J.P. Hambleton

Prediction of the bullet effect for rockfall barriers: a scaling approach

The so-called “bullet effect” refers to the perforation of a rockfall protection mesh by impact of a small block, which has a kinetic energy lower than the design value, where the design value is determined through tests with relatively large blocks. Despite playing a key role in the overall performance of a flexible rockfall barrier, this phenomenon is still poorly understood at present. An innovative approach for quantitatively characterizing this effect based on dimensional analysis is proposed in this paper. The analysis rests on a hypothesis that the relevant variables in the impact problem can be combined into three strongly correlated dimensionless parameters. The relationship between these dimensionless parameters (i.e., the scaling relationship) is subsequently investigated and validated by means of data generated with a finite element model. The validation process shows that the dimensionless parameters are apt and that the proposed scaling relationship characterizes the bullet effect with a reasonable level of accuracy. An example from the literature involving numerical simulation of a full rock barrier is considered, and satisfactory agreement between the calculated performance of the barrier and that predicted by the established scaling relationship is observed.

Perforation of Flexible Rockfall Barriers by Normal Block Impact

Flexible rockfall barriers are a common form of protection against falling blocks of rock and rock fragments (rockfall). These barriers consist of a system of cables, posts, and a mesh, and their capacity is typically quantified in terms of the threshold of impact (kinetic) energy at which the barrier fails. This threshold, referred to here as the “critical energy,” is often regarded as a constant. However, several studies have pointed out that there is no single representative value of critical energy for a given barrier. Instead, the critical energy decreases as the block size decreases, a phenomenon referred to as the “bullet effect.” In this paper, we present a simple analytical model for determining the critical energy of a flexible barrier. The model considers a block that impacts normally and centrally on the wire mesh, and rather than incorporate the structural details of the cables and posts explicitly, the supporting elements are replaced by springs of representative stiffness. The analysis reveals the dependence of the critical energy on the block size, as well as other relevant variables, and it provides physical insight into the impact problem. For example, it is shown that bending of the wire mesh during impact reduces the axial force that can be sustained within the wires, thus reducing the energy that can be absorbed. The formulas derived in the paper are straightforward to use, and the analytical predictions compare favorably with data available in the literature.

Anisotropic elastic, strength, and fracture properties of Marcellus shale

Abstract Shales are considered to be both source and cap rocks, and play an important role in various geotechnical applications including oil and gas exploration and production. A deep understanding of the mechanical properties of shale is essential. In this work, deformability, strength, and fracturability of Marcellus shale were investigated through experimental studies, and a database of its mechanical properties was generated. Uniaxial compression, direct tension, and Brazilian tests were performed on Marcellus shale specimens in various bedding plane orientations with respect to loading directions to study the static mechanical properties of the material and their anisotropy. The experimental results revealed that the transversely isotropic model is applicable for describing the elastic behavior of Marcellus shale in pure tension and compression. However, the elastic properties measured from these two experiments were not exactly the same. In addition, differences exist between Brazilian and direct tensile strengths, both of which varied with bedding plane orientations and loading directions, and were associated with different failure modes. The deformability of Marcellus shale was also studied through seismic velocity measurements, as a means for comparison with the static measurements. Finally, a series of three-point-bending tests were conducted on specimens of increasing size in three different principal notch orientations to investigate the fracture properties of the material. It was found that there exists a significant size effect on the fracture properties calculated from the measured peak loads using Linear Elastic Fracture Mechanics (LEFM). The fracture properties of the material calculated by using Bažants Size Effect Law (SEL) were independent of the testing method and were found to be anisotropic.

Predicting wheel forces using bearing capacity theory for general planar loads

This paper assesses the applicability of bearing capacity theory for evaluating the forces generated on wheels operating on clay under steady rolling conditions. Considering advances in bearing capacity theory, in particular the interaction diagrams developed for general loading, a theoretical model for computing the horizontal force or torque from fundamental input parameters such as the vertical force (weight), wheel diameter, and undrained shear strength of the soil is presented. The predictions are compared with existing analytical solutions and data from laboratory testing, and reasonable agreement is demonstrated. The newly proposed model provides a means to predict wheel forces analytically under any operating condition (driven, braked, or towed), provided the contact length and so-called contact angle, which defines the position of the contact interface, can be estimated. The model provides a rigorous, convenient framework for evaluating wheel forces under arbitrary loading and enables a natural physical interpretation of the mobility problem.

Analysis of Inclined Strip Anchors in Sand Based on the Block Set Mechanism

Anchors are widely used in foundation systems for structures requiring uplift resistance. As demonstrated by numerous theoretical and experimental studies on the subject, uncertainty remains as to both the theoretical uplift capacity of anchors in idealised soils and the suitability of the various modelling assumptions in capturing the responses observed during tests. This study, which deals exclusively with the theoretical uplift capacity, presents newly obtained predictions of uplift capacities and the corresponding collapse mechanisms for inclined strip anchors in sand. The analysis is based on the upper bound (kinematic) method of limit analysis and the so-called block set mechanism, in which a collapse mechanism consisting of sliding rigid blocks is optimised with respect to interior angles and edges of the blocks. It is demonstrated that the method provides lower (better) estimates of uplift capacity in some cases compared to previous upper bound calculations. Also, variations in the predicted collapse mechanism with changes in embedment and inclination are assessed in detail.

R-adaptivity in Limit Analysis

Direct methods aim to find the maximum load factor that a domain made of a plastic material can sustain before undergoing full collapse. Its analytical solution may be posed as a constrained maximisation problem, which is computationally solved by resorting to appropriate discretisation of the relevant fields such as the stress or velocity fields. The actual discrete solution is though strongly dependent on such discretisation, which is defined by a set of nodes, elements, and the type of interpolation. We here resort to an adaptive strategy that aims to perturb the positions of the nodes in order to improve the solution of the discrete maximisation problem. When the positions of the nodes are taken into account, the optimisation problem becomes highly non-linear. We approximate this problem as two staggered linear problems, one written in terms of the stress variable (lower bound problem) or velocity variables (upper bound problem), and another with respect to the nodal positions. In this manner, we show that for some simple problems, the computed load factor may be further improved while keeping a constant number of elements.

A Simplified Kinematic Method for 3D Limit Analysis

The kinematic (upper bound) method of limit analysis is a powerful technique for evaluating rigorous bounds on limit loads that are often very close to the true limit load. While generalized computational techniques for two-dimensional (e.g., plane strain) problems are well established, methods applicable to three-dimensional problems are relatively underdeveloped and underutilized, due in large part to the cumbersome nature of the calculations for analytical solutions and the large computation times required for numerical approaches. This paper proposes a simple formulation for three-dimensional limit analysis that considers material obeying the Mohr-Coulomb yield condition and collapse mechanisms consisting of sliding rigid blocks separated by planar velocity discontinuities. A key advantage of the approach is its reliance on a minimal number of unknowns, can dramatically reduce processing time. The paper focuses specifically on tetrahedral blocks, although extension to alternative geometries is straightforward. For an arbitrary but fixed arrangement of blocks, the procedure for computing the unknown block velocities that yield the least upper bound is expressed as a second-order cone programming problem that can be easily solved using widely available optimization codes. The paper concludes with a simple example and remarks regarding extensions of the work.