Section modulus is an
important property used in beam design. 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, i.e. sections ability to resist stress. 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, i.e. the
top and bottom fibres of the section. The distance of the fibres from top to
bottom is therefore built into the calculation.
Section modulus is the ratio of a
section's MI to the distance of extreme fiber (under compression/tension) from
the neutral axis.
The neutral axis is the axis through which there are no longitudinal
stresses or strains. For a symmetric section, N.A coincides with centroid or
center of mass.
It means capacity of a cross-section to resist rotation about a particular axis. Greater MI implies more capacity to resist rotation that is more force required to rotate the particular section. Moment of Inertia is a measure of stiffness
In other words, it is a measure of
the 'efficiency' of a shape to resist bending caused by loading. A beam tends
to change its shape when loaded. We can say, it 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.
Moment of Inertia is also called second moment of area.
i.e. I = Ay2 where I = Moment of Inertia;
A = Area of the section;
y= distance of centroid/center of mass from that particular axis.
This means that Area is multiplied with square of distance. i.e. why the unit is length^4.
Slenderness in actuality is supposed to give a measure of how much is length as compared to Slenderness is measured as length/Rg. The length here is the unsupported length i.e. there should be no support in the whole length which restricts the member from moving.
The
Radius of Gyration is used to describe the distribution of cross sectional area in a column around
its centroidal
axis. The radius of gyration is given by the following formulae:
Where I is the second moment of area and A is the
total cross-sectional area.
Radius of Gyration (Rg) is a property
of a section and it measures how much area is distributed to resist rotation
about an axis.
In other words, it is a measure of
the resistance of a cross-section to lateral buckling. The radius of gyration
is related to the size and shape of the cross-section.
For example, if all the material of a
hollow section is concentrated to produce a solid section with the same
cross-sectional area, it will have a smaller radius of gyration than a hollow
member where the material is distributed further away from the centre of the
cross section. Therefore, hollow sections have more moment resisting capacity
than the solid sections, although the cross-sectional area is same.
See the illustration, in each case,
hollow cross-section will have a larger radius of gyration than solid one and
the buckling strength is therefore increased.
Slenderness ratio is always used for direct compression members like column, but it is equally important for any part of member which gets compressive forces.
The gyration radius is useful in
estimating the stiffness of a column. It should also be noted that when a
column of rectangular cross-section is loaded it will buckle in the direction
of the smaller dimension in cross-section. A column of square or circular
cross-section will be equally prone to buckling in both directions. This is
because the cross section will offer equal resistance to buckling in the
direction x and y. That’s why these shapes are ideal 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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