# Impact of Mining Disturbance on Highway Sustainability:
### Surface Curvature  
Surface curvature is a tensor quantity that is characterized by three components: two normal curvatures and a twist. The normal curvature (\(\mathrm{K}\)) is the second derivative of subsidence, or the rate of tilt change. It is calculated as the difference in surface tilt between two neighboring sections on the profile line divided by the initial distance between their centers. Normal curvature denotes the natural surface bending or flexure and is expressed in \(1/\kilometers\) (km\({}^{-1}\)).  
Figure 1: Subsidence and slope: 1, 2, 3—subsidence curves for subcritical, critical, and supercritical subsidence troughs; 4, 5—slope curves for subcritical and critical subsidence troughs.  
Curvature is usually expressed as its reciprocal, the radius of curvature (R), in kilometers (km), which signifies the circular arc radius approximating the curve at that point. Curvature is convex over the goaf edges and concave toward the bottom of the subsidence trough. According to the convention, convex curvature is positive and concave curvature is negative.  
The first derivative of the subsidence curve equation \(U(\zeta)\) is the slope \(N=U^{\prime}(x)\) while the second defines the curvature. When the subsidence trough equation \(U(x,y)\) is known, curvature at any point \(M(x,y)\) is calculated as follows [69]:  
\[x_{M}=x_{A}+\xi_{M}\ \cos\varphi,\text{ and }y_{M}=y_{A}+\xi_{M}\ \sin\varphi.\]  
\[If:\ K_{x}=\frac{\partial^{2}U}{\partial x^{2}},\ K_{xy}=\frac{\partial^{2}U} {\partial x\partial y},K_{y}=\frac{\partial^{2}U}{\partial y^{2}},\]  
it follows that  
\[K_{M}(\varphi)=\frac{1}{2}\big{(}K_{x}+K_{y}\big{)}+\frac{1}{2}\big{(}K_{x}-K_ {y}\big{)}\ \cos 2\varphi+K_{xy}\sin 2\varphi, \tag{9}\]  
whereby \(K_{x}\) and \(K_{y}\) are the curvatures at the point \(M(x,y)\) for profiles parallel to the coordinate axes. In practice, the extreme values of curvatures \(K_{I}\) and \(K_{II}\) at a point \(M(x,y)\) are  
\[K_{I}=\frac{1}{2}\big{(}K_{x}+K_{y}\big{)}+\frac{1}{2}\sqrt{\big{(}K_{x}-K_{ y}\big{)}^{2}+4K_{xy}^{2}}, \tag{10}\]  
whereby \(K_{I}\) is the highest and \(K_{II}\) the lowest value of the main curvatures [82].  
The third component of surface curvature, twist, is the surface torsion in plan, determined as the sum of the change rates of two orthogonal normal curvatures perpendicular to their own axes.  
The four points, denoted as \(A\), \(B\), \(C\), and \(D\) on a plane \(z=H\), constitute the vertices of a rectangle, with sides \(AB=CD=a\) and \(BC=AD=b\) forming an angle \(\varphi\) with axis \(O_{x}\) and \(O_{y}\) (Figure 2).  
The differences in subsidence at these points, denoted as \(U_{A}\), \(U_{B}\), \(U_{C}\), and \(U_{D}\), determine the slopes \(N_{AD}=\frac{U_{D}-U_{A}}{b}\) and \(N_{BC}=\frac{U_{C}-U_{B}}{b}\), which are generally unequal. The discrepancy \(\Delta N=N_{BC}-N_{AB}\
eq 0\) defines the angle at which the line segment AB twists. The average value per unit length,  
\[K_{AB}=\frac{\Delta N}{a}, \tag{11}\]  
Figure 2: Surface twist around the x- and y-axes.  
is referred to as twist. Twist is measured in units of 1/kilometers (km\({}^{-1}\)) or millimeters per meter (mm/m), and its reciprocal, in kilometers (km), is also used.