Analytical solution for dynamic response of asphalt pavement with subgrade modulus varying with depth
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摘要: 推导了落锤式弯沉仪（FWD）荷载作用下考虑路面结构黏弹性、横观各向同性、层间连续条件以及路基模量沿深度方向非均匀分布的沥青路面动力响应解析解。首先，在考虑横观各向同性轴对称动力基本方程基础上，借助Hankel-Laplace积分变换建立了路面结构部分的常系数常微分方程组以及路基结构部分的变系数常微分方程组；然后，应用Frobenius法以及刚度矩阵法获得了沥青路面动力响应解析解，并编制数值计算程序；最后，结合ABAQUS有限元结果以及文献对比验证了解析解的可靠性，并就路面结构以及路基结构参数进行讨论。结果表明，路面结构中的层间接触条件和横观各向同性能对弯沉产生显著影响，同时路基模量沿深度的非均匀分布对弯沉的影响也不容忽视，建议在实际的FWD道路检测以及路面力学分析中给予充分考虑。Abstract: The analytical solution for the dynamic response of asphalt pavement under falling weight deflectometer (FWD) load is obtained considering the viscoelasticity, transverse isotropy, interlayer contact and non-uniform distribution of subgrade modulus along depth. Firstly, based on the transversely isotropic axisymmetric dynamic equations, the constant coefficient ordinary differential equations for pavement structure and the variable coefficient ordinary differential equations for subgrade are established through the Hankel-Laplace transform. Then, combined with the Frobenius method and the stiffness matrix method, the analytical solution of dynamic response for the asphalt pavement is obtained, and the numerical program is also compiled. Then, the reliability of analytical solution is verified by comparison and the results of finite element method. Finally, the influences of interlayer condition, modulus ratio and distribution of subgrade modulus on the road deflection are analyzed. The results show that the interlayer contact and transverse isotropy in the pavement structure have a significant impact on the road deflection. The influences of the subgrade modulus distribution along depth also cannot be ignored. It is suggested that the above characteristics of pavement structure and subgrade should be fully considered in the FWD tests and mechanical analysis of the pavement.
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0. 引言
