半無限體的應(yīng)力-粘彈性分析

半無限體的應(yīng)力-粘彈性分析

0土體粘彈性與樁基分析的關(guān)系在計算建筑物的基本沉降時，應(yīng)首先計算開挖體中的應(yīng)力和位移。土壤中電壓的解通?；诓嘉髂人箍斯剑⒏鶕?jù)公共布負荷哈希公式計算電壓，但波斯尼哈希分解基本上是基于衰減半空間表面的負荷。通過該方法計算的沉降物理值往往大于實際沉降觀測值。Mindlin早在1936年就給出了半無限體內(nèi)部作用有豎向集中力和水平向集中力的空間問題彈性解;徐志英以Mindlin公式為依據(jù),推導(dǎo)出半無限體內(nèi)部豎向均布荷載作用下的土中豎向應(yīng)力公式;此后還有一些學(xué)者將Mindlin公式應(yīng)用于樁基分析。袁聚云系統(tǒng)研究了豎向均布荷載作用在土體內(nèi)部、水平向均布荷載作用在土體內(nèi)部及豎向線荷載和條形均布荷載作用在土體內(nèi)部時的土中應(yīng)力公式。鑒于天然土體所固有的粘彈性性質(zhì),許多學(xué)者針對粘彈性問題進行了研究,文獻則研究了半空間Burgers體在法向集中力或切向集中力作用下的粘彈性解;文獻以布西奈斯克(Boussinesq)豎向位移解為依據(jù),運用彈性-粘彈性對應(yīng)原理,研究了矩形均布荷載作用下建筑物基礎(chǔ)沉降的粘彈性計算方法。但對于半無限體內(nèi)部作用集中力的空間粘彈性問題的解答,截止目前,作者尚沒有檢索到較為系統(tǒng)的研究成果。雖然工程實踐證明,以Mindlin解為依據(jù)導(dǎo)出的樁基礎(chǔ)以及其他深基礎(chǔ)的沉降計算公式,往往與實測值吻合較好,但Mindlin計算理論并沒有考慮土體的粘彈性特征。建筑物的持續(xù)沉降通常與地基土的流變性質(zhì)有關(guān),因此,考慮地基土的粘彈性特征,研究半無限體內(nèi)部作用集中力的粘彈性解,對于較準確的計算樁基位移與建筑物的沉降,無疑具有非常重要的理論價值和實際應(yīng)用價值。1半無限空間的粘合彈性解1.1u3000laplace本構(gòu)方程為了研究問題的方便,在進行理論分析之前,首先對空間半無限體做如下假設(shè):1)假定半無限體是線性粘彈介質(zhì);2)半無限體是均勻各向同性的連續(xù)變形體,在深度和水平方向上無限延伸;3)半無限在內(nèi)部集中力作用下的應(yīng)力為三維應(yīng)力狀態(tài),應(yīng)力球張量和應(yīng)變球張量之間符合彈性關(guān)系,而應(yīng)力偏張量和應(yīng)變偏張量之間符合Kelvin粘彈性本構(gòu)方程。對時間t進行Laplace變換后的復(fù)雜應(yīng)力狀態(tài)下本構(gòu)方程為:ˉΡ1(s)ˉsij=ˉQ1(s)ˉeij,ˉΡ2(s)ˉσkk=ˉQ2(s)ˉεkk(1)其中,ˉsij、ˉeij分別為應(yīng)力、應(yīng)變偏張量的Laplace變換;ˉσkk、ˉεkk為應(yīng)力、應(yīng)變球張量的Laplace變換;ˉΡ1、ˉQ1和ˉΡ2、ˉQ2的表達式為ˉΡ1=m∑k=0p′ksk,ˉQ1=n∑k=0q′ksk,ˉΡ2=m∑k=0p″ksk,ˉQ2=n∑k=0q″ksk。如圖1所示,對Kelvin模型,則有ˉΡ1(s)=1,ˉQ1(s)=2Gk+2ηksˉΡ2(s)=1,ˉQ2(s)=3Κ}(2)式中,K為體積彈性模量;Gk、ηk分別為Kelvin模型系數(shù)。由式(2)可得:ˉE(s)=18Κ(Gk+ηks)/(6Κ+2Gk+2ηks)ˉμ(s)=(3Κ-2Gk-2ηks)/(6Κ+2Gk+2ηks)ˉG(s)=ˉQ1(s)/[2ˉΡ1(s)]=Gk+2ηks}(3)1.2土的剪切模量計算假定粘彈性空間半無限體內(nèi)部深度h處受突加豎向集中力P(t)=P0H(t)作用,對P(t)求Laplace變換為:ˉΡ(s)=Ρ0/s(4)由文獻,半無限彈性體內(nèi)部作用有豎向集中力時,半無限體內(nèi)部任一點M(x,y,z)的Mindlin解答為:σx=Ρ8π{-b1(z-h)R31+b23x2(z-h)R51-b13(z-h)R32+b44(z+h)R32+b230hx2z(z+h)R72+b33x2(z-h)R52-b16h(z+h)zR52+b5?12h2(z+h)R52+b6[4R2(R2+z+h)?