Internal force diagrams are crucial tools for understanding how structures handle loads. They show shear forces, axial forces, and bending moments acting within beams and other elements. These diagrams help engineers spot critical points and design safer, more efficient structures.

In this part of the chapter, we'll learn how to create and interpret these diagrams. We'll explore different types of loads, sign conventions, and the relationships between load, shear, and moment. This knowledge is key for analyzing statically determinate structures.

Types of Internal Force Diagrams

Shear, Axial, and Bending Moment Diagrams

• Shear force diagram represents the internal shear forces acting perpendicular to the beam's axis
  • Illustrates the variation of shear force along the length of the beam
  • Positive shear forces typically shown above the reference line, negative below
• Axial force diagram depicts the internal forces acting parallel to the beam's longitudinal axis
  • Displays tension (positive) or compression (negative) forces along the beam
  • Crucial for analyzing columns and truss members
• Bending moment diagram shows the internal bending moments along the beam
  • Represents the tendency of the beam to rotate about a point
  • Positive moments usually cause tension in the bottom fibers, compression in the top

Diagram Construction and Interpretation

• Shear force diagram constructed by calculating the algebraic sum of vertical forces from one end
  • Jumps occur at point loads, slope changes at distributed loads
• Axial force diagram typically constant for beams without inclined supports or applied axial loads
  • Changes occur at connection points or where axial loads are applied
• Bending moment diagram derived from the shear force diagram
  • Area under the shear force curve between two points equals the change in bending moment
• Diagrams help identify critical sections where maximum internal forces occur
  • Essential for designing beams, columns, and other structural elements

Loading Conditions

Types of Loads and Their Effects

• Distributed loads spread over a finite length or area of the structure
  • Uniform distributed loads have constant intensity (kN/m or lb/ft)
  • Non-uniform distributed loads vary in intensity along their length
  • Represent scenarios like snow on a roof or self-weight of a beam
• Point loads concentrated at specific locations on the structure
  • Modeled as forces acting at a single point
  • Can be vertical, horizontal, or inclined
  • Represent scenarios like vehicle wheels on a bridge or column loads
• Moment loads apply rotational forces to the structure
  • Measured in units of force times distance (kN·m or lb·ft)
  • Can be applied at any point along the beam
  • Represent scenarios like wind loads on tall structures or connections in frames

Load Combinations and Analysis

• Real structures often experience combinations of different load types
  • Superposition principle allows adding effects of individual loads
• Load paths describe how forces are transferred through the structure
  • Distributed loads typically converted to equivalent point loads for analysis
• Dynamic loads vary with time (wind, earthquakes) but often simplified as static loads
  • Amplification factors used to account for dynamic effects in static analysis
• Load factors and load combinations specified by building codes
  • Account for uncertainties and ensure structural safety

Diagram Characteristics

Sign Conventions and Coordinate Systems

• Sign conventions crucial for consistent interpretation of diagrams
  • Positive shear typically causes clockwise rotation of the beam segment
  • Positive moment usually causes compression in top fibers, tension in bottom
  • Positive axial force generally indicates tension, negative for compression
• Global coordinate system defines overall structure orientation
  • Local coordinate systems used for individual members or elements
• Consistent use of sign conventions essential for equilibrium equations
  • Helps in identifying tension and compression zones in the structure

Discontinuities and Critical Points

• Discontinuities in diagrams occur at specific locations
  • Point loads cause jumps in shear force diagrams
  • Support reactions create discontinuities in shear and moment diagrams
  • Changes in cross-section can lead to stress concentrations
• Critical points identified where internal forces reach maximum values
  • Often occur at supports, load application points, or section changes
  • Crucial for design as they determine required member strengths
• Zero points in diagrams provide valuable information
  • Zero shear indicates maximum or minimum moment
  • Zero moment indicates inflection points where curvature changes

Relationships Between Load, Shear, and Moment

• Fundamental relationships link load, shear, and moment diagrams
  • Rate of change of shear force equals the negative of the applied load
  • Rate of change of bending moment equals the shear force
• Mathematical expressions of relationships:
  • $\frac{dV}{dx} = -w(x)$ where V is shear and w(x) is distributed load
  • $\frac{dM}{dx} = V$ where M is moment and V is shear
• Area method uses these relationships for diagram construction
  • Area under the load diagram between two points equals change in shear
  • Area under the shear diagram between two points equals change in moment
• Understanding these relationships allows quick sketching of diagrams
  • Helps in visualizing behavior of structures under various loading conditions