Elastic–brittle–plastic analysis of circular openings in Hoek–Brown media

A closed-form solution is presented in this paper for the prediction of displacements around circular openings in a brittle rock mass subject to a hydrostatic stress field. The rock mass is assumed to be governed by Hoek–Brown yield criterion and a non-associated flow rule is used. For the elastic–brittle–plastic analysis of circular openings in an infinite Hoek–Brown medium, the existing analytical solutions were found to be incorrect. The present closed-form solution is based on a theoretically consistent method and the solution does not require the use of any numerical method.The present closed-form solution was validated by using the finite element method. In the finite element analysis, the infinite boundary was simulated “exactly” by using the newly developed elastic support method. Several cases were analyzed and the present closed-form solutions for stresses and displacements were found to be in an excellent agreement with those obtained by using the finite element method.     

Upper bound solution for ultimate bearing capacity with a modified Hoek–Brown failure criterion

Conventional calculations of ultimate bearing capacity are formulated in terms of a linear Mohr–Coulomb (MC) failure criterion. However, experimental data shows that the strength envelops of almost all types of rocks are nonlinear over the wide range of normal stresses. In this paper, the strength envelope of rock masses is considered to follow a modified Hoek–Brown failure criterion that is a nonlinear failure criterion. Two different kinds of techniques are used to develop the ultimate bearing capacity in the framework of limit analysis in plasticity.The first technique is the generalized tangential technique proposed by the authors. Based on a multi-wedge translation failure mechanism, a generalized tangential technique is used to formulate the bearing capacity problem as a classical optimization problem where the objective function, which is to be minimized with respects to the parameters of failure mechanism and the location of tangency point, corresponds to the dissipated power. The minimum solution is obtained by optimization. Using the technique, the effects of rock weight under the base of the footing and surcharge load can be considered. The second technique, “tangential” line technique, was originally used to analyze slope stability with a nonlinear failure criterion. In order to assess the validity of the proposed method, the “tangential” line technique is extended to evaluate the bearing capacity factor with the nonlinear failure criterion, where the effects of self-weight and surcharge load on the bearing capacity cannot considered. The second technique, however, has to utilize the previously calculated ultimate bearing capacity factor with a linear MC failure criterion. Numerical results are compared and presented for practical use in rock engineering.     

A Hoek–Brown criterion with intrinsic material strength factorization

Probably the most common failure criterion for rock masses is the Hoek–Brown (HB) failure criterion. The HB criterion is an empirical relation that extrapolates the strength of intact rock to that of rock masses. For design purposes, the HB criterion is often fitted using equivalent Coulomb failure lines. However, equivalent Mohr–Coulomb (MC) shear strength parameters cannot yield the same failure characteristics as the HB criterion. The curvilinear HB criterion automatically accommodates changing stress fields; the MC criterion does not. The extended HB criterion proposed in this paper provides a solution to this problem by incorporating an intrinsic material strength factorization scheme. The original HB criterion is additionally enhanced by adopting the spatial mobilized plane (SMP) concept, first introduced by Matsuoka and Nakai (MN). The SMP concept accounts for the experimentally proven, influence of intermediate principal stresses on failure, which is disregarded in the original HB criterion. A small set of examples provided at the end of the article gives a good indication of the merits of using the extended HB criterion in practical applications.     

A semi-analytical method for the elastic–plastic large deflection analysis of stiffened panels under combined biaxial compression/tension, biaxial in-plane bending, edge shear, and lateral pressure loads

In the earlier publications [Paik JK, Thayamballi AK, Lee SK, Kang SJ. A semi-analytical method for the elastic-plastic large deflection analysis of welded steel or aluminium plating under combined in-plane and lateral pressure loads. Thin-Walled Struct 2001;39;125–52; Paik JK, Thayamballi AK. Ultimate limit state design of steel-plated structures. Chichester: Wiley; January 2003], the author presented a semi-analytical method for the elastic–plastic large deflection analysis of unstiffened plates under biaxial loads, edge shear, biaxial in-plane bending and out-of-plane (lateral) pressure loads until the ultimate strength is reached. In the present paper, a similar method is applied to stiffened panels subjected to the same type of loading. The effect of initial imperfections in the form of initial deflection and welding residual stresses is accounted for in the calculations. The validity of the developed method is demonstrated by comparing with existing theoretical and numerical results where relevant. The present theory can be useful for ultimate strength analysis of plates and stiffened panels made of steel or aluminium alloys.     

