• Structural Engineering and Mechanics
• /
• v.28 no.2
• /
• pp.129-152
• /
• 2008

This paper considers the solution of the stochastic differential equations (SDEs) with random operator and/or random excitation using the spectral SFEM. The random system parameters (involved in the operator) and the random excitations are modeled as second order stochastic processes defined only by their means and covariance functions. All random fields dealt with in this paper are continuous and do not have known explicit forms dependent on the spatial dimension. This fact makes the usage of the finite element (FE) analysis be difficult. Relying on the spectral properties of the covariance function, the Karhunen-Loeve expansion is used to represent these processes to overcome this difficulty. Then, a spectral approximation for the stochastic response (solution) of the SDE is obtained based on the implementation of the concept of generalized inverse defined by the Neumann expansion. This leads to an explicit expression for the solution process as a multivariate polynomial functional of a set of uncorrelated random variables that enables us to compute the statistical moments of the solution vector. To check the validity of this method, two applications are introduced which are, randomly loaded simply supported reinforced concrete beam and reinforced concrete cantilever beam with random bending rigidity. Finally, a more general application, randomly loaded simply supported reinforced concrete beam with random bending rigidity, is presented to illustrate the method.

• KOREAN JOURNAL OF CROP SCIENCE
• /
• v.35 no.2
• /
• pp.156-164
• /
• 1990

A field experiment was conducted to study both the distribution characteristics of photosynthetically active radiation (PAR) in the soybean canopy and their relationships with dry matter production. The soybean cultivars 'Hwanggeumkong' and 'Paldalkong' were sown with the spaces of 60$\times$15cm and 30$\times$15cm at Suwon on May 20 and on June 20 in 1989. The ratio of PAR to the total shortwave radiation was estimated by the empirical equation derived from sunshine hours and direct incoming radiation. The functional relationships between the PAR interception and the leaf area index were expressed as a function of Beer's law. The extinction coefficients(k) in the functions ranged from 0.77 to 0.92. The values of k were greater at higher planting density, but they were affected neither by planting dates nor by varieties. The reflection ratio of PAR($\alpha$) was determined by the exponential function as below; $\alpha$=$\alpha$p-($\alpha$p-$\alpha$o) exp(-kㆍLAI) where $\alpha$p was the reflectance at the maximum LAI and $\alpha$o was that of the bare soil. The ap ranged from 0.025 to 0.035 and $\alpha$o ranged from 0.11 to 0.12, respectively. The reflected PAR ranged from 0.049 to 0.064 and the transmitted PAR ranged from 0.168 to 0.340 until maximum dry weights were observed. The slope from the linear regression of dry matter on absorbed PAR, conversion efficiency, ranged from 1.30 to 2.3g MJ$^{-1}$ during the growing season until maximum dry weight was reached. The total dry matter yield above ground (TDM) increased with the increases in the conversion efficiency. TDM was higher in Hwanggeumkong than Paldalkong and higher in the space of 30$\times$15cm than 60$\times$15cm, Paldalkong showed higher harvest index than Hwanggeumkong. than Hwanggeumkong.