $A_i$ and the centroid coordinates $(\bar{x}_i, \bar{y}_i)$ of each part relative to the reference axes
- Apply the composite centroid formula above
Example: An I-section is divided into three rectangles (web and two flanges)

Centroid of Standard Steel Sections:

The centroid locations for standard rolled steel sections (I-beams, channels, angles, tees) are tabulated in steel section handbooks (like IS 808 or AISC manuals)
Key Point: For symmetrical sections (I-beam, channel about one axis), the centroid lies at the intersection of the axes of symmetry
For unsymmetrical sections (angle, channel about both axes), the centroid must be looked up or calculated
Example: For an unequal leg angle section (L 150 x 90 x 10), the centroid is not at the intersection of the center lines of the legs
It is closer to the larger leg

3. Moment of Inertia (Second Moment of Area)

Definition: The Moment of Inertia (

I

) of a plane area about an axis is a measure of its geometric stiffness or resistance to bending and deflection about that axis

It depends on both the shape of the area and the axis about which it is computed

Physical Significance:

It quantifies how the area of a cross-section is distributed relative to a given axis
A larger moment of inertia for a given area means the material is spread farther from the axis, resulting in greater bending stiffness
In the flexure formula, bending stress $\sigma$ is inversely proportional to $I$
$\sigma = \frac{My}{I}$
Where $M$ is bending moment and $y$ is distance from neutral axis

Formulas:

About Centroidal Axes: For a shape with area $A$
Rectangle (width $b$ , depth $d$ )
- $I_{xx} = \frac{bd^3}{12}$ (about horizontal centroidal axis)
- $I_{yy} = \frac{db^3}{12}$ (about vertical centroidal axis)
Circle (diameter $D$ )
- $I_{xx} = I_{yy} = \frac{\pi D^4}{64}$
Triangle (base $b$ , height $h$ )
- $I_{base} = \frac{bh^3}{12}$ (about base)

Parallel Axis Theorem (Transfer Formula):

Purpose: To calculate the moment of inertia about any axis parallel to a known centroidal axis
Formula: $I = I_c + A d^2$
Where:
- $I$ = Moment of inertia about the new parallel axis
- $I_c$ = Moment of inertia about the centroidal axis parallel to the new axis
- $A$ = Area of the shape
- $d$ = Perpendicular distance between the two parallel axes
Application: This theorem is essential for finding the moment of inertia of composite/built-up sections

Moment of Inertia of Composite/Built-up Sections:

Procedure:
- Divide the composite section into simple parts
- Find the centroid of the entire composite section (as described in Section 2)
- For each part, calculate its moment of inertia about its own centroidal axis parallel to the desired composite axis
- Use the Parallel Axis Theorem to transfer this moment of inertia to the centroidal axis of the composite section
- $I_{part} = I_{c,part} + A_{part} d_{part}^2$
- Where $d_{part}$ is the distance between the centroid of the part and the centroid of the composite section
- Sum the contributions from all parts
- $I_{total} = \sum (I_{c,part} + A_{part} d_{part}^2)$

Polar Moment of Inertia (J):

Definition: Moment of inertia about an axis perpendicular to the plane (the z-axis)
Formula: $J = I_{xx} + I_{yy}$
Significance: Measures resistance to torsion (twisting)
$\tau = \frac{T r}{J}$
Where $T$ is torque

4. Radius of Gyration

Definition: The Radius of Gyration (

k

) is a distance that represents how far from the axis the entire area of the section could be concentrated and still have the same moment of inertia

Mathematical Definition: It is the root-mean-square distance of the area elements from the axis

$k = \sqrt{\frac{I}{A}}$
Where $I$ is the moment of inertia about the axis, and $A$ is the total area

Interpretation:

A larger radius of gyration indicates that the area is distributed farther from the axis, leading to greater bending stiffness and buckling resistance
It is a measure of the effectiveness of the cross-sectional shape in resisting buckling

Application in Column Design:

The slenderness ratio of a column, which determines its buckling load (Euler's formula), is defined as
$\lambda = \frac{L_e}{k_{min}}$
Where $L_e$ is the effective length of the column, and $k_{min}$ is the minimum radius of gyration (about the axis with the smallest $k$ )
Key Insight: For a given area, a shape with a higher minimum $k$ will have a lower slenderness ratio and thus a higher buckling load

Radius of Gyration for Common Sections:

Rectangle (about centroidal axis parallel to depth)
- $k_{xx} = \frac{d}{\sqrt{12}}$
Circle
- $k = \frac{D}{4}$

The geometric properties of sections—centroid, moment of inertia, and radius of gyration—are not merely abstract calculations

They are the fundamental link between the physical shape of a structural member and its engineering performance

The ability to compute these properties for complex, built-up sections using the principles of composite areas and the parallel axis theorem is an essential skill for designing efficient and safe beams, columns, and other structural elements