Stiffness

Stiffness or rigidity is the extent to which an object resists deformation in response to an applied force. The inverse of stiffness is flexibility or pliability: the more flexible an object is, the less stiff it is. Other terms with related meanings are compliance and elasticity.

Stiffness or rigidity is the extent to which an object resists deformation in response to an applied force. The inverse of stiffness is flexibility or pliability: the more flexible an object is, the less stiff it is. Other terms with related meanings are compliance and elasticity.

Calculations

The stiffness, k, of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single degree of freedom (DOF) (for example, stretching or compression of a rod), the stiffness is defined as k = \frac {F}{\delta} where, F is the force on the body \delta is the displacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring) Stiffness is usually defined under quasi-static conditions, but sometimes under dynamic loading. In the International System of Units, stiffness is typically measured in newtons per meter ( \mathrm{N} / \mathrm{m} ). In Imperial units, stiffness is typically measured in pounds (lbs) per inch. Generally speaking, deflections (or motions) of an infinitesimal element (which is viewed as a point) in an elastic body can occur along multiple DOF (maximum of six DOF at a point). For example, a point on a horizontal beam can undergo both a vertical displacement and a rotation relative to its undeformed axis. When there are M degrees of freedom a M \times M matrix must be used to describe the stiffness at the point. The diagonal terms in the matrix are the direct-related stiffnesses (or simply stiffnesses) along the same degree of freedom and the off-diagonal terms are the coupling stiffnesses between two different degrees of freedom (either at the same or different points) or the same degree of freedom at two different points. In industry, the term influence coefficient is sometimes used to refer to the coupling stiffness. It is noted that for a body with multiple DOF, the equation above generally does not apply since the applied force generates not only the deflection along its direction (or degree of freedom) but also those along with other directions. For a body with multiple DOF, to calculate a particular direct-related stiffness (the diagonal terms), the corresponding DOF is left free while the remaining should be constrained. Under such a condition, the above equation can obtain the direct-related stiffness for the degree of unconstrained freedom. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses. The elasticity tensor is a generalization that describes all possible stretch and shear parameters. A single spring may intentionally be designed to have variable (non-linear) stiffness throughout its displacement.

Rotational stiffness

A body may also have a rotational stiffness, k, given by k = \frac{M}{\theta} where M is the applied moment \theta is the rotation angle In the SI system, rotational stiffness is typically measured in newton-metres per radian. In the SAE system, rotational stiffness is typically measured in inch-pounds per degree. Further measures of stiffness are derived on a similar basis, including: shear stiffness - the ratio of applied shear force to shear deformation torsional stiffness - the ratio of applied torsion moment to the angle of twist k = E \cdot \frac{A}{L} where E is the (tensile) elastic modulus (or Young's modulus), A is the cross-sectional area, L is the length of the element. Similarly, the torsional stiffness of a straight section is k = G \cdot \frac{J}{L} where G is the rigidity modulus of the material, J is the torsion constant for the section. The torsional stiffness has dimensions \mathrm{[force]} \cdot \mathrm{[length]} / \mathrm{[angle]} , so that its SI units are \mathrm{N} \cdot \mathrm{m} / \mathrm{rad} . For the special case of unconstrained uniaxial tension or compression, Young's modulus be thought of as a measure of the stiffness of a structure.

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