1.0 PROPERTIES OF SECTIONS
1.1 Introduction
The strength of a structural component is dependent on the geometrical properties of its cross-section in
addition to its material and other properties. For example, a beam with a large cross-section will,
generally, be able to resist a bending moment more readily than a beam with a smaller cross-section.
Typical cross-sections of structural members are shown in Figure 6.1 below.
Figure 6.1: Typical cross-sections of structural components
The cross-section of Figure 6.1 (c) is used extensively in structural engineering. It is quite common to make
cross-sections of metal structural members in the form of the cross-sections in Figure 6.1 (c), (d), and (e)
because these cross-sections are structurally more efficient in bending than cross-sections in Figure 6.1
(a) and (b).
Wooden beams are usually rectangular cross-section because they have grain and will have lines of
weakness along their grain if constructed in the other shapes as shown in Figure 6.1 (c), (d) and (e).
1.2 Centre of Gravity
The centre of gravity (CoG) of a body is the point through which the whole weight of the body acts. A
body has only one centre of gravity for all positions of the body.
1.3 Centroid
The centroid is the point at which the total area of a plane figure (e.g. rectangle, square, triangle, circle
etc.) is assumed to be concentrated. The centroid is also represented as CoG or simply G. it is important
to note that the centroid and the centre of gravity are at the same point.
1.4 Centroid or centre of gravity of simple plane figures
(i) CoG of a uniform rod lies at its middle point.
(ii) CoG of a triangle lies at the point where the three medians of the triangle meet.
(iii) CoG of a rectangle or of a parallelogram is at the point where its diagonals meet each other. It is
also the point of intersection of the lines joining the middle points of the opposite sides.
(iv) CoG of a circle is at its centre.
,1.5 Centroid/Centre of gravity of areas of plane figures by Method of Moments
Figure 6.2 shows a plane figure of total area A whose centre of gravity is to be determined. Let A be
composed of a number of small areas 𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 … 𝑒𝑡𝑐
Figure 6.2
𝑠𝑜, 𝐴 = 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + ⋯
𝐿𝑒𝑡 𝑥1 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎1 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
𝑥2 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎2 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
𝑥3 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎3 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
𝑥4 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎4 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
… 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛
The moments of all small areas about the y-axis will be given by:
𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + 𝑎4 𝑥4 + ⋯ (𝑖)
Let CoG be the centre of gravity of the total area A whose distance from y-axis is 𝑥̅ .
𝑇ℎ𝑒𝑛 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 𝑎𝑏𝑜𝑢𝑡 𝑦 − 𝑎𝑥𝑖𝑠 = 𝐴𝑥̅ … (𝑖𝑖)
The moments of all small areas about y-axis must be equal to the moment of total areas about the same
axis. Hence equating (i) and (ii) we get:
𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + 𝑎4 𝑥4 + ⋯ = 𝐴𝑥̅
𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + 𝑎4 𝑥4 + ⋯
∴ 𝑥̅ = … (𝑖𝑖𝑖)
𝐴
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + ⋯
, Similarly, taking moments about the x-axis and also the moment of total area about x-axis, we will get:
𝑎1 𝑦1 + 𝑎2 𝑦2 + 𝑎3 𝑦3 + 𝑎4 𝑦4 + ⋯
𝑦̅ = … (𝑖𝑣)
𝐴
𝑤ℎ𝑒𝑟𝑒: 𝑦̅ = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝐶𝑜𝐺 𝑡𝑜 𝑥 − 𝑎𝑥𝑖𝑠
𝑦1 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎1 𝑡𝑜 𝑥 − 𝑎𝑥𝑖𝑠
𝑦2 , 𝑦3 , 𝑦4 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎2 , 𝑎3 , 𝑎4 𝑡𝑜 𝑥 − 𝑎𝑥𝑖𝑠 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
1.6 Important Points
(i) The axis, about which moments of areas are taken, is known as the axis of reference e.g. y-axis or
x-axis.
(ii) The axis of reference, or plane figures is generally taken as the lowest line of the figure for
determining 𝑦̅, and the left-most line of the figure for calculating 𝑥̅ .
(iii) If the given section is symmetrical about the x-x- axis or the y-y axis, then the CoG of the section
will lie on the axis of symmetry.
(iv) The centre of gravity of composite bodies or section like T-section, I-section, L-section etc. are
obtained by splitting the respective figures into rectangular components then equations (iii) and
(iv) are applied.
1.7 Area Moment of Inertia
Consider a thin lamina (plate) of area A as shown in Figure 6.3 below:
Figure 6.3
𝐿𝑒𝑡 𝑥 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝐴 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
𝑦 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝐴 𝑓𝑟𝑜𝑚 𝑥 − 𝑎𝑥𝑖𝑠
The moment of area about the y-axis will be equal to:
1.1 Introduction
The strength of a structural component is dependent on the geometrical properties of its cross-section in
addition to its material and other properties. For example, a beam with a large cross-section will,
generally, be able to resist a bending moment more readily than a beam with a smaller cross-section.
