飽和土體瞬態響應有限元分析①

邱流潮， 賀向麗， 盧 海

(1.中國農業大學水利與土木工程學院，北京 100083； 2.云南省交通規劃設計研究院,云南 昆明 650011)

飽和土體瞬態響應有限元分析①

邱流潮1， 賀向麗1， 盧 海2

(1.中國農業大學水利與土木工程學院，北京 100083； 2.云南省交通規劃設計研究院,云南 昆明 650011)

給出基于Biot 多孔介質理論分析飽和土體在動載荷作用下瞬態響應的有限元公式，數值計算部分采用本文有限元法分別計算一維飽和土柱在兩種不同類型動載荷作用下的瞬態響應，并將數值計算結果與文獻中的解析解進行比較，二者結果十分吻合，從而驗證本文方法的可行性。

飽和土體； 瞬態響應； 有限元法； 數值模擬

0 引言

飽和多孔土體動力響應問題的研究在巖土工程以及地震工程等領域有著非常廣泛的應用價值，是土動力學中的重要研究課題。自從Biot[1-2]提出描述飽和多孔介質波動理論的基本方程以來，國內外眾多學者[3-4]對飽和多孔介質動力學問題進行了研究。de Boer[5]、黃茂松等[6]和Schanz等[7]對飽和多孔介質動力學方面的研究成果做了比較詳細的綜述。目前飽和多孔介質動力學分析常用的方法有解析法、有限元方法及邊界元法。有限元方法由于能夠適應任意復雜幾何形體、邊界條件以及不同的材料模型而被廣泛采用。本文將給出基于Biot 多孔介質理論分析飽和土體在動載荷作用下瞬態響應的有限元公式，并對一維飽和土柱在兩種不同類型動載荷作用下的瞬態響應進行數值分析。

1 運動方程

基于Biot理論并忽略孔隙流體的慣性效應后，多孔介質土體運動方程可表示為[8]

(1)

孔隙流體連續方程為[9]：

(2)

(3)

2 有限元公式

對位移變量u和孔隙水壓力p引入插值近似：

(4)

(5)

式中，U和P包含u 和p離散變量(節點值)；Nu和Np為形函數。式(1)、(2)的有限元離散公式為：

(6)

式中，t+Δt表示當前時間步；Δt為時間步長。式中各矩陣和向量計算如下：

式中，B為 位移梯度矩陣；D為固體骨架的本構矩陣；α和β為Rayleigh 阻尼系數[10]。

3 數值計算

基于上述有限元公式，分析如圖1所示的一維飽和土柱在兩種不同類型動載荷作用下的瞬態響應，包括骨架位移和孔隙水壓力。

圖1 一維飽和土柱幾何圖Fig.1 Geometric drawing of the 1D saturated soil column

計算中為了模擬一維問題，將固體運動和流體運動限制在豎向(z向)，飽和土柱表面載荷σ(z=0，t)=f(t)，表面孔隙水壓力為零。這里主要計算

f(t)為正弦載荷和階躍載荷(step load)兩種情況下的動力響應。該問題已在文獻[11]中給出了解析解。為方便進行比較，本文所有計算條件均與文獻[11]相同。數值計算中取土柱長10 m，寬0.5 m，用四邊形單元離散。圖2(a)為正弦載荷作用下的動力響應，圖2(b)為階躍載荷作用下的動力響應，從圖中可以看出，本文有限元計算結果與文獻[9]中的解析解十分吻合。

4 結論

本文給出了基于Biot 多孔介質理論分析飽和土體在動載荷作用下瞬態響應的有限元公式，并以此分別計算一維飽和土柱在兩種不同類型動載荷作用下的瞬態響應，最后將數值計算結果與文獻中的解析解進行比較。結果顯示用本文方法計算飽和土體瞬態響應是可行的。需要說明的是本文方法中沒有考慮孔隙水的慣性效應。

圖2 動力響應Fig.2 Dynamic response

References)

[1] Biot M A.Theory of Propagation of Elastic Waves in a Fluid-saturated Porous Solid Ⅰ：Low-frequency Range[J].Journal of the Acoustical Society of America，1956，28：168-178.

[2] Biot M A.Theory of Propagation of Elastic Waves in a Fluid-saturated Porous Solid Ⅱ：Higher Frequency Range[J].Journal of the Acoustical Society of America，1956，28：179-191.

[3] 路江鑫，孫立強，曲京輝，等.地震荷載作用下飽和粉土地基液化深度試驗研究[J].地震工程學報，2014，36(3)：544-548.LU Jiang-xin，SUN Li-qiang，QU Jing-hui，et al.Experimental Study on Liquefaction Depth of Saturated Silty Soil Ground under Seismic Loading[J].China Earthquake Engineering Journal，2014，36(3)：544-548.(in Chinese)

[4] 王相寶，李亮，崔智謀，等.基于流固耦合兩相介質動力模型的飽和土體-地下結構體系地震反應研究[J].地震工程學報，2014，36(2)：228-232.WANG Xiang-bao，LI Liang，CUI Zhi-mou，et al.Study on the Seismic Response of Saturated Soil：Underground Structure Based on Dynamic Model of Fluid-solid Coupling Media[J].China Earthquake Engineering Jounal，2014，36(2)：228-232.(in Chiniese)

[5] De Boer R.Highlights in the Historical Development of the Porous Media Theory-toward a Consistent Macroscopic Theory[J].Applied Mechanics Reviews，1996，49：201-262.

