🧱Structural Analysis Unit 5 – Frame Analysis

Key Concepts and Definitions

• Frame analysis involves the study of the behavior, stability, and strength of structural frames under various loading conditions
• Frames are composed of interconnected members (beams, columns, trusses) that form a stable structure capable of resisting loads
• Loads can be classified as dead loads (permanent, fixed weights), live loads (temporary, movable weights), or environmental loads (wind, snow, seismic)
• Reactions are the forces and moments that develop at the supports of a frame to maintain equilibrium under applied loads
  • Supports can be fixed (no translation or rotation), pinned (allows rotation), or roller (allows translation and rotation)
• Internal forces in frame members include axial forces (tension or compression), shear forces, and bending moments
• Stability refers to a frame's ability to maintain its equilibrium position under applied loads without collapsing or experiencing excessive deformation
• Determinacy relates to the number of equilibrium equations available compared to the number of unknown reactions and internal forces in a frame
  • Statically determinate frames have sufficient equations to solve for all unknowns
  • Statically indeterminate frames have more unknowns than available equations and require additional compatibility conditions

Types of Frames and Their Components

• Rigid frames consist of members with fixed connections that resist rotation and provide high stability
  • Examples include portal frames, multi-story frames, and bridge frames
• Pin-jointed frames have members connected by frictionless pins that allow rotation but not translation
  • Trusses are a common type of pin-jointed frame
• Gabled frames feature sloped roof members connected to vertical columns, often used in residential and industrial buildings
• Vierendeel frames have rigid joints and no diagonal members, relying on the bending resistance of the members for stability
• Braced frames incorporate diagonal members (braces) to provide lateral stability and resist wind and seismic loads
• Moment-resisting frames rely on the bending strength of the members and rigid connections to resist lateral loads without additional bracing
• Connections in frames can be welded, bolted, or riveted, depending on the material and design requirements
  • Welded connections provide a rigid, continuous joint
  • Bolted and riveted connections allow for easier assembly and disassembly

Forces and Loads on Frames

• Dead loads are permanent, constant forces acting on a frame, such as the self-weight of the structure and fixed equipment
• Live loads are variable, temporary forces caused by occupancy, movable equipment, or storage
  • Examples include people, furniture, vehicles, and stored materials
• Environmental loads are forces caused by natural phenomena, such as wind, snow, rain, and earthquakes
  • Wind loads are lateral forces that vary with height and exposure
  • Snow loads are vertical forces that depend on the geographical location and roof geometry
  • Seismic loads are dynamic forces caused by ground motion during earthquakes
• Impact loads are sudden, short-duration forces caused by collisions or explosions
• Thermal loads result from temperature changes that cause expansion or contraction of frame members
• Combination loads consider the simultaneous occurrence of different load types, using load factors to account for the probability and severity of each load
• Distributed loads are forces applied over a length or area of a frame member (uniform, triangular, or trapezoidal)
• Concentrated loads are forces applied at a specific point on a frame member

Methods of Frame Analysis

• The force method (flexibility method) involves solving for redundant forces by considering the compatibility of deformations
  • Releases are introduced to make the structure statically determinate
  • Compatibility equations are written to solve for the redundant forces
• The displacement method (stiffness method) solves for unknown displacements by considering the equilibrium of forces
  • Member stiffness matrices relate end forces to end displacements
  • Global stiffness matrix is assembled from member stiffness matrices
  • Equilibrium equations are solved for unknown displacements
• The moment distribution method is an iterative procedure for analyzing indeterminate frames
  • Fixed-end moments are calculated for each member
  • Moments are distributed to adjacent members based on their relative stiffness
  • Process is repeated until convergence is achieved
• The slope-deflection method relates member end moments to end rotations and displacements
  • Slope-deflection equations are derived from moment-curvature relationships
  • Equilibrium and compatibility conditions are applied to solve for unknown rotations and displacements
• Influence lines represent the variation of a specific response (reaction, shear, moment) at a point in a frame as a unit load moves along the structure
  • Used to determine the critical position of moving loads for maximum response
• Computer-based methods, such as finite element analysis (FEA), discretize the frame into smaller elements and solve for displacements, stresses, and strains using numerical techniques

Stability and Determinacy of Frames

• Stable frames maintain equilibrium under applied loads without collapsing or experiencing excessive deformation
  • Stability requires sufficient reactions to prevent rigid body motion (translation and rotation)
• Unstable frames have insufficient reactions to maintain equilibrium and may collapse or deform excessively under load
• Determinate frames have enough equilibrium equations to solve for all unknown reactions and internal forces
  • Statically determinate frames satisfy the equation: $r = 3j - 3$, where $r$ is the number of reactions and $j$ is the number of joints
• Indeterminate frames have more unknown reactions and internal forces than available equilibrium equations
  • Statically indeterminate frames require additional compatibility conditions to solve for the unknowns
  • Degree of indeterminacy is the difference between the number of unknowns and the number of equilibrium equations
• Kinematically indeterminate frames have more unknown joint displacements than available compatibility equations
• Stability and determinacy are independent concepts
  • A frame can be stable and determinate, stable and indeterminate, or unstable
• Geometric stability refers to a frame's ability to maintain its shape under loading without experiencing large deformations or buckling
  • Slender members and frames with high height-to-width ratios are more susceptible to geometric instability

