5.3 Analysis of rigid frames

Rigid frame analysis is crucial for understanding how structures respond to loads. This section covers methods like the portal and cantilever approaches, which simplify complex frames for quick estimates. We'll also look at exact techniques like moment distribution and slope deflection.

These analysis methods help engineers determine shear forces, moments, and axial loads in frame members. By mastering these tools, you'll be able to design safer, more efficient structures that can withstand various loading conditions.

Approximate Analysis Methods

Portal Method and Cantilever Method

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• Portal method assumes hinges form at midpoints of beams and columns
  • Divides multi-story frame into separate portal frames
  • Simplifies analysis by treating each story independently
  • Calculates shear distribution based on relative stiffness of columns
• Cantilever method treats entire frame as a vertical cantilever beam
  • Assumes inflection points occur at midheight of columns
  • Distributes shear forces proportionally to column areas
  • Works well for tall, slender structures (height-to-width ratio > 4)
• Both methods provide quick estimates for preliminary design
  • Accuracy generally within 10-15% of exact methods for regular frames
  • Less accurate for irregular or highly indeterminate structures

Sway and Sidesway Analysis

• Sway refers to lateral displacement of a structure due to horizontal loads
  • Caused by wind, seismic activity, or unbalanced vertical loads
  • Affects stability and serviceability of buildings
• Sidesway analysis evaluates frame behavior under lateral loads
  • Considers P-Delta effects (secondary moments due to axial loads)
  • Assesses frame stiffness and drift limitations
• Methods to analyze sway include:
  • Approximate hand calculations (portal and cantilever methods)
  • Computer-based finite element analysis
  • Pushover analysis for nonlinear behavior
• Importance of sway control in structural design
  • Ensures occupant comfort and prevents damage to non-structural elements
  • Critical for tall buildings and structures in high wind or seismic zones

Exact Analysis Methods

Moment Distribution Method

• Iterative technique developed by Hardy Cross for indeterminate structures
  • Begins with assumed fixed-end moments
  • Distributes unbalanced moments at joints using distribution factors
  • Continues until moment balance is achieved within acceptable tolerance
• Steps in moment distribution:
  1. Calculate fixed-end moments
  2. Determine distribution factors based on member stiffnesses
  3. Distribute unbalanced moments
  4. Carry over 50% of distributed moments to far ends
  5. Repeat steps 3-4 until convergence
• Advantages include:
  • Can handle complex, highly indeterminate structures
  • Provides insight into load path and member behavior
  • Adaptable to various loading conditions and support types

Slope Deflection Method

• Based on force-displacement relationships of structural members
  • Expresses end moments in terms of rotations and displacements
  • Forms a system of equations solved simultaneously
• Key equations for slope deflection: $M_{AB} = \frac{2EI}{L}(2\theta_A + \theta_B - 3\psi) + FEM_{AB}$ $M_{BA} = \frac{2EI}{L}(\theta_A + 2\theta_B - 3\psi) + FEM_{BA}$ Where:
  • E = modulus of elasticity
  • I = moment of inertia
  • L = member length
  • θ = end rotations
  • ψ = relative displacement
  • FEM = fixed-end moments
• Process involves:
  1. Write slope deflection equations for each member
  2. Apply equilibrium conditions at each joint
  3. Solve resulting system of equations for unknown rotations and displacements
  4. Calculate member end moments and reactions

Fixed-End Moments and Their Applications

• Fixed-end moments represent end moments in a fully fixed beam
  • Depend on loading condition and beam geometry
  • Serve as starting point for moment distribution method
• Common fixed-end moment formulas:
  • Uniformly distributed load: $FEM = \frac{wL^2}{12}$
  • Concentrated load at midspan: $FEM = \frac{PL}{8}$
  • Triangular load: $FEM = \frac{wL^2}{30}$ (at heavily loaded end)
• Applications in structural analysis:
  • Used in moment distribution and slope deflection methods
  • Help in quick estimation of moments in continuous beams
  • Provide basis for simplified analysis of indeterminate structures

Frame Analysis Results

Shear and Moment Diagrams

• Shear diagram represents internal shear force distribution along a member
  • Plotted on the tension side of the member
  • Jumps occur at points of concentrated loads or reactions
• Moment diagram shows bending moment variation along a member
  • Plotted on the compression side of the member
  • Parabolic for uniformly distributed loads, linear for point loads
• Relationship between shear and moment diagrams:
  • Slope of moment diagram equals shear at any point
  • Area under shear diagram between two points equals change in moment
• Importance in structural design:
  • Identifies critical sections for member sizing
  • Helps determine required reinforcement in concrete members
  • Guides placement of splices and connections in steel structures

Axial Force Diagrams and Their Interpretation

• Axial force diagram shows distribution of normal forces along member axis
  • Tension forces typically shown as positive, compression as negative
  • Constant for prismatic members under pure axial load
• Interpretation of axial force diagrams:
  • Indicates load transfer path through the structure
  • Helps identify members prone to buckling (long compression members)
  • Guides selection of appropriate cross-sections and materials
• Combined with moment diagrams for beam-column design
  • Interaction diagrams used to check capacity under combined loading
  • Critical for design of columns in multi-story frames

Influence Lines for Frames

• Influence lines show effect of a unit load at any position on a specific response
  • Can be drawn for reactions, shear forces, moments, or displacements
  • Useful for determining worst-case loading scenarios
• Construction of influence lines for frames:
  1. Apply unit load at various positions along the frame
  2. Analyze structure for each load position
  3. Plot desired response (reaction, moment, etc.) vs. load position
• Applications in frame analysis:
  • Determine critical load positions for maximum effects
  • Analyze effects of moving loads (bridges, crane runways)
  • Assess impact of settlement or support movement on frame behavior
• Interpretation requires understanding of structural behavior
  • Peaks indicate most influential load positions
  • Sign changes reveal load positions causing reversal of forces or moments