劉林超，閆啟方，劉 滕

(信陽師范學(xué)院 土木工程學(xué)院，河南 信陽 464000)

上覆分?jǐn)?shù)階粘彈性飽和場地土位移地震放大系數(shù)

劉林超，閆啟方，劉 滕

(信陽師范學(xué)院 土木工程學(xué)院，河南 信陽 464000)

考慮土體液相和固相的耦合作用，將基巖上覆場地土視為兩相飽和多孔介質(zhì)。為了考慮飽和場地土的粘彈性特性，其固相土骨架的應(yīng)力應(yīng)變關(guān)系利用分?jǐn)?shù)階Kelvin粘彈性模型來描述，建立了上覆分?jǐn)?shù)階粘彈性飽和場地土在簡諧地震波作用下的運(yùn)動控制方程。運(yùn)用分?jǐn)?shù)導(dǎo)數(shù)的性質(zhì)并考慮上覆場地土的邊界條件和透水性條件求解了上覆分?jǐn)?shù)階粘彈性場地土在簡諧地震波作用下的振動問題，得到了飽和場地土的位移地震放大系數(shù)。采用數(shù)值算例分析討論了分?jǐn)?shù)導(dǎo)數(shù)的階數(shù)、液固耦合系數(shù)、土體模型參數(shù)、基巖土體剪切模量比等參數(shù)對位移地震放大系數(shù)的影響。研究結(jié)果表明，分?jǐn)?shù)導(dǎo)數(shù)的階數(shù)、液固耦合系數(shù)、土體模型參數(shù)、基巖土體剪切模量比對飽和場地土的地震響應(yīng)有較大的影響，通過壓實場地土，可以達(dá)到增大液固耦合系數(shù)減小地震響應(yīng)的作用，通過增大飽和場地土的粘性和剪切模量也可以減小地震反應(yīng)。

飽和土；分?jǐn)?shù)導(dǎo)數(shù)；地震波；地震放大系數(shù)；液固耦合系數(shù)

地震往往會造成建筑物和人員的傷亡，在進(jìn)行建筑抗震設(shè)計時要選擇有利的建筑場地，這是因為不同的場地條件和場地土的動力學(xué)特性對地震的放大效應(yīng)不同，且十分敏感。因此，研究不同場地條件下場地土的振動特性和地震反應(yīng)成為地震和巖土工程領(lǐng)域研究的一個重點(diǎn)和難點(diǎn)，且受到了該領(lǐng)域?qū)＜业淖銐蛑匾?。?0世紀(jì)60年代開始，不少學(xué)者就采用各種模型、方法對地震激勵作用下場地土的動力學(xué)特性進(jìn)行了研究，Idriss等[1]在一維剪切梁模型的基礎(chǔ)上研究了剪切模量為常數(shù)和沿深度按冪函數(shù)變化的場地土的動力特性和地震反應(yīng)；Wolf[2]將土層簡化為豎桿，研究了剪切模量按指數(shù)函數(shù)隨深度增大的勻質(zhì)自由場地土層的豎向振動；Zhao[3-4]對剪切模量沿深度按冪函數(shù)分布的場地土進(jìn)行了地面振動分析和橫向振動研究；高玉峰[5]在時間域內(nèi)給出了基巖任意輸入地震作用下一維土層地震反應(yīng)的解析解；欒茂田等[6]基于一維剪切梁模型，運(yùn)用分離變量方法研究了各層土的剪切模量沿深度按冪函數(shù)規(guī)律變化的水平成層非均質(zhì)地基的自振特性和地震動力反應(yīng)。尚守平等[7]基于一維波動模型，假設(shè)土層剪切模量沿深度按指數(shù)規(guī)律增長，研究了基巖上作用豎直向上傳播的穩(wěn)態(tài)剪切地震波激振的場地土的橫向自由振動。Chen等[8]在凍融情況下研究了土層的地震反應(yīng)。Nimtaj等[9]利用時域-頻域混合方法研究了土層的非線性地震反應(yīng)。這些研究對研究場地土的動態(tài)特性起到了重要作用。然而，由于實際工程中土體是由固液氣組成的三相多孔介質(zhì)，且具有粘彈性性質(zhì)，將其視為單相彈性介質(zhì)勢必與工程實際不符?？紤]土體的粘彈性特性，李剛等[10]采用工程波動理論，考慮簡諧SH波誘發(fā)的水平振動和場地土的材料阻尼，假定場地土體系處于反平面應(yīng)變狀態(tài)，研究半空間上SH波激勵下上覆粘彈性場地土的自由場動力反應(yīng)；劉林超等[11]借助于一維波動模型和分?jǐn)?shù)導(dǎo)數(shù)粘彈性本構(gòu)關(guān)系，分析了在豎直向上傳播的剪切地震波激勵作用下，基巖上分?jǐn)?shù)導(dǎo)數(shù)粘彈性場地土的橫向振動問題；段瑋瑋等[12]基于Biot飽和土理論和分?jǐn)?shù)階導(dǎo)數(shù)理論研究了飽和分?jǐn)?shù)導(dǎo)數(shù)型粘彈性土層的豎向振動放大效應(yīng)。為了考慮液相的影響，筆者基于飽和土的多孔介質(zhì)理論，將基巖上覆土體視為兩相飽和多孔介質(zhì)，利用分?jǐn)?shù)導(dǎo)數(shù)粘彈性模型描述固相土骨架的應(yīng)力應(yīng)變關(guān)系，研究簡諧地震波作用下上覆分?jǐn)?shù)階粘彈性飽和場地土的地震反應(yīng)與振動特性。

