小應(yīng)變硬化模型(HSS)在Rayleigh波作用下場(chǎng)地響應(yīng)分析中的應(yīng)用

施有志, 林樹枝, 楊榮華

(1.廈門理工學(xué)院土木工程與建筑學(xué)院,福建 廈門 361021; 2.上海交通大學(xué)船舶海洋與建筑工程學(xué)院,上海 200240;3.廈門市建設(shè)局,福建 廈門 361003)

小應(yīng)變硬化模型(HSS)在Rayleigh波作用下場(chǎng)地響應(yīng)分析中的應(yīng)用

施有志1,2, 林樹枝3, 楊榮華1

(1.廈門理工學(xué)院土木工程與建筑學(xué)院,福建 廈門 361021; 2.上海交通大學(xué)船舶海洋與建筑工程學(xué)院,上海 200240;3.廈門市建設(shè)局,福建 廈門 361003)

小應(yīng)變硬化模型(HSS); 瑞利阻尼; Rayleigh波; 地震動(dòng)力響應(yīng); 有限元?jiǎng)恿Ψ治?/p>

0 引言

土體受沖擊荷載或地震作用時(shí),巨大的能量在地基中產(chǎn)生強(qiáng)烈的震動(dòng),并以波動(dòng)的形式從震源向四周傳播,其中包括壓縮波P波、剪切波S波和以Rayleigh波為主的面波組成。體波(包括P波和S波)從波源沿著一個(gè)半球波陣面徑向向外傳播,而Rayleigh波則沿著一個(gè)圓柱波徑向向外傳播。Miller等[1]曾研究了三種彈性波占地震總輸入能量的百分比,發(fā)現(xiàn)Rayleigh波占67.3%,S波占25.8%,P波占6.9%。由于Rayleigh波能量分布一般僅限于距離半空間自由表面兩倍其波長(zhǎng)范圍的巖土層內(nèi),因此它對(duì)淺埋地下結(jié)構(gòu)的地震反應(yīng)具有重要影響。李恒等[2]、陳永新等[3]研究地表巖土層特點(diǎn)對(duì)地震動(dòng)特性的影響,但未對(duì)Rayleigh波的影響進(jìn)行探討。目前對(duì)Rayleigh波的研究主要體現(xiàn)在巖土參數(shù)反演及物探等方面的應(yīng)用[4-5];另有學(xué)者研究沖擊荷載產(chǎn)生的Rayleigh波的作用機(jī)制[7-8];Makris[9-10]研究Rayleigh波作用下樁的響應(yīng);蔣東旗等[11-12]研究遠(yuǎn)場(chǎng)地震動(dòng)對(duì)樁基響應(yīng)的影響。上述研究都將Rayleigh波考慮為簡(jiǎn)諧振動(dòng)輸入進(jìn)行分析。岳慶霞等[13]提出了近似Rayleigh地震波的概念,利用已有的地震記錄,將之視為水平方向的波動(dòng),而后利用傅里葉變換得到豎直方向的波動(dòng),并考慮沿深度方向的衰減,從而得到整個(gè)位移場(chǎng);羅韜[14]通過(guò)小波變換和傅里葉變換對(duì)地震波進(jìn)行低頻重構(gòu),得到整個(gè)Rayleigh波場(chǎng)。上述構(gòu)建的Rayleigh波不是通過(guò)波的干涉形成,與實(shí)際的地震動(dòng)有一定的差異。有限元?jiǎng)恿Ψ治鋈允茄芯縍ayleigh波對(duì)土體動(dòng)力響應(yīng)的主要手段。土體由于土粒間的摩擦、孔隙水和空氣的黏滯性,需要引入阻尼來(lái)模擬土體在循環(huán)加載作用下的阻尼特性。在各種阻尼模型中使用最廣泛的是Rayleigh阻尼[15],研究者們圍繞著Rayleigh阻尼的取值,亦目標(biāo)頻率的選取,提出了多種不同的方法[16-19],但尚未形成統(tǒng)一的認(rèn)識(shí)。

