引言

关于场地地震反应的分析已有大量研究成果(Hashash等,2010),研究表明土壤在地震作用下会表现出材料非线性效应(Joyner等,1975Huang等,2001Arslan等,2006Hosseini等,2012)。等效线性化方法(Schnabel等,1972Idriss等,1992Bardet等,2000王笃国等,2016)是一种频域方法,通过在不同土体应变条件下选择等效阻尼比和剪切模量,将非线性问题转化为线性问题。当采用材料非线性本构模型描述土体非线性时,需采用时间积分算法求解非线性动力有限元方程。时间积分算法可分为隐式方法和显式方法(Crisfield,1991Chopra,2009)。隐式算法(栾茂田等,1992Chopra,2009)每时刻需求解线性代数方程组,计算效率相对较低,如Wilson-θ法和Newmark法等。显式算法无需求解线性代数方程组,适合于强非线性和自由度数目较大的问题。研究者已提出多种显式时间积分算法(Chung等,1994王进廷等,2002Belytschko等,2014)。作者Zhao等(2019)近期提出一种二阶精度的单步显式算法,该算法适合变时步问题,在线弹性范围内稳定性较好。本文将该算法推广至求解非线性动力有限元方程中,并将其应用于地震波垂直入射时非线性地震反应分析。

1 非线性动力有限元方程的显式时间积分算法

设已知非线性体系第ti时步的受力状态,求解第ti+1时步的非线性结构动力学方程:

$ \mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\ddot u}}_{i + 1}} + \mathit{\boldsymbol{C}}{\mathit{\boldsymbol{\dot u}}_{i + 1}} + \mathit{\boldsymbol{f}}_{i + 1}^S = {\mathit{\boldsymbol{f}}_{i + 1}} $ (1)

式中MCf Sf分别表示非线性体系的质量矩阵、阻尼矩阵、内力向量和外荷载向量;u表示位移,点号对时间t求导,i+1表示第ti+1时刻。第i+1时刻时间步长为:

$ \Delta {t_i} = {t_{i + 1}} - {t_i} $ (2)

文献(Zhao等,2019)显式方法求解非线性方程(1)的过程如下,第i+1时刻位移ui+1为:

$ {\mathit{\boldsymbol{u}}_{i + 1}} = {\mathit{\boldsymbol{u}}_i} + \Delta {t_i}{\mathit{\boldsymbol{\dot u}}_i} + \frac{{\Delta t_i^2}}{2}{\mathit{\boldsymbol{\ddot u}}_i} $ (3)

i+1时刻位移增量Δui、内力增量ΔfiS和内力全量fi+1S分别为:

$ {\Delta {\mathit{\boldsymbol{u}}_i} = {\mathit{\boldsymbol{u}}_{i + 1}} - {\mathit{\boldsymbol{u}}_i}} $ (4)
$ {\Delta \mathit{\boldsymbol{f}}_i^S = \mathit{\boldsymbol{f}}(\Delta {\mathit{\boldsymbol{u}}_i})} $ (5)
$ {\mathit{\boldsymbol{f}}_{i + 1}^S = \mathit{\boldsymbol{f}}_i^S + \Delta \mathit{\boldsymbol{f}}_i^S} $ (6)

i+1时刻预估速度${{\mathit{\boldsymbol{\dot {\tilde u}}}}_{i + 1}}$、预估加速度${{\mathrm{\ddot{\tilde{\mathit{\boldsymbol{u}}}}}}_{i+1}}$、速度${{\mathit{\boldsymbol{\dot u}}}_{i + 1}}$和加速度${{\mathit{\boldsymbol{\ddot u}}}_{i + 1}}$分别为