Typical cross-sections of structural members are shown in Figure 6.1 below.
Figure 6.1: Typical cross-sections of structural components
The cross-section of Figure 6.1 (c) is used extensively in structural engineering. It is quite common to make
cross-sections of metal structural members in the form of the cross-sections in Figure 6.1 (c), (d), and (e)
because these cross-sections are structurally more efficient in bending than cross-sections in Figure 6.1
(a) and (b).
Wooden beams are usually rectangular cross-section because they have grain and will have lines of
weakness along their grain if constructed in the other shapes as shown in Figure 6.1 (c), (d) and (e).
1.2 Centre of Gravity
The centre of gravity (CoG) of a body is the point through which the whole weight of the body acts. A
body has only one centre of gravity for all positions of the body.
1.3 Centroid
The centroid is the point at which the total area of a plane figure (e.g. rectangle, square, triangle, circle
etc.) is assumed to be concentrated. The centroid is also represented as CoG or simply G. it is important
to note that the centroid and the centre of gravity are at the same point.
1.4 Centroid or centre of gravity of simple plane figures
(i) CoG of a uniform rod lies at its middle point.
(ii) CoG of a triangle lies at the point where the three medians of the triangle meet.
(iii) CoG of a rectangle or of a parallelogram is at the point where its diagonals meet each other. It is
also the point of intersection of the lines joining the middle points of the opposite sides.
(iv) CoG of a circle is at its centre.
,1.5 Centroid/Centre of gravity of areas of plane figures by Method of Moments
Figure 6.2 shows a plane figure of total area A whose centre of gravity is to be determined. Let A be
composed of a number of small areas 𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 … 𝑒𝑡𝑐
Figure 6.2
𝑠𝑜, 𝐴 = 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + ⋯
𝐿𝑒𝑡 𝑥1 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎1 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
𝑥2 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎2 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
𝑥3 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎3 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
𝑥4 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎4 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
… 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛
The moments of all small areas about the y-axis will be given by:
𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + 𝑎4 𝑥4 + ⋯ (𝑖)
Let CoG be the centre of gravity of the total area A whose distance from y-axis is 𝑥̅ .
𝑇ℎ𝑒𝑛 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 𝑎𝑏𝑜𝑢𝑡 𝑦 − 𝑎𝑥𝑖𝑠 = 𝐴𝑥̅ … (𝑖𝑖)
The moments of all small areas about y-axis must be equal to the moment of total areas about the same
axis. Hence equating (i) and (ii) we get:
𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + 𝑎4 𝑥4 + ⋯ = 𝐴𝑥̅
𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + 𝑎4 𝑥4 + ⋯
∴ 𝑥̅ = … (𝑖𝑖𝑖)
𝐴
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + ⋯
, Similarly, taking moments about the x-axis and also the moment of total area about x-axis, we will get:
𝑎1 𝑦1 + 𝑎2 𝑦2 + 𝑎3 𝑦3 + 𝑎4 𝑦4 + ⋯
𝑦̅ = … (𝑖𝑣)
𝐴
𝑤ℎ𝑒𝑟𝑒: 𝑦̅ = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝐶𝑜𝐺 𝑡𝑜 𝑥 − 𝑎𝑥𝑖𝑠
𝑦1 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎1 𝑡𝑜 𝑥 − 𝑎𝑥𝑖𝑠
𝑦2 , 𝑦3 , 𝑦4 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑎2 , 𝑎3 , 𝑎4 𝑡𝑜 𝑥 − 𝑎𝑥𝑖𝑠 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
1.6 Important Points
(i) The axis, about which moments of areas are taken, is known as the axis of reference e.g. y-axis or
x-axis.
(ii) The axis of reference, or plane figures is generally taken as the lowest line of the figure for
determining 𝑦̅, and the left-most line of the figure for calculating 𝑥̅ .
(iii) If the given section is symmetrical about the x-x- axis or the y-y axis, then the CoG of the section
will lie on the axis of symmetry.
(iv) The centre of gravity of composite bodies or section like T-section, I-section, L-section etc. are
obtained by splitting the respective figures into rectangular components then equations (iii) and
(iv) are applied.
1.7 Area Moment of Inertia
Consider a thin lamina (plate) of area A as shown in Figure 6.3 below:
Figure 6.3
𝐿𝑒𝑡 𝑥 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝐴 𝑓𝑟𝑜𝑚 𝑦 − 𝑎𝑥𝑖𝑠
𝑦 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶𝑜𝐺 𝑜𝑓 𝑎𝑟𝑒𝑎 𝐴 𝑓𝑟𝑜𝑚 𝑥 − 𝑎𝑥𝑖𝑠
The moment of area about the y-axis will be equal to:
