Lower-bound Theorem Research Articles

LOWER BOUND THEOREMS AND A GENERALIZED LOWER BOUND CONJECTURE FOR BALANCED SIMPLICIAL COMPLEXES

A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated Lower Bound Theorem for pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; we propose the balanced analog of the Generalized Lower Bound Conjecture and establish some related results. We close with constructions of balanced manifolds with few vertices.

Low velocity impact and quasi-static in-plane loading on a graded honeycomb structure; experimental, analytical and numerical study

Through the increasing development of technology in different industries, and the integral requirement of energy absorption, light shock absorbers such as honeycomb structure under in-plane and out-of-plane loads have been in the center of attention. The purpose of this research is to analyze the behavior of graded honeycomb structure (GHS) under low-velocity impact and quasi-static loading. To begin with using the lower-bound theorem, an analytical equation for plateau stress is represented, taking power hardening model into consideration. To compare the acquired analytical equations, empirical tests are conducted on test specimens made of aluminum 6061-O, under previously mentioned loading. Uniaxial tensile tests on each row material are performed to collect data on material properties. The low-velocity and quasi-static tests are conducted with Drop-weight and Santam compression machines, respectively. The quasi-static test is conducted to study the strain rate effect on behavior of the structure. Two experimental tests are simulated in ABAQUS/CAE. Based on the conducted comparisons, the numerical and analytical results indicate a satisfactory agreement with experimental results. Given the performed comparison between experimental and numerical mode shapes, a “V” deformation mode is distinguished for test specimen.

Separation index of graphs and stacked 2-spheres

In 1987, Kalai proved that stacked spheres of dimension d≥3 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d=2. In this article, we give a characterisation of stacked 2-spheres using what we call the separation index. Namely, we show that the separation index of a triangulated 2-sphere is maximal if and only if it is stacked. In addition, we prove that, amongst all n-vertex triangulated 2-spheres, the separation index is minimised by some n-vertex flag sphere for n≥6.Furthermore, we apply this characterisation of stacked 2-spheres to settle the outstanding 3-dimensional case of the Lutz–Sulanke–Swartz conjecture that “tight-neighbourly triangulated manifolds are tight”. For dimension d≥4, the conjecture has already been proved by Effenberger following a result of Novik and Swartz.

Bearing Capacity Factors for Ring Foundations

Bearing capacity factors because of the components of cohesion, surcharge, and unit weight, respectively, have been computed for smooth and rough ring footings for different combinations of r(i)= r(o) and. by using lower and upper bound theorems of the limit analysis in conjunction with finite elements and linear optimization, where r(i) and r(o) refer to the inner and outer radii of the ring, respectively. It is observed that for a smooth footing with a given value of r(o), the magnitude of the collapse load decreases continuously with an increase in r(i). Conversely, for a rough base, for a given value of r(o), hardly any reduction occurs in the magnitude of the collapse load up to r(i)= r(o) approximate to 0.2, whereas for r(i)= r(o) > 0.2, the magnitude of the collapse load, similar to that of a smooth footing, decreases continuously with an increase in r(i)= r(o). The results from the analysis compare reasonably well with available theoretical and experimental data from the literature. (C) 2015 American Society of Civil Engineers.

The degree of point configurations: Ehrhart theory, Tverberg points and almost neighborly polytopes

The degree of a point configuration is defined as the maximal codimension of its interior faces. This concept is motivated from a corresponding Ehrhart-theoretic notion for lattice polytopes and is related to neighborly polytopes, to the Generalized Lower Bound Theorem and, by Gale duality, to Tverberg theory.The main results of this paper are a complete classification of point configurations of degree 1, as well as a structure result on point configurations whose degree is less than a third of the dimension. Statements and proofs involve the novel notion of a weak Cayley decomposition, and imply that the m-core of a set S of n points in Rr is contained in the set of Tverberg points of order (3m−2(n−r)) of S.

