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Axial stiffness measures resistance to extension or shortening along a specified loading axis. For a proportional response, $F=K_{\mathrm{ax}}\Delta L$; for a nonlinear response, specify whether the reported value is a secant stiffness or tangent stiffness. A straight uniform bar in small-strain linear elasticity haswhere $E$ is Young's modulus, $A$ is the load-bearing cross-sectional area and $L$ the deforming length.^[ref] Axial rigidity $EA$ has units of force; axial stiffness $EA/L$ has units of force per length.

An original stepped-bar example shows why a single area is sometimes insufficient. Join two ideal straight segments end to end, with no distributed axial loading. Both carry the same force $F$, so their extensions add:Choose $E_1=E_2=70\mathrm{GPa}$, $L_1=20\mathrm{mm}$, $L_2=30\mathrm{mm}$, $A_1=1\mathrm{m}{\mathrm{m}}^2$ and $A_2=2\mathrm{m}{\mathrm{m}}^2$. The total mechanical compliance is $0.0005\mathrm{mm}{\mathrm{N}}^{-1}$, giving $K_{\mathrm{ax}}=2000\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$. A $100\mathrm{N}$ force extends the bar by $0.050\mathrm{mm}$, divided into $0.0286$ and $0.0214\mathrm{mm}$ in the two segments. The thinner segment extends more despite being shorter.

In parallel, components sharing the same extension instead add their axial stiffnesses, as in a straight unbonded material-fibre bundle with every strand engaged. These are different constraints, described by series and parallel spring stiffness.

A physical test should measure extension between defined specimen points. Crosshead displacement can include machine compliance and grip motion, so the slope of the complete apparatus is not automatically the specimen's axial stiffness. Structural geometry and loading constraints must be separated from the material's Young's modulus.

A force-displacement curve records how a measured force varies with a specified displacement during loading or unloading. The choice of displacement matters: it can describe specimen extension, motion of a machine crosshead or another reference. A curve can reveal resistance to motion, nonlinear response and differences between loading and unloading.

A concrete apparatus uses a moving crosshead, specimen fixtures and a load cell. The University of Illinois describes an Instron 4483 frame whose ball screws move the crosshead, whose load cell measures force, and whose encoder records crosshead position; an extensometer can separately measure specimen length change. The acquisition system converts those signals into engineering units and records them.^[ref]

Those displacement channels need not agree. Crosshead motion can include mechanical compliance of the machine, grip seating and deformation elsewhere in the load train. A specimen extensometer therefore answers a different geometrical question from the crosshead encoder.^[ref]

For a monotonic loading branch, the local slope $dF/dx$ is a tangent stiffness, while the ratio $F/x$ from a chosen zero point is a secant stiffness. These coincide for the linear model $F=kx$ but can differ for a nonlinear curve. The force here is the applied loading force in the positive displacement direction; a spring's restoring force has the opposite sign.

The work supplied along this one-dimensional branch isFor an own linear example, $k=200\mathrm{N}{\mathrm{m}}^{-1}$ and $x_1=0.05\mathrm{m}$ give final force $10\mathrm{N}$ and workThe nonlinear model $F=c{x}^3$, with $c=80000\mathrm{N}{\mathrm{m}}^{-3}$, reaches the same final force but givesIts final secant stiffness is $200\mathrm{N}{\mathrm{m}}^{-1}$, while its tangent stiffness is $3c{x}_1^2=600\mathrm{N}{\mathrm{m}}^{-1}$. A final force and displacement therefore do not determine either the full curve or its work integral.

When loading and unloading follow different branches, a single-valued relation between force and displacement may be insufficient. Hysteresis records that path dependence; relating a cycle's work to dissipation also requires an energy balance and the relevant state assumptions. Report the measured displacement channel, sign convention and loading path before treating a curve as a material property independent of its apparatus.

Gear mesh stiffness describes the force response of the engaged tooth contacts to a specified relative approach along the mesh direction. In the linear parallel-contact model underlying gear tooth load sharing, all active pairs have the same approach $\delta$ andThe sum assumes compatible displacement coordinates. Adding a bending stiffness, a torsional stiffness and a normal contact stiffness without transforming their coordinates would be dimensionally and mechanically inconsistent.

