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:
Calculate fixed-end moments
Determine distribution factors based on member stiffnesses
Distribute unbalanced moments
Carry over 50% of distributed moments to far ends
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 A B = 2 E I L ( 2 θ A + θ B − 3 ψ ) + F E M A B M_{AB} = \frac{2EI}{L}(2\theta_A + \theta_B - 3\psi) + FEM_{AB} M A B = L 2 E I ( 2 θ A + θ B − 3 ψ ) + FE M A B
M B A = 2 E I L ( θ A + 2 θ B − 3 ψ ) + F E M B A M_{BA} = \frac{2EI}{L}(\theta_A + 2\theta_B - 3\psi) + FEM_{BA} M B A = L 2 E I ( θ A + 2 θ B − 3 ψ ) + FE M B A
Where:
E = modulus of elasticity
I = moment of inertia
L = member length
θ = end rotations
ψ = relative displacement
FEM = fixed-end moments
Process involves:
Write slope deflection equations for each member
Apply equilibrium conditions at each joint
Solve resulting system of equations for unknown rotations and displacements
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: F E M = w L 2 12 FEM = \frac{wL^2}{12} FEM = 12 w L 2
Concentrated load at midspan: F E M = P L 8 FEM = \frac{PL}{8} FEM = 8 P L
Triangular load: F E M = w L 2 30 FEM = \frac{wL^2}{30} FEM = 30 w L 2 (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:
Apply unit load at various positions along the frame
Analyze structure for each load position
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