Lateral load distribution is a critical aspect of structural engineering, ensuring that buildings can withstand forces such as wind and seismic activity. In frame buildings, especially those with moment-resisting frames, understanding how lateral loads are distributed is essential for stability and safety. This article explores the degrees of freedom in frame structures and various methods used for lateral load analysis.
Degrees of Freedom in Frame Buildings
In a two-dimensional moment-resisting frame, each joint typically exhibits three degrees of freedom: horizontal displacement, vertical displacement, and rotation. The total degrees of freedom in such a frame is calculated as 3Nj, where Nj represents the number of joints.
In practical scenarios, axial deformation in beams is often negligible, ensuring uniform horizontal displacement across beam-level joints. Similarly, in mid-rise buildings, axial deformation of columns is minimal, reducing the degrees of freedom to only one horizontal displacement and one rotation per joint.
For dynamic analysis, further simplifications can be made by reducing degrees of freedom through static condensation, resulting in one degree of freedom per storey. In three-dimensional frames, each joint can theoretically have six degrees of freedom, but typically, three degrees of freedom per floor are considered. The building’s free vibration analysis is conducted by solving a (3N × 3N) Eigenvalue problem, where N is the number of storeys. Once the natural frequency and mode shape are determined, the maximum seismic force at each storey level can be computed.
Lateral Load Analysis of Moment-Resisting Frames
Once lateral loads are determined, the next step is analyzing the frame for member forces. This can be done either through computer-based methods or approximate methods, depending on the design stage and project requirements. Approximate methods are often used in preliminary design or to validate computational results.
Methods of Lateral Load Analysis
A. Portal Frame Method
The portal frame method is widely used for analyzing lateral loads in moment-resisting frames. These frames serve as primary load-bearing elements, commonly found at bridge entrances and high-rise buildings. They transfer lateral forces from the top of the frame to the foundation, resisting forces such as wind, earthquakes, and traffic loads.
1. Pin-Supported Portal Frames
- These frames are statically indeterminate to the first degree.
- To simplify analysis, an inflection point is assumed at the midpoint of the girder, reducing the system to a determinate structure.
- Horizontal reactions at the column bases are equal, allowing for reaction and moment calculations using statics.
2. Fixed-Supported Portal Frames
- These frames are statically indeterminate to the third degree.
- Inflection points are assumed at the midpoints of all three members.
- Reactions and moment diagrams are determined by dismembering the frame at these points and solving statically.
3. Partially Fixed Portal Frames
- These frames have supports that allow slight rotations due to construction constraints.
- Engineers assume inflection points at h/3 of column height, along with hinges at the girder’s center.
- This approach provides practical and conservative estimates for real-world designs.
4. Trussed Frames
- Trusses in portal frames enhance structural efficiency over large spans.
- Analysis follows similar principles as basic portal frames, with pin-supported or fixed-supported columns.
- Inflection points play a key role in determining axial and lateral force distribution.
B. Cantilever Method
The cantilever method models a building frame as a long cantilever beam subjected to transverse loading. This approach is especially useful for tall and slender buildings.
- Assumptions include placing inflection points in columns and beams.
- Axial forces are approximated based on column geometry and position.
- The method is ideal for frames with non-uniform column sections.