7-77 Draw The Shear And Moment Diagrams For The Beam

7-77: Drawing Shear and Moment Diagrams for a Simply Supported Beam

This comprehensive guide will walk you through the process of drawing shear and moment diagrams for a simply supported beam, specifically addressing problem 7-77 (assuming this refers to a standard structural mechanics textbook problem). While I cannot access specific problem numbers from textbooks, I will provide a detailed, general method applicable to most simply supported beam problems with various loading conditions. Understanding this method will allow you to solve problem 7-77 and similar problems effectively.

Understanding Shear and Moment Diagrams

Before diving into the steps, let's clarify the significance of shear and moment diagrams. These diagrams are crucial tools in structural analysis, providing a visual representation of the internal forces within a beam.

• Shear Force Diagram (SFD): This diagram shows the variation of the internal shear force along the length of the beam. The shear force at any point represents the algebraic sum of the vertical forces acting on either side of that point. A positive shear force typically indicates upward shear on the left side of the section.

• Bending Moment Diagram (BMD): This diagram illustrates the variation of the bending moment along the beam's length. The bending moment at any point is the algebraic sum of the moments of all forces about that point. A positive bending moment usually indicates sagging (curvature downwards).

Steps to Construct Shear and Moment Diagrams

The process typically involves these steps:

1. Identify Supports and Loads

Carefully examine the beam and identify the type of supports (e.g., simply supported, cantilever, fixed). Note the location and magnitude of all external loads, including concentrated loads (point loads), uniformly distributed loads (UDLs), and triangularly distributed loads.

2. Calculate Reactions at Supports

For a simply supported beam, use the equilibrium equations (ΣFx = 0, ΣFy = 0, ΣM = 0) to determine the vertical reactions at the supports. This step is crucial as the reactions are the starting point for drawing the shear and moment diagrams. Remember to choose a convenient point to sum the moments around to simplify calculations.

3. Draw the Shear Force Diagram (SFD)

• Start at one end: Begin at one support and move along the beam.

• Concentrated Loads: At the point of a concentrated load, the shear force will experience a sudden jump equal to the magnitude of the load (positive upward, negative downward).

• Uniformly Distributed Loads (UDLs): For a UDL, the shear force will change linearly. The slope of the shear force diagram will be equal to the intensity of the UDL (negative for downward loads).

• Triangularly Distributed Loads: The shear force will change parabolically.

• Points of Zero Shear: Locate the points where the shear force is zero. These points are crucial as they often correspond to the location of maximum bending moment.

4. Draw the Bending Moment Diagram (BMD)

• Start at one end (with the known reaction): Begin at a support and move along the beam.

• Areas under the SFD: The change in bending moment between two points is equal to the area under the shear force diagram between those two points. This is a fundamental relationship between shear and moment.

• Concentrated Moments: If there are any concentrated moments applied to the beam, the bending moment diagram will have a sudden jump equal to the magnitude of the moment.

• Positive and Negative Bending Moments: Remember that positive bending moments cause sagging (concave downwards), while negative bending moments cause hogging (concave upwards).

• Maximum Bending Moment: Identify the point where the bending moment is maximum (often at a point of zero shear). This is critical for determining the beam's strength and stability.

Example: A Simply Supported Beam with a Concentrated Load

Let's consider a simply supported beam of length L with a concentrated load P at a distance 'a' from the left support.

1. Reactions: Using equilibrium equations, the reactions at the left (RA) and right (RB) supports are:

   RA = P(L-a)/L RB = Pa/L

2. SFD:

   • Starting from the left support, the shear force is initially RA (positive).
   • At the point of the concentrated load (distance 'a' from the left support), the shear force suddenly drops by P, becoming RA - P = -Pa/L.
   • The shear force remains constant at -Pa/L until the right support.

3. BMD:

   • Starting from the left support, the bending moment is initially zero.
   • The bending moment increases linearly from 0 to a maximum value at the point of the concentrated load. The maximum bending moment (Mmax) is calculated as: Mmax = RA * a = P(L-a)a/L
   • After the concentrated load, the bending moment decreases linearly to zero at the right support.

Example: A Simply Supported Beam with a Uniformly Distributed Load (UDL)

Consider a simply supported beam of length L with a uniformly distributed load (w) over its entire length.

1. Reactions: The reactions at both supports are equal and are: RA = RB = wL/2

2. SFD:

   • The shear force starts at wL/2 (positive) at the left support.
   • It decreases linearly with a slope of -w until it reaches -wL/2 at the right support.
   • The point of zero shear is at the midpoint of the beam (L/2).

3. BMD:

   • The bending moment starts at zero at the left support.
   • It increases parabolically, reaching its maximum value at the midpoint (L/2). The maximum bending moment is given by: Mmax = wL²/8
   • After the midpoint, the bending moment decreases parabolically to zero at the right support.

Advanced Considerations and Problem-Solving Strategies

• Multiple Loads: For beams with multiple concentrated loads and/or UDLs, apply the principles described above sequentially. The shear force changes abruptly at concentrated loads and linearly or parabolically for distributed loads. The bending moment changes are determined by the area under the shear force diagram.

• Different Support Conditions: The approach changes slightly for different support conditions like cantilever beams or fixed beams. Equilibrium equations need to be adjusted accordingly.

• Using Software: Structural analysis software can significantly simplify the process, particularly for complex loading scenarios. However, understanding the fundamental principles remains essential for interpreting the results and validating software outputs.

• Sign Conventions: Maintain consistent sign conventions throughout the calculations and diagram drawing. The sign conventions described earlier (positive shear upward and positive moment for sagging) are common but ensure consistency.

• Checking your work: Verify your reactions and diagram accuracy. The shear and bending moment should be zero at the free end of a cantilever beam. For simply supported beams, they should be zero at the supports.

Conclusion

Drawing accurate shear and moment diagrams is a fundamental skill in structural analysis. By systematically applying the principles of statics and the relationships between shear and moment, you can effectively solve problems such as problem 7-77 and analyze the internal forces within any beam subjected to various loading conditions. Remember to practice consistently and familiarize yourself with various load cases to master this crucial skill. The ability to interpret and create these diagrams is vital for engineers and designers to ensure structural integrity and safety. By using this detailed guide, you are equipped to tackle a range of structural analysis problems confidently. Remember to always double-check your work and consider using software to verify your calculations, especially in complex scenarios.