SAP2000 has always offered both exact Eigen vectors and Load-dependent Ritz (LDR vectors). As stated in the CSI Analysis Reference Manual that is included with SAP2000:
- Eigenvector analysis determines the undamped free-vibration modes shapes and frequencies of the system. These natural modes provide an excellent insight into the behavior of the structure.
- Ritz-vector analysis seeks to find modes that are excited by a particular loading. Ritz vectors can provide a better basis that do eigenvectors when used for response-spectrum or time-history analyses that are based on modal superposition.
We always recommend performing an Eigen analysis of the structure, since it can help to check behavior and find modeling problems. The natural frequency information can also be important for understanding where resonance can be expected with different types of loads.
For determining the response to horizontal ground acceleration, eigen modes are generally quite efficient, although it may be necessary to include a missing-mass (residual-mass) mode to account for high-frequency effects that are missed by the Eigen modes. Most common structural analysis programs have this capability, as does SAP2000.
For vertical ground acceleration, or localized loading such as by machine vibration, eigen modes may be much less efficient, and determining whether or not you have enough modes can be difficult. Load-dependent Ritz vectors are very effective for this purpose, and have the advantage of always including the missing-mass modes automatically. Even for horizontal ground excitation, Ritz vectors are more efficient and widely used.
Mass participation is a common measure for determining whether or not you have enough modes. However, it is only useful for ground acceleration, and does not give information about localized response. SAP2000 provides static and dynamic participation measures for other types of loading as well. When determining convergence of localized response with respect to the number of modes, Ritz vectors converge much faster and more uniformly than do Eigen vectors.
Load dependent Ritz vectors are well documented in the open literature (e.g., Wilson et al), in standard finite-element textbooks (e.g., Cook et al), and in standard structural dynamics textbooks (e.g., Chopra). Please let us know if you need detailed citations.
SAP2000 lets you, the engineer, determine the type of modes you feel are most appropriate. In fact, you can calculate both types of modes simultaneously, in the same model, even in the same run, and compare their behavior. SAP2000 offers much flexibility in the calculation of Eigen or Ritz modes, including the consideration of P-delta and other nonlinear effects, modes at different stages of construction, and frequency shifts for specialized loading.
When using Eigen vectors, you control the convergence tolerance. Orthogonality is strictly maintained to within the accuracy of the machine (15 decimal digits). Sturm sequence checks are performed and reported to avoid missing any Eigen vectors when using shifts. Internal accuracy checks are performed and used to control the solution automatically. SAP2000 will detect and report ill-conditioned systems, but will still produce eigen vectors that you can use to trace the source of the modeling problem. Ritz vectors are not subject to convergence questions, but strict orthogonality of the vectors is maintained as it is for eigen vectors.
