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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
Back
Introduction
Centre of area (centroid)
Second moment of area ( I )
Calculating the moment of inertia ( I )
Changing to significant figures
Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Introduction
Section properties involve the mathematical properties of structural shapes. They are of
great use in structural analysis and design. Note that these properties have nothing to do
with the strength of the material, but are based solely on the shape of the section. It
explains why some shapes are more efficient at supporting loads than others.
These properties are looked at in more detail following:
centre of area (or centroid)
second moment of area or moment of inertia (I)
section modulus (Z)
radius of gyration (r)
All of these values are generally calculated for both axes of symmetry (x-x and y-y on the
standard coordinate plane). These axes are shown in the following diagram.
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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
Back
Introduction
Centre of area (centroid)
Second moment of area ( I)
Calculating the moment of inertia ( I)
Changing to significant figures Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
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Calculating the radius of gyration
Centre of area (centroid)
The centroid of a section is important in structural design and is like the centre ofgravity of the shape. Most structural shapes have their centroid tabulated by the
manufacturer. Many other simple shapes are symmetrical about the x-x and y-y axes andthe centroid can easily be seen. Occasionally a structural shape is made up of more than
one structural section. In these cases a calculation is needed to work out the position of
the centroid so that further structural design and analysis can be carried out.
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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
Back
Introduction
Centre of area (centroid)
Second moment of area ( I)
Calculating the moment of inertia ( I )
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Changing to significant figures
Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Second moment of area (I)
The second moment of area is also known as the moment of inertia of a shape. It isdirectly related to the area of material in the cross-section and the displacement of that
area from the centroid. Once the centroid is located, the more important structural
properties of the shape can be calculated. The axis that determines the centroid is also
known as the neutral axis (N/A).
The second moment of area is a measure of the 'efficiency' of a shape to resist bending
caused by loading. A beam tends to change its shape when loaded. The second moment
of area is a measure of a shape's resistance to change.
Certain shapes are better than others at resisting bending as demonstrated in the diagram.Clearly, the orientation of the shape also influences bending.
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Section properties
Back
Introduction
Centre of area (centroid)
Second moment of area ( I)
Calculating the moment of inertia ( I )
Changing to significant figures
Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Calculating the moment of inertia (I)
For simple shapes such as squares, rectangles and circles, simple formulas have been
worked out and the values must be calculated for each case.
Circle
Rectangle
Square
Because millimetres are used, large numbers are generated in the calculation.
A standard method of denoting moment of inertia is to write the values as: number x106
mm4.
The 106 factor removes unwanted digits from the value.
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The calculation is usually worked using foursignificant figures, so some rounding off is
required and the decimal point may need to be moved to use the factor of 106.
Example: 73,473,278 mm4 is written as 73.47 x 106 mm4
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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
Back
Introduction
Centre of area (centroid)
Second moment of area ( I )
Calculating the moment of inertia ( I) Changing to significant figures
Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Comparing beam A and beam BA value forI is required about the x-x axis for a simple rectangle like the beam in the
diagram which is shown in two orientations.
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Calculating I
Use the formula: where b = breadth (width) and d = depth (height) to calculatethe moment of inertia about the x-x axis for beams A and B using information from the
diagram.
View the text alternative. (RTF 71 KB)
a
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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
Back
Introduction Centre of area (centroid)
Second moment of area ( I)
Calculating the moment of inertia ( I )
Changing to significant figures
Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Comparing beam A and beam B
A value forI is required about the x-x axis for a simple rectangle like the beam in thediagram which is shown in two orientations.
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Calculating I
Use the formula: where b = breadth (width) and d = depth (height) to calculate
the moment of inertia about the x-x axis for beams A and B using information from the
diagram.
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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
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Back
Introduction
Centre of area (centroid)
Second moment of area ( I )
Calculating the moment of inertia ( I) Changing to significant figures
Comparing beam A and beam B Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Section modulus (Z)
Another property used in beam design is section modulus (Z).
The section modulus of the cross-sectional shape is of significant importance in designing
beams. It is a direct measure of the strength of the beam. A beam that has a larger
section modulus than another will be stronger and capable of supporting greater loads.
It includes the idea that most of the work in bending is being done by the extreme fibresof the beam, ie the top and bottom fibres of the section. The distance of the fibres from
top to bottom is therefore built into the calculation.
The elastic modulus is denoted by Z. To calculate Z, the distance (y) to the extremefibres from the centroid (or neutral axis) must be found as that is where the maximum
stress could cause failure.
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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
Back
Introduction Centre of area (centroid)
Second moment of area ( I)
Calculating the moment of inertia ( I )
Changing to significant figures
Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Calculating the section modulus
To calculate the section modulus, the following formula applies:
where I = moment of inertia, y = distance from centroid to top or bottom edge of
the rectangle
For symmetrical sections the value ofZ is the same above or below the centroid.
For asymmetrical sections, two values are found: Z max and Z min.
To calculate the value ofZ for a simple symmetrical shape such as a rectangle:
where
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and y =
This gives the formula forZ as:
Note: The standard form of writing the value ofZ is to write it as a number x 103 mm3, eg
a value of 2,086 is written as 2.086 x 103.
Calculating Z
View the text alternative. (RTF 71 KB)
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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
Back
Introduction Centre of area (centroid)
Second moment of area ( I)
Calculating the moment of inertia ( I )
Changing to significant figures
Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Radius of gyration (r)
This is another property of a section and is also a function ofthe second moment of area. Put in general terms, the radius of
gyration can be considered to be an indication of the stiffness
of a section based on the shape of the cross-section when used
as a compression member (for example a column).
The diagram indicates that this member will bend in the
thinnest plane.
The radius of gyration is used to compare how various
structural shapes will behave under compression along an axis.It is used to predict buckling in a compression member or
beam.
The formula for the radius of gyration r is:
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where I = second moment of area
A = area of material in the cross section
Note: The unit of measurement for the radius of gyration is mm.
The smallest value of the radius of gyration is used for structural calculations as this is
the plane in which the member is most likely to buckle. Square or circular shapes areideal choices for columns as there is no smallest radius of gyration. They have the same
value because the radius is constant.
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BCGBC4010A > Structural principles - Properties > Section properties
Section properties
Back
Introduction
Centre of area (centroid) Second moment of area ( I)
Calculating the moment of inertia ( I)
Changing to significant figures
Comparing beam A and beam B
Section modulus (Z)
Calculating the section modulus
Radius of gyration (r)
Calculating the radius of gyration
Calculating the radius of gyration
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To calculate the radius of gyration for the cross-section of the beam in the
diagram, start with the values ofI that were calculated earlier.
Ixx = 33.3 x 106 mm4
Iyy = 2.08 x 106
mm4
Refer to the diagram for the values of b and d that are used in the calculationof A.
A = Area of cross-section = 50 mm x 200 mm = 10,000 mm2
Substitute I and A into the formula forr to give:
This is the value of the radius of gyration about the x-x axis.
Calculating r
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