Calculating Reactions and Internal Forces

• Reactions are determined by applying equilibrium equations (ΣFx = 0, ΣFy = 0, ΣMz = 0) to the entire frame
  • Free body diagrams are used to visualize the external forces and reactions acting on the frame
• Internal forces (axial, shear, and moment) are calculated by analyzing each member individually
  • Axial forces are determined using the method of sections and applying equilibrium equations
  • Shear forces are calculated by considering the equilibrium of a portion of the member and constructing shear force diagrams
  • Bending moments are determined by analyzing the equilibrium of a portion of the member and constructing bending moment diagrams
• Sign conventions are used to maintain consistency in the direction and sense of internal forces and moments
  • Tension is typically considered positive for axial forces
  • Counterclockwise moments are usually taken as positive
• Shear and moment diagrams provide a graphical representation of the variation of internal forces along the length of a member
  • Diagrams are constructed by analyzing the member at critical points (supports, load points, and cross-section changes)
• Maximum and minimum values of internal forces are identified from the shear and moment diagrams
  • Location and magnitude of maximum moments are critical for designing the member's cross-section
• Influence lines can be used to determine the maximum internal forces caused by moving loads
  • Influence lines are constructed by placing a unit load at various positions along the member and calculating the corresponding internal force at a specific point

Deformation and Displacement in Frames

• Deformation refers to the change in shape or size of a frame or its members under applied loads
• Displacements are the translations and rotations of joints or specific points in a frame
• Axial deformation is the change in length of a member due to tension or compression forces
  • Axial deformation is calculated using Hooke's law: $δ = (PL)/(AE)$, where $P$ is the axial force, $L$ is the member length, $A$ is the cross-sectional area, and $E$ is the modulus of elasticity
• Flexural deformation is the bending of a member due to transverse loads and moments
  • Flexural deformation is characterized by the deflection (vertical displacement) and slope (rotation) of the member
  • Deflection is calculated using the moment-area method or integration of the moment-curvature relationship: $EI(d^2v/dx^2) = M(x)$
• Shear deformation is the change in shape of a member due to shear forces
  • Shear deformation is usually negligible compared to flexural deformation in slender members
• Torsional deformation is the twisting of a member due to torsional moments
• Combined deformation considers the simultaneous occurrence of different types of deformation (axial, flexural, shear, and torsional)
• Serviceability limit states are design criteria that ensure the functionality and comfort of a structure under normal use
  • Deflection limits are imposed to prevent excessive deformation that may cause damage to finishes, partitions, or equipment
  • Vibration limits are set to avoid discomfort or disturbance to occupants
• Deformation compatibility ensures that the deformations of connected members are consistent and do not result in gaps or overlaps
  • Compatibility equations relate the displacements and rotations at the ends of connected members

Real-World Applications and Examples

• Building frames, such as multi-story steel or concrete structures, are designed to resist gravity and lateral loads
  • Moment-resisting frames and braced frames are commonly used in high-rise buildings to provide lateral stability
  • Serviceability limits on deflection and vibration are critical for occupant comfort and the performance of non-structural components
• Bridge frames, including girder bridges and truss bridges, are designed to span long distances and carry heavy traffic loads
  • Influence lines are used to determine the critical position of moving vehicle loads for maximum internal forces
  • Fatigue and dynamic load effects are important considerations in bridge frame design
• Industrial frames, such as crane gantries and conveyor supports, are subject to large concentrated loads and impact forces
  • Stiffness and strength requirements govern the design of industrial frames to ensure safe and efficient operation
• Transmission towers and telecommunication masts are tall, slender frames that must resist wind and ice loads
  • Geometric nonlinear analysis may be necessary to account for the effects of large deflections and P-delta moments
• Scaffolding and temporary support structures are designed to be easily assembled and dismantled while providing safe access and support for construction activities
  • Connections and bracing systems are critical for the stability and load-carrying capacity of temporary frames
• Aerospace structures, such as aircraft fuselages and wings, are lightweight frames that must withstand complex loading conditions
  • Advanced materials, such as composites and high-strength alloys, are used to optimize strength-to-weight ratios
  • Finite element analysis is extensively used to model the behavior of aerospace frames under various flight scenarios
• Automotive frames, including car chassis and roll cages, are designed to protect occupants and maintain structural integrity during crashes
  • Crashworthiness and energy absorption are key considerations in the design of automotive frames
  • Computer simulations and physical testing are used to validate the performance of automotive frames under impact loads