Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 296

Figure 8.9. Illustration of the peak factor effect

On the other hand, the G X bandwidth of the X(t) double peak process will be
definitely wider. It can be estimated owing to the formula:

1/ 2

ª

2

º

į Į8/ʌİ with: Į = ȦȦ/ ȦȦ |

[8.43]

x

¬ ¼

n m n m

valid in the case: H 1 = H 2 = HoG 1 = G 2 = G = (4H / S) 1/2 1.

For example: f 1 = 4 Hz, f 2 = 6 Hz, H = 0.01 oG X = 0.26 and G = 0.11.

In Figure 8.9, we represented the evolution of the peak factor according to the
bandwidth, for the response process of 1-dof and 2-dof oscillators. The curves allow
us to determine the P 1, P 2 and P X peak factors. In the previous example, we have, for
100 half-cycles carried out with an average 5 Hz frequency, P 1 |P 2 | 2.5 and P X |
3.0, that is, a 1.20 ratio. This ratio represents the error made when carelessly
applying the maximum quadratic combination formula. This error always tends to be
underestimated. In our example, it is equal to 20%.

There exist correction processes of the complete quadratic combination formula
that consist of introducing a multiplying function determined from abacuses into the
cross-terms of the sum (see for instance [GIB 88]).