# MIDDLE THIRD & MIDDLE QUARTER RULE CLASS NOTES

**MIDDLE THIRD & MIDDLE **

**QUARTER RULE CLASS NOTES**

## It is applicable when bending is combined with axial loading.

## Eccentric loading also cause bending and axial loading combined.

## In such cases, maximum and minimum stresses will co-exist in the

## outermost fibers.

**Maximum stress**

#### σ_{max} = M/Z + P/A= Maximum compressive

**Minimum stress**

#### σ_{min} = M/Z – P/A+ can be compressive, zero or tensile depending upon the relative magnitude of M/Z and P/A

#### Case 1

#### If M/Z < P/A, net compressive stress will be the minimum stress

#### If M/Z = P/A, net stress will be zero

#### If M/Z > P/A, net stress will be tensile.

#### Most of applications under combined bending and axial loading use brittle materials which are very weak in tension. Therefore as far as possible, such applications should be designed for NO TENSION.

**ECCENTRIC LOADING**

### ANOTHER IMPORTANT APPLICATION OF BENDING AND AXIAL LOAD COMBINED

#### It is a case of eccentric loading. In almost every item of daily use (chair, table, stool, welded joints, riveted joints etc.) load is eccentrically applied. An eccentric load is equal to sum of bending and axial load combined. Eccentric loading will cause simultaneously two stresses, one having additive effect while the other is having opposite effect. This will reduce the load carrying capacity of a component. Since most of the practical applications have eccentric loading, it is of great interest to limit this eccentricity for zero tension in the various cross sections. It is Middle Third Rule for a rectangular section and Middle Quarter Rule for circular sections.

**Middle third rule for a rectangular section**

#### Case 1 with respect to x-axis

#### b is width and d is depth

#### Width’b’ is parallel to x-axis

#### Depth ‘d’ is perpendicular to x-axis

#### In these cases M/Z –P/A =0

#### M=P e, P/bd

#### Z = (1/6)bd^{2}

#### Pe/(1/6)bd^{2}^{ }–P/bd = 0

#### e = d/6

#### This can be on either side of x-axis.

#### Therefore total d/6 + d/6 = d/3= Middle third of ‘d’

#### Case 2 with respect to y axis

#### M/Z–P/A=0

#### M=P e, P/bd

#### Z = (1/6)db^{2}

#### Pe/(1/6)db^{2}^{ }–P/bd = 0

#### e = b/6

#### This can be on either side of y-axis.

#### Therefore total b/6 + b/6 = b/3= Middle third of ‘b’

#### Similarly Middle quarter rule is for a circular section.

**Q. Middle Quarter Rule and Middle Third Rule For Eccentric Loading **

**MIDDLE QUARTER RULE **

** the eccentricity of the load must lie in the middle quarter of the circular section to avoid tension.**

**Eccentric loading is when load is not acting along the axis. It is at some distance from the axis. Eccentric loading becomes equivalent to axial loading plus bending loading. **

** There is a MIDDLE THIRD RULE for a rectangular section under eccentric loading. The eccentricity of the load must lie in the middle third of the rectangular section to avoid tension.**

https://www.mesubjects.net/wp-admin/post.php?post=7433&action=edit MCQ Bending & Axial Loading Combined