土體硬化模型(HSS)是高級(jí)本構(gòu)模型,能夠考慮土體在循環(huán)加載下的滯回效應(yīng),在動(dòng)力計(jì)算中,HSS模型的滯回行為就會(huì)引起阻尼。為了研究HSS模型中本身的滯回環(huán)特性,本文以廈門地區(qū)淺層的素填土及粉質(zhì)黏土為研究對(duì)象,采用有限元?jiǎng)恿Ψ治?輸入變化的小應(yīng)變參數(shù),考察HSS模型的小應(yīng)變參數(shù)對(duì)場(chǎng)地動(dòng)力響應(yīng)的影響,并與土體采用摩爾-庫(kù)倫模型結(jié)合Rayleigh阻尼的計(jì)算結(jié)果進(jìn)行對(duì)比。研究成果有利于利用HSS模型研究Rayleigh波對(duì)淺埋結(jié)構(gòu)物的動(dòng)力響應(yīng)分析以及阻尼的設(shè)置方法。

1 HSS模型簡(jiǎn)介

HSS模型是以HS模型為基礎(chǔ),考慮應(yīng)變歷史的影響并結(jié)合修正的Hardin-Drnevich[20]剪切模量關(guān)系式提出的一種能反應(yīng)土體小應(yīng)變特性的本構(gòu)模型。小應(yīng)變土體硬化模型(HSS)能夠考慮土體在循環(huán)加載下的滯回效應(yīng),滯回環(huán)的大小取決于相應(yīng)的應(yīng)變幅值的大小。在動(dòng)力計(jì)算中,HSS模型的滯回行為就會(huì)引起阻尼,滯回阻尼量取決于施加的荷載幅值和相應(yīng)的應(yīng)變幅值。HSS模型產(chǎn)生的最大滯回阻尼則取決于G0/Gur,其中G0為初始剪切剛度,Gur為卸載重加載剪切模量,該剛度比越大,則最大滯回阻尼也越大。

通常而言,地震能在土體中引起小應(yīng)變,土體表現(xiàn)出高剪切剛度G0,當(dāng)剪應(yīng)變?chǔ)梅翟黾佣鼓芰肯脑黾訒r(shí)剛度會(huì)降低。要考慮上述表征材料小應(yīng)變行為的三個(gè)參數(shù),需要使用基于土體硬化模型(Hardening Soil model,HS)拓展而來(lái)的小應(yīng)變土體硬化模型(HS small model,HSS)。HS模型已經(jīng)基于冪率參數(shù)m考慮了土體剛度的應(yīng)力相關(guān)性,在此基礎(chǔ)上,HSS模型引入了另外兩個(gè)參數(shù),即表征小應(yīng)變水平下的高剛度特性的參數(shù)G0和表征G衰減至初始G0的70%時(shí)的剪應(yīng)變的參數(shù)γ0.7。

HSS模型中土體的應(yīng)力相關(guān)性采用下式表示:

(1)

土體典型的滯回行為如圖1所示。Gs為割線模量;Gt為切線剪切模量。可見初始加載曲線的初始切線和割線剛度與最大剪切剛度G0一致。隨著剪應(yīng)變?cè)龃?剛度發(fā)生衰減。當(dāng)加載方向發(fā)生反轉(zhuǎn),剛度從相同的G0出發(fā),然后降低至下一次加載反轉(zhuǎn)。

圖1 HSS模型的滯回行為Fig.1 Hysteretic behavior of HSS model

2 廈門地區(qū)典型土層HSS模型中阻尼參數(shù)研究

2.1 模型建立

建立有限元模型如圖2。沿地表距沖擊荷載作用點(diǎn)分別為40、50、60、70及80 m處設(shè)5個(gè)位移監(jiān)測(cè)點(diǎn)(圖3)。

模型對(duì)稱軸上受到一個(gè)集中荷載作為動(dòng)力輸入。動(dòng)力荷載采用隨時(shí)間按三角形變化的荷載來(lái)模擬,自0.05 s之后開始施加荷載,荷載持續(xù)時(shí)間取0.025 s,荷載振幅取50 kN(圖4)。

