Isogeometric Analysis of Longitudinal Displacement of a Simplified Tunnel Model Based on Elastic Foundation Beam

Serious uneven settlement of the tunnel may directly cause safety problems.At this stage,the deformation of the tunnel is predicted and analyzed mainly by numerical simulation,while the commonly used finite element method(FEM)uses low-order continuous elements.Therefore,the accuracy of tunnel settlement prediction is not enough.In this paper,a method is proposed to study the vertical deformation of the tunnel by using the combination of isogeometric analysis(IGA)and Bézier extraction operator.Compared with the traditional IGA method,this method can be easily integrated into the existing FEM framework,and ensure the same accuracy.A numerical example of an elastic foundation beam subjected to uniformly distributed load and an engineering example of an equivalent elastic foundation beamof the tunnel are given.The results show that the solution of the IGA method is closer to the theoretical solution of the initial-parameter method than the FEM,and the accuracy and reliability of the proposedmodel are verified.Moreover,it not only provides some theoretical support for the longitudinal design of the tunnel,but also provides a new way for the application and popularization of IGA in tunnel engineering.

Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation

The quasi-Green＇s function method is used to solve the free vibration problem of clamped thin plates on the Winkler foundation. Quasi-Green＇s function is established by the fundamental solution and the boundary equation of the problem. The function satisfies the homogeneous boundary condition of tile problem. The mode-shape differential equation of the free vibration problem of clamped thin plates on the Winkler foundation is reduced to the Fredholm integral equation of the second kind by Green＇s formula. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The numerical results show the high accuracy of the proposed method.

Temperature stress analysis for bi-modulus beam placed on Winkler foundation

The materials with different moduli in tension and compression are called bi-modulus materials. Graphene is such a kind of materials with the highest strength and the thinnest thickness. In this paper, the mechanical response of the bi-modulus beam subjected to the temperature effect and placed on the Winkler foundation is studied. The differential equations about the neutral axis position and undetermined parameters of the normal strain of the bi-modulus foundation beam are established. Then, the analytical expressions of the normal stress, bending moment, and displacement of the foundation beam are derived. Simultaneously, a calculation procedure based on the finite element method （FEM） is developed to obtain the temperature stress of the bi-modulus struc- tures. It is shown that the obtained bi-modulus solutions can recover the classical modulus solution, and the results obtained by the analytical expressions, the present FEM proce- dure, and the traditional FEM software are consistent, which verifies the accuracy and reliability of the present analytical model and procedure. Finally, the difference between the bi-modulus results and the classical same modulus results is discussed, and several reasonable suggestions for calculating and optimizing the certain bi-modulus member in practical engineering are presented.

BENDING OF RECTANGULAR THIN PLATES WITH FREE EDGES LAID ON TENSIONLESS WINKLER FOUNDATION

In this paper, the bending problem of rectangular thin plates with free edges laid on tensionless Winkler foundation has been solved by employing Fourier series with supplementary terms. By assuming proper form of series for deflection, the basic differential equation with given boundary conditions can be transformed into a set of infinite algebraic equations. Because the boundary of contact region cannot bedetermined in advance, these equations are weak nonlinear ones. They can be solved by using iterative procedures.

VIBRATIONS OF STEPPED RECTANGULAR THIN PLATES ON WINKLER’S FOUNDATION

Differential equations of free/forc ed vibrations of stepped rectangular thin plates on Winkler's foundation are estab lished by using singular functions, and their general solutions are also solved for expression of vibration mode function and frequency equations on usual suppo rts derived with W operator, as well as forced responses of such plates unde r different_type loads disc ussed with Fourier expansion of generalized functions.

VIBRATIONS OF STEPPED ONE-WAY THIN RECTANGULAR PLATES SUBJECTED TO IN-PLANE TENSILE/ COMPRESSIVE FORCE IN Y-DIRECTION ON WINKLER'S FOUNDATION

Differential equations of free/forced vibrations of n_step one_way thin rectangular plates subjected to in_plane tensile/compressive force in y_direction on Winkler's foundation are established by using singular functions, their general solutions solved for, expression of vibration mode function and frequency equation on usual supports derived with W operator. Influence functions for various cases deduced here may also be used to solve problems of static buckling or stability for beams and plates in relevant circumstances.

VIBRATIONS OF ONE-WAY RECTANGULAR STEPPEDTHIN PLATES ON WINKLER'S FOUNDATION

Differential equations of free forced vibrations of one -way rectangular stepped thin plated on Winkler 's foundation are established by using singular functions ,their general solutions are solved :exprssion of vibration mode function and frequency equations on usual supports are derived from W operator :forced responses of such plates under different -type loads are discussed with generalized functions .

Bending the Foundation Beam on Elastic Base by Two Reaction Coefficient of Winkler’s Subgrade

A new method for analysis of counter beams is presented in the paper. The analysis has taken into account their stiffness EI, Winkler’s space with modulus of subgrade reaction k and equality deformities of the foundation beam with the ground. The solution is found by using the numerical analysis of the Winkler’s model, with variation of different moduli of the subgrade reaction k2 outside the force zone r, while under the force P exists the modulus of the subgrade reaction k, up to the definition of minimum bending moments. The exponential function k2(r), as the geometric position of the minimum moments is approximately assumed. From the potential energy conditions of the reciprocity of displacement and reaction, the width of the zone r and the modulus of the subgrade reaction k2 are explicitly determined, introducing in the calculation initial and calculation soil displacement wsi successively. At the end of the paper, it presented numerical example in which the influence of k and k2 values on bending moments of the counter beam is analyzed. The essential idea of this paper is to decrease the quantity of the reinforcement in the foundations, beams, i.e. to obtain a cost-efficient foundation construction.

Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis

This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method. The analyses are worked out considering the First-order Shear Deformation Theory (FSDT) for the above mentioned moderately thick structural elements. The effect of the shell curvatures is included from the beginning of the theory formulation in the kinematic model. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The discretization of the system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. Comparisons between the present formulation and the Reissner-Mindlin theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs. Very good agreement is observed. Finally, new results are presented in order to investtigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the behavior of laminated shells of revolution.

Winkler地基梁动力学系统的无量纲化与参数识别

为简化Winkler地基梁动力学系统的双参数识别计算,提出一种新无量纲方法,通过对系统时间、空间坐标进行线性变换,实现动力学方程系数的彻底归一化,得到与系统参数解耦的广义频率方程,发现频率、频率比仅由无量纲梁长决定的本质.提出基于频率比互等关系的双参数识别算法,该算法通过对广义频率方程进行一次求解即可在相应边界条件下得到频率、频率比关于无量纲梁长的预解集,在得到该系统任意两阶实测频率后,即可依托于时间、空间还原系数所建立的线性转换关系实现对双系统参数的定解.较之于传统双参数识别算法,该算法具有两个特点:(1)识别计算仅涉及单变量超越方程的求解与线性转换,避免了双参数超越方程组的非线性迭代问题,可使识别计算得到有效简化.(2)任意系统参数值的变化仅影响时空、空间还原系数的大小,预解集具有适用于系统参数值任意变化的一般性,可有效避免因系统参数值改变而导致重复迭代的情况,实现了解的一般化.

Seismic response of bridge pier on rigid caisson foundation in soil stratum

An analytical method to study the seismic response of a bridge pier supported on a rigid caisson foundation embedded in a deep soil stratum underlain by a homogeneous half space is developed. The method reproduces the kinematic and inertial responses, using translational and rotational distributed Winkler springs and dashpots to simulate the soil-caisson interaction. Closed-form solutions are given in the frequency domain for vertical harmonic S-wave excitation. Comparison with results from finite element （FE） analysis and other available solutions demonstrates the reliability of the model. Results from parametric studies are given for the kinematic and inertial responses. The modification of the fundamental period and damping ratio of the bridge due to soil-structure interaction is graphically illustrated.

Analytical solution for evaluating deformation response of existing metro tunnel due to excavation of adjacent foundation pit

The excavation of foundation pit generates soil deformation around existing metro tunnel with shield driving method,which may lead to the deformation of tunnel lining.It is a challenge to evaluate the deformation of shield tunnel accurately and take measures to reduce the tunnel upward displacement as much as possible for geotechnical engineers.A new simplified analytical method is proposed to predict the longitudinal deformation of existing metro tunnel due to excavation unloading of adjacent foundation pit in this paper.Firstly,the additional stress of soils under vertical axisymmetric load in layered soil is obtained by using elastic multi-layer theory.Secondly,the metro tunnel is regarded as a Timoshenko beam supported by Winkler foundation so that the shear effect of tunnels can be taken into account.The additional stress acting on the tunnel due to excavation unloading in layered soil are compared with that in homogeneous soil.Additionally,the effectiveness of the analytical solution is verified via two actual cases.Moreover,parametric analysis is conducted to investigate the responses of the metro tunnel by considering such factors as the variation of subgrade coefficient,offset distance from the excavation center to tunnel longitudinal axis as well as equivalent shear stiffness.The proposed method can be used to provide theoretical basis for similar engineering project.

