Shear and Moment Diagrams: An Introduction

Internal Forces (V and M)

• Internal forces in beams: Shear Force (V) and Bending Moment (M)
• Importance: Knowing the value of V and M at every point for beam design

Plotting Shear Force and Bending Moment

• To determine V and M across a beam, graph them along the beam's length
• Purpose: Identifying maximum shear force and bending moment for safe design

Key Concepts:

Related Graphs

• V and M graphs should be aligned vertically; each graph is the integral of the one above it:
  • Load Curve
  • Shear Force Curve
  • Bending Moment Curve
• Order of Lines:
  • Concentrated Load -> Horizontal Line -> Slope -> Parabolic -> Cubic

Problem Solving Steps

Step 1: Global Equilibrium

• Identify support reactions (e.g., pin, roller)
• Calculate reaction forces:
  • Sum of Forces = 0
  • Sum of Moments = 0
• Example: Problem with 18 kN, 12 kN, and 6 kN loads over discrete intervals
• Solve for reaction forces at supports: B_Y and A_Y

Step 2: Discontinuities

• Identify and draw discontinuities where loads change (concentrated loads, boundary conditions)
• Correct discontinuities back to distributed loads if required for accurate plotting

Step 3: Plotting the V Diagram

• Plot the shear force (V) across the beam:
  • Start from zero, apply loads, and maintain the integral relationship
  • Ensure summation returns to zero at the end
• Example values are given to illustrate the change in V across intervals

Tips for Accurate Graphing

• Ensure all calculations are correct; discrepancies indicate mistakes in global equilibrium
• Analysis through analogy (walking across the beam with a backpack): load additions affect the shear force step by step
• Mark plus/minus to indicate the upward/downward slope on M diagram

Plotting the M Diagram

• Areas under V diagram give moments:
  • Height in kN and width in meters
  • Area calculated as ": height * width = kN * meter = moment"
• Plot Bending Moment (M) based on area calculations, ensuring slope consistency
  • Example: Plot calculations for specific beam segments, e.g., 28, 40, 12 kN.m
• Ensure bending moment also returns to zero, validating correctness

Conclusion

• Maximum Shear Force: 16 kN
• Maximum Bending Moment: 28 kN.m
• Importance of consistent and careful plotting for designing safe structures

Next Steps

• Look forward to advanced topics like concentrated moments in future presentations