Shear force and bending moment diagrams show internal forces in beams. They help us see where a beam might break or bend too much. These diagrams are key tools for engineers designing safe structures.

Building these diagrams involves calculating reactions, plotting equations, and finding critical points. Understanding how loads affect shear and moment helps us interpret the diagrams and predict beam behavior under different conditions.

Shear Force and Bending Moment Diagrams

Graphical Representations and Internal Forces

• Shear force and bending moment diagrams are graphical representations of the internal forces and moments acting on a beam along its length
• Shear force is the internal force that acts perpendicular to the beam's axis
• Bending moment is the internal moment that causes the beam to bend
• Concentrated loads (point loads), distributed loads, and moments can be applied to a beam
• The magnitudes and locations of these loads and moments affect the shear force and bending moment diagrams

Constructing Diagrams and Sign Conventions

• The sign convention for shear force and bending moment diagrams follows the right-hand rule
  • Positive shear force acts upward
  • Positive bending moment causes compression on the top of the beam
• Constructing shear force and bending moment diagrams involves:
  • Calculating the reactions at the supports
  • Determining the equations for shear force and bending moment along the beam
  • Plotting these equations
• Discontinuities in the shear force diagram occur at concentrated loads and reactions
• Discontinuities in the bending moment diagram occur at concentrated moments
• The slope of the shear force diagram represents the distributed load acting on the beam
• The slope of the bending moment diagram represents the shear force

Critical Points on Diagrams

Locations and Significance of Critical Points

• Critical points on shear force and bending moment diagrams include:
  • Locations of zero shear force
  • Maximum or minimum shear force
  • Zero bending moment (inflection points)
  • Maximum or minimum bending moment
• Points of zero shear force indicate the locations of maximum or minimum bending moment
  • The slope of the bending moment diagram changes sign at these points
• Points of maximum or minimum shear force occur at:
  • Concentrated loads
  • Reactions
  • Discontinuities in the distributed load
• Points of zero bending moment (inflection points) indicate the locations where the beam changes from positive to negative curvature or vice versa

Identifying Critical Points

• Points of maximum or minimum bending moment occur where the shear force diagram crosses the zero axis
  • The slope of the bending moment diagram is zero at these points
• To identify critical points:
  • Analyze the shear force diagram for zero crossings, maxima, and minima
  • Examine the bending moment diagram for zero crossings, maxima, minima, and inflection points
  • Consider the locations of concentrated loads, reactions, and discontinuities in the distributed load

Interpretation of Diagrams

Insights from Shape and Magnitude

• The shape and magnitude of shear force and bending moment diagrams provide insights into the internal forces and moments acting on the beam
• They help identify critical regions that may require special attention in design
• The magnitude of the shear force at any point represents the net internal shear force acting at that cross-section of the beam
• The magnitude of the bending moment at any point represents the net internal moment causing bending at that cross-section of the beam
• Regions with high magnitudes of shear force or bending moment indicate areas of the beam that experience significant internal stresses

Deflected Shape and Curvature

• The curvature of the bending moment diagram indicates the beam's deflected shape
  • Positive curvature corresponds to sagging (beam bending downward)
  • Negative curvature corresponds to hogging (beam bending upward)
• Sudden changes in the slope of the shear force or bending moment diagrams indicate the presence of:
  • Concentrated loads
  • Reactions
  • Moments acting on the beam
• Analyzing the curvature and slope changes helps understand the beam's behavior under loading

Load, Shear, and Moment Relationships

Interconnections and Derivatives

• The load, shear force, and bending moment diagrams are interconnected
• Understanding their relationships is crucial for analyzing beam behavior
• The distributed load acting on the beam is equal to the slope of the shear force diagram at any point
• The shear force at any point is equal to the slope of the bending moment diagram at that point
• The second derivative of the bending moment diagram with respect to the beam's length is equal to the distributed load acting on the beam

Discontinuities and Integration

• Concentrated loads and reactions appear as discontinuities or jumps in the shear force diagram
  • The magnitude of the jump is equal to the load or reaction
• Concentrated moments appear as discontinuities or jumps in the bending moment diagram
  • The magnitude of the jump is equal to the applied moment
• By integrating the shear force diagram, the bending moment diagram can be obtained
• By integrating the bending moment diagram, the beam's deflection can be determined
• These relationships allow for the calculation of internal forces, moments, and deflections at any point along the beam