Static Limit Analysis Research Articles

Generalized Thrust Network Analysis of Triangular Masonry Cross Vaults Inspired by Musmeci

A triangular parabolic cross vault, that was designed by Musmeci in the 1950s as a reinforced concrete structure but remained unbuilt, is revisited from the original perspective of its reinvention as a masonry structure. In the framework of static limit analysis under classical Heyman’s assumptions, a generalized thrust network analysis is adopted for a structural safety assessment. The performances of the vault, subject to its self-weight, are investigated through minimum-thrust and minimum-thickness analyses by conforming to the original geometry and assuming the vault thickness as the only design parameter. Further insight is achieved by exploring a more general class of triangular parabolic masonry cross vaults, whose rise-to-span ratio is an additional design parameter. The static efficiency of the smart and unconventional geometry proposed by Musmeci is thus proven, motivating the possibility of bringing it to new life in the form of a masonry structure.

An isogeometric collocation method for the static limit analysis of masonry domes under their self-weight

A novel isogeometric collocation method is proposed for the static limit analysis of axially-symmetric masonry domes subject to their self-weight. A shell-based static formulation is employed, alongside Heyman’s assumptions on masonry, to characterize the equilibrated and statically admissible stress states in the dome. As a distinctive feature of the approach, a vector stress function is introduced, generating point-wise self-equilibrated shell stress resultants in the dome. Accordingly, the classical minimum-thrust problem is formulated in terms of the unknown vector stress function, and the static admissibility conditions are enforced as the only optimization constraint. NURBS-based isogeometric analysis is adopted to accomplish the need for an accurate geometric description of the dome and a high-order continuous interpolation of the vector stress function. A discrete minimum-thrust problem is derived as a linear programming problem, with the static admissibility conditions checked at suitable collocation points. Instrumental to its solution is the computation of a particular solution of the equilibrium equations, which is obtained by an isogeometric collocation method. By a mechanical interpretation of the dual optimization problem, the settlement mechanism of the dome corresponding to its minimum-thrust state is also computed. Numerical results, dealing with a thorough convergence analysis, parametric analyses on spherical and ogival domes with parameterized geometry, and the real case of the Taj Mahal central dome are presented to prove the computational merit of the proposed approach.

Limit and shakedown analysis of double-layer tube considering different material properties

This paper is devoted to the maximum bearing capacity of double-layer tubes subjected to monotonic and cyclic loading, within the framework of static limit and shakedown analysis. The double-layer tube comprises inner and outer layers made of different materials, both obeying the von Mises criterion. The effects of different geometric dimensions (thickness and volume fraction) and material properties (elastic modulus, Poisson’s ratio, and yield stress) on the effective mechanical behavior are studied analytically and numerically. The results indicate that the bearing capacity of the double-layer tube under monotonic load reaches its maximum value when both inner and outer layers reach their elastic or plastic limits simultaneously. The computation of plastic and shakedown limits are presented, divided into different cases by considering the volume fraction of the inner and outer layers. The double-layer structure under cyclic loads may fail due to fatigue and excessive deformation mechanisms of the inner or outer layers at the first cycle. Only when both layers fail due to fatigue simultaneously, the shakedown limit of the structure reaches its maximum value. Compared to a single-layer tube, the long-term strength of the double-layer tube can be significantly improved by applying appropriate geometric dimensions and materials in fabrication.

Airy-based static limit analysis of structures using stabilized radial point interpolation method

This paper presents a novel formulation for static limit analysis of structures, for which the Airy stress function is approximated using stabilized Radial Point Interpolation Mesh-free method (RPIM). The stress field is determined as second-order derivatives of the Airy function, and the equilibrium equations are automatically satisfied a priori. The so-called Stabilized Conforming Nodal Integration (SCNI) is employed to ensure a present method is truly a mesh-free approach, meaning that all constraints in problems are only enforced at nodes. With the use of the Airy function, SCNI, and Second-Order Cone Programming (SOCP), the size of the resulting problem is kept to be minimum. Several benchmark problems having arbitrary geometries and boundary conditions are investigated. The obtained numerical solutions are compared with those available in other studies to perform the computational aspect of the proposed method.