For an original example, two zero-gap pairs with $k_1=80$ and $k_2=120N/\mu \mathrm{m}$ give $k_{\mathrm{mesh}}=200N/\mu \mathrm{m}$. A $1000\mathrm{N}$ load produces $5\mu \mathrm{m}$ normal approach. If only the first pair is engaged, the same ideal stiffness model instead gives $12.5\mu \mathrm{m}$.

With gaps or nonlinear response, specify whether the reported value is secant stiffness $F/\delta$ from a reference or the local gear mesh tangent stiffness $dF/d\delta$. Supposewhere $\delta$ is measured in micrometres and the result in newtons. The second pair starts carrying load at $\delta =3\mu \mathrm{m}$, corresponding to $F=240\mathrm{N}$. At $F=720\mathrm{N}$, approach is $5.4\mu \mathrm{m}$, so the origin-based secant stiffness is about $133.3N/\mu \mathrm{m}$ while tangent stiffness is $200N/\mu \mathrm{m}$. At the ideal corner itself, a unique two-sided derivative does not exist.

A small-perturbation experiment can hold contact position and mean load approximately fixed, apply a controlled force increment and measure the corresponding relative displacement. Shaft and support mechanical compliance must be separated or included explicitly in the reported system response. Rotating the mesh changes contact position, so one scalar does not necessarily describe a whole cycle.

This matters dynamically: changes in gear mesh stiffness can alter oscillatory response and gear tooth load fractions. The primary NASA analysis used an improved tooth-stiffness model coupled to dynamic-load calculations, rather than prescribing stiffness independently of the load solution.^[ref] The elementary parallel-spring sum explains the linear limit, while real contacts require a stated engagement and deformation model.

The gear mesh tangent stiffness is the local derivative of total mesh force with respect to a specified relative approach, with contact position and other state variables held fixed:It predicts a sufficiently small incremental force $\Delta F\simeq k_{\tan }\Delta \delta$. Unlike a reference-based secant stiffness $F/\delta$, it describes the local response rather than the whole loading interval.

For $F(\delta )={\sum }_j{k}_j\max (\delta -e_j,0)$ away from any gear tooth engagement threshold,Only members of the gear mesh active-contact set contribute. With $k_1=80$, $k_2=120N/\mu \mathrm{m}$ and gaps $0$ and $3\mu \mathrm{m}$, the gear mesh tangent stiffness is $80N/\mu \mathrm{m}$ before the second contact and $200N/\mu \mathrm{m}$ after it. At approach $4.8\mu \mathrm{m}$, an additional $0.1\mu \mathrm{m}$ gives $20\mathrm{N}$ more force within the same branch.

The ideal transition at $3\mu \mathrm{m}$ has distinct left and right derivatives. A centered finite difference with any sufficiently small positive $h$ giveswhere numerical displacements are expressed in micrometres. The value $140$ is the mean of the two branch slopes; it does not establish a unique derivative at the corner, even as $h$ shrinks.

The measured displacement channel also matters. If a contact stiffness $200N/\mu \mathrm{m}$ is in series with support stiffness $300N/\mu \mathrm{m}$, their mechanical compliances add, givingA measurement spanning the whole assembly would therefore not isolate the contact stiffness.

For a perturbation experiment, report the mean load, mesh position, displacement endpoints and whether the increment crosses a contact transition. A primary NASA analysis used a coupled tooth-stiffness/dynamic-load calculation; the derivative above explains a local quasistatic limit, rather than replacing that dynamical model.^[ref]

The secant stiffness of a one-dimensional force-displacement curve, relative to a chosen reference $(x_0,F_0)$, isIt is the slope of the secant line joining the reference to the point of interest. With an unloaded reference $x_0=0$, $F_0=0$, this becomes $k_{\sec }=F/x$. Its units are force per displacement. Unlike tangent stiffness, it describes the accumulated change from the reference rather than the local response to an arbitrarily small change.

