Contents here is detailed explanation of our paper¹⁾ on rational complex modulus. It is composed of three collums, which are Complex modulus is SHAKE, explained here, and Lysmer's proposal and its discussion issue, and rational complex modulus.

Complex modulus used in SHAKE²⁾ is explained in Complex modulus in SHAKE in detail. After the release of original SHAKE, Prof. Lysmer, one of the developer of SHAKE, proposed a new complex modulus³⁾. Since he suggest to replace original complex modulus into his new complex modulus for the people who bought SHAKE from the University of California, this new modulus was eployed a new version of SHAKE⁴⁾ as well as many computer programs. Howerver, there is only few comment in the Lysmer's memo, mechanical properties are not well documented. In addidion, as shown here, it has irrational matters. They are discussed in detail.

1. Lysmer's proposal

Lysmer's proposal is based on the vibration problem of one-degree of freedom sysmtem. Three methods are discussed here. One is the case using Voigt model, the second one is based on complex moduli by means of Sorokin model, and the third one is based on Lysmer model.

1.1 Solution of the Voigt model

Equation of motion of one-degree of freedom system is laready explained in Issue related to damping, which is

\(\hspace{20mm}\)\(m\ddot{\overline{u}}+c\dot{\overline{u}}+k\overline{u}=e^{i\omega t}\) (1)

Here, external force (right hand side of Eq. (1)) is replaced to \(e^{i\omega t}\) from \(f(x)\), since we solve the behavoir uder harmonic loading. The right hand side is external load applied at the mass with magnitude 1. It is also noted that displacenet is set a complex number.

When we use the relationships between natural circular frequency \(\omega_0=\sqrt{k/m}\) and critical damping ratio \(\beta_1\) in one-degree of freedom system expressed as

\(\hspace{20mm}\)\(2\beta_1=\cfrac{c}{\sqrt{mk}} \)(2)

we can obtain steady state solution of Eq. (1) as

\(\hspace{20mm}\)\(\overline{u}=\cfrac{1}{\omega_0^2-\omega^2+2i\beta_1\omega\omega_0}\cfrac{e^{i\omega t}}{k}=\cfrac{1}{1-\alpha^2+2i\beta_1\alpha}\cfrac{e^{i\omega t}}{k} \)(3)

We used critical damping ratio as \(\beta\), but as is explained in damping ratio of SHAKE, damping ratio discussed for one-degree of freedom and that for complex moludus in seismic behavior of ground are different. Therefore, in order to distinguish them, damping ratio used in one-degree of freedom system is denoted as \(\beta_1\). In addition, \(\alpha=\omega/\omega_0\) is called as tuning ratio, which is a ratio of circular frequency of applied load and natural circular frequency. Real part of the solution yields

\(\hspace{20mm}\)\(u=\cfrac{\cos{(\omega t-\phi)}}{k\sqrt{(1-\alpha^2)^2+(2\alpha\beta_1)^2}}\)，\(\tan{\phi}=\cfrac{1\alpha\beta_1}{1-\alpha^2}\)(4a,b)

Here, \(\phi\) denotes phase lag. Since Voigt model is used here, we will refer this solution as Voigt model.

1.2 Solution based on Sorokin model

The complex modlus by Sorolin is aready shown in Eq. (20) in Complex modulus in SHAKE. The shear modulus \(G\) is replaced into spring constant \(k\) for the discussion here, equation of motion and complex mudulus becomes

\(\hspace{20mm}\)\(m\dot{\overline{u}}+k^*\overline{u}=e^{i\omega t} \)(5)

Steady state sulution of thie equation motion yields

\(\hspace{20mm}\)\(\overline{u}=\cfrac{1}{\omega_0^2-\omega^2+2i\beta_1\omega_0^2}\cfrac{e^{i\omega t}}{m} \)(6)

Therefore, steady state solution is obtained by taking real part, resulting in

\(\hspace{20mm}\)\(u=\cfrac{1}{k\sqrt{(1-\alpha^2)^2+(2\beta_1)^2}}\cos{(\omega t-\phi)}\)，\(\tan\phi=\cfrac{2\beta_1}{1-\alpha^2}\)(7)

This solution is different from Eq. (4a)

1.3 Lysmer's proposal (Lysmer model)

　Lysmer proposed the following complex modulus³⁾.

\(\hspace{20mm}\)\(k^*=k(1-2\beta_1^2+2i\beta\sqrt{1-\beta_1^2})\)(8)

Displacement is evaluated as follows by using the same procedure in the preceding.

