Analytical solutions for Euler‐Bernoulli Beam on Pasternak foundation subjected to arbitrary dynamic loads

In this paper, the dynamic response of an infinite beam resting on a Pasternak foundation and subjected to arbitrary dynamic loads is developed in the form of analytical solution. The beam responses investigated are deflection, velocity, acceleration, bending moment, and shear force. The mechanical resistance of the Pasternak foundation is modeled using two parameters, that is, one accounts for soil resistance due to compressive strains in the soil and the other accounts for the resistance due to shear strains. Because the Winkler model only represents the compressive resistance of soil, comparatively, the Pasternak model is more realistic to consider shear interactions between the soil springs. The governing equation of the beam is simplified into an algebraic equation by employing integration transforms, so that the analytical solution for the dynamic response of the beam can be obtained conveniently in the frequency domain. Both inverse Laplace and inverse Fourier transforms combined with convolution theorem are applied to convert the solution into the time domain. The solutions for several special cases, such as harmonic line loads, moving line loads, and travelling loads are also discussed and numerical examples are conducted to investigate the influence of the shear modulus of foundation on the beam responses. The proposed solutions can be an effective tool for practitioners. Copyright © 2017 John Wiley & Sons, Ltd.     

Load–Deflection response of transversely isotropic piles under lateral loads

In general, pile materials are assumed to be isotropic during the analysis of the load–deflection response of piles under lateral loads. However, commonly used materials such as reinforced concrete and timber as well as potentially promising new pile materials such as fiber reinforced polymers are typically transversely isotropic materials. Experimental studies have shown that transversely isotropic materials have a high ratio of section longitudinal modulus to the section in‐plane shear modulus (E_zz/G_xz) compared to the value for isotropic materials. The high modulus ratio leads to a more significant shear deformation effect in beam bending. To account for the shear deformation effect, the Timoshenko Beam Theory has been adopted in deriving the solutions for the load–deflection response of transversely isotropic piles under lateral loads instead of the Classical (Euler–Bernoulli) Beam Theory. The load–deflection responses depend on the shear effect coefficient, the lateral soil resistance, the embedment ratio, and the boundary conditions. The deflection of the pile, if the shear deformation effect is considered, is always larger than if it is neglected. Copyright © 2000 John Wiley & Sons, Ltd.     

Simplified uncertainties analysis of continuous buried steel pipes on an elastic foundation in the presence of low stiffness zones

In conventional design, continuous buried steel pipes are designed based on their transverse behavior where low stiffness zones of soil, uncertainties in soil and structure properties are not considered. Winkler’s analytical model and the FOSM method are used to estimate uncertainties in terms of the coefficient of variation in the differential settlement (CV_Δ) and the bending moment (CV_M) as the function of the uncertainties in the subgrade reaction modulus (k_s) and the low stiffness zone length beneath the pipe. Results show that the values of CV_Δ and CV_M are very different depending on the uncertainties of k_s and the length of the low stiffness zone.     

钢筋混凝土梁在低速撞击下的动力响应

研究分析了在半无限粘弹性Winkler地基上的钢筋混凝土梁受低速刚性体撞击的塑性动力响应问题。认为低速撞击下梁的计算必须考虑撞击时局部损伤的影响,研究梁结构整体破坏,从而建立了局部损伤和整体变形的关系式,推导出了钢筋混凝土梁在低速中心撞击时弹塑性阶段撞击力和撞击处的横向位移的计算公式。     

Time-Variant Reliability Analysis of Unsaturated Soil Slopes Under Rainfall

Time-variant reliability analysis for a typical unsaturated soil slope is performed. Eight rainfall conditions are considered, and three slope models are set up for studying the influence of shear strength parameters, hydraulic conductivity parameters, rainfall intensity and duration on the reliability of the soil slope. Sensitivity analysis shows that when the saturated hydraulic conductivity (k _s) is very small, the variation of hydraulic conductivity has little effect on the reliability index (β). For saving the computation effort, only the shear strength parameters are needed in performing the reliability analysis in this condition. With the increase of k _s, the importance of hydraulic conductivity becomes large. The reliability index of the soil slope is changing with time (t), and the shape of β–t curves for different slope model is quite different for they depend on the value of k _s. When k _s is very small, β keeps decreasing for all the 18 simulation days. With the increase of k _s, β decreases to its minimum value at about the cessation day of rainfall events, and it then increases gradually due to the redistribution of suction in the soil slope.     

Load–settlement behavior of rock-socketed drilled shafts using Osterberg-Cell tests

Osterberg-Cell (O-Cell) tests are widely used to predict the load–settlement behavior of large-diameter drilled shafts socketed in rock. The loading direction of O-Cell tests for shaft resistance is opposite to that of conventional downward load tests, meaning that the equivalent top load–settlement curve determined by the summation of the mobilized shaft resistance and end bearing at the same deflection neglects the pile-toe settlement caused by the load transmitted along the pile shaft. The emphasis is on quantifying the effect of coupled shaft resistance, which is closely related to the ratios of pile diameter to soil modulus (D/E_s) and total shaft resistance to total applied load (R_s/Q) in rock-socketed drilled shafts, using the coupled load-transfer method. The proposed analytical method, which takes into account the effect of coupled shaft resistance, was developed using a modified Mindlin’s point load solution. Through comparisons with field case studies, it was found that the proposed method reasonably estimated the load-transfer behavior of piles and coupling effects due to the transfer of shaft shear loading. These results represent a significant improvement in the prediction of load–settlement behaviors of drilled shafts subjected to bi-directional loading from the O-Cell test.     

Analysis of coupled axial and lateral deformation of roots in soil

Plant roots can help to stabilise slopes. Existing analytical models to predict their mechanical contribution are however limited: they typically focus on the ultimate limit state, employ various empirical factors, and ignore much of the underlying root-soil interaction. A new model was developed based on large deflection Euler-Bernoulli elastic beam theory that can be used to study the mobilisation of root strength under various loading conditions (direct shear and pull-out). Both lateral and axial loading of the root by the soil were incorporated, based on existing methodologies for foundation piles (p-y and t-z curves). The model is able to take the key parameters into account (root biomechanical properties, root architectural properties, and soil properties) while remaining quick to solve using a numerical boundary value problem solver. The model was compared with experimental direct shear test data using various root analogues (rubber, plastic, and wood) in dry sand with various densities and effective stress levels and was able to accurately predict the measured shear force-displacement behaviour. Comparison with experimentally measured pull-out force-displacement curves using rubber and wooden root analogues with various architectures in dry and partially saturated sands was also satisfactory. In the future, this model can aid with addressing long-standing problems in the root-reinforcement community: quantifying the effect of (sequential) mobilisation of root strength in direct shear, the effect of the angle at which the root crosses a shear plane, the effect of root topology on root-reinforcement or the effect of root bending, and root shear shear forces on root-reinforcement.     

Non‐linear consolidation analysis of soil with variable compressibility and permeability under cyclic loadings

In this paper, a simple semi‐analytical method has been developed to solve the one‐dimensional non‐linear consolidation problems by considering the changes of compressibility and permeability of the soil layer, subjected to complicated time‐dependent cyclic loadings at the ground surface. The solution presented here takes into account e ～ lg k_v and e ～ lg σ′ linear responses. With c_k the slope of the e ～ lg k_v line and c_c as the slope of the e ～ lg σ′ line, the identified parameter c_c/c_k is found to control the rate of consolidation. Using the solutions obtained, some diagrams are prepared and the relevant behaviours of one‐dimensional non‐linear consolidation of saturated soft soil under cyclic loadings are discussed. The method in this paper does not require any special data; conventional oedometer data can be used. Therefore, the method is particularly efficient and convenient for engineering practice. Copyright © 2006 John Wiley & Sons, Ltd.     

桩基承台梁受力分析的幂级数解答

将桩基承台梁视为置于弹性地基上的有限长梁,将竖向桩体及承台梁下桩间土体视为刚度不同的弹簧系列,基于Winkler弹性地基梁理论,推导出考虑桩土共同工作的承台梁竖向位移控制微分方程,并给出其幂级数半解析解,进而导得了在集中荷载、外加弯矩及分布荷载共同作用下桩基承台梁的竖向位移、转角、弯矩及剪力的计算公式。最后通过与链杆法、Newmark法的比较,验证了本文幂级数解答的正确性。在此基础上,探讨分析了基桩差异性、承台梁下土体作用、桩径及荷载形式等因素对桩基承台梁受力变形的影响。研究表明：当考虑上述因素影响时,桩基承台梁的竖向变形、弯矩及桩顶反力均发生不同程度的变化,因此,在实际的设计计算中应予以考虑。     