Influence of water-rich tunnel by shield tunneling on existing bridge pile foundation in layered soils

At present,shield tunneling often needs to pass through a large number of bridge pile foundations.However,there are few studies on the influence of shield tunneling on adjacent pile foundations by combining with groundwater seepage.Based on Winkler model,the calculation equations of shield tunneling on vertical and horizontal displacement of adjacent bridge pile are derived.Meanwhile,full and part three-dimensional finite element models are established to analyze the trend of bridge pier settlement,ground surface settlement trough,vertical and horizontal displacement of the pile and pile stress under three calculation conditions,i.e.,not considering groundwater effect,considering stable groundwater effect and fluid-soil interaction.The results show that the calculated value is small when the effect of groundwater is not considered;the seepage velocity of the soil above the excavation face is faster than that of the surrounding soil under fluid-soil interaction,and after the shield passing,the groundwater on both sides shows a flow trend of“U”shape on the ground surface supplying to the upper part of the tunnel;the vertical displacement of the pile body is bounded by the horizontal position of the top of the tunnel,the upper pile body settles,and the lower pile body deforms upward.The horizontal displacement of pile body presents a continuous“S”shape distribution,causing stress concentration near the tunnel.The calculated results of fluid-soil interaction are in good agreement with the field measured data and accord with the actual situation.

Calculation of foundation pit deformation caused by deep excavation considering influence of loading and unloading

A new analytical solution for ground surface settlement induced by deep excavation is proposed based on the elastic half space Melan’s solution,and the analytical model is related to the physical and mechanical properties of soil with the loading and unloading action during excavation process.The change law of earth pressure of the normal consolidation soil after the foundation pit excavation was analyzed,and elastic displacement calculation methods of analytic solution were further established given the influence of excavation and unloading.According to the change of stress state in the excavation process of foundation pit,the planar mechanical analysis model of the foundation excavation problem was established.By combining this model with the physical equations and geometric equations of plane strain problem with consideration of the loading and unloading modulus of soil,constitutive equation of the plane strain problem was also established.The loading and unloading modulus formula was obtained by using the parameter calculation method in Duncan-Chang curve model.The constitutive equation obtained from the model was used to calculate the soil stress state of each point to determine its loading and unloading modulus.Finally,the foundation pit displacement change after excavation was calculated,and thus the soil pressure distribution after retaining structure deformation.The theoretical results calculated by making corresponding programs were applied to engineering practice.By comparing the conventional calculation results with monitoring results,the practicability and feasibility of the calculation model were verified,which should provide a theoretical basis for similar projects.

集中力下的Winkler地基上双层弹塑性梁

为了完善弹性地基上弹塑性梁的理论基础,假设梁弯曲符合理想弹塑性模型,双层梁层间为竖向弹簧连接,根据受力平衡和本构关系建立Winkler地基上双层弹塑性梁的微分方程并给出通解.以无限长双层梁中心处作用集中荷载为例,研究同质同宽双层梁的加载全过程,分析梁厚、层间竖向模量等参数对梁极限承载力的影响.结果表明:当上、下层梁厚度相同或下层梁更薄时,不论层间竖向模量取多大都是上层梁先屈服;层间竖向模量趋近于无穷时,上下层梁的挠曲线相同,极限承载力趋近于双层叠合梁的极限承载力,此时结构退化成双层叠合梁;当层间竖向模量与地基反应模量之比大于5000时,上、下层梁竖向拉压变形对梁极限承载力的影响超过3%,应予以考虑,宜采用广义模量计入梁截面竖向拉压变形的影响.