Stability of elastic–plastic conical shells under shear loading

A method of determination of critical loads for thin-walled conical shells loaded by shear forces developed by moment of twist is presented. The three governing equations of neutral equilibrium with respect to basic displacement vector components u, v, and w are used. It is assumed that effective stress in the prebuckling state of stress in the shell can exceed the yield limit of the shell material. The use is made both of physical relations of Nadai–Hencky small elastic–plastic deformation theory of plasticity, and Prandtl–Reuss J₂ incremental plastic flow theory. Also, a bilinear stress–strain material model, material compressibility and Shanley approach will be accepted in the analysis. Galerkin method is applied to solve the problem equations and iterative techniques are accepted in numerical algorithm to determine critical loads for elastic–plastic shells.     

Critical load of a bilayered orthotropic elastic–plastic conical shell with the change of the shell basic surface location

The paper presents the derivation of stability equations and the comparison of critical loads for an orthotropic elastic–plastic conical shell with proper location of the shell basic surface. Prandtl–Reuss plastic flow theory is accepted in the analysis. To derive the stability equations the variational method was accepted and Ritz method was used to solve the equations. Numerical results were obtained by the use of a special iterative algorithm of the elastic–plastic analysis implemented on the PC.     

Linearization of the Hoek and Brown rock failure criterion for tunnelling in elasto-plastic rock masses

We present a novel methodology for estimation of equivalent Mohr–Coulomb strength parameters that can be used for design of supported tunnels in elasto-plastic rock masses satisfying the non-linear empirical Hoek–Brown failure criterion. We work with a general adimensional formulation of the Hoek–Brown failure criterion in the space of normalized Lambe's variables for plane stress, and we perform linearization considering the stress field in the plastic region around the tunnel. The procedure is validated using analytical solutions to a series of benchmark test cases. Numerical solutions are also employed to validate the procedure in cases for which analytical solutions are not available. Results indicate that the stress field in the plastic region around the tunnel, as well as the linearization method employed and the quality of the rock mass, has a significant impact on computed estimates of equivalent Mohr–Coulomb strength parameters. Results of numerical analyses also show that our proposed linearization method can be employed to estimate loads and moments on the tunnel support system. We recommend the equating model responses (EMR) method to compute equivalent Mohr–Coulomb strength parameters when the tunnel support pressure is accurately known, and we further show that our newly introduced linearization method can be employed as an alternative to the best fitting in the existing stress range (BFe) and best fitting in an artificial stress range (BFa) methods, providing performance estimates that are generally better than estimates of the BFe and BFa methods when differences with the response of the Hoek–Brown rock mass are of engineering significance (say more than 10%).     

A theoretical solution for analysis of tunnels below groundwater considering the hydraulic–mechanical coupling

An analytical solution for the analysis of tunnels below groundwater table in plane strain axisymmetric condition is presented. Seepage body force and secondary permeability of the rock mass due to the mechanical–hydraulic coupling are taken into account. The strain-softening behavior model and Hoek–Brown empirical strength criterion for the rock mass are used in the analysis. As the derived analytical equations do not have closed form solutions, a computer program has been prepared for solving the corresponding equations numerically and examining the analysis. It is shown that the tunnel stability depends on the seepage and the pore water pressure particularly in the case of high pore pressure gradient.     

A semi-analytical method for the elastic-plastic large deflection analysis of welded steel or aluminum plating under combined in-plane and lateral pressure loads

The aim of the present paper is to develop a semi-analytical method which can quickly and accurately compute the elastic–plastic large deflection response of welded steel or aluminum plating under a combination of biaxial compression/tension, biaxial in-plane bending, edge shear and lateral pressure loads, until the ultimate limit state is reached. The post-weld initial imperfections (i.e. initial deflection and residual stresses) are included in the method as parameters of influence. It is assumed that the plating is simply supported at all (four) edges which are kept straight. A unique feature of the developed method is that geometric nonlinearity associated with large deflection response of plating under combined loads is treated by analytically solving the nonlinear governing differential equations of the elastic large deflection plate theory, while material nonlinearity due to plasticity is dealt with implicitly by a numerical procedure. This approach reduces the magnitude of numerical computations, resulting in a saving of modeling effort and computing time. As another contribution, this paper investigates and discusses the ultimate strength characteristics of plating, by varying the plate properties and load combinations, based on elastic–plastic large deflection analysis using the developed method.     

