Vertical Point Load Research Articles

Sandbox Testing for Structural Assessment of Plastic Manholes

Structural behavior of flexible plastic manholes is very different from rigid concrete products. Although this is state-of-the-art knowledge, no international or EN standard exists on structural design of these products. Therefore, traditional soil models and soil pressure coefficients are usually applied if calculations for design checks are required. These models are based on rigid wall structures and pipe design methods and do not adequately account for the flexibility and soil interaction of the manholes in the subsurface 3D-space. Recently a working group was established in Germany to develop a design standard for concrete and plastic manholes. In order to calibrate the design equations for the flexible manholes on experimental data and field experience, an extensive in-situ testing of a PP manhole (DN 1000) in a sandbox was conducted. The comparison of deflection readings and calculation result reveals a great difference. Evidently, the assumption of horizontal soil pressure on plastic manholes is wrong. Only soil pressure coefficients of less than 0.1 give realistic results. This is significantly less than the coefficient for active soil pressure, which is widely used today. Short Summary: The paper presents results from sandbox testing of plastic manholes. It focuses on horizontal deflections resulting from wheel loads at the surface. Explanation is given on why most part of vertical point loads adjacent to flexible manholes is not acting on these products. Theoretical background for this specific behaviour is the load distribution of vertical soil pressure in the subsurface 3D-space. Based on these findings, a range for realistic soil pressure coefficients for actions on plastic manholes is presented.

Sandbox Testing for Structural Assessment of Plastic Manholes

Structural behavior of flexible plastic manholes is very different from rigid concrete products. Although this is state-of-the-art knowledge, no international or EN standard exists on structural design of these products. Therefore, traditional soil models and soil pressure coefficients are usually applied if calculations for design checks are required. These models are based on rigid wall structures and pipe design methods and do not adequately account for the flexibility and soil interaction of the manholes in the subsurface 3D-space. Recently a working group was established in Germany to develop a design standard for concrete and plastic manholes. In order to calibrate the design equations for the flexible manholes on experimental data and field experience, an extensive in-situ testing of a PP manhole (DN 1000) in a sandbox was conducted. The comparison of deflection readings and calculation result reveals a great difference. Evidently, the assumption of horizontal soil pressure on plastic manholes is wrong. Only soil pressure coefficients of less than 0.1 give realistic results. This is significantly less than the coefficient for active soil pressure, which is widely used today. Short Summary: The paper presents results from sandbox testing of plastic manholes. It focuses on horizontal deflections resulting from wheel loads at the surface. Explanation is given on why most part of vertical point loads adjacent to flexible manholes is not acting on these products. Theoretical background for this specific behaviour is the load distribution of vertical soil pressure in the subsurface 3D-space. Based on these findings, a range for realistic soil pressure coefficients for actions on plastic manholes is presented

Kinematic response of pile groups and piled rafts to a distant stationary or moving harmonic load via the FEM

Using the finite-element method (FEM), this study presents results on piled foundations excited by a stationary or moving harmonic vertical point load acting on the soil surface at a given distance. For the stationary loading case, typical configurations of pile groups and corresponding piled rafts are examined in the frequency domain. The contribution of the raft alone in the overall response is also explored. The soil is modelled as a linear-elastic continuum with hysteretic damping. A wide range of frequencies from 8 to 64 Hz is considered. Of interest is the resulting vertical oscillation mode. The influence of the number of pile rows in the direction of wave propagation on the resulting vibration reduction is assessed by monitoring a reference pile at the furthest back pile row and an observation point at the free-field behind. The associated wave-passage effect is quantified through appropriate transfer functions for the piles. Subsequently, the case of a harmonic load travelling with a constant speed parallel to the pile group is investigated. Most importantly, it is revealed that a good approximation for this case is a stationary load located on the moving load path at the shortest distance from the pile group. This has important implications in the vibration protection practice, as detailed modelling of moving loads is then not required. The methodology presented can be extended to arbitrary foundation geometries and layered soil profiles.

Understanding the force motion trade off of rigid and hinged floating platforms for marine renewables.