The Stability of Cylindrical Shells Containing an FGM Layer Subjected to Axial Load on the Pasternak Foundation

In this study, the stability of cylindrical shells that composed of ceramic, FGM, and metal layers subjected to axial load and resting on Winkler-Pasternak foundations is investigated. Material properties of FGM layer are varied continuously in thickness direction according to a simple power distribution in terms of the ceramic and metal volume fractions. The modified Donnell type stability and compatibility equations on the Pasternak foundation are obtained. Applying Galerkin’s method analytic solutions are obtained for the critical axial load of three-layered cylindrical shells containing an FGM layer with and without elastic foundation. The detailed parametric studies are carried out to study the influences of thickness variations of the FGM layer, radius-to-thickness ratio, material composition and material profile index, Winkler and Pasternak foundations on the critical axial load of three-layered cylindrical shells. Comparing results with those in the literature validates the present analysis.

Influence of elastic foundations and carbon nanotube reinforcement on the hydrostatic buckling pressure of truncated conical shells

In this study,the effects of elastic foundations(EFs)and carbon nanotube(CNT)reinforcement on the hydrostatic buckling pressure(HBP)of truncated conical shells(TCSs)are investigated.The first order shear deformation theory(FOSDT)is generalized to the buckling problem of TCSs reinforced with CNTs resting on the EFs for the first time.The material properties of composite TCSs reinforced with CNTs are graded linearly according to the thickness coordinate.The Winkler elastic foundation(W-EF)and Pasternak elastic foundation(P-EF)are considered as the EF.The basic relations and equations of TCSs reinforced with CNTs on the EFs are obtained in the framework of the FOSDT and solved using the Galerkin method.One of the innovations in this study is to obtain a closed-form solution for the HBP of TCSs reinforced with CNTs on the EFs.Finally,the effects of the EFs and various types CNT reinforcements on the HBP are investigated simultaneously.The obtained results are compared with the results in the literature,and the accuracy of results is confirmed.

有限深度介质上梁非线性能量汇减振的Winkler地基实现

土-结构相互作用对结构动力学特性的影响是一种具有非线性能量汇特征的效应。利用考虑土体质量的Winkler地基梁理论,将有限深度弹性介质等效为非线性能量汇系统的附加质量,建立简谐激励下弹性介质上简支梁系统的非线性动力学模型。采用Galerkin方法和增量谐波平衡法分析了弹性介质上简支梁的非线性动力响应。利用数值计算方法验证了理论结果的正确性,并分析了非线性能量汇的有效性。通过参数优化和分析,揭示了不同参数范围内Winkler地基的减振效果,讨论了其最佳参数范围。研究结果表明:在合理的参数范围内,弹性介质对其支承梁的动力响应有良好的抑制作用,能够快速有效地吸收共振条件下的振动能量,并具有良好的鲁棒性。优化后的非线性能量汇可使梁的共振幅值降低超95%,且具有较宽的减振频带。研究成果从非线性能量汇的角度展现了土-结构相互作用效应的减振机理,为基于弹性地基设计的结构振动抑制提供了理论依据。

Vibration suppression of magnetostrictive laminated beams resting on viscoelastic foundation

The vibration suppression analysis of a simply-supported laminated composite beam with magnetostrictive layers resting on visco-Pasternak’s foundation is presented.The constant gain distributed controller of the velocity feedback is utilized for the purpose of vibration damping.The formulation of displacement field is proposed according to Euler-Bernoulli’s classical beam theory(ECBT),Timoshenko’s first-order beam theory(TFBT),Reddy’s third-order shear deformation beam theory,and the simple sinusoidal shear deformation beam theory.Hamilton’s principle is utilized to give the equations of motion and then to describe the vibration of the current beam.Based on Navier’s approach,the solution of the dynamic system is obtained.The effects of the material properties,the modes,the thickness ratios,the lamination schemes,the magnitudes of the feedback coefficient,the position of magnetostrictive layers at the structure,and the foundation modules are extensively studied and discussed.

Local buckling of thin plate on tensionless elastic foundations under interactive uniaxial compression and shear

This paper uses a mathematical method to develop an analytical solution to the local buckling behaviour of long rectangular plates resting on tensionless elastic Winkler foundations and under combined uniform longitudinal uniaxial compressive and uniform in-plane shear loads. Fitted formulas are derived for plates with clamped edges and simplified supported edges. Two examples are given to demonstrate the application of the current method： one is a plate on tensionless spring foundations and the other is the contact between the steel sheet and elastic solid foundation. Finite element （FE） analysis is also conducted to validate the analytical results. Good agreement is obtained between the current method and FE analysis.