Sections and cross-sections#

Introduction#

Concept and process of computing the axial-force and moment of the cross-section under a given distribution of strain is described.

Sections#

General#

Sections consist of:

The characteristics of geometry and material are combined to computed the axial force, the lever arm and the moment corresponding with an applied Curvature or constant strain.

Procedure#

Todo

add figure for better description of the procedure

To compute axial force, lever arm and moment of a section first the following steps are conducted, assuming that curvature \(\kappa\) and neutral-axis \(z_\mathrm{n}\) are known.

  1. mapping strain-distribution onto the section considering curvature \(\kappa\) and \(z_\mathrm{n}\)

  2. split section at material curve points into sub-sections

  3. map stresses on top- and bottom-edge of the sub-section

As the procedure above ensures that no material points - and therefore no change in stress-strain slope - are within each sub-section a linear relationship between vertical position \(z\) and stress \(\sigma\) is present. This allows to determine a Linear stress-distribution over height of each sub-section.

Linear stress-distribution over height#

The linear stress-distribution over the height of a sub-section is computed by formula (1).

(1)#\[\sigma(z) = m_\mathrm{\sigma} \cdot \varepsilon + c_\mathrm{\sigma}\]

where the slope \(m_\mathrm{\sigma}\) and the interception \(c_\mathrm{\sigma}\) is computed as follows assuming two stress-vertical-position points (\(\sigma_\mathrm{top}\) | \(z_\mathrm{top}\) ) and (\(\sigma_\mathrm{bottom}\) | \(z_\mathrm{bottom}\) ).

(2)#\[m_\mathrm{\sigma} = \frac{\sigma_\mathrm{bottom} - \sigma_\mathrm{top}}{z_\mathrm{bottom} - z_\mathrm{top}}\]
(3)#\[c_\mathrm{\sigma} = \sigma_\mathrm{top} - m_\mathrm{\sigma} \cdot z_\mathrm{top}\]

Linear distributions allow easy integration, giving the opportunity to determine Axial force, Lever arm and Moment in conclusion.

Axial force#

The axial force \(N_i\) of each rectangular sub-section with index \(i\) is computed according to formula (4). It assumes that the stress distribution over its height is distributed linearly.

(4)#\[N_i = \int_{z_\mathrm{top}}^{z_\mathrm{bottom}} \sigma(z) \cdot b(z) dz = \int_{z_\mathrm{top}}^{z_\mathrm{bottom}} (m_\mathrm{\sigma} \cdot \varepsilon + c_\mathrm{\sigma}) \cdot (m_\mathrm{b} \cdot z + c_\mathrm{b}) dx\]

with \(\sigma(z)\) given in formula (1) and \(b(z)\) by formula (1) in Rectangle. The axial-force \(N_i\) of the sub-section is achieved by integration between the vertical position of the top-edge \(z_\mathrm{top}\) and the vertical position of the bottom-edge \(z_\mathrm{bottom}\).

Due to the very limited size of the Circle its axial force \(N_i\) is computed according Formula (5).

(5)#\[N_i = A_\mathrm{circle} \cdot \sigma(z_\mathrm{circle-centroid})\]

where \(A_\mathrm{circle}\) is the cross-sectional area of the circle (see formula (4) in Circle) and the stress is obtained at the vertical position of the circle’s centroid \(z_\mathrm{circle-centroid}\).

Similar procedure is conducted for the Lever arm and the Moment.

Lever arm#

The distance between the origin and the vertical point of the axial force is the lever arm of a sub-section \(r_i\) under a given stress-distribution \(\sigma(z)\).

(6)#\[r_i = \frac{1}{N_i} \int_{z_\mathrm{top}}^{z_\mathrm{bottom}} \sigma(z) \cdot b(z) \cdot z~dz\]

where \(N_i\) is the axial force of the sub-section (see Formula (4)), \(\sigma(z)\) (see Formula (1)) and \(b(z)\) the width of the geometry at the vertical position \(z\) (see formula (1) in Rectangle).

For circles the lever arm applies to the geometrical centroid in vertical direction, that is an input-parameter.

(7)#\[r_i = z_\mathrm{centroid}\]

The lever arm is used to compute the moment the sub-section contributes to the overall moment of the cross-section under a given stress-distribution.

Moment#

The moment of each sub-section \(M_i\) is computed taking the Axial force and the Lever arm into account as given in formula (8).

(8)#\[M_i = N_i \cdot r_i\]

Implementation#

These functionalities are part of ComputationSection, ComputationSectionCurvature and ComputationSectionStrain.

Cross-section#

A cross-section consist of a number of sections, that result during computation under a given strain-distribution to a number of sub-sections (index \(i\)). Total axial force \(N_\mathrm{cs}\) and moment \(M_\mathrm{cs}\) under a given cross-section are therefore computed by summing axial forces \(N_i\) and moment \(M_i\) of the given sub-section up as shown in formulas (9) and (10).

(9)#\[N_\mathrm{cs} = \sum_i N_i\]
(10)#\[M_\mathrm{cs} = \sum_i M_i\]

Summary#

For computation of a cross-section’s axial force \(N_\mathrm{cs}\) and moment \(M_\mathrm{cs}\) each section is split into appropriate sub-sections considering the stress-strain-relationship of its material. The split is conducted in a way that each sub-section has a linear stress-distribution allowing to normalize the process of computing axial force, lever-arm and moment of each sub-section and in turn of the overall cross-section.

The computation of a cross-section under a given strain-distribution is required for Strain based design.