摘要
采用连续排水边界的必要性及软黏土中水的渗流存在着起始水力坡降(i0)的现象已逐渐被认识,但变荷载下同时考虑连续排水边界条件和起始水力坡降的一维固结解析解还鲜见报道。针对外荷载随时间逐渐增加的实际情况,建立变荷载下同时考虑连续排水边界和起始水力坡降的固结模型。采用傅里叶变换及拉普拉斯变换获得模型的近似解析解,利用该解答分析动边界移动规律、超静孔隙水压力随时间的消散规律及平均固结度的增长规律。结果表明:在加载速率不变的情况下,起始水力坡降下排水面透水情况对固结性状的影响与其在达西定律下相同,透水情况越好,孔压消散速率越快;透水情况越差,超静孔隙水压力消散速度越慢。与完全透水边界条件下相比,连续排水条件下起始水力坡降i0对固结性状影响无明显改变,i0越大,移动边界到达土层底部的时间越长,固结完成时土中残留的超静孔压越大,按孔压定义的平均固结度越小;i0值越小,移动边界到达土层底部的时间越短,固结完成时土中残留的超静孔压越小,按孔压定义的平均固结度越大。在起始水力坡降和边界排水条件不变的情况下,随着加载时间的增加,超静孔隙水压力峰值越小,超静孔隙水压力达到峰值的时间越长,但加载时间对超静孔压残留值及按孔压定义的最终平均固结度无影响。
自太沙基创立一维固结理论以来,固结理论得到了快速发
目前,已经开展对连续排水边界下土体一维固结理论的相关研
考虑实际工程中施工荷载均随时间逐渐增加的事实,对变荷载下同时考虑连续排水边界和起始水力坡降的软土地基一维固结理论展开研究,建立其固结模型并获得近似解答,基于该解答着重分析边界系数b和起始水力坡降i0对固结性状的影响及不同工况下固结性状的异同。
如

图1 软土地基一维固结模型
Fig. 1 One-dimensional consolidation model ofsoft ground
梅国雄
(1) |
式中:b为界面系数,与界面材料性质有关,可通过实测孔压反演得到;u为超静孔隙水压力;t为时间;e为常数;为积分变量;为变荷载的一阶导数;为变荷载的初始值。超静孔压的初始值为。
由于起始水力坡降的存在,渗流过程中存在着移动边界问题,在渗流移动边界处及以下的超静孔压没有变化,故移动边界处的超静孔压应满
(2) |
(3) |
式中:z为深度;h(t)为t时刻移动边界距离透水面的距离;为变荷载;为起始水力坡降;为水的重度。土中水的渗流遵循起始水力坡降的渗流模型,其表达
(4) |
式中:v为黏土层中水的流速;kv为黏土层的渗透系数;i为水力坡降。以
(5) |
式中:为固结系数,cv=kvEs/γw。
前面已建立了变荷载下考虑起始水力坡降的单面排水固结控制方程及其初始条件与边界条件。为得到解析解答做如下变量代换,令
(6) |
式中:w(z,t)为关于深度z与时间t的待定函数。
控制方程(5)变为
(7) |
式中:为定解方程中关于外部荷载的待定函数,其表达式为
(8) |
其定解条件变为
(9) |
(10) |
(11) |
将定解函数和进行傅里叶展开
(12) |
(13) |
式中:,1、2、3…
(14) |
求解
(15) |
代入
(16) |
将代入定解条件中,根据正弦函数的正交关系
得
(17) |
故渗流前锋未到达土层底面的超静孔隙水压力的表达式为
(18) |
式中:
;
根据
(19) |
渗流前锋到达土层底部时孔压表达式为
(20) |
式中:
单级加载为变荷载达到稳定值qc前荷载大小与时间成正比例函数关系,荷载达到最大值后保持恒定不变。单级加载随时间变化的函数关系为
(21) |
式中:tc为外荷载达到最大值的时间。
当0<t<tc时,外荷载处于加荷载阶段,将
(22) |
在处理移动边界问题时,需要将t时刻下移动边界距离排水面的位置假定为常数h,对
(23) |
再利用Laplace逆变换对
(24) |
式中:
将
(25) |
式中:
若该时段内移动边界未到达土层底面,则移动边界位置变化规律应满足
(26) |
将
(27) |
式中:,,,,。
若该时段内渗流前锋能到达土层底面,设移动边界到达土层底部的时间为tH,则当tH<t<tc时,移动边界处于土层底部且不再变化,即h(t)=H,此时超静孔压表达式为
(28) |
式中:
水力坡降i0的存在导致土中渗流不满足达西定律,故令i0(R)=0,则模型可以退化为仅考虑连续排水边界的渗流模型。
1)当时,超静孔压的表达式退化为
(49) |
式中:
沉降量为
(50) |
2)当时,该时段下外荷载达到峰值且不再变化(),此时孔压表达式退化为
(51) |
式中:
沉降量的表达式退化为
(52) |
影响固结模型性状的主要因素是变荷载的加载速度、起始水力坡降及边界透水性能3个方面,具体体现为无量纲时间因子Tvc、无量纲变量R及无量纲变量B对固结性状的影响,相关计算参数见
由于起始水力坡降i0的存在,导致土层在固结过程中存在移动边界现象,并且土中的超静孔压在固结完成后并不能完全消散。若移动边界能到达土层底面,则整个土层均会发生渗流固结;若渗流锋面最终不能到达土层底面,则仅移动边界之上的土层才发生固结,移动边界之下的土层中超静孔压则保持与外荷载大小一致。