(1-x2R2(R2+z+h)-x2R22)]}(5a)σy=Ρ8π{-b1(z-h)R31+b23y2(z-h)R51-b13(z-h)R32+b44(z+h)R32+b230hy2z(z+h)R72+b33y2(z-h)R52-b16hz(z+h)R52+b512h2(z+h)R52+b64R2(R2+z+h)?[1-y2R2(R2+z+h)-y2R22]}(5b)σz=Ρ8π[b1(z-h)R31+b23(z-h)3R51-b1(z-h)R32+b230hz(z+h)3R72+b33z(z+h)2R52-b23h(z+h)(5z-h)R52](5c)τyz=Ρ8π[b1yR31-b1yR32+b23y(z-h)2R51+b230hz(z+h)2R72+b33z(z+h)R52-b13h(3z+h)R52](5d)τxz=Ρ8π[b11R31-b21R32+b23(z-h)2R51+b230hz(z+h)2R72+b33z(z+h)R52-b23h(3z+h)R52](5e)τxy=Ρ8π{b23xy(z-h)R51+b33xy(z-h)R52-b64R22(R2+z+h)[1R2(R2+z+h)-1R2]+b230hz(z+h)R72}(5f)ux=Ρ16π[b9(z-h)xR31+b7x(z-h)R32-b104xR2(R2+z+h)+b96xhz(z+h)R32](6a)uy=Ρ16π[b9y(z-h)R31+b7y(z-h)R32-b104yR2(R2+z+h)+b96yhz(z+h)R32](6b)uz=Ρ16π[b71R1+b88R2-b71R2+b9(z-h)2R31+b7(z+h)2R32-b92hzR32+b96hz(z+h)2R52](6c)其中,r為集中力的作用線到計算點的水平距離,r=√x2+y2;R1=√r2+(z-h)2;R2=√r2+(z+h)2;μ為土的泊松比;G為土的剪切模量,G=E/[2(1+μ)];E為土的彈性模量;且有:b1=(1-2μ)/(1-μ)b2=1/(1-μ)b3=(3-4μ)/(1-μ)b4=μ(1-2μ)/(1-μ)b5=μ/(1-μ)b6=1-2μb7=(3-4μ)/[G(1-μ)]b8=(1-μ)/Gb9=1/[G(1-μ)]b10=(1-2μ)/G}(7)對式(7)取關(guān)于時間t的Laplace變換,可得式(8):ˉb1(s)=[1-2ˉμ(s)]/[1-ˉμ(s)]ˉb2(s)=1/[1-ˉμ(s)]ˉb3(s)=[3-4ˉμ(s)]/[1-ˉμ(s)]}(8)ˉb4(s)=ˉμ(s)[1-2ˉμ(s)]/[1-ˉμ(s)]ˉb5(s)=ˉμ(s)/[1-ˉμ(s)]ˉb6(s)=1-2ˉμ(s)ˉb7=[3-4ˉμ(s)]/{ˉG(s)[1-ˉμ(s)]}ˉb8(s)=[1-ˉμ(s)]/ˉG(s)ˉb9(s)=1/{ˉG(s)[1-ˉμ(s)]}ˉb10(s)=[(1-2ˉμ(s)]/ˉG(s)}(8)根據(jù)準靜態(tài)彈性-彈粘性對應(yīng)原理,先對式(5)~式(6)進行關(guān)于時間t的Laplace變換,并將式(4)、式(8)代入,再進行關(guān)于時間t的Laplace逆變換,可得空間半無限粘彈性體的應(yīng)力和位移解答為:σx=Ρ08π{[v1(1-a1(t))+32a1(t)]?[-(z-h)R31-3(z-h)R32-6h(z+h)zR52]+[v2(1-a1(t))+12a1(t))][3x2(z-h)R51+30hx2z(z+h)R72]+[v3(1-a1(t))+72a1(t)]?3x2(z-h)R52+[v4-v5a2(t)+v6a1(t)]?4(z+h)R32+[v7(1-a1(t))-12a1(t)]?12h2(z+h)R52+[v8(1-a2(t))+3a2(t)]?[4R2(R2+z+h)(1-x2R2(R2+z+h)-x2R22)]}(9a)σy=Ρ08π{[v1(1-a1(t))+32a1(t)][-(z-h)R31-3(z-h)R32-6h(z+h)zR52]+[v2(1-a1(t))+12a1(t)][3y2(z-h)R51+30hy2z(z+h)R72]+[v3(1-a1(t))+72a1(t)]3y2(z-h)R52+[v4-v5a2(t)+v6a1(t)]4(z+h)R32+[v7(1-a1(t))-12a1(t)]12h2(z+h)R52+[v8(1-a2(t))+3a2(t)][4R2(R2+z+h)?