Slip#

General#

For determination of the slip \(s\) the following equation needs to be solved.

(1)#\[\vec{M}_\mathrm{R} - \vec{M}_\mathrm{E} = 0\]

Where \(\vec{M}_\mathrm{R}\) describes the resisting moments of the cross-section and \(\vec{M}_\mathrm{E}\) stands for the external moments that occur if a beam is loaded perpendicular to the beam axis. The \(\vec{\text{arrow}}\) above the moment symbols indicates a vector or a single columned matrix. Each entry in the vector stands for one computed point along the beam.

The computation of the external Moment \(\vec{M}_\mathrm{E}\) is technical mechanics (Force times lever arm) and described under Loading.

The resisting Moment \(\vec{M}_\mathrm{R}\) is computed as described in the following.

Computations#

General#

In case the cross-section consists of two parts that move relatively to each other and that are connected by shear connectors the resisting moment \(\vec{M}_\mathrm{R}\) depends on the Strain-difference \(\vec{\varepsilon_\mathrm{\Delta}}\) and the axial force along the beam \(N\). The computation of this values is described hereafter.

Strain-difference#

The strain difference at a specific point along the beam \(\vec{\varepsilon_{\mathrm{\Delta}i}}\) depends on the slip \(s\) and is computed as follows.

(2)#\[\varepsilon_{i} = \frac{s_{i}}{x_{i} - x_\mathrm{s=0}}\]

Where \(s_{i}\) is the slip at a position \(x_{i}\) along the beam and \(x_\mathrm{s=0}\) is the position where the slip is zero.

../_images/theory_strain_difference-dark.svg
../_images/theory_strain_difference-light.svg

Axial- and Shear-Forces at Shear connectors#

The axial-force \(N_{i}\) at a position \(x_{i}\) results from the sum of the transferred shear forces by the shear connectors \(P_{j}\). The relevant shear connectors \(P_{j}\) are those between the position \(x_\mathrm{s=0}\) where the slip is zero and the given position.

\[N_{i} = \sum_{j=1}^{i} P_{j}\]

The transferred shear force of a shear connector at a position along the beam \(P_{i}\) depends on the slip at this position.

../_images/theory_load_slip_relationship-dark.svg
../_images/theory_load_slip_relationship-light.svg

Exemplary load-slip relationship of a shear connector#

Determination of the resisting moment#

The resisting moment \(M_\mathrm{R}\) at a position \(i\) is determined by finding the moment that is associated with the axial-force \(N_{i}\) and the strain-difference \(\vec{\varepsilon_\mathrm{\Delta}}\) in the corresponding \(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-relationship.

Note

For computation of the \(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-relationship please refer to M-N-\kappa-\varepsilon_\mathrm{\Delta}-curve.

Assuming new distribution of slip#

The above given computations base on the slip along the beam \(\vec{s}\). At the beginning of the computation \(\vec{s}\) is unknown. The aim of the computation is to find a distribution of slip along the beam that leads to a similar distribution of moment like the moment from the applied load as indicated in Formula (1).

Therefore, between each iteration a new guess is made on the slip along the beam \(\vec{s}\) using the Levenberg-Marquardt algorithm. Therefore one iteration-step looks as given in Formula (3).

(3)#\[\vec{s}_\mathrm{n+1} = \vec{s}_\mathrm{n} - \alpha_{n} \left( (\mathbf{J}_\mathrm{n})^\mathrm{T} \mathbf{J}_\mathrm{n} + \lambda~\mathrm{diag}(\mathbf{J}_\mathrm{n}^\mathrm{T} \mathbf{J}_\mathrm{n}) \right)^{-1} (\mathbf{J}_\mathrm{n})^\mathrm{T} f(\vec{s}_\mathrm{n})\]

where \(s_\mathrm{n+1}\) describes the new guess of the distribution of slip. The subscript \(\mathrm{n}\) stands for the current iteration-step. Therefore, \(s_\mathrm{n}\) is the slip at the current step of the iteration-process. \(\alpha_{n}\) controls the size of the increment and must be greater zero (\(\alpha_{n} > 0\)). \(\mathbf{J}_\mathrm{n}\) is the Jacobian-Matrix, that includes all first-order partial derivates of the function \(f(s_\mathrm{n})\) as indicated in Formula (4). \(\lambda\) is the damping-factor, that is multiplied by the diagonal of \(\mathbf{J}_\mathrm{n}^\mathrm{T}\mathbf{J}_\mathrm{n}\).