圖2 有限元模型Fig.2 Finite element model

圖3 位移監(jiān)測(cè)點(diǎn)布置Fig.3 Distribution of displacement monitoring points

圖4 沖擊荷載定義Fig.4 Definition of impact loading

在有限元?jiǎng)恿τ?jì)算中,為了盡量避免應(yīng)力波在模型邊界上的反射導(dǎo)致計(jì)算結(jié)果失真,需要將模型邊界取得足夠遠(yuǎn),但這會(huì)大大增加單元數(shù)量和計(jì)算成本,因此需引入人工邊界。本文在模型底部和Xmax、Ymax邊界上引入吸收邊界,來(lái)確保模型邊界上的應(yīng)力波被吸收而不發(fā)生反彈。

2.2 參數(shù)取值

以廈門地區(qū)淺層土層為例,概化出素填土及粉質(zhì)黏土兩種土層為研究對(duì)象。土體本構(gòu)采用HSS模型,模型基本輸入?yún)?shù)見表1。

表1 兩類典型土體HSS模型基本輸入?yún)?shù)

表中參數(shù)的定義如下:

γ:天然重度;

φ′:有效內(nèi)摩擦角;

γ0.7:當(dāng)割線剪切模量Gsecant衰減為0.7倍的初始剪切模量G0時(shí)對(duì)應(yīng)的剪應(yīng)變。

當(dāng)土體本構(gòu)采用摩爾-庫(kù)倫模型時(shí),參數(shù)取值如下:素填土剪切波波速vS=135 m/s,壓縮波波速vP=325 m/s,重度γ=18 kN/m3,黏聚力c=15 kN/m2,摩擦角φ=25°;粉質(zhì)黏土剪切波波速vS=102 m/s,壓縮波波速vP=227 m/s,重度γ=18.4 kN/m3,黏聚力c=37 kN/m2,摩擦角φ=15°。

2.3 計(jì)算方案

C=αM+βK

式中:α和β分別為質(zhì)量比例阻尼系數(shù)和剛度比例阻尼系數(shù)。

將“MC+Rayleigh阻尼”與“HSS+滯回環(huán)阻尼”計(jì)算結(jié)果進(jìn)行對(duì)比分析。針對(duì)素填土和粉質(zhì)黏土的計(jì)算方案如表2所列。

表2 計(jì)算方案

圖5 HSS模型小應(yīng)變阻尼曲線Fig.5 Small strain damping curve of HSS model

圖6 Rayleigh阻尼曲線Fig.6 Rayleigh damping curve

2.4 計(jì)算結(jié)果

(1) 素填土

以距沖擊荷載水平距離50 m處的地表點(diǎn)豎向位移為例,對(duì)場(chǎng)地動(dòng)力響應(yīng)特征進(jìn)行分析,各計(jì)算方案下該點(diǎn)豎向位移-時(shí)間曲線如圖7～圖9所示。

圖10所示為素填土計(jì)算方案3與方案6得出的不同時(shí)刻網(wǎng)格變形形態(tài)圖。由圖10可以看出:使用“HSS+滯回環(huán)”與使用“MC+Rayleigh阻尼”得到的波速是基本一致的,在加卸載循環(huán)中MC模型表現(xiàn)出了彈性行為,反彈較為明顯,HSS模型則因其滯回環(huán)效應(yīng)累積了塑性變形,對(duì)材料阻尼的考慮更為符合實(shí)際。

(2) 粉質(zhì)黏土

以距沖擊荷載水平距離50 m處的地表點(diǎn)豎向位移為例,對(duì)場(chǎng)地動(dòng)力響應(yīng)特征進(jìn)行分析,各計(jì)算方案下該點(diǎn)豎向位移-時(shí)間曲線如圖11所示。

從圖11可以看到,采用“MC+Rayleigh阻尼”時(shí),波動(dòng)很快趨于穩(wěn)定,幾乎沒有發(fā)生振蕩;但采用帶有滯回環(huán)的HSS模型時(shí),位移波動(dòng)出現(xiàn)大幅明顯振蕩,波動(dòng)平穩(wěn)下來(lái)要比Rayleigh阻尼慢得多,且存在殘余變形。由于HSS模型的小應(yīng)變剛度只能考慮滯回環(huán)阻尼,仍然不能完全體現(xiàn)材料阻尼的影響,因此其使得振幅較MC更大。