\(\hspace{20mm}\)\(u=\cfrac{1}{k\sqrt{(1-\alpha^2)^2+(2\beta_1)^2}}\cos{(\omega t-\phi)}\)，\(\tan\phi=\cfrac{2\beta_1\sqrt{1-\beta_1^2}}{1-\alpha^2-2\beta_1^2}\)(9)

Displacement amplitde in Eq. (7) and Eq. (9) are same. However, since phase lag is different to eath other, restorning force caharacteristics is not the same.

Explanation up to here is seen in several books such as ref. 5) and 6) or technical paper such as ref. 7). Accoring to the books^5),6), this proposal is not well regared because difference between Sorokin model and Lysmer model is small at small damping ratio (\(h\leq 0.3\)). It looks that they feel Sorokin model is sufficient. In addition, problems that \(G^*=-G\) at \(\beta=1\) and real part becomes negative are pointed out.

2 Applicability of Lysmer model and discuttion

Equation (8) appears suddenly and its mechanical property is not expained. Here a detailed investigation is made on the background of the model.

2.1 Soltions other than Lysmer model

Lysmer's proposal is based on the setting the displacement amplitude same with Voigt model in one-degree of freedom system. Howerver, solution under this condition is not one, but there are many solutions. For example, let's set complex modulus as \(k^*=k(k_r+ik_i)\) and substitute this modulus into Eq. (5) and using the condition that displacement amplitude is same with Eq. (4a), we obtain

\(\hspace{20mm}\)\(1-2\alpha^2+4\alpha^2\beta_1^2=k_r^2-2k_r\alpha^2+k_i^2\)(10)

It means that a combination of \(k_r\) and \(k_i\) that satisfy this condition gives same displacement amplitude with that of Voigt model. As an example, let's set \(k_r=1-2\beta_1^2\), then we can get \(k_i=2\beta_1\sqrt{1-\beta_1^2}\). On the other hand, in order to get the same hysteretic absorbing energy, setting \(k_i=2\beta_1^2\), then we can get \(k_r=\alpha^2\pm\sqrt{(1-\alpha^2)(1-\alpha^2-4\beta_1^2)}\). Since this \(k_r\) includes the tuning ratio, material property is frequency dependent nature, therefore it is not a good solution. Possibly the Lysmer model is the only solution that does not include tuning ratio\(\alpha\). It is a surprizing finding.

As can be understood above discussion, it is impossible to develop a model that agrees in Voigt model with displacement energy and phase lag. In this meaning, it does not seem rational to get same displacement amplitude because hysteretic energy depends on phase lag.

2.2 Steasy state behavior under harmonic external load

Generally speaking, discussion on damping constant or critical damping ratio is discussed in the damped free vibration condition, whereas only amplification characteristics is discussed under the harmonic external loading conditions. Amplification characteristics shown below if frequently appears in the textbook on vibration.

\(\hspace{20mm}\)

In the followings, the same problem is discussed from the different viewpoint, i.e., viewpoint of damping. The basic concept is same with the one that Lysmer considered, but we discuss only in real number. Equation of motion is as follows.

\(\hspace{20mm}\)\(m\ddot{u}+c\dot{u}+ku=\cos{\omega t}\) (11)

Restoring force denoted as \(Q\), is obtained by using the procedure same with preceding as

\(\hspace{20mm}\)\(Q=-m\ddot{u}+\cos{\omega t}=C\dot{u}+ku=\cfrac{2\beta_1k}{\omega_0}\dot{u}+ku\) (12)

Frequency of displacement is same with that of external load under steasy state condition, although there may be some phase lag. Therefore, displacement can be set as

\(\hspace{20mm}\)\(u=u_0\cos{(\omega t+\phi)} \)(13)

Substituting of Eq. (12) and using the relationship \(\sin{(\omega t+\phi)}=\pm\sqrt{1-\cos{(\omega t+\phi)}}\), we obtain

\(\hspace{20mm}\)\(Q=ku\mp 2\beta_1k\cfrac{\omega}{\omega_0}\sqrt{u_0^2-u^2} \)(14)

Then hysteretic absorbing energy \(\Delta W\) and elastic energy are calculated as

\(\hspace{20mm}\)\(\Delta W=2k\beta_1\cfrac{\omega}{\omega_0}\pi u_0^2 \)(15)

Therefore, damping constant yields

\(\hspace{20mm}\)\(h=\cfrac{1}{3\pi}\cfrac{\Delta W}{W}=\beta_1\cfrac{\omega}{\omega_0}\)(16)

In other words, the result \(\beta_1=h\) is obtained only when \(\omega=\omega_0\), i.e., resonat frequency. This also inticate that Lysmer's proposal under which discussion is made under the condition \(\beta_1=h\) is poor mechanical meaning.