[6] 黃茂松，李進軍.飽和多孔介質土動力學理論與數值解法[J].同濟大學學報：自然科學版， 2004，32(7)：851-856.HUANG Mao-song，LI Jin-jun.Dynamics of Fluid-saturated Porous Media and its Numerical Solution[J].Journal of Tongji University，2004，32(7)：851-856.(in Chinese)

[7] Schanz M.Poroelastodynamics：Linear Models，Analytical Solutions，and Numerical Methods[J].Applied Mechanics Reviews，2009，62：1-15.

[8] Hassanizadeh S M，Gray W G.High Velocity Flow in Porous Media[J].Transport in Porous Media，1987，2：521-531.

[9] Chen Zhiyun，Steeb Holger，Diebels Stefan.Analysis of Wave Propagation in the Fluid-saturated Porous Media[J].Proc Appl Math Mech，2006，6：429-430.

[10] Bathe K J.Finite Element Procedures[M].Englewood Cliffs，New Jersey：Prentice-Hall，1996.

[11] De Boer R，Ehlers W，Liu Z.One-dimensional Transient Wave Propagation in Fluid Saturated Incompressible Porous Media[J].Arch Appl Mech，1993，63：59-72.

中國地震局發布尼泊爾8.1級地震烈度圖

發布時間：2015-05-01 20:35:29

2015年4月25日，尼泊爾發生8.1級強震,波及尼泊爾、中國、印度、孟加拉等國。此次地震災區最高烈度為Ⅸ度及以上，等震線長軸總體呈NWW走向，Ⅵ度區及以上總面積約為214 700 km2，其中Ⅸ度區及以上面積約8 300 km2，長軸155 km，短軸63 km；Ⅷ度區面積約20 500 km2，長軸260 km，短軸135 km；Ⅶ度區面積約45 000 km2，長軸383 km，短軸236 km；Ⅵ度區面積約140 900 km2，長軸588 km，短軸470 km,其地震烈度圖如圖1。

Finite Element Analysis of the Transient Response of Saturated Soils

QIU Liu-chao1， HE Xiang-li1， LU Hai2

(1.CollegeofWaterResources&CivilEngineering，ChinaAgriculturalUniversity，Beijing100083，China；2.YunnanInstituteofTrafficPlanningandDesign，Kunming，Yunan650011，China)

An investigation of the dynamic response of saturated soil plays an important role in classical application fields such as soil mechanics，hydrology，ocean engineering and so on.Furthermore，it is essential to the development of emerging sciences and technologies，such as the mechanical characteristic of skin and soft tissue in biology.Therefore，it is important to provide appropriate theoretical analyses and numerical simulation methods.In addition，the transient response of saturated soil is also essential to the understanding of deformation and the pore water pressures generated by ground motion.This response is a key factor in the dynamic analysis of building foundations，offshore structures，and wave propagation in geological medium during blasts or earthquakes.Saturated soil is one that exhibits a solid faction and a porous space filled with a viscous fluid on a microscopic scale.Two approaches are possible for addressing the description of such a soil.The first approach is at the microscopic scale.Here，the “solid elastic” and “viscous fluid” phases each constitute distinct geometric domains.A geometric point is found at a given instant in one of these two clearly identifiable phases.The second approach considers the problem from the macroscopic level.The elementary volume is considered to be the superposition of two material particles with different kinematics occupying the same geometric points at the same instant.Thus，the saturated soil is considered as a two-phase continuum；the skeleton particle is constituted by the solid matrix and connected porous space，and the fluid particle is formed from the fluid saturating this connected porous space.There are many theories describing the characteristics of saturated soils，e.g.，Biot Theory，porous media theory，hybrid mixture theory，and so on.Most of the transient response studies for saturated soils are solved by numerical methods such as the finite element method (FEM)and finite difference method (FDM).Compared to the FDM，the most attractive feature of the FEM is its ability to handle nonlinear material and complicated geometries (and boundaries)with relative ease.In this investigation，based on Biot Theory，a mathematical model of a two-dimensional saturated elastic soil is established，and a time-domain FEM for analyzing the transient dynamic response of saturated soil under cyclic loading is presented.To verify the efficiency and accuracy of the proposed method，a one-dimensional saturated soil column subject to two different surface loadings was simulated.The first numerical example models the transient response of the saturated soil column due to sine wave loading.The second case is for the dynamic response of the soil column subject to step loading.For both numerical examples，the solid displacement history and pore pressure history are presented and compared with analytical solutions.Good agreement between the computed results and analytical solutions show the efficiency and accuracy of the proposed method.

saturated soil； transient response； finite element method； numerical simulation

圖1 2015年4月25日尼泊爾8.1級地震烈度圖Fig.1 Seismic intensity map of the Nepal MS8.1 earthquake on April 25, 2015

2014-08-20

國家自然科學基金面上項目(11172321)；國家自然科學基金青年基金項目(51109212)

邱流潮(1971-)，男，博士，副教授，主要從事水利工程數值仿真研究.E-mail：qiuliuchao@cau.edu.cn

TU45

A

1000-0844(2015)02-0472-04

10.3969/j.issn.1000-0844.2015.02.0472