(1-y2R2(R2+z+h)-y2R22)]}(9b)σz=Ρ08π{[v1(1-a1(t))+32a1(t)](z-hR31-z-hR32)+[v2(1-a1(t))+32a1(t)][3(z-h)3R51-3h(z+h)(5z-h)R52+30hz(z+h)3R72]+[v3(1-a1(t))+72a1(t)]3z(z+h)2R52}(9c)τyz=Ρ8π{[v1(1-a1(t))+32a1(t)]?[yR31-yR32-3h(3z+h)R52]+[v2(1-a1(t))+12a1(t)]?[3y(z-h)2R51+30hz(z+h)2R72]+[v3(1-a1(t))+72a1(t)]3z(z+h)R52}(9d)τxz=Ρ8π{[v1(1-a1(t))+32a1(t)][1R31-1R32]+[v2(1-a1(t))+12a1(t)][3(z-h)2R51+30hz(z+h)2R72-3h(3z+h)R52]+[v3(1-a1(t))+72a1(t)]3z(z+h)R52}(9e)τxy=Ρ08π{[v1(1-a1(t))+12a1(t)][3xy(z-h)R51+30hz(z+h)R72]+[v3(1-a1(t))+72a1(t)]?3xy(z-h)R52-[v8(1-a2(t))+32a2(t)]?4R22(R2+z+h)(1R2(R2+z+h)-1R2)}(9f)ux=Ρ016π{[v3Gk-2Gka3(t)-v9a1(t)]x(z-h)R32+[v2Gk-2Gka3(t)+v9a1(t)][(z-h)xR31+6xhz(z+h)R32]-[v10-v10a2(t)]?4xR2(R2+z+h)}(10a)uy=Ρ016π{[v3Gk-2Gka3(t)-v9a1(t)]y(z-h)R32+[v2Gk-2Gka3(t)+v9a1(t)][(z-h)yR31+6yhz(z+h)R32]-v10[1-a2(t)]?4yR2(R2+z+h)}(10b)uz=Ρ016π{[1R1-1R2+(z+h)2R32]?[v3Gk-2Gka3(t)-v9a1(t)]+[(z-h)2R31-2hzR32+6hz(z+h)2R52]?[v2Gk-2Gka3(t)+v9a1(t)]+8R2[1Gkv2-12Gka3(t)-v102a2(t)]}(10c)式(9)~式(10)中,v1=6Gk3Κ+4Gk,v2=6Κ+2Gk3Κ+4Gk,v3=6Κ+14Gk3Κ+4Gk,v4=6Gk(3Κ-2Gk)6Κ+2Gk,v5=9Κ3Κ+Gk,v6=9Κ2(3Κ+4Gk),v7=3Κ-2Gk3Κ+4Gk,v8=6Gk6Κ+2Gk,v9=63Κ+4Gk,v10=33Κ+3Gk,a1(t)=e-3Κ+4Gk4ηkt,a2(t)=e-6Κ+2Gk2ηkt,a3(t)=e-Gkηkt。1.3hzz+hs15-3xr35.若粘彈性空間半無限體內(nèi)部深度h處受突加水平向集中力P(t)=P0H(t)作用,P(t)的Laplace變換同樣為式(4)。若假定半無限土體的球張量之間符合彈性關(guān)系,偏張量之間符合Kelvin模型,根據(jù)文獻,則半無限彈性體內(nèi)部作用有水平向集中力時,當(dāng)集中力作用方向與x軸正向一致時,半無限體內(nèi)部任一點M(x,y,z)的Mindlin應(yīng)力和位移的彈性解為:σx=Ρ0x8π{b1(s)[1R13-5R23]+b2(s)[3x2R15-18h2R25+18h(z+h)R25-30hx2zR27]+b3(s)3x2R25+b4(s)4R23-b5(s)12h(z+h)R25+b6(s)4R2(R2+z+h)2?[3-x2(3R2+z+h)R22(R2+z+h)]}(11a)σy=Ρx8π{b1(s)[6h(z+h)R25-1R13-3R23]+b2(s)?[3y2R15-6h2R25-30hy2zR27]+b3(s)3y2R25+b4(s)4R23+b6(s)4R2(R2+z+h)2?[1-y2(3R2+z+h)R22(R2+z+h)]}(11b)σz=Ρx8π{b1[1R23-1R13-6h(z+h)R25]+b2[3(z-h)2R15-6h2R25-30hz(z+h)2R27]+b33(z+h)2R25}(11c)τyz=Ρxy8π{b2[3(z-h)R15-30hz(z+h)R27]+b33(z+h)R25-b16hR25}(11d)τzx=Ρ8π{b1(z-hR13-z-hR23-6hx2R25)+b2[3x2(z-h)R15+6hz(z+h)R25-30x2hz(z+h)R22]+b33x2(z+h)R25}(11e)τxy=Ρy8π{b1[1R13-1R23]+b2[3x2R15-6hzR25(1-5x2R22)]+b33x2R25+b64R2(R2+z+h)2?