All about Section Properties – Area | Centroid | Moment of Inertia | Section modulus | Radius of Gyration

Construction is a vital engineering which creates buildings, roads, bridges and other spaces that connect the world. Concrete and steel are the main construction materials. These materials are used in various shapes for building any structure. These shapes and proportions of each cross section is critical in any structural design and defines its sectional properties. For example, to keep bending and shear stresses within allowable limits largely depends on the cross section of the beam. 

Different concrete sections are available. They can be rectangular, square, T-section, I-section or circle. These concrete sections can be precast or cast in situ concrete. 

Similarly, various steel sections are available. They are mainly manufactured in the industry with some specific dimensions. These steel sections can be angle sections, channel sections, T- sections, I-sections, round bars, hollow structural sections (HSS) and square bars.

Steel Construction Manual by American Institute of Steel Construction (AISC) specifies some standard dimensions and section properties for W, M, S, HP or bearing piles, channel section, angle sections, double angle sections, W and S shapes with cap channels, HSS, pipes and different other shapes. ASTM A1085 also specifies the section properties of some HSS steel sections. 

Different types of structural steel sections
Different types of structural steel sections – (Source: https://www.montanstahl.com/) 
Different types of concrete sections
Different types of concrete sections –  (Source: https://skyciv.com) 

For structural analysis and design different section properties are required which are based on the shapes of the structural sections. They are not dependent on the strength of the materials. They are as follows

  • Cross sectional area
  • Centroid 
  • Second moment of area or moment of inertia
  • Radius of gyration
  • Section modulus (elastic)
  • Section modulus (plastic)

Cross Sectional Area – Section Properties

Cross sectional area implies the total area of a cross section which is a two dimensional section. If the structural section is of complex shapes, it can be divided into simple shapes and the cross sectional area will be the summation of the area of individual sections. The value of the cross sectional area is required to calculate the stress of any structural member corresponding to the applied loads. In case of using steel as reinforcement bars, the cross sectional areas are specified and those bars are manufactured according to those specifications. 

The cross-sectional area of reinforcement bars based on the number of bars being used
The cross-sectional area of reinforcement bars – (Source: https://structx.com/)

Centroid – Section Property

Centroid simply means the central point of any cross section which is similar to the centroid of gravity of a body. But calculation of centroid is dependent on the geometrical shape of the area and it only can be implied for a two dimensional section. The position of the centroid can be either inside or outside of the structural section. Following are some characteristics of a centroid. 

  • If a section has one line of symmetry, the centroid will lie on that line of symmetry.
  • If a section has two lines of symmetry, the centroid will be at the intersection point of the two lines of symmetry. 
  • The axis used for determining the centroid is called the neutral axis (N/A).
Centroid of different structural sections
Centroid of different structural sections – (Source: https://www.dlsweb.rmit.edu.au)

Centroid of complex sections are calculated by dividing it into simple sections whose centroids are known and then taking moments of all areas about top or bottom of the section for y axis and left or right of the section for x axis. The equations are as follows. 

Centroid of complex sections
Centroid of complex sections – (Source: https://homework.study.com)

Centroid to x axis of the above figure can be calculated by 

Centroid formula

Where

Similarly, centroid to y axis of any figure can be calculated by 

Second Moment of Area or Moment of Inertia

Second moment of area, which is also called moment of area, measures a beam’s ability to resist bending when load is applied. It is denoted by I. It is used for calculating stresses and deflection of beams, the buckling of columns and the torsion of shafts. There is a relationship between the second moment of area and deflection of any member. Greater moment of inertia means smaller deflection.  It is always calculated with respect to an axis like the centroid. The unit of the second moment of area is length to the fourth power (m4 or ft4). 

Second Moment of Area or Moment of Inertia
Second Moment of Area or Moment of Inertia – (Source: https://lorisweb.com/)

The moment of area of the above figure for the entire area A with respect to x and y axis can be calculated by the following formulas. 

Following figure shows the second moment of area of some common geometric figures about the centroid. 

Moment of area of Standard Shapes
Moment of area of Standard Shapes – (Source: https://slideplayer.com)

Parallel axis theorem is used to calculate moment of inertia about any given axis. It is the summation of the moment of area about the centroid and the product of area and square of perpendicular distance between centroidal axis and the parallel axis. Thus the formula for the moment of area about any axis can be expressed as follows. 

moment of area about any axis can be expressed as follows

Where dy is the distance between centroid and y axis and sx is the distance between centroid and x axis. 

The bending stress of a beam can be showed by the following formula. 

Where M= Calculated bending moment 

           Y = Vertical distance away from neutral axis

           I = Moment of inertia about neutral axis 

The transverse shear stress of a beam can be showed by following formula. 

Where V = Shear force 

            Q = First moment of area 

             t = Width of section at point of shear stress

Radius of Gyration

Radius of gyration is defined as the radial distance from the centroid of the section which would have the same second moment of area as the body’s actual distribution of mass, if the total mass of the body were concentrated there. It is a useful parameter to estimate the stiffness of a column thus it can predict the buckling in a compression member. It can be calculated by the following formula. 

The smallest value of radius of gyration indicates the axis around which the compression member is most likely to buckle. So the smallest value is used for structural design. It is often recommended to use a square or circular shape column as the radius of gyration for any plane is same for those shapes. 

Possible buckling directions for a simply supported column
Figure: Possible buckling directions for a simply supported column – (Source: https://www.ecourses.ou.edu) 

Elastic Section modulus

Maximum compressive and tensile stress causes maximum bending moment either at the top or bottom fiber of the section. The distance from the extreme fiber to the neutral axis is used to calculate the elastic section modulus of any cross section which is expressed by 

If the cross section is symmetrical then the value of y is the same for top and bottom. 

Symmetrical Rectangular and T sections having the same value for y
Symmetrical Rectangular and T sections having the same value for y- (Source: https://www.dlsweb.rmit.edu.au)

When the structural design is done using material behavior till the yield point, elastic section modulus is used. A beam having greater section modulus will be capable of supporting greater loads. 

Plastic Section Modulus 

While designing structures, if plastic behavior is dominant, plastic section modulus is used. It is calculated based on PNA (Plastic Neutral Axis). This axis indicates the line where compression force from the area of compression and tensile force from the area of tension are equal. Plastic section modulus is basically the first moment of area about the PNA. It can be expressed as 

Where AC = Area of cross-section under compression

            AT = Area of cross-section under tension  

            YC = Distance from PNA to the centroid of the areas under compression
            YT = Distance from PNA to the centroid of areas under tension

Plastic section modulus Figure
Plastic section modulus Figure – (Source: https://mechcontent.com/section-modulus/)

As maximum tensile and compressive stress depends on the section modulus, the position of the structural cross section can affect the value. Following diagram would clarify this statement. 

difference between the elastic and plastic section modulus
Difference between plastic section modulus with rotation – (Source: https://boeingconsult.com/) 

The main difference between the elastic and plastic section modulus is in elastic section modulus, the section is assumed to remain elastic but in plastic section modulus, the entire section is assumed to be yielded. 

Stress strain diagram of steel
Stress strain diagram of steel-(Source: https://www.youtube.com/)

Conclusion: 

Section properties are very crucial parameters for designing any infrastructure. So care should be taken during manufacturing steel sections or during casting of concrete sections to ensure the safety of the public.