7-70 Draw The Shear And Moment Diagrams For The Beam

7-70: Drawing Shear and Moment Diagrams for Beams

Determining shear and moment diagrams is fundamental in structural analysis. Understanding these diagrams is crucial for designing safe and efficient beams, ensuring they can withstand anticipated loads. This comprehensive guide will walk you through the process of drawing shear and moment diagrams for a beam, using a step-by-step approach, focusing on problem 7-70 as a practical example (although the specific problem statement of 7-70 isn't provided, we will use a generalized approach applicable to various beam scenarios). We will cover different loading conditions and the underlying principles involved.

Understanding Shear and Moment Diagrams

Before diving into the drawing process, let's clarify what shear and moment diagrams represent:

• Shear Diagram: A shear diagram graphically illustrates the internal shear forces acting along the length of a beam at any given point. The vertical distance from the baseline of the diagram represents the magnitude of the shear force. A positive shear force indicates upward shear on the left side of the section, while a negative shear force indicates downward shear.

• Moment Diagram: A moment diagram depicts the internal bending moments acting along the beam's length. The vertical distance from the baseline represents the magnitude of the bending moment. A positive bending moment causes compression on the top and tension on the bottom of the beam, while a negative bending moment does the opposite.

These diagrams are essential because they directly inform engineers about:

• Maximum Shear and Moment: Identifying the locations and magnitudes of maximum shear and bending moment is vital for designing a beam that can withstand the most significant stresses.
• Beam Design: Knowing the shear and moment values helps in selecting appropriate beam size, material, and reinforcement (if applicable).
• Potential Failure Points: The diagrams help pinpoint areas prone to failure, allowing for careful design modifications.

Steps for Drawing Shear and Moment Diagrams

The process of constructing shear and moment diagrams generally follows these steps:

1. Determine Reactions: Begin by calculating the support reactions at the beam's ends or supports. This involves using equilibrium equations (∑F_x = 0, ∑F_y = 0, and ∑M = 0). Properly determining the reactions is crucial for accurate shear and moment diagrams. Consider the type of support (pin, roller, fixed) when applying the equilibrium equations.

2. Draw the Shear Diagram: Starting from one end of the beam, move along its length and account for each load and support reaction.

   • Concentrated Loads: Concentrated loads cause a sudden change (a jump) in the shear diagram. The magnitude of the jump is equal to the magnitude of the concentrated load (positive for downward loads, negative for upward loads).

   • Uniformly Distributed Loads (UDLs): UDLs create a linearly changing shear force. The slope of the shear diagram is equal to the intensity of the UDL (positive slope for downward UDLs).

   • Uniformly Varying Loads (UVLs): UVLs cause a parabolically changing shear force. The rate of change of the slope is related to the intensity of the UVL.

3. Draw the Moment Diagram: To draw the moment diagram, consider the shear diagram and utilize the following relationship:

   The slope of the moment diagram at any point is equal to the shear at that point (dM/dx = V). This means:

   • Constant Shear: A constant shear results in a linearly varying moment.
   • Linearly Varying Shear: A linearly varying shear results in a parabolically varying moment.
   • Parabolically Varying Shear: A parabolically varying shear results in a cubic varying moment.

4. Check for Accuracy: Once you've drawn both diagrams, verify their accuracy by checking for consistency between the shear and moment diagrams. The area under the shear diagram between any two points should equal the difference in moment values between those two points. This provides an excellent method to confirm accuracy.

Example: Applying the Steps (Generalized Approach)

Let's apply these steps to a generalized beam problem, mirroring the kind of problem you might encounter as 7-70. Remember that 7-70's specific details are not provided, so this is a versatile example illustrating the process.

Consider a simply supported beam of length L with the following loads:

• A concentrated load P at a distance 'a' from the left support.
• A uniformly distributed load (UDL) 'w' extending from the left support to a distance 'b'.

Step 1: Determining Reactions:

• Let R₁ and R₂ be the reactions at the left and right supports, respectively.
• Sum of vertical forces (∑F_y = 0): R₁ + R₂ = P + w*b
• Sum of moments about the left support (∑M_left = 0): R₂L = P(a) + wb(b/2)

Solving these two simultaneous equations gives the values of R₁ and R₂.

Step 2: Drawing the Shear Diagram:

• Start at the left support with a shear value equal to R₁.
• Move to the point where the concentrated load P acts. The shear value suddenly drops by P.
• Then the shear varies linearly due to the UDL w, with a slope equal to -w.
• At distance 'b', the linear variation due to the UDL stops. The shear will be constant from there until the right support, which will show the value of R₂, confirming the calculation.

Step 3: Drawing the Moment Diagram:

• The moment at the left support is zero.
• The moment diagram varies linearly between the left support and the concentrated load, with a slope equal to the shear at that point.
• After the concentrated load, there's a discontinuity, indicating a sudden change in the slope. It follows a parabolic shape due to the linear variation of the shear created by the UDL.
• Finally the moment at the right support is zero.

Step 4: Checking for Accuracy:

The area under the shear diagram from the left support to any point should equal the moment at that point. This is a powerful tool to cross-check your calculations and ensure accuracy.

Advanced Considerations

This process can be extended to more complex loading scenarios such as:

• Multiple Concentrated Loads: Simply add the effect of each load individually on the shear diagram, causing jumps at each location.
• Combinations of Loads: Combine the methods for concentrated, distributed, and varying loads. The shear diagram will be a combination of linear and parabolic segments, and the moment diagram will have corresponding higher-order curve segments.
• Overhanging Beams: Overhanging beams extend beyond their supports, leading to negative moments in the overhanging section.
• Cantilever Beams: Cantilever beams are fixed at one end and free at the other. The process is very similar, but reactions and moments at the fixed end are more complex to calculate.

Software Tools

While manual calculation is important for understanding the underlying principles, software tools can significantly streamline the process, especially for complex beams. These tools can automate the calculations and provide accurate diagrams. However, understanding the manual process remains crucial for interpreting the results and troubleshooting potential errors.

Conclusion

Drawing accurate shear and moment diagrams is a critical skill for structural engineers. The step-by-step method outlined above, combined with a good understanding of the underlying principles, empowers you to analyze various beam structures accurately. Remember to carefully determine reactions, track changes in shear due to each load, and use the relationship between shear and moment to construct accurate diagrams. Mastering this skill is vital for ensuring the safety and efficiency of your designs. By consistently practicing and applying this method to various load conditions, you’ll build a solid foundation in structural analysis.