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)bd²

Pe/(1/6)bd²  –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)db²

Pe/(1/6)db²  –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.

 

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