Dimensionless Charts Research Articles

Analytical solutions of stresses and displacements for deeply buried twin tunnels in viscoelastic rock

A set of analytical solutions is presented to calculate the stresses and displacements generated when two closely located circular tunnels are sequentially excavated in viscoelastic rock. The solutions are provided for circular tunnels excavated in time dependent rock for any type of linear viscoelastic model, e.g. Burgers and Poyting-Thomson model. In the presented solutions the sequential excavation of the tunnels is also accounted for. The radii of the twin tunnels are time-dependent functions reflecting the excavation process adopted.The solutions are provided as analytical expressions in integral form. These were obtained by extending the principle of correspondence to solid media with time varying boundaries. A comparison of the stresses and displacements predicted by the analytical solutions and FEM analyses for an example case of twin tunnels excavated in a generalized Kelvin medium shows a good agreement between the two methods. Then, a parametric analysis was performed to investigate the influence of the tunnel spacing on displacements and stresses for various excavation processes. Several dimensionless charts summarizing the result of the parametric analysis are provided for the benefit of practitioners.

Active and passive arching stresses in c′-ϕ′ soils: A sensitivity study using computational limit analysis

Vertical soil arching, commonly known as the “trapdoor mechanism,” is a pervasive phenomenon in various geotechnical applications that can be evaluated through a variety of analytical and numerical approaches, most of which exist due to a variety of proposed arching mechanisms, many of which are focused on either purely frictional or cohesive soils. This study investigates the realized arching mechanisms and associated loads for trapdoors under both active and passive arching conditions in c′-ϕ′ soils through a series of dimensionless charts using both upper and lower bound limit analyses. An associated sensitivity analysis demonstrates that arching loads are highly dependent on collapse mechanism, a function of not only geometry, but soil shear strength, with cohesion affecting the realized mechanism and arching loads.

Numerical study of contact stresses under foundations resting on cohesionless soil: Effects of foundation rigidity and applied stress level

The effects of foundation rigidity and applied stress on the distributions of contact stresses beneath circular foundations resting on sandy soil were investigated in this paper. Three foundation cases were considered in this study; one case considered rigid foundations behavior with Kr value of 1 (Kr is the relative soil/foundation stiffness index) and two cases considered non-rigid foundations behavior with Kr values of 0.05 and 0.005. The built-in material models of the Finite Element (FE) program ABAQUS were used to capture the stresses at the contact surfaces between the soil and the foundation at different applied loading levels (from 5%-100% of the ultimate bearing capacity of the foundation) for the three foundation cases. The FE analysis results indicated that the contact stresses (qc) are strongly dependent on the relative soil/foundation stiffness index (Kr) and the ratio (q*) of the applied stress on the foundation (q) relative to the ultimate bearing capacity of the foundation (qu). For a rigid foundation, the contact stress distribution was found to have either a parabolic shape at the ultimate (q* = 1) with peak contact stress of 1.4q or a saddle shape for q* < 1 with peak contact stress that ranges from 1.3–1.5 q. For a non-rigid foundation, contact stresses distribution has a saddle shape at all q* values with a non-zero stresses at the edges and peak stress that ranges from 1.1–1.38 q. The numerical simulations results were then interpreted and presented in form of dimensionless charts, which in turn, will enable the estimation of the contact stress at any point within the contact area between the foundation and the soil.

Load-Settlement Mechanism of Shallow Foundations Rested on Saturated Sand with Upward Seepage

AbstractIn this study, the ultimate bearing capacity and settlement of square and strip foundations in the presence of upward seepage were assessed using physical modeling. The experiments were conducted in two series with different loading values of excess pore-water-pressure paths. The experimental results demonstrate significant degradation of bearing capacity resulting from excess pore-pressure development; however, a remarkable amount of soil strength was observed when the excess pore-pressure ratio was unity in the free-field ground. A correction coefficient (fγ) for the bearing-capacity factor Nγ was estimated based on the results. This factor is a function of upward-seepage magnitude. Based on the experimental results, dimensionless charts are presented to predict the settlement of square and strip footings rested on saturated sand with varied amounts of excess pore-water pressure. Results show strong dependency of settlement on the bearing-capacity safety factor presumed for the foundation design.