$ {{\mathit{\boldsymbol{\dot {\tilde u}}}}_{i + 1}} = {{\mathit{\boldsymbol{\dot u}}}_i} + \Delta {t_i}{{\mathit{\boldsymbol{\ddot u}}}_i} $ (7)
$ {{\mathrm{\ddot{\tilde{\mathit{\boldsymbol{u}}}}}}_{i+1}} = {\mathit{\boldsymbol{M}}^{ - 1}}({\mathit{\boldsymbol{f}}_{i + 1}} - \mathit{\boldsymbol{C}}{{\mathit{\boldsymbol{\dot {\tilde u}}}}_{i + 1}} - \mathit{\boldsymbol{f}}_{i + 1}^S) $ (8)
$ {{\mathit{\boldsymbol{\dot u}}}_{i + 1}} = {{\mathit{\boldsymbol{\dot u}}}_i} + \frac{{\Delta {t_i}}}{2}({{\mathit{\boldsymbol{\ddot u}}}_i} + {{\mathrm{\ddot{\tilde{\mathit{\boldsymbol{u}}}}}}_{i+1}}) $ (9)
$ {{\mathit{\boldsymbol{\ddot u}}}_{i + 1}} = {\mathit{\boldsymbol{M}}^{ - 1}}({\mathit{\boldsymbol{f}}_{i + 1}} - \mathit{\boldsymbol{C}}{{\mathit{\boldsymbol{\dot u}}}_{i + 1}} - \mathit{\boldsymbol{f}}_{i + 1}^S) $ (10)

式(3)—式(10)为求解式(1)的显式算法。算法中需由位移增量计算内力增量,目前常用的应力计算方法包括向前欧拉法、向后欧拉法和完全隐式计算法等(Sloan等,1992Sloan等,2001Ahadi等,2003)。下面给出式(5)由位移增量计算内力增量的过程,即一种带误差控制的修正欧拉算法。

对于每个有限单元,由位移增量Δuie计算应变增量Δεie的表达式为:

$ \Delta \mathit{\boldsymbol{\varepsilon }}_i^e = {\mathit{\boldsymbol{B}}^e}\Delta \mathit{\boldsymbol{u}}_i^e $ (11)

式中Be为应变矩阵。将ti时刻单元应变增量Δεie赋值给子步应变增量Δεseti时刻单元应力σie赋值给σi+1e,初始化子步应变增量和应力状态分别为:

$ {\Delta \mathit{\boldsymbol{\varepsilon }}_s^e \leftarrow \Delta \mathit{\boldsymbol{\varepsilon }}_i^e} $ (12)
$ {\mathit{\boldsymbol{\sigma }}_{i + 1}^e \leftarrow \mathit{\boldsymbol{\sigma }}_i^e} $ (13)

每个子步中应力增量计算思路见图 1,具体计算公式如下:


图 1 修正欧拉算法计算应力增量 Fig. 1 Modified Euler algorithm to calculate stress increment
$ {\mathit{\boldsymbol{D}}_1^e = \mathit{\boldsymbol{D}}(\mathit{\boldsymbol{\sigma }}_{i + 1}^e)} $ (14)
$ {\Delta \sigma _1^e = \mathit{\boldsymbol{D}}_1^e\Delta \mathit{\boldsymbol{\varepsilon }}_s^e} $ (15)
$ {\mathit{\boldsymbol{D}}_2^e = \mathit{\boldsymbol{D}}(\mathit{\boldsymbol{\sigma }}_{i + 1}^e + \Delta \mathit{\boldsymbol{\sigma }}_1^e)} $ (16)
$ {\Delta \mathit{\boldsymbol{\sigma }}_2^e = \mathit{\boldsymbol{D}}_2^e\Delta \mathit{\boldsymbol{\varepsilon }}_s^e} $ (17)
$ \Delta \mathit{\boldsymbol{\sigma }}_s^e = \frac{{\Delta \mathit{\boldsymbol{\sigma }}_1^e + \Delta \mathit{\boldsymbol{\sigma }}_2^e}}{2} $ (18)

式中De为单元应力-应变关系矩阵。判断每个子步中应力增量Δσ s是否符合精度要求的误差判断式为:

$ {e_r} = \frac{{\left\| {\Delta \mathit{\boldsymbol{\sigma }}_1^e - \Delta \mathit{\boldsymbol{\sigma }}_2^e} \right\|}}{{\left\| {\mathit{\boldsymbol{\sigma }}_{i + 1}^e + \Delta \mathit{\boldsymbol{\sigma }}_s^e} \right\|}} $ (19)

判断误差er是否小于预先给定的判断值st,条件不满足时,缩小子步应变增量为:

$ \Delta \mathit{\boldsymbol{\varepsilon }}_s^e \leftarrow A\sqrt {{s_t}/{e_r}} \Delta \mathit{\boldsymbol{\varepsilon }}_s^e $ (20)

式中A为误差峰值系数。采用缩小的子步应变增量重新进行式(14)—式(19)的计算与判断,循环直至满足精度要求,更新剩余应变增量和应力状态分别为:

$ {\Delta \mathit{\boldsymbol{\varepsilon }}_i^e \leftarrow \Delta \mathit{\boldsymbol{\varepsilon }}_i^e - \Delta \mathit{\boldsymbol{\varepsilon }}_s^e} $ (21)
$ {\mathit{\boldsymbol{\sigma }}_{i + 1}^e \leftarrow \mathit{\boldsymbol{\sigma }}_{i + 1}^e + \Delta \mathit{\boldsymbol{\sigma }}_s^e} $ (22)

利用更新剩余应变增量和应力状态循环执行式(14)—式(20),直至剩余应变增量小于等于零结束。

利用求得的第i+1时刻单元应力可得到单元应力增量和内力增量分别为:

$ {\Delta \mathit{\boldsymbol{\sigma }}_i^e = \mathit{\boldsymbol{\sigma }}_{i + 1}^e - \mathit{\boldsymbol{\sigma }}_i^e} $ (23)
$ {\Delta \mathit{\boldsymbol{f}}_i^S = \sum {\int {{\mathit{\boldsymbol{B}}^{e{\rm{T}}}}} } \Delta \mathit{\boldsymbol{\sigma }}_i^e{\bf{d}}A} $ (24)
2 地震波垂直入射时场地非线性地震反应分析

本节将上述非线性有限元方程的显式时间积分算法应用于地震波垂直入射时场地非线性地震反应分析中。假定基岩为线弹性半空间,考虑基岩上覆土层的材料非线性,不考虑土体阻尼。在土层下部设置黏性边界条件模拟半空间基岩的辐射阻尼,并在该处以等效结点力的方式实现地震动输入。

计算模型见图 2,选取A点作为观测点。土体非线性材料本构模型选取邓肯-张模型(Duncan等,1970),土体线弹性参数见表 1杜修力等,2016),杜修力等(2016)未给出702震灾防御技术14卷配套的非线性参数,故算例中的非线性参数参考实际情况选取,后续研究中将使用更真实表现土体非线性行为的本构模型及真实工程场地参数。算例中的大气压参数取100kPa,内摩擦角增量取0°。入射地震动分别选取狄拉克脉冲和实测地震动(Gilroy Array #3,Coyote Lake, 1979)。入射狄拉克脉冲见图 3,观测点结果见图 4,实测地震动见图 5,观测点结果见图 6图 4图 6中给出采用中心差分法的计算结果作为参考解,由图 4图 6可知,本文算法与中心差分法计算结果吻合较好,说明本文算法的有效性。


图 2 大开车站沿线土层纵断面构造 Fig. 2 Site condition of the Daikai subway station in vertical direction
表 1 土层参数 Table 1 Parameters of soils

图 3 狄拉克脉冲速度和加速度时程图 Fig. 3 Velocity and acceleration time history of the Dirac pulse

图 4 狄拉克脉冲入射时场地反应分析结果 Fig. 4 Results of site analysis under the incident of Dirac pulse

图 5 实测地震动速度和加速度时程图 Fig. 5 Velocity and acceleration time history of the seismic motion

图 6 实测地震动入射时场地反应分析结果 Fig. 6 Results of site reaction analysis under the incident of the seismic motion

表 1ρcsvENRfcθ为模型参数,分别表示密度、剪切波速、泊松比、无量纲幂次、破坏比、土的内聚力、土的摩擦角。DF为试验常数。

3 结论

本文发展一种求解材料非线性结构动力学方程的显式时间积分算法,并应用于地震波竖直入射时非线性地震反应分析中,通过算例验证了该方法的有效性。该显式算法具有无需对角阻尼矩阵、单步、稳定性良好等优点。本文考虑了邓肯-张非线性弹性本构模型,下步研究可考虑将该显式算法扩展到弹塑性本构模型及更能反映土层真实变形的本构模型中。

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