Collapse capacity of masonry domes under horizontal loads: A static limit analysis approach

A static limit analysis approach is proposed for assessing the collapse capacity of axisymmetric masonry domes subject to horizontal forces. The problem formulation is based on the sound theoretical framework provided by the classical statics of shells. After introducing the shell stress resultants on the dome mid-surface, integral equilibrium equations are enforced for its typical part. Heyman’s assumptions of infinite compressive and vanishing tensile strengths are made, with cohesive-frictional behavior governing the shear strength, to characterize the admissible stress states in the dome. An original computational strategy is developed to address the resulting static limit analysis problem, involving the introduction of a mesh on the dome mid-surface, the interpolation of the physical components of the shell stress tensors on the element boundaries, and the imposition of equilibrium and admissibility conditions respectively for the elements and at the nodes of the mesh. The descending discrete second-order cone programming problem is solved by standard and effective optimization tools, automatically providing collapse multiplier of horizontal forces, incipient collapse mechanism and expected crack pattern. Convergence analysis, validation with experimental results available in the literature, and parametric analyses with respect to geometric parameters and material shear response, are presented for spherical and ellipsoidal masonry domes, proving the reliability of the proposed approach for estimating the pseudo-static seismic resistance of masonry domes. Finally, as an application to a real structure, the pseudo-static seismic assessment of the central dome of the Taj-Mahal is presented.

Collapse analysis of soils by adaptive pseudo-static limit analysis

The prediction of failure load factor and stress distribution at limit state plays an important role in geo-technical engineering design. This paper develops a pseudo-lower bound numerical procedure for adaptive limit analysis of geo-mechanics problems governed by the Mohr–Coulomb failure criterion. Equilibrium equations for the residual stress are enforced in a weak manner, allowing the use of displacement-based elements within the static limit analysis framework. The method does not require the construction of stress-based elements, but the stress fields at limit state can still be obtained. Non-linear failure criterion is cast in the form of a standard conic constraint, enabling the use of the highly efficient second-order cone based solvers. A refinement indicator based on Mohr–Coulomb yield function values is proposed, and adaptive mesh refinement is performed to speed up the computational progress. The proposed method is applied to various soil engineering problems, providing accurate bearing capacity together with stress distribution at limit state.

A finite difference method for the static limit analysis of masonry domes under seismic loads

The static limit analysis of axially symmetric masonry domes subject to pseudo-static seismic forces is addressed. The stress state in the dome is represented by the shell stress resultants (normal-force tensor, bending-moment tensor, and shear-force vector) on the dome mid-surface. The classical differential equilibrium equations of shells are resorted to for imposing the equilibrium of the dome. Heyman’s assumptions of infinite compressive and vanishing tensile strength, alongside with cohesive-frictional shear response, are adopted for imposing the admissibility of the stress state. A finite difference method is proposed for the numerical discretization of the problem, based on the use of two staggered rectangular grids in the parameter space generating the dome mid-surface. The resulting discrete static limit analysis problem results to be a second-order cone programming problem, to be effectively solved by available convex optimization softwares. In addition to a convergence analysis, numerical simulations are presented, dealing with the parametric analysis of the collapse capacity under seismic forces of spherical and ogival domes with parameterized geometry. In particular, the influence that the shear response of masonry material and the distribution of horizontal forces along the height of the dome have on the collapse capacity is explored. The obtained results, that are new in the literature, show the computational merit of the proposed method, and quantitatively shed light on the seismic resistance of masonry domes.

