8 Cross-sectional Properties
Designing Structural elements requires the knowledge of applied external loads and internal reactions, material strengths, and cross-sectional properties. The geometrical properties of a structural element are critical in keeping axial, shear, and bending stresses within allowable limits and moderating the amount of deflection. The following demonstrations show how the shape of the cross-sections affects their stiffness.
Figure 8-1: Relation between the cross-sectional depth (rise) of a beam and its stiffness
Chapter 7 discussed that “area” is one of the cross-sectional properties, which is important to reduce the amount of stress in beams and columns. This chapter focuses on the shape of the cross-sectional area and its distribution about the neutral axis of beams. In the following, some cross-sectional properties that will be necessary to calculate beams and columns are introduced.
Center of gravity
The center of gravity of a body is the point about which the mass of the body is balanced or the point through which the weight of the body acts. When the density of a body is uniform throughout, the center of gravity and the centroid (geometric center) of the body are at the same point.
- The centroid of a rectangle is defined as the center point where all the diagonals intersect each other.
- A centroid of a triangle is the point where the three medians of the triangle meet.
Figure 8-2: The center of gravity of some basic geometrical shapes
Video 8-1: Neutral axis in a beam (https://www.youtube.com/watch?v=BthnS6LJt8s&t=1s)
You can use the following equation to find the center of Gravity of a compounded shape:
[latex]\bar{X}= \frac{\sum ({Area \times d_x})}{\sum{Area}}[/latex]
[latex]\bar{Y} = \frac{\sum( {Area \times d_y})}{\sum{Area}}[/latex]
Example
1st Moment of Area
Video 8-2: A demonstration of the moment of areas of two rods (https://www.youtube.com/watch?v=m9weJfoW5J0)
By definition, the tendency of an area alone to rotate about an axis in the plane of that area.
Q = A[latex]\bar{x}[/latex]
At the Neutral/Centroid axis:
A1 x1 = A2x2
Figure 8-3: The difference of 1st moment of area in two beams with different cross sections
2nd Moment of Area/Inertia
By definition, the 2nd moment of area is the distance of force distribution from the neutral axis. The 2nd moment of area is a geometric property that describes how the area of a cross-section is distributed around the neutral axis, essentially measuring how resistant a shape is to bending forces and deflection. The 2nd moment of area involved the first moment of area multiply by a second moment arm. The second moment arm is the distance between the centroid of the force distribution and the neutral axis.
Figure 8-4: The pattern of force distribution in relation to the distance from the neutral axis
The second moment of area of a beam with a rectangular section can be calculated using the following equation:
Ix = [latex]\frac{bh^3}{12}[/latex]
Where:
Ix = Second moment of area
h = depth of the beam
b = with of the beam
Section Modulus
By definition, the section modulus (Sx) of a beam with a symmetric section equals its second moment of area divided by half its depth at the extreme fiber.
Sx = [latex]\frac{I_x}{c}[/latex]
Where:
Sx = Section modulus
Ix = Second moment of area
c= h/2 at extreme fibers of a symmetric section
h = depth of the beam
The section modulus will help determine the cross-section shape of a beam as discussed in the Chapter 9.
Geometrical properties of steel beam cross-sections
Tables of design dimensions, detailing dimensions, axial flexure, strong-axis flexure, and weak-axis flexure of steel beams are provided in the Steel Construction Manual published by the American Institute of Steel Construction (AISC). You may find the cross-sectional area (A), depth of the beam (d), Ix, and Sx.
Figure 8-5: Example of steel profiles listed in Steel Construction Manual [15]
