General

Introduction

Here basic formulas are listed to compute the Curvature, Strain at a given vertical position, Position of a given strain and the Neutral axis. The given formulas define the basis for this piece of software.

Curvature

Formula (1) computes the curvature \(\kappa\) given a strain \(\varepsilon\) at a position \(z\) and the neutral axis \(z_\mathrm{m}\).

(1)\[\kappa = \frac{\varepsilon}{z - z_\mathrm{m}}\]

Given two points strain-position points \((z_1 | \varepsilon_\mathrm{1}), (z_2 | \varepsilon_\mathrm{2})\) the curvature is computed as given in formula (2).

(2)\[\kappa = \frac{\varepsilon_\mathrm{1} - \varepsilon_\mathrm{2}}{z_1 - z_2}\]

Strain

The strain \(\varepsilon\) a given position \(z\) is computed by formula (3), that is a rearrangement of formula (1).

(3)\[\varepsilon = \kappa \cdot (z - z_\mathrm{m})\]

Position

The vertical position \(z\) of a given strain \(\varepsilon\) and the vertical position of the neutral axis \(z_\mathrm{m}\) is computed using formula (4).

(4)\[z = \frac{\varepsilon}{\kappa} + z_\mathrm{m}\]

Neutral axis

The neutral axis \(z_\mathrm{m}\) under a given curvature \(\kappa\) and a strain \(\varepsilon\) at a position \(z\) is computed by formula (5).

(5)\[z_\mathrm{m} = z - \frac{\varepsilon}{\kappa}\]