Secant stiffness
= Secant stiffness
{title2=$k_{\rm sec}=F/x$}
The \b[secant stiffness] of a one-dimensional <force-displacement curve>, relative to a chosen reference $(x_0,F_0)$, is
$$
k_{\rm sec}(x)=\frac{F(x)-F_0}{x-x_0},\qquad x\ne x_0.
$$
It 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 $k_{\rm sec}=F/x$. 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,
$$
F(x)=ax+bx^3,\qquad
a=100\,\mathrm{N\,m^{-1}},\quad
b=1000\,\mathrm{N\,m^{-3}},
$$
take $x=0.10\,\mathrm m$. The force is $11\,\mathrm N$ and the origin-based <secant stiffness> is $110\,\mathrm{N\,m^{-1}}$. The <tangent stiffness> at the same point is
$$
\frac{dF}{dx}=a+3bx^2=130\,\mathrm{N\,m^{-1}}.
$$
For 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.<Structural Stiffness Concept by TU Delft OpenCourseWare>[Force-deflection slope and axial stiffness EA/L]{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 $117.5\,\mathrm{N\,m^{-1}}$, rather than $110\,\mathrm{N\,m^{-1}}$.
Nor does an endpoint <secant stiffness> determine the <work> needed to get there. Here the exact loading work is
$$
W=\int_0^xF(s)\,ds=\frac{ax^2}{2}+\frac{bx^4}{4}=0.525\,\mathrm J.
$$
Approximating it by $\tfrac12 k_{\rm sec}x^2$ would instead give $0.550\,\mathrm J$. The endpoint ratio does not replace the complete <force-displacement curve>.