Draw The Shear And Moment Diagrams For The Overhang Beam

Drawing Shear and Moment Diagrams for Overhang Beams: A Comprehensive Guide

Overhang beams, characterized by their extended portion beyond the support, present a unique challenge in structural analysis. Understanding how to accurately draw their shear and moment diagrams is crucial for ensuring the structural integrity of any design incorporating these elements. This comprehensive guide will walk you through the process step-by-step, covering various load scenarios and providing practical tips to enhance your understanding.

Understanding Shear and Moment Diagrams

Before diving into the specifics of overhang beams, let's establish a foundational understanding of shear and moment diagrams.

Shear Force: This represents the internal forces within a beam resisting vertical loads. Imagine slicing through the beam at a point; the shear force is the upward or downward force required to maintain equilibrium on either side of the cut. A positive shear force typically indicates an upward force on the left side of the section.

Bending Moment: This reflects the internal resisting couples within a beam subjected to bending. It's the moment (force multiplied by distance) needed to balance the external moments acting on a section of the beam. A positive bending moment generally corresponds to a sagging bending condition (like a U-shape).

Analyzing Overhang Beams: A Systematic Approach

The key to successfully analyzing overhang beams lies in a systematic approach. Here's a detailed breakdown:

1. Determine Reactions

The first step involves calculating the reactions at the supports. This is achieved through the principles of statics:

• ΣFy = 0: The sum of vertical forces must equal zero.
• ΣMx = 0: The sum of moments about a chosen point must equal zero.

For an overhang beam, you'll typically have two reactions: a reaction force and a reaction moment at the fixed or pinned support and a reaction force at the free end (for simple overhangs).

Example: Consider an overhang beam with a length of 10 meters. A 5 kN point load is located 2 meters from the fixed support and a 2 kN/m uniformly distributed load (UDL) spans the entire 10-meter length. To solve for the reaction forces, we will use the equations of static equilibrium:

ΣFy = R1 + R2 - 5kN - (2kN/m * 10m) = 0 ΣM1 = (5kN * 2m) + (2kN/m * 10m * 5m) - (R2 * 10m) = 0

Solving these simultaneous equations will provide values for R1 (reaction force at the fixed support) and R2 (reaction force at the free end).

2. Construct the Shear Force Diagram (SFD)

Once the reactions are known, the next step is creating the shear force diagram. This is done by progressing along the beam, noting how the shear force changes at each load point and support:

• Concentrated Loads: At a point load, the shear force jumps discontinuously by the magnitude of the load. An upward load leads to an increase in shear force, while a downward load leads to a decrease.
• Uniformly Distributed Loads (UDL): With a UDL, the shear force changes linearly. The slope of the shear force diagram equals the magnitude of the UDL.
• Supports: At the supports, the shear force must equal the reaction force at that support.

Remember: The shear force diagram will cross the zero axis where the bending moment is maximum (or minimum if it is a negative bending moment).

3. Construct the Bending Moment Diagram (BMD)

The bending moment diagram is created using the shear force diagram. The key concept is that the slope of the bending moment diagram at any point equals the shear force at that point.

• Linear Shear: If the shear force is constant (or linear), the bending moment will change linearly (or quadratically).
• Points of zero shear: The bending moment is either maximum or minimum at points where the shear force is zero.
• Concentrated Moments: Concentrated moments introduce a sudden jump or drop in the bending moment diagram. The jump is directly equal to the magnitude of the concentrated moment, with a positive moment causing an upward jump.

Note: Pay close attention to the signs of the shear force and bending moments as they reflect the nature of the internal stresses within the beam. Remember positive shear implies upward force on the left, and positive bending moment signifies sagging.

Special Considerations for Overhang Beams

Overhang beams introduce some unique characteristics in their shear and moment diagrams:

• Multiple Points of Zero Shear: The presence of overhangs can frequently lead to the shear force crossing the zero line at multiple locations. This indicates multiple points of maximum (or minimum) bending moment.
• Negative Bending Moments: Overhangs commonly lead to regions of negative bending moments, meaning hogging. This is often manifested as a curve in the opposite direction to the sagging sections of the beam.
• Maximum Moment in Overhang: The maximum bending moment in an overhang can occur at the point of support. The moment is zero at the free end of the overhang.

Illustrative Examples

Let's examine a few examples to solidify our understanding.

Example 1: Simple Overhang with Point Load

A beam of 6m has a 10kN point load at 2m from the free end of a 2m overhang. The shear force diagram will show a jump of -10kN at the point load, then a constant value until the fixed support (the reaction). The bending moment diagram will start at 0 at the free end, follow a linear relationship to a maximum moment value at the load, and then continue with a slope equal to the reaction until the support.

Example 2: Overhang with UDL

A beam of 8m with a 4m overhang carries a 5kN/m UDL across its entire length. The shear force diagram will show a linear decrease across the entire span due to the UDL. The bending moment diagram will be a parabolic curve, with the maximum moment likely occurring either near or at the support.

Example 3: Combination of Loads

A beam with both point loads and UDLs on the main span and an overhang would involve applying the principles discussed above sequentially, taking into account how each load contributes to the overall shear and bending moment.

Practical Tips and Considerations

• Accurate Scaling: Ensure your diagrams are accurately scaled to accurately reflect the magnitudes of the shear and bending moments.
• Clear Labeling: Clearly label all critical points on the diagrams, including values of shear force and bending moment, support reactions, and positions of loads.
• Check for Equilibrium: After drawing the diagrams, verify that the area under the shear force diagram correlates to the change in bending moment. This is a crucial check to ensure the accuracy of your calculations.
• Software Tools: While manual calculation and sketching are crucial for understanding the principles, consider using structural analysis software for complex beam scenarios. This allows verification of your manual calculations and enhances efficiency when dealing with numerous loads and supports.

Conclusion

Drawing shear and moment diagrams for overhang beams may seem daunting at first, but by systematically following the steps outlined and practicing with varied examples, you’ll master this essential skill in structural analysis. Remember that accurate diagrams are vital for the safe and efficient design of structures, facilitating the correct selection of beam sizes and materials. Mastering this skill allows for the safe and efficient design of structures. The combination of theoretical understanding, consistent practice, and potentially utilizing engineering software ensures that you develop a robust competency in structural analysis.