7-80 Draw The Shear And Moment Diagrams For The Beam

Shear and moment diagrams are essential tools in structural engineering, providing a visual representation of the internal shear forces and bending moments within a beam subjected to various loads. These diagrams are crucial for understanding the behavior of beams under stress and for ensuring their structural integrity. Accurately drawing these diagrams allows engineers to determine the maximum shear and moment values, which are critical for designing beams that can safely withstand applied loads. This detailed guide provides a comprehensive understanding of how to draw shear and moment diagrams for beams, complete with step-by-step instructions and illustrative examples.

Understanding Shear and Moment Diagrams

Before diving into the process of drawing shear and moment diagrams, it’s important to understand what these diagrams represent:

• Shear Diagram: A shear diagram illustrates the variation of internal shear force along the length of the beam. The shear force at any point on the beam is the algebraic sum of all the vertical forces acting to the left (or right) of that point.
• Moment Diagram: A moment diagram illustrates the variation of internal bending moment along the length of the beam. The bending moment at any point on the beam is the algebraic sum of the moments of all the forces acting to the left (or right) of that point about that location.

These diagrams are used to:

• Identify the location and magnitude of maximum shear force and bending moment, which are essential for design.
• Check the beam’s capacity to resist shear and bending stresses.
• Ensure the structural stability and safety of the beam.

Prerequisites

Before you start drawing shear and moment diagrams, ensure you have the following:

• Beam Configuration: Understand the beam's support conditions (e.g., simply supported, cantilever, fixed).
• Load Types: Identify all loads acting on the beam (e.g., concentrated loads, uniformly distributed loads, moments).
• Reactions: Calculate the support reactions at each support. This is a fundamental step, as the accuracy of your diagrams depends on these values.

Step-by-Step Guide to Drawing Shear and Moment Diagrams

Follow these steps to accurately draw shear and moment diagrams for a beam:

1. Calculate Support Reactions

The first step is to determine the support reactions. This involves applying the equations of static equilibrium:

• ΣF_x = 0 (Sum of horizontal forces equals zero)
• ΣF_y = 0 (Sum of vertical forces equals zero)
• ΣM = 0 (Sum of moments equals zero)

Example:

Consider a simply supported beam with a length L, carrying a concentrated load P at the mid-span (L/2).

1. Free Body Diagram (FBD): Draw the beam with all applied loads and support reactions.

2. Equilibrium Equations:

   • ΣF_y = R_A + R_B - P = 0
   • ΣM_A = R_B * L - P * (L/2) = 0

   Solving these equations, we get:

   • R_A = P/2
   • R_B = P/2

2. Draw the Shear Diagram

The shear diagram plots the shear force along the length of the beam. Here’s how to construct it:

1. Establish a Baseline: Draw a horizontal line representing zero shear force. This line should be the same length as the beam.

2. Consider Forces from Left to Right: Start from the left end of the beam and move towards the right, accounting for each force.

   • Upward Force: An upward force (like a support reaction) causes an upward jump in the shear diagram. The magnitude of the jump equals the magnitude of the force.
   • Downward Force: A downward force (like an applied load) causes a downward jump in the shear diagram. The magnitude of the jump equals the magnitude of the force.
   • Uniformly Distributed Load (UDL): A UDL causes a linear (straight line) change in the shear diagram. If the UDL is downward, the shear diagram slopes downward with a slope equal to the magnitude of the UDL.
   • No Load: If there is no load between two points, the shear diagram remains constant (horizontal line).

3. Close the Diagram: Ensure the shear diagram returns to zero at the right end of the beam. If it doesn't, there is likely an error in your calculations.

Example (Continuing from above):

• At the left support (A), the reaction force R_A = P/2 causes an upward jump of P/2 in the shear diagram.
• From A to the mid-span, there is no load, so the shear diagram remains constant at P/2.
• At the mid-span, the concentrated load P causes a downward jump of P. So, the shear force changes from P/2 to P/2 - P = -P/2.
• From the mid-span to the right support (B), there is no load, so the shear diagram remains constant at -P/2.
• At the right support (B), the reaction force R_B = P/2 causes an upward jump of P/2, bringing the shear diagram back to zero.

3. Draw the Moment Diagram

The moment diagram plots the bending moment along the length of the beam. Here’s how to construct it:

1. Establish a Baseline: Draw a horizontal line representing zero bending moment. This line should be the same length as the beam.

2. Consider the Area Under the Shear Diagram: The change in bending moment between any two points is equal to the area under the shear diagram between those points.

   • Positive Shear Area: A positive area in the shear diagram indicates an increase in bending moment.
   • Negative Shear Area: A negative area in the shear diagram indicates a decrease in bending moment.
   • Concentrated Moment: A concentrated moment applied to the beam causes a direct jump in the moment diagram.

3. Determine the Shape of the Moment Diagram:

   • Constant Shear: A constant shear force results in a linear (straight line) change in the moment diagram.
   • Linear Shear: A linear change in shear force results in a parabolic (curved) change in the moment diagram.
   • Parabolic Shear: A parabolic change in shear force results in a cubic change in the moment diagram.

4. Boundary Conditions:

   • For a simply supported beam, the bending moment at the supports is zero.
   • For a cantilever beam, the bending moment at the free end is zero.

5. Close the Diagram: The moment diagram should start and end at zero for simply supported beams. For cantilever beams, the moment at the free end should be zero.

Example (Continuing from above):

• At the left support (A), the bending moment is zero.
• From A to the mid-span, the shear force is constant at P/2. The area under the shear diagram is (P/2) * (L/2) = PL/4. Therefore, the bending moment increases linearly from 0 to PL/4 at the mid-span.
• From the mid-span to the right support (B), the shear force is constant at -P/2. The area under the shear diagram is (-P/2) * (L/2) = -PL/4. Therefore, the bending moment decreases linearly from PL/4 to 0 at the right support.
• The maximum bending moment occurs at the mid-span and is equal to PL/4.

