Page 359 - Structural Steel Designers Handbook AISC, AASHTO, AISI, ASTM, and ASCE-07 Design Standards

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LATERAL-FORCE DESIGN

LATERAL-FORCE DESIGN 8.13

8.5 DYNAMIC METHOD OF SEISMIC LOAD DISTRIBUTION

The equivalent static-force method (Art. 8.4) 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 con-
sistent 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, that is, soft stories.
• 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.
• The in-plane offset of the lateral-force-resisting elements is greater than the length of these elements.
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 follow-
ing 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 in effective diaphragm 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 analy-
sis, 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. Nonlinear pushover analyses are also used with increasing frequency to
evaluate the inelastic behavior and inelastic seismic design in irregular or unusual structures.
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 accelera-
tion a g may be expressed as

2
dx dx
m + c + kx = − ma g (8.13)
dt 2 dt
2
2
where d x/dt is the acceleration of the structure, dx/dt is the velocity relative to the ground 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 (8.13) can be solved by a number of
methods.
The maximum acceleration is often expressed as a function of the fundamental period of vibra-
tion 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 earth-
quakes and actual structural characteristics.

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