\(M\)-\(\kappa\)-curve#

Definition#

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add figure of a Moment-Curvature-Curve

The \(M\)-\(\kappa\)-curve shows the relationship between the moment \(M\) and the curvature \(\kappa\) of a given cross-section. The curve considers the non-linear stress-strain relationships of the Materials of the cross-section. Therefore, each \(M\)-\(\kappa\)-point is computed using Strain based design.

The following sections outline the process of computing the \(M\)-\(\kappa\)-curve as implemented.

Procedure#

The overall procedure to compute the \(M\)-\(\kappa\)-curve is as follows:

  1. compute the \(M\)-\(\kappa\)-point under failure

  2. determine all position-strain-values between the failure-curvature and zero curvature

  3. use these position-strain-values to compute all intermediate \(M\)-\(\kappa\)-points

The outlined process is applied for positive and negative curvatures.

Failure#

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create figure

The curvature at failure \(\kappa_\mathrm{fail}\) is crucial for an efficient computation of the full \(M\)-\(\kappa\)-curve. The position-strain-pair at failure is a value of the cross-section and therefore computed during determination of the Boundary values.

In case the position-strain-pair at failure is known it is used to compute \(\kappa_\mathrm{fail}\) by Iteration.

Intermediate values#

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create figure

Between the curvature at failure \(\kappa_\mathrm{fail}\) and no curvature of the cross-section at all, various position-strain-points are passed. These position-strain-points are determined and allow therefore a computation of a holistic of the full \(M\)-\(\kappa\)-curve also by Iteration.

The full \(M\)-\(\kappa\)-curve is used to compute the non-linear load-deformation-behaviour of a beam.