Secondary stresses in thin-walled beams with closed cross sections

An accurate method of determining secondary stresses in thin-walled, uniform beams of closed cross-section is herein presented. The cross-sections are assumed to be preserved by closely spaced rigid diaphragms. In part I the integro-differential equation governing axial displacements is formulated and solved for a beam without longitudinal stiffeners. In Part II the corresponding summation-difference equation is developed and solved for a beam with stiffeners (flanges and stringers). The cross-section, loading distribution and end conditions are assumed to be arbitrary. By introducing generalized difference equations the mathematical analysis for the stiffened beam may be performed in a manner exactly analogous to the process used for the unstiffened beam. A separation of variables in the homogeneous equation leads to the natural stress or displacement modes for a cross-section. The solution of the non-homogeneous equation is then expressed as an expansion in terms of the natural stress modes. Particular attention is given to cross-sections with single symmetry and double symmetry

No effect of seed source on multiple aspects of ecosystem functioning during ecological restoration: cultivars compared to local ecotypes of dominant grasses

Genetic principles underlie recommendations to use local seed, but a paucity of information exists on the genetic distinction and ecological consequences of using different seed sources in restorations. We established a field experiment to test whether cultivars and local ecotypes of dominant prairie grasses were genetically distinct and differentially influenced ecosystem functioning. Whole plots were assigned to cultivar and local ecotype grass sources. Three subplots within each whole plot were seeded to unique pools of subordinate species. The cultivar of the increasingly dominant grass, Sorghastrum nutans, was genetically different than the local ecotype, but genetic diversity was similar between the two sources. There were no differences in aboveground net primary production, soil carbon accrual, and net nitrogen mineralization rate in soil between the grass sources. Comparable productivity of the grass sources among the species pools for four years shows functional equivalence in terms of biomass production. Subordinate species comprised over half the aboveground productivity, which may have diluted the potential for documented trait differences between the grass sources to influence ecosystem processes. Regionally developed cultivars may be a suitable alternative to local ecotypes for restoration in fragmented landscapes with limited gene flow between natural and restored prairie and negligible recruitment by seed