Using the Maximum Effective Moment Criterion to Interpret Quartz <c>‐Fabric Patterns

The Maximum Effective Moment (MEM) criterion predicts that the initial orientation of ductile shear zones and shear bands is ~55° relative to the maximum principal stress axis (σ1) and that the kinematic vorticity number (Wk) is ~0.94. These preferred orientations should be reflected in the pattern of quartz -fabrics in shear zones and shear bands. Common quartz -fabrics in plane strain can be divided into low-temperature (L) and high-temperature (H) fabrics, with each group showing three patterns. A steady flow with a constant value of Wk≈0.94 gives rise to L-1 and H-1 patterns, which are commonly characterized by a single axis girdle normal to the shear zone and a single -point maximum parallel to the shear zone.Once the conjugate set develops, L-1 and H-1 have opening angles of ~70° and ~110°, respectively. L-2 and H-2 are asymmetric patterns associated with variable deformation partitioning and vorticity values of 0< Wk<0.94. In contrast, L-3 and H-3 are symmetric patterns associated with 100% deformation partitioning and Wk=0. The opening angle in quartz -fabrics is implicitly linked to the temperature during deformation. The opening angle is ~70° at low temperature and ~110° at high temperature. However, a linear correction between the opening angle and the temperature cannot be established. During deformation partitioning, synthetic shear bands form earlier than antithetic bands and are more easily developed. This may result in opening angles of <70° for low-temperature fabrics and of >110° for high-temperature fabrics. The following criteria can be used to recognize reworked shear zones that have experienced multiple orogenic phases and changes in the stress state: 1) the initial Wk is larger or smaller than ~0.94; 2) the change in Wk is abrupt, rather than progressive; 3) inconsistent shear senses are inferred for the different phases of deformation; and 4) a negative value of Wk is found in reworked shear zones.     

地表临时堆载诱发下既有盾构隧道纵向变形分析

违规临时地表堆载将引起地层附加应力,对既有盾构隧道产生不利的影响,严重者将导致隧道结构破坏。现有方法多是将隧道简化为搁置于Winkler地基的Euler-Bernoulli梁,不能考虑隧道的剪切变形和隧道埋深对基床反力系数的影响。针对既有研究的不足,提出考虑隧道剪切效应和隧道埋深的地表堆载下既有盾构隧道变形和受力的简化解析解。将既有盾构隧道简化为搁置于Winkler地基的Timoshenko梁,地基反力系数考虑隧道埋深的影响。通过三维有限元模型和已发表工程案例的实测数据,验证所提方法的正确性及适用性。通过参数分析发现,在荷载中心与隧道中心距离较近情况下,浅埋盾构隧道将发生较大的沉降变形;提高等效抗弯刚度和基床反力系数可以减少隧道沉降变形。而增大等效剪切刚度对隧道的沉降变形贡献较小,但是可以明显减小管片之间的错台变形。该研究成果可为合理预测地表堆载对既有盾构隧道的影响提供一定的理论支持。     

边坡框架梁文克尔简化计算方法及工程应用

为了改进边坡框架梁的计算方法,基于文克尔地基梁假定,考虑框架梁与土体的互相作用,通过关键点拟合的方法,把指数函数和三角函数简化为一元二次函数,推导出边坡框架梁的简化计算公式。该方法被应用于福建一处边坡支护工程,采用锚索框架梁支护,施工过程中对边坡进行位移监测。研究结果表明,传统的文克尔地基梁计算需要求解微分方程,计算过程比较复杂,不适合设计中使用,而简化后的计算方法容易掌握,可以方便计算出框架梁的端部弯矩和跨中弯矩。边坡位移监测结果表明该简化计算方法安全可行。研究结果对边坡框架梁的工程设计有一定的参考作用。      

双层地基水平受荷桩受力变形分析

基于双层地基中的水平受荷桩的特性,对其受力变形进行了分析。将水平受荷桩视为竖直放置的弹性地基梁,基于Winkler弹性地基梁理论,考虑桩土共同工作得到水平受荷桩位移控制微分方程及其幂级数解答,进而根据内力与位移的连续条件得到了由桩顶受力及变形条件表示任一深度处桩身的水平位移、转角、弯矩及剪力的计算矩阵表达式。通过一具体算例将幂级数解计算结果与《公路桥涵地基与基础设计规范》推荐的简化计算公式计算结果进行了比较。结果表明：当第1层地基土的厚度在某一定值时,《规范》简化计算方法所得结果与幂级数解接近;但当层厚不在该值附近时,两个方法计算结果存在差异。     

