• Coupled systems mechanics
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• v.12 no.3
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• pp.199-220
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• 2023

This article presents a new analytical model to study the effect of porosity on the shear correction factors (SCFs) of functionally graded porous beams (FGPB). For this analysis, uneven and logarithmic-uneven porosity functions are adopted to be distributed through the thickness of the FGP beams. Critical to the application of this theory is a determination of the correction factor, which appears as a coefficient in the expression for the transverse shear stress resultant; to compensate for the assumption that the shear strain is uniform through the depth of the cross-section. Using the energy equivalence principle, a general expression is derived from the static SCFs in FGPB. The resulting expression is consistent with the variationally derived results of Reissner's analysis when the latter are reduced from the two-dimensional case (plate) to the one-dimensional one (beam). A convenient algebraic form of the solution is presented and new study cases are given to illustrate the applicability of the present formulation. Numerical results are presented to illustrate the effect of the porosity distribution on the (SCFs) for various FGPBs. Further, the law of changing the mechanical properties of FG beams without porosity and the SCFare numerically validated by comparison with some available results.

• Structural Engineering and Mechanics
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• v.89 no.1
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• pp.1-11
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• 2024

This article introduces a novel analytical model for examining the impact of porosity on shear correction factors (SCFs) in functionally graded porous beams (FGPB). The study employs uneven and logarithmic-uneven modified porosity-dependent power-law functions, which are distributed throughout the thickness of the FGP beams. Additionally, a modified exponential-power law function is used to estimate the effective mechanical properties of functionally graded porous beams. The correction factor plays a crucial role in this analysis as it appears as a coefficient in the expression for the transverse shear stress resultant. It compensatesfor the assumption that the shear strain is uniform across the depth of the cross-section. By applying the energy equivalence principle, a general expression for static SCFs in FGPBs is derived. The resulting expression aligns with the findings obtained from Reissner's analysis, particularly when transitioning from the two-dimensional case (plate) to the one-dimensional case (beam). The article presents a convenient algebraic form of the solution and provides new case studies to demonstrate the practicality of the proposed formulation. Numerical results are also presented to illustrate the influence of porosity distribution on SCFs for different types of FGPBs. Furthermore, the article validates the numerical consistency of the mechanical property changesin FG beams without porosity and the SCF by comparing them with available results.

• Journal of Educational Research in Mathematics
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• v.17 no.1
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• pp.67-90
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• 2007

The purpose of this study is to understand students' thinking process in the algebra problem solving, on the base of the works of Vinner(1997a, 1997b). Thus, two middle school students were evaluated in this case study to examine how they think to solve algebra word problems. The following question was considered to analyze the thinking process from the similarity-based perspective by focusing on the process of solving algebra word problems; What is the relationship between similarity and the characteristics of thinking process at the time of successful and unsuccessful problem solving? The following results were obtained by analyzing the success or failure in problem solving based on the characteristics of thinking process and similarity composition. Successful problem solving can be based on pseudo-analytical thought and analytical thought. The former is the rule applied in the process of applying closed formulas that is constructed structural similarity not related with the situations described in the text. The latter means that control and correction occurred in all stages of problem solution. The knowledge needed for solutions was applied with the formulation of open-end formulas that is constructed structural similarity in which memory and modification with the related principles or concepts. In conclusion, the student's perception on the principles involved in a solution is very important in solving algebraic word problems.

• School Mathematics
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• v.18 no.3
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• pp.589-609
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• 2016

In this study, researchers have modernly reinterpreted geometric solving of cubic equations presented by an arabic mathematician, Omar Khayyam in medieval age, and have considered the pedagogical significance of geometric solving of the cubic equations using two conic sections in terms of analytic geometry. These efforts allow to analyze educational application of mathematics instruction and provide useful pedagogical implications in school mathematics such as 'connecting algebra-geometry', 'induction-generalization' and 'connecting analogous problems via analogy' for the geometric approaches of cubic equations: $x^3+4x=32$, $x^3+ax=b$, $x^3=4x+32$ and $x^3=ax+b$. It could be possible to reciprocally convert between algebraic representations of cubic equations and geometric representations of conic sections, while geometrically approaching the cubic equations from a perspective of connecting algebra and geometry. Also, it could be treated how to generalize solution of cubic equation containing variables from geometric solution in which coefficients and constant terms are given under a perspective of induction-generalization. Finally, it could enable to provide students with some opportunities to adapt similar solving procedures or methods into the newly-given cubic equation with a perspective of connecting analogous problems via analogy.

• Journal of the Korean Institute of Telematics and Electronics S
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• v.35S no.12
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• pp.109-120
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• 1998

An image segmentation approach based on the fuzzy worm searching and MIN-MAX clustering algorithm is proposed in this paper. This algorithm deals with fuzzy worm value and min-max node at a gross scene level, which investigates the edge information including fuzzy worm action and spatial relationship of the pixels as the parameters of its objective function. But the conventional segmentation methods for edge extraction generally need the mask information for the algebraic model, and take long run times at mask operation, whereas the proposed algorithm has single operation according to active searching of fuzzy worms. In addition, we also propose both genetic fuzzy worm searching and genetic min-max clustering using genetic algorithm to complete clustering and fuzzy searching on grey-histogram of image for the optimum solution, which can automatically determine the size of ranges and has both strong robust and speedy calculation. The simulation results showed that the proposed algorithm adaptively divided the quantized images in histogram region and performed single searching methods, significantly alleviating the increase of the computational load and the memory requirements.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.7 no.2
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• pp.156-162
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• 1995

A numerical method for a 2-D oil boom model considering the flexibility of skirt has been developed The neater is assumed rigid and the skirt is tensioned membrane having a point mass at its end The fluid motion is potential. The kinematic condition which demands the continuity of the displacement is imposed at the joint between the floater and the skirt. The dynamic condition for the point mass is imposed at the bottom end of the skirt. The numerical method is based on the Green's function method in the frame of linear potential theory. It finds it's solution simultaneously from the total system of three equations, integral equation, the equation of motion of the floater and the equilibrium equation of the deformation of the skirt. Integral equation is derived by applying the Green's theorem to radiation potential and Green's function. Proper descretization of those three equations leads to the system of a linear algebraic equation. Due to the flexibility of skirt the motion of floater can be diminished in some range of wave frequency and furthermore the mechanism of resonance of the oil boom can be changed. The motion responses of various oil booms have been compared varying the length of the skirt and the point mass. The numerical method has been validated indirectly from the good correspondence between the motion responses of the flexible skirt model and the rigid skirt model in low frequency limit.