Page 360 - Structural Steel Designers Handbook AISC, AASHTO, AISI, ASTM, and ASCE-07 Design Standards

Brockenbrough_Ch08.qxd 9/29/05 5:21 PM Page 8.14

LATERAL-FORCE DESIGN

8.14 CHAPTER EIGHT

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

+
+
[ ]{} [ ]{} [ ]{} = − [ ]{ }a g (8.14)
Mx ˙˙
C ˙
K
x
x
B
M
where [M], [C], and [K] are n × n square matrices of the mass, damping, and stiffness, and {¨x},
˙
{x}, and {x} are column vectors of the acceleration, relative velocity, and relative displacement.
The column vector {B} defines the direction of the ground acceleration relative to the orientation
of the mass matrix. The multidegree-of-freedom equations are coupled. They can be solved simul-
taneously by a number of methods. However, the single-degree-of-freedom response spectrum
method is also commonly used for multidegree-of-freedom systems. The solution is assumed to be
separable and the n eigenvalues (natural frequencies) ω i and eigenvectors (mode shapes) {Φ i } are
found. The solutions for the relative displacements, relative velocities, and accelerations are then
for i equals 1 to n:
n
i ∑
{} = {Φ j } ( ) (8.15a)
ft
x
j
j=1
n
i ∑
˙
{˙ } = {Φ j } () (8.15b)
ft
x
j
j=1
i ∑
{˙˙ } = n {Φ j } () (8.15c)
˙˙
ft
x
j
j=1
The mode shapes are orthogonal with respect to the mass and the stiffness matrix. This orthogo-
nality uncouples the equations of motion if the damping matrix is a diagonal matrix or proportional
T
T
to a combination of the mass and stiffness matrix; that is, {Φ j } [M]{Φ i } and {Φ j } [K]{Φ i } are zero
if i ≠ j and scalar numbers if i = j.
The response-spectrum technique can then be used to find the maximum values of f j (t) for each
mode of vibration. Figure 8.4 shows a typical design response spectrum as produced by ASCE 7. The
response is based on calculations of the single-degree-of-freedom elastic response for a range of
earthquake acceleration records. Given the frequencies of the modes of vibration for a multidegree-
of-freedom system, a spectral acceleration for each mode, S ai , can be determined from the response
spectrum. The base shear V i acting in each mode can then be determined from
({Φ } [ ]{ }) 2
T
MB
V = j S ai (8.16)
i
Φ
Φ
T
{} [ ]{}
M
i
i
The distribution of this maximum base shear over the structure is
Φ
T
MB
{} = ({ } [ ]{ }) []{ }S (8.17)
i
F
MB
Φ
Φ
i
T
M
{} [ ]{} ai
i
i
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.
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