7-53 Draw The Shear And Bending-moment Diagrams For The Beam

7-53: Drawing Shear and Bending Moment Diagrams for a Simply Supported Beam

Determining the shear and bending moment diagrams for a beam is a crucial step in structural analysis. These diagrams visually represent the internal forces within a beam under load, allowing engineers to assess stress levels and ensure the structural integrity of the design. This comprehensive guide focuses on problem 7-53, a common example involving a simply supported beam with various loading conditions. We'll break down the process step-by-step, explaining the underlying principles and providing a detailed solution.

Understanding Shear and Bending Moment

Before diving into problem 7-53, let's refresh our understanding of shear force and bending moment.

Shear Force

Shear force is the internal force acting parallel to the cross-section of the beam. It's a measure of the transverse force trying to cause one section of the beam to slide past another. A positive shear force is typically defined as upward on the left side of the section.

Bending Moment

Bending moment is the internal moment acting about the cross-section of the beam. It's a measure of the bending effect on the beam, causing it to curve. A positive bending moment is typically defined as causing concave upwards curvature (like a smiley face).

Problem 7-53 Setup (Assumed)

Since the problem statement only provides the number "7-53," we'll need to assume a specific beam configuration to proceed. Let's assume the following scenario for problem 7-53:

A simply supported beam of length L = 10 meters is subjected to the following loads:

• A uniformly distributed load (UDL) of w = 2 kN/m acting over the entire length.
• A concentrated load of P = 5 kN acting at a distance of 3 meters from the left support.

This setup provides a realistic and challenging example that showcases various aspects of shear and bending moment calculations.

Step-by-Step Solution:

Here's a detailed step-by-step guide to drawing the shear and bending moment diagrams for our assumed problem 7-53 scenario:

1. Reactions at the Supports

The first step is to determine the reactions at the supports (R_A and R_B). We'll use the equations of equilibrium:

• ΣF_y = 0: R_A + R_B - wL - P = 0
• ΣM_A = 0: R_BL - wL(L/2) - P(3) = 0

Substituting the given values (L = 10m, w = 2 kN/m, P = 5 kN):

• R_A + R_B - (2)(10) - 5 = 0 => R_A + R_B = 25 kN
• 10R_B - (2)(10)(5) - 5(3) = 0 => 10R_B = 115 kN => R_B = 11.5 kN

Therefore, R_A = 25 kN - 11.5 kN = 13.5 kN

2. Shear Force Diagram

To draw the shear force diagram, we'll analyze the shear force at different points along the beam:

• At x = 0 (left support): V = R_A = 13.5 kN (positive)
• At x = 3m (point load): V = R_A - P = 13.5 kN - 5 kN = 8.5 kN
• At x = 10m (right support): V = R_A - P - wL = 13.5 kN - 5 kN - (2 kN/m)(10m) = -11.5 kN (negative, this should match -R_B)

The shear force diagram will be a straight line with a slope equal to the uniformly distributed load (-w). There will be a sudden drop of 5 kN at x = 3m due to the concentrated load. The diagram starts at +13.5kN, drops to +8.5kN at x=3m, then decreases linearly to -11.5kN at x=10m.

3. Bending Moment Diagram

The bending moment diagram is determined using the shear force diagram. The slope of the bending moment diagram at any point is equal to the shear force at that point.

• At x = 0: M = 0 (simply supported)
• At x = 3m: M = R_A(3) - P(0) = 13.5kN * 3m = 40.5 kNm
• At x = 10m: M = 0 (simply supported)

The bending moment equation can be derived considering the different sections of the beam. For example, from x=0 to x=3m the bending moment equation would be: M(x) = R_Ax - wx²/2 And from x=3m to x=10m: M(x) = R_Ax - wx²/2 - P(x-3)

The bending moment diagram will have a parabolic shape due to the uniformly distributed load. The maximum bending moment will occur where the shear force is zero. This point can be found by setting the shear force equation to zero and solving for x. The maximum bending moment can then be calculated by substituting this x value into the bending moment equation.

Finding the point of zero shear:

The shear force equation between x=3m and x=10m is: V(x) = 8.5 - 2x. Setting V(x) = 0: 8.5 - 2x = 0 => x = 4.25m

Calculating the maximum bending moment:

Substituting x = 4.25m into the bending moment equation (from x=3 to x=10): M_max = 13.5(4.25) - 2(4.25)²/2 - 5(4.25-3) = 43.1875 kNm

The bending moment diagram will start at 0, reach a maximum positive value at approximately x=4.25m, then decrease parabolically to 0 at x=10m.

Drawing the Diagrams

After calculating the critical points, we can draw the shear and bending moment diagrams. Use a clear scale to represent the values. Label all key points: reactions, loads, maximum bending moment, and zero-shear points. Accurate diagrams are essential for effective structural analysis. Remember to use proper notation, indicating positive and negative regions. Computer-aided design (CAD) software can assist in generating accurate diagrams.

Advanced Considerations:

• Different Load Combinations: Problem 7-53 could involve a different combination of loads (point loads, UDL, couples, etc.). The process remains similar, but the calculations become more complex.
• Overhanging Beams: Overhanging beams introduce additional complexity due to the possibility of negative shear forces and moments at different locations.
• Statically Indeterminate Beams: If the beam is statically indeterminate (more supports than needed for equilibrium), you'll need to use additional methods, such as the method of superposition or the flexibility method.

Conclusion:

Drawing accurate shear and bending moment diagrams is a cornerstone of structural analysis. This guide provides a detailed step-by-step process for a typical simply supported beam problem. While we've used an assumed configuration for problem 7-53, the underlying principles remain the same for various loading and support conditions. Remember to always carefully analyze the problem statement, apply the principles of statics, and accurately represent the results graphically. Mastering this skill is critical for anyone involved in structural engineering. Understanding how to interpret and use these diagrams is vital to ensuring the safety and stability of any structure. Thorough calculations and clear visualization are keys to successful structural analysis.