• 대한수학회논문집
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• 제33권2호
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• pp.581-590
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• 2018

In this paper, we consider surfaces in the 3-dimensional Euclidean space $\mathbb{E}^3$ which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the third fundamental form. We present an important family of surfaces, namely, tubes in $\mathbb{E}^3$. We show that tubes are of infinite III-type.

• 대한수학회지
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• 제53권4호
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• pp.909-928
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• 2016

Chen and Ishikawa studied the surfaces of revolution of the polynomial and the rational kind of finite type in Euclidean 3-space $E^3$ [10]. Here, the pedal of revolution surfaces of the polynomial and the rational kind are discussed. Also, as a special case of general revolution surfaces, the sphere and catenoid are studied for the kind of finite type.

• 대한수학회지
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• 제32권1호
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• pp.151-160
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• 1995

The notion of finite type submanifold was introduced by B.-Y. Chen [1]. A lot of works were done in this field of study by many authors. B.-Y. Chen also extended this notion to pseudo-Riemannian submanifold of pseudo-Euclidean space ([2]).

• 대한수학회지
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• 제44권2호
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• pp.407-442
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• 2007

The study of Euclidean submanifolds with finite type "classical" Gauss map was initiated by B.-Y. Chen and P. Piccinni in [11]. On the other hand, it was believed that for spherical sub manifolds the concept of spherical Gauss map is more relevant than the classical one (see [20]). Thus the purpose of this article is to initiate the study of spherical submanifolds with finite type spherical Gauss map. We obtain several fundamental results in this respect. In particular, spherical submanifolds with 1-type spherical Gauss map are classified. From which we conclude that all isoparametric hypersurfaces of $S^{n+1}$ have 1-type spherical Gauss map. Among others, we also prove that Veronese surface and equilateral minimal torus are the only minimal spherical surfaces with 2-type spherical Gauss map.

• 대한수학회보
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• 제48권3호
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• pp.601-609
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• 2011

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ to have pointwise 1-type Gauss map.

• Geomechanics and Engineering
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• 제12권2호
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• pp.295-312
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• 2017

Cast in situ and grouted concrete helical piles with 150-200 mm diameter half cylindrical ribs have become an economical and effective choice in Shanghai, China for uplift piles in deep soft soils. Though this type of pile has been successful used in practice, the reinforcing mechanism and the contribution of the ribs to the total resistance is not clear, and there is no clear guideline for the design of such piles. To study the inclusion of ribs to the contribution of shear resistance, the shear behaviour between silty sand and concrete slabs with parallel ribs at different spacing and angles were tested in a large direct shear box ($600mm{\times}400mm{\times}200mm$). The front panels of the shear box are detachable to observe the soil deformation after the test. The tests were modelled with three-dimensional finite element method in ABAQUS. It was found that, passive zones can be developed ahead of the ribs to form undulated failure surfaces. The shear resistance and failure mode are affected by the ratio of rib spacing to rib diameter. Based on the shape and continuity of the failure zones at the interface, the failure modes at the interface can be classified as "punching", "local" or "general" shear failure respectively. With the inclusion of the ribs, the pull out resistance can increase up to 17%. The optimum rib spacing to rib diameter ratio was found to be around 7 based on the observed experimental results and the numerical modelling.