The four section modulus, Zxx (t), Zxx (c) for position (a) and Zyy (t) and Zyy (c) for position (b), can be found by dividing the particular moment of inertia (I) by the distance a, b, c and d respectively.Īs can be seen the the compression is greater than the tension. This is shown in the figure below.įor each axis (x-x and y-y) exists one moments of inertia (Ixx and Iyy) and as the distance to the outer fibre is different in angle position (a) and (b) there are two section modulus for each axis (x-x and y-y). I of a rectangular section is equal to the breadth times depth cubed over 12.įor asymmetric section the distance to the neutral axis is not the same to the extreme outer fibre and therefore two values for each axis exists. In the section shown opposite the section moduli of the point are:Īs the maximum stress occurs in the extreme outer fibres, this distance is needed for the calculation of the maximum bending stress.įollowing from this we can conclude that for a rectangular section y' should be substituted with d/2 and x' with b/2 respectively in the above formula to find the stress in the extreme outer fibres. To calculate the bending stress in structural members (beams), a property called SECTION MODULUS is used to express the bending moment/stress relationship.Įach point within a cross-section of a beam will have a section modulus, this being the ratio of the second moment of inertia to the distance between a point within the section and the relevant axis.
The formula for a rectangular section is: However, a simple formula has been derived for a rectangular section, which is the most important section in this subject. Both board have the same cross-sectional area, but the area distributed differently about the horizontal axis.Ĭalculus is usually used to find the moment of inertia (I) of an irregular section. The board that is supported on its 50 mm edge is considerably stiffer than that supported along its 200 mm edge. If two boards with actual dimensions of 200 by 50 mm were laid side by side - one on the 50mm side and the other on the 200 mm side. Let us look at two boards to intuitively determine which will deflect more and why. The second moment of area (I) is an important figure that is used to determine the stress in a section, to calculate the resistance to buckling, and to determine the amount of deflection in a beam. = the distance between the centroid of the object and the = the second moment of area (moment of inertia) around the xx-axis The second moment of area or moment of inertia (I) is expressed mathematically as: The second moment of area (I) about a given axis is the sum product of the area and the square of the distance from the centroid to the axis. Second moment of area (I) or moment of inertia The centre of gravity is found by dividing the specific area moment (x- and y-direction) by the total area. The easiest way is to organise all data in a table as shown below: Dimensions
Determine the area (or volume) of each part.Divide the body into several parts (A1&A2).The position of the centre of gravity of a compound body can be found by dividing the body into several parts where the centre of gravity of the individual parts are known. Note: The centre of gravity is not necessarily within the body of the shape, it can fall outside as with most angular shapes.Ī more precise procedure to find the centre of gravity is the first moment of area method. The principle is shown in the Figure below. The line of action will always pass through the centre of gravity of the particular shape. The centre of different shapes cut out from a card board can be found by hanging it from a string. A useful analogy that helps understanding this idea may be found by considering the centre of gravity or centre of mass. If it is supported at this point it is in a state of equilibrium and should not fall off. The centre of an area, or centroid, of a shape is the point at which it is in equilibrium.
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