1 模型與控制方程

(1)

圖1 基巖上覆分?jǐn)?shù)導(dǎo)數(shù)粘彈性飽和場地土Fig.1 Fractional derivative viscoelastic saturated site soil on a bedrock

(2)

(3)

(4)

為了更加合理的考慮上覆飽和場地土的粘彈性特性，采用分?jǐn)?shù)階Kelvin粘彈性本構(gòu)模型來描述固相土骨架的應(yīng)力-位移關(guān)系，即[14]

(5)

將式(5)代入式(1)，可得簡諧地震波作用下上覆分?jǐn)?shù)階粘彈性飽和場地土的運(yùn)動控制方程為

(6)

(7)

(8)

2 簡諧地震波作用下上覆分?jǐn)?shù)階粘彈性飽和場地土振動求解

(9)

(10)

(11)

(12)

(13)

(14)

將式(13)代入式(12)整理得

(15)

(16)

求解式(16)有

(17)

由式(13)、(16)、(17)可得

(18)

(19)

式(17)、(18)、(19)中，C1、C2、C3、C4、C5為待定系數(shù)，可以從邊界條件得到。

3 上覆分?jǐn)?shù)階粘彈性飽和場地土的位移地震放大系數(shù)

將基巖看作彈性體，由基巖的入射地震波位移表達(dá)式(1)和基巖剪力與位移的關(guān)系可知地震波產(chǎn)生的基巖剪應(yīng)力幅值為

(20)

(21)

C1+C3=v0

(22)

(23)

(24)

(25)

(26)

求解式(22)～(26)可得

C1=a1v0，C2=a2v0，C3=a3v0，

C4=a4v0，C5=a5v0

(27)

式中：

(28)

由式(17)可得固相土骨架的位移為

(29)

引入分?jǐn)?shù)導(dǎo)數(shù)粘彈性飽和場地土的地震位移放大系數(shù)[9]

(30)