隧道开挖引起邻近单桩水平反应分析

隧道开挖会降低邻近桩基承载力,如何更为合理评价桩基水平附加响应是需要解决的问题。基于Pasternak双参数地基模型和三折线弹塑性荷载传递模型,采用两阶段分析法,并考虑侧向土体作用及地基土层的非均质特性,提出了更符合实际的单桩水平反应简化分析方法。通过与Winkler地基梁法及边界元法的对比分析,验证了方法的合理性。结合对单桩水平反应的多种影响因素进行参数分析,通过各因素相应的修正系数来对基准工况中单桩最大水平反应进行修正,得到计算工况中单桩的最大水平位移和最大弯矩。分析结果表明,桩基水平位移计算时可忽略侧向土体作用,而弯矩计算时应予以考虑;桩基计算工况的最大水平位移 最大弯矩 与平均地层损失比 呈现线性关系,而与隧道半径R、隧道轴线深度H、桩距隧道中心线距离x及桩身柔度系数 均呈现非线性关系。     

Method of initial parameters for laterally loaded piles embedded in layered soils

The method of initial parameters (MIP) was originally developed to solve the problem of a beam on an elastic foundation with applied concentrated forces along the span, which introduce discontinuities in the mathematical formulation of the problem. MIP is modified in this paper so that it can be used for solving the problem of a laterally loaded pile with discontinuities due to soil layering along the length of the pile. In this paper, the basis of MIP is outlined, and its use to find the deflection, slope, bending moment and shear force of laterally loaded piles is illustrated. Example problems of laterally loaded piles embedded in multi-layered soil media are provided.     

水平受荷长桩弹塑性计算解析解

当考虑桩侧土体非线性本构关系时对水平受荷桩的计算一般需采用数值方法,解析结果相对较少。基于Winkler地基模型和桩侧土体简化的弹塑性本构关系,对均质地基中水平荷载作用下桩头嵌固的长桩进行了解析推导,得到了桩身最大挠度及最大弯矩与荷载关系的统一解析表达式,并采用相同的方法求得高桩情形下桩头挠度的计算式。计算表明,联合荷载作用下桩身泥面处的挠度和转角不等于单个荷载作用时的线性叠加,采用常规的线性叠加法计算将偏于不安全。所求解析式借助计算器即可进行最大挠度和最大弯矩的计算,大大方便了工程的计算应用。     

Analysis of laterally loaded piles with rectangular cross sections embedded in layered soil

An analysis is developed to determine the response of laterally loaded rectangular piles in layered elastic media. The differential equations governing the displacements of the pile–soil system are derived using variational principles. Closed‐form solutions of pile deflection, the slope of the deflected curve, the bending moment and the shear force profiles can be obtained by this method for the entire pile length. The input parameters needed for the analysis are the pile geometry and the elastic constants of the soil and pile. The new analysis allows insights into the lateral load response of square, rectangular and circular piles and how they compare. Copyright © 2007 John Wiley & Sons, Ltd.     

Axial kinematic response of end‐bearing piles to P waves

Kinematic pile–soil interaction under vertically impinging seismic P waves is revisited through a novel continuum elastodynamic solution of the Tajimi type. The proposed model simulates the steady‐state kinematic response of a cylindrical end‐bearing pile embedded in a homogeneous viscoelastic soil stratum over a rigid base, subjected to vertically propagating harmonic compressional waves. Closed‐form solutions are obtained for the following: (i) the displacement field in the soil and along the pile; (ii) the kinematic Winkler moduli (i.e., distributed springs and dashpots) along the pile; (iii) equivalent, depth‐independent, Winkler moduli to match the motion at the pile head. The solution for displacements is expressed in terms of dimensionless transfer functions relating the motion of the pile head to the free‐field surface motion and the rock motion. It is shown that (i) a pile foundation may significantly alter (possibly amplify) the vertical seismic excitation transmitted to the base of a structure and (ii) Winkler moduli pertaining to kinematic loading differ from those for inertial loading. Simple approximate expressions for kinematic Winkler moduli are derived for use in applications. Copyright © 2013 John Wiley & Sons, Ltd.