Evaluation of Energy Dissipation Using Different Drop Broken‐Back Culverts Under Pressure Flow Conditions

Hydraulic jumps formed in broken‐back culverts were investigated experimentally by using energy dissipation devices. Additional examination measured the reduction in scour downstream of different‐height broken‐back culverts by forming a hydraulic jump inside the culverts. A broken‐back culvert model in the laboratory represents 150 ft in the field. The results were analyzed in terms of the inlet Froude number and different drop heights of broken‐back culverts. The calculated efficiency of energy dissipation in the culverts ranged between 55 and 90%. The dimensionless charts were created on the basis of data collection showing the relationship between the different relative drops and optimal sill height, optimal sill location, total head losses, optimal end sill, Froude number, and efficiency of hydraulic jump. The objective of this research was to obtain the maximum energy dissipation in different drop heights using analysis of dimensionless relationship charts.

Control of Seepage through Earth Dams Based on Pervious Foundation Using Toe Drainage Systems

Many dangerous effects arise from seepage through earth dams based on pervious layer. Therefore, the dam embankment must be provided with seepage control measures to avoid such effects. In the present work, different control methods were used such as flat slopes, toe drainage systems, and a catch drain in the tail water. The hydraulic performance of each control measure was evaluated using the analytical solutions, previously developed, to estimate the seepage quantity (q), the height of seepage surface (h3), and the coordinates of the free surface (hx). Study was conducted on a physical model for a dam embankment having a top width (b) = 10.0 meter, height (Hd) = 30.0 meter, and slope factor (m) = 1.5. The obtained results were analyzed and presented in dimensionless charts. Results showed that, the used control measures possess a great effect on the characteristics of seepage through earth dams based on pervious foundations. A comparative study was conducted between the studied toe drainage systems to enable the designers the better choice for design purposes.

On dimensionless loading parameters for close-in blasts

Close-range blasts pose a threat through severe damage to structures and injury or death. In this work, the spatial and temporal descriptions of a localised blast load are presented using 6 non-dimensional parameters. These are found to be solely functions of the charge stand-off distance to diameter ratio for a cylindrically-shaped charge. Numerical simulations of a localised blast are performed using AUTODYN, where the pressure variation on a rigid barrier for various charge stand-off/diameter combinations is obtained. The least-square regression is then utilised to obtain the relationship between stand-off/diameter ratio and dimensionless loading parameters. The relevant expressions and dimensionless charts are presented. The proposed equations are verified by comparing experimental data with numerical results obtained by finite element analysis (FEA) of blast loaded steel plates (using the user-defined subroutine VDLOAD implemented in the FEA package ABAQUS/Explicit). Excellent correlation of the measured permanent displacement with numerically predicted results is obtained.

Reliability of Harvested Rainfall as an Auxiliary Source of Non-potable Water

Rainwater harvesting is an ancient practice that involves collecting, storing, and using precipitationto meeton-site water needs. This paper develops and demonstrates guidelines for sizing the capacity of storage tanks to provide areliable continuous supply of harvested rainwater for residential households. Operation of the rainwater harvesting system is simulated with a stochastic mass balance performed on an Excel spreadsheet. The daily volume of rainwater in an unbounded tank is tracked to determine the maximum accumulated deficit on a monthly basis. Results are summarized in dimensionless charts showing the minimum size of a rainwater storage tankneeded to meet water demands at specified levels of reliability.