Ultimate load prediction of MMNCs structures

Abstract The paper proposes a nonlocal limit analysis procedure to evaluate the load bearing capacity of metal matrix nanocomposites (MMNCs) structural elements. The promoted procedure rephrases and extends, in a nonlocal context, a static limit analysis numerical method, known in literature as elastic compensation method. Due to the ductility and the intrinsic nonlocal behaviour of MMNCs materials, a nonlocal elastic-perfectly plastic constitutive behaviour is hypothesized with the further hypothesis that nonlocality belongs only to the elastic phase. An associative von Mises type yield condition together with a nonlocal elasticity model of integral type are adopted. A nonlocal formulation of the finite element method is used throughout and a numerical application is presented and critically discussed with the aim to test the capability of the method in capturing the main features of the novel nonlocal limit analysis approach.

Static limit analysis and strength of porous solids with hill orthotropic matrix

The present study deals with a strength criterion for ductile porous materials consisting in a Hill type orthotropic matrix containing spherical voids. The originality of the study lies into an attempt to develop an approximate static Limit Analysis for this class of materials, based on the recent work of Cheng et al. (2014) initially proposed for isotropic von Mises matrix. To this end, we considered, in the framework of a statical limit analysis framework, a trial stress field complying with the boundary conditions of the homogenization problem. Interestingly, the proposed procedure delivers an anisotropic macroscopic criterion which is not only pressure dependent, but exhibits an original sensitivity to the sign of the third invariant of the stress deviator. The obtained results are discussed and compared to existing theoretical models, to numerical bounds and to recently available Finite Element results. Finally, we provide the plastic strain rate equations and the void evolution law which are crucial for formulating the failure of anisotropic ductile metals. The influence of the plastic anisotropy on these constitutive equations is illustrated.

Static Limit Analysis of Reinforced Soil Structures by a Simple Finite Element and Second-Order Cone Programming

TO DISCRETIZE REINFORCED SOIL STRUCTURES IN PLANE STRAIN AND PREDICT LOWER BOUNDS ON THEIR COLLAPSE LOAD, A SIMPLE THREE-NODE TRIANGULAR FINITE ELEMENT IS FORMULATED. THE ELEMENT SATISFIES THE EQUILIBRIUM EQUATIONS AND THE MECHANICAL BOUNDARY CONDITIONS IN A WEAK SENSE. A MODIFIED MOHR-COULOMB YIELD SURFACE IS ADOPTED TO DESCRIBE THE REINFORCED SOIL BEHAVIOR FROM A MACROMECHANICS POINT OF VIEW. IT IS ALSO TAKEN INTO ACCOUNT THE POSSIBILITY OF TENSION FAILURE OF THE REINFORCEMENT AND FAILURE OF THE REINFORCEMENT INTERFACE. THE STATED NONLINEAR CONVEX OPTIMIZATION PROBLEM IS CAST AS SECOND-ORDER CONE PROGRAMMING. NUMERICAL EXAMPLES ILLUSTRATE THE PREDICTIVE ACCURACY OF THE ABOVE SCHEME AS WELL AS THE EFFICIENCY AND SPEED OF AN INTERIOR-POINT METHOD TO REACH OPTIMAL SOLUTIONS.

Airy-based equilibrium mesh-free method for static limit analysis of plane problems

This paper presents a numerical procedure for lower bound limit analysis of plane problems governed by von Mises yield criterion. The stress fields are calculated based on the Airy function which is approximated using the moving least squares technique. With the use of the Airy-based equilibrium mesh-free method, equilibrium equations are ensured to be automatically satisfied a priori, and the size of the resulting optimization problem is reduced significantly. Various plane strain and plane stress with arbitrary geometries and boundary conditions are examined to illustrate the performance of the proposed procedure.