4 位移地震放大系數(shù)分析與討論

針對由式(30)得到的上覆分?jǐn)?shù)階粘彈性飽和場地土地震位移放大系數(shù)，采用數(shù)值算例的形式來分析討論分?jǐn)?shù)導(dǎo)數(shù)的階數(shù)α、液固耦合系數(shù)Sv、土體模型參數(shù)κ、基巖土體剪切模量比GJ/G0對地震位移放大系數(shù)的影響。在未作說明的情況下，有關(guān)參數(shù)的取值為α=0.5、nS=0.67、nF=0.33、Sv=0.5、κ=0.8、GJ/G0=1 000、ρJ/ρ0=2。圖2～6為上覆分?jǐn)?shù)階粘彈性飽和場地土地震位移放大系數(shù)隨無量綱頻率的變化曲線。很明顯，曲線存在著波峰和波谷，也就是說，在簡諧地震波作用下，系統(tǒng)將產(chǎn)生共振現(xiàn)象。分?jǐn)?shù)導(dǎo)數(shù)的階數(shù)α對地震位移放大系數(shù)的影響見圖2和圖3，隨著分?jǐn)?shù)導(dǎo)數(shù)的階數(shù)α的增大，上覆分?jǐn)?shù)階粘彈性飽和場地土地震位移放大系數(shù)將減小，且逐漸退化到經(jīng)典粘彈性飽和土的情形。由圖1、圖2和圖3可以看出，飽和土液固耦合系數(shù)Sv對上覆分?jǐn)?shù)階粘彈性飽和場地土地震位移放大系數(shù)的影響相當(dāng)明顯(圖4)。當(dāng)液固耦合系數(shù)Sv較小時，地震位移放大系數(shù)隨頻率變化曲線的峰值越大且越尖，隨著液固耦合系數(shù)Sv的增大，地震位移放大系數(shù)隨頻率變化曲線的峰值將明顯減小?？梢姡趯嶋H工程中，對飽和的軟土需要進(jìn)行輾壓，這樣將使其液固耦合系數(shù)增大，從而減小其地震反應(yīng)。隨著飽和場地土土體模型參數(shù)κ(即粘性)的增長，場地土的耗散地震能量的能力將增大，此時，地震位移放大系數(shù)將減小(圖5)?；鶐r土體剪切模量比GJ/G0對地震位移放大系數(shù)的影響見圖6，隨著基巖土體剪切模量比GJ/G0的減小，也即土體剪切模量G0的增大，地震位移放大系數(shù)減小，這是因為土體剪切模量越大，相應(yīng)的剪應(yīng)力將增大，可見在實際工程中提高基巖上覆土的剪切模量對提高地基的抗震性能有利。

圖2 分?jǐn)?shù)導(dǎo)數(shù)的階數(shù)α對位移地震放大系數(shù)的影響(Sv=0.05)Fig.2 The influence of the order of fractional derivative on displacement seismic amplification coefficient

圖3 分?jǐn)?shù)導(dǎo)數(shù)的階數(shù)α對位移地震放大系數(shù)的影響(Sv=0.5)Fig.3 The influence of the order of fractional derivative on displacement seismic amplification coefficient

圖4 液固耦合系數(shù)Sv對位移地震放大系數(shù)的影響Fig.4 The influence of fluid and solid coupling coefficient on displacement seismic amplification coefficient

圖5 土體模型參數(shù)κ對位移地震放大系數(shù)的影響Fig.5 The influence of soil model parameter on displacement seismic amplification coefficient

圖6 基巖土體剪切模量比GJ/G0對位移地震放大系數(shù)的影響Fig.6 The influence of shear modulus ratio of rock and soil on displacement seismic amplification coefficient