[1-x2(3R2+z+h)R22(R2+z+h)]+b26hzR25(1-5x2R22)}(11f)ux=Ρ16π{b7(1R1+x2R23)+b9[1R2+x2R13+2hzR23?(1-3x2R22)]+b104R2+z+h?[1-x2R2(R2+z+h)]}(12a)uy=Ρxy16π[b9(1R13-6hzR25)+b71R23-b104R2(R2+z+h)](12b)uz=Ρx16π[b9(z-hR13-6hz(z+h)R25)+b7(z-h)R23+b104R2(R2+z+h)](12c)以上式中r、R1、R2的表達式與式(5)~式(6)相同。根據(jù)準靜態(tài)彈性-彈粘性對應(yīng)原理,在相同荷載條件下,首先對式(11)~式(12)進行關(guān)于時間t的Laplace變換,并將式(4)、式(8)代入,再進行關(guān)于時間t的Laplace逆變換,可得Kelvin空間半無限粘彈性體的應(yīng)力和位移分量的解答:σx=Ρ0x8π{[v1(1-a1(t))+32a1(t)][1R13-5R23]+[v2(1-a1(t))+12a1(t)][3x2R15-18h2R25+18h(z+h)R25-30hx2zR27]+[v3(1-a1(t))+72a1(t)]3x2R25+[v4-v5a2(t)+v6a1(t)]4R23-[v7(1-a1(t))-12a1(t)]12h(z+h)R25+[v8(1-a2(t))+32a2(t)]4R2(R2+z+h)2[3-x2(3R2+z+h)R22(R2+z+h)]}(13a)σy=Ρ0x8π{[v1(1-a1(t))+32a1(t)][1R13-5R23]+[v2(1-a1(t))+12a1(t)][3y2R15-18h2R25+18h(z+h)R25-30hy2zR27]+[v3(1-a1(t))+72a1(t)]3y2R25+[v4-v5a2(t)+v6a1(t)]4R23-[v7(1-a1(t))-12a1(t)]12h(z+h)R25+[v8(1-a2(t))+32?a2(t)]4R2(R2+z+h)2[3-y2(3R2+z+h)R22(R2+z+h)]}(13b)σz=Ρ0x8π{[v1(1-a1(t))+32a1(t)][1R23-1R13-6h(z+h)R25]+[v2(1-a1(t))+12a1(t)]?[3(z-h)2R15-6h2R25-30hz(z+h)2R27]+[v3(1-a1(t))+72a1(t)]3(z+h)2R25}(13c)τyz=Ρxy8π{-[v1(1-a1(t))+32a1(t)]6hR25+[v2(1-a1(t))+12a1(t)][3(z-h)R15-30hz(z+h)R27]+[v3(1-a1(t))+72a1(t)]3(z+h)R25}(13d)τzx=Ρ8π{[v1(1-a1(t))+32a1(t)](z-hR13-z-hR23-6hx2R25)+[v2(1-a1(t))+12a1(t)]?[3x2(z-h)R15+6hz(z+h)R25-30x2zh(z+h)R27]+[v3(1-a1(t))+72a1(t)]3x2(z+h)R25}(13e)τxy=Ρy8π{[v1(1-a1(t))+32a1(t)][1R13-1R23]+[v2(1-a1(t))+12a1(t)][3x2R15-6hzR25(1-5x2R22)]+[v3(1-a1(t))+72a1(t)]3x2R25+[v8(1-a2(t))+32a2(t)]4R2(R2+z+h)2?[1-x2(3R2+z+h)R22(R2+z+h)]}(13f)ux=Ρ16π{[v3Gk-2Gka3(t)-v9a1(t)](1R1+x2R23)+[v2Gk-2Gka3(t)+v9a1(t)][1R2+x2R13+2hzR23?(1-3x2R22)]+[v10-v10a2(t)e-6Κ+2Gk2ηkt]?4R2+z+h[1-x2R2(R2+z+h)]}(14a)uy(x,y,z,t)=Ρxy16π{[v3Gk-2Gka3(t)-v9a1(t)]?1R23+[v2Gk-2Gka3(t)+v9a1(t)]?(1R13-6hzR25)-[