Analytical Solutions for Tunnels of Elliptical Cross-Section in Rheological Rock Accounting for Sequential Excavation

AbstractTime dependency in tunnel excavation is mainly due to the rheological properties of rock and sequential excavation. In this paper, analytical solutions for deeply buried tunnels with elliptical cross-section excavated in linear viscoelastic media are derived accounting for the process of sequential excavation. For this purpose, an extension of the principle of correspondence to solid media with time varying boundaries is formulated for the first time. An initial anisotropic stress field is assumed. To simulate realistically the process of tunnel excavation, solutions are developed for a time-dependent excavation process with the major and minor axes of the elliptical tunnel changing from zero until a final value according to time-dependent functions specified by the designers. In the paper, analytical expressions in integral form are obtained assuming the incompressible generalized Kelvin viscoelastic model for the rheology of the rock mass, with Maxwell and Kelvin models solved as particular cases. An extensive parametric analysis is then performed to investigate the effects of various excavation methods and excavation rates. Also the distribution of displacements and stresses in space at different times is illustrated. Several dimensionless charts for ease of use of practitioners are provided.

Finite strain elastoplastic solutions for the undrained ground response curve in tunnelling

SummaryThe instantaneous response of saturated low permeability grounds to tunnel excavation is important for deformations and stability close to the tunnel face. It is characterized by zero volume change in combination with the development of excess pore pressures. In tunnelling through poor quality ground under great depth of cover and high in situ pore pressure, heavily squeezing conditions (characterized by very large convergences) may occur soon after excavation. This paper presents exact finite strain analytical solutions for the undrained ground response around cylindrical and spherical openings that are unloaded from uniform and isotropic initial stress states, on the basis of the Modified Cam Clay (MCC) model and the Mohr–Coulomb (MC) model. The solution for a Drucker–Prager material is also given as it requires only a very small modification to the MC solution. The so‐called ground response curve, that is, the relationship between the support pressure and the cavity wall displacement, is derived in closed form for the MC model. The solution for the MCC problem is semi‐analytical in that it uses the trapezium rule for the computation of a definite integral. The influence of the significant parameters of the problem on the predicted deformation behaviour is shown by means of dimensionless charts. Finally, the practical usefulness of the solutions presented is illustrated by applying them to the breccia zones of the planned Gibraltar Strait tunnel – an extreme case of weak, low permeability ground under high pore pressure. The solutions can serve as a trustworthy benchmark for numerical procedures that incorporate material and geometric nonlinearities. Copyright © 2014 John Wiley &amp; Sons, Ltd.

HYDRAULIC HEAVE – DESIGN CHARTS AND DESIGN FORMULA FOR THE REQUIRED EMBEDDED LENGTH

For the determination of the required embedded length for the safety against hydraulic heave several approximate solutions exist. However, most of these solutions do not take into account the geometrical boundary conditions such as width B and length L of the excavation as well as the thickness of the aquifer S. Thus, values obtained by such simplified approximate solutions can easily lead to either uneconomical or unsafe design. For this reason investigations on the safety against hydraulic heave have been carried out at the Chair of Geotechnical Engineering at RWTH Aachen University. Based on the results of numerous calculations dimensionless design charts have been generated. With the help of these design charts the required embedded length T can be determined quite easily taking into account the difference of the ground water level H, the Thickness of the aquifer S, the geometrical dimensions B and L of the excavation and the unit weight of submerged soil γ′. In addition to these design charts a formula has been developed. By use of this design formula the required embedded length can directly be determined taking into account the before mentioned boundary conditions.

Exact and Approximate Solutions for Seepage through Semipermeable Cutoff Walls

AbstractAn explicit analytical solution is developed for the problem of steady, two-dimensional seepage in the vertical plane through a fully penetrating, semipermeable cutoff wall. The flow is driven by a head drop across an overlying dam, whose width is equal to the width of the wall. Results including the discharge beneath the dam, the differential head across the wall, the hydraulic efficiency of the wall, and the exit gradient along the bed of the downstream reservoir are presented as dimensionless charts. Rigorous comparisons are made with solutions obtained by approximate methods, which are applicable to more-general problems. The analytical solutions and approximate methods are used to estimate the discharge and the differential head at the Shikwamkwa Dam in Ontario, Canada, and to assess the need for more-detailed seepage modeling.

