• Geomechanics and Engineering
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• v.11 no.3
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• pp.439-455
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• 2016

This paper presented solutions of displacement and stress for a circular opening which is reinforced with grouted rock bolt. It satisfies the Mohr-Coulomb (M-C) or generalized Hoek-Brown (H-B) failure criterion, and exhibits elastic-brittle-plastic or strain-softening behavior. The numerical stepwise produce for strain-softening rock mass reinforced with grouted rock bolt was developed with non-associative flow rules and two segments piecewise linear functions related to a principle strain-dependent plastic parameter, to model the transition from peak to residual strength. Three models of the interaction mechanism between grouted rock bolt and surrounding rock proposed by Fahimifar and Soroush (2005) were adopted. Based on the axial symmetrical plane strain assumption, the theoretical solution of the displacement and stress were proposed for a circular tunnel excavated in elastic-brittle-plastic and strain-softening rock mass compatible with M-C or generalized H-B failure criterion, which is reinforced with grouted rock bolt. It showed that Fahimifar and Soroush's (2005) solution is a special case of the proposed solution for n = 0.5. Further, the proposed method is validated through example comparison calculated by MATLAB programming. Meanwhile, some particular examples for M-C or generalized H-B failure criterion have been conducted, and parametric studies were carried out to highlight the influence of different parameters (e.g., the very good, average and very poor rock mass). The results showed that, stress field in plastic region of surrounding rock with considering the supporting effectiveness of the grouted rock bolt is more than that without considering the effectiveness of the grouted rock bolt, and the convergence and plastic radius are reduced.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.853-855
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• 1994

The object of present note is to give a very short proof of Stirling's formula which uses only a formula for the generalized zeta function. There are several proofs for this formula. For example, Dr. E. J. Routh gave an elementary proof using Wallis' theorem in lectures at Cambridge ([5, pp.66-68]). We can find another proof which used the Maclaurin summation formula ([5, pp.116-120]). In [1], they used the Central Limit Theorem or the inversion theorem for characteristic functions. In [2], pp. Diaconis and D. Freeman provided another proof similarly as in [1]. J. M. Patin [7] used the Lebesgue dominated convergence theorem.

• Honam Mathematical Journal
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• v.34 no.3
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• pp.467-472
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• 2012

The aim of this paper is to provide two interesting summation formulae with the argument unity for the generalized hypergeometric function of higher order. The results are obtained with the help of two new summation formulae very recently obtained by Kim et al.. Summation formulae obtained earlier by Carlson and re-derived by Exton turn out to be special cases of our main findings.

• Communications of the Korean Mathematical Society
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• v.29 no.4
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• pp.519-526
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• 2014

A large number of summation and transformation formulas involving (generalized) hypergeometric functions have been developed by many authors. Here we aim at establishing two (presumably) new general hypergeometric transformations. The results are derived by manipulating the involved series in an elementary way with the aid of certain hypergeometric summation theorems obtained earlier by Rakha and Rathie. Relevant connections of certain special cases of our main results with several known identities are also pointed out.

• Wind and Structures
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• v.21 no.2
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• pp.119-158
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• 2015

The characteristics of the coherence functions of X axial, Y axial, and RZ axial (i.e., body axis) wind forces on the Shanghai World Trade Centre - a 492 m super-tall building with section varying along height are studied via a synchronous multi-pressure measurement of the rigid model in wind tunnel simulating of the turbulent, and the corresponding mathematical expressions are proposed there from. The investigations show that the mathematical expressions of coherence functions in across-wind and torsional-wind directions can be constructed by superimposition of a modified exponential decay function and a peak function caused by turbulent flow and vortex shedding respectively, while that in along-wind direction need only be constructed by the former, similar to that of wind speed. Moreover, an inductive analysis method is proposed to summarize the fitted parameters of the wind force coherence functions of every two measurement levels of altitudes. The comparisons of the first three order generalized force spectra show that the proposed mathematical expressions accord with the experimental results well. Later, the influences of coherence functions on wind-induced dynamic responses are analyzed in detail based on the proposed mathematical expressions and the frequency-domain method of random vibration theory.

• Geomechanics and Engineering
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• v.12 no.3
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• pp.347-360
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• 2017

The problem of cylindrical cavity expansion incorporating deformation dependent of intermediate principal stress in rock or soil mass is investigated in the paper. Assumptions that the initial axial total strain is a non-zero constant and the axial plastic strain is not zero are defined to obtain the numerical solution of strain which incorporates deformation-dependent intermediate principal stress. The numerical solution of plastic strains are achieved by the 3-D plastic potential functions based on the M-C and generalized H-B failure criteria, respectively. The intermediate principal stress is derived with the Hook's law and plastic strains. Solution of limited expansion pressure, stress and strain during cylindrical cavity expanding are given and the corresponding calculation approaches are also presented, which the axial stress and strain are incorporated. Validation of the proposed approach is conducted by the published results.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.439-446
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• 2015

The main objective of this paper is to establish certain explicit expressions for the Humbert functions ${\Phi}_2$(a, a + i ; c ; x, -x) and ${\Psi}_2$(a ; c, c + i ; x, -x) for i = 0, ${\pm}1$, ${\pm}2$, ..., ${\pm}5$. Several new and known summation formulas for ${\Phi}_2$ and ${\Psi}_2$ are considered as special cases of our main identities.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.65-74
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• 2014

The well known quadratic transformation formula due to Gauss: $$(1-x)^{-2a}{_2F_1}\[{{a,b;}\\\hfill{21}{2b;}}\;-\frac{4x}{(1-x)^2}\]={_2F_1}\[{{a,a-b+\frac{1}{2};}\\\hfill{65}{b+\frac{1}{2};}}\;x^2\]$$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series $_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.191-200
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• 2015

Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. Exton [13] presented a very general double generating relation involving products of two Laguerre polynomials. Motivated essentially by Exton's derivation [13], the authors aim to show how one can obtain nineteen new generating relations associated with products of two Laguerre polynomials in the form of a single result. We also consider some interesting and potentially useful special cases of our main findings.

• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.297-301
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• 2013

When certain general single or multiple hypergeometric functions were introduced, their reduction formulas have naturally been investigated. Here, in this paper, we aim at presenting a very interesting reduction formula for the Srivastava's triple hypergeometric function $F^{(3)}[x,y,z]$ by applying the so-called Beta integral method to the Henrici's triple product formula for hypergeometric series.