Draw typical S.F. and B.M. diagrams for a cantilever with UDL

Short Answer:

A cantilever beam with uniformly distributed load (UDL) is a beam fixed at one end and free at the other, with a load distributed evenly over its entire length. This type of loading causes both shear force and bending moment to vary gradually along the beam’s length.

The Shear Force Diagram (S.F.D.) for a cantilever with UDL is a straight line that decreases linearly from the fixed end to the free end. The Bending Moment Diagram (B.M.D.) is a parabolic curve that is maximum at the fixed end and zero at the free end. These diagrams help engineers understand internal forces developed in the beam.

Detailed Explanation:

Typical S.F. and B.M. Diagrams for a Cantilever with UDL

A cantilever beam is a structural member that is fixed at one end and free at the other end. When it is subjected to a uniformly distributed load (UDL) over its entire length, the beam experiences internal shear forces and bending moments due to the external load. The UDL acts vertically downward and is evenly spread, creating continuous and gradually varying internal reactions within the beam.

The study of Shear Force (S.F.) and Bending Moment (B.M.) diagrams helps in understanding how these internal forces change along the beam. These diagrams are essential in mechanical and civil engineering design because they determine the strength and stability requirements for beams under load.

Assumptions for Analysis

Before drawing the S.F. and B.M. diagrams, the following assumptions are made:

  1. The beam is horizontal and perfectly fixed at one end.
  2. The load is uniformly distributed over the entire span with an intensity of w (kN/m).
  3. The beam is homogeneous and elastic, obeying Hooke’s law.
  4. The self-weight of the beam is included in the UDL.
  5. The load acts perpendicular to the longitudinal axis of the beam.

Given Data

Let:

  • Intensity of UDL = w (kN/m)
  • Length of beam = L (m)
  • Fixed end = A
  • Free end = B

The UDL acts throughout the beam from A to B.

Step 1: Reaction at Fixed End

Since the beam is fixed at A, it provides both a vertical reaction and a moment reaction to resist the load.
The total load acting on the beam due to UDL is:

This total load acts at the center of the beam, i.e., at a distance of  from the fixed end.

Hence,

  • Vertical reaction at A:
  • Fixed end moment at A:

Step 2: Shear Force (S.F.) Calculation

At any section of the beam at a distance x from the free end B, the shear force is given by:

At different points:

  • At the free end (x = 0):
  • At the fixed end (x = L):

Thus, the Shear Force Diagram (S.F.D.) is a straight line starting from zero at the free end and increasing linearly to  at the fixed end. The negative sign indicates that the shear force acts downward on the left-hand side of the beam.

Shape of S.F.D.:
A straight inclined line decreasing linearly from fixed end (maximum) to free end (zero).

Step 3: Bending Moment (B.M.) Calculation

At any section of the beam at a distance x from the free end B, the bending moment is given by:

At different points:

  • At the free end (x = 0):
  • At the fixed end (x = L):

Hence, the Bending Moment Diagram (B.M.D.) is a parabolic curve that starts from zero at the free end and reaches the maximum value  at the fixed end. The negative sign indicates a hogging moment, meaning the beam bends concave downward.

Shape of B.M.D.:
A parabolic curve starting from zero at the free end and curving down to maximum negative value at the fixed end.

Step 4: Nature of Diagrams

  1. Shear Force Diagram (S.F.D.):
  • Shape: Straight, sloping line.
  • Maximum value at fixed end = .
  • Zero at free end.
  • Linear variation due to constant UDL.
  1. Bending Moment Diagram (B.M.D.):
  • Shape: Parabolic curve.
  • Maximum bending moment at fixed end = .
  • Zero at free end.
  • Parabolic variation due to linear change in shear force.

These diagrams represent the internal reactions within the beam under the action of a uniformly distributed load.

Step 5: Important Observations

  1. The maximum shear force occurs at the fixed end, where the beam resists the entire load.
  2. The maximum bending moment also occurs at the fixed end, where the load has the greatest leverage.
  3. Both shear force and bending moment decrease progressively toward the free end, becoming zero at that point.
  4. The diagrams are continuous and smooth, indicating uniform distribution of load and internal stresses.
  5. The slope of the bending moment diagram at any point is equal to the shear force at that point.

Example Calculation

Let’s assume:

Then,

  1. Maximum Shear Force at fixed end =
  2. Maximum Bending Moment at fixed end =

Hence, the S.F.D. will be a straight line from 0 to 8 kN, and the B.M.D. will be a downward parabolic curve reaching 16 kN·m at the fixed end.

Conclusion

For a cantilever beam with uniformly distributed load (UDL), the shear force varies linearly from zero at the free end to maximum at the fixed end, while the bending moment varies parabolically from zero at the free end to maximum negative value at the fixed end. The S.F.D. is a straight line, and the B.M.D. is a parabolic curve. These diagrams are fundamental tools for engineers to analyze and design beams for strength, stiffness, and safety under distributed loading conditions.