This is easy explanation of our proposal on the complex modulus¹⁾. This paper is originally written in Japanese, then, transrated into English (JAEE permits English paper same with Japane paer only when they are identical.)

I alreasy showed two issues, Complex modulus in SHAKE and Lysmer's proposal and its discussion issue on the complex modulus used in the equivalene linear analysis of ground. The followings are conclusions

1)SHAKE²⁾ uses a complex modulus, named Sorokin model. This model reproduce damping ratio same with the laboratory test result (cyclic shear test) or so called i-\(\gamma\) relationships. However, it overestimates maximum stress \(\sqrt{1+4h^2}\) times larger
2)Lysmer model³⁾
Lysmer's proposal ,named Lysmer model, is based on a signle-degree-of-freedom (SDOF) system and the complex modulus is determined so that the model gives the same displacement of the viscous damping in which Voilgt model is used. Howerver, behaviors of a SDOF system and that of soil are are different; they should not be compared. This indicates that Lysmer's discussion is not rational.
3) Lysmer gives the same maximum shear stress with given G-\(\gamma\) relationships, but damping ratio is underestimated.

1. YAS model

Figure 1 shows the schematic figure of the hysteresis curves by the test and models. As shown in the preceding, the complex modulus used in SHAKE (Soroin model) gives the same damping ratio, but maximu shear stress is evaluated \(\sqrt{1+4h^2}\) times larger. The YAS model is a model which gives ths same maximum shear stress and the damping ratio with test as shown in Figure 1.

Let us consider a model that gives the damping ratio same with test. As shown in Eq. (14) in Complex modulus in SHAKE, hysteretic absorption energy \(\Delta W\) depends on the imaginary part only. Therefore, the complex modulus is expressed as

\(\hspace{20mm}\)\(\overline{G}_Y^*=G_r+2ihG \)(1)

\(\hspace{20mm}\)\(\overline{\gamma}=\gamma e^{i\omega t} \)(2)

Then the stress-strain relationships result in

\(\hspace{20mm}\)\(\overline{\tau}=G_Y^*\overline{\gamma}=(G_r+2iGh)\gamma_0e^{i\omega t}\)(3)

Then, stress-strain relationships is obtained by using the real part as

\(\hspace{20mm}\)\(\tau=\mathrm{Re}{(\overline{\tau})}=\gamma_0(G_r\cos{\omega t}-2Gh\sin{\omega t})=\gamma_0\sqrt{G_r^2+4G^2h^2}\cos{(\omega t+\phi)} \)

\(\hspace{20mm}\)\(\tan{\phi}=2Gh/G_r \)(4)

\(\hspace{20mm}\)\(G_r=\sqrt{1-4h^2} \)(5)

Therefore, finally we obtain the complex modulus of the YAS model as

\(\hspace{20mm}\)\(\overline{G}_Y^*=G(\sqrt{1-4h^2}+2ih)\)(6)

Finally we obtains stress-strain relationships of the YAS model as

\(\hspace{20mm}\)\(\tau=G\gamma_0(\sqrt{1-4h^2}\cos{\omega t}+2h\sin{\omega t})=G\gamma_0\cos{(\omega t+\phi)}\)
\(\hspace{20mm}\)\(\tan{\phi}=\cfrac{2h}{\sqrt{1-4h^2}}\)(7)

2. Comparison with other complex moduli

Three complex moduli have been introduced. They are

\(\hspace{20mm}\)Sorokin model: \(G^*=G(1+2ih)\) (the model used in the original SHAKE)

\(\hspace{20mm}\)Lysmer model: \(G^*=G(1-2\beta^2+2i\beta\sqrt{1-\beta^2})\) (proposed by LYSMER)

\(\hspace{20mm}\)YAS model: \(\overline{G}_Y^*=G(\sqrt{1-4h^2}+2ih)\) (Proposed by us)

It is noted that \(\beta\) is used for the Lysmer model, whereas \(h\) is used for other models. Since \(h=\beta\) in the Sorokin and YAS models, both parameters can be used; mechanically important prameter, damping ratio \(h\) is used in the above equation. Howerver, since \(\beta\) is different from \(h\), \(\beta\) is used to make the difference clear.

It is already shown that the relation between \(\beta\) and \(h\) is expressed as

\(\hspace{20mm}\)\(h=\beta\sqrt{1-\beta^2} \)(8)

When \(\beta\) is calculated from this equation, and substituted into the Lysmer model, one can obtain the complex modulus same with the YAS model. In other words, it is his mistake to believe \(h=\beta\). If he caucllated \(h\) from \(\beta\), he could obtain the result same with us.

Hysteresis curves of three models are shown in Figure 3 for \(\beta =\)0.1, 0.2, and 0.3. Values of h for corresponding \(\beta\) is also shown in the figure for the Lyamer model. The difference between (\beta\) and \(h\) is a little less than 5% at \(h\)=0.3.

The real and imaginary parts of the complex moduli are shown in Figure 4 with respect to the damping ratio and the damping parameter. The real part of the Sorokin model is a straight line (constant value), and the imaginary part is also a linear line. Imaginary part of the YAS model is same with that of the Sorokin model, which indicates that both the Sorokin and YAS models absorb the same hysteretic energy. However, the real part of the YAS model becomes smaller as \(\beta\) increases, which indicates that stiffness of the linear part becomes smaller as \(\beta\) increases. Since the shape of the hysteretic curve is an eliptic shape in the complex moduli, in order to get the same maximum stress, the slope of the linear part is to become smaller as \(\beta\). This discussion is the same with the Lysmer model. In addition, Imaginary part becomes smaller as \(\beta\) in the Lysmer model, which results in smaller hysteretic absorption energy than that of the test. The error is, however, small; it is about 5% at \(h=30\)%. If this error is negrigible, use of the Lysmer model is possible, but use of the YAS model is reccommended because it is mechanically rational.

Figure 3. Hysteresis curves for three models

Figure 4. Real and Imaginary parts of three complex moduli as functions with respect to damping ratio \(h\) and damping parameter \(\beta\)

1)Yoshida, N. and Adachi, K.; Complex moduli for seismic response analysis of ground, Journal of JAEE (Japanese version will be printed soon)
2)Schnabel, P. B., Lysmer, J. and Seed, H. B. (1972): SHAKE A Computer program for earthquake response analysis of horizontally layered sites, Report No. EERC72-12, University of California, Berkeley
3) Lysmer, J.: Modal damping and complex stiffness, University of California Lecture note, University of California, Berkeley, 1973.