In this work, we compare the motion and structural response of a rigid and hinged floating structure subject to regular waves. We do this to understand better whether what is the best option for floating marine renewable installations. The hinged structure has two hinges and three pontoons, whilst the rigid structure is made by replacing the hinges with rigid steel bars. We instrument the pontoons with motion detection spheres and with strain gauges to measure vertical point loads. We find that the motion response of the platforms is similar between hinged and rigid at low and high frequencies. However, at intermediate frequency waves, single and triple sagging occur for rigid and hinged structures, respectively. We find significant load alleviation for the hinged structure in the range of frequencies where sagging behaviour occurs. These insights reveal that hinged design can contribute to long term survivability by reducing loads in the structure, whilst identification of motion patterns and natural frequencies are necessary to select operating modes for marine renewable generators mounted on the platforms.

Semi-analytical solution for the Lamb’s problem in second gradient elastodynamics

In the present paper, we consider the Lamb-type problem of the dynamic response of isotropic half-space subjected to the time-harmonic vertical point load within the simplified strain gradient elasticity theory (SGET). The classical elastodynamics solution of this problem predicts singular displacements and stresses at the point of the load application. In the present study, we use the technique of Lamé potentials and Hankel transform to solve the high-order motion equations of SGET. An explicit solution accounting for traction and double traction boundary conditions is obtained in the transformed domain. The inverse transformation is performed numerically with accurate capturing of the solution behavior around the Rayleigh pole, which position is defined by the modified scale-dependent Rayleigh equation. Based on obtained results, it is shown that SGET solution is bounded at the epicenter of the load, and that it takes into account the spatial dispersion effects for the propagated waves. These effects are important for the analysis of the high-frequency processes in the structured materials. The intensity of non-classical effects in the solution is controlled by the length scale parameters of gradient model. Based on numerical calculations, we derive the universal non-dimensional relations between the amplitudes of the applied load and maximum displacements on the surface, which can be used to identify the length scale parameters of SGET based on the atomistic calculations or experimental data.

Solution and sensitivity analysis of nonlinear equations using a hypercomplex-variable Newton-Raphson method

The classical Newton-Raphson (NR) method for solving nonlinear equations is enhanced in two ways through the use of hypercomplex variables and algebra. In particular, i) the Jacobian is computed in a highly accurate and automated way, and ii) the derivative of the solution to the nonlinear equations is computed with respect to any parameter contained within the system of equations. These advances provide two significant enhancements in that it is straightforward to provide an accurate Jacobian and to construct a reduced order model (ROM) of arbitrary order with respect to any parameter of the system. The ROM can then be used to approximate the solution for other parameter values without requiring additional solutions of the nonlinear equations. Several case studies are presented including 1D and 2D academic examples with fully functioning Python code provided. Additionally, a case of study of the catenary of an elastic cable subject to its own weight and a vertical point load. Derivatives up to 10th order were computed with respect to material, loading, and geometrical parameters. The derivatives were used to generate reduced order models of the cable deformation and reaction forces at its ends with respect to multiple input parameters. Results show that from a single hypercomplex evaluation of the cable under a single vertical point load, it is possible to generate an accurate reduced order model capable of predicting the cable deformation with 1.5 times the load in the opposite direction and with 3.5 times the load in the same direction without resolving the system of equations.

Damage identification in cable-stayed bridges based on the behavior of neutral axis under traffic loading

A beam’s Neutral Axis (NA) is a parameter that has been utilized to detect damage. Its position depends on the geometry of the cross-section and also a member’s axial loads. Shifts in its position can indicate changes in either load distribution or geometric characteristics. When considering structures such as cable-stayed bridges where the inclined cables induce significant axial forces, the effect of axial loads on NA position is significant. In this research, a simplified analytical model was developed consisting of the key elements of a cable-stayed bridge (beam, cable, and bearing). The model was loaded with a vertical point load representing a vehicle load. The NA position was calculated beneath the load as it moves over the length of the beam. By changing element properties, one can observe how the NA position behaves in different locations of the beam, due to simulated damage in the beam, cable, or bearing. It was found that changes in NA position can capture not only whether damage has occurred, but also in which component of the structure the damage is located (beam, cable, or bearing). This ability to use NA position for damage identification renders it an excellent parameter for monitoring cable-stayed bridges.

Innovative FRP-Reinforced Self-Bearing Arches

The architectural beauty of curved structures is recognized worldwide. Owing to their balanced form, vaults, domes, and arches were largely adopted over the last centuries. Currently, the erection of new curved masonry structures is dramatically reduced. However, strengthening existing arches is often required to prevent their collapse under static or seismic force. Present limitations regarding such constructions include the antiquated and expensive erection technique employed for curved members. Moreover, the assembly of provisional supporting structures (scaffolding) is generally time- and cost-consuming. Accordingly, the following research provides an innovative construction method that focuses on overcoming the drawbacks of the scaffolding-based approach. A fast lift-up technique was developed to reduce the time and costs of traditional construction techniques. Specifically, the required number of blocks that constitute the arch in this study was aligned side by side on the floor and then connected using a bonded strip of composite material (i.e., FRP—fiber-reinforced polymer). The middle block was then lifted. Thus, each corresponding block could be rotated around the contact point with the adjacent block. When all blocks left the floor, an arch shape was achieved. To validate the proposed method in terms of mechanical performance, an experiment was conducted, as reported herein. A traditional masonry arch built using scaffolding was tested for comparison, and two types of reinforced arches were tested until failure. The first was traditionally built and then reinforced, whereas the second was built using the proposed innovative technique. The specimens were instrumented and tested under a centered or eccentric vertical point load. In all cases, the composite application prevented failure of the hinged joints, as is typical of unreinforced arches, resulting in significantly increased loadbearing capacity, which registered scatters of |11|% and |28|% for the centered and eccentric loads, respectively, when comparing the traditional reinforced arch with the innovative solution.

High efficiency generation of S-wave via a transmissive binary coding metasurface based on machine learning approach

The well-known Lamb’s problem deals with the generation of surface and cylindrical body waves in an elastic half-space due to vertical point excitations on its surface. Owing to the potentials of S-waves in various engineering applications, the physical response of Lamb’s problem has inspired the authors to innovate a new idea for the generation of such S-waves by applying vertical excitations on an elastic half-space. An analytical model based on Lamb’s problem is established to study the elastic half-space wave motions under a vertical point load. Due to mathematical complexities in attaining the corresponding displacement fields, a machine learning approach is proposed to analyze the entire wavefield of the elastic half-space. Through a trained neural network, it is discovered that exertion of multiple vertical excitations with opposite magnitudes can induce beam splitting of transmitted S-waves. By optimizing the parameters, low R-waves and P-waves but high S-waves are observed in the wavefields predicted by the trained neural network. Analogous to excitations with opposite magnitudes, “0” and “1” coding elements that have out-of-phase response are introduced as a binary coding metasurface to investigate the feasibility of a binary coding metasurface in realizing beam splitting of transmitted S-waves via an input of P-waves. The beam splitting of transmitted S-waves is observed in both aluminum and elastomer thin plates. The outcome is significant and it proves that wave mode conversion of P-waves to S-waves is actualizable in both hard and soft materials through a transmissive binary coding metasurface, thus providing an avenue for various shear wave applications.

Settlement and Bearing Capacity of Rectangular Footing in Reliance on the Pre-Overburden Pressure of Soil Foundation

This article presents a solution for the quantitative evaluation of the stress–strain state (SSS) and the bearing capacity of rectangular foundations, factoring in the unit weight of the soil mass and different values of pre-overburden pressure (POP). In order to assess the SSS of the soil subgrade below a rigid rectangular footing under a uniformly distributed load, the authors applied the Boussinesq basic solution for an elastic half-space subjected to a vertical point load on its surface. As a result, the formulas for vertical stress, mean stress, shear strain, and volumetric strain for any point in Cartesian coordinates (x, y, z) and foundation settlement were determined. Additionally, the application of Hencky’s system of physical equations, with non-linear dependencies between mean stress and volumetric strain as well as deviator stress and shear strain, along with the experimental curves, depicts the relationships between bulk modulus and volume stress, and shear modulus and shear stress. The authors point out the non-linear behavior of the subgrade soil and propose a method for estimating the bearing capacity of a rigid rectangular foundation.

