Understanding Cantilever Beams and Shear Force

A cantilever beam is a structural element fixed at one end and unsupported at the other (the free end). In this problem, we have a cantilever beam with a span of 4 meters.

The beam supports a gradually varying load. This means the load intensity changes along the length of the beam. Specifically, the load intensity is zero at the free end and increases linearly to 3 KN/m at the fixed end. This type of loading is also known as a triangular load.

We need to determine the magnitude of shear force at the fixed end of the beam.

Calculating Shear Force at the Fixed End

For a cantilever beam, the shear force at any section is the algebraic sum of all vertical forces acting on the beam to the left or right of that section. At the fixed end, the shear force is equal to the total vertical load acting on the entire beam.

Load Distribution

The load is distributed triangularly over the span ($L = 4$ m).

• Load intensity at the free end ($x=0$): $w(0) = 0$ KN/m
• Load intensity at the fixed end ($x=L$): $w(L) = 3$ KN/m

The load intensity $w(x)$ at a distance $x$ from the free end can be represented as:

$$ w(x) = \frac{w_{max}}{L} x $$

Substituting the given values:

$$ w(x) = \frac{3 \text{ KN/m}}{4 \text{ m}} x $$

Total Load Calculation

The total load acting on the beam is the area under the load distribution diagram. For a triangular load, this is the area of a triangle:

$$ \text{Total Load} = \frac{1}{2} \times \text{Base} \times \text{Height} $$

Here, the base is the span of the beam ($L = 4$ m), and the height is the maximum load intensity ($w_{max} = 3$ KN/m).

$$ \text{Total Load} = \frac{1}{2} \times L \times w_{max} $$

$$ \text{Total Load} = \frac{1}{2} \times 4 \text{ m} \times 3 \text{ KN/m} $$

$$ \text{Total Load} = 2 \times 3 \text{ KN} $$

$$ \text{Total Load} = 6 \text{ KN} $$

Shear Force at Fixed End

The shear force ($V$) at the fixed end is equal to the total load on the beam.

$$ V_{\text{fixed end}} = \text{Total Load} $$

$$ V_{\text{fixed end}} = 6 \text{ KN} $$

Conclusion

The magnitude of the shear force at the fixed end of this cantilever beam carrying a gradually varying load from 0 to 3 KN/m over a span of 4 m is 6 kN.