Elastic Foundation Beam Research Articles

Application and Effect Evaluation of Elastic Foundation Beam Method in Deformation Analysis of Underground Passage Underneath Excavated Tunnel

Tunneling under existing underground buildings has become a common issue in densely populated urban areas. The current elastic analysis for ground displacement from new tunnel construction focuses on circular tunnels. However, theoretical analysis for non-circular chamber excavations is lacking. This paper aims to develop an elastic analysis for ground deformation from non-circular chamber excavations using the ‘equivalent radius’ theory, Verruijt’s formula, and Park’s model. It adjusts the ellipticization parameter δ and uniform radial displacement ε in Verruijt’s formula. Additionally, equivalent stiffness values for diverse existing hollow structures are considered using various approaches. A comparison of the theoretical results from field measurements and simulations shows the modified equations predict ground deformation well. Finally, parametric analysis explores the impacts of non-circular cavern excavations on existing structures under different factors.

Deformation and Fracture Mechanisms of Thick Hard Roofs in Upward Mining Coalfaces: A Mechanical Model and Its Validation

The safety and efficiency of underground coal mining are threatened by thick hard roofs characterized by large overhang areas, problematic spontaneous caving, and high dynamic load upon their breakage. In this study, a mechanical model of the bearing capacity of thick hard roofs in upward mining coalfaces associated with mining activities is built based on bending theories for beams with single generalized displacement and the elastic foundation beam theory. Using this method, we analyze the deformation and fracture mechanisms of a thick hard roof during upward mining. We further derive the mechanical equations of rotational angle, bending moment, shear force, and deflection of the free overhang and coal-bearing zone in the thick hard roof and an equation for calculating the limiting span. The mechanical behaviors of the thick hard roof bearing state are analyzed under different parameters. The results show that the foundation coefficient, roof thickness, and angle of upward mining have little influence on the roof bending moment but are positively correlated to the limiting span. Roof load and overhang length have a significant influence on the roof bending moment. They are negatively and positively correlated with the limiting span, respectively. Finally, a case study is performed on the Ш601 upward mining coalface in the Zhuzhuang Coal Mine. The distribution characteristics of the bending moment of the thick hard roof at different extraction stages are analyzed. At each stage, the limiting spans of the thick hard roof upon breaking were calculated as 13.18, 18.82, and 22.50 m, respectively, being close to the on-site measured periodic weighting lengths of 13.33, 19.33 m, and 22.67 m. This close fit proves the feasibility and accuracy of the developed mechanical model. The present study offers theoretical guidance for estimating the weighting length of thick hard roofs in coalfaces and for engineering technology control in similar scenarios.

Modeling mechanical behavior of buried pipe suffering from frost-heaving force based on the Winkler Elastic Foundation Beam Theory

Modeling mechanical behavior of buried pipe suffering from frost-heaving force based on the Winkler Elastic Foundation Beam Theory

Experimental investigation on shear fatigue behavior of perfobond strip connectors made of high strength steel and ultra-high performance concrete

This paper explores the fatigue performance of perfobond strip connectors (PBL) made from high-strength steel (HSS) and ultra-high performance concrete (UHPC) by fabricating and testing eight push-out specimens. The failure modes, load-slip curves, stiffness degradation, and residual strength of the PBLs are presented and discussed. The experimental results show that fatigue loads caused connector’s crack formation and stiffness degradation, leading to concrete dowel shear fracture and transeverse rebar bending-shear rupture damage. Compared to the static performance, the fatigue residual strength of the PBL subjected to 0.5 × 106, 1.25 × 106, and 2.0 × 106 loading cycles decreases by 1.2 %, 3.6 %, and 14.5 %, respectively, with corresponding reductions in ductility of 14.7 %, 19.4 %, and 4.7 %. Under the same loading cycles, the fatigue residual strength and ductility of the PBL with a maximum repeated load of 0.65Pu are 13.0 % and 3.9 %, respectively, smaller than those of 0.55Pu. A semi-empirical model based on the elastic foundation beam theory and linear cumulative damage model is developed to predict the residual strength of the PBLs using HSS and UHPC. The accuracy and applicability of the model are calibrated using the experimental results. The findings of this study can serve as an experimental foundation and theoretical reference for the fatigue design of PBLs made with high-strength materials.

Analysis of the underlying shield tunnel deformation caused by a newly built channel

As the development of underground space progresses, newly constructed waterways inevitably intersect with existing tunnels, posing significant challenges to the safety of underlying tunnels and making deformation control a critical issue in engineering design and construction. This study investigates the impact of waterway foundation pit excavation on the deformation of underlying tunnels. The research considers not only the vertical unloading at the bottom of the pit but also the horizontal unloading from the four side walls, which induces additional vertical stress fields in the underlying soil. These additional stresses are applied to the tunnel structure, modeled as a Pasternak elastic foundation beam, and the vertical tunnel deformation due to excavation unloading is calculated using the finite difference method. To validate the analytical results, a three-dimensional numerical model is employed. Furthermore, the study analyzes deformation trends under varying excavation sizes and the relative positions of the foundation pit and tunnel. The findings demonstrate that the proposed method provides a highly accurate, cost-effective, and efficient solution for predicting tunnel deformation during the excavation of small foundation pits. This research offers valuable insights for the design and assessment of underground structures in urban environments.

Subsidence Prediction Method Based on Elastic Foundation Beam and Equivalent Mining Height Theory and Its Application

Grouting technology in overburden separation is recognized as an effective method to prevent surface subsidence and reuse solid waste. This study used mechanical analysis to explore deflection characteristics of key strata and accurately predict and control surface subsidence. Conceptualizing the coal–rock mass beneath the key strata as an elastic foundation, we developed a method to calculate the elastic foundation coefficients for various regions and established an equation for key strata deflection, validated through discrete element numerical simulations. This simulation also examined subsidence behavior under different grout injection–extraction ratios. Additionally, combining the equivalent mining height theory with the probability integral method, we formulated a predictive model for surface subsidence during grouting. Applied to the 8006 working face of the Wuyang Coal Mine, this model was supported by numerical simulations and field data, which showed a maximum surface subsidence of 546 mm at a 33% injection–extraction ratio, closely matching the theoretical value of 557 mm and demonstrating a nominal error of 2%. Post-grouting, the surface tilt was reduced to below 3 mm/m, meeting regulatory standards and eliminating the need for ongoing surface structure maintenance. These results confirm the model’s effectiveness in forecasting and controlling surface subsidence with grouting. The study can provide a basis for determining the grouting injection–extraction ratios and evaluating the effectiveness of surface subsidence control in grouting into overburden separation projects.

A sandwich-type Winkler-Pasternak double elastic foundation beam model for analysis of nut column load transfer systems of shiplifts

The rack and pinion shiflifts feature its capability of fortification against extreme accidents of unbalance of ship chamber. This requires the load transfer systems of nut columns to be able to transfer the huge and centralized loads from the safety mechanisms to the tower columns in a distributive bearing way. Aiming at exploring a design-oriented computation method for the nut column load transfer systems (NCLTSs), a sandwich-type semi-infinite coupling Winkler-Pasternak double elastic foundation beam model (DEFBM) has been established, with a distributive assumption of the axial constrain force on the interfaces between the nut columns, mortar beds, adjusting beams and the second stage concrete based on the physical model test, has been proposed. A set of formulars for describing the distribution of deflections, internal forces and stress of nut columns and adjusting beams have been derived. A case study of the Three-Gorges shiplift has been conducted to demonstrate the approach of calculation of internal forces and stress by applying the formulas based on sandwich-type Winkler-Pasternak DEFBM, showing the convenience and rationality of the method to parametric layout design of the nut columns and adjusting beams in NCLTSs. Not only to NCLTSs and rack load transfer systems of shiplifts, the sandwich-type Winkler-Pasternak DEFBM also supplies the reference to the modeling and analysis of the rail-embedment structural systems broadly applied in hydraulic engineering.

Analytical Solution for Longitudinal Seismic Responses of Circular Tunnel Crossing Fault Zone

ABSTRACTThis paper proposes a simplified analytical solution for longitudinal seismic responses of a circular tunnel crossing a fault zone under longitudinally propagating shear waves. The transmissions and reflections of shear waves at two geological interfaces between the fault zone and intact rock are considered when calculating the free‐field displacement. An improved elastic foundation beam model considering different tangential contact conditions at the tunnel‒rock interface is also adopted. According to the continuous conditions at the two geological interfaces, explicit expressions for the tunnel displacement, bending moment, and shearing force are given. The effectiveness of the proposed analytical solution is validated via numerical simulations, and the importance of accounting for tangential contact conditions at the tunnel‒rock interface is emphasized. Moreover, parametric studies are performed to investigate the effects of the fault zone width, rock conditions, tunnel lining stiffness, tangential contact conditions, and earthquake frequency on the deformation and internal forces of tunnels subjected to seismic waves. This novel analytical solution can be utilized to quickly estimate the longitudinal seismic responses of circular tunnels crossing fault zones subjected to longitudinally propagating shear waves, particularly in the preliminary engineering design, and can be extended to geological conditions with multiple interfaces.

Shear behavior of multi-row perforated GFRP rib shear connectors in GFRP-concrete hybrid beams/desks

Perforated GFRP ribs (PFRs) and epoxy bonding are commonly used shear connections ensuring a full composite action in FRP-concrete hybrid beam/deck. This study demonstrates test results of 27 push-out specimens to compare the failure modes, shear load-slip (P-S) curves, and shear performance of PFRs and epoxy bonding. The variables included the sanding depth on GFRP profiles (0.5 mm/1.0 mm) and the rows of PFRs (1/2/3/5). The push-out tests highlighted that the PFR specimens with 0.5 mm or 1.0 mm sanding depth experienced interface debonding failure and shearing failure of PFR, respectively. The epoxy bonding specimens with 0.5 mm or 1.0 mm sanding depth suffered adhesive failure and cohesive failure, respectively. Increasing the sanding depth from 0.5 mm to 1.0 mm is an effective approach to prevent the debonding failure. The typical shear load-slip (P-S) curves for PFRs exhibit the micro-slipping stage (slip≤0.2 mm) and the slipping stage (slip>0.2 mm). The P-S curves of epoxy bonding showed only the elastic stage. The shear capacity of the double-row PFR connector was close to epoxy bonding. The shear capacity and stiffness of the PFRs increased 37.2∼47.3 % and 99.7∼119.7 % with the sanding depth improved from 0.5 mm to 1.0 mm, respectively. Increasing the sanding depth would improve the shear stiffness of epoxy bonding. The shear capacity and stiffness of PFRs decreased with the rows of PFRs increased. Calculation equations for the shear capacity of epoxy bonding and PFRs were evaluated by comparing the test results collected from the literature and calculated results. Based on the elastic foundation beam theory, a calculation equation for the shear stiffness of PFRs was proposed and a theoretical bi-linear model was suggested to predict the load-slip curves.

Slip modulus of inclined screw connectors in glued laminated bamboo and wood-concrete composite beams

ABSTRACT The existing research has predominantly focused on the shear behavior of inclined screws in wood-concrete composite beams. However, there is a significant research gap regarding the slip modulus of inclined screw connectors in glued laminated bamboo and wood (GLBW)-concrete composite beam. To address this gap, nine groups of shear tests were performed to study the load-slip behavior of inclined screw connectors in GLBW-concrete composite beams. This study analyzed the impacts of screw embedding length, screw diameter, screw embedding angle, and bamboo board thickness on the slip modulus of inclined screw connectors. The results showed that in comparison to inclined screw connectors with a 90 mm embedding length, the slip modulus of screw connectors with 110 and 130 mm embedding lengths increased by 11% and 19%, respectively. As the screw embedding angle decreased, the slip modulus of inclined screw connectors increased. For bamboo thicknesses of 30, 60, and 300 mm, the slip modulus of inclined screws was improved by 40%, 78%, and 89%, respectively, compared to the screws in pure wood beam. This study presented an analytical model for predicting the slip modulus of inclined screws in GLBW-concrete composite beam based on the layering method and elastic foundation beam theory.

Improved calculation method for corresponding characteristics of foundation pit excavation on the diaphragm wall

The excavation of the pit causes displacement of the surrounding soil, and the excessive deformation causes damage to the existing support structure, which in turn affects the safety of the pit. Reasonable calculation of structural deformation and internal force is crucial for design and construction. Most of the existing theoretical methods simplify the diaphragm wall(DW) as an Euler-Bernoulli beam acting on the Winkler foundation, consider beam-soil interaction, and simplify the soil as isotropic and continuous. However, the shear effects due to the differential deformation of the structure, the unloading stresses acting on the structure due to foundation excavation, and the discontinuous nature of the multilayered soil are neglected. In this paper, an improved analysis method is proposed based on the elastic foundation beam theory. The DW is simplified as a Timoshenko beam and the foundation is simplified as a Vlasov two-parameter model, and a proposed model considering the shear effect of the DW and the interaction of adjacent springs is established, and the proposed method for the deformation and internal force of the DW is obtained by the finite-difference method. The correctness and applicability of the proposed method are verified by numerical simulation and field monitoring data. The effects of equivalent bending stiffness, equivalent shear stiffness, soil elastic modulus, and excavation depth on the deformation and internal force of the DW were further analyzed. The results show that the proposed method can accurately solve the deformation and internal force of the DW, and the maximum errors between the proposed method and the numerical simulation results are only 4.5 % and 1.3 %, respectively. The equivalent bending stiffness of the DW and the elastic modulus of the soil have more significant effects on the horizontal deformation and internal force. The excavation depth is more sensitive to the deformation of the DW, and there is an exponential decay trend between the two. When the equivalent shear and bending stiffnesses reach 6.8×107 kN·m2 and 2.9×107 kN/m, the effect on the horizontal deformation is no longer obvious. The proposed method in this paper can accurately calculate the internal force and deformation of the DW.

Study on the Impact of Deep Foundation Excavation of Reclaimed Land on the Deformation of Adjacent Subway Tunnels

The objective of this research is to investigate the characteristics of the deformation response in adjacent subway tunnels caused by deep foundation excavation of reclaimed land. Focusing on a deep foundation excavation project situated in proximity to Line 11 of the subway in Shenzhen, this study employs theoretical analysis, numerical simulation, and on-site measurements to thoroughly investigate the deformation issues induced by the unloading of the excavation. The research results are as follows: using the energy method to calculate the uneven deformation of adjacent subway tunnels caused by the excavation can overcome the limitations of traditional algorithms, which treat the subway tunnel as a uniformly elastic foundation beam, resulting in more reasonable calculation results. Increasing the self-stiffness (EI)eq of the tunnel can effectively reduce the maximum displacement (wmax) of the tunnel, and as (EI)eq increases, its “weakening effect” on wmax gradually diminishes. Underground continuous walls can effectively control tunnel deformation, with tunnel displacement decreasing as the thickness and concrete strength of the continuous walls increase. “Long excavation” deep foundation excavations can impact the displacement and uplift range of the tunnel, with the maximum tunnel displacement showing a nonlinear decrease with increasing excavation depth. Tunnel displacement decreases as geotechnical parameters (elastic modulus E, internal friction angle φ, and cohesion C) increase, with the elastic modulus being the most sensitive parameter. The research findings can be applied to tunnel construction, maintenance, and safety evaluations, providing valuable references for similar engineering projects in the future.

Research on the Correction of Soil Pressure Distribution Pattern in Circular Working Wells in Soft Soil Areas

This paper investigates the distribution characteristics of earth pressure on circular caisson foundations in soft soil areas based on practical engineering considerations. Utilizing a three-dimensional numerical analysis model and a displacement-based method for earth pressure calculation, the lateral earth pressure on the retaining structure is examined. Adjustments are made to the distribution pattern of earth pressure on circular caisson foundations in soft soil areas. The study demonstrates that the radius of the caisson significantly influences the spatial distribution of soil pressure. Additionally, empirical formulas for the depth and radius of the reduction in soil pressure around circular caisson foundations are obtained through the research. Comparative analysis reveals that the modified earth pressure planar foundation beam method, which accounts for the arching effect, is simpler, features more mature parameter selection, and is more cost-effective than the three-dimensional finite element method. In comparison to the standard planar elastic foundation beam method for earth pressure, it aligns more closely with the actual direction of displacement.

Analysis of mechanical response and failure characteristics of tunnels crossing active faults considering axial force and geometrical nonlinearity

Tunnels subjected to active fault dislocation may experience significant damage. This paper establishes a novel methodology and solution procedure for analyzing the mechanical response and failure characteristics of tunnels subjected to active fault dislocation based on an elastic foundation beam model. The proposed methodology includes axial, transverse, and vertical soil-tunnel interaction terms, in addition to geometrical nonlinearity and axial force terms in the governing equation. This approach has a significantly extended application range, effectively addressing the problems encountered in tunnels crossing active faults with diverse crossing angles, dip angles, and fault types. The proposed methodology is verified by comparison to a 3D FEM model with various fault types, experimental tests, and on-site case, and the results are in excellent quantitative and qualitative agreement with the numerical, experimental, and on-site results. When fault displacement is below 0.5 m, disregarding geometric nonlinearity results in calculation errors of approximately 10% to 17% for peak axial force (Nmax), 18% to 22% for peak shear force (Vmax), and 20% to 30% for peak bending moment (Mmax). Finally, the responses caused by different factors, i.e., fault type, fault displacement, tunnel stiffness, and tunnel diameter, are investigated in detail to better understand tunnels crossing active faults. The results show that amongst the various fault types, the Vmax and the Mmax experienced by tunnels subjected to oblique slip-fault dislocation surpass those of other fault types, accompanied by the most extensive failure range. The augmentation of fault displacement, tunnel stiffness, and tunnel diameter precipitates a corresponding escalation in the Nmax, Vmax, Mmax, and failure range. Under oblique-slip fault dislocation, the tunnel undergoes an initial phase of shear failure, followed by tension-bending failure, delineated by distinct fault displacement thresholds of 0.1 m and 0.2 m, respectively. The proposed methodology provides the advantage of reliable stability analysis and design of tunnels crossing active faults.

Waveform conversion characteristic analysis of elastic waves in a hull grillage structure

Waveform conversion is closely related to energy transmission and redistribution in a coupled structure. In this research, a modified spectral element method (SEM) is used to analyze the characteristics of waveform conversion of finite-size coupled hull grillage based on the Timoshenko elastic foundation beam theory. Based on a theoretical derivation, the energy transfer coefficient of the coupling point and the energy density of the energy efferent and afferent structure are calculated by adding a non-reflecting boundary in an SEM program to research the parameter effects on waveform conversion. This research provides useful guidance for structural design for vibration reduction at a high frequency.

Research on the widths of protective coal pillars in main roadway based on the damaged foundation beam model

Long-term stability of the protective coal pillars in coal mines can ensure normal and safe use of roadways; in turn, the integrity of the roadway plays an important role in safe and efficient mining in coal mines. The initial damage and rheological damage caused by coal-seam excavation disturbance have important influences on the selection of reasonable widths of the protective coal pillars. Based on damage mechanics and the theory of elastic foundation beams, a coupled damage constitutive relationship of rock-like materials is proposed in this work by considering the influence of threshold and residual strength to establish the mechanical model of the elastic damage foundation beam. By theoretical analysis and numerical calculation, the differential equation of the deflection curve of the beam is solved, the influence range of the advance abutment pressure under the condition of coupling damage is obtained, and the width of the protective coal pillar is determined. The results show that compared with the elastic foundation beam model, after considering the coupling damage of coal, the influence range of the advance abutment pressure increases with time over a certain duration. At the end of this duration, the influence range of the advance abutment pressure remains almost unchanged. For the case considered in this paper, the long-term stability of the roadway is guaranteed when the width of the coal pillar is 120 m. The potential interference or change related to the specific situation of the site can be studied by changing the relevant parameters in the model to study different working conditions, or the damage of the roof rock beam during the mining process of the working face can be considered, which will be the subject of further research in the future.

Investigating the influence of excavating a tunnel undercrossing an existing tunnel at zero distance.

In urban areas with limited underground space, the new tunnel construction introduces additional loads and displacements to existing tunnels, raising serious safety concerns. These concerns become particularly pronounced in the case of closely undercrossing excavation at zero-distance. The conventional elastic foundation beam model, which assumes constant reaction coefficients for the subgrade, fails to account for foundation loss. In this study, the existing tunnel is modeled as an Euler-Bernoulli beam supported by the Pasternak elastic foundation, and the foundation loss caused by zero-distance undercrossing excavations is considered. Furthermore, an analytical solution is proposed to evaluate the mechanical response in segments, by establishing governing differential equations and boundary conditions for the excavation and neutral zones, and underpinning loads are also considered. The analytical solution is validated in two case studies. Finally, a parametric analysis is performed to explore the influence of various parameters on the mechanical response of the existing tunnel.

Mitigating interfacial stresses in strip-bonded beams by appending end-anchored unbonded segments

Bonding strips is widely used for the strengthening of structures. This paper addresses the challenge of stress concentration in the strip-strengthened beams, where interfacial stresses at the end of the strip are usually large enough to causes premature brittle debonding failure. Lengthening bonded strips and thickening adhesive layer are effective solutions to decrease the stresses but sometimes difficult to implement. For example, obstacles under the beam may stop the strip from being lengthened with bonding. This paper proposes a feasible solution for this case that is extending the strip without bonding and then fixing it with additional anchors at the end. Analytical expressions are derived for interfacial stresses in strip-bonded beams with the end-anchored unbonded segment (EAUS). The deformation coordination relationship between EAUS and the beam is considered. The anchor bolts are analogous to elastic foundation beams to obtain their shear stiffness which will greatly influence the tension at the strip end. The theoretical results are then verified by finite element simulations. Moreover, parameter studies are performed to explore the effects of various strengthening configurations such as the length of the bonded and unbonded segments on interfacial stresses. The investigations show that the presence of the EAUS could substantially alleviate the concentration of interfacial stresses. The findings of this study provide a new approach on how to control stress concentration in the strip-strengthened beams.

An Improved Winkler Foundation Modulus for a Beam in a Full Space

The Winkler foundation modulus is key to evaluating the response of underground structures using the elastic foundation beam model. In this paper, an improved formula of the Winkler foundation modulus for a beam embedded in a full space is proposed to overcome the limitation of inconsistent assumptions in previous studies. To achieve this goal, the bending responses of the beam are obtained using the elastic foundation beam model and three-dimensional elastic continuum model, respectively, wherein a consistent assumption is proposed that tangential interactions at the beam–ground interface are ignored in the two models. In addition, as deformation of the site is an important source of the underground structure response, the beam is applied to standard soil displacement of the free field on the Winkler foundation to improve the accuracy of the Winkler modulus obtained by fitting solutions based on the concentrated force on the beam. The formula for the Winkler foundation is obtained by equating the first zero of the bending moment in the two models. The Winkler foundation modulus is verified by comparing the results with numerical solutions and previous studies.

State-of-the-art on the anchorage performance of rock bolts subjected to shear load

AbstractRock bolts are extensively utilized in underground engineering as a means of offering support and stability to rock masses in tunnels, mines, and other underground structures. In environments of high ground stress, faults or weak zones can frequently arise in rock formations, presenting a significant challenge for engineering and potentially leading to underground engineering collapse. Rock bolts serve as a crucial structural element for the transmission of tensile stress and are capable of withstanding shear loads to prevent sliding of weak zones within rock mass. Therefore, a complete understanding of the behavior of rock bolts subjected to shear loads is essential. This paper presents a state-of-the-art review of the research progress of rock bolts subjected to shear load in three categories: experiment, numerical simulation, and analytical model. The review focuses on the research studies and developments in this area since the 1970s, providing a comprehensive overview of numerous factors that influence the anchorage performance of rock bolts. These factors include the diameter and angle of the rock bolt installation, rock strength, grouting material, bolt material, borehole diameter, rock bolt preload, normal stress, joint surface roughness and joint expansion angle. The paper reviews the improvement of mechanical parameter setting in numerical simulation of rock bolt shear. Furthermore, it delves into the optimization of the analytical model concerning rock bolt shear theory, approached from the perspectives of both Elastic foundation beam theory coupled with Elastoplasticity theory and Structural mechanic methods. The significance of this review lies in its ability to provide insights into the mechanical behavior of rock bolts. The paper also highlights the limitations of current research and guidelines for further research of rock bolts.