Mechanical analysis of circular liners with particular reference to composite supports. For example, liners consisting of shotcrete and steel sets

This paper describes a methodology for the mechanical analysis of composite supports, such as liners consisting of shotcrete and steel sets. The methodology presented here is based on an established technique of structural analysis commonly referred to as the ‘equivalent section’ approach. This technique consists in treating the composite section of a straight beam as a homogenized section of equivalent mechanical properties. The equations presented in this paper have been derived from application of the theory of elastic shells (or curved beams) and therefore are more appropriate for the analysis of circular tunnel liners. The proposed methodology for the design of liners is based on the construction of capacity diagrams, another established technique of structural analysis and concrete design that can be conveniently extended to the analysis of composite sections for tunnel liners. When applying the theory of elastic shells to derive the equations that conform to the proposed methodology, the problem of determining the mechanical response of semi-circular arches treated with the theory of thin and thick formulations has been re-visited. Observations of practical interest arising from the comparison of results obtained with both approaches are discussed.     

Analytical theory for an approach calculation of non-balanced composite box beams

For aeroelastic problems, optimization of the behavior of box beams, carried out in composite materials, can lead to the construction of structures whose longitudinal axis is not necessarily the orthotropic axis of the materials. These beams present couplings such as Flexion–Torsion or Traction–Torsion. In this study, we propose an analytical theory which allows these structures to be dimensioned with extreme accuracy and without using complicated calculations. The method developed, based on a weak hypothesis on the field of deformations, makes it possible to obtain from simple analytical calculations, the stresses and displacement in a cross-section for normal load , flexion moments and torsion moments . It is then possible using the laminated plates theory, to determine the stresses in each layer. The results obtained correspond perfectly to those found in a 3D Finite Element model, calculated using CATIA–ELFINI software. On the central part of the beams, the relative differences noticed between these two methods on the calculation of stress, strain and rigidities are negligible. Near the embedded section, warping is very important and the relative error is great.     

On wind–rain induced vibration of cables of cable-stayed bridges based on quasi-steady assumption

Wind–rain induced vibration of cables of cable-stayed bridges is presently a problem of great concern. Similar to the classical galloping theory, this paper adopts quasi-steady assumption to study wind–rain induced vibration of cables. A wind tunnel test was first made to measure wind pressures and thus wind forces acting on a 3-D cable model and the upper artificial rivulet model. A new theoretical model for instability of a 2-D sectional rigid model with a moving artificial rivulet is then established and the instability criterion is proposed. The instability criterion is verified through wind tunnel test on a 2-D rigid sectional cable model with a moving artificial rivulet. Finally, theoretical models of wind–rain induced vibration of 3-D sectional cables and 3-D continuous cables are, respectively, developed based on the measured mean wind forces mentioned above, and the vibration characteristics are investigated as well as an explanation of the mechanism of wind–rain induced vibration of stay cables is made.     

Fracture prediction in 4-point bending of an extruded aluminum panel

Currently, the ICE cab cars in Europe are of all-aluminum construction for durability and low weight. Several serious train collisions involving this type of train raised questions about their integrity under accidental loads. A typical aluminum cab car is assembled from extruded aluminum panels, which are MIG welded, forming long seam lines. In this work, crushing characteristics of full scale sandwich panels under a 4-point bending load configuration were studied experimentally and numerically. Tests were run all the way to first fracture which occurred inside buckling induced dimples on the compressive side. The force–deflection response and fracture initiation were predicted by the finite element method and compared with experimental results. The location of the first fracture and the corresponding load was predicted with great accuracy using a newly developed fracture criterion. Based on a simplified beam model, a closed form analytical solution of the force–deflection response for the pre-fracture stage was also developed.     

An elastic–plastic analysis of large deflection of thin shell structure using a delta-sequence function

The analysis of thin shell under large deflection has complex problem associated with the geometrical and material nonlinearities, in which the solutions for stress and deformation are desired to obtain the same level accuracy. One of ordinary and powerful method for this subject is a finite element method that has generally inconvenience to be necessarily extensive calculation due to large number of freedom.This paper is concerned with the elastic–plastic analysis of thin shell structures by the hybrid method using a functional for the principle of modified complementary energy. For elastic–plastic materials, the numerical calculation could be well executed by the way that the stress distribution across the panel thickness is expressed as continuous function using the delta-sequence function. This approach introduces a considerable way for the reduction of computing volume. The proposed method was applied to discuss the resisting mechanism of thin shell structure under large deflection.     

