Beam-spring Model Research Articles

Displacement-based Design of Shield Tunnel Lining Considering Mobilized Strength of Undrained Clay

This paper presents a displacement-performance design method for the shield tunnel based on the mobilized strength design (MSD) theory. Firstly, the nonlinear relationship between the ground volume loss caused by tunnel excavation and the earth pressure was constructed to obtain the real earth pressure on the tunnel lining. Secondly, the nonlinear relationship between the convergence displacement of the lining and the ground resistance was built to acquire the subgrade reaction working on the tunnel lining. Finally, the proposed method was verified by orthogonal numerical experiments and compared with the finite element method (FEM) results. Compared with the FEM, the average deviation of the bending moment and the axial force are about 9% and 1%. The average variation of the bending moment can be reduced to 1% after calibration. The results calculated by this developed method are better than the traditional beam-spring model. Its accuracy is close to the FEM based on the strata-structure concept. Therefore, the proposed method based on MSD theory in this paper has the characteristics of high efficiency and simplicity, which provides a new auxiliary design approach for the structural design of shield tunnels in clay areas.

Random analysis method for nonlinear interaction between shield tunnel and spatially variable soil

The spatial variability of soil properties is likely to have a remarkable influence on the longitudinal responses of shield tunnels. This paper proposes a one-dimensional random analysis method (RAM) for the interaction between the shield tunnel with circumferential joints and the spatially variable soil with nonlinear elastoplastic behaviors. A deterministic short beam-spring model with longitudinal variation in the initial subgrade reaction coefficient and the ultimate soil resistance is developed and solved by the finite difference method. The random field of the soil properties discretized by the Karhunen-Loeve expansion method is used to establish the RAM. A hypothetical example is further presented to demonstrate the applicability of the proposed method. The results imply that the soil nonlinearity may lead to a negative effect on the reliability control of the tunnel structure and should be considered to obtain more reasonable probabilistic evaluations of the tunnel deformations. The increasing joint rotational stiffness or shearing stiffness results in the lower variabilities of the maximum tunnel settlement, and reduces the probabilities of the maximum joint opening or joint dislocation exceeding the allowable value, respectively.

Comparative Analysis and Safety Evaluation of Shield Segment Structure Model under Surcharge Loading.

In shield tunneling projects, the selection of an accurate model to calculate the mechanical response of segment structure plays a crucial role in the design and cost of the project. The shell-spring and beam-spring models are two widely used methods for this purpose. However, it is still not clear how accurate and different these models are in calculation results under surcharge load. Therefore, to accurately calculate the internal forces and deformation of the segment structure and clarify the difference between the two models' results, the shell-spring and beam-spring models were established based on a subway shield tunnel project in Zhengzhou city. The reliability of the models was verified by comparing and analyzing the differences in deformation results between the models and field measurements. Furthermore, the safety of the segment structure was evaluated according to the ultimate bearing capacity of the normal section. The results declare that: (1) In the shell-spring model, the internal force gradually reduces from the edges towards the center of the segment width, and the shield segment exhibits a prominent non-plane strain state. (2) The internal force of the beam-spring model is larger than that of the shell-spring model. The axial force difference between the two models is relatively small; meanwhile, there is a larger disparity in the bending moment. However, with an increase in surcharge loading, the discrepancy in internal forces between the two models gradually decreases. (3) The calculation results of the shell-spring model are close to the field-measured values and the shield tunnel model test values, which verifies the accuracy and reliability of the shell-spring model. Therefore, it is more reasonable to use the shell-spring model to calculate the mechanical response of the segment structure. (4) With an increase in surcharge loading, the safety of the shield tunnel decreases gradually. Therefore, surcharge loading above the shield tunnel should be reasonably controlled to meet the requirements of the normal use of the shield segment. This manuscript aims to provide a reference for the future design and optimization of the shield tunnels' lining structure.

