Admissible Stress Fields Research Articles

A new non-intrusive technique for the construction of admissible stress fields in model verification

In this paper, we investigate a new procedure for constructing admissible stress tensors that are needed in some methods for robust global/goal-oriented error estimation. This procedure is based on properties of the approximate finite element solution which are applied through prolongation conditions. It involves the calculation of equilibrated tractions along element boundaries, leading to the determination of admissible stress tensors at the element level. The main idea of the proposed approach is to use the partition of unity method for the calculation of the equilibrated tractions, thus defining local problems on patches of elements which can be solved in an automatic and non-intrusive manner. Therefore, this hybrid procedure leads to reasonable computational costs and can be easily implemented into existing finite element codes. Two-dimensional experiments illustrate the capabilities of the method as compared to alternative approaches.

Elastic Solutions for Masonry-like Panels

The problem of equilibrium for normal elastic no-tension materials can be formulated as a minimum problem for the complementary energy functional over the set of statically admissible stress fields ℍ. Masonry facades are often formed by an assembly of units in the form of rectangular panels. A special class of mixed boundary value problem for such units is solved by minimizing the complementary energy over the subset \(\tilde{\mathbb {H}}\) of ℍ defined by the admissible stress fields that are of rank one. These minimal uniaxial stress states are often the exact solutions of such BVP’s as verified by deriving the corresponding fracture strains through the compatibility equations and checking the boundary conditions for the associated displacements. The novelty of the proposed approach, besides the generality of the formulation leading to the uniaxial stress solution (i.e., the compression ray distribution) in all the possible combinations of given rigid displacements of the bases, resides in the analytical description of the associated anelastic strain.

Simplified Limit Load Determination Using the mα-Tangent Method

Limit load determination of mechanical components and structures by the mα-tangent method is proposed herein. The proposed technique is a simplified method that enables rapid determination of limit loads for a general class of mechanical components and structures. The method makes use of statically admissible stress field based on a linear elastic finite element analysis to estimate the limit loads. The method is applied to a number of mechanical component configurations and the results compare well with those obtained by the corresponding elastic-plastic finite element analyses results.

Integration of measures and admissible stress fields for masonry bodies

We study the compatibility of loads for bodies made of a no-tension (masonry) material. Loads are defined as weakly compatible if they can be equilibrated by an admissible stress field represented by a tensor valued measure, and strongly compatible if they can be equilibrated by a square integrable function. In the present study, we examine situations in which weak compatibility implies strong compatibility. For families of loads that depend on a parameter and the families of measures that equilibrate these loads, we find that, under some conditions, averaging with respect to the parameter leads to a measure with a square integrable density that equilibrates the loads. We illustrate the procedure on twodimensional rectangular panels free from gravity, clamped at the bottom, and subjected to various loads on the free part of the boundary.

剛塑性解析法による強度分布最適化を援用した最適形状に関する一考察

Estimation of optimum shape of solids subjected to external loads in the context of rigid-plastic analysis is investigated in this paper. Instead of solving optimum shape directly, rigid-plastic strength distribution optimization method is used to obtain areas where their material sterengths are zero. Since it is able to constuct a statically admissible stress field without such zero-strength-areas, it can be interpreted that a certain optimum shape can be obtained by removing those zero-strength-areas. To ensure the validity of a proposed method, Bishop's complete stress solution of a punching problem and a numerical solution is compared.

Limit states for brick masonry based on homogenization approach

In this paper, a procedure is developed for assessing the strength of brick masonry based on homogenization theory. The approach invokes a lower bound analysis whereby plastically admissible stress fields are constructed in the constituents involved, subject to periodic boundary conditions and static equilibrium requirements. The critical load is obtained by solving a constrained optimization problem. The analysis employs a set of specific loading histories such as axial tension, pure shear and biaxial tension–compression at different orientations of bed joints. The performance of this approach is verified against numerical solutions based on finite element analysis. In the second part of this paper, a methodology is outlined for identification of a macroscopic failure criterion that incorporates a critical plane approach. A quantitative verification of this criterion is carried out for different loading conditions and the results are compared with the experimental data available in the literature.

