In our previous topics, we were discussing the basic concept of direct stresses and bending stresses, where we have discussed that if
a body is subjected to axial tensile or compressive loads and also to bending
moments, in that situation there will be produced direct stress and bending
stress in the body.

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We have determined there the resultant stress when a
column will be subjected with an eccentric load and load was eccentric with
respect to single axis i.e. YY axis.

Today we will continue here with one very important
topic in strength of material i.e. difference between direct stress and bending stress and also we will see here resultant stress when a column will be
subjected with an eccentric load and load will be eccentric with respect to
both axes i.e. XX and YY axis.

Let us go ahead step by step for easy understanding,
however if there is any issue we can discuss it in comment box which is
provided below this post.

Before going ahead, let us first brief here the
basics of direct stresses and bending stresses for easy understanding and after
that we will see here the determination of resultant stress when a column will
be subjected with an eccentric load and load will be eccentric with respect to
both axes i.e. XX and YY axis.

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.

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**Direct
stress**

Let us consider one column, as displayed in
following figure, which is fixed at one end and let us apply one load P axially
to the other end of the column. In simple, we can say that column will be
subjected to compressive load and as load is applied axially, there will be
developed direct compressive stress only in the body and intensity of this direct
compressive stress will be uniform across the cross section of the column.

We have following information from above figure

P = Axial applied compressive load which is acting
on the column through its axis

A= Area of cross section of the given column

Area of cross section of the given column = b x d,
(For rectangular cross section)

Ïƒ

_{d }= Direct compressive stress developed in the column due to axial applied compressive load
b= Width of the cross section of the given
column

d= height or depth of the cross section of the given
column

Direct compressive stress developed in the column

_{ }= Axial applied compressive load/ Area of cross section of the given column
Ïƒ

_{d }= P/ A
Ïƒ

_{d }= P/ (b x d)
Unit of Direct compressive stress = N/mm

^{2}###
**Bending
stress**

Now let us consider one column, as displayed in
following figure, which is fixed at one end and let us apply one load P to the
other end of the column at a distance e from the axis of the column. In simple,
we can say that column will be subjected to an eccentric load and line of
action of this load will be at a distance e from the axis of the column.

Distance between the axis of the column and line of
action of load i.e. e will be termed as eccentricity of the load and such load
will be termed as eccentric load. There will be produced direct stress and
bending stress in the column due to this eccentric load.

Unit of bending stress = N/mm

^{2}
Recall the concept of bending
stress and we will write here the expression for the bending stress
developed in the body.

Where,

I is the area moment of inertia of the column
rectangular section across the axis YY

I = db

^{3}/12
M = Moment formed by the load P

M= P x e

P = Load applied with an eccentricity e

y = Distance of the point from neutral axis where
bending stress is to be determined

Let us come to the main topic
i.e. resultant
stress when a column will be subjected with an eccentric load and load will be
eccentric with respect to both axes i.e. XX and YY axis.

Let us see the following figure, where one column is
subjected with a load which is eccentric with respect to XX axis and YY axis. We
have following information from this figure as mentioned here.

P = Eccentric load
applied on column

b = Width of the column

d = depth of the column

e

_{x}= Eccentricity of load with respect to XX axis
e

_{Y}= Eccentricity of load with respect to YY axis
Ïƒ

_{d }= Direct stress = P/ (b x d)
Ïƒ

_{bX}= Bending stress due to eccentricity of load with respect to XX axis
Ïƒ

_{bY}= Bending stress due to eccentricity of load with respect to YY axis
M

_{X}, Moment of eccentric load about XX axis = P x e_{x}
M

_{Y}, Moment of eccentric load about YY axis = P x e_{Y}
I

_{X}, Moment of inertia about XX axis = bd^{3}/12
I

_{Y}, Moment of inertia about YY axis = db^{3}/12
We will calculate here
the direct stress, bending stress due to eccentricity of load with respect to
XX axis and bending stress due to eccentricity of load with respect to YY axis
and finally we will add these stresses algebraically to secure the resultant
stress.

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*Direct
stress*

*Direct stress*

Direct stress is already calculated
above and it is as mentioned here

Ïƒ

_{d }= P/ (b x d)####
*Bending
stress due to eccentricity of load with respect to XX axis*

*Bending stress due to eccentricity of load with respect to XX axis*

Ïƒ

_{bX}= (M_{X }. y )/I_{X}
Ïƒ

_{bX}= (P . e_{x }. y )/I_{X}
Value of I

_{X}is already determined above, bd^{3}/12
Value of y will be in the
range of (- d/2) to (+ d/2)

####
*Bending
stress due to eccentricity of load with respect to YY axis*

*Bending stress due to eccentricity of load with respect to YY axis*

Ïƒ

_{bY}= (M_{X }. x )/I_{Y}
Ïƒ

_{bY}= (P . e_{Y }. x )/I_{Y}
Value of I

_{Y}is already determined above, db^{3}/12
Value of y will be in the
range of (- b/2) to (+ b/2)

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*Resultant
stress (Ïƒ*_{R}) will be calculated as mentioned here

*Resultant stress (Ïƒ*

_{R}) will be calculated as mentioned here
Let us determine the resultant
stress at each point

####
*Point
A, resultant stress *

*Point A, resultant stress*

As we can observe here
that value of co-ordinates X and Y will be negative here and therefore
resultant stress will be minimum at this point

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*Point
B, resultant stress *

*Point B, resultant stress*

As we can observe here
that value of co-ordinate X will be positive and value of co-ordinate Y will be
negative here and therefore resultant stress will be written as mentioned here

(Ïƒ

_{R})_{B}= Ïƒ_{d }+ Ïƒ_{bX}- Ïƒ_{bY}####
*Point
C, resultant stress *

*Point C, resultant stress*

As we can observe here
that value of co-ordinates X and Y will be positive here and therefore
resultant stress will be maximum at this point

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*Point
D, resultant stress *

*Point D, resultant stress*

As we can observe here
that value of co-ordinate X will be negative and value of co-ordinate Y will be
positive here and therefore resultant stress will be written as mentioned here

(Ïƒ

_{R})_{B}= Ïƒ_{d }- Ïƒ_{bX}+ Ïƒ_{bY}
Please comment your feedback and suggestions in
comment box provided at the end of this post. We will discuss another topic in
our next post.

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**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

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