Dynamic response of unsaturated full-space caused by a circular tunnel subjected to a vertical harmonic point load

This study proposes a model to analyze a deep tunnel embedded in unsaturated soil. Key feature is the consideration of the coupled soil water and air. The analysis emphasizes on the dynamic responses of wave field induced by a unit vertical harmonic point load on the invert level of tunnel. The tunnel is modelled as an infinitely long shell with a circular section. The soil is simulated as an unsaturated poroelastic full-space in which the solid matrix is filled with the coupled soil water and air. Stress compatibility and displacement continuity are used to describe the interaction between the tunnel and the soil. A good agreement regarding the results of soil displacements between the present model and existing models for saturated soil and elastic media confirms the validity of the proposed model. The numerical analysis investigates the effects of water saturation on soil displacements and matric suction under different load frequencies. The results reveal that the dynamic responses of unsaturated soil are different from those in saturated soil. Increased water saturation can vary the value of displacements and matric suction of the soil, but not their distribution pattern around the tunnel. Displacements and matric suction distribution curves along the circumferences of the tunnel have more peak points in the case of higher load frequencies.

A free, square, point-loaded ice sheet: A finite element-discrete element approach

Deflection of a free, thin, square, elastic, point-loaded ice sheet is studied. A three-dimensional, combined finite element-discrete element method (FEM-DEM) is applied. Examined is a set of square, self-similar (plan view) ice sheet samples with the side lengths of L=20, 40, 80, and 160 m; thicknesses of h=0.5, 1.0, and 1.5 m; and the average discrete element sizes of l=2h and 3h. The samples are generated and meshed via a centroidal Voronoi tessellation procedure. Each mesh is unstructured and consists of convex polyhedra as the discrete elements and a Delaunay-triangulated, in-plane beam lattice of co-rotational, viscously damped Timoshenko beam finite elements. Two load cases are considered: a sheet sample subjected to i) a vertical, centrally applied point load; and ii) a vertical point load on an edge. In both load cases, a good agreement is found between the deflections computed with the FEM-DEM approach and with FEM. Maximum, non-dimensionalized deflections w¯max (w¯max=q/Fz, where q=q(wmax) is an approximate, resultant reaction force due to the buoyancy, wmax the computed maximum deflection, and Fz the applied point load) scale, in each load case, either by a power-law, w¯max∝hm1(L), or exponentially, w¯max∝10m2(h)L, depending on whether computed over h (for each L) or vice versa. The exponents m1 and m2 are not constants, but functions of L and h, respectively, and approximately equal in either load case. In-plane size estimates are sought for an FEM-DEM sheet sample to represent either a) an infinite, elastic, point-loaded ice sheet; or b) a semi-infinite, elastic, point-loaded (with the load on the free edge) ice sheet with a free edge. A free, square, point-loaded FEM-DEM sheet sample well approximates both a) and b), if L/2>≈5lch, where lch denotes a characteristic length.

Assessment of cyclic response to suction caisson in clay using a three-dimensional displacement approach

ABSTRACTAn investigation was made to present analytical solutions of cyclic response to suction caisson subjected to inclined cyclic loadings in clay using a three-dimensional displacement approach. A model representing the relationship between vertical load and vertical displacement and that between lateral load and lateral displacement along the skirt of suction caisson subjected to cyclic loadings is proposed for overconsolidated clay. For the effect of vertical load on cyclic load capacity of suction caisson, using the Mindlin solution in the case of a vertical point load, the vertical stress of soil under the base of suction caisson is presented. For the stress state of soil beneath the base of suction caisson subjected to cyclic loading, the Mohr–Coulomb failure line and critical state line are presented and the relationship between total stress, effective mean principal stress, stress difference, and pore-pressure is elucidated. The comparison of results predicted by the present method for a suction caisson subjected to cyclic loadings in clay has shown good agreement with those obtained from field tests. Cyclic behavior of clay up to failure is made clear from the relationship between cyclic tensile load, vertical and lateral displacements, and rotation and that between depth, vertical, and lateral pressures.

