Direct and Bending Stresses

Most often, a structural member is subjected to different types of stresses that acts simultaneously. Such stresses are axial, shear, flexure, and torsion. Superposition method is used to determine the combined effect of two or more stresses acting over the cross-section of the member.

Combined or compound stresses 

Combined stress is defined as any possible combinations of direct stress (tensile,compressive,shear) and indirect stress (bending,torsional,thermal) developed inside the body. i.e., Combined Stress = Direct Stress + Indirect Stress.
Possible combinations are as follows:
1. axial and shear
2. axial and flexural
3. axial and torsional
4. torsional and flexural
5. torsional and shear
6. flexural and shear
7. axial, torsional, and flexural
8. axial, torsional, and shear
9. axial, flexural, and shear
10. torsional, flexural, and shear
11. axial, torsional, shear, and flexural

Column with Uniaxial Eccentric Loading

When vertical loads do not coincide with center of gravity of column cross section, but rather act eccentrically either on X or Y axis of the column cross section, then it is called uniaxially eccentric loading column.

Column with Biaxial Eccentric Loading

When vertical load on the column is not coincide with center of gravity of column cross section and does not act on either axis (X and Y axis), then the column is called biaxially eccentric loaded column.

Direct and Bending Stresses : 

Direct Stresses alone is produced in a body when it is subjected to an axial tensile or compressive load, and Bending stress is produced in the body when it is subjected to bending moment. 

But if a body is subjected to axial loads and bending moments, then both the stresses will be produced in the body. Therefore, 

The Direct stress (or) Axial stress σa = Load/ Area 

σa =P/A 

The Bending stress 

σb = + Moment at the section / Section Modulus 

σb = + M/Z 

Where e is eccentricity of load P, M is bending moment and Z is the section modulus about bending axis. 

when a column of rectangular section is subjected to an eccentric load the section is subjected both direct stress and bending stress. the Resultant stresses due to the combined Bending and Direct Stress are, 

σ = σa + σb 

σ = P/A + M/Z 

σmax = P/A + M/Z 

σmin = P/A - M/Z

The stress distribution form face A to face B as shown in the figure. 

σmax = σa + σb

σmin = σa - σb

Note: 

1.Eccentric load produces direct stress as well as bending stress.

2. If direct stress is more than bending stress (σa greater than σb),then the stress throughout the section will be compressive

3.If direct stress is equal to bending stress(σa= σb),then the tensile stress will be zero.

4.If direct stress is less than bending stress (σa less than σb),then there will be tensile stress.

5.Hence for no tensile stress in the section, the direct stress will be greater than or equal to bending stress.

Core or Kernel of the section :

The core or Kernel of the section is the area with in which the line of action of the eccentric load may be applied, so as not to produce tensile stress in any part of the section.

1.Middle Third Rule for a rectangular section : 

For a rectangular section there will be no tensile stress if the load is on either axis within the middle third of the section.

The minimum stress (σMin) must be greater or equal to zero for no tensile stress at any point along the width of the column.

Therefore we can say that if load will be applied with an eccentricity equal to or less than b/6 from the axis YY and on any side of the axis YY then there will not be any tensile stress developed in the column. Hence range within which load could be applied without developing any tensile stress at any point of the section along the width of the column will be b/3 or middle third of the base.

Similarly in order to not develop any tensile stress at any point in the section along the depth of the column, eccentricity of the load must be less than or equal to (d/6) with respect to axis XX.Therefore we can say that if load will be applied with an eccentricity equal to or less than d/6 from the axis XX and on any side of the axis XX then there will not be any tensile stress developed in the column.

2.Middle Quarter Rule for Circular Sections:

For a circular section there will be no tensile stress if the load is on either axis within the a circle of diameter equal to one-forth of the main section diameter.

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.