Understanding Shear Stress in Rectangular Sections

Shear stress is a stress component parallel to a cross-section of a material, caused by forces that tend to make one section of the material slide past an adjacent section. In beams, shear stress arises due to the transverse shear force.

For a beam with a rectangular cross-section subjected to a transverse shear force, the shear stress distribution is not uniform across the depth. It is zero at the top and bottom surfaces (where y is maximum) and maximum at the neutral axis (where y=0).

Calculating Average Shear Stress

The average shear stress ($\tau_{avg}$) on a cross-section is defined as the total shear force (V) acting on the section divided by the area (A) of the cross-section.

Mathematically, the average shear stress is given by:

\(\tau_{avg} = \frac{V}{A}\)

Calculating Maximum Shear Stress in Rectangular Sections

For a rectangular cross-section of width 'b' and depth 'd', the shear stress distribution is parabolic. The maximum shear stress ($\tau_{max}$) occurs at the neutral axis.

The formula for maximum shear stress in a rectangular section is:

\(\tau_{max} = \frac{3}{2} \frac{V}{A}\)

Here, A is the area of the rectangular section, which is \(A = b \times d\).

Comparing Maximum and Average Shear Stress

From the formulas, we can see a direct relationship between the maximum shear stress and the average shear stress for a rectangular section:

\(\tau_{max} = \frac{3}{2} \left(\frac{V}{A}\right)\)

Since \(\tau_{avg} = \frac{V}{A}\), we can substitute this into the equation for \(\tau_{max}\):

\(\tau_{max} = \frac{3}{2} \tau_{avg}\)

This means the maximum shear stress is 1.5 times the average shear stress in a rectangular cross-section.

Calculating the Percentage Greater Than Average Shear Stress

To find out how much greater the maximum shear stress is than the average shear stress as a percentage, we use the formula:

Percentage greater = \(\left(\frac{\text{Maximum shear stress} - \text{Average shear stress}}{\text{Average shear stress}}\right) \times 100\%\)

Substitute the relationship \(\tau_{max} = 1.5 \tau_{avg}\):

Percentage greater = \(\left(\frac{1.5 \tau_{avg} - \tau_{avg}}{\tau_{avg}}\right) \times 100\%\)

Percentage greater = \(\left(\frac{(1.5 - 1) \tau_{avg}}{\tau_{avg}}\right) \times 100\%\)

Percentage greater = \(\left(\frac{0.5 \tau_{avg}}{\tau_{avg}}\right) \times 100\%\)

Percentage greater = \(0.5 \times 100\%\)

Percentage greater = \(50\%\)

Therefore, the maximum shear stress in a rectangular cross-section is 50 per cent greater than the average shear stress on the cross-section.

Let's summarize the key values:

The calculation shows that \(\tau_{max}\) is 1.5 times \(\tau_{avg}\). This represents an increase of \(1.5 - 1.0 = 0.5\) times the average stress. As a percentage, this is \(0.5 \times 100\% = 50\%\).

So, the maximum shear stress is 50 per cent greater than the average shear stress in a rectangular cross section.