# Dynamic Method of Seismic Load Distribution

The ‘‘Uniform Building Code’’ static-force method (Art. 9.4) is based on a single-mode response with approximate load distributions and corrections for higher-mode response.

These simplifications are appropriate for simple. regular structures. However, they do not consider the full range of seismic behavior in complex structures. The dynamic method of seismic analysis is required for many structures with unusual or irregular geometry, since it results in distributions of seismic design forces that are consistent with the distribution of mass and stiffness of the frames, rather than arbitrary and empirical rules. Irregular structures include frames with any of the following characteristics:

The lateral stiffness of any story is less than 70% of that of the story above or less than 80% of the average stiffness for the three stories above The mass of any story is more than 150% of the effective mass for an adjacent story,

except for a light roof above The horizontal dimension of the lateral-force-resisting system in any story is more than 130% of that of an adjacent story The story strength is less than 80% of the story above Frames with a story strength that is less than 80% of that of the story above must be designed with consideration of the P effects caused by gravity loading combined with the seismic loading.

Frames with horizontal irregularities place great demands on floors acting as diaphragms and the horizontal load-distribution system. Special care is required in their design when any of the following conditions exist:

The maximum story drift due to torsional irregularity is more than 1.2 times the average story drift for the two ends of the structure.

There are reentrant corners in the plan of the structure with projections more than 15% of the plan dimension

The diaphragms are discontinuous or have cutouts or openings totaling more than 50% of the enclosed area or changes of stiffness of more than 50%

There are discontinuities in the lateral-force load path Irregular structures commonly require use of a variation of the dynamic method of seismic analysis, since it provides a more appropriate distribution of design loads. Many of these structures should also be subjected to a step-by-step dynamic analysis (linear or nonlinear) for specific accelerations to check the design further.

The dynamic method is based on equations of motion for linear-elastic seismic response.

The equation of motion for a single-degree-of-freedom system subjected to a seismic ground acceleration ag may be expressed as

motion, and x is the displacement from an equilibrium position. The coefficients m, c, and k are the mass, damping, and stiffness of the system, respectively. Equation (9.12) can be solved by a number of methods.

The maximum acceleration is often expressed as a function of the fundamental period of vibration of the structure in a response spectrum. The response spectrum depends on the acceleration record. Since response varies considerably with acceleration records and structural period, smoothed response spectra are commonly used in design to account for the many uncertainties in future earthquakes and actual structural characteristics.

Most structures are multidegree-of-freedom systems. The n equations of motion for a system with n degrees of freedom are commonly written in matrix form as

and is normalized by the zone factor Z used in the static-force method. Given the modes of vibration for a multidegree-of-freedom system, a spectral acceleration for each mode, Sai , can be determined from the response spectra. The base shear Vi acting in each mode can then be determined from

Other response characteristics for each mode can be calculated from similar equations.

The maximum response in each mode does not occur at the same time for all modes. So some form of modal combination technique is used. The complete quadratic combination (CQC) method is one commonly used method for rationally combining these modal contributions.

(E. L. Wilson et al., ‘‘A Replacement for the SRSS Method in Seismic Analysis,’’ Earthquake Engineering and Structural Dynamics, vol. 9, pp. 187–194, 1981.) The method degenerates into a variation of the square root of the sum of the squares (SRSS) method when the modes of vibration are well-separated. The summation must include an adequate number of modes to assure that at least 90% of the mass of the structure is participating in the seismic loading.

The total seismic design force and the force distribution over the height and width of the structure for each mode can be determined by this method. The combined force distribution takes into account the variation of mass and stiffness of the structure, unusual aspects of the structure, and the dynamic response in the full range of modes of vibration, rather than the single mode used in the static-force method. The combined forces are used to design the structure, often reduced by R in accordance with the ductility of the structural system. In many respects, the dynamic method is much more rational than the static-force method, which involves many more assumptions for computing and distributing design forces. The dynamic method sometimes permits smaller seismic design forces than the static-force method. However, while it offers many rational advantages, the dynamic method is still a linear-elastic approximation to an inelastic-design method. As a result, it assumes that the inelastic response is distributed throughout the structure in the same manner as predicted by the elastic-mode shapes. This assumption may be inadequate if there is a brittle link in the system.