圖7 γ0.7不變?cè)龃髸r(shí)的場(chǎng)地動(dòng)力響應(yīng)(素填土)Fig.7 Site dynamic response when γ0.7 remains and rises (plain fill)

圖不變?chǔ)?.7變化時(shí)的場(chǎng)地動(dòng)力響應(yīng)(素填土)Fig.8 Site dynamic response when γ0.7 changes and remains (plain fill)

圖9 “HSS +滯回環(huán)”與“MC+Rayleigh阻尼”動(dòng)力響應(yīng)曲線對(duì)比(素填土)Fig.9 “HSS+hysteresis loop” and “MC+Rayleigh damping” dynamic response curve comparison(plain fill)

圖10 “MC+Rayleigh阻尼”與“HSS+滯回環(huán)”場(chǎng)地動(dòng)力響應(yīng)對(duì)比(素填土)Fig.10 “MC+Rayleigh damping” and “HSS+hysteresis loop” site dynamic response comparison (plain fill)

圖11 “HSS +滯回環(huán)”與“MC+Rayleigh阻尼”動(dòng)力響應(yīng)曲線對(duì)比(粉質(zhì)黏土)Fig.11 “HSS+hysteresis loop” and “MC+Rayleigh damping” dynamic response curve comparison(silty clay)

3 結(jié) 語(yǔ)

本文以廈門地區(qū)受Rayleigh波影響顯著的場(chǎng)地淺層典型土體素填土和粉質(zhì)黏土為研究對(duì)象,采用有限元?jiǎng)恿Ψ治?土體本構(gòu)采用小應(yīng)變硬化模型(HSS),利用模型本身的滯回環(huán)特性,輸入變化的小應(yīng)變參數(shù),考察HSS模型的小應(yīng)變參數(shù)對(duì)場(chǎng)地動(dòng)力響應(yīng)的影響,并與土體采用摩爾-庫(kù)倫模型結(jié)合Rayleigh阻尼(“MC+Rayleigh阻尼”)的計(jì)算結(jié)果進(jìn)行對(duì)比,得出如下結(jié)論:

(2) 與“MC模型+Rayleigh阻尼”動(dòng)力響應(yīng)結(jié)果相比,“HSS模型+滯回環(huán)”得到的動(dòng)力響應(yīng)波動(dòng)更加劇烈,更重要的是HSS模型能夠給出殘余變形量;而MC模型+Rayleigh阻尼則由于MC模型為理想彈塑性模型,在卸載重加載條件下表現(xiàn)為純彈性行為,無(wú)法反映出卸載重加載過(guò)程中塑性應(yīng)變的積累及其累積阻尼效應(yīng),這是即便使用Rayleigh阻尼也無(wú)法考慮的情況。

(3) HSS模型還不能夠全面反映循環(huán)加載作用下塑性體積應(yīng)變的累積。在小幅振動(dòng)情況下,即使采用小應(yīng)變土體硬化模型也不能完全體現(xiàn)材料阻尼,因此在考慮滯回阻尼的基礎(chǔ)上,仍然建議借助Rayleigh阻尼來(lái)更加全面地模擬土體的實(shí)際阻尼特性。

References)

[1] Miller G F,Pursey H.On the Partition of Energy between Elastic Waves in a Semi-infinite Solid[J].Proceedings of the Royal Society A,1955,233(233):55-69.

[2] 李恒,張靜波,吳建超.軟夾層和硬夾層對(duì)地表地震動(dòng)特性的影響[J].地震工程學(xué)報(bào),2014,36(3):441-445. LI Heng,ZHANG Jing-bo,WU Jian-chao.Effects of Soft and Hard Interlayers on Ground Motion Characteristics[J].China Earthquake Engineering Journal,2014,36(3):441-445.(in Chinese)[3] 陳永新,遲明杰,李小軍.地表巖土層對(duì)地震動(dòng)特性的影響分析[J].地震工程學(xué)報(bào),2015,37(3):743-747. CHEN Yong-xin,CHI Ming-jie,LI Xiao-jun.Effect of over Laying Rock and Soil Layers on Ground Motion Characteristics[J].China Earthquake Engineering Journal,2015,37(3):743-747.(in Chinese)