Coefficient charts for active earth pressures under combined loadings

Rankine's theory of earth pressure cannot be directly employed to c-<TEX>${\phi}$</TEX> soils backfill with a sloping ground subjected to complex loadings. In this paper, an analytical solution for active earth pressures on retaining structures of cohesive backfill with an inclined surface subjected to surcharge, pore water pressure and seismic loadings, are derived on the basis of the lower-bound theorem of limit analysis combined with Rankine's earth pressure theory and the Mohr-Coulomb yield criterion. The generalized active earth pressure coefficients (dimensionless total active thrusts) are presented for use in comprehensive design charts which eliminate the need for tedious and cumbersome graphical diagram process. Charts are developed for rigid earth retaining structures under complex environmental loadings such as the surcharge, pore water pressure and seismic inertia force. An example is presented to illustrate the practical application for the proposed coefficient charts.

Computational Methods for Masonry Vaults: A Review of Recent Results

The present paper makes a critical review of some methods and models, now available in the technical litera-ture and commonly used in the analysis of masonry vaults up to their collapse, by highlighting advantages and drawbacks of each approach. All methods adopted to describe the mechanical behavior of masonry structures, in order to be reliable, must take into account the distinctive aspects of masonry, namely the scarce (or zero) tensile strength, the good resistance in compression and the occurrence of failure mechanisms through rotation-translation of rigid macro-blocks. Classic no-tension material models disregard the small existing tensile strength and make the assumption of (1) infinitely elastic be-havior in compression and (2) isotropy, giving thus the possibility to deal with either semi-analytical approaches (espe-cially for arches) or robust numerical procedures. More advanced but rather complex models are nowadays able to deal al-so with anisotropy induced by texture, small tensile strength and softening in tension, as well as by finite strength in com-pression. Traditionally – and nowadays it is still an opinion commonly accepted, in contrast with step by step complex procedures, Limit Analysis has proved to be the most effective Method for a fast and reliable evaluation of the load bear-ing capacity of vaulted masonry structures: classic lower and upper bound theorems recall respectively the concepts of equilibrium and occurrence of failure mechanisms with rigid elements. The so-called Thrust Network Method moves its steps from lower bound theorems, whereas FE limit analysis approaches with infinitely resistant elements and dissipation on interfaces take inspiration from the upper bound point of view. An alternative to Limit Analysis is represented by tradi-tional FEM combined with either elastic-plastic or damaging models with softening, commonly used for other materials but recently adapted also to masonry. They are able to provide a large set of output numerical information but further studies are still needed to ensure their proper application.

A Numerical Approach for Lower Bound Limit Analysis

In this paper, a numerical procedure for plastic limit analysis of 3-D elastic-perfectly plastic bodies under complex loads is presented. The method is based on the lower-bound limit theorem and von Mises yield criterion so that the lower-bound limit analysis can be conducted by solving a nonlinear mathematical programming problem. A SQP algorithm and a dimension reduction-based technique are used to solve the discretized finite element optimization formulation. A conception of active constraint set is introduced, so that the number of constraints can be reduced greatly. The basis vectors of reduced residual stress spaces are constructed by performing an equilibrium iteration procedure of elasto-plastic finite element analysis. The numerical procedure is applied to carry out the plastic limit analysis of pipelines with part-through slots under internal pressure, bending moment and axial force. The effects of different sizes of part-through slots on the limit loads of pipelines are studied.

Ultimate bearing capacity of shallow strip and circular foundations by using limit analysis, finite elements, and optimization

This paper presents a review of the works done mainly at Indian Institute of Science on the determination of the ultimate bearing capacity of shallow strip and circular foundations with the usages of lower and upper bound theorems of the limit analysis. After providing an outline of the historical developments in the field of limit analysis, a few important problems, illustrating the recent advancements in determining the bearing capacity of shallow foundations, have been addressed. The effects of (i) footing–soil interface roughness, (iii) size of footing, (iii) seismic forces, and (iv) interference of two closely spaced footings, on the bearing capacity have been presented.

Seismic stability of earth-rock dams using finite element limit analysis

In this study, a finite element limit analysis method is developed to assess the seismic stability of earth-rock dams. A pseudo-static approach is employed within the limit analysis framework to determine the lower and upper bounds on the critical seismic coefficients of dams. The interlocking force in the soil is considered, and the rockfill material is assumed to follow the Mohr–Coulomb failure criterion and an associated flow rule. Based on the native form of the failure criterion, the lower and upper bound theorems are formulated as second-order cone programming problems. The nonlinear shear strength properties of rockfill materials are also considered. The developed finite element limit analysis is applied to two different types of earth-rock dams. The results indicate that the rigorous lower and upper bounds are very close even for rockfill materials with large internal friction angles. The failure surfaces are easily predicted using the contour of the yield function and the displacement field obtained by the limit analysis method. In addition, the pore water pressures are modelled as external forces in the limit analysis to assess the seismic stability of earth-rock dams in the reservoir filling stage.