Borne supérieure du domaine de résistance macroscopique d'un matériau hétérogène aléatoire

We prove here that the admissible stress fields, through any representative volume of a heterogeneous random material, must verify stability, strength and periodicity requirements concerning mean microstructural stress fields. From the above-mentioned conditions a method to compute an upper bound for the effective yield strength is deduced. In the particular case of periodic materials, this method coincides with the classical theory of homogenization method when dealing with the calculation at failure. The pertinence of the theory is tested on a fissured material. To cite this article: B. Guessab, S. Turgeman, C. R. Mecanique 335 (2007).

Torsion analysis of thin-walled beams including shear deformation effects

The first part of the paper develops a theory for the torsional analysis for open thin-walled beams of general cross-sections which accounts for shear deformation effects. Statically admissible stress fields are postulated in agreement with those resulting from the Vlasov thin-walled beam theory. The principle of stationary complementary energy is then adopted to formulate the governing field compatibility condition under the stress fields postulated. The naturally arising boundary terms are found to relate the warping deformations to the internal force fields. A torsion beam example is solved using the new theory in order to illustrate its applicability to practical problems. The second part of the paper implements the solution numerically in a force-based finite element context. Two finite elements are developed by assuming linear and hyperbolic bimoment fields. The FEA solutions are shown to provide lower bound representations of the stiffness when compared to those based on conventional beam theories founded on postulated kinematic assumptions.

An Analytical Method for Interlaminar Stresses Due to Global Effects of Ply Drop-Offs

An analytical method to determine the interlaminar stresses in laminates with ply drop-offs is presented. The mechanisms that cause interlaminar stresses are identified and divided into effects due to global and local discontinuities. The former type of discontinuity arises due to the overall laminate construction, such as the differences in laminate properties from the undropped region to the dropped region and due to the termination of the plies. The latter type of discontinuity arises due to the geometric and related details of the ply drop-offs in the drop-off region. In the current work, the interlaminar stresses due to the global effects of ply drop-offs are obtained via consideration of a simplified model of a laminate with ply drop-offs based on the assumption that the drop-off zone has no effective length and that the load transfer from the terminated plies to the continuous plies occurs completely in the undropped region. Mathematically, closed-form expressions for the interlaminar stresses are obtained by assuming a priori a set of admissible stress fields that depends on known functions in the thickness direction and arbitrary functions in the lengthwise direction. The principle of minimum complementary energy along with appropriate boundary conditions are then used to obtain the final form of the interlaminar stresses. The results from the current work are validated through comparisons with results from previous finite element analysis from the literature for a free-edge problem, which the current method can also be used to analyze, and for laminates with external and internal ply drop-offs. The current method allows one to obtain the overall stress field efficiently and, thus, can be used to examine many different laminate configurations. The contribution to the overall stress field due to the global effects of ply drop-offs and its importance as well as possible extensions to obtain more accurate local stress fields are discussed.

Admissible stress fields and arching in piles of sand

Stress distributions under conical and prismatic heaps of sand can exhibit a local minimum at the centre of the base. Occurrence of the ‘dip’ in the stress distribution is affected by the sand deposition history and the deflection of the base. The dip is due to arching of sand over the centre core of the heap. Radial stress fields in prismatic piles of sand are constructed, and distributions with a dip indicative of arching are found. Arching is promoted in stress fields where the sections adjacent to the sloping surfaces are in the yielding state whereas the inner core of the heap is in the elastic state. The static approach of limit analysis is used to find that the tendency to arching increases with an increase in the horizontal component of the reaction under half of the mound.

Admissible stress fields and arching in piles of sand

Stress distributions under conical and prismatic heaps of sand can exhibit a local minimum at the centre of the base. Occurrence of the ‘dip’ in the stress distribution is affected by the sand deposition history and the deflection of the base. The dip is due to arching of sand over the centre core of the heap. Radial stress fields in prismatic piles of sand are constructed, and distributions with a dip indicative of arching are found. Arching is promoted in stress fields where the sections adjacent to the sloping surfaces are in the yielding state whereas the inner core of the heap is in the elastic state. The static approach of limit analysis is used to find that the tendency to arching increases with an increase in the horizontal component of the reaction under half of the mound.

Lower bound solutions for bearing capacity of jointed rock

This paper applies numerical limit analysis to evaluate the bearing capacity of strip footings on an anisotropic, homogenous material. While solutions exist for footings on jointed rock [Proceedings of the Third International Conference on Computational Plasticity 2 (1992) 935; Proceedings of the International Conference on Structural Foundations on Rock 3 (1980) 83] little work has been done on the effect of variation of joint strength and relative joint orientation (for cases with two or more joint sets). From a macroscopic point of view, many jointed materials such as rock may be assumed to have anisotropic homogenous properties. The overall behaviour of a jointed rock mass is controlled by the mechanical properties of the intact rock as well as the strength and orientation of the discontinuities. The formulation presented here assumes plane strain conditions, and makes use of the Mohr–Coulomb failure criterion. In order to utilise the lower bound theorem of classical plasticity two basic assumptions must be made. Firstly the material is assumed to exhibit perfect plasticity and obey an associated flow rule without strain hardening or softening. Secondly, it is assumed that the body undergoes only small deformations at the limit load so that the effect of geometry changes is small. By using a Mohr–Coulomb approximation of the yield surfaces, the proposed numerical procedure computes a statically admissible stress field via linear programming and finite elements. The stress field is modelled using linear three-noded triangular elements and allows statically admissible stress discontinuities at the edges of each triangle. By imposing equilibrium, yield and stress boundary conditions on the unknown stresses, an expression of the collapse load is formed which can be maximized subject to a number of linear constraints on the nodal stresses. As all the requirements are met for a statically admissible stress field, the solution obtained is a rigorous lower bound on the actual collapse load. An extensive parametric analysis is presented to investigate the effects of joint orientation and strength properties on the overall bearing capacity of jointed rock. The analysis overcomes the limitations of previous solutions in that non-orthogonal joint sets are considered. Consequently this work represents an invaluable tool for designers.

Influence of a nonlinear failure criterion on the bearing capacity of a strip footing resting on rock mass using a lower bound approach

The strength envelope of almost all geomaterials is nonlinear when one considers a wide range of stresses. Therefore, a nonlinear failure criterion needs to be used in stability analysis whenever the effects of nonlinearity are too significant to be neglected. This paper presents a lower bound solution to the bearing capacity calculation of a strip footing resting on a homogenous weightless rock mass using the nonlinear Hoek–Brown (HB) failure criterion. Two types of admissible stress fields are used to develop solutions. The first stress field has three stress legs. The second stress field has a spiral-like shape with n stress legs, where n may vary from 9 to 18 000 (even to infinity). Using two admissible stress fields, equations are derived and used for the calculation of lower bound bearing capacity values of a strip footing on rock mass. The influences of the admissible stress leg number n and the material parameter s in the nonlinear HB failure criterion are investigated. It is found that the lower bound bearing capacity calculated using the spiral-like shape admissible stress field approaches to the true optimum value as the stress leg number n increases, and the nonlinear material parameter s has a strong influence on the bearing capacity of the footing.Key words: HB failure criterion, bearing capacity, stress leg, lower bound theorem, rock.

Macroscopic strain potentials in nonlinear porous materials

By taking a hollow sphere as a representative volume element (RVE), the macroscopic strain potentials of porous materials with power-law incompressible matrix are studied in this paper. According to the principles of the minimum potential energy in nonlinear elasticity and the variational procedure, static admissible stress fields and kinematic admissible displacement fields are constructed, and hence the upper and the lower bounds of the macroscopic strain potential are obtained. The bounds given in the present paper differ so slightly that they both provide perfect approximations of the exact strain potential of the studied porous materials. It is also found that the upper bound proposed by previous authors is much higher than the present one, and the lower bounds given by Cocks is much lower. Moreover, the present calculation is also compared with the variational lower bound of Ponte Castaneda for statistically isotropic porous materials. Finally, the validity of the hollow spherical RVE for the studied nonlinear porous material is discussed by the difference between the present numerical results and the Cocks bound.

Static Load Dispersion Analysis of Solid-Air Composites (Simplified Model of Interfacial Load Transmission)

The solid-air composites will be proposed to make a new movement of the composite material design. The purpose of the present study is to investigate the static load dispersion function caused by the internal hydrostatic pressures by applying numerical analysis. To simplify the evaluation of the static load dispersion, it was supposed the admissible stress field, where each stress field in closed cells is individual and two different stress fields being in both sides of a cell wall satisfy the equilibrium of forces through the cell wall. Non-linear deformation under in-plane indentation of the solid-air composites was incrementally simulated by using the finite element formulation based on displacement method. It was shown that the assumption of admissible stress field on the interfacial load transmission is useful in evaluating the static load dispersion because the numerical results were well in agreement with the experimental ones.

固‐気体複合材料の静的荷重分散解析―薄膜接着が荷重分散に及ぼす影響―

In order to propose a new movement of the composite material design, the load dispersion of the solid-air composites caused by the hydrostatic pressures is investigated by applying numerical analysis. To simplify the evaluation of the load dispersion, the admissible stress field is supposed, where each stress field of closed cell is independent and that two different fields being in both sides of a cell wall satisfy the equilibrium of forces through the cell wall. The non-linear deformation under indentation was incrementally simulated by using the finite element formulation based on displacement method. The effect of adhesive film on the load dispersion is considered in the numerical model. It was shown that the assumption of admissible stress field on the interfacial load transmission is useful in evaluating the static load dispersion because the numerical results were well in agreement with the experimental ones. And it was clarified that a few structural design parameters become effective for the load dispersion.

A lumped stress method for plane elastic problems and the discrete-continuum approximation

This paper proposes a rational method to approximate a plane elastic body through a latticed structure composed of truss elements. The method is based on the introduction of a relaxed stress energy that allows an extension of the original problem to a larger space of admissible stress fields, including stresses concentrated along lines. Use is made of polyhedral approximations of the Airy stress function. The truss analogy is employed to obtain a displacement formulation. The paper includes several numerical applications of the method to sample problems, a numerical convergence study and comparisons with exact solutions and standard finite element approximations.

Analytical solution of passive earth pressure

INTRODUCTION It is well recognised that, when wall friction is present, a non-uniform stress field arises as well as a non-planar failure surface. This renders the problem of computing exact values of earth pressures non-trivial, and analytical solutions are not available in this case. In particular, when dealing with passive earth pressure, current practice relies on solutions provided by limit equilibrium methods with a curved (typically log-spiral) surface, but as these procedures are essentially of kinematical nature they are not conservative. In fact, should the assumed mechanism be admissible in kinematics terms, these solutions represent an upper bound of the exact solution. For this reason it is of interest to search for a statically admissible stress field, because this approach provides a conservative answer or the exact one (Calladine, 1985). In this respect, the numerical solution obtained by Sokolowski (1965), based on the method of characteristics, is actually of major interest, and the one most commonly used by designers is that of Caquot & Kerisel (1948) or Kerisel & Absi (1990). This paper is intended to contribute to this problem by providing an analytical solution for earth pressure coefficients, based on the lower bound theorem of plasticity. The expression obtained for the passive earth pressure coefficient is

Stability Analysis of Complex Soil Slopes using Limit Analysis

The limit equilibrium method is commonly used for slope stability analysis. Limit equilibrium solutions, however, are not rigorous because neither static nor kinematic admissibility conditions are satisfied. Limit analysis takes advantage of the lower- and upper-bound theorems of plasticity theory to provide rigorous bounds on the true solution of a stability problem. In this study, finite- element models are used to construct both statically admissible stress fields for lower-bound analysis and kinematically admissible velocity fields for upper-bound analysis of soil slopes. While limit analysis of relatively simple slopes, typically homogeneous and of simple geometry, has been done previously, limit analysis of slopes with complex geometries, soil profiles, and groundwater patterns could not be effectively done in the past. In this paper, the theoretical basis and procedure for limit analysis of such slopes is presented. Various examples of slopes are selected from the literature and analyzed using both limit equilibrium and limit analysis. Factors of safety from limit equilibrium and limit analysis are compared. A comparison is also made, for each example, between the critical slip surfaces from limit equilibrium with the velocity field and plastic zone from the upper-bound solution and with the stress field from the lower-bound solution.

A new multi-solution approach suitable for structural identification problems

In this paper, a multi-solution strategy aimed at greatly decreasing the cost of structural identification problems is proposed. It is based on the large time increment method and on the possibility offered by this approach to obtain a family of admissible displacement and stress fields in order to solve a nonlinear time-dependent problem. This technique relies on a time and space decomposition of the solution. It has been extended to solve at a low cost, the inverse problem of identification, for various techniques. In particular, it has been adapted to the minimization of the gradient of the gap between test and simulation. In this first attempt, it is applied to the simple case of bending of an elastic–viscoplastic beam. The issue of the numerical efficiency of this type of strategy is studied.