Stiffness

For pain and/or loss of range of motion of a joint, see joint stiffness. For the term regarding the stability of a differential equation, see stiff equation.

"Flexibility" redirects here. For other uses, see Flexibility (disambiguation).

Stiffness is the rigidity of an object — the extent to which it resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.

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 (for example, stretching or compression of a rod), the stiffness is defined as

$k={\frac {F}{\delta }}$

where

F is the force applied on the body

δ is the displacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring)

In the International System of Units, stiffness is typically measured in newtons per metre. In English Units, stiffness is typically measured in pound force (lbf) 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 degrees of freedom (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 x 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 own direction (or degree of freedom), but also those along other directions.

For a body with multiple DOF, in order 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 be used to obtain the direct-related stiffness for the degree of freedom which is unconstrained. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses.

The inverse of stiffness is compliance, typically measured in units of metres per newton. In rheology it may be defined as the ratio of strain to stress, and so take the units of reciprocal stress, e.g. 1/Pa.

Rotational stiffness

A body may also have a rotational stiffness, k, given by

$k={\frac {M}{\theta }}$

where

M is the applied moment

θ is the rotation

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 - ratio of applied shear force to shear deformation
• torsional stiffness - ratio of applied torsion moment to angle of twist

Relationship to elasticity

In general, elastic modulus is not the same as stiffness. Elastic modulus is a property of the constituent material; stiffness is a property of a structure. That is, the modulus is an intensive property of the material; stiffness, on the other hand, is an extensive property of the solid body dependent on the material and the shape and boundary conditions. For example, for an element in tension or compression, the axial stiffness is

$k={\frac {AE}{L}}$

where

A is the cross-sectional area,

E is the (tensile) elastic modulus (or Young's modulus),

L is the length of the element.

Similarly, the rotational stiffness is

$k={\frac {nEI}{L}}$

where

"I" is the moment of inertia,

"n" is an integer depending on the boundary condition (=4 for fixed ends)

For the special case of unconstrained uniaxial tension or compression, Young's modulus can be thought of as a measure of the stiffness of a material.

Use in engineering

The stiffness of a structure is of principal importance in many engineering applications, so the modulus of elasticity is often one of the primary properties considered when selecting a material. A high modulus of elasticity is sought when deflection is undesirable, while a low modulus of elasticity is required when flexibility is needed.

See also

• Elasticity (physics)
• Elastic modulus
• Mechanical impedance
• Hardness
• Hooke's law
• Moment of inertia
• Stiffness (mathematics)
• Young's modulus
• Compliant mechanism