Uplift Resistance of Long Pipelines in the Presence of Seismic Forces

The vertical uplift resistance of long pipes buried in sands and subjected to pseudostatic seismic forces has been computed by using the lower-bound theorem of the limit analysis in conjunction with finite elements and nonlinear optimization. The soil mass is assumed to follow the Mohr-Coulomb failure criterion and an associated flow rule. The failure load is expressed in the form of a nondimensional uplift factor Fγ. The variation of Fγ is plotted as a function of the embedment ratio of the pipe, horizontal seismic acceleration coefficient (kh), and soil friction angle (ϕ). The magnitude of Fγ is found to decrease continuously with an increase in the horizontal seismic acceleration coefficient. The reduction in the uplift resistance becomes quite significant, especially for greater values of embedment ratios and lower values of friction angle. The predicted uplift resistance was found to compare well with the existing results reported from the literature.

Seismic Bearing Capacity of Shallow Embedded Foundations on a Sloping Ground Surface

By using the lower-bound theorem of the limit analysis in conjunction with finite elements and nonlinear optimization, bearing-capacity factors, N-c and N-gamma q, with an inclusion of pseudostatic horizontal seismic body forces, have been determined for a shallow embedded horizontal strip footing placed on sloping ground surface. The variation of N-c and N-gamma q with changes in slope angle (beta) for different values of seismic acceleration coefficient (k(h)) has been obtained. The analysis reveals that irrespective of ground inclination and the embedment depth of the footing, the factors N-c and N-gamma q decrease quite considerably with an increase in k(h). As compared with N-c, the factor N-gamma q is affected more extensively with changes in k(h) and beta. Unlike most of the results reported in literature for the seismic case, the present computational results take into account the shear resistance of soil mass above the footing level. An increase in the depth of the embedment leads to an increase in the magnitudes of both N-c and N-gamma q. (C) 2014 American Society of Civil Engineers.

Limit analysis for reinforced concrete rectangular members under bending, shear and torsion

The ultimate strength of reinforced concrete (RC) rectangular members subjected to combined bending, shear and torsion is obtained from the limit analysis proposed in the present paper. Based on a warped failure surface determined by external loads, and a reasonable assumed stress distribution balancing external loads but not violating the yield condition, the bending-shear-torsion interaction can be derived from equilibrium conditions. According to the definition of lower-bound theorem in limit analysis, the calculated ultimate loads will be carried safely by the structure. The present method is a simple approach to obtain carrying capacities for RC elements under complex external loads. After comparing with the test results, a good agreement has been observed. The present method can be extended to explain the failure mechanism of RC members subjected to axial loads, and it is possible to develop a simple unified theory of RC members for engineering.

Vertical uplift resistance of two closely spaced horizontal strip anchors embedded in cohesive–frictional weightless medium

The vertical uplift resistance of two closely spaced horizontal strip plate anchors has been investigated by using lower and upper bound theorems of the limit analysis in combination with finite elements and linear optimization. The interference effect on uplift resistance of the two anchors is evaluated in terms of a nondimensional efficiency factor (ηc). The variation of ηcwith changes in the clear spacing (S) between the two anchors has been established for different combinations of embedment ratio (H/B) and angle of internal friction of the soil ([Formula: see text]). An interference of the anchors leads to a continuous reduction in uplift resistance with a decrease in spacing between the anchors. The uplift resistance becomes a minimum when the two anchors are placed next to each other without any gap. The critical spacing (Scr) between the two anchors required to eliminate the interference effect increases with an increase in the values of both H/B and [Formula: see text]. The value of Scrwas found to lie approximately in the range 0.65B–1.5B with H/B = 1 and 11B–14B with H/B = 7 for [Formula: see text] varying from 0° to 30°.