(a) R≤1

(b) R>1
图2 R对X-Tv曲线的影响
Fig. 2 Influences of R on the X-Tv curves

图3 R对u/qc-z/H曲线的影响(Tv=0.35)
Fig. 3 Influences of R on the u/qc-z/H curves (Tv=0.35)
由

图4 R对u/qc-Tv曲线的影响(z/H=0.7)
Fig. 4 Influences of R on the u/qc-Tv curves (z/H=0.7)

图5 R对Up-Tv曲线的影响
Fig. 5 Influences of R on the Up-Tv curves

图6 R对S-Tv曲线的影响
Fig. 6 Influences of R on the S-Tv curves
由
由于超静孔隙水压力的消散,渗流前锋会随着固结时间的推移而逐渐下移。如

图7 B对渗流移动边界的影响
Fig. 7 Influences of B on the moving boundary of seepage flow

图8 B对u/qc-z/H曲线的影响(Tv=0.35)
Fig. 8 Influences of B on the u/qc-z/H curves (Tv=0.35)

图9 B对u/qc-Tv曲线的影响
Fig. 9 Influences of B on the u/qc-Tv curves

图10 B对固结度Up的影响
Fig. 10 Influences of B on the consolidation degree Up

图11 B对沉降量S的影响
Fig. 11 Influences of B on the steelement amount S
针对外荷载加载速率对固结性状的影响,选取不同Tvc值对超静孔压及按空压定义的平均固结度进行固结性状分析。
如

图12 Tvc 对u/qc-Tv曲线的影响
Fig. 12 Influences of Tvc on the u/qc-Tv curves

图13 Tvc 对Up-Tv曲线的影响
Fig. 13 Influences of Tvc on the Up-Tv curves

图14 Tvc 对S-Tv曲线的影响
Fig. 14 Influences of Tvc on the S-Tv curves
考虑实际中的变荷载、土中存在的起始水力坡降及不同透水性能的排水边界条件,以传统Terzaghi一维固结理论为基础,重新建立并推导单级荷载下考虑起始水力坡降和连续排水边界的一维固结控制方程及其解析解,结论如下:
1)给出了基于变荷载下连续排水边界和起始水力坡降的软土一维固结的解析解答,并给出了特殊的单级加载下该模型的固结解析解。目前考虑起始水力坡降在完全透水边界下的固结解和达西定律下考虑连续排水边界的固结解均是本文解析解的特例。
2)本文解为变荷载下同时考虑连续排水边界和起始水力坡降的固结问题提供了固结计算的理论支撑。
3)与在完全透水边界下相比,连续排水边界下起始水力坡降和加载速率对软黏土固结性状的影响并未发生明显改变。在考虑连续排水边界的情况下,与完全透水边界下相比,加载速率的大小对软土固结性状的影响并未发生明显改变。连续排水边界对起始水力坡降所引起的移动边界下移速度影响较大,其透水性越好,移动边界下移速度越快。
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