2.3 Comparison with other models

Lysmer model is compared with other model. Here, it is noted that all models except Voigt model is used for the seismic analysis of ground, which indicates frequency independent stress-strain model whereas comparison here is made based on one-degree of freedom system in which damping parameter is used for a critical damping ratio. Two cases are compared in the below figure, tuning ratio\(\alpha\) = 0.5 and 2.0. The case \(\alpha=0.5\) means \(\omega/\omega_0=0.5\), Therefore frequency of external load is less than natural frequency, i.e., slow external force. On the other hand, the case \(\alpha=2.0\) is fast loading. vertical and horizontal axes in the left figures are displacement and time although it is modofied a little, and those in the right figures are force and displacement (force-displacement relationships). Here the Voigt model is a target model that Lysmer discussed. Sorokin model is the complex modulus used in the original SHAKE, and YAS model is our proposal which is more rational property and is explained in Rational complex modulus.

\(\hspace{20mm}\)

Looking at the time histories, Solid black line (Voigt model) is located left side, which means phas lag (\(\phi\) in Eq. (9)) is small, resulting is small energy absorption by hysteresis loop. If \(\phi=0\), external force and displacement moves in the same manner without phase lag. This can be seen i the right figures.

These figures uses \(\beta=0.3\), i.e., damping ratio is 30%. In our paper ¹⁾, \(\beta=0.2\) is used, so if you are interested in the effect of \(beta\), see our paper as well. Displacement of Voigt これを見ると，Voigt and Lysmer model are same, but response is quite different. This means that Lysmer's proposal is not impactive.

3 Mechanical property of Lymser model

Some stories above are show in some books as decrived before. Howerver, mechanical property of Lysmer model is not know at present. So here, we change our interests from vibration of one-degree of freedom system to frequency indepedent material property. Therefore, \(\beta_1\) is replaced to \(\beta\), and spring constant \(k, \overline{k}^*\) is replaced to \(G, \overline{G}_L^*\). Thes complex stress-strain relationships, real number stress-strain relationships are as folloew.

\(\hspace{20mm}\)\(\overline{\tau}=\overline{G}_L^*\overline{\gamma}=G\gamma_0(1-2\beta^2+2i\beta\sqrt{1-\beta^2})(\cos{\omega t}+i\sin{\omega t}) \)(17)

\(\hspace{20mm}\)\(\tau=G\{(1-2\beta^2)\gamma\pm 2\beta\sqrt{1-\beta^2}\sqrt{\gamma_0^2-\gamma^2}=G\gamma_0\cos{(\omega t+\phi)} \)

\(\hspace{20mm}\)\(\tan{\phi}=\cfrac{2\beta\sqrt{1-\beta^2}}{1-2\beta^2} \)(18)

There is no decription in Lysmer's proposal, but it is clear that maximum shear stress same with that by the laboratory test. This is one of the important feature of the Lysmer model. Next energy absorbed by hysteresis loop becomes

\(\hspace{20mm}\)\(\Delta _W=2\beta G\gamma_0^2\pi\sqrt{1-\beta^2} \)(19)

Then damping ratio \(h\) is calculated, which result in

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

All the people have considered that \(\beta=h\), but they were all wrong. As seen in the equaiton, \(h\) is smaller than \(\beta\). This is big disadvantege of the Lysmer model. Therefore, computer programs that uses \(\beta\) as damping ratio, damping ratio is underestimated

As is discueed, Lysmer model is not a rational complex model for the seismic response analysis of ground, because discussion is mede without disringushing one-degree of freedom and stress-strain relationships of soil, and discussion by Lysmer is not sufficient from both viewpoint. Then we propose a more rational complex modulus model for the seismic response analysis, which is named as YAS (Yoshida-Adachi-Sorolin) model. This is explained in rational complex modulus in detail.

1)吉田望，安達健司：地盤の地震応答解析のための複素剛性法，日本地震学会論文集（投稿中）
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.
4) Idriss, I.M. and Sun, J.L.: User's manual for SHAKE91, A computer program for conducting equivalent linear seismic response analysis of horizontally layered soil deposits, University of California, Davis, 1992
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7) Udaka, T., Oshima, K., Watanabe, T. and Nakama, T.: Increasing the accuracy of responses through improving equivalent linear analysis in comparison to using nonlinear analysis, Japanese Geotechnical Journal, Vol. 9, No. 2, pp. 185–202, 2004. (in Japanese)