Biaxial capacity of rigid footings: Simple closed-form equations and experimental results

This manuscript first presents an analytical procedure to derive dimensionless charts for the analysis and design of rigid rectangular foundations under axial load and biaxial moment. It then shows that conditions of symmetry in the normalized domain of the problem lead to practical closed-form equations that provide the entire coupled axial-load-biaxial-bending capacity envelope of shallow rectangular footings. The resulting relations may find direct application in the performance-based seismic analysis/design of soil-structure systems for which the foundations are vulnerable (i.e., prone to uplifting and/or soil plastification). It is demonstrated that the equations are a generalized version of the widely used equivalent width concept proposed by Meyerhof. Results from an experimental program involving 19 tests with 200mm square and 200×300mm rectangular foundations models under biaxial loading are presented to show that the proposed simple equations provide reasonable estimates of the measured capacity. Further comparisons with large-scale and small-scale foundation models available in the literature suggest that, similar to Meyerhod’s equivalent width concept, the proposed formulation is relatively independent of scale effects.

An analytical approach for the sequential excavation of axisymmetric lined tunnels in viscoelastic rock

The main factors for the observed time dependency in tunnel construction are due to the sequence of excavation, the number of liners and their times of installation and the rheological properties of the host rock. A generalized derivation procedure for any viscoelastic model is presented accounting for the sequential excavation of a circular tunnel supported by any number of liners of different thicknesses and stiffnesses installed at different times in a viscoelastic surrounding rock under a hydrostatic stress field under plane strain axisymmetric conditions. Sequential excavation was accounted for assuming the radius of the tunnel growing from an initial value to a final one according to a time-dependent function to be prescribed by the designer. The effect of tunnel advancement was also considered. For the generalized Kelvin viscoelastic model, explicit analytical closed form solutions were presented, which can be reduced to the solutions for the Maxwell and Kelvin models.An extensive parametric analysis was then performed to investigate the effect of the excavation process adopted, the rheological properties of the rock, stiffness, thickness and installation times of the liners on tunnel convergence, pressure on the liners and on the stress field in the rock for a support system made of three liners. Several dimensionless charts for ease of use of practitioners were provided.

Reassessing rock mass properties and slope instability triggering conditions in Valles Marineris, Mars

The rock walls of the Valles Marineris valleys (VM) in the equatorial area of Mars exhibit several gravitational failures which resulted in a series of large landslides up to several hundred cubic kilometers in volume. Questions arise as to forces at play and rock strength in the stability of the walls of VM. In this work we address the stability analysis of the walls of VM by considering the strength of the materials of the walls and the causes of landslides. Using finite element calculations and the limit analysis upper bound method, we explore the range of cohesion and friction angle values associated with realistic failure geometries, and compare predictions with the more classical Culmannʼs translational failure model. Our analysis is based both on synthetic, simplified slope profiles, and on the real shape of the walls of VM taken from the MOLA topographic data. Validation of the calibrated cohesion and friction angle values is performed by comparing the computed unstable cross sectional areas with the observed pre- and post-failure profiles, the estimated failure surface geometry and ridge crest retreat. This offers a link between rock mass properties, slope geometry and volume of the observed failure, represented in dimensionless charts. The role of groundwater flow and seismic action on the decrease of slope stability is also estimated. Pseudo-static seismic analyses provide another set of dimensionless charts and show that low seismicity events induced by meteoroid impacts, consistent with the size of craters, could be a cause for some of the observed landslides, if poor rock properties for VM are assumed. Analyses suggest that rock mass properties are more similar to their earth equivalents with respect to what has been previously supposed.

Stack ventilation in multi-storey atrium buildings: A dimensionless design approach

Using a simplified mathematical model, a preliminary design strategy for steady stack ventilation in multi-storey atrium buildings is developed. By non-dimensionalising the governing equations of flow, two key dimensionless parameters are identified – a ventilation performance indicator, λ, and atrium enhancement parameter, E – which quantify the performance of the ventilation system and the effectiveness of the atrium in assisting flows. Analytical expressions are determined to inform the vent sizes needed to provide the desired balance between indoor air temperature, ventilation flow rate and heat inputs for any distribution of occupants within the building, and also to ensure unidirectional flow. Dimensionless charts for determining the required combination of design variables are presented with a view to informing first-order design guidance for naturally ventilated buildings.

