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Section Modulus
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Section Modulus

When Designing a Beam another property used is Section Modulus (Z) also known as Sx.

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 fibres of 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 or Sx. To calculate Z, the distance (y) to the extreme fibres from the centroid (or neutral axis) must be found as that is where the maximum stress could cause failure.

In structural engineering, the section modulus of a beam is the ratio of a cross section's second moment of area to the distance of the extreme compressive fibre from the neutral axis.

For symmetrical sections this will mean the Zx max and Zx min are equal. For unsymmetrical sections (T-beam for example) the section modulus used will differ depending on whether the compression occurs in the web or flange of the section.

The section modulus is directly related to the strength of a corresponding beam. It is expressed in units of volume, such as, inches3, m3, mm3.

For general design, the Elastic section modulus is used, applying up to the Yield point for most metals and other common materials.

Beam Bending Loading Beam Bending Loading
The Beams Shown are a Wood and Steel Beams





The following are some typical section and equations for some of the most common members:


Typical Sections

Assume that the Square Shape is a 6" x 6", therefore, to find the Section Modulus (Sx):
Sx = d 3 / 6 = 6 3 / 6 = 36 inch 3 , for xx through the Center.

Once we know the Section Modulus, the Moment of Inertia can be computer using the Equation:
Ix = Sx x c = 36 x 3 = 108 inch 4
or the Equation Ix = d 4 / 12 may be used.

or
Sx = d 3 / 3 = 6 3 / 3 = 72 inch 3 , for xx through the Bottom Edge.



Typical Sections

Assume that the Square Shape is a 4" wide x 6" high, therefore, to find the Section Modulus (Sx):
Sx = bd 2 / 6 = 4 x 6 2 / 6 = 24 inch 3 , for xx through the Center.

Once we know the Section Modulus, the Moment of Inertia can be computer using the Equation:
Ix = Sx x c = 24 x 3 = 72 inch 4
or the Equation Ix = bd 3 / 12 may be used.

or
Sx = bd 2 / 3 = 4 x 6 2 / 3 = 48 inch 3 , for xx through the Bottom Edge.




Typical Sections

Assume that the Hollow Retangular Shape is a 4" (b) wide x 6" (d) high, with a 2" (b1) x 4" (d1) Hollow Center,
therefore, to find the Section Modulus (Sx):
Sx = (bd 3 )-(b1d1 3 ) / 6d =
(4 x 6 3 ) - (2 x 4 3 ) / 6 x 6 =
20.44 inch 3 , for xx through the Center.



Application of Section Modulus (Sx)


The following are Pratical Examples of Loads and Calculations being applied to Retangular Members

Example of a Simple Supported Beam with a Uniformly Distributed Load with Equations and Solutions:
Uniformly Distributed Load (100 lb per ft) on a 2 x 10 Beam Loading


Example of a Simple Supported Beam with a Point Concentrated Load with Equations and Solutions:
Point Concentrated Load (600 lb) on a 2 x 12 Beam Loading


Example of a Simple Supported Beam with a Point Concentrated Load with Equations and Solutions:
Point Concentrated Load (1200 lb) on a 4 x 12 Beam Loading