Advanced Scenarios and Considerations

1. Overhanging Beams

For overhanging beams (beams that extend beyond their supports), the process remains the same. Calculate the support reactions and then proceed to draw the shear and moment diagrams, paying close attention to the areas and directions of the loads.

2. Cantilever Beams

Cantilever beams are fixed at one end and free at the other. The fixed end provides both a reaction force and a reaction moment.

• Support Reactions: Calculate the vertical reaction and the moment reaction at the fixed end.
• Shear Diagram: Draw the shear diagram as described above, starting from the free end.
• Moment Diagram: The bending moment at the free end is zero. Calculate the bending moment along the beam, considering the area under the shear diagram. The maximum bending moment will occur at the fixed end.

3. Multiple Loads

When a beam is subjected to multiple loads (various combinations of concentrated loads, UDLs, and moments), the process is the same, but requires careful attention to detail.

• Calculate Support Reactions: Sum all vertical forces and moments to find reactions.
• Shear Diagram: Account for each load as you move from left to right. Remember that UDLs create slopes, while concentrated loads create jumps.
• Moment Diagram: Determine the bending moment by calculating the area under the shear diagram. Ensure you consider the shape of the shear diagram when determining the shape of the moment diagram.

4. Internal Hinges

Internal hinges in a beam introduce additional considerations. An internal hinge can only transmit shear force but not bending moment.

• Location of Zero Moment: The bending moment at an internal hinge is always zero.
• Analysis: Analyze each section of the beam separately up to the hinge, then continue the analysis on the next section, ensuring continuity in shear but a reset of the moment to zero at the hinge.

5. Variable Distributed Loads

For beams with variable distributed loads (where the load intensity changes along the length), the shear and moment diagrams can be more complex.

• Load Function: Define the load as a function of position along the beam (w(x)).
• Shear Force: The shear force at any point is the integral of the load function: V(x) = ∫ w(x) dx.
• Bending Moment: The bending moment at any point is the integral of the shear function: M(x) = ∫ V(x) dx.
• Calculus: Use calculus to find the shear and moment equations, and then plot the diagrams.

Common Mistakes to Avoid

When drawing shear and moment diagrams, avoid these common mistakes:

• Incorrect Support Reactions: Always double-check your support reaction calculations. Errors here will propagate through the entire process.
• Sign Conventions: Be consistent with your sign conventions for shear and moment.
• Area Calculations: Ensure you accurately calculate the areas under the shear diagram. Remember to account for the shapes (rectangles, triangles, parabolas).
• Boundary Conditions: Verify that your moment diagram satisfies the boundary conditions (e.g., zero moment at simple supports).
• Units: Keep track of your units throughout the calculations and diagrams.

Practical Tips for Accuracy

• Draw Clear Diagrams: Use a ruler and graph paper to draw neat and accurate diagrams.
• Label Diagrams: Clearly label all key points on the shear and moment diagrams, including the values of shear force and bending moment.
• Check Equilibrium: After drawing the diagrams, check that the equations of equilibrium are satisfied.
• Use Software: Utilize structural analysis software to verify your hand calculations and diagrams. Software like AutoCAD, Civil 3D, and specialized structural analysis tools can provide accurate results and visual representations.

Examples of Shear and Moment Diagrams

Here are a few additional examples to illustrate the concepts discussed:

Example 1: Simply Supported Beam with UDL

Consider a simply supported beam of length L with a uniformly distributed load w (force per unit length) over the entire span.

1. Support Reactions: R_A = R_B = wL/2

2. Shear Diagram:

   • Starts at +wL/2 at the left support.
   • Decreases linearly to -wL/2 at the right support.
   • Passes through zero at the mid-span.

3. Moment Diagram:

   • Starts at zero at the left support.
   • Increases parabolically to a maximum value at the mid-span.
   • The maximum bending moment is M_max = (wL^2)/8.
   • Decreases parabolically to zero at the right support.

Example 2: Cantilever Beam with Concentrated Load at Free End

Consider a cantilever beam of length L with a concentrated load P at the free end.

1. Support Reactions:

   • Vertical reaction at the fixed end: R = P (upward)
   • Moment reaction at the fixed end: M = PL (counter-clockwise)

2. Shear Diagram:

   • Constant shear force of -P along the entire beam.

3. Moment Diagram:

   • Starts at zero at the free end.
   • Decreases linearly to -PL at the fixed end.

Example 3: Overhanging Beam with Multiple Loads

Consider an overhanging beam with a support at point A and point B, where A is at the left end and B is at a distance L1 from A. The beam extends a distance L2 beyond point B with a concentrated load P at the end of the overhang.

1. Support Reactions: Calculate R_A and R_B using equilibrium equations.

2. Shear Diagram:

   • Begins with R_A.
   • Remains constant until R_B.
   • Changes at R_B and then remains constant until the end of the overhang.
   • Ends with a change due to the concentrated load P.

3. Moment Diagram:

   • Starts at zero.
   • Changes linearly up to R_B.
   • Changes linearly again up to the end of the overhang, ending at zero due to the concentrated load P.

Conclusion

Drawing shear and moment diagrams is a fundamental skill for structural engineers. It requires a clear understanding of statics, load types, and support conditions. By following the step-by-step guide outlined in this article, and practicing with various examples, you can develop the proficiency needed to accurately analyze and design beams for structural applications. Remember to avoid common mistakes, double-check your calculations, and utilize software tools to verify your results. With practice and attention to detail, you can master the art of drawing shear and moment diagrams, ensuring the safety and stability of your structural designs.