DIFFERENCE BETWEEN DIRECT STRESS AND BENDING STRESS

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.

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.

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)

σ_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

σ_d = P/ A

σ_d = P/ (b x d)

Unit of Direct compressive stress = N/mm²

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/mm²

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 = db³/12

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 

σ_d = Direct stress = P/ (b x d)

σ_bX = Bending stress due to eccentricity of load with respect to XX axis

σ_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 = bd³/12

I_Y, Moment of inertia about YY axis = db³/12

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.

Direct stress

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

σ_d = P/ (b x d)

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

σ_bX = (M_X . y )/I_X

σ_bX = (P . e_x . y )/I_X

Value of I_X is already determined above, bd³/12

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

σ_bY = (M_X . x )/I_Y

σ_bY = (P . e_Y . x )/I_Y

Value of I_Y is already determined above, db³/12

Value of y will be in the range of (- b/2) to (+ b/2)

Resultant stress (σ_R) will be calculated as mentioned here

Let us determine the resultant stress at each point

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

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

(σ_R)_B = σ_d + σ_bX - σ_bY

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

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

(σ_R)_B = σ_d - σ_bX + σ_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.

Reference:

Strength of material, By R. K. Bansal

Image Courtesy: Google

Also read

Shear force and bending moment diagrams for a simply supported beam with a point load acting at midpoint of the loaded beam

Shear force and bending moment diagrams for a simply supported beam with uniform distributed load

Different types of load acting on the beam

Assumptions made in the theory of simple bending