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The following assumptions are made while deriving Euler's formula: [3] The material of the column is homogeneous and isotropic. The compressive load on the column is axial only. The column is free from initial stress. The weight of the column is neglected. The column is initially straight (no eccentricity of the axial load).
A special type of area density is called column density (also columnar mass density or simply column density), denoted ρ A or σ. It is the mass of substance per unit area integrated along a path; [ 1 ] It is obtained integrating volumetric density ρ {\displaystyle \rho } over a column: [ 2 ] σ = ∫ ρ d s . {\displaystyle \sigma =\int \rho ...
The progression of thicknesses is clear in column 3 (U.S. standard for sheet and plate iron and steel 64ths inch (delta)). The thicknesses vary first by 1 ⁄ 32 inch in higher thicknesses and then step down to increments of 1 ⁄ 64 inch, then 1 ⁄ 128 inch, with the final increments at decimal fractions of 1 ⁄ 64 inch.
The slenderness ratio is an indicator of the specimen's resistance to bending and buckling, due to its length and cross section. If the slenderness ratio is less than the critical slenderness ratio, the column is considered to be a short column. In these cases, the Johnson parabola is more applicable than the Euler formula. [5]
Examining the density formula, we see that the mass of a beam depends directly on the density. Thus if a beam's cross-sectional dimensions are constrained and weight reduction is the primary goal, performance of the beam will depend on Young's modulus divided by density.
A steel column is extended by welding or bolting splice plates on the flanges and webs or walls of the columns to provide a few inches or feet of load transfer from the upper to the lower column section. A timber column is usually extended by the use of a steel tube or wrapped-around sheet-metal plate bolted onto the two connecting timber sections.
Plates are defined as plane structural elements with a small thickness compared to the planar dimensions. [1] The typical thickness to width ratio of a plate structure is less than 0.1. [ citation needed ] A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two ...
For example, when the material should be both stiff and light, for a rod a combination of high Young's modulus and low density indicates the best material, whereas for a plate the cube root of stiffness divided by density / is the best indicator, since a plate's bending stiffness scales by its thickness cubed.