In our previous topics, we have seen
some important concepts such as Concepts of direct and bending stresses,
Middle third rule for rectangular section, bending
stress in beams,Â basic
concept of shear force and bending moment,Â strain
energy stored in body,Â beam
bending equation,Â bending
stress of composite beam,Â shear
stress distribution diagramÂ for various
sections etc.

Now we will be concentrated here another very
important topic i.e. Middle quarter rule for circular sections with the help of
this post.

Cast iron and Cement concrete columns are weak under
tensile load and therefore we must be sure that there should not be any tensile
load anywhere in the section and hence load must be applied in such a way that
there will be no tensile load in the section of cement concrete columns.

As we have discussed that when a body
will be subjected with an axial tensile or axial compressive load, there will
be produced only direct stress in the body. Similarly, when a body will be
subjected to a bending moment there will be produced only bending stress in the
body.

Now let us think that a body is
subjected to axial tensile or compressive loads and also to bending moments, in
this situation there will be produced direct stress and bending stress in the
body.

If a column will be subjected with an
eccentric load then there will be developed direct stresses and bending
stresses too in the column and we will determine the resultant stress developed
at any point in the column by adding direct and bending stresses algebraically.

###
**Principle used **

We will consider here compressive stress
as positive and tensile stress as negative and we will have the value of resultant
stress at any point in the column section. There will be maximum stress and
minimum stress in the section of column as mentioned here.

Ïƒ

_{Max}Â = Direct stress + Bending stress
Ïƒ

_{Max}Â = Ïƒ_{dÂ }+ Ïƒ_{b}
Ïƒ

_{Min}Â = Direct stress - Bending stress
Ïƒ

_{Min}Â =Â Â Ïƒ_{dÂ }- Ïƒ_{b}
If minimum stress Ïƒ

_{Min}Â = 0, it indicates that there will be no stress at the respective point in the section
If minimum stress Ïƒ

_{Min}Â = Negative, it indicates that there will be tensile stress at the respective point in the section
If minimum stress Ïƒ

_{Min}Â = Positive, it indicates that there will be compressive stress at the respective point in the section###
**
Let us come to the main subject i.e.Â ****Middle quarter rule for circular
sections**

Let us consider a circular section of area A and of diameter
d as displayed in following figure. Let us consider that an eccentric load P is
acting over the circular section with eccentricity e with respect to axis YY.

Let us find here first direct stress and it could be
written as displayed here in following figure.

Now we will determine here the bending stress and we
can easily determine bending stress by considering the following steps as
displayed here.

Minimum stress at any point in the section will be
given by following formula as mentioned here.

As we have seen above the various conditions of
minimum stress values and their importance and therefore we can easily say that
minimum stress (Ïƒ

_{Min}) must be greater or equal to zero for no tensile stress at any point and on any side of the centre of the circle.
Let us analyze the above equation and we will
conclude that in order to not develop any tensile stress at any point and on
any side of the centre of the circle, eccentricity of the load must be less
than or equal to d/8.

Therefore we can say that if load will be applied
with an eccentricity equal to or less than d/8 from the axis YY and on any side
of the axis YY then there will not be any tensile stress developed in the circular
section.

Similarly, we can also say that if load will be
applied with an eccentricity equal to or less than d/8 from the axis XX and on
any side of the axis XX then there will not be any tensile stress developed in
the circular section.

Hence range within which load could be applied
without developing any tensile stress at any point of the section will be d/4
or middle quarter of the main circular section.

Area of the circle of diameter d/4 within which load
could be applied without developing any tensile stress at any point of the
section will be termed as Kernel of the section.

We will start new topic in the category of strength
of material in our next post.

Please
comment your feedback and suggestions in comment box provided at the end of
this post.

###
**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

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