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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).
Stiffness is the extent to which an object resists deformation in response to an applied force. [ 1 ] The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law: . Young's modulus E describes the material's strain response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire or putting a weight on top of a column, with the wire getting longer and the column losing height),
Young's modulus is the slope of the linear part of the stress–strain curve for a material under tension or compression.. Young's modulus (or Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise.
However, in cases of non-prismatic members, such as the case of the tapered beams or columns or notched stair stringers, the flexural rigidity will vary along the length of the beam as well. The flexural rigidity, moment, and transverse displacement are related by the following equation along the length of the rod, x {\displaystyle x} :
Backward differentiation formula, a family of implicit methods especially used for the solution of stiff differential equations; Condition number; Differential inclusion, an extension of the notion of differential equation that allows discontinuities, in part as way to sidestep some stiffness issues; Explicit and implicit methods
The equation interpolates between the yield stress of the material to the critical buckling stress given by Euler's formula relating the slenderness ratio to the stress required to buckle a column. Buckling refers to a mode of failure in which the structure loses stability. It is caused by a lack of structural stiffness. [1]
The bending stiffness is the resistance of a member against bending deflection/deformation. It is a function of the Young's modulus E {\displaystyle E} , the second moment of area I {\displaystyle I} of the beam cross-section about the axis of interest, length of the beam and beam boundary condition.