An assessment of the role of the third stress invariant in the Gurson approach for ductile fracture

The present paper is concerned with the effects of the Lode angle (or the third stress invariant) in the yielding of porous materials. This is addressed in the framework of Gurson's analysis of voided materials. It is shown first that without the approximations operated by Gurson, the Lode angle of the macroscopic strain rate is naturally involved in the analysis and consequently the third stress invariant affects the yield criterion. Pushing forward Gurson's analysis without his approximations and still considering his axisymmetric trial velocity field , changes on the predicted yield locus are observed but only in the intermediate range of triaxialities. The same predictions are obtained for very small and very large stress triaxialities . A more carefull inspection of the results shows in fact that most of the changes are second order effects and direct effects of the Lode angle are hardly visible. Pursuing the analysis with a new class of trial velocity fields due to Si et al. (2007) (containing the Gurson field) and including non-axisymmetric contributions, changes on the prediction of the yielding behaviour are observed at the low triaxiality regime. Again, effects of the Lode angle are seen to be rather small at least in the range of porosities pertinent to ductile fracture . However, inclusion of Lode angle brings changes on the shape but also on the symmetries of the yield domain. The exact hydrostatic prediction obtained in the Gurson model is also maintained here. Comparison with the numerical simulations presented by Thoré et al. (2011) using optimization algorithms in the context of limit analysis show a very good agreement with the results and in particular that these results fall well between the static and kinematic limit analysis bounds that they furnished. • We introduce consistently Lode angle in Gurson's analysis. • We improve predictions of the Gurson approach at Small triaxialities by using a new velocity trial field. • We assess the effects of Lode angle in the yield criterion. • We compare all results to simulations based on limit analysis.

Plastic limit pressure of spherical vessels with combined hardening involving large deformation

The paper aims to investigate plastic limit pressure of spherical vessels of nonlinear combined isotropic/kinematic hardening materials. The Armstrong-Frederick kinematic hardening model is adopted and the Voce hardening law is incorporated for isotropic hardening behavior. Analytically, we extend sequential limit analysis to deal with combined isotropic/kinematic hardening materials. Further, exact solutions of plastic limit pressure were developed analytically by conducting both static and kinematic limit analysis. The onset of instability was also derived and solved iteratively by Newton's method. Numerically, elastic–plastic analysis is also performed by the commercial finite-element code ABAQUS incorporated with the user subroutine UMAT implemented with user materials of combined hardening. Finally, the problem formulation and the solution derivations presented here are validated by a very good agreement between the numerical results of exact solutions and the results of elastic–plastic finite-element analysis by ABAQUS.

Exact solutions of sequential limit analysis of pressurized cylinders with combined hardening based on a generalized Hölder inequality: Formulation and validation

The paper aims to investigate nonlinear combined isotropic/kinematic hardening cylinders under internal proportional pressure by sequential limit analysis. The Armstrong–Frederick kinematic hardening model is adopted and the Voce hardening law is incorporated for isotropic hardening behavior. In particular, we establish the kinematic formulation of sequential limit analysis from the corresponding static formulation by a generalized Hölder inequality. Especially, it is found that the derived kinematic formulation involving combined isotropic/kinematic hardening is equivalent to that by the bipotential concept. Further, exact solutions of plastic limit pressure were developed analytically by performing both static and kinematic limit analysis. Finally, the problem formulation and the solution derivations presented here are validated by a very good agreement between the numerical result of exact solutions of the present work and the upper bounds from the kinematic formulation available in literature.

Exact solutions for plastic responses of orthotropic strain-hardening rotating hollow cylinders

This paper presents exact solutions for plastic responses of plastically orthotropic strain-hardening rotating hollow cylinders. Hill’s yield criterion and the Voce hardening law were adopted. The concept of sequential limit analysis was employed to deal with the strain-hardening behavior by updating the yield criterion and the deformed configuration sequentially. First, exact solutions of the plastic angular velocity were acquired by solving both static and kinematic limit analysis problems. The existence of closed-form exact solutions for some cases was also illustrated. Moreover, the global hardening behavior was analytically evaluated. The corresponding distributions of plastic stress components and equivalent plastic strain were also developed. Second, the elastic–plastic capability of the commercial finite-element code ABAQUS was employed for rigorous validations of analytical solutions. Finally, good agreement is observed between analytical solutions and numerical results.

