1.4 Geometric Properties of Sections

1.4 Geometric Properties of Sections

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

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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)

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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

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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

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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

    • $k = \frac{D}{4}$

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• 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