Shear Force and Bending Moment Diagrams

Introduction

• Key graphical methods for mechanical and civil engineers
• Used to analyze how a beam is loaded

Internal Forces in Beams

Types of Internal Forces

• Shear Forces: Oriented vertically
• Normal Forces: Oriented along the beam axis
  • Compressive: Top of the beam (shortens)
  • Tensile: Bottom of the beam (lengthens)

Representation

• Internal forces can be represented by two resultants:
  • Shear Force: Resultant of vertical internal forces
  • Bending Moment: Resultant of normal internal forces

Drawing Diagrams

Shear Force and Bending Moment Diagrams

• Dependent on loads acting and beam supports

Types of Loads

• Concentrated forces
• Distributed forces
• Concentrated moments

Types of Supports

• Pinned Supports: Prevent vertical/horizontal displacement, allow rotation
• Roller Supports: Prevent vertical displacement, allow horizontal displacement/rotation
• Fixed Supports: Prevent all displacements and rotation

Reaction Forces and Moments

• Corresponding reaction forces/moments at support locations based on the degree of restraint
• Pinned support example: prevents vertical/horizontal displacement (reaction forces), allows rotation (no reaction moment)

Determining Shear Forces and Bending Moments

Steps

1. Draw Free Body Diagram (FBD): Shows applied and reaction loads
2. Calculate Reaction Forces/Moments: Use equilibrium equations

• Equilibrium: Forces/moments must cancel out
• Statically Determinate: All reaction loads can be calculated with equilibrium equations
• Statically Indeterminate: More unknowns than equilibrium equations (need additional methods)

3. Calculate Internal Forces/Moments: For each beam location, cut the beam and maintain equilibrium

Sign Convention

• Applied Forces: Positive if downward
• Shear Forces/Bending Moments: Defined differently for left/right side of the cut
  • Positive bending moments cause sagging, negative cause hogging

Example: Beam with Pinned and Roller Supports

• Loaded by two concentrated forces
• Steps:
  1. Draw Free Body Diagram
  2. Use equilibrium equations to find reaction forces at Points A and B
  3. Draw Shear Force and Bending Moment Diagrams

Calculations

• Vertical forces and moments around a point must be zero
• Example values: R-A = 12 kN, R-B = 9 kN
• Determine shear forces/bending moments by moving along beam length

More Complex Loading

• Relationships: Between distributed force, shear force graph, and bending moment graph
  • Differential relationships show the slope of shear force and bending moment curves
  • Integration used to find changes over a beam section
  • Example: Bending moment quadratic equation differentiated to find shear force/distributed force equations

Area Method

• Area under shear force curve for a section equals change in bending moment
• Useful for sense-checking diagrams

Example: Cantilever with Concentrated Moment and Distributed Force

• Fully fixed support gives both reaction forces and a reaction moment

• Equilibrium equations used to find reaction forces/moments

• Steps:

  1. Draw free body diagram
  2. Calculate shear forces and bending moments moving left to right

  Special Considerations

  • Uniformly distributed force can be replaced by an equivalent concentrated force
  • Resulting shear force and bending moment equations derived using normal methods

Predicting Beam Shape

• Bending moment diagram can indicate sagging, hogging, and straight sections

Conclusion

• Understanding these diagrams helps in analyzing how beams are loaded and predicting their deformations
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