Observations by Evaluating the Uncertainty of Stress Distribution in Truss Structures Based on Probabilistic and Possibilistic Methods

Load-bearing mechanical structures like trusses face uncertainty in loading along with uncertainty in stress and strength, which are due to uncertainty in their development, production, and usage. According to the working hypothesis of the German Collaborative Research Center SFB 805, uncertainty occurs in processes that are not or only partial deterministic and can only be controlled in processes. The authors classify, compare, and evaluate four different direct methods to describe and evaluate the uncertainty of normal stress distribution in simple truss structures with one column, two columns, and three columns. The four methods are the direct Monte Carlo (DMC) simulation, the direct quasi-Monte Carlo (DQMC) simulation, the direct interval, and the direct fuzzy analysis with α-cuts, which are common methods for data uncertainty analysis. The DMC simulation and the DQMC simulation are categorized as probabilistic methods to evaluate the stochastic uncertainty. On the contrary, the direct interval and the direct fuzzy analysis with α-cuts are categorized as possibilistic methods to evaluate the nonstochastic uncertainty. Three different truss structures with increasing model complexity, a single-column, a two-column, and a three-column systems are chosen as reference systems in this study. Each truss structure is excited with a vertical external point load. The input parameters of the truss structures are the internal system properties such as geometry and material parameters, and the external properties such as magnitude and direction of load. The probabilistic and the possibilistic methods are applied to each truss structure to describe and evaluate its uncertainty in the developing phase. The DMC simulation and DQMC simulation are carried out with full or “direct” sample sets of model parameters such as geometry parameters and state parameters such as forces, and a sensitivity analysis is conducted to identify the influence of every model and state input parameter on the normal stress, which is the output variable of the truss structures. In parallel, the direct interval and the direct fuzzy analysis with α-cuts are carried out without altering and, therefore, they are direct approaches as well. The four direct methods are then compared based on the simulation results. The criteria of the comparison are the uncertainty in the deviation of the normal stress in one column of each truss structure due to varied model and state input parameters, the computational costs, as well as the implementation complexity of the applied methods.

Behaviour of Post-tensioned Interlocking Bricks for Masonry Wall Subjected to Out-of-plane Loads

Objectives: To investigate the post-tensioning behavior of the soil-cement interlocking bricks used in masonry wall subjected to out-of-plane loads. Methods/Statistical Analysis: Three types of wall specimen were tested with post-tensioned forces of 50% and 65% of maximum post-tension force. The wall specimens were tested horizontally after applying the post-tensioning force by applying two vertical point loads distributed equally along the wall. Findings: Results showed that service load and ultimate load for W1 are 6.2 KN and 9.5 KN respectively. The mentioned loads for W2 were 7 KN and 11.8 KN respectively. The maximum deflection occurred in W1 and W2 was 69 mm and 32 mm respectively. On the other hand, the service load and ultimate load for the post-tensioned wall with two tendons were found to be 14 KN and 20 KN respectively. The deflection in both cases was approximately same which it reaches up to 59mm. It can be concluded that post-tensioning level of interlocking bricks should be always from 60% to 65% in order to get optimum results for service ability and ultimate limit states. Application/Improvements: The post-tensioned wall developed using soil-cement interlocking blocks can be used in reinforced and post-tensioned masonry applications instead of conventional brick masonry. The post tensioning enhances the performance of brick masonry.

Analytical formulas and design charts for transversely isotropic half-spaces subject to linearly distributed pressures

We provide analytical formulas and design charts for computing displacements, strains and stresses in transversely isotropic half-spaces subject to linearly distributed vertical pressures. The domain integrals extended to the loaded region, resulting from the solution associated with a vertical point load, are first transformed into boundary integrals. For polygonal domains, the boundary integrals are further reduced to algebraic sums depending upon the loading function and the position vertices of the loaded region. A detailed account of this transformation is reported for all integrals in order to highlight the singularities to be coped with and how they can be circumvented. Design charts for the vertical stress are reported in order to validate the proposed formulation. Finally, the values of displacements and stresses underneath a foundation of arbitrary shape comparatively show the influence of modelling half-spaces as isotropic or transversely isotropic media.

