Shear Force and Bending Moment Diagrams

Introduction

• Important graphical methods for mechanical and civil engineers.
• Used to analyze beams under loading.
• Objective: Understand shear forces and bending moments.

Definitions

• Shear Forces: Vertical internal forces that develop to maintain equilibrium.
• Bending Moments: Resultant of tensile and compressive normal forces.
  • Top of the Beam: Compressive forces (shorter).
  • Bottom of the Beam: Tensile forces (longer).

Internal Forces Representation

• Internal forces can be represented with two resultants:
  • Shear Force (vertical internal forces).
  • Bending Moment (normal internal forces).

Types of Loads on Beams

1. Concentrated Forces
2. Distributed Forces
3. Concentrated Moments

Types of Supports

• Pinned Supports: Prevent vertical and horizontal displacements; allow rotation.
• Roller Supports: Prevent vertical displacement; allow horizontal movement and rotation.
• Fixed Supports: Prevent all displacements and rotation.

Steps to Determine Shear Forces and Bending Moments

1. Draw Free Body Diagram: Show all applied and reaction loads.
2. Calculate Reaction Forces/Moments: Use equilibrium equations to maintain balance of forces and moments.
3. Determine Internal Forces: Cut the beam at various points and calculate shear forces and bending moments.

Equilibrium Conditions

• For equilibrium:
  • Sum of vertical/horizontal forces = 0
  • Sum of moments at any point = 0
• Statically Determinate Beams: All reactions can be calculated.
• Statically Indeterminate Beams: More unknowns than equations.

Sign Convention

• Applied Forces: Positive if acting downwards.
• Shear Forces:
  • Pointing downwards (left side of cut) = Positive
  • Pointing upwards (right side of cut) = Positive
• Bending Moments:
  • Sagging = Positive
  • Hogging = Negative

Example Calculation

• Beams with pinned and roller supports and concentrated forces:
  • Calculate reaction forces at supports using equilibrium equations.
  • Draw shear force and bending moment diagrams based on calculations.

Complex Loading and Relationships

• Distributed Forces: Can be replaced with equivalent concentrated forces.
• Relationships:
  • D-V/D-X = - Distributed Force
  • D-M/D-X = Shear Force
• Integrating these equations gives insight into changes in shear force and bending moment.

Example with Distributed Force and Concentrated Force

• Analyze bending moment curve using quadratic equations.
• Verify accuracy using derivatives and area under curves.

Final Example - Cantilever Beam

1. Draw free body diagram including reaction forces and moments.
2. Apply equilibrium equations:
   • Vertical forces and moments at point A.
3. Analyze shear forces and bending moments as loads are applied.
4. Predict deformed shape based on bending moment distribution:
   • Sagging where moments are positive, hogging where negative.

Conclusion

• Understanding shear forces and bending moments assists in beam analysis.
• Essential concepts for structural engineering and design.