The Two-Colour Rado Number for the Equation ax + by = (a + b)z

Gupta, Swati; Rangan, J. Thulasi; Tripathi, A. N.

doi:10.1007/s00026-015-0269-6

Cited by 4 publications

(2 citation statements)

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“…Since the 1980s, Professor Collins of University of Toronto in Canada proposed the compression field theory for shear strength of concrete structures and then proposed the modified compression field theory (MCFT), which provided a new way to calculate the shear strength of RC members, and the accuracy of MCFT has been widely recognized internationally. Hsu and Mo put forward the softened truss model, which was modified by Gupta and Rangan . In this model, the stress distribution of the central panel is assumed to be even, and the compressive stress flow is idealized by the parallel compressive struts.…”

Section: Introductionmentioning

confidence: 99%

Shear strength prediction of reinforced concrete walls

Cao

et al. 2019

Structural Design Tall Build

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Summary Most of existing codes ignore the effect of arch action on shear strength of reinforced concrete (RC) walls, and it leads to more conservative results. A new model considering the influence of the truss and arch on shear strength of RC walls is proposed in this paper. The shear strength provided by the truss is obtained by the modified compression field theory, and the shear strength provided by the arch is obtained by the shear deformation coordination between the truss and arch, which has an upper bound value. Experimental data of 78 RC walls failed in shear are collected to evaluate the adequacy of the proposed method by comparing with the models of Hwang and Lee, Wood, ACI 318‐11，, EC8, MCBC‐04, and JGJ 3‐2010, respectively. The results show that the average value and coefficient of variation of the experimental‐to‐calculated strength ratios obtained by the proposed model are 1.01 and 0.23, which shows better accuracy and reliability. Therefore, the proposed model can predict the shear strength of RC walls failed in shear accurately.

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Section: Introductionmentioning

confidence: 99%

Shear strength prediction of reinforced concrete walls

Cao

et al. 2019

Structural Design Tall Build

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“…such that for a given positive integer r, there exists a least positive integer n = R(S ; r) such that every r-coloring of the integers in the interval [1, n] yields a monochromatic solution to the set S . There has been a growing interest in the determination of the Rado numbers R(S ; r), particularly when S is a single equation and r = 2; for instance, see [1,3,4,5,6,7,9,13,14,15]. When r = 2, we denote this number simply by R(S ).…”

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confidence: 99%

On Disjunctive Rado Numbers for Some Sets of Equations

Dileep,

Moondra,

Tripathi

2024

Electron. J. Combin.

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Given a set of linear equations $S$ with positive integral parameters $a_1,\ldots,a_k$, $k \ge 2$, the disjunctive Rado number for the set $S$ is the least positive integer $R=R_d(S)$, if it exists, such that every $2$-coloring $\chi$ of the integers in $[1, R]$ admits a monochromatic solution to at least one equation in $S$. We give conditions for the existence of $R_d(S)$, and also give general upper and lower bounds on $R_d(S)$, when $S$ is a set of additive equations $\{y=x+a_1,\ldots,y=x+a_k\}$. We also determine $R_d(S)$ when $\max a_i$ is large enough, or when $a_1,\ldots,a_k$ form an arithmetic or geometric progression. We also give conditions for the existence of $R_d(S)$ when $S$ is a set of multiplicative equations $\{y=a_1 x,\ldots,y=a_k x\}$. Further, we give a general search-based algorithm to determine $R_d(S)$ when $S$ is a system of equations in two variables, given an upper bound on $R_d(S)$ and an algorithm to determine solutions to $S$. This algorithm runs in time $O(ka_k \log a_k)$ for the case of additive equations, which is exponentially better than the brute-force algorithm for the problem.

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