半无限体的应力-粘弹性分析

半无限体的应力-粘弹性分析

0土体粘弹性与桩基分析的关系在计算建筑物的基本沉降时，应首先计算开挖体中的应力和位移。土壤中电压的解通常基于布西娜斯克公式，并根据公共布负荷哈希公式计算电压，但波斯尼哈希分解基本上是基于衰减半空间表面的负荷。通过该方法计算的沉降物理值往往大于实际沉降观测值。Mindlin早在1936年就给出了半无限体内部作用有竖向集中力和水平向集中力的空间问题弹性解;徐志英以Mindlin公式为依据,推导出半无限体内部竖向均布荷载作用下的土中竖向应力公式;此后还有一些学者将Mindlin公式应用于桩基分析。袁聚云系统研究了竖向均布荷载作用在土体内部、水平向均布荷载作用在土体内部及竖向线荷载和条形均布荷载作用在土体内部时的土中应力公式。鉴于天然土体所固有的粘弹性性质,许多学者针对粘弹性问题进行了研究,文献则研究了半空间Burgers体在法向集中力或切向集中力作用下的粘弹性解;文献以布西奈斯克(Boussinesq)竖向位移解为依据,运用弹性-粘弹性对应原理,研究了矩形均布荷载作用下建筑物基础沉降的粘弹性计算方法。但对于半无限体内部作用集中力的空间粘弹性问题的解答,截止目前,作者尚没有检索到较为系统的研究成果。虽然工程实践证明,以Mindlin解为依据导出的桩基础以及其他深基础的沉降计算公式,往往与实测值吻合较好,但Mindlin计算理论并没有考虑土体的粘弹性特征。建筑物的持续沉降通常与地基土的流变性质有关,因此,考虑地基土的粘弹性特征,研究半无限体内部作用集中力的粘弹性解,对于较准确的计算桩基位移与建筑物的沉降,无疑具有非常重要的理论价值和实际应用价值。1半无限空间的粘合弹性解1.1u3000laplace本构方程为了研究问题的方便,在进行理论分析之前,首先对空间半无限体做如下假设:1)假定半无限体是线性粘弹介质;2)半无限体是均匀各向同性的连续变形体,在深度和水平方向上无限延伸;3)半无限在内部集中力作用下的应力为三维应力状态,应力球张量和应变球张量之间符合弹性关系,而应力偏张量和应变偏张量之间符合Kelvin粘弹性本构方程。对时间t进行Laplace变换后的复杂应力状态下本构方程为:ˉΡ1(s)ˉsij=ˉQ1(s)ˉeij,ˉΡ2(s)ˉσkk=ˉQ2(s)ˉεkk(1)其中,ˉsij、ˉeij分别为应力、应变偏张量的Laplace变换;ˉσkk、ˉεkk为应力、应变球张量的Laplace变换;ˉΡ1、ˉQ1和ˉΡ2、ˉQ2的表达式为ˉΡ1=m∑k=0p′ksk,ˉQ1=n∑k=0q′ksk,ˉΡ2=m∑k=0p″ksk,ˉQ2=n∑k=0q″ksk。如图1所示,对Kelvin模型,则有ˉΡ1(s)=1,ˉQ1(s)=2Gk+2ηksˉΡ2(s)=1,ˉQ2(s)=3Κ}(2)式中,K为体积弹性模量;Gk、ηk分别为Kelvin模型系数。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土的剪切模量计算假定粘弹性空间半无限体内部深度h处受突加竖向集中力P(t)=P0H(t)作用,对P(t)求Laplace变换为:ˉΡ(s)=Ρ0/s(4)由文献,半无限弹性体内部作用有竖向集中力时,半无限体内部任一点M(x,y,z)的Mindlin解答为: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为集中力的作用线到计算点的水平距离,r=√x2+y2;R1=√r2+(z-h)2;R2=√r2+(z+h)2;μ为土的泊松比;G为土的剪切模量,G=E/[2(1+μ)];E为土的弹性模量;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)对式(7)取关于时间t的Laplace变换,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)根据准静态弹性-弹粘性对应原理,先对式(5)~式(6)进行关于时间t的Laplace变换,并将式(4)、式(8)代入,再进行关于时间t的Laplace逆变换,可得空间半无限粘弹性体的应力和位移解答为: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.若粘弹性空间半无限体内部深度h处受突加水平向集中力P(t)=P0H(t)作用,P(t)的Laplace变换同样为式(4)。若假定半无限土体的球张量之间符合弹性关系,偏张量之间符合Kelvin模型,根据文献,则半无限弹性体内部作用有水平向集中力时,当集中力作用方向与x轴正向一致时,半无限体内部任一点M(x,y,z)的Mindlin应力和位移的弹性解为: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的表达式与式(5)~式(6)相同。根据准静态弹性-弹粘性对应原理,在相同荷载条件下,首先对式(11)~式(12)进行关于时间t的Laplace变换,并将式(4)、式(8)代入,再进行关于时间t的Laplace逆变换,可得Kelvin空间半无限粘弹性体的应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)-[