(4)#\[\begin{split}\mathrm{J} = \begin{bmatrix} \frac{\partial f_{1}}{\partial s_{1}} & \dots & \frac{\partial f_{1}}{\partial s_{i}} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_{j}}{\partial s_{1}} & \dots & \frac{\partial f_{j}}{\partial s_{i}} \end{bmatrix}\end{split}\]
Determining the Jacobian-Matrix using numerical differentiation

As the determination of the resistance Moment \(M_\mathrm{R}\) from the \(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-relationship depends on a number of factors that depend for themselves again on various factors a deterministic formula is assumed to be not available. Or at least possible only under high effort.

Therefore, the Jacobian Matrix is determined by numerical differentiation. The forward difference quotient is used as given in Formula eq:forward_differentiation_coefficient.

(5)#\[\frac{f(s_{i} + h) - f(s_{i})}{h}\]

where \(f(s_{i})\) describes the function the difference quotient is to be determined of and \(h\) is the interval length.

The theory is that in case \(h\) is sufficiently small the difference quotient denotes to the derivate of \(f\) at the distribution of slip \(s_i\) as indicated in Formula (6).

(6)#\[f'(s_{i}) = \lim_{h \rightarrow 0} \frac{f(s_{i} + h) - f(xs_{i})}{h}\]
Levenberg-Marquardt algorithm

For each step in the iteration-process the last term in Formula (3) is solved on its own as follows.

(7)#\[\Delta s_{n} = \left[ (\mathbf{J}_\mathrm{n})^\mathrm{T} \mathbf{J}_\mathrm{n} + \lambda~\mathrm{diag}(\mathbf{J}_\mathrm{n}^\mathrm{T} \mathbf{J}_\mathrm{n}) \right]^{-1} (\mathbf{J}_\mathrm{n})^\mathrm{T} f(\vec{s}_\mathrm{n})\]

To compute \(\Delta s_{n}\) Formula (7) is transformed to match a linear system of equations (\(\mathbf{A}x = \vec{b}\)).

(8)#\[\left[(\mathbf{J}_\mathrm{n})^\mathrm{T} \mathbf{J}_\mathrm{n} + \lambda~\mathrm{diag}(\mathbf{J}_\mathrm{n}^\mathrm{T} \mathbf{J}_\mathrm{n})\right] \Delta s_{n} = (J_\mathrm{n})^\mathrm{T} f(\vec{s}_\mathrm{n})\]

where \(\mathbf{A} = (\mathbf{J}_\mathrm{n})^\mathrm{T} \mathbf{J}_\mathrm{n}+ \lambda~\mathrm{diag}(\mathbf{J}_\mathrm{n}^\mathrm{T} \mathbf{J}_\mathrm{n})\), \(b = (J_\mathrm{n})^{T} f(\vec{s}_\mathrm{n})\) and \(x = \Delta s_{n}\).

This form allows QR-decomposition to solve this set of linear equations.

The overall procedure is given in the following section.

Procedure#

The computation of the slip in the composite joint requires the following preparations:

  • Definition of cross-section, load-slip relationship of the shear connector(s), positioning, beam-length, loading by the user

  • Computation of the \(M\)-\(N\)-\(\kappa\)-\(\varepsilon_\mathrm{\Delta}\)-relationships of the given cross-section at each position

  • Computation of the moment \(M_\mathrm{E}\) by the applied load along the beam

The procedure to compute the slip in the composite joint along the of a beam considering slip in the composite joint is as follows.

  1. assume distribution of slip \(s\) along the beam

  2. compute the theory.slip.strain_difference \(\varepsilon_\mathrm{\Delta}\) at each position considering the slip \(s\)

  3. compute the transferred shear force of each shear connector \(P_{i}\) and sum it up to compute the resulting axial forces \(N_{i}\) at each position

  4. use axial force \(N\) and strain-difference \(\varepsilon_\mathrm{\Delta}\) to determine the resisting moment \(M_\mathrm{R}\) at each position

  5. compare \(\vec{M}_\mathrm{R}\) and applied Moment \(\vec{M}_\mathrm{E}\)

  6. if the difference of \(\vec{M}_\mathrm{R}\) and \(\vec{M}_\mathrm{E}\) are outside a given tolerance then restart procedure by assuming a new distribution of slip, else the correct distribution of slip \(s\) is found

The correct distribution of slip is then used to compute curvature \(\kappa\) and subsequently the deformation of the beam under the given load.

The following figure describes the process of computing the slip as it is described above graphically.

../_images/theory_slip_flowchart-dark.svg
../_images/theory_slip_flowchart-light.svg

Procedure to compute the deformation considering slip#