Time-harmonic response of saturated porous transversely isotropic half-space under surface tractions

Summary An analytical investigation is here presented for dynamic analysis of a half-space containing saturated porous transversely isotropic material under surface tractions. The solid displacement-pore fluid pressure formulation of Biot’s theory known as u - p formulation is accepted as the governing equations for the whole half-space, which is the domain of the problem. The free surface of the half-space is considered completely permeable. Two scalar potential functions are for the first time introduced to uncouple the governing system of partial differential equations. The potential functions are introduced in such a way that the governing operators for the potential functions to be physically meaningful. By applying Fourier and Hankel integral transforms, the potential functions are determined by solving two ordinary differential equations. Then, the displacements, stresses and pore fluid pressure are presented as the solutions for the boundary value problem involved in this paper in terms of some line integrals that are evaluated numerically. These responses are derived for general patch load, and presented for horizontal and vertical circular loads, vertical ring load and vertical point load as well as special cases. In addition, the degeneration of the proposed solution to thermoelastodynamics is shown. Selected numerical results for a half-space subjected to uniform horizontal load applied on a circular disc are illustrated.

Ring foundation on elastic subgrade: an analytical solution for computer modelling using the Lagrangian multiplier method

In the practice of geotechnical engineering, the case of a ring footing carrying a set of concentrated point loads is a common problem. At times, the induced vertical and angular displacements for the ring footing need to be evaluated at a relatively precise level. By making use of the governing set of equations derived for the case of a general curved beam, expressions that can be easily implemented in modern computing software are derived for the vertical and angular displacements of a ring footing of rectangular cross section as functions of the radial position. The loading case considered is a vertical point load, and the soil is modelled as elastic. Estimates of the displacements have been shown for a common range of practical applications. The behaviour for a set of concentrated loads may be evaluated using the derived equations through direct superposition. Nonlinear finite element analysis is used to evaluate the vertical deflection and angular twist of the ring foundation. Numerical analysis performed for three ring foundations with different radii and cross sections is reported to validate the accuracy of the derived analytical solution. Copyright © 2016 John Wiley & Sons, Ltd.

Dynamic response of axisymmetric transversely isotropic viscoelastic continuously nonhomogeneous half-space

An analytical derivation is presented for dynamic response of axisymmetric transversely isotropic linear viscoelastic continuously nonhomogeneous half-space subjected to vertical either point or circular patch load. The material coefficients are considered to vary in terms of depth as bounded exponentially functions, and the mass density is assumed to be constant. Hankel integral transforms accompanied with Frobenius series method are applied to solve the boundary value problem. The unknown constants are determined by satisfying boundary conditions and regularity conditions at infinity, after which the displacement and stress fields are specified in Hankel space. The inverse Hankel integral transforms are utilized to specify the displacements and stresses in real domain. It is shown that inhomogeneity parameters affect the dynamic response of the half-space considerably, especially at the vicinity of the free-surface. Moreover, viscoelastic behavior of the half-space is parametrically studied assuming several damping ratios, and it is seen that it can considerably change the dynamic responses especially for long horizontal distances from the excitation point.

Testing and analysis of pultruded GFRP continuous beams for the deflection serviceability limit state

An investigation of the deformation response of an unequal two-span pultruded glass fibre reinforced polymer (GFRP) wide flange (WF) beam up to the deflection serviceability limit is described. The beam was subjected to vertical point loading at the centre of the longer span. Mid-span deflections, support rotations and outer surface flange strains recorded during major- and minor-axis flexure tests on the beam are presented and shown to be both repeatable and linear. New closed-form shear deformation equations are presented for the forces and displacements of two-span continuous beams of arbitrary span ratio with the longer span subjected to a vertical mid-span point load. The equations have been used to predict the mid-span deflections, support rotations and surface strains recorded during the flexure tests. It is shown that the equations are able to predict the experimental deflections accurately. Depending on the particular support, the rotations (particularly the minor-axis rotations) are slightly under/over-estimated and, in general, the surface strains are over-estimated. It is concluded that this investigation provides further confirmation of the utility of shear deformation continuous beam equations for predicting the deformations of pultruded GFRP beams up to the deflection serviceability limit.