An analytical closed-form solution for the in-plane elastic modulus of plain-woven textile composites

Engineering application of textile composites needs simple relations for the estimation of their mechanical properties. Many homogenization and simplified models have been already proposed for the analytical characterization of the elastic moduli of this class of materials. Along this route, the current study introduces an analytical solution for the longitudinal elastic modulus of plain-woven textile composites. A hybrid beam-spring model of the representative unit cell (RUC) is postulated consisting of a horizontal beam connected to an inclined beam, two supporting springs beneath the free end of the horizontal beam and the common end of the horizontal and inclined beams, and a torsion spring at the free end of the inclined beam. The mechanical and geometrical properties of the beams and springs are calculated from the weave structure and the symmetry conditions. A reduction technique from the matrix analysis of structures is borrowed to consider the effect of internal degrees of freedom on the in-plane stiffness coefficient and a closed-form formula is extracted for the elastic modulus. The proposed analytical formula is finally verified against available experimental data and a wide range of FE predictions for the GFRP and CFRP plain-woven composites.

梁ばねモデルによる杭の水平抵抗解析に用いる水平地盤反力

In this study, evaluation methods of subgrade reaction for horizontal resistance analysis of piles using the beam-spring model were studied by simulation analysis of horizontal loading test results of pile. As a result, the analysis for pile head horizontal load obtained results on the safe side when the method of the guideline was used for an evaluation of subgrade reaction. On the other hand, the analysis using an evaluation method of coefficient of subgrade reaction proposed by the authors and an evaluation method of plastic subgrade reaction proposed by Kishida-Nakai produced results that generally corresponded to the measured values.

<b>Seismic Design Method for Shield Tunnels in Ground Conditions Subject to Change in the Longitudinal Direction</b>

The authors of this paper investigated the effect of seismic motion on shield tunnels that are located in areas with varying ground conditions. This research revealed that the larger the inclination angle of a boundary between a basic layer and a soft layer becomes, the larger the sectional forces, such as axial force, bending moment and shear force become. In addition, the sectional force obtained by an equivalent stiffness beam model was found to be larger than that obtained by a beam-spring model. This research also confirmed that results using a beam-spring model when the inclination angle is 5.9 degrees satisfy verification requirements.

Multiband topologically protected states realized by elastic honeycomb structures based on fundamental mechanical elements

Topological metamaterials with novel properties of topologically protected states have been developed vigorously inspired by the recent emergence of topological phases in electronic systems. In mechanical community, however, the vast majority of such investigations have either focused on discrete mass–spring systems, whose operating bands are single and are difficult to be transformed into practice, or on various carefully designed continuous elastic systems, suffering from the drawbacks of structural complexity and the presence of modal hybridization. In this study, we design an elastic system composed of clamped-free Euler–Bernoulli (EB) beams coupled via slender rods capable of supporting multiband uni-mode topological states, which overcomes the shortcomings of both discrete lattice and continuous structure to a great extent. Through a theoretical beam-spring model together with finite element analysis, we reveal that the symmetry of the underlying lattice structure and the nature of multi-order natural modes of the beam ensure the deterministic existence of multiple Dirac points. Splittings of these degeneracies induced by the mirror symmetry breaking give rise to two kinds of valley Hall phases characterized by opposite valley Chern numbers. Numerical simulations demonstrate that the domain walls formed by structures with distinct topological indices support elastic interface waves at multiple frequency ranges, which are topologically protected manifested by the features of being immune to backscattering of sharp corners and defects. In addition, we show that the frequency range of interest can be easily tuned by varying strength of symmetry breaking. Our results provide an entirely novel paradigm to design topological mechanical systems with multiple working frequencies. Moreover, thanks to the structural simplicity, the proposed elastic system can be exploited in practice as an excellent and versatile platform to explore other intriguing wave phenomena brought about by richer topological notions.

Analytical approach for segmental tunnels considering nonlinear longitudinal joints in soft soils

As a discontinuous ring structure, joints are the weakness of a tunnel lining, which have a significant influence on the tunnel deformations and internal forces. Since the deformations at the joints are discontinuous, joint behaviors are treated as nonlinear rotational springs. Besides, the tunnel-soil interactions are modelled as a series of radial and circumferential nonlinear Winker springs. Thereafter, an analytical approach based on the beam-spring model is proposed. The reliability of this method is verified with three cases available in the literatures. It indicates that the proposed method is reasonable to estimate the tunnel deformations. Furthermore, parametric studies are carried out to reveal the influence of the coefficient of earth pressure at-rest, the soil stiffness and local surcharge loadings on the tunnel responses. It has been shown that increasing the coefficient of earth pressure and soil stiffness is beneficial for reducing tunnel convergences. Under the earth pressures, cracks are most likely to appear at the crown, waist, and bottom of the tunnel. For the metro tunnel in Shanghai, when the relative local surcharge loading is larger than 0.3 (the corresponding convergence is about 14‰D, D is the tunnel external diameter), joints subjected to the negative moment are susceptible to reaching the ultimate state.

Multiband Pure Topological States in Elastic Structures

Inspired by notions of topological physics, recent years have witnessed the rapid development of mechanical metamaterials with novel properties of topological states. However, most of the current investigations have either focused on discrete mass-spring lattices, with topological states limited to a single operating band, or on various elaborate continuous elastic systems, enduring the drawbacks of modal couplings. It remains largely unexplored how to design topological elastic systems that naturally possess multiple operating bands and are free from modal couplings. In this study, we design an elastic system based on fundamental mechanical elements (beams, rods and nuts), which is capable of supporting multiband pure topological states. Through an equivalent beam-spring model with lumped masses together with finite element analysis, we demonstrate that our proposed structure exhibits multiple Dirac points (DPs) at different frequencies. We show that simply adjusting the heights of nuts fastened on beams can lift the degeneracies, giving rise to two kinds of valley Hall phases characterized by opposite valley Chern numbers. The dispersion diagram of the supercell formed by unit cells with different topological indices shows that there simultaneously exist perfectly pure interface modes (i.e., no other modes coexist) within two frequency ranges. Furthermore, numerical simulations demonstrate that the domain wall formed by structures with distinct topological properties supports topologically protected interface waves over dual frequency ranges. Our results have potential for the design of mechanical systems that need to work under changeable working frequencies and may have significant impact on many diverse fields such as vibration control, energy harvesting and seismic isolation.

Numerical Investigation of the Segmental Lining Performance for a Shield Tunnel

The performance of segmental linings in China's Yellow River Crossing Tunnel is studied as a case study, and a complete three-dimensional numerical model is developed for the segmental joint to perform a bending test and reproduce the joint rotational performance. Besides, an improved three-dimensional solid-spring model together with a straightforward two-dimensional beam-spring model are presented for the segmental lining to take the influences of the joint into account. The bending test shows that the rotation stiffness of joints presents complex and nonlinear characteristics. The segmental lining simulation reveals that the segmental joint has little impact on the axial forces of the lining. However, the bending rigidity of the segmental lining corresponding to the joint location is weakened, which leads to discontinuous stress distribution, decreased bending moments and increased deformations. According to the comparison, the proposed three-dimensional solid-spring model excluding contact has the advantage of mesh generation, and it is capable of reproducing the three-dimensional segmental lining effects. The proposed two-dimensional beam-spring model combined with the calculation methods of all the spring stiffnesses is an effective and further simplified method for the segmental lining modeling, which incorporates the influences of the joint and staggered format.

Algorithm for an Effective Ratio of the Transverse Bending Rigidity Based on the Segment Joint Bending Stiffness

An algorithm for calculating the effective ratio of the transverse bending rigidity is established based on the segment longitudinal joint bending stiffness. With the knowledge of this effective ratio, the bending rigidity of a modified uniform rigidity ring is fully defined. To verify this developed algorithm, the effective ratios and convergence deformations of the modified uniform rigidity rings obtained with different methods are compared. Moreover, the responses of the modified uniform rigidity ring model under loading obtained from this algorithm are compared to those obtained with the existing generally accepted beam-spring model. The results show that although the bending moments obtained from these two models are different, the axial forces, horizontal convergent deformations, and vertical convergent deformations are quite consistent with each other. The modified uniform rigidity ring model built on the developed effective ratio algorithm is applicable for the analysis of the tunnel convergence deformation and the interaction between the tunnel structure and the ground during operation. This modified uniform rigidity ring model is simpler and easier to use than the beam-spring model; thus, the significance of the developed algorithm for the effective ratio of the transverse bending rigidity is demonstrated.

The effect of geologic conditions on the fire behavior of tunnels considering soil-structure interaction

This paper compares the fire performance of bored reinforced concrete tunnel lining sections under four ground conditions, ranging from shallow soft soil to deep rock. The fire performance is analyzed using a verified finite element beam-spring model. The investigated lining sections are adopted from realistic high-profile tunnels. A series of analyses are conducted to evaluate the performance of tunnel sections under a wide range of fire scenarios and ground conditions. First, the fire behavior of the four tunnel sections under the RABT-train fire curve is simulated during heating and cooling, and the results are reported in terms of structural performance and sectional stress–strain response. Second, the time to reach a limit state for the four tunnel sections is quantified under an extended RWS fire curve. Finally, a simplified methodology is proposed to include temperature-dependent soil properties, including the heat induced excess pore pressure and change in subgrade reaction modulus under elevated temperatures. The influence of temperature-dependent soil properties on the structural fire performance of a shallow tunnel section in soft clay is studied. The results show that bored reinforced concrete linings could experience significant irrecoverable damage from exposure to major fires. The moderate-depth tunnel section in soft soil and the deep tunnel section in high lateral pressure rock are the two critical cases, due to the large deformations and high internal forces, respectively. The importance of including temperature-dependent soil properties in tunnel fire analyses is demonstrated.

Comparison of 3D Solid and Beam–Spring FE Modeling Approaches in the Evaluation of Buried Pipeline Behavior at a Strike-Slip Fault Crossing

Validated 3D solid finite element (FE) models offer an accurate performance of buried pipelines at earthquake faults. However, it is common to use a beam–spring model for the design of buried pipelines, and all the design guidelines are fitted to this modeling approach. Therefore, this study has focused on (1) the improvement of modeling techniques in the beam–spring FE modeling approach for the reproduction of the realistic performance of buried pipelines, and (2) the determination of an appropriate damage criterion for buried pipelines in beam–spring FE models. For this paper, after the verification of FE models by pull-out and lateral sliding tests, buried pipeline performance was evaluated at a strike-slip fault crossing using nonlinear beam–spring FE models and nonlinear 3D solid FE models. Material nonlinearity, contact nonlinearity, and geometrical nonlinearity effects were considered in all analyses. Based on the results, pressure and shear forces caused by fault movement and pipeline deformation around the high curvature zone cause local confinement of the soil, and soil stiffness around the high curvature zone locally increases. This increases the soil–pipe interaction forces on pipelines in high curvature zones. The beam–spring models and design guidelines, because of the uniform assumption of the soil spring stiffness along the pipe, do not consider this phenomenon. Therefore, to prevent the underestimation of forces on the pipeline, it is recommended to consider local increases in soil spring stiffness around the high curvature zone in beam–spring models. Moreover, drastic increases in the strain responses of the pipeline in the beam–spring model is a good criterion for a damage evaluation of the pipeline.

Active length of a continuous pipe or tunnel subjected to reverse faulting

Numerous researches were conducted to explore the response of a pipe or tunnel to fault-induced deformation, most of which concentrated on strain distribution and failure mechanism. In this study, both centrifuge and numerical modeling are performed to investigate the active lengths of stainless steel pipes and concrete tunnels subjected to reverse faulting. The active length is assumed as the part of a pipe or tunnel with different displacements from the surrounding soil. Two centrifuge tests of pipes are conducted to validate the three-dimensional beam-spring model. Parametric studies are performed to investigate the active length of a structure (continuous pipe or tunnel) in reverse fault and its influence factors. The results showed that the active length depends on fault deformation, size of the structure (diameter and diameter to thickness ratio), material properties of structure (Young's modulus and yielding stress), burial depth, and frictional angle of sand. The active length has a positive correlation with relative structure-soil stiffness. The relationship of active length and relative structure-soil stiffness is obtained by a comprehensive parametric study with 960 finite elements.

3D modelling of concrete tunnel segmental joints and the development of a new bolt-spring model

In a segmental metro tunnel, it is widely observed that cracks, water leakage and other structural defects usually appear at segmental joint section of a smaller stiffness than main tunnel segment section. In this study, a 3D finite element analysis was conducted to simulate a typical concrete segmental joint explicitly using 3D continuum elements, and the performance of 3D continuum model was validated against laboratory tests available in literature. Since such continuum model may cost excessive computational time when a large-scale tunnel structure is analysed, a new bolt-spring model was developed to simplify the structural features of concrete segmental joint. In the bolt-spring model, the interaction between bolt and segment was modelled by a set of normal springs and shear springs: the stiffness of the normal springs is mainly determined by the bolt itself, whilst the shear springs take account of bolt-segment friction, interaction and shear resistance of the bolt. In the meanwhile, the interaction between two adjoining tunnel segments is explicitly modelled using contact elements. The proposed bolt-spring model is able to simulate the details of joint deformation and contact pressure between segments more realistically than previously available by conventional methods (e.g. continuous ring, beam-spring model (BSM)), where segment-segment interaction is not explicated modelled. Compared to the continuum model, the bolt-spring model saves up to 90% computational time without compromising numerical accuracy. Furthermore, this paper compared the mechanical behaviour of a concrete joint against that of a cast-iron one with particular emphasis on the development of different bolt-spring models.

Influence of Rotational Stiffness Modeling on the Joint Behavior of Quasi-Rectangular Shield Tunnel Linings

A beam-spring model with constant rotational stiffness is a practical tool for the prediction of the general deformations and bending moments in circular tunnel linings. However, in reality, the rotational stiffness of a segmental joint is not constant, due to nonlinear deformations and local yielding in the vicinity of the joint. These are a result of the specific geometry at the joint, which is related to water-tightness measures and buildability issues. For quasi-rectangular tunnels this nonlinearity should not be neglected, as the bending component in the lining is significantly larger compared to circular linings. To date, there are only few studies that have investigated a calculation method for consideration of the joint’s nonlinear moment-axial force and shear-axial force interaction behavior and its consequences on the calculated lining behavior. In this paper, an iterative incremental method is proposed to tackle this issue, based on rotational stiffness curves derived from 3D nonlinear finite element modelling of the joints, and substantiated by testing. The significance of the variable rotational stiffness is highlighted through a comparison with results based on a constant stiffness assumption. Further, using the proposed calculation method, the effects of the circumferential joints, the bending moment transmission and several other parameters on the full-ring behavior of quasi-rectangular tunnels are discussed for a wide interval of design parameters. The results provide some new insights into the behavior of this non-traditional tunnel type. Although the presented results are related to specific overall and local geometries, the presented method is considered to be useful for the design of other special tunnel geometries.

Generalized beam–spring model for predicting elastic behavior of partially composite concrete sandwich wall panels

Partially composite sandwich wall panels (SWP) have been used in the construction industry for at least twenty years. Currently there is limited codified guidance for designers of partially composite concrete SWP, but they are being designed safely and routinely. Design is often guided by the composite connector manufacturers who sell proprietary composite connectors often using a variation of truss-type matrix methods for prediction of elastic behavior of SWPs. The purpose of this paper is to quantify the accuracy of such a model in a uniform manner by developing a generalized version of these matrix models called the beam-spring model (BSM). The primary benefits of the model are its simplicity and versatility making it suitable for design and variations of this model are already in use today. Its accuracy was established in this paper by comparing to 51 specimens from the literature and found to be similar to that of other methods related to strength and cracking of concrete members in the literature. These results indicate that this BSM, or the very similar methods that are already standard practice, provides an adequately safe elastic analysis. On average, the BSM predicted cracking load within 2% with a coefficient of variation (COV) of 7%, out-of-plane panel stiffness within 3% with a COV of 17%, and cracking deflection within 2% with a COV of 18%. These results also indicate that the United States precast industry’s use of 7.5fc' for predicting the cracking moment is acceptable for partially composite precast panels. A parametric study performed with the BSM presents several different design situations and can be used as a rudimentary design tool for preliminary behavior and arbitrary connectors.

Longitudinal beam response of concrete segmental linings simultaneously backfilled with bicomponent grouts

The longitudinal beam response of concrete segmental linings (CSLs) to modern shield tunnelling with tunnel boring machines (TBMs) can impair ring joint performance. The opening and shearing of ring joints may lead to groundwater ingress. The differential displacements between rings, owing to either longitudinal shear forces or the ovalisation of rings subjected to unevenly distributed longitudinal forces, can increase the risk of structural damage around ring joint connections. Sequential beam-spring models have been deployed in the past for the prediction of the longitudinal response of CSLs during construction. However, the calibration of the models was limited to CSLs simultaneously backfilled with mortar grouts and usually required the amendment of the lining bending stiffness to obtain good approximations to the field data. In this paper, a sequential beam-spring model based on Bogaards’ and Bakker’s (1999) approach is proposed for the construction loads and boundary conditions representative of CSLs simultaneously backfilled with bicomponent grouts (BGs). The model is applied to the Crossrail’s Thames tunnel (CTT) case study to provide the complete history of the CTT longitudinal behaviour near the instrumented rings during tunnelling and the residual response as a result of tunnelling. The calibration is conducted satisfactorily without the need for tweaking the lining bending stiffness. It is shown that the lining responds to the TBM steering around curves mainly through longitudinal bending and to cumulative uplift forces through shear. The long term longitudinal internal forces are highly dependent on the sequence of TBM moments and TBM transverse actions near the tail skin and may lead to deferred structural damage with the progressive loss of longitudinal tunnel prestressing. An assessment method for the early detection of tunnel sections susceptible to ring joint damage is proposed for asset owners of future tunnels.

Comparative Investigations on Internal Force Distribution Characteristics on Concrete Segments in Qiongzhou Strait Sub-Sea Tunnel

In the design of shield tunnel concrete segments, the calculation of internal forces on segment sections is an important part. In order to study the influence of combining calculation of water and soil pressure and separate calculation of water and soil pressure towards internal forces on segment sections, taking the Qiongzhou Strait sub-sea tunnel as the background, ANSYS software is used to establish the equivalent stiffness circle model, hinged model and beam spring model, and the internal forces on segment are calculated by two methods of combining calculation of water and soil pressure and separate calculation of water and soil pressure. The results show that the difference of the bending moment between the two methods is great while the difference of the axial force is small, the maximum bending moment of combining calculation of water and soil pressure is 2.2∼2.9 times larger than that of separate calculation of water and soil pressure, and the axial force is 11%∼16% larger than separate calculation of water and soil pressure.

Root branching affects the mobilisation of root-reinforcement in direct shear

The contribution of roots to the mechanical behaviour of soil has typically only been studied for the ultimate limit state. In these approaches, roots are typically modelled as straight and unbranched structures. This approach overlooks the fact that roots may have to deform significantly to mobilise their strength, a process that will be influenced by root architecture effects such as branching, amongst others. Sequential mobilisation of roots affects the peak root-reinforcement, thus differences in mobilisation are important to consider when quantifying root-reinforcement. In this paper, the effect of root branching was modelled using a large-deformation Euler-Bernoulli beam-spring model. The effect of soil was incorporated using non-linear springs, similar to p-y and t-z theory used for foundation piles. By connecting multiple beams together (i.e. applying appropriate linked boundary conditions at root connection points) the effect of branching could be analysed. A soil displacement profile corresponding with direct shear loading was then imposed and the response of the root analysed. It was shown that adding branches led to a quicker mobilisation of root-reinforcement. Branches increased the axial resistance to root displacement and changed the shape of the deformed roots. The presence of branching counteracted root slippage, and thus increased reinforcement. Larger branching densities increased this effect. This analysis demonstrated that the architecture of the root system has a strong effect on the mobilisation of root strength, which directly affects the maximum amount of reinforcement the roots will provide. Future modelling of root-reinforcement, both at the ultimate and serviceable limit state, should account for this effect.