Analytical solution for a circular opening in an elastic–brittle–plastic rock

This paper deals with the analytical solutions for the prediction of displacements around a circular opening in an elastic–brittle–plastic rock mass compatible with a linear Mohr–Coulomb or a nonlinear Hoek–Brown yield criterion. Three different cases of definitions for elastic strains in the plastic region, used in the existing solutions, are mentioned. The closed-form analytical solutions for the displacement in the plastic region are derived on a theoretically consistent way for all the cases by employing a non-associated flow rule. The results of the dimensionless displacements are compared using the data of the soft and hard rocks to investigate the effect of different definitions for elastic strains with the dilation angle.     

对加劲板在局部和整体弯曲作用下的后屈曲及强度的半解析分析

对经受平面内双轴及剪切荷载作用下的带缺陷加劲板的预先和后屈曲分析采用半解析方法分析。结合了Rayleigh-Ritz方法的大变形理论,可以分析板局部及整体的平衡。采用vonMises屈服标准预测的极限强度作为膜压力的破坏标准。采用Fortran程序计算的结果也被非线性有限元分析的结果证实。该法因此也适用于结构优化设计以及可靠性研究。     

Flexural–torsional behavior of thin-walled composite beams

This paper presents a flexural–torsional analysis of I-shaped laminated composite beams. A general analytical model applicable to thin-walled I-section composite beams subjected to vertical and torsional load is developed. This model is based on the classical lamination theory, and accounts for the coupling of flexural and torsional responses for arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric. Governing equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for thin-walled composites under vertical and torsional loading, addressing the effects of fiber angle, and laminate stacking sequence.     

A nonlinear strength criterion for rock-like materials based on fracture mechanics

A nonlinear strength criterion for rock-like materials is developed in this paper. Taking α as an angle of micro-failure orientation in rock-like materials, a formulation between α and load is derived from a mixed-mode fracture criterion based on linear elastic fracture mechanics. According to micro-failure experimental phenomena of rock-like materials, a failure characteristic parameter under triaxial compression condition is chosen, which is relevant to confining pressure and is an invariant. A theoretical nonlinear strength criterion is also derived, which is exactly in the same mathematical form as the original Hoek–Brown empirical strength criterion. In addition, it is also found that the coefficient m in the Hoek–Brown criterion has physical meaning which is related to the ratio between the uniaxial compressive strength and the uniaxial tensile strength.     

Obtaining dynamic complete stress–strain curves for rock using the Split Hopkinson Pressure Bar technique

The feasibility of using the Split Hopkinson Pressure Bar (SHPB) technique to obtain complete dynamic stress–strain curves for rock is established in the laboratory. The SHPB test system, in conjunction with a mean strain hypothesis, can be used not only for obtaining the rock’s constitutive curve before the peak strength but also after the peak strength — and so it is possible to analyze and characterize the post-failure behaviour of rock with the SHPB method. Some typical complete dynamic curves for marble and granite are given in this paper, together with an interpretative discussion on the shapes of the curves.     

Thermal buckling analysis of rectangular composite plates with temperature-dependent properties based on a layerwise theory

Thermal buckling analysis of rectangular composite multilayered plates under uniform temperature rise is investigated using a layerwise plate theory. von Karman strain–displacement equations are employed to account for large deflections occurrence. It is already proven that the layerwise theory results are compatible with the three-dimensional theory of elasticity results. The accuracy of the present results is increased by substituting each layer by many virtual sub-layers. The final governing equations are not simplified or linearized. Material properties are assumed to vary with temperature. Hermitian finite element formulation is used to ensure a C¹ continuity for the lateral deflections. No semi-analytic solution is employed to reduce the problem to an eigenvalue one. Layerwise formulations are usually displacement-based. Therefore, force or moment boundary conditions (e.g. simply supported boundary condition), are approximately satisfied. A FEM algorithm is presented to exactly incorporate the boundary conditions. A proposed numerical scheme and a modified Budiansky instability criterion presented by the author are used to determine the buckling temperature in a computerized solution. Finally, results of the present techniques are compared with the results of the high-order theories presented by some well-known researchers and the influences of various geometric and mechanical properties parameters of the composite plate on the buckling temperature are studied.