Geometry

Todo

theory - geometries: draw figure(s)

Introduction

Within this section the computation of the width \(b(x)\) of rectangles and the area of circles \(A_\mathrm{circle}\) is presented. Combined with stress-values from Materials the axial force \(N_i\), the lever arm \(z_i\) and the moment \(M_i\) are computed (see Sections and cross-sections).

Rectangle

Width

For computation of arbitrary small layers of an rectangle or a trapezoid their width \(b\) at each vertical position is defined as given in formula (1).

(1)\[b(z) = m_\mathrm{b} \cdot z + c_\mathrm{b}\]

The slope \(m_\mathrm{b}\) in (1) is therefore computed using formula (2). Where \(b_\mathrm{top}\) is the width on the top-edge, \(b_\mathrm{bottom}\) is the width on the bottom-edge, \(z_\mathrm{top}\) is the vertical position of the top-edge and \(z_\mathrm{bottom}\) is the vertical position of the bottom-edge of rectangle or trapezoid.

(2)\[m_\mathrm{b} = \frac{b_\mathrm{top} - b_\mathrm{bottom}}{z_\mathrm{top} - z_\mathrm{bottom}}\]

In case of a rectangle the width-slope \(m_\mathrm{b} = 0.0\).

The interception-value \(c_\mathrm{b}\) is therefore determined by rearranging formula (1) to formula (3) and applying known values, like \(b(z_\mathrm{top}) = b_\mathrm{top}\).

(3)\[c_\mathrm{b} = b(z) - m_\mathrm{b} \cdot z\]

In case of rectangles \(c_\mathrm{b}\) by formula (3) applies to \(c_\mathrm{b} = b_\mathrm{top}\).

Implementation

The above given formulas are implemented in Rectangle and Trapezoid and used to compute axial-force, the lever-arm of the axial-force and therefore the moment of arbitrary small layers of the rectangle or the trapezoid.

Circle

Area

The circles are intended for use as reinforcement-bars and therefore in most applications small in comparison with other geometric instances. Therefore, computation is reduced to the cross-sectional area of the circle \(A_\mathrm{circle}\) as shown in (4).

(4)\[A_\mathrm{circle} = \frac{\pi \cdot d}{4}\]

Where \(d\) is the diameter of the reinforcement bar.

Implementation

Formula (4) is implemented in Circle. Combined with the vertical position of the centroid the moment may be computed as well.