A constrained non‐linear system approach for the solution of an extended limit analysis problem

Tangaramvong, Sawekchai; Tin‐Loi, F.

doi:10.1002/nme.2796

Cited by 14 publications

(12 citation statements)

References 30 publications

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“…Bolzon and Corigliano, 1997;Cocchetti and Maier, 2003;Tangaramvong and Tin-Loi, 2010a). This is eminently suitable since plastic strains will localize strongly in a limited number of fixed critical zones, whilst the remaining part of the structure can be considered to be still in the elastic regime.…”

Section: Discrete Structural Modelmentioning

confidence: 98%

Mathematical programming approaches for the safety assessment of semirigid elastoplastic frames

Tangaramvong

Tin‐Loi

2011

International Journal of Solids and Structures

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a b s t r a c tThis paper presents two complementary mathematical programming based approaches for the accurate safety assessment of semirigid elastoplastic frames under quasistatic loads. The inelastic behavior of the flexible connections and material plasticity are accommodated through piecewise linearized nonlinear yield surfaces. As is necessary for this class of structures, geometric nonlinearity is taken into account. Moreover, only a 2nd-order geometric approximation is included as this is sufficiently accurate for practical structures. The work described has a twofold contribution. First, we develop an algorithm that can robustly and efficiently process the complete (path-dependent) nonholonomic response of the structure in a stepwise (path-independent) holonomic fashion. The governing formulation is cast in mixed statickinematic variables and leads naturally to what is known in the mathematical programming literature as a mixed complementarity problem (MCP). The novelty of the proposed algorithm is that it processes the MCP directly without using some iterative (and often cumbersome) predictor-corrector procedure. Second, in the spirit of simplified analyses, the classical limit analysis approach is extended to compute the limit load multiplier under the simultaneous influence of joint flexibility, material and geometric nonlinearities, and limited ductility. Our formulation is an instance of the challenging class of optimization problems known as a mathematical program with equilibrium constraints (MPEC). Various nonlinear programming based algorithms are proposed to solve the MPEC. Finally, four numerical examples, concerning practical structures and benchmark cases, are provided to illustrate application of the analyses as well as to validate the accuracy and robustness of the proposed schemes.

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Section: Discrete Structural Modelmentioning

confidence: 98%

Mathematical programming approaches for the safety assessment of semirigid elastoplastic frames

Tangaramvong

Tin‐Loi

2011

International Journal of Solids and Structures

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“…For a generic 2D truss element i , there are two generalized stress components for the second‐order geometrically nonlinear analysis

{boldq}^{i} = ([], q_{normala}^{i}) \in frakturℜ

{boldq}_{boldg}^{i} = ([], q_{normalg}^{i}) \in frakturℜ

where q i collects the axial component of the elemental generalized stress, and

{boldq}_{boldg}^{i}

collects the generalized stress in the transverse direction, which is particularly employed for describing the second‐order geometric nonlinearity condition. The corresponding generalized strains are defined as

{bolde}^{i} = ([], e_{normala}^{i}) \in frakturℜ

{bolde}_{boldg}^{i} = ([], e_{normalg}^{i}) \in frakturℜ

where e i is the axial component of the elemental generalized strain, and

{bolde}_{normalg}^{i}

corresponds to the transverse component.…”

Section: Deterministic Linear Bifurcation Buckling Analysismentioning

confidence: 99%

“…In contrast to the 2D truss element, there are four generalized stress/strain components involved in the second‐order geometrically nonlinear 2D frame element. The axial and two end rotational components are adopted from linear analysis such that

{boldq}^{i} = ([], \begin{matrix} q_{1}^{i} \\ q_{2}^{i} \\ q_{3}^{i} \end{matrix}) \in {frakturℜ}^{3}

{bolde}^{i} = ([], \begin{matrix} e_{1}^{i} \\ e_{2}^{i} \\ e_{3}^{i} \end{matrix}) \in {frakturℜ}^{3}

where the additional transverse component is employed for the purpose of the second‐order geometrically nonlinear analysis and takes the form of

{boldq}_{boldg}^{i} = ([], q_{normalg}^{i}) \in frakturℜ

{bolde}_{boldg}^{i} = ([], e_{boldg}^{i}) \in frakturℜ

Therefore, the equilibrium condition of the i th element for the second‐order geometrically nonlinear analysis is

([], \begin{matrix} \cos θ & - \sin θ / L_{i} & - \sin θ / L_{i} \\ \sin θ & \cos θ / L_{i} & \cos θ / L_{i} \\ 0 & 1 & 0 \\ - \cos θ & \sin θ / L_{i} & \sin θ / L_{i} \\ - \sin θ & - ... \end{matrix})

…”

Section: Deterministic Linear Bifurcation Buckling Analysismentioning

confidence: 99%

Robust stability analysis of structures with uncertain parameters using mathematical programming approach

Gao

Tangaramvong

et al. 2014

Int. J. Numer. Meth. Engng

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SUMMARYThis paper presents a novel mathematical programming approach for the static stability analysis of structures with uncertainties within the framework of FEM. The considered uncertain parameters are material properties, geometry of element cross section, and loading conditions, all of which are described by an interval model. The proposed method formulates the two cases of interest, namely, worst and best buckling load calculation, into a pair of mathematical programming problems. Two straightforward advantages are exhibited by such formulations. The first advantage is that the proposed formulation can overcome the interference on the sharpness of bounds of the buckling load due to the interval dependence issue. The second benefit is that the information of uncertain parameters causing the extremities of buckling load can always be retrieved as by‐products of the uncertain stability analysis. Some numerical examples are presented to illustrate the capability of the proposed method on various structures and the sharpness of the bounds of the buckling load factors. The efficiency and effectiveness of the proposed method are also demonstrated through comparison with the classical Monte Carlo simulation method. Copyright © 2014 John Wiley & Sons, Ltd.

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“…This approach was first proposed by Maier 1. Mathematical programs include a large class of problems related to optimization, including convex and non‐convex optimization (example 2–4) complementarity problems (example 5–9), second‐order cone complementarity problems (example 10, 11), and mathematical programs with equilibrium constraints (example 12, 13). In his seminal paper, Maier 1 describes the mathematical program formulations of three classes of problems:(1) incremental state update, (2) limit analysis, and (3) shakedown.…”

Section: Introductionmentioning

confidence: 99%

“…Monographs 14, 15 provide an overview of the main ideas in all three classes. We refer the reader to References13, 16–23 as examples of recent work related to the latter two classes. Our focus in the present paper is on the first class—incremental state update.…”

Section: Introductionmentioning

confidence: 99%

Complementarity framework for non‐linear dynamic analysis of skeletal structures with softening plastic hinges

Sivaselvan

2010

Numerical Meth Engineering

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SUMMARYIn this paper, we describe an algorithm for the incremental state update of elasto-plastic systems with softening. The algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem. In developing the algorithm, we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance, and hence the ability to compute numerical solutions to practical size problems. For example, the notion of a tangent stiffness matrix arises. Numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm. The numerical examples also raise some interesting questions about multiplicity of the solutions.

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A constrained non‐linear system approach for the solution of an extended limit analysis problem

Cited by 14 publications

References 30 publications

Mathematical programming approaches for the safety assessment of semirigid elastoplastic frames

Mathematical programming approaches for the safety assessment of semirigid elastoplastic frames

Robust stability analysis of structures with uncertain parameters using mathematical programming approach

Complementarity framework for non‐linear dynamic analysis of skeletal structures with softening plastic hinges

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