Strain based design

Introduction

Sections and cross-sections indicates how axial force and moment are computed for a Sections and Cross-section under a given strain-distribution.

Aim

The aim is to find a distribution of strains over a beam cross-section leading to an equilibrium of the horizontal forces \(H\) (see Formula (1)). In consequence the summarized axial forces of all sub-sections \(N_i\) must become zero. This may also consider an applied axial-force \(N_\mathrm{applied}\).

(1)\[H = \sum N_i - N_\mathrm{applied} = 0\]

The strain-distribution leading to equilibrium of the axial-forces is found by iteration.

Boundary values

Curvature

Computation of maximum possible curvature

Before the iteration is started boundary-values are determined. In case the iteration aims to determine the neutral-axis \(z_\mathrm{n}\) and the curvature \(\kappa\) this means a maximum curvature \(\kappa_\mathrm{max}\) and a minimum curvature \(\kappa_\mathrm{min}\).

For the maximum curvature \(\kappa_\mathrm{max}\) (positive or negative) first the maximum and minimum strains at bottom- and top-edge of each section are determined. By computing the curvature from these (position | strain)-values with the input (position | strain)-value the maximum positive or the maximum negative curvature may be determined by taking the minimum positive or the maximum negative curvature.

The minimum curvature is defined in general as zero curvature. In case this zero curvature leads to a strain in the several sections that is above the maximum or below the minium strain then the minimum curvature is adapted appropriately.

Position of neutral axis

Nevertheless if the curvature \(\kappa\) is given, a prevention from exceeding maximum positive and negative strains is needed. This is the case if the overall cross-section consists of two subsidiary cross-sections, i.e. concrete-slab and steel-girder, and one cross- section has already been computed and therefore a curvature is available.

For this case another boundary function is implemented that considers the maximum and minimum positive and negative strains. Taking into account a given positive curvature the boundary neutral axes are:

• the lower position of the neutral axis from the curvature and maximum positive strains

• the higher position of the neutral from the curvature and the maximum negative strains

Boundary values for the neutral axis \(z_\mathrm{m}\) under positive curvature

In case the given curvature is negative it is vice versa and the boundary neutral axes are:

• the lower position of the neutral axis from the curvature and maximum negative strains

• the higher position of the neutral from the curvature and the maximum positive strains

The resulting neutral axis must lie between these computed boundary neutral axes. Otherwise, no equilibrium of forces may be found.

Finding equilibrium of axial forces

To find the equilibrium of axial forces a Newton-algorithm is used as given in formula (2). The neutral axis \(z_\mathrm{n}\) is used as variable \(x\).

(2)\[x_\mathrm{n+1} = x_\mathrm{n} - \frac{f(x_\mathrm{n})}{f'(x_\mathrm{n})}\]

where \(x_\mathrm{n}\) is the original variable value and \(x_\mathrm{n+1}\) the new iteration value, i.e. the new value of the neutral axis \(z_\mathrm{n}\).

In case the Newton-Algorithm does not find a solution a Bisection-algorithm is implemented as fallback.

(3)\[x_\mathrm{n+1} = \frac{x_{f(x)>0} + x_{f(x)<0}}{2}\]

where \(x_\mathrm{n+1}\) is the new iteration value, \(x_{f(x)>0}\) is the variable-value with the smallest resulting value \(f(x)>0\) greater zero and \(x_{f(x)<0}\) is the variable-value with the smallest absolut resulting value smaller zero. \(f(x)\) may therefore be given as axial force depending on neutral axis \(N(z_\mathrm{n})\).

The fallback-mechanism also strikes in case the Newton-Algorithm computes the same value twice.

The Boundary values are used on the one hand as starting values and on the other hand to make sure that strains stay within the minimal and maximal strains of the material models.