For an original nonlinear model,take $x=0.10\mathrm{m}$. The force is $11\mathrm{N}$ and the origin-based secant stiffness is $110\mathrm{N}{\mathrm{m}}^{-1}$. The tangent stiffness at the same point isFor a linear curve through the reference, both reduce to the same constant stiffness. The interpretation of a measured force-deflection slope as structural stiffness also incorporates geometry, rather than specifying a material constant alone.^[ref]

Changing the reference changes the secant. In the same model, $F(0.05\mathrm{m})=5.125\mathrm{N}$, so the chord from $0.05$ to $0.10\mathrm{m}$ has slope $117.5\mathrm{N}{\mathrm{m}}^{-1}$, rather than $110\mathrm{N}{\mathrm{m}}^{-1}$.

Nor does an endpoint secant stiffness determine the work needed to get there. Here the exact loading work isApproximating it by $\frac{1}{2}{k}_{\sec }{x}^2$ would instead give $0.550\mathrm{J}$. The endpoint ratio does not replace the complete force-displacement curve.

Stiffness describes the resistance of a specified mechanical system to displacement under an applied force. For a linear one-dimensional response measured from an unloaded reference,the stiffness $k$ is constant and has units $\mathrm{N}{\mathrm{m}}^{-1}$. A nonlinear force-displacement curve requires a more specific definition: secant stiffness compares a finite change from a chosen reference, while tangent stiffness is the local derivative $dF/dx$.

Stiffness depends on geometry as well as material. For a straight, uniform bar under axial linear elasticity,where $E$ is Young's modulus, $A$ is cross-sectional area and $L$ is length.^[ref] Unlike the structural stiffness, Young's modulus relates stress to strain and is expressed in units of pressure.

For an original example, take $E=200\mathrm{GPa}$, $A=10\mathrm{m}{\mathrm{m}}^2$ and $L=1\mathrm{m}$. ThenA $1000\mathrm{N}$ axial load gives $x=0.0005\mathrm{m}=0.5\mathrm{mm}$. Doubling only the length halves the stiffness and doubles this extension. The material's Young's modulus has remained the same.

Series and parallel spring stiffness depends on how the components are connected. Two identical ideal springs of stiffness $k$ carrying the same force in series have total extension $2F/k$, so their combined stiffness is $k/2$. In parallel, equal displacement gives total force $2kx$ and combined stiffness $2k$. These relationships assume the stated force and displacement constraints.

The reciprocal $C=1/k$ is mechanical compliance in the scalar linear case. For a response involving several coupled displacements and forces, a stiffness matrix records the additional relationships; one scalar cannot describe every loading direction.

The tangent stiffness at a point on a differentiable force-displacement curve is its local slope,It governs the first-order response to a small displacement change:The graphical interpretation is the slope of the tangent line. Structural stiffness measured from force-deflection behaviour includes geometry as well as material response.^[ref]

For the illustrative model $F(x)=ax+b{x}^3$, choose $a=100\mathrm{N}{\mathrm{m}}^{-1}$ and $b=1000\mathrm{N}{\mathrm{m}}^{-3}$. At $x_0=0.10\mathrm{m}$, the tangent stiffness is $130\mathrm{N}{\mathrm{m}}^{-1}$. An additional $\Delta x=0.001\mathrm{m}$ therefore predicts a force increase of $0.130\mathrm{N}$ to first order. Direct substitution gives $0.130301\mathrm{N}$; the difference is the higher-order contribution omitted by the tangent approximation. The secant stiffness from the origin is only $110\mathrm{N}{\mathrm{m}}^{-1}$ at this point.

In a displacement-controlled experiment, one can estimate the local slope with a central finite difference,Making $h$ smaller reduces the approximation error for a smooth exact curve, but measured differences can become dominated by measurement uncertainty. The force channel, displacement channel and loading branch must therefore be specified. Crosshead movement can include machine mechanical compliance, while an extensometer records a selected specimen extension.^[ref]

For a curve with hysteresis, loading and unloading can have different local slopes at the same displacement. At a sharp corner a unique derivative may not exist, and a finite-interval slope should be reported as such.