Shakedown Analysis of Framed Structures: Strong Duality and Primal-dual Analysis

The paper is aimed to illustrate the strong duality between the lower and upper bound formulations of shakedown analysis in a novel way. By the lower or upper bound theorem, shakedown analysis is a well-known direct method to evaluate the load carrying capacity of a structure subjected to cyclic loads. In the paper, the Hölder inequality is uniquely utilized to establish the upper bound formulation from the lower bound formulation. Accordingly, the strong duality between them is revealed by duality theorems. Following that, shakedown analysis is performed by the primal-dual algorithm provided by the computing tool MATLAB. Moreover, elastic-plastic analysis is also conducted for comparisons and validations using the commercial finite-element code ABAQUS. Finally, comparisons with good agreement validate the numerical results presented in the paper.

Vertical Uplift Resistance of Two Interfering Horizontal Anchors in Clay

AbstractThe vertical uplift resistance of two interfering rigid strip plate anchors embedded horizontally at the same level in clay has been examined. The lower and upper bound theorems of the limit analysis in combination with finite-elements and linear optimization have been employed to compute the failure load in a bound form. The analysis is meant for an undrained condition and it incorporates the increase of cohesion with depth. For different clear spacing (S) between the anchors, the magnitude of the efficiency factor (ηcγ) resulting from the combined components of soil cohesion (c) and soil unit weight (γ), has been computed for different values of embedment ratio (H/B), the rate of linear increase of cohesion with depth (m) and normalized unit weight (γH/c). The magnitude of ηcγ has been found to reduce continuously with a decrease in the spacing between the anchors, and the uplift resistance becomes minimum for S/B=0. It has been noted that the critical spacing between the anchors required to eli...

Bearing Capacity of Circular Footings on Reinforced Soils

A method is presented for determining the ultimate bearing capacity of a circular footing reinforced with a horizontal circular sheet of reinforcement placed over granular and cohesive-frictional soils. It was assumed that the reinforcement sheet could bear axial tension but not the bending moment. The analysis was performed based on the lower-bound theorem of the limit analysis in combination with finite elements and linear optimization. The present research is an extension of recent work with strip foundations reinforced with different layers of reinforcement. To incorporate the effect of the reinforcement, the efficiency factors eta(gamma) and eta(c), which need to be multiplied by the bearing capacity factors N-gamma and N-c, were established. Results were obtained for different values of the soil internal friction angle (phi). The optimal positions of the reinforcements, which would lead to a maximum improvement in the bearing capacity, were also determined. The variations of the axial tensile force in the reinforcement sheet at different radial distances from the center were also studied. The results of the analysis were compared with those available from literature. (C) 2014 American Society of Civil Engineers.

Lower Bound Limit Analysis of Masonry Arches with CFRP Reinforcements: A Numerical Method

This paper presents a numerical method to predict the ultimate load of masonry arches strengthened with carbon-fiber-reinforced polymer (CFRP) strips bonded to the intrados. The voussoirs of the arch and the CFRP strip, ideally divided into the same number of parts as the voussoirs, are modeled as rigid blocks. A finite set of stress resultants represents the stress state acting on interfaces of the rigid blocks. The local failure modes at the block interfaces are defined according to experimental evidence. The model is developed within an associated framework in such a way that the normality rule is satisfied: the upper- and lower-bound theorems of classical limit analysis apply. The ultimate load is predicted by a lower bound approach. The feasible domain is defined by the equilibrium equations and by the linear constraints imposed on the stress resultants. All the relations defining the model are linear, so that a linear programming problem is imposed. The predictions of the numerical model compare well with experimental results.

The short toric polynomial

We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining Stanley's toric polynomials may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric $h$-vector in terms of the $cd$-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric $h$-vector of a dual simplicial Eulerian poset in terms of its $f$-vector. This formula implies Gessel's formula for the toric $h$-vector of a cube, and may be used to prove that the nonnegativity of the toric $h$-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for simplicial polytopes.

Face numbers of pseudomanifolds with isolated singularities

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower and upper bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the $f$-vector can be completely characterized are described. We also show that the Hilbert function of a generic Artinian reduction of the face ring of a simplicial complex $\Delta$ with isolated singularities minus the $h$-vector of $\Delta$ is a PL-topological invariant.