[4] 卞鵬,王媛,王篤強(qiáng).有障礙物地基瑞利波頻散曲線的特征研究[J].山東大學(xué)學(xué)報(bào):工學(xué)版,2012,42(1):99-103. BIAN Peng,WANG Yuan,WANG Du-qiang.Research on the Frequency Dispersion Curve of the Rayleigh Wave in the Foundation with an Obstacle[J].Journal of Shandong University:Engineering Science,2012,42(1):99-103.(in Chinese)

[5] 唐平祥.瑞利波法在公路地基中的應(yīng)用[J].公路工程,2012,37(3):209-212. TANG Ping-xiang.Rayleigh Wave Method for Soil Characterization[J].Highway Engineering,2012,37(3):209-212.(in Chinese)

[6] 黃達(dá),金華輝.土石比對(duì)碎石土強(qiáng)夯地基加固效果影響規(guī)律瑞利波檢測(cè)分析[J].巖土力學(xué),2012,33(10):3067-3072. HUANG Da,JIN Hua-hui.Influences of Soil-rock Ratio on Foundation with Detritus Soil under Dynamic Compaction Based on Rayleigh Wave Detection[J].Rock and Soil Mechanics,2012,33(10):3067-3072.(in Chinese)

[7] 牛志榮,楊桂通.沖擊荷載下土體位移特征研究[J].巖土力學(xué),2005,26(11):52-57. NIU Zhi-rong,YANG Gui-tong.Studies on the Displacement of Soils Subjected to the Impact Loading[J].Rock and Soil Mechanics,2005,26(11):52-57.(in Chinese)

[8] 牛志榮,路國(guó)運(yùn).土體受沖擊時(shí)Rayleigh波作用機(jī)制探討[J].巖土力學(xué),2009,30(6):1583-1589. NIU Zhi-rong,LU Guo-yun.Discussion on Mechanism and Effect of Rayleigh Wave on Soil Subjected to Impact Loading[J].Rock and Soil Mechanics,2009,30(6):1583-1589.(in Chinese)

[9] Makris N.Soil-pile Interaction During the Passage of Rayleigh Waves:An Analytical Solution[J].Earthquake Engineering and Structural Dynamics,1994,23(2):153-167.

[10] Makris N,Badoni D.Seismic Response of Pile Groups under Oblique-shear and Rayleigh Waves[J].Earthquake Engineering & Structural Dynamics,1995,24(4):517-532.

[11] 蔣東旗,王立忠,陳云敏.遠(yuǎn)場(chǎng)地震引起的單樁橫向位移和內(nèi)力[J].巖土工程學(xué)報(bào),2003,25(2):174-178. JIANG Dong-qi,WANG Li-zhong,CHEN Yun-min.Lateral Displacement and Internal Force of Single Pile Induced by Far-fieldearthquake[J].Chinese Jounal of Geotechnical Engineering,2003,25(2):174-178.(in Chinese)

[12] 蔣東旗,郝玉龍,陳云敏.遠(yuǎn)場(chǎng)地震作用下樁間橫向動(dòng)力相互作用的研究[J].地震工程與工程振動(dòng),2004,24(4):170-176. JIANG Dong-qi,HAO Yu-long,CHEN Yun-min.Lateral Dynamic Pile-pile Interaction Induced by Far-field Earthquake[J].Earthquake Engineering and Engineering Vibration,2004,24(4):170-176.(in Chinese)

[13] 岳慶霞,李杰.近似Rayleigh地震波作用下地下綜合管廊響應(yīng)分析[J].防災(zāi)減災(zāi)工程學(xué)報(bào),2008,28(4):409-416. YUE Qing-xia,LI Jie.Response Analysis of Utility Tunnel in Earthquake of Approximate Rayleigh Waves[J].Journal of Disaster Prevention and Mitigation Engineering,2008,28(4):409-416.(in Chinese)

[14] 羅韜.基于小波變換的Rayleigh地震波及地下綜合管廊地震響應(yīng)研究[D].濟(jì)南:山東建筑大學(xué)土木工程學(xué)院,2013. LUO Tao.Rayleigh Wave Research Based on Wavelet Transform and Response of the Utility Tunnel[D].Jinan:Shandong Jianzhu University,School of Architecture and Construction,2013.(in Chinese)

[15] 潘旦光.直接確定Rayleigh阻尼系數(shù)的一種優(yōu)化方法[J].工程力學(xué),2013,30(9):16-21. PAN Dan-guang.An Optimization Method for the Direct Determination of Rayleigh Damping Coefficients[J].Engineering Mechanics,2013,30(9):16-21.(in Chinese)

[16] Duhee P,Youssef M A.Soil Damping Formulation in Nonlinear Time Domain Site Response Analysis[J].Journal of Earthquake Engineering,2004,8(2):249-274.

