Understanding Shear Stress in Beams

When a beam is subjected to a transverse shear force, internal shear stresses are developed across its cross-section. These shear stresses are not uniformly distributed but vary along the depth of the beam's cross-section. The distribution of shear stress depends on the shape of the beam's cross-section. The maximum shear stress is a critical parameter in the design and analysis of beams, especially simply supported beams, to ensure they can safely carry the applied loads without failing due to shear.

Shear Stress Distribution in a Triangular Cross-Section

For a beam with a triangular cross-section of altitude 'h', the distribution of shear stress is unique. The formula for shear stress ($\tau$) at any level 'y' from the neutral axis is given by:

$$ \tau = \frac{VQ}{Ib} $$

Where:

• \(V\) is the shear force at the section.
• \(Q\) is the first moment of area of the section above (or below) the level where shear stress is being calculated, taken about the neutral axis.
• \(I\) is the moment of inertia of the entire cross-section about the neutral axis.
• \(b\) is the width of the cross-section at the level where shear stress is being calculated.

For a triangular cross-section with its base at the bottom and apex at the top (as is typically assumed unless stated otherwise for a simply supported beam), the key locations are:

• The neutral axis (NA) is located at a distance of \(\frac{h}{3}\) from the base, or equivalently, \(\frac{2h}{3}\) from the apex (top point) of the triangle.
• Unlike rectangular or circular cross-sections where maximum shear stress occurs at the neutral axis, for a triangular cross-section (base at bottom), the maximum shear stress does not occur at the neutral axis.

Maximum Shear Stress Location

For a triangular cross-section with its base at the bottom and its apex pointing upwards (as is commonly encountered), the maximum shear stress occurs at a specific location along its altitude 'h'.

The location of maximum shear stress in such a triangular cross-section is at a distance of \(\frac{h}{2}\) from the apex (top point) of the triangle.

Now, let's determine the distance of this point from the neutral axis:

1. Distance of the neutral axis from the apex (top) of the triangle = \(\frac{2h}{3}\).
2. Distance of the point of maximum shear stress from the apex (top) of the triangle = \(\frac{h}{2}\).
3. The distance from the neutral axis to the point of maximum shear stress is the difference between these two distances: $$ \text{Distance} = \frac{2h}{3} - \frac{h}{2} $$ $$ \text{Distance} = \frac{4h - 3h}{6} $$ $$ \text{Distance} = \frac{h}{6} $$

Therefore, the maximum shear stress in a simply supported beam with a triangular cross-section (altitude h) occurs at a distance of \(\frac{h}{6}\) from the neutral axis.

Analyzing the Options

Let's evaluate the given options based on our understanding of shear stress in a triangular cross-section:

• Option 1: h/3 from bottom of beam
  This is the location of the neutral axis for a triangular section with the base at the bottom. The maximum shear stress is not at the neutral axis for this shape.
• Option 2: h/3 from top of the beam
  This location is \(\frac{h}{3}\) from the apex. The neutral axis is at \(\frac{2h}{3}\) from the apex. The maximum shear stress is at \(\frac{h}{2}\) from the apex. So, this option is incorrect.
• Option 3: h/6 from neutral axis
  As derived above, the maximum shear stress occurs at a distance of \(\frac{h}{6}\) from the neutral axis for a triangular cross-section (base at bottom). This matches our calculation.
• Option 4: h/5 from top the beam
  This location does not correspond to the point of maximum shear stress or any other significant point in the triangular cross-section.

Based on the analysis, the maximum shear stress for a triangular cross-section (altitude h) occurs at a distance of \(\frac{h}{6}\) from the neutral axis.