Uniformly Distributed Load Sample Problems With Solutions

Uniformly Distributed Load Sample Problems with Solutions

Understanding uniformly distributed loads (UDLs) is crucial in structural engineering and mechanics. A uniformly distributed load is a load that is spread evenly across a beam or structure's length. This article provides several sample problems involving UDLs, complete with detailed solutions, helping you grasp the fundamental concepts and calculations involved. This will cover topics like calculating reactions, shear forces, and bending moments.

What is a Uniformly Distributed Load?

A uniformly distributed load (UDL) is represented as a load acting uniformly along a length, often denoted by 'w' (in units of force per unit length, e.g., N/m or lb/ft). Unlike concentrated loads, which act at a specific point, a UDL is spread across the entire span of the beam or structure. Understanding how to analyze structures subjected to UDLs is essential for ensuring safety and stability in various engineering applications.

Sample Problem 1: Simply Supported Beam with UDL

Problem: A simply supported beam of length 10 meters carries a uniformly distributed load of 5 kN/m. Calculate the reactions at the supports.

Solution:

1. Draw a free body diagram: This is the first and most crucial step. Sketch the beam, indicating the supports (usually denoted by A and B), the length (10m), and the uniformly distributed load (5 kN/m).

2. Apply equilibrium equations: For a statically determinate beam like this, we use the equations of equilibrium:

   • ΣFx = 0 (Sum of horizontal forces equals zero) – In this case, there are no horizontal forces.
   • ΣFy = 0 (Sum of vertical forces equals zero)
   • ΣM = 0 (Sum of moments equals zero)

3. Calculate reactions:

   • ΣFy = RA + RB - (5 kN/m * 10 m) = 0 (RA and RB are reactions at supports A and B)
   • ΣM at A = RB * 10 m - (5 kN/m * 10 m * 5 m) = 0 (Taking moments about point A)

Solving these two equations simultaneously, we get:

• RB = 25 kN
• RA = 25 kN

Therefore, the reactions at both supports are 25 kN each. This is expected for a symmetrically loaded simply supported beam.

Sample Problem 2: Cantilever Beam with UDL

Problem: A cantilever beam of length 6 meters carries a uniformly distributed load of 2 kN/m. Determine the reactions (shear force and bending moment) at the fixed support.

Solution:

1. Free body diagram: Draw the cantilever beam with the UDL and the fixed support.

2. Equilibrium equations:

   • ΣFy = R - (2 kN/m * 6 m) = 0 (R is the vertical reaction at the fixed support)
   • ΣM at fixed support = M + (2 kN/m * 6 m * 3 m) = 0 (M is the bending moment at the fixed support; the moment arm for the UDL is half the length)

Solving these equations:

• R = 12 kN (Vertical reaction)
• M = -36 kNm (Bending moment; negative sign indicates clockwise moment)

The fixed support experiences a vertical reaction of 12 kN and a counter-clockwise bending moment of 36 kNm.

Sample Problem 3: Overhanging Beam with UDL

Problem: An overhanging beam with a length of 8 meters (4m supported span and 4m overhang) has a UDL of 3 kN/m over the entire length. Calculate the reactions at the supports.

Solution: This problem is slightly more complex due to the overhang. The method remains the same: draw a free body diagram and apply the equilibrium equations. The solution will involve solving simultaneous equations to find the reactions at the two supports. The key difference here lies in calculating the moment caused by the UDL on the overhanging portion.

Understanding Shear Force and Bending Moment Diagrams

For more complex scenarios, drawing shear force and bending moment diagrams is essential. These diagrams graphically represent the variation of shear force and bending moment along the beam's length. They are crucial for determining the maximum shear force and bending moment, which are used in structural design to ensure the beam's safety and stability. Creating these diagrams typically involves calculating shear force and bending moment at various points along the beam and plotting these values.

Conclusion:

These sample problems illustrate the basic principles of analyzing structures under uniformly distributed loads. Remember that the free body diagram is the starting point for any structural analysis. By systematically applying equilibrium equations, you can successfully solve a wide range of problems involving UDLs and other loading conditions. Further exploration of beam analysis techniques, including more complex loading scenarios and different support conditions, will build a solid foundation in structural mechanics.