Understanding Melt Index and ASTM D1238

Abstract In plastics manufacturing, the melt flow index (MFI) is used as a routine indicator of rheological behavior when more expensive and laborious determinations of well-defined material functions are impractical. The MFI is the mass flow rate in a pressure driven flow through a standardized abrupt cylindrical contraction into a short tube performed under a standardized combination of pressure drop and temperature. In this paper, we use a finite element model to explore the connections between rheological properties and melt index. We explore the role of shear thinning by modeling the flow through the melt indexer using the Bird-Carreau model. We then explore the role of melt viscoelasticity in the MFI using the corotational Maxwell model. We present our results in dimensionless charts designed to help plastics engineers specify the MFI of a plastic for an industrial manufacturing process of known material functions. Worked examples are included to show how to use the results.

Vertical stress averaging over a layer depth down the axis of symmetry of uniformly loaded circular regime: an analytical cum graphical solution

Estimation of average vertical stress over a compressible layer lying under loaded footings is of great significance in predicting its consolidation settlement. The analytical solutions and dimensionless design charts developed so far cannot be used directly for averaging vertical stress over a stratum. An analytical expression is deduced in this study that does the job of averaging the vertical stress over a layer depth along the vertical axis of symmetry of a uniform circular load. The solution has been depicted in the form of a non-dimensional design chart to facilitate easier practices. Current solution is free from repetitive calculations and avoids the use of any other design chart or elastic equation as otherwise needed. Current piece of work may be used in circumstances where the maximum settlement is desired below uniformly loaded circular footings.

VERTICAL VIBRATIONS OF BASE ISOLATED MACHINEFOUNDATIONS

Vibration of base isolated machine foundations has been studied using the ScaledBoundary Finite Element Method (SBFEM) and the cone model method. Thedynamic stiffness of soil supporting rigid massless foundation was determined. Thisstiffness is of complex value. The real part represents the reflected energy of therestoring and inertial forces while the imaginary part represents the energy dissipatedwithin the endless extent of the soil as a geometric damping. The effect of geometricand material properties of soil upon the real and imaginary parts of the dynamicstiffness was determined and represented in terms of dimensionless charts for thefrequency range of interest. Results have shown that increasing the embedment ratiohas a significant effect on the dynamic stiffness, it increases the dynamic stiffnessconsiderably. The effect of stiffness ratio(stiffness of isolator/ stiffness of soil) wasdemonstrated for isolated machine foundations. The use of soft isolators reduces thedynamic response of foundation and the soil reaction

Influence of Spatially Variable Side Friction on Single Drilled Shaft Resistance and LRFD Resistance Factors

Load and resistance factor design (LRFD) is a method that aims at meeting specified target reliabilities (probabilities of failure) of engineered systems. The present work focuses on ultimate side friction resistance for axial loads on single cylindrical drilled shaft foundations in the presence of spatially variable rock/soil strength. Core sample data are assumed to provide reliable information about local strength in terms of mean, coefficient of variation and spatial correlation structure (variogram) at a site. The geostatistical principle of support up-scaling is applied to quantify the reduction in variability between local strength and the average ultimate shaft side friction resistance without having to recur to lengthy stochastic finite difference/element simulations. Site and shaft specific LRFD resistance factors (Φ values) are given based on the assumption of lognormal load and resistance distributions and existing formulas recommended by the Federal Highway Administration. Results are efficiently represented in dimensionless charts for a wide range of target reliabilities, shaft dimensions, and geostatistical parameters including nested variograms of different types with geometric and/or zonal anisotropies. Field data of local rock strength is used to demonstrate the method and to evaluate the sensitivity of obtained resistance factors to potentially uncertain variogram parameters.