Building frames are integral to the structural integrity of a wide range of construction projects, especially in multi-story buildings. Their design and analysis are vital to ensure safety, stability, and durability. Building frames are typically designed to have rigid beam-column joints, making them strong and capable of bearing various loads. In structural engineering, different analysis methods are used to determine the internal forces, reactions, and displacements in building frames. The method of analysis chosen depends on factors like the type of frame, its configuration (portal bay or multi-bay), and the degree of indeterminacy of the structure. In this article, we explore the various methods used to analyze building frames, including the force method, displacement method, approximate methods, and Kani’s method.
Types of Building Frames
Building frames come in many shapes and sizes, but the most common form is the reinforced concrete multistory frame. These frames are designed to provide a rigid connection between beams and columns, ensuring that the entire structure behaves as a unified whole under various loads. The design of these frames typically assumes that the joints between beams and columns are rigid, meaning that the angles between the members do not change when the structure is loaded.
In practice, the design and analysis of building frames depend heavily on the frame’s geometry, the number of bays, the load distribution, and the degree of indeterminacy (whether the internal forces can be determined directly from the external loads and boundary conditions).
Analysis Methods for Building Frames
- Force Method (Flexibility Method)
The force method, also known as the flexibility method or method of consistent deformation, is primarily used to analyze statically indeterminate structures. This method focuses on calculating the internal forces and reactions of a structure that cannot be determined by static equilibrium alone.
In the force method, the frame is first transformed into a statically determinate system by removing the redundant forces or constraints. Then, the magnitude of the redundant forces required to restore the structure’s boundary conditions is calculated. The approach works by relating the deformations to the forces, providing a means to solve for the unknown internal forces and reactions.
The force method is particularly suitable for analyzing single-storey frames and frames with uncommon geometries, such as gabled frames. While effective, it can become complex when dealing with large, multi-storey structures.
- Displacement Method
The displacement method is a more general approach, which focuses on determining the displacements in a structure due to applied loads. The basic principle behind the displacement method is to express the unknown displacements as a function of the loads using the load-displacement relationship. Once the displacements are determined, the unknown forces or reactions can be obtained from the compatibility equations.
Several methods fall under the displacement method, each suited to different types of frames and structures:
2.1 Slope Deflection Method
The slope deflection method is often used to analyze both statically determinate and indeterminate frames and beams. This method assumes that the deformations of the structure are solely due to bending, neglecting the influence of axial and shear stresses. Additionally, it assumes that all joints in the frame are rigid, meaning the angles between the members remain unchanged under loading. The slope deflection method can be effective for analyzing frames with multiple degrees of freedom.
2.2 Moment Distribution Method
The moment distribution method is a step-by-step iterative process that distributes and balances internal moments at the joints of a structure until the structure reaches equilibrium. The process begins by assuming that all joints are fixed, and then each joint is unlocked and locked in succession. The internal moments are distributed and balanced at each stage to determine the final joint rotations. This method is known for its accuracy and flexibility, especially in the analysis of continuous frames.
2.3 Direct Stiffness Method
The direct stiffness method is a matrix-based approach where equilibrium equations are formulated into a single matrix relationship. It is particularly useful for analyzing large, complex structures because it allows for automatic solving of the system of equations. The method directly relates the displacements of each joint to the applied loads, and is commonly used in modern computer-based structural analysis software.
- Approximate Methods
In some cases, especially during the preliminary design phase, approximate methods can be used to quickly estimate the internal forces and moments in a frame. These methods involve making realistic assumptions about the behavior of the structure, simplifying the calculations while still providing useful results for early-stage designs.
3.1 Portal Method
The portal method is primarily used to analyze frames subjected to horizontal loads, such as wind or seismic forces. Key assumptions of the portal method include the location of points of inflection at the mid-height of columns and mid-span of beams. Additionally, the total horizontal shear at each floor is assumed to be distributed among the columns, with exterior columns carrying half the load of the interior columns. While the portal method simplifies the analysis, it is most effective for frames with regular geometry and consistent load distribution.
3.2 Cantilever Method
The cantilever method is typically applied to high-rise structures. This method assumes that inflection points occur at the midpoint of each girder and at mid-height of each column. It also assumes that the intensity of axial stress in a column is proportional to its horizontal distance from the center of gravity of all the columns in a storey. Although the cantilever method is an approximation, it can provide a quick and reasonable estimate of internal forces in high-rise frames.
3.3 Points of Inflection Method
The points of inflection method is used to analyze frames subjected to vertical loads, such as dead and live loads. This method simplifies the frame by introducing points of inflection, reducing the structure to a statically determinate form. Key assumptions in this method include the location of inflection points and the assumption that axial forces in beams are negligible. This approach is often used in preliminary design stages to estimate the behavior of frames under vertical loads.
- Kani’s Method
Kani’s method is an iterative approach used to analyze indeterminate frames by distributing the unknown fixed-end moments to adjacent joints in order to satisfy continuity conditions. The most important feature of Kani’s method is its self-correcting nature—any errors made during the iteration process are automatically corrected in subsequent steps. This method is particularly useful for frames with multiple bays and complicated geometries.
The method is effective for structures where the internal moments are not easily calculated using other methods, and it provides a way to refine the analysis by progressively distributing moments to reach a more accurate solution.