[17] Hudson M,Idriss I M,Beikae M.User Manual for QUAD4M:A Computer Program to Evaluate the Seismic Response of Soil Structures Using Finite Element Procedures and Incorporating a Compliant Base[D].Berkeley:University of California,1994,52-80.

[18] Annie O L,Jonathan P S,Youssef M A,et al.Use of Exact Solutions of Wave Propagation Problems to Guide Implementation of Nonlinear Seismic Ground Response Analysis Procedures[J].Journal of Geotechnical and Geoenvironmental Engineering,2007,133(11):1385-1398.

[19] Nozomu Y,Satoshi K,Iwao S,et al.Equivalent Linear Method Considering Frequency Dependent Characteristics of Stiffness and Damping[J].Soil Dynamics and Earthquake Engineering,2002,22(3):205-222.

[20] Hardin B O,Drnevich V P.Shear Modulus and Damping in Soils[J].Journal of the Soil Mechanics and Founda-tions Division,1972,98(7):667-692.

Application of Hardening Small Strain Model in the Site Response Analysis under Rayleigh Wave Excitation

SHI You-zhi1,2, LIN Shu-zhi3, YANG Rong-hua1

(1.SchoolofCivilEngineeringandArchitecture,XiamenUniversityofTechnology,Xiamen361021,Fuyuan,China; 2.SchoolofNavalArchitecture,Ocean&CivilEngineering,ShanghaiJiaotongUniversity,Shanghai200240,China; 3.XiamenConstructionBureau,Xiamen361003,Fujian,China)

This study examines the dynamic response characteristics of surface waves mainly comprising Rayleigh waves in surface soils under impact and seismic loads. Moreover, the study analyzes, through numerical simulation, the settling methods of soil layer damping. The research area for this study is the shallow plain fill and silty clay of the Xiamen area. The model was generated using finite element dynamic analysis, with a hardening small strain model (HSS) for soil constitutive. The hysteresis characteristics of the model allowed the input of different small strain parameters to study the influence of HSS model's small strain parameters on site dynamic response. Results were then compared with the combined Mohr-Coulomb (MC) model and Rayleigh damping model (MC + Rayleigh damping). The study shows that when using HSS model with hysteresis, the wave speed increases with increases in the initial shear modulus; however, amplitude and residual deformation decline. While the HSS model can reach the residual deformation value, "MC + Rayleigh damping" cannot. This reflects the accumulation of plastic strain and the damping effect during the process of unloading and reloading as the constitutive model is an ideal elastic-plastic model that demonstrates pure elastic behavior under the conditions of unloading and reloading. However, as the HSS model cannot fully reflect the accumulation of plastic volume strain under the effect of reloading and taking hysteresis damping into consideration, the use of the Rayleigh damping model is advised to allow full simulation of the real damping characteristics of the soil.

hardening small strain model; Rayleigh damping; Rayleigh wave; seismic dynamic response; finite element dynamic analysis

2016-04-13 基金項(xiàng)目：福建省自然科學(xué)基金資助項(xiàng)目(2016J01271);福建省住房和城鄉(xiāng)建設(shè)廳科學(xué)技術(shù)項(xiàng)目(2015-K-38);福建省住房和城鄉(xiāng)建設(shè)廳科學(xué)技術(shù)項(xiàng)目(2016-K-26)

施有志(1976-),男,福建晉江人,博士,副教授,主要從事巖土工程、地下工程等領(lǐng)域的教學(xué)與科研工作。 E-mail:2013110907@xmut.edu.cn。

TU435

A

1000-0844(2016)06-0896-07

10.3969/j.issn.1000-0844.2016.06.0896