在水利工程建设与湖泊、河道的清淤疏浚过程中会产生大量的高含水率疏浚淤泥。在工程上因其力学性质差难以直接利用,通常进行抛泥处理,造成大量的资源浪费和严重的环境污染。水泥固化法处理高含水率淤泥,在填海工程中被大量使用[1]。在水泥固化土的研究中,有学者就水泥固化土强度影响参数方面展开了研究,也有学者为提高水泥土材料的力学特性,研究了掺入其它材料的影响[2-4]。
另一方面,中国每年产生的废弃砖块约占建筑垃圾总量的30%～50%。关于废砖细骨料再生研究中,Letelier等[5]利用再生骨料和废砖粉作为水泥替代品,研究了结构混凝土的力学性能。Kumar等[6]利用废砖细骨料、混凝土细骨料和pozzol烷材料制备砌块,测试了砌块养护28 d后湿压强度、吸水率和吸湿率等特性。中国目前仍存在建筑垃圾排放量大,回收利用率低等问题[7]。
在疏浚土等不良土的处理方法中,还可掺混不同粒径的砂土,通过改变粒径级配达到改善不良土力学特性的目的[8]。基于此,本文在传统水泥固化土方法基础上提出用水泥-废砖细骨料双掺固化处理高含水率黏土的方法,通过测定不同龄期和不同配合比试样的无侧限抗压强度,分析了双掺固化土的应力-应变关系、抗压强度-破坏应变关系及废砖细骨料的掺入对强度的影响。
1. 试验材料与方法
1.1 试验材料
（1）通过预试验确定本试验所用细骨料的粒径范围为2～5 mm,密度1.306 g/cm3,吸水率为10.57%。
（2）所取原状土的物理力学性质指标见表1,通过加入水使其达到本文所设计的含水率72.4%。
表 1 黏土的物理力学性质指标Table 1. Physical and mechanical properties of clay含水率/% 孔隙比 液限/% 塑限/% 液性指数 塑性指数 32.06 0.397 55.11 15.11 0.42 40.00 （3）采用工程上常用的普通硅酸盐水泥,即P.O 42.5R水泥。
1.2 试验方法
考虑废砖细骨料掺量分别为0%,8%,10%和12%,水泥掺量分别为6%,8%和10%（均为黏土干质量的百分比）等多种情况,设置7 d和28 d两种养护龄期。每组配合比条件下分别制作3个压缩试样,测定其无侧限抗压强度。试样的制备步骤如下：
（1）混合底泥进行搅拌。加入计算所需的相应固化剂和细骨料,使用搅拌器匀速搅拌5 min制备一定含水率的黏土-水泥-废砖细骨料混合物,搅拌均匀后制成混合泥浆。
（2）开展试样制作。为方便后期脱模,在装入混合料前,在模具（直径为3.91 cm,高度为8 cm）内壁均匀涂上一层凡士林。将制备好的混合泥浆,分3次延模具壁一侧缓缓滑入,一次倒入1/3模具容积,每次倒入后作一段时间振捣,使小气泡从表面破出,避免内部气泡间隙对试样强度的影响。灌制满后,用刮刀进行刮平,铺垫保鲜膜后封盖。
（3）开展试样养护。将试样密封后置于充满水的水箱中,并放置在标准养护室（20±3℃,湿度>95%）内,养护至设计龄期。
2. 试验结果与分析
2.1 废砖细骨料掺量对强度的影响
图1为废砖细骨料掺量与无侧限抗压强度在水泥掺量在10%条件下的关系曲线图。由图1可知：当废砖细骨料掺量从10%增加到12%,试样强度均有了较大幅度的提升；但养护龄期为28 d增长率比7 d时略小。分析认为：当养护龄期达到28 d时,近似认为废砖细骨料中的水分达到饱和,此时细骨料的湿润度与周围水泥土湿润度相当,根据再生废砖骨料的吸水返水特性[9]分析可知,此时细骨料的返水能力比吸水能力强,双掺固化土中的水分会有所增加,故出现龄期为28 d的水泥-废砖细骨料双掺固化土的强度增长速率较7 d变缓的现象；在相同废砖细骨料掺量情况下,双掺固化土抗压强度随试样养护龄期的增加而增大,且28 d无侧限抗压强度相较7 d无侧限抗压强度平均提升了1.63倍。
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图 1 废砖细骨料掺量对强度的影响Figure 1. Influences of amount of waste brick fine aggregate on strength2.2 水泥掺量对强度的影响
图2为废砖细骨料掺量在10%条件下,水泥掺量与无侧限抗压强度的关系曲线图。由图2可知：当养护龄期为7 d时,试样强度随水泥掺量的增加成线性增长；养护龄期为28 d,当水泥掺量大于8%时,强度增长速率有减小的趋势。分析认为：这一现象与废砖骨料的吸水返水特性有关；在相同水泥掺量情况下,双掺固化土抗压强度随试样养护龄期的增加而增大,且28 d无侧限抗压强度相较7 d无侧限抗压强度平均提升了约1.44倍。
2.3 双掺固化土与水泥固化土的对比
