🧱Structural Analysis Unit 3 – Truss Analysis

Key Concepts and Definitions

• Trusses consist of straight members connected at joints, forming a stable geometric configuration
• Members are connected by frictionless pins, allowing rotation but preventing translation
• Loads and reactions are applied only at the joints
• Trusses are designed to carry loads primarily through axial forces (tension or compression) in the members
  • Axial forces act along the longitudinal axis of the member
  • Tension forces tend to elongate the member
  • Compression forces tend to shorten the member
• Trusses are assumed to be statically determinate, meaning the unknown forces can be determined using equilibrium equations alone
• The term "truss" is derived from the Old French word "trousse," which means "collection of things bound together"

Types of Trusses

• Planar trusses have all members and loads lying in a single plane (2D)
  • Common planar trusses include Warren, Pratt, Howe, and Bailey trusses
• Space trusses have members and loads in three dimensions (3D)
  • Examples of space trusses include tetrahedral and octahedral configurations
• Simple trusses are supported by pinned joints at one end and roller supports at the other end
• Compound trusses are formed by connecting two or more simple trusses
• Bridge trusses are designed to carry loads across spans (rivers, valleys, or roadways)
  • Examples include through trusses, deck trusses, and arch trusses
• Roof trusses are used to support roofs and transfer loads to the walls or columns
  • Common roof truss configurations include King Post, Queen Post, and Fink trusses

Assumptions in Truss Analysis

• Trusses are composed of straight members connected at their ends by frictionless pins
• Loads and reactions act only at the joints, not along the length of the members
• Members are connected in a way that allows rotation but prevents translation at the joints
• All members are two-force members, meaning they have forces acting only at their ends
• The weight of the members is negligible compared to the applied loads
• Axial forces (tension or compression) are constant along the length of each member
• Trusses are assumed to be statically determinate for analysis purposes
• Deformations in the truss are small enough to not significantly affect the geometry or force distribution

Method of Joints

• The Method of Joints is used to determine the forces in truss members by considering the equilibrium of each joint
• Begin by drawing a free-body diagram of the entire truss, showing all external loads and reactions
• Identify a joint with at least one known force and two unknown member forces
  • Start at a joint with the most known forces or simplest geometry
• Draw a free-body diagram of the joint, showing all forces acting on it
• Apply the equations of equilibrium (ΣFx = 0 and ΣFy = 0) to solve for the unknown member forces
  • Use trigonometry to determine the components of the member forces in the x and y directions
• Repeat the process for the next joint, using the previously calculated member forces as known values
• Continue until all member forces have been determined

Method of Sections

• The Method of Sections is used to determine the force in a specific member without analyzing the entire truss
• Begin by drawing a free-body diagram of the entire truss, showing all external loads and reactions
• Imagine cutting the truss into two sections, with the desired member being one of the cut members
• Choose the section that has fewer unknown forces for analysis
• Draw a free-body diagram of the chosen section, showing all external loads, reactions, and member forces at the cut
  • Indicate the assumed directions of the unknown member forces (tension or compression)
• Apply the equations of equilibrium (ΣFx = 0, ΣFy = 0, and ΣM = 0) to solve for the desired member force
  • Use a moment equation to solve for the force in the desired member directly
• Repeat the process for any other members of interest

Zero-Force Members

• Zero-force members are truss members that do not carry any load under a given loading condition
• Identifying zero-force members can simplify the analysis by reducing the number of unknown forces
• Two-force members connected to the truss at only one joint are always zero-force members
• Members that connect two joints with no external loads or reactions are zero-force members
• In a symmetric truss with symmetric loading, members that are symmetric about the center are zero-force members
• Removing a zero-force member from the truss does not affect the forces in the remaining members
  • However, removing a zero-force member may affect the stability of the truss

Stability and Determinacy

• Stability refers to a truss's ability to maintain its shape and support loads without collapsing
• Determinacy refers to the ability to determine all member forces using the equations of equilibrium alone
• Stable and determinate trusses have a unique solution for member forces under a given loading condition
• Unstable trusses have insufficient members or support conditions to maintain equilibrium
  • Unstable trusses may collapse or have infinitely many solutions for member forces
• Indeterminate trusses have more unknown forces than available equilibrium equations
  • Indeterminate trusses require additional compatibility equations or methods (like the force method) for analysis
• The determinacy of a truss can be assessed using the equation: m + r = 2j, where m is the number of members, r is the number of reaction components, and j is the number of joints
  • If m + r < 2j, the truss is unstable
  • If m + r > 2j, the truss is indeterminate
  • If m + r = 2j, the truss is stable and determinate

Real-World Applications

• Bridges: Trusses are widely used in bridge construction to span large distances and support heavy loads (vehicular traffic, trains, or pedestrians)
  • Examples include the Golden Gate Bridge (San Francisco), Sydney Harbour Bridge (Australia), and Quebec Bridge (Canada)
• Roofs: Trusses are used to support roofs in residential, commercial, and industrial buildings
  • Roof trusses allow for large open spaces beneath, such as in auditoriums, gymnasiums, and warehouses
• Cranes and towers: Trusses are used in the construction of cranes and towers to provide strength and stability
  • Examples include tower cranes used in construction sites and communication towers
• Aircraft: Trusses are used in aircraft structures, particularly in wings and fuselages, to provide lightweight and strong support
  • The Wright brothers' first successful airplane, the Wright Flyer, used a truss structure for its wings
• Space structures: Trusses are used in the construction of space structures, such as the International Space Station (ISS), to provide a rigid and lightweight framework
  • The ISS's main truss structure spans over 100 meters and supports solar arrays, radiators, and other essential components