5 結(jié) 論

為了使對基巖上覆場地土動力特性的研究更加符合工程實際，必須要考慮液相影響和土體的粘彈性特性。筆者在飽和土理論、粘彈性理論、分?jǐn)?shù)導(dǎo)數(shù)理論等理論的基礎(chǔ)上，研究了上覆分?jǐn)?shù)階粘彈性飽和場地土的地震位移響應(yīng)。與將上覆土視為單相彈性介質(zhì)一樣，上覆分?jǐn)?shù)階粘彈性飽和場地土在簡諧地震激勵的作用下同樣存在有共振現(xiàn)象。通過分析發(fā)現(xiàn)，飽和土的液固耦合系數(shù)和土體模型參數(shù)對場地土的地震位移放大系數(shù)有較大的影響，可見，忽略液相和固相的影響以及土體的粘性來研究場地土的地震響應(yīng)問題將與工程實際存在差異。為了降低地震造成的地震響應(yīng)放大效應(yīng)的影響，需要對場地土進(jìn)行壓實，達(dá)到增大液固耦合系數(shù)和土體剪切模量進(jìn)而減小地震響應(yīng)的目的。

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(編輯 胡英奎)

Displacement seismic amplification coefficient of overlying fractional derivative viscoelastic saturated site soil

LiuLinchao，YanQifang，LiuTeng

(School of Civil Engineering,Xinyang Normal University,Xinyang 464000,Henan,P. R. China)

Considering the coupling effect between fluid phase and solid phase,the site soil on bedrock is considered as two-phase saturated porous media. In order to consider the viscoelastic characteristics of the saturated site soil,the stress-strain relationship of soil skeleton is described by fractional derivative Kelvin viscoelastic model,and the motion control equations of overlying fractional derivative viscoelastic saturated site soil under the action of harmonic earthquake wave are established. Using the properties of fractional derivative and considering the boundary conditions and permeable conditions,the vibration problem of overlying fractional derivative viscoelastic saturated site soil under the action of harmonic earthquake wave is solved,and the seismic displacement amplification coefficient of the saturated site soil is also obtained. The influences of the order of fractional derivative,liquid and solid coupling coefficient,model parameters of soil,and bedrock and soil shear modulus ratio on the seismic displacement amplification coefficient are analyzed and discussed by numerical examples. The results show that the order of fractional derivative,liquid and coupling coefficient,model parameters of soil,bedrock and soil shear modulus ratio have great effects on the seismic response of the site soil. The fluid and solid coupling coefficient could the increa of be increased and the seismic response can be decreased by compacting the saturated soil. Meanwhile,the viscosity and shear modulus of the saturated site soil can also reduce the seismic response.

saturated soil; fractional derivative; seismic wave; seismic amplification coefficient; fluid and soil coupling coefficient

10.11835/j.issn.1674-4764.2015.05.007

2015-07-16 基金項目：國家自然科學(xué)基金(U1504505)；河南省科技發(fā)展計劃項目(142102210063)；河南省高等學(xué)校重點(diǎn)科研項目(15A560036)；河南省高等學(xué)校青年骨干教師資助計劃項目(2013GGJS-121)；信陽師范學(xué)院重大課題預(yù)研項目(2013ZDYY19)；信陽師范學(xué)院青年科研基金項目(2014-QN-063)；信陽師范學(xué)院青年骨干教師資助計劃(2012007)。

劉林超(1979-)，副教授，博士，主要從事多孔介質(zhì)理論、粘彈性理論、樁基動力學(xué)研究，(E-mail)llc109@126.com。

Foundation item：National Natural Science Foundation of China(No.U1504505); Development of Science and Technology Plan Project of Henan Province(No.142102210063); Key Scientific Research Project of Henan Province(15A560036); Youth Backbone Teacher Plan Project of Colleges and Universities in Henan Province(2013GGJS-121); Major Beforehand Research Project of Xinyang Normal University(2013ZDYY19); Scientific Research Fund for Young of Xinyang Normal University(2014-QN-063); Youth Backbone Teacher Plan Project of Xinyang Normal University(2012007).

TU573.12

A

1674-4764(2015)05-0048-06

Received:2015-07-16

Author brief：Liu Linchao(1979-),associate professor,PHD,main research interests: theory of porous medium,viscoelastic theory,pile dynamics,(E-mail) llc109@126.com.