通过对两种固化土的强度特性进行对比分析（图3）发现,龄期为7 d的水泥固化土,随水泥掺量的增加成非直线增长,这与郑少辉等[3]分析不同水灰比固化土的强度所得研究结果相近,即当水泥剂量小于16%时无侧限抗压强度随水泥剂量的增加呈非线性增长。在两种养护龄期下,均出现双掺固化土强度的总体增长速率比水泥固化土强度增长速率高的现象。分析可知,再生废砖细骨料具有孔隙率高、吸水性强等特征,能够吸收土体中部分多余水分,且废砖细骨料含量越多吸水性越强,从而有效降低土体含水率,进而随之强化水泥在低含水率下的固化效率,加快了双掺固化土强度的形成。对7 d龄期条件,当水泥剂量大于等于8%时,废砖细骨料的掺入,明显提高了固化土的强度,说明要使废砖细骨料在改善固化土强度方面发挥作用,对水泥掺量存在一个最低剂量要求。
废砖细骨料为颗粒状,在固化土体中可视为游离状态,在制作无侧限抗压试样时,由于分层振捣处理导致废砖颗粒分布不均,形成的受力骨架也有所差异,故测出的强度不一,导致随着龄期和废砖细骨料含量的增长,强度的变异系数明显变大。
综上所述,在水泥剂量满足最低要求（本文测的最低剂量为8%）的情况下,废砖细骨料掺入和龄期增长都有利于固化土强度的提升；废砖细骨料的掺入,在增大固化土强度的同时也会增大固化土的变异性。
2.4 双掺固化土的破坏形态
图4为双掺固化土无侧限抗压强度试验的破坏形态。试样受压破坏后出现多条裂缝,主裂缝不突出不明显,破坏后试样破碎成块状,为塑性剪切破坏。故水泥-废砖细骨料双掺固化土的破坏形态主要表现为塑性剪切破坏。
2.5 压缩变形特性
图5为用水泥-废砖细骨料双掺法处理高含水率黏土的固化土应力-应变曲线图。由图可见其破坏应变分布在2.5%～3%,与水泥固化土的破坏应变一般介于0.5%～2%的认识[8, 11-14]有一定的偏差。分析其原因有两点：①由于废砖骨料在试样中成悬浮分布状态,当其掺量较小时,颗粒之间并没有形成骨架；②当骨料的湿润度与周围水泥土湿润度相当时,骨料表现出返水能力比吸水能力强的特性,使土体的水分略微增加所致。
图6为双掺固化土破坏应变与抗压强度的关系曲线。由图6可知,破坏应变随着抗压强度增大呈先略微减小后明显增大的趋势,这与其他学者得出的破坏应变随抗压强度增大而减小的试验结果[8, 10-14]有一定的偏差。分析其原因,主要是废砖的掺入使固化土的韧性在一定程度上得到提升。
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图 6 固化土破坏应变与抗压强度的关系Figure 6. Relationship between failure strain and compressive strength of DMSC samples3. 结论
（1）废砖细骨料对高含水率水泥固化土的强度有显著的提升效果,且早期强度增长速率比后期快。
（2）要发挥废砖细骨料的作用,水泥掺量需满足最低剂量8%的要求；掺入废砖细骨料在提高固化土强度的同时,也增加了固化土的变异性。控制变异性可提高其在工程建设上应用的安全可靠性。
（3）双掺法处理高含水率黏土固化土的破坏形态主要表现为塑性剪切破坏,其破坏应变在2.5%～3%,韧性比一般固化土的韧性好。
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图 9 路基变模量参数–弯沉峰值云图(r=0.0 m)
Figure 9. Cloud map of non-uniform distribution coefficient and deflection peak of subgrade (r=0.0 m)
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图 11 路基模量比neN影响探究(r=0.0 m)
Figure 11. Researches on influence of subgrade modulus ratio neN (r=0.0 m)
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图 12 弯沉盆曲线(路基模量比影响)
Figure 12. Curves of deflection basin under influence of subgrade modulus ratio
表 1 路面结构层材料参数
Table 1 Material parameters of pavement structure
层位 垂直模量/MPa ne(Eh/Ev) ?v(?h) 密度/(kg∙m-3) 厚度/m 层间滑移系数αx 面层 Burgers
模型0.2 0.25 2400 0.18 0.8 基层 910 0.4 0.30 2300 0.35 0.7 底基层 350 0.5 0.35 2200 0.20 0.0 路基 300~
200e-0.2z1.0 0.35 1600 ∞ — 
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