Uplift Capacity of Axially Loaded Piles in Clays

The present study aims at determining the uplift capacity of axially loaded piles in clays whose undrained cohesion increases linearly with depth. With the application of the axisymmetric static limit analysis approach, proposed recently by the writers, the variation of the nondimensional uplift factors with respect to changes in the embedment ratio (H/B) has been obtained for several rates of increase of soil cohesion with depth defined in terms of a nondimensional factor m for given values of soil cohesion (c0) along the ground surface and diameter (B) of pile. The uplift resistance has been evaluated in the form of the uplift factors, Fcb and Fct due to the components of the shaft resistance and the total resistance, respectively. For the given values of c0 and B, the magnitude of the uplift resistance increases continuously with an increase in the value of m. Furthermore, the Fcb values are found to remain unaffected with the variation in the shaft adhesion, whereas, the Fct values increase continuously with an increase in the adhesion factor of the pile shaft.

Stability of an unsupported vertical circular excavation in clays under undrained condition

By using the axisymmetric finite elements static limit analysis formulation, proposed recently by the authors, the stability numbers ( γH/ c o ) for an unsupported vertical circular excavation in clays, whose cohesion increases with depth, have been determined under undrained condition; γ = unit weight, H = height of the excavation and c o = cohesion along ground surface. The results are obtained for various values of H/ b and m; where b = the radius of the excavation and m = a non-dimensional parameter which accounts for the rate of the increase of cohesion with depth. The values of the stability numbers increase continuously both with increases in H/ b and m. The results obtained in this study compare well with those available in literature.

Vertical uplift resistance of circular plate anchors in clays under undrained condition

The vertical uplift resistance of circular plate anchors, embedded horizontally in a clayey stratum whose cohesion increases linearly with depth, has been obtained under undrained ( ϕ = 0) condition. The axi-symmetric static limit analysis formulation in combination with finite elements proposed recently by the authors has been employed. The variation of the uplift factor ( F c ) with changes in the embedment ratio ( H/ B) has been computed for several rates of increases of soil cohesion with depth. It is noted that in all the cases, the magnitude of F c increases continuously with depth up to a certain value of H cr/ B beyond which the uplift factor becomes essentially constant. The proposed static limit analysis formulation is seen to provide acceptable results even for the two other simple chosen axi-symmetric problems.

Static and kinematic limit analysis of orthotropic strain-hardening pressure vessels involving large deformation

This paper analytically investigates plastic limit pressure of orthotropic strain-hardening cylindrical vessels under internal pressure. It is an interesting problem to illustrate the interaction between strengthening and weakening behavior during the deformation process. The Voce hardening law and Hill's yield criterion were adopted in the paper. A sequence of static and kinematic limit analysis problems were performed by updating the yield criterion and the deformed configuration. The equality relation between the greatest lower bound and the least upper bound was confirmed explicitly. Accordingly, exact solutions of plastic limit pressure were developed with an integral term. Particularly, exact closed-form solutions were obtained for certain values of the hardening exponent of the Voce hardening law. Finally, numerical efforts were also made for rigorous validations.

Plastic limit analysis of ductile composite structures from micro- to macro-mechanical analysis

The load-bearing capacity of ductile composite structures comprised of periodic composites is studied by a combined micro/macromechanical approach. Firstly, on the microscopic level, a representative volume element (RVE) is selected to reflect the microstructures of the composite materials and the constituents are assumed to be elastic perfectly-plastic. Based on the homogenization theory and the static limit theorem, an optimization formulation to directly calculate the macroscopic strength domain of the RVE is obtained. The finite element modeling of the static limit analysis is formulated as a nonlinear mathematical programming and solved by the sequential quadratic programming method, where the temperature parameter method is used to construct the self-stress field. Secondly, Hill’s yield criterion is adopted to connect the micromechanical and macromechanical analyses. And the limit loads of composite structures are worked out on the macroscopic scale. Finally, some examples and comparisons are shown.