5.3 Analysis of rigid frames
4 min read•august 9, 2024
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 .
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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- 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
- 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
- refers to lateral displacement of a structure due to horizontal loads
- Caused by wind, seismic activity, or unbalanced vertical loads
- Affects and of buildings
- evaluates frame behavior under lateral loads
- Considers (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
- 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
- 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 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:
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 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
- Common fixed-end moment formulas:
- Uniformly distributed load:
- Concentrated load at midspan:
- Triangular load: (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 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
- 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
Key Terms to Review (28)
ACI 318: ACI 318 is the American Concrete Institute's Building Code Requirements for Structural Concrete, which provides guidelines and standards for the design and construction of reinforced concrete structures. It serves as a foundational document that influences not only the analysis of rigid frames but also the overall compliance with building codes and design standards in structural engineering.
AISC Code: The AISC Code, set forth by the American Institute of Steel Construction, is a standard that provides guidelines for the design and construction of steel structures. It ensures that these structures are safe, stable, and efficient by offering a consistent framework for analyzing loads, material properties, and design methods, particularly in the context of rigid frames.
Axial Load: An axial load is a force that acts along the longitudinal axis of a structural member, typically causing tension or compression. This type of loading is critical in determining the stability and strength of structures, influencing how members respond under different conditions. Understanding axial loads is essential for analyzing various structural forms, as it helps predict deformations, stresses, and potential failure modes within systems such as trusses and frames.
Bending Moment: A bending moment is a measure of the internal moment that induces bending in a structural element due to external loads applied to it. It is crucial in understanding how beams respond to various types of loads and supports, which directly influences the design and analysis of structures.
Cantilever Method: The cantilever method is a structural analysis technique used to evaluate the behavior and response of structures that have elements fixed at one end and free at the other. This approach is particularly useful for analyzing rigid frames and understanding the effects of applied loads, as it allows for the simplification of complex structures by isolating segments for easier analysis. The cantilever method is essential in both determining internal forces and designing stable frames, especially in scenarios where side sway may or may not occur.
Deflection: Deflection refers to the displacement of a structural element from its original position due to applied loads. It is a crucial concept in understanding how structures respond to forces, influencing the design and performance of various structural elements under different loading conditions.
Equilibrium: Equilibrium refers to a state in which all the forces and moments acting on a structure are balanced, resulting in no net movement or rotation. This fundamental condition is crucial for maintaining the stability and integrity of various structures, ensuring that they can withstand applied loads without deforming or collapsing.
Euler-Bernoulli Beam Theory: Euler-Bernoulli Beam Theory is a fundamental theory in structural analysis that describes the relationship between the bending of beams and the applied loads on them. This theory assumes that plane sections of the beam remain plane and perpendicular to the neutral axis after deformation, allowing for the calculation of deflections, shear forces, and moments. It simplifies the analysis of rigid frames by providing a basis for understanding how beams behave under various loading conditions.
Finite Element Analysis: Finite Element Analysis (FEA) is a computational technique used to obtain approximate solutions to complex structural engineering problems by breaking down structures into smaller, simpler parts called finite elements. This method allows engineers to analyze the behavior of structures under various loads, enabling effective design and optimization.
Fixed-end Moments: Fixed-end moments are the bending moments that occur at the ends of a beam or frame when it is fixed in place and subjected to external loads. These moments are crucial in analyzing structures because they represent the internal stresses that resist the applied loads, helping to determine how the structure will behave under various loading conditions.
Lateral load: Lateral load refers to the forces acting horizontally on a structure, such as wind, earthquakes, or other environmental factors. These loads can significantly impact the structural integrity and stability of buildings and frames, necessitating careful consideration during the design and analysis processes to ensure that structures can withstand such forces without experiencing failure or excessive deformation.
Load Path: Load path refers to the route through which loads are transferred through a structure to the ground. Understanding load paths is crucial for ensuring that structural elements effectively support and distribute forces, which impacts stability, safety, and overall performance. The concept of load paths is intertwined with the analysis of structural components, helping engineers identify critical elements and potential failure points within various types of structures.
Masonry shear wall equations: Masonry shear wall equations are mathematical expressions used to analyze and design masonry shear walls, which are structural elements that resist lateral forces such as wind and seismic loads. These equations help in calculating the shear capacity, flexural strength, and overall stability of the wall, taking into account factors like material properties and wall dimensions. Understanding these equations is crucial for ensuring the structural integrity of buildings that utilize masonry shear walls in their design.
Method of Joints: The Method of Joints is a technique used to analyze trusses by isolating each joint to solve for the forces in the members connected to that joint. This method is fundamental in understanding how loads are transferred through a truss structure and relies on the assumption that all joints are pin-connected, allowing for equilibrium conditions to be applied at each joint to determine internal member forces.
Method of Sections: The method of sections is a technique used in structural analysis to determine the internal forces in a truss by cutting through the truss and analyzing the equilibrium of one of the resulting sections. This method allows for direct calculation of member forces without needing to analyze every joint, making it particularly useful for large or complex truss structures.
Moment Distribution Method: The moment distribution method is a structural analysis technique used to analyze indeterminate structures by distributing moments at the joints until equilibrium is achieved. This method allows for the consideration of both fixed and pinned supports, enabling engineers to solve for internal forces and moments in continuous beams and frames effectively.
Moment-Resisting Frame: A moment-resisting frame is a structural system that provides stability and strength by allowing beams and columns to resist bending moments and lateral forces without the need for additional bracing. This type of frame is designed to maintain its shape and integrity under loads, making it ideal for buildings that experience dynamic forces like wind or earthquakes.
P-delta effects: P-delta effects refer to the secondary moments that arise in a structure due to the displacement of vertical loads caused by lateral displacements, like sway. This effect is critical for analyzing the stability and strength of structures, particularly in frames subjected to lateral loads, as it can significantly amplify the internal forces and moments in the members. Understanding p-delta effects is essential for ensuring that structures can safely withstand the combined effects of gravity and lateral forces.
Plastic hinge: A plastic hinge is a localized region in a structural member where plastic deformation occurs, allowing the member to rotate without further increase in moment. This concept is crucial in analyzing rigid frames because it signifies a shift from elastic behavior to plastic behavior, affecting the overall stability and load-carrying capacity of the structure during extreme loading conditions.
Portal Method: The portal method is a simplified analysis technique used for calculating the forces and moments in rigid frames, particularly those that have a symmetric layout and experience lateral loads. It focuses on transforming the frame into a series of individual portal frames to help simplify the calculation of reactions, internal forces, and moments. This method is particularly useful when analyzing structures that may or may not exhibit sidesway, allowing engineers to derive approximate solutions efficiently.
Pushover analysis: Pushover analysis is a nonlinear static analysis method used to evaluate the seismic performance of structures by applying incremental lateral loads until failure. This technique helps engineers understand how a structure behaves under seismic forces and identify potential weaknesses or failure modes, making it essential for the analysis of rigid frames.
Redundancy: Redundancy refers to the presence of more structural elements than are necessary to maintain stability and carry loads in a structure. This concept is essential in engineering design, as it provides additional safety and resilience against unforeseen forces or failures. When considering redundancy, engineers ensure that if one component fails, the structure can still function, maintaining overall integrity and safety.
Rigid Frame: A rigid frame is a structural system where members are connected in such a way that they resist deformation under loads, maintaining their shape and stability. This type of frame can handle both vertical and lateral loads effectively, making it suitable for various applications, including buildings and bridges. Its design allows for the transfer of loads through the frame itself without relying solely on supports or foundations.
Serviceability: Serviceability refers to the ability of a structure to perform its intended function without experiencing unacceptable levels of deformation or discomfort to its occupants. It focuses on the structure’s performance under normal use, ensuring that it remains functional and aesthetically pleasing while minimizing excessive deflection and vibrations that could lead to dissatisfaction or damage.
Sidesway analysis: Sidesway analysis is the evaluation of lateral displacements and deflections in structures, particularly in rigid frames under lateral loads like wind or seismic forces. This analysis is crucial for ensuring stability and serviceability of the structure, as excessive sidesway can lead to failure or functionality issues. It assesses how much a frame will sway and helps engineers design appropriate bracing or support systems to mitigate movement.
Slope deflection method: The slope deflection method is a structural analysis technique used to determine the internal forces and displacements of indeterminate beams and frames by relating the slopes and deflections at the supports and joints to the bending moments. This method incorporates the principles of equilibrium, compatibility, and material behavior, allowing for the analysis of structures that have more supports or constraints than can be solved using simple methods. It is particularly useful in analyzing rigid frames and in dealing with statically indeterminate structures where traditional methods fall short.
Stability: Stability refers to the ability of a structure to maintain its equilibrium and resist collapse under applied loads or forces. It involves ensuring that structures can withstand various conditions without experiencing excessive deformation or failure, which is crucial for safety and functionality in engineering designs.
Sway: Sway refers to the lateral movement or deflection of a structure, especially under load or external forces like wind or seismic activity. In the context of rigid frames, sway is crucial as it affects the overall stability and strength of the structure, leading to potential failures if not properly accounted for during design and analysis.