Stresses in linings of bored tunnels

An analytical procedure is presented for evaluating the stresses in linings of bored tunnels caused by kinematic soil–lining interaction. Three cases of plane–strain interaction are treated as produced by (1) relaxation of in situ soil stresses near a lining following its installation, (2) overburden pressure at the soil surface, and (3) two-dimensional free-field soil response normal to the lining axis as produced by an earthquake. The procedure is applied separately to a steel lining and a concrete lining for a site located at the lower end of Market Street in San Francisco through which a BART (Bay Area Rapid Transit) tunnel passes. © 1998 John Wiley & Sons, Ltd.     

An approximate method for soil-structure interaction for sh-waves—the born approximation

An approximate method is proposed for the scattering of SH-waves by foundations of irregular shape and the resulting soil-structure interaction problems. The scattering of elastic waves by the rigid foundation embedded in half-space is solved approximately by using integral representation of the wave equation. The procedure is the Born approximation which has been widely used in quantum mechanics for collision and scattering theory though not well-known in elastodynamics. This paper extends the previous work of the authors on the scattering of waves to account for soil-structure interaction. The motion of the foundation is evaluated by the balance of momentum under stresses due to the incident waves as well as the waves generated by its own motion and the forces coming from the superstructure. The model investigated consists of an infinitely long elastic shear wall of height H and thickness h erected on a rigid infinitely long foundation. Results are presented for the cases with circular, elliptical and rectangular foundations. For a circular foundation, excellent agreement is found with the exact solutions for the foundation displacement and the relative displacement between the top and bottom of the structure for the entire range of wave numbers. For an elliptical foundation, accuracy decreases with increasing wave numbers. Foundation displacements are compared for foundation shapes that are shallow elliptical, deep elliptical, rectangular and circular. It is observed that foundation displacements are dependent on the angle of incidence except for a semi-circle. The results on the details of the scattered field are, however, not as accurate.     

Dynamic stiffness matrices for two circular foundations

A systematic procedure is presented for generating dynamic stiffness matrices for two independent circular foundations on an elastic half-space medium. With the technique reported in References 1–3, the analytic solution of three-dimensional (3D) wave equations satisfying the prescribed traction due to the vibration of one circular foundation can be found. Since there are two analytic solutions for two prescribed tractions due to the vibrations of two circular foundations, the principle of superposition must be used to obtain the total solution. The interaction stresses (prescribed tractions) are assumed to be piecewise linear in the r-directions of both cylindrical co-ordinates for the two circular foundations. Then, the variational principle and the reciprocal theorem are employed to generate the dynamic stiffness matrices for the two foundations. In the process of employing the variational principle, a co-ordinate transformation matrix between two cylindrical co-ordinate systems is introduced. Some numerical results of dynamic stiffness matrices for the interaction of two identical rigid circular foundations are presented in order to show the effectiveness and efficiency of the present method, and some elaborations for its future extensions are also discussed.     

Bridge pier scour mitigation under steady and unsteady flow conditions

Watercourse morphology is affected by local scouring when the flow interferes with anthropic structures. Controlling the scour hole size is of predominant importance to guarantee bridge safety as well as to limit the variations of river morphology. A combined countermeasure against bridge pier scour is proposed and tested in order to reduce the maximum scour depth and deviate it away from the bridge foundation. In the first part of the laboratory campaign, combination of two countermeasures (bed-sill and collar) was evaluated for a circular pier under clear-water and live-bed steady flow conditions. The proposed combined countermeasure exhibited an efficiency of about 64% in terms of scour depth reduction. Afterwards, it was tested in unsteady flow conditions, first for a circular pier, then in the case of a rectangular pier with round nose and tail, two circular in-line piers and two rectangular in-line piers, under a hydrograph with a peak flow velocity slightly above the threshold condition of sediment motion. Results showed that the combined countermeasure had an efficiency of about 63% for a single circular pier; however, higher efficiency (about 75%) was obtained in applications to rectangular pier and two in-line circular or rectangular piers.     

Dynamic response of twin circular tunnels due to incident SH-waves

The steady-state solution for two parallel under ground tunnels of circular cross-section subjected to incident plane harmonic SH-waves is obtained in closed form, by using the method of wave function expansion. The image technique is employed to account for the reflection of waves at the ground surface. Numerical studies are carried out to determine the influence of spacing between the tunnels on the shear stresses in concrete and steel linings.     

Effect of pier shape and pier alignment on the equilibrium scour depth at single piers

The equilibrium scour depth at uniform single bridge piers depends on a large number of variables,including the pier horizontal cross-section shape and its alignment angle towards the flow direction.The influence of these variables has been studied by only a few researchers,mostly,on the basis of tests that were far from approaching equilibrium.This experimental study aims at revisiting the influence of piers' shape and alignment on local scouring for length-width ratios smaller than or equal to 4,by increasing the experimental evidence.Fifty five long-duration laboratory tests were run under steady,clear-water flow,close to the threshold for initiation of sediment motion.Five pier shapes were considered:circular,rectangular square-nosed,rectangular round-nosed,oblong,and zero-spacing(packed) pile-groups;the tested skew-angles were 0°,30°,45°,60°,and 90°.It was concluded that i) the shape factor can be taken as 1.0,for rectangular round-nosed and oblong cross-section piers,and as 1.2,for rectangular squarenosed and packed pile-group cross-section piers,ii) the shape factor does not vary significantly with the duration of tests,this way confirming the robustness of the shape factors reported to date,iii) the effect of shape is present at skewed piers although the associated coefficients remain in the narrow range of1.0-1.2,and iv) for length-width ratios smaller than 4,the shape factor is of the same order of magnitude as the skew angle factor and should not be neglected.     

Boundary differential equation method: simplified dynamic soil stiffnesses for embedded rigid foundations

With a simplified model and Galerkin's weighted residual procedure, two simple differential equations of dynamic behavior of a bounded rectangular medium are established along the boundaries in the x- and y-direction in the medium. Solutions of these equations yield closed form expressions of soil stiffnesses for various cases of a partially embedded rigid foundation, including the stiffnesses per depth of foundation with rectangular base area and the stifnesses of strip foundation. The developed procedure provides the definition of the weight functions, which are used in Galerkin's method for weighted residual. In addition to these weight functions, their conjugators are also suitable for weight functions. When the soil depth is finite, the original weight functions fail to produce physically meaningful results in some frequency range but the conjugators do not fail at any frequencies. The developed equations to compute soil stiffnesses for embedded foundations are simple yet capable of calculating the responses close to those computed by the much more elaborated finite element method.     

Approximate evaluation of stresses in degraded tunnel linings

This paper uses numerical and analytical methods to examine the static and seismic response of tunnels with intact and degraded segmental concrete tunnel liners. Concrete degradation is simulated using a non-linear finite element (FE) model that accounts for soil-structure interaction and the non-linear stress–strain response of the soil and concrete. The non-linear FE model is used to calculate radial stresses in tunnel linings with local concrete delaminations and that are subject to both static and seismic loads. Then, the FE results are compared with an analytical solution for jointed tunnel linings in order to assess the accuracy of the solution for predicting stresses in degraded liners. The analyses and results presented in this paper illustrate a simple method for estimating and evaluating the effect of concrete degradation on the distribution of thrust and moment in segmental tunnel linings subject to either static or seismic loads.     

Dynamic behaviour of rigid foundations of arbitrary shape on a halfspace

An approximate method for computation of the compliance functions of rigid plates resting on an elastic or visco-elastic halfspace excited by forces and moments in all degrees of freedon is presented. The method is based on a Green's function approach. These functions are given for all degrees of freedom in form of well-behaved integrals. The numerical procedure is described and is used to evaluate the vertical, horizontal, rocking and torsion compliance functions of rectangular plates with side ratios 1 ≤ b/a ≤ 10 and non-dimensional frequency 0≤a₀≤10. It is shown how this method can be extended to problems concerning a linear visco-elastic halfspace and a halfspace with variable stiffness.     

Dynamic analysis of flexible rectangular fluid containers subjected to horizontal ground motion

In this paper, an analytical method is proposed to determine the dynamic response of 3‐D rectangular liquid storage tanks with four flexible walls, subjected to horizontal seismic ground motion. Fluid–structure interaction effects on the dynamic responses of partially filled fluid containers, incorporating wall flexibility, are accounted for in evaluating impulsive pressure. The velocity potential in which boundary conditions are satisfied is solved by the method of separation of variables using the principle of superposition. The impulsive pressure distribution is then computed. Solutions based on 3‐D modeling of the rectangular containers are obtained by applying the Rayleigh–Ritz method using the vibration modes of flexible plates with suitable boundary conditions. Trigonometrical functions that satisfy boundary conditions of the storage tank such that the flexibility of the wall is thoroughly considered are used to define the admissible vibration modes. The analysis is then performed in the time domain. Moreover, an analytical procedure is developed for deriving a simple formula that evaluates convective pressure and surface displacements in a similar rigid tank. The variation of dynamic response characteristics with respect to different tank parameters is investigated. A mechanical model, which takes into account the deformability of the tank wall, is developed. The parameters of such a model can be obtained from developed charts, and the maximum seismic loading can be predicted by means of a response spectrum characterizing the design earthquake. Accordingly, a simplified but sufficiently accurate design procedure is developed to improve code formulas for the seismic design of liquid storage tanks. Copyright © 2013 John Wiley & Sons, Ltd.     

Problem on the vibrations of plates

Summary In this paper, the problem on the vibrations of rectangular, elliptical and circular plates of non-isotropic material have been solved by using Galerkin's method. With the help of this method, the same problem for an isotropic circular plate has also been considered at the end.     

A simulation experiment for optimal design hyetograph selection

The aim of this work is to assess the accuracy of literature design hyetographs for the evaluation of peak discharges during flood events. Five design hyetographs are examined in a set of simulations, based upon the following steps: (i) an ideal river basin is defined, characterized by a Beta distribution shaped unit hydrograph (UH); (ii) 1000 years of synthetic rainfall are artificially generated; (iii) a discharge time‐series is obtained from the convolution of the rainfall time‐series and the UH, and the reference T‐years flood is computed from this series; (iv) for the same return period T, the parameters of the intensity–duration–frequency (IDF) curve are estimated from the 1000 years of synthetic rainfall; (v) five design hyetographs are determined from the IDF curves and are convolved with the discrete UH to find the corresponding design hydrographs; (vi) the hydrograph peaks are compared with the reference T‐years flood and the advantages and drawbacks of each of the five approaches are evaluated. The rainfall and UH parameters are varied, and the whole procedure is repeated to assess the sensitivity of results to the system configuration. We found that all design hyetographs produce flood peak estimates that are consistently biased in most of the climatic and hydrologic conditions considered. In particular, significant underestimation of the design flood results from the adoption of any rectangular hyetograph used in the context of the rational formula. In contrast, the Chicago hyetograph tends to overestimate peak flows. In two cases it is sufficient to multiply the result by a constant scaling factor to obtain robust and nearly unbiased estimates of the design floods. Copyright © 2007 John Wiley & Sons, Ltd.     

Elastk-plastic dynamic analysis of structures using known elastic solutions

A numerical method is shown to analyse the dynamic elastic-plastic responses of those structures with known elastic solutions. The displacement at one point at time t caused by a unit load applied at another point at zero time, called dynamic influence coefficient, is calculated from the known elastic solutions. Incremental plastic strain is accounted for by a set of additional incremental loads, so the stiffness matrix and the eigenvectors do not vary with time. From the incremental load including that caused by the incremental plastic strain, the displacement vs. time of the structure is obtained. This method is applied to simply supported beams with bilinear stress-strain relations with different strain-hardening rates and to a simply supported elastic-ideally plastic rectangular plate. This procedure can be extended to structures with no available known analytical elastic solutions. For these structures, the elastic solutions can be obtained by the finite element method.     

Topology of Rayleigh–Bénard convection and magnetoconvection in plane layer

ABSTRACT

The present study aims to link the dynamics of geophysical fluid flows with their vortical structures in physical space and to study the transition of these structures due to the control parameters. The simulations are carried in a rectangular box filled with liquid gallium for three different cases, namely, Rayleigh–Bénard convection (RBC), magnetoconvection (MC) and rotating magnetoconvection (RMC). The physical setup and material properties are similar to those considered by Aurnou and Olson in their experimental work. The simulated results are validated with theoretical results of Chandrasekhar and experimental results of Aurnou and Olson. The results are also topologically verified with the help of Euler number given by Ma and Wang. For RBC, the onset is obtained at Ra greater than 1708 and at this Ra, the symmetric rolls are orientated in/along a horizontal axis. As the value of Ra increases further, the width of the horizontal rolls starts to amplify. It is observed that these two-dimensional rolls are nothing but the cross-sections of three-dimensional (3D) cylindrical rolls with wave structures. When the vertically imposed magnetic field is added to RBC, the onset of convection is delayed due to the effect of Lorentz force on the thermal buoyancy force. The presence of 3D rectangular structures is highlighted and analysed. When the magnetically influenced rectangular box rotates about vertical axis at low rotation rates in magnetoconvection model, the onset of convection gets further delayed by magnetic field, which is in general agreement with the theoretical predictions. The critical Ra increases linearly with magnetic field intensity. Coherent thermal oscillations are detected near the onset of convection, at moderate rotation rates.     

Free vibration and transient response of thick and thin plates using the finite element method

An isoparametric quadrilateral plate bending element is used for the free and forced vibration analysis of both thick and thin plates. Plates of rectangular, circular and triangular planform are analysed and excellent results are obtained. The element performance is assessed by comparison with analytical solutions based on Mindlin's thick plate theory, three-dimensional elasticity solutions and solutions based on thin plate theory.     

Comparison between Shannon and Tsallis entropies for prediction of shear stress distribution in open channels

The concept of Tsallis entropy was applied to model the probability distribution functions for the shear stress magnitudes in circular channels (with filling ratios of 0.506, 0.666, 0.826), circular with flat bed (filling ratios of 0.333, 0.666), rectangular channel (1.34, 2, 3.94, 7.37 aspect ratios) and compound channel (with relative depths of 0.324, 0.46). The equation for the shear stress distribution was obtained according to the entropy maximization principle, and is able to estimate the shear stress distribution as much on the walls as the channel bed. The approach is also compared with the predictions obtained based on the Shannon entropy concept. By comparing the two prediction models, this study highlights the application of Tsallis entropy to estimate the shear stress distribution of open channels. Although the results of the two models are similar in the circular cross-section, the differences between them are more significant in circular with flat bed and rectangular channels. For a wide range of filling ratio values, experimental data are used to illustrate the accuracy and reliability of the proposed model.     

The dynamics of a rigid foundation on the surface of an elastic half-space

The response of a rigid rectangular foundation block resting on an elastic half-space has been determined by considering first the displacement functions for any position on the surface of an unloaded half-space due to a harmonic point force. The influence of the foundation has been taken into account by assuming a relaxed condition at the interface, i.e. a uniform displacement under the foundation and that the sum of the point forces must equal the total applied force. The three motions of vertical, horizontal and rocking have been considered and numerical values for the in-phase and the quadrature components of the displacement functions are presented for a Poisson's ratio of 0.25. The effect of the mass and inertia of the foundation can be allowed for by an impedance matching technique. Response curves and non-dimensional resonant frequency curves are given for a square and a rectangular foundation for different mass and inertia ratios and for several values of Poisson's ratio. These curves are for design purposes and are an addition to similar published curves for circular and infinitely long rectangular foundations. Some of the calculated results have been verified by a laboratory experiment.     

Fluid–structure interaction analysis of 3-D rectangular tanks by a variationally coupled BEM–FEM and comparison with test results

A variationally coupled BEM–FEM is developed which can be used to analyse dynamic response, including free-surface sloshing motion, of 3-D rectangular liquid storage tanks subjected to horizontal ground excitation. The tank structure is modelled by the finite element method and the fluid region by the indirect boundary element method. By minimizing a single Lagrange function defined for the entire system, the governing equation with symmetric coefficient matrices is obtained. To verify the newly developed method, the analysis results are compared with the shaking-table test data of a 3-D rectangular tank model and with the solutions by the direct BEM–FEM. Analytical studies are conducted on the dynamic behaviour of 3-D rectangular tanks using the method developed. In particular, the characteristics of the sloshing response, the effect of the rigidity of adjacent walls on the dynamic response of the tanks and the orthogonal effects are investigated. © 1998 John Wiley & Sons, Ltd.     

Equivalent mechanical models of sloshing fluid in arbitrary‐section aqueducts

This paper reports on a semi‐analytical/numerical method to model sloshing water in an arbitrarily shaped aqueduct. The water motion is assumed to be inviscid, compressible, and linear (small displacement). The transverse sloshing fluid in an aqueduct is equivalently simplified as a fixed rigid mass M₀ and a mass–spring system (M₁, K₁). According to a rule that the actual fluid (computed with finite element model) and its equivalent mechanical model have the same first sloshing frequency and acting effects on the aqueduct, the analytical solutions of the fixed (impulsive) mass M₀, sloshing (convective) massM₁, spring stiffness K₁, and their locations in the aqueduct body are acquired by the least squares (curve fitting) algorithm. Applying this equivalent principle, the equivalent mechanical models are respectively obtained for the sloshing water in rectangular, semicircular, U‐shaped, and trapezoid aqueducts. The equivalent principle and fluid models are validated through comparison investigations involving rectangular and U‐shaped aqueducts. The dynamic properties and seismic responses of the original and equivalent systems are simulated, compared, and discussed for a U‐shaped aqueduct bridge. The main purpose of this paper is to provide a simplified model of sloshing fluid for the seismic/wind‐resistant computation of the support structures of the aqueduct bridge. Copyright © 2011 John Wiley & Sons, Ltd.     

Curve‐Number/Green–Ampt mixed procedure for streamflow predictions in ungauged basins: Parameter sensitivity analysis

A procedure combining the Soil Conservation Service‐Curve Number (SCS‐CN) method and the Green–Ampt (GA) infiltration equation was recently developed to overcome some of the drawbacks of the classic SCS‐CN approach when estimating the volume of surface runoff at a sub‐daily time resolution. The rationale of this mixed procedure, named Curve Number for Green–Ampt (CN4GA), is to use the GA infiltration model to distribute the total volume of the net hyetograph (rainfall excess) provided by the SCS‐CN method over time. The initial abstraction and the total volume of rainfall given by the SCS‐CN method are used to identify the ponding time and to quantify the hydraulic conductivity parameter of the GA equation. In this paper, a sensitivity analysis of the mixed CN4GA parameters is presented with the aim to identify conditions where the mixed procedure can be effectively used within the Prediction in Ungauged Basin perspective. The effects exerted by changes in selected input parameters on the outputs are evaluated using rectangular and triangular synthetic hyetographs as well as 100 maximum annual storms selected from synthetic rainfall time series. When applied to extreme precipitation events, which are characterized by predominant peaks of rainfall, the CN4GA appears to be rather insensitive to the input hydraulic parameters of the soil, which is an interesting feature of the CN4GA approach and makes it an ideal candidate for the rainfall excess estimation at sub‐daily temporal resolution at